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abstract: 'We derive the Lorentz self-force for a charged particle in arbitrary non-relativistic motion via averaging the retarded fields. The derivation is simple and at the same time pedagogically accessible. We obtain the radiation reaction for a charged particle moving in a circle. We pin down the underlying concept of mass renormalization.'
author:
- Asrarul Haque
title: 'A Simple Derivation of Lorentz Self-Force'
---
Introduction
============
The electromagnetic field goes to infinity at the position of a point charge. The electrostatic field at the position of the charged particle $$\vec E(\vec r) = \frac{1} {{4\pi \varepsilon _0 }}\frac{q} {{r^3
}}\vec r\xrightarrow{{r \to 0}}\infty$$ and the self-energy of the point charge in the rest frame of the charged particle $$U = \frac{{\varepsilon _0 }}{2}\int {E^2 } d^3 r =\frac{1}{{32\pi ^2
\varepsilon _0 }}\int\limits_0^\pi {\sin \theta d\theta
\int\limits_0^{2\pi } {d\phi \int\limits_0^\infty {\frac{{q^2
}}{{r^2 }}dr} } } = - \frac{{q^2 }}{{8\pi\varepsilon _0 }}\left[
{\frac{1}{r}} \right]_{r = 0}^{r = \infty } \to \infty$$ turn out divergent. The concept of the point-like (structureless or dimensionless object) charge may be an idealization. The measurement of the anomalous magnetic moment of an electron based on the quantum field theoretic calculations leads to the following upper bound on the size of the electron [@kin] $l:~l\le 10^{-17}cm $. Thus, a charged particle might be considered as an extended object with a finite size.\
In order to circumvent the problem of divergence of the field at the point charge, it is plausible either: (1) to consider the averaged value of the field in the suitably small closed region surrounding the point charge as the value of the field under consideration at the position of the point charge or (2) to think a charged particle as an extended object having small dimension with a charge distribution. Our derivation of the self-force is based on the former consideration.\
If an extended charged particle moves with non-uniform velocity, the charge elements comprising such charge distribution, begin to exert forces on one another. However, these forces do not cancel out due to retardation giving rise to a net force known as the self-force. Thus, the radiating extended charged particle experiences a self-force which acts on the charge particle itself. The Lorentz self-force [@jack1 p. 753] arising due to a point charge conceived as a uniformly charged spherical shell of radius $s$ is given by $$\vec F_{self} = - \frac{2}{3}\frac{{q^2 }}{{4\pi \varepsilon_0
c^2s }}\dot {\vec v}(t) + \frac{2}{3}\frac{{q^2 }}{{4\pi
\varepsilon_0 c^3 }}\ddot{\vec v}(t)+ O(s)~~\textup{with}~~|\vec s|=
s \label {ccc}$$ where,
- the quantity $\frac{2}{3}\frac{{q^2 }}{{4\pi \varepsilon
_0 c^2s }}$ in the first term stands for electromagnetic mass and becomes divergent as $s\to 0^+$,
- the second term represents the radiation reaction and is independent of the dimension of the charge distribution and
- the third term corresponds to the first finite size correction and is proportional to the radius of the shell $s$.
The usual method [@jack1] of computing the self-force (\[ccc\]) involves rather cumbersome calculation. Boyer [@boy] has obtained the expression for the self-force using the averaged value of the retarded field for a charged particle in uniform circular motion. Boyer’s derivation of the self-force for charged particle in uniform circular motion involves an unsophisticated calculation.\
In this article, we derive the expression for the self-force for a point charge in arbitrary non-relativistic motion by averaging the retarded field in rather neat and sophisticated way. Our derivation of the self-force unlike Boyer’s derivation pertaining to the specific context, leads to the general (non-relativistic) expression for the self-force. In the following section, we shall define the self-force in terms of the averaged retarded field over the surface of the spherical shell.
The Self-Force
==============
We shall be considering for our purpose, an average field on the surface of a spherical shell of radius $s$ due to a point charge $q$ sitting at the center of the shell. An average field [@cla] over the surface of a spherical shell of radius $s$ is defined by $$\overline{\vec E}_q(t) =
%\mathop{Lim}\limits_{s \to 0^+}
\frac{1}{{4\pi s^2 }}\int\limits_{\sum} {dA} \vec E(\vec r,t)$$ where $\sum$ is the surface of the spherical shell of radius $s$. We now define the self-force as $$\vec F_{Self} = q\mathop {Lim}\limits_{s \to 0^+} \overline{\vec
E}_q(t) = q\vec E_{Self}$$ where the field ${\vec E}(\vec r,t)$ depends upon the position and motion of the charge particle at the retarded time. The field due to an accelerated charged particle is the sum of the *velocity fields* as well as the *acceleration fields*: $$\vec E(\vec r,t) = \vec E^{Vel}(\vec r,t) + \vec E^{Acc}(\vec r,t).$$ The average field contribution from the *velocity fields* (for $v<<c$) $\overline{ \vec E_{q}^{Vel}}(t)$ on the surface of a spherical shell due to a charged particle at its center gets filtered out of $\overline{\vec E}_q(t)$ because for each spatial point ( say ($s_x,s_y,s_z$)) on the surface there exists a corresponding point ($-s_x,-s_y,-s_z$) on the surface.\
However, the average field contribution from the *acceleration fields* $\overline{ \vec E_{q}^{Acc}}(t)$ over the surface of a spherical shell due to a charged particle situated at the center of the shell turns out nonzero because the *acceleration fields* involve a term $\vec s(\vec s.\vec a)$ which is even in the vector $\vec s$. Moreover, $\overline{ \vec E_{q}^{Acc}}(t)$ leads to radiation reaction as well as a term associated with the electromagnetic contribution to mass. We shall now obtain the derivation of the self-force in the following section.
Radiation Reaction From a Point Charge in Arbitrary Non-relativistic Motion
===========================================================================
In order to obtain the expression for the self-force, we begin with the field [@jack1 p. 664] due to a point charge located at $\vec r_q(t)$ at time $t$ in an arbitrary motion which is given by $$\vec E(\vec r,t) = \left[ { \frac{q {\left( {\vec s - \frac{{\vec
v}}{c}s} \right)\left( {1 - \frac{{v^2 }}{{c^2 }}} \right) }}{{4\pi
\varepsilon _0 \left| {\vec s - \frac{{\vec v}}{c}s} \right|^3}} +
\frac{q\left ( \vec s \times \left( {\vec s - \frac{{\vec v}}{c}s}
\right) \times \frac{{\vec a}}{{c^2 }}\right)}{{4\pi \varepsilon _0
\left| {\vec s - \frac{{\vec v}}{c}s} \right|^3}} } \right]_{t =
t_{{\mathop{\rm Re}\nolimits} t} }$$ where, $$\vec s = \vec r - \vec
r_q(t_{Ret}) = (x - x_q(t_{Ret}))\hat i + (y - y_q(t_{Ret}))\hat j +
(z - z_q(t_{Ret}))\hat k,$$ and $t_{Ret}=t-s/c$. The quantities $\vec s,~\vec v$ and $\vec a$ in the square brackets are evaluated at the retarded time $t_{Ret}$. The first term represents the *velocity fields* $\vec
E^{Vel}(\vec r,t)$ and is the independent of the acceleration of the point charge. The second term represents *acceleration fields* $\vec E^{Acc}(\vec r,t)$ and exists only when $\vec a =\frac{{d\vec v}}{{dt}} \ne 0$. Thus, for $\vec a=0$, there will be no radiation. For sufficiently small speed of the point charge ( in the limit $v/c\to 0$ ) the field takes the form $$\begin{aligned}
\vec E (\vec r,t) &=& \frac{q}{{4\pi \varepsilon _0 }}\frac{\vec s
}{{\left| {\vec s} \right|^3 }}+ \frac{q}{{4\pi \varepsilon _0
}}\frac{1}{{\left| {\vec s} \right|^3 }}\left[ {\vec s \times \left(
{\vec s \times \frac{{\vec a}}{{c^2 }}} \right)}
\right]\nonumber\\
&=&\frac{q}{{4\pi \varepsilon _0 }}\frac{\vec s }{{\left| {\vec s}
\right|^3 }}+ \frac{q}{{4\pi \varepsilon _0 }}\frac{1}{{c^2
s^3}}\left[ {\vec s(\vec s.\vec a) - s^2\vec a} \right]\end{aligned}$$
The velocity and corresponding acceleration of the point charge are given by $$\begin{aligned}
\vec v &=& \dot{\vec r}_q(t_{Ret}) = \dot x_q(t_{Ret})\hat i + \dot
y_q(t_{Ret})\hat j + \dot
z_q(t_{Ret})\hat k \nonumber\\
\vec a &=& \ddot{\vec r}_q(t_{Ret}) = \ddot x_q(t_{Ret})\hat i +
\ddot y_q(t_{Ret})\hat j + \ddot z_q(t_{Ret})\hat k\end{aligned}$$ Now, the average field over the surface of sphere (as shown in the FIG. \[casa\]) is $$\begin{aligned}
\overline{\vec E}_q(t) &=& \frac{1}{{4\pi s^2 }} \iint { d\theta d\phi s^2 \sin \theta}
\vec E (\vec r,t) \nonumber\\
&=& \frac{q}{{4\pi \varepsilon _0 }}\frac{1}{{c^2 }}\frac{1}{{4\pi s^3 }} \iint {
d\theta d\phi \sin \theta } \left[ {c^2\vec s + \vec s(\vec s.\vec a) - s^2\vec a}
\right]
\end{aligned}$$ where $\vec s(\vec s.\vec a)$ may be expressed as $$\begin{aligned}
\vec s(\vec s.\vec a) &=& \left[ {\vec s(s_x a_x + s_y a_y + s_z a_z )} \right]\nonumber \\
&=& (s_x^2 a_x + s_x s_y a_y + s_x s_z a_z )\hat i \nonumber\\
&+& (s_y s_x a_x + s_y^2 a_y + s_y s_z a_z )\hat j \nonumber\\
&+& (s_z s_x a_x + s_z s_y a_y + s_z^2 a_z )\hat k.\end{aligned}$$ The vector $\vec s$ in the spherical polar coordinates is given by $$\vec s = (s_x,s_y,s_z)= (s\sin \theta \cos \phi, s\sin \theta \sin\phi, s\cos
\theta)$$ We can show that $$\begin{aligned}
\iint {s_i \sin \theta d\theta d\phi } &=& 0\\
\iint {s_i
s_j\sin\theta d\theta d\phi} &=& \frac{{4\pi }}{3}s^2 \delta _{ij}\end{aligned}$$ where $\delta _{ij}=1$ for $i=j$ and $0$ otherwise with $i,j =
x,y,z$. The average field now becomes $$\overline{\vec E}_q(t) = \frac{q}{{4\pi \varepsilon _0 c^2
s}}\left[ { \frac{1}{3}\vec a(t_{Ret} ) - \vec a(t_{Ret} )} \right]
= \frac{q}{{4\pi \varepsilon _0 c^2 s}}\left[ { - \frac{2}{3}\vec
a(t_{Ret} )} \right].$$ We note that the velocity fields contribution to $\overline{\vec
E}_q(t) $ vanishes for a charged particle in arbitrary non-relativistic motion. In fact, the vanishing contribution of the velocity fields apparently brings out the error in Boyer’s [@boy] calculation for the contribution of the velocity fields. In the limit $s \to 0^+$, we get $$\vec a(t_{Ret}) = \vec a(t - s/c) = \vec a(t) - \frac{s}{c}\dot {\vec a}(t) + O(s^2 )$$ Now, $$\overline{\vec E}_q(t)
= - \frac{2}{3}\frac{q}{{4\pi \varepsilon _0 c^2 }}\frac{{\vec a(t)}}{s}
+ \frac{2}{3}\frac{q}{{4\pi \varepsilon _0 c^3 }}\dot{ \vec a}(t) + O(s )$$ Thus, the self-field $\vec E_{Self}$ yields $$\begin{aligned}
\vec E_{Self}&=& \mathop {Lim}\limits_{s \to 0^+}\overline{\vec E}_q(t) \nonumber \\% = \frac{q}{{4\pi \varepsilon _0 c^2 s}}\left[ { - \frac{2}{3}\vec a(t) + \frac{2}{3}\frac{s}{c}\dot {\vec a}(t)} \right]+ O(s ) )\nonumber \\
&=& - \mathop {Lim}\limits_{s \to 0^+}\left(\frac{2}{3}\frac{q}{{4\pi \varepsilon _0 c^2s }}
\right)\vec a(t) + \frac{2}{3}\frac{q}{{4\pi \varepsilon _0 c^3 }}\dot{ \vec a}(t)% + O(s )
\end{aligned}$$ Now, the self-force for the point charge limit is given as: $$\vec F_{Self} = q\vec E_{Self} = - \mathop {Lim}\limits_{s \to
0^+}\left(\frac{2}{3}\frac{q^2}{{4\pi \varepsilon_0 c^2s
}}\right)\vec a(t) + \frac{2}{3}\frac{q^2}{{4\pi \varepsilon_0 c^3
}}\dot {\vec a}(t)%+ O(s)$$ which is same as equation (\[ccc\]) in the limit $s \to 0^+$. The self-force can be expressed in terms of $r_q$ and $\theta_q$ variables as $$\begin{aligned}
\vec F_{Self} &=&
- \frac{2}{3}\frac{q}{{4\pi \varepsilon _0 c^2 s}}\left[ {(\ddot r_q - r_q\dot {\theta}_q ^2 )\hat r_q +
(r_q\ddot {\theta}_q + 2\dot r_q\dot {\theta}_q )\hat {\theta}_q } \right] \nonumber\\
&+& \frac{2}{3}\frac{q}{{4\pi \varepsilon _0 c^3 }}\left[ {( \dddot r_q- 3\dot r_q\dot {\theta}_q ^2
- 3r_q\dot {\theta}_q \ddot {\theta}_q )\hat r_q + (r_q\dddot {\theta}_q + 3\dot r_q\ddot {\theta}_q
+ 3\ddot r_q\dot {\theta}_q
- r_q\dot {\theta}_q ^3 )\hat {\theta}_q } \right]\end{aligned}$$We shall now study the following illustrative examples pertaining to radiation reaction.
Radiation Reaction From a Charged Particle Executing Simple Harmonic Motion
---------------------------------------------------------------------------
Consider a charged particle $q$ of mass $m$ executing simple harmonic motion along X-axis with frequency $\omega$, its displacement from equilibrium is $$x(t) = x_0\sin\omega t$$ and its acceleration is $$a(t) =\ddot x = -x_0\omega^2 \sin\omega t.$$ The charged particle having nonzero acceleration will radiate and will therefore experience the self-force (Please see the equation (\[sine\]) ) given by $$\begin{aligned}
F^{Oscillator}_{Self} &=& \mathop {Lim}\limits_{s \to 0^ + } \left(
{ - \frac{2}{3} \frac{{q^2 }}{{4\pi \varepsilon _0 c^2 s}}}
\right)\left[ { - x_0 \omega ^2
\sin \omega \left( {t - \frac{s}{c}} \right)} \right]\nonumber \\
& =& \mathop {Lim}\limits_{s \to 0^ + } \frac{2}{3}\frac{{q^2 }}{{4\pi
\varepsilon _0 c^2 s}}x_0 \omega ^2 \sin \omega t - \frac{2}{3}
\frac{{q^2 }}{{4\pi \varepsilon_0 c^3 }}x_0 \omega^3 \cos \omega t
\end{aligned}$$ Thus, the self-force is the sum of the finite piece $(- \frac{2}{3}
\frac{{q^2 }}{{4\pi \varepsilon_0 c^3 }}x_0 \omega^3 \cos \omega t)$ and the divergent piece $(\mathop {Lim}\limits_{s \to 0^ + }
\frac{2}{3}\frac{{q^2 }}{{4\pi
\varepsilon _0 c^2 s}}x_0 \omega ^2 \sin \omega t)$.
Radiation Reaction From a Charged Particle Moving in a Circle
--------------------------------------------------------------
Suppose a charged particle is moving in a circle of radius $R$ with uniform angular speed $\omega$ (as shown in the FIG. \[casa9\]). The charge particle will experience the centripetal force $(-m\omega^2 R\hat r_q)$ acting towards the center.
Now, $$\vec s(t_{Ret}) = (x - R\cos \omega t_{Ret } )\hat i + (y - R\sin
\omega t_{Ret })\hat j + z\hat k$$ The velocity and acceleration are $$\begin{aligned}
\dot{\vec r}_q(t_{Ret}) &=& -R\omega \sin \omega t_{Ret} \hat i +R\omega co{\mathop{\rm s}\nolimits} \omega t_{Ret} \hat j \\
\vec a(t_{Ret}) &=& \ddot{\vec r}_q(t_{Ret}) = -R\omega ^2 \cos \omega t_{Ret} \hat i - R\omega ^2 \sin \omega t_{Ret} \hat j\end{aligned}$$ The acceleration $ \vec a(t_{Ret})$ (Please see the Appendix) in the limit $s \to 0^+$ yields, $$\begin{aligned}
\vec a(t_{Ret}) &=& -R\omega ^2 \cos \omega \left( {t - \frac{s}{c}} \right)\hat i - R\omega ^2 \sin \omega \left( {t - \frac{s}{c}} \right)\hat j \nonumber \\
% &=&R\omega ^2 \left[ {\cos \omega t\cos \left( {\frac{{\omega s}}{c}} \right)\hat i + \sin \omega t\sin \left( {\frac{{\omega s}}{c}} \right)\hat i} \right] \\
% &+& R\omega ^2 \left[ {\sin \omega t\cos \left( {\frac{{\omega s}}{c}} \right)\hat j - \cos \omega t\sin \left( {\frac{{\omega s}}{c}} \right)\hat j} \right] \\
% &=& R\omega ^2 \left[ {\cos \omega t\hat i + \sin \omega t\frac{{\omega s}}{c}\hat i + \sin \omega t\hat j - \cos \omega t\frac{{\omega s}}{c}\hat j} \right] \\
&=& -R\omega ^2 \left[ {\hat r_q - \frac{{\omega s}}{c}\hat {\theta}_q } \right]\end{aligned}$$ The self-force is given by $$\vec F^{Circle}_{self}= \mathop
{Lim}\limits_{s \to 0^ + } \frac{2}{3}\frac{q^2R}{{4\pi \varepsilon
_0 c^2 s}} \omega ^2 \hat r_q - \frac{2}{3}\frac{q^2R}{{4\pi
\varepsilon _0 c^3 }}\omega ^3 \hat {\theta}_q$$ The self-force experienced by the charged particle picks up both the tangential component which is responsible for the radiation reaction as well as the radial component which displays the singular behavior in the limit $s \to 0^+$.
Mass Renormalization
====================
The equation of motion for radiating charged particle is given as: $$\begin{aligned}
m_B \dot {\vec v} &=& \vec F_{Ext} + \vec F_{Self} \nonumber\\%=\vec F_{Ext} + \vec F^{Div}_{Self}+\vec F^{Rad}\nonumber \\
&=& \vec F_{Ext} - \left(\frac{2}{3}\frac{{q^2 }}{{4\pi \varepsilon _0 c^2s }}\right)\dot{ \vec v}(t) +
\frac{2}{3}\frac{{q^2 }}{{4\pi \varepsilon _0 c^3 }}\ddot{ \vec v}(t)\nonumber \\
&=& \vec F_{Ext} - m_{Em} \dot {\vec v}(t) + \frac{2}{3}\frac{{q^2 }}{{4\pi \varepsilon _0 c^3 }}\ddot
{\vec v}(t)\label
{cc}
\end{aligned}$$ where $m_B$ corresponds to the mass of the charged particle that is not associated with the radiation reaction and is called bare mass. The bare mass $m_B$ refers to the physical phenomena at arbitrary short distance surrounding the point charge. The bare mass is not directly related to quantities that one measures at finite spatial length from the charged particle. However, $m_{Em}=\frac{2}{3}\frac{{q^2 }}{{4\pi \varepsilon _0 c^2s }}$, defined as the electromagnetic mass, arises due to the presence of the electromagnetic field. The electromagnetic mass $m_{Em}$ is divergent for the point charge ($s\to 0^+$). Now, we may rewrite equation (\[cc\]) as $$(m_B + m_{Em} )\dot {\vec v}(t) = \vec F_{Ext} +
\frac{2}{3}\frac{{q^2 }}{{4\pi \varepsilon_0 c^3 }}\ddot {\vec
v}(t)$$ In order to tame the divergence, the process of renormalization is implemented as follows: Since a point charge causes an infinite electromagnetic mass, we assume it to be ($+\infty + m_R$) so its bare mass must be postulated to be minus infinite ($-\infty $) so as to render the observable physical (renormalized) mass $$m_R = m_B + m_{Em} = m_B + \frac{2}{3}\frac{{q^2 }}{{4\pi \varepsilon _0 c^2s }}
=\textup{Finite}$$ finite. This shift is known as mass renormalization. The bare mass and the electromagnetic mass are themselves not physical observables.\
In the case of charged particle executing simple harmonic motion, the divergence piece of the self force $ m_{Em}x_0 \omega ^2 \sin
\omega t$ acts away from the equilibrium position. Whereas for the charged particle moving in a circle, the divergent piece of the self-force appears in the form of the centrifugal force $m_{Em}
\omega ^2 R \hat r_q$. To have a sensible theory, these infinities are made to absorb via mass renormalization to obtain the physically observable mass.
Conclusion
==========
We derive the expression for the self-force for a charged particle in arbitrary non-relativistic motion in rather neat and sophisticated way than that presented by Boyer [@boy] in the specific context of a charged particle in uniform circular motion. We discuss illustrative examples pertaining to radiation reaction and obtain explicitly the divergence pieces in the expressions of their respective self-forces.We discuss the concept of mass renormalization which implements the renormalization prescription as to how to tame the divergence.
The author would like to thank the anonymous reviewers for their helpful comments and suggestions.
Calculation of $\vec a(t_{Ret})$ for a Charge Particle Moving in a Circle
=========================================================================
In the limit $s\to 0^+$, we have $$\begin{aligned}
\cos \left( {\omega t - \frac{{\omega s}}{c}} \right)= \cos \omega t\cos\frac{{\omega s}}{c} + \sin\frac{{\omega s}}{c}\sin \omega t\cong \cos \omega t + \frac{{\omega s}}{c}\sin \omega t \\
\sin \left( {\omega t - \frac{{\omega s}}{c}} \right)= \sin \omega t\cos\frac{{\omega s}}{c}-\cos \omega t\sin\frac{{\omega s}}{c} \cong \sin \omega t - \frac{{\omega s}}{c}\cos \omega
t\label{sine}
\end{aligned}$$ The unit vectors $\hat r_q(t_{Ret})$ and $\hat {\theta}_q (t_{Ret})$ in the limit $s\to 0^+$ may be expressed as $$\begin{aligned}
\hat r_q(t_{Ret} ) = \cos \left( {\omega t - \frac{{\omega s}}{c}} \right)\hat i + \sin \left( {\omega t -
\frac{{\omega s}}{c}} \right)\hat j \cong \hat r_q(t) - \frac{{\omega s}}{c}\hat {\theta}_q (t) \\
\hat {\theta}_q (t_{Ret} ) = -\sin \left( {\omega t - \frac{{\omega s}}{c}} \right)\hat i +
\cos \left( {\omega t - \frac{{\omega s}}{c}} \right)\hat j \cong \frac{{\omega s}}{c}\hat r_q(t) + \hat {\theta}_q (t)\end{aligned}$$ The velocity $\dot{\vec r}_q(t_{{\mathop{\rm Re}\nolimits} t} )$ is: $$\begin{aligned}
\dot{\vec r}_q(t_{{\mathop{\rm Re}\nolimits} t} ) &=&R\omega \hat {\theta}_q (t_{{\mathop{\rm Re}\nolimits} t} )
= R\omega \hat {\theta}_q (t - \frac{s}{c})\nonumber \\
&=& R\omega \left[ {\frac{{\omega s}}{c}\hat r_q(t) + \hat {\theta}_q (t)} \right] + O(s^2 )
\end{aligned}$$ The acceleration $ \vec a(t_{{\mathop{\rm Re}\nolimits} t} )$ reads $$\begin{aligned}
\vec a(t_{{\mathop{\rm Re}\nolimits} t} ) &=& \ddot{\vec r}_q(t_{{\mathop{\rm Re}\nolimits} t} ) =
-R\omega ^2 \hat r_q(t_{{\mathop{\rm Re}\nolimits} t} ) \nonumber\\
&=& -R\omega ^2 \left[ {\hat r_q(t) - \frac{{\omega s}}{c}\hat {\theta}_q (t)} \right] + O(s^2 )
\end{aligned}$$
[99]{} T. Kinoshita, *Quantum Electrodynamics* ( T.K. Editor, World Scientific, 1990), p. 471. J. D. Jackson, *Classical Electrodynamics* (New York: John Wiley & Sons, 2003). T. H. Boyer, “Mass Renormalization and Radiation Damping for a Charged Particle in Uniform Circular Motion,” Am. J. Phys. **40**, 1843–1846 (1972). See, for a rigorous definition and the criteria as to how to choose the small surface surrounding the charge, Claudio Teitelboim, “Radiation Reaction as a Retarded Self-Interaction,” Phys. Rev. D **4** (2), 345–347 (1971).
|
---
abstract: |
We explore the connection between absorption by neutral gas and extinction by dust in mid-infrared (IR) selected luminous quasars. We use a sample of 33 quasars at redshifts $0.7< z \lesssim 3$ in the 9 deg$^2$ Boötes multiwavelength survey field that are selected using [*Spitzer Space Telescope*]{} Infrared Array Camera colors and are well-detected as luminous X-ray sources (with $>$150 counts) in [*Chandra*]{} observations. We divide the quasars into dust-obscured and unobscured samples based on their optical to mid-IR color, and measure the neutral hydrogen column density $N_{\rm H}$ through fitting of the X-ray spectra. We find that all subsets of quasars have consistent power law photon indices $\Gamma\approx1.9$ that are uncorrelated with $N_{\rm H}$. We classify the quasars as gas-absorbed or gas-unabsorbed if $N_{\rm H} > 10^{22} \;{\rm
cm}^{-2}$ or $N_{\rm H} < 10^{22} \;{\rm cm}^{-2}$, respectively. Of 24 dust-unobscured quasars in the sample, only one shows clear evidence for significant intrinsic $N_H$, while 22 have column densities consistent with $N_{\rm H} < 10^{22} \;{\rm cm}^{-2}$. In contrast, of the nine dust-obscured quasars, six show evidence for intrinsic gas absorption, and three are consistent with $N_{\rm H} <
10^{22} \;{\rm cm}^{-2}$. We conclude that dust extinction in IR-selected quasars is strongly correlated with significant gas absorption as determined through X-ray spectral fitting. These results suggest that obscuring gas and dust in quasars are generally co-spatial, and confirm the reliability of simple mid-IR and optical photometric techniques for separating quasars based on obscuration.
author:
- 'S.M. Usman'
- 'S.S. Murray'
- 'R.C. Hickox'
- 'M. Brodwin'
title: OBSCURATION BY GAS AND DUST IN LUMINOUS QUASARS
---
Introduction
============
In the past decade, sensitive mid-infrared (IR) observations with the [*Spitzer Space Telescope*]{} and [*Wide-Field Infrared Explorer*]{} (WISE) have forever changed our understanding of BH growth by unveiling, at last, large populations of obscured quasars. Many luminous type 2 active galactic nuclei (AGNs) have been identified from narrow optical emission lines (@zaka03 [@zaka04; @zaka05; @reye08qso2]), radio luminosity (e.g., @mcca93highzradio [@mart06; @seym07radiohosts]), or X-ray properties (e.g, @alex01xfaint [@ster02; @trei04; @vign06qso2; @vign09qso2]), but the most efficient techniques for finding obscured quasars employ mid-IR photometry. Pioneering work with [*Spitzer*]{} showed that obscured quasars have similar mid-IR SEDs to their unobscured counterparts, but are dominated by host galaxy light in the optical (@lacy04 [@ster05; @rowa05; @poll06; @alon06; @donl08spitz; @hick07abs], hereafter H07). In the mid-IR, obscured quasars can be reliably selected using simple color criteria, in particular being very red in \[3.6\]–\[4.5\] (characteristic of a “hot” mid-IR SED) and very red in $R-[4.5]$, indicating a faint optical counterpart (Fig.1). Mid-IR studies find roughly equal numbers of obscured and unobscured quasars (e.g., ).
Obscured quasars therefore represent a large fraction of the massive BH growth in the Universe, but their precise nature remains a mystery. In particular, what is the origin of the obscuring material, and what role do these objects play in the evolution of black holes and galaxies? The simple “unified model” for AGN attributes obscuration to the orientation of a gas- and dust-rich torus intrinsic to the central engine (e.g., @anto93 [@ball06b]), but it is not clear whether this model applies to objects with quasar luminosities. A competing hypothesis is that quasars are fueled by major mergers of galaxies that drive gas and dust clouds to the nucleus, obscuring the central engine, as suggested in models of BH-galaxy co-evolution (e.g., @sand88 [@hopk08frame1]).
One clue about the nature of the obscuring material is the connection between obscuration by [*dust*]{} which is manifested through extinction of rest-frame ultraviolet and optical nuclear light, and absorption by neutral [*gas*]{} which is detectable by its effect on the observed X-ray spectrum. In the simplest unified scenarios, neutral gas and dust are co-spatial and so both types of obscuration should be observed in the same systems. This is in fact what is observed in most obscured AGNs, however $\sim$30% of moderate-luminosity AGNs show a mismatch between optical and X-ray classification criteria [e.g., @tozz06; @trou09optx], suggesting some deviation from the simplest unified scenario.
![$L_{4.5 \mu m} vs. L_{R}$ (calculated as $\nu L_\nu$ in the observed frame) for IR-selected quasars. The selection boundary of $log(L_{R}/L_{4.5\mu})=-0.4$ (corresponding to $R-[4.5] = 6.1$ in Vega magnitudes) is shown as a dashed line with IRQSO-1s and IRQSO-2s residing above and below the line, respectively. Contours are derived from IR source density. The filled blue stars (XQSO-1), purple squares (XQSO-1.5), and red circles (XQSO-2) are the X-ray classifications for our X-ray spectroscopic sample, discussed in §3.\[fig:1\]](f1-eps-converted-to.pdf){width="\columnwidth"}
Obscuration is particularly interesting for powerful quasars (given the expected importance of merger fueling) but the obscuration properties of luminous AGN have been less well studied due to their relative rarity. Furthermore, it is also interesting to study the obscuration in luminous quasars given the reported decreasing trend of obscured AGN at increasing luminosities (e.g., @ued03 [@has05; @merl14agnobs]). A recent study of X-ray selected AGNs in the COSMOS field @merl14agnobs found that at quasar luminosities ($L_X> 10^{44}$ erg s$^{-1}$) $\sim$80% of AGNs have classifications for dust and gas obscuration that agree, with the majority of the “mismatch” corresponding to optically-unobscured but gas-obscured sources. Previous studies of such objects have suggested they are preferentially hosted by more rapidly star-forming galaxies, indicating that the X-ray absorbing gas arises from galaxy-scale structures that are far larger than the putative torus, and may be associated with an outflowing wind [@page04submm]. However, @merl14agnobs found that X-ray obscured, optically-unobscured quasars have spectra and photometric properties that are identical to their X-ray unobscured counterparts, suggesting that the absorption may instead be due to small-scale clouds within the putative torus.
In this study, we further explore the connection between gas and dust obscuration in luminous quasars, using observations from the wide-field (9 deg$^2$) Boötes multiwavelength survey area that enable efficient selection of rare, luminous sources. We analyze X-ray spectra of 33 high-luminosity IR-selected quasars in the XBoötes Deep Survey. We find a strong correspondence between dust and gas obscuration in luminous quasars, consistent with the predictions of unified models as well as simple evolutionary scenarios in which the gas and dust are co-spatial.
Observations and quasar sample
==============================
Our sample of quasars is drawn from , who selected 1479 luminous AGNs based on the mid-IR color criteria of @ster05. The vast majority of these sources have estimates of bolometric luminosity $L_{\rm bol} > 10^{12} L_{\sun}$ corresponding to the commonly-used division between “Seyfert galaxies” and “quasars” [e.g., @hopk09fueling], so for the remainder of this paper we refer to the sample objects as quasars. The mid-IR observations come from the Spitzer IRAC Shallow Survey [@eise04; @brod06] and optical photometry from the NOAO Deep Wide Field Survey @jann99, and is limited to redshifts $0.7 < z < 3$ determined by optical spectroscopy from the AGN and Galaxy Evolution Survey (AGES; @koch12ages) or using photometric redshifts from @brod06. found that the full IR-selected quasar sample could be easily divided into dust-obscured and unobscured sources based on their optical to mid-IR color, as they exhibited a bimodal color distribution with a boundary at $R-[4.5] = 6.1$ (or equivalently $\log(L_{R}/L_{{\rm 4.5\mu m}})=-0.4$, where $L_{R}$ and $L_{{\rm 4.5\mu m}}$ are the luminosities in the [ *observed*]{}-frame $R$ and 4.5 micron bands; see Figure \[fig:1\]). We also include estimates of quasar bolometric luminosity derived from fitting mid-IR spectral energy distributions, as described by @hick11qsoclust.
showed that the objects with blue $R-[4.5]$ colors are dominated in the optical by unobscured light from the quasar nucleus, while the nucleus is obscured in the redder sources such that the optical light is dominated by the host galaxy. also showed via an X-ray stacking analysis that the average X-ray spectra of unobscured quasars were consistent with no absorption by neutral gas, while the obscured quasars had harder average X-ray spectra indicating significant gas absorption with $N_{\rm H} \sim
3\times10^{22}$ cm$^{-2}$. However, given the shallow (5 ks; @murr05) X-ray observations available in the analysis, this comparison was only possible for [ *average*]{} X-ray hardness ratios and could not explore variations in X-ray spectra between individual sources.
This study expands on that work by analyzing deeper [*Chandra*]{} ACIS observations (including from the XBoötes Deep Survey) of quasars in the sample. The deeper observations consist of 36 exposures between 2001 and 2012 with an average exposure time of 41.6 ks, of which 13 are with the ACIS-I array and 13 are with ACIS-S (Table \[tab:1\]). The data were reprocessed using the CHAV (v.2011.01.25) and CIAO 4.1.2 (CALDB 4.1.4) packages. To obtain sufficient quality X-ray spectra we limit our analyses to objects with $>$150 counts in the 0.2-7 keV band. We match the sources to the 1479 IR-selected quasars from using the TOPCAT [@tayl05topcat] cone-search algorithm with a 10 radius. A total of 33 quasars were selected, with an average of $\approx\;300$ counts per source.
Using the $R-[4.5]$ color criterion, 24 of the IR-selected quasars in our sample are classified as unobscured (IRQSO-1) while 9 are obscured (IRQSO-2). Our bright X-ray spectroscopic sample contains a low fraction of IRQSO-2s (9/33 or 27%) that is significantly smaller than the 43% obscured fraction in the full IRQSO sample from . This is due to the fact that X-ray QSO2s are generally fainter in X-rays; the fraction of IRQSOs with X-ray counterparts in original shallow (5 ks) XBoötes survey is $\approx$60% for IRQSO-1s and only $\approx$30% for IRQSO-2s, while the average X-ray flux of the undetected sources, determined from stacking, is also systematically lower for the IRQSO-2s . The fainter X-ray emission from IRQSO-2s is most likely due to higher levels of gas obscuration that not only hardens the X-ray spectrum but reduces the observed flux [e.g., @alex08compthick]. This fact reflects a correspondence between gas and dust obscuration in these quasars, which we aim to test further by direct measurement of the obscuring column density ($N_{\rm H}$) in the X-ray spectra of the bright sources for which these measurements are possible.
{width="7.0in"}
X-Ray Analysis
==============
We extract source and background spectra and produce response functions using standard CIAO software. Sources and background spectra were extracted from hand-selected circular regions with mean radii of 11.5 pixels and 100.1 pixels, respectively. The resulting spectra were fit using SHERPA (4.1.2) with a one-dimensional power law (xspowerlaw.p1) convolved with two neutral hydrogen absorption laws, one for the Milky Way (xswabs.abs1), and one for the host galaxy (xszwabs.abs2) absorption. The Milky Way absorption column density was frozen at the mean of all 33 quasars, $N_{\rm H-MW} = 1.05 \times
10^{20}$ cm$^{-2}$, derived from the Chandra X-ray Center’s COLDEN calculator. The host galaxy redshifts are fixed at values taken either from AGES spectroscopic measurements [@koch12ages] or photometric redshifts [@brod06], as described in . The power-law index $\Gamma$, normalization, and intrinsic $N_{\rm H}$ were allowed to vary in the spectral fitting. Examples of the X-ray spectral fits are shown in Figure \[fig:2\] for one source showing no evidence for gas absorption, and another with detectable $N_{\rm H}$.
[ccccccccc]{}
SDWFS J142942.63+335654.94 & 47.1 & 1.1251 & 607 & 17.3 & 45.1 & $1.77 \pm 0.09$ & $0+0.07$ & 1\
SDWFS J142810.31+353847.31 & 37.9 & 0.8028 & 852 & 16.5 & 44.7 & $1.80 \pm 0.07$ & $0+0.02$ & 1\
SDWFS J143520.20+340929.20 & 42.6 & 1.0972 & 182 & 1.75 & 44.1 & $2.13 \pm 0.20$ & $0+0.08$ & 1\
SDWFS J143520.60+340514.68 & 42.6 & 0.7973 & 217 & 3.38 & 44.0 & $1.59 \pm 0.14$ & $0+0.06$ & 1\
SDWFS J143513.41+350053.77 & 44.0 & 1.1471 & 325 & 3.50 & 44.4 & $1.81 \pm 0.13$ & $0+0.15$ & 1\
SDWFS J143520.14+350413.23 & 44.0 & 1.0512 & 339 & 1.10 & 43.8 & $2.30 \pm 0.18$ & $0+0.13$ & 1\
SDWFS J142922.94+351517.74 & 42.0 & 0.9041 & 408 & 10.4 & 44.6 & $1.87 \pm 0.14$ & $0+0.09$ & 1\
SDWFS J142634.05+351602.56 & 14.9 & 1.1055 & 177 & 6.61 & 44.7 & $1.61 \pm 0.17$ & $0+0.17$ & 1\
SDWFS J143651.98+350537.97 & 54.3 & 0.864 & 200 & 0.54 & 43.3 & $2.49 \pm 0.21$ & $0+0.05$ & 1\
SDWFS J142651.47+351924.40 & 29.7 & 1.756 & 833 & 12.4 & 45.4 & $1.83 \pm 0.07$ & $0+0.07$ & 1\
SDWFS J142839.20+353455.55 & 37.9 & 1.0693 & 228 & 5.31 & 44.5 & $1.70 \pm 0.16$ & $0+0.35$ & 1\
SDWFS J142829.91+342758.68 & 68.6 & 1.1401 & 301 & 3.31 & 44.4 & $1.93 \pm 0.16$ & $0+0.41$ & 1\
SDWFS J143245.88+333758.45 & 33.8 & 1.0816 & 192 & 3.43 & 44.3 & $2.13 \pm 0.26$ & $0+0.34$ & 1\
SDWFS J142917.20+342130.31 & 28.7 & 1.2734 & 306 & 6.06 & 44.8 & $2.07 \pm 0.18$ & $0+0.56$ & 1\
SDWFS J142607.71+340426.61 & 51.2 & 4.32 & 468 & 7.27 & 46.1 & $2.67 \pm 0.12$ & $0+0.94$ & 1\
SDWFS J143310.25+335421.99 & 27.7 & 0.8792 & 238 & 9.62 & 44.6 & $1.79 \pm 0.33$ & $0.57+0.75$ & 1.5\
SDWFS J143153.67+344138.17 & 50.2 & 0.8254 & 481 & 8.16 & 44.4 & $1.72 \pm 0.17$ & $1.31\pm 0.30$ & 2\
SDWFS J142658.70+324003.78 & 42.0 & 1.7737 & 212 & 5.45 & 45.1 & $1.56 \pm 0.25$ & $0+1.67$ & 1.5\
SDWFS J142915.19+343820.18 & 29.7 & 2.3515 & 202 & 3.87 & 45.2 & $2.12 \pm 0.27$ & $0+1.83$ & 1.5\
SDWFS J142622.68+334202.41 & 33.8 & 1.3517 & 182 & 4.27 & 44.7 & $1.94 \pm 0.31$ & $0.42+1.41$ & 1.5\
SDWFS J143434.40+330549.14 & 56.8 & 2.0656 & 154 & 0.49 & 44.2 & $2.39 \pm 0.45$ & $0.76+2.38$ & 1.5\
SDWFS J142859.55+350349.11 & 42.0 & 1.6126 & 165 & 2.89 & 44.7 & $2.19 \pm 0.45$ & $1.28+2.21$ & 1.5\
SDWFS J143431.07+332825.51 & 39.9 & 1.0554 & 155 & 4.70 & 44.5 & $1.76 \pm 0.38$ & $1.45+1.68$ & 1.5\
SDWFS J143450.01+352520.65 & 30.7 & 0.9629 & 184 & 8.17 & 44.6 & $2.56 \pm 0.42$ & $1.98\pm 1.09$ & 1.5\
SDWFS J143210.97+343957.89 & 81.9 & 0.83 & 204 & 1.93 & 43.8 & $1.66 \pm 0.24$ & $0+0.66$& 1\
SDWFS J142824.97+352842.63 & 37.9 & 1.49 & 217 & 4.90 & 44.8 & $1.98 \pm 0.20$ & $0+0.71$& 1\
SDWFS J142845.07+350903.30 & 42.0 & 0.81 & 463 & 9.52 & 44.5 & $1.87 \pm 0.20$ & $0.72 \pm 0.42$ & 1.5\
SDWFS J142813.98+325502.82 & 33.8 & 1.54 & 264 & 10.8 & 45.2 & $1.08 \pm 0.28$ & $2.21+3.35$& 1.5\
SDWFS J143502.04+330556.51 & 56.8 & 1.58 & 299 & 5.09 & 44.9 & $1.84 \pm 0.25$ & $3.07 \pm 1.62$& 2\
SDWFS J143503.49+340241.92 & 42.6 & 1.06 & 219 & 2.54 & 44.2 & $2.85 \pm 0.44$ & $3.41 \pm 1.35$& 2\
SDWFS J143359.09+331301.06 & 42.0 & 0.92 & 330 & 17.3 & 44.9 & $1.64 \pm 0.31$ & $3.97 \pm 1.49$& 2\
SDWFS J142916.10+335537.36 & 24.6 & 0.98 & 220 & 7.30 & 44.6 & $2.36 \pm 0.34$ & $4.34 \pm 1.21$ & 2\
SDWFS J142707.05+325214.17 & 42.0 & 2.28 & 244 & 5.65 & 45.4 & $1.97 \pm 0.28$ & $5.66 \pm 2.98$& 2
The spectral fit parameters ($\Gamma$ and intrinsic $N_{\rm H}$) and $1\sigma$ errors along with other quasar properties are shown in Table 1. In many cases, the observed $N_{\rm H}$ is consistent with zero and only upper limits are obtained. We classify the quasars as gas-absorbed (XQSO-2) or gas-unabsorbed (XQSO-1) if $N_{\rm H} > 10^{22} \;{\rm cm}^{-2}$ or $N_{\rm H} < 10^{22} \;{\rm
cm}^{-2}$, respectively, following the convention used widely in the literature (e.g., @tozz06 [@lan13; @ued14]). For several sources (XQSO-1.5) the classification is ambiguous, with $N_{\rm H}$ consistent with either subset within the uncertainties. (We note that a division at $N_{\rm H}=10^{21.5}$ cm$^{-2}$, as in @merl14agnobs, produces more ambiguous classifications due to the uncertainties in our $N_{\rm H}$ measurements, but results in the same qualitative conclusions.) In Figure \[fig:1\] we show the IR and optical luminosities of the quasars in different X-ray classes, and in Figure \[fig:gamma\_nh\] we show the distribution in $\Gamma$ and $N_{\rm H}$ for all the sources. We note that there is no clear correlation between $\Gamma$ and $N_{\rm H}$, verifying that we have sufficient counts in each source to sufficiently break the degeneracy between those parameters. The error-weighted average value of $\Gamma$ is $1.89\pm0.03$ for the full sample, and for the subsets is $1.90\pm0.03$ (XQSO-1), $1.83\pm0.10$ (XQSO-1.5) and $1.90\pm0.11$ (XQSO-2), all consistent with previous measurements for the intrinsic photon index for AGNs and quasars [e.g., @tozz06; @xue11cdfs].
![X-ray spectral fit parameters for the full sample of 33 IR- and X-ray selected quasars. The main panel shows $\Gamma$ and $N_{\rm H}$ with uncertainties, highlighting the populaton of XQSO-1.5s with ambiguous classifications, and showing that there is no clear correlation between $\Gamma$ and $N_{\rm H}$. Dust unobscured (IRQSO-1) sources are marked with open symbols. The top panel shows the fractional distribution of the best-fit values of $N_{\rm H}$ for the IRQSO-1s and 2s, respectively, showing the clear correlation between dust and gas obscuration. The other two panels show the distributions in best-fit $N_{\rm H}$ and $\Gamma$ for the XQSO-1s, 1.5s, and 2s separately. \[fig:gamma\_nh\]](f3-eps-converted-to.pdf){width="\columnwidth"}
The primary result of this analysis is the correspondence between the optical/mid-IR and X-ray obscuration criteria. Among the 24 dust-unobscured (IRQSO-1) quasars, we find 15 XQSO-1s, 8 XQSO-1.5s (of which 7 have $N_{\rm H}$ consistent with zero), and only one XQSO-2. In contrast, the 9 dust-obscured quasars (IRQSO-2s) comprise only 2 XQSO-1s, 2 XQSO-1.5s (of which one has $N_{\rm H}$ consistent with zero) and 5 XQSO 2s. We thus obtain a strong, if not perfect, correlation between absorption by gas and dust, with only a few examples showing clearly anomalous gas absorption for the observed optical and mid-IR properties (broadly consistent with the results of @merl14agnobs. The largest $N_{\rm
H}$ observed in our XQSO-2 sample is relatively modest (only a few $\times10^{22}$ cm$^{-2}$); it is likely that more heavily obscured sources ($N_{\rm H}>10^{23}$) are excluded from our X-ray spectroscopic sample because their observed fluxes are too faint to yield the required numbers of counts, as discussed in § 2.
Discussion and conclusions
==========================
Overall, these results indicate that obscuration by gas and dust are strongly correlated in IR-selected luminous quasars. This correspondence indicates that gas and dust obscuration generally arise in the same structures, consistent with the predictions of the unified model but also with the simplest evolutionary models. We thus cannot draw robust conclusions about whether the obscuring material is in a small-scale torus or due to larger-scale galactic structures. A recent study of their far-IR properties using 250 $\mu$m data from [*Herschel*]{} suggests that the IRQSO-2s have higher average rates of star formation than IRQSO-1s (Chen et al. 2014, in prep), suggesting that at least some quasars are obscured by galaxy-scale material associated with rapid star formation. However, only 4 of our 33 objects are detected at 250 $\mu$m with [*Herschel*]{} and of these two show neither gas nor dust obscuration, while the other two are obscured in both classifications, so it is unclear whether large-scale material is responsible for any of the observed obscuration.
We find that only a small fraction of dust-unobscured quasars show [*clear*]{} X-ray absorption signatures. Of the quasars in our X-ray spectroscopic sample, $\approx$17% are IRQSO-1s with best-fit values of $N_{\rm H}>10^{22}$ cm$^{-2}$, similar to the fraction of such mismatches found by @merl14agnobs. We note however that all but one of these objects are consistent with having little or no gas absorption, and their distribution in the plane IR and optical luminosity (Figure 1) is indistinguishable to IRQSO-1s with no X-ray absorption. We therefore caution that the observed incidence of these mismatches must be treated as an upper limit, and that the X-ray absorbed broad-line quasars may in fact be rare in the full quasar population; an accurate census of quasars with modest obscuration provides strong motivation for future high-throughput X-ray observatories.
Overall our results serve to verify that obscured quasar selection based on optical to mid-IR color preferentially identifies systems that show evidence for obscuration at other wavelengths. Even in our bright X-ray spectroscopic sample, for which the effective flux limit biases us [*against*]{} heavily X-ray obscured sources as discussed in § 2, we still find that the majority of dust-obscured quasar candidates show clear evidence for X-ray absorption. This indicates that the full population of dust-obscured quasars likely has a very high incidence of corresponding gas absorption. These results confirm that optical/mid-IR color selection is effective in selecting even moderately obscured quasars at the highest luminosities, providing a strong basis for future large statistical studies of obscured quasars selected based on WISE and optical photometry.
This work was supported by NASA through ADAP award NNX12AE38G and by the National Science Foundation through grant number 1211096.This work is based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. This work is based on observations with the Chandra X-ray Telescope, which is operated by SAO under a contract with NASA NAS8-03060.
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abstract: 'This work addresses the problem of semantic image segmentation of nighttime scenes. Although considerable progress has been made in semantic image segmentation, it is mainly related to daytime scenarios. This paper proposes a novel method to *progressive* adapt the semantic models trained on daytime scenes, along with large-scale annotations therein, to nighttime scenes via the bridge of twilight time — the time between dawn and sunrise, or between sunset and dusk. The goal of the method is to alleviate the cost of human annotation for nighttime images by transferring knowledge from standard daytime conditions. In addition to the method, a new dataset of road scenes is compiled; it consists of 35,000 images ranging from daytime to twilight time and to nighttime. Also, a subset of the nighttime images are densely annotated for method evaluation. Our experiments show that our method is effective for knowledge transfer from daytime scenes to nighttime scenes, without using extra human annotation.'
author:
- 'Dengxin Dai$^{1}$ and Luc Van Gool$^{1,2}$[^1][^2]'
bibliography:
- 'IEEEfull.bib'
title: '**Dark Model Adaptation: Semantic Image Segmentation from Daytime to Nighttime** '
---
INTRODUCTION
============
Autonomous vehicles will have a substantial impact on people’s daily life, both personally and professionally. For instance, automated vehicles can largely increase human productivity by turning driving time into working time, provide personalized mobility to non-drivers, reduce traffic accidents, or free up parking space and generalize valet service [@autonomous:vehicle:guide:policymakers]. As such, developing automated vehicles is becoming the core interest of many, diverse industrial players. Recent years have witnessed great progress in autonomous driving [@drive:surroundview:route:planner], resulting in announcements that autonomous vehicles have driven over many thousands of miles and that companies aspire to sell such vehicles in a few years. All this has fueled expectations that fully automated vehicles are coming soon. Yet, significant technical obstacles must be overcome before assisted driving can be turned into full-fletched automated driving, a prerequisite for the above visions to materialize.
While perception algorithms based on visible light cameras are constantly getting better, they are mainly designed to operate on images taken at daytime under good illumination [@vision:atmosphere; @semantic:foggy:scene]. Yet, outdoor applications can hardly escape from challenging weather and illumination conditions. One of the big reasons that automated cars have not gone mainstream yet is because it cannot deal well with nighttime and adverse weather conditions. Camera sensors can become untrustworthy at nighttime, in foggy weather, and in wet weather. Thus, computer vision systems have to function well also under these adverse conditions. In this work, we focus on semantic object recognition for nighttime driving scenes.
Robust object recognition using visible light cameras remains a difficult problem. This is because the structural, textural and/or color features needed for object recognition sometimes do not exist or highly disbursed by artificial lights, to the point where it is difficult to recognize the objects even for human. The problem is further compounded by camera noise [@nighttime:noise:reduction:16] and motion blur. Due to this reason, there are systems using far-infrared (FIR) cameras instead of the widely used visible light cameras for nighttime scene understanding [@night:vision:pedestrian:05; @day:night:16]. Far-infrared (FIR) cameras can be another choice [@night:vision:pedestrian:05; @day:night:16]. They, however, are expensive and only provide images of relatively low-resolution. Thus, this work adopts visible light cameras for semantic segmentation of nighttime road scenes. Another reason of this choice is that large-scale datasets are available for daytime images by visible light cameras [@Cityscapes]. This makes model adaptation from daytime to nighttime feasible.
High-level semantic tasks is usually tackled by learning from many annotations of real images. This scheme has achieved a great success for good weather conditions at daytime. Yet, the difficulty of collecting and annotating images for all other weather and illumination conditions renders this standard protocol problematic. To overcome this problem, we depart from this traditional paradigm and propose another route. Instead, we choose to *progressively* adapt the semantic models trained for daytime scenes to nighttime scenes, by using images taken at the twilight time as intermediate stages. The method is based on progressively self-learning scheme, and its pipeline is shown in Figure \[fig:pipeline\].
\[fig:pipeline\]
The main contributions of the paper are: 1) a novel model adaptation method is developed to transfer semantic knowledge from daytime scenes to nighttime scenes; 2) a new dataset, named *Nighttime Driving*, consisting of images of real driving scenes at nighttime and twilight time, with $35,000$ unlabeled images and $50$ densely annotated images. These contributions will facilitate the learning and evaluation of semantic segmentation methods for nighttime driving scenes. *Nighttime Driving* is available at <http://people.ee.ethz.ch/~daid/NightDriving/>.
Related Work {#sec:related}
============
Semantic Understanding of Nighttime Scenes
------------------------------------------
A lot of work for nighttime object detection/recognition has focused on human detection, by using FIR cameras [@night:vision:pedestrian:05; @pedestrian:detection:tracking:night:09] or visible light cameras [@cnn:human:detection:nighttime:17], or a combination of both [@nighttime:pedestrian:detection:08]. There are also notable examples for detecting other road traffic objects such as cars [@nighttime:object:proposal:18] and their rear lights [@night:rear:lights:16]. Another group of work is to develop methods robust to illumination changes for robust road area detection [@road:detection:illumination:invariant] and semantic labeling [@outdoor:transformation:labeling:iv15]. Most of the research in this vein had been conducted before deep learning was widely used.
Semantic understanding of visual scenes have recently undergone rapid growth, making accurate object detection feasible in images and videos in daytime scenes [@DomainAdaptiveFasterRCNN; @refinenet]. It is natural to raise the question of how to extend those sophisticated methods to other weather conditions and illumination conditions, and examine and improve the performance therein. A recent effort has been made for foggy weather [@semantic:foggy:scene]. This work would like to initiate the same research effort for nighttime.
Model Adaptation
----------------
The concurrent work in [@SynRealDataFogECCV18] on adaptation of semantic models from clear weather condition to light fog then to dense fog is closely related to ours.
Road Scene Understanding
------------------------
Road scene understanding is a crucial enabler for applications such as assisted or autonomous driving. Typical examples include the detection of roads [@recent:progress:lane], traffic lights [@traffic:light:survey:16], cars and pedestrians [@Cityscapes; @semantic:foggy:scene], and tracking of such objects [@vehicles:road:survey:13; @pathtrack]. We refer the reader to the excellent surveys [@looking:at:human]. The aim of this work is to extend/adapt the advanced models developed recently for road scene understanding at daytime to nighttime, without manually annotating nighttime images.
Approach {#sec:approach}
========
Training a segmentation model with large amount of human annotations should work for nighttime images, similar to what has been achieved for daytime scene understanding [@MastRCNN; @refinenet]. However, applying this protocol to other weather conditions and illumination conditions is problematic as it is hardly affordable to annotate the same amount of data for all different conditions and their combinations. We depart from this protocol and investigate an automated approach to transfer the knowledge from existing annotations of daytime scenes to nighttime scenes. The approach leverages the fact that illumination changes continuously between daytime and nighttime, through the twilight time. Twilight is the time between dawn and sunrise, or between sunset and dusk. Twilight is defined according to the solar elevation angle, which is the position of the geometric center of the sun relative to the horizon [@twilight:definition]. See Figure \[fig:twillight\] for an illustration.
During a large portion of twilight time, solar illumination suffices enough for cameras to capture the terrestrial objects and suffices enough to alleviate the interference of artificial lights to a limited amount. See Figure \[fig:pipeline\] for examples of road scenes at twilight time. These observations lead to our conjecture that the domain discrepancy between daytime scenes and twilight scenes, and the the domain discrepancy between twilight scenes and nighttime scenes are both smaller than the domain discrepancy between daytime scenes and nighttime scenes. Thus, images captured during twilight time can serve our purpose well — transfer knowledge from daytime to nighttime. That is, twilight time constructs a bridge for knowledge transfer from our source domain daytime to our target domain nighttime.
![Twilight is defined according to the solar elevation angle and is categorized into three subcategories: civil twilight, nautical twilight, and astronomical twilight. (picture is from wikipedia).[]{data-label="fig:twillight"}](Twilight2.png){width="0.9\linewidth"}
In particular, we train a semantic segmentation model on daytime images using the standard supervised learning paradigm, and apply the model to a large dataset recorded at civil twilight time to generate the class responses. The three subgroups of twilight are used: civil twilight, nautical twilight, and astronomical twilight [@twilight:definition]. Since the domain gap between daytime condition and civil twilight condition is relatively small, these class responses, along with the images, can then be used to fine-tune the semantic segmentation model so that it can adapt to civil twilight time. The same procedure is continued through nautical twilight and astronomical twilight. We then apply the final fine-tuned model to nighttime images.
This learning approach is inspired by the stream of work on model distillation [@hinton2015distilling; @dai:metric:imitation; @supervision:transfer]. Those methods either transfer supervision from sophisticated models to simpler models for efficiency [@hinton2015distilling; @dai:metric:imitation], or transfer supervision from the domain of images to other domains such as depth maps [@supervision:transfer]. We here transfer the semantic knowledge of annotations of daytime scenes to nighttime scenes via the unlabeled images recorded at twilight time.
Let us denote an image by $\mathbf{x}$, and indicate the image taken at *daytime*, *civil twilight time*, *nautical twilight time*, *astronomical twilight time* and *nighttime* by $\mathbf{x}^0$, $\mathbf{x}^1$, $\mathbf{x}^2$, $\mathbf{x}^3$, and $\mathbf{x}^4$, respectively. The corresponding human annotation for $\mathbf{x}^0$ is provided and denoted by $\mathbf{y}^0$, where $\mathbf{y}^0(m,n) \in\{1, ..., C\}$ is the label of pixel $(m,n)$, and $C$ is the total number of classes. Then, the training data consist of labeled data at daytime $\mathcal{D}^0 =\{(\mathbf{x}^0_i, \mathbf{y}^0_{i})\}_{i=1}^{l^0}$, and three unlabeled datasets for the three twilight categories: $\mathcal{D}^1=\{\mathbf{x}^1_{j}\}_{j=1}^{l^1}$, $\mathcal{D}^2=\{\mathbf{x}^2_{k}\}_{k=1}^{l^2}$, and $\mathcal{D}^3=\{\mathbf{x}^3_{q}\}_{q=1}^{l^3}$, where $l^0$, $l^1$, $l^2$, and $l^3$ are the total number of images in the corresponding datasets. The method consists of eight steps and it is summarized below.
1. train a segmentation model with daytime images and the human annotations: $$\min_{\phi^0} \frac{1}{l^0}\sum_{i=1}^{l^0} L(\phi^0(\mathbf{x}^0_i), \mathbf{y}^0_i),$$ where $L(.,.)$ is the cross entropy loss function; \[item1\]
2. apply segmentation model $\phi^0$ to the images recorded at civil twilight time to obtain “noisy” semantic labels: $\hat{\mathbf{y}}^1 = \phi^0(\mathbf{x}^1)$, and augment dataset $\mathcal{D}^1$ to $\hat{\mathcal{D}}^1$: $\hat{\mathcal{D}}^1=\{(\mathbf{x}^1_j, \hat{\mathbf{y}}^1_j)\}_{j=1}^{l^1}$; \[item2\]
3. instantiate a new model $\phi^1$ by duplicating $\phi^0$, and then fine-tune (retrain) the semantic model on $\mathcal{D}^0$ and $\hat{\mathcal{D}}^1$: $$\phi^1 \leftarrow \phi^0,$$ and $$\min_{\phi^1} \Big(\frac{1}{l^0}\sum_{i=1}^{l^0} L(\phi^1(\mathbf{x}^0_i), \mathbf{y}^0_i) + \frac{\lambda^1 }{l^1}\sum_{j=1}^{l^1} L(\phi^1(\mathbf{x}^1_j), \hat{\mathbf{y}}^1_j) \Big),
\label{eq:step3}$$ where $\lambda^1$ is a hyper-parameter balancing the weights of the two data sources; \[item3\]
4. apply segmentation model $\phi^1$ to the images recorded at nautical twilight time to obtain “noisy” semantic labels: $\hat{\mathbf{y}}^2 = \phi^1(\mathbf{x}^2)$, and augment dataset $\mathcal{D}^2$ to $\hat{\mathcal{D}}^2$: $\hat{\mathcal{D}}^2=\{(\mathbf{x}^2_k, \hat{\mathbf{y}}^2_k)\}_{k=1}^{l^2}$; \[item4\]
5. instantiate a new model $\phi^2$ by duplicating $\phi^1$, and fine-tune (train) semantic model on $\mathcal{D}^0$, $\hat{\mathcal{D}}^1$ and $\hat{\mathcal{D}}^2$: $$\phi^2 \leftarrow \phi^1,$$ and then $$\begin{gathered}
\min_{\phi^2} \Big( \frac{1}{l^0}\sum_{i=1}^{l^0} L(\phi^2(\mathbf{x}^0_i), \mathbf{y}^0_i) + \frac{\lambda^1}{l^1}\sum_{j=1}^{l^1} L(\phi^2(\mathbf{x}^1_j), \hat{\mathbf{y}}^1_j) \\
+ \frac{\lambda^2}{l^2}\sum_{k=1}^{l^2} L(\phi^2(\mathbf{x}^2_k), \hat{\mathbf{y}}^2_k) \Big),
\label{eq:step5}\end{gathered}$$ where $\lambda^1$ and $\lambda^2$ are hyper-parameters regulating the weights of the datasets; \[item5\]
6. apply segmentation model $\phi^2$ to the images recorded at astronomical twilight data to obtain “noisy” semantic labels: $\hat{\mathbf{y}}^3 = \phi^2(\mathbf{x}^3)$, and augment dataset $\mathcal{D}^3$ to $\hat{\mathcal{D}}^3$: $\hat{\mathcal{D}}^3=\{(\mathbf{x}^3_q, \hat{\mathbf{y}}^3_q)\}_{q=1}^{l^3}$; ; \[item6\]
7. instantiate a new model $\phi^3$ by duplicating $\phi^2$, and fine-tune (train) the semantic model on all four datasets $\mathcal{D}^0$, $\hat{\mathcal{D}}^1$, $\hat{\mathcal{D}}^2$ and $\hat{\mathcal{D}}^3$: $$\phi^3 \leftarrow \phi^2,$$ and then $$\begin{gathered}
\min_{\phi^3} \Big( \frac{1}{l^0}\sum_{i=1}^{l^0} L(\phi^3(\mathbf{x}^0_i), \mathbf{y}^0_i) + \frac{\lambda^1}{l^1}\sum_{j=1}^{l^1} L(\phi^3(\mathbf{x}^1_j), \hat{\mathbf{y}}^1_j) \\
+ \frac{\lambda^2}{l^2}\sum_{k=1}^{l^2} L(\phi^3(\mathbf{x}^2_k), \hat{\mathbf{y}}^2_k) + \frac{\lambda^3}{l^3}\sum_{q=1}^{l^3} L(\phi^3(\mathbf{x}^3_q), \hat{\mathbf{y}}^3_q) \Big),
\label{eq:step7}\end{gathered}$$ where $\lambda^1$, $\lambda^1$ and $\lambda^3$ are hyper-parameters regulating the weights of the datasets; \[item7\]
8. apply model $\phi^3$ to nighttime images to perform the segmentation: $\hat{\mathbf{y}}^4 = \phi^3(\mathbf{x}^4)$.
We term our method Gradual Model Adaptation. During training, in order to balance the weights of different data sources (in Equation \[eq:step3\], Equation \[eq:step5\] and Equation \[eq:step7\]), we empirically give equal weight to all training datasets. An optimal value can be obtained via cross-validation. The optimization of Equation \[eq:step3\], Equation \[eq:step5\] and Equation \[eq:step7\] are implemented by feeding to the training algorithm a stream of hybrid data, for which images in the considered datasets are sampled proportionally according to the parameters $\lambda^1$, $\lambda^2$, and $\lambda^3$. In this work, they all set to $1$, which means all datasets are sampled at the same rate.
Rather than applying the model trained on daytime images directly to nighttime images, Gradual Model Adaptation breaks down the problem to three progressive steps to adapt the semantic model. In each of the step, the domain gap is much smaller than the domain gap between daytime domain and nighttime domain. Due to the unsupervised nature of this domain adaptation, the algorithm will also be affected by the noise in the labels. The daytime dataset $\mathcal{D}^1$ is always used for the training, to balance between noisy data of similar domains and clean data of a distinct domain.
Experiments {#sec:experiment}
===========
Data Collection
---------------
*Nighttime Driving* was collected during 5 rides with a car inside multiple Swiss cities and their suburbs using a GoPro Hero 5 camera. We recorded 5 large video sequence with length of about 2 hours. The video recording starts from daytime, goes through twilight time and ends at full nighttime. The video frames are extracted at a rate of one frame per second, leading to 35,000 images in total. According to [@twilight:definition] and the sunset time of each recording day, we partition the dataset into five parts: daytime, civil twilight time, nautical twilight time, astronomical twilight time, and nighttime. They consist of 8000, 8750, 8750, 8750, and 9500 images, respectively.
We manually select 50 nighttime images of diverse visual scenes, and construct the test set of *Nighttime Driving* therefrom, which we term *Nighttime Driving-test*. The aforementioned selection is performed manually in order to guarantee that the test set has high diversity, which compensates for its relatively small size in terms of statistical significance of evaluation results. We annotate these images with fine pixel-level semantic annotations using the 19 evaluation classes of the Cityscapes dataset [@Cityscapes]: *road*, *sidewalk*, *building*, *wall*, *fence*, *pole*, *traffic light*, *traffic sign*, *vegetation*, *terrain*, *sky*, *person*, *rider*, *car*, *truck*, *bus*, *train*, *motorcycle* and *bicycle*. In addition, we assign the *void* label to pixels which do not belong to any of the above 19 classes, or the class of which is uncertain due to insufficient illumination. Every such pixel is ignored for semantic segmentation evaluation.
Experimental Evaluation
-----------------------
Our model of choice for experiments on semantic segmentation is the RefineNet [@refinenet]. We use the publicly available *RefineNet-res101-Cityscapes* model, which has been trained on the daytime training set of Cityscapes. In all experiments of this section, we use a constant base learning rate of $5\times{}10^{-5}$ and mini-batches of size 1.
Our segmentation experiment showcases the effectiveness of our model adaptation pipeline, using twilight time as a bridge. The models which are obtained after the initial adaptation step are further fine-tuned on the union of the daytime Cityscapes dataset and the previously segmented twilight datasets, where the latter sets are labeled by the adapted models one step ahead.
We evaluate four variants of our method and compare them to the original segmentation model trained on daytime images directly. Using the pipeline described in Section \[sec:approach\], three models can be obtained, in particular $\phi^1$, $\phi^2$, and $\phi^3$.
We also compare to an alternative adaptation approach which generates labels (by using the original model trained on daytime data) for all twilight images at once and fine-tunes the original daytime segmentation model once. To put in another word, the three-step progressive model adaptation is reduced to a one-step progressive model adaptation.
Model Fine-tuning on twilight data Mean IoU
------------------------ ---------------------------------------------------------------------------------- ----------
Refinenet [@refinenet] — 35.2
Refinenet $\phi^1$ ($\rightarrow$ civil) 38.6
Refinenet $\phi^2$ ($\rightarrow$ civil $\rightarrow$ nautical) 39.9
Refinenet $\phi^3$ ($\rightarrow$ civil $\rightarrow$ nautical $\rightarrow$ astronomical) **41.6**
Refinenet $\rightarrow$ all twilight (1-step adaptation) 39.1
: Performance comparison between the variants of our method to the original segmentation model.[]{data-label="table:experiments"}
**Quantitative Results**. The overall intersection over union (IoU) over all classes of the semantic segmentation by all methods are reported in Tables \[table:experiments\]. The table shows that all variants of our adaptation method improve the performance of the original semantic model trained with daytime data. This is mainly due to the fact that twilight time fall into the middle ground of daytime and nighttime, so the domain gaps from twilight to the other two domains are smaller than the direct domain gap of the two.
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\(b) ground truth
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\[fig:sem:seg1\]
Also, it can be seen from the table that our method benefits from the progressive adaptation in three steps, i.e. from daytime to civil twilight, from civil twilight to nautical twilight, and from nautical twilight to astronomical twilight. The complete pipeline outperforms the two incomplete alternatives. This means that the gradual adaptation closes the domain gap progressively. As the model is adapted one more step forward, the gap to the target domain is further narrowed. Data recorded through twilight time constructs a trajectory between the source domain (daytime) and the target domain (nighttime) and makes daytime-to-nighttime knowledge transfer feasible.
Finally, we find that our three-step progressive pipeline outperforms the one-step progressive alternative. This is mainly due to the unsupervised nature of the model adaptation: the method learns from generated labels for model adaptation. This means that the accuracy of the generated labels directly affect the quality of the adaptation. The one-step adaptation alternative proceeds more aggressively and in the end learns from more noisy generated labels than than our three-step complete pipeline. The three-step model adaptation method generate labels only on data which falls slightly off the training domain of the previous model. Our three-step model adaptation strikes a good balance between computational cost and quality control.
**Qualitative Results**. We also show multiple segmentation examples by our method (the three-step complete pipeline) and the original daytime RefineNet model in Figure \[fig:sem:seg1\]. From the two figures, one can see that our method generally yields better results than the original RefineNet model. For instance, in the second image of Figure \[fig:sem:seg1\], the original RefineNet model misclassified some *road* area as *car*.
While improvement has been observed, the performance of for nighttime scenes is still a lot worse than that for daytime scenes. Nighttime scenes are indeed more challenging than daytime scenes for semantic understanding tasks. There are more underlying causal factors of variation that generated night data, which requires either more training data or more intelligent learning approaches to disentangle the increased number of factors. Also, the models are adapted in an unsupervised manner. Introducing a reasonable amount of human annotations of nighttime scenes will for sure improve the results. This constitutes our future work.
**Limitation**. Many regions in nighttime images are uncertain for human annotators. Those areas should be treated as a separate, special class; algorithms need to be trained to predict this special class as well. It is misleading to assign a class label to those areas. This will be implemented in our next work. We also argue that street lamps should be considered as a separate class in addition to the classes considered in Cityscapes’ daytime driving.
CONCLUSIONS {#sec:conclusion}
===========
This work has investigated the problem of semantic image segmentation of nighttime scenes from a novel perspective. This paper has proposed a novel method to *progressive* adapts the semantic models trained on daytime scenes to nighttime scenes via the bridge of twilight time — the time between dawn and sunrise, or between sunset and dusk. Data recorded during twilight times are further grouped into three subgroups for a three-step progressive model adaptation, which is able to transfer knowledge from daytime to nighttime in an unsupervised manner. In addition to the method, a new dataset of road driving scenes is compiled. It consists of 35,000 images ranging from daytime to twilight time and to nighttime. Also, 50 diverse nighttime images are densely annotated for method evaluation. The experiments show that our method is effective for knowledge transfer from daytime scenes to nighttime scenes without using human supervision.
**Acknowledgement** This work is supported by Toyota Motor Europe via the research project TRACE-Zurich.
[^1]: $^{1}$Dengxin Dai and Luc Van Gool are with the Toyota TRACE-Zurich team at the Computer Vision Lab, ETH Zurich, 8092 Zurich, Switzerland [firstname.lastname@vision.ee.ethz.ch ]{}
[^2]: $^{2}$Luc Van Gool is also with the Toyota TRACE-Leuven team at the Dept of Electrical Engineering ESAT, KU Leuven 3001 Leuven, Belgium
|
---
abstract: |
Our existence on this planet is heavily reliant on animals. It is our ethical obligation to improve their well-being by understanding their needs. Several studies show that animal needs are often expressed through their faces and mammalian brains are capable enough to decode social signals from fellow animal faces. Though remarkable progress has been made towards the automatic understanding of human faces, this has regrettably not been the case with animal faces. There exists significant room and appropriate need to develop automatic systems capable of interpreting animal faces. Among many transformative impacts, such a technology will foster better and cheaper animal healthcare, and further advance animal psychology understanding.
We believe the underlying research progress is mainly obstructed by the lack of an adequately annotated dataset of animal faces, covering a wide spectrum of animal species. To this end, we introduce a large-scale, hierarchical annotated dataset of animal faces, featuring 21.9K faces captured ‘in-the-wild’ conditions. These faces belong to 334 diverse species, while covering 21 different animal orders across biological taxonomy. Each face is consistently annotated with 9 landmarks on key facial features. It is structured and scalable by design; its development underwent four systematic stages involving rigorous, manual annotation effort of over 6K man-hours. We benchmark the proposed dataset for face alignment using the existing art under two new problem settings. Results showcase its challenging nature, unique attributes and present definite prospects for novel, adaptive, and generalized face-oriented CV algorithms. We further benchmark the dataset across related tasks, namely face detection and fine-grained recognition, to demonstrate multi-task applications and opportunities for improvement. Experimental evaluation indicates that this dataset will push the algorithmic advancements across many related CV tasks and encourage the development of novel systems for animal facial behaviour monitoring. We will make the dataset publicly available.
author:
- Muhammad Haris Khan
- John McDonagh
- Salman Khan
- Muhammad Shahabuddin
- Aditya Arora
- Fahad Shahbaz Khan
- Ling Shao
- Georgios Tzimiropoulos
bibliography:
- 'egbib.bib'
date: 'Received: date / Accepted: date'
title: 'AnimalWeb: A Large-Scale Hierarchical Dataset of Annotated Animal Faces'
---
Introduction {#intro}
============
Animals are a fundamental part of our world. It is our moral duty to improve the condition and well-being of animals in labs, farms and homes by understanding their needs and requirements often expressed through their faces. Behavioural and neurophysiological literature have shown that mammalian brains can interpret social signals on fellow animal’s faces and have developed specialized skills to process facial features. Therefore, the study of animal faces is of prime importance.
![*AnimalWeb:* We introduce a large-scale, hierarchical dataset of annotated animal faces featuring diverse species while covering a broader spectrum of animal biological taxonomy. The dataset exhibits unique challenges e.g., large biodiversity in species, high variations in pose, scale, appearance, deformations and backgrounds. Further, it offers unique attributes like class imbalance (CI), multi-task applications (MTA), and zero-shot face alignment (ZFA). Facial landmarks shown in blue and the images belong to classes with identical color in the hierarchy. []{data-label="fig:teaser_fig"}](figures/teaser1.pdf){width="\linewidth"}
Facial landmarks can help us better understand animals and foster their well-being via deciphering their facial expressions. Facial expressions reflect the internal emotions and psychological state of an animal being. As an example, animals with different anatomical structure (such as mice, horses, rabbits and sheep), show a similar grimace expression when in pain i.e., tighten eyes and mouth, flatten cheeks and unusual ear postures. Understanding abnormal animal expressions and behaviours with visual imagery is a much cheaper and quicker alternative to clinical examinations and vital signs monitoring. Encouraging indicators show such powerful technologies could indeed be possible, e.g., fearful cows widen their eyes and flatten their ears [@kutzer2015habituation], horses close eyes in depression [@fureix2012towards], sheep positions its ears backward when facing unpleasant situations [@boissy2011cognitive], and rats ear change colors and shape when in joy [@finlayson2016facial]. Furthermore, large-scale annotated datasets of animal faces can help advance the animal psychology understanding to a new level. For example, for non-primate animals, the scientific understanding of animal expressions is generally limited to the development of only pain coding systems. However, other expressions could be equally important to understand e.g., sadness, boredom, hunger, anger and fear.
We believe the research progress towards automatic understanding of animal facial behaviour is largely hindered by the lack of sufficiently annotated animal faces, covering a wide spectrum of animal species. In comparison, significant progress has been made towards automatic understanding and interpretation of human faces [@xiong2013supervised; @cao2014face; @tzimiropoulos2015project; @trigeorgis2016mnemonic; @bulat2017far; @masi2016we; @wang2017face], while animal face analysis is largely unexplored in vision community [@yang2016human; @rashid2017interspecies]. There is a plenty of room for new algorithms and a pressing need to develop computational tools capable of understanding animal facial behavior. To this end, we introduce a large-scale, hierarchical dataset of annotated animal faces, termed AnimalWeb, featuring diverse species while covering a broader spectrum of animal biological taxonomy. Fig. \[fig:teaser\_fig\] provides a holistic overview of the dataset key features.
AnimalWeb construction follows the well established hierarchy of animals biological classification. In animal kingdom, the tree begins from Phylum and boils down to Class, Order, Family, Genus, and Species. Every image in the dataset has been labelled with the genus-species i.e. the leaf of this classification tree. Image collection is driven by the motivation to offer complete in-the-wild conditions (such as pose, expression, illumination, and occlusions) and diverse coverage of orders in the animal kingdom. **Contributions:** To our knowledge, we build and annotate the largest dataset of animal faces captured under altogether in-the-wild conditions. It encompasses 21 different orders across animal biological taxonomy. Each order probes various families (ranging from 1 to 12), and each family further explores an average of 8 genuses. This diverse coverage makes up a total of 334 different animal species resulting in a count of 21.9K animal faces. Each face is consistently annotated with 9 fiducial landmarks centered around key facial components such as eyes and mouth. Finally, the dataset design and development followed four systematic stages involving an overall, rigorous, manual labelling effort of 6,833 man-hours by experts and trained volunteers.
We benchmark AnimalWeb for face alignment with the state-of-the-art human face alignment algorithms [@bulat2017far; @xiong2017combining]. Results indicate that the dataset is challenging for current best methods developed for human face alignment particularly due to biodiversity, specie imbalance, and adverse in-the-wild conditions (e.g., extreme poses). We show results under two different settings, namely known species evaluation and unknown species evaluation. These settings reveal the capability of the proposed dataset for testing under two novel problem settings: few-shot face alignment and zero-shot face alignment. Further, we demonstrate related applications possible with this dataset, in particular, animal face detection and fine-grained specie recognition. Experimental results signal that the dataset is a strong experimental base for algorithmic advances in computer vision. For instance, the development of novel, adaptive, and generalized facial alignment algorithms towards the betterment of society and economy.
{width="\linewidth"}
Related Datasets
================
Owing to ever-growing interest in automatic face analysis, several face alignment datasets mainly targeting human faces have been published [@gross2010multi; @sagonas2013300; @shen2015first; @deng2018menpo]. However, there has been little to no progress towards creating datasets for animal faces at a comparable scale [@yang2016human; @rashid2017interspecies]. In this section, we categorize existing human and animal face alignment benchmarks according to their level of difficulty and briefly overview each category.
Human Face Alignment
--------------------
**Low Difficulty Datasets:** Since the seminal work of Active Appearance Models (AAMs) [@cootes1998active], various 2D datasets featuring human face landmark annotations have been proposed. Among these, the prominent ones are XM2VTS [@messer1999xm2vtsdb], BioID [@jesorsky2001robust], FRGC [@phillips2005overview], and Multi-PIE [@gross2010multi]. These datasets were collected under constrained environments with limited expression, frontal pose, and normal lighting variations. Following them, few datasets were proposed with faces showing occlusions and other variations such as COFW [@burgos2013robust; @ghiasi2015occlusion] and AFW [@zhu2016face].
**Moderate Difficulty Datasets:** 300W [@sagonas2013300] is considered a popular dataset amongst several others in human face alignment. It has been widely adopted both by scientific community as well as industry [@trigeorgis2016mnemonic; @xiong2013supervised; @ren2014face; @zhu2016unconstrained]. This benchmark was developed for the 300W competition held in conjunction with ICCV 2013. 300W benchmark originated from LFPW [@belhumeur2013localizing], AFW [@zhu2016face], IBUG [@sagonas2013300], and 300W private [@sagonas2016300] datasets. In total, it provides 4,350 images with faces annotated using the 68 landmark frontal face markup scheme. In pursuit of promoting face tracking research, 300VW [@shen2015first] is introduced featuring 114 videos. Such datasets paced research progress towards human face alignment in challenging conditions.
**High Difficulty Datasets:** More recently, efforts are directed to manifest greater range of variations. For instance, Annotated Facial Landmarks in the wild (AFLW) [@koestinger2011annotated] proposed a collection of 25K annotated human faces with up to 21 landmarks. It, however, excluded locations of invisible landmarks. Zhu [*et al*. ]{}[@zhu2016unconstrained] provided manual annotations for invisible landmarks, but there are no landmark annotations along the face contour. Along similar lines, Zhu [*et al*. ]{}[@zhu2016face] developed a large scale training dataset by synthesizing profile views from 300W dataset using a 3D Morphable Model (3DMM). Though it could serve as a large training set, the synthesized profile faces have artifacts that can hurt fitting accuracy. Jeni [*et al*. ]{}[@jeni2016first] reported a dataset introduced in a competition held along ECCV 2016; it typically consisted of images photographed in controlled conditions or are produced synthetically.
Lately, Menpo benchmark [@deng2018menpo] was released as part of competitions held along ICCV 2017. It contains landmarks annotations both from 2D and 3D perspectives and exhibits large variations in pose, expression, illumination and occlusions. Faces are also classified into semi-frontal and profile based on their orientation and annotated accordingly. Menpo 2D benchmark contains 7,576 and 7,281 annotated training and testing images, respectively, taken from AFLW and FDDB.
Animal Face Alignment
---------------------
Despite scientific value, pressing need and direct impact on animal health and welfare, only little attention has been paid in developing an annotated dataset of animal faces [@yang2016human; @rashid2017interspecies]. Although datasets such as ImageNet [@deng2018menpo] and iNaturalist [@van2018inaturalist] offer reasonable species variety, they are targeted at image-level classification and region-level detection tasks. The two animal face alignment datasets are reported in [@yang2016human] and [@rashid2017interspecies]. Yang [*et al*. ]{}[@yang2016human] collected 600 sheep faces and annotated them with 8 fiducial landmarks. Similarly, Rashid [*et al*. ]{}[@rashid2017interspecies] reported a collection of 3717 horse faces with points marked around 8 facial features. These datasets are severely limited in terms of biodiversity, size, and range of possible real-world conditions. To the best of our knowledge, the proposed dataset is a first large-scale, hierarchical collection of annotated animal faces with 9 landmarks. It possess real-world properties e.g., large variations in pose, scale and appearance as well as unique attributes such as species imbalance, multi-task applications, and zero-shot face alignment. Next, we introduce our proposed dataset.
Dataset Properties
==================
AnimalWeb has been constructed following the animal biological taxonomy. It populates faces from 334 different species spread over 21 different animal orders. Below, we highlight some of the unique aspects of this newly introduced dataset (Fig. \[fig:main\_fig\]).
![Distribution of faces per specie in AnimalWeb. We see that 29% of the total species contain 65% of the total faces. The dataset shows the natural occurrence patterns of different species.[]{data-label="fig:species_dist"}](figures/species_dist.png){width="0.8\linewidth"}
**Scale:** The proposed dataset is aimed at offering a large-scale and diverse coverage of annotated animal faces. It contains 21.9K annotated faces, offering 334 different animal species with variable number of animal faces in each species. Fig. \[fig:species\_dist\] shows the distribution of faces per specie. We see that 29% of the total species contain 65% of the total faces. Also, the maximum and minimum number of faces per specie are 241 and 1, respectively. Both these statistics highlight the large imbalance between species and high variability in the instance count for different species. This marks the conformity with the real-world where different species are observed with varying frequencies.
Offered species in AnimalWeb cover 21 different orders from animal classification tree. An average of 3 families have been covered in each order. Similarly, on average 8 genuses have been explored in each family. To the best of our knowledge, AnimalWeb is the first large-scale dataset of annotated animal faces that is easily scalable to offer greater biodiversity coverage in a principled way. It can be highly impactful, for instance, annotated faces could play a vital role in interpreting greater variety of animal expressions not possible with the current approaches based solely on pain coding systems. Tab. \[tab:comp\_datasets\] draws a comparison between AnimalWeb and various popular datasets for face alignment. We see that AnimalWeb is bigger (in face count) compared to 80% of datasets targeted at human face alignment. Importantly, very little or rather no attention is subjected towards constructing annotated animal faces dataset mimicking real-world properties, and the existing ones are limited to only single species.
**Diversity:** Robust computational tools aimed at detecting/tracking animal facial behaviour in open environments are difficult to realize without observations that can exhibit real-world scenarios as much as possible. We therefore aim at ensuring diversity along two important dimensions, **(1)** imaging variations in scale, pose, expression, and occlusion, **(2)** species coverage in the animal biological taxonomy. Fig. \[fig:main\_fig\] shows some example variations captured in the dataset. We observe that animal faces exhibit great pose variations and their faces are captured from very different angles (e.g., top view) that are quite unlikely for human faces. In addition, animal faces can show great range of pose and scale variations.
Fig. \[fig:pose\_diversity\] (top row) reveals that faces in AnimalWeb exhibits much greater range of shape deformations. Each image is obtained by warping all possible ground truth shapes to a reference shape, thereby removing similarity transformations. The bottom row in Fig. \[fig:pose\_diversity\] attempts to demonstrate image diversification in AnimalWeb and other datasets. We observe that it comprises more diversified images than other commonly available human face alignment datasets.
To gauge scale diversity, we plot the distribution of normalized face sizes for AnimalWeb in Fig. \[fig:face\_dist\] and popular human face alignment datasets. AnimalWeb offers 32% more range of small face sizes ($<0.2$) in comparison to competing datasets for human face alignment.
![Top: AnimalWeb covers significantly larger space of deformations compared to popular human face alignment datasets. Bottom: It offers more diversity - large variability in appearances, viewpoints, poses, clutter and occlusions resulting in the blurriest mean image with the smallest lossless JPG file size when compared to popular human face alignment datasets.[]{data-label="fig:pose_diversity"}](figures/crop_pose_diversity3.pdf){width="\linewidth"}
![Face sizes distribution in AnimalWeb and popular human face alignment datasets. AnimalWeb offers 32% more range of small face sizes ($<0.2$) in comparison to competing datasets. Face sizes along x-axis are normalized by images size.[]{data-label="fig:face_dist"}](figures/face_dist_all_crop.png){width="0.8\linewidth"}
Fig. \[fig:species\_miniature\] provides a miniature view of the hierarchical nature, illustrating diversity of the dataset. Two different orders, Primates and Carnivora, have been shown with randomly chosen 8 and 5 families along with some of their respective genuses. It can be seen that AnimalWeb exhibits hierarchical structure with variable number of children nodes for each parent node. We refer to Tab. \[tab:dist\_families\] for the count of families, genuses, species, and finally faces in every order present in the dataset. There exists a total of 21 orders and each order explores on average 3 families, 8 genuses, and 1024 faces. Primates and Carnivora orders populate maximum number of families i.e. 12 among others. We see a similar trend further down the hierarchy. Both aforementioned orders also comprise maximum count of genuses, species, and faces.
![A miniature glimpse of the hierarchical nature of AnimalWeb. Two different orders, Primates and Carnivora, have been shown with 8 and 5 families along with some of their respective genuses.[]{data-label="fig:species_miniature"}](figures/species_diagram.png){width="\linewidth"}
Constructing AnimalWeb
======================
In this section, we detail four important steps followed towards the construction of the proposed dataset (see Fig. \[fig:dataset\_lifecycle\]). These steps include image collection, workflow development, facial point annotation, and annotation refinement. We elaborate these further below.
Image Collection {#sec:img_collection}
----------------
To achieve image collection, we first developed a taxonomic framework to realise a structured, scalable dataset design followed by a detailed collection protocol to ensure real-world conditions before starting image collection process.
**Taxonomic Framework Development.** We develop a taxonomic framework for the AnimalWeb dataset. A simple, hierarchical tree-like data structure is designed following the well established biological animal classification. The prime motivation for this is to carry out image collection - the next step in dataset construction - in a structured and principled way. The obvious other advantage for this methodology lies in recording the various statistics such as image count at different nodes of the tree.
**Data Collection Protocol.** Starting from animal kingdom we restricted ourselves to vertebrates group (phylum), and further within vertebrates to Mammalia class. We wanted those animals whose faces exhibit roughly regular and identifiable face structure. Some excluded animal examples are insects and worms that possibly violate this condition. Given these restrictions, 21 orders were shortlisted for collection task, whom scientific names are depicted in Tab. \[tab:dist\_families\].
Finally, we set the bound for number of images to be collected per genus-species between 200-250. This would increase the chances of valuable collection effort to be spent in exploring the different possible species - improving biodiversity - rather than heavily populating a few (commonly seen). With this constraint, we ended up with an average of 65 animal faces per specie. **Image Source.** The Internet is the only source used for collecting images for this dataset. Other large-scale computer vision datasets such as ImageNet [@deng2009imagenet] and MS COCO [@lin2014microsoft] have also relied on this source to achieve the same. Specifically, we choose Flickr*[^1]*, which is a large image hosting website, to search first, then select, and finally download relevant animal faces.
**Collection.** We use both common and scientific names of animal species from the taxonomic framework (described earlier) to query images. Selection is primarily based on capturing various in-the-wild conditions e.g. various face poses. A team of 3 trained volunteers completed the image collection process under the supervision of an expert. For each worker, it took an average of 100 images per hour amounting to a total of $\sim$250 man-hours. After download, we collected around 25K candidate images. Finally, a visual filtering step helped removing potential duplicates across species in 43.8 man-hours.
Workflow Development {#subsection:workflow_dev}
--------------------
Annotating faces can be regarded as the most important, labour-intensive and thus a difficult step towards this dataset construction. To actualize this, we leveraged the great volunteers resource from a large citizen science web portal, called Zooniverse [^2]. It is home to many successful citizen science projects. We underwent the following stages to accomplish successful project launch through this portal.
**Project Review.** This is the *first* stage and it involves project design and review. The project is only launched once it gets reviewed by Zooniverse experts panel whom main selection criterion revolves around gauging the impact of a research project.
**Workflow design and development.** Upon clearing review process, in the *second* phase, the relevant image metadata is uploaded to the server and an annotator interface (a.k.a workflow) is developed. The workflow is first designed for annotating points and is then thoroughly verified. Two major quality checks are 1) its ease of use for a large volunteer group, bearing different domain expertise, and 2) its fitness towards the key project deliverables. In our case, the workflow defines ’order’ and ’name’ for each facial point. Further, it also comprises a clear action-plan in case of ambiguities (e.g., invisible landmarks) by linking a professionally developed help page. It shows instructions and illustrations to annotate points across all possible species across diverse poses. Lastly, our workflow is thoroughly tested by a 5-member team of experts and it took 20 man-hours of effort.
**9 pts. markup scheme.** The annotator interface in our case required annotators to adhere to the 9 landmarks markup scheme as shown in Fig. \[fig:markup\_scheme\]. We believe that 9 landmarks provide good trade-off between annotation effort and facial features coverage.
Facial Point Annotation {#subsection:fp_annot}
-----------------------
After workflow development, the project is exposed to a big pool of Zooniverse volunteers for annotating facial landmarks. These volunteers have a prior experience of annotating many different successful citizen science projects related to animals. Every face is annotated by at least 5 different volunteers and this equals a labour-intensive effort of $\sim$5408 man-hours in total. Multiple annotations of a single face improves the likelihood of recovering annotated points closer to the actual location of facial landmarks, provided more than half of these multiple annotations qualify this assumption. To this end, we choose to take median value of multiple annotations of a single face.
The annotation portal allows annotators to raise a query with the experts throughout the annotation life cycle. This also helps in removing many different annotation ambiguities for other volunteers as well who might experience the same later in time. The whole exercise of Zooniverse crowdsourcing took 80 man-hours of experts’ time.
{width="0.6\linewidth"}
Refining Annotations {#subsection:ref_annot}
--------------------
Annotations performed by zooniverse volunteers can be inaccurate and missing for some facial points. Further they could be inconsistent, and unordered. Unordered point annotations result if, for instance, left eye landmark is swapped with right eye. Above mentioned errors are in some sense justifiable since point annotations on animal faces, captured in real-world settings, is a complicated task.
We hired a small team of 4 trained volunteers for refinement. This team task was to perform manual corrections and it was supervised by an expert. The refinement completed in two passes listed below and took 438 man-hours of manual effort.
**First pass.** In the first pass, major errors were rectified e.g., correcting points ordering. This refinement proceeded species-wise to enforce consistency in annotations across every possible species in the dataset. A total of 548 man-hours were spent in the first pass.
**Second pass.** In the second pass, pixel perfect annotations were ensured by cross-annotator review. For instance, the refinements on the portion of the dataset done by some member in the first pass is now reviewed and refined by another member of the team.
Benchmarking AnimalWeb
======================
We extensively benchmark AnimalWeb for face alignment task. In addition, we demonstrate multi-task applications by demonstrating experimental results for two other related tasks: face detection and fine-grained image recognition.
Animal Facial Point Localization
--------------------------------
We select the state-of-the-art method in 2D human face alignment for evaluating the proposed dataset. Specifically, we take Hourglass (HG) deep learning based architecture; it has shown excellent results on a range of challenging 2D face alignment datasets [@bulat2017far; @tang2018quantized] and competitions [@xiong2017combining].
**Datasets.** 300W-public, 300W-private, and COFW are deemed the most popular and challenging benchmarks for 2D human face alignment, and are publicly available. 300W-public contains 3148 training images and 689 testing images. 300W-private comprises 600 images for testing only. We only use COFW for testing purposes; its testing set contains 507 images.
**Evaluation Metric.** We use Normalized Mean Error (NME) as the face alignment evaluation metric, $$\text{NME} = \frac{1}{N} \sum_{i=1}^{N}\sum_{l=1}^{L}(\frac{\parallel x{i}^{'}(l) - x{i}^{g}(l)\parallel }{d_{i}}).$$ It calculates the Euclidean distance between the predicted and the ground truth point locations and normalizes by $d_{i}$. We choose ground truth face bounding box size as $d_{i}$, as other measures such as Interocular distance could be biased for profile faces [@ramanan2012face]. In addition to NME, we report results using Cumulative Error Distribution (CED) curves, Area Under Curve (AUC) @0.08 (NME) error, and Failure Rate (FR) @0.08 (NME) error.
**Training Details.** For all our experiments, we use the settings described below to train HG networks both for human datasets and AnimalWeb. Note, these are similar settings as described in [@tang2018quantized; @xiong2017combining] to obtain top performances on 2D face alignment datasets. We set the initial learning rate to $10^{-4}$ and used a mini-batch of 10. During the process, we divide the learning rate by 5, 2, and 2 at 30, 60, and 90 epochs, respectively, for training a total of 110 epochs. We also applied random augmentation: rotation (from -$30^{o}$ to $30^{o}$), color jittering, scale noise (from 0.75 to 1.25). All networks were trained using RMSprop [@tieleman2012divide].
AnimalWeb is assessed under two different train/test splits. The first setting randomly takes 80% images for training and the rest 20% for testing purposes from each specie. [^3] We term this as ‘*Known species evaluation*’ since during training the network sees examples from every species expected upon testing phase. This setting can also be regarded as so-called ‘*few-shot face alignment*’.
The second setting randomly divides all species into 80% for training and 20% for testing. We term it as ‘*Unknown species evaluation*’ as the species encountered in testing phase are not available during training. This setting can also be deemed as so-called ‘*zero-shot face Alignment*’ (ZFA). Unknown species evaluation is, perhaps, more akin to real-world settings than its counterpart. This is because it is quite likely for a deployed facial behaviour monitoring system to experience some species that were unavailable at training. This setting is also more challenging compared to the first because facial appearance of species encountered during testing can be quite different to the ones available at training time.
**Known Species Evaluation.** Tab. \[tab:all\_results\_alignment\] reveals comparison between AnimalWeb and 3 different human face alignment benchmarks, 300W-public, 300W-private, and COFW, when stacking 2 and 3 modules of HG network. Human face alignment results are shown both in terms of 68 pts. and 9 pts. To make fair comparison, the 9 pts. chosen on human faces are the same as for animal faces. Further, 9 pts. results correspond to the model trained with 9 pts. on human faces. We see a considerable gap (NME error difference) between all the results for human face alignment datasets and AnimalWeb. For instance, the NME error difference between COFW tested using HG-2 network is $\sim 1$ unit with AnimalWeb under the known species evaluation protocol. We observe a similar trend in the CED curves displayed in Fig. \[fig:ced\_hg\]. Performance of COFW dataset, the most challenging among human faces, is 15% higher across the whole spectrum of pt-pt-error. Finally, we display some example fittings under known species evaluation settings in Fig. \[fig:known\_species\_first\]. We see that the existing best method struggles under various in-the-wild situations exhibited in AnimalWeb.
{width="1.0\linewidth"}
\[fig:pr\_cruve\]
Fig. \[fig:classwise\] depicts specie-wise testing results for AnimalWeb. For each specie, results are averaged along the number of instances present in it. We observe poorer performance for some species compared to others. This is possibly due to large intra-specie variations coupled with the scarcity of enough training instances relative to others. For instance, *stripedneckedmongoose* species have only 8 training samples compared to *silvesteriswildcat* species populated with 26 training examples.
We report pose-wise results based on yaw angle in Tab. \[tab:pose\_wise\]. It can be seen that AnimalWeb is challenging for large poses. The performance drops as we move towards the either end of (shown) yaw angle spectrum from $[-45^{o},45^{o}]$ range. Further, Tab. \[tab:facesize\_wise\] shows results for AnimalWeb under different face sizes. We observe room for improvement across a wide range of face sizes.
**Unknown Species Evaluation.** Here, we report results under unknown species settings. Note, we randomly choose 80% of the species for training and the rest 20% for testing. Tab. \[tab:all\_results\_alignment\] draws comparison between unknown species settings and its counterpart. As expected, accuracy is lower for unknown case versus the known case. For example, HG-2 displays $\sim1$ unit poor performance under unknown case in comparison to known. Animal faces display much larger inter-species variations between some species. For example, *adeliepenguins* and *giantpandas* whom face appearances are radically different (see 5th row in Fig. \[fig:known\_species\_first\]). Fig. \[fig:unknown\_species\_first\] displays example fittings under this setting. We see that the fitting quality is low for a few frontal poses since the face appearance of species seen during training could be very different to species encountered when testing.
Low performance of existing face alignment algorithms under unknown species setting present obvious opportunities for the design and development of so-called ’zero-shot face alignment algorithms’ that are robust to unseen facial appearance patterns. For instance, novel methods that can better leverage shared prior knowledge and similarities across seen species to perform satisfactorily under unknown species.
{width="0.96\linewidth"}
Animal Face Detection
---------------------
We evaluate the performance of animal face detection using a Faster R-CNN [@ren2015faster] baseline. Our ground-truth is a tightly enclosed face bounding box for each animal face, that is obtained by fitting the annotated facial landmarks. We first evaluate our performance on the face localization task. We compare our dataset with one of the most challenging human face detection dataset WIDER Face [@yang2016wider] in terms of Precision-Recall curve (Fig. \[fig:pr\_cruve\]). Note that WIDER Face is a large-scale dataset with $393,703$ face instances in 32K images and introduces three protocols for evaluation namely ‘easy’, ‘medium’ and ‘hard’ with the increasing level of difficulty. The performance on our dataset lies close to that of medium curve of WIDER Face, which shows that there exists a reasonable margin of improvement for animal face detection. We also compute overall class-wise detection scores where the Faster R-CNN model achieves a mAP of 0.636. Some qualitative examples of our animal face detector are shown in Fig. \[fig:qual\_det\_results\].
Fine-grained species recognition
--------------------------------
Since our dataset is labeled with fine-grained species, one supplementary task of interest is the fine-grained classification. We evaluate the recognition performance on our dataset by applying Residual Networks [@he2016deep] with varying depths (18, 34, 50 and 101). Results are reported in Tab. \[tab:fine-grained\_re\]. We can observe a gradual boost in top-1 accuracy as the network capacity is increased. Our dataset shows a similar difficulty level in comparison to other fine-grained datasets of comparable scale, e.g., CUB-200-2011 [@dataset_cub] and Stanford Dogs [@dataset_dogs] with 200 and 120 classes, respectively. A ResNet50 baseline on CUB-200 and Stanford Dogs achieve an accuracy of 81.7% and 81.1% [@sun2018multi], while the same network achieves an accuracy of 80.04% on AnimalWeb.
Conclusion
==========
In this paper, we introduce a large-scale, hierarchical dataset, named AnimalWeb, of annotated animal faces. It features 21.9K faces from 334 diverse animal species while exploring 21 different orders across animal biological taxonomy. Each face is consistently annotated with 9 fiducial landmarks centered around key facial components. It is structured and scalable by design. Benchmarking AnimalWeb under new settings for face alignment, employing current state-of-the-art method, reveal its challenging nature. It conjectures that existing best methods for (human) face alignment are suboptimal for this task, highlighting the need for specialized and robust algorithms to analyze animal faces. We also show the applications of the dataset for related tasks, specifically face detection and fine-grained recognition. Results conclude that the proposed dataset is a good experimental foundation for algorithmic advances in CV and the resulting technology for the betterment of society and economy.
[^1]: https://www.flickr.com/
[^2]: https://www.zooniverse.org/
[^3]: For validation, we recommend using 10% of the data from the training set.
|
---
abstract: 'Since the 1940s, population projections have in most cases been produced using the deterministic cohort component method. However, in 2015, for the first time, in a major advance, the United Nations issued official probabilistic population projections for all countries based on Bayesian hierarchical models for total fertility and life expectancy. The estimates of these models and the resulting projections are conditional on the UN’s official estimates of past values. However, these past values are themselves uncertain, particularly for the majority of the world’s countries that do not have longstanding high-quality vital registration systems, when they rely on surveys and censuses with their own biases and measurement errors. This paper is a first attempt to remedy this for total fertility rates, by extending the UN model for the future to take account of uncertainty about past values. This is done by adding an additional level to the hierarchical model to represent the multiple data sources, in each case estimating their bias and measurement error variance. We assess the method by out-of-sample predictive validation. While the prediction intervals produced by the current method have somewhat less than nominal coverage, we find that our proposed method achieves close to nominal coverage. The prediction intervals become wider for countries for which the estimates of past total fertility rates rely heavily on surveys rather than on vital registration data.'
author:
- |
Peiran Liu[^1] and Adrian E. Raftery[^2]\
University of Washington
bibliography:
- 'References.bib'
title: Accounting for Uncertainty About Past Values In Probabilistic Projections of the Total Fertility Rate for All Countries
---
[***Keywords:*** Bayesian hierarchical model, Markov chain Monte Carlo, Measurement error, Population projection, Total fertility rate, Vital registration]{}
Introduction {#sec:intro}
============
Population projections or forecasts consist of forecasts of future population numbers and also the components of population change, namely births, deaths and migration, broken down by age and sex, and possibly also by other categories such as race. They are used by governments at all levels (local, regional, state, national and international) for planning and policy decision-making, since knowing the future numbers of people is key to government policy-making. They are also used by the private sector for strategic decisions, and by researchers in the health and social sciences.
The most widely used population projections for many individual countries are produced by their national statistical agency, such as the U.S. Census Bureau in the case of the United States [@USCensus]. The United Nations publishes projections of population by age and sex, and mortality, fertility and migration rates for all countries by five-year age-groups in five-year periods to the year 2100, updated every two years in the UN’s [*World Population Prospects*]{}, whose most recent edition was published in 2017 [@UN2017]. The UN’s population projections are widely viewed as the gold standard regularly updated projections for all countries [@LutzSamir2010].
Since the 1940s, population projections have in most cases been produced by a deterministic method called the cohort-component method . This is based on the [*demographic balancing equation*]{}, namely $$\mbox{Population}_{t+1} = \mbox{Population}_t + \mbox{Births}_t
- \mbox{Deaths}_t + \mbox{Immigrants}_t - \mbox{Emigrants}_t ,$$ where Population refers to the number at time $t$, and Births, Deaths, Immigrants and Emigrants refer to the numbers in the time interval from time $t$ to time $t+1$. The cohort-component method uses an age-structured version of this, of which a simple form is $$\begin{aligned}
\mbox{Population}_{a+1,t+1} &=& \mbox{Population}_{a,t}
* \mbox{Survival Rate}_{a,t} + \mbox{Net Migration}_{a,t} , \\
\mbox{Population}_{0,t+1} &=&
\sum_a \mbox{Women}_{a,t} * \mbox{Fertility Rate}_{a,t} .\end{aligned}$$
This method is simple to implement, but it requires assumptions about future fertility, mortality and migration rates by age and sex. These are typically produced subjectively by experts, either in-house experts working at the agency producing the projections, or a panel of outside experts assembled by the agency. Uncertainty is communicated by scenarios; for example the UN traditionally published High, Medium and Low variants, in which the total fertility rates (TFR) for all countries and all future periods were increased or decreases by half a child per woman. This deterministic approach has been extensively criticized on the grounds that it has no probabilistic basis, and it can give implausible results over multiple projection periods [@Keyfitz1981; @Stoto1983; @LeeTuljapurkar1994]; for a review and summary of this literature see the National Research Council report on the topic [@LeeBulatao2000].
Methods for probabilistic forecasting of future fertility rates have been proposed by [@lee1993modeling], [@alders2007assumptions]), [@alho2006new], [@alho2008uncertain] and [@booth2009stochastic], in each case either for individual countries or groups of countries, typically in the developed world, but these were not easily applied to the U.N.’s task of producing forecasts for all countries.
In 2015, the U.N. adopted a different method for their official population projections for all countries [@UN2015]. This method was probabilistic and statistically-based, replacing the previous deterministic method, thus responding to the critiques. They used Bayesian hierarchical models to produce probabilistic projections of the total fertility rate , and life expectancy . These projections were then simulated from, and each simulated trajectory was translated into age-specific fertility and mortality rates, which in turn were input into the cohort-component method to yield many possible future population trajectories of all countries .
This method indicated that world population was likely to be higher than had previously been thought, reaching 11.2 billion (95% prediction interval 9.5 to 13.2 billion) in 2100, from 7.4 billion now . The main reason for this is that fertility in high-fertility countries, many of them in Sub-Saharan Africa, has been declining more slowly than experts had expected, and the statistical approach took this into account more fully than the expert-based assumptions.
Although the new U.N. method takes account of uncertainty more systemmatically than previous methods, there are still sources of uncertainty that it does not account for. The Bayesian hierarchical model used by the U.N. is conditional on estimates of present and past population, and fertility and mortality rates. In countries with long-established high quality vital registration systems, and hence accurate counts of births and deaths, this is not a large source of uncertainty; this is the case for about 80 of the world’s 200 or so countries. However, the remaining 120 or so countries do not have longstanding high quality vital registration systems, and there fertility and mortality rates are typically estimated from surveys that can be subject to poor coverage in time and space, biases and measurement error. For example, the Demographic and Health Surveys (DHS) are one of the most important and reliable sources of data on fertility rates in countries without good vital registration [@DHS2008], but they have suffered from large underestimates of TFR in some countries in Sub-Saharan Africa, according to [@schoumaker2010reconstructing; @schoumaker2011omissions; @schoumaker2014quality] and [@pullum2013assessment].
Thus the estimated present and past vital rates and population numbers for these countries are not exact, and the uncertainty about them is not accounted for in the projections. This may lead uncertainty in the projections to be underestimated . Demographers have developed methods for correcting estimates of TFR for specific forms of bias, such as recall errors, developing indirect estimation methods for this purpose [@brass1964uses; @brass2015demography]. Bias and uncertainty of past and present estimates were modeled by [@alkema2012estimating], using multiple data quality indicators, such as source of the data, estimation method (e.g. direct or indirect), recall time for retrospective birth history surveys, and so on. But these methods have not been used to account for the uncertainty in population projections that is due to uncertainty about past and present values.
In this paper we extend the UN probabilistic projection method to account for uncertainty about past and present total fertility rates, which may be the most important remaining unaccounted for source of uncertainty. This is made possible by the recent publication of a new dataset by the U.N. Population Division that contains not just estimates of past and present fertility rates for all countries, but also the data from all the data sources on which the estimates are based, including censuses, vital registration systems, partial and sample vital systems, international surveys such as the DHS and the Multiple Indicator Cluster Surveys, or MICS [@UNICEF2015], and national, regional and local surveys [@UN2015WFD]. We do this by developing a new Bayesian hierarchical model that extends the U.N. model to account for bias and measurement error in the different information sources.
The article is organized as follows. The data and proposed methodology are described in Section \[sec:method\]. In Section \[sec:outofsample\] we report the method’s performance using out-of-sample predictive validation. We then provide more detail in Section \[sec:case\], which is a case study of how the method works for Nigeria, which is one of the most important countries for uncertainty about future world population, because it is the most populous country in Africa, has very high fertility, and does not have a long-established high-quality vital registration system. We conclude with a discussion in Section \[sec:discussion\].
Method {#sec:method}
======
Notation
--------
We restrict our attention to estimation of the TFR of each country. The TFR is a period measure, defined as the number of children a woman would bear if she survived to the end of the reproductive interval and at each age she experienced the age-specific fertility rates prevalent in the period to which it refers. It is defined in units of children per woman.
We use the symbol $y$ to denote TFR estimates from different data sources and the symbol $f$ to denote the true (unobserved) TFR. Although the U.N. Population Division’s estimates of past TFR values do contain error, we assume that they are unbiased, in the sense that the errors do not tend to be systematically in one direction or the other; for discussion of this assumption see [@alkema2012estimating]. These official U.N. estimates of past TFR values will be denoted by $u$. All of these parameters will be indexed by country $c$ and time $t$. Data from different sources $y$ are also indexed by their source, denoted by $s$, and the bias and measurement error variance of these estimates are denoted by $\delta$ and $\rho^2$, respectively. The quantities of interest are the unknown past, present and future TFR $f$. We estimate past TFR for the time period $[t_0, t_1]$, while prediction will be for the period $[t_1,t_2]$. In practice in this article, $t_0=1950$, $t_1=2015$ and $t_2=2100$.
The three-phase Bayesian hierarchical model of [@alkema2011probabilistic] will be used to model the total fertility rates. For describing the Bayesian hierarchical model, country-specific parameters controlling the shape of total fertility rates of country $c$ are denoted by $\theta_c$, and the global parameters are denoted by $\psi$. In constructing the probabilistic projections of TFR for all countries, we are also interested in the country-specific parameters $\theta_c$.
Data
----
We use the World Fertility Data 2015 [@UN2015WFD] from the U.N. Population Division for 201 countries in the world. This database is publicly available and includes estimates of TFR from surveys, censuses and sample or partial vital registration data for countries without high-quality vital registration systems. It includes data available as of November 2015 and covers the time period from 1950 to 2015. These data were used to produce the estimates of past TFR in the United Nations World Population Prospects ([wpp]{}) 2015 Revision. These estimates were in turn part of the basis for the U.N.’s 2015 population projections for all countries.
We use TFR estimates from national and international surveys, indirect estimates and vital registration for all 201 countries to estimate the bias and variance of different data sources. We take the estimates in the [wpp]{} 2015 revision as a baseline, assuming that they are unbiased (but not that they are without error). This assumption, also used by [@alkema2012estimating], is made because the analysts producing past estimates were often aware of sources of bias in datasets and corrected for them. While this assumption is not perfect, it seems reasonable to argue that [wpp]{} provides the least biased set of estimates available. In the 2015 revision of the [wpp]{}, the U.N. estimated the five-year average TFR, $u_{c,t}$, for country $c$ in time period $(t, t+5)$, for each five-year period from 1950 to 2015. The outcome in each five-year period $(t, t+5)$ is an estimate of the average TFR between July 1 of year $t$ and July 1 of year $t+5$, and so centered on January 1 of year $t+3$. We construct trajectories and estimations in five-year intervals, and project TFR up to year 2100 probabilistically according to these estimated trajectories of the past.
Model
-----
#### Three Phase Bayesian Hierarchical Model:
Our methodology builds on that of [@alkema2011probabilistic] and [@raftery2014bayesian], as implemented by . This divides the evolution of TFR in a country into three phases: pre-demographic transition, transition and post-transition, as illustrated in Figure \[plot:three\_phase\].
During the fertility transition or decline phase (Phase II), the total fertility rate is modeled as a random walk with negative drift, namely $$\begin{aligned}
\label{phaseII}
f_{c,t} = f_{c,t-5} - g(f_{c,t-5}|\theta_c) + \varepsilon_{c,t} ,\end{aligned}$$ where $g(\cdot|\theta_c)$ is the expected five-year decrement in the TFR over the next period, modeled by a double logistic function governed by the country-specific parameter vector $\theta_c = (\Delta_{c1}, \Delta_{c2}, \Delta_{c3}, \Delta_{c4}, d_c)$, and $\varepsilon_{c,t}$ is random noise around the expected decrement.
During the post-transition phase (Phase III), the total fertility rate is modeled by a Bayesian Hierarchical Autoregressive Model as: $$\begin{aligned}
\label{phaseIII}
f_{c,t} = \mu_c + \rho_c(f_{c,t-5} - \mu_c) + \varepsilon_{c,t} ,\end{aligned}$$ where $\mu_c$ is the long-term mean of the TFR for country $c$, and $\varepsilon_{c,t}$ is the random noise similar to that in phase II.
Since all or almost all countries have already started the fertility transition, modeling the TFR during the pre-demographic transition Phase I was not necessary for projection purposes in previous work. However, for constructing probabilistic estimation of past TFR from 1950 to 2015, we do need to model the Phase I data. They are modeled by a random walk model from year 1950 to the start of fertility transition as: $$\begin{aligned}
\label{phaseI}
f_{c,t} = f_{c,t-5} + \varepsilon_{c,t}\, .\end{aligned}$$
The country-specific parameters in all three phases, $(\theta_c, \mu_c)$, follow a world distribution, which is governed by world parameters $\psi$, and these in turn have a prior distribution. The start and end of the fertility transition (phase II) are defined based on the UN estimates $u_{c,t}$, by rules given in [@alkema2011probabilistic].
#### Model of Imperfect Data: {#method:3.2}
The TFR estimates from different data sources $y_{c,t,s}$ are modeled based on the unobserved true value $f_{c,t}$. Building on [@alkema2012estimating], we distinguish between the bias and measurement error variance in our model. The estimated TFR are modeled by a conditional normal distribution as: $$\begin{aligned}
\label{observed}
& y_{c,t,s} | f_{c,t} = \mathcal{N}(f_{c,t} + \delta_{c,s}, \rho_{c,s}^2)\,,\\
& \mathbb{E}[\delta_{c,s}] = \bm{x}_{c,s}\bm{\beta} \,,\label{eq:5} \\
& \mathbb{E}[\rho_{c,s}] = \bm{x}_{c,s}\bm{\gamma}\,.\label{eq:6}\end{aligned}$$ The bias and measurement error variance, $\delta_{c,s}$ and $\rho_{c,s}$, are estimated using data quality indicators, denoted by $x_{c,s}$. The estimation process is described in the following sections.
#### Complete Model Layout: {#sec:completemodel}
We combine the three-phase Bayesian hierarchical model and imperfect data model into a four-level Bayesian hierarchical model with an additional level for the data sources. Estimation and prediction is then equivalent to getting the posterior distribution of the unknown TFR values $f_{c,t}$ in the estimation period $[t_0, t_1]$ and the prediction period $[t_1, t_2]$, based on the observed TFR estimates from different data sources.
The observed estimates of TFR can be measured for any time between $t_0$ and $t_1$. However, we seek estimates of the average over five-year periods. We approximate the true TFR at any time by assuming that the TFR evolves linearly between the centers of any two successive five-year intervals. This is a reasonable assumption because most demographic quantities, including TFR, typically evolve relatively smoothly over time. Specifically, for any $t \in [t_{\ell}, t_{\ell}+5]$, where $t_{\ell}$ and $t_{\ell}+5$ are the centers of two successive five-year periods, we assume that $$\begin{aligned}
& f_{c,t} = \frac{1}{5}[(t_{l+t}-t) f_{c,t_{l}} + (t - t_l)f_{c,t_{l+5}}].\end{aligned}$$
Then, we model the observed TFR estimates in Level 1, conditional on the true total fertility rates, which are modeled with the extant three-phase BHM in Level 2, conditional on the country-specific parameters. The country-specific parameters are then modeled conditionally on the global parameters in Level 3, which have a prior distribution specified by hyperparameters (Level 4). The overall model is specified as follows: $$\begin{aligned}
\label{eq:model}
\text{Level 1: }
& y_{c,t,s} | f_{c,t} \sim \mathcal{N}(f_{c,t} + \delta_{c,s}, \rho_{c,s}^2)\, , \\
& \mathbb{E}[\delta_{c,s}] = \bm{x}_{c,s}\bm{\beta} \,,\\
& \mathbb{E}[\rho_{c,s}] = \bm{x}_{c,s}\bm{\gamma}\,,\\
& f_{c,t} = \frac{1}{5}[(t_{l+t}-t) f_{c,t_{l}} + (t - t_l)f_{c,t_{l+5}}]\text{ for }t \in [t_l, t_{l+5}]\,;\\
\text{Level 2: }
&\text{Phase I: }f_{c,t} = f_{c,t-5} + \varepsilon_{c,t} \,, \\
&\text{Phase II: }f_{c,t} = f_{c,t-5} - g(f_{c,t-5}|\theta_c) + \varepsilon_{c,t} \,, \\
&\text{Phase III: }f_{c,t} = \mu_c + \rho_c(f_{c,t-5} - \mu_c) + \varepsilon_{c,t} \,,\\
& \varepsilon_{c,t} \sim \mathcal{N}(0, \sigma_{c,t}^2) \,;\\
\text{Level 3: }
& \theta_c \sim h(\cdot | \psi)\,, \\
& \mu_c \sim \mathcal{N}(\bar{\mu}, \sigma_\mu^2)\,, \\
& \rho_c \sim \mathcal{N}(\bar{\rho}, \sigma_\rho^2)\,;\\
\text{Level 4: }
& \psi, \bar{\mu}, \sigma_\mu, \bar{\rho}, \sigma_\rho \sim \pi(\cdot)\, .\end{aligned}$$
Here, $g$ denotes the double logistic function, and $h$ and $\pi$ denote the conditional and unconditional distributions of the parameters of interest, respectively. The parameter $\theta_c$ controls the shape of the double logistic curve. The functional form of the prior distribution $\pi(\cdot)$ is as specified by [@alkema2011probabilistic]. A complete specification of the model, including some further details, is given in the Supplementary Information.
Inference is based on the joint posterior distribution of $(f_{c,t}, \theta_c)$. The model is summarized graphically in Figure \[model:layout\].
(f1) [$f_{t_0}$]{}; (y1) \[above right =of f1\][$y_{t_0 + \Delta_{t,1}, s_1}$]{}; (f2) \[below right =of y1\] [$f_{t_0 + 5}$]{}; (y2) \[above right =of f2\][$y_{t_0 + 5 + \Delta_{t,2}, s_2}$]{}; (f3) \[below right =of y2\] [$f_{t_0 + 10}$]{}; (y3) \[above right =of f3\][$y_{t_0 + 10 + \Delta_{t,3}, s_3}$]{}; (f4) \[below right =of y3\] [$f_{t_0 + 15}$]{}; (dots) \[right =of f4\] [$\dots$]{}; (f13) \[right =of dots\] [$f_{t_1}$]{}; (theta) \[below =of f3\][$\theta_c, \mu_c, \rho_c$]{}; (psi) \[below =of theta\][$\psi$]{}; (pi) \[right =of psi\][$\pi(\cdot)$]{}; (f1) to\[bend right=0\] (y1); (f1) to\[bend right=0\] (f2); (f2) to\[bend right=0\] (f3); (f2) to\[bend right=0\] (y1); (f2) to\[bend right=0\] (y2); (f3) to\[bend right=0\] (y2); (f3) to\[bend right=0\] (y3); (f4) to\[bend right=0\] (y3); (f3) to\[bend right=0\] (f4); (f4) to\[bend right=0\] (dots); (dots) to\[bend right=0\] (f13); (theta) to\[bend right=0\] (f1); (theta) to\[bend right=0\] (f2); (theta) to\[bend right=0\] (f3); (theta) to\[bend right=0\] (f4); (theta) to\[bend right=0\] (f13); (theta) to\[bend right=0\] (f13); (psi) to\[bend right=0\] (theta); (pi) to\[bend right=0\] (psi);
Estimation
----------
#### Estimation of Bias and Measurement Error Variance: {#sec:biasvariance}
The bias $\delta_{c,s}$, and measurement error variance, $\rho_{c,s}^2$, of the observed TFR estimates are estimated in a first stage, as input to the Bayesian hierarchical model, building on the method of [@alkema2012estimating].
We first estimate the bias of TFR. As we discussed in Section \[method:3.2\], the UN estimates will be treated as unbiased but not error-free, providing a baseline reference. Then, for each observation $y_{c,t,s}$, we have $$\begin{aligned}
& \mathbb{E}[y_{c,t,s} - u_{c,t}] = f_{c,t} + \delta_{c,s} - f_{c,t} = \delta_{c,s} .\end{aligned}$$ Thus we can use the difference between observed TFR and UN estimates, $(y_{c,t,s} - f_{c,t})$, as samples for our estimation of the bias and measurement error variance of each source. The parameters $\bm{\beta}$ are estimated by linear regression on data quality indicators $\bm{x}_{c,s}$, as in equation (\[eq:5\]). The estimated biases $\hat{\delta}_{c,s}$ are then equal to the fitted values $\bm{x}_{c,s}\hat{\beta}$.
We estimate the source-specific measurement error variance of the TFR estimates by regression on the data quality covariates $\bm{x}_{c,s}$ of the plug-in estimate $\rho_{c,s} = \sqrt{\frac{\pi}{2}} \mathbb{E}|z_{c,t,s} - u_{c,t}|$, where the unobserved true values $f_{c,t}$ are replaced by the UN estimates of TFR (taken to be unbiased), where the fitted values are used.
#### Estimation of the Complete Model:
Given the estimated bias $\hat{\delta}_{c,s}$ and measurement error variance $\hat{\rho}_{c,s}^2$, we estimate the Bayesian hierarchical model for TFR using a purpose-built Markov Chain Monte Carlo (MCMC) algorithm, specially coded in R. The roughly 3,600 parameters and unknown TFR values are updated one at a time, using Gibbs steps, Metropolis-Hastings steps or slice sampling [@neal2003slice] for each parameter as appropriate. We monitored convergence by inspecting trace plots and using standard convergence diagnostics [@gelman1992inference; @raftery1995number].
We thin enough for the thinned sample to be roughly independent. In practice, for the final results we ran 3 chains, each of length 12,000 iterations with a burn-in of 2,000, and we thinned the resulting chains by 10, to obtain a final, approximately independent sample of size 1,000 from the posterior distribution. More information about the convergence diagnostics used is provided in the Supplementary Information.
Prediction of Future TFR
------------------------
Unlike the projection process developed by [@alkema2011probabilistic] and used by the U.N., we have probabilistic rather than point TFR estimates of past rates over the time period $[t_0, t_1]$. Thus, instead of just sampling from the posterior trajectories of country-specific parameters obtained from estimation process, we also generate posterior trajectories of past TFR values. We proceed by repeating the following process many times. We first select a joint sample of model parameters and past and present TFR for all countries from the posterior distribution. Then, given the sampled model parameters and past and present TFR values, we simulate a trajectory of future TFR values, from 2015 to 2100 using the model specified by (\[phaseII\]) and (\[phaseIII\]). This yields a sample from the joint posterior predictive distribution of future TFR in all countries and time periods considered, taking account of uncertainty about past values.
Our method also differs slightly from the extant method in the way the end of the fertility transition, at which the model shifts from that for Phase II to that for Phase III, is determined. The current U.N. method uses deterministic rules based on the U.N. estimates [@alkema2011probabilistic], and does not account for uncertainty about when the fertility transition ended. In our method, we retain the deterministic rules, but apply them separately to each sampled trajectory of past TFR values. Thus our method takes account of uncertainty about when the fertility transition ended in a particular country, and hence which phase the country is in at the end of the estimation period.
Results {#sec:outofsample}
=======
We assess the predictive performance of our model using out-of-sample predictive validation, used for probabilistic forecasts, for example, by . We include all countries and regions in our validation exercise.
Study Design
------------
The data we have cover the period from 1950 to 2015. We split this into the estimation period, $[t_0 = 1950, t_1 = 2005]$, and the prediction period, $[t_1 = 2005, t_2 = 2015]$. The inputs to our method consist of all TFR estimates from different sources referring to the estimation period.
For the U.N. estimates used as a reference, we take the values published in the [wpp]{} 2008 revision [@UN2008]. The U.N. estimates of the past have been refined since then as more data have become available, but we deliberately do not take advantage of this in our estimation. This makes our validation exercise more analogous to the real prediction task at hand, for which we are using U.N. estimates in the [wpp]{} 2015 revision of past TFR values up to 2015 to predict values past 2015. It can be expected that these estimates of TFR values up to 2015 will become more accurate in the future as data accumulate, but we are not able to take advantage of this for the present purpose.
We are making probabilistic projections, and so we evaluate not only the point predictions, but also the predictive intervals. Our aim is to account for an important source of uncertainty ignored by the present state of the art method, so the accuracy of the prediction intervals may be even more important than the point predictions. If our method is working well, we would expect the current state of the art intervals to have less than nominal coverage, and our method to give coverage closer to nominal. To evaluate our method, we compare our probabilistic projections with those produced by the U.N. in [wpp]{} 2015.
Our out-of-sample validation experiment proceeds as follows.
1. Choose the subset of the original data set $\mathcal{D}$ with TFR observed before year $2005$ as the training data $\mathcal{D}_{\text{train}}$. We remove those observations before 2005 for those estimates in studies that provide series of estimates ending end after 2005. For example, if a study lasts for 20 years and ends in 2008, yielding TFR observations for 1988 to 2008, we remove all observations from this study even though some of the estimates are for years before 2005.
2. Estimate bias and measurement error variance for all data points using UN estimates $u_{c,t}$ from [wpp]{} 2008 revision as the reference.
3. Draw a sample from the joint posterior distribution of model parameters and past TFR values for 1950 to 2005, using MCMC.
4. For each sampled trajectory including the unobserved past TFR values and the model parameters, determine the TFR phase of country $c$ for each time period for this trajectory, and make probabilistic projections for the projection period $[2005, 2015]$.
Out of Sample Validation Results
--------------------------------
We produce results for all countries using our method. For comparison, we also produce results using the method of [@alkema2011probabilistic], which underlies the current U.N. methodology and does not take account of uncertainty about past TFR values.
We summarize the results in Table \[out\_of\_sample\]. This is based on the predictive intervals for each of the 201 countries and for both of the periods $[2005, 2010]$ and $[2010, 2015]$, so that each entry in Table \[out\_of\_sample\] is an average over $201 \times 2 = 402$ values. For each TFR value to be predicted, we take the predictive median as the point estimate, and we compute the quantile-based 80% and 95% prediction intervals. The table shows the mean absolute error (MAE) of the point estimates (the smaller the better), and the coverage of the prediction intervals (the closer to the nominal value the better).
Current Method Proposed Method
-------------------------- ---------------- -----------------
Mean Absolute Error 0.250 0.242
Coverage of 80% interval 74.0% 79.6%
Coverage of 95% interval 86.7% 94.5%
: Mean Absolute Error and Coverage of Out of Sample TFR Point and Interval Predictions for Current Method [@alkema2011probabilistic] and Proposed Method[]{data-label="out_of_sample"}
The proposed method improves the point predictions slightly over the current method, as measured by the MAE. However, it improves the coverage of the prediction intervals substantially. Under the current method, the coverage of the prediction intervals is somewhat below the nominal level, indicating that some of the uncertainty is being missed. Under the proposed method, the coverage of the prediction intervals is much closer to the nominal level, suggesting that the new method is capturing most of the missed uncertainty by taking account of uncertainty in past TFR values.
For illustration, results of the out-of-sample validation exercise are shown in Figure \[out\_of\_sample\_plots\] for Argentina, Botswana, Nigeria and the United States. Of these, only the United States has had a high-quality vital registration system for the entire period, while Argentina has a vital registration that was of lower quality in the early years, and the other two countries have no comprehensive vital registration systems, relying instead on censuses and periodic surveys.
![\[out\_of\_sample\_plots\] Out of Sample Validation Results for Argentina, Botswana, Nigeria and the United States. Estimates of past TFR values for $[1950, 2005]$ are shown by dots, with different sources corresponding to different colors, as described in the side captions. The UN estimates are shown in black. The posterior distributions of past values for $[1950, 2005]$ are shown in orange, with the posterior median as the solid line, the posterior 80% interval as the dark shaded region, and the posterior 95% intervals as the light shaded region. The corresponding posterior predictive distributions for $[2005, 2015]$ based on data up to 2005, are shown in blue.](Paper_Plots/Out_of_sample_validation/Argentina.pdf "fig:"){width="7cm" height="3.5cm"} ![\[out\_of\_sample\_plots\] Out of Sample Validation Results for Argentina, Botswana, Nigeria and the United States. Estimates of past TFR values for $[1950, 2005]$ are shown by dots, with different sources corresponding to different colors, as described in the side captions. The UN estimates are shown in black. The posterior distributions of past values for $[1950, 2005]$ are shown in orange, with the posterior median as the solid line, the posterior 80% interval as the dark shaded region, and the posterior 95% intervals as the light shaded region. The corresponding posterior predictive distributions for $[2005, 2015]$ based on data up to 2005, are shown in blue.](Paper_Plots/Out_of_sample_validation/Nigeria.pdf "fig:"){width="7cm" height="3.5cm"}
![\[out\_of\_sample\_plots\] Out of Sample Validation Results for Argentina, Botswana, Nigeria and the United States. Estimates of past TFR values for $[1950, 2005]$ are shown by dots, with different sources corresponding to different colors, as described in the side captions. The UN estimates are shown in black. The posterior distributions of past values for $[1950, 2005]$ are shown in orange, with the posterior median as the solid line, the posterior 80% interval as the dark shaded region, and the posterior 95% intervals as the light shaded region. The corresponding posterior predictive distributions for $[2005, 2015]$ based on data up to 2005, are shown in blue.](Paper_Plots/Out_of_sample_validation/Botswana.pdf "fig:"){width="7cm" height="3.5cm"} ![\[out\_of\_sample\_plots\] Out of Sample Validation Results for Argentina, Botswana, Nigeria and the United States. Estimates of past TFR values for $[1950, 2005]$ are shown by dots, with different sources corresponding to different colors, as described in the side captions. The UN estimates are shown in black. The posterior distributions of past values for $[1950, 2005]$ are shown in orange, with the posterior median as the solid line, the posterior 80% interval as the dark shaded region, and the posterior 95% intervals as the light shaded region. The corresponding posterior predictive distributions for $[2005, 2015]$ based on data up to 2005, are shown in blue.](Paper_Plots/Out_of_sample_validation/United_States_of_America.pdf "fig:"){width="7cm" height="3.5cm"}
It can be seen that the posterior intervals of past TFR values are very narrow for the United States, reflecting the high quality vital registration data available for the entire period, while for Argentina they are somewhat wider. For both Botswana and Nigeria the intervals are far wider, reflecting the much lower quality of the available data. For the earlier years, from the 1950s to the 1970s, the intervals for Botswana and Nigeria were especially wide, reflecting the sparsity of the data for these decades. The predictive distributions cover the observations in all cases, although in some cases they lie towards the edge of the intervals, as expected if the intervals are well calibrated.
Case Study: TFR Estimation and Projection For Nigeria {#sec:case}
=====================================================
In this section, we illustrate the method by producing probabilistic forecasts of the TFR of Nigeria from 2015 to 2100, using data available up to 2015. As we have discussed, the method first estimates the bias and measurement error variance of the different data sources. It then estimates the uncertainty about past TFR values, and takes this uncertainty into account when making probabilistic projections.
Estimation of Bias and Measurement Error Variance of Different Data Sources {#sec:biasvariance_results}
---------------------------------------------------------------------------
From 1950 to 2015, according to the U.N.’s [wpp]{} 2015 revision, the TFR in Nigeria reached its peak around 1980 at about 6.7 children per woman. It then declined slowly, reaching about 5.7 in 2015. However, the data on which these estimates are based are surprisingly noisy, as can be seen in Figure \[plot:nig\_estimates\].
These data come from several sources, including national censuses, which are comprehensive but sparse in time and have issues of coverage. The other sources are mostly surveys, including the internationally organized Demographic and Health Surveys (DHS), the Multiple Indicator Cluster Surveys (MICS) run by UNICEF, and the Malaria Indicators Survey, or MIS, also run by DHS. There are also several occasional national cross-sectional and panel surveys. Some of the surveys, notably DHS and MICS, collect birth histories, which allow one survey to generate estimates for several past years, in some cases using indirect methods.
We first estimate the bias and measurement error variance of the different data sources using the approach outlined in Section \[method:3.2\]. From each observed TFR estimate $f_{c,t,s}$ we subtract the corresponding UN TFR estimate to obtain an estimate of the bias for that source, country and time, namely $z_{c,t,s} = f_{c,t,s} - u_{c,t}$. As data quality covariates, $\bm{x}_{c,s}$, we use the source of the data and whether the estimate is direct or indirect. We then estimate the bias $\delta_{c,s}$ for country $c$ and data source $s$ as the fitted value from a regression of the $z_{c,t,s}$ on the data quality covariates $\bm{x}_{c,s}$, as in [@alkema2012estimating].
The U.N. TFR estimates are for five-year periods, and we treat them as referring to the middle of the period. Thus, for example, we treat estimates for the 2010–2015 period as referring to the beginning of 2013. An observed TFR estimate can refer to any year between 1950 and 2015, and we use the convex combination of the two U.N. estimates closest to the time to which it refers as the corresponding U.N. estimate, $u_{c,t}$.
Similarly, after we get the fitted value of the bias estimates $\hat{\delta}_{c,s}$, we obtain the measurement error standard deviation estimates by regressing $|z_{c,t,s} - \hat{\delta}_{c,s}|$ on the same data quality covariates. The fitted biases and measurement error standard deviations are summarized in Table \[table:estimates\].
[ccccccccc]{}\
& Source & Estimate Type & $\mu(\delta)$ & $\sigma(\delta)$ & $\hat{\delta}$ & $\hat{\sigma}(\delta) = \hat{\rho}$ & RMSE & $n$\
1 & DHS & D & 0.04 & 0.48 & 0.11 & 0.38 & 0.40 & 28\
2 & DHS & C & -0.26 & 0.51 & -0.48 & 0.46 & 0.66 &10\
3 & Census & D & 0.00 & 0.91 & -0.43 & 0.50 & 0.66 & 2\
4 & Census & C & -1.46 & 0.43 & -1.02 & 0.58 &1.17 & 2\
5 & MICS & D & -1.10 & 1.03 & -0.33 & 0.81 & 0.87 & 2\
6 & MICS & C & -0.79 & 0.18 & -0.92 & 0.89 & 1.28 & 2\
7 & MICS & I & 0.29 & 1.64 & 0.20 & 1.35 & 1.36 & 15\
8 & MIS & D & 0.70 & 0.48 & 0.22 & 0.56 & 0.60 & 5\
9 & MIS & I & 0.68 & 1.37 & 0.75 & 1.09 & 1.32 & 30\
10 & Survey & D & -0.50 & 0.58 & -0.47 & 0.42 & 0.63 & 4\
11 & Survey & C & -1.18 & 0.95 & -1.06 & 0.49 & 1.17 & 8\
12 & Survey & I & 0.14 & 0.98 & 0.06 & 0.95 & 0.95 & 15\
13 & Survey-NR & D & -0.40 & 0.18 & -0.60 & 0.21 & 0.64 & 3\
14 & Survey-NR & C & -1.48 & 0.18 & -1.18 & 0.29 & 1.22 & 2\
We can see from Table \[table:estimates\] that direct estimates from the DHS are the highest quality estimates as measured by estimated mean squared error (equal to $\sqrt{\hat{\delta}^2 + \hat{\rho}^2}$). Direct estimates generally have smaller variances than indirect estimates. Figure \[plot: bv\_estimates\] plots the fitted biases and measurement error standard deviations against the observed ones; the model fit seems reasonably good.
Estimation of Past and Projection of Future TFR {#sec:pastpresenttfr}
-----------------------------------------------
The fertility transition, or Phase II, started in 1980 in Nigeria, according to the definition of [@alkema2011probabilistic]. We initialize the MCMC algorithm with a warm start, simulating the starting values for the global parameters $\psi$ and the country-level parameters $\theta_c$ from their posterior distribution from the model that does not take account of uncertainty about past TFR values [@alkema2011probabilistic; @raftery2014bayesian]. The true past fertility rates are initialized as the U.N. estimates.
The results are shown in Figure \[plot: tfr estimates\]. This is based on data up to 2015, and can be compared with Figure \[out\_of\_sample\_plots\](c), which is based on data up to 2005. The posterior distribution for the 2000-2005 period is tighter, because more data relevant to this period were available in 2015 than in 2005. The posterior distibution widens slightly for the past period, 2010–2015, again reflecting the relative paucity of data relevant to this period by 2015. One could expect that this posterior distribution will tighten as more data relevant to 2010–2015 become available in the future.
We make projections in two steps. In the first step, we will sample one trajectory from the MCMC results obtained in Section \[sec:pastpresenttfr\]. Then given the sampled trajectory, the phase of the most recent year is determined by this trajectory, and then future TFR is sampled according to the country-specific parameters of this trajectory. The resulting projection is summarized in Figure \[plot: projection\]:
The projections of future TFR from 2015 to 2100, taking account of uncertainty about the past, are shown in Figure \[plot: projection\]. The black solid and dotted curves show the U.N.’s 2015 probabilistic projection (not taking account of uncertainty about the past), while the blue line and shaded region shows the projection from our method. Both project that Nigeria’s TFR will likely decline, with a great deal of uncertainty about how fast this will happen. Our proposed method yields a similar predictive median to the current U.N. method, but somewhat wider prediction intervals. As we saw in the out-of-sample validation study, these wider intervals do incorporate an important additional source of uncertainty, and, on average, take the intervals from undercovering the truth to some extent, to close to nominal coverage.
Model Validation: Simulation Study
----------------------------------
We now run a simulation study with input data on past TFRs chosen to resemble the Nigerian data, to see how accurately the proposed method captures past TFR values. For each simulation, we sampled one TFR trajectory from the posterior distribution of our previous analysis as the true (unobserved) TFR. Then we randomly generated TFR estimates from the normal distribution in Level 1 of the model, by assuming the bias of data points are the previous estimated bias ($\hat{\delta}_{c,t,s}$), and measurement error variances are the previous estimated variances ($\hat{\rho}_{c,s}$). We then treated sampled data points as the input data for the estimation process. We still treated the U.N. estimates as unbiased, as before. We repeated the simulation process 1000 times. The estimation results of one simulation are shown in Figure \[plot: simulation\].
If we take the median of posterior TFR estimates as the point estimate, the mean absolute error (MAE) for all 13 time periods is 0.157. Breaking it down by the 13 time periods, the result are shown in Table \[tbl:simulation\].
80% Interval Coverage 95% Interval Coverage Mean Absolute Error
--------------------- ----------------------- ----------------------- ---------------------
$f_{Nigeria,1953}$ 0.865 0.917 0.259
$ f_{Nigeria,1958}$ 0.888 0.976 0.263
$f_{Nigeria,1963}$ 0.895 0.982 0.222
$f_{Nigeria,1968}$ 0.794 0.914 0.177
$f_{Nigeria,1973}$ 0.847 0.967 0.139
$f_{Nigeria,1978}$ 0.718 0.898 0.152
$f_{Nigeria,1983}$ 0.797 0.933 0.118
$f_{Nigeria,1988}$ 0.926 0.979 0.103
$f_{Nigeria,1993}$ 0.936 0.980 0.097
$f_{Nigeria,1998}$ 0.952 0.981 0.116
$f_{Nigeria,2003}$ 0.848 0.940 0.090
$f_{Nigeria,2008}$ 0.893 0.963 0.103
$f_{Nigeria,2013}$ 0.805 0.935 0.198
: Simulation Coverage and Mean Absolute Errors for 13 Estimation Periods, .[]{data-label="tbl:simulation"}
The overall coverage rate of the 80% interval was 85.9%, and the overall coverage rate of the 95% interval is 95.1%. The overall coverage rate was close to the nominal rate, and the MAE was also low. Thus the model gave accurate point and intervals estimates of past values in the simulation study.
Discussion {#sec:discussion}
==========
Since 2015, the U.N. has been producing probabilistic projections of the total fertility rate as part of their official population projections for all countries of the world, using the Bayesian hierarchical model of [@alkema2011probabilistic] and [@raftery2014bayesian]. However, one important source of uncertainty has not been accounted for so far in these projections, namely uncertainty about past fertility levels. This uncertainty is small for countries with long-standing vital registration systems; this is the case for less than half of the world’s roughly 200 countries. For the other countries, however, this uncertainty can be considerable.
We have developed a new method for projecting the total fertility rate probabilistic for all countries that extends the U.N. method to take account of uncertainty about past TFR values. In a validation experiment, we found that the existing U.N. method leads to prediction intervals whose coverage is somewhat lower than nominal, while for our new method the coverage is close to nominal. For the countries with the highest quality data on past rates, mostly in Europe and North America, our method gives results that are similar to the current method. However, for countries with lower quality data where TFR estimates have been based on surveys for at least part of the past 60 years, our method gives intervals that are noticeably wider than the current ones.
The long-term implications of these results could be far reaching. The countries with the most uncertainty about past TFR values are also largely those with the highest current fertility levels and the greatest uncertainty about future levels, many of which are in Sub-Saharan Africa. Not surprisingly, therefore, our method indicates that these are also the countries for which the understatement of uncertainty was greatest. Thus our TFR results could lead to a considerable increase in uncertainty about long-term population in these countries, especially as the effects of differences in TFR compound over generations. The population of Sub-Saharan Africa is currently around 1 billion, and current projections are that it will increase to between 3.4 and 4.8 billion in 2100 with 80% probability . This interval will be wider still once uncertainty about past TFR has been factored in, with even more dramatic implications for future population levels in Africa, and hence for the world as a whole.
Our method is in two stages. In the first stage we estimate the bias and measurement error variance of the different data sources by country using a classical analysis of variance method. In the second stage we estimate a Bayesian hierarchical model taking the point estimates from the first stage as input. In principle it would be possible to unify these two stages by including the estimation of the bias and variance of the different sources in the Bayesian hierarchical model. However, this would complicate the model considerably, making it harder to specify, code, debug and interpret, and it seems unlikely that it would change the results appreciably. We feel that our modeling decision strikes a reasonable balance between complexity and performance. This is supported by the good assessment of predictive uncertainty provided by our method.
To use these projections of total fertility in population projections, one must convert them to age-specific fertility rates. The U.N. currently does this using the methodology described by . Each simulated future TFR value is converted to a corresponding age-specific fertility pattern, which is used with age-specific mortality and migration rates in the cohort-component projection method to project the corresponding future population by age and sex. A subtle point is that this takes account of uncertainty about future [*total*]{} fertility, but not about future [*age-specific*]{} fertility given total fertility, i.e. about the number but not the timing of future births. Because the age pattern of births is relatively concentrated regardless of their number, this is a much smaller source of uncertainty than uncertainty about the number of births. Nevertheless, it should be addressed in future research.
We have produced results for all the world’s countries with populations over 100,000 as of 2015, except for one: China, the most populous country. We did not include China in our analysis because the estimates for its TFR suffer from a unique form of bias, which would require a different kind of analysis. This is due to the One Child Policy, introduced in 1979. As a result of this policy, many Chinese families did not report births to the authorities, with the hope of being able to circumvent the policy and have additional children. The underreporting was particularly severe in the late 1990s, and [@goodkind2004china] has argued that this was because the 1991 Decree pushed the responsibility of implementing family planning rules, especially the one-child policy, to local governments, giving them a greater incentive to underreport the number of births.
There have been many efforts to correct for this underreporting. For example, [@yi1996fertility; @retherford2005far; @cai2008assessment] and [@merli2000births] attempted to correct estimates of TFR in 2000. The clearest evidence of this underreporting comes from primary school enrollments several years later, which were typically substantially larger than the reported number of births during this period. [@zhai2007analysis] used these enrollment data to correct the TFR estimates for the late 1990s. The U.N. has also been using enrollment data to correct available estimates. Our method would not be sufficient to give good estimates of China’s TFR in the period of severe underreporting. Instead, for China it would be desirable to extend our method to include enrollment data, taking account of uncertainty in the enrollment data in the model. A simpler approach would be to include enrollment-corrected survey and census estimates as inputs to our method, but we felt that a more comprehensive approach was desirable given the great demographic importance of China and the unique data issues it presents, and so we omitted China from the present analysis.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by NICHD grants R01 HD054511 and R01 HD070936. Raftery’s research was also partly supported by the Center for Advanced Study in the Behavioral Sciences (CASBS) at Stanford University. This work was carried out in cooperation with the United Nations Population Division, and in particular we thank Kirill Andreev, Ann Biddlecom, and Vladimirá Kantovorá, Stephen Kisambira and other colleagues from the Population Division for preparing and making available the U.N. World Fertility Data, an invaluable resource for researchers. We also thank Hana Ševčíková for methodological support, and Patrick Gerland and Laina Mercer for helpful comments.
[^1]: Peiran Liu is Ph.D. Student, Department of Statistics, University of Washington, Seattle, WA, 98195 (Email: [prliu@uw.edu](prliu@uw.edu))
[^2]: Adrian E. Raftery is Professor of Statistics and Sociology, University of Washington, Seattle, WA, 98195 (Email: [raftery@u.washington.edu](raftery@u.washington.edu))
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abstract: 'We provide an explanation of recent experimental results of Xue et al. [@susi], where full revivals in a time-dependent quantum walk model with a periodically changing coin are found. Using methods originally developed for “electric” walks with a space-dependent, rather than a time-dependent coin, we provide a full explanation of the observations of Xue et al. We extend the analysis from periodic time-dependence to quasi-periodic behaviour with periods incommensurate to the step size. Spectral analysis, one of the principal tools for the study of electric walks, fails for time-dependent systems, but we find qualitative propagation behaviour of the time-dependent system in close analogy to the electric case.'
author:
- 'C. Cedzich'
- 'R. F. Werner'
bibliography:
- 'peng.bib'
title: 'Revivals in Quantum Walks with quasi-periodically time-dependent coin'
---
Introduction
============
Quantum walks are fundamental dynamical systems, involving a walking particle with internal degrees of freedom moving in discrete time steps on a lattice [@Ambainis2001; @Grimmett; @SpaceTimeCoinFlux; @GenMeasuringDevice]. They have become an important test bed for many complex quantum phenomena, being well accessible both to experimental and theoretical investigation. In particular, they have recently attracted much attention as a computational resource [@AmbainisKR05; @Magniez:2011ke; @Anonymous:rSnqW2vc; @Lovett:2010ff; @SearchOnFractalLattice; @Quenching]. They exhibit a rich variety of quantum effects such as Landau-Zener tunneling [@landauzehner], the Klein paradox [@kurzwel:2008dm] and Bloch oscillations [@Gensketal]. By taking into account on-site interactions between two particles performing a quantum walk, the formation of molecules has also been established [@molecules; @moleculestheothers; @moleculestheothers2]. Recently, a complete topological classification of quantum walks obeying a set of discrete symmetries has been obtained [@Kita; @Kita2; @KitaObservation; @Asbo1; @Asbo2; @Asbo3; @Asbo4; @UsOnTop; @LongVersion]. Quantum walks have been experimentally realized in such diverse physical systems as neutral atoms in optical lattices [@Karski:2009], trapped ions [@Zaehringer:2010bs; @Schmitz2009], wave guide lattices [@Peruzzo:2010co; @PhysRevLett.108.010502] and light pulses in optical fibres [@Schreiber:2010cl; @Schreiber:2012ix] as well as single photons in free space [@PhysRevLett.104.153602]. On the other hand, one can also observe a growing interest in quantum walks in mathematical literature where they are viewed as physical realizations of the more abstract concept of CMV matrices, the unitary analog to Jacobi matrices [@CMVoriginal; @CGMV; @recurrence; @QDapproach; @Damanik].
It is easy to make the coin operation of a quantum walk depend on either the location of the walker or the time (number of time step) or both. A complete analysis for the randomly time-dependent case was given in Ref. [@TRcoin], even when the coin choice is driven by an external Markov process, hence allowing for correlated coin choices. In this case, the ballistic propagation ($x(t)\sim t$) of the non-random system reverts to diffusive propagation $x(t)\sim t^{1/2}$, i.e., Gaussian spreading with a momentum-dependent diffusion constant. In sharp contrast, disorder in space (i.e., a space-dependent set of coins fixed throughout the evolution) leads to Anderson localization [@Localization; @dynlocalain], i.e., purely discrete spectrum with exponentially localized eigenfunctions, and no propagation. Combining both kinds of disorder [@TRcoin] again leads to diffusive scaling, so adding temporal disorder will slow down propagation in a non-random system but will speed up an Anderson localized one.
It is well-known that quasi-periodic potentials share the possibility of Anderson localization with disordered ones. For quantum walks this has been analyzed in detail in [@electric; @InhomogeneousWalkFillman]. In [@electric] the critical parameter is the electric field $\Phi$. For rational fields $\Phi=2\pi n/m$ one observes sharp revivals after $m$ or $2m$ steps, which are exponentially sharp as a function of $m$. Hence, somewhat paradoxically, the revival is the sharper the longer it takes. On the other hand, the evolution does not become exactly periodic, and small errors accumulate over revival cycles leading ultimately to ballistic transport. For irrational fields sufficiently close to a rational, i.e., up to ${\mathcal O}(1/m^2)$ as for the continued fraction convergents, one also sees the revivals. However, depending on the infinite sequence of convergents, the long term evolution may be quite different. It may involve further, yet sharper revivals on larger time scales, but typically (with probability one) localization-like behaviour.
Looking at quasi-periodic temporal modulations of the dynamics as in [@susi] is a natural question. However, judging from the experience with random choices one would hardly expect the same methods to apply. Yet this is the case, as we show in this letter. The core of the argument is again a revival statement for the rational case, based on a trace formula established in [@electric]. However, the reason for the appearance of continued fraction approximations is different in the two cases. For example, in the temporal case analyzed in this note the revival statement holds uniformly for all initial wave functions, whereas in [@electric] we had to restrict the initial support region. Another marked difference is that in the temporal case we are not repeatedly applying the same unitary operator, so there is no operator of which we could gather (discrete vs. continuous) spectral information. Therefore, the interplay between spectral properties and propagation properties, which is typical for autonomous (not explicitly time-dependent) evolutions, has no analogue in the temporal case.
The system
==========
We consider, as in [@susi], a translation invariant quantum walk on the 1D lattice $\Ir$ with local spin-1/2 degree of freedom. Basis vectors of the system Hilbert space are thus denoted by $\ket{x,s}$ with $x\in\Ir$ and $s=\pm1$. The standard state-dependent shift $S$ acts as $S\ket{x,s}=\ket{x+s,s}$ and $C(t)$ is a time-dependent coin acting solely on $s$, the internal degree of freedom. The concrete model given in [@susi] is given by $$\label{eq:coin}
C(t)=R_x(t \EF)R_y(\theta),$$ with $R_{x,y}(\theta)$ the rotation around the $x$ and $y$ axis in spin space, respectively. The $t^{\rm th}$ time step of the walk is then given by the unitary operator $\We(t)=SC(t)$. Note that when $\Phi/(2\pi)=n/m$ is rational, we have $W(t+k*m)=W(t)$ so the evolution is periodic. Otherwise, it is quasi-periodic. In either case we use $$\label{Wm}
\We^{[m,1]}=\We(m)\We(m-1)\cdots\We(1)$$ as a short hand for the first $m$ steps of the walk. For the rest of the paper we will generalize the above coin operator by allowing instead of $R_y(\theta)$ a slightly more general unitary coin such that the system under consideration becomes $$\label{eq:we}
\We(t)=SR_x(t\EF)\begin{pmatrix}a&b\\-b^*&a^* \end{pmatrix},$$ where $\abs{a}^2+\abs{b}^2=1$. All results remain valid if instead of $R_x(t\EF)$ we would have chosen any unitary $R$ with $R^m=\idty$ for some $m\in\mathbb N$. The choice $R=R_x(\EF)$ is made to retain analogy with [@susi].
The electric walk turns out to be closely related. It has no time dependence in the coin operation (so $\Phi=0$ in ). Instead, after each step the wave function is multiplied by the $x$-dependent phase $\exp(i \Phi x)$, where $\Phi$ plays the role of an electric field [@electric].
The basic observation in [@susi] is that for certain fine-tuned rational values of $\EF,\theta$ and a specific initial state there are revivals (see [@susi Table I, Table II] for theoretical and experimental results, respectively). This observation will be generalized in this manuscript to almost all values of $\EF,\theta$. These revivals will no longer be exact, so that, in contrast to [@susi] the evolved state will not be exactly periodic. When $\EF/(2\pi)$ is rational the denominator of $\EF/(2\pi)$ sets the time for the revival, which will even be exponentially sharp in the denominator. Hence even for a moderately large denominator the time evolution will be periodic for all practical purposes. For irrational parameters one typically still gets an infinite hierarchy of revivals governed by the continued fraction expansion of $\EF/(2\pi)$. Remarkably, these qualitative features are independent of $\theta$ and the initial state. Similar behaviour is known in the case where $\EF$ is an electric field [@electric], and indeed the analysis of the rational case uses a formula originally developed for that case.
Revivals and a trace formula
============================
We begin the analysis with the rational case $\EF=2\pi\nr/\dr$ with $\nr$ and $\dr$ coprime. In this case $R_x^t(\EF)$ is periodic in $t$ with $R_x^m(\EF)=\idty$. Our main result is the so called *revival theorem* similar to that in [@electric]. It states that for $\dr$ odd $\We^{[2\dr,1]}$ and for $\dr$ even $\We^{[\dr,1]}$ are norm-close to the identity with quality exponentially good in $m$: $$\begin{aligned}
{2}\label{reviveOdd_app}
\opNorm{\,\We^{[2\dr,1]}+\idty\,}&= &\: \mathcal O(\abs{\widetilde\alpha}^m)\qquad&\dr\text{ odd} \\
\opNorm{\,\We^{[\dr,1]}+(-1)^{\dr/2}\idty\,}&= &\: \mathcal O(\abs{\widetilde\alpha}^m)\qquad &\dr\text{ even}\;
\label{reviveEven_app}\end{aligned}$$ where $\widetilde\alpha$ depends solely on the coin parameters. The exponentially good quality of the estimates depends on $\abs{\widetilde\alpha}<1$. For the Hadamard walk with $a=b=1/\sqrt{2}$ the deviation from a perfect reproduction of the initial state is $2^{-m/2+1}$. The exponentially good quality of the revivals for this choice of coin are illustrated in Figure \[fig:revivals\] for $m=310$. In sharp contrast, for coin parameters $a=i/\sqrt{2}$ and $b=1/\sqrt{2}$ we find $\abs{\widetilde\alpha}=1$ and hence no revival predictions at all. Also, the difference in behavior for $\dr$ even and odd is understood intuitively, since the probability to find the particle at the origin is non-zero only after an even number of steps. To prove the above revival theorem note that the “temporally regrouped” walk $\We^{[\dr,1]}$ is independent of time. Hence we can apply the standard theory of translation invariant walks (see, e.g. [@Grimmett; @SpaceTimeCoinFlux; @TRcoin]) and consider the Fourier-transformed operator $$\begin{aligned}
\label{WEnp}
\We^{[\dr,1]}(k)&=&S(k)R_x(\EF)CS(k)R_x^2(\EF)\dots R_x^m(\EF)C\end{aligned}$$ in momentum space with dispersion relation $$\label{eq:disrel}
2\cos\omega_\pm(k)=\tr\We^{[\dr,1]}(k).$$
![ Position distributions for $\EF=2\pi/155$ and $\phi=\pi/4$. Note the expected revival at $t=310$ and the Bloch oscillations described in [@Gensketal].[]{data-label="fig:revivals"}](barsrat.png){width=".85\columnwidth"}
To get an explicit expression for $\omega_\pm$ we thus need to evaluate the trace $\tr\We^{[\dr,1]}(k)$ for which we adapt a result from [@electric], called the *trace formula*. This provides an expression for traces of the type $\tau_m(M) = \tr(MR^0MR^1\cdots MR^{m-1})$ where $R^m=\idty$ with $R$ unitary and $M=\left(\begin{smallmatrix} \alpha & \beta\\ \gamma & \delta \end{smallmatrix}\right)$ is a general $2\times2$ matrix. For $m$ odd we get: $$\label{trodd}
\tau_m(M)=\widetilde\alpha^m+\widetilde\delta^m\;,$$ and for $m$ even: $$\label{treven}
\tau_m(M)=-\bigl(\widetilde \alpha^{m}+\widetilde \delta^{m}\bigr)+2(-1)^{m/2}\bigl((\widetilde \alpha\widetilde \delta)^{m/2}-\det(M)^{m/2}\bigr).$$ Here, $\widetilde\alpha =(BMB^*)_{11}$ and $\widetilde\delta =(BMB^*)_{22}$ and $B$ is the unitary diagonalizing $R$.
We apply the trace formula to the walk with rational $\EF/(2\pi)=\tfrac\nr\dr$ such that $R_x^\dr(\EF)=\idty$. In and we let $M\mapsto CS(k)$ which corresponds to replacing $\alpha\mapsto ae^{ik},\beta\mapsto be^{-ik},\gamma\mapsto -b^*e^{ik},\delta\mapsto a^*e^{-ik}$ and $\det M\mapsto1$. Note that $R\propto R_x=\exp(i\sigma_x)$ in the trace formula requires $B=\tfrac1{\sqrt{2}}\left(\begin{smallmatrix}1&1\\1&-1\end{smallmatrix}\right)$ which yields $2\widetilde\alpha=\alpha+\beta+\gamma+\delta$ and $2\widetilde\delta=\alpha-\beta-\gamma+\delta$. Writing $a=\abs ae^{ik_a}$ and $b=\abs be^{ik_b}$ in polar form then results in $$\widetilde\alpha =\widetilde\delta^*=\abs{a}\cos(k_a+k) + i\abs{b}\sin(k_b-k)
$$ such that by writing $\widetilde\alpha=\abs{\widetilde\alpha}\exp(i\theta_{\widetilde\alpha})$ the dispersion relation reads $$\begin{aligned}
\label{dispersEpaps}
&\cos\omega_\pm(k)=\\
&=\begin{cases}\abs{\widetilde\alpha}^m\cos(m\theta_{\widetilde\alpha})& m\mbox{\ odd}\\
-\abs{\widetilde\alpha}^m\cos(m\theta_{\widetilde\alpha})+2(-1)^{\frac m2+1}\left(1-\abs{\widetilde\alpha}^m\right) & m\mbox{ even}.
\end{cases} \nonumber\end{aligned}$$ Using this expression in the proof of the revival theorem in [@electric] yields Eqs. and .
The irrational case
===================
The next step is to consider irrational values for $\EF/(2\pi)$. Here, the spectral picture breaks down due to the time-dependence of the operator $\We(t)$ as there is no concatenation of $\We(t)$ which is periodic in $t$. However, one may still classify such systems by their long-time propagation behaviour. We distinguish between two different regimes of irrationality depending on the approximability of $\EF/(2\pi)$ by continued fractions. Denoting by $c_\kk$ the continued fraction coefficients of an irrational number $x$ and by $\nr_\kk/\dr_\kk$ its continuants we have $\abs{x-\nr_\kk/\dr_\kk}<c_{\kk+1}^{-1}\dr_\kk^{-2}$ [@Hardy] and it is this quadratic quality of approximation in $\dr_\kk$ which is crucial for our result. We then distinguish two regimes by how rapidly we can approximate $x$ depending on the sequence of continued fraction coefficients $c_i$. The two regimes are irrationals the sequence of $c_i$ of which is bounded or unbounded, respectively. Independent of this distinction we may estimate the norm difference of two time-dependent walks with fields $\EF,\EF'$ by $\opNorm{\We(t)-W_{\EF'}(t)}\leq t\abs{\EF-\EF'}$ such that, irrespective of the initial state, after $t$ steps we find $$\opNorm{\We^{[t,1]}-W_{\EF'}^{[t,1]}}\leq \frac t2(t+1)\abs{\EF-\EF'}.$$ Taking $\EF'/(2\pi)=\nr_i/\dr_i$ to be a continuant of $\EF$ we find, due to the quadratic quality of the approximation of $\EF$ in $\dr_i$, $$\begin{aligned}\label{reviveirr}
\opNorm{\,\We^{[2{\dr}_{\kk},1]}+\idty\,}&\leq \frac{4\pi}{c_{\kk+1}}+\Order\left(\tfrac1{{\dr}_{\kk}}\right) \quad\dr_{\kk} \text{ odd} \\
\opNorm{\,\We^{[\dr_{\kk},1]}+(-1)^{\dr_{\kk}/2}\idty\,}&\leq \frac{\pi}{c_{\kk+1}}+\Order\left(\tfrac1{\dr_{\kk}}\right) \quad\dr_{\kk}\text{ even}.
\end{aligned}$$
Thus, for irrational numbers the sequence of continued fraction of which coefficients diverges, i.e., $c_i\rightarrow\infty$, we get an infinite sequence of sharper and sharper revivals followed by farther and farther excursions. These revivals are, in contrast to [@electric], independent of the initial state. Depending on the parity of the denominator of the continuants of $\EF/(2\pi)$, these revivals occur at times $\dr_i$ and $2\dr_i$ which grow at least exponentially [@Hardy].
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{height=".22\textheight"} {height=".22\textheight"}
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For numbers the sequence of continued fraction coefficients of which is bounded, however, does not predict any revivals. The best known and worst approximable irrational is the Golden Ratio $\EF/(2\pi)=\varphi=(\sqrt{5}-1)/2$, which has constant continued fraction coefficients $c_i=1$. Numerical simulations suggest that such systems do not show transport at all - a conjecture similar to that in [@electric] the analytical proof of which is work in progress. Figure \[fig:irrat\] shows the trajectory of the initial vector $\psi=\ket0\otimes\ket0$ on the Bloch sphere and the position distribution for $\EF/(2\pi)=\varphi$. The trajectory of the Bloch vector at $x=0$ follows a closed curve which, after delving into the interior of the Bloch sphere approaches the initial vector $\psi$ arbitrarily close at times independent of any continued fraction of $\varphi$ due to the quasi-periodic dependence of the walk on $\EF$. These erratic revivals of arbitrarily good quality (Fig. \[fig:irrat\], left) and such behaviour in time-independent systems would be a signature of pure point spectrum with exponentially decaying eigenfunctions, in the literature referred to as *Anderson localization*. However, as noted above, such a spectral treatment is meaningless in the time-dependent case. Additionally, irrationals with bounded continued fraction coefficients are of measure zero [@Hardy], such that even in the irrational case we proved the occurrence of revivals at times equal to the denominators of the continuants for almost all $\EF$.
Impurities in the choice of $\EF$
=================================
As experimental implementations never are perfect but always contain impurities, let us briefly comment on the validity of the results above in the presence of (linear) noise in $\EF$. Exact rationality of this parameter seems necessary for the appearance of the revivals as this exactness is indeed for stating the revival theorem. However, as already may be inferred from the estimate for irrational values of $\EF$, there is some stability of the revivals against noise in $\EF$. Let us model fluctuations by $$\EF_\epsilon(x_t)=\EF+\epsilon x_t,\qquad x_t\in[0,1],$$ where $x_t\in[-1,1]$ is chosen randomly in each time step. Using the approximation above with $\EF'=\EF_\epsilon(x_t)$ we find $$\opNorm{\We^{[t,1]}-W_{\EF'}^{[t,1]}}\leq \tfrac t2(t+1)\epsilon.$$ Thus in analogy with for $\EF/(2\pi)=\nr/\dr$ $$\begin{aligned}\label{eq:noiseestimates}
\opNorm{\,W_{\EF'}^{[2\dr,1]}+\idty\,}&\leq\dr(2\dr+1)\epsilon+\Order(\widetilde\alpha^\dr) \\
\opNorm{\,W_{\EF'}^{[\dr,1]}+(-1)^{\dr/2}\idty\,}&\leq\tfrac\dr2(\dr+1)\epsilon+\Order(\widetilde\alpha^\dr)
\end{aligned}$$ for $\dr$ odd and even, respectively. Hence if random fluctuations can be controlled on the order of $\epsilon=\Order(\dr^{-2})$ signatures of revivals are found, see Figure \[fig:probsgrid\].
Quite striking is the observation that for irrational values of $\EF$ for which no transport is observed in clean systems, such as the Golden Ratio, the presence of noise makes the walk propagate. This “noise-induced transport” is a reminiscent of the fact that the set of values showing no transport has measure zero such that independent of $\epsilon>0$ with probability one in each step the random variable $\EF_\epsilon$ induces transport (Figure \[fig:probsgrid\], right).
Gauge-equivalence between walks quasiperiodic in space and time
===============================================================
The similarity of the results in the body of the paper and those for electric walks [@electric] strongly suggest a relation between the two models. In continuous time on the lattice systems with linear potential and systems with a uniform time-dependent vector potential are gauge-equivalent.Led by this example we here examine the possibility of transforming the spatial dependence of electric walks to a temporal one by a local gauge transformation $G_t=\bigoplus_xG_{x,t}$. The electric walk model is defined by $$\label{eq:welectric}
W^E_\EF=e^{i\EF\hat x}CS=:\bigoplus_x C_xS,$$ where $\hat x$ denotes the position operator. To establish a gauge equivalence between $W^E_\EF$ and an explicitly time-dependent walk $W(t)$ (like the one in ) we have to find $G_{x,t}$ such that $$W(t):=G_{t}W^E_\EF G_{t-1}^*$$ is uniform in space, i.e. translation invariant, but explicitly time-dependent. By unitarity of $G_t$ we find $$W^{[t,1]}=W(t)\dots W(1)=G_t(W^E_\EF)^tG_0.$$ Demanding locality of the $G_{x,t}$ and a shift-coin decomposition of $W(t)$ forces $G_{t}$ and $S$ to commute, which is guaranteed by choosing the ansatz $G_{x,t}=e^{-i\EF tx}\idty$. Then we find the time-dependent and translation invariant walk $$\label{eq:gauged}
W(t)=Ce^{-i\EF(t-1)\sigma_z}S.$$ Comparing this time-dependent walk with we find that though the models are not exactly gauge-equivalent, the operators implementing time-dependence $e^{-i\EF(t-1)\sigma_z}$ in and $R_x(t\EF)$ in are unitarily equivalent, since the occurrence of revivals qualitatively does not depend on the explicit form of the time-dependent operator $R$ but only on the condition $R^\dr=\idty$ for some $\dr\in\mathbb N$. Quantitatively, however, the revival structure does depend on $R$ via $\widetilde\alpha$. The revival predictions for electric walks hence agree with those for but not for those for . Note that these revivals for in the irrational case become independent of the support of initial states in sharp contrast to [@electric]. This is a reminiscent of the spatial dependence of the $G_{x,t}$.
![Return probabilities $p(t)$ for $\EF_\epsilon$ chosen randomly in each time step. Left: $\EF=2\pi/100$ where the green (solid), blue (dashed) and red (dotted) lines correspond to noise $\epsilon=10^{-4},5\times10^{-4}$ and $\epsilon=10^{-3}$, respectively. The revivals at $t=100$ in the first two cases as well as the absence of any revival for $\epsilon=10^{-3}>\Order(100^{-2})$ are predicted by . Right: $\EF=2\pi(\sqrt{5}-1)/2$. Whereas $p(t)$ for $\epsilon=0$ (red, solid) is bounded from below, it converges to zero for $\epsilon=10^{-3}$ (green, dashed), thereby implying transport.[]{data-label="fig:probsgrid"}](probsgrid2.png){width="\columnwidth"}
Conclusion and Outlook
======================
In spite of the fact that the modification of quantum walks by randomness in time and by randomness in space leads to qualitatively very different phenomena we have established a case where quasi-periodic modifications in space and time, respectively, lead to very similar behaviour, especially with regard to revivals. This holds for both the commensurate and the incommensurate case and revival signatures are shown to be stable with respect to noise in the quasi-periodic parameter. The models studied here are based on coined quantum walks with qubit coins, a fact that is used in an essential way. There appears to be no general mapping allowing such conclusions in more general cases. Indeed some similarities observed numerically, such as the appearance of a “smooth” trajectory in Fig. \[fig:irrat\] (left), which is well understood in the space-quasi-periodic case await an analytic explanation. It would also be very interesting to establish connections for systems with higher dimensional lattices and coin spaces.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank Referee A for his suggestion to consider a gauge equivalence between the model in this paper and the electric walk model. We acknowledge support from the ERC grant DQSIM and the European project SIQS.
APPENDIX: Detailed investigation of [@susi Table I]
===================================================
We discuss the findings of [@susi Table I] by means of the more general theory laid out in the body of the paper. In [@susi] the coin is given by $C=R_y(\theta)$ and results are given for $\theta=0$ and $\theta=\pi/2$ [@susi Table I]. These choices lead to $C=\idty$ and $C=-i\sigma_y$, respectively, for which the revival theorem predicts perfect revivals as we show below. Let us first examine the case $C=\idty$. Then $$\abs{\widetilde\alpha}^2=\cos^2(k),\qquad \cos(m\theta_{\widetilde\alpha})=1$$ such that the revival theorem and results in $$\begin{aligned}
\opNorm{\We^{2m}+\idty}&=2,\quad &m\:\text{odd}, \\
\opNorm{\We^m+(-1)^{m/2}\idty}&=0,\quad &m\:\text{even}.\end{aligned}$$ Hence, contrary to the electric case where $C=\idty$ shows no revivals at all, for $m$ even periodically occurring perfect revivals are predicted. For $m$ odd no revivals are predicted which agrees with [@susi Table I] where only even values for $m$ are considered. Choosing $C=i\sigma_y$ yields $$\abs{\widetilde\alpha}^2=\sin^2(k),\quad \cos(m\theta_{\widetilde\alpha})=\begin{cases}1 & m\mod 4=0 \\ 0 & m\:\text{odd} \\ -1 & m \mod4=2\end{cases}$$ such that in either case $$\begin{aligned}
\opNorm{\We^{2m}+\idty}&=0,\quad &m\:\text{odd}, \\
\opNorm{\We^m+(-1)^{m/2}\idty}&=0,\quad &m\:\text{even}\end{aligned}$$ giving revival predictions independent of the parity of $m$. This agrees with[@susi Table I] where for $\theta=\pi/2$ also odd values for $m$ are admissible.
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abstract: 'Compact group galaxies often appear unaffected by their unusually dense environment. Closer examination can, however, reveal the subtle, cumulative effects of multiple galaxy interactions. Hickson Compact Group (HCG) 59 is an excellent example of this situation. We present a photometric study of this group in the optical ([*HST*]{}), infrared ([*Spitzer*]{}) and X-ray ([*Chandra*]{}) regimes aimed at characterizing the star formation and nuclear activity in its constituent galaxies and intra-group medium. We associate five dwarf galaxies with the group and update the velocity dispersion, leading to an increase in the dynamical mass of the group of up to a factor of 10 (to $2.8\times10^{13}~$), and a subsequent revision of its evolutionary stage. Star formation is proceeding at a level consistent with the morphological types of the four main galaxies, of which two are star-forming and the other two quiescent. Unlike in some other compact groups, star-forming complexes across HCG 59 closely follow mass-radius scaling relations typical of nearby galaxies. In contrast, the ancient globular cluster populations in galaxies HCG 59A and B show intriguing irregularities, and two extragalactic H [ii]{} regions are found just west of B. We age-date a faint stellar stream in the intra-group medium at $\sim1~$Gyr to examine recent interactions. We detect a likely low-luminosity AGN in HCG 59A by its $\sim10^{40}$ [erg s$^{-1}$]{} X-ray emission; the active nucleus rather than star formation can account for the UV+IR SED. We discuss the implications of our findings in the context of galaxy evolution in dense environments.'
author:
- 'I. S. Konstantopoulos, S. C. Gallagher, K. Fedotov, P. R. Durrell, P. Tzanavaris, A. R. Hill, A. I. Zabludoff, M. L. Maier, D. M. Elmegreen, J. C. Charlton, K. E. Johnson, W. N. Brandt, L. M. Walker, M. Eracleous, A. Maybhate, C. Gronwall, J. English, A. E. Hornschemeier, J. S. Mulchaey'
bibliography:
- 'references.bib'
title: 'The merger history, AGN and dwarf galaxies of Hickson Compact Group 59'
---
Introduction {#sec:intro}
============
Compact galaxy groups populate the high density tail of the galaxy number density distribution. The systems catalogued by @hickson82 [Hickson Compact Groups, or HCGs] exhibit some features, such as dynamical and evolutionary states, elliptical fractions and X-ray properties of the intra-group medium (IGM) similar to galaxy clusters, their massive, more populous counterparts. In contrast to the well-studied cluster galaxies, however, the specific effects of the compact group environment on the evolution of its galaxies are not yet clear.
HCGs are defined through criteria of isolation and surface brightness[^1] that give rise to self-gravitating, dense groupings of a few (typically four) main members. Because of their masses, these galaxies orbit around the group barycenter rather sluggishly, with velocity dispersions on the order of $\sigma_\textup{\scriptsize CG}\sim250$ [@tago08; @asqu], [*cf.*]{} galaxy cluster dispersions of $\sigma_\textup{\scriptsize cluster}\sim750~$[@binggeli87; @the86; @asqu]. This trait makes HCGs valuable laboratories for galaxy evolution: the low velocity dispersions force some galaxies into strong, prolonged interactions while others appear undisturbed but are apparently undergoing enhanced secular evolution. That is to say, this latter population is affected by gravitational interplay with their neighbors, but evolve more subtly, without obvious, strong interactions [@isk10].
Relating the various observational characteristics of compact groups to those of clusters is important for understanding whether they constitute their own class, or if they are simply mini-clusters. Perhaps more appropriately, structures like compact groups may be considered plausible building blocks of clusters at higher $z$ [[e.g.]{} @fujita04; @rudick06]. Revealing past investigations of HCGs as a class have focussed on gas content. Their members are typically deficient in [H [i]{}]{}gas when compared to galaxies of similar morphological types and masses [[e.g.]{} the sample of isolated galaxies in @haynes84]. @vm01 proposed an evolutionary sequence based on mapping the spatial distribution of H[[i]{}]{} across a large sample of HCGs. @johnson07 [hereafter J07] added to this investigation by quantifying the gas richness of twelve groups with the relation of H[[i]{}]{}-to-dynamical mass, $\log(M_\textup{\scriptsize
H{\,{\sc i}}})/\log(M_\textup{\scriptsize dyn}$). This gave rise to the hypothesis of an alternate, two-pronged evolutionary diagram for HCGs, which we explored in @isk10. In one path, the galaxies have strong interactions before exhausting their cold gas reservoirs for star formation, in the other, gas is processed by star formation within individual galaxies prior to late-stage dry mergers.
Furthermore, the mid-IR colors of HCG galaxies show an interesting bimodal distribution that distinguishes star-forming from quiescent systems. @walker10 interpret this statistically significant gap as evidence for accelerated galaxy evolution in the compact group environment. Their similar mid-IR color distributions relate HCGs to the infall regions of clusters and set them apart from any other galaxy sample compared, interacting or quiescent. This theme was expanded by @tzanavaris10 who found this gap apparent also in the distribution of specific star formation rates for HCG galaxies. These observations together point to compact groups as local examples of the plausible building blocks of clusters in the early universe.
In addition, HCGs, which are isolated by selection, could potentially help explain the evolutionary history of some field ellipticals. For example, @rubin90 originally proposed (see also [gallagher08]{}) that HCG 31 will evolve into a single, field elliptical through a wet merger (one where gas is still available during the interaction). ‘Fossil groups’, the [probable]{} ultimate fate of isolated groupings, were examined by @jones03, who defined a criterion of diffuse X-ray emission in excess of $0.5\times10^{42}h_{70}^{-2}~$erg s$^{-1}$ for such a classification. This arises from the processing of a group’s IGM during a merger (or series of mergers), but the low total mass of most local compact groups suggests their potential well lacks the depth required to heat the IGM to X-ray detectable levels [@mz98]. Using multiple mergers as a vehicle toward a fossil group end-state maps one path of galaxy evolution from the ‘blue cloud’ of star-forming disk galaxies to the ‘red sequence’ of quiescent bulge-dominated galaxies [@bell04].
Fossil group formation may provide an analogy to cluster centers or sub-clumps where the buildup of cD galaxies occurs. If this turns out to be valid, the study of compact groups could also help illuminate morphological transformations in the innermost cores of clusters. Exploring these different scenarios may prove fruitful for our understanding of galaxy evolution and the buildup of stellar mass in the universe. Making meaningful progress in this area requires detailed multi-wavelength studies in order to map the range of physical processes affecting galaxies that are found in these environments, determine their histories, and project their evolution.
A consistent treatment of a large sample of HCGs is therefore in order. In this work we continue the series of @gallagher10 and @isk10 and provide a comprehensive, multi-wavelength study of HCG 59. We will look at the current state of the group through its star formation and nuclear activity; investigate its past through the star cluster populations; try to unravel the history of mergers in the group; examine its dwarf galaxy system; and place it in the context of HCGs in general.
The core of HCG 59 consists of four giant galaxies, [a typical number for HCGs in general]{}. The group lies at a distance of 60 Mpc, based on a recession velocity of $v_R=4047~$ [@hickson92 corrected to the reference frame defined by the 3K Microwave Background] and $H_0=73~$Mpc$^{-1}$. Three of the galaxies, A (type Sa), B (E0), and C (Sc), have seemingly undisturbed morphologies, and the fourth (D, Im) is an unusually large irregular with a normal, peaked light profile. The total stellar mass of the group is [from the 2MASS $K_s$-band luminosities; @tzanavaris10], while the H[[i]{}]{} mass of is comparable to the value expected for the morphological types and stellar masses of the member galaxies, according to @vm01. This is therefore a somewhat gas-rich compact group, given that HCGs typically contain only about a third of the H[[i]{}]{} expected. On the other hand, the J07 scheme classifies the H[[i]{}]{} content of the galaxy group as a Type II, [*i.e.*]{} intermediate in gas content, according to its ratio of gas-to-dynamical mass of $\log(M_\textup{\scriptsize
H{\,{\sc i}}})/\log(M_\textup{\scriptsize dyn}) = 0.81\pm 0.05$. These classifications are based on different criteria and the disparity can thus be reconciled. Table \[tab1\] summarizes some of the general characteristics of the four galaxies, while Table \[tab2\_dlx\] presents some derived and literature values of the mass content and nuclear identifications in the four galaxies.
This paper is organized in the following way: Section \[sec:obs\] presents the optical, IR, and X-ray datasets used throughout this work. Section \[sec:clusters\] provides a full account of the young and old star cluster populations, which we use as our prime diagnostics of current star formation and ancient interactions. In Section \[sec:results\] we discuss the main findings of this work. Finally, in Section \[sec:summary\] we summarize the work presented and offer ties to previous and future work in this series.
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Observations {#sec:obs}
============
[*HST*]{} optical imaging {#sec:obs-hst}
-------------------------
The analysis presented in this paper is based largely on [*HST*]{}-ACS/WFC multi-band data. Images were taken in the *F435W, F606W F814W* bands in two pointings to cover all known giant group members. We will refer to these filters as [$B_{435}$]{}, [$V_{606}$]{}, [$I_{814}$]{} (and the set as *BVI*) to denote the closest matches in the Johnson photometric system. The notation does not, however, imply a conversion between the two systems. The observations were executed on 2007 February 24, as part of GO program 10787 (PI: Jane Charlton). The exposure times were 1710, 1230 and 1065 seconds in the *BVI* bands respectively. Three equal sub-exposures were taken with each filter with a three-point dither pattern (sub-pixel dithering). Images were reduced ‘on the fly’ to produce combined, geometrically corrected, cosmic-ray cleaned images. For the analysis of point sources, we used the standard [*HST*]{} pipeline products with a nominal pixel scale of $0{\mbox{\ensuremath{.\!\!^{\prime\prime}}}}05$ per pixel. For analysis of the extended sources, we ran `MultiDrizzle` [@multidrizzle] with the pixel scale set to $0{\mbox{\ensuremath{.\!\!^{\prime\prime}}}}03$ per pixel to improve the spatial resolution. The absolute image astrometry was checked with the world coordinate system of the Two Micron All-Sky Survey catalog [2MASS; @2mass] by identifying four unsaturated point sources in common; the average offset was $\sim0.01$ in RA and Dec. The four main galaxy [$I_{814}$]{}-band light profiles were fit with Sérsic profiles using GALFIT [@sersic; @peng]; the best-fitting centroid positions are given in Table \[tab1\].
We used the images, presented in Figure \[fig:finder-hst\], to characterize the optical morphology of the galaxies, and to detect and photometer star clusters and cluster complexes. All reported magnitudes are in the Vega magnitude system. In Section \[sec:clusters\], we present the analysis of these two scales (clusters and complexes) of the star formation hierarchy and also distinguish between young massive clusters (YMCs) and globular clusters (GCs).
Optical point source photometry {#sec:sc-phot}
-------------------------------
We follow the same rationale applied in our previous work [@gallagher10; @isk10] and use star clusters to infer the star formation activity and history in each of the HCG 59 galaxies. At the adopted distance to HCG 59 of 60 Mpc, we expect some contamination by supergiant stars, which can have absolute $V$-band magnitudes as bright as $-8.5$ [@efremov86]. At this distance, one ACS pixel measures $\sim13$ pc, [[*cf.*]{} the average star cluster radius of $\sim4$ pc; [e.g.]{} @remco07], meaning that clusters are at most marginally resolved and can be considered point sources for the purposes of selection and photometry. We select clusters using the method described in @gallagher10; in brief, we perform the initial selection on median-divided images, require selection in all three bands, and filter the resulting catalog using point spread function (PSF) photometry. Our PSF filtering applied the following criteria from the output of the `ALLSTAR` routine in IRAF[^2]: $\chi$ values below 3.0; a sharpness in the range $[-2.0,2.0]$; and a photometric error less than $0.3~$mag. Aperture corrections are first measured between 3 and 10 pixels and then added to the @sirianni05 corrections to infinity. Finally, foreground (Galactic) extinction with $E(B-V)=0.037$ is accounted for using the standard Galactic extinction law [a correction of $A_V\sim0.12$ mag; @schlegel98].
In order to fortify the selection against stars, we apply a conservative absolute magnitude cut at $M_V<-9$ mag, which produces the high-confidence sample. We do, however, define a larger sample by relaxing the magnitude cut and applying stricter PSF-fitting criteria to detect globular clusters, which are expected to be fainter and point-like. In order to minimize contamination from marginally resolved sources such as compact background galaxies, we follow @rejkuba05 and apply hyperbolic filters (starting narrow for bright sources and widening for fainter sources) with a maximum cut-off at the above mentioned criteria of magnitude error, $\chi$ and sharpness. We will refer to this as the extended sample.
The application of these criteria assigns 240 bright star cluster candidates (SCCs) to the high-confidence sample and 948 to the extended sample. Specifically, the numbers of detected SSCs (extended sample numbers in parentheses) in galaxies A through D are 7 (29), 77 (213), 13 (63) and 65 (217), with a further 78 (426) objects coincident with what would be the intra-group medium. We will provide a full analysis of these cluster populations in Section \[sec:scs\].
In order to test the completeness of the final list of SCCs, we used `ADDSTAR` to add 3000 artificial stars to the image (over the entire field, including the galaxies) in the apparent magnitude range 24–28, [*i.e.*]{} absolute magnitudes of $(-9.89, -5.89)$. [Because the final catalogue only contains sources detected in all three filters, we include this effect by calculating completeness fractions based only on artificial stars detected in all three bands [[e.g.]{} @darocha02].]{} The limiting magnitudes for the 90% and 50% recovery rates are (26.56, 27.25), (26.51, 27.19) and (26.47, 27.15) in the $B_{435}$ , $V_{606}$ and $I_{814}$ bands respectively (after photometric corrections are applied). For the distance modulus used of 33.89 mag, $m=26.5$ mag corresponds to $M\simeq-7.4$ mag.
[Our assessment of the state of star formation in HCG 59 is not limited to star clusters. Star cluster complexes represent a larger scale of star formation, as the optically blended concentrations of gas, stars, and dust that make up small star-forming regions, and likely include groups of clusters. In contrast to star clusters, these can be resolved to even greater distances than studied here, as the fractal distribution of gas about a galaxy gives rise to such structures at all scales [[e.g.]{} as demonstrated for M33 by @bastian07]. ]{}
Globular Cluster Candidate Selection {#sec:gc-selec}
------------------------------------
Globular clusters are also selected from the [*HST*]{} images. Since the process is tuned to the color distributions found in HCG 59, we provide a full account below. As contamination from supergiants is not a problem for objects with GC-like colors, we adopt a fainter magnitude limit to select old GC candidates than we used for SCCs. We have chosen a cutoff at $V_{606}=26$, which corresponds to $M_V\sim
-7.7$ at our adopted distance modulus for HCG 59, or slightly more luminous than the expected peak in the globular cluster luminosity function (GCLF) at $M_V\sim -7.4$ [e.g. @az98; @harris01]. The majority of GC candidates brighter than this limit lie above the 90% photometric completeness level in all 3 filters. Assuming a Gaussian GC luminosity function with a peak at $M_V=-7.4\pm 0.2$ and a dispersion $\sigma=1.2\pm 0.2$, our faint-end cutoff then samples $39\pm 8 \%$ of the entire GCLF.
We have selected GC candidates (GCCs) according to the color-space distribution of Milky Way GCs. We de-reddened the colors of globular clusters from the @harris96 catalog by their listed $E(B-V)$ values, and then defined a parallelogram based on the intrinsic $(B-V)-(V-I)$ color distribution of the MW GCs. This parallelogram was then converted to the ACS filters using the ‘synthetic’ transformations in @sirianni05. The color selection region adopted here is 0.10 mag wider in $(V-I)$ than that used in the analysis of HCG 7 [@isk10], but still does not exceed the boundaries of the @harris96 MW GCs. To quantify, 95 of 97 MW GCs from the @harris96 catalog (those with [*BVI*]{} information) lie within this box. All point sources in the HCG 59 fields with 1$\sigma$ error bars that overlap our color selection region are considered GC candidates, and are plotted in Figure \[fig:gc-loc\].
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Due to the close (projected) proximity of the galaxies in the group, it is likely that the halo GCs in each system will appear superposed. In an attempt to quantify the GCCs in each galaxy, we use the relationship between the galactic mass and the radial extent of the GC systems in galaxies of @rhode07. To compute the expected size of each halo, we have adopted the mass-to-light conversions in that work, although we stress the general conclusions we reach are not dependent on the detailed size of any given halo. As the predicted masses of all of the group galaxies are just below the lowest mass galaxies in the @rhode07 sample, we adopted a radial extent of 15 kpc (or 56 at the assumed distance to HCG 59) for each of galaxies A, B and C. Galaxy D is a lower luminosity system, and as seen in Figure \[fig:gc-loc\], GCCs in this object already lie within the projected halo of GCCs in galaxy A. Discussion of the individual GC systems in each group galaxy follows in Section \[sec:gcs\].
Background and Foreground Contamination
---------------------------------------
Contamination in our color-selected sample of GC candidates is expected from a variety of sources, including foreground Milky Way halo stars, reddened young clusters, and unresolved background galaxies. With the small number of GC candidates present in some of the HCG 59 galaxies (discussed below), contamination can be significant.
Predictions from Milky Way star count models [the Besançon model of @robin03] suggest that only 3–4 foreground Milky Way stars will appear in the magnitude and color ranges for expected GCs in each of our ACS fields. Determining the contamination from younger, reddened clusters is more difficult, particularly in the central regions of the late-type galaxies C and D, where such objects could be present. Unresolved background galaxies are not likely contributing in any significant way to the numbers of objects in our fields; analyses of the background objects (with GC-like colors) in HCG 7 [@isk10] showed that the predicted foreground Milky Way star counts were similar to the observed putative background contamination, leaving little room for background galaxies to contribute significantly.
[We also consider the @pirzkal05 analysis of stars in the Hubble Ultra-Deep Field (HUDF). Within the range of colors shown in Figure \[fig:colors-all\], they found the main contaminant of ‘void sky’ to be M-stars, however, with a [$V_{606}$]{}-[$I_{814}$]{} of $\sim2.0$, they are too red to be considered in our analysis. All Main Sequence stars detected in the HUDF are too bright to be mistaken for star clusters by our detection algorithm. In fact, the only class of stellar object that can be found in the color-space occupied by our cluster candidates is white dwarfs, which @pirzkal05 find to have a density of $1.1\pm0.3 \times 10^{-2}~$pc$^{-3}$. The maximum Galactic volume covered by our two pointings is a cube of $\sim8000~$px on a side, or $\sim0.24~$pc$^{-3}$, assuming a scale height of 400 pc (the maximum height quoted by @pirzkal05). Such a volume might be expected to host $\sim2.6\times10^{-3}$ white dwarfs. Therefore, we consider this potential source of contamination to be negligible.]{}
The dominant source of contamination to our GCC sample will actually be stars from the Sagittarius dwarf galaxy. Our ACS fields (at $l
\sim 254^\circ$, $b\sim +69^\circ$) are superposed on a rather dense part of the tidally induced (and bifurcated) leading arm [stream ‘A’ from @bel06] of Sgr [e.g., @maj03; @bel06; @new07; @yan09]; the presence of such streams is not accounted for in traditional Milky Way star count models. To investigate the impact of Sgr Stream stellar populations, in Figure \[fig:gc-diag\] we have overlaid 12 Gyr isochrones of @marigo08 with a range of metallicities expected for the Sgr leading arm, $[\textup{M/H}]\sim -1 \pm
0.5$ [e.g. @chou2007; @yan09], at distances between 26 and 36 kpc onto color-magnitude diagrams of the point sources in both of our ACS fields [assuming $d\sim 31$ kpc, with a spread of $\pm 5$ kpc; @new07; @no10; @correnti2010]. From this, we see that stars just below the Sgr Stream main sequence turnoff do have colors and magnitudes similar to that of the brighter ($V_{606} < 25$) GC candidates in our study, making some contamination likely.
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To estimate the [*total*]{} contamination in our GCC sample, we assume those GC candidates that lie far outside the GC system halos (as shown in Figure \[fig:gc-loc\]) are instead contaminating sources. The one exception to this is a region (called ‘outer B’ in Figure \[fig:gc-loc\]) that lies outside the GC system of galaxy B, opposite to the direction of galaxy A. We will return to this feature below. There are a total of 18 objects in 8.4 arcmin$^2$, or a background surface density $\Sigma_{back}=2.3 \pm
0.5~\textup{arcmin}^{-2}$. This is much higher than the predicted surface density of MW halo stars from the Besançon model ($\Sigma_\textup{\scriptsize MW}\sim 0.4~\textup{arcmin}^{-2}$), indicating that Sgr leading arm stars are the dominant foreground source of contamination in our sample.
Of course, for the above analysis we are making the assumption that these contaminating objects are not *bona-fide* ‘intra-group’ GCs that lie far outside the main galaxies of the group. To test this, we compare the $V_{606}$ luminosity function for the background source sample with the luminosity function of the large GC candidate population surrounding galaxy B. We show this in the right panel of Figure \[fig:gc-diag\]. The extrahalo luminosity function does not show a sharp rise with increasing magnitude as expected of a GC luminosity function and exhibited by the GCCs in galaxy B. Although a definitive comparison is not possible with so few objects in the IGM area, this is consistent with these IGM objects being contaminants and not a part of a diffuse population of intragroup GCs. Thus we adopt the surface density above as that of the ‘background’ in the analyses that follow.
Las Campanas wide-field imaging: low surface brightness light and dwarf galaxies {#sec:campanas}
--------------------------------------------------------------------------------
We extend the coverage of the [*HST*]{} observations through wide-field imaging with the Las Campanas Observatory (LCO) 2.5-meter telescope. We took $B$- and $R$-band images of a 25 diameter around the group with the Wide Field Reimaging CCD Camera (WFCCD). The data were obtained on as part of an imaging campaign that covers all 12 HCGs in the J07 sample. The $B$ and $R$ filter exposure times were 300 s and 600 s, respectively.
These images allow for the detection of low surface brightness features, such as the signatures of past interactions, over a large area. We present this analysis in Figure \[fig:bplusr\], where we stacked the $B$ and $R$ images and applied a Gaussian smoothing filter to the result. This image shows only features and $R$-band contours that register at least $3\,\sigma$ above the background. We find two faint features, a ‘bridge’ that appears to connect galaxies A and B and an arc extending from B toward a compact structure to its north-west (we will later refer to this as the ‘B-I arc’). There is another compact, extended source in the space between galaxies C and D.
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The original purpose of the LCO observing program was to prepare a sample of dwarf galaxy candidates for spectroscopic follow-up. Though our redshift survey has yet to cover HCG 59, it is covered by the Sloan Digital Sky Survey [SDSS; @sdss]: a spectroscopic search sweeping a radius of 30 arcminutes around the nominal center of the group (the geometric center of the region enveloping the four known members) yields seven spectra with redshifts in the range 0.01–0.02: galaxies C and D and five compact galaxies. We therefore consider the membership of SDSS galaxies J114817.89+124333.1 and J114813.50+123919.2, which are covered by our wide-field imaging and J114930.72+124037.5, J114940.11+122338.6 and J114912.21+123753.8 which lie at projected distances greater than 13 arcminutes from the group center. The first of these galaxies is also present in the [*HST*]{}imaging, but lies partly in the ACS chip gap. In Section \[sec:dwarfs\] we attempt to determine whether these are HCG 59 members through a phase-space analysis.
The @hickson82 naming convention assigns letters in order of brightness. Since our imaging does not cover all five dwarf candidates, we used the SDSS $r$-band photometry to consistently classify the galaxies as HCG 59 F through J. We have omitted the letter E, as it was assigned in the original catalog to a background galaxy. We attempted to measure stellar masses for these galaxies using 2MASS $K$-band images [@2mass], however, they are below the detection limit of that survey. Table \[tab:dwarfs\] summarizes all of the information presented in this section: measured and SDSS photometry, radial velocities, galaxy morphologies and projected distances from the group barycenter. The latter two properties will be discussed in Section \[sec:dwarfs\]
[*Spitzer*]{} observations: infrared spectral energy distributions {#sec:obs-spit}
------------------------------------------------------------------
The optical imaging was complemented by [*Spitzer*]{} imaging in the mid-infrared (IRAC 3.6–8 and MIPS 24 observations) presented in J07 and shown in Figure \[fig:finder-spit\]. In addition to the Rayleigh-Jeans tail of stellar photospheric emission, the IRAC bands probe the presence of hot dust and polycyclic aromatic hydrocarbons (PAHs), while the $24~\mu$m observations trace cooler thermal dust emission. The dust and PAH emission are both stimulated by star formation activity. The harder spectra of active galactic nuclei typically destroy PAH molecules while heating dust to hotter temperatures than found in galaxies with star formation alone. At low AGN luminosities, the IR SEDs are often ambiguous [particularly in the presence of star formation; e.g., @gallagher08].
{width="60.00000%"} {width="50.00000%"}
The [*Spitzer*]{} images were combined with *JHK$_S$* observations from [[2MASS]{}]{} [@2mass] to plot the IR spectral energy distribution (SED) of each galaxy (following J07), presented in the frequency-space plot of Figure \[fig:seds\]. We have used the @silva98 templates for galaxies of various morphological types. These map the SED of different galaxies as the sum of starlight and gas and dust emission from star formation and interstellar cirrus. We calculate the spectral index of the SED within the IRAC bands through a simple power-law fit. This was defined by @gallagher08 as [$\alpha_\textup{\scriptsize IRAC}$]{}, and it serves as a measure of star formation activity. In logarithmic frequency units, the flux difference from 8 to 4.5$\,\mu$m leads to a positive gradient in quiescent environments, while star formation registers as a negative slope. For HCGs, the steepness of the slope is sensitive to the specific star formation rate [@tzanavaris10].
In brief, the SEDs of galaxies B and C follow their morphological types of E/S0 and Sc, while the irregular nature of D does not allow for a template to be assigned (we use Sc in Figure \[fig:seds\]). Galaxy A, nominally an Sa, shows strong excess light at wavelengths associated with PAH and/or hot dust emission ($\lambda\geq5.8~\mu$m). If the flux in this region is associated with star formation, an Sc template would be more appropriate. This is, however, inconsistent with the optical morphology of A. Furthermore, the shape of this emission is also consistent with the AGN dust bump [e.g., @elvis94; @gallagher08], so in Section \[sec:hcg59a\] we will examine the nuclear activity of this galaxy.
![Near-to-mid IR SEDs for the primary HCG 59 members. The photometric data, shown as diamonds, are drawn from [[2MASS]{}]{} (*JHK$_s$*) and [*Spitzer*]{} MIPS/IRAC [3.5–24 $\mu$m]{} (from J07). We annotate each panel with the morphological type from the literature (top row), as well as the type chosen to most accurately represent the data (bottom row). The bracketed number quotes the age in Gyr of the template for the elliptical, or the inclination in degrees of the spirals. We also cite the value of the [$\alpha_\textup{\scriptsize IRAC}$]{} diagnostic [@gallagher08], which is fit on the wavelength range indicated by the dashed line. The solid curves represent GRASIL templates [@silva98] appropriate for the morphological type of each galaxy (see text). Galaxies B and C are well represented by their nominal templates, while A and D deviate: in the case of galaxy D this is due to its irregular nature, while we consider the nuclear activity of A to be the source of the excess mid-IR emission (see Section \[sec:hcg59a\] for details). []{data-label="fig:seds"}](f6.eps "fig:"){width="1.1\linewidth"}\
The SED-fitting process could not be carried out for the new dwarf galaxies catalogued in Section \[sec:campanas\] due to the limited spatial coverage of the [*Spitzer*]{} imaging and the faintness of the dwarfs (they were not detected in 2MASS). As of July 2011, the Wide-field Infrared Survey Explorer (WISE) photometric catalogue did not cover HCG 59.
The high energy picture: [*Chandra*]{}-ACIS observations {#sec:xrays}
--------------------------------------------------------
The dataset is completed by [*Chandra*]{}-ACIS data in the 0.5–8.0 keV range. Results reported here are drawn from the work by Tzanavaris [et al.]{} (2011 in prep.). HCG 59 was observed by [*Chandra*]{} between 2008-04-12 and 2008-04-13 at the aim point of the back-illuminated S3 CCD of ACIS in very faint mode with an exposure time of 39 ks (observation ID 9406, sequence number 800743, PI S. Gallagher). The data were processed using standard [*Chandra*]{} X-ray Center aspect solution and grade filtering, from which the level 2 events file was produced. Figure \[fig:xray\] shows an adaptively smoothed 3-band X-ray image, with optical ([$I_{814}$]{}) contours overplotted for comparison.
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`Wavdetect`, the `CIAO 4.1.2`[^3] wavelet detection tool [@freeman2002] was used in the soft (0.5–2.0 keV), hard (2.0–8.0 keV) and full (0.5–8.0 keV) bands to detect candidate point sources. The 1024$\times$1024 S3 chip field was searched with `wavdetect` at the $10^{-5}$ false-probability threshold. Wavelet scales used were 1, 1.414, 2, 2.828, 4, 5.657 and 8.0 pixels. Source lists produced by `wavdetect` for each band were matched against each other by means of custom-made scripts (K. D. Kuntz, priv. comm.) to calculate a unique position for each candidate point source, taking into account the varying size of the [*Chandra*]{} PSF across the S3 CCD.
Point source photometry was carried out for the objects in the source list using `ACIS Extract`[^4] [@broos2010] which takes into account the varying [*Chandra*]{} PSF accross the CCD. Poisson $\pm1\,\sigma$ errors on net counts were calculated by means of the approximations of @gehrels1986. Sources with measured net counts smaller than the $2\,\sigma$ error were flagged as non-detections. Note that this method produces very similar results to choosing a binomial probability threshold of 0.004 in `ACIS Extract` [@xue].
As sources have too few counts for reliable spectral fitting, we apply the method of @gallagher2005 to obtain a rough estimate of the spectral shape by using hardness ratios, defined as HR $\equiv (H-S) /
(H+S)$, where $H$ and $S$ represent the counts in the hard and soft bands, respectively. Briefly, we use the X-ray spectral modeling tool `XSPEC` [@arnaud1996], version 12.5.0, to simulate the instrumental response and transform observed HR values into an effective power law index $\Gamma$ (where ), and also obtain associated X-ray fluxes and luminosities. This modeling includes neutral absorption from the Galactic $N_{\rm H}$ of $2.6\times10^{20}$ cm$^{-2}$ [@nh_ref2].
We estimate flux limits of $f_X \gtrsim 2.7 \times 10^{-16}$ [erg cm$^{-2}$ s$^{-1}$]{}(0.5 - 2.0 keV) and $ f_X \gtrsim 1.6 \times 10^{-15}$ [erg cm$^{-2}$ s$^{-1}$]{} (2.0 - 8.0 keV), corresponding to luminosity limits of $L_X=1.3\times10^{38}$ [erg s$^{-1}$]{} (0.5 - 2.0 keV) and $L_X=7.7\times10^{38}$ [erg s$^{-1}$]{} (2.0 - 8.0 keV). Assuming these limits, we use the $\log N - \log S$ relation of @cappelluti2007 to estimate the number of background sources we would expect to detect in the HCG 59 field over the $1.96\times
10^{-2}$ square degree area of the S3 chip. This number is $\sim 25$ and $\sim 20$ in the soft and hard bands, respectively, with a $\sim
20$% uncertainty. In the much smaller area ($3.66 \times 10^{-4}$ square degrees) covered by our galaxies, we expect $< 1$ background source in each band.
We detect a total of 40 sources in the soft band and 33 sources in the hard band over the ACIS S3 field. We thus expect about 15 soft and 13 hard sources to be point sources associated with HCG 59. We find that 9 soft and 3 hard sources are located inside the boundaries of the MIR-based HCG 59 galaxy regions of J07. We note that 11 sources that have only hard-band emission are located far from the HCG 59 galaxies and are thus likely background AGN.
In the central regions of two group galaxies, there are three notable X-ray point sources detected with high significance. As these point sources are all with 5 of the [*Chandra*]{} optical axis, the X-ray positions are the ACIS Extract “mean positions of events within the extraction regions” [^5] with intrinsic positional uncertainties of a few tenths of an arcsecond. There is some additional uncertainty from matching the absolute reference frames of [*Chandra*]{} and [*HST*]{}, but this is expected to be small as both are consistent with 2MASS at the $\sim0.1\arcsec$ level.
The first source, in galaxy A, has a full-band luminosity of $L_{X
({\rm 0.5-8.0})} = 1.1\times10^{40}$ [erg s$^{-1}$]{} and an estimated $\Gamma
= 1.3\pm0.3$. The X-ray position of this source is 0.7 from our quoted optical position (Table 1). This isolated point source in the nuclear region of galaxy A has a luminosity that is consistent with known low-luminosity AGN [e.g., @Ho2001] and significantly higher than individual, luminous X-ray binaries. The two point sources found in the central region of galaxy B have X-ray positions 0.2 and 1.3 from the [*HST*]{}$i$-band galaxy centroid position (Table 1). Unfortunately, these sources have fewer than 10 counts in each band, precluding even a rough $\Gamma$ estimate. Their full-band luminosities are $L_{X ({\rm
0.5-8.0})} = (1.7, 1.4)\times10^{39}$ [erg s$^{-1}$]{}, respectively.
The [*Chandra*]{} images are also sensitive to diffuse emission from MK degree gas. As can be seen in our adaptively smoothed image, some soft, diffuse emission is detected in galaxy D (Im), likely associated with star formation, as well as in galaxy A. In both cases, the diffuse emission covers an area several times the size of the [*Chandra*]{} PSF at that location. We obtain an upper limit on the IGM surface brightness as follows. We calculate the count rate in a source-free region between group galaxies, and estimate the corresponding flux for $kT = 0.5$ keV thermal emission using PIMMS.[^6] We thus estimate the IGM surface brightness to be $\lesssim 7.3 \times
10^{-17}$ [erg cm$^{-2}$ s$^{-1}$]{} arcsec$^{-2}$. Finally, we note that dwarf galaxy I is not coincident with any detected X-ray sources; none of the other new dwarf members of HCG 59 are within the [*Chandra*]{} field of view.
We will discuss the implications of these observations further in Section 4.
The young and old star cluster populations: star formation over a Hubble time {#sec:clusters}
=============================================================================
Young star clusters and the past $\sim\,$Gyr of star formation {#sec:scs}
--------------------------------------------------------------
The population of star clusters is representative of star formation as a whole in any system [[e.g.]{} @bressert10]. They are formed [*en masse*]{} after large events [e.g., @gelys07b; @isk08; @isk09a; @isk09b; @bastian09antennae] and at a slower pace at all times when a galaxy is forming stars [@ladalada03]. The extreme brightness of young clusters makes them detectable to large distances and therefore a reliable tracer of the star formation history of their host galaxy over a Gyr or so. Beyond that point in time they are referred to as intermediate age clusters and eventually globular clusters. As a whole, the cluster population of a galaxy can reveal its star-forming history over a Hubble time.
In this section, we analyze the cluster populations of the four main galaxies in HCG 59 using the [*HST*]{} PSF photometry described in Section \[sec:sc-phot\]. We use color-color (CC) and color-magnitude (CMD) diagrams to roughly age-date the clusters by comparing them to evolutionary tracks. The [*BVI*]{} filter combination of our [*HST*]{} images lacks coverage below $\sim4000~$Å, which is crucial to breaking the age-reddening degeneracy (owing to the inclusion of the Balmer jump and near-UV continuum). However, it is still possible to infer the passing of intense bursts of star formation via the clumping of data-points along the evolutionary track, and unreddened young clusters are clearly evident.
Figures \[fig:colors-all\] through \[fig:cmd\] show the high-confidence sample ([*i.e.*]{} clusters with $M_V<-9$ mag; see Section \[sec:sc-phot\]) as solid dots, while the extended sample is marked with open triangles. The solid and dashed red lines (running top to bottom) show @marigo08 evolutionary tracks for simple stellar populations (SSP) of $\slantfrac{1}{5}$ and . This is slightly different from our previous work, where we used @bc03 models. We made this change because the @marigo08 models seem to provide a good fit to both young clusters and globulars, unlike other model suites which focus on one part of the cluster population. The green lines that run more or less horizontally show Starburst99 [SB99; @sb99] tracks of the same metallicities. These also include nebular emission, which we expect to often be present during the first $\sim10$ Myr of evolution. At this state the cluster is still surrounded by residual gas from the time of its formation ionized by UV photons. This short-lived phase ends when the first stars evolve and explode in supernovae that expel the gas. Clusters with colors redder than 0.8 in both axes are most likely GCs, although they might be highly reddened young clusters.
When contrasting the extended and high-confidence samples, the former appears to spread more in color-space. This reflects a mass effect intrinsic to star clusters, rather than indicating contamination. Two recent studies, @silvavilla11 and @popescu10, independently reached the conclusion that lower mass (fainter) clusters, $M\lesssim10^4~$, often exhibit deviations from the theoretical model tracks. In this mass regime, the underlying stellar initial mass functions (IMF) are under-sampled and stochastic effects dominate the overall light. Given that the IMF is populated randomly, it is physically equivalent to creating either one high mass star, or $\sim100$ low mass stars (given the IMF slope). In contrast, a high-mass cluster will populate the IMF fully. Consequently, a population of high-mass clusters will have smaller intrinsic photometric dispersion. A cluster with lower mass will run out of material before the IMF becomes fully sampled, thus leading to the presence of gaps and spikes. Because of this effect, the presence of a high mass star in a low mass cluster will make the cluster appear to have a larger photometrically derived mass than one with only lower mass stars.
We now treat the population of each galaxy individually. In the following paragraphs, all star and globular cluster candidates discussed are taken from the high-confidence sample. The distinctions of young, intermediate, and globular are inferred from the locations of candidates along evolutionary tracks within the [$B_{435}$]{}–[$V_{606}$]{}vs. [$V_{606}$]{}–[$I_{814}$]{} color-color space.
[**Galaxy A**]{} (Sa) hosts a very small detectable population of five GCCs and two intermediate-age cluster candidates. The CMD shows them all to be consistent with having high masses, with . The lack of young clusters, combined with the high masses of the ones detected, indicates that galaxy A has stopped forming stars at a high rate. Because the mass-to-light ratio of a stellar population increases with age ([*i.e.*]{}, the older a cluster becomes, the higher its mass must be for detection), young clusters should dominate the cluster population when the SFR is high. The non-detection of [*any*]{} young clusters is further evidence that the UV+IR emission is dominated by AGN continuum, and the inferred SFR of 5 yr$^{-1}$ is severely overestimated.
[**Galaxy B**]{} (E/S0) hosts mostly red clusters; the GCCs vastly outnumber young cluster candidates in this system. In fact, the four SCCs in the high-confidence sample do not strictly belong to galaxy B, but are found in the IGM: two nebular sources are located in a clump to the west of the galaxy, which we are treating as part of a stream that connects this galaxy to the dwarf HCG 59I in our field of view. We will return to these H[[ii]{}]{} regions in Section \[sec:bplusr\]. The two non-nebular SCCs are distinct clusters in dwarf galaxy I. This means that the presence of patchy dust (as seen in the [*HST*]{} images of 59B) by itself is not indicative of star formation at a high enough level to produce massive clusters, consistent with its SFR of 0.02 yr$^{-1}$.
[**Galaxy C**]{} (Sc) shows a continuous star formation history, evident in the smooth distribution of datapoints along the evolutionary track. The presence of nebular sources indicates some current star formation, while the low masses ($\approx10^{4}$ ) derived throughout the sample imply an overall low level of star formation over time. This is in accord with the low value of 0.16 yr$^{-1}$ for the SFR of the galaxy. There is no pronounced GC population.
[**Galaxy D**]{} (Im) is an unusually large irregular galaxy. It shows a continuous star formation history through to the present. The CMD shows a handful of SCCs with . At 0.48 yr${^-1}$, this galaxy has the largest SFR of those with young clusters. There is no old component in the cluster distribution, no evident halo of GCCs, again possibly due to the low mass of the system which implies a small globular cluster system. Furthermore, the extended sample does not reveal a tight, correlated color distribution characteristic of GCs. The youngest clusters lie at a typical color-space distance of $\sim0.3$ mag away from the nebular model track along the reddening vector, indicating the presence of dusty star-forming regions.
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Star cluster complexes {#sec:complexes}
----------------------
One step above star clusters in the star formation hierarchy is cluster complexes, large agglomerates of young stars, arranged in a fractal distribution that follows the collapse of the progenitor gas. These structures can be used to understand the global star formation activity in a galaxy. They are found to be more compact at higher redshifts than in the local universe [with two to five times higher mass surface density, as found by @elmegreen09], although in one local interacting compact group, HCG 31, we find complexes to be similar to those at intermediate redshifts [@gallagher10]. We argue in this series of papers that compact groups might process gas more efficiently when interacting than most other environments apart from the infall regions of galaxy clusters [@walker10]. HCG 31 fits well in that context.
Star-forming complexes are extended amorphous regions, with dust scattered across their surfaces. Their boundaries were identified by eye and measured by contours down to a limiting surface brightness about 10$\sigma$ above the background. We single out 30 star-forming complexes in this compact group, three in galaxy C, and 27 in D. We find no complexes in the E/S0 galaxy B or A, the evolved (Sa) spiral. The sizes of these complexes are comparable to those in local systems [Figure \[fig:complexes\]; [*cf.*]{} @elmegreen94; @elmegreen96] and consistent with HCG 7, a relatively inactive compact group [@isk10]. Their colors are indicative of star formation and nebular emission, as expected of star-forming regions (Figure \[fig:complexes\], right). In all, the star-forming complexes in HCG 59 are comparable to their counterparts in local star-forming galaxies. They follow very closely a relation between brightness and size, much more so than we found in our previous study of HCG 7. We show this correlation in Figure \[fig:complexes\] (right) where the line is a simple linear fit to the data. The linear fit implies a similar surface brightness for complexes across the group, unlike in HCG 7, where we found significantly more scatter.
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The ancient globular cluster systems of HCG 59 {#sec:gcs}
----------------------------------------------
Globular clusters represent the earliest eras of star formation in a given galaxy. Their color distributions carry the imprints of interactions and mergers and thus may probe the history of their hosts over a very long timescale.
In this section we provide a full analysis of the number and color distributions of the GC populations in HCG 59 and complement the star formation and interaction histories we began to explore in Section \[sec:scs\]. This analysis is built around the plots of Figure \[fig:gcs\]. The top row has a color-color plot (left) and histogram of the $(B-I)_0$ distributions (right) in each galaxy. (The color-magnitude diagram is shown in the left panel Figure \[fig:gc-diag\].) Owing to their projected proximity, the GC systems of galaxies A and D overlap, and so we cannot provide separate analyses. However, given the relative masses of the two galaxies and their evolutionary stages, it is safe to assume that the population is dominated by A.
{width="33.00000%"} {width="31.00000%"} {width="33.00000%"}
We find a fairly large population of globular clusters across this compact group, the vast majority of which are found in and around B. Recall that given the fading of clusters with age, GCs need extremely high masses, $M\gtrsim10^5~$, to be detected to such large distances. We also find some clusters at large radial distances from the center of this galaxy, including some found along the stellar stream that appears to connect galaxies A and B (see Section \[sec:bplusr\]) and its projection on the far side of galaxy B. Galaxies A and C host small populations. To quantify these populations, we derive the specific frequency, $S_N$, a measurement of the number of clusters per unit galaxy luminosity, for each galaxy. First, we correct the observed number of GC candidates by the background correction noted above, and then calculate the total number of GCs expected around each galaxy by first adopting a photometric completeness fraction of $f=0.9\pm 0.1$ for objects with $V_{606}<26$, and correcting for the expected fraction of GCs that lie below this magnitude limit. The $S_N$ values are 0.3, 7.7 and 0.1 for galaxies A, B and C respectively, assuming that all bright GCCs consistent with the halo of A are actually bound to A. The measured and derived numbers are collected in Table \[tab-gc\].
The size of the population in galaxy C is consistent with its Sc morphological type, while B has a tremendously rich system, about twice the number of GCCs expected. Conversely, the GC population of galaxy A is much poorer than expected, compared to the values of $S_N\sim 1$ typically seen in Sa spirals [@chandar2004]. Many of the detected GCCs appear to lie in a ring just outside of the bulge of the galaxy. [Comparing to other GC populations in HCGs, our specific frequency for HCG 59B of $S_N=7.7\pm 3.0$ is larger than (yet still consistent within the uncertainties) those observed so far in other large elliptical galaxies in HCGs: global values of $S_N=3.6 \pm
1.8$ for HCG 22A from @darocha02,[^7] and $S_N=4.4\pm 1.3$ for HCG 90C from @barkhouse01.]{}
Inspection of the region marked ‘outer B’ in Figure 3 shows a possible excess of GC candidates over the background level described above in Section 2.4, perhaps tracing an intra-group stellar population. [This is not unusual, as compact groups by their very nature are likely to promote interactions. [@white03] detected diffuse intragroup emission in HCG 90 accounting for up to half the light of the group. Two further studies, @darocha05 and @darocha08, found intra-group light in another six groups: HCGs 15, 35, 51, 79, 88, and 95. ]{} In HCG 59, a total of $N=15\pm 4$ GCCs lie within the 2.7 arcmin$^2$ region, where we expect a background contribution of $6\pm 2$ objects. Assuming only Poisson noise, this suggests a $\sim 2\sigma$ excess of objects in this part of HCG 59. This possible excess could be due to variations in the stellar density in the Galactic halo or the Sgr Stream. However, the luminosity function of this small number of sources in the ‘outer B’ region (shown as the solid line in the right panel of Fig. 3) is weighted towards the faint end, suggesting that at least some of these objects are indeed true GCCs located far ($\sim
25-50$ kpc) from galaxy B. The unusually large population of GCCs associated with B, the anomalously small population in A, and the possible population of GCs in the IGM (presumably stripped from a member galaxy), all indicate that this compact group environment may have redistributed GCs between member galaxies and/or to the IGM.
[In particular, possible interactions in the recent history of galaxy B might have redistributed its GC system. With this in mind, we compare the azimuthally averaged radial profile of the spatial distribution of the GC system with the surface brightness profile of the galaxy, as derived through GALFIT [@galfit]. The galaxy is best fit by a single-component Sérsic profile of shape $n=3.1$. Interestingly, we find the Sérsic profile of the galaxy to provide a better description of the GC system than the best fitting power-law profile of index $-1.3$. This is contrary to the finding of power-law GC system distributions around loose group member NGC 6868 and HCG 22 A. If that is to be considered the norm for compact groups, then perhaps the recent interaction activity about galaxy B has changed the shape of its GC halo. ]{}
The bottom panel of Figure \[fig:gcs\] shows the $(B-I)_0$ color distribution (converted to the Johnson photometric system for direct comparison to Galactic globulars) of the clusters in each galaxy. The shaded area in the plot of A/D shows clusters that are clearly part of A, [*i.e.*]{} omits the ones that are projected upon the body of D. This does not alter the distribution, strengthening our assumption of a small population in D. Galaxy C also hosts a very small population, as expected due to its low mass.
The color distribution can act as a proxy of metallicity for GCs and we take advantage of that to compare the two populous distributions of galaxies A and B. The color distribution of galaxy A seems fairly flat and the low numbers do not allow for a statistical treatment. Galaxy B, however, provides a large enough population to perform a test for bimodality, using the KMM algorithm of @ashman94. This returns no evidence for a composite distribution in the metallicity distribution.
Discussion {#sec:results}
==========
Extending the membership of HCG 59 {#sec:dwarfs}
----------------------------------
In Section \[sec:campanas\] we introduced a search for dwarf galaxies in HCG 59, which we continue here. Since all objects we are considering here are covered by SDSS, we will not provide images and spectra here. More information can be obtained from the SDSS database using the plate IDs, Modified Julian Dates (MJDs), and fiber IDs given in Table \[tab:dwarfs\].
Regarding the morphologies of the new members, I, the candidate covered by our [*HST*]{} imaging, seems irregular, with a peaked light profile. This agrees with its spectrum, which shows clear emission lines and a continuum shape typical of a spiral. We classify it as dIm. The rest of the galaxies are not covered with high-resolution imaging, so we are more conservative in classifying them. F shows quite clear spiral structure in the SDSS images, and is therefore given an Sd type. We note that its star formation seems to be declining, given that H$\alpha$ is the only detectable emission line. G and H appear quite irregular and elongated, with spectra exhibiting blended emission and absorption. We assign them dIr types. Finally, J shows a morphology closer to spherical and weak emission lines (although the S/N does not allow for a confident determination); we assign it a dE type.
Regarding their roles as group members, four of the five galaxies do not appear to be interacting with any other members, as might be expected from their locations far from the group core (Figure \[fig:pspace\], left). It is only I that shows some evidence of an interaction with galaxy B, in the form of the ‘B-I arc’ described in the Section \[sec:campanas\]. This stellar stream includes the large star-forming region we find to the west of galaxy B, which is part of the analysis of Section \[sec:scs\].
In order to assess whether these galaxies belong to the group, we perform a phase-space analysis, following the statistical studies of @mz98 and @zm00. This is shown in Figure \[fig:pspace\], right panel, where each datapoint represents a dwarf galaxy. The x-axis measures the distance of a galaxy from the group centroid, calculated as the mass-weighted average position of the four main members (A through D). The y-axis shows the offset of a galaxy’s radial velocity from the group mean, normalized to the core group (A through D) velocity dispersion of $314\,$.
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We find all galaxies to satisfy our membership constraints. Four of five spectroscopically detected galaxies lie within the boundary set by the four main members; galaxy G, which has the largest offsets in physical and velocity space, is moving with a radial velocity offset less than three times the group dispersion. We therefore consider all five galaxies under consideration here to be members of HCG 59, based on a strict 3$\sigma$-clip, as demonstrated in Figure \[fig:pspace\]. The inclusion of new members updates the velocity dispersion of the group to $335~$. Based on this value, we derive a dynamical mass for HCG 59 of $M_{dyn}=2.8\times10^{13}~$, a $\sim10$-fold increase with respect to the main members alone. This in fact changes the J07 evolutionary stage of the group from Type II (intermediate) to Type III (gas-poor), as it yields a ratio of H[[i]{}]{}-to-dynamical mass of 0.71, with the caveat that the measured H[[i]{}]{} mass is likely underestimated because the dwarfs at large group radii are not included.
It is interesting to find dwarf galaxies at large distances from the center of the group. This lack of barycentric clustering is also observed in the Local Group, where it is seen as a morphology-density relation: dwarf irregulars (dIr) are found at larger distances from the group center [@grebel99] than the quiescent dwarf spheroidals (dSph) and dwarf ellipticals (dE). This may indicate that some dIrs are galaxies experiencing their first infalls to the group center. Such a situation could explain the relatively large velocity offset and the star-forming nature of galaxy G.
[The membership of galaxy G seems the most uncertain of the five galaxies discussed above, given the marginal agreement with the $3\,\sigma$ velocity cut, and the large projected barycentric distance. This is important, as its inclusion does affect the updated dynamical properties, due to the large change in group radius. We quantify this in Table \[tab5\], where we summarize the dynamical properties of HCG 59. Those numbers show a change of mass by a factor of $\sim4$ or $\sim10$ with and without galaxy G, while the J07 type changes from II to III regardless of the inclusion of G. The velocity dispersion is most affected: including G increases the value to 336 from the original 314, while excluding G significantly reduces the dispersion to 208, which is more consistent with the evolved state suggested by our analysis. ]{}
The current state of star formation {#sec:SF}
-----------------------------------
We have presented several diagnostics of star formation activity across HCG 59. The @tzanavaris10 SFRs of galaxies A through D are $\simeq[4.99, 0.02, 0.16, 0.48]~$ yr$^{-1}$. These are determined from the combination of UV and IR light assuming that all of the light emanates from star-forming regions. The presence of young star clusters in galaxies C and D shows that stars are forming at a fair pace, in accord with the star formation rates quoted in Table \[tab1\]. Overall, the SFRs are consistent with the infrared SEDs of these galaxies with the notable exception of A which is almost certainly strongly contaminated by AGN emission and shows no evidence of ongoing star formation from the other evidence on hand. There are several soft X-ray point sources throughout the group, which are likely to probe compact stellar remnants local to HCG 59 galaxies. There is also soft, diffuse X-ray emission confined to the galaxies, which probes $\sim10^6~$K gas heated by star formation (stellar winds and SNe). The IR images do not show much that is surprising: emission along the spiral arms of C and in the star-forming clumps embedded in D. Four of the five dwarf galaxies show star formation activity, J being the exception. They are found to be star-forming based on either their bright emission lines or blue continua. Galaxies A and B, on the other hand, exhibit quiescent or even extinguished star formation. In the case of A, this is inferred by the absence of young star clusters.
In all, the group does not appear to be undergoing a burst or any other event notable in terms of current star formation. With the exception of the two most massive galaxies, the group is forming stars at a regular pace. This is also exemplified by the behavior of star cluster complexes across the group, which follow very closely after a brightness-radius relationship consistent with typical nearby galaxies, and in contrast to HCGs 7 and 31 [@isk10; @gallagher10].
Signatures of interactions in the intra-group medium {#sec:bplusr}
----------------------------------------------------
Major interaction or merger events are very often accompanied by bursts of star and cluster formation. The examination of star clusters in HCG 59 presented in Section \[sec:scs\] did not show any evidence of such events in the last few Gyr. Given the high mass-detection limit for star clusters at this distance, they cannot be used to trace minor dynamical events. We therefore search for such evidence in the lowest surface brightness features detectable in our LCO images. In Section \[sec:campanas\], we reported the detection of a low surface brightness stream of material in the projected area between galaxies A and B. This is visible at low surface brightness in our [*HST*]{} imaging and quite pronounced in the wide-field images from Las Campanas, at the $>3\,\sigma$ level. We also detected an arc of luminous material to the west of galaxy B, perhaps connecting it to compact galaxy I, which we discuss in Section \[sec:dwarfs\].
The low surface brightness and limited extent of the ‘B-I arc’ preclude precision photometry. We can therefore only pursue an in-depth analysis of the bridge between A and B. In order to derive the photometric properties of this feature, we first drew a color-map to look for an evident [*$B-R$*]{} gradient. Unfortunately, the stream is not bright enough to clearly dominate the image background. We thus conducted photometry of the area and the outskirts of the two galaxy that bracket it. We used large apertures of radius 75 pc (8 px) to reduce the background noise. The results are plotted in Figure \[fig:bridge\]; on the left panel we compare the measured photometry with the evolution of the [*$B-R$*]{} color, according to the @marigo08 model tracks of three metallicities. We find an intermediate value between the colors of the outskirts of the two galaxies that define this region (plotted as yellow dashed lines), cautioning that the emission in this region might be affected by the two galaxy light-envelopes to some extent.
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The origin of the bridge is not clear and its faintness makes it difficult to ascertain the dominant source of emission. It could consist either of stripped stars, or stars that formed [*in situ*]{} from stripped gas. If this is mixed stellar material from the two galaxies, we cannot study it in any more detail. We can, however, develop the [*in situ*]{} formation scenario further, by treating the bridge as a simple stellar population. In this case, the color-magnitude diagram of Figure \[fig:bridge\] (right), provides an age estimate of about 1 Gyr, depending on metallicity. In addition, the CMD plotted in this figure provides an estimate of the stellar mass contained: with $10^7~$ of $\sim\,1~$Gyr old stars in each aperture, we extrapolate a mass in the order of $\sim10^8$ , [*i.e.*]{} a density of $\sim100~$ pc$^{2}$. If the stars here are stripped from a galaxy, the overall mass will be higher, as the $M/L$ of simple stellar populations increases with time.
HCG 59B as a merger remnant {#sec:hcg59b}
---------------------------
HCG 59B, the E/S0 galaxy on the west side of the group, seems quite regular at first glance, however, a close inspection of the low-level light reveals some interesting features. While *ELLIPSE* fitting shows an overall smooth isophotal structure in [$B_{435}$]{} light, there is severe isophotal twisting in the central regions. This is spatially coincident with several patches of extinction we detect in the [*HST*]{} images; they are most pronounced in [$B_{435}$]{}, observable in [$V_{606}$]{} and hardly detectable in [$I_{814}$]{}, implying a thin column of dust. In the [*Spitzer*]{} bands, the fits indicate very symmetric structure in the 3.6 and $4.5\,\mu$m bands, but the 5.8 and $8.0\,\mu$m fits show a faint cusp some $\sim7~$px, or $\sim2.5~$kpc from the $r^{-1/4}$ surface brightness profile peak. The color-composite IRAC image (Figure \[fig:finder-spit\]) shows hints of structure in galaxy B; however, an evolved elliptical/lenticular with near-zero star formation should present a smooth isophotal profile in all bands. Nonetheless, its SED (Figure \[fig:seds\]) shows a gradual decline, indicative of emission from stellar photospheres, rather than the heated dust associated with star formation.
To investigate these irregularities further, we take advantage of the high resolution of the [*HST*]{} images. We construct pixel-by-pixel color maps of B, with the aim of tracing the exact location of the extinction patches. If the underlying stellar population is evolved to the same stage ([*i.e.*]{} an evolved SSP), then extinction will be the only source of discrepancies in color. There are, in fact, three possible sources of color variations in this filter combination: (i) extinction across a similarly colored stellar population (mixed or coeval); (ii) spatially separated stellar populations of various ages and/or metallicities; or (iii) the presence of gas and star formation – [*i.e.*]{} H[[ii]{}]{} region emission lines.
The maps, shown in Figure \[fig:colmapb\], cover the three possible filter combinations; for reference, the [$V_{606}$]{} image is also shown. The $B-I$ map is the one most sensitive to extinction. To quantify, the @cardelli89 extinction law assigns almost twice the extinction in the [$B_{435}$]{} as it does to the [$I_{814}$]{}, $A_{435}/A_{814} = 1.85$ ($A_{435}/A_V = 1.13$, [*cf.*]{} $A_{814}/A_V =
0.61$). This is therefore the map we use to detect patches of low extinction, of order 0.3 mag, in three fingers extending approximately eastwards from the north-south line through the nucleus. This faint structure is seen in all three colormaps, but not in the [$V_{606}$]{} image. The *F606W* filter covers various emission lines, including H$\alpha$, \[N[[ii]{}]{}\], \[S[[ii]{}]{}\], H$\beta$, and \[O[[iii]{}]{}\]. Interestingly, @martinez10 found H$\alpha$, \[NII\] and \[SII\] emission in the spectrum of B, at relative intensities consistent with a composite H[[ii]{}]{} region plus AGN emission. From the concentrated, blue core of the $V-I$ image, this line emission appears to be spatially coincident with the nucleus.
{width="\textwidth"}
Circumstantial evidence for a close encounter in the recent history of galaxy B is provided by the uneven distribution of its GC system. GCs normally form spherical haloes, however, here we find GCs at large radii, many concentrated along the stellar stream that seemingly connects galaxies A and B, and its extension across the far side of galaxy B. Furthermore, the overabundance of GCs in galaxy B is matched by a severe dearth of clusters in A. Given the possibility that the two galaxies interacted $\sim1$ Gyr ago, a scenario whereby GCs are transferred between the two systems is not out of the question. It is unclear from a dynamical perspective why in the process of an interaction the GCs would flow from A, the more massive entity, to B. In a simple thought experiment, we move as many clusters from B to A as are required to level the $S_N$ of A to the nominal value for an Sa. This still leaves an excess of GCCs in B relative to normal. However, the factor-of-two uncertainties involved in the determination of $S_N$ do not rule out this scenario of GC ‘swapping’.
The X-ray map of this galaxy, as described in Section \[sec:xrays\], reveals two distinct X-ray sources in the nuclear region. Unfortunately, with full-band luminosities of $L_X = (1.4, 1.7)\times10^{39}~$erg s$^{-1}$, neither is sufficiently luminous be identified as an unambiguous AGN – a possibility that the optical spectroscopy of @martinez10 has indicated. Given the lack of ongoing star formation in the region, the sources are unlikely to be high mass X-ray binaries, though luminous low mass X-ray binaries (associated with older stellar populations) or groups of them unresolved at the distance of HCG 59 are plausible. In addition, due to the uncertainty in matching X-ray sources to optical imaging, we cannot confidently derive a one-to-one correlation between the optical clumps and these sources, although the correlation is confirmed to within the X-ray positional uncertainties. We note however that, as discussed in Section 2.7, the nuclear X-ray sources are spatially close to the optical center of the galaxy.
The excess of globular clusters and uneven dust distribution in the nuclear regions of B hint at some interaction in the more distant ($>$ Gyr) past, but the lack of additional evidence for structural disturbances limits our ability to infer more. We do detect a few young clusters in galaxy B and a non-zero (though low) SFR, and therefore some reservoir of cold gas is present. Accretion of a satellite galaxy is therefore a possibility. Furthermore, the unimodal GC color distribution does not favor a gas-rich, major merger in the past. This conclusion follows the paradigm of @muratov10, who attribute the known bimodality of GC colors to late-epoch mergers. It is also a reasonable assumption that the implied interaction did not feature a major merger with a gas-rich system, as that would have enhanced the young and intermediate-age cluster populations.
Nuclear activity in HCG 59A {#sec:hcg59a}
---------------------------
In Section \[sec:obs-spit\] we reported that the IR emission in galaxy A is consistent with being dominated by an AGN, rather than star formation. This is based on the disparity between the galaxy’s morphological type of Sa and the high UV+IR SFR suggested by interpreting the emission as related to SF. Furthermore, the lack of young massive clusters is inconsistent with a SFR of 5 $\,\textup{yr}^{-1}$.
This hypothesis is supported further by the finding of a hard X-ray source in the nuclear region of galaxy A, with $L_X =
1.1\times10^{40}~$erg s$^{-1}$ as reported in Section \[sec:xrays\]. The spectroscopic AGN survey of HCGs [@martinez10] places the galaxy at the interface of the H[[ii]{}]{}and AGN zones in the ‘BPT’ diagram [@bpt], based on optical emission-line ratios. The 2[$^{\prime\prime}$]{}-wide slit they used encompasses a wide region (effective aperture of 0.58 kpc), therefore the signal is most likely diluted by circumnuclear and disk light. Visual inspection of the central region reveals asymmetric structure, resembling a second nuclear source of comparable [$I_{814}$]{} luminosity to the nucleus. As in the previous section, we employ [*HST*]{} color maps to take advantage of the spatial resolution of $\simeq12~$pc per pixel. The maps, shown in Figure \[fig:colmap\], reveal a cone of blue light at $(x, y) \simeq (-0.1, 0.1)$, with colors of $B_{435}-V_{606}\simeq0.9$, $V_{606}-I_{814}\simeq0.6$, and $B_{435}-I_{814}\simeq1.6$. The complexity of the central region inhibits easy interpretation, but one clue are the emission lines covered in the three bands. [$V_{606}$]{} covers H$\beta$, \[O[[iii]{}]{}\], H$\alpha$ and \[N[[ii]{}]{}\], which can be associated with star formation and/or AGN activity. In this scenario, the blue cone could stand out as a result of geometry, perhaps being located in a break in the dust distribution. A simple explanation could relate this feature to an unreddened line of sight through the inner spiral structure of A. In that case, however, we would expect to see bright, young star clusters, as they are known to shine through thick columns of dust, let alone relatively dust-free regions [[e.g.]{} Region B in M82; @smith07regb; @isk08].
{width="\textwidth"}
A different interpretation can relate this structure to an AGN. The lack of symmetry could suggest a small narrow-line region, photoionized by the AGN continuum, with projection effects and obscuration hiding the cone on the far side. This would produce strong \[O[[iii]{}]{}\] and \[N[[ii]{}]{}\] emission, the presence of which was reported by @martinez10. This geometry is consistent with the inclination of galaxy A of no more than $30^{\circ}$ (assuming the AGN and galaxy share the same inclination angle).
Combining the pieces of evidence collected from the X-ray, optical and MIR emission, we propose that the nuclear emission in HCG 59A is dominated by a low-luminosity AGN with a photoionized narrow-line region. The onset of activity may be related to a possible encounter with galaxy B about 1 Gyr ago, as tentatively dated from the colors of the bridge connecting the two galaxies (Section \[sec:bplusr\]).
Summary {#sec:summary}
=======
We have presented an analysis of HCG 59, a compact group comprising four main galaxies and at least five newly discovered dwarfs at the $M_r<-15.0~$mag level. Our results are based on multi-wavelength observations and continue a series of papers that have followed two different approaches: on the one hand we have treated the overall properties of Hickson compact groups [@johnson07; @gallagher08; @tzanavaris10; @walker10]; on the other hand, we have surrounded our [*HST*]{} observations with a multi-wavelength dataset to pursue in-depth investigations of individual CGs, one at a time [@palma02; @gallagher10; @isk10].
Compared to HCGs 7 and 31, two compact groups previously studied in this series, HCG 59 presents something of an intermediate step: where HCG 7 was found to be interacting solely in the dynamical sense ([*i.e.*]{}currently in the absence of direct hit encounters), HCG 59 shows evidence for stronger interactions in the recent past. There is evidence for star formation in the intragroup medium in the H [ii]{} regions to the Northwest of galaxy B, in sharp contrast to HCG 7 where the star-formation associated with each galaxy was self-contained. In the context of the evolutionary sequence we proposed in @isk10, it occupies a stage further along than HCG 7. It has begun building an IGM, as testified by some amount of intra-group light, but has yet to build up a large elliptical fraction. Its classification as a relatively unevolved group (J07) is in accord with standard diagnostics such as its velocity dispersion of $\sim335~$ and lack of diffuse, extended X-ray emission.
Through the use of SDSS data, we associated five dwarf galaxies to HCG 59 for the first time. Their inclusion updated the velocity dispersion and dynamical mass of the group and changed its J07 evolutionary type to an evolved group of Type III (originally Type II, or intermediate). The star-forming nature of these dwarfs, their radial velocities, and distances from the group core seem to suggest that some may be infalling for the first time. The noted relation between morphological type and barycentric distance follows the one observed in the Local Group and may be considered – conversely – as an indication of the ‘young’ dynamical state of HCG 59. The above information highlights the importance of studying the dwarf galaxy contingent of compact groups.
Star formation is proceeding at a regular pace in this CG, certainly at a rate consistent with the morphologies of the member galaxies. The regularity of star cluster complexes agrees with this image. The star cluster population does not show evidence of major, gas-rich, interactions in the past $\sim \textup{Gyr}$ and the IR SEDs are generally as expected. One exception to this rule is galaxy D. Given its large size, the morphological regularity of its neighbors, and the lack of evidence of recent interactions, it is not clear why it has such an irregular structure and why it is forming stars at the rate that it is.
Where the information is unclear for galaxy D, B is evidently in the midst of at least one dynamical process. This probably started more than a Gyr ago given the lack of tidal features such as shells and tails commonly observed in such events in the optical [@schweizer98; @rogues; @mullan11]. It is likely physically associated with an arc of star formation that seemingly connects it to dwarf galaxy I (the arc that hosts the discovered extragalactic H[[ii]{}]{} regions), and 59B lies at one end of a low surface brightness bridge of stellar material, which might physically connect it with A. We found this feature to be no older than $\sim1~$Gyr by age-dating its stellar population. Perhaps most intriguingly, the globular cluster population of B is anomalously large, with significantly more globular cluster candidates than A, despite its lower stellar mass. The origin of this discrepancy is unclear, but hints at an additional event in the more distant past.
In one sense, this group is typical of early or intermediate-stage HCGs. There are plenty of dynamical processes at play, however they are all proceeding at a low level and centered around one or two objects. Galaxy B is at the focus of all such processes that our diagnostics can reach, like galaxy B in HCG 7 (also an early-type galaxy). In addition, both these groups may feature an active nucleus host galaxy (the dominant galaxy A in both groups, albeit the detection is not certain in HCG 7).
We find the presence of a low-luminosity AGN in A to be likely given the spatial coincidence of the galaxy centroid with a $\sim10^{40}$ [erg s$^{-1}$]{} X-ray point source. The inferred SFR from the IR+UV luminosity is not supported given the lack of young star clusters, and therefore likely is overestimated because of significant AGN contamination. If the A-B bridge constitutes a physical connection between the two galaxies, a causal connection between the interaction and the AGN is possible. In this case, we can constrain the timescale since the onset to no more than one Gyr.
In the introduction, we discussed HCGs as a potentially special environment in terms of galaxy evolution. The apparent duality of ‘modes’ in which HCG galaxies are found – either star-forming or quiescent – and the evident lack of an intermediate stage population are in accord with the mid-IR color-space ‘gap’ discussed in our previous work [@johnson07; @tzanavaris10; @walker10]. The impact of the group environment will be the topic of the next paper in this series. There we will treat HCGs 16, 22 and 42 with a goal of understanding the evolutionary processes at play in compact groups and relating their galaxies to those found at other levels of galaxy clustering.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank the referee, Cristiano Da Rocha, for his constructive criticism of the manuscript and suggested additions that elevated the work. ISK thanks Ranjan Vasudevan and Matt Povich for educational discussions on the X-ray properties of AGN and star-forming regions. We thank Gordon Garmire for his contribution in obtaining the X-ray dataset. Support for this work was provided by NASA through grant number HST-GO-10787.15-A from the Space Telescope Science Institute which is operated by AURA, Inc., under NASA contract NAS 5-26555 and through Chandra Award No. GO8–91248 issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory under NASA contract NAS8-03060. SCG, KF, and ARH thank the National Science and Engineering Council of Canada and the Ontario Early Researcher program. Funding was provided by the National Science Foundation under award 0908984. PRD would like to acknowledge support from [*HST*]{} grant HST-GO-10787.07-A. AIZ acknowledges support from the NASA Astrophysics Data Analysis Program through grant NNX10AE88G. KEJ gratefully acknowledges support for this work provided by NSF through CAREER award 0548103 and the David and Lucile Packard Foundation through a Packard Fellowship. PT acknowledges support through a NASA Postdoctoral Program Fellowship at Goddard Space Flight Center, administered by Oak Ridge Associated Universities through a contract with NASA. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
[*Facilities:*]{} , , ,
[^1]: $\theta_N\geq 3~\theta_G$, [*i.e.*]{} a circular area defined by three galaxy-mean-radii about the group is devoid of galaxies of comparable brightness. A group surface brightness of $\mu<26.0~$mag defines galaxy density.
[^2]: 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.
[^3]: http://cxc.harvard.edu/ciao
[^4]: http://www.astro.psu.edu/xray/docs/TARA/AE.html
[^5]: See Section 5.3 of the ACIS Extract User Manual – http://www2.astro.psu.edu/xray/docs/TARA/ae\_users\_guide.html
[^6]: http://heasarc.nasa.gov/Tools/w3pimms.html
[^7]: The lower $S_N\sim
1.7\pm 1.0$ from @barkhouse01 for HGC 22 A is due to their larger $M_V$ for the galaxy – both studies find a very similar total number of globular clusters.
|
---
author:
- 'C. M. Fotopoulou, K. M. Dasyra, F. Combes, P. Salomé'
- 'M. Papachristou'
bibliography:
- 'AA\_34416.bib'
title: |
Complex molecular gas kinematics in the inner 5kpc of 4C12.50\
as seen by ALMA
---
=24.8cm
Introduction {#sec:intro}
============
Galactic winds that are driven by the feedback of active galactic nuclei (AGNs) or young stars and that are detected in molecular gas tracers are now considered common [@sakamoto06; @sakamoto09; @leon07; @feruglio10; @fischer10; @fluetsch19; @alatalo11; @rangwala11; @sturm11; @krips11; @dasyra_combes11; @dasyra_combes12; @aalto12; @tsai12; @morganti13a; @combes13; @spoon13; @veilleux13; @cicone14; @garciaburillo14; @george14; @sakamoto14; @alatalo15; @tombesi15; @aalto16; @stone16; @gonzalez-alfonso17; @pereira18]. In galaxies with active nuclei, the momentum rate of the molecular winds is often considerably higher, that is, about 20 times higher, than the pressure exerted by the AGN radiation [e.g., @cicone14; @carniani15]. Multiple photon scatterings [@ishibashi_fabian15] and, more frequently, an energy-conserving expansion of an ionized gas bubble that leads to momentum boosting [@king11; @faucher12; @zubovas14] have been evoked to justify the high momentum rates of molecular winds. Radiation pressure can drive such winds during specific phases of the ionized medium expansion: when the expansion happens so rapidly compared to the radiative cooling that it is nearly adiabatic (as in the Sedov-Taylor phase of supernovae). Radio jets, when powerful or nearly relativistic, can efficiently drive adiabatically expanding bubbles as they rapidly deposit energy in the interstellar medium (ISM) for most expansion phases [@wagner16]. Indeed, several of the molecular winds in the above-mentioned studies were detected in galaxies with AGN jets.
@dasyra16 and @oosterloo17 examined what happens in the molecular gas when a radio jet impacts clouds. Focusing on the nearby galaxy IC5063, they found that the flux ratios of CO lines reveal the presence of highly excited and optically thin gas in the wind. This result indicated that jet–ISM interactions can leave traces in the kinematics and/or in the excitation of the impacted gas. In previous work, @dasyra14 reported heating of the molecular gas in the wind of another radio galaxy, 4C12.50. This result emerged from the comparison of two molecular gas probes: the fraction of accelerated cold ($<$25K) gas in CO data was less than one third of the total reservoir, which corresponded to the fraction of accelerated warm ($\sim$400K) gas in data [from the [[*Spitzer*]{}]{} Space Telescope; @dasyra14].
The system 4C12.50, also known as IRAS13451+1217 and PKS1345+12, is a good candidate for further studies of the impact of AGN feedback on the ISM. This system is an ultraluminous infrared galaxy (ULIRG) in the local Universe, which originated from the merger of other galaxies (Fig. \[fig:optical\]). It has two nuclei 4.4 kpc away: a western nucleus, from which the radio emission emerges, and an eastern nucleus. We refer to these nuclei as main or primary, and secondary, respectively. In optical wavelengths, the continuum emission of the primary nucleus is comparable to (i.e., lower by a factor of 1.3 than) that of the secondary nucleus within a radius of 1$\arcsec$. The K- and L-band images with Subaru show that the primary galaxy is roughly twice as massive [@imanishi14]. Contrarily, the primary nucleus has a significantly greater [\]]{} emission than the secondary nucleus (i.e., greater by a factor of three within the same radius), as derived from the subtraction of two images (Fig. \[fig:optical\]). This indicates that the gas transfer toward the main nucleus has significantly progressed. The [\]]{} emission reveals nonregular structures, such as a filamentary ridge to the northwest of the main nucleus. Traces of this ridge can also be observed in the continuum emission, indicating the presence of stars. Several shells and tidal tails that are caused by the merger are seen at distances greater than 2. Some of them are visualized in another optical, H$\alpha,$ and continuum image, from which we subtracted a model of one bulge and two disks to bring up underlying structures [@dasyra11]. Further larger-scale tails are seen by [@emonts16]. @stanghellini93 suggested that the merging system is part of a poor cluster. A third, smaller galaxy is seen 18 kpc northwest of the main nucleus, which could potentially be part of a merging group.
The two unambiguously merging nuclei are embedded in an X-ray bubble of about 30$\times$15 kpc (Fig. \[fig:radio\_emission\]; @siemiginowska08). Another, discrete X-ray component is detected along the radio jet axis further to the south. Radio emission is detected in two very different scales. A superluminal jet is seen within 100 pc from the main nucleus at 2 to 6 cm wavelengths [@stanghellini97; @lister03]. More diffuse emission, extending out to 100 kpc, is seen at 1.36 and 1.66 GHz (see Fig. \[fig:radio\_emission\]; @stanghellini05). This indicates the presence of a jet that restarted recently - as recently as a few thousand years [@lister03]. Tilts of the radio emission in all scales indicate that the jet is precessing and that it could impact the ISM at various locations. Indeed, an outflow of atomic gas had been seen in absorption by @morganti04 [@morganti13b] in clouds at the tip of the small-scale radio emission. The outflow had long been studied in its ionized phase [@holt08; @holt09; @holt11]. @dasyra_combes11, @guillard12, and @dasyra_combes12 reported the outflow detection in rotational lines and in CO(2$-$3) absorption. The outflow had yet to be seen in CO emission.
To further investigate the physical conditions of the low-temperature molecular gas in the wind of 4C12.50 (including its location, mass, and excitation), we acquired new millimeter interferometric data of CO (1$-$0), (3$-$2), and (4$-$3). For simplicity, we adopted a $\Lambda$CDM cosmology with H$_0$=70 Mpc$^{-1}$, $\Omega_{M}$=0.3, and $\Omega_{\Lambda}$=0.7, which yields 2.2kpc per arcsecond and a luminosity distance of $D_L=570$ Mpc [@wright06] at the redshift of the source.
The ALMA data and their reduction {#sec:data}
=================================
The Atacama Large Millimeter Array (ALMA) observed 4C12.50 for the cycle 2 program 2013.1.00180.S (PI Dasyra) in band 3 (July 2015), and in bands 7 and 8 (May 2015). At the time, the array had 37 operational antennas. In all bands, four spectral windows of 1.875GHz bandwidth were employed. In band 3, we used one spectral window to cover the velocity range -3300 to 2100 around the line, and one spectral window to cover the velocity range -3800 to 1100 around the HCS$^+$(3-2) line. The other two spectral windows targeted the continuum at rest frame 2.641($\pm$0.026)mm and 2.364($\pm$0.021)mm. In band 7, we used two partially overlapping spectral windows to cover the velocity range -2600 to 900 around the line, and two spectral windows to obtain the restframe 896($\pm3$) and 902($\pm3$) continuum. In band 8, we used two spectral windows to cover the velocity range to 840 around the line, one spectral window to target the line, and one spectral window to obtain the continuum at restframe 668.4($\pm$1.7). For the line observations, we typically chose a velocity bin of 5($\pm$1), which is well suited for the detection of lines originating from small cloud ensembles (either in emission or in absorption in front of a background continuum). For the continuum observations, we chose coarser velocity bins of $\sim$20 up to $\sim$90.
To create the cubes, we reran the Common Astronomy Software Applications (CASA) reduction routines delivered by ESO. The calibration routines were run with CASA version 4.3.1, 4.2.2, and 4.3.1 for bands 3, 7, and 8, respectively. The automated pipeline results were restored for bands 3 and 7. For band 8, the calibrations were fine tuned: the data from the antennas DA34 and DV02 were flagged due to high system temperatures caused by an error in the antenna position database. After a first basic flagging, the calibration pipeline computed the water vapor, system temperature, and antenna position calibration solutions and applied them to the measurement set (MS), that is, to the visibilities of all baselines, spectral windows, and target fields. It then extracted the pertinent spectral windows and continued with the masking of antenna-shadowed baselines and edge channels. The assignment of a flux model for the calibrator, and the computation of analytic expressions for the bandpass (phase and amplitude vs. frequency) and gain (phase and amplitude vs. time) calibrations were then done. The flux calibration solution was found using 3C273 in band 7, and using Titan in bands 3 and 8. Some band 8 baselines had to be excluded due to the extent of Titan. In all bands, the bandpass calibration solution was found using the QSO J1256-0547 (3C279). The phase was self-calibrated on 4C12.50 using additional observations dedicated for this purpose.
Once the calibrations were applied, the visibilities of the selected fields were converted into image cubes. For the imaging part of the pipeline, CASA version 5.4 was used. The image reconstruction was executed with the routine tclean that also performs the beam cleaning. The science-intended data were merged with the calibration-intended data of 4C12.50 during the image reconstruction phase for the sake of maximum uv plane coverage, even though the addition of the calibration-intended data had a small contribution to the total exposure time. As science target, 4C12.50 was observed for 24.4, 5.2, and 24 minutes, and as phase calibrator, 4C12.50 was observed for 2.0, 0.8, and 2.0 minutes in bands 3, 7, and 8, respectively. For the image reconstruction, we used Briggs weighting of the visibilities with robustness parameter of 2 for band 3, and 1 for bands 7 and 8. A robustness parameter of 2 corresponds to natural weighting, most appropriate for high-sensitivity detection experiments. Indeed, a robustness parameter below 2 led to considerable flux losses in band 3. In bands 7 and 8, a robustness parameter of either 1 or 2 led to identical flux measurements. In these cases, the lower value was selected, as it leads to a narrower beam with smaller sidelobes. During the image reconstruction, a primary beam correction was also applied. This procedure was repeated for continuum-free cubes, produced by the uvcontsub routine. First-order polynomials were employed for the modeling and the subtraction of the continuum in all spectral windows with one exception: in band 8, we had to use a zero-order polynomial as the main spectral window extended from -400 to 840 of the line center, and only the right-hand side of the continuum could be sampled. Contrarily, in band 3 more than 1000 were available on the left side of the line and a few hundred on the right side of the line enabling the continuum determination. Access to the continuum flux on both sides of the line reduces the uncertainty in the continuum slope. The small number of continuum-free channels on the right side of the line and the dynamic range of the observations add to the uncertainty in the continuum level.
In all bands, the configuration of the antennas led to a sub-arcsecond beam. The beam was 0.62 $\times$ 0.56 at a position angle (PA) of 14$\degr$ in band 3, 0.61 $\times$ 0.47 at a PA of 53$\degr$ in band 7, and 0.43 $\times$ 0.36 at a PA of -1$\degr$ in band 8. At a common pixel scale of 0.08 and at a common velocity resolution of 20, the noise level is at 0.40 mJy/beam for , at 1.1 mJy/beam for , and at 1.2 mJy/beam for . For the examination of flux ratios, we produced a second set of datacubes, by convolving the data to a common beam of 0.63$\times$0.63 using the routine ia.convolve2d. In this fully homogenized dataset the noise level is at 0.41 mJy/beam, 1.2 mJy/beam, and 1.4 mJy/beam for , , and , respectively.
To create the deepest possible cubes for each line, we also looked into the archive for additional CO line data from observations of 4C12.50 as a calibrator for other programs. Indeed, the line was observed with 38 antennas for 13.6 mins as a calibrator for the program 2013.1.00976.S. We calibrated the measurement set using the delivered calibration pipeline with CASA 4.7.0, and then we merged the calibrated measurement set with ours using the routine concat in CASA 5.4 to create a final cube with $\sim$40 mins of on-source exposure. As before, we used tclean to reconstruct this cube, which has a new (common) beam of 0.81$\times$0.74at a PA of -22$\degr$. The common pixel scale of 0.08 was selected for comparison purposes. This cube reached noise levels of 0.17 mJy at a spectral resolution of 100. Likewise, the line was observed with 27 antennas for 5.8 mins as a calibrator for the program 2012.1.00797.S. We calibrated the data in CASA 4.1, and we merged the calibrated measurement set with ours in CASA 5.4. We again adopted the common pixel scale of 0.08 for comparison purposes. The beam was 0.63$\times$0.48 at a PA of 51$\degr$. This cube reached noise levels of 1.1 mJy/beam at a spectral resolution of 20and 0.57 mJy at a spectral resolution of $\sim$100 . No further data were found for . From here onwards, we refer to the concatenated low-spatial-resolution cubes as deep cubes.
Results {#sec:results}
=======
Line fluxes {#sec:basic}
-----------
Spectra of all detected lines are shown in Figs. \[fig:galaxy\_spectra\] and \[fig:spectra\]. The CO (1$-$0), (3$-$2), and (4$-$3) lines that we detected provided an average redshift $z$ of 0.122($\pm$0.001) when fitted by Gaussian profiles. At that redshift, 1 corresponds to 2.2kpc. The integrated CO line fluxes from the high-spatial-resolution cubes are 13.6($\pm$1.5) Jy, 66($\pm$2) Jy and 70($\pm$2) Jy for CO (1$-$0), (3$-$2), and (4$-$3), respectively. The integrated CO(1$-$0) flux from the deep concatenated data, which have a high coverage of the uv-plane in short baselines, is 13.5($\pm$1.2) Jy. The results from the data are in good agreement with the results of [@evans99] yielding a flux of 14($\pm$4) Jy. Previous interferometric observations taken with the IRAM Plateau de Bure (PdB) array yielded a flux of 18.2($\pm$1.5) Jy at a resolution of 4.0$\times$ 3.8 [@dasyra14]. The integrated CO(3$-$2) flux from the deep concatenated cube is 67($\pm$2) Jy. For comparison, the line flux is comparable to the value we had previously measured with the IRAM 30m single dish telescope: 50($\pm$8) Jy [@dasyra_combes12]. This means that not only did the band 7 data not miss extended emission, but that they were more sensitive than the 30m telescope data. As in both bands the high-resolution data reach comparable flux levels to the deep and the ancillary data, we perform all of our calculations with them.
Additionally, and a line ($\nu$=0 4\[2,3\] - 3\[3,0\]) at restframe 448.00108 GHz were detected in band 8. The emission, a shock tracer, is nearly unresolved, originating from the vicinity of the main nucleus of 4C12.50. The ion emission tracing the AGN radiation is also circumnuclear with a low-signal extension towards the northwest. The overall fluxes of the and lines are 3.5($\pm$0.2) and 2.1($\pm$0.2) Jy, respectively. HCS$^+$(3$-$2) was not detected in band 3. Unfortunately, we were unable to test the CO(2$-$3) absorption previously reported by @dasyra_combes12. The concatenation of the two spectral windows of the band 7 observations at about -900 (indicated in Fig. \[fig:spectra\]), together with bandpass calibration and continuum subtraction uncertainties, led to significant differences in the continuum level and slope between the spectral window edges. These differences can mimic absorption or emission features or remove real absorption or emission features near the concatenation region. In the nuclear spectrum of for example, the continuum to the right of the line is higher than the continuum to the left of the line (Fig. \[fig:spectra\]).
Disk and ambient gas emission {#sec:disk}
------------------------------
The CO data indicate an overall gas extent that exceeds $\sim$12($\pm$2) kpc along the east–west axis (Fig. \[fig:disk\_detections\]). Its shape is determined by the superposition of spatially resolved structures and its total mass in , as inferred from the high-resolution data, is 7.7($\pm$0.8)$\times$10$^9$ (Fig. \[fig:galaxy\_spectra\]). The mass is computed as in @solomon97. $$\label{eq:mass}
M_{H_2}=3.25\times10^7 \alpha S _{CO(1-0)} \Delta V \nu_{obs}^{-2}D_L^2/(1+z) {\,\hbox{$M_{\odot}$}},$$ where $\alpha$ is the CO intensity to mass conversion factor, $S _{CO(1-0)} \Delta V$ is the integrated line flux in Jy, $\nu_{obs}$ is the observed frequency in GHz, and $D_L$ is the source luminosity distance in megaparsec. Throughout our work we adopted an $\alpha$ value of 0.8/(Kpc$^2$) for consistency with the literature assuming a lower conversion factor for ULIRGs than for the Milky Way [@downes98].
A dynamically settled disk is detected around the main nucleus, from which the radio jet is launched. Its mass within 0.8 (radius 1.8kpc) is 3.8($\pm$0.4)$\times$10$^9$. Compared to the emission, the or the emission in the disk is more nucleated (Fig. \[fig:disk\_detections\]). In all CO lines, the disk is centered at the main nucleus, and it follows a progression from the northeast to the southwest with increasing velocity (from negative to positive). Depending on the emission line and the galaxy side examined for the kinematics, the projected disk circular velocity is 150-200. The disk rotation pattern is best revealed by the fitting of a Gaussian line profile in every spatial pixel of the ALMA data (see Fig. \[fig:momenta\] for the momenta maps). The rotation is clearly seen for the inner 2 kpc, and it agrees with the results of @imanishi16 [@imanishi18] for comparable or smaller scales. Counter-rotating blobs or high-velocity dispersion regions are also seen in the momenta maps, for example to the west of the nucleus.
A blueshifted stream from the primary nucleus reaches, in the plane of the sky, the secondary nucleus (see Fig. \[fig:disk\_detections\]; -300$<$V$<$-100 panel). This stream could either be a true bridge or a tail of the main nucleus in front of the secondary nucleus, similarly to the tail in projection between, for example, M51 and NGC5195 [@toomre72]. In this scenario, the system originates from the major merger of a gas-rich galaxy with an elliptical galaxy that has a (potentially small) disk. The companion elliptical has gone through a series of turn-arounds and pericenters, creating loops and spiral extensions at each pericenter. Another scenario is that the system originates from the major merger of two gas-rich galaxies with bulges, and that gas stripping led to a ratio of secondary-disk flux over primary-disk flux that is low for the [\]]{} and even lower for the CO.
Galactic components (out of dynamical equilibrium) south of the primary nucleus {#sec:southern_structure}
-------------------------------------------------------------------------------
A tail-like structure south of the main nucleus is the brightest extra-nuclear region detected in our data (Figs. \[fig:south\_pos\], \[fig:south\_neg\]). Its total flux, as deduced from the high-resolution cube is 3.1($\pm$0.6) Jy, which corresponds to a mass of 1.7($\pm$0.3)$\times$10$^9$.
This structure is detected in all CO(1$-$0), (3$-$2), and (4$-$3) lines and is brightest in . The detected emission is so bright that it dominates the kinematics of the region: its velocity dispersion is low, typical of that of a spiral arm (60-80), while the velocity dispersion of the disk exceeds 120 (Fig. \[fig:momenta\]). Its mean velocity is high (200-300) compared to that of the disk, meaning that its redshifted component dominates.
The redshifted emission primarily originates from a structure that spirals out from the nucleus to the southwest. In its inner part, the gas reaches velocities of $\sim$300. Further away, towards the southwest, the gas velocity exceeds 500 (Fig. \[fig:south\_pos\]). The structure is either a spiral arm that has conserved its initial angular momentum, or a tidal tail, or material that was accreted in a polar ring geometry during the merger. In any case, it can be related to the large-scale structure seen by @dasyra14 at scales $>$4 kpc. Adding its mass to the above calculated mass of the tail-like structure, the mass of the south structure reaches 1.8($\pm$0.4)$\times$10$^9$. A secondary, fainter structure with different orientation and angular momentum vector, which spirals out from the nucleus to the southeast is also seen in the inner part of the emission. The same applies to the inner part of the blueshifted emission partly coinciding with the redshifted emission (Fig. \[fig:south\_neg\]). It might be yet another spiral arm or twisted tidal tail, or even part of a second bridge to the secondary nucleus. Due to the spatial overlap of the blueshifted and the redshifted components, we cannot reliably identify wind candidates or ascribe masses to individual components.
Other structures as wind components or candidates {#sec:outflow_emission}
-------------------------------------------------
### Nuclear wind in CO {#sec:res_nuclear}
A tentative wind detection in the nuclear region is seen in the data (Fig. \[fig:cen\_wind\_detection\]). Components at multiple velocities are seen due to line-of-sight effects and/or to an intrinsic distribution of velocities (potentially caused by the different efficiency of acceleration of the molecular gas at different densities). The molecular wind terminal velocity exceeds -1800 for the blueshifted wing and 1400for the redshifted wing. @holt03, in their optical spectroscopic study of 4C12.50, also found a very broad wind in the nucleus of the system. The [\]]{} line profile required three individual Gaussian components to be fit; one of them was centered at -1980, with a width of 1944. The terminal velocity of the [\]]{} was thus -2950. A Gaussian fit to the data provides the residual emission underlying the disk emission (Fig. \[fig:cen\_wind\_detection\]), indicating a residual flux of 0.94($\pm$0.12) Jy. This corresponds to a nuclear wind mass of 5.3($\pm$0.7)$\times$10$^8$ .
Despite the strong continuum and the high dynamic range (of order 1000; Fig. \[fig:spectra\]), the wind detection survives the test of altering pipeline parameter values (e.g., first- and second-order continuum fit, different spectral window ranges for the continuum fit) in both datasets, that is, in our high-resolution and in the deep cubes. In previous work carried out by the team using Plateau de Bure (PbD) data [@dasyra14; @dasyra_combes12] this nuclear emission was not detected because the spectral window employed was not sufficient (the PdB data reached -1500 , whereas the detection in the current ALMA data exceeds -1800 ; Fig. \[fig:cen\_wind\_detection\]) and the PdB sensitivity was not adequate to detect it: the PdB data were sensitive down to 1.1mJy/beam.
### Extended wind: high-velocity CO in radially extending filaments {#sec:res_ridge}
[*Herschel*]{} observations indicated the possible existence of an extended molecular wind in 4C12.50: unlike in most other ULIRGs, the outflowing OH molecules in 4C12.50 are primarily seen in emission [@spoon13]. This result indicated that many OH molecules are located in lines of sight free of background emission or that the background emission is low compared to the OH emission. A region in which an extended outflow is likely to be detected in 4C12.50 is north of the main nucleus [@zaurin07]. There, a very distinct [\]]{} ridge is seen in optical data, referred to as an “arc” by @tadhunter18. Our ALMA data indicate that the molecular gas disk is rather abruptly cut along this ridge at low velocities (see CO line contours near systemic velocity; Fig. \[fig:disk\_detections\]). This result could be indicative of a turnover of a tidal tail or stream or of an interface where a gas phase transition from molecular to ionized occurs. The transition could be due to the deposition of energy by some mechanism, for example, stellar or AGN radiation, collision-related shocks, or jet. Filaments that radially extend from the nucleus towards the ridge are additionally seen in the [\]]{} image. Two of them are also distinct in the stellar continuum image. @holt03 found kinematically distorted [\]]{} emission along a 160-position-angle slit that contained such filaments.
High-velocity clouds are detected tentatively along these filaments in the ALMA data, with signal-to-noise ratios between 3 and 5 (Fig. \[fig:radial\_outflows\]). In a filament along a position angle of -10, designated as F1 in Fig. \[fig:radial\_outflows\] and prominently seen in [\]]{}, we see at about -2200. The total gas mass in region F1 is 0.5($\pm$0.1)$\times$10$^8$.
In a second filament along a position angle of -30 that is designated as F2 in Fig. \[fig:radial\_outflows\] and that is prominently seen in the stellar continuum (Fig. \[fig:outflow\_tot\]), we again detect molecular gas. This time, we see emission with velocities of about -1200, -2200 , and with a corresponding mass equal to 1.3($\pm$0.2)$\times$10$^8$. The gas in or near this filament is counter-rotating in all CO lines. Moreover, this is the only extra-nuclear region with detection of gas out of dynamical equilibrium; this filament is very nicely outlined in the third panel of the last row of Fig. \[fig:disk\_detections\] for gas counter-rotating with respect to the local disk velocity field. The molecular gas mass in F1 and F2 is 1.8($\pm$0.2)$\times$10$^8$.
In both filaments F1 and F2, we also observe marginal emission at positive velocities. Focusing on velocities near 1370 we see that the redshifted emission has the shape of a ring, which is centered at the nucleus but occupies the whole disk. The mass inside it, excluding the nuclear component contribution, is calculated using the integrated flux of the area between the ellipses of the left panel of the third row of Fig. \[fig:radial\_outflows\]: namely 1.8($\pm$0.4)$\times$10$^8$ . To ensure that all extranuclear components are accounted for, we extracted the spectrum from a larger area around the nucleus as shown in the last panels of Fig. \[fig:radial\_outflows\]. Subtracting the spectrum of the nuclear wind and integrating the flux in the velocity range: -980$<$V$<$-400 we calculate the mass of the blueshifted extended emission to be: 6.0($\pm$1.0)$\times$10$^8$.
Adding the mass probed by the redshifted emission disk-wide and the blueshifted extended emission to the mass probed by F1 and F2, we find that the total mass of the extended wind reaches 1.0($\pm$0.1)$\times$10$^9$ and tentatively spreads over an area of roughly 40kpc$^2$.
Several arguments are in favor of the credibility of the wind detections in the filaments and in the extended disk-wide region despite their low S/N ratios. At $\sim$0.7 blueshifted and redshifted emission partially overlaps in the disk-wide region. The detections are located along a line that connects the main nucleus to the region of highest-velocity dispersion, as shown in Fig. \[fig:extended\_over\_maps\]. This is observed in both and maps. The shapes and the loci of the high-velocity blobs F1 and F2 coincide with those of their optical filamentary counterparts (in either the [\]]{} or stellar continuum image; Fig. \[fig:outflow\_tot\]). The emission in F1 and F2 has components in two or more velocity ranges. Additionally, F2 is detected in both and . Some of the blobs also trace the excess emission that we see after the subtraction of a bulge and two disks from the +continuum image (Fig. \[fig:outflow\_tot\]). The detection of stars along some regions also indicates two potential origin mechanisms for the accelerated gas: either it is related to a locally generated stellar wind or to an AGN generated nuclear wind. We evaluate the most likely scenario in Section \[sec:discussion\], noting that the conclusions we draw about the origin of localized winds hold even if some detections turn out not to be real.
Gas excitation {#sec:res_excitation}
--------------
\[sec:excitation\]
Because the accelerated gas can have different excitation from that of the ambient gas, CO flux ratios are examined as wind indicators. Observed flux ratios, computed from the data at the common beam resolution, are shown in Fig. \[fig:disk\_excitation\_highres\]. Clear differences are seen between the excitation of the nucleus and that of the southern tidal structure. The gas in the southern structure is subthermally excited. A range of excitations is seen between this region and the disk. Contrarily, higher excitation is observed near the nucleus. At positive velocities, some gas displays a $/$ ratio above the value of 1.78, which is the upper limit in case of optically thick gas emission. Excitation temperatures from these maps will be presented in a forthcoming paper (Paraschos et al. 2019, in prep.).
Potential excitation differences can also be revealed by the comparison of multi-wavelength, multi-temperature gas probes. From previous [[*Spitzer*]{}]{} data, we know that in the warm ($\sim$400$K$) phase, the outflow has a mass of 5.2$\times$10$^7$[@dasyra11], whereas the disk has a mass of 1.4$\times$10$^8$[@dasyra11]. Therefore, the fraction of warm in the wind exceeds 30%. From our new ALMA data, we find that the mass of the wind is as high as 1.5$\times$10$^9$ whereas that of the ambient gas is 7.7$\times$10$^{9}$. Therefore, the fraction of cold in the wind is less than 20%. This confirms our past findings of heating of the accelerated gas. It is noteworthy that some of the cold gas mass in the wind may be unaccounted for due to the spatial overlap of multiple structures out of dynamical equilibrium in our ALMA data. Still, the mass loss cannot be as high as required to alter our initial conclusion. The effects of gas acceleration and heating could possibly delay star formation temporarily.
Discussion {#sec:discussion}
==========
Energy output of the galaxy at radio and infrared wavelengths {#sec:energetics}
-------------------------------------------------------------
To evaluate whether the jet, the AGN radiation pressure, or the starburst can sustain winds, we first need to calculate the energy output of the galaxy at radio and infrared wavelengths. To obtain the jet power, $P_j$, we fitted the SED at radio, IR, optical, UV, and X-ray wavelengths. Because the jet and the dust can emit at common frequencies, we simultaneously fitted their SEDs using a standard $\chi ^2$ minimization method [@lmfit]. For the dust emission, we used a modified black-body law at long wavelengths, coupled with a power law at short wavelengths [@casey12]. We found the dust temperature to be $55 \, \mathrm{K}$ and to be 2.5$\times 10^{12}$ (in good agreement with @dasyra14). For $P_j$, we fitted the synchrotron radiation using a broken power law with an exponential cut off [@ryb12]. At radio wavelengths, up to 4$\times10^{11}$Hz, this model is well determined from our data (Fig. \[fig:radio\_spectrum\_fit\]). The integral of the fit provides the lower limit for the synchrotron power, which is 9.1$\times 10^{43}\,\mathrm{erg}/\mathrm{s}$. At higher frequencies however, the dust, stellar, and AGN emission outshine the jet emission. To find an upper limit for the synchrotron power, we fitted the data under the extreme assumption that 0.6% of the 6$\times10^{14}$Hz emission originates from the jet. The fraction 0.6% corresponds to the part of the 5092 flux that is enclosed in a radius of 150pc in the data; 150pc is the distance at which the radio jet interacts with ISM clouds per @morganti13b. In this model (Fig. \[fig:radio\_spectrum\_fit\]), $P_j$ is 6.2$\times 10^{44} \,\mathrm{erg}/\mathrm{s}$. The X-rays may also be bolstered by a contribution from various Compton components related to the jet or magnetic fields [@finke16]. To obtain the maximum possible contribution of the jet to the X-ray emission, we again adopted a power law with a cut-off that was steeper than before due to the absence of pertinent data above $10^{19}$Hz. The maximum power output in the X-rays is 8.4$\times 10^{43} \, \mathrm{erg}/\mathrm{s}$. In summary, we find the jet power to be in the range $10^{44}-8\times10^{44}\,\mathrm{erg}/\mathrm{s}$. For comparison, @guillard12 calculated the power of the radio-jet of 4C12.50 from the monochromatic 178 MHz flux, following @punsly05. They adopted a calibration of the monochromatic flux to the bolometric output that takes into account X-rays and the plasma thermal energy in the lobes. Their approach yielded $P_j$=3$\times$10$^{45}$erg$\,s^{-1}$. Likewise, using the calibration between the monochromatic 1.4 GHz luminosity and the bolometric jet luminosity following [@sulentic10], $P_j$ can be as high as 10$^{46}$erg$\,s^{-1}$ .
To obtain the force exerted on the gas due to AGN radiation pressure we assume that the AGN luminosity is well described by its value, as the flux that was absorbed and re-emitted by the dust is higher than that seen in the optical. Following @veilleux09, we ascribe half the to the AGN. Based on our previous analysis, the AGN-related part of the is 4.7$\times$10$^{45}$ergs$^{-1}$. The force exerted on the gas due to the AGN radiation pressure is $L_{AGN}/c$= 1.6$\times$10$^{35}$ergcm$^{-1}$.
To evaluate whether or not the starburst can sustain winds, we need to calculate the energy released by SNe and the force exerted on the gas due to stellar radiation pressure. We assume that for every 100 formed, there is one SN. We also assume that each SN ejects material with a kinetic energy of $10^{51} erg$. Given a star formation rate (SFR) of $\sim$200yr$^{-1}$, computed from the [@kennicutt98] formula for /2, the power released by the SN is 6$\times$$10^{43}$ergs$^{-1}$. Also taking into account the fraction of the total star formation that can be ascribed to any given area (as explained in Sections \[sec:disc\_ridge\] and \[sec:disc\_nuclear\]) we calculate the power released locally by the SN. Furthermore, using the fraction of the total $/$2 that can be attributed to a local starburst, we calculate the force exerted on the gas due to stellar radiation pressure.
Extended wind energetics: gas acceleration not sustained by a local starburst {#sec:disc_ridge}
-----------------------------------------------------------------------------
We begin our energetics study of the extended wind by ruling out that the low-S/N CO detections in filaments are accelerated by the local starburst. For this purpose, we attribute a part of the SFR or of to each ellipse of F1 and F2 of Fig. \[fig:radial\_outflows\] using the ratio of the locally enclosed emission over the total emission in our high-resolution data. The filament F1, for example, comprises roughly 8% of the total CO emission. Based on this fraction, the local is 3.8$\times$10$^{44}$ergs$^{-1}$ and the local SFR is $\sim$15yr$^{-1}$. In this case, the mechanical energy from the SN is 4.8$\times$10$^{42}$ergs$^{-1}$ and the force due to stellar radiation pressure is 1.3$\times$10$^{34}$ergcm$^{-1}$. To calculate the energetics of the gas in F1, we need to use the distance $d$ of the accelerated gas from the spot of the wind generation. For this purpose, we use the mean value of the ellipse semi-axes (0.4$\arcsec$) as the distance from the star-forming region. Using the mass and the mean velocity presented in Section \[sec:res\_ridge\], we find that the wind kinetic luminosity, $L_{kin}$, computed as $(1/2) MV^{3}d^{-1}$, is 2.0$\times$10$^{44}$ergs$^{-1}$. The wind momentum rate, $\dot{M}V$ [@combes13], is 1.8$\times$10$^{36}$ ergcm$^{-1}$. In this formula, $\dot{M}$ is the mass-flow rate of the accelerated gas, equal to $MVd^{-1}$. We find that neither the SN nor the stellar radiation pressure in the area can drive the wind along the filament, each being at least two orders of magnitude short of the required levels.
The same calculation for the filament F2 shows that it comprises roughly 9% of the total CO emission. Therefore, the local is 4.2$\times$10$^{44}$ergs$^{-1}$ and the local SFR is $\sim$20yr$^{-1}$. The mechanical energy released by the SN is 5.4$\times$10$^{42}$ergs$^{-1}$ and the force due to stellar radiation pressure is 1.4$\times$10$^{34}$ergcm$^{-1}$. The distance $d$ is 0.7and calculating the wind energetics as in the case of F1 we find that $L_{kin}$ is 1.4$\times$10$^{44}$ergs$^{-1}$ and $\dot{M}V$ is 1.5$\times$10$^{36}$ ergcm$^{-1}$. In this case neither the SN nor the stellar radiation pressure in the area are adequate drivers of the wind.
Inversely, how high does the local SFR need to be to sustain the wind? If the SN were a local generation mechanism, the SFR would need to be higher than 630yr$^{-1}$ and 440yr$^{-1}$ in the areas of F1 and F2, respectively, in order to sustain the wind. Therefore, the local SNe cannot drive the flow by themselves in the areas of the filaments. However, given that the velocity in the filaments is the highest that we have observed, the local starburst is likely to assist in their acceleration. Still, the main driver of all wind candidates needs to be sought in a central mechanism.
Overall wind energetics: sustainability by a central mechanism {#sec:disc_nuclear}
--------------------------------------------------------------
To evaluate whether or not a power source at the nucleus (i.e., within our beam) could sustain the wind, we evaluate whether the wind can be sustained via the nuclear starburst or via the jet and the AGN. For the nuclear starburst, we compare the energy deposited by SNe to the wind kinetic luminosity and, as above, the force exerted on the gas due to stellar radiation pressure to the wind momentum rate. As explained in Section \[sec:energetics\] the power released by the SN is 6$\times$$10^{43}$ergs$^{-1}$. Taking further into account that the nuclear SN can only be ascribed 10-20% of the total star formation, then the total SN power is 6-12$\times$10$^{42}$ergs$^{-1}$. The fraction ascribed to nuclear star formation is computed from the fraction of emission within the radius of a beam (0.3$\arcsec$). Furthermore, taking into account that 10-20% of /2 can be ascribed to the nuclear starburst, that is, 4.7-9.4$\times$10$^{44}$ergs$^{-1}$, the force exerted on the gas due to stellar radiation pressure, $L_{SB}/c$, is then 1.6-3$\times$10$^{34}$ergcm$^{-1}$. Alternatively, the force exerted on the gas due to the AGN radiation pressure is $L_{AGN}/c$= 1.6$\times$10$^{35}$ergcm$^{-1}$.
To find the central mechanism that could sustain the extended wind, we need to compute the respective wind kinetic luminosity and momentum rate assuming the distance of the accelerated clouds from the main nucleus. For this purpose, we compute $L_{kin}$ and $\dot{M}V$ for each individual region of Fig. \[fig:radial\_outflows\] using its distance from the nucleus. We sum the results for the extended regions with the F1, the F2 (using the two velocity ranges of F2 as two different regions), the redshifted disk-wide and the extended blueshifted detections from Table \[tab:extended\_wind\]. We find that the mass-flow rate $\dot{M}$ carried by the extended wind is as high as $\sim$500yr$^{-1}$. Then, $L_{kin}$ (extended) is 3.8$\times$10$^{44}$ergs$^{-1}$ and $\dot{M}V$(extended) is 4.4$\times$10$^{36}$ ergcm$^{-1}$. Therefore, the mechanical power of the SN is insufficient to drive the flow, considering that it is low and that only a part of it is deposited in the ISM. Likewise, the stellar radiation pressure within the central 0.3$\arcsec$ cannot drive the wind, since the momentum rate of the wind is two orders of magnitude higher than the force exerted on gas by the stellar radiation pressure. As this momentum rate is also an order of magnitude higher than $L_{AGN}/c$, the AGN radiation pressure is unable to drive this wind, unless an energy-conserving expansion has significantly boosted the momentum rate of the gas (i.e., by a factor of 30). Such a boost is high, but not impossible, as shown by [@cicone14].
To add to the above numbers those for the circumnuclear wind, we set $d$ equal to 0.3$\arcsec$, which is approximately half of the semi-major axis of the ellipse in Fig. \[fig:cen\_wind\_detection\]. The mass of the circumnuclear wind is 5.3($\pm$0.7)$\times$10$^8$, and the mean velocity of either the blueshifted or the redshifted component is $\sim$800. Using these numbers and assuming that the clouds and the CO clouds have similar spatial and velocity distributions, we find that the kinetic luminosity of the nuclear wind is 1.3$\times$10$^{44}$ergs$^{-1}$ and the momentum rate is 3.3$\times$10$^{36}$ergcm$^{-1}$. Adding these numbers to the values we calculated for the extended wind, the total kinetic luminosity and the total momentum rate of the outflow reach 5.1$\times$10$^{44}$ergs$^{-1}$ and 7.7$\times$10$^{36}$ergcm$^{-1}$, respectively. These numbers may yet increase, considering that part of the circumnuclear wind is unresolved and thus the used radius may be overestimated.
Overall, a combination of the jet, radiation pressure, and SN mechanical power needs to be invoked. The jet is the most likely generation mechanism given that it carries the most power. The jet alone could have driven the flow if a past event, which has been recorded for this system, were found to have deposited enough energy onto the ISM. In this case we are observing a fossil outflow which expands into an already carved cavity [@fluetsch19]. This could explain the radial symmetry of the wind even though the radiation pressure is equally well-suited to explain it. However, the force due to the radiation pressure cannot drive the outflow by itself, but could assist it. The same applies to the local generation mechanism along the filaments: a local starburst could further accelerate the gas that has already been accelerated.
Conclusions {#sec:conclusions}
===========
We obtained ALMA data of the radio galaxy 4C12.50 in order to determine the millimeter properties of its outflow, which was previously known to exist from large- and small-angular-resolution observations at other wavelengths. We mapped the CO distribution, kinematics, and excitation with ALMA at a resolution of $\sim$0.5 and found the following results.
- The main gaseous disk is rather compact, extending to radii of $\sim$6 kpc in CO (1$-$0) and $\sim$2 kpc in CO (4$-$3). Some co-rotating CO emission is also seen in the secondary nucleus and in a bridge connecting the nuclei.
- A shock-tracer, , and a hard-ionization-field tracer, are seen in the main nucleus.
- Several extranuclear structures are seen. These include a prominent tidal tail south of the main nucleus with gas seen in negative and positive velocities.
- The CO emission is abruptly cut along an [\]]{} ridge north of the main nucleus, where no CO is detected. This indicates a phase transition of the molecular gas.
- Extended wind components are tentatively seen in the line. The detections are also seen in regions with [\]]{} and stellar continuum emission. They include high-velocity (-2000 ) filaments that extend radially from the nucleus to the [\]]{} ridge, a redshifted disk-wide emission (V$\sim$1350 ) and an extended blueshifted emission (V$\sim$-700 ). The total mass is 1.0($\pm$0.1)$\times$10$^9$.
- A circumnuclear wind candidate is detected in emission, in , through broad blueshifted and redshifted line wings. The emission peaks within 200 pc from the radio core, and it coincides with a region of high CO excitation (within the velocity range of the disk), as indicated by its / line ratio. The mass of the circumnuclear wind is equal to 5.3($\pm$0.7)$\times$10$^8$.
- The total mass of the wind is as high as 1.5($\pm$0.1)$\times$10$^9$.
- Both at the nucleus and at the extra-nuclear regions, the wind can be sustained mainly by the jet. The radiation pressure of the AGN can help, in particular near the nucleus. It is plausible that the starburst also contributes, in particular for any extended wind components. However, the starburst cannot drive the wind alone based on its radiation pressure or its energy release by SN remnants.
This paper makes use of the ALMA data ADS/JAO.ALMA 2013.1.00180.S, 2013.1.00976.S, 2012.1.00797.S. 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. K. M. Dasyra acknowledges financial support by the Hellenic Foundation for Research and Innovation (HFRI) and the General Secretariat For Research and Technology, under grant number 1882. We would like to thank the referee, B. Emonts, for detailed comments which led to a significant improvement of the paper.
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---
abstract: 'From the very first multimessenger event of GW170817, clean robust constraints can be obtained for the tidal deformabilities of the two stars involved in the merger, which provides us unique opportunity to study the equation of states (EOSs) of dense stellar matter. In this contribution, we employ a model from the quark level, describing consistently a nucleon and many-body nucleonic system from a quark potential. We check that our sets of EOSs are consistent with available experimental and observational constraints at both sub-nuclear saturation densities and higher densities. The agreements with ab-initio calculations are also good. Especially, we tune the density dependence of the symmetry energy (characterized by its slope at nuclear saturation $L$) and study its influence on the tidal deformability. The so-called $QMF18$ EOS is named after the case of $L=40~\rm MeV$, and it gives $M_{\rm TOV} =2.08~M_\odot$ and $R= 11.77~\rm km$, $\Lambda=331$ for a $1.4\,M_\odot$ star. The tidal signals are demonstrated to be insensitive to the uncertainty on the crust-core matching, despite the good correlation between the symmetry energy slope and the radius of the star.'
author:
- Ang Li
- 'Zhen-Yu Zhu'
title: Note on neutron star equation of state in the light of GW170817
---
INTRODUCTION
============
Thanks to the development of many-body theories of nuclear matter, there is a possibility that the long-standing, open problem of the equation of state (EOS) of dense matter can be understood, by confronting laboratory measurements of nuclear properties & reactions (e.g., RIBLL at HIRFL, HIRA at NSCL, CEE at HIAF, CBM at FAIR) and observations in astronomy (e.g., HXMT [@hxmt], eXTP [@extp1; @extp2], SVOM [@svom], SKA [@ska], FAST [@fast], Urumqi, Lijiang [@lijiang], NICER [@nicer]), especially after the recent multimessenger observations of neutron-star (NS) merger GW170817 [@2017PhRvL.119p1101A; @2018PhRvL.121p1101A].
The tidal deformability $\Lambda$ describes the amount of induced mass quadrupole moment when reacting to a certain external tidal field [@1992PhRvD..45.1017D; @2009PhRvD..80h4035D]. It is normalized with a factor of $R^5$ from the second Love number $k_2$, being $R$ the NS radius. $k_2$ also has a dependence on $R$ (see, e.g., [@2010PhRvD..81l3016H; @2010PhRvD..82b4016P; @2018ApJ...862...98Z; @2018PhRvD..97h3015Z]). Following the tidal deformability observation of the GW170817 event at the premerger stage, robust lower limits can be put on the radius of merging stars’ radii, which are around 10.7 km [@2017ApJ...850L..34B]. A more detailed result from LIGO-Virgo Collaboration is $11.9^{+1.4}_{-1.4}$ km at the $90\%$ level [@2018PhRvL.121p1101A].
On the other hand, it has been established that the star radius is rather sensitive to the symmetry energy (essentially its slope $L$) with the maximum mass only slightly modified (see, e.g., [@2001ApJ...550..426L; @2004Sci...304..536L; @2006PhLB..642..436L]). A smaller $L$ (softer symmetry energy) leads to a smaller radius. Therefore, it is meaningful to study the relation of the tidal deformability with the uncertain $L$ parameter, i.e., the symmetry energy slope at nuclear matter saturation. We will examine in details in this contribution how the presently uncertain symmetry energy slope influences the tidal deformability of GW170817-like events. We use the matter state model from the quark level and suppose the merging stars are both NSs without strangeness phase transitions. The employed model has the advantage to tune consistently only the slope of interest with the other saturation properties fixed at their empirical values, respectively. Brief discussions are also made on the effect of crust-core matching and the possibility of drawing information on (inner-)crust EOS from GW signals.
MODEL
=====
{width="18pc"}
{width="18pc"}
The theoretical model we employ here is the quark mean-field (QMF) model (see e.g., [@2018ApJ...862...98Z; @2000PhRvC..61d5205S; @2002NuPhA.707..469S; @2014PhRvC..89b5802H; @2014PTEP.2014a3D02H; @2016PhRvC..94d4308X; @2018PhRvC..97c5805Z] and its recent extension [@2018arXiv180504678Z]). The model starts with a flavor independent two-parameter potential, $U(r)=\frac{1}{2}(1+\gamma^0)(ar^2+V_0)$, confining the constituent quarks inside a nucleon. The Dirac equation of the confined quarks is written as $$\begin{aligned}
[\gamma^{0}(\epsilon_{q}-g_{\omega q}\omega-\tau_{3q}g_{\rho q}\rho)-\vec{\gamma}\cdot\vec{p} -(m_{q}-g_{\sigma q}\sigma)-U(r)]\psi_{q}(\vec{r})=0,\end{aligned}$$ where $\psi_{q}(\vec{r})$ is the quark field, $\sigma$, $\omega$, and $\rho$ are the classical meson fields. $g_{\sigma q}$, $g_{\omega q}$, and $g_{\rho q}$ are the coupling constants of $\sigma, ~\omega$, and $\rho$ mesons with quarks, respectively. $\tau_{3q}$ is the third component of isospin matrix. This equation can be solved exactly and its ground state solution for energy is $$\begin{aligned}
(\mathop{\epsilon'_q-m'_q})\sqrt{\frac{\lambda_q}{a}}=3,\end{aligned}$$ where $\lambda_q=\epsilon_q^\ast+m_q^\ast,\ \mathop{\epsilon'_q}=\epsilon_q^\ast-V_0/2,\ \mathop{m'_q}=m_q^\ast+V_0/2$. The effective single quark energy is given by $\epsilon_q^*=\epsilon_{q}-g_{q\omega}\omega-\tau_{3q}g_{q\rho}\rho$ and the effective quark mass by $m_q^\ast = m_q-g_{\sigma q}\sigma$ with the quark mass $m_q$ = 300 MeV. The zeroth-order energy of the nucleon core $E_N^0=\sum_q\epsilon_q^\ast$ can be obtained by solving Eq. (1). Corrections due to center-of-mass motion $\epsilon_{c.m.}$, quark-pion coupling $\delta M_N^\pi$, and one gluon exchange $(\Delta E_N)_g$ are included to obtain the nucleon mass, see details in [@2018ApJ...862...98Z]. With these corrections on the energy, we can then determine the mass of nucleon in medium: $$\begin{aligned}
M^\ast_N=E^{0}_N-\epsilon_{c.m.}+\delta M_N^\pi+(\Delta E_N)_g.\end{aligned}$$ The nucleon radius is written as $$\begin{aligned}
\langle r_N^2\rangle = \frac{\mathop{11\epsilon'_q + m'_q}}{\mathop{(3\epsilon'_q + m'_q)(\epsilon'^2_q-m'^2_q)}}.\end{aligned}$$ The potential parameters ($a$ and $V_0$) are determined from reproducing the nucleon mass and radius in free space, namely $M_N = 939$ MeV and $r_N = 0.87$ fm.
Then nuclear matter is described by point-like nucleons and mesons interacting through exchange of $\sigma,~\omega,~\rho$ mesons. The cross coupling from $\omega$ meson and $\rho$ meson is also included. The calculation of confined quarks gives the relation of effective nucleon mass as a function of $\sigma$ field, which defines the $\sigma$ coupling with nucleons (depending on the parameter $g_{\sigma q}$). The meson coupling constants are fitted by reproducing the empirical saturation properties of nuclear matter. $m_{\sigma} = 510~\rm{MeV}$, $m_{\omega}=783~\rm{MeV}$, and $m_{\rho}=770~\rm{MeV}$ are the meson masses. The QMF framework describes consistently a nucleon and many-body nucleonic system from a quark potential.
The Lagrangian for describing nuclear matter is written as: $$\begin{aligned}
\mathcal{L}& = & \overline{\psi}\left(i\gamma_\mu \partial^\mu - M_N^\ast - g_{\omega N}\omega\gamma^0 - g_{\rho N}\rho\tau_{3}\gamma^0\right)\psi -\frac{1}{2}(\nabla\sigma)^2 - \frac{1}{2}m_\sigma^2 \sigma^2 - \frac{1}{3} g_2\sigma^3 - \frac{1}{4}g_3\sigma^4 \nonumber \\
& & + \frac{1}{2}(\nabla\rho)^2 + \frac{1}{2}m_\rho^2\rho^2 + \frac{1}{2}(\nabla\omega)^2 + \frac{1}{2}m_\omega^2\omega^2 + \frac{1}{2}g_{\rho N}^2\rho^2 \Lambda_v g_{\omega N}^2\omega^2, \end{aligned}$$ where $g_{\omega N}$ and $g_{\rho N}$ are the nucleon coupling constants for $\omega$ and $\rho$ mesons. From the quark counting rule, we obtain $g_{\omega N}=3g_{\omega q}$ and $g_{\rho N}=g_{\rho q}$. There are six parameters ($g_{\sigma q}, g_{\omega q}, g_{\rho q}, g_2, g_3, \Lambda_v$) in the Lagrangian. Recently, several new QMF parameter sets have been fitted by reproducing the saturation density $n_0=0.16~\rm fm^{-3}$ and the corresponding values at saturation point for the binding energy $E/A=-16~\rm MeV$, the incompressibility $K=240~\rm MeV$ [@2006EPJA...30...23S; @2010JPhG...37f4038P], the symmetry energy $E_{\rm sym}=~31~\rm MeV$ [@2013PhLB..727..276L], the symmetry energy slope $L= 20-80~\rm MeV$ [@2013PhLB..727..276L; @2009PhRvL.102l2502C] and the effective mass $M_N^\ast=0.77$. We refer to [@2018ApJ...862...98Z] for the detailed values of model parameters. We report in Figure 1 and Figure 2 the resulting EOS in symmetric nuclear matter (SNM) and the symmetry energy, respectively. We see overall good agreements of them with various laboratory nuclear experiments. The slope $L$ changes evidently the density dependence of the symmetry energy, and different outcomes deviate more at higher densities. The current laboratory constraints seem favor the cases around $L = 40$ MeV.
![Pressure of neutron star matter as a function of both energy density and number density, with different values of symmetry energy slope $L$. The colour coding is the same with Fig 2. The insets show the crust-core matching details. The core EOS is from the present QMF calculation. The inner (outer) crust EOS is the usual NV (BPS) one [@bps; @nv]. We keep the energy density as increasing functions of $P$, using simple horizontal cutoff if necessary, for example in the cases of $L= 20,~25~\rm MeV$.](art/np){width="300pt"}
RESULTS AND DISCUSSIONS
=======================
For calculating the EOS of NS matter, we introduce beta-equilibrium and charge neutrality condition between nucleons and leptons. Figure 3 presents the resulting EOS of NS matter with different values of symmetry energy slope $L$. We glue different core EOSs to the same NV + BPS crust EOS [@bps; @nv], keeping the pressure $P$ as increasing functions of the energy density $\rho$. The matching procedure can be different as discussed in [@2016PhRvC..94c5804F], and the proper way to do is using Maxwell construction, guaranteeing that the pressure is an increasing function of both the density and the chemical potential. We plan to further improve this part after finishing developing the unified EOS in our model. Presently it is important to note that there may be no thermodynamic consistency, or the speed of sound may be unphysical in some $L$ cases. Since it is the $P(\rho)$ function enters the TOV equations, we make use of the present collections of the star EOSs, and move forwards to discuss the resulting mass-radius relations, as well as the tidal deformability of merging system, focusing on the effects of both the slope parameter and the behaviour of the matching interface.
![NS EOSs within QMF with different values of symmetry energy slope $L$, to be compared with the favored regions from ab-initio calculations at subsaturation density in chiral effective field theory and at very high density in perturbative QCD. The blue region at lower densities is from the calculation of pure neutron matter incorporated with beta equilibrium. The lighter blue region is the envelope of its general polytropic extensions that are causal and support a neutron star of $\sim 2 M_{\odot}$ [@2010Natur.467.1081D; @2013Sci...340..448A; @2016ApJ...832..167F]. They are both from Hebeler et al. 2013 [@2013ApJ...773...11H]. Other regions are from Annala et al. 2018 [@2018PhRvL.120q2703A], with color coding same with [@2018PhRvL.120q2703A] and will be explained later in Figure 5. ](art/eos){width="320pt"}
Before doing so, we further compare our star EOSs with first-principle calculations of nuclear EOS in chiral effective field theory and in perturbative QCD. Various preferred regions are included in our Figure 4, from Hebeler et al. 2013 [@2013ApJ...773...11H] and Annala et al. 2018 [@2018PhRvL.120q2703A]. We see immediately that the very small and very large cases of $L= 20, 25, 80 \rm MeV$ are not compatible with the lower density band based on chiral effective field theory [@2013ApJ...773...11H]. The allowed $L$ values in our QMF model may be in the range of $\sim 30-60~\rm MeV$ from this neutron matter constraint. And the cases of $L=30-60~\rm MeV$ also locate within the uncertainty bands of its general polytropic extensions over the entire density range. The slope parameter affects more evidently the low-density ranges than the high-density ranges. The four cases of $L=30-60~\rm MeV$ lie within the green $\Lambda < 400$ region. Those can be more clearly seen in Figure 5 in the plots of the NS mass-radius relations. It is expected that the TOV mass of the star hardly changes with changing $L$, but there is a strong positive correlation between the slope parameter and the radius of a $1.4\,M_\odot$ star (see more discussion in e.g., [@2018PhRvC..98f5804H; @2018PhRvL.121f2701L] and a small dependence is found, however, in [@2018PhRvC..98f5804H]). $L$ parameter does not affect much $\Lambda$, if we consider the preferred cases of $L=30-60~\rm MeV$.
![Mass-radius relation of neutron star with different values of symmetry energy slope $L$, to be compared with the results from Annala et al. 2018 [@2018PhRvL.120q2703A]. $M_{\rm max}$ stands for the maximum gravitational mass in static, or the TOV mass. $\Lambda$ is the dimensionless tidal deformability for a $1.4\,M_\odot$ star. The mass measurements of two heavy pulsars are also shown [@2013Sci...340..448A; @2010Natur.467.1081D; @2016ApJ...832..167F], as well as the specific values of $\Lambda$ for four $L$ cases. The shaded regions show the black hole limit, the Buchdahl limit and the causality limit, respectively. Adapted from [@2018ApJ...862...98Z].](art/mr){width="320pt"}
{width="21pc"}
To understand better the relation between $L$ and $\Lambda$, we present in Figure 5, for more $L$ values, the results of both $\Lambda$ and the measured mass-weighed tidal deformability ($\tilde{\Lambda}$). A chirp mass of 1.188$\,M_\odot$ and mass ratio of 0.7 [@2017PhRvL.119p1101A] are employed for the calculation of $\tilde{\Lambda}$ in a binary system. Since the mass-weighed tidal deformability is expected to be very weakly dependent on the mass ratio (see e.g., [@2018ApJ...852L..29R]), considering one mass ratio case should be representative enough for analysis. In the more reliable ranges of $L=30-60~\rm MeV$ within QMF, neither $\Lambda(1.4)$ nor $\tilde{\Lambda}$ shows good correlation with $L$. Similar results are also found by Lim & Holt 2018 [@2018PhRvL.121f2701L]. Previously, we discussed this unexpected point in [@2018ApJ...862...98Z] mainly using the relevance of a crust of a small star with a mass $1.4\,M_\odot$ and the possibility of drawing information on the (inner-)crust EOS from GW signals. However, the analysis of Gamba et al. [@2019arXiv190204616G] argued negative, where they sampled the symmetry energy (its slope) in the interval of $30-40~\rm MeV$ ($30-70~\rm MeV$), reanalyzed the data of GW170817 by adding the level of uncertainty coming from the choice of the crust structure model. In their study, it is also true that the choice of the crust EOS affects the radii of the NSs in the coalescence ($\sim 3\%$, and about 0.3 km), but the tidal parameters are not sensitive to the EOS at low crust densities. The GW measurements mainly probe the high density EOS in NSs’ cores. The low sensitivity of $L$ parameter to the tidal signal can then be understood, considering it only affects the lower density EOS as seen in Fig 4.
We have proposed a new $QMF18$ EOS in our previous study [@2018ApJ...862...98Z], corresponding to the case of $L=40~\rm MeV$. and it gives $M_{\rm TOV} =2.08~M_\odot$ and $R= 11.77~\rm km$, $\Lambda=331$ for a $1.4\,M_\odot$ star. They are also demonstrated in Fig 5 to fulfill the black hole limit, the Buchdahl limit and the causality limit. We become aware that it agrees perfectly with some latest results within other theoretical framework, see, e.g., [@2018PhRvC..97b5806M; @2018ApJ...860..139B; @2019arXiv190104371M].
SUMMARY AND PERSPECTIVE
=======================
In summary, in this contribution, we continued our recent work [@2018ApJ...862...98Z] on NS EOS from the quark level in the light of GW170817. We confront our results with ab-initio calculations and find satisfying agreements. Important constraints on the parameter space of our model can be made especially from the chiral effective field study of neutron matter, namely the slope of the symmetry energy at saturation is found to be in the range of $30-60~\rm MeV$ within QMF.
We also pay attention to the choice of crust-core matching and its possible influences on the GW tidal signals. We made plots for different $L$ values on the pressure as functions of the density. The over-simplified treatment of our work for the matching procedures, gluing different core EOSs to one crust model, prevents us discussing in a consistent way the effect of crust EOS. However, the present calculations seem to clearly demonstrate that any claims regarding constraining the symmetry energy parameters with GW tidal signals should be considered with caution, although it may be safe to translate constraints on tidal deformability to constraints on the radius of merging stars.
For future plans, along this line, we can make detailed studies for tidal deformability on the interplay of the saturation parameters with various possible strangeness phase transition [@2018ApJ...860..139B; @2018PhRvD..97h4038P; @2018Most; @2018arXiv180901116B; @2018PhRvD..98f3020Z; @2019PhRvD..99b3009C; @2019xia] at higher densities (usually above $2\rho_0$). An extend $QMF18$ EOS with unified crust and core properties will be useful as well for supernova simulation or pulsar studies. The pulsar properties can be predicted [@2016ApJS..223...16L] and updated studies can be done for short gamma-ray bursts [@2016PhRvD..94h3010L; @2017ApJ...844...41L].
Besides the proposed NS model of GW170817 in this contribution, the possibility of strange star merger for GW170817 has been tested and brings many new perspectives not only for this single event, but for the fundamental strong interaction, see, e.g., [@2018PhRvD..97h3015Z; @2018RAA....18...24L; @2018ApJ...852L..32D; @2015ApJ...804...21G; @2017IJMPS..4560042P]. Also, supramassive/hypermassive magnetars as the remnants of binary mergers if confirmed might put severe challenges to NS model since it requires the TOV mass of the underlying EOS should be no less than $\sim 2.3 M_{\odot}$, see discussions for example in [@2017ApJ...850L..19M; @2018ApJ...852L..25R; @2019MNRAS.483.1912P; @2019ApJ...872..114S; @2019arXiv190301466K], although more decisive analysis from the electromagnetic counterparts are still necessary. Realistic nuclear EOSs seem hard to be beyond this value, see discussions in e.g., [@esym; @eos; @2018RPPh...81e6902B]. Our QMF model only gives a TOV mass around $2.1~M_{\odot}$, slightly lower than the APR one which is around $2.2~M_{\odot}$ [@apr]. It might be high time to resolve these tensions between microscopic many-body calculations of nuclear matter (with or without QCD transition [@2015PhRvD..91d5003K; @2018PhRvD..97b3018B]) and astrophysical NS merger observations.
ACKNOWLEDGMENTS
===============
We thank Constança Provid[ê]{}ncia and James Lattimer for enlightening discussions during the Xiamen-CUSTIPEN Workshop on the EOS of Dense Neutron-Rich Matter in the Era of Gravitational Wave Astronomy, Jan. 3-7, Xiamen, China. This work was supported in part by the National Natural Science Foundation of China (Grant No. 11873040).
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|
---
abstract: 'The effect of rotation on the boundary layers (BLs) in a Rayleigh-Bénard (RB) system at a relatively low Rayleigh number, i.e. $Ra = 4\times10^7$, is studied for different $Pr$ by direct numerical simulations and the results are compared with laminar BL theory. In this regime we find a smooth onset of the heat transfer enhancement as function of increasing rotation rate. We study this regime in detail and introduce a model based on the Grossmann-Lohse theory to describe the heat transfer enhancement as function of the rotation rate for this relatively low Ra number regime and weak background rotation $Ro\gtrsim 1$. The smooth onset of heat transfer enhancement observed here is in contrast to the sharp onset observed at larger $Ra \gtrsim 10^8$ by Stevens [*[et al.]{}*]{} \[Phys. Rev. Lett. [**[103]{}**]{}, 024503, 2009\], although only a small shift in the $Ra-Ro-Pr$ phase space is involved.'
author:
- 'Richard J.A.M. Stevens$^1$'
- 'Herman J.H. Clercx$^{2,3}$'
- Detlef Lohse$^1$
title: 'Boundary layers in rotating weakly turbulent Rayleigh-Bénard convection.'
---
Introduction
============
Normally the transition between different turbulent states is smooth, because the large random fluctuations that characterize the turbulent flow make sure that the entire phase space is explored and therefore the transitions between different states, that are explored as a control parameter is changed, are washed out. A classical system to study turbulence is Rayleigh Bénard (RB) convection [@ahl09; @ahl09b; @loh10]. For given aspect ratio $\Gamma\equiv D/L$ ($D$ is the cell diameter and $L$ its height) and given geometry, its dynamics are determined by the Rayleigh number $Ra=\beta g\Delta L^3 /(\kappa \nu)$ and the Prandtl number $Pr=\nu/\kappa$. Here $\beta$ is the thermal expansion coefficient, $g$ the gravitational acceleration, $\Delta$ the temperature difference between the plates, and $\nu$ and $\kappa$ are the kinematic and thermal diffusivity, respectively. The heat transfer in a RB system can satisfactory be described by the Grossmann-Lohse (GL) theory [@ahl09; @gro00; @gro01; @gro02; @gro04] and shows that RB convection has different turbulent regimes in the $Ra-Pr$ phase space (see Fig. 3 of Ref. [@ahl09]). The case where the RB system is rotated around a vertical axis, i.e. Rotating Rayleigh-Bénard (RRB) convection, at an angular speed $\Omega$ is interesting for industrial applications and problems in geology, oceanography, climatology, and astronomy. The rotation rate of the system is non-dimensionalized in the form of the Rossby number $Ro=\sqrt{\beta g \Delta/L}/(2\Omega)$. The dynamics of RRB convection are thus determined by three control parameters, i.e. $Ra$, $Pr$, and $Ro$, and this leads to a huge $Ra-Pr-Ro$ phase space, see Fig. \[fig:Phase space\].
It is widely understood [@cha81] that rotation suppresses convective flow, and with it convective heat transport, when the rate of rotation is sufficiently large. However, experimental [@liu97; @liu09; @ros69; @pfo84; @bou86; @bou90; @zho93; @kin09; @zho09b; @ste09] and numerical [@ore07; @jul96; @spr06; @kun08b; @kin09; @sch09; @zho09b; @ste09; @ste09d] studies on RRB convection have shown that rotation can also enhance the heat transport with respect to the non-rotating case. This heat transport enhancement is caused by Ekman pumping [@ros69; @zho93; @jul96; @vor98; @vor02; @kun08b; @kin09; @zho09b; @ste09; @ste09d] and its efficiency depends strongly on the combination of $Ra$, $Pr$, and $Ro$ [@zho09b; @ste09; @ste09d]. In this paper we will discuss the results of Direct Numerical Simulations (DNS) that show that this heat transfer enhancement as function of the Ro number is smooth for relatively low Ra number, here $Ra=4\times 10^7$ (see Fig. \[Fig\_SCL09\_highRa\_a\]a), while experimental and numerical data for $Ra=2.73 \times 10^8$ and $Pr=6.26$ show a sharp onset for the heat transport enhancement, see Fig. \[Fig\_SCL09\_highRa\_a\]b. This difference is remarkable since only a small shift in the $Ra-Pr-Ro$ phase space is involved (see Fig. \[fig:Phase space\]).
In this paper we will first describe the flow characteristics found in the simulations. We will show that there is a smooth transition from one turbulent regime to another for the relatively low Ra number regime whereas a sharp transition is found for higher $Ra$. In section \[Sec2\] we will discuss the properties of the BLs found in the DNS in detail. Subsequently the laminar BL theory for flow over an infinitely large rotating disk will be discussed in section \[Sec3\] in order to explain the BL properties found in the DNS. The derived scaling laws from this theory will be used in a model based on the GL theory to describe the heat transfer enhancement as function of $Ro$ for the relatively low $Ra$ number regime with weak background rotation, see section \[Sec4\].
\[t\]
Numerical results for boundary layers in RRB convection {#Sec2}
=======================================================
The flow characteristics of RRB convection for $Ra=4\times 10^7$, $1<Ro<\infty$, and $0.2<Pr<20$, are obtained from solving the three-dimensional Navier-Stokes equations within the Boussinesq approximation:
$$\begin{aligned}
\frac{D\textbf{u}}{Dt} &=& - \nabla P + \left( \frac{Pr}{Ra} \right)^{1/2} \nabla^2 \textbf{u} + \theta \textbf{$\widehat{z}$}- \frac{1}{Ro} \widehat{z} \times \textbf{u}, \\
\frac{D\theta}{Dt} &=& \frac{1}{(PrRa)^{1/2}}\nabla^2 \theta ,\end{aligned}$$
with $\nabla \cdot \textbf{u} = 0$. Here **$\widehat{z}$** is the unit vector pointing in the opposite direction to gravity, $D/Dt = \partial_t + \textbf{u} \cdot \nabla $ the material derivative, $\textbf{u}$ the velocity vector (with no-slip boundary conditions at all walls), and $\theta$ the non-dimensional temperature, $0\leq \theta \leq 1$. Finally, $P$ is the reduced pressure (separated from its hydrostatic contribution, but containing the centripetal contributions): $P=p - r^2/(8Ro^2)$, with $r$ the distance to the rotation axis. The equations have been made non-dimensional by using, next to $L$ and $\Delta$, the free-fall velocity $U=\sqrt{\beta g \Delta L}$. A constant temperature boundary condition is used at the bottom and top plate and the side wall is adiabatic. Further details about the numerical procedure can be found in Refs. [@ver96; @ver99; @ver03].
The first set of simulations is used to study the $Ro$ number dependence of the following quantities: the normalized heat transfer, the thickness of the thermal BL, and the normalized averaged root mean square (rms) vertical velocity fluctuations. Here we have simulated RRB convection at several $Ro$ numbers for three different $Pr$ numbers ($Pr=0.7$, $Pr=6.4$, and $Pr=20$). All these simulations are performed on a grid with $65 \times 193 \times 129$ nodes, respectively, in the radial, azimuthal, and vertical directions, allowing for sufficient resolution in the bulk and the BL according the resolution criteria set in Ref. [@ste09b]. The Nusselt number is calculated in several ways as is discussed in detail in Ref. [@ste09b] and its statistical convergence has been controlled. Some data for $Pr=6.4$ have already been published in Ref. [@ste09]. There the average was over $4000$ dimensionless time units. The new results for $Pr=0.7$ and $Pr=20$ are averaged over $2500$ dimensionless time units. Note that we simulated the flow for a large number of eddy turnover times to reduce the statistical error in the obtained Nusselt number results and to prevent the influence of transient effects. This is necessary to accurately resolve the transition regime where the heat transfer starts to increase and to accurately determine the flow statistics. Furthermore, we note that all simulations are started from a new flow field in order to rule out hysteresis effects.
The second set of simulations is used to study the $Pr$ number dependence of the same set of quantities. Here we simulated RRB for several $Pr$ numbers and three different $Ro$ numbers ($Ro=\infty$, $Ro=3$, and $Ro=1$). The simulations for $Pr\geqq 0.70$ are performed on a $97 \times 257\times193$ and the simulations at $Pr=0.25$ and $Pr=0.45$ are performed on a $129 \times 385 \times257$ grid. The finer resolution is needed here as the structure of the flow changes and the Reynolds number based on the LSC increases for lower $Pr$. For most cases the flow is simulated for $400$ dimensionless time units and $200$ dimensionless time units were simulated before data are collected to prevent any influence of transient effects [@ste09b]. For $Pr=2$ and $Pr=4.4$ we averaged over $1200$ dimensionless time units in order to obtain more accurate statistics on the velocity field. Note that the second set of simulations is partially overlapping with the first set of simulations. We find that results obtained on the different grids, are very similar, i.e. the difference is generally between $0.5\%$ and $1\%$, see Figs. \[Fig\_SCL09\_highRa\_a\]a, \[Fig\_SCL09\_model\]a and \[Fig\_SCL09\_model\_Pr\]a. Again all simulations are started from a new flow field in order to rule out any hysteresis effects.
Fig. \[Fig\_SCL09\_profile\_temp\] shows the azimuthally averaged temperature profile at the cylinder axis for different system parameters. In previous numerical studies concerning (rotating) RB convection the thermal BL thickness is usually defined by either looking at the maximum rms value of the temperature fluctuations or by considering the BL thickness based on the slope of the mean temperature profile. In the latter case it is usually assumed that no mean temperature gradient exists in the bulk (the BL thickness according to this assumption is denoted by $\lambda^{sl-ng}_{\theta}$). The temperature gradient in the bulk is, however, strongly influenced by rotation [@zho09b; @kun09], and when rotation is present also by $Pr$, see Fig. \[Fig\_SCL09\_gradients\]. We prefer to define the thermal BL thickness $\lambda_{\theta}^{sl}$ as the intersection point between the linear extrapolation of the temperature gradient at the plate with the behaviour found in the bulk, see Fig. \[Fig\_SCL09\_profile\_temp\]a. From now on this definition of the thermal BL thickness will be used here.
For the relatively low $Ra$ number regime, here $Ra=4\times10^7$, the heat transfer enhancement as function of $Ro$ is smooth, see Fig. \[Fig\_SCL09\_model\]a. Note that although the behaviour of $\lambda_{\theta}^{sl}$ (see fig. \[Fig\_SCL09\_model\]b), i.e. the horizontally averaged value of the radially dependent thermal BL thickness ($\lambda_\theta^{sl}(r)$), as function of $Ro$ is similar for all $Pr$ the behaviour of $Nu$ is very different. This is due to the influence of $Pr$ on the effect of Ekman pumping [@ste09d; @zho09b]. At low $Pr$ the larger thermal diffusivity limits the effect of Ekman pumping and causes a larger destabilizing temperature gradient in the bulk [@zho09b; @ste09d], see Fig. \[Fig\_SCL09\_gradients\]. Due to the limited effect of Ekman pumping there is no heat transport enhancement for low $Pr$, see Fig. \[Fig\_SCL09\_model\].
Fig. \[Fig\_SCL09\_flow\] shows that the volume averaged $Re_{z,rms}$ (dimensionless rms velocity of the axial velocity fluctuations), which is a measure for the strength of the LSC, decreases strongly for strong enough rotation. The vertical dashed lines in Fig. \[Fig\_SCL09\_flow\] indicate the position where $Re_{z,rms}(\Omega)$/$Re_{z,rms}(0)$ becomes smaller than $1$, which we use to indicate the point at which the LSC strength starts to decrease. This value is determined by extrapolating the behaviour observed at low $Ro$ numbers to reduce the effect of the uncertainty in single data points. In Ref. [@ste09] (see figure 3 of that paper) we also used this method to indicate the position of the onset of the heat transfer enhancement in the high $Ra$ number regime. However, for this lower $Ra$ number we do not find any evidence for a sudden onset around this point, see Fig. \[Fig\_SCL09\_model\] where the vertical lines are plotted at the same positions as in Fig. \[Fig\_SCL09\_flow\]. In contrast to the decrease in the volume averaged value of $Re_{z,rms}$ the horizontally averaged value of $Re_{z,rms}$ at the edge of the thermal BL (thus at the distance $\lambda^{sl}_{\theta}(r)$ from either the top or bottom boundary) increases. This indicates that Ekman pumping, which is responsible for the increase in $Nu$, sets in and no sign of Ekman pumping prior to the decrease in LSC strength is found. Fig. \[Fig\_SCL09\_flow\] thus shows that the flow makes a transition between two different turbulent states, i.e. a transition from a LSC dominated regime to an Ekman pumping dominated regime [@ste09]. Fig. \[Fig\_SCL09\_flow\]a shows no increase in the horizontally averaged value of $Re_{z,rms}$ at the edge of the thermal BL for $Pr=0.7$, because the flow is suppressed for higher $Ro$, i.e. lower rotation rate, when $Pr$ is lower, see the discussion in Ref. [@ste09d].
The $Pr$ number dependence of the $Nu$ number and the thickness of the thermal BL is shown in Fig. \[Fig\_SCL09\_model\_Pr\]. From Fig. \[Fig\_SCL09\_model\_Pr\]a we can conclude that hardly any $Pr$ number dependence on the $Nu$ number exists in the weak rotating regime ($Ro=3$). However, a strong $Pr$ number effect appears for stronger rotation rates, where Ekman pumping is the dominant effect [@zho09b]. Fig. \[Fig\_SCL09\_model\_Pr\]b shows that the effect of weak background rotation on the thermal BL thickness is largest for $Pr \approx 2$. The $Pr$ number dependence of $Re_{z,rms}$ is shown in Fig. \[Fig\_SCL09\_flow\_Pr\]. The difference between the data points obtained for the bottom and top BL in Fig. \[Fig\_SCL09\_flow\_Pr\] indicate the uncertainty in the results. Increasing the averaging time, which we checked for $Pr=2$ and $Pr=4.4$, reduces the differences for the data points obtained for the bottom and top BL. Furthermore, we note that it is important to take the radial thermal BL dependence ($\lambda_\theta^{sl}(r)$) into account for lower $Pr$, where the radial BL dependence is strongest, and we excluded the region close to the sidewall ($0.45<r<0.5$) from the horizontal averaging in order to eliminate the effect of the sidewall.
The computation of the kinetic BL thickness can be either based on the position of the maximum rms value of the azimuthal velocity fluctuations [@ste09; @kun10], or on the position of the maximum value of $\epsilon_u^":= \bf{u} \cdot \nabla^2 \bf{u}$, i.e. two times the height at which this quantity is highest, as shown in [@ste09b]. In ref. [@lak10] we will compare in detail the profiles of $\epsilon_u$ and $\epsilon_u^"$ and will show that the latter is indeed suited to define the BL thickness. Here we first average $\epsilon_u^"$ horizontally in the range $0.05\le r \le 0.45$ before we determine the position of the maximum. This $r$ range has been taken to exclude the region close to the sidewall, where $\epsilon_u^"$ misrepresents the kinetic BL thickness due to the rising plumes, and the region close to the cylinder axis, since there it is numerically very difficult to reliably calculate $\epsilon_u^"$, due to the singularity in the coordinate system. When the radially dependent kinetic BL thickness ($\lambda_u^{\epsilon_u^"}(r)$) is horizontally averaged a small difference, depending on the averaging time, is observed between the bottom and top because of the specific orientation of the LSC. We note that the same quantity $\epsilon_u^"$ is used in Ref. [@ste09b], where it is shown that the kinetic BL thickness based on $\epsilon_u^"$ represents the BL thickness better than the one considering the maximum rms velocity fluctuations, which is normally used in the literature. The volume averaged value of $\epsilon_u^"$ is the same as the volume averaged kinetic energy dissipation rate $\epsilon_u$ (although it differs locally), which can easily be derived using Gauss’s theorem [@lak10]. Fig. \[Fig\_SCL09\_profile\_epsilonu\] shows the azimuthally averaged profiles for $\epsilon_u^"$ at $r=0.25L$ and in Fig. \[Fig\_SCL09\_kineticBL\] the kinetic BL thickness based on the position of the maximum kinetic dissipation rate is shown as function of $Ro$ and $Pr$. To compare the relative changes in the kinetic BL thicknesses the values are normalized by values for the non-rotating case. For all $Pr$ numbers there is a change in the BL behaviour at the point where the LSC decreases in strength (vertical dashed lines in Fig. \[Fig\_SCL09\_kineticBL\]a).
We conclude this section with a brief summary of the results obtained for the high $Ra$ number regime [@zho09b; @ste09]. For $Ra\gtrsim 1\times10^8$ a sudden onset at $Ro=Ro_c$ in the heat transport enhancement occurs, see Fig. \[Fig\_SCL09\_highRa\_a\]b, where we find a smooth transition at lower $Ra$, see Fig. \[Fig\_SCL09\_highRa\_a\]a. Fig. \[Fig\_SCL09\_highRa2\]a shows the volume averaged ratio $Re_{z,rms}(\Omega)$/$Re_{z,rms}(0)$. The behaviour is similar to the one observed at lower $Ra$, see Fig. \[Fig\_SCL09\_flow\]. For the high $Ra$ number regime the onset at $Ro_c$ is defined as the point where the ratio $Re_{z,rms}(\Omega)$/$Re_{z,rms}(0)$ becomes smaller than $1$. For the low $Ra$ number this point indicates the position where the LSC strength starts to decrease and just as for the relatively low $Ra$ number regime, Ekman pumping in the high $Ra$ number regime is indicated by an increase of the horizontally averaged $Re_{z,rms}$ value at the edge of the thermal BL. Although the two cases, i.e. the relatively low $Ra$ number regime and the high $Ra$ number regime both, show a transition between two different turbulent states the important difference between the two is that the transition is sharp in the high $Ra$ number regime and smooth in the relatively low $Ra$ number regime. The onset in the high $Ra$ number regime is also observed in the behaviour of the BLs. Fig. \[Fig\_SCL09\_highRa2\]b shows that the kinetic BL thickness does not change below onset ($Ro> Ro_c$) and above onset the BL behaviour is dominated by rotational effects and thus Ekman scaling (proportional to $Ro^{1/2}$) is observed. This scaling factor is also found in the laminar BL theory, which will be discussed in the next section. Finally, fig. \[Fig\_SCL09\_highRa2\]c shows the thermal BL thickness $\lambda_\theta^{sl}$.
Boundary layer theory for weak background rotation {#Sec3}
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In the previous section we observed that there is a smooth increase in the heat transfer as function of the $Ro$ number when the $Ra$ number is relatively low. In this section we set out to account for the increase in the heat transfer as function of the rotation rate within a model, which extends the ideas of the GL-theory to the rotating case. In the GL theory the Prandtl-Blasius BL theory for laminar flow over an infinitely large plate was employed in order to estimate the thicknesses of the kinetic and thermal BLs, and the kinetic and thermal dissipation rates. These results were then connected with the $Ra$ and $Pr$ number dependence of the Nusselt number. In perfect analogy, in the present paper we apply laminar BL theory for the flow over an infinitely large rotating plate to study the effect of rotation on the scaling laws. We stress that employing the results of laminar BL theory over an infinite rotating disk to the rotating RB case in a closed cylinder is fully analogous to employing Prandtl-Blasius BL theory for flow over an infinite plate to the standard RB case without rotation, where the method was very successful [@gro00; @gro01; @gro02; @gro04; @ahl09].
In both cases the equations used to derive the scaling laws are time independent and therefore the resulting solutions are associated with laminar flow. However, evidently, high Rayleigh number thermal convection is time dependent. Therefore one wonders whether the derived scaling laws still hold for time-dependent flow over an infinite rotating disk. We will show that the $Ro$ and $Pr$ scaling that is derived is not changed when temporal changes are included. This is again in perfect analogy to the Prandtl-Blasius BL case where the scaling laws also hold for time dependent flow provided that the viscous BL does not break down [@gro04]. Indeed, recent experiments and numerical simulations [@sun08; @zho09c; @zho10] have shown that in non-rotating RB the BLs scaling wise behave as in laminar flow and therefore we feel confident to assume the same for the weakly rotating case. The basic idea of the model we introduce is to combine the effect of the LSC roll, which is implemented in the GL theory by the use of laminar Prandtl-Blasius BL theory over an infinitely large plate, and the influence of the rotation on the thermal BL.
The system we are analyzing to study the influence of rotation on the thermal BL thickness above a heated plate is schematically shown in Fig. \[Figure\_ste08\_setup\]a. It is the laminar flow of fluid over an infinite rotating disk. The disk rotates with an angular velocity $\Omega_D$ and the fluid at infinity with angular velocity $\Omega_F=s\Omega_D$, with $s<1$. Fig. \[Figure\_ste08\_setup\]b shows that a positive radial velocity is created due to the action of the centrifugal force. Because of continuity there is a negative axial velocity, i.e. fluid is flowing towards the disk.
The system has been analyzed before in the literature, e.g. Refs. [@loi67; @rog60; @sch79; @wij85; @zan77]. Here we will briefly summarize the procedure.
The system is analyzed by using the Navier-Stokes equations in cylindrical coordinates and assuming a steady stationary, axial symmetric solution. To reduce the Navier-Stokes equations to a set of ordinary differential equations (ODE) we employ self-similarity. The first step is to determine the dimensionless height in the system just as in the Prandtl-Blasius approach, in which the BL thickness scales as [@ll87] $\delta \sim \sqrt{\nu x/U}$. For the case of a large rotating disk in a fluid rotating around an axis perpendicular to the disk, the thickness of the BL can be estimated by replacing $U$ by $\Omega_D x$ [@ll87]. The thickness then scales as $\delta \sim \sqrt{\nu/\Omega_D}$. The similarity variable in the system is the dimensionless height
$$\label{Eq dimensionless height}
\zeta=z \sqrt{\frac{\Omega_D}{\nu}}.$$
According to von Kármán, the following self-similarity ansatz for the velocity components and the pressure can be taken [@loi67; @rog60; @sch79; @wij85; @zan77] $$\begin{aligned}
\label{Eq assumption radial velocity}
u &=& r \Omega_D F(\zeta),\\
\label{Eq assumption tangential velocity}
v &=& r \Omega_D G(\zeta),\\
\label{Eq assumption axial velocity}
w &=& \sqrt{\nu \Omega_D} H(\zeta),\\
\label{Eq assumption pressure}
p &=& \rho \nu \Omega_D P(\zeta) + \frac{1}{2} \rho s^2 \Omega_D^2 r^2.\end{aligned}$$ After substitution into the Navier-Stokes equations one obtains a system of four coupled ODEs, $$\begin{aligned}
\label{Eq radial momentum}
&&F^2 + F^{'}H - G^{2} -F^{''} + s^2 = 0, \\
\label{Eq tangential momentum}
&&2FG + HG^{'} - G^{''} = 0, \\
\label{Eq pressure}
&&P^{'}+HH^{'}-H^{''} = 0,\\
\label{Eq continuity}
&&2F + H^{'} = 0,\end{aligned}$$ where the prime indicates differentiation with respect to $\zeta$. This set of ODEs must be supplemented by the boundary conditions $$\begin{aligned}
\label{Eq boundary condition dimensional z0}
u = 0,\hspace{3mm} v= r \Omega_D &,\hspace{3mm} w=0 \hspace{3mm} &\mbox{ for } z=0,\\
\label{Eq boundary condition dimensional zinfty}
u = 0,\hspace{3mm} v=s r \Omega_D &\hspace{3mm} &\mbox{ for } z=\infty.\end{aligned}$$ When substituting the self similarity ansatz (\[Eq assumption radial velocity\]) - (\[Eq assumption pressure\]) into these boundary conditions one obtains $$\begin{aligned}
\label{Eq boundary condition nondimensional z0}
F = 0,\hspace{3mm} G=1 &,\hspace{3mm} H=0 \hspace{3mm} & \mbox{ for } \zeta=0,\\
\label{Eq boundary condition nondimensional zinfty}
F = 0,\hspace{3mm} G=s &\hspace{3mm} & \mbox{ for } \zeta=\infty.\end{aligned}$$ Note that the boundary condition at infinity together with the continuity equation (\[Eq continuity\]) gives $H^{'}(\zeta\rightarrow\infty)=0$. One can further simplify the set of ODEs by realizing that the ODE for the pressure (\[Eq pressure\]) is decoupled from the ODEs determining the velocity profiles by using the continuity equation (\[Eq continuity\]) in (\[Eq pressure\]) and subsequently integrating this relation. The velocity profiles, for the von Kármán case ($s=0$), are shown in Fig. \[Figure\_ste08\_setup\]b. The inset shows that the dimensionless kinetic BL thickness $\overline{\lambda}_u \equiv {\lambda_u}/\delta$ decreases with increasing relative rotation rate $s$ of the fluid at infinity. This is due to the decreasing effect of the centrifugal force. We determined $\overline{\lambda}_u^{99\%}$, the dimensionless kinetic BL thickness at which the velocity has achieved $99\%$ of the outer flow velocity, using the tangential velocity profile, i.e. when $G(\zeta)=s+0.01(1-s)$. Additionally, we calculated $\overline{\lambda}_u^{sl}$, the dimensionless kinetic BL thickness based on the slope of the tangential velocity at the disk. The scaling of the kinetic BL predicted by the above rotating BL theory, i.e. $Ro^{1/2}$, is the classical Ekman scaling. In the simulations of RRB we find the same scaling of the kinetic BL once the flow is dominated by rotational effect, i.e. $Ro \lesssim Ro_c$, see Fig. \[Fig\_SCL09\_highRa2\]b.
The GL theory heavily builds on laminar Prandtl-Blasius BL theory, which describes the laminar flow over an infinite plate. In the Prandtl-Blasius theory the temperature field is assumed to be passive to derive the $Pr$ number scaling. As we want to extend the GL theory to the rotating case we keep this analysis analogous to the Prandtl-Blasius theory. Therefore, we assume the temperature field to be passive in order to derive the scaling laws as function of the $Pr$ number. We non-dimensionalize the temperature by
$$\label{Eq definition dimensionless temperature}
\widetilde{\theta}(\zeta)= \frac{\theta-\theta_\infty}{\theta_{b}-\theta_{\infty}},$$
where $\theta_b$ is the temperature of the bottom disk, and $\theta_\infty<\theta_b$ is the ambient temperature. Then one obtains the following ODE describing the temperature field [@spa59; @vir80; @mil52] $$\label{Eq ODE energy equation}
\widetilde{\theta}^{''} = Pr H(\zeta) \widetilde{\theta}',$$ where the prime indicates a differentiation with respect to $\zeta$. The boundary conditions are $$\begin{aligned}
\label{Eq boundary condition thermal z0}
\widetilde{\theta} = 1 & \mbox{ for } & \zeta = 0,\\
\label{Eq boundary condition thermal zinfty}
\widetilde{\theta} = 0 & \mbox{ for } & \zeta = \infty.\end{aligned}$$ The resulting system of ODEs subjected to the boundary conditions, is solved numerically with a fourth order Runge-Kutta method using a Newton-Raphson root finding method to find the initial conditions. One can take the analytic solution for the Ekman case ($s\approx 1$), see appendix \[appA\], or the known solution for the Bödewadt case, see for example [@rog60; @sch79], as one of the starting cases to determine the solutions over the whole parameter range in $s$ and $Pr$.
In this way we obtain the full temperature profile for all $s$ and $Pr$. For the heat transfer the most relevant quantity is the thermal BL thickness $\lambda_\theta$, which we non-dimensionalized by $\delta$, thus $\overline{\lambda}_\theta \equiv \lambda_\theta/\delta$. One can distinguish between $\overline{\lambda}_{\theta}^{sl}$ and $\overline{\lambda}_{\theta}^{99\%}$, the dimensionless BL thickness based on the $99\%$ criterion, thus when $\widetilde{\theta}(\zeta)=0.01$. Fig. \[Figure\_ste08\_thermalBL\]a shows that the asymptotic scaling of $\overline{\lambda}_{\theta}^{99\%}$ and $\overline{\lambda}_{\theta}^{sl}$ is the same. Furthermore, the figure shows that rotation does not influence the scaling of the thermal BL thickness in the high $Pr$ regime, because the same scaling, namely proportional to $Pr^{-1/3}$, is found as for the Prandtl-Blasius case. However, the rotation does influence the scaling in the low $Pr$ regime, where now $\overline{\lambda}_{\theta} \propto Pr^{-1}$ instead of $\overline{\lambda}_{\theta} \propto Pr^{-1/2}$ as found in the Prandtl-Blasius case. Notice that $\overline{\lambda}_{\theta}^{99\%} > \overline{\lambda}_{\theta}^{sl}$, which is due to the decreasing temperature gradient with increasing height.
In Fig. \[Figure\_ste08\_thermalBL\]b we show the effective power-law exponent $\gamma= (d \log \overline{\lambda}_\theta)/(d \log Pr)$ of an assumed effective power law $\overline{\lambda}_\theta \sim Pr^{\gamma}$. It confirms that the effective scaling in the high $Pr$ regime is the same for the Prandtl-Blasius (no rotation) and the von Kármán case ($s=0$), but already at $Pr=1$ a significant difference is observed.
The temperature advection equation (\[Eq ODE energy equation\]) directly suggests the following relation between the scaling of the thermal BL thickness $\overline{\lambda}_{\theta}^{sl}$ and the scaling of the axial velocity at the edge of the thermal BL, $H_{BL}\sim 1/Pr \overline{\lambda}_{\theta}^{sl}$. This immediately implies for the low $Pr$ regime, with $\overline{\lambda}_{\theta}^{sl} \sim Pr^{-1}$, that $H_{BL}$ is independent of $Pr$. For the high $Pr$ regime, with $\overline{\lambda}_{\theta}^{sl} \sim Pr^{-1/3}$, it gives $H_{BL}(Pr) \sim Pr^{-2/3}$. The scaling of the thermal BL thickness in the low $Pr$ regime can be understood on physical grounds. In this regime $\overline{\lambda}_\theta \gg \overline{\lambda}_u$ and, the kinetic BL is fully submerged in the thermal BL. The axial velocity at the edge of the kinetic BL (\[Eq boundary condition nondimensional zinfty\]) is $H_{BL}=H(\zeta \rightarrow \infty$), as can be shown by applying mass conservation expressed by Eq. (\[Eq continuity\]). As a consequence, in the low $Pr$ regime the axial velocity is constant in almost the whole thermal BL. Then Eq. (\[Eq ODE energy equation\]) can trivially be integrated and immediately gives $\overline{\lambda}_{\theta}^{sl} \sim Pr^{-1}$. This derivation is valid for all $s$, i.e. the scaling in the low $Pr$ regime does not depend on the rotation of the fluid at infinity. The scaling in the high $Pr$ regime is also independent of the rotation of the fluid at infinity. In the Ekman case $s \approx 1$, see appendix \[appA\], this $1/3$ scaling regime shifts towards very large $Pr$.
The equations (\[Eq radial momentum\])-(\[Eq continuity\]) are time independent and therefore the resulting solutions are understood to describe laminar flow. Temporal changes can easily be included by adding $\partial_{\tilde{t}} \tilde{u}_{\theta}$, $\partial_{\tilde{t}} \tilde{u}_{r}$, and $\partial_{\tilde{t}} \tilde{u}_{z}$ where $\tilde{t}=t\Omega_D$ without changing the $Ro$ and $Pr$ scaling discussed above, i.e. the derived scaling laws above still hold for time dependent flow provided that the viscous BL does not break down. This is in perfect analogy to the Prandtl-Blasius BL case where the scaling laws also hold for time dependent flow [@gro04].
To investigate the crossover between the high and the low $Pr$ regime we define $Pr_{cross}$ as the crossover point. $Pr_{cross}$ is calculated by determining the intersection between the asymptotic behaviour of the high and the low $Pr$ regime. To calculate $Pr_{cross}$ we considered $\overline{\lambda}_{\theta}^{sl}(Pr)$ and $\overline{\lambda}_{\theta}^{99\%}(Pr)$. The inset of Fig. \[Figure\_ste08\_thermalBL\]a shows that the low $Pr$ regime becomes more favored when the rotation of the fluid at infinity becomes stronger. Then the kinetic BL thickness becomes thinner and the axial velocity decreases, i.e. the thermal BL thickness increases. Because the thermal BL thickness decreases with increasing $Pr$ the crossover shifts towards higher $Pr$ as is shown in the inset of Fig. \[Figure\_ste08\_thermalBL\]a. This effect is visible in RRB as shown in Fig. 5b of Ref. [@ste09d]. Here it is shown that for the non-rotating case $\lambda_u < \lambda_\theta^{sl}$ when $Pr\lesssim 1$. When the rotation rate is increased, i.e. $Ro$ is lowered, this transition shifts towards higher $Pr$ and for $Ro=0.1$ $\lambda_u < \lambda_\theta^{sl}$ when $Pr\lesssim 9$.
In summary, the laminar rotating BL theory explains the $Ro^{1/2}$ scaling of the kinetic BL thickness in RRB convection and the shift of the position where $\lambda_u=\lambda_\theta$ towards higher $Pr$ when the flow is dominated by rotational effects.
Model for smooth onset in RRB convection {#Sec4}
========================================
In this section we will introduce a model in the spirit of the GL approach in order to describe the smooth increase in the heat transfer as a function of $Ro$ that is observed for relatively low $Ra$ and weak background rotation. Since the GL theory assumes smooth transitions between different turbulent states the model is limited to the relatively low $Ra$ number regime, since a sharp onset as function of $Ro$ is found for $Ra\gtrsim 1\times10^8$. Furthermore, we assume that then the LSC, a basic ingredient of the GL model, is still present. In this simple model we neglect the influence of Ekman pumping, because it is a local effect that is rather insignificant at weak background rotation. This is supported by the results in Fig. \[Fig\_SCL09\_flow\] and the EPAPS document of Ref. [@ste09], where we find no evidence for Ekman pumping at weak background rotation. When strong rotation is applied Ekman pumping is the dominant effect and the validity regime of our model is thus restricted to weak background rotation. The basic idea of the model is to combine the effect of the LSC roll, which is implemented in the GL theory by the use of the laminar Prandtl-Blasius theory over an infinitely large plate, and the influence of rotation on the thermal BL.
Applying laminar BL theory requires that the viscous BL above the flat rotating plate does not brake down. This assumption will now be verified. The stability for the von Kármán flow has been studied theoretically and experimentally by Lingwood [@lin95; @lin96; @lin97], showing that instability occurs at $Re \approx 510$ where $Re$ is defined as $Re=r\sqrt{\Omega/\nu}$ (with $r$ the distance to the rotation axis). Lingwood also pointed out that other experimental studies show the same transition point within a very narrow Reynolds number range $Re = 513 \pm 15$, see [@lin95] and references therein. More recent experiments show similar results [@col99; @zou03]. For the Ekman case the instability occurs at $Re\approx200$, see [@lin97]. For the case under consideration it can be shown that $Re\lesssim 55$ (with $r=12.5~{\rm{cm}}$, $\Omega\approx 0.20~{\rm{rad/s}}$, and $\nu=1\times 10^6~{\rm{m}}^2/{\rm{s}}$) [@zho09b]. It can safely be conjectured that laminar BL theory can be applied as the estimated Reynolds number is an order of magnitude smaller than the critical Reynolds number. For further discussion on the stability of the rotational flow we refer to the classic Refs. [@cha81; @bus70].
We introduce $\lambda_{\theta R}=\overline{\lambda}_\theta^{sl}(Pr)\sqrt{\nu/\Omega}$, see $\overline{\lambda}_\theta^{sl}(Pr)$ in Fig. \[Figure\_ste08\_thermalBL\]a, as the thermal BL thickness based on the background rotation and $\lambda_{\theta C}$ as the BL thickness based on the LSC roll. Furthermore, $\Gamma$ is the diameter-to-height aspect ratio of the RB cell and we set the radial length $r=(\Gamma L)/2$. Note that this is analogous to the length $L$ which is introduced in the GL theory for the length of the plate. Thus the Reynolds number based on the background rotation is
$$Re_R=\frac{\Omega L^2 \Gamma^2}{4 \nu} \propto \frac{1}{Ro}~.$$
To calculate $\lambda_{\theta R}$ we used $\nu = 1 \times 10^{-6} m^2/s$ (water) and $\Gamma=1$ and we set $\lambda_{\theta R}=\lambda_{\theta C}$ at $Ro =\infty$. The strength of the LSC roll is taken constant and $\lambda_{\theta C}$ is known from $\lambda_{\theta C}/L$ = $(2 Nu)^{-1}$.
We model the increase of $Nu$ as a crossover between a convection role dominated BL and a rotation dominated BL. Thus without rotation the BL thickness is determined by the LSC roll, i.e. $\lambda_{\theta}= \lambda_{\theta C}$, and when rotation becomes dominant the BL thickness is determined by the rotating BL thickness, i.e. $\lambda_{\theta}= \lambda_{\theta R}$. We now model the crossover between these two limiting cases as $$\label{Eq crossover}
\frac{\lambda_{\theta}}{L}=\frac{\sqrt{Re_R}\lambda_{\theta R}+\alpha^* \sqrt{Re_C}\lambda_{\theta C}}{(\sqrt{Re_R} + \alpha^* \sqrt{Re_C})L}.$$ Here the square root of Reynolds has been chosen since the dimensionless BL thicknesses scale with $1/\sqrt{Re}$. We rewrite the above equation in terms of $Ro$, using $Re_R \propto Re_C/Ro$, $$\label{Eq model}
\frac{\lambda_\theta}{L} =\frac{\frac{1}{\alpha \sqrt{Ro}} \lambda_{\theta R} + \lambda_{\theta_C}}{\left(\frac{1}{\alpha\sqrt{Ro}}+1 \right)L}$$ and we use the free parameter $\alpha$ to fit the model with the numerical data shown in Fig. \[Fig\_SCL09\_model\]. It can be concluded that the presented model, based on the approach of the GL theory, indeed reflects the increase of $Nu$ (as compared to the case without rotation) observed at relatively low $Ra$ number ($Ra=4\times10^7$) when the LSC is still present. Furthermore, the thickness of the thermal BLs is also reflected correctly by the model. Note that the large value of the parameter $\alpha=55$ indicates that the influence of the rotation is rather weak before Ekman pumping sets in and it also explains that for higher $Ra$ ($Ra\gtrsim 1\times10^8$) no heat transfer enhancement is observed below onset. The sudden (instead of smooth) transition is then fully determined by the rotation rate where Ekman pumping sets in. This may be because at higher $Ra$ the thermal BL is already much thinner due to the stronger LSC and therefore the effect of weak rotation is not sufficient to result in a significant thinner thermal BL. When $Ro < Ro_c$ the model cannot be used, since Ekman pumping is dominant in this regime which is responsible for the strong increase observed in $Nu$ when $Ro < Ro_c$.
Conclusions
===========
To summarize, we have studied the effect of rotation on the RB system at relatively low $Ra$ number, i.e. $Ra=4\times10^7$ by using DNS. We find a smooth increase of the heat transfer as function of the rotation rate when weak rotation is applied. To describe this heat transfer enhancement we have extended the GL theory to the rotating case by studying the influence of rotation on the scaling of the thermal BL thickness. It is based on a similar approach as in the laminar Prantl-Blasius BL theory over an infinitely large plate, as we analyzed the flow over an infinitely large rotating disk where the fluid at infinity is allowed to rotate. Just as in the Prantdl-Blasius BL theory we used a passive temperature field to calculate the characteristics of the thermal BL. It turns out that weak background rotation does not influence the scaling of the BL thickness in the high $Pr$ regime, because again $Pr^{-1/3}$ scaling is found. However, rotation does influence the scaling in the low $Pr$ regime where we find a scaling of $Pr^{-1}$ instead of $Pr^{-1/2}$ found in the Prandtl-Blasius BL theory. With our model for the thermal BL thickness, see Eq. (\[Eq model\]), we can explain the increased heat transfer observed in the relatively low $Ra$ number regime before the strength of the LSC decreases. The model neglects the effect of Ekman pumping as this effect is rather insignificant before the strength of the LSC decreases, i.e. the regime to which the model is applied. This means that the model cannot predict the heat transfer enhancement that is observed at moderate rotation rates where Ekman pumping is the dominant mechanism. The contrast between the smooth onset at $Ra=4\times10^7$ and the sharp onset at $Ra\gtrsim 1\times10^8$ is remarkable since only a small shift in the $Ra-Pr-Ro$ phase space is involved.\
*Acknowledgments*: We thank R. Verzicco for providing us with the numerical code and F. Fontenele Araujo, F. Busse, G.J.F. van Heijst, C. Sun, and L. van Wijngaarden for discussions. The work is sponsored by the Foundation for Fundamental Research on Matter (FOM) and the National Computing Facilities (NCF), both sponsored by NWO. The numerical calculations have been performed on the Huygens cluster of SARA in Amsterdam.
Ekman boundary layer theory {#appA}
===========================
In this appendix the results obtained from the model with weak background rotation will be compared with analytic results obtained from Ekman BL theory [@gre90], which uses a rotating reference frame. In the Ekman case the fluid at infinity is rotating at almost the same velocity as the disk, i.e. the limiting case of the model will be checked. We will indicate all quantities calculated in the rotating reference frame with an asterisk.
We will use the BEK model, presented in [@fal91; @lin97; @jas05], to derive a similar ODE as in section \[Sec3\] for the temperature advection equation in the rotating reference frame. In the BEK model the following self-similarity assumption for the axial velocity is proposed:
$$\label{Eq assumption axial velocity BEK}
w=\sqrt{\nu \Omega^{*}} Ro^* H^*(\xi),$$
where $\xi= z \sqrt{\Omega^*/\nu}$. Here, $\Omega^{*}$ is a system rotation rate, and $Ro^*$ is a constant determined by the rotation rate. We call this still the Rossby number since it also represents a dimensionless inverse rotation. In the BEK model $Ro^*$, $\Delta \Omega$, and $\Omega^{*}$, respectively, are defined as $$\label{Eq Rossbystar}
Ro^*=\frac{\Delta \Omega}{\Omega^{*}},$$ $$\label{Eq deltaOmega}
\Delta \Omega = \Omega_F - \Omega_D,$$ $$\label{Eq Omegastar}
\Omega^{*}=\frac{\Omega_F}{2-Ro^*}+\frac{\Omega_D}{2+Ro^*}.$$ We obtain the following temperature advection equation in the BEK model $$\label{Eq ODE energy equation rotating reference frame}
\widetilde{\theta}^{''} = Pr H^{*}(\xi) Ro^* \widetilde{\theta}'.$$ From Eqs. (\[Eq Rossbystar\])-(\[Eq Omegastar\]) one obtains, after some algebra, $$s=\frac{\Omega_F}{\Omega_D}= \left[\frac{2+Ro^*-Ro^{*2}}{2-Ro^*-Ro^{*2}}\right].$$ This means for the Ekman case ($s \approx 1$, i.e. $Ro^* \rightarrow 0 $) that $\Omega_F \simeq \Omega_D (1+Ro^*)$. Thus when the fluid at infinity is rotating slower than the disk $Ro^*$ is negative. From now on we assume $Ro^*$ to be negative, i.e. $s<1$.
Using Ekman BL theory [@gre90] one can derive analytic solutions for the radial and tangential velocity profiles in the rotating reference frame. These analytical solutions read [@gre90]: $$\begin{aligned}
\label{Eq solution Ekman radial}
u_E &=& - \Delta \Omega r e^{-\zeta} \sin \zeta = r\Omega^* Ro^* F^*(\zeta)~,\\
\label{Eq solution Ekman tangential}
v_E &=& \Delta \Omega r(1 - e^{-\zeta} \cos \zeta) = r\Omega^* Ro^* G^*(\zeta)~,\end{aligned}$$ with $\zeta$ as in (\[Eq dimensionless height\]). With the analytic expression for the radial velocity (\[Eq solution Ekman radial\]) and the continuity equation one obtains $$\begin{aligned}
\label{Eq solution Ekman axial}
w_E & = & \Delta \Omega \sqrt{\frac{\nu}{\Omega_D}}\left(1 - e^{-\zeta} \left[\sin \zeta + \cos \zeta \right]\right)\nonumber\\
\nonumber\\
& = & \sqrt{\nu \Omega^*} Ro^* H^*(\zeta)~.\end{aligned}$$ In the case $Ro^* \rightarrow 0$ it is found that $\Omega_D \approx \Omega^*$, thus $\xi \approx \zeta$. In particular, the expression for the axial flow reduces to $H^*(\xi)=\sqrt{\Omega^*/\Omega_D}(1-e^{-\xi} \left[\sin \xi + \cos \xi \right])$, where $\sqrt{\Omega^*/\Omega_D} \approx 1$. We find that the analytic expressions and the above numerical solutions are identical within numerical accuracy for the limiting case (Ekman solution, $s\approx1, Ro^* \rightarrow 0$). (Note that a coordinate transformation has to be applied as the Ekman solution is expressed in the corotating reference frame, whereas the numerical solution has been defined in the laboratory frame.)
Now we use this approach to calculate the BL characteristics for the Ekman layer. With the temperature advection equation (\[Eq ODE energy equation rotating reference frame\]) and the analytic expression $H^*(\xi)$ for the axial velocity we determine the effective scaling exponent $\gamma$ in $\overline{\lambda}_\theta \sim Pr^{\gamma}$ as function of $Pr$ for $Ro^*=-10^{-3}$ and $Ro^*=-10^{-5}$. Fig. \[Figure\_ste08\_thermalBL\] shows that when $Ro^*$ goes to zero the low $Pr$ regime ($\overline{\lambda}_\theta \gg \overline{\lambda}_u$) is extended to higher $Pr$, because the kinetic BL thickness becomes thinner and the axial velocity becomes smaller, i.e. the thermal BL becomes thicker.
Substitution of $H^*(\xi)$ in (\[Eq ODE energy equation rotating reference frame\]) yields the following expression for the temperature gradient in the Ekman layer $$\label{Eq temperature gradient Ekman BL}
\frac{d\widetilde{\theta}}{d\xi}=C_1 e^{\left[ \left(e^{-\xi}\cos \xi +\xi \right)Pr Ro^* \right]},$$ where $C_1$ is a constant of integration which does not depend on $\xi$. To determine the constant $C_1$, and thereby $Nu$, one needs to integrate Eq. (\[Eq temperature gradient Ekman BL\]). An analytic result can be derived by substituting $\cos \xi = (e^{i\xi}+e^{-i\xi})/2$, $A=PrRo^*$, $z=A\xi$, $B=(i-1)/A$ (and $\overline{B}$ the complex conjugate of $B$), and evaluate the integral
$$\label{Eq AB}
\widetilde{\theta}(z) = \frac{C_1}{A}\int e^z e^{\frac{1}{2}Ae^{Bz}} e^{\frac{1}{2}Ae^{\overline{B}z}}dz + C_2~.$$
The integration constants $C_1$ and $C_2$ are determined by the boundary conditions $\widetilde{\theta}(\xi=0)=1$ and $\widetilde{\theta}(\xi\rightarrow \infty)=0$: $C_2=0$ and, for small $A$,
$$\label{Eq coeff1}
\frac{1}{C_1}\approx \frac{1}{A}-\frac{1}{2}A-\frac{3}{16}A^2+{\mathcal{O}}(A^3)~.$$
The thermal BL thickness scales according to $\overline{\lambda}_\theta\propto e^{-A}/C_1$. With (\[Eq coeff1\]) we immediately see that in the small Prandtl number limit: $\overline{\lambda}_\theta\propto Pr^{-1}$. Comparison with Fig. \[Figure\_ste08\_thermalBL\]b reveals that the analytic results are in good agreement with the numerical data for $\overline{\lambda}_\theta$ represented by the solid lines. Moreover, it predicts the scaling for $Ro^*=-1$ (the von Kármán case, thus far outside the regime of applicability of the Ekman analysis) surprisingly well, see Fig. \[Figure\_ste08\_thermalBL\]b.
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---
abstract: 'The simplest interpretation of the [[Bicep2]{}]{} result is that the scalar primordial power spectrum is slightly suppressed at large scales [@Contaldi:2014zua; @Contaldi:2014rna]. These models result in a large tensor-to-scalar ratio $r$. In this work we show that the type of inflationary trajectory favoured by [[Bicep2]{}]{} also leads to a larger non-Gaussian signal at large scales, roughly an order of magnitude larger than a standard slow-roll trajectory.'
author:
- 'Jonathan S. Horner'
- 'Carlo R. Contaldi'
bibliography:
- 'paper.bib'
title: 'BICEP’s bispectrum'
---
0.2in
Introduction
============
The recent results from [[Bicep2]{}]{} [@Ade:2014xna], hinting at a detection of primordial $B$-mode power in the Cosmic Microwave Background (CMB) polarisation, place the inflationary paradigm on much firmer footing. This result, in combination with the [[Planck]{}]{} total intensity measurement [@Ade:2013ktc], imply that primordial perturbations are generated from an almost de-Sitter like phase of expansion early in the Universe’s history before the standard big bang scenario.
At first glance there is potential tension between the polarisation measurements made by [[Bicep2]{}]{} and [[Planck]{}]{}’s total intensity measurements. [[Planck]{}]{}’s power spectrum is lower than the best-fit $\Lambda$CDM models at multipoles $\ell \lesssim 40$ and [[Bicep2]{}]{}’s high $B$-mode measurement exacerbates this since tensor modes also contribute to the total intensity. The tension is indicated by the difference in the $r\sim 0.2$ value implied by [[Bicep2]{}]{}’s measurements and the 95% limit of $r < 0.1$ implied by the [[Planck]{}]{} data for $\Lambda$CDM models. Many authors have pointed out how the tension can be alleviated by going beyond the primordial power-law, $\Lambda$CDM paradigm by allowing running of the spectral indices, enhanced neutrino contributions (see for examples [@Czerny:2014wua; @Contaldi:2014zua; @Gong:2014cqa; @Zhang:2014dxk]) or more exotic scenarios [@Anchordoqui:2014dpa]. However the simplest explanation, that also fits the data best, is one where there is a slight change in acceleration trajectory during the inflationary phase when the largest modes were exiting the horizon. This was shown by [@Contaldi:2014zua] where a specific model was used to generate a slightly faster rolling trajectory at early times. The effect of such a “slow-to-slow-roll” transition is to result in a slightly suppressed primordial, scalar power spectrum that fits the [[Planck]{}]{} data despite the large tensor contribution required by [[Bicep2]{}]{}. In [@Contaldi:2014rna] the author analyses generalised accelerating, or inflating, trajectories that fit the combination of [[Bicep2]{}]{} and [[Planck]{}]{} data and conclude that the suppression is required at a significant level and the best-fit trajectories are all of the form where the acceleration has a slight enhancement at early times.
An alternative explanation is that the $B$-mode power observed by [[Bicep2]{}]{} is not due to foregrounds and is not primordial. This possibility has been discussed by various authors [@Mortonson:2014bja; @Flauger:2014qra] who point out that more measurements on the frequency dependence of the signal are required to definitively state whether we have detected the signature of primordial tensor modes. These measurements will be provided in part by the [[Planck]{}]{}polarisation analysis and [[Bicep2]{}]{}’s cross-correlation with further KECK data [@Ade:2014xna].
If the [[Bicep2]{}]{} result stands the test of time then the signal we point out in the analysis below is expected to be present if the simplest models of inflation driven by a single, slow-rolling scalar field are the explanation behind the measurements. In this case a measurement of tensor mode amplitude, or $r$, is a direct measurement of the background acceleration since $r \sim 16\epsilon$ and the tension between [[Bicep2]{}]{} polarisation and [[Planck]{}]{} total intensity measurements implies a change in the acceleration at early times. In turn, the change in acceleration enhances the non-Gaussianity on scales that were exiting the horizon while the acceleration was changing.
In this [*paper*]{} we construct a simple toy-model inspired by the best fitting trajectories found in [@Contaldi:2014rna] and calculate its bispectrum numerically. At small scales, as one would expect, the non-Gaussianity is small $\mathcal{O}(10^{-2})$ [@Maldacena:2002vr; @Creminelli:2004yq] but at large scales, where the scalar power spectrum is suppressed, the non-Gaussianity can be significantly larger, $\mathcal{O}(10^{-1})$. The results are compared against the slow-roll approximation in the equilateral configuration and the squeezed limit consistency relation. Whilst at small scales there is exceptional agreement with the slow-roll approximation, at large scales the results can deviate by up to 10%.
This [*paper*]{} is organised as follows. We outline the calculation of the scalar and tensor power spectra in Section \[power\_spec\] and summarise the calculation of the bispectrum in Section \[bispectrum\]. Our results are presented in Section \[results\] and we discuss their implications in Section \[conclusion\].
Computation of the scalar power spectrum {#power_spec}
========================================
The calculation is best performed in a gauge where all the scalar perturbations are absorbed into the metric such that $g_{ij} =
a^{2}\,(t)e^{2\zeta(t, \mathbf{x})}\delta_{ij}$ and the inflaton perturbation $\delta\phi(t, \mathbf{x}) = 0$. The primordial power spectrum is then simply given by: $$\label{PowerSpectrum}
\langle\zeta^{}_{k_{1}}\zeta^\star_{k_{2}}\rangle = (2\pi)^{3}\delta^{(3)}(\mathbf{k}_{1}+\mathbf{k}_{2})P_\zeta(k_1)\,,$$ where $\mathbf{k}$ is the Fourier wavevector and $k\equiv |\mathbf{k}|$. The mode $\zeta_{k}(t)$ satisfies the Mukhanov-Sasaki equation [@Mukhanov:1985rz; @Sasaki:1986hm] $$\label{Mukh}
\frac{\mathrm{d}^{2}\zeta_{k}}{\mathrm{d}N^{2}} + (3 + \epsilon - 2\eta)\frac{\mathrm{d}\zeta_{k}}{\mathrm{d}N} + \frac{k^{2}}{a^{2}H^{2}}\zeta_{k} = 0\,.$$ In the above $N$ is the number of $e-$folds which increases with time or alternatively $$H = \frac{\dot{a}}{a} = \frac{dN}{dt}\,,$$ and $\epsilon$ and $\eta$ are the usual slow-roll variables defined by $$\epsilon = -\frac{\dot{H}}{H^{2}},\,\,\,\,\,\,\,\, \eta = \epsilon - \frac{1}{2H}\frac{d\ln\epsilon}{dt}\,.$$
Outside the horizon $\zeta_{k}$ quickly goes to a constant and the power spectrum is then related to the freeze-out value of $\zeta_k$ on scales $k\ll aH$ $$P_\zeta(k) = \left|\zeta_{k\ll aH}\right|^{2}\,.$$ The initial conditions for the solutions to (\[Mukh\]) can be set when the mode is much smaller than the horizon $k\gg aH$ and takes on the Bunch-Davies form [@Bunch:1978yq] $$\label{Initial_zeta}
\zeta_{k} \to \frac{1}{M_{\rm pl}}\,\frac{e^{-ik\tau}}{2a\sqrt{k\epsilon}}\,,$$ where $\tau$ is conformal time defined by $\mathrm{d}N/\mathrm{d}\tau =
aH$.
An identical calculation can be performed for the tensor power spectrum $P_{h}(k) = \left|h_{k\ll aH}\right|^{2}$ with $h_{k}$ satisfying the following differential equation $$\label{tensors}
\frac{\mathrm{d}^{2}h_{k}}{\mathrm{d}N^{2}} + (3 -
\epsilon)\frac{\mathrm{d}h_{k}}{\mathrm{d}N} +
\frac{k^{2}}{a^{2}H^{2}}h_{k} = 0\,,$$ with initial condition $$h_{k} \to \frac{1}{M_{\rm pl}}\,\frac{e^{-ik\tau}}{a\sqrt{2k}}\,,$$ in the limit where $k \gg aH$. Solving for $P_{\zeta}(k)$ and $P_{h}(k)$ numerically we can calculate $n_{s}, r$ and $n_{t}$ directly from their definitions: $$\begin{aligned}
\label{nsr_numeric}
n_{s}(k_\star) & = & 1 + \left.\frac{\mathrm{d}\ln \left[k^{3}P_{\zeta}(k)\right]}{\mathrm{d}\ln k}\right|_{k = k_\star}\,\\
r(k_\star) & = & 8\,\frac{P_h(k_\star)}{P_{\zeta}(k_\star)}\,\nonumber\\\nonumber
n_{t}(k_\star) & = & \left.\frac{\mathrm{d}\ln \left[k^{3}P_{h}(k)\right]}{\mathrm{d}\ln k}\right|_{k = k_\star}\,\\\nonumber\end{aligned}$$ The factor of 8 comes from how the tensor perturbations are normalised in the second order action.
The above procedure outlines the general calculation of the primordial power spectrum from inflation. In this work we are interested in specifying a background model favoured by the recent [[Bicep2]{}]{} + [[Planck]{}]{}data. In particular we choose a function for $\epsilon$, then $\eta$ and $H$ are easily obtained by its derivative and integral respectively.
Instead of a direct function of time or $N$ though we specify $\epsilon(x)$ where $x = \ln (k'/k_{\rm min})$. $k'$ is the mode crossing the horizon at $e-$foldings $N$ ($k' = aH$) and $k_{\rm min} \sim
10^{-5} (\text{Mpc})^{-1}$ is the largest scale observable today. In addition to being proportional to $r$ this condition allows one to easily specify how the background should evolve in our observational window. For concreteness we require $\epsilon$ to be relatively large, but still satisfying the slow-roll limit, at large scales and then to flatten out into another slow-roll regime with a smaller value. To this end we adopt a simple toy-model for $\epsilon$ as a function of x $$\label{toy_model}
\epsilon = \left\{\epsilon_1 \tanh\left[(x - x_0)\right] + \epsilon_2\right\}\left(1 + m x\right)\,,$$ where the coefficients $\epsilon_1$, $\epsilon_2$, $m$, and $x_0$ are chosen to give a final power spectrum with the required suppression and position ($\sim 26\%$ and 1.5$\times 10^{-3}$ Mpc$^{-1}$ respectively [@Contaldi:2014zua]) and $n_s\sim 0.96$ on small scales. Fig. \[fig:background\] shows $\epsilon$ and $\eta$ as a function of $N$ for this toy-model and the resulting power spectra are shown in Fig. \[fig:ps\].
![ Background functions $\epsilon$ (red, solid) and $\eta$ (blue, dashed) of our toy-model plotted as a function of $e-$folds $N$. The grey vertical line indicates roughly the time when the first observable mode crosses the horizon.[]{data-label="fig:background"}](fig){width="8.5cm"}
-- --
-- --
Computation of the bispectrum {#bispectrum}
=============================
The largest contribution to primordial non-Gaussianity will come from the bispectrum of the curvature perturbation $$\label{3rd}
\langle\zeta_{k_{1}}\zeta_{k_{2}}\zeta_{k_{3}}\rangle = (2\pi)^{3}\delta^{(3)}(\mathbf{k}_{1}+\mathbf{k}_{2}+\mathbf{k}_{3})B(k_{1}, k_{2}, k_{3})\,.$$ The quantity that is often quoted in observational constraints is the dimensionless, reduced bispectrum $$\begin{aligned}
\label{fnl_def}
f_{\mathrm{NL}}(k_{1}, k_{2}, k_{3}) &=& \frac{5}{6}\,B(k_{1}, k_{2},
k_{3}) / \left(|\zeta_{k_{1}}|^{2}|\zeta_{k_{2}}|^{2}+\right.\nonumber\\
&&\left.|\zeta_{k_{1}}|^{2}|\zeta_{k_{3}}|^{2}+|\zeta_{k_{2}}|^{2}|\zeta_{k_{3}}|^{2}\right)\,,\end{aligned}$$ The analytical calculation is much simpler if we consider the equilateral configuration $f_{\text{NL}}(k,k,k)$ however this is not a directly observed quantity as the estimator requires $B(k_{1},
k_{2}, k_{3})$ to be factorizable [@Creminelli:2005hu]. This is not true for the general case, which we are considering. However the overall amplitude of the reduced bispectrum gives a good indication of the size of the expected observable $f_{\mathrm{NL}}$.
All theories of inflation will produce a non-zero bispectrum. This is simply because gravity coupled to a scalar field is a non-linear theory and will contain interaction terms for the primordial curvature perturbation $\zeta(t,\textbf{x})$. These interaction terms will source the bispectrum with the largest contributors coming from tree-level diagrams associated with the cubic interaction terms. The bispectrum can then be calculated using the “in-in” formalism [@Adshead:2009cb; @Maldacena:2002vr; @Seery:2005gb], which to tree level becomes $$\label{in_in}
\langle\zeta^{3}(t)\rangle = -i\int_{-\infty}^{t}\mathrm{d}t'\langle\left[\zeta^{3}(t), H_{\text{int}}(t')\right]\rangle\,,$$ where $H_{\text{int}}$ is the interaction Hamiltonian associated with the following third order action $$\begin{aligned}
\label{action_final}
S_{3} &=& \!\!\int d^4x\, a^{3}\epsilon \left[\left(2\eta - \epsilon\right)\zeta \dot{\zeta}^{2} + \frac{1}{a^{2}}\epsilon\zeta(\partial\zeta)^{2}\right.\nonumber\\
& &\!\!\!\!\left. - (\epsilon - \eta)\zeta^{2}\partial^{2}\zeta - 2\epsilon\left(1 - \frac{\epsilon}{4}\right)\dot{\zeta}\partial_{i}\zeta\partial_{i}\partial^{-2}\dot{\zeta}\right.\nonumber\\
& & \!\!\!\!\left. + \frac{\epsilon^{2}}{4}\partial^{2}\zeta\partial_{i}\partial^{-2}\dot{\zeta}\partial_{i}\partial^{-2}\dot{\zeta}\right]\,,\end{aligned}$$
The numerical calculation of the bispectrum is technically challenging and is described in more detail in [@Horner:2013sea]. Briefly, for the equilateral configuration it requires the calculation of the following integral $$\label{fnl_prelim}
f_{\mathrm{NL}} = \frac{1}{3|\zeta|^{4}}\times{\cal I}\left[\zeta^{*3}\int_{N_{0}}^{N_{1}} dN\, (f_{1}\zeta^{3} + f_{2}\zeta\zeta^{\prime 2})\right]\,,$$ where $\zeta = \zeta_{k}$, $\zeta^{\prime} =
\mathrm{d}\zeta/\mathrm{d}N$, and $\cal I$ represents the imaginary part. The background functions $f_{i}$ are given by $$\begin{aligned}
\!\!\!\!\!f_{1} & = & \frac{5k^{2}a\epsilon}{H}(2\eta - 3\epsilon)\,,\nonumber\\
\!\!\!\!\!f_{2} & = & -5Ha^{3}\epsilon\left(4\eta - \frac{3}{4} \epsilon^{2}\right)\,.\end{aligned}$$ The times $N_{0}$ and $N_{1}$ correspond to when the mode is sufficiently sub- and super-horizon respectively. For calculating the shape dependence we restrict ourselves to the case of isosceles triangles so we parametrise our modes in the following way. $|\textbf{k}_{1}| = |\textbf{k}_{2}| = k, |\textbf{k}_{3}| =
\beta k $. This covers most configurations of interest ($\beta = 0$ is squeezed, $\beta = 1$ is equilateral, $\beta = 2$ is folded) and is simple to interpret.
-- --
-- --
Results
=======
For the toy-model given in (\[toy\_model\]) the non-Gaussianity amplitude is plotted in Fig. \[fig:fnl1\]. For comparison, as well as a consistency check, we plot the full-numerical calculation (blue-dashed) as well as the the slow-roll approximation (red-solid) which, in the equilateral limit, is given by [@Maldacena:2002vr] $$\label{SRapprox}
f_{\mathrm{NL}}(k) = \frac{5}{12}\left(n_{s}(k) - 1 + \frac{5}{6}n_{t}(k)\right)\,.$$
In applying this formula we used the exact values of $n_{s}$ and $n_{t}$ given by equations \[nsr\_numeric\]. As can be seen from Fig. \[fig:fnl1\], if values close to $r\sim 0.2$ are confirmed from polarisation measurements, the non-Gaussianity on large scales are likely to be an order of magnitude larger than expected. This is simply because $r \propto \epsilon$ but on smaller scales $\epsilon$ is constrained to be lower by the total intensity measurements. The only way to reconcile the two regimes is by having $\epsilon$ change to a lower value at later times and this results in an enhancement of non-Gaussianity being generated as the value is changing. Fig. \[fig:fnl1\] also shows that, even with strong scale dependence, there is remarkable agreement between the full numerical results and the Maldacena formula, with deviations only occurring at the largest scales. Fig. \[fig:fnl2\] shows the complete scale and shape dependence of [$f_{\rm NL}$]{}.
![ [$f_{\rm NL}$]{}as a function of scale $k$ and shape $\beta$. There is a mild peak in the equilateral limit, $\beta = 1$. For all shapes the non-Gaussianity peaks around the scales corresponding to the size of the horizon at the time when the background acceleration is changing.[]{data-label="fig:fnl2"}](fnl){width="8.5cm"}
Discussion {#conclusion}
==========
Models of inflation that contain a feature causing the background acceleration to change can reconcile [[Planck]{}]{} and [[Bicep2]{}]{} observations of the CMB total intensity and polarisation power spectra. We have shown that these models result in enhanced non-Gaussianity at scales corresponding to the size of the horizon at the time when the acceleration is changing. The level of non-Gaussianity at these scales is an order of magnitude larger than what is expected in the standard case with no feature and is strongly scale dependent.
Whilst the effect was illustrated using a simple toy-model of the background evolution $H(t), \epsilon(t)$, etc, we expect the non-Gaussian enhancement to be present in any model where the acceleration changes relatively quickly in order to fit the [[Planck]{}]{} and [[Bicep2]{}]{} combination. The exact form of non-Gaussianity will obviously be model dependent.
It is not clear that this level of non-Gaussianity will be observable since it corresponds to scales $\ell \sim 2\to 80$ where there may not be a sufficient number of CMB modes on the sky to ever constrain [$f_{\rm NL}$]{}to ${\cal O}(10^{-1})$. However cross-correlation with other surveys of large scale structure may help to constrain non-Gaussianity on these scales. In particular it may be possible to detect any anomalous correlation of modes induced by the non-Gaussianity.
The biggest question at this time however is whether or not the claimed detection of primordial tensor modes by [[Bicep2]{}]{} is correct. This will be addressed in the near future as the polarisation signal is observed at more frequencies at the same signal-to-noise levels reached by the [[Bicep2]{}]{} experiment.
We thank Marco Peloso for useful discussions. JSH is supported by a STFC studentship. CRC and JSH acknowledge the hospitality of the Perimeter Institute for Theoretical Physics and the Canadian Institute for Theoretical Astrophysics where some of this work was carried out.
|
---
author:
- |
[^1]\
School of Physics, KIAS, Seoul 130-722, Korea\
E-mail:
title: Electroweak symmetry breaking and cold dark matter from strongly interacting hidden sector
---
Introduction
============
Revealing the origin of the electroweak symmtry breaking (EWSB) is the most pressing question in particle physics in the era of CERN Large Hadron Collider (LHC). Another important problem in particle astrophysics and cosmology is to identify the nature of cold dark matter (CDM). Also there is a more speculative issue about the existence of a new hidden sector, which is generic in supersymmetric (SUSY) model buildings or superstring theories. In this talk, I would like to consider three seemingly unrelated questions:
- Can all the masses arise (mostly) from quantum mechanics, as in massless QCD ?
- What is the nature of CDM ? Is it possible to have all the global symmetry as accidental symmetries, as in the standard model (SM) ?
- What would be the phenomenological consequences, if there is a hidden sector ?
I will present models with a hidden sector where these seemingly unrelated questions are in fact closely connected with each other. More details and complete list of references can be found in Ref.s [@ko1; @ko2]. Let me remind you that there is a good old example, namely quantum chromodynamics(QCD), where we can learn many lessons related with the issues listed above. QCD has many nice features: renormalizability, asymptotic freedom, confinement and chiral symmetry breaking, dynamical generation of hadron masses, natural hierarchy between the Planck scale and the QCD scale $\Lambda_{\rm
QCD}$. In addition pions are stable if electroweak interactions are switched off. It would be nice if we could have a model for EWSB in the same manner as the dimensional transmutation in QCD, and CDM is stable as pions are stable under strong interaction.
The basic features of our models are the following. We assume a vectorlike confining gauge theory such as QCD in the hidden sector. Then dimensional transmutation will occur in the hidden sector, and this scale is transmitted to the SM by a messenger, and triggers EWSB. And the lightest mesons in the hidden sector becomes a CDM.
Model I
=======
Let us assume that there is a new strong interaction that is described by $SU(N_{h,C})$ guage theory with vectorlike quarks ${\cal Q}_i$ and $\overline{\cal Q}_i$ with $N_{h,f}$ flavors, such as QCD with the confinement scale $\Lambda_h$. This scale is presumed to be higher than the electroweak scale by at least an order of magnitude. $${\cal L}_{hidden} = - {1\over 4}{\cal G}_{\mu\nu} {\cal G}^{\mu\nu} +
\sum_{k=1}^{N_{HF}}
\overline{\cal Q}_k ( i {\cal D} \cdot \gamma - M_k ) {\cal Q}_k$$ Then this new strong interaction will trigger chiral symmetry breaking due to nonzero $\langle {\cal Q \overline{Q}} \rangle
\equiv \Lambda_{H,\chi}^3$. For illustration, we assume that there is an approximate $SU(2)_L \times SU(2)_R$ global symmetry in the hidden sector that breaks down to $SU(2)_V$ spontaneously. Then at low energy scale, massless Nambu-Goldstone bosons will appear, which we call hidden sector pions $\pi_h$’s. Also there would be a scalar resonance like the ordinary $\sigma$, and we call it $\sigma_h$, and $\vec{\pi}_h$ and $\sigma_h$ will form $SU(2)_L \times SU(2)_R$ bidoublet (denoted as $H_2$) and the low energy effective theory will be the same as the Gelmann-Levy’s linear $\sigma$ model, except that the mesons ($\pi_h$’s) are all in the hidden sector, so that they are all SM singlets.
The potential for the SM Higgs and the hidden sector $H_2$ is given by $$V ( H_1 , H_2 ) =
-\mu^2_1 (H^\dagger_1
H_1)+\frac{\lambda_1}{2} (H^\dagger_1 H_1)^2 -\mu^2_2 (H^\dagger_2
H_2) \nonumber + \frac{\lambda_2}{2} (H^\dagger_2 H_2)^2
+
\lambda_3 (H^\dagger_1 H_1)(H^\dagger_2 H_2) + \frac{a
v^3_2}{2}\sigma_h$$ This looks like the potential in the 2-Higgs doublet model, but there are important differences. First, $H_2$ is a SM singlet, not a SM doublet. $W$ and $Z^0$ get masses entirely from $H_1$ VEV. And the $a$ term is new in our model, and necessary to generate the mass for the hidden sector pion. Note that the $\lambda_3$ term connects the SM and the hidden sector, and originates from nonrenormalizable interactions between two sectors, or by some messengers.
It is straightforward to analyze phenomenology from this scalar potential. The generic features of our models can be summarized as follows. The origin of the EWSB, namely the negative Higgs mass$^2$ parameter could be the chiral symmetry breaking in the strongly interacting hidden sector. The electroweak precision test does not put strong constraints unlike in the ordinary technicolor models, since $H_2$ does not contribute to the $W$ and $Z^0$ masses at tree level. And no Higgs-mediated FCNC problem since $H_2$ does not couple to the SM fermions. There are more than one neutral Higgs-like scalar bosons, and they can decay into the $\pi_h$ with a large invisible branching ratio. This makes relatively difficult to produce and discover these Higgs-like neutral scalars at colliders. See Fig. 1 (a) and (b). Finally, the hidden sector pion ($\pi_h$) is stable due to the flavor conservation of hidden sector strong interaction, and could be a good CDM candidate. Direct detection rate of the $\pi_h$ is promisingly within the sensitivity of the current/future DM detection experiments.
![\[fig:br-t1\] Branching ratios of (a) $h$ and (b) $H$ as functions of $m_{\pi_h}$ for $\tan\beta = 1$, $m_h = 120$ GeV and $m_H = 300$ GeV.](h120-300.eps "fig:"){width="6cm"} ![\[fig:br-t1\] Branching ratios of (a) $h$ and (b) $H$ as functions of $m_{\pi_h}$ for $\tan\beta = 1$, $m_h = 120$ GeV and $m_H = 300$ GeV.](hh120-300-1-1.eps "fig:"){width="6cm"}
![\[fig2\] $\sigma_{SI} (\pi_h p \rightarrow \pi_h p )$ as functions of $m_{\pi_h}$ for (a) $\tan\beta = 1$ in Model I, and (b) Model II. ](dr-1.eps "fig:"){width="6cm"} ![\[fig2\] $\sigma_{SI} (\pi_h p \rightarrow \pi_h p )$ as functions of $m_{\pi_h}$ for (a) $\tan\beta = 1$ in Model I, and (b) Model II. ](both-X-section.eps "fig:"){width="6cm"}
Model II with classical scale invariance
========================================
The Model I has a few drawbacks, since the hidden sector quark masses $M_k$’s are given by hand, and the Model I is not renormalizable. These can be cured by introducing a real singlet scalar $S$ and making the following replacement, $M_k \rightarrow
\lambda_k S$ in Eq. (1). Then ${\cal L}_{\rm hidden}$ has classical scale symmetry. With a real singlet $S$, the SM lagrangian is implemented into $$\label{eq:sm}
{\cal L}_{\rm SM} = {\cal L}_{\rm kin} + {\cal L}_{\rm Yukawa}
- {\lambda_{H} \over 4}~( H^{\dagger} H )^2 - {\lambda_{SH} \over
2}~S^2 ~ H^{\dagger} H - {\lambda_S \over 4}~S^4$$ assuming classical scale symmetry. Since there are no mass parameters in this lagrangian, this is a suitable starting point to investigate if it is possible to have all the masses from quantum mechanical effects. Note that the real singlet scalar $S$ plays the role of messenger connecting the SM Higgs sector and the hidden sector quarks.
Dimensional transmutation in the hidden sector will generate the hidden QCD scale and chiral symmetry breaking with developing nonzero $\langle \bar{\cal Q}_k {\cal Q}_k \rangle$. Then the $\lambda_k S $ term generate the linear potential for the real singlet $S$, leading to nonzero $\langle S \rangle$. This in turn generates the hidden sector current quark masses through $\lambda_k$ terms as well as the EWSB through $\lambda_{SH}$ term. The $\pi_h$ will get nonzero masses, and becomes a good CDM candidate. Due to the presence of the messenger $S$, the CDM pair annihilation into the SM particles occurs more efficiently in Model II than in Model I, and it is easy to accommodate the WMAP data on $\Omega_{\rm CDM} h^2$. Direct detection rates are in the interesting ranges (see Fig. 2 (b)). All the qualitative features of this model is similar to the Model I. See Ref. [@ko2] for more details.
Conclusions
===========
In this talk, I presented models where the origin of EWSB and CDM lie in the hidden sector technicolor interaction. In the Model II, all the masses including the CDM mass arise quantum mechanically from dimensional transmutation in the hidden sector. One can enjoy many variations of these models by considering different gauge groups and matter fields in the hidden sector. If we include the radiative corrections to the scalar potential, the details could change, but the qualitative features described in this talk would remain untouched.
Acknowledgments {#acknowledgments .unnumbered}
===============
I am grateful to Taeil Hur, D.W. Jung and J.Y. Lee for collaborations.
[0]{} T. Hur, D. W. Jung, P. Ko and J. Y. Lee, arXiv:0709.1218 \[hep-ph\]. S. Baek, T. Hur and P. Ko, In preparation.
[^1]: This work was supported in part by Korea Neutrino Research Center (KNRC) of Seoul National University through National Research Foundation of Korea Grant.
|
---
abstract: 'We use numerical simulations to study the dynamics of surface discharges, which are common in high-voltage engineering. We simulate positive streamer discharges that propagate towards a dielectric surface, attach to it, and then propagate over the surface. The simulations are performed in air with a two-dimensional plasma fluid model, in which a flat dielectric is placed between two plate electrodes. Electrostatic attraction is the main mechanism that causes streamers to grow towards the dielectric. Due to the net charge in the streamer head, the dielectric gets polarized, and the electric field between the streamer and the dielectric is increased. Compared to streamers in bulk gas, surface streamers have a smaller radius, a higher electric field, a higher electron density, and higher propagation velocity. A higher applied voltage leads to faster inception and faster propagation of the surface discharge. A higher dielectric permittivity leads to more rapid attachment of the streamer to the surface and a thinner surface streamer. Secondary emission coefficients are shown to play a modest role, which is due to relatively strong photoionization in air. In the simulations, a high electric field is present between the positive streamers and the dielectric surface. We show that the magnitude and decay of this field are affected by the positive ion mobility.'
address: |
$^1$Xi’an Jiaotong University, Xi’an, China\
$^2$Centrum Wiskunde & Informatica, Amsterdam, The Netherlands\
$^3$Centre for Mathematical Plasma-Astrophysics, KU Leuven, Belgium
author:
- 'Xiaoran Li$^1$, Anbang Sun$^{1}$, Guanjun Zhang$^1$, Jannis Teunissen$^{2,3}$'
bibliography:
- 'references.bib'
title: A computational study of positive streamers interacting with dielectrics
---
Introduction {#sec:introduction}
============
Electric discharges in electronic devices and HV (high-voltage) equipment often occur along dielectric materials. In the regions of HV stress around an insulator, electron avalanches and streamer discharges can develop. These partial discharges may eventually result in surface flashover of the insulator, i.e., electric breakdown. In [@cookson1970] it was found that around atmospheric pressure, surface flashover voltages were 10%-50% lower than flashover voltages in pure gas gaps. A dielectric present in the vicinity of the electrodes not only modifies the fields between the electrodes, but also serves as a possible source or sink of electrons during the breakdown process. Studying the interaction between dielectrics and streamer discharges is therefore important to understand surface flashover.
Early studies of surface discharges focused on the measurement of flashover voltage [@cookson1981; @sudarshan1986]. In the past few decades, the use of high-speed cameras has revealed more details about the early stages of surface discharges. In several experiments, streamer discharges were observed to have an affinity to propagate along dielectric surfaces rather than through the background gas only [@sobota2008; @trienekens2014]. This affinity for a dielectric surface was reported to depend on the discharge gap geometry [@sobota2009], gas composition, pressure [@dubinova2016], and dielectric properties [@allen1999; @meng2015].
To gain more insight into the physics of surface discharges, different types of numerical simulations have been performed, see e.g. [@dubinova2016; @jorgenson2003; @jansky2010; @babaeva2016; @georghiou2005; @sun2018; @sima2016]. Studies on the interaction between plasmas and dielectrics have often been performed at lower pressure and in noble gases, where the discharge mechanisms are relatively well understood [@zhang2018b; @sun2018a]. Several authors have also studied surface discharges in atmospheric air. An incomplete list is given below.
Jorgenson *et al*. [@jorgenson2003] investigated the role of photoemission in the surface breakdown process. With Monte Carlo simulations, they concluded that photoemission plays a role at low field values near the breakdown threshold. Celestin *et al*. [@celestin2009] studied dielectric barrier discharges in air both experimentally and computationally, and highlighted the importance of surface charge. Jánský *et al*. [@jansky2010] presented simulations of an air plasma discharge at atmospheric pressure, initiated by a needle anode set inside a dielectric capillary tube. Babaeva *et al*. [@Babaeva_2015; @babaeva2016] performed a computational investigation of nanosecond pulsed surface discharges of positive and negative polarity. A hybrid fluid-Monte Carlo model was used to more accurately capture secondary electron emission caused by positive ions and photons. [Sima *et al*. [@sima2016] presented 2D axisymmetric fluid simulations of discharges spreading radially over a dielectric surface in a N$_{2}$/O$_{2}$ mixture.]{} Furthermore, several computational studies of plasma-liquid interaction and plasma-tissue interaction have been performed at atmospheric pressure, see for example [@babaeva2013; @Tian_2014]. In such studies, the liquid or skin is often modeled as a dielectric, sometimes with a finite conductivity.
The studies mentioned above have greatly improved our understanding of surface discharges in air. Here, this past work is extended in several ways. We consider a different geometry, namely a flat dielectric placed between parallel-plate electrodes, see figure \[fig:figure-2\]. This geometry resembles some actual HV insulation applications, and its simplicity makes it suitable for numerically studying surface discharges. Our focus here is on positive surface streamers, which can be computationally expensive to simulate. We have therefore developed an efficient fluid model with adaptive mesh refinement. For simplicity and efficiency, 2D simulations are used here, as full 3D simulations would still be very costly. Another novel aspect is that secondary emission due to both ions and photons is considered in the model.
The content of the paper is as follows. The simulation model is described in section \[sec:simulation-model\]. In section \[sec:interaction-dielectric\], we focus on the attraction of streamers to dielectrics, and we look at the differences between surface and gas-phase streamers. Afterwards, several discharge parameters are varied, to study their effect on the streamer’s inception time, propagation velocity and morphology:
- The applied voltage in section \[sec:effect-appl-volt\]
- The permittivity ($\varepsilon$) of the dielectric material in section \[sec:effect-perm-diel\]
- The secondary electron emission coefficients (for positive ions and photons) in section \[sec:effect-electron-emission\]
- The positive ion mobility in section \[sec:effect-positive-ion\].
Simulation Model {#sec:simulation-model}
================
The 2D fluid model used in this paper is based on Afivo-streamer [@teunissen2017; @teunissen2018], which is an open-source plasma fluid code for streamer discharges that features adaptive mesh refinement (AMR), geometric multigrid methods for Poisson’s equation, and OpenMP parallelism. For a recent comparison of six streamer simulation codes, including Afivo-streamer, see [@bagheri2018]. We have made several modifications to be able to simulate surface streamers:
- Electrons, ions and photons can be absorbed by dielectric surfaces.
- Surface densities and fluxes are stored separately from their equivalents in the gas.
- The electric field computation takes the surface charge into account.
- A new Monte Carlo photoemission module was implemented, and the photoionization routines were adjusted to account for the dielectric.
These changes are described in more detail below.
With our 2D model, we effective simulate planar surface discharges. This leads to some differences compared to a full 3D description. First, the electric fields and charge densities in 3D are typically higher, as the streamer heads have a stronger curvature. Second, it is often observed that both surface and gas streamers are present in experiments [@allen1999; @meng2017]. We do not observe these two components in our 2D model, but have seen them in preliminary 3D simulations that are still under development.
Fluid Model {#sec:fluid-model}
-----------
The fluid model used here is of the drift-diffusion-reaction type with the local field approximation [@luque2012]. The model keeps track of the electron density *n*$_e$, the positive ion density *n*$_{i}^{+}$ and the negative ion density *n*$_{i}^{-}$, which involve in time as $$\begin{aligned}
\frac{\partial n_e}{\partial t}= -\nabla \cdot {\mathbf{\Gamma}}_e + S_{i}-S_a + S_{pi}+S_{pe},\label{eq:electron-ddt}\\
\qquad {\mathbf{\Gamma}}_e = -n_e\mu_e {\mathbf{E}} - D_e\nabla n_e,\nonumber\\
\frac{\partial {n_{i}}^{+}}{\partial t}=-\nabla \cdot {\mathbf{\Gamma}}_i^+ + S_{i}+S_{pi},\\
\qquad {\mathbf{\Gamma}}_i^+ = {n_{i}}^{+}\mu_i^{+} {\mathbf{E}},\nonumber\\
\frac{\partial {n_{i}}^{-}}{\partial t}=-\nabla \cdot {\mathbf{\Gamma}}_i^-+S_{a},\\
\qquad {\mathbf{\Gamma}}_i^- = -{n_{i}}^{-} \mu_i^{-} {\mathbf{E}},\nonumber\end{aligned}$$ Here, fluxes are indicated by a ${\mathbf{\Gamma}}$, ${\mu}_e$ is the electron mobility, $D_e$ the electron diffusion coefficient, ${\mathbf{E}}$ the electric field, and $\mu_i^\pm$ the positive/negative ion mobilities. Furthermore, several source terms are present. The electron impact ionization and electron attachment terms are given by $S_i = \alpha \mu_e |{\mathbf{E}}| n_e$ and $S_a = \eta \mu_e |{\mathbf{E}}| n_e$, respectively, where $\alpha$ and $\eta$ are the ionization and attachment coefficients. The production of photoelectrons from photoionization is included with the term $S_{pi}$, and secondary electron emission is accounted for by $S_{pe}$.
The local field approximation is used, so that ${\mu}_e$, $D_e$, $\alpha$ and $\eta$ are functions of the local electric field strength. Electron transport and reaction coefficients for air ($1 \, \textrm{bar}$, $300 \, \textrm{K}$) were generated with Monte Carlo particle swarm simulations (see e.g. [@Rabie_2016a]), using Phelps’ cross sections [@Phelps_1985]. The positive ion mobility $\mu_i^+ = 3 \times 10^{-4}\,\mathrm{m}^{2}/\mathrm{Vs}$ is here considered to be constant, but in section \[sec:effect-positive-ion\] it is varied to investigate its effect on surface discharges. For simplicity, the negative ion mobility is set to zero ($\mu_i^- = 0$) throughout the paper.
We assume that electrons and ions attach to the surface when they flow onto a dielectric. They do not move or react on the surface, but secondary electron emission from the surface is taken into account. For the impact of positive ions, a SEE (secondary electron emission) coefficient ${\gamma}_{i}$ is used. When a photon hits a dielectric surface, we assume that the photon is absorbed, and a SEE photoemission coefficient ${\gamma}_{pe}$ is used. The effect of these SEE coefficients is studied in section \[sec:effect-electron-emission\], elsewhere they are set to zero. Secondary emission leaves behind positive surface charge on the dielectric. Therefore, the surface charge density ${\sigma}_{s}$ changes in time as $$\partial_t \sigma_{s}=-e(\Gamma_e+{\Gamma_i^{-}})
+ e(1+\gamma_{i})\Gamma_{i}^{+} + e\gamma_{pe}\Gamma_{pe},$$ where $e$ is the elementary charge and the other terms correspond to the fluxes towards the gas-dielectric interface: $\Gamma_e$ for electrons,[$\Gamma_{i}^{-}$ for negative ions,]{} $\Gamma_{i}^{+}$ for positive ions, and $\Gamma_{pe}$ for photons. We study positive streamers, which means that electrons generally move away from dielectrics. Therefore, an accurate description of the electron flux towards the surface [@Hagelaar_2000] is not required here.
Electric Field {#sec:electric-field}
--------------
The electric field ${\mathbf{E}}$ is calculated by first solving Poisson’s equation for the electric potential $\varphi$: $$\begin{aligned}
\nabla \cdot \left(\varepsilon \nabla \varphi \right)=-(\rho +\delta_{s}\sigma_{s}),\end{aligned}$$ where $\varepsilon$ is the dielectric permittivity, ${\rho}$ is the volume charge density, and $\delta_{s}$ maps the surface charge $\sigma_s$ on the gas-dielectric interface to the grid cells adjacent to the dielectric. Afterwards, the electric field is computed as $${\mathbf{E}}=-\nabla \varphi.
\label{equ:solve E}$$ At the dielectric interface, we ensure that the normal component of the electric field satisfies the classic jump condition $$\varepsilon_{1}E_{1}- \varepsilon_{2}E_{2}=\sigma_{s}.
\label{equ:solve E at boundary}$$ Details about the numerical implementation, which is compatible with adaptive mesh refinement, will be presented in a forthcoming paper.
Photoionization and Photoemission {#sec:photoionization-emission}
---------------------------------
![Illustration of photoionization and photoemission mechanisms in air. Two types of photons are considered. High-energy photons can generate photoionization and photoemission, whereas low-energy photons are not absorbed by the gas and only contribute to photoemission. []{data-label="fig:photo-processes"}](pictures/new_photoionization_photoemission.jpg){width="1.0\linewidth"}
Positive streamer discharges need a source of free electrons ahead of them in order to propagate. Photons can generate such free electrons through photoionization in the gas or photoemission from a dielectric surface. In $\mathrm{N}_{2}$–$\mathrm{O}_{2}$ mixtures, non-local photoionization can take place when an excited nitrogen molecule emits a UV photon in the 98 to 102.5 nm range, which has enough energy to ionize an oxygen molecule. Photoionization often plays an important role in electrical discharges, see e.g. [@pancheshnyi2005; @Pancheshnyi_2014].
The role of photoemission in surface discharges is less well understood. Photons can be emitted from several excited states. The probability of photoemission not only depends on the photon energy, but also on the surface properties [@jorgenson2003]. For simplicity, we consider only two types of photons in this paper: high-energy photons, which can generate photoionization and photoemission, and low-energy photons, which can only contribute to photoemission and are not absorbed in the gas. These processes are illustrated in figure \[fig:photo-processes\].
The low-energy and high-energy photons are modeled with a Monte Carlo (MC) method. The basic functionality of this method is described in [@Bagheri_2019] and chapter 11 of [@teunissen]. Discrete photons are produced stochastically, and their direction is isotropically distributed. A photoionization event occurs when a high-energy photon is absorbed by the gas, in which case the electron and positive ion density are locally increased. The absorption length of high-energy photons is sampled from the absorption function for air, see e.g. [@Bagheri_2019].
Photons hitting a dielectric are absorbed and contribute to the local photoemission flux at the surface. This photoemission flux increases the electron density in the first grid cell next to the dielectric. For the low-energy and high-energy photons, photoemission coefficients ${\gamma}_{peL}$ and ${\gamma}_{peH}$ are used, respectively. The effect of photoemission is investigated in section \[sec:effect-electron-emission\]; elsewhere in the paper photoemission is not taken into account (so that ${\gamma}_{peL} = {\gamma}_{peH} = 0$).
The production of photons is handled per grid cell. The average number of high-energy photons produced within a time step is proportional to the number of impact ionization events that have occurred. The proportionality factor is here set to $\xi p_q / (p + p_q)$, where $\xi = 0.05$ is a proportionality factor, $p = 1 \, \textrm{bar}$ is the gas pressure and $p_q = 40 \, \textrm{mbar}$ is the collisional quenching pressure, see [@Bagheri_2019] for details. For simplicity, the generation of low-energy photons is here assumed to be equal to the number of high-energy photons. As the low-energy photons only contribute to photoemission, their effect can be controlled through the corresponding photoemission coefficient.
Since simulations are here performed in 2D, the discrete photons cannot correspond to (single) physical photons. Instead, the total photon number is fixed, so that the MC method always uses $10^5$ photons.
Computational domain and initial conditions {#sec:parameters}
-------------------------------------------
![The computational domain. A parallel-plate geometry is used, with a flat dielectric present on the left. Discharges start from the ionized seed present close to the top electrode, as described in the text.[]{data-label="fig:figure-2"}](pictures/small_simulation_setup.jpg){width="0.6\linewidth"}
We use a parallel-plate electrode geometry with a flat dielectric in between, as shown in Figure \[fig:figure-2\]. The computational domain measures (40 mm)$^{2}$, and a dielectric is present on the left side with a width of $10 \, \mathrm{mm}$. The dielectric permittivity is set to $\varepsilon = 2$, but in section \[sec:effect-perm-diel\] it is varied to investigate its effect on surface discharges.
As a gas, artificial air (80% N$_{2}$ and 20% O$_{2}$) at 1 bar and 300 K is used. For the electric potential, Dirichlet boundary conditions are applied at the upper and lower boundaries, and Neumann zero boundary conditions on the left and right side. A voltage of $100 \, \textrm{kV}$ is applied. In section \[sec:effect-appl-volt\], this voltage is varied. The background densities of electrons and positive ions are set to 10$^{10}$ m$^{{-}3}$ [@sun2014].
To start a discharge, the background field has to be locally enhanced. We do this by placing an ionized seed of about $2 \, \mathrm{mm}$ long with a radius of about 0.4 mm. The electron and positive ion density are 5${\times}$10$^{18}$ m$^{{-}3}$ at the center, and they decay at distances above $d=0.2 \, \mathrm{mm}$ with a so-called smoothstep profile: $1-3x^{2}+2x^{3}$ up to $x = 1$, where $x = (d-0.2 \, \mathrm{mm})/0.2 \, \mathrm{mm}$. When the electrons from a seed drift upwards, the electric field at the bottom of the seed is enhanced so that a positive streamer can form.
Results & discussion {#sec:results-discussion}
====================
Section d (mm) $U$ (kV) $\varepsilon_r$ $\gamma_{i}$ $\gamma_{pe}$ $\mu_i^+$ (m$^{2}$/Vs)
--------- --------------- ---------------- ----------------- -------------- --------------- -------------------------------
3.1 $(0.5,1,2,5)$ 100 2 0 0 $3\times10^{-4}$
3.2 0.5 $(92,100,112)$ 2 0 0 $3\times10^{-4}$
3.3 0.5 100 $(2,3,5)$ 0 0 $3\times10^{-4}$
3.4.1 0.4 100 2 $(0,0.5)$ 0 $3\times10^{-4}$
3.4.2 $(0.5,1)$ 100 2 0 $(0,0.5)$ $3\times10^{-4}$
3.5 0.5 100 2 0 0 $(0, 1, 5, 10) \times10^{-4}$
In section \[sec:interaction-dielectric\], the initial seed is placed at different distances from the dielectric to study the streamer-dielectric interaction. We also point out the main differences between *surface* and *gas* streamers. Next, we systematically study the effect of several parameters on the surface discharges: the applied voltage (section \[sec:effect-appl-volt\]), the dielectric permittivity (section \[sec:effect-perm-diel\]), the secondary electron emission coefficients (section \[sec:effect-electron-emission\]), and finally the ion mobility (section \[sec:effect-positive-ion\]). [The parameters investigated and their values in each section are shown in table \[parameter\_table\].]{}
Streamer-dielectric interaction {#sec:interaction-dielectric}
-------------------------------
Previous experiments have revealed that dielectrics attract positive streamers, see e.g. [@sobota2008; @trienekens2014]. [This attraction is also present in our numerical model. Figure \[fig:process\_d1\] shows the evolution of the electron density for an initial ionized seed placed $1 \, \mathrm{mm}$ away from the dielectric. It can be seen that the streamer start to grow in air and then gradually develops towards the dielectric. After connecting with the dielectric, the streamer propagates down over its surface. The evolution shown here is in qualitative agreement with the discharge photographs in [@sobota2008].]{}
![The streamer development process between $10 \, \mathrm{ns}$ and $18 \, \mathrm{ns}$ for seed placed at $1 \, \mathrm{mm}$ from the dielectric[]{data-label="fig:process_d1"}](pictures/process_d1.jpg){width="1.0\linewidth"}
To study the streamer-dielectric attraction, we have placed the initial ionized seed at different distances from the dielectric. [Figures \[fig:diff seed position\]a–d show the electron density for seeds placed at $0.5 \, \mathrm{mm}$, $1 \, \mathrm{mm}$, $2 \, \mathrm{mm}$ and $5 \, \mathrm{mm}$ from the dielectric. For comparison, the electron density at $20 \, \mathrm{ns}$ in the absence of a dielectric is also shown. Figure \[fig:diff seed position\]a shows the electron density at $15 \, \mathrm{ns}$, for the other cases, which develop more slowly, results at $20 \, \mathrm{ns}$ are shown.]{} The closer the streamer is located to the dielectric, the stronger the attraction to the dielectric becomes. It can also be seen that a nearby dielectric increases the streamer’s velocity in the gas, and that streamers here propagate faster on the surface than in the gas.
![The electron density for streamers starting from different locations. For panels a–d, the initial seed was placed at $0.5 \, \mathrm{mm}$, $1 \, \mathrm{mm}$, $2 \, \mathrm{mm}$ and $5 \, \mathrm{mm}$ from the dielectric. For comparison, a streamer in bulk gas is shown in panel e. Results are shown at $20 \, \mathrm{ns}$, except for panel a, which has the fastest propagation.[]{data-label="fig:diff seed position"}](pictures/ne_diffd.jpg){width="1.0\linewidth"}
### Attraction to the dielectric
{width="0.9\linewidth"}
As photoemission is disabled here (see table \[parameter\_table\]), the attraction of the streamer to the dielectric is purely electrostatic. The net charge in the streamer head polarizes the dielectric, which increases the electric field between the streamer and the dielectric. This effect is illustrated in figure \[fig:Ex for diff seeds\], which shows the horizontal electric field ($E_{x}$) around the streamer heads. With a dielectric present, $|E_{x}|$ increases on the dielectric side, and it is reduced on the other side. The closer the streamer is to a dielectric, stronger this effect becomes. Eventually, the streamer will turn into a surface streamer. As shown in figure \[fig:diff seed position\], such a surface streamer is thinner and quite asymmetric compared to a gas streamer.
We remark that the attraction to the dielectric is here due to the space charge from the streamer itself. Without space charge, the electric field has no horizontal component in our plate-to-plate geometry. In previous research, there was often also static field enhancement between pointed electrodes and dielectrics, see e.g. [@sobota2008; @sobota2009].
### Effect on streamer velocity
Several experimental studies have found that surface discharges are faster than bulk gas streamers [@trienekens2014; @allen1999; @akyuz2001]. Their increased velocity was attributed to increased ionization rates near the dielectric, accumulated negative charge and electron emission from the dielectric surface. Electron emission from the dielectric is not included here (it is in section \[sec:effect-electron-emission\]), but we still find that the surface streamers are significantly faster. Figure \[fig:v-maxE-y\] shows the streamer velocity and its maximal electric field for the case $d = 1 \, \textrm{mm}$. Note that both the velocity and the maximal electric field increase when the surface streamer forms, at around $y = 35 \, \textrm{mm}$. Even though the higher field is mostly in the horizontal direction, see figure \[fig:Ex for diff seeds\], it still contributes to a faster vertical growth.
As can be seen in figure \[fig:diff seed position\], the electron density inside a surface streamer ($\sim 10^{21} \, \mathrm{m}^{-3}$) is higher than in a gas streamer ($\sim 10^{18} \, \mathrm{m}^{-3}$). There seem to be several related effects that lead to the increased surface streamer velocity. The strong electric field between a surface streamers and a dielectric pulls surface streamers towards the dielectric. This reduces their radius [(as shown in figure \[fig:diff seed position\])]{}, and results in an asymmetric streamer head shape, which also leads to stronger electric field enhancement. The result is that the ionization rate is increased, that the streamer has a higher degree of ionization, and that it propagates faster. This behavior is quite distinct from gas streamers, which typically propagate faster when they have a larger radius [@Briels_2008].
![The maximum electric field and streamer velocity versus the vertical position of the streamer head.[]{data-label="fig:v-maxE-y"}](pictures/maxEandv_y_d1_v2.jpg){width="0.9\linewidth"}
### Cathode sheath {#sec:cathode-sheath}
As shown in figure \[fig:floating\_streamer\], the surface streamer ‘hovers’ over the dielectric surface without fully connecting to it. This phenomenon was also observed in simulations of dielectric barrier discharges [@babaeva2011; @babaeva2016; @stepanyan2014; @soloviev2017], and it only occurs for positive streamers. The reason is that positive streamers grow from incoming electron avalanches, but such avalanches require sufficient distance before they reach ionization levels comparable to the discharge. Positive streamers can therefore not immediately connect to the dielectric surface. Due to the net charge in the streamer head, a very high electric field is present in the narrow gap between streamer and dielectric. The effect of secondary electron emission on these dynamics is studied in section \[sec:effect-electron-emission\], and the role of the positive ion mobility is investigated in section \[sec:effect-positive-ion\].
![The electron density and the electric field around the positive streamer head at $20 \, \mathrm{ns}$, for an initial seed at $1 \, \mathrm{mm}$ from the dielectric. Note the gap between the streamer and the dielectric.[]{data-label="fig:floating_streamer"}](pictures/cathode_sheath_v2.jpg){width="0.95\linewidth"}
We remark that the maximum electric field of a positive surface streamer can rapidly rise to very high values, as shown in figures \[fig:v-maxE-y\]a and \[fig:floating\_streamer\]. These high-field areas typically contain a low electron density, but they still pose a problem for plasma fluid simulations. In our model, transport coefficients go up to $35 \, \mathrm{kV/mm}$; for higher fields, the values at $35 \, \mathrm{kV/mm}$ are used. More generally, the validity of the local field approximation is questionable when there are such high electric fields (and corresponding strong gradients). For future studies in such ultra-high electric fields, particle-in-cell simulations could therefore be more suitable, as was also observed in [@Babaeva_2015]. Finally, we remark that a potential physical limitation for this maximum electric field is field emission of electrons from the surface.
Effect of applied voltage {#sec:effect-appl-volt}
-------------------------
To study the effect of the applied voltage on surface discharges, we have performed simulations for applied voltages of $92 \, \mathrm{kV}$, $100 \, \mathrm{kV}$ and $112 \, \mathrm{kV}$, which correspond to background electric fields of $2.3 \, \mathrm{kV/mm}$, $2.5 \, \mathrm{kV/mm}$ and $2.8 \, \mathrm{kV/mm}$, respectively. In all cases, the initial seed was located at $0.5 \, \mathrm{mm}$ from the dielectric, and the evolution up to $15 \, \mathrm{ns}$ was simulated. Figure \[fig:maxE\_t\_diffE\] shows the maximum electric field versus time, and [figure \[fig:2D\_diffE\] shows the electron density for three cases at $7 \, \mathrm{ns}$ and $9 \, \mathrm{ns}$. Both figures reveal the following stages in the streamer’s development:]{}
1. The inception stage, in which the maximum electric field is from 0 to about $9 \, \mathrm{kV/mm}$ in our setup and the streamer is hardly propagating, [as shown in figure \[fig:2D\_diffE\]a.]{}
2. The gas-propagation stage, in which streamers propagate in the gas with a maximum electric field below $12.5 \, \mathrm{kV/mm}$. [This stage is visible in figures \[fig:2D\_diffE\]a and figure \[fig:2D\_diffE\]b. ]{}
3. The transition stage from a gas streamer to a surface streamer, in which the maximum electric field increases sharply. The streamer also loses its rounded head shape, [as shown in figures \[fig:2D\_diffE\]b and \[fig:2D\_diffE\]c.]{}
4. The surface propagation stage. The growth of the maximum electric field is slowing down, and the streamer propagates along the dielectric, as shown in figure [\[fig:2D\_diffE\]c.]{}
Figure \[fig:maxE\_t\_diffE\] shows that when the voltage is changed, the streamers still exhibit similar behavior in these four stages. The main difference is that the inception stage becomes shorter. For background fields of $2.3 \, \mathrm{kV/mm}$, $2.5 \, \mathrm{kV/mm}$ and $2.8 \, \mathrm{kV/mm}$, the inception stages last $8.5 \, \mathrm{ns}$, $5.5 \, \mathrm{ns}$ and $3.75 \, \mathrm{ns}$, respectively. The second stage also becomes slightly shorter for a higher applied voltage.
![The maximum electric field versus time for streamers in background electric fields of $2.3 \, \mathrm{kV/mm}$, $2.5 \, \mathrm{kV/mm}$ and $2.8 \, \mathrm{kV/mm}$ between. The indicated stages are I: initial stage, II: gas propagation, III: transition towards a surface streamer, IV: stable surface propagation.[]{data-label="fig:maxE_t_diffE"}](pictures/maxE_t_diffE.jpg){width="1.0\linewidth"}
![The electron density for streamers in different background electric fields ($2.3 \, \mathrm{kV/mm}$, $2.5 \, \mathrm{kV/mm}$ and $2.8 \, \mathrm{kV/mm}$) at $7 \, \mathrm{ns}$ and $9 \, \mathrm{ns}$.[]{data-label="fig:2D_diffE"}](pictures/diffvoltage_7ns_9ns_v2.jpg){width="1.0\linewidth"}
[Figure \[fig:v\_y\_diffE\] shows the streamer velocities versus their vertical location for the three applied voltages. As expected, a higher background electric field leads to a higher streamer velocity for streamers of the same length, in agreement with the experimental results of [@meng2015].]{}
![Streamer velocities versus their vertical location for different background electric fields.[]{data-label="fig:v_y_diffE"}](pictures/v_y_diffE.jpg){width="1.0\linewidth"}
Effect of dielectric permittivity {#sec:effect-perm-diel}
---------------------------------
The relative permittivity $\varepsilon$ of dielectric materials varies over a wide range. To study how $\varepsilon$ affects surface discharges, we have performed simulations with $\varepsilon$ set to 2, 3 and 5. As before, the initial seed was placed at $0.5 \, \mathrm{mm}$ from the dielectric, and simulations were performed up to $15 \, \mathrm{ns}$.
![The streamers’ maximal electric fields versus time for dielectrics with relative permittivities $\varepsilon_r$ of 2, 3 and 5.[]{data-label="fig:maxE-t-diffe"}](pictures/maxE_t_diffeps.jpg){width="1.0\linewidth"}
The maximum electric field versus time for the three permittivities is shown in figure \[fig:maxE-t-diffe\]. The main difference we observe is that the second and third stages are shorter for a higher permittivity. A higher $\varepsilon$ means the dielectric polarizes more strongly, which leads to a stronger attraction of streamers to the dielectric. Streamers therefore start the surface propagation stage earlier, and their maximum electric field increases more rapidly. Note that their maximum electric field is also higher during the surface propagation stage. In summary, we can conclude that a higher permittivity leads to a faster transition into a surface streamer, and a higher maximum field during the surface propagation stage.
[Figure \[fig:2D\_diffeps\] shows the electron density for these three cases when all the streamers are at $y = 36 \, \mathrm{mm}$. It can clearly be seen that the streamers attach more rapidly to the dielectric with a higher permittivity. Notice also that the surface streamer’s radius is smaller wither a higher dielectric permittivity, a result of the stronger electrostatic attraction.]{}
![The streamers’ electron density for dielectrics with relative permittivities $\varepsilon_r$ of 2, 3 and 5. Results are shown at different times, at the moment when the streamer heads are at $y = 36 \, \textrm{mm}$.[]{data-label="fig:2D_diffeps"}](pictures/2D_diffeps.jpg){width="1.0\linewidth"}
[Figure \[fig:v\_y\_diffeps\] shows the velocity versus the streamer’s vertical position for the different $\varepsilon_r$. The permittivity has only a small effect on the streamer’s velocity, in agreement with the experimental observations of [@sobota2009]. In contrast, a negative correlation between the permittivity and the streamer velocity was found in [@meng2015]. The discrepancy could come from the different geometry that was used, in which multiple surface and gas streamers propagated next to a cylindrical dielectric.]{}
![Streamer velocities versus their vertical location for different relative dielectric permittivities.[]{data-label="fig:v_y_diffeps"}](pictures/v_y_diffeps.jpg){width="1.0\linewidth"}
Effect of Secondary Electron emission from Dielectrics {#sec:effect-electron-emission}
------------------------------------------------------
Electron emission from dielectrics may influence streamer velocities [@tan2007] and affect the high electric field in the dielectric-plasma gap [@campanell2016]. In this section, we study how secondary electron emission affects surface streamers in our computational geometry. Both ion-induced secondary emission (ISEE) and photo-emission are considered.
### Ion-induced secondary electron emission {#sec:electron-emission-ions}
The ion-induced secondary electron emission (ISEE) yield $\gamma_{i}$ can vary over a wide range [@motoyama2006; @motoyama2004; @tschiersch2017]. Here we consider two cases, $\gamma_{i} = 0.5$ and $\gamma_{i} = 0$ (i.e., no secondary emission). [In this section, the initial ionized seed’s center is placed $0.4 \, \mathrm{mm}$ away from the dielectric, so that streamers start directly next to the dielectric.]{} Figure \[fig:isse\] shows the electron density and electric field distribution at $15 \, \mathrm{ns}$ for both ISEE coefficients. It can be seen that ISEE here has little effect on the streamer length and the electric field. The electron density in the streamer-dielectric gap is slightly higher behind the head for the streamer with $\gamma_{i} = 0.5$, but this has negligible influence on the electric field in the gap. That the ISEE yield hardly affects the streamer’s propagation and head shape makes sense: since positive ions need some time to reach the dielectric, ISEE does not release electrons from the dielectric at the streamer head.
![The electron density and electric field at $15 \, \mathrm{ns}$ with the ion-induced secondary electron emission coefficient $\gamma_{i}$ set to 0 (left) and $0.5$ (right).[]{data-label="fig:isse"}](pictures/diff_ISEE.jpg){width="1.0\linewidth"}
### Photoemission {#sec:electron-emission-photons}
The photo-emission coefficient ${\gamma}_{pe}$ for typical dielectric materials varies between 10$^{-4}$ to 10$^{-1}$ for photon energies of 5–20 eV [@fujihira1972; @buzulutskov1997]. This yield can be higher if the material contains stains or defects, or when it is negatively charged [@jorgenson2003; @motoyama2006]. The measurement of ${\gamma}_{pe}$ of dielectrics in air is often quite challenging [@dubinova2016a]. We here use several values for ${\gamma}_{pe}$ to demonstrate how photoemission affects positive streamers.
As discussed in section \[sec:photoionization-emission\], we consider low-energy and high-energy photons, with the main distinction that high-energy photons can be absorbed in the gas. The following four cases are considered for the photoemission coefficients ${\gamma}_{peL}$ and ${\gamma}_{peH}$ for low-energy and high-energy photons:
1. ${\gamma}_{peH}$=0, ${\gamma}_{peL}$=0
2. ${\gamma}_{peH}$=0.5, ${\gamma}_{peL}$=0
3. ${\gamma}_{peH}$=0, ${\gamma}_{peL}$=0.5
4. ${\gamma}_{peH}$=0.5, ${\gamma}_{peL}$=0.5
In the simulations, streamers start near the dielectric, with the seed placed at either $0.5 \, \mathrm{mm}$ or $1 \, \mathrm{mm}$ from the dielectric.
{width="0.9\linewidth"}
![a) The electron density for simulations with different photoemission coefficients at $11 \, \mathrm{ns}$. b) Zoom of the electron density at $y = 36.4 \, \textrm{mm}$.[]{data-label="fig:diff_phse"}](pictures/ne_diffphse_t11_v2.jpg "fig:"){width="1\linewidth"} ![a) The electron density for simulations with different photoemission coefficients at $11 \, \mathrm{ns}$. b) Zoom of the electron density at $y = 36.4 \, \textrm{mm}$.[]{data-label="fig:diff_phse"}](pictures/ne_x_diffphse_t11y364_v2.jpg "fig:"){width="1\linewidth"}
As shown in figure \[fig:diff\_phse\_inception\], photoemission by low-energy photons helps to start a discharge near a dielectric. [At $5.5 \, \mathrm{ns}$, the ${\gamma}_{peL} = 0.5$ cases show the streamer already bending towards the dielectric with a sharp tip, due to photoemission.]{} However, we remark that the inception time for these four cases is the same when the seeds are placed at $1 \, \mathrm{mm}$ away from the dielectric. The secondary electrons from the dielectric then need more time to reach the streamer, and the streamers have already started due to the photoionization they generate. We conclude that photoemission can be important for discharges close to dielectrics and for discharges in gases with less photoionization than air.
[Figure \[fig:diff\_phse\]a shows the electron density distribution for the above four cases at $11 \, \mathrm{ns}$. The streamers with ${\gamma}_{peL} = 0.5$ are longer than the other two, since they start earlier. Photoemission also causes them to attach to the dielectric more rapidly. Another difference is that the narrow gap between streamer and dielectric is smaller with more photoemission. This happens because photoemission provides seed electrons in the gap, which allows the streamer to get closer to the dielectric. To see this more clearly, the electron density distributions at $y = 36.4 \, \mathrm{mm}$ (the dashed line in figure \[fig:diff\_phse\]a) are shown for these four cases in figure \[fig:diff\_phse\]b. Without photoemission, the electron density has a wider profile with a lower maximum, and it is located farther from the dielectric. When photoemission is included, the effect of the low-energy photons (i.e., ${\gamma}_{peL} = 0.5$) is most important here.]{}
![Streamer velocities versus their vertical location for different photoemission coefficients.[]{data-label="fig:v_y_diffphse"}](pictures/v_y_diffphse_v2.jpg){width="1.0\linewidth"}
[Figure \[fig:v\_y\_diffphse\] shows the streamer velocities versus their vertical position for all four cases. The photoemission coefficients have only a small effect on the velocity, which is a little higher with $\gamma_{peL} = 0.5$. We think this is somewhat surprising. A possible explanation is that photoemission mostly leads to growth towards the dielectric, whereas photoionization in the gas contributes most of the free electrons that cause growth parallel to the dielectric. Another effect in the simulations presented here is that high-energy photons contribute less to a streamer’s growth very close to a dielectric. There are two reasons for this. First, these photons are absorbed at shorter distances if they hit a dielectric. Second, their photoemission coefficient is here less than one ($\gamma_{peH} = 0.5$), whereas in the gas they always lead to photoionization. ]{}
Effect of positive ion mobility {#sec:effect-positive-ion}
-------------------------------
![a) The electric field at $5.5 \, \mathrm{ns}$ after streamer inception for a positive ion mobility of 3${\times}$10$^{-4 }$m$^{2}$/Vs. The electron and positive ion dynamics in the streamer-dielectric gap are illustrated. b) The $E_{x}$ field at the point $(x, y) = (10.01 \, \mathrm{mm}, 36 \, \mathrm{mm})$ versus time for streamers with different positive ion mobilities. Here $t=0$ corresponds to the streamers’ respective inception times, which vary by less than a nanosecond for the four cases. []{data-label="fig:ion_mobility_gap"}](pictures/observation_point_v2.jpg "fig:"){width="0.8\linewidth"} ![a) The electric field at $5.5 \, \mathrm{ns}$ after streamer inception for a positive ion mobility of 3${\times}$10$^{-4 }$m$^{2}$/Vs. The electron and positive ion dynamics in the streamer-dielectric gap are illustrated. b) The $E_{x}$ field at the point $(x, y) = (10.01 \, \mathrm{mm}, 36 \, \mathrm{mm})$ versus time for streamers with different positive ion mobilities. Here $t=0$ corresponds to the streamers’ respective inception times, which vary by less than a nanosecond for the four cases. []{data-label="fig:ion_mobility_gap"}](pictures/Ex_t_diffui_v2.jpg "fig:"){width="1.0\linewidth"}
The positive ion mobility $\mu_i^+$ can affect surface streamers in two ways. First, a higher ion mobility increases the amount of ion-induced secondary electron emission (ISEE). However, since ISEE was found to play a negligible role in section \[sec:electron-emission-ions\], its role is not further studied here, and we set the ISEE yield to zero (i.e., $\gamma_i = 0$).
A second effect is that a higher ion mobility increases the conductivity of the discharge, in particular in regions where the ion density is high compared to the electron density. For positive surface streamer discharges, such a region is present in the streamer-dielectric gap. This gap typically contains a high electric field, especially close to the streamer head, see section \[sec:cathode-sheath\]. Electrons rapidly drift away from the surface, whereas positive ions move from the high-density discharge region towards the surface, as illustrated in figure \[fig:ion\_mobility\_gap\]a.
To investigate how the positive ion mobility ($\mu_i^+$) affects the decay of the high electric field in the streamer-dielectric gap, we have performed simulations with positive ion mobilities of 0, 1${\times}$10$^{-4 }$m$^{2}$/Vs, 5${\times}$10$^{-4 }$m$^{2}$/Vs and 1${\times}$10$^{-3}$ m$^{2}$/Vs, using a seed placed $0.5 \, \mathrm{mm}$ from the dielectric. For these simulations, we have recorded $E_{x}$ in the middle of the gap at the point indicated in figure \[fig:ion\_mobility\_gap\]a. The recorded fields are shown versus time in figure \[fig:ion\_mobility\_gap\]b. The maximum electric field occurs when the streamer heads pass by the observation point indicated in figure \[fig:ion\_mobility\_gap\]a. The decay of the peak in $E_{x}$ is faster for higher $\mu_i^+$, which is most clearly visible for the $\mu_i^+ = 5 \times 10^{-4} \, \mathrm{m}^{2}/\mathrm{Vs}$ and $\mu_i^+ = 1 \times 10^{-3} \, \mathrm{m}^{2}/\mathrm{Vs}$ cases. Note that the field also decays when the ions are immobile. This mainly happens because the amount of net space charge is lower behind the streamer head, but electron avalanches in the gap (due to e.g. photoionization) also contribute.
Conclusions {#sec:conclusions}
===========
In this paper, we have studied positive surface streamers with numerical simulations. A 2D fluid model for surface discharges based on the Afivo-streamer code [@teunissen2017] was developed. The model includes a Monte Carlo procedure for secondary electron emission (from both high and low energy photons) and support for dielectric surfaces. These new features are compatible with the adaptive mesh refinement and the parallel multigrid solver provided by the underlying Afivo framework [@teunissen2018].
We have used the new model to investigate the interaction positive streamers and dielectrics. We considered a parallel-plate geometry, with a flat dielectric between the two electrodes. Positive streamer discharges started from an ionized seed that was placed near the dielectric and the positive electrode. The effect of several parameters was investigated: the applied voltage, the dielectric permittivity, secondary electron emission caused by ions and photons, and the mobility of positive ions. Our main findings are summarized below:
1. A narrow gap forms between positive streamers and dielectrics, as was also observed in earlier work [@babaeva2016; @stepanyan2014; @soloviev2017]. A very high electric field can be present in this so-called ‘cathode sheath’.
2. The attraction of positive streamers to the dielectric was found to be mostly electrostatic. In our geometry, this attraction was caused by the net charge in the streamer head, which polarized the dielectric, increasing the field between the streamer and the dielectric. A higher dielectric permittivity led to a more rapid attachment of the streamer to the dielectric.
3. Compared to gas streamers, surface streamers had a smaller radius, a higher electric field, and a higher electron density. In our simulations, this gave surface streamers a higher propagation velocity than gas streamers.
4. A higher applied voltage caused the positive surface discharges to start earlier, but they behaved qualitatively similar. A higher dielectric permittivity also accelerated the formation of surface streamers.
5. Photoemission can accelerate streamer inception near dielectrics. However, photoemission hardly increases the velocity of surface streamers. A possible reasons is that photoemission mostly leads to growth towards the surface, whereas photoionization contributes more to the growth parallel to the surface.
6. The positive ion mobility affects the decay of the high electric field in the streamer-dielectric gap.
Acknowledgment {#acknowledgment .unnumbered}
==============
This project was supported by the State Key Laboratory of Electrical Insulation and Power Equipment (EIPE18203), the National Natural Science Foundation of China (51777164) and the Fundamental Research Funds for the Central Universities of China (xtr042019009).
Availability of model and data {#availability-of-model-and-data .unnumbered}
==============================
The source code and input files for the model used in this paper are available at https://gitlab.com/MD-CWI-NL/afivo-streamer (git commit `08f1c828`).
References {#references .unnumbered}
==========
|
---
abstract: 'We present our the construction of an atom interferometer for inertial sensing in microgravity, as part of the I.C.E. (*Interférométrie Cohérente pour l’Espace*) collaboration. On-board laser systems have been developed based on fibre-optic components, which are insensitive to mechanical vibrations and acoustic noise, have sub-MHz linewidth, and remain frequency stabilised for weeks at a time. A compact, transportable vacuum system has been built, and used for laser cooling and magneto-optical trapping. We will use a mixture of quantum degenerate gases, bosonic $^{87}$Rb and fermionic $^{40}$K, in order to find the optimal conditions for precision and sensitivity of inertial measurements. Microgravity will be realised in parabolic flights lasting up to 20s in an Airbus. We show that the factors limiting the sensitivity of a long-interrogation-time atomic inertial sensor are the phase noise in reference frequency generation for Raman-pulse atomic beam-splitters and acceleration fluctuations during free fall.'
author:
- 'R. A. Nyman'
- 'G. Varoquaux'
- 'F. Lienhart'
- 'D. Chambon'
- 'S. Boussen'
- 'J.-F. Clément'
- 'T. Muller'
- 'G. Santarelli'
- 'F. Pereira Dos Santos'
- 'A. Clairon'
- 'A. Bresson'
- 'A. Landragin'
- 'P. Bouyer'
date: 'Received: date / Revised version: date'
title: 'I.C.E.: a Transportable Atomic Inertial Sensor for Test in Microgravity'
---
Introduction
============
Intense research effort has focussed on the study of degenerate quantum gases and macroscopic matter waves since their first observation in 1995. Atom interferometers benefit from the use of trapped ultracold atomic gases, gaining good signal-to-noise ratios due to the high atomic densities, and the coherence required for the visibility of interference patterns due to the low temperatures[@Bermann97]. The sensitivity of an interferometric measurement also depends on the interrogation time, the time during which the sample freely evolves. This time is limited by both the free-fall of the atomic cloud, requiring tall vacuum chambers, and by its free expansion, demanding extra-sensitive detection systems for extremely dilute clouds. Ultralow temperatures further reduce the expansion.
In conceiving the next generation of extreme-precision atom interferometers, there is much to be gained by performing experiments in microgravity [@Sleator99; @Bongs04]. Free-fall heights of more than 100m, corresponding to durations of 5 seconds or more are available either in a drop tower (e.g. ZARM Bremen, Germany) or in a parabolic flight in an aeroplane. Laboratory experiments are limited to about 300ms of free fall. The sensitivity of an interferometric accelerometer increases quadratically with time, and thus one can expect to gain more than two orders of magnitude in having a transportable, drop-compatible device.
There remain questions over the best method to perform atom interferometry. Bosons suffer from interaction shifts leading to systematic errors such as the clock shift, a problem not apparent in ultracold fermions[@Gupta03]. However, degenerate fermions have an intrinsically broad momentum distribution due to Pauli blocking, limiting the visibility of interference patterns. Furthermore, to achieve quantum degeneracy, fermions must be cooled using a buffer gas, typically an ultracold gas of bosons, thus complicating experiments using fermions. Pairs of fermions (molecules or Cooper pairs[@Regal04]) can be created by applying a homogeneous magnetic field (Feshbach resonances[@LENS_Feshbach]), offering yet more possible candidate species for atom interferometers.
A further bonus to free-fall is the possibility of using weaker confining forces for the atoms, since gravity need not be compensated with additional levitation forces[@Leanhardt03]. Temperatures achieved by evaporative cooling and adiabatic expansion are lowered as the trapping potential is reduced. Not only does the sensitivity of an interferometric measurement benefit, but also new phases of matter may be observed if the kinetic energy can be made smaller than the interatomic potential. A reduced-gravity environment will permit study of new physical phenomena, e.g. spin dynamics and magnetic ordering (see for example [@Schmaljohann04] and references therein).
This article presents our design for a transportable, boson-fermion mixture, atom interferometer, compatible with a parabolic flight in an aeroplane. We describe our laser systems: a temporary bench for ground-based development, and the rack-mounted transportable system, based on frequency-doubled telecommunications lasers. We then explain our vacuum system and optics for atomic manipulation, and the accompanying support structure. Finally we describe the Raman-transition based atom-interferometric accelerometer, and show that the limits to in-flight performance are vibrations (acceleration fluctuations) and phase-noise on the Raman laser frequency difference.
Overview of the Experiment
--------------------------
The central components of this project are the atomic-physics vacuum system, the optics, and their supports. The atomic manipulation starts with alkali-metal vapour dispensers for rubidium and potassium[@SAES_Getters]. A slow jet of atoms is sent from the collection chamber by a dual-species, two-dimensional, magneto-optical trap (2D-MOT) to the trapping chamber, for collection and cooling in a 3D-MOT. Atoms are then be transferred to a conservative, far-off-resonance optical-dipole trap (FORT) for further cooling towards degeneracy. The sample is then ready for coherent manipulation in an atom-interferometer. Raman two-photon transition will be used as atomic beam-splitters and mirrors. Three-pulse sequences ($\pi/2 - \pi - \pi/2$) will be used for accelerometry.
All light for the experiment arrives by optical fibres, making the laser sources independent of the vacuum system. Transportable fibred laser sources for laser cooling and trapping have been fabricated with the required frequency stability. The techniques for mechanically-stable power distribution by free-space fibre couplers function according to specifications. The vacuum chamber is compatible with the constraints of microgravity in an Airbus parabolic flight. Such a flight permits total interrogation times up to 7s, giving a potential sensitivity of better than $10^{-9}\,\rm{m}\,\rm{s}^{-2}$ per shot, limited by phase noise on the frequency reference for the Raman transitions.
Laser Systems
=============
Ground-based laser diodes for Potassium and Rubidium cooling
------------------------------------------------------------
Our test laser system is not intended to fly, but nonetheless represents several technical achievements, detailed in Ref. [@Nyman06]. All of the lasers and optical amplifiers for trapping and cooling light are built around commercial semiconductor elements (*Eagleyard*) with home-made mounts and drive electronics. Semiconductor technology is one of the candidates for atomic-physics lasers in micro-gravity experiments: the chips are small, lightweight and robust, with low power consumption.
Extended-cavity grating-diode lasers (based on a design by Arnold et al.[@Arnold98]) are locked to atomic transitions (the hyperfine structure of the D2 lines of $^{87}$Rb and $^{39}$K, as appropriate), frequency shifted by acousto-optical modulators, injected into tapered amplifiers, then input to the optical fibres. We produce more than 200mW of useful light (out of the fibres) for trapping and cooling each species for both the 2D-MOT and the 3D-MOT.
One major difficulty was in making the master oscillator at 766.5nm (potassium D2 transition, wavelength in air). Semiconductor lasers at 780nm (rubidium D2 line) have been available for some time[@Wieman91], but are less easily found at short wavelengths. We pulled a 780nm diode to 766.5nm using very weak feedback, by anti-reflection coating the output face, and ensuring low reflectance from the grating (which was optimised for UV not visible light). Decreasing the feedback increases the threshold current, which increases the number of carriers in the active region, increasing the energy of the lasing transitions, thus giving gain at relatively short wavelengths. The tapered amplifiers we use work equally well for the two wavelengths.
Continuous-Wave Fibre-Laser Source at 780 nm for Rubidium Cooling {#source}
-----------------------------------------------------------------
An entirely pigtailed laser source is particularly appropriate in our case as it does not suffer from misalignments due to environmental vibrations. Moreover, telecommunications laser sources in the C-band (1530–1570 nm) have narrow linewidths ranging from less than 1MHz for laser diodes, down to a few kHz for Erbium doped fibre lasers. By second-harmonic generation (SHG) in a nonlinear crystal, these $1.56\mu$m sources can be converted to 780nm sources [@Mahal96; @Thompson03; @Dingjan06]. Such devices avoid having to use extended cavities as their linewidths are sufficiently narrow to satisfy the requirements of laser cooling.
Our laser setup is sketched in Figure \[fig:scheme\]. A 1560nm Erbium doped fibre laser is amplified by a 500mW polarisation-maintaining (PM) Erbium-doped fibre amplifier (EDFA). A 90/10 PM fibre-coupler directs $10\%$ of the pump power to a pigtailed output. $90\%$ of light is then sent into a periodically-poled Lithium-Niobate Waveguide (PPLN-WG). This crystal is pigtailed on both sides with 1560nm single-mode fibres. The input fibre is installed in a polarisation loop system in order to align the electric field with principal axes of the crystal. A fibre-coupler which is monomode at 780nm, filters pump light after the crystal and sends half of the 780nm light into a saturated- absorption spectroscopy device for frequency servo-control. The other half is the frequency-stabilised pigtailed output. The whole device, including the frequency control electronics was implemented in a rack for ease of transport. Typical output from the first generation device was 500$\mu$W of 780nm light, with more than 86dB attenuation of 1560nm light after 3m of monomode fibre. A more recent version ($>50$mW) has been used to power a magneto-optical trap.
Two PPLN-WGs from HC-Photonics were tested. Both have a poling period appropriate for SHG at 780nm. They have the same quasi-phase matching temperature of $63^\circ$C. The first is 13mm long, doped with $1\%$ MgO, and is used in our laser source. The second is 30mm long, doped with $5\%$ MgO. Figure \[fig:PPLNWG\] gives the output power as a function of the pump power. The 13mm long crystal has a fibre-to-fibre efficiency of $10\%/$W. The fit curve corresponds to the non-depleted pump regime. Photorefractive effects appear around 10mW of 780nm light. In practice the laser is run with 100mW pump power. Power fluctuations in this crystal are due to two phenomenon: first the input fibre does not maintain polarisation, and polarisation fluctuations lead to a variation of the output power. Secondly the output fibre of the crystal is not single mode at 780nm. Thus the power distribution in the fundamental mode varies with time, leading to power fluctuations when the crystal is pigtailed to a single-mode fibre at 780nm. The second crystal has a fibre-to-fibre efficiency of $120\%/$W for low pump power. The fit curve corresponds to a depleted regime. Photorefractive threshold is estimated around 60mW of second harmonic. The input fibre is still not polarisation maintaining, leading to output power drifts, but the output fibre is PM and single mode at 780nm, which greatly reduces power fluctuations.
### Frequency Stabilisation:
A Doppler-free saturated-absorption spectroscopy system without polarisation sensitive elements provides the frequency reference signal. The frequency of the laser has been tested by locking to a crossover of $^{85}$Rb. The laser frequency is oscillated over a few 100kHz by modulating the piezoelectric element of the fibre Bragg grating of the pump laser. The modulation frequency is 1.3kHz, permitting long-term drifts to be compensated without significantly broadening the laser linewidth. . The spectroscopic signal is demodulated by a phase-sensitive detection and fed back to the piezo. Figure \[fig:freqcontrol\] presents the spectral density of noise with and without frequency stabilisation. Noise up to 1.6Hz is attenuated, a frequency corresponding to the low-pass filter bandwidth of the demodulation. Points below $7\rm{kHz}/\sqrt{\rm{Hz}}$ are not represented because they are below the measurement noise. The r.m.s. frequency excursion in the band 0–20Hz is less than 200kHz.
The laser remains frequency locked even with strong mechanical disturbances (hand claps, knocks on the rack ...), but cannot withstand even small variations of the ambient temperature. The fibre source at 1560nm, though temperature controlled, suffers frequency drifts due to temperature changes of the fibre. Small fluctuations are compensated by the frequency loop but long-term drifts are beyond the range of the piezoelectric servo-loop, so the laser jumps out of lock. An integrated circuit based on a PIC 16F84 micro-controller was developed: the output voltage of the regulator is monitored by the micro-controller, and, when fixed boundaries are exceeded, the set temperature of the laser controller is adjusted. This additional loop prevents the frequency control from unlocking without modifying the frequency properties of the source. The laser typically stays locked for up to three weeks.
Fibre Power Splitters
---------------------
The optical bench and the vacuum chamber are not rigidly connected to each other, and laser light is transported to the vacuum chamber using optical fibres. Stability in trapping and coherent atom manipulation is assured by using only polarisation maintaining fibres. Six trapping and cooling laser beams are needed for the 3D-MOT and five for the 2D-MOT, with relative power stability better than a few percent. We have developed fibre beam-splitters based on polarising cubes and half-wave plates with one input fibre and the relevant number of output fibres. The stability of the beam splitters has been tested by measuring the ratio of output powers between different outputs as a function of time. Fluctuations are negligible on short time scales (less than $10^{-4}$ relative intensity over 1s), and very small over typical periods of experimental operation (less than 1% over a day). Even over months, drifts in power distribution are only a few percent, which is sufficient for this experiment.
Mechanical and Vacuum Systems
=============================
The mechanical construction of the apparatus is critical to any free-fall experiment. Atomic-physics experiments require heavy vacuum systems and carefully aligned optics. Our design is based around a cuboidal frame of foam-damped hollow bars with one face being a vibration-damped optical breadboard: see Figs. \[fig:chamber model\] and \[fig:chamber photo\]. The outside dimensions are 1.2m $\times$ 0.9m $\times$ 0.9m, and the total weight of the final system is estimated to be 400kg (excluding power supplies, lasers, control electronics, air and water flow). The frame provides support for the vacuum system and optics, which are positioned independently of one another. The heavy parts of the vacuum system are rigged to the frame using steel chains and high-performance polymer slings under tension, adjusted using turnbuckles; most of the equipment being standard in recreational sailing or climbing. The hollow bars have precisely positioned grooves which permit optical elements to be rigidly fixed (bolted and glued) almost anywhere in the volume within the frame. An adaptation for transportability will be to enclose the frame in a box, including acoustic and magnetic shielding, temperature control, air overpressure (dust exclusion), as well as ensuring safety in the presence of the high-power lasers.
The vacuum chamber has three main parts: the collection chamber (for the 2D-MOT), the trapping chamber (for the 3D-MOT and the FORT) and the pumps (combined ion pump and titanium sublimation pump) Between the collection and trapping chambers there is an orifice and a getter pump, allowing for a high differential pressure, permitting rapid collection by the 2D-MOT but low trap losses in the 3D-MOT and FORT. The magnetic coils for the 2D-MOT are under vacuum, and consume just 5W of electrical power.
The main chamber has two very large viewports as well as seven side windows (and one entry for the atoms from the 2D-MOT). Thus there is plenty of optical access for the 3D-MOT, the FORT, imaging and interferometry. To preserve this optical access, the magnetic coils are outside of the chamber, although this markedly increases their weight and power consumption.
To avoid heating due to vibrations in the FORT optics, or measurement uncertainties due to vibrations of the imaging system, the trapping chamber is as close to the breadboard as possible. For laboratory tests, the breadboard is lowest, and the 2D-MOT arrives at 45$^\circ$ to the vertical, leaving the vertical axis available for addition of interferometry for precise measurements, e.g a standing light wave. Around the main chamber, large electromagnet coils in Helmholtz-configuration will be added, to produce homogeneous, stable fields up to 0.12T (1200G), or gradients up to 0.6T/m (60G/cm).
2D-MOT
------
The 2D-MOT is becoming a common source of cold-atoms in two-chamber atomic-physics experiments[@Dieckmann98], and is particularly efficient for mixtures [@Ospelkaus06] of $^{40}$K and $^{87}$Rb, if isotopically enriched dispensers are used. Briefly, a 2D-MOT has four sets of beams (two mutually orthogonal, counter-propagating pairs) transverse to the axis of the output jet of atoms, and a cylindrical-quadrupole magnetic field generated by elongated electromagnet pairs (one pair, or two orthogonal pairs). Atoms are cooled transverse to the axis, as well as collimated. Implicitly, only slow atoms spend enough time in the 2D-MOT to be collimated, so the output jet is longitudinally slow. The number of atoms in the jet can be increased by the addition of the push beam, running parallel to the jet: a 2D-MOT$^+$. Typically the output jet has a mean velocity below 30ms$^{-1}$, with up to 10$^{10}$ at.s$^{-1}$ of $^{87}$Rb and 10$^{8}$at.s$^{-1}$ of $^{40}$K.
Our design uses 40mW per species for each of the four transverse beams, each divided into two zones of about 20mm using non-polarising beam-splitter cubes, corresponding to about three times the saturation intensity for the trapping transitions. The push beam uses 10mW of power, and is about 6mm in diameter. Each beam comes from an individual polarisation-maintaining optical fibre, with the light at 766.5nm and 780nm being superimposed on entry to the fibres. The 2D-MOT is seen as two bright lines of fluorescence in the collection chamber.
At the time of writing we do not have much quantitative data for the performance of our $^{87}$Rb 2D-MOT. One interesting test we have performed is spectroscopy of the confined cloud, using a narrow probe beam parallel to the desired output jet (replacing the push beam): see Fig \[fig:2D-MOT spectrum\]. We detect a significant number of atoms in the 2D-MOT with velocities at or below 20ms$^{-1}$ (the output jet should have a similar velocity distribution). More sensitive spectroscopy is difficult, since the probe beam must be smaller than the transverse dimension of the atom cloud (less than 0.5mm) and much less than saturation intensity (1.6mWcm$^{-2}$), so as not to excessively perturb the atoms. We used a lock-in detection (modulation-demodulation-integration) method, averaging over many spectra. A saturated-absorption spectroscopy signal was used for calibration. We have not yet tested a $^{39}$K or $^{40}$K 2D-MOT.
3D-MOT and Optical-Dipole Trap
------------------------------
The atomic jet from the 2D-MOT is captured by the 3D-MOT in the trapping chamber. At the time of writing, we have observed the transfer and capture of atoms, significantly increased by the addition of the push beam[@MOT_performance]. The 3D-MOT uses one polarisation-maintaining fibre input per species. Beams are superimposed and split into 6 arms (on a small optical breadboard fixed near one face of the frame) for the three, orthogonal, counter-propagating beam pairs. Once enough number of atoms are collected in the 3D-MOT, the 2D-MOT is to be turned off, and the 3D-MOT optimised for transfer to the FORT.
The FORT will consist of two, nearly-orthogonal ($70^\circ$) beams making a crossed, dipole trap using 50W of light at 1565nm. We will have rapid control over intensity using an electro-optical modulator, and beam size using a mechanical zoom, after the design of Kinoshita et al.[@Kinoshita05]. Optimisation of transfer from the 3D-MOT to the FORT, and the subsequent evaporative cooling will require experiments. Strong, homogeneous, magnetic fields will be used to control interspecies interactions via Feshbach resonances[@LENS_Feshbach], to expedite sympathetic cooling of $^{40}$K by $^{87}$Rb.
We can expect to load the 3D-MOT during less than 5s, then cool to degeneracy in the optical-dipole trap in around 3–10s. Thus we will be able to prepare a sample for interferometry in less than the free-fall time of a parabolic flight (around 20s).
Performance
===========
Coherent Raman-pulse Interferometer
-----------------------------------
The acceleration measurement is based on an atomic interferometer using light pulses as beam splitters[@Chu91; @Borde91], a technique which has demonstrated best performance for atomic inertial sensors. Three Raman pulses ($\pi/2 - \pi - \pi/2$) to generate respectively the beam splitter, the mirror and the beam re-combiner of the atomic interferometer. Two counter-propagating lasers (Raman lasers) drive coherent transitions between the two hyperfine ground states of the alkaline atoms. Two partial wave-packets are created with differing momenta, due to absorption and stimulated emission of photons in the Raman lasers. The differences in momenta correspond to velocity differences of 1.2cms$^{-1}$ for $^{87}$Rb and 2.6cms$^{-1}$ for $^{40}$K for Raman lasers tuned close to the D2 lines. Finally, fluorescence detection gives a measurement of the transition probability from one hyperfine level to the other, given by $P=\frac{1}{2}(1-\cos(\Phi))$, where $\Phi$ being the interferometric phase difference. It can be shown[@Antoine03] that the interferometric phase difference depends only on the difference of phase between the Raman lasers at the classical position of the centre of the atomic wave-packets at the time of the pulses. In the case of an experiment in free fall, with no initial velocity of the atoms, the interferometric phase depends only on the average relative acceleration of the experimental apparatus with respect to the centre of mass of the free-falling atoms, taken along the direction of propagation of the Raman lasers. We neglect here the effects gradients of gravity on expanding and separating wave-packets, which cause small changes to the final fringe visibility.
As the measurement is performed in time domain with pulses of finite duration $\tau_R-2\tau_R-\tau_R$ separated by a free evolution time $T$, it is also sensitive to fluctuations of the relative phase of the Raman lasers between pulses. Moreover, as the measurement is not continuous but has dead time, the sensitivity of the interferometer is limited by an aliasing effect similar to the Dick effect in atomic clocks[@Dick87]. Thus, the sensitivity of the interferometer also depends on vibrations and on the phase noise on the beat note between the Raman lasers at multiples of the cycling frequency $T_c$. The effects of these noise sources is calculated[@Cheinet06] using the sensitivity function which gives the influence of the fluctuations of the Raman phase on the transition probability, and thus on the interferometric phase.
Influence of Phase Noise
------------------------
The sensitivity of the interferometer can be characterised by the Allan variance of the interferometric phase fluctuations, $\sigma^{2}(\tau)$, defined by: $$\begin{aligned}
\sigma_{\Phi}^{2}(\tau)&=&\frac{1}{2}\langle(\bar{\delta \Phi}_{k+1}-\bar{\delta \Phi}_{k})^{2}\rangle \nonumber\\
&=&\frac{1}{2}\lim_{n\rightarrow \infty}\left\{
\frac{1}{n}\sum_{k=1}^{n}(\bar{\delta \Phi}_{k+1}-\bar{\delta \Phi}_{k})^{2}\right\}
\label{eq:variance_allan}\end{aligned}$$ where $\delta \Phi$ is the fluctuation of the phase measured at the output of the interferometer, $\bar{\delta \Phi}_{k}$ is the average value of $\delta \Phi$ over the interval from $t_{k}$ to $t_{k+1}$ (of duration $\tau$). For an interferometer operated sequentially at a rate $f_c=1/T_{\rm{c}}$, $\tau$ is a multiple of $T_c$, $\tau=m T_c$.
When evaluating the stability of the interferometric phase $\Phi$, one should take into account the fact that the measurement is pulsed. The sensitivity of the interferometer is limited only by the phase noise at multiples of the cycling frequency weighted by the Fourier components of the transfer function. For large averaging times ($\tau \gg T_C$), the Allan variance of the interferometric phase is given by $$\label{eq:dick} \sigma^{2}_{\Phi}(\tau)={\frac{1}{\tau}}\sum_{n=1}^{\infty}|H(2\pi n f_{\rm{c}})|^2
S_{\phi}({2\pi n f_{\rm{c}}})$$ where $S_{\phi}$ is the spectral power density of the phase difference between the Raman lasers.
Assuming square Raman pulses, the transfer function $H(f)$ of the Raman laser phase fluctuations to the interferometric phase is [@Cheinet06]:
$$\left|H(f) \right|^2 =
\left|
-\frac{4 \Omega \omega}{\omega^2-\Omega^2}\sin \left(\omega \frac{T+2\tau_R}{2}\right)
\left(\sin\left(\omega \frac{T+2\tau_R}{2}\right) +
\frac{\Omega}{\omega}\sin\left(\omega\frac{T}{2}\right)\right)
\right|^{2} \label{eq:Transfunct}$$
6 pt where $\omega = 2\pi f$ and $\Omega$ is the Rabi oscillation frequency, taken in such way that the Raman $\pi$-pulses have the ideal transfer efficiency: $\Omega = \pi/2\tau_R$. The transfer function is characterised by zeroes at multiples of 1/(T+2$\tau_R$) and decreases as 1/$\Omega^2$ for frequencies higher than the Rabi frequency, as illustrated in Figure \[transfer\].
For white phase noise $S_{\phi}^0$, and to first order in $\tau_R/T$, the phase stability is given by: $$\label{whiteeq1} \sigma^{2}_{\Phi}(\tau)=\frac{\pi \Omega}{2}
S_{\phi}^0\frac{T_c}{\tau}.$$ Thus the transfer function filters such noise for frequencies greater than the Rabi frequency: the shorter the pulse duration $\tau_R$, and thus the greater the Rabi frequency, the greater the interferometer noise. However, longer-duration pulses interact with fewer atoms (smaller velocity distributions) leading to an pulse duration, around $10\mu$s. More quantitatively, a desired standard deviation of interferometer phase below 1 mrad per shot, with pulse duration $\tau_R=10\mu$s, demands white phase noise of $4\times10^{-12}$ rad$^2$/Hz or less.
Generation of a Stable Microwave Source for Atom Interferometry
---------------------------------------------------------------
### The 100 MHz Source Oscillator:
The frequency difference between the Raman beams needs to be locked to a very stable microwave oscillator, whose frequency is close to the hyperfine transition frequency, $f_{MW}=6.834$ GHz for $^{87}$Rb, and $1.286$ GHz for $^{40}$K. The reference frequency will be delivered by a frequency chain, which transposes an RF source (typically a quartz oscillator) into the microwave domain, retaining the low level of phase noise. With degradation-free transposition the phase noise power spectral density of the RF oscillator, of frequency $f_{RF}$, is multiplied by $(f_{MW}/f_{RF})^2$.
No single quartz oscillator fulfills the requirements of very low phase noise over a sufficiently large frequency range. We present in figure \[fig:specquartz\] the specifications of different high stability quartz oscillators: a Premium 10 MHz-SC from Wenzel, a BVA OCXO 8607-L from Oscilloquartz, and a Premium 100 MHz-SC quartz from Wenzel. The phase noise spectral density is shown as transposed to 100 MHz, for fair comparison of the different oscillators.
The 100 MHz source we plan to develop for the ICE project will be a combination of two phase-locked quartz oscillators: one at 100 MHz locked onto one of the above-mentioned high-stability 10 MHz reference oscillators. The bandwidth of the lock corresponds to the frequency below which the phase noise of the reference oscillator is lower than the noise of the 100 MHz oscillator.
The phase noise properties of such a combined source can be seen in Figure \[fig:specquartz\], where we also show (solid line) the performance of the 100 MHz source developed by THALES for the PHARAO space clock project. This combined source has been optimized for mimimal phase noise at low frequency, where it reaches a level of noise lower than any commercially available quartz oscillator. An atomic clock is indeed mostly limited by low-frequency noise, so the requirements on the level of phase noise at higher frequency ($f>1$kHz) are less stringent than for an atom interferometer. A medium performance 100 MHz oscillator is thus sufficient.
Using a simple model for the phase-lock loop, we calculated the phase noise spectral power density of the different combined sources we can make by locking the Premium 100 MHz-SC either to the Premium 10 MHz-SC (Source 1), or the BVA (source 2), or even the PHARAO source (source 3). We then estimated the impact on the interferometer of the phase noise of the 100 MHz source, assuming we are able to transpose the performance of the source at 6.8 GHz without degradation. The results presented in Table \[tb:noise\] were calculated using Equation \[eq:dick\] for the Allan standard deviation of the interferometric phase fluctuations for the different configurations and various interferometer parameters.
--------------- -------------- -------------------------------- -------------------------------- ------------------------- ----------------------------- ---------------------------------- --
Source 1 Source 2 Source 3 Best Source Best Source
[**$T_c$**]{} [**$2T$**]{} [**$\sigma_{\Phi}$($T_c$)**]{} [**$\sigma_{\Phi}$($T_c$)**]{} [**$\sigma_{\Phi}$**]{} [**$\sigma_a$($T_c$)**]{} [**$\sigma_a$(1s)** ]{}
[**(s)**]{} [**(s)**]{} [**(mrad)**]{} [**(mrad)**]{} [**(mrad)**]{} [**(m.s$^{-2}$) / shot**]{} [**(m.s$^{-2}$.Hz$^{-1/2}$)**]{}
0.25 0.1 1.2 3.5 2.2 3x10$^{-8}$ 1.5x10$^{-8}$
10 2 22 8.8 4.6 1.1x10$^{-9}$ 3.6x10$^{-9}$
10 5 55 20 10 9.9x10$^{-11}$ 3.1x10$^{-10}$
15 10 110 37 19 4.7x10$^{-11}$ 1.8x10$^{-10}$
--------------- -------------- -------------------------------- -------------------------------- ------------------------- ----------------------------- ---------------------------------- --
For short interrogation times, such as $2T=100$ ms (the maximum interrogation time possible when the experiment is tested on the ground), Source 1 is best, whereas for long interrogation times, where the major contribution to the noise comes from the lowest frequencies (0.1–10 Hz), Sources 2 and 3 are better.
We are currently using a source based on the design of Source 1 for the gravimeter experiment at SYRTE [@Cheinet01]. Its performance is about 10% better than predicted, as the reference oscillator phase noise level is lower than the specifications. Considering that the interferometer is intended for a zero-g environment, we plan to build a source based on Source 2.
We have assumed here that for any source, the phase noise below 1 Hz is accurately described as flicker noise, for which the spectral density scales as $S_{\phi}(f)= S_{\phi}(1\rm{Hz})/\it{f}^3$. If the phase noise behaves as pure flicker noise over the whole frequency spectrum, the Allan standard deviation of the interferometer phase scales as $T$. We note that the observer behaviour of the gravimeter is consistent with Table \[tb:noise\].
The sensitivity of the accelerometer improves with the square of the interrogation time, $T^2$. For example, for $2T=10$s and $T_c=15$ s, the phase noise of Source 3 would limit the acceleration sensitivity of the interferometer to $4.6\times10^{-11}\rm{ms}^{-2}$ per shot for $^{87}$Rb. As the hyperfine splitting of $^{40}$K is five times smaller, the transposed phase noise is lower, and the measurement limit with $^{40}K$ decreases to $8.7\times10^{-12}\rm{ms}^{-2}$ per shot.
### The Frequency Chain:
The microwave signal is generated by multiplication of the 100 MHz source. We have developed a synthesis chain whose principle is shown in figure \[fig:setupchain\].
The source is first frequency doubled, the 200 MHz output is filtered, amplified to 27 dBm, and sent to a Step Recovery Diode (SRD), which generates a comb of frequencies, at multiples of 200 MHz. An isolator is placed after the SRD in order to prevent back reflections to damage the SRD. The 35th harmonic (7 GHz) is then filtered (passed) using a passive filter. A dielectric resonator oscillator (DRO) is then phase locked onto the 7 GHz harmonic, with an adjustable offset frequency provided by direct digital synthesis. A tunable microwave source is thus generated which copies the phase noise of the 7 GHz tooth of the comb, within the bandwidth of the DRO phase-lock loop (about 500 kHz). The noise added by the frequency chain was measured by mixing the outputs of two identical chains, with a common 100 MHz source; this noise is weaker than the noise due to the 100 MHz source.
The derived contribution to the phase noise of a $^{87}$Rb interferometer is 0.6 mrad per shot for $\tau_R=10 \mu$s, $2T=10$s and $T_c=15$s. The sensitivity limit due to the frequency synthesis is almost negligible for the $^{40}K$. In conclusion, the limit to sensitivity comes predominantly from the phase noise of the low frequency oscillator. This contribution could be further reduced by the use of cryogenic sapphire oscillator [@Mann01].
Zero-Gravity Operation
----------------------
In this section, we estimate the possible limitations of the interferometer when used in a parabolic flight, by calculating the effect of residual acceleration in the Airbus (the proposed test vehicle for this experiment) during a parabola. During a typical flight the residual acceleration can be of the order of 0.1ms$^{-2}$, with fluctuations of acceleration of the same order (Fig. \[airacc\]).
To determine the influence of environmental noise on the acceleration measurement, one uses the transfer function $H(f)$ for the phase (Equation \[eq:Transfunct\]). Phase noise is equivalent to position noise, since the phase of the Raman beams is $\Delta\phi = k_L\delta z$, where $k_L$ is the wave-vector of the laser, $\Delta z$ the position difference along the laser path, and position is the second integral of acceleration over time. The variance of the fluctuation of the phase shift at the output of the interferometer is: $$\sigma^2= \langle\left|\delta\left(\Delta\phi\right)\right|^2\rangle = k_L^2 \int_0^{\infty}S_a (f)\left|H(f)\right|^2/\omega^4 \,df
\label{var}$$ where $S_a (f)$ is the acceleration-noise power density which corresponds to the Fourier transform of the temporal fluctuation.
From the residual acceleration curves for the Airbus, one can deduce the acceleration noise power in a bandwidth from 0.05 to 10Hz, giving an estimation of the noise on the measurement of acceleration. A spectral acceleration-noise power density curve for the useful low-noise part of a parabola is shown in Figure \[spec2\], and is converted to interferometric phase noise power spectral density by multiplication by $k_L^2 |H|^2/ \omega^4$.
The vibration noise results in a substantial residual phase noise which is incompatible with the operation of the accelerometer. Calculating the variance of the fluctuations from Equation \[var\], one obtains a variance $\sigma_{\Phi}\sim 10^7$rad, which corresponds to acceleration noise $\sigma_{a}\sim 1$ms$^{-2}$ where $\sigma_{\Phi}=k_L \sigma_{\rm a}T^2$ with $T=1$s. Thus a vibration isolation system will be required, reducing the noise by 60–80dB around 0.5 Hz, about 40dB at 50 Hz and less than 10dB beyond 1kHz. The situation can be more favourable if one restricts the measurements in the middle of the parabola, as indicated on the Figure \[airacc\].
Conclusions
===========
We have shown our design for a transportable atom interferometer for parabolic flights in an Airbus. The device is built in two main parts, the laser systems and the atomic physics chamber. We have made major technical advances: high-stability frequency synthesis for coherent atom manipulation, flight-compatible laser sources and fibre power splitters, as well as a rugged atomic-physics chamber.
We have analysed the possibility of using this device in the micro-gravity environment of a parabolic flight, as a high-precision accelerometer, taking advantage of the long interrogation times available to increase the sensitivity to accelerations. We conclude that the limits to measurement under such conditions come from acceleration fluctuations and from phase noise in the frequency synthesis, and thus both aspects are to be minimised. Sensitivity of better than $10^{-9}\,\rm{m}\,\rm{s}^{-2}$ per shot is predicted. Comparisons of acceleration measurements made using two different atomic species (K and Rb) are possible.
The I.C.E. collaboration is funded by the CNES, as is RAN’s fellowship. Further support comes from the European Union STREP consortium FINAQS.
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|
---
abstract: |
The decays of $J/\psi\to \eta\phi f_0(980)~(\eta\to \gamma\gamma,
\phi \to K^+K^-, f_0(980)\to\pi^+\pi^-)$ are analyzed using a sample of $5.8 \times 10^{7}$ $J/\psi$ events collected with the BESII detector at the Beijing Electron-Positron Collider (BEPC). A structure at around $2.18~$GeV/$c^2$ with about $5\sigma$ significance is observed in the $\phi f_0(980)$ invariant mass spectrum. A fit with a Breit-Wigner function gives the peak mass and width of $m=2.186\pm 0.010~(stat)\pm 0.006~(syst)~$GeV/$c^2$ and $\Gamma=0.065\pm 0.023~(stat)\pm 0.017~(syst)~$GeV/$c^2$, respectively, which are consistent with those of $Y(2175)$, observed by the BaBar collaboration in the initial-state radiation (ISR) process $e^+e^-\to\gamma_{ISR}\phi f_0(980)$. The production branching ratio is determined to be $Br(J/\psi\to\eta
Y(2175))\cdot Br(Y(2175)\to\phi f_0(980))\cdot
Br(f_0(980)\to\pi^+\pi^-)=(3.23\pm 0.75~(stat)\pm0.73~(syst))\times
10^{-4}$, assuming that the $Y(2175)$ is a $1^{--}$ state.
author:
- |
M. Ablikim$^{1}$, J. Z. Bai$^{1}$, Y. Bai$^{1}$, Y. Ban$^{11}$, X. Cai$^{1}$, H. F. Chen$^{16}$, H. S. Chen$^{1}$, H. X. Chen$^{1}$, J. C. Chen$^{1}$, Jin Chen$^{1}$, X. D. Chen$^{5}$, Y. B. Chen$^{1}$, Y. P. Chu$^{1}$, Y. S. Dai$^{18}$, Z. Y. Deng$^{1}$, S. X. Du$^{1}$, J. Fang$^{1}$, C. D. Fu$^{14}$, C. S. Gao$^{1}$, Y. N. Gao$^{14}$, S. D. Gu$^{1}$, Y. T. Gu$^{4}$, Y. N. Guo$^{1}$, Z. J. Guo$^{15}$$^{a}$, F. A. Harris$^{15}$, K. L. He$^{1}$, M. He$^{12}$, Y. K. Heng$^{1}$, J. Hou$^{10}$, H. M. Hu$^{1}$, T. Hu$^{1}$, G. S. Huang$^{1}$$^{b}$, X. T. Huang$^{12}$, Y. P. Huang$^{1}$, X. B. Ji$^{1}$, X. S. Jiang$^{1}$, J. B. Jiao$^{12}$, D. P. Jin$^{1}$, S. Jin$^{1}$, Y. F. Lai$^{1}$, H. B. Li$^{1}$, J. Li$^{1}$, R. Y. Li$^{1}$, W. D. Li$^{1}$, W. G. Li$^{1}$, X. L. Li$^{1}$, X. N. Li$^{1}$, X. Q. Li$^{10}$, Y. F. Liang$^{13}$, H. B. Liao$^{1}$$^{c}$, B. J. Liu$^{1}$, C. X. Liu$^{1}$, Fang Liu$^{1}$, Feng Liu$^{6}$, H. H. Liu$^{1}$$^{d}$, H. M. Liu$^{1}$, J. B. Liu$^{1}$$^{e}$, J. P. Liu$^{17}$, H. B. Liu$^{4}$, J. Liu$^{1}$, Q. Liu$^{15}$, R. G. Liu$^{1}$, S. Liu$^{8}$, Z. A. Liu$^{1}$, F. Lu$^{1}$, G. R. Lu$^{5}$, J. G. Lu$^{1}$, C. L. Luo$^{9}$, F. C. Ma$^{8}$, H. L. Ma$^{2}$, L. L. Ma$^{1}$$^{f}$, Q. M. Ma$^{1}$, M. Q. A. Malik$^{1}$, Z. P. Mao$^{1}$, X. H. Mo$^{1}$, J. Nie$^{1}$, S. L. Olsen$^{15}$, R. G. Ping$^{1}$, N. D. Qi$^{1}$, H. Qin$^{1}$, J. F. Qiu$^{1}$, G. Rong$^{1}$, X. D. Ruan$^{4}$, L. Y. Shan$^{1}$, L. Shang$^{1}$, C. P. Shen$^{15}$, D. L. Shen$^{1}$, X. Y. Shen$^{1}$, H. Y. Sheng$^{1}$, H. S. Sun$^{1}$, S. S. Sun$^{1}$, Y. Z. Sun$^{1}$, Z. J. Sun$^{1}$, X. Tang$^{1}$, J. P. Tian$^{14}$, G. L. Tong$^{1}$, G. S. Varner$^{15}$, X. Wan$^{1}$, L. Wang$^{1}$, L. L. Wang$^{1}$, L. S. Wang$^{1}$, P. Wang$^{1}$, P. L. Wang$^{1}$, W. F. Wang$^{1}$$^{g}$, Y. F. Wang$^{1}$, Z. Wang$^{1}$, Z. Y. Wang$^{1}$, C. L. Wei$^{1}$, D. H. Wei$^{3}$, Y. Weng$^{1}$, N. Wu$^{1}$, X. M. Xia$^{1}$, X. X. Xie$^{1}$, G. F. Xu$^{1}$, X. P. Xu$^{6}$, Y. Xu$^{10}$, M. L. Yan$^{16}$, H. X. Yang$^{1}$, M. Yang$^{1}$, Y. X. Yang$^{3}$, M. H. Ye$^{2}$, Y. X. Ye$^{16}$, C. X. Yu$^{10}$, G. W. Yu$^{1}$, C. Z. Yuan$^{1}$, Y. Yuan$^{1}$, S. L. Zang$^{1}$$^{h}$, Y. Zeng$^{7}$, B. X. Zhang$^{1}$, B. Y. Zhang$^{1}$, C. C. Zhang$^{1}$, D. H. Zhang$^{1}$, H. Q. Zhang$^{1}$, H. Y. Zhang$^{1}$, J. W. Zhang$^{1}$, J. Y. Zhang$^{1}$, X. Y. Zhang$^{12}$, Y. Y. Zhang$^{13}$, Z. X. Zhang$^{11}$, Z. P. Zhang$^{16}$, D. X. Zhao$^{1}$, J. W. Zhao$^{1}$, M. G. Zhao$^{1}$, P. P. Zhao$^{1}$, Z. G. Zhao$^{1}$$^{i}$, H. Q. Zheng$^{11}$, J. P. Zheng$^{1}$, Z. P. Zheng$^{1}$, B. Zhong$^{9}$ L. Zhou$^{1}$, K. J. Zhu$^{1}$, Q. M. Zhu$^{1}$, X. W. Zhu$^{1}$, Y. C. Zhu$^{1}$, Y. S. Zhu$^{1}$, Z. A. Zhu$^{1}$, Z. L. Zhu$^{3}$, B. A. Zhuang$^{1}$, B. S. Zou$^{1}$\
(BES Collaboration)\
date: 'Nov.28, 2007'
title: 'Observation of the $Y(2175)$ in $J/\psi\rightarrow \eta\phi f_0(980)$'
---
A new structure, denoted as $Y(2175)$ and with mass $m=2.175\pm0.010\pm0.015$ GeV/$c^2$ and width $\Gamma=58\pm16\pm20$ MeV/$c^2$, was observed by the BaBar experiment in the $e^+e^-\to\gamma_{ISR}\phi f_0(980)$ initial-state radiation (ISR) process [@babary21752006; @babary21752007]. This observation stimulated some theoretical speculation that this $J^{PC}=1^{--}$ state may be an $s$-quark version of the $Y(4260)$ since both of them are produced in $e^+e^-$ annihilation and exhibit similar decay patterns [@babar4260]. There have been a number of different interpretations proposed for the $Y(4260)$, including: a $c \bar cg$ hybrid [@y4260hybrid1; @y4260hybrid2; @y4260hybrid3]; a $4^3S_1$ $c\bar{c}$ state [@y4260-4s]; a $[cs]_S [\bar{c}\bar{s}]_S$ tetraquark state [@y4260-4quark]; or baryonium [@baryonium]. Likewise a $Y(2175)$ has correspondingly been interpreted as: a $s \bar s g$ hybrid [@hybrid]; a $2^3D_1~s\bar s$ state [@ssbar]; or a $s \bar ss\bar s$ tetraquark state [@tetraquark]. As of now, none of these interpretations have either been established or ruled out by experiment.\
In this letter we report the observation of the $Y(2175)$ in the decays of $J/\psi\to\eta\phi f_0(980)$, with $\eta\to
\gamma\gamma,~\phi \to K^+K^-,~f_0(980)\to\pi^+\pi^-$, using a sample of $5.8 \times 10^{7}$ $J/\psi$ events collected with the upgraded Beijing Spectrometer (BESII) detector at the Beijing Electron-Positron Collider (BEPC).\
BESII is a large solid-angle magnetic spectrometer that is described in detail in Ref. [@BESII]. Charged particle momenta are determined with a resolution of $\sigma_p/p=1.78\%\sqrt{1+p^2}$ in a 40-layer cylindrical drift chamber. Particle identification is accomplished using specific ionization ($dE/dx$) measurements in the main drift chamber (MDC) and time-of-flight (TOF) measurements in a barrel-like array of 48 scintillation counters. The $dE/dx$ resolution is $\sigma_{dE/dx}=8.0\%$, and the TOF resolution is $\sigma_{TOF}=180$ ps for Bhabha tracks. Outside of the time-of-flight counters is a 12-radiation-length barrel shower counter (BSC) comprised of gas tubes interleaved with lead sheets. The BSC measures the energies and directions of photons with resolutions of $\sigma_E/E\simeq 21\%/\sqrt{E(\mbox{GeV})}$, $\sigma_{\phi} = 7.9$ mrad, and $\sigma_{z}$ = 2.3 cm. The iron flux return of the magnet is instrumented with three double layers of counters that are used to identify muons.\
In this analysis, a GEANT3-based Monte Carlo (MC) package with detailed consideration of the detector performance is used. The consistency between data and MC has been validated using many high purity physics channels [@simbes]. For $J/\psi \to
\eta Y(2175)(Y(2175) \to \phi f_0(980), f_0(980) \to \pi^+\pi^-)$, a Monte-Carlo generator that assumes the $Y(2175)$ quantum numbers to be $J^{PC} =1^{--}$ and considers the angular distributions for $1^{--} \to 0^{-+} + 1^{--}$; $1^{--}\to 1^{--}+ 0^{++}$ is used to determine the detection efficiency.\
For a candidate event, we require four good charged tracks with zero net charge. A good charged track is one that can be well fitted to a helix within the polar angle region $|\cos \theta|<0.8$ and has a transverse momentum larger than $70$ MeV/$c$. For each charged track, the TOF and $dE/dx$ information are combined to form particle identification confidence levels for the $\pi$, $K$ and $p$ hypotheses; the particle type with the highest confidence level is assigned to each track. The four charged tracks are required to consist of an unambiguously identified $K^+ K^-\pi^+\pi^-$ combination. Candidate photons are required to have an energy deposited in the BSC that is greater than 60 MeV and to be isolated from charged tracks by more than $5^{\circ}$; at least two photons are required. A four-constraint (4C) energy-momentum conservation kinematic fit is performed to the $K^+ K^-\pi^+\pi^-\gamma\gamma$ hypothesis and the $\chi^{2}_{4C}$ is required to be less than 15. For events with more than two selected photons, the combination with the smallest $\chi^{2}$ is chosen. An $\eta$ signal is evident in the $\gamma\gamma$ invariant mass spectrum (Fig. \[draft-mass\](a)); $\eta\to\gamma\gamma$ candidates are defined as $\gamma$-pairs with $|M_{\gamma\gamma}-0.547|<0.037$ GeV/$c^2$. A $\phi$ signal is distinct in the $K^+K^-$ invariant mass spectrum (Fig. \[draft-mass\](b)), and for these candidates, we require $|m_{K^+K^-}-1.02|<0.019$GeV/$c^2$. In the $\pi^+\pi^-$ invariant mass spectrum, candidate $f_0(980)$ mesons are defined by $|m_{\pi^+\pi^-}-0.980|<0.060$GeV/$c^2$ (Fig. \[draft-mass\](c)). The $\phi f_0(980)$ invariant mass spectrum for the selected events is shown in Fig. \[draft-mass2\](a), where a clear enhancement is seen around $2.18~$ GeV/$c^2$.\
The Dalitz plot of $m^2_{\eta f_0(980)}$ versus $m^2_{\eta\phi}$ for the selected events is shown in Fig. \[draft-mass2\](b), where a diagonal band can be seen. This band corresponds to the structure observed around 2.18 GeV/$c^2$ in the $\phi f_0(980)$ invariant mass spectrum shown in Fig. \[draft-mass2\](a).\
![(a) The $\gamma\gamma$ invariant mass spectrum. (b) The $K^+K^-$ invariant mass spectrum. (c) The $\pi^+\pi^-$ invariant mass spectrum. The solid arrows in each plot show the cuts imposed for $\eta$, $\phi$ and $f_0$ selection. The dashed arrows show the sideband regions used to estimate background levels.[]{data-label="draft-mass"}](draft-mass-ch2.eps){width="7.5cm" height="14cm"}
![(a) The $\phi f_0(980)$ invariant mass spectrum. The open histogram is data and the shaded histogram is $J/\psi~\to~\eta\phi f_0(980)$ phase-space MC events (with arbitrary normalization). (b) The Dalitz plot of $m^2_{\eta f_0(980)}$ versus $m^2_{\eta\phi}$. The ellipse shows the resonance band in $\phi f_0(980)$ invariant mass spectrum. []{data-label="draft-mass2"}](draft-mass2.eps "fig:"){width="6cm" height="4cm"} ![(a) The $\phi f_0(980)$ invariant mass spectrum. The open histogram is data and the shaded histogram is $J/\psi~\to~\eta\phi f_0(980)$ phase-space MC events (with arbitrary normalization). (b) The Dalitz plot of $m^2_{\eta f_0(980)}$ versus $m^2_{\eta\phi}$. The ellipse shows the resonance band in $\phi f_0(980)$ invariant mass spectrum. []{data-label="draft-mass2"}](draft-daliz2.eps "fig:"){width="6cm" height="6cm"}
To clarify the origin of the observed structure, we have made extensive studies of potential background processes using both data and MC. Non-$\eta$ or non-$f_0(980)$ processes are studied with $\eta$-$f_0(980)$ mass sideband events (0.074 GeV/$c^2<|M_{\gamma\gamma}-0.547|<0.111$GeV/$c^2$ or 0.090 GeV/$c^2<|m_{\pi^+\pi^-}-0.980|<0.150$GeV/$c^2$). Non-$\phi$ processes are studied with $\phi$ mass sideband events (0.038 GeV/$c^2<(m_{K^+K^-}-1.02)<0.057$GeV/$c^2$ or -0.038 GeV/$c^2<(m_{K^+K^-}-1.02)<-0.019$GeV/$c^2$). The scaled $M_{\pi^+\pi^-K^+K^-}$ distribution for the summed total of sideband events (minus double counting) are shown as a shaded histogram in Fig. \[draft-sideband\]. No structure around $2.18~$GeV/$c^2$ is evident. In addition, we also checked for possible backgrounds from various $J/\psi$ decays using Monte-Carlo simulation, and no evidence of a structure at $2.18 ~$GeV/$c^2$ is observed.\
![The $\phi f_0(980)$ invariant mass spectrum. The open histogram is data and the shaded histogram shows the sideband-determined background.[]{data-label="draft-sideband"}](draft-sideband.eps){width="6.5cm" height="5cm"}
We fit the $\phi f_0(980)$ invariant mass spectrum (see Fig. \[draft-mass2\](a)) and the total sidebands (see Fig. \[draft-sideband\]) simultaneously. The procedure is as follows: First we fit the sideband distribution with a 3rd-order polynomial. Next we use the polynomial shape as the background function for both the $\phi f_0(980)$ invariant mass spectrum histogram and the total sideband histogram, and the signal and background normalizations are allowed to float. In this fit, the normalization for the background polynomial is constrained to be the same for both the signal and sideband histograms. We use a constant-width Breit-Wigner (BW) convolved with a Gaussian mass resolution function (with $\sigma$ = $12$MeV/$c^2$) to represent the $Y(2175)$ signal. The mass and width obtained from the fit (shown as smooth curves in Fig. \[draft-fit\]) are $m=2.186\pm
0.010~(stat)~$GeV/$c^2$ and $\Gamma=0.065\pm
0.023~(stat)~$GeV/$c^2$. The fit yields $52\pm12$ signal events and $-2lnL$ ($L$ is the likelihood value of the fit) $=~78.6$. A fit to the mass spectrum without a BW signal function returns $-2lnL~=~116.0$. The change in $-2lnL$ with a change of degrees of freedom $=$ 3 corresponds to a statistical significance of $5.5~\sigma$ for the signal.\
Using the MC-determined selection efficiency of $1.44\%$, we find the product branching ratio to be:
![The top panel shows the fit (solid curve) to the data (points with error bars); the dashed curve indicates the background function. The bottom panel shows the simultaneous fit to the sideband events (points with error bars) with the same background function. The background normalizations for the two plots are constrained to be equal.[]{data-label="draft-fit"}](draft-fit-v2.eps "fig:"){width="6cm" height="4cm"} ![The top panel shows the fit (solid curve) to the data (points with error bars); the dashed curve indicates the background function. The bottom panel shows the simultaneous fit to the sideband events (points with error bars) with the same background function. The background normalizations for the two plots are constrained to be equal.[]{data-label="draft-fit"}](draft-fit2-v2.eps "fig:"){width="6cm" height="4cm"}
$Br(J/\psi \to \eta Y(2175))\cdot Br(Y(2175)\to \phi
f_0(980))Br(f_0(980)\to \pi^+\pi^-)=(3.23\pm 0.75)\times
10^{-4}$.
Fits that use different treatments for the background are also tried. If the background is fitted as a 3rd-order polynomial with all parameters allowed to float, the signal yield is $61 \pm 14$ events, with mass and width of $m=2.182\pm
0.010~(stat)~$GeV/$c^2$ and $\Gamma=0.073\pm
0.024~(stat)~$GeV/$c^2$, respectively. The statistical significance is $4.9 ~\sigma$. If the background shape is fixed to the shape of phase space, the fit yields $57 \pm 13$ signal events, with a statistical significance of $5.3 ~\sigma$. The mass and width obtained are $m=2.182\pm 0.009~(stat)~$GeV/$c^2$ and $\Gamma=0.069\pm 0.022~(stat)~$GeV/$c^2$. For all of the background shapes considered, the fitted masses and widths of the signal are consistent with each other. We take the results with the background shape fixed to the sideband shape as the central values.
We determine the systematic uncertainties of the mass and width measurements by varying the functional form used to represent the background, the fitting range of the invariant mass spectrum, the bin width of the invariant mass spectrum, allowing the sideband and signal background normalizations to differ, and including possible fitting biases. The latter are estimated from the differences between the input and output mass and width values from a MC study. Adding each contribution in quadrature, the total systematic errors on the mass and width are 6 MeV/$c^2$ and 17 MeV/$c^2$, respectively. The systematic error on the branching ratio measurement comes mainly from the uncertainties in the MDC simulation (including systematic uncertainties of the tracking efficiency and the kinematic fits), the photon detection efficiency, the particle identification efficiency, the $\eta$ decay branching ratio to $\gamma\gamma$ and the $\phi$ decay branching ratio to $K^+K^-$, the background function, the fitting range of the invariant mass spectrum, the bin width of the invariant mass spectrum, the fitting method and the total number of $J/\psi $ events [@ssfang]. Adding all contributions in quadrature gives a total systematic error on the product branching ratio of $22.7\%$.\
We studied the small peak near 2.47 GeV/$c^2$ in the $\phi f_0(980)$ invariant mass spectrum (see Fig. \[draft-mass2\](a)), which was also noted by BaBar [@babary21752007]. A fit was made to the $\phi f_0(980)$ invariant mass spectrum using two non-interfering Breit-Wigner functions with mass and width of the second peak fixed to the BaBar fitted results: $2.47~$GeV/$c^2$ and $0.077~$GeV/$c^2$ [@babary21752007], respectively. The fit results indicate a significance for the first peak of $5.8 ~\sigma$, with a mass and width of $m=2.186\pm 0.010~(stat)~$GeV/$c^2$ and $\Gamma=0.065\pm 0.022~(stat)~$GeV/$c^2$, respectively. The statistical significance of the second peak is only $2.5~\sigma$.\
In summary, the $J/\psi \to \eta \phi f_0(980)$ decay process with $\eta \to \gamma \gamma$, $\phi \to K^+ K^-$, and $f_0(980) \to \pi^+\pi^-$ has been analyzed. A structure, the $Y(2175)$, is observed with about $5\sigma$ significance in the $\phi f_0(980)$ invariant mass spectrum. From a fit with a Breit-Wigner function, the mass is determined to be $M=2.186\pm 0.010~(stat)\pm
0.006~(syst)~$GeV/$c^2$ , the width is $\Gamma=0.065\pm
0.023~(stat)\pm 0.017~(syst)~$GeV/$c^2$ and the product branching ratio is $Br(J/\psi \to \eta Y(2175))\cdot Br(Y(2175)\to \phi
f_0(980))\cdot Br(f_0(980)\to\pi^+\pi^-)=(3.23\pm 0.75~(stat)\pm
0.73~(syst))\times 10^{-4}$. The mass and width are consistent with BaBar’s results. The identification of the precise nature of the $Y(2175)$ requires measurements of additional decay channels [@hybrid; @ssbar]. This is the subject of the work that is currently in progress.\
The BES collaboration thanks the staff of BEPC and computing center for their hard efforts. This work is supported in part by the National Natural Science Foundation of China under contracts Nos. 10491300, 10225524, 10225525, 10425523, 10625524, 10521003, the Chinese Academy of Sciences under contract No. KJ 95T-03, the 100 Talents Program of CAS under Contract Nos. U-11, U-24, U-25, and the Knowledge Innovation Project of CAS under Contract Nos. U-602, U-34 (IHEP), the National Natural Science Foundation of China under Contract No. 10225522 (Tsinghua University), and the Department of Energy under Contract No.DE-FG02-04ER41291 (U. Hawaii).
[99]{} BaBar Collaboration, B. Aubert [*et al.*]{}, Phys. Rev. D [**74**]{}, 091103(R) (2006). BaBar Collaboration, B. Aubert [*et al.*]{}, Phys. ReV. D [**76**]{}, 031102 (2007). BaBar Collaboration, B. Aubert [*et al.*]{}, Phys. Rev. Lett. [**95**]{}, 142001 (2005) S.L. Zhu, Phys. Lett. B [**625**]{}, 212 (2005). F.E. Close and P.R. Page, Phys. Lett. B [**628**]{}, 215 (2005). E. Kou and O. Pene, Phys. Lett. B [**631**]{}, 164 (2005). F.J. Llanes-Estrada, Phys. Rev. D [**72**]{}, 031503 (2005). L. Maiani, V. Riquer, F. Piccinini and A.D. Polosa, Phys. Rev. D [**72**]{}, 031502 (2005) C. F. Qiao, Phys. Lett. B [**639**]{} 263,(2006). Gui-Jun Ding, Mu-lin Yan, Phys. Lett. B [**650**]{}, 390-400 (2007). Gui-Jun Ding, Mu-lin Yan, hep-ph/0701047 Zhi-Gang Wang, Nucl. Phys. A [**791**]{}, 106-116 (2007). BES Collaboration, J.Z. Bai [*et al.*]{}, Nucl. Instr. Meth. A [**458**]{}, 627 (2001). BES Collaboration, M. Ablikim [*et al.*]{}, Nucl. Instr. Meth. A [**552**]{}, 344 (2005). S.S. Fang [*et al.*]{}, High Energy Physics and Nuclear Physics [**27**]{}, 277 (2003).
|
---
author:
- 'B. L. Alterman'
- 'Justin C. Kasper'
title: Helium Variation Across Two Solar Cycles
---
|
---
abstract: 'A family of implicit-in-time mixed finite element schemes is presented for the numerical approximation of the acoustic wave equation. The mixed space discretization is based on the displacement form of the wave equation and the time-stepping method employs a three-level one-parameter scheme. A rigorous stability analysis is presented based on energy estimation and sharp stability results are obtained. A convergence analysis is carried out and optimal a priori $L^\infty(L^2)$ error estimates for both displacement and pressure are derived.'
author:
- 'Samir Karaa[^1]'
title: 'A Priori Error Estimates for Mixed Finite Element $\theta$-Schemes for the Wave Equation[^2]'
---
[**Key words**]{} - [wave equation, mixed finite elements, error estimation, energy technique]{} [**Mathematical subject codes**]{} 65L05, 65M12, 65M60, 65M15
Introduction
============
The acoustic wave equation is used to model the effects of wave propagation in heterogeneous media. Solving this equation efficiently is of fundamental importance in many real-life problems. In geophysics, it helps for instance in the interpretation of the seismograph field data and to predict damage patterns due to earthquakes. Using finite element methods for its approximation is attractive because of the ability to handle complex discretizations and design adaptive grid refinement strategies based on error indicators.
Previous attempts on wave simulation by finite elements have used continuous Galerkin methods [@Baker-3; @Bao; @CJRT; @Dupont-15; @Marfurt; @Rauch], mixed finite element methods [@CDW-12; @CDW-13; @Geveci-19; @JRW; @Pani-2001; @Pani-2004; @VT-2008], and discontinuous Galerkin methods [@GS-2009; @Johnson; @RW-27; @RW-28]. In a mixed finite element formulation both displacements and stresses are approximated simultaneously. This approach provides higher-order approximations to the stresses. This property is important in many problems, in particular in modeling boundary controlability of the wave equation [@GKW-89]. One of the main difficulties of the mixed finite element techniques is the requirement of compatibility of the approximating spaces for convergence and stability.
Given a bounded convex polygonal domain $\Omega$ in $\mathbb{R}^m$, $m=2,\,3$, with boundary $\partial\Omega=\Gamma_D\cup\Gamma_N$, and unit outward normal ${\pmb \nu}$, the general form of the wave equation is $$\begin{aligned}
\rho{{\pmb u}}_{tt} +{\nabla\cdot}{\pmb{\tilde \tau}}&=&{{\pmb f}}\qquad \mbox{ in } \Omega\times
(0,T),\label{eq:o1}\\
{\nabla\cdot}{{\pmb u}}&=&0 \qquad \mbox{ on } \Gamma_D\times (0,T),\label{eq:o2}\\
{{\pmb u}}\cdot{\pmb \nu} &=&0 \qquad \mbox{ on } \Gamma_N\times (0,T),\label{eq:o3}\\
{{\pmb u}}(\cdot,0)&=&{{\pmb u}}^0 \qquad \mbox{ in } \Omega,\label{eq:o4}\\
{{\pmb u}}_t(\cdot,0)&=&{{\pmb v}}^0 \qquad \mbox{ in } \Omega,\label{eq:o5}\end{aligned}$$ where ${{\pmb u}}$ is the displacement, $\rho$ is the density, and ${\pmb{\tilde\tau}}$ is the stress tensor given by the generalized Hooke’s law ${\pmb{\tilde\tau}}=\lambda({\nabla\cdot}{{\pmb u}}){\pmb{\tilde I}}+
\mu(\nabla{{\pmb u}}+(\nabla{{\pmb u}})^T)$. Here $\lambda>0$ and $\mu$ are the Lamé coefficients characterizing the material. The function ${{\pmb f}}$ represents a general source term and ${{\pmb u}}^0$ and ${{\pmb v}}^0$ are initial conditions on displacements and velocities. We assume that ${{\pmb f}}$, ${{\pmb u}}^0$ and ${{\pmb v}}^0$ are smooth enough so that there is a unique solution ${{\pmb u}}\in{\cal C}^2((0,T)\times \Omega)$ to (\[eq:o1\])-(\[eq:o5\]), see [@Knops-Payne-1971].
The limiting case of (\[eq:o1\]) with $\mu=0$ is referred to as the acoustic wave equation, which is $$\label{eq:wave1}
\rho{{\pmb u}}_{tt} +{\nabla\cdot}(\lambda({\nabla\cdot}{{\pmb u}}){\pmb{\tilde I}})={{\pmb f}}.$$ It is assumed that $\rho$ and $\lambda$ are bounded below and above by the positive constants $\rho_0$, $\rho_1$, $\lambda_0$, and $\lambda_1$, respectively. This vector equation is equivalent to the scalar wave equation after making the substitution $p=\lambda{\nabla\cdot}·u$. The mixed method is established by using this relationship, leading to the coupled system $$\begin{aligned}
\rho{{\pmb u}}_{tt} -\nabla p&=&{{\pmb f}}\qquad \qquad \mbox{ in } \Omega\times
(0,T),\label{eq:oo1}\\
\lambda^{-1}p&=&{\nabla\cdot}{{\pmb u}}\qquad \mbox{ in } \Omega\times(0,T),\label{eq:oo2}\end{aligned}$$ with the appropriate boundary and initial conditions.
A priori error estimates for solving (\[eq:oo1\])-(\[eq:oo2\]) were obtained in [@CDW-12; @CDW-13; @Geveci-19; @JRW]. In [@Geveci-19], Geveci derived $L^\infty$-in-time, $L^2$-in-space error bounds for the continuous-in-time mixed finite element approximations of velocity and stress. In [@CDW-12; @CDW-13], a priori error estimates were obtained for the mixed finite element approximation of displacement which requires less regularity than was needed in [@Geveci-19]. Stability for a family of discrete-in-time schemes was also demontratred. In [@JRW], an alternative mixed finite element displacement formulation was proposed reducing requirement on the regularity on the displacement variable. For the explicit discrete-in-time problem, stability results were established and error estimates were obtained. The effectiveness of the method analyzed in [@JRW] was demonstrated in [@Jenkins] by providing simulations using both lowest-order and next-to-lowest-order Raviart–-Thomas elements on rectangles [@R16].
The purpose of this paper is to analyze an implicit time-stepping method combined with the mixed finite element discretization proposed in [@JRW]. We prove the stability of the proposed method by using energy estimation, and show in particular that it conserves certain energy. We also invertigate the convergence of the method and prove optimal a priori $L^\infty(L^2)$ error estimates for both displacement and pressure. The rest of the paper is organized as follows. In sections 2 and 3, we introduce notations and describe the weak formulation of the problem. The fully discrete mixed finite element method is presented in section 4. Stability results are established in section 4 and optimal a priori error estimates are obtained in section 5. Conclusions are given in the last section.
Notation {#sect:Notation}
========
We shall use the following inner products and norms in this paper. The $L^2$-inner product over $\Omega$ is defined by $$(u,v)=\int_\Omega uv\,d\Omega,$$ inducing the $L^2$-norm over $\Omega$, $||v||_{L^2(\Omega)}=(v,v)^{1/2}$. The inner product over the boundary $\partial \Omega$ is denoted by $$\langle u,v\rangle=\int_{\partial \Omega} uv\,d\Omega$$ for $u$, $v\in H^{{\frac{1}{2}}+\varepsilon}(\Omega)$ with $\varepsilon>0$. We introduce the time-space norm: $$||v||_{L^2(0,T;L^2(\Omega))}=||v||_{L^2(L^2)}=\left(\int_0^T||v||^2_{L^2(\Omega)}dt
\right)^{{\frac{1}{2}}}.$$ The time-space norm $||\cdot||_{L^\infty(L^2)}$ is similarly defined. In addition to the $L^2$ spaces, we use the standard Sobolev space for mixed methods: $${{\pmb H}}(\Omega,\mbox{div})=\{{{\pmb v}}:{{\pmb v}}\in(L^2(\Omega))^m,{\nabla\cdot}{{\pmb v}}\in L^2(\Omega)\},$$ with associated norm $$||{{\pmb v}}||_{{{\pmb H}}(\Omega,\mbox{div})}= ||{{\pmb v}}||_{L^2(\Omega)}+||{\nabla\cdot}{{\pmb v}}||_{L^2(\Omega)},$$ where $$||{{\pmb v}}||_{L^2(\Omega)}=\left(\sum_{i=1}^m||v_i||_{L^2(\Omega)}^2\right)^{\frac{1}{2}}.$$ For the time discretization, we adopt the following notation. Let $N$ be a positive integer, ${{\Delta t}}=T/N$, and $t^n=n{{\Delta t}}$. For any function $v$ of time, let $v^n$ denote $v(t^n)$. We shall use this notation for functions defined for all times as well as those defined only at discrete times. Set $$v^{n+{\frac{1}{2}}}={\frac{1}{2}}\left(v^{n+1}+v^n\right),$$ $${\bar{\partial}_t}v^{n+{\frac{1}{2}}}=\frac{1}{{{\Delta t}}}\left(v^{n+1}-v^n\right),$$ $${\bar{\partial}_t}v^{n}=\frac{1}{2{{\Delta t}}}\left(v^{n+1}-v^{n-1}\right),$$ $${\bar{\partial}_{tt}}v^{n}=\frac{1}{{{\Delta t}}^2}\left(v^{n+1}-2v^n+v^{n-1}\right),$$ $$v^{n;\theta}=\theta v^{n+1}+(1-2\theta)v^n+\theta v^{n-1},$$ where $0\leq\theta\leq 1$. We also define the discrete $l^\infty$-norm for time-discrete functions by $$||v||_{l^\infty_{{\Delta t}}(0,T;L^2(\Omega))}=||v||_{l^\infty(L^2)}=
\max_{0\leq n\leq N}||v^n||_{L^2(\Omega)}.$$
Weak Formulation
================
The finite element approximation of the wave problem is based on its weak formulation which is derived in the usual manner. Integrating by parts and using the data on the boundary of $\Omega$, we obtain the weak formulation [@JRW]: For any $t\geq 0$, find $({{\pmb u}}(t),p(t))\in {{\pmb V}}\times W$ such that $$\begin{aligned}
({{\pmb u}}(0),{{\pmb v}})&=&({{\pmb u}}^0,{{\pmb v}})\qquad \forall {{\pmb v}}\in {{\pmb V}},\label{eq:w1-a}\\
({{\pmb u}}_t(0),{{\pmb v}})&=&({{\pmb v}}^0,{{\pmb v}})\qquad \forall {{\pmb v}}\in {{\pmb V}},\label{eq:w1-b}\\
(\lambda^{-1}p(0),w)&=&(\nabla\cdot {{\pmb u}}^0,w)\qquad \forall w\in W,\label{eq:w1-c}\\
(\rho{{\pmb u}}_{tt}(t) ,{{\pmb v}})+(p(t),{\nabla\cdot}{{\pmb v}})&=&({{\pmb f}}(t),{{\pmb v}})\qquad \forall {{\pmb v}}\in {{\pmb V}},
\quad t>0,\label{eq:w1}\\
(\lambda^{-1}p(t),w)-({\nabla\cdot}{{\pmb u}}(t),w)&=&0\qquad \forall w\in W,\quad t>0, \label{eq:w2}\end{aligned}$$ where ${{\pmb V}}$ and $W$ are given by $${{\pmb V}}=\{{{\pmb v}}\in {{\pmb H}}(\Omega,\mbox{div})\,:\,{{\pmb v}}\cdot{\pmb\nu}|_{\Gamma_N}=0\},$$ $$W=H^{{\frac{1}{2}}+\varepsilon}(\Omega) \mbox{ for any } \varepsilon>0.$$ The present formulation requires less regularity on the displacement than standard approaches. For instance in [@CDW-12; @CDW-13] it is necessary that $\nabla p\in {{\pmb H}}(\Omega,\mbox{div})$ so that ${\nabla\cdot}{{\pmb u}}\in H^2(\Omega)$. Here, it is only required that ${\nabla\cdot}{{\pmb u}}\in H^{\frac{1}{2}}$, and it can be verified that the solution ${{\pmb u}}$ of problem (\[eq:o1\])-(\[eq:o5\]) with $p=\lambda{\nabla\cdot}{{\pmb u}}$ is a solution to (\[eq:w1\])-(\[eq:w2\]), see [@JRW].
Differentiate (\[eq:w2\]) with respect to time to obtain $$\label{eq:w3}
(\lambda^{-1}p_t,w)-({\nabla\cdot}{{\pmb u}}_t,w)=0\qquad \forall w\in W.$$ We next assume ${{\pmb f}}=0$ and choose ${{\pmb v}}={{\pmb u}}_t$ and $w=p$ in (\[eq:w1\]) and (\[eq:w3\]), respectively, so that $$\begin{aligned}
(\rho{{\pmb u}}_{tt} ,{{\pmb u}}_t)+(p,{\nabla\cdot}{{\pmb u}}_t)&=&0,\label{eq:w4}\\
(\lambda^{-1}p_t,p)-({\nabla\cdot}{{\pmb u}}_t,p)&=&0 \label{eq:w5}.\end{aligned}$$ By adding the two equations, we find that $$\label{eq:w6}
(\rho{{\pmb u}}_{tt} ,{{\pmb u}}_t)+(\lambda^{-1}p_t,p)=0,$$ or $${\frac{1}{2}}\frac{d}{dt}\left|\left|\rho^{\frac{1}{2}}{{\pmb u}}_{t} \right|\right|^2_{L^2(\Omega)}
+{\frac{1}{2}}\frac{d}{dt}\left|\left|{\lambda}^{-{\frac{1}{2}}} p \right|\right|^2_{L^2(\Omega)}=0.$$ Thus, in the absence of forcing, the (continuous) energy $$\label{eq:w7}
{\frac{1}{2}}\left|\left|\rho^{\frac{1}{2}}{{\pmb u}}_{t} \right|\right|^2_{L^2(\Omega)}
+{\frac{1}{2}}\left|\left|{\lambda}^{-{\frac{1}{2}}} p \right|\right|^2_{L^2(\Omega)}$$ is conserved for all time. It will be shown that a similar form of energy is conserved by the numerical solution of the wave problem.
Finite Element Approximation
============================
For the finite element approximation, we let $\{{\cal E}_h\}_{h>0}$ be a quasi-uniform family of finite element partitions of $\Omega$, where $h$ is the maximum element diameter. Let ${{\pmb V}}_h\times W_h$ be any of the usual mixed finite element approximating subspaces of ${{\pmb V}}\times W$, that is, the Raviart-Thomas-Nedelec spaces [@R15; @R16], Brezzi-Douglas-Marini spaces [@R5], or Brezzi-Douglas-Fortin-Marini spaces [@R4]. For each of these mixed spaces there is a projection $\Pi_h:{{\pmb H}}(\Omega,\mbox{div})\rightarrow {{\pmb V}}_h$ such that for any ${{\pmb z}}\in {{\pmb H}}(\Omega,\mbox{div})$ $$\label{eq:w8}
({\nabla\cdot}\Pi_h{{\pmb z}},w)=({\nabla\cdot}{{\pmb z}},w)\quad \forall w\in W_h.$$ We have the property that, if ${{\pmb z}}\in {{\pmb H}}(\Omega,\mbox{div})\cap {{\pmb H}}^k(\Omega)$, then $$\label{eq:w9}
||\Pi_h{{\pmb z}}-{{\pmb z}}||_0\leq C h^j||{{\pmb z}}||_j, \quad 1\leq j\leq k,$$ where $k$ is associated with the degree of polynomial and $||\cdot||_s$ is the standard Sobolev norm on $(H^s(\Omega))^m$. Here and in what follows, $C$ is a generic positive constant which is independent of $h$ and ${{\Delta t}}$.
For $\phi\in W$, we denote by ${\cal P}_h\phi$ the $L^2$-projection of $\phi$ onto $W_h$ defined by requiring that $$\label{eq:w10}
({\cal P}_h\phi,w)=(\phi,w)\quad\forall w\in W_h.$$ If $\phi\in W\cap H^k(\Omega)$, then we also have $$\label{eq:w11}
||{\cal P}_h\phi-\phi||_s\leq C h^{j-s}||\phi||_j, \quad 0\leq s\leq k,\quad 0\leq j\leq k.$$ The semidiscrete mixed finite element approximation to $({{\pmb u}}(t),p(t))$ is to seek $({{\pmb U}}(t),P(t))\in {{\pmb V}}_h\times W_h$ satisfying $$\begin{aligned}
({{\pmb U}}(0),{{\pmb v}})&=&(\Pi_h{{\pmb u}}^0,{{\pmb v}})\quad \forall {{\pmb v}}\in {{\pmb V}}_h,\label{eq:ww1}\\
({{\pmb U}}_t(0),{{\pmb v}})&=&(\Pi_h{{\pmb v}}^0,{{\pmb v}})\quad \forall {{\pmb v}}\in {{\pmb V}}_h,\label{eq:ww2}\\
(P(0),w)&=&(p(0),w)\quad \forall w\in W_h,\label{eq:ww3}\\
(\rho{{\pmb U}}_{tt}(t),{{\pmb v}})+(P(t),{\nabla\cdot}{{\pmb v}}) &=&({{\pmb f}}(t),{{\pmb v}})\quad \forall {{\pmb v}}\in
{{\pmb V}}_h,\quad t>0,\label{eq:ww4}\\
(\lambda^{-1}P(t),w)-({\nabla\cdot}{{\pmb U}}(t),w)&=&0\qquad\qquad \forall w\in W_h,\quad
t>0.\label{eq:ww5}\end{aligned}$$ Existence and uniqueness of a solution $({{\pmb U}}(t),P(t))$ to the variational problem (\[eq:ww1\])-(\[eq:ww5\]) is shown in [@JRW].
The fully discrete mixed finite element $\theta$-scheme is then defined by finding a sequence of pairs $({{\pmb U}}^{n},P^{n})\in {{\pmb V}}_h\times W_h$, $0\leq n\leq N$, such that $$\begin{aligned}
({{\pmb U}}^0,{{\pmb v}})&=&(\Pi_h {{\pmb u}}^0,{{\pmb v}})\quad \forall {{\pmb v}}\in {{\pmb V}}_h,\label{eq:www1}\\
(P^0,w)&=&(p^0,w)\quad \forall w\in W_h,\label{eq:www2}\\
\left(\rho{\bar{\partial}_t}{{\pmb U}}^{\frac{1}{2}},{{\pmb v}}\right)+\theta^2{{\Delta t}}\left({\bar{\partial}_t}P^{\frac{1}{2}},{\nabla\cdot}{{\pmb v}}\right)
+\frac{{{\Delta t}}}{2}(P^0,{\nabla\cdot}{{\pmb v}})&=&\left(\frac{{{\Delta t}}}{2}{{\pmb f}}^0+\theta{{\Delta t}}^2{\bar{\partial}_t}{{\pmb f}}^{\frac{1}{2}},{{\pmb v}}\right)\nonumber\\
&& +\left(\rho\Pi_h{{\pmb v}}^0,{{\pmb v}}\right)\quad \forall {{\pmb v}}\in
{{\pmb V}}_h, \label{eq:www3}\\
(\rho{\bar{\partial}_{tt}}{{\pmb U}}^n,{{\pmb v}})+(P^{n;\theta},{\nabla\cdot}{{\pmb v}}) &=&({{\pmb f}}^{n;\theta},{{\pmb v}})\quad \forall
{{\pmb v}}\in {{\pmb V}}_h,\label{eq:www4}\\
(\lambda^{-1}P^{n+1/2},w)-({\nabla\cdot}{{\pmb U}}^{n+1/2},w)&=&0\quad \forall w\in
W_h.\label{eq:www5}\end{aligned}$$ Equation (\[eq:www3\]) is derived from the following expansion: $${{\pmb u}}^{1}={{\pmb u}}^0+{{\Delta t}}{{\pmb v}}^0+{{\Delta t}}^2\left[\theta
{{\pmb u}}^{1}_{tt}+\left(\frac{1}{2}-\theta\right){{\pmb u}}^{0}_{tt}\right]+{\cal O}({{\Delta t}}^3).$$ The present $\theta$-scheme is explicit in time if $\theta=0$ and implicit otherwise. The existence and uniqueness of a solution to the resulting linear system for a nonzero value of $\theta$ follows from the unisolvancy of the mixed formulation of the following elliptic problem: $$\begin{aligned}
{\nabla\cdot}(\lambda\nabla \phi)+\frac{1}{\theta{{\Delta t}}^2}\rho \phi &=&0 \qquad \mbox{ in } \Omega,\\
\phi &=&0 \qquad \mbox{ on } \partial\Omega.\end{aligned}$$ The explicit case has been considered in [@JRW]. As expected from an explicit scheme, the method is conditionally stable. As a stability constraint, it requires to choose $${{\Delta t}}={\cal O}(h).$$ In the next sections, stability and convergence properties of the proposed $\theta$-scheme are analyzed.
Stability Analysis
==================
We derive sharp stability bounds based on the energy technique and show that the proposed scheme conserves certain energy. We consider (\[eq:www4\]) and (\[eq:www5\]) for the homogeneous case $$\begin{aligned}
(\rho{\bar{\partial}_{tt}}{{\pmb U}}^n,{{\pmb v}})+({P^{n;\theta}},{\nabla\cdot}{{\pmb v}})&=&0\qquad \forall {{\pmb v}}\in {{\pmb V}}_h,\label{eq:a1}\\
(\lambda^{-1}P^{n+1/2},w)-({\nabla\cdot}{{\pmb U}}^{n+1/2},w)&=&0\qquad \forall w\in W_h \label{eq:a2}.\end{aligned}$$ We will make use of the [*inverse assumption*]{}, which states that there exists a constant $C_0$ independent of $h$, such that $$\label{eq:ii}
||{\nabla\cdot}\phi||_{L^2(\Omega)}\leq C_0 h^{-1}||\phi||_{L^2(\Omega)}$$ for all $\phi\in W_h$. The following stability result holds.
\[th:1\]The fully discrete scheme $(\ref{eq:www1})$-$(\ref{eq:www5})$ is stable if $$\label{eq:cfl-1}
{{\Delta t}}^2\left(\frac{1}{4}-\theta\right)\frac{C_0^2\lambda_1}{h^2\rho_0}\leq 1,
$$ and conserves the discrete energy $$\label{eq:cfl-ee}
E_h^{n+{\frac{1}{2}}}=\frac{1}{2}\left[||\rho^{{\frac{1}{2}}}{\bar{\partial}_t}{{\pmb U}}^{n+{\frac{1}{2}}}||^2+
{{\Delta t}}^2\left(\theta-\frac{1}{4}\right)
||\lambda^{-{\frac{1}{2}}}{\bar{\partial}_t}P^{n+{\frac{1}{2}}}||^2+||\lambda^{-{\frac{1}{2}}}P^{n+{\frac{1}{2}}}||^2\right].$$ The scheme is unconditionally stable if $\theta\geq 1/4$.
If we subtract (\[eq:a2\]) from itself, with $n+1/2$ replaced by $n-1/2$, we find that $$\label{eq:s1}
(\lambda^{-1}(P^{n+1}-P^{n-1},w)-(\nabla\cdot(U^{n+1}-U^{n-1}),w)=0.$$ As (\[eq:a1\]) holds for all ${{\pmb v}}\in {{\pmb V}}_h$ and (\[eq:s1\]) holds for all $w\in W_h$, we choose ${{\pmb v}}={\bar{\partial}_t}{{\pmb U}}^n$ and $w=\frac{{P^{n;\theta}}}{2{{\Delta t}}}$ so that $$\begin{aligned}
(\rho{\bar{\partial}_{tt}}{{\pmb U}}^n,{\bar{\partial}_t}{{\pmb U}}^n)+({P^{n;\theta}},{\nabla\cdot}{\bar{\partial}_t}{{\pmb U}}^n)&=&0,\label{eq:s2}\\
(\lambda^{-1}{\bar{\partial}_t}P^n,{P^{n;\theta}})- ({\nabla\cdot}{\bar{\partial}_t}{{\pmb U}}^n,{P^{n;\theta}})&=&0\label{eq:s3}.\end{aligned}$$ By adding (\[eq:s2\]) and (\[eq:s3\]) we obtain $$\label{eq:s4}
(\rho{\bar{\partial}_{tt}}{{\pmb U}}^n,{\bar{\partial}_t}{{\pmb U}}^n)+(\lambda^{-1}{\bar{\partial}_t}P^n,{P^{n;\theta}})=0.$$ Note that $$\begin{aligned}
\label{eq:s5}
{P^{n;\theta}}&=&{{\Delta t}}^2\theta{\bar{\partial}_{tt}}P^n+P^n\nonumber\\
&=&{{\Delta t}}^2\left(\theta-\frac{1}{4}\right){\bar{\partial}_{tt}}P^n+\frac{1}{2}\left(P^{n+{\frac{1}{2}}}+P^{n-{\frac{1}{2}}}\right).\end{aligned}$$ Hence, (\[eq:s4\]) can be rewritten as $$\label{eq:s6}
(\rho{\bar{\partial}_{tt}}{{\pmb U}}^n,{\bar{\partial}_t}{{\pmb U}}^n)+
{{\Delta t}}^2\left(\theta-\frac{1}{4}\right)(\lambda^{-1}{\bar{\partial}_{tt}}P^n,{\bar{\partial}_t}P^n)+
\frac{1}{2}\left(\lambda^{-1}(P^{n+{\frac{1}{2}}}+P^{n-{\frac{1}{2}}}),{\bar{\partial}_t}P^n\right)=0.$$ Using that $$\bar\partial_{t}{{\pmb U}}^n=\frac{{\bar{\partial}_t}{{\pmb U}}^{n+{\frac{1}{2}}}+{\bar{\partial}_t}{{\pmb U}}^{n-{\frac{1}{2}}} }{2},\qquad
{\bar{\partial}_{tt}}{{\pmb U}}^n=\frac{{\bar{\partial}_t}{{\pmb U}}^{n+{\frac{1}{2}}}-{\bar{\partial}_t}{{\pmb U}}^{n-{\frac{1}{2}}}}{{{\Delta t}}},$$ we deduce that $$\begin{aligned}
(\rho{\bar{\partial}_{tt}}{{\pmb U}}^n,{\bar{\partial}_t}{{\pmb U}}^n) &=&
\frac{1}{2{{\Delta t}}}\,(\rho{\bar{\partial}_t}{{\pmb U}^{n+\frac{1}{2}}}-\rho{\bar{\partial}_t}{{\pmb U}^{n-\frac{1}{2}}},{\bar{\partial}_t}{{\pmb U}^{n+\frac{1}{2}}}+{\bar{\partial}_t}{{\pmb U}^{n-\frac{1}{2}}})\\
&=&
\frac{1}{2{{\Delta t}}}\left[ (\rho{\bar{\partial}_t}{{\pmb U}^{n+\frac{1}{2}}},{\bar{\partial}_t}{{\pmb U}^{n+\frac{1}{2}}})- (\rho{\bar{\partial}_t}{{\pmb U}^{n-\frac{1}{2}}},{\bar{\partial}_t}{{\pmb U}^{n-\frac{1}{2}}}) \right],\end{aligned}$$ and similarly $$(\lambda^{-1}{\bar{\partial}_{tt}}P^n,{\bar{\partial}_t}P^n) = \frac{1}{2{{\Delta t}}}
\left[ (\lambda^{-1}{\bar{\partial}_t}{P^{n+\frac{1}{2}}},{\bar{\partial}_t}{P^{n+\frac{1}{2}}})- (\lambda^{-1}{\bar{\partial}_t}{P^{n-\frac{1}{2}}},{\bar{\partial}_t}{P^{n-\frac{1}{2}}}) \right].$$ We also have $$\begin{aligned}
\left(\lambda^{-1}({P^{n+\frac{1}{2}}}+{P^{n-\frac{1}{2}}}),{\bar{\partial}_t}P^n\right) &=&
\frac{1}{{{\Delta t}}}\,(\lambda^{-1}{P^{n+\frac{1}{2}}}+\lambda^{-1}{P^{n-\frac{1}{2}}},{P^{n+\frac{1}{2}}}-{P^{n-\frac{1}{2}}})\\
&=&
\frac{1}{{{\Delta t}}}\left[ (\lambda^{-1}{P^{n+\frac{1}{2}}},{P^{n+\frac{1}{2}}})-(\lambda^{-1}{P^{n-\frac{1}{2}}},{P^{n-\frac{1}{2}}}) \right].\end{aligned}$$ Hence, (\[eq:s6\]) is equivalent to $$\frac{1}{{{\Delta t}}}\left(E_h^{n+{\frac{1}{2}}}-E_h^{n-{\frac{1}{2}}}\right)=0,$$ where $E^{n+{\frac{1}{2}}}_h$ is the quantity defined by (\[eq:cfl-ee\]). This relation indicates that $E_h^{n+{\frac{1}{2}}}$ is conserved for all time, which guarantees the stability of the scheme if and only if $E_h^{n+{\frac{1}{2}}}$ defines a positive energy. A sufficient condition is that $$\left|\left|\rho^{{\frac{1}{2}}}{\bar{\partial}_t}{{\pmb U}^{n+\frac{1}{2}}}\right|\right|^2+{{\Delta t}}^2\left(\theta-\frac{1}{4}\right)
\left|\left|\lambda^{-{\frac{1}{2}}}{\bar{\partial}_t}{P^{n+\frac{1}{2}}}\right|\right|^2\geq 0$$ for all $n\geq 0$. Clearly, the scheme is unconditionally stable when $\theta\geq 1/4$. Now, using Cauchy-Schwarz inequality and the inverse assumption (\[eq:ii\]), we obtain $$\begin{aligned}
\left(\lambda^{-1}{\bar{\partial}_t}{P^{n+\frac{1}{2}}},w\right)&=&\left({\nabla\cdot}{\bar{\partial}_t}{{\pmb U}^{n+\frac{1}{2}}},w\right)\\
&\leq& \left|\left|{\nabla\cdot}{\bar{\partial}_t}{{\pmb U}^{n+\frac{1}{2}}}\right|\right|_{L^2(\Omega)}||w||_{L^2(\Omega)}\\
&\leq&
\frac{C_0}{h}\left|\left|{\bar{\partial}_t}{{\pmb U}^{n+\frac{1}{2}}}\right|\right|_{L^2(\Omega)}||w||_{L^2(\Omega)}.\end{aligned}$$ By setting $w={\bar{\partial}_t}{P^{n+\frac{1}{2}}}$, we see that $$\begin{aligned}
\left|\left|\lambda^{-{\frac{1}{2}}}{\bar{\partial}_t}{P^{n+\frac{1}{2}}}\right|\right|^2_{L^2(\Omega)}
&\leq& \frac{C_0}{h}\left|\left|{\bar{\partial}_t}{{\pmb U}^{n+\frac{1}{2}}}\right|\right|_{L^2(\Omega)}
\left|\left|{\bar{\partial}_t}{P^{n+\frac{1}{2}}}\right|\right|_{L^2(\Omega)}\\
&\leq& \frac{C_0 \lambda_1^{{\frac{1}{2}}}}{h\rho_0^{\frac{1}{2}}}\left|\left|\rho^{\frac{1}{2}}{\bar{\partial}_t}{{\pmb U}^{n+\frac{1}{2}}}\right|\right|_{L^2(\Omega)}
\left|\left|\lambda^{-{\frac{1}{2}}}{\bar{\partial}_t}{P^{n+\frac{1}{2}}}\right|\right|_{L^2(\Omega)},\end{aligned}$$ or $$\left|\left|\lambda^{-{\frac{1}{2}}}{\bar{\partial}_t}{P^{n+\frac{1}{2}}}\right|\right|_{L^2(\Omega)}\leq
\frac{C_0
\lambda_1^{{\frac{1}{2}}}}{h\rho_0^{\frac{1}{2}}}\left|\left|\rho^{{\frac{1}{2}}}{\bar{\partial}_t}{{\pmb U}^{n+\frac{1}{2}}}\right|\right|_{L^2(\Omega)}.$$ Hence, a sufficient condition for stability is given by $$||\rho^{{\frac{1}{2}}}{\bar{\partial}_t}{{\pmb U}^{n+\frac{1}{2}}}||^2+{{\Delta t}}^2\left(\theta-\frac{1}{4}\right)
\frac{C_0^2{\lambda}_1}{h^2\rho_0}||\rho^{{\frac{1}{2}}}{\bar{\partial}_t}{{\pmb U}^{n+\frac{1}{2}}}||^2\geq 0,$$ which completes the proof.
The case with $\theta=1/4$ is interesting because the form of the discrete energy in this case is similar to that of the continuous problem. In addition, one can verify that the time truncation error is minimized over the set of all $\theta\geq 1/4$ when $\theta=1/4$.
Convergence Analysis
====================
In this section, we prove optimal convergence of the fully discrete finite element solution in the $L^\infty(L^2)$ norm. Some of the techniques used in the proofs can be found in previous works [@Kar-FE-Theta; @Karaa-2012]. In order to estimate the errors in the finite element approximation, we define the auxiliary functions $${{\pmb \chi}}^n={{\pmb U}}^n-\Pi_h{{\pmb u}}^n,\quad {{\pmb \eta}}^n={{\pmb u}}^n-\Pi_h{{\pmb u}}^n, \qquad {\xi}^n=P^n-{\cal P}_hp^n,
\qquad \zeta=p^n-{\cal P}_hp^n,$$ where $\Pi_h$ and ${\cal P}_h$ are defined in Section 4. From (\[eq:w1\])-(\[eq:w2\]) and (\[eq:www4\])-(\[eq:www5\]), and the properties of the projections $\Pi_h$ and ${\cal P}_h$, we arrive at $$\begin{aligned}
(\rho{\bar{\partial}_{tt}}{{\pmb \chi}}^n,v)+({\xi}^{{n;\theta}},{\nabla\cdot}{{\pmb v}})&=& (\rho{\bar{\partial}_{tt}}{{\pmb \eta}}^n,{{\pmb v}})+({\pmb r}^n,{{\pmb v}})\qquad
\forall {{\pmb v}}\in {{\pmb V}}_h,\quad n\geq 1,\label{eq:b1}\\
(\lambda^{-1}{\xi}^{n+1/2},w)-({\nabla\cdot}{{\pmb \chi}}^{n+1/2},w)&=&(\lambda^{-1}\zeta^{n+1/2},w)\quad
\qquad\forall w\in W_h,\quad n\geq 0,\label{eq:b2}\end{aligned}$$ where ${\pmb r}^n=\rho ({{\pmb u}}^{{n;\theta}}_{tt}-{\bar{\partial}_{tt}}{{\pmb u}}^n)$. Another equation has to be derived for the initial errors ${{\pmb \chi}}^1$ and ${\xi}^1$. Consider (\[eq:w1\]) at $n=0$ and $n=1$, respectivey, and subtract the resulting equations so that $$\label{eq:cv2}
\left(\rho{\bar{\partial}_t}{{\pmb u}}_{tt}^{\frac{1}{2}},{{\pmb v}}\right)+\left({\bar{\partial}_t}p^{\frac{1}{2}},{\nabla\cdot}{{\pmb v}}\right)=
\left({\bar{\partial}_t}{{\pmb f}}^{\frac{1}{2}},{{\pmb v}}\right).$$ A use of Taylor’s formula with integral remainder yields $$\label{eq:cv1}
{\bar{\partial}_t}{{\pmb u}}^{\frac{1}{2}}={{\pmb v}}^0+\frac{{{\Delta t}}}{2}{{\pmb u}}_{tt}^0+\frac{1}{2{{\Delta t}}}
\int_0^{{\Delta t}}({{\Delta t}}-t)^2\frac{\partial^3 {{\pmb u}}}{\partial t^3}(t)\,dt.$$ Using (\[eq:cv2\]) and (\[eq:cv1\]), we readily obtain $$\begin{aligned}
\label{eq:cv3}
\left(\rho{\bar{\partial}_t}{{\pmb u}}^{\frac{1}{2}},{{\pmb v}}\right)+\theta{{\Delta t}}^2\left({\bar{\partial}_t}p^{\frac{1}{2}},{\nabla\cdot}{{\pmb v}}\right)&=&
\theta{{\Delta t}}^2\left({\bar{\partial}_t}{{\pmb f}}^{\frac{1}{2}},{{\pmb v}}\right)-\theta{{\Delta t}}^2
\left(\rho{\bar{\partial}_t}{{\pmb u}}_{tt}^{\frac{1}{2}},{{\pmb v}}\right)\nonumber\\
&&+(\rho{{\pmb v}}^0,{{\pmb v}})+\frac{{{\Delta t}}}{2}(\rho{{\pmb u}}_{tt}^0,{{\pmb v}})\nonumber\\
&&+\frac{1}{2{{\Delta t}}}
\int_0^{{\Delta t}}({{\Delta t}}-t)^2\left(\rho\frac{\partial^3 {{\pmb u}}}{\partial t^3},{{\pmb v}}\right)\,dt.\end{aligned}$$ Subtracting (\[eq:cv3\]) from (\[eq:www3\]) and taking into account (\[eq:www2\]) and (\[eq:w1\]) to arrive that $$\label{eq:cv5}
\begin{split}
\left(\rho{\bar{\partial}_t}{{\pmb \chi}}^{\frac{1}{2}},{{\pmb v}}\right)+\theta{{\Delta t}}^2\left({\bar{\partial}_t}{\xi}^{\frac{1}{2}},{\nabla\cdot}{{\pmb v}}\right)
&+\frac{{{\Delta t}}}{2}\left({\xi}^0,{\nabla\cdot}{{\pmb v}}\right)=
\left(\rho{\bar{\partial}_t}{{\pmb \eta}}^{\frac{1}{2}},{{\pmb v}}\right)+\left(\rho(\Pi_h{{\pmb v}}^0-{{\pmb v}}^0),{{\pmb v}}\right)\\
&
+\theta{{\Delta t}}^2\left(\rho{\bar{\partial}_t}{{\pmb u}}_{tt}^{\frac{1}{2}},{{\pmb v}}\right)
-\frac{1}{2{{\Delta t}}}
\int_0^{{\Delta t}}({{\Delta t}}-t)^2\left(\rho\frac{\partial^3 {{\pmb u}}}{\partial t^3},{{\pmb v}}\right)\,dt.
\end{split}$$ Note that ${\xi}^0=0$ and ${{\pmb \chi}}^0=0$. We now state and prove our convergence result.
\[Th:1\] If ${{\pmb u}}\in L^\infty({\pmb H}(\Omega;{\rm div}))$, $\frac{\partial^3 {{\pmb u}}}{\partial t^3}\in L^1({\pmb L}^2(\Omega))$, $\frac{\partial^4 {{\pmb u}}}{\partial t^4}\in L^\infty({\pmb L}^2(\Omega))$, and $p\in L^\infty(L^2(\Omega))$, then for $\{{{\pmb U}}^n,P^n\}$ defined by $(\ref{eq:www1})$-$(\ref{eq:www5})$ there exists a constant $C$ independent of $h$ and ${{\Delta t}}$ such that if $$\label{eq:cfl2}
{{\Delta t}}^2\left(\frac{1}{4}-\theta\right)\frac{\lambda_1C_0^2}{\rho_0h^2}<\frac{1}{2},$$ then the following a priori error estimate holds: $$\label{eq:cv7}
\begin{split}
\left|\left|\rho^{\frac{1}{2}}({{\pmb u}}-{{\pmb U}})\right|\right|_{l^\infty(L^2)}+&
\left|\left|{\lambda}^{-{\frac{1}{2}}}(p-P)\right|\right|_{l^\infty(L^2)}\\
&
\leq C(h^r+{{\Delta t}}^2)\left(||{{\pmb u}}||_{L^\infty(H^r)}+
\left|\left|\frac{\partial^3 {{\pmb u}}}{\partial t^3} \right|\right|_{L^\infty(L^2)}+
||p||_{L^\infty(L^2)}\right),
\end{split}$$ where $r$ is associated with the degree of the finite element polynomial.
We first rearrange (\[eq:b1\]) in the form $$\label{eq:b3}
(\rho{\bar{\partial}_{tt}}{{\pmb \chi}}^n,{{\pmb v}})+{{\Delta t}}^2\left(\theta-\frac{1}{4}\right) ({\bar{\partial}_{tt}}{\xi}^n,{\nabla\cdot}{{\pmb v}})
+{\frac{1}{2}}\left({\xi^{n+\frac{1}{2}}}+{\xi^{n-\frac{1}{2}}},{\nabla\cdot}{{\pmb v}}\right)
= (\rho{\bar{\partial}_{tt}}{{\pmb \eta}}^n,v)+({\pmb r}^n,{{\pmb v}}).$$ Summing over time levels and multiplying through by ${{\Delta t}}$ yields $$\label{eq:b4}
\begin{split}
(\rho{\bar{\partial}_t}{{\pmb\chi}^{n+\frac{1}{2}}}-\rho{\bar{\partial}_t}&{{\pmb \chi}}^{{\frac{1}{2}}},{{\pmb v}})+{{\Delta t}}^2\left(\theta-\frac{1}{4}\right)
\left({\bar{\partial}_t}{\xi^{n+\frac{1}{2}}}-{\bar{\partial}_t}{\xi}^{\frac{1}{2}},{\nabla\cdot}{{\pmb v}}\right)\\
&+\frac{{{\Delta t}}}{2}\sum_{i=1}^{n}\left({\xi}^{i+{\frac{1}{2}}}+{\xi}^{i-{\frac{1}{2}}},{\nabla\cdot}{{\pmb v}}\right)
=\left(\rho{\bar{\partial}_t}{{\pmb \eta}}^{n+{\frac{1}{2}}}-\rho{\bar{\partial}_t}{{\pmb \eta}}^{{\frac{1}{2}}},{{\pmb v}}\right)+\left({{\Delta t}}\sum_{i=1}^{n}{\pmb r}^i,v\right).
\end{split}$$ Upon defining $$\phi^0=0, \qquad \phi^n={{\Delta t}}\sum_{i=0}^{n-1}{\xi}^{i+{\frac{1}{2}}},$$ we verify that $$\phi^{n+{\frac{1}{2}}}=\frac{{{\Delta t}}}{2} {\xi}^{\frac{1}{2}}+\frac{{{\Delta t}}}{2}
\sum_{i=1}^{n}\left({\xi}^{i+{\frac{1}{2}}} +{\xi}^{i-{\frac{1}{2}}}\right).$$ Taking into account (\[eq:cv5\]) and that ${\bar{\partial}_t}{\xi}^{\frac{1}{2}}=\frac{2}{{{\Delta t}}}{\xi}^{\frac{1}{2}}$, (\[eq:b4\]) becomes $$\label{eq:b5}
(\rho{\bar{\partial}_t}{{\pmb\chi}^{n+\frac{1}{2}}},{{\pmb v}})+{{\Delta t}}^2\left(\theta-\frac{1}{4}\right)
\left({\bar{\partial}_t}{\xi^{n+\frac{1}{2}}},{\nabla\cdot}{{\pmb v}}\right)+
\left(\phi^{n+{\frac{1}{2}}},{\nabla\cdot}{{\pmb v}}\right)=\left(\rho{\bar{\partial}_t}{{\pmb \eta}}^{n+{\frac{1}{2}}},{{\pmb v}}\right)
+\left({{\pmb R}}^n,{{\pmb v}}\right),$$ where $$\begin{aligned}
{{\pmb R}}^n={{\Delta t}}\sum_{i=1}^{n}{\pmb r}^i+\rho(\Pi_h{{\pmb v}}^0-{{\pmb v}}^0)+\theta{{\Delta t}}^2
\rho{\bar{\partial}_t}{{\pmb u}}_{tt}^{\frac{1}{2}}-\frac{1}{2{{\Delta t}}}
\int_0^{{\Delta t}}({{\Delta t}}-t)^2\rho\frac{\partial^3 {{\pmb u}}}{\partial t^3}\,dt.\end{aligned}$$ Since ${\bar{\partial}_t}\phi^{n+{\frac{1}{2}}}={\xi^{n+\frac{1}{2}}}$, (\[eq:b2\]) reads $$\label{eq:b6}
\left(\lambda^{-1}{\bar{\partial}_t}\phi^{n+{\frac{1}{2}}},w\right)-\left({\nabla\cdot}{{\pmb\chi}^{n+\frac{1}{2}}},w\right)=
\left(\lambda^{-1}{\zeta^{n+\frac{1}{2}}},w\right).$$ Choosing $v={{\pmb\chi}^{n+\frac{1}{2}}}$ and $w={\phi^{n+\frac{1}{2}}}$ in (\[eq:b5\]) and (\[eq:b6\]), respectively, and adding the resulting equations, we arrive at $$\label{eq:b7}
\begin{split}
(\rho{\bar{\partial}_t}{{\pmb\chi}^{n+\frac{1}{2}}},{{\pmb\chi}^{n+\frac{1}{2}}})+&{{\Delta t}}^2\left(\theta-\frac{1}{4}\right)
\left({\bar{\partial}_t}{\xi^{n+\frac{1}{2}}},{\nabla\cdot}{{\pmb\chi}^{n+\frac{1}{2}}}\right)+\left(\lambda^{-1}{\bar{\partial}_t}{\phi^{n+\frac{1}{2}}},{\phi^{n+\frac{1}{2}}}\right)\\
&
=\left(\rho{\bar{\partial}_t}{{\pmb \eta}}^{n+{\frac{1}{2}}},{{\pmb\chi}^{n+\frac{1}{2}}}\right)+\left({{\pmb R}}^n,{{\pmb\chi}^{n+\frac{1}{2}}}\right)+\left(\lambda^{-1}{\zeta^{n+\frac{1}{2}}},{\phi^{n+\frac{1}{2}}}\right).
\end{split}$$ Again, choose $w={\bar{\partial}_t}{\xi^{n+\frac{1}{2}}}$ in (\[eq:b2\]) so that $$\left({\bar{\partial}_t}{\xi^{n+\frac{1}{2}}},{\nabla\cdot}{{\pmb\chi}^{n+\frac{1}{2}}}\right)=\left(\lambda^{-1}{\xi^{n+\frac{1}{2}}},{\bar{\partial}_t}{\xi^{n+\frac{1}{2}}}\right)-\left(\lambda^{-1}{\zeta^{n+\frac{1}{2}}},
{\bar{\partial}_t}{\xi^{n+\frac{1}{2}}}\right).$$ Substitution into (\[eq:b7\]) yields $$\label{eq:b8}
\begin{split}
(\rho{\bar{\partial}_t}{{\pmb\chi}^{n+\frac{1}{2}}},{{\pmb\chi}^{n+\frac{1}{2}}})+&{{\Delta t}}^2\left(\theta-\frac{1}{4}\right)
\left(\lambda^{-1}{\bar{\partial}_t}{\xi^{n+\frac{1}{2}}},{\xi^{n+\frac{1}{2}}}\right)
+\left(\lambda^{-1}{\bar{\partial}_t}{\phi^{n+\frac{1}{2}}},{\phi^{n+\frac{1}{2}}}\right)\\
=&{{\Delta t}}^2\left(\theta-\frac{1}{4}\right) \left(\lambda^{-1}{\zeta^{n+\frac{1}{2}}},{\bar{\partial}_t}{\xi^{n+\frac{1}{2}}}\right)+
\left(\rho{\bar{\partial}_t}{{\pmb \eta}}^{n+{\frac{1}{2}}},{{\pmb\chi}^{n+\frac{1}{2}}}\right)\\
&+\left({{\pmb R}}^n,{{\pmb\chi}^{n+\frac{1}{2}}}\right)+\left(\lambda^{-1}{\zeta^{n+\frac{1}{2}}},{\phi^{n+\frac{1}{2}}}\right).
\end{split}$$ The terms on the right-hand side of (\[eq:b8\]) are bounded using Cauchy-Schwarz inequality as $$\begin{aligned}
\left(\lambda^{-1}{\zeta^{n+\frac{1}{2}}},{\bar{\partial}_t}{\xi^{n+\frac{1}{2}}}\right)&\leq&
\left|\left|\lambda^{-{\frac{1}{2}}}{\zeta^{n+\frac{1}{2}}}\right|\right|_{L^2(\Omega)}
\left|\left|\lambda^{-{\frac{1}{2}}}{\bar{\partial}_t}{\xi^{n+\frac{1}{2}}}\right|\right|_{L^2(\Omega)}\\
\left(\rho{\bar{\partial}_t}{{\pmb \eta}}^{n+{\frac{1}{2}}},{{\pmb\chi}^{n+\frac{1}{2}}}\right)&\leq&
\left|\left|\rho {\bar{\partial}_t}{{\pmb \eta}}^{n+{\frac{1}{2}}} \right|\right|_{L^2(\Omega)}
\left|\left| {{\pmb\chi}^{n+\frac{1}{2}}}\right|\right|_{L^2(\Omega)}\\
\left({{\pmb R}}^n,{{\pmb\chi}^{n+\frac{1}{2}}}\right)&\leq&
\left|\left| {{\pmb R}}^n \right|\right|_{L^2(\Omega)}
\left|\left| {{\pmb\chi}^{n+\frac{1}{2}}}\right|\right|_{L^2(\Omega)}\\
\left(\lambda^{-1}{\zeta^{n+\frac{1}{2}}},{\phi^{n+\frac{1}{2}}}\right)&\leq&
\left|\left| \lambda^{-{\frac{1}{2}}}{\zeta^{n+\frac{1}{2}}}\right|\right|_{L^2(\Omega)}
\left|\left| \lambda^{-{\frac{1}{2}}}{\phi^{n+\frac{1}{2}}}\right|\right|_{L^2(\Omega)}.\end{aligned}$$ We now distinguish the cases where $\theta\geq \frac{1}{4}$ and $\theta< \frac{1}{4}$. In the first case, we sum on (\[eq:b8\]) over time levels and multiply through by $2{{\Delta t}}$. This results in $$\label{eq:b10}
\begin{split}
&\left|\left|\rho^{\frac{1}{2}}{{\pmb \chi}}^{n+1}\right|\right|^2_{L^2(\Omega)}-
\left|\left|\rho^{\frac{1}{2}}{{\pmb \chi}}^0\right|\right|^2_{L^2(\Omega)}+
\left|\left|{\lambda}^{-{\frac{1}{2}}}\phi^{n+1}\right|\right|^2_{L^2(\Omega)}-
\left|\left|{\lambda}^{-{\frac{1}{2}}}\phi^{0}\right|\right|^2_{L^2(\Omega)} \\
&\qquad
+{{\Delta t}}^2\left(\theta-\frac{1}{4}\right)\left(
\left|\left|{\lambda}^{-{\frac{1}{2}}}{\xi}^{n+1}\right|\right|^2_{L^2(\Omega)}-
\left|\left|{\lambda}^{-{\frac{1}{2}}}{\xi}^{0}\right|\right|^2_{L^2(\Omega)} \right)\\
&\leq2{{\Delta t}}^2\left(\theta-\frac{1}{4}\right)\sum_{i=0}^n\left|\left|{\lambda}^{-{\frac{1}{2}}}\zeta^{i+{\frac{1}{2}}}
\right|\right|_{L^2(\Omega)}
\left( \left|\left|{\lambda}^{-{\frac{1}{2}}}{\xi}^{i+1} \right|\right|_{L^2(\Omega)}+
\left|\left|{\lambda}^{-{\frac{1}{2}}}{\xi}^{i} \right|\right|_{L^2(\Omega)}\right)\\
&\qquad+2{{\Delta t}}\sum_{i=0}^n\left|\left|{{\pmb \chi}}^{i+{\frac{1}{2}}} \right|\right|_{L^2(\Omega)}
\left(\left|\left|\rho {\bar{\partial}_t}{{\pmb \eta}}^{i+{\frac{1}{2}}} \right|\right|_{L^2(\Omega)}+
\left|\left|{{\pmb R}}^i\right|\right|_{L^2(\Omega)}\right)\\
&\qquad+2{{\Delta t}}\sum_{i=0}^n\left|\left|\lambda^{-{\frac{1}{2}}} \zeta^{i+{\frac{1}{2}}}
\right|\right|_{L^2(\Omega)}
\left|\left|\lambda^{-{\frac{1}{2}}} \phi^{i+{\frac{1}{2}}}\right|\right|_{L^2(\Omega)}.
\end{split}$$ Since $\left|\left|{\lambda}^{-{\frac{1}{2}}}\xi^{i} \right|\right|_{L^2(\Omega)}
\leq \left|\left|{\lambda}^{-{\frac{1}{2}}}{\xi}\right|\right|_{l^\infty(L^2)}$ and $\left|\left|\rho^{\frac{1}{2}}{{\pmb \chi}}^{i+{\frac{1}{2}}} \right|\right|_{L^2(\Omega)}
\leq \left|\left|\rho^{\frac{1}{2}}{{\pmb \chi}}\right|\right|_{l^\infty(L^2)}$, then $$\label{eq:b11}
\begin{split}
&\left|\left|\rho^{\frac{1}{2}}{{\pmb \chi}}^{n+1}\right|\right|^2_{L^2(\Omega)}+
\left|\left|{\lambda}^{-{\frac{1}{2}}}\phi^{n+1}\right|\right|^2_{L^2(\Omega)}
+{{\Delta t}}^2\left(\theta-\frac{1}{4}\right)
\left|\left|{\lambda}^{-{\frac{1}{2}}}{\xi}^{n+1}\right|\right|^2_{L^2(\Omega)}\\
&\leq4{{\Delta t}}^2\left(\theta-\frac{1}{4}\right)
\left|\left|{\lambda}^{-{\frac{1}{2}}}{\xi}\right|\right|_{l^\infty(L^2)}
\left(\sum_{i=0}^n\left|\left|{\lambda}^{-{\frac{1}{2}}}\zeta^{i+{\frac{1}{2}}}
\right|\right|_{L^2(\Omega)}\right)
\\
&\qquad+\frac{2{{\Delta t}}}{\rho_0^{\frac{1}{2}}}\left|\left|\rho^{\frac{1}{2}}{{\pmb \chi}}\right|\right|_{l^\infty(L^2)}
\left( \sum_{i=0}^n\left|\left|\rho {\bar{\partial}_t}{{\pmb \eta}}^{i+{\frac{1}{2}}} \right|\right|_{L^2(\Omega)}+
\sum_{i=0}^n\left|\left|{{\pmb R}}^i\right|\right|_{L^2(\Omega)}\right)\\
&\qquad+2{{\Delta t}}\left|\left|\lambda^{-{\frac{1}{2}}}\phi\right|\right|_{l^\infty(L^2)}
\left(\sum_{i=0}^n\left|\left|\lambda^{-{\frac{1}{2}}} \zeta^{i+{\frac{1}{2}}} \right|
\right|_{L^2(\Omega)}\right).
\end{split}$$ Applying the algebraic inequality: $ab\leq \frac{\epsilon}{2} a^2+\frac{1}{2\epsilon} b^2$ to the right-hand side of (\[eq:b11\]) shows that $$\label{eq:b12}
\begin{split}
&\left|\left|\rho^{\frac{1}{2}}{{\pmb \chi}}^{n+1}\right|\right|^2_{L^2(\Omega)}+
\left|\left|{\lambda}^{-{\frac{1}{2}}}\phi^{n+1}\right|\right|^2_{L^2(\Omega)}
+{{\Delta t}}^2\left(\theta-\frac{1}{4}\right)
\left|\left|{\lambda}^{-{\frac{1}{2}}}{\xi}^{n+1}\right|\right|^2_{L^2(\Omega)}\\
&\leq\frac{1}{2}{{\Delta t}}^2\left(\theta-\frac{1}{4}\right)
\left|\left|{\lambda}^{-{\frac{1}{2}}}{\xi}\right|\right|_{l^\infty(L^2)}^2
+8\left(\theta-\frac{1}{4}\right)
\left({{\Delta t}}\sum_{i=0}^{N-1}\left|\left|{\lambda}^{-{\frac{1}{2}}}\zeta^{i+{\frac{1}{2}}}
\right|\right|_{L^2(\Omega)}\right)^2
\\
&\qquad+\frac{1}{2}\left|\left|\rho^{\frac{1}{2}}{{\pmb \chi}}\right|\right|^2_{l^\infty(L^2)}+
C{{\Delta t}}^2 \left(\sum_{i=0}^{N-1}\left|\left|\rho {\bar{\partial}_t}{{\pmb \eta}}^{i+{\frac{1}{2}}} \right|\right|_{L^2(\Omega)}+
\sum_{i=0}^{N-1}\left|\left|{{\pmb R}}^i\right|\right|_{L^2(\Omega)}\right)^2\\
&\qquad+\frac{1}{2}\left|\left|\lambda^{-{\frac{1}{2}}}\phi\right|\right|^2_{l^\infty(L^2)}
+4\left({{\Delta t}}\sum_{i=0}^{N-1}\left|\left|\lambda^{-{\frac{1}{2}}} \zeta^{i+{\frac{1}{2}}} \right|
\right|_{L^2(\Omega)}\right)^2.
\end{split}$$ If we take the supremum over $n$ on the left-hand side and use the fact that $N{{\Delta t}}=T$, we conclude that $$\label{eq:b13}
\begin{split}
&\left|\left|\rho^{\frac{1}{2}}{{\pmb \chi}}\right|\right|^2_{l^\infty(L^2)}+
\left|\left|{\lambda}^{-{\frac{1}{2}}}\phi\right|\right|^2_{l^\infty(L^2)}
\leq C\left|\left|\lambda^{-{\frac{1}{2}}}\zeta\right|\right|^2_{l^\infty(L^2)}\\
&\qquad+
C{{\Delta t}}^2\left( \sum_{i=0}^{N-1}\left|\left|\rho {\bar{\partial}_t}{{\pmb \eta}}^{i+{\frac{1}{2}}}
\right|\right|_{L^2(\Omega)}\right)^2+C{{\Delta t}}^2\left(\sum_{i=0}^{N-1}
\left|\left|{{\pmb R}}^i\right|\right|_{L^2(\Omega)}\right)^2.
\end{split}$$ For the case $\theta<\frac{1}{4}$, we can follow the analysis presented in [@Kar-FE-Theta; @Karaa-2012] to derive error estimates similar to (\[eq:b13\]) under condition (\[eq:cfl2\]).
To complete the proof, we need to bound each term on the right-hand side of (\[eq:b13\]). The first term can be bounded using the approximation properties. Similarly, we have $${{\Delta t}}\sum_{i=0}^{N-1}\left|\left|\rho{{\pmb \eta}}^{i+{\frac{1}{2}}}\right|\right|_{L^2(\Omega)}
\leq C\left(h^k||{{\pmb u}}||_{L^\infty(H^k(\Omega))}+{{\Delta t}}^2
\left|\left|\frac{\partial^3{{\pmb u}}}{\partial
t^3}\right|\right|_{L^1(0,T;L^2(\Omega))}\right).$$ For the last term on the right-hand side of (\[eq:b13\]), we have $$\begin{aligned}
{{\Delta t}}\sum_{i=0}^{N-1}||{{\pmb R}}^i||_{L^2(\Omega)}
&\leq& C||{{\pmb R}}||_{l^\infty(L^2)}\\
&\leq& C{{\Delta t}}\sum_{i=1}^{N-1}||{\pmb r}^i||_{L^2(\Omega)}+C
||\rho(\Pi_h{{\pmb v}}^0-{{\pmb v}}^0)||_{L^2(\Omega)}\\
&&
+C\theta{{\Delta t}}^2\left|\left|\rho{\bar{\partial}_t}{{\pmb u}}_{tt}^{\frac{1}{2}}\right|\right|_{L^2(\Omega)}
+C\left|\left|\frac{1}{2{{\Delta t}}}\int_0^{{\Delta t}}\rho({{\Delta t}}-t)^2\frac{\partial^3{{\pmb u}}}{\partial
t^3}(t)\,dt\right|\right|_{L^2(\Omega)}.\end{aligned}$$ To estimate $||{\pmb r}^i||_{L^2(\Omega)}$, we make use of the identity $$\label{eq:t11}
{\bar{\partial}_{tt}}{{\pmb u}}^n=u_{tt}^n+\frac{1}{6{{\Delta t}}^2}\int_{-{{\Delta t}}}^{{{\Delta t}}}({{\Delta t}}-|s|)^3
\frac{\partial^4{{\pmb u}}}{\partial t^4}(t^n+s)ds.$$ From the Taylor’s expansions of ${{\pmb u}}_{tt}^{n+1}$ and ${{\pmb u}}_{tt}^{n-1}$ about ${{\pmb u}}_{tt}^{n}$; $${{\pmb u}}^{n+1}_{tt}={{\pmb u}}^n_{tt}+{{\Delta t}}{{\pmb u}}_{ttt}^n+\int_{0}^{{{\Delta t}}}({{\Delta t}}-|s|)\frac{\partial^4{{\pmb u}}}{\partial t^4}(t^n+s)\,ds,$$ and $${{\pmb u}}^{n-1}_{tt}={{\pmb u}}^n_{tt}-{{\Delta t}}{{\pmb u}}_{ttt}^n+\int_{-{{\Delta t}}}^{0}({{\Delta t}}-|s|)\frac{\partial^4{{\pmb u}}}{\partial t^4}(t^n+s)\,ds,$$ we obtain $$\label{eq:t12}
{{\pmb u}}^{n;\theta}_{tt}={{\pmb u}}_{tt}^n+\theta\int_{-{{\Delta t}}}^{{{\Delta t}}}({{\Delta t}}-|s|)
\frac{\partial^4{{\pmb u}}}{\partial t^4}(t^n+s)\,ds.$$ Subtracting (\[eq:t11\]) from (\[eq:t12\]) yields $${{\pmb u}}^{n;\theta}_{tt}-{\bar{\partial}_{tt}}{{\pmb u}}^n=
\frac{1}{6{{\Delta t}}^2}\int_{-{{\Delta t}}}^{{{\Delta t}}}(|s|-{{\Delta t}})^3\frac{\partial^4{{\pmb u}}}{\partial t^4}(t^n+s)\,ds
-\theta\int_{-{{\Delta t}}}^{{{\Delta t}}}(|s|-{{\Delta t}})\frac{\partial^4{{\pmb u}}}{\partial t^4}(t^n+s)\,ds.$$ Hence, $$||{\pmb r}^i||_{L^2(\Omega)}^2=
||\rho\left({{\pmb u}}^{n;\theta}_{tt}-{\bar{\partial}_{tt}}{{\pmb u}}^n\right)||_{L^2}
\leq C{{\Delta t}}^3
\int_{-{{\Delta t}}}^{{{\Delta t}}}
\left|\left|\rho^{\frac{1}{2}}\frac{\partial^4{{\pmb u}}}{\partial t^4}
(t^n+s)\right|\right|_{L^2(\Omega)}^2\,ds\leq C{{\Delta t}}^4\left|\left|\rho^{\frac{1}{2}}\frac{\partial^4{{\pmb u}}}{\partial t^4}
\right|\right|_{L^\infty(L^2)}^2,$$ and therefore $${{\Delta t}}\sum_{i=1}^{n}||{\pmb r}^i||_{L^2(\Omega)}\leq
C{{\Delta t}}^2\left|\left|\rho^{\frac{1}{2}}\frac{\partial^4{{\pmb u}}}{\partial
t^4}\right|\right|_{L^\infty(L^2)}\sum_{i=1}^{n}{{\Delta t}}\leq CT{{\Delta t}}^2\left|\left|^2\rho^{\frac{1}{2}}\frac{\partial^4{{\pmb u}}}{\partial t^4}
\right|\right|_{L^\infty(L^2)}.$$ Similarly, we have $$\left|\left|\rho{\bar{\partial}_t}{{\pmb u}}_{tt}^{\frac{1}{2}}\right|\right|_{L^2(\Omega)}^2=
\left|\left|{{\Delta t}}\int_0^{{\Delta t}}\rho\frac{\partial^3{{\pmb u}}}{\partial
t^3}(t)\,dt\right|\right|_{L^2(\Omega)}^2
\leq C{{\Delta t}}^3\int_0^{{\Delta t}}\left|\left|\rho^{\frac{1}{2}}\frac{\partial^3{{\pmb u}}}{\partial
t^3}\right|\right|_{L^2(\Omega)}^2dt
\leq C{{\Delta t}}^4
\left|\left|\rho^{\frac{1}{2}}\frac{\partial^3{{\pmb u}}}{\partial
t^3}\right|\right|_{L^\infty(L^2)}^2,$$ and $$\left|\left|\frac{1}{2{{\Delta t}}}\int_0^{{\Delta t}}\rho({{\Delta t}}-t)^2\frac{\partial^3{{\pmb u}}}{\partial
t^3}(t)\,dt\right|\right|_{L^2(\Omega)}^2\leq C{{\Delta t}}^3\int_0^{{\Delta t}}\left|\left|\rho^{\frac{1}{2}}\frac{\partial^3{{\pmb u}}}{\partial
t^3}\right|\right|_{L^2(\Omega)}^2dt
\leq C{{\Delta t}}^4
\left|\left|\rho^{\frac{1}{2}}\frac{\partial^3{{\pmb u}}}{\partial
t^3}\right|\right|_{L^\infty(L^2)}^2.$$ Finally, using the approximation property (\[eq:w9\]) and combining all the bounds, we arrive at $${{\Delta t}}\sum_{i=0}^{N-1}||{{\pmb R}}^i||_{L^2(\Omega)}\leq C(h^k+{{\Delta t}}^2),$$ which completes the proof of the desired estimate.
[**Remarks.**]{} It is worthwhile to mention that the time discretization method is fourth-order accurate when $\theta=1/12$. To preserve the temporal accuracy of the finite element scheme one has to modify (\[eq:www3\]) carefully to obtain an appropriate initial value ${{\pmb U}}^1$. The analysis presented in [@Kar-FE-Theta] can be used to derive optimal a priori error estimates in this case.
Conclusions
===========
We proposed and analyzed a family of fully discrete mixed finite element schemes for solving the acoustic wave equation. We derived stability conditions for conditionally implicit stable schemes covering the explicit case treated by Jenkins, Rivière and Wheeler [@JRW]. The error estimates established provided optimal convergence rates for the use of mixed finite elements methods in solving the acoustic wave equation.
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[^1]: Department of Mathematics and Statistics, Sultan Qaboos University, P. O. Box 36, Al-Khod 123, Muscat, Sultanate of Oman ([skaraa@squ.edu.om]{}).
[^2]: This research was supported by Sultan Qaboos University under Grant IG/SCI/DOMS/13/02.
|
---
abstract: |
In this paper, a new method is developed to obtain explicit and integral expressions for the kernel of the $(\kappa, a)$-generalized Fourier transform for $\kappa =0$. In the case of dihedral groups, this method is also applied to the Dunkl kernel as well as the Dunkl Bessel function. The method uses the introduction of an auxiliary variable in the series expansion of the kernel, which is subsequently Laplace transformed. The kernel in the Laplace domain takes on a much simpler form, by making use of the Poisson kernel. The inverse Laplace transform can then be computed using the generalized Mittag-Leffler function to obtain integral expressions. In case the parameters involved are integers, explicit formulas are obtained using partial fraction decomposition.
New bounds for the kernel of the $(\kappa, a)$-generalized Fourier transform are obtained as well.
\
[*Keywords*]{}: Dunkl kernel, Generalized Fourier transform, Dihedral group, Bessel function, Poisson kernel \
[*Mathematics Subject Classification:*]{} 33C52, 43A32, 42B10
author:
- 'Denis Constales$^{1}$ [^1]'
- 'Hendrik De Bie$^{1}$[^2]'
- 'Pan Lian$^{2}$[^3]'
date: |
\
\
\
title: 'Explicit formulas for the Dunkl dihedral kernel and the $(\kappa, a)$-generalized Fourier kernel'
---
Introduction
============
Recently, a lot of attention has been given to various generalizations of the Fourier transform. This paper focusses on two in particular, namely the Dunkl transform [@D1; @deJ] and the $(\kappa, a)$-generalized Fourier transform [@SKO]. Both transforms depend on a number of parameters, and as such it is an open problem, except for certain special cases, to find concrete formulas for their integral kernels.
Our aim in this paper is to develop a new method for finding explicit expressions as well as integral expressions for these kernels. Explicit expressions can be obtained when some of the arising parameters take on rational or integer values. The integral expressions we will obtain are valid in full generality and are expressed in terms of the generalized Mittag-Leffler function (see [@MH] or the subsequent Definition \[DefML\]).
Essentially our method works as follows. Consider the following series expansion, for $x,y \in {\mathbb{R}}^m$ $$K^{m}(x,y)=2^{\lambda}\Gamma(\lambda+1) \sum_{j=0}^{\infty} (-i)^j\frac{\lambda+j}{\lambda}z^{-\lambda}J_{j+\lambda}(z)C^{\lambda}_{j}(\xi)$$ with $\lambda=(m-2)/2$, $z=|x||y|$, $\xi=\langle x,y\rangle/z$, $J_{j+\lambda}(z)$ the Bessel function and $C_{j}^{\lambda}(\xi)$ the Gegenbauer polynomial. It is not so easy to recognize that this is the classical Fourier kernel $e^{-i\langle x, y\rangle}$.
However, when we introduce an auxiliary variable $t$ in the kernel as follows $$K^{m}(x,y,t)=2^{\lambda}\Gamma(\lambda+1) \sum_{j=0}^{\infty}(-i)^j \frac{\lambda+j}{\lambda}z^{-\lambda}J_{j+\lambda}(t z)C^{\lambda}_{j}(\xi)$$ we can take the Laplace transform in $t$ of $K^{m}(x,y,t)$. Simplifying the result by making use of the Poisson kernel (see subsequent Theorem \[pe\]) then yields $$\mathcal{L}(K^{m}(x,y,t))=\Gamma(\lambda+1)\frac{1}{(s+i \langle x, y\rangle)^{\lambda+1}}.$$ of which we immediately compute the inverse Laplace transform as $$K^{m}(x,y,t) = t^{\frac{m-2}{2}} e^{-i t \langle x, y\rangle}$$ and the classical Fourier kernel is recovered by putting $t=1$.
We develop this method in full generality for the Dunkl kernel related to dihedral groups, as well as for the $(\kappa, a)$- generalized Fourier transform when $\kappa =0$. The restriction to dihedral groups is necessary, because only then the Poisson kernel for the Dunkl Laplace operator is known, see [@DX] or subsequent Theorem \[DunklPoisson\].
Let us describe our main results. The Laplace transform of the $(0,a)$-generalized Fourier transform is obtained in Theorem \[laplradial\]. When $a = 2/n$ and $m$ is even, the result is a rational function and we can apply partial fraction decomposition to obtain an explicit expression, see Theorem \[th3\]. We prove that the kernel for $a = 2/n$ is bounded by 1 in Theorem \[th7\], for both even and odd dimensions. For arbitrary $a$, the integral expression in terms of the generalized Mittag-Leffler function is given in Theorem \[ga\].
The Laplace transform of the Dunkl kernel for dihedral groups is obtained in Theorem \[ld1\]. Two alternative integral expressions for the Dunkl kernel, again in terms of the generalized Mittag-Leffler function, are given in Theorem \[m1\] and \[m2\].
The paper is organized as follows. After the necessary preliminaries in section \[prelim\], we first study the $(\kappa,a)$-generalized Fourier transform for $\kappa =0$ in section \[radialFT\]. In section \[dunklFT\] we then study the Dunkl kernel for dihedral groups. We also show how our methods can be applied to the Dunkl Bessel function.
Preliminaries {#prelim}
=============
In this section, we give a brief overview of the theory of Dunkl operators, the $(\kappa, a)$-generalized Fourier transform and the Laplace transform. Most of these results are taken from [@DX], [@rm] and [@SKO]. We use the notation $\langle \cdot,\cdot \rangle$ for the standard inner product on $\mathbb{R}^{m}$ and $|\cdot|$ for the associated norm. For a non-zero vector $\alpha\in \mathbb{R}^{m}$, the reflection $r_{\alpha}$ with respect to the hyperplane orthogonal to $\alpha$ is defined by $$r_{\alpha}(x)=x-2\frac{\langle \alpha, x \rangle}{|\alpha|^{2}}\alpha.$$ A reduced root system $\mathcal{R}$ is a finite set of non-zero vectors in $\mathbb{R}^{m}$ such that $r_{\alpha} \mathcal{R}=\mathcal{R}$ and $\mathbb{R} \alpha \cap \mathcal{R}= \{\pm\alpha\}$ for all $\alpha\in \mathcal{R}$. The finite reflection group generated by $\{r_{\alpha}: \alpha\in \mathcal{R}\}$ is a subgroup of the orthogonal group $O(m)$ which is called a Coxeter group. Three infinite families of root systems are $A_{n-1}$, $B_{n}$ and the root system associated to the dihedral groups. We give the latter as an example which will be used later.
In the Euclidean space $\mathbb{R}^{2}$, let $d\in O(2, \mathbb{R})$ be the rotation over $2\pi/k$ and $e$ the reflection at the $y$-axis. The group $I_{k}$ generated by $d$ and $e$ consists of all orthogonal transformations which preserve a regular $k$-sided polygon centered at the origin. The group $I_{k}$ is a finite reflection group which is usually called dihedral group.
We define the action of $G$ on functions by $$(g\cdot f)(x):=f(g^{-1}\cdot x),\qquad x\in \mathbb{R}^{m}, g\in G.$$ A multiplicity function $\kappa: \mathcal{R} \rightarrow \mathbb{C}$ is a function invariant under the action of $G$. Furthermore, set $\mathcal{R}_{+}:=\{\alpha\in \mathcal{R}:\langle\alpha, \beta \rangle>0\}$ for some $\beta \in \mathbb{R}^{m}$ such that $\langle\alpha, \beta \rangle \neq 0$ for all $\alpha\in \mathcal{R}$. From now on, fix the positive subsystem $\mathcal{R}_{+}$ and the multiplicity function $\kappa$. The Dunkl operator $T_{i}$ associated to $\mathcal{R}_{+}$ and $\kappa$ is then defined by $$T_{i}f(x)=\frac{\partial f}{\partial x_{i}}+\sum_{\alpha\in \mathcal{R}_{+}}\kappa(\alpha) \alpha_{i}\frac{f(x)-f(r_{\alpha}(x))}{\langle \alpha, x\rangle}, \qquad f\in C^{1}(\mathbb{R}^{m})$$ where $\alpha_{i}$ is the $i$-th coordinate of $\alpha$. All the $T_{i}$ commute with each other. They reduce to the corresponding partial derivatives when $\kappa=0$. The Dunkl Laplacian $\Delta_{\kappa}$ is then defined as $\Delta_{\kappa}=\sum_{i=1}^{m}T_{i}^2$. The weight function associated with the root system $\mathcal{R}$ and the multiplicity function $\kappa$ is given by $$\upsilon_{\kappa}(x):=\prod_{\alpha\in \mathcal{R}_{+}}|\langle x,\alpha\rangle|^{2\kappa(\alpha)}.$$ It is $G$-invariant and homogeneous of degree $2\langle \kappa\rangle$ where the index $\langle \kappa \rangle$ of the multiplicity function $\kappa$ is defined as $$\langle \kappa\rangle :=\sum_{\alpha\in \mathcal{R}_{+}}\kappa_{\alpha}=\frac{1}{2}\sum_{\alpha\in \mathcal{R}}\kappa_{\alpha}.$$ We also denote by $\mathcal{H}_{j}(\upsilon_{\kappa})$ the space of Dunkl harmonics of degree $j$, i.e. $\mathcal{H}_{j}(\upsilon_{\kappa})=\mathcal{P}_{j}\cap \rm{ker} \Delta_{\kappa}$ with $\mathcal{P}_{j}$ the space of homogeneous polynomials of degree $j$. There exists a unique linear and homogeneous isomorphism on polynomials which intertwines the algebra of Dunkl operators and the algebra of usual partial differential operators, i.e. $V_{\kappa}(\mathcal{P}_{j})=\mathcal{P}_{j}, \quad V_{\kappa}|_{\mathcal{P}_{0}}=id$ and $T_{\xi} V_{\kappa}=V_{\kappa} \partial_{\xi}$ for all $\xi \in \mathbb{R}^{m}$. In the following, we denote by $P_{j}(G; x,y)$ the reproducing kernel of $\mathcal{H}_{j}(\upsilon_{\kappa})$ and $P(G; x,y)$ the Poisson kernel. For $j \in \mathbb{N}$ and $|y|\le |x|=1$, we have [@DX] $$\begin{aligned}
\label{pd} P_{j}(G; x,y)=\frac{j+\lambda_{\kappa}}{\lambda_{\kappa}} V_{\kappa}[C_{j}^{\lambda_{\kappa}}(\langle \cdot, \frac{y}{|y|}\rangle)](x)|y|^{j},\end{aligned}$$ and $$\begin{aligned}
\label{pois}P(G;x,y)=\sum_{j=0}^{\infty} P_{j}(G;x, y)=\sum_{j=0}^{\infty} P_{j}(G;x,\frac{y}{|y|})|y|^{j}=V_{\kappa}\biggl(\frac{1-|y|^2}{(1-2\langle \cdot, y\rangle +|y|^2)^{\lambda_{\kappa}+1}}\biggr)(x)\end{aligned}$$ where $\lambda_{\kappa}= \langle \kappa\rangle +\frac{m-2}{2}$. Rösler [@rm1] proved there exists a unique positive probability-measure $\mu_{x}(\xi)$ on $\mathbb{R}^{m}$ such that $$V_{\kappa}f(x)=\int_{\mathbb{R}^{m}}f(\xi)d\mu_{x}(\xi)$$ for the positive multiplicity function. In [@SKO], Dunkl’s interwining operator $V_{\kappa}$ was extended to $C(B)$ with $B$ the closed unit ball in $\mathbb{R}^m$ for the regular values of $\kappa$. Denoting $$\tilde{(V_{\kappa}}h):=(V_{\kappa}h_{y})(x)=\int_{\mathbb{R}^{m}}h(\langle \xi, y\rangle)d\mu_{x}(\xi),$$ this operator satisfies $$\begin{aligned}
\label{sv1}||\tilde{V_{\kappa}}h||_{L^{\infty}(B\times B)}\le ||h||_{L^{\infty}([-1,1])}.\end{aligned}$$
It is known that the operators $T_{j}$ have a joint eigenfunction $E_{\kappa}(x,y)$ satisfying $$T_{j}E_{\kappa}(x,y)=-iy_{j}E_{\kappa}(x,y), \qquad j=1,\ldots,m.$$ The function $E_{\kappa}(x,y)$ is called the Dunkl kernel, which is the exponential function $e^{-i\langle x, y\rangle}$ when $\kappa=0$. This kernel together with the weight function is used to define the so-called Dunkl transform $$\mathcal{F}_{\kappa}: L^{1}(\mathbb{R}^{m}, \upsilon_{\kappa})\rightarrow C(\mathbb{R}^{m})$$ by $$\mathcal{F}_{\kappa}f(y):=c_{\kappa}\int_{\mathbb{R}^{m}}f(x)E(x,y)\upsilon_{\kappa}(x)dx \quad(y\in \mathbb{R}^{m})$$ with $c_{\kappa}^{-1}=\int_{\mathbb{R}^{m}} e^{-|x|^{2}/2} \upsilon_{\kappa}(x)dx$ the Mehta constant associated to $G$. Again, when $\kappa=0$, we recover the classical Fourier transform. The Dunkl transform shares many properties with the Fourier transform. In [@HR], using the harmonic oscillator $-(\Delta-|x|^2)/2$, Howe found the spectral description of the Fourier transform and its eigenfunctions forming the basis of $L^{2}(\mathbb{R}^{m})$: $$\mathcal{F}=e^{\frac{i\pi m}{4}}e^{\frac{i\pi}{4}(\Delta-|x|^2)}$$ with $\Delta$ the Laplace operator. Similarly, the Dunkl transform also has the exponential notation $$\mathcal{F}_{\kappa}=e^{\frac{i\pi \mu}{4}}e^{\frac{i\pi}{4}(\Delta_{\kappa}-|x|^2)}$$ where $\mu=m+2\langle \kappa \rangle$, see [@Said]. In [@SKO], the authors defined a radially deformed Dunkl-type harmonic oscillator $$\Delta_{\kappa,a}=|x|^{2-a}\Delta_{\kappa}-|x|^{a},\qquad a>0.$$ Then the $(\kappa,a)$-generalized Fourier transform is defined by $$\mathcal{F}_{\kappa,a}=e^{\frac{i\pi}{2a}(m-2+2\langle \kappa\rangle+a)}e^{\frac{i\pi}{2a} \Delta_{\kappa,a}}$$ in $L^{2}(\mathbb{R}^{m}, |x|^{a-2}\upsilon_{\kappa}(x) )$. We write the $(\kappa,a)$-generalized Fourier transform as an integral transform $$\mathcal{F}_{\kappa,a}f(y)=c_{\kappa,a}\int_{\mathbb{R}^{m}}B_{\kappa,a}(x,y)f(x)|x|^{a-2}\upsilon_{\kappa}(x)dx$$ where $c_{\kappa,\alpha}^{-1}=\int_{\mathbb{R}^{m}} e^{-|x|^{a}/a}|x|^{a-2}\upsilon_{\kappa}(x)dx$. The series expression of $B_{\kappa,a}(x,y)$ is given in [@SKO] as follows,
\[the1\] For $x, y\in \mathbb{R}^{m}$ and $a>0$, we have $$B_{\kappa,a}(x,y)=a^{\frac{2\langle \kappa\rangle+m-2}{a}}\Gamma\biggl(\frac{2\langle \kappa\rangle+m+a-2}{a}\biggr) \sum_{j=0 }^{\infty} B_{\kappa,a}^{(j)}(z) P_{j}(G;\omega,\eta)$$ where $x=|x| \omega,$ $y=|y|\eta$, $z=|x||y|$, $\lambda_{\kappa,a,j}=\frac{2j+2\langle \kappa\rangle+m-2}{a}$, $$B_{\kappa, a}^{(j)}(z)=e^{-i\frac{\pi}{2}\frac{j}{a}} z^{-\langle \kappa\rangle-m/2+1}J_{\lambda_{\kappa,a,j}}\biggl(\frac{2}{a}z^{a/2}\biggr),$$ and $$P_{j}(G;\omega,\eta):=\biggl(\frac{\langle \kappa\rangle+j+\frac{m-2}{2}}{\langle \kappa\rangle+\frac{m-2}{2}}\biggr)V_{\kappa}[C_{j}^{\lambda_{\kappa}}(\langle \cdot, \eta\rangle)](\omega),$$ is the reproducing kernel of the space of spherical $\kappa$-harmonic polynomials of degree $j$.
This transform recovers the Dunkl transform when $a=2$, the Fourier transform when $a=2$ and $\kappa=0$. The operator $\mathcal{F}_{0,1}$ is the unitary inversion operator of the Schrödinger model of the minimal representation of the group $O(m+1,2)$ [@KM]. The explicit expression of the Dunkl kernel is only known for the groups $\mathbb{Z}_{2}^{m}$, the root systems $A_{2}$, $B_{2}$ and some dihedral groups with integer muliplicity function $\kappa$, see[@amr], [@Ade], [@D1], [@DX] and [@DDY]. For the integral kernel $B_{\kappa,a}(x,y)$, except the already known Dunkl kernel, closed expressions have been found when $\kappa=0$ and $a=\frac{2}{n}$ with $ n\in \mathbb{N}$ in dimension 2, see [@Rad2]. For higher even dimension, an iterative procedure using derivatives is given there as well. Pitt’s inequalities and uncertainty principles for the $(\kappa, a)$-generalized Fourier transform have been established in [@J1; @Go] .
The Laplace transform is an integral transform which takes a function of a positive real variable $t$ to a function of a complex variable $s$. For a function $f(t)$ which has an exponential growth $|f(t)|\le Ce^{\alpha t}, t\ge t_{0}$, the Laplace transform is defined as $$F(s)=\mathcal{L}(f(t))(s)=\int_{0}^{\infty}e^{-st}f(t)dt.$$ The inverse Laplace transform is given by the Bromwich integral or the Post’s inversion formula. In practice, it is typically more convenient to decompose a Laplace transform into known transforms of functions obtained from a table, for example [@E2]. For more details on the Laplace transform, we refer to [@DG].
The kernel of the $(\kappa,a)$-generalized Fourier transform {#radialFT}
============================================================
Explicit expression of the kernel when $a=\frac{2}{n}$ and $m$ even
-------------------------------------------------------------------
In this section, we first establish the connection between the kernel of the $(0,a)$-generalized Fourier kernel and the Poisson kernel for the unit ball by introducing an auxiliary variable in the kernel and using the Laplace transform. Then we give the explicit formula for the kernel when $a=\frac{2}{n}$ and $m$ even.
The kernel $K^{m}_{a}(x,y)=B_{0, a}(x,y)$ for $a>0$ is given in Theorem \[the1\] (see also [@Rad2], [@SKO]) $$K_{a}^{m}(x,y)=a^{2\lambda/a}\Gamma\biggl(\frac{2\lambda+a}{a}\biggr)\sum_{j=0}^{\infty}e^{-\frac{i\pi j}{a}}\frac{\lambda+j}{\lambda}z^{-\lambda}J_{\frac{2(j+\lambda)}{a}}\biggl(\frac{2}{a}z^{a/2}\biggr)C^{\lambda}_{j}(\xi)$$ with $\lambda=(m-2)/2$, $z=|x||y|$, $\xi=\langle x,y\rangle/z$ and $C_{j}^{\lambda}(\xi)$ the Gegenbauer polynomial. We introduce an auxiliary variable $t$ in the kernel as follows $$\begin{aligned}
\label{ke1}K_{a}^{m}(x,y,t)=a^{2\lambda/a}\Gamma\biggl(\frac{2\lambda+a}{a}\biggr)\sum_{j=0}^{\infty}e^{-\frac{i\pi j}{a}}\frac{\lambda+j}{\lambda}z^{-\lambda}J_{\frac{2(j+\lambda)}{a}}\biggl(\frac{2}{a}z^{a/2} t\biggr)C^{\lambda}_{j}(\xi).\end{aligned}$$ Before we take the Laplace transform, we give the expansion of the Poisson kernel in terms of Gegenbauer polynomials.
\[pe\][@DX] For $x,y \in \mathbb{R}^{m}$ and $ |y|\le |x|=1$, the Poisson kernel for the unit ball is $$\begin{aligned}
P(x,y)=\frac{1-|y|^2}{|x-y|^{m}}=\frac{1-|y|^2}{(1-2\xi|y|+|y|^2)^{m/2}}=\sum_{j=0}^{\infty}\frac{j+m/2-1}{m/2-1}C_{j}^{m/2-1}(\xi)|y|^{j}, \quad \xi=\langle x, \frac{y}{|y|}\rangle. \end{aligned}$$
This result can be extended for $\lambda>0,$ we have $$\begin{aligned}
\label{ac1}\frac{1-|y|^2}{(1-2\xi|y|+|y|^2)^{\lambda+1}}=\sum_{j=0}^{\infty}\frac{j+\lambda}{\lambda}C_{j}^{\lambda}(\xi)|y|^{j}. \end{aligned}$$ It is still valid for $z\in \mathbb{C}$, $|z|<1$ and $|\xi|<1$, (see [@olf]) $$\begin{aligned}
\label{ac2}\frac{1-z^2}{(1-2\xi z+z^2)^{\lambda+1}}=\sum_{j=0}^{\infty}\frac{j+\lambda}{\lambda}C_{j}^{\lambda}(\xi)z^{j}. \end{aligned}$$ To establish the validity of the analytic continuation of (\[ac1\]) to (\[ac2\]), note that the left-hand side of (\[ac2\]) is analytic in $z$ in any disk centered at the origin of the complex plane that does not contain any zero of the denominator, hence analytic in $0\le |z|<1$. By the estimate $$|C_{j}^{\lambda}(\xi)|\le C_{j}^{\lambda}(1)=\frac{(2\lambda)_{j}}{j!},$$ the right-hand side of (\[ac2\]) will certainly converge to an analytic continuation of that of (\[ac1\]) for all $z$ satisfying $|z|\le |y|<1$, hence for the whole unit disk.
By Theorem \[pe\] and the formula from [@E2] $$\begin{aligned}
\label{l1} \mathcal{L}(J_{\nu}(bt))=\frac{1}{\sqrt{s^2+b^2}} \biggl(\frac{b}{s+\sqrt{s^2+b^2}}\biggr)^{\nu}, \qquad \mbox{Re}\,\nu >-1, \mbox{Re} \, s>|\mbox{Im}\, b|
,\end{aligned}$$ we take the Laplace transform with respect to $t$ in (\[ke1\]). With $u_{R}=e^{\frac{-i\pi}{a}} (\frac{2z^{a/2}}{aR})^{2/a}$, $r=\sqrt{s^{2}+(\frac{2}{a}z^{a/2})^{2}}$, $R=s+r$, $\lambda=(m-2)/2$, $z=|x||y|$ and $\xi=\langle x,y\rangle/z$, for ${\rm Re} \, s$ big enough, we obtain $$\begin{aligned}
\label{rl}
&&\mathcal{L}(K_{a}^{m}(x,y,t))\nonumber\\&=&2^{2\lambda/a}\Gamma\biggl(\frac{2\lambda+a}{a}\biggr)\frac{1}{r}\biggl(\frac{1}{R}\biggr)^{2\lambda/a}\frac{1-u_{R}^{2}}{(1-2\xi u_{R}+u_{R}^2)^{\lambda+1}} \nonumber\\
&=&2^{2\lambda/a}\Gamma\biggl(\frac{2\lambda+a}{a}\biggr)\frac{1}{r}\frac{R^{2/a}-\frac{e^{-2i\pi/a}(2/a)^{4/a}z^{2}}{R^{2/a}}}{(R^{2/a}-2\xi e^{-i\pi/a}(2/a)^{2/a}z+\frac{e^{-2i\pi/a}(2/a)^{4/a}z^{2}}{R^{2/a}} )^{\lambda+1}}
\nonumber
\\&=&2^{2\lambda/a}\Gamma\biggl(\frac{2\lambda+a}{a}\biggr)\frac{1}{r}\frac{(s+r)^{2/a}-e^{-2i\pi/a}(r-s)^{2/a}}{((s+r)^{2/a}-2\xi e^{-i\pi/a}(2/a)^{2/a}z+e^{-2i\pi/a}(r-s)^{2/a})^{\lambda+1}}.\end{aligned}$$
The validity of transforming term by term in (\[ke1\]) is guaranteed by the following theorem.
[@DG] Let the function $F(s)$ be represented by a series of $\mathcal{L}$-transforms $$F(s)=\sum_{v=0}^{\infty}F_{v}(s), \quad F_{v}(s)=\mathcal{L}(f_{v}(t)),$$ where all integrals $$\mathcal{L}(f_{v})=\int_{0}^{\infty}e^{-st}f_{v}(t)dt=F_{v}(s), \quad(v=0,1,\cdots)$$ converge in a common half-plane ${\rm Re}\,s \ge x_{0}$. Moreover, we require that the integrals $$\mathcal{L}(|f_{v}|)=\int_{0}^{\infty}e^{-st}|f_{v}(t)|dt=G_{v}, \quad(v=0,1,\cdots)$$ and the series $$\sum_{v=0}^{\infty}G_{v}(x_{0})$$converge which implies that $\sum_{v=0}^{\infty}F_{v}(s)$ converges absolutely and uniformly in ${\rm Re} \,s\ge x_{0}$. Then $\sum_{v=0}^{\infty}f_{v}(t)$ converges, absolutely, towards a function $f(t)$ for almost all $t\ge 0$; this f(t) is the original function of $F(s)$; $$\mathcal{L}\biggl(\sum_{v=0}^{\infty}f_{v}(t)\biggr)=\sum_{v=0}^{\infty}F_{v}(s).$$
Hence we can summarize our results as follows,
\[laplradial\] The kernel of the deformed Fourier transform in the Laplace domain is $$\begin{aligned}
\label{rl2}
&&\mathcal{L}(K_{a}^{m}(x,y,t))\nonumber
\nonumber
\\&=&2^{2\lambda/a}\Gamma\biggl(\frac{2\lambda+a}{a}\biggr)\frac{1}{r}\frac{(s+r)^{2/a}-e^{-2i\pi/a}(r-s)^{2/a}}{((s+r)^{2/a}-2\xi e^{-i\pi/a}(2/a)^{2/a}z+e^{-2i\pi/a}(r-s)^{2/a})^{\lambda+1}}\end{aligned}$$ where $r=\sqrt{s^{2}+(\frac{2}{a}z^{a/2})^{2}}.$
By direct computation, we have the following simpler expression when $m > 2$.
When $\lambda>0$, the kernel of the deformed Fourier transform in the Laplace domain is $$\begin{aligned}
&&\mathcal{L}(K_{a}^{m}(x,y,t))\nonumber\nonumber
\\&=&-2^{2\lambda/a}\Gamma\biggl(\frac{2\lambda}{a}\biggr)\frac{d}{d s}\biggl(\frac{1}{((s+r)^{2/a}-2\xi e^{-i\pi/a}(2/a)^{2/a}z+e^{-2i\pi/a}(r-s)^{2/a})^{\lambda}}\biggr)\end{aligned}$$ where $r=\sqrt{s^{2}+(\frac{2}{a}z^{a/2})^{2}}.$
Let us now look at a few special cases. When $a=1$, (\[rl2\]) reduces to $$\mathcal{L}(K_{1}^{m}(x,y,t))=\Gamma(2\lambda+1)\frac{s}{(s^2+2z+2\xi z)^{\lambda+1}}.$$ Using the formula in [@E2] $$\begin{aligned}
\label{nf1}\mathcal{L}^{-1}(2^{\nu+1}\pi^{-1/2}\Gamma(\nu+3/2)a^{\nu}\sqrt{s^2+a^2}^{-2\nu-3}s )=t^{\nu+1}J_{\nu}(at),\qquad \mbox{Re}\,\nu >-1, \mbox{Re}\, s>|\mbox{Im} \, a| \end{aligned}$$ and then setting $t=1$ in $K_{1}^{m}(x,y,t)$, we reobtain the kernel $$K_{1}^{m}(x,y)=\Gamma(\lambda+1/2)\tilde{J}_{\frac{m-3}{2}}(\sqrt{2(|x||y|+\langle x,y\rangle)})$$ with $\tilde{J}_{\nu}(z)=J_{\nu}(z)(z/2)^{-\nu}$, see [@Rad3].
When $a=2$, (\[rl2\]) reduces to $$\mathcal{L}(K_{2}^{m}(x,y,t))=\Gamma(\lambda+1)\frac{1}{(s+i\xi z)^{\lambda+1}}.$$ By the inverse transform formula in [@E2] $$\begin{aligned}
\mathcal{L}\biggl(\frac{t^{k-1}e^{-\alpha t}}{\Gamma(k)}\biggr)=\frac{1}{(s+\alpha)^{k}} \qquad k>0,\end{aligned}$$ and then putting $t=1$ in $K_{2}^{m}(x,y,t)$, we get the classical Fourier kernel $$K_{2}^{m}(x,y)=e^{-i\langle x, y\rangle}.$$
We are interested in the case when $a=\frac{2}{n}$, because it has a close relationship with the Dunkl kernel and Dunkl Bessel function associated with dihedral groups which we will discuss in Section 4. When $a=\frac{2}{n}$, the Fourier kernel in the Laplace domain is $$\begin{aligned}
\label{rl3} \mathcal{L}(K_{\frac{2}{n}}^{m}(x,y,t))= \Gamma(n\lambda+1)\frac{Q_{n-1}(s)}{P_{n}(s)^{\lambda+1}} ,\end{aligned}$$ with $$\begin{aligned}
Q_{n-1}(s)&=&\frac{(s+r)^{n}-e^{-in\pi}(r-s)^{n}}{2^{n}r},\\ P_{n}(s)&=&\frac{(s+r)^{n}-2\xi e^{-in\pi/2}(n)^{n}z+e^{-in\pi}(r-s)^{n}}{2^{n}}.\end{aligned}$$ By direct computation, we have $$\begin{aligned}
\label{dr1}\frac{d }{d s}P_{n}(s)=nQ_{n-1}(s),\end{aligned}$$ and $$\begin{aligned}
\label{dl1} \mathcal{L}(K_{\frac{2}{n}}^{m}(x,y,t))= \Gamma(n\lambda+1)\frac{\frac{d }{d s}P_{n}(s)}{n(P_{n}(s))^{\lambda+1}}
=-\Gamma(n\lambda)\frac{d}{ds}\frac{1}{P_{n}(s)^{\lambda}} ,\end{aligned}$$ when $\lambda>0$.
We can investigate both functions $Q_{n-1}(s)$ and $P_{n}(s)$ in more detail. This is done in the following lemma.
\[lem1\] The function $P_{n}(s)$ is a polynomial of degree $n$ in $s$ with the factorization $$P_{n}(s)=\prod_{l=0}^{n-1}\biggl(s+inz^{1/n} \cos\biggl(\frac{q+2\pi l}{n}\biggr)\biggr),$$ where $q=\arccos(\xi)$, $\xi=\frac{\langle x, y\rangle}{|x||y|}$. The function $Q_{n-1}(s)$ is a polynomial of degree $n-1$ in $s$. When $n$ is odd, $Q_{n-1}(s)$ has the factorization $$Q_{n-1}(s)=\prod_{l=1}^{n-1}\biggl(s-inz^{1/n}\cos\biggl(\frac{l\pi}{n}\biggr)\biggl).$$ When $n$ is even, $Q_{n-1}(s)$ has the factorization$$Q_{n-1}(s)=\prod_{l=0,l\neq\frac{n}{2}}^{n-1}\biggl(s-inz^{1/n}\sin\biggl(\frac{l\pi}{n}\biggr)\biggl).$$
1. We show that $P_{n}(s)$ is a polynomial of degree $n$ in $s$, $$\begin{aligned}
2^{n}P_{n}(s)&=&(s+r)^{n}-2\xi e^{-in\pi/2}(n)^{n}z+e^{-in\pi}(r-s)^{n}\\&=& (s+r)^{n}+(-1)^{n}(r-s)^{n}-2\xi e^{-in\pi/2}(n)^{n}z\\&=&\sum_{k=0}^{n}\binom{n}{k}s^{n-k}r^{k}+(-1)^{n}\sum_{k=0}^{n}\binom{n}{k}(-1)^{n-k}s^{n-k}r^{k}-2\xi e^{-in\pi/2}(n)^{n}z\\&=&\biggl(\sum_{k=0}^{n}\binom{n}{k}s^{n-k}r^{k}(1+(-1)^{k})\biggr)-2\xi e^{-in\pi/2}(n)^{n}z
\\&=&2\sum_{k=0}^{ \lfloor n/2 \rfloor}\binom{n}{2k}s^{n-2k}(s^2+(nz^{1/n})^2)^{k}-2\xi e^{-in\pi/2}(n)^{n}z.
\end{aligned}$$ Hence $2^{n}P_{n}(s)$ is a polynomial of degree $n$ in $s$. The coefficient of $s^{n}$ is $2\sum_{k=0}^{ \lfloor n/2 \rfloor}\binom{n}{2k}=2^{n}.$
2. We verify $2^{n}P_{n}(s_{l})=0$ with $s_{l}=-inz^{1/n} \cos(\frac{q+2\pi l}{n}),\quad l=0,\cdots, n-1$. Denote $\xi=\cos (q)=\frac{e^{iq}+e^{-iq}}{2}$. When $\sin(\frac{q+2\pi l}{n})\ge 0$, we have $$\begin{aligned}
2^{n}P_{n}(s_{l})&=&(-inz^{1/n})^{n}\biggl[\biggl(\cos\biggl(\frac{q+2\pi l}{n}\biggr)+i\sin\biggl(\frac{q+2\pi l}{n}\biggr)\biggr)^{n}-2\xi\\&&+\biggl(\cos\biggl(\frac{q+2\pi l}{n}\biggr)-i\sin\biggl(\frac{q+2\pi l}{n}\biggr)\biggr)^{n}\biggr]\\&=&(-inz^{1/n})^{n}\biggl(e^{iq}-2\biggl(\frac{e^{iq}+e^{-iq}}{2}\biggr)+e^{-iq}\biggr)\\&=&0.
\end{aligned}$$ Similarly, we have $2^{n}P_{n}(s_{l})=0$ when $\sin(\frac{q+2\pi l}{n})<0$. Hence, $s_{l}, l=0,\cdots, n-1$ are all roots of $2^{n}P_{n}$ and we get the factorization $$P_{n}(s)=\prod_{l=0}^{n-1}\biggl(s+inz^{1/n} \cos\biggl(\frac{q+2\pi l}{n}\biggr)\biggr)
.$$
3. For $2^{n}Q_{n-1}(s)$, we have $$\begin{aligned}
2^{n}Q_{n-1}(s)&=&\frac{(s+r)^{n}-e^{-in\pi}(r-s)^{n}}{r}\\
&=&\frac{1}{r}((s+r)^{n}-(-1)^{n}(r-s)^{n})
\\&=&\frac{1}{r}\sum_{k=0}^{n}\binom{n}{k}s^{n-k}r^{k}(1-(-1)^{n}(-1)^{n-k})
\\&=&\frac{2}{r}\sum_{k=0}^{\lfloor n/2 \rfloor }\binom{n}{2k+1}s^{n-2k-1}r^{2k+1}
\\&=&2\sum_{k=0}^{\lfloor n/2 \rfloor }\binom{n}{2k+1}s^{n-2k-1}(s^{2}+(nz^{1/n})^{2})^{k}.
\end{aligned}$$ So $2^{n}Q_{n-1}(s)$ is a polynomial of degree $n-1$ in $s$.
4. When $n$ is odd, $s_{l}=inz^{1/n}\cos(\frac{l\pi}{n})=inz^{1/n}\sin(\frac{\pi}{2}+\frac{l\pi}{n})$, $l=0,\cdots, n-1$ are $n$ roots of $(2^{n}rQ_{n-1})(s)=0$. Indeed, we have $r_{l}=\sqrt{s_{l}^2+(nz^{1/n})^2}=-nz^{1/n}\cos(\frac{\pi}{2}+\frac{l\pi}{n})$ and $$\begin{aligned}
2^{n}r_{l}Q_{n-1}(s_{l})&=&(s_{l}+r_{l})^{n}-e^{-in\pi}(r_{l}-s_{l})^{n}\\&=&(s_{l}+r_{l})^{n}+(r_{l}-s_{l})^{n}\\&=&(-nz^{1/n})^{n}(e^{-i\frac{\pi n}{2}-il\pi}+e^{i\frac{\pi n}{2}+il\pi})\\&=&0\end{aligned}$$ because $n$ is odd. Note that $r_{l}= 0$ if and only if when $l=0$. So $s_{l}$, $l=1,\cdots, n-1$ are the $n-1$ roots of the polynomial $Q_{n-1}(s)$. Hence, we have $$\begin{aligned}
Q_{n-1}(s)&=&\prod_{l=1}^{n-1}\biggl(s-inz^{1/n}\sin\biggl(\frac{\pi}{2}+\frac{l\pi}{n}\biggr)\biggl).\end{aligned}$$ When $n$ is even, we verify $2^{n}r_{l}Q_{n-1}(s_{l})=0$ with $s_{l}=inz^{1/n}\sin(\frac{l\pi}{n})$, $l=0, \cdots, n-1$. For $l\le \frac{n}{2}$,$$\begin{aligned}
2^{n}r_{l}Q_{n-1}(s_{l})&=&(s_{l}+r_{l})^{n}-e^{-in\pi}(r_{l}-s_{l})^{n}\\&=&(s_{l}+r_{l})^{n}-(r_{l}-s_{l})^{n}\\&=&(nz^{1/n})^{n}(e^{il\pi}-e^{-il\pi})\\&=&0.\end{aligned}$$ Similarly, for $l>\frac{n}{2}$, we have $2^{n}r_{l}Q_{n-1}(s_{l})=0$. Moreover, we have $r_{l}=0$ if and only if $l=\frac{n}{2}.$ So $s_{l}$, $l\neq\frac{n}{2}$ are the $n-1$ roots of the polynomial $Q_{n-1}(s)$. Hence we have $$\begin{aligned}
Q_{n-1}(s)&=&\prod_{l=0,l\neq\frac{n}{2}}^{n-1}\biggl(s-inz^{1/n}\sin\biggl(\frac{l\pi}{n}\biggr)\biggl).\end{aligned}$$
We now have all the tools necessary to compute the inverse Laplace transform. First we treat the case of dimension 2.
\[co\] For $a=\frac{2}{n}, n\in \mathbb{N}$ and $m=2$, we have $$K_{\frac{2}{n}}^{2}(x,y)=\frac{1}{n}\sum_{l=0}^{n-1}e^{-inz^{1/n} \cos(\frac{q+2\pi l}{n})}.$$
We have, using (\[rl3\]) and (\[dr1\]) $$\begin{aligned}
\mathcal{L}(K^{2}_{\frac{2}{n}}(x,y,t))=\frac{Q_{n-1}(s)}{P_{n}(s)}=\frac{1}{n}\frac{\frac{d}{d s}P_{n}(s)}{P_{n}(s)}=\frac{1}{n}\sum_{l=0}^{n-1}\frac{1}{s+inz^{1/n} \cos(\frac{q+2\pi l}{n})}
.\end{aligned}$$ Taking the inverse Laplace transform and putting $t=1$ yields the result.
This result was previously obtained in [@Rad2] in a different way, using series multisection.
When the dimension $m>2$, we first use (\[dl1\]) to obtain $$\begin{aligned}
\label{kl1} K_{\frac{2}{n}}^{m}(x,y,t)
=-\Gamma(n\lambda)\mathcal{L}^{-1}\biggl(\frac{d}{ds}\frac{1}{P_{n}(s)^{\lambda}}\biggr).\end{aligned}$$ The inverse Laplace transform can be computed using the property of the Laplace transform $$\begin{aligned}
\mathcal{L}^{-1}\biggl(-\frac{d}{ds}\mathcal{L}(f(t))\biggr)=tf(t) \end{aligned}$$ and the partial fraction decomposition $$\begin{aligned}
\label{kl2}
\mathcal{L}^{-1}\biggl(\frac{1}{P_{n}(s)^{\lambda}}\biggr)=\sum_{k=1}^{n}\sum_{l=1}^{\lambda} \frac{\Phi_{kl}(a_{k}) }{(\lambda-l)!(l-1)!} t^{\lambda-l}e^{a_{k}t}\end{aligned}$$ with $a_{k}=-inz^{1/n} \cos(\frac{q+2\pi k}{n})$, $q=\arccos(\xi)$ and $$\Phi_{kl}(s)=\frac{d^{l-1}}{d s^{l-1}}\biggl[\biggl(\frac{s-a_{k}}{P_{n}(s)}\biggr)^{\lambda}\biggr].$$ Putting $t=1$ in (\[kl1\]) and (\[kl2\]), then yields
\[th3\] When $a=\frac{2}{n}, n\in \mathbb{N}$, the kernel of the $(0,a)$-generalized Fourier transform in even dimension $m>2$ is given by $$K_{\frac{2}{n}}^{m}(x,y)=\Gamma(n\lambda)\sum_{k=1}^{n}\sum_{l=1}^{\lambda} \frac{\Phi_{kl}(-inz^{1/n} \cos(\frac{q+2\pi k}{n})) }{(\lambda-l)!(l-1)!}e^{-inz^{1/n} \cos(\frac{q+2\pi k}{n})}.$$
As we have given the factored form of $P_{n}(s)$ in Lemma \[lem1\], it is possible to give an explicit formula of $\Phi_{kl}(s)$ by the following result from [@Bo].
\[pfd\] Suppose $\phi(s)$ is a proper rational function having $m$ zeros $ -\sigma_{h}$ of multiplicity $M_{h}$ and $n$ poles $-s_{k}$ of multiplicity $N_{k}$, $$\phi(s)=\frac{\prod_{h=1}^{m}(s+\sigma_{h})^{M_{h}}}{\prod_{k=1}^{n}(s+s_{k})^{N_{k}}}.$$ Define the functions $$f_{k}(s)=\phi(s)(s+s_{k})^{N_{k}}=\frac{\prod_{h=1}^{m}(s+\sigma_{h})^{M_{h}}}{\prod_{\substack{k'=1,\\k'\neq k}}^{n}(s+s_{k'})^{N_{k'}}}, \qquad k=1,2,\cdots, n,$$ obtained from $\phi(s)$ by removing the factor $(s+s_{k})^{N_{k}}.$ The first derivative of $f_{k}(s)$ is given by $$f_{k}^{(1)}(s)=f_{k}(s)g_{k}(s)$$ with $$g_{k}(s)=\sum_{h=1}^{m}\frac{M_{h}}{s+\sigma_{h}}-\sum_{\substack{k'=1,\\k'\neq k}}^{n}\frac{N_{k'}}{s+s_{k'}} .$$ The $r$-th derivative of $g_{k}$ is given by $$g_{k}^{(r)}(s)=(-1)^{r}r!\biggl[\sum_{h=1}^{m}\frac{M_{h}}{(s+\sigma_{h})^{r+1}} -\sum^{n}_{\substack{k'=1,\\k'\neq k}}\frac{N_{k'}}{(s+s_{k'})^{r+1}}\biggr].$$ The $i$-th derivative of $f_{k}(s)$ can be expressed by $$\begin{aligned}
&&f_{k}^{(i)}=(-1)^{i-1}f_{k}^{(0)}\\&&
\begin{vmatrix}-1 & 0&0& \cdots&0&0&g_{k}^{(0)} \\
g_{k}^{(0)} & -1&0& \cdots&0&0&g_{k}^{(1)}\\
2g_{k}^{(1)} & g_{k}^{(0)}&-1& \cdots&0&0&g_{k}^{(2)}\\
&&&\cdots&&&\\ (i-1)g_{k}^{(i-2)} & \binom{i-1}{2}g_{k}^{(i-3)}&\binom{i-1}{3}g_{k}^{(i-4)}& \cdots&(i-1)g_{k}^{(1)}&g_{k}^{(0)}&g_{k}^{(i-1)} \end{vmatrix}
.\end{aligned}$$
Generating function when $a=\frac{2}{n}$ and $m$ even
-----------------------------------------------------
For fixed $a=\frac{2}{n}$ and $n\in \mathbb{N}$, we define the formal generating function of the $(0,a)$-generalized Fourier kernel of even dimension by $$G_{\frac{2}{n}}(x,y,\varepsilon)=\sum_{\lambda=0}^{\infty}\frac{1}{2^{n\lambda} \Gamma(n\lambda+1)}(-2e^{-in\pi/2}(n)^{n}z\varepsilon)^{\lambda}K_{\frac{2}{n}}^{m}(x,y)
.$$ We introduce an auxiliary variable $t$ in the generating function as $$G_{\frac{2}{n}}(x, y, \varepsilon, t)=\sum_{\lambda=0}^{\infty}\frac{1}{2^{n\lambda} \Gamma(n\lambda+1)}(-2e^{-in\pi/2}(n)^{n}z\varepsilon)^{\lambda}K_{\frac{2}{n}}^{m}(x,y,t).$$ Then we compute the Laplace transform of $G_{\frac{2}{n}}(x, y, \varepsilon, t)$, and get $$\begin{aligned}
\mathcal{L}(G_{\frac{2}{n}}(x, y, \varepsilon, t))&=&\sum_{\lambda=0}^{\infty}
\frac{1}{r}\frac{((s+r)^{n}-e^{-in\pi}(r-s)^{n})(-2e^{-in\pi/2}(n)^{n}z\varepsilon)^{\lambda}}{((s+r)^{n}-2\xi e^{-in\pi/2}(n)^{n}z+e^{-in\pi}(r-s)^{n})^{\lambda+1}}
\\&=&\frac{1}{r}\frac{(s+r)^{n}-e^{-in\pi}(r-s)^{n}}{(s+r)^{n}-2(\xi+\varepsilon)e^{-in\pi/2}(n)^{n}z+e^{-in\pi}(r-s)^{n}}.\end{aligned}$$ Comparing with Theorem \[co\], we find the only difference is that $\xi$ in the latter becomes $\xi+\varepsilon$. Now we can give the generating function by taking the inverse Laplace transform and setting $t=1$.
Let $a=2/n$, with $n\in \mathbb{N}$. Then the formal generating function of the $(0,a)$-generalized Fourier kernel of even dimension is $$\begin{aligned}
G_{\frac{2}{n}}(x, y, \varepsilon)&=&\sum_{\lambda=0}^{\infty}\frac{1}{2^{n\lambda} \Gamma(n\lambda+1)}(-2e^{-in\pi}(n)^{n}z\varepsilon)^{\lambda}K_{\frac{2}{n}}^{m}(x,y)\\&=&\frac{1}{n}\sum_{l=0}^{n-1}e^{-inz^{1/n} \cos(\frac{\tilde{q}+2\pi l}{n})},\end{aligned}$$ with $\tilde{q}=\arccos (\xi+\varepsilon)$.
By taking consecutive derivatives with respect to $\varepsilon$, we can get an alternative expression for the even dimensional kernel $K_{\frac{2}{n}}^{m}(x,y)$. This coincides with Proposition 2 in [@Rad2] and Theorem 1 in [@Dn].
The bounds of the kernel when $a=\frac{2}{n}$ and $m\ge 2$
----------------------------------------------------------
In this section, we prove the boundedness of the kernel $K_{\frac{2}{n}}^{m}(x,y)$, $m\ge 2$. This is not obvious from the explicit expansion in Theorem \[th3\] as we don’t know the bounds of $\Phi_{kl}(a_{k})$ in (\[kl2\]). We first establish a technical lemma. Let us recall the convolution formula of the Laplace transform. Denoting $\mathcal{L}(g(t))=G(s)$ and $\mathcal{L}(f(t))=F(s)$, we have $$\begin{aligned}
\label{lc}\mathcal{L}^{-1}(G(s)F(s))&=&\int_{0}^{t}g(t-\tau)f(\tau)d\tau.\end{aligned}$$
\[le1\] For $a_{j}\in \mathbb{R}, j=1,\cdots,n,$ and $ k>0$, put $$F_{n, k}(s)=\displaystyle\frac{1}{\prod_{j=1}^{n}(s+ia_{j})^{k}}$$ with inverse Laplace transform $$f_{n, k}(t)=\mathcal{L}^{-1}(F_{n,k}(s)).$$Then $$|f_{n, k}(t)|\le \frac{t^{nk-1}}{\Gamma(nk)}, \qquad\forall t\in ]0,\infty[.$$
We prove it by induction. By the Laplace transform formula $$\begin{aligned}
\mathcal{L}\biggl(\frac{t^{k-1}e^{-\alpha t}}{\Gamma(k)}\biggr)=\frac{1}{(s+\alpha)^{k}}, \qquad k>0,\end{aligned}$$ we have $$f_{1, k}(t)=\frac{t^{k-1}}{\Gamma(k)}e^{-ia_{1}t},$$ so $$|f_{1, k}(t)|\le \frac{t^{k-1}}{\Gamma(k)}.$$ When $n=2$, by the convolution formula (\[lc\]), we have $$\begin{aligned}
|f_{2, k}(t)|&=&\biggl|\int_{0}^{t}\frac{(t-\tau)^{k-1}e^{-ia_{1}(t-\tau)}}{\Gamma(k)} f_{1, k}(\tau)d\tau\biggr|\le\int_{0}^{t}\frac{(t-\tau)^{k-1}}{\Gamma(k)}|f_{1, k}(\tau)|d\tau\\
&\le&\frac{1}{\Gamma(k)^{2}}\int_{0}^{t}(t-\tau)^{k-1}\tau^{k-1}d\tau
=\frac{t^{2k-1}}{\Gamma(k)^{2}}\int_{0}^{1}(1-x)^{k-1}x^{k-1}dx\\&=&\frac{t^{2k-1}}{\Gamma(2k)}\end{aligned}$$ where we have substituted $\tau=tx$ in the third integral. We assume $$\begin{aligned}
\label{mi1}|f_{n-1, k}(t)|\le \frac{t^{(n-1)k-1}}{\Gamma((n-1)k)}.\end{aligned}$$ Then by the convolution formula (\[lc\]) and (\[mi1\]), we have $$\begin{aligned}
|f_{n, k}(t)|&\le& \int_{0}^{t}\frac{(t-\tau)^{k-1}e^{-ia_{n}(t-\tau)}}{\Gamma(k)}|f_{n-1}(\tau)|d\tau
\\&\le&\int_{0}^{t}\frac{(t-\tau)^{k-1}}{\Gamma(k)}\frac{\tau^{(n-1)k-1}}{\Gamma((n-1)k)}d\tau
\\&\le&\frac{t^{nk-1}}{\Gamma(k)\Gamma((n-1)k)}\int_{0}^{1}x^{(n-1)k-1}(1-x)^{k-1}dx
\\&\le&t^{nk-1}\frac{B((n-1)k,k)}{\Gamma(k)\Gamma((n-1)k)}\\&=&\frac{t^{nk-1}}{\Gamma(nk)}.\end{aligned}$$ where we used the same substitution as before, and with $B(u,v)$ the beta function.
By (\[dl1\]), when $\lambda>0$, $$\begin{aligned}
\label{lk1} \mathcal{L}(K_{\frac{2}{n}}^{m}(x,y,t))=-\Gamma(n\lambda)\frac{d}{ds}\frac{1}{(P_{n}(s))^{\lambda}}
=-\Gamma(n\lambda)\frac{d}{ds}\frac{1}{\biggl(\prod_{l=0}^{n-1}\biggl(s+inz^{1/n} \cos\biggl(\frac{q+2\pi l}{n}\biggr)\biggr)\biggr)^{\lambda}}. \end{aligned}$$ Setting $t=1$, we get $$\begin{aligned}
K_{\frac{2}{n}}^{m}(x,y)=\Gamma(n\lambda)f_{n, \lambda}(1)\end{aligned}$$ with $a_{l}=nz^{1/n} \cos\biggl(\frac{q+2\pi l}{n}\biggr)$ in $f_{n,\lambda}(t)$. The problem of finding an integral expression of $K_{\frac{2}{n}}^{m}(x,y)$ thus reduces to finding an integral expression of the function $f_{n, \lambda}(t)$.
From the Laplace transform table [@E2], we have $$\begin{aligned}
\mathcal{L}^{-1}\biggl(\frac{d}{ds}\biggl(\frac{1}{((s+ib)(s-ib))^{\nu+1/2}}\biggr)\biggr)
=\frac{\sqrt{\pi}}{2^{\nu}\Gamma(\nu+1/2)}t^{\nu+1}\frac{J_{\nu}(b t)}{b^{\nu}}, \qquad {\rm Re}\, \nu>-1, {\rm Re}\,s>|{\rm Im}\, b|.\end{aligned}$$ Compared with (\[lk1\]), the Fourier kernel $K_{\frac{2}{n}}^{m}(x,y)$ and $f_{n, k}(t)$ could be thought of as a generalization of the Bessel function. We will see similar behavior in the Dunkl case, see Section 4.
By the inverse Laplace formula from [@E2], $$\begin{aligned}
\mathcal{L}^{-1}\biggl(\frac{\Gamma(\nu)}{(s+a)^{\nu}(s+b)^{\nu}}\biggr)=\sqrt{\pi}\biggl(\frac{t}{a-b}\biggr)
^{\nu-1/2}e^{-\frac{(a+b)t}{2}}I_{\nu-1/2}\biggl(\frac{a-b}{2}t\biggr), \qquad {\rm Re}\,\nu>0.\end{aligned}$$ we can express $f_{n, \lambda}(t)$ as the convolution of Bessel functions and exponential functions, using (\[lc\]).
In particular, when $n=3$, and ${\rm Re}\,s >0$, we have $$\begin{aligned}
\label{I3} f_{3, k}(t)=\mathcal{L}^{-1}(F_{3, k}(s))&=&\mathcal{L}^{-1}\biggl(\displaystyle\frac{1}{\prod_{j=1}^{3}(s+ia_{j})^{k}}\biggr)
\nonumber\\&=&\frac{t^{3k-1}}{\Gamma(3k)}e^{ia_{1}t}\Phi_{2}(k,k;3k; i(a_{1}-a_{2})t, i(a_{1}-a_{3})t )\end{aligned}$$ where $\Phi_{2}(c_{1},c_{2};c_{3};w,z)=\sum_{k,l=0}^{\infty}\frac{(c_{1})_{k}(c_{2})_{l}}{(c_3)_{k+l}}\frac{w^{k}z^{l}}{k!l!},$ see [@PB]. Another derivation of the expression obtained here without using Laplace transform is given in [@DDa].
Now we can give the main result of this subsection,
\[th7\] For $n\in \mathbb{N}$ and $m\ge 2$, the kernel of the $(0,2/n)$-generalized Fourier transform satisfies $$|K^{m}_{\frac{2}{n}}(x,y)| \le 1.$$
When $a=\frac{2}{n}$, the Laplace transform of the $(0,a)$-generalized Fourier kernel is $$\begin{aligned}
&&\mathcal{L}(K_{\frac{2}{n}}^{m}(x,y,t))= \Gamma(n\lambda+1)G_{1}(s)G_{2}(s)\end{aligned}$$ with $$\begin{aligned}
&&G_{1}(s)=\frac{Q_{n-1}(s)}{\prod_{l=0}^{n-1}(s+inz^{1/n} \cos(\frac{q+2\pi l}{n})) }, \\ &&G_{2}(s)=\frac{1}{(\prod_{l=0}^{n-1}(s+inz^{1/n} \cos(\frac{q+2\pi l}{n})))^{\lambda}}.\end{aligned}$$ Denote $g_{j}(t)=\mathcal{L}^{-1}(G_{j}), j=1,2.$ By Lemma \[le1\], we know that the inverse Laplace transform $g_{2}(t)$ of $G_{2}(s)$ is bounded by $\frac{t^{n\lambda-1}}{\Gamma(n\lambda)}$. By Theorem \[co\], we know that $g_{1}(t)=K_{\frac{2}{n}}^{2}(x,y,t)$ is bounded by $1$ for any $t\in \mathbb{R}$. Using the convolution formula (\[lc\]) again, then setting $t=1$, we have $$\begin{aligned}
|K_{\frac{2}{n}}^{m}(x,y)|&=& \Gamma(n\lambda+1)\biggl|\int_{0}^{1}g_{1}(1-\tau) g_{2}(\tau)d\tau \biggr|
\\& \le& \Gamma(n\lambda+1) \int_{0}^{1}\frac{\tau^{n\lambda-1}}{\Gamma(n\lambda)}d\tau
\\&=&\frac{ \Gamma(n\lambda+1)}{\Gamma(n\lambda) n\lambda}\\&=&1.\end{aligned}$$
With this result, valid for both even and odd dimension, we could get the bound of the $(\kappa, a)$-generalized Fourier kernel for any reduced root system with positive multiplicity function $\kappa$ and some $\alpha$. Theorem \[th7\] greatly extends the applicability of the uncertainty principle and generalized translation operator in [@J1] and [@Go].
Integral expression of the kernel for arbitrary $a>0$
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In Theorem \[th7\], we have shown that the Fourier kernel $K_{\frac{2}{n}}^{m}(x,y)$ when $m\ge 2$ is the Laplace convolution of the Fourier kernel when $m=2$ and the function $f_{n, k}(t)$ in Lemma \[le1\]. In this subsection we give the integral expression of the Fourier kernel of $K_{a}^{m}(x,y)$ for $m\ge 2$ and $a>0$.
For general $a>0$ and $m\ge 2$, the Fourier kernel in the Laplace domain can be written as $$\begin{aligned}
&&\mathcal{L}(K_{a}^{m}(x,y,t))\nonumber\\&=&2^{2\lambda/a}\Gamma\biggl(\frac{2\lambda+a}{a}\biggr)\frac{1}{r}\biggl(\frac{1}{R}\biggr)^{2\lambda/a}\frac{1-u_{R}^{2}}{(1-2\xi u_{R}+u_{R}^2)^{\lambda+1}} \nonumber\\&=&2^{2\lambda/a}\Gamma\biggl(\frac{2\lambda+a}{a}\biggr)\frac{1}{r}\biggl(\frac{r-s}{(\frac{2}{a}z^{a/2})^2}\biggr)^{2\lambda/a}
\frac{1-u_{R}^{2}}{((u_{R}-e^{i\varrho})(u_{R}-e^{-i\varrho}))^{\lambda+1}},\end{aligned}$$ where $u_{R}=e^{\frac{-i\pi}{a}} (\frac{2z^{a/2}}{aR})^{2/a}$, $r=\sqrt{s^{2}+(\frac{2}{a}z^{a/2})^{2}}$, $R=s+r$ and $\xi=\frac{e^{i\varrho}+e^{-i\varrho}}{2}$.
It is possible to give an integral expression of this kernel in terms of the generalized Mittag-Leffler function. We give the definition and its Laplace transform here, see also Chapter 2 in [@MH].
\[DefML\] The generalized Mittag-Leffler function is defined by $$E_{\epsilon,\gamma}^{\delta}(z):=\sum_{n=0}^{\infty}\frac{(\delta)_{n}z^{n}}{\Gamma(\epsilon n+\gamma)n!},$$ where $\epsilon, \gamma, \delta \in \mathbb{C}$ with ${\rm Re}\, \epsilon >0.$ For $\delta=1,$ it reduces to the Mittag-Leffler function.
The Laplace transform of the generalized Mittag-Leffler function is $$\mathcal{L}(t^{\gamma-1}E^{\delta}_{\epsilon, \gamma}(bt^{\epsilon}))=\frac{1}{s^{\gamma}}\frac{1}{(1-bs^{-\epsilon})^{\delta}}$$ where ${\rm Re}\, \epsilon>0$, ${\rm Re}\, \gamma>0$, ${\rm Re}\, s>0$ and $s>|b|^{1/({\rm Re}\, \epsilon)}$, see [@MH].
Now, we give the integral expression of the $(0, a)$-generalized Fourier kernel as follows.
\[ga\] Let $b_{\pm}=e^{\pm i\varrho}e^{i\pi/a}(\frac{2}{a})^{2/a}z$ and $$h(t)=z^{-2(\lambda+1)}\int_{0}^{t}\zeta^
{\frac{2}{a}(\lambda+1)-1}E^{\lambda+1}_{\frac{2}{a}, \frac{2}{a}(\lambda+1)}(b_{+}\zeta^{\frac{2}{a}})
(t-\zeta)^
{\frac{2}{a}(\lambda+1)-1}E^{\lambda+1}_{\frac{2}{a}, \frac{2}{a}(\lambda+1)}(b_{-}(t-\zeta)^{\frac{2}{a}})d\zeta
.$$ Then for $a>0$ and $m\ge 2$, the kernel of the $(0,a)$-generalized Fourier transform is $$\begin{aligned}
K_{a}^{m}(x,y)&=&c^{m}_{a}\int_{0}^{1}\biggl((1+2\tau)^{-\frac{\lambda}{a}}
J_{\frac{2\lambda}{a}}\biggl(\frac{2}{a}z^{a/2}\sqrt{1+2\tau}\biggr)\\&&-e^{-i\frac{2\pi}{a}}(1+2\tau)^{-\frac{\lambda+2}{a}}
J_{\frac{2\lambda+4}{a}}\biggl(\frac{2}{a}z^{a/2}\sqrt{1+2\tau}\biggr)\biggr)h(\tau)d\tau.\end{aligned}$$ with $c^{m}_{a}=2^{-(2\lambda+4)/a}\Gamma\biggl(\frac{2\lambda+a}{a}\biggr)e^{-i\frac{2\pi(\lambda+1)}{a}}a^{4(\lambda+1)/a}$.
Denote $\mathcal{L}(K_{a}^{m}(x,y,t))=H_{1}(s)H_{2}(s)$ where $$\begin{aligned}
H_{1}(s)&=&\frac{1}{(u_{R}-e^{i\varrho})^{\lambda+1}}\cdot\frac{1}{(u_{R}-e^{-i\varrho})^{\lambda+1}}, \\H_{2}(s)&=&2^{2\lambda/a}\Gamma\biggl(\frac{2\lambda+a}{a}\biggr)\frac{1}{r}\biggl(\frac{1}{R}\biggr)^{2\lambda/a}(1-u_{R}^{2}).\end{aligned}$$ By direct computation, we have $$H_{1}(s)=e^{-i\frac{2\pi(\lambda+1)}{a}}\biggl(\biggl(\frac{a}{2}\biggr)^{2/a}z^{-1}\biggr)^{2(\lambda+1)}\biggl[\frac{1}{(\varpi^{2/a}-b_{+})^{\lambda+1}}\cdot\frac{1}{(\varpi^{2/a}-b_{-})^{\lambda+1}}\biggr]$$ with $\varpi=r-s$.
Using the generalized Mittag-Leffler function, we have $$\begin{aligned}
\label{leff}\mathcal{L}^{-1}\biggl(\frac{1}{(s^{2/a}-b)^{\lambda+1}}\biggr)=t^
{\frac{2}{a}(\lambda+1)-1}E^{\lambda+1}_{\frac{2}{a}, \frac{2}{a}(\lambda+1)}(bt^{\frac{2}{a}}).\end{aligned}$$ Now by the inverse Laplace transform formula from [@PB] $$\begin{aligned}
\label{lb}\mathcal{L}^{-1}\biggl(\frac{(\sqrt{s^2+a^2}-s)^{\nu}}{\sqrt{s^2+a^2}}F(\sqrt{s^2+a^2}-s)\biggr)=(a^2t)^{\nu/2}\int_{0}^{t}(t+2\tau)^{-\nu/2}J_{\nu}(a\sqrt{t^2+2\tau t})f(\tau)d\tau\end{aligned}$$ where $\mathcal{L}(f(t))=F(s),$ ${\rm Re} \,\nu>-1$ and ${\rm Re} \, s>|{\rm Im}\, a|$ and the Laplace convolution formula (\[lc\]), we get the result.
Dunkl kernel associated to the dihedral group {#dunklFT}
=============================================
Integral expression of the kernel
---------------------------------
The dihedral group $I_{k}$ is the group of symmetries of the regular $k$-gon. We use complex coordinates $z_{0}=x+iy$ and identify $\mathbb{R}^{2}$ with $\mathbb{C}$. For a fixed $k$ and $j\in\{0,1,\cdots,k-1\}$, the rotations in $I_{k}$ consist of $z_{0}\rightarrow z_{0}e^{2ij\pi/k}$ and the reflections in $I_{k}$ consist of $z_{0}\rightarrow \bar{z}_{0}e^{2ij\pi/k}$. In particular, we have $I_{1}=\mathbb{Z}_{2}$ and $I_{2}=\mathbb{Z}_{2}^{2}$. The weight function associated with $I_{2k}$ and $\kappa=(\alpha,\beta)$ is given by $$\upsilon_{\kappa}(z_{0})=\biggl|\frac{z_{0}^{k}-\overline{z_{0}}^{k}}{2i}\biggr|^{2\alpha}\biggl|\frac{z_{0}^{k}+\overline{z_{0}}^{k}}{2}\biggr|^{2\beta}.$$ The weight function $\upsilon_{\kappa}(z_{0})$ associated with the group $I_{k}$, when $k$ is an odd integer, is the same as the weight function $\upsilon_{(\alpha,\beta)}(z_{0})$ associated with the group $I_{2k}$ with $\beta=0$, i.e. $$\upsilon_{\kappa}(z_{0})=\biggl|\frac{z_{0}^{k}-\overline{z_{0}}^{k}}{2i}\biggr|^{2\alpha}.$$ We also put $P_{j}(G;x,y)$ the reproducing kernel of $\mathcal{H}_{j}(\upsilon_{\kappa})$ and by $P(G;x,y)$ the Poisson kernel, see (\[pd\]) and (\[pois\]). We denote by $$d\mu_{\gamma}(w)=c_{\gamma}(1+w)(1-w^2)^{\gamma-1}dw$$ with $c_{\gamma}=[B(\frac{1}{2}, \gamma)]^{-1}.$ It was proved that finding a closed formula of the Poisson kernel which reproduces any $h$-harmonic in the disk reduces to the cases $k=1$ and $k=2$, see [@D; @DX].
[@DX] \[DunklPoisson\] For each weight function $\upsilon_{\kappa}(z)$ associated with the group $I_{2k}$, the Poisson kernel is given by $$P(I_{2k}; z_{1}, z_{2})=\frac{1-|z_{1}|^{2}|z_{2}|^{2}}{1-|z_{1}^{k}|^{2}|\overline{z_{2}}^{k}|^{2}} \frac{|1-z_{1}^{k}\overline{z_{2}}^{k}|^{2}}{|1-\overline{z_{1}}z_{2}|^{2}} P(I_{2}; z_{1}^{k}, z_{2}^{k}),$$ where the Poisson kernel $P(I_{2}; z_{1}, z_{2} )$ associated with $\upsilon_{\kappa}(x+iy)=|y|^{2\alpha}|x|^{2\beta}$ is given by $$\begin{aligned}
P(I_{2}; z_{1}, z_{2})&=&
\int_{-1}^{1}\int_{-1}^{1}\frac{1-|z_{1}z_{2}|^{2}}{[1-2({\rm Im} \, z_{1})({\rm Im} \, z_{2})u -2({\rm Re} \, z_{1}) ({\rm Re} \, z_{2}) v+|z_{1}z_{2}|^{2} ]^{\alpha+\beta+1}}d\mu_{\alpha}(u)d\mu_{\beta}(v).\end{aligned}$$ For each weight function $\upsilon_{\kappa}(z)$ associated with odd-$k$ dihedral group $I_{k}$, the Poisson kernel is given by $$P(I_{k};z_{1}, z_{2})=\frac{1-|z_{1}|^2|z_{2}|^2}{1-|z_{1}^{k}|^2|\overline{z_{2}}^{k}|^2}\frac{|1-z_{1}^{k}\overline{z_{2}}^{k}|^{2}}{|1-z_{1}\overline{z_{2}}|^2}P(I_{1};z_{1}^{k},z_{2}^{k})$$ where the Poisson kernel $P(I_{1}; z_{1}, z_{2}) $ associated with $\upsilon_{\kappa}(x+iy)=|y|^{2\alpha}$ is given by $$P(I_{1};z_{1},z_{2})=\int_{-1}^{1}\frac{1-|z_{1}z_{2}|^2}{(1-2({\rm Im}\, z_{1})({\rm Im}\, z_{2})u-2({\rm Re}\, z_{1})({\rm Re} \,z_{2})+|z_{1}z_{2}|^2)^{\alpha+1}}d\mu_{\alpha}(u).$$
In the following, we write $z_{1}=|z_{1}|\omega, z_{2}=|z_{2}|\eta \in \mathbb{C}$ and $b=|z_{1}||z_{2}|$. Based on the $\mathfrak{sl}_{2}$ relation of $\Delta_{\kappa}, |x|^2$ and the Euler operator, an orthonormal basis of $L^{2}(\mathbb{R}^{m}, \upsilon_{\kappa}(x)dx)$ for the general Dunkl case and a series expansion of the Dunkl kernel was constructed in [@Said; @SKO]. In particular, the Dunkl kernel $E_{\kappa}(z_{1}, z_{2})=B_{\kappa, 2}(x,y)$ associated with the dihedral group $I_{k}$ has the following series expansion (see also Theorem \[the1\]) $$\begin{aligned}
\label{fe1}
E_{\kappa}(z_{1}, z_{2})= 2^{\langle \kappa \rangle}\Gamma(\langle \kappa\rangle +1)\sum_{j=0}^{\infty}(-i)^{j} b^{-\langle \kappa \rangle} J_{j+\langle \kappa \rangle}(b) P_{j}(I_{k};\omega,\eta)\end{aligned}$$ with $$\langle \kappa \rangle=\left\{
\begin{array}{ll}
(\alpha+\beta)k/2, & \hbox{when $k$ is even;} \\
k\alpha, & \hbox{ when $k$ is odd.}
\end{array}
\right.$$ We introduce an auxiliary variable $t$ in (\[fe1\]) as follows $$\begin{aligned}
E_{\kappa}(z_{1}, z_{2}, t)= 2^{\langle \kappa \rangle}\Gamma(\langle \kappa\rangle +1)\sum_{j=0}^{\infty}(-i)^{j} b^{-\langle \kappa \rangle} J_{j+\langle \kappa \rangle}(bt) P_{j}(I_{k};\omega,\eta).\end{aligned}$$ Then fixing $z_{1}, z_{2}\in \mathbb{C}$, we take the Laplace transform with respect to $t$. Using (\[l1\]), $r=(s^2+b^2)^{1/2}$ and $R=s+r$, for ${\rm Re} \,s$ big enough, we have $$\begin{aligned}
\mathcal{L}(E_{\kappa}(z_{1}, z_{2}, t))\nonumber&=&\frac{2^{\langle \kappa \rangle}\Gamma(\langle \kappa\rangle +1)}{rR^{\langle \kappa\rangle}}\sum_{j=0}^{\infty} \biggl(\frac{-ib}{R}\biggr)^{j} P_{j}(I_{k}; \omega,\eta)\\&=&\frac{2^{\langle \kappa \rangle}\Gamma(\langle \kappa\rangle +1)}{rR^{\langle \kappa \rangle}}P\biggl(I_{k}; \omega, \frac{-ib}{R}\eta\biggr)\end{aligned}$$ where $P(I_{k}; \omega, z_{0}\eta),$ $|z_{0}|<1$ is the analytic continuation of the Poisson kernel $P(I_{k}; \omega, b\eta)$ obtained by acting with the intertwining operator $V_{\kappa}$ on $x$ on both sides of (\[ac2\]).
In order to get the integral expression of the Dunkl kernel, we first denote and simplify $$\begin{aligned}
f_{I_{2k}}(s)&=&\frac{2^{k(\alpha+\beta)}}{rR^{k(\alpha+\beta)}}\frac{1-(\frac{-ib}{ R})^2}{1-(\frac{-ib}{ R})^{2k}}\frac{1-2(\frac{-ib}{ R})^{k}{\rm Re}\,(\omega^{k}\bar{\eta}^{k})+(\frac{-ib}{ R})^{2k}}{1-2{\rm Re}\,(\omega\bar{\eta})(\frac{-ib}{ R})+(\frac{-ib}{ R})^2}\\
&&\times \frac{1-(\frac{-ib}{ R})^{2k}}{(1-2(\frac{-ib}{ R})^{k}(({\rm Im}\, \omega^{k})({\rm Im} \,\eta^{k})u+({\rm Re}\, \omega^{k})({\rm Re} \, \eta^{k})v)+(\frac{-ib}{ R})^{2k})^{\alpha+\beta+1}}\\
&=& \frac{2^{k(\alpha+\beta)}}{r}\frac{\displaystyle \biggl(R+\frac{b^2}{R}\biggr)\biggl(R^{k}-2(-ib)^{k}{\rm Re}\,(\omega^{k}\bar{\eta}^{k})+\displaystyle \frac{(-ib)^{2k}}{R^{k}}\biggr)}{\displaystyle R-2(-ib){\rm Re}\,(\omega\bar{\eta})+\frac{(-ib)^2}{R}}
\\&&\times\frac{1}{\displaystyle \biggl(R^{k}-2(-ib)^{k}(({\rm Im} \,\omega^{k})({\rm Im} \,\eta^{k})u+({\rm Re}\, \omega^{k})({\rm Re}\, \eta^{k})v)+\frac{(-ib)^{2k}}{R^{k}} \biggr)^{\alpha+\beta+1}}\end{aligned}$$ and $$\begin{aligned}
g_{I_{k}}(s)&=&\frac{2^{k\alpha}}{rR^{k\alpha}}\frac{1-(\frac{-ib}{ R})^2}{1-(\frac{-ib}{ R})^{2k}}\frac{1-2(\frac{-ib}{ R})^{k}\mbox{Re}\,(\omega^{k}\bar{\eta}^{k})+(\frac{-ib}{ R})^{2k}}{1-2\mbox{Re}\,(\omega\bar{\eta})(\frac{-ib}{ R})+(\frac{-ib}{ R})^2}
\\&&\times\frac{1-(\frac{-ib}{ R})^{2k}}{(1-2(\frac{-ib}{ R})^{k}((\mbox{Im}\, \omega^{k})(\mbox{Im} \, \eta^{k})u+(\mbox{Re}\, \omega^{k})(\mbox{Re}\, \eta^{k}))+(\frac{-ib}{ R})^{2k})^{\alpha+1}}
\\&=& \frac{2^{k\alpha}}{r}\frac{\displaystyle \biggl(R+\frac{b^2}{R}\biggr)\biggl(R^{k}-2(-ib)^{k}\mbox{Re}\,(\omega^{k}\bar{\eta}^{k})+\displaystyle \frac{(-ib)^{2k}}{R^{k}}\biggr)}{\displaystyle R-2(-ib)\mbox{Re}\,(\omega\bar{\eta})+\frac{(-ib)^2}{R}}
\\&&\times\frac{1}{\displaystyle \biggl(R^{k}-2(-ib)^{k}((\mbox{Im} \,\omega^{k})(\mbox{Im} \,\eta^{k})u+(\mbox{Re}\, \omega^{k})(\mbox{Re}\, \eta^{k}))+\frac{(-ib)^{2k}}{R^{k}} \biggr)^{\alpha+1}}.\end{aligned}$$ By $R=s+r=s+\sqrt{s^2+b^2}$ and $\displaystyle\frac{1}{R}=\frac{1}{s+\sqrt{s^2+b^2}}=\frac{\sqrt{s^2+b^2}-s}{b^2}$, we get $$\begin{aligned}
&&R+\frac{b^2}{R}=s+r+b^2\frac{r-s}{b^2}=2r \\&&R+\frac{(-ib)^2}{R}=s+r-b^2\frac{r-s}{b^2}=2s \end{aligned}$$ and $$R^{k}+\frac{(-ib)^{2k}}{R^{k}}=(s+r)^{k}+(-1)^{k}(r-s)^{k}=\sum_{j=0}^{k}\binom{k}{j}(1+(-1)^{k+j})s^{j}r^{k-j}$$ which means that $R^{k}+\frac{(-ib)^{2k}}{R^{k}}$ is always a polynomial in $s$ as $k$ is a positive integer. We can apply Lemma \[lem1\] because $|({\rm Im} \,\omega^{k})({\rm Im} \,\eta^{k})u+({\rm Re}\, \omega^{k})({\rm Re}\, \eta^{k})v|\le 1$, for $u,v \in [-1, 1]$. Hence $f_{I_{2k}}(s)$ and $ g_{I_{k}}(s)$ have the following factorization,
\[lem3\] Let $$A(s,q)=\prod_{l=0}^{k-1}\biggl(s+ib\cos\biggl(\frac{q+2\pi l}{k}\biggr)\biggr),$$ $$B(s)=(s+ib{\rm Re}\,(\omega\bar{\eta})).$$ Then $f_{I_{2k}}(s)$ has the following factorization $$\begin{aligned}
f_{I_{2k}}(s)&=&\frac{A(s,q(1,1))}{B(s)[A(s,q(u,v))]^{\alpha+\beta+1}}
=\frac{1}{B(s)[A(s,q(u,v))]^{\alpha+\beta}}+\frac{(-ib)^{k}\cos(q(u-1,v-1))}{2^{k-1}B(s)[A(s,q(u,v))]^{\alpha+\beta+1}},\end{aligned}$$ and $g_{I_{k}}(s)$ has the following factorization$$\begin{aligned}
g_{I_{k}}(s)&=&\frac{A(s,q(1,1))}{B(s)[A(s,q(u,1))]^{\alpha+1}}
=\frac{1}{B(s)[A(s,q(u,1))]^{\alpha}}+\frac{(-ib)^{k}\cos(q(u-1,0))}{2^{k-1}B(s)[A(s,q(u,1))]^{\alpha+1}}.\end{aligned}$$ where $q(u, v)=\arccos(({\rm Im} \,\omega^{k})({\rm Im} \,\eta^{k})u+({\rm Re}\, \omega^{k})({\rm Re}\, \eta^{k})v)$.
For the first equality, we only need to show that $q(1, 1)= \arccos({\rm Re}\,(\omega^{k}\bar{\eta}^{k}))$, i.e. $${\rm Re}\,(\omega^{k}\bar{\eta}^{k})=({\rm Im} \,\omega^{k})({\rm Im} \,\eta^{k})+({\rm Re}\, \omega^{k})({\rm Re}\, \eta^{k})$$ which follows by expanding the left-hand side. For the second equality, we have used $$2^{k}A(s,q(u,v))= R^{k}-2(-ib)^{k}((\mbox{Im} \,\omega^{k})(\mbox{Im} \,\eta^{k})u+(\mbox{Re}\, \omega^{k})(\mbox{Re}\, \eta^{k}))+\frac{(-ib)^{2k}}{R^{k}}.$$
Now, we have our first main result in this section
\[ld1\] For the even dihedral group $I_{2k}$, the radial Laplace transform of the Dunkl kernel is $$\begin{aligned}
\mathcal{L}(E_{\kappa}(z_{1}, z_{2},t))&=&\Gamma(k(\alpha+\beta) +1)
\int_{-1}^{1}\int_{-1}^{1}f_{I_{2k}}(s)d\mu_{\alpha}(u)d\mu_{\beta}(v).\end{aligned}$$ For odd-$k$ dihedral group $I_{k}$, the radial Laplace transform of the Dunkl kernel $E_{\kappa}(z_{1}, z_{2}, t)$ is $$\begin{aligned}
\mathcal{L}(E_{\kappa}(z_{1}, z_{2}, t))&=&\Gamma(k\alpha +1)
\int_{-1}^{1}
g_{I_{k}}(s)d\mu_{\alpha}(u).\end{aligned}$$
For any dihedral group, when the multiplicity function $\kappa$ takes integer values, we know from Lemma \[lem3\] that $f_{I_{2k}}(s)$ and $g_{I_{k}}(s)$ are rational functions. So then the Dunkl kernel can be obtained by the inverse Laplace transform through partial fraction decomposition using Theorem \[ld1\] and \[pfd\].
It is known that the Dunkl kernel for positive integer $\kappa$ can in principle be expressed as elementary functions, see [@O1] and [@DJ]. However, this is not made concrete there. In [@DDY], the authors use the shift principle of [@O1] and act with multiple combinations of the Dunkl operators on the Dunkl Bessel function to derive the Dunkl kernel in the dihedral setting. However, there the Dunkl Bessel function was only known in a few cases. In subsection 4.2, we will give the integral expression of the generalized Bessel function using the Laplace transform. Also, acting multiple combinations of the Dunkl operators turns out not to be feasible in practice.
When the multiplicity function $\kappa$ is not integer valued, we can still derive integral formulas for the kernel using Theorem \[ld1\]. First denote $$\begin{aligned}
g_{\alpha}(t, q(u,v))&=&\mathcal{L}^{-1}\biggl(\frac{1}{A(s,q(u,v))^{\alpha}}\biggr)=\mathcal{L}^{-1}\biggl(\frac{1}{\prod_{l=0}^{k-1}\biggl(s+ib\cos\biggl(\frac{q(u,v)+2\pi l}{k}\biggr)\biggr)^{\alpha}}\biggr)
\\&=&\mathcal{L}^{-1}\biggl(\frac{2^{k\alpha}}{
\biggl(R^{k}-2(-ib)^{k}(({\rm Im} \,\omega^{k})({\rm Im} \,\eta^{k})u+({\rm Re}\, \omega^{k})({\rm Re}\, \eta^{k})v)+\frac{(-ib)^{2k}}{R^{k}} \biggr)^{\alpha}}\biggr)
\\&=&\mathcal{L}^{-1}\biggl(\frac{2^{k\alpha-1}e^{i k\alpha\pi}\varpi_{0}^{k\alpha}}{r}\biggl(\frac{b^2}{\varpi_{0}}+\varpi_{0}\biggr)\cdot \frac{1}{(\varpi_{0}^{k}-e^{iq(u,v)}(e^{i\frac{\pi}{2}}b)^{k})^{\alpha}(\varpi_{0}^{k}-e^{-iq(u,v)}(e^{i\frac{\pi}{2}}b)^{k})^{\alpha}}
\biggr)\end{aligned}$$ where $\varpi_{0}=r-s$. Using the same method as in Theorem \[ga\], by formula (\[leff\]) and (\[lb\]), we have $$\begin{aligned}
g_{\alpha}(t, q(u,v))&=&2^{k\alpha-1}e^{ik\alpha\pi}b^{k\alpha+1}\int_{0}^{t}\biggl[J_{k\alpha-1}(b\sqrt{t^2-2\tau t})\\&&+t(t+2\tau)^{-1}J_{k\alpha+1}(b\sqrt{t^2+2\tau t})\biggr]t^{\frac{k\alpha-1}{2}}(t+2\tau)^{-\frac{k\alpha-1}{2}}\tilde{h}_{\alpha}(\tau)d\tau,\end{aligned}$$ where $\tilde{h}_{\alpha}(t)$ is the convolution of two generalized Mittag-Leffler functions, $$\begin{aligned}
\tilde{h}_{\alpha}(t)=\int_{0}^{t}\zeta^
{k\alpha-1}E^{\alpha}_{k, k\alpha}(e^{iq(u,v)}(e^{i\frac{\pi}{2}}b)^{k}\zeta^{k})
(t-\zeta)^{k\alpha-1}E^{\alpha}_{k, k\alpha}(e^{-iq(u,v)}(e^{i\frac{\pi}{2}}b)^{k}(t-\zeta)^{k})d\zeta.\end{aligned}$$ Now, by the convolution formula (\[lc\]), we have
\[m1\] Let $a_{u,v}^{l}$ be the $k+1$ roots of $B(s)A(s,q(u,v))$, i.e. $a^{l}_{u,v}=\displaystyle -ib\cos\biggl(\frac{q+2\pi l}{k}\biggr)$, $l=0,\cdots, k-1$ and $a^{k}_{u,v}=-ib {\rm Re}\,(\omega \bar{\eta})$. Then for each dihedral group $I_{2k}$ and positive multiplicity function $\kappa$, the Dunkl kernel is given by $$\begin{aligned}
E_{\kappa}(z_{1}, z_{2})&=&\Gamma(k(\alpha+\beta)+1)
\int_{-1}^{1}\int_{-1}^{1}\int_{0}^{1}\biggl[\sum_{l=0}^{k}\frac{A(s,q(1,1))(s-a_{u,v}^{l})}{B(s)A(s,q(u,v))}\biggl|_{s=a_{u,v}^{l}}e^{a^{l}_{u,v}(1-\tau)}\biggr]
\\&&g_{\alpha+\beta}(\tau, q(u,v))
d\tau d\mu_{\alpha}(u)d\mu_{\beta}(v).\end{aligned}$$ For each odd-$k$ dihedral group $I_{k}$ and positive multiplicity function $\kappa$, the Dunkl kernel is $$E_{\kappa}(z_{1}, z_{2})=\Gamma(k\alpha +1)\int_{-1}^{1}\int_{0}^{1}\biggl[\sum_{l=0}^{k}\frac{A(s,q(1,1)(s-a_{u,1}^{l}))}{B(s)A(s,q(u,1))}\biggl|_{s=a_{u,1}^{l}}e^{a^{l}_{u,1}(1-\tau)}\biggr]g_{\alpha}(\tau, q(u,1))d\tau d\mu_{\alpha}(u),$$ where $q(u, v)=\arccos(({\rm Im} \,\omega^{k})({\rm Im} \,\eta^{k})u+({\rm Re}\, \omega^{k})({\rm Re}\, \eta^{k})v)$.
We only prove the odd dihedral group $I_{k}$ cases. We write $g_{I_{k}}$ as
$$\begin{aligned}
\label{fi}g_{I_{k}}(s)
=\frac{A(s,q(1,1))}{B(s)[A(s,q(u,1))]}\cdot\frac{1}{[A(s,q(u,1))]^{\alpha}}.\end{aligned}$$
The inverse Laplace transform of the second factor on the right-hand side of (\[fi\]) is $g_{\alpha}(t, q(u,1))$. The first factor on the right-hand side of (\[fi\]) is inversed by partial fraction decomposition. Then by the Laplace convolution formula (\[lc\]), we get the result.
Using the second equality in Lemma \[lem3\], the integral expression of the Dunkl kernel also reduces to the integral expression of $f_{n,k}(t)$ in Lemma \[le1\]. Indeed, put $$h_{\alpha}(t, q(u,v))=\mathcal{L}^{-1}\biggl(\frac{1}{B(s)A(s,q(u,v))^{\alpha}}\biggr)=\mathcal{L}^{-1}\biggl(\frac{1}{(s+ib{\rm Re}\,(\omega\bar{\eta}))\prod_{l=0}^{k-1}\biggl(s+ib\cos\biggl(\frac{q(u,v)+2\pi l}{k}\biggr)\biggr)^{\alpha}}\biggr),$$ which is the convolution of $g_{\alpha}(t, q(u,v))$ and $e^{-ib{\rm Re}\,(\omega\bar{\eta})}$. Then we have
\[m2\] For each dihedral group $I_{2k}$ and positive multiplicity function $\kappa$, the Dunkl kernel is given by $$\begin{aligned}
E_{\kappa}(z_{1}, z_{2})&=&\Gamma(k(\alpha+\beta)+1)
\int_{-1}^{1}\int_{-1}^{1}[h_{\alpha+\beta}(1, q(u,v))\\&&+2^{1-k}(-ib)^{k}\cos(q(u-1,v-1))h_{\alpha+\beta+1}(1, q(u,v))]d\mu_{\alpha}(u)d\mu_{\beta}(v).\end{aligned}$$ For each odd-$k$ dihedral group $I_{k}$ and positive multiplicity function $\kappa$, the Dunkl kernel is $$E_{\kappa}(z_{1}, z_{2})=\Gamma(k\alpha +1)\int_{-1}^{1}[h_{\alpha}(1, q(u,1))+2^{1-k}(-ib)^{k}\cos(q(u-1,0))h_{\alpha+1}(1, q(u,1))]d\mu_{\alpha}(u),$$ where $q(u, v)=\arccos(({\rm Im} \,\omega^{k})({\rm Im} \,\eta^{k})u+({\rm Re}\, \omega^{k})({\rm Re}\, \eta^{k})v)$.
Let us now look at a few special cases. When $k=1$ and any positive $\alpha$, $g_{I_{1}}(s)$ becomes $$\begin{aligned}
\label{k1}g_{I_{1}}(s)&=& \frac{1}{(s+ib((\mbox{Im}\, \omega)(\mbox{Im}\, \eta)u+(\mbox{Re}\, \omega)(\mbox{Re}\, \eta)))^{\alpha+1}}.\end{aligned}$$ We take the inverse Laplace transform of (\[k1\]) and set $t=1$, then we reobtain the Dunkl kernel for $I_{1}$, which is $$E_{ \kappa}(z_{1},z_{2})=\int_{-1}^{1}e^{-\displaystyle i(u \mbox{Im}\,z_{1}\mbox{Im}\,z_{2}+\mbox{Re}\,z_{1} \mbox{Re}\, z_{2} )}d\mu_{\alpha}(u).$$ It coincides with the known result of the integral representation of the intertwining operator of the rank $1 $ case, for $\mbox{Re} \,\alpha>0$, $$V_{\alpha}p(x)=\int_{-1}^{1}p(xu)d\mu_{\alpha}(u),$$ which can be found in [@DX]. Similarly, we reobtain the Dunkl kernel for $I_{2}$, which is $$E_{\kappa}(z_{1},z_{2})=\int_{-1}^{1}\int_{-1}^{1}e^{-\displaystyle i(u\mbox{Im}\,z_{1}\mbox{Im}\,z_{2}+v\mbox{Re}\,z_{1} \mbox{Re}\, z_{2} )}d\mu_{\alpha}(u)d\mu_{\beta}(v)$$ which coincides with the result obtained using the intertwining operator for $\mathbb{Z}_{2}^{2}$.
For the dihedral group $I_{3}$ and $I_{6}$, we can get the integral expression of the Dunkl kernels by (\[I3\]) as both of them are related to the function $f_{3, k}(t)$.
For the dihedral group $I_{4},$ we have $$\begin{aligned}
f_{I_{4}}(s)=\frac{s^2+b^2\biggl(\frac{1+{\rm Re}\,\omega^2 \bar{\eta}^2}{2}\biggr)}{(s+ib{\rm Re}\, \omega\bar{\eta} )\biggl(s^2+b^2\biggl(\frac{1+({\rm Im}\,\omega^2 )({\rm Im} \,\eta^2) u+({\rm Re}\,\omega^2) ({\rm Re}\, \eta^2) v}{2}\biggr)\biggr)^{\alpha+\beta+1}}.\end{aligned}$$ We take the inverse Laplace transform and set $t=1$. We get the Dunkl kernel for $I_{4}$, using Theorem \[m1\], $$\begin{aligned}
E_{\kappa}(z_{1}, z_{2})&=&\frac{\sqrt{\pi}\Gamma(2(\alpha+\beta)+1)}{2^{\alpha+\beta-1/2}\Gamma(\alpha+\beta)}
\int_{-1}^{1}\int_{-1}^{1}\int_{0}^{1}
\frac{1}{\theta_{2}^{2}-\theta_{3}^{2}}\biggl(e^{-ib\theta_{3}\tau}(\theta_{1}^2-\theta_{3}^{2})
+(\theta_{1}^2-\theta_{2}^{2})\biggl(\frac{i\theta_{3}}{\theta_{2}}\sin(b\theta_{2}\tau)-\cos(b\theta_{2}\tau)\biggr)\biggr)\\&&
(1-\tau)^{\alpha+\beta-1/2}\frac{J_{\alpha+\beta-1/2}(b\theta_{2}(1-\tau) )}{(b\theta_{2})^{\alpha+\beta-1/2}}
d\tau d\mu_{\alpha}(u) d\mu_{\beta}(v),\end{aligned}$$ or using Theorem \[m2\], $$\begin{aligned}
E_{\kappa}(z_{1}, z_{2})&=&\frac{\sqrt{\pi}\Gamma(2(\alpha+\beta)+1)}{2^{\alpha+\beta-1/2}\Gamma(\alpha+\beta)}
c_{\alpha}c_{\beta}\int_{-1}^{1}\int_{-1}^{1}\int_{0}^{1}
e^{-ib\theta_{3}\tau}\biggl(
(1-\tau)^{\alpha+\beta-1/2}\frac{J_{\alpha+\beta-1/2}(b\theta_{2}(1-\tau) )}{(b\theta_{2})^{\alpha+\beta-1/2}}\\&&+
\frac{b^{2}(\theta_{1}^2-\theta_{2}^{2})}{2(\alpha+\beta)}(1-\tau)^{\alpha+\beta+1/2}\frac{J_{\alpha+\beta+1/2}(b\theta_{2}(1-\tau) )}{(b\theta_{2})^{\alpha+\beta+1/2}}\biggr)
d\tau d\mu_{\alpha}(u) d\mu_{\beta}(v),\end{aligned}$$ where $\theta_{1}=\sqrt{\frac{1+({\rm Re}\,\omega^2 \bar{\eta}^2)}{2}}$, $\theta_{2}=\sqrt{\frac{1+({\rm Im}\,\omega^2 )({\rm Im} \,\eta^2) u+({\rm Re}\,\omega^2)( {\rm Re}\, \eta^2 )v}{2}}$ and $\theta_{3}={\rm Re}\, \omega\bar{\eta}$.
The kernel of the $(\kappa,a)$-generalized Fourier transform with dihedral symmetry can be obtained similarly.
Dunkl Bessel function
---------------------
Following [@DX], we define the Dunkl Bessel function by $$D_{\kappa}(z_{1}, z_{2})=\frac{1}{|I_{k}|}\sum_{g\in I_{k}}E_{\kappa}(z_{1}, g\cdot z_{2}).$$ Let $z_{1}=|z_{1}|e^{i\phi_{1}}$, $z_{2}=|z_{2}|e^{i\phi_{2}}$, $\phi_{1}, \phi_{2} \in [1,\pi/2k]$ and $b=|z_{1}||z_{2}|$. Then the Dunkl Bessel function associated to $I_{2k}, k\ge 2$ is given by (see [@Dn]) $$D_{\kappa}(|z_{1}|, \phi_{1}, |z_{2}|, \phi_{2} )=c_{k,\kappa}\biggl(\frac{2}{b}\biggr)^{\langle \kappa \rangle}\sum_{j= 0}^{\infty} i^{2kj+\langle \kappa\rangle } J_{2kj+\langle \kappa\rangle}(b)p_{j}^{\alpha-1/2,\beta-1/2}(\cos(2k\phi_{1})) p_{j}^{\alpha-1/2,\beta-1/2}(\cos(2k\phi_{2}))$$ where $p_{j}^{\alpha-1/2,\beta-1/2}$ is the $j$-th orthonormal Jacobi polynomial of parameters $(\alpha-1/2,\beta-1/2)$ and $$c_{k,\kappa}=2^{\alpha+\beta}\frac{\Gamma(\langle \kappa\rangle +1)\Gamma(\alpha+1/2)\Gamma(\beta+1/2) }{\Gamma(\alpha+\beta+1)}.$$
With the Dijksma-Koornwinder product formula for the Jacobi polynomial, the Dunkl Bessel function becomes $$\begin{aligned}
\label{B1}D_{\kappa}(|z_{1}|, \phi_{1}, |z_{2}|, \phi_{2} )&=&\Gamma(\langle \kappa\rangle +1)\int_{-1}^{1}\int_{-1}^{1}\biggl(\frac{2}{b}\biggr)^{\langle \kappa \rangle}\sum_{j= 0}^{\infty}\frac{(2j+\alpha+\beta)}{\alpha+\beta} i^{2kj+\langle \kappa\rangle } J_{2kj+\langle \kappa\rangle}(b)\nonumber\\&&C_{2j}^{\alpha+\beta}(z_{k\phi_{1},k\phi_{2}}(u,v))\mu^{\alpha}(du)\mu^{\beta}(dv) \end{aligned}$$ where $\mu^{\alpha}$ is the symmetric beta probability measure $$\mu^{\alpha} (du)=\frac{\Gamma(\alpha+1/2)}{\sqrt{\pi}\Gamma(\alpha)}(1-u^2)^{\alpha-1}du, \quad \alpha>-1,$$ and $$z_{\phi_{1},\phi_{2}}(u,v)=u\cos \phi_{1}\cos \phi_{2}+v\sin\phi_{1}\sin \phi_{2},$$ and $C_{2j}^{\alpha}(x)$ the Gegenbauer polynomial. Now the integrand of (\[B1\]) equals $\displaystyle\frac{f_{2k}^{+}+f_{2k}^{-}}{2}$ with $$\begin{aligned}
f_{2k}^{\pm}(b, \xi)=\Gamma(k(\alpha+\beta)+1)\biggl(\frac{2}{b}\biggr)^{k(\alpha+\beta)}\sum_{j=0}^{\infty}\frac{(j+\alpha+\beta)}{\alpha+\beta} (\pm1)^{j} e^{i\frac{\pi}{2}k(j+\alpha+\beta) } J_{k(j+\alpha+\beta)}(b)C_{j}^{\alpha+\beta}(z_{k\phi_{1},k\phi_{2}}).\end{aligned}$$ As before, we introduce an auxiliary variable $t$ in the series $$\begin{aligned}
f_{2k}^{\pm}(b, \xi, t)=\Gamma(k(\alpha+\beta)+1)\biggl(\frac{2}{b}\biggr)^{k(\alpha+\beta)}\sum_{j=0}^{\infty}\frac{(j+\alpha+\beta)}{\alpha+\beta} (\pm1)^{j} e^{i\frac{\pi}{2}k(j+\alpha+\beta) } J_{k(j+\alpha+\beta)}(bt)C_{j}^{\alpha+\beta}(z_{k\phi_{1}, k\phi_{2}}).\end{aligned}$$ and take the Laplace transform term by term. This yields $$\begin{aligned}
\label{bes}
\mathcal{L}(f_{2k}^{\pm})&=&\Gamma(k(\alpha+\beta)+1)\frac{(2e^{i\frac{\pi}{2}})^{k(\alpha+\beta)}}{r}\frac{R^{k}-\frac{(-1)^{k}b^{2k}}{R^{k}}}{(R^{k}-2(\pm (ib)^{k})z_{k\phi_{1},k\phi_{2}}+\frac{(-1)^{k}b^{2k}}{R^{k}} )^{\alpha+\beta+1}}
\nonumber\\&=&\Gamma(k(\alpha+\beta)+1)\frac{(2e^{i\frac{\pi}{2}})^{k(\alpha+\beta)}}{r}\frac{(r+s)^{k}-(-1)^{k}(r-s)^{k}}{((r+s)^{k}-2(\pm (ib)^{k})z_{k\phi_{1},k\phi_{2}}+(-1)^{k}(r-s)^{k} )^{\alpha+\beta+1}}\end{aligned}$$ where $r=\sqrt{r^2+b^2}$, $R=s+r$. Comparing (\[bes\]) with (\[rl3\]), and using the same method as in Theorem \[th7\], we get $|f_{2k}^{\pm}|\le 1$. Then we have $$|D_{\kappa}(z_{1},z_{2})|=\biggl|\int_{-1}^{1}\int_{-1}^{1} \frac{f_{2k}^{+}+f_{2k}^{-}}{2} \mu^{\alpha}(du)\mu^{\beta}(dv)\biggr|\le 1$$ because $\int_{-1}^{1}\int_{-1}^{1}\mu^{\alpha}(du)\mu^{\beta}(dv)=1$, giving an alternative and direct proof of the boundedness of the Dunkl Bessel function. Also, using (\[bes\]) and (\[rl3\]), it is now in principle possible to find an integral expression for the Dunkl Bessel function. We illustate this for the dihedral group $I_{4}$. In that case, we have $$\begin{aligned}
\mathcal{L}(f_{4}^{\pm})&=&\Gamma(2(\alpha+\beta)+1)\frac{(2e^{i\frac{\pi}{2}})^{2(\alpha+\beta)}}{r}\frac{(r+s)^{2}-(-1)^{2}(r-s)^{2}}{((r+s)^{2}-2(\pm (ib)^{2})z_{2\phi_{1}, 2\phi_{2}}+(-1)^{2}(r-s)^{2} )^{\alpha+\beta+1}}
\\&=&\Gamma(2(\alpha+\beta)+1)e^{i\pi (\alpha+\beta)}\displaystyle\frac{s}{\biggl(s^2+b^2\biggl(\frac{1-(\pm z_{2\phi_{1},2\phi_{2}})}{2}\biggr)\biggr)^{\alpha+\beta+1}}.\end{aligned}$$ Using the inverse Laplace transform formula (\[nf1\]), we have, after evaluating at $t=1$, $$\begin{aligned}
f_{4}^{+}+f_{4}^{-}&=&e^{i\pi(\alpha+\beta)}\frac{\sqrt{\pi}\Gamma(2(\alpha+\beta)+1)}
{\Gamma(\alpha+\beta+1)2^{\alpha+\beta+1/2}}\biggl(\frac{J_{\alpha+\beta-1/2}(b_{1})}{b_{1}^{\alpha+\beta-1/2}}
+\frac{J_{\alpha+\beta-1/2}(b_{2})}{b_{2}^{\alpha+\beta-1/2}}\biggr)\\
&=&e^{i\pi(\alpha+\beta)}2^{\alpha+\beta-1/2}
\Gamma(\alpha+\beta+1/2)\biggl(\frac{J_{\alpha+\beta-1/2}(b_{1})}{b_{1}^{\alpha+\beta-1/2}}
+\frac{J_{\alpha+\beta-1/2}(b_{2})}{b_{2}^{\alpha+\beta-1/2}}\biggr)\end{aligned}$$ where $b_{1}=\biggl(b\sqrt{\frac{1-z_{2\phi_{1}, 2\phi_{2}}}{2}}\biggr)$, $b_{2}=\biggl(b\sqrt{\frac{1+ z_{2\phi_{1}, 2\phi_{2}}}{2}}\biggr)$. In the second equality, we have used the Gauss duplication formula $$\sqrt{\pi}\Gamma(2v)=2^{2v-1}\Gamma(v)\Gamma(v+1/2).$$ Hence for $I_{4}$, the Dunkl Bessel function is given by
$$D_{\kappa}(z_{1},z_{2})=e^{i\pi(\alpha+\beta)}2^{\alpha+\beta-3/2}
\Gamma(\alpha+\beta+1/2)\int_{-1}^{1}\int_{-1}^{1} \biggl(\frac{J_{\alpha+\beta-1/2}(b_{1})}{b_{1}^{\alpha+\beta-1/2}}
+\frac{J_{\alpha+\beta-1/2}(b_{2})}{b_{2}^{\alpha+\beta-1/2}}\biggr) \mu^{\alpha}(du)\mu^{\beta}(dv).$$
When $\alpha+\beta$ is integer, the integral expression of the Dunkl Bessel function associated to $I_{4}$ was obtained in [@Dn]. Our result hence extends this result to arbitrary $\alpha, \beta>0$.
For odd dihedral groups, the integral expression of the Dunkl Bessel function is computed in a similar way.
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[^1]: E-mail: [denis.constales@ugent.be]{}
[^2]: E-mail: [hendrik.debie@ugent.be]{}
[^3]: E-mail: [pan.lian@outlook.com]{}
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abstract: 'Quantum-dynamical full-dimensional (15D) calculations are reported for the protonated water dimer () using the multiconfiguration time-dependent Hartree (MCTDH) method. The dynamics is described by curvilinear coordinates. The expression of the kinetic energy operator in this set of coordinates is given and its derivation, following the polyspherical method, is discussed. The PES employed is that of Huang [*et al.*]{} . A scheme for the representation of the potential energy surface (PES) is discussed which is based on a high dimensional model representation scheme (cut-HDMR), but modified to take advantage of the mode-combination representation of the vibrational wavefunction used in MCTDH. The convergence of the PES expansion used is quantified and evidence is provided that it correctly reproduces the reference PES at least for the range of energies of interest. The reported zero point energy of the system is converged with respect to the MCTDH expansion and in excellent agreement ($16.7$ below) with the diffusion Monte Carlo result on the PES of Huang [*et al*]{}. The highly fluxional nature of the cation is accounted for through use of curvilinear coordinates. The system is found to interconvert between equivalent minima through wagging and internal rotation motions already when in the ground vibrational-state, i.e., T=0. It is shown that a converged quantum-dynamical description of such a flexible, multi-minima system is possible.'
author:
- Oriol Vendrell
- Fabien Gatti
- David Lauvergnat
- 'Hans-Dieter Meyer'
title: 'Full Dimensional (15D) Quantum-Dynamical Simulation of the Protonated Water-Dimer I: Hamiltonian Setup and Analysis of the Ground Vibrational State'
---
Introduction
============
The protonated water-dimer (Zundel cation) is the smallest system in which an excess proton is shared between two water molecules. Much effort has been devoted to this system due to the importance that the solvated proton has in several areas of chemistry and biology. In recent years a fast development of the spectroscopical techniques available to probe the vibrational dynamics of ionic species in the gas phase has taken place, and several studies have appeared around the system [@asm03:1375; @fri04:9008; @hea04:11523; @ham05:244301] and other more complex clusters and molecules [@hea05:1765; @ros07:249]. In order to achieve a satisfactory understanding of the spectroscopy and dynamics accurate theoretical results are needed in parallel and several works have appeared to fill this gap which are based on different theoretical approaches, from classical- to quantum-dynamical methods [@ven01:240; @dai03:6571; @hua05:044308; @ham05:244301; @sau05:1706; @kal06:2933]. Accurate quantum-dynamical simulations of the dynamics and spectroscopy of the system require of an accurate reference potential energy surface (PES). Very accurate PES and dipole moment surfaces (DMS) have been recently produced by Huang et al. [@hua05:044308] for . They are based on several ten-thousands of coupled-cluster calculations combined with a clever fitting algorithm which is based on a redundant set of coordinates, namely all atom-atom distances. The PES is able to describe the floppy motions occurring at typical energies of excitation in the linear IR regime and dissociates correctly [@hua05:044308]. Several works have already appeared in which the PES and DMS of Huang et al. [@hua05:044308] were used [@mc05:061101; @ham05:244301; @kal06:2933]. In Ref. [@mc05:061101] full-dimensional vibrational calculations for were undertaken using diffusion Monte Carlo (DMC) and variational wavefunction methods. The zero point energy (ZPE), some vibrational excited-state energies and properties of the ground vibrational state of the system are reported and discussed. The PES of Huang et al. was also used in Ref. [@ham05:244301] where the experimental IR spectrum of the system was reported together with the calculation of relevant excited-vibrational states.
As will be later discussed, features several symmetry-equivalent minimum energy structures and large amplitude motions between different regions connected by low energy barriers. For such systems curvilinear coordinates, e.g. bond-bond angle, dihedral angle, or internal rotation coordinates, offer an advantage over rectilinear coordinates since they provide a physically meaningful description of the different large-amplitude molecular motions. As a remark, an attempt was made to describe the system by a set of rectilinear coordinates, since they result in a simple expression of the kinetic energy operator (KEO). The resulting Hamiltonian was unsuccessful at describing of variuos large-amplitude displacements, which resulted in abandonement of such an approach and introduction of a curvilinear-coordinates based Hamiltonian. When deriving the KEO for a Hamitonian in curvilinear coordinates one may define those in terms of rectilinear atom motions and use standard differential calculus to obtain the desired expression [@pod28:812]. This approach becomes extremely complex and error prone already for systems of a few atoms. Another possibility is to use an approximate KEO where some cross-terms are neglected, thus simplifying the derivation. In this case, however, an uncontrolled source of error is introduced in the calculation. These drawbacks are overcome by employing the polyspherical approach [@gat98:8804; @gat99:7225; @gat01:8275] when defining the coordinate set and deriving the corresponding KEO. In the polyspherical method the system is described by a set of vectors of any kind (e.g. Jacobi, valence, …). The kinetic energy is given in terms of the variation in length and orientation of these vectors, the latter is defined by angles with respect to a body fixed frame. Following the polyspherical method the KEO of the system at hand is derived in a systematic way without use of differential calculus. The polyspherical approach has already been discussed in a general framework as well as for some molecular systems (see for instance [@gat98:8804; @gat99:7225; @gat01:8275; @gat03:507]). In this paper the full derivation of the KEO in a set of polyspherical coordinates is illustrated for the cation and all the key steps are carefully discussed. The derivation is followed step-by-step for this rather complex system, thus providing the basic guidelines to be followed for the derivation of KEOs for complex molecular systems and clusters.
In quantum-dynamical wavefunction-based studies, as the one being introduced here, it is convenient to represent the operators on a discrete variable representation (DVR) grid, which in the multidimensional case is the direct product of one-dimensional DVRs. The potential operator is then given by the value of the PES at each point of the grid. In the present case, however, the resulting primitive grid has more than $10^{15}$ points. This number makes clear that the potential must be represented in a more compact form to make the calculations feasible. Algorithms exist which take advantage of the fact that correlation usually involves the concerted evolution of only a small number of coordinates as compared to the total number. Hierarchical representations of the PES are then constructed including potential-function terms up to a given number of coordinates [@bow03:533; @rab99:197; @li01:1]. We discuss here a variation of a hierarchical representation of the PES in which the coordinates are grouped together into modes, and the modes are the basic units that define the hierarchical expansion. It will be shown that this approach allows for a fast convergence of the PES representation if the modes are defined as groups of the most strongly correlated coordinates.
The present work provides new reference results on the properties of the ground vibrational state of . The convergence of the given results is established by comparison to the DMC results [@mc05:061101] on the reference PES of Huang et al. [@hua05:044308]. These results, together with the Hamiltonian defined here, set also the theoretical and methodological framework used in the companion paper [@ven07:paperII] where the IR spectrum and vibrational dynamics of are analyzed.
The paper is organized as follows. Section \[sec:MCTDH\] presents a brief description of the multiconfiguration time-dependent Hartree (MCTDH) method. Section \[sec:Hamil\] details the construction of the Hamiltonian operator for the system. The derivation of the KEO is discussed in \[H5O2+\], while the construction of the potential is detailed in \[sec:cluster\]. Section \[sec:ZPE\] discusses the calculation of the ground vibrational state of the system and gives a comparison to available results. In section \[sec:wavefunction\] some properties of the system in relation to its ground vibrational-state wavefunction are discussed. In section \[sec:quality\] the quality of the potential expansion is analyzed and discussed. Section \[sec:conclusions\] provides some general conclusions.
Brief description of the MCTDH method {#sec:MCTDH}
=====================================
The *Multiconfiguration time-dependent Hartree* (MCTDH) method [@mey90:73; @man92:3199; @bec00:1; @mey03:251] is a general algorithm to solve the time-dependent Schrödinger equation. The MCTDH wave function is expanded in a sum of products of so–called *single–particle functions* (SPFs). The SPFs, $\varphi(Q,t)$, may be one– or multi–dimensional functions and, in the latter case, the coordinate $Q$ is a collective one, $Q=(q_k,\cdots,q_l)$. As the SPFs are time–dependent, they follow the wave packet and often a rather small number of SPFs suffices for convergence.
The *ansatz* for the MCTDH wavefunction reads $$\begin{aligned}
\label{eq:ansatz}
\Psi(q_1,\cdots,q_f,t) & \equiv & \Psi(Q_1,\cdots,Q_p,t) \nonumber \\
& = & \sum_{j_1}^{n_1} \cdots \sum_{j_p}^{n_p} A_{j_1,\cdots,j_p}(t) \,
\prod_{\kappa=1}^{p} \varphi^{(\kappa)}_{j_\kappa}(Q_\kappa,t) \\
& = & \sum_J A_J \, \Phi_J \nonumber \; ,\end{aligned}$$ where $f$ denotes the number of degrees of freedom and $p$ the number of MCTDH *particles*, also called *combined modes*. There are $n_\kappa$ SPFs for the $\kappa$’th particle. The $A_J \equiv A_{j_1 \ldots j_f}$ denote the MCTDH expansion coefficients and the configurations, or Hartree-products, $\Phi_J$ are products of SPFs, implicitly defined by Eq. (\[eq:ansatz\]). The SPFs are finally represented by linear combinations of time-independent primitive basis functions or DVR grids.
The MCTDH equations of motion are derived by applying the Dirac-Frenkel variational principle to the *ansatz* Eq. (\[eq:ansatz\]). After some algebra, one obtains $$\begin{aligned}
\label{eq:odea}
i \dot{A}_J & = & \sum_L \, \langle \Phi_J \! \mid \! H \! \mid \!
\Phi_L \rangle \, A_L \; , \\
\label{eq:odephi}
i \dot{\bm{\varphi}}^{(\kappa)} & = & \left( 1-P^{(\kappa)} \right)
\left( \bm{\rho}^{(\kappa)}
\right)^{-1} \! \langle \mathbf{H} \rangle^{(\kappa)}
\bm{\varphi}^{(\kappa)} \; ,\end{aligned}$$ where a vector notation, $\bm{\varphi}^{(\kappa)}=(\varphi^{(\kappa)}_1,
\cdots,\varphi^{(\kappa)}_{n_\kappa})^T$, is used. Details on the derivation, the definitions of the mean-field $\langle \mathbf{H} \rangle^{(\kappa)}$, the density $\bm{\rho}^{(\kappa)}$ and the projector $P^{(\kappa)}$, as well as more general results, can be found in Refs. [@man92:3199; @bec00:1; @mey03:251]. Here and in the following (except for Section \[sec:Hamil\]) we use a unit system with $\hbar=1$.
The MCTDH equations conserve the norm and, for time-independent Hamiltonians, the total energy. MCTDH simplifies to Time-Dependent Hartree when setting all $n_\kappa=1$. Increasing the $n_\kappa$ recovers more and more correlation, until finally, when $n_\kappa$ equals the number of primitive functions, the standard method (i. e. propagating the wave packet on the primitive basis) is used. It is important to note, that MCTDH uses variationally optimal SPFs, because this ensures early convergence.
The solution of the equations of motion requires to build the mean–fields at every time–step. A fast evaluation of the mean–fields is hence essential. To this end the Hamiltonian is written as a sum of products of mono–particle operators: $$\label{eq:prod-form}
H = \sum_{r = 1}^s c_r \prod_{\kappa = 1}^p h_r^{(\kappa)} \, ,$$ where $h_r^{(\kappa)}$ operates on the $\kappa$-th particle only and where the $ c_r$ are numbers. In this case the matrix-elements of the Hamiltonian can be expressed by a sum of products of mono–mode integrals, the evaluation of which is fast. $$\label{eq:prod-form-mat}
\langle\Phi_J|H|\Phi_L\rangle = \sum_{r = 1}^s c_r \prod_{\kappa = 1}^p
\langle\varphi_{j_\kappa}|h_r^{(\kappa)}|\varphi_{l_\kappa}\rangle \, ,$$ Similar expressions apply to the matrix of mean-fields $\langle \mathbf{H} \rangle^{(\kappa)}$. For further information on the MCTDH method, see http://www.pci.uni-heidelberg.de/tc/usr/mctdh/.
System Hamiltonian {#sec:Hamil}
==================
Kinetic energy operator for {#H5O2+}
----------------------------
The derivation of the exact kinetic energy operator for H$_5$O$_2^+$ is based on a polyspherical approach which has been devised in previous articles [@gat98:8804; @gat98:8821; @iun99:3377; @gat99:7225; @gat01:8275; @gat03:507]. This approach can be seen as a very efficient way to obtain kinetic energy operators for the family of polyspherical coordinates. The formalism can be applied whatever the number of atoms and whatever the set of vectors: Jacobi, Radau, valence, satellite, etc. The formalism is not restricted to total J=0 and hence the operator may include overall rotation and Coriolis coupling.\
The protonated water dimer system is described by six Jacobi vectors. The choice of the Jacobi vectors is not unique and several clustering schemes are possible, one natural choice for is given in Figure \[fig:Vect\].
[**FIGURE \[fig:Vect\] AROUND HERE**]{}
The polyspherical approach straightforwardly provides the expression of the kinetic energy operator in terms of the angular momenta associated with the vectors describing the system. Here we use the technique of “separation into two subsystems" which is described in Ref. (see Eq. (37) there). The full kinetic energy operator reads (we use the notation $\partial_x = \frac{\partial}{\partial x}
$ and $\partial^2_x = \frac{\partial^2}{\partial x^2}$ throughout the paper):
$$\begin{aligned}
\label{Eq:48}
&\hat{ T}& = (-\frac{\hbar^2} {2 \mu_R} \frac{1} {R} {\partial^2_R} R) +
\frac{{({\vec{J}^{\dagger}} \cdot {\vec{J}}+
{({\vec{L}}_A+{\vec{L}}_B+{\vec{l}})}^2}
- 2 { ({\vec{L}}_A+{\vec{L}}_B+{\vec{l}})
\cdot {\vec{J}})}_{E2}} {2 \mu_R R^2} \nonumber \\
& + & \sum_{i=1}^{2} (-\frac{\hbar^2} {2 \mu_{iA}} \frac{1} {R_{iA}}
{\partial^2_{R_{iA}}} R_{iA} ) +
\frac{{({{\vec{L}_A}^2} +
{\vec{L}_{1A}^{\dagger}} \cdot
\vec{L}_{1A} - 2 \vec{L}_A \cdot
{\vec{L}_{1A}})}_{BFA}} {2 \mu_{2A} R_{2A}^2}
+ \frac{{({{\vec{L}_{1A}}^{\dagger}} \cdot
{\vec{L}_{1A}})}_{BFA}} { 2 \mu_{1A} R_{1A}^2} \nonumber \\
& + & \sum_{i=1}^{2} (-\frac{\hbar^2} {2 \mu_{iB}} \frac{1} {R_{iB}}
{\partial^2_{R_{iB}}} R_{iB} ) + \frac{{({{\vec{L}_B}^2}
+ {{\vec{L}_{1B}}^{ \dagger}} \cdot {\vec{L}_{1B}} - 2 {\vec{L}_B}
\cdot {\vec{L}_{1B}})}_{BFB}} {2 \mu_{2B} R_{2B}^2}
+ \frac{{({{\vec{L}_{1B}}^{\dagger}} \cdot {\vec{L}_{1B}})}_{BFB}}
{2 \mu_{1B} R_{1B}^2} \nonumber \\
& - & \frac{\hbar^2}{2m} \frac{1}{r}{\partial^2_{r}} r
+ \frac{{({\vec{l}}^2)}_{E2}}{2 m r^2} \nonumber \\\end{aligned}$$
with
- R, R$_{1A}$, R$_{2A}$, R$_{1B}$, R$_{2B}$, r, are the lengths of the vectors $\vec{R}$, $\vec{R}_{1A}$, $\vec{R}_{2A}$, $\vec{R}_{1B}$, $\vec{R}_{2B}$, $\vec{r}$, respectively.
- $\mu_{1 \, A (B)}$ = $\frac{m_O 2 m_H}{m_O + 2 m_H}$, $\mu_{2 \, A (B)}$ = $\frac{m_H}{2}$, $\mu_R$ = $\frac{m_O + 2 m_H}{2}$, and $m = \frac{2 m_H (m_O + 2 m_H)}
{2 m_O + 5 m_H} $.
- ${\vec{L}}_{1 \, A (B)} $ is the angular momentum associated with $\vec{R}_{1 \, A (B)}$.
- ${\vec{L}}_{2 \, A (B)} $ is the angular momentum associated with $\vec{R}_{2 \, A (B)}$ and ${\vec{L}}_{A (B)} $ (= ${\vec{L}}_{1 \, A (B)}
+ {\vec{L}}_{2 \, A (B)}$) is the total angular momentum associated with monomer A (B).
- ${\vec{l}}$ is the angular momentum associated with the proton.
- ${\vec{J}}$ is the total angular momentum of the system. ${\vec{J}}$ = ${\vec{l}}$ + ${\vec{L}}_{A} $ + ${\vec{L}}_{B} $ + ${\vec{L}}_{R}$ and ${\vec{L}}_{R}$ is the angular momentum associated with $\vec{R}$.
- E2 is the frame resulting from the two first Euler rotations starting from the space fixed frame, such that the z$^{E2}$ axis lies parallel to $\vec{R}$. In other words, the two first Euler angles $\alpha$ and $\beta$ are identical to the two spherical angles of $\vec{R}$ in the space fixed frame.
- BFA (B) is the body fixed frame associated with monomer A (B). The two first Euler angles $\alpha_{A (B)}$ and $\beta_{A (B)}$ of monomer A (B) are defined such as z$^{BFA (B)}$ lies parallel to $\vec{R}_{2 \, A (B)}$ (in other words they are identical to the two spherical angles of $\vec{R}_{2 \, A (B)}$ in the E2 frame). The third Euler angle $\gamma_{A (B)}$ is defined such as $\vec{R}_{1 \, A (B)}$ remains parallel to the (${(xz)}^{BFA (B)}$, $x^{BFA (B)} > 0$) half plane. This definition of BFA(B) is similar to the one recently used to describe the water dimer (see e.g. Ref [@lef02:8710]). Note however that we have changed the definition of the z$^{BFA (B)}$ axis which is now parallel to the vector between the two hydrogen atoms and not to the vector joining the center of mass of H$_2$ to the oxygen atom as in Ref. [@lef02:8710]. This change was performed to avoid the singularity which appears in the KEO when $\vec{R}$ and z$^{BFA (B)}$ are parallel.
It should be emphasized that all the angular momenta appearing in the kinetic energy operator are all [*computed*]{} in the space fixed frame but [*projected*]{} onto the axes of different frames. This last point, as well as the properties of the corresponding projections, is addressed in detail in Ref. [@gat03:167]. We recall here only the main point, i.e. the fact that if a vector is not involved in the definition of a frame F, the expression of the projections of the corresponding angular momentum onto the F-axes in terms of the coordinates in this frame, is identical to the usual one in a space fixed frame. This very useful property will be utilized several times in the following for instance to obtain Eqs. (\[Eq:44455ter\],\[Eq:44455bis\],\[Eq:44455\],\[Eq:455\],\[Eq:proton\],\[Eq:47\]).\
We now slightly change the volume element for the lengths of the vectors (as usual), and assume J=0. The operator can then be recast in the following form:
$$\begin{aligned}
\label{Eqs:48}
&\hat{ T}& = (-\frac{\hbar^2} {2 \mu_R} {\partial_R^2} ) +
\frac{ {( {\vec{L}}_A+{\vec{L}}_B+{\vec{l}} )}^2_{E2} }
{2 \mu_R R^2}
- \frac{\hbar^2}{2m} {\partial_{r}^2}
+ \frac{{{({\vec{l}}^2)}_{E2}}}{2 m r^2}
\nonumber \\
& + & \sum_{i=1}^{2} (-\frac{\hbar^2} {2 \mu_{iA}}
{\partial_{R_{iA}}^2} ) +
\frac{{({{\vec{L}_A}^2} +
{{\vec{L}_{1A}}^{\dagger}} \cdot {\vec{L}_{1A}} - 2 {\vec{L}_A}
\cdot {\vec{L}_{1A}})}_{BFA}} {2 \mu_{2A} R_{2A}^2} +
\frac{{({{\vec{L}_{1A}}^{\dagger}}
\cdot {\vec{L}_{1A}})}_{BFA}} { 2 \mu_{1A} R_{1A}^2} \nonumber \\
& + & \sum_{i=1}^{2} (-\frac{\hbar^2} {2 \mu_{iB}}
{\partial_{R_{iB}}^2} ) +
\frac{{({{\vec{L}_B}^2} + {{\vec{L}_{1B}}^{ \dagger}}
\cdot {\vec{L}_{1B}} - 2 {\vec{L}_B}
\cdot {\vec{L}_{1B}})}_{BFB}} {2 \mu_{2B} R_{2B}^2} +
\frac{{({{\vec{L}_{1B}}^{\dagger}} \cdot {\vec{L}_{1B}})}_{BFB}}
{2 \mu_{1B} R_{1B}^2} \nonumber \\\end{aligned}$$
which is to be used with the volume element :
$$dR dR_{1A} dR_{2A} dR_{1B} dR_{2B} dr d\alpha_A \sin \beta_A d\beta_A d\gamma_A
d\alpha_B \sin \beta_B d\beta_B d\gamma_B \sin \theta d\theta d\varphi_H
\sin \theta_{1A} d\theta_{1A} \sin \theta_{1B} d\theta_{1B}$$
with
$$\begin{array} {ccccc}
0 & \leq & R, R_{1A}, R_{2A}, R_{1B}, R_{2B}, r & < &\infty \nonumber \\
0 & \leq & \beta_A, \beta_B, \theta, \theta_{1A}, \theta_{1B} & \leq &\pi \nonumber \\
0 & \leq & \alpha_A, \gamma_A, \alpha_B, \gamma_B, \varphi_H & < &2 \pi \nonumber \\
\end{array}$$
We have 16 degrees of freedom instead of 15 since we have not defined the third Euler angle yet. The angles are depicted in Figure \[fig:Coord\]:
[**FIGURE \[fig:Coord\] AROUND HERE**]{}
- $\theta $ and $\varphi_H$ are the spherical angles of the proton in the E2 frame: $\theta $ is the angle between z$^{E2}$ and the vector $\vec{r}$, $\varphi_H$ describes the rotation of $\vec{r}$ around z$^{E2}$.
- $\beta_A$ and $\alpha_A$ are the spherical angles of $\vec{R}_{2A}$ in the E2 frame. The angle $\beta_A$ describes the rocking motion of water A fragment while $\alpha_A$ describes the rotation of the water A fragment around the $\vec{R}\equiv\vec{z}_{E2}$ axis (the same for B).
- $\theta_{1A}$ and $\gamma_A$ are the spherical angles of $\vec{R}_{1A}$ in the E2A frame, the E2A frame is the frame obtained after the two Euler rotations $\alpha_A$ and $\beta_A$ with respect to the E2 frame. Consequently, $\theta_{1A}$ is the angle between $\vec{R}_{1A}$ and $\vec{R}_{2A}$ and $\gamma_A$ describes the rotation of $\vec{R}_{1A}$ around $\vec{R}_{2A}$. Hence $\gamma_A$ describes the wagging motion of the water A fragment (the same for B). The fact that we have used the separation into two subsystems for the two water monomers (see Eq. (37) in Ref. [@gat99:7225]) explains why the angles of $\vec{R}_{1A (B)}$ are defined with respect to $\vec{R}_{2A (B)}$ and not with respect to $\vec{R}$ as in the standard formulation of the polyspherical approach (see Eq. (65) in Ref. [@gat98:8804]). This allows us to have purely intramonomer angles instead of angles mixing the intramonomer and intermonomer motions.
Since ${({{\vec{L}_{A (B)}}^2})}_{BFA (B)}$ = ${({{\vec{L}_{A (B)}}^2})}_{E2}$ (here we can use ${{\vec{L}_{A (B)}}^2}$ rather than ${\vec{L}_{A (B)}}^\dagger {\vec{L}_{A (B)}}$ since the projections of $\vec{L}_{A (B)}$ onto the axes of the E2 or BFA (B) frames are hermitian: see Eqs. (\[Eq:44455ter\],\[Eq:44455\],\[Eq:455\],\[Eq:47\]) below), we can rewrite the operator as:
$$\begin{aligned}
\label{Eq:482}
&\hat{ T}& = (-\frac{\hbar^2} {2 \mu_R}
{\partial_R^2} )
+ \sum_{i=1}^{2} (-\frac{\hbar^2} {2 \mu_{iA}}
{\partial_{R_{iA}}^2} )
+ \sum_{i=1}^{2} (-\frac{\hbar^2} {2 \mu_{iB}}
{\partial_{R_{iB}}^2} )
- \frac{\hbar^2}{2m} {\partial_{r}^2}
+ \frac{{{({l}^2)}_{E2}}}{2 m r^2} \nonumber \\
& + & {({{\vec{L}_A}^2})}_{BFA}
(\frac{1}{2 \mu_R R^2}+\frac{1}{2 \mu_{2A} R_{2A}^2})
+ {({{\vec{L}_B}^2})}_{BFB}
(\frac{1}{2 \mu_R R^2}+\frac{1}{2 \mu_{2B} R_{2B}^2}) \nonumber \\
& + & {({{\vec{L}_{1A}}^{\dagger}} \cdot
{\vec{L}_{1A}})}_{BFA} (\frac{1} {2 \mu_{1A} R_{1A}^2} +
\frac{1} {2 \mu_{2A} R_{2A}^2})
+ {({{\vec{L}_{1B}}^{\dagger}} \cdot {\vec{L}_{1B}})}_{BFB}
(\frac{1} {2 \mu_{1B} R_{1B}^2} +
\frac{1} {2 \mu_{2B} R_{2B}^2}) \nonumber \\
& - & \frac{{ ({\vec{L}_A} \cdot
{\vec{L}_{1A}})}_{BFA}} { \mu_{2A} R_{2A}^2}
- \frac{{ ({\vec{L}_B} \cdot
{\vec{L}_{1B}})}_{BFB}} { \mu_{2B} R_{2B}^2}
\nonumber \\
& + &
\frac{ { ({\vec{L}_A \cdot \vec{L}_B})}_{E2} }
{ \mu_R R^2}
+ \frac{{(\vec{l}^2)}_{E2}} {2 \mu_R R^2}
+ \frac{ {( \vec{L}_A+\vec{L}_B) \cdot
\vec{l}_{E2}} }
{ \mu_R R^2} \nonumber \\ \end{aligned}$$
This operator could be straightforwardly used along with an adequate primitive basis set of spherical harmonics and Wigner rotation-matrix elements. However, in this work, we prefer to employ a direct product primitive basis such as a DVR for all the degrees of freedom (only because the latter basis set is numerically more efficient for our problem). We have then to explicitly express of the angular momenta in terms of the coordinates, i.e. we have to detail the following terms:
- \(1) ${({{\vec{L}_A}^2})}_{BFA}$, ${({{\vec{L}_B}^2})}_{BFB}$, $ {({{\vec{L}_{1A}}^{\dagger}}
\cdot {\vec{L}_{1A}})}_{BFA}$, ${({{\vec{L}_{1B}}^{\dagger}}
\cdot {\vec{L}_{1B}})}_{BFB}$
- \(2) ${({\vec{L}_A} \cdot
{\vec{L}_{1A}})}_{BFA}$, ${({\vec{L}_B} \cdot
{\vec{L}_{1B}})}_{BFB}$
- \(3) ${({\vec{L}}_A \cdot {\vec{L}}_B)}_{E2}$
- \(4) ${\vec{l}}^2_{E2}$
- \(5) ${({\vec{L}}_A+{\vec{L}}_B)
\cdot {\vec{l}}}_{E2}$
but before doing so we have to further specify the coordinates:
- \(i) to avoid the $1/r^2$ singularity we use the BF Cartesian coordinates $x$, $y$, and $z$ for the proton (the definition of BF frame is given in (ii)). Hence we use, $\partial_{\varphi_H^{BF}} = x \partial_y - y
\partial_x$.
- \(ii) we slightly change the coordinate system (from E2 to BF) and explicitly define the third Euler angle $\gamma$ of the system and then the BF frame. We define $\gamma$ such that it is identical to the first Euler angle of monomer A (here, we slightly break the symmetry between the two monomers). Consequently, we have:
$$\begin{aligned}
\label{Eq:485}
\gamma & = & \alpha_A \nonumber \\
\alpha & = & \alpha_B - \alpha_A \nonumber \\
\varphi_H^{BF} & = & \varphi_H^{E2} - \alpha_A \nonumber \\\end{aligned}$$
since J=0, we can then put:
$$\label{Eq:486}
\partial_\gamma = 0$$
and find
$$\begin{aligned}
\label{Eq:487}
\partial_{\alpha_A} & = & -\partial_{\alpha} -\partial_{\varphi_H^{BF}} =
-\partial_{\alpha} - (x \partial_{y} -y \partial_{x}) \nonumber \\
\partial_{\alpha_B} & = & \partial_{\alpha} \nonumber \\\end{aligned}$$
Note that this transformation does not affect the other angles.
- \(iii) in order to have hermitian conjugate momenta for the angles, we use u$_{\beta_A}$ = $ \cos{\beta_A}$, u$_{\beta_B}$ = $ \cos{\beta_B}$, u$_{\theta_{1A}}$ = $\cos{ \theta_{1A} }$, u$_{\theta_{1B}}$ = $\cos{ \theta_{1B} }$.\
For each angle $\eta$, we have (with u$_\eta$ = $\cos \eta$): $\partial_{\eta} = - \sin \eta \partial_{u_\eta}
= - \sqrt{1-u_\eta^2} \partial_{u_\eta}$ and $\partial_{\eta}^\dagger = - \partial_{u_\eta} \sqrt{1-u_\eta^2}$.
We have now 15 degrees of freedom: $$\begin{array} {ccccc}
0 & \leq & R, R_{1A}, R_{2A}, R_{1B}, R_{2B} & < & \infty \nonumber \\
-1 & \leq & u_{\beta_A}, u_{\beta_B}, u_{\theta_{1A}}, u_{\theta_{1B}} & \leq & 1 \nonumber \\
0 & \leq & \alpha, \gamma_A, \gamma_B & < & 2 \pi \nonumber \\
- \infty & < & x, y, z & < & \infty \nonumber \\
\end{array}$$ Let us now detail all the terms appearing in the kinetic energy operator:
- \(1) For $\vec{L}_{B \, BFB}$, the situation is simple since monomer B is not involved in the definition of the body fixed frame of the whole system. Thus, we have : $$\label{Eq:44455ter}
\begin{array} {l}
{{L}}_{B \, x^{BFB}} =
i \hbar \frac{\cos{\gamma_B}} {\sin \beta_B}
{ \partial_{\alpha_B}}
+ i \hbar \sin \gamma_B \sin \beta_B
{ \partial_{u_{\beta B}}}
-i \hbar \frac{u_{\beta B}} {\sin \beta_B} { \cos
\gamma_B}
{ \partial_{{\gamma}_B}}\\
{{L}}_{B \, y^{BFB}} = -i \hbar \frac{\sin \gamma_B}
{\sin \beta_B} { \partial_{\alpha_B}}
+ i \hbar \cos \gamma_B \sin \beta_B { \partial_{u_{\beta B}} }
+ i \hbar \frac{u_{\beta B}} {\sin \beta_B} \sin \gamma_B
{ \partial_{{\gamma}_B}} \\
{{L}}_{B \, z^{BFB}} = -i \hbar
{ \partial_{\gamma_B}}
\end{array}$$ and $$\begin{aligned}
\label{Eq:44455bis}
{{\vec{L}_{B\;BFB}}^2} & = &
- \hbar^2 (\partial_{u_{\beta_B}} {(1-u_{\beta_B}^2)} \partial_{u_{\beta_B}}
+ \frac{1} {1-u_{\beta_B}^2} (\partial_{\alpha_B}^2 +
\partial_{\gamma_B}^2 - 2 u_{\beta_B} \partial_{\alpha_B} \partial_{\gamma_B}))\end{aligned}$$
Similarly ${{\vec{L}_{A\;BFA}}^2}$ can be seen as the total angular momentum of monomer A projected onto the axes of the Body fixed frame of the monomer. We obtain the usual expression:
$$\label{Eq:44455}
\begin{array} {l}
{{L}}_{A \, x^{BFA}} =
i \hbar \frac{\cos{\gamma_A}} {\sin \beta_A}
{ \partial_{\alpha_A}}
+ i \hbar \sin \gamma_A \sin \beta_A
{ \partial_{u_{\beta A}}}
-i \hbar \frac{u_{\beta A}} {\sin \beta_A} { \cos
\gamma_A}
{ \partial_{{\gamma}_A}}\\
{{L}}_{A \, y^{BFA}} = -i \hbar \frac{\sin \gamma_A}
{\sin \beta_A} { \partial_{\alpha_A}}
+ i \hbar \cos \gamma_A \sin \beta_A { \partial_{u_{\beta A}} }
+ i \hbar \frac{u_{\beta A}} {\sin \beta_A} \sin \gamma_A
{ \partial_{{\gamma}_A}} \\
{{L}}_{A \, z^{BFA}} = -i \hbar
{ \partial_{\gamma_A}}
\end{array}$$
and $$\begin{aligned}
\label{Eqs:1}
{\vec{L}_{A\;BFA}}^2 & = & - \hbar^2
(\partial_{u_{\beta_A}} {(1-u_{\beta_A}^2)} \partial_{u_{\beta_A}}
+ \frac{1} {1-u_{\beta_A}^2} (\partial_{\alpha_A}^2 +
\partial_{\gamma_A}^2 - 2 u_{\beta_A} \partial_{\alpha_A} \partial_{\gamma_A}))\end{aligned}$$ However, since monomer A is involved in the definition of the third Euler angle of the system, we apply the change of coordinates, Eqs. (\[Eq:485\]) and (\[Eq:487\]): $$\begin{aligned}
\label{Eq:4871}
(\partial_{\alpha_A}^2 & + &
\partial_{\gamma_A}^2 - 2 u_{\beta_A} \partial_{\alpha_A} \partial_{\gamma_A})
= \nonumber \\
(\partial_{\alpha}^2 & + & x^2 \partial_y^2
+ y^2 \partial_x^2 -x \partial_x \partial_y y
-y \partial_y \partial_x x \nonumber \\
& + & 2 x \partial_y \partial_{\alpha} - 2 y \partial_x \partial_{\alpha}
+ \partial_{\gamma_A}^2 + 2 u_{\beta_A} \partial_{\alpha} \partial_{\gamma_A}
\nonumber \\
& + & 2 u_{\beta_A} \partial_{\gamma_A} x \partial_y -
2 u_{\beta_A} \partial_{\gamma_A}
y \partial_x) \nonumber \\\end{aligned}$$
For the projections of $\vec{R}_{1 A (B)}$ onto to the BFA (B) axes, the situation is not straightforward since $\vec{R}_{1 A (B)}$ is involved in the definition of BFA (B). However, since we have used the same convention for the BF frames as in our previous articles, the expressions of these projections is already known (identical to Eq. (47a) in Ref. [@gat98:8804] for instance):
$$\label{Eq:4455}
\begin{array} {l}
{L}_{1 A \, x^{BFA}}
=
i \hbar \frac{u_{\theta 1 A}} {\sin{\theta_{1A}}}
{ \partial_{\gamma_A}}
\\
{L}_{1 A \, y^{BFA}} = i \hbar
\sin{\theta_{1A}} {\partial_{u_{\theta 1 A}}}
\\
{L}_{1 A \, z^{BFA}} = -i \hbar
{ \partial_{\gamma_A}}
\end{array}$$
(This expression can be obtained starting from the E2A frame which is the frame resulting from the two first Euler rotations $\gamma_A$ and $\beta_A$. The expression of the projections of $\vec{L}_{1 A}$ onto this E2A frame are the usual ones since $\vec{R}_{1 \, A}$ is not involved in the definition of this frame. Applying then the third Euler rotation and the coordinate transformation leading to the BFA coordinates yields Eq. (\[Eq:4455\])). Note that ${L}_{1 A \, y^{BFA}}$ is not hermitian (this is due to the fact that $\vec{R}_{1 A}$ is involved in the definition of BFA). However, at the end we obtain the usual expression:
$$\begin{aligned}
{({{\vec{L}_{1A}}^{\dagger}}
\cdot {\vec{L}_{1A}})}_{BFA}
& = & - \hbar^2 (\partial_{u_{\theta{1A}}} {(1-u_{\theta_{1A}}^2)}
\partial_{u_{\theta{1A}}} + \frac{1}
{1-u^2_{\theta_{1A}}} \partial_{\gamma_A}^2)\end{aligned}$$
and exactly the same for B:
$$\begin{aligned}
{({{\vec{L}_{1B}}^{\dagger}}
\cdot {\vec{L}_{1B}})}_{BFB}
& = & - \hbar^2 (\partial_{u_{\theta{1B}}} {(1-u_{\theta_{1B}}^2)}
\partial_{u_{\theta{1B}}} +
\frac{1} {1-u^2_{\theta_{1B}}} \partial_{\gamma_B}^2) \end{aligned}$$
- \(2) The terms in ${({\vec{L}_{A (B)}} \cdot
{\vec{L}_{1A (B)}})}_{BFA (B)}$ are not obvious since $\vec{R}_{1 \, A (B)}$ is involved in the definition of BFA (B). First, let us explicitly highlight the hermiticity of the term:
$$\begin{aligned}
\label{Eq:491}
{({\vec{L}_A} \cdot {\vec{L}_{1A}})}_{BFA} & = &
1/2 {({\vec{L}_A}^\dagger \cdot {\vec{L}_{1A}} +
{\vec{L}_{1A}}^\dagger \cdot {\vec{L}_{A}})_{BFA}} \nonumber \\\end{aligned}$$
Combining Eq. (\[Eq:4455\]) and Eq. (\[Eq:44455\]) (as well as Eq. (\[Eq:485\]) and Eq. (\[Eq:487\])), we obtain: $$\begin{aligned}
\label{Eq:493}
& & {({\vec{L}_A} \cdot {\vec{L}_{1A}})}_{BFA} = \nonumber \\
& - & \frac{\hbar^2}{2} [- (\partial_{\alpha} +
x \partial_y - y \partial_x)
\frac{u_{\theta_{1A}}}{\sin \theta_{1A}}
\frac{\cos \gamma_A}{\sin \beta_A} \partial_{\gamma_A} \nonumber \\
& + & \partial_{u_{\beta_A}} \frac{u_{\theta_{1A}}}{\sin \theta_{1A}}
\sin \beta_A \sin \gamma_A \partial_{\gamma_A}
- 2 \partial_{\gamma_A} \frac{u_{\beta_A}}{\sin \beta_A}
\cos \gamma_A \frac{u_{\theta_{1A}}}{\sin \theta_{1A}} \partial_{\gamma_A}
\nonumber \\
& + & 2 \partial_{\gamma_A}^2
+ (\partial_{\alpha} + x \partial_y - y
\partial_x)
\frac{\sin \gamma_A}{\sin \beta_A} \sin \theta_{1A} \partial_{u_{\theta_{1A}}}
\nonumber \\
& + & \partial_{u_{\beta_A}} \sin \beta_A \cos \gamma_A
\sin \theta_{1A} \partial_{u_{\theta_{1A}}}
+ \partial_{\gamma_A} \frac{u_{\beta_A}}{\sin \beta_A}
\sin \gamma_A \sin \theta_{1A} \partial_{u_{\theta_{1A}}} \nonumber \\
& - & \partial_{\gamma_A} \frac{u_{\theta_{1A}}}{\sin \theta_{1A}}
\frac{\cos \gamma_A}{\sin \beta_A}
(\partial_{\alpha} + x \partial_y - y
\partial_x)
+ \partial_{\gamma_A} \frac{u_{\theta_{1A}}}{\sin \theta_{1A}}
\sin \beta_A \sin \gamma_A \partial_{u_{\beta_A}} \\
& + & \partial_{u_{\theta_{1A}}}
\frac{\sin \gamma_A}{\sin \beta_A} \sin \theta_{1A}
(\partial_{\alpha} + x \partial_y - y
\partial_x) \nonumber \\
& + & \partial_{u_{\theta_{1A}}}
\sin \beta_A \cos \gamma_A \sin \theta_{1A}
\partial_{u_{\beta_A}} + \partial_{u_{\theta_{1A}}}
\frac{u_{\beta_A}}{\sin \beta_A}
\sin \gamma_A \sin \theta_{1A} \partial_{\gamma_A}] \nonumber \\\end{aligned}$$ and for B (the situation is a little bit simpler since monomer B is not involved in the definition of the third Euler angle see Eq. (\[Eq:487\])): $$\begin{aligned}
\label{Eq:492}
& & {({\vec{L}_B} \cdot {\vec{L}_{1B}})}_{BFB} = \nonumber \\
& - & \frac{\hbar^2}{2} [\partial_{\alpha} \frac{u_{\theta_{1B}}}{\sin \theta_{1B}}
\frac{\cos \gamma_B}{\sin \beta_B}
\partial_{\gamma_B}
+ \partial_{u_{\beta_B}} \frac{u_{\theta_{1B}}}{\sin \theta_{1B}}
\sin \beta_B \sin \gamma_B \partial_{\gamma_B}
- 2 \partial_{\gamma_B} \frac{u_{\theta_{1B}}}{\sin \theta_{1B}}
\cos \gamma_B \frac{u_{\beta_B}}{\sin \beta_B}
\partial_{\gamma_B} \nonumber \\
& + & 2 \partial_{\gamma_B}^2 - \partial_{\alpha}
\frac{\sin \gamma_B}{\sin \beta_B} \sin \theta_{1B}
\partial_{u_{\theta_{1B}}} + \partial_{u_{\beta_B}} \sin \beta_B \cos \gamma_B
\sin \theta_{1B} \partial_{u_{\theta_{1B}}}
+ \partial_{\gamma_B} \frac{u_{\beta_B}}{\sin \beta_B}
\sin \gamma_B \sin \theta_{1B} \partial_{u_{\theta_{1B}}} \nonumber \\
& + & \partial_{\gamma_B} \frac{u_{\theta_{1B}}}{\sin \theta_{1B}}
\frac{\cos \gamma_B}{\sin \beta_B} \partial_{\alpha}
+ \partial_{\gamma_B} \frac{u_{\theta_{1B}}}{\sin \theta_{1B}}
\sin \beta_B \sin \gamma_B \partial_{u_{\beta_B}} \nonumber \\
& - & \partial_{u_{\theta_{1B}}}
\frac{\sin \gamma_B}{\sin \beta_B} \sin \theta_{1B}
\partial_{\alpha} + \partial_{u_{\theta_{1B}}}
\sin \beta_B \cos \gamma_B \sin \theta_{1B}
\partial_{u_{\beta_B}} \nonumber \\
& + & \partial_{u_{\theta_{1B}}}
\frac{u_{\beta_B}}{\sin \beta_B}
\sin \gamma_B \sin \theta_{1B} \partial_{\gamma_B}] \nonumber \\\end{aligned}$$
- \(3) For the term in $1/2 {( {\vec{L}}_A^\dagger \cdot {\vec{L}}_B )}_{E2}$, we note that the expression of the projections of ${\vec{L}}_{A (B)}$ onto the E2 can be seen as those of total angular momentum of a system onto the axes of the Space fixed frame. Indeed, since monomers A and B are not involved in the definition of E2, we have (similar to Eq. (18) in Ref. [@gat98:8804]):
$$\label{Eq:455}
\begin{array} {l}
{{L}}_{A \, x^{E2}} =
i \hbar \cos \alpha_A \frac{u_A} {\sin \beta_A} { \partial_{\alpha_A}}
- i \hbar \sin \alpha_A \sin \beta_A { \partial_{u_{\beta A}} }
-i \hbar \frac{ \cos
\alpha_A} { \sin \beta_A} { \partial_{{\gamma}_A}}\\
{{L}}_{A \, y^{E2}} = i \hbar \sin \alpha_A
\frac{u_A} {\sin \beta_A} { \partial_{\alpha_A}}
+ i \hbar \cos \alpha_A \sin \beta_A {\partial_{u_{\beta A}} }
- i \hbar \frac{ \sin \alpha_A} { \sin \beta_A} {\partial_{{\gamma}_A}} \\
{{L}}_{A \, z^{E2}} = -i \hbar
{\partial_{\alpha_A}}
\end{array}$$
After, noticing that $$\begin{aligned}
\label{Eq:484}
{( {\vec{L}}_A \cdot {\vec{L}}_B)}_{E2} & = &
\frac{1}{2} {( {\vec{L}}_A^\dagger \cdot {\vec{L}}_B +
{\vec{L}}_B^\dagger \cdot {\vec{L}}_A )}_{E2} \nonumber \\\end{aligned}$$
we obtain after using Eq. (\[Eq:487\]):
$$\begin{aligned}
\label{Eq:490}
& & \frac{1}{2} {( {\vec{L}}_A^\dagger \cdot {\vec{L}}_B )}_{E2} =
- \frac{\hbar^2}{2} [ - \partial_{\alpha}
\cos{\alpha} \frac{u_{\beta_A}}{\sin \beta_A}
\frac{u_{\beta_B}}{\sin \beta_B} \partial_{\alpha} \nonumber \\
& - & (x \partial_y - y
\partial_x)
\cos{\alpha} \frac{u_{\beta_A}}{\sin \beta_A}
\frac{u_{\beta_B}}{\sin \beta_B} \partial_{\alpha} \nonumber \\
& + & \partial_{\gamma_A} \frac{\cos{\alpha}}{\sin \beta_A \sin \beta_B}
\partial_{\gamma_B}
+ \partial_{\alpha} \sin{\alpha} \frac{u_{\beta_A}}{\sin \beta_A}
\sin \beta_B \partial_{u_{\beta_B}}
+ (x \partial_y - y
\partial_x)
\sin{\alpha} \frac{u_{\beta_A}}{\sin \beta_A}
\sin \beta_B \partial_{u_{\beta_B}}
\nonumber \\
& + & \partial_{\alpha} \cos{\alpha} \frac{u_{\beta_A}}{\sin \beta_A}
\frac{1}{\sin \beta_B} \partial_{\gamma_B}
+ \partial_{u_{\beta_A}} \sin{\alpha} \sin \beta_A
\frac{u_{\beta_B}}{\sin \beta_B} \partial_{\alpha} \nonumber \\
& + & (x \partial_y - y
\partial_x) \cos{\alpha} \frac{u_{\beta_A}}{\sin \beta_A}
\frac{1}{\sin \beta_B} \partial_{\gamma_B} \nonumber \\
& + & \partial_{u_{\beta_A}} \cos{\alpha} \sin \beta_A
\sin \beta_B \partial_{u_{\beta_B}}
- \partial_{u_{\beta_A}} \sin{\alpha}
\frac{\sin \beta_A}{\sin \beta_B} \partial_{\gamma_B} \nonumber \\
& - & \partial_{\gamma_A} \cos{\alpha} \frac{1}{\sin \beta_A}
\frac{u_{\beta_B}}{\sin \beta_B} \partial_{\alpha}
+ \partial_{\gamma_A} \sin{\alpha}
\frac{\sin \beta_B}{\sin \beta_A} \partial_{u_{\beta_B}}
- \partial^2_{\alpha}
- \partial_{\alpha} (x \partial_y - y
\partial_x) ] \nonumber \\\end{aligned}$$
- \(4) Since the proton is not involved in the definition of the whole body fixed frame, the expression of the projections of $\hat{\vec{l}}$ onto the E$_2$ (or BF) axes is the usual one. At the end, we obtain:
$$\begin{aligned}
\label{Eq:proton}
{({\vec{l}})}^2_{E2} & = & {({\vec{l}})}^2_{BF} =
- \hbar^2
(y^2 \partial_z^2 + z^2 \partial_y^2
+ z^2 \partial_x^2 + x^2 \partial_z^2
+ x^2 \partial_y^2 + y^2 \partial_x^2 \nonumber \\
& & -y \partial_y \partial_z z - \partial_y
y z \partial_z - \partial_x x z \partial_z
-x \partial_x \partial_z z - x \partial_x \partial_y y
- \partial_x x y \partial_y)\end{aligned}$$
- \(5) For the last term ${({\vec{L}}_A+{\vec{L}}_B)
\cdot {\vec{l}}}_{E2}
=
{{({\vec{L}}_A+{\vec{L}}_B)^\dagger \cdot {\vec{l}} +
{\vec{l}}^\dagger \cdot ({\vec{L}}_A+{\vec{L}}_B)}_{BF}}$, we need to know the expression of the projections of ${\vec{L}}_{A (B)}$ onto the body fixed axes. For ${\vec{L}}_{B}$, there is no particular problem since B is not involved in the definition of $\gamma$ and thus of the BF frame:
$$\label{Eq:47}
\begin{array} {l}
{{L}}_{B \, x^{BF}} = i \hbar
\frac{u_{\beta B}}{\sin \beta_B} \cos \alpha {\partial_{\alpha}}
- i \hbar \sin \alpha \sin \beta_B { \partial_{u_{\beta B}}}
- i \hbar \frac{\cos \alpha}{\sin \beta_{\beta B}} { \partial_{{\gamma}_B}}\\
{{L}}_{B \, y^{BF}} = i \hbar \sin \alpha \frac{u_{\beta B}} {\sin \beta_B}
{ \partial_{\alpha}}
+i \hbar \cos \alpha \sin \beta_B { \partial_{u_{\beta B}}}
-i \hbar \frac{\sin \alpha}{\sin \beta_B} {\partial_{{\gamma}_B}} \\
{{L}}_{B \, z^{BF}} = -i \hbar
{ \partial_{\alpha}}
\end{array}$$
For ${\vec{L}}_{A}$, it is less straightforward. However, applying the third Euler rotation to Eq. (\[Eq:455\]) and using Eq. (\[Eq:487\]) yields:
$$\label{Eq:466}
\begin{array} {l}
{{L}}_{A \, x^{BF}} =
- i \frac{u_{\beta A}}{\sin \beta_A} ( {\partial_{\alpha}}
+ x \partial_{y} -y \partial_{x})
- \frac{i}{\sin \beta_A} {\partial_{{\gamma}_A}}\\
{{L}}_{A \, y^{BF}} = i \sin \beta_A
{ \partial_{u_{\beta A}}} \\
{{L}}_{A \, z^{BF}} = i ({\partial_{\alpha}}
+ x \partial_{y} -y \partial_{x})
\end{array}$$
and thus: $$\begin{aligned}
{({\vec{L}}_A+{\vec{L}}_B) \cdot {\vec{l}}}_{E2}
& = & \frac{1}{2}
{({({\vec{L}}_A+{\vec{L}}_B)^\dagger \cdot {\vec{l}} +
{\vec{l}}^\dagger \cdot ({\vec{L}}_A+{\vec{L}}_B))}_{BF}}
= \frac{\hbar^2} {2} [ (-
2 \frac{u_{\beta A}}{\sin \beta_A} {\partial_{\alpha}}
- 2 \frac{1}{\sin \beta_A} {\partial_{{\gamma}_A}}
\nonumber \\
& + & \frac{u_{\beta B}}{\sin \beta_B} \cos \alpha
{\partial_{\alpha}}
+ { \partial
\alpha} \frac{u_{\beta B}}{\sin \beta_B} \cos \alpha
- \sin \alpha \sin \beta_B {\partial_{u_{\beta B}}}
\nonumber \\
& - & {\partial_{u_{\beta B}}} \sin \alpha \sin \beta_B
- 2 \frac{\cos \alpha}{\sin \beta_B}
{\partial_{{\gamma}_B}}
) (y { \partial_z} - z { \partial_y }) \nonumber \\
& + & ( \sin \beta_A { \partial_{u_A}}
+ {\partial_{u_{\beta A}}} \sin \beta_A
\nonumber \\
& + & \sin \alpha \frac{u_{\beta B}}{\sin \beta_B} {\partial_{\alpha}}
+ {\partial_{\alpha}} \sin \alpha \frac{u_B}{\sin \beta_B}
+ \cos \alpha \sin \beta_B { \partial_{u_{\beta B}}}
\nonumber \\
& + & { \partial_{u_{\beta B}}} \cos \alpha \sin \beta_B
- 2 \frac{\sin \alpha} {\sin \beta_B}
{ \partial_{{\gamma}_B}}) (z { \partial_x} - x
{ \partial_z} )\nonumber \\
& + & 2 (x^2 \partial_y^2 + y^2 \partial_x^2 -
x \partial_x \partial_y y - \partial_x x y \partial_y)] \nonumber \\
&-& {\hbar^2} [ \frac{u_A}{\sin \beta_A}
(x \partial_y y \partial_z + x y \partial_y \partial_z
-2 x \partial_y^2 z -2 \partial_x y^2 \partial_z
+ \partial_x y \partial_y z + \partial_x \partial_y y z)]
\nonumber \\\end{aligned}$$
The final expression of the operator in Eq. (\[Eq:482\]) is recast as $\hat{T} = \hat{T}_1 + \hat{T}_2 + \hat{T}_3 + \hat{T}_4$ with
$$\begin{aligned}
\label{Eqs:482}
&\hat{ T}_1& = (-\frac{\hbar^2} {2 \mu_R}
\frac{\partial^2}{\partial_R^2} )
+ \sum_{i=1}^{2} (-\frac{\hbar^2} {2 \mu_{iA}}
\frac{\partial^2 }{\partial_{R_{iA}}^2} )
+ \sum_{i=1}^{2} (-\frac{\hbar^2} {2 \mu_{iB}}
\frac{\partial^2 }{\partial_{R_{iB}}^2} )
- \frac{\hbar^2}{2m} \frac{\partial^2}{\partial_{r}^2}
+ \frac{{{({l}^2)}_{E2}}}{2 m r^2} \nonumber \\
& + & {({{\vec{L}_A}^2})}_{BFA}
(\frac{1}{2 \mu_R R^2}+\frac{1}{2 \mu_{2A} R_{2A}^2})
+ {({{\vec{L}_B}^2})}_{BFB}
(\frac{1}{2 \mu_R R^2}+\frac{1}{2 \mu_{2B} R_{2B}^2}) \nonumber \\
& + & {({{\vec{L}_{1A}}^{\dagger}} \cdot
{\vec{L}_{1A}})}_{BFA} (\frac{1} {2 \mu_{1A} R_{1A}^2} +
\frac{1} {2 \mu_{2A} R_{2A}^2})
+ {({{\vec{L}_{1B}}^{\dagger}} \cdot {\vec{L}_{1B}})}_{BFB}
(\frac{1} {2 \mu_{1B} R_{1B}^2} +
\frac{1} {2 \mu_{2B} R_{2B}^2}) \nonumber \\\end{aligned}$$
$$\begin{aligned}
\label{Eqs:4821}
&\hat{ T}_2 & =
- \frac{{ ({\vec{L}_A} \cdot
{\vec{L}_{1A}})}_{BFA}} { \mu_{2A} R_{2A}^2}
- \frac{{ ({\vec{L}_B} \cdot
{\vec{L}_{1B}})}_{BFB}} { \mu_{2B} R_{2B}^2}
\nonumber \\\end{aligned}$$
$$\begin{aligned}
\label{Eqs:4822}
&\hat{ T}_3 & =
\frac{ { ({{\vec{L}}_A \cdot {\vec{L}}_B})}_{E2} }
{ \mu_R R^2}\end{aligned}$$
$$\begin{aligned}
\label{Eqs:4823}
&\hat{ T}_4 & =
\frac{{({\vec{l}}^2)}_{E2}} {2 \mu_R}
+ \frac{ {( {\vec{L}}_A + {\vec{L}}_B)
\cdot {\vec{l}}}_{E2} }
{ \mu_R R^2} \nonumber \\\end{aligned}$$
The expression of the operator in terms of the 15 degrees of freedom: $R, R_{1A}, R_{2A}, R_{1B}, R_{2B}, x, y ,z, \alpha, u_{\beta_A},
\gamma_A,
u_{\beta_B}, \gamma_B,
u_{\theta_{1A}}, u_{\theta_{1B}}$ reads: (note that the hermiticity clearly appears)
$$\begin{aligned}
\label{Eq:494}
&\hat{ T}_1 & = (-\frac{\hbar^2} {2 \mu_R} {\partial_R^2} )
+ \sum_{i=1}^{2} (-\frac{\hbar^2} {2 \mu_{iA}} {\partial_{R_{iA}}^2} )
+ \sum_{i=1}^{2} (-\frac{\hbar^2} {2 \mu_{iB}} {\partial_{R_{iB}}^2} )
- \frac{\hbar^2}{2m} ({\partial_{x}^2}
+ {\partial_{y}^2} +
{\partial_{z}^2})
\nonumber \\
& - & \hbar^2 (\partial_{u_{\beta_A}} {(1-u_{\beta_A}^2)} \partial_{u_{\beta_A}}
+ \frac{1} {1-u_{\beta_A}^2} (
\partial_{\alpha}^2 + x^2 \partial_y^2 + y^2
\partial_x^2 -x \partial_x \partial_y y
-y \partial_y \partial_x x \nonumber \\
& + & 2 x \partial_y \partial_{\alpha} - 2 y \partial_x \partial_{\alpha}
+ \partial_{\gamma_A}^2 + 2 u_{\beta_A} \partial_{\alpha} \partial_{\gamma_A}
\nonumber \\
& + & 2 u_{\beta_A} \partial_{\gamma_A} x \partial_y
- 2 u_{\beta_A} \partial_{\gamma_A}
y \partial_x))
(\frac{1}{2 \mu_R R^2}+\frac{1}{2 \mu_{2A} R_{2A}^2}) \nonumber \\
& - & \hbar^2 (\partial_{u_{\beta_B}} {(1-u_{\beta_B}^2)} \partial_{u_{\beta_B}}
+ \frac{1} {1-u_{\beta_B}^2} (\partial_{\alpha}^2 +
\partial_{\gamma_B}^2 - 2 u_{\beta_B} \partial_{\alpha} \partial_{\gamma_B}))
(\frac{1}{2 \mu_R R^2}+\frac{1}{2 \mu_{2B} R_{2B}^2}) \nonumber \\
& - & \hbar^2 (\partial_{u_{\theta{1A}}} {(1-u_{\theta_{1A}}^2)}
\partial_{u_{\theta{1A}}} + \frac{1} {1-u_{\theta_{1A}}} \partial_{\gamma_A}^2)
(\frac{1} {2 \mu_{1A} R_{1A}^2} +
\frac{1} {2 \mu_{2A} R_{2A}^2}) \nonumber \\
& - & \hbar^2 (\partial_{u_{\theta{1B}}} {(1-u_{\theta_{1B}}^2)}
\partial_{u_{\theta{1B}}} + \frac{1} {1-u_{\theta_{1B}}} \partial_{\gamma_B}^2)
(\frac{1} {2 \mu_{1B} R_{1B}^2} +
\frac{1} {2 \mu_{2B} R_{2B}^2}) \nonumber \\\end{aligned}$$
$$\begin{aligned}
\hat{T}_2 =
& & \frac{ \hbar^2} {2 \mu_{2A} R_{2A}^2}
[- (\partial_{\alpha} + (x \partial_y -y \partial_x))
\frac{u_{\theta_{1A}}}{\sin \theta_{1A}}
\frac{\cos \gamma_A}{\sin \beta_A} \partial_{\gamma_A} \nonumber \\
& + & \partial_{u_{\beta_A}} \frac{u_{\theta_{1A}}}{\sin \theta_{1A}}
\sin \beta_A \sin \gamma_A \partial_{\gamma_A}
- 2 \partial_{\gamma_A} \frac{u_{\beta_A}}{\sin \beta_A}
\cos \gamma_A \frac{u_{\theta_{1A}}}{\sin \theta_{1A}} \partial_{\gamma_A}
\nonumber \\
& + & 2 \partial_{\gamma_A}^2 + (\partial_{\alpha} + (x \partial_y
-y \partial_x))
\frac{\sin \gamma_A}{\sin \beta_A} \sin \theta_{1A} \partial_{u_{\theta_{1A}}}
\nonumber \\
& + & \partial_{u_{\beta_A}} \sin \beta_A \cos \gamma_A
\sin \theta_{1A} \partial_{u_{\theta_{1A}}}
+ \partial_{\gamma_A} \frac{u_{\beta_A}}{\sin \beta_A}
\sin \gamma_A \sin \theta_{1A} \partial_{u_{\theta_{1A}}} \nonumber \\
& - & \partial_{\gamma_A} \frac{u_{\theta_{1A}}}{\sin \theta_{1A}}
\frac{\cos \gamma_A}{\sin \beta_A}
(\partial_{\alpha} + (x \partial_y - y \partial_x)) \nonumber \\
& + & \partial_{\gamma_A} \frac{u_{\theta_{1A}}}{\sin \theta_{1A}}
\sin \beta_A \sin \gamma_A \partial_{u_{\beta_A}} \\
& + & \partial_{u_{\theta_{1A}}}
\frac{\sin \gamma_A}{\sin \beta_A} \sin \theta_{1a}
(\partial_{\alpha} + (x \partial_y -y \partial_x)) \nonumber \\
& + & \partial_{u_{\theta_{1a}}}
\sin \beta_A \cos \gamma_A \sin \theta_{1A}
\partial_{u_{\beta_A}} + \partial_{u_{\theta_{1A}}}
\frac{u_{\beta_A}}{\sin \beta_A}
\sin \gamma_A \sin \theta_{1A} \partial_{\gamma_A}] \nonumber \\
& + & \frac{\hbar^2} { 2 \mu_{2B} R_{2B}^2}
[\partial_{\alpha} \frac{u_{\theta_{1B}}}{\sin \theta_{1B}}
\frac{\cos \gamma_B}{\sin \beta_B}
\partial_{\gamma_B}
+ \partial_{u_{\beta_B}} \frac{u_{\theta_{1B}}}{\sin \theta_{1B}}
\sin \beta_B \sin \gamma_B \partial_{\gamma_B}
\nonumber \\
& - & 2 \partial_{\gamma_B} \frac{u_{\theta_{1B}}}{\sin \theta_{1B}}
\cos \gamma_B \frac{u_{\beta_B}}{\sin \beta_B}
\partial_{\gamma_B} \nonumber \\
& + & 2 \partial_{\gamma_B}^2 - \partial_{\alpha}
\frac{\sin \gamma_B}{\sin \beta_B} \sin \theta_{1B}
\partial_{u_{\theta_{1B}}} + \partial_{u_{\beta_B}} \sin \beta_B \cos \gamma_B
\sin \theta_{1B} \partial_{u_{\theta_{1B}}}
\nonumber \\
& + & \partial_{\gamma_B} \frac{u_{\beta_B}}{\sin \beta_B}
\sin \gamma_B \sin \theta_{1B} \partial_{u_{\theta_{1B}}} \nonumber \\
& + & \partial_{\gamma_B} \frac{u_{\theta_{1B}}}{\sin \theta_{1B}}
\frac{\cos \gamma_B}{\sin \beta_B} \partial_{\alpha}
+ \partial_{\gamma_B} \frac{u_{\theta_{1B}}}{\sin \theta_{1B}}
\sin \beta_B \sin \gamma_B \partial_{u_{\beta_B}} \nonumber \\
& - & \partial_{u_{\theta_{1B}}}
\frac{\sin \gamma_B}{\sin \beta_B} \sin \theta_{1B}
\partial_{\alpha} + \partial_{u_{\theta_{1B}}}
\sin \beta_B \cos \gamma_B \sin \theta_{1B}
\partial_{u_{\beta_B}} \nonumber \\
& + & \partial_{u_{\theta_{1B}}}
\frac{u_{\beta_B}}{\sin \beta_B}
\sin \gamma_B \sin \theta_{1B} \partial_{\gamma_B}] \nonumber \\\end{aligned}$$
$$\begin{aligned}
\hat{T}_3 & = & \frac{\hbar^2}{2} [ - 2\partial_{\alpha}
\cos{\alpha} \frac{u_{\beta_A}}{\sin \beta_A}
\frac{u_{\beta_B}}{\sin \beta_B} \partial_{\alpha} \nonumber \\
& - & (x \partial_y -y \partial_x)
\cos{\alpha} \frac{u_{\beta_A}}{\sin \beta_A}
\frac{u_{\beta_B}}{\sin \beta_B} \partial_{\alpha} \nonumber \\
& - & \partial_{\alpha}
\cos{\alpha} \frac{u_{\beta_A}}{\sin \beta_A}
\frac{u_{\beta_B}}{\sin \beta_B} (x \partial_y -y \partial_x) \nonumber \\
& + & 2 \partial_{\gamma_A} \frac{\cos{\alpha}}{\sin \beta_A \sin \beta_B}
\partial_{\gamma_B}
+ \partial_{\alpha} \sin{\alpha} \frac{u_{\beta_A}}{\sin \beta_A}
\sin \beta_B \partial_{u_{\beta_B}} \nonumber \\
& + & \partial_{u_{\beta_B}} \sin{\alpha} \frac{u_{\beta_A}}{\sin \beta_A}
\sin \beta_B \partial_{\alpha}
+ (x \partial_y -y \partial_x)
\sin{\alpha} \frac{u_{\beta_A}}{\sin \beta_A}
\sin \beta_B \partial_{u_{\beta_B}} \nonumber \\
& + & \partial_{u_{\beta_B}} \sin{\alpha} \frac{u_{\beta_A}}{\sin \beta_A}
\sin \beta_B (x \partial_y - y \partial_x)
\nonumber \\
& + & \partial_{\alpha} \cos{\alpha} \frac{u_{\beta_A}}{\sin \beta_A}
\frac{1}{\sin \beta_B} \partial_{\gamma_B}
+ \partial_{\gamma_B} \cos{\alpha} \frac{u_{\beta_A}}{\sin \beta_A}
\frac{1}{\sin \beta_B} \partial_{\alpha} \nonumber \\
& + & \partial_{u_{\beta_A}} \sin{\alpha} \sin \beta_A
\frac{u_{\beta_B}}{\sin \beta_B} \partial_{\alpha}
+ \partial_{\alpha} \sin{\alpha} \sin \beta_A
\frac{u_{\beta_B}}{\sin \beta_B} \partial_{u_{\beta_A}}
\nonumber \\
& + & 2 (x \partial_y -y \partial_x) \cos{\alpha} \frac{u_{\beta_A}}{\sin \beta_A}
\frac{1}{\sin \beta_B} \partial_{\gamma_B} \nonumber \\
& + & \partial_{u_{\beta_A}} \cos{\alpha} \sin \beta_A
\sin \beta_B \partial_{u_{\beta_B}}
+ \partial_{u_{\beta_B}} \cos{\alpha} \sin \beta_A
\sin \beta_B \partial_{u_{\beta_A}} \nonumber \\
& - & \partial_{u_{\beta_A}} \sin{\alpha}
\frac{\sin \beta_A}{\sin \beta_B} \partial_{\gamma_B}
- \partial_{\gamma_B} \sin{\alpha}
\frac{\sin \beta_A}{\sin \beta_B} \partial_{u_{\beta_A}} \nonumber \\
& - & \partial_{\gamma_A} \cos{\alpha} \frac{1}{\sin \beta_A}
\frac{u_{\beta_B}}{\sin \beta_B} \partial_{\alpha}
- \partial_{\alpha} \cos{\alpha} \frac{1}{\sin \beta_A}
\frac{u_{\beta_B}}{\sin \beta_B} \partial_{\gamma_A} \nonumber \\
& + & \partial_{\gamma_A} \sin{\alpha}
\frac{\sin \beta_B}{\sin \beta_A} \partial_{u_{\beta_B}}
+ \partial_{u_{\beta_B}} \sin{\alpha}
\frac{\sin \beta_B}{\sin \beta_A} \partial_{\gamma_A} \nonumber \\
& - & 2 \partial^2_{\alpha}
- 2 \partial_{\alpha} (x \partial_y -y \partial_x) ] \nonumber \\\end{aligned}$$
$$\begin{aligned}
\hat{T}_4 & = & \frac{1} {2 \mu_R R^2}
(y^2 \partial_z^2 + z^2 \partial_y^2 + z^2 \partial_x^2
+ x^2 \partial_z^2 + x^2 \partial_y^2 + y^2 \partial_x^2 \nonumber \\
& & -y \partial_y \partial_z z - \partial_y y z \partial_z
- \partial_x x z \partial_z -x \partial_x \partial_z z -
x \partial_x \partial_y y - \partial_x x y \partial_y)
\nonumber \\
& + & \frac{1}{2 \mu_R R^2} [ (-
2 \frac{u_{\beta A}}{\sin \beta_A} { \partial_{\alpha}}
- 2 \frac{1}{\sin \beta_A} {\partial_{{\gamma}_A}}
\nonumber \\
& + & \frac{u_{\beta B}}{\sin \beta_B} \cos \alpha { \partial_{\alpha}}
+{ \partial_{\alpha}} \frac{u_{\beta B}}{\sin \beta_B} \cos \alpha
- \sin \alpha \sin \beta_B { \partial_{u_{\beta B}}}
- { \partial_{u_{\beta B}}} \sin \alpha \sin \beta_B
- 2 \frac{\cos \alpha}{\sin \beta_B} {\partial_{{\gamma}_B}}
) \nonumber \\
& & (y
{ \partial_z} - z { \partial_y}) \nonumber \\
& + & ( \sin \beta_A { \partial_{u_{\beta A}}}
+ { \partial_{u_{\beta A}}} \sin \beta_A
\nonumber \\
& + & \sin \alpha \frac{u_{\beta B}}{\sin \beta_B}
{ \partial_{\alpha}}
+ { \partial_{\alpha}} \sin \alpha
\frac{u_{\beta B}}{\sin \beta_B}
+ \cos \alpha \sin \beta_B {\partial_{u_{\beta B}}}
+ { \partial_{u_{\beta B}}} \cos \alpha \sin \beta_B
- 2 \frac{\sin \alpha} {\sin \beta_B}
{ \partial_{{\gamma}_B}}) \nonumber \\
& & (z { \partial_x} - x
{ \partial_z} )\nonumber \\
& + & 2 (x^2 \partial_y^2 + y^2 \partial_x^2
- x \partial_x \partial_y y - \partial_x x y \partial_y)] \nonumber \\
&-& \frac{1}{2 \mu_R R^2} [ \frac{u_{\beta A}}{\sin \beta_A}
(x \partial_y y \partial_z + x y \partial_y \partial_z
-2 x \partial_y^2 z -2 \partial_x y^2 \partial_z +
\partial_x y \partial_y z + \partial_x \partial_y y z)]
\nonumber \\\end{aligned}$$
This completes the derivation of the kinetic energy operator of H$_5$O$_2^+$. We emphasize again that this operator is exact. Furthermore, the correctness of the derivation and implementation of the KEO was checked by comparing it with numerical results provided by the program TNUM [@lau02:8560]. A KEO can formally be written as . TNUM computes $G$, $F$ and $V_{extra}$ numerically. We have checked that the numerical values of all the functions $G_{ij}(\mathbf{q})$ at several non-symmetrical grid points $\mathbf{q}$ agree with those provided by the program TNUM. The functions $F_j(\mathbf{q})$ are determined through the hermiticity of the KEO. As our KEO is obviusly hermitian, there is no need to check the $F_j$.
Hierarchical Representation of the Potential using Mode-Combination {#sec:cluster}
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The exact PES ($v(\mathbf{q})$) for is a function of the 15 internal coordinates previously defined, where “exact" refers to the full dimensional PES of Bowman and collaborators [@hua05:044308]. The calculation of vibrational levels or the IR spectrum requires a high accuracy in the representation of the Hamiltonian. The exact and trivial representation of $v(\mathbf{q})$ on a product grid is unfortunately beyond current computational capabilities: in this case the potential would be given on a grid of $\approx10^{15}$ points ($\approx10^{4}$ TB of disk space). A direct use of the potential is hence impossible and in particular it is impossible to convert the potential to MCTDH product form by using the potfit algorithm [@jae96:7974; @jae98:3772] since the full product grid is needed for such a transformation. The problem of representing a high-dimensional PES for quantum-dynamical computation has already been considered in the context of Multimode simulations [@bow03:533], as well as in the more general context of the high dimensional model representation (HDMR) [@rab99:197; @ali01:127; @li06:2474; @man06:084109].
In such hierarchical representations a multidimensional function dependent on $f$ variables is approximated as: $$\label{eq:hdmr}
\tilde{v}(\mathbf{q}) = v^{(0)} + \sum_{\alpha=1}^f v^{(1)}_{\alpha}(q_\alpha) +
\sum_{\alpha<\beta}^f v^{(2)}_{\alpha\beta}(q_\alpha,q_\beta)
+ \sum_{\alpha<\beta<\gamma}^f v^{(3)}_{\alpha\beta\gamma}(q_\alpha,q_\beta,q_\gamma) \cdots$$ where $\mathbf{q}$ is a coordinate vector and $\tilde{v}(\mathbf{q})$ denotes the hierarchical approximation to $v(\mathbf{q})$. The component functions in Eq. (\[eq:hdmr\]) can be determined by minimizing the functional [@ali01:127; @man06:084109] $$\label{eq:hdmrErr}
\int_D \left[ v(\mathbf{q}) - \tilde{v}(\mathbf{q}) \right]^2 w(\mathbf{q})d\mathbf{q},$$ where $w(\mathbf{q})$ is some weight function on the integration domain that determines the form of the component functions. Taking $w(\mathbf{q})=\prod_{\kappa}^f w_\kappa(q_\kappa)$ leads to component functions of the form [@ali01:127; @wan03:4707; @li06:2474; @man06:084109]
\[eq:hdmrComp\] $$\begin{aligned}
v^{(0)} & = &
\int_D \prod_{\kappa=1}^f w_\kappa(q_\kappa) v(\mathbf{q})\,d\mathbf{q} \\
v^{(1)}_{\alpha}(q_\alpha) & = &
\int_{D^{f-1}} \prod_{\kappa\neq\alpha}^f w_\kappa(q_\kappa)
v(\mathbf{q})\,d\mathbf{q}^{\alpha} - v^{(0)} \\
v^{(2)}_{\alpha\beta}(q_\alpha,q_\beta) & = &
\int_{D^{f-2}} \prod_{\kappa\neq\alpha,\beta}^f w_\kappa(q_\kappa)
v(\mathbf{q})\,d\mathbf{q}^{\alpha\beta}
- v^{(1)}_{\alpha}(q_\alpha) - v^{(1)}_{\alpha}(q_\alpha) - v^{(0)}\\
\nonumber \cdots & &
\end{aligned}$$
There $\mathbf{q}^{\alpha}$ represents a vector of coordinates in which the $\alpha$-th component has been removed, $\mathbf{q}^{\alpha\beta}$ represents a vector in which $\alpha$ and $\beta$ components have been removed, and so on. Non-separable weights have also been considered, which lead to more complicated expressions for component functions (see for example the Appendix in Ref. ). It is possible to evaluate the integrals in Eq. (\[eq:hdmrComp\]) by random sampling of the coordinate space, leading to the so called RS-HDMR [@ali01:127; @wan03:4707; @li06:2474; @man06:084109]. However, the cumbersome multi-dimensional integrals can be solved trivially by choosing the particular weight $$\label{eq:hdmrCutDelta}
w(\mathbf{q}) = \prod_{\alpha=1}^f \delta(q_\alpha-a_\alpha)$$ which corresponds to the cut-HDMR approximation [@bow03:533; @rab99:197; @li01:1; @ali01:127]. There $a_\alpha$ is the $\alpha$-th component of point $\mathbf{a}$, the reference expansion point in coordinate space. Using the definition of $w(\mathbf{q})$ in Eq. (\[eq:hdmrCutDelta\]), the different terms in Eq. (\[eq:hdmr\]), which we will also refer to as uncombined clusters (UC), are given up to second order by [@li01:1]
\[eq:hdmr-terms\] $$\begin{aligned}
v^{(0)} &=& v(\mathbf{a}) \\
v^{(1)}_\alpha(q_\alpha) &=& v(q_\alpha;\mathbf{a}^\alpha) - v^{(0)} \\
v^{(2)}_{\alpha\beta}(q_\alpha,q_\beta)
&=& v(q_\alpha,q_\beta;\mathbf{a}^{\alpha\beta})
- v^{(1)}_\alpha(q_\alpha) - v^{(1)}_\beta(q_\beta) - v^{(0)},
\end{aligned}$$
where higher orders follow trivially. Cut-HDMR representations of a PES – also called n-mode representation – have been already used successfully to accurately compute vibrational energy-levels of molecular systems [@bow03:533; @cha04:2071] and reaction rates for molecule-surface scattering [@kro07:334]. Unfortunately, the use of a HDMR representation of the PES of the form of Eq. (\[eq:hdmr\]) leads to a combinatorial increase in the number of terms as the order of correlation increases. At the same time, higher order terms are given on grids with an exponentially increasing number of points, which leads to quick stagnation in the maximum correlation order that can be practically included.
Instead of directly adopting the expression in Eq. (\[eq:hdmr\]) for $v(\mathbf{q})$ we make use of the fact that the wavefunction in Eq. (\[eq:ansatz\]) is given in terms of combined modes. Hence we define the potential also as a function of the combined modes. The use of combined modes instead of coordinates as base of the hierarchical expansion attenuates the combinatorial increase in complexity found when using Eq. (\[eq:hdmr\]) while still retaining the simple evaluation of the expansion terms given by Eq. (\[eq:hdmr-terms\]) and the inclusion of high-order correlations.
For $f$ coordinates $\mathbf{q}=[q_1,\ldots, q_f]$, $p$ particles or combined modes $\mathbf{Q}=[Q_1,\ldots, Q_p]$ are defined, such that $Q_i = [q_1^{(i)},\ldots,q_{f_i}^{(i)}]$ and $\sum_{i=1}^p f_i = f$. The reference potential can be equivalently given as a function of the combined modes, $V(\mathbf{Q}) \equiv v(\mathbf{q})$. The general hierarchical expansion of Eq. (\[eq:hdmr\]) is now written in terms of the combined modes $Q_i$, instead of coordinates $q_\alpha$, which up to second order reads: $$\label{eq:hdmr-VQ}
\tilde{V}(\mathbf{Q}) = V_0 + \sum_{i} V_i^{(1)}(Q_i)
+ \sum_{ij} V_{ij}^{(2)}(Q_i,Q_j).$$ First order $V_i^{(1)}$ and second order $V_{ij}^{(2)}$ combined clusters (CC) are defined in an analogous way to Eq. (\[eq:hdmr-terms\]):
\[eq:VQ-terms\] $$\begin{aligned}
V^{(0)} &=& V(\mathbf{a}) \\
V_i^{(1)}(Q_i) &=& V(Q_i;\mathbf{a}^i) - V_0 \\
V_{ij}^{(2)}(Q_i,Q_j) &=&
V(Q_i,Q_j;\mathbf{a}^{ij})
- V_i^{(1)}(Q_i) - V_j^{(1)}(Q_j) - V_0
\end{aligned}$$
First order CC can be given directly in multidimensional product-grids which are direct products of the corresponding 1D DVR grids since the wavepacket is expanded in sums of products of SPFs which are defined on the same combined grids as used in the cluster expansion. Second order and higher CC are conveniently and accurately represented as sums of products of multidimensional mode-functions by employing the potfit-algorithm [@jae96:7974; @jae98:3772].
On the other hand, first order CC can be exactly represented as a cut-HDMR expansion of the form of Eq. (\[eq:hdmr\]) up to order $f_i$ in terms of the UC made of the coordinates $[q_1^{(i)},q_2^{(i)},\ldots]$ in $Q_i$, while analogously, second order CC can be given by a cut-HDMR expansion up to order $f_i+f_j$ in terms of the corresponding UC. When inspecting the expansion in Eq. (\[eq:hdmr-VQ\]) one realizes that the correlation between three coordinates belonging to different modes, i.e. $[q_\alpha^{(i)},q_\beta^{(j)},q_\gamma^{(k)}]$, is accounted for up to second order with respect to the uncombined representation since only the UC $v^{(2)}_{\alpha\beta}(q_\alpha^{(i)},q_\beta^{(j)})$, $v^{(2)}_{\alpha\gamma}(q_\alpha^{(i)},q_\gamma^{(k)})$ and $v^{(2)}_{\beta\gamma}(q_\beta^{(j)},q_\gamma^{(k)})$ are contained in the second-order CC $V_{ij}^{(2)}(Q_i,Q_j)$, $V_{ik}^{(2)}(Q_i,Q_k)$ and $V_{jk}^{(2)}(Q_j,Q_k)$ respectively. Thus, using a second order expansion of CC one implicitely introduces a cluster-selection scheme of higher (up to $f_i+f_j$) order UC. The higher-order cluster-selection scheme is determined by how coordinates are grouped together into combined modes. The same reasoning can be extended to higher than second order CC. By combining coordinates which are strongly coupled it is then possible to get a high-accuracy potential while still using a reduced number of CC.
[**FIGURE \[fig:mode-comb\] AROUND HERE**]{}
Assuming that the grid representation of each coordinate uses on average $N$ points, the number of points in coordinate-space, $N_{tot}$, used to define the UC, i.e. without mode combination, is given by $$\label{eq:mode-comb}
N_{tot} = \sum_{\alpha=0}^{h} \binom{f}{\alpha} N^{\alpha},$$ where $h$ is the maximum allowed order for the clusters and $f$ is the number of degrees of freedom. In case the coordinates are combined into modes with $m$ coordinates each, the number of grid points needed to represent the clusters is given by $$\label{eq:mode-comb2}
N_{tot} = \sum_{i=0}^{h/m} \binom{f/m}{i} N^{\,i\cdot m}.$$ The number of points $N$ for different values of the parameters defining different clustering schemes is depicted in Fig. \[fig:mode-comb\]. The horizontal axis represents $h$, the maximum order of clustering in terms of the uncombined coordinates, i.e., the maximum order of the UC in the expansion. A total of $f=15$ coordinates is assumed. A horizontal line is drawn at $10^8$, which, tentatively, constitutes a practical limit to the total number of grid points that can be used both concerning the generation of the clusters and their subsequent use in the dynamical calculations. The example in Fig. \[fig:mode-comb\] assumes that there are $N=10$ points per coordinate. Using 3D modes it is possible to include 6th order UC with around $10^7$ grid points, which is of the order of the PES presented in section \[sec:PES\]. The inclusion of up to 5th order UC without mode-combination would require more than $10^8$ points while the inclusion of 6th order UC would require around $10^{10}$ points. We emphasize again, however, that the representations up to order $h$ with and without mode-combination are not equivalent. The representation without mode-combination contains [*all*]{} the possible UC of coordinates up to order $h$. In the case of using CC one is implicitly selecting a subset of UC. A definition of meaningful mode combinations should however be possible in most cases. Such a definition would be based on chemical common-sense, e.g., coordinates belonging to the same chemical group or to the same molecule in a cluster are good candidates to be combined. As a final remark, we note that there is ongoing effort by other groups [@li06:2474; @man06:084109] to use parametrized functions instead of grids to describe the clusters. This approach may turn out to be more efficient than a grid representation.
Potential Energy Surface for the cation {#sec:PES}
---------------------------------------
In order to construct the PES for the cation employing the approach described above one must start by defining the combined modes that are going to be used. In the present case the following five multidimensional modes are selected: $Q_1 = [z,\alpha,x,y]$, $Q_2 = [\gamma_A,\gamma_B]$, $Q_3 = [R, u_{\beta_{A}}, u_{\beta_{B}}]$, $Q_4 = [R_{1 A}, R_{2 A}, u_{\theta_{1 A}}]$ and $Q_5 = [R_{1 B}, R_{2 B}, u_{\theta_{1 B}}]$. It is convenient that coordinates $x$, $y$ and $\alpha$ are grouped together due to symmetry conserving reasons which are exposed below. Modes $Q_2$ and $Q_3$ contain the wagging and rocking coordinates, respectively. Modes $Q_4$ and $Q_5$ contain the Jacobi coordinates which represent the internal configuration of each water molecule. Coordinates $z$ and $R$ are good candidates to be combined together, as will be discussed in Section \[sec:quality\]. They are not combined here since this would require the definition of a 6th mode. We have, in fact, started to do some tests with a 6 particle mode-combination, and we give some brief remark on this below. The definition of the underlying 1D grids is provided in Table \[tab:1Dgrids\].
[**TABLE \[tab:1Dgrids\] AROUND HERE**]{}
Following the procedure outlined above one may select a reference point in coordinate space and proceed straightforwardly to the computation of the clusters defining the cluster expansion. Instead of this, the PES expansion is defined in terms of $M=10$ reference points and the weight in Eq. (\[eq:hdmrErr\]) takes the form $$\label{eq:weight}
w(\mathbf{q}) = \frac{1}{M} \sum_{l=1}^M \delta(\mathbf{q}-\mathbf{a}_l).$$ The reference points $\mathbf{a}_l$ are located on or very close to stationary points in the lowest energy regions of the PES. After Eq. (\[eq:weight\]) the PES expansion is given by $$\label{eq:VtotQ}
\tilde{V}_{tot}(\mathbf{Q}) = \frac{1}{M} \sum_{l=1}^{M} \tilde{V}_l({\mathbf{Q}}).$$ The $\tilde{V}_l({\mathbf{Q}})$ terms are given by Eqs. (\[eq:hdmr-VQ\]) and (\[eq:VQ-terms\]). The specific form of $\tilde{V}_l({\mathbf{Q}})$ that has been used here is given by $$\label{eq:VtotQ2}
\tilde{V}_l({\mathbf{Q}}) = V_l^{(0)}
+ \sum_{i=1}^5 V_{l,i}^{(1)}(Q_i)
+ \sum_{i=1}^4 \sum_{j=i+1}^5 V_{l,ij}^{(2)}(Q_i,Q_j)
+ V_{l,z23}^{(3)}(z,Q_2,Q_3),$$ where the modes $Q_1\cdots Q_5$ have been defined above. The $V_l^{(0)}$ term is the energy at the reference geometry $l$. The $V_{l,i}^{(1)}$ terms are the intra-group potentials obtained by keeping the coordinates in other groups at the reference geometry $l$, while the $V_{l,ij}^{(2)}$ terms account for the group-group correlations. The potential with up to second-order terms gives already a very reasonable description of the system. The $V_{l,z23}^{(3)}$ term accounts for three-mode correlations between the displacement of the central proton, the distance between both water molecules and the angular wagging and rocking motions. Note that the primitive grids in each coordinate are the same irrespective of the reference point used to expand the potential. This means that the average, Eq. (\[eq:VtotQ\]), can be carried out before the dynamical calculations by summing over all the generated grids of the same coordinates for each reference geometry, involving no extra cost for the dynamics. The justification for the multi-reference approach lies on the nature of . The protonated water-dimer is a very floppy system featuring several equivalent minima and large amplitude motions that traverse low potential energy barriers. Thus, the amount of configurational space available to the system at low vibrational energies is already large and then it is not well covered by a single reference point. The property that, for a single reference point, the PES expansion is exact at the reference point and hypersurfaces involving the displacement of up to $h_m$ modes is lost after averaging over several reference geometries. However, the overall mean error is reduced by the averaging.
The use of several reference points has a further implication which is related to the symmetry properties of the system Hamiltonian. In the case of a possible single reference geometry to define the PES expansion would be one of the eight equivalent absolute minima which belong to the ${\mathcal C}_{2}$ point groups. This choice, however, breaks the total symmetry of the Hamiltonian. We have seen that cut-HDMR is exact at the reference point and all hypersurfaces in which up to $h_m$ modes have been displaced from the reference point, where $h_m$ is the expansion order in terms of CC. The description of the rest of ${\mathcal C}_{2}$ points when one is used as a reference is thus not equivalent and the whole symmetry is broken. A possible solution would be to use as reference one of the two equivalent ${\mathcal D}_{2d}$ stationary points, which lie around 300 above the absolute ${\mathcal C}_{2}$ minima. They correspond to $\alpha=90$ or $270$ degrees, respectively. Using only one of the ${\mathcal D}_{2d}$ points as reference results in a similar breakage of the total symmetry of the Hamiltonian. The same happens again if one of the two equivalent ${\mathcal D}_{2h}$ ($\alpha=0$ or $180$ deg.) stationary points is used, which are even higher in energy. One should note that this is a highly symmetrical, multiminima system in which several permutations of identical particles are possible through crossing of low energy conformational barriers. The vibrational levels of can be labeled according to the permutation-inversion symmetry group ${\mathcal G}_{16}$ [@bunker-book:symmetry; @wal99:10403], which contains the ${\mathcal D}_{2d}$ point group as a subgroup, but additionally allows to permute the H-atoms of either of the two monomers. The two ${\mathcal D}_{2d}$ and eight equivalent ${\mathcal C}_{2}$ geometries which are used as reference points are depicted in Fig. \[fig:refGeos\]. The difference between the structures at the left and right columns is a 180 degrees rotation of $\alpha$, or equivalently the permutation of the two hydrogens of one of the water moieties. The way out of the symmetry breakage problem is to use all the structures depicted in Fig. \[fig:refGeos\] as reference points of the cluster expansion. The final potential is given then, as discussed above, by the average with respect to all the reference points. By using this set of reference points the symmetry of the original PES is maintained. Indeed, it will be maintained as long as, for an arbitrary selected reference geometry, all the symmetry equivalent points generated by the permutations-inversions of the ${\mathcal G}_{16}$ group are also considered as reference points.
[**FIGURE \[fig:refGeos\] AROUND HERE**]{}
Results and Discussion
======================
Quantum Dynamical Calculations {#sec:ZPE}
------------------------------
The kinetic energy and potential energy operators already discussed are used to compute the zero point energy (ZPE) of the system and the corresponding ground-state vibrational-wavefunction. The algorithm that implements the computation of eigenvalues and eigenfunctions of the system Hamiltonian within the MCTDH program is called [*improved relaxation*]{} and is described elsewhere [@mey06:179]. This algorithm is essentially a multiconfiguration self-consistent field approach that takes advantage of the MCTDH machinery. All the reported simulations were performed with the Heidelberg MCTDH package of programs [@mctdh:package].
The comparison between the largest, converged MCTDH calculation and other reported results on the same PES is given in Table \[tab:zpe\]. As a reference we take the given diffusion Monte Carlo (DMC) result [@mc05:061101] which has an associated statistical uncertainty of 5 . A simple normal-modes analysis (NMA) with normal modes constructed from the Hessian matrix taken at the $\mathcal{C}_{2}$ minimum yields a ZPE of $12\,635$ , [*only*]{} $242$ larger than the DMC result. We believe that this surprisingly good result arises from fortuitous error cancellation. As will be discussed below, 3 out of the 15 internal degrees of freedom ($\gamma_a$, $\gamma_b$ and $\alpha$) are not described by a single-well in the lowest energy region of the potential, while the proton-transfer motion ($z$) features a nearly quartic potential which is strongly coupled to the water-water stretching ($R$). The most comprehensive calculations on the vibrational ground state based on a wavefunction approach to date are those of Bowman and collaborators [@mc05:061101] using the Multimode program [@bow03:533]. The vibrational configuration interaction (VCI) results, both using the single reference (SR) and reaction path (RP) variants are found in Table \[tab:zpe\]. These calculations use a normal-mode based Hamiltonian. They incorporate correlation between the different degrees of freedom due to the cluster expansion of the potential [@bow03:533] and the use of a CI wavefunction. The best reported VCI result for the ZPE lies still 104 above the DMC value. It is worth to mention that before switching to a Hamiltonian based on polyspherical coordinates we tried a Hamiltonian expressed in rectilinear-coordinates and obtained results similar to those of Bowman and collaborators.
[**TABLE \[tab:zpe\] AROUND HERE**]{}
The MCTDH converged result for the ZPE is given in Table \[tab:zpe\]. The obtained value for the ZPE is $12\,376.3$ , $16.7$ below the DMC value. Table \[tab:zpe-mctdh\] contains ZPE values obtained using an increasing number of configurations. According to these results the MCTDH reported values are assumed to be fully converged with respect to the number of configurations. The deviation from the DMC result must be attributed to the cluster expansion of the potential, Eqs. (\[eq:VtotQ\],\[eq:VtotQ2\]).
[**TABLE \[tab:zpe-mctdh\] AROUND HERE**]{}
We give here some technical details regarding the largest MCTDH calculation with $10\,500\,000$ configurations, which is probably one of the largest MCTDH calculations performed to date in terms of required computational resources. The calculation was made using the recent parallel version of the MCTDH code which is still under development in our group. The calculation was run on a 8-processor, shared-memory machine with Intel Itanium2 processors, and used a total amount of $13.5$ Gb of main memory. 4 Gb were devoted to storage of the Krylov vectors and Krylov times Hamiltonian vectors of the Davidson diagonalization procedure. The rest was needed for the representation of the Hamiltonian and the mean fields, the vector of coefficients, SPFs and work arrays. Each step of the wavefunction relaxation lasted around $13.5$ hours of wall-clock time, and consisted of the aforementioned Davidson diagonalization of the system Hamiltonian in the current basis of SPFs followed by a 2 fs imaginary-time propagation of the SPFs. The whole procedure was iterated until self-consistency of the expansion coefficients and SPFs was reached.
In contrast, the smallest calculation reported in Table \[tab:zpe-mctdh\], which used $172\,800$ configurations, consumed a total amount of memory of around $263$ Mb. Each diagonalization of the system Hamiltonian plus imaginary time propagation of the SPFs lasted $10$ minutes of wall clock time using two processors in parallel on the same Intel Itanium2 cluster. The comparison of the two aforementioned simulations illustrates one of the major strengths of MCTDH, namely, the usage of variationally optimal coefficients and orbitals leads to an early convergence with respect to the size of the multiconfigurational expansion. Relatively good results (only $7.4$ energy difference between the largest and smallest calculation) are already obtained using very moderate computational resources.
Properties of the ground state of the system {#sec:wavefunction}
--------------------------------------------
The probability density of the ground state wavefunction with respect to some selected coordinates and integration over the remaining coordinates is given in Fig. \[fig:wavefunc\]. Fig. \[fig:wavefunc\]a shows the density along the proton-transfer coordinate $z$. The probability density is non negligible in a range spanning about 1 bohr. Fig. \[fig:wavefunc\]b depicts the density along the $\alpha$ internal rotation coordinate. Along this coordinate the system interconverts between two equivalent regions of configurational space. The barrier corresponds to planar configurations of the whole system and is about 300 high depending on the configuration of the rest of coordinates. The system can interconvert between both halves even when in the ground vibrational state. The dotted curve in Fig. \[fig:wavefunc\]b depicts the density at a 10 times enlarged scale and clearly shows a non vanishing density for $\alpha=0,\pi$. The splitting state arising from the barrier along $\alpha$ has also been computed and the splitting energy has been found to be 1 . The small asymmetry observed in the density in Fig. \[fig:wavefunc\]b has been analyzed. It arises from the fact that the position of the proton is defined relative to one of the two water monomers, monomer A, in order to have 15 internal coordinates (see Eq. (\[Eq:485\]) and related text). Consequently, a change in $\alpha$ rotates the central proton so that its relative position to monomer A remains unaltered. In contrast, in order for the relative position of the central proton and monomer B to remain unaltered during a change in $\alpha$, coordinates $x$ and $y$ must change accordingly. We emphasize that the kinetic operator is still exact and not affected by this. However, $\alpha$, $x$ and $y$ coordinates are defined on non-matching grids. The total symmetry between monomers A and B with respect to the central proton is conserved in the case of a continuous configurational space instead of a discretized one. Thus, symmetry is slightly broken due to discretization of the configurational space. Energetically, the effect of this symmetry breakage must be well below 1 which is the splitting caused by the barrier along $\alpha$, since a larger perturbation would localize the density on one side of the barrier breaking the double well feature. A perfect symmetry after discretization of the coordinates would probably be obtained if one switches to cylindrical coordinates ($z$, $\rho$, $\phi$) for the proton and uses identical grids for the angles $\alpha$ and $\phi$.
Fig. \[fig:wavefunc\]c shows the probability density along the wagging coordinates. It consists of four equivalent maxima, each of them centered at around $\pm 30$ degrees from the planar water configuration, so that both water molecules are in pyramidal configuration. The probability density corresponding to one of the two water molecules in a planar configuration is however quite large and indicates a high probability of exchange between equivalent configurations of the system in which the water molecules switch between pyramidal geometries. Each of these four density maxima corresponds roughly to one of the ${\mathcal C}_{2}$ equivalent minima on the PES. A total of 8 equivalent ${\mathcal C}_{2}$ minima are present since the barrier along coordinate $\alpha$ divides the configurational space in two equivalent halves. When both monomers are in planar configuration the system interconverts between both ${\mathcal D}_{2d}$ and ${\mathcal D}_{2h}$ configurations by rotation along $\alpha$.
[**FIGURE \[fig:wavefunc\] AROUND HERE**]{}
Only after the introduction of a Hamiltonian based on polyspherical coordinates a satisfactory description the cation was possible. The use of polyspherical coordinates allows for the characterization of the system in terms of well defined stretching, bending, rocking, wagging and internal rotation motions, each of which corresponds to [*a single*]{} coordinate. This fact keeps the correlation between coordinates in the PES relatively small, and the relation between different coordinates can be usually understood in simple physical terms. The representation of the WF given in these coordinates converges then more quickly than a WF constructed from rectilinear coordinates and can be more compact. The price to pay, however, is a much more complicated expression for the kinetic energy operator.
Quality of the PES expansion {#sec:quality}
----------------------------
In order to illustrate the convergence of the mode-combination based cluster-expansion PES introduced in Section \[sec:PES\] the expectation values of the different terms of the potential are calculated for the ground vibrational state. These values are given in in Table \[tab:cluExpect\]. The sum of the first order $\langle\Psi_0|V^{(1)}(Q_i)|\Psi_0\rangle$ terms is close to 6800 , half the ZPE, indicating that they carry the major weight in the description of the PES. The second order clusters introduce the missing correlation between modes. They have expectation values one order of magnitude smaller than the first order terms with one exception, the matrix element arising from the $V^{(2)}(Q_1,Q_2)$ potential. This can be easily understood by noting that modes $Q_1$ and $Q_2$ contain coordinates $z$ and $R$, respectively. These two coordinates are strongly correlated and indeed they would be good candidates to be put in the same mode in an alternative mode-combination scheme. The only third order term that was introduced presents a rather marginal contribution to the potential energy of the system. These values prove that the PES representation used is of a good quality and rather well converged with respect to the reference PES, at least for the energy domain of interest. The square root of the expectation value of the potential squared is depicted in the third column. It is a measure of the dispersion around the expectation value and should also ideally vanish. The values indicate that the PES representation is good, albeit not yet fully converged. Some more terms of higher than second order may be added in the future as computational resources allow for it.
[**TABLE \[tab:cluExpect\] AROUND HERE**]{}
To finish the discussion it is worth mentioning that we have recently started to test a new mode-combination. It is based on 6 modes instead of 5. The definition of the modes in terms of the coordinates is as follows: $\tilde{Q}_1 = [z,R]$, $\tilde{Q}_2 = [x,y,\alpha]$, $\tilde{Q}_3 = [\gamma_A,\gamma_B]$, $\tilde{Q}_4 = [u_{\beta_A}, u_{\beta_B}]$, $\tilde{Q}_5 = [R_{1 A}, R_{2 A}, u_{\theta_{1 A}}]$ and $\tilde{Q}_6 = [R_{1 B},R_{2 B}, u_{\theta_{1 B}}]$. At the beginning we started with 5 modes in order to keep the number of configurations as small as possible. This has the drawback that one of the combined modes, $Q_1=[z,\alpha,x,y]$ is very large, thus its SPFs are harder to propagate. However, in the reported calculations we use the parallel version of the MCTDH code, and the parts which can be parallelized most efficiently are those involved with the vector of coefficients. Thus, a 6-modes scheme seems to be more efficient when the parallel code is used. In the 6-modes scheme we include all the first, second and some selected third order clusters. The obtained values of the ZPE are still preliminary but located in the region of -10 with respect to DMC, in good accordance to the results of the 5-mode scheme. Such results indicate indeed that the reported simulations are robust with respect to the mode-combination scheme.
Summary and Conclusion {#sec:conclusions}
======================
The protonated water-dimer () is studied in its full dimensionality (15D) by quantum-dynamical wavefunction-methods, using the MCTDH approach. A set of curvilinear coordinates is used which accounts for the fluxional, multi-minima nature of the cation. An exact expression for the kinetic energy operator of the system is derived using the polyspherical method. A discussion of the different steps involved in the derivation is given. The set of polyspherical coordinates introduced allows the description of the different motions of the system, namely proton transfer, water-water stretching, OH stretchings, bendings and internal rotation by a single coordinate each which also have clear geometrical meanings in terms of angles or distances. The PES used in the calculations is that of Huang et al., the most accurate PES available for this system to date. The PES must be represented in a numerically and computationally adequate way for the quantum-dynamical simulations on a discrete multidimensional grid to be feasible. To this end, a variation of the hierarchical cut-HDMR method is presented which takes advantage of the mode-combination strategy used to represent the wavefunction. Combined modes, instead of the coordinates, are used to define the hierarchical expansion. The expansion of the PES is seen to converge quickly with the number of clusters in the expansion, which can be attributed to two factors:
- A large amount of the correlation is already captured within the modes, which leads to quick convergence of the PES with respect to the number of clusters in the expansion.
- The set of polyspherical coordinates used allows for the definition of physically meaningful modes avoiding at the same time artificial correlations (which are due to inadequate coordinates) to appear both in the potential energy and wave function.
The ZPE of the system is calculated and the results obtained show an excellent agreement with previous DMC calculations on the same PES. The reported ZPE with MCTDH lies only $16.7$ below the reported DMC result. This deviation from the “exact" DMC result (statistical error 5 ) must be due to the potential representation, as the KEO is exact and as it is shown, the reported ZPE is converged with respect to the number of configurations in the MCTDH expansion. A fully converged MCTDH calculation needs a rather large amount of computational resources, however, it is shown that reasonably good results are obtained with a much smaller configurational space due to the optimality of both the coefficients and the SPFs in the MCTDH expansion. The properties of the ground vibrational state are analyzed. The central proton delocalizes on a range of about 1 bohr along the water-water axis. The fluxional nature of the system is exemplified in the probability density along the wagging $\gamma_A$ and $\gamma_B$ coordinates. The 2D space spanned by these coordinates presents four density maxima which roughly correspond to the $\mathcal{C}_{2}$ minima of the system. In going between different higher density regions the system switches between pyramidal conformations of the water molecules. These conformational changes take place along low potential-energy barriers. The system is completely delocalized over these low barriers leading to a highly symmetric ground-state wave function. The internal rotation coordinate $\alpha$ is seen to divide the configurational space in two equivalent halves. The probability density for $\alpha=0$ and $\alpha=\pi$ (planar $\mathcal{D}_{2h}$) is non-negligible for the ground vibrational state. The tunneling splitting arising from the barrier along $\alpha$ is computed to be 1 . The convergence of the PES expansion is monitored with respect to the expectation values of the potential-energy terms which define it, i.e. by inspecting $\langle \Psi_0 | V_{ij\cdots}| \Psi_0 \rangle$ and $\langle \Psi_0 | V_{ij\cdots}^2| \Psi_0 \rangle^{1/2}$. A good convergence is observed at second order with respect to the combined modes. The only third order term present has a rather marginal contribution to the energy with respect to $| \Psi_0 \rangle$. Only after switching to curvilinear coordinates the fluxional motion of the highly symmetrical cation was correctly accounted for. The present paper has focused on the definition of an adequate set of coordinates and the derivation of the expression of the kinetic energy using the polyspherical method. A convenient way to represent the PES for high-dimensional quantum-dynamical simulation has been also discussed. The validity of the Hamiltonian setup has been established by comparison to available DMC results in the literature. Moreover, the properties of the ground vibrational state have been investigated. Our study provides a picture of the system, in which the cluster has to be viewed as highly anharmonic, flexible, multi-minima, coupled system. We show that a converged quantum-dynamical description of such a complex molecular system can still be achieved. The companion paper following this one focuses on the infrared spectroscopy and dynamics of .
Acknowledgments
===============
The authors thank Prof. J. Bowman for providing the potential-energy routine, M. Brill for the help with the parallelized code, and the Scientific Supercomputing Center Karlsruhe for generously providing computer time. O. V. is grateful to the Alexander von Humboldt Foundation for financial support. Travel support by the Deutsche Forschungsgemeinschaft (DFG) is also gratefully acknowledged.
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Coord. $N$ $x_i$ $x_f$ DVR
-------------------- ----- ----------- ----------- -----
$z$ 27 -1.8 1.8 HO
$\alpha$ 21 0 $2\pi$ exp
$x$ 5 -0.9 0.9 HO
$y$ 5 -0.9 0.9 HO
$R$ 16 4.2 6.5 HO
$u_{\beta_{A}}$ 7 -0.5 0.5 sin
$u_{\beta_{B}}$ 7 -0.5 0.5 sin
$\gamma_A$ 19 $\pi$-1.8 $\pi$+1.8 sin
$\gamma_B$ 19 -1.8 1.8 sin
$R_{1 A}$ 9 0.5 1.8 HO
$R_{2 A}$ 9 2.2 3.8 HO
$u_{\theta_{1 A}}$ 7 -0.5 0.5 sin
$R_{1 B}$ 9 0.5 1.8 HO
$R_{2 B}$ 9 2.2 3.8 HO
$u_{\theta_{1 B}}$ 7 -0.5 0.5 sin
: Definition of the one-dimensional grids. $N$ denotes the number of grid points and $x_i$, $x_f$ the location of first and last point. The DVRs are defined in Appendix B of Ref. [@bec00:1]. []{data-label="tab:1Dgrids"}
Method ZPE() $\Delta$()
---------- ------------- ------------
DMC $12\,393$ $0$
harmonic $12\,635$ $242$
VCI-SR $12\,590$ $197$
VCI-RP $12\,497$ $104$
MCTDH $12\,376.3$ $-16.7$
: Comparison of the zero point energy (ZPE) of the cation calculated by various approaches on the PES by Huang et. al.[@hua05:044308]: diffusion Monte-Carlo (DMC), normal mode analysis (harmonic), vibrational CI single reference (VCI-SR) and reaction path (VCI-RP) as published in [@mc05:061101] and MCTDH results. $\Delta$ denotes the difference to the DMC result. The converged MCTDH result is obtained with $10\,500\,000$ configurations. Compare with Table \[tab:zpe-mctdh\]. []{data-label="tab:zpe"}
SPFs per Mode N configs. ZPE() $\Delta$()
-------------------- ---------------- ------------- ------------
$(20,20,12,6,6)$ $172\,800$ $12\,383.7$ $-9.3$
$(35,25,15,8,8)$ $840\,000$ $12\,378.5$ $-14.5$
$(40,40,20,8,8)$ $2\,048\,000$ $12\,377.8$ $-15.2$
$(60,40,20,8,8)$ $3\,072\,000$ $12\,376.7$ $-16.3$
$(70,50,30,10,10)$ $10\,500\,000$ $12\,376.3$ $-16.7$
: Comparison of the zero point energy (ZPE) of the cation between different MCTDH calculations with ascending number of configurations. The $\Delta$ values are given with respect to the diffusion Monte Carlo result, $12\,393$ [@mc05:061101]. []{data-label="tab:zpe-mctdh"}
$\langle\Psi_0|V|\Psi_0\rangle$ $\langle\Psi_0|V^2|\Psi_0\rangle^{1/2}$
---------------------- --------------------------------- -----------------------------------------
$V^{(1)}(Q_1)$ 1293.6 1807.7
$V^{(1)}(Q_2)$ 750.6 966.9
$V^{(1)}(Q_3)$ 171.5 266.9
$V^{(1)}(Q_4)$ 2293.2 3062.8
$V^{(1)}(Q_5)$ 2293.1 3062.8
$V^{(2)}(Q_1,Q_2)$ -526.9 1037.2
$V^{(2)}(Q_1,Q_3)$ -78.8 290.2
$V^{(2)}(Q_1,Q_4)$ -27.5 231.8
$V^{(2)}(Q_1,Q_4)$ -27.4 231.7
$V^{(2)}(Q_2,Q_3)$ -10.5 37.6
$V^{(2)}(Q_2,Q_4)$ -24.7 117.5
$V^{(2)}(Q_2,Q_5)$ -24.7 117.9
$V^{(2)}(Q_3,Q_4)$ -18.8 180.9
$V^{(2)}(Q_3,Q_5)$ -18.8 180.9
$V^{(2)}(Q_4,Q_5)$ 1.2 9.9
$V^{(3)}(z,Q_2,Q_3)$ 1.0 50.4
: Expectation value of the different terms of the potential expansion (central column) and square root of the expectation value of the potential squared (right column). All energies in . The combined modes read: $Q_1 = [z,\alpha,x,y]$, $Q_2 = [\gamma_A,\gamma_B]$, $Q_3 = [R, u_{\beta_{A}}, u_{\beta_{B}}]$, $Q_4 = [R_{1 A}, R_{2 A}, u_{\theta_{1 A}}]$ and $Q_5 = [R_{1 B}, R_{2 B}, u_{\theta_{1 B}}]$. []{data-label="tab:cluExpect"}
Figure Captions {#figure-captions .unnumbered}
===============
[ [**Figure \[fig:Vect\]:**]{} Jacobi description of the H$_5$O$_2^+$ system. The vector $\vec{R}$ connects the two centers of mass of the water monomers. The vector $\vec{r}$ connects the center of mass of the water dimer with the central proton. ]{}
[ [**Figure \[fig:Coord\]:**]{} Definition of the angles for the H$_5$O$_2^+$ system. The angles $\alpha_A$ and $\alpha_B$ describe the rotation of the water monomers around the vector $\vec{R}$, or equivalently around the z-axis of the E2- or BF-frame. ]{}
[ [**Figure \[fig:mode-comb\]:**]{} Number of grid points needed for the representation of the clusters of the PES expansion. $m$ is the number of coordinates making a mode. 10 grid points per coordinates and 15 coordinates are assumed. A horizontal line is drawn at $10^8$, which tentatively signals the maximum practical number of points both regarding their calculation and the use of the grids in the dynamical calculations. ]{}
[**Figure \[fig:refGeos\]:**]{} Geometries of the 10 reference points used in the PES expansion. The view is along the O-H-O axis. Hence only the closest of the two oxygen and the four hydrogens can be seen. The difference between the geometries in the left column and each geometry at the right column is a rotation of $\pi$ along $\alpha$. Equivalently, the pairs of structures (a,b), (c,j), (e,h), (g,f), (i,d) are related by a permutation of hydrogen atoms of one of the monomers. The following coordinates are identical for all reference points: $R=4.70$ au, $x,y,z=0$, $R_{1(A,B)}=1.07$ au, $R_{2(A,B)}=2.98$ au, $\theta_{1A(1B)}=0$, $u_{\beta_{A(B)}}=0$. Only coordinates $\alpha$, $\gamma_A$ and $\gamma_B$ differ at the 10 reference points.
[**Figure \[fig:wavefunc\]:**]{} For the ground vibrational state probability density along selected coordinates and integration over the rest: probability density along the $z$ proton-transfer coordinate (a), along the $\alpha$ internal rotation coordinate (b) and on the 2D space spanned by the wagging $\gamma_A$ and $\gamma_B$ coordinates (c). The dotted line in (b) corresponds to a 10 times enlarged scale. It indicates that the probability density at $\alpha=\pi$ is not vanishing.
![[]{data-label="fig:Vect"}](fig1.eps){width="8.5cm"}
![[]{data-label="fig:Coord"}](fig2.eps){width="8.5cm"}
![[]{data-label="fig:mode-comb"}](fig3.eps){width="8.5cm"}
![[]{data-label="fig:refGeos"}](fig4.eps){width="8.5cm"}
![[]{data-label="fig:wavefunc"}](fig5.eps){width="12.75cm"}
|
NUMERICAL SOLUTION OF THE BELTRAMI EQUATION
R. Michael Porter[^1]
Department of Mathematics,\
Centro de Investigación y de Estudios Avanzados del I.P.N.,\
Apdo. Postal 14-740, 07000 México, D.F., Mexico
Introduction
============
The Beltrami equation $$\frac{ \partial f(z)/ \partial\conj{z} }
{ \partial f(z)/ \partial z } = \mu(z)$$ determines a unique normalized quasiconformal self mapping $f$ of the unit disk $\D=\{z:|z|<1\}$ in the complex plane. Here $\mu$ is a given measurable function in $\D$ with $\|\mu\|_\infty<1$, and is called the Beltrami derivative (or complex dilatation) of $f$. We say that $f$ is $\mu$-conformal.
The Beltrami equation has been studied intensively in large measure due to its importance in the theory of deformations of Kleinian groups and their applications to Teichmüller spaces [@Harv],[@Lehto]. Other applications of the Beltrami equation are mentioned in the introduction to [@Dar-fastB]. Some more recent applications, such as mapping of the cerebral cortex use the Beltrami equation in the spirit of its original application, dating back to Gauss, of finding a conformal mapping from a surface in 3-space onto a planar region; this is done effectively in [@AHTK] without explicit use of the Beltrami derivative $\mu$.
With the increasing use computer studies it has become of great interest to solve the Beltrami equation numerically. One method for doing this is suggested naturally by the classical existence proof given by Mori-Boyarskii-Ahlfors-Bers [@Harv],[@Lehto]. For this method one must evaluate singular integrals of the form $$T_mg(z) = \int\!\!\int_{\D} \frac{g(\zeta)}{(\zeta-z)^m} \,d\xi d\eta
,\quad m=1,2$$ (defined as Cauchy Principal Values when $m=2$), and calculate sums of Newmann series of the form $$\sum \mu T_2(\mu T_2(\dots(\mu T_2 ( 1 ) ) .$$ A related approach involving the singular integrals was developed by Daripa and Mashat [@Dar-Mash]. Instead of summing the Newmann series, their method involves iteration towards a solution of a related Dirichlet problem. Their work incudes refinement of the technique of evaluation of the singular integrals via FFT, which is of interest in itself.
G. B. Williams [@BW] proposed an alternative method of solving the Beltrami equation, based on conformal welding of circle packings. A sequence of circle packings is produced whose corresponding affine mapping converge to the $\mu$-conformal mapping. The emphasis in [@BW] is on investigating properties of circle packings and on proving they can be constructed to converge to a solution of the Beltrami equation, and little information is given towards the point of view of numerical analysis.
The present approach to the numerical solution of the Beltrami equation is in the spirit of [@BW], but with the aim of applying conformal mapping methods other than just circle packing. We start from the observation that the general solution of the Beltrami equation for constant $\mu$, $|\mu|<1$, is an affine linear mappping of the form $$f(z) = a( z + \mu\conj{z}) + b$$ for complex constants $a,b$ with $a\not=0$. Thus general quasiconformal mappings are approximated locally by such real linear affine maps. The essential idea of our method is as follows. In each element of a mesh division of $\D$, a first attempt to construct $f$ may be carried out by using such affine maps. Suppose that the piecewise linear $\mu$-conformal mapping $f_0$ which results from combining these local mappings is continuous and in fact one-to-one from $\D$ to a domain $D=f(\D)$. Let $h:D\to\D$ be a conformal mapping, the existence being guaranteed by the Riemann mapping theorem. Then the composition $f=h\circ f_0$ is a $\mu$-conformal self mapping of $\D$. For numerical work one must usually work with Beltrami derivatives which are continuous or at least piecewise continuous.
The problem of calculating a $\mu$-conformal self-mapping of the sphere $S^2$ can also be solved numerically via the Disk Algorithm, by using domain and image disks covering all but an insignificant part of the Riemann sphere. Consider also the problem of constructing a conformal mapping from a surface in Euclidean space $\R^3$ to $\D$. This can be accomplished by first transforming it nonconformally to a disk in $\R^2$ and then applying the Disk Algorithm to the induced Beltrami equation.
In Section \[secprelim\] we gather the basic facts we will need about quasiconformal mappings. In Section \[secalg\] we define the Disk Algorithm for solving the Beltrami equation, and prove that it converges to the desired function when the parameters are chosen appropriately. In Section \[seccompu\] we discuss the choice of conformal mapping algorithm to be plugged into the Disk Algorithm, and the corresponding computational cost. Numerical examples are provided in Section \[secnumres\].
\[secprelim\]Preliminaries
==========================
First we discuss affine linear quasiconformal mappings. Let $\mu$, $a$, $b$ be complex constants, $a\not=0$, $|\mu|<1$, and consider the mappings $$\begin{aligned}
L[\mu](z) &=& z + \mu \conj{z} \label{defL}\\ A[a,b](z) &=&
az+b \label{defA}\end{aligned}$$ for $z\in\C$. Thus $L[\mu]$ is $\mu$-conformal and linear, while $A[a,b]$ is conformal and affine linear. Further, the images of two points, $w_i=A[a,b](z_i)$, $i=1,2$, determine the coefficients $a$ and $b$. All $\mu$-conformal affine linear mappings are of the form $A[a,b]\circ L[\mu]$, and this decomposition is unique. In particular we have the following.
\[propBdef\] Given $z_1$, $z_2$ distinct and $w_1$, $w_2$ distinct, together with $|\mu|<1$, there is a unique $\mu$-conformal affine linear mapping $B[\mu;\,z_1,z_2;\,w_1,w_2]$ sending $z_1$ to $w_1$ and $z_2$ to $w_2$.
We will be interested in mappings of triangles, which can be described as follows.
\[propAdef\] Given $z_1$, $z_2$, $z_3$ noncollinear and $w_1$, $w_2$, $w_3$ noncollinear, there is a unique affine linear mapping $T[z_1,z_2,z_3;\, w_1,w_2,w_3]$ which sends $z_i$ to $w_i$ ($i=1,2,3$). Its Beltrami derivative is equal to $$\label{muA}
-\frac{(z_2-z_1)(w_3-w_1) - (z_3-z_1)(w_2-w_1)}
{(\conj{z_2}-\conj{z_1})(w_3-w_1) -
(\conj{z_3}-\conj{z_1})(w_2-w_1)}.$$
The map $T=T[z_1,z_2,z_3;\,w_1,w_2,w_3]$ is well-defined because each of the pairs $(z_2-z_1,z_3-z_1)$, $(w_2-w_1,w_3-w_1)$ is a basis of plane over the field of real numbers. To verify that the Beltrami derivative of $T$ is (\[muA\]), first consider the function $$f(\zeta)=T[0,1,z;\,0,1,w](\zeta) =
\frac{1}{z-\conj{z}}
\left( (w-\conj{z})\zeta + (w-z)\conj{\zeta} \right)$$ for $z,w$ fixed. The Beltrami derivative of $f$ is $\mu=-(w-\conj{z})/(w-z)$. The conformal linear map $h_1=A[1/(z_2-z_1),-z_1/(z_2-z_1)]$ takes $z_1,z_2,z_3$ to $0,1,z$ where $z=(z_3-z_1)/(z_2-z_1)$. An analogous conformal linear mapping $h_2$ takes $w_1,w_2,w_3$ to $0,1,w$ when $w$ is chosen appropriately. Then $A$ is the composition $h_2^{-1}\circ f\circ h_1$ and its Beltrami derivative $$(\mu\circ h_1)\frac{\,\conj{h_1'}\,}{h_1'}$$ as obtained from the Chain Rule, is seen to be equal to (\[muA\]).
The conformal mapping (\[defA\]) sends triangles to similar triangles. A Beltrami derivative (\[muA\]) can be regarded as a measure of how much the triangles with vertices $z_1,z_2,z_3$ and $w_1,w_2,w_3$ fail to be similar. This observation will be used in the proof of the following technical result, which in turn will be used in the proof of convergence of the Disk Algorithm.
\[lemmtriangle\] Fix $z_0=r_0e^{i\theta}$ with $r_0>0$, and fix $\mu_0$, $|\mu_0|<1$. Let $\epsilon>0$. Then there exists $\delta>0$ such that the following holds. Suppose $r_1>r_0$ and $\theta_0\le\theta_0'<\theta_1\le\theta_1'<\theta_2$, and write $$z_k = r_0 e^{i\theta_k},\quad
z_k' = r_1 e^{i\theta_k'},\quad k=1,2.$$ Further, suppose that $|w_0|=|w_1|=|w_2|=r>0$ and $|\mu_1|,|\mu_2|<1$, and set $$\begin{aligned}
w_1' &=& B[\mu_1;\, z_0, z_1;\, w_0, w_1]( z_1' ),\\
w_2' &=& B[\mu_2;\, z_1, z_2;\, w_1, w_2]( z_2' ) .\end{aligned}$$ Suppose that $z_1,z_2,z_1',z_2'$ are within $\delta$ of $z_0$, that $\mu_1,\mu_2$ are within $\delta$ of $\mu_0$, and that the ratios $$\label{thrats}
\frac{\theta_2-\theta_1}{\theta_1-\theta_0}, \quad
\frac{\theta_2'-\theta_1'}{\theta_1-\theta_0}, \quad
\frac{w_2-w_1}{w_1-w_0}$$ are within $\delta$ of $1$. Let $\mu$ be the Beltrami derivative of the affine linear mapping $T[z_1',z_1,z_0;\, w_1',w_1,w_0]$. Then $$|\mu-\mu_0| < \epsilon.$$
(20,11)(-2,-1) (2,0)
(0,0) (0,0) (5,0) (10,0) (0,0)[(1,0)[10]{}]{} (2.5,5) (7.5,5) (2.5,5)[(1,0)[5]{}]{} (0,0)[(1,2)[2.5]{}]{} (5,0)[(-1,2)[2.5]{}]{} (5,0)[(1,2)[2.5]{}]{} (10,0)[(-1,2)[2.5]{}]{} (-1,-1)[$z_2$]{} (4.5,-1)[$z_1$]{} (10,-1)[$z_0$]{} (1.5,5.5)[$z_2'$]{} (7.2,5.5)[$z_1'$]{}
(17,1)
(0,0) (0,0) (5,0) (10,0) (0,0)[(1,0)[10]{}]{} (2.5,7.5) (7.5,7.5) (2.5,7.5)[(1,0)[5]{}]{} (0,0)[(1,3)[2.5]{}]{} (5,0)[(-1,3)[2.5]{}]{} (5,0)[(1,3)[2.5]{}]{} (10,0)[(-1,3)[2.5]{}]{} (-1,-1)[$w_2$]{} (4.5,-1)[$w_1$]{} (10,-1)[$w_0$]{} (1.5,8.2)[$w_2'$]{} (7.2,8.2)[$w_1'$]{}
The positions of the points involved are represented schematically in Figure \[figsimil\]. First suppose that $z_0,z_1,z_2$, rather than lying on the circle $|z|=r_0$, are instead collinear and further are equally spaced, and likewise for $w_0,w_1,w_2$; also suppose that the segment $z_1',z_2'$ is parallel to $z_0,z_1,z_2$ and that $|z_2'-z_1'|=|z_2-z_1|$. Suppose further that $\mu_1=\mu_2=\mu_0$. One may calculate from Proposition \[propAdef\] that the Beltrami differential of $T[z_1',z_1,z_0;\, w_1',w_1,w_0]$ must be equal to $\mu_0$. Alternatively, one may simply observe that the triangles $w_1',w_1,w_0$ and $w_2',w_2,w_1$ are similar because the triangles $z_1',z_1,z_0$ and $z_2',z_2,z_1$, are similar and have parallel bases. Then by elementary geometry the triangle $w_1,w_1',w_2'$ is similar to $w_1',w_1,w_0$. Since $z_1,z_1',z_2'$ is similar to $z_1',z_1,z_0$, it follows again that the Beltrami derivative of the affine mapping is $\mu_0$.
Returning now to the hypothesis that $|z_k|=r_0$, by taking $\delta$ sufficiently small, which implies that $r_1$ is close to $r_0$, one can assure that the arcs of the circles of radii $r_0$, $r_1$ containing the $z_k$, $z_k'$ will be approximated by straight lines, so the Beltrami derivative $\mu^*$ of the affine mapping between the middle triangles can be made to satisfy $|\mu^*-\mu_0|<\epsilon/2$. Up to now we have referred only to equally spaced points, that is, when the ratios (\[thrats\]) are all equal to $1$. In general, since $\mu$ is a continuous function of $r_1,\theta_1,\theta_2,\theta_1',\theta_2', \mu_1$, and $\mu_2$, we see that by taking $\delta$ sufficiently small we can assure that $|\mu-\mu^*|<\epsilon/2$. This completes the proof.
We recall some well known properties of quasiconformal mappings [@AhlLQM], [@Lehto], [@LV]. If $f_1,f_2$ are quasiconformal mappings in the same planar domain and have the same Beltrami derivative, then $f_2=h\circ f_1$ where $h$ is a conformal mapping from the image of $f_1$ to the image of $f_2$. If $\mu$ is measurable in $\D=\{z\in\C\colon|z|<1\}$ and $\|\mu\|_\infty<1$, then there is a unique $\mu$-conformal mapping $f\colon\D\to\D$ satisfying the normalization $$f(0)=0,\quad f(1)=1.$$ In this paper all mappings will be piecewise smooth.
\[secalg\]Disk Algorithm for Solving Beltrami Equations
=======================================================
In this section we will describe the main algorithm for solving the Beltrami equation. By assumption a formula or program is provided for calculating the proposed Beltrami derivative $\mu(z)$ in all of $\D$.
Ring Extension procedure
------------------------
First we describe the basic step. Let $z_1,z_2,\dots,z_N$ be the vertices of a positively oriented simple closed polygon, lying within another such polygon whose vertices are $z'_1,z'_2,\dots,z'_N$. We suppose that the interiors of the triangles $D_k$ with vertices $z_k,z_{k+1},z'_{k+1}$ ($k=1,\dots,N$) are disjoint (indices are taken modulo $N$). Let $|\mu(z)|<1$ in the doubly connected domain $R$ between these two polygons. Let $w_1,w_2,\dots,w_N$ also form a simple closed polygon in $\C$.
The Ring Extension produces $w'_1,w'_2,\dots,w'_N$ as follows. For each $k$ let $\mu_k$ be the average value of $\mu$ over the triangle $D_k$; this exists because prescribed Beltrami differentials are by hypothesis integrable. Define $$\label{ringalg}
w'_k = B[\mu_k;\, z_k, z_{k+1};\, w_k, w_{k+1} ](z'_k),
\quad k=1,\dots,N,$$ where $B$ is as given in Proposition \[propBdef\]. We will say that the Ring Extension is *applicable* for the given data if the interiors of the triangles $w_k,w_{k+1},w'_{k+1}$ are disjoint.
(0,4.2)(5,.5) (7.1 ,0)[![Elements of the Ring Extension.[]{data-label="figringalg"}](figringstep.eps "fig:")]{}
(0,0) (5.5,4)[$z$]{} (10.7,4)[$w$]{}
Disk algorithm
--------------
We now define the main iteration of the algorithm for solving the Beltrami equation in a disk.
The unit disk $\D$ is subdivided by a mesh with vertices $z_{jk}$, where for each $j$, the points $z_{j1},z_{j2},\dots,z_{jN}$ form a positively oriented polygon, with $$|z_{jk}|=r_j,\quad j=1,\dots,M,\quad 0<r_1<\cdots<r_M=1.$$ The arguments $$\theta_{jk}=\arg z_{jk}, \quad k=1,\dots,N,$$ of the vertices are determined modulo $2\pi$. In what follows we will take these arguments to be uniformly spaced, though this simplification is not strictly necessary for the basic idea of the algorithm.
Fix $j$, and assume that the images $w_{jk}$ have been calculated for a $\mu$-conformal mapping $w=f(z)$ from the subdisk $\{|z|<r_{j}\}$ to $\{|w|<r_{j}\}$ normalized by $f(0)=0$. We wish to calculate the images $w_{j+1,k}=f(z_{j+1,k})$.
For convenience of notation, we will suppose that $\theta_{j+1,k}$ is close (modulo $2\pi$) to $(\theta_{j,k}+\theta_{j,k+1})/2$ (recall Figure \[figsimil\]). In all of our applications of the method this will be taken to be exactly so, by using the definition $$\label{zjk}
z_{jk} = r_j e^{2\pi i(k+j/2)/N}.$$ If (\[zjk\]) does not hold, then, one must modify slightly the statements and proofs given below. The term $j/2$ in the exponent of (\[zjk\]) has been included in order to relate the indices to Figure \[figsimil\] and to formula (\[ringalg\]). The Disk Algorithm requires carrying out the following for $j=1,\dots,M$.
DISK ALGORITHM: $j$-th iteration.
- **Step 1.** Assume that the Ring Extension with Beltrami derivative $\mu$ is applicable to the doubly connected region $R_j$ between $z_{j1},z_{j2},\dots,z_{jN}$ and $z_{j+1,1},z_{j+1,2},\dots,z_{j+1,N}$, mapping the inner points to the vertices $w_{j1},w_{j2},\dots,w_{jN}$. Let the resulting points be called $$\tilde w_{j+1,1},\ \tilde w_{j+1,2},\ \dots,\ \tilde w_{j+1,N}.$$ According to (\[ringalg\]), these are specifically $$\tilde w_{j+1,k} = B[\mu_{jk};\, z_{j,k}, z_{j,k+1};\,
w_{jk}, w_{j,k+1} ](z_{j+1,k})$$ for $k=1,\dots,N$.
- **Step 2.** The points $\tilde w_{j+1,1}, \tilde
w_{j+1,2},\dots,\tilde w_{j+1,N}$, joined in order, bound a simply connected polygonal domain $D_{j+1}$. Let $$h_{j+1}\colon
D_{j+1}\to\{|z|<r_{j+1}\}$$ be a conformal mapping satisfying $h_{j+1}(0)=0$, calculated by any appropriate known method. Define $$w_{j+1,k} = h_{j+1}(\tilde w_{j+1,k}).$$
- **Step 3.** Apply $h_{j+1}$ to each $w_{j',k}$ for $j'\le j$ and all $k$. For simplicity of notation we will call the images $w_{j',k}$ as well.
Some details remain to be specified to make this general scheme precise and to complete the Disk Algorithm. One must decide how to start the algorithm by finding the first polygon $w_{11},w_{12,},\dots,w_{1N}$. We have done this using an explicit formula [@Sz] for the conformal mapping to $\D$ from an ellipse with semimajor and semiminor axes $a$, $b$ ($a^2-b^2=1$) and foci $\pm1$, $$\label{ellipticint}
w=\sqrt{k}\, {\rm sn}\,(\frac{2K}{\pi} \sin^{-1} u ; k^2)$$ where the Jacobi modulus $k$ is related to the complete elliptic integral $K$ and the Jacobi theta functions by $$q=(a+b)^{-4}=e^{-\frac{\pi K(1-m)}{K(m)}}$$ $$k = \sqrt{m} = \left( \frac{\theta_2}{\theta_3}\right)^2$$ (notation from [@WW]). Note that the image of the circle $|z|=r_1$ under the mapping $L[\mu]$ is an ellipse with axes $r_1(1+|\mu|)$, $r_1(1-|\mu|)$ slanted in the directions $(1/2)\arg \mu$, $(1/2)(\arg\mu+\pi)$ respectively, modulo $\pi$. This ellipse is sent by the affine map $A[1/(2r_1\sqrt{\mu}),0]$ to an ellipse with axes $a,b$. Then via (\[ellipticint\]) this is transformed conformally to the unit disk and can then be rescaled to any desired radius, completing the first step of the Disk Algorithm.
A more crucial decision is the choice of conformal mapping procedure in Step 2 of the Disk Algorithm. This will be discussed in Section \[seccompu\].
Convergence of Disk Algorithm
-----------------------------
Once the Disk Algorithm has been executed, the collection $\{w_{jk}\}_{j,k}$ defines in a natural way a piecewise linear mapping of the polygon $\{z_{Mk}\}_k$ to the polygon $\{w_{Mk}\}_k$, which sends $z_{jk}$ to $w_{jk}$. If desired, this mapping may be completed to a self mapping of $\D$ by extrapolating to the remaining points near the circumference but for large $M$ this portion of the domain is insignificant. We now justify that this piecewise linear mapping is a valid approximation for the $\mu$-conformal mapping of $\D$.
For purposes of normalization in the the following theorem, note that $z_{j,-j/2}$ given by (\[zjk\]) is positive real when $j$ is even. We write $[j/2]$ for the greatest integer no greater than $j/2$.
Let $|\mu(z)|\le\kappa<1$ for all $z\in\D$, where $\mu$ is piecewise continuous in $\D$. Let $M_n,N_n\to\infty$, and for each $n$, let $0=r_{n0}<r_{n1}<r_{n2}<\cdots<r_{nM_n}=1$ be chosen in such a way that $\sup_j(r_{nj}-r_{n,j-1})\to0$ as $n\to\infty$. Suppose further that the Ring Extension is always applicable so that Step 1 can be carried out for all $j$. Let $f_n$ be the piecewise linear self mapping of $\D$ produced by the Disk Algorithm, normalized so that $w_{j,-[j/2]}$ is positive real. Then $\{f_n\}$ converges uniformly on compact subsets of $\D$ to a $\mu$-conformal self mapping $f$ of $\D$.
Let $\kappa<\kappa'<1$. By construction, on each outward pointing triangle of the mesh (such as $z_1',z_1,z_0$ in Figure \[figsimil\]), $f_n$ is an affine linear quasiconformal mapping whose dilatation $|\mu_n|$ is no greater than $\kappa$. Then by Proposition \[lemmtriangle\] we have $|\mu_n|<\kappa'$ on the inward pointing triangles (such as $z_1,z_1',z_2'$ in Figure \[figsimil\]) if the mesh is fine enough. Therefore $f_n$ is $\kappa'$-quasiconformal for sufficiently large $n$. By compactness of bounded families of uniformly quasiconformal mappings [@LV], there is a limit mapping $f$ which is also $\kappa$-quasiconformal (it is not constant because $f(0)=0$, $f(1)=1$). Further, $\mu_n$ converges uniformly to the Beltrami differential of $f$ almost everywhere, so it follows that $f$ is $\mu$-conformal.
Condition to guarantee applicability
------------------------------------
We must now show how to choose radii $r_j$ so that the Ring Algorithm will be applicable in Step 1. The matter is to avoid crossing of images of adjacent triangles as in Figure \[figcross\]. In the context of Figure \[figsimil\], let us say that a mapped triangle $w_0,w_1,w_1'$ is *skewed* when one of the base angles is greater than $\pi/2$, thus, when the perpendicular from the apex $w_1'$ to the line containing the base $w_0,w_1$ of the triangle does not pass through the base. It is clear that if none of the triangles mapped in Step 1 is skewed, then the argument of each $\tilde w_{j+1,k}$ will lie between those of $w_{jk}$ and $w_{j+1,k}$, so the Ring Extension will be applicable. In general, given $N$, we will have to take $M$ sufficiently large, taking $\sup|\mu|$ into account as well. This is reflected in the following result.
(0,5)(5,.5) (8 ,0)[![Crossing triangles in the (failed) Ring Extension. The outer polygon (dotted line) is not simple. []{data-label="figcross"}](figcross.eps "fig:")]{}
(0,0)
\[theononskew\] Suppose that the inequality $$\label{nonskewcond}
\frac{r_{j+1}}{r_j} < \cos\frac{2\pi}{N} +
\frac{(1-|\mu(z)|)^2}{2|\mu(z)|}
\sin\frac{2\pi}{N}$$ is satisfied for all $z$ in the ring $r_j\le|z|\le r_{j+1}$. Then all of the $N$ outward-pointing image triangles produced in the $j$-th application of Step 1 of the algorithm are non-skewed.
(0,7.5)(5,.5) (6,1.5)[![Proof of Theorem \[theononskew\] []{data-label="fignonskew"}](fignonskew1.eps "fig:")]{}
(0,6)(5,.5) (12,0.5)[![Proof of Theorem \[theononskew\] []{data-label="fignonskew"}](fignonskew2.eps "fig:")]{}
(0,6)(5,.5) (8,-3)[![Proof of Theorem \[theononskew\] []{data-label="fignonskew"}](fignonskew3.eps "fig:")]{}
(0,0)(0,-.5) (1.2,1.1)[$0$]{} (1,6.2)[$z_1$]{} (5.3,3.3)[$z_0$]{} (4.5,6.7)[$z'_1$]{} (3.8,5.2)[$z$]{} (1.65,1.95)[$\alpha$]{} (6.5,4.5)[$iz\sin\alpha$]{} (9.4,3.1)[$0$]{} (10,2.)[$-iz\sin\alpha$]{} (10.8,5.5)[$\beta'z $]{} (2,-1.25)[$-\sin\alpha$]{} (8.5,-1.25)[$\sin\alpha$]{} (6,-1.25)[$0$]{} (7,1.45)[$-i\beta'c$]{}
In the context of Figure \[figsimil\], let us suppose that $z_0=e^{-i\alpha}z$, $z_1=e^{i\alpha}z$, and $z_1'=\beta z$, where $r_j=|z|$, $r_{j+1}=\beta r_j$, and $2\alpha=\theta_1-\theta_0$. We normalize the figure by subtracting the midpoint $(\cos\alpha)z$ of the segment from $z_0$ to $z_1$, as shown in Figure \[fignonskew\], in which $\beta'=\beta-\cos\alpha$. Carrying out Step 1, we take an average value for $\mu$ within the triangle and apply $L[\mu]$. For convenience we follow this by composition with the conformal mapping $A[-i/(z-\mu\conj{z}),0]$ (recall the formulas (\[defL\]),(\[defA\])). It is thus seen that the condition for the image not to be skewed is $$|\re i\beta' c| < \sin\alpha$$ where $$c = \frac{z+\mu\conj{z}}{z-\mu\conj{z}} =
\frac{1+d}{1-d} ,\quad d=\mu\,\conj{z}/z .$$ Since $|\re ic| = |\im c| = 2|\im
d|/|1-d|^2 \le 2|d|/|1-d|^2$, our condition is satisfied when $$\frac{\beta'}{\sin\alpha} < \frac{|1-d|^2}{2|d|}$$ But $\alpha=2\pi/N$ and $|d|=|\mu|$, so we are done.
The condition in this theorem is appropriate for when one is not going to take into account precise information regarding the distribution of $|\mu|$ within $\D$, and in such cases it is reasonable to make the ratio $\beta=r_{j+1}/r_j$ independent of $j$. Note that $\beta=1+(\pi/N)\tan a$ where $a$ is the base angle of the isosceles mesh triangles. Since $r_M=1$, the radii are determined as $r_j=\beta^{M-j}$. It is also reasonable to take the central “hole” in the mesh not much bigger than the largest (i.e., outer) triangles, which have a base of $2\pi/N$. Since $r_1=\beta^{1-M}$, this suggests defining $M$ at least as large as $$\label{Mdef}
M \approx \frac{ \log(N/(2\pi)) }{ \log\beta }$$ if it is desired to assure $(\ref{nonskewcond})$ and to reasonably cover $\D$ with the mesh. (See Table \[tabratios\].) Of course, larger values of $M$ may be used if a finer mesh is desired.
$N$ $\mu=0.1$ $\mu=0.2$ $\mu=0.3$ $\mu=0.4$ $\mu=0.5$
----- ------------ ------------- ------------ ------------- --------------
32 2.561 (5) 1.605 (9) 1.299 (15) 1.156 (27) 1.078 (52)
64 1.789 (8) 1.309 (18) 1.155 (32) 1.083 (58) 1.044 (107)
128 1.396 (16) 1.156 (37) 1.079 (70) 1.043 (127) 1.023 (230)
256 1.198 (34) 1.078 (80) 1.04 (154) 1.022 (279) 1.012 (504)
512 1.099 (71) 1.039 (175) 1.02 (339) 1.011 (614) 1.006 (1108)
: \[tabratios\]Ratios $r_{j+1}/r_j$ required by the estimate (\[nonskewcond\]) for given values of $N$ and $\|\mu\|_\infty$. The corresponding values of $M$ prescribed by (\[Mdef\]) are shown in parentheses.
\[seccompu\]Conformal mapping methods and computational cost of the Disk Algorithm
==================================================================================
The vast literature developed during the past century on methods for mapping a simply connected domain conformally to a disk (see [@Delil0], [@Gaier], [@Henr], [@Weg] and the references therein), gives an ample range of options for implementing Step 2 of the Disk Algorithm.
A natural choice is to apply an osculation method [@Henr], for two reasons: the convergence is guaranteed regardless of the complexity of the original configuration, and most common osculation methods provide an approximation of the mapping from non-circular domain to a disk (rather than the inverse direction). However, it is well known that osculation methods tend to converge slowly.
The fastest available conformal mapping methods, many of which cost $O(N\log N)$ computations per iteration and converge linearly or in some cases quadratically, arrive quickly at good approximations for the boundary maps $z\mapsto w$, and would be applicable in the Disk Algorithm because the domains involved to be mapped are are close to circular. However, such methods provide directly only the boundary values of the conformal mapping, so one must then use integration to calculate each inner value as a boundary integral. Further, it is the inverse mapping $w\mapsto z$ which is needed for the Disk Algorithm. Thus additional programming would be required. Since $N$ may be potentially large, we have ruled out Schwarz-Christoffel methods [@BV], [@DT].
In the examples computed in Section \[secnumres\] we have used the Zipper Algorithm [@MR]. Briefly, this algorithm approximates a conformal mapping from a the interior of an $N$-point Jordan curve polygon to the unit disk, as the composition of $N$ partial zipper mappings, each of which is in turn a composition of a few Möbius transformations and square roots. To find each partial zipper mapping requires requires $O(N)$ arithmetic operations, thus giving an operation count of $O(N^2)$ for calculating the zipper mapping. The Zipper Algorithm is similar to the classical osculation mappings in the sense that it provides an explicit formula for the approximated conformal mapping in the interior as well as on the boundary of the domain. It is different in that for fixed $N$, it provides a single mapping rather than an infinite convergent sequence of mappings.
The calculation of the computational cost of the Disk Algorithm with zipper mappings is as follows. In the $j$th iteration, Step 1 requires $O(N)$ arithmetic operations. Step 2 requires $O(N^2)$ operations to find the conformal mapping, and in the process produces the $N$ image points of the boundary. Over $M$ values of $j$, Steps 1 and 2 cost $O(MN^2)$. In Step 3, the mapping, which costs $O(N)$ operations to map a single point, must be applied to $j$ collections of $N$ points each, making $O(jN^2)$ operations. Summing these costs over $1\le j\le M$ gives $O(M^2N^2)$ as the total operation count of Step 3.
The preceding analysis is for the cost applied to the complete triangular mesh in $\D$. However, there are situations when one may only be interested in calculating $f(z)$ for a reduced subset of interior points; then it is not necessary to apply the conformal mappings to the mesh points other than these. Then the cost of Step 3 may be much lower. For example, with a submesh of $M'$ concentric rings of $N'$ points ($M'<M$, $N'<N$), the calculation count is reduced to $O(MN^2)+O(M'^2N'N)$. As an extreme case, when one only wants to find the boundary map $f|_{\partial\D}$, it is not necessary to keep any of the inner image data; in other words, Step 3 may be suppressed entirely, and the operation count is then $O(MN^2)$. Examples of proper submeshes $(M',N')$ will be given in Section \[secnumerical\].
Corresponding operation counts may be obtained similarly if other conformal mapping algorithms are used in Step 2 of the Disk Algorithm.
\[secnumres\]Numerical Results\[secnumerical\]
==============================================
The Disk Algorithm was programmed in Mathematica (Wolfram), on a standard laptop computer of approximately 1GH, with the conformal mapping step being carried out by calling Fortran routines obtained from the website of [@MR].
The following examples illustrate the results of the Disk Algorithm for a variety of prescribed Beltrami derivatives $\mu$.
\(1) *Constant $\mu$.* We take $\mu(z)=0.4$. The Disk Algorithm was applied for a mesh of equilateral triangles with $(M,N)=(64,128)$. Figure \[figconst\] shows the image of a subdomain with $(M,N)=(32,32)$.
Note that the image triangles in Figure \[figconst\] are not similar, even though $\mu$ is constant, since the similarity class of an image triangle depends upon both the value of $\mu$ and the slope of the base of the domain triangle.
(0,6)(5,.5) (5.2,0)[![Image for constant Beltrami derivative $\mu=0.4$. The original triangulation of unit disk is shown at left.[]{data-label="figconst"}](figconstz.eps "fig:")]{}
(0,6)(5,.5) (11.7,0)[![Image for constant Beltrami derivative $\mu=0.4$. The original triangulation of unit disk is shown at left.[]{data-label="figconst"}](figconst.eps "fig:")]{}
Note that constant $\mu$ doesn’t imply that all the triangles are skewed the same way, it depends on the position of the domain triangle too.
\(2) *Radial quasiconformal mapping.* Let $\varphi\colon[0,1]\to[0,1]$ be an increasing diffeomorphism of the unit interval. Then the radially symmetric function $$f(z) = \varphi(|z|)e^{i \arg z}$$ has Beltrami derivative equal to $$\mu(z) = \frac{|z|(\varphi'(z)/\varphi(z)) - 1 }
{|z|(\varphi'(z)/\varphi(z)) + 1 }$$ when $z\not=0$. As an illustration we will take $$\varphi(r) = 1 + \frac{1}{2}\sin^2 \pi(r-\frac{1}{2})$$ as in Figure \[figradial\].
(0,6)(5,.5) (6,1)[![Radial function $\varphi$ of example 2 (left), together with rotationally symmetric image domain.[]{data-label="figradial"}](figphiradial.eps "fig:")]{}
(0,5)(5,.5) (11.6,0)[![Radial function $\varphi$ of example 2 (left), together with rotationally symmetric image domain.[]{data-label="figradial"}](figradial.eps "fig:")]{}
The resulting Beltrami derivative satisfies $\|\mu\|=0.33$ approximately (check). The Disk Algorithm was applied with $(M,N)=(256,128)$, with only the real values being conserved in Step 3. These real values were compared with the true values $\varphi(r)$ and found to have an error no greater than $0.0025$. The image region is depicted in Figure \[figradial\].
\(3) *Sectorial quasiconformal mapping.* Let $\psi\colon[0,2\pi]\to[0,2\pi]$ be an increasing diffeomorphism. Write $\widetilde\psi(e^{i\theta})=e^{i\psi(\theta)}$. Then the sectorially symmetric function $$f(z) = |z|\,\widetilde\psi\left(\frac{z}{|z|} \right)$$ has Beltrami derivative equal to $$\mu(z) = \frac{1-\psi'(\theta)}{1+\psi'(\theta)}\,
\frac{\conj{z}}{z}$$ when $z\not=0$. As an example we will take $$\psi(\theta) = \left\{ \begin{array}{ll}
\frac{\theta}{2}, \quad& 0\le\theta\le\pi,\\
\frac{\pi}{2}+\frac{3(\theta-\pi)}{2},& \pi\le\theta\le2\pi.
\end{array} \right.$$ The Disk Algorithm was applied with $(M,N)=(256,128)$, conserving no interior values in Step 3. The final boundary values were compared with the true values $\psi(\theta)$ and found to have an error no greater than $0.055$. The image region is depicted in Figure \[figsectorial\].
(0,6)(5,.5) (6,1)[![Angular function $\psi$ of example (3 (left), together with image domain under sectorial mapping.[]{data-label="figsectorial"}](figpsisectorial.eps "fig:")]{}
(0,5)(5,.5) (11.6,0)[![Angular function $\psi$ of example (3 (left), together with image domain under sectorial mapping.[]{data-label="figsectorial"}](figsectorial.eps "fig:")]{}
\(4) *Exterior mappings.* In Daripa [@Dar-fastB], quasiconformal mappings from $\D$ to the exterior of an ellipse (the origin being sent to $\infty$) are calculated with the following two sample Beltrami derivatives, $$\begin{aligned}
\mu_1(z) &=& |z|^2 e^{0.65(iz^5-2.0)}, \\
\mu_2(z) &=& \frac{1}{2}|z|^2 \sin(5\re z) .\end{aligned}$$ Daripa uses $M$ radii equally spaced in $[0,1]$, in contrast to the exponential spacing we have been using. The exterior mapping results can be related to those of the Disk Algorithm by use of the formula $$h(z) = \frac{(1+\alpha)-(1-\alpha)z^2}{2\alpha z}$$ which transforms $\D$ conformally to the exterior of an ellipse with aspect ratio $\alpha$. Composition of $h$ following the quasiconformal self mapping of $D$ provides a mapping to the exterior of the ellipse with the same Beltrami derivative.
(0,5.3)(5,.5) (4.8,0)[![Self mapping of unit disk with Beltrami derivative $\mu_1$ (left), and image of grid near unit circle under exterior conformal mapping for ellipse. Above: Beltrami derivative $\mu_1$; below: Beltrami derivative $\mu_2$.[]{data-label="figdar1"}](figdar1.eps "fig:")]{}
\
(0,5.7)(5,.5) (4.8,0)[![Self mapping of unit disk with Beltrami derivative $\mu_1$ (left), and image of grid near unit circle under exterior conformal mapping for ellipse. Above: Beltrami derivative $\mu_1$; below: Beltrami derivative $\mu_2$.[]{data-label="figdar1"}](figdar2.eps "fig:")]{}
To match the examples in [@Dar-fastB], $\alpha=0.6$ is specified. (However, the inner ellipses in [@Dar-fastB] appear to have aspect ratios of approximately 0.47; axes are not drawn.) Our results are depicted in Figure \[figdar1\]. The images we obtain with $(M,N)=(64,64)$ and $(M,N)=(256,256)$ turn out to be indistinguishable. These images appear a bit different from the figures shown in [@Dar-fastB], and it is not clear that the discrepancies can be accounted for simply by a different selection of level curves.
Computation times are reported in [@Dar-fastB] for $N=64$ as approximately 8.5 seconds of CPU on a MIPS computer described as “approximately 15 times slower than the CRAY-YMP at Texas A & M University.” Our laptop CPU times, using Mathematica/Fortran, were approximately 85 seconds.
(0,8)(5,.5) (7.5,0)[![Self mapping of unit disk with Beltrami derivative defined by the Klein modular function. Computations were made with $(M,N)=(256,512)$, $(M',N')=(128,64)$. The standard fundamental domain in the upper half plane has been mapped to the unit disk (lighter contour); the image under the quasiconformal mapping is superimposed.[]{data-label="figmodular"}](figmodular.eps "fig:")]{}
\(5) *Quasiconformal deformation of Fuchsian groups.* In this example $\mu$ is taken to be a quadratic differential for a Fuchsian group $G$ (see [@Lehto] for definitions). The elliptic modular group is the collection of conformal transformations of the upper half plane generated by the two mappings $\tau\mapsto\tau+1$, $\tau\mapsto-1/\tau$. This Fuchsian group has a standard fundamental domain $\{\tau:
\im\tau>0,\ |\re\tau|<1/2,\ |\tau|>1\}$. The Klein modular function $J$ satisfies the invariance relations $J(\tau) = J(\tau+1) = J(1/\tau)$, from which it follow that $$J'(\gamma(\tau))\gamma'(\tau) = J'(\tau)$$ for every element $\tau$ in the elliptic modular group. The upper half plane is transformed into the unit disk $\D$ via the elementary conformal mapping $$z = h(\tau) = i\frac{1-\tau}{1+\tau}.$$ Let $G$ denote the corresponding group of self mappings of $\D$. Let $\varphi(z) = J(\tau).$ Then for $0<\alpha<1$ the function $$\mu(z) = \alpha \frac{\conj{\varphi'(z)}}{\varphi'(z)}$$ is a Beltrami differential for the group $G$; that is, $\mu(\gamma(z))\conj{\gamma(z)}/\gamma(z) = \mu(z)$ for every $\gamma$ in $G$. This implies that any $\mu$-conformal self mapping $f\colon\D\to\D$ satisfies $f(\gamma(z))=\gamma_\mu(f(z))$ where $\gamma\mapsto\gamma_\mu$ is an isomorphism from $G$ to another Fuchsian group $G_\mu$.
We have taken $\alpha=0.4$. Figure \[figmodular\] shows part of the tesselation of $\D$ by fundamental domains of $G$ and their images, which are fundamental domains of $G_\mu$. Some inaccuracies are clearly visible inasmuch as the image curves must be hyperbolic geodesics.
Discussion and Conclusions
==========================
It is stated in Daripa [@Dar-fastB] that prior to that article there were no constructive methods published for solving the Beltrami equation numerically. We discuss here some aspects of [@Dar-fastB] in relation to the Disk Algorithm.
As we mentioned in the Introduction, the method of [@Dar-fastB] is based on evaluation of singular integrals. That method is presented in the context of finding a $\mu$-conformal mapping to a prescribed star-shaped domain, and is in some ways reminiscent of the classical method of Theodorsen [@Henr] for conformal mappings. Convergence proofs are not supplied.
Daripa’s main algorithm requires evaluation of the $\partial/\partial\overline{z}$ derivatives which appear in the singular integrals. A variant is also proposed which does not require these derivatives; however this is not used in the numerical examples provided. The operation count of one iteration of Daripa’s method is $O(MN\log N)$. This should be multiplied by the average number of iterations required, which depends on how refined the mesh is and how much accuracy is desired.
The examples we have given in Section \[secnumres\] show that our method begins to have difficulties when $|\mu|$ is large as $0.5$. Likewise, in the examples which we have taken from [@Dar-fastB], $\|\mu\|_\infty$ is approximately $0.5$, but it should be noted that $|\mu(z)|$ is in fact bounded by 0.12 for $|z|<0.5$, and by 0.05 for $|z|<0.3$. In fact, an important limitation stated in [@Dar-fastB] is that the Beltrami derivative $\mu$ must be Hölder continuous. Further, it is recommended that $\mu$ vanish at least as fast as $|z|^3$ at the origin. The Disk Algorithm, in contrast, is not subject to any continuity requirement on $\mu$.
We believe that the Disk Algorithm for solving the Beltrami equation is conceptually simpler than other methods which have been presented, and is easier to implement.
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[^1]: Research partially supported by CONACyT grants 46936 and 60160
|
---
abstract: |
Cognitive engineering is a multi-disciplinary field and hence it is difficult to find a review article consolidating the leading developments in the field. The incredible pace at which technology is advancing pushes the boundaries of what is achievable in cognitive engineering. There are also differing approaches to cognitive engineering brought about from the multi-disciplinary nature of the field and the vastness of possible applications. Thus research communities require more frequent reviews to keep up to date with the latest trends. In this paper we shall discuss some of the approaches to cognitive engineering holistically to clarify the reasoning behind the different approaches and to highlight their strengths and weaknesses. We shall then show how developments from seemingly disjointed views could be integrated to achieve the same goal of creating cognitive machines. By reviewing the major contributions in the different fields and showing the potential for a combined approach, this work intends to assist the research community in devising more unified methods and techniques for developing cognitive machines.
***artificial intelligence, cognitive architecture, bio-inspired***
author:
- |
Jarryd Son\
Electrical Engineering Department\
University of Cape Town, South Africa\
Email: jdsonza@gmail.com
- |
Amit Kumar Mishra\
Electrical Engineering Department\
University of Cape Town, South Africa\
Email: akmishra@ieee.org
bibliography:
- 'ref.bib'
title: A Survey of Brain Inspired Technologies for Engineering
---
Introduction
============
The functioning of the brain has intrigued researchers since the beginning of scientific endeavours. The introduction of computers saw the advent of exciting developments which has culminated in the development of the new discipline of artificial intelligence (AI). Within the field of AI there has been a divided opinion on what the best approach is to create cognitive machines [@intelligence2003modern]. This division is based primarily on how information is claimed to be processed in the brain [@smolensky1987; @HARNAD1990335]. On one side is the symbolic approach and the other is the sub-symbolic approach.
Symbolic approaches such as cognitive architectures have a long history in AI and their developments have been devoted towards creating computational models that formalize the structure of the human brain [@langley2009cognitive]. Cognitive architectures such as Soar [@laird1987soar] and ACT-R [@anderson2013architecture] have been under development for many decades and have been successfully applied in various studies.
There have been arguments against the use of symbol systems because they oversimplify the underlying mechanisms required for cognition [@smolensky1987]. The alternative is to mimic the biology of brains, which gives rise to the sub-symbolic, connectionist approach. Connectionist approaches such as artificial neural networks (ANN’s) have been through several generations of major developments and have become the leading technology in AI in recent years [@lowry2016visual].
The work discussed in this paper is focussed on how the mammalian brain can provide insights into creating appropriate models for designing cognitive machines. Furthermore, this work suggests unifying the differing approaches to create a holistic model rather than narrowing in on specific features. This work contributes to the engineering community by exploring useful technologies that could assist in creating more intelligent machines. Machine learning and AI has had a strong focus in tasks such as image recognition, language translation and financial analytics, however applications of such technology for machines that interact with the physical world has been less prominent.
The paper is organized as follows: Section II looks at symbolic AI approaches, Section III looks at the increasingly popular connectionist approaches, Section IV identifies the advantageous qualities of a hybrid approach, Section V reviews advances in creating specialised hardware that mimics biology of the brain, and lastly the paper concludes in Section VI.
Symbolic Approaches
===================
Robotics is often cited as a field where AI and machine learning technology can be used, however many of the attempts focus on perceptual systems and ignore high-level cognitive capabilities [@hanford2011cognitive]. Much of the early success in achieving such capabilities was through the use of symbolic AI systems. AI pioneers, Alan Newell and Herbert A. Simon formulated the physical symbol system hypothesis that claims that, “a physical symbol system has the necessary and sufficient means for intelligent action" [@intelligence2003modern]. Their work culminated in the creation of many impressive AI systems including the creation of the Soar cognitive architecture [@laird1987soar].
Soar is one of many cognitive architectures that aims to create a formal, structured model of a cognitive system [@langley2009cognitive]. Figure \[fig:Soar\_block\] is an illustration of how Soar is composed and should clarify what a cognitive architecture entails.
![A simplified block diagram of the Soar cognitive architecture. [@laird2008extending][]{data-label="fig:Soar_block"}](./Pictures/SoarDiagram){width="0.85\linewidth"}
Formal structure plays an important role not only in Soar but all cognitive architectures as they must define features that remain constant in a cognitive agent [@langley2009cognitive]. The necessary components for cognition are defined by various cognitive models created by cognitive scientists. These components are designed as individual modules that can be interconnected to form a complete architecture. These models are typically based on the physical symbol system hypothesis, however there has been a shift towards hybrid-like approaches such as the CLARION [@langley2009cognitive] and Sigma [@rosenbloom2013sigma] architectures.
Cognitive architectures aim to utilise the knowledge contained in each of the different modules in a coherent and unified manner to produce cognitive behaviour [@langley2009cognitive]. The benefit of taking this approach is that agents can be designed with specific features that are well defined and can be understood by people. This is particularly important for high level cognitive capabilities that are easier to understand at a symbolic level than at a sub-symbolic level [@smolensky1990tensor]. From a scientific and engineering point of view this is an important attribute that is missing in connectionist models because they are so complex that they become incomprehensible for human interpretation (see Section \[sec:connectionist\]).
Autonomous vehicles are a popular application of AI in engineering and even though there has been great success in recent years the technology used still lack the capabilities most associated with intelligence such as problem solving and decision making [@hanford2011cognitive]. These capabilities are useful for machines operating in dynamic, unknown and uncertain environments and cognitive architectures bring these capabilities to the engineering community.
A particularly interesting feature of humans is the ability to recall sequences of historical events that can be applied to the current situation rather than having to perform a new process to achieve the same outcome. This type of memory, known as episodic memory, has been introduced in Soar and it offers some impressive capabilities as shown by Nuxoll and Laird [@nuxoll2012enhancing]. This ability to retain sequences of actions and events could be useful in mobile robot navigation tasks where navigation becomes a recollection of movements and not an entirely new navigation process (which is often computationally expensive).
Episodic memory is just one example of many high-level cognitive capabilities where the classical approach to AI has been at the forefront [@langley2009cognitive]. Cognitive architectures allow for easier implementation of the complex structures required for high-level cognitive capabilities compared to non-symbolic approaches that obscure these formal structures. While it is definitely possible for connectionist models to implement these exact structures (this must be true because the human brain is a biological connectionist system) the techniques and methods in achieving this seem out of reach for now [@haykin2009neural].
There are many cognitive architectures in development which makes it overwhelming to choose one to focus on. Additionally there is a steep learning curve required to use each of them proficiently, which is not helped by the lack of learning resources. Where resources are available they are often limited to “toy" examples or are outdated. This presents a stumbling block for development especially when connectionist models have highly active communities and many resources with real-world examples that makes it easier to get involved.
These issues may seem trivial in the broader scheme of things, however, they underline one of the major downfalls of using these approaches - rigidity. The formal structure of cognitive architectures confines designers to specific tools and methods, whereas connectionist approaches follow the same guiding principles. Cognitive architectures are also reliant on humans to encode much of the necessary knowledge which creates many practical and theoretical problems [@HARNAD1990335] that will not be discussed in this paper.
There is certainly a place for symbol system approaches in equipping machines with high-level cognitive capabilities. Symbolic AI systems also offer the advantage of providing insight and understanding that can guide cognitive machine design. Unfortunately this is provided at the expense of requiring greater human effort and more rigid structures. The connectionist views as explained in Section \[sec:connectionist\] allow for greater autonomy which results in less human effort and a more flexible structure.
Sub-symbolic Approaches {#sec:connectionist}
=======================
An alternative to symbolic AI is the connectionist approach that does away with formal processing blocks that model cognition in favour of an approach inspired by neurobiology [@smolensky1987]. Instead of relying on hand-engineered features and symbolic data structures connectionist models, such as artificial neural networks (ANN’s), rely on the processing power of having many simple, interconnected processing units that allow for massively parallel computing [@haykin2009neural]. An illustration of the analogy between the artificial and biological neuron is provided in Figure \[fig:neurons\].
![An artificial neuron mimics the structure of a biological neuron. Neurons take in multiple weighted inputs, add them together and pass them through an activation function that determines the output.[]{data-label="fig:neurons"}](./Pictures/neurons){width="0.8\linewidth"}
These elementary building blocks allow ANN’s to embed information in all of the weights across the network. An important feature is that the activation functions are mostly non-linear and are therefore useful for solving non-linear problems, whereas traditional approaches often involve the linearisation of problems so that linear techniques can be applied
The information contained in a neural network is a result of learning and not through direct encoding by a human [@smolensky1987]. Learning is often performed in a supervised manner (although unsupervised learning is improving) where the neural network is provided with an input and a desired (labelled) output. The input data is passed through the network to produce the actual output which can be compared to the desired output [@haykin2009neural]. Various algorithms exist to perform this comparison and use this to update the weights and biases in the network. Through appropriate training it is possible to create a network that can perform a desired task with little human involvement. Unlike the rigid structure of cognitive architectures, connectionist models are adaptive by nature [@haykin2009neural].
The beauty of ANN’s is that one can use them to accomplish many different tasks even though the principles and methods will remain the same [@haykin2009neural]. So whether one is interpreting radio data, extracting features from range finding sensors or even performing image compression, the fundamental principles remain the same. From a practical perspective this flexibility extends to how the same models can be built using completely different programming languages and hardware, whereas each cognitive architecture has their own rules for creating models and are mostly restricted to CPUs. The development of neuromorphic hardware as described in Section \[sec:neuromorphic\] makes these models even more attractive.
Recent developments have seen an increase in the number of neural networks with recurrent connections that essentially act as some form of memory [@lipton2015critical]. Instead of only seeing a snapshot of data at each time step the network is capable of using the data from previous time steps to assist in processing the current data [@lipton2015critical]. Training of these recurrent neural networks (RNN’s) posed issues for many years, however new techniques have helped solve this [@hochreiter1997long]. For example a popular RNN model and learning algorithm known as long short-term memory (LSTM) introduces specialised gates that control the flow of information to allow for the learning of long-term dependencies [@hochreiter1997long]. In this model the basic building block is no longer just a neuron, but rather what is known as a “memory cell" [@hochreiter1997long]. Figure \[fig:LSTM\_architecture\] illustrates the complexity of these models and hopefully reveals the difficulties associated with using them.
![This diagram illustrates how an LSTM memory cell can be“unfolded" over time. Where $x_t$ indicates the current input, and $h_t$ indicates the cell’s current state. There are also various activation blocks that control what information gets stored in the cell (input gate) and when information is accessed from memory (output gate). [@Olah2015][]{data-label="fig:LSTM_architecture"}](./Pictures/lstm){width="0.75\linewidth"}
This increased complexity in RNN’s makes them capable of learning more advanced features at the expense of becoming more difficult to train and compute [@Schmidhuber2015DLReview]. These extended capabilities are particularly attractive for engineering applications that rely on sequences of data such as interpreting motions recorded by an inertial measurement unit or range sensor data from a mobile robot.
A major stumbling block in creating large neural networks, however, is the sheer quantity of model parameters that make it impossible for a human to comprehend [@smolensky1987]. This leads to difficulty in predicting the outcome of a given network. Other popular black-box models, such as transfer functions, differ from connectionist models in that there are well defined methods for analysing them. It is easier to predict how the model will behave and what changes need to be made to obtain the desired performance. Unfortunately connectionist models do not have such methods yet. This is problematic because engineering design concerns itself with understanding how design choices affect the performance of the system.
This approach has worked well for perceptual tasks such as classification and recognition of patterns, however, this is a small piece of the cognition and there is still difficulty in learning complex representations necessary for high-level cognitive functions [@sun1999artificial]. Connectionist models have the potential to match and even enhance the capabilities of pure symbol systems, but these developments are likely still far away.
A problem with having to compute large connectionist networks is that they require specialised hardware to compute efficiently. GPUs are better suited to the massively parallel computation required in neural networks compared to CPUs, however their power usage is still far beyond that of the human brain [@haykin2009neural]. This is not a problem for applications that can run on desktop computers or servers where there is sufficient power. It becomes an issue when creating mobile devices where power is limited. Alternative computing hardware could solve this, as discussed in Section \[sec:neuromorphic\].
Having explored the two different approaches to AI it is suggested that a third option be looked at - a hybrid approach. [@HARNAD1990335] suggests that this is a viable and attractive solution to many problems that each individual approach is faced with and that it may be necessary to accomplish cognitive machines.
Hybrid Approaches {#sec:hybrid}
=================
As mentioned before, symbolic and connectionist models are each suited to specific levels of cognitive capabilities. Symbolic models are able to perform high-level cognitive tasks such as reasoning and planning while lacking sufficient capabilities of handling low-level tasks such as perception and action. Connectionist approaches have been extremely successful in perceptual tasks and are useful in adaptive control but so far lack the high-level capabilities that are necessary for complex tasks [@HARNAD1990335; @smolensky1990tensor].
In the current applications that have made neural networks popular, such as image recognition, there has been no real need for high-level capabilities. In the case of machines such robots, there is a need for strong perception and action capabilities because the agents must interact with the physical world [@haykin2012cognitive] and there is also the need to include problem solving and decision making capabilities for the robotic agents to operate without human intervention [@haykin2012cognitive]. Various industries already employ robots to minimize the need for humans to perform dangerous tasks or tasks that require extremely high precision and accuracy. Some applications have been out of reach due to the lack of high-level cognitive capabilities.
It is possible to bring cognitive machines closer to realisation by combining the strengths of symbol systems with connectionist models, as well as other non-symbolic approaches. Hybrid cognitive robotic architectures have been explored before in [@kelley2009hybrid] and [@hanford2011cognitive] but there remains a wealth of untapped capabilities such as the use of episodic memory.
The SS-RICS architecture in [@kelley2009hybrid] used a common robotics approach for generating a map for navigation that utilizes metric information from sensor data. They encountered various issues with this in that the classification of intersections based on sensor data was often incorrect and compounded as the robot continued its task. They argue that without a useful perceptual system the higher-level capabilities can never be realised because there would be difficulty in creating meaningful symbolic relationships [@kelley2009hybrid]. The CRS architecture used in [@hanford2011cognitive] used fuzzy logic to improve the classification of intersections but odometery errors meant that the robot mistakenly identified the same intersections as different ones.
In both SS-RICS and the CRS architectures the majority of the faults were with the perceptual systems used. Despite this, both attempts showed some useful results from their experiments that showed a glimpse of what could be possible with a hybrid system should the perceptual systems have been up to the task.
Thankfully perceptual systems have improved substantially as mentioned in Section \[sec:connectionist\]. An example of where a cognitive machine could leverage a hybrid cognitive architecture is in robotic mapping. Traditional mapping techniques such as building occupancy grids caused issues in SS-RICS and the CRS as mentioned above. Even though they both mention that such techniques are not cognitive processes they continue to use them as a step towards providing semantic labels for a high-level symbol system [@kelley2009hybrid; @hanford2011cognitive].
This paper proposes a hybrid architecture intended for use in a mobile robot that can be realised by combining the different approaches in a hierarchical fashion as shown in Figure \[fig:hybrid\_hierarchy\].
![A cognitive architecture can be used as a deliberative layer because of its high-level cognitive capabilities. The middle executive layer would control the flow of information between the deliberative and reactive layer. An ANN would make an appropriate reactive layer for perception and action.[]{data-label="fig:hybrid_hierarchy"}](./Pictures/hybrid_hierarchy){width="0.55\linewidth"}
The proposed architecture could be applied to robot navigation where it could use RNNs to utilize sequential sensor data to construct useful local representations of the local environment. The advantage of using ANNs is that they are capable of extracting better features than those that are hand-engineered. The cognitive architecture could use that local information to construct topological maps which allow for easier path planning, decision making and problem solving compared to metric maps [@hanford2011cognitive]. Topological maps are also far more compact and will have less memory requirements. There could also be a global feedback loop that can augment the perceptual system for enhanced capabilities i.e. it may be possible to use the cognitive architecture to control what features the perceptual system should focus on.
The tight integration between the different layers is something that will need to be looked at carefully. The hybrid model needs to be designed in such a way that the addition of a structured symbol system does not inhibit the flexible nature of the connectionist system and that the connectionist system is capable of forming meaningful symbolic relationships.
Neuromorphic Emulation {#sec:neuromorphic}
======================
As mentioned before in Section \[sec:connectionist\] there is a need for specialised hardware to implement connectionist models. The Human Brain Project has a platform that aims to emulate the functioning inside the human brain. They provide a review of neuromorphic technology in [@calimera2013human] where they provide a breakdown of neuromorphic hardware as shown in Figure \[fig:Neuromophic\_hardware\_tree\].
![Neuromorphic hardware can be divided into emulations and simulations that describe in what manner the neural networks are implemented.[]{data-label="fig:Neuromophic_hardware_tree"}](./Pictures/Neuromophic_hardware_tree){width="0.75\linewidth"}
Simulating large scale neural networks using von Neumann architectures is inefficient and would require incredibly large amounts of power [@benjamin2014neurogrid; @merolla2014million; @furber2014spinnaker]. One of the primary bottlenecks to overcome is the inefficient movement of data that occurs between processors and memory in traditional von Neumann architectures [@merolla2014million]. In order to emulate neural networks there needs to be a tighter integration between processors and memory to form the individual neurons. Various approaches to this idea have been undertaken by research teams from around the world. Most notable are the works done by the University of Manchester with their SpiNNaker project [@furber2014spinnaker], IBM with their TrueNorth architecture[@merolla2011digital], Stanford University and their Neurogrid architecture [@benjamin2014neurogrid], and a team from the University of California at Santa Barbara [@prezioso2015training].
The University of Manchester have done work on designing and implementing a neurally inspired computational hardware as part of the Neural Computing Platform for the Human Brain Project. The architecture employed in their SpiNNaker project utilizes processing nodes consisting of 18 general purpose ARM968 cores and extra memory for each node [@furber2014spinnaker]. The block diagram for one node is provided in Figure \[fig:SpiNNaker\_Node\].
![A SpiNNaker node consists of interconnected blocks of existing digital components. [@furber2014spinnaker][]{data-label="fig:SpiNNaker_Node"}](Pictures/SpiNNaker){width="0.4\linewidth"}
They then place multiple processing nodes on a single PCB with FPGA’s for high-speed interconnectivity between the nodes. They have adopted a simplistic approach to achieving massively parallel computing by opting to use large quantities of existing processors rather than designing custom circuitry to emulate individual neurons [@furber2013overview]. The use of existing technology allows for quicker prototyping and construction (especially when considering the scale to which they are aiming to achieve). Much of the alternative research involves the design and implementation of custom circuitry to emulate neurons more closely to achieve better efficiency.
At IBM they have developed what they call digital neurosynaptic cores as the fundamental building blocks of their TrueNorth architecture [@merolla2011digital; @merolla2014million]. They are able to implement spiking neural networks by using existing digital electronics “blocks" such as decoders, encoders and SRAM in a mesh structure to emulate the axons, neurons and synapses respectively. The structure of their implementation is shown in Figure \[fig:Neurosynaptic\_core\].
![The TrueNorth architecture utilizes a custom arrangement of digital circuits to emulate a mesh of neurons. [@merolla2011digital][]{data-label="fig:Neurosynaptic_core"}](Pictures/TrueNorth){width="0.7\linewidth"}
This approach does not use existing computer architectures such as the ARM cores used in the SpiNNaker architecture. The benefit of this is an increased number of neurons and synapses per chip as well as improved efficiency. TrueNorth has 1 million neurons and 256 million synapses [@merolla2014million] compared to SpiNNaker’s approximate number of 18 thousand neurons and 18 million synapses per node [@furber2013overview]. Despite this achievement there are still further developments in replicating the efficiency of neurons by delving into analog electronics.
The team working at Stanford are working on a mixed analog-digital hardware platform for neural computing called Neurogrid [@benjamin2014neurogrid]. Their aim is to reduce the power requirements for neural computing as much as possible by using sub-threshold analog electronics. Rather than relying on digital memory to store synaptic weights the Neurogrid allows for these values to be stored directly in the electronic make-up of the neurons. An example of an analog silicon neuron is provided in Figure \[fig:analogneuron\].
![Analog elements are able to directly emulate the function of a biological neuron from physical principles. [@benjamin2014neurogrid][]{data-label="fig:analogneuron"}](Pictures/analog_neuron){width="0.6\linewidth"}
Their technology has proved to be very efficient [@benjamin2014neurogrid] and is a highly promising project.
At the University of California at Santa Barbara they have been working on creating neural networks that use the well suited properties of memristors [@prezioso2015training]. Memristors have a fundamental property that is very much like the synaptic connection between neurons [@thomas2013memristor]. Where an increase in flux in one direction causes the resistance to increase and flux in the opposite direction causes resistance to decrease [@thomas2013memristor]. In their paper [@prezioso2015training] they were able to train a single-layer perceptron network to classify a 3x3-pixel image without the use of any CMOS components - only memristors. The simplicity of the circuit makes it a very intriguing prospect and certainly a more accurate model of a neuron.
Each of the mentioned projects were chosen to provide an overview of the range of approaches one could use to design neurmorphic chips. The technology is no doubt still in the early stages of development but it is clear that cognitive machines will rely on such technology in the future.
Conclusion
==========
This paper was intended to provide a brief review of technologies that can assist in enabling the creation of cognitive machines. In the relatively brief history of AI much has changed over the years due to new scientific insights and rapid technological growth. The symbolic AI systems that excelled early on were stunted by the oversimplification of cognitive mechanisms [@intelligence2003modern]. In an opposite trend the early abandonment of connectionist models has been reversed in astounding fashion due to technological advancements that have made massively parallel computation more feasible [@intelligence2003modern].
A hybrid cognitive architecture utilizing state-of-the-art techniques from both approaches is a viable option for creating machines that are enhanced with cognitive capabilities. Prior attempts at creating hybrid approaches have not integrated the absolute best of both worlds. As mentioned by Smolensky [@smolensky1990tensor] it is ill-advised for the two camps (symbolic and connectionist) to ignore each other, however major developments in hybrid models have fallen behind in comparison to developments in the individual fields. While the ultimate goal may be to have a full hardware realization of a neural network, a hybrid cognitive model may allow for sufficient capabilities to outperform existing machines in the mean time. The technological landscape is changing and cognitive engineering is very much at the forefront.
|
---
abstract: 'Let $M$ be an orientable, simply-connected, closed, non-spin 4-manifold and let ${\mathcal{G}}_k(M)$ be the gauge group of the principal $G$-bundle over $M$ with second Chern class . It is known that the homotopy type of ${\mathcal{G}}_k(M)$ is determined by the homotopy type of ${\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)$. In this paper we investigate properties of ${\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)$ when $G=SU(n)$ that partly classify the homotopy types of the gauge groups.'
address: 'Mathematical Sciences, University of Southampton, SO17 1BJ, UK'
author:
- Tseleung So
title: 'Homotopy types of $SU(n)$-gauge groups over non-spin 4-manifolds'
---
[^1]
Introduction
============
Let $G$ be a simple, simply-connected, compact Lie group and let $M$ be an orientable, simply-connected, closed 4-manifold. Then a principal $G$-bundle $P$ over $M$ is classified by its second Chern class $k\in{\mathbb{Z}}$. The associated gauge group ${\mathcal{G}}_k(M)$ is the topological group of $G$-equivariant automorphisms of $P$ which fix $M$.
When $M$ is a spin 4-manifold, topologists have been studying the homotopy types of gauge groups over $M$ extensively over the last twenty years. On the one hand, Theriault showed that [@theriault10a] there is a homotopy equivalence $${\mathcal{G}}_k(M)\simeq{\mathcal{G}}_k(S^4)\times\prod^d_{i=1}\Omega^2G,$$ where $d$ is the second Betti number of $M$. Therefore to study the homotopy type of ${\mathcal{G}}_k(M)$ it suffices to study ${\mathcal{G}}_k(S^4)$. On the other hand, many cases of homotopy types of ${\mathcal{G}}_k(S^4)$’s are known. For examples, there are 6 distinct homotopy types of ${\mathcal{G}}_k(S^4)$’s for $G=SU(2)$ [@kono91], and 8 distinct homotopy types for $G=SU(3)$ [@HK06]. When localized rationally or at any prime, there are 16 distinct homotopy types for $G=SU(5)$ [@theriault15] and 8 distinct homotopy types for $G=Sp(2)$ [@theriault10b].
When $M$ is a non-spin 4-manifold, the author showed that [@so16] there is a homotopy equivalence $${\mathcal{G}}_k(M)\simeq{\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)\times\prod^{d-1}_{i=1}\Omega^2G,$$ so the homotopy type of ${\mathcal{G}}_k(M)$ depends on the special case ${\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)$. Compared to the extensive work on ${\mathcal{G}}_k(S^4)$, only two cases of ${\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)$ have been studied, which are the $SU(2)$- and $SU(3)$-cases [@KT96; @theriault12]. As a sequel to [@so16], this paper investigates the homotopy types of ${\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)$’s in order to explore gauge groups over non-spin 4-manifolds.
A common approach to classifying the homotopy types of gauge groups is as follows. Atiyah, Bott and Gottlieb [@AB83; @gottlieb72] showed that the classifying space $B{\mathcal{G}}_k(M)$ is homotopy equivalent to the connected component ${\textrm{Map}}_k(M, BG)$ of the mapping space ${\textrm{Map}}(M, BG)$ containing the map $k\alpha\circ q$, where $q:M\to S^4$ is the quotient map and $\alpha$ is a generator of $\pi_4(BG)\cong{\mathbb{Z}}$. The evaluation map $ev:B{\mathcal{G}}_k(M)\to BG$ induces a fibration sequence $$\label{fib_Gk(M) ev}
{\mathcal{G}}_k(M)\longrightarrow G\overset{\partial_k}{\longrightarrow}{\textrm{Map}}^*_0(M, BG)\longrightarrow B{\mathcal{G}}_k(M)\overset{ev}{\longrightarrow}BG.$$ For $M=S^4$, the order of $\partial_1:G\to\Omega^3_0G$ helps determine the classification of ${\mathcal{G}}_k(S^4)$’s by the following theorem. The first part is due to [@theriault10b] and the second is due to [@KKT14].
\[thm\_counting lemma S4\] Let $m$ be the order of $\partial_1$. Denote the $p$-component of $a$ by $\nu_p(a)$ and the greatest common divisor of $a$ and $b$ by $(a, b)$.
1. If $(m,k)=(m,l)$, then ${\mathcal{G}}_k(S^4)$ is homotopy equivalent to ${\mathcal{G}}_l(S^4)$ when localized rationally or at any odd prime.
2. If ${\mathcal{G}}_k(S^4)$ is homotopy equivalent to ${\mathcal{G}}_l(S^4)$ and $G$ is of low rank (for details please see [@KKT14]), then $\nu_p(m,k)=\nu_p(m,l)$ for any odd prime $p$.
Therefore the classification problem reduces to calculating the order $m$ of $\partial_1$. Known examples are $m=12$ for $G=SU(2)$ [@kono91], $m=24$ for $G=SU(3)$ [@HK06], $m=120$ for $G=SU(5)$ [@theriault15] and $m=40$ for $G=Sp(2)$ [@theriault10b]. For most cases of $G$, the exact value of $m$ is difficult to compute, but we are still able to obtain partial results. When $G$ is $SU(n)$, the order of $\partial_1$ and $n(n^2-1)$ have the same odd primary components if $n<(p-1)^2+1$ [@KKT14; @theriault17]. Moreover, Hamanaka and Kono showed a necessary condition $(n(n^2-1),k)=(n(n^2-1),l)$ for a homotopy equivalence ${\mathcal{G}}_k(S^4)\simeq{\mathcal{G}}_l(S^4)$ [@HK06].
In this paper we consider gauge groups over ${\mathbb{C}}{\mathbb{P}}^2$. Take $M={\mathbb{C}}{\mathbb{P}}^2$ in (\[fib\_Gk(M) ev\]) and denote the boundary map by $\partial'_k:G\to{\textrm{Map}}^*_0({\mathbb{C}}{\mathbb{P}}^2, BG)$. Since ${\textrm{Map}}^*_0({\mathbb{C}}{\mathbb{P}}^2, BG)$ is not an H-space, $[G,{\textrm{Map}}^*_0({\mathbb{C}}{\mathbb{P}}^2,BG)]$ is not a group so the order of $\partial'_k$ makes no sense. However, we can still define an “order” of $\partial'_k$ [@theriault12], which will be mentioned in Section 2. We show that the “order” of $\partial'_1$ helps determine the homotopy type of ${\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)$ like part (1) of Theorem \[thm\_counting lemma S4\].
Let $m'$ be the “order” of $\partial'_1$. If $(m', k)=(m', l)$, then ${\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)$ is homotopy equivalent to ${\mathcal{G}}_l({\mathbb{C}}{\mathbb{P}}^2)$ when localized rationally or at any prime.
In Section 4, we study the $SU(n)$-gauge groups over ${\mathbb{C}}{\mathbb{P}}^2$ and use unstable $K$-theory to give a lower bound on the “order” of $\partial'_1$.
\[thm\_main thm\] When $G$ is $SU(n)$, the “order” of $\partial'_1$ is at least $\frac{1}{2}n(n^2-1)$ for $n$ odd, and $n(n^2-1)$ for $n$ even.
In Section 5, we prove a necessary condition for the homotopy equivalence similar to that in [@HK06].
\[thm\_necessary condition\] Let $G$ be $SU(n)$. If ${\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)$ is homotopy equivalent to ${\mathcal{G}}_l({\mathbb{C}}{\mathbb{P}}^2)$, then $$\begin{cases}
(\frac{1}{2}n(n^2-1),k)=(\frac{1}{2}n(n^2-1),l), &\text{for $n$ odd};\\
(n(n^2-1),k)=(n(n^2-1),l), &\text{for $n$ even}.
\end{cases}$$
Some facts about boundary map $\partial'_1$
===========================================
Take $M$ to be $S^4$ and ${\mathbb{C}}{\mathbb{P}}^2$ respectively in fibration (\[fib\_Gk(M) ev\]) to obtain fibration sequences $$\label{fib_Gk(S4)}
{\mathcal{G}}_k(S^4)\longrightarrow G\overset{\partial_k}{\longrightarrow}\Omega^3_0G\longrightarrow B{\mathcal{G}}_k(S^4)\overset{ev}{\longrightarrow}BG$$ $$\label{fib_Gk(CP2)}
{\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)\longrightarrow G\overset{\partial'_k}{\longrightarrow}{\textrm{Map}}^*_0({\mathbb{C}}{\mathbb{P}}^2,BG)\longrightarrow B{\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)\overset{ev}{\longrightarrow}BG.$$ There is also a cofibration sequence $$\label{cofib_CP2}
S^3\overset{\eta}{\longrightarrow}S^2\longrightarrow{\mathbb{C}}{\mathbb{P}}^2\overset{q}{\longrightarrow}S^4,$$ where $\eta$ is Hopf map and $q$ is the quotient map. Due to the naturality of $q^*$, we combine fibrations (\[fib\_Gk(S4)\]) and (\[fib\_Gk(CP2)\]) to obtain a commutative diagram of fibration sequences $$\label{digm_dfn of tilde partial}
\xymatrix{
{\mathcal{G}}_k(S^4)\ar[d]^{q^*}\ar[r] &G\ar@{=}[d]\ar[r]^{\partial_k} &\Omega^3_0G\ar[d]^{q^*}\ar[r] &B{\mathcal{G}}_k(S^4)\ar[d]^{q^*}\ar[r] &BG\ar@{=}[d]\\
{\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)\ar[r] &G\ar[r]^-{\partial'_k} &{\textrm{Map}}^*_0({\mathbb{C}}{\mathbb{P}}^2, BG)\ar[r] &B{\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)\ar[r] &BG
}$$ It is known that [@lang73] $\partial_k$ is triple adjoint to Samelson product $${\langle{k\imath,{\mathds{1}}}\rangle}:S^3\wedge G\overset{k\imath\wedge{\mathds{1}}}{\longrightarrow}G\wedge G\overset{{\langle{{\mathds{1}},{\mathds{1}}}\rangle}}{\longrightarrow}G,$$ where $\imath:S^3\to SU(n)$ is the inclusion of the bottom cell and ${\langle{{\mathds{1}},{\mathds{1}}}\rangle}$ is the Samelson product of the identity on $G$ with itself. The order of $\partial_k$ is its multiplicative order in the group $[G, \Omega^3_0G]$.
Unlike $\Omega^3_0G$, ${\textrm{Map}}^*_0({\mathbb{C}}{\mathbb{P}}^2, BG)$ is not an H-space, so $\partial'_k$ has no order. In [@theriault12], Theriault defined the “order” of $\partial'_k$ to be the smallest number $m'$ such that the composition $$G\overset{\partial_k}{\longrightarrow}\Omega^3_0G\overset{m'}{\longrightarrow}\Omega^3_0G\overset{q^*}{\longrightarrow}{\textrm{Map}}^*_0({\mathbb{C}}{\mathbb{P}}^2, BG)$$ is null homotopic. In the following, we interpret the “order” of $\partial'_k$ as its multiplicative order in a group contained in $[{\mathbb{C}}{\mathbb{P}}^2\wedge G, BG]$.
Apply $[-\wedge G, BG]$ to cofibration (\[cofib\_CP2\]) to obtain an exact sequence of sets $$[\Sigma^3G, BG]\overset{(\Sigma\eta)^*}{\longrightarrow}[\Sigma^4G, BG]\overset{q^*}{\longrightarrow}[{\mathbb{C}}{\mathbb{P}}^2\wedge G, BG].$$ All terms except $[{\mathbb{C}}{\mathbb{P}}^2\wedge G, BG]$ are groups and $(\Sigma\eta)^*$ is a group homomorphism since $\Sigma\eta$ is a suspension. We want to refine this exact sequence so that the last term is replaced by a group. Observe that ${\mathbb{C}}{\mathbb{P}}^2$ is the cofiber of $\eta$ and so there is a coaction $\psi:{\mathbb{C}}{\mathbb{P}}^2\to{\mathbb{C}}{\mathbb{P}}^2\vee S^4$. We show that the coaction gives a group structure on $Im(q^*)$.
\[lemma\_dfn Im group\] Let $Y$ be a space and let $A\overset{f}{\to}B\overset{g}{\to}C\overset{h}{\to}\Sigma A$ be a cofibration sequence. If $\Sigma A$ is homotopy cocommutative, then $Im(h^*)$ is an abelian group and $$[\Sigma B, Y]\overset{(\Sigma f)^*}{\longrightarrow}[\Sigma A, Y]\overset{h^*}{\longrightarrow}Im(h^*)\longrightarrow0$$ is an exact sequence of groups and group homomorphisms.
Apply $[-,Y]$ to the cofibration to get an exact sequence of sets $$\label{exact seq_set exact seq}
[\Sigma B, Y]\overset{(\Sigma f)^*}{\longrightarrow}[\Sigma A, Y]\overset{h^*}{\longrightarrow}[C, Y].$$ Note that $[\Sigma B, Y]$ and $[\Sigma A, Y]$ are groups, and $(\Sigma f)^*$ is a group homomorphism. We will replace $[C, Y]$ by $Im(h^*)$ and define a group structure on it such that $h^*:[\Sigma A, Y]\to Im(h^*)$ is a group homomorphism.
For any $\alpha$ and $\beta$ in $[\Sigma A, Y]$, we define a binary operator $\boxtimes$ on $Im(h^*)$ by $$h^*\alpha\boxtimes h^*\beta=h^*(\alpha+\beta).$$ To check this is well-defined we need to show $h^*(\alpha+\beta)\simeq h^*(\alpha'+\beta)\simeq h^*(\alpha+\beta')$ for any $\alpha,\alpha',\beta,\beta'$ satisfying $h^*\alpha\simeq h^*\alpha'$ and $h^*\beta\simeq h^*\beta'$.
First we show $h^*(\alpha+\beta)\simeq h^*(\alpha'+\beta)$. By definition, we have $$h^*(\alpha+\beta)=(\alpha+\beta)\circ h=\triangledown\circ(\alpha\vee\beta)\circ\sigma\circ h,$$ where $\sigma:\Sigma A\to\Sigma A\vee\Sigma A$ is the comultiplication and $\triangledown:Y\vee Y\to Y$ is the folding map. Since $C$ is a cofiber, there is a coaction $\psi:C\to C\vee\Sigma A$ such that $\sigma\circ h\simeq(h\vee{\mathds{1}})\circ\psi$. $$\xymatrix{
C\ar[r]^-{\psi}\ar[d]^-{h} &C\vee\Sigma A\ar[d]^-{h\vee{\mathds{1}}}\\
\Sigma A\ar[r]^-{\sigma} &\Sigma A\vee\Sigma A
}$$ Then we obtain a string of equivalences $$\begin{aligned}
h^*(\alpha+\beta)
&=&\triangledown\circ(\alpha\vee\beta)\circ\sigma\circ h\\
&\simeq&\triangledown\circ(\alpha\vee\beta)\circ(h\vee{\mathds{1}})\circ\psi\\
&\simeq&\triangledown\circ(\alpha'\vee\beta)\circ(h\vee{\mathds{1}})\circ\psi\\
&\simeq&\triangledown\circ(\alpha'\vee\beta)\circ\sigma\circ h\\
&=&h^*(\alpha'+\beta)\end{aligned}$$ The third line is due to the assumption $h^*\alpha\simeq h^*\alpha'$. Therefore we have $h^*(\alpha+\beta)\simeq h^*(\alpha'+\beta)$. Since $\Sigma A$ is cocommutative, $[\Sigma A, Y]$ is abelian and $h^*(\alpha+\beta)\simeq h^*(\beta+\alpha)$. Then we have $$h^*(\alpha+\beta)\simeq h^*(\beta+\alpha)\simeq h^*(\beta'+\alpha)\simeq h^*(\alpha+\beta').$$ This implies $\boxtimes$ is well-defined.
Due to the associativity of $+$ in $[\Sigma A, Y]$, $\boxtimes$ is associative since $$\begin{aligned}
(h^*\alpha\boxtimes h^*\beta)\boxtimes h^*\gamma
&=&h^*(\alpha+\beta)\boxtimes h^*\gamma\\
&=&h^*((\alpha+\beta)+\gamma)\\
&=&h^*(\alpha+(\beta+\gamma))\\
&=&h^*\alpha\boxtimes h^*(\beta+\gamma)\\
&=&h^*\alpha\boxtimes (h^*\beta\boxtimes h^*\gamma).\end{aligned}$$ Clearly the trivial map $\ast:C\to Y$ is the identity of $\boxtimes$ and $h^*(-\alpha)$ is the inverse of $h^*\alpha$. Therefore $\boxtimes$ is indeed a group multiplication.
By definition of $\boxtimes$, $h^*:[\Sigma A, Y]\to Im(h^*)$ is a group homomorphism, and hence an epimorphism. Since $[\Sigma A, Y]$ is abelian, so is $Im(h^*)$. We replace $[C,Y]$ by $Im(h^*)$ in (\[exact seq\_set exact seq\]) to obtain a sequence of groups and group homomorphisms $$[\Sigma B, Y]\overset{(\Sigma f)^*}{\longrightarrow}[\Sigma A, Y]\overset{h^*}{\longrightarrow}Im(h^*)\longrightarrow0.$$ The exactness of (\[exact seq\_set exact seq\]) implies $ker(h^*)=Im(\Sigma f)^*$, so the sequence is exact.
Applying Lemma \[lemma\_dfn Im group\] to cofibration $\Sigma^3G\to\Sigma^2G\to{\mathbb{C}}{\mathbb{P}}^2\wedge G$ and the space $Y=BG$, we obtain an exact sequence of abelian groups $$\label{exact seq_general q^* refined}
[\Sigma^3G, BG]\overset{(\Sigma\eta)^*}{\longrightarrow}[\Sigma^4G, BG]\overset{q^*}{\longrightarrow}Im(q^*)\longrightarrow0.$$ In the middle square of (\[digm\_dfn of tilde partial\]) $\partial'_k\simeq q^*\partial_k$, so $\partial'_k$ is in $Im(q^*)$. For any number $m$, , so the “order” of $\partial'_k$ defined in [@theriault12] coincides with the multiplicative order of $\partial'_k$ in $Im(q^*)$. The exact sequence (\[exact seq\_general q\^\* refined\]) allows us to compare the orders of $\partial_1$ and $\partial'_1$.
\[lemma\_partial’\] Let $m$ be the order of $\partial_1$ and let $m'$ be the order of $\partial'_1$. Then $m$ is $m'$ or $2m'$.
By exactness of (\[exact seq\_general q\^\* refined\]), there is some $f\in[\Sigma^3G, BG]$ such that $(\Sigma\eta)^*f\simeq m'\partial_1$. Since $\Sigma\eta$ has order 2, $2m'\partial_1$ is null homotopic. It follows that $2m'$ is a multiple of $m$. Since $m$ is greater than or equal to $m'$, $m$ is either $m'$ or $2m'$.
When $G=SU(2)$, the order $m$ of $\partial_1$ is 12 and the order $m'$ of $\partial'_1$ is 6 [@KT96]. When , $m=24$ and $m'=12$ [@theriault12]. It is natural to ask whether $m=2m'$ for all $G$. However, this is not the case. In a preprint by Theriault and the author, we showed that $m=m'=40$ for $G=Sp(2)$.
In the $S^4$ case, part (1) of Theorem \[thm\_counting lemma S4\] gives a sufficient condition for ${\mathcal{G}}_k(S^4)\simeq{\mathcal{G}}_l(S^4)$ when localized rationally or at any prime. In the ${\mathbb{C}}{\mathbb{P}}^2$ case, Theriault showed a similar counting statement, in which the sufficient condition depends on the order of $\partial_1$ instead of $\partial'_1$.
\[thm\_stephen counting lemma\] Let $m$ be the order of $\partial_1$. If $(m,k)=(m,l)$, then ${\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)$ is homotopy equivalent to ${\mathcal{G}}_l({\mathbb{C}}{\mathbb{P}}^2)$ when localized rationally or at any prime.
Lemma \[lemma\_partial’\] can be used to improve the sufficient condition of Theorem \[thm\_stephen counting lemma\].
Let $m'$ be the order of $\partial'_1$. If $(m',k)=(m',l)$, then ${\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)$ is homotopy equivalent to ${\mathcal{G}}_l({\mathbb{C}}{\mathbb{P}}^2)$ when localized rationally or at any prime.
By Lemma \[lemma\_partial’\], $m$ is either $m'$ or $2m'$. If $m=m'$, then the statement is same as Theorem \[thm\_stephen counting lemma\]. If we localize rationally or at any odd prime, then $(m,k)=(m',k)$ for any $k$, so a homotopy equivalence ${\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)\simeq{\mathcal{G}}_l({\mathbb{C}}{\mathbb{P}}^2)$ follows by Theorem \[thm\_stephen counting lemma\]. It remains to consider the case where $m=2m'$ when localized at 2.
Assume $m=2^n$ and $m'=2^{n-1}$. For any $k$, $(2^{n-1},k)=2^i$ where $i$ an integer such that $0\leq i\leq n-1$. If $i\leq n-2$, then $k=2^it$ for some odd number $t$ and $(2^{n-1},k)=2^i$. The sufficient condition $(2^{n-1},k)=(2^{n-1},l)$ is equivalent to $(2^n,k)=(2^n,l)$. Again the homotopy equivalence ${\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)\simeq{\mathcal{G}}_l({\mathbb{C}}{\mathbb{P}}^2)$ follows by Theorem \[thm\_stephen counting lemma\]. If $i=n-1$, then $(2^n,k)$ is either $2^n$ or $2^{n-1}$. We claim that ${\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)$ has the same homotopy type for both $(2^n,k)=2^n$ or $(2^n,k)=2^{n-1}$.
Consider fibration (\[fib\_Gk(CP2)\]) $${\textrm{Map}}^*_0({\mathbb{C}}{\mathbb{P}}^2, G)\longrightarrow{\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)\longrightarrow G\overset{\partial'_k}{\longrightarrow}{\textrm{Map}}^*_0({\mathbb{C}}{\mathbb{P}}^2,BG).$$ If $(2^n,k)=2^n$, then $k=2^nt$ for some number $t$. By linearity of Samelson products, $\partial_k\simeq k\partial_1$. Since $\partial'_k\simeq q^*k\partial_1\simeq q^*2^nt\partial_1$ and $\partial_1$ has order $2^n$, $\partial'_k$ is null homotopic and we have $${\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)\simeq G\times{\textrm{Map}}^*_0({\mathbb{C}}{\mathbb{P}}^2,G).$$ If $(2^n,k)=2^{n-1}$, then $k=2^{n-1}t$ for some odd number $t$. Writing $t=2s+1$ gives . Since $\partial'_k\simeq q^*k\partial_1\simeq q^*(2^ns+2^{n-1})\partial_1\simeq q^*2^{n-1}\partial_1$ and $\partial'_1$ has order $2^{n-1}$, $\partial'_k$ is null homotopic and we have $${\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)\simeq G\times{\textrm{Map}}^*_0({\mathbb{C}}{\mathbb{P}}^2,G).$$ The same is true for ${\mathcal{G}}_l({\mathbb{C}}{\mathbb{P}}^2)$ and hence ${\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)\simeq{\mathcal{G}}_l({\mathbb{C}}{\mathbb{P}}^2)$.
Plan for the proofs of Theorems \[thm\_main thm\] and \[thm\_necessary condition\]
==================================================================================
From this section onward, we will focus on $SU(n)$-gauge groups over ${\mathbb{C}}{\mathbb{P}}^2$. There is a fibration $$\label{exact seq_U fibration}
SU(n)\longrightarrow SU(\infty)\overset{p}{\longrightarrow}W_n,$$ where $p:SU(\infty)\to W_n$ is the projection and $W_n$ is the symmetric space $SU(\infty)/SU(n)$. Then we have $$\begin{aligned}
{\tilde{H}^{*}}(SU(\infty))&=&\Lambda(x_3,\cdots,x_{2n-1},\cdots),\\
{\tilde{H}^{*}}(SU(n))&=&\Lambda(x_3,\cdots,x_{2n-1}),\\
{\tilde{H}^{*}}(BSU(n))&=&{\mathbb{Z}}[c_2,\cdots,c_n],\\
{\tilde{H}^{*}}(W_n)&=&\Lambda(\bar{x}_{2n+1},\bar{x}_{2n+3},\cdots),\end{aligned}$$ where $x_{2n+1}$ has degree $2n+1$, $c_i$ is the $i^{\text{th}}$ universal Chern class and $x_{2i+1}=\sigma(c_{i+1})$ is the image of $c_{i+1}$ under the cohomology suspension $\sigma$, and $p^*(\bar{x}_{2i+1})=x_{2i+1}$. Furthermore, $H^{2n}(\Omega W_n)\cong{\mathbb{Z}}$ and $H^{2n+2}(\Omega W_n)\cong{\mathbb{Z}}$ are generated by $a_{2n}$ and $a_{2n+2}$, where $a_{2i}$ is the transgression of $x_{2i+1}$.
The $(2n+4)$-skeleton of $W_n$ is $\Sigma^{2n-1}{\mathbb{C}}{\mathbb{P}}^2$ for $n$ odd, and is $S^{2n+3}\vee S^{2n+1}$ for $n$ even, so its homotopy groups are as follows: $$\label{table_pi Wn}
\begin{array}{c|c c c c}
&\multicolumn{4}{c}{\pi_i(W_n)}\\ \hline
i &\leq 2n &2n+1 &2n+2 &2n+3\\ \hline
n\text{ odd} &0 &{\mathbb{Z}}&0 &{\mathbb{Z}}\\
n\text{ even} &0 &{\mathbb{Z}}&{\mathbb{Z}}/2{\mathbb{Z}}&{\mathbb{Z}}\oplus{\mathbb{Z}}/2{\mathbb{Z}}\\ \end{array}$$
The canonical map $\epsilon:\Sigma{\mathbb{C}}{\mathbb{P}}^{n-1}\to SU(n)$ induces the inclusion of the generating set. Let $C$ be the quotient ${\mathbb{C}}{\mathbb{P}}^{n-1}/{\mathbb{C}}{\mathbb{P}}^{n-3}$ and let $\bar{q}:\Sigma{\mathbb{C}}{\mathbb{P}}^{n-1}\to\Sigma C$ be the quotient map. Then there is a diagram $$\xymatrix{
[\Sigma C, SU(n)]\ar[r]^-{(\partial'_k)_*}\ar[d]^-{\bar{q}^*} &[\Sigma C, {\textrm{Map}}^*({\mathbb{C}}{\mathbb{P}}^2,BSU(n))]\ar[r]\ar[d]^-{\bar{q}^*} &[\Sigma C, B{\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)]\ar[d]^-{\bar{q}^*}\\
[\Sigma{\mathbb{C}}{\mathbb{P}}^{n-1}, SU(n)]\ar[r]^-{(\partial'_k)_*} &[\Sigma{\mathbb{C}}{\mathbb{P}}^{n-1}, {\textrm{Map}}^*({\mathbb{C}}{\mathbb{P}}^2,BSU(n))]\ar[r] &[\Sigma{\mathbb{C}}{\mathbb{P}}^{n-1}, B{\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)],
}$$ where $(\partial'_k)_*$ sends $f$ to $\partial'_k\circ f$ and the rows are induced by fibration (\[fib\_Gk(CP2)\]). In particular, in the second row the map $\epsilon:\Sigma{\mathbb{C}}{\mathbb{P}}^{n-1}\to SU(n)$ is sent to $(\partial'_k)_*(\epsilon)=\partial'_k\circ\epsilon$. In Section 4, we use unstable $K$-theory to calculate the order of $\partial'_1\circ\epsilon$, giving a lower bound on the order of $\partial'_1$. Furthermore, in [@HK06] Hamanaka and Kono considered an exact sequence similar to the first row to give a necessary condition for ${\mathcal{G}}_k(S^4)\simeq{\mathcal{G}}_l(S^4)$. In Section 5 we follow the same approach and use the first row to give a necessary condition for ${\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)\simeq{\mathcal{G}}_l({\mathbb{C}}{\mathbb{P}}^2)$.
We remark that it is difficult to use only one of the two rows to prove both Theorems \[thm\_main thm\] and \[thm\_necessary condition\]. On the one hand, $\partial'_1\circ\epsilon$ factors through a map $\bar{\partial}:\Sigma C\to{\textrm{Map}}^*({\mathbb{C}}{\mathbb{P}}^2, BSU(n))$. There is no obvious method to show that $\bar{\partial}$ and $\partial'_1\circ\epsilon$ have the same orders except direct calculation. Therefore we cannot compare the orders of $\bar{\partial}$ and $\partial'_1$ to prove Theorem \[thm\_main thm\] without calculating the order of $\partial'_1\circ\epsilon$. On the other hand, applying the method used in Section 5 to the second row gives a much weaker conclusion than Theorem \[thm\_necessary condition\]. This is because $[\Sigma C,B{\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)]$ is a much smaller group than $[\Sigma{\mathbb{C}}{\mathbb{P}}^{n-1},B{\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)]$ and much information is lost by the map $\bar{q}^*$.
A lower bound on the order of $\partial'_1$
===========================================
The restriction of $\partial_1$ to $\Sigma{\mathbb{C}}{\mathbb{P}}^{n-1}$ is $\partial_1\circ\epsilon$, which is the triple adjoint of the composition $${\langle{\imath,\epsilon}\rangle}:S^3\wedge\Sigma{\mathbb{C}}{\mathbb{P}}^{n-1}\overset{\imath\wedge\epsilon}{\longrightarrow}SU(n)\wedge SU(n)\overset{{\langle{{\mathds{1}},{\mathds{1}}}\rangle}}{\longrightarrow}SU(n).$$ Since $SU(n)\simeq\Omega BSU(n)$, we can further take its adjoint and get $$\rho:\Sigma S^3\wedge\Sigma{\mathbb{C}}{\mathbb{P}}^{n-1}\overset{\Sigma\imath\wedge\epsilon}{\longrightarrow}\Sigma SU(n)\wedge SU(n)\overset{[ev,ev]}{\longrightarrow}BSU(n),$$ where $[ev,ev]$ is the Whitehead product of the evaluation map $$ev:\Sigma SU(n)\simeq\Sigma\Omega BSU(n)\to BSU(n)$$ with itself. Similarly, the restriction $\partial'_1\circ\epsilon$ is adjoint to the composition $$\rho':{\mathbb{C}}{\mathbb{P}}^2\wedge\Sigma{\mathbb{C}}{\mathbb{P}}^{n-1}\overset{q\wedge{\mathds{1}}}{\longrightarrow}S^4\wedge\Sigma{\mathbb{C}}{\mathbb{P}}^{n-1}\overset{\Sigma\imath\wedge\epsilon}{\longrightarrow}\Sigma SU(n)\wedge SU(n)\overset{[ev,ev]}{\longrightarrow}BSU(n).$$ Since we will frequently refer to the facts established in [@HK03; @HK06], it is easier to follow their setting and consider its adjoint $$\gamma=\tau(\rho'\circ T):{\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1}\to SU(n),$$ where $T:\Sigma{\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1}\to{\mathbb{C}}{\mathbb{P}}^2\wedge\Sigma{\mathbb{C}}{\mathbb{P}}^{n-1}$ is the swapping map and $\tau:[\Sigma{\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1},BSU(n)]\to[{\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1},SU(n)]$ is the adjunction. By adjunction, the orders of $\partial'_1\circ\epsilon,\rho'$ and $\gamma$ are the same. We will calculate the order of $\gamma$ using unstable $K$-theory to prove Theorem \[thm\_main thm\].
Apply $[{\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1},-]$ to fibration (\[exact seq\_U fibration\]) to obtain the exact sequence $${\tilde{K}}^0({\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1})\overset{p_*}{\longrightarrow}[{\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1},\Omega W_n]\longrightarrow[{\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1},SU(n)]\longrightarrow{\tilde{K}}^1({\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1}).$$ Since ${\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1}$ is a CW-complex with even dimensional cells, ${\tilde{K}}^1({\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1})$ is zero. First we identify the term $[{\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1},\Omega W_n]$.
\[lemma\_\[Sigma CP\^n-1,Omega W\_n\]\] We have the following:
- $[\Sigma^{2n-4}{\mathbb{C}}{\mathbb{P}}^2,\Omega W_n]\cong{\mathbb{Z}}$;
- $[\Sigma^{2n-3}{\mathbb{C}}{\mathbb{P}}^2,\Omega W_n]=0$ for $n$ odd;
- $[\Sigma^{2n-2}{\mathbb{C}}{\mathbb{P}}^2,\Omega W_n]\cong{\mathbb{Z}}\oplus{\mathbb{Z}}$.
First, apply $[\Sigma^{2n-4}-, \Omega W_n]$ to cofibration (\[cofib\_CP2\]) to obtain the exact sequence $$\pi_{2n}(W_n)\longrightarrow\pi_{2n+1}(W_n)\longrightarrow[\Sigma^{2n-4}{\mathbb{C}}{\mathbb{P}}^2,\Omega W_n]\longrightarrow\pi_{2n-1}(W_n).$$ We refer to Table (\[table\_pi Wn\]) freely for the homotopy groups of $W_n$. Since $\pi_{2n-1}(W_n)$ and $\pi_{2n}(W_n)$ are zero, $[\Sigma^{2n-4}{\mathbb{C}}{\mathbb{P}}^{n-1},\Omega W_n]$ is isomorphic to $\pi_{2n+1}(W_n)\cong{\mathbb{Z}}$.
Second, apply $[\Sigma^{2n-3}-, \Omega W_n]$ to (\[cofib\_CP2\]) to obtain $$\pi_{2n+2}(W_n)\longrightarrow[\Sigma^{2n-3}{\mathbb{C}}{\mathbb{P}}^2,\Omega W_n]\longrightarrow\pi_{2n}(W_n).$$ Since $\pi_{2n}(W_n)$ and $\pi_{2n+2}(W_n)$ are zero for $n$ odd, so is $[\Sigma^{2n-3}{\mathbb{C}}{\mathbb{P}}^2,\Omega W_n]$.
Third, apply $[\Sigma^{2n-2}-, \Omega W_n]$ to (\[cofib\_CP2\]) to obtain $$\pi_{2n+2}(W_n)\overset{\eta_1}{\longrightarrow}\pi_{2n+3}(W_n)\longrightarrow[\Sigma^{2n-2}{\mathbb{C}}{\mathbb{P}}^2,\Omega W_n]\overset{j}{\longrightarrow}\pi_{2n+1}(W_n)\overset{\eta_2}{\longrightarrow}\pi_{2n+2}(W_n),$$ where $\eta_1$ and $\eta_2$ are induced by Hopf maps $\Sigma^{2n}\eta:S^{2n+3}\to S^{2n+2}$ and , and $j$ is induced by the inclusion $S^{2n+1}\hookrightarrow\Sigma^{2n-2}{\mathbb{C}}{\mathbb{P}}^2$ of the bottom cell. When $n$ is odd, $\pi_{2n+2}(W_n)$ is zero and $\pi_{2n+1}(W_n)$ and $\pi_{2n+3}(W_n)$ are ${\mathbb{Z}}$, so $[\Sigma^{2n-2}{\mathbb{C}}{\mathbb{P}}^{n-1},\Omega W_n]$ is ${\mathbb{Z}}\oplus{\mathbb{Z}}$. When $n$ is even, the $(2n+4)$-skeleton of $W_n$ is $S^{2n+1}\vee S^{2n+3}$. The inclusions $$\begin{array}{c c c}
i_1:S^{2n+1}\to S^{2n+1}\vee S^{2n+3}
&\text{and}
&i_2:S^{2n+3}\to S^{2n+1}\vee S^{2n+3}
\end{array}$$ generate $\pi_{2n+1}(W_n)$ and the ${\mathbb{Z}}$-summand of $\pi_{2n+3}(W_n)$, and the compositions $$\begin{array}{c c c}
j_1:S^{2n+2}\overset{\Sigma^{2n-1}\eta}{\longrightarrow}S^{2n+1}\overset{i_1}{\longrightarrow}W_n
&\text{and}
&j_2:S^{2n+3}\overset{\Sigma^{2n}\eta}{\longrightarrow}S^{2n+2}\overset{\Sigma^{2n-1}\eta}{\longrightarrow}S^{2n+1}\overset{i_1}{\longrightarrow}W_n
\end{array}$$ generate $\pi_{2n+2}(W_n)$ and the ${\mathbb{Z}}/2{\mathbb{Z}}$-summand of $\pi_{2n+3}(W_n)$ respectively. Since $\eta_1$ sends $j_1$ to $j_2$, the cokernel of $\eta_1$ is ${\mathbb{Z}}$. Similarly, $\eta_2$ sends $i_1$ to $j_1$, so $\eta_2:{\mathbb{Z}}\to{\mathbb{Z}}/2{\mathbb{Z}}$ is surjective. This implies the preimage of $j$ is a ${\mathbb{Z}}$-summand. Therefore $[\Sigma^{2n-2}{\mathbb{C}}{\mathbb{P}}^2,\Omega W_n]\cong{\mathbb{Z}}\oplus{\mathbb{Z}}$.
Let $C$ be the quotient ${\mathbb{C}}{\mathbb{P}}^{n-1}/{\mathbb{C}}{\mathbb{P}}^{n-3}$. Since $\Omega W_n$ is $(2n-1)$-connected, $[{\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1},\Omega W_n]$ is isomorphic to $[{\mathbb{C}}{\mathbb{P}}^2\wedge C,\Omega W_n]$ which is easier to determine.
\[lemma\_\[CP CP\] is free\] The group $[{\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1},\Omega W_n]\cong[{\mathbb{C}}{\mathbb{P}}^2\wedge C,\Omega W_n]$ is isomorphic to ${\mathbb{Z}}^{\oplus3}$.
When $n$ is even, $C$ is $S^{2n-2}\vee S^{2n-4}$. By Lemma \[lemma\_\[Sigma CP\^n-1,Omega W\_n\]\], $[{\mathbb{C}}{\mathbb{P}}^2\wedge C, \Omega W_n]$ is $[\Sigma^{2n-2}{\mathbb{C}}{\mathbb{P}}^2, \Omega W_n]\oplus[\Sigma^{2n-4}{\mathbb{C}}{\mathbb{P}}^2, \Omega W_n]\cong{\mathbb{Z}}^{\oplus3}$.
When $n$ is odd, $C$ is $\Sigma^{2n-6}{\mathbb{C}}{\mathbb{P}}^2$. Apply $[\Sigma^{2n-6}{\mathbb{C}}{\mathbb{P}}^2\wedge-, \Omega W_n]$ to cofibration (\[cofib\_CP2\]) to obtain the exact sequence $$[\Sigma^{2n-3}{\mathbb{C}}{\mathbb{P}}^2,\Omega W_n]\longrightarrow[\Sigma^{2n-2}{\mathbb{C}}{\mathbb{P}}^2,\Omega W_n]\longrightarrow[\Sigma^{2n-6}{\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^2,\Omega W_n]\longrightarrow$$ $$\longrightarrow[\Sigma^{2n-4}{\mathbb{C}}{\mathbb{P}}^2,\Omega W_n]\longrightarrow[\Sigma^{2n-3}{\mathbb{C}}{\mathbb{P}}^2,\Omega W_n]$$ By Lemma \[lemma\_\[Sigma CP\^n-1,Omega W\_n\]\], the first and the last terms $[\Sigma^{2n-3}{\mathbb{C}}{\mathbb{P}}^2, \Omega W_n]$ are zero, while the second term $[\Sigma^{2n-2}{\mathbb{C}}{\mathbb{P}}^2, \Omega W_n]$ is ${\mathbb{Z}}\oplus{\mathbb{Z}}$ and the fourth $[\Sigma^{2n-4}{\mathbb{C}}{\mathbb{P}}^2, \Omega W_n]$ is ${\mathbb{Z}}$. Therefore is ${\mathbb{Z}}^{\oplus3}$.
Define $a:[{\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1}, \Omega W_n]\to H^{2n}({\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1})\oplus H^{2n+2}({\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1})$ to be a map sending $f\in[{\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1}, \Omega W_n]$ to $a(f)=f^*(a_{2n})\oplus f^*(a_{2n+2})$. The cohomology class $\bar{x}_{2n+1}$ represents a map $\bar{x}_{2n+1}:W_n\to K({\mathbb{Z}},2n+1)$ and $a_{2n}=\sigma(\bar{x}_{2n+1})$ represents its loop . Similarly $a_{2n+2}=\sigma(\bar{x}_{2n+3})$ represents a loop map. This implies $a$ is a group homomorphism. Furthermore, $a_{2n}$ and $a_{2n+2}$ induce isomorphisms between $H^i(\Omega W_n)$ and $H^i(K(2n,{\mathbb{Z}})\times K(2n+2,{\mathbb{Z}}))$ for $i=2n$ and $2n+2$. Since is a free ${\mathbb{Z}}$-module by Lemma \[lemma\_\[CP CP\] is free\], $a$ is a monomorphism. Consider the diagram $$\label{digm_injective Phi}
{\footnotesize\xymatrix{
{\tilde{K}}^0({\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1})\ar[r]^-{p_*}\ar@{=}[d] &[{\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1}, \Omega W_n]\ar[r]\ar[d]^-{a} &[{\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1}, SU(n)]\ar[d]^-{b}\ar[r] &0\\
{\tilde{K}}^0({\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1})\ar[r]^-{\Phi} &H^{2n}({\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1})\oplus H^{2n+2}({\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1})\ar[r]^-{\psi} &Coker(\Phi)\ar[r] &0
}}$$ In the left square, $\Phi$ is defined to be $a\circ p^*$. In the right square, $\psi$ is the quotient map and $b$ is defined as follows. Any $f\in[{\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1},SU(n)]$ has a preimage $\tilde{f}$ and $b(f)$ is defined to be $\psi(a(\tilde{f}))$. An easy diagram chase shows that $b$ is well-defined and injective. Since $b$ is injective, the order of $\gamma\in[{\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1}, SU(n)]$ equals the order of $b(\gamma)\in Coker(\Phi)$. In [@HK03], Hamanaka and Kono gave an explicit formula for $\Phi$.
\[thm\_Phi\] For any $f\in{\tilde{K}}^0(Y)$, we have $$\Phi(f)=n!ch_{2n}(f)\oplus(n+1)!ch_{2n+2}(f),$$ where $ch_{2i}(f)$ is the $2i^{\text{th}}$ part of $ch(f)$.
Let $u$ and $v$ be the generators of $H^2({\mathbb{C}}{\mathbb{P}}^2)$ and $H^2({\mathbb{C}}{\mathbb{P}}^{n-1})$. For $1\leq i\leq n-1$, denote $L_i$ and $L'_i$ as the generators of ${\tilde{K}}^0({\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1})$ with Chern characters $ch(L_i)=u^2(e^v-1)^i$ and $ch(L'_i)=(u+\frac{1}{2}u^2)\cdot(e^v-1)^i$. By Theorem \[thm\_Phi\] we have $$\begin{aligned}
\Phi(L_i)
&=&n(n-1)A_iu^2v^{n-2}+n(n+1)B_iu^2v^{n-1},\\
\Phi(L'_i)&=&\frac{n(n-1)}{2}A_iu^2v^{n-2}+nB_iuv^{n-1}+\frac{n(n+1)}{2}B_iu^2v^{n-1},\end{aligned}$$ where $$\begin{array}{c c c}
A_i=\sum^i_{j=1}(-1)^{i+j}\binom{i}{j}j^{n-2}
&\text{and}
&B_i=\sum^i_{j=1}(-1)^{i+j}\binom{i}{j}j^{n-1}.
\end{array}$$
Write an element $xu^2v^{n-2}+yuv^{n-1}+zu^2v^{n-1}\in H^{2n}({\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1})\oplus H^{2n+2}({\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1})$ as $(x, y, z)$. Then the coordinates of $\Phi(L_i)$ and $\Phi(L'_i)$ are $(n(n-1)A_i, 0, n(n+1)B_i)$ and $(\frac{n(n-1)}{2}A_i, nB_i, \frac{n(n+1)}{2}B_i)$ respectively.
\[lemma\_simplify span of Im Phi\] For $n\geq3$, $Im(\Phi)$ is spanned by $(\frac{n(n-1)}{2},n,\frac{n(n+1)}{2})$, $(n(n-1),0,0)$ and $(0,2n,0)$.
By definition, $Im(\Phi)=span\{\Phi(L_i),\Phi(L'_i)\}^{n-1}_{i=1}$. For $i=1$, $A_1=B_1=1$. Then $$\begin{aligned}
\Phi(L_1)
&=&(n(n-1), 0, n(n+1))\\
&=&2(\frac{1}{2}n(n-1), n, \frac{1}{2}n(n+1))-(0,2n,0)\\
&=&2\Phi(L'_1)-(0,2n,0)\end{aligned}$$ Equivalently $(0,2n,0)=2\Phi(L'_1)-\Phi(L_1)$, so $span\{\Phi(L_1),\Phi(L'_1)\}=span\{\Phi(L'_1),(0,2n,0)\}$. For other $i$’s, $$\begin{aligned}
\Phi(L_i)
&=&(n(n-1)A_i, 0, n(n+1)B_i)\\
&=&2(\frac{1}{2}n(n-1)A_i, nB_i, \frac{1}{2}n(n+1)B_i)-(0,2nB_i,0)\\
&=&2\Phi(L'_i)-B_i(0,2n,0)\end{aligned}$$ is a linear combination of $\Phi(L'_i)$ and $(0, 2n, 0)$, so $Im(\Phi)=span\{\Phi(L'_1),\cdots,\Phi(L'_{n-1}),(0,2n,0)\}$.
We claim that $span\{\Phi(L'_i)\}^{n-1}_{i=1}=span\{\Phi(L'_1), (n(n-1),0,0)\}$. Observe that $$\begin{aligned}
\Phi(L'_i)
&=&(\frac{n(n-1)}{2}A_i,nB_i,\frac{n(n+1)}{2}B_i)\\
&=&(\frac{n(n-1)}{2}B_i,nB_i,\frac{n(n+1)}{2}B_i)+(\frac{n(n-1)}{2}(A_i-B_i),0,0)\\
&=&B_i\Phi(L'_1)+\frac{A_i-B_i}{2}\cdot(n(n-1),0,0).\end{aligned}$$ The difference $$\begin{aligned}
A_i-B_i
&=&\sum^i_{j=1}(-1)^{i+j}\binom{i}{j}j^{n-2}-\sum^i_{j=1}(-1)^{i+j}\binom{i}{j}j^{n-1}\\
&=&\sum^i_{j=1}(-1)^{i+j+1}\binom{i}{j}(j^{n-1}-j^{n-2})\\
&=&\sum^i_{j=1}(-1)^{i+j+1}\binom{i}{j}(j-1)j^{n-2}\end{aligned}$$ is even since each term $(j-1)j^{n-2}$ is even and $n\geq3$. Therefore $\frac{A_i-B_i}{2}$ is an integer and $\Phi(L'_i)$ is a linear combination of $\Phi(L'_1)$ and $(n(n-1),0,0)$.
Furthermore, $$\begin{aligned}
\Phi(L'_2)
&=&B_2\Phi(L'_1)+(A_2-B_2)(\frac{n(n-1)}{2},0,0)\\
&=&B_2\Phi(L'_1)-2^{n-3}(n(n-1),0,0)\end{aligned}$$ and $$\begin{aligned}
\Phi(L'_3)
&=&B_3\Phi(L'_1)+(A_3-B_3)(\frac{n(n-1)}{2},0,0)\\
&=&B_3\Phi(L'_1)-(3^{n-2}-3\cdot2^{n-3})(n(n-1),0,0).\end{aligned}$$ Since $2^{n-3}$ and $3^{n-2}-3\cdot2^{n-3}$ are coprime to each other, there exist integers $s$ and $t$ such that $2^{n-3}s+(3^{n-2}-3\cdot2^{n-3})t=1$ and $$(n(n-1),0,0)=(sB_2+tB_3)\Phi(L'_1)-s\Phi(L'_2)-t\Phi(L'_3).$$ Therefore $(n(n-1),0,0)$ is a linear combination of $\Phi(L'_1),\Phi(L'_2)$ and $\Phi(L'_3)$. This implies $span\{\Phi(L'_1),(n(n-1),0,0)\}=span\{\Phi(L'_i)\}^{n-1}_{i=1}$.
Combine all these together to obtain $$\begin{aligned}
Im(\Phi)
&=&span\{\Phi(L_i),\Phi(L'_i)\}^{n-1}_{i=1}\\
&=&span\{\Phi(L'_1),(n(n-1),0,0),(0,2n, 0)\}\\
&=&span\{(\frac{n(n-1)}{2},n,\frac{n(n+1)}{2}),(n(n-1),0,0),(0,2n,0)\}.\end{aligned}$$
Back to diagram (\[digm\_injective Phi\]). The map $\gamma$ has a lift $\tilde{\gamma}:{\mathbb{C}}{\mathbb{P}}^2\wedge{\mathbb{C}}{\mathbb{P}}^{n-1}\to\Omega W_n$. By exactness, the order of $\gamma$ equals the minimum number $m$ such that $m\tilde{\gamma}$ is contained in $Im(p_*)$. Since $a$ and $b$ are injective, the order of $\gamma$ equals the minimum number $m'$ such that $m'a(\tilde{\gamma})$ is contained in $Im(\Phi)$.
\[lemma\_lifting\] Let $\alpha:\Sigma X\to SU(n)$ be a map for some space $X$. If $\alpha':{\mathbb{C}}{\mathbb{P}}^2\wedge X\to SU(n)$ is the adjoint of the composition $${\mathbb{C}}{\mathbb{P}}^2\wedge\Sigma X\overset{q\wedge{\mathds{1}}}{\longrightarrow}\Sigma S^3\wedge\Sigma X\overset{\Sigma\imath\wedge\alpha}{\longrightarrow}\Sigma SU(n)\wedge SU(n)\overset{[ev,ev]}{\longrightarrow}BSU(n),$$ then there is a lift $\tilde{\alpha}$ of $\alpha'$ such that $\tilde{\alpha}^*(a_{2i})=u^2\otimes\Sigma^{-1}\alpha^*(x_{2i-3})$, where $\Sigma$ is the cohomology suspension isomorphism. $$\xymatrix{
&\Omega W_n\ar[d]\\
{\mathbb{C}}{\mathbb{P}}^2\wedge X\ar[r]^-{\alpha'}\ar@{-->}[ur]^-{\tilde{\alpha}} &SU(n)
}$$
In [@HK03; @HK06], Hamanaka and Kono constructed a lift $\Gamma:\Sigma SU(n)\wedge SU(n)\to W_n$ of $[ev,ev]$ such that $\Gamma^*(\bar{x}_{2i+1})=\sum_{j+k=i-1}\Sigma x_{2j+1}\otimes x_{2k+1}$. Let $\tilde{\Gamma}$ be the composition $$\tilde{\Gamma}:{\mathbb{C}}{\mathbb{P}}^2\wedge\Sigma X\overset{q\wedge{\mathds{1}}}{\longrightarrow}\Sigma S^3\wedge\Sigma X\overset{\Sigma\imath\wedge\alpha}{\longrightarrow}\Sigma SU(n)\wedge SU(n)\overset{\Gamma}{\longrightarrow}W_n.$$ Then we have $$\begin{aligned}
\tilde{\Gamma}^*(\bar{x}_{2i+1})
&=&(q\wedge{\mathds{1}})^*(\Sigma\imath\wedge\alpha)^*\Gamma^*(\bar{x}_{2i+1})\\
&=&(q\wedge{\mathds{1}})^*(\Sigma\imath\wedge\alpha)^*\left(\sum_{j+k=i-1}\Sigma x_{2j+1}\otimes x_{2k+1}\right)\\
&=&(q\wedge{\mathds{1}})^*(\Sigma u_3\otimes\alpha^*(x_{2i-3}))\\
&=&u^2\otimes\alpha^*(x_{2i-3}),\end{aligned}$$ where $u_3$ is the generator of $H^3(S^3)$.
Let $T:\Sigma{\mathbb{C}}{\mathbb{P}}^2\wedge X\to{\mathbb{C}}{\mathbb{P}}^2\wedge\Sigma X$ be the swapping map and let $\tau:[\Sigma{\mathbb{C}}{\mathbb{P}}^2\wedge X,W_n]\to[{\mathbb{C}}{\mathbb{P}}^2\wedge X,\Omega W_n]$ be the adjunction. Take $\tilde{\alpha}:{\mathbb{C}}{\mathbb{P}}^2\wedge X\to\Omega W_n$ to be the adjoint of $\tilde{\Gamma}$, that is $\tilde{\alpha}=\tau(\tilde{\Gamma}\circ T)$. Then $\tilde{\alpha}$ is a lift of $\alpha'$. Since $$(\tilde{\Gamma}\circ T)^*(\bar{x}_{2i+1})=T^*\circ\tilde{\Gamma}^*(\bar{x}_{2i+1})=T^*(u^2\otimes\alpha^*(x_{2i-3}))=\Sigma u^2\otimes\Sigma^{-1}\alpha^*(x_{2i-3}),$$ we have $\tilde{\alpha}^*(a_{2i})=u^2\otimes\Sigma^{-1}\alpha^*(x_{2i-3})$.
\[lemma\_tilde gamma coordinate\] In diagram (\[digm\_injective Phi\]), $\gamma$ has a lift $\tilde{\gamma}$ such that $a(\tilde{\gamma})=u^2v^{n-2}\oplus u^2v^{n-1}$.
Recall that $\gamma$ is the adjoint of the composition $$\rho':{\mathbb{C}}{\mathbb{P}}^2\wedge\Sigma{\mathbb{C}}{\mathbb{P}}^{n-1}\overset{q\wedge{\mathds{1}}}{\longrightarrow}\Sigma S^3\wedge{\mathbb{C}}{\mathbb{P}}^{n-1}\overset{\Sigma\imath\wedge\epsilon}{\longrightarrow}\Sigma SU(n)\wedge SU(n)\overset{[ev,ev]}{\longrightarrow}BSU(n).$$ Now we use Lemma \[lemma\_lifting\] and take $\alpha$ to be $\epsilon:\Sigma{\mathbb{C}}{\mathbb{P}}^{n-1}\to SU(n)$. Then $\gamma$ has a lift $\tilde{\gamma}$ such that $\tilde{\gamma}^*(a_{2i})=u^2\otimes\Sigma^{-1}\epsilon^*(x_{2i-3})=u^2\otimes v^{i-2}$. This implies $$a(\tilde{\gamma})=\tilde{\gamma}^*(a_{2n})\oplus\tilde{\gamma}^*(a_{2n+2})=u^2v^{n-2}\oplus u^2v^{n-1}.$$
Now we can calculate the order of $\partial'_1\circ\epsilon$, which gives a lower bound on the order of $\partial'_1$.
\[thm\_order of gamma\] When $n\geq3$, the order of $\partial'_1\circ\epsilon$ is $\frac{1}{2}n(n^2-1)$ for $n$ odd and $n(n^2-1)$ for $n$ even.
Since $\partial'_1\circ\epsilon$ is adjoint to $\gamma$ , it suffices to calculate the order of $\gamma$. By Lemma \[lemma\_simplify span of Im Phi\], $Im(\Phi)$ is spanned by $(\frac{1}{2}n(n-1),n,\frac{1}{2}n(n+1)),(n(n-1),0,0)$ and $(0,2n,0)$. By Lemma \[lemma\_tilde gamma coordinate\], $a(\tilde{\gamma})$ has coordinates $(1,0,1)$. Let $m$ be a number such that $ma(\tilde{\gamma})$ is contained in $Im(\Phi)$. Then $$m(1, 0, 1)=s(\frac{1}{2}n(n-1),n,\frac{1}{2}n(n+1))+t(n(n-1),0,0)+r(0,2n,0)$$ for some integers $s, t$ and $r$. Solve this to get $$\begin{array}{c c c}
m=\frac{1}{2}tn(n^2-1),
&s=-2r,
&s=t(n-1).
\end{array}$$ Since $s=-2r$ is even, the smallest positive value of $t$ satisfying $s=t(n-1)$ is 1 for $n$ odd and 2 for $n$ even. Therefore $m$ is $\frac{1}{2}n(n^2-1)$ for $n$ odd and $n(n^2-1)$ for $n$ even.
For $SU(n)$-gauge groups over $S^4$, the order $m$ of $\partial_1$ has the form $m=n(n^2-1)$ for $n=3$ and $5$ [@HK06; @theriault15]. If $p$ is an odd prime and $n<(p-1)^2+1$, then $m$ and $n(n^2-1)$ have the same $p$-components [@KKT14; @theriault17]. These facts suggest it may be the case that $m=n(n^2-1)$ for any $n>2$. In fact, one can follow the method Hamanaka and Kono used in [@HK06] and calculate the order of $\partial\circ\epsilon$ to obtain a lower bound $n(n^2-1)$ for $n$ odd. However, it does not work for the $n$ even case since $[S^4\wedge{\mathbb{C}}{\mathbb{P}}^{n-1},\Omega W_n]$ is not a free ${\mathbb{Z}}$-module. An interesting corollary of Theorem \[thm\_order of gamma\] is to give a lower bound on the order of $\partial_1$ for $n$ even.
When $n$ is even and greater than 2, the order of $\partial_1$ is at least $n(n^2-1)$.
The order of $\partial'_1\circ\epsilon$ is a lower bound on the order of $\partial'_1$, which is either the same as or half of the order of $\partial_1$ by Lemma \[lemma\_partial’\]. The corollary follows from Theorem \[thm\_order of gamma\].
A necessary condition for ${\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)\simeq{\mathcal{G}}_l({\mathbb{C}}{\mathbb{P}}^2)$
========================================================================================================================
In this section we follow the approach in [@HK06] to prove Theorem \[thm\_necessary condition\]. The techniques used are similar to that in Section 4, except we are working with the quotient $\Sigma C=\Sigma{\mathbb{C}}{\mathbb{P}}^{n-1}/\Sigma{\mathbb{C}}{\mathbb{P}}^{n-1}$ instead of $\Sigma{\mathbb{C}}{\mathbb{P}}^{n-1}$. When $n$ is odd, $C$ is $\Sigma^{2n-6}{\mathbb{C}}{\mathbb{P}}^2$, and when $n$ is even, $C$ is $S^{2n-2}\vee S^{2n-4}$. Apply $[\Sigma C,-]$ to fibration (\[fib\_Gk(CP2)\]) to obtain the exact sequence $$[\Sigma C, SU(n)]\overset{(\partial'_k)_*}{\longrightarrow}[\Sigma C, {\textrm{Map}}^*_0({\mathbb{C}}{\mathbb{P}}^2,BSU(n))]\longrightarrow[\Sigma C, B{\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)]\longrightarrow[\Sigma C, BSU(n)],$$ where $(\partial'_k)_*$ sends $f\in[\Sigma C, SU(n)]$ to $\partial'_k\circ f\in[\Sigma C, {\textrm{Map}}^*_0({\mathbb{C}}{\mathbb{P}}^2, BSU(n))]$. Since $BSU(n)\to BSU(\infty)$ is a $2n$-equivalence and $\Sigma C$ has dimension $2n-1$, $[\Sigma C, BSU(n)]$ is ${\tilde{K}}^0(\Sigma C)$ which is zero. Similarly, $[\Sigma C, SU(n)]\cong[\Sigma^2C, BSU(n)]$ is ${\tilde{K}}^0(\Sigma^2C)\cong{\mathbb{Z}}\oplus{\mathbb{Z}}$. Furthermore, by adjunction we have $[\Sigma C, {\textrm{Map}}^*_0({\mathbb{C}}{\mathbb{P}}^2,BSU(n))]\cong[\Sigma C\wedge{\mathbb{C}}{\mathbb{P}}^2, BSU(n)]$. The exact sequence becomes $$\label{exact seq_necessary Sigma C}
{\tilde{K}}^0(\Sigma^2C)\overset{(\partial'_k)_*}{\longrightarrow}[\Sigma C\wedge{\mathbb{C}}{\mathbb{P}}^2, BSU(n)]\longrightarrow[\Sigma C, B{\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)]\longrightarrow0.$$ This implies $[\Sigma C, B{\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)]\cong[C, {\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)]$ is $Coker(\partial'_k)_*$. Also, apply $[{\mathbb{C}}{\mathbb{P}}^2\wedge C, -]$ to fibration (\[exact seq\_U fibration\]) to obtain the exact sequence $$\label{exact seq_necessary CP2 C}
[{\mathbb{C}}{\mathbb{P}}^2\wedge C, \Omega SU(\infty)]\overset{p_*}{\longrightarrow}[{\mathbb{C}}{\mathbb{P}}^2\wedge C, \Omega W_n]\longrightarrow[{\mathbb{C}}{\mathbb{P}}^2\wedge C, SU(n)]\longrightarrow[{\mathbb{C}}{\mathbb{P}}^2\wedge C, SU(\infty)].$$
Observe that $[{\mathbb{C}}{\mathbb{P}}^2\wedge C, \Omega SU(\infty)]\cong{\tilde{K}}^0({\mathbb{C}}{\mathbb{P}}^2\wedge C)$ is ${\mathbb{Z}}^{\oplus4}$ and is zero. Combine exact sequences (\[exact seq\_necessary Sigma C\]) and (\[exact seq\_necessary CP2 C\]) to obtain the diagram $$\xymatrix{
&{\tilde{K}}^0({\mathbb{C}}{\mathbb{P}}^2\wedge C)\ar[d]^-{p_*}\ar[dr]^-{\Phi} & &\\
&[{\mathbb{C}}{\mathbb{P}}^2\wedge C, \Omega W_n]\ar[d]\ar[r]^-{a} &H^{2n}({\mathbb{C}}{\mathbb{P}}^2\wedge C)\oplus H^{2n+2}({\mathbb{C}}{\mathbb{P}}^2\wedge C) &\\
{\tilde{K}}^0(\Sigma^2C)\ar[r]^-{(\partial'_k)_*} &[{\mathbb{C}}{\mathbb{P}}^2\wedge C, SU(n)]\ar[r]\ar[d] &[C, B{\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)]\ar[r] &0\\
&0 & &
}$$ where $a(f)=f^*(a_{2n})\oplus f^*(a_{2n+2})$ for any $f\in[{\mathbb{C}}{\mathbb{P}}^2\wedge C, \Omega W_n]$, and $\Phi$ is defined to be $a\circ p_*$. By Lemma \[lemma\_\[CP CP\] is free\] $[{\mathbb{C}}{\mathbb{P}}^2\wedge C,\Omega W_n]$ is free. Following the same argument in Section 4 implies the injectivity of $a$.
Our strategy to prove Theorem \[thm\_necessary condition\] is as follows. If ${\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)$ is homotopy equivalent to ${\mathcal{G}}_l({\mathbb{C}}{\mathbb{P}}^2)$, then $[C, {\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)]\cong[C, {\mathcal{G}}_l({\mathbb{C}}{\mathbb{P}}^2)]$ and exactness in (\[exact seq\_necessary CP2 C\]) implies that $Im(\partial'_k)_*$ and $Im(\partial'_l)_*$ have the same order in $[{\mathbb{C}}{\mathbb{P}}^2\wedge C, SU(n)]$, resulting in a necessary condition for a homotopy equivalence ${\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)\simeq{\mathcal{G}}_l({\mathbb{C}}{\mathbb{P}}^2)$. To calculate the order of $Im(\partial'_k)_*$, we will find a preimage $\tilde{\partial}_k$ of $Im(\partial'_k)_*$ in $[{\mathbb{C}}{\mathbb{P}}^2\wedge C,\Omega W_n]$. Since $a$ is injective, we can embed $\tilde{\partial}_k$ into $H^{2n}({\mathbb{C}}{\mathbb{P}}^2\wedge C)\oplus H^{2n+2}({\mathbb{C}}{\mathbb{P}}^2\wedge C)$ and work out the order of $Im(\partial'_k)_*$ there.
Let $u, v_{2n-4}$ and $v_{2n-2}$ be generators of $H^2({\mathbb{C}}{\mathbb{P}}^2)$, $H^{2n-4}(C)$ and $H^{2n-2}(C)$. Then we write an element $xu^2v_{2n-4}+yuv_{2n-2}+zu^2v_{2n-2}\in H^{2n}({\mathbb{C}}{\mathbb{P}}^2\wedge C)\oplus H^{2n+2}({\mathbb{C}}{\mathbb{P}}^2\wedge C)$ as $(x, y, z)$. First we need to find the submodule $Im(a)$.
\[lemma\_im lambda\] For $n$ odd, $Im(a)$ is $\{(x, y, z)|x+y\equiv z\pmod{2}\}$, and for $n$ even, $Im(a)$ is $\{(x, y, z)|y\equiv0\pmod{2}\}$.
When $n$ is odd, $C$ is $\Sigma^{2n-6}{\mathbb{C}}{\mathbb{P}}^2$ and the $(2n+3)$-skeleton of $\Omega W_n$ is $\Sigma^{2n-2}{\mathbb{C}}{\mathbb{P}}^2$. To say $(x, y, z)\in Im(a)$ means there exists $f\in[{\mathbb{C}}{\mathbb{P}}^2\wedge C,\Omega W_n]$ such that $$\label{equation in proof of lemma Im a}
\begin{array}{c c c}
f^*(a_{2n})=xu^2v_{2n-4}+yuv_{2n-2}
&\text{and}
&f^*(a_{2n+2})=zu^2v_{2n-2}.
\end{array}$$ Reducing to homology with ${\mathbb{Z}}/2{\mathbb{Z}}$-coefficients, we have $$\begin{array}{c c c}
Sq^2(u)=u^2,
&Sq^2(v_{2n-4})=v_{2n-2},
&Sq^2(a_{2n})=a_{2n+2}.
\end{array}$$ Apply $Sq^2$ to (\[equation in proof of lemma Im a\]) to get $x+y\equiv z\pmod{2}$. Therefore $Im(a)$ is contained in $\{(x, y, z)|x+y\equiv z\pmod{2}\}$. To show that they are equal, we need to show that $(1, 0, 1),(0, 1, 1)$ and $(0, 0, 2)$ are in $Im(a)$. Consider maps $$\begin{array}{l}
f_1:{\mathbb{C}}{\mathbb{P}}^2\wedge C\overset{q_1}{\longrightarrow}S^4\wedge C\simeq\Sigma^{2n-2}{\mathbb{C}}{\mathbb{P}}^2\hookrightarrow\Omega W_n\\
f_2:{\mathbb{C}}{\mathbb{P}}^2\wedge C\overset{q_2}{\longrightarrow}{\mathbb{C}}{\mathbb{P}}^2\wedge S^{2n-2}\hookrightarrow\Omega W_n\\
f_3:{\mathbb{C}}{\mathbb{P}}^2\wedge C\overset{q_3}{\longrightarrow}S^{2n+2}\overset{\theta}{\longrightarrow}\Omega W_n
\end{array}$$ where $q_1,q_2$ and $q_3$ are quotient maps and $\theta$ is the generator of $\pi_{2n+3}(W_n)$. Their images are $$\begin{array}{c c c}
a(f_1)=(1, 0, 1)
&a(f_2)=(0, 1, 1)
&a(f_3)=(0, 0, 2)
\end{array}$$ respectively, so $Im(a)=\{(x, y, z)|x+y\equiv z\pmod{2}\}$.
When $n$ is even, $C$ is $S^{2n-2}\vee S^{2n-4}$ and the $(2n+3)$-skeleton of $\Omega W_n$ is $S^{2n+2}\vee S^{2n}$. Reducing to homology with ${\mathbb{Z}}/2{\mathbb{Z}}$-coefficients, $Sq^2(v_{2n-4})=0$ and $Sq^2(a_{2n})=0$. Apply $Sq^2$ to (\[equation in proof of lemma Im a\]) to get $y\equiv 0\pmod{2}$. Therefore $Im(a)$ is contained in $\{(x, y, z)|y\equiv0\pmod{2}\}$. To show that they are equal, we need to show that $(1, 0, 0),(0, 2, 0)$ and $(0, 0, 1)$ are in $Im(a)$. The maps $$\begin{array}{l}
f'_1:{\mathbb{C}}{\mathbb{P}}^2\wedge C\overset{q'_1}{\longrightarrow}S^4\wedge(S^{2n-2}\vee S^{2n-4})\overset{p_1}{\longrightarrow}S^4\wedge S^{2n-4}\hookrightarrow\Omega W_n\\
f'_2:{\mathbb{C}}{\mathbb{P}}^2\wedge C\overset{q'_2}{\longrightarrow}S^4\wedge(S^{2n-2}\vee S^{2n-4})\overset{p_2}{\longrightarrow}S^4\wedge S^{2n-2}\hookrightarrow\Omega W_n
\end{array}$$ where $q'_1$ and $q'_2$ are quotient maps and $p_1$ and $p_2$ are pinch maps, have images $a(f'_1)=(1, 0, 0)$ and $a(f'_2)=(0, 0, 1)$. To find $(0, 2, 0)$, apply $[-\wedge S^{2n-2},\Omega W_n]$ to cofibration (\[cofib\_CP2\]) to obtain the exact sequence $$\pi_{2n+3}(W_n)\longrightarrow[{\mathbb{C}}{\mathbb{P}}^2\wedge S^{2n-2},\Omega W_n]\overset{i^*}{\longrightarrow}\pi_{2n+1}(W_n)\overset{\eta^*}{\longrightarrow}\pi_{2n+2}(W_n)$$ where $i^*$ is induced by the inclusion $i:S^2\hookrightarrow{\mathbb{C}}{\mathbb{P}}^2$ and $\eta^*$ is induced by Hopf map $\eta$. The third term $\pi_{2n+1}(W_n)\cong{\mathbb{Z}}$ is generated by $i':S^{2n+1}\to W_n$, the inclusion of the bottom cell, and the fourth term $\pi_{2n+2}(W_n)\cong{\mathbb{Z}}/2{\mathbb{Z}}$ is generated by $i'\circ\eta$, so $\eta^*:{\mathbb{Z}}\to{\mathbb{Z}}/2{\mathbb{Z}}$ is a surjection. By exactness $[{\mathbb{C}}{\mathbb{P}}^2\wedge S^{2n-2},\Omega W_n]$ has a ${\mathbb{Z}}$-summand with the property that $i^*$ sends its generator $g$ to $2i'$. Therefore the composition $$f'_3:{\mathbb{C}}{\mathbb{P}}^2\wedge(S^{2n-2}\vee S^{2n-4})\overset{pinch}{\longrightarrow}{\mathbb{C}}{\mathbb{P}}^2\wedge S^{2n-2}\overset{g}{\longrightarrow}\Omega W_n$$ has image $(0,2,0)$. It follows that $Im(a)=\{(x, y, z)|y\equiv0\pmod{2}\}$.
Now we split into the $n$ odd and $n$ even cases to calculate the order of $Im(\partial'_k)_*$.
The order of $Im(\partial'_k)_*$ for $n$ odd
--------------------------------------------
When $n$ is odd, $C$ is $\Sigma^{2n-6}{\mathbb{C}}{\mathbb{P}}^2$. First we find $Im(\Phi)$ in $Im(a)$. For $1\leq i\leq4$, let $L_i$ be the generators of ${\tilde{K}}^0({\mathbb{C}}{\mathbb{P}}^2\wedge C)\cong{\mathbb{Z}}^{\oplus 4}$ with Chern characters $$\begin{array}{l l}
ch(L_1)=(u+\frac{1}{2}u^2)\cdot(v_{2n-4}+\frac{1}{2}v_{2n-2})
&ch(L_2)=(u+\frac{1}{2}u^2)v_{2n-2}\\[15pt]
ch(L_3)=u^2(v_{2n-4}+\frac{1}{2}v_{2n-2})
&ch(L_4)=u^2v_{2n-2}.
\end{array}$$ By Theorem \[thm\_Phi\], we have $$\begin{aligned}
\Phi(L_1)&=&\frac{n!}{2}u^2v_{2n-4}+\frac{n!}{2}uv_{2n-2}+\frac{(n+1)!}{4}u^2v_{2n-2}\\
\Phi(L_2)&=&n!uv_{2n-2}+\frac{(n+1)!}{2}u^2v_{2n-2}\\
\Phi(L_3)&=&n!u^2v_{2n-4}+\frac{(n+1)!}{2}u^2v_{2n-2}\\
\Phi(L_4)&=&(n+1)!u^2v_{2n-2}.\end{aligned}$$
By Lemma \[lemma\_im lambda\], $Im(a)$ is spanned by $(1, 0, 1), (0, 1, 1)$ and $(0, 0, 2)$. Under this basis, the coordinates of the $\Phi(L_i)$’s are $$\begin{array}{l l}
\Phi(L_1)=(\frac{n!}{2},\frac{n!}{2},\frac{(n-3)\cdot n!}{8}),
&\Phi(L_2)=(0,n!,\frac{(n-1)\cdot n!}{4}),\\[15pt]
\Phi(L_3)=(n!,0,\frac{(n-1)\cdot n!}{4}),
&\Phi(L_4)=(0,0,\frac{(n+1)!}{2}).
\end{array}$$ We represent their coordinates by the matrix $$M_{\Phi}=L
\begin{pmatrix}
\frac{n(n-1)}{2} &\frac{n(n-1)}{2} &\frac{n(n-1)(n-3)}{8}\\
0 &n(n-1) &\frac{n(n-1)^2}{4}\\
n(n-1) &0 &\frac{n(n-1)^2}{4}\\
0 &0 &\frac{n(n^2-1)}{2}
\end{pmatrix},$$ where $L=(n-2)!$. Then $Im(\Phi)$ is spanned by the row vectors of $M_{\Phi}$.
Next, we find a preimage of $Im(\partial'_k)_*$ in $[{\mathbb{C}}{\mathbb{P}}^2\wedge C,\Omega W_n]$. In exact sequence (\[exact seq\_necessary Sigma C\]) ${\tilde{K}}^0(\Sigma^2C)$ is ${\mathbb{Z}}\oplus{\mathbb{Z}}$. Let $\alpha_1$ and $\alpha_2$ be its generators with Chern classes $$\begin{array}{l l}
c_{n-1}(\alpha_1)=(n-2)!\Sigma^2v_{2n-4} &c_n(\alpha_1)=\frac{(n-1)!}{2}\Sigma^2v_{2n-2}\\
c_{n-1}(\alpha_2)=0 &c_n(\alpha_2)=(n-1)!\Sigma^2v_{2n-2}.
\end{array}$$
\[lemma\_lift of alpha\] For $i=1,2$, $\xi_k(\alpha_i)$ has a lift $\tilde{\alpha}_{i,k}:{\mathbb{C}}{\mathbb{P}}^2\wedge C\to\Omega W_n$ such that $$a(\tilde{\alpha}_{i,k})=ku^2\otimes\Sigma^{-2}c_{n-1}(\alpha_i)\oplus ku^2\otimes\Sigma^{-2}c_{n}(\alpha_i).$$
For dimension and connectivity reasons, $\alpha_i:\Sigma^2C\to BSU(\infty)$ lifts through $BSU(n)\to BSU(\infty)$. Label the lift $\Sigma^2C\to BSU(n)$ by $\alpha_i$ as well. Let $\alpha'_i:\Sigma C\to SU(n)$ be the adjoint of $\alpha_i$. Then $(\partial'_k)_*(\alpha_i)$ is the adjoint of the composition $${\mathbb{C}}{\mathbb{P}}^2\wedge\Sigma C\overset{q\wedge{\mathds{1}}}{\longrightarrow}\Sigma S^3\wedge\Sigma C\overset{\Sigma k\imath\wedge\alpha'_i}{\longrightarrow}\Sigma SU(n)\wedge SU(n)\overset{[ev,ev]}{\longrightarrow}BSU(n).$$ By Lemma \[lemma\_lifting\], $(\partial'_k)_*(\alpha_i)$ has a lift $\tilde{\alpha}_{i,k}$ such that $\tilde{\alpha}_{i,k}^*(a_{2j})=ku^2\otimes\Sigma^{-1}(\alpha')^*(x_{2j-3})$. Since $\sigma(c_{j-1})=x_{2j-3}$, we have $\tilde{\alpha}_{i,k}^*(a_{2j})=ku^2\otimes\Sigma^{-2}c_{j-1}(\alpha_i)$ and $$a(\tilde{\alpha}_{i,k})=ku^2\otimes\Sigma^{-2}c_{n-1}(\alpha_i)\oplus ku^2\otimes\Sigma^{-2}c_{n}(\alpha_i).$$
By Lemma \[lemma\_lift of alpha\], $(\partial'_k)_*(\alpha_1)$ and $(\partial'_k)_*(\alpha_2)$ have lifts $$\begin{array}{c c c}
\tilde{\alpha}_{1,k}=(n-2)!ku^2v_{2n-4}+\frac{(n-1)!}{2}ku^2v_{2n-2}
&\text{and}
&\tilde{\alpha}_{2,k}=(n-1)!ku^2v_{2n-2}.
\end{array}$$ We represent their coordinates by the matrix $$M_{\partial}=kL
\begin{pmatrix}
1 &0 &\frac{n-3}{4}\\
0 &0 &\frac{n-1}{2}
\end{pmatrix}.$$ Let $\tilde{\partial}_k=span\{\tilde{\alpha}_{1,k}, \tilde{\alpha}_{2,k}\}$ be the preimage of $Im(\partial'_k)_*$ in $[{\mathbb{C}}{\mathbb{P}}^2\wedge C,\Omega W_n]$. Then $\tilde{\partial}_k$ is spanned by the row vectors of $M_{\partial}$.
\[lemma\_im xi 4m+3\] When $n$ is odd, the order of $Im(\partial'_k)_*$ is $$|Im(\partial'_k)_*|=\frac{\frac{1}{2}n(n^2-1)}{(\frac{1}{2}n(n^2-1), k)}\cdot\frac{n}{(n,k)}.$$
Suppose $n=4m+3$ for some integer $m$. Then $$M_{\Phi}=(4m+3)L
\begin{pmatrix}
2m+1 &2m+1 &2m^2+m\\
0 &4m+2 &4m^2+4m+1\\
4m+2 &0 &4m^2+4m+1\\
0 &0 &8m^2+12m+4
\end{pmatrix}$$ and $$M_{\partial}=kL
\begin{pmatrix}
1 &0 &m\\
0 &0 &2m+1
\end{pmatrix}.$$ Transform $M_{\Phi}$ into Smith normal form $$A\cdot M_{\Phi}\cdot B=(4m+3)L
\begin{pmatrix}
(2m+1) & &\\
&(2m+1) &\\
& &(2m+1)(4m+4)\\
& &0
\end{pmatrix},$$ where $$\begin{array}{c c c}
A=
\begin{pmatrix}
1 &0 &0 &0\\
-2 &0 &1 &0\\
4m+2 &1 &-(2m+1) &0\\
4 &-2 &-2 &1
\end{pmatrix}
&\text{and}
&B=
\begin{pmatrix}
1 &-m &-(2m+1)\\
0 &0 &1\\
0 &1 &2
\end{pmatrix}.
\end{array}$$ The matrix $B$ represents a basis change in $Im(a)$ and $A$ represents a basis change in $Im(\Phi)$. Therefore $[{\mathbb{C}}{\mathbb{P}}^2\wedge C, SU(n)]$ is isomorphic to $$\frac{{\mathbb{Z}}}{\frac{1}{2}(4m+3)!{\mathbb{Z}}}\oplus\frac{{\mathbb{Z}}}{\frac{1}{2}(4m+3)!{\mathbb{Z}}}\oplus\frac{{\mathbb{Z}}}{\frac{1}{2}(4m+4)!{\mathbb{Z}}}.$$ We need to find the representation of $\tilde{\partial}_k$ under the new basis represented by $B$. The new coordinates of $\tilde{\alpha}_{1,k}$ and $\tilde{\alpha}_{2,k}$ are the row vectors of the matrix $$M_{\partial}
\cdot
\begin{pmatrix}
1 &-m &-(2m+1)\\
0 &0 &1\\
0 &1 &2
\end{pmatrix}
=
\begin{pmatrix}
kL &0 &-kL\\
0 &(2m+1)kL &(4m+2)kL
\end{pmatrix}.$$ Apply row operations to get $$\begin{pmatrix}
1 &0\\
4m+2 &1
\end{pmatrix}
\cdot
\begin{pmatrix}
kL &0 &-kL\\
0 &(2m+1)kL &(4m+2)kL
\end{pmatrix}
=
\begin{pmatrix}
kL &0 &-kL\\
(4m+2)kL &(2m+1)kL &0
\end{pmatrix}.$$ Let $\mu=(kL,0,-kL)$ and $\nu=((4m+2)kL, (2m+1)kL,0)$. Then $$\tilde{\partial}_k=\{x\mu+y\nu\in[{\mathbb{C}}{\mathbb{P}}^2\wedge C,\Omega W_n]|x, y\in{\mathbb{Z}}\}.$$ If $x\mu+y\nu$ and $x'\mu+y'\nu$ are the same in $Im(\Phi)$, then we have $$\left\{
\begin{array}{r c l l}
xkL+(4m+2)ykL &\equiv &x'kL+(4m+2)y'kL &\pmod{(2m+1)(4m+3)L}\\
(2m+1)ykL &\equiv &(2m+1)y'kL &\pmod{(2m+1)(4m+3)L}\\
xkL &\equiv &x'kL &\pmod{(2m+1)(4m+3)(4m+4)L}
\end{array}\right.$$ These conditions are equivalent to $$\left\{
\begin{array}{r c l l}
xk &\equiv &x'k &\pmod{(2m+2)(4m+3)(4m+2)}\\
yk &\equiv &y'k &\pmod{(4m+3)}
\end{array}\right.$$ This implies that there are $\displaystyle{\frac{(2m+2)(4m+3)(4m+2)}{((2m+2)(4m+3)(4m+2), k)}}$ distinct values of $x$ and $\displaystyle{\frac{4m+3}{(4m+3,k)}}$ distinct values of $y$, so we have $$|Im(\partial'_k)_*|=\frac{(2m+2)(4m+3)(4m+2)}{((2m+2)(4m+3)(4m+2), k)}\cdot\frac{4m+3}{(4m+3,k)}.$$
When $n=4m+1$, we can repeat the calculation above to obtain $$|Im(\partial'_k)_*|=\frac{2m(4m+2)(4m+1)}{(2m(4m+2)(4m+1), k)}\cdot\frac{4m+1}{(4m+1,k)}.$$
The order of $Im(\partial'_k)_*$ for $n$ even
---------------------------------------------
When $n$ is even, $C$ is $S^{2n-2}\vee S^{2n-4}$. For $1\leq i\leq4$, let $L_i$ be the generators of ${\tilde{K}}^0({\mathbb{C}}{\mathbb{P}}^2\wedge C)\cong{\mathbb{Z}}^{\oplus 4}$ with Chern characters $$\begin{array}{l l}
ch(L_1)=(u+\frac{1}{2}u^2)v_{2n-4}
&ch(L_2)=u^2v_{2n-4}\\[15pt]
ch(L_3)=(u+\frac{1}{2}u^2)v_{2n-2}
&ch(L_4)=u^2v_{2n-2}.
\end{array}$$ By Theorem \[thm\_Phi\], we have $$\begin{aligned}
\Phi(L_1)&=&\frac{n!}{2}u^2v_{2n-4}\\
\Phi(L_2)&=&n!u^2v_{2n-4}\\
\Phi(L_3)&=&n!uv_{2n-2}+\frac{(n+1)!}{2}u^2v_{2n-2}\\
\Phi(L_4)&=&(n+1)!u^2v_{2n-2}.\end{aligned}$$
By Lemma \[lemma\_im lambda\], $Im(a)$ is spanned by $(1, 0, 0), (0, 2, 0)$ and $(0, 0, 1)$. Under this basis, the coordinates of the $\Phi(L_i)$’s are $$\begin{array}{l l}
\Phi(L_1)=(\frac{n!}{2},0,0),
&\Phi(L_2)=(n!,0,0),\\[15pt]
\Phi(L_3)=(0,\frac{n!}{2},\frac{(n+1)!}{2}),
&\Phi(L_4)=(0,0,(n+1)!).
\end{array}$$ We represent the coordinates of $\Phi(L_i)$’s by the matrix $$M_{\Phi}=\frac{n(n-1)}{2}L
\begin{pmatrix}
1 &0 &0\\
2 &0 &0\\
0 &1 &n+1\\
0 &0 &2n+2
\end{pmatrix}$$ Then $Im(\Phi)$ is spanned by the row vectors of $M_{\Phi}$.
In exact sequence (\[exact seq\_necessary Sigma C\]) ${\tilde{K}}^0(\Sigma^2C)$ is ${\mathbb{Z}}\oplus{\mathbb{Z}}$. Let $\alpha_1$ and $\alpha_2$ be its generators with Chern classes $$\begin{array}{l l}
c_{n-1}(\alpha_1)=(n-2)!\Sigma^2v_{2n-4} &c_n(\alpha_1)=0\\
c_{n-1}(\alpha_2)=0 &c_n(\alpha_2)=(n-1)!\Sigma^2v_{2n-2}.
\end{array}$$ By Lemma \[lemma\_lift of alpha\], $(\partial'_k)_*(\alpha_1)$ and $(\partial'_k)_*(\alpha_2)$ have lifts $$\begin{array}{c c c}
\tilde{\alpha}_{1, k}=(n-2)!ku^2v_{2n-4}
&\text{and}
&\tilde{\alpha}_{2, k}=(n-1)!ku^2v_{2n-2}.
\end{array}$$ We represent their coordinates by a matrix $$M_{\partial}=kL
\begin{pmatrix}
1 &0 &0\\
0 &0 &n-1
\end{pmatrix}.$$ Then the preimage $\tilde{\partial}_k=span\{\tilde{\alpha}_{1, k}, \tilde{\alpha}_{2, k}\}$ of $Im(\partial'_k)_*$ is spanned by the row vectors of $M_{\partial}$. We calculate as in the proof of Lemma \[lemma\_im xi 4m+3\] to obtain the following lemma.
\[lemma\_im xi even\] When $n$ is even, the order of $Im(\partial'_k)_*$ is $$|Im(\partial'_k)_*|=\frac{\frac{1}{2}n(n-1)}{(\frac{1}{2}n(n-1), k)}\cdot\frac{n(n+1)}{(n(n+1),k)}.$$
Proof of Theorem \[thm\_necessary condition\]
---------------------------------------------
Before comparing the orders of $Im(\partial'_k)_*$ and $Im(\partial'_k)_*$, we prove a preliminary lemma.
\[lemma\_gcd prime argument\] Let $n$ be an even number and let $p$ be a prime. Denote the $p$-component of $t$ by $\nu_p(t)$. If there are integers $k$ and $l$ such that $$\nu_p(\frac{1}{2}n,k)\cdot\nu_p(n,k)=\nu_p(\frac{1}{2}n,l)\cdot\nu_p(n,l),$$ then $\nu_p(n, k)=\nu_p(n,l)$.
Suppose $p$ is odd. If $p$ does not divide $n$, then $\nu_p(n,k)=\nu_p(n,l)=1$, so the lemma holds. If $p$ divides $n$, then $\nu_p(\frac{1}{2}n,k)=\nu_p(n,k)$. The hypothesis becomes $\nu_p(n,k)^2=\nu_p(n,l)^2$, implying that $\nu_p(n, k)=\nu_p(n,l)$.
Suppose $p=2$. Let $\nu_2(n)=2^r$, $\nu_2(k)=2^t$ and $\nu_2(l)=2^s$. Then the hypothesis implies $$\label{eqn_nu lemma}
min(r-1,t)+min(r,t)=min(r-1,s)+min(r,s).$$ To show $\nu_2(n,k)=\nu_2(n,l)$, we need to show $min(r,t)=min(r,s)$. Consider the following cases: (1) $t,s\geq r$, (2) $t,s\leq r-1$, (3) $t\leq r-1,s\geq r$ and (4) $s\leq r-1,t\geq r$.
Case (1) obviously gives $min(r,t)=min(r,s)$. In case (2), when $t,s\leq r-1$, equation (\[eqn\_nu lemma\]) implies $2t=2s$. Therefore $t=s$ and $min(r,t)=min(r,s)$.
It remains to show cases (3) and (4). For case (3) with $t\leq r-1,s\geq r$, equation (\[eqn\_nu lemma\]) implies $$2t=min(r-1, s)+r.$$ Since $s\geq r$, $min(r-1,s)=r-1$ and the right hand side is $2r-1$ which is odd. However, the left hand side is even, leading to a contradiction. This implies that this case does not satisfy the hypothesis. Case (4) is similar. Therefore $\nu_2(n, k)=\nu_2(n, l)$ and the asserted statement follows.
In exact sequence (\[exact seq\_necessary Sigma C\]), $[C,{\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)]$ is $Coker(\partial'_k)_*$. By hypothesis, ${\mathcal{G}}_k({\mathbb{C}}{\mathbb{P}}^2)$ is homotopy equivalent to ${\mathcal{G}}_l({\mathbb{C}}{\mathbb{P}}^2)$, so $|Im(\partial'_k)_*|=|Im(\partial'_k)_*|$. The $n$ odd and $n$ even cases are proved similarly, but the even case is harder.
When $n$ is even, by Lemma \[lemma\_im xi even\] the order of $Im(\partial'_k)_*$ is $$|Im(\partial'_k)_*|=\frac{\frac{1}{2}n(n-1)}{(\frac{1}{2}n(n-1),k)}\cdot\frac{n(n+1)}{(n(n+1),k)},$$ so we have $$\label{eqn_necessary pf condition even}
(\frac{1}{2}n(n-1),k)\cdot(n(n+1),k)=(\frac{1}{2}n(n-1),l)\cdot(n(n+1),l).$$ We need to show that $$\label{eqn_necessry pf target even}
\nu_p(n(n^2-1),k)=\nu_p(n(n^2-1),l)$$ for all primes $p$. Suppose $p$ does not divide $\frac{1}{2}n(n^2-1)$. Equation (\[eqn\_necessry pf target even\]) holds since both sides are 1. Suppose $p$ divides $\frac{1}{2}n(n^2-1)$. Since $n-1$, $n$ and $n+1$ are coprime, $p$ divides only one of them. If $p$ divides $n-1$, then $\nu_p(\frac{1}{2}n,k)=\nu_p(n,k)=\nu_p(n+1,k)=1$. Equation (\[eqn\_necessary pf condition even\]) implies $\nu_p(n-1,k)=\nu_p(n-1,l)$. Since $$\nu_p(n(n^2-1),k)=\nu_p(n-1,k)\cdot\nu_p(n,k)\cdot\nu_p(n+1,k),$$ this implies equation (\[eqn\_necessry pf target even\]) holds. If $p$ divides $n+1$, then equation (\[eqn\_necessry pf target even\]) follows from a similar argument. If $p$ divides $n$, then equation (\[eqn\_necessary pf condition even\]) implies $\nu_p(\frac{1}{2}n,k)\cdot\nu_p(n,k)=\nu_p(\frac{1}{2}n,l)\cdot\nu_p(n,l)$. By Lemma \[lemma\_gcd prime argument\] $\nu_p(n, k)=\nu_p(n,l)$, so equation (\[eqn\_necessry pf target even\]) holds.
When $n$ is odd, by Lemma \[lemma\_im xi 4m+3\] the order of $Im(\partial'_k)_*$ is $$|Im(\partial'_k)_*|=\frac{\frac{1}{2}n(n^2-1)}{(\frac{1}{2}n(n^2-1),k)}\cdot\frac{n}{(n,k)},$$ so we have $$(\frac{1}{2}n(n^2-1),k)\cdot(n,k)=(\frac{1}{2}n(n^2-1),l)\cdot(n,l).$$ We can argue as above to show that for all primes $p$, $$\nu_p(\frac{1}{2}n(n^2-1),k)=\nu_p(\frac{1}{2}n(n^2-1),l).$$
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[^1]:
|
---
abstract: 'The idea of changing our energy system from a hierarchical design into a set of nearly independent microgrids becomes feasible with the availability of small renewable energy generators. The smart microgrid concept comes with several challenges in research and engineering targeting load balancing, pricing, consumer integration and home automation. In this paper we first provide an overview on these challenges and present approaches that target the problems identified. While there exist promising algorithms for the particular field, we see a missing integration which specifically targets smart microgrids. Therefore, we propose an architecture that integrates the presented approaches and defines interfaces between the identified components such as generators, storage, smart and devices.'
author:
- |
Anita Sobe, Wilfried Elmenreich\
Institute of Networked and Embedded Systems,\
Alpen-Adria Universität Klagenfurt, Austria\
anita.sobe@aau.at, wilfried.elmenreich@aau.at
bibliography:
- 'itg.bib'
title: 'Smart Microgrids: Overview and Outlook'
---
Introduction
============
The trend towards distributed renewable energy production leads to new challenges. Renewable energy sources typically rely on the weather and thus lead to variable energy production which is hard to manage. However, they are an important part of future smart grids and therefore there are a lot of efforts to make these sources more efficient. Future smart grids will not only have to integrate distributed renewable energy sources, but will also have to integrate information and communication technologies (ICT) for management and control [@Farhangi2010]. Currently, ICT integration is done by installing smart meters, which opens a wide area of new applications.
Future efforts target the increase of manageability and efficiency by dividing the smart grid into sub-systems [@Mohn2011a]. Such sub-systems are called *smart microgrids* and consist of energy consumers and producers at a small scale and are able to manage themselves. Examples for smart microgrids are households, villages, industry sites, or a university campus. A smart microgrid can either be connected to the backbone grid, to other microgrids or it can run in a so called island mode. Dynamic islanding is one of the main solutions to overcome faults and voltage sags [@Lasseter2011]. According to Mohn and Piasecky in [@Mohn2011a] smart microgrids need to be controlled on two levels, (1) analog-centric control for power stability and (2) digital-centric control for system automation. We are specifically interested in the second level, which is responsible for calculating the need for energy based on its price, reliability, and current system state.
In more detail we will give an overview on dynamic pricing, Smart Home automation in combination with Demand Response (DR) and power load balancing in island mode. These topics target improved reliability, better management of distributed resources, and higher power efficiency, but are typically isolated research efforts. We want to subsume these topics and strive to give an outlook on a holistic approach of smart microgrids.
Dynamic Pricing
===============
Consumer Integration and Home Automation
========================================
Load Balancing in Island Mode
=============================
Conclusion and Outlook
======================
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by Lakeside Labs GmbH, Klagenfurt, Austria and funding from the European Regional Development Fund and the Carinthian Economic Promotion Fund (KWF) under grant KWF-$20214|22935|24445$.
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abstract: 'We present a microscopic study on current generation in graphene in response to an electric field. While scattering is generally considered to reduce the current, we reveal that in graphene Auger processes give rise to a current enhancement via a phenomenon we denote dark carrier multiplication. Based on a microscopic approach, we show that, if other scattering channels are absent, this prevents the carrier distribution to reach a stationary value. Taking into account scattering with phonons a finite current is restored, however its value exceeds the stationary current without scattering.'
author:
- Roland Jago$^1$
- Florian Wendler$^2$
- Ermin Malic$^1$
title: 'Current enhancement due to field-induced dark carrier multiplication in graphene'
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Transport properties of graphene, in particular the exceptionally high electrical conductivity even at room temperature, have been intensively studied since its discovery [@Geim2007; @Morozov2008; @CastroNeto2009; @DasSarma2011Review]. In early graphene samples, electrical transport was limited by disorder resulting in mobilities of up to $\unit[20000]{cm^{2}V^{-1}s^{-1}}$ at low temperatures [@Novoselov2005; @Zhang2005; @Mariani2008]. However, it was demonstrated that eliminating the extrinsic disorder, the fundamental limit of the mobility at room-temperature is considerably higher [@Morozov2008; @Du2008; @Bolotin2008]. Depending on the sample, flexural or in-plane phonons are considered to be responsible for limiting the intrinsic conductivity [@Katsnelson2008; @Mariani2008; @Mariani2010_PHONONS; @Castro2010]. Moreover, in contrast to conventional materials with a parabolic band structure, carrier-carrier scattering has an impact on the current [@Kashuba2008; @Fritz2008; @Gornyi2012; @Sun2012]. While most studies in literature focus on the linear response and deploy the Drude approach for the conductivity, there are only a few studies addressing the non-linear response of graphene to an electric field [@Bistritzer2009; @Balev2009; @Dora2010; @Rosenstein2010; @Tani2012].
In this work, we provide a microscopic access to the time- and momentum-dependent carrier dynamics in graphene in a constant in-plane electric field. Using the density matrix formalism, we calculate all intrinsic carrier-phonon and carrier-carrier scattering channels within the second-order Born-Markov approximation. This allows us to investigate the temporal evolution of the carrier density as well as the generation and the dynamics of the electrical current giving new insights into carrier transport in graphene. In a many-particle process that we denote as dark carrier multiplication (dark CM), Coulomb-induced processes bridging the valence and the conduction band (Auger scattering) significantly increase the carrier density in response to the electric field. Signatures of this effect have already been found in near-infrared transient absorption measurements under high electric fields using THz excitation pulses [@Tani2012]. Furthermore, a carrier density increase has been discussed in literature as a consequence of the radiative coupling [@Balev2009] or the Schwinger mechanism [@Dora2010; @Rosenstein2010]. The latter two effects are much weaker at room temperature than Auger processes, which have been demonstrated to be extremely efficient in graphene [@Winzer2010_Multiplication; @Winzer2012; @Brida2013; @Ploetzing2014; @05_Mittendorff_Auger_NatPhys_2014; @02_Wendler_CM_NatCommun_2014; @Gierz2015]. The aim of this work is to investigate the impact of the field-induced dark CM on the generation and enhancement of electric currents in graphene.
![Carrier occupation near the Dirac point in response to an in-plane electric field. The blue filling represents the electron occupation, while the white region in the valence band describes the hole occupation. The electric field accelerates electrons and holes (orange arrows), which opens up relaxation channels including Auger (red arrows) and phonon-induced scattering (green arrows). []{data-label="fig:sketch"}](dark-fig2-scheme){width="1\columnwidth"}
When graphene is placed in an external in-plane electric field $\mathbf{E}$, its electrons are accelerated in the direction anti-parallel to the field which we set to be the negative x-direction $\mathbf{E}=(-E,0,0)$. This is illustrated in Fig. \[fig:sketch\] showing the Dirac cone of graphene including resulting Coulomb- and phonon-induced scattering processes. Since we consider neutral graphene characterized by a vanishing chemical potential, the electron-hole symmetry allows us to focus on the dynamics of electrons in the conduction band. Nevertheless, it is instructive to note that electrons and holes appearing on the same side of the Dirac cone, move into opposite directions in real space, since the electron group velocity is given by $\mathbf{v}_{\mathbf{k}}^{\lambda}=\lambda v_{\text{F}}\mathbf{k}/|\mathbf{k}|$ with the Fermi velocity $v_{\text{F}}$ and the band index $\lambda=\pm1$ denoting the conduction ($\lambda=+1$) and the valence ($\lambda=-1$) band, respectively. Therefore, electrons at $\mathbf{k}$ move into the $\lambda\mathbf{k}$-direction, and, since a hole is nothing else than a missing electron, it can be represented by an electron moving into the $-\lambda\mathbf{k}$-direction, which corresponds to a positively charged hole moving into the $\lambda\mathbf{k}$-direction. Hence, the group velocities of electrons and holes are the same, both changing sign when the band is switched. Consequently, the shift of electron and hole occupations to the right in k space, cf. Fig. \[fig:sketch\], means that electrons and holes move into opposite directions in real space.
![(a) Dynamics of the current density $j(t)$ (in field direction) in graphene subject to an in-plane dc electric field at room temperature. The black line represents the result in the absence of many-particle scattering processes and the dashed gray line illustrates the saturation current density given by Eq. (\[eq:j\_max\]). The red-shaded area shows the region, where current density enhancement takes place. (b) Equilibrium current density for varying electric fields $E$ (blue line). Here, the dashed red line is a fit to the linear region, where the slope corresponds to the conductivity $\sigma$. The black and yellow lines describe the current density without taking into account any scattering processes or neglecting the dark carrier multiplication (dCM), respectively. (c) Field-dependent dCM.[]{data-label="fig:current+CM(field)"}](dark-fig1-grad){width="1\columnwidth"}
As electrons are accelerated in the electric field, the magnitude of their velocity remains constant and only the direction of their motion changes. Consequently, if scattering channels are switched off, the current density saturates to a finite value corresponding to the situation, in which all electrons move into the direction anti-parallel to the field. The resulting current density dynamics exhibits an initial increase until the equilibrium current density is reached after a few picoseconds, cf. the black line in Fig. \[fig:current+CM(field)\] (a). While the time scale on which the saturation takes place is given by the field strength $E$, the saturation current density only depends on the initial temperature-dependent carrier density, cf. the black line in Fig. \[fig:current+CM(field)\](b).
The current density dynamics at room temperature including all scattering channels is shown in Fig. \[fig:current+CM(field)\](a) for different field strengths $E$. For $E<\unit[0.22]{V \mu m^{-1}}$, the current density saturates, as expected, below the value without scattering suggesting that a certain non-zero resistivity has been introduced. Surprisingly, we reveal that for sufficiently high $E$ many-particle scattering processes can even enhance the saturation current density, i.e. they introduce a scattering-induced current density amplification, cf. the red-shaded area in Fig. \[fig:current+CM(field)\](a). This can be traced back to Auger scattering, which give rise to a significant increase of the charge carrier density. This process bridging the valence and the conduction band (Fig.\[fig:sketch\](a)) corresponds to the creation of an additional electron-hole pairs which indirectly – via an enhanced carrier density – boosts the generated current density. In analogy to the regular carrier multiplication (CM) induced by an optical excitation [@Winzer2010_Multiplication], we define the field-induced dark CM by the ratio of the equilibrium carrier density with ($n$) and without ($n_{0}$) the electric field: $\text{dCM}=n/n_{0}$. The generated dark CM is shown in Fig. \[fig:current+CM(field)\](c) as a function of the electric field. Its increased efficiency explains the observed enhancement of the current density at high fields above the saturation value without scattering.
Figure \[fig:current+CM(field)\](b) demonstrates that Ohm’s law ${\bf j=\sigma{\bf E}}$ is valid in graphene for small fields, cf. the linear increase with a slope corresponding to a conductivity of $\sigma=\unit[1.1]{k\Omega^{-1}}$. This is larger than experimentally obtained values [@Novoselov2005; @Zhang2005; @Morozov2008], since we consider a perfect graphene sample and an idealized situation without any negative influences of the environment. To illustrate that the large conductivity results from the strong impact of the dark CM, we approximate the current density excluding the influence of the dark CM $j_{\text{no-dCM}}=j-j_{\text{dCM}}$, cf. the yellow line in Fig. \[fig:current+CM(field)\](b). This approximation is obtained using the relation $j=e_{0}n\, v_{\text{drift}}$ with the drift velocity $v_{\text{drift}}$ (average velocity of charged carriers). Now, the initial carrier density $n_{0}$ is separated from the dark CM-induced density $n_{\text{dCM}}$ resulting in $j_{\text{no-dCM}}=e_{0}n_{0}\, v_{\text{drift}}$. In the absence of dark CM, the current density is much smaller and shows only a minimal increase with the field strength. Most importantly, it always stays well below the saturation current density in the case without scattering.
The mobility $\mu$ can be estimated via the slope of the drift velocity $v_{\text{drift}}$ plotted over the electric field yielding in the linear region a value of $\mu\approx\unit[4,000]{cm^{2}V^{-1}s^{-1}}$. It cannot be inferred from the usual relation $\mu=\sigma/(e_{0}n)$, where $e_{0}$ is the elementary charge, since now the carrier density $n$ depends on the electric field due to the appearance of dark CM. Instead, using $j=e_{0}n\, v_{\text{drift}}$ and the generalized definitions of the mobility and the conductivity $\mu=\text{d}v_{\text{drift}}/\text{d}E$ and $\sigma=\text{d}j/\text{d}E$, respectively, the relation $\mu=\sigma/(e_{0}n)-(v_{\text{drift}}/n)(\text{d}n/\text{d}E)$ is found. The reason for the comparably small value of the mobility (despite the rather large conductivity), is the relatively high temperature ($T=\unit[300]{K}$) corresponding to a large carrier density, which is even enhanced by the dark CM.
Before we go further into detail of the carrier dynamics leading to the enhancement of the current density via many-particle scattering, we first briefly introduce the applied microscopic approach. The many-particle Hamilton operator $H=H_{\text{0}}+H_{\text{c-c}}+H_{\text{c-ph}}+H_{\text{c-f}}$ consists of the (i) free carrier and phonon contribution $H_{\text{0}}$, (ii) carrier-carrier $H_{\text{c-c}}$ and (iii) carrier-phonon $H_{\text{c-ph}}$ interaction accounting for Coulomb- and phonon-induced scattering, and (iv) the interaction of an external electric field with carriers. The latter contribution reads for electrons in second quantization [@Meier1994] $$\begin{aligned}
H_{\text{c-f}} & =-ie_{0}\mathbf{E}\cdot\sum_{\mathbf{k}}a_{\mathbf{k}\lambda}^{\dagger}\nabla{}_{\mathbf{k}}a_{\mathbf{k}\lambda},\label{cfH}\end{aligned}$$ where $a{}_{\mathbf{k}\lambda}^{\dagger}$, $a{}_{\mathbf{k}\lambda}$ denote creation and annihilation operators for electrons in the band $\lambda$ and with the momentum **$\mathbf{k}$**. Evaluating the Heisenberg’s equation of motion and applying the second-order Born-Markov approximation [@KochBuch; @Kira2006PQE; @MalicBuch], we obtain the graphene Bloch equations explicitly including the impact of an electric field: $$\begin{aligned}
\dot{\rho}_{\mathbf{k}}^{\lambda}(t) & =\Gamma_{\mathbf{k}\lambda}^{\text{in}}\,\big(1-\rho_{\mathbf{k}}^{\lambda}\big)-\Gamma_{\mathbf{k}\lambda}^{\text{out}}\,\rho_{\mathbf{k}}^{\lambda}-\frac{e_{0}}{\hbar}\mathbf{E}\cdot\nabla_{\mathbf{k}}\rho_{\mathbf{k}}^{\lambda},\label{eq:rho}\\[10pt]
\dot{n}_{\mathbf{q}}^{j}(t) & =\Gamma_{\mathbf{q}j}^{\text{em}}\,\big(n_{\mathbf{q}}^{j}+1\big)-\Gamma_{\mathbf{q}j}^{\text{abs}}\, n_{\mathbf{q}}^{j}-\gamma_{\text{ph}}\,\big(n_{\mathbf{q}}^{j}-n_{\mathbf{q},\text{B}}^{j}\big).\label{eq:n}\end{aligned}$$ The equations describe the coupled dynamics of the electron occupation probability $\rho_{\mathbf{k}}^{\lambda}=\langle a_{\mathbf{k}\lambda}^{\dagger}a^{\phantom{\dagger}}_{\mathbf{k}\lambda}\rangle$ in the state $(\mathbf{k},\lambda)$ and the phonon number $n_{\mathbf{q}}^{j}=\langle b_{\mathbf{q}j}^{\dagger}b^{\phantom{\dagger}}_{\mathbf{q}j}\rangle$ in the mode $j$ with the momentum $\mathbf{q}$. The time- and momentum-dependent in- and out-scattering rates $\Gamma_{\mathbf{k}\lambda}^{\text{in/out}}(t)$ contain all relevant carrier-carrier and carrier-phonon scattering channels including acoustic $\Gamma\text{LA}$, $\Gamma\text{TA}$ and optical $\Gamma\text{LO}$, $\Gamma\text{TO}$, KLO, KTO phonons. The dynamics of the phonon number $n_{\mathbf{q}}^{j}$ is determined by the emission and absorption rates $\Gamma_{\mathbf{q}j}^{\text{em/abs}}(t)$ driving $n_{\mathbf{q}}^{j}$ towards the initial Bose-distribution $n_{\mathbf{q},\text{B}}^{j}$. The phonon decay rate $\gamma_{\text{ph}}$ is adjusted to the experimentally obtained value [@Kang2010]. For more details on the microscopic origin of the rates, we refer to Refs. .
The presence of an electric field enters the Bloch equations through the scalar product between the electrical field ${\bf E}$ and the gradient $\nabla_{\mathbf{k}}$ of the carrier occupation $\rho_{\mathbf{k}}^{\lambda}$, cf. Eq. (\[eq:rho\]). It stems from the commutation of the carrier-field Hamilton operator from Eq. (\[cfH\]) with $\rho_{\mathbf{k}}^{\lambda}$. Within the applied Born approximation, the electric field does not influence the lattice positions, hence there is no electric field contribution in Eq. (\[eq:n\]). Using the coordinate transformation [@Meier1994] $\mathbf{k}\rightarrow\mathbf{k}-e_{0}\mathbf{E}\, t/\hbar$, one can transform the Bloch equations according to $d/dt\rightarrow d/dt-e_{0}\mathbf{E}\cdot\nabla_{\mathbf{k}}/\hbar$ resulting in regular Bloch equations without the electric field. Here, the field-induced dynamics is hidden in the new field- and time-dependent momenta $\mathbf{k}-e_{0}\mathbf{E}\, t/\hbar$. This means that we consider a moving coordinate frame describing the electric field-induced shift of the carrier occupation $\rho_{\mathbf{k}}^{\lambda}$ in momentum space. In case of a vanishing electric field $E=0$, the equilibrium distributions of electrons and phonons $\rho_{\mathbf{k}}^{\lambda}$ and $n_{\mathbf{q}}^{j}$ are given by Fermi-Dirac and Bose-Einstein distributions, respectively. When this equilibrium is disturbed by a field $E\neq0$, many-particle scattering occurs until new equilibrium distributions emerge. In conventional semiconductors, the carrier densities of the two equilibrium distributions (with and without the field) are the same. In contrast, strongly efficient Auger scattering in graphene results in a carrier density increase, which is quantified by the dark CM $n=\text{dCM}\cdot n_{0}$ [^1].
Numerical evaluation of the graphene Bloch equations provides a microscopic access to the coupled time- and momentum-resolved carrier and phonon dynamics under the influence of an electric field and its interplay with Coulomb- and phonon-induced scattering channels. The main observable in our study is the intraband current density $j(t)$ which reads under electron-hole symmetry [@MalicBuch] $$\begin{aligned}
\mathbf{j}(t) & =-\frac{8e_{0}}{L^{2}}\sum_{\mathbf{k}}\rho_{\mathbf{k}}^{\text{c}}(t)\frac{\nabla_{\mathbf{k}}\varepsilon_{\mathbf{k}}^{\text{c}}}{\hbar}=e_{0}n(t)\,\mathbf{v}_{\text{drift}}(t).\label{eq:j(t)}\end{aligned}$$ The appearing area of the graphene flake $L^{2}$ cancels out after performing the sum over all momenta ${\bf k}$. The factor 8 describes the spin and valley degeneracy of the electronic states as well as the electron-hole symmetry. The current density can be rephrased in terms of carrier density $n(t)=(8/L^{2})\sum_{\mathbf{k}}\rho_{\mathbf{k}}^{\text{c}}(t)$ and drift velocity $\mathbf{v}_{\text{drift}}(t)=-(8/L^{2}n(t))\sum_{\mathbf{k}}\rho_{\mathbf{k}}^{\text{c}}(t)\mathbf{v}_{\mathbf{k}}^{\text{c}}$, where $\mathbf{v}_{\mathbf{k}}^{\text{c}}=\nabla_{\mathbf{k}}\varepsilon_{\mathbf{k}}^{\text{c}}/\hbar$ is the electron group velocity. In conventional materials with a parabolic band structure with the effective mass $m$, the group velocity reads $\mathbf{v}_{\mathbf{k}}^{\text{c}}=\hbar\mathbf{k}/m$, whereas in graphene, it is given by $\mathbf{v}_{\mathbf{k}}^{\text{c}}=v_{\text{F}}\mathbf{e}_{\mathbf{k}}$ due to its linear bandstructure $\varepsilon_{\mathbf{k}}^{\text{c}}=\hbar v_{\text{F}}|\mathbf{k}|$. Consequently, only the angle between the electronic momentum $\mathbf{k}$ and the applied electrical field $\mathbf{E}$ determines the contribution of the carrier occupation $\rho_{\mathbf{k}}^{\text{c}}(t)$ to the current density in graphene, while the magnitude of $\mathbf{k}$ is irrelevant in this regard.
![Carrier occupation at different times (with respect to the switch-on of the electric field) for a field strength of $E=\unit[0.32]{V\mu m^{-1}}$ and a temperature of $T=\unit[300]{K}$ including (a) no scattering contributions, (b) only carrier-carrier scattering, (c) only carrier-phonon scattering, and (d) full carrier dynamics.[]{data-label="fig:occupations"}](dark-fig3-rho){width="1\columnwidth"}
The spectral distribution of the electron occupation $\rho_{\mathbf{k}}^{\text{c}}(t)$ is shown in Fig. \[fig:occupations\] at room temperature and along the $k_{x}$-direction (corresponding to the direction of the applied electric field) at different times and including different scattering channels. The case without scattering demonstrates that the in-plane electric field accelerates the available charge carriers in the k space into the opposite field direction ($-\mathbf{E}$), cf. Fig. \[fig:occupations\](a). Initially without the field, the carrier occupation is fully symmetric in $k$, hence the group velocities of the charge carriers point in different directions resulting in zero drift velocity and current density, cf. Eq. (\[eq:j(t)\]). Switching on the electric field (at $t=0$), the carriers are shifted resulting in an asymmetric carrier distribution and thus a current density is generated. It increases until the velocity of all charge carriers aligns with the field. The saturation current density sets in already at picosecond time scales and long before the charge carriers leave the linear region of the Brillouin zone, cf. the black line in Fig. \[fig:current+CM(field)\] (a). The generated saturation current density is independent of the strength of the electric field (Fig. \[fig:current+CM(field)\](b)) and scales with $T^{2}$ $$\begin{aligned}
j_{\text{sat}}=\frac{8e_{0}v_{F}}{L^{2}}\sum_{\mathbf{k}}\rho_{\mathbf{k},0}^{\text{c}}=e_{0}v_{F}n_{0}=\frac{e_{0}\pi k_{B}^{2}}{3\,\hbar^{2}v_{F}}T^{2},\label{eq:j_max}\end{aligned}$$ where $k_{B}$ is the Boltzmann constant.
Next, we include all carrier-carrier scattering channels and observe that the distribution is still slightly shifted with respect to the Dirac point and becomes spectrally broader due to Auger scattering, cf. Fig. \[fig:occupations\](b). While carrier-carrier scattering does not influence the current in conventional materials, in graphene its impact on the current is twofold: (i) it redistributes charge carriers and thereby induces a resistivity, i.e. current reduction, and (ii) it results in a carrier density increase via Auger processes (i.e. dark CM) giving rise to a current enhancement. While mechanism (i) leads to a carrier distribution resembling a Fermi distribution, which is shifted and distorted along the field direction, mechanism (ii) causes a steady increase of the carrier density without reaching an equilibrium. In contrast, including only carrier-phonon scattering channels, cf. Fig. \[fig:occupations\](c), an equilibrium carrier distribution is reached, however, it considerably deviates from a Fermi distribution. While low energetic acoustic phonons play only a minor role for energy relaxation [@Malic2011; @Winnerl2011], they have a strong impact on momentum relaxation and thus on the current density, cf. dashed lines in Fig. \[fig:occupations\](c). The complete dynamics including both carrier-carrier and carrier-phonon scattering is displayed in Fig. \[fig:occupations\](d). Here, the distribution resembles the case of purely Coulomb-induced dynamics. However, now it exhibits an equilibrium due to carrier-phonon scattering competing with Auger processes and stabilizing the carrier density.
As a result, evaluating the graphene Bloch equations, the current density dynamics is revealed on a microscopic footing. It results from the interplay of the field-induced acceleration of charge carriers and Coulomb- and phonon-induced scattering processes. We find current-reducing carrier-carrier and carrier-phonon scattering channels as well as current-enhancing Auger scattering, a specific Coulomb channel giving rise to a dark CM. It is small at low electric fields $E$, but causes a current density increase above the saturation value for $E>\unit[0.22]{V\mu m^{-1}}$, cf. Fig. \[fig:current+CM(field)\](b).
![Current density and dark carrier multiplication in dependence of temperature $T$ and dielectric background constant $\varepsilon_{\text{bg}}$ with otherwise the same parameters as in Fig. \[fig:occupations\]. The thin black (yellow) lines show the behavior without many-particle scattering (excluding the impact of the dark CM). The blue-shaded areas illustrate the region, where current density enhancement takes place. []{data-label="fig:current(temperature,substrate)"}](dark-fig42-results-grad){width="1\columnwidth"}
Next, we investigate the dependence of the equilibrium current density on the strength of the carrier phonon scattering and the Coulomb interaction, which can be controlled by varying temperature and substrate, respectively. The substrate is assumed to be only on one side of graphene resulting in an averaged background dielectric constant $\varepsilon_{\text{bg}}=(\varepsilon_{\text{s}}+1)/2$. The standard substrate used in Figs. \[fig:sketch\]- \[fig:occupations\] is silicon carbide with a static dielectric constant $\varepsilon_{\text{s}}=9.66$ [@Patrick1970]. The crucial point in the temperature- and substrate-dependence of the current density is their influence on the dark CM, cf. blue and yellow lines in Fig. \[fig:current(temperature,substrate)\](a) and (c), respectively.
The temperature dependence of the dark CM is determined by Pauli blocking, which is small for narrow carrier distributions at low temperatures, cf. Fig. \[fig:current(temperature,substrate)\](b). To understand the influence of the dark CM on the current density (Fig. \[fig:current(temperature,substrate)\](a)), we first consider the case without scattering (black line), where the current density is found to scale with $T^{2}$ according to Eq. (\[eq:j\_max\]). Switching on the scattering channels but suppressing the carrier density increase due to the dark CM (yellow line) the current density is strongly reduced, and owing to an enhanced acoustic phonon scattering this reduction increases with the temperature. Due to the pronounced dark CM at low $T$ (Fig. \[fig:current(temperature,substrate)\](b)), a significant current density amplification occurs up to the room temperature (blue-shaded region). For higher temperatures, the dark CM becomes negligible and the efficient scattering with acoustic phonons leads to a saturation of the current density.
The substrate dependence reveals a clear enhancement of the current density and the dark CM at higher dielectric background constants $\varepsilon_{bg}$, cf. Figs. \[fig:current(temperature,substrate)\](c) and (d). A larger $\varepsilon_{bg}$ screens the Coulomb potential and reduces the efficiency of carrier-carrier scattering. This leads to an enhanced dark CM, since the time window for Auger processes is increased. This resembles the increase of the conventional carrier multiplication at low pump fluences, cf. Ref. . Moreover, at low dielectric constants the Coulomb-induced redistribution of charge carriers competing with Auger scattering is suppressed resulting in improved conditions for the dark CM.
In conclusion, we have investigated the impact of the time- and momentum-resolved carrier dynamics on the generation of currents in graphene in the presence of an in-plane electric field. We show that field-induced acceleration of charge carriers provides excellent conditions for Auger scattering, which give rise to a dark carrier multiplication resulting in a significant enhancement of the generated currents. The presented insights are applicably to the entire class of Dirac materials.
This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 696656 (Graphene Flagship). Furthermore, we acknowledge support from the Swedish Research Council (VR) and the Deutsche Forschungsgemeinschaft through SFB 658 and SPP 1459. The computations were performed on resources at Chalmers Centre for Computational Science and Engineering (C3SE) provided by the Swedish National Infrastructure for Computing (SNIC). Finally, we thank Andreas Knorr (TU Berlin) for inspiring discussions.
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[^1]: Note that for an infinitely extended graphene sheet in an in-plane electric field, electron-hole pairs can also be created due to the Schwinger mechanism [@Schwinger1951; @Dora2010; @Rosenstein2010]. Here, we restrict our investigation to finite temperatures of $T\geq\unit[100]{K}$, where the field-induced acceleration of thermal charge carriers gives rise to a dark CM that prevails over the Schwinger effect.
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abstract: 'We define the algebraic part of the motivic cohomology group with compact supports $H_c^{m}(X,{ {\mathbb Z} }(n))$ of an arbitrary connected scheme $X$ over an algebraically closed field. Our definition is a generalization of the classical notion of the algebraic part of Chow groups. For the algebraic part of motivic cohomology, we define and study regular homomorphisms with targets in the category of semi-abelian varieties. We give a criterion for the existence of universal regular homomorphisms and show their existence for $m\leq n+2$ if $X$ is an arbitrary connected scheme, and for $(m,n)=(2\dim X,\dim X)$ if $X$ is smooth. Note that these indices include those corresponding to algebraic cycles of codimensions $1,2$ and $\dim X.$ For an arbitrary smooth connected scheme $X,$ we show that the universal regular homomorphism in codimension $\dim X$ is nothing but the Albanese map in the sense of Serre. We also identify the target of the universal regular homomorphism in codimension one of an arbitrary smooth connected scheme with a variant of Picard varieties and conclude that it is an isomorphism.'
address: 'Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan'
author:
- Tohru Kohrita
title: Algebraic part of motivic cohomology
---
Introduction
============
We define the algebraic part $H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))$ (Definition \[defn: algebraic part\]) of the motivic cohomology group with compact supports $H_c^{m}(X,{ {\mathbb Z} }(n))$ of an arbitrary connected scheme $X$ of dimension $d_X$ over an algebraically closed field $k.$ If the scheme $X$ is smooth and proper and $(m,n)=(2r,r),$ it is canonically isomorphic to the algebraic part (see the equation (\[equation: classical algebraic part\]) in Section \[section: The case of smooth proper schemes\] for the definition) of the Chow group $CH^r(X)$ (Proposition \[prop: comparison with the classical algebraic part\]). If $X$ is smooth and $(m,n)=(2d_X,d_X),$ our algebraic part agrees with the degree zero part of the Suslin homology (Proposition \[prop: algebraic part of zero cycles\], Remark \[rem: up to p-torsion without resolution\]) via the Friedlander-Voevodsky duality isomorphism.
As an analogue of the classical theory, we define regular homomorphisms for $H_{c,alg}^{m}(X,{ {\mathbb Z} }(n)).$ A regular homomorphism is a group homomorphism from $H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))$ to the group of rational points of a semi-abelian variety $S$ that is “continuous" in the sense of Definition \[defn: regular homomorphism\]. We say that a regular homomorphism $\phi: H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))\longrightarrow S(k)$ is universal if, given any regular homomorphism $\phi':H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))\longrightarrow S'(k),$ there is a unique homomorphism of semi-abelian varieties $a: S\longrightarrow S'$ such that $a\circ\phi=\phi'$ holds.
We prove the following existence theorem.
Let $X$ be any connected scheme of dimension $d_X$ over an algebraically closed field $k.$ Then, there exists a universal regular homomorphism $$\Phi_{c,X}^{m,n}:H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))\longrightarrow Alg_{c,X}^{m,n}(k)$$ if $m\leq n+2.$ If $X$ is smooth, the universal regular homomorphism exists for $(m,n)=(2d_X,d_X)$ as well.
This generalizes the existence known for smooth proper connected schemes for $(m,n)=(2,1)$ and $(4,2).$ If $X$ is smooth, proper and connected, the target $Alg_{c,X}^{2,1}$ of the universal regular homomorphism $\Phi_{c,X}^{2,1}$ is the Picard variety $Pic_{X,red}^0$ of $X$ and $\Phi_{c,X}^{4,2}$ is the algebraic representative in codimension two constructed by Murre ([@Murre Theorem A]).
For an arbitrary smooth connected scheme $X,$ we show under the assumption of resolution of singularities that $Alg_{c,X}^{2d_X,d_X}$ is canonically isomorphic to Serre’s generalized Albanese variety (Proposition \[prop: agrees with Serre’s\]). As a corollary, or more precisely, as a corollary to the homological version (which does not depend on resolution of singularities) of Proposition \[prop: agrees with Serre’s\], the covariant functoriality of Albanese varieties with respect to scheme morphisms extends to that with respect to morphisms of motives (Corollary \[cor: covariant for DM morphisms\]). Along the way, we shall also give a motivic proof of the classical fact (Corollary \[cor: Weil’s theorem\]) that algebraic equivalence relation of cycles can be defined by parametrization by abelian varieties.
For the case where $(m,n)=(2,1)$ and $X$ is smooth, we relate the universal regular homomorphism with the relative Picard group of a good compactification of $X$ (Definition \[defn: good compactification\]). Suppose that $X$ is a smooth connected scheme over $k$ with a good compactification $\bar X$ with the non-empty boundary divisor $Z,$ and let $Pic_{\bar X,Z,red}^0$ be the reduction of the identity component of the group scheme representing the functor that sends $T\in Sch/k$ to the relative Picard group $Pic(T\times \bar X,T\times Z)$ considered in [@Barbieri-Viale-Srinivas] (see Proposition \[prop: representability of the relative Picard functor\]). We obtain the following theorem.
Assume resolution of singularities. If $X$ is as above, then there is a canonical regular homomorphism $$\phi_0: H_{c,alg}^2(X,{ {\mathbb Z} }(1))\longrightarrow Pic_{\bar X,Z,red}^0(k).$$ It is universal and, in fact, it is an isomorphism.
This, in particular, implies the independence of $Pic_{\bar X,Z,red}^0$ from the choice of a good compactification of $X.$
With this interpretation of $Alg_{c,X}^{2,1},$ we show, in Theorem \[cor: max abelian subvariety\], that the maximal abelian subvariety of $Alg_{c,X}^{2,1}$ is uniquely isomorphic to $Alg_{c,\mathfrak P,X}^{2,1},$ the target of the variant of the universal regular homomorphism defined by “proper parametrization" (Definitions \[defn: algebraic part by proper parametrization\] and \[defn: algebraic representative by proper parametrization\]).
[*Convention.*]{} Schemes are assumed separated and of finite type over some field, and morphisms of schemes are those over the base field. A scheme $X$ pointed at $x$ means a pair of scheme $X$ and a rational point $x$ on $X.$ A group scheme is considered pointed at the unit unless otherwise noted. A curve means a connected scheme of pure dimension one over a field. The letter $k$ stands for an algebraically closed field unless otherwise noted.
For an arbitrary scheme $X$ over $k$ and integers $m$ and $n,$ by motivic homology and cohomology with or without compact supports, we mean the following four theories introduced in [@VSF5] (but we follow the notation in [@MVW]):
- [**motivic homology**]{}: $H_m(X,{ {\mathbb Z} }(n)):=Hom_{DM_{Nis}^{-}(k)}({ {\mathbb Z} }(n)[m],M(X)),$
- [**motivic cohomology**]{}: $H^m(X,{ {\mathbb Z} }(n)):=Hom_{DM_{Nis}^{-}(k)}(M(X),{ {\mathbb Z} }(n)[m]), $
- [**motivic homology with compact supports**]{}: $H_m^{BM}(X,{ {\mathbb Z} }(n)):=Hom_{DM_{Nis}^{-}}({ {\mathbb Z} }(n)[m],M^c(X)),$
- [**motivic cohomology with compact supports**]{}: $H_c^m(X,{ {\mathbb Z} }(n)):=Hom_{DM_{Nis}^{-}(k)}(M^c(X),{ {\mathbb Z} }(n)[m]).$
The index $m$ is called a degree and $n$ a twist. We simply write $DM$ for $DM_{Nis}^-(k).$
By resolution of singularities, we mean that the base field $k$ “admits resolution of singularities" in the sense of [@Friedlander-Voevodsky Definition 3.4]:
- For any scheme $X$ over $k,$ there is a proper surjective morphism $Y\longrightarrow X$ such that $Y$ is a smooth scheme over $k.$
- For any smooth scheme $X$ over $k$ and abstract blow-up $q:X'\longrightarrow X,$ there exists a sequence of blow-ups with smooth centers $p:X_n\longrightarrow\cdots\longrightarrow X_1=X$ such that $p$ factors through $q.$
These conditions are satisfied over any field of characteristic zero (Ibid. Proposition 3.5) by Hironaka’s resolution of singularities ([@Hironaka]).
[*Acknowledgements.*]{} This paper is based on the author’s doctoral dissertation at Nagoya University. The author wishes to thank his advisors Thomas Geisser and Hiroshi Saito for their constant advice and encouragements. He would also like to thank Shane Kelly, Hiroyasu Miyazaki, Shuji Saito and Rin Sugiyama for helpful conversations about the topic of this paper. Thanks also go to Ryo Horiuchi and Takashi Maruyama for many discussions during the author’s graduate study.
Universal regular homomorphisms {#chapter: Algebraic representatives}
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In this section, we define the algebraic part of motivic cohomology with compact supports of an arbitrary connected scheme and study its basic properties. We then consider regular homomorphisms, a version of rational homomorphisms in [@Samuel Section 2.5] (see also [@Murre Definition 1.6.1]) in our context. We prove the existence of universal regular homomorphisms in the indices including those corresponding to zero cycles and cycles of codimensions one and two. To motivate our discussion, let us review the case of smooth proper schemes.
The case of smooth proper schemes {#section: The case of smooth proper schemes}
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For a smooth proper connected scheme $X$ over an algebraically closed field $k,$ $Z^r(X)$ denotes the free abelian group generated by the set of cycles of codimension $r.$ The Chow group $CH^r(X)$ of $X$ in codimension $r$ is defined as the group $Z^r(X)/_{\sim_\mathrm{rat}}$ of rational equivalence classes of cycles [@Fulton Sections 1.3 and 1.6]. There is another coarser equivalence relation on $Z^r(X)$ called algebraic equivalence [@Fulton Section 10.3]. The cycles algebraically equivalent to zero form a subgroup of $Z^r(X),$ and its image $A^r(X)$ in $CH^r(X)$ is called the algebraic part of $CH^r(X).$ It is by definition that we have the equality $$\label{equation: classical algebraic part}
A^r(X)=\bigcup_{\substack{T,~{\text{smooth, proper}}\\ {\text{connected}}}}\{CH_0(T)^0\times CH^r(T\times X)\longrightarrow CH^r(X)\},$$ where the map sends a pair $(\sum_i n_it_i,Y)\in CH_0(T)^0\times CH^r(T\times X)$ to $\sum n_i Y_{t_i}\in CH^r(X).$ Here, $Y_{t_i}$ is the pullback of $Y$ along $t_i\times id_X: X\cong { {\mathrm{Spec~}} }k\times X\longrightarrow T\times X,$ i.e. the image of the intersection of cycles $(t_i\times X)\cdot Y\in CH^{\dim T+r}(T\times X)$ under the proper pushforward along the projection $T\times X\longrightarrow X.$
A regular homomorphism relates $A^r(X)$ with an abelian variety. It is a group homomorphism from $A^r(X)$ to the group of rational points $A(k)$ of an abelian variety $A$ such that any family of cycles in $A^r(X)$ parametrized by a smooth proper scheme $T$ gives rise to a scheme morphism from $T$ to $A.$ More precisely,
\[defn: classical regular homomorphism\] Let $A$ be an abelian variety over $k.$ A group homomorphism $\phi:A^r(X)\longrightarrow A(k)$ is called [**regular**]{} if, for any smooth proper connected scheme $T$ over $k$ pointed at a rational point $t_0,$ and for any cycle $Y\in CH^r(T\times X),$ the composition $$T(k)\buildrel w_Y\over\longrightarrow A^r(X)\buildrel\phi\over\longrightarrow A(k),$$ where $w_Y$ maps $t\in T(k)$ to $Y_t-Y_{t_0},$ is induced by a scheme morphism $T\longrightarrow A.$
A regular homomorphism $\phi:A^r(X)\longrightarrow A(k)$ is said [**universal**]{} ([@Samuel Section 2.5, Remarque (2)]) if for any regular homomorphism $\phi':A^r(X)\longrightarrow A'(k),$ there is a unique homomorphism of abelian varieties $a: A\longrightarrow A'$ such that $a\circ\phi=\phi'.$ The universal regular homomorphism, if it exists, is called the [**algebraic representative**]{} of $A^r(X)$ (or of $X$ in codimension $r$) and written as $$\Phi^r_X: A^r(X)\longrightarrow Alg_X^r(k).$$ We also refer to the target abelian variety $Alg_X^r$ itself as the algebraic representative.
\[rem: new remark\] For any smooth proper connected scheme $X$ of dimension $d_X,$ the algebraic representative exists if $r=1,2$ or $d_X.$
For $r=1,$ it is given by the isomorphism $$w_{\mathcal P}^{-1}: A^1(X)\buildrel\cong\over\longrightarrow Pic_{X,red}^0(k),$$ where $\mathcal P\in CH^1(Pic_{X,red}^0\times X)$ is the divisor corresponding to the Poincaré bundle on $Pic_{X,red}^0\times X.$
The case $r=d_X$ coincides with the Albanese map (see Proposition \[prop: agrees with Serre’s\] for a motivic treatment of this fact) $$alb_X: A^{d_X}(X)\longrightarrow Alb_X(k),$$ i.e. the map sending $\sum_i n_i\cdot x_i\in A^{d_X}(X)$ to $\sum_i n_i a_p(x_i)\in Alb_X(k),$ where $a_p:X\longrightarrow Alb_X$ is the canonical map that sends $p\in X(k)$ to the unit $0\in Alb_X(k).$ As $a_p=a_q+a_p(q)$ for all rational points $p$ and $q$ of $X$ by the universality of Albanese varieties, the Albanese map $alb_X$ is independent of the choice of $p.$
The algebraic representatives carry information on the algebraic parts of Chow groups in the following sense.
\[thm: classical Rojtman-type\] Let $X$ be a smooth proper connected scheme over $k.$ The algebraic representative $$\Phi_X^r:A^r(X)\longrightarrow Alg_X^r(k)$$ is an isomorphism if $r=1$ and induces an isomorphism on torsion if $r=\dim X.$ If $k={ {\mathbb C} },$ then it is an isomorphism on torsion for $r=2$ as well.
The case $r=1$ is the theory of Picard varieties (see [@Kleiman Proposition 9.5.10]). The case $r=\dim X$ is known as Rojtman’s theorem [@Rojtman; @Bloch; @Milne]. (In these references, schemes are assumed projective, but the claim for proper schemes follows from the projective case by Chow’s lemma because both Chow groups of zero cycles and Albanese varieties of smooth proper schemes are birational invariants.) The codimension $2$ case (over ${ {\mathbb C} }$) is due to Murre [@Murre Theorem C]. While the surjectivity is immediate from the construction, the injectivity is proved by relating the torsion part of $Alg_X^2({ {\mathbb C} })$ with étale cohomology by Artin’s comparison theorem ([@SGA4 Exposé XI, Théorème 4.4]) with the aid of Griffiths’s intermediate Jacobian and then using the theorem of Merkurjev-Suslin on norm residue homomorphisms ([@Merkurjev-Suslin]).
The algebraic part {#section: The algebraic part}
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We define the algebraic part of motivic cohomology groups with compact supports $H_c^{m}(X,{ {\mathbb Z} }(n)).$ We shall simply write $DM$ for Voevodsky’s tensor triangulated category $DM_{Nis}^{-}(k)$ of motives over $k$ ([@MVW p.110]).
Let $X$ and $T$ be schemes over $k.$ We consider the map $$P_T: H_0(T,{ {\mathbb Z} })\times Hom_{DM}(M(T)\otimes M^c(X),{ {\mathbb Z} }(n)[m])\longrightarrow H_c^{m}(X,{ {\mathbb Z} }(n))$$ that sends a pair $(Z,Y)$ with $Z\in H_0(T,{ {\mathbb Z} })\buildrel\text{def}\over= Hom_{DM}({ {\mathbb Z} },M(T))$ and $Y \in Hom_{DM}(M(T)\otimes M^c(X),{ {\mathbb Z} }(n)[m])$ to the composition $$Y\circ (Z\otimes id_{M^c(X)}): M^c(X)\cong{ {\mathbb Z} }\otimes M^c(X)\buildrel Z\otimes id_{M^c(X)}\over\longrightarrow M(T)\otimes M^c(X)\buildrel Y\over\longrightarrow { {\mathbb Z} }(n)[m]$$ in $Hom_{DM}(M^c(X),{ {\mathbb Z} }(n)[m])\buildrel\text{def}\over=H_c^{m}(X,{ {\mathbb Z} }(n)).$
The structure morphism of $T$ induces the degree map $deg:H_0(T,{ {\mathbb Z} })\longrightarrow H_0(k,{ {\mathbb Z} })\cong{ {\mathbb Z} }.$ We set $H_0(T,{ {\mathbb Z} })^0:=\ker(deg).$ Below, we shall identify the zero-th motivic homology $H_0(T,{ {\mathbb Z} })$ with the Suslin homology $H_0(C_*{ {\mathbb Z} }_{tr}(T)(k))$ by the canonical isomorphism.
\[defn: algebraic part\] Let $X$ be a connected scheme over $k$ and $\mathfrak T$ be a class of connected $k$-schemes. The [**algebraic part by $\mathfrak T$-parametrization**]{} of the motivic cohomology group with compact supports $H^{m}_c(X,{ {\mathbb Z} }(n))$ is defined as $$H^{m}_{c,\mathfrak T}(X,{ {\mathbb Z} }(n)):=\bigcup_{T\in\mathfrak T}\mathrm{im}\{H_0(T,{ {\mathbb Z} })^0\times Hom_{DM}(M(T)\otimes M^c(X),{ {\mathbb Z} }(n)[m])\buildrel P_T\over\longrightarrow H_c^{m}(X,{ {\mathbb Z} }(n))\}.$$
If $\mathfrak T$ is the class of connected smooth schemes, $H^{m}_{c,\mathfrak T}(X,{ {\mathbb Z} }(n))$ is written as $H^{m}_{c,alg}(X,{ {\mathbb Z} }(n))$ and simply called the [**algebraic part**]{} of $H^{m}_{c}(X,{ {\mathbb Z} }(n)).$
\[prop: algebraic part\] Let $X$ be a connected scheme over $k.$ If a class $\mathfrak T$ of connected schemes is closed under product (for example, the class of smooth connected schemes), then $H^{m}_{c,\mathfrak T}(X,{ {\mathbb Z} }(n))$ is a subgroup of $H^{m}_c(X,{ {\mathbb Z} }(n)).$
We need to show that $H^{m}_{c,\mathfrak T}(X,{ {\mathbb Z} }(n))$ is closed under addition and taking inverses. For taking inverses, let $x\in H^{m}_{c,\mathfrak T}(X,{ {\mathbb Z} }(n)).$ Then, there is a connected scheme $T,$ $Z\in H_0(T,{ {\mathbb Z} })^0$ and $Y\in Hom_{DM}(M(T)\otimes M^c(X),{ {\mathbb Z} }(n)[m])$ such that $x=Y\circ (Z\otimes id_{M^c(X)}).$ Now, by the additivity of the category $DM,$ we have $-x=Y\circ (-Z\otimes id_{M^c(X)})\in H^{m}_{c,\mathfrak T}(X,{ {\mathbb Z} }(n)).$
For the closedness under addition, take another $x'\in H^{m}_{c,\mathfrak T}(X,{ {\mathbb Z} }(n)),$ and choose a connected scheme $T',$ an element $Z'\in H_0(T,{ {\mathbb Z} })^0$ and $Y'\in Hom_{DM}(M(T')\otimes M^c(X),{ {\mathbb Z} }(n)[m])$ such that $x'= Y'\circ (Z'\otimes id_{M^c(X)}).$ It is clear that $x+x'$ belongs to $H^{m}_{c,\mathfrak T}(X,{ {\mathbb Z} }(n))$ if $Y=Y'.$ We shall reduce the general case to this.
Let us write $Z=\sum_i n_i t_i$ and $Z'=\sum_i n_i' t_i'$ with $n_i,n_i'\in{ {\mathbb Z} },$ $t_i\in T(k)$ and $t_i'\in T'(k),$ and choose $s\in T(k)$ and $s'\in T'(k).$ Define $$\begin{aligned}
Y''&:=& Y\circ(p\otimes id_{M^c(X)})+Y'\circ(p'\otimes id_{M^c(X)})\nonumber\\
&\in& Hom_{DM}({ {\mathbb Z} }\otimes M^c(X),{ {\mathbb Z} }(n)[m])\nonumber\\
&\cong& Hom_{DM}(M^c(X),{ {\mathbb Z} }(n)[m])\nonumber,\end{aligned}$$ where $p:M(T\times T')\longrightarrow M(T)$ and $p':M(T\times T')\longrightarrow M(T')$ are the morphisms induced by the projections. Consider the diagram $$\xymatrix{ & & & M(T)\otimes M^c(X) \ar[dr]^-{Y} \\
M^c(X)\cong{ {\mathbb Z} }\otimes M^c(X) \ar[rr]^-{\sum n_i(t_i\times s')\otimes id} \ar@/^1pc/[urrr]^-{\sum n_i t_i\otimes id} \ar@/_1pc/[drrr]_{\sum n_i s'\otimes id=0} & & M(T\times T')\otimes M^c(X) \ar[ur]_-{p\otimes id} \ar[dr]^-{p'\otimes id} \ar[rr]^-{Y''} & & { {\mathbb Z} }(n)[m] \\
& & & M(T')\otimes M^c(X) \ar[ur]_-{Y'}}$$ with $\sum n_i s'\otimes id=0$ because $\sum_i n_i=0.$ We have $$\begin{aligned}
x&=& Y \circ\big(\sum_i n_i t_i\otimes id_{M^c(X)}\big)\nonumber\\
&=& Y \circ\big(\sum_i n_i t_i\otimes id_{M^c(X)}\big) +Y' \circ\big(\sum_i n_i s'\otimes id_{M^c(X)}\big)\nonumber\\
&=& Y\circ (p\otimes id_{M^c(X)})\circ \big(\sum_i n_i(t_i\times s')\otimes id_{M^c(X)}\big) +Y'\circ (p'\otimes id_{M^c(X)})\circ \big(\sum_i n_i(t_i\times s')\otimes id_{M^c(X)}\big)\nonumber\\
&=& \big(Y\circ (p\otimes id_{M^c(X)}) +Y'\circ (p'\otimes id_{M^c(X)})\big)\circ \big(\sum_i n_i(t_i\times s')\otimes id_{M^c(X)}\big)\nonumber\\
&=& Y''\circ \big(\sum_i n_i(t_i\times s')\otimes id_{M^c(X)}\big).\nonumber\\\end{aligned}$$ Similarly, we have $$x'=Y''\circ\big(\sum_i n_i'(s\times t_i')\otimes id_{M^c(X)}\big).$$ Therefore, we conclude that $$x+x'=Y''\circ \big(\big(\sum_i n_i(t_i\times s') + \sum_i n_i'(s\times t_i')\big)\otimes id_{M^c(X)}\big) \in H^{m}_{c,\mathfrak T}(X,{ {\mathbb Z} }(n)).$$
For smooth proper schemes, our definition of algebraic part agrees with the classical notion.
\[lem: curves are enough\] Let $\mathfrak T_{(1)}$ be the subclass of $\mathfrak T$ consisting of schemes of dimension one. If $\mathfrak T$ is either the class of connected smooth schemes or that of connected smooth proper schemes, we have $$H^{m}_{c,\mathfrak T}(X,{ {\mathbb Z} }(n))= H^{m}_{c,\mathfrak T_{(1)}}(X,{ {\mathbb Z} }(n)).$$
The group $H^{m}_{c,\mathfrak T}(X,{ {\mathbb Z} }(n))$ is clearly generated by the set $$G:=\bigcup_{T\in\mathfrak T}\mathrm{im}\{D_T\times Hom_{DM}(M(T)\otimes M^c(X),{ {\mathbb Z} }(n)[m])\buildrel P_T\over\longrightarrow H_c^{m}(X,{ {\mathbb Z} }(n))\},$$ where $D_T$ is the subset of $H_0(T,{ {\mathbb Z} })^0$ consisting of the classes of zero cycles of the form $t_1-t_2$ ($t_1,t_2\in T(k)$).
Suppose that $x\in G.$ Then, it is the image of some $t_1-t_2\in D_T$ and $Y\in Hom_{DM}(M(T)\otimes M^c(X),{ {\mathbb Z} }(n)[m])$ under $P_T.$ By [@Mumford Chapter II, Section 6, Lemma], there is a smooth connected curve $C$ (which can be chosen to be proper if $T$ is proper) with a morphism $f: C\longrightarrow T$ whose image contains $t_1$ and $t_2.$ By the commutativity of the diagram $$\xymatrix{ H_0(C,{ {\mathbb Z} })^0 \ar[d]_{f_*} &\times& Hom_{DM}(M(C)\otimes M^c(X),{ {\mathbb Z} }(n)[m]) \ar[r]^-{P_C} & H_c^{m}(X,{ {\mathbb Z} }(n)) \ar@{=}[d]\\
H_0(T,{ {\mathbb Z} })^0 &\times& Hom_{DM}(M(T)\otimes M^c(X),{ {\mathbb Z} }(n)[m]) \ar[u]^{f^*} \ar[r]^-{P_T} & H_c^{m}(X,{ {\mathbb Z} }(n)),}$$ we can see that $x$ is the image of the pair $c_1-c_2\in H_0(C,{ {\mathbb Z} })^0$ ($c_i$ is any preimage of $t_i$) and $f^*(Y)\in Hom_{DM}(M(C)\otimes M^c(X),{ {\mathbb Z} }(n)[m])$ under $P_C.$ Therefore, we have the inclusion $G\subset H^{m}_{c,\mathfrak T_{(1)}}(X,{ {\mathbb Z} }(n)).$
Thus, it remains to show that $H^{m}_{c,\mathfrak T_{(1)}}(X,{ {\mathbb Z} }(n))$ is closed under taking inverses and addition. By the same argument as the proof of Proposition \[prop: algebraic part\], we can show that it is indeed closed under taking inverses and that, for any $x$ and $y\in H^{m}_{c,\mathfrak T_{(1)}}(X,{ {\mathbb Z} }(n)),$ there are $T\in \mathfrak T$ (in this case, $T$ may be taken to be a product of two curves in $\mathfrak T_{(1)}$), $Z\in H_0(T,{ {\mathbb Z} })^0$ and $Y\in Hom_{DM}(M(T)\otimes M^c(X),{ {\mathbb Z} }(n)[m])$ such that $x+y$ is the image of the pair $(Z,Y)$ under $$P_T: H_0(T,{ {\mathbb Z} })^0\times Hom_{DM}(M(T)\otimes M^c(X),{ {\mathbb Z} }(n)[m])\longrightarrow H_c^{m}(X,{ {\mathbb Z} }(n)).$$
We would like to replace $T$ with a curve in $\mathfrak T_{(1)}.$ Since $Z$ is of degree zero, it is the class of a zero cycle of the form $$t_1+t_2+\cdots+t_r-t_{r+1}-\cdots-t_{2r},$$ with $t_i\in T(k)$ for which we allow repetition. If $r=0,$ there is nothing to prove because then $x+y=0.$ We may thus assume that $r\geq1.$ Consider the $r$-th power $T^r=T\times\cdots\times T\in\mathfrak T,$ the $i$-th projections $p_i:T^r\longrightarrow T$ and rational points $t_+:=(t_1,\cdots,t_r)$ and $t_-:=(t_{r+1},\cdots,t_{2r})$ on $T^r.$ Pulling back $Y$ along each $p_i$ and summing them up, we obtain the morphism of motives $$\sum_{i=1}^r Y\circ (p_i\otimes id_{M^c(X)}): M(T^r)\otimes M^c(X)\buildrel p_i\otimes id_{M^c(X)}\over\longrightarrow M(T)\otimes M^c(X)\buildrel Y\over\longrightarrow { {\mathbb Z} }(n)[m].$$ Now, the image of the pair $t_+-t_-\in H_0(T^r,{ {\mathbb Z} })^0$ and this morphism under $$P_{T^r}: H_0(T^r,{ {\mathbb Z} })^0\times Hom_{DM}(M(T^r)\otimes M^c(X),{ {\mathbb Z} }(n)[m])\longrightarrow H_c^{m}(X,{ {\mathbb Z} }(n))$$ is nothing but $x+y.$
Find a curve $C'\in \mathfrak T_{(1)}$ with a morphism $f':C'\longrightarrow T^r$ such that the image contains $t_+$ and $t_-$ as in the second paragraph of the present proof and proceed similarly. We may see that $x+y$ is in the image of $$P_{C'}: H_0(C',{ {\mathbb Z} })^0\times Hom_{DM}(M(C')\otimes M^c(X),{ {\mathbb Z} }(n)[m])\longrightarrow H_c^{m}(X,{ {\mathbb Z} }(n)).$$ Therefore, $H^{m}_{c,\mathfrak T_{(1)}}(X,{ {\mathbb Z} }(n))$ is closed under addition.
\[prop: comparison with the classical algebraic part\] Suppose that $X$ is a smooth proper connected scheme over $k.$ Then, there is a natural isomorphism $$H_{c,alg}^{2r}(X,{ {\mathbb Z} }(r))\buildrel\cong\over\longrightarrow A^r(X).$$
The isomorphism is given by restricting the natural isomorphism for smooth $X$ $$H_{c}^{2r}(X,{ {\mathbb Z} }(r))=H^{2r}(X,{ {\mathbb Z} }(r))\buildrel F \over\longrightarrow CH^r(X)$$ in [@MVW Theorem 19.1].
Recall that, by definition, $$A^r(X):=\bigcup_{\substack{T,~{\text{sm, conn}}\\ {\text{proper}}}}\mathrm{im}\{CH_0(T)^0\times CH^r(T\times X)\buildrel\text{pullback}\over\longrightarrow CH^r(X)\},$$ where the map sends a pair of $\sum_i n_i t_i\in CH_0(X)^0$ and $Y\in CH^r(T\times X)$ to $\sum_i n_i Y_{t_i}.$ Here, $Y_{t_i}$ is the pullback of $Y$ along $t_i\times id_X.$ The (contravariant) naturality of the comparison map $F$ implies the commutativity of the diagram $$\xymatrix{ H_0(T,{ {\mathbb Z} })^0 \ar@{=}[d] &\times & Hom_{DM}(M(T)\otimes M^c(X),{ {\mathbb Z} }(r)[2r]) \ar[d]_F^\cong \ar[r]^-{P_T} & H_c^{2r}(X,{ {\mathbb Z} }(r)) \ar[d]_F^\cong \\
CH_0(T)^0 &\times & CH^r(T\times X) \ar[r]^-{\text{pullback}} & CH^r(X)}$$ for all smooth and proper schemes $X$ and $T.$ Therefore, $F$ induces an isomorphism $$H_{c,\mathfrak P}^{2r}(X,{ {\mathbb Z} }(r))\longrightarrow A^r(X),$$ where $\mathfrak P$ is the class of smooth proper connected schemes.
We claim that $H_{c,\mathfrak P}^{2r}(X,{ {\mathbb Z} }(r))=H_{c,alg}^{2r}(X,{ {\mathbb Z} }(r)).$ By definition, we have the inclusion $H_{c,\mathfrak P}^{2r}(X,{ {\mathbb Z} }(r))\subset H_{c,alg}^{2r}(X,{ {\mathbb Z} }(r)).$ For the other inclusion $H_{c,\mathfrak P}^{2r}(X,{ {\mathbb Z} }(r))\supset H_{c,alg}^{2r}(X,{ {\mathbb Z} }(r)),$ by Lemma \[lem: curves are enough\], it is enough to prove $$H_{c,\{\text{smooth proper curves}\}}^{2r}(X,{ {\mathbb Z} }(r))\supset H_{c,\{\text{smooth curves}\}}^{2r}(X,{ {\mathbb Z} }(r)).$$
It is enough to observe the surjectivity of $i^*$ in the following commutative diagram for a smooth curve $C$ and its smooth compactification $i:C\hookrightarrow\bar C:$ $$\xymatrix{ H_0(C,{ {\mathbb Z} })^0 \ar[d]_{i_*} &\times& Hom_{DM}(M(C)\otimes M^c(X),{ {\mathbb Z} }(r)[2r]) \ar[r]^-{P_C} & H_c^{2r}(X,{ {\mathbb Z} }(r)) \ar@{=}[d]\\
H_0(\bar C,{ {\mathbb Z} })^0 &\times& Hom_{DM}(M(\bar C)\otimes M^c(X),{ {\mathbb Z} }(r)[2r]) \ar[u]^{i^*} \ar[r]^-{P_{\bar C}} & H_c^{2r}(X,{ {\mathbb Z} }(r)).}$$ But the map $i^*$ is surjective because there is a commutative diagram $$\xymatrix{ Hom_{DM}(M(\bar C)\otimes M^c(X),{ {\mathbb Z} }(r)[2r]) \ar[r]^{i^*} \ar[d]_F^\cong & Hom_{DM}(M(C)\otimes M^c(X),{ {\mathbb Z} }(r)[2r]) \ar[d]_F^\cong\\
CH^r(C\times X) \ar@{->>}[r]^{i^*} & CH^r(\bar C\times X)}$$
If $(m,n)\neq (2r,r),$ then $H_{c,alg}^m(X,{ {\mathbb Z} }(n))$ and $H_{c,\mathfrak P}^m(X,{ {\mathbb Z} }(n))$ do not agree even for smooth proper $X$ (see Remark \[rem: interesting example one\]). We study more about algebraic part by $\mathfrak P$-parametrization in Subsection \[subsection: Algebraic part by proper parametrization\].
For smooth proper connected schemes $X$ over $k,$ we have $A^{d_X}(X)=CH_0(X)^0.$ This extends to our non-proper setting.
\[prop: algebraic part of zero cycles\] Let $X$ be a smooth connected scheme of dimension $d_X$ over $k.$ Under the assumption of resolution of singularities, there is a canonical morphism (induced by the Friedlander-Voevodsky duality isomorphism) $$H_{c,alg}^{2d_X}(X,{ {\mathbb Z} }(d_X))\cong H_0(X,{ {\mathbb Z} })^0.$$
The duality isomorphism for smooth $X$ ([@VSF5 Theorem 4.3.7 (3)]) $$M^c(X)^*\cong M(X)(-d_X)[-2d_X]$$ gives the isomorphism $$H_{c,alg}^{2d_X}(X,{ {\mathbb Z} }(d_X))\cong\bigcup_{\substack{T,~{\text{smooth}}\\ {\text{connected}}}}\mathrm{im}\{H_0(T,{ {\mathbb Z} })^0\times Hom_{DM}(M(T),M(X))\buildrel \text{composition} \over\longrightarrow H_0(X,{ {\mathbb Z} })\}.$$
Since $X$ is smooth and connected, we have (the first inclusion is via the above isomorphism) $$\begin{aligned}
H_{c,alg}^{2d_X}(X,{ {\mathbb Z} }(d_X))&\supset&\mathrm{im}\{H_0(X,{ {\mathbb Z} })^0\times Hom_{DM}(M(X),M(X))\buildrel \text{composition} \over\longrightarrow H_0(X,{ {\mathbb Z} })\}\nonumber\\
&\supset&\mathrm{im}\{H_0(X,{ {\mathbb Z} })^0\times \{id_{M(X)}\} \longrightarrow H_0(X,{ {\mathbb Z} })\}\nonumber\\
&=& H_0(X,{ {\mathbb Z} })^0.\nonumber\end{aligned}$$
For the other inclusion, we need to show that for any smooth connected scheme $T,$ any $a\in Hom_{DM}({ {\mathbb Z} },M(T))$ with $str_T\circ a=0$ ($str_T:M(T)\to{ {\mathbb Z} }$ is the morphism induced by the structure morphism of $T$) and any $b\in Hom_{DM}(M(T),M(X)),$ the composition $b\circ a$ belongs to $H_0(X,{ {\mathbb Z} })^0,$ i.e. the large triangle of the diagram in $DM$ $$\xymatrix{ { {\mathbb Z} }\ar[r]^a \ar[dr]_0 & M(T) \ar[d]^{str_T} \ar[r]^b & M(X) \ar[dl]^{str_X}\\
& { {\mathbb Z} }}$$ is commutative if the left triangle is commutative.
Since $T$ is smooth and connected, the group $Hom_{DM}(M(T),{ {\mathbb Z} })\cong{ {\mathbb Z} }$ is generated by $str_T.$ Thus, there is an integer $n$ such that $str_X\circ b=n\cdot str_T.$ Hence, $str_X\circ b\circ a=n\cdot str_T\circ a=0.$
\[rem: up to p-torsion without resolution\] Note that Proposition \[prop: algebraic part of zero cycles\] is valid up to $p$-torsion even without resolution of singularities by the Friedlander-Voevodsky duality with ${ {\mathbb Z} }[\frac{1}{p}]$-coefficients proved in [@Kelly Theorem 5.5.14 (3)].
\[prop: algebraic part is divisible\] Let $X$ be a connected scheme over $k.$ Then, the group $H^{m}_{c,alg}(X,{ {\mathbb Z} }(n))$ is divisible.
Any element of $H^{m}_{c,alg}(X,{ {\mathbb Z} }(n))$ is the image of some element of $H_0(C,{ {\mathbb Z} })^0$ for some smooth curve $C$ by Lemma \[lem: curves are enough\]. Thus, it suffices to show that $H_0(C,{ {\mathbb Z} })^0$ is divisible. If $C$ is proper, it is a consequence of the Abel-Jacobi theorem. If $C$ is not proper, take the smooth compactification $C\hookrightarrow \bar C$ with $Z:=\bar C\setminus C$ endowed with the induced reduced structure. The localization sequence for motivic cohomology with compact supports yields $$\cdots\longrightarrow\bigoplus k^*\longrightarrow H_0(C,{ {\mathbb Z} })\buildrel f\over\longrightarrow H_0(\bar C,{ {\mathbb Z} })\longrightarrow 0.$$ This gives the short exact sequence $$0\longrightarrow \ker f\longrightarrow H_0(C,{ {\mathbb Z} })^0\longrightarrow H_0(\bar C,{ {\mathbb Z} })^0\longrightarrow 0$$ The kernel of $f$ is divisible because it is an image of $\bigoplus k^*,$ and $H_0(\bar C,{ {\mathbb Z} })^0$ is also divisible by the smooth proper case. Hence, the middle group is divisible as well.
In the rest of this section, we show that certain classes of algebraic groups are enough to define algebraic part.
\[prop: parametrization by semiabelian varieties\] Let $X$ be a connected scheme over $k$ and $\mathfrak S$ be the class of semi-abelian varieties over $k.$ Then, $$H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))=H_{c,\mathfrak S}^{m}(X,{ {\mathbb Z} }(n)).$$
The inclusion “$\supset$" is obvious. We show the other inclusion. By Lemma \[lem: curves are enough\], it is enough to show that $$H^{m}_{c,\{{\text{smooth~curves}}\}}(X,{ {\mathbb Z} }(n))\subset H^{m}_{c,\mathfrak S}(X,{ {\mathbb Z} }(n)).$$
Let $x\in H^{m}_{c,\{{\text{smooth curves}}\}}(X,{ {\mathbb Z} }(n)).$ Then there are a smooth affine curve $C,$ $Z\in H_0(C,{ {\mathbb Z} })^0$ and $Y\in Hom_{DM}(M(C)\otimes M^c(X),{ {\mathbb Z} }(n)[m])$ such that $x=Y\circ (Z\otimes id_{M^c(X)}).$ (If we find a proper curve $C$, just remove one point not supporting the divisor $Z.$) In other words, we have the equality $$H_{c,\{\text{smooth~curves}\}}^{m}(X,{ {\mathbb Z} }(n))=H^{m}_{c,\{{\text{smooth~affine~curves}}\}}(X,{ {\mathbb Z} }(n)).$$
Let $C^l$ be the $l$-th power and $C^{(l)}$ the $l$-th symmetric power of the curve $C.$ Write the quotient morphism as $f:C^l\longrightarrow C^{(l)}$ and the diagonal as $\Delta: C\longrightarrow C^l.$ Let us also write $\alpha, \beta$ and $\gamma$ for $P_{C^{(l)}}, P_{C^l}$ and $P_C,$ respectively. Consider the following diagram.
$$\xymatrix{ H_0(C^{(l)},{ {\mathbb Z} })^0 & \times & Hom_{DM}(M(C^{(l)})\otimes M^c(X),{ {\mathbb Z} }(n)[m]) \ar[d]_{-\circ f} & \buildrel\alpha\over\longrightarrow & H_c^{m}(X,{ {\mathbb Z} }(n)) \ar@{=}[d] \\
H_0(C^{l},{ {\mathbb Z} })^0 \ar[u]^{f\circ -} & \times & Hom_{DM}(M(C^{l})\otimes M^c(X),{ {\mathbb Z} }(n)[m]) \ar@/_/[u]_{-\circ^t\Gamma_f} \ar[d]_{-\circ \Delta} &\buildrel\beta\over\longrightarrow & H_c^{m}(X,{ {\mathbb Z} }(n)) \ar@{=}[d] \\
H_0(C,{ {\mathbb Z} })^0 \ar[u]^{\Delta\circ-}& \times & Hom_{DM}(M(C)\otimes M^c(X),{ {\mathbb Z} }(n)[m]) \ar@/_/[u]_{-\circ P} &\buildrel\gamma\over\longrightarrow & H_c^{m}(X,{ {\mathbb Z} }(n)).}$$
Here, $P:=\sum_{i=1,\cdots, l}p_i,$ where $p_i:C^l\longrightarrow C$ is the $i$-th projection and the summation is taken in $Hom_{DM}(M(C^l),M(C)).$ The transpose $^t\Gamma_f$ of the graph $\Gamma_f$ of $f$ is a finite correspondence from $C^{(l)}$ to $C^l$ because $f:C^l\longrightarrow C^{(l)}$ is a finite surjective morphism ([@MilneJV Propositions 3.1 and 3.2]).
By inspection, we may see that $^t\Gamma_f\circ f\circ\Delta=^t\Gamma_f\circ (f\circ\Delta) =a\cdot\Delta$ in $Cor_k(C,C^l)$ ($a$ is the intersection multiplicity). Since $P\circ\Delta=l\cdot id_C,$ we have $P\circ ^t\Gamma_f\circ f\circ\Delta =P\circ (a\cdot\Delta)=a\cdot l\cdot id_C.$
Therefore, since $\gamma(Z,Y)=x,$ the commutativity of the straight (uncurved) arrows and $``="$ in the diagram gives $$\begin{aligned}
&&\alpha(f\circ\Delta\circ Z, Y\circ(P\otimes id_{M^c(X)})\circ (^t\Gamma_f\otimes id_{M^c(X)}))\nonumber\\
&=&\gamma(Z,Y\circ(P\otimes id_{M^c(X)})\circ (^t\Gamma_f\otimes id_{M^c(X)})\circ(f\otimes id_{M^c(X)})\circ(\Delta\otimes id_{M^c(X)})) \nonumber\\
&=&\gamma(Z, Y\circ ((P\circ ^t\Gamma_f\circ f\circ\Delta)\otimes id_{M^c(X)}))\nonumber\\
&=&\gamma(Z, l\cdot a\cdot Y)\nonumber\\
&=& l\cdot a\cdot x.\nonumber \end{aligned}$$
Thus, we have a commutative diagram $$\xymatrix{ H_0(C^{(l)},{ {\mathbb Z} })^0 \ar[rr]^{\alpha_{Y\circ P\circ ^t\Gamma_f}} && H_c^{m}(X,{ {\mathbb Z} }(n)) \\
H_0(C,{ {\mathbb Z} })^0 \ar[u]^{f\circ\Delta\circ-} \ar[urr]_{\gamma_{l\cdot a\cdot Y}}}$$ where $\alpha_{Y\circ P\circ ^t\Gamma_f}:=\alpha(-, Y\circ(P\otimes id_{M^c(X)})\circ (^t\Gamma_f\otimes id_{M^c(X)})$ and $\gamma_{l\cdot m\cdot Y}:=\gamma(-,l\cdot m\cdot Y).$ Therefore, we have $$\mathrm{im}(\alpha_{Y\circ P\circ ^t\Gamma_f})\supset \mathrm{im}(\gamma_{l\cdot a\cdot Y})= l\cdot a\cdot\mathrm{im}(\gamma_Y)=\mathrm{im}(\gamma_Y).$$ The last equality holds because $H_0(C,{ {\mathbb Z} })^0$ is divisible. Since this is true for all $Y\in Hom_{DM}(M(C)\otimes M^c(X),{ {\mathbb Z} }(n)[m]),$ we conclude that $\mathrm{im}(\alpha)\supset\mathrm{im}(\gamma).$
Now, since $C$ is an affine curve, $C^{(l)}$ is an affine bundle over the smooth connected commutative algebraic group $Pic^0(C^+)$ if $l$ is sufficiently large by [@Wickelgren Appendix] ($C^+:=\bar C\coprod_{\bar C\setminus C}{ {\mathrm{Spec~}} }k$ for a smooth compactification $\bar C$ of $C$). By Chevalley’s theorem (see [@NeronBook Chapter 9, Section 2, Theorem 1]), there is a smooth connected affine commutative algebraic subgroup $L$ of ${Pic}^0(C^+).$ By [@Borel Theorem 10.6 (i) and (ii)], $L$ has a connected unipotent algebraic subgroup $U$ such that the quotient $L/U$ is a torus. Hence, the quotient ${Pic}^0(C^+)/U$ is a semi-abelian variety. Moreover, by [@Borel Corollary 15.5 (ii)], there is a composition series consisting of connected algebraic subgroups $$U=U_0\supset U_1\supset\cdots\supset U_n=\{e\}$$ such that each quotient algebraic group $U_i/U_{i+1}$ is isomorphic to $\mathbb{G}_a.$ We are given with the exact sequence $$0\longrightarrow \mathbb G_a(\cong U_{n-1})\longrightarrow Pic^0(C^+)\longrightarrow Pic^0(C^+)/U_{n-1}\longrightarrow 0$$ of algebraic groups. Now, $Pic^0(C^+)$ is a $\mathbb G_a$-torsor (in the fppf topology) over $Pic^0(C^+)/U_{n-1}.$ Since $\mathbb G_a$-torsors are representable by [@Milne; @etale Chapter III, Theorem 4.3(a)] and the canonical map $$H_{Zar}^1(Pic^0(C^+)/U_{n-1},\mathbb G_a)\longrightarrow H_{fppf}^1(Pic^0(C^+)/U_{n-1},\mathbb G_a)$$ is an isomorphism by [@Milne; @etale Chapter III, Proposition 3.7], $Pic^0(C^+)$ is a $\mathbb G_a$-torsor over $Pic^0(C^+)/U_{n-1}$ in the Zariski topology as well.
Therefore, by the Mayer-Vietoris exact triangle ([@MVW (14.5.1)]) and the ${ {\mathbb A} }^1$-homotopy invariance in $DM,$ we see that the canonical map $$M(q): M(Pic^0(C^+))\longrightarrow M(Pic^0(C^+)/U_{n-1})$$ is an isomorphism. Repeating this process, we obtain the canonical isomorphism $$M(Pic^0(C^+))\buildrel\cong\over\longrightarrow M(Pic^0(C^+)/U).$$
Now, the isomorphisms $$M(C^{(l)})\buildrel \cong\over\longrightarrow M(Pic^0(C^+))\buildrel\cong\over\longrightarrow M(Pic^0(C^+)/U)$$ mean that $$\mathrm{im}(\alpha)=\mathrm{im}\{H_0(Pic^0(C^+)/U,{ {\mathbb Z} })^0 \times Hom_{DM}(M(Pic^0(C^+)/U)\otimes M^c(X),{ {\mathbb Z} }(n)[m])\longrightarrow H_c^{m}(X,{ {\mathbb Z} }(n)) \}.$$ This equality holds for all smooth affine curves $C.$ Since $Pic^0(C^+)/U$ is a semi-abeian variety, we conclude $$H_{c,\mathfrak S}^{m}(X,{ {\mathbb Z} }(n))\supset H_{c,\{\text{smooth~affine~curves}\}}^{m}(X,{ {\mathbb Z} }(n))=H^{m}_{c,\{{\text{smooth~curves}}\}}(X,{ {\mathbb Z} }(n)).$$
As a corollary, we recover the following well-known fact (see, for example, [@Lang p.60, Theorem 1]) on algebraic equivalence of cycles.
\[cor: Weil’s theorem\] Let $X$ be a smooth proper connected scheme over $k$ and $\mathfrak A$ be the class of abelian varieties over $k.$ Then, $$A^r(X)=\bigcup_{A\in\mathfrak A} \mathrm{im}\{CH_0(A)^0\times CH^r(A\times X)\buildrel\text{pullback}\over\longrightarrow CH^r(X)\}.$$
The right hand side is canonically isomorphic to $H_{c,\mathfrak A}^{2r}(X,{ {\mathbb Z} }(r))$ under the comparison map [@MVW Theorem 19.1]. By Propositions \[prop: comparison with the classical algebraic part\] and \[prop: parametrization by semiabelian varieties\], the left hand side is isomorphic to $H_{c,\mathfrak S}^{2r}(X,{ {\mathbb Z} }(r)).$ Thus, it is enough to prove that $$\begin{aligned}
&&\bigcup_{S\in\mathfrak S}\mathrm{im}\{H_0(S,{ {\mathbb Z} })^0 \times Hom_{DM}(M(S)\otimes M(X),{ {\mathbb Z} }(r)[2r])\buildrel P_S\over\longrightarrow H^{2r}(X,{ {\mathbb Z} }(r))\}\nonumber\\
&=&\bigcup_{A\in\mathfrak A}\mathrm{im}\{H_0(A,{ {\mathbb Z} })^0 \times Hom_{DM}(M(A)\otimes M(X),{ {\mathbb Z} }(r)[2r])\buildrel P_A\over\longrightarrow H^{2r}(X,{ {\mathbb Z} }(r))\}.\nonumber\end{aligned}$$
We claim that for a semi-abelian variety $S$ with the Chevalley decomposition $0\to\mathbb G_m^s\to S\to A\to0,$ there is an inclusion $$\begin{aligned}
&&\mathrm{im}\{H_0(S,{ {\mathbb Z} })^0 \times Hom_{DM}(M(S)\otimes M(X),{ {\mathbb Z} }(r)[2r])\buildrel P_S\over\longrightarrow H^{2r}(X,{ {\mathbb Z} }(r))\}\nonumber\\
&\subset&\mathrm{im}\{H_0(A,{ {\mathbb Z} })^0 \times Hom_{DM}(M(A)\otimes M(X),{ {\mathbb Z} }(r)[2r])\buildrel P_A\over\longrightarrow H^{2r}(X,{ {\mathbb Z} }(r))\}.\nonumber\end{aligned}$$
We show this by induction on the torus rank $s.$ If $s=0,$ the claim is trivial. Suppose that the claim is true for semi-abelian varieties of torus rank $s-1$ and let $S$ be a semi-abelian variety with torus rank $s.$ Now, there is a short exact sequence of algebraic groups $$0\longrightarrow \mathbb G_m\longrightarrow S\longrightarrow S'\longrightarrow 0$$ with a semi-abelian variety $S'$ of torus rank $s-1.$
By a similar argument as in the proof of Proposition \[prop: parametrization by semiabelian varieties\]—this time with Hilbert’s Satz 90 ([@Milne; @etale Chapter III, Proposition 4.9]) instead of \[Ibid., Chapter III, Proposition 3.7\]—we can see that $S$ is a $\mathbb G_m$-torsor over $S'$ in the Zariski topology. Hence, there is an associated line bundle $p:E\longrightarrow S'$ with the zero section $s: S'\longrightarrow E$ such that $E\setminus s(S')\cong S.$ By the ${ {\mathbb A} }^1$-homotopy invariance, $p$ induces an isomorphism of motives $M(E)\buildrel\cong\over\longrightarrow M(S').$
Hence, we have the commutative diagram $$\xymatrix{ H_0(S,{ {\mathbb Z} })^0 \ar[d]_{inc_*}& \times & Hom_{DM}(M(S)\otimes M(X),{ {\mathbb Z} }(r)[2r]) & \buildrel P_S\over\longrightarrow & H^{2r}(X,{ {\mathbb Z} }(r)) \ar@{=}[d] \\
H_0(E,{ {\mathbb Z} })^0 \ar[d]_{p_*}^\cong & \times & Hom_{DM}(M(E)\otimes M(X),{ {\mathbb Z} }(r)[2r]) \ar@{->>}[u]^{-\circ (inc\otimes id_{M(X)})} &\buildrel P_E\over\longrightarrow & H^{2r}(X,{ {\mathbb Z} }(r)) \ar@{=}[d] \\
H_0(S',{ {\mathbb Z} })^0 & \times & Hom_{DM}(M(S')\otimes M(X),{ {\mathbb Z} }(r)[2r]) \ar[u]^{-\circ (p\otimes id_{M(X)})}_\cong &\buildrel P_{S'}\over\longrightarrow & H^{2r}(X,{ {\mathbb Z} }(r)).}$$ The upper middle map is surjective because it can be identified with the pullback of Chow groups $$CH^r(E\times X)\longrightarrow CH^r(S\times X)$$ along the open immersion $S\times X\hookrightarrow E\times X.$ The commutativity of the diagram implies that $$\mathrm{im}(P_S)\subset\mathrm{im}(P_{S'}).$$ Since the torus rank of $S'$ is $s-1,$ the induction hypothesis gives the inclusion $$\mathrm{im}(P_{S'})\subset \mathrm{im}\{H_0(A,{ {\mathbb Z} })^0 \times Hom_{DM}(M(A)\otimes M(X),{ {\mathbb Z} }(r)[2r])\buildrel P_A\over\longrightarrow H^{2r}(X,{ {\mathbb Z} }(r))\}.$$ Therefore, the claim is proved.
Regular homomorphisms {#section: regular homomorphism}
---------------------
The classical definition of regular homomorphisms (Definition \[defn: classical regular homomorphism\]) readily generalizes to our setting.
\[defn: regular homomorphism\] Let $X$ be a connected scheme over $k$ and let $S$ be a semi-abelian variety over $k.$ A group homomorphism $\phi: H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))\longrightarrow S(k)$ is called [**regular**]{} if for any smooth connected scheme $T$ pointed at $t_0\in T(k)$ and for any $Y\in Hom_{DM}(M(T)\otimes M^c(X),{ {\mathbb Z} }(n)[m]),$ the composition $$T(k)\buildrel w_Y\over\longrightarrow H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))\buildrel\phi\over\longrightarrow S(k)$$ is induced by some scheme morphism $T\longrightarrow S.$ Here, the map $w_Y$ sends $t\in T(k)=Hom_{Sch/k}({ {\mathrm{Spec~}} }k,T)$ to the morphism of motives $Y\circ(t\otimes id_{M^c(X)})-Y\circ(t_0\otimes id_{M^c(X)}):$ $$M^c(X)\cong{ {\mathbb Z} }\otimes M^c(X)\buildrel t\otimes id_{M^c(X)}-t_0\otimes id_{M^c(X)}\over\longrightarrow M(T)\otimes M^c(X) \buildrel Y\over\longrightarrow { {\mathbb Z} }(n)[m],$$ where $t$ and $t_0$ are regarded as morphisms from ${ {\mathbb Z} }\buildrel\text{def}\over=M(k)$ to $M(T)$ in $DM.$
A regular homomorphism $\phi:H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))\longrightarrow S(k)$ is said [**universal**]{} if for any regular homomorphism $\phi':H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))\longrightarrow S'(k),$ there is a unique homomorphism of semi-abelian varieties $a: S\longrightarrow S'$ such that $a\circ\phi=\phi'.$
\[defn: algebraic representative\] The universal regular homomorphism, if it exists, is written as $$\Phi^{m,n}_{c,X}: H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))\longrightarrow Alg_{c,X}^{m,n}(k).$$ The target semi-abelian variety $Alg_{c,X}^{m,n}$ is called the [**semi-abelian representative**]{} of $H_{c,alg}^{m}(X,{ {\mathbb Z} }(n)).$
For smooth proper schemes $X,$ under the assumption of resolution of singularities, the universal regular homomorphism $\Phi_{c,X}^{2r,r}$ agrees with the algebraic representative $\Phi_X^r$ for the Chow group reviewed in Subsection \[section: The case of smooth proper schemes\] (Proposition \[prop: comparison with the classical algebraic representatives\]).
\[rem: surjection from a semiabelian variety\] Let $X$ be a connected scheme over $k.$ Then, given a regular homomorphism $$\phi:H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))\longrightarrow S(k),$$ there is a semi-abelian variety $S_0$ (pointed at the unit) and $Y_0\in Hom_{DM}(M(S_0)\otimes M^c(X),{ {\mathbb Z} }(n)[m])$ such that $\mathrm{im}(\phi\circ w_{Y_0})=\mathrm{im}(\phi).$
We follow the method of [@Murre Proof of Lemma 1.6.2 (i)]. Consider the diagram $$S'(k)\buildrel w_{Y'}\over\longrightarrow H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))\buildrel\phi\over\longrightarrow S(k)$$ where $S'$ is a semi-abelian variety (pointed at the unit) and $Y'\in Hom_{DM}(M(S')\otimes M^c(X),{ {\mathbb Z} }(n)[m]).$ Since the composition is induced by a homomorphism of semi-abelian varieties, the image of $\phi\circ w_{Y'}$ has a structure of a semi-abelian variety.
Choose $S_0$ and $Y_0$ such that the dimension of $\mathrm{im}(\phi\circ w_{Y_0})$ is maximal among such diagrams. We claim that they have the desired property $\mathrm{im}(\phi\circ w_{Y_0})=\mathrm{im}(\phi).$
Suppose that $\mathrm{im}(\phi\circ w_{Y_0})\neq\mathrm{im}(\phi).$ Then, there is an element $x\in H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))$ such that $\phi(x)\notin\mathrm{im}(\phi\circ w_{Y_0}).$ Using Proposition \[prop: parametrization by semiabelian varieties\], we can find a semi-abelian variety $S_1$ and $Y_1\in Hom_{DM}(M(S_1)\otimes M^c(X),{ {\mathbb Z} }(n)[m])$ such that $\phi(x)\in\mathrm{im}(\phi\circ w_{Y_1}).$
Now, let us put $$S_2:=S_0\times S_1$$ and $$Y_2:=Y_0\circ(p_0\otimes id_{M^c(X)})+Y_1\circ(p_1\otimes id_{M^c(X)}),$$ where $p_i:M(S_0\times S_1)\longrightarrow M(S_i)$ ($i=0, 1$) is the morphism induced by the $i$-th projection. Also, let $e:H_0(S_0,{ {\mathbb Z} })^0\longrightarrow H_0(S_0\times S_1,{ {\mathbb Z} })^0$ be the map that sends the class of $\sum_i n_i x_i$ ($n_i\in { {\mathbb Z} }$ and $x_i\in S_0(k)$) to that of $\sum_i n_i(x_i\times 0),$ where $0$ denotes the basepoint (i.e. the unit) of $S_1.$ (Note that the product $S_2:=S_0\times S_1$ is a semi-abelian variety. Indeed, if it is not semi-abelian, then $S_0\times S_1$ contains $\mathbb G_a$ as an algebraic subgroup. However, if neither $S_0$ nor $S_1$ allows imbedding $\mathbb G_a\hookrightarrow S_i,$ there cannot be a nontrivial imbedding $\mathbb G_a\hookrightarrow S_0\times S_1.$) Observe that we are given with the commutative diagram $$\xymatrix{H_0(S_0,{ {\mathbb Z} })^0 \ar@/^1pc/[drr]^-{w_{Y_0}} \ar[dr]_-e\\
& H_0(S_2,{ {\mathbb Z} })^0 \ar[r]_-{w_{Y_2}} & H_{c,alg}^{m}(X,{ {\mathbb Z} }(n)) \ar[r]_-\phi & S(k)}$$ Therefore, $\mathrm{im}(\phi\circ w_{Y_2})\supset\mathrm{im}(\phi\circ w_{Y_0}).$ Similarly, we have $\mathrm{im}(\phi\circ w_{Y_2})\supset\mathrm{im}(\phi\circ w_{Y_1}).$ Since $\phi(x)\in \mathrm{im}(\phi\circ w_{Y_1})$ but $\phi(x)\not\in \mathrm{im}(\phi\circ w_{Y_0})$ and $\mathrm{im}(\phi\circ w_{Y_2})$ and $\mathrm{im}(\phi\circ w_{Y_0})$ are semi-abelian subvarieties (in particular, closed and irreducible subvarieties) of $S,$ these inclusions imply that $\dim\mathrm{im}(\phi\circ w_{Y_2})>\dim\mathrm{im}(\phi\circ w_{Y_0}).$ This is a contradiction.
\[lem: the image is semiabelian\] If $\phi:H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))\longrightarrow S(k)$ is a regular homomorphism, then the image of $\phi$ has a structure of a semi-abelian subvariety of $S.$
By Proposition \[rem: surjection from a semiabelian variety\], there are semi-abelian variety $S_0$ and morphism $Y_0\in Hom_{DM}(M(S_0)\otimes M^c(X),{ {\mathbb Z} }(n)[m])$ such that $\mathrm{im}(\phi)=\mathrm{im}(\phi\circ w_{Y_0}).$ Since $\phi\circ w_{Y_0}$ is a homomorphism between semi-abelian varieties, its image $\mathrm{im}(\phi)$ is a semi-abelian variety.
Even if we had allowed regular homomorphisms to take a larger class of algebraic groups for the targets in Definition \[defn: regular homomorphism\], Corollary \[lem: the image is semiabelian\] still holds because it basically depends only on Proposition \[prop: parametrization by semiabelian varieties\]. Thus, considering only the class of semi-abelian varieties in defining regular homomorphisms is not a restriction.
\[prop: an algebraic representative is surjective\] Suppose $\Phi^{m,n}_{c,X}: H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))\longrightarrow Alg_{c,X}^{m,n}(k)$ is the universal regular homomorphism. Then, it is surjective, and it also induces a surjective homomorphism on the torsion parts.
The surjectivity of $\Phi^{m,n}_{c,X}$ is immediate from Corollary \[lem: the image is semiabelian\]. As for the claim on the torsion parts, by Proposition \[rem: surjection from a semiabelian variety\], there is a semi-abelian variety $S_0$ and a surjective homomorphism, say, $f: S_0(k)\longrightarrow Alg_{c,X}^{m,n}(k)$ that factors through $H_{c,alg}^{m}(X,{ {\mathbb Z} }(n)).$ Since the kernel of $f$ is an extension of a finite group by a divisible group, we have $\mathrm{ker}(f)\otimes { {\mathbb Q} }/{ {\mathbb Z} }=0.$ This implies that $f$ induces a surjection on the torsion parts.
\[rem: homological approach\] There is a homological analogue of Subsections \[section: The algebraic part\] and \[section: regular homomorphism\]. For motivic homology $H_i(X,{ {\mathbb Z} }(j))$ of any connected scheme $X,$ we define the algebraic part by $$H_i^{alg}(X,{ {\mathbb Z} }(j))=\bigcup_{\substack{T,~{\text{smooth}}\\ {\text{connected}}}}\mathrm{im}\{H_0(T,{ {\mathbb Z} })^0\times Hom_{DM}(M(T)(j)[i],M(X))\longrightarrow H_i(X,{ {\mathbb Z} }(j))\},$$ where the map sends $z\in H_0(T,{ {\mathbb Z} })^0$ and $Y'\in Hom_{DM}(M(T)(j)[i],M(X))$ to the composition $$Y'\circ (z\otimes{ {\mathbb Z} }(j)[i]): { {\mathbb Z} }\otimes { {\mathbb Z} }(j)[i]\buildrel z\otimes{ {\mathbb Z} }(j)[i]\over\longrightarrow M(T)\otimes{ {\mathbb Z} }(j)[i]\buildrel Y'\over\longrightarrow M(X).$$
By a regular homomorphism $\phi': H_i^{alg}(X,{ {\mathbb Z} }(j))\longrightarrow S(k)$ ($S$ is a semi-abelian variety), we mean a group homomorphism such that the composition $$T(k)\buildrel {v_{Y'}}\over\longrightarrow H_i^{alg}(X,{ {\mathbb Z} }(j))\buildrel\phi'\over\longrightarrow S(k)$$ (The map $v_{Y'}$ sends $t\in T(k)$ to $Y'\circ ((t-t_0)\otimes{ {\mathbb Z} }(j)[i])$) is induced by a scheme morphism for any smooth connected scheme $T$ pointed at $t_0$ and for any morphism $Y'\in Hom_{DM}(M(T)(j)[i],M(X)).$ The initial regular homomorphism $\Phi_{i,j}^{X}:H_i^{alg}(X,{ {\mathbb Z} }(j))\longrightarrow Alg_{i,j}^X(k)$ among all regular homomorphisms for a fixed $H_i^{alg}(X,{ {\mathbb Z} }(j))$ is, if it exists, said universal as before.
The theory for motivic cohomology with compact supports and that for motivic homology are equivalent for smooth schemes via the Friedlander-Voevodsky duality ([@VSF5 Theorem 4.3.7 (3)]). We have already seen a special case in the proof of Proposition \[prop: algebraic part of zero cycles\]. Let $X$ be a smooth connected scheme of dimension $d.$ Suppose $m+i=2d$ and $n+j=d.$ Then, with Voevodsky’s cancellation theorem ([@Voevodsky; @cancellation]), the Friedlander-Voevodsky duality isomorphism induces an isomorphism $$f:H_{c,alg}^m(X,{ {\mathbb Z} }(n))\buildrel\cong\over\longrightarrow H_i^{alg}(X,{ {\mathbb Z} }(j))$$ under resolution of singularities, or unconditionally up to $p$-torsion by [@Kelly Theorem 5.5.14 (3)].
Moreover, a map $$\phi': H_i^{alg}(X,{ {\mathbb Z} }(j))\longrightarrow S(k)$$ is regular if and only if $$\phi:=\phi'\circ f: H_{c,alg}^m(X,{ {\mathbb Z} }(n))\longrightarrow S(k)$$ is regular as can be seen from the commutativity of the following diagram for all smooth connected pointed schemes $T:$ $$\xymatrix{ T(k) \ar[r]^-{v_{Y'}} \ar@{=}[d] & H_i^{alg}(X,{ {\mathbb Z} }(j)) \ar[r]^-{\phi'} & S(k) \ar@{=}[d] \\
T(k) \ar[r]^-{w_Y} & H_{c,alg}^m(X,{ {\mathbb Z} }(n)) \ar[u]^f_{\cong} \ar[r]^-\phi & S(k)}$$ where $Y\in Hom_{DM}(M(T)\otimes M^c(X),{ {\mathbb Z} }(n)[m])$ and $Y'\in Hom_{DM}(M(T)(j)[i],M(X))$ are any morphisms corresponding to each other under the duality isomorphism and the cancellation theorem.
Therefore, we may conclude that $\phi$ is universally regular if and only if $\phi'$ is universally regular. In particular, $Alg_{c,X}^{m,n}$ is isomorphic to $Alg_{i,j}^X$ by a unique isomorphism.
Existence of universal regular homomorphisms
--------------------------------------------
We prove the existence of universal regular homomorphisms $\Phi_{c,X}^{m,n}:H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))\longrightarrow Alg_{c,X}^{m,n}(k)$ for $m\leq n+2$ for all connected schemes $X,$ and for $(m,n)=(2\dim X,\dim X)$ for all smooth connected schemes $X.$ We use the methods of Serre [@Serre], Saito [@Saito] and Murre [@Murre].
We consider an analogue of a maximal morphism ([@Serre Définition 2]), which we shall call a maximal homomorphism (Definition \[defn: maximal homomorphism\]). We characterize the universal regular homomorphism as the maximal homomorphism whose target semi-abelian variety has the maximal dimension (Proposition \[prop: existence criterion\]). This is a generalization of [@Saito Theorem 2.2] as presented in [@Murre Proposition 2.1] to our context (cf. [@Serre Théorème 2]). With this criterion, it then remains to bound the dimensions of the targets of maximal homomorphisms to obtain the existence of universal regular homomorphisms. To achieve this, we use the Beilinson-Lichtenbaum conjecture which is now a theorem by the work of Rost, Voevodsky and others.
Throughout this subsection, $X$ is an arbitrary connected scheme over $k.$
\[defn: maximal homomorphism\] A regular homomorphism $\phi: H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))\longrightarrow S(k)$ is called [**maximal**]{} if it is surjective and for any factorization $$\xymatrix{& S'(k) \ar[d]^-{\forall\pi,~{\text{isogeny}}}\\
H_{c,alg}^{m}(X,{ {\mathbb Z} }(n)) \ar[r]_-\phi \ar[ur]^-{\forall~{\text{regular}}} & S(k),}$$ $\pi$ is an isomorphism.
\[lem: maximal homomorphism factorization\] Let $\phi: H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))\longrightarrow S(k)$ be a regular homomorphism. Then, there is a factorization $$\xymatrix{ H_{c,alg}^{m}(X,{ {\mathbb Z} }(n)) \ar[rr]^-\phi \ar[dr]_-g && S(k) \\
& S'(k) \ar[ur]_-h}$$ where $g$ is a maximal homomorphism and $h$ is a finite morphism.
We follow the proof of [@Serre Théorème 1]. By Corollary \[lem: the image is semiabelian\], we may assume that $\phi$ is surjective. If $\phi$ is maximal, there is nothing to prove.
If $\phi$ is not maximal, there is a factorization $$\xymatrix{& S_1(k) \ar[d]^-{\pi_1}\\
H_{c,alg}^{m}(X,{ {\mathbb Z} }(n)) \ar@{->>}[r]_-\phi \ar@{->>}[ur]^-{\phi_1,~{\text{regular}}} & S(k),}$$ where $\pi_1$ is an isogeny that is not an isomorphism. Note that $\phi_1$ is surjective because $\pi_1$ is an isogeny. If $\phi_1$ is maximal, there is nothing more to do.
Repeat this process. Suppose that we obtain an infinite tower $$\xymatrix{& \vdots\ar[d]^{\pi_3,~{\text{isog., not an isom.}}}\\
& S_2(k) \ar[d]^{\pi_2,~{\text{isog., not an isom.}}}\\
& S_1(k) \ar[d]^{\pi_1,~{\text{isog, not an isom.}}}\\
H_{c,alg}^{m}(X,{ {\mathbb Z} }(n)) \ar@{->>}[r]_<<<<\phi \ar@{->>}[ur]|-{\phi_1,~{\text{reg.}}} \ar@{->>}[uur]|-{\phi_2,~{\text{reg.}}} & S(k).}$$ By Proposition \[rem: surjection from a semiabelian variety\], choose a semi-abelian variety $S_0$ and $Y_0\in Hom_{DM}(M(S_0)\otimes M^c(X),{ {\mathbb Z} }(n)[m])$ such that $\phi\circ w_{Y_0}$ is surjective. Then, since $\pi_i$’s are isogeny, $\phi_i\circ w_{Y_0}$ is surjective for all $i.$
Then, we obtain the diagram of function fields $$\xymatrix{& \vdots \\
& K(S_2) \ar@{_{(}->}[ddl] \ar@{^{(}->}[u]_{\text{not an isom.}}\\
& K(S_1) \ar@{_{(}->}[dl] \ar@{^{(}->}[u]_{\text{not an isom.}}\\
K(S_0) & K(S) \ar@{_{(}->}[l] \ar@{^{(}->}[u]_{\text{not an isom.}}}$$ Therefore, we have $$K(S_0)\supset \bigcup_{i\geq1}K(S_i)\supset K(S),$$ where the extension $\bigcup_{i\geq1}K(S_i)/ K(S)$ is not finitely generated. However, the extension $K(S_0)/K(S)$ is finitely generated. Since a subextension of a finitely generated field extension is finitely generated ([@Serre Lemme 1]), this is a contradiction.
We need one more lemma.
\[lem: existence implies uniqueness\] Let $\phi: H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))\longrightarrow S(k)$ be a surjective regular homomorphism and $\phi': H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))\longrightarrow S'(k)$ be any regular homomorphism. Then, there is at most one scheme morphism $f:S\longrightarrow S'$ that makes the following diagram commute: $$\label{diagram: uniqueness}
\xymatrix{ H_{c,alg}^{m}(X,{ {\mathbb Z} }(n)) \ar@{->>}[r]^<<<<<\phi \ar[dr]_<<<<<<<<<{\phi'} & S(k) \ar[d]^f \\
& S'(k).}$$
Choose a semi-abelian variety $S_0$ and $Y_0\in Hom_{DM}(M(S_0)\otimes M(X),{ {\mathbb Z} }(n)[m])$ as in Proposition \[rem: surjection from a semiabelian variety\]. Then, since $\phi\circ w_{Y_0}: S_0\longrightarrow S$ is a morphism between connected smooth schemes and all fibers have the same dimension, it is flat by [@Matsumura Corollary to Theorem 23.1]. It is also surjective, so it is a strict epimorphism. In particular, $$i: Hom_{Sch}(S,S')\buildrel -\circ\phi\circ w_{Y_0}\over\longrightarrow Hom_{Sch}(S_0,S')$$ is injective.
Suppose that there is another $f':S\longrightarrow S'$ that makes the diagram (\[diagram: uniqueness\]) commute. Then, $f$ and $f'$ are sent to the same element under the injection $i:$ $f\circ\phi\circ w_{Y_0}=\phi'\circ w_{Y_0}=f'\circ\phi\circ w_{Y_0}.$ Hence, $f=f'.$
\[prop: existence criterion\] There is a universal regular homomorphism for $H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))$ if and only if there is a constant $c$ such that the inequality $\dim S\leq c$ holds for any maximal homomorphism $$\phi: H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))\longrightarrow S(k).$$
In fact, the maximal homomorphism with a maximal dimensional target is the universal regular homomorphism.
$``\Rightarrow"$ is clear. We prove the converse by combining the arguments for [@Serre Théorème 2] and [@Murre Proposition 2.1].
Let $\phi_0: H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))\longrightarrow S_0(k)$ be a maximal homomorphism with the maximal dimensional target $S_0.$ Suppose that $\phi: H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))\longrightarrow S(k)$ is a regular homomorphism.
Now, since $\phi_0\times\phi: H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))\longrightarrow (S_0\times S)(k)$ is also regular, by Lemma \[lem: maximal homomorphism factorization\], there is a factorization $$\phi_0\times\phi: H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))\buildrel g,~{\text{max.}}\over\longrightarrow S_1(k)\buildrel i,~{\text{fin.}}\over\longrightarrow (S_0\times S)(k)$$ with some maximal homomorphism $g.$
Consider the commutative diagram $$\xymatrix{& & S_0(k) \\
H_{c,alg}^{m}(X,{ {\mathbb Z} }(n)) \ar@{->>}[r]^-{g,~{\text{max.}}} \ar@/^/@{->>}[urr]^-{\phi_0,~{\text{max.}}} \ar@/_/[drr]_-\phi & S_1(k) \ar[r]^-i \ar[ur]^{\text{isom.}} \ar[dr] & (S_0\times S)(k) \ar[u]_-{p_0} \ar[d]^-p\\
& & S(k)}$$
Since $p_0\circ i$ is surjective and $S_0$ has the maximal dimension, we must have $\dim S_0=\dim S_1.$ Hence, $p_0\circ i$ is an isogeny. Since $\phi_0$ is a maximal homomorphism, $p_0\circ i$ is an isomorphism. We put $$r:=(p_0\circ i)^{-1}:S_0\longrightarrow S_1,$$ and define $h:=p\circ i\circ r: S_0\longrightarrow S.$ Then, $$\begin{aligned}
h\circ\phi_0 &=& p\circ i\circ r\circ \phi_0 \nonumber\\
&=& p\circ i\circ g\nonumber\\
&=& \phi.\nonumber\end{aligned}$$
By Lemma \[lem: existence implies uniqueness\], $h$ is the only scheme morphism for which $\phi=h\circ\phi_0$ holds. Therefore, $\phi_0$ is the universal regular homomorphism for $H_{c,alg}^{m}(X,{ {\mathbb Z} }(n)).$
\[thm: existence in codimensions\] Let $X$ be any connected scheme over $k.$ Then, there is a universal regular homomorphism for $H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))$ if $m\leq n+2.$
By Proposition \[prop: existence criterion\], it suffices to show that the dimensions of the target semi-abelian varieties of surjective regular homomorphisms are bounded.
Let $\phi: H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))\longrightarrow S(k)$ be a surjective regular homomorphism. By Proposition \[rem: surjection from a semiabelian variety\], we may choose a semi-abelian variety $S_0$ and $Y_0\in Hom_{DM}(M(S_0)\otimes M^c(X),{ {\mathbb Z} }(n)[m])$ such that the composition $$f: S_0(k)\buildrel w_{Y_0}\over\longrightarrow H_{c,alg}^{m}(X,{ {\mathbb Z} }(n))\buildrel\phi\over\longrightarrow S(k)$$ is a surjective homomorphism. We also write the corresponding homomorphism of semi-abelian varieties by the same symbol $f.$
Let $l$ be a prime relatively prime to the characteristic of the base field $k$ and the index $(\ker f:\ker f^0),$ where $\ker f^0$ is the identity component (it is a semi-abelian variety) of the group scheme $\ker f.$
Let us look at the $l$-torsion part: $$\xymatrix{_l S_0(k) \ar@/_2pc/[rrrr]_{_l f} \ar[rr]^-{_l w_{Y_0}} && _l H_{c,alg}^{m}(X,{ {\mathbb Z} }(n)) \ar[rr]^-{_l\phi} && _l S(k).}$$ We claim that $_l\phi$ is surjective. In fact, $_l f$ is surjective. For this, by the snake lemma, it is enough to show that $\ker f/l\cdot\ker f$ vanishes. Since $\ker f^0$ is a semi-abelian variety and $(\ker f:\ker f^0)$ is prime to $l,$ another application of the snake lemma to the diagram $$\xymatrix{ 0\ar[r] & \ker f^0 \ar[r] \ar[d]^{l} & \ker f \ar[r] \ar[d]^{l} & \ker f/\ker f^0 \ar[r] \ar[d]^{l} & 0\\
0\ar[r] & \ker f^0 \ar[r] & \ker f \ar[r] & \ker f/\ker f^0 \ar[r] & 0}$$ yields the desired result.
Now, we have $$\label{diagram: 1}
\xymatrix{ H_{c}^{m-1}(X,{ {\mathbb Z} }/l(n)) \ar@{->>}[r] & _l H_{c}^{m}(X,{ {\mathbb Z} }(n)) \supset {_l}H_{c,alg}^{m}(X,{ {\mathbb Z} }(n)) \ar@{->>}[r]^<<<<{_l\phi} & _l S(k).}$$ Since the dimension of $S$ is less than or equal to the $l$-rank of $S,$ it is now enough to prove that $H_{c}^{m-1}(X,{ {\mathbb Z} }/l(n))$ is finite for any connected scheme $X$ if $m\leq n+2.$
Let $X\hookrightarrow \bar X$ be a Nagata compactification of $X$ with $Z:=\bar X\setminus X.$ Since $l\neq\mathrm{char}~k,$ there is a localization sequence ([@Kelly Proposition 5.5.5]) $$\cdots\longrightarrow H^{m-2}(Z,{ {\mathbb Z} }/l(n)) \longrightarrow H^{m-1}_c(X,{ {\mathbb Z} }/l(n)) \longrightarrow H^{m-1}(\bar X,{ {\mathbb Z} }/l(n)) \longrightarrow\cdots.$$ Therefore, it suffices to show the finiteness of $H^{m-1}(Y,{ {\mathbb Z} }/l(n))$ ($m\leq n+2$) for all connected proper schemes $Y.$ This follows from the case where $Y$ is irreducible. Indeed, this can be seen by induction on the number of irreducible components and on the dimension of $Y$ using the abstract blow-up sequence for motivic cohomology with ${ {\mathbb Z} }/l$-coefficients ([@Kelly Proposition 5.5.4]) associated with the abstract blow-up square $$\xymatrix{ Y_1\times(\bigcup_{i\neq 1}Y_i) \ar[r] \ar[d] & \bigcup_{i\neq1} Y_i \ar[d]^-{\text{inc.}}\\
Y_1 \ar[r]_-{\text{inc.}} & Y=\bigcup_{i=1,\cdots,r} Y_i,}$$ where $Y_i$’s are the irreducible components of $Y$ and $Y_1$ has the minimal dimension among $Y_i$’s.
Suppose that $Y$ is an irreducible proper scheme. By de Jong’s alteration ([@de; @Jong Theorem 4.1]), there is a smooth projective scheme $Y'$ and a proper surjective morphism $h:Y'\longrightarrow Y$ with an open dense subscheme $U$ of $Y$ such that the pullback $h':U'\longrightarrow U$ of $h$ over $U$ is finite and étale of degree $\delta$ prime to $l$ (choose a different $l$ if necessary), i.e. there is a pullback square: $$\xymatrix{Y' \ar[d]_h & U' \ar@{_{(}->}[l]_{\text{dense}} \ar[d]^{h',~\text{fin. \'et. of degree $\delta$}}\\
Y & U \ar@{_{(}->}[l]^{\text{dense}}}$$ The localization sequence associated with the dominant morphism $U\hookrightarrow Y$ $$\cdots\longrightarrow H_c^{m-1}(U,{ {\mathbb Z} }/l(n)) \longrightarrow H^{m-1}_c(Y,{ {\mathbb Z} }/l(n)) \longrightarrow H_c^{m-1}(Y\setminus U,{ {\mathbb Z} }/l(n)) \longrightarrow\cdots$$ implies that the finiteness of $H^{m-1}_c(Y,{ {\mathbb Z} }/l(n))$ follows from the finiteness of $H_c^{m-1}(U,{ {\mathbb Z} }/l(n))$ by induction on dimensions because $\dim Y\setminus U<\dim Y.$ This, in turn, follows from the finiteness of $H_c^{m-1}(U',{ {\mathbb Z} }/l(n))$ because the composition $$H_c^{m-1}(U,{ {\mathbb Z} }/l(n))\buildrel h'^*\over \longrightarrow H_c^{m-1}(U',{ {\mathbb Z} }/l(n))\buildrel h'_*\over\longrightarrow H_c^{m-1}(U,{ {\mathbb Z} }/l(n))$$ is the multiplication by $\delta,$ which is an isomorphism as $\delta$ is prime to $l.$
Now, the induction on dimension with the localization sequence associated with the dominant morphism $U'\hookrightarrow Y'$ $$\cdots\longrightarrow H_c^{m-2}(Y'\setminus U',{ {\mathbb Z} }/l(n)) \longrightarrow H^{m-1}_c(U',{ {\mathbb Z} }/l(n)) \longrightarrow H_c^{m-1}(Y',{ {\mathbb Z} }/l(n)) \longrightarrow\cdots$$ implies that the finiteness of $H^{m-1}_c(U',{ {\mathbb Z} }/l(n))$ follows from that of $H_c^{m-1}(Y',{ {\mathbb Z} }/l(n))=H^{m-1}(Y',{ {\mathbb Z} }/l(n))$ (the equality holds because $Y'$ is proper).
By the theorem of Rost and Voevodsky, the Geisser-Levine étale cycle map ([@Geisser-Levine]) $$H^{m-1}(Y',{ {\mathbb Z} }/l(n))\longrightarrow H_{\acute et}^{m-1}(Y',\mu_l^{\otimes n})$$ is injective for $m\leq n+2.$ Since the target étale cohomology group is finite by [@Milne; @etale Chapter VI, Corollary 2.8], we obtain the finiteness of $H^{m-1}(Y',{ {\mathbb Z} }/l(n)).$
Let us deal with the zero cycle case. We assume that the scheme $X$ is smooth to use the Friedlander-Voevodsky duality theorem.
\[thm: existence in zero dim\] If $X$ is a smooth connected scheme of dimension $d,$ then $H^{2d}_{c,alg}(X,{ {\mathbb Z} }(d))$ has a universal regular homomorphism.
Proceeding as in the proof—particularly the diagram (\[diagram: 1\])—of Theorem \[thm: existence in codimensions\], we need to show the finiteness of $H_c^{2d-1}(X,{ {\mathbb Z} }/l(d)).$ Since $X$ is smooth and $l$ is prime to the characteristic of $k,$ this group is isomorphic to the motivic homology group $H_1(X,{ {\mathbb Z} }/l)$ by the Friedlander-Voevodsky duality ([@Kelly Theorem 5.5.14]; cf. [@VSF5 Theorem 4.3.7 (3)] under resolution of singularities). But $H_1(X,{ {\mathbb Z} }/l)$ is finite because its dual is isomorphic to the finite group $H^1_{{ {\acute et} }}(X,{ {\mathbb Z} }/l)$ by a theorem of Suslin-Voevodsky ([@MVW Theorem 10.9]; cf. [@Suslin-Voevodsky-Inventiones Corollary 7.8] under resolution of singularities).
\[prop: agrees with Serre’s\] Assume resolution of singularities. For any smooth connected scheme $X$ of dimension $d$ over $k,$ the semi-abelian representative $Alg_{c,X}^{2d,d}$ is canonically isomorphic to Serre’s generalized Albanese variety. In fact, the Albanese map $alb_X$ is the universal regular homomorphism for $H_{c,alg}^{2d}(X,{ {\mathbb Z} }(d))$ (up to the duality isomorphism $H_{c,alg}^{2d}(X,{ {\mathbb Z} }(d))\cong H_0(X,{ {\mathbb Z} })^0$).
Let $id'\in Hom_{DM}(M(X)\otimes M^c(X),{ {\mathbb Z} }(d)[2d])$ be the element corresponding to $id\in Hom_{DM}(M(X),M(X))$ under the Friedlander-Voevodsky duality ([@VSF5 Theorem 4.3.7 (3)]). The composition $$X(k)\buildrel w_{id'} \over\longrightarrow H_{c,alg}^{2d}(X,{ {\mathbb Z} }(d))\buildrel\Phi_{c,X}^{2d,d}\over\longrightarrow Alg_{c,X}^{2d,d}(k)$$ has a structure of a scheme morphism because $\Phi_{c,X}^{2d,d}$ is regular. Thus, the universality of generalized Albanese varieties gives the dotted arrow $$\xymatrix{X(k) \ar[r]^-{w_{id}} \ar@{=}[d] & H_0(X,{ {\mathbb Z} })^0 \ar[r]^-{alb_X} & Alb_X(k) \ar@{..>}[d]^{\exists !} \\
X(k) \ar[r]^-{w_{id'}} & H_{c,alg}^{2d}(X,{ {\mathbb Z} }(d)) \ar[u]_\cong^f \ar[r]^-{\Phi_{c,X}^{2d,d}} & Alg_{c,X}^{2d,d}(k)}$$ where the middle vertical isomorphism is the one in Proposition \[prop: algebraic part of zero cycles\].
Conversely, we show that $alb_X\circ f$ is regular, or equivalently, the composition $$\label{equation: our goal}
T(k) \buildrel (Y\circ-) -(Y\circ t_0) \over\longrightarrow H_0(X,{ {\mathbb Z} })^0 \buildrel alb_X\over\longrightarrow Alb_X(k)$$ is induced by a scheme morphism for any smooth connected scheme $T$ pointed at $t_0$ and for any morphism $Y\in Hom_{DM}(M(T),M(X)).$
For Serre’s Albanese variety $Alb_S,$ $\widetilde{Alb_S}:=Hom_{Sm/k}(-,Alb_S)$ is a homotopy invariant presheaf with transfers by [@Spiess-Szamuely Lemma 3.2]. (In [*loc. cit.,*]{} the symbol $\widetilde{Alb_S}$ stands for Ramachandran’s Albanese sheaf, but the lemma is applicable since its proof deals with arbitrary commutative group schemes.) Note that $\widetilde{Alb_S}$ is a sheaf with respect to the Nisnevich topology by [@Milne; @etale Chapter II, Corollary 1.7].
Let $a_{S,p}:S\longrightarrow Alb_S$ be the universal morphism that sends a rational point $p\in S(k)$ to the unit of $Alb_S.$ Since $\widetilde{Alb_S}$ is a presheaf with transfers, by Yoneda’s lemma, there is a corresponding map $$\tilde a_{S,p}:\mathbb Z_{tr}(S)\longrightarrow \widetilde{Alb_S}$$ of presheaves with transfers. Taking $C_*$ and composing with the canonical map $C_*\widetilde{Alb_S}\longrightarrow \widetilde{Alb_S},$ which exists by the homotopy invariance of $\widetilde{Alb_S},$ we obtain $$\label{eqn: SS}
\tilde{\tilde a}_{S,p}:C_*\mathbb Z_{tr}(S)\buildrel \tilde a_{S,p} \over\longrightarrow C_*\widetilde{Alb_S}\buildrel{\text{canon.}}\over\longrightarrow \widetilde{Alb_S}.$$
Composing the map (\[eqn: SS\]) with the inclusions $$\tilde S:=Hom_{Sm/k}(-,S)\buildrel\text{inc.}\over\longrightarrow { {\mathbb Z} }_{tr}(S)\buildrel\text{$id$ in deg. 0}\over\longrightarrow C_*{ {\mathbb Z} }_{tr}(S)$$ of complexes of presheaves on $Sm/k$, we obtain $$\tilde S\longrightarrow C_*{ {\mathbb Z} }_{tr}(S)\buildrel \tilde{\tilde a}_{S,p}\over\longrightarrow \widetilde{Alb_S}.$$ By taking the zero-th cohomology presheaves, this induces the maps $$\tilde S\longrightarrow H^0(C_*{ {\mathbb Z} }_{tr}(S))\buildrel H^0(\tilde{\tilde a}_{S,p})\over \longrightarrow \widetilde{Alb_S}$$ of presheaves on $Sm/k.$ By Nisnevich sheafification, this gives $$\label{map: Albanese sheaf}
\tilde S\longrightarrow H^0(C_*{ {\mathbb Z} }_{tr}(S))_{Nis}\buildrel H^0(\tilde{\tilde a}_{S,p})_{Nis}\over \longrightarrow \widetilde{Alb_S}$$
Now, let $\mathcal Y: C_*{ {\mathbb Z} }_{tr}(T)\longrightarrow C_*{ {\mathbb Z} }_{tr}(X)$ be the morphism in $D^-(Sh_{Nis}(Cor_k))$ corresponding to $Y\in Hom_{DM}(M(T),M(X))$ under the canonical isomorphisms $$\begin{aligned}
Hom_{DM}(M(T),M(X))&\cong& Hom_{DM}(C_*{ {\mathbb Z} }_{tr}(T),C_*{ {\mathbb Z} }_{tr}(X))\nonumber\\
&\cong& Hom_{D^-(Sh_{Nis}(Cor_k))}(C_*{ {\mathbb Z} }_{tr}(T),C_*{ {\mathbb Z} }_{tr}(X)) \text{ ~~(because $C_*{ {\mathbb Z} }_{tr}(X)$ is $\mathbb A^1$-local)}.\nonumber\end{aligned}$$ Passing $\mathcal Y$ to the zero-th cohomology Nisnevich sheaves, we obtain the genuine map of sheaves on $(Sm/k)_{Nis}$ $$\label{map: map between H0}
H^0(\mathcal Y)_{Nis}: H^0(C_*{ {\mathbb Z} }_{tr}(T))_{Nis}\longrightarrow H^0(C_*{ {\mathbb Z} }_{tr}(X))_{Nis}.$$
With the maps (\[map: Albanese sheaf\]) for $S=X$ and $T$ (respectively pointed at $x_0$ and $t_0$) and the map (\[map: map between H0\]), we obtain the diagram with solid arrows: $$\xymatrixcolsep{4pc}\xymatrix{\tilde X \ar[r] & H^0(C_*{ {\mathbb Z} }_{tr}(X))_{Nis} \ar[r]^-{H^0(\tilde{\tilde a}_{X,x_0})} & \widetilde{Alb_X}\\
\tilde T \ar[r]_-i & H^0(C_*{ {\mathbb Z} }_{tr}(T))_{Nis} \ar[r]_-{H^0(\tilde{\tilde a}_{T.t_0})} \ar[u]^{H^0(\mathcal Y)_{Nis}} & \widetilde{Alb_T} \ar@{..>}[u]_{\exists!~\tilde h}}$$ Note that the composition $\tilde T\longrightarrow \widetilde{Alb_T}$ of the bottom row is nothing but the map induced by $a_{T,t_0}:T\longrightarrow Alb_T.$ The universality of Serre’s Albanese variety and Yoneda’s lemma implies that there is a unique scheme morphism $h:Alb_T\longrightarrow Alb_X$ that induces $\tilde h$ in the above diagram such that $$\tilde h\circ H^0(\tilde{\tilde a}_{T,t_0})\circ i = H^0(\tilde{\tilde a}_{X,x_0})\circ H^0(\mathcal Y)_{Nis}\circ i.$$
Taking the section over ${ {\mathrm{Spec~}} }k,$ we obtain the diagram $$\xymatrixcolsep{4pc}\xymatrix{ & H_0(X,{ {\mathbb Z} }) \ar[r]^-{alb_{X,x_0}} & Alb_X(k)\\
T(k) \ar@/_2pc/[rr]_-{a_{T,t_0}} \ar[r]^-{\iota} & H_0(T,{ {\mathbb Z} }) \ar[r]^-{alb_{T.t_0}} \ar[u]^{Y\circ-} & Alb_T(k) \ar[u]_{h}}$$ (where $\iota$ is the canonical map that sends a rational point of $T$ to its class in the Suslin homology and $alb_{T,t_0}:=H^0(\tilde{\tilde a}_{X,x_0})(k)$ and similarly for $alb_{X,x_0}$) such that the equality $$\label{equation: equality}
h\circ a_{T,t_0}=alb_{X,x_0}\circ (Y\circ-)\circ\iota$$ holds. The equation (\[equation: equality\]) implies that $$\begin{aligned}
alb_X (Y\circ\iota(t) - Y\circ\iota(t_0))&=& alb_{X,x_0}\circ(Y\circ\iota(t))- alb_{X,x_0}\circ(Y\circ\iota(t_0))\nonumber\\
&=& h\circ a_{T,t_0}(t) - h\circ a_{T,t_0}(t_0)\nonumber\\
&=& h\circ a_{T,t_0}(t) - h(0)\nonumber\end{aligned}$$ for all $t\in T(k).$ In other words, the diagram $$\xymatrixcolsep{4pc}\xymatrixrowsep{3pc}\xymatrix{ T(k) \ar[drr]_{a_{T,t_0}} \ar[r]^-{(Y\circ-)-(Y\circ t_0)} & H_0(X,{ {\mathbb Z} })^0 \ar[r]^-{alb_X} & Alb_X(k) \\
& & Alb_T(k) \ar[u]_{h(-)-h(0)}}$$ commutes. Thus, the composition (\[equation: our goal\]) is induced by a scheme morphism.
Proposition \[prop: agrees with Serre’s\] says that, by Remark \[rem: homological approach\], $Alg_{0,0}^X$ agrees with $Alb_X$ under the assumption of resolution of singularities. However, resolution of singularities is unnecessary for this agreement. Indeed, it was needed only to use the Friedlander-Voevodsky duality theorem. Working throughout with the homological formulation explained in Remark \[rem: homological approach\], we can prove the homological version of the criterion (Proposition \[prop: existence criterion\]). Namely,
For any connected scheme $X$ over $k,$ the universal regular homomorphisms $\Phi_{i,j}^X: H_i^{alg}(X,{ {\mathbb Z} }(j))\longrightarrow Alg_{i,j}^X(k)$ exists if and only if there is a constant $c$ such that for any maximal homomorphism $\phi: H_i^{alg}(X,{ {\mathbb Z} }(j))\longrightarrow S(k),$ we have the inequality $\dim S\leq c.$
Therefore, as in the proof of Theorem \[thm: existence in zero dim\], the existence of $\Phi_{0,0}^X: H_0^{alg}(X,{ {\mathbb Z} })\longrightarrow Alg_{0,0}^X(k)$ follows, once one shows that the Suslin homology $H_1(X,{ {\mathbb Z} }/l)$ is finite for primes $l\neq\mathrm{char}~k.$ If $X$ is smooth, this follows from the finiteness of $H_{\acute et}^1(X,{ {\mathbb Z} }/l)$ by [@MVW Theorem 10.9]. Now, $Alg_{0,0}^X$ agrees with $Alb_X$ by the very argument after the diagram (\[equation: our goal\]) in the proof of Proposition \[prop: agrees with Serre’s\]—the part which does not use resolution of singularities. (If we assume resolution of singularities, the homological universal regular homomorphism $\Phi_{0,0}^X$ exists for all connected schemes $X.$ It can be seen by reducing the finiteness of $H_1(X,{ {\mathbb Z} }/l)$ to the case where $X$ is smooth by induction on the dimension of $X$ using the blow-up exact sequence for Suslin homology associated with a resolution of singularities of $X.$)
Since $Alg_{0,0}^X$ is by definition covariantly functorial in $X$ with respect to morphisms in $DM,$ we conclude that the covariant functoriality of Albanese varieties with respect to scheme morphisms extends to that with respect to morphisms of motives:
\[cor: covariant for DM morphisms\] Let $X$ and $Y$ be any smooth connected schemes. Then, any morphism $M(X)\longrightarrow M(Y)$ in $DM$ functorially induces a homomorphism of semi-abelian varieties $Alb_X\longrightarrow Alb_Y$ in the manner compatible with the usual functoriality of Serre’s Albanese varieties.
Study in codimension one
========================
As we mentioned in Remark \[rem: new remark\], the algebraic representatives in codimension one of smooth proper schemes are Picard varieties. We give a similar interpretation to the semi-abelian representative $Alg_{c,X}^{2,1}$ of an arbitrary smooth scheme $X.$
Motivic cohomology with compact supports as cdh hypercohomology {#subsection: 3.1}
---------------------------------------------------------------
For our purpose, we interpret motivic cohomology with compact supports as cdh hypercohomology. Let us review the definition of cdh cohomology with compact supports. Let $Sch/k$ be the category of schemes over $k.$ The cdh-topology is the minimal Grothendieck topology on $Sch/k$ generated by Nisnevich coverings and abstract blow-up squares ([@Friedlander-Voevodsky Definition 3.2]). The derived category $D^-(Sh_{cdh}(Sch/k))$ of bounded above complexes of cdh sheaves on $Sch/k$ is simply denoted by $D^-_{cdh}.$ By cdh sheafification, we mean the composition of the functors $$Sh_{Zar}(Sm/k)\buildrel\text{$t$-sheafification}\over\longrightarrow Sh_t(Sm/k)\longrightarrow Sh_{cdh}(Sch/k),$$ where $t$ signifies the Grothendieck topology obtained by restricting the cdh topology on $Sch/k$ to $Sm/k.$ The second functor is the left adjoint to the forgetful functor $Sh_{cdh}(Sch/k)\longrightarrow Sh_t(Sm/k);$ it is an equivalence of categories under resolution of singularities because, then, any scheme has a smooth cdh cover (\[Ibid., proof of Lemma 3.6\]).
\[defn: cdh with compact supports\] For a bounded above complex $\mathcal F$ of cdh sheaves on $Sch/k,$ the [**cdh cohomology with compact supports of $X\in Sch/k$ with coefficients in $\mathcal F$**]{} is defined as $$H_c^m(X_{cdh},\mathcal F):=Hom_{D^-_{cdh}}({ {\mathbb Z} }^c(X)_{cdh},\mathcal F[m]),$$ where ${ {\mathbb Z} }^c(X)_{cdh}$ is the cdh sheafification of the presheaf that sends an irreducible scheme $U$ to the free abelian group ${ {\mathbb Z} }^c(X)(U)$ generated by closed subschemes $Z$ of $U\times X$ such that the projection $Z\longrightarrow U$ is an open immersion.
\[prop: motivic cohomology as cdh cohomology\] Under resolution of singularities, for any $X\in Sch/k$ and any non-negative integers $m$ and $n,$ there is an isomorphism $$H_c^m(X,{ {\mathbb Z} }(n))\buildrel\cong\over\longrightarrow H_c^m(X_{cdh},{ {\mathbb Z} }(n)_{cdh}),$$ where ${ {\mathbb Z} }(n)_{cdh}$ is the cdh sheafification of Voevodsky’s motivic complex ${ {\mathbb Z} }(n)\buildrel\text{def}\over=C_*{ {\mathbb Z} }_{tr}(\mathbb G_m^{\wedge n})[-n]$ ([@MVW Definition 3.1]).
If $X$ is proper, this is [@MVW Theorem 14.20] (see also [@Suslin-Voevodsky Theorem 5.14]).
For a non-proper $X,$ choose a compactification $X\hookrightarrow \bar X$ with $Z:=\bar X\setminus X.$ For any scheme $S\in Sch/k,$ consider the composition $f_{S}^{m,n}$ of maps $$\begin{aligned}
Hom_{DM}(M^c(S),{ {\mathbb Z} }(n)[m]) &\buildrel (a)\over=& Hom_{D^-(Sh_{Nis}^{tr}(Sm/k))}(z_{equi}(S,0),{ {\mathbb Z} }(n)[m])\nonumber\\
&\buildrel (b)\over\to& Hom_{D^-(Sh_{Nis}(Sm/k))}({ {\mathbb Z} }^c(S),{ {\mathbb Z} }(n)[m]) \nonumber\\
&\buildrel (c)\over\to& Hom_{D^-(Sh_{t}(Sm/k))}({ {\mathbb Z} }^c(S)_t,{ {\mathbb Z} }(n)_t[m]) \nonumber\\
&\buildrel (d)\over\cong& Hom_{D^-(Sh_{cdh}(Sch/k))}({ {\mathbb Z} }^c(S)_{cdh},{ {\mathbb Z} }(n)_{cdh}[m]).\nonumber\end{aligned}$$ Here, (a) is an equality by the Nisnevich variant of [@MVW Lemma 9.19] because ${ {\mathbb Z} }(n)\buildrel\text{def}\over=C_*{ {\mathbb Z} }_{tr}(\mathbb G_m^{\wedge n})[-n]$ is an $\mathbb A^1$-local object by [@MVW Corollary 14.9]. (b) is induced by the inclusion ${ {\mathbb Z} }^c(S)\hookrightarrow z_{equi}(S,0).$ (c) is induced by the $t$-sheafification, and (d) is due to the equivalence of categories between $Sh_{t}(Sm/k)$ and $Sh_{cdh}(Sch/k).$ By the construction, $f_{S}^{m,n}$ is functorial in $S$ with respect to pushforwards along proper morphisms and pullbacks along flat morphisms. Therefore, $f_{S}^{m,n}$ is a chain map between the long exact sequence induced by the localization triangle $$M^c(Z)\longrightarrow M^c(\bar X)\longrightarrow M^c(X)\buildrel [+1]\over\longrightarrow$$ in $DM$ ([@MVW Theorem 16.15]) and that induced by the short exact sequence $$\label{ses: localization sequence of cdh sheaves}
0\longrightarrow { {\mathbb Z} }(Z)_{cdh}\longrightarrow { {\mathbb Z} }(\bar X)_{cdh}\longrightarrow { {\mathbb Z} }^c(X)_{cdh}\longrightarrow 0,$$ in $D^-(Sh_{cdh}(Sch/k))$ ([@Friedlander-Voevodsky Corollary 3.9]): $$\xymatrix{\ar[r] & H^{m-1}(\bar X,{ {\mathbb Z} }(n)) \ar[r] \ar[d]_{f_{\bar X}^{m-1,n}}^\cong & H^{m-1}(Z,{ {\mathbb Z} }(n)) \ar[r] \ar[d]_{f_{Z}^{m-1,n}}^\cong & H_c^{m}(X,{ {\mathbb Z} }(n)) \ar[d]_{f_{X}^{m,n}} \\
\ar[r] & H_{cdh}^{m-1}(\bar X,{ {\mathbb Z} }(n)_{cdh}) \ar[r] & H_{cdh}^{m-1}(Z,{ {\mathbb Z} }(n)_{cdh}) \ar[r] & H_c^m(X_{cdh},{ {\mathbb Z} }(n)_{cdh})}$$ $$\xymatrix{ \ar[r] & H^m(\bar X,{ {\mathbb Z} }(n)) \ar[r] \ar[d]_{f_{\bar X}^{m,n}}^\cong & H^m(Z,{ {\mathbb Z} }(n)) \ar[d]_{f_{Z}^{m,n}}^\cong \ar[r] &\\
\ar[r] & H_{cdh}^m(\bar X,{ {\mathbb Z} }(n)_{cdh}) \ar[r] & H_{cdh}^m(Z,{ {\mathbb Z} }(n)_{cdh}) \ar[r] &}$$ The four arrows between the cohomology groups without compact supports in the above diagram are isomorphisms by the proper case. Hence, the middle map is also an isomorphism.
\[rem: explicit cdh cohomology with compact supports\] Let $\mathcal F$ be a bounded above complex of cdh sheaves on $Sch/k$ and let $\mathcal I^\bullet$ be the total complex of a Cartan-Eilenberg resolution of $\mathcal F$ in $Sh_{cdh}(Sch/k).$ In view of the short exact sequence (\[ses: localization sequence of cdh sheaves\]) in the proof of Proposition \[prop: motivic cohomology as cdh cohomology\], we can express the cdh cohomology with compact supports more explicitly as $$H_c^m(X_{cdh},\mathcal F)\cong H^n(cone(\mathcal I^\bullet(\bar X)\longrightarrow \mathcal I^\bullet(Z))[-1]).$$
Now, our interest in this section is the cohomology group $H_c^2(X,{ {\mathbb Z} }(1))$ for a smooth scheme $X.$ In order to study this group with Proposition \[prop: motivic cohomology as cdh cohomology\], we would like to know ${ {\mathbb Z} }(1)_{cdh}$ explicitly.
\[lem: cdh sheafification of Gm\] Suppose that $X\in Sch/k$ is a simple normal crossing divisor on some smooth scheme. Then, under resolution of singularities, the restriction ${ {\mathbb Z} }(1)_{cdh,X}$ of ${ {\mathbb Z} }(1)_{cdh}$ to the small Zariski site on $X$ is quasi-isomorphic to the Zariski sheaf $\mathbb G_{m,X}[-1]$ of units on $X.$
There is a quasi-isomorphism ${ {\mathbb Z} }(1)\buildrel qis\over\longrightarrow{ {\mathbb G_m} }[-1]$ of complexes of presheaves on $Sm/k$ ([@MVW Theorem 4.1]). Therefore, we need to show that the restriction of the cdh sheafification of ${ {\mathbb G_m} }$ to the small Zariski site on $X$ agrees with ${ {\mathbb G_m} }_{,X},$ i.e. the canonical map $a: \mathbb G_{m,X}\longrightarrow \mathbb G_{m,cdh,X}$ on $X_{Zar}$ is an isomorphism. If $X$ is smooth, this follows from [@MVW Proposition 13.27] since ${ {\mathbb G_m} }$ has the structure of a homotopy invariant Nisnevich sheaf with transfers.
For the injectivity, it is enough to show that for any affine open subscheme $U={ {\mathrm{Spec~}} }A$ of $X,$ the map $a$ induces an injection $a_U:{ {\mathbb G_m} }_{,X}(U)\hookrightarrow { {\mathbb G_m} }_{,cdh,X}(U).$ Let $U_j$’s be the irreducible components of $U$ corresponding to the minimal ideals $\mathfrak p_j$’s of $A.$ Note that $\{U_j\longrightarrow U\}_j$ is a cdh cover by smooth schemes $U_j$ as $X$ is a strict normal crossing divisor. Now, consider the composition $$A^*={ {\mathbb G_m} }_{,X}(U)\buildrel a_U\over\longrightarrow { {\mathbb G_m} }_{,cdh,X}(U)\buildrel res\over \hookrightarrow \prod_j{ {\mathbb G_m} }_{,cdh,X}(U_j)=\prod (A/{\mathfrak p_j})^*.$$ The last equality follows from the smooth case. Suppose that $s\in{ {\mathbb G_m} }_{,X}(U)=A^*$ is mapped to the unit under $a_U.$ Then, the image of $s$ under the above composition of maps is also, of course, the unit $1.$ This means that $s-1\in\bigcap_j \mathfrak p_j=\sqrt{(0)}.$ Since $U$ is reduced, we conclude that $s=1.$
For the surjectivity, first note that $X$ is equidimensional and the lemma is true if $d=0$ or $r=1$ by the case where $X$ is smooth. We proceed by induction on the number $r$ of irreducible components of $X$ and the dimension $d$ of $X.$
Suppose that the lemma holds for $r\leq r_0$ and for dimensions less than that of $X.$ We prove the surjectivity of $a$ for a strict normal crossing divisor $X$ with $r_0+1$ irreducible components $X_0,X_1,\cdots,X_{r_0}.$ Let us put $Y:=X_1\cup\cdots\cup X_{r_0}$ and consider the abstract blow-up with all arrows closed immersions $$\xymatrix{ X_0\cap Y \ar[r]^{i'} \ar[d]_{p'} & Y \ar[d]^p\\
X_0 \ar[r]_i & X}$$ Put $f:=p\circ i'.$ There is a commutative diagram of Zariski sheaves on $X:$ $$\label{diagram: cdh on snc}
\xymatrix{ 0\ar[r] & { {\mathbb G_m} }_{,cdh,X}\ar[r]^-{(p^\sharp,i^\sharp)} & p_*{ {\mathbb G_m} }_{,cdh,Y} \oplus i_*{ {\mathbb G_m} }_{,cdh,{X_0}} \ar[r]^-{p'^\sharp\over i'^\sharp} \ar@{=}[d]^{\text{ ind. hypo.}} & f_*{ {\mathbb G_m} }_{,cdh,{X_0}\cap Y} \ar@{=}[d]^{\text{ smaller dim. case}}\\
& { {\mathbb G_m} }_{,X} \ar[r]^-{(p^\sharp,i^\sharp)} \ar@{_{(}->}[u]_a & p_*{ {\mathbb G_m} }_{,Y} \oplus i_*{ {\mathbb G_m} }_{,{X_0}} \ar[r]^-{p'^\sharp\over i'^\sharp} & f_*{ {\mathbb G_m} }_{,{X_0}\cap Y}}$$ where the upper row is exact because the blow-up square is a cdh cover.
For the surjectivity of $a,$ it suffices to show the exactness of the lower row in the diagram (\[diagram: cdh on snc\]). We may do this at the stalks. Let $x\in X$ be a closed point of $X$ and $R:=\mathcal O_{X,x}$ be the stalk of the structure sheaf at $x.$ Let us only deal with the case where $x$ lies in $X_0\cap Y$ because the other case where $x\not\in X_0\cap Y$ is simpler. (An argument as below shows that, if $x\not\in X_0\cap Y,$ the map $ (p^\sharp,i^\sharp): { {\mathbb G_m} }_{,X} \longrightarrow p_*{ {\mathbb G_m} }_{,Y} \oplus i_*{ {\mathbb G_m} }_{,{X_0}}$ induces an isomorphism on stalks.)
In this case, since $X$ is a simple normal crossing divisor, $\mathcal O_{X_0.x}=R/(f),$ where $f$ is the defining equation of $X_0,$ and $\mathcal O_{Y_x}=R/(g)$ for $g:=\prod_{i=1,\cdots s} g_i$ where $g_i$ is the defining equations of the irreducible components of $Y$ passing through $x.$ We need to show that $$R^*\longrightarrow (R/(f))^* \oplus (R/(g))^* \longrightarrow (R/(f,g))^*$$ is exact. Suppose that $(\bar t,\bar t')\in (R/(f))^* \oplus (R/(g))^*$ is mapped to the unit in $(R/(f,g))^*.$ Since $R$ is a local ring, $\bar t$ and $\bar t'$ are respectively represented by units $t$ and $t'$ in $R.$ Therefore, there exist elements $a$ and $b$ in $R$ such that $t/t'-1=af+bg.$ Hence, we have $t-t'af=t'+ t'bg=:t_0.$ The element $t_0$ is invertible in the local ring $R$ because $t$ is invertible and $t'af$ belongs to the maximal ideal. Since $t_0\in R^*$ is mapped to $(\bar t,\bar t')$ under the first arrow, the exactness follows.
Relative Picard groups
----------------------
We need the following results on relative Picard groups.
For $X\in Sch/k$ and a closed subscheme $Z\buildrel i \over\hookrightarrow X,$ the [**relative Picard group**]{} $Pic(X,Z)$ is the group consisting of isomorphism classes of pairs $(\mathcal L, u),$ where $\mathcal L$ is a line bundle on $X$ and $u$ is a trivialization $u:\mathcal L|_Z\buildrel\cong\over\longrightarrow\mathcal O_Z.$ The group structure is given by the tensor product. The pair $(\mathcal L,u)$ is called a line bundle on $(X,Z).$
\[lem: relative Picard as sheaf cohomology\] Suppose that $i:Z\hookrightarrow X$ is a closed subscheme of a scheme $X$ over $k.$ Then, there is a canonical isomorphism $$Pic(X,Z)\cong H^1_{Nis}(X,cone({ {\mathbb G_m} }_{,X}\longrightarrow i_*{ {\mathbb G_m} }_{,Z})[-1]).$$
By [@Suslin-Voevodsky-Inventiones Lemma 2.1], we have a canonical isomorphism $$Pic(X,Z)\cong H^1_{Zar}(X,cone({ {\mathbb G_m} }_{,X}\longrightarrow i_*{ {\mathbb G_m} }_{,Z})[-1])\buildrel \cong\over\longrightarrow H^1_{\acute et}(X,cone({ {\mathbb G_m} }_{,X}\longrightarrow i_*{ {\mathbb G_m} }_{,Z})[-1]).$$ induced by the change of sites. Consider the factorization $$\xymatrixcolsep{4pc}\xymatrixrowsep{3pc}\xymatrix{ H^1_{Zar}(X,cone({ {\mathbb G_m} }_{,X}\longrightarrow i_*{ {\mathbb G_m} }_{,Z})[-1]) \ar[d]_{\text{change of sites}} \ar[r]^-\cong & H^1_{\acute et}(X,cone({ {\mathbb G_m} }_{,X}\longrightarrow i_*{ {\mathbb G_m} }_{,Z})[-1])\\
H^1_{Nis}(X,cone({ {\mathbb G_m} }_{,X}\longrightarrow i_*{ {\mathbb G_m} }_{,Z})[-1]) \ar[ur]_-{\text{change of sites}}}$$ Since the change of sites maps of a sheaf cohomology in degree one are injective, all the arrows in the diagram are isomorphisms.
Let $X$ and $Z$ be as above. The [**relative Picard functor of the pair $(X,Z)$**]{} is the functor $$Pic_{X,Z}:Sch/k\longrightarrow Ab$$ that sends $T\in Sch/k$ to the relative Picard group $Pic(T\times X,T\times Z).$
On the representability of the relative Picard functor, the following is known.
\[prop: representability of the relative Picard functor\] Let $X$ be a connected smooth proper scheme over $k$ and let $Z$ be a non-empty simple normal crossing divisor on $X.$ Then, the relative Picard functor $Pic_{X,Z}$ is representable by a group scheme locally of finite type over $k.$
The group scheme representing the relative Picard functor $Pic_{X,Z}$ is denoted by the same symbol $Pic_{X,Z}.$ The identity component $Pic_{X,Z}^0$ has the following structure.
\[prop: structure of a relative Picard variety\] Let $X$ be a connected smooth proper scheme over $k$ and let $Z$ be a non-empty simple normal crossing divisor with irreducible components $Z_i$ on $X.$ Then, the maximal reduced subscheme $Pic_{X,Z,red}^0$ of the identity component of $Pic_{X,Z}$ is a semi-abelian variety over $k$ such that there is an exact sequence $$0\longrightarrow T_{X,Z} \longrightarrow Pic_{X,Z,red}^0 \longrightarrow A_{X,Z} \longrightarrow 0,$$ where $T_{X,Z}$ is the torus over $k$ representing the functor $$Sch/k\ni T\mapsto \mathrm{coker}\{{ {\mathbb G_m} }(T\times X)\longrightarrow { {\mathbb G_m} }(T\times Z)\}\in Ab,$$ and $A_{X,Z}$ is the abelian variety $(\mathrm{ker}\{Pic_{X}^0\longrightarrow \bigoplus_i Pic_{Z_i}^0 \})_{red}^0.$
Universal regular homomorphisms in codimension one
--------------------------------------------------
By relative Nisnevich cohomology, we mean the following.
Let $\mathcal F$ be a bounded above complex of Nisnevich sheaves on $Sch/k.$ For a closed immersion $Z\hookrightarrow X,$ the [**relative Nisnevich cohomology of the pair $(X,Z)$ with coefficients in $\mathcal F$**]{} is defined as $$H_{Nis}^m(X,Z;\mathcal F):=H^m(cone(\mathcal I^\bullet(X)\longrightarrow \mathcal I^\bullet(Z))[-1]),$$ where $\mathcal I^\bullet$ is the total complex of a Cartan-Eilenberg resolution of $\mathcal F$ in $Sh_{Nis}(Sch/k).$
It is clear from the definition that there is a long exact sequence of cohomology groups $$\cdots\longrightarrow H_{Nis}^m(X,\mathcal F) \longrightarrow H_{Nis}^m(Z,\mathcal F) \longrightarrow H_{Nis}^{m+1}(X, Z;\mathcal F) \longrightarrow H_{Nis}^{m+1}(X,\mathcal F) \longrightarrow\cdots.$$
Let us interpret relative Picard groups in terms of relative Nisnevich cohomology.
\[prop: Picard group as relative Nisnevich cohomology\] Let $i:Z\hookrightarrow X$ be a closed subscheme of $X$ over $k.$ Then, there is a canonical isomorphism $$Pic(X,Z)\cong H^1_{Nis}(X,Z;{ {\mathbb G_m} }).$$
Let ${ {\mathbb G_m} }\longrightarrow\mathcal I^\bullet$ be an injective resolution in $Sh_{Nis}(Sch/k).$ Then, its restriction ${ {\mathbb G_m} }_{,X}\longrightarrow \mathcal I^\bullet_X$ to the small Nisnevich site $X_{Nis}$ on $X$ is also an injective resolution, and similarly, so is ${ {\mathbb G_m} }_{,Z}\longrightarrow\mathcal I^\bullet_Z$ on $Z_{Nis}.$
Since $i:Z\hookrightarrow X$ is a closed immersion, $i_*: Sh(Z_{Nis})\longrightarrow Sh(X_{Nis})$ is exact and preserves injectives (for its left adjoint $i^*$ is exact). Therefore, $i_*{ {\mathbb G_m} }_{,Z}\longrightarrow i_*\mathcal I^\bullet_Z$ is an injective resolution on $X_{Nis}.$ Hence, $$\begin{aligned}
Pic(X,Z)&\cong& H^1_{Nis}(X,cone({ {\mathbb G_m} }_{,X}\longrightarrow i_*{ {\mathbb G_m} }_{,Z})[-1]) {\text{~~(by Lemma~\ref{lem: relative Picard as sheaf cohomology})}}\nonumber\\
&\cong & H^1_{Nis}(X,cone(\mathcal I^\bullet_X \longrightarrow i_*\mathcal I^\bullet_Z)[-1])\nonumber\\
&=& H^1(cone(\mathcal I^\bullet(X) \longrightarrow \mathcal I^\bullet(Z))[-1])\nonumber\\
&=& H^1_{Nis}(X,Z;{ {\mathbb G_m} }).\nonumber\end{aligned}$$
\[defn: good compactification\] A smooth connected scheme $X$ over $k$ is said to have a [**good compactification**]{} if there is a smooth proper scheme $\bar X$ with an open immersion $X\hookrightarrow \bar X$ such that $Z:=\bar X\setminus X$ is a simple normal crossing divisor on $\bar X.$
Suppose that $\bar X$ is a smooth proper scheme and $Z$ is a simple normal crossing divisor on $\bar X.$ We now give a motivic interpretation of the relative Picard functor $Pic_{\bar X,Z}$ restricted to $Sm/k$ (Proposition \[prop: relative Picard functor and cdh sheaves\]). Let us start with a lemma.
\[lem: parametrization and cdh sheaves\] Under resolution of singularities, for arbitrary schemes $X$ and $T,$ there is a canonical isomorphism natural in $T$ $$Hom_{DM}(M(T)\otimes M^c(X),{ {\mathbb Z} }(n)[m])\buildrel\cong\over\longrightarrow Hom_{D^-(Sh_{cdh}(Sch/k))}({ {\mathbb Z} }(T)_{cdh}\otimes{ {\mathbb Z} }^c(X)_{cdh},{ {\mathbb Z} }(n)_{cdh}[m]).$$
For any scheme $S\in Sch/k,$ there is a composition, which we call $f_{T,S}^{m,n},$ of canonical maps $$\begin{aligned}
Hom_{DM}(M(T)\otimes M^c(S),{ {\mathbb Z} }(n)[m]) &\buildrel \text{def}\over= & Hom_{DM}({ {\mathbb Z} }_{tr}(T)\otimes { {\mathbb Z} }_{tr}^c(S),{ {\mathbb Z} }(n)[m])\nonumber\\
&\buildrel (a)\over=& Hom_{D^-(Sh_{Nis}^{tr}(Sm/k))}({ {\mathbb Z} }_{tr}(T)\otimes z_{equi}(S,0),{ {\mathbb Z} }(n)[m])\nonumber\\
&\buildrel (b)\over\to& Hom_{D^-(Sh_{Nis}(Sm/k))}({ {\mathbb Z} }(T)\otimes { {\mathbb Z} }^c(S),{ {\mathbb Z} }(n)[m]) \nonumber\\
&\buildrel (c)\over\to& Hom_{D^-(Sh_{t}(Sm/k))}({ {\mathbb Z} }(T)_t\otimes { {\mathbb Z} }^c(S)_t,{ {\mathbb Z} }(n)_t[m]) \nonumber\\
&\buildrel (d)\over\cong& Hom_{D^-(Sh_{cdh}(Sch/k))}({ {\mathbb Z} }(T)_{cdh}\otimes { {\mathbb Z} }^c(S)_{cdh},{ {\mathbb Z} }(n)_{cdh}[m]).\nonumber\end{aligned}$$ Here, (a) is an equality by the Nisnevich variant of [@MVW Lemma 9.19] because ${ {\mathbb Z} }(n)$ is an $\mathbb A^1$-local object by [@MVW Corollary 14.9]. (b) is induced by the inclusions ${ {\mathbb Z} }(T)\hookrightarrow { {\mathbb Z} }_{tr}(T)$ and ${ {\mathbb Z} }^c(S)\hookrightarrow z_{equi}(S,0).$ (c) is induced by the $t$-sheafification, and (d) is due to the equivalence of categories between $Sh_{t}(Sm/k)$ and $Sh_{cdh}(Sch/k)$ (see the first paragraph of Subsection \[subsection: 3.1\]). By the construction, $f_{T,S}^{m,n}$ is functorial in $T$ with respect to pushforwards along all morphisms and functorial in $S$ with respect to pushforwards along proper morphisms and pullbacks along flat morphisms. Therefore, the localization triangles in $DM$ ([@MVW Theorem 16.15]) and in $D^-(Sh_{cdh}(Sch/k))$ ([@Friedlander-Voevodsky Corollary 3.9]) associated with a good compactification $X\hookrightarrow\bar X$ of $X$ with the boundary divisor $Z$ give rise to the commutative diagram (Note that $DM$ is a tensor triangulated category ([@MVW p.110]), and $-\otimes{ {\mathbb Z} }(T)_{cdh}$ is exact in $Sh_{cdh}(Sch/k)$ because ${ {\mathbb Z} }(T)$ is a presheaf of free abelian groups and sheafification is exact.) $$\xymatrix{\ar[r] & H^{m-1}(T\times \bar X,{ {\mathbb Z} }(n)) \ar[r] \ar[d]_{f_{T,\bar X}^{m-1,n}}^\cong & H^{m-1}(T\times Z,{ {\mathbb Z} }(n)) \ar[r] \ar[d]_{f_{T, Z}^{m-1,n}}^\cong & Hom_{DM}(M(T)\otimes M^c(X),{ {\mathbb Z} }(n)[m]) \ar[d]_{f_{T,X}^{m,n}} \\
\ar[r] & H_{cdh}^{m-1}(T\times \bar X,\widetilde{{ {\mathbb Z} }(n)}) \ar[r] & H_{cdh}^{m-1}(T\times Z,\widetilde{{ {\mathbb Z} }(n)}) \ar[r] & Hom_{D^-_{cdh}}(\widetilde{{ {\mathbb Z} }(T)}\otimes\widetilde{{ {\mathbb Z} }^c(X)},\widetilde{{ {\mathbb Z} }(n)}[m]) }$$ $$\xymatrix{ \ar[r] & H^m(T\times \bar X,{ {\mathbb Z} }(n)) \ar[r] \ar[d]_{f_{T,\bar X}^{m,n}}^\cong & H^m(T\times Z,{ {\mathbb Z} }(n)) \ar[d]_{f_{T,Z}^{m,n}}^\cong \ar[r] &\\
\ar[r] & H_{cdh}^m(T\times \bar X,\widetilde{{ {\mathbb Z} }(n)}) \ar[r] & H_{cdh}^m(T\times Z,\widetilde{{ {\mathbb Z} }(n)}) \ar[r] &}$$ where $``\sim"$ stands for the cdh sheafification. The four arrows between the cohomology groups are isomorphisms by [@MVW Theorem 14.20], so the middle map is also an isomorphism.
\[prop: relative Picard functor and cdh sheaves\] Let $X$ and $T$ be smooth schemes over $k$ and let $\bar X$ be a good compactification of $X$ with the boundary divisor $Z.$ Under resolution of singularities, there is an isomorphism natural in $T\in Sm/k$ $$F: Pic(T\times \bar X,T\times Z)\buildrel \cong\over\longrightarrow Hom_{DM}(M(T)\otimes M^c(X),{ {\mathbb Z} }(1)[2])$$ such that $F$ fits in the commutative diagram $$\xymatrix{ H_{Nis}^0(T\times Z,{ {\mathbb G_m} }) \ar[r] \ar[d]_\cong^{\text{change of sites}} & Pic(T\times\bar X, T\times Z) \ar[r] \ar[d]_F & H_{Nis}^1(T\times \bar X,{ {\mathbb G_m} }) \ar[d]_\cong^{\text{change of sites}}\\
H_{cdh}^1(T\times Z,{ {\mathbb Z} }(1)_{cdh}) \ar[r] & Hom_{DM}(M(T)\otimes M^c(X),{ {\mathbb Z} }(1)[2]) \ar[r] & H_{cdh}^2(T\times\bar X,{ {\mathbb Z} }(1)_{cdh}).}$$
Let ${ {\mathbb G_m} }_{,cdh}\longrightarrow \mathcal I^\bullet$ be an injective resolution in $Sh_{cdh}(Sch/k)$ and ${ {\mathbb G_m} }\longrightarrow \mathcal J^\bullet$ in $Sh_{Nis}(Sch/k).$ Since $\mathcal I^\bullet$ is still a complex of injective sheaves when restricted to the Nisnevich site, there is an augmentation-preserving chain map (unique up to chain homotopy) $f:\mathcal J^\bullet\longrightarrow \mathcal I^\bullet$ of complexes of Nisnevich sheaves on $Sch/k.$
The short exact sequence $$0\longrightarrow { {\mathbb Z} }(T)_{cdh}\otimes { {\mathbb Z} }(Z)_{cdh} \longrightarrow { {\mathbb Z} }(T)_{cdh}\otimes { {\mathbb Z} }(\bar X)_{cdh} \longrightarrow { {\mathbb Z} }(T)_{cdh}\otimes { {\mathbb Z} }^c(X)_{cdh}\longrightarrow0$$ in $Sh_{cdh}(Sch/k)$ induces the top horizontal sequence which is exact in each degree in the following commutative diagram of complexes of *presheaves* in $T\in Sm/k$ with values in $Ab$ (where $``\sim"$ stands for the cdh sheafification): $$\xymatrixcolsep{1.0pc}\xymatrix{0 \ar[r] & Hom_{Sh_{cdh}}(\widetilde{{ {\mathbb Z} }(T)}\otimes\widetilde{{ {\mathbb Z} }^c(X)},\mathcal I^\bullet) \ar[r]& Hom_{Sh_{cdh}}(\widetilde{{ {\mathbb Z} }(T\times \bar X)},\mathcal I^\bullet) \ar[r] & Hom_{Sh_{cdh}}(\widetilde{{ {\mathbb Z} }(T\times Z)},\mathcal I^\bullet) \ar[r] & 0\\
& & Hom_{Sh_{Nis}}({ {\mathbb Z} }(T\times\bar X),\mathcal I^\bullet) \ar[u]^{\text{adjunction}}_\cong \ar[r] & Hom_{Sh_{Nis}}({ {\mathbb Z} }(T\times Z),\mathcal I^\bullet) \ar[u]^{\text{adjunction}}_\cong\\
& cone(h)[-1]\ar[r] \ar@{..>}[uu]^{\exists~k{\text{ in }} D^-(PSh(Sm/k))}& Hom_{Sh_{Nis}}({ {\mathbb Z} }(T\times\bar X),\mathcal J^\bullet) \ar[u]^{f\circ-} \ar[r]_h & Hom_{Sh_{Nis}}({ {\mathbb Z} }(T\times Z),\mathcal J^\bullet) \ar[u]^{f\circ-}}$$ The dotted arrow $k$ exists in the derived category $D^-(PSh(Sm/k))$ of presheaves on $Sm/k.$ Now, taking cohomology groups, we obtain the following commutative diagram natural in $T\in Sm/k:$ $$\label{diagram: the thing we want}
\xymatrix{ H_{Nis}^0(T\times\bar X,{ {\mathbb G_m} }) \ar[r] \ar[d]^a_\cong & H_{Nis}^0(T\times Z,{ {\mathbb G_m} }) \ar[r] \ar[d]^b_\cong & H_{Nis}^1(T\times\bar X, T\times Z;{ {\mathbb G_m} }) \ar[d]^{k',~\text{induced by $k$}} \\
H_{cdh}^0(T\times\bar X,\widetilde{{ {\mathbb G_m} }}) \ar[r] & H_{cdh}^0(T\times Z,\widetilde{{ {\mathbb G_m} }}) \ar[r] & Hom_{D^-_{cdh}}(\widetilde{{ {\mathbb Z} }(T)}\otimes\widetilde{{ {\mathbb Z} }^c(X)},\widetilde{{ {\mathbb G_m} }}[1])}$$ $$\xymatrix{\ar[r] & H_{Nis}^1(T\times\bar X,{ {\mathbb G_m} }) \ar[r] \ar[d]^c_\cong & H_{Nis}^1(T\times Z,{ {\mathbb G_m} }) \ar[d]^d \ar[r] &\\
\ar[r] & H_{cdh}^1(T\times\bar X,\widetilde{{ {\mathbb G_m} }}) \ar[r] & H_{cdh}^1(T\times Z,\widetilde{{ {\mathbb G_m} }}) \ar[r] &}$$ where all solid vertical arrows induced by the change of sites. Note that the naturality in $T$ follows because the map $k$ was constructed in the derived category of presheaves.
The maps $a$ and $c$ are isomorphisms by [@MVW Theorem 14.20] and $b$ is by Lemma \[lem: cdh sheafification of Gm\]. In view of Proposition \[prop: Picard group as relative Nisnevich cohomology\] and Lemma \[lem: parametrization and cdh sheaves\], it remains to show that the map $k'$ is an isomorphism. For this, it is enough to prove that $d$ is injective.
Observe that it suffices to prove that the composition $$f: H^1_{Zar}(T\times Z,{ {\mathbb G_m} })\buildrel {\text{change of sites}}\over\hookrightarrow H^1_{Nis}(T\times Z,{ {\mathbb G_m} }) \buildrel d\over\longrightarrow H^1_{cdh}(T\times Z,\widetilde{{ {\mathbb G_m} }})$$ is injective because the first injective map is actually an isomorphism by Hilbert’s Satz 90 ([@Milne; @etale Chapter III, Proposition 4.9]).
Now, by the construction of $d,$ the map $f$ factors through $H^1_{Zar}(T\times Z,\widetilde{{ {\mathbb G_m} }})$ as $$\xymatrix{ H_{Zar}^1(T\times Z,{ {\mathbb G_m} }) \ar[d]_i^\cong \ar[r]^f & H_{cdh}^1(T\times X,\widetilde{{ {\mathbb G_m} }}) \\
H_{Zar}^1(T\times Z,\widetilde{{ {\mathbb G_m} }}) \ar@{^{(}->}[ur]_-{~~~g,~\text{change of sites}}}$$ where $i$ is an isomorphism by Lemma \[lem: cdh sheafification of Gm\] and $g$ is injective because it is a change of sites in degree one. Therefore, $d$ is injective.
We are ready to compare the semi-abelian representative $Alg_{c,X}^{2,1}$ of a smooth scheme $X$ with the relative Picard variety $Pic_{\bar X,Z, red}^0$ of a good compactification $(\bar X,Z)$ of $X.$
\[prop: relative Picard is regular\] Assume resolution of singularities. For any connected smooth scheme $X$ over $k$ with a good compactification $\bar X$ with the non-empty boundary divisor $Z,$ the canonical homomorphism $$H_{c,alg}^2(X,{ {\mathbb Z} }(1)) \buildrel i:=inc\over\hookrightarrow H_c^2(X,{ {\mathbb Z} }(1)) \buildrel g\over\longrightarrow Pic(\bar X, Z) \buildrel\psi\over\cong Pic_{\bar X,Z}(k)$$ ($g$ is the inverse of $F$ in Proposition \[prop: relative Picard functor and cdh sheaves\] evaluated at $T={ {\mathrm{Spec~}} }k,$ and $\psi$ is as in Proposition \[prop: representability of the relative Picard functor\]) factors through $Pic_{\bar X,Z,red}^0(k)$ as $$\xymatrix{ H_{c,alg}^2(X,{ {\mathbb Z} }(1)) \ar[dr]_{\phi_0} \ar[rr]^{\psi\circ g\circ i} && Pic_{\bar X, Z}(k) \\
& Pic_{\bar X,Z, red}^0(k) \ar[ur]}$$ and the homomorphism $\phi_0$ is regular.
It is enough to show that for any smooth connected scheme $T$ over $k$ pointed at $t_0\in T(k)$ and for any $Y\in Hom_{DM}(M(T)\otimes M^c(X),{ {\mathbb Z} }(1)[2]),$ the composition $$T(k)\buildrel w_Y\over\longrightarrow H_{c,alg}^2(X,{ {\mathbb Z} }(1)) \buildrel \psi\circ g\circ i\over\longrightarrow Pic_{\bar X,Z}(k)$$ is induced by a scheme morphism, where $w_Y$ is the map defined in Definition \[defn: regular homomorphism\]. (It is because, then, the image of $T$ is connected and contains the identity. And as we already know, $Pic_{\bar X,Z,red}^0$ is a semi-abelian variety by Proposition \[prop: structure of a relative Picard variety\].)
Observe that there is a commutative diagram $$\xymatrix{
T(k) \ar@{=}[dd] \ar[r]^<<<<<{w_Y} & H_{c,alg}^2(X,{ {\mathbb Z} }(1)) \ar@{_{(}->}[d]^i \\
& H_c^2(X,{ {\mathbb Z} }(1)) \ar[d]^g \\
T(k) \ar[r]^<<<<<<<{B_{F^{-1}(Y)}} & Pic(\bar X,Z) \ar[r]^\psi & Pic_{\bar X,Z}(k).}$$ Here, $F$ is the map defined in Proposition \[prop: relative Picard functor and cdh sheaves\]. Let $(\mathcal L,u:\mathcal L|_{T\times Z}\cong\mathcal O_{T\times Z})$ be the line bundle on the pair $(T\times\bar X, T\times Z)$ that represents $F^{-1}(Y).$ $B_{F^{-1}(Y)}$ is defined as the map that sends $t\in T(k)$ to $(\mathcal L_t\otimes\check{\mathcal L}_{t_0}, u_t\otimes\check u_{t_0}),$ where $\mathcal L_t$ means the pullback of the line bundle along $t:{ {\mathrm{Spec~}} }k\longrightarrow T,$ $u_t$ is the restriction of $u$ to $\{t\}\times X,$ and $\check-$ signifies the dual of invertible sheaves. The commutativity of the diagram follows from the naturality of $F.$
Now, let $(\mathcal P, p)$ be the Poincaré bundle, which is by definition the line bundle $(\mathcal P,p)$ representing the class in $Pic(Pic_{\bar X,Z}\times\bar X,Pic_{\bar X,Z}\times Z)$ corresponding to the identity in $Hom_{Sch/k}(Pic_{\bar X,Z},Pic_{\bar X,Z}).$ The representability of the relative Picard functor means that $F^{-1}(Y)$ is the pullback of $(\mathcal P,p)$ along some scheme morphism $h: T\longrightarrow Pic_{\bar X,Z}.$ Hence, we are given with the commutative diagram $$\label{diagram: Poincare}
\xymatrixcolsep{3pc}\xymatrix{ T(k) \ar[dr]_-{h} \ar[r]^{B_{F^{-1}(Y)}} & Pic(\bar X,Z) \ar[rr]^\psi && Pic_{\bar X,Z}(k) \\
& Pic_{\bar X,Z}(k) \ar[u]_{B_{(\mathcal P,p)}} \ar[urr]_{id-h(t_0)}}$$ where $Pic_{\bar X,Z}$ is pointed at $h(t_0)$ in defining $B_{(\mathcal P,p)}.$
Since $\psi\circ B_{F^{-1}(Y)}=(id-h(t_0))\circ h,$ we conclude that $\psi\circ B_{F^{-1}(Y)}$ is induced by a scheme morphism.
\[thm: main theorem in codim one\] Assume resolution of singularities. If $X$ is a smooth connected scheme over $k$ with a good compactification $\bar X$ with the non-empty boundary divisor $Z,$ then the regular homomorphism in Proposition \[prop: relative Picard is regular\] $$\phi_0: H_{c,alg}^2(X,{ {\mathbb Z} }(1))\longrightarrow Pic_{\bar X,Z,red}^0(k)$$ is an isomorphism. In particular, it is the universal regular homomorphism for $H_{c,alg}^2(X,{ {\mathbb Z} }(1)).$
The injectivity of $\phi_0$ follows because $\psi\circ g:H_c^2(X,{ {\mathbb Z} }(1))\longrightarrow Pic_{\bar X, Z}(k)$ is an isomorphism. For the surjectivity, observe that $$\mathrm{im}(g\circ i)=\bigcup_{\substack{T,~{\text{smooth}}\\ {\text{connected}}}}\mathrm{im}\{H_0(T,{ {\mathbb Z} })^0 \times Pic(T\times\bar X, T\times Z)\buildrel\text{pullback}\over\longrightarrow Pic(\bar X, Z)\}=:Pic_{alg}(\bar X, Z).$$ Thus, it remains to show that the elements of $Pic_{alg}(\bar X, Z)$ correspond to the rational points on $Pic_{\bar X,Z,red}^0.$ The proof is a modification (in fact, simpler) of that of [@Kleiman Proposition 9.5.10]. We include the proof for the convenience of the reader.
Let us consider $(\mathcal L,u)\in Pic(\bar X, Z)$ and the corresponding rational point $\alpha\in Pic_{\bar X, Z}(k).$
Suppose that $(\mathcal L, u)$ belongs to $Pic_{alg}(\bar X, Z).$ Then, there are a smooth connected scheme $T,$ rational points $t_0, t_1\in T(k)$ and $(\mathcal M, v)\in Pic(T\times \bar X, T\times Z)$ (see the proof of Lemma \[lem: curves are enough\]) such that $$(\mathcal L,u)=(\mathcal M_{t_1},v_{t_1})-(\mathcal M_{t_0},v_{t_0}).$$ Now, $(\mathcal M, v)$ defines a morphism $\tau: T\longrightarrow Pic_{\bar X, Z},$ and we have $\alpha=\tau(t_1)-\tau(t_0).$ Since $\tau$ is continuous, $T$ is connected, and the image of the map $\sigma:=\tau(-)-\tau(t_0): T(k)\longrightarrow Pic_{\bar X, Z}(k)$ contains the identity, the image of $\sigma$ is contained in the identity component of $Pic_{\bar X.Z}.$ In particular, $$\alpha=\tau(t_1)-\tau(t_0)\buildrel\text{def}\over=\sigma(t_1)\in Pic_{\bar X,Z}^0(k).$$
Conversely, suppose that $\alpha\in Pic_{\bar X, Z}^0(k).$ The inclusion $Pic_{\bar X, Z}^0\hookrightarrow Pic_{\bar X, Z}$ corresponds to an element $(\mathcal N,w)\in Pic(Pic_{\bar X, Z}^0\times\bar X, Pic_{\bar X, Z}^0\times Z).$ Then, $(\mathcal N_\alpha, w_\alpha)=(\mathcal L, u)$ and $(\mathcal N_0,w_0)=(\mathcal O_{\bar X}, id_{\mathcal O_Z})$ (The subscript $0$ signifies the unit of $Pic_{\bar X,Z}^0$) because for any rational point $s\in Pic_{\bar X, Z}^0(k),$ the diagram $$\xymatrixcolsep{0.1pc}\xymatrix{ \mathcal N\in Pic(Pic_{\bar X, Z}^0\times\bar X, Pic_{\bar X, Z}^0\times Z) \ar[d]_{(-)_s} & \cong & Hom_{Sch/k}(Pic_{\bar X, Z}^0,Pic_{\bar X, Z})\ni inc \ar[d]^{-\circ s}\\
\mathcal N_s\in Pic({ {\mathrm{Spec~}} }k\times\bar X,{ {\mathrm{Spec~}} }k\times Z) & \cong & Hom_{Sch/k}({ {\mathrm{Spec~}} }k, Pic_{\bar X,Z})\ni s}$$ commutes. Therefore, we have $$(\mathcal L, u)= (\mathcal L,u)-(\mathcal O_{\bar X}, id_{\mathcal O_Z})=(\mathcal N_\alpha,w_\alpha)-(\mathcal N_0,w_0)\in Pic_{alg}(\bar X,Z).$$ Thus, the surjectivity of $\phi_0$ is shown.
The last assertion of the theorem follows from the first part because there is a commutative diagram: $$\xymatrix{ H_{c,alg}^2(X,{ {\mathbb Z} }(1)) \ar@{->>}[r]^-{\Phi_{c,X}^{2,1}} \ar[dr]^{\phi_0}_{\text{isom.}} & Alg_{c,X}^{2,1}(k) \ar@{..>}[d]^{\exists!~\text{by the universality}}\\
& Pic_{\bar X,Z,red}^0(k)}$$
Algebraic part by proper parametrization {#subsection: Algebraic part by proper parametrization}
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In this subsection, we study the relation between the universal regular homomorphism and its variant defined by “proper parametrization". In the case of codimension one, we conclude under the assumption of resolution of singularities that this variant is nothing but the maximal abelian subvariety of the semi-abelian representative $Alg_{c,X}^{2,1}$ if the scheme $X$ is smooth (Theorem \[cor: max abelian subvariety\]).
\[defn: algebraic part by proper parametrization\] Let $X$ be a connected scheme over $k.$ With the notation in Definition \[defn: algebraic part\], if $\mathfrak T=\mathfrak P$ is the class of smooth proper connected schemes over $k,$ $$H^m_{c,\mathfrak P}(X,{ {\mathbb Z} }(n))\buildrel\text{def}\over=\bigcup_{T\in\mathfrak P}\mathrm{im}\{H_0(T,{ {\mathbb Z} })^0\times Hom_{DM}(M(T)\otimes M^c(X),{ {\mathbb Z} }(n)[m])\buildrel P_T\over\longrightarrow H_c^{m}(X,{ {\mathbb Z} }(n))\}$$ is called the [**algebraic part by proper parametrization**]{} of $H^m_{c}(X,{ {\mathbb Z} }(n)).$
\[rem: interesting example one\] We have already seen this group in the proof of Proposition \[prop: comparison with the classical algebraic part\]. There, we proved the equality $$H^{2r}_{c,\mathfrak P}(X,{ {\mathbb Z} }(r))=H^{2r}_{c,alg}(X,{ {\mathbb Z} }(r))$$ for any smooth proper scheme $X$ over $k$ and any integer $r.$
This equality does not hold in general if $(m,n)\neq (2r,r).$ For example, consider any smooth proper connected scheme $X$ over $k.$ By definition, the algebraic part by proper parametrization of $H^1_{c}(X,{ {\mathbb Z} }(1))$ is $$\begin{aligned}
H^1_{c,\mathfrak P}(X,{ {\mathbb Z} }(1))&\buildrel\text{def}\over=&\bigcup_{T\in\mathfrak P}\mathrm{im}\{H_0(T,{ {\mathbb Z} })^0\times Hom_{DM}(M(T)\otimes M^c(X),{ {\mathbb Z} }(1)[1])\buildrel P_T\over\longrightarrow H_c^{1}(X,{ {\mathbb Z} }(1))\}\nonumber\\
&\cong&\bigcup_{T\in\mathfrak P}\mathrm{im}\{H_0(T,{ {\mathbb Z} })^0\times \mathcal O^*(T\times X)\buildrel p\over\longrightarrow \mathcal O^*(X)\},\nonumber\end{aligned}$$ where $p$ is the pullback of an invertible regular function on $T\times X$ along zero cycles of degree zero on $T.$ Since $T\times X$ is proper and any regular function on a proper connected scheme is constant, $p$ is the zero map. Hence, we have $$H^1_{c,\mathfrak P}(X,{ {\mathbb Z} }(1))=0.$$
On the other hand, the parametrization by non-proper schemes picks up more elements—in fact, all elements in this case—of $H_c^1(X,{ {\mathbb Z} }(1)).$ Indeed, we have $$\begin{aligned}
H^1_{c,alg}(X,{ {\mathbb Z} }(1))&\buildrel\text{def}\over=&\bigcup_{T,\text{sm. conn.}}\mathrm{im}\{H_0(T,{ {\mathbb Z} })^0\times Hom_{DM}(M(T)\otimes M^c(X),{ {\mathbb Z} }(1)[1])\buildrel P_T\over\longrightarrow H_c^{1}(X,{ {\mathbb Z} }(1))\}\nonumber\\
&\supset&\mathrm{im}\{H_0(\mathbb G_m,{ {\mathbb Z} })^0\times Hom_{DM}(M(\mathbb G_m)\otimes M^c(X),{ {\mathbb Z} }(1)[1])\buildrel P_{\mathbb G_m}\over\longrightarrow H_c^{1}(X,{ {\mathbb Z} }(1))\}\nonumber\\
&\cong&\mathrm{im}\{H_0(\mathbb G_m,{ {\mathbb Z} })^0\times \mathcal O^*(\mathbb G_m\times X)\buildrel p\over\longrightarrow \mathcal O^*(X)\}\nonumber\\
&=&\mathcal O^*(X).\nonumber\end{aligned}$$ The last equality holds because the map $p$ sends a pair $$(g-1, \mathbb G_m\times X\buildrel \text{proj.}\over\longrightarrow \mathbb G_m)\in H_0(\mathbb G_m,{ {\mathbb Z} })^0\times \mathcal O^*(\mathbb G_m\times X)$$ to $g\in\mathcal O^*(X)=k^*.$ Therefore, we must have $H^1_{c,alg}(X,{ {\mathbb Z} }(1))=H_c^{1}(X,{ {\mathbb Z} }(1))\cong k^*.$
For any index $(m,n),$ $H^m_{c,\mathfrak P}(X,{ {\mathbb Z} }(n))$ is a subgroup of $H^m_{c}(X,{ {\mathbb Z} }(n))$ by Proposition \[prop: algebraic part\], and it is a divisible group because any element is in the image of the degree zero part of the zeroth Suslin homology of some smooth projective curve (see the first two lines of the proof of Proposition \[prop: algebraic part is divisible\]).
\[defn: algebraic representative by proper parametrization\] Let $X$ be a connected scheme over $k$ and let $S$ be a semi-abelian variety over $k.$ A group homomorphism $\phi:H^m_{c,\mathfrak P}(X,{ {\mathbb Z} }(n))\longrightarrow S(k)$ is called [**$\mathfrak P$-regular**]{} if for any smooth proper connected scheme $T$ over $k$ pointed at $t_0\in T(k)$ and for any $Y\in Hom_{DM}(M(T)\otimes M^c(X),{ {\mathbb Z} }(n)[m]),$ the composition $$T(k)\buildrel w_Y\over\longrightarrow H^m_{c,\mathfrak P}(X,{ {\mathbb Z} }(n))\buildrel\phi\over\longrightarrow S(k)$$ is induced by some scheme morphism $T\longrightarrow S.$
The universal $\mathfrak P$-regular homomorphism (defined similarly to Definition \[defn: regular homomorphism\]), if it exists, is written as $$\Phi^{m,n}_{c,\mathfrak P,X}: H_{c,\mathfrak P}^{m}(X,{ {\mathbb Z} }(n))\longrightarrow Alg_{c,\mathfrak P,X}^{m,n}(k).$$
\[prop: image is abelian\] If $\phi:H_{c,\mathfrak P}^m(X,{ {\mathbb Z} }(n))\longrightarrow S(k)$ is a $\mathfrak P$-regular homomorphism, then the image of $\phi$ has a structure of an abelian subvariety of $S.$ In particular, if $Alg_{c,\mathfrak P,X}^{m,n}$ exists, it is an abelian variety.
Choose a pointed smooth proper connected scheme $T$ over $k$ and a morphism $Y\in Hom_{DM}(M(T)\otimes M^c(X),{ {\mathbb Z} }(n)[m])$ such that the dimension of $\mathrm{im}(\phi\circ w_Y)$ is maximal. Then, by the same argument as in the proof of Proposition \[rem: surjection from a semiabelian variety\], we may prove that $\mathrm{im}(\phi)=\mathrm{im}(\phi\circ w_Y).$ (This time, we do not need to use Proposition \[prop: parametrization by semiabelian varieties\] so that $\mathrm{im}(\phi\circ w_Y)$ would be semi-abelian. Instead, we may simply use the fact that the image of a proper irreducible scheme $T$ is closed and irreducible to derive that $\mathrm{im}(\phi\circ w_Y)$ is closed and irreducible in $S.$)
Now, since the image of the group homomorphism $\phi$ is the image of the smooth proper connected scheme $T$ under a scheme morphism, it has a structure of an algebraic subgroup of the semi-abelian variety $S$ that is proper over $k.$ Thus, it is an abelian variety.
As Definition \[defn: maximal homomorphism\], a surjective $\mathfrak P$-regular homomorphism $\phi:H^m_{c,\mathfrak P}(X,{ {\mathbb Z} }(n))\longrightarrow S(k)$ (the target $S$ is automatically an abelian variety by Proposition \[prop: image is abelian\]), is called [**maximal**]{} if given any factorization $$\xymatrix{& S'(k) \ar[d]^-{\forall\pi,~{\text{isogeny}}}\\
H_{c,\mathfrak P}^{m}(X,{ {\mathbb Z} }(n)) \ar[r]_-\phi \ar[ur]^-{\forall~{\text{$\mathfrak P$-regular}}} & S(k),}$$ $\pi$ is an isomorphism.
Any $\mathfrak P$-regular homomorphism $\phi$ can be factored as $$\xymatrix{ H_{c,\mathfrak P}^{m}(X,{ {\mathbb Z} }(n)) \ar[rr]^-\phi \ar[dr]_-g && S(k) \\
& S'(k) \ar[ur]_-h}$$ with a maximal $\mathfrak P$-regular homomorphism $g$ and a finite morphism $h.$ Since we know that the image of $\phi$ has a structure of a scheme by Proposition \[prop: image is abelian\], this can be proved as Lemma \[lem: maximal homomorphism factorization\].
We also have a version of Lemma \[lem: existence implies uniqueness\]. Namely, for $\mathfrak P$-regular homomorphisms $\phi$ and $\phi',$ there is at most one scheme morphism $f$ that makes the diagram $$\xymatrix{ H_{c,\mathfrak P}^{m}(X,{ {\mathbb Z} }(n)) \ar@{->>}[r]^-\phi \ar[dr]_-{\phi'} & S(k) \ar[d]^-f \\
& S'(k)}$$ commute. This can be seen by flat descent (see the proof of Lemma \[lem: existence implies uniqueness\]) from the commutative diagram $$\xymatrix{ Alb_T(k) \ar@{->>}@/^2pc/[rrr]^-{\exists!,\text{ faithfully flat}} & T(k) \ar[l]_-{a_{t_0}} \ar[r]^-{w_Y} & H_{c,\mathfrak P}^{m}(X,{ {\mathbb Z} }(n)) \ar@{->>}[r]_-\phi \ar[dr]_-{\phi'} & S(k) \ar[d]^-f\\
&&& S'(k)}$$ where the smooth proper connected scheme $T$ pointed at $t_0$ and $Y\in Hom_{DM}(M(T)\otimes M^c(X),{ {\mathbb Z} }(n)[m])$ are such that the composition $\phi\circ w_Y: T(k)\longrightarrow S(k)$ is surjective (as in the proof of Proposition \[prop: image is abelian\]), and $a_{t_0}:T\longrightarrow Alb_T$ is the canonical morphism that sends $t_0$ to the unit of $Alb_T.$
With these properties at hand, the same argument for the proof of Proposition \[prop: existence criterion\] yields:
\[prop: criterion for alg-prop\] Suppose that $X$ is a connected scheme over $k.$ Then, there is a universal $\mathfrak P$-regular homomorphism for $H_{c,\mathfrak P}^{m}(X,{ {\mathbb Z} }(n))$ if and only if there is a constant $c$ such that $\dim S\leq c$ for any maximal $\mathfrak P$-regular homomorphism $$\phi: H_{c,\mathfrak P}^{m}(X,{ {\mathbb Z} }(n))\longrightarrow S(k).$$
In fact, the maximal $\mathfrak P$-regular homomorphism with a maximal dimensional target is the universal $\mathfrak P$-regular homomorphism $\Phi_{c,\mathfrak P,X}^{m,n}.$
With this criterion, we obtain the following existence theorem.
Let $X$ be a connected scheme of dimension $d$ over $k.$ Then, there is a universal $\mathfrak P$-regular homomorphism $$\Phi_{c,\mathfrak P,X}^{m,n}: H_{c,\mathfrak P}^{m}(X,{ {\mathbb Z} }(n))\longrightarrow Alg_{c,\mathfrak P,X}^{m,n}(k)$$ if $m\leq n+2.$ If, in addition, $X$ is smooth, the universal $\mathfrak P$-regular homomorphism exists for $(m,n)=(2d,d)$ as well.
Let $\phi: H_{c,\mathfrak P}^{m}(X,{ {\mathbb Z} }(n))\longrightarrow S(k)$ be a surjective $\mathfrak P$-regular homomorphism. Then, there are smooth proper connected scheme $T$ pointed at $t_0\in T(k)$ and $Y\in Hom_{DM}(M(T)\otimes M^c(X),{ {\mathbb Z} }(n)[m])$ such that the composition $$\phi\circ w_Y:T(k)\longrightarrow S(k)$$ is surjective. Suppose now that $a_{t_0}: T\longrightarrow Alb_T$ is the canonical map that sends $t_0$ to the unit of $Alb_T.$ The universality of $a_{t_0}$ induces the (surjective) homomorphism $f: Alb_T\longrightarrow S$ such that $f\circ a_{t_0}=\phi\circ w_Y.$
The maps $w_Y: T(k)\longrightarrow H_{c,\mathfrak P}^{m}(X,{ {\mathbb Z} }(n))$ and $a_{t_0}: T(k)\longrightarrow Alb_T(k)$ factor through $H_0(T,{ {\mathbb Z} })^0$ and yields the following commutative diagram $$\xymatrix{T(k) \ar[dr] \ar@/^1pc/[drr]^-{w_Y} \ar@/_1pc/[ddr]_-{a_{t_0}} \\
& H_0(T,{ {\mathbb Z} })^0 \ar[r] \ar[d]^-{alb_T} & H_{c,\mathfrak P}^{m}(X,{ {\mathbb Z} }(n)) \ar[r]^-\phi & S(k) \\
& Alb_T(k) \ar@{->>}@/_1pc/[urr]_f}$$ where $T(k)\longrightarrow H_0(T,{ {\mathbb Z} })^0$ sends $t\in T(k)$ to the class of $t-t_0\in H_0(T,{ {\mathbb Z} })^0,$ and $H_0(T,{ {\mathbb Z} })^0\longrightarrow H_{c,\mathfrak P}^{m}(X,{ {\mathbb Z} }(n))$ sends $Z: { {\mathbb Z} }\longrightarrow M(T)\in H_0(T,{ {\mathbb Z} })^0$ to $${ {\mathbb Z} }\otimes M^c(X)\buildrel Z\otimes id_{M^c(X)}\over\longrightarrow M(T)\otimes M^c(X)\buildrel Y\over\longrightarrow { {\mathbb Z} }(n)[m].$$
We claim that the $l$-torsion part $_l\phi: _l H_{c,\mathfrak P}^{m}(X,{ {\mathbb Z} }(n)) \longrightarrow _l S(k)$ is surjective for almost all prime $l.$ Since the $l$-torsion part of the Albanese map $_l alb_T$ is surjective by Rojtman’s theorem (Theorem \[thm: classical Rojtman-type\]), this follows from the surjectivity of $_l f$ for $l$ relatively prime to the index $(\mathrm{ker}f:\mathrm{ker}f^0).$ The surjectivity of $_l f$ follows from the divisibility of the group of rational points of an abelian variety (see the proof of Theorem \[thm: existence in codimensions\] for details).
Therefore, by Proposition \[prop: criterion for alg-prop\], it is enough to show the finiteness of $_l H_{c,\mathfrak P}^{m}(X,{ {\mathbb Z} }(n))$ for one of the above $l.$ Let us assume that $l\neq\mathrm{char}~k.$ As explained in the proof of Theorems \[thm: existence in codimensions\], this follows from the finiteness of $H_c^{m-1}(X,{ {\mathbb Z} }/l(n)).$ But this has already been shown in the proofs of Theorems \[thm: existence in codimensions\] and \[thm: existence in zero dim\].
For smooth proper connected schemes $X,$ the representatives $Alg_X^r,$ $Alg_{c,X}^{2r,r}$ and $Alg_{c,\mathfrak P,X}^{2r,r}$ coincide (at least under the assumption of resolution of singularities).
\[prop: comparison with the classical algebraic representatives\] Let $X$ be a connected proper scheme over $k.$ If there exist algebraic representative (Subsection \[section: The case of smooth proper schemes\]) and universal $\mathfrak P$-regular homomorphism, then $Alg_X^{r}$ and $Alg_{c,\mathfrak P,X}^{2r,r}$ are canonically isomorphic.
If, in addition, $X$ is smooth and if $Alg_{c,X}^{2r,r}$ also exists, then the canonical map $Alg_{c,\mathfrak P,X}^{2r,r}\longrightarrow Alg_{c,X}^{2r,r}$ is an isomorphism under resolution of singularities.
There is a unique isomorphism between $Alg_X^r$ and $Alg_{c,\mathfrak P,X}^{2r,r},$ one for each direction, by the universality of these objects with respect to $\mathfrak P$-regular homomorphisms and by Proposition \[prop: comparison with the classical algebraic part\].
For the second assertion, note that there is a canonical map given by the universal property of $Alg_{c,\mathfrak P,X}^{2r,r}:$ $$\xymatrixcolsep{4pc}\xymatrix{ H_{c,\mathfrak P}^{2r}(X,{ {\mathbb Z} }(r)) \ar[r]^-{\Phi_{c,\mathfrak P,X}^{2r,r}} \ar@{=}[d] & Alg_{c,\mathfrak P,X}^{2r,r} \ar@{..>}[d]^{\exists!}\\
H_{c,alg}^{2r}(X,{ {\mathbb Z} }(r)) \ar[r]_-{\Phi_{c,X}^{2r,r}} & Alg_{c,X}^{2r,r}}$$ We need to construct the inverse of the dotted arrow. For this, it suffices to show that $\Phi_{c,\mathfrak P,X}^{2r,r}$ is regular (not only $\mathfrak P$-regular).
Suppose that $T$ is a connected smooth scheme over $k$ and $Y$ is a morphism in $Hom_{DM}(M(T)\otimes M^c(X),{ {\mathbb Z} }(r)[2r]).$ We need to show that the composition $$T(k)\buildrel w_Y\over\longrightarrow H_{c,\mathfrak P}^{2r}(X,{ {\mathbb Z} }(r)) \buildrel \Phi_{c,\mathfrak P,X}^{2r,r}\over\longrightarrow Alg_{c,\mathfrak P,X}^{2r,r}(k)$$ comes from a scheme morphism. But this is clear because $w_Y$ factors as $$\xymatrix{ T(k) \ar[dr]_{inc.} \ar[rr]^-{w_Y} && H_{c,\mathfrak P}^{2r}(X,{ {\mathbb Z} }(r))\\
& \bar T(k) \ar[ur]_-{w_{\bar Y}}}$$ where $\bar T$ is a smooth compactification of $T$ which exists by resolution of singularities and $\bar Y$ is the preimage of $Y$ under the surjective map $$Hom_{DM}(M(\bar T)\otimes M^c(X),{ {\mathbb Z} }(r)[2r])\cong CH^r(\bar T\times X)\longrightarrow CH^r(T\times X)\cong Hom_{DM}(M(T)\otimes M^c(X),{ {\mathbb Z} }(r)[2r]).$$ Here, we used the smoothness of $X$ to compare motivic cohomology with Chow groups by [@MVW Theorem 19.1].
\[rem: interesting counterexample\] Suppose that $X$ is a smooth proper connected scheme over $k.$ Since $H_{c,\mathfrak P}^1(X,{ {\mathbb Z} }(1))$ is a trivial group by Remark \[rem: interesting example one\], the universal $\mathfrak P$-regular homomorphism is clearly the trivial isomorphism $$H_{c,\mathfrak P}^1(X,{ {\mathbb Z} }(1))\buildrel \cong\over\longrightarrow ({ {\mathrm{Spec~}} }k)(k).$$
On the other hand, we claim that the universal regular homomorphism for $H_{c,alg}^1(X,{ {\mathbb Z} }(1))$ is the isomorphism $$v: H_{c,alg}^1(X,{ {\mathbb Z} }(1))\buildrel \cong\over\longrightarrow \mathbb G_m(k)$$ that sends an invertible regular function $f\in\mathcal O^*(X)\cong H_{c,alg}^1(X,{ {\mathbb Z} }(1))$ to its (unique) value (in $k^*$). Since we already know the existence of the universal regular homomorphism $\Phi_{c,X}^{1,1}$ (Theorem \[thm: existence in codimensions\]) and the map $v$ is an isomorphism (Remark \[rem: interesting example one\]), it is enough to show the regularity of $v.$ Suppose $T$ is a smooth connected scheme pointed at $t_0$ and $Y$ is an element of $\mathcal O^*(T\times X)\cong Hom_{DM}(M(T)\otimes M^c(X),{ {\mathbb Z} }(1)[1]).$ Then, we have the commutative diagram $$\xymatrix{ T(k) \ar[r]^-{w_Y} \ar[d]_-{id\times \{x\}} & H_{c,alg}^1(X,{ {\mathbb Z} }(1)) \ar[r]^-v & \mathbb G_m(k) \\
T(k)\times \{x\} \ar@{^{(}->}[r]_-{\text{inc.}} & T(k)\times X(k) \ar[r]_-{Y} & \mathbb G_m(k) \ar[u]_-{-\times (v(Y|_{\{t_0\}\times X}))^{-1}}}$$ where $x$ is any rational point on $X.$ Therefore, the composition $v\circ w_Y$ is induced by a scheme morphism.
Now, suppose that $X$ is an arbitrary connected scheme over $k.$ If the universal regular homomorphisms $\Phi_{c,X}^{m,n}$ and the universal $\mathfrak P$-regular homomorphism $\Phi_{c,\mathfrak P,X}^{m,n}$ exist, there is always a canonical morphism from the abelian variety $Alg_{c,\mathfrak P,X}^{m,n}$ to the semi-abelian variety $Alg_{c,X}^{m,n}$ induced by the universality: $$\label{diagram: ab to semiab}
\xymatrixcolsep{4pc}\xymatrix{ H_{c,\mathfrak P}^m(X,{ {\mathbb Z} }(n)) \ar@{_{(}->}[d]_-{inc.} \ar@{->>}[r]^-{\Phi_{c,\mathfrak P,X}^{m,n}} & Alg_{c,\mathfrak P,X}^{m,n}(k) \ar@{..>}[d]^{\exists!}\\
H_{c,alg}^m(X,{ {\mathbb Z} }(n)) \ar@{->>}[r]_-{\Phi_{c,X}^{m,n}} & Alg_{c,X}^{m,n}(k)}$$ For smooth $X$ and $(m,n)=(2,1),$ we know the precise relation between $Alg_{c,\mathfrak P,X}^{m,n}$ and $Alg_{c,X}^{m,n}.$
\[cor: max abelian subvariety\] Assume resolution of singularities. If $X$ is a smooth connected scheme over $k,$ the abelian variety $Alg_{c,\mathfrak P,X}^{2,1}$ is canonically isomorphic to the maximal abelian subvariety of $Alg_{c,X}^{2,1}$ under the canonical homomorphism in the diagram (\[diagram: ab to semiab\]).
If $X$ is proper, it is already dealt with in Proposition \[prop: comparison with the classical algebraic representatives\]. Below, we assume that $X$ is not proper.
We consider the diagram $$\xymatrixcolsep{4pc}\xymatrix{ H_{c,\mathfrak P}^2(X,{ {\mathbb Z} }(1)) \ar@{_{(}->}[d]_{inc} \ar@{->>}[r]^-{\Phi_{c,\mathfrak P,X}^{2,1}} & Alg_{c,\mathfrak P,X}^{2,1}(k) \ar@{..>}[d]^{\exists !~a} \\
H_{c,alg}^2(X,{ {\mathbb Z} }(1)) \ar[r]^-{\Phi_{c,X}^{2,1}}_-{\text{isom.}} & Alg_{c,X}^{2,1}(k)}$$ By Theorem \[thm: main theorem in codim one\], $\Phi_{c,X}^{2,1}$ is an isomorphism. The commutativity of the diagram implies that $a$ is injective. Let $A^{max}$ be the maximal abelian subvariety of $Alg_{c,X}^{2,1}.$ We need to show that the image of $a$ is $A^{max}(k).$ Equivalently, we need to prove $\mathrm{im}(\Phi_{c,X}^{2,1}\circ inc)=A^{max}(k).$ Since $\mathrm{im}(\Phi_{c,X}^{2,1}\circ inc)=\mathrm{im}(a)$ is an abelian variety, we have $\mathrm{im}(\Phi_{c,X}^{2,1}\circ inc)\subset A^{max}(k).$
For the other inclusion, let $\bar X$ be a good compactification of $X$ with $Z:=\bar X\setminus X.$ Suppose $(\mathcal P^0,p^0)\in Pic(Pic_{\bar X,Z,red}^0\times\bar X, Pic_{\bar X,Z,red}^0\times Z)$ is the element corresponding to the canonical map $Pic_{\bar X,Z,red}^0\longrightarrow Pic_{\bar X,Z}$ under Proposition \[prop: representability of the relative Picard functor\], and $(\mathcal Q,q)\in Pic(A^{max}\times\bar X, A^{max}\times Z)$ is the one corresponding to the canonical map $A^{max}\hookrightarrow Alg_{c,X}^{2,1}\cong Pic_{\bar X,Z,red}^0\longrightarrow Pic_{\bar X,Z}.$ Then, there is a commutative diagram (we use the same notation as in Proposition \[prop: relative Picard is regular\] but this time all the semi-abelian varieties are pointed at the identity elements, and all $``\hookrightarrow"$ stand for the inclusions) $$\xymatrix{ A^{max}(k) \ar@{_{(}->}[d] \ar[r]^-{w_{F(\mathcal Q,q)}} & H_{c,\mathfrak P}^2(X,{ {\mathbb Z} }(1)) \ar@{^{(}->}[r]^-{inc} & H_{c,alg}^2(X,{ {\mathbb Z} }(1)) \ar[r]^-{\Phi_{c,X}^{2,1}} \ar@{_{(}->}[d]^i \ar[dr]^{\phi_0} & Alg_{c,X}^{2,1}(k) \ar[d]_\cong^{\text{Theorem~\ref{thm: main theorem in codim one}}} \\
Alg_{c,X}^1(k) \ar[d]_\cong^{\text{Theorem~\ref{thm: main theorem in codim one}}} & & H_{c}^2(X,{ {\mathbb Z} }(1)) \ar[d]^g & Pic_{\bar X,Z,red}^0(k) \ar@{_{(}->}[d]\\
Pic_{\bar X,Z,red}^0(k) \ar[rr]^{B_{(\mathcal P^0,p^0)}} \ar@{_{(}->}[d] & & Pic(\bar X,Z) \ar[r]^\psi & Pic_{\bar X,Z}(k) \\
Pic_{\bar X,Z}(k) \ar[urr]_{B_{(\mathcal P,p)}} \ar@/_2pc/[urrr]_{id} }$$ The commutativity of the diagram implies that the composition of maps in the top row is injective. Hence, we have $\mathrm{im}(\Phi_{c,X}^{2,1}\circ inc)\supset A^{max}(k).$
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---
abstract: 'Given $n$-copies of unknown bipartite ( possiblly mixed ) state, our task is to test whether the state is a pure state of not. Allowed to use the global operations, optimal one-sided error test is the projection onto the symmetric subspace, obviously. Is it possible to approximate the globally optimal measurement by LOCC when $n$ is large?'
author:
- |
Keiji Matsumoto\
National Institute of Informatics, Tokyo, Japan, and JST, Tokyo, Japan
title: Test of Purity by LOCC
---
Introduction
============
Given $n$-copies of unknown bipartite ( possiblly mixed ) state, our task is to test whether the state is a pure state of not. Allowed to use the global operations, optimal one-sided error test is the projection onto the symmetric subspace, obviously. Is it possible to approximate the globally optimal measurement by LOCC when $n$ is large?
A standard form of an ensemble of identical bipartite pure states
=================================================================
Suppose we are given $n$-copies of unknown pure bipartite state $\left\vert
\phi\right\rangle \in\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, which is unknown. Here we assume $\mathcal{H}_{A}\simeq\mathcal{H}_{B}\simeq\mathcal{H}$ and $\dim\mathcal{H}=d$. It is known that $\left\vert \phi\right\rangle ^{\otimes
n}$ has the standard form defined as follows.
Note $|\phi\rangle^{\otimes n}$ is invariant by the reordering of copies, or the action of the permutation $\sigma$ in the set $\{1,\ldots n\}$ such that $$\bigotimes_{i=1}^{n}|h_{i,A}\rangle|h_{i,B}\rangle\mapsto\bigotimes_{i=1}^{n}|h_{\sigma^{-1}(i),A}\rangle|h_{\sigma^{-1}(i),B}\rangle,\label{sym}$$ where $|h_{i,A}\rangle\in\mathcal{H}_{A}$ and $|h_{i,B}\rangle\in
\mathcal{H}_{B}\;$. Action of the symmetric group occurs a decomposition of the tensored space $\mathcal{H}^{\otimes n}$ [@GW], $$\mathcal{H}^{\otimes n}=\bigoplus_{\lambda}\mathcal{W}_{\lambda},\;\mathcal{W}_{\lambda}:=\mathcal{U}_{\lambda}\otimes\mathcal{V}_{\lambda},$$ where $\mathcal{U}_{\lambda}$ and $\mathcal{V}_{\lambda}$ is an irreducible space of the tensor representation of $\mathrm{SU}(d)$, and the representation (\[sym\]) of the symmetric group, respectively, and $$\lambda=(\lambda_{1},\ldots,\lambda_{d}),\quad\;\lambda_{i}\geq\lambda
_{i+1}\geq0,\,\sum_{i=1}^{d}\lambda_{i}=n$$ is called *Young index*, which $\mathcal{U}_{\lambda}$ and $\mathcal{V}_{\lambda}$ uniquely corresponds to. We denote by $\mathcal{U}_{\lambda,A}$, $\mathcal{V}_{\lambda,A}$, and $\mathcal{U}_{\lambda,B}$, $\mathcal{V}_{\lambda,B}$ the irreducible component of $\mathcal{H}_{A}^{\otimes n}$ and $\mathcal{H}_{B}^{\otimes n}$ , respectively. Also, $\mathcal{W}_{\lambda,A}:=\mathcal{U}_{\lambda,A}\otimes\mathcal{V}_{\lambda,A}$, $\mathcal{W}_{\lambda,B}:=\mathcal{U}_{\lambda,B}\otimes\mathcal{V}_{\lambda,B}$.
Due to [@MatsumotoHayashi2], in terms of this decomposition, $\left\vert
\phi\right\rangle ^{\otimes n}$ can be written as $$\left\vert \phi\right\rangle ^{\otimes n}=\bigoplus_{\lambda}a_{\lambda
}\left\vert \phi_{\lambda}\right\rangle \left\vert \Phi_{\lambda}\right\rangle
,\label{decomposition}$$ where $\left\vert \phi_{\lambda}\right\rangle \in\mathcal{U}_{\lambda
,A}\otimes\mathcal{U}_{\lambda,B}$, and $\left\vert \Phi_{\lambda
}\right\rangle \in\mathcal{V}_{\lambda,A}\otimes\mathcal{V}_{\lambda,B}$. While $a_{\lambda}$ and $\left\vert \phi_{\lambda}\right\rangle $ are dependent on $\left\vert \phi\right\rangle $, $\left\vert \Phi_{\lambda
}\right\rangle $ is a maximally entangled state which does not depend on $\left\vert \phi\right\rangle $,$$\left\vert \Phi_{\lambda}\right\rangle :=\frac{1}{\sqrt{d_{\lambda}}}\sum_{i=1}^{d_{\lambda}}\left\vert f_{i}\right\rangle \left\vert
f_{i}\right\rangle ,$$ with $\left\{ \left\vert f_{i}\right\rangle \right\} $ being an orthonormal complete basis of $\mathcal{V}_{\lambda}$, and $d_{\lambda}:=\dim
\mathcal{V}_{\lambda}$.
Observe that linear span of the state vectors in the of (\[decomposition\]) is the symmetric subspace of $\left( \mathcal{H}_{A}\otimes\mathcal{H}_{B}\right) ^{\otimes n}$. Therefore, denoting the projector on this subspace by $\Pi^{n}$, we have $$\Pi^{n}=\bigoplus_{\lambda}\mathcal{U}_{\lambda,A}\otimes\mathcal{U}_{\lambda,B}\otimes\left\vert \Phi_{\lambda}\right\rangle \left\langle
\Phi_{\lambda}\right\vert .\label{project-sym}$$
Optimal LOCC of maximally entangled state
=========================================
[@Tsuda] treats the problem of testing whether the given state $\rho$ is the $d$dimensional maximally entangled state $$\left\vert \Phi\right\rangle :=\frac{1}{\sqrt{d}}\sum_{i=1}^{d}\left\vert
f_{i}\right\rangle \left\vert f_{i}\right\rangle ,$$ and found out a protocol whose the probability $P_{acc}$ of accepting the hypothesis equals $$P_{acc}=\frac{\left\langle \Phi\right\vert \rho\left\vert \Phi\right\rangle
+\frac{1}{\left( d\right) ^{2}}}{1+\frac{1}{\left( d\right) ^{2}}}.\label{tsuda-acc}$$
When $d$ is very large, $$P_{acc}\approx\left\langle \Phi\right\vert \rho\left\vert \Phi\right\rangle ,$$ the RHS of which is the accepting probability of globally optimal one-sided test.
Protocol
========
Observe the globally optimal test, $\Pi^{n}$, is equivalent to the composition of the projector $\mathcal{W}_{\lambda,A}\otimes\mathcal{W}_{\lambda,B}$ followed by $\mathbf{I}_{\mathcal{U}_{\lambda,A}\otimes\mathcal{U}_{\lambda
,B}}\otimes\left\vert \Phi_{\lambda}\right\rangle \left\langle \Phi_{\lambda
}\right\vert $. While the former is done by an LOCC, the latter cannot be implemented by LOCC. Hence, instead, we perform the asymptotically optimal test of the maximally entangled state in [@Tsuda]. So, our protocol is:
(i)
: A and B applies the projective measurement $\left\{
\mathcal{W}_{\lambda,A}\right\} _{\lambda}$ and $\left\{ \mathcal{W}_{\lambda,B}\right\} _{\lambda}$, respectively.
(ii)
: Do the test for maximally entangled state to $\mathrm{tr}_{\mathcal{U}_{\lambda,A}}\rho_{n,\lambda}$, where $p_{\lambda}:=\mathrm{tr}\,\rho^{\otimes n}\,\mathcal{W}_{\lambda,A}\otimes\mathcal{W}_{\lambda,B}$ and $\rho_{n,\lambda}:=\frac{1}{p_{\lambda}}\mathcal{W}_{\lambda,A}\otimes\mathcal{W}_{\lambda,B}\,\rho^{\otimes n}\,\mathcal{W}_{\lambda,A}\otimes\mathcal{W}_{\lambda,B}$.
Peformance of the protocol
--------------------------
In this subsection, it is proved that our protocol is asymptotically as good as globally optimal test, $\Pi^{n}$. If the given state is a pure state, obviously the acceptance probability $P_{opt}^{n}$ of the test $\Pi^{n}$ is 1. If the input is not a pure state, due to \[grep-type-2\], we have $$\begin{aligned}
-\lim_{n\rightarrow\infty}\frac{1}{n}\log P_{opt}^{n} & =D\left( \,\left(
1,0,\cdots,0\right) \,|\,|\,\boldsymbol{p}\right) \\
& =-\log p_{1}.\end{aligned}$$ Also, by (\[project-sym\]), when the given state is $\rho^{\otimes n}$, $$\begin{aligned}
P_{opt}^{n} & :=\sum_{\lambda}\mathrm{tr}\rho^{\otimes n}\mathcal{U}_{\lambda,A}\otimes\mathcal{U}_{\lambda,B}\otimes\left\vert \Phi_{\lambda
}\right\rangle \left\langle \Phi_{\lambda}\right\vert \\
& =\sum_{\lambda}p_{\lambda}\mathrm{tr}\left\langle \Phi_{\lambda}\right\vert
\rho_{n,\lambda}\left\vert \Phi_{\lambda}\right\rangle .\end{aligned}$$
Below, we will show our LOCC test is asymptotically equivalent to this globally optimal test. On the other hand, due to \[tsuda-acc\], our test will accept the input $\rho_{n}$ with the probability$$P_{\ast}^{n}:=\sum_{\lambda}p_{\lambda}\frac{\mathrm{tr}\left\langle
\Phi_{\lambda}\right\vert \rho_{n,\lambda}\left\vert \Phi_{\lambda
}\right\rangle +\frac{1}{\left( d_{\lambda}\right) ^{2}}}{1+\frac{1}{\left(
d_{\lambda}\right) ^{2}}}.$$ If the given state $\rho$ is a pure state, $$P_{\ast}^{n}=\sum_{\lambda}p_{\lambda}\frac{1+\frac{1}{\left( d_{\lambda
}\right) ^{2}}}{1+\frac{1}{\left( d_{\lambda}\right) ^{2}}}=1.$$ Suppose $\rho$ is not a pure state. Observe $$\begin{aligned}
P_{\ast}^{n} & \leq\sum_{\lambda}p_{\lambda}\left( \mathrm{tr}\left\langle
\Phi_{\lambda}\right\vert \rho_{n,\lambda}\left\vert \Phi_{\lambda
}\right\rangle +\frac{1}{\left( d_{\lambda}\right) ^{2}}\right) \\
& =P_{opt}^{n}+\sum_{\lambda}\frac{p_{\lambda}}{\left( d_{\lambda}\right)
^{2}},\end{aligned}$$ where $$\begin{aligned}
\sum_{\lambda}\frac{p_{\lambda}}{\left( d_{\lambda}\right) ^{2}} &
=\sum_{\lambda}\frac{\mathrm{tr}\,\rho^{\otimes n}\,\mathcal{W}_{\lambda
,A}\otimes\mathcal{W}_{\lambda,B}}{\left( d_{\lambda}\right) ^{2}}\\
& \leq\sum_{\lambda}\frac{p_{1}^{n}\left( \dim\,\mathcal{W}_{\lambda
,A}\right) ^{2}}{\left( d_{\lambda}\right) ^{2}}\\
& =p_{1}^{n}\sum_{\lambda}\left( \dim\,\mathcal{U}_{\lambda,A}\right)
^{2}\\
& =p_{1}^{n}\sum_{\lambda}\left( \frac{\prod_{i<j}\left( \lambda
_{i}-\lambda_{j}-i+j\right) }{\prod_{i=1}^{d-1}\left( d-i\right) !}\right)
^{2}\\
& \leq p_{1}^{n}\left( n+1\right) ^{d}n^{d^{2}}.\end{aligned}$$ (Also, one may use the relation $$\begin{aligned}
& \sum_{\lambda}\left( \dim\,\mathcal{U}_{\lambda,A}\right) ^{2}\\
& =\dim\left( \text{symmetric subspace of }\left(
\mathbb{C}
^{d^{2}}\right) ^{\otimes n}\right) \\
& \leq\left( n+1\right) ^{d^{2}}$$ )
Therefore, even if the state $\rho$ is not a pure state, $$-\lim_{n\rightarrow\infty}\frac{1}{n}\log P_{\ast}^{n}\geq-\lim_{n\rightarrow
\infty}\frac{1}{n}\log P_{acc}^{n}.$$ Since the other side of inequality is trivial, we have $$-\lim_{n\rightarrow\infty}\frac{1}{n}\log P_{\ast}^{n}=-\lim_{n\rightarrow
\infty}\frac{1}{n}\log P_{acc}^{n}.$$
Therefore, regardless $\rho$ is pure or not, our LOCC protocol very closely approximates the globally optimal protocol when $n$ is large.
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M. Hayashi, K. Matsumoto, and Y. Tsuda, “A study of LOCC-detection of a maximally entangled state using hypothesis testing”, J. of Phys. A, 39 14427-14446 (2006).
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Group representation theory {#appendixA}
===========================
\[lem:decohere\] Let $U_{g}$ and $U_{g}^{\prime}$ be an irreducible representation of $G$ on the finite-dimensional space $\mathcal{H}$ and $\mathcal{H}^{\prime}$, respectively. We further assume that $U_{g}$ and $U_{g}^{\prime}$ are not equivalent. If a linear operator $A$ in $\mathcal{H}\oplus\mathcal{H}^{\prime}$ is invariant by the transform $A\rightarrow U_{g}\oplus U_{g}^{\prime}AU_{g}^{\ast}\oplus U_{g}^{^{\prime
}\ast}$ for any $g$, $\mathcal{H}A\mathcal{H^{\prime}}=0$ [@GW].
\[lem:shur\] (Shur’s lemma [@GW]) Let $U_{g}$ be as defined in lemma \[lem:decohere\]. If a linear map $A$ in $\mathcal{H}$ is invariant by the transform $A\rightarrow U_{g}AU_{g}^{\ast}$ for any $g$, $A=c\mathrm{Id}_{\mathcal{H}}$.
Representation of symmetric group and SU
========================================
Due to [@GW], we have $$\begin{aligned}
\dim\mathcal{U}_{\lambda} & =\frac{\prod_{i<j}\left( l_{i}-l_{j}\right)
}{\prod_{i=1}^{d-1}\left( d-i\right) !},\label{dim-representation-1}\\
d_{\lambda} & =\dim\mathcal{V}_{\lambda}=\frac{n!}{\prod_{i=1}^{d}\left(
\lambda_{i}+d-i\right) !}\prod_{i<j}\left( l_{i}-l_{j}\right)
,\label{dim-representation-2}$$ with $l_{i}:=\lambda_{i}+d-i$. It is easy to show$$\log\dim\mathcal{U}_{\lambda}\leq d^{2}\log n.\label{dim-zero-rate}$$
Below, $$\left\vert \phi\right\rangle =\sum_{i=1}^{d}\sqrt{p_{i}}\left\vert
e_{i}\right\rangle \left\vert e_{i}\right\rangle ,$$ where $\left\{ \left\vert e_{i}\right\rangle \right\} _{i}$ is an orthonormal basis of $\mathcal{H}$. With $a_{\lambda}^{\phi}=\mathrm{Tr}\left\{ \mathcal{W}_{\lambda,A}\left( \mathrm{Tr}_{B}|\phi\rangle\langle
\phi|\right) ^{\otimes n}\right\} $, $$\begin{aligned}
\left\vert \frac{\log d_{\lambda}}{n}-\mathrm{H}\left( \frac{\lambda}{n}\right) \right\vert & \leq\frac{d^{2}+2d}{2n}\log
(n+d),\label{grep-type-1}\\
\sum_{\frac{\lambda}{n}\in\mathrm{R}}a_{\lambda}^{\phi} & \leq\left(
n+1\right) ^{d\left( d+1\right) /2}\exp\left\{ -n\min_{\boldsymbol{q}\,\in\mathrm{R}}\mathrm{D}\left( \boldsymbol{q}||\boldsymbol{p}\right)
\right\} ,\label{grep-type-2}$$ where $\mathrm{R}$ is an arbitrary closed subset [@MatsumotoHayashi].
|
---
abstract: 'The flexoelectric effect refers to polarization induced in an insulator when a strain gradient is applied. We have developed a first-principles methodology based on density-functional perturbation theory to calculate the elements of the bulk clamped-ion flexoelectric tensor. In order to determine the transverse and shear components directly from a unit cell calculation, we calculate the current density induced by the adiabatic atomic displacements of a long-wavelength acoustic phonon. Previous implementations based on the charge-density response required supercells to capture these components. At the heart of our approach is the development of an expression for the current-density response to a generic long-wavelength phonon perturbation that is valid for the case of nonlocal pseudopotentials. We benchmark our methodology on simple systems of isolated noble gas atoms, and apply it to calculate the clamped-ion flexoelectric constants for a variety of technologically important cubic oxides.'
author:
- 'Cyrus E. Dreyer'
- Massimiliano Stengel
- David Vanderbilt
bibliography:
- 'flexo.bib'
title: 'Current-density implementation for calculating flexoelectric coefficients'
---
\#1 \#1 \#1 \#1
Introduction
============
The flexoelectric (FxE) effect, where polarization is induced by a strain gradient, is universal in all insulators. As devices shrink to the micro and nano scale, large strain gradients can occur, and therefore the FxE effect can play a significant role in the properties of such devices, influencing the so-called dielectric dead layer[@Majdoub2009], domain walls and domain structure[@Lee2011; @Yudin2012; @Borisevich2012], relative permittivity and Curie temperature[@Catalan2004; @Catalan2005], critical thickness of films to exhibit switchable polarization[@Zhou2012], and spontaneous polarization in the vicinity of twin and antiphase boundaries[@Morozovska2012]. Also, the FxE effect can be exploited for novel device design paradigms, such as piezoelectric “meta-materials” constructed from nonpiezoelectric constituents[@Zhu2006; @Bhaskar2016], or mechanical switching of ferroelectric polarization [@Lu2012; @Gruverman2003].
One of the crucial limitations to understanding and exploiting the FxE effect is the lack of a clear experimental and theoretical consensus on the size and sign of the FxE coefficients, even in commonly studied materials such at SrTiO$_3$ and BaTiO$_3$[@Zubko2013; @Yudin2013]. A key element to forming this understanding is the development of an efficient first-principles methodology to calculate all of the components of the bulk FxE tensor. Recently, Stengel, [@Stengel2013] and Hong and Vanderbilt[@Hong2011; @Hong2013] (HV), developed the formalism for calculating the full bulk FxE tensor from first principles. [^1]
Each element of the FxE tensor has a “clamped-ion” (CI) contribution, arising from the effect of the strain gradient on the valence electrons in the crystal, and a “lattice-mediated” (LM) contribution, arising from internal relaxations induced by the applied strain and strain gradient [@Stengel2013; @Hong2013]. In Refs. and , HV described an implementation for calculating the bulk CI and LM longitudinal FxE coefficients (i.e., the coefficients relating the induced polarization in direction $\alpha$ to a gradient of uniaxial strain $\varepsilon_{\alpha\alpha}$, also in direction $\alpha$). Their methodology involved using density functional theory (DFT) to calculate the real-space response of the charge density to atomic displacements in a simple $N\times 1\times 1$ bulk supercell containing $N$ repitions of the primitive bulk cell.
In Ref. , Stengel developed a strategy that allowed a calculation of the full FxE response for cubic SrTiO$_3$ based in part on the charge-density response to a long-wavelength acoustic phonon, and in part on large slab supercell calculations (repeated slabs separated by vacuum). The first part of this methodology allowed the LM contributions to all bulk FxE tensor elements, as well as the CI contributions to the longitudinal coefficients, to be determined from linear-response calculation on a single unit cell using density-functional perturbation theory (DFPT) [@Baroni2001]. However, the “transverse” and “shear” CI contributions [@Hong2013; @Stengel2014; @Stengel2013natcom] had to be calculated indirectly by relating them to the open-circuit electric field appearing across the slab when a long wavelength acoustic phonon was applied to the slab supercell as a whole. As a result, this implementation required DFPT calculations to be performed on large slab supercells.
The implementation described in Ref. thus provides a methodology for calculating the full FxE tensor for a given material. However, the reliance on computationally intensive slab supercell calculations for the transverse and shear CI coefficients represents a significant limitation to efficient calculation, especially in complex materials. Therefore, it is highly desirable to develop an approach that allows the full bulk FxE tensor, including its longitudinal, transverse, and shear components, to be obtained from DFPT calculations on single unit cells.
The essential problem is that single-unit-cell DFPT calculations that determine only the charge-density response to a long-wavelength phonon, as in Ref. , are incapable of revealing the transverse and shear CI contributions, since the induced charge is proportional to the *divergence* of the polarization, which is absent for transverse phonons. To go further, it is necessary to compute the induced *polarization itself*. Unfortunately, the well-known Berry-phase formulation [@KingSmith1993; @Resta1994] of the electric polarization is useless here, since it provides only the total polarization, which averages to zero over a phonon wavelength. Instead, we need access to the spatially resolved polarization on the scale of the wavelength. The only clear path to obtaining this local polarization is via its relation to the adiabatic current density [@Hong2013; @Stengel2013; @StengelUNPUB]. Thus, the desired methodology is one that computes the spatially resolved *current density* induced by a strain gradient perturbation [@Hong2013; @Stengel2013; @StengelUNPUB] in the context of long-wavelength longitudinal *and transverse* phonons.
The microscopic current density is, of course, just proportional to the quantum-mechanical probability current, as discussed in any standard textbook [@Sakuri]. However, this standard formula assumes a local Hamiltonian of the form $H=p^2/2m+V$ with a local potential $V$. Thus, it becomes problematic if the Hamiltonian of interest contains *nonlocal* potentials, as the probability current no longer satisfies the continuity equation[@Li2008]. This issue is very relevant in the context of DFT, since most popular implementations make use of a plane-wave basis set with a pseudopotential approximation to reduce the size of the basis set by avoiding an explicit description of the core electrons. Virtually all modern pseudopotential implementations contain nonlocal potentials in the form of projectors that operate on the wavefunctions [@Vanderbilt1990; @Hamann1979; @Kleinman1982; @Blochl1994]. Therefore, the standard formula for the current density is not a fit starting point for the current-response theory that we have in mind (we expand on these considerations in Sec. \[curden\]).
The definition and calculation of the microscopic current density in a nonlocal pseudopotential context is a rather general problem that has received considerable previous attention [@Umari2001; @Li2008; @Vignale1991; @ICL2001; @Pickard2003; @Mauri1996; @Mauri1996_nmr] in view of its application to the calculation of magnetic susceptibility [@Vignale1991; @ICL2001; @Pickard2003; @Mauri1996; @Mauri1996_nmr], nuclear magnetic resonance chemical shifts [@Pickard2001], electron paramagnetic resonance $g$ tensors[@Pickard2002], and so forth. Unfortunately a general, systematic solution that is appropriate to our scopes has not emerged yet. To see why this is challenging, it is important to note that the continuity equation is only one of the criteria that must be satisfied by a physically meaningful definition of the current density. Two other criteria are important. First, the formula must also reduce to the textbook expression in regions of space that lie outside the range of the nonlocal operators (pseudopotentials are typically confined to small spheres surrounding the atoms). Second, it must reduce to the well-known expressions for the macroscopic current in the long-wavelength limit. The approaches that have been proposed so far have either been specialized to a certain physical property (e.g., dielectric [@Umari2001] or diamagnetic [@Pickard2003] response), or limited in scope to a subset of the above criteria. For example, Li [*et al.*]{} [@Li2008] proposed a strategy that guarantees charge continuity by construction but does not satisfy the two additional criteria, as we shall see in Sec. \[curden\].
In addition to the technical challenges related to nonlocal pseudopotentials, there is another complication associated with the calculation of the flexoelectric coefficients using the current density in bulk. Namely, the bulk nonlongitudinal responses contain a contribution coming from the gradients of the local rotations in the crystal. This “circulating rotation-gradient” (CRG) contribution, derived in Ref. (where it is referred to as a “dynamic” or “gauge-field” term), must be treated carefully when comparing our calculations with previous results. We will discuss this point in Sec. \[diamag\].
In this work we develop a first-principles methodology based on DFT to calculate the full bulk CI FxE tensor from a single unit cell. At the heart of our technique lies the introduction of a physically sound microscopic current-density operator in the presence of nonlocal pseudopotentials that fulfills all criteria that we stated in the above paragraphs: (i) it satisfies the continuity equation; (ii) the contribution of the nonlocal pseudopotentials is correctly confined to the atomic spheres; and (iii) it reduces to the macroscopic velocity operator in the long-wavelength limit. We will discuss our approach for calculating the current density in the context of earlier works, and how it applies to the problem of calculating bulk FxE coefficients. Finally, we will demonstrate that the results for the CI FxE coefficients from our current-density implementation are in excellent agreement with the previous charge-density-based DFT implementations described above [@Hong2013; @Stengel2014], confirming that it is an accurate and efficient method for calculating the FxE response of materials.
The paper is organized as follows. In Sec. \[Approach\] we outline the general approach to determining FxE coefficients; in Sec. \[Form\] we give the formalism used in our calculations of the current density; in Sec. \[Imp\] we provide details of the implementation of the formalism; Sec. \[Res\] presents benchmark tests for the simple case of isolated noble gas atoms, and results for several technologically important, cubic oxide compounds; in Sec. \[Disc\], we discuss some technical issues that are associated with the current density in the presence of nonlocal pseudopotentials; and we conclude the paper in Sec. \[Con\].
Approach\[Approach\]
====================
The goal of this work is to calculate the bulk CI flexoelectric tensor elements $$\label{muII}
\mu^{\text{I}}_{\alpha\beta,\omega\nu}=\frac{d
P_\alpha}{d\eta_{\beta,\omega\nu}},$$ where $P_\alpha$ is the polarization in direction $\alpha$, and $$\eta_{\beta,\omega\nu}=\frac{\partial^2u_\beta}{\partial
r_\omega\partial r_\nu}$$ is the strain gradient tensor, where $u_\beta$ is the $\beta$ component of the displacement field. The superscript “I” indicates that the tensor elements are defined with respect to the unsymmetrized displacements [@Nye1985]; superscripts “II” will be used to indicate tensor elements defined with respect to symmetrized strain.
Calculating the polarization in Eq. (\[muII\]) is tricky from a quantum-mechanical standpoint, as it does not correspond to the expectation value of a well-defined operator. As mentioned above, the Berry-phase method [@KingSmith1993; @Resta1994] can be used to obtain the formal macroscopic polarization averaged over the cell. However, we require access to the local polarization *density* $P_\alpha(\textbf{r})$. Although the static microscopic polarization density is not well defined in a quantum mechanical context, at the linear-response level the *induced* polarization $P_{\alpha,\lambda}(\textbf{r})=\partial
P_\alpha(\textbf{r})/\partial\lambda$ resulting from a small change in parameter $\lambda$ can be equated to the local current flow via $\partial P_\alpha(\textbf{r})/\partial\lambda=
\partial J_\alpha(\textbf{r})/\partial\dot{\lambda}$, where $\dot{\lambda}$ is the rate of change of the adiabatic parameter, $\lambda$. Following the approach of Ref. , we now consider an adiabatic displacement of sublattice $\kappa$ (i.e., a given atom in the unit cell along with all of its periodic images) of a crystal in direction $\beta$ as given by $$\label{phon}
u_{\kappa\beta}(l,t)=\lambda_{\kappa\beta\textbf{q}}(t)e^{i\textbf{q}\cdot\textbf{R}_{l\kappa}},$$ where $l$ is the cell index. In this case, the induced local polarization density $P_{\alpha,\kappa\beta\textbf{q}}(\textbf{r})$ in direction $\alpha$ induced by mode $\kappa\beta$ of wavevector $\textbf{q}$ is $$\label{Jrt-dv}
P_{\alpha,\kappa\beta\textbf{q}}(\textbf{r})=
\frac{\partial J_{\alpha}(\textbf{r})}{\partial\dot{\lambda}_{\kappa\beta\textbf{q}}} .$$ Using the fact that the linearly induced current will be modulated by a phase with the same wavevector as the perturbation in Eq. (\[phon\]), we can define $$\label{Jrt}
P^{\textbf{q}}_{\alpha,\kappa\beta}(\textbf{r})=
P_{\alpha,\kappa\beta\textbf{q}}(\textbf{r})
e^{-i\textbf{q}\cdot\textbf{r}},$$ which is therefore a lattice-periodic function. This quantity, the *cell-periodic part of the first-order induced polarization density*, will play a central role in our considerations. It is also convenient to define $$\label{Pbar}
\overline{P}_{\alpha,\kappa\beta}^{\textbf{q}} \equiv
\frac{1}{\Omega}\int_{\text{cell}}
P_{\alpha,\kappa\beta}^{\textbf{q}}(\textbf{r}) d^3r,$$ where $\Omega$ is the cell volume, as the cell average of this response. In Ref. it was shown that the CI flexoelectric tensor elements are given by the second wavevector derivatives of $\overline{P}_{\alpha,\kappa\beta}^{\textbf{q}}$ via $$\label{muI}
\begin{split}
\mu^{\text{I}}_{\alpha\beta,\omega\nu}&= -\frac{1}{2}\sum_\kappa\frac{\partial^2\overline{P}_{\alpha,\kappa\beta}^{\textbf{q}}}{\partial
q_\omega\partial q_\nu}\Bigg\vert_{\textbf{q}=0}.
\end{split}$$
This formulation suggests that it may be possible to compute the polarization responses $\overline{P}^{\textbf{q}}_{\alpha,\kappa\beta}$ entirely from a single-unit-cell calculation, similar to the way that phonon responses are computed in DFPT. In fact, this is the case. The formalism necessary to compute these responses at the DFT level will be presented in the next sections, giving access to an efficient and robust means to compute the flexoelectric coefficients through Eq. (\[muI\]).
Formalism\[Form\]
=================
Given a time-dependent Hamiltonian with a single-particle solution $\Psi(t)$, the current density at a point **r** in Cartesian direction $\alpha$ can be written $$J_\alpha(\textbf{r})=
\langle\Psi(t)\vert\hat{\mathcal{J}}_\alpha(\textbf{r})\vert\Psi(t)\rangle
\label{Js}$$ where $\hat{\mathcal{J}}_\alpha(\textbf{r})$ is the current-density operator (a caret symbol over a quantity will indicate an operator). We will first address how to treat the time-dependent wavefunctions (Sec. \[adpert\]), and then discuss the form of the current-density operator in (Sec. \[curden\]) .
Adiabatic density-functional perturbation theory\[adpert\]
----------------------------------------------------------
### Adiabatic response
We write the time-dependent Schrödinger equation as $$\label{seq}
i\frac{\partial}{\partial t}\vert\Psi\rangle=\hat{H}(\lambda(t))\vert\Psi\rangle.$$ where $\hat{H}(\lambda(t))$ is the Hamiltonian, and $\lambda$ parametrizes the time-dependent atomic motion. Since we are interested in the current density resulting from adiabatic displacements, we expand the wavefunction $\vert\Psi(t)\rangle$ to first order in the velocity, $\dot{\lambda}$: [@Messiah1981; @Thouless1983; @Niu1984] $$\label{psiad}
\vert\Psi(t)\rangle \simeq e^{i\gamma(t)}e^{i\phi(\lambda(t))}[\vert\psi(\lambda(t))\rangle+\dot{\lambda}(t)\vert\delta\psi(\lambda(t))\rangle],$$ where $\vert\psi(\lambda)\rangle$ is the lowest-energy eigenfunction of the time-independent Hamiltonian at a given $\lambda$, and $\vert\delta\psi(\lambda)\rangle$ is the first order adiabatic wavefunction \[defined by Eq. (\[psiad\])\]; $\gamma(t)=-\int_0^t
E(\lambda(t^\prime))d t^\prime$ is the dynamic phase, with $E(\lambda)$ being the eigenenergy of $\vert\psi(\lambda)\rangle$; $\phi(\lambda(t))=\int_0^t \langle\psi(\lambda(t^\prime))\vert
i\partial_t \psi(\lambda(t^\prime))\rangle d t^\prime$ is the geometric Berry phase [@Berry1984] (we have used the shorthand $\partial_t=\partial/\partial t$). We work in the parallel-transport gauge, $\langle\psi(\lambda)\vert i\partial_\lambda
\psi(\lambda)\rangle=0$, so the Berry phase contribution vanishes.
Equation (\[psiad\]) is written assuming a single occupied band, but in the multiband case we shall let the evolution be guided by multiband parallel transport instead. In this case, the first-order wavefunctions, $\delta\psi_n$, given by adiabatic perturbation theory[@Messiah1981; @Thouless1983; @Niu1984], are $$\label{deltapsi}
\vert\delta\psi_n\rangle=-i\sum_{m}^\text{unocc}\vert\psi_m\rangle\frac{\langle\psi_m\vert\partial_\lambda\psi_n\rangle}{\epsilon_n-\epsilon_m},$$ where $\epsilon_n$ is the eigenvalue of the $n$th single particle wavefunction, and $\partial_\lambda$ is shorthand for $\partial/\partial\lambda$. The wavefunction $\vert\partial_\lambda\psi_n\rangle$ is the first-order wavefunction resulting from the *static* perturbation $$\label{delpsi}
\vert\partial_\lambda\psi_n\rangle=\sum_m^\text{unocc}\vert\psi_m\rangle\frac{\langle\psi_m\vert\partial_\lambda\hat{H}\vert\psi_n\rangle}{\epsilon_n-\epsilon_m},$$ which is the quantity calculated in conventional DFPT implementations [@Baroni2001; @Gonze1997].
### Density functional theory
We will implement the calculations of the current density in the context of plane-wave pseudopotential DFT, so the single-particle wavefunctions we will use in Eq. (\[deltapsi\]) are solutions to the Kohn-Sham equation for a given band $n$ and wavevector **k**, $$\label{KSeq}
\hat{H}_{\text{KS}}\vert\psi_{n\textbf{k}}\rangle=\epsilon_{n\textbf{k}}\vert\psi_{n\textbf{k}}\rangle.$$ where the Kohn-Sham Hamiltonian is $$\label{HKS}
\hat{H}_{\text{KS}}=\hat{T}_{\text{s}}+\hat{V}_{\text{H}}+\hat{V}_{\text{XC}}+\hat{V}_{\text{ext}}^{\text{loc}}+\hat{V}_{\text{ext}}^{\text{nl}}.$$ Here $\hat{T}_{\text{s}}$ is the single-particle kinetic energy, $\hat{V}_{\text{H}}$ is the Hartree potential, $\hat{V}_{\text{XC}}$ is the exchange correlation potential, and the external potential contains both a local and nonlocal part (last two terms). We will consider norm-conserving, separable, Kleinmann-Bylander type [@Kleinman1982] pseudopotentials. The form of the nonlocal potential (henceforth referred to as $\hat{V}^{\text{nl}}$) is given by Eq. (\[VNL\]). We will drop the “KS” subscript from here on. Note that, although we focus on norm-conserving pseudopotentials in this work, the issues pertaining to nonlocal potentials that will be discussed in Sec. \[curden\] would apply to ultrasoft [@Vanderbilt1990] and projector augmented wave (PAW) [@Blochl1994] potentials as well.
### Polarization response
Using the expansion in Eq. (\[psiad\]), the first-order one-particle density matrix is $$\label{denmat}
\delta\hat{\rho}=\dot{\lambda}\frac{2}{N_k}\sum_{n\textbf{k}}\left(\vert\delta\psi_{n\textbf{k}}\rangle\langle\psi_{n\textbf{k}}\vert+\vert\psi_{n\textbf{k}}\rangle\langle\delta\psi_{n\textbf{k}}\vert\right)$$ where the factors $(2/N_k)\sum_{n\textbf{k}}$ take care of the spin degeneracy, sum over occupied Bloch bands, and average over the Brillouin zone. A monochromatic perturbation such as that of Eq. (\[phon\]) always comes together with its Hermitian conjugate, coupling states at **k** with those at $\textbf{k}\pm\textbf{q}$, so that each perturbed wavefunction has two components that we refer as $\delta\psi_{n,\textbf{k}+\textbf{q}}$ and $\delta\psi_{n,\textbf{k}-\textbf{q}}$ respectively. We wish to select the cross-gap response at $+\textbf{q}$, so we project onto this component of the density matrix to obtain [@Adler1962] $$\label{denmat2}
\delta\hat{\rho}_\textbf{q}=\dot{\lambda}\frac{2}{N_k}\sum_{n\textbf{k}}\left(
\vert\delta\psi_{n,\textbf{k}+\textbf{q}}\rangle\langle\psi_{n\textbf{k}}\vert
+
\vert\psi_{n\textbf{k}}\rangle\langle\delta\psi_{n,\textbf{k}-\textbf{q}}\vert
\right).$$ Specializing now to the perturbation of Eq. (\[phon\]), the corresponding polarization response is $$\label{Plambda1}
\begin{split}
P_{\alpha,\kappa\beta\textbf{q}}(\textbf{r})
&=\frac{2}{N_k}\sum_{n\textbf{k}}\Big[\langle\psi_{n\textbf{k}}\vert\hat{\mathcal{J}}_\alpha(\textbf{r})\vert\delta\psi_{n\textbf{k},\textbf{q}}^{\kappa\beta}\rangle
\\
&\phantom{=\frac{2}{N_k}\sum_{n\textbf{k}}\Big[}+\langle\delta\psi_{n\textbf{k},-\textbf{q}}^{\kappa\beta}\vert \hat{\mathcal{J}}_\alpha(\textbf{r})\vert\psi_{n\textbf{k}}\rangle\Big].
\end{split}$$ Using Eqs. (\[deltapsi\]) and (\[delpsi\]), the needed first-order wave functions are $$\label{pertwf}
\vert\delta\psi^{\kappa\beta}_{n\textbf{k},\textbf{q}}\rangle=-i\sum_{m}^{\text{unocc}}\vert\psi_{m\textbf{k}+\textbf{q}}\rangle\frac{\langle\psi_{m\textbf{k}+\textbf{q}}\vert\partial_{\lambda_{\kappa\beta\textbf{q}}}\hat{H}\vert\psi_{n\textbf{k}}\rangle}{(\epsilon_{m\textbf{k}+\textbf{q}}-\epsilon_{n\textbf{k}})^2}.$$
For Eq. (\[muI\]), we require the cell-average of the $\textbf{q}$-dependent polarization response \[Eq. (\[Pbar\])\]. Defining the operator $$\label{Jq0}
\hat{\mathcal{J}}_\alpha(\textbf{q})=\frac{1}{\Omega}
\int_{\rm cell} d^3r\,e^{-i\bf q\cdot r}\,\hat{\mathcal{J}}_\alpha(\textbf{r}),$$ Eq. (\[Pbar\]) can be written $$\begin{split}
\label{Pq}
\overline{P}_{\alpha,\kappa\beta}^{\textbf{q}}&=\frac{2}{N_k}\sum_{n\textbf{k}} \Big[ \langle \psi_{n\textbf{k}}\vert\hat{\mathcal{J}}_\alpha(\textbf{q})\vert\delta \psi_{n\textbf{k},\textbf{q}}^{\kappa\beta}\rangle
\\
&\phantom{\frac{2}{N_k}\sum_{n\textbf{k}} \Big[}+\langle \delta \psi_{n\textbf{k},-\textbf{q}}^{\kappa\beta}\vert\hat{\mathcal{J}}_\alpha(\textbf{q})\vert \psi_{n\textbf{k}}\rangle \Big].
\end{split}$$
The ground-state and first-order wavefunctions can be expressed in terms of cell-periodic Bloch functions in the normal way: $$\langle\textbf{s}\vert\psi_{n\textbf{k}}\rangle=u_{n\textbf{k}}(\textbf{s})e^{i\textbf{k}\cdot\textbf{s}}, \;\;\langle\textbf{s}\vert \delta \psi_{n\textbf{k},\textbf{q}}^{\kappa\beta}\rangle=\delta u^{\kappa\beta}_{n\textbf{k},\textbf{q}}(\textbf{s})e^{i(\textbf{k}+\textbf{q})\cdot\textbf{s}}.$$ (Indices $\bf s$ and ${\bf s}'$ are not to be confused with the point **r** at which the current density is evaluated.) Using this notation, the cell-periodic first-order static wavefunction is written $\vert\partial_{\lambda}u^{\kappa\beta}_{n\textbf{k},\textbf{q}}\rangle$, which is equivalent to $\vert
u_{n\textbf{k},\textbf{q}}^{\tau_{\kappa\beta}}\rangle$ in the notation of Gonze and Lee [@Gonze1997] and $\vert \Delta
u_n^{\textbf{k}+\textbf{q}}\rangle$ in the notation of Baroni *et al.* [@Baroni2001]
By factoring out the phases with wavevector **k** and **q**, we can ensure that we only consider cell-periodic quantities, and therefore all calculations can be performed on a unit cell. [@Baroni2001] To this end, we define a cell-periodic operator [^2] $$\label{Jkqdef}
\hat{\mathcal{J}}_\alpha^{\textbf{k},\textbf{q}}=
e^{-i\textbf{k}\cdot\hat{\textbf{r}}} \hat{\mathcal{J}}_\alpha(\textbf{q})
e^{i(\textbf{k}+\textbf{q})\cdot\hat{\textbf{r}}} .$$ Using the fact that $\hat{\mathcal{J}}_\alpha(\textbf{q})=
\hat{\mathcal{J}}^\dagger_\alpha(-\textbf{q})$ it follows that $\left(\hat{\mathcal{J}}_\alpha^{\textbf{k},-\textbf{q}}\right)^\dagger=
e^{-i(\textbf{k}-\textbf{q})\cdot\hat{\textbf{r}}}
\hat{\mathcal{J}}_\alpha(\textbf{q})
e^{i\textbf{k}\cdot\hat{\textbf{r}}}$ so that Eq. (\[Pq\]) can be written as $$\begin{split}
\label{Pq2}
\overline{P}_{\alpha,\kappa\beta}^{\textbf{q}}&=\frac{2}{N_k}\sum_{n\textbf{k}} \Big[ \langle u_{n\textbf{k}}\vert\hat{\mathcal{J}}_\alpha^{\textbf{k},\textbf{q}}\vert\delta u_{n\textbf{k},\textbf{q}}^{\kappa\beta}\rangle
\\
&\phantom{=\frac{2}{N_k}\sum_{n\textbf{k}} \Big[}+\langle \delta u_{n\textbf{k},-\textbf{q}}^{\kappa\beta}\vert\left(\hat{\mathcal{J}}_\alpha^{\textbf{k},-\textbf{q}}\right)^\dagger\vert u_{n\textbf{k}}\rangle \Big].
\end{split}$$
In this work, we shall limit our focus to materials with time-reversal symmetry (TRS); then we have $$\label{TReq}
\langle \textbf{s} \vert u_{n\textbf{k}} \rangle= \langle u_{n-\textbf{k}} \vert\textbf{s} \rangle, \;\;\langle\textbf{s}\vert \delta u_{n\textbf{k},\textbf{q}}^{\kappa\beta}\rangle= -\langle \delta u_{n\,-\textbf{k},-\textbf{q}}^{\kappa\beta}\vert \textbf{s}\rangle,$$ where the negative sign in the second expression is a result of the $-i$ in the first-order adiabatic wavefunction \[see Eq. (\[deltapsi\])\]. Assuming that the current operator has the correct “TRS odd” nature, i.e., $\Big( \langle {\bf
s}|\hat{\mathcal{J}}^{\bf k,-q}_\alpha |{\bf s}' \rangle \Big)^* =
-\langle {\bf s}|\hat{\mathcal{J}}^{\bf -k,q}_\alpha |{\bf s}'
\rangle$, Eq. (\[Pq2\]) simplifies to $$\begin{split}
\label{PqTR}
\overline{P}_{\alpha,\kappa\beta}^{\textbf{q}}
&=\frac{4}{N_k}\sum_{n\textbf{k}} \langle u_{n\textbf{k}}\vert\hat{\mathcal{J}}_\alpha^{\textbf{k},\textbf{q}}
\vert\delta u_{n\textbf{k},\textbf{q}}^{\kappa\beta}\rangle.
\end{split}$$
Current-density operator\[curden\]
----------------------------------
We now consider the form of the current-density operator. If particle density is conserved, any physically meaningful definition of current density must satisfy the continuity condition $$\label{conteq}
\nabla\cdot\textbf{J}(\textbf{r})=-\frac{\partial \rho(\textbf{r})}{\partial t},$$ where $\rho$ is the particle density. In a quantum mechanical treatment[@Sakuri], $\rho(\textbf{r})=\vert\Psi(\textbf{r})\vert^2$, where $\Psi$ is the solution to the time-dependent Schrödinger equation. Combining Eq. (\[seq\]) with its complex conjugate gives $$\label{conteq2}
\frac{\partial}{\partial t}\rho(\textbf{r})=-i\langle\Psi\vert\left[\vert\textbf{r}\rangle\langle\textbf{r}\vert,\hat{H}\right]\vert\Psi\rangle=-i\langle\Psi\vert\left[\hat{\rho}(\textbf{r}),\hat{H}\right]\vert\Psi\rangle,$$ where $\hat{\rho}(\textbf{r})$ is the particle density operator. (We use atomic units throughout with an electron charge of $-1$.) In terms of the first-order adiabatic expansion of Eq. (\[psiad\]), we can use Eq. (\[conteq2\]) to write the induced density from an adiabatic perturbation parameterized by $\lambda$ as $$\label{conteqlam}
\begin{split}
\rho_\lambda(\textbf{r})=-i\Big(&\langle\psi\vert\left[\hat{\rho}(\textbf{r}),\hat{H}\right]\vert\delta\psi\rangle+\langle\delta\psi\vert\left[\hat{\rho}(\textbf{r}),\hat{H}\right]\vert\psi\rangle\Big) .
\end{split}$$
### Local potentials\[locpot\]
Consider the simplest case of a Hamiltonian of the form $\hat{H}^{\text{loc}}=\hat{\textbf{p}}^2/2 + \hat{V}^{\text{loc}}$ where $\hat{\textbf{p}}$ is the momentum operator and $\hat{V}^{\text{loc}}=\int\hat{\rho}(\textbf{r})V(\textbf{r})d^3r$ is a local scalar potential. The local potential commutes with the density operator, so the only contribution to the current is from the momentum operator. Comparing Eqs. (\[conteq\]) and (\[conteq2\]) results in the textbook form of the current-density operator $$\label{jloc}
\begin{split}
\hat{\mathcal{J}}_\alpha^{\text{loc}}(\textbf{r})&=-\frac{1}{2}\left(\vert\textbf{r}\rangle\langle\textbf{r}\vert\hat{p}_\alpha+\hat{p}_\alpha\vert\textbf{r}\rangle\langle\textbf{r}\vert\right)
\\
&=-\frac{1}{2}\left\{\hat{\rho}(\textbf{r}),\hat{p}_\alpha\right\}.
\end{split}$$ Using Eq. (\[Jq0\]), we have $$\label{jqloc}
\begin{split}
\hat{\mathcal{J}}_\alpha^{\text{loc}}(\textbf{q})&=-\frac{1}{2}\left(e^{-i\textbf{q}\cdot\hat{\textbf{r}}}\hat{p}_\alpha+\hat{p}_\alpha e^{-i\textbf{q}\cdot\hat{\textbf{r}}}\right),
\end{split}$$ which gives the cell-periodic operator (Appendices \[sepICL\] and \[Jloc\]) $$\label{jkqloc}
\begin{split}
\hat{\mathcal{J}}_\alpha^{\textbf{k},\textbf{q},\text{loc}}&=-\left(\hat{p}_\alpha^\textbf{k}+\frac{q_\alpha}{2}\right),
\end{split}$$ where $\hat{p}_\alpha^{\textbf{k}}=-i\hat{\nabla}_\alpha+\hat{k}_\alpha$ is the cell-periodic momentum operator ($\hat{\nabla}_\alpha$ is a spatial derivative in the $\alpha$ direction, and the overall minus sign is from the electron charge).
### Continuity condition and nonlocal potentials\[contsec\]
As mentioned above, *nonlocal* potentials are ubiquitous in modern pseudopotential implementations of DFT [@Vanderbilt1990; @Hamann1979; @Kleinman1982; @Blochl1994]. When nonlocal potentials are present in the Hamiltonian, the current density in Eq. (\[jloc\]) does not satisfy the continuity equation.
To see this, consider a Hamiltonian with a nonlocal potential: $\hat{H}^{\text{nl}}=\hat{\textbf{p}}^2/2 +
\hat{V}^{\text{loc}}+\hat{V}^{\text{nl}}$ with $\hat{V}^{\text{nl}}=\int d^3r\int d^3r^\prime
\hat{\rho}(\textbf{r},\textbf{r}^\prime)V(\textbf{r},\textbf{r}^\prime)$ where $\hat{\rho}(\textbf{r},\textbf{r}^\prime)=\vert\textbf{r}\rangle\langle\textbf{r}^\prime\vert$. In this case, there is a term in the induced density \[Eq. (\[conteqlam\])\] resulting from the nonlocal potential: $$\begin{split}
\label{rhoNL}
\rho^{\text{nl}}_\lambda(\textbf{r})=-i\Big(&\langle\psi\vert\left[\hat{\rho}(\textbf{r}),\hat{V}^{\text{nl}}\right]\vert\delta\psi\rangle
\\
&+\langle\delta\psi\vert\left[\hat{\rho}(\textbf{r}),\hat{V}^{\text{nl}}\right]\vert\psi\rangle\Big),
\end{split}$$ If we write the total induced current as the sum of contributions from the local and nonlocal parts, $\textbf{J}=\textbf{J}^{\text{loc}}+\textbf{J}^{\text{nl}}$, then we have $$\label{conteq3}
\nabla\cdot\textbf{J}^{\text{nl}}(\textbf{r})=-\rho^{\text{nl}}_\lambda(\textbf{r}).$$ This “nonlocal charge,” $\rho^{\text{nl}}_\lambda$, measures the degree to which the continuity equation, Eq. (\[conteq\]), breaks down if Eq. (\[jloc\]) is used in a nonlocal pseudopotential context.
Li *et al.*[@Li2008] argued that such nonlocal charge could be used to reconstruct the nonlocal contribution to the current density via a Poisson equation. Indeed, Eq. (\[conteq3\]) indicates that the irrotational part of $\textbf{J}^{\text{nl}}$ can be determined by calculating Eq. (\[rhoNL\]). Their approach yields a conserved current by construction, but there are two additional requirements that a physically meaningful definition of the quantum-mechanical electronic current should satisfy:
- The nonlocality of the Hamiltonian should be confined to small spheres surrounding the ionic cores. In the interstitial regions, the nonlocal part of the pseudopotentials vanish, and the Hamiltonian operator is local therein. Thus, the current-density operator should reduce to the simple textbook formula outside the atomic spheres. The corollary is that $\textbf{J}^{\text{nl}}(\textbf{r})$ must vanish in the interstitial regions.
- The macroscopic average of the microscopic current should reduce to the well-known expression $\hat{v}_\alpha= -i[\hat{r}_\alpha,\hat{H}]$ for the electronic velocity operator [@Starace1971; @Hybertsen1987; @Giannozzi1991; @DalCorso1994]. This is routinely used in the context of DFPT, e.g., to calculate the polarization response to ionic displacements needed for the Born effective charge tensor.
The strategy proposed by Li [*et al.*]{} [@Li2008] falls short of fulfilling either condition. Regarding the first (spatial confinement), note that the nonlocal charge associated to individual spheres generally has a nonzero dipole (and higher multipole) moments. Therefore, even if the nonlocal charge is confined to the sphere, an irrotational field whose divergence results in such a charge density will generally have a long-ranged character and propagate over all space.
Regarding the relation to the macroscopic particle velocity, note that the construction proposed by Li [*et al.*]{} [@Li2008] in practice discards the solenoidal part of the nonlocal current and hence fails at describing its contribution to the transverse polarization response. This is precisely the quantity in which we are interested in the context of flexoelectricity, and is also crucial for obtaining other important quantities, such as the Born charge tensor, that are part of standard DFPT implementations.
Therefore, a calculation of Eqs. (\[rhoNL\]) does not contain the necessary information to determine $\textbf{J}^{\text{nl}}$, and an alternative derivation to the textbook one outlined in Sec. \[locpot\] is required.
### Current-density operator generalized for nonlocal potentials\[secJ\]
In light of the previous section, we will now focus on determining an expression for $\hat{\mathcal{J}}_\alpha$ that is applicable when nonlocal potentials are present in the Hamiltonian. For the case of a perturbation that is uniform over the crystal, corresponding to the long wavelength $\textbf{q}=0$ limit of Eq. (\[phon\]), it is well known that the momentum operator should be replaced with the canonical velocity operator $\hat{v}_\alpha$ [@Starace1971; @Hybertsen1987; @Giannozzi1991; @DalCorso1994] in order to determine the *macroscopic* current.
In Ref. , the expression for the *microscopic* current operator that was used to calculate the current induced by a uniform electric field was Eq. (\[jloc\]) with $\hat{p}_\alpha$ replaced by $\hat{v}_\alpha$. Although this treatment will result in the correct current when averaged over a unit cell, this operator does not satisfy the continuity condition in Eq. (\[conteq\]) except in the special case of a Hamiltonian with only local potentials, where it reduces to Eq. (\[jloc\]).
Since we shall be treating a long wavelength acoustic phonon in this study, and we require the polarization response be correct at least to second order in **q** \[*cf.* Eq. (\[muI\])\], we require a version of $\hat{\mathcal{J}}_\alpha$ that is designed to handle spatially varying perturbations. Therefore, for our purposes, we need an alternative starting point for the derivation of a current-density expression, different from the one based on the continuity condition that led to, e.g., Eq. (\[jloc\]).
In general, for an arbitrary electronic Hamiltonian $\hat{H}^{\textbf{A}}$ coupled to a vector potential $\textbf{A}(\textbf{r})$, the most general form for the current-density operator is $$\label{dHdA}
\hat{\mathcal{J}}_\alpha(\textbf{r})=-\frac{\partial\hat{H}^{\textbf{A}}}{\partial A_\alpha(\textbf{r})} .$$ Our strategy will be to use a vector potential to probe the response to the strain gradient, which will give us the current density via Eq. (\[dHdA\]). Since we are treating the strain gradient in terms of a long-wavelength acoustic phonon of wavevector $\bf q$, and we are interested in the response occurring at the same wavevector $\bf q$, it is useful to define $$\begin{aligned}
\hat{\mathcal{J}}_\alpha(\textbf{r})&=\sum_{\textbf{G}}\hat{\mathcal{J}}_\alpha(\textbf{G}+\textbf{q})e^{i(\textbf{G}+\textbf{q})\cdot\textbf{r}},
\label{fft-J}
\\
A_\alpha(\textbf{r})&=\sum_{\textbf{G}}A_\alpha(\textbf{G}+\textbf{q})e^{i(\textbf{G}+\textbf{q})\cdot\textbf{r}},
\label{fft-A}
\\
P_{\alpha,\kappa\beta\textbf{q}}(\textbf{r})&=\sum_{\textbf{G}}
P_{\alpha,\kappa\beta\textbf{q}}(\textbf{G}+\textbf{q})e^{i(\textbf{G}+\textbf{q})\cdot\textbf{r}}.
\label{fft-dlP}\end{aligned}$$ With these definitions, Eq. (\[dHdA\]) becomes $$\label{dHdAGq}
\hat{\mathcal{J}}_\alpha(\textbf{G}+\textbf{q})=-\frac{\partial\hat{H}^{\textbf{A}}}{\partial A^*_\alpha(\textbf{G}+\textbf{q})}$$ and the desired operator for Eq. (\[Pq\]) is $$\label{dHdAq}
\hat{\mathcal{J}}_\alpha({\textbf{q}})=
-\frac{\partial\hat{H}^{\textbf{A}}}{\partial A^*_\alpha(\textbf{q})}.$$
Again, if the Hamiltonian of interest had the form of $H^{\text{loc}}=(\hat{\textbf{p}}+\hat{\textbf{A}})^2/2 +
\hat{V}^{\text{loc}}$, where the scalar potential is local and $\hat{\textbf{A}}=\int\hat{\rho}(\textbf{r})\textbf{A}(\textbf{r})d^3r$ is a local vector potential, then $\hat{\mathcal{J}}^{\text{loc}}_\alpha(\textbf{r})=-\frac{1}{2}
\left\{\hat{\rho}(\textbf{r}),(\hat{p}_\alpha+\hat{A}_\alpha)\right\}$. However, for our implementation, we are considering the case where the potential $\hat{V}$ is nonlocal, so we must determine how to couple a generally nonlocal Hamiltonian to a spatially nonuniform vector potential field (which will be the case for a finite **q** perturbation).
The standard strategy for describing the coupling to the vector potential is to multiply the nonlocal operator by a complex phase containing the line integral of the vector potential **A**[@ICL2001; @Pickard2003; @Essin2010]; in the real-space representation: $$\label{Aphase}
\mathcal{O}^{\textbf{A}}(\textbf{s},\textbf{s}^\prime)=\mathcal{O}(\textbf{s},\textbf{s}^\prime)e^{-i\int_{\textbf{s}^\prime\rightarrow\textbf{s}}\textbf{A}\cdot d\ell}.$$ The different methods that have been proposed for coupling **A** to a nonlocal Hamiltonian amount to applying the complex phase in Eq. (\[Aphase\]) to either the entire Hamiltonian[@Essin2010] or just the nonlocal potential[@ICL2001; @Pickard2003], and choosing either a straight-line path[@ICL2001; @Essin2010] or a path that passes through the centers of the atoms[@Pickard2003] to perform the line integral.
### Straight-line path \[formICL\]
Using Feynman path integrals, Ismail-Beigi, Chang, and Louie [@ICL2001] (ICL) derived the following form of a nonlocal Hamiltonian coupled to a vector potential field: $$\label{HICL}
\begin{split}
\hat{H}^{\textbf{A}}_{\text{ICL}}&=\frac{1}{2}(\hat{\textbf{p}}+\hat{\textbf{A}})^2+\hat{V}^{\text{loc}}
\\
&+\int d^3s \int d^3s^\prime\hat{\rho}(\textbf{s},\textbf{s}^\prime)V^{\text{nl}}(\textbf{s},\textbf{s}^\prime)e^{-i\int_{\textbf{s}^\prime}^{\textbf{s}}\textbf{A}\cdot d\ell},
\end{split}$$ where the line integral is taken along a straight path from **s** to $\textbf{s}^\prime$. Since the approach used in Ref. to perform the minimal substitution $\hat{\textbf{p}}\rightarrow\hat{\textbf{p}}+\hat{\textbf{A}}$ is general, applying to both local and nonlocal Hamiltonians, this approach is equivalent to the approach of Essin *et al.*, where the coupled Hamiltonian is written as $$\label{HA1}
H^{\textbf{A}}(\textbf{s},\textbf{s}^\prime)=H(\textbf{s},\textbf{s}^\prime)e^{-i\int_{\textbf{s}^\prime}^{\textbf{s}}\textbf{A}\cdot d\ell},$$ i.e., all of the **A** dependence is contained in the complex phase, and the line integral is also taken along a straight path from **s** to $\textbf{s}^\prime$.
Expanding Eq. (\[HA1\]) to first order gives $$\label{HA2}
\begin{split}
H^{\textbf{A}}(\textbf{s},\textbf{s}^\prime)&= H(\textbf{s},\textbf{s}^\prime)-iH(\textbf{s},\textbf{s}^\prime)\int_{\textbf{s}^\prime}^{\textbf{s}}\textbf{A}\cdot d\ell+\cdots.
\end{split}$$ We would like to evaluate Eq. (\[dHdAq\]) for this form of the Hamiltonian. Since $\textbf{A}(\textbf{r})$ is real we can write Eq. (\[fft-A\]) as $A_\alpha(\textbf{r})=A^*_\alpha(\textbf{r})
=A^*_\alpha(\textbf{q})e^{-i\textbf{q}\cdot\textbf{r}}$ so that the integral over **A** for the ICL[@ICL2001] path is $$\label{AICL}
\begin{split}
\int_{\textbf{s}^\prime}^{\textbf{s}}\textbf{A}\cdot d\ell&=\int_0^1d\tau\textbf{A}[\textbf{s}^\prime+\tau(\textbf{s}-\textbf{s}^\prime)]\cdot(\textbf{s}-\textbf{s}^\prime)
\\
&=\textbf{A}^*(\textbf{q})\cdot(\textbf{s}-\textbf{s}^\prime)\int_0^1d\tau e^{-i{\textbf{q}}\cdot[\textbf{s}^\prime+\tau(\textbf{s}-\textbf{s}^\prime)]}
\\
&=-\textbf{A}^*(\textbf{q})\cdot(\textbf{s}-\textbf{s}^\prime)\frac{e^{-i\textbf{q}\cdot\textbf{s}}-e^{-i\textbf{q}\cdot\textbf{s}^\prime}}{i\textbf{q}\cdot(\textbf{s}-\textbf{s}^\prime)}
\end{split}$$ Therefore, from Eqs. (\[HA2\]) and (\[dHdAq\]), $$\label{JqSL}
\langle\textbf{s}\vert\hat{\mathcal{J}}_{\alpha}^{\text{ICL}}(\textbf{q})\vert\textbf{s}^\prime\rangle=-iH(\textbf{s},\textbf{s}^\prime)(s_\alpha-s_\alpha^\prime)\frac{e^{-i\textbf{q}\cdot\textbf{s}}-e^{-i\textbf{q}\cdot\textbf{s}^{\prime}}}{i\textbf{q}\cdot(\textbf{s}-\textbf{s}^\prime)}.$$ In practice we shall normally work in terms of the cell-periodic current operator of Eq. (\[Jkqdef\]), whose position representation follows as $$\label{JkqICL}
\langle\textbf{s}\vert\hat{\mathcal{J}}_{\alpha}^{\textbf{k},\textbf{q},\text{ICL}}\vert\textbf{s}^\prime\rangle=-iH^{\textbf{k}}(\textbf{s},\textbf{s}^\prime)(s_\alpha-s_\alpha^\prime)\frac{e^{-i\textbf{q}\cdot(\textbf{s}-\textbf{s}^\prime)}-1}{i\textbf{q}\cdot(\textbf{s}-\textbf{s}^\prime)}.$$
We can see that the current operator of Eq. (\[JqSL\]) satisfies the continuity condition of Eq. (\[conteq\]) as follows. In reciprocal space the continuity equation becomes $i\textbf{q}\cdot[-\hat{\mathcal{J}}^{\text{ICL}}(\textbf{q})]=
-\partial\hat{\rho}_{\textbf{q}}/\partial t$, where $\hat{\rho}_{\textbf{q}}=e^{-i\textbf{q}\cdot\hat{\textbf{r}}}$ is the $\textbf{G}=0$ particle density operator for a given **q**, and the negative sign in front of the current operator reflects the sign of the electron charge. But from Eq. (\[JqSL\]) it quickly follows that $$\label{JqSLp}
-i\textbf{q}\cdot\langle\textbf{s}\vert\hat{\mathcal{J}}_{\alpha}^{\text{ICL}}(\textbf{q})\vert\textbf{s}^\prime\rangle=
i\langle\textbf{s}\vert\left[\hat{\rho}_\textbf{q},\hat{H}\right]\vert\textbf{s}^\prime\rangle$$ which, using the Ehrenfest theorem, is nothing other than $-\partial
\hat{\rho}_{\textbf{q}}/\partial t$ in the position representation.
In the case that only local potentials are present, only the kinetic term in the Hamiltonian contributes to $\hat{\mathcal{J}}_{\alpha}^{\text{ICL}}(\textbf{q})$. We show in Appendix \[sepICL\] that the current operator then reduces to the form of Eq. (\[jqloc\]). The fact that the local and nonlocal parts can be separated confirms the equivalence of the ICL \[Eq. (\[HICL\])\] and Essin *et al.* \[Eq. (\[HA1\])\] approaches.
In the case that nonlocal potentials are present, we show in Appendix \[sepICL\] that, for $\textbf{q}=0$, Eq. (\[JqSL\]) reduces to the well-known expression for the canonical velocity operator[@Starace1971; @Hybertsen1987; @Giannozzi1991; @DalCorso1994] $\hat{\mathcal{J}}^{\text{ICL}}_{\alpha}(\textbf{q}=0)=-\hat{v}_{\alpha}=i\left[\hat{r}_\alpha,\hat{H}\right]$, where the $-1$ comes from the electron charge. We discuss the case of nonlocal potentials and finite **q** perturbations in Sec. \[longwave\].
### Path through atom center\[formPM\]
Subsequently, Pickard and Mauri [@Pickard2003] (PM) proposed using a path from **s** to the atom center, **R**, and then to $\textbf{s}^\prime$, which was constructed explicitly to give better agreement for magnetic susceptibility between pseudopotential and all-electron calculations. This approach can be regarded as a generalization to spatially nonuniform fields of the gauge-including projector augmented-wave (GIPAW) method [@Pickard2001; @Pickard2003], where the PAW transformation is modified with a complex phase in order to ensure that the pseudowavefunction has the correct magnetic translational symmetry.
The coupled Hamiltonian used in Ref. is of the form $$\label{HPM}
\begin{split}
\hat{H}^{\textbf{A}}_{\text{PM}}&=\frac{1}{2}(\hat{\textbf{p}}+\hat{\textbf{A}})^2+\hat{V}^{\text{loc}}+\sum_{\zeta=1}^N\int d^3s \int d^3s^\prime
\\
&\times \hat{\rho}(\textbf{s},\textbf{s}^\prime)V^{\text{nl}}_\zeta(\textbf{s},\textbf{s}^\prime)e^{-i\int_{\textbf{s}^\prime\rightarrow\textbf{R}_\zeta\rightarrow\textbf{s}}\textbf{A}\cdot d\ell},
\end{split}$$ where $N$ is the number of atoms in the cell, $\textbf{R}_\zeta$ is the position of atom $\zeta$, and $V_\zeta^{\text{nl}}$ is the nonlocal potential for that atom. The PM approach explicitly splits the nonlocal contribution from **A** into contributions from each atomic sphere centered at $\textbf{R}_\zeta$. [^3] Therefore, the total current operator is $$\begin{split}
\label{JkqPM}
\hat{\mathcal{J}}_{\alpha}^{\textbf{k},\textbf{q},\text{PM}}=-\left(\hat{p}^{\textbf{k}}_\alpha+\frac{q_\alpha}{2}\right)+\sum_{\zeta=1}^N \hat{\mathcal{J}}_{\alpha,\zeta}^{\textbf{k},\textbf{q},\text{PM,nl}},
\end{split}$$ where the superscript “nl” and the subscript $\zeta$ emphasize that each item in the summation describes the contribution to the current from the nonlocal potential of the atom $\zeta$; it is obvious from Eqs. (\[HPM\]) and (\[JkqPM\]) that $\hat{\mathcal{J}}_\alpha^{\text{loc}}$ will be recovered in the case of a local potential.
For an atom at position $\textbf{R}_\zeta$, the line integral in Eq. (\[HPM\]) is $$\begin{split}
\int_{\textbf{s}^\prime\rightarrow\textbf{R}_\zeta \rightarrow\textbf{s}}\textbf{A}\cdot d\ell&=-\textbf{A}^*(\textbf{q})\cdot(\textbf{R}_\zeta -\textbf{s}^\prime)\frac{e^{-i\textbf{q}\cdot\textbf{R}_\zeta }-e^{-i\textbf{q}\cdot\textbf{s}^\prime}}{i\textbf{q}\cdot(\textbf{R}_\zeta -\textbf{s}^\prime)}-\textbf{A}^*(\textbf{q})\cdot(\textbf{s}-\textbf{R}_\zeta )\frac{e^{-i\textbf{q}\cdot\textbf{s}}-e^{-i\textbf{q}\cdot\textbf{R}_\zeta }}{i\textbf{q}\cdot(\textbf{s}-\textbf{R}_\zeta )}.
\end{split}$$ Therefore we have $$\label{JqPMNL}
\begin{split}
\langle\textbf{s}\vert\hat{\mathcal{J}}_{\alpha,\zeta}^{\text{PM},\text{nl}}(\textbf{q})\vert\textbf{s}^\prime\rangle&=-iV_\zeta^{\text{nl}}(\textbf{s},\textbf{s}^\prime)\bigg[(R_{\alpha,\zeta}-s^\prime_\alpha)\frac{e^{-i\textbf{q}\cdot\textbf{R}_\zeta }-e^{-i\textbf{q}\cdot\textbf{s}^\prime}}{i\textbf{q}\cdot(\textbf{R}_\zeta -\textbf{s}^\prime)}+(s_\alpha-R_{\alpha,\zeta})\frac{e^{-i\textbf{q}\cdot\textbf{s}}-e^{-i\textbf{q}\cdot\textbf{R}_\zeta }}{i\textbf{q}\cdot(\textbf{s}-\textbf{R}_\zeta )}\bigg],
\end{split}$$ so the cell-periodic operator is $$\begin{split}
\label{JkqPMNL}
\langle\textbf{s}\vert\hat{\mathcal{J}}_{\alpha,\zeta}^{\textbf{k},\textbf{q},\text{PM,nl}}\vert\textbf{s}^\prime\rangle=-iV^{\text{nl}}_\zeta(\textbf{s},\textbf{s}^\prime)&\bigg[(R_{\alpha,\zeta}- s^\prime_\alpha)\frac{e^{-i\textbf{q}\cdot(\textbf{R}_\zeta-\textbf{s}^\prime)}-1}{i\textbf{q}\cdot(\textbf{R}_\zeta-\textbf{s}^\prime)}
+(s_\alpha-R_{\alpha,\zeta})\frac{e^{-i\textbf{q}\cdot(\textbf{s}-\textbf{s}^\prime)}-e^{-i\textbf{q}\cdot(\textbf{R}_\zeta-\textbf{s}^\prime)}}{i\textbf{q}\cdot(\textbf{s}-\textbf{R}_\zeta)}\bigg].
\end{split}$$
From Eqs. (\[JqPMNL\]) and (\[rhoNL\]), we see that $i\textbf{q}\cdot[-\hat{\mathcal{J}}^{\text{PM},\text{nl}}(\textbf{q})]=i\left[e^{-i\textbf{q}\cdot\hat{\textbf{r}}},\hat{V^{\text{nl}}}\right]=-\hat{\rho}_{\lambda}^{\text{nl}}$. Therefore, Eq. (\[JkqPM\]) satisfies the continuity condition. Also, in the case of a $\textbf{q}=0$ perturbation, $\hat{\mathcal{J}}^{\text{PM},\text{nl}}_{\alpha}(\textbf{q}=0)=i\left[\hat{r}_\alpha,\hat{V}^{\text{nl}}\right]$, which is the nonlocal contribution to $-\hat{v}_\alpha$, as expected. We discuss the case of nonlocal potentials and finite **q** perturbations in the next section.
Finally, we see that for the longitudinal response (where $\textbf{q}=q_\alpha\hat{\alpha}$), the ICL and PM approaches produce identical operators. This is expected, since they both satisfy the continuity equation. Only circulating currents (e.g., transverse or shear FxE components) may exhibit path dependence.
Long wavelength expansion \[longwave\]
--------------------------------------
Recall that only the induced polarization up to second order in $\textbf{q}$ is required for the FxE coefficients \[*cf.* Eq. (\[muI\])\]. Therefore, instead of attempting to calculate Eq. (\[PqTR\]) with either Eq. () or (\[JkqPM\]) directly, we will expand these expressions for the current-density operator to second order in **q**.
Considering the Hamiltonian in Eq. (\[HKS\]), there are contributions to $\hat{\mathcal{J}}_\alpha^{\textbf{q}}$ from the kinetic energy and nonlocal part of the pseudopotential. We show in Appendix \[sepICL\] \[Eq. (\[LocNL4\])\] that the kinetic energy only contributes up to first order in **q**, and for a local Hamiltonian, the current operator reduces to the form of Eq. (\[jkqloc\]).
The nonlocal potential will, however, contribute at all orders. As mentioned in Sec. \[formICL\] and \[formPM\], for $\textbf{q}=0$, both the ICL and PM approaches give $\hat{\mathcal{J}}_\alpha^{\textbf{k},\textbf{q}=0}=-\hat{v}_\alpha^{\textbf{k}}=i[\hat{r}_\alpha,\hat{H}^{\textbf{k}}]=-\hat{p}_\alpha^{\textbf{k}}+\hat{\mathcal{J}}_\alpha^{\textbf{k},\text{nl}(0)}$, where we have defined $\hat{\mathcal{J}}_\alpha^{\textbf{k},\text{nl}(0)}\equiv
i[\hat{r}_\alpha,\hat{V}^{\textbf{k}\text{,nl}}]$. At higher orders in **q**, and for nonlongitudinal response, the ICL and PM approaches may no longer agree.
Up to second order in **q**, the current operator can be written as $$\begin{split}
\label{JqExpand} \hat{\mathcal{J}}_{\alpha}^{\textbf{k},\textbf{q}}&\simeq-\left(\hat{p}_\alpha^{\textbf{k}}+\frac{q_\alpha}{2}\right) +\hat{\mathcal{J}}_\alpha^{\textbf{k},\text{nl}(0)}
\\
&\phantom{=}+ \frac{q_\gamma}{2} \hat{\mathcal{J}}_{\alpha,\gamma}^{\textbf{k},\text{nl}(1)} +\frac{q_\gamma q_\xi}{6}\hat{\mathcal{J}}_{\alpha,\gamma \xi}^{\textbf{k},\text{nl}(2)}.
\end{split}$$ where the higher order terms in **q** ($\hat{\mathcal{J}}_{\alpha,\gamma}^{\textbf{k},\textbf{q},\text{nl}(1)}$ and $\hat{\mathcal{J}}_{\alpha,\gamma\xi}^{\textbf{k},\textbf{q},\text{nl}(2)}$) are the result of the nonlocal part of the Hamiltonian *and* the fact that the monochromatic perturbation is nonuniform (i.e, finite **q**). Expressions for these last two terms in Eq. (\[JqExpand\]) are derived in Appendix \[JNL\] for the ICL path \[Eqs. (\[ICL1\]) and (\[ICL2\])\] and PM path \[Eqs. (\[PM1\]) and (\[PM2\])\].
Plugging the current operator from Eq. (\[JqExpand\]) into Eq. (\[PqTR\]), readily yields the induced polarization, $$\overline{P}_{\alpha,\kappa\beta}^{\textbf{q}} = \overline{P}^{\textbf{q},\text{loc}}_{\alpha,\kappa\beta} + \overline{P}^{\textbf{q},\text{nl}}_{\alpha,\kappa\beta},
\label{PqExpand}$$ where we have separated the contribution of the local current operator (loc) from the nonlocal (nl) part. The exact expression for $\overline{P}^{\textbf{q},\text{loc}}_{\alpha,\kappa\beta}$ is derived in Appendix \[Jloc\], yielding Eq. (\[pkq\]); the approximate (exact only up to second order in ${\bf q}$) expression for $\overline{P}^{\textbf{q},\text{nl}}_{\alpha,\kappa\beta}$ is derived in Appendix \[JNL\] \[see Eq. (\[Pqexpand2\])\].
Circulating rotation-gradient contribution and diamagnetic susceptibility \[diamag\]
-------------------------------------------------------------------------------------
Transverse or shear strain gradients result in rigid rotations of unit cells which must be treated carefully in order to calculate physically meaningful values of the flexoelectric tensor. This issue can be loosely compared to the well-known distinction between the proper and improper piezoelectric tensor, [@Martin1972; @Vanderbilt2000] but, in the case of strain gradients, it is complicated by the fact that different parts of the sample typically rotate by different amounts. The reader is referred to Ref. for a complete discussion; only the results of that work necessary for our purposes will be reproduced here.
Larmor’s theorem states that the effects of a uniform rotation and those of a uniform magnetic field are the same to first order in the field/angular velocity. Therefore, the local rotations of the sample dynamically produce circulating diamagnetic currents that will contribute to the bulk flexoelectric coefficients as defined in Eq. (\[muI\]). As was shown in Ref. (see also Appendix \[Appdiamag\] for an abridged derivation), this circulating rotation-gradient (CRG) [^4] contribution only concerns the nonlongitudinal components and is proportional to the diamagnetic susceptibility of the material, $\chi_{\gamma\lambda}=\partial
M_\gamma/\partial H_\lambda$, where $M$ is the magnetization and $H$ the magnetic field. Specifically, $$\label{pchi}
\begin{split}
\overline{P}^{(2,\omega\nu),\text{CRG}}_{\alpha,\beta}&=\sum_{\gamma\lambda}\left(\epsilon^{\alpha\omega\gamma}\epsilon^{\beta\lambda\nu}+\epsilon^{\alpha\nu\gamma}\epsilon^{\beta\lambda\omega}\right)\chi_{\gamma\lambda},
\end{split}$$ where $\epsilon$’s are the Levi-Civita symbols.
The CRG contribution represents a physical response of the bulk material to the rotations resulting from such nonlongitudinal strain gradients. However, in the context of calculating FxE coefficients, it is useful to remove this contribution. The reasoning for doing this is based on the fact that, as shown in Ref. , the diamagnetic circulating currents from the CRG contribution are divergentless, and therefore do not result in a build up of charge density anywhere in the crystal. Therefore, for the experimentally relevant case of a *finite* crystal, where the polarization response is completely determined by the induced charge density, the CRG contribution will not produce an electrically measurable response.
The fact that the CRG does contribute to the bulk FxE coefficients, but not to the measurable response of a finite sample, highlights the fact that, for flexoelectricity, the bulk and surface response are intertwined[@Stengel2014; @StengelUNPUB; @StengelChapter]. Indeed, it was determined in Ref. that there is a surface CRG contribution that will exactly cancel the bulk one \[Eq. (\[pchi\])\]. Thus removing the CRG contribution from the bulk coefficients simply corresponds to a different way of partitioning the response between the bulk and the surface. In this work we are focused on the bulk response, and are free to choose a convention for this partition. In order to make a more direct connection with experiments, and to be able to directly compare with charge-density-based calculations [@Stengel2014], we choose to remove the CRG contribution from our calculated $\overline{P}^{(2,\omega\nu)}_{\alpha,\kappa\beta}$.
To calculate $\chi_{\gamma\lambda}$, there is again a subtlety involved in the use of nonlocal pseudopotentials. Conventional calculations of the diamagnetic susceptibility involve applying a vector potential perturbation and calculating the current response [@ICL2001; @Pickard2001; @Pickard2003; @Vignale1991; @Mauri1996]. In the case of a local Hamiltonian the aforementioned rotational field is indistinguishable from an electromagnetic vector potential, and the expression for $\chi_{\gamma\lambda}$ is identical to the diamagnetic susceptibility. However, in the case of a nonlocal Hamiltonian this is no longer true. In that case, the perturbation remains the *local* current operator, $\hat{\mathcal{J}}^{\text{loc}}$, while the current response is evaluated using the total (local plus nonlocal) $\hat{\mathcal{J}}$ (*cf.* Appendix \[Appdiamag\]). This difference indicates that Larmor’s theorem may break down for nonlocal potentials. This is discussed further in Sec. \[Disc\].
Implementation\[Imp\]
=====================
The procedure for calculating the FxE coefficients using the formalism in Sec. \[Form\] is as follows. We first perform conventional DFPT phonon calculations \[displacing sublattice $\kappa$ in direction $\beta$, as in Eq. (\[phon\])\] at small but finite wavevectors **q** to obtain the static first-order wavefunctions $\vert\partial_{\lambda}
u^{\kappa\beta}_{n\textbf{k},\textbf{q}}\rangle$. We choose $\vert
q\vert < 0.04$, where here and henceforth we express $q$ in reduced units of $2\pi/a$ ($a$ is the cubic lattice constant). To avoid the sum over empty states in Eq. (\[deltapsi\]), we determine the first-order adiabatic wavefunctions by solving the Sternheimer equation $$\label{deltastern}
(H_{\textbf{k}}-\epsilon_{n\textbf{k}})\vert\delta u^{\kappa\beta}_{n\textbf{k},\textbf{q}}\rangle=-i\mathcal{Q}_{c,\textbf{k}+\textbf{q}}\vert\partial_{\lambda} u^{\kappa\beta}_{n\textbf{k},\textbf{q}}\rangle$$ where $\epsilon_{n\textbf{k}}$ is the eigenvalue of band $n$ and $k$-point **k** and $\mathcal{Q}_{c,\textbf{k}+\textbf{q}}$ is the projector over conduction band states (implemented as one minus the projector over valence states). Then we apply the current operator in Eq. (\[JqExpand\]) to obtain $\overline{P}_{\alpha,\kappa\beta}^{\textbf{q}}$ from Eq. (\[PqTR\]) (see Appendices \[Jloc\] and \[JNL\] for details).
As will be discussed in Sec. \[Bench\], we will use the ICL path for most of the calculations in this study, so the explicit expression for this case is provided in this section. The local contribution to $\overline{P}_{\alpha,\kappa\beta}^{\textbf{q}}$ is derived in Appendix \[Jloc\], leading to Eq. (\[pkq\]). The three terms in the small-${\bf q}$ expansion of the nonlocal part are determined in Appendix \[ICL\] by combining Eqs. (\[JkqICL\]) and (\[PqTR\]), and expanding in powers of **q**, leading to Eq. (\[Pqexpand2\]). Combining Eq. (\[Pqexpand2\]) with Eqs. (\[ICL0\])-(\[ICL2\]) and adding Eq. (\[pkq\]), we have
$$\begin{split}
\label{ICLimp}
\overline{P}_{\alpha,\kappa\beta}^{\textbf{q},\text{ICL}}&=-\frac{4}{N_k}\sum_{n\textbf{k}}\Bigg[\langle u_{n\textbf{k}}\vert\hat{p}_\alpha^{\textbf{k}}+\frac{q_\alpha}{2}\vert\delta u^{\kappa\beta}_{n\textbf{k},\textbf{q}}\rangle+\langle u_{n\textbf{k}}\vert\frac{\partial \hat{V}^{\textbf{k}{,\text{nl}}}}{\partial k_\alpha}\vert\delta u^{\kappa\beta}_{n\textbf{k},\textbf{q}}\rangle
\\
&+\frac{1}{2}\sum_{\gamma=1}^3 q_\gamma\langle u_{n\textbf{k}}\vert\frac{\partial^2 \hat{V}^{\textbf{k},\text{nl}}}{\partial k_\alpha\partial k_\gamma}\vert\delta u^{\kappa\beta}_{n\textbf{k},\textbf{q}}\rangle
+\frac{1}{6}\sum_{\gamma=1}^3\sum_{\xi=1}^3q_\gamma q_\xi\langle u_{n\textbf{k}}\vert\frac{\partial^3 \hat{V}^{\textbf{k},\text{nl}}}{\partial k_\alpha\partial k_\gamma\partial k_\xi}\vert\delta u^{\kappa\beta}_{n\textbf{k},\textbf{q}}\rangle\Bigg],
\end{split}$$
where we have again assumed TRS \[*cf.* Eq. \[PqTR\]\]. A similar equation can be obtained for the PM path using the first- and second-order current operators derived in Appendix \[PM\] \[Eqs. (\[PM1\]) and (\[PM2\])\].
In order to obtain $\overline{P}^{(2,\omega\nu)}_{\alpha,\kappa\beta}$, we calculate numerical second derivatives with respect to $q_\omega$ and $q_\nu$ yielding the needed flexoelectric coefficients $\mu^{\text{I}}_{\alpha\beta,\omega\nu}$ via Eq. (\[muI\]). Note that, in addition to the explicit factors of $q$ multiplying the last two terms, each term has an implicit $q$ dependence through $\delta
u^{\kappa\beta}_{n\textbf{k},\textbf{q}}$ so all terms may contribute to the second derivative.
Since we will consider cubic materials there are three independent FxE coefficients [@Hong2013; @Stengel2013]: $$\label{mu}
\begin{split}
&\mu_{\text{L}}=\mu^{\text{II}}_{11,11}=\mu^{\text{I}}_{11,11},
\\
&\mu_{\text{S}}=\mu^{\text{II}}_{12,12}=\mu^{\text{I}}_{11,22},
\\
&\mu_{\text{T}}=\mu^{\text{II}}_{11,22}=2\mu^{\text{I}}_{12,12}-\mu^{\text{I}}_{11,22},
\end{split}$$ where L stands for longitudinal, S for shear, and T for transverse.
Electrostatic boundary conditions\[electroBC\]
----------------------------------------------
The current response to a phonon perturbation, just like other response properties, displays a strongly nonanalytic behavior in a vicinity of the $\Gamma$ point (${\bf q}=0$), so some care is required when taking the long-wavelength expansions described in the previous Sections. A long-wavelength phonon naturally imposes “mixed” electrical (ME) boundary conditions: [@Hong2013] Along the longitudinal direction ($\hat{\bf q}$) the electric displacement field, ${\bf D}$, must vanish (${\bf D}\cdot \hat{\bf q}=0)$; conversely, periodicity is preserved in the planes that are normal to $\hat{\bf q}$, resulting in a vanishing electric field therein. In general, the bulk FxE tensor needs to be defined under isotropic “short-circuit” (SC) boundary conditions, which implies that the problematic longitudinal ${\bf
E}$-fields must be suppressed. In our calculations, this goal can be achieved using the procedure of Refs. and , where the $\textbf{G}=0$ component of the self-consistent first-order potential is removed in the DFPT calculation of $\partial_{\lambda}u^{\kappa\beta}_{n\textbf{k},\textbf{q}}$ \[Eq. (\[deltastern\])\]. We will use this procedure for the calculations of cubic oxides in Sec. \[Cub\].
For several reasons, one may sometimes be interested in calculating the flexoelectric coefficients under mixed electrical boundary conditions; in such a case, of course, the $\textbf{G}=0$ component of the self-consistent first-order potential should not be removed. Then, however, one must keep in mind that the long-wavelength expansion of the polarization response is only allowed along a fixed direction in reciprocal space. (This implies performing the calculations at points ${\bf q}= q
\hat{\bf q}$, and subsequently operating the Taylor expansion as a function of the one-dimensional parameter $q$.) In crystals where the macroscopic dielectric tensor is isotropic and $\hat{\bf q}$ corresponds to a high-symmetry direction, the longitudinal coefficients for mixed electrical boundary conditions are simply related to the short circuit ones by the dielectric constant, $\epsilon$, $$\label{BCs}
\mu_{\rm L}^{\rm SC} = \epsilon \mu_{\rm L}^{\rm ME}.$$
We will use mixed electrical boundary conditions for our benchmark calculations of noble gas atoms in Sec. \[Bench\] since, in this particular system, $\mu_{\rm L}^{\rm ME}$, rather than $\mu_{\rm
L}^{\rm SC}$, can be directly compared to the moments of the real-space charge density [@Hong2013], as discussed in Sec. \[IRCmod\].
Magnetic susceptibility contribution\[mag\]
-------------------------------------------
In Sec. \[diamag\], we explained that the diamagnetic susceptibility is required in order to correct for the CRG contribution to the FxE coefficients. To avoid the sum over states in Eq. (\[udyn\]), we solve the Sternheimer equation $$\label{diamagstern}
(\hat{H}_{\textbf{k}}-\epsilon_{n\textbf{k}})\vert\partial_{\dot{\alpha}} u^\alpha_{n\textbf{k},\textbf{q}}\rangle=\mathcal{Q}_{c,\textbf{k}+\textbf{q}}\left(\hat{p}_\alpha^{\textbf{k}}+\frac{q_\alpha}{2}\right)\vert u_{n\textbf{k}}\rangle.$$ Recall that $-\left(\hat{p}_\alpha^{\textbf{k}}+\hat{q}_\alpha/2\right)$ is the cell-averaged current operator in the case of a local potential. We then apply the *full* current operator \[Eq. (\[JqExpand\])\] to obtain Eq. (\[pdyn\]) at several small but finite $q$ (as above, $\vert q\vert < 0.04$) in order to perform a numerical second derivative and obtain $\overline{P}^{(2,\omega\nu),\text{
CRG}}_{\alpha,\beta}$ from Eq. (\[pchi\]).
For the case of a material with cubic symmetry, where $\chi_{\alpha\beta}=\chi_{\text{mag}}\delta_{\alpha\beta}$, we see from Eq. (\[pchi\]) that there will be two nonzero elements of the CRG contribution: $\overline{P}^{(2,22),\text{
CRG}}_{1,1}=2\chi_{\text{mag}}$ and $\overline{P}^{(2,12),\text{
CRG}}_{1,2}=-\chi_{\text{mag}}$. Therefore, the CI FxE constants with the CRG contribution removed, $\mu^\prime$, are given by [@StengelUNPUB] $$\label{mucorr}
\begin{split}
&\mu_{\text{L}}^\prime=\mu_{\text{L}},
\\
&\mu_{\text{S}}^\prime=\mu_{\text{S}}-\chi_{\text{mag}},
\\
&\mu_{\text{T}}^\prime=\mu_{\text{T}}+2\chi_{\text{mag}},
\end{split}$$ for cubic materials.
Rigid-core correction \[RCC\]
-----------------------------
It was demonstrated in Ref. that the CI FxE constants depend on the treatment of the core density, which will be different for a different choice of pseudopotential. This dependence is exactly canceled when the surface contribution is calculated consistently with the same pseudopotentials [@Stengel2013natcom; @StengelChapter]. In order to report more “portable” values for the bulk FxE coefficients, we apply the rigid-core correction (RCC) of Refs. and : $$Q^{\text{RCC}}_\kappa=4\pi\int dr r^4 \left[\rho_\kappa^{\text{AE}}(\textbf{r})-\rho_\kappa^{\text{PS}}(\textbf{r})\right],$$ where $\rho_\kappa^{\text{AE}}(r)$ is the all-electron density of the free atom of type $\kappa$, and $\rho_\kappa^{\text{PS}}(r)$ is the corresponding pseudocharge density. In Table \[RCCtab\] we list $Q^{\text{RCC}}$ for the various atoms that we will require for the cubic oxides reported below (no RCC is included for the noble gas atoms in Sec. \[Bench\]). Specifically, for short circuit boundary conditions, $\epsilon\sum_\kappa Q^{\text{RCC}}_{\kappa}/6\Omega$ must be added to $\mu_{\text{L}}$ and $\mu_{\text{T}}$ [@StengelChapter].
\[RCCtab\]
$Q^{\text{RCC}}$ $Q^{\text{RCC}}$
---- ------------------ ---- ------------------
Sr $-5.93$ Ba $-13.39$
Ti $-0.54$ Zr $-4.55$
O $-0.01$ Pb $-15.16$
Mg $-4.85$
: $Q^{\text{RCC}}$ for the various atoms in the materials in Sec. \[Cub\] in units of e Bohr$^2$.
Computational details
---------------------
We have implemented the procedure for calculating the FxE coefficients in the [abinit]{} code [@Abinit_1]. The PBE generalized gradient approximation functional [@pbe] is used throughout. The conventional phonon and dielectric constant calculations are carried out using the DFPT implementation available in the code [@Abinit_phonon_1; @Gonze1997]. In order to solve the nonselfconsistent Sternheimer Eqs. (\[diamagstern\]) and (\[deltastern\]), [abinit]{}’s implementation of the variational approach of Ref. is used.
The nuclei and core electrons are described with optimized norm-conserving Vanderbilt pseudopotentials [@Hamann2013] provided by [abinit]{}. For the cubic oxides, an $8\times8\times8$ Monkhorst-Pack [@Monkhorst1976] $k$-point mesh is used to sample the Brillouin zone, and the plane-wave energy cutoff is set of 60 Ha. For the isolated atoms, a $2\times2\times2$ $k$-point mesh is used, and the plane-wave energy cutoff is set of 70 Ha.
Results\[Res\]
==============
Benchmark test: Isolated noble gas atoms\[Bench\]
-------------------------------------------------
### Isolated rigid charge model\[IRCmod\]
In order to test the implementation described in Sec. \[Imp\], we consider the toy model of a material made of rigid noninteracting spherical charge distributions arranged in a simple cubic lattice, as explored in Refs. , , and . We shall refer to this henceforth as the “isolated rigid charge" (IRC) model. Of course, such a material is fictitious, since it would have no interatomic forces to hold it together; even so, it serves as an interesting test case since its FxE properties can be determined analytically and compared to our numerical calculations. In this section, we will briefly summarize the expectations of the IRC model (see Refs. and for a more complete discussion).
For the IRC “material,” there is only one sublattice per cell. Each “atom” is represented by a spherically symmetric charge density $\rho_{\text{IRC}}(r)$ that falls to zero beyond a cutoff $r_c$ chosen small enough to ensure that the atomic spheres do not overlap. The atoms are assumed to be neutral, $\int_0^{r_c}\rho_{\text{IRC}}(r)\,r^2\,dr=0$. It was shown in Ref. that the longitudinal and shear coefficients for the IRC model calculated from the induced current-density are $$\label{muIRC}
\mu_{\text{L,IRC}}=\mu_{\text{S,IRC}}=\frac{Q_{\text{IRC}}}{2\Omega},$$ where $\Omega=a^3$ is the cell volume, and $$\label{QIRC}
Q_{\text{IRC}}=\int d^3r \rho_{\text{IRC}}(r)x^2$$ is the quadrupolar moment of the atomic charge density (of course the direction $x$ is arbitrary since the charge density is spherically symmetric).
The FxE constants in Eq. (\[muIRC\]) include the CRG contribution to the current discussed in Sec. \[diamag\][@StengelChapter; @StengelUNPUB; @Stengel2014]. Removing this contribution from our bulk coefficients \[see Eq. (\[mucorr\])\] results in the primed coefficients for the IRC model[@StengelUNPUB] $$\label{muprimeIRC}
\mu_{\text{L,IRC}}^\prime=\frac{Q_{\text{IRC}}}{2\Omega},\;\;\;\mu_{\text{S,IRC}}^\prime=0,$$ where the CRG contribution is given by $$\label{XIRC}
\chi_{\text{mag,IRC}}=\mu_{\text{S,IRC}}= \frac{Q_{\text{IRC}}}{2\Omega}$$ If we assume that Larmor’s theorem holds (i.e., that the CRG contribution is identical to the magnetic susceptibility), Eq. (\[XIRC\]) is just a statement of the Langevin theory of diamagnetism, which relates the magnetic susceptibility to the quadrupole moment of a spherical atomic charge (see Sec. \[Disc\]).
### Noble gas atoms\[noble\]
In the following subsections (\[RSmoment\], \[tstlong\], \[tstshear\]), we will compare the behavior of this model with the results of DFT calculations on isolated noble gas atoms. Several points should be considered when comparing the results of such calculations to the expectations of the IRC model (relations in Sec. \[IRCmod\]).
Firstly, the noble gas atoms in our DFT calculations are slightly polarizable, i.e., not perfectly described by rigid charge densities. For this reason the longitudinal FxE coefficient will depend on the choice of electrostatic boundary conditions (see Sec. \[electroBC\]). We will use mixed electrical boundary conditions, where we should find \[analogously to Eq. (\[muIRC\])\] $$\label{muNG}
\mu_{\text{L,NG}}^{\text{ME}}=\frac{Q_{\text{NG}}}{2\Omega},$$ where the subscript “NG” indicates a DFT calculation on a noble gas atom, and $Q_{\text{NG}}$ is the quadropole moment of the unperturbed, ground-state charge density of the noble gas atom. If we had used short circuit boundary conditions, there would have been a factor of $\epsilon$ on the right-hand side of Eq. (\[muNG\]). Of course, in the IRC model, the “atoms” are neutral, rigid, and spherical, so $\epsilon=1$, and, from Eq. (\[BCs\]), short circuit and mixed electric boundary conditions give the same FxE coefficients.
Also, since our noble-gas-atom calculations will use nonlocal pseudopotentials, the equality of $\mu_{\text{S,NG}}$ and $Q_{\text{NG}}/2\Omega$ is not guaranteed; in fact, we will see in Sec. \[tstshear\] that they are not equal. This will be discussed further in Sec. \[Disc\] in the context of the expected symmetry of the charge response. Similarly, we will find that $\chi_{\text{mag}}$ does not equal $Q_{\text{NG}}/2\Omega$ \[*cf*. Eq. (\[XIRC\])\], indicating that Larmor’s theorem breaks down for our form of the current in the presence of nonlocal pseudopotentials (discussed in Sec. \[Disc\]).
Note that, as with the IRC model, we will drop the $\kappa$ subscript when discussing the noble gas atoms since the “crystals” that we are considering have only a single sublattice. Also, as all calculations will use mixed electrical boundary condition, we will drop the explicit “ME” labels.
### Computational strategy: Real-space moments of the charge density\[RSmoment\]
In addition to the relations in Eqs. (\[muIRC\]), (\[muprimeIRC\]), and (\[XIRC\]) of Sec. \[IRCmod\] and Eq. (\[muNG\]) of Sec. \[noble\], we can perform specific tests of the components of our implementation by exploiting the correspondence between two methods of calculating the FxE coefficients: (i) the long-wavelength expansion in reciprocal space of the polarization induced by a phonon \[i.e., Eq. (\[muI\])\] that we have described so far in this work, and (ii) the computation of the real-space moments of the induced microscopic polarization or charge density from the displacement of an isolated atom in a crystal [@Stengel2013; @Hong2013]. For the case of the isolated noble gas atoms, displacing the entire sublattice (i.e., applying a **q**=0 acoustic phonon perturbation) is equivalent to displacing a single atom.
It is particularly useful to compare our methodology to the real-space moments of the induced charge density, since they can be readily calculated from a conventional, DFPT phonon calculation (with $\textbf{q}=0$). Specifically, the longitudinal noble-gas response in direction $\alpha$ is [@Stengel2013; @Hong2013] $$\begin{split}
\label{comp}
\mu_{\text{L,NG}}&=-\frac{1}{2}\frac{\partial^2\overline{P}_{\alpha,\alpha}^{\textbf{q},\text{NG}}}{\partial q_\alpha^2}\Bigg\vert_{\textbf{q}=0}
=\frac{1}{6\Omega}\int_{\text{cell}}d^3r\rho^{\text{NG}}_{\alpha\textbf{q}=0}(\textbf{r})r_\alpha^3.
\end{split}$$ where $\rho^{\text{NG}}_{\alpha\textbf{q}}(\textbf{r})\equiv \partial
\rho^{\text{NG}}(\textbf{r})/\partial \lambda_{\alpha\textbf{q}}$ is the first-order induced charge density from a phonon with wavevector $\textbf{q}$ and noble gas atoms displaced in the $\alpha$ direction. $\overline{P}_{\alpha,\alpha}^{\textbf{q}}$ is calculated with mixed electrical boundary conditions. As mentioned in Sec. \[noble\], the right-hand side of Eq. (\[comp\]) equals $Q_{\text{NG}}/2\Omega$. Recall that, since the charge density is related to the divergence of the polarization, it only gives the longitudinal FxE coefficient. Therefore, we can only use an expression like the one in Eq. (\[comp\]) to test our implementation of $\mu_{\text{L}}$.
In general (i.e., not specific to the case of the isolated noble gas atoms), the induced charge density can be split into contributions from the local and nonlocal parts of the Hamiltonian, as we did for the polarization in Eq. (\[PqExpand\]). Using the continuity condition, we can write the first-order charge as $$\label{delchg}
\rho_{\alpha\textbf{q}}(\textbf{G}+\textbf{q})=-i(\textbf{G}+\textbf{q})\cdot\textbf{P}^{\text{loc}}_{\alpha\textbf{q}}(\textbf{G}+\textbf{q})+\rho_{\alpha\textbf{q}}^{\text{nl}}(\textbf{G}+\textbf{q}) .$$ Here $\textbf{P}^{\text{loc}}_{\alpha\textbf{q}}$ is the “local” part of the induced polarization and $\rho_{\alpha\textbf{q}}^{\text{nl}}$ is the nonlocal charge introduced in Sec. \[contsec\]. Using the reciprocal-space version of Eq. (\[jloc\]), the local induced polarization is (assuming TRS) $$\label{Ploc}
\begin{split}
P^{\text{loc}}_{\alpha,\alpha\textbf{q}}(\textbf{G}+\textbf{q})=-\frac{2}{N_k}\sum_{n\textbf{k}}\langle \psi_{n\textbf{k}}\vert \left\{ e^{-i(\textbf{G}+\textbf{q})\cdot\hat{\textbf{r}}}, \hat{p}_\alpha\right\}\vert\delta \psi^{\alpha}_{n\textbf{k},\textbf{q}}\rangle
\end{split}$$ and the nonlocal charge density from Eq. (\[rhoNL\]) is given (in reciprocal space) by $$\label{rhonl}
\begin{split}
\rho^{\text{nl}}_{\alpha\textbf{q}}(\textbf{G}+\textbf{q})=-\frac{4i}{N_k}\sum_{n\textbf{k}}\langle \psi_{n\textbf{k}}\vert \left[ e^{-i(\textbf{G}+\textbf{q})\cdot\hat{\textbf{r}}}, \hat{V}^{\text{nl}}\right]\vert\delta \psi^{\alpha}_{n\textbf{k},\textbf{q}}\rangle
\end{split}$$ The first-order charge on the left-hand side of Eq. (\[delchg\]) can be obtained from a conventional DFPT phonon calculation, and thus Eq. (\[delchg\]) allows for several tests of our methodology.
A simple test of the nonlocal contribution at $\textbf{q}=0$ is to compare the dipole moment of the nonlocal charge with $\overline{P}_{\alpha,\alpha}^{\textbf{q}\text{,nl}(0)}$ \[i.e., the second term in Eq. (\[ICLimp\])\], which should give the nonlocal contribution to the Born effective charge $$\label{Ztst}
Z^*_{\alpha\beta,\text{nl}}=\overline{P}_{\alpha,\beta}^{\textbf{q}=0,\text{nl}}=\int_{\text{cell}}d^3r\rho^{\text{nl}}_{\beta\textbf{q}=0}(\textbf{r})r_\alpha.$$ Again, this relation is generally applicable. For cubic symmetry, the Born effective charge tensor has only one independent element, which we write as $Z^*\equiv Z^{*\text{NG}}_{\alpha\alpha}$. Of course, for the case of the noble gas atom “material,” there is only one sublattice, so the sum of the nonlocal contribution with the local part (including the ionic charge) will vanish due to the acoustic sum rule (ASR) [@Pick1970].
For the case of the isolated noble gas atoms, we can use Eqs. (\[comp\]) and (\[delchg\]) to relate the real-space octupole moment of $\rho^{\text{nl}}_{\alpha\textbf{q}=0}(\textbf{r})$ \[Fourier transform of Eq. (\[rhonl\])\] averaged over the cell, to the second **q** derivative of $\overline{P}_{\alpha,\alpha}^{\textbf{q}\text{,nl}}$ \[see Eq. (\[Pqexpand2\])\] evaluated at $\textbf{q}=0$. Specifically, we should find that [@Hong2013; @Stengel2013] $$\begin{split}
\label{NLcomp}
-\frac{1}{2}\frac{\partial^2\overline{P}_{\alpha,\alpha}^{\textbf{q},\text{nl,NG}}}{\partial q_\alpha^2}\Bigg\vert_{\textbf{q}=0}=\frac{1}{6\Omega}\int_{\text{cell}}d^3r\rho^{\text{nl,NG}}_{\alpha\textbf{q}=0}(\textbf{r})r_\alpha^3,
\end{split}$$ and similarly for the local part, $$\label{loccomp}
-\frac{1}{2}\frac{\partial^2\overline{P}_{\alpha,\alpha}^{\textbf{q},\text{loc,NG}}}{\partial q_\alpha^2}\Bigg\vert_{\textbf{q}=0}=\frac{1}{6\Omega}\int_{\text{cell}}d^3r\left[-\nabla\cdot\textbf{P}^{\text{loc,NG}}_{\alpha\textbf{q}=0}(\textbf{r})\right]r_\alpha^3,$$ where we again perform the reciprocal space calculations using mixed electrical boundary conditions.
The comparisons in Eqs. (\[NLcomp\]) and (\[loccomp\]) test both the long-wavelength expansion of the current operator (local and nonlocal), and the accuracy of the adiabatic first-order wavefunction at finite **q**.
### Test of implementation: Longitudinal response\[tstlong\]
To test $P^{\text{loc}}_{\alpha,\alpha\textbf{q}=0}$ and $\delta
\psi^{\alpha}_{n\textbf{k},\textbf{q}=0}$, we calculate the first-order charge \[left-hand side of Eq. (\[delchg\])\] from a $\textbf{q}=0$ phonon by conventional DFPT, and compare to what we obtain for the right-hand side of Eq. (\[delchg\]) calculated using Eqs. (\[Ploc\]) and (\[rhonl\]) (with $\textbf{q}=0$). We Fourier transform the quantities in Eq. (\[delchg\]) to real space and plot their planar averages in Fig. \[NLchg\] for He, Ne, Ar, and Kr atoms in $16\times16\times16$ Bohr cells. Summing the contributions from the nonlocal charge (blue dashed curves) and the gradients of the local induced polarization (green dot-dashed) gives the red solid curves in Fig. \[NLchg\]. As expected from Eq. (\[delchg\]), the red curve lies on top of the black circles, which correspond to the first-order charge from the $\textbf{q}=0$ DFPT phonon calculations.
![\[NLchg\] (Color online) Planar average of the local \[Eq. (\[Ploc\]), green dot-dashed curve\], nonlocal \[Eq. (\[rhonl\]), blue dashed\], and total \[Eq. (\[delchg\]), red solid\] first-order charge for noble gas atoms displaced in the $x$ direction by a $\textbf{q}=0$ phonon. The black circles correspond to the first-order charge calculated using a conventional, static, DFPT calculation. The box size is $16\times16\times16$ Bohr, but zoomed in to only show $\pm 5$ Bohr.](./16x16x16.pdf){width="\columnwidth"}
Now we can take the real-space moments of the curves in Fig. \[NLchg\] and compare them with the results of our reciprocal space expansion. As discussed in Sec. \[RSmoment\], the first moment of the blue dashed curves gives the nonlocal contribution to the Born effective charge, which should correspond to $\overline{P}_{\alpha,\alpha}^{\textbf{q}=0,\text{nl}}$ \[Eq. (\[Ztst\])\]. In Table \[NLchgtab\] we give the nonlocal contribution to $Z^*$ for the noble gas atoms in $14\times14\times14$ Bohr boxes. The ASR requires that the total $Z^*$ vanishes; for our noble gas atoms, we calculate the magnitude of the total $Z^*$ to be less than $10^{-4}$ e, so the “local” part (including the contribution from the ionic charge) is the same magnitude but opposite sign as the numbers in the second and third columns of Table \[NLchgtab\]. The second column of Table \[NLchgtab\], labeled $P^{\text{nl}}$, is calculated using the reciprocal space current and the third column (labeled $\rho^{\text{nl}}$) is from the real-space dipole moment of the charge density. We see that there is excellent agreement between the two methods, indicating that $\overline{P}_{\alpha,\alpha}^{\textbf{q}=0,\text{nl}}$ is accurately calculated.
---- ----------------- -------------------- ------------------ --------------------- ----------------- --------------------
$P^\textrm{nl}$ $\rho^\textrm{nl}$ $P^\textrm{loc}$ $\rho^\textrm{loc}$ $P^\textrm{nl}$ $\rho^\textrm{nl}$
He $-0.027$ $-0.027$ $-0.470$ $-0.470$ $0.004$ $0.004$
Ne $-0.155$ $-0.155$ $-1.872$ $-1.872$ $0.028$ $0.028$
Ar $1.556$ $1.556$ $-4.620$ $-4.623$ $0.073$ $0.072$
Kr $-0.214$ $-0.214$ $-5.878$ $-5.874$ $-0.099$ $-0.099$
---- ----------------- -------------------- ------------------ --------------------- ----------------- --------------------
: Calculation of the Born effective charge and $\mu_{\text{L}}$ using the moments of the local and nonlocal charge (columns labeled $\rho$) compared to the current-density implementation (columns labeled $P$) for atoms in a $14\times14\times14$ Bohr box. Mixed electrical boundary conditions are used.[]{data-label="NLchgtab"}
It is also clear from Fig. \[NLchg\] and Table \[NLchgtab\] that the nonlocal correction to the Born effective charge can be very large, on the order of one electron for Ar. We see a similarly large contribution for atoms with empty $3d$ shells (but projectors in this channel) such as a Ca atom or Ti$^{4+}$ ion (not shown).
Now we would like to test the accuracy of our long-wavelength expansion of the current operator (Sec. \[longwave\]) for calculating $\mu_{\text{L}}$. In Table \[NLchgtab\] we give both the local and nonlocal contributions to $\mu_{\text{L}}$ using the right-hand side of Eqs. (\[NLcomp\]) and (\[loccomp\]) (labeled as $\rho^{\text{loc}}$ and $\rho^{\text{nl}}$), compared to those calculated from our current-density implementation \[left-hand side of Eqs. (\[NLcomp\]) and (\[loccomp\]), labeled as $P^{\text{loc}}$ and $P^{\text{nl}}$\]. The agreement between the real-space moments and reciprocal-space derivatives of the expansion in Eq. (\[ICLimp\]) is excellent. Also, we can see that even though the nonlocal contribution to the Born effective charge is large for Ar, the first-order nonlocal charge is almost purely dipolar, with the third moment being almost two orders of magnitude smaller than the contribution of the local part.
Also, from Table \[IRCtab\] and Fig. \[IRCplot\], we see that $\mu_{\text{L}}= Q_{\text{NG}}/2\Omega$ \[consistent with Eq. (\[muNG\])\] quite accurately for sufficiently large simulation cells.
### Test of implementation: Shear response\[tstshear\]
In Table \[IRCtab\] we give the longitudinal and shear FxE coefficients, as well as $\chi_{\text{mag}}$ and $Q_{\text{NG}}/2\Omega$, for noble gas atoms in $14\times14\times14$ Bohr boxes. For $\mu_{\text{S}}$ and $\chi_{\text{mag}}$, we give values using the ICL and PM paths for the nonlocal correction. In Fig. \[IRCplot\], we show the dependence of these quantities on the box size.
\[IRCtab\]
$\mu_{\text{L}}$ $\mu_{\text{S}}^{\text{ICL}}$ $\mu_{\text{S}}^{\text{PM}}$ $\chi_{\text{mag}}^{\text{ICL}}$ $\chi_{\text{mag}}^{\text{PM}}$ $Q_{\text{NG}}/2\Omega$
---- ------------------ ------------------------------- ------------------------------ ---------------------------------- --------------------------------- ------------------------- --
He $-0.468$ $-0.467$ $-0.464$ $-0.468$ $-0.464$ $-0.466$
Ne $-1.840$ $-1.693$ $-1.655$ $-1.692$ $-1.655$ $-1.845$
Ar $-4.545$ $-5.008$ $-5.086$ $-5.013$ $-5.081$ $-4.554$
Kr $-5.968$ $-5.901$ $-5.917$ $-5.903$ $-5.921$ $-5.990$
: Longitudinal and shear (ICL and PM path) FxE coefficients for noble gas atoms in $14\times14\times14$ Bohr boxes, as well as the diamagnetic susceptibility correction, $\chi_{\text{mag}}$ (ICL and PM path), and the quadrupole moment of the unperturbed charge density divided by two times the volume \[*cf.* Eqs. (\[muIRC\]) and (\[QIRC\])\]. All quantities are in units of pC/m, and mixed electrical boundary conditions used.
![\[IRCplot\] (Color online) The longitudinal (red squares) and shear (blue diamonds) FxE coefficients, as well as the diamagnetic susceptibility correction (black circles) and $Q_{\text{NG}}/2\Omega$, for (a) He, (b) Ne, (c) Ar, and (d) Kr atoms in cells with various lattice constants. All quantities are multiplied by the cell volume, $\Omega$. ](./AtDiamag.pdf){width="\columnwidth"}
From Table \[IRCtab\] and Fig. \[IRCplot\], we see that $\mu_{\text{S}}=\chi_{\text{mag}}$ (consistent with the isotropic symmetry of the atoms) for sufficiently large simulation cells. However, for atoms other than He, $\chi_{\text{mag}}$ is noticeably different from $Q_{\text{NG}}/2\Omega$, even for large box sizes. This discrepancy demonstrates that either Larmor’s theorem or the Langevin theory of diamagnetism breaks down when nonlocal pseudopotentials are present (see Sec. \[Disc\] for further discussion).
When we compare the two path choices, PM (Sec. \[formPM\]) and ICL (Sec. \[formICL\]), we find slight quantitative differences for the shear component and diamagnetic correction. However, the differences between the paths vanishes for $\mu_{\text{S}}^\prime$ \[see Eq. (\[mucorr\])\], indicating that although the CRG contribution is path-dependent, the “true” shear response (which is vanishing for spherical symmetry) is not for this system. This result is an excellent test that our implementation is sound. Indeed, for a cubic solid, all three components of the electronic flexoelectric tensor $\bm{\mu}'$ can be related to the surface charge accumulated via the mechanical deformation of a finite crystallite; thus, they should not depend on the aforementioned path choice. As the path choice is irrelevant in our context, in the next Section we shall perform our calculations on cubic oxides using the ICL path. In Sec. \[Disc\] we shall provide a critical discussion of the ICL and PM prescriptions from a more general perspective, and leave a detailed comparison of the two approaches for a future work.
Cubic oxides\[Cub\]
-------------------
We now apply our methodology to calculate the bulk, CI FxE coefficients for several technologically important cubic oxides. As mentioned before, we will be using short circuit boundary conditions and the ICL path for the nonlocal contribution.
As an example of a typical calculation, in Fig. \[STOPq\] we plot the induced polarization \[Eq. (\[ICLimp\])\] versus $\textbf{q}=(q_x,0,0)$ for cubic SrTiO$_3$, both for polarization direction and atomic displacement $\alpha=\beta=x$ and $\alpha=\beta=y$. As expected, the dependence on $q$ is quadratic (there is no linear term since cubic SrTiO$_3$ is not piezoelectric [@Hong2013; @Stengel2013]), and $\overline{P}^{\textbf{q}}=0$ at $\textbf{q}=0$, which is required by the ASR condition that the sum of the Born effective charges should vanish[@Pick1970]. By taking the second derivative of the black (red) dashed curves in Fig. \[STOPq\], we can obtain $\mu_{11,11}^{\text{I}}$ ($\mu_{11,22}^{\text{I}}$). The remaining coefficient $\mu_{12,12}^{\text{I}}$ is obtained by calculating $\overline{P}^{\textbf{q}}_{12}$ at various $\textbf{q}=(q_x,q_y,0)$, and performing a numerical mixed derivative $\partial^2/\partial
q_x\partial q_y$ (not shown).
![\[STOPq\] (Color online) Induced polarization vs $\textbf{q}=(q_x,0,0)$ for cubic SrTiO$_3$. The black (red) points correspond to the $x$ ($y$) component of the polarization for atomic displacements of the atoms in the $x$ ($y$) direction. Dashed curves are quadratic fits. Units are with respect to the calculated SrTiO$_3$ lattice constant $a=7.435$ Bohr.](./STOPq.pdf){width="\columnwidth"}
In Table \[cubtab\], we give the FxE coefficients corrected for the CRG contribution \[*cf.* Eq. (\[mucorr\])\] and the RCC (Sec. \[RCC\]). As discussed above, the RCC is added to the longitudinal and transverse coefficients [@StengelChapter]. Note that the reported $\chi_{\text{mag}}$ is given in pC/m, whereas other quantities are in nC/m, so this correction is quite small for the materials calculated. The contribution of the nonlocal potentials to the FxE coefficients in Table \[cubtab\], which are computed using the ICL path of Appendix \[ICL\], represents a more significant correction than was the case in Sec. \[Bench\]: they are in the range of $0.03$ to $0.12$ nC/m for the longitudinal and transverse coefficients, and in the range of $-0.02$ to $0.008$ nC/m for the shear coefficients.
\[cubtab\]
$a$ (Bohr) $\epsilon$ RCC $\mu^\prime_{\text{L}}$ $\mu^\prime_{\text{T}}$ $\mu^\prime_{\text{S}}$ $\chi_{\text{mag}}\times 10^{3}$
----------- ------------ ------------ ---------- -------------------------- ------------------------- ------------------------- ----------------------------------
SrTiO$_3$ 7.435 6.191 $-0.049$ $-0.87$ ($-0.9$,$-0.88$) $-0.84$ ($-0.83$) $-0.08$($-0.08$) $-7.3$
BaTiO$_3$ 7.601 6.657 $-0.107$ $-1.01$ ($-1.1$) $-0.99$ $-0.08$ $-1.7$
SrZrO$_3$ 7.882 4.558 $-0.049$ $-0.63$ $-0.58$ $-0.05$ $-36.0$
PbTiO$_3$ 7.496 8.370 $-0.158$ $-1.39$ ($-1.5$) $-1.35$ $-0.09$ $-22.4$
MgO 8.058 3.148 $-0.015$ $-0.28$ ($-0.3$) $-0.30$ $-0.07$ $-66.1$
The only material for which first-principles calculations of the transverse and shear coefficients are available (in parentheses in Table \[cubtab\]) is SrTiO$_3$, and our values are in excellent agreement with those previous calculations [@Stengel2014].
For all of the materials, the longitudinal and transverse responses are of similar magnitude, and the shear response is significantly smaller. This is a similar trend to that of the isolated noble gas atoms and of the IRC model \[*cf.* Eq. (\[muprimeIRC\])\], suggesting that the response is dominated by the “spherical” contribution. The behavior of the cubic oxides differ significantly from the IRC model, however, when it comes to the contribution of the CRG correction $\chi_{\text{mag}}$. For isolated atoms, $\chi_{\text{mag}}$ is equal to $\mu_{\text{IRC,S}}$, and is of the same order as $\mu^\prime_{\text{IRC,L}}$; therefore, a vanishing value of $\mu^\prime_{\text{IRC,S}}$ is only obtained after removing the CRG contribution \[Eq. (\[mucorr\])\]. In the case of the cubic oxides, the CRG correction is only a minor contribution to $\mu^\prime_{\text{S}}$, and $\chi_{\text{mag}}$ is two orders of magnitude smaller than $\mu^\prime_{\text{L}}$. In fact, $\chi_{\text{mag}}$ for the cubic oxides is comparable to that of the isolated atoms, while the FxE coefficients for the cubic oxides are two orders of magnitude larger. This indicates that although the bonding of atoms in the cubic compounds significantly enhances the FxE coefficients, it does not have a large effect on the CRG correction.
It should be noted that the value of $\chi_{\text{mag}}$ for SrTiO$_3$ ($-2.28\times10^{-7}$ cm$^3$/g after unit conversion) is in fair agreement with the measured diamagnetic susceptibility of around $-1\times10^{-7}$ cm$^3$/g from Ref. .
Discussion\[Disc\]
==================
Before closing, it is useful to recap the technical issues that are associated with the calculation of the current density response in a nonlocal pseudopotential context, and critically discuss them in light of the result presented in this work. In particular, it is important to clarify whether our proposed approach matches the expectations, especially regarding the known transformation properties of the current density upon rototranslations, or whether there is any deviation that needs to be kept in mind when computing flexoelectric coefficients and other current-related linear-response properties.
As we have already discussed at length in the earlier Sections, our definition of the current density (i) satisfies the continuity equation by construction, (ii) correctly reduces to the textbook formula in the region of space where the Hamiltonian is local, and (iii) is consistent with the known formula for the macroscopic current operator. However, we have not yet discussed some additional properties of the current density that were established in earlier works, that might be used as “sanity checks” of our implementation:
- Translational invariance of the charge-density response: As established by Martin [@Martin1972], simultaneous uniform translation of all atoms in the crystal must yield the same variation in charge density at every point as if the static charge density were rigidly shifted. Therefore, if the whole crystal undergoes a translation with uniform velocity ${\bf v}$, the current density in the laboratory frame must be $${\bf J}({\bf r}) = {\bf v} \rho({\bf r}),
\label{transl}$$ where $\rho({\bf r})$ is the static charge density.
- Larmor’s theorem: The circulating currents generated in a crystallite by a uniform rotation with constant angular velocity $\bm{\omega}$ (as observed in the frame of the rotating material) are, in the linear limit of small velocities, identical to the orbital currents that would be generated by an applied (and constant in time) ${\bf B}$-field. As a corollary, the rotational $g$-factor of closed-shell molecules corresponds to their paramagnetic susceptibility.
- Langevin’s diamagnetism: The magnetic susceptibility of a spherically symmetric atom is proportional to the quadrupolar moment of its ground-state charge density.
In the following, we shall analyze how our formalism stands in relationship to these latter “weak” \[compared to the “strong” conditions (i-iii) above\] criteria of validity. (By “weak” we mean not required for a physically sound calculation of the flexoelectric tensor, but possibly necessary for a wider range of physical properties.)
Translational invariance of the charge-density response
-------------------------------------------------------
Based on our results of Table III, we can safely conclude that both flavors of the current-density operator (ICL and PM) break translational invariance, Eq. (\[transl\]). To see this, consider the shear flexoelectric coefficient of an isolated atom in a box, (e.g., $\mu_{\rm S,NG}$). This quantity can be defined in real space as the second moment of the microscopic current-density response to the displacement of an isolated atom, $$\mu_{\rm S} = \frac{1}{2\Omega} \int d^3 r \frac{\partial J_y({\bf r})}{\partial \dot{\lambda}_y} x^2,
\label{mus-jr}$$ where $\dot{\lambda}_y$ stands for the velocity of the atom along $y$. This formula, as it stands, is not very practical for calculations: our implementation does not allow for a fully microscopic calculation of ${\bf J}({\bf r})$, and therefore we had to replace Eq. (\[mus-jr\]) with computationally more tractable small-${\bf q}$ expansions. Still, Eq. (\[mus-jr\]) is quite useful for our purposes, as it allows us to draw general conclusions about ${\bf J}({\bf r})$ without the need for calculating it explicitly. In particular, if translational invariance \[Eq. (\[transl\])\] were satisfied, then we could plug Eq. (\[transl\]) into Eq. (\[mus-jr\]) and use Eq. (\[QIRC\]) to obtain $\mu_{\rm S,NG}
= \frac{1}{2\Omega}\int d^3r \rho(\textbf{r}) x^2
=Q_{\text{NG}}/2\Omega$. \[This equality is a necessary but not sufficient condition for the validity of Eq. (\[transl\]).\] As we can see from Table III, $\mu_{\rm S,NG}$ is only approximately equal to $Q_{\text{NG}}/2\Omega$ for both the ICL and PM flavors of the current-density operator. This implies that neither approach is able to guarantee translational invariance.
Similarly, the data we have in hand does not allow us to establish a clear preference between the PM and ICL recipes, as the discrepancies between the two are typically much smaller (and devoid of a systematic trend) than their respective failure at satisfying $\mu_{\rm S,NG} = Q_{\text{NG}}/2\Omega$. Note that the discrepancy strictly consists of *solenoidal* (i.e., divergenceless) contributions to the current response; the longitudinal components are exactly treated, as one can verify from the excellent match between the longitudinal coefficient, $\mu_{\text{L}}$, and the quadrupolar estimate in Table III.
Langevin diamagetism and Larmor’s theorem
-----------------------------------------
We come now to the assessment of the Larmor and Langevin results. One of the virtues of the PM recipe resides in its superior accuracy when comparing the orbital magnetic response to all-electron data. Indeed, in the context of our discussion, one can verify that it exactly complies with Langevin’s theory of diamagnetism in the case of isolated spherical atoms. [^5] The situation, however, is not so bright regarding Larmor’s theorem. If the latter were satisfied, then the “rotational orbital susceptibility” $\chi_{\rm mag}$ would match Langevin’s quadrupolar expression, as we know that Langevin’s result holds in the case of a “true” ${\bf B}$-field. By looking, again, at Table III, we clearly see that this is not the case – again, there is a discrepancy between the last column (based on the static quadrupole) and the calculated values of $\chi_{\rm mag}$. Since the deviations in $\chi_{\rm mag}$ and $\mu_{\rm S}$ are essentially identical in the limit of an isolated atom in a box, it is reasonable to assume that the underlying factors are similar.
It should be noted that our value for Ne (after unit conversion, ICL path) is $\chi_{\text{mag}}^{\text{ICL}}=-7.29\times10^{-6}$ cm$^3$/mole, which is fairly close in magnitude to previously calculated values of the diamagnetic susceptibility of Ne: $-7.75\times10^{-6}$ cm$^3$/mole[@ICL2001] and $-7.79\times10^{-6}$ cm$^3$/mole[@Mauri1996].
Unphysical spatial transfer resulting from nonlocal pseudopotentials
--------------------------------------------------------------------
The reason why the current density violates both translational invariance and Larmor’s theorem has to be sought in the unphysical transfer of density that can result from the presence of a nonlocal potential. That is, a nonlocal operator may project the wavefunction (and therefore the particle amplitude) from a point ${\bf
r}$ to a distant point ${\bf r}'$ in a discontinuous manner, such that no current flows through a given surface surrounding ${\bf
r}$ even though the charge density within that surface changes. Of course, this is just a conceptual way of describing the violation of the continuity equation, discussed in Sec. \[curden\].
Taking the example of a single atom placed at $\textbf{R}=0$ and using the PM approach, it is shown in Appendix \[appDiv\] that the current density can be written as $${\bf J}^{\rm nl}({\bf r}) \sim \frac{\hat{\bf r} C(\hat{\bf r})}{r^2} .$$ where $C(\hat{\textbf{r}})$ is a direction-dependent constant that depends on the nonlocal charge \[Eq. (\[Cr\])\]. Therefore, the current-density field diverges near the atomic site, $\textbf{r}\rightarrow0$, and such a divergence can have a different prefactor and sign depending on the direction.
A diverging ${\bf J}$-field is problematic to deal with and unphysical. One can easily realize that this characteristic is incompatible, for example, with the correct transformation laws of ${\bf J}$ under rigid translations. In particular, the electronic charge density is always finite in a vicinity of the nucleus, even in the all-electron case where the corresponding potential does, in fact, diverge. This implies that Eq. (\[transl\]) cannot be satisfied by a diverging ${\bf J}$-field.
For the ICL path, the nonlocal current does not have such a simple relation to the nonlocal charge as in the case of the PM path \[Eq. (E4)\]; therefore a similar derivation as in Appendix E may not be possible for the ICL case. However, our numerical results in Table III are sufficient to conclude that the ICL path violates translational symmetry as well. The extent of the violation can be quantified by looking at the discrepancy between $\mu_{\text{L}}$ and $\mu_{\text{S}}$, which is comparably large in the PM and ICL cases—recall that these two values should, in principle, coincide for the isolated spherical atoms model.
At present it is difficult to predict whether it might be possible to cure the above drawbacks by simply choosing a different path for the definition of the current operator, or whether these difficulties may require a deeper revision of the nonlocal pseudopotential theory in contexts where the microscopic current density is needed. In any case, the flexoelectric coefficients we calculated in this work for cubic materials are unaffected by these issues: Once the “diamagnetic” contribution has been removed, the three independent coefficients are all well defined in terms of the charge-density response. Nonetheless, the above caveats should be kept in mind when using the present current-density implementation to access flexoelectric coefficients in less symmetric materials, or other response properties that depend on the microscopic current response.
Conclusions\[Con\]
==================
We have developed a DFPT implementation for calculating the bulk CI flexoelectric tensor from a single unit cell. Therefore, we have overcome the limitations of previous implementations (Refs. and ), which required supercells to calculate the transverse and shear CI FxE coefficients.
Our implementation is based on calculating the microscopic current density resulting from the adiabatic atomic displacements of a long-wavelength acoustic phonon. We have determined a form for the current-density operator that satisfies the continuity condition in the presence of nonlocal, norm-conserving pseudopotentials, and reduces to the correct form in the limit of a uniform, macroscopic perturbation, and/or when only local potentials are present.
In order to benchmark our methodology, we have used noble gas atoms to model systems of noninteracting spherical charge densities. The tests demonstrate the accuracy of our nonlocal correction to the current operator, as well as the calculated CRG corrections derived in Ref. . For our form of the current density, we demonstrate that nonlocal pseudopotentials result in a violation of translational invariance and Larmor’s theorem, though this does not affect our FxE coefficients after the CRG contribution has been removed. Finally, we have applied our methodology to several cubic oxides, all of which show similar trends in that the longitudinal and transverse responses are similar ($\sim1$ nC/m), and the shear response is an order of magnitude smaller.
Combining the methodology of this paper with DFPT implementations for calculating the lattice-mediated contribution to the bulk FxE coefficients [@Stengel2013; @Stengel2014], and the surface contribution [@Stengel2014], will allow for efficient calculation of the full FxE response for a variety of materials.
We are grateful to K. M. Rabe, D. R. Hamann, B. Monserrat, H. S. Kim, A. A. Schiaffino, C. J. Pickard, and S. Y. Park for useful discussions. CED and DV were supported by ONR Grant N00014-16-1-2951. MS acknowledges funding from from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 724529), and from Ministerio de Economía, Industria y Competitividad (MINECO-Spain) through Grants No. MAT2016-77100-C2-2-P and SEV-2015-0496, and from Generalitat de Catalunya through Grant No. 2017 SGR1506.
Essin *et al.* approach and the Ismail-Beigi, Chang, and Louie straight-line path\[sepICL\]
===========================================================================================
Here we perform a long-wavelength expansion of the current operator using the approach of Essin *et al.*[@Essin2010], and confirm that the approach is equivalent to that of ICL [@ICL2001]. We start from Eq. (\[JqSL\]) and rewrite it as $$\label{LocNL1}
\begin{split}
\langle\textbf{s}\vert\hat{\mathcal{J}}_{\alpha}^{\text{ICL}}(\textbf{q})\vert\textbf{s}^\prime\rangle&=-iH(\textbf{s},\textbf{s}^\prime)(s_\alpha-s_\alpha^\prime)\frac{e^{-i\textbf{q}\cdot\textbf{s}}-e^{-i\textbf{q}\cdot\textbf{s}^{\prime}}}{i\textbf{q}\cdot(\textbf{s}-\textbf{s}^\prime)}
\\
&=-i\left[\hat{r}_\alpha,\hat{H}\right]_{\textbf{s}\textbf{s}^\prime}\frac{e^{-i\textbf{q}\cdot\textbf{s}}-e^{-i\textbf{q}\cdot\textbf{s}^{\prime}}}{i\textbf{q}\cdot(\textbf{s}-\textbf{s}^\prime)}
\\
&=-\left(i\left[\hat{r}_\alpha,\hat{T}\right]_{\textbf{s}\textbf{s}^\prime}+i\left[\hat{r}_\alpha,\hat{V}^{\text{nl}}\right]_{\textbf{s}\textbf{s}^\prime}\right)\frac{e^{-i\textbf{q}\cdot\textbf{s}}-e^{-i\textbf{q}\cdot\textbf{s}^{\prime}}}{i\textbf{q}\cdot(\textbf{s}-\textbf{s}^\prime)},
\end{split}$$ where $\hat{T}$ is the kinetic energy operator and $\hat{V}^{\text{nl}}$ is the nonlocal part of the potential (the local part of the potential does not contribute). We now factor out a $e^{-i\textbf{q}\cdot\textbf{s}^\prime}$ and then expand the term outside of the parentheses $$\label{LocNL2}
\begin{split}
\langle\textbf{s}\vert\hat{\mathcal{J}}_{\alpha}^{\text{ICL}}(\textbf{q})\vert\textbf{s}^\prime\rangle&=-\left(i\left[\hat{r}_\alpha,\hat{T}\right]_{\textbf{s}\textbf{s}^\prime}+i\left[\hat{r}_\alpha,\hat{V}^{\text{nl}}\right]_{\textbf{s}\textbf{s}^\prime}\right)e^{-i\textbf{q}\cdot\textbf{s}^\prime}\left(-1+\frac{i\textbf{q}\cdot(\textbf{s}-\textbf{s}^\prime)}{2}+\frac{[\textbf{q}\cdot(\textbf{s}-\textbf{s}^\prime)]^2}{6}+...\right).
\end{split}$$ As mentioned in Sec. \[formICL\], if $\textbf{q}=0$, then $\hat{\mathcal{J}}_{\alpha}^{\text{ICL}}(\textbf{q}=0)=i\left[\hat{r}_\alpha,\hat{H}\right]=-\hat{v}_\alpha$, the velocity operator.
Consider the case of a Hamiltonian with a local potential, so the only term in Eq. (\[LocNL2\]) is the commutator of the position operator with the kinetic part of the Hamiltonian. We can rewrite this term as $$\label{LocNL3}
\begin{split}
\langle\textbf{s}\vert\hat{\mathcal{J}}_{\alpha}^{\text{loc}}(\textbf{q})\vert\textbf{s}^\prime\rangle&=-\left(-i\left[\hat{r}_\alpha,\hat{T}\right]_{\textbf{s}\textbf{s}^\prime}-\sum_{\gamma=1}^3 \frac{q_\gamma}{2}\left[\hat{r}_\gamma,\left[\hat{r}_\alpha,\hat{T}\right]\right]_{\textbf{s}\textbf{s}^\prime}+\sum_{\gamma=1}^3\sum_{\xi=1}^3 \frac{iq_\gamma q_\xi}{6}\left[\hat{r}_\xi,\left[\hat{r}_\gamma,\left[\hat{r}_\alpha,\hat{T}\right]\right]\right]_{\textbf{s}\textbf{s}^\prime}+...\right)e^{-i\textbf{q}\cdot\textbf{s}^\prime}.
\end{split}$$ The term at zeroth order in $q$ is simply the momentum operator: $\hat{p}_\alpha=-i\left[\hat{r}_\alpha,\hat{T}\right]$; at first order in $q$, we have $\hat{q}_\alpha/2$ (the nested commutator is simply the Kroneker delta function $-\delta_{\alpha\gamma}$); higher order terms vanish. So in the case of a Hamiltonian that only has a local potential, $$\label{LocNL4}
\begin{split}
\hat{\mathcal{J}}_{\alpha}^{\textbf{q},\text{loc}}=-\left(\hat{p}_\alpha+\frac{q_\alpha}{2}\right),
\end{split}$$ which is the cell-periodic momentum operator for the case of local potentials, as we derive in Appendix \[Jloc\]. Therefore the local and nonlocal components can be cleanly separated. The nonlocal part of the potential in Eq. \[LocNL1\] is addressed in Appendix \[ICL\].
Note that the approach of Essin *et al.* does not work for an arbitrary choice of path. Specifically, if we were to use Eq. (\[HA1\]) with the PM path choice $\textbf{s}^\prime\rightarrow\textbf{R}\rightarrow\textbf{s}$, the expression would not reproduce the correct form of the current for local potentials (except for the case of the longitudinal response). Of course, in the PM form of the coupled Hamiltonian in Eq. (\[HPM\]), the current in the case of only local potentials trivially reduces to the correct form $\hat{\mathcal{J}}_\alpha^{\text{loc}}(\textbf{r})=-\frac{1}{2}\left\{\hat{\rho}(\textbf{r}),(\hat{p}_\alpha+\hat{A}_\alpha)\right\}$.
Derivation of induced polarization: Local potentials\[Jloc\]
============================================================
In this section we derive $P^{\textbf{q},\text{loc}}_{\alpha,\kappa\beta}$ for Eq. (\[PqExpand\]). This is a straightforward generalization of what was derived by Umari, Dal Corso, and Resta [@Umari2001] to finite **q** perturbations, and has been derived previously in other contexts (e.g. for determining magnetic[@Mauri1996] or dielectric[@Adler1962] susceptibility, and in the context of phonon deformation potentials[@Khan1984]).
Using the adiabatic expansion of the time-dependent wavefunction \[Eqs. (\[psiad\]) and (\[deltapsi\])\], to first order in $\dot{\lambda}$ we can write the density matrix as $$\begin{split}
\label{rho}
\rho(t)&=-\frac{2}{N_k}\sum_{n\textbf{k}}\vert\Psi_{n\textbf{k}}(\lambda(t))\rangle\langle\Psi_{n\textbf{k}}(\lambda(t))\vert\simeq-\frac{2}{N_k}\sum_{n\textbf{k}}\left[\vert\psi_{n\textbf{k}}\rangle\langle\psi_{n\textbf{k}}\vert+\dot{\lambda}(\vert\delta\psi_{n\textbf{k}}\rangle\langle\psi_{n\textbf{k}}\vert+\vert\psi_{n\textbf{k}}\rangle\langle\delta\psi_{n\textbf{k}}\vert)\right]
\end{split}$$ where the factor of two is assuming spin degeneracy. If we apply the local current-density operator \[Eq. (\[jloc\])\], retaining terms only to linear order in $\dot{\lambda}$, and take the derivative with respect to $\dot{\lambda}$ we obtain the induced polarization $$\begin{split}
P_\alpha^{\text{loc}}(\textbf{r})=-\frac{1}{N_k}\sum_{n\textbf{k}}\big[\langle\psi_{n\textbf{k}}\vert\textbf{r}\rangle\langle\textbf{r}\vert\hat{p}_\alpha\vert\delta\psi_{n\textbf{k}}\rangle
+\langle\delta\psi_{n\textbf{k}}\vert\textbf{r}\rangle\langle\textbf{r}\vert\hat{p}_\alpha\vert\psi_{n\textbf{k}}\rangle+\langle\psi_{n\textbf{k}}\vert\hat{p}_\alpha\vert\textbf{r}\rangle\langle\textbf{r}\vert\delta\psi_{n\textbf{k}}\rangle
+\langle\delta\psi_{n\textbf{k}}\vert\hat{p}_\alpha\vert\textbf{r}\rangle\langle\textbf{r}\vert\psi_{n\textbf{k}}\rangle\big] .
\end{split}
\label{pr}$$
Now consider the perturbation in Eq. (\[phon\]): the displacement of a sublattice $\kappa$ in direction $\beta$ modulated by a phase with wavevector **q**. We begin with the real-space expression for the polarization induced by this perturbation: $$P^{\textbf{q},\text{loc}}_{\alpha,\kappa\beta}(\textbf{r})=-\frac{1}{N_k}\sum_{n\textbf{k}}\left[\langle\psi_{n\textbf{k}}\vert\textbf{r}\rangle\langle\textbf{r}\vert\hat{p}_\alpha\vert\delta\psi_{n\textbf{k},\textbf{q}}^{\kappa\beta}\rangle
+\langle\delta\psi_{n\textbf{k},-\textbf{q}}^{\kappa\beta}\vert\textbf{r}\rangle\langle\textbf{r}\vert\hat{p}_\alpha\vert\psi_{n\textbf{k}}\rangle
+\langle\psi_{n\textbf{k}}\vert\hat{p}_\alpha\vert\textbf{r}\rangle\langle\textbf{r}\vert\delta\psi_{n\textbf{k},\textbf{q}}^{\kappa\beta}\rangle
+\langle\delta\psi_{n\textbf{k},-\textbf{q}}^{\kappa\beta}\vert\hat{p}_\alpha\vert\textbf{r}\rangle\langle\textbf{r}\vert \psi_{n\textbf{k}}\rangle
\right]
\label{pqr}$$ where the subscript **q** in $\delta\psi_{n\textbf{k},\pm\textbf{q}}^{\kappa\beta}$ indicates that the perturbation couples states at **k** to those at $\textbf{k}\pm\textbf{q}$. If we assume TRS \[see Eq. (\[TReq\])\], then we have $$P^{\textbf{q},\text{loc}}_{\alpha,\kappa\beta}(\textbf{r})=-\frac{2}{N_k}\sum_{n\textbf{k}}\left[\langle\psi_{n\textbf{k}}\vert\textbf{r}\rangle\langle\textbf{r}\vert\hat{p}_\alpha\vert\delta\psi_{n\textbf{k},\textbf{q}}^{\kappa\beta}\rangle
+\langle\psi_{n\textbf{k}}\vert\hat{p}_\alpha\vert\textbf{r}\rangle\langle\textbf{r}\vert\delta\psi_{n\textbf{k},\textbf{q}}^{\kappa\beta}\rangle
\right]
\label{pqrTRS}$$ We Fourier transform Eq. (\[pqrTRS\]) to reciprocal space and consider the cell periodic part $$\begin{split}
P^{\text{loc}}_{\alpha,\kappa\beta}(\textbf{G}+\textbf{q})&=-\frac{2}{N_k}\sum_{n\textbf{k}}\int d^3r\Big[\langle\psi_{n\textbf{k}}\vert\textbf{r}\rangle e^{-i(\textbf{G}+\textbf{q})\cdot\textbf{r}} \langle\textbf{r}\vert\hat{p}_\alpha\vert\delta\psi^{\kappa\beta}_{n\textbf{k},\textbf{q}}\rangle
+\langle\psi_{n\textbf{k}}\vert\hat{p}_\alpha\vert\textbf{r}\rangle e^{-i(\textbf{G}+\textbf{q})\cdot\textbf{r}}\langle\textbf{r}\vert\delta\psi^{\kappa\beta}_{n\textbf{k},\textbf{q}}\rangle\Big].
\end{split}$$ We now explicitly insert the expansion of the wavefunctions in terms of plane waves $$\begin{split}
\psi_{\textbf{k}}(\textbf{s})=\sum_m c_{\textbf{k},\textbf{G}_m}e^{i(\textbf{G}_m+\textbf{k})\cdot\textbf{s}}
\\
\delta\psi^{\kappa\beta}_{n\textbf{k},\textbf{q}}(\textbf{s})=\sum_m \delta c_{\textbf{k}+\textbf{q},\textbf{G}_m}e^{i(\textbf{G}_m+\textbf{k}+\textbf{q})\cdot\textbf{s}},
\end{split}$$ where we have dropped the band index and the $\kappa\beta$ indices for the expansion coefficients $c$ and $\delta c$, and $m$ indexes a reciprocal lattice vector $\textbf{G}_m$. Then, applying the momentum operator, $$\begin{split}
P^{\text{loc}}_{\alpha,\kappa\beta}(\textbf{G}+\textbf{q})
&=-\frac{2}{N_k}\sum_{\textbf{k}}\sum_{m,m^\prime}\int d^3r c^*_{\textbf{k},\textbf{G}_m}\delta c_{\textbf{k}+q_\alpha,\textbf{G}_{m^\prime}}\Big[(k_\alpha+q_\alpha+G_{\alpha,m^\prime}) e^{-i(\textbf{G}+\textbf{G}_m-\textbf{G}_{m^\prime})\cdot\textbf{r}}
\\
&\phantom{=-\frac{2}{N_k}\sum_{\textbf{k}}\sum_{m,m^\prime}\int d^3r c^*_{\textbf{k},\textbf{G}_m}\delta c_{\textbf{k}+q_\alpha,\textbf{G}_{m^\prime}}\Big[}+(k_\alpha+G_{\alpha,m}) e^{-i(\textbf{G}+\textbf{G}_m-\textbf{G}_{m^\prime})\cdot\textbf{r}}\Big]
\\
&=-\frac{4}{N_k}\sum_{\textbf{k}}\sum_{m} c^*_{\textbf{k},\textbf{G}_m}\delta c_{\textbf{k}+q_\alpha,\textbf{G}_{m}+\textbf{G}}\left(k_\alpha+G_{\alpha,m}+\frac{q_\alpha+G_\alpha}{2}\right)
\\
&=-\frac{2}{N_k}\sum_{n\textbf{k}}\langle u_{n\textbf{k}}\vert e^{-i\textbf{G}\cdot\hat{\textbf{r}}}\left(\hat{p}^{\textbf{k}}_\alpha+\frac{q_\alpha}{2}\right) +\left(\hat{p}^{\textbf{k}}_\alpha+\frac{q_\alpha}{2}\right) e^{-i\textbf{G}\cdot\hat{\textbf{r}}}\vert\delta u^{\kappa\beta}_{n\textbf{k},\textbf{q}}\rangle,
\end{split}$$ where, in the last line, we have restored the band and $\kappa\beta$ indices, $\hat{p}_\alpha^{\textbf{k}}=-i\hat{\nabla}_\alpha+k_\alpha$ is the cell-periodic momentum operator ($\hat{\nabla}_\alpha$ is a spatial derivative in the $\alpha$ direction), and we have used that $\psi_{n\textbf{k}}(\textbf{s})=u_{n\textbf{k}}(\textbf{s})e^{i\textbf{k}\cdot\textbf{s}}$. In Sec. \[Bench\], we use this result to calculate real-space moments of the local contribution to the FxE coefficient. Otherwise, we are usually interested in the $\textbf{G}=0$ term: $$\begin{split}
\label{pkq}
\overline{P}^{\textbf{q},\text{loc}}_{\alpha,\kappa\beta}
&=-\frac{4}{N_k}\sum_{n\textbf{k}}\langle u_{n\textbf{k}}\vert\left(\hat{p}^{\textbf{k}}_\alpha+\frac{q_\alpha}{2}\right)\vert\delta u^{\kappa\beta}_{n\textbf{k},\textbf{q}}\rangle.
\end{split}$$
Current density in the presence of nonlocal pseudopotentials\[JNL\]
===================================================================
Here we derive the contributions to the current from the nonlocal potentials \[$P^{\textbf{q},\text{nl}}_{\alpha,\kappa\beta}$ in Eq. (\[PqExpand\])\], which we obtain by expanding the nonlocal current-density operator up to second order in **q** \[Eq. (\[JqExpand\])\], $$\label{Pqexpand2}
\begin{split}
\overline{P}_{\alpha,\kappa\beta}^{\textbf{q},\text{nl}}&\simeq \frac{4}{N_k}\sum_{n\textbf{k}}\Bigg[\langle u_{n\textbf{k}}\vert \hat{\mathcal{J}}_\alpha^{\textbf{k},\text{nl}(0)} \vert\delta u^{\kappa\beta}_{n\textbf{k},\textbf{q}}\rangle
+\frac{1}{2}\sum_{\gamma=1}^3 q_\gamma\langle u_{n\textbf{k}}\vert \hat{\mathcal{J}}_{\alpha,\gamma}^{\textbf{k},\text{nl}(1)} \vert\delta u^{\kappa\beta}_{n\textbf{k},\textbf{q}}\rangle
+\frac{1}{6}\sum_{\gamma=1}^3\sum_{\xi=1}^3q_\gamma q_\xi\langle u_{n\textbf{k}}\vert \hat{\mathcal{J}}_{\alpha,\gamma \xi}^{\textbf{k},\text{nl}(2)} \vert\delta u^{\kappa\beta}_{n\textbf{k},\textbf{q}}\rangle\Bigg],
\end{split}$$
The nonlocal potential that we are interested in is that of the norm-conserving pseudopotential. In reciprocal space, the nonlocal potential in the separable Kleinman-Bylander[@Kleinman1982] form is given by[@Martin2004] $$\label{VNL}
V^{\text{nl}}(\textbf{K},\textbf{K}^\prime)=\sum_{\zeta}e^{-i\textbf{K}\cdot\textbf{R}_\zeta}\left(\sum_{lm}\frac{Y_{\zeta lm}^*(\hat{\textbf{K}})T_{\zeta l}^{*}(\vert \textbf{K}\vert)\times T_{\zeta l}(\vert \textbf{K}^\prime\vert)Y_{\zeta lm}(\hat{\textbf{K}}^\prime)}{E^{\text{KB}}_{\zeta l}}\right)e^{i\textbf{K}^\prime \cdot\textbf{R}_\zeta}$$ where $\textbf{K}=\textbf{G}+\textbf{k}$; **R**$_\zeta$ is the atomic position of atom $\zeta$; $Y_{\zeta lm}$ is the spherical harmonic for the $lm$ angular momentum channel; $T_{\zeta l}(K)$ is the Fourier transform of the radial function, $\widetilde{\psi}_{\zeta
l}(r)V_{\zeta l}(r)$, where $V_{\zeta l}(r)$ are the pseudopotentials and $\widetilde{\psi}_{\zeta l}(r)$ the pseudoorbitals; $E^{\text{KB}}_{\zeta
l}=\langle\widetilde{\psi}_{\zeta l}\vert
\hat{V}_l\vert\widetilde{\psi}_{\zeta l}\rangle$ is the Kleinman-Bylander energies. The term in the parentheses is the nonlocal form factor, and the phase factors surrounding it are the structure factors. We define $$\langle\textbf{K}\vert \phi_{\zeta lm}\rangle=e^{i\textbf{K}\cdot\textbf{R}_\zeta}Y_{\zeta lm}(\hat{\textbf{K}})T_{\zeta l}(\vert \textbf{K}\vert)$$ so $$\label{NLop}
\hat{V}^{\text{nl}}=\sum_{\zeta lm}\frac{\vert \phi_{\zeta lm}\rangle\langle \phi_{\zeta lm}\vert}{E^{\text{KB}}_{\zeta l}} .$$
Ismail-Beigi, Chang, and Louie straight-line path\[ICL\]
--------------------------------------------------------
For the straight-line path of Essin *et al.*[@Essin2010] and Ismail-Beigi, Chang, and Louie,[@ICL2001] we combine Eq. (\[JkqICL\]) and (assuming we have TRS) Eq. (\[PqTR\]). Since we have already addressed the local part in Appendix \[Jloc\], we only consider the nonlocal part of the Hamiltonian, defining the operator $$\begin{split}
\langle\textbf{s}\vert\hat{\mathcal{J}}_{\alpha}^{\textbf{k},\textbf{q},\text{ICL,nl}}\vert\textbf{s}^\prime\rangle&=-iV^{\textbf{k},\text{nl}}(\textbf{s},\textbf{s}^\prime)(s_\alpha-s_\alpha^\prime)\left[\frac{e^{-i\textbf{q}\cdot(\textbf{s}-\textbf{s}^\prime)}-1}{i\textbf{q}\cdot(\textbf{s}-\textbf{s}^\prime)}\right].
\end{split}$$ Expanding the term in square brackets in powers of **q** gives $$\begin{split}
\langle\textbf{s}\vert\hat{\mathcal{J}}_{\alpha}^{\textbf{k},\textbf{q},\text{ICL,nl}}\vert\textbf{s}^\prime\rangle&=iV^{\textbf{k},\text{nl}}(\textbf{s},\textbf{s}^\prime)(s_\alpha-s_\alpha^\prime)\left[1-\frac{i\textbf{q}\cdot(\textbf{s}-\textbf{s}^\prime)}{2}+\frac{[i\textbf{q}\cdot(\textbf{s}-\textbf{s}^\prime)]^2}{6}-\cdots\right].
\\
&=i\left[\hat{r}_\alpha,\hat{V}^{\textbf{k},\text{nl}}\right]_{\textbf{s}\textbf{s}^\prime}
-\frac{1}{2}\sum_{\gamma=1}^3 q_\gamma\left[\hat{r}_\gamma,\left[\hat{r}_\alpha,\hat{V}^{\textbf{k},\text{nl}}\right]\right]_{\textbf{s}\textbf{s}^\prime}
-\frac{i}{6}\sum_{\gamma=1}^3\sum_{\xi=1}^3 q_\gamma q_\xi\left[\hat{r}_\xi,\left[\hat{r}_\gamma,\left[\hat{r}_\alpha,\hat{V}^{\textbf{k},\text{nl}}\right]\right]\right]_{\textbf{s}\textbf{s}^\prime}+\cdots,
\end{split}$$ so we can write the operator as $$\begin{split}
\label{kgradop}
\hat{\mathcal{J}}_{\alpha}^{\textbf{k},\textbf{q},\text{ICL,nl}}&=\sum_{\gamma_1\cdots \gamma_n}\frac{q_{\gamma_1}\cdots q_{\gamma_n}}{(n+1)!} \hat{\mathcal{J}}_{\alpha,\gamma_1\cdots\gamma_n}^{\textbf{k},\text{ICL,nl}(n)}, \qquad
\hat{\mathcal{J}}_{\alpha,\gamma_1\cdots\gamma_n}^{\textbf{k},\text{ICL,nl}(n)} = -\frac{\partial^{n+1}\hat{V}^{\textbf{k},{\text{nl}}}}{\partial k_\alpha \partial k_{\gamma_1}\cdots \partial k_{\gamma_n}}.
\end{split}$$ In terms of the cell-periodic projectors $\phi_{\zeta
lm}^\textbf{k}(\textbf{s})=e^{-i\textbf{k}\cdot\textbf{s}}\phi_{\zeta
lm}(\textbf{s})$ \[see Eq. (\[NLop\])\], the lowest-order terms in Eq. (\[kgradop\]), to be incorporated into Eq. (\[Pqexpand2\]), are $$\label{ICL0}
\hat{\mathcal{J}}_\alpha^{\textbf{k},\text{nl}(0)} =-\sum_{\zeta lm}\frac{1}{E^{\text{KB}}_{\zeta l}}\left(\vert \phi^{\textbf{k}}_{\zeta lm}\rangle\langle\partial_\alpha \phi^{\textbf{k}}_{\zeta lm}\vert+\vert\partial_\alpha \phi^{\textbf{k}}_{\zeta lm}\rangle\langle \phi^{\textbf{k}}_{\zeta lm}\vert\right),$$ $$\label{ICL1}
\begin{split}
\hat{\mathcal{J}}_{\alpha,\gamma}^{\textbf{k},\text{ICL,nl}(1)}
=-\sum_{\zeta lm}\frac{1}{E^{\text{KB}}_{\zeta l}}\left(\vert\partial_\gamma \phi^{\textbf{k}}_{\zeta lm}\rangle\langle\partial_\alpha \phi^{\textbf{k}}_{\zeta lm}\vert
+\vert \phi^{\textbf{k}}_{\zeta lm}\rangle\langle\partial_\alpha\partial_\gamma \phi^{\textbf{k}}_{\zeta lm}\vert
+\vert\partial_\alpha\partial_\gamma \phi^{\textbf{k}}_{\zeta lm}\rangle\langle \phi^{\textbf{k}}_{\zeta lm}\vert
+\vert \partial_\alpha \phi^{\textbf{k}}_{\zeta lm}\rangle\langle\partial_\gamma \phi^{\textbf{k}}_{\zeta lm}\vert\right),
\end{split}$$ and $$\label{ICL2}
\begin{split}
\hat{\mathcal{J}}_{\alpha,\gamma\xi}^{\textbf{k},\text{ICL,nl}(2)}
=-\sum_{\zeta lm}\frac{1}{E^{\text{KB}}_{\zeta l}}&\big(\vert\partial_\xi\partial_\gamma \phi^{\textbf{k}}_{\zeta lm}\rangle\langle\partial_\alpha \phi^{\textbf{k}}_{\zeta lm}\vert
+\vert\partial_\xi \phi^{\textbf{k}}_{\zeta lm}\rangle\langle\partial_\gamma\partial_\alpha \phi^{\textbf{k}}_{\zeta lm}\vert
+\vert \partial_\gamma \phi^{\textbf{k}}_{\zeta lm}\rangle\langle\partial_\alpha\partial_\xi \phi^{\textbf{k}}_{\zeta lm}\vert
+\langle u_{n\textbf{k}}\vert \phi^{\textbf{k}}_{\zeta lm}\rangle\langle\partial_\gamma\partial_\alpha\partial_\xi \phi^{\textbf{k}}_{\zeta lm}\vert
\\
&+\vert\partial_\gamma\partial_\alpha\partial_\xi \phi^{\textbf{k}}_{\zeta lm}\rangle\langle \phi^{\textbf{k}}_{\zeta lm}\vert
+\langle u_{n\textbf{k}}\vert\partial_\alpha\partial_\xi \phi^{\textbf{k}}_{\zeta lm}\rangle\langle\partial_\gamma \phi^{\textbf{k}}_{\zeta lm}\vert
+\vert \partial_\gamma\partial_\alpha \phi^{\textbf{k}}_{\zeta lm}\rangle\langle\partial_\xi \phi^{\textbf{k}}_{\zeta lm}\vert
+\vert \partial_\alpha \phi^{\textbf{k}}_{\zeta lm}\rangle\langle\partial_\gamma\partial_\xi \phi^{\textbf{k}}_{\zeta lm}\vert\big).
\end{split}$$ These correspond to last three terms in Eq. (\[JqExpand\]), here specialized to the ICL path. Note that $\hat{\mathcal{J}}_\alpha^{\textbf{k},\text{nl}(0)} = -{\partial
\hat{V}^{\textbf{k},\text{nl}}}/{\partial k_\alpha}$ represents the well-known nonlocal correction to the Born effective charge (with an overall negative sign from the electron charge), which, combined with the local part \[Eq. (\[pkq\])\] yields the velocity operator $\hat{v}_\alpha^{\textbf{k},\textbf{q}}$ and should be unsensitive to the path choice.
Pickard and Mauri path through atom center\[PM\]
------------------------------------------------
The PM [@Pickard2003] path goes through the center of the atom. For simplicity of the derivation, we consider a single atom positioned at the origin ($\textbf{R}=0$); the generalization to an atom not at the origin simply involves an extra phase in the structure factors in Eq. (\[VNL\]). Then Eq. (\[JqPMNL\]) becomes $$\begin{split}
\langle\textbf{s}\vert\hat{\mathcal{J}}_{\alpha}^{\text{PM,nl}}(\textbf{q})\vert\textbf{s}^\prime\rangle=-iV^{\text{nl}}(\textbf{s},\textbf{s}^\prime)&\bigg[s^\prime_\alpha\frac{1-e^{-i\textbf{q}\cdot\textbf{s}^\prime}}{i\textbf{q}\cdot\textbf{s}^\prime}+s_\alpha\frac{e^{-i\textbf{q}\cdot\textbf{s}}-1}{i\textbf{q}\cdot\textbf{s}}\bigg].
\end{split}$$ Following the same steps as in Appendix \[ICL\], we arrive at slightly different current operators for the terms to first and second order in **q** (the zeroth order term is the same as for the ICL path, as expected), $$\begin{split}
\label{PM1}
\hat{\mathcal{J}}^{\textbf{k},\text{PM,nl}(1)}_{\alpha,\gamma}=-\sum_{\zeta lm}\frac{1}{E^{\text{KB}}_{\zeta l}}\big(2\vert\partial_\alpha\phi^{\textbf{k}}_{\zeta lm}\rangle\langle\partial_\gamma\phi^{\textbf{k}}_{\zeta lm}\vert
+\vert\phi^{\textbf{k}}_{\zeta lm}\rangle\langle\partial_\alpha\partial_\gamma\phi^{\textbf{k}}_{\zeta lm}\vert
+\vert\partial_\alpha\partial_\gamma\phi^{\textbf{k}}_{\zeta lm}\rangle\langle\phi^{\textbf{k}}_{\zeta lm}\vert\big),
\end{split}$$ $$\begin{split}
\label{PM2}
\hat{\mathcal{J}}^{\textbf{k},\text{PM,nl}(2)}_{\alpha,\gamma\xi}=-\sum_{\zeta lm}\frac{1}{E^{\text{KB}}_{\zeta l}}\big(3\vert\partial_\alpha\partial_\gamma\phi^{\textbf{k}}_{\zeta lm}\rangle\langle\partial_\xi\phi^{\textbf{k}}_{\zeta lm}\vert
+3\vert\partial_\alpha\phi^{\textbf{k}}_{\zeta lm}\rangle\langle\partial_\gamma\partial_\xi\phi^{\textbf{k}}_{\zeta lm}\vert
+\vert\phi^{\textbf{k}}_{\zeta lm}\rangle\langle\partial_\alpha\partial_\gamma\partial_\xi\phi^{\textbf{k}}_{\zeta lm}\vert
+\vert\partial_\alpha\partial_\gamma\partial_\xi\phi^{\textbf{k}}_{\zeta lm}\rangle\langle\phi^{\textbf{k}}_{\zeta lm}\vert\big).
\end{split}$$ We see immediately that, for the case of a longitudinal perturbation, Eqs. (\[PM1\]) and (\[PM2\]) are identical to their ICL counterparts \[*cf.* Eq. (\[ICL1\]) and Eq. (\[ICL2\])\].
Diamagnetic correction\[Appdiamag\]
===================================
In this section we provide some details about the calculation of the CRG contribution to the transverse and shear FxE coefficients, which is related to the diamagnetic susceptibility. We refer the reader to Ref. for a complete discussion.
For the case of a small deformation **u** that is applied to the atoms of a crystal adiabatically through the perturbation parameter $\lambda(t)$, the CRG contribution to linear order in the velocity is $$\label{hdyn}
\dot{\lambda}\hat{H}^{(\dot{\lambda})}=-\frac{1}{2}\left(\hat{\textbf{A}}\cdot\hat{\textbf{p}}+\hat{\textbf{p}}\cdot\hat{\textbf{A}}\right).$$ Here $\bf A$ is not the vector potential of electromagnetism, but one that emerges when transforming from the static reference frame to the CRG one. For a monochromatic perturbation, it becomes just $\textbf{A}=\dot{\lambda}\textbf{u}=\dot{\lambda}e^{i\textbf{q}\cdot\textbf{r}}$, so $$\begin{split}
\label{hdyn2}
\hat{H}^{(\dot{\lambda}_\beta)}(\textbf{q})&=-e^{i\textbf{q}\cdot\hat{\textbf{r}}}\left(\hat{p}_\beta+\frac{q_\beta}{2}\right)
\end{split}$$ which we recognize as the local current operator \[*cf.* Eq. (\[LocNL4\]) or (\[pkq\])\]. Therefore, the first-order, cell-periodic wavefunctions with respect to this perturbation are $$\label{udyn}
\vert\partial_{\dot{\lambda}_\beta}u^\beta_{n\textbf{k},\textbf{q}}\rangle=\sum_m^{\text{unocc}}\frac{\vert u_{m\textbf{k},\textbf{q}}\rangle\langle u_{m\textbf{k},\textbf{q}}\vert\left(\hat{p}_\beta^{\textbf{k}}+q_\beta/2\right) \vert u_{n\textbf{k}}\rangle}{\epsilon_{m\textbf{k},\textbf{q}}-\epsilon_{n\textbf{k}}},$$ and the (cell averaged) induced polarization from the CRG part of the metric perturbation is $$\label{pdyn}
\begin{split}
\overline{P}^{\textbf{q},\text{ CRG}}_{\alpha,\beta}&=\frac{4}{N_k}\sum_{n\textbf{k}} \langle u_{n\textbf{k}}\vert\hat{\mathcal{J}}^{\textbf{k},\textbf{q}}_\alpha\vert\partial_{\dot{\lambda}_\beta}u^\beta_{n\textbf{k},\textbf{q}}\rangle.
\end{split}$$ The contribution to the FxE coefficient is determined by taking the second derivative of $\overline{P}^{\textbf{q},\text{CRG}}_{\alpha,\beta}$ with respect to $q$: $$\label{pdyn2}
\begin{split}
\overline{P}^{(2,\omega\nu),\text{ CRG}}_{\alpha,\beta}&=\frac{\partial^2\overline{P}^{\textbf{q},\text{CRG}}_{\alpha,\beta}}{\partial q_\omega \partial q_\nu}\Bigg\vert_{\textbf{q}=0}.
\end{split}$$ The CRG contribution is closely related to the diamagnetic susceptibility, $\chi_{\alpha\beta}$. In fact, in the case where only local potentials are present in the Hamiltonian \[so that $\hat{\mathcal{J}}^{\textbf{k},\textbf{q}}_\beta=-(\hat{p}_\beta^{\textbf{k}}+q_\beta/2)$ in Eq. (\[pdyn\])\], Eq. (\[pdyn2\]) has the same form as the expressions for the magnetic susceptibility derived in, e.g., Refs. and \[*cf.* Eq. (11) and Eq. (9) in those works, respectively\].
The magnetic susceptibility relates the magnetization, ${\bf M}$, to the external magnetic field, ${\bf B}$, via $M_\gamma=\chi_{\gamma\lambda}^{\text{mag}}B_\lambda$. This can be rewritten to relate the bound currents to the vector potential, $$J_\alpha=\epsilon^{\alpha\zeta\gamma}\nabla_\zeta\chi_{\gamma\lambda}\epsilon^{\lambda\rho\beta}\nabla_\rho A_\beta,$$ so that $$\overline{P}^{\textbf{q},\text{ CRG}}_{\alpha,\beta}\sim\epsilon^{\alpha\zeta\gamma}q_\zeta\chi_{\gamma\lambda}\epsilon^{\beta\lambda\rho}q_\rho,$$ where we have expressed the spatial derivatives in reciprocal space and canceled the resulting negative sign by permutating the second Levi-Civita symbol. Performing the ${\bf q}$-derivatives in Eq. (\[pdyn2\]) gives $$\label{apchi}
\begin{split}
\overline{P}^{(2,\omega\nu),\text{ CRG}}_{\alpha,\beta}&=\sum_{\gamma\lambda}\left(\epsilon^{\alpha\omega\gamma}\epsilon^{\beta\lambda\nu}+\epsilon^{\alpha\nu\gamma}\epsilon^{\beta\lambda\omega}\right)\chi_{\gamma\lambda}.
\end{split}$$
In the case that nonlocal potentials are present in the Hamiltonian, a calculation of the magnetic susceptibility would involve replacing the “displacement velocity” operator, $-\left(\hat{p}_\beta^{\textbf{k}}+q_\beta/2\right)$, in Eq. (\[udyn\]) with the full electromagnetic current operator from Eq. (\[JqExpand\]), as well as evaluating extra terms originating from the second-order Hamiltonian [@ICL2001; @Pickard2001; @Pickard2003]. This is in contrast to the case of the CRG contribution we would like to calculate, where the only change in the case of nonlocal potentials is replacing $\hat{\mathcal{J}}^{\textbf{k},\textbf{q}}_\alpha$ in Eq. (\[pdyn\]) with the full current operator from Eq. (\[JqExpand\]); Eqs. (\[hdyn2\]) and (\[udyn\]) are unchanged. Therefore, Eq. (\[pdyn\]) does not strictly correspond to the magnetic susceptibility in this case. However, we show in Sec. \[Disc\] that the numerical values are quite similar to previously calculated diamagnetic susceptibilities .
Divergence of the current at the atomic site for the PM path\[appDiv\]
======================================================================
To illustrate the point that nonlocal pseudopotentials allow unphysical transfer of charge between **r** and $\textbf{r}^\prime$, we shall consider the PM[@Pickard2003] definition of the current density, which provides a particularly transparent manifestation of such unphysical behavior. For simplicity, we focus our attention on a single atomic sphere \[so we drop the $\zeta$ index of Eq. (\[HPM\])\], and we set the corresponding nuclear site as the coordinate origin. (There is no approximation here, as the contributions from different sites are spatially separated and additive.) Now suppose we wish to evaluate the nonlocal current density at the point ${\bf r}_0$. We need then to calculate Eq. (\[dHdAq\]) with Eq. (\[HPM\]), using a Dirac delta as a vector potential, $${\bf A}({\bf r}) = A \hat{\bf r}_0 \, \delta ({\bf r - r}_0) = A \hat{\bf r}_0\delta(\hat{\textbf{r}}-\hat{\textbf{r}}_0)\frac{\delta(r-r_0)}{4\pi r^2},$$ where the caret above the position variable denotes a direction (not to be confused with the position operator), and in the second equality we have written the Dirac delta function in spherical coordinates. We choose the vector potential to be oriented along the radial direction, as this is the only allowed component within the PM theory: it is easy to see that a purely tangential ${\bf A}$ field yields a vanishing nonlocal contribution to the current \[see Eq. (\[HPM\])\]. Then, the line integral needed for the first order term in Eq. (\[HA2\]) is $$\begin{split}
\int_{\textbf{s}^\prime\rightarrow 0 \rightarrow\textbf{s}}\textbf{A}\cdot d\ell &=\int_0^1\textbf{A}(\tau\textbf{s})\cdot\textbf{s}d\tau-\int_0^1 \textbf{A}(\tau\textbf{s}^\prime)\cdot\textbf{s}^\prime d\tau
\\
&=A\hat{\textbf{r}}_0\cdot\left[\delta(\hat{\textbf{s}}-\hat{\textbf{r}}_0)\textbf{s}\int^1_0 \frac{\delta(\tau s-r_0)}{4\pi (s\tau)^2}d\tau-\delta(\hat{\textbf{s}}^\prime-\hat{\textbf{r}}_0)\textbf{s}^\prime\int^1_0 \frac{\delta(\tau s^\prime-r_0)}{4\pi (s^\prime\tau)^2}d\tau\right]
\\
&=\frac{A}{4\pi r_0^2}\left[\delta(\hat{\bf s} - \hat{\bf r}_0)\theta(s-r_0)-\delta(\hat{\bf s}^\prime - \hat{\bf r}_0)\theta(s^\prime-r_0)\right],
\end{split}$$ where $\theta$ is the Heaviside step function. Therefore, we can write the current-density operator as (recall that the tangential components vanish, so the current is purely radial) $$\label{radJ}
\langle\textbf{s}\vert\hat{\mathcal{J}}(\textbf{r})\vert\textbf{s}^\prime\rangle=\frac{iV^{\text{nl}}(\textbf{s},\textbf{s}^\prime)}{4\pi r^2}\left[\delta(\hat{\bf s} - \hat{\bf r})\theta(s-r)-\delta(\hat{\bf s}^\prime - \hat{\bf r})\theta(s^\prime-r)\right] .$$ Considering a general time-dependent wavefunction as in Eq. (\[Js\]), the current density is $$\label{JdivPM}
\begin{split}
\textbf{J}^{\text{nl}}(\textbf{r},t)&=\frac{i\hat{\textbf{r}}}{4\pi r^2}\int d^3s \int d^3s^\prime \Psi^*(\textbf{s},t)\Psi(\textbf{s}^\prime,t)V^{\text{nl}}(\textbf{s},\textbf{s}^\prime)\left[\delta(\hat{\bf s} - \hat{\bf r})\theta(s-r)-\delta(\hat{\bf s}^\prime - \hat{\bf r})\theta(s^\prime-r)\right]
\\
&=\frac{i\hat{\textbf{r}}}{r^2}\sum_{lm}\int_r^\infty ds\left[\langle\phi_{lm}\vert\Psi(t)\rangle \Psi^*(s\hat{\textbf{r}},t)\phi_{lm}(s\hat{\textbf{r}})-\langle\Psi(t)\vert\phi_{lm}\rangle\Psi(s\hat{\textbf{r}},t)\phi_{lm}^*(s\hat{\textbf{r}})\right]s^2
\\
&=\frac{\hat{\textbf{r}}}{r^2}\int_r^\infty ds \,\rho^{\text{nl}}(s\hat{\textbf{r}})s^2
\end{split}$$ where we have identified the nonlocal charge $\rho^{\text{nl}}(\textbf{r})=-i\langle\Psi\vert\left[\vert
\textbf{r}\rangle\langle
\textbf{r}\vert,\hat{V}^{\text{nl}}\right]\vert\Psi\rangle$ \[*cf.* Eq. (\[rhoNL\])\]. Note that the upper limit of the integral can be set to $r_{\rm c}$, i.e., the core radius that was used in the generation of the pseudopotential (the nonlocal current density field is strictly contained within a sphere of radius $r_{\rm c}$). This shows that, in the special case of the Pickard-Mauri theory, the nonlocal density does, in fact, provide complete information about the current density.
Unfortunately, a consequence of the above derivations is that the nonlocal current density *diverges* as $|{\bf r - R}|^{-2}$ in the vicinity of an atomic site ${\bf R}$. To see this it suffices to observe that the integral in the above equation tends, for $r \rightarrow 0$, to a direction-dependent constant, $$\label{Cr}
\int_0^{+\infty} s^2 ds \, \rho^{\rm nl}(s \hat{\bf r}) = C(\hat{\bf r}).$$ Thus, the current-density field diverges near the atomic site as $${\bf J}^{\rm nl}({\bf r}) \sim \frac{\hat{\bf r} C(\hat{\bf r})}{r^2} .$$
[^1]: The FxE response of any finite crystal also has an important contribution from the surface, as discussed in Refs. and , and calculated using density-functional theory for SrTiO$_3$ in . In this work, we will exclusively focus on bulk contribution, which poses a more significant challenge for a computational treatment.
[^2]: Note that the definition of Eq. (\[Jkqdef\]) involves a choice of convention in that the exponential factor $e^{i\textbf{q}\cdot\textbf{r}}$ is placed to the right of $\hat{\mathcal{J}}_\alpha(\textbf{q})$ as opposed to the left. Choosing the opposite convention would simply switch the operators between the two terms in Eq. (\[Pq2\]).
[^3]: In contrast to the ICL straight-line path, Eq. (\[HA1\]) using the PM $\textbf{s}^\prime\rightarrow\textbf{R}_\zeta\rightarrow\textbf{s}$ path \[i.e., the phase in Eq. (\[HPM\]) multiplying the entire Hamiltonian instead of just $V^{\text{nl}}(\textbf{s},\textbf{s}^\prime)$\] does *not* recover $\hat{\mathcal{J}}_\alpha^{\text{loc}}$ for local potentials.
[^4]: Recall that in Ref. , this contribution is referred to as the “dynamic” gauge-field term.
[^5]: This can be deduced from Eq.(12) and (13) of Ref. : By placing a single spherical pseudoatom in the gauge origin, all nonlocal contributions vanish by construction as they are multiplied by ${\bf R}$; thus, the applied magnetic field enters the Hamiltonian via the usual substitution ${\bf p} \rightarrow {\bf p} + {\bf A}$. Then, the first order Hamiltonian is the angular momentum operator, which commutes with the ground-state density matrix and yields a vanishing linear response, and the second-order piece picks the quadrupolar moment of the ground-state density, as in the local case.
|
---
author:
- |
Lin-Tian Luh\
Department of Mathematics, Providence University\
Shalu Town, Taichung County\
Taiwan\
Email: ltluh@pu.edu.tw
title: The Mystery of the Shape Parameter IV
---
[**Abstract**]{}. This is the fourth paper of our study of the shape parameter c contained in the famous multiquadrics $(-1)^{\lceil \beta\rceil}(c^{2}+\|x\|^{2})^{\beta},\ \beta>0$, and the inverse multiquadrics $(c^{2}+\|x\|^{2})^{\beta},\ \beta<0$. The theoretical ground is the same as that of [@Lu6]. However we extend the space of interpolated functions to a more general one. This leads to a totally different set of criteria of choosing c.\
\
[**keywords**]{}: radial basis function, multiquadric, shape parameter, interpolation.
Introduction
============
Again, we are going to adopt the radial function $$\begin{aligned}
h(x):=\Gamma(-\frac{\beta}{2})(c^{2}+|x|^{2})^{\frac{\beta}{2}},\ \beta\in R\backslash 2N_{\geq 0},\ c>0\end{aligned}$$ , where $|x|$ is the Euclidean norm of $x$ in $R^{n},\ \Gamma$ is the classical gamma function, and $c,\beta$ are constants. This definition looks more complicated than the ones mentioned in the abstract. However it will simplify the Fourier transform of $h$ and our analysis of some useful results.
In order to make this paper more readable, we review some basic ingredients mentioned in the previous papers, at the cost of wasting a few pages.
For any interpolated function $f$, our interpolating function will be of the form $$\begin{aligned}
s(x):=\sum_{i=1}^{N}c_{i}h(x-x_{i})+p(x)\end{aligned}$$ where $p(x)\in P_{m-1}$, the space of polynomials of degree less than or equal to $m-1$ in $R^{n}, X=\{x_{1},\cdots,x_{N}\}$ is the set of centers(interpolation points). For $m=0,\ P_{m-1}:=\{0\}$. We require that $s(\cdot )$ interpolate $f(\cdot )$ at data points $(x_{1},f(x_{1})),\cdots,(x_{N},f(x_{N}))$. This results in a linear system of the form $$\begin{aligned}
\sum_{i=1}^{N}c_{i}h(x_{j}-x_{i})+\sum_{i=1}^{Q}b_{i}p_{i}(x_{j})=f(x_{j}) & & ,j=1,\cdots,N \nonumber \\
\\
\sum_{i=1}^{N}c_{i}p_{j}(x_{i})=0 & & ,j=1,\cdots,Q \nonumber\end{aligned}$$ to be solved, where $\{p_{1},\cdots,p_{Q}\}$ is a basis of $P_{m-1}$.
This linear system is solvable because $h(x)$ is conditionally positive definite(c.p.d.) of order $m=max\{ \lceil \frac{\beta}{2}\rceil , 0\}$ where $\lceil \frac{\beta}{2}\rceil \}$ denotes the smallest integer greater than or equal to $\frac{\beta}{2}$.
Besides the linear system, another important object is the function space. Each function of the form (1) induces a function space called [**native space**]{} denoted by ${\cal C}_{h,m}(R^{n})$, abbreviated as ${\cal C}_{h,m}$, where $m$ denotes its order of conditional positive definiteness. For each member $f$ of ${\cal C}_{h,m}$ there is a seminorm $\|f\|_{h}$, called the $h$-norm of $f$. The definition and characterization of the native space can be found in [@Lu1], [@Lu2], [@Lu3-1], [@MN1], [@MN2] and [@We]. In this paper all interpolated functions belong to the native space.
Although our interpolated functions are defined in the entire $R^{n}$, interpolation will occur in a simplex. The definition of simplex can be found in [@Fl]. A 1-simplex is a line segment, a 2-simplex is a triangle, and a 3-simplex is a tetrahedron with four vertices.
Let $T_{n}$ be an n-simplex in $R^{n}$ and $v_{i},\ 1\leq i\leq n+1$ be its vertices. Then any point $x\in T_{n}$ can be written as convex combination of the vertices: $$x=\sum_{i=1}^{n+1}c_{i}v_{i},\ \sum_{i=1}^{n+1}c_{i}=1,\ c_{i}\geq 0.$$ The numbers $c_{1},\cdots ,c_{n+1}$ are called the barycentric coordinates of $x$. For any n-simplex $T_{n}$, the [**evenly spaced points**]{} of degree $l$ are those points whose barycentric coordinates are of the form $$(\frac{k_{1}}{l},\frac{k_{2}}{l},\cdots,\frac{k_{n+1}}{l}),\ k_{i}\ nonnegative\ integers\ with\ \sum_{i=1}^{n+1}k_{i}=l.$$ It’s easily seen that the number of evenly spaced points of degree $l$ in $T_{n}$ is exactly $$N=dimP_{l}^{n}=\left( \begin{array}{c}
n+l \\
n
\end{array} \right)$$ where $P_{l}^{n}$ denotes the space of polynomials of degree not exceeding $l$ in n variables. Moreover, such points form a determining set for $P_{l}^{n}$, as is shown in [@Bo].
In this paper the evaluation argument $x$ will be a point in an n-simplex, and the set $X$ of centers will be the evenly spaced points in that n-simplex.
Fundamental Theory
==================
Before introducing the main theorem, we need to define two constants.
Let $n$ and $\beta$ be as in (1). The numbers $\rho$ and $\Delta_{0}$ are defined as follows.
[()]{}[ ]{}
Suppose $\beta <n-3$. Let $s=\lceil \frac{n-\beta -3}{2}\rceil $. Then
[()]{}[ ]{}
if $\beta <0,\ \rho=\frac{3+s}{3}\ and\ \Delta_{0}=\frac{(2+s)(1+s)\cdots 3}{
\rho^{2}};$
if $\beta >0,\ \rho=1+\frac{s}{2\lceil \frac{\beta}{2}\rceil +3} \ and \ \Delta_{0}=\frac{(2m+2+s)(2m+1+s)\cdots (2m+3)}{\rho^{2m+2}}$\
where $ m=\lceil \frac{\beta}{2}\rceil$.
Suppose $n-3\leq \beta <n-1$. Then $\rho=1$ and $\Delta_{0}=1$.
Suppose $\beta \geq n-1$. Let $s=-\lceil \frac{n-\beta -3}{2}\rceil $. Then $$\rho =1\ and \ \Delta_{0}=\frac{1}{(2m+2)(2m+1)\cdots (2m-s+3)} \ where \ m=\lceil \frac{\beta}{2}\rceil.$$
The following theorem is the cornerstone of our theory. We cite it directly from [@Lu3] with a slight modification to make it easier to understand.
Let $h$ be as in (1). For any positive number $b_{0}$, let $C=\max \left\{ \frac{2}{3b_{0}},8\rho\right\}$ and $\delta_{0}=\frac{1}{3C}$. For any n-simplex $Q$ of diameter $r$ satisfying $\frac{1}{3C}\leq r\leq \frac{2}{3C}$(note that $\frac{2}{3C}\leq b_{0}$), if $f\in {\cal C}_{h,m}$, $$\begin{aligned}
|f(x)-s(x)|\leq 2^{\frac{n+\beta-7}{4}}\pi^{\frac{n-1}{4}}\sqrt{n\alpha_{n}}c^{\frac{\beta}{2}-l}\sqrt{\Delta_{0}}\sqrt{3C}\sqrt{\delta}(\lambda')^{\frac{1}{\delta}}\|f\|_{h}\end{aligned}$$ holds for all $x\in Q$ and $0<\delta<\delta_{0}$, where $s(x)$ is defined as in (2) with $x_{1},\cdots ,x_{N}$ the evenly spaced points of degree $l$ in $Q$ satisfying $\frac{1}{3C\delta}\leq l\leq \frac{2}{3C\delta}$. The constant $\alpha_{n}$ denotes the volume of the unit ball in $R^{n}$, and $0<\lambda'<1$ is given by $$\lambda'=\left(\frac{2}{3}\right)^{\frac{1}{3C}}$$ which only in some cases mildly depends on the dimension n.
[**Remark**]{}:(a)Note that the right-hand side of (4) approaches zero as $\delta\rightarrow 0^{+}$. This is the key to understanding Theorem2.2. The number $\delta$ is in spirit equivalent to the well-known fill-distance. Although the centers $x_{1},\cdots,x_{N}$ are not purely scattered, the shape of the simplex is controlled by us. Hence the distribution of the centers is practically quite flexible. (b)In (4) the shape parameter c plays a crucial role and greatly influences the error bound. This provides us with a theoretical ground of choosing the optimal c. However we need further work before presenting useful criteria.
In this paper all interpolated functions belong to a kind of space defined as follows.
For any positive number $\sigma$, $$E_{\sigma}:=\left\{ f\in L^{2}(R^{n}):\ \int |\hat{f}(\xi)|^{2}e^{\frac{|\xi|^{2}}{\sigma}}d\xi<\infty \right\}$$ where $\hat{f}$ denotes the Fourier transform of $f$. For each $f\in E_{\sigma}$, its norm is $$\|f\|_{E_{\sigma}}:=\left\{ \int|\hat{f}(\xi)|^{2}e^{\frac{|\xi|^{2}}{\sigma}}d\xi\right\}^{1/2}$$.
The following lemma is cited from [@Lu5].
Let $h$ be as in (1). For any $\sigma>0$, if $\beta<0$, $|n+\beta|\geq 1$ and $n+\beta+1\geq 0$, then $E_{\sigma}\subseteq {\cal C}_{h,m}(R^{n})$ and for any $f\in E_{\sigma}$, the seminorm $\|f\|_{h}$ of $f$ satisfies $$\|f\|_{h}\leq 2^{-n-\frac{1+\beta}{4}}\pi^{-n-\frac{1}{4}}c^{\frac{1-n-\beta}{4}}\left\{ (\xi^{*})^{\frac{n+\beta+1}{2}}e^{c\xi^{*}-\frac{(\xi^{*})^{2}}{\sigma}}\right\}^{1/2}\|f\|_{E_{\sigma}}$$ where $$\xi^{*}:=\frac{c\sigma+\sqrt{c^{2}\sigma^{2}+4\sigma(n+\beta+1)}}{4}$$.
Under the conditions of Theorem2.2, if $f\in E_{\sigma},\ \beta<0,\ |n+\beta|\geq 1$ and $n+\beta+1\geq 0$, (4) can be transformed into $$\begin{aligned}
|f(x)-s(x)|\leq 2^{-\frac{3n}{4}-2}\pi^{-\frac{3}{4}n-\frac{1}{2}}\sqrt{n\alpha_{n}}\sqrt{\Delta_{0}}\sqrt{3C}c^{\frac{\beta-n+1-4l}{4}}\left\{(\xi^{*})^{\frac{n+\beta+1}{2}}e^{c\xi^{*}-\frac{(\xi^{*})^{2}}{\sigma}}\right\}^{1/2}\sqrt{\delta}(\lambda')^{\frac{1}{\delta}}\|f\|_{E_{\sigma}}\end{aligned}$$ where $$\xi^{*}:=\frac{c\sigma+\sqrt{c^{2}\sigma^{2}+4\sigma(n+\beta+1)}}{4}$$.
[**Proof**]{}. This is an immediate result of Theorem2.2 and Lemma2.4. $\sharp$\
\
Note that Corollary2.5 covers the very useful case $\beta=-1,\ n\geq 2$. However the case $\beta=-1,\ n=1$ is excluded. For this case we need a different approach.
Let $\sigma>0,\ \beta=-1$ and $n=1$. For any $f\in E_{\sigma}$, $$\|f\|_{h}\leq 2^{-(n+\frac{1}{4})}\pi^{-1}\left\{ \frac{1}{ln2}+2\sqrt{3}M(c)\right\}^{1/2}\|f\|_{E_{\sigma}}$$ where $M(c):=e^{1-\frac{1}{c^{2}\sigma}}$ if $c\leq \frac{2}{\sqrt{3\sigma}}$ and $M(c):=g(\frac{c\sigma+\sqrt{c^{2}\sigma^{2}+4\sigma}}{4})$ if $c>\frac{2}{\sqrt{3\sigma}}$, where $g(\xi):=\sqrt{c\xi}e^{c\xi-\frac{\xi^{2}}{\sigma}}$.
[**Proof**]{}. This is just Theorem2.5 of [@Lu5]. $\sharp$
Let $\sigma>0,\ \beta=-1$ and $n=1$. Under the conditions of Theorem2.2, if $f\in E_{\sigma}$, (4) can be transformed into $$\begin{aligned}
|f(x)-s(x)|\leq 2^{\frac{\beta-3n}{4}-2}\pi^{\frac{n-5}{4}}\sqrt{n\alpha_{n}}\sqrt{\Delta_{0}}\sqrt{3C}c^{\frac{\beta}{2}-l}\left\{ \frac{1}{ln2}+2\sqrt{3}M(c)\right\}^{1/2}\sqrt{\delta}(\lambda')^{\frac{1}{\delta}}\|f\|_{E_{\sigma}}\end{aligned}$$ where $M(c)$ is defined as in Lemma2.6.
[**Proof**]{}. This is an immediate result of Theorem2.2 and Lemma2.6. $\sharp$\
\
Now we have dealt with the most useful cases for $\beta<0$. The next step is to treat $\beta>0$.
Let $\sigma>0,\ \beta>0$ and $n\geq 1$. For any $f\in E_{\sigma}$, $$\|f\|_{h}\leq d_{0}c^{\frac{1-\beta-n}{4}}\left\{ \frac{(\xi^{*})^{\frac{1+\beta+n}{2}}e^{c\xi^{*}}}{e^{\frac{(\xi^{*})^{2}}{\sigma}}}\right\} ^{1/2}\|f\|_{E_{\sigma}}$$ where $\xi^{*}=\frac{c\sigma+\sqrt{c^{2}\sigma^{2}+4\sigma(1+\beta+n)}}{4}$ and $d_{0}$ is a constant depending on $n,\ \beta$ only.
[**Proof**]{}. This is just Theorem2.8 of [@Lu5]. $\sharp$
Let $\sigma>0,\ \beta>0$ and $n\geq 1$. If $f\in E_{\sigma}$, (4) can be transformed into $$\begin{aligned}
|f(x)-s(x)|\leq 2^{\frac{n+\beta-7}{4}}\pi^{\frac{n-1}{4}}\sqrt{n\alpha_{n}}\sqrt{\Delta_{0}}\sqrt{3C}d_{0}c^{\frac{1+\beta-n-4l}{4}}\left\{ \frac{(\xi^{*})^{\frac{1+\beta+n}{2}}e^{c\xi^{*}}}{e^{\frac{(\xi^{*})^{2}}{\sigma}}}\right\} ^{1/2}\sqrt{\delta}(\lambda')^{\frac{1}{\delta}}\|f\|_{E_{\sigma}}\end{aligned}$$ where $d_{0},\ \xi^{*}$ are as in Lemma2.8.
[**Proof**]{}. This is an immediate result of Theorem2.2 and Lemma2.8. $\sharp$
Criteria of Choosing c
======================
Note that in (5),(6) and (7), there is a main function of c. As in [@Lu5], let’s call this function the MN function, denoted by $MN(c)$, and its graph the MN curve. The optimal choice of c is then the number minimizing $MN(c)$. However, unlike [@Lu5], the range of c is the entire interval $(0,\infty)$, rather than a proper subset of $(0,\infty)$.
We now begin our criteria.\
\
[**Case1**]{}. Let $f\in E_{\sigma}$ and $h$ be as in (1). Under the conditions of Theorem2.2, for any fixed $\delta$ satisfying $0<\delta<\delta_{0}$, the optimal value of c in $(0,\infty)$ is the number minimizing $$MN(c):=c^{\frac{\beta-n+1-4l}{4}}\left\{(\xi^{*})^{\frac{n+\beta+1}{2}}e^{c\xi^{*}-\frac{(\xi^{*})^{2}}{\sigma}}\right\}^{1/2}$$ where $$\xi^{*}=\frac{c\sigma+\sqrt{c^{2}\sigma^{2}+4\sigma(n+\beta+1)}}{4}$$.\
\
[**Reason**]{}: This is a direct consequence of (5). $\sharp$\
\
[**Remark**]{}:(a)It’s easily seen that $MN(c)\rightarrow\infty$ as $c\rightarrow\infty$. Also, if $n+\beta+1>0,\ MN(c)\rightarrow\infty$ as $c\rightarrow 0^{+}$. (b)Case1 covers the frequently seen case $\beta=-1,\ n\geq 2$. (c)The number c minimizing $MN(c)$ can be easily found by Mathematica or Matlab.\
\
[**Numerical Results**]{}:\
\
[**Case2**]{}. Let $f\in E_{\sigma}$ and $h$ be as in (1). Under the conditions of Theorem2.2, for any fixed $\delta$ satisfying $0<\delta<\delta_{0}$, the optimal value of c in $(0,\infty)$ is the number minimizing $$MN(c):=c^{\frac{\beta}{2}-l}\left\{\frac{1}{ln2}+2\sqrt{3}M(c)\right\}^{1/2}$$ where $$M(c):=\left\{ \begin{array}{ll}
e^{1-\frac{1}{c^{2}\sigma}} & \mbox{if $0<c\leq \frac{2}{\sqrt{3\sigma}}$,} \\
g(\frac{c\sigma+\sqrt{c^{2}\sigma^{2}+4\sigma}}{4}) & \mbox{if $\frac{2}{\sqrt{3\sigma}}<c$}
\end{array} \right.$$, $g$ being defined by $g(\xi):=\sqrt{c\xi}e^{c\xi-\frac{\xi^{2}}{\sigma}}$.\
\
[**Reason**]{}: This is a direct result of (6). $\sharp$\
\
[**Remark**]{}: Note that $MN(c)\rightarrow \infty$ both as $c\rightarrow \infty$ and $c\rightarrow 0^{+}$. Now let’s see some numerical examples.
[**Case3**]{}. Let $f\in E_{\sigma}$ and $h$ be as in (1). Under the conditions of Theorem2.2, for any fixed $\delta$ satisfying $0<\delta<\delta_{0}$, the optimal value of c in $(0,\infty)$ is the number minimizing $$MN(c):=c^{\frac{1+\beta-n-4l}{4}}\left\{\frac{(\xi^{*})^{\frac{1+\beta+n}{2}}e^{c\xi^{*}}}{e^{\frac{(\xi^{*})^{2}}{\sigma}}}\right\}^{1/2}$$ , where $$\xi^{*}=\frac{c\sigma+\sqrt{c^{2}\sigma^{2}+4\sigma(1+\beta+n)}}{4}$$.\
\
[**Reason**]{}: This follows from (7). $\sharp$\
\
[**Remark**]{}: By observing that $$c\xi^{*}-\frac{(\xi^{*})^{2}}{\sigma}=\frac{1}{16}\left[ 2c^{2}\sigma+2c\sqrt{c^{2}\sigma^{2}+4\sigma(n+\beta+1)}-(4n+\beta+1)\right]$$ , we can easily obtain useful results as follows. (a)If $1+\beta-n-4l>0$, $\lim_{c\rightarrow0^{+}}MN(c)=0$. (b)If $1+\beta-n-4l<0,\ \lim_{c\rightarrow 0^{+}}MN(c)=\infty$. (c)If $1+\beta-n-4l=0,\ \lim_{c\rightarrow 0^{+}}MN(c)$ is a finite positive number. (d)$\lim_{c\rightarrow \infty}MN(c)=\infty$.\
\
[**Numerical Results**]{}: For simplicity, we offer results for $n=1$ only. In fact for $n\geq 1$ similar results can be presented without slight difficulty.
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|
---
abstract: 'We show via tensor network methods that the Harper-Hofstadter Hamiltonian for hard-core bosons on a square geometry supports a topological phase realizing the $\nu=1/2$ fractional quantum Hall effect on the lattice. We address the robustness of the ground state degeneracy and of the energy gap, measure the many-body Chern number, and characterize the system using Green functions, showing that they decay algebraically at the edges of open geometries, indicating the presence of gapless edge modes. Moreover, we estimate the topological entanglement entropy by taking a combination of lattice bipartitions that reproduces the topological structure of the original proposals by Kitaev and Preskill, and Levin and Wen. The numerical results show that the topological contribution is compatible with the expected value $\gamma = 1/2$. Our results provide extensive evidence that FQH states are within reach of state-of-the-art cold atom experiments.'
author:
- 'M. Gerster'
- 'M. Rizzi'
- 'P. Silvi'
- 'M. Dalmonte'
- 'S. Montangero'
bibliography:
- 'references.bib'
title: Fractional quantum Hall effect in the interacting Hofstadter model via tensor networks
---
Introduction
============
The Harper-Hofstadter model [@hofstadter1976energy] plays an archetypical role in the current understanding of topological quantum matter on a lattice. It encompasses the basic coupling between particles and a background magnetic field, and supports topological bands with finite Chern number for a broad range of fluxes and tunneling rates [@bernevig2013topological]. Those remarkable properties have motivated proposals [@Dalibard2011; @*Jaksch2003; @*Goldman2014] and recent experiments in both solid-state [@dean2013hofstadter] and cold atom systems [@aidelsburger2013realization; @miyake2013realizing; @kennedy2015observation; @aidelsburger2015measuring; @Mancini2015; @*Stuhl2015], which have extensively investigated its non-interacting limit, including the observation of its fractal spectrum [@dean2013hofstadter] and the measurement of a finite Chern number [@aidelsburger2015measuring] in some of its bands. More recently, experiments using bosonic atoms in optical lattices have shown impressive capabilities to approach the [*strongly interacting*]{} regime [@kennedy2015observation], with the ultimate goal of stabilizing lattice analogues of fractional quantum Hall (FQH) states [@bernevig2013topological; @fradkin2013field]. However, despite promising small system results based on exact diagonalization [@Sorensen2005; @Hafezi2007; @moller2009composite; @sterdyniak2012particle], in the large flux regime available to experiments – when the magnetic length is of order of the lattice spacing – theoretical evidence of such states at large scales has been lacking, especially regarding smoking guns of topological order – gapless edge modes, entanglement properties, and many-body Chern numbers.
![\[fig:sketch\]Sketch of the system: $N$ hard-core bosons (red balls) are hopping on a $L\times L$ lattice, perpendicular to an external magnetic field. The magnetic field gives rise to a flux $\phi$ through each plaquette, which in Landau gauge leads to the indicated phase factors in the hopping amplitudes $t$. The fluxes in the topmost row and rightmost column (faded) are only present in case of periodic boundary conditions (PBC). For PBC, additional phase twists $\theta_x$ and $\theta_y$ can be introduced (see text), allowing for the definition of topological invariants.](fig_sketch){width="0.75\columnwidth"}
In this work, we study the strongly-interacting bosonic Hofstadter model, and show that it supports a FQH ground state (GS) akin to the $\nu=1/2$ Laughlin state in the continuum [@fradkin2013field; @kalmeyer1987equivalence; @regnault2011fractional]. Our analysis is based on a combination of diagnostics, including GS degeneracies on different topologies, the measurement of the many-body Chern number, Green functions’ decay at the edge and in the bulk, and a direct measurement of the topological entanglement entropy. The corresponding results serve as a quantitative guideline to address the stability of such phases against temperature: crucially, we show how the spectral gap of the bulk excitations is of the order of $10\%$ of the tunneling rates, showing how FQH states are within reach for temperatures available to current experiments.
The enabling tool of our analysis are numerical simulations based on the tree-tensor network ansatz [@Tagliacozzo2009; @Murg2010; @*Nakatani2013; @Gerster2014]. This class of tailored variational wave functions extends the matrix-product state (MPS) ansatz – the tensor network class at the heart of the density-matrix renormalization group (DMRG) [@White1992; @Schollwock2011]. To obtain the presented results on square lattices with sizes of up to $32 \times 32$ sites, we exploit the reduced scaling of computational costs of the loopless geometry of TTNs, a characteristic not shared by other network structures such as PEPS and MERA [@Verstraete_2006; @*Vidal_2007]. Additionally, differently from typical DMRG approaches, TTN warrant direct access to the reduced density matrix of a variety of lattice bipartitions. In the following, we show how this feature enables the direct evaluation of the topological entanglement entropy using a specific combination of various partitions, in parallel to the original proposals of Refs. , which are instead not immediately applicable to DMRG studies.
The paper is structured as follows: First, we define the model (Sec. \[sec:model\]) and describe the employed numerical method (Sec. \[sec:ttn\]). Then, we proceed to present the numerical evidence for a FQH GS in Sec. \[sec:evidence\], based on the low-energy spectrum (Sec. \[sec:en-spectra\]), the many-body Chern number (Sec. \[sec:mbcn\]), the correlation functions (Sec. \[sec:corr-func\]), and the topological entanglement entropy (Sec. \[sec:tee\]). In Sec. \[sec:transition\] we analyze the robustness of the FQH state upon introducing a superlattice potential. Finally, we conclude our work in Sec. \[sec:conclusion\].
Model Hamiltonian {#sec:model}
=================
We study spinless bosonic particles hopping on a $L \times L$ square lattice under the influence of an external magnetic field, as illustrated in Fig. \[fig:sketch\]. In the Landau gauge, the system is described by the following Hamiltonian [@Sorensen2005] $$\begin{aligned}
H &=&U \sum_{x,y} n_{x,y}(n_{x,y} - 1) -t \sum_{x,y} \left\{ a^\dagger_{x+1,y} a_{x,y} \, \mathrm{e}^{-i\, 2\pi \delta_{xL} \theta_x} +\right.\nonumber\\
&+ & \left.a^\dagger_{x,y+1} a_{x,y} \, \mathrm{e}^{i\, 2\pi (\phi x - \delta_{yL} \theta_y)} + \mathrm{h.c.}\right\}
\label{eq:ham}\end{aligned}$$ of bosonic particles, $[a_{x,y}, a_{x',y'}^{\dagger}] = \delta_{xx'} \delta_{yy'}$, $n_{} = a^{\dagger} a$. Here, $\phi$ is the magnetic flux through each plaquette (resulting in a magnetic filling factor $\nu=N/(\phi L^2)$, with $N$ the number of bosons in the system) and $\theta_x$, $\theta_y$ implement the twists in the boundary condition [@Hafezi2007]. In the dilute limit (small densities and small fluxes) the lattice physics approaches the one of the continuum [^1]. However, in the large flux limit, available in cold gases experiments, the phase diagram is not set. On small systems, it has been shown by exact diagonalization (ED) that the GS of the model described by Eq. at filling factor $\nu=1/q$ (where $q$ is an even integer) is compatible with a lattice analogue of the (bosonic) Laughlin wave function [@Hafezi2007; @Laughlin1983; @moller2009composite; @sterdyniak2012particle; @Hugel:2016aa], exhibiting topological GS degeneracy and a non-zero Chern number [@Niu1985; @*Tao1986]. However, the overlap with the exact Laughlin wave functions rapidly degrades with system size already for small systems of 6 particles. From a complementary viewpoint, the ladder version of the Hofstadter model has also been shown to share similarities with FQH states [@Budich:2017aa; @Strinati:2016aa; @Petrescu:2016aa; @Haller2017]. Very recently, iDMRG results on cylinders have shown strong signatures of integer quantum Hall states, and have reported fractional current quantization in regimes different from the one we consider here [@He2017]. Throughout, we focus on the strongly interacting case $U\rightarrow\infty$ (hard-core bosons) with flux values $\phi=1/8$ and $1/16$ respectively, which correspond to flux setups that are experimentally available. Finally, we fix the energy scale by setting $t=1$.
![\[fig:ttn\]Binary tree-tensor network ansatz for a $L\times L$ lattice. The blue dots are the physical sites with local dimension $d$ ($d=2$ for hard-core bosons). Each tensor groups two sites to one virtual site (gray dots), leading to a hierarchical tree structure. In order to capture the 2D lattice geometry, the grouping is performed in $x$- and $y$-direction, alternating from level to level. The dimension of the virtual sites in the level (counting from below) is $\min(d^{2^l}, m)$, where $m$ is the bond dimension of the TTN. The additional cyan link at the top left tensor has dimension one and selects the global particle number $N$ [@Singh2011].](fig_ttn.pdf){width="0.8\columnwidth"}
{width="\textwidth"}
Tree-tensor network ansatz {#sec:ttn}
==========================
We employ a tree-tensor network (TTN) ansatz [@Shi2006] for the GS and the two lowest excited states to verify the properties discussed above. The specific binary TTN used in this work is illustrated in Fig. \[fig:ttn\], where the standard graphical notation for tensor networks (TN) is employed [@Orus2014; @Silvi2017]: tensors are drawn as cubes, with attached lines symbolizing tensor indices (links). Links that are shared by two tensors are contracted, which in TN language means that over their corresponding mutual indices is to be summed. The dimension of each link in the TTN is upper bounded by a constant $m$ (bond dimension), which serves as the refinement parameter of the ansatz: the larger $m$, the more accurate the true many-body state can be approximated. In a binary TTN each tensor has at most three links; therefore, the scaling of the computational resources with the bond dimension is moderate in algorithms using this class of TN states [@Gerster2014]. Furthermore, we exploit particle number conservation by restricting the ansatz to the $N$ particle symmetry sector [@Singh2011].
While it is known that a 2D-TTN is not compatible with the area law for the entanglement entropy [@Tagliacozzo2009; @Ferris2013], it possesses several beneficial features which make it a promising tool for the study of intermediate system sizes: (a) The existence of a numerically stable search algorithm for eigenstates [@Murg2010; @*Nakatani2013; @Gerster2014; @Silvi2017]; (b) a low-order polynomial scaling $\mathcal{O}(m^4 L^2)$ of the computational cost; (c) easy interchange of various boundary conditions (open, periodic, twisted); (d) access to the entanglement entropy for bipartition shapes that enable the determination of the topological entanglement entropy (TEE) [@Kitaev2006; @*Levin2006]. In what follows, we will exploit these properties to gather a number of numerical pieces of evidence supporting a FQH GS of the model Eq. in the case of filling $\nu=1/2$.
Numerical evidence for FQH ground state {#sec:evidence}
=======================================
Low-energy spectra {#sec:en-spectra}
------------------
As first evidence we verify the GS degeneracy, intimately connected to the topological order of the system [@Wen1990; @fradkin2013field]. On a torus geometry the GS degeneracy at $\nu=1/2$ filling is expected to be twofold (independent of the twist angles $\theta_x$, $\theta_y$), while the first excited state is to be separated by a bulk gap. We determined the three lowest-energy eigenstates, reported in Fig. \[fig:chernnum\]a: We clearly observe a finite energy gap $E_2-E_0\approx 0.1t$ which is typically more than two orders of magnitude larger than the energy difference $E_1-E_0$ between the two states manifold (multiplet) in the GS for the system sizes considered here. Conversely, we verified that for open boundary conditions (OBC) the quasi-degeneracy is removed and we observe that $E_2-E_1 \approx E_1-E_0$.
Many-body Chern number {#sec:mbcn}
----------------------
As second evidence we determine the many-body Chern number (MBCN) [@Niu1985; @*Tao1986], which is a direct signature of topological order. For the numerical calculation of the MBCN we follow the prescription put forward by Hatsugai [@Hatsugai_2004; @*Hatsugai_2005], which requires the knowledge of the GS manifold on a 2D grid of twist angles $(\theta_x, \theta_y) \in [0,1]\times [0,1]$ (note that this grid also has a torus geometry). From a practical point of view, this method relies on the calculation of overlaps, a task which can be easily accomplished with TN states: More specifically, we need to choose two reference multiplets $\Phi$, $\Phi^\prime$ which have to be non-parallel but otherwise can be arbitrary. (Commonly, one picks two GS multiplets at twist angles far from each other [@Hafezi2007a].) This corresponds to two different gauge references, which can be used to define two scalar fields: $\Lambda_{\Phi^{(\prime)}}=\det \langle \Phi^{(\prime)}_j | P(\theta_x,\theta_y) | \Phi^{(\prime)}_k \rangle$, where the determinant runs over the indexes $j,k \in \{0,1\}$ of the GS multiplet and $$\begin{aligned}
P(\theta_x,\theta_y) &=& | \Psi_0(\theta_x,\theta_y) \rangle \langle \Psi_0 (\theta_x,\theta_y) | +\\
&+& | \Psi_1(\theta_x,\theta_y) \rangle \langle \Psi_1 (\theta_x,\theta_y) |\nonumber\end{aligned}$$ is the projector on the GS manifold at $(\theta_x,\theta_y)$. If the gauge has been fixed appropriately, $\Lambda_\Phi$ is well-defined (i.e. non-vanishing) where $\Lambda_{\Phi^\prime}$ is not and vice versa, thus forming two complementary regions on the boundary condition torus (an example is shown in Fig. \[fig:chernnum\]b-c). Finally, the MBCN is given by the number of branch vortices in the argument field $$\Omega_{\Phi\rightarrow\Phi^\prime} = \arg\left( \det \langle \Phi^\prime_j | P(\theta_x,\theta_y) | \Phi_k \rangle \right)$$ to be counted (with sign) in any of these two regions [@Hatsugai_2004; @*Hatsugai_2005; @Hafezi2007a]. We performed this procedure using TTN states; the result is shown in Fig. \[fig:chernnum\]d-e. We obtain a MBCN of 1, clearly demonstrating the topological order in the GS of the model. Combined with the two-fold degeneracy discussed above, this shows that the system has a Chern number per state $C = 1/2$, as expected for the $\nu=1/2$ Laughlin state.
 Green functions along the $y$-direction for PBC (a1) and OBC (a2), for $L=16$, $N=8$, $\phi=1/16$. In the PBC case the decay is exponential, irrespective of the choice of $x$. In contrast, for OBC, the decay at the edges ($x\approx 1$ or $y\approx L$) is much slower, signaling the presence of edge modes with algebraic correlations. (b) Current in the $x$-direction. (b1): comparison between PBC (no current) and OBC (currents at the edges). (b2): OBC current for $L=32$, $\phi=1/32$, and (b3): for $\phi=1/8$, $L=16$, each with two different bond dimensions $m$. All panels use the same color code for $x$, which is illustrated in (c). The cartoon in (c) shows the edge current (red arrow) and demonstrates why $\mathcal{I}_x$ vanishes in the bulk (dark-shaded background), while at the edges (bright background) it is nonzero whenever $y \approx 1$ or $y \approx L$. Moreover it is obvious that the sign of $\mathcal{I}_x$ at $y \approx 1$ is opposite to the sign of $\mathcal{I}_x$ at $y \approx L$.](fig_correlations){width="\columnwidth"}
Correlation functions and edge modes {#sec:corr-func}
------------------------------------
Indirect evidence of a fractional quantum Hall state can also be gathered by monitoring the behavior of the Green function in the system $\mathcal{G}(x,x';y,y') = \langle a^\dagger_{x,y}a_{x',y'}\rangle$. The Green function in periodic systems contains information about bulk excitations: in our case, it is expected to decay exponentially as a function of distance. In contrast, for open boundaries the Green function is expected to reveal the existence of gapless edge modes. Indeed, in the case of $\nu=1/2$, the phase field operator of the corresponding chiral Luttinger liquid edge mode is expected to have considerable overlap with the creation and annihilation operators on the lattice. In Fig. \[fig:corr\]a, we plot the normalized $\mathcal{G}(x,x;1,y)$ as a function of $y$ both for PBC and OBC. In the case of PBC, the results clearly show that $\mathcal{G}$ decays exponentially as a function of distance, with a correlation length of approximately 2 sites - and thus, considerably smaller than our system sizes. Since for OBC translational invariance is broken, we consider different values of $x$ (marked by different colors as also illustrated in Fig. \[fig:corr\]c). Here, two distinct regimes are visible: along the edge (here for $x\lesssim 4$) the correlation decays as a power law, indicating a gapless mode localized close to the boundary. In sharp contrast, far from the edges ($x \simeq L/2$) the decay becomes exponential, consistent with the PBC results. Our results on the correlation functions thus strongly confirm a finite bulk gap, coexisting with gapless edge modes.
An alternative route to detect chiral edge modes, likely more accessible in experiments, is given by the particle current density. Specifically, we consider here the current in the $x$-direction $$\mathcal{I}_x(x,y) = i\, \langle a^\dagger_{x+1,y} a_{x,y} - a_{x+1,y} a^\dag_{x,y} \rangle,$$ again both for PBC and OBC. In Fig. \[fig:corr\]b we show the normalized $\mathcal{I}_x(x,y)$ as a function of $y$ and for different values of $x$ (again with the same color code). The results are in strong agreement with the ones obtained from the Green function analysis: For PBC, where there is only bulk, the current vanishes throughout the system, while for OBC we observe strongly enhanced currents along the edges, i.e. at $y\approx1$ or $y\approx L$. In Fig. \[fig:corr\]c we provide an illustration of the edge current present in an OBC lattice, giving an intuitive (albeit over-simplified) explanation for the behavior of the currents plotted in Fig. \[fig:corr\]b.
 TEE $\gamma$ obtained from an extrapolation in $m$: Lines are linear fits through data points with $\ell>10$, for $L=16$, $N=16$, $\phi=1/8$. The resulting $\gamma$ is compatible with $1/2$. (b) Arrangement of square partitions for the extraction of $\gamma$ [@Kitaev2006; @*Levin2006] and resulting TEE (c) as a function of the block size $a$. For large enough $a$, the data is compatible with $\gamma=1/2$.](fig_entropy){width="0.9\columnwidth"}
Topological entanglement entropy {#sec:tee}
--------------------------------
An unambiguous indicator for topological order is the topological entanglement entropy (TEE) $\gamma$, as introduced in Refs. . A finite value of $\gamma$ signals the presence of anyonic excitations above the GS manifold - remarkably, without having to directly access excited states. In particular, $\gamma$ is directly related to the quantum dimensions of the excitations. Extracting the TEE in numerical simulations is however challenging. System sizes available to exact diagonalization are typically too small to measure $\gamma$ in a meaningful fashion. DMRG simulations can reach considerable system sizes: in that context, one tries to extract the TEE $\gamma$ as a correction to the perimeter-law scaling of the entanglement entropy: $S(\ell) = c \, \ell - \gamma$, where $S(\ell)$ is the von Neumann entropy $S = - \mathrm{tr}[ \rho \log_2 \rho]$ for a spatial partition of perimeter $\ell$. In Fig. \[fig:entropy\]a, we show our results for $S(\ell)$ for PBC with $L=16$. In order to avoid strong finite size effects, we excluded from our fits the data with $\ell< 10$. After extrapolation to $m\rightarrow\infty$, we get $\gamma=0.47\pm0.19$, which is close to the value obtained with our maximum bond dimension. Despite the large error bar, mostly due to the fact that we have only access to few points in $\ell$, our result show that the system has $\gamma>0$, and the result is compatible with the expected TEE for a Laughlin state with $\nu=1/q$, $\gamma=\log_2 \sqrt{q}$. While this procedure has been well tested in several models [@depenbrock2012nature; @Jiang_2012], it is desirable to apply a method for obtaining the TEE which follows directly the original prescription: Such technique operates on regions with different shapes in order to exclude spurious effects. Within our TN ansatz this is possible considering the geometry in Fig. \[fig:entropy\]b, where it can be shown that the following relation holds: $$\begin{aligned}
-\gamma &=& S_A + S_B + S_C + S_D + \\
&-& S_{AB} - S_{BC} - S_{CD} - S_{DA} + S_{ABCD} \; . \nonumber\end{aligned}$$ The basic building block of this procedure are square partitions of size $a\times a$. The results of the corresponding simulations, extrapolated in $m\rightarrow\infty$, are shown in Fig. \[fig:entropy\]c: for $a=4$ the results indicate a finite TEE of $\gamma\simeq 0.5$, again with rather large error bars (owing to the uncertainty in the $m$ extrapolation), but otherwise in good agreement with the expected behavior for $\nu=1/2$ FQH states. We note that for $a=1,2$ the results are far from this prediction, as expected, since for those values the partition size is smaller than the correlation length $\xi$ as computed from the Green function decay.
![\[fig:suplatt\] Left column: on-site occupations $\langle n_{x,y} \rangle$ of the lattice sites for different values of $\mu$, showing how the particles gradually localize at the energetically favored sites of the superlattice. Right column: plots of $\Gamma_\Phi$ for different values of $\mu$. For $\mu \gtrsim -1.0$ there exist regions (visible as black areas enclosed by cyan lines) where $\Gamma_\Phi$ vanishes, indicating topologically non-trivial character. System parameters are $L=16$, $N=8$, $\phi=1/16$.](fig_suplatt){width="1\columnwidth"}
Phase transition from FQH state to trivial insulator {#sec:transition}
====================================================
Our diagnostics also allows for the detection of possible phase transitions. Here we demonstrate this by investigating the robustness of the FQH state upon introducing a superlattice potential into the model from Eq. , which, in turn, can be engineered in cold atoms experiments [@Sebby-Strabley2006; @*Foelling2007]. We consider the Hamiltonian $$\tilde{H} = H + \mu \sum_{(x,y) \in \, \mathrm{sup.latt.}} n_{x,y} \; ,
\label{eq:ham_suplatt}$$ where $\mu \leq 0$ denotes the depth of the superlattice potential. We focus on the case where the number of sites of the superlattice is equal to the number of bosons $N$ in the system. For sufficiently large $|\mu|$, the GS of $\tilde{H}$ is a product state with the particles pinned at the superlattice minima (see Fig. \[fig:suplatt\]) which is, by construction, topologically trivial. The transition between the two qualitatively different GSs occurs at some critical potential depth $\mu_c < 0$. In order to detect this transition we use the energy spectra on the boundary condition torus, and the MBCN, as shown in Figs. \[fig:mu\_gaps\] and \[fig:mu\_mbcn\]. With increasing depth of the potential, we observe how the GS quasi-degeneracy is removed (see Fig. \[fig:mu\_gaps\]).
 Three lowest eigenenergies of $\tilde{H}$ as a function of $\mu$, at $\theta_x=\theta_y=0$. (b) Low-energy spectra on the whole boundary condition torus for $\mu=-1.0$ and $\mu=-2.0$. System parameters are $L=16$, $N=8$, $\phi=1/16$.](fig_mu_gaps){width="0.9\columnwidth"}
As long as the GS multiplet is still quasi-degenerate, i.e. separated from the first excited state ($E_2 - E_1 \gtrsim E_1 - E_0$) on the whole torus of twist angles, we measure a MBCN of one (see Fig. \[fig:mu\_mbcn\]).
 MBCN as a function of $\mu$, displaying a transition at $\mu_c \approx -1$. (b) Determination of MBCN for $\mu=-1.0$, resulting in $\mathrm{MBCN}=1$. System parameters are $L=16$, $N=8$, $\phi=1/16$.](fig_mu_mbcn){width="0.95\columnwidth"}
At the critical point $-1.3 \lesssim \mu_c \lesssim -1.0$ this condition is no longer met and the MBCN vanishes, signaling a disappearance of the topological order. The topologically trivial character of such a GS can be evidenced by considering the quantity $$\Gamma_\Phi(\theta_x, \theta_y) = \langle \Phi | \Psi_0(\theta_x, \theta_y) \rangle \langle \Psi_0(\theta_x, \theta_y) | \Phi \rangle \; ,$$ where $\Phi$ represents a gauge reference, formed by a single GS at an (arbitrary) reference point $(\theta^{[r]}_x, \theta^{[r]}_y)$. If the GS of the system is in fact topologically trivial, $\Gamma_\Phi$ is well-defined (i.e. non-vanishing) on the whole boundary condition torus. On the contrary, this is not the case if the GS has non-trivial topological character: in that case there is a finite region on the torus where $\Gamma_\Phi$ vanishes. Whenever this happens, the GS is quasi-degenerate and Hatsugai’s method for determining the MBCN (as described in Sec. \[sec:mbcn\]) is applicable. In Fig. \[fig:suplatt\] we show $\Gamma_\Phi$ for different values of $\mu$, demonstrating how it gradually becomes well-defined on the whole boundary condition torus as the depth of the superlattice potential is increased.
Conclusions and outlook {#sec:conclusion}
=======================
We have presented extensive numerical evidence supporting the existence of a fractional quantum Hall phase in the Harper-Hofstadter Hamiltonian for hard-core bosons on a square lattice. Our analysis considered a wide range of independent diagnostics, including spectral properties, the many-body Chern number, correlation functions, currents, and entanglement entropies, and shows how TTN algorithms provide a flexible tool to address the interplay of gauge fields and interactions in two-dimensional systems. The results indicate that the correlation length is typically of the order of a few lattice sites, with corresponding gaps of order $0.1t$: these signatures point toward the fact that lattice fractional quantum Hall states can be realized in present cold atom experiments, as well as potentially in cavity array experiments [@Kapit:2014aa; @Owens:2017aa]. The temperatures and sizes required match those already available in experiments, and for which adiabatic state preparation protocols have very recently been shown to be applicable [@He2017; @Motruk:2017aa].
We thank M. Burrello and A. Sterdyniak for a careful reading of the manuscript, J. Budich and H.-H. Tu for discussions, and F. Tschirsich for contributing numerical libraries. Numerical calculations have been performed with the computational resources provided by the bwUniCluster project [^2], the JUSTUS project, CINECA via the TEDDI project, and the MOGON cluster at the JGU Mainz. We acknowledge financial support from EU projects RYSQ and UQUAM, the German Research Foundation (DFG) through the SFB/TRR21, OSCAR and TWITTER, and the Baden-Württemberg Stiftung via Eliteprogramm for postdocs. S.M. gratefully acknowledges the support of the DFG via a Heisenberg fellowship.
{width="\textwidth"}
Algorithm for ground and low excited states with tree-tensor networks
=====================================================================
The ground state search is performed according to a variational scheme, first developed in the context of DMRG/MPS ground state algorithms [@Schollwock2011], and which in the meantime has been generalized to arbitrary loopless TN geometries [@Murg2010; @*Nakatani2013; @Gerster2014; @Silvi2017]. The basic motivation is the following: Although the TN representation of the many-body state already reduces dramatically the number of coefficients in the state vector (as compared to the full Hilbert space dimension), solving the minimization problem $$E_0 = \langle \Psi_0 | H | \Psi_0 \rangle \stackrel{!}{=} \mathrm{min.}
\label{eq:gs_energy}$$ for the entire TN state $|\Psi_0\rangle$ directly is in practice not feasible, because the number of coefficients in $|\Psi_0\rangle$ is typically still far too large to be handled by numerical eigensolvers. Instead, one applies an iterative strategy: out of all the tensors, which together constitute the TN state, only the coefficients of one (or two, dependent on the update scheme [@Hubig2015; @Silvi2017]) tensor(s) are considered variational, while the others are taken to be fixed. This allows one to contract the physical Hamiltonian $H$ to a reduced, effective Hamiltonian $H_\mathrm{eff}$, only acting on the degrees of freedom of the variational tensor(s). For loopless TNs the resulting reduced optimization problem can again be formulated as a standard eigenvalue problem, whose size is now manageable by an eigensolver. One then successively targets all the tensors in the TN, thereby gradually decreasing the energy expectation value $\langle \Psi_0 | H | \Psi_0 \rangle$. This procedure is called [*sweeping*]{}. A sweep is completed after all the tensors in the TN have been updated once. For the TTN architecture employed here (sketched in Fig. \[fig:ttn\]), the ansatz contains $N_s-2$ tensors ($N_s=L^2$: number of lattice sites) and all involved contractions (both computation of $H_\mathrm{eff}$ and solving the reduced eigenvalue problem) have computational complexity $\mathcal{O}(m^4)$ or less. Therefore, the GS algorithm runtime scales as $\mathcal{O}(N_s \, m^4)$. Convergence of the GS energy is typically reached after less than ten sweeps.
To determine excited states, we employ a very similar algorithm to the one just described, with the small modification that orthogonality to all previously determined eigenstates is enforced. This can be achieved by penalizing overlap with these eigenstates; more in detail, in order to obtain the $n$-th excited state of $H$ (orthogonal to all lower-lying eigenstates $|\Psi_k \rangle$, $k \in [0, \, n-1]$) we solve the optimization problem $$E_n = \langle \Psi_n | H | \Psi_n \rangle + \sum_{k=0}^{n-1} \epsilon_k \langle \Psi_n | P_k | \Psi_n \rangle \stackrel{!}{=} \mathrm{min.} \; ,
\label{eq:ex_energy}$$ where $P_k=|\Psi_k \rangle \langle \Psi_k |$ is the projector on the $k$-th eigenstate and $\epsilon_k$ is as an energy penalty which has to be chosen large enough, which means larger than the energy difference $|E_k - E_n|$ to the target state. Of course this energy difference is not known at the start of the algorithm, but in practice one can simply estimate a value which is guaranteed to be large enough, e.g. one can set $\epsilon_k$ to be one order of magnitude larger than a typical energy scale in the system. The scaling of the computational complexity of the algorithm is not changed by the additional projective terms in Eq. : In complete analogy to the effective Hamiltonian these terms lead to effective projectors, which only contribute a (typically small) overhead to the algorithm. In particular, the runtime scaling for determining excited states remains at $\mathcal{O}(N_s \, m^4)$.
![\[fig:m\_energy\]GS energy $E_0$ as a function of bond dimension $m$ for two different system sizes $L=16$ (a) and $L=32$ (b). Also shown are two different $m \rightarrow \infty$ extrapolations, a simple one linear in $1/m$ and a more versatile one where the exponent of $1/m$ is allowed to be adjusted by the fit. This procedure can be used to estimate the ground state energies and corresponding errors to be $E_0 = -29.015 \pm 0.016 $ (a), and $E_0= -60.84 \pm 0.03$ (b).](fig_m_energy){width="\columnwidth"}
![\[fig:m\_corr\]Bond dimension convergence of the Green functions from Fig. \[fig:corr\]a. Different colors denote different bond dimensions, while the two different symbols refer to two different choices of $x$ (circles: $x=2$, crosses: $x=8$). For PBC the different choices of $x$ result in the same curves, as expected for a system with translational invariance; only at $y\approx L/2$ translational invariance is slightly broken, indicating a finite bond dimension error of the order of $10^{-2}$.](fig_m_corr){width="\columnwidth"}
Procedure for determining the location of branch vortices
=========================================================
As described in Sec. \[sec:mbcn\], we obtain the branch vortex count of a point $(\theta_x, \theta_y)$ on the boundary condition grid by summing up the angle differences $\Delta \Omega$ of the argument field $\Omega_{\Phi\rightarrow\Phi^\prime}$ along the four nearest neighbors of $(\theta_x, \theta_y)$. This procedure, following Refs. , is exemplified in Fig. \[fig:vortices\] for four different illustrative configurations of $\Omega_{\Phi\rightarrow\Phi^\prime}$, displaying different branch vorticities. Moreover, in Fig. \[fig:vortices\]e we provide an illustration for the color code used in Figs. \[fig:chernnum\]d and \[fig:mu\_mbcn\]b.
Bond dimension convergence of observables
=========================================
We exemplify the convergence of the energy expectation value as a function of the bond dimension $m$ in Fig. \[fig:m\_energy\]. Estimates for finite bond dimension errors can be obtained by extrapolating to the limit $m\rightarrow \infty$. For the system sizes considered here, we reached bond dimensions of up to $m\approx 500$, typically displaying truncation errors of order of $10^{-6}$. In Fig. \[fig:m\_corr\] we demonstrate bond dimension convergence for the Green functions shown in Fig. \[fig:corr\]a.
[^1]: Notice that the continuum limit is also recovered by adding tailored long-range hoppings, which effectively flatten the lowest band [@kapit2010exact].
[^2]: [b]{}wUniCluster: funded by the Ministry of Science, Research and Arts and the universities of the state of Baden-W[ü]{}rttemberg, Germany, within the framework program bwHPC.
|
---
abstract: 'To make the development of efficient multi-core applications easier, libraries, such as Grand Central Dispatch, have been proposed. When using such a library, the programmer writes so-called [*blocks*]{}, which are chunks of codes, and dispatches them, using [ *synchronous*]{} or [*asynchronous*]{} calls, to several types of waiting queues. A scheduler is then responsible for dispatching those blocks on the available cores. Blocks can synchronize via a global memory. In this paper, we propose Queue-Dispatch Asynchronous Systems as a mathematical model that faithfully formalizes the synchronization mechanisms and the behavior of the scheduler in those systems. We study in detail their relationships to classical formalisms such as pushdown systems, Petri nets, fifo systems, and counter systems. Our main technical contributions are precise worst-case complexity results for the Parikh coverability problem and the termination question for several subclasses of our model. We give an outlook on extending our model towards verifying input-parametrized fork-join behaviour with the help of abstractions.'
author:
- Gilles Geeraerts
- Alexander Heußner
- |
Jean-François Raskin\
Université Libre de Bruxelles – Belgium
title: 'Queue-Dispatch Asynchronous Systems'
---
Introduction\[sec:introduction\]
================================
The computing power delivered by computers has followed an exponential growing rate the last decades. One of the main reasons was the steady increase of the CPU clock rates. This growth, however, has come to an end a few years ago, because further increasing the clock rate would incur major engineering challenges related to power dissipations. In order to overcome this and meet the continuous need for more computing power, multi-core CPU’s have been introduced and are now ubiquitous. However, in order to harness the power of multiple cores, software applications need to be fundamentally modified and the programmers now have to write programs with parallelism in mind. But writing parallel programs is a notoriously difficult and error prone task. Also, writing [*efficient*]{} and [*portable*]{} parallel code for multi-core platforms is difficult, as the number of available cores will vary greatly from one platform to another, and might also depend on the current load, the energy management policy, and so forth.
In order to alleviate the task of the programmer, several high level programming interfaces have been proposed, and are now available on several operating systems. A popular example is [*Grand Central Dispatch*]{}, [<span style="font-variant:small-caps;">Gcd</span>]{}for short, a technology that is present in MacOS X (since 10.6), iOS (since version 4), and FreeBSD. In [<span style="font-variant:small-caps;">Gcd</span>]{}, the programmer writes so-called [*blocks*]{} which are chunks of codes, and send them to [*queues*]{}, together with several dependency constraints between those blocks (for instance, one block cannot start before the previous one in the queue has finished). The scheduler is then responsible for dispatching those blocks on the available cores, through a thread pool that the scheduler manages (thereby avoiding the explicit and costly creation/destruction of threads by the programmer that is in addition extremely error-prone).
So far, to the best of our knowledge, no formal model has been proposed for systems relying on [<span style="font-variant:small-caps;">Gcd</span>]{}or similar technologies, making those programs [*de facto*]{} out of reach of current verification methods and tools. This is particularly unfortunate as the control structure of such programs is rich and may exhibit complex behaviors. Indeed, the state-space of such programs is infinite even when types of variables are abstracted to finite domains of values. This is not surprising as asynchronous calls and recursive synchronous calls can send an unbounded number of blocks to queues. Also, those programs are, as any parallel program, subject to concurrency bugs that are difficult to detect using testing only.
\
#### [**Contributions**]{}
In this paper, we introduce [*Queue-Dispatch Asynchronous Systems*]{}, [<span style="font-variant:small-caps;">Qdas</span>]{}for short, as a formal model for programs written using libraries such as [<span style="font-variant:small-caps;">Gcd</span>]{}. Our model is composed of [*blocks*]{}, that are finite transition systems with finite data-domain variables that can do [*asynchronous*]{} (non-blocking) and [*synchronous*]{} (blocking) calls to other blocks (possibly recursively). However, a call does not immediately trigger the execution of the callee: the block is inserted into a queue that can be either [ *concurrent*]{} or [*serial*]{}. In concurrent queues, several blocks can be taken from the queue and executed in parallel, while in serial queues, a block can be dequeued only if the previous block in the queue has completed its execution. Queues are maintained with a [fifo]{}policy. To formalize configurations of such systems, our formal semantics relies on [*call task graph*]{}, [$\textsc{Ctg}$]{}for short, in which nodes model tasks that are either in queues or executing, and edges model dependencies between tasks and within queues.
We then study the decidability border for the [*Parikh coverability problem*]{} and the *termination problem* on several subclasses of [<span style="font-variant:small-caps;">Qdas</span>]{}. Our results are summarized in Table\[tab:decidability\]. The [*Parikh image*]{} of a [$\textsc{Ctg}$]{}is an abstraction that counts for each type and state of blocks the number of occurrences in the [$\textsc{Ctg}$]{}and the [*Parikh coverability*]{} problem asks for the reachability of a [$\textsc{Ctg}$]{}that contains at least a given number of blocks of each type that are in a given set of states. Not surprisingly, this problem is undecidable for [<span style="font-variant:small-caps;">Qdas</span>]{}, but we identify several subclasses for which the problem is decidable. For those decidable cases, we characterize the exact complexity of the problem.
The main positive decidability results with precise complexity are as follows: First, we show that [<span style="font-variant:small-caps;">Qdas</span>]{}with [*only*]{} synchronous calls are essentially equivalent to pushdown systems with finite domain data-variables, and we show that the Parikh coverability problem is [<span style="font-variant:small-caps;">ExpTime-C</span>]{}for synchronous concurrent [<span style="font-variant:small-caps;">Qdas</span>]{}(Theorem\[thm:syncqdas\]). Second, for synchronous [<span style="font-variant:small-caps;">Qdas</span>]{}with only serial queues, the problem is [<span style="font-variant:small-caps;">PSpace-C</span>]{}(Theorem\[thm:serialsyncqdas\]). Third, we show that [<span style="font-variant:small-caps;">Qdas</span>]{}with [*only*]{} asynchronous calls and [*only*]{} concurrent queues are essentially equivalent to lossy Petri nets and show that the Parikh coverability problem is [<span style="font-variant:small-caps;">ExpSpace-C</span>]{}for that class (Theorem\[thm:concasyncqdas\]). This decidability border is precise as we show that if we allow either $(i)$ asynchronous calls with synchronous queues, or $(ii)$ synchronous and asynchronous calls with concurrent queues, then the Parikh coverability problem becomes undecidable (Theorem\[the:async-seri-undec\] and Theorem\[thm:concqdasundec\]). The previous proof’s ideas allow to derive similar results for termination wrt. the subclasses of [<span style="font-variant:small-caps;">Qdas</span>]{}. The *termination* problem asks given a [<span style="font-variant:small-caps;">Qdas</span>]{}whether all its executions are finite.
We enhance up our results by presenting an extension of [<span style="font-variant:small-caps;">Qdas</span>]{}with an explicit fork/join construct that, in addition, is parametrized by the input. As Parikh coverability and termination lifted to this setting are undecidable, we propose two over-approximations that allow for solutions in practice.
[*Remark*]{}: Due to the lack of space, detailed formal proofs are deferred to the appendix.
#### [**Related Works**]{}
The basic model checking result for asynchronous programs is the [<span style="font-variant:small-caps;">ExpSpace</span>]{}-hardness for the control-state reachability problem obtained by making formal a link with *multi-set pushdown systems* ([<span style="font-variant:small-caps;">Mpds</span>]{}). The underlying two basic ideas are : $(i)$ to untangle the call stack and the storage of pending asynchronous calls by imposing that the next call in a serialized execution-equivalent program is only processed when the call stack is empty; and $(ii)$ to only count the number of pending calls for each block while the call stack is non-empty. The original reduction in [@sen-k-2006-300-a] is based on Parikh’s theorem and derives the lower bound from a Petri net reachability problem [@esparza-j-1998-374-a]. A Parikh-less reduction was presented in [@jhala-r-2007-339-a] that relied on the convergence of an over- and under-approximation derived from interprocedural dataflow analysis.
The close relation between asynchronous programs and Petri nets can also be used to prove additional decidability results for liveness questions [@ganty-p-2009-102-a; @ganty-p-2010--a]. The following results are based on a (polynomial-time) reduction of asynchronous systems to an “equivalent” Petri net or extension thereof: *fair* termination (i.e., testing whether each dispatched call terminates) is complete in [<span style="font-variant:small-caps;">ExpSpace</span>]{}, the boundedness question is decidable in [<span style="font-variant:small-caps;">ExpSpace</span>]{}(i.e., asking whether we can bound the number of pending calls), fair non-starvation (i.e., asking, when assuming fairness on runs, whether every pending call is eventually dispatched) is decidable. The authors also consider extensions of asynchronous programs with cancellation (i.e., an additional operation removing all pending instances of a block) and testing whether there is *no* pending instance of a given block. In the first case, they show reduction to the model to Petri nets with transfer arcs or reset arcs, in the second case they show reduction to Petri nets with one inhibitor arc. Multi-set pushdown automata are subsumed by *well-structured transition systems with auxiliary storage* and inherit their decidability results presented in [@chadha-r-2007-136-a; @chadha-r-2009-4169-a]. Analogously, one can show that termination, control-state maintainability, and simulation with respect to finite state systems are decidable for asynchronous programs.
All the models considered in the aforementioned publications do not consider causality constraints on the sequence of asynchronous dispatch calls, as would be necessary to model the [fifo]{}policies of [<span style="font-variant:small-caps;">Gcd</span>]{}. However, this is possible with [<span style="font-variant:small-caps;">Qdas</span>]{}. A more detailed look on the differences between the model of [@ganty-p-2010--a] and the ([fifo]{}-less) subclass of asynchronous serial [<span style="font-variant:small-caps;">Qdas</span>]{}is presented in Section\[sec:parikh-cover-probl\].
A series of parallel programming libraries and techniques is formalized in [@bouajjani-a-2012--a] with the help of *recursively parallel programs*. These allow to model fork/join based parallel computations based on a reduction to recursive vector addition systems with states. With respect to [<span style="font-variant:small-caps;">Qdas</span>]{}and asynchronous programming, recursively parallel programs only cover the classical asynchronous models presented above and not the advanced scheduling strategies for different queues that introduce more sophisticated behaviours.
Preliminaries {#sec:prelims}
=============
***Grand Central Dispatch*** ([<span style="font-variant:small-caps;">Gcd</span>]{}) is a technology developed by Apple [@apple--2010--a; @apple--2011--a] that is publicly available at <http://libdispatch.macosforge.org/> under a free license. [<span style="font-variant:small-caps;">Gcd</span>]{}is the main inspiration for the formal model of queue-dispatch asynchronous systems. In the following, we often present our examples as pseudo code using a syntax inspired by [<span style="font-variant:small-caps;">Gcd</span>]{}. In the [<span style="font-variant:small-caps;">Gcd</span>]{}framework, the programmer has to organize his code into *blocks*. During the execution of a [<span style="font-variant:small-caps;">Gcd</span>]{}program, one or several *tasks* run in parallel, each executing a given block (initially, only the `main` block is running). Tasks can call (or *dispatch* in the [<span style="font-variant:small-caps;">Gcd</span>]{}vocabulary) other blocks, either *synchronously* (the call is blocking), or *asynchronously* (the call is not blocking). A *dispatch* consists in inserting the block into a [fifo]{}*queue*. In our examples, we use the keywords ${\ensuremath{{\ensuremath{\mathtt{dispatch_a}}\xspace}}}$ and ${\ensuremath{{\ensuremath{\mathtt{dispatch_s}}\xspace}}}$ to refer to asynchronous and synchronous dispatches respectively. At any time, the scheduler can decide to *dequeue* blocks from the queues and to assign them to tasks for execution. All queues ensure that the blocks are dequeued in [fifo]{}order, however the actual scheduling policy depends on the type of queue. [<span style="font-variant:small-caps;">Gcd</span>]{}supports two types of queues: *concurrent queues* allow several tasks from the same queue to run in parallel, whereas *serial queues* guarantee that *at most one* task from this queue is running. In our examples, concurrent (or serial) queues are declared as global variables of type `c_queue` (`s_queue`). In addition, all blocks have access to the same set of *global variables* (in this work, we assume that the variables range over finite domains).
\[ex:gcd-matrixmult\] Let us consider the pseudo code in Fig. \[fig:example-gcd\] that computes the product of two integer matrices $\texttt{matrix1}$ and $\texttt{matrix2}$ of constant size ($\texttt{l,m,n}$) in a matrix $\texttt{matrix}$. The `main` task forks a series of `one_cell` blocks. Each `one_cell` computes the value of a single cell of the result. The parallelism is achieved via the [<span style="font-variant:small-caps;">Gcd</span>]{}scheduler, thanks to *asynchronous dispatches* on the *concurrent* queue `workqueue`. Asynchronous dispatches are needed to make sure that `main` is not blocked after each dispatch, and a concurrent queue allow all the `one_cell` block to run in parallel. The variable `count` is incremented each time the computation of a cell is finished and acts as a semaphore for the $\texttt{main}$ block, to ensure that `matrix` contains the final result. As only reading and writing to a variable are atomic, we need to guarantee exclusive access of two consecutive operations on `count` (line \[lst:count\]). This is achieved by a dedicated block `increase` that is dispatched to the *serial* queue `semaphore`. As only `increase` blocks can increase `count`, this queue implicitly locks the access to the variable. Moreover, the synchronous dispatch in line \[lst:syncdisp\] guarantees that a block terminates only after it has increased `count`.
global int const l,m,n (*@\label{lst:const}@*)
global int[l][m] matrix1, int[m][n] matrix2, int[l][n] matrix
global c_queue workqueue, s_queue semaphore, int count (*@\mbox{}\\[0.5ex]@*)
block increase():
count = count + 1 (*@\label{lst:count}\\[0.5ex]@*)
block one_cell(int i, int j):
for k in range(m):
matrix[i][j]+= matrix1[i][k] * matrix2[k][j]
dispatch_s(semaphore,increase()) (*@\label{lst:syncdisp}\\[0.5ex]@*)
def main():
// read input matrix1, matrix2
count = 0
for i in range(l): (*@\label{lst:fork-start}@*)
for j in range(n):
dispatch_a(workqueue,one_cell(i,j))
wait(count = l*n) (*@\label{lst:join}@*)
// print the result
#### **Basic Notations:**
Given a set $S$, let ${{\ensuremath{\left| S \right|}}}$ denote its cardinality. For an $I$-indexed family of sets $(S_i)_{i \in I}$, we write elements of $\prod_{i \in I} S_i$ in bold face, i.e., $\vec{s}\in\prod_{i\in I} S_i$. The $i$-component of $\vec{s}$ is written $s_i \in S_i$, and we identify $\vec{s}$ with the indexed family of elements $(s_i)_{i \in I}$. We use $\operatorname{\mathaccent\cdot\cup}$ to denote the disjoint union of sets. An *alphabet* $\Sigma$ is a finite set of *letters*. We write $\Sigma^*$ for the set of all *finite words*, over $\Sigma$ and denote the empty word by $\varepsilon$. The concatenation of two words $w,w'$ is represented by $w\cdot w'$. For a letter $\sigma\in\Sigma$ and a word $w\in\Sigma^*$, let ${\ensuremath{\left| w \right|}}_\sigma$ be the number of occurrences of $\sigma$ in $w$. We use standard complexity classes, e.g., polynomial time ([<span style="font-variant:small-caps;">PTime</span>]{}) or deterministic exponential time ([<span style="font-variant:small-caps;">ExpTime</span>]{}), and mark completeness by appending “-C” ([<span style="font-variant:small-caps;">PSpace-C</span>]{}).
Let ${\ensuremath{\mathbb{D}}\xspace}$ be a finite ***data domain*** with an *initial element* $d_0\in{\ensuremath{\mathbb{D}}\xspace}$, and let ${\ensuremath{\mathcal{X}}\xspace}$ be a finite set of variables ranging over ${\ensuremath{\mathcal{D}}\xspace}$. A *valuation* of the variables in ${\ensuremath{\mathcal{X}}\xspace}$ is a function $\mathbf{d}: {\ensuremath{\mathcal{X}}\xspace}\rightarrow {\ensuremath{\mathbb{D}}\xspace}$. An *atom* is an expression of the form $x=d$ or $x\neq d$, where $x\in{\ensuremath{\mathcal{X}}\xspace}$ and $d\in {\ensuremath{\mathbb{D}}\xspace}$. A *guard* if a finite conjunction of atoms. An assignment is an expression of the form $x\gets v$, where $x\in {\ensuremath{\mathcal{X}}\xspace}$ and $v\in {\ensuremath{\mathbb{D}}\xspace}$. Let ${\ensuremath{\mathsf{guards}\left({\ensuremath{\mathcal{X}}\xspace}\right)}}$, ${\ensuremath{\mathsf{assign}\left({\ensuremath{\mathcal{X}}\xspace}\right)}}$ and ${\ensuremath{\mathsf{vals}\left({\ensuremath{\mathcal{X}}\xspace}\right)}}$ denote respectively the sets of all guards, assignments and valuations over variables from ${\ensuremath{\mathcal{X}}\xspace}$. Guards, atoms and valuations have their usual semantics: for all valuations ${\ensuremath{\vec{d}}}$ of ${\ensuremath{\mathcal{X}}\xspace}$ and all $g\in{\ensuremath{\mathsf{guards}\left({\ensuremath{\mathcal{X}}\xspace}\right)}}$, we write ${\ensuremath{\vec{d}}}\models g$ iff ${\ensuremath{\vec{d}}}$ satisfies $g$.
A ***pushdown system with data*** is a pushdown system (see [@bouajjani-a-1997-135-a] for details) equipped with a finite set of variables ${\ensuremath{\mathcal{X}}\xspace}$ over a finite domain ${\ensuremath{\mathbb{D}}\xspace}$. A configuration of a [<span style="font-variant:small-caps;">Pds</span>]{}with data is a pair $(s,w,{\ensuremath{\vec{d}}})$ where $s$ is a control state, $w$ is the stack content, and ${\ensuremath{\vec{d}}}$ is a valuation of the variables
[proposition]{}[propreachpdsdata]{}\[prop:reach\_pds\_data\] The reachability problem is [<span style="font-variant:small-caps;">ExpTime-C</span>]{}for [<span style="font-variant:small-caps;">Pds</span>]{}with data.
A ***Petri net*** ([<span style="font-variant:small-caps;">Pn</span>]{}) is a tuple $N=\langle P, T, m_0 \rangle$ where $P$ is a finite set of places, a *marking* of the places is function $m:P\rightarrow {\ensuremath{\mathbb{N}}\xspace}$ that associates, to each place $p\in P$ a number $m(p)$ of tokens, $T$ is finite set of transitions, each transition $t\in T$ is a pair $(I_t,
O_t)$ where $I_t: P\rightarrow \{0,1\}$ and $O_t:
P\rightarrow \{0,1\}$ are respectively the *input* and *output functions* of $t$, and $m_0$ is the *initial marking*. Given two markings $m_1$ and $m_2$, we let $m_1{\preceq}m_2$ iff $m_1(p)\leq m_2(p)$ for all $p\in P$. Given a marking $m$, a transition $t=(I_t, O_t)$ is *enabled* in $m$ iff $m(p)\geq
I_t(p)$ for all $p\in P$. When $t$ is enabled in $m$, one can *fire* the transition $t$ in $m$, which produces a new marking $m'$ s.t. $m'(p)=m(p)-I_t(p)+O_t(p)$ for all $p$. This is denoted , or simply $m\rightarrow m'$ when the transition identity is irrelevant. A *run* is a finite sequence $m_0m_1\ldots m_n$ s.t. for all $1\leq i\leq n$: $m_{i-1}\rightarrow
m_i$. For a [<span style="font-variant:small-caps;">Pn</span>]{}$N$, we denote by $\operatorname{\textit{Reach}}(N)$ (resp. ${\ensuremath{\textit{Cover}}}(N)$) the *reachability (coverability) set* of $N$, i.e. the set of all markings $m$ s.t. there exists a run $m_0m_1\ldots m_n$ of $N$ with $m=m_n$ ($m{\preceq}m_n$). The *coverability problem* asks, given a [<span style="font-variant:small-caps;">Pn</span>]{}$N$ and a marking $m$, whether $m\in{\ensuremath{\textit{Cover}}}(N)$. It is [<span style="font-variant:small-caps;">ExpSpace</span>]{}-complete [@esparza-j-1998-374-a]. The termination problem, i.e., whether all executions of the Petri net are finite, is decidable in [<span style="font-variant:small-caps;">ExpSpace-C</span>]{} [@lipton; @rackoff].
Queue-dispatch asynchronous systems
===================================
#### **Syntax:**
We now define our formal model for queue-dispatch asynchronous systems. Let ${\ensuremath{\mathbb{D}}\xspace}$ be a finite data domain containing an *initial value* $d_0$. A *queue-dispatch asynchronous system* ([<span style="font-variant:small-caps;">Qdas</span>]{}) ${\ensuremath{\mathcal{A}}\xspace}$ is a tuple ${\langle {CQID\xspace}, {SQID\xspace}, \Gamma, main, {\ensuremath{\mathcal{X}}\xspace}, \Sigma,
({\ensuremath{\mathcal{TS}}\xspace}_\gamma)_{\gamma\in\Gamma}\rangle\xspace}$ where:
${CQID\xspace}$ and ${SQID\xspace}$ are respectively sets of *(c)oncurrent* and *(s)erial queues*;
$\Gamma$ is the finite set of *blocks* and $main\in\Gamma$ the *initial block*. Each block $\gamma\in\Gamma$ is a tuple $\langle S_\gamma, s^0_\gamma,
f_\gamma, \Sigma, \Delta_\gamma \rangle$ where $\langle S_\gamma,
s^0_\gamma, \Sigma, \Delta_\gamma\rangle$ is an [<span style="font-variant:small-caps;">Lts</span>]{}and $f_\gamma\in S$ a distinct final state;
${\ensuremath{\mathcal{X}}\xspace}$ is a finite set of ${\ensuremath{\mathbb{D}}\xspace}$-valued variables;
$\Sigma$ is the set of *actions*, with $\Sigma=(\{{\ensuremath{{\ensuremath{\mathtt{dispatch_s}}\xspace}}},{\ensuremath{{\ensuremath{\mathtt{dispatch_a}}\xspace}}}\}\times ({CQID\xspace}\cup{SQID\xspace})$$\times
\Gamma\setminus\{main\}) \cup {\ensuremath{\mathsf{guards}\left({\ensuremath{\mathcal{X}}\xspace}\right)}} \cup {\ensuremath{\mathsf{assign}\left({\ensuremath{\mathcal{X}}\xspace}\right)}}$.
We assume that ${SQID\xspace}, {CQID\xspace}, \Gamma, {\ensuremath{\mathcal{X}}\xspace}$, and all $S_\gamma$ for $\gamma\in\Gamma$ are disjoint from each other. Let $S=\operatorname{\ooalign{$\bigcup$\cr\hfill$\cdot$\hfill}}_{\gamma\in\Gamma}S_\gamma$, $F=\operatorname{\ooalign{$\bigcup$\cr\hfill$\cdot$\hfill}}_{\gamma\in\Gamma} \{f_\gamma\}$, $\Delta=\operatorname{\ooalign{$\bigcup$\cr\hfill$\cdot$\hfill}}_{\gamma\in\Gamma}\Delta_\gamma$, and ${QID\xspace}={SQID\xspace}\operatorname{\mathaccent\cdot\cup}{CQID\xspace}\operatorname{\mathaccent\cdot\cup}\{\imath\}$ (where $\imath\notin{SQID\xspace}\cup{CQID\xspace}$). We further assume that ${\ensuremath{\varepsilon}\xspace}\notin\Sigma$.
#### **Call-task graphs:**
We formalize the semantics of [<span style="font-variant:small-caps;">Qdas</span>]{}using the notion of *call-task graph* ([$\textsc{Ctg}$]{}) to describe the system’s global configurations.
A configuration of a [<span style="font-variant:small-caps;">Qdas</span>]{}(see Fig.\[fig:example-ctg\] for an example) contains a set of running tasks, represented by *task vertices* (depicted by round nodes), a set of called but unscheduled blocks, represented by *call vertices* (square nodes). Call vertices are held by queues, and the linear order of each queue is represented by *queue edges* (solid edges). Synchronous calls add an additional dependency (the caller is waiting for the termination of the callee) that is represented by a *wait edge* (dashed edges) between the caller and the callee. Wait edges are also inserted between the head of a *serial* queue and the running task that has been extracted from this queue (if it exists) to indicate that the task has to terminate before a new block can be dequeued. Note that only vertices without outgoing edges can execute a computation step, the others are currently blocked. Each node $v$ is labeled by a block $\lambda(v)$, an by the identifier ${\ensuremath{\textit{queue}}}(v)$ of the queue that contains it (for call vertices) or that contained it (for task vertices). Task vertices are labeled by their current state ${\ensuremath{\textit{state}}}(v)$ (for convenience, we also label call vertices by the initial state of their respective blocks – not shown in the figure).
The [$\textsc{Ctg}$]{}in Fig.\[fig:example-ctg\] depicts a configuration of a [<span style="font-variant:small-caps;">Qdas</span>]{}with two queues. Queue $q_2$ is serial (note the outgoing wait edge to the running task) and contains $\gamma_2\gamma_2\gamma_2$, and $q_1$ is parallel with content $\gamma_1\gamma_2$. There are 4 active tasks, two of them (`main` and the task running $\gamma_1$) are blocked. The task running $\gamma_3$ has been dequeued from $q_2$ and is currently at location $s$.
\[zstd/.style=[state,font=,inner sep=1pt,minimum size=15pt]{}, zstd2/.style=[zstd,rectangle]{}, lab/.style=[font=,inner sep=1pt]{}, anchor=west\]
(0,0) node\[zstd2\] (00) [$\gamma_1$]{}; (00.south east) node\[lab,anchor=south west\] [$q_1$]{}; (00)+(1,0) node\[zstd2\] (01) [$\gamma_2$]{}; (01.south east) node\[lab,anchor=south west\] [$q_1$]{}; (00) edge\[->\] (01);
(4,0) node\[zstd2\] (10) [$\gamma_2$]{}; (10.south east) node\[lab,anchor=south west\] [$q_2$]{}; (10)+(1,0) node\[zstd2\] (11) [$\gamma_3$]{}; (11.south east) node\[lab,anchor=south west\] [$q_2$]{}; (11)+(1,0) node\[zstd2\] (12) [$\gamma_2$]{}; (12.south east) node\[lab,anchor=south west\] [$q_2$]{}; / in [10/11,11/12]{} () edge\[->\] ();
(12)+(1.5,0) node\[zstd\] (13) [$\gamma_3$]{}; (13.south east) node\[lab,anchor=north west\] [$q_2$]{}; (13.north east) node\[lab,anchor=south west\] [$s$]{}; (12) edge\[->\] (13);
(0.5,1) node\[zstd\] (main) [[$\textit{main}$]{}]{}; (main.south east) node\[lab,anchor=north west\] [$\imath$]{}; (main.north east) node\[lab,anchor=south west\] [$s$]{}; (main)+(1,0) node\[zstd\] (one) [$\gamma_1$]{}; (one.south east) node\[lab,anchor=north west\] [$q_1$]{}; (one.north east) node\[lab,anchor=south west\] [$s'$]{}; (main) edge\[->\] (one) (one) edge\[->,out=0,in=90\] (11);
(2.5,0.1) node\[zstd\] (two) [$\gamma_2$]{}; (two.south east) node\[lab,anchor=north west\] [$q_1$]{}; (two.north east) node\[lab,anchor=south west\] [$s''$]{};
[background]{} (01)+(0.5,0) coordinate (dummy); node\[fit=(00) (01) (dummy),fill=gray!20,inner sep=4pt\] (c1) ; (c1.north west) node\[anchor=south west,inner sep=2pt,font=\] [queue $q_1$]{}; (12)+(0.5,0) coordinate (dummy); node\[fit=(10) (12) (dummy),fill=gray!20\] (c1) ; (c1.north west) node\[anchor=south west,inner sep=2pt,font=\] [queue $q_2$]{};
\(13) +(0.5,0.8) node\[draw, rectangle callout,callout absolute pointer=[(13.east)+(0.1,0)]{},font=, fill=gray!5,text width=1.8cm, align=center\] [$\lambda(v)=\gamma_3$ ${\ensuremath{\textit{state}}}(v)=s$ ${\ensuremath{\textit{queue}}}(v)=q_2$]{};
Formally, given a [<span style="font-variant:small-caps;">Qdas</span>]{}${\ensuremath{\mathcal{A}}\xspace}={\langle {CQID\xspace}, {SQID\xspace}, \Gamma, main,
{\ensuremath{\mathcal{X}}\xspace}, \Sigma, ({\ensuremath{\mathcal{TS}}\xspace}_\gamma)_{\gamma\in\Gamma}\rangle\xspace}$, a *call-task graph* over ${\ensuremath{\mathcal{A}}\xspace}$ is a tuple ${\ensuremath{G}}_{\ensuremath{\mathcal{A}}\xspace}={\langleV,E,\lambda,{\ensuremath{\textit{queue}}},{\ensuremath{\textit{state}}}\rangle\xspace}$ where: $V=V_C\operatorname{\mathaccent\cdot\cup}V_T$ is a finite set of *vertices*, partitioned into a set $V_C$ of *call vertices* and a set $V_T$ of *task vertices*; $E\subseteq V\times V$ is a set of *edges*; $\lambda: V\rightarrow \Gamma$ labels each vertex by a block; ${\ensuremath{\textit{queue}}}: V\rightarrow {QID\xspace}\cup\{\imath\}$ associates each vertex to a queue identifier (or $\imath$); and ${\ensuremath{\textit{state}}}: V\rightarrow S$ associates each vertex to a [<span style="font-variant:small-caps;">Lts</span>]{}state. For each $q\in{QID\xspace}$, let $V_q={\left\{v\in V\,\vert\,{\ensuremath{\textit{queue}}}(v)=q\right\}}$. The set $E$ is partitioned into the set $E_W$ of *wait edges* and the set $E_Q=\operatorname{\ooalign{$\bigcup$\cr\hfill$\cdot$\hfill}}_{q\in{QID\xspace}} E_q$ of *queue edges* where, for each $q\in{QID\xspace}$, $E_q=E\cap (V_q\times V_q)$.
A [$\textsc{Ctg}$]{}is *empty* iff $V=\emptyset$. The *Parikh image* ${\ensuremath{\mathsf{Parikh}}}({\ensuremath{G}})$ of a [$\textsc{Ctg}$]{}${\ensuremath{G}}$ of ${\ensuremath{\mathcal{A}}\xspace}$ is a function $f:S\rightarrow {\ensuremath{\mathbb{N}}\xspace}$, s.t. for all $s\in S$: $f(s)=|{\left\{v\in
V\,\vert\,{\ensuremath{\textit{state}}}(v)=s\right\}}|$. Given two Parikh images ${\ensuremath{\mathsf{Parikh}}}(G)$ and ${\ensuremath{\mathsf{Parikh}}}(G')$, we let ${\ensuremath{\mathsf{Parikh}}}(G){\preceq}{\ensuremath{\mathsf{Parikh}}}(G')$ iff for all $s\in
S$: ${\ensuremath{\mathsf{Parikh}}}(G)(s)\leq{\ensuremath{\mathsf{Parikh}}}(G')(s)$. A *path* (of length $n$) in ${\ensuremath{G}}_{\ensuremath{\mathcal{A}}\xspace}$ is a sequence of vertices $v_0,v_1,\ldots, v_n$ s.t. for all $1\leq i\leq n$: $(v_{i-1},v_i)\in E$. Such a path is *simple* iff $v_i\neq v_j$ for all $1\leq i<j\leq n$. The *restriction* of ${\ensuremath{G}}_{\ensuremath{\mathcal{A}}\xspace}$ to $V'\subseteq V$ is the [$\textsc{Ctg}$]{}${\ensuremath{G}}_{\ensuremath{\mathcal{A}}\xspace}'={\langleV', E',\lambda',
{\ensuremath{\textit{queue}}}', {\ensuremath{\textit{state}}}'\rangle\xspace}$, where $E'=E\cap (V'\times V')$, and $\lambda'$, ${\ensuremath{\textit{queue}}}'$ and ${\ensuremath{\textit{state}}}'$ are respectively the restrictions of $\lambda$, ${\ensuremath{\textit{queue}}}$ and ${\ensuremath{\textit{state}}}$ to $V'$.
In the rest of the paper, we assume that all the [$\textsc{Ctg}$]{}we consider are *well-formed*, i.e., they fulfill the following requirements:
1. For each $v\in V_T$: ${\ensuremath{\textit{state}}}(v)\in S_{\lambda(v)}$ where $S_{\lambda(v)}$ are the states of ${\ensuremath{\mathcal{TS}}\xspace}_{\lambda(v)}$.
2. Each *call* vertex has at most one outgoing (queue or wait) edge, at most one incoming *wait* edge, and at most one incoming *queue* edge. Each *task* vertex has at most one outgoing, and at most one incoming *wait* edge.
3. For each $q\in {QID\xspace}$, the restriction of ${\ensuremath{G}}_{\ensuremath{\mathcal{A}}\xspace}$ to $V_q$ is either empty or contains one and only one simple path of length $|V_q|-1$. Intuitively, this ensures the well-formedness of the queues.
4. For each $q\in {SQID\xspace}$, there is at most one task vertex $v$ s.t. ${\ensuremath{\textit{queue}}}(v)=q$. This ensures that queues in ${SQID\xspace}$ indeed force the serial execution of its members.
For convenience, we also introduce the following notations. Let ${\ensuremath{G}}_{\ensuremath{\mathcal{A}}\xspace}$ be a [$\textsc{Ctg}$]{}, and let $q$ be a queue identifier of ${\ensuremath{\mathcal{A}}\xspace}$. Then, ${\ensuremath{\textit{head}}}(q,{\ensuremath{G}}_{\ensuremath{\mathcal{A}}\xspace})$ and ${\ensuremath{\textit{tail}}}(q,{\ensuremath{G}}_{\ensuremath{\mathcal{A}}\xspace})$ denote respectively the head and the tail of $q$ in the configuration described by ${\ensuremath{G}}_{\ensuremath{\mathcal{A}}\xspace}$, that is, ${\ensuremath{\textit{head}}}(q,{\ensuremath{G}}_{\ensuremath{\mathcal{A}}\xspace})$ is the call vertex $v\in V_q$ that has no incoming queue edge, or $\bot$, if such a vertex does not exist; and ${\ensuremath{\textit{head}}}(q,{\ensuremath{G}}_{\ensuremath{\mathcal{A}}\xspace})$ is the call vertex $v\in V_q$ that has no outgoing *queue* edge (but possibly an outgoing *wait* edge), or $\bot$, if such a vertex does not exist. Remark that, when they exist, these vertices are necessarily unique because of the well-formedness assumptions. Finally, we say that a vertex $v$ is *unblocked* iff it has no outgoing edge, and that it is *final* iff $(i)$ $v$ is an *unblocked task* vertex and $(ii)$ ${\ensuremath{\textit{state}}}(v)=f_\lambda(v)$ (that is, $v$ represents a task that has reached the final state of its transition system and is not waiting on another task).
Let us now define several operations on [$\textsc{Ctg}$]{}. We will rely on these operations when defining the formal semantics of [<span style="font-variant:small-caps;">Qdas</span>]{}. Let ${\ensuremath{\mathcal{A}}\xspace}$ be a [<span style="font-variant:small-caps;">Qdas</span>]{}and ${\ensuremath{G}}_{\ensuremath{\mathcal{A}}\xspace}={\langleV,E,\lambda,\allowbreak {\ensuremath{\textit{queue}}},{\ensuremath{\textit{state}}}\rangle\xspace}$ be a [$\textsc{Ctg}$]{}for ${\ensuremath{\mathcal{A}}\xspace}$. Then:
for all $v\in V$: ${\ensuremath{G}}\setminus v$ is the restriction of ${\ensuremath{G}}$ to $V\setminus\{v\}$.
for all $\gamma\in\Gamma$ and $q\in{QID\xspace}$, ${\ensuremath{\textsf{enqueue}}}(q,\gamma)({\ensuremath{G}}_{\ensuremath{\mathcal{A}}\xspace})$ is the [$\textsc{Ctg}$]{}${\langleV', E',
\lambda', {\ensuremath{\textit{queue}}}', {\ensuremath{\textit{state}}}\rangle\xspace}$ where: $V'=V\cup\{v'\}$, $v'$ is a fresh queue vertex, $\lambda(v')=\gamma$, ${\ensuremath{\textit{queue}}}(v')=q$, ${\ensuremath{\textit{state}}}(v')=s^0_\gamma$, and for all $v\in V$: $\lambda'(v)=\lambda(v)$ and ${\ensuremath{\textit{queue}}}'(v)={\ensuremath{\textit{queue}}}(v)$. Finally, $E'=E\cup E_1\cup E_2$, where: $(i)$ $E_1=\{(v',{\ensuremath{\textit{tail}}}({\ensuremath{G}}_{\ensuremath{\mathcal{A}}\xspace},q))\}$ if ${\ensuremath{\textit{tail}}}({\ensuremath{G}}_{\ensuremath{\mathcal{A}}\xspace},q)\neq\bot$, and $E_1=\emptyset$ otherwise, and $(ii)$ if $v\in V$ is a *task* node s.t. ${\ensuremath{\textit{queue}}}(v)=q\in{SQID\xspace}$, then $E_2=\{(v',v)\}$, otherwise $E_2=\emptyset$. Intuitively, this operation inserts a call to $\gamma$ in the queue $q$, by creating a new vertex $v'$ and adding an edge to maintain the FIFO ordering, if necessary (set $E_1$). In the case of a *serial* queue that was empty before the enqueue, a supplementary edge (in set $E_2$) might be necessary to ensure that $v'$ is blocked by a currently running $v$ which has been extracted from $q$.
for all $q\in{QID\xspace}$, if ${\ensuremath{\textit{head}}}(q)$ is different from $\bot$ and *unblocked*, then ${\ensuremath{\textsf{dequeue}}}(q)({\ensuremath{G}}_{\ensuremath{\mathcal{A}}\xspace})$ is the [$\textsc{Ctg}$]{} ${\langleV_C'\operatorname{\mathaccent\cdot\cup}V'_T, E', \lambda, {\ensuremath{\textit{queue}}}, {\ensuremath{\textit{state}}}\rangle\xspace}$ where $V'_C=V_C\setminus\{{\ensuremath{\textit{head}}}(q)\}$ and $V'_T=V'_T\cup\{{\ensuremath{\textit{head}}}(q)\}$. Otherwise, ${\ensuremath{\textit{head}}}(q)=\bot$ and ${\ensuremath{\textsf{dequeue}}}(q)({\ensuremath{G}}_{\ensuremath{\mathcal{A}}\xspace})$ is undefined. Intuitively, this operation removes the first (with respect to the FIFO ordering) block from $q$ and turns the corresponding *call* vertex ${\ensuremath{\textit{head}}}(q)$ into a *task* vertex, meaning that the block is now running as a task.
for all $\delta=(s,a,s')\in\Delta$, ${\ensuremath{\textsf{step}}}(\delta)({\ensuremath{G}}_{\ensuremath{\mathcal{A}}\xspace})$ is a *set* of [$\textsc{Ctg}$]{}defined as follows. ${\langleV, E, \lambda,
{\ensuremath{\textit{queue}}}, {\ensuremath{\textit{state}}}'\rangle\xspace}\in{\ensuremath{\textsf{step}}}(\delta)({\ensuremath{G}}_{\ensuremath{\mathcal{A}}\xspace})$ iff there exists an *unblocked* $v\in V_T$ s.t. ${\ensuremath{\textit{state}}}(v)=s$, ${\ensuremath{\textit{state}}}'(v)=s'$ and for all $v'\neq v$: ${\ensuremath{\textit{state}}}'(v')={\ensuremath{\textit{state}}}(v')$. Remark that ${\ensuremath{\textsf{step}}}(\delta)({\ensuremath{G}}_{\ensuremath{\mathcal{A}}\xspace})$ can be empty. Intuitively, each graph in ${\ensuremath{\textsf{step}}}(\delta)({\ensuremath{G}}_{\ensuremath{\mathcal{A}}\xspace})$ corresponds to the firing of an $a$-labeled transition by a task that is not blocked.
for all unblocked $v\in V\cup\{\bot\}$, all $v'\in V$: ${\ensuremath{\textsf{letwait}}}(v,v')({\ensuremath{G}}_{\ensuremath{\mathcal{A}}\xspace})$ is either the [$\textsc{Ctg}$]{}${\ensuremath{G}}_{\ensuremath{\mathcal{A}}\xspace}$ if $v=\bot$, or the [$\textsc{Ctg}$]{}${\langleV, E\cup (v,v'),\lambda, {\ensuremath{\textit{queue}}},
{\ensuremath{\textit{state}}}\rangle\xspace}$ if $v\neq \bot$. Intuitively, this operation adds a wait edge between nodes $v$ and $v'$ when $v\neq\bot$, and does not modify the [$\textsc{Ctg}$]{}otherwise.
#### **Semantics of [<span style="font-variant:small-caps;">Qdas</span>]{}:**
For a [<span style="font-variant:small-caps;">Qdas</span>]{}${\ensuremath{\mathcal{A}}\xspace}$ with set of variables ${\ensuremath{\mathcal{X}}\xspace}$, a *configuration* is a pair $({\ensuremath{G}},{\ensuremath{\vec{d}}})$, where ${\ensuremath{G}}$ is a [$\textsc{Ctg}$]{}of ${\ensuremath{\mathcal{A}}\xspace}$ and ${\ensuremath{\vec{d}}}\in{\ensuremath{\mathsf{vals}\left({\ensuremath{\mathcal{X}}\xspace}\right)}}$. The operational semantics of ${\ensuremath{\mathcal{A}}\xspace}$ is given as a transition system ${\ensuremath{\llbracket {\ensuremath{\mathcal{A}}\xspace}\rrbracket}}$ whose states are configurations of ${\ensuremath{\mathcal{A}}\xspace}$; and whose transitions reflect the semantics of the actions labeling the transitions of the [<span style="font-variant:small-caps;">Qdas</span>]{}. Formally, given a [<span style="font-variant:small-caps;">Qdas</span>]{}${\ensuremath{\mathcal{A}}\xspace}={\langle{CQID\xspace},
{SQID\xspace}, \Gamma, main, {\ensuremath{\mathcal{X}}\xspace}, \Sigma, ({\ensuremath{\mathcal{TS}}\xspace}_\gamma)_{\gamma\in\Gamma}\rangle\xspace}$, ${\ensuremath{\llbracket {\ensuremath{\mathcal{A}}\xspace}\rrbracket}}$ is the labeled transition system $\langle C, c^0, {\ensuremath{\smash{\widetilde{\Sigma}}}},
\Longrightarrow \rangle$ where: $(i)$ $C$ contains all the pairs $({\ensuremath{G}},{\ensuremath{\vec{d}}})$ where ${\ensuremath{\vec{d}}}\in{\ensuremath{\mathsf{vals}\left({\ensuremath{\mathcal{X}}\xspace}\right)}}$, and ${\ensuremath{G}}$ is a [$\textsc{Ctg}$]{}of ${\ensuremath{\mathcal{A}}\xspace}$, $(ii)$ $c^0=({\ensuremath{G}}^0,{\ensuremath{\vec{d}}}^0)$ with ${\ensuremath{\vec{d}}}^0(x)=d_0$ for all $x\in{\ensuremath{\mathcal{X}}\xspace}$, and ${\ensuremath{G}}^0={\langle\{v^0\},\emptyset,\lambda,
{\ensuremath{\textit{queue}}},{\ensuremath{\textit{state}}}\rangle\xspace}$, where $v^0$ is a *task node*, $\lambda(v^0)=main$, ${\ensuremath{\textit{state}}}(v^0)=s^0_{\text{main}}$ and ${\ensuremath{\textit{queue}}}(v^0)=\imath$, $(iii)$ ${\ensuremath{\smash{\widetilde{\Sigma}}}}=\Sigma\operatorname{\mathaccent\cdot\cup}\{{\ensuremath{\varepsilon}\xspace}\}$ and $(iv)$ $\big(({\ensuremath{G}},{\ensuremath{\vec{d}}}),a,({\ensuremath{G}}',{\ensuremath{\vec{d}}}')\big)\in\Longrightarrow$ iff one of the following holds:
Async. dispatch:
: $a={\ensuremath{{\ensuremath{\mathtt{dispatch_a}}\xspace}}}(q,\gamma)$, ${\ensuremath{\vec{d}}}'={\ensuremath{\vec{d}}}$, and there are $\delta=(s,a,s')\in \Delta$ and ${\ensuremath{G}}''\in{\ensuremath{\textsf{step}}}(\delta)({\ensuremath{G}})$ s.t.: ${\ensuremath{G}}'={\ensuremath{\textsf{enqueue}}}(q,\gamma)({\ensuremath{G}}'')$.
Sync. dispatch:
: $a={\ensuremath{{\ensuremath{\mathtt{dispatch_s}}\xspace}}}(q,\gamma)$, ${\ensuremath{\vec{d}}}'={\ensuremath{\vec{d}}}$ and there are $\delta=(s,a,s')\in \Delta$ and ${\ensuremath{G}}''\in{\ensuremath{\textsf{step}}}(\delta({\ensuremath{G}}))$ s.t.: ${\ensuremath{G}}'={\ensuremath{\textsf{letwait}}}(v,v')\big({\ensuremath{\textsf{enqueue}}}(q,\gamma)({\ensuremath{G}}'')\big)$ where $v$ is the node whose ${\ensuremath{\textit{state}}}$ has changed during the ${\ensuremath{\textsf{step}}}$ operation, and $v'$ is the fresh node that has been created by the ${\ensuremath{\textsf{enqueue}}}$ operation. That is, a queue vertex $v'$ labeled by $\gamma$ is added to $q$ and a *wait* edge is added between the node $v$ representing the task that performs the *synchronous* dispatch, and $v'$, as the dispatch is *synchronous*.
Test:
: $a=g\in{\ensuremath{\mathsf{guards}\left({\ensuremath{\mathcal{X}}\xspace}\right)}}$, ${\ensuremath{\vec{d}}}'={\ensuremath{\vec{d}}}$, ${\ensuremath{\vec{d}}}\models g$, and there is $\delta=(s,a,s')\in \Delta$ s.t. ${\ensuremath{G}}'\in{\ensuremath{\textsf{step}}}(\delta)({\ensuremath{G}})$.
Assignment:
: $a=x\gets v\in{\ensuremath{\mathsf{assign}\left({\ensuremath{\mathcal{X}}\xspace}\right)}}$, ${\ensuremath{\vec{d}}}'(x)=v$, for all $x'\neq x$: ${\ensuremath{\vec{d}}}'(x)={\ensuremath{\vec{d}}}(x)$ and there is $\delta=(s,a,s')\in \Delta$ s.t. ${\ensuremath{G}}'\in{\ensuremath{\textsf{step}}}(\delta)({\ensuremath{G}})$.
Scheduler action:
: $a={\ensuremath{\varepsilon}\xspace}$, ${\ensuremath{\vec{d}}}'={\ensuremath{\vec{d}}}$ and:
- either there is a final vertex $v$ s.t. ${\ensuremath{G}}'={\ensuremath{G}}\setminus v$;
- or there is $q\in{CQID\xspace}$ s.t. ${\ensuremath{\textit{head}}}(q,{\ensuremath{G}})\neq\bot$ and ${\ensuremath{G}}'={\ensuremath{\textsf{dequeue}}}(q)({\ensuremath{G}})$. That is, the scheduler schedules a block (represented by $v$) from a concurrent queue.
- or there is $q\in {SQID\xspace}$ s.t. ${\ensuremath{\textit{head}}}(q,{\ensuremath{G}})=v$, $v$ is *unblocked*, as well as ${\ensuremath{G}}'={\ensuremath{\textsf{letwait}}}({\ensuremath{\textit{head}}}(q,G''),v)(G'')$ and $G''={\ensuremath{\textsf{dequeue}}}(q)({\ensuremath{G}})$. That is, the scheduler schedules a block (represented by $v$) from the serial queue $q$. As the queue is serial, a *wait* edge is inserted between the next waiting block in $q$ (now represented by ${\ensuremath{\textit{head}}}(q,G'')$) and $v$.
A *run* $\rho$ of a [<span style="font-variant:small-caps;">Qdas</span>]{}is an alternating sequence $c_0
a_1 c_1 a_2\dots a_n c_n$ of configurations and actions where $(c_i,a_{i+1},c_{i+1})\in\Longrightarrow$ for all $0\leq i < n$ and $c_0=c^0$. A run is *finite* if this sequence is finite. A configuration $c$ is *reachable* in ${\ensuremath{\mathcal{A}}\xspace}$ iff there exists a finite run $c_0 a_1 c_1 a_2\dots a_n c_n$ of ${\ensuremath{\mathcal{A}}\xspace}$ s.t. $c_n=c$. We denote by $\operatorname{\textit{Reach}}({\ensuremath{\mathcal{A}}\xspace})$ the set of all reachable configurations of ${\ensuremath{\mathcal{A}}\xspace}$.
The decision problem on [<span style="font-variant:small-caps;">Qdas</span>]{}we mainly consider in this work is the *Parikh coverability problem*: given a [<span style="font-variant:small-caps;">Qdas</span>]{}${\ensuremath{\mathcal{A}}\xspace}$ with set of locations $S$ and a function $f:S\mapsto {\ensuremath{\mathbb{N}}\xspace}$, it asks whether there is $c=({\ensuremath{G}},{\ensuremath{\vec{d}}})\in\operatorname{\textit{Reach}}({\ensuremath{\mathcal{A}}\xspace})$ s.t. $f{\preceq}{\ensuremath{\mathsf{Parikh}}}({\ensuremath{G}})$. When the answer to this question is ‘yes’, we say that $f$ is *Parikh-coverable* in ${\ensuremath{\mathcal{A}}\xspace}$. It is well-known that meaningful verification questions can be reduced to this problem. For instance, consider a *mutual exclusion* question, asking whether it is possible to reach, in a [<span style="font-variant:small-caps;">Qdas</span>]{}${\ensuremath{\mathcal{A}}\xspace}$, a configuration in which at least two tasks are executing the same block $\gamma$ and are in the same control state $s$. If yes, the mutual exclusion (of control state $s$) is violated. This can be encoded into an instance of the Parikh coverability problem, where $f(s)=2$ and $f(s')=0$ for all $s'\neq s$, and would allow, for example, to verify if there are more than one block of type `increase` running in Example\[ex:gcd-matrixmult\].
In addition, we look at the *(universal) termination problem*: given a [<span style="font-variant:small-caps;">Qdas</span>]{}${\ensuremath{\mathcal{A}}\xspace}$, it asks whether all executions of ${\ensuremath{\mathcal{A}}\xspace}$ are finite, i.e., there is no infinite run of ${\ensuremath{\mathcal{A}}\xspace}$. Regarding Example\[ex:gcd-matrixmult\], this permits to test whether the `main` task terminates, i.e., all dispatched blocks terminate.
From the Parikh coverability problem to Termination\[sec:parikh-cover-probl\]
=============================================================================
Before regarding the termination problem, we first study in this section the Parikh coverability problem from a computational point of view. As expected, this problem is undecidable in general. However, when restricting the types of queues and dispatches that are allowed, it is possible to retain decidability. In these cases, we characterize the complexity of the problem. Formally, we consider the following subclasses of [<span style="font-variant:small-caps;">Qdas</span>]{}. A [<span style="font-variant:small-caps;">Qdas</span>]{}${\ensuremath{\mathcal{A}}\xspace}$ with set of transitions $\Delta$, set of serial queues ${SQID\xspace}$ and set of concurrent queues ${CQID\xspace}$, is *synchronous* iff there exists no $(s,a,s')\in\Delta$ with $a\in\{{\ensuremath{\mathtt{dispatch\_a}}\xspace}\}\times {QID\xspace}\times \Gamma$; it is *asynchronous* iff there exists no $(s,a,s')\in\Delta$ with $a\in\{{\ensuremath{\mathtt{dispatch\_s}}\xspace}\}\times {QID\xspace}\times \Gamma$; it is *concurrent* iff ${SQID\xspace}=\emptyset$ and ${CQID\xspace}\neq\emptyset$; it is *serial* iff ${CQID\xspace}=\emptyset$ and ${SQID\xspace}\neq\emptyset$; it is *queueless* iff ${CQID\xspace}={SQID\xspace}=\emptyset$.
#### **Queueless [<span style="font-variant:small-caps;">Qdas</span>]{}:**
In a queueless [<span style="font-variant:small-caps;">Qdas</span>]{}, there is no dispatch possible, so the only task that can execute at all time is the `main` one. Thus, configurations of queueless [<span style="font-variant:small-caps;">Qdas</span>]{}can be encoded as tuples $(s,{\ensuremath{\vec{d}}})$, where $s$ is a state of `main`, and ${\ensuremath{\vec{d}}}$ is a valuation of the variables. Hence queueless [<span style="font-variant:small-caps;">Qdas</span>]{}are essentially [<span style="font-variant:small-caps;">Lts</span>]{}with variables over a finite data domain, thus:
\[prop:queueless\] The Parikh coverability is [<span style="font-variant:small-caps;">PSpace-C</span>]{} for queueless [<span style="font-variant:small-caps;">Qdas</span>]{}.
#### **Synchronous [<span style="font-variant:small-caps;">Qdas</span>]{}:**
(a)
/ in [0/0,1/1]{} (1\*,0) node\[circle,draw\] () ; (2,0) node (2) […]{}; (3,0) node\[circle,draw,inner sep=1pt\] (3) ; (4,0) node\[rectangle,inner sep=5pt,draw\] (4) ; / in [0/1,1/2,2/3,3/4]{} () edge \[->\] ();
(b)
/ in [0/0,1/1]{} (1\*,0) node\[circle,draw\] () ; (3,0) node\[circle,draw\] (3) ; (1\*2,0) node (2) […]{}; / in [0/1,1/2,2/3]{} () edge \[->\] ();
In synchronous [<span style="font-variant:small-caps;">Qdas</span>]{}, there is no concurrency in the sense there is at most one running task that can fire an action at all times. All the other tasks have necessarily performed a *synchronous* dispatch and are thus blocked. More precisely, in every reachable configuration $({\ensuremath{G}},{\ensuremath{\vec{d}}})$ of a synchronous [<span style="font-variant:small-caps;">Qdas</span>]{}, ${\ensuremath{G}}$ is of one of the forms depicted in Fig. \[fig:config-sync\] (i.e. $v_0,\dots,v_{n-1}\in V_T$ and either $v_n\in V_T$ or $v_n\in
V_C$). When the current [$\textsc{Ctg}$]{}is of the form Fig. \[fig:config-sync\](a), the only possible action is that the scheduler starts running $v_n$’s block and we obtain a graph of the form Fig. \[fig:config-sync\](b). In the case where the [$\textsc{Ctg}$]{}is of the form (a), either $v_n$ terminates, which removes $v_n$ from the [$\textsc{Ctg}$]{}, or $v_n$ executes an internal action, which does not change the shape of the [$\textsc{Ctg}$]{}, or $v_n$ does a synchronous call, which adds a call vertex as successor of $v_n$ which will be directly scheduled. W.l.o.g., we assume in the following that for synchronous [<span style="font-variant:small-caps;">Qdas</span>]{}the combined action of [${\ensuremath{\mathtt{dispatch_s}}\xspace}$]{}and scheduling the dispatched block is atomic.
For a [$\textsc{Ctg}$]{}${\ensuremath{G}}$ and $w\in S^*$, we write ${\ensuremath{G}}\triangleright w$ iff for all $0\leq i\leq n$: $w_i={\ensuremath{\textit{state}}}(v_i)$ and the empty [$\textsc{Ctg}$]{}is mapped to the empty word ${\ensuremath{\varepsilon}\xspace}$. Given a synchronous [<span style="font-variant:small-caps;">Qdas</span>]{}${\ensuremath{\mathcal{A}}\xspace}$ with set of local states $S$ as before, we can build a pushdown system with data ${\ensuremath{\mathcal{P}}\xspace}_{\ensuremath{\mathcal{A}}\xspace}$ such that, at all times, the current location of $P_{\ensuremath{\mathcal{A}}\xspace}$ encodes the current location of the (single) running block in ${\ensuremath{\mathcal{A}}\xspace}$, and the stack content records the sequence of synchronous dispatches, as described above. A guard or assignment in ${\ensuremath{\mathcal{A}}\xspace}$ is kept as is in ${\ensuremath{\mathcal{P}}\xspace}_{\ensuremath{\mathcal{A}}\xspace}$. A synchronous dispatch $(s,{\ensuremath{{\ensuremath{\mathtt{dispatch_s}}\xspace}}}(q,\gamma),s')$ in ${\ensuremath{\mathcal{A}}\xspace}$ is simulated by a push of $s'$ (to record the local state that has to be reached when the callee terminates) and moves the current state of ${\ensuremath{\mathcal{P}}\xspace}_{\ensuremath{\mathcal{A}}\xspace}$ to the initial state of $\gamma$. The termination of a block is simulated by a pop (and we encode the termination of $main$ in testing the stack’s emptiness).
[proposition]{}[propsyncqdaspds]{} \[prop:sync\_qdas\_pds\] Given a synchronous [<span style="font-variant:small-caps;">Qdas</span>]{}${\ensuremath{\mathcal{A}}\xspace}$, then we can construct a pushdown system with data ${\ensuremath{\mathcal{P}}\xspace}_{\ensuremath{\mathcal{A}}\xspace}$ such that the following holds: for any run $\rho=c_0a_1c_1\dots a_nc_n$ of ${\ensuremath{\mathcal{A}}\xspace}$, there exists a run $\pi=x_0a_1x_1\dots a_nx_n$ in ${\ensuremath{\mathcal{P}}\xspace}_{\ensuremath{\mathcal{A}}\xspace}$ such that for all $c_i=({\ensuremath{G}}_i,{\ensuremath{\vec{d}}}_i)$ and $x_i=(s_i,w_i,{\ensuremath{\vec{d}}}'_i)$ we have ${\ensuremath{\vec{d}}}_i={\ensuremath{\vec{d}}}_i'$ and ${\ensuremath{G}}_i\triangleright w_i$ ($0\leq i \leq n$), and vice versa.
The previous proposition allows to derive results on the reachability problem. However, we are interested in the Parikh coverability problem. Let $f$ be a Parikh image of ${\ensuremath{\mathcal{A}}\xspace}$. Then, by Proposition \[prop:syncqdas\_parikh\_sim\], looking for a reachable configuration of ${\ensuremath{\mathcal{A}}\xspace}$ that covers $f$ amounts to finding a reachable configuration $(s_i,w_i,{\ensuremath{\vec{d}}}_i)$ of ${\ensuremath{\mathcal{P}}\xspace}_{\ensuremath{\mathcal{A}}\xspace}$ s.t. the Parikh image $P$ of $w_i$ is s.t. $f{\preceq}P$ (as the [$\textsc{Ctg}$]{}is encoded by the stack content $w_i$). To achieve this, we augment ${\ensuremath{\mathcal{P}}\xspace}_{\ensuremath{\mathcal{A}}\xspace}$ with a *widget* that works as follows. In any location of ${\ensuremath{\mathcal{P}}\xspace}_{\ensuremath{\mathcal{A}}\xspace}$, we can jump non-deterministically to the widget. Then, the widget pops all the values from the stack, and checks that at least $f(s)$ symbols $s$ are present on the stack. The widget jumps to an accepting state iff it is the case. We call ${\ensuremath{\mathcal{P}}\xspace}_{{\ensuremath{\mathcal{A}}\xspace},f}$ the resulting [<span style="font-variant:small-caps;">Pds</span>]{}. Clearly, one can build such a widget for all $f$, and this effectively reduces the Parikh coverability problem of [<span style="font-variant:small-caps;">Qdas</span>]{}to the location reachability problem of [<span style="font-variant:small-caps;">Pds</span>]{}. Moreover, for all $f$, the widget is of size exponential in $\vert S\vert$ and exponential in the binary encoding of $max_{s\in S}f(s)$. Hence, building ${\ensuremath{\mathcal{P}}\xspace}_{{\ensuremath{\mathcal{A}}\xspace},f}$ requires exponential time:
[proposition]{}[propsyncqdasparikhsim]{} \[prop:syncqdas\_parikh\_sim\] Given a synchronous [<span style="font-variant:small-caps;">Qdas</span>]{}${\ensuremath{\mathcal{A}}\xspace}$ with states $S$ and a function , then one can generate a [<span style="font-variant:small-caps;">Pds</span>]{} ${\ensuremath{\mathcal{P}}\xspace}_{{\ensuremath{\mathcal{A}}\xspace},f}$ of size exponential in ${\ensuremath{\mathcal{A}}\xspace}$ and a state $s$ of ${\ensuremath{\mathcal{P}}\xspace}_{{\ensuremath{\mathcal{A}}\xspace},f}$, s.t. ${\ensuremath{\mathcal{P}}\xspace}_{{\ensuremath{\mathcal{A}}\xspace},f}$ reaches $s$ iff $f$ is Parikh coverable in ${\ensuremath{\mathcal{A}}\xspace}$.
As testing emptiness of a pushdown system without data is [<span style="font-variant:small-caps;">PTime-C</span>]{} [@bouajjani-a-1997-135-a], the Parikh coverability problem is in [<span style="font-variant:small-caps;">ExpTime</span>]{}for *synchronous* [<span style="font-variant:small-caps;">Qdas</span>]{}(with both types of queues). A matching lower bound is obtained by reducing the reachability question of [<span style="font-variant:small-caps;">Pds</span>]{}with data (see Proposition \[prop:reach\_pds\_data\]). This reduction requires only one *concurrent* queue, so the Parikh reachability problem is [<span style="font-variant:small-caps;">ExpTime</span>]{}-hard for *synchronous concurrent* [<span style="font-variant:small-caps;">Qdas</span>]{}. Hence we derive the following:
\[thm:syncqdas\] The Parikh coverability problem is [<span style="font-variant:small-caps;">ExpTime-C</span>]{}for synchronous and for synchronous concurrent [<span style="font-variant:small-caps;">Qdas</span>]{}.
Let us take a closer look on the dispatches that happen in runs of synchronous [<span style="font-variant:small-caps;">Qdas</span>]{}that have only *serial* queues. Here, each task except the `main` task blocks the queue it is started from. Hence, any other block dispatched to these already blocked queues deadlocks. Thus, all reachable [$\textsc{Ctg}$]{}have at most $\vert SQID\vert +2$ vertices. Hence, the pushdown systems used in all previous constructions have bounded stack height, and we can apply test on a finite transition system. The lower bound can be derived from Proposition\[prop:queueless\]. by testing the emptiness of the intersection of $n$ finite processes, that is ${\textsc{PSpace}\xspace}$-complete [@kozen-d-1977-254-a].
\[thm:serialsyncqdas\] The Parikh coverability problem is [<span style="font-variant:small-caps;">PSpace-C</span>]{}for serial synchronous [<span style="font-variant:small-caps;">Qdas</span>]{}.
#### **Concurrent asynchronous [<span style="font-variant:small-caps;">Qdas</span>]{}:**
Let us now establish a relationship between *concurrent asynchronous [[<span style="font-variant:small-caps;">Qdas</span>]{}]{}* and Petri nets that proves that the Parikh coverability problem is [<span style="font-variant:small-caps;">ExpSpace</span>]{}-complete. We first show how to reduce the [<span style="font-variant:small-caps;">Qdas</span>]{}Parikh coverability problem to the Petri net coverability problem. Given a concurrent asynchronous [<span style="font-variant:small-caps;">Qdas</span>]{}${\ensuremath{\mathcal{A}}\xspace}$, we construct a Petri net $N_{\ensuremath{\mathcal{A}}\xspace}$ as follows: The places of $N_{\ensuremath{\mathcal{A}}\xspace}$ are $({\ensuremath{\mathcal{X}}\xspace}\times{\ensuremath{\mathbb{D}}\xspace})\cup S$. Each place $s\in S$ counts how many blocks are currently running and are in state $s$. Each place $(x,d)$ encodes the fact that variable $x$ contains value $d$ in the current valuation. Remark that we have no place to encode the contents of the queue, as the dispatch of block $\gamma$ directly creates a new token in $s^0_\gamma$. This encoding is, however, correct with respect to to the *Parikh coverability problem*, as ${\ensuremath{\mathsf{Parikh}}}(G)$ does not distinguish between a block $\gamma$ that is waiting in a queue, and a task executing $\gamma$ in its initial state. Thus:
[proposition]{}[propfromqdastopn]{} \[prop:from-qdas-to-pn\] For all concurrent asynchronous [<span style="font-variant:small-caps;">Qdas</span>]{}${\ensuremath{\mathcal{A}}\xspace}$ with set of location $S$, we can build, in polynomial time, a Petri net $N_{\ensuremath{\mathcal{A}}\xspace}$ s.t. $f$ is Parikh-coverable in ${\ensuremath{\mathcal{A}}\xspace}$ iff $m\in{\ensuremath{\textit{Cover}}}(N_{\ensuremath{\mathcal{A}}\xspace})$, where $m$ is the marking s.t. for all $s\in S$: $m(s)=f(s)$ and for all $p\in
P\setminus S$: $m(p)=0$.
Let us now reduce the Petri net coverability problem to the [<span style="font-variant:small-caps;">Qdas</span>]{}Parikh coverability problem. Let $N={\langleP,T,m_0\rangle\xspace}$ be a Petri net. We associate to $N$ the concurrent asynchronous [<span style="font-variant:small-caps;">Qdas</span>]{}${{\ensuremath{\mathcal{A}}\xspace}}_N={\langle{CQID\xspace}, \emptyset, \Gamma, \mathtt{main}, {\ensuremath{\mathcal{X}}\xspace}, \Sigma,
({\ensuremath{\mathcal{TS}}\xspace}_\gamma)_{\gamma\in\Gamma}\rangle\xspace}$, on the finite domain ${\ensuremath{\mathbb{D}}\xspace}=\{0,1\}$, where ${CQID\xspace}=\{C\}$, $\Gamma=\{main,trans\}\cup P$, ${\ensuremath{\mathcal{X}}\xspace}=\{v_p\mid p\in P\}$ and $({\ensuremath{\mathcal{TS}}\xspace}_\gamma)_{\gamma\in\Gamma}$ is given by the pseudo-code in Fig. \[fig:simulossyPN\] (this construction is an extension of a construction found in [@ganty-p-2010--a]). We assume that, for $\gamma\in\{\mathtt{trans},\mathtt{main}\}$ ${\ensuremath{s^{\ell}_{\mathtt{\gamma}}}}$ is the location of $\gamma$’s [<span style="font-variant:small-caps;">Lts</span>]{}that is reached when the control reaches line $\ell$. Let $G={\langleV, E,
\lambda, {\ensuremath{\textit{queue}}}, {\ensuremath{\textit{state}}}\rangle\xspace}$ be a [$\textsc{Ctg}$]{}for ${\ensuremath{\mathcal{A}}\xspace}_N$, and let $m$ be a marking of $N$. Then, we say that *$G$ encodes $m$*, written $G\rhd m$ iff $(i)$ ${\ensuremath{\mathsf{Parikh}}}(G)({\ensuremath{s^{14}_{\mathtt{trans}}}})={\ensuremath{\mathsf{Parikh}}}(G)({\ensuremath{s^{8}_{\mathtt{main}}}})=1$, $(ii)$ for all $p\in P$: ${\ensuremath{\mathsf{Parikh}}}(G)(s^0_p)=m(p)$ and $(iii)$ for all $p\in P$, for all $s\in S_p\setminus\{s^0_p\}$: ${\ensuremath{\mathsf{Parikh}}}(G)(s)=0$. Thus, intuitively, a [$\textsc{Ctg}$]{}$G$ encodes a marking $m$ iff `main` is at line 8, `trans` is at line 14, $m(p)$ counts the number of `p` blocks that are either in $C$ or executing but at their initial state, and there are no `p` blocks that are in state $s_p^{mid}$ or $s_p^{fin}$.
``` {numberblanklines="false"}
def main():
for each $p\in P$:
$v_p$ := 0
select $k_p\in\{0,\ldots,m_0(p)\}$
for i = 0...$k_p$:
dispatch_a($C$, p())
dispatch_a($C$, trans())
while(true): do nothing (*@\mbox{ }\\[-0.5ex]@*)
block p(): // For all $p\in P$
while($v_p = 0$): do nothing
$v_p$ := 0
```
``` {startFrom="last"}
block trans():
while(true):
select $t=(I_t,O_t)\in T$
for each $p\in P$ s.t. $I_t(p)=1$:
$v_p$ := true
while($\exists p\in P$: $v_p=1$): do nothing
for each $p\in P$ s.t. $O_t(p)=1$:
dispatch_a($C$, p())
```
The intuition behind the construction is as follows. Each run of the [<span style="font-variant:small-caps;">Qdas</span>]{}${\ensuremath{\mathcal{A}}\xspace}_N$ starts with an initialization phase, where `main` initializes all the $v_p$ variables to $0$ and dispatches, for all $p\in P$, $k_p$ blocks `p` with $k_p\leq m_0(p)$, then dispatches a call to `trans`. At that point, the only possible action is that the scheduler dequeues all the blocks. All the `p` tasks are then blocked, as they need that $v_p=1$ to proceed and terminate. Then, `trans` cyclically picks a transition $t$, sets to $1$ all the variables $v_p$ s.t. $t$ consumes a token in $p$, and waits that all the $v_p$ variables return to $0$. This can only happen because *at least* $I_t(p)$ `p` tasks have terminated, for all $p\in P$. So, when `trans` reaches line 19, the encoded marking has been decreased by *at least* $I_t$. Remark that more than $I_t(p)$ `p` tasks could terminate, as they run concurrently, and the lines 11 and 12 do not execute atomically. Then, `trans` dispatches one new `p` block iff $t$ produces a token in $p$. This increases the encoded marking by $O_t$, so the effect of one iteration of the main `while` loop of `trans` is to simulate the effect of $t$, plus a possible token loss. Hence, the resulting marking is guaranteed to be in ${\ensuremath{\textit{Cover}}}(N)$ (but maybe not in $\operatorname{\textit{Reach}}(N)$). This is formalized by the following proposition:
[proposition]{}[propfrompntoqdas]{}\[prop:from-pn-to-qdas\] For all Petri nets $N$, we can build, in polynomial time, a concurrent asynchronous [<span style="font-variant:small-caps;">Qdas</span>]{}${\ensuremath{\mathcal{A}}\xspace}_N$ s.t. $m\in{{\ensuremath{\textit{Cover}}}}(N)$ iff there exists $(G,{\ensuremath{\vec{d}}})\in\operatorname{\textit{Reach}}({\ensuremath{\mathcal{A}}\xspace}_N)$ with $G\rhd m$.
\[thm:concasyncqdas\] The Parikh coverability problem is [<span style="font-variant:small-caps;">ExpSpace</span>]{}-complete for concurrent asynchronous [<span style="font-variant:small-caps;">Qdas</span>]{}.
#### **Asynchronous Serial [<span style="font-variant:small-caps;">Qdas</span>]{}:**
Let us show that for the class of [<span style="font-variant:small-caps;">Qdas</span>]{}with one serial queue, and where asynchronous dispatches are allowed, the Parikh coverability problem is *undecidable*. We establish this by a reduction from the control-state reachability problem in a [fifo]{}system which is known to be undecidable [@brand-d-1983-323-a].
Intuitively, we use the serial queue to model the unbounded, reliable fifo queue where sending a message $m$ is encoded as asynchronously dispatching a block $\gamma_m$. This block $\gamma_m$ contains the control-flow of receiving $m$, i.e., that will resume the [fifo]{}system’s execution directly after receiving $m$. The [fifo]{}system’s global state is guarded in a global variable. Receiving a certain message $m$ is encoded as terminating the currently running task and assuring (via a global variable) that the succeeding task’s type is the one of the expected message.
\[the:async-seri-undec\] The Parikh coverability problem is undecidable for asynchronous [<span style="font-variant:small-caps;">Qdas</span>]{}with at least one serial queue.
#### **Concurrent [<span style="font-variant:small-caps;">Qdas</span>]{}:**
Let us show that, once we allow both synchronous and asynchronous dispatches in a *concurrent* [<span style="font-variant:small-caps;">Qdas</span>]{}, the Parikh coverability problem becomes undecidable. For that purpose, we reduce the reachability problem of two counter systems.
The crux of the construction is the use of variables, i.e., global memory, to implement a rendez-vous synchronization. Given two distinct tasks, one can use their nested access to two lock variables to guard a shared data variable by assuring that a value written to the variable must be read before it is overwritten.
Let us give the construction’s intuition: Each counter is encoded similarly to the construction for synchronous [<span style="font-variant:small-caps;">Qdas</span>]{}as pushdown stack over a singleton alphabet, i.e., a sequence of nested synchronous dispatched blocks, these are controlled via rendez-vous from the [$\textit{main}$]{}task that in the beginning asynchronously dispatched the two counters.
\[thm:concqdasundec\] The Parikh coverability problem is undecidable for concurrent [<span style="font-variant:small-caps;">Qdas</span>]{}that use both synchronous and asynchronous dispatches.
#### **Termination Problem:**
We use the previous constructions to directly lift the undecidability results from the Parikh coverability problem to the termination problem. The close connection of synchronous [<span style="font-variant:small-caps;">Qdas</span>]{}with [<span style="font-variant:small-caps;">Pds</span>]{}(with data) allows to directly derive an [<span style="font-variant:small-caps;">ExpTime</span>]{}algorithm for the termination problem from the emptiness testing of Büchi [<span style="font-variant:small-caps;">Pds</span>]{} [@esparza-j-2000-232-a]. Up to our knowledge, no completeness result is known for the latter problem, thus leaving a gap to the directly derivable [<span style="font-variant:small-caps;">PSpace</span>]{}-hardness via finite systems. The result for asynchronous concurrent [<span style="font-variant:small-caps;">Qdas</span>]{}directly follows from Petri nets [@lipton; @rackoff].
\[thm:termination\] The termination problem is [<span style="font-variant:small-caps;">PSpace-C</span>]{}for synchronous serial [<span style="font-variant:small-caps;">Qdas</span>]{}, it is in [<span style="font-variant:small-caps;">ExpTime</span>]{}and [<span style="font-variant:small-caps;">PSpace</span>]{}-hard for synchronous [<span style="font-variant:small-caps;">Qdas</span>]{}, and it is [<span style="font-variant:small-caps;">ExpSpace-C</span>]{}for asynchronous concurrent [<span style="font-variant:small-caps;">Qdas</span>]{}. It is undecidable for asynchronous serial [<span style="font-variant:small-caps;">Qdas</span>]{}, and [<span style="font-variant:small-caps;">Qdas</span>]{}that use both synchronous and asynchronous dispatches.
Extending QDAS with Fork/Join\[sec:forkjoin\]
=============================================
We return to the introductory matrix multiplication example. The crux of the algorithm is the parallel for-loop that *forks* a finite number of subtasks and waits for their termination (*join*). The latter had to be implemented via a global semaphore which $(i)$ restricts the number of forkable tasks by the underlying finite value domain, and $(ii)$ needs to be properly guarded by the programmer for access outside fork and join. In the following we thus want to extend [<span style="font-variant:small-caps;">Qdas</span>]{}by an explicit fork/join construct (which also exists in GCD). Further, the given matrix multiplication algorithm depended on an a priori fix size for the factor matrices, however, in practice, one wants to verify the algorithm for any possible (correct) input of any size. Thus, we need to consider the verification of extended [<span style="font-variant:small-caps;">Qdas</span>]{}where the number of forked tasks is parametrized by the input.
As fork/join behaviour relies on asynchronously dispatching tasks on a concurrent queue, we ignore in the following synchronous dispatches and serial queues, thus also partially avoiding the previous basic undecidability results. Note that asynchronous concurrent [<span style="font-variant:small-caps;">Qdas</span>]{}can be regarded as over-approximations of all other classes of [<span style="font-variant:small-caps;">Qdas</span>]{}.
#### **QDAS extended by fork/join**
An *[<span style="font-variant:small-caps;">Qdas</span>]{}extended by fork/join ([<span style="font-variant:small-caps;">eQdas</span>]{})* is a tuple $\langle {CQID\xspace}, \emptyset,
\Gamma, main, {\ensuremath{\mathcal{X}}\xspace}, \Sigma, ({\ensuremath{\mathcal{TS}}\xspace}_\gamma)_{\gamma\in\Gamma}\rangle$ that is equivalent to a [<span style="font-variant:small-caps;">Qdas</span>]{}except that we replace in $\Sigma$ the synchronous dispatch by the following action: $\{{\ensuremath{{\ensuremath{\mathtt{forkjoin}}\xspace}}\xspace}\}\times {CQID\xspace}\times \Gamma \times ({\ensuremath{\mathbb{N}}\xspace}\cup\{\ast\})$. The *parameter* of a [${\ensuremath{\mathtt{forkjoin}}\xspace}$]{}action is the last value of the tuple. An [<span style="font-variant:small-caps;">eQdas</span>]{}is *$\ast$-free* if in all ${\ensuremath{\mathcal{TS}}\xspace}_\gamma$ for $\gamma\in\Gamma$ the parameter of the [${\ensuremath{\mathtt{forkjoin}}\xspace}$]{}action is not $\ast$.
The semantics of an [<span style="font-variant:small-caps;">eQdas</span>]{} is given analogous to standard [<span style="font-variant:small-caps;">Qdas</span>]{}as transition system $\langle C, c^0,$ $ \Sigma, \Longrightarrow \rangle$ where we additionally extend the transition relation $\Longrightarrow$ given by tuples $\big(({\ensuremath{G}},{\ensuremath{\vec{d}}}),a,({\ensuremath{G}}',{\ensuremath{\vec{d}}}')$ by the following case:
Fork/join:
: $a={\ensuremath{{\ensuremath{\mathtt{forkjoin}}\xspace}}\xspace}(q,\gamma,p)$ with $p\in({\ensuremath{\mathbb{N}}\xspace}\cup\{\ast\})$, ${\ensuremath{\vec{d}}}'={\ensuremath{\vec{d}}}$ and there are $\delta=(s,a,s')\in \Delta$, and ${\ensuremath{G}}''\in{\ensuremath{\textsf{step}}}(\delta({\ensuremath{G}}))$ such that: if $p=\ast$ then we choose non-deterministically an $n\in{\ensuremath{\mathbb{N}}\xspace}$, else $n=p$, so that ${\ensuremath{G}}'={\ensuremath{G}}''_n$ where ${\ensuremath{G}}''_0={\ensuremath{G}}''$ and for $0<i\leq n$ we define ${\ensuremath{G}}''_{i+1}={\ensuremath{\textsf{letwait}}}(v,v_{i+1}')\big({\ensuremath{\textsf{enqueue}}}(q,\gamma_{i+1})({\ensuremath{G}}''_i)\big)$ where $v$ is the node whose ${\ensuremath{\textit{state}}}$ has changed during the ${\ensuremath{\textsf{step}}}$ operation, and $v'_{i+1}$ is the fresh node that has been created by the ${\ensuremath{\textsf{enqueue}}}$ operation.
Intuitively, a [${\ensuremath{\mathtt{forkjoin}}\xspace}$]{}action appends a sequence of blocks to a queue by additionally adding a wait edge to each newly create node. Hence, the join is modeled by a separate action that is taken by the scheduler after deleting the wait edges.
The *extended Parikh coverability problem* asks, given an [<span style="font-variant:small-caps;">eQdas</span>]{}${\ensuremath{\mathcal{A}}\xspace}$ with locations ${\ensuremath{\mathcal{S}}\xspace}$ and a mapping $f:{\ensuremath{\mathcal{S}}\xspace}\rightarrow{\ensuremath{\mathbb{N}}\xspace}$, whether there exists $c=({\ensuremath{G}},{\ensuremath{\vec{d}}})\in\operatorname{\textit{Reach}}({\ensuremath{\mathcal{A}}\xspace})$ with $f{\preceq}{\ensuremath{\mathsf{Parikh}}}({\ensuremath{G}})$. The *extended termination problem* asks, given an [<span style="font-variant:small-caps;">eQdas</span>]{}${\ensuremath{\mathcal{A}}\xspace}$ whether there is no infinite run possible in ${\ensuremath{\mathcal{A}}\xspace}$.
As [${\ensuremath{\mathtt{forkjoin}}\xspace}$]{}actions with parameter $1$ are semantically equivalent to a synchronous dispatch action, we can directly reduce the two counter machine simulation from the proof of Theorem\[thm:concqdasundec\] to [<span style="font-variant:small-caps;">eQdas</span>]{}.
\[thm:forkjoin\_undec\] Both the extended Parikh coverability and extended termination problem are undecidable.
Consequently, we focus on two distinct over-approximations for [<span style="font-variant:small-caps;">eQdas</span>]{}in the following that allow us to give approximative answers to our verification problems.
#### **$\ast$-free [<span style="font-variant:small-caps;">eQdas</span>]{}:**
Given an [<span style="font-variant:small-caps;">eQdas</span>]{}${\ensuremath{\mathcal{A}}\xspace}$ that is $\ast$-free. We construct a Petri net $N_{\ensuremath{\mathcal{A}}\xspace}^\times$ by extending the previous construction from asynchronous concurrent [<span style="font-variant:small-caps;">Qdas</span>]{}to Petri nets as follows: As in the [<span style="font-variant:small-caps;">eQdas</span>]{}semantics we split a single [${\ensuremath{\mathtt{forkjoin}}\xspace}$]{}action of a block $\gamma$ on a queue $q$ with parameter $n\in{\ensuremath{\mathbb{N}}\xspace}$ into $(i)$ a fork transition that creates $n$ new tokens in $s_\gamma^0$, and $(ii)$ a subsequent join transition that depends on taking $n$ tokens from the place representing $f_\gamma$. Analogous to the proof of Proposition\[prop:from-qdas-to-pn\] we can show the following:
[proposition]{}[propastfreeapprox]{} \[prop:astfreeapprox\] For all $\ast$-free [<span style="font-variant:small-caps;">eQdas</span>]{}with set of location ${\ensuremath{\mathcal{S}}\xspace}$, we can build in polynomial time a Petri net $N_{\ensuremath{\mathcal{A}}\xspace}^\times$ st. $f$ is Parikh-coverable in ${\ensuremath{\mathcal{A}}\xspace}$ if $m\in{\ensuremath{\textit{Cover}}}(N_{\ensuremath{\mathcal{A}}\xspace}^\times)$, where $m$ is the marking s.t. for all $s\in S$: $m(s)=f(s)$ and for all $p\in
P\setminus S$: $m(p)=0$. Further, if $N_{\ensuremath{\mathcal{A}}\xspace}^\times$ terminates, then ${\ensuremath{\mathcal{A}}\xspace}$ is guaranteed to terminate.
As coverability and termination are decidable for Petri nets, we can decide extended Parikh coverability and extended termination on this over-abstraction.
#### **[<span style="font-variant:small-caps;">eQdas</span>]{}with $\ast$ parametrized fork/join:**
Given an [<span style="font-variant:small-caps;">eQdas</span>]{}${\ensuremath{\mathcal{A}}\xspace}$ that is not $\ast$-free, we construct a Petri net $N_{\ensuremath{\mathcal{A}}\xspace}^\ast$ as follows starting from the construction for asynchronous concurrent [<span style="font-variant:small-caps;">Qdas</span>]{}: For [${\ensuremath{\mathtt{forkjoin}}\xspace}$]{}actions whose parameter is not $\ast$, we proceed as in the above construction for $\ast$-free [<span style="font-variant:small-caps;">eQdas</span>]{}. However, we need to model the forking of an arbitrary number of blocks when the parameter of the [${\ensuremath{\mathtt{forkjoin}}\xspace}$]{}action equals $\ast$. For this, we use Petri nets extended with $\omega$-arcs. An outgoing arc of a transition labeled with $\omega$ adds an arbitrary number of tokens to the corresponding place, thus, we translate the fork of block $\gamma$ into an $\omega$-transition leading to place $s_\gamma^0$. The join is approximated by a transition that non-deterministically chose to advance the original workflow, ignoring not already terminated forked tasks. Thus by extending the proof of Proposition\[prop:from-qdas-to-pn\]:
[proposition]{}[propwithastapprox]{} \[prop:withastapprox\] For all [<span style="font-variant:small-caps;">eQdas</span>]{}with set of location ${\ensuremath{\mathcal{S}}\xspace}$, we can build in polynomial time a Petri net $N_{\ensuremath{\mathcal{A}}\xspace}^\ast$ st. $f$ is Parikh-coverable in ${\ensuremath{\mathcal{A}}\xspace}$ if $m\in{\ensuremath{\textit{Cover}}}(N_{\ensuremath{\mathcal{A}}\xspace}^\ast)$, where $m$ is the marking s.t. for all $s\in S$: $m(s)=f(s)$ and for all $p\in
P\setminus S$: $m(p)=0$. Further, if $N_{\ensuremath{\mathcal{A}}\xspace}^\ast$ terminates, then ${\ensuremath{\mathcal{A}}\xspace}$ is guaranteed to terminate.
We have recently shown that the termination problem is decidable for Petri nets with $\omega$-arcs [@geeraerts-g-2012--a]. Hence, also extended termination is decidable on the previous abstraction.
With respect to coverability, we can replace the $\omega$-arcs of $N_{\ensuremath{\mathcal{A}}\xspace}^\ast$ by a non-deterministic loop that adds an arbitrary number of tokens to the original arc’s target place. Note that this simple trick does not work for verifying termination. Consequently, we can use the known algorithms for coverability on this polynomially larger standard Petri net, and hence the extended Parikh coverability problem is decidable on this abstraction.
Conclusion & Outlook
====================
We introduce the, up to our knowledge, first formal model that grasps the core of [<span style="font-variant:small-caps;">Gcd</span>]{}, and that allows to derive basic results on the decidability of verification question thereupon. Due to the obvious undecidability issues of the model, we currently focus on several under- and over-approximative approaches (e.g., language bounded verification, graph minor based abstractions, novel Petri net extensions [@geeraerts-g-2012--a]) as well as enhancements for additional [<span style="font-variant:small-caps;">Gcd</span>]{}features like task groups, priorities, and timer events.
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Proof for Section \[sec:prelims\]
=================================
For the upper bound, we generate a reachability-equivalent [<span style="font-variant:small-caps;">Pds</span>]{}(without data) by encoding all possible data valuations into the pushdown system’s states. This leads to an exponential blowup of the state space. The lower bound can be derived from the reduction of the emptiness test of the intersection of a context-free language with $n$ regular languages that is known to be [<span style="font-variant:small-caps;">ExpTime</span>]{}-hard (hardness follows easily by a reduction from linearly bounded alternating Turing machines; a closely related problem, the reachability of pushdown systems with checkpoints, is shown to be [<span style="font-variant:small-caps;">ExpTime</span>]{}-hard in (\*).
[(\*) Javier Esparza, Antonín Kučera, and Stefan Schwoon: *Model checking LTL with regular valuations for pushdown systems*, in [ Information and Computation]{}, 186(2):355–376, 2003. ]{}
Proofs of Section\[sec:parikh-cover-probl\]
===========================================
#### **Synchronous [<span style="font-variant:small-caps;">Qdas</span>]{}:**
Let ${\ensuremath{\mathcal{A}}\xspace}$ be a synchronous [<span style="font-variant:small-caps;">Qdas</span>]{}with a set of locations $S$, a set of rules $\Delta$, a set of final states $F$, and set of queues ${SQID\xspace}$. Let ${\ensuremath{G}}$ be a [$\textsc{Ctg}$]{}of one of the forms given in Fig. \[fig:config-sync\], and let $w=w_0w_1\cdots w_n$ be a word in $S^*$. Then, *${\ensuremath{G}}$ is encoded by $w$*, written ${\ensuremath{G}}\triangleright w$, iff for all $0\leq i\leq n$: $w_i={\ensuremath{\textit{state}}}(v_i)$ and the empty [$\textsc{Ctg}$]{}is mapped to the empty word ${\ensuremath{\varepsilon}\xspace}$.
Given a synchronous [<span style="font-variant:small-caps;">Qdas</span>]{}${\ensuremath{\mathcal{A}}\xspace}={\langle {CQID\xspace}, {SQID\xspace}, \Gamma, main,
{\ensuremath{\mathcal{X}}\xspace}, \Sigma, ({\ensuremath{\mathcal{TS}}\xspace}_\gamma)_{\gamma\in\Gamma}\rangle\xspace}$ with set of local states $S$ as before, we build a pushdown system with data ${\ensuremath{\mathcal{P}}\xspace}_{\ensuremath{\mathcal{A}}\xspace}={\langleY,{\ensuremath{\mathcal{X}}\xspace},y^0,S,\Sigma_{\ensuremath{\mathcal{P}}\xspace},\Delta_{\ensuremath{\mathcal{P}}\xspace}\rangle\xspace}$ where:
the set of states is $Y=S\cup\{{\ensuremath{\varepsilon}\xspace}\}$ and the initial state is $y^0=s^0_{{\ensuremath{\textit{main}}\xspace}}$
$\Sigma_{\ensuremath{\mathcal{P}}\xspace}=\left(\{{\ensuremath{{\ensuremath{\mathtt{push}}\xspace}}},{\ensuremath{{\ensuremath{\mathtt{pop}}\xspace}}}\}\times S \right) \cup
\{{\ensuremath{{\ensuremath{\mathtt{empty?}}\xspace}}}\} \cup {\ensuremath{\mathsf{guards}\left({\ensuremath{\mathcal{X}}\xspace}\right)}}\cup{\ensuremath{\mathsf{assign}\left({\ensuremath{\mathcal{X}}\xspace}\right)}}$
a tuple $(y,a,y')$ is a transition rule in $\Delta_{\ensuremath{\mathcal{P}}\xspace}\subseteq
Y\times \Sigma_{\ensuremath{\mathcal{P}}\xspace}\times Y$ iff
- $a\in{\ensuremath{\mathsf{guards}\left({\ensuremath{\mathcal{X}}\xspace}\right)}}\cup{\ensuremath{\mathsf{assign}\left({\ensuremath{\mathcal{X}}\xspace}\right)}}$ and $(y,a,y')\in\Delta$
- $a={\ensuremath{{\ensuremath{\mathtt{push}}\xspace}}}(s')$, $(s,{\ensuremath{{\ensuremath{\mathtt{dispatch_s}}\xspace}}}(q,\gamma),s')\in\Delta$ and $y'=s^0_\gamma$
- $a={\ensuremath{{\ensuremath{\mathtt{pop}}\xspace}}}(s)$, $y\in F$ and $y'=s$
- $a={\ensuremath{{\ensuremath{\mathtt{empty?}}\xspace}}}$, $y=f_{main}$, and $y'={\ensuremath{\varepsilon}\xspace}$.
Thus, at all times, the current location of $P_{\ensuremath{\mathcal{A}}\xspace}$ encodes the current location of the (single) running block in ${\ensuremath{\mathcal{A}}\xspace}$, and the stack content records the sequence of synchronous dispatches, as described above. A guard or assignment in ${\ensuremath{\mathcal{A}}\xspace}$ is kept as is in $P_{\ensuremath{\mathcal{A}}\xspace}$. A synchronous dispatch $(s,{\ensuremath{{\ensuremath{\mathtt{dispatch_s}}\xspace}}}(q,\gamma),s')$ in ${\ensuremath{\mathcal{A}}\xspace}$ is simulated by a push of $s'$ (to record the local state that has to be reached when the callee terminates) and moves the current state of $P_{\ensuremath{\mathcal{A}}\xspace}$ to the initial state of $\gamma$. The termination of a block is simulated by a pop (and we use the ${\ensuremath{{\ensuremath{\mathtt{empty?}}\xspace}}}$ action for the termination of $main$).
We assert that the semantics of ${\ensuremath{\mathcal{P}}\xspace}_{\ensuremath{\mathcal{A}}\xspace}$ is the usual semantics for pushdown systems with data, i.e., an infinite transition system with configurations $c=(y,w,d)\in Y\times S^* \times {\ensuremath{\mathbb{D}}\xspace}^{\ensuremath{\mathcal{X}}\xspace}$. Thus, we can interpret configurations also as follows: $(x,d)\in S^*\times {\ensuremath{\mathbb{D}}\xspace}^{\ensuremath{\mathcal{X}}\xspace}$ with $x=w\cdot y\in S^*\cdot (S\cup\{{\ensuremath{\varepsilon}\xspace}\}$.
Let $({\ensuremath{G}},{\ensuremath{\vec{d}}})\in\operatorname{\textit{Reach}}({\ensuremath{\mathcal{A}}\xspace})$ be reachable by a run $({\ensuremath{G}}_0,{\ensuremath{\vec{d}}}_0)a_1({\ensuremath{G}}_1,{\ensuremath{\vec{d}}}_1)a_2\dots a_n({\ensuremath{G}}_n,{\ensuremath{\vec{d}}}_n)$. Then we can induce a run $(x_0,{\ensuremath{\widehat{\vec{d}}}}_0)a_1(x_1,{\ensuremath{\widehat{\vec{d}}}}_1)a_2\dots
a_n(x_n,{\ensuremath{\widehat{\vec{d}}}}_n)$ in ${\ensuremath{\mathcal{P}}\xspace}_{\ensuremath{\mathcal{A}}\xspace}$ such that ${\ensuremath{\vec{d}}}_i={\ensuremath{\widehat{\vec{d}}}}_i$ and ${\ensuremath{G}}_i\triangleright x_i$ for $0\leq i \leq n$.
By construction of ${\ensuremath{\mathcal{P}}\xspace}_{\ensuremath{\mathcal{A}}\xspace}$, $x_0\triangleright G_0$ and ${\ensuremath{\vec{d}}}_0={\ensuremath{\widehat{\vec{d}}}}_0$. We now assume that there exists a prefix of the [<span style="font-variant:small-caps;">Qdas</span>]{}’s run of length $0\leq
j\leq n$ of the form $({\ensuremath{G}}_0,{\ensuremath{\vec{d}}}_0)\dots
({\ensuremath{G}}_j,{\ensuremath{\vec{d}}}_j)$ such that there exists a run of the pushdown system $(x_0,{\ensuremath{\widehat{\vec{d}}}}_0)\dots(x_j,{\ensuremath{\widehat{\vec{d}}}}_j)$ that fullfills the induction hypothesis. We now consider the outcome of a [<span style="font-variant:small-caps;">Qdas</span>]{}transition labeled $a_{j+1}$. We know that ${\ensuremath{G}}_j$ must be a path of vertices $v_0\dots v_n$ connected by wait edges.
Sync. dispatch:
: dispatching a block $\gamma$ on queue $q$ leads to $({\ensuremath{G}}_{j+1},{\ensuremath{\vec{d}}}_{j+1})$ with ${\ensuremath{\vec{d}}}_{j}={\ensuremath{\vec{d}}}_{j+1}$ and ${\ensuremath{G}}_{j+1}$ is a path graph $v_0 v_1\dots v_n v_{n+1}$ with new distinct vertex $v_{n+1}$ where ${\ensuremath{\textit{state}}}(v_{n+1})=v^0_\gamma$. We mapped the dispatch rule to a ${\ensuremath{{\ensuremath{\mathtt{push}}\xspace}}}$ of the current state to the pushdown and jumping to the new initial state, i.e., we go from $(x_j,{\ensuremath{\widehat{\vec{d}}}}_j)$ to $(x_{j+1},{\ensuremath{\widehat{\vec{d}}}}_{j+1})$ where ${\ensuremath{\widehat{\vec{d}}}}_j={\ensuremath{\widehat{\vec{d}}}}_{j+1}$ and $x_{j+1}=x_j\cdot s^0_\gamma$. Obviously, ${\ensuremath{G}}_{j+1}\triangleright x_{j+1}$.
Test/Assignment:
: ${\ensuremath{G}}_{j+1}$ equals ${\ensuremath{G}}_{j}$ except for ${\ensuremath{\textit{state}}}_j(v_n)=s$ and ${\ensuremath{\textit{state}}}_{j+1}(v_n)=s'$ and a possible change of ${\ensuremath{\vec{d}}}_{j+1}$ according to the underlying data action. Executing the same action on ${\ensuremath{\mathcal{P}}\xspace}_{\ensuremath{\mathcal{A}}\xspace}$ assures that ${\ensuremath{\widehat{\vec{d}}}}_{j+1}={\ensuremath{\vec{d}}}_{j+1}$ and changing the control state of the pushdown only changes $x_j=w\cdot s$ to $x_{j+1}=w\cdot s'$; thus, ${\ensuremath{G}}_{j+1}\triangleright x_{j+1}$.
Termination:
: To apply the action ${\ensuremath{G}}_j$ consists of a (non-empty) path ending in $v$ with ${\ensuremath{\textit{state}}}_j(v)\in F$ and ${\ensuremath{G}}_{j+1}={\ensuremath{G}}_{j}\setminus v$, and ${\ensuremath{\vec{d}}}_j={\ensuremath{\vec{d}}}_{j+1}$. Note that ${\ensuremath{G}}_{j+1}$ could be possibly empty. Given a $(x_j,{\ensuremath{\widehat{\vec{d}}}}_j)$ according to the induction hypothesis, then we have to consider two cases: either $x_j=w_j\cdot y_j$ with $w_j\in S^+$ and $y_j\in S$ (i.e., there is at least one element on the stack), or $x_j=y_j\in S$ (i.e., stack is empty). In the second case, we know that $x_j\in S_{\ensuremath{\textit{main}}\xspace}$ and by the induction hypothesis, that $x_j=s^0_{\ensuremath{\textit{main}}\xspace}$ and ${\ensuremath{G}}_j$ a path of length 1. Now, ${\ensuremath{\mathcal{P}}\xspace}_{\ensuremath{\mathcal{A}}\xspace}$ takes the ${\ensuremath{{\ensuremath{\mathtt{empty?}}\xspace}}}$ transition leading to the (bottom) state ${\ensuremath{\varepsilon}\xspace}$, i.e., $x_{j+1}={\ensuremath{\varepsilon}\xspace}$, hence ${\ensuremath{G}}_{j+1}$ is empty and ${\ensuremath{G}}_{j+1}\triangleright {\ensuremath{\varepsilon}\xspace}$. If the stack is not empty, then we can take a [${\ensuremath{\mathtt{pop}}\xspace}$]{}transition such that $x_{j+1}=w\in S^+$ for $x_j=w\cdot s$, hence ${\ensuremath{G}}_{j+1}\triangleright
x_j$. Obviously ${\ensuremath{\vec{d}}}_{j+1}={\ensuremath{\vec{d}}}_j={\ensuremath{\widehat{\vec{d}}}}_j={\ensuremath{\widehat{\vec{d}}}}_{j+1}$.
(Recall that we asserted dispatch and scheduling/dequeueing to be atomic, so we do not need to consider other actions of the scheduler.)
The reverse direction follows analogously as the previous inductive construction used necessary *sufficient* steps. [$\Box$]{}
We assert that the semantics of ${\ensuremath{\mathcal{P}}\xspace}_{\ensuremath{\mathcal{A}}\xspace}$ is the usual semantics for pushdown systems with data, i.e., an infinite transition system with configurations $c=(y,w,d)\in Y\times S^* \times {\ensuremath{\mathbb{D}}\xspace}^{\ensuremath{\mathcal{X}}\xspace}$. Thus, we can interpret configurations also as follows: $(x,d)\in S^*\times {\ensuremath{\mathbb{D}}\xspace}^{\ensuremath{\mathcal{X}}\xspace}$ with $x=w\cdot y\in S^*\cdot (S\cup\{{\ensuremath{\varepsilon}\xspace}\}$.
Let $({\ensuremath{G}},{\ensuremath{\vec{d}}})\in\operatorname{\textit{Reach}}({\ensuremath{\mathcal{A}}\xspace})$ be reachable by a run $({\ensuremath{G}}_0,{\ensuremath{\vec{d}}}_0)a_1({\ensuremath{G}}_1,{\ensuremath{\vec{d}}}_1)a_2\dots a_n({\ensuremath{G}}_n,{\ensuremath{\vec{d}}}_n)$. Then we can induce a run $(x_0,{\ensuremath{\widehat{\vec{d}}}}_0)a_1(x_1,{\ensuremath{\widehat{\vec{d}}}}_1)a_2\dots
a_n(x_n,{\ensuremath{\widehat{\vec{d}}}}_n)$ in ${\ensuremath{\mathcal{P}}\xspace}_{\ensuremath{\mathcal{A}}\xspace}$ such that ${\ensuremath{\vec{d}}}_i={\ensuremath{\widehat{\vec{d}}}}_i$ and ${\ensuremath{G}}_i\triangleright x_i$ for $0\leq i \leq n$.
By construction of ${\ensuremath{\mathcal{P}}\xspace}_{\ensuremath{\mathcal{A}}\xspace}$, $x_0\triangleright G_0$ and ${\ensuremath{\vec{d}}}_0={\ensuremath{\widehat{\vec{d}}}}_0$. We now assume that there exists a prefix of the [<span style="font-variant:small-caps;">Qdas</span>]{}’s run of length $0\leq
j\leq n$ of the form $({\ensuremath{G}}_0,{\ensuremath{\vec{d}}}_0)\dots
({\ensuremath{G}}_j,{\ensuremath{\vec{d}}}_j)$ such that there exists a run of the pushdown system $(x_0,{\ensuremath{\widehat{\vec{d}}}}_0)\dots(x_j,{\ensuremath{\widehat{\vec{d}}}}_j)$ that fullfills the induction hypothesis. We now consider the outcome of a [<span style="font-variant:small-caps;">Qdas</span>]{}transition labeled $a_{j+1}$. We know that ${\ensuremath{G}}_j$ must be a path of vertices $v_0\dots v_n$ connected by wait edges.
Sync. dispatch:
: dispatching a block $\gamma$ on queue $q$ leads to $({\ensuremath{G}}_{j+1},{\ensuremath{\vec{d}}}_{j+1})$ with ${\ensuremath{\vec{d}}}_{j}={\ensuremath{\vec{d}}}_{j+1}$ and ${\ensuremath{G}}_{j+1}$ is a path graph $v_0 v_1\dots v_n v_{n+1}$ with new distinct vertex $v_{n+1}$ where ${\ensuremath{\textit{state}}}(v_{n+1})=v^0_\gamma$. We mapped the dispatch rule to a ${\ensuremath{{\ensuremath{\mathtt{push}}\xspace}}}$ of the current state to the pushdown and jumping to the new initial state, i.e., we go from $(x_j,{\ensuremath{\widehat{\vec{d}}}}_j)$ to $(x_{j+1},{\ensuremath{\widehat{\vec{d}}}}_{j+1})$ where ${\ensuremath{\widehat{\vec{d}}}}_j={\ensuremath{\widehat{\vec{d}}}}_{j+1}$ and $x_{j+1}=x_j\cdot s^0_\gamma$. Obviously, ${\ensuremath{G}}_{j+1}\triangleright x_{j+1}$.
Test/Assignment:
: ${\ensuremath{G}}_{j+1}$ equals ${\ensuremath{G}}_{j}$ except for ${\ensuremath{\textit{state}}}_j(v_n)=s$ and ${\ensuremath{\textit{state}}}_{j+1}(v_n)=s'$ and a possible change of ${\ensuremath{\vec{d}}}_{j+1}$ according to the underlying data action. Executing the same action on ${\ensuremath{\mathcal{P}}\xspace}_{\ensuremath{\mathcal{A}}\xspace}$ assures that ${\ensuremath{\widehat{\vec{d}}}}_{j+1}={\ensuremath{\vec{d}}}_{j+1}$ and changing the control state of the pushdown only changes $x_j=w\cdot s$ to $x_{j+1}=w\cdot s'$; thus, ${\ensuremath{G}}_{j+1}\triangleright x_{j+1}$.
Termination:
: To apply the action ${\ensuremath{G}}_j$ consists of a (non-empty) path ending in $v$ with ${\ensuremath{\textit{state}}}_j(v)\in F$ and ${\ensuremath{G}}_{j+1}={\ensuremath{G}}_{j}\setminus v$, and ${\ensuremath{\vec{d}}}_j={\ensuremath{\vec{d}}}_{j+1}$. Note that ${\ensuremath{G}}_{j+1}$ could be possibly empty. Given a $(x_j,{\ensuremath{\widehat{\vec{d}}}}_j)$ according to the induction hypothesis, then we have to consider two cases: either $x_j=w_j\cdot y_j$ with $w_j\in S^+$ and $y_j\in S$ (i.e., there is at least one element on the stack), or $x_j=y_j\in S$ (i.e., stack is empty). In the second case, we know that $x_j\in S_{\ensuremath{\textit{main}}\xspace}$ and by the induction hypothesis, that $x_j=s^0_{\ensuremath{\textit{main}}\xspace}$ and ${\ensuremath{G}}_j$ a path of length 1. Now, ${\ensuremath{\mathcal{P}}\xspace}_{\ensuremath{\mathcal{A}}\xspace}$ takes the ${\ensuremath{{\ensuremath{\mathtt{empty?}}\xspace}}}$ transition leading to the (bottom) state ${\ensuremath{\varepsilon}\xspace}$, i.e., $x_{j+1}={\ensuremath{\varepsilon}\xspace}$, hence ${\ensuremath{G}}_{j+1}$ is empty and ${\ensuremath{G}}_{j+1}\triangleright {\ensuremath{\varepsilon}\xspace}$. If the stack is not empty, then we can take a [${\ensuremath{\mathtt{pop}}\xspace}$]{}transition such that $x_{j+1}=w\in S^+$ for $x_j=w\cdot s$, hence ${\ensuremath{G}}_{j+1}\triangleright
x_j$. Obviously ${\ensuremath{\vec{d}}}_{j+1}={\ensuremath{\vec{d}}}_j={\ensuremath{\widehat{\vec{d}}}}_j={\ensuremath{\widehat{\vec{d}}}}_{j+1}$.
(Recall that we asserted dispatch and scheduling/dequeueing to be atomic, so we do not need to consider other actions of the scheduler.)
The reverse direction follows analogously as the previous inductive construction used necessary *sufficient* steps. [$\Box$]{}
[lemma]{}[lengenparikautomata]{}\[lem:genparikautomata\] Given a finite set $S$ and a function $f:S\rightarrow {\ensuremath{\mathbb{N}}\xspace}$, then there exists a finite automaton ${\ensuremath{\mathcal{F}}\xspace}_f$ with alphabet $S$ of size exponential in $\vert S\vert$ and polynomial in (in the binary encoding of) $max_{s\in S}f(s)$ such that ${\ensuremath{\mathcal{L}}\xspace}({\ensuremath{\mathcal{F}}\xspace}_f)=\{w \in S^* : |w|_s\geq f(s) \text{ for all } s\in
S\}$.
Given a set $S$ and a function $f:S\mapsto {\ensuremath{\mathbb{N}}\xspace}$. Let $k=max_{s\in S}f(s)$ (which must exists as $S$ is finite). Then ${\ensuremath{\mathcal{F}}\xspace}_f$ is the finite automaton ${\langleQ,S,q^0,\Delta,q^f\rangle\xspace}$ with states $Q=S\times\{0\dots k\}$ (interpreted as an $S$-indexed vector of values in $0\dots k$), an action alphabet $S$, the initial state is $q^0$ where $q^0(s)=f(s)$, the finial state is $q^f$ where $q^f(s)=0$. The transitions of ${\ensuremath{\mathcal{F}}\xspace}_f$ are defined as follows: $(q,s,q')\in\Delta$ iff $q'(s)=q(s)-1$ for $q(s)>1$, else $q'(s)=q(s)$, and for all $t\in S\setminus\{s\}$ we have $q'(t)=q(t)$. Thus each transition labeled by an action $s$ reduces the “counter” $q(s)$ by one until zero and once arrived at zero, the counter $q(s)$ remains zero for any further $s$ action. Further, the control structure of ${\ensuremath{\mathcal{F}}\xspace}_f$ is acyclic (except for the loops at $q^f$), thus each run can visit each state in $Q\setminus \{q^f\}$.
If $w=a_1\dots a_n\in{\ensuremath{\mathcal{L}}\xspace}({\ensuremath{\mathcal{A}}\xspace})$ then it was accepted by a run $q_0a_1q_1\dots
a_nq_n$ where $q_0=q^0$ and $q_n=q^f$. Due to our construction of $\Delta$, it holds for $w=a_1\dots a_n$ that $|w|_s\geq q_0(s)=f(s)$ for all $s\in S$. If $w\notin {\ensuremath{\mathcal{L}}\xspace}({\ensuremath{\mathcal{A}}\xspace})$ then there exists a run $q_0 a_1q_1\dots a_nq_n$ where $q_0=q^0$ and for $q_n\neq q^f$ it holds that there exists at least one $s\in S$ such that $q_n(s)>0$, each transition $(q_{i-1},a_i,q_i)$ assures that $q_{i-1}(s)\geq q_i(s)$, hence $|w|_s<f(s)$ for at least one $s\in S$.[$\Box$]{}
First, we construct the [<span style="font-variant:small-caps;">Pds</span>]{}with data ${\ensuremath{\mathcal{P}}\xspace}_{\ensuremath{\mathcal{A}}\xspace}$ and states $S$ as mentioned before. Then, we translate the [<span style="font-variant:small-caps;">Pds</span>]{}with data to a bisimilar [<span style="font-variant:small-caps;">Pds</span>]{}without data $\widehat{{\ensuremath{\mathcal{P}}\xspace}_{\ensuremath{\mathcal{A}}\xspace}}={\langle\widehat{Y},\widehat{y^0},
\widehat{\Phi},\widehat{\Sigma}, \widehat{\Delta}\rangle\xspace}$ by encoding all possible valuations of variables into the [[<span style="font-variant:small-caps;">Pds</span>]{}]{}’s states by the standard product construction, i.e., $\widehat{Y}=S\times\left({\ensuremath{\mathcal{X}}\xspace}\times{\ensuremath{\mathbb{D}}\xspace}\right)$. Given $y\in\widehat{Y}$, let $S(y)\in S$ denote the original state component. Note: $\widehat{{\ensuremath{\mathcal{P}}\xspace}_{\ensuremath{\mathcal{A}}\xspace}}$ is at most exponentially larger as ${\ensuremath{\mathcal{P}}\xspace}_{\ensuremath{\mathcal{A}}\xspace}$ and this construction does not change the pushdown system’s behaviour with respect to the stack but only internal actions.
Second, from the function $f$, we construct the automaton ${\ensuremath{\mathcal{F}}\xspace}_f={\langleQ,S,q^0,\Delta_{\ensuremath{\mathcal{F}}\xspace},q^f\rangle\xspace}$ analogous to Lemma\[lem:genparikautomata\].
Finally, we define the [<span style="font-variant:small-caps;">Pds</span>]{} ${\ensuremath{\mathcal{P}}\xspace}_{{{{\ensuremath{\mathcal{A}}\xspace},f}}}={\langleY,y^0,\Phi,\Sigma,\Delta_{{{\ensuremath{\mathcal{A}}\xspace},f}}\rangle\xspace}$ as follows
states are $Y=\widehat{Y}\operatorname{\mathaccent\cdot\cup}Q$ (assuring disjointness by relabeling when necessary)
$y^0=\widehat{y^0}$ is the initial state
$\Phi=\widehat{\Phi}$ is the stack alphabet (where $\widehat{\Phi}=S$ due to the above construction)
$\Sigma=\widehat{\Sigma} \cup\{{\ensuremath{\varepsilon}\xspace}\}$
a tuple $(y,a,y')$ is a rule in $\Delta_{{{\ensuremath{\mathcal{A}}\xspace},f}}\subseteq
Y\times\Sigma\times Y$ iff one of the following holds
$(y,a,y')\in\widehat{\Delta}$ (include all transition rules of $\widehat{{\ensuremath{\mathcal{P}}\xspace}_{\ensuremath{\mathcal{A}}\xspace}}$);
$a={\ensuremath{{\ensuremath{\mathtt{pop}}\xspace}}}(s)$ for $s\in\Phi$ and $(q,s,q')\in\Delta_{\ensuremath{\mathcal{F}}\xspace}$ (include rules of ${\ensuremath{\mathcal{F}}\xspace}_f$ and change an $s$ action to [[${\ensuremath{\mathtt{pop}}\xspace}$]{}(s)]{} for $s\in S$);
$y\in \widehat{Y}$, $a={\ensuremath{{\ensuremath{\mathtt{push}}\xspace}}}(z)$ for $z=S(y)$, and $y'=q^0$ (connect all states in $\widehat{Y}$ with the initial state of ${\ensuremath{\mathcal{F}}\xspace}_f$, additionally stocking the current “state”-component on the stack).
Note that ${\ensuremath{\mathcal{P}}\xspace}_{{{{\ensuremath{\mathcal{A}}\xspace},f}}}$ is of size exponential with respect to both the [<span style="font-variant:small-caps;">Qdas</span>]{} and $f$ due to serial composition.
We now have to show that if there is a run in ${\ensuremath{\mathcal{P}}\xspace}_{{{\ensuremath{\mathcal{A}}\xspace},f}}$ that reaches the state $q^f$, then there exists configuration $c=({\ensuremath{G}},{\ensuremath{\vec{d}}})$ of ${\ensuremath{\mathcal{A}}\xspace}$ such that $f{\preceq}{\ensuremath{\mathsf{Parikh}}}(G)$.
Assert that there exists a run of ${\ensuremath{\mathcal{P}}\xspace}_{{{\ensuremath{\mathcal{A}}\xspace},f}}$ reaching $q^f$, then it must be of the following form $\langle x_0, a_1, x_1, \dots, a_k,
x_k, a_{k+1}, x_{k+1}, a_{k+2},\dots, a_n, x_n\rangle$ where $x_i=(y_i,w_i)\subseteq Y\times S^*$ are the corresponding infinite transition systems configurations. Further, $y_0=y^0$, $y_n=q^f$, $y_{k+1}=q^0$, and $\langle y_1\dots
y_k\rangle$ is a subrun that only uses states in $\widehat{Y}$ as well as transitions in $\widehat{{\ensuremath{\mathcal{P}}\xspace}_{\ensuremath{\mathcal{A}}\xspace}}$; $\{y_{k+1},\dots,y_{n}\}\subseteq Q$ and the corresponding transitions are derived from $\Delta_{\ensuremath{\mathcal{F}}\xspace}$, as well as $a_{k+1}={\ensuremath{{\ensuremath{\mathtt{push}}\xspace}}}(S(y_k))$.
Let us take a closer look on the first part of the run: $\langle y_0, a_1,\dots, a_n, x_k\rangle$ is equivalent to a run of $\widehat{{\ensuremath{\mathcal{P}}\xspace}_{\ensuremath{\mathcal{A}}\xspace}}$ that reaches a configuration $x_k$. The latter is, following Propositions\[prop:sync\_qdas\_pds\] and \[prop:pds\_sync\_qdas\], similar to a run of the original [<span style="font-variant:small-caps;">Qdas</span>]{}${\ensuremath{\mathcal{A}}\xspace}$ that reaches a configuration $c=({\ensuremath{G}},{\ensuremath{\vec{d}}})$ where ${\ensuremath{G}}\triangleright y_k\cdot S(y_k)$. Thus, $c\in\operatorname{\textit{Reach}}({\ensuremath{\mathcal{A}}\xspace})$.
The transition $(x_k, {\ensuremath{{\ensuremath{\mathtt{push}}\xspace}}}(S(y_k)), x_{k+1})$ now transfers the encoding of ${\ensuremath{G}}$ to the stack, i.e., $w_{k+1}=y_k\cdot S(y_k)$. All other information on data encoded in $y_k$ is lost in this step.
Now, by Lemma\[lem:genparikautomata\] we know that the subrun $\langle x_{k+1}, a_{k+2}, \dots, a_n, x_n\rangle$ leading to the final state of ${\ensuremath{\mathcal{F}}\xspace}_f$ assures that $|w_{k+1}|_s\geq f(s)$ for all $s\in S$. Hence, for the previously found $c=({\ensuremath{G}},{\ensuremath{\vec{d}}})\in\operatorname{\textit{Reach}}({\ensuremath{\mathcal{A}}\xspace})$ it holds that $f{\preceq}{\ensuremath{\mathsf{Parikh}}}(G)$.[$\Box$]{}
Let us take a closer look on the dispatches that happen in runs of synchronous [<span style="font-variant:small-caps;">Qdas</span>]{}that have only *serial* queues. Assume a run of such a [<span style="font-variant:small-caps;">Qdas</span>]{}, and suppose the first dispatch performed along this run (by `main`) is ${\ensuremath{{\ensuremath{\mathtt{dispatch_s}}\xspace}}}(q,\gamma)$. As the dispatch is synchronous, `main` is blocked, and the scheduler has to dequeue $\gamma$ to let the system progress. Cleraly, if $\gamma$ performs a synchronous dispatch ${\ensuremath{{\ensuremath{\mathtt{dispatch_s}}\xspace}}}(q,\gamma')$ to the same queue $q$, we reach a deadlock. Indeed, the task running $\gamma$ is blocked by the synchronous dispatch of $\gamma'$, but we need to wait for the termination of $\gamma$ to be able to dequeue $\gamma'$ from $q$ (because $q$ is serial). So, $\gamma$ has to dispatch its blocks to other queues. For the same reason, we also reach a deadlock if a block called by $\gamma$ performs a synchronous dispatch into $q$. We conclude that, in all reachable [$\textsc{Ctg}$]{}, the following holds for all queues: either the queue contains one block and there is no running task from this queue, or the queue is empty, and there is at most one running task from this queue. Hence, all the reachable [$\textsc{Ctg}$]{}have at most $\vert SQID\vert +2$ vertices. Thus, the pushdown systems used in all previous constructions have bounded stack height and we can apply the emptiness test on a finite state system when proving Proposition\[prop:syncqdas\_parikh\_sim\]. The lower bound can be derived from Proposition\[prop:queueless\]. Thus we can derive:
### From [<span style="font-variant:small-caps;">Pds</span>]{}to [<span style="font-variant:small-caps;">Qdas</span>]{}
Given a [<span style="font-variant:small-caps;">Pds</span>]{}${\ensuremath{\mathcal{P}}\xspace}$, we construct a synchronous [<span style="font-variant:small-caps;">Qdas</span>]{}${\ensuremath{\mathcal{A}}\xspace}_{\ensuremath{\mathcal{P}}\xspace}$ as shown in Figure\[fig:sim\_pds\]. The underlying idea is the inverse of the above simulation: we map a [${\ensuremath{\mathtt{push}}\xspace}$]{}action of a letter $\phi$ to synchronous dispatch call of a block $\phi$ and simulate the stack contents in the [$\textsc{Ctg}$]{}such that we can only map a [${\ensuremath{\mathtt{pop}}\xspace}$]{}action to a task’s termination if we match the topmost letter of the stack, encoded in the block name.
global state := $x^0$
global c_queue q
def $\phi$(): // for each $\phi\in\Phi\dotcup\{\main\}$
while(true):
select $(s,a,s')\in\Delta_\Pp$ where state=$s$
if $a=\push(\phi')$ :
state := $s'$
dispatch_s(q,$\phi'$)
if $a=\pop(\phi)$ and $\phi=\phi'$ :
state := $s'$
terminate
\[zstd/.style=[state,font=]{},anchor=west\]
(-0.5,1.2) node\[anchor=west\] (dummy)[[$\textsc{Ctg}$]{}in $\operatorname{\textit{Reach}}({\ensuremath{\mathcal{A}}\xspace}_{\ensuremath{\mathcal{P}}\xspace})$:]{}; (0,0.5) node\[zstd\] (00) [${\ensuremath{\textit{main}}\xspace}$]{}; (00)+(1,0) node\[zstd\] (01) [$\phi_1$]{}; (01)+(1,0) node\[font=,anchor=west\] (02) […]{}; (02)+(0.5,0) node\[zstd\] (03) [$\phi_k$]{}; (00) edge\[->\] (01) (01) edge\[->\] (02) (02) edge\[->\] (03);
[background]{} node\[fit=(00) (03),fill=gray!20\] (c1) ; (c1.north east) node\[anchor=south east,inner sep=0pt,font=\] [stack]{};
The control state of the [<span style="font-variant:small-caps;">Pds</span>]{}is stored in the variable `state` and the behaviour of the control structure of [$\mathcal{P}$]{}is encoded as non-determinstic choice (line $7$) that assures that reaching the dispatch and termination actions (lines $11$/$15$) demands that the selected transition rule harmonizes with the current change of the variable `state` from $s$ to $s'$ and that a ${\ensuremath{{\ensuremath{\mathtt{push}}\xspace}}}(\phi')$ action is only possible if the currently running task is labeled by the blockname $\phi'$ (line $13$).
A reachable configuration of ${\ensuremath{\mathcal{A}}\xspace}_{\ensuremath{\mathcal{P}}\xspace}$ is given by $({\ensuremath{G}},{\ensuremath{\vec{d}}})$ where ${\ensuremath{G}}$ is—as discussed before—a path of vertices $v_0v_1\dots v_k$. As before, synchronous dispatch calls assure there is no more than one task active at the same time. Given $c=({\ensuremath{G}},{\ensuremath{\vec{d}}})\in\operatorname{\textit{Reach}}({\ensuremath{\mathcal{A}}\xspace}_{\ensuremath{\mathcal{P}}\xspace})$ and a configuration $y=(x,w)\in X\times \Phi^*$ that is reachable in ${\ensuremath{\mathcal{P}}\xspace}$; then $c$ is represented by $y$, written $c \triangleright y$, iff ${\ensuremath{\vec{d}}}(\texttt{state})=x$ and for $w=w_1\dots w_k$ $\lambda(v_i)=w_i$ for $1\leq i\leq k$ and $\lambda(v_0)={\ensuremath{\textit{main}}\xspace}$. Hence, the state of the [<span style="font-variant:small-caps;">Pds</span>]{}is stored in the variable `state`, and the path $v_1\dots v_k$ encodes in the underlying task’s blocks the stack content, where the empty stack is represented by a single vertex labeled by [$\textit{main}$]{}.
[proposition]{}[propsyncpdsqdas]{} \[prop:sync\_pds\_qdas\] Given a pushdown system ${\ensuremath{\mathcal{P}}\xspace}$, then we can generate a synchronous [<span style="font-variant:small-caps;">Qdas</span>]{}${\ensuremath{\mathcal{A}}\xspace}_{\ensuremath{\mathcal{P}}\xspace}$ such that the following holds: for any run $\pi=y_0a_1y_1\dots a_ny_n$ in ${\ensuremath{\mathcal{P}}\xspace}$ there exists a run $\rho=c_0a_1c_1\dots a_nc_n$ of ${\ensuremath{\mathcal{A}}\xspace}_{\ensuremath{\mathcal{P}}\xspace}$ such that for all $c_i\triangleright x_i$ ($0\leq i \leq n$), and vice versa.
Given a run $\rho=y_0 a_1 y_1 \dots a_k y_k$ of the [<span style="font-variant:small-caps;">Pds</span>]{}${\ensuremath{\mathcal{P}}\xspace}$. W.l.o.g. let us consider in the following underlying sequence of configurations and fired transition rules $y_0 \delta_1 y_1 \dots \delta_k y_k$ where $\delta_i=(x_i,a_i,x_i')\in\Delta_{\ensuremath{\mathcal{P}}\xspace}$ for $1\leq i \leq k$.
We show inductively how ${\ensuremath{\mathcal{A}}\xspace}_{\ensuremath{\mathcal{P}}\xspace}$ generates a run that simulates $\rho$.
For the initial configuration of ${\ensuremath{\mathcal{P}}\xspace}$ $y_0=(x^0,{\ensuremath{\varepsilon}\xspace})$ and the initial configuration $c_0=({\ensuremath{G}},{\ensuremath{\vec{d}}})$ with ${\ensuremath{G}}$ consists of a single node $v_0$ with $\lambda(v_0)={\ensuremath{\textit{main}}\xspace}$ and ${\ensuremath{\vec{d}}}(state)=x^0$ it holds that $c_0 \triangleright y_0$.
Now assert that the [<span style="font-variant:small-caps;">Pds</span>]{}${\ensuremath{\mathcal{P}}\xspace}$ reached configuration $y_i$ ($0\leq i \leq k$) such that ${\ensuremath{\mathcal{A}}\xspace}_{\ensuremath{\mathcal{P}}\xspace}$ simulated the prefix of the run until $c_i=({\ensuremath{G}}_i,{\ensuremath{\vec{d}}}_i)$ with $c_i\triangleright y_i$. Assert that ${\ensuremath{G}}_i$ is a path $v_0 v_1\dots v_l$. We do a case-by-case analysis with respect to $\delta_{i+1}=(x,a,x')$ that leads to $y_{i+1}$:
only the task corresponding to $v_l$ is active and the only way to exit its `while` loop is via the lines $11$ and $15$, that assure that line $7$ selected $\delta=(x,a,x')\in\Delta_p$ with ${\ensuremath{\vec{d}}}_i(state)=x$, and that we set ${\ensuremath{\vec{d}}}_{i+1}(state)=x'$;;
if $a={\ensuremath{{\ensuremath{\mathtt{push}}\xspace}}}(\phi)$ for $\phi\in\Phi$, then we fire the synchronous dispatch that leads to ${\ensuremath{G}}_{i+1}=
v_0\dots v_l v_{l+1}$ with $\lambda(v_{l+1})=\phi$, thus $({\ensuremath{G}}_{i+1},{\ensuremath{\vec{d}}}_{i+1})\triangleright y_{i+1}$;
if $a={\ensuremath{{\ensuremath{\mathtt{pop}}\xspace}}}(\phi)$ for $\phi\in\Phi$ and we left the while loop then $\lambda(v_l)= \phi$ (by line $13$), and ${\ensuremath{G}}_{i+1}$ equals $v_0 \dots v_{l-1}$, thus $({\ensuremath{G}}_{i+1},{\ensuremath{\vec{d}}}_{i+1})\triangleright y_{i+1}$.
The reverse direction follows analogously by considering lines $10;11$ and $14;15$ as atomic actions (i.e., setting the `state` variable and changing the call graph of the [<span style="font-variant:small-caps;">Qdas</span>]{}).
Asynchronous Concurrent [<span style="font-variant:small-caps;">Qdas</span>]{}
------------------------------------------------------------------------------
The proof of the proposition relies on the following lemma, showing that $N_{\ensuremath{\mathcal{A}}\xspace}$ can simulate precisely the sequence of Parikh images that are reachable in ${\ensuremath{\mathcal{A}}\xspace}$. Let $(G,{\ensuremath{\vec{d}}})$ be a configuration of ${\ensuremath{\mathcal{A}}\xspace}$, and let $m$ be marking of $N_{\ensuremath{\mathcal{A}}\xspace}$. We say that *$m$ encodes $(G,{\ensuremath{\vec{d}}})$*, written $m\rhd (G,{\ensuremath{\vec{d}}})$ iff: $(i)$ for all $x\in{\ensuremath{\mathcal{X}}\xspace}$: $m(x,{\ensuremath{\vec{d}}}(x))=1$, $(ii)$ for all $x\in{\ensuremath{\mathcal{X}}\xspace}$: for all $d\in{\ensuremath{\mathbb{D}}\xspace}\setminus\{{\ensuremath{\vec{d}}}(x)\}$: $m(x,d)=0$ and $(iii)$ for all $s\in
S$ $m(s)={\ensuremath{\mathsf{Parikh}}}(G)(s)$. Then:
\[lemma:from-qdas-to-pn\] Let ${\ensuremath{\mathcal{A}}\xspace}$ be a concurrent asynchronous [<span style="font-variant:small-caps;">Qdas</span>]{}with set of variables ${\ensuremath{\mathcal{X}}\xspace}$ and set of locations $S$, and let $N_{\ensuremath{\mathcal{A}}\xspace}$ be its associated [<span style="font-variant:small-caps;">Pn</span>]{}. Then, for all $(G,{\ensuremath{\vec{d}}})\in\operatorname{\textit{Reach}}({\ensuremath{\mathcal{A}}\xspace})$ there is $m\in\operatorname{\textit{Reach}}(N_{\ensuremath{\mathcal{A}}\xspace})$ s.t. $m\rhd (G,{\ensuremath{\vec{d}}})$ and for all $m\in\operatorname{\textit{Reach}}(N_{\ensuremath{\mathcal{A}}\xspace})$, there is $(G,{\ensuremath{\vec{d}}})\in\operatorname{\textit{Reach}}({\ensuremath{\mathcal{A}}\xspace}) $ s.t. $m\rhd
(G,{\ensuremath{\vec{d}}})$.
We prove the two statements separately.
Let $(G,{\ensuremath{\vec{d}}})$ be a configuration in $\operatorname{\textit{Reach}}({\ensuremath{\mathcal{A}}\xspace}_N)$, and let $(G_0,{\ensuremath{\vec{d}}}_0)a_0(G_1,{\ensuremath{\vec{d}}}_1)a_1\cdots a_{n-1}(G_n,{\ensuremath{\vec{d}}}_n)$ be a run s.t. $(G,{\ensuremath{\vec{d}}})=(G_n,{\ensuremath{\vec{d}}}_n)$. Let us build, inductively, a run $m_0m_1\cdots m_k$ of $N_{\ensuremath{\mathcal{A}}\xspace}$ s.t. $m_k\rhd (G,{\ensuremath{\vec{d}}})$. The induction is on the length $n$ of the [<span style="font-variant:small-caps;">Qdas</span>]{}run.
**Base case $n=0$.** It is easy to check that $m_0\rhd
(G_0,{\ensuremath{\vec{d}}}_0)$.
**Inductive case $n=\ell$.** Let us assume that $m_0m_1\cdots
m_j$ is a run of $N_{\ensuremath{\mathcal{A}}\xspace}$ s.t. $m_j\rhd (G_{\ell-1},{\ensuremath{\vec{d}}}_{\ell-1})$, and let us show how to complete it, if needed. We consider several case depending on $a_{n-1}$. In the case where $a_{n-1}=\varepsilon$ and the scheduler action consists in dequeueing a block from a queue, we have ${\ensuremath{\mathsf{Parikh}}}(G_{\ell-1})={\ensuremath{\mathsf{Parikh}}}(G_\ell)$ and ${\ensuremath{\vec{d}}}_\ell={\ensuremath{\vec{d}}}_{\ell-1}$. By induction hypothesis $m_j\rhd
(G_{\ell-1},{\ensuremath{\vec{d}}}_{\ell-1})$, hence $m_j\rhd (G_\ell, {\ensuremath{\vec{d}}}_\ell)$, and we do not add elements to the run built so far. In the case where $a_{\ell-1}={\ensuremath{{\ensuremath{\mathtt{dispatch_a}}\xspace}}}(\gamma,q)$, we assume $(s,a_{\ell-1},s')\in\Delta$ is the corresponding [<span style="font-variant:small-caps;">Lts</span>]{} transition. Clearly, ${\ensuremath{\mathsf{Parikh}}}(G_\ell)(s')={\ensuremath{\mathsf{Parikh}}}(G_{\ell-1})(s')+1$, ${\ensuremath{\mathsf{Parikh}}}(G_\ell)(s)={\ensuremath{\mathsf{Parikh}}}(G_{\ell-1})(s)-1$, ${\ensuremath{\mathsf{Parikh}}}(G_\ell)(s^0_\gamma)={\ensuremath{\mathsf{Parikh}}}(G_{\ell-1})(s^0_\gamma)+1$ and for all other location $s$: ${\ensuremath{\mathsf{Parikh}}}(G_\ell)(s)={\ensuremath{\mathsf{Parikh}}}(G_{\ell-1})(s)$. It is easy to check that the [<span style="font-variant:small-caps;">Pn</span>]{}transition $t$ s.t. $I(t)(p)=1$ iff $p=s$ and $O(t)(p)=1$ iff $p\in\{s',s^0_\gamma\}$ is fireable from $m_j$ (as $m_j\rhd (G_{\ell-1},{\ensuremath{\vec{d}}}_{\ell-1})$ by induction hypothesis) and yields the same effect, i.e. the marking $m$ with $m_j\xrightarrow{t}m$ is s.t. $m\rhd (G_\ell,{\ensuremath{\vec{d}}}_\ell)$. All the other cases (test, assignment and task termination) are treated similarly.
Now, let $m_0m_1\cdots m_n$ be a run of $N_{\ensuremath{\mathcal{A}}\xspace}$ and let us build, inductively, a run $(G_0,{\ensuremath{\vec{d}}}_0)a_0(G_1,{\ensuremath{\vec{d}}}_1)a_1\cdots
a_{k-1}(G_k,{\ensuremath{\vec{d}}}_k)$ s.t. $m_n\rhd (G_k,{\ensuremath{\vec{d}}}_k)$ *and* all the queues are empty in $G_n$. The induction is on the length $n$ of the [<span style="font-variant:small-caps;">Pn</span>]{}run.
**Base case $n=0$.** It is easy to check that $m_0\rhd
(G_0,{\ensuremath{\vec{d}}}_0)$.
**Inductive case $n=\ell$.** Let us assume that $(G_0,{\ensuremath{\vec{d}}}_0)a_0\cdots a_{j-1}(G_j,{\ensuremath{\vec{d}}}_j)$ is a run of ${\ensuremath{\mathcal{A}}\xspace}$ s.t. $m_{\ell-1}\rhd (G_j,{\ensuremath{\vec{d}}}_j)$ and all the queues are empty in $G_j$. Let $t$ be the [<span style="font-variant:small-caps;">Pn</span>]{}transition s.t. $m_{\ell-1}\xrightarrow{t}m_\ell$ and let us show how we can extend the run of ${\ensuremath{\mathcal{A}}\xspace}$. We consider several cases. If $t$ is a transition that corresponds to an asynchronous dispatch, then there are $s$, $s'$, $\gamma$ and $q$ s.t. $I_t(p)=1$ iff $p=s$ and $O_t(p)=1$ iff $p\in\{s', s^0_\gamma\}$. By definition of $N_{\ensuremath{\mathcal{A}}\xspace}$, there is a transition $(s,{\ensuremath{{\ensuremath{\mathtt{dispatch_a}}\xspace}}}(\gamma,q),s')$ in ${\ensuremath{\mathcal{A}}\xspace}$. Moreover, $m_{\ell-1}(s)\geq 1$, since $t$ is fireable from $m_{\ell-1}$. As $m_{\ell-1}\rhd (G_j,{\ensuremath{\vec{d}}}_j)$, the $(s,{\ensuremath{{\ensuremath{\mathtt{dispatch_a}}\xspace}}}(\gamma,q),s')$ is fireable from $(G_j,{\ensuremath{\vec{d}}}_j)$, and leads to a configuration $(G_{j+1}, {\ensuremath{\vec{d}}}_{j+1})$, where a $\gamma$ block has been enqueued in $q$, hence ${\ensuremath{\vec{d}}}_{j+1}={\ensuremath{\vec{d}}}_j$, ${\ensuremath{\mathsf{Parikh}}}(G_{j+1})(s)={\ensuremath{\mathsf{Parikh}}}(G_j)(s)-1$, ${\ensuremath{\mathsf{Parikh}}}(G_{j+1})(s')={\ensuremath{\mathsf{Parikh}}}(G_j)(s')+1$, ${\ensuremath{\mathsf{Parikh}}}(G_{j+1})(s^0_\gamma)={\ensuremath{\mathsf{Parikh}}}(G_j)(s^0_\gamma)+1$ and for all other state $s''$: ${\ensuremath{\mathsf{Parikh}}}(G_{j+1})(s'')={\ensuremath{\mathsf{Parikh}}}(G_j)(s'')$. It is easy to check that $m_\ell\rhd (G_{j+1},{\ensuremath{\vec{d}}}_{j+1})$, however, queue $q$ contains a call to $\gamma$ in $G_{j+1}$ and is thus the only non-empty queue in this [$\textsc{Ctg}$]{}. Thus, from $(G_{j+1},{\ensuremath{\vec{d}}}_{j+1})$, we execute the scheduler action that dequeues from $q$. This has no effect on the Parikh image of the [$\textsc{Ctg}$]{}. Thus, we reach $(G_{j+2},{\ensuremath{\vec{d}}}_{j+2})$ s.t. ${\ensuremath{\vec{d}}}_{j+1}={\ensuremath{\vec{d}}}_{j+2}$, ${\ensuremath{\mathsf{Parikh}}}(G_{j+1})={\ensuremath{\mathsf{Parikh}}}(G_{j+2})$, hence $m\ell\rhd (G_{j+2},{\ensuremath{\vec{d}}}_{j+2})$ too, and all the queues are empty in $G_{j+2}$, which concludes the induction step. All the other cases are treated similarly.[$\Box$]{}
We can now prove Proposition \[prop:from-qdas-to-pn\]:
It is easy to check that the construction of $N_{\ensuremath{\mathcal{A}}\xspace}$, as described above, is polynomial. Then, assume $f$ is Parikh coverable in ${\ensuremath{\mathcal{A}}\xspace}$, i.e. there is $(G,{\ensuremath{\vec{d}}})\in \operatorname{\textit{Reach}}({\ensuremath{\mathcal{A}}\xspace})$ s.t. $f{\preceq}{\ensuremath{\mathsf{Parikh}}}(G)$. By Lemma \[lemma:from-qdas-to-pn\], there is $m'\in\operatorname{\textit{Reach}}(N_{\ensuremath{\mathcal{A}}\xspace})$ s.t. $m'\rhd (G,{\ensuremath{\vec{d}}})$. Hence, for all $s\in S$: $m'(s)={\ensuremath{\mathsf{Parikh}}}(G)(s)$. So, for all $s\in S$: $m(s)=f(s)\leq {\ensuremath{\mathsf{Parikh}}}(G)(s)=m'(s)$. Hence, $m{\preceq}m'$ (as $m(p)=0$ for all $p\not\in S$). Since $m'\in\operatorname{\textit{Reach}}(N_{\ensuremath{\mathcal{A}}\xspace})$, we conclude that $m\in{\ensuremath{\textit{Cover}}}(N_{\ensuremath{\mathcal{A}}\xspace})$. On the other hand, assume $m\in{\ensuremath{\textit{Cover}}}(N_{\ensuremath{\mathcal{A}}\xspace})$, with $m(p)=0$ for all $p\not\in S$, and let $f$ be s.t. for all $s\in S$: $f(s)=m(s)$. Since $m\in{\ensuremath{\textit{Cover}}}(N_{\ensuremath{\mathcal{A}}\xspace})$, there is $m'\in\operatorname{\textit{Reach}}(N_{\ensuremath{\mathcal{A}}\xspace})$ s.t. $m{\preceq}m'$. By Lemma \[lemma:from-qdas-to-pn\], there is $(G,{\ensuremath{\vec{d}}})\in\operatorname{\textit{Reach}}({\ensuremath{\mathcal{A}}\xspace})$ s.t. $m'\rhd (G,{\ensuremath{\vec{d}}})$. Thus, by definition of $\rhd$, for all $s\in S$: $m'(s)={\ensuremath{\mathsf{Parikh}}}(G)(s)$. Thus, since $m{\preceq}m'$ and by definition of $f$, we conclude that for all $s\in S$: $f(s)=m(s)\leq m'(s)={\ensuremath{\mathsf{Parikh}}}(G)(s)$. Hence, $f$ is Parikh-coverable in ${\ensuremath{\mathcal{A}}\xspace}$.[$\Box$]{}
The proof of Proposition \[prop:from-pn-to-qdas\] is split into two lemmata, given hereunder. They rely on an alternate characterization of ${\ensuremath{\textit{Cover}}}(N)$. That is, $m\in{\ensuremath{\textit{Cover}}}(N)$ iff $m$ is reachable by a so-called *lossy* run of $N$, i.e. a sequence of markings $m_0'm_1'\cdots m_n'$ s.t. $m_0'{\preceq}m_0$ and for all $0\leq i\leq
n-1$: there is ${\ensuremath{\overline{m}}\xspace}_{i+1}$ and a transition $t_i$ s.t. $m_i'\xrightarrow{t_i}{\ensuremath{\overline{m}}\xspace}_{i+1}$ and $m_{i+1}'{\preceq}{\ensuremath{\overline{m}}\xspace}_{i+1}$. Intuitively, a lossy run corresponds to firing a transition of the PN, and then spontaneously losing some tokens. The proof of these lemmata also assumes that each $p\in P$, the [<span style="font-variant:small-caps;">Lts</span>]{}${\ensuremath{\mathcal{TS}}\xspace}_p={\langle\{s^0_P, s^{mid}_p,s^{fin}_p\}, s^0_p,
\Sigma,\Rightarrow\rangle\xspace}$ is as depicted in Fig. \[fig:lts-p\].
![The [<span style="font-variant:small-caps;">Lts</span>]{}of bloc [p]{}.[]{data-label="fig:lts-p"}](lts-p)
Let $N={\langleP,T,m_0\rangle\xspace}$ be a [<span style="font-variant:small-caps;">Pn</span>]{}, and let ${\ensuremath{\mathcal{A}}\xspace}_N={\langle{CQID\xspace},
\emptyset, \Gamma, \mathtt{main}, {\ensuremath{\mathcal{X}}\xspace}, \Sigma,
({\ensuremath{\mathcal{TS}}\xspace}_\gamma)_{\gamma\in\Gamma}\rangle\xspace}$ be its corresponding [<span style="font-variant:small-caps;">Qdas</span>]{}. **If** $m\in{\ensuremath{\textit{Cover}}}(N)$ **then** there exists $(G,{\ensuremath{\vec{d}}})\in\operatorname{\textit{Reach}}({\ensuremath{\mathcal{A}}\xspace}_N)$ s.t. $G\rhd m$.
Let $m$ be a marking from ${\ensuremath{\textit{Cover}}}(N)$. and let $m_0'm_1'\cdots m_n'$ be a lossy [<span style="font-variant:small-caps;">Pn</span>]{}run s.t. $m=m_n$. The proof is by induction on the length of the run. More precisely, we show that, for all $0\leq
i\leq n$, there is a reachable configuration $(G_i,{\ensuremath{\vec{d}}}_i)\in\operatorname{\textit{Reach}}({\ensuremath{\mathcal{A}}\xspace}_N)$ s.t.: for all $p\in P$: ${\ensuremath{\vec{d}}}_i(v_p)=0$, $G_i={\langleV^i, E^i,
\lambda^i,{\ensuremath{\textit{queue}}}^i,{\ensuremath{\textit{state}}}^i\rangle\xspace}$, $G_i\rhd m$, $m{\preceq}m_i'$ and $E^i=\emptyset$.
**Base case: $m_0'$**. Let us consider the run of ${\ensuremath{\mathcal{A}}\xspace}_N$ that consists in: $(a)$ executing block `main` up to line 8, then $(b)$ emptying the queue $C$. The execution of $(a)$ has the effect that: $(i)$ all $v_p$ variables are initialized to $0$ and keep this value, $(ii)$ for all place $p$: *at most* $m_0(p)$ copies of block `p` are asynchronously dispatched in queue $C$ and $(iii)$ one copy of block `trans` is dispatched in $C$. Then, the execution of $(b)$ creates one running task for each block that is present in $C$. Thus, the execution of $(a)$ followed by $(b)$ reaches a configuration $(G_0,{\ensuremath{\vec{d}}}_0)$ with $G_0={\langleV^0=V_T^0\operatorname{\mathaccent\cdot\cup}V_C^0,E^0,\lambda^0, {\ensuremath{\textit{queue}}}^0,
{\ensuremath{\textit{state}}}^0\rangle\xspace}$ s.t. $V_C^0=\emptyset$ (the queue has been emptied), for all $p$: $|{\left\{v\in V_T^0\,\vert\,\lambda(v)=\mathtt{p}\right\}}|=m_0(p)$, $|{\left\{v\in V_T^0\,\vert\,\lambda(v)=\mathtt{trans}\right\}}|=1$ and $E^0=\emptyset$ (the queue is empty and all the calls are asynchronous). Moreover, ${\ensuremath{\textit{state}}}$ is such that each task running a `p` block is still in its initial state $s^0_p$, hence $G_0\rhd m_0$. Similarly, the task running the `trans()` block is about to enter the `while` loop at line 14. Finally, as the variables have been initialized to $0$ and not modified, we have ${\ensuremath{\vec{d}}}_0(v_p)=0$ for all $p\in P$.
**Inductive case: $m_i$** Let us assume there exist $(G_{i-1},{\ensuremath{\vec{d}}}_{i-1})\in\operatorname{\textit{Reach}}({\ensuremath{\mathcal{A}}\xspace}_N)$ that respects all the conditions given at the beginning of the proof (in particular $G_{i-1}\rhd m_{i-1}$). Let $t_i$ and ${\ensuremath{\overline{m}}\xspace}_i$ be the [<span style="font-variant:small-caps;">Pn</span>]{}transition and marking s.t. and $m_i{\preceq}{\ensuremath{\overline{m}}\xspace}_i$ and let us show that ${\ensuremath{\mathcal{A}}\xspace}_N$ can simulate it. This is achieved by the following sequence of actions in ${\ensuremath{\mathcal{A}}\xspace}_N$. First, the block executing `trans` enters the `while` loop at line 14 and selects $t_i$ as transition $t$. Then, it sets all the variables $v_p$ s.t. $I_{t_i}(p)=1$ to $1$. Thus, at that point $v_p$ contains $1$ iff $I_{t_i}(p)=1$, since all $v_p$ variables were equal to $0$ by induction hypothesis. Then, the task executing `trans` is blocked as it need to wait up to the point were all $v_p$ are equal to $0$. Since $G_{i-1}\rhd m_{i-1}$ by induction hypothesis, we know that there are, in $G_{i-1}$, $m_{i-1}(p)$ tasks executing block $\mathtt{p}$, for all $p\in P$. However, $t_i$ is fireable from $m_{i-1}$, and a loss of ${\ensuremath{\overline{m}}\xspace}_i-m_i$ token is still possible after the firing. Hence, $m_{i-1}(p)\geq
(I_{t_i}(p)+{\ensuremath{\overline{m}}\xspace}_i(p)-m_i(p))$ for all $p$. Thus, for all $p$, there is at least $(I_{t_i}(p)+{\ensuremath{\overline{m}}\xspace}_i(p)-m_i(p))$ tasks executing $\mathtt{p}$ in $G_{i-1}$. Thus, we complete the run of ${\ensuremath{\mathcal{A}}\xspace}_N$ by letting, for all $p$, $(I_{t_i}(p)+{\ensuremath{\overline{m}}\xspace}_i(p)-m_i(p))$ `p` task execute lines 11 in turn one after the other. Then, letting them all execute line 12, and reach their final state (Remark that all the `p` task must first execute line 11 before one of them can execute line 12, as this sets $v_p$ to $0$ and would prevent other tasks to execute line 11). This is possible because none of those tasks are blocked, since the [$\textsc{Ctg}$]{}contains no edge, by induction hypothesis. At that point, ${\ensuremath{\mathcal{A}}\xspace}_N$ has reached a configuration $(G',{\ensuremath{\vec{d}}}')$ s.t. ${\ensuremath{\vec{d}}}'(v_p)=0$ for all $p\in P$ (by line 12) and where $G'\rhd
m_{i-1}-(I_{t_i}+{\ensuremath{\overline{m}}\xspace}_i-m_i)$. Moreover, $G'$ still respects all the other hypothesis as no new dispatch have been performed. Then, the simulation of $t_i$ proceeds by letting the `trans` task finish the current iteration of the main `while` loop. This consists in executing the `for` loop of line 19, which dispatches one `p` block in $C$ iff $O_{t_i}(p)=1$, i.e., the effect of $t_i$ is to add a token to $p$. Finally, the scheduler empties queue $C$ and creates tasks for all the blocks that have just been added to $C$. It also kills all the `p` tasks that have reached their final state. As a consequence, the configuration that is reached is $(G_i, {\ensuremath{\vec{d}}}_i)$, where $G_i\rhd
m_{i-1}-(I_{t_i}+{\ensuremath{\overline{m}}\xspace}_i-m_i)+O_{t_i} =
(m_{i-1}-I_{t_i}+O_{t_i})-{\ensuremath{\overline{m}}\xspace}_i+m_i = {\ensuremath{\overline{m}}\xspace}_i-{\ensuremath{\overline{m}}\xspace}_i+m_i = m_i$ and ${\ensuremath{\vec{d}}}_i$ is s.t. ${\ensuremath{\vec{d}}}_i(v_p)=0$ for all $p\in P$. Moreover, since the queue has been emptied by the scheduler, $G_i$ contains only task nodes and no edge, as all the calls are asynchronous. The task executing `trans` is still active and at line 14, and all the `p` tasks are in their initial state.[$\Box$]{}
Let $N={\langleP,T,m_0\rangle\xspace}$ be a [<span style="font-variant:small-caps;">Pn</span>]{}, and let ${\ensuremath{\mathcal{A}}\xspace}_N={\langle{CQID\xspace},
\emptyset, \Gamma, \mathtt{main}, {\ensuremath{\mathcal{X}}\xspace}, \Sigma,
({\ensuremath{\mathcal{TS}}\xspace}_\gamma)_{\gamma\in\Gamma}\rangle\xspace}$ be its corresponding [<span style="font-variant:small-caps;">Qdas</span>]{}. **If** there are $(G,{\ensuremath{\vec{d}}})\in\operatorname{\textit{Reach}}({\ensuremath{\mathcal{A}}\xspace}_N)$ and $m$ s.t. $G\rhd m$ **then** $m(G)\in{{\ensuremath{\textit{Cover}}}}(N)$.
For a [$\textsc{Ctg}$]{}$G$ of ${\ensuremath{\mathcal{A}}\xspace}_N$ with set of vertices $V$, we denote by $M(G)$ the marking of $N$ s.t. for all $p\in P$: $M(G)(p)=|{\left\{v\in
V\,\vert\,{\ensuremath{\textit{state}}}(v)=s^0_p\right\}}|$. Thus, in the case where $G$ encodes a configuration s.t. `trans` is at line 14, `main` is at line 8, and all the `p` blocks are in their initial state, then $G\rhd M(G)$.
In order to establish the lemma, we prove a stronger statement: every time we reach, along a run, a configuration $(G,{\ensuremath{\vec{d}}})$ s.t. `trans` is at line 14, then $M(G)\in{{\ensuremath{\textit{Cover}}}}(N)$. Formally, let $\rho=(G_0,{\ensuremath{\vec{d}}}_0)a_0(G_1,{\ensuremath{\vec{d}}}_1)a_1(G_2,{\ensuremath{\vec{d}}}_2)\cdots(G_n,{\ensuremath{\vec{d}}}_n)$ be a run of ${\ensuremath{\mathcal{A}}\xspace}_N$, where, for all $0,\leq i\leq n$: $G_i={\langleV_i,E_i,\lambda_i,{\ensuremath{\textit{queue}}}_i,{\ensuremath{\textit{state}}}_i\rangle\xspace}$. Let $\pi:\{0,\ldots, k\}\rightarrow \{0,\ldots,n\}$ be the monotonically increasing function s.t. $k\leq n$ and for all $0\leq j\leq n$: there exists $v\in V_j$ with ${\ensuremath{\textit{state}}}_i(v)={\ensuremath{s^{14}_{\mathtt{trans}}}}$ iff there is $0\leq \ell\leq k$ with $k=\pi(\ell)$. That is the sequence $\pi(1), \pi(2),\ldots, \pi(k)$ identifies the indexes of all the configurations of the run where `trans` is at line 14. Let us show, by induction on $i$ that all the $M(G_{\pi(i)})$’s are reachable *in the lossy semantics* of $N$.
**Base case $i=0$** Let us show that $M(G_{\pi(0)})=m_0$, i.e., that the first time `trans` reaches line 14, $M(G_{\pi(0)})$ is the initial marking of $N$. Observe that the prefix of the run must have the following form. Initially, only the `main` block is executing: it first sets all the variables $v_p$ to $0$, then dispatches asynchronously at most $m_0(p)$ calls to each `p` block (for all $p\in P$), then finally dispatches an asynchronous call to `trans` and reaches line 8. Along this execution, the scheduler might decide to pick up some `p` blocks from $C$. However, as long as the scheduler has not scheduled the call to `trans`, the [$\textsc{Ctg}$]{}met along the run do not encode any marking, by definition of $\rhd$. When the scheduler starts a task to run the `trans` block, we thus reach a configuration $(G,{\ensuremath{\vec{d}}})$ where: $(i)$ the queue $C$ is empty, as dequeueing the `trans` block is possible only if all the `p` blocks have been dequeued, and no other dispatch has been performed; $(ii)$ all the `p` tasks are blocked in their initial state as ${\ensuremath{\vec{d}}}(v_p)=0$ for all $p\in P$; and $(iii)$ `main` is still blocked in the infinite loop at line $8$. Since the scheduler has just dequeued `trans` from $C$, $G$ is necessarily the first [$\textsc{Ctg}$]{}to encode a marking, so $G=G_{\pi(0)}$. Moreover, by the loop at line 4, it is clear that $G\rhd m$ with $m{\preceq}m_0$.
**Inductive case $i=\ell\geq 1$** The induction hypothesis is that $M(G_{\pi(i-1})\in {{\ensuremath{\textit{Cover}}}}(N)$. Let us consider the $\rho'=(G_{\pi(\ell-1)},{\ensuremath{\vec{d}}}_{\pi(\ell-1)})\cdots(G_{\pi(\ell)},{\ensuremath{\vec{d}}}_{\pi(\ell)})
$, i.e. the portion of $\rho$ that allows to reach $(G_{\pi(\ell)},{\ensuremath{\vec{d}}}_{\pi(\ell)})$ from $(G_{\pi(\ell-1)},{\ensuremath{\vec{d}}}_{\pi(\ell-1)})$. We consider two cases:
1. Either **trans** has not performed an iteration of its main `while` loop along $\rho'$. In this case, the only actions that can occur along $\rho'$ are scheduler actions consisting in dequeueing `p` blocks or the termination of some `p` tasks that where still in state $s^{mid}_p$. In both cases, this does not modify the value of $M(G)$, so $M(G_{\pi(i)})=M(G_{\pi(i-1})\in {{\ensuremath{\textit{Cover}}}}(N)$.
2. Or `trans` has performed a complete iteration of its main `while` loop possibly interleaved with the dequeue of `p` blocks and the termination of `p` tasks. Since the dequeues and terminations have no influence on the value of $M(G)$ as argued above, let us focus on the effect of executing one iteration of the `while` loop. The iteration first selects a [<span style="font-variant:small-caps;">Pn</span>]{}transition $t$ and sets all the variables $v_p$ s.t. $I_t(p)=1$ to $1$. The reached configuration is then $(G,{\ensuremath{\vec{d}}})$ where $M(G)=M(G_{\pi(i-1)})$, as these operations do not manipulate `p` blocks or tasks. Then, `trans` is blocked by the test at line 18. As only `p` blocks can set $v_p$ variables to $0$, we are sure that, when `trans` reaches line 19, *at least* $I_t(p)$ `p` blocks have left their initial state, for all $p\in P$. Thus, when `trans` is at line 19, the configuration is $(G',{\ensuremath{\vec{d}}}')$, where for all $p\in P$: $M(G')(p)\leq M(G)(p)-I_t(p)=M(G_{\pi(i-1)})-I_t(p)$. Afterwards, `trans` terminates the iteration of the `while` loop by dispatching $O_t(p)$ `p` blocks for all $p\in P$, and reaches line 14, which finishes $\rho'$. Hence, we reach $(G_{\pi(i)},{\ensuremath{\vec{d}}}_{\pi(i)})$, where for all $p\in P$: $M(G_{\pi(i)})(p)\leq M(G_{\pi(i-1)})-I_t(p)+O_t(p)$. Since $M(G_{\pi(i-1)})\in{{\ensuremath{\textit{Cover}}}}(N)$ by induction hypothesis, we conclude that $M(G_{\pi(i)})\in{{\ensuremath{\textit{Cover}}}}(N)$ too.[$\Box$]{}
Asynchronous Serial [<span style="font-variant:small-caps;">Qdas</span>]{}
--------------------------------------------------------------------------
We establish the undecidability for asynchronous serial [<span style="font-variant:small-caps;">Qdas</span>]{}by a reduction from the control-state reachability problem in a [fifo]{}system. Let $F={\langleS_F,s_F^0,M,\Delta_F\rangle\xspace}$ be a [fifo]{}system and let $c\in S_F$ be a control state whose reachability has to be tested. We build the asynchronous serial [<span style="font-variant:small-caps;">Qdas</span>]{}${\ensuremath{\mathcal{A}}\xspace}_F={\langle \emptyset, \{q\}, \Gamma, \mathtt{main}, {\ensuremath{\mathcal{X}}\xspace}, \Sigma,
({\ensuremath{\mathcal{TS}}\xspace}_\gamma)_{\gamma\in\Gamma}\rangle\xspace}$ on domain ${\ensuremath{\mathbb{D}}\xspace}=M\cup
S_F\cup\{{\ensuremath{\varepsilon}\xspace}\}$, where $\Gamma=M\cup\{{\ensuremath{\varepsilon}\xspace},\mathtt{main}\}$, ${\ensuremath{\mathcal{X}}\xspace}=\{\mathtt{state},\mathtt{head}\}$ and the ${\ensuremath{\mathcal{TS}}\xspace}_{\gamma}$ are given by the pseudo code in Fig. \[fig:fifo-to-qdas\].
global state, head
global s_queue q
def main():
state := $s^0_F$
head := $\e$
dispatch_a(q, $\e$)
while(true): do nothing
node\[anchor=north west,draw, rectangle callout,callout relative pointer=[(1.2,-0.08)]{},font=, fill=gray!5,text width=4.7cm\] [ Note that the reachability of a state $c$ of the [fifo]{}system is explicitely coded into the control structure.]{};
``` {startFrom="last"}
def m(): //for all $m\in M\cup\{\e\}$
if (head$\neq m$): goto 20
while(true):
if (state = $c$): goto 21
select $(s,a,s')\in\Delta_y$
if ($s\neq$state): goto 20
state := $s'$
if ($a=!n$): dispatch_a(q, $n$)
else if ($a=?n$):
head := $n$
terminate
while(true): do nothing // wrong guess
while(true): do nothing // $c$ is reached
```
\[zstd/.style=[state,font=]{},zstd2/.style=[zstd,rectangle]{},anchor=west\]
(-0.5,0.6) node\[anchor=west\] (dummy)[[$\textsc{Ctg}$]{}type (a):]{}; (0,0) node\[zstd2\] (00) [$m_1$]{}; (00)+(0.7,0) node\[zstd2\] (01) [$m_2$]{}; (01)+(0.7,0) node\[font=,anchor=west\] (02) […]{}; (02)+(0.7,0) node\[zstd2\] (03) [$m_n$]{}; (03)+(0.7,0) node\[zstd\] (04) [[$\textit{main}$]{}]{}; (00) edge\[->\] (01) (01) edge\[->\] (02) (02) edge\[->\] (03);
[background]{} node\[fit=(00) (03),fill=gray!20\] (c1) ; (c1.north east) node\[anchor=south east,inner sep=0pt,font=\] [queue $q$]{};
\[zstd/.style=[state,font=]{},zstd2/.style=[zstd,rectangle]{},anchor=west\]
(-0.5,0.6) node\[anchor=west\] (dummy)[[$\textsc{Ctg}$]{}type (b):]{}; (0,0) node\[zstd2\] (00) [$m_1$]{}; (00)+(0.7,0) node\[zstd2\] (01) [$m_2$]{}; (01)+(0.7,0) node\[font=,anchor=west\] (02) […]{}; (02)+(0.7,0) node\[zstd2\] (03) [$m_n$]{}; (03)+(0.7,0) node\[zstd\] (04) [$m$]{}; (04)+(0.4,0) node\[zstd\] (05) [[$\textit{main}$]{}]{}; (00) edge\[->\] (01) (01) edge\[->\] (02) (02) edge\[->\] (03) (03) edge\[->,dashed\] (04);
[background]{} node\[fit=(00) (03),fill=gray!20\] (c1) ; (c1.north east) node\[anchor=south east,inner sep=0pt,font=\] [queue $q$]{};
\[fig:shapes-aaf\]
Intuitively, runs of ${\ensuremath{\mathcal{A}}\xspace}_F$ simulate the runs of $F$, by encoding the current state of $F$ in variable `state` and the content of $F$’s queue into the content of the serial queue `q`. More precisely, it easy to check that, once `main` has reached line 8, all the [$\textsc{Ctg}$]{}that are reached in ${\ensuremath{\mathcal{A}}\xspace}_F$ are of either shapes depicted in Fig. \[fig:shapes-aaf\], for $\{m_1,\ldots,
m_n,m\}\subseteq M\cup\{{\ensuremath{\varepsilon}\xspace}\}$. That is, there are at most two running tasks: `main` and possibly one task running a $m$ block (for $\texttt{m}\in M\cup\{{\ensuremath{\varepsilon}\xspace}\}$), that has to terminate to allow a further dequeue from `q`. This is because `q` is a serial queue and all the dispatches are asynchronous. When the [$\textsc{Ctg}$]{}is of shape , the duty of the running $m$ block is to simulate a run of $F$. It runs an infinite `while` loop (line 11 onwards – ignore the test at line 10 for the moment), that $(i)$ tests whether $c$ has been reached (line 12) and jumps to line 20 if it is the case; $(ii)$ guesses a transition $(s,a,s')$ of $F$; and $(iii)$ checks that the guessed transition is indeed fireable from the current configuration of $F$, and, if yes, simulate it. This consists in, first testing that $s$ is the current state (line 14). If not, the block jumps to the infinite loop of line 19, which ends the simulation. Otherwise, the current state is update to $s'$, and the channel operation is then simulated. A send of message $m$ is simulated (line 16) by an asynchronous dispatch of block $m$ to `q`. The simulation of a receive of $m$ from `q` is more involved, as only the scheduler can decide to dequeue a block from `q`, and this can happen only if the current running block terminates (line 19). Still, we have to check that message $m$ is indeed in the head of `q`. This is achieved by setting global variable `head` to $m$, and letting the next dequeues block check that itself encodes the value stored into `head`. This is performed at line 10. If this test is not satisfied, the block jumps to the infinite loop of line 20, and the simulation ends. Otherwise, it proceeds with the simulation. Thus, in all reachable configurations of ${\ensuremath{\mathcal{A}}\xspace}_F$, a block $m$ (with $m\in M\cup\{{\ensuremath{\varepsilon}\xspace}\}$ will reach line 21 iff $c$ is reachable in $F$. This effectively reduces the control location reachability of [fifo]{}systems to the Parikh coverability problem of serial asynchronous [<span style="font-variant:small-caps;">Qdas</span>]{}.
The proof of Theorem \[the:async-seri-undec\] relies on the next Lemma, that formalizes the relationship between reachable configurations of ${\ensuremath{\mathcal{A}}\xspace}_F$ and reachable configurations of $F$.
For all $\gamma\in\Gamma$, we denote by ${\ensuremath{s^{\ell}_{\mathtt{\gamma}}}}$ the location of ${\ensuremath{\mathcal{TS}}\xspace}_\gamma$ that corresponds to line $\ell$ in Fig. \[fig:fifo-to-qdas\]. Then, we say that a configuration $({\ensuremath{G}},{\ensuremath{\vec{d}}})$ of ${\ensuremath{\mathcal{A}}\xspace}_F$ encodes a configuration $(s,w)$ of $F$, written $({\ensuremath{G}},{\ensuremath{\vec{d}}})\rhd(s,w)$ iff: $(i)$ $s={\ensuremath{\vec{d}}}(\mathtt{state})$, $(ii)$ ${\ensuremath{G}}$ is of either shapes in Fig. \[fig:shapes-aaf\] with $w=m_0m_1\cdots m_n$, $(iii)$ ${\ensuremath{\mathsf{Parikh}}}(G)({\ensuremath{s^{8}_{\mathtt{{\tt main}}}}})=1$ and $(iv)$ there exists $m\in
M\cup\{{\ensuremath{\varepsilon}\xspace}\}$ s.t. ${\ensuremath{\mathsf{Parikh}}}(G)({\ensuremath{s^{12}_{\mathtt{m}}}})=1$. That is, $s$ and $w$ are encoded as described above, `main` is at line 8, and the running $\mathtt{m}$ block is at line 12. Then:
\[lem:from-fifo-to-qdas\] Let $F$ be a FIFO system, let $c$ be a configuration of $F$, and let ${\ensuremath{\mathcal{A}}\xspace}_F$ be its associated [<span style="font-variant:small-caps;">Qdas</span>]{}. For all run $(s_0,w_0)(s_1,w_1)\cdots(s_n,w_n)$ of $F$ s.t. for all $0\leq i<n$: $s_i\neq c$, there exists $(G,{\ensuremath{\vec{d}}})\in\operatorname{\textit{Reach}}({\ensuremath{\mathcal{A}}\xspace}_F)$ s.t. $(G,{\ensuremath{\vec{d}}})\rhd(s_n,w_n)$. Moreover, for all $(G,{\ensuremath{\vec{d}}})\in\operatorname{\textit{Reach}}({\ensuremath{\mathcal{A}}\xspace}_F)$ and for all configuration $(s,w)$ of $F$: $(G,{\ensuremath{\vec{d}}})\rhd(s,w)$ implies $(s,w)\in\operatorname{\textit{Reach}}(F)$
First, we consider a run $(s_0,w_0)(s_1,w_1)\cdots(s_n,w_n)$ of $F$ s.t. for all $0\leq i<n$: $s_i\neq c$, and build a run $({\ensuremath{G}}_0,{\ensuremath{\vec{d}}}_0)a_0({\ensuremath{G}}_1,{\ensuremath{\vec{d}}}_1)a_1\cdots
a_{k-1}({\ensuremath{G}}_k,{\ensuremath{\vec{d}}}_k)$ of ${\ensuremath{\mathcal{A}}\xspace}_F$ s.t. $({\ensuremath{G}}_k,{\ensuremath{\vec{d}}}_k)\rhd(s_n,w_n)$, by induction on the length of $F$’s run.
**Base case $n=0$:** Consider the run of ${\ensuremath{\mathcal{A}}\xspace}_F$ that consists in executing lines 5, 6, 7 of `main` (which sets the `head` variable to ${\ensuremath{\varepsilon}\xspace}$), then dequeueing the ${\ensuremath{\varepsilon}\xspace}$ block from the queue, then executing lines 10 and 11 of ${\ensuremath{\varepsilon}\xspace}$. Remark that the test at line 10 is not satisfied, as $\mathtt{head}={\ensuremath{\varepsilon}\xspace}$, and that the queue is now empty. Clearly, the resulting configuration $({\ensuremath{G}},{\ensuremath{\vec{d}}})\rhd(s^0_F,w_0)$ as $w_0={\ensuremath{\varepsilon}\xspace}$.
**Inductive case $n=\ell$**. Let us assume that there is a reachable configuration $({\ensuremath{G}}, {\ensuremath{\vec{d}}})$ of ${\ensuremath{\mathcal{A}}\xspace}_F$ s.t. $({\ensuremath{G}},{\ensuremath{\vec{d}}})\rhd (s_{\ell-1}, w_{\ell-1})$, and let us build a sequence of ${\ensuremath{\mathcal{A}}\xspace}_F$ transitions that is fireable from $({\ensuremath{G}},{\ensuremath{\vec{d}}})$ and reaches a configuration encoding $(s_\ell,w_\ell)$. In $({\ensuremath{G}},{\ensuremath{\vec{d}}})$, there is, by definition of $\rhd$, a task running a $b$ block, for $b\in M\cup\{{\ensuremath{\varepsilon}\xspace}\}$, that is at line 12. Moreover, ${\ensuremath{\vec{d}}}(\mathtt{state})=s_{\ell-1}$. Let $\delta$ be the transition of $F$ s.t. $(s_{\ell-1},w_{\ell-1})\xrightarrow{\delta}(s_\ell,w_\ell)$. By hypothesis, $s_{\ell-1}\neq c$, hence, we let $b$ execute line 12; select $\delta=(s_{\ell-1},a,s_\ell)$ at line 13; execute line 14, where the condition of the `if` is not satisfied as $s=s_{\ell-1}=\mathtt{state}$; and execute line 16, which reaches a configuration $({\ensuremath{G}}',{\ensuremath{\vec{d}}}')$ where ${\ensuremath{\vec{d}}}'(\mathtt{state})=s_\ell$. We consider three cases to complete the simulation of $\delta$ in ${\ensuremath{\mathcal{A}}\xspace}_F$. If $a=!n$, the $b$ task performs an asynchronous dispatch of $n$ to `q`, and jumps to line 11, then 12. Clearly, the resulting configuration $({\ensuremath{G}}'',{\ensuremath{\vec{d}}}'')$ is s.t. $({\ensuremath{G}}'',{\ensuremath{\vec{d}}}'')\rhd(s_\ell,w_\ell)$ (in particular, the dispatch has correctly updated the content of the queue). If $a={\ensuremath{\varepsilon}\xspace}$, the $b$ tasks jumps directly to line 11, then to line 12. Again, the resulting configuration $({\ensuremath{G}}'',{\ensuremath{\vec{d}}}'')$ is s.t. $({\ensuremath{G}}'',{\ensuremath{\vec{d}}}'')\rhd(s_\ell,w_\ell)$, as the content of the queue has not been modified. Finally, if $a=!n$, the running $b$ block sets `head` to $n$ and terminates. Let $({\ensuremath{G}}'',{\ensuremath{\vec{d}}}'')$ be the ${\ensuremath{\mathcal{A}}\xspace}_F$ configuration reached at that point. As $\delta$ is fireable from $(s_{\ell-1},w_{\ell-1})$ in $F$, since $({\ensuremath{G}}, {\ensuremath{\vec{d}}})\rhd(s_{\ell-1},w_{\ell-1})$, and as the content of the queue has not been modified since then, the head of `q` is necessarily an $n$ block in ${\ensuremath{G}}''$. Moreover, ${\ensuremath{\vec{d}}}''(\mathtt{head})=n$ and ${\ensuremath{\vec{d}}}''(\mathtt{state})=s_\ell$. Thus, we let the scheduler dequeue this $n$ block, and we let the task running it execute line 10 (where the condition of the `if` is not satisfied), then line 11. Clearly, the resulting configuration encodes $(s_\ell, w_\ell)$.
Now, let $\rho=({\ensuremath{G}}_0,{\ensuremath{\vec{d}}}_0)a_0({\ensuremath{G}}_1,{\ensuremath{\vec{d}}}_1)a_1\cdots a_{n-1}
({\ensuremath{G}}_n,{\ensuremath{\vec{d}}}_n)$ be a run of ${\ensuremath{\mathcal{A}}\xspace}_F$ s.t. there is $(s,w)$ with $({\ensuremath{G}}_n,{\ensuremath{\vec{d}}}_n)\rhd(s,w)$, and let us build, by induction on the length of this run, a run $(s^0_F,w_0)(s_1,w_1)\cdots(s_k,w_k)$ a run of $F$ s.t. $(s_k,w_k)=(s,w)$.
Let $K=|{\left\{({\ensuremath{G}}_i,{\ensuremath{\vec{d}}}_i)\,\vert\,{\ensuremath{\mathsf{Parikh}}}({\ensuremath{G}}_i)({\ensuremath{s^{12}_{\mathtt{m}}}})=1\textrm{
for }m\in M\cup\{{\ensuremath{\varepsilon}\xspace}\}\right\}}|$, i.e., $K$ is the number of times an $m$ block reaches line $12$ along $\rho$. Let us consider the increasing monotonic function $\rho:\{1,\ldots,K\}\rightarrow\{0,\ldots,n\}$ s.t. for all $0\leq
i\leq n$: there exists $m\in M\cup\{{\ensuremath{\varepsilon}\xspace}\}$ s.t. ${\ensuremath{\mathsf{Parikh}}}({\ensuremath{G}}_i)({\ensuremath{s^{12}_{\mathtt{m}}}})=1$ iff there is $1\leq
j\leq K$ s.t. ${\ensuremath{G}}_i=\rho(j)$, that is, $\rho(i)$ is the index, in $\rho$ of the $i$th time a configuration is reached where an $\mathtt{m}$ block is at line 12. Clearly, by definition of $\rhd$ only the $({\ensuremath{G}}_{\rho(j)},{\ensuremath{\vec{d}}}_{\rho(j)})$ configurations (for $1\leq j\leq K$) can encode a configuration of $F$, as no $m$ block is at line 12 in the other configurations of $\rho$. So, it is sufficient to show that all those $({\ensuremath{G}}_{\rho(j)},{\ensuremath{\vec{d}}}_{\rho(j)})$ configurations encode a reachable configuration of $F$. We proceed by induction on $j$, and show that: for all $1\leq j\leq K$: $({\ensuremath{G}}_{\rho(j)},{\ensuremath{\vec{d}}}_{\rho(j)})$ encodes a reachable configuration of $F$ and ${\ensuremath{G}}_{\rho(j)}$ contains exactly one $m$ task (for $m\in M\cup\{e\}$), that has been dequeued from `q`.
**Base case $j=0$:** Observe that the subrun $({\ensuremath{G}}_0,{\ensuremath{\vec{d}}}_0)a_0\cdots
a_{\rho(1)-1}({\ensuremath{G}}_{\rho(1)},{\ensuremath{\vec{d}}}_{\rho(1)})$ is necessarily an initialization phase where `main` sets `state` to $s_F^0$, `head` to ${\ensuremath{\varepsilon}\xspace}$, dispatches an ${\ensuremath{\varepsilon}\xspace}$ block, and reaches line 8, where it will stay forever. Then, the scheduler dequeues the ${\ensuremath{\varepsilon}\xspace}$ block, which empties the queue. The ${\ensuremath{\varepsilon}\xspace}$ task then traverses line 10 (as `head`$={\ensuremath{\varepsilon}\xspace}$) and 11 and reaches line 12. So, clearly $({\ensuremath{G}}_{\rho(0)},{\ensuremath{\vec{d}}}_{\rho(0)})\rhd (s^0_F,{\ensuremath{\varepsilon}\xspace})$ and contains exactly one $m$ task (for $m\in M\cup\{e\}$), that has been dequeued from `q`.
**Inductive case $j=\ell$:** Let us assume that $({\ensuremath{G}}_{\rho(\ell-1)},{\ensuremath{\vec{d}}}_{\rho(\ell-1)})$ encodes a reachable configuration $(s_{\ell-1}, w_{\ell-1})$ of $F$. We consider several cases. If $({\ensuremath{G}}_{\rho(\ell-1)},{\ensuremath{\vec{d}}}_{\rho(\ell-1)}) =
({\ensuremath{G}}_{\rho(\ell)},{\ensuremath{\vec{d}}}_{\rho(\ell)})$ we are done. Otherwise, we have necessarily performed one iteration (possibly interrupted at line 12, 14 or 19) of the while loop at line 11 between $({\ensuremath{G}}_{\rho(\ell-1)},{\ensuremath{\vec{d}}}_{\rho(\ell-1)})$ and $({\ensuremath{G}}_{\rho(\ell)},{\ensuremath{\vec{d}}}_{\rho(\ell)})$, as, by induction hypothesis, ${\ensuremath{G}}_{\rho(\ell-1)}$ contains exactly one $m$ task (with $m\in M\cup\{{\ensuremath{\varepsilon}\xspace}\}$) that blocks `q`, and `main` can only loop at line 8, which does not modify the current configuration. Then, observe that the conditions of the `if` at lines 12 and 14 were necessarily false during the iteration. Otherwise, $m$ would have reached line 21, from which it cannot escape. From that point, no configuration is reachable where an $\mathtt{m}$ block is at line 12 , and $({\ensuremath{G}}_{\rho(ell)},
{\ensuremath{\vec{d}}}_{\rho(\ell)})$ cannot exist. Thus, we consider three cases:
- If we have entered the `if` at line 16 during the iteration, then a transition of the form $(s,!n,s')$ has been guessed, with $\texttt{state}=s$ and a dispatch of $n$ has been performed into $q$. As $({\ensuremath{G}}_{\rho(\ell-1)},{\ensuremath{\vec{d}}}_{\rho(\ell-1)})\rhd (s_{\ell-1},
w_{\ell-1})$ by induction hypothesis, $s_{\ell-1}=s$, and thus $(s,!n,s')$ is fireable from $(s_{\ell-1}, w_{\ell-1})$ and reaches $(s',n\cdot w_{\ell-1})$. Clearly, this configuration is encoded by $({\ensuremath{G}}_{\rho(\ell)},{\ensuremath{\vec{d}}}_{\rho(\ell)})$.
- If we have entered the `else if` at line 17 during the iteration, then a transition of the form $(s,?n,s')$ has been guessed, with $\texttt{state}=s$, `head` has been set to $n$, the current $m$ block has been terminated, a new block $m'$ has been dequeued by the scheduler (as there is necessarily a running `m` block in ${\ensuremath{G}}_{\rho(\ell)}$). Moreover $m'=n$, because $m'$ has to be at line 12 in ${\ensuremath{G}}_{\rho(\ell)}$, so the test of line 10 had to be false to allow $m'$ to reach line 12. As $({\ensuremath{G}}_{\rho(\ell-1)},{\ensuremath{\vec{d}}}_{\rho(\ell-1)})\rhd (s_{\ell-1},
w_{\ell-1})$ by induction hypothesis, $s_{\ell-1}=s$. As a dequeue of a block $m'=n$ has been performed, $w_{\ell-1}$ is of the form $w\cdot n$. Thus, $(s,?m,s')$ is fireable from $(s_{\ell-1},
w_{\ell-1})$ and reaches $(s',w)$. Clearly, this configuration is encoded by $({\ensuremath{G}}_{\rho(\ell)},{\ensuremath{\vec{d}}}_{\rho(\ell)})$.
- Finally, if neither the `if` nor the `else if` have been entered during the iteration, then a transition of the form $(s,{\ensuremath{\varepsilon}\xspace},s')$ has been guessed, with $\texttt{state}=s$. As $({\ensuremath{G}}_{\rho(\ell-1)},{\ensuremath{\vec{d}}}_{\rho(\ell-1)})\rhd (s_{\ell-1},
w_{\ell-1})$ by induction hypothesis, $s_{\ell-1}=s$, and thus $(s,{\ensuremath{\varepsilon}\xspace},s')$ is fireable from $(s_{\ell-1}, w_{\ell-1})$ and reaches $(s',w_{\ell-1})$. Clearly, this configuration is encoded by $({\ensuremath{G}}_{\rho(\ell)},{\ensuremath{\vec{d}}}_{\rho(\ell)})$.[$\Box$]{}
We can now prove Theorem \[the:async-seri-undec\]:
Let $F$ be a FIFO system, with set of messages $M$ and associated serial asynchronous [<span style="font-variant:small-caps;">Qdas</span>]{}${\ensuremath{\mathcal{A}}\xspace}_F$ and let $c$ be a control location of $F$. For all $m\in M\cup\{{\ensuremath{\varepsilon}\xspace}\}$, let $f_m$ be the Parikh image s.t. $f_m({\ensuremath{s^{21}_{\mathtt{{\tt main}}}}})=1$ and $f_m(s)=0$ for all $s\neq
{\ensuremath{s^{21}_{\mathtt{{\tt main}}}}}$. Remark that there are only finitely many such $f_m$. Then, we show that $c$ is reachable in $F$ iff there exists $m\in M\cup\{{\ensuremath{\varepsilon}\xspace}\}$ s.t. $f_m$ is Parikh-coverable in ${\ensuremath{\mathcal{A}}\xspace}_F$.
Assume $c$ is reachable in $F$, and let $(c,w)$ be a configuration in $\operatorname{\textit{Reach}}(F)$. Without loss of generality, assume $c$ is reachable by run that visits $c$ only once. By Lemma \[lem:from-fifo-to-qdas\], there is $({\ensuremath{G}},{\ensuremath{\vec{d}}})\in\operatorname{\textit{Reach}}({\ensuremath{\mathcal{A}}\xspace}_F)$ s.t. $({\ensuremath{G}},{\ensuremath{\vec{d}}})\rhd(c,w)$. Hence, in $({\ensuremath{G}},{\ensuremath{\vec{d}}})$, there is a task running an $m$ block (for $m\in M\cup\{{\ensuremath{\varepsilon}\xspace}\}$) that is at line 12, and ${\ensuremath{\vec{d}}}(\mathtt{state})=c$. Thus, $m$ can execute one step and reach line 21, so $f_m$ is Parikh coverable in ${\ensuremath{\mathcal{A}}\xspace}_F$.
For the reverse direction, assume there is $m\in M\cup\{{\ensuremath{\varepsilon}\xspace}\}$ that is Parikh-coverable in ${\ensuremath{\mathcal{A}}\xspace}_F$. Hence, there is $({\ensuremath{G}},{\ensuremath{\vec{d}}})\in\operatorname{\textit{Reach}}({\ensuremath{\mathcal{A}}\xspace}_F)$ where a task running block $m$ is at line 21. The only way for that block to reach line 21 is from line 12, with a valuation ${\ensuremath{\vec{d}}}'$ s.t. ${\ensuremath{\vec{d}}}'(\mathtt{state})=c$. Thus, there is, in $\operatorname{\textit{Reach}}({\ensuremath{\mathcal{A}}\xspace}_F)$ a configuration $({\ensuremath{G}}',{\ensuremath{\vec{d}}}')$ with ${\ensuremath{\vec{d}}}'(\mathtt{state})=c$, a task running an $m$ block at line 12, and necessarily `main` at line 8 (otherwise, only `main` would be running). Hence, $({\ensuremath{G}}',{\ensuremath{\vec{d}}}')$ is a reachable configuration of ${\ensuremath{\mathcal{A}}\xspace}_F$ s.t. $({\ensuremath{G}}',{\ensuremath{\vec{d}}}')\rhd (c,w)$ for some queue content $w$. Thus, by Lemma \[lem:from-fifo-to-qdas\], $(c,w)\in\operatorname{\textit{Reach}}(F)$, and $c$ is reachable in $F$.
We have thus reduced the control location reachability problem of FIFO systems to the Parikh coverability problem of serial asynchronous [<span style="font-variant:small-caps;">Qdas</span>]{}(using only one serial queue). The former is undecidable. Hence the theorem.[$\Box$]{}
Concurrent [<span style="font-variant:small-caps;">Qdas</span>]{}
-----------------------------------------------------------------
We reduce the reachability problem of two counter systems. Let us give the intuition of the construction. For each ${\ensuremath{\mathcal{P}}\xspace}$, we construct a [<span style="font-variant:small-caps;">Qdas</span>]{}${\ensuremath{\mathcal{A}}\xspace}_{\ensuremath{\mathcal{P}}\xspace}$ s.t. all reachable [$\textsc{Ctg}$]{}in ${\ensuremath{\mathcal{A}}\xspace}_{\ensuremath{\mathcal{P}}\xspace}$ encode configurations of ${\ensuremath{\mathcal{P}}\xspace}$ and are of the form depicted in Fig.\[fig:sim\_counter\]. That is, (after an initialization phase), there are always three tasks that are unblocked: a [$\textit{main}$]{}task to simulate ${\ensuremath{\mathcal{P}}\xspace}$’s control structure, and, for each $i=\{1,2\}$, either a task $eins(i)$ or a task $null(i)$. If the task $null(i)$ is unblocked, then counter $i$ is zero in the current configuration of ${\ensuremath{\mathcal{P}}\xspace}$. Otherwise, the current valuation of counter $i$ is encoded by the number of $eins(i)$ tasks in the [$\textsc{Ctg}$]{}. Remark that, as in the case of synchronous [<span style="font-variant:small-caps;">Qdas</span>]{}, the parts of the [$\textsc{Ctg}$]{}that encode each counter behave as pushdown stacks. Finally, the control location of ${\ensuremath{\mathcal{P}}\xspace}$ is recorded in global variable `state`.
global state
global $\ell_1^1$, $\ell_2^1$, $x^1$ // rdvz channel 1
global $\ell_1^2$, $\ell_2^2$, $x^2$ // rdvz channel 2
global c_queue q
def main():
foreach i in {1,2}:
dispatch_a(q, null(i))
i?ack
state := $x^0$
while(true):
select $(s,a,s')\in\Delta_\Pp$ where state=$s$
if $a=\incr(1)$ :
1!$\incr$
1?$\ack$
state:=$s'$
\\ other actions analogous
...
(0,0.3) coordinate (base); (1.5,0) node\[zstd\] (1) [[1]{}]{}; (1)+(-0.5,0) node (s) (s) edge\[->\] (1); (3,0) node\[zstd\] (2) [2]{}; (4,0) node\[zstd\] (f) ; (f) circle (3pt); (0,-1) node\[zstd\] (3) [3]{}; (1.5,-1) node\[zstd\] (4) [4]{}; (3,-1) node\[zstd\] (5) [5]{}; (3) edge\[->,bend left=11\] node\[above,font=\][$i!\operatorname{{\ensuremath{\mathtt{ack}}\xspace}}$]{} (4); (4) edge\[->,bend left=11\] node\[below,font=\][$i?\operatorname{{\ensuremath{\mathtt{is\_zero}}\xspace}}$]{} (3); (1) edge\[->\] node\[left,font=,pos=0.3\][$i!\operatorname{{\ensuremath{\mathtt{ack}}\xspace}}$]{} (4); (4) edge\[->\] node\[below,font=\][$i?\operatorname{{\ensuremath{\mathtt{incr}}\xspace}}$]{}(5); (5) edge\[->\] node\[right,font=\][${\ensuremath{{\ensuremath{\mathtt{dispatch_s}}\xspace}}}(q,eins(i))$]{} (2); (2) edge\[->\] node\[left,pos=0.3,font=\][$i!\operatorname{{\ensuremath{\mathtt{ack}}\xspace}}$ ]{}(4);
(0,0) node [$null(i)$:]{};
\[zstd/.style=[state,font=]{}\] (1.5,0) node\[zstd\] (1) [1]{}; (1)+(-0.5,0) node (s) (s) edge\[->\] (1); (3,0) node\[zstd\] (2) [2]{}; (0,-1) node\[zstd\] (3) ; (3) circle (3pt); (1.5,-1) node\[zstd\] (4) [4]{}; (3,-1) node\[zstd\] (5) [5]{}; (4) edge\[->\] node\[below,font=\][$i?\operatorname{{\ensuremath{\mathtt{decr}}\xspace}}$]{} (3); (1) edge\[->\] node\[left,font=,pos=0.3\][$i!\operatorname{{\ensuremath{\mathtt{ack}}\xspace}}$]{} (4); (4) edge\[->\] node\[below,font=\][$i?\operatorname{{\ensuremath{\mathtt{incr}}\xspace}}$]{}(5); (5) edge\[->\] node\[right,font=\][${\ensuremath{{\ensuremath{\mathtt{dispatch_s}}\xspace}}}(q,eins(i))$]{} (2); (2) edge\[->\] node\[left,pos=0.3,font=\][$i!\operatorname{{\ensuremath{\mathtt{ack}}\xspace}}$ ]{}(4);
(0,0) node [$eins(i)$:]{};
\[zstd/.style=[state,font=]{},anchor=west\]
(-0.5,1.2) node\[anchor=west\] (dummy)[[$\textsc{Ctg}$]{}in $\operatorname{\textit{Reach}}({\ensuremath{\mathcal{A}}\xspace}_{\ensuremath{\mathcal{P}}\xspace})$:]{}; (0,0.7) node\[zstd\] (0) [[$\textit{main}$]{}]{}; (0,0) node\[zstd\] (00) [null(1)]{}; (00)+(1,0) node\[zstd\] (01) [eins(1)]{}; (01)+(1,0) node\[font=,anchor=west\] (02) […]{}; (02)+(0.5,0) node\[zstd\] (03) [eins(1)]{}; (0,-0.8) node\[zstd\] (11) [null(2)]{}; (11)+(1,0) node\[zstd\] (12) [eins(2)]{}; (12)+(1,0) node\[font=,anchor=west\] (13) […]{}; (13)+(1.2,0) node\[zstd\] (14) [eins(2)]{}; (00) edge\[->\] (01) (01) edge\[->\] (02) (02) edge\[->\] (03) (11) edge\[->\] (12) (12) edge\[->\] (13) (13) edge\[->\] (14);
[background]{} node\[fit=(00) (03),fill=gray!20\] (c1) ; node\[fit=(11) (14),fill=gray!20\] (c2); (c1.north east) node\[anchor=south east,inner sep=0pt,font=\] [counter 1]{}; (c2.north east) node\[anchor=south east,inner sep=0pt,font=\] [counter 2]{};
The actual operations on the counters will be simulated by the $eins(i)$ and $null(i)$ running tasks. As [$\textit{main}$]{}simulates the control structure, we need to synchronize [$\textit{main}$]{}with those $eins(i)$ and $null(i)$ tasks. Let us explain intuitively how we can achieve *rendezvous* synchronization between running tasks using global variables of [<span style="font-variant:small-caps;">Qdas</span>]{}. Consider a [<span style="font-variant:small-caps;">Qdas</span>]{}with three global variables $\ell_1$, $\ell_2$ ranging over Boolean and $X$ over a finite set of ‘messages’ $M$. Let $\gamma_1$ and $\gamma_2$ be two blocks whose [<span style="font-variant:small-caps;">Lts</span>]{}are:\
$\gamma_1$:
in [0,1,2,3,4,5]{} (1.5\*,0) node\[circle,draw,inner sep=4pt\] () ; (0) node\[font=\][$s_0$]{}; (5) node\[font=\][$s_5$]{}; (0) edge\[->\] node\[above,font=\][$\ell_1 = 1$]{} (1); (1) edge\[->\] node\[above,font=\][$x\gets m$]{} (2); (2) edge\[->\] node\[above,font=\][$\ell_2\gets 1$]{} (3); (3) edge\[->\] node\[above,font=\][$\ell_1=0$]{} (4); (4) edge\[->\] node\[above,font=\][$\ell_2\gets 0$]{} (5); (0)+(-0.7,0) node (z) (z) edge\[->\](0);
(for $m\in M$)\
$\gamma_2$:
in [0,1,2,3,4,5]{} (1.5\*,0) node\[circle,draw,inner sep=4pt\] () ; (0) node\[font=\][$s_0'$]{}; (5) node\[font=\][$s_5'$]{}; (0) edge\[->\] node\[above,font=\][$\ell_1\gets 1$]{} (1); (1) edge\[->\] node\[above,font=\][$\ell_2=1$]{} (2); (2) edge\[->\] node\[above,font=\][$x=m$]{} (3); (3) edge\[->\] node\[above,font=\][$\ell_1\gets 0$]{} (4); (4) edge\[->\] node\[above,font=\][$\ell_2= 0$]{} (5); (0)+(-0.7,0) node (z) (z) edge\[->\](0);
\
Assume a configuration $c$ of the [<span style="font-variant:small-caps;">Qdas</span>]{}where $\ell_1=\ell_2=0$ and where two distinct tasks are running $\gamma_1$ and $\gamma_2$, are unblocked, and are in $s_0$ and $s_0'$ respectively. Assume that no other task can access $\ell_1$, $\ell_2$ and $m$. It is easy to check that, from $c$, there is only one possible interleaving of the transitions of$\gamma_1$ and $\gamma_2$. So if $\gamma_2$ reaches $s_5'$ from $c$, then $\gamma_1$ must have reached $s_5$, and the $x=m$ test in $\gamma_1$ has been fired *after* the $x\gets m$ assignment in $\gamma_2$. This achieves rendezvous synchronisation between $\gamma_1$ and $\gamma_2$, with the passing of message $m$. This can easily be extended to rendezvous via different “channels”, by adding extra global variables. So, we extend the syntax of [<span style="font-variant:small-caps;">Qdas</span>]{}by allowing transitions of the form $(s_0,c!m,s_5)$ and $(s_0',c?m,s_5')$ (for $m\in M$) to denote respectively a send and a receive of message $m$ on a rendezvous channel $c$.
We rely on this mechanism to let [$\textit{main}$]{}send operations to be performed on the counters to the $null(i)$ and $eins(i)$ running tasks. More precisely, for a [2<span style="font-variant:small-caps;">Cs</span>]{}${\ensuremath{\mathcal{P}}\xspace}={\langleX,x^0,\Sigma_{\ensuremath{\mathcal{P}}\xspace},\Delta_{\ensuremath{\mathcal{P}}\xspace}\rangle\xspace}$, we build the [<span style="font-variant:small-caps;">Qdas</span>]{}${\ensuremath{\mathcal{A}}\xspace}_{\ensuremath{\mathcal{P}}\xspace}={\ensuremath{{\langle{CQID\xspace}, \emptyset,\allowbreak \Gamma,
\allowbreak \mathtt{main},\allowbreak {\ensuremath{\mathcal{X}}\xspace},\allowbreak \Sigma,\allowbreak
({\ensuremath{\mathcal{TS}}\xspace}_\gamma)_{\gamma\in\Gamma}\rangle\xspace}}}$ where ${CQID\xspace}=\{q\}$, $\Gamma=(\{null,eins\}\times\{1,2\})\cup\{{\ensuremath{\textit{main}}\xspace}\}$, ${\ensuremath{\mathcal{X}}\xspace}=\{\ell_1^1,\ell_2^1,x^1,\ell_1^2,\ell_2^2,x^2\}$ where $x^1,x^2$ range over the domain $\{\operatorname{{\ensuremath{\mathtt{incr}}\xspace}},\operatorname{{\ensuremath{\mathtt{decr}}\xspace}},\operatorname{{\ensuremath{\mathtt{is\_zero}}\xspace}},\operatorname{{\ensuremath{\mathtt{ack}}\xspace}}\}$, and the transition systems are given in Fig.\[fig:sim\_counter\]. The variables ${\ensuremath{\mathcal{X}}\xspace}$ encode two channels that we call $1$ and $2$ in the pseudo code of Fig. \[fig:sim\_counter\]. The [$\textit{main}$]{}task runs an infinite `while` loop (line 12 onwards) that consists in guessing a transition $(s,a,s')$ of $F$ and synchronising, via *rendezvous* on the channels $1$ and $2$, with the relevant $null$ or $eins$ unblocked task, to let it execute the operation on the counter. When a $null(i)$ or $eins(i)$ receives an $\operatorname{{\ensuremath{\mathtt{incr}}\xspace}}$ message, it performs an asynchronous dispatch of $eins(i)$ into $q$ to increment counter $i$, and acknowledges the operation to [$\textit{main}$]{}, thanks to message $\operatorname{{\ensuremath{\mathtt{ack}}\xspace}}$. When an $eins$ block receives a $\operatorname{{\ensuremath{\mathtt{decr}}\xspace}}$ message, it terminates, which decrements the counter. $null$ blocks cannot receive $\operatorname{{\ensuremath{\mathtt{decr}}\xspace}}$ messages, so, if [$\textit{main}$]{}requests a $\operatorname{{\ensuremath{\mathtt{decr}}\xspace}}$ operation when the counter is zero, [$\textit{main}$]{}gets blocked. This means that the guessed transition was not fireable in the currently simulated [2<span style="font-variant:small-caps;">Cs</span>]{}configuration, and ends the simulation. Finally, only $null$ blocks can receive and acknowledge $\operatorname{{\ensuremath{\mathtt{is\_zero}}\xspace}}$ messages, so, again, [$\textit{main}$]{}is blocked after sending $\operatorname{{\ensuremath{\mathtt{is\_zero}}\xspace}}$ to a non-zero counter. Note that we need both *asynchronous* calls to start two counters in parallel, and *synchronous* calls to encode the counter values. The result of Theorem\[thm:concqdasundec\] follows directly from:
[proposition]{}[propsimpdsrdvzqdas]{} \[prop:sim\_pds\_rdvz\_qdas\] Given a [2<span style="font-variant:small-caps;">Cs</span>]{}, then we can reduce its reachability question to the Parikh coverability question for a concurrent [<span style="font-variant:small-caps;">Qdas</span>]{}that demands both synchronous and asynchronous dispatch actions.
As discussed before, we can separate each ${\ensuremath{G}}$ for $({\ensuremath{G}},{\ensuremath{\vec{d}}})\in\operatorname{\textit{Reach}}({\ensuremath{\mathcal{A}}\xspace}_{\ensuremath{\mathcal{C}}\xspace})$ into three components, one consisting only of a vertex $v_0$ with $\lambda(v_0)={\ensuremath{\textit{main}}\xspace}$ and two paths $v_1 v_2 \dots v_k$ and $v_1'v_2'\dots v_l'$ which we will call $counter1$ and $counter2$ in the following.
As before, we define a relation between configurations of the [2<span style="font-variant:small-caps;">Cs</span>]{}${\ensuremath{\mathcal{C}}\xspace}$ and the [<span style="font-variant:small-caps;">Qdas</span>]{}${\ensuremath{\mathcal{A}}\xspace}_{Cc}$. For $c=({\ensuremath{G}},{\ensuremath{\vec{d}}})\in\operatorname{\textit{Reach}}({\ensuremath{\mathcal{A}}\xspace}_{\ensuremath{\mathcal{C}}\xspace})$ and $y=(x,k,l)\in\operatorname{\textit{Reach}}({\ensuremath{\mathcal{C}}\xspace})\subseteq X\times {\ensuremath{\mathbb{N}}\xspace}\times {\ensuremath{\mathbb{N}}\xspace}$ we write $c\triangleright y$ if ${\ensuremath{\vec{d}}}(state)=x$, ${\ensuremath{\left| counter1 \right|}}=k$, and ${\ensuremath{\left| counter2 \right|}}=l$.
The rendezvous assures a unique interleaving of actions of [$\textit{main}$]{}, $null(1)$, and $null(2)$ until [$\textit{main}$]{}reaches line $12$. Let us in the following consider the reached configuration $c^0=({\ensuremath{G}}^0,{\ensuremath{\vec{d}}}^0)$ with ${\ensuremath{\vec{d}}}^0(state)=x^0$, ${\ensuremath{\vec{d}}}^0(\ell_0)={\ensuremath{\vec{d}}}^0(\ell_1)=0$ and ${\ensuremath{G}}^0$ with\
\[zstd/.style=[state,font=,inner sep=1pt,minimum size=15pt]{}, zstd2/.style=[zstd,rectangle]{}, lab/.style=[font=,inner sep=1pt]{}, anchor=west\] node\[zstd\] (13) [${\ensuremath{\textit{main}}\xspace}$]{}; (13.south east) node\[lab,anchor=north west\] [$q$]{}; (13.north east) node\[lab,anchor=south west\] [$s_{12}$]{};
(1.5,0) node\[zstd\] (13) [$null(1)$]{}; (13.south east) node\[lab,anchor=north west\] [$q$]{}; (13.north east) node\[lab,anchor=south west\] [$4$]{};
(3.5,0) node\[zstd\] (13) [$null(2)$]{}; (13.south east) node\[lab,anchor=north west\] [$q$]{}; (13.north east) node\[lab,anchor=south west\] [$4$]{};
\
(where $s_{12}$ is the state of [$\textit{main}$]{}in line $12$) as “initial” configuration of the [<span style="font-variant:small-caps;">Qdas</span>]{}.
Note that $counter1$ and $counter2$ are independent, i.e., they do not synchronize except via ${\ensuremath{\textit{main}}\xspace}$. Further, there is no more than one task active in $counter1$ and $counter2$. The unique tasks $zero(1)$ and $zero(2)$ never terminate. The rendezvous synchronization assures that there is only *one* possible interleaving between the [$\textit{main}$]{}task and the currently running tasks in $counter1$ and $counter2$:
[$\textit{main}$]{}does loops of the form\
in [0,1,2,3,4]{} (1.8\*,0) node\[circle,draw,inner sep=4pt\] () ;
\(0) node\[font=\][$s_0$]{}; (4) node\[font=\][$s_0$]{}; (0) edge\[->\] node\[above,font=\][$state = s$]{} (1); (1) edge\[->\] node\[above,font=\][$i!a$]{} (2); (2) edge\[->\] node\[above,font=\][$i?ack$]{} (3); (3) edge\[->\] node\[above,font=\][$state\leftarrow s'$]{} (4);
(for $i\in \{1,2\},a\in\Sigma_{\ensuremath{\mathcal{C}}\xspace}$)\
which leads to the following interleaving of actions of [$\textit{main}$]{} with actions of the $i$-th counter component.\
in [0,1,2,3,4,5,6,7]{} (1.7\*,0) node\[circle,draw,inner sep=4pt\] () ;
\(0) edge\[->\] node\[above,font=\][$state = s$]{} (1); (1) edge\[->\] node\[above,font=\][$i!a$]{} (2); (2) edge\[->\] node\[above,font=\][$i?a$]{} (3); (3) edge\[->,dotted\] node\[above,font=\][$c_i(a)$]{} (4); (4) edge\[->\] node\[above,font=\][$i!ack$]{} (5); (5) edge\[->\] node\[above,font=\][$i?ack$]{} (6); (6) edge\[->\] node\[above,font=\][$state\leftarrow s'$]{} (7);
\
where
node\[circle,draw,inner sep=2pt\] (a) ; +(1,0) node\[circle,draw,inner sep=2pt\] (b) ; (a) edge\[->,dotted\] node\[above,font=\][$c_i(a)$]{} (b);
translates the sent action $a$ to a meta-action $c_i(a)$ of the $i$-th counter as follows:\
an action $\operatorname{{\ensuremath{\mathtt{incr}}\xspace}}$ is mapped to the action ${\ensuremath{{\ensuremath{\mathtt{dispatch_s}}\xspace}}}(q,eins(i))$ and the activation of the dispatched task
an action $\operatorname{{\ensuremath{\mathtt{decr}}\xspace}}$ is mapped to the termination of the current task; which is only possible if the current task is a block $eins(i)$
the test for empty stack is mapped to an epsilon action; this action is only possible in $null(i)$.
Note that if $c_i(a)$ is not possible, then there will be no acknowledgement, hence ${\ensuremath{\mathcal{A}}\xspace}_{\ensuremath{\mathcal{C}}\xspace}$ blocks.
Thus we can cut a run of ${\ensuremath{\mathcal{A}}\xspace}_{\ensuremath{\mathcal{C}}\xspace}$ into (an initial phase and) a sequence of phases of the above form that will be abbreviated $trans(s,a,s')$ in the following.
Let ${\ensuremath{\mathcal{C}}\xspace}$ be a [2<span style="font-variant:small-caps;">Cs</span>]{}and ${\ensuremath{\mathcal{A}}\xspace}_{{\ensuremath{\mathcal{C}}\xspace}}$ the associated [<span style="font-variant:small-caps;">Qdas</span>]{}, if $y\in\operatorname{\textit{Reach}}({\ensuremath{\mathcal{C}}\xspace})$ then there exists $c\in\operatorname{\textit{Reach}}({\ensuremath{\mathcal{A}}\xspace}_{{\ensuremath{\mathcal{C}}\xspace}})$ such that $c\triangleright y$. Further, if $c=({\ensuremath{G}},{\ensuremath{\vec{d}}})\in\operatorname{\textit{Reach}}({\ensuremath{\mathcal{A}}\xspace}_{\ensuremath{\mathcal{C}}\xspace})$ where ${\ensuremath{\vec{d}}}$ valuates ${\ensuremath{\vec{d}}}(\ell_0)={\ensuremath{\vec{d}}}(\ell_1)=0$, then there exists $y\in\operatorname{\textit{Reach}}({\ensuremath{\mathcal{C}}\xspace})$ with $c\triangleright y$.
Given a run $x_0 \delta_1 x_1 \delta_2 \dots \delta_k x_k$ of ${\ensuremath{\mathcal{C}}\xspace}$, then there exists a run of ${\ensuremath{\mathcal{A}}\xspace}_{\ensuremath{\mathcal{C}}\xspace}$ that can be cut into phases $t_1,\dots,t_k$ where $t_i=trans(s_{i-1},a_i,s_i)$ where $\delta_i=(s_{i-1},a_i,s_i)$ for $1\leq i \leq k$. Obviously $c^0\triangleright x^0$ and ${\ensuremath{\vec{d}}}^0(\ell_0)={\ensuremath{\vec{d}}}^0(\ell_1)=0$. Hence, the reverse direction follows by a straightforward inductive argument.
|
=5000
Strongly correlated electronic systems often exhibit stripe phases [@emery:prl00]. In two-dimensional electron systems (2DES) such a stripe phase is believed to be at the origin of the recently observed electronic transport anisotropy at half-fillings of high Landau levels [@horst:aps93; @lilly:prl99a; @du:ssc99; @pan:prl99; @lilly:prl99b; @shayegan:physicae99; @pan:prl00]. At Landau level filling factors $\nu$ = 9/2, 11/2, 13/2, etc. the magneto-resistance is a maximum along one current direction, whereas it is a minimum when the current direction is rotated by 90$^{\circ}$ within the plane of the sample. In a purely perpendicular magnetic field ($B$) the direction of anisotropy is pinned to the crystal lattice [@lilly:prl99a; @du:ssc99], but re-orients itself when an in-plane $B$ field ($B_{ip}$) is added by tilting the sample. At large $B_{ip}$ the easy-axis of anisotropy in the plane of the sample (the direction of minimum resistance) is [*always*]{} perpendicular to $B_{ip}$ [@pan:prl99; @lilly:prl99b]. Although the nature of this anisotropy remains uncertain, experimental data [@horst:aps93; @lilly:prl99a; @du:ssc99; @pan:prl99; @lilly:prl99b; @shayegan:physicae99; @pan:prl00] and theoretical models [@koulakov:prl96; @moessner:prb96; @fradkin:prb99; @fertig:prl99; @rezayi:prl99; @simon:prl99; @jungwirth:prb99; @philip:prl00; @fradkin:prl00; @maeda:prb00; @macdonald:prb00; @vonoppen:prl00; @cote:prb00] point to the formation of a unidirectional charge density wave, often referred to as the “stripe phase”, or to a state akin to a liquid crystal phase [@fradkin:prb99]. A very similar anisotropy is also observed at $\nu$ = 5/2 and $\nu$ = 7/2 in the second Landau level under large $B_{ip}$ [@pan:prl99; @lilly:prl99b]. Modeling [@rezayi:prl00] suggests that an electronic anisotropic phase, not unlike the one at half-fillings of higher Landau levels, has been induced by the in-plane $B$ field.
So far, anisotropy has only been observed at [*half-filled*]{} Landau levels. In this letter, we present data that show strong electronic transport anisotropies at [*fully filled*]{} Landau levels. They are created by the very strong in-plane $B$ fields at very large tilt in the regime of the integral quantum Hall effect (IQHE) at $\nu$ = 4, 6, and 8. The origin of these anisotropies is unknown, although, phenomenologically, they resemble the anisotropies at half-filled Landau levels: the magneto-resistance is a minimum when the current is perpendicular to $B_{ip}$ and a maximum when the current is along $B_{ip}$. A striped spin density wave phase may be at the origin of these new observations.
Our sample consists of a 350Å wide GaAs quantum well embedded into Al$_{.24}$Ga$_{.76}$As and delta-doped from both sides at a distance of 490Å. The specimen has a size of 5mm $\times$ 5mm and is contacted via eight indium contacts placed symmetrically around the perimeter. The electron density is established after illuminating the sample with a red light-emitting diode at $\sim$ 4.2K and, within limits, the density can be tuned by exposure time. At an electron density of $n = 4.2\times10^{11}$ cm$^{-2}$ two electrical subbands are populated having densities $n_0 \sim 3.1\times10^{11}$ cm$^{-2}$ and $n_1 \sim 1.1\times10^{11}$ cm$^{-2}$ as determined by Fourier analysis of the low-field Shubnikov-de Haas oscillations. All angular dependent measurements are carried out in a dilution refrigerator equipped with an [*in-situ*]{} rotator placed inside a 33 Tesla resistive magnet. We define the axis of rotation as the [*y*]{}-axis. Consequently, the in-plane field, $B_{ip}$, extends along the [*x*]{}-axis when the sample is rotated.
We have measured $R_{xx}$ and $R_{yy}$, which differ only in the in-plane current direction, at more than 10 tilt angles ($\theta$) between 0$^{\circ}$ and 90$^{\circ}$. $R_{xx}$ represents the direction for which, under tilt, the current runs along $B_{ip}$. Figure 1 shows data at five selected angles, from $\theta$ = 81.1$^{\circ}$ to 84.4$^{\circ}$. At $\theta$ = 0$^{\circ}$ (not shown) both $R_{xx}$ and $R_{yy}$ vanish at $\nu$ = 6 as expected for an isotropic quantum Hall state. As $\theta$ is increased towards 81.1$^{\circ}$, both $R_{xx}$ and $R_{yy}$ remain vanishingly small at $\nu$ = 6, although the widths of the resistance minima and of the Hall plateau shrink with increasing $\theta$. Very generally, such an angular dependence is readily understood for the spin-unpolarized $\nu$ = 6 state. While the $\nu$ = 6 state always occurs at the same perpendicular magnetic field, $B_{perp}$, the total magnetic field at tilt angle, $\theta$, increases as $B_{tot}=B_{perp}$/cos($\theta$). Since the electron spin experiences $B_{tot}$, the Zeeman splitting of all Landau levels increases with increasing $\theta$. This leads to a reduction of the energy gap at $\nu$ = 6 and a shrinking width and depth (not visible on the linear scale of Fig. 1) of the $R_{xx}$ and $R_{yy}$ minima. Eventually, this leads to a collapse and disappearance of the $\nu$ = 6 IQHE state. Indeed, at $\theta$ = 83.3$^{\circ}$, $R_{xx}$ has turned from a deep minimum into a [*strong peak*]{} and the usual Hall plateau has vanished. Therefore, the disappearance of $R_{xx}$ can be rationalized as the closing of the $\nu$ = 6 energy gap. However, very surprisingly, the electrical transport turns out to be [*strongly anisotropic*]{}. In contrast to $R_{xx}$, which shows a strong [*maximum*]{} at this angle and filling factor, $R_{yy}$, continues to shows a strong [*minimum*]{} at $\nu$ = 6. Just as in the case of half-fillings [@pan:prl99; @lilly:prl99b] the easy-axis of this anisotropy at full-filling factors is perpendicular to $B_{ip}$. The direction of anisotropy is not dependent on the orientation of the crystallographic axis with respect to the in-plane field, as we determined by performing the same experiments on the same specimen mounted in a configuration rotated by 90$^{\circ}$ about the sample normal. Furthermore, none of the resistance measurements showed any hysteresis as a function of the sweep direction of the $B$ field. Finally, in the anisotropic regime, the generally strong Hall plateau at $\nu$ = 6 disappears for both directions of current.
This is the first time that such an anisotropy has been observed in a state as robust as an IQHE state. To learn more about this anisotropic state we perform $T$-dependent studies of $R_{xx}$ and $R_{yy}$. For comparisons, we choose $\theta$ = 81.1$^{\circ}$, where the electronic transport is isotropic, and $\theta$ = 83.3$^{\circ}$, where transport is strongly anisotropic. in Figure 2a and Figure 2b we show three representative traces of $R_{xx}$ and $R_{yy}$. At $\theta$ = 81.1$^{\circ}$, $R_{xx}$ and $R_{yy}$ exhibit the usual activated behavior: the value of both resistances increases with increasing $T$. On the other hand, at $\theta$ = 83.3$^{\circ}$, $R_{xx}$ and $R_{yy}$ behave oppositely: $R_{xx}$ decreases whereas $R_{yy}$ increases with increasing $T$. The $T$-dependencies are quantified in Figure 2c and Figure 2d, where $R_{xx}$ and $R_{yy}$ are shown on Arrhenius plots. At $\theta$ = 81.1$^{\circ}$, $R_{xx}$ and $R_{yy}$ show well-behaved activated behavior yielding a single energy gap of $\Delta \sim$ 1K for both current directions [@footnote1]. On the other hand, the data for $R_{xx}$ and $R_{yy}$ at $\theta$ = 83.3$^{\circ}$ show no longer activated behavior. $R_{xx}$ and $R_{yy}$ appear to start from similar values at high temperature but then diverge from each other roughly exponentially with exponents of similar magnitude but [*opposite sign*]{}. At the lowest temperatures both resistances assume an approximately $T$-independent behavior. This dependence is qualitatively the same as the $T$-dependence of the anisotropic state at $\nu$ = 9/2 [@du:ssc99; @fradkin:prl00].
The remarkable anisotropy found in the IQHE is not limited to the $\nu$ = 6 state. Similar anisotropies are observed at filling factors $\nu$ = 4 and $\nu$ = 8. Figure 3 shows the $\nu$ = 8 and $\nu$ = 4 anisotropy in the same sample at slightly different densities, tuned by applying different doses of light. We have not performed a systematic study of these states.
The cause of the anisotropy at integral quantum Hall states is unknown. Before speculating about the origin of this new phenomenon it is instructive to consider in more detail the single particle states in this two-electric subband specimen. Figure 4a shows the usual Landau fan diagram for a density of 4.2 $\times 10^{11}$ cm$^{-2}$. The Zeeman splitting is enhanced by a factor of 10 to be visible. The position of the Fermi level, $E_f$, is indicated by a heavy line. Clearly, in the vicinity of $\nu$ = 4, 6 and 8, Landau levels from both electric subbands contribute and $E_f$ jumps between levels of different origin. Using such a simple single-particle picture and a 2DES of zero thickness with densities appropriate for the data of Figs. 1 and 3, one would expect the gaps at $\nu$ = 4, 6 and 8 to close at $\theta$ = 83.4$^{\circ}$, 88.4$^{\circ}$ and 88.3$^{\circ}$, respectively. These values differ from experiment, especially in the case of the $\nu$ = 6 and $\nu$ = 8 states.
The discrepancy is largely the result of the neglect of exchange and of the thickness of the wave function. In the remainder we focus on the state at $\nu$ = 6, which we studied most extensively and which shows the strongest anisotropy in experiment. We expect similar arguments to hold for $\nu$ = 4 and $\nu$ = 8. Figure 4b shows the result of a self-consistent local-density-approximation calculation [@jungwirth] performed for a density $n = 4.2\times 10^{11}$ cm$^{-2}$ at a filling factor $\nu$ = 6 as a function of $B_{ip}$. The gap at $\nu$ = 6 (shaded region) undergoes strong variations, comes almost to a close at $B_{ip} \sim 2.5$T (not shown), and vanishes at $B_{ip} \sim 18.5$T due to level crossing. The experimental value of $B_{ip}$ for the strong anisotropy is $\sim$25T. However, we consider the theoretical result of $\sim$18.5T to be sufficiently close to $\sim$25T to attribute the disappearance of the energy gap at $\nu$ = 6 in Fig. 1 to the crossing of spin-split Landau levels originating from different electrical subbands ([*i*]{} = 1, 2). This provides a rational for the appearance of novel features in the data at this filling factor and angle. However, none of such level crossing considerations can explain the observed [*anisotropy*]{}, which represents the remarkable finding in our data. The origin of this phenomenon must be the result of correlated electron behavior.
Previously, large electrical anisotropies have only been observed at half-filled Landau levels [@horst:aps93; @lilly:prl99a; @du:ssc99; @pan:prl99; @lilly:prl99b; @shayegan:physicae99; @pan:prl00]. It is believed that there the electron system spontaneously breaks into striped domains of alternating filling factors such as $\nu$ = 4 and $\nu$ = 5 around $\nu$ = 9/2 [@koulakov:prl96; @moessner:prb96; @fradkin:prb99; @fertig:prl99; @rezayi:prl99; @simon:prl99; @jungwirth:prb99; @philip:prl00; @fradkin:prl00; @maeda:prb00; @macdonald:prb00; @vonoppen:prl00; @cote:prb00]. Given the similarity of the observed properties of the anisotropic phases around $\nu$ = 9/2 and $\nu$ = 6 one might speculate on a similar underlying striped geometry. The driving force behind the phase separation in the $\nu$ = 9/2 case is exchange. The energetic gain from breaking into domains of $\nu$ = 4 and $\nu$ = 5 is counteracted by a strong electrostatic cost for creating an inhomogeneous charge distribution. This is the reason for the formation of very narrow stripes of $\nu$ = 4 and $\nu$ = 5 states, which are only a few magnetic lengths wide. A phase, consisting of stripes around $\nu$ = 6, would carry a much smaller, electrostatic burden.
At the point of collapse of the $\nu$ = 6 energy gap in Fig. 4b two electronic configurations are degenerate. At $B_{ip}$ smaller than the level crossing in Fig. 4b the electrons occupy three spin-unpolarized levels emanating from the lowest three Landau levels ([*N*]{} = 0, 1, and 2) of the lower electronic subband, [*i*]{} = 1 [@footnote2]. (Note, an earlier anti-crossing at $B_{ip} \sim 2.5$T exchanges states [*i*]{} = 1, [*N*]{} = 2 and [*i*]{} = 2, [*N*]{} = 0). The total system is spin-unpolarized (3 spin-up, 3 spin-down). At $B_{ip}$ larger than the level crossing in Fig. 4b the electrons occupy only two spin-unpolarized levels emanating from the lowest two Landau levels ([*N*]{} = 0, 1) of the lower electronic subband ([*i*]{} = 1). In addition, they occupy the spin-up states (solid lines) of two levels emanating from the [*i*]{} = 1, [*N*]{} = 2 and the [*i*]{} = 2, [*N*]{} = 0 states. There, the total system is [*partially spin-polarized*]{} (4 spin-up, 2 spin-down). In the vicinity of the level crossing in Fig. 4b, a phase separation of the electronic system [@giuliani:prb85; @yalagadda:prb91] into spin-unpolarized and partially spin-polarized domains may occur driven by exchange. A very small gain in exchange energy may suffice, since the charge density in both configurations is identical and, to first order, there is no associated electrostatic cost.
Such a pattern resembles the pattern of a spin-density wave, SDW. The existence of an in-plane magnetic field and the so-induced coupling of spin and orbital motion will energetically favor a given orientation of the stripes with respect to $B_{ip}$. The resulting stripe phase of alternating IQHE configurations is bound to have one-dimensional edge-states along its interface between neighboring domains, which carry the electric current in a highly anisotropic fashion. This transport pattern would be analogous to the pattern invoked in the stripe phases that are believed to form at half-fillings of Landau levels, such as $\nu$ = 9/2 and 13/2 and believed to be responsible for the anisotropic electronic behavior. However, without the application of other experimental techniques and without a detailed theoretical investigation this picture remains speculative.
In summary, we have observed strongly anisotropic transport under high in-plane magnetic field in the regime of the IQHE in a quantum well sample with two occupied electrical subbands. Phenomenologically, the data have much in common with the previously discovered anisotropy at half-fillings of high Landau levels. From a simple level crossing picture we conjecture that a novel striped spin-density wave may be at the origin of this phenomenon.
We would like to thank E. Palm and T. Murphy for experimental assistance, E. P. De Poortere, S. P. Shukla, and E. Tutuc for the help in numerical calculation, and N. Bonesteel, R. R. Du, A.H. MacDonald, N. Read, and K. Yang for useful discussion. We are indebted to T. Jungwirth for providing the energy level scheme of Figure 4b. A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by NSF Cooperative Agreement No. DMR-9527035 and by the State of Florida. D.C.T. and W.P. are supported by the DOE and the NSF.
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$R_{xx}$ is about three times larger than $R_{yy}$ at the same temperature. This level of initial anisotropy is not uncommon in 2DES, whether they are patterned or cleaved into squares. It is probably due to small density variations leading to current inhomogeneities. However, it cannot be the origin of the strong anisotropy since the strong anisotropy observed at $\nu$ = 4, 6, and 8 is pegged to $B_{ip}$ and not to any particular sample direction.
T. Jungwirth, unpublished.
Strictly speaking, the separation into subband levels and Landau levels only holds at zero tilt angle. At non-zero tilt they are mixed and one can use this nomenclature only approximately.
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abstract: 'In this paper, we continue our previous study of the low energy effective theory for D4-brane in the large $C$-field background. The gauge field part of the effective action was found in an earlier work. In this paper, we focus on the matter field part of the action and the supersymmetry transformation. Moreover, we calculate the central charges of super algebra and extensively study BPS solutions of this effective theory. The BPS states considered in this paper include light-like gauge field configurations, the F1 ending on D4 solution, tilted D4-brane, BPS solution with two types of magnetic charges (D2 ending on D4), holomorphic embedding of D4-brane and the intersection of two D4-branes along a 2-brane.'
---
.2in
**Supersymmetry and BPS States on.5cm D4-brane in Large $C$-field Background**
.5in [Chen-Te Ma$^{a,}$[^1] andChi-Hsien Yeh$^{a,b, }$[^2]]{}\
\
Introduction {#1}
============
The descriptions of M-theory are five superstring theories and 11-dimensional supergravity. They are related to each other from duality and dimension reduction [@Witten:1995ex]. For example, the superstring theory can be corresponded to the 11-dimensional supergravity compactified on $S^{1}$ circle [@Townsend:1995kk]. The solitonic solutions of 11-dimensional supergravity are called M2 and M5 branes, which play important roles as D-branes in string theory. Although the description of M-theory is not known totally, the low energy effective theory of two and five branes are known to be.
A recent progress was the construction of the low energy effective theory for a single M5-brane in the large $C$-field background [@M51; @M52; @Ho:2009zt] [^3] and its 1/2 BPS states [@Ho:2012dn]. We call this theory to be the Nambu-Poisson M5-brane theory or NP M5 theory in short and all known 1/2 BPS states have their counterparts in the absence of $C$-field background. From the NP M5-brane theory, we can derive the low energy effective theory for D4-brane in the large $C$-field background from double dimensional reduction (DDR) method [@Ho].
This D4-brane effective action for a large $C$-field background is different from the original DBI description [@Leigh]. This new D4-brane action has Nambu-Poisson structure as the NP M5-brane theory. As it is well known, the effective action of D-brane in the large NS-NS $B$-field background should be described by gauge theories on noncommutative space [@ChuHo; @Schomerus; @Seiberg:1999vs]. A similar phenomenon also occurs in the low energy effective theory for D4-brane in the large $C$-field background. The gauge symmetry of an NP M5-brane is the volume-preserving diffeomorphism (VPD) [@Matsuo], which is described by the Nambu-Poisson bracket. After doing DDR on the worldvolume direction which is different with the $C$-field background directions, we get an effective D4-brane with the $C$-field. Since the $C$-field background is parallel to the D4-brane after DDR, it is natural to expect that the D4-brane inherits the VPD symmetry. It is also expected that the geometry of this theory is equipped with a 3-bracket structure [@Chu:2009iv; @Huddleston:2010cx].
One purpose of this paper is to study more details of this effective action with matter fields. The supersymmetry transformation of all fields and the modification of field strengths from matter fields will be given in this paper. In the previous paper [@Ho], we did not write down the manifest supersymmetry transformation of all fields. The reason is that we did not only do DDR from the NP M5-brane theory but also do electric-magnetic dual transformation of this theory. The supersymmetry transformation of new fields in dual action cannot be directly obtained from DDR on the NP M5-brane. Hence, we need to completely calculate the supersymmetry transformation of this dual action, and find the supersymmetry transformation of these new fields to make the dual action invariant. In this paper, we give all supersymmetry transformation of fields for this effective theory, and also hope to give a starting point on the construction of D$p$-brane in the R-R $(p-1)$-form field background.
In this paper, we also want to find 1/2 BPS states on D4-brane in the large $C$-field background. Most of solutions on D4-brane are easy to be expected from M5-brane. To simplify, we do not write down all details for these solutions. The BPS configurations considered in this paper include light-like solutions, the F1 string ending on D4-brane, tilted D4-brane, BPS solution with two types of magnetic charges, holomorphic embedding of D4-brane and the intersection of two D4-branes along a 2-brane.
The plan of this paper is as follows. We review the D4-brane in the large $C$-field background in Sec. \[2\] and the gauge symmetry and supersymmetry in Sec. \[3\]. In Sec. \[4\] we show BPS configurations on D4-bane. But we do not show the details on the solutions derived from the NP M5-brane theory for brevity. Finally, in Sec. \[5\] we conclude.
D4-Brane in Large $C$-field Background {#2}
======================================
To carry out DDR on the NP M5-brane theory along the $x^2$-direction [^4], we set x\^2 \~x\^2 + 2R, and let all other fields to be independent of $x^2$. As a result we can set $\del_2$ to be zero when it acts on any fields. Here $R$ is the radius of the circle of compactification and we should take $R \ll 1$ to let M5-brane reduce to D4-brane.
After we perform DDR on the $x^2$ direction, we need to re-interpret the gauge fields $b_{\mu\dm}$ and $b_{\dm\dn}$ in the NP M5-brane theory from the viewpoint of D4-brane. The gauge field $b_{\m\dm}$ becomes two kinds of fields $\{b_{2\dm},b_{\a\dm}\}$ after DDR, where $\a=0,1$. We should identity $b_{2\dm}$ as components of the one-form potential on the D4-brane, b\_[2]{}a\_. We use the same chirality condition with the NP M5-brane theory, \^[012]{}= , \^[012]{}=-.
Action
------
After performing DDR on the $x^2$ direction, we get the low energy effective description for D4-brane in the large $C$-field background [@Ho]. The action of the gauge fields is S\^[D4]{}\_[gauge]{}&=&d\^[5]{}x { -\_\^[2]{} -\_[2]{}\^2 .\
&&-\_\^2 . +\^\_a\_B\_\^[ ]{} +\^F\_B\_\^[ ]{}B\_\^[ ]{}},\[D4inCgauge\] where we use the definition, $\eps^{\a\b2}\equiv\eps^{\a\b}$.
The action of the scalar fields $X^{I} ~(I=6,7,8,9,10)$ is S\^[D4]{}\_[X]{}&=&d\^[5]{}x { -\_X\^[I]{}[D]{}\^X\^[I]{} -\_X\^[I]{}\^X\^[I]{}+gB\_\^[ ]{}\_X\^[I]{}\^X\^[I]{} .\
&&- B\_\^[ ]{}B\^\_[ ]{} \_X\^[I]{}\^X\^[I]{} -\^\_F\_F\^ \_X\^[I]{}\^X\^[I]{}\
&&.--{X\^,X\^[I]{},X\^[J]{}}\^[2]{} -{X\^[I]{},X\^[J]{},X\^[K]{}}\^[2]{}}.\[D4inCX\] The action of the fermionic field $\psi$ is S\^[D4]{}\_&=&d\^[5]{}x {|\^\_+|\^[D]{}\_+g|\^[2]{}\^F\_\_-g|\^B\_\^[ ]{}\_.\
&&.+g\^2|\_\^[I]{}{X\^,X\^[I]{},} -g\^2|\^[IJ]{} \_{X\^[I]{},X\^[J]{},}}.\[D4inCPsi\]
The Nambu-Poisson bracket $\{\cdot, \cdot, \cdot\}$ is used to define the algebraic structure for gauge symmetry. In general, it satisfies Leibniz rule and fundamental identity. It is defined by {f, g, h} = \^\_f \_g \_h.
In the above, we used the notation B\_\^ && \^\_b\_,\
b\^ && \^ b\_,\
X\^ && + b\^.
The covariant derivatives are defined by \_&&(\_ - gB\_\^\_) , (= X\^I, ) \[dmu\]\
[D]{}\_&&\_ {X\^,X\^,}, and the field strengths are defined by $$\begin{aligned}
{\cal H}_{\a\dm\dn} &\equiv&
\eps_{\dm\dn\dlam}(\del_{\a}b^{\dlam}-B_{\a}{}^{\dlam}-g\del_{\ds}{b}^{\dlam}B_{\a}{}^{\ds}),\label{h12def}\\
{\cal H}_{\dot1\dot2\dot3}
&\equiv&g^2\{X^{\dot1},X^{\dot2},X^{\dot3}\}-\frac{1}{g}
\nonumber\\
&=&H_{\dot1\dot2\dot3}
+\frac{g}{2}
(\partial_{\dot\mu}b^{\dot\mu}\partial_{\dot\nu}b^{\dot\nu}
-\partial_{\dot\mu}b^{\dot\nu}\partial_{\dot\nu}b^{\dot\mu})
+g^2\{b^{\dot1},b^{\dot2},b^{\dot3}\}.
\label{h30def}\end{aligned}$$
We also use the notation of Abelian field strength, F\_=\_a\_-\_a\_. In fact, the field strength of D4-brane is deformed by the $C$-field background. The field strength ${\cal H}_{2\dm\dn}$ can be understood as the deformed field strength on D4-brane, \_[2]{}\_. The remaining two field strengths ${\cal F}_{\a\dm}$ and ${\cal F}_{\a\b}$ are also deformed by the $C$-field background. Before discussing this issue, we should study how to find the remaining one-form degree $a_{\a}$ in this theory.
Electric-Magnetic Duality Transformation
----------------------------------------
From the viewpoint of D4-brane, there are one-form gauge potential $a_{\dm}$ and $a_{\a}$. The one-form gauge field $a_{\a}$ can be understood as the EM duality of $b_{\a\dm}$. After 3-dimensional EM dual transformation in $x^{\dm}$ spaces, the d.o.f of gauge field $b_{\a\dm}$ (one form in $x^{\dm}$ space) can be re-interpreted into the d.o.f of gauge field $a_{\a}$ (0-form in $x^{\dm}$ space). Moreover, the field $B_{\a}^{~\dm}$ can be understood as a new independent field $\breve{B}_{\a}^{~\dm}$ which is not divergenceless. The $\breve{B}_{\a}^{~\dm}$ is at most quadratic so it can be integrated out and re-expressed by the other fields ($b^{\dm}, a_{\dm}, a_{\a}, X^{I}, \Psi$) from the equations of motion.
The complete dual action is, $$\begin{aligned}
\label{eq:2.1}
S[b^{\dm},a_A,\breve{B}_{\a}^{~\dm},X^{I},\Psi]&=&
\int d^{5}x\left\{
-\frac{1}{2}{\cal D}_{\dm}X^{I}{\cal D}^{\dm}X^{I} -\frac{1}{2}\del_{\a}X^{I}\del^{\a}X^{I}+g\breve{B}_{\a}^{~\dm}\del_{\dm}X^{I}\del^{\a}X^{I} \right.
\nn\\
&&- \frac{g^{2}}{2}\breve{B}_{\a}^{~\dm}\breve{B}^{\a}_{~\dn}
\del_{\dm}X^{I}\del^{\dn}X^{I}
-\frac{g^{2}}{8}\eps^{\dm\dr\dt}\eps_{\dn\ds\dd}F_{\dr\dt}F^{\ds\dd}
\del_{\dm}X^{I}\del^{\dn}X^{I}
\nn\\
&&-\frac{g^{4}}{4}\{X^{\dm},X^{I},X^{J}\}^{2}
-\frac{g^{4}}{12}\{X^{I},X^{J},X^{K}\}^{2}
\nn \\
&&+\frac{i}{2}\bar{\Psi}\Gamma^{\a}\del_{\a}\Psi
+\frac{i}{2}\bar{\Psi}\Gamma^{\dr}{\cal D}_{\dr}\Psi
+g\frac{i}{4}\bar{\Psi}\Gamma^{2}\eps^{\dm\dn\dr}F_{\dn\dr}\del_{\dm}\Psi
-g\frac{i}{2}\bar{\Psi}\Gamma^{\a}\breve{B}_{\a}^{~\dm}\del_{\dm}\Psi
\nn\\
&&+g^2\frac{i}{2}\bar{\Psi}\Gamma_{\dm} \G^I \{X^{\dm},X^{I},\Psi\}
-g^2\frac{i}{4}\bar{\Psi}\Gamma^{IJ}
\Gamma_{\dot{1}\dot{2}\dot{3}}\{X^{I},X^{J},\Psi\}
\nn\\
&&-\frac{1}{2g^{2}} -\frac{1}{2}({\cal H}_{\dot{1}\dot{2}\dot{3}})^{2}
-\frac{1}{4}{\cal F}_{\dn\dr}{\cal F}^{\dn\dr}
-\frac{1}{4}(\eps_{\dm\dn\dr}
(\del_{\a}b^{\dm}-V_{\ds}^{~\dm}\breve{B}_{\a}^{~\ds}))^2
\nn \\
&&\left.
+\eps^{\a\b}\del_{\b}a_{\dm}\breve{B}_{\a}^{~\dm}
+\frac{g}{2}\eps^{\a\b}F_{\dm\dn}\breve{B}_{\a}^{~\dm}\breve{B}_{\b}^{~\dn}-\eps^{\a\b}\del_{\dm}a_{\b}\breve{B}_{\a}^{~\dm}
\right\}.\end{aligned}$$ Here, we define $V_{\dm}{}^{\ds}\equiv g \del_{\dm}X^{\ds}$.
$\breve{B}_{\a}^{~\dm}$ in the dual action is replaced by its EOM solution. The equation of motion for $\breve{B}_{\a}^{~\dm}$ is V\_\^[ ]{}(\^b\_-V\^\_[ ]{}\^\_[ ]{}) +\^F\_+g\^F\_\_\^[ ]{} +g\_X\^[I]{}\^X\^[I]{}-g|\^\_-g\^[2]{}\^\_[ ]{}\_X\^[I]{}\^X\^[I]{}=0. \[r2\] Its solution is \_\^[ ]{}&=& (**M**\^[-1]{})\^\_[ ]{} (V\_\^[ ]{}\^b\_+\^F\_ +g\_X\^[I]{}\^X\^[I]{}-g|\^\_)\
&&(**M**\^[-1]{})\^\_[ ]{}W\_\^,\[hatB\] where **M**\_\^[ ]{} (V\_V\_\^[ ]{}+g\^[2]{}\_X\^[i]{}\_X\^[i]{})\^ -g\^F\_ and $(\textbf{M}^{-1})^{\dm\dn}_{~~~\a\b}$ is defined by (**M**\^[-1]{})\^\_[ ]{}**M**\_\^[ ]{} =\^\_[ ]{}\^[ ]{}\_.
In this dual action, the field $\breve{B}_{\a}{}^{\dm}$ is not divergenceless. We expect that there are remaining terms proportional to $\del_{\dm}\breve{B}_{\a}{}^{\dm}$ after we do gauge and supersymmetry transformation except for $a_{\a}$ and $\breve{B}_{\a}{}^{\dm}$. (We use dimensional reduction to obtain the transformation laws of other fields from M5-brane to D4-brane.) To make the dual action being invariant, we should find gauge and supersymmetry transformation of $a_{\a}$ and $\breve{B}_{\a}{}^{\dm}$ to cancel these remaining terms. In next section, we show the details of gauge and supersymmetry transformation of $a_{\a}$ and $\breve{B}_{\a}{}^{\dm}$, which are not directly obtained from DDR on the NP M5-brane.
Symmetry {#3}
========
Gauge Symmetry
--------------
The gauge transformation of fields is \_&=&g \^\_(= X\^I, ),\
\_ b\_ &=&\_\_ -\_\_+g\^\_b\_ +g(\_\^)b\_,\
\_ b\_[2]{} &=&-\_\_[2]{} +g\^\_b\_[2]{} +g(\_\^)b\_[2]{},\[gt5\]\
\_ b\^&=& \^+g\^\_b\^.\[transf-bdm\] We expect that $U(1)$ gauge symmetry on D4-brane can be examined from the gauge transformations on the NP M5-brane theory [@M51; @M52]. The gauge transformation parameter $\Lambda_2$ shall be identified with the $U(1)$ gauge transformation parameter. This is consistent with the identification of $a_{\dm}$ with $b_{\dm 2}$. The gauge symmetry parameterized by $\Lambda_{\dm}$, i.e. volume-preserving diffeomorphism (VPD) symmetry still appears in the D4-brane. Hence, we have gauge transformation on $a_{\dm}$, \_ a\_&=& \_+ g(\^\_a\_+a\_\_\^).\[transf-adm\] The gauge symmetry combines $U(1)$ gauge symmetry and VPD symmetry.
### Gauge Transformation of $a_{\a}$
The field $a_{\a}$ was introduced by the dual transformation and its gauge transformation rule has to be solved from the requirement that the dual action to be invariant. First, we need to realize that Chern-Simons term must be gauge invariant by itself. The gauge transformation [^5] of the Chern-Simons term is &&\_(\^\_a\_\_\^[ ]{} +\^F\_\_\^[ ]{}\_\^[ ]{} -\^\_a\_\_\^[ ]{})\
&=&\_\_\^[ ]{}\^. Hence we get \_a\_=\_+g(\^\_a\_+a\_\_\^). \[transf-aa\]
In our formulation of the self dual gauge fields $b$, the components $b_{\mu\nu}$ do not explicitly appear in the action. In [@Pasti; @Furuuchi:2010sp], the components $b_{\mu\nu}$ are used to explicitly exhibit the self duality of the gauge field, and their gauge transformation are given by \_b\_=\_\_-\_\_ +g. Identifying $b_{\b 2}$ with $a_{\b}$ and setting $\del_2 = 0$ from DDR, we get exactly the same gauge transformation rule as eq.(\[transf-aa\]) with $\Lambda_2 = \lam$.
We find the gauge transformation of $a_{\dm}$ (eq.(\[transf-adm\])) and $a_{\a}$ (eq.(\[transf-aa\])) are of the same form ($A=\a,\dm$) \_ a\_[A]{}= \_[A]{}+g(\^\_a\_[A]{}+a\_\_[A]{}\^).\[transf-a\]
Let us also give the gauge transformation of $V_{\dn}{}^{\dm}$, $\textbf{M}_{\dm\dn}{}^{\a\b}$, $W_{\dm}{}^{\a}$ and $\hat{B}_{\a}^{~\dm}$, \_ V\_\^ &=& g\^\_V\_\^ + g(\_\^) V\_\^,\
\_ **M**\_\^ &=& g,\
\_ W\_\^&=&\_\^**M**\_\^+ g,\
\_ \_\^[ ]{} &=& \_\^+ g(\^\_\_\^[ ]{}- \_\^[ ]{}\_\^).
### Covariant Variable with $U(1)$ and VPD Symmetry
In the original NP M5-brane theory, we have the covariant field strengths [^6] \_&=& \_b\^+g (\_b\^\_b\^-\_b\^\_b\^) +g\^2 {b\^,b\^,b\^},\
[F]{}\_&& [H]{}\_[2]{} = F\_+g .
The covariant version of $F_{\a\dm}$ can be defined as \_ \_\_[H]{}\^. This is motivated by the intuition that ${\cal F}_{\a\dm}$ corresponds to ${\cal H}_{\a\dm 2}$ in the NP M5-brane theory. Replacing $B_{\a}{}^{\dm}$ by $\hat{B}_{\a}{}^{\dm}$, we can rewrite ${\cal H}^{\b\dn\dlam}$ from Eq.(\[h12def\]) by the other fields. (That is, we avoided to use $\del_{\a} b^{\dm}$ directly. The dependence on $\del_{\a} b^{\dm}$ only appears through $\hat{B}_{\a}{}^{\dm}$.) As a result, we have \_\_\^[ ]{}{F\_+ g-g\^[2]{}\_\^\_\_X\^[I]{}\^X\^[I]{}}. This is also in agreement with the definition of ${\cal H}_{\mu\nu\dm}$ defined in [@Pasti; @Furuuchi:2010sp].
By inspection, we can guess the covariant form of $F_{\a\b}$. \_&=& F\_+g,\[Fab\] where F\_[AB]{} \_A a\_B - \_B a\_A. Unlike ${\cal F}_{\dm\dn}$ and ${\cal F}_{\a\dm}$, the components ${\cal F}_{\a\b}$ cannot be directly matched with the field strength ${\cal H}_{\a\b 2}$ on M5-brane theory because it involves the fields that does not exist in D4-brane.
Supersymmetry Transformation
----------------------------
The supersymmetry transformation of fields (except for $a_{\a}$ and $\breve{B}_{\b}^{~\dn}$) on D4-brane from DDR is $$\begin{aligned}
\label{eq:3.1}
\delta_{\eps} X^I &=&i{\overline}\epsilon\Gamma^I\Psi,\\
\delta_{\eps} \Psi
&=&{\cal D}_\a X^I\Gamma^\a\Gamma^I\epsilon
+{\cal D}_{\dot\mu}X^I\Gamma^{\dot\mu}\Gamma^I\epsilon+\frac{1}{2}g\eps^{\dm\dn\dr}F_{\dn\dr}\del_{\dm}X^{I}\G^{2}\G^{I}\epsilon
\nonumber\\&&
-\frac{1}{2}{\cal F}_{\dn\dr}\G^{2}\G^{\dn\dr}\epsilon-\frac{1}{2}
{\cal H}_{\a\dot\nu\dot\rho}
\Gamma^\a\Gamma^{\dot\nu\dot\rho}\epsilon
-\left(\frac{1}{g}+{\cal H}_{\dot1\dot2\dot3}\right)
\Gamma_{\dot1\dot2\dot3}\epsilon
\nonumber \\&&
-\frac{g^2}{2}\{X^{\dot\mu},X^I,X^J\}
\Gamma_{\dot\mu}\Gamma^{IJ}\epsilon
+\frac{g^2}{6}\{X^I,X^J,X^K\}
\Gamma^{IJK}\Gamma^{\dot1\dot2\dot3}\epsilon,\\
\delta_{\eps} b_{\dot\mu\dot\nu}
&=&-i({\overline}\epsilon\Gamma_{\dot\mu\dot\nu}\Psi),\\
\delta_{\eps} a_{\dot\mu}
&=&i({\overline}\epsilon\Gamma_2\Gamma_{\dot\nu}\Psi)(\delta_{\dm}{}^{\dn}+g\del_{\dm}b^{\dn})
-ig({\overline}\epsilon\Gamma_2\Gamma^I\Gamma_{\dot1\dot2\dot3}\Psi)
\partial_{\dot\mu}X^I.\end{aligned}$$
### Supersymmetry Transformation of ${\a_{\alpha}}$ and $\breve{B}_{\beta}^{~\dm}$
We cannot obtain the supersymmetry transformation for ${\a_{\alpha}}$ and $\breve{B}_{\beta}^{~\dm}$ directly from DDR. But we find -$\eps^{\a\b}\del_{\dm}a_{\b}\breve{B}_{\a}^{~\dm}$ this term in action (Eq.(\[eq:2.1\])) can give us some information to determine the supersymmetry transformation for $\breve{B}_{\beta}^{~\dm}$. Firstly, the supersymmetry variation of $-\eps^{\a\b}\del_{\dm}a_{\b}\breve{B}_{\a}^{~\dm}$ will become $$\begin{aligned}
\label{eq:3.2}
-\eps^{\a\b}\del_{\dm}\delta_{\eps} a_{\b}\breve{B}_{\a}^{~\dm}-\eps^{\a\b}\del_{\dm}a_{\b}\delta_{\eps}\breve{B}_{\a}^{~\dm}.\end{aligned}$$ Because the other terms in action do not create $a_{\a}$ to cancel the second term of Eq.(\[eq:3.2\]) from supersymmetry transformation, the only one way is to use partial integration by part to delete it (so it needs $\del_{\dm}\delta_{\eps}\breve{B}_{\a}^{~\dm}=0$). Fortunately, we have one simple candidate for $\delta_{\epsilon}\breve{B}_{\mu}^{~\dm}$ is $\epsilon^{\dot{\mu}\dot{\nu}\dot{\rho}}\partial_{\dot{\nu}}\delta_{\epsilon}b_{\mu\dot{\lambda}}$ (We already know supersymmetry transformation for $b_{\mu\dot{\lambda}}$ on M5-brane.). From the above discussion, we know $$\begin{aligned}
\label{eq:3.3}
\delta_{\eps}\breve{B}_{\a}^{~\dm}=-i\eps^{\dm\dn\dr}{\overline}\epsilon\G_{\a}\G_{\dlam}\del_{\dn}\Psi(\d_{\dr}{}^{\dlam}+g\del_{\dr}b^{\dlam})+ig\eps^{\dm\dn\dr}{\overline}\epsilon\G_{\a}\G^{I}\Gamma_{\dot1\dot2\dot3}\del_{\dn}\Psi
\partial_{\dr}X^I.\end{aligned}$$ After many trivial but long calculations, we calculate $$\begin{aligned}
\label{eq:3.4} \delta_{\epsilon}S&=&-\frac{1}{2}ig\delta_{\eps}{\overline}\Psi\G^{\a}\Psi\del_{\dm}\breve{B}_{\a}^{~\dm}-ig\eps^{\a\g}{\overline}\epsilon\Gamma_2\Gamma_{\dot\nu}\Psi\del_{\g}b^{\dn}\del_{\dm}\breve{B}_{\a}^{~\dm}\nn\\
&&+ig\eps^{\a\g}{\overline}\epsilon\Gamma_2\Gamma^I\Gamma_{\dot1\dot2\dot3}\Psi
\partial_{\g}X^I\del_{\dm}\breve{B}_{\a}^{~\dm}\nn\\
&&+\eps^{\a\g}\delta_{\eps} a_{\g}\del_{\dm}\breve{B}_{\a}^{~\dm}.\end{aligned}$$ Hence, we can get supersymmetry transformation of $a_{\beta}$, $$\begin{aligned}
\delta_{\eps} a_{\b}&=&-\frac{1}{2}ig\delta_{\eps}{\overline}\Psi\G^{\a}\Psi\eps_{\a\b}+ig{\overline}\epsilon\Gamma_2\Gamma_{\dot\nu}\Psi\del_{\b}b^{\dn}\nn\\
&&-ig{\overline}\epsilon\Gamma_2\Gamma^I\Gamma_{\dot1\dot2\dot3}\Psi\partial_{\b}X^I,\end{aligned}$$ where $$\begin{aligned}
\delta_{\eps} {\overline}\Psi
&=&{\overline}\epsilon\Gamma^I\Gamma^A{\cal D}_A X^I
+\frac{1}{2}g{\overline}\epsilon\G^{I}\G^{2}\eps^{\dm\dn\dr}F_{\dn\dr}\del_{\dm}X^{I}
\nonumber\\&&
-\frac{1}{2}{\overline}\epsilon\G^{\dn\dr}\G^{2}{\cal F}_{\dn\dr}-\frac{1}{2}{\overline}\epsilon\Gamma^{\dot\nu\dot\rho}\Gamma^\a
{\cal H}_{\a\dot\nu\dot\rho}
-{\overline}\epsilon\Gamma_{\dot1\dot2\dot3}\left(\frac{1}{g}+{\cal H}_{\dot1\dot2\dot3}\right)
\nonumber\\&&
-\frac{g^2}{2}{\overline}\epsilon\Gamma^{IJ}\Gamma_{\dot\mu}\{X^{\dot\mu},X^I,X^J\}
+\frac{g^2}{6}{\overline}\epsilon\Gamma^{\dot1\dot2\dot3}\Gamma^{IJK}\{X^I,X^J,X^K\}.\end{aligned}$$
### Linearized Supersymmetry Transformation
This theory has $16$ additional fermionic symmetries $\delta_{\chi}$, which shifts the fermion by a constant spinor $$\delta_{\chi}\Psi=\chi,\quad
\delta_{\chi}X^I=\delta_{\chi}b^{\dot\mu}
=\delta_{\chi}a_{\dm}=0.$$ We also get this fermionic transformation of $a_{\alpha}$, $$\label{eq:3.6}
\delta_{\chi}a_{\a}=-\frac{i}{2}g{\overline}\chi\G^{\b}\Psi\eps_{\a\b}.$$ These two supersymmetry transformations ($\d_{\eps}$, $\d_{\chi}$) are all nonlinear, which means that the supersymmetry transformations have constant spinor terms. When all fields vanish, the transformation does not vanish. This kind of SUSY is actually a broken SUSY. This result comes from the background effect. After excluding the background effect, we can linearize the supersymmetry transformation, $$\label{eq:3.7}
\delta\equiv\frac{1}{g}\delta_{\chi\rightarrow\Gamma_{\dot1\dot2\dot3}\eps}+\delta_{\eps},$$ then we get linearized supersymmetry transformation of $a_{\a}$, $$\label{eq:3.8}
\delta a_{\a}=\delta a_{\dm\rightarrow\a}+\frac{1}{2}\delta{\overline}\Psi\G^{\b}\Psi\eps_{\a\b}.$$ On the other hand, the $\d\Psi$ do not have $\frac{1}{g}$ term, &=&[D]{}\_X\^I\^\^I+[D]{}\_X\^I\^\^I+g\^F\_\_X\^[I]{}\^[2]{}\^[I]{}\
&& -\_\^[2]{}\^-\_ \^\^-[H]{}\_[123]{}\_[123]{}\
&& -{X\^,X\^I,X\^J} \_\^[IJ]{}+{X\^I,X\^J,X\^K} \^[IJK]{}\^[123]{}. \[dPsi\]
### Supersymmetry Transformation of $\hat{B}_{\a}{}^{\dm}$ Field
Since Eq.(\[eq:2.1\]) is at most quadratic in $\breve{B}_{\a}{}^{\dm}$ so we can do Gaussian integration. Classically, it is equivalent to replace the field $\breve{B}_{\a}{}^{\dm}$ in action by its solution of EOM ($\hat{B}_{\a}{}^{\dm}$). Now we worry the supersymmetry transformation of $\hat{B}_{\a}{}^{\dm}$ may be different. In fact, $\d_{\eps}\hat{B}_{\a}{}^{\dm}$ is equal to $\d_{\eps}\breve{B}_{\a}{}^{\dm}$ with the additional terms proportional to EOM of some fields. The difference what we get is, \_\_\^ &=& \_\_\^ - 2 (**M**\^[-1]{})\^\_[ ]{}(\_|\_)\_\^( ),\[dhatB\] where (\_|\_)\_\^ \^\^\_V\_\^-g\^[I]{}\^\_X\^[I]{}, which is the coefficient of $\hat{B}_{\dn}{}^{\b}$ in $\d_{\eps}{\overline}\Psi$.\
The EOM of ${\overline}\Psi$ is ( )&=&\^\_+g\^[2]{}\^F\_\_-g\^\_\^\_\
&&+\^[D]{}\_+g\^[2]{}\_\^[I]{}{X\^,X\^[I]{},}-g\^2\^[IJ]{}\^{X\^[I]{},X\^[J]{},}.\
This relation (\[dhatB\]) also implies \_(**M**\_\^[ ]{}\_\^-W\_\^)=0, which means that we can write down a simpler form than Eq.(\[dhatB\]). This simpler form can be understood as the $\d_{\eps}\breve{B}_{\a}{}^{\dm}$ with the additional terms proportional to EOM of fermionic field. On the other hand, the another fermionic symmetry of $\hat{B}_{\a}{}^{\dm}$ is easy to calculate, \_\_\^=0.
After integrating out $\breve{B}_{\a}{}^{\dm}$ field, the supersymmetry transformation of fermion is just to replace $\breve{B}_{\a}{}^{\dm}$ with $\hat{B}_{\a}{}^{\dm}$. Now we get all supersymmetry transformation of fields on D4-brane in the large $C$-field background. We hope to use the handle to open the new directions for the researches of D$p$-brane in the large $(p-1)$-form background in the future.
Super Algebra and Central Charges
---------------------------------
To compute super algebra, we start to calculate the super charge of the D4-brane in the large $C$-field background [^7] [^8]. The super charge $Q$ is calculated from the spatial integral of the time component of supercurrent $J^{0}$, where $J^{0}$ is defined by this way ${\overline}{\eps}J^{0}=-\d{\overline}{\Psi}\G^{0}\Psi$. The anticommutator of the supercharges is {Q,Q}=2d\^[4]{}x T\_[00]{}+\_[n=0]{}\^[5]{}d\^[4]{}xZ\_[n]{}. Here, we divide it into two parts: the energy part and the central charges part.
The energy part is T\_[00]{}&=&\_[0]{}X\^I[D]{}\_[0]{}X\^I +([D]{}\_[a]{}X\^I)\^2+g\^[2]{} F\_F\^\_X\^[I]{}\^X\^[I]{}+g\^[2]{}F\_F\^\_X\^[I]{}\^X\^[I]{}\
&& +\_[0]{}[H]{}\_[0]{}\^+([H]{}\_[1]{})\^2 + ([F]{}\_)\^2 +([H]{}\_[123]{})\^2\
&& +{X\^, X\^I, X\^J}\^2 +{X\^I, X\^J, X\^K}\^2.
We classify the central charges part according to the number of the scalar fields we choose to turn on. In order to compare with the BPS solutions in the next section, the momentum term ($T_{0a}, a=(1,\dot\mu)$) is included in the central charges. Let us now describe each terms of $Z_n$. The convention of indices here are $\bar{a}=(0, \dot\mu)$. Z\_[0]{}=[H]{}\_[0]{}[H]{}\^[a]{} \^0\_[a]{} -[H]{}\_[0]{}[F]{}\^\^2\^0 +[F]{}\_[H]{}\_[1]{}\^\^\^[12]{}. We find ${\cal H}$${\cal H}$ on M5-brane becomes ${\cal F}$${\tilde{\cal F}}$ and ${\cal H}$${\cal H}$ on D4-brane. From the D4-brane perspective, ${\cal F}$${\tilde{\cal F}}$ can be thought as the charge of a D0-brane within the worldvolume of the D4-brane and ${\cal H}$${\cal H}$ can be thought as a pp-wave intersected a D4-brane.
Next, we have Z\_[1]{}&=&-g\^F\_\_X\^ID\_0X\^I\^2\^0+ 2D\_0X\^ID\_[a]{}X\^I\^0\^[a]{}+D\_[a]{}X\^I[H]{}\_[bcd]{}\^[abcd]{}\^I\
&&+\^F\_\_X\^I[H]{}\_[bcd]{}\^[2bcd]{}\^I +D\_[a]{}X\^I[F]{}\_\^[a 2]{}\^I\
&&+D\_X\^I[H]{}\_[0]{}\^\^[21]{}\^I. The $({\cal D}X){\cal H}$ term corresponds to the charge of F1 ending on D4 solution. If we consider one scalar field is active and assume that this scalar is a function of only four of the spatial worldvolume coordinates of the D4-brane. On the other hand, $\frac{g}{6}\epsilon^{\dot\mu\dot\nu\dot\rho}F_{\dot\nu\dot\rho}\partial_{\dot\mu}X^I{\cal H}_{bcd}$ and $D_{\dot\mu}X^I{\cal F}_{\dot\nu\dot\rho}$ terms correspond to our BPS solution with two types of magnetic charges.
When two scalar fields are turned on, we will need to consider $Z_{0}$, $Z_{1}$ and $Z_{2}$, where $Z_{2}$ is defined by Z\_[2]{}&=&D\_aX\^ID\_bX\^J\^[ab]{}\^[IJ]{} +g\^F\_\_X\^ID\_bX\^J\^[2b]{}\^[IJ]{} +2g\^2D\_X\^I{X\^,X\^I,X\^J}\^J\
&&+2g\^2D\_0X\^I{X\^,X\^I,X\^J}\^0\_\^J +\_[0]{}{X\_,X\^J,X\^K}\^\^[JK]{}\^0\
&&-g\^2[H]{}\_[1]{}{X\^,X\^I,X\^J}\^[1]{}\^[IJ]{} -g\^2[F]{}\_{X\^,X\^I,X\^J}\^[2]{}\^[IJ]{}\
&&-g\^2[H]{}\_[123]{}{X\^,X\^I,X\^J}\_\^[IJ]{}\^[123]{}. We see the first term ${\cal D}_a X^I{\cal D}_b X^J$ corresponds to the charge of the 2-brane vortex living on the D4-brane worldvolume.
We also have Z\_[3]{}&=&g\^2D\_[a]{}X\^[I]{}{X\^,X\^J,X\^K}\^[a]{}\_\^[IJK]{} +\^ F\_\_X\^I{X\^,X\^J,X\^K}\^[2]{}\_\^[IJK]{}\
&&-g\^2D\_[|[a]{}]{}X\^I{X\^I,X\^J,X\^K}\^[|[a]{}]{}\^[JK]{}\^[123]{} -\_[1]{}{X\^I,X\^J,X\^K}\^[1]{} \^[IJK]{}\^[123]{}\
&&-\_{X\^I,X\^J,X\^K}\^[2]{}\^[IJK]{}\_[123]{} +g\^4{X\^I,X\^J,X\^}{X\^I,X\^K,X\^}\_\^[JK]{}\
and Z\_4&=&-D\_1X\^I{X\^J,X\^K,X\^L}\^1\^[IJKL]{}\^[123]{}\
&&-\^F\_\_X\^I{X\^J,X\^K,X\^L}\^2 \^[IJKL]{}\_[123]{}\
&&+g\^4{X\^,X\^I,X\^J}{X\^I,X\^K,X\^L}\_\^[JKL]{}\_[123]{}\
&&-{X\^,X\^I,X\^J}{X\_,X\^K,X\^L}\^[IJKL]{}. The charge $\frac{g^2}{3}{\cal D}_1 X^I\{X^J, X^K, X^L\}$ is equipped with 3-bracket on D4-brane. The geometry of the M5-brane is similar with the situation of the $C$-field modified Basu-Harvey equation as a boundary condition of the multiple M2-brane theory [@Chu:2009iv].
Finally, the last of $Z_{n}$ is Z\_[5]{}=-{X\^I, X\^J, X\^K}{X\^I, X\^L, X\^M}\^[JKLM]{}. This term is relevant only if we turn on all scalars $X^{6},\cdots, X^{10}$.
BPS Solutions {#4}
=============
In this paper, we only consider pure bosonic solitons, namely those with the fermion field $\Psi = 0$. The BPS condition is therefore simply that the supersymmetry transformation of $\Psi$ (eq.(\[dPsi\])) vanishes for some supersymmetry parameters $\eps$. We systematically study BPS solutions by classifying them according to the number of scalars that are turned on.
Most of these BPS solutions are directly derived from the NP M5-brane by DDR. So we do not write down the details of these solutions. We divide two parts of this section here. In the first part, we mention the BPS solutions on D4-brane which are derived from the NP M5-brane directly and discuss the instanton solution particularly. The next part is to discuss the BPS solutions after we turn on one scalar field ($X^{6}$). This solution is called BPS solution with two types of magnetic charges, which can be understood as D2 ending on D4 in geometric viewpoint. This solution can be related to self-dual string solution on M5-brane from non-linear superpositions. In order to emphasize it, we put this solution in a new section.
Solutions via DDR from the NP M5-brane BPS States
-------------------------------------------------
Some solutions are easy to be derived from the NP M5-brane by DDR, so we do not write down the details from these solutions. From the NP M5-brane theory, we have obtained M-waves, the self-dual string (M2 ending on M5), tilted M5-brane, holomorphic embedding of M5-brane and the intersection of two M5-branes along a 3-brane BPS solutions. We classify these solutions in the previous paper [@Ho:2012dn] with the number of scalar fields we turn on. When we turn off all scalar fields, we get M-waves (Light-Like) BPS solutions. If we turn on one scalar field $X^{6}$, we get the self-dual string (M2 ending on M5) and tilted M5-brane BPS solutions. When we turn on two scalar fields $X^6$ and $X^7$, we get holomorphic embedding of M5-brane and the intersection of two M5-branes along a 3-brane BPS solutions.
After doing double dimensional reduction, it directly obtains these BPS solutions on D4-brane from the above solutions. We can obtain pp-wave, F1 ending on D4 solution, tilted D4-brane, holomorphic embedding of D4-brane and the intersection of two D4-branes along a 2-brane. The light-like BPS solutions in M5-brane theory give the pp-wave BPS solutions and the trivial instanton solution on D4-brane [^9]. The BPS solutions with turning on one scalar field $X^6$ on M5-brane can be related directly to the F1 ending on D4 and the tilted D4-brane BPS solutions in D4-brane theory. Finally, the BPS solutions with two scalar fields on M5-brane can be reduced to the holomorphic embedding of D4-brane and the intersection of two D4-branes along a 2-brane BPS solutions in D4-brane theory.
### Instanton Solution
Even if these solutions can easily be derived from BPS solutions in the NP M5-brane theory. There is still an interesting issue about the BPS stares of D0-brane (instanton) [^10] exists whether or not. In these solutions, we do not have a non-trivial instanton solution in the large $C$ field background after we do DDR. To emphasize the reason, we show the detail calculation on this solution. The instanton solution on D4-brane can be related to the light-like BPS solution in the previous paper [@Ho:2012dn]. The BPS solution satisfies the BPS condition $\G^{02} \eps = \pm \eps$. After DDR, the BPS conditions are &[H]{}\_[0]{} = \_,\
&[H]{}\_[1]{} = 0,\
&[H]{}\_ = 0. This solution preserves $\frac{1}{2}$ SUSY, which can be thought as a D0-brane. The energy density is bound by $\mid\frac{1}{2}{\cal H}_{0\dot\mu\dot\nu}{\cal F}^{\dot\mu\dot\nu}\mid$, and it is consistent with the central charge $Z_{0}$.
Firstly, we impose a gauge fixing condition b\^[1]{}=b\^[2]{}=b\^[3]{}=0. From ${\cal H}_{1\dot\mu\dot\nu}$=0, we obtain \_[1]{}\^=0. From ${\cal H}_{0\dot\mu\dot\nu}=\pm{\cal F}_{\dot\mu\dot\nu}$, we obtain \_[0]{}\^=\^ F\_.
Let us consider the equation of motion eq.(\[r2\]) without turning on scalar, -\^\_+\^F\_+g\^F\_ \_\^=0. This implies \_[0]{}=-F\_[1]{}. We can get the following relation, &&F\_[11]{}=F\_[23]{},\
&&F\_[12]{}=F\_[31]{},\
&&F\_[13]{}=F\_[21]{}. As $U(1)$ gauge theory, we do not have non-trivial instanton solutions [^11]. So we do not have non-trivial instanton solutions in the large $C$-field background [^12].
This BPS solution of M5-brane is a wave which travels at the speed of light on $x_2$ direction. If we do DDR on this direction, we will obtain the trivial solution on D4-brane in the large C-field background. However, we propose the next possible instanton solution from combining $\G^{02}\eps=\pm\eps$ and $\G^{026}=\pm\eps$. This solution preserves $\frac{1}{4}$ SUSY with D0-brane and F1-string ending on D4-brane interpretation.
BPS Solution with Two Types of Magnetic Charges
-----------------------------------------------
After turning on one scalar $X^{6}$, we have a new solution which is related to the solution of the NP M5-brane theory with the non-linear way. The BPS condition which we consider is $\Gamma^{016}\epsilon$=$\pm$$\epsilon$. This solution preserves $\frac{1}{2}$ SUSY, and the geometric picture is D2 ending on D4. The energy density is bounded by \^([F\_]{}D\_X\^[6]{}-gF\_[H]{}\_\_X\^[6]{}), and it is consistent with the central charges on D4-brane.
This solution up to first g order is [^13], X\^[6]{}&=&,\
b\^&=&-x\^+g( -+)x\^+[O]{}(g\^[2]{}), \[p2\]\
F\_&=&-\_x\^+[O]{}(g\^[2]{}), where the notation $a$ is $\sqrt{x_{\dot1}^2 + x_{\dot2}^2 +x_{\dot3}^2 }$.
From this solution, we can know it contains two types of magnetic charges. The one ($Q_{M2}$=-4$\pi m_{2}$, where $m_{2}$=$\frac{k_{2}}{(2\pi)^{\frac{3}{2}}T_{D_{4}}^{\frac{2}{5}}}$) is from $F^{\dot\mu\dot\nu}$ and another ($Q_{M1}$=-4$\pi m_{1}$, where $m_{1}$=$\frac{k_{1}}{(2\pi)^{\frac{3}{2}}T_{D_{4}}^{\frac{3}{5}}}$) is from ${\cal H}_{\dot{1}\dot{2}\dot{3}}$. And we see two charges have the interaction from the results of the first-order expansion. The reason is due to ${\cal F}_{\dot\nu\dot\rho}$. This field strength offers interaction between $a^{\dot{\mu}}$ and $b^{\dot{\nu}}$ from the first-order term. This interaction is due to the strength of Nambu-Poisson bracket so this interaction disappears if $g$=0. We also find an interesting connection between this solution and solution of Ref. [@Furuuchi], which is called Furruchi-Takimi (FT) solution in this paper. We find this solution is just a non-linear superposition of FT solution, if we integrate FT solution with respect to $x_{2}$ [^14], X\_[(0)]{}\^[6(FT)]{}dx\_[2]{}&=&dx\_[2]{}=,\
\_[(0)]{}\^[(FT)]{}dx\_[2]{}&=&dx\_[2]{}=0, where $X_{(0)}^{6}{}^{(FT)}$ and ${\cal H}_{(0)}^{(FT)}$ are the zero-order of FT solutions. When $m_{2}$=$m\pi$, two solutions are the same. We can interpret our zero-order solutions are just linear superposition of the zero-order of FT solutions. If we examine the first-order solutions, we find two solutions cannot be the same from integration. Due to the strength of Nambu-Poisson bracket breaks the linear superposition effect, we interpret this soliton solution is a non-linear superposition of FT solution.
We also find an ansatz from the perturbative solutions. b\^&=&f( a ) x\^,\
X\^[6]{}&=&h( a ),\
F\^&=&\^x\_, where $C$ is an arbitrary constant. From the perturbation, we observe $F^{\dot\mu\dot\nu}$ should be a function of $a$. And we have $dF=0$ this restriction. The only one possible ansatz should be monopole solution for field strength. It creates a source on the origin. The function of $f(a)$ and $h(a)$ need to satisfy &=&-,\
&=&. If we want $f$ and $h$ are smooth function, we need $gf(0)$+1=0 this constraint on the origin. We find the only one smooth solution on this boundary condition, but it does not have finite energy. b\^&=&-x\^,\
X\^6&=&+,\
F\^&=&\^x\_, where $E$ is an arbitrary constant. We think this ansatz is not good enough to probe the non-perturbative soliton solution and this solution should be the tilted D4-brane BPS solution. But it is still an interesting problem to find an exact solution to describe this soliton solution near origin in the future.
Conclusion and Discussion {#5}
=========================
In this paper, we give more details of the low energy effective theory for D4-brane in the large $C$-field background. When theory couples with matter fields, there are not only U(1) and VPD gauge symmetry but also supersymmetry. We obtain the supersymmetry transformation of all fields in the low energy effective theory. In the previous paper [@Ho], we did not point out how to calculate the supersymmetry transformation law after the duality transformation. After the duality transformation, the interpretation of $B_{\a}{}^{\dm}$ field was changed. We should treat the field $B_{\a}{}^{\dm}$ to be the new field $\breve{B}_{\a}{}^{\dm}$ in the dual action. So the divergence of $\breve{B}_{\a}{}^{\dm}$ does not vanish. ($\del_{\dm}\breve{B}_{\a}{}^{\dm} \neq 0$.) This property helps us to find the supersymmetry transformation of $a_{\a}$, which comes from the duality transformation. After obtaining all supersymmetry transformation laws, we also check the Lagrangian being supersymmetrical invariant. Now, the full supersymmetry of this effective action is completely understood in this work.
Moreover, we are also interested in the topological quantities of this theory so we calculated the central charges from supercurrent. These central charges let us know the possible topological solutions. In the last section, we studied BPS solutions of the effective field theory for D4-brane in the large $C$-field background. The large $C$-field background turns on new interactions on the D4-brane worldvolume through the Nambu-Poisson structure, and modifies some of the BPS configurations. Most of them correspond to the double dimensional reduction of the BPS solutions in the NP M5-brane theory so we just mention them without details. On the other hand, we also found a new perturbative solution which was not directly related to the self-dual string BPS solution in the NP M5-brane theory [@Ho:2012dn; @Furuuchi] after we do DDR. It is related to the self-dual string solution with non-linear superpositions. This geometric picture can be easily understood as D2 ending on D4.
We did not find the instanton solutions in this effective theory. Originally, we are interested in the topological quantities because of the well-known $U(1)$ instanton solution of D-brane in the large NS-NS $B$-field background [@Seiberg:1999vs; @Nekrasov]. We wonder if there is a similar $U(1)$ instanton solution in the large $C$-field background. However, we cannot find it in $\frac{1}{2}$ BPS states. But this solution may survive in $\frac{1}{4}$ BPS solutions.
Finally, it is still an open question: how to generalize our work to all D$p$-brane in all R-R field backgrounds with matter fields. If the generalization is successful, it is possible to find new BPS states. These work help us to understand more about the geometrical structure of the Nambu-Poisson gauge theory and open the new direction on D$p$-brane in the R-R field background.
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors thank Prof. Pei-Ming Ho. This paper can be finished because of his encouragement and many useful suggestions. The authors also thank Wei-Ming Chen, Kazuyuki Furuuchi, Hiroshi Isono, Sheng-Lan Ko and Tomohisa Takimi for useful discussions. Without their discussions, it is possible to lose some interesting ideas. The authors are supported in part by the National Science Council, Taiwan, R.O.C.
.8cm
[99]{}
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[^1]: e-mail address: yefgst@gmail.com
[^2]: e-mail address: d95222008@ntu.edu.tw
[^3]: The large $C$-field limit for M5-brane was first considered in [@Berman].
[^4]: The NP M5-brane theory is defined on six dimensional worldvolume, and we use the notation $\{x^{0},x^{1},x^{2};x^{\dot{1}},x^{\dot{2}},x^{\dot{3}}\}\equiv\{x^{\mu};x^{\dm}\}$ to label these worldvolume coordinates. The $C$-field background is given by $C_{\dot 1\dot 2\dot 3}dx^{\dot1}dx^{\dot2}dx^{\dot3}$.
[^5]: The gauge transformation of $\breve{B}_{\a}^{~\dm}$ is the same as $B_{\a}^{~\dm}$. ($~\d_{\Lambda} \breve{B}_{\mu}{}^{\dm} =
\del_{\mu}\kappa^{\dm} + g\kappa^{\dn}\del_{\dn}\breve{B}_{\mu}{}^{\dm}
- g(\del_{\dn}\kappa^{\dm})\breve{B}_{\mu}{}^{\dn}$.)
[^6]: A field $\hat{\Phi}$ is covariant if its gauge transformation is $\delta_{\Lambda} \hat{\Phi} = g\kappa^{\dm}\del_{\dm}\hat{\Phi}$.
[^7]: In this section, we follow the methods mentioned in the papers [@Lambert; @Low].
[^8]: In the papers [@Low; @Low2], the author calculates the central charges from the superalgebra of the NP M5-brane theory, and performs DDR to get the central charges of D4-brane theory in the large $B$-field background. He also provides more details with the interpretation of these central charges.
[^9]: The reason for this instanton solution is trivial will be given latter.
[^10]: We call “instanton” because we use the same terminology of D(-1) solutions in D3-brane case.
[^11]: In five dimensional theory, these relations can be understood as self-duality equations with static gauge fields and temporal gauge $a_{0}=0$.
[^12]: In fact, our effective theory is well defined in a special scaling limit [@Ho], it is still possible that instanton solutions exist in other limits.
[^13]: Here, $\breve{B}_{\a}{}^{\dm}=0$.
[^14]: We do integration before we perform DDR on $x_2$ this direction so the range of $x_{2}$ is from $-\infty$ to $\infty$.
|
---
abstract: 'We present explicit expressions of the helicity conservation in nematic liquid crystal flows, for both the Ericksen-Leslie and Landau-de Gennes theories. This is done by using a minimal coupling argument that leads to an Euler-like equation for a modified vorticity involving both velocity and structure fields (e.g. director and alignment tensor). This equation for the modified vorticity shares many relevant properties with ideal fluid dynamics and it allows for vortex filament configurations as well as point vortices in 2D. We extend all these results to particles of arbitrary shape by considering systems with fully broken rotational symmetry.'
author:
- |
François Gay-Balmaz$^1$ and Cesare Tronci$^2$\
\
[$^1$ *Control and dynamical systems, California Institute of Technology*]{}\
[$^2$ *Section de Mathématiques, École Polytechnique Fédérale de Lausanne, Switzerland*]{}
title: The helicity and vorticity of liquid crystal flows
---
Introduction
============
Several studies on nematic liquid crystal flows have shown high velocity gradients and led to the conclusion that the coupling between the velocity $\boldsymbol{u}(\mathbf{x},t)$ and structure fields is a fundamental feature of liquid crystal dynamics [@ReyDenn02]. This conclusion has been reached from different viewpoints and by using different theories, such as the celebrated Ericksen-Leslie (EL) and the Landau-de Gennes (LdG) theories [@Chonoetal; @Tothetal; @TaoFeng; @BlSvetal]. Evidence of high velocity gradients also emerged [@KuKaDe] by using kinetic approaches based on the Doi model [@DoEd1988]. The essential difference between EL and LdG theories resides in the choice of the order parameter: while EL theory for rod-like molecules considers the dynamics of the director field $\mathbf{n}(\mathbf{x},t)$ and it is successful in the description of low molar-mass nematics, the LdG theory generalizes to variable molecule shapes by considering a traceless symmetric tensor field ${\sf Q}(\mathbf{x},t)$. In the presence of high disclination densities the LdG theory is more reliable, since molecules may easily undergo phase transitions (e.g. from uniaxial to biaxial order) that are naturally incorporated in the theory. However, the dynamical LdG theory is not completely established and different versions are available in the literature [@BeEd1994; @Lubensky2003; @QiSh1998]. Here, we shall adopt the formulation of Qian and Sheng [@QiSh1998], which will be simply referred to as LdG theory.
This paper considers both EL and LdG theories and it shows how the strong interplay between velocity and order parameter field reflects naturally in the helicity conservation for nematics. In this paper, the term “helicity” stands for the hydrodynamic helicity and *not* the helicity of the single liquid crystal molecule. The helicity conservation for incompressible liquid crystal flows arises from the simple velocity transformation $\boldsymbol{u}\to\boldsymbol{\mathcal{C}}=\boldsymbol{u}+\mathbf{J}$, where the vector $\mathbf{J}$ depends only on the order parameter field. The covariant vector $\boldsymbol{\mathcal{C}}$ is the total circulation (momentum per unit mass of fluid) and $\mathbf{J}$ is the circulation associated with entrainment of fluid due to its local interaction with the nematic order parameter field. We shall show how this change of velocity variable takes the equation for the ordinary vorticity $\boldsymbol\omega=\nabla\times\boldsymbol{u}$ into an Euler-like equation for the modified vorticity $\overline{\boldsymbol\omega}=\nabla\times\boldsymbol{\mathcal{C}}$, thereby extending many properties of ordinary ideal fluids to nematic liquid crystals. The helicity is then given by $$\mathcal{H}=\int \left( \overline{\boldsymbol{ \omega }} \cdot
\boldsymbol{\mathcal{C}} \right) \,\mathrm{d}^3\mathbf{x},$$ and this quantity naturally extends the usual expression $\int
\left(\boldsymbol{ \omega } \cdot \boldsymbol{ u} \right)\,
\mathrm{d}^3\mathbf{x}$ for the helicity of three dimensional ideal flows. We recall that in Hamiltonian fluid dynamics the conservation of the hydrodynamic helicity is strictly associated to the Hamiltonian structure of the equations and holds for any Hamiltonian. This point will be further developed in the last part of the paper, where all the results will be derived directly from the Hamiltonian structure of the liquid crystal equations, see [@GayBalmazTronci]. Invariant functions like the helicity are called Casimir and are of fundamental importance for the study of nonlinear stability. For two dimensional flows, such invariant quantities are given by $$\label{Casimir2D}
\int \Phi(\overline{ { \omega }})\ \mathrm{d}^2\mathbf{x},$$ where $\Phi$ is an arbitrary smooth function. As we shall see, the same circulation concept leading to hydrodynamic helicity applies quite generally in complex fluid theory and is related to an analogy between complex fluids and non-Abelian Yang-Mills fluid plasmas [@GiHoKu1983; @HoKu1988].
In addition to helicity conservation, we present the existence of vortex-like configurations for the modified vorticity $
\overline{\boldsymbol\omega}$. Vortex structures are well known to arise in superfluid flows and their behavior is often reminiscent of disclination lines in liquid crystals. However, here we shall consider vortices that are characterized by a combination of velocity and structures fields. After extending these results to fluids with molecules of arbitrary shapes, the end of this paper discusses the geometric basis of the present treatment.
Director formulation
====================
In the context of EL theory, disclinations are singularities of the director field and thus their dynamics is related to the evolution of the gradient $\nabla\mathbf{n}$. This relation has been encoded by Eringen [@Eringen] in the *wryness tensor* $$\label{GammaDef}
\gamma_\text{\tiny
EL}=\mathbf{n}\times\nabla\mathbf{n}\,,\quad
\text{ or }\quad
\left(\gamma_\text{\tiny
EL}\right)^a_i=\varepsilon^{abc\,}{n}_b\times\partial_i{n}_c\,,$$ which identifies the amount by which the director field rotates under an infinitesimal displacement $\mathrm{d}\mathbf{x}$. Thus, the EL wryness tensor $\gamma_\text{\tiny EL}$ determines the spatial rotational strain [@Ho2002]. In this paper we shall investigate the role of the EL wryness tensor in helicity conservation and vorticity dynamics in the EL theory. For this purpose, we ignore dissipation and concentrate on nonlinearity. This simplifies the resulting formulas. We also restrict to incompressible flows to ignore ordinary fluid thermodynamics.
Upon denoting by $J$ the microinertia constant [@Eringen], we introduce the angular momentum variable $$\label{SigmaDef}
\boldsymbol\sigma_\text{\tiny EL}=J\mathbf{n}\times D_t{\mathbf{n}}$$ that is associated to the director precession. While $D_t{\mathbf{n}}=\partial_t{\mathbf{n}}+\boldsymbol{u}\cdot\nabla{\mathbf{n}}$ denotes material time-derivative, the spatial derivatives of the director field will be denoted equivalently by $\partial \mathbf{n}/\partial
x^i=\partial_i\mathbf{n}=\mathbf{n}_{,i}$ depending on convenience. Upon using Einstein’s summation convention, one can express the Ericksen-Leslie equations as [@Ho2002; @GayBRatiu; @GayBalmazTronci] $$\begin{aligned}
\label{EL1}
\partial_t \boldsymbol{u}+\boldsymbol{u}\!\cdot\!\nabla \boldsymbol{u} &=-\partial_i\!\left(\nabla\mathbf{n}^T\!\cdot\!\frac{\partial F}{\partial\mathbf{n}_{,i}}\right)-\nabla p\,,\quad \ \nabla\cdot\boldsymbol{u} =0
\\
\partial_t{\boldsymbol{\sigma}}_\text{\tiny EL}+\boldsymbol{u} \cdot\!\nabla\boldsymbol{\sigma}_\text{\tiny EL}&=\mathbf{h}\times\mathbf{n}\,,\hspace{.7cm} \partial_t{\mathbf{n}}+\boldsymbol{u} \cdot\!\nabla\mathbf{n}=J^{-1}\boldsymbol{\sigma}_\text{\tiny EL}\times\mathbf{n}
\label{EL2}\end{aligned}$$ where $p$ is the hydrodynamic pressure by which incompressibility is imposed and $\mathbf{h}$ is the thermodynamics derivative representing the First Law response in energy to changes in the director field $$\mathbf{h}:=\frac{\partial
F}{\partial\mathbf{n}}-\partial_{i\!}\left(\frac{\partial
F}{\partial\mathbf{n}_{,i}}\right)\,.$$ The quantity $F$ is taken to be the Oseen-Zöcher-Frank free energy $$\label{standard_energy}
F=K_1(\operatorname{div}\mathbf{n})^2+K_2(\mathbf{n}\cdot\nabla\times\mathbf{n})^2+K_3|\mathbf{n}\times\nabla\times\mathbf{n}|^2\,.$$ Of course, this choice is not a limitation, because our considerations apply to a generic form of $F$. For example, effects of external electric and magnetic fields may be taken into account with easy modifications. The Ericksen-Leslie fluid equations follow immediately from equations and , as shown in [@GayBRatiu].
In Ericksen-Leslie nematodynamics, the quantity $\boldsymbol{\sigma}_\text{\tiny
EL}\cdot\gamma_\text{\tiny EL}$ denotes the vector of momentum per unit mass, with components ${\left(\boldsymbol{\sigma}_\text{\tiny
EL}\cdot\gamma_\text{\tiny EL}\right)_i}={\left({\sigma}_\text{\tiny
EL}\right)_a\left(\gamma_\text{\tiny EL}\right)_i^a}$. We consider the vector $\boldsymbol{\mathcal{C}}_\text{\tiny
EL}$ defined as the sum $$\label{Cdef}
\boldsymbol{\mathcal{C}}_\text{\tiny
EL}:=\boldsymbol{u}+\boldsymbol{\sigma}_\text{\tiny
EL}\cdot{\gamma}_\text{\tiny
EL}=\boldsymbol{u}+\boldsymbol{\sigma}_\text{\tiny
EL}\cdot\mathbf{n}\times\nabla\mathbf{n}
\,,$$ reminiscent of the minimal coupling formula in electromagnetic gauge theory. We observe the following equation of motion [@GayBalmazTronci] (see appendix \[appendix1\]): [^1] $$\label{Cequation}
\partial_t\boldsymbol{\mathcal{C}}_\text{\tiny
EL}-\boldsymbol{u}\times\nabla\times\boldsymbol{\mathcal{C}}_\text{\tiny
EL} =-\nabla\!\left(
\phi+\boldsymbol{u}\cdot\boldsymbol{\mathcal{C}}_\text{\tiny
EL}\right) \,,$$ where $$\phi=p+F-\frac12\left|\mathbf{u}\right|^2-\frac1{2J}\left|\boldsymbol{\sigma}_\text{\tiny
EL}\right|^2 \,.$$ At this point, taking the curl of equation yields the Euler-like equation $$\label{EulerForNematics}
\partial_t\overline{\boldsymbol{\omega}}_\text{\tiny
EL}+\nabla\times\left(\boldsymbol{u}\times\overline{\boldsymbol{\omega}}_\text{\tiny
EL}\right)=0\,$$ for the modified vorticity $$\label{NewVort}
\overline{\boldsymbol{\omega}}_\text{\tiny
EL}:=\nabla\times\boldsymbol{\mathcal{C}}_\text{\tiny
EL}=\boldsymbol{\omega}+\nabla\times\left(\boldsymbol{\sigma}_\text{\tiny
EL}\cdot{\gamma}_\text{\tiny EL}\right)
\,.$$ Notice that the velocity $\boldsymbol{u}$ can be expressed as $$\label{untangled-u}
\boldsymbol{u}=-\nabla\times\boldsymbol\psi=-\nabla\times\Delta^{-1}\overline{\boldsymbol\omega}_\text{\tiny
EL}+\boldsymbol{\sigma}_\text{\tiny
EL}\cdot{\gamma}_\text{\tiny EL}+\nabla\varphi \,,$$ where $\boldsymbol\psi=\Delta^{-1}\boldsymbol{\omega}$ denotes the velocity potential, which is given by the convolution of the vorticity $\boldsymbol{\omega}$ with the Green’s function of the Laplace operator (analogously for $\Delta^{-1}\overline{\boldsymbol\omega}_\text{\tiny EL}$). Here the pressure-like quantity $\varphi$ is a scalar function arising from the term $
{\nabla\times\nabla\times\Delta^{-1}\left(\boldsymbol{\sigma}_\text{\tiny
EL}\cdot{\gamma}_\text{\tiny
EL}\right)}={{\boldsymbol{\sigma}_\text{\tiny
EL}\cdot{\gamma}_\text{\tiny EL}}+\nabla\varphi}$ and whose only role is to keep the velocity $\boldsymbol{u}$ divergence free, so that $
{\nabla\cdot(\boldsymbol{\sigma}_\text{\tiny
EL}\cdot{{\gamma}_\text{\tiny EL}})}=-\Delta\varphi
$. The relation can be inserted into equations so to express the EL equations in terms of the modified vorticity $\overline{\boldsymbol\omega}_\text{\tiny EL}$. An explicit expression of the quantity $\boldsymbol{\sigma}_\text{\tiny
EL}\cdot{\gamma}_\text{\tiny EL}$ arises from the definitions and : $
{\boldsymbol{\sigma}_\text{\tiny
EL}\cdot{\gamma}_\text{\tiny EL}}={J\,\nabla\mathbf{n}\cdot
D_t\mathbf{n}}
$.
At this point we recognize that equation possesses all the usual properties of Euler’s equation. For example, Ertel’s commuting relation $$\label{ertel}
\big[{D_{t\,}},\,\overline{\boldsymbol{\omega}}_\text{\tiny
EL}\cdot\nabla\big]\,\alpha={D_t}\!\left(\overline{\boldsymbol{\omega}}\cdot\nabla\alpha\right)-\overline{\boldsymbol{\omega}}\cdot\nabla\!
\left({D_{t\,}}\alpha\right)=0$$ follows easily by direct verification, for any scalar function $\alpha(\mathbf{x},t)$. Moreover, one has the following Kelvin circulation theorem [@GayBalmazTronci] $$\label{kelvin}
\frac{d}{dt}\oint_{\Gamma(t)}
\!\big(\boldsymbol{u}+\boldsymbol{\sigma}_\text{\tiny
EL}\cdot\gamma_\text{\tiny
EL}\big)\cdot\mathrm{d}\mathbf{x}=0\,,$$ where the line integral is calculated on a loop $\Gamma(t)$ moving with velocity $\boldsymbol{u}$. Also, conservation of the helicity [@GayBalmazTronci] $$\mathcal{H}_\text{\tiny
EL}=\!\int\overline{\boldsymbol{\omega}}_\text{\tiny
EL}\cdot\boldsymbol{\mathcal{C}}_\text{\tiny EL}\
\mathrm{d}^3\mathbf{x}
=\!\int\!\left(\boldsymbol{u}+\boldsymbol{\sigma}_\text{\tiny
EL}\cdot\gamma_\text{\tiny
EL}\right)\cdot\nabla\times\left(\boldsymbol{u}+\boldsymbol{\sigma}_\text{\tiny
EL}\cdot\gamma_\text{\tiny
EL}\right)
\mathrm{d}^3\mathbf{x}$$ follows from the relation $$\label{h-equation}
\partial_t\left(\boldsymbol{\mathcal{C}}_\text{\tiny
EL}\cdot\overline{\boldsymbol{\omega}}_\text{\tiny
EL}\right)+\nabla\cdot\big(\left(\boldsymbol{\mathcal{C}}_\text{\tiny
EL}\cdot\overline{\boldsymbol{\omega}}_\text{\tiny
EL}\right)\boldsymbol{u}\big)=-\,\overline{\boldsymbol{\omega}}_\text{\tiny
EL}\cdot\nabla \phi
\,,$$ which is obtained by using equations and . Integrating equation over the fluid volume yields $$\label{h-equation-int}
\frac{d}{dt}\mathcal{H}_\text{\tiny EL}=-\varoiint_S\, \phi \
\overline{\boldsymbol{\omega}}_\text{\tiny
EL}\cdot\mathrm{d}\mathbf{S}-\varoiint_S\left(\boldsymbol{\mathcal{C}}_\text{\tiny
EL}\cdot\overline{\boldsymbol{\omega}}_\text{\tiny
EL}\right)\boldsymbol{u}\cdot\mathrm{d}\mathbf{S}$$ where $S$ is the surface determined by the fluid boundary. Consequently, the right hand side of equation vanishes when $\overline{\boldsymbol{\omega}}_\text{\tiny EL}$ and $\boldsymbol{u}$ are both tangent to the boundary, thereby producing conservation of $\mathcal{H}_\text{\tiny EL}$. Remarkably, one can show that helicity conservation persists for *any* free energy $F(\mathbf{n},\nabla\mathbf{n})$, that is the helicity $\mathcal{H}_\text{\tiny EL}$ is a Casimir for EL dynamics, see [@GayBalmazTronci]. At this point, a question about boundary conditions arises: while the condition of velocity tangent to the boundary is the usual condition in hydrodynamics, the condition $$\label{b-cond}
\nabla\times\left(\boldsymbol{u}+\boldsymbol{\sigma}_\text{\tiny
EL}\cdot\mathbf{n}\times\nabla\mathbf{n}\right)\cdot\mathrm{d}\mathbf{S}=0$$ emerges here for the first time. Upon denoting $\boldsymbol\pi=\boldsymbol\sigma_\text{\tiny EL}\times\mathbf{n}$, one has ${\nabla\times\left(\boldsymbol{u}+\boldsymbol{\sigma}_\text{\tiny
EL}\cdot\mathbf{n}\times\nabla\mathbf{n}\right)}={\boldsymbol\omega+\nabla\pi_a\times\nabla n_a}$, so that the boundary condition reads as ${\nabla\pi_a\times\nabla n_a\cdot\mathrm{d}\mathbf{S}}={-\,\boldsymbol\omega\cdot\mathrm{d}\mathbf{S}}$. This relation evidently differs from the usual “anchoring” boundary conditions (see e.g. [@Stewart]) that are widely used in the literature and for which the director alignment at the surface is insensitive to the flow. Indeed, the physical relevance of the boundary condition resides in the fact that it involves *both* fluid and field variables, contrarily to other commonly available boundary conditions. The complete physical justification of , however, requires more study in the future.
One of the most relevant consequences of equation is the existence of singular vortex-like configurations in Ericksen-Leslie nematodynamics. In two dimensions, equation has the usual point vortex solution $$\overline{\omega}_\text{\tiny
EL}(x,y,t)=\sum_{i=1}^Nw_i\,\delta(x-X_i(t))\,\delta(y-Y_i(t)) \,,$$ where $(X_i,Y_i)$ are canonically conjugate variables with respect to the Hamiltonian $\psi=\sum(\Delta^{-1}\omega)(X_i,Y_i)$. Here $\psi$ is the potential of the velocity $\boldsymbol{u}$, satisfying EL equation . Upon using relation and suppressing the EL label for convenience, one expresses the Hamiltonian as $$\begin{aligned}
\nonumber
\psi(X_i,Y_i,\boldsymbol{\sigma},\mathbf{n})=
&-\frac1{4\pi}\sum_{h}\left(\sum_kw_h\,w_k\log|(X_h-X_k,Y_h-Y_k)|\right.
\\
&+\left.w_h\!\int\!\big\{\sigma_a,\gamma^a\big\}(x',y')\,\log|(x'-X_h,y'-Y_h)|\,\mathrm{d}x'\,\mathrm{d}y'\right)
,\end{aligned}$$ where $|(x,y)|=\sqrt{x^2+y^2}$ and $\{\cdot,\cdot\}$ denotes the canonical Poisson bracket in $(x,y)$ coordinates, arising from the 2D relation $
{\nabla\times\left(\boldsymbol{\sigma}\cdot{\gamma}\right)}
=
{\nabla\times\left(\nabla\mathbf{n}\cdot\boldsymbol{\pi}\right)}
=
{\big\{\pi_a,\,{n}_{a}\big\}}
$, where $\boldsymbol\pi= \boldsymbol\sigma\times\mathbf{n}$.
Other vortex-like structures are also allowed by equation , e.g. vortex cores and patches. In three dimensions, the vortex filament $$\overline{\boldsymbol{\omega}}_\text{\tiny
EL}(\mathbf{x},t)=\int\frac{\partial\mathbf{R}(s,t)}{\partial s}\
\delta(\mathbf{x}-\mathbf{R}(s,t))\ \mathrm{d}s$$ is also a solution of , with $\partial_t\mathbf{R}=\boldsymbol{u}(\mathbf{R},t)$. The existence of these vortex structures (including vortex sheets) rise natural stability questions concerning possible equilibrium vortex configurations. Instead of pursuing this direction, which will be the subject of our future work, the next sections will show how all the above observations also hold in the LdG theory and for fluid molecules of arbitrary shape.
To conclude this section, we emphasize that in all the above discussion the velocity and the structure fields are strongly coupled together. Indeed, the singular vortex structures only exist for the vorticity $\overline{\boldsymbol{\omega}}_\text{\tiny
EL}=\nabla\times\boldsymbol{\mathcal{C}}_\text{\tiny EL}$, while there is no way for the ordinary vorticity $\boldsymbol{\omega}=\nabla\times\boldsymbol{u}$ or the ‘director vorticity’ $\nabla\times(\boldsymbol{\sigma}_\text{\tiny
EL}\cdot\boldsymbol{\gamma}_\text{\tiny EL})$ to be singular. This strong interplay between the macro- and micromotion is the same that emerges in many of the experiments and simulations reviewed in [@ReyDenn02].
The alignment tensor
====================
In the preceding section, we investigated the hydrodynamics of a uniaxial nematic liquid crystal. At this point, it is natural to argue that in the presence of disclinations the molecules can change the configuration of their order parameter (e.g. from uniaxial to biaxial) and the EL equations cannot be used as a faithful model, which is rather given by the LdG theory based on the alignment tensor $\sf Q$. Several dynamical fluid models for the evolution of the alignment tensor $\mathsf{Q}$ are found in the literature [@BeEd1994; @Lubensky2003; @QiSh1998]. In this paper we shall show how the ideal Qiang-Sheng (QS) model [@QiSh1998] for the LdG tensor order parameter also allows for helicity conservation, in analogy to EL theory.
The ideal QS model reads as $$\begin{aligned}
\label{QS1}
&\partial_t \boldsymbol{u}+\boldsymbol{u}\!\cdot\!\nabla
\boldsymbol{u} = -\partial_l\!\left(\frac{\partial
\mathcal{F}}{\partial\mathsf{Q}_{ij\,,l}}\nabla\mathsf{Q}_{ij}\right)-\nabla
p\,,\ \ \nabla\cdot\boldsymbol{u} =0
\\ \label{QS2}
&\partial_t\mathsf{Q}+\boldsymbol{u} \cdot\!\nabla
\mathsf{Q}=J^{-1}\, \mathsf{P}
\\
&\partial_t\mathsf{P}+\boldsymbol{u} \cdot\!\nabla\mathsf{P}=
-\frac{\partial\mathcal{F}}{\partial
\mathsf{Q}}+\partial_i\frac{\partial\mathcal{F}}{\partial
\mathsf{Q}_{,i}}-\lambda \mathbf{I}
\label{QS3}\end{aligned}$$ where $\mathsf{P}$ is conjugate to $\mathsf{Q}$ and $\mathbf{I}$ is the identity matrix, while $\lambda$ is a Lagrange multiplier arising from the condition $\operatorname{Tr}\mathsf{Q}=0$. Here the free energy $\mathcal{F}(\mathsf{Q},\nabla\mathsf{Q})$ contains the Landau-de Gennes free energy [@deGePr1993] as well as interaction terms with external fields. Notice that the molecular field ${\partial\mathcal{F}/\partial
\mathsf{Q}}-{\partial_i(\partial\mathcal{F}/\partial \mathsf{Q}_{,i})}$ is always symmetric, so that $\mathsf{P}$ is also symmetric at all times.
The circulation vector $\boldsymbol{\mathcal{C}}_\text{\tiny
QS}$ for the above system is defined by $$\boldsymbol{\mathcal{C}}_\text{\tiny
QS}:=\boldsymbol{u}+\mathsf{P}_{ij}\nabla\mathsf{Q}_{ij} \,.$$ Indeed, a direct verification shows that the above vector satisfies equation , that is (cf appendix \[appendix2\]) $$\label{Ceq-Qtensor}
\partial_t\boldsymbol{\mathcal{C}}_\text{\tiny QS}+\nabla\left(\boldsymbol{u}\cdot\boldsymbol{\mathcal{C}}_\text{\tiny QS}\right)-\boldsymbol{u}\times\nabla\times\boldsymbol{\mathcal{C}}_\text{\tiny QS}
=-\nabla \phi
\,,$$ with $$\phi=p+\mathcal{F}-\frac12\left|\boldsymbol{u}\right|^2-\frac1{2J}\mathsf{P}_{ij}\mathsf{P}_{ij}
\,.$$ Thus, the Euler-like equation $$\label{LdGEuler}
\partial_t\overline{\boldsymbol{\omega}}_\text{\tiny QS}+\nabla\times\left(\boldsymbol{u}\times\overline{\boldsymbol{\omega}}_\text{\tiny QS}\right)=0$$ holds for the modified vorticity $\overline{\boldsymbol{\omega}}_\text{\tiny
QS}=\nabla\times\boldsymbol{\mathcal{C}}_\text{\tiny QS}$. The circulation theorem $$\frac{d}{dt}\oint_{\Gamma(t)}
\!\left(\boldsymbol{u}+\mathsf{P}_{ij}\nabla\mathsf{Q}_{ij}\right)\cdot\mathrm{d}\mathbf{x}=0\,,$$ and the helicity conservation (for $\overline{\boldsymbol{\omega}}_\text{\tiny QS}$ and $\boldsymbol{u}$ both tangent to the boundary) $$\frac{d}{dt}\int\boldsymbol{\mathcal{C}}_\text{\tiny
QS}\cdot\overline{\boldsymbol{\omega}}_\text{\tiny QS}\
\mathrm{d}^3\mathbf{x}=0$$ are a natural consequence of the Euler-like equation for the QS model of LdG theory. Again Ertel’s commutation relation for $\overline{\boldsymbol{\omega}}_\text{\tiny QS}$ follows easily from equation . Moreover, vortex structures similar to those appearing in EL theory also exist in the LdG formulation.
At this point, one can ask about other LdG formulations and in particular one wonders whether the latter also exhibit vortex structures and conservation of hydrodynamic helicity. Among the LdG formulations of liquid crystal dynamics, the one by Beris and Edwards [@BeEd1994] is probably among the most common, although it is not known to possess helicity conservation. In particular, this theory treats the order parameter as a “conformation tensor field”, so that the symmetric matrix $\sf Q$ is replaced by a symmetric tensor field on physical space. This deep geometric difference is probably responsible for the absence of the hydrodynamic helicity in the Beris-Edwards formulation. In this sense, the peculiarity of the QS model for the LdG tensor dynamics resides in exhibiting helicity conservation and its associated vorticity dynamics. These quantities both involve coupling between velocity and structure fields. The next section shows how this is actually a situation common to all fluid systems exhibiting rotational symmetry breaking.
Completely broken symmetries
============================
The tensor order parameter $\sf Q$ arises as usual from the broken rotational symmetry that is typical of liquid crystal materials. When this rotational symmetry is fully broken, one needs to account for the dynamics of the whole particle orientation, which is determined by an orthogonal matrix $O$ (such that $O^{-1}=O^T$). This is a situation occurring, for example, in spin glass dynamics [@Fischer; @DzVo; @IsKoPe]. Eringen’s wryness tensor (here denoted by $\boldsymbol{\kappa}$) is written in terms of $O$ as [@Eringen] $$\label{NewWryness}
\kappa^s_i=\frac12\,\varepsilon^{mns} \,O_{mk}\, \partial_i O_{nk}\,
=\frac12\,\varepsilon^{mns} \, \partial_i O_{nk}\, O_{km}^{-1}
\,,\quad\text{ (sum over repeated indexes})$$ (In this section we suppress labels such as EL or QS, in order to better adapt to the tensor index notation.) Although orthogonal matrices are difficult to work with analytically and the use of quaternions could be preferable, the correspondence between quaternions and rotation matrices is not unique. Thus, following Eringen’s work [@Eringen], we identify molecule orientations with orthogonal matrices.
It is the purpose of this section to show how equation is not peculiar of nematic liquid crystals. Rather, equations of this form are peculiar of all systems with broken rotational symmetries. In particular, equation also holds in the case of complete symmetry breaking, for a vorticity variable $\overline{\boldsymbol\omega}$ depending on the fluid velocity $\boldsymbol{u}$, on the full particle orientation $O$ and on the angular momentum vector $$\label{NewSigma}
\sigma_r=\varepsilon_{rmn} \,O_{mk}\, \Psi_{nk}\, =\varepsilon_{rmn}
\, \Psi_{nk}\, O_{km} ^{-1}$$ where $\Psi$ is the variable conjugate to $O$. In the well-known spin glass theory of Halperin and Saslow [@HaSa], the rotation matrix is small and thus it is replaced by its infinitesimal rotation angle $\boldsymbol\theta$, where $\exp(\boldsymbol\theta)=O$. Then, $\boldsymbol\theta$ and $\boldsymbol\sigma$ become canonically conjugate variables, as shown in [@Fischer]. Here we consider the whole matrix $O$ to account for arbitrary rotations.
In the case of fully broken symmetry, the (incompressible) equations of motion can be written for an arbitrary energy density $\mathcal{E}(\boldsymbol{\sigma},O)$ as [@IsKoPe] $$\begin{aligned}
\label{SG1}
&\partial_t \boldsymbol{u}+\boldsymbol{u}\!\cdot\!\nabla
\boldsymbol{u} =-
\sigma_r\nabla\frac{\partial\mathcal{E}}{\partial\sigma_r}+\frac{\partial\mathcal{E}}{\partial
O_{mn}}\nabla O_{mn}
-\nabla p
\\ \label{SG2}
&\partial_t{\sigma}_r+\boldsymbol{u}
\cdot\!\nabla\sigma_r=\varepsilon_{r\!ji}\left(\sigma_i\frac{\partial\mathcal{E}}{\partial
\sigma_j} - O_{ih}\frac{\partial\mathcal{E}}{\partial O_{jh}}\right)
\\
&\partial_tO_{mn}+\boldsymbol{u} \cdot\!\nabla
O_{mn}=-\varepsilon_{mkj}\frac{\partial\mathcal{E}}{\partial
\sigma_j}O_{kn} \label{SG3}\end{aligned}$$ Notice that the validity of the above set of equations is completely general. Indeed, the above system is derived in [@IsKoPe] in a general fashion, under the only hypothesis that the broken symmetry group is $SO(3)$. More general broken symmetries can be certainly treated in the same way, although this paper focuses only on rotational symmetries.
In this context, equation generalizes immediately by considering the wryness tensor in . Thus, the new circulation vector is defined by $$\label{NewC}
\boldsymbol{\mathcal{C}}:=\boldsymbol{u}+\boldsymbol{\sigma}\cdot{\kappa}=\boldsymbol{u}+\frac12\,\varepsilon_{mns}\,\sigma_s\,
O_{mk}\nabla O_{nk}\,,$$ where $\boldsymbol{u}$, $\boldsymbol{\sigma}$ and $O$ satisfy equations , and .
At this point it is natural to ask whether the new $\boldsymbol{\mathcal{C}}$ satisfies equation . Remarkably, a positive answer again arises from a direct calculation by using the ordinary properties of the Levi-Civita symbol. One obtains (see appendix \[appendix3\]) $$\label{Ceq-Omodel}
\partial_t\boldsymbol{\mathcal{C}}+\nabla\left(\boldsymbol{u}\cdot\boldsymbol{\mathcal{C}}\right)-\boldsymbol{u}\times\nabla\times\boldsymbol{\mathcal{C}}
=-\nabla \left(p-\frac12\left|\boldsymbol{u}\right|^2\right) \,.$$ Consequently, the Euler-like equation holds also in this case when the rotational symmetry is completely broken. Explicitly one writes $$\label{NewCequation}
\partial_t (\nabla\times\boldsymbol{\mathcal{C}})+
\nabla\times\big(\boldsymbol{u}\times\nabla\times\boldsymbol{\mathcal{C}}\big)=0
\,.$$ In turn, equations and imply the circulation law $$\label{NewCirculation}
\frac{d}{dt}\oint_{\Gamma(t)}
\!\left(\boldsymbol{u}+\frac12\,\varepsilon_{mns}\,\sigma_s\,
O_{mk}\nabla O_{nk}\right)\cdot\mathrm{d}\mathbf{x}=0\,,$$ and the helicity conservation $$\frac{d}{dt}\int_V\boldsymbol{\mathcal{C}}\cdot\overline{\boldsymbol{\omega}}\
\mathrm{d}^3\mathbf{x}=0\,,$$ with $\overline{\boldsymbol{\omega}}=\nabla\times\boldsymbol{\mathcal{C}}$. Thus, the existence of vortex configurations is independent of the type of symmetry breaking characterizing the fluid. Therefore, such vortices may exist in liquid crystals independently of the choice of order parameter. However, one should also emphasize that the energy conserving assumption may fail in several situations and one would then be forced to consider viscosity effects. Moreover, polymeric liquid crystals do not seem to fit easily into the present framework; rather their description requires other liquid crystal theories such as the celebrated Doi theory [@DoEd1988].
Geometric origin of the helicity invariant
==========================================
In the previous sections, the helicity and vorticity of various systems with broken symmetry have been presented. However, the explicit formulation of these results still lack some more justification that can be found in the deep geometric nature of these systems, as it was emphasized in [@GayBalmazTronci]. This section aims to give a brief overview of the geometric setting of the liquid crystal equations that eventually leads to the explicit formulation of their helicity and vorticity. This will show how these quantities can be found without any of the calculation presented in the Appendix, by simply relying on geometric symmetry concepts. The reader is also addressed to [@MaRa; @Ho2002].
As our starting point, we write the total Poisson bracket for a general (incompressible) fluid system with broken symmetry, involving an order parameter space $M$. In this case, the dynamical variables consist of the fluid momentum $\mathbf{m(x)}$, the order parameter state $\mathcal{Q}(\mathbf{x})\in M$ and its conjugate variable $\mathcal{P}(\mathbf{x})$, so that $(\mathcal{Q}(\mathbf{x}),\mathcal{P}(\mathbf{x}))\in T^*M$. For simplicity, we restrict to consider the case when $M$ is a matrix vector space. The total Poisson bracket reads as $$\begin{aligned}
\{F,G\}=&\int\mathbf{m}\cdot\left[\frac{\delta F}{\delta \mathbf{m}},\frac{\delta G}{\delta \mathbf{m}}\right]\mathrm{d}^3x+\int\operatorname{Tr}\!\left(\left(\frac{\delta F}{\delta \mathcal{Q}}\right)^{\!T}\frac{\delta G}{\delta \mathcal{P}}-\left(\frac{\delta F}{\delta \mathcal{P}}\right)^{\!T}\frac{\delta G}{\delta \mathcal{Q}}\right)\mathrm{d}^3x
\nonumber
\\
&
+
\left\langle \frac{\delta
F}{\delta (\mathcal{Q},\mathcal{P})}\,,\pounds_{\frac{\delta G}{\delta \mathbf{m}}} \left(\mathcal{Q},\mathcal{P}\right) \right\rangle
-
\left\langle\frac{\delta
G}{\delta (\mathcal{Q},\mathcal{P})}\,, \pounds_{\frac{\delta F}{\delta \mathbf{m}}}\left(\mathcal{Q},\mathcal{P}\right) \right\rangle,
\label{PB-general}\end{aligned}$$ where the angle bracket denotes the pairing $$\begin{aligned}
\left\langle\frac{\delta
G}{\delta (\mathcal{Q},\mathcal{P})}\,, \pounds_{\frac{\delta F}{\delta \mathbf{m}}}\left(O,\mathcal{P}\right) \right\rangle=
\int\operatorname{Tr}\!\left(\left(\frac{\delta
G}{\delta \mathcal{Q}}\right)^{\!T\!}\pounds_{\frac{\delta F}{\delta \mathbf{m}}}\mathcal{Q}+\left(\frac{\delta
G}{\delta \mathcal{P}}\right)^{\!T\!}\pounds_{\frac{\delta F}{\delta \mathbf{m}}}\mathcal{P}\right).\end{aligned}$$ The above bracket is derived from the relabeling symmetry that characterizes all fluid systems. In particular, this bracket characterizes all Hamiltonian fluid systems with broken symmetry. The relabeling symmetry carried by the fluid emerges mathematically as an invariance property of the Hamiltonian functional $\mathscr{H}:T^{*\!}\operatorname{Diff}(\Bbb{R}^3)\times T^*C^\infty(\Bbb{R}^3,M)\to\Bbb{R}$ under the diffeomorphism group $\operatorname{Diff}(\Bbb{R}^3)$ of smooth invertible maps. Here the notation $C^\infty(\Bbb{R}^3,M)$ stands for the space of $M$-valued scalar functions, i.e. the space of order parameter fields. The reduction process induces a reduced Hamiltonian ${H:\mathfrak{X}^*(\Bbb{R}^3)\times T^*C^\infty(\Bbb{R}^3,M)\to\Bbb{R}}$, where $\mathfrak{X}^*(\Bbb{R}^3)$ denotes the space of differential one-forms, i.e. the space of fluid momentum vectors $\mathbf{m(x)}$. This process leading to the reduced Hamiltonian $H=H(\mathbf{m},\mathcal{Q},\mathcal{P})$ is widely explained in [@MaRa; @Ho2002; @KrMa1987].
Each term in the above Poisson bracket possesses a precise geometric meaning. While the first term coincides with the Poisson bracket for ordinary fluids, the second term is the canonical bracket for the order parameter field $\mathcal{Q}(\mathbf{x})$ and its conjugated momentum $\mathcal{P}(\mathbf{x})$. Moreover, the whole second line contains the two terms arising from the action of the relabeling symmetry group $\operatorname{Diff}(\Bbb{R}^3)$ on the canonical order parameter variables $(\mathcal{Q}(\mathbf{x}),\mathcal{P}(\mathbf{x}))$. Poisson brackets of this form were applied in different contexts, from electromagnetic charged fluids [@Ho1986; @Ho1987] to superfluid dynamics [@HoKu1982], [and even to superfluid plasmas [@HoKu1987].]{}
At this point, upon following the Hamiltonian version of Noether’s theorem (see [@MaRa]), one can construct the total momentum $$\boldsymbol{\mathcal{C}}=\mathbf{m}+\mathbf{J}(\mathcal{Q},\mathcal{P})$$ where $\mathbf{J}(\mathcal{Q},\mathcal{P})$ is the (cotangent-lift) momentum map of components $${J}_i(\mathcal{Q},\mathcal{P})=\operatorname{Tr}\!\left(\mathcal{P}^T\partial_i\mathcal{Q}\right).$$ The geometric meaning of this momentum shift by a momentum map is best explained in [@KrMa1987]. In Lie derivative notation, the dynamics of the circulation quantity reads as $$\label{C-general}
\left(\frac{\partial}{\partial t}+\pounds_{\textstyle\frac{\delta H}{\delta \mathbf{m}}}\right)\boldsymbol{\mathcal{C}}=-\nabla\phi$$ thereby yielding Noether’s conservation relation $$\frac{d}{dt}\oint_{\Gamma(t)} \boldsymbol{\mathcal{C}}\cdot\operatorname{d}\mathbf{x}=0$$ which then arises naturally as the circulation conservation determined by the relabeling symmetry of the system. The explicit proof of circulation theorems of this kind can be found in many works in geometric fluid dynamics; see [@HoMaRa1998] for a modern reference. After recalling that for incompressible flows $\mathbf{m}=\boldsymbol{u}$, it is easy to recognize that replacing $M$ by the space $\operatorname{Sym}_0(3)$ of traceless symmetric matrices transforms the relation exactly into the relation , which then produces the results in Section 3. Moreover, the corresponding vorticity relation for $\overline{\boldsymbol\omega}=\mathbf{d} \boldsymbol{\mathcal{C}}$ is easily obtained by taking the exterior differential of equation and recalling that this operation commutes with Lie derivative. Then, one obtains $\left(\partial_t+\pounds_{\boldsymbol{u}}\right)\mathbf{d} \boldsymbol{\mathcal{C}}=0$. The form of the helicity is also easily derived from the above arguments, upon recalling an old result in [@KrMa1987]. In particular, if $\mathcal{H}({\bf m})$ denotes ordinary Euler’s helicity, then $\mathcal{H}(\boldsymbol{\mathcal{C}})$ is a Casimir invariant of the Poisson bracket . Notice that all the above relations hold for an arbitrary manifold $M$ other than a matrix space. This only requires using the appropriate pairing between vectors and co-vectors.
So far, we only used the cotangent-lift momentum map, which can be found for all the cases when the dynamics involve conjugate variables in a cotangent bundle $T^*M$. However, this does not appear to be the case for the discussion in Section 4, where $M=SO(3)$ and $\mathcal{Q}=O$. This apparent contradiction is easily solved by noticing that $$\operatorname{Tr}\!\left(\mathcal{P}^T\,\partial_iO\right)
=
\operatorname{Tr}\!\left((\mathcal{P}O^{-1})^T\partial_iO\,O^{-1}\right).$$ Then, upon denoting $\hat{\sigma}=\mathcal{P}O^{-1}$ and $\hat{\kappa_i}=\partial_iO\,O^{-1}$, the usual isomorphism between antisymmetric matrices in the Lie algebra $\mathfrak{so}(3)$ and vectors in $\mathbb{R}^3$ yields the term $\boldsymbol\sigma\cdot\kappa$ in the circulation quantity . Then, upon repeating the same steps as above, the momentum map $\operatorname{Tr}\!\left(\mathcal{P}^T\partial_iO\right)=\boldsymbol\sigma\cdot\kappa$ returns exactly the same results as in Section 4.
The case of nematic liquid crystals treated in Section 2 can be also obtained by a direct computation, upon setting $M=S^2/\Bbb{Z}_2$, which is the director space. Upon denoting $\boldsymbol{\pi}=JD_t{\mathbf{n}}$ the corresponding conjugate variable, it is easy to see that $\nabla\mathbf{n}\cdot\boldsymbol{\pi}=\boldsymbol\sigma\cdot\gamma_\text{\tiny
EL}$. However, the geometric meaning of this simple step requires more basis that can be found in [@GayBalmazTronci], where this last relation is justified by Lagrangian reduction.
At this point, it is clear that the above arguments ensure the results in this paper without any need for further discussion. Nevertheless, the Appendix gives explicit proofs that can be followed without previous knowledge in geometric mechanics.
Conclusions
===========
This paper provided explicit expressions for the helicity conservation in liquid crystals, in both EL and LdG theories. This conservation arises from an Euler-like equation that allows for singular vortex structures in any dimension. Some of the ideal fluid properties were extended to liquid crystal flows, e.g. Ertel’s commutation relation. These results were also shown to hold for molecules of arbitrary shapes, by considering fully broken rotational symmetries occurring in some spin glass dynamics. All of the results were eventually justified by geometric symmetry arguments.
The energy-Casimir method can then be applied to study nonlinear stability properties of these systems, see [@HoMaRaWe1985] for several examples of how this method applies to many types of fluids. This can be used, for example, to explore the coupled macro- and micro-motion of the stationary (generalized Beltrami) solutions. While the 3D stability analysis is limited by the fact that the helicity is the only Casimir invariant, the 2D case is much reacher because the whole family of Casimir invariants becomes available.
One more remark concerns physically observable effects. More particularly, one wonders how conservation of total circulation causes observable effects. Even more, one would like to observe these effects in a particular experiment. A simple technique that could be used to this purpose is the use of external electric fields that drive the order-parameter variables, thereby generating fluid circulation by conservation of the total circulation. Then, if one applies an external field to a trivial motionless liquid crystal, the director alignment caused by the field would result in the generation of fluid motion.
Other physical questions also arise about the nature of vortex solutions, which evidently represent much more that simply disclinations dragged around by a smooth flow. It is possible that these solutions share many analogies with superfluid vortices in He$^3$-A, whose order parameter space is again the whole group $SO(3)$. However, the nature of these singular solutions is left open for future investigations, together with their stability properties.
#### Acknowledgements.
We are indebted with Darryl Holm for his keen remarks about the relation between the modified vorticity and helicity conservation in Ericksen-Leslie theory. Some of this work was carried out while visiting him at Imperial College London.
Appendix
========
Derivation of equations and \[appendix1\]
-----------------------------------------
Upon using the notation $\pounds_{\boldsymbol{u}}$ for the Lie derivative with respect to the velocity vector field $\boldsymbol{u}$ [@MaRa], we can rewrite equations - as $$\begin{aligned}
\label{EL1-bis}
\left(\frac{\partial}{\partial t}
+\pounds_{\boldsymbol{u}}\right)\boldsymbol{u}
&=\nabla\mathbf{n}\cdot\mathbf{h}
-\nabla\left(p+F-\frac12\left|\mathbf{u}\right|^2\right)
\\
\left(\frac{\partial}{\partial t}
+\pounds_{\boldsymbol{u}}\right)\boldsymbol{\sigma}&=\mathbf{h}\times\mathbf{n}\,,\hspace{.7cm}
\left(\frac{\partial}{\partial t}
+\pounds_{\boldsymbol{u}}\right)\mathbf{n}=J^{-1}\boldsymbol{\sigma}\times\mathbf{n}.
\label{EL2-bis}\end{aligned}$$ Then, one simply calculates $$\begin{aligned}
\left(\frac{\partial}{\partial t}
+\pounds_{\boldsymbol{u}}\right)\boldsymbol{\mathcal{C}}_\text{\tiny
EL}
=&
\left(\frac{\partial}{\partial t}
+\pounds_{\boldsymbol{u}}\right)\big(\boldsymbol{u}+\boldsymbol{\sigma}_\text{\tiny
EL}\cdot\mathbf{n}\times\nabla\mathbf{n}\big)
\\
=&\ \nabla\mathbf{n}\cdot\mathbf{h} -
\nabla\left(p+F-\frac12\left|\mathbf{u}\right|^2\right) +
\mathbf{h}\times\mathbf{n}\cdot\mathbf{n}\times\nabla\mathbf{n}
\\
& - J^{-1}\boldsymbol{\sigma}_\text{\tiny
EL}\cdot\nabla\mathbf{n}\times\left(\boldsymbol{\sigma}_\text{\tiny
EL}\times\mathbf{n}\right) + J^{-1}\boldsymbol{\sigma}_\text{\tiny
EL}\cdot\mathbf{n}\times\left(\nabla\boldsymbol{\sigma}_\text{\tiny
EL}\times\mathbf{n}\right)
\\
&+ J^{-1}\boldsymbol{\sigma}_\text{\tiny
EL}\cdot\mathbf{n}\times\left(\boldsymbol{\sigma}_\text{\tiny
EL}\times\nabla\mathbf{n}\right).\end{aligned}$$ At this point, standard vector identities yield $$\begin{aligned}
\mathbf{h}\times\mathbf{n}\cdot\mathbf{n}\times\nabla\mathbf{n}&=\left(\mathbf{h}\cdot\mathbf{n}\right)\left(\nabla\mathbf{n}\cdot\mathbf{n}\right)-\left(\nabla\mathbf{n}\cdot\mathbf{h}\right)\left(\mathbf{n}\cdot\mathbf{n}\right)
\\
& = -\nabla\mathbf{n}\cdot\mathbf{h}
\\
\nabla\mathbf{n}\times\left(\boldsymbol{\sigma}_\text{\tiny
EL}\times\mathbf{n}\right)&=\left(\nabla\mathbf{n}\cdot\mathbf{n}\right)\boldsymbol{\sigma}_\text{\tiny
EL}-\left(\nabla\mathbf{n}\cdot\boldsymbol{\sigma}_\text{\tiny
EL}\right)\mathbf{n}
\\
&=-\left(\nabla\mathbf{n}\cdot\boldsymbol{\sigma}_\text{\tiny
EL}\right)\mathbf{n}
\\
\mathbf{n}\times\left(\nabla\boldsymbol{\sigma}_\text{\tiny
EL}\times\mathbf{n}\right)&=\left(\mathbf{n}\cdot\mathbf{n}\right)\nabla\boldsymbol{\sigma}_\text{\tiny
EL}-\left(\nabla\boldsymbol{\sigma}_\text{\tiny
EL}\cdot\mathbf{n}\right)\mathbf{n}
\\
& =\nabla\boldsymbol{\sigma}_\text{\tiny
EL}-\left(\nabla\boldsymbol{\sigma}_\text{\tiny
EL}\cdot\mathbf{n}\right)\mathbf{n}
\\
\mathbf{n}\times\left(\boldsymbol{\sigma}_\text{\tiny
EL}\times\nabla\mathbf{n}\right)&=\left(\nabla\mathbf{n}\cdot\mathbf{n}\right)\boldsymbol{\sigma}_\text{\tiny
EL}-\left(\boldsymbol{\sigma}_\text{\tiny
EL}\cdot\mathbf{n}\right)\nabla\mathbf{n}
\\
&= 0\,,\end{aligned}$$ where we have made use of the relations $|\mathbf{n}|^2=1$ and $\boldsymbol{\sigma}_\text{\tiny EL}\cdot\mathbf{n}=0$. Therefore equation follows directly from $$\left(\frac{\partial}{\partial t}
+\pounds_{\boldsymbol{u}}\right)\boldsymbol{\mathcal{C}}_\text{\tiny
EL}=-\nabla\left(p+F-\frac12\left|\mathbf{u}\right|^2-\frac1{2J}|\boldsymbol{\sigma}_\text{\tiny
EL}|^2\right) \,.$$ The equation follows by taking the curl of the above equation, upon recalling that the curl is given by the exterior differential, so that $\mathbf{d}\!\left(\boldsymbol{\mathcal{C}}_\text{\tiny
EL}\cdot\mathrm{d}\mathbf{x}\right)=(\nabla\times\boldsymbol{\mathcal{C}}_\text{\tiny
EL})\cdot\mathrm{d}\mathbf{S}$. Since the differential always commutes with the Lie derivative [@MaRa], equation follows immediately. It is also easy to see that equation arises by calculating $
\left(\partial_t
+\pounds_{\boldsymbol{u}}\right)\left(\boldsymbol{\mathcal{C}}_\text{\tiny
EL}\cdot\overline{\boldsymbol{\omega}}_\text{\tiny EL}\right)
=\overline{\boldsymbol{\omega}}_\text{\tiny EL}\cdot\nabla\phi$.
Derivation of equations and \[appendix2\]
-----------------------------------------
Upon introducing the Lie derivative notation, the equations -- read as $$\begin{aligned}
\label{QS1-bis}
&\left(\frac{\partial}{\partial t} +\pounds_{\boldsymbol{u}}\right)
\boldsymbol{u} = \mathsf{h}_{ij}\nabla\mathsf{Q}_{ij}-\nabla\left(
p+\mathcal{F} -\frac12|\boldsymbol{u}|^2\right)\,,\ \ \nabla\cdot\boldsymbol{u}
=0
\\ \label{QS2-bis}
&\left(\frac{\partial}{\partial t} +\pounds_{\boldsymbol{u}}\right)
\mathsf{Q}=J^{-1}\, \mathsf{P}
\\
&\left(\frac{\partial}{\partial t}
+\pounds_{\boldsymbol{u}}\right)\mathsf{P}=-\mathsf{h} -\lambda
\mathbf{I}
\label{QS3-bis}\end{aligned}$$ where we have denoted the molecular field by $$\mathsf{h}=\frac{\partial\mathcal{F}}{\partial
\mathsf{Q}}-\partial_i\frac{\partial\mathcal{F}}{\partial
\mathsf{Q}_{,i}} \,.$$ Then, one simply calculates $$\begin{gathered}
\left(\frac{\partial}{\partial t}
+\pounds_{\boldsymbol{u}}\right)\boldsymbol{\mathcal{C}}_\text{\tiny
QS} = \left(\frac{\partial}{\partial t}
+\pounds_{\boldsymbol{u}}\right)\Big(\boldsymbol{u}+\mathsf{P}_{ij}\nabla\mathsf{Q}_{ij}\Big)
\\
=\mathsf{h}_{ij}\nabla\mathsf{Q}_{ij}-\nabla\left(
p+\mathcal{F} -\frac12|\boldsymbol{u}|^2\right)-\mathsf{h}_{ij}\nabla\mathsf{Q}_{ij}-\lambda\,\delta_{ij}\nabla\mathsf{Q}_{ij}+\frac{1}{J}\mathsf{P}_{ij}
\nabla \mathsf{P}_{ij} \,,\end{gathered}$$ which becomes $$\left(\frac{\partial}{\partial t}
+\pounds_{\boldsymbol{u}}\right)\boldsymbol{\mathcal{C}}_\text{\tiny
QS} =
-\nabla\!\left(p+\mathcal{F}+\lambda\,\delta_{ij\,}\mathsf{Q}_{ij}-\frac12|\boldsymbol{u}|^2-\frac{1}{2J}\mathsf{P}_{ij}\mathsf{P}_{ij}\right)
\,.$$ Finally, taking the curl of the above equation returns .
Derivation of equation \[appendix3\]
------------------------------------
Upon using the Lie derivative notation, equations -- may be written as $$\begin{aligned}
\label{SG1-bis}
&\left(\frac{\partial}{\partial t} +\pounds_{\boldsymbol{u}}\right)
\boldsymbol{u} =-
\sigma_r\nabla\frac{\partial\mathcal{E}}{\partial\sigma_r}+\frac{\partial\mathcal{E}}{\partial
O_{mn}}\nabla O_{mn}
-\nabla\!\left(p-\frac12\left|\boldsymbol{u}\right|^2\right)
\\ \label{SG2-bis}
&\left(\frac{\partial}{\partial t}
+\pounds_{\boldsymbol{u}}\right)\sigma_r=\varepsilon_{r\!ji}\left(\sigma_i\frac{\partial\mathcal{E}}{\partial
\sigma_j} - O_{ih}\frac{\partial\mathcal{E}}{\partial O_{jh}}\right)
\\
&\left(\frac{\partial}{\partial t} +\pounds_{\boldsymbol{u}}\right)
O_{mn}=-\varepsilon_{mkj}\frac{\partial\mathcal{E}}{\partial
\sigma_j}O_{kn} \label{SG3-bis}\end{aligned}$$ so that, upon denoting $\phi=p-|\boldsymbol{u}|^2/2$, one computes $$\begin{aligned}
\left(\frac{\partial}{\partial t}
+\pounds_{\boldsymbol{u}}\right)\boldsymbol{\mathcal{C}}
= & \left(\frac{\partial}{\partial t}
+\pounds_{\boldsymbol{u}}\right)\left(\boldsymbol{u}+\frac12\,\varepsilon_{mns}\,\sigma_s\,
O_{mk}\nabla O_{nk}\right)
\\
=&-
\sigma_r\nabla\frac{\partial\mathcal{E}}{\partial\sigma_r}+\frac{\partial\mathcal{E}}{\partial
O_{mn}}\nabla O_{mn}
-\nabla\phi
\\
&+
\frac12\varepsilon_{mns}\left(\varepsilon_{sji}\,\sigma_i\frac{\partial\mathcal{E}}{\partial
\sigma_j}-\varepsilon_{sji}\,O_{ih}\frac{\partial\mathcal{E}}{\partial
O_{jh}}\right)O_{mk}\nabla O_{nk}
\\
&-
\frac12\varepsilon_{mns}\,\sigma_s\,\varepsilon_{mhj}\,\frac{\partial\mathcal{E}}{\partial
\sigma_j}O_{hk}\nabla O_{nk}
\\
&-
\frac12\varepsilon_{mns}\,\sigma_s\,O_{mk}\,\varepsilon_{nhj}\left(\nabla\frac{\partial\mathcal{E}}{\partial
\sigma_j}\,O_{hk}+\frac{\partial\mathcal{E}}{\partial
\sigma_j}\nabla O_{hk}\right) .\end{aligned}$$ At this point we observe that, since $O_{mk}O_{hk}=O_{mk}O_{kh}^{-1}=\delta_{mh}$, then $$\begin{aligned}
-
\sigma_r\nabla\frac{\partial\mathcal{E}}{\partial\sigma_r}-\frac12\,\varepsilon_{mns}\,\sigma_s\,O_{mk}\,\varepsilon_{nhj}\nabla\frac{\partial\mathcal{E}}{\partial
\sigma_j}\,O_{hk}=& -
\sigma_r\nabla\frac{\partial\mathcal{E}}{\partial\sigma_r}-\frac12\,\varepsilon_{hns}\,\varepsilon_{nhj}\,\sigma_s\nabla\frac{\partial\mathcal{E}}{\partial
\sigma_j}
\\
&= -
\sigma_r\nabla\frac{\partial\mathcal{E}}{\partial\sigma_r}+\delta_{sj}\,\sigma_s\nabla\frac{\partial\mathcal{E}}{\partial
\sigma_j}=0.\end{aligned}$$ Moreover, the sum of all terms in $\partial\mathcal{E}/\partial\boldsymbol{\sigma}$ can be written as $$\begin{aligned}
&\sigma_s\,\frac{\partial\mathcal{E}}{\partial\sigma_j}\left(\varepsilon_{ijs}\,\varepsilon_{mni}\,O_{mk}\nabla
O_{nk}-\varepsilon_{ims}\,\varepsilon_{inj}\,O_{nk}\nabla
O_{mk}-\varepsilon_{mis}\,\varepsilon_{inj}\nabla
O_{nk}\,O_{mk}\right)
\\
= &\
-\sigma_s\,\frac{\partial\mathcal{E}}{\partial\sigma_j}\,\varepsilon_{ijs}\,\varepsilon_{imn}\,O_{mk}\nabla
O_{nk}
\\
=& \
-\sigma_s\,\frac{\partial\mathcal{E}}{\partial\sigma_j}\left(\delta_{jm}\,\delta_{sn}-\delta_{jn}\,\delta_{sm}\right)O_{mk}\nabla
O_{nk}
\\
=&\
-\sigma_s\,\frac{\partial\mathcal{E}}{\partial\sigma_j}\left(\nabla
O_{sk}\,O_{kj}^{-1}+O_{sk}\nabla O_{kj}^{-1}\right)
\\
=&-\sigma_s\,\frac{\partial\mathcal{E}}{\partial\sigma_j}\,\nabla\!\left(O_{sk}\,O_{kj}^{-1}\right)=0.\end{aligned}$$ In addition, we calculate $$\begin{aligned}
&\frac{\partial\mathcal{E}}{\partial O_{mn}}\,\nabla
O_{mn}-\frac12\,\varepsilon_{sji}\,\varepsilon_{mns}\,O_{ih}\,\frac{\partial\mathcal{E}}{\partial
O_{jh}}\,O_{mk}\,\nabla O_{nk}
\\
=&\frac{\partial\mathcal{E}}{\partial O_{mn}}\,\nabla
O_{mn}+\frac12\left(\delta_{mj}\,\delta_{ni}-\delta_{mi}\,\delta_{nj}\right)O_{ih}\,\frac{\partial\mathcal{E}}{\partial
O_{jh}}\,O_{mk}\,\nabla O_{nk}
\\
=& \frac{\partial\mathcal{E}}{\partial O_{mn}}\,\nabla
O_{mn}+\frac12\left(O_{ih}\,O_{jk}\,\nabla
O_{ik}-O_{ih}\,O_{ik}\,\nabla
O_{jk}\right)\frac{\partial\mathcal{E}}{\partial O_{jh}}
\\
=& \frac{\partial\mathcal{E}}{\partial O_{mn}}\,\nabla
O_{mn}+\frac12\left(O_{hi}^{-1}\,\nabla
O_{ik}\,O_{kj}^{-1}-\delta_{hk}\,\nabla
O_{jk}\right)\frac{\partial\mathcal{E}}{\partial O_{jh}}
\\
=& \frac{\partial\mathcal{E}}{\partial O_{mn}}\,\nabla
O_{mn}-\frac12\left(\nabla O_{hj}^{-1}+\nabla
O_{jh}\right)\frac{\partial\mathcal{E}}{\partial O_{jh}}
\\
=& \frac{\partial\mathcal{E}}{\partial O_{mn}}\,\nabla O_{mn}-\nabla
O_{jh}\frac{\partial\mathcal{E}}{\partial O_{jh}}=0.\end{aligned}$$ Thus, we have proved the relation $$\begin{aligned}
\left(\frac{\partial}{\partial t}
+\pounds_{\boldsymbol{u}}\right)\boldsymbol{\mathcal{C}}=-\nabla\phi
\,,\end{aligned}$$ whose curl yields the corresponding Euler-like equation , thereby recovering the corresponding helicity conservation. Notice that the above result also holds in the case of explicit dependence of the free energy $\mathcal{E}$ on the gradient $\nabla O$ of the orientational order parameter.
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[^1]: [ The same idea has been applied in superfluid plasmas, that is, in superfluid solutions whose charged condensates are coupled electromagnetically [@HoKu1987]. However, ]{} the similarity with gauge theory does not end with the electromagnetic analogy. The equation corresponding to (\[Cequation\]) also follows by inspection for a Yang-Mills fluid plasma (chromohydrodynamics, cf. [@GiHoKu1983]) either from equation (2.35) or (2.49) of [@HoKu1988]. By this observation, chromohydrodynamics acquires a circulation theorem and the theory of complex fluids inherits an analogy with Yang-Mills fluid plasma, first noticed in [@Ho2002]. This minimal coupling argument requires the wryness tensor $\gamma_\text{\tiny EL}$ to be a connection one-form: although this is not the case for the expression $\mathbf{n}\times\nabla\mathbf{n}$, a connection one-form can be obtained by the addition of terms parallel to $\mathbf{n}$. [By good fortune, these extra terms make no contribution in (\[Cdef\]) because $\boldsymbol{\sigma}_\text{\tiny EL}\cdot\mathbf{n}=0$.]{}
|
---
abstract: 'We find that [*singlet*]{} superconducting pairing can lead to Majorana fermions in three dimensional Dirac superconductors (3DDS) if the pairing order parameter is a pseudo-scalar, i.e. it changes sign under mirror reflection. The pseudo-scalar superconducting order parameter, $\Delta_5$ can close and reopen the spectral gap caused by the scalar Dirac mass $m$ in a three-dimensional Dirac material (3DDM), giving rise to a two-dimensional Majorana sea (2DMS) at the plane of the gap kink. By bringing the Hamiltonian into a canonical form which then gives the winding number, we show that this system belongs to the DIII class of topological superconductors. We calculate the transport signature of 2DMS, namely a perfect Andreev-Klein transmission that manifests in a robust peak in the differential conductance. Further, we find the $4\pi$ periodicity in the $\Delta_5|m|\Delta_5$ Josephson junctions. Gauge transformed version of the present scenario implies that the interface of a conventional s-wave superconductor with a peculiar 3DDM whose mass is a pseudo-scalar, $m_5$ also hosts a 2DMS.'
author:
- Morteza Salehi
- 'S. A. Jafari'
title: 'Sea of Majorana fermions from pseudo-scalar superconducting order in three dimensional Dirac materials'
---
*Introduction*.– Band topology in insulators and superconductors is connected with the change in the sign of the gap parameter which in turn gives rise to zero energy states at the location of gap kink. This mechanism in the case of insulators gives rise to gapless surface modes protected by a topological invariant [@BernevigBook; @Zhang2011RMP; @ShenBook2013]. When the spectral gap is of the superconducting (pairing) nature, these topologically protected modes will be Majorana zero modes, which are their own anti-particles [@Alicea; @Beenakker2013rev]. To realize Majorana fermions (MFs) various scenarios have been proposed which involve closing and re-opening the superconducting gap in one way or another. Gaping chiral modes of topological insulators (TIs) by Zeeman field and superconducting pairing gives rise to MFs in the interface region where these two gapping mechanisms are of comparable strength [@Fu2008PRL; @Fu2009PRB]. In one-dimensional nano-wires this can be achieved by the competition between a polarizing Zeeman field, and depolarizing spin-orbit interaction [@Sau; @vonOppen]. Alternative scenario is to generate MFs involve driving supercurrent through a superconductor (SC) next to a TI in a pairing dominated gap regime where twist due to supercurrent reduces the phase space for Cooper pairing and gives way to Zeeman dominated gap [@Romito]. In two-dimensions, the vortex core of a p-wave SC binds a MF [@Stone2004]. The required p-wave SC can be engineered on the surface of a TI [@Fu2008PRL] by proximity to a conventional s-wave SC. The above scenarios are first of all limited to low dimensions, and secondly require a TR breaking by a Zeeman field.
In this letter we propose yet another [*three dimensional*]{} system that admits a two dimensional sheet of MFs without requiring a Zeeman field. We find that a peculiar [*pseudo-scalar*]{} (odd-parity) superconducting order parameter, $\Delta_5$ can give rise to a two-dimensional sheet of MFs when it is interfaced with a three-dimensional Dirac material (3DDM). To set the stages for our finding, let us start by noting that in a one-band situation described by a parabolic band dispersion, the strength of the gap is characterized by a (scalar) gap parameter. However in 3DDM where the relevant degrees of freedom are described by the Dirac equation [@Fuseya2011PRL], new possibilities can arise. We assume a 3DDM which has a single Dirac point in the $\Gamma$ point of it’s Brillouin zone [@Zhang2009NP; @Narayan2014PRL; @Liu2014Science]. For such a 3DDM, the low-energy degrees of freedom are described by 4-component Dirac spinors and hence the most general pairing is $\bar\psi\hat{\Delta}_S\psi_c=\psi^\dagger\gamma^0\hat{\Delta}_S\psi_c$ where $\psi_c$ satisfies the same Dirac equation as $\psi$, but with opposite charge and is covariant under Lorentz transformation. The $4\times 4$ pairing matrix $\hat{\Delta}_S$ can be expanded in terms of a basis composed of $\mathbbm{1}$, four $\gamma^\mu$, $\gamma^5$, four $\gamma^5\gamma^\mu$ and six anti-commutators $\sigma^{\mu\nu}=i[\gamma^\mu,\gamma^\nu]/2$ [@ZeeBookQFT] as $\hat{\Delta}_S=\Delta_0\mathbbm{1}+\Delta_\mu\gamma^\mu+\Delta_5\gamma^5+\Delta_{5\mu}\gamma^5\gamma^\mu+\Delta_{\mu\nu}\sigma^{\mu\nu}$. Then the pairing $\Delta_0$ will be the conventional scalar pairing, while $\Delta_5$ will be a pseudo-scalar pairing under the Lorentz transformation. Similarly $\Delta_\mu,\Delta_{5\mu}, \Delta_{\mu\nu}$, will be vector, pseudo-vector and tensor superconducting pairings. We have examined all the above $16$ pairing channels and find that the pseudo-scalar superconducting gap $\Delta_5$ works in an opposite direction to the normal Dirac gap $m\gamma^0$. Therefore placing a $m$ dominated region (i.e. a 3DDM) next to a $\Delta_5$ dominated region denoted as $m|\Delta_5$, gives rise to a two-dimensional sheet of Majorana fermions. This mechanism, [*does not require p-wave pairing, nor a magnet to break time-reversal*]{}. Instead, it requires a peculiar form of pseudo-scalar superconducting order parameter that breaks the mirror symmetry. We then proceed to show that such a system belongs to the DIII topological class allowing $Z$-number classification which in turn guarantees the existence of two-dimensional Majorana sea (2DMS) on the region where the strength of $m$ and $\Delta_5$ are equal. We further corroborate our results by showing a perfect Andreev reflection and robust zero-mode resonance peak in differential conductance and fractional supercurrent in $\Delta_5|m|\Delta_5$ Josephson junction.
[*Model:*]{} We consider a Dirac material with a single Dirac cone, $$\mathcal{H}(\textbf{k})= v_F \gamma_0\left( \hbar \boldsymbol{\gamma}.\mathbf{k}+ m v_F\right).
\label{Eq.01}$$ We use the representation $\gamma^0=\tau_3 \otimes \sigma_0$ and $\gamma^j=\tau_1 \otimes i \sigma_j$ for the Clifford algebra where $\sigma_j$ and $\tau_j$ are Pauli matrices acting on spin and band spaces, respectively. Also, $\sigma_0$ and $\tau_0$ are the $2 \times2$ unit matrices. The $\textbf{k}$ is the wave vector of excitations. In four space-time dimensions one can also construct $\gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3$ which will be very essential for our discussion in this paper. In Eq.(\[Eq.01\]), the mass term is of the ordinary $m\gamma^0$ nature and is responsible for the band gap, and $v_F$ is the Fermi velocity. The pairing Hamiltonian for such a system is: $$H_{\textit{BCS}}=\frac{1}{2}\int d\textbf{r}\left(\begin{array}{cc}
\psi^\dagger & \psi_c^\dagger
\end{array}\right)H_{\rm DBdG}\left(\begin{array}{c}
\psi \\
\psi_c
\end{array}\right),
\label{Eq.BCS}$$ corresponding to which the Dirac-Bogoliubov-deGennes (DBdG) equation in **k**-space is, $$\left( \begin{array}{cc}
\mathcal{H}(\textbf{k})-\mu & \gamma_0\hat{\Delta}_S e^{i \phi}\\
\hat{\Delta}^\dagger_S\gamma_0 e^{-i \phi} & \mu+\mathcal{C}\mathcal{H}(\textbf{k})\mathcal{C}^{-1} \\
\end{array}\right)
\left( \begin{array}{c}
u \\
v\\
\end{array}\right)= \varepsilon \left(\begin{array}{c}
u \\
v \\
\end{array} \right),
\label{Eq.DBdG}$$ where $\varepsilon$ is the energy of eigenstate with respect to the chemical potential $\mu$. Here $\phi$ is the macroscopic phase of superconductor, $u (v)$ is the electron (hole) part of BdG wave function in Nambu space. The anti-unitary operator $\mathcal{C}=\gamma_2\gamma_0 K$ is the charge-conjugation of Eq.(\[Eq.01\]), where $K$ is the complex-conjugation: This means that Lorentz transformation for $\bar\psi_c=\psi_c^\dagger\gamma^0$ is the inverse of the Lorentz transformation for $\psi$. Therefore requiring the superconducting pairing to be Lorentz invariant [@ZeeBookQFT; @Capelle1999PRB2], the scalar (conventional) superconducting pairing is given by $\bar\psi_c\Delta_s\mathbbm{1}\psi=\psi_c^\dagger\gamma^0\Delta_s\mathbbm{1}\psi\sim v\gamma^0\Delta_s u$. Similarly pseudo-scalar pairing is $\bar\psi_c\Delta_5\gamma^5\psi=\psi_c^\dagger \Delta_5\gamma^0\gamma^5\psi\sim v \Delta_5 \gamma^0\gamma^5 u$, etc. We have examined all of the above 16 possible superconducting pairing channels. We find that only the $\Delta_5$ parameter gives rise to a gap closing when both $m$ and $\Delta_5$ are present. To see this in $\Delta_5$ channel, let us for clarity of notation set $\hbar$ and $v_F$ to $1$ which gives the eigenvalues of Eq. (\[Eq.DBdG\]) as $\varepsilon=\pm\sqrt{A\pm2 \sqrt{B}}$, where $A=k^2+m^2+\mu^2+\Delta_5^2$ and $B=\Delta_5^2 m^2+(m^2+k^2)\mu^2$. When the Dirac mass $m$ dominates the spectral gap ($\Delta_5\to 0$) the above eigenvalues reduce to $ \varepsilon=\pm \mu \pm\sqrt{k^2+\Delta_5^2}$. In this case each eigenvalue is doubly (spin) degenerate [@Supplement]. Deep in the 3DDS region where $\Delta_5$ dominates the eigenvalues acquire the following structure $\varepsilon=\pm\sqrt{(k\pm \mu)^2+\Delta_5^2}$ which corresponds to a pairing gap at the Fermi level. In a region where both $m$ and $\Delta_5$ are non-zero the nature of gap can be more clearly seen if we look at $\mu=0$ where dispersion becomes $\varepsilon=\pm\sqrt{k^2+(m\pm \Delta_5)^2}$ and the band gap is determined by $\Delta_\pm=(m\pm\Delta_5)$ which clearly indicates the competition between the Dirac mass $m$ and the pseudo-scalar superconducting parameter $\Delta_5$. Fig. \[Fig.TopologicalPhaseTransition\] summarizes the closing and reopening of the spectral gap as one moves from $m$ dominated region to $\Delta_5$ dominated region.
![(Color online) (a) Dispersion relation of 3DDM regime. (b) Gap closing and topological phase transition for one band whereas the other one remains trivial. (c) Dispersion relation of 3DDS regime. In these results we set $\mu=0$. The length scale $l_D=\hbar/ m v_F$ is set by the energy gap of the 3DDM.[]{data-label="Fig.TopologicalPhaseTransition"}](TPT.jpg)
The closing of the Dirac gap and re-opening of it in the form of a pseudo-scalar superconducting gap in this system does not require any magnetic field. This implies that when a $\Delta_5$ 3DDS is brought next to a normal 3DDM with gap parameter $m$, such that $\Delta_5>m$, the induced $\Delta_5$ on the normal 3DDM side decays towards the bulk of 3DDM and crosses $m$ somewhere in the interface where the excitations become gapless.
Let us now prove that when $\Delta_5=m$ a two dimensional Majorana sea appears. The pseudo-scalar character of $\Delta_5$ pairing can be interpreted as a [*singlet*]{} superconductor whose mirror image has an opposite sign. To understand further the properties of such a pairing, let us now explicitly construct the anti-unitary operators corresponding to particle-hole, and time-reversal transformation for Eq. : Let $\eta_j$ be set of Pauli matrices in the Nambu space. Then the particle-hole and time-reversal symmetries in this space can be defined as $PH=i\eta_2\otimes\gamma^0\gamma^5\gamma^2 K$ and $TR=\eta_0\otimes \gamma^0\gamma^1\gamma^3K$, respectively. Owing to $PH^2=1$ and $TR^2=-1$, the chiral symmetry $SL=PH*TR$ satisfies $SL^2=1$ which places the present system in the DIII class of topological superconductors which can be classified with winding number. We explicitely obtain the topoligical charge [@Supplement], $$%\mathcal{Q}=\frac{1}{2}\left(\frac{m+\Delta_5}{|m+\Delta_5|}
%-\frac{m-\Delta_5}{|m-\Delta_5|}\right).
{\cal Q}=\Theta(|\Delta_5|-|m|)~\text{sign}(\Delta_5)
\label{Eq.WindingNumber}$$ which clearly shows that ${\cal Q}$ can be $0,\pm 1$, and hence a $Z$ number classification [@Schnyder2008PRB; @Chiu2016RMP].
![(Color online) (a) Schematic illustration of 3DDM$|$3DDS junction ($m|\Delta_5$). We assume a 3DDM can be superconductor by proximity effect. (b) The Josephson junction of 3DDS$|$3DDM$|$3DDS ($\Delta_5|m|\Delta_5$).[]{data-label="Fig.structure"}](Model.jpg)
[*Transport signatures of Majorana sea*]{}. To further corroborate our central result concerning a sea of Majorana fermions in the present system, let us now focus on the transport properties arising from 2DMS which can be directly accessed in experiments. Let us begin by looking into a single $m|\Delta_5$ junction. The lateral coordinates in the plane of junction are $(x,y)$ confined within a lateral dimensions $(W_x,W_y)$, which means the corresponding components of wave vector in these directions are quantized as $k_{x(y)}=(n_{x(y)}+1/2)\pi/W_{x(y)}$. Each mode can be identified with a set of quantum number $n=(n_x,n_y)$. In the 3DDM side of the junction ($z<0$), there are eight components of wave functions, $\Psi^{M,\pm}_{e(h) \uparrow(\downarrow)}$ where the indices $e$ ($h$) characterize electron (hole)-like quasi-particles, $\uparrow (\downarrow)$ denotes its spin direction with respect to $z$-axis and $\pm$ indicates the right or left mover character of carriers [@Supplement]. The interface between $m$, ($z<0$) and $\Delta_5$ ($z > 0$) regions can reflect an incident electron as a hole. In this Andreev reflection process, the missing charge of $2e$ is absorbed by superconductor as Cooper pair [@Andreev1964JETP]. Typically the s-wave assumption of superconductivity imposes that the reflected hole from an $\uparrow$-spin incident electron must be in a $\downarrow$-spin state and vice versa. However, due to strong spin-orbit coupling encoded in the Dirac Hamiltonian of a 3DDM, when an electron in a given spin direction hits a 3DDM, the spin of the electron transmitted into 3DDM can be flipped [@Salehi2015AOP]. Therefore in 3DDM$|$3DDS junction we also take into account the possibility of an anomalous – i.e. spin flipped – Andreev reflection of the reflected hole. The boundary condition for an $\uparrow$ spin electron hitting the interface is given by, $$\begin{array}{rl}
\Psi_{e,\uparrow}^{M,+} & +\sum_{\nu=\uparrow,\downarrow}r_{N,\alpha}\Psi_{e,\nu}^{M,-}+\sum_{\nu=\uparrow,\downarrow}r_{A,\nu}\Psi_{h,\nu}^{M,-}\\
& =\sum_{\kappa=1,2}t_{e,\kappa}\Psi_{e,\kappa}^{S,+}+\sum_{\kappa=1,2}t_{h,\kappa}\Psi_{h,\kappa}^{S,+}
\end{array}.
\label{BoundaryCondition}$$ The left hand side is related to the wave function in 3DDM side, and the right hand side describes the wave equation in 3DDS side. Here $r_{N,\uparrow}$ and $r_{N,\downarrow}$ are the amplitudes of conventional and spin-flipped normal reflection, respectively. The $r_{A,\downarrow}$ and $r_{A,\uparrow}$ are the amplitudes of conventional and anomalous Andreev reflection, respectively. Similar processes take place for $\downarrow$ spin and hole carriers as well [@Supplement]. The probability of these reflections vs. $\theta$, the polar angle of incidence, are depicted in Fig. (\[PvsAngle\]). Because of the conservation of parallel component of wave vector, $k_n=\sqrt{k_x^2+k_y^2}$ at the scattering process, the angle of propagation for reflected hole ($\theta'$) has a critical value $\theta_C=\sin^{-1}((\mu-\varepsilon)^2-\Delta_D^2)/(\mu+\varepsilon)^2-\Delta_D^2))$ beyond which the reflected hole can not contribute to transport. For zero modes, $\varepsilon= 0$, the conventional and spin-flipped normal reflections would disappear and we are left with conventional and anomalous Andreev reflection given by, $$r_{A,\downarrow}=-e^{-i \phi} \cos\theta,~~~~
r_{A,\uparrow}= e^{i \alpha-i\phi}\sin\theta
\label{Eq.08zero-energyAR}$$ where, $\alpha=\arctan(k_y/k_x)$ is azimuthal angle. From Eq. (\[Eq.08zero-energyAR\]), it is obvious that for a zero-energy incident electron at any angle of propagation we have a perfect Andreev reflection, $|r_{A,\downarrow}|^2+|r_{A,\uparrow}|^2=1$. This is a transport signature of Majorana fermions [@Beenakker2013rev; @Knez2012PRL]. This effect is robust against changing the chemical potential and angles of incidence ($\alpha, \theta$). The BTK formula for the differential conductance of the junction [@Blonder1982PRB] will be,
![(Color online) Probability of Andreev and normal reflections vs polar angle of incident. The conventional (spin-flipped) normal reflection is shown by solid (dashed) blue line. Also, the conventional (anomalous) Andreev reflection is prevaricated by solid (dashed)red line. (a) For $\varepsilon= \Delta_5$, the Andreev-Klein tunneling occurs in $\theta=0$. (b) In $\varepsilon=0$, the perfect Andreev reflection occurs for all angles of incidence. The input values are $ \mu=m, \Delta_5=5m$.[]{data-label="PvsAngle"}](PvsAngle.jpg){width="6" height="2.5"}
$$\frac{G(\varepsilon)}{G_0}=\sum_n\left[1+\sum_{\nu}\left|r_{A,\nu}(\varepsilon,k_n)\right|^2-\left|r_{N,\nu}(\varepsilon,k_n)\right|^2\right]
\label{Eq.Conductance}$$
where $G_0=e^2/h $ is the quantum of conductance. Note that the summation over $\nu$ includes spin-flipped contributions as well. When the linear dimensions of the interface are much larger than the superconducting coherence length, $\xi_s=\hbar v_F/\Delta_5$, i.e. $W_x,W_y\gg \xi_s$, the summation over mode indices $n$ in Eq. (\[Eq.Conductance\]) can be replaced by an integral. The conductance for various values of chemical potential has been plotted in Fig. (\[Fig.conductance\]) as a function of energy. Note that the height of the resonance peak at zero energy is pinned at $2G_0$ which indeed arises from the 2DMS.
![(Color online) Conductance of the $m|\Delta_5$ system. We set $\Delta_5=5 m$ and $\mu/m=\{2,3,4,5\} $ for $\{a, b, c, d\}$ curves respectively.[]{data-label="Fig.conductance"}](Conductance.jpg){width="5cm" height="3.25cm"}
Now let us move to the next transport signature of 2DMS, namely the fractional Josephson current. Consider the geometry depicted in part(b) of Fig. (\[Fig.structure\]) and assume that $L$ is the length of junction and $\delta\phi=\phi_R-\phi_L$ is the phase difference between the two $\Delta_5$ superconductors. In th short junction limit, $L \ll \xi_S$, the Andreev bound states (ABS) responsible for carrying the super-current between two 3DDS region are given by [@Supplement], $$\epsilon_{\pm,n}(\delta\phi)=\pm\Delta_S\sqrt{\tau_n}\cos(\delta\phi/2).
\label{Eq.12}$$ where $\tau_n$ is the normal transmission probability of the junction, $\tau_n=\left(\cosh^2\kappa_nL+\mu^2\sinh^2\kappa_nL/\kappa_n^2\right)^{-1}$, with $\kappa_n=\sqrt{\Delta_D^2+k_n^2-\mu^2}$. Despite that the Hamiltonian in Eq. (\[Eq.DBdG\]) is invariant under $2\pi$ phase shift, $\phi\rightarrow\phi+2\pi$, the ABS in Eq.(\[Eq.12\]) clearly has a $4\pi$ period. Using Eq. (\[Eq.12\]), the corresponding Josephson current becomes, $I_\pm(\delta\phi) =(e\Delta_S/\hbar)\sum_{n}\partial\epsilon_{\pm,n}(\delta\phi)/\partial\delta\phi=\pm I_c\sin(\delta\phi/2)$, where $I_c=(e\Delta_S/2\hbar)\sum_{n}\sqrt{\tau_n}$ is the critical value of the Josephson current. In the absence of perturbations which violate fermion parity conservation, such a form of fractional Josephson current is a signature of 2DMS on the surface of 3DDS [@Fu2009PRB].
![(Color online) The Andreev bound states obtained for Josephson junction of part (b) in Fig.(\[Fig.structure\]). These states show $4\pi$ periodicity with respect to $\delta\phi$ which is an another direct signature of 2DMS. ](ABS.jpg)
To summarize we have shown that a 2DMS can be obtained from singlet pairing as well, provided the superconducting order is pseudo-scalar, i.e. it changes sign under mirror reflection. In this extent, the present system is a singlet cousin in the family of odd-parity superconductors [@Fu2010PRL]. Our scenario for Majorana surface states does not require proximity to any magnet, as no triplet pairing is involved here. Perfect Andreev reflection and fractional Josephson current as two hallmarks of the ensuing Majorana sea leave clear transport footprints. The junction between such a superconductor and conventional $s$ and $d$ wave superconductors similar to the odd-parity p-wave superconductors may provide anomalous flux quantization in units of $h/4e$ [@Fu2010PRL]. Our analysis depends on the existence of the $\gamma^5$ matrix which is a unique opportunity in even space-time dimensions. Therefore in 1+1 dimension as well, Majorana bound states are expected from a competition between the Dirac (Peierls) mass $m$ and $\Delta_5$ [*singlet*]{} superconductivity. Moreover, gauge transformation can place the odd-parity on the Dirac mass. Therefore for a peculiar Dirac Hamiltonians where the mass term $m_5$ carries the minus sign needed in making of the odd-parity, Majorana fermions can arise in $m_5|\Delta_0$ interfaces where $\Delta_0$ is a conventional s-wave superconductors.
[*Acknowledgment:*]{} SAJ was supported by Alexander von Humboldt fellowship for experienced researchers.
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[**Supplementary material**]{}
In this supplementary material we expand the details of calculations referenced in the main letter text.
A. Pairing under $\Delta_5$ order parameter
===========================================
The wave functions satisfying the Dirac equation $$H_D(\mathbf{k})|\psi\rangle=\epsilon |\psi\rangle
\label{E.01}$$ is a four component spinor as, $$H_{D}(\mathbf{k})=\left(
\begin{array}{cccc}
m-\mu & 0 & i k_z & i (k_x-i k_y)\\
0 & m-\mu & i (k_x+i k_y) & -ik_z \\
-i k_z & -i (k_x-ik_y) & -m-\mu & 0\\
-i (k_x+i k_y) & i k_z & 0 & -m-\mu \\
\end{array}
\right)\left(\begin{array}{c}
\phi_\uparrow \\
\phi_\downarrow \\
\chi_\uparrow\\
\chi_\downarrow\\
\end{array}\right)=\epsilon \left(\begin{array}{c}
\phi_\uparrow \\
\phi_\downarrow \\
\chi_\uparrow\\
\chi_\downarrow\\
\end{array}\right)
\label{E.02}$$ where $\phi$ and $\chi$ correspond to the conduction and valence band, respectively. Also, $\uparrow (\downarrow)$ correspons to the spin directions of carrier with respect to $z$-axis. The hole part of DBdG equation is obtained by operating with the charge conjugation on the Hamiltonina and its spinor of Eq.(\[E.01\]). It consists in the complex conjugation operator $K$ and multiplication by some $4\times 4$ matrxi the form of which depends on the representation of $\gamma$ matrices and in our case is [@ZeeBookQFT], $$\mathcal{C}=\left(\begin{array}{cccc}
0 & 0 & 0& -1 \\
0 & 0& 1 & 0 \\
0 & 1 & 0 & 0 \\
-1 & 0& 0& 0 \\
\end{array}\right)K
\label{E.03}$$ such that, $$\mathcal{C}H_D(\mathbf{k})\mathcal{C}^{-1}{\mathcal C}|\psi\rangle=\epsilon{\mathcal C} |\psi\rangle
\label{E.04}$$ which can be recast into the form, $$H_D(-\mathbf{k})|\psi_c\rangle=-\epsilon |\psi_c\rangle
\label{E.05}$$ where the charge conjugated state $|\psi_c\rangle$ is explicitely given by [@ZeeBookQFT], $$|\psi_c\rangle=\mathcal{C}|\psi\rangle=\left(\begin{array}{c}
-\chi_\downarrow^*\\
\chi_\uparrow^* \\
\phi_\downarrow^*\\
-\phi_\uparrow^*
\end{array}\right)
\label{E.06}$$
From $|\psi\rangle$ and $|\psi_c\rangle$ we can construct a Nambu spinor $|\mathbf{\Psi}\rangle$ that satisfies DBdG equation in the direct product space of Eqs. (\[E.04\]) and (\[E.06\]), $$H_{DBdG}\left|\mathbf{\Psi}\right\rangle=\epsilon\left|\mathbf{\Psi}\right\rangle$$ The explicit representation of the DBdG Hamiltonian matrix in the pseudo-scalar pairing channel becomes, $$\footnotesize
\left(
\begin{array}{cccccccc}
m-\mu & 0 & i k_z & i (k_x-i k_y) & 0 & 0 & i \Delta_5 & 0 \\
0 & m-\mu & i (k_x+i k_y) & -ik_z & 0 & 0 & 0 & i \Delta_5 \\
-i k_z & -i (k_x-ik_y) & -m-\mu & 0 & i\Delta_5 & 0 & 0 & 0 \\
-i (k_x+i k_y) & i k_z & 0 & -m-\mu & 0 & i \Delta_5 & 0 & 0 \\
0 & 0 & -i \Delta_5 & 0 & \mu -m & 0 & i k_z & i (k_x-ik_y) \\
0 & 0 & 0 & -i \Delta_5 & 0 & \mu -m & i (k_x+ik_y) & -i k_z \\
-i \Delta_5 & 0 & 0 & 0 & -i k_z & -i (k_x-ik_y) & m+\mu & 0 \\
0 & -i \Delta_5 & 0 & 0 & -i (k_x+ik_y) & i k_z & 0 &m+\mu \\
\end{array}
\right)\left(\begin{array}{c}
\phi_\uparrow \\
\phi_\downarrow \\
\chi_\uparrow \\
\chi_\downarrow \\
-\chi_\downarrow^* \\
\chi_\uparrow ^* \\
\phi_\downarrow^* \\
-\phi_\uparrow^* \\
\end{array}\right)=\epsilon \left(\begin{array}{c}
\phi_\uparrow \\
\phi_\downarrow \\
\chi_\uparrow \\
\chi_\downarrow \\
-\chi_\downarrow^* \\
\chi_\uparrow ^* \\
\phi_\downarrow^* \\
-\phi_\uparrow^* \\
\end{array}\right)
\label{E.07}$$
To clarify the total spin of the pairing symmetry let extract the pairing part of $\left\langle \mathbf{\Psi}\right| H_{DBdG}\left|\mathbf{\Psi}\right\rangle$ which in terms of $\{\phi_i, \chi_i \}$ becomes, $$\footnotesize
\left(\begin{array}{c}
\phi_\uparrow^* \\
\phi_\downarrow^* \\
\chi_\uparrow^* \\
\chi_\downarrow^* \\
-\chi_\downarrow \\
\chi_\uparrow \\
\phi_\downarrow \\
-\phi_\uparrow \\
\end{array}\right)^T
\left(
\begin{array}{cccccccc}
0 & 0 & 0 & 0 & 0 & 0 & i \Delta_5 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & i \Delta_5 \\
0 & 0 & 0 & 0 & i\Delta_5 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & i \Delta_5 & 0 & 0 \\
0 & 0 & -i \Delta_5 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -i \Delta_5 & 0 & 0 & 0 & 0 \\
-i \Delta_5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & -i \Delta_5 & 0 & 0 & 0 & 0 & 0 & 0 \\
\end{array}
\right)\left(\begin{array}{c}
\phi_\uparrow \\
\phi_\downarrow \\
\chi_\uparrow \\
\chi_\downarrow \\
-\chi_\downarrow^* \\
\chi_\uparrow ^* \\
\phi_\downarrow^* \\
-\phi_\uparrow^* \\
\end{array}\right)
\label{E.088}$$ and simplifies to, $$i \Delta_5 \left( \phi_\uparrow^* \phi_\downarrow^*-\phi_\downarrow^*\phi_\uparrow^*+\phi_\uparrow\phi_\downarrow-\phi_\downarrow\phi_\uparrow\right)+
i \Delta_5 \left( \chi_\downarrow^*\chi_\uparrow^*-\chi_\uparrow^*\chi_\downarrow^*+\chi_\downarrow\chi_\uparrow-\chi\uparrow\chi_\downarrow\right)$$ Which clearly shows the singlet nature of pairing as is expected from singlet character of a pseudo-scalar pairing.
B. Calculation of the winding number
====================================
To calculate the topological charge of our Hamiltonian we need to construct the particle-hole ($PH$), time-reversal ($TR$), and sublattice ($SL$) transformations in the Nambu space.
$B_1$. Symmetry operators
-------------------------
In the main text we introduces the above three operators the explicit form of which is given by, $$PH=-i \eta_2\otimes\tau_0\otimes\sigma_2 K=\left(
\begin{array}{cccccccc}
0 & 0 & 0 & 0 & 0 & i & 0 & 0 \\
0 & 0 & 0 & 0 & -i & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & i \\
0 & 0 & 0 & 0 & 0 & 0 & -i & 0 \\
0 & -i & 0 & 0 & 0 & 0 & 0 & 0 \\
i & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -i & 0 & 0 & 0 & 0 \\
0 & 0 & i & 0 & 0 & 0 & 0 & 0 \\
\end{array}
\right)K,
\label{Eq.01}$$
$$TR=i \eta_0\otimes\tau_3\otimes\sigma_2 K=\left(
\begin{array}{cccccccc}
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
\end{array}
\right)K,
\label{Eq.02}$$
and $$SL=\eta_2\otimes\tau_3\otimes\sigma_0=\left(
\begin{array}{cccccccc}
0 & 0 & 0 & 0 & -i & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & -i & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & i & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & i \\
i & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & i & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & -i & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -i & 0 & 0 & 0 & 0 \\
\end{array}
\right),
\label{Eq.03}$$ where $\eta_\nu$, $\tau_\nu$ and $\sigma_\nu$ with $\nu=\{0,...,3\}$ are Pauli matrices which act on Nambu, band and spin spaces, respectively. Also, $\nu=0$ corresponds to 2 by 2 unit matrix.
$B_2$. Chiral DBdG equation
---------------------------
A Hamiltonian with chiral symmetry can be rotated to block off-diagonal form. The unitary operator constructed from bases of chiral matrix can convert the original DBdG to block off-diagonal matrix (Chiral DBdG). The eigenvalues and eigenvectors of Eq. (\[Eq.03\]) are, $$\left(
\begin{array}{ll}
\lambda_1= -1 & u_1=\{0,0,0,-i,0,0,0,1\} \\
\lambda_2= -1 & u_2=\{0,0,-i,0,0,0,1,0\} \\
\lambda_3= -1 & u_3=\{0,i,0,0,0,1,0,0\} \\
\lambda_4= -1 & u_4=\{i,0,0,0,1,0,0,0\} \\
\lambda_5= 1 & u_5=\{0,0,0,i,0,0,0,1\} \\
\lambda_6= 1 & u_6=\{0,0,i,0,0,0,1,0\} \\
\lambda_7= 1 & u_7=\{0,-i,0,0,0,1,0,0\} \\
\lambda_8= 1 & u_8=\{-i,0,0,0,1,0,0,0\} \\
\end{array}
\right),
\label{Eq.04}$$ which gives the unitary matrix $U_c$ as, $$U_c=\left(
\begin{array}{cccccccc}
0 & 0 & 0 & \frac{i}{\sqrt{2}} & 0 & 0 & 0 & -\frac{i}{\sqrt{2}} \\
0 & 0 & \frac{i}{\sqrt{2}} & 0 & 0 & 0 & -\frac{i}{\sqrt{2}} & 0 \\
0 & -\frac{i}{\sqrt{2}} & 0 & 0 & 0 & \frac{i}{\sqrt{2}} & 0 & 0 \\
-\frac{i}{\sqrt{2}} & 0 & 0 & 0 & \frac{i}{\sqrt{2}} & 0 & 0 & 0 \\
0 & 0 & 0 & \frac{1}{\sqrt{2}} & 0 & 0 & 0 & \frac{1}{\sqrt{2}} \\
0 & 0 & \frac{1}{\sqrt{2}} & 0 & 0 & 0 & \frac{1}{\sqrt{2}} & 0 \\
0 & \frac{1}{\sqrt{2}} & 0 & 0 & 0 & \frac{1}{\sqrt{2}} & 0 & 0 \\
\frac{1}{\sqrt{2}} & 0 & 0 & 0 & \frac{1}{\sqrt{2}} & 0 & 0 & 0 \\
\end{array}
\right).
\label{Eq.05}$$ This transformation brings the DBdG Hamiltonian, Eq. (\[E.07\]) to the canonical form, $$\begin{array}{rl}
U_c^{-1} H_{BD}U_c
& =\left(
\begin{array}{cccccccc}
0 & 0 & 0 & 0 & m+\mu & 0 & i k_z-\Delta_5 & k_y-i k_x \\
0 & 0 & 0 & 0 & 0 & m+\mu & -i k_x-k_y & -i k_z-\Delta_5 \\
0 & 0 & 0 & 0 & \Delta_5-i k_z & i k_x-k_y & \mu -m & 0 \\
0 & 0 & 0 & 0 & i k_x+k_y & i k_z+\Delta_5 & 0 & \mu -m \\
m+\mu & 0 & i k_z+\Delta_5 & k_y-i k_x & 0 & 0 & 0 & 0 \\
0 & m+\mu & -i k_x-k_y & \Delta_5-i k_z & 0 & 0 & 0 & 0 \\
-i k_z-\Delta_5 & i k_x-k_y & \mu -m & 0 & 0 & 0 & 0 & 0 \\
i k_x+k_y & i k_z-\Delta_5 & 0 & \mu -m & 0 & 0 & 0 & 0 \\
\end{array}
\right)
\end{array}
\label{Eq.07}$$
$B_3$. Projector Matrix
-----------------------
In the presence of translational invariance, the ground states of 3D Dirac superconductor can be constructed as a Fermi sea that gets then gapped out by superconducting pairing which separates the filled and empty states in 3D Brillouin zone (BZ). This enables to define a projector operator as, $$P(k)=\sum_{a\in{\rm filled}}\left|u_a (k)\right\rangle \left\langle u_a(k)\right|.
\label{Eq.08}$$ To construct the projector operator, at first step, we must calculate the wave functions of Eq. (\[Eq.07\]) for energy eigenvalues of the filled states which are given by, $$\epsilon_{1,2}=-\sqrt{k_x^2+k_y^2+k_z^2+(m-\Delta_5)^2},~~~~
\epsilon_{3,4}=-\sqrt{k_x^2+k_y^2+k_z^2+(m+\Delta_5)^2}
\label{Eq.09}$$ The wave functions corresponding to the eigenvalues in Eq. (\[Eq.09\]) are, $$\begin{array}{l}
u_1(k)=\left(-\frac{1}{2},0,\frac{1}{2},0,\frac{1}{2} (\delta_m-i k_m \cos (\theta )),\frac{1}{2} i e^{-i \alpha } k_m \sin (\theta ),\frac{1}{2} (\delta_m-i k_m \cos (\theta )),\frac{1}{2} i e^{-i \alpha } k_m \sin (\theta )\right)^T\\
u_2(k)=\left(0,-\frac{1}{2},0,\frac{1}{2},\frac{1}{2} i e^{i \alpha } k_m \sin (\theta ),\frac{1}{2} (\delta_m+i k_m \cos (\theta )),\frac{1}{2} i e^{i \alpha } k_m \sin (\theta ),\frac{1}{2} (\delta_m+i k_m \cos (\theta ))\right)^T\\
u_3(k)=\left(\frac{1}{2},0,\frac{1}{2},0,-\frac{1}{2} \delta_p+i k_p \cos (\theta )),\frac{1}{2} i e^{-i \alpha } k_p \sin (\theta ),\frac{1}{2} (\delta_p+i k_p \cos (\theta )),\frac{1}{2} (-i) e^{-i \alpha } k_p \sin (\theta )\right)^T\\
u_4(k)=\left(0,\frac{1}{2},0,\frac{1}{2},\frac{1}{2} i e^{i \alpha } k_p \sin (\theta ),-\frac{1}{2} (\delta_p-i k_p \cos (\theta )),\frac{1}{2} (-i) e^{i \alpha } k_p \sin (\theta ),\frac{1}{2} (\delta_p-i k_p \cos (\theta ))\right)^T
\end{array}
\label{Eq.10}$$ where we have used the notations, $$\begin{array}{l}
k=\sqrt{k_x^2+k_y^2+k_z^2}\\
\xi_m=\sqrt{k^2+(m-\Delta_5)^2}\\
\xi_p=\sqrt{k^2+(m+\Delta_5)^2}\\
\delta_m=(m-\Delta_5)/\xi_m\\
\delta_p=(m+\Delta_5)/\xi_p\\
k_p=k/\xi_p\\
k_m=k/\xi_m\\
\end{array}
\label{Eq.11}$$ The polar and azimuthal angels in the $k$-space are denoted by $(\theta, \alpha)$. The projector operator can be calculated by plugging the wave functions of Eq. (\[Eq.10\]) into Eq. (\[Eq.08\]) which gives, $$P(k)=\frac{1}{2}\left(
\begin{array}{cc}
\mathbf{1} & q\\
q^\dagger & \mathbf{1}\\
\end{array}\right)$$ where $\mathbf{1}$ and $q$-matrix are 4 by 4 matrices. Then the $Q$-matrix is obtained as: $$Q(k)=2P(k)-\mathbf{1}=\left(\begin{array}{cc}
0 & q \\
q^\dagger & 0
\end{array}
\right)$$ Here $\mathbf{1}$ is a unit matrix with 8 by 8 dimension. The $q$-matrix which is the off diagonal block of projector is essential for calculation of the topologial index, and is given by, $$\footnotesize
q(k)=
\left(
\begin{array}{cccc}
-\frac{1}{2} (\delta_m+\delta_p+i(k_m-k_p) \cos \theta ) & \frac{1}{2} i e^{i \alpha } (k_m-k_p) \sin \theta & \frac{1}{2} (-\delta_m+\delta_p-i (k_m+k_p) \cos \theta ) & \frac{1}{2} i e^{i \alpha } (k_m+k_p) \sin \theta \\
\frac{1}{2} (k_m-k_p)i e^{-i \alpha} \sin \theta & \frac{1}{2} (-\delta m-\delta p+i (k_m-k_p) \cos \theta ) & \frac{1}{2} (k_m+k_p) i e^{-i\alpha} \sin \theta & \frac{1}{2} (-\delta m+\delta p+i (k_m+k_p) \cos \theta ) \\
\frac{1}{2} (\delta m-\delta p+i (k_m+k_p) \cos \theta ) & -\frac{1}{2} i e^{i \alpha } (k_m+k_p) \sin \theta & \frac{1}{2} (\delta m+\delta p+i (k_m-k_p) \cos \theta ) & -\frac{1}{2} i e^{i \alpha } (k_m-k_p) \sin \theta \\
-\frac{1}{2} i e^{-i \alpha } (k_m+k_p) \sin \theta & \frac{1}{2} (\delta m-\delta p-i (k_m+k_p) \cos \theta ) & -\frac{1}{2} i e^{-i \alpha } (k_m-k_p) \sin \theta & \frac{1}{2} (\delta m+\delta p-i (k_m-k_p) \cos \theta ) \\
\end{array}
\right)
\label{Eq.14}$$
$B_4$. Topological invariant
----------------------------
We are now set to calculate the topological invariant from $q(k)$ matrix that is given by the integral $$\mathcal{Q}\left[q\right]=\int \frac{d^3k}{24 \pi^2}\epsilon^{\mu\nu\rho}tr\left[\left(q^{-1}\partial_\mu q\right)\left(q^{-1}\partial_\nu q\right)\left(q^{-1}\partial_\rho q\right)\right],
\label{Eq.15}$$ Here the $\epsilon^{\mu\nu\rho}$ is the Levi-Civita tensor and $\mu, \nu, \rho$ run over 3 directions $\{k_x, k_y, k_z\}$. By plugging Eq. (\[Eq.14\]) into Eq. (\[Eq.15\]) we find, $$\mathcal{Q}=\frac{1}{2}\left(\frac{m+\Delta_5}{\left|m+\Delta_5\right|}-\frac{m-\Delta_5}{\left|m-\Delta_5\right|}\right).$$ For 3D Dirac materials (non superconducting) case where $(\Delta_5=0, m\neq 0$, the winding number will be $\mathcal{Q}=0$ while for 3D Dirac superconductor $(\Delta_5\neq 0, m=0)$ the Winding number will be $\mathcal{Q}={\rm sign}(\Delta_5)$. The above topological index therefore shows a clear change by movign from $\Delta_5$ dominated gapped state to $m$ dominated gapped state in a $m|\Delta_5$ region.
C. Transport
============
In this section we obtain the transport coefficients in normal and Andreev channels.
$C_1$. Wave functions of 3DDM region
------------------------------------
In 3DDM region, the superconducting gap is zero $\Delta_5=0$, which reduces the Dirac-Bogoliubov-deGennes to, $$\left( \begin{array}{cc}
\mathcal{H}(\textbf{k})-\mu & 0\\
0 & \mu+\mathcal{C}\mathcal{H}(\textbf{k})\mathcal{C}^{-1} \\
\end{array}\right)
\left( \begin{array}{c}
u \\
v\\
\end{array}\right)= \epsilon \left(\begin{array}{c}
u \\
v \\
\end{array} \right)
\label{3DDMBdG}$$ The electron part of Eq. (\[3DDMBdG\]) is completely decoupled from the hole one. Each part has four components which can be defined as right or left mover with respect to $z$-axis. Also, each states has two spin orientations, $\{\uparrow, \downarrow\}$. For electron part with wave vector $k=|\mathbf{k}|=\sqrt{(\epsilon+\mu)^2-m^2}$, we can define the longitudinal and transverse components of wave vector as: $$\begin{cases}
k_z=|\mathbf{k}|\cos\theta\\
k_{n}=\sqrt{k_x^2+k_y^2}=|\mathbf{k}|\sin\theta\\
\end{cases}$$ As mentioned in the main text, the transverse components of wave vector are quantized due to the finite width of junction and are labaled by $n=(n_x, n_y)$. So the wave functions can be written as, $$\begin{array}{l}
\Psi_{e,\uparrow}^{M,\pm}=\left(
\begin{array}{ccccc}
1, & 0,& \mp i \eta_e\cos\theta,& -i \eta_e e^{i \alpha} \sin\theta, & \mathbf{0}^4
\end{array}\right)^Te^{\pm i k_zz+ i \vec{k}_{n}.\vec{r}_{||}},\\
\Psi_{e,\downarrow}^{M,\pm}=\left(
\begin{array}{ccccc}
0, & 1,& -i \eta_e e^{-i \alpha} \sin\theta, & \pm i \eta_e\cos\theta, & \mathbf{0}^4
\end{array}\right)^Te^{\pm i k_zz+ i \vec{k}_{n}.\vec{r}_{||}},\\
\end{array}
\label{WaveFunctions1}$$ where $\mathbf{0}^4$, is the $1\times4$ vector with zero entities, and $\eta_e=\sqrt{(\epsilon+\mu-m)/(\epsilon+\mu+m)}$. In order to conserve thei current, the wave functions of Eq. (\[WaveFunctions1\]) need the normalization factor, $1/\sqrt{2\eta_e \cos\theta}$. For the wave functions of hole part we obtain, $$\begin{array}{l}
\Psi_{h,\uparrow}^{M,\pm}=\left(
\begin{array}{ccccc}
\mathbf{0}^4,& i \eta_h e^{-i \alpha}\sin\theta', & \mp i \eta_h \cos\theta', & 0, & 1\\
\end{array}\right)^Te^{\pm i k_z'z+i \vec{k}_{n}.\vec{r}_{||}},\\
\Psi_{h,\downarrow}^{M,\pm}=\left(
\begin{array}{ccccc}
\mathbf{0}^4,& \pm i \eta_h \cos\theta', & i \eta_h e^{i \alpha}\sin\theta', & 1, & 0
\end{array}\right)^Te^{\pm i k_z'z+ i \vec{k}_{n}.\vec{r}_{||}},
\end{array}
\label{WaveFunction2}$$ where $\eta_h=\sqrt{(\epsilon-\mu-m)/(\epsilon-\mu+m)}$ and $k'=|\mathbf{k'}|=\sqrt{(\epsilon-\mu)^2-m^2}$. The polar angle of propagation for hole-like quasi-particle is $\cos\theta'=k_{n}/k'$. Owing to translation invariance in the transverse directions, $k_{||}$ is conserved at the scattering processes. The $k_z'$ can be written as, $$k_z'=\sqrt{(\epsilon-\mu)^2-m^2)-k_{n}^2}.$$ This equation implies that for incident angle ($\theta$) beyond critical value of, $$\theta_c=\arcsin\left((\sqrt{\frac{(\epsilon-\mu)^2-m^2}{(\epsilon+\mu)^2-m^2}}\right),$$ the perpendicular component of the wave vector for hole-like wave functions will be imaginary. This means hole-like states beyond these propagation angles can not contribute to transport. The normalization factor for wave functions of Eq. (\[WaveFunction2\]) is $1/\sqrt{2\eta_h\cos\theta'}$.
$C_2$. Wave functions of 3DDS region
------------------------------------
In superconducting region the pairing matrix $\hat{\Delta}_S=\Delta_5\gamma^5$ and we have $\mu_S\gg m$. In this regime the DBdG Hamiltonian will be: $$\left( \begin{array}{cc}
\mathcal{H}(\textbf{k})-\mu_S & \hat{\Delta}_S \gamma_0 e^{i \phi}\\
\gamma_0\hat{\Delta}^\dagger_S e^{-i \phi} & \mu_S+\mathcal{C}\mathcal{H}(\textbf{k})\mathcal{C}^{-1} \\
\end{array}\right)
\left( \begin{array}{c}
u \\
v\\
\end{array}\right)= \epsilon \left(\begin{array}{c}
u \\
v \\
\end{array} \right).
\label{Eq.DBdG}$$ The eigenvalues of Eq.(\[Eq.DBdG\]) are, $$\epsilon= \pm \sqrt{(|\mathbf{k}| \pm \mu_S)^2+\Delta_5^2}.
\label{Eq.Seigenvalues}$$ Each eigenvalues are doubly degenerate because of spin. The wave vectors of electron-like and hole-like quasi-particles corresponding to the eigenvalues of Eq.(\[Eq.Seigenvalues\]) are, $$\begin{cases}
|\mathbf{k}_e^S|=\left(\mu_S+\sqrt{\epsilon^2-\Delta_S^2}\right)/\hbar v_F\\
|\mathbf{k}_h^S|=\left(\mu_S-\sqrt{\epsilon^2-\Delta_S^2}\right)/\hbar v_F\\
\end{cases}$$ with corresponding wave functions: $$\begin{array}{l}
\Psi_{e, \kappa=1}^{S, \pm}=
\left( \begin{array}{cccccccc}
1, & 0, & \mp i\cos\theta_S, & -ie^{i\alpha}\sin\theta_S, & \mp e^{-i\phi-i\beta}\cos\theta_S, & -e^{-i\phi +i\alpha-i\beta}\sin\theta_S,& -i e^{-i \phi-i \beta}, & 0
\end{array}\right)^T e^{\pm i k_z^Sz}\\
\Psi_{e, \kappa=2}^{S, \pm}=
\left( \begin{array}{cccccccc}
0, & 1, & -ie^{-i\alpha}\sin\theta_S, & \pm i \cos\theta_S, & - e^{-i\phi-i\alpha-i\beta}\sin\theta_S, & \pm e^{-i\phi-i\beta}\cos\theta_S,& 0, & -i e^{-i\phi-i\beta}
\end{array}\right)^T e^{\pm i k_z^Sz}\\
\Psi_{h, \kappa=1}^{S, \pm}=
\left( \begin{array}{cccccccc}
\mp i \cos\theta_S, & i e^{i\alpha}\sin\theta_S, & 1, & 0, & -i e^{i \beta-i \phi}, & 0,& \mp e^{-i\phi+i\beta}\cos\theta_S, & e^{i\alpha+i\beta-i\phi}\sin\theta_S
\end{array}\right)^T e^{\pm i k_z'^Sz}\\
\Psi_{h, \kappa=2}^{S, \pm}=
\left( \begin{array}{cccccccc}
i e^{-i\alpha}\sin\theta_S, & \pm i \cos\theta_S, & 0, & 1, & 0, & -i e^{-i\phi+i\beta}, & e^{-i\alpha-i\phi+i\beta}\sin\theta_S, & \pm e^{-i\alpha+i\beta-i\phi}\sin\theta_S
\end{array}\right)^T e^{\pm i k_z'^Sz}
\end{array}
\label{Wavefunctions3}$$ Note that all wave functions in Eq. (\[Wavefunctions3\]) must be multiplied by a the Bloch phase of parallel component of wave vector, $\exp(i \mathbf{k}_{n}.\mathbf{r}_{||})$ and the normalization factor $1/2\sqrt{\cos\theta_S}$.
$C_3$. Boundary condition
-------------------------
Using wave functions of Eqs. (\[WaveFunctions1\]), (\[WaveFunction2\] and (\[Wavefunctions3\]), one can solve the boundary conditions of Eq. (4) in the main text to complete the construction of scattering states. The reflection amplitudes for an $\uparrow$-spin incident electron are derived as, $$\begin{array}{l}
r_{N,\uparrow}=\frac{\left(\omega_e^{c^2}-\omega_e^{s^2}-1 \right)\left(\chi_h^{+}\cos 2\beta+2i \omega_h^{c}\sin 2\beta\right)-\left(\chi_h^{-}\left(\omega_e^{c^2}-\omega_e^{s^2}+1 \right)+4 \omega_e^s\omega_h^s\right)}{\left(\chi_e^{+}\chi_h^{+}+4\omega_e^c\omega_c^c\right)\cos 2\beta+2i\left(\chi_h^{+}\omega_e^c+\chi_e^{+}\omega_h^c\right)\sin 2\beta-\left(\chi_e^{-}\chi_h^{-}-4\omega_e^s\omega_h^s\right)},\\
\\
r_{N,\downarrow}=\frac{2e^{i\alpha}\omega_e^c\{\left(\omega_e^s(\chi_h^{-}-\chi_h^{+}\cos 2\beta)-2i\omega_e^s\omega_h^s\sin 2\beta-2 \omega_h^s\right)\}}{\left(\chi_e^{+}\chi_h^{+}+4\omega_e^c\omega_c^c\right)\cos 2\beta+2i\left(\chi_h^{+}\omega_e^c+\chi_e^{+}\omega_h^c\right)\sin 2\beta-\left(\chi_e^{-}\chi_h^{-}-4\omega_e^s\omega_h^s\right)},\\
\\
r_{A,\uparrow}=\frac{2i e^{i\alpha-i\phi}\sqrt{\omega_e^c\omega_h^c}\left(2(\omega_e^s+\omega_h^s)\cos \beta+2i(\omega_e^s\omega_h^c+\omega_e^c\omega_h^s)\sin\beta\right)}{\left(\chi_e^{+}\chi_h^{+}+4\omega_e^c\omega_c^c\right)\cos 2\beta+2i\left(\chi_h^{+}\omega_e^c+\chi_e^{+}\omega_h^c\right)\sin 2\beta-\left(\chi_e^{-}\chi_h^{-}-4\omega_e^s\omega_h^s\right)},\\
\\
r_{A,\downarrow}=\frac{-4i e^{-i\phi}\sqrt{\omega_e^c\omega_h^c}\left((\omega_e^c+\omega_h^c)\cos\beta-i(1+\omega_e^c\omega_h^c-\omega_e^s\omega_h^s)\sin\beta\right)}{\left(\chi_e^{+}\chi_h^{+}+4\omega_e^c\omega_c^c\right)\cos 2\beta+2i\left(\chi_h^{+}\omega_e^c+\chi_e^{+}\omega_h^c\right)\sin 2\beta-\left(\chi_e^{-}\chi_h^{-}-4\omega_e^s\omega_h^s\right)},
\end{array}
\label{Eq.ReflectionAmplitudes}$$ where the notation is: $$\begin{cases}
\omega_{e(h)}^s=\eta_{e(h)}\sin\theta (\theta')\\
\omega_{e(h)}^c=\eta_{e(h)}\cos\theta (\theta')\\
\chi_{e(h)}^{\pm}=\eta_{e(h)}\pm 1
\end{cases}.
\label{Eq.ProbabilitiesVairables}$$ In zero mode regime, $\epsilon \rightarrow 0$, these reflection amplitudes reduces to Eq. (6) of the main text.
D. Josephson Current
====================
In order to obtain the supercurrent for $\Delta_5 | m | \Delta_5$ Josephson junction we use transfer matrix method to calculate the energy dependence of the Andreev bound states (ABS)[@Titov2006PRB]. In the short junction limit, $L \ll \xi_S$, when an $\uparrow$-spin electron hits the right interface of $\Delta_5|m|\Delta_ 5$ junction at $z=L$, the reflection amplitudes of Eq. (\[Eq.ReflectionAmplitudes\]) reduce to: $$\begin{cases}
r_{N,\uparrow\uparrow}^{(1)}=\frac{\cos (\beta ) \left(\eta ^2 \cos (2 \theta )-1\right)}{\left(\eta ^2+1\right) \cos (\beta )+2 i \eta \sin (\beta ) \cos (\theta )}\\
\\
r_{N,\uparrow\downarrow}^{(1)}=-\frac{2 e^{i \alpha } \eta ^2 \cos (\beta ) \sin (\theta ) \cos (\theta )}{\left(\eta ^2+1\right) \cos (\beta )+2 i \eta \sin (\beta ) \cos (\theta )}\\
\\
r_{A,\uparrow\uparrow}^{(1)}=\frac{2 i \eta e^{i (\alpha -\phi )} \sin (\theta ) \cos (\theta )}{\left(\eta ^2+1\right) \cos (\beta )+2 i \eta \sin (\beta ) \cos (\theta )}\\
\\
r_{A,\uparrow\downarrow}^{(1)}=-\frac{2 i \eta e^{-i \phi } \cos ^2(\theta )}{\left(\eta ^2+1\right) \cos (\beta )+2 i \eta \sin (\beta ) \cos (\theta )}\\
\\
\end{cases}.
\label{Eq.ShortJunctionFor1}$$ Here, $\eta=\sqrt{(\mu-m)/(\mu+m)}$. We use superscript $(2)$ when an $\downarrow$-spin electron hits the interface. Similarly, $(3)$ and $(4)$ are using for $\uparrow$-spin and $\downarrow$-spin hole, respectively. The first index in the doulbe spin indices indicates the spin configuration of incoming particle and the second one denotes the outgoing one. The primarily matrices $M_1$ and $M_2$ can be constructed by these 16 amplitudes as [@Titov2006PRB]: $$\begin{aligned}
M_1=
\left(
\begin{array}{cccc}
0 & 0 & -r_{A,\uparrow\uparrow}^{(3)} & -r_{A,\downarrow\uparrow}^{(4)} \\
0 & 0 & -r_{A,\uparrow\downarrow}^{(3)} & -r_{N,\downarrow\downarrow}^{(4)} \\
1 & 0 & -r_{N,\uparrow\uparrow}^{(3)} & -r_{N,\downarrow\uparrow}^{(4)} \\
0 & 1 & -r_{N,\uparrow\downarrow}^{(3)} & -r_{N,\downarrow\downarrow}^{(4)} \\
\end{array}
\right),\\
M_2=\left(
\begin{array}{cccc}
r_{N,\uparrow\uparrow}^{(1)} & r_{N,\downarrow\uparrow}^{(2)} & -1 & 0 \\
r_{N,\uparrow\downarrow}^{(1)} & r_{N,\downarrow\downarrow}^{(2)} & 0 & -1 \\
r_{A,\uparrow\uparrow}^{(1)} & r_{A,\downarrow\uparrow}^{(2)} & 0 & 0 \\
r_{A,\uparrow\downarrow}^{(1)} & r_{A,\downarrow\downarrow}^{(2)} & 0 & 0 \\
\end{array}
\right).
\label{Eq.PrimarilyTransferMatrix}\end{aligned}$$
Using these two matrices, the transfer matrix at the right interface which convert electron to hole wave functions can be calculated as: $$M_{e\rightarrow h}=M_1^{-1}M_2=\left(\begin{array}{cccc}
M_{11} & M_{12} & M_{13} & M_{14}\\
M_{21} & M_{22} & M_{23} & M_{24}\\
M_{31} & M_{32} & M_{33} & M_{34}\\
M_{41} & M_{42} & M_{43} & M_{44}\\
\end{array} \right)
\label{Eq.MeToh}$$ With the similar procedure, one can construct the transfer matrix in the left interface which can convect hole to electron wave function, $M'_{h\rightarrow e}$. Also, there are two another transfer matrices, $\{T_e, T_h\}$, which can be used for transferring electron or hole wave function from left interface to the right one. The energy of ABSs is then obtained from the condition, $$\det[\mathbf{1}-M'_{h\rightarrow e}T_h^{-1}M_{e\rightarrow h}T_e]=0.
\label{Eq.ABS01}$$ After some straightforward algebra we obtain the energy of ABSs as given in Eq. (8) of the main text. The derivation of Josephson current from ABSs is explained in the main text as well.
[99]{}
M. Titov and C. W. J. Beenakker, Phys. Rev. B. **74**, 041401 (2006).
|
---
abstract: 'The concurrence versus participation ratio phase diagram of the eigenstates of the quantum infinite range Heisenberg spin glass shows two distinct separate clouds. We show that the ‘special states’ that agglomerate away from the main one, are precisely those that are obtained by ‘promoting’ the eigenstates of a lower number sector of the Hamiltonian to particle added eigenstates of a higher number sector of the Hamiltonian. We compare the properties of these states with states of similar structure constructed from GOE random matrices that are easier to understand. In particular, we obtain the scaling behaviour of average entanglement of these special states with system size. By studying a power-law decay Hamiltonian, we see a merger of the main cloud into these special states as we go away from the infinite-range model and move towards short-range models. This could indicate that short-range quantum spin glasses are essentially different from the infinite range model.'
author:
- Arun Kannawadi
- Auditya Sharma
- Arul Lakshminarayan
bibliography:
- 'ref2014.bib'
title: 'Localized eigenstates with enhanced entanglement in quantum Heisenberg [spin-glasses ]{}'
---
Introduction
============
The last decade or so has seen a proliferation of interest in understanding quantum condensed matter systems from an entanglement perspective; for a somewhat early review see [@Amico08]. The notion of entanglement [@epr_paradox; @Schrodinger_Entanglement] and how to measure it has developed through the aid of a number of works within the general framework of quantum information. The condensed matter systems studied have concentrated around “clean" systems that display quantum phase transitions. Exceptions include work on the von Neumann, or block, entropy in disordered models where it has been shown that the kind of scaling found in corresponding clean critical chains ($\sim \log L$, for a chain of length $L$) persist and may even be enhanced in the presence of quenched disorder [@RefaelMoore09].
Another somewhat different but related tack of research has been on disordered systems from the point of view of quantum chaos and random matrix theories, for example [@Brown08]. Here the applicability of measures conventionally used for few-body quantized classically chaotic systems has been sought to be applied to many-body systems without an apparent classical limit, chaotic or otherwise. The rationale being that rather than a putative classical limit, it is the nonintegrable nature of the models that determine if random matrix theories maybe applicable. Nonintegrability could naturally be found in clean systems as well and such systems display some signatures conventionally attributed to quantum chaos, for example see [@Lak05; @Karthik07]. However the well-developed theories of random matrices [@Mehtabook; @Haakebook] especially the Gaussian ensembles, are applicable if there are many-body interactions that tend to make the Hamiltonians full matrices rather than the typical sparse matrices that arise out of the typically two-body interactions of many-body systems. The notion of “two-body-random ensembles" and “embedded ensembles" developed mostly within nuclear physics was precisely to plug this lacuna [@Kota01]. However its applicability to condensed matter systems is largely unexplored. It has also been pointed out that enhanced multipartite entanglement in disordered spin systems is possible that can be useful for multiport quantum dense coding [@Prabhu11].
The present work maybe seen in this context as an exploration of entanglement in disordered many body spin 1/2 particles. The disorder is via the interaction that is two-body type and long-ranged. The features in these systems maybe compared to what is expected from conventional random matrix theories. Also unlike most studies related to condensed matter this work looks at not just ground states but indeed excited states as well. In particular we will study single and two-particle (or magnon) sectors. There have been many interesting studies related to quantum communication across spin chains where such subspaces have played a dominant role, see for an overview [@Sougato07]. Being the simplest subspaces we also concentrate on these, although they may not contain the ground state. The kind of systems we study may then be either classified simply as long-ranged disordered Heisenberg models or quantum Heisenberg spin-glasses. Although spin glasses have been around for four decades, most studies have focussed on the classical version. See Talagrand [@talagrand2011mean] and references therein. Quantum spin glasses have also naturally been considered, for example [@antoine00]. A reason why quantum spin glasses have received less attention though is that numerical techniques to study quantum problems are less developed, as opposed to the classical world, where Monte Carlo methods are very advanced [@newmanbarkema]. The infinite-range Sherrington-Kirkpatrick spin glass is the prototype model in classical spin glasses [@sherrington75], the quantum version of which is the starting point of our study here.
Definite-particle states are natural to quantum systems that are pure and conserve particle number or total spin. A [$m-$particle$ $ ]{}state lies in the subspace of the Hilbert space which is spanned by the basis vectors that have $m$-number of ‘$1$’s (or equivalently ‘$0$’s) when expressed in spin-$z$ basis. In one-particle states (one spin up or one spin down) there is a clear monotonic relation [@ArulSub; @EntLoc1; @EntLoc2] between localization, for example as measured by the participation ratio, and the inter-spin entanglement as measured say by concurrence or tangle: the more the localization, the lesser the entanglement. While this has long been appreciated, the case of higher particle numbers or spins is more complicated. There is no such monotonic relationship between localization and entanglement even for two-particle states. To study this in the simplest statistical context, random definite particle states were studied using an ensemble that was uniformly distributed in such subspaces [@Vikram11]. From this it is known for example that while the expected entanglement between two spins (or qubits in the language of quantum information) for one particle states having $L$ qubits scales as $1/L$ and that of two-particle states scale as $1/L^2$, in the case of three or more particles entanglement is practically absent and is (super) exponentially small in $L$. This is consistent with such states having larger multipartite entanglement and the fact that entanglement moves away from being locally shared. In some sense the “environment" seen in such cases, by for example two given spins, is too large for entanglement to remain intact.
Here we make use of concurrence as a measure of inter-spin entanglement, and with the aid of numerical exact diagonalization and some analytical calculation, point out and explain a striking class of entangled states that arise from the consideration of the *quantum Heisenberg* [spin-glass ]{}. Our motivation comes from trying to understand some of these striking features observed in a recent hitherto unreported study [@Arun:2011] of entanglement in quantum [spin-glasses ]{}. In particular the two-particle sector states separate into two classes, whose entanglement scales very differently with the total number of spins $L$. Even those with a smaller amount of entanglement are still larger than those expected of random states. Thus these highly disordered Hamiltonians are still quite far from behaving as random states that are uniformly distributed in the definite particle subspace. Fig. \[fig0\] shows up-front two diagrams, one featuring ‘average concurrence’ and the other ‘particpation ratio’ of the ‘${\ensuremath{N_{\uparrow}}}=2$’ sector of the eigenstates of quantum Heisenberg infinite range spin glass. The illumination of the why-and-how and the consequences of these spikes observed is the core message of this paper.
(a){width="0.9\columnwidth"} (b){width="0.9\columnwidth"}
The structure of the paper is as follows. In Sec. \[sec:formulation\], we describe the [spin-glass ]{}Hamiltonian under study and the quantum properties of the system we are interested in. We also highlight the existence of a special class of eigenstates, which we call as ‘promoted states’. These states form the subject of this work. In Sec. \[sec:promoted\], we obtain a few analytical results for the quantities described in Sec. \[sec:formulation\] when the system is in a ‘promoted’ state, with a number of assumptions and compare with two specific [spin-glass ]{}models namely the Infinite range and [Nearest-Neighbour ]{}models. In Sec. \[sec:powerlawdecay\], we show systematically that the ‘promoted’ states, that were distinguishable from the rest of the eigenstates from an entanglement perspective in the case of the Infinite range [spin-glass ]{}model, become indistiguishable as the range of interaction between the spins becomes smaller. The last section summarizes the conclusions of our work, and offers an outlook for future work.
Formulation of the problem {#sec:formulation}
==========================
Symmetries of the [Spin-glass ]{}Hamiltonian
--------------------------------------------
We start by considering a large number $(L)$ of qubits, labelled arbitrarily from $1$ to $L$. The generic quantum Heisenberg model is given by $$\begin{aligned}
\label{eq:generic_hamiltonian}
{\ensuremath{H}}= \sum_{i<j}J_{ij}[\sigma_{i}^{x}\sigma_{j}^{x}+\sigma_{i}^{y}\sigma_{j}^{y}+\sigma_{i}^{z}\sigma_{j}^{z}] = \sum_{i<j} J_{ij} \vec{\sigma_i}.\vec{\sigma_j}\end{aligned}$$ where $J_{ij}$ represents the ‘interaction strength’ between the qubits $i$ and $j$. If all the $J_{ij}$ s are negative(positive), then the system is a ferromagnet(anti-ferromagnet). If $J_{ij}$ s have mixed signs, then the system is said to be ‘frustrated’ and no long-range order maybe found as in [@longrangeorder_absence], with a few exceptions, see for example [@longrangeorder_present].
A useful way to think of this Hamiltonian is to replace the $\vec{\sigma_i}.\vec{\sigma_j}$ with $2\hat{S}_{ij} - \mathbf{1}$, where $\hat{S}_{ij}$ is the swap operator that interchanges the $z$-component of the spins of qubits $i$ and $j$. It is immediately clear that the Hamiltonian connects only states of the same particle number (number of spin-up qubits). Thus, when expressed in the $\sigma_z$ basis (or Fock state basis), the Hamiltonian takes a block diagonal form, where each block is characterized by a definite “particle number" [$N_{\uparrow}$]{}, which is nothing but the total number of spins up in the $z$ direction. This is a great simplification since we can study only one or a few definite particle sectors at a time and the associated Hilbert space ${\ensuremath{\mathcal{H}}}_{{\ensuremath{N_{\uparrow}}}}$ is much smaller and grows only polynomially with $L$, the number of qubits. This definite particle structure is of course a manifestation of the fact that the system can have any of the $L+1$ allowed values for the total $z$-spin, but once a value is assumed, it is conserved until acted by a random external magnetic field. Mathematically, it is the consequence of rotational symmetry and in particular, of the fact that $\sigma_z= \sum_i \sigma_{i}^z$ commutes with the Hamiltonian. Considering the isotropy of the Hamiltonian, it also follows that the operators $\sigma^{\pm} = \sum_{i} \sigma_{i}^{\pm}$ too commute with the Hamiltonian. This implies that if $|\psi\rangle$ is an eigenstate of ${\ensuremath{H}}$, then $\sigma^{\pm}|\psi\rangle$ must also be an eigenstate of ${\ensuremath{H}}$ with the same eigenvalue. The particle added state $\sigma^+ |\psi\rangle$ is referred to as a “promoted" state corresponding to ${
\ensuremath{\left|\psi \right\rangle}
}$.
For any definite-particle sector of the Hamiltonian, one can show (readily from the swap operator form) that the state with equal coefficients for all the basis states is necessarily an eigenstate, with eigenvalue $S_J := \sum_{i<j} J_{ij}$. This corresponds to states promoted progressively from the [$0-$particle$ $ ]{}state ${
\ensuremath{\left|0 \right\rangle}
}^{\otimes L}$ and we will refer to such states as ‘all-one’ states of the appropriate particle number.
The two models of [spin-glass ]{}that we use intensively in this work are
1. Infinite-range model where $J_{ij} \sim \mathcal{N}(0,1)$ $\forall i<j$.
2. [Nearest-Neighbour ]{}model where $J_{ij} \sim \mathcal{N}(0,1)$ iff $j=i-1$ else 0, with a periodic condition such that $j=0$ corresponds to $j=L$.
Here, $\mathcal{N}(0,1)$ stands for the standard normal distribution. Also, we briefly study a one-dimensional power law decay model that contains the above two models as special cases.
Entanglement & Localization
---------------------------
As stated in the introduction, for [$1-$particle$ $ ]{}states, there is a direct relation between the localization of a state and the average entanglement between qubits [@ArulSub; @EntLoc1; @EntLoc2] while no definite relation is known to exist in the case of higher particle states. We will be interested in concurrence as a measure of bipartite entanglement and participation ratio (PR) as a measure of localization.
Concurrence is the entanglement between any two two-level systems in a mixed or pure state. Therefore it is a good measure of entanglement between any two qubits or spin-1/2 particles in a general many-body state. For definite particle states, it is known that the reduced density matrix of any two spins takes the following form [@Connor2001]: $$\begin{aligned}
\rho =
\begin{pmatrix}
v & 0 & 0 & 0\\
0 & w & z & 0\\
0 & z^{*} & x & 0\\
0 & 0 & 0 & y
\end{pmatrix}
\label{eq:rho}\end{aligned}$$ With $\rho$ of such a form, concurrence has a simple expression [@Connor2001]: $$\begin{aligned}
\label{eq:concurrence}
C(\rho) = \max\left({2(|z|-\sqrt{vy}),0}\right). \end{aligned}$$ Appendix \[app:algo\] describes a simple algorithm to do a fast computation of $C$ for Hamiltonians with this symmetry. Next, we recall that the inverse participation ratio is a basis dependent quantity that is defined as follows. If a state $|\psi\rangle = \sum_{i}a_{i}|i\rangle$, where $|i\rangle$ are the kets in the computational basis, then ${\text{IPR } }= \sum_{i} a_{i}^{4},$ from which the participation ratio $\text{PR} = \frac{1}{{\text{IPR } }}$ is immediately obtained. It is typical to consider the $S_{z}$ basis for spin-systems, and we do the same.
During our study of entanglement in the [spin-glass ]{}systems [@Arun:2011], in addition to the high-PR-high-concurrence all-one state, several other states with high concurrence were observed. But these states did not stand out when a similar plot of PR was made. Refer Fig. \[fig0\]. It is clear that the eigenstates would clearly separate out into two distinct clouds in a phase diagram where we plot the average concurrence against PR ($+$ points in Fig. \[fig1\]). The spikes at $E-S_J = 0$ in both Fig. \[fig0\] (a) and (b) correspond to the [$2-$particle$ $ ]{}all-one state. While no other significant spikes are found in Fig. \[fig0\](a), several such spikes are found in Fig. \[fig0\](b), with some of them being greater than the spike for all-one state. The authors were motivated by this observation to investigate further, which resulted in this work.
Note that the [$2-$particle$ $ ]{}all-one state is not the state with the highest average concurrence, while the [$1-$particle$ $ ]{}all-one state, with pairwise concurrence between any two qubits being $2/L$, has the maximum possible average pairwise entanglement [@ArulSub]. Next, we show that the these spikes are in fact the set of all ‘promoted’ eigenstates, arising due to the isotropy of the Hamiltonian.
Promoted States {#sec:promoted}
===============
Recollect that promoted states are of the form $\sigma^+{
\ensuremath{\left|\psi \right\rangle}
}$. It is easy to show that the operator $\sigma^+\sigma^-$ leaves the promoted states invariant, upto a scale factor. We mark those states that have this property in Fig \[fig1\] with boxes. Also we include data for random states and ‘promoted’ random states.
![Scatter plot the average concurrence vs. the participation ratio (PR)of all the eigenstates of the ${\ensuremath{N_{\uparrow}}}=2$ sector in infinite range quantum Heisenberg spin glass for $L=25$. \[fig1\] Data for *random* states and *random-promoted* states are included.](SG_Random_genuine_promo_L25.eps){width="\columnwidth"}
It is known [@Vikram11] that concurrence of a random [$2-$particle$ $ ]{}state ($N_{\uparrow}=2$) is given by $\frac{16}{L^{2}\pi^{3/2}}$, which with $L=25$ gives $\sim 0.005$. We infer that the eigenstates of these Hamiltonians tend to have higher concurrence and are more localized than random states (circles). Clearly, the distribution of points for the eigenstates arising from the Hamiltonian is very different from what is expected for random states. This immediately suggests that entanglement properties of random states, such as scaling behavior [@Vikram11] may not be found for eigenstates of spin systems, at least those with two-body interactions [@Arun:2011].
The most striking feature in the infinite-range model is that the eigenstates cluster into two separate clouds. From Fig. \[fig1\], we see that it is the promoted states, identified and labelled as boxes (over $+$ points), that stand out and form a separate cloud. These promoted states also tend to be distinct when we consider 2-qubit and 3-qubit entanglement in [$3-$particle$ $ ]{}sector as we verified (figure not included).
Random Promoted States {#section: Analytical}
----------------------
![$P(C>0)$ plotted against $L$. The dashed lines are the fits taking into account of these errorbars and the dotted-dashed lines are fits without taking the errorbars into acccount. Points with $L <8$ are not considered for finding fit parameters. []{data-label="fig:probability"}](probability_gnuplot.eps){width="\columnwidth"}
![Scaling behaviour of the promoted states compared against the scaling behaviour expected for random [$2-$particle$ $ ]{}states, which is $\frac{16}{L^2\pi^{3/2}}$ and random promoted [$2-$particle$ $ ]{}states and its estimate. ‘Analytical’ refers to curve $0.465/L$. The solid line corresponds to the upper bound given in Sec. \[subsec:heuristic\]. The fit parameters are given in Table \[table:table2\]. The y-axis is in logscale for better visibility.[]{data-label="fig:scaling"}](entanglement_scaling_gnuplot.eps){width="\columnwidth"}
The correlations in the matrix elements and the structure of the Hamiltonian make any analytical study of the [spin-glass ]{}eigenstates complicated. In order to understand why the ‘promoted’ nature makes states special, we study *random* promoted states which are amenable to an analytical approach. From the orthogonality condition applied with respect to the ‘all-one’ eigenstate, for all other eigenstates, the coefficients in the standard basis must add to zero. We will enforce this property extensively on the random promoted states to simplify our expressions. Random [$1-$particle$ $ ]{}states, by definition, span the surface of the unit sphere in the associated Hilbert space uniformly. In addition, we impose the constraint that the states must lie in the hyperplane orthogonal to the all-one state, to mimic the promoted states arising in [spin-glass ]{}systems. Thus, if we denote an unnormalized random [$1-$particle$ $ ]{}state as $${
\ensuremath{\left|\psi_{1p} \right\rangle}
} = \sum_i a'_i{
\ensuremath{\left|i \right\rangle}
},$$ where ${
\ensuremath{\left|i \right\rangle}
}$ refers to the [$1-$particle$ $ ]{}basis state where only the qubit at $i$ is in ${
\ensuremath{\left|1 \right\rangle}
}$ state and the rest in ${
\ensuremath{\left|0 \right\rangle}
}$, then the $a'_{i}$ s come from a j.p.d.f containing $\delta\left({\sum_i a_i'}\right)$ but we shall still consider them as i.i.d variables for all practical purposes. Isotropy in the Hilbert space is achieved when their marginals happen to be the standard normal distribution $\mathcal{N}(0,1)$, see for example [@WoottersRandom]. The normalization is assumed to only set the scale of the coefficients and the normalized coefficients can continued to be treated independent of each other [@Vikram11]. Although we restrict ourselves to the subspace that satisfies $\sum_{i}a_{i} = 0$ for theoretical analyses that follow, in practice, we do not include this condition in the Monte Carlo simulations.
The action of the ‘promotion-operator’, $\sigma^{+}$, on a [$1-$particle$ $ ]{}state is given by \_k a\_k [ $\left|k \right\rangle$ ]{} \_[i<j]{} a\_[ij]{} [ $\left|ij \right\rangle$ ]{}, a\_[ij]{} \~a\_i+a\_j. Here, $|ij\kt$ refers to the [$2-$particle$ $ ]{}basis state where the qubits at $i$ and $j$ are in ${
\ensuremath{\left|1 \right\rangle}
}$ state and rest in ${
\ensuremath{\left|0 \right\rangle}
}$. The [ ]{}of the promoted [$2-$particle$ $ ]{}state can be expressed in terms of the [ ]{}of the corresponding [$1-$particle$ $ ]{}state when $\sum_i a_i = 0$: $$\label{eq:ipr_relation}
\sum_{i<j}a_{ij}^4 = \frac{1}{(L-2)^2}\left((L-8) \sum_i a_i^4 + 3\right),$$ where $a_{ij} = (a_i+a_j)/\sqrt{L-2}$, are the normalized coefficients. In finding the normalization, the condition $\sum_i a_i = 0$ is used, which is the case for eigenstates of [spin-glass ]{}Hamiltonian. In a more general case, the normalization depends on the coefficients themselves. We observe in passing the somewhat amusing fact that whatever maybe the [$1-$particle$ $ ]{}state, when $L=8$ the promoted [$2-$particle$ $ ]{}state has an [ ]{}of exactly $1/12$, provided the coefficients sum to zero.
For random $N$ dimensional states, the average [ ]{}$\br I \kt \sim 3/N$ [@brody81]. Using this in Eq. \[eq:ipr\_relation\] for random [$2-$particle$ $ ]{}promoted states, the average [ ]{} I\_ \~,same as what we expect for “genuine” [$2-$particle$ $ ]{}random states, as the dimensionality of such a space is $\sim L^2/2$, and is confirmed by Fig. \[fig1\]. Thus as far as localization is concerned there is no difference, typically, between promoted and genuine [$2-$particle$ $ ]{}states. We find a very different situation regarding quantum correlations such as entanglement to which we now turn.
The elements of the two-spin reduced density matrix that are involved in the entanglement between them, as quantified by concurrence, are $z$, $v$, $y$ (refer to Eqs. \[eq:rho\] and \[eq:concurrence\]). For [$2-$particle$ $ ]{}states, when $\rho$ is the density matrix of spins at positions 1 and 2 (which we consider for simplicity and without any loss of generality), these elements are y=a\_[12]{}\^2 ,z= \_[k=3]{}\^[L]{} a\_[2k]{} a\_[1k]{} , v=\_\^[L]{}a\_[kl]{}\^2. Note that we are considering *real* state ensembles. We may expect that while all the three quantities are random variables, $y$ being a single term has more fluctuation than the other two which are sums. For generic random [$2-$particle$ $ ]{}states $\br |z|^2 \kt \sim 4/L^3$ and $\br |z| \kt^2=(2/\pi) \br |z|^2 \kt $ [@Vikram11].
However for [$2-$particle$ $ ]{}states promoted from [$1-$particle$ $ ]{}states obeying $\sum_i a_i = 0 $, it is straightforward to show that, up to the leading order, y (a\_1+a\_2)\^2/L, z (1+L a\_1 a\_2)/L, v 1. \[promoZY\] Therefore although $z$ appears as a sum of order $L$ number of terms, it simplifies for promoted states to this simple form which implies that both $|z|$ and $\sqrt{y}$ are of the same order of magnitude, namely $1/L$. This follows since $a_i \sim 1/\sqrt{L}$. As $v ={\cal O}(1)$, the concurrence in promoted [$2-$particle$ $ ]{}states is always in a fine balance between the two competing terms $|z|$ and $\sqrt{y}$. In contrast, for generic [$2-$particle$ $ ]{}states, the order of $\sqrt{y}$ is $1/L$ which is much larger than the order of $|z|$ which is $1/L^{3/2}$, resulting in the probability that the concurrence is nonzero decreasing as $1/\sqrt{L}$ [@Vikram11].
The probability that the concurrence is nonzero for promoted [$2-$particle$ $ ]{}states is now estimated. This is the same as $P(z^2>y)$, which using Eq. (\[promoZY\]) results in the following where $x_i = \sqrt{L}a_i$ are independently distributed according to ${\cal N}(0,1)$.
P(C>0)=P\[ (1-x\_1\^2)(1-x\_2\^2)>0\]=\
\^2 ( )+\^2 ( ) 0.566.
Note that as $v<1$ in reality, the above can be expected to underestimate the actual probability. It is also worth recounting that random [$1-$particle$ $ ]{}states have a probability 1 that the concurrence is nonzero.
Fig. \[fig:probability\] shows how $P(C>0)$ varies with $L$ for different systems. We fit a model of the form $p+q/L^r$ to the data points from Monte Carlo simulations for random promoted states (without enforcing $\sum_i a_i =0$), and the values for the parameters obtained are tabulated in Table \[table:table1\].
Parameter Estimate
----------- -------------------
$p$ $0.564 \pm 0.002$
$q$ $0.426 \pm 0.032$
$r$ $0.754 \pm 0.042$
: Best fit curve $p+q/L^r$ to $P(C>0)$ vs. $L$ for random promoted [$2-$particle$ $ ]{}states. Fits have been made for $L\ge8$.[]{data-label="table:table1"}
The probability of non-zero concurrence approaches from above the theoretical asymptotic value of $0.566$ slower than $1/L$. In sharp contrast, the probability of non-zero concurrence *increases* with $L$ for the [spin-glass ]{}systems, with the [nearest-neighbour ]{}model increasing faster than the infinite-range model. The $P(C>0)$ curve is described an exponential curve of the form $p - q \exp{\left(-L/r\right)}$. The functional forms are purely data driven and are not motivated by any physical argument.
Parameter Infinite range Nearest-Neighbour
----------- -------------------- --------------------
$p$ $0.660 \pm 0.004 $ $0.834 \pm 0.003$
$q$ $0.106 \pm 0.003$ $ 0.402 \pm 0.004$
$r$ $59.565 \pm 5.365$ $21.891 \pm 0.585$
: Best fit curves $p-q\exp{\left(-L/r\right)}$ to $P(C>0)$ vs. $L$ for promoted [$2-$particle$ $ ]{}states of Infinite range and [Nearest-Neighbour ]{}[spin-glass ]{}models. Fits have been made for $L \ge 8$.[]{data-label="table:SG_NN_probability"}
The average concurrence of random promoted [$2-$particle$ $ ]{}states may also be estimated as
C2 \_[\_i(1-x\_i\^2) >0]{} (|z|-)e\^[-(x\_1\^2+x\_2\^2)/2]{} dx\_1 dx\_2 \[eq:integral\]\
,
where $x_i = \sqrt{L}a_i$ and Eq. \[promoZY\] and the assumption of independent marginals have been used. The final result was obtained by setting $v=1$ and factoring out the $L$ dependence and the $L$-independent integral was evaluated numerically to obtain $0.465$. The $1/L$ behaviour is to be compared with generic [$2-$particle$ $ ]{}states that have an average concurrence $\sim 1/L^2$ [@Vikram11], and that for generic random [$1-$particle$ $ ]{}states which goes as $\sim 1/L$ [@ArulSub]. The promoted states have a smaller entanglement than [$1-$particle$ $ ]{}states, however they are much larger than what maybe expected for generic [$2-$particle$ $ ]{}states.
Interestingly, for the promoted [$0-$particle$ $ ]{}state i.e. the all-one state in the [$1-$particle$ $ ]{}([$N_{\uparrow}$]{}=1) sector, the concurrence between any two spins is $2/L$ and in [$2-$particle$ $ ]{}([$N_{\uparrow}$]{}=2) sector, it is $$C = \frac{2}{\binom{L}{2}} \left( L-2 -\sqrt{\frac{(L^2-5L+6)}{2}} \right),$$ which also scales as $1/L$.
Fig. \[fig:scaling\] shows how in our Monte Carlo simulations the average concurrence scales with the size for different systems. The exponent is smaller for the promoted [$2-$particle$ $ ]{}eigenstates when compared to promoted [$2-$particle$ $ ]{}random states. We fit a model $\langle C \rangle = b/L^a$ to the points ($L\ge8$) and obtain parameters displayed in Table \[table:table2\].
Model $a$ $b$
------------------------ -------------------- --------------------
Infinite Range $1.063 \pm 0.008$ $ 0.832 \pm 0.023$
[Nearest-Neighbour ]{} $0.758 \pm 0.003 $ $0.570 \pm 0.006$
Random Promo $1.138 \pm 0.004$ $0.900 \pm 0.014 $
: Best fit curve $b/L^a$ to $\langle C \rangle$ vs $L$. Fits have been made for $L\ge8$.[]{data-label="table:table2"}
Thus the promoted random [$2-$particle$ $ ]{}states retain the larger entanglement present in [$1-$particle$ $ ]{}states while being delocalized like [$2-$particle$ $ ]{}states. It is interesting that this feature of random states is present intact in the eigenstates of quantum spin glass Hamiltonians with long range interactions.
Heuristic understanding of the deviations {#subsec:heuristic}
-----------------------------------------
From Fig \[fig:scaling\], it is clear that the infinite range [spin-glass ]{}Hamiltonian and [nearest-neighbour ]{}(NN) Hamiltonian differ from each other and from the random promoted states. The average entanglement in the [spin-glass ]{}eigenstates is higher than the entanglement in random states, which can be related to the higher probability of non-zero concurrence. We can understand this as follows: In the case of [$1-$particle$ $ ]{}sector, eigenstates of the Hamiltonian are typically more localized than random states and as a result, less entangled. When promoted to the [$2-$particle$ $ ]{}sector, these states tend to be more entangled than the random promoted states.
Random states may be viewed as the ensemble of eigenstates of random matrices that belong to the Gaussian Orthogonal Ensemble (GOE). GOE matrices are symmetric, with off-diagonal terms being i.i.d and drawn from $\mathcal{N}(0,1)$ and the diagonal terms drawn from $\mathcal{N}(0,2)$, where as for the [spin-glass ]{}Hamiltonians under consideration, the variance of the diagonal terms is much larger when compared to that of the off-diagonal terms. The variance of the diagonal terms in the [$1-$particle$ $ ]{}sector of the [spin-glass ]{}Hamiltonian matrices are $ \sim L^2/2 $ times bigger than that of the non-zero off-diagonal elements. As a result, the diagonal elements end up being typically much larger in magnitude when compared to the off-diagonal terms and hence are ‘close’ to being a diagonal matrix, with the [Nearest-Neighbour ]{}model being ‘closer’ because of its sparse structure.
For a diagonal matrix, the basis vectors are its eigenvectors (in the non-degenerate case). Thus, we expect the energy eigenstates of the [spin-glass ]{}systems to be typically localized. As a toy problem, consider a [$1-$particle$ $ ]{}basis state i.e. where only one of the coefficients is non-zero and hence $1$ (upto a sign). The condition $\sum_i a_i =0$ is no longer applicable and the corresponding promoted state is given by $a_{ij} = (a_i + a_j)/\sqrt{L-1}$. We can factor out the state of the qubit with up spin, implying that that particular qubit is not entangled to the remaining $L-1$ qubits, that are in the maximally entangled (on average) all-one [$1-$particle$ $ ]{}state. Given such a promoted localized state, the probability of non-zero concurrence is $\binom{L-1}{2} / \binom{L}{2} \sim 1$ and the concurrence between any two pair of qubits, when not zero is $2/(L-1)$. Thus, the average concurrence $\langle C \rangle = \frac{2}{L-1}\frac{L-2}{L}$. This is plotted as the solid line in Fig. \[fig:scaling\]
However, in the [Nearest-Neighbour ]{}model, for large values of $L$, the diagonal elements tend to assume the eigenvalue corresponding to the all-one state. This leads to a high degeneracy and as a result, we have a high dimensional eigen sub-space corresponding to the eigenvalue $\sum_{i<j}J_{ij}$. Since the quantities of interest are basis dependent, it is no longer meaningful to discuss about the average concurrence and participation ratio of the eigenstates for large values of $L$.
The one-dimensional Power-Law Decay Model {#sec:powerlawdecay}
=========================================
(a){width="0.9\columnwidth"}(b){width="0.9\columnwidth"}\
(c){width="0.9\columnwidth"}(d){width="0.9\columnwidth"}\
(e){width="0.9\columnwidth"}(f){width="0.9\columnwidth"}
To study the differences between the infinite-range and [nearest-neighbour ]{}models, let us introduce a notion of distance that have been absent so far. Consider a family of Hamiltonians with the spins being put on a closed chain and where the interactions fall with distance as a power law, with perdioic boundary conditions imposed. The distance between the $i^{th}$ and $j^{th}$ spins is taken to be the length of the chord between the two sites, when all the sites are put on a circle [@sy:2011] given by $$r_{ij} = \frac{L}{\pi}\sin\Big[{\frac{\pi}{L}(i-j)}\Big].$$ The $J_{ij}$s defined in Eq. \[eq:generic\_hamiltonian\] obey $\mathcal{N}(0,1/r_{ij}^{\sigma})$. We obtain the infinite range Heisenberg [spin-glass ]{}when $\sigma=0$ and recover the $1$-dimensional nearest neighbour model asymptotically as $\sigma \rightarrow \infty$. The cloud of non-promoted eigenstates has a tendency to merge with that of promoted states as we increase $\sigma$ as shown in Fig \[fig:varying\_sigma\]. In the study of the same model with classical spins [@sy:2011], the regime $\sigma=(0,0.5)$ was identified as the infinite-range universality class, the regime $\sigma=(0.5,0.67)$ as the mean-field universality class and $\sigma=(0.67,1)$ as short-range. So, in that case anything greater than $\sigma=1$ is already in the super-short range regime, and the system should already have characteristics of the nearest-neighbor model. But in the quantum version of the model, even at $\sigma=2$, a faint separation of the clouds can still be discerned, and only at $\sigma \sim 2.5$, the figure is for all practical purposes, is same as what we obtain for nearest neighbour spin chain. Here we note that it is the lower cloud that merges into the cloud of promoted states. The position of the promoted eigenstates in the Concurrence-PR plot is relatively stable. Thus, the properties of these states are relatively robust and must not be too sensitive to our assumption of the model. This could be advantageous in some eventual quantum information application, since defects in physical realization of spin glass wouldn’t affect these states much.
There are two interesting aspects that come out from the phase diagram for large $\sigma$ (short-range). One is that apparently there are no distinct clouds. The second is that inspite of the absence of a clear separation between clouds, the ‘special eigenstates’ continue to occupy the higher concurrence - higher $PR$ region. A purely numerical approach can perhaps never really tell whether the merger is a true merger or if the clouds continue to exist, but only get arbitrarily close.
Conclusions and Future Work
===========================
We have discovered that in the general [spin-glass ]{}Hamiltonian, a special class of eigenstates form a distinct cloud in the concurrence-PR phase diagram. These special eigenstates, which we show to be ‘promoted-eigenstates’, display signfincantly higher concurrence compared with the rest of the eigenstates. As a first step to understand the peculiarities of ‘promoted-eigenstates’, we have studied the properties of random promoted states by an analytical approach. We find that the random promoted states are quite different from the promoted eigenstates of the Hamiltonian, which we atribute to the additional structures in the Hamiltonian. While we understand some deviations qualitatively, a lot of them still remain a mystery.
By considering a power-law decay model in $1$-d, we have shown that the ‘regular eigenstates’ tend to merge with the ‘special eigenstates’ as we move further and further away from the infinite-range case and move towards the short-range model. This is a significant observation because it could indicate that there is something inherently deferent between the infinite-range and short-range models in the quantum version of the model. If one speculates that there is some direct connection between the quantum and classical version of the Heisenberg spin glass, this could indicate that perhaps the RSB picture which is valid for the SK model, may not be applicable for the short-range models. The application of a uniform magnetic field to the quantum [spin-glass ]{}, in spite of breaking the symmetry with regard to the the $\sigma^{\pm}$ operators, preserves the set of eigenstates of the zero-field Hamiltonian, and therefore the promoted eigenstates continue to exist. However, if the applied external magnetic field is random over different sites, there would be no special promoted eigenstates. Some of our preliminary checks on the infinite-range Hamiltonian suggest that in fact, the clouds remain in tact in the presence of a uniform field, whereas they disappear with random fields. In a recent work [@sy:2010], it was shown that the so called Almeida-Thouless line of phase transitions exists for the infinite-range *classical* vector spin glass under the application of *random* fields. It would therefore be exciting to investigate if and what consequence the disappearance of these special promoted eigenstates has to the Almeida-Thouless line in *quantum* Heisenberg spin glasses. The AT line lies at the heart of the RSB-versus-droplet-picture debate in classical [spin-glasses ]{}, and it would be a significant avenue of research to study the same with quantum spins. Another future work [@Arun:2011] of interest for us involves the study of the impact of these promoted states in inhibiting the transition from power-law to exponential scaling as reported in Vijayaraghavan et al [@Vikram11].
We end this paper by suggesting an experimental technique to attain these promoted [$m-$particle$ $ ]{}states for applications that may required localized yet highly entangled states. Starting from the [$0-$particle$ $ ]{}eigenstate, one can adiabatically flip any one qubit and the system would be in the all-one state of the [$1-$particle$ $ ]{}sector. Now holding the particle number constant, we can ‘heat up’ or ‘cool down’ the system so that it is in any other [$1-$particle$ $ ]{}state. Now again, we could adibatically flip one of the qubits and obtain a promoted [$2-$particle$ $ ]{}state. By repeating the process of adibatic flipping and heating/cooling, we can obtain a promoted [$m-$particle$ $ ]{}state.
We thank Dr. Subrahmanyam for useful discussions. The Hamiltonian matrices are diagonalized using `Eigen` package [@eigenweb]. Curve fitting to find optimal parameters, including the errorbars, were done using SciPy’s `curve_fit` routine.
Algorithm for concurrence {#app:algo}
=========================
Computing the average concurrence of a given state turns out to be computationally the most expensive part of our code[^1]. Here we give a brief description of the algorithm we used to compute the essential elements of the reduced-density-matrix-of-any-two-spins $\rho$ en-route to computing the concurrence for systems with definite-particle-symmetry for any particle number.
Let us say we are interested in the concurrence between sites $i,j$ of some particular eigenstate $|\psi\rangle = \sum_k a_k {
\ensuremath{\left|k \right\rangle}
}$, where $|k\rangle$ are the basis states. For a given pair $i,j$, we first initialize all the elements of the reduced density matrix (which is real in our case) to $0$ and then loop over all $k$. For each state $|k\rangle$, we first find the states of the $i$ and $j$ sites. If both $i$ and $j$ have up(down) spins we add $a_{k}^{2}$ to $v$($y$). The off-diagonal element is a bit more complicated and requires another loop and a search algorithm within, but not difficult. If $i$ has an up(down) spin and $j$ has a down(up) spin, we do the following. We find all states in the basis which have a *down*(*up*) spin at $i$ and an *up*(*down*) spin at $j$, but are exactly identical to the state $|k\rangle$ at every other site. Let us call such states $|l\rangle$, which thus have coefficients $a_{l}$ in the eigenstate $|\psi\rangle$ of interest. We just add $a_k^2$ to $w (x)$ and add sum of the products $a_{k}a_{l}$ for all $l$ to $z$. Although, $w$ and $x$ are not required to compute concurrence, they may be required for other measures of entanglement, such as log-negativity or the Von-Neumann entropy that measures how the qubits $i$ and $j$ are entangled to rest of the system. Having thus computed $v$,$y$, and $z$, the concurrence is trivially obtained by the formula of Connor and Wootters [@Connor2001].
For the [$2-$particle$ $ ]{}states, this algorithm loops over all possible pairs of spins $(\sim \mathcal{O}(L^2))$, over all basis vectors $(\sim \mathcal{O}(L^2))$ and for $(L-2)$ cases out of $\binom{L}{2}$, it compares the $k^{th}$ basis with all other basis, which goes as $\mathcal{O}(L^2)$. In short, calculating the average concurrence for any given [$2-$particle$ $ ]{}state is $\mathcal{O}(L^5)$. A full analysis of a [$2-$particle$ $ ]{}states will involve diagonalization of [$N_{\uparrow}$]{}=2 sector of the Hamiltonian which is $\mathcal{O}(L^6)$. But it would be wise to obtain the eigenstates of [$N_{\uparrow}$]{}=1 sector of the Hamiltonian, which is $\mathcal{O}(L^3)$, and then ‘promote’ them to [$2-$particle$ $ ]{}states. One might also consider computing concurrence between randomly selected pairs of spins instead of all spins.
The most compact way to represent the basis vectors is to denote them by the positions of up spins, as denoted by ${
\ensuremath{\left|ij \right\rangle}
}$. But we quickly realized that generalizing it to an arbitrary definite-particle state is tedious. Instead, the basis are represented by integers and the binary strings of the basis vectors are encoded in bit representation of integers. In order to be able to use large values of $L$, the 128 bit integers in `C++ Boost`[^2] libraries were used. This is generic and any [$m-$particle$ $ ]{}basis vectors can be easily created.
[^1]: <https://github.com/arunkannawadi/spinglass>
[^2]: <http://www.boost.org>
|
---
abstract: 'In the ground state of [$\rm Ho_2Ti_2O_7$]{} spin ice, the disorder of the magnetic moments follows the same rules as the proton disorder in water ice. Excitations take the form of magnetic monopoles that interact via a magnetic Coulomb interaction. Muon spin rotation has been used to probe the low-temperature magnetic behaviour in single crystal [Ho$_{2-x}$Y$_{x}$Ti$_2$O$_7$]{} ($x=0$, 0.1, 1, 1.6 and 2). At very low temperatures, a linear field dependence for the relaxation rate of the muon precession $\lambda(B)$, that in some previous experiments on $\rm Dy_2Ti_2O_7$ spin ice has been associated with monopole currents, is observed in samples with $x=0$, and 0.1. A signal from the magnetic fields penetrating into the silver sample plate due to the magnetization of the crystals is observed for all the samples containing Ho allowing us to study the unusual magnetic dynamics of Y doped spin ice.'
author:
- 'L. J. Chang'
- 'M. R. Lees'
- 'G. Balakrishnan'
- 'Y. -J. Kao'
- 'A. D. Hillier'
title: 'Low-temperature muon spin rotation studies of the monopole charges and currents in Y doped Ho$_2$Ti$_2$O$_7$'
---
In the spin ice materials [$\rm R_2Ti_2O_7$]{} (R = Ho, Dy) [@Ramirez; @Harris; @Bramwell1] a large ($\sim\!10\mu_B$) magnetic moment on the R$^{3+}$ ions giving a strong, but at low temperature almost completely screened dipole-dipole interaction, together with a local Ising-like anisotropy leads to an effective nearest-neighbour frustrated ferromagnetic interaction between the magnetic moments. The organizing principles of the magnetic ground-state in spin ice, or “ice rules", require that two R$^{3+}$ spins should point in and two out of each elementary tetrahedron in the [$\rm R_2Ti_2O_7$]{} pyrochlore lattice [@Harris; @Bramwell2; @Hertog; @Yavorskii; @Isakov]. Excitations above the ground state manifold, which locally violate the ice rules, can be viewed as magnetic monopoles of opposite “magnetic charge" connected by Dirac strings [@Castelnovo; @Ryzhkin; @Jaubert]. Evidence of magnetic monopoles in spin ice has recently been observed in several experiments [@Fennell; @Morris; @Kadowaki].
Given the existence of magnetic monopoles, it is logical to consider the nature of the magnetic charges and any associated currents or “magnetricity". Bramwell *et al*. used transverse-field muon spin-rotation (TF-$\mu$SR) to investigate the magnitude and dynamics of the magnetic charge in [$\rm Dy_2Ti_2O_7$]{} spin ice [@Bramwell3]. In these experiments the equivalence of electricity and magnetism proposed in Ref. [@Castelnovo] was assumed and Onsager’s theory [@Onsager], which describes the nonlinear increase with applied field in the dissociation constant of a weak electrolyte (second Wien effect), was applied to the problem of spin ice. It was argued that in spin ice, if the magnetic field $B$ is changed, the relaxation of the magnetic moment $\nu_{\mu}$ occurs at the same rate as that of the monopole density and so in the weak field limit, $\nu_{\mu}(B)/\nu_{\mu}(0)=\kappa(B)/\kappa(0)=1+b/2$, where $\kappa$ is the magnetic conductivity and $b=\mu_0Q^3B/8\pi k_B^2T^2$ with a magnetic charge $Q$ [@Bramwell3]. At low temperature, the fluctuating local fields lead to a de-phasing of the muon precession and an exponential decay in the oscillatory muon polarization as a function of time $t$ $$A(t)=A_0\cos(2\pi \upsilon t)\exp(-\lambda t),
\label{Exponential decay}$$ where $A_0$ is the initial muon asymmetry, $\upsilon=\gamma_{\mu}B/2\pi$ is the frequency of the oscillations, and $\gamma_{\mu}$ is the gyromagnetic ratio. With $\nu_{\mu}(B)/\nu_{\mu}(0)=\lambda(B)/\lambda(0)$ one can directly infer the magnetic monopole charge. These measurements have proven intriguing and controversial. Dunsiger *et al*. [@Dunsiger] contend that the TF-$\mu$SR data never takes a form where $\lambda\propto\nu$ (see however [@Bramwell6]). It has also been suggested that the magnetic field at any muon implantation site in [$\rm Dy_2Ti_2O_7$]{} is likely to take a range of values up to 0.5 T [@Dunsiger; @Lago; @Blundell]. If this is the case it is difficult to understand how the fields of 1-2 mT used in Ref. [@Bramwell3] could lead to a precession signal. Both Dunsiger *et al*. [@Dunsiger] and later Blundell [@Blundell] have suggested that the signals seen in the $\mu$SR data in Ref. [@Bramwell3] originate from outside the sample. In their reply to this suggestion, Bramwell *et al*. [@Bramwell5] acknowledged that their experiments exploited both muons implanted in the sample (interior muons) and muons decaying outside the sample (exterior muons), with the aim of separating near and far field contributions to the signal. They went on to note that the signal at higher temperatures is dominated by muons implanted in the silver backing plate. This possibility was not discussed in their original paper [@Bramwell3]. Nevertheless, they continued to insist that the signal at low temperature ($0.4>T>0.07$ K) cannot be explained by exterior muons and that the Wien effect signal originates from muons within the sample or muons sufficiently close to the surface of the sample so as to probe the monopolar far field.
Results {#results .unnumbered}
=======
Fig. \[Fig1\] shows a TF-$\mu$SR time spectrum collected at 150 mK in a field of 2 mT for a pure [$\rm Ho_2Ti_2O_7$]{} sample. This curve is representative of the data collected during this study. A rapid loss in asymmetry from an initial value of $\sim0.22$ occurs outside the time window of the MuSR spectrometer [@Bramwell3; @Lago; @Blundell]. The slowly relaxing component of the data were fit using Eq. \[Exponential decay\].
![\[Fig1\]**TF-$\boldsymbol{\mu}$SR time spectrum collected at 150 mK in a field of 2 mT for a pure Ho$\boldsymbol{_{2}}$Ti$\boldsymbol{_2}$O$\boldsymbol{_7}$ sample.** These results are representative of the data collected during this study.](Chang_Figure1.eps){width="0.7\columnwidth"}
Fig. \[Fig2\] shows the temperature dependence of the muon relaxation rate $\lambda(T)$ for [Ho$_{2-x}$Y$_{x}$Ti$_2$O$_7$]{} extracted from fits to $\mu$SR time data collected in 2 mT, (see methods and [@SuppNote]). For all the samples containing Ho, a nearly $T$ independent $\lambda(T)$ is observed at low-temperature. As the temperature is raised there is a rapid increase in $\lambda(T)$ at some crossover temperature $T_{CR}$. This $T_{CR}$ increases from $\sim0.4$ K for the crystals with $x=1.6$ and 1.0 (data not shown) to 0.5 K for the samples with $x=0.1$ and 0.0. Above $T_{CR}$ the relaxation rate decreases with increasing temperature and has a similar $T$ dependence for all four samples containing Ho that were studied. For two samples ($x=0.1$ and 1.6) we also collected field-cooled-cooling data. In both cases a divergence between the zero-field-cooled warming (ZFCW) and the field-cooled cooling (FCC) curves appears at $T_{CR}$. For pure [$\rm Y_2Ti_2O_7$]{} a temperature independent relaxation rate is measured for the whole temperature range (0.05 to 5 K) studied.
![\[Fig2\] **Temperature dependence of the muon relaxation rate $\boldsymbol{\lambda(T)}$ extracted the fits to the TF-$\boldsymbol{\mu}$SR time spectra collected in 2 mT for samples of Ho$\boldsymbol{_{2-x}}$Y$\boldsymbol{_{x}}$Ti$\boldsymbol{_2}$O$\boldsymbol{_7}$ with $\boldsymbol{x=0}$, 0.1, 1.6 and 2.0.** The closed symbols show the zero-field-cooled warming data and the open symbols show the field-cooled cooling data.](Chang_Figure2.eps){width="0.7\columnwidth"}
In order to better understand the origins of these signals we have also collected relaxation data as a function of temperature in 2 mT for the pure [$\rm Ho_2Ti_2O_7$]{} sample discussed above, covered with a silver foil 0.25 mm thick. This thickness of foil is expected to stop all the muons before they reach the sample. Muons implanted in silver have a negligible relaxation and so any relaxation must result from a combination of the externally applied field and/or field lines originating from the sample penetrating into the silver. The $\lambda(T)$ curve obtained in this way (see [@SuppNote]) is very similar to the signal from the pure [$\rm Ho_2Ti_2O_7$]{} shown in Fig. \[Fig2\]a and demonstrates that at least some of the signal come from fields within the silver, but that these fields are the result of the magnetic properties of the sample [@SuppNote].
As a next step we then investigated the magnetic field dependence of muon relaxation rate. Fig. \[Fig3\] shows $\lambda(B)$ for a sample with $x=0$ at selected temperatures. Studies were also made for samples with $x=0.1$, 1, 1.6 and 2. Following Bramwell *et al*., linear fits to the $\lambda(B)$ data were made at each temperature. Using the gradient and intercept extracted from each fit, the effective magnetic charge $Q_{\rm{eff}}$ was obtained from $Q_{\rm{eff}}=2.1223m^{1/3}T^{2/3}$, where $m=(d\lambda(B)/dB)/\lambda_0$ [@Bramwell3]. For samples with $x=0$ and 0.1 the resulting values of $Q_{\rm{eff}}$ range from 4.5 to $7.5~\mu_B$Å$^{-1}$ in the temperature regime in which Onsager’s theory is expected to be valid, but increase rapidly as the temperatures increase outside this range (see Fig. \[Fig4\]).
![\[Fig3\]**Magnetic field dependence of the muon relaxation rate $\boldsymbol{\lambda(B)}$ for pure Ho$\boldsymbol{_{2}}$Ti$\boldsymbol{_2}$O$\boldsymbol{_7}$ at three different temperatures.** The values for $m=(d\lambda(B)/dB)/\lambda_0$ and the effective magnetic charge $Q_{\rm{eff}}$ shown in figure \[Fig4\] have been obtained from the straight line fits to the data.](Chang_Figure3.eps){width="0.6\columnwidth"}
![\[Fig4\]**$\boldsymbol{Q_{\rm{eff}}}$ versus $\boldsymbol{1/T}$ for samples of Ho$\boldsymbol{_{2-x}}$Y$\boldsymbol{_{x}}$Ti$\boldsymbol{_2}$O$\boldsymbol{_7}$ with $\boldsymbol{x=0}$ and 0.1.** The vertical dashed lines indicate the high and low temperature limits between which the Onsager theory is expected to be valid [@Bramwell3] and the horizontal line marks the value for $Q_{\rm{eff}}=4.6~\mu_B$Å$^{-1}$ [@Castelnovo]. The inset shows $m(T)$ for the same data; the solid line shows $m=Q_{\rm{eff}}^3/T^{2}$ with $Q_{\rm{eff}}=5~\mu_B$Å$^{-1}$. Also shown in both plots are the data of Bramwell *et al*. from Ref. [@Bramwell3].](Chang_Figure4.eps){width="0.7\columnwidth"}
At high temperature, a linear field dependence for $\lambda(B)$ is also observed for the two samples with a much higher yttrium doping ($x=1$ and 1.6) but the calculated $Q_{\rm{eff}}$ is always greater that $\sim10~\mu_B$Å$^{-1}$. For $x=1$ and 1.6 in the low-temperature regime $T < T_{CR}$ there is no systematic linear field dependence in $\lambda(B)$ and no signal that can be associated with magnetricity.
We have also looked for a linear magnetic field dependence in $\lambda(B)$ for the pure [$\rm Ho_2Ti_2O_7$]{} sample covered in a thick (0.25 mm) silver foil. At higher temperatures $T > T_{CR}$ we observed a linear behaviour leading to a large $Q_{\rm{eff}}$ (i.e. $Q_{\rm{eff}}> 10~\mu_B$Å$^{-1}$), but at low temperatures $T < T_{CR}$ we found no signature of magnetricity and could not obtain reliable linear fits to the $\lambda(B)$ data or physically acceptable values for $Q_{\rm{eff}}$.
Discussion {#discussion .unnumbered}
==========
We can draw a number of important conclusions from our work. Our results indicate that at higher temperatures, as suggested previously [@Dunsiger; @Blundell; @Bramwell5], the dominant contribution to the $\lambda(T)$ signal arises from stray fields from the magnetized spin ice that penetrate into the silver sample plate. The observation of a signal in a sample covered with thick Ag foil adds weight to this hypothesis. The sample coverage of the Ag backing plates used in our experiments was always approximately 50%. It will be interesting to explore how this signal changes as this coverage is varied. It may also be important to consider the ratio between the surface area and the volume of the spin ice in these and other experiments. Differences between the bulk and surface conductivity of water ice are well documented [@PetrenkoWhitworth] and it is likely that analogous processes operate in spin ice. In reply to the comments on their work, however, Bramwell *et al*. [@Bramwell5] make the point that a signal from muons implanted in the sample plate may not negate the important findings of their study. Our data are consistent with the suggestion made in Ref. [@Bramwell5] that the Wien effect signal may arise from inside the sample or from within the Ag sample plate but at distances very close to the spin ice sample surface. We will return to this point later. First we note that the $\lambda(T)$ curve for pure [$\rm Ho_2Ti_2O_7$]{} follows closely the form expected for the magnetization of pure spin ice [@Snyder2] supporting the view that $\lambda(T)$ reflects the magnetization in all the samples studied. This then raises an interesting question concerning the low-temperature magnetic dynamics of spin ice.
Recently there have been a number of experimental reports on the magnetic dynamics of spin ice (see for example [@Giblin; @Slobinsky; @Yaraskavitch; @Erfanifam; @Petrenko]). In addition to the discussion of magnetic monopoles and the Wien effect [@Castelnovo; @Ryzhkin; @Jaubert; @Bramwell3] authors have also considered the effects of thermal quenching [@Castelnovo2]. A key component of the current theories of spin ice, is that the magnetic response at low temperatures and small applied fields is limited to monopole motion. So as the monopole density decreases the characteristic time scales become longer. This view has recently been called into question following new low-temperature AC susceptibility measurements that exhibit an activated behaviour with energy barriers that are inconsistent with the present understanding of monopoles in spin ice [@Yaraskavitch; @Matsuhira; @Quilliam]. Our results for the $x=1.6$ material, showing the survival of ZFCW-FCC splitting in a sample with only 15% Ho add a further twist to this puzzle. Given the large number of non-magnetic “defects" on the corners of many of the tetrahedra in this diluted material, it is not easy to attribute the slow relaxation to a low monopole density. At such low concentrations of magnetic ions even the concepts of a spin ice and monopoles are questionable.
It is conceivable that single ion physics plays a more important role in the behaviour of the diluted materials. Our diffuse neutron-scattering studies of single-crystal [Ho$_{2-x}$Y$_{x}$Ti$_2$O$_7$]{} showed that at low temperature the scattering patterns are characteristic of a dipolar spin ice and appear to be unaffected by Y doping up to at least $x=1.0$ [@Chang]. One possible scenario is that effects, such as distortions in the local environment due to the variation in the size of the Ho$^{3+}$/Y$^{3+}$ ions [@Snyder1], produce energy barriers at low-$T$ that exceed the cost of an isolated monopole. The slow dynamics and the ZFCW-FCC hysteresis at low temperatures would thus cross over from a regime where this behaviour is attributed to low monopole density to a regime where it is due to exceedingly slow single ion physics. Alternatively, the long-range nature of the dipolar interactions may give rise to collective effects beyond the monopole description which introduce new energy barriers to spin flipping at very low temperatures that occur in both undiluted and diluted systems. The same qualitative form for the $\lambda(T)$ data for samples with $x=0.1$ and 1.6 indicate that additional ingredients may be required to explain the low $T$ behaviour in spin ice and that further studies on diluted samples are needed to fully understand the role played by factors such as impurities, dislocations, and surface effects on the low-temperature dynamics of spin ice.
Returning to the question of magnetricity in spin ice we note that in our $\mu$SR data the low-temperature signal that has previously been interpreted as a signature of magnetricity is seen in the $x=0$ and 0.1 samples and is not observed in the more dilute [Ho$_{2-x}$Y$_{x}$Ti$_2$O$_7$]{} materials. Within the $T$ range indicated by the dashed lines in Fig. \[Fig4\], where the theory presented by Bramwell *et al*. is expected to be valid, the value of $Q_{\rm{eff}}$ agrees with expectations. Following Blundell [@Blundell] we also plot $m$ versus $T$. We see that the expected $m\propto T^{-2}$ only holds for the same narrow $T$ range. Our experiments, including two separate runs on pure [$\rm Ho_2Ti_2O_7$]{} carried out three months apart, demonstrate the reproducibility of the data (see Fig. \[Fig2\]a). A realignment of the [$\rm Ho_2Ti_2O_7$]{} disks between runs also shows that the results are not particularly sensitive to the exact details of the sample geometry. Our results for the samples with a higher Y content and with the thick Ag foil demonstrate that the behaviour cannot be attributed to instrumental effects. The samples were made at Warwick [@Balakrishnan] and are Ho rather than Dy based pyrochlores, eliminating the possibility of material specific results.
In summary transverse-field $\mu$SR experiments on [Ho$_{2-x}$Y$_{x}$Ti$_2$O$_7$]{}, including measurements on non-magnetic [$\rm Y_2Ti_2O_7$]{} and a sample of [$\rm Ho_2Ti_2O_7$]{} covered in thick silver foil, suggest that the majority signal in the $\lambda(T)$ response comes from stray fields due the sample magnetization penetrating into the silver sample plate [@Dunsiger; @Blundell]. The results for [$\rm Ho_2Ti_2O_7$]{} are comparable with those observed for [$\rm Dy_2Ti_2O_7$]{}. The low-temperature ($T< T_{CR}$) linear field dependence in $\lambda(B)$ is only observed in samples with $x= 0$ and 0.1. In this low-temperature regime the value of $Q_{\rm{eff}}$ agrees quantitatively with the theory presented in Ref. [@Bramwell3]. The low-temperature hysteresis in $\lambda(T)$ for the magnetically dilute material ($x= 1.6$) appears inconsistent with the current understanding of monopoles in spin ice.
Methods {#methods .unnumbered}
=======
Single crystals of [Ho$_{2-x}$Y$_{x}$Ti$_2$O$_7$]{} ($x=0$, 0.1, 1, 1.6 and 2) were grown in an image furnace using the floating zone technique [@Balakrishnan]. The single crystal disks were glued on to a silver plate and covered with a thin (0.01 mm) sheet of silver foil to improve thermal conductivity. The plate was then attached to the cold stage of an Oxford Instruments $^3$He/$^4$He dilution refrigerator. Transverse-field muon spin-rotation experiments were performed using the MuSR spectrometer at the ISIS pulsed muon facility, Rutherford Appleton Laboratory, UK. The magnetic field was applied along the \[001\] direction, perpendicular to the initial direction of the muon spin polarization which was along a \[110\] axis. Measurements were carried out as a function of applied field at fixed temperature and as a function of temperature in a fixed magnetic field. See [@SuppNote] for full details of the measurement protocols.
[100]{}
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V., Moessner, R. & Sondhi, S. L. Why spin ice obeys the ice rules. *Phys. Rev. Lett.* **95**, 217201 (2005). Castelnovo, C., Moessner, R. & Sondhi, S. L. Magnetic monopoles in spin ice. *Nature* **451**, 42-45 (2008). Ryzhkin, I. A. Magnetic relaxation in rare-earth oxide pyrochlores. *J. Exp. Theor. Phys.* **101**, 481-486 (2005). Jaubert, L. D. C. & Holdsworth, P. C. W. Signature of magnetic monopole and Dirac string dynamics in spin ice. *Nature Phys.* **5**, 258-261 (2009). Fennell, T. *et al*. Magnetic Coulomb phase in the spin ice Ho$_{2}$Ti$_{2}$O$_{7}$. *Science* **326**, 415-417 (2009). Morris, D. J. P. *et al*. Dirac strings and magnetic monopoles in the spin Ice Dy$_{2}$Ti$_{2}$O$_{7}$. *Science* **326**, 411-414 (2009). Kadowaki, H. *et al*. Observation of magnetic monopoles in spin ice. *J. Phys. Soc. Jpn.* **78**, 103706 (2009). Bramwell, S. T. *et al*. Measurement of the charge and current of magnetic monopoles in spin ice. *Nature* **461**, 956-959 (2009). Onsager, L. Deviations from Ohm’s law in weak electrolytes. *J. Chem. Phys.* **2**, 599-615 (1934). Dunsiger, S. R. *et al*. Spin ice: magnetic excitations without monopole signatures using muon spin rotation. *Phys. Rev. Lett.* **107**, 207207 (2011). Bramwell, S. T., Generalized longitudinal susceptibility for magnetic monopoles in spin ice, *Phil. Trans. R. Soc. A.* **370**, 5738-5766 (2012) Lago, J., Blundell, S. J. & Baines, C. $\mu$SR investigation of spin dynamics in the spin-ice material Dy$_{2}$Ti$_{2}$O$_{7}$. *J. Phys.: Condens. Matter* **19**, 326210 (2007). Blundell, S. J. Monopoles, magnetricity, and the stray field from spin ice. *Phys. Rev. Lett.* **108**, 147601 (2012). See supplementary material for a complete description of the sample preparation, measurement protocols, and additional experimental results. Petrenko, V. F. and Whitworth, R. W. *Physics of Ice*, (Oxford University Press, Oxford, 1999). Bramwell, S. T. & Giblin, S. R. 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Balakrishnan, G., Petrenko, O. A., Lees, M. R. & Paul, D. M. Single crystal growth of rare earth titanate pyrochlores. *J. Phys.: Condens. Matter* **10**, L723-L725 (1998).
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by the EPSRC, United Kingdom (EP/I007210/1) and the National Science Council, Taiwan (grant no. NSC 101-2112-M-006-010-MY3). Some of the equipment used in this research was obtained through the Science City Advanced Materials project: Creating and Characterizing Next Generation Advanced Materials project, with support from Advantage West Midlands (AWM) and part funded by the European Regional Development Fund (ERDF). We would like to thank Stephen Blundell, Steve Bramwell, Claudio Castelnovo, and Sean Giblin for useful discussions.
Author contributions {#author-contributions .unnumbered}
====================
L.J.C and M.R.L. conceived of the project. G.B. prepared the samples. A.D.H, L.J.C, and M.R.L. planned and carried out the experiments. M.R.L, A.D.H, Y.J.K, and L.J.C helped to analyse the data, draft the paper, and prepare the figures. All the authors reviewed the manuscript.
Additional information {#additional-information .unnumbered}
======================
Competing financial interests: The authors declare no competing financial interests.
Supplementary Information {#supplementary-information .unnumbered}
=========================
Sample Preparation {#sample-preparation .unnumbered}
------------------
Single crystals of [Ho$_{2-x}$Y$_{x}$Ti$_2$O$_7$]{} ($x=0$, 0.1, 1, 1.6 and 2) were grown in an image furnace using the floating zone technique [@Balakrishnan1]. The cylindrical crystals were cut into circular disks $\sim\!6$ mm in diameter and $\sim\!1$ mm thick. These disks were oriented using the Laue x-ray diffraction technique and then glued, using GE varnish, in a circular pattern on to a silver sample plate as shown in Fig. \[Fig1\_Suppl\]. The samples were covered with a thin (0.01 mm) sheet of silver foil to improve thermal conductivity and mounted on the cold stage of an Oxford Instruments $^3$He/$^4$He dilution refrigerator.
![\[Fig1\_Suppl\] **Schematic of the experimental sample geometry used for the transverse-field $\boldsymbol{\mu}$SR measurements.** The single crystal disks were glued to a silver sample plate with GE varnish. The transverse field (TF) $H$ was applied along a \[001\] direction perpendicular to the initial direction of the muon spin polarization which was along a \[110\] axis.](Chang_Suppl_Figure1.eps){width="0.5\columnwidth"}
$\mathbf{\mu}$SR Experiments {#mathbfmusr-experiments .unnumbered}
----------------------------
Muon spin rotation ($\mu$SR) experiments were performed using the MuSR spectrometer at the ISIS pulsed muon facility, Rutherford Appleton Laboratory, United Kingdom.
For the magnetic field sweeps at fixed temperature the samples were zero-field cooled to a temperature well below the eventual measuring temperature, thermalised, and then slowly warmed in zero field to the required measuring temperature. A transverse external field was then applied. For these measurements each field point took approximately 15 minutes to collect. At the end of each field sweep the magnetic field was reduced to zero and the sample warmed to 4 K. Note, during zero-field cooling, the stray fields at the sample position were cancelled to less than 3 $\mu$T by three pairs of coils forming an active compensation system.
The temperature sweeps were always made following a field sweep measurement at base temperature. This means that in practice the samples were zero-field-cooled to the base temperature of the cryostat, thermalised and a field of 2 mT was applied in steps of 0.25 mT over a period of at least two hours. Data were then collected in zero-field-cooled warming (ZFCW) mode by warming the sample to each measuring temperature up to maximum of 4 K and then in field-cooled cooling (FCC) mode on subsequent cooling to each measuring temperature. Each point in these temperature scans took around 15 minutes to collect. Due to the large low-temperature hyperfine contribution to the specific heat for the samples containing holmium, the effective base temperature of the dilution refrigerator for these samples was limited to 100 mK.
Ho$_2$Ti$_2$O$_7$ covered in thick silver foil {#ho_2ti_2o_7-covered-in-thick-silver-foil .unnumbered}
==============================================
![\[Fig2\_Suppl\] **Temperature dependence of the muon relaxation rate $\boldsymbol{\lambda(T)}$ extracted from fits to the TF-$\boldsymbol{\mu}$SR time spectra collected in 2 mT during ZFCW for a sample of Ho$\mathbf{_{2}}$Ti$\mathbf{_{2}}$O$\mathbf{_{7}}$ covered with 0.25 mm thick silver foil**. At low temperature a fit using Equation \[Exponential decay\] gives $\lambda$. At higher temperature a two component fit using Equation \[Two Exponential decay\] gives $\lambda$ and $\lambda_2$.](Chang_Suppl_Figure2.eps){width="0.7\columnwidth"}
Muon spin rotation spectra for a sample of pure [$\rm Ho_2Ti_2O_7$]{} covered with a silver foil 0.25 mm thick were collected at fixed temperature in 2 mT. The temperature dependence of the muon relaxation rate $\lambda(T)$ extracted from fits to this data are shown in Fig. \[Fig2\_Suppl\].
For the low-temperature data ($T<T_{CR}$) the data were fit using $$A(t)=A_0\cos(2\pi \upsilon t)\exp(-\lambda t),
\label{Exponential decay}$$ where $A_0$ is the initial muon asymmetry, $\upsilon=\gamma_{\mu}B/2\pi$ is the frequency of the oscillations, and $\gamma_{\mu}$ is the gyromagnetic ratio.
In order to obtain satisfactory fits to the data above $T_{CR}$ the modified expression. $$A(t)=A_0\cos(2\pi \upsilon t)\exp(-\lambda t)+ A_2\exp(-\lambda_2 t)
\label{Two Exponential decay}$$ was used. The additional $A_2\exp(-\lambda_2 t)$ term is required to take account of the larger range stray fields within the thick silver foil. This is because the muon facility at ISIS has a significant momentum bite and so the implantation distance for the lower energy muons will be less than those with a higher energy.
[100]{}
Balakrishnan, G., Petrenko, O. A., Lees, M. R. & Paul, D. M. Single crystal growth of rare earth titanate pyrochlores. *J. Phys.: Condens. Matter* **10**, L723-L725 (1998).
|
---
abstract: 'We review the nature of the oscillations of main-sequence and supergiant stars of spectral type B. Seismic tuning of the interior structure parameters of the $\beta\,$Cep stars has been achieved since three years. The results are based on frequencies derived from long-term monitoring and progress in this area is rapid. Oscillations in mid-B stars as well as Be stars are well established by now, but we lack good mode identification to achieve seismic modelling. We provide recent evidence of g-mode pulsations in supergiant B stars. The spherical wavenumbers of their modes are yet unidentified, preventing seismic probing of such evolved hot stars at present. Improving the situation for the three groups of g-mode oscillators requires multi-site long-term high-resolution spectroscopy in combination with either space photometry or ground-based multicolour photometry. The CoRoT programme and its ground-based programme will deliver such data in the very near future.'
author:
- 'C. Aerts'
title: The pulsations and potential for seismology of B stars
---
Introduction
============
A large fraction of the stars of spectral type B is known to be variable. Since more than a century now, these variables have been divided in different classes, according to their periods and morphology of the lightcurves. In this review, we concentrate on those classes of variable B stars with established periodic variability resulting from stellar oscillations and situated near or above the main sequence. This concerns the classes of the $\beta\,$Cep stars, the slowly pulsating B stars, the pulsating Be stars and the pulsating supergiant B stars. For a review on the oscillations of subdwarf B stars, we refer to the paper by Fontaine (these proceedings).
Large inventories of pulsating B stars were established during the first part of the 20[*th*]{} century. These were mainly based on photographic spectroscopy (see \[1\] for one of the earliest review papers). The introduction of photo-electric photometry in the second half of the 20[*th*]{} century allowed much larger systematic survey campaigns, resulting in fainter class members among them cluster stars. The Hipparcos mission subsequently allowed the discovery of more than 100 bright periodic B stars \[2\]. Still today, new pulsating B stars are found, mainly from large-scale surveys, as we will discuss below for each class separately. These early survey works resulted in a fairly good statistics of the frequencies and amplitudes of the oscillations, but not beyond that.
As of the 1970s, the research of pulsating B stars extended towards the area of mode identification from observations. The motivation for this was that, at that time, the samples of pulsating B stars were large enough to delineate the observational instability strips, but no instability mechanism was known to explain the oscillations. Identification of the mode wavenumbers $(\ell,m)$ could therefore help to discover such a mechanism and to understand the mode selection. Mode identification was first mainly attempted from multicolour photometry using the method introduced by \[3\] and based on previous theoretical works by \[4\] and \[5\], \[6\]. The degree of the oscillation modes can be identified from amplitude ratios and/or phase differences (see, e.g., \[7\] for a review of this method and \[8\] for a recent improvement). Later on, from the mid 1980s, the possibility of performing high-resolution spectroscopy emerged from improved instrumental technology. This, in combination with the suggestion of \[9\] that one can compute theoretical line profiles for various kinds of nonradial oscillations, initiated a series of still ongoing efforts to obtain high spatial- and time-resolution spectroscopic observations of pulsating stars B with the specific aim to perform mode identification.
Meanwhile, the instability mechanism is well known. It is the $\kappa$-mechanism acting in the partial ionisation zones of the iron-like elements (see \[10\] for an excellent review). The mode selection, however, is still totally unknown to us.
It was only a few years ago that accurate enough frequencies, combined with unambiguous mode identification, became available for several nonradial modes in a few selected B stars which had been monitoring since many years. In this paper, we report on the current status of B star asteroseismology, highlighting the recent successes in the seismic interpretation of the interior structure parameters of the $\beta\,$Cep stars, and pointing out the difficulties yet to overcome to achieve the same success for other B-type pulsators.
$\beta\,$Cep stars
==================
The $\beta\,$Cep stars are a well-established group of near-main sequence pulsating stars. They have masses between 8 and about 18M$_\odot$ and oscillate in low-order p and g modes with periods between about 2 and 8h excited by the $\kappa\,$mechanism acting in the partial ionisation zones of iron-group elements \[11\]. The agreement between observed $\beta\,$Cep stars and the theoretical instability strip is very satisfactory for the class as a whole, although the blue part of the strip is not well populated \[12\]. Most of the $\beta\,$Cep stars show multiperiodic light and line profile variations. The majority of the $\beta\,$Cep stars rotate at only a small fraction of their critical velocity. An recent overview of the observational properties of the class is available in \[13\].
Recently, numerous new candidate members have been found from large-scale surveys, in the LMC and SMC \[14\] as well as in our own Galaxy \[15\], \[16\]. Assuming that all these faint variable stars are indeed $\beta\,$Cep stars more than doubles the number of class members to over 200. The occurrence of so many $\beta\,$Cep stars in environments with very low metallicity implies new unanticipated challenges to the details of the mode excitation, which relies heavily on the iron opacity.
The amplitudes and frequencies of the $\beta\,$Cep stars seem quite stable, although very few dedicated long-term studies are available. The B2III star 12Lac, e.g., was known to have six oscillation modes from photometry \[17\] and these same modes were recovered in high-resolution spectroscopy more than a decade later \[18\] and yet again, together with many more modes, in a recent multisite campaign \[19\]. The B3V star HD129929, on the other hand, was monitored during 21 years in 3-week campaigns from La Silla with one and the same high-precision photometer attached to the 0.70-m Swiss telescope \[20\]. This also led to the detection of six independent oscillation modes, with very small amplitude variability for the triplet frequencies only, if any. Suggestions for evolutionary frequency changes from O-C diagrams have been made, but we regard these as premature.
Significant progress in the detailed seismic modelling of the $\beta\,$Cep stars has occurred since a few years. While such modelling was already attempted a decade ago for the stars 16Lac \[21\] and 12Lac \[22\], doubtful mode identification prevented quantitative results. It took until the exploitation of the 21-yr single-site multi-colour data set of the star HD129929 to discover that standard stellar models are unable to explain that star’s oscillation behaviour. Indeed, from the modelling of three identified $m=0$ modes, \[20\] derived a core overshoot parameter of $0.10\pm 0.05$H$_{\rm p}$ (with H$_{\rm p}$ the local pressure scale height) and proved the star to undergo non-rigid internal rotation from the splitting within an $\ell=2$ and an $\ell=1$ mode, with the core rotating four times faster than the envelope. For details, we refer to \[23\] and \[24\].
This modelling result was soon followed by the one derived for the B2III star $\nu\,$Eri, which was the target of a 5-month multisite photometric and spectroscopic campaign. Numerous new frequencies were found and identified compared to the four known before the start of the campaign \[25\], \[26\], \[27\]. The modelling was done by two independent teams using different evolution and oscillation codes. This led to different results depending on the number of fitted $m=0$ components (three $m=0$ modes were fitted by \[28\] while four by \[29\]). The main and far most important conclusion was, however, the same for both studies: current seismic models do not predict all the observed modes of $\nu\,$Eri to be excited. One needs a factor four enhancement in the iron opacity, either locally in the driving region, or globally in the star, to solve this excitation problem. This led to the suggestion to include radiative diffusion in the models to solve this outstanding issue, in analogy to the subdwarf B pulsators \[30\]. Promising first attempts to compute main-sequence B-star models including diffusion were made by \[31\]. They found that the diffusion effects do not alter the frequency values in a significant way, but have indeed the potential to solve $\nu\,$Eri’s excitation problem (or, better phrased: our inability to explain its mode excitation …).
Meanwhile, two more $\beta\,$Cep stars were modelled seismically, each of them having two well-identified frequencies. The example of $\beta\,$CMa is illustrative of the power of asteroseismology: having two well-identified oscillation modes in a slow rotator is sufficient to derive a quantitative estimate of the core overshoot parameter, which was found to be $d_{\rm ov}= 0.15\pm 0.05$H$_{\rm p}$ for this somewhat evolved B2III $\beta\,$Cep star. The way this is achieved, is illustrated nicely in Fig.\[anwesh\], taken from the paper by \[32\]. Because the frequency spectra of $\beta\,$Cep stars are so sparse for low-order p and g modes, one does not have many degrees of freedom to fit the well-identified modes. This is why we can put limits on internal structure parameters as shown in Fig.\[anwesh\], of course assuming that the input physics of the models is the correct one. A similar, but less stringent constraint was derived for the B2IV star $\delta\,$Ceti from a combination of MOST space photometry and archival ground-based spectroscopy \[33\].
Additional multisite campaigns have been done for the stars $\theta\,$Oph \[34\], \[35\], 12Lac \[19\] and V2052Oph (Handler, unpublished). These have a somewhat higher projected rotation velocity, and it would be interesting to know if the range of values found so far for the core overshoot parameter and the level of non-rigidity of the internal rotation remains valid for them. The modelling is ongoing at present.
Slowly pulsating B stars
========================
The term “slowly pulsating B stars” (SPB stars) was introduced by \[36\], after years of photometric monitoring of variable mid-B stars with multiperiodic brightness and colour variations. After a few years, the Hipparcos mission led to a tenfold increase in the number of class members \[2\]. Subsequent huge long-term multicolour photometric and high-resolution spectroscopic follow-up campaigns concentrated on the brightest new class members found from Hipparcos \[37\], \[38\] and resulted in a much better understanding of the pulsational and rotational behaviour of the class members \[39\]. Accurate frequencies and mode identification are available for some 15 members \[40\], \[41\]. The mode identification results are in excellent agreement with theoretical computations made by \[42\] predicting mainly dipole modes to be excited. All confirmed SPB stars are slow rotators \[39\].
In Fig.\[ovel\] we show as an illustration the frequency spectrum of the Geneva $B$ and Hipparcos light, and radial velocity variations of the brightest among the SPB stars, $o\,$Vel (B3IV). Despite the long-term monitoring of almost two decades in photometry, \[40\] found only four independent frequencies for this star. This is typical for single-site ground-based data of main-sequence stars with gravity modes, because the latter have periodicities ranging from 0.8 to 3d. This leads to severe alias problems, as illustrated in Fig.\[ovel\] where the confusion between frequencies $f$ and $1-f$ is prominent. Only with multisite data, or, even better, with uninterrupted data from space, can one avoid such confusion. This is illustrated nicely by the MOST light curve (reproduced in Fig.\[hd163830\]) of the new SPB star HD163830 discovered by that mission \[43\]. This lightcurve implied a five-fold of the number of gravity modes in one star compared to the best ground-based datasets for such pulsators.
As for the $\beta\,$Cep stars, numerous new SPB stars (some 70) were discovered in the Magellanic Clouds from OGLE and MACHO data \[14\]. The number of class members is therefore about 200 at the time of writing (assuming all the Magellanic Clouds variables to have been classified correctly). Trustworthy mode identification is only available for the highest-amplitude frequency of a handful of SPB stars, however, and it concerns only the spherical wavenumbers of the dominant mode \[41\]. This is why seismic tuning of the interior structure of SPB stars has not been achieved so far.
Pulsating Be stars
==================
Be stars are Population I B stars close to the main sequence that show, or have shown in the past, Balmer line emission in their photospheric spectrum. This excess is attributed to the presence of a circumstellar equatorial disk. See the review on Be stars by \[44\] for general information on this rather inhomogeneous class of stars. Magnetic fields \[45\] and nonradial oscillations \[46\] have been detected in some Be stars. It is unclear at present if these mechanisms are able to explain a disk for the whole class of Be stars.
Be stars show variability on very different time scales and with a broad range of amplitudes. \[47\] studied a subclass of the Be stars showing one dominant period between 0.5 and 2d in their photometric variability, with amplitudes of a few tens of a mmag which he termed the $\lambda\,$Eri variables. He provided extensive evidence of a clear correlation between the photometric period and the rotational period of the $\lambda\,$Eri stars and interpreted that correlation in terms of rotational modulation. When observed spectroscopically, several of the $\lambda\,$Eri stars turn out to have complex line profile variations with travelling sub-features similar to those observed in the rapidly rotating $\beta\,$Cep stars, except for the much longer periods (days versus hours). This rather seems to suggest oscillations as origin of this complex spectroscopic variability.
Nonradial oscillations were already discovered in the Be star $\omega\,$CMa \[48\], a star listed among the $\lambda\,$Eri variables in \[47\]’s list. An extensive summary of the detection of short-period line profile variations due to oscillations in hot Be stars is provided in \[46\]. They monitored 27 early-type Be stars spectroscopically during six years and found 25 of them to be line profile variables at some level. Some of their data are shown in a grey-scale plot in Fig.\[rivi\]. For several of their targets the variability was interpreted in terms of nonradial oscillations with $\ell=m=+2$. Almost all stars in the sample also show traces of outburst-like variability rather than a steady star-to-disk mass transfer. The authors interpreted the disk formation in terms of multimode beating in combination with fast rotation.
The view on pulsating Be stars became more complicated when \[49\] introduced the class of $\zeta\,$Oph variables. These are late-O type stars with clear complex multiperiodic line profile variations which he attributed to high-degree nonradial oscillations. They are named after the prototypical O9.5V star $\zeta\,$Oph, whose rotation is very close to critical and whose photometric variability was recently firmly established by the MOST space mission. \[50\] disentangled a dozen significant oscillation frequencies in the 24-d photometric light curve assembled from space. These frequencies range from 1 to 10d$^{-1}$ and clearly indicate the star’s relationship to the $\beta\,$Cep stars.
Multiperiodic oscillations were recently also reported in the rapidly rotating B5Ve star HD 163868 from a 37-d MOST light curve. \[51\] derived a rich frequency spectrum, with more than 60 significant peaks, resembling that of an SPB star and termed the star an SPBe star in view of its Be nature. They interpreted the oscillation periods between 7 and 14h as high-order prograde sectorial g modes and those of several days as Rossby modes (e.g. \[52\] for a good description of such modes). There is remaining periodicity above 10d which cannot be explained at present. Finally, nonradial oscillations at low amplitude were also detected in the bright B8Ve star $\beta\,$CMi \[53\].
As for the SPB stars, seismic modelling of the interior structure of Be stars has not yet been achieved, in this case by lack of enough frequencies, of frequency accuracy, of unambiguous mode identification and of appropriate stellar models for rapid rotators.
Pulsating B supergiants
=======================
Oscillations have not yet been firmly established in luminous stars with $\log
L/L_{\odot}>5$ and $M>20$M$_{\odot}$, although they are predicted in that part of the HR diagram. \[12\] and \[54\] predicted SPB-type g modes to be unstable at such high luminosities for respectively pre- and post-TAMS models (Fig.\[karolien\]).
\[2\] discovered a sample of B supergiants to be periodically variable with SPB-type periods from the Hipparcos mission. These stars, and additional similar ones, were subjected to detailed spectroscopic and frequency analyses by \[55\], who found their masses to be below 40M$_\odot$ and photometric periods between 1 and 25d. The stars were found to be situated at the high-gravity limit of $\kappa$-driven pre-TAMS g-mode instability strip (\[12\], see Fig.\[karolien\]). This implies that the interpretation of their variability in terms of nonradial g-mode oscillations excited by the $\kappa\,$mechanism, as first suggested by \[2\], is plausible.
A new step ahead in the understanding of these stars was achieved by \[54\], who detected both p and g modes in the B2Ib/II star HD163899 from MOST space-based photometry. The authors deduced 48 frequencies below 2.8d$^{-1}$ with amplitudes below 4mmag and computed post-TAMS stellar models and their oscillation frequencies which turn out to be compatible with the observed ones.
Further research is needed to evaluate if seismic modelling in terms of internal physics evaluation of these SPB supergiants, as \[54\] termed their target, is feasible.
Discussion and Future Prospects
===============================
The classes of the $\beta\,$Cep and SPB stars are now well established, containing more than 200 members each. Four of the brightest and slowest rotators among the $\beta\,$Cep stars have been modelled seismically since 2003, resulting in stringent constraints on the core overshoot parameter of $d_{\rm
ov}\in [0.05\pm 0.05,0.20\pm 0.05]$H$_{\rm p}$. Note that this range is lower than the one found from a handful of eclipsing binaries with a B-type star \[56\], implying that the latter probably also experience rotational mixing near their core, which mimics additional core overshoot. In two stars (besides the Sun), seismic evidence for non-rigid internal rotation was established. Both these stars have a core spinning faster than the envelope, one with a factor three and the other one with a factor four. This was derived from the computation of the Ledoux splitting coefficients, after successful seismic modelling of the zonal components of observed frequency multiplets, and a confrontation with the high-precision observed values of these coefficients. We conclude that asteroseismology of $\beta\,$Cep stars has been highly successful during the past few years, and its future looks very promising given that several multisite campaigns of moderate rotators have been done but are not yet exploited and CoRoT will be launched very soon.
Between one and five frequencies of g modes have been established in the brightest among the SPB stars, from long-term photometric and spectroscopic campaigns. This is rather disappointing, given the large observational effort that went into this result. The example of the SPB star HD163830 observed by MOST makes it clear that one needs photometry from space with a high duty cycle to make efficient progress in the detection of frequencies for these stars. The same holds true for the g modes in Be stars and B supergiants. We are eagerly awaiting the results from CoRoT in this respect.
The oscillations detected in Be stars and very-late Oe stars show a multitude of different behaviour, which is in full accordance with the one of $\beta\,$Cep stars and SPB stars. It seems that pulsating Be stars are complicated analogues of the SPB stars, while the $\zeta\,$Oph stars undergo the same oscillations than $\beta\,$Cep stars, but the members of both these classes having emission lines in their spectrum rotate typically above half of the critical velocity, with some rotating very close to critical velocity. It remains to be studied what the role of the oscillations is in the disk formation for the class of Be stars as a whole.
Probing of B supergiant models has recently come within view, with the discovery of nonradial g modes in such a star by the MOST mission. This case study is complemented by the interpretation of the variability of the Hipparcos lightcurves of a sample of some 40 B supergiants in terms of g modes. These two entirely independent studies open the upper part of the HR diagram for seismic tuning of stellar evolution models of supergiant stars, which are the precursors of stellar black holes. At present, none of the existing analysis codes include the effects of a radiation-driven stellar wind, which would be the next step towards apropriate modelling of detected oscillation frequencies in such stars.
By far the largest stumbling block in the application of asteroseismology to g-mode pulsators among the B stars is the lack of unambiguous mode identification and good models including rotation in a consistent way. On the observational side, this can only be resolved from coordinated initiatives, because it requires long-term multisite multitechnique campaigns, including multicolour photometry and high-resolution spectroscopy. Space photometry has the potential of detecting a much higher number of oscillations than ground-based photometry, as the MOST mission has shown us and will hopefully continue to do so. However, it cannot deliver the badly needed mode identification, because we do not have the comfort of dealing with frequency spacings as in solar-like oscillators. Moreover, the rotational splitting is of the same order or even larger than the separation between zonal g-mode frequencies of subsequent radial order, implying that the measured frequency spectrum is insufficient to unravel the nature of the detected modes. On the theoretical side, it is fair to state that we do not have appropriate seismic models for stars rotating at a considerable fraction of their critical velocity. Moreover, it was recently discovered that half of the SPB stars turn out to have a magnetic field \[57\], such that not only the Coriolis is important for such pulsators, but likely also the Lorentz force.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author is supported by the Fund for Scientific Research of Flanders (FWO) under grant G.0332.06 and by the Research Council of the University of Leuven under grant GOA/2003/04. She is very grateful to the organisers for giving her the opportunity to present this work at the meeting.
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---
abstract: 'W. B. Jordan’s conclusion that the quadruple principal value integral in problem 89-2 vanishes does not hold. The error sneaks in through a contribution of a subintegral which impedes some sign symmetry with respect to the master parameter (the Fermi radius) and which was overlooked in the published solution. In summary, the original problem of solving the quadruple integral remains unsolved.'
author:
- 'Richard J. Mathar'
bibliography:
- 'all.bib'
title: 'Erratum to “Solutions problem 89-2: On the principal value of a quadruple integral”, SIAM Rev. 32 (1990) 143.'
---
Statement of the Task
=====================
The manuscript is concerned with the evaluation of the principal value of the quadruple integral [@GlasserSIAMR31] $$F(a) = P\int_{-\infty}^\infty \frac{dx}{x^2}
\int_{-\infty}^\infty \frac{dx'}{x'}
\int_{-1}^1 dy
\int_{-1}^1 dy'
\frac{x-x'}{(x-x')^2+(y+y')^2}\Delta(x,y)\Delta(x',y')
.
\label{eq.Fdef}$$ The function $\Delta(x,y)$ is defined to be $+1$ in the moon-shaped region inside the unit circle centered at $x=a$, and it is $-1$ in the moon-shaped mirror region inside the unit circle centered at $x=-a$. In the infinite exterior region and in the lens-shaped region around the center, where the two circles overlap, $\Delta(x,y)$ is zero. The two connected regions that contribute to the integral are illustrated in Figure \[fig.circ\]. They touch each other at two common points on the vertical axis.
![The two moon-shaped regions of integration for $x'$ and $y'$ and for $x$ and $y$, one filled with bubbles, the other with hatching depending on the sign. The circles intersect the vertical axis at $y,y'=\pm \sqrt{1-a^2}$. []{data-label="fig.circ"}](circ)
The two circles represent Fermi disks in the application to solid state physics [@MaldagueSSC26; @RajagopalPRB15; @GeldartCJP48_155]; this is not relevant to the further calculation.
The effect of switching the sign of the parameter $a$ is $$\Delta(x,y) \to -\Delta(x,y);\quad
\Delta(x',y') \to -\Delta(x',y').$$ So the product $\Delta(x,y)\Delta(x',y')$ is invariant towards changing the sign of $a$, and $$F(a)=F(-a)
.$$ In that sense one only needs to consider $a\ge 0$ for the rest of the calculation.
Criticism of Jordan’s conclusion
================================
Jordan’s calculation [@JordanSIAMR32] argues that the double inner integral $I(x,y)$ of $F$, an integral over the full right circle of $x'$ and $y'$ in Figure \[fig.circ\], is an even function of $a$. This would cause the entire integral to vanish at the time when the sum over both circles is involved, because $\Delta(x',y')$ is an odd function of $a$.
The error appears implicitly in the step $$\begin{aligned}
I_2/\pi &=& \int_0^a (A-Be^{-u})\operatorname{csch}u dr + \int_a^1 B dr
\\
&=& \int_0^a (A\operatorname{csch}u -B \coth u)dr +\int_0^1 Bdr,\end{aligned}$$ in the 4-th but last equation on page 144, which splits $\int_0^a B dr$ off the left integral and unites it with the right integral. Although formally correct, this step *only* applies to the cases where $a\ge 0$. If $a$ is *negative*, the upper limit of the first and the lower limit of the second integral in the first of these two lines must be clamped to zero; this is basically a consequence of the role of $r$ as a radial circular coordinate which cannot become negative.
By an equivalent reasoning, the step is not correct if $a>1$, because then the upper limit on $r$ is 1 and the second integral $\int_a^1 Bdr$ should not contribute at all.
We see that for negative $a$ the first integral $\int_0^a (A-Be^{-u})\operatorname{csch}u dr$ should not contribute at all and the second be changed to $\int_0^1 Bdr$, so the contribution of $\log[(p+2a)/p']$ to $I_2/\pi$ in the last equation of page 144 must be dropped for negative $a$.
As a consequence, Jordan’s final equation $$I=\frac{\pi}{2}\log(2x)^4[(x+iy)^2+1-a^2][(x-iy)^2+1-a^2]$$ is invalid whenever $a$ is negative. Although $I$ in that form is an even function of $a$, its validity is restricted to $a>0$—and for the congruential reason concerning the upper limit also to $a<1$. The cancellation claimed by Jordan when the values at positive and negative $a$ are subtracted while calculating the double integral over $x'$ and $y'$ is simply inhibited because the values of $I_2$ at negative $a$ are not those obtained by symmetric (even) extrapolation.
Glasser’s quest [@GlasserSIAMR31] of obtaining values of the quadruple integral—perhaps not analytically but merely with satisfactory numerical methods—remains unanswered so far.
|
---
abstract: 'In star forming galaxies, dust plays a significant role in shaping the ultraviolet (UV) through infrared (IR) spectrum. Dust attenuates the radiation from stars, and re-radiates the energy through equilibrium and non-equilibrium emission. Polycyclic aromatic hydrocarbons (PAH), graphite, and silicates contribute to different features in the spectral energy distribution; however, they are all highly opaque in the same spectral region – the UV. Compared to old stellar populations, young populations release a higher fraction of their total luminosity in the UV, making them a good source of the energetic UV photons that can power dust emission. However, given their relative abundance, the question of whether young or old stellar populations provide most of these photons that power the infrared emission is an interesting question. Using three samples of galaxies observed with the Spitzer Space Telescope and our dusty radiative transfer model, we find that young stellar populations (on the order of 100 million years old) dominate the dust heating in star forming galaxies, and old stellar populations (13 billion years old) generally contribute less than 20% of the far-IR luminosity.'
author:
- 'Ka-Hei Law'
- 'Karl D. Gordon'
- 'K. A. Misselt'
title: 'Young, UV-bright stars dominate dust heating in star forming galaxies'
---
INTRODUCTION {#intro}
============
The infrared radiation from most star-forming galaxies is dominated by emission from dust grains heated by absorbed stellar energy. Dust emission is powered by absorption of radiation from ionizing and non-ionizing stars. Dust is most efficient at absorbing photons in the ultraviolet (UV) as the relative optical depth of dust is the highest in the UV [@Gordon2003]. Only early type (O and B) stars produce significant amounts of UV photons; however, these hot massive stars have short lifetimes (less than 100 million years) and are formed in relatively small numbers compared to less massive, less luminous, and cooler stars that produce very few UV photons. Given the initial mass function and evolutionary history, this implies that star-forming galaxies have a small mass fraction of UV bright, young stars as compared to UV faint, old stars. Thus, the question arises: which population of stars dominates the dust heating in star-forming clouds? The less numerous but much brighter in the UV young stars ($<
100$ Myr) or the numerous but much fainter in the UV old stars? How does this answer change when we consider the emission at specific IR wavelengths?
The majority of the IR energy from star-forming galaxies is emitted at far-IR ([$\scriptstyle\sim$]{}100 ) wavelengths. Historically, this far-IR emission has been identified as infrared cirrus emission from dust heated by non-ionizing populations [@Helou1994], which are older than [$\scriptstyle\sim$]{}10 Myr. @Cirrus1987 interpreted the far-IR emission from spiral disks in terms of two thermal components with different temperatures and found that the cirrus component contributes more than half of the total far-IR flux. However, as old stars emit very few of the UV photons that power dust emission, it is possible that UV-bright young stars could dominate cirrus emission. For example, a small number of young stars embedded in a large optically thin cloud can result in a dilute radiation field, cold dust temperature and therefore cold cirrus emission.
Star formation rate (SFR) indicators are important observational probes of the star formation histories of galaxies. They are usually single-band or wavelength-integrated quantities that are presumed to trace a specific regime of recent star formation in a region or galaxy [@Kennicutt1998]. The most common SFR indicators include the H$\alpha$ flux (tracing unobscured ionizing stars, $< 10$ Myr), UV (tracing unobscured ionizing and UV-bright, non-ionizing stars, $< 100$ Myr), and total infrared (TIR, tracing obscured star formation).
The UV and H$\alpha$ flux are heavily attenuated by dust, with a typical extinction of 0-4 mag and 0-2 mag respectively [@Kennicutt2009]. Since the UV and H$\alpha$ flux only trace the stellar light unabsorbed by dust, an accurate estimation of star formation activity requires a correction factor to account for the effect of dust. For starburst galaxies, emission-line diagnostics and UV colors may be used to such purpose, but they are often difficult to obtain or highly uncertain [@Kennicutt2009]. An alternative way is to look at the IR flux, which accounts for the energy missing in the UV and the optical [@Calzetti2007]. @Kennicutt2009 found that the combination of H$\alpha$ and TIR provides a robust SFR measurement. @Leroy2008 and @Bigiel2008 used the 24 flux combined with the GALEX far-UV to study the SFR in nearby galaxies.
TIR alone as a SFR indicator suffers from a number of problems. It does not trace unobscured star formation, and for the typical amount of dust in galaxies, a non-trivial amount of energy escapes in the UV and optical. Recalling the “cirrus emission" problem, TIR can only work well as an SFR indicator if the total infrared emission correlates well with newly formed stars ($< 100$ Myr), instead of old stars that formed long ago. Therefore, understanding whether young stars or old stars dominate the TIR dust emission is an important step in interpreting the TIR SFR indicator. In addition, TIR is often integrated over a sparsely sampled wavelength interval in intermediate or high-redshift galaxies, which may introduce many uncertainties [@Calzetti2007]. @Sauvage1992 found a systematic decrease of the L(far-IR)/L(H-alpha) ratio from early- to late-type spirals, and suggested a systematically varying cirrus fraction as the explanation, accounting for 86 percent of L(far-IR) for Sa galaxies to about 3 percent for Sdm galaxies. @Xu1996 also found that the heating of the diffuse dust is dominated by optical radiation from stars at least a billion years old in M31. With these results, it would be important to subtract the cirrus emission from the total infrared luminosity in the calculation of star formation rate.
In the simplest model, dust emission can be thought of as modified blackbody radiation at a certain temperature plus emission features. However, equilibrium dust emission is not the only possible emission path. @Duley1973 pointed out that non-equilibrium dust emission should also be important and this has been confirmed observationally [e.g. @Sellgren1984]. Upon the absorption of an energetic photon, a large molecule or small dust grain can attain a very high temperature for a short period of time before it cools. In many situations, if non-equilibrium heating is not considered, the observed mid-IR luminosity implies an unrealistically high equilibrium dust temperature. While older stars can rarely excite dust grains to produce non-equilibrium heating, this type of heating clearly dominates the mid-IR emission from star-forming galaxies.
Recent interesting results from the Herschel observatory have begun exploring the problem of the source of dust heating using the far-IR and sub-mm. Specifically, in a study of M81, @Bendo2010 found that the far-IR to sub-mm (160-500 ) emission is dominated by dust heated by evolved stars while the 70 emission is caused by the active galactic nucleus and young stars in star forming regions. From a study of 51 nearby galaxies, @Boselli2010 found that the warm dust (f60/f100) correlates with star formation, while cold dust (f350/f500) anti-correlates. Our study does not probe cold dust in the submillimeter range, but we plan to do so in a later paper.
In this paper, we examine the relationship between IR luminosity and the age of stellar populations. Through the use of a self-consistent radiative transfer and dust emission model, observed dust properties, and stellar and dust geometries designed to represent galactic environments, we compare our results to observed galaxies (the SINGS sample, @SINGS, the starburst galaxies in @Engelbracht2008, and the LVL sample, @LVL) and attempt to give a quantitative answer on the age of the stellar populations that dominates the IR emission in star forming galaxies.
METHOD
======
Model
-----
DIRTY (DustI Radiative Transfer, Yeah!) is a self-consistent Monte Carlo radiative transfer model [@DIRTY1; @DIRTY2]. Due to its Monte Carlo nature, it allows for arbitrary dust and stellar distributions. It computes emission from the three standard dust grain components [@Weingartner2001] - carbon grains, silicates, and polycyclic aromatic hydrocarbons (PAH). One of the key strengths of DIRTY is that it is completely self-consistent. It avoids assumptions whenever possible; the radiation field is directly calculated from radiative transfer, instead of the usual method of scaling the standard solar neighborhood radiation field [@Draine2009]. The initial radiation field is obtained from an external spectral evolutionary synthesis model. Both equilibrium and non-equilibrium emission are calculated from the radiation field and their contributions to the radiation field itself are iteratively taken into account. DIRTY has been used to model a variety of situations, for example, the dusty starburst nucleus of M33 [@Gordon1999] and the general behavior of galaxies [@Witt2000]. Figure \[fig:dirtysed\] shows examples of global spectral energy distributions (SED) from DIRTY.
Parameter Values
-------------------------------------- ----------------------------------------------------------------------------------------------------
Optical depth at V band ($\tau_{v}$) 0.2 - 5.0
Stellar age (Myr) 1 - 13000
Radius (pc) 100 - 10000
Dust model Milky Way dust (with $R_V = 3.1$, $b_C = 6 \times 10^{-5}$) and SMC Bar dust from @Weingartner2001
Stellar model PEGASE 2 [@PEGASE] with instantaneous bursts
Metallicity Solar (0.02) and 1/5 Solar (0.004)
\[table:params\]
Young and old galaxies are best described by different dust geometries, which can lead to different dust absorption efficiencies and effective optical depths [@Witt1992]. Following the notation of @Witt2000, we use the CLOUDY and SHELL geometries. In the CLOUDY geometry, stars are uniformly distributed in a spherical volume, and a dusty core is located within 0.69 times of the radius of the stellar distribution; whereas in the SHELL geometry, stars extend only to 0.3 times of the outer radius and are surrounded by a concentric shell of dust from 0.3 to 1.0 of the outer radius. See Figure 1 of @Witt2000 for a graphical description of the CLOUDY and SHELL geometries.
For both geometries, the stellar density per unit volume is constant. The CLOUDY model reproduces the general characteristics of old stellar populations in galaxies with strong central bulges, where the majority of stars are found outside of the dust. The SHELL model mimics a young star cluster with surrounding clouds. It has been shown that starbursts require a SHELL geometry to explain various color-color plots [@Gordon1997] and the attenuated-to-intrinsic ratios of hydrogen lines [@Calzetti2001]. In this study, we use SHELL for stars that are 100 Myr old or younger and CLOUDY for older ones. Other model parameters can be found in Table \[table:params\].
@Witt1996 found that it is necessary to model interstellar dust as a multi-phase medium. To achieve this, a small fraction (15 % total filling factor) of clumps of high density dust is randomly placed into a field of low density dust with 100 times lower density in the global dust geometries mentioned previously. Without the clumpiness, the effectiveness of dust absorption would be overestimated. In this study, we exclusively use clumpy dust distributions. The optical depth averaged over all sightlines (from infinity to the center of the geometry) is normalized to the desired value of [$\tau_{v}$]{}.
We use the spectral evolutionary synthesis (SES) model PEGASE 2 [@PEGASE] as the stellar input to our model. We model starbursts of various ages with solar and 1/5 solar metallicity and the Padova evolutionary tracks. An old galaxy may contain both old stellar populations and young stellar populations, although the latter is expected to be much less abundant considering that the typical timescale for gas depletion is about 3 Gyr [@Pflamm2009]. The time-dependent profile of star formation could be modeled as an exponentially decaying burst [@exp_time1973; @exp_time2003]. However, since we want to model the characteristics (the luminosity ratios) of stellar populations at a certain age, we use instantaneous starbursts. To bracket the possible real world scenarios, we model the extreme cases of very young (1 Myr) and very old (13 Gyr) starburst populations. Gas continuum and emission of recombination lines, as calculated by PEGASE, are included. See @Gordon1999 for a detailed discussion on how we use spectral evolutionary synthesis models with our dusty radiative transfer model.
The dust extinction curve for stars in the Milky Way (MW), Small and Large Magellanic Clouds (SMC and LMC) are found to have overall similar shapes with significant variation in the UV [@Gordon2003]. There are two distinct features that differentiate the different types of dust. The first one is the 2175 Å bump, which is found to vary in strength on average between the MW diffuse ISM/LMC general (strong bump), LMC2 (near 30 Dor)/SMC Wing (weak bump), and SMC Bar (no bump). The second is the far UV rise, which generally varies in strength inversely with the 2175 Åbump. Dust properties can be influenced by star forming activity and metallicity. In particular, @Gordon1997 found that the SMC Bar type dust (lacking a 2175 Å bump) describes the starburst galaxies better than either the LMC or MW type dust. From the GMASS survey, @Noll2009 found that there is a wide range of UV dust properties, including those that are intermediate between SMC Bar and LMC2 type dusts. Modeling results show that the type of dust can affect the strength of PAH features [@Draine2007], which in turn affects the IR observations. As a result, it is important to choose an appropriate dust type for our study, and to explore the sensitivity of our results to the dust type. To span the whole range of known dust properties, we use SMC Bar type dust and Milky Way type dust. For a fair comparison, we use the dust models of @Weingartner2001 for both the SMC and Milky Way type dust.
Radius and stellar mass are not independent dimensions in our model. Since the radiation intensity drops as the square of the radius, an increase in the model radius by a factor of $x$ can be compensated by an increase in the stellar mass by a factor of $x^{2}$. The two cases give the same radiation intensity in each grid cell in the model, and therefore the same dust temperature. The resultant SEDs will have the same shape, with the only difference being the lower overall luminosity in the latter case. This has been confirmed by test runs of our model; for example, a stellar population with $10^{11}$ solar masses and 10 kpc radius gives the same SED as the one with $10^9$ solar masses and 1 kpc radius, when the luminosity in the latter is scaled up by a factor of 100 (all the other parameters are unchanged). Since our study is based on the ratios of luminosities, a larger, more luminous galaxy and a smaller, dimmer galaxy with same radiation field intensity give the same result.
Due to the Monte Carlo and iterative nature of our model, the run time varies significantly depending on the parameters. The fastest models take about 4 CPU hours on a 2.33 GHz Intel Xeon processor, while the slowest ones often need about 2 days. The average run time is about 20 hours. Each model uses a single thread and we launch multiple processes in parallel to utilize all the CPU cores in multi-core processors. We iterate the radiative transfer and dust emission processes until the global energy conservation error is within 5%. The resultant uncertainty of flux at each wavelength is usually within 1%.
Luminosity Ratios {#lratio}
-----------------
To compare the modeled SED with infrared observations, we calculate the luminosities in the IRAC and MIPS bands with calibrated response functions [@IRACresp; @MIPS24; @MIPS70; @MIPS160]. Since the galaxies in our sample have a very wide range of total luminosity, we study luminosity ratios instead of raw luminosities. We normalize the mid- and far-IR luminosities by the IRAC1 (3.6 ) luminosity, and the resulting ratios are the characteristics of the dust and stellar populations that we compare with our model. These luminosity ratios are measures of the efficiency of the stellar and dust distribution at producing radiation in a certain wavelength regime (relative to IRAC1). At 3.6 , the IRAC1 band is only mildly contaminated by dust emission (e.g., the 3.3 aromatic/PAH feature), and is less affected by dust extinction than the shorter wavelengths. The IRAC1 luminosity can be considered as a proxy of stellar luminosity and an approximate measure of mass, although the mass-to-light ratio depends on their underlying stellar populations and the infrared colors [@Bell2001]. The combination of the PEGASE SES and DIRTY models account for stellar age, dust emission, and absorption self-consistently. As we see in later discussions, the luminosity ratios are robust against changes in radius or total stellar luminosity.
Data
----
From the SINGS (Spitzer Nearby Galaxies Survey) dataset [@SINGS], we have good measurements of the UV, visible and IR fluxes for a number of nearby galaxies. They range from spiral to elliptical to irregular galaxies, and are a good representation of a wide range of galaxies. We removed 4 galaxies (M81 Dwarf A, NGC 3034, Holmberg IX, and DDO 154) for which the IRAC or MIPS band fluxes are not well measured (either an upper limit or saturated). For the remaining 71 galaxies, we calculate the luminosity from the flux data from @SINGSData and distance data from @SINGS. The metallicity is calculated from the average of the “high" and “low" oxygen abundance values from @Moustakas2010; the same method was used in @Calzetti2010. Figure \[fig:sings\] shows the wide range of galaxies types in the SINGS sample, where we have used Eq. 22 from @Draine2007b to compute $L_{TIR}$.
To expand the range of galaxies studied, we include the starburst galaxies from @Engelbracht2008 and the Local Volume Legacy (LVL) sample from @LVL. The starburst galaxies have a higher fraction of recent star formation and UV-bright young stars. Their data include imaging and spectroscopy from the Spitzer Space Telescope, as well as ground-based near-infrared imaging. The Engelbracht sample consists of 66 local star forming galaxies of which 65 have high quality IRAC and MIPS data available; UM 420 was dropped from our sample because the MIPS data were not available. For this sample we take all flux, distance and metallicity values from @Engelbracht2008. On the other hand, the LVL is a statistically unbiased sample of 258 galaxies in the local universe out to 11 Mpc which consists mainly of dwarf galaxies, and we take flux and distance values from @LVL and metallicity values from @Marble2010. We removed galaxies without good IRAC and/or MIPS measurements (no data or flux available only as an upper bound) and galaxies without metallicity data, and arrived at a sample of 194 galaxies.
Dataset Count IRAC2/IRAC1 IRAC3/IRAC1 IRAC4/IRAC1 MIPS24/IRAC1 MIPS70/IRAC1 MIPS160/IRAC1
------------------------- ------- ------------------- ------------------- ------------------- ------------------- ------------------- -------------------
SINGS 71 0.433 $\pm$ 0.073 0.411 $\pm$ 0.274 0.438 $\pm$ 0.362 0.110 $\pm$ 0.151 0.131 $\pm$ 0.122 0.049 $\pm$ 0.031
H II nuclei 13 0.469 $\pm$ 0.147 0.567 $\pm$ 0.402 0.604 $\pm$ 0.329 0.140 $\pm$ 0.181 0.141 $\pm$ 0.058 0.066 $\pm$ 0.017
Starburst nuclei 9 0.441 $\pm$ 0.036 0.499 $\pm$ 0.314 0.562 $\pm$ 0.457 0.249 $\pm$ 0.273 0.218 $\pm$ 0.181 0.053 $\pm$ 0.032
LINER/Seyfert nuclei 21 0.408 $\pm$ 0.029 0.317 $\pm$ 0.127 0.307 $\pm$ 0.211 0.055 $\pm$ 0.077 0.086 $\pm$ 0.128 0.037 $\pm$ 0.028
Others 28 0.434 $\pm$ 0.045 0.381 $\pm$ 0.244 0.419 $\pm$ 0.406 0.091 $\pm$ 0.093 0.133 $\pm$ 0.105 0.049 $\pm$ 0.034
Type I 9 0.465 $\pm$ 0.041 0.282 $\pm$ 0.186 0.226 $\pm$ 0.288 0.138 $\pm$ 0.224 0.145 $\pm$ 0.121 0.035 $\pm$ 0.019
Type E 6 0.397 $\pm$ 0.034 0.232 $\pm$ 0.163 0.188 $\pm$ 0.271 0.050 $\pm$ 0.095 0.061 $\pm$ 0.104 0.014 $\pm$ 0.021
Type S0 7 0.507 $\pm$ 0.200 0.616 $\pm$ 0.622 0.594 $\pm$ 0.630 0.245 $\pm$ 0.262 0.247 $\pm$ 0.207 0.047 $\pm$ 0.036
Type SA 19 0.420 $\pm$ 0.035 0.418 $\pm$ 0.208 0.451 $\pm$ 0.325 0.074 $\pm$ 0.116 0.100 $\pm$ 0.087 0.053 $\pm$ 0.031
Type SAB 18 0.418 $\pm$ 0.023 0.434 $\pm$ 0.150 0.515 $\pm$ 0.294 0.082 $\pm$ 0.064 0.113 $\pm$ 0.072 0.059 $\pm$ 0.026
Type SB 12 0.431 $\pm$ 0.041 0.431 $\pm$ 0.255 0.495 $\pm$ 0.342 0.137 $\pm$ 0.143 0.164 $\pm$ 0.143 0.059 $\pm$ 0.033
Engelbracht 65 0.518 $\pm$ 0.184 0.578 $\pm$ 0.395 0.738 $\pm$ 0.632 0.636 $\pm$ 0.612 0.439 $\pm$ 0.345 0.094 $\pm$ 0.163
LVL 194 0.436 $\pm$ 0.037 0.287 $\pm$ 0.158 0.237 $\pm$ 0.225 0.061 $\pm$ 0.104 0.097 $\pm$ 0.076 0.033 $\pm$ 0.022
\[table:stat\]
In Table \[table:stat\] we list the statistics of the band ratios of our sample. The SINGS galaxies are classified into different nuclei and morphological types according to @SINGS. We keep the Engelbracht sample (as a group of starburst galaxies) and the LVL sample (representing the dwarf galaxies) separate from the SINGS’ categories for a clear statistical comparison. Each number in the table is the average (plus or minus the standard deviation) of the ratio of the given band luminosity to the IRAC1 luminosity. Figure \[fig:stat\] shows the table entries graphically. Among the SINGS galaxies, the starburst and H II nuclei tend to have higher luminosity ratios while those without active nuclei tend to have lower luminosity ratios. Galaxies with Seyferts (Sy) and LINERS (L) nuclei are in between the two extremes. While our model does not simulate Seyfert (Sy) or LINERS (L) nuclei, the luminosity ratios of these galaxies are within the range of the other galaxies, and we keep them in our study. Focusing on the MIPS160 column, we notice that the luminosity ratio for elliptical galaxies is significantly lower than the other types of galaxies. On the other hand, the @Engelbracht2008 starburst galaxies show a much higher ratio in all the MIPS bands, meaning that they are more efficient in producing far-infrared emission.
RESULTS
=======
Dust Type {#dust}
---------
In Figure \[fig:dust\_type\], we plot the MIPS24 to IRAC1 luminosity ratio for our galaxy sample, together with models with SMC Bar (left) and Milky Way type dust (right). Different models on the same curve have different stellar mass, characterized by their different IRAC1 luminosity on the x-axis. Here we use $\tau_{v} = 1.0$ and radius = 10 kpc, but we see similar trends with other parameters. We note that the curves are relatively flat until they tick up in the regime of very high IRAC1 luminosity. This is because dust emission in the MIPS24 band is dominated by non-equilibrium heating for young stellar populations, or stellar continuum for old stellar populations. In either case, the fraction of MIPS24 to IRAC1 is a constant. Only at very high luminosity, does the dust temperature become high enough for equilibrium heating to make a comparable contribution and the curves start to turn up. We discuss the effects of non-equilibrium emission in section \[noneq\].
From 1 Myr old to 13 Gyr old, the models with SMC Bar type dust roughly span the whole range of luminosity ratios for the galaxies. On the other hand, the models with MW type dust are unable to cover all the galaxies. Even at the youngest age of 1 Myr, the model curve is below some galaxies (including most of the @Engelbracht2008 starburst galaxies). At the other extreme, both types of dust do equally well for low MIPS24 to IRAC1 luminosity ratios. None of the galaxies have luminosity ratios lower than our oldest models (13 Gyr, close to the age of the universe). We see a similar distinction in the MIPS70 to IRAC1 luminosity ratios. From this perspective, the SMC Bar type dust is a better choice for our study, although this is different from @Draine2007 who found that the SINGS galaxy sample has similar dust-to-gas ratio and similar PAH abundance to MW type dust. It is possible that the different results from the dust types are due to dust processing. If the MW type dust is more susceptible to destruction in a UV radiation field compared to the SMC Bar type dust, the former will produce less far-IR emission per unit IRAC1 luminosity and therefore a lower luminosity ratio. Unless otherwise specified, discussions in the following sections refer to SMC Bar type dust. Despite the differences we discuss here, the conclusion we draw about the age of stellar populations heating the dust (see section \[constrain\]) is independent of the type of dust.
Age
---
To study what stellar age would best reproduce the observed luminosity ratios, we fix the other model parameters and see how the model results change as a function of stellar age. Knowing that $\tau_{v}$ is on the order of unity for normal disk galaxies [@Holwerda2007], we assume an optical depth of $\tau_{v} = 1.0$. @Dale2006 also found that the average attenuation is $A_v = 1.0$ (which equals $\tau_{v} = 0.92$) for a large portion of SINGS and some archival sources from ISO and Spitzer. We set the radius to be 10 kpc, a reasonable size of a galaxy. Our results do depend on the choice of optical depth and radius, but we first see what we find with these values. In section \[tvradius\] we examine the effect of varying the optical depth and radius.
In Figure \[fig:panel\], we plot the luminosity ratios versus IRAC1 luminosity for IRAC and MIPS bands; the five curves correspond to five different stellar ages. The younger three (1, 10 and 100 Myr) are modeled with the SHELL geometry and the older two (1 and 13 Gyr) are modeled with the CLOUDY geometry. Younger populations are more efficient in producing IR due to being more embedded in the dust and their much higher intrinsic L(UV)/L(IRAC1) ratios, and the difference is most pronounced in the mid-IR. Taking the IRAC4/IRAC1 vs IRAC1 plot as an example, we see that the 10 Myr line matches the data the best (among the models with SMC type dust). The 1 Myr old models produce too much 8 flux (per unit 3.6 flux, on average), while the older models produce too little. For the MIPS24/IRAC1 plot, a combination of the 10 Myr old and 100 Myr old models match the median of the data, while for the MIPS160/IRAC1 plot, a combination of the 100 Myr old and 1 Gyr old models would do. This shows that the age of the stellar populations that dominates IR emission is on the order of 100 million years. The difference from IRAC4 to MIPS160 shows that the importance of younger stellar populations (less than 100 Myr) is relatively higher at shorter wavelengths.
As we vary the age, the IRAC2-to-IRAC1 plot shows a different trend from the other plots. Both the data and the models fall into a very narrow range ([$\scriptstyle\sim$]{}0.5 dex). This is because this color is dominated by stars only and the colors of populations of stars in the mid-IR are relatively constant; it is much less sensitive to dust compared to the longer wavelengths.
Note that the data points that have very low luminosity ratios in the MIPS bands are elliptical galaxies. The elliptical galaxies are shown as red symbols in Figure \[fig:sings\], and they show up in the lower right corner in the MIPS70 and MIPS160 plots. From our models, older stellar populations produce less dust emission per unit near-IR luminosity, and therefore should be better at explaining elliptical galaxies. However, the luminosity ratios of these galaxies are so low that even the oldest models (at about the age of the universe) cannot explain them. Noting that we have used $\tau_{v} = 1.0$ and 10 kpc radius in Figure \[fig:panel\], this result suggest that the optical depth of elliptical galaxies could be lower than the average galaxy, or that their radii are larger than our modeled value. The former is the most probable explanation, as the zero-dust case for our oldest model gives a value of [$\scriptstyle\sim$]{}0.0005 for MIPS160/IRAC1, low enough to explain the elliptical galaxies.
Optical depth and radius {#tvradius}
------------------------
To illustrate the effect of the optical depth and radius of the model region on the results, we fix the stellar age at 100 Myr and vary the optical depth and radius in Figure \[fig:tauradius\]. Dust with higher optical depth absorbs more energy and emits more far-IR radiation, and therefore gives a higher ratio in the plots. A larger radius dilutes the radiation field, lowers the dust temperature and shifts the equilibrium dust emission peak to a longer wavelength in the SED. In our luminosity ratio diagrams, it shifts the curves horizontally to the right. Because of the behavior of our model as explained in Section \[model\], we may also interpret the effect with mass surface density ($M/\pi r^2$). With the same reasoning, as the mass surface density decreases, the curves shift to the right.
As we study the effects of variations in optical depth and radius, we find a noticeable difference for the IRAC bands and the MIPS bands. For MIPS bands, changing the two parameters can change the curves significantly. This is evident on the plot for MIPS160 (Figure \[fig:tauradius\], right). However, in the IRAC bands, the curves are flatter and span a narrower range. As explained in section \[dust\], this is because the dust emission in the IRAC bands is dominated by non-equilibrium heating. And since the curves are flat, shifting them horizontally makes no difference in our results; they are not very sensitive to the choice of radius. See section \[noneq\] for a discussion on equilibrium vs non-equilibrium dust emission.
From Figure \[fig:tauradius\] we can see that old stellar populations could possibly reproduce the observed luminosity ratios in Figure \[fig:panel\] only if the optical depth is very high, say 5 or 10. This is not realistic, as @Holwerda2007 has shown that for normal disk galaxies [$\tau_{v}$]{} is on the order of unity. In addition, if [$\tau_{v}$]{} has such a high value, the optical depth at shorter wavelengths will be even greater and we wouldn’t be able to observe these galaxies in UV.
Equilibrium and non-equilibrium emission {#noneq}
----------------------------------------
To help interpret our results, we examine the fraction of luminosity due to non-equilibrium heating. The plots in Figure \[fig:noneq\] shows the different behavior of the fraction in different IR bands.
In the far-IR (MIPS70 to SPIRE500), non-equilibrium emission is negligible compared to equilibrium emission for a wide range of IRAC1 luminosity. Only at very low IRAC1 luminosity and large ($>10$ kpc) radii does non-equilibrium emission contribute significantly in the far-IR. When the flux is dominated by equilibrium emission, it depends on the dust temperature and therefore model parameters like the radius, and this can be seen in the MIPS160 curves of Figure \[fig:tauradius\].
On the other hand, the mid-IR bands (e.g. IRAC) exhibit different behavior. When modeled with young stellar populations (1 - 100 Myr old), the flux is dominated by non-equilibrium emission because of the large amount of highly energetic UV photons. When modeled with older stellar populations, the fraction of non-equilibrium emission is small, so equilibrium emission is the major source of the IR flux from dust. However, the dust temperature is not high enough to emit significant energy in mid-IR, so the flux is dominated by the stellar continuum. Either way, equilibrium emission does not play a significant role in the mid-IR bands, except at extremely high luminosities. Therefore, the radius parameter (which affects the dust temperature) does not change the result as much as they do for the far-IR bands.
Non-equilibrium emission is dominated by very high energy UV photons. As opacity goes up with the energy of the photon, most high energy UV photons are likely to be absorbed even in a low $\tau_{v}$ environment. IRAC4 is dominated by non-equilibrium emission, therefore further increases in $\tau_{v}$ do not result in more high energy UV photons being absorbed and so the IRAC4 to IRAC1 luminosity ratio would not change significantly. If we go back to Figure \[fig:tauradius\], we see that the luminosity ratio is less sensitive to the optical depth in IRAC4 then in MIPS160. MIPS24 is the turnover point for the two behaviors. This explains why the curves behave so differently in IRAC plots and MIPS plots as we change the model parameters in the previous sections.
Constraints on the fraction of luminosity from old stars {#constrain}
--------------------------------------------------------
In this section, we attempt to calculate the fraction of IR luminosity that could be due to old stars using simple assumptions. Assume there are two non-interacting populations of stars, one younger and one older. When we look at models with the same IRAC1 luminosity as an observed galaxy, if the observed luminosity (say in MIPS160) is in between the younger (higher) and older (lower) luminosities, there exists a fraction $x$ of old stars of which the combination of the two model stellar populations reproduces the observed luminosity. We can write the luminosity of the $i^{th}$ galaxy at wavelength $\lambda$, $L_{i}(\lambda)$ as
$$L_{i}(\lambda) = [1 - x_{i}(\lambda)] Y_{i}(\lambda) + x_{i}(\lambda) O_{i}(\lambda)
\label{eqn:init}$$
where $x_{i}$ is the fraction of old stars, and $Y_{i}$ and $O_{i}$ are the luminosities of the young and old models (e.g. 1 Myr and 13 Gyr) that have the same IRAC1 luminosity as the $i^{th}$ galaxy, respectively. The two terms represent the contribution of luminosity from the two populations. Eq. \[eqn:init\] can be inverted to solve for $x_{i}$ to yield
$$x_{i}(\lambda) =
\frac{L_{i}(\lambda)-Y_{i}(\lambda)}{O_{i}(\lambda)-Y_{i}(\lambda)}
\label{eqn:x}$$
When we use the formula for galaxies that have a luminosity (per unit IRAC1 luminosity) lower than the one given by the old model, $x_i$ will be greater than 1, which is not physical. This corresponds to the case where the real galaxy is less efficient than the old model in producing dust emission. Such a situation can arise if, for example, the galaxy has a lower average optical depth than that assumed in our model. As a simplification we simply set $x_i = 1$ in these cases. On the other hand, for galaxies with the luminosity per unit IRAC1 luminosity higher than our young model, $x_i$ will be negative and we set $x_i = 0$. After solving for the fraction of old stars $x_i$, we can calculate the fraction of luminosity due to old stars $f_{i}$:
$$f_{i}(\lambda) = \frac{x_{i}(\lambda) O_{i}(\lambda)}{L_{i}(\lambda)}
\label{eqn:f}$$
----------- ----------- ------------------- ------------------------ -------------------- --------------------
Dust Type Flux band Fraction of Fraction of luminosity Number of galaxies Number of galaxies
old stars $x$ from old stars $f$ with $x_i = 1$ with $x_i = 0$
MW IRAC4 0.854 $\pm$ 0.018 0.285 $\pm$ 0.078 5 1
MW MIPS24 0.611 $\pm$ 0.102 0.068 $\pm$ 0.017 1 9
MW MIPS70 0.569 $\pm$ 0.110 0.145 $\pm$ 0.071 3 8
MW MIPS160 0.572 $\pm$ 0.107 0.181 $\pm$ 0.062 4 7
SMC Bar IRAC4 0.736 $\pm$ 0.055 0.220 $\pm$ 0.061 2 1
SMC Bar MIPS24 0.942 $\pm$ 0.008 0.050 $\pm$ 0.007 1 1
SMC Bar MIPS70 0.964 $\pm$ 0.001 0.120 $\pm$ 0.053 3 1
SMC Bar MIPS160 0.931 $\pm$ 0.004 0.203 $\pm$ 0.055 3 1
----------- ----------- ------------------- ------------------------ -------------------- --------------------
\[table:constrain\]
----------- ----------- ------------------- ------------------------ -------------------- --------------------
Dust Type Flux band Fraction of Fraction of luminosity Number of galaxies Number of galaxies
old stars $x$ from old stars $f$ with $x_i = 1$ with $x_i = 0$
MW IRAC4 0.758 $\pm$ 0.049 0.191 $\pm$ 0.058 1 1
MW MIPS24 0.184 $\pm$ 0.101 0.021 $\pm$ 0.006 1 40
MW MIPS70 0.174 $\pm$ 0.083 0.022 $\pm$ 0.014 1 36
MW MIPS160 0.516 $\pm$ 0.105 0.088 $\pm$ 0.016 1 8
SMC Bar IRAC4 0.590 $\pm$ 0.097 0.139 $\pm$ 0.040 1 6
SMC Bar MIPS24 0.630 $\pm$ 0.118 0.017 $\pm$ 0.003 1 7
SMC Bar MIPS70 0.830 $\pm$ 0.039 0.020 $\pm$ 0.005 1 2
SMC Bar MIPS160 0.913 $\pm$ 0.013 0.097 $\pm$ 0.009 1 1
----------- ----------- ------------------- ------------------------ -------------------- --------------------
\[table:constrain\_engel\]
----------- ----------- ------------------- ------------------------ -------------------- --------------------
Dust Type Flux band Fraction of Fraction of luminosity Number of galaxies Number of galaxies
old stars $x$ from old stars $f$ with $x_i = 1$ with $x_i = 0$
MW IRAC4 0.950 $\pm$ 0.006 0.512 $\pm$ 0.102 35 1
MW MIPS24 0.775 $\pm$ 0.050 0.081 $\pm$ 0.013 2 9
MW MIPS70 0.512 $\pm$ 0.076 0.031 $\pm$ 0.012 2 25
MW MIPS160 0.803 $\pm$ 0.029 0.155 $\pm$ 0.029 4 1
SMC Bar IRAC4 0.887 $\pm$ 0.021 0.410 $\pm$ 0.083 16 1
SMC Bar MIPS24 0.954 $\pm$ 0.005 0.061 $\pm$ 0.008 1 1
SMC Bar MIPS70 0.945 $\pm$ 0.008 0.028 $\pm$ 0.009 2 1
SMC Bar MIPS160 0.969 $\pm$ 0.003 0.128 $\pm$ 0.022 4 1
----------- ----------- ------------------- ------------------------ -------------------- --------------------
\[table:constrain\_lvl\]
Using 1 Myr (young) models and 13 Gyr (old) models, we calculate the fractions $x$ and $f$ for the 3 samples of galaxies, and tabulate the results for the IRAC4, MIPS24, MIPS70 and MIPS160 bands in Tables \[table:constrain\]-\[table:constrain\_lvl\]. Again, [$\tau_{v}$]{}= 1 and 10 kpc radius are used. The fractions are shown here as the average plus or minus the standard deviation. The number of galaxies with out-of-range $x_i$ in each sample is given in columns 4 and 5. They are adjusted to $x_i$ = 0 or 1 as explained above.
While the fraction of old stars $x$ can exceed 90% (as in some of the calculation for models with SMC Bar dust), the fraction of the luminosity produced by old stars, $f$, is much lower; it is generally lower than 20%, with the exception of the IRAC4 band for the LVL galaxies. Although $f$ is high for LVL/IRAC4, the remaining bands in the LVL sample yield lower values of $f$, consistent with the other two galaxy samples. If we take the LVL/IRAC4 combination out of the picture and restrict our results to SMC Bar dust, the highest value of $f$ is 22.0%. Or, if we use 10 (100) Myr old stars instead of 1 Myr old stars as the younger population, the highest fraction becomes 27.9% (25.2%), again with MW/IRAC4/SINGS. On the other hand, if we keep the 1 Myr old stars but change the older population to 1 Gyr old stars, the highest fraction becomes 42.8%.
The fraction of luminosity $f$ is generally higher for MW type dust, but the choice of dust does not affect our conclusion that dust emission is dominated by young stars. It is remarkable that the values of $f$ computed from the MIPS luminosities are similar for both types of dust even when the fractions of old stars $x$ is consistently higher for SMC Bar type dust (this is generally true for different stellar ages as well). This could be attributed to the steeper far-UV rise in the SMC Bar extinction curve; for the same value of [$\tau_{v}$]{}, the SMC Bar type dust is more effective in absorbing far-UV photons, and therefore requires fewer young stars to produce the same IR luminosity. The higher number of galaxies with $x_i = 0$ for MW type dust can be understood by looking at Figure \[fig:panel\]; the MW 1 Myr old line is below a significant number of galaxies.
If we do a more complex study to include more than two stellar ages in this analysis, the fraction $f$ will only be lower. We assume a linear contribution from the 2 stellar populations, which is not necessarily accurate because the effect of dust temperature on dust emission is non-linear. The more well-mixed the young and old stars are, the more non-linear their contribution will be. However, as older stars release many fewer UV photons compared to younger stars, it is almost certain that increasing the fraction of old stars would decrease the far-IR/IRAC1 ratios. Therefore, the far-IR/IRAC1 ratios are monotonically decreasing functions of the fraction of old stars (when all other parameters are fixed). Real stellar populations include a wide range of stellar ages, and with our analysis we can see whether such populations are closer to the “old" population or the “young" population. While this simple analysis cannot tell us what the best stellar age for the observed galaxies is, and the calculated fraction of old stars may suffer from non-linear effects, it is evident that dust emission is dominated by young stars.
Metallicity {#metal}
-----------
In stars with high metallicity, absorption lines form a continuum that leads to the production of a soft UV radiation field. This stellar atmosphere effect is known as “line blanketing". In contrast, stars with lower metallicity produce a harder radiation field. Empirically, metallicity correlates with the amount of dust (and therefore a low metallicity is often associated with a high gas-to-dust ratio). In the previous sections, we use a fixed (solar) metallicity for our models. However, the combination of a harder radiation field and a lower optical depth may give different results.
Metallicity has a somewhat phase-transition like effect. For the relatively high metallicities observed in the LMC and the Milky Way, observables are qualitatively similar. However, when we go to the low metallicity found in SMC, we find qualitatively different results, such as the lack of aromatic emission. For example, @Engelbracht2005 found an abrupt change in the 8-to-24 color at around 1/4 solar metallicity. @Calzetti2010 choose a metallicity of $log(O/H)=8.1$ as a rough dividing line for the two behaviors. We use the same value to divide our sample into two groups: the high metallicity galaxies with metallicity higher than 8.1, and the low metallicity galaxies with metallicity lower than 8.1.
In Fig. \[fig:metal\], we compare the high metallicity galaxies with the low metallicity ones. We plot the high metallicity galaxies with solar metallicity models (as in Fig. \[fig:panel\]), but 1/5 solar metallicity models for the low metallicity galaxies. The low metallicity sample has lower luminosity ratios in both IRAC4 and MIPS160 bands, showing that metallicity does have an effect on dust heating. They also have lower IRAC1 luminosity on average, consistent with the view that the big luminous galaxies are more evolved and have more metals, while the less luminous galaxies have relatively more young stars and lower metallicity. Calculations of the fraction of luminosity from old stars ($f$) shows that the results for the MIPS bands are qualitatively the same for the two metallicity groups; but for the IRAC4 band, $f$ of the low metallicity group can be as high as two times that of the high metallicity group. For SINGS and LVL, $f$ of the low metallicity group is higher than 0.5 for IRAC4. It suggests that the non-starburst, low metallicity galaxies may have lower PAH abundance than our dust model. Since there is no significant trend in the MIPS bands, metallicity does not affect our main conclusion.
Sub-mm Predictions {#herschel}
------------------
Data Sample Dust Type $f(100 \micron)$ $f(250 \micron)$ $f(350 \micron)$ $f(500 \micron)$
------------- ----------- ------------------- ------------------- ------------------- -------------------
Engelbracht MW 0.046 $\pm$ 0.006 0.136 $\pm$ 0.028 0.176 $\pm$ 0.036 0.213 $\pm$ 0.044
Engelbracht SMC Bar 0.032 $\pm$ 0.001 0.227 $\pm$ 0.028 0.349 $\pm$ 0.039 0.463 $\pm$ 0.044
SINGS MW 0.129 $\pm$ 0.059 0.237 $\pm$ 0.067 0.277 $\pm$ 0.072 0.308 $\pm$ 0.077
SINGS SMC Bar 0.111 $\pm$ 0.052 0.355 $\pm$ 0.053 0.477 $\pm$ 0.049 0.572 $\pm$ 0.045
LVL MW 0.105 $\pm$ 0.025 0.311 $\pm$ 0.037 0.397 $\pm$ 0.040 0.463 $\pm$ 0.042
LVL SMC Bar 0.068 $\pm$ 0.021 0.375 $\pm$ 0.029 0.566 $\pm$ 0.025 0.706 $\pm$ 0.020
\[table:herschel\]
The Herschel Space Observatory was recently launched in May 2009. The far infrared imaging camera of the Spectral and Photometric Imaging Receiver (SPIRE) has 3 photometric bands centered at 250, 350 and 500 . In addition, the Photodetector Array Camera and Spectrometer (PACS) has a photometric band at 100 that MIPS does not have. It is interesting to see, from a model point of view, the contribution of luminosity of old stars at these wavelengths.
Table \[table:herschel\] shows the fraction of luminosity ($f$) due to old stars at 100, 250, 350 and 500 , calculated with Equation \[eqn:herschel\]. For the $i$th galaxy, we use the fraction of old stars ($x_i$) calculated with Equation \[eqn:x\] for MIPS160 and the luminosity from the corresponding old ($O_i$) and young models ($Y_i$) to estimate $f_i$. The denominator is the estimated luminosity at the SPIRE wavelengths, and the numerator is the luminosity due to old stars only.
$$f_{i}(\lambda) =
\frac{x_i(160 \micron) O_i(\lambda)}{x_i(160 \micron) O_i(\lambda) + (1 - x_i(160 \micron)) Y_i(\lambda)}
\label{eqn:herschel}$$
We convolve our model SEDs with the relative spectral response functions of the SPIRE bands (SPIRE Observers’ Manual, 2010) and the PACS 100 band (PACS Observer’s Manual, 2010) to compute the band integrated luminosity. We calculate the average and standard deviation for each combination of data sample (SINGS, Engelbracht and LVL) and model dust type (MW, SMC bar). We choose MIPS160 for the estimation of $x$ because it is the longest wavelength in this study. As we go from 100 to 500 , $f$ increases, showing that old stars are increasing in importance at longer wavelengths.
Moreover, $f$ in general has an increasing trend from the Engelbracht to the SINGS and the LVL sample. We have similar trends in some of the other bands but the trend in the SPIRE bands is much clearer, especially in the longest wavelength. This shows the order of the importance of the old stars in the 250-500 regime for the three catalogs. It is easy to understand the lower $f$ in the starburst sample as it has more recent star formation activity. So for starburst galaxies, young stars still contribute more luminosity in the sub-mm range (up to 500 ) compared to old stars. The difference between $f$ of LVL and $f$ of SINGS is likely due to a combination of their stellar populations, galaxy type and composition. We will continue to explore this wavelength regime and have a better understanding when Herschel data is available.
CONCLUSION
==========
Using our dusty radiative transfer model, together with IRAC and MIPS observations on the SINGS galaxies, the starburst galaxies in @Engelbracht2008 and the LVL galaxies, we have studied the effect of stellar age on infrared luminosity. We found that MW type dust tends to produce too little infrared luminosity for some of the galaxies (especially for the starburst galaxies), and so SMC Bar type dust is a more appropriate choice. However, we also note that we have similar results with both types of dusts when we calculate the luminosity due to old stars.
From an analysis of the IRAC4/IRAC1 and MIPS160/IRAC1 luminosity ratios vs IRAC1 luminosity plots, we found that the observed luminosity cannot be produced by 13 Gyr old stellar populations alone. The stellar age that dominates dust heating is on the order of 100 Myr. However, a small number of galaxies - the elliptical galaxies - did not fit well into our analysis. Their lower far-IR to IRAC1 ratios could be attributed to their deficiency of dust.
We found that the models are more sensitive to changes in parameters (such as stellar age, radius and optical depth) in the far-IR bands compared to the mid-IR bands. This can be explained by the fact that non-equilibrium emission dominates mid-IR, but is mostly negligible in far-IR. When non-equilibrium emission dominates, the luminosity ratio is less dependent on dust temperature and is therefore less affected by changes in the radius and stellar mass; it is also less sensitive to optical depth $\tau_{v}$ because the extinction for highly energetic photons saturates.
With the simplistic assumption that the observed galaxies are composed of two stellar populations of different ages, we found that the fraction of far-IR luminosity from 13 Gyr old stars is generally less than 20%. The result does not depend on the metallicity. Therefore, cold does not necessarily mean old; our study shows that far-IR radiation is dominated by a small number of younger stars.
We are currently building a large grid of dusty radiation models and spectral evolutionary synthesis models. With the model grid we will attempt to further confirm this study by fitting each galaxy individually and derive properties such as stellar age. We will compare the statistics of the resultant properties to the results in this paper and explain any differences or new features observed.
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|
---
abstract: 'This study is motivated by the observation, based on photographs from the Cassini mission, that Saturn’s rings have a fractal structure in radial direction. Accordingly, two questions are considered: (1) What Newtonian mechanics argument in support of that fractal structure is possible? (2) What kinematics model of such fractal rings can be formulated? Both challenges are based on taking Saturn’s rings’ spatial structure as being statistically stationarity in time and statistically isotropic in space, but statistically non-stationary in space. An answer to the first challenge is given through the calculus in non-integer dimensional spaces and basic mechanics arguments (Tarasov (2006) *Celest. Mech. Dyn. Astron.* **94**). The second issue is approached in Section 3 by taking the random field of angular velocity vector of a rotating particle of the ring as a random section of a special vector bundle. Using the theory of group representations, we prove that such a field is completely determined by a sequence of continuous positive-definite matrix-valued functions defined on the Cartesian square $F^{2}$ of the radial cross-section $F$ of the rings, where $F$ is a fat fractal.'
author:
- 'Anatoliy Malyarenko[^1]'
- 'Martin Ostoja-Starzewski[^2]'
bibliography:
- 'Saturn.bib'
title: 'Saturn rings: fractal structure and random field model[^3]'
---
Introduction
============
A recent study of the photographs of Saturn’s rings taken during the Cassini mission has demonstrated their fractal structure [@Li2015]. This leads us to ask these questions:
Q1: What mechanics argument in support of that fractal structure is possible?
Q2: What kinematics model of such fractal rings can be formulated?
These issues are approached from the standpoint of Saturn’s rings’ spatial structure having (i) statistical stationarity in time and (ii) statistical isotropy in space, but (iii) statistical non-stationarity in space. The reason for (i) is an extremely slow decay of rings relative to the time scale of orbiting around Saturn. The reason for (ii) is the obviously circular, albeit disordered and fractal, pattern of rings in the radial coordinate. The reason for (iii) is the lack of invariance with respect to arbitrary shifts in Cartesian space which, on the contrary and for example, holds true for a basic model of turbulent velocity fields. Hence, the model we develop is one of rotational fields of all the particles, each travelling on its circular orbit whose radius is dictated by basic orbital mechanics.
The Q1 issue is approached in Section 2 from the standpoint of calculus in non-integer dimensional space, based on an approach going back to @doi:10.1142/S0217979205032656 [@MR2210182]. We compare total energies of two rings — one of non-fractal and another of fractal structure, both carrying the same mass — and infer that the fractal ring is more likely. We also compare their angular momenta.
The Q2 issue is approached in Section 3 in the following way. Assume that the angular velocity vector of a rotating particle is a single realisation of a random field. Mathematically, the above field is a random section of a special vector bundle. Using the theory of group representations, we prove that such a field is completely determined by a sequence of continuous positive-definite matrix-valued functions $\{\,B_{k}(r,s)\colon k\geq 0\,\}$ with $$\sum_{k=0}^{\infty }\operatorname{tr}(B_{k}(r,r))<\infty,$$where the real-valued parameters $r$ and $s$ run over the radial cross-section $F$ of Saturn’s rings. To reflect the observed fractal nature of Saturn’s rings, [@MR610942] and independently [@MR665254] supposed that the set $F$ is a *fat fractal subset* of the set $\mathbb{R}$ of real numbers. The set $F$ itself is not a fractal, because its Hausdorff dimension is equal to $1$. However, the topological boundary $\partial F$ of the set $F$, that is, the set of points $x_{0}$ such that an arbitrarily small interval $(x_{0}-\varepsilon,x_{0}+\varepsilon)$ intersects with both $F$ and its complement, $\mathbb{R}\setminus F$, is a fractal. The Hausdorff dimension of $\partial F$ is not an integer number.
Mechanics of fractal rings
==========================
Basic considerations
--------------------
We begin with the standard gravitational parameter, $\mu =GM_{\mathrm{Saturn}}$; its value for Saturn ($\mu =37,931,187$ $km^{3}/s^{2}$) is known but will not be needed in the derivations that follow. For any particle of mass $m$ located within the ring, we take $m\ll M_{\mathrm{Saturn}}$ with dimensions also much smaller than the distance to the center of Saturn. Each particle is regarded as a rigid body, with its orbit about the spherically symmetric Saturn being circular. We are using the cylindrical coordinate system $\left( r,\theta ,z\right) $, such that the $z$-axis is aligned with the normal to the plane of rings, Fig. \[fig:1\]. The particle’s orbital frame of reference with the origin $O$ at its center of mass is made of three axes: $a_{1}$ in the radial direction, $a_{2}$ tangent to the orbit in the direction of motion, and $a_{3}$ normal to the orbit plane. All the particles orbit around Saturn in the same plane. The attitude of any given particle is described by the vector of body axes $\left\{ \mathbf{x}\right\}
^{T}=\{x_{1},x_{2},x_{3}\}^{T}$, which are related to the vector $\left\{
\mathbf{a}\right\} $ in the orbital frame of reference of the particle by$$\left\{ \mathbf{x}\right\} =\left[ \mathbf{l}\right] \left\{ \mathbf{a}\right\} .$$Here $\left[ \mathbf{l}\right] $ is the matrix of direction cosines $l_{i}$, $i=1,2,3$.
![The planar ring of particles, adapted from [@Li2015 Fig. 5(b)], showing the Saturnian (Cartesian and cylindrical) coordinate systems as well as the orbital frame of reference $\mathbf{(}a_{1},a_{2},a_{3})$ and the body axes $\mathbf{(}x_{1},x_{2},x_{3})$ of a typical particle.[]{data-label="fig:1"}](rings.png){width="\columnwidth"}
Henceforth, we consider two rings: Euclidean (i.e. non-fractal) and a fractal one; both rings are planar, Fig. \[fig:1\]. Hereinafter the subscript $_{\mathbb{E}}$ denotes any Euclidean object. Next, we must consider the mass of a Euclidean ring (body $B_{\mathbb{E}}$) versus a fractal ring (body $B_{\alpha }$). From a discrete system point of view, the ring is made of $I$ particles $\{i=1,...,I\}$, each with a respective mass $m_{i}$, moment of inertia $\mathbf{j}_{i}$, and positions $\mathbf{x}_{i}$.
The mass of a Euclidean ring $B_{\mathbb{E}}$, with radius $r\in \lbrack
R_{D},R]$ and thickness $h$ in $z$-direction, is now taken in a continuum sense $$\begin{array}{c}
M_{\mathbb{E}}=\sum_{i=1}^{I}m_{i}\rightarrow \int_{B_{\mathbb{E}}}\rho dB_{\mathbb{E}}=h\int_{R_{D}}^{R}\int_{0}^{2\pi }\rho _{\mathbb{E}}hdS_{2} \\
=2\pi h\rho _{\mathbb{E}}\int_{R_{D}}^{R}rdr=\rho _{\mathbb{E}}h\pi \left(
R^{2}-R_{D}^{2}\right) .\end{array}$$In the above we have assumed the mass to be homogeneously distributed throughout the ring with a mass density $\rho _{\mathbb{E}}$. To get quantitative results, one may take: $R=140\times 10^{6}m$ as the outer radius of Saturn’s F ring, $R_{D}=74.5\times 10^{6}m$ as the radius of the (inner) D ring, and the rings’ thickness $h=100m$.
Mass densities
--------------
All the rings constituting the fractal ring $B_{\alpha }$ are embedded in $\mathbb{R}^{3}$, also with radius $r\in \lbrack R_{D},R]$ and thickness $h$ in $z$-direction. The parameter $\alpha $ ($<1$) denotes the fractal dimension in the radial direction, i.e. on any ray(any because the ring is axially symmetric about $z$). Thus, the (planar) fractal dimension, such as seen and measured on photographs, is $D=\alpha +1<2$, consistent with the fact that Saturn’s rings are partially plane-filling if interpreted as a planar body. In order to do any analysis involving $B_{\alpha }$, in the vein of @doi:10.1142/S0217979205032656 [@MR2210182], we employ the integration in non-integer dimensional space. That is, we take the infinitesimal element $dB_{\alpha }$ of $B_{\alpha }$ according to [@PhysRevE.88.057001]: $$dB_{\alpha }=h\text{ }dS_{\alpha },\text{ \ \ }dS_{\alpha }=\alpha \left(
\frac{r}{R}\right) ^{\alpha -1}dS,\text{ \ \ }dS=rdrd\theta .$$Now, the mass of a fractal ring is $$\begin{array}{c}
M_{\alpha }=\sum_{i=1}^{I}m_{i}\rightarrow \int_{B}\rho _{\alpha }dB_{\alpha
}=h\int_{R_{D}}^{R}\int_{0}^{2\pi }\rho _{\alpha }dS_{\alpha } \\
=\displaystyle2\pi h\rho _{\alpha }\int_{R_{D}}^{R}\alpha \left( \frac{r}{R}\right) ^{\alpha -1}rdr=2\pi h\rho _{\alpha }\frac{\alpha }{\alpha +1}\left(
R^{2}-\frac{R_{D}^{\alpha +1}}{R^{\alpha -1}}\right) ,\end{array}$$which involves an effective mass density $\rho _{\alpha }$ of a fractal ring. Note that the above correctly reduces to (1) for $\alpha \rightarrow 1$. Since the rings in both interpretations must have the same mass, requiring $M_{\alpha }=M_{\mathbb{E}}$ for any $\alpha $, gives $$\rho _{\alpha }=\frac{\alpha +1}{2\alpha }\rho _{\mathbb{E}},$$which is a decreasing function of $\alpha $ (i.e. we must have $\rho
_{\alpha }>\rho _{\mathbb{E}}$ for $\alpha <1$) and which correctly gives $\lim_{\alpha \rightarrow 1}\rho _{\alpha =1}=\rho _{\mathbb{E}}$ for $\alpha
=1$, i.e. when the fractal ring becomes non-fractal. Thus, a fractal ring has a higher effective mass density than the homogeneous Euclidean ring of the same overall dimensions.
Moments of inertia
------------------
The moment of inertia of the Euclidean ring ($r\in \lbrack 0,R]$ and thickness $h$ in $z$-direction), assuming $\rho _{\mathbb{E}}=\mathrm{const}$, is$$I_{\mathbb{E}}=\frac{1}{2}\pi h\rho _{\mathbb{E}}\left(
R^{4}-R_{D}^{4}\right) =\frac{1}{2}M\left( R^{2}+R_{D}^{2}\right) ,$$while the moment of inertia of a fractal ring is$$\begin{array}{c}
I_{\alpha }=h\int_{B}\rho _{\alpha
}r^{2}dB_{E}=h\int_{R_{D}}^{R}\int_{0}^{2\pi }r^{2}\rho _{\alpha
}hdS_{\alpha } \\
\displaystyle2\pi h\rho _{\alpha }\int_{R_{D}}^{R}r^{2}\alpha \left( \frac{r}{R}\right) ^{\alpha -1}rdr=2\pi h\rho _{\alpha }\frac{\alpha }{\alpha +3}\left( R^{4}-\frac{R_{D}^{\alpha +3}}{R^{\alpha -1}}\right) .\end{array}$$Now, take the limit $\alpha \rightarrow 1$:$$\lim_{\alpha \rightarrow 1}I_{\alpha }=\frac{1}{2}\pi h\rho _{\mathbb{E}}\left( R^{4}-R_{D}^{4}\right) =I_{\mathbb{E}},$$as expected. Note that $I_{\alpha }$ is an increasing function of $\alpha $(i.e. we must have $I_{\alpha }<I_{\mathbb{E}}$ for $\alpha <1$) and which correctly gives $\lim_{\alpha \rightarrow 1}I_{\alpha }=I_{\mathbb{E}}$ for $\alpha =1$. We also observe from (6) that a fractal ring has a lower moment of inertia than the homogeneous Euclidean ring with the same overall dimensions.
Energies
--------
Since for an object of mass $m$ on a circular orbit the total energy is $E=-\mu /2r$, the total energy (sum of kinetic and potential) of the Euclidean ring is$$\begin{array}{c}
\displaystyle E_{\mathbb{E}}=-\sum_{i=1}^{I}\frac{\mu m_{i}}{2r_{i}}\rightarrow -\int_{B}\frac{\mu \rho _{E}}{2r}\text{ }dB \\
=\displaystyle-\frac{1}{2}h\mu \rho _{\mathbb{E}}\int_{0}^{R}\int_{0}^{2\pi
}r^{-1}rdrd\theta =-\pi h\mu \rho _{\mathbb{E}}\left( R-R_{D}\right) .\end{array}$$On the other hand, the total energy of the fractal ring $B_{\alpha }$ is \[again with $dS_{\alpha }=\alpha \left( \frac{r}{R}\right) ^{\alpha
-1}rdrd\theta $\]$$\begin{array}{c}
E_{\alpha }=\displaystyle-\sum_{i=1}^{I}\frac{\mu m_{i}}{2r_{i}}\rightarrow
-\int_{B}\frac{\mu \rho _{\alpha }}{2r}\text{ }dB=-\int_{R_{D}}^{R}\frac{1}{2}h\mu \rho _{\alpha }\frac{\alpha +1}{2\alpha }r^{-1}dS_{\alpha } \\
=\displaystyle-\int_{R_{D}}^{R}\int_{0}^{2\pi }\frac{1}{2}h\mu \rho _{\alpha
}\alpha \frac{\alpha +1}{2\alpha }r^{-1}\left( \frac{r}{R}\right) ^{\alpha
-1}rdrd\theta =-\pi h\mu \rho _{\alpha }\frac{\alpha +1}{2\alpha }\left(
R-R_{D}\right) .\end{array}$$Now, take the limit $\alpha \rightarrow 1$:$$\lim_{\alpha \rightarrow 1}E_{\alpha }=\frac{1}{2}\pi h\rho _{\mathbb{E}}\left( R-R_{D}\right) =E_{\mathbb{E}},$$as expected.
Comparing $E_{\alpha }$ with $E_{\mathbb{E}}$, gives $$E_{\alpha }=\frac{\alpha +1}{2\alpha }E_{\mathbb{E}},$$which is a decreasing function of $\alpha $. Thus, given the minus sign in (8) and (9), the fractal ring has a lower total energy than the homogeneous Euclidean ring with the same overall dimensions and the same mass. In other words, with reference to question Q1 in the Introduction, the ring having a fractal structure is more likely than that with a non-fractal one.
The foregoing argument extends the approach of @yang2007applied, who showed that a Euclidean ring has a lower energy than a Euclidean spherical shell, which in turn is lower than that of a Euclidean ball. Putting all the inequalities together, we have$$E_{\alpha }\leq E_{\mathbb{E}}\leq E_{\mathrm{shell}}\leq E_{\mathrm{ball}}.$$
Angular Momenta
---------------
For any particle of velocity $v$ on a circular orbit of radius $r$ around a planet:$$\mu =rv^{2}=r^{3}\Omega ^{2}=4\pi ^{2}r^{3}/T^{2},$$where $\Omega $ is the angular velocity and $T$ is the period. This implies: $$v=\sqrt{\mu /r}\text{\ \ \ and \ \ }\Omega =\sqrt{\mu /r^{3}}.$$
For the Euclidean ring ($r\in \lbrack 0,R]$ and thickness $h$ in $z$-direction), the angular momentum is$$\begin{array}{c}
\displaystyle H_{\mathbb{E}}=\sum_{i=1}^{I}m_{i}r_{i}v_{i}\rightarrow
h\int_{R_{D}}^{R}\int_{0}^{2\pi }\rho _{\mathbb{E}}rv\text{ }rdrd\theta \\
=\displaystyle h\int_{R_{D}}^{R}\int_{0}^{2\pi }\rho _{\mathbb{E}}r\sqrt{\mu
/r}\text{ }rdrd\theta =2\pi h\rho _{\mathbb{E}}\sqrt{\mu }\frac{2}{5}\left(
R^{5/2}-R_{D}^{5/2}\right) ,\end{array}$$while for the fractal ring $B_{\alpha }$, the angular momentum is$$\begin{array}{c}
H_{\alpha }=\sum_{i=1}^{I}m_{i}r_{i}v_{i}\rightarrow
h\int_{0}^{R}\int_{R_{D}}^{2\pi }\rho (r)rv\text{ }dS_{\alpha }=\displaystyle2\pi h\rho _{\alpha }\int_{R_{D}}^{R}r\sqrt{\frac{\mu }{r}}\alpha \left(
\frac{r}{R}\right) ^{\alpha -1}\text{ }rdr \\
=\displaystyle2\pi h\rho _{\alpha }\sqrt{\mu }\frac{\alpha }{\alpha +3/2}R^{1-\alpha }\left( R^{3/2+\alpha }-R_{D}^{3/2+\alpha }\right) .\end{array}$$This correctly reduces to $H_{\mathbb{E}}$ above for $\alpha \rightarrow 1$.
Comparing $H_{\alpha }$ with $H_{\mathbb{E}}$, shows that $H_{\alpha }$ is an increasing function of $\alpha $ and this correctly gives $\lim_{\alpha
\rightarrow 1}H_{\alpha =1}=H_{\mathbb{E}}$, i.e. the fractal ring has a lower angular momentum than the homogeneous Euclidean ring with the same overall dimensions.
At this point, we note that in inelastic collisions the momentum is conserved (just as in elastic collisions), but the kinetic energy is not as it is partially converted to other forms of energy. If this argument is applied to the rings, one may argue that $H_{\alpha }=H_{\mathbb{E}}$should hold for any $\alpha $, which can be satisfied by accounting for the angular momentum of particles due to rotation about their own axes . Thus, instead of (13), writing $j_{i}$ for the moment of inertia of the particle $i$, we have the contribution of the angular momentum of that rotation in terms of the Euler angle $\phi $ about the $a_{3}$ axis:$$H_{\mathbb{E}}=\sum_{i=1}^{I}m_{i}r_{i}v_{i}+\sum_{i\in I}j_{i}\omega
_{zi}\rightarrow h\int_{R_{D}}^{R}\int_{0}^{2\pi }\rho _{\mathbb{E}}rv\text{
}rdrd\theta +h\int_{R_{D}}^{R}\int_{0}^{2\pi }j\phi \text{ }rdrd\theta .$$The first integral can be calculated as before, while in the second one we could assume $j=const$ although this would still leave the microrotation $\omega _{z}$ as an unknown function of $r$. Turning to the fractal ring we also have two terms$$H_{\alpha }=\sum_{i=1}^{I}m_{i}r_{i}v_{i}+\sum_{i\in I}j_{i}\omega
_{zi}\rightarrow h\int_{R_{D}}^{R}\int_{0}^{2\pi }\rho _{E}rv\text{ }dS_{\alpha }+h\int_{R_{D}}^{R}\int_{0}^{2\pi }j\phi \text{ }dS_{\alpha },$$showing that the statistics $\omega _{z}\left( r\right) $ needs to be determined. At this point we turn to the question Q2.
A stochastic model of kinematics
================================
First, we consider the particles in Saturn’s rings at a time moment $0$.
Introduce a spherical coordinate system $(r,\varphi ,\theta )$ with origin $O $ in the centre of Saturn such that the plane of Saturn’s rings corresponds to the polar angle’s value $\theta =\pi /2$. Let $\overline{\bm{\omega}}(r,\varphi )\in \mathbb{R}^{3}$ be the angular velocity vector of a rotating particle located at $(r,\varphi )$. We assume that $\overline{\bm{\omega}}(r,\varphi )$ is a *single realisation of a random field*.
To explain the exact meaning of this construction, we proceed as follows. Let $(x,y,z)$ be a Cartesian coordinate system with origin in the centre of Saturn such that the plane of Saturn’s rings corresponds to the $xy$-plane, Fig. \[fig:1\]. Let $O(2)$ be the group of real orthogonal $2\times 2$ matrices, and let $SO(2)$ be its subgroup consisting of matrices with determinant equal to $1$. Put $G=O(2)\times SO(2)$, $K=O(2)$. The homogeneous space $C=G/K=SO(2)$ can be identified with a circle, the trajectory of a particle inside rings.
Consider the real orthogonal representation $U$ of the group $O(2)$ in $\mathbb{R}^3$ defined by $$\label{eq:1}
g=
\begin{pmatrix}
g_{11} & g_{12} \\
g_{21} & g_{22}\end{pmatrix}
\mapsto U(g)=
\begin{pmatrix}
g_{11} & g_{12} & 0 \\
g_{21} & g_{22} & 0 \\
0 & 0 & \det g\end{pmatrix}
.$$ Introduce an equivalence relation in the Cartesian product $G\times\mathbb{R}^3$: two elements $(g_1,\mathbf{x}_1)$ and $(g_2,\mathbf{x}_2)$ are equivalent if and only if there exists an element $g\in O(2)$ such that $(g_2,\mathbf{x}_2)=(g_1g,U(g^{-1})\mathbf{x}_1)$. The *projection map* maps an element $(g,\mathbf{x})\in G\times\mathbb{R}^3$ to its equivalence class and defines the quotient topology on the set $E_U$ of equivalence classes. Another projection map, $$\pi\colon E_U\to C,\qquad\pi(g,\mathbf{x})=gK,$$ determines a *vector bundle* $\xi=(E_U,\pi,C)$.
The topological space $R=\mathbb{R}^2\setminus\{\mathbf{0}\}$ is the union of circles $C_r$ of radiuses $r>0$. Every circle determines the vector bundle $\xi_r=(E_{Ur},\pi_r,C_r)$. Consider the vector bundle $\eta=(E,\pi,R) $, where $E$ is the union of all $E_{Ur}$, and the restriction of the projection map $\pi$ to $E_{Ur}$ is equal to $\pi_r$. The random field $\overline{\bm{\omega}}(r,\varphi)$ is a *random section* of the above bundle, that is, $\overline{\bm{\omega}}(r,\varphi)\in\pi^{-1}(r,\varphi)=\mathbb{R}^3$. In what follow we assume that the random field $\overline{\bm{\omega}}(r,\varphi)$ is *second-order*, i.e., $\mathsf{E}[\|\overline{\bm{\omega}}(r,\varphi)\|^2]<\infty$ for all $(r,\varphi)\in R$.
There are at least three different (but most probably equivalent) approaches to the construction of random sections of vector bundles, the first by @MR2737761, the second by @MR2884225 [@MR2977490], and the third by @MR3170229. In what follows, we will use the second named approach. It is based on the following fact: the vector bundle $\eta=(E,\pi,R)$ is *homogeneous* or *equivariant*. In other words, the action of the group $O(2)$ on the bundle base $R$ induces the action of $O(2)$ on the total space $E$ by $(g_0,\mathbf{x})\mapsto(gg_0,\mathbf{x})$. This action identifies the spaces $\pi^{-1}(r_0,\varphi)$ for all $\varphi\in[0,2\pi)$, while the action of the multiplicative group $\mathbb{R}^+$ on R, $\lambda(r,\varphi)=(\lambda r,\varphi)$, $\lambda>0$, identifies the spaces $\pi^{-1}(r,\varphi_0)$ for all $r>0$. We suppose that the random field $\overline{\bm{\omega}}(r,\varphi)$ is *mean-square continuous*, i.e., $$\lim_{\|\mathbf{x}-\mathbf{x}_0\|\to 0}\mathsf{E}[\|\overline{\bm{\omega}}(\mathbf{x})-\overline{\bm{\omega}}(\mathbf{x}_0))\|^2]=0$$ for all $\mathbf{x}_0\in R$.
Let $\langle \overline{\bm{\omega}}(\mathbf{x})\rangle =\mathsf{E}[\overline{\bm{\omega}}(\mathbf{x})]$ be the one-point correlation vector of the random field $\overline{\bm{\omega}}(\mathbf{x})$. On the one hand, under rotation and/or reflection $g\in O(2)$ the point $\mathbf{x}$ becomes the point $g\mathbf{x}$. Evidently, the axial vector $\overline{\bm{\omega}}(\mathbf{x})$ transforms according to the representation and becomes $U(g)\overline{\bm{\omega}}(g\mathbf{x})$. The one-point correlation vector of the so transformed random field remains the same, i.e., $$\langle \overline{\bm{\omega}}(g\mathbf{x})\rangle =U(g)\langle \overline{\bm{\omega}}(\mathbf{x})\rangle .$$On the other hand, the one-point correlation vector of the random field $\overline{\bm{\omega}}(r,\varphi )$ should be independent upon an arbitrary choice of the $x$- and $y$-axes of the Cartesian coordinate systems, i.e., it should not depend on $\varphi $. Then we have $$\langle \overline{\bm{\omega}}(\mathbf{x})\rangle =U(g)\langle \overline{\bm{\omega}}(\mathbf{x})\rangle$$for all $g\in O(2)$, i.e., $\langle \overline{\bm{\omega}}(\mathbf{x})\rangle $ belongs to a subspace of $\mathbb{R}^{3}$ where a trivial component of $U$ acts. Then we obtain $\langle \overline{\bm{\omega}}(\mathbf{x})\rangle =\mathbf{0}$, because $U$ does not contain trivial components.
Similarly, let $\langle\overline{\bm{\omega}}(\mathbf{x}),\overline{\bm{\omega}}(\mathbf{y})\rangle=\mathsf{E}[\overline{\bm{\omega}}(\mathbf{x}) \otimes\overline{\bm{\omega}}(\mathbf{y})]$ be the two-point correlation tensor of the random field $\overline{\bm{\omega}}(\mathbf{x})$. Under the action of $O(2)$ we should have $$\langle\overline{\bm{\omega}}(g\mathbf{x}),\overline{\bm{\omega}}(g\mathbf{y})\rangle =(U\otimes U)(g)\langle\overline{\bm{\omega}}(\mathbf{x}),\overline{\bm{\omega}}(\mathbf{y})\rangle.$$ In other words, the random field $\overline{\bm{\omega}}(\mathbf{x})$ is *wide-sense isotropic* with respect to the group $O(2)$ and its representation $U$.
Consider the restriction of the field $\overline{\bm{\omega}}(\mathbf{x})$ to a circle $C_r$, $r>0$. The spectral expansion of the field $\{\,\overline{\bm{\omega}}(r,\varphi)\colon\varphi\in C_r\,\}$ can be calculated using @MR2884225 [Theorem 2] or @MR2977490 [Theorem 2.28].
The representation $U$ is the direct sum of the two irreducible representations $\lambda_-(g)=\det g$ and $\lambda_1(g)=g$. The vector bundle $\eta$ is the direct sum of the vector bundles $\eta_-$ and $\eta_1$, where the bundle $\eta_-$ (resp. $\eta_1$) is generated by the representation $\lambda_-$ (resp. $\lambda_1$). Let $\mu_0$ be the trivial representation of the group $SO(2)$, and let $\mu_k$ be the representation $$\mu_k(\varphi)=
\begin{pmatrix}
\cos(k\varphi) & \sin(k\varphi) \\
-\sin(k\varphi) & \cos(k\varphi)\end{pmatrix}
.$$ The representations $\lambda_-\otimes\mu_k$, $k\geq 0$ are all irreducible orthogonal representations of the group $G=O(2)\times SO(2)$ that contain $\lambda_-$ after restriction to $O(2)$. The representations $\lambda_1\otimes\mu_k$, $k\geq 0$ are all irreducible orthogonal representations of the group $G=O(2)\times SO(2)$ that contain $\lambda_1$ after restriction to $O(2)$. The matrix entries of $\mu_0$ and of the second column of $\mu_k$ form an orthogonal basis in the Hilbert space $L^2(SO(2),\mathrm{d}\varphi)$. Their multiples $$e_k(\varphi)=
\begin{cases}
\frac{1}{\sqrt{2\pi}}, & \mbox{if } k=0, \\
\frac{1}{\sqrt{\pi}}\cos(k\varphi), & \mbox{if } k\leq-1 \\
\frac{1}{\sqrt{\pi}}\sin(k\varphi), & \mbox{if } k\geq 1\end{cases}$$ form an orthonormal basis of the above space. Then we have $$\label{eq:3}
\overline{\bm{\omega}}(r,\varphi)=\sum_{k=-\infty}^{\infty}e_k(\varphi)\mathbf{Z}^k(r),$$ where $\{\,\mathbf{Z}^k(r)\colon k\in\mathbb{Z}\,\}$ is a sequence of centred stochastic processes with $$\begin{aligned}
\mathsf{E}[\mathbf{Z}^k(r)\otimes\mathbf{Z}^l(r)]&=\delta_{kl}B^{(k)}(r),\\
\sum_{k\in\mathbb{Z}}\operatorname{tr}(B^{(k)}(r))&<\infty. \end{aligned}$$
It follows that $$\mathbf{Z}^k(r)=\int_{0}^{2\pi}\overline{\bm{\omega}}(r,\varphi)e_k(\varphi)\,\mathrm{d}\varphi.$$ Then we have $$\label{eq:2}
\mathsf{E}[\mathbf{Z}^k(r)\otimes\mathbf{Z}^l(s)]=\int_{0}^{2\pi}\int_{0}^{2\pi} \mathsf{E}[\overline{\bm{\omega}}(r,\varphi_1)\otimes\overline{\bm{\omega}}(s,\varphi_2)] e_k(\varphi_1)\,\mathrm{d}\varphi_1e_l(\varphi_2)\,\mathrm{d}\varphi_2.$$ The field is isotropic and mean-square continuous, therefore $$\mathsf{E}[\overline{\bm{\omega}}(r,\varphi_1)\otimes\overline{\bm{\omega}}(s,\varphi_2)] =B(r,s,\cos(\varphi_1-\varphi_2))$$ is a continuous function. Note that $e_k(\varphi)$ are spherical harmonics of degree $|k|$. Denote by $\mathbf{x\cdot y}$ the standard inner product in the space $\mathbb{R}^d$, and by $\mathrm{d}\omega(\mathbf{y})$ the Lebesgue measure on the unit sphere $S^{d-1}=\{\,\mathbf{x}\in\mathbb{R}^d\colon\|\mathbf{x}\|=1\,\}$. Then $$\int_{S^{d-1}}\,\mathrm{d}\omega(\mathbf{x})=\omega_d=\frac{2\pi^{d/2}}{\Gamma(d/2)},$$ where $\Gamma$ is the Gamma function.
Now we use the Funk–Hecke theorem, see @MR1688958. For any continuous function $f$ on the interval $[-1,1]$ and for any spherical harmonic $S_k(\mathbf{y})$ of degree $k$ we have $$\int_{S^{d-1}}f(\mathbf{x\cdot y})S_k(\mathbf{x})\,\mathrm{d}\omega(\mathbf{x})=\lambda_kS_k(\mathbf{y}),$$ where $$\lambda_k=\omega_{d-1}\int_{-1}^{1}f(u)\frac{C^{(n-2)/2}_k(u)}{C^{(n-2)/2}_k(1)} (1-u^2)^{(n-3)/2}\,\mathrm{d}u,$$ $d\geq 3$, and $C^{(n-2)/2}_k(u)$ are Gegenbauer polynomials. To see how this theorem looks like when $d=2$, we perform a limit transition as $n\downarrow 2$. By @MR1688958 [Equation 6.4.13’], $$\lim_{\lambda\to 0}\frac{C^{\lambda}_k(u)}{C^{\lambda}_k(1)}=T_k(u),$$ where $T_k(u)$ are Chebyshev polynomials of the first kind. We have $\omega_1=2$, $\mathbf{x\cdot y}$ becomes $\cos(\varphi_1-\varphi_2)$, and $\mathrm{d}\omega(\mathbf{x})$ becomes $\mathrm{d}\varphi_1$. We obtain $$\int_{0}^{2\pi}B(r,s,\cos(\varphi_1-\varphi_2))e_k(\varphi_1)\,\mathrm{d}\varphi_1 =B^{(k)}(r,s)e_k(\varphi_2),$$ where $$B^{(k)}(r,s)=2\int_{-1}^{1}B(r,s,u)T_{|k|}(u)(1-u^2)^{-1/2}\,\mathrm{d}u,$$ Equation becomes $$\mathsf{E}[\mathbf{Z}^k(r)\otimes\mathbf{Z}^l(s)]=\int_{0}^{2\pi}B^{(k)}(r,s) e_k(\varphi_2)e_l(\varphi_2)\,\mathrm{d}\varphi_2=\delta_{kl}B^{(k)}(r,s).$$ In particular, if $k\neq l$, then the processes $\mathbf{Z}^k(r)$ and $\mathbf{Z}^l(r)$ are uncorrelated.
Calculate the two-point correlation tensor of the random field $\overline{\bm{\omega}}(r,\varphi)$. We have $$\label{eq:4}
\begin{aligned}
\mathsf{E}[\overline{\bm{\omega}}(r,\varphi_1)\otimes\overline{\bm{\omega}}(s,\varphi_2)]
&=\sum_{k=-\infty}^{\infty}e_k(\varphi_1)e_k(\varphi_2)B^{(k)}(r,s)\\
&=\frac{1}{2\pi}B^{(0)}(r,s)+\frac{1}{\pi}\sum_{k=1}^{\infty}
\cos(k(\varphi_1-\varphi_2))B^{(k)}(r,s). \end{aligned}$$
Now we add a time coordinate, $t$, to our considerations. A particle located at $(r,\varphi)$ at time moment $t$, was located at $(r,\varphi-\sqrt{GM}t/r^{3/2})$ at time moment $0$. It follows that $$\overline{\bm{\omega}}(t,r,\varphi)=\overline{\bm{\omega}}\left(r,\varphi-\frac{\sqrt{GM}t}{r^{3/2}}\right),$$ where $G$ is Newton’s gravitational constant and $M$ is the mass of Saturn. Equation gives $$\label{eq:5}
\overline{\bm{\omega}}(t,r,\varphi)=\sum_{k=-\infty}^{\infty}
e_k\left(\varphi-\frac{\sqrt{GM}t}{r^{3/2}}\right)\mathbf{Z}^k(r),$$ while Equation gives $$\begin{aligned}
\mathsf{E}[\overline{\bm{\omega}}(t_1,r,\varphi_1)\otimes\overline{\bm{\omega}}(t_2,s,\varphi_2)] &=\frac{1}{2\pi}B^{(0)}(r,s)\\
&\quad+\frac{1}{\pi}\sum_{k=1}^{\infty}
\cos\left(k\left(\varphi_1-\varphi_2-\frac{\sqrt{GM}(t_1-t_2)}{r^{3/2}}\right)\right)B^{(k)}(r,s). \end{aligned}$$
Conversely, let $\{\,B^{(k)}(r,s)\colon k\geq 0\,\}$ be a sequence of continuous positive-definite matrix-valued functions with $$\label{eq:6}
\sum_{k=0}^{\infty}\operatorname{tr}(B^{(k)}(r,r))<\infty,$$ and let $\{\,\mathbf{Z}_k(r)\colon k\in\mathbb{Z}\,\}$ be a sequence of uncorrelated centred stochastic processes with $$\mathsf{E}[\mathbf{Z}^k(r)\otimes\mathbf{Z}^l(s)]=\delta_{kl}B^{(|k|)}(r,s).$$ The random field may describe rotating particles inside Saturn’s rings, if all the functions $B^{(k)}(r,s)$ are equal to $0$ outside the rectangle $[R_0,R_1]^2$, where $R_0$ (resp. $R_1$) is the inner (resp. outer) radius of Saturn’s rings.
To make our model more realistic, we assume that all the functions $B^{(k)}(r,s)$ are equal to $0$ outside the Cartesian square $F^{2}$, where $F $ is a *fat fractal* subset of the interval $[R_{0},R_{1}]$, see @PhysRevLett.55.661. @MR665254 calls these sets *dusts of positive measure*. Such a set has a positive Lebesgue measure, its Hausdorff dimension is equal to $1$, but the Hausdorff dimension of its boundary is not an integer number.
A classical example of a fat fractal is a *fat Cantor set*. In contrast to the ordinary Cantor set, where we delete the middle one-third of each interval at each step, this time we delete the middle $3^{-n}$th part of each interval at the $n$th step.
To construct an example, consider an arbitrary sequence of continuous positive-definite matrix-valued functions $\{\,B^{(k)}(r,s)\colon k\geq
0\,\} $ satisfying of the following form: $$B^{(k)}(r,s)=\sum_{i\in I_k}\mathbf{f}_{ik}(r)\mathbf{f}^{\top}_{ik}(s),$$ where $\mathbf{f}_{ik}(r)\colon[R_0,R_1]\to\mathbb{R}^3$ are continuous functions, satisfying the following condition: for each $r\in[R_0,R_1]$ the set $I_{kr}=\{\,i\in I_k\colon f_i(r)\neq 0\,\}$ is as most countable and the series $$\sum_{i\in I_{kr}}\|\mathbf{f}_i(r)\|^2$$ converges. The so defined function is obviously positive-definite. Put $$\tilde{B}^{(k)}(r,s)=\sum_{i\in I_k}\tilde{\mathbf{f}}_{ik}(r)\tilde{\mathbf{f}}^{\top}_{ik}(s),\qquad r,s\in F.$$ The functions $\tilde{B}^{(k)}(r,s)$ are the restrictions of positive-definite functions $B^{(k)}(r,s)$ to $F^2$ and are positive-definite themselves. Consider the centred stochastic process $\{\,\tilde{\mathbf{Z}}^k(r)\colon r\in F\,\}$ with $$\mathsf{E}[\tilde{\mathbf{Z}}^k(r)\otimes\tilde{\mathbf{Z}}^l(s)]
=\delta_{kl}\tilde{B}^{(|k|)}(r,s),\qquad r,s\in F.$$ Condition guarantees the mean-square convergence of the series $$\overline{\bm{\omega}}(t,r,\varphi)=\sum_{k=-\infty}^{\infty}
e_k\left(\varphi-\frac{\sqrt{GM}t}{r^{3/2}}\right)\tilde{\mathbf{Z}}^k(r)$$ for all $t\geq 0$, $r\in F$, and $\varphi\in[0,2\pi]$.
Closure
=======
This paper reports an investigation of the fractal character of Saturnian rings. First, working with the calculus in a non-integer dimensional space, by energy arguments, we infer that the fractally structured ring is more likely than a non-fractal one. Next, we develop a kinematics model in which angular velocities of particles form a random field.
[^1]: Mälardalen University, Sweden
[^2]: University of Illinois at Urbana-Champaign, USA
[^3]: This material is based upon the research partially supported by the NSF under grant CMMI-1462749.
|
---
abstract: 'Semiconductor nanowires offer the possibility to grow high quality quantum dot heterostructures, and in particular CdSe quantum dots inserted in ZnSe nanowires have demonstrated the ability to emit single photons up to room temperature. In this letter, we demonstrate a bottom-up approach to fabricate a photonic fiber-like structure around such nanowire quantum dots by depositing an oxide shell using atomic layer deposition. Simulations suggest that the intensity collected in our NA=0.6 microscope objective can be increased by a factor 7 with respect to the bare nanowire case. Combining micro-photoluminescence, decay time measurements and numerical simulations, we obtain a 4-fold increase in the collected photoluminescence from the quantum dot. We show that this improvement is due to an increase of the quantum dot emission rate and a redirection of the emitted light. Our ex-situ fabrication technique allows a precise and reproducible fabrication on a large scale. Its improved extraction efficiency is compared to state of the art top-down devices.'
author:
- Mathieu Jeannin
- Thibault Cremel
- Teppo Häyrynen
- Niels Gregersen
- 'Edith Bellet-Amalric'
- Gilles Nogues
- Kuntheak Kheng
title: 'Enhanced photon extraction from a nanowire quantum dot using a bottom-up photonic shell'
---
[^1]
[^2]
Introduction
============
Controlling and enhancing the spontaneous emission of quantum emitters is one of the current key issues in the field of nanophotonics. Semiconductor quantum dots (QDs) are considered as promising and efficient single-photon emitters for quantum optics applications. [@Michler2000; @Santori2001; @Santori2002; @Zrenner2002; @Akopian2006; @Shields2007] Over the past few years, several approaches have been pursued to control their emission properties, from the use of photonic crystals [@Viasnoff-Schwoob2005; @Lund-Hansen2008] to top-down photonic wires [@Claudon2010; @Heinrich2010; @Bleuse2011] and trumpets.[@Munsch2012; @Munsch2013] These strategies are based on the early work of Purcell[@Purcell1946] which demonstrated that the spontaneous emission of an emitter can be modified by engineering its electromagnetic environment. They rely on a waveguiding approach to increase the coupling between a well-defined propagating optical mode and the QD while simultaneously reducing the coupling between the QD and background radiation modes, offering control of both the optical mode profile and the QD spontaneous emission rate.
In this context, the interest of the dot-in-a-nanowire configuration fabricated using bottom-up methods naturally arises because it provides a simple way to ensure the centering of a single quantum emitter in the photonic structure.[@Reimer2012; @Bulgarini2012; @Bulgarini2014] The bottom-up fabrication method also avoids heavy processing, like etching the semiconducting material, that is often detrimental to the QDs optical properties. However, the main realizations up to now concern III-V semiconductors, [@Reimer2012; @Bulgarini2012; @Bulgarini2014] limiting the operation range to the cryogenic temperature. Tackling this issue, the potential of II-VI materials, in particular CdSe QDs inserted inside ZnSe nanowires (NWs) has been demonstrated in previous studies. They allow for robust high temperature single-photon emission using heteroepitaxial [@Tribu2008] or homoepitaxial [@Bounouar2012a] nanowire growth. Contrary to all the aforementioned systems where the photonic wire structure has a diameter comparable to the wavelength $\lambda/n$ of the guided light which allows for highly efficient coupling to the HE$_{11}$ mode[@Nowicki2008], the diameter of the II-VI NW embedding the QD ($\sim$) is much smaller than the wavelength of the emitted light (). It leads to light emission predominantly into non-guided radiation modes and a low collection efficiency. An additional fabrication effort has thus to be made to ensure an efficient coupling to the collection optics.
In a previous report[@Cremel2014] we have theoretically investigated the potential of using an oxide shell deposition on a bare ZnSe NW to form a thick photonic wire structure. In this article, we experimentally demonstrate the use of atomic layer deposition (ALD) to fabricate a conformal aluminum oxide (Al$_2$O$_3$) shell around ZnSe NWs containing a single CdSe QD. We show that the oxide shell drastically enhances the light intensity emitted by the QD, and we use time-resolved microphotoluminescence to systematically study the effect of the shell thickness on the nanowire quantum dot (NWQD) emission rate. Our results are compared to numerical simulations accounting for the real NW geometry, evidencing the different physical mechanisms leading to the enhancement of the spontaneous emission from the QD and to the improved light collection from the emitting structure.
Principles of operation
=======================
To illustrate the effect of the NW and its surrounding medium on the QD emission rate, let us consider a QD placed inside an infinitely long cylinder as illustrated in Fig. \[fig:Semianalytical\](a) radiating a field at a wavelength $\lambda$. The cylinder is made of a dielectric material (refractive index $n$) and has a diameter $d$. We first consider a dipole orientation perpendicular to the NW axis in order to use the NW as a propagation medium for the emitted light. In the limit where $d \ll
\lambda/n$, the dielectric screening effect[@Bleuse2011] reduces the spontaneous emission rate $\gamma$ by a factor: $$\frac{\gamma}{\gamma_0}=\frac{4}{n(n^2+1)^2},\label{eq:screening}$$ where $\gamma_0$ is the radiative emission rate in the bulk material of index $n$.[@ClaudonGerard_HarnessingLightwith_13] For a ZnSe cylinder ($n_{\rm ZnSe}$ = 2.68 at $\lambda$=), the screening factor is $\sim$1/45. If the NW is surrounded by a shell of refractive index $n_s$ instead of vacuum, equation \[eq:screening\] remains valid by replacing $n$ with the index contrast $n/n_s$. For an surrounding medium ($n_s=$1.77), the screening factor becomes $\sim$1/4.1, resulting in an order of magnitude larger radiative rate.
![(a) Geometry of the infinite NW. (b) Total spontaneous emission rate (black $+$) and spontaneous emission rate into the first guided mode HE$_{11}$ (red ) as a function of shell radius for a radial dipole. (c) Fraction $\beta$ of power radiated into the HE$_{11}$ mode. []{data-label="fig:Semianalytical"}](Semianalytical){width="8.6cm"}
In addition to changing the dielectric screening, the shell also influences the guiding of light along the NW. We have computed the total emission rate $\gamma$ and the emission rate $\gamma_{\rm HE11}$ into the fundamental HE$_{11}$ waveguide mode from a radial dipole as function of the shell thickness $t_s$ \[see Fig. \[fig:Semianalytical\](a)\] using a semi-analytical approach[@Yariv1997] combined with an efficient non-uniform discretization scheme in k space.[@HaeyrynenGregersen_OpengeometryFourier_16] The results are plotted in Figure \[fig:Semianalytical\](b). We observe that the shell thickness of $\sim$ not only leads to an increased total emission rate, it also allows for confinement of the fundamental HE$_{11}$ mode to the core-shell NW leading to a preferential coupling of the emitted light to this mode. Figure \[fig:Semianalytical\](c) presents the spontaneous emission $\beta$ factor representing the fraction $\beta=\gamma_{\rm HE11}/\gamma$ of emitted light coupled to the HE$_{11}$ mode. We observe indeed that up to 71% of the emitted light is coupled to this mode for $t_s$=. The dipole thus becomes coupled to the equivalent of a monomode photonic wire[@Claudon2010; @Heinrich2010; @Bleuse2011; @Reimer2012; @Bulgarini2012; @Bulgarini2014] paving the way to the control of its far-field radiation pattern.
Sample fabrication
==================
Our emitters are CdSe QDs embedded inside a ZnSe NW with a thin, epitaxial passivation Zn$_{0.83}$Mg$_{0.17}$Se shell grown around the NW. They are grown by molecular beam epitaxy on a GaAs(111)B substrate. A ZnSe buffer layer is first grown on the GaAs substrate after which a thin layer of Au (less than one monolayer thick) is evaporated on the sample surface and dewetted at to form small ($\sim$ diameter) Au droplets that serve as a catalyst for the NW growth. The substrate temperature is then set at and a flux of Zn and Se atoms with an excess of Se is used, inducing preferential growth of vertical ZnSe NWs. The NWs are in wurtzite phase and their diameter is the same as the droplet ($\sim$ diameter). The thickness of the initial Au layer is chosen to ensure a low NW density ($\leq$ 1 NW per ). After the growth of a high NW, the atom fluxes are stopped to allow the evacuation of residual Se atoms inside the droplet. Then, the QD is grown under a flux of Cd and Se atoms for . The fluxes are interrupted again before the ZnSe growth is resumed, resulting in an expected QD height of 2- inserted in a $\sim$ high NW. Finally, an epitaxial Zn$_{0.83}$Mg$_{0.17}$Se shell ( thick) is grown around the NW at . A scanning electron microscope (SEM) image of such a CdSe/ZnSe/ZnMgSe core/shell NWQD system is presented in Figure \[fig:NWPresentation\](a). The flag-shape termination of the NW is formed during the growth of the ZnMgSe shell. It is present in some NWs.
![ (a) SEM image of a standing ZnSe/ZnMgSe NW embedding a CdSe QD. The QD position is marked by the red square. (b,c) Tilted SEM image of a ZnSe NW after a and thick Al$_2$O$_3$ shell deposition respectively. The NW is sketched on the SEM image. Note the circular shape of the shell as well as its hemispherical termination above the NW apex. (d) Sketch of the NWQD geometry, indicating the QD height (2-), the NW diameter ($\simeq$), the epitaxial shell thickness ($\simeq$) and the ALD shell thickness $t_s$. []{data-label="fig:NWPresentation"}](NWPresentation){width="7cm"}
The higher bandgap of Zn$_{0.83}$Mg$_{0.17}$Se shell prevents the charge carriers to recombine non-radiatively on the ZnSe NW sidewall and hence improves the quantum yield of the CdSe emitter. In principle, it could directly be used to grow a photonic wire of diameter $\sim \lambda / n_{\rm ZnSe}$ around the NWQD. However, during the epitaxial shell growth two phenomena are competing: the radial growth of the shell around the wurtzite NWs, and the vertical growth of a 2D Zn$_{0.83}$Mg$_{0.17}$Se layer on the sample surface. The radial shell growth rate is very low because the growth of ZnSe on WZ surfaces is not favourable. Because of this low shell growth rate, a trade-off has to be found to avoid burying the NWs in a Zn$_{0.83}$Mg$_{0.17}$Se matrix. As a result, only thin epitaxial shells can be fabricated.
The complexity of creating a thick epitaxial shell is one of the reasons why we fabricate the photonic structure by depositing an oxide shell around the NW using ALD. Another reason is that, since this process step can be done *separately* from the NW growth process, it allows to tune ex situ the shell parameters after a first optical characterization of the QD. Indeed, due to its slow deposition rate, the ALD process allows to precisely control the deposited thickness, which can also be finally verified using scanning electron microscopy. We have tested several oxide materials, and selected Al$_2$O$_3$ because it produced very smooth and conformal, amorphous shells. Figure \[fig:NWPresentation\](b) and (c) show two SEM images of the resulting oxide shell deposition ( and ), and the complete structure is sketched in Fig. \[fig:NWPresentation\](d). We note that the conformal deposition allows to end the NW+shell structure by an almost perfect half-sphere as can be seen in Fig. \[fig:NWPresentation\](b,c). ALD also buries the Au droplet under the shell. The latter might interact with the field emitted by the QD through its localized plasmon resonance. Considering its small diameter it will essentially absorb the incoming field. Moreover the guided HE11 mode profile presents a minimum on the NW axis. This is why we neglect the droplet influence in the following.
Experimental results
====================
![ (a) PL spectrum of a NWQD with thick photonic shell for different pumping powers. It shows a exciton (X), charged exciton (CX) and biexciton (XX) lines. The corresponding pumping powers are reported in panel (b). The black rectangle indicates the integration bandwidth used to extract the total exciton emission intensity (X line). (b) Integrated exciton emission intensity as a function of pumping power, in a log-log scale. (c) Blue crosses: Total exciton emission intensity for different NWQDs as a function of the oxide shell radius. Red diamonds: average of the experimental data points. Data are normalized to the average intensity at $t_s=$ Black lines: results of the numerical simulations for a radial (solid line) and an axial (dashed line) dipole. They are normalized to the axial intensity at $t_s=$ []{data-label="fig:IVsThick"}](IVsThick){width="8cm"}
A sample from a single epitaxial growth process is cut in pieces, and photonic structures with different oxide shell thicknesses are fabricated. Taking advantage of the low NW density, individual structures are optically characterized directly on the growth substrate. The samples are mounted on the cold finger of a He-flux cryostat and cooled down to . Individual photonic structures are probed using confocal microphotoluminescence (PL). They are excitated by a supercontinuum pulsed laser (Fianium WhiteLase, pulse duration, repetiton rate ) and a spectrometer selecting a bandwidth centered around . This excitation energy, below the ZnSe gap, allows us to induce crossed transitions between delocalized states in the NW 1D continuum and a discrete confined 0D state in the NWQD band structure[@VasanelliBastard_ContinuousAbsorptionBackground_02]. In this configuration, the NW axis is aligned with the optical axis and emission from the QD is collected by a $NA=0.6$ objective. A typical NWQD spectrum is presented in Figure \[fig:IVsThick\](a) as a function of the pump laser power. Three lines can be identified and are attributed to the exciton (X), the charged exciton (CX) and the bi-exciton (XX) respectively. The total emission intensity of the X line as a function of the pump power is reported in Figure \[fig:IVsThick\](b). It shows a linear increase at low pumping power, and a constant plateau at high pumping powers corresponding to the saturation of the exciton level.[^3] Under pulsed excitation, we note that changing the shell thickness might modify the laser power in the NW and the excitation probability of the QD. Hence if affects the slope at low power in Fig. \[fig:IVsThick\](b). It has however no effect on the saturation plateau which only depends on the QD emission rate and light collection efficiency. This allows us to compare statistical sets of nanostructures with different oxide shell thicknesses. The total integrated emission at saturation as a function of the oxide shell thickness is reported in blue markers for each NWQD in Figure \[fig:IVsThick\](c). The values have been normalized to the average intensity at $t_s=$. For NWs without an oxide shell, the luminescence intensity is very low and we were never able to reach the saturation regime, this is why we do not report the corresponding points in Fig. \[fig:IVsThick\](c). For each shell thickness, we observe a large spread in exciton saturation intensity. However, we note a general trend of increasing saturation intensity with increasing shell thickness, as demonstrated by the red markers which show the position of the average intensity of our measurements for each shell thickness. On average, the deposition of a thick shell results in the experiments in an almost 4-fold enhancement of the collected intensity with respect to the thick shell case. The semi-analytical calculations show that this enhancement is 10-fold when we compare to a NW without oxide shell.
![(a) Example of TRPL signal versus time for 2 NWs with a $t_s$= shell. Background counts are measured for $t<0$ and substracted. Amplitude of counts are normalized to 1 to compare the 2 datasets. Red lines are mono-exponential fits, whose corresponding points in (b) are showm by arrows. (b) Blue crosses: experimental exciton decay times for several QDs as a function of the oxide shell radius. The vertical error bars represent the fit error. Black lines: numerical simulation results for a radial dipole (solid line), or an axial dipole (dashed line). Red dashed-dotted line: Semi-analytical calculations for the infinite NW. []{data-label="fig:TRPLVsThick"}](TRPLVsThick){width="8.6cm"}
The observed increase in intensity at saturation corresponds to the combination of improved collection efficiency through light redirection from the structure and enhancement of the spontaneous emission rate. In the latter case, a modification of the QD dynamics is expected to be detected by measuring the exciton decay rate. Time-resolved measurements were carried out using a low pump power as compared to the exciton saturation power to avoid any repopulation of the X level. The measured decay transients are thus monoexponential. The fitted decay constant is the total exciton decay time $\tau$. The experiment was carried out in another setup on a different set of photonic structures compared to the one of figure \[fig:IVsThick\](c). The same excitation laser was used, the QD fluorescence was spectrally filtered in a spectrometer (500gr/mm grating) and integrated on an avalanche photodiode in a photon correlation setup, using the exit slit of the spectrometer as a spectral bandpass filter. The results of these measurements, presented in Figure \[fig:TRPLVsThick\] show also a great dispersion in decay time. One observes however that longer lifetimes are observed for smaller shell thickness (up to ). Increasing the shell thickness leads to an overall decrease in the measured exciton lifetime, hence an enhancement of the exciton decay rate in agreement with the results of the numerical simulations. For systems without an oxide shell, only a few NWQDs give a large enough signal to be properly measured. They yield a much smaller dispersion of short decay times.
Discussion and comparison to numerical simulations
==================================================
Dispersion of the results
-------------------------
For each oxide shell thickness, the large variations of the experimental results in both Figs. \[fig:IVsThick\](c) and \[fig:TRPLVsThick\] have several possible origins. First, the presence of non-radiative recombination channels can reduce the intensity at saturation and change the decay time. The non-radiative recombination rate can vary from QD to QD because of fabrication inhomogeneities, leading to a spread in the measured values.[@Stepanov2015] Second, variations in the QD aspect ratio and piezoelectric fields induced internal strain applied by both the ZnSe core and the Zn$_{0.83}$Mg$_{0.17}$Se shell lead to different overlap of electron and hole wavefunctions and hence different exciton oscillator strengths. Finally, considering the QD aspect ratio and internal strain, we expect a heavy-hole exciton type for our QDs.[@Eshelby1957; @Eshelby1959; @Zielinski2013; @Ferrand2014] Heavy-hole exciton recombination results in a mixture of circularly polarized emission, composed of two degenerate out-of-phase radial dipoles. However, strain and confinement effects might lead to valence band mixing between light hole and heavy hole levels,[@Karlsson2006; @Tonin2012; @JeanninNogues_Lightholeexciton_17] resulting in an emission composed of a mixture between axial and radial dipoles and hence to a spread in total emitted intensity, as we discuss later. Additional measurements on NWQDs grown in similar conditions and mechanically dispersed on a substrate (i.e. lying horizontally on it) revealed that one NWQD out of 6 emit light polarized along the NW axis, while others emit light polarized perpendicularly to the NW axis, evidencing the presence of both kinds of dipoles. Due to the Zn$_{0.83}$Mg$_{0.17}$Se shell and low temperature of observation, we expect that non-radiative effects play a minor role. The epitaxial shell prevents non-radiative decay channels owing to surface traps. Additional measurements as a function of temperature show that both the emission intensity and the decay time do not change significantly up to 150- (not presented here). This indicates that the non-radiative effects are not dominating at low temperature, as in the present experiment. While we cannot yet completely rule out the contribution of non-radiative effects, we think that the major effect to explain the dispersion of the results comes from variations in valence band mixing and oscillator strength due to the local environment of the QD. Finally let us stress that the shortest decay times (1-) we measure remain longer than the decay time of CdSe self-assembled QD embedded in bulk ZnSe (<)[@Bacher99]. The reduction of the dielectric screening effect is a main effect we evidence.
Collected intensity and radiative lifetime
------------------------------------------
To better understand the effect of the shell deposition on the NWQD emission, we perform numerical simulations of the photonic structure formed by the full NW + oxide shell geometry \[see Fig. \[fig:NWPresentation\](d)\]. It takes into account the presence of the ZnSe substrate, and the shell and layer deposited on the NWs and substrate. The QD is modeled as an oscillating electric dipole, either in the axial direction (along the NW axis) or in the radial direction (orthogonal to the NW axis). We perform finite-element method simulations (Comsol v4.1) to compute the total field radiated by the dipole.[@JeanninNogues_Lightholeexciton_17] For each shell thickness and dipole orientation, we evaluate the power radiated towards the objective by computing the flux of the Poynting vector over a surface limited by its numerical aperture (NA=0.6) in a region far from the NW where near field can be neglected. The results of these simulations are reported in Figure \[fig:IVsThick\](c) in black lines for an axial (dashed line) or radial (solid line) dipole. The results are normalized to the axial intensity at $t_s=$. Comparing the simulated integrated intensity in the case of a and reveals an enhancement factor less than 2-fold for an axial dipole and almost 4-fold for a radial dipole. The 4-fold enhancement observed in our measurements suggests that on average, the dominant emitting dipole in our structure is radial, in good agreement with the recombination of a heavy hole exciton.
The theoretical limits for the radiative lifetimes is extracted from the numerical simulations by integrating the total power radiated over every direction for the two dipole orientations (radial and axial) $P$. We normalize this value by the same quantity computed for a dipole in bulk $P_0$. For a purely radiative system we have $P/P_0 =
\gamma/\gamma_0=\tau_0/\tau$ [@Novotny2012], where $\tau$ and $\tau_0$ are the radiative lifetime for the nanostructure and for bulk respectively. Radiative times are presented in black lines in Figure \[fig:TRPLVsThick\], where we have chosen $\tau_0=\unit{300}{\pico\second}$ in good agreement with previously reported radiative lifetime of CdSe QD in bulk ZnSe[@FlissikowskiHenneberger_PhotonBeatsfrom_01]. The axial dipole radiates with an almost constant decay time as a function of the oxide shell thickness, while the radial dipole decay time strongly decreases with increasing oxide shell thickness $t_s$. Additionally, we compare the decay time for the radial dipole computed for the full geometry to the semi-analytical calculations for the infinite NW presented in fig. \[fig:Semianalytical\](b) with the same $\tau_0$ value. The agreement is excellent indicating that interference effects due to reflections from the substrate and from the top hemispherical termination are negligible.
Comparing the trends of the simulations, we can confirm that our emitters bear a strong radial dipole character. The measurements dispersion can be well understood by considering that the real emitters are a mixture of radial and axial dipoles radiating with a characteristic decay time comprised between the simulated lifetimes of the pure radial and axial dipole. We do not observe long decay time for NWQDs without an oxide shell in Fig. \[fig:TRPLVsThick\]. For these systems, it is very difficult to find emitters which are bright enough to be detected is because both the laser absorption and the emission rate of a radial dipole are very weak for such small NW diameters. We think that the emitters which have been selected correspond to NWQDs having a large fraction of axial dipole character as they are the brightest ones when no oxide shell is present.
Radiation pattern
-----------------
To analyze the mechanisms leading to the increase in collected intensity with increasing shell thickness, we present in Figure \[fig:FarField\] several simulated radiation patterns. They are represented as polar plots of the far-field intensity $I(\theta)$ in the top $(x,z)$ plane, $\theta$ is the angle between the direction of observation and the vertical $z$ axis. Simulations are made using respectively a radial dipole \[along $x$, Figures \[fig:FarField\](a-c)\] or an axial dipole \[along $z$, Figures \[fig:FarField\](d-f)\].
Figures \[fig:FarField\](a,d) show the effect of the NW structure alone (no oxide shell being present) on such dipoles by comparing it to the case of a free standing dipole in vacuum above the same ZnSe substrate. One can see that the presence of the NW does not affect the shape of radiation diagram, which is essentially determined by the interferences between the directly radiated field and its reflection on the substrate. Most remarkably, in the case of the radial dipole the presence of the NW dramatically reduces the emission intensity through the dielectric screening effect discussed earlier. Simulations show a radiative rate reduction by a factor $\sim 1/16\simeq n_{\rm ZnSe}/45$ in agreement with the dielectric screening value predicted by Eq. . In contrast, in the case of the axial dipole it can be seen that the presence of the NW only slightly increases the emitted intensity.
Figures \[fig:FarField\](b,e) show the computed radiation patterns of the NWQD for increasing oxide shell thickness $t_s$. In the case of the radial dipole, the shell first reduces the index contrast between the NW and the surrounding medium (cf. Eq. \[eq:screening\]), resulting in a strong reduction of the emitter lifetime and thus in an increased total emitted intensity as seen in in Fig. \[fig:IVsThick\](c) and Fig. \[fig:TRPLVsThick\]. Note that the the intensity pattern shown in the polar plot must be multiplied by the solid angle $\sin\theta \mathrm{d}\theta$ if one wants to evaluate the power radiated in the numerical aperture. This is why the intensity for an axial dipole can be larger than for a radial one, as seen in Fig. \[fig:IVsThick\](c). Second, as shown in Figure \[fig:Semianalytical\](c), the shell presence ensures preferential emission into the guided HE$_{11}$ mode for increasing shell thickness. As a consequence a near-Gaussian far-field emission pattern corresponding to the far-field emission profile of the HE$_{11}$ mode[@GregersenMoerk_Controllingemissionprofile_08] is observed for $t_s$=, contrary to the structures with a smaller oxide shell thickness where one observes the presence of two closely-spaced lobes at small emission angles ($\pm\unit{10}{\degree}$ with respect to the $z$-axis). The resulting emission into the 0.6 NA cone is maximum for $t_s=$, where the emission into the HE$_{11}$ mode is nearly maximum \[cf. Fig. \[fig:Semianalytical\](b)\]. The effect of the oxide shell thickness on the axial dipole is completely different. While the total emitted intensity does not vary much, and hence the emitter lifetime stays constant (as noted in Fig. \[fig:TRPLVsThick\]), the light emitted by the axial dipole does not couple to the HE$_{11}$ mode but is emitted exclusively into radiation modes. Thus the fraction of intensity emitted towards the collection lens increases only slightly as the oxide shell thickness increases \[cf. Fig.\[fig:FarField\](e)\]. This intensity increase for the axial dipole also presented in Fig.\[fig:IVsThick\](c) is not due to a change in the spontaneous emission rate of the emitter, but rather to a slight redirection of the emitted light.
Finally, Figures \[fig:FarField\](c,f )compare the actual hemispherical geometry of the oxide shell termination to the flat end of a simple lateral shell. They show that the presence of the hemisphere is beneficial to the radiation pattern for both kinds of dipole. For the radial dipole, the hemisphere enables a near-adiabatic expansion of the HE$_{11}$ mode[@GregersenMoerk_Controllingemissionprofile_08] leading to a narrowing of the far-field emission pattern and an increased collection by the numerical aperture. The axial dipole benefits less from the hemispherical termination of the photonic structure since no light from this dipole is coupled to the HE11 mode. We also note that half of the emitted light propagates towards the growth substrate and due to the index-matching condition between the NW and the substrate, this light is predominantly lost.
In order to assess the performances of our device we compute the ratio $\eta$ between the power radiated into a 0.6 NA to the one radiated into the top air side hemisphere. This parameter is a good figure of merit for the antenna redirection effect although it cannot be directly related to the overall collection efficiency because of the power lost in the substrate. For our full photonic structure and a radial dipole one has $\eta\simeq$80% for $t_s$=. This value reduces to $\simeq$66% for a flat terminated core-shell photonic wire illustrating the importance of the adiabatic expansion of the HE$_{11}$ guided mode at the end of the wire. For the dipole in the NW without shell $\eta\simeq$55%. We have also simulated a structure inspired by state-of-the-art devices fabricated by top-down methods in Ref. [@Claudon2010]. In this case we simulate a oxide shell photonic wire where the hemispherical termination is replaced by a conical tapper of whose radius progressively decreases from 120 to in . In this case one has $\eta\simeq$94%, showing that although beneficial our hemispherical termination is not optimal.
Conclusion
==========
In summary, we have presented a bottom-up approach to fabricate a dielectric antenna around a QD inserted inside a NW. This method allows for both reproducible and very precise fabrication of the structure on a large ensemble of emitters at once. It is based on the deposition of a thick oxide shell around the NW using atomic layer deposition. Experiments show a 4-fold enhancement of the QD photoluminescence shown in Fig. \[fig:IVsThick\](c) between a and a thick shell. Semi-analytical calculations and numerical simulations of the structure reveal that the oxide shell thickness strongly acts on the radial dipole emission through two main phenomena: the reduction of the dielectric screening, which increases the spontaneous emission rate from the QD, and the redirection of light through a waveguiding effect. Simulations suggest that the collected intensity is multiplied by a factor 7 with respect to the bare NW case. The fabrication process of the photonic shell is very simple and can be applied to QDs emitting single photons up to room temperature. Although not optimal, the resulting structure is a step towards the best nanowire single photon sources operating at low temperature[@Claudon2010]. Dielectric screening could be further reduced by growing an oxide shell of higher index matching $n_{\rm ZnSe}$ like . We note also that in our system a large fraction of the emitted power is radiated in the substrate. This loss channel could be reduced by having a mirror at the bottom of the structure.[@FriedlerRobert-Philip_Efficientphotonicmirrors_08; @Reimer2012] Moreover, to fully benefit from the waveguiding approach, a better control on the intrinsic QD properties has to be reached to ensure the presence of radial dipoles, which radiate more efficiently in the experimental collection aperture.
This work was supported by the French National Research Agency under the contract ANR-10-LABX-51-01 and the Danish Research Council for Technology and Production (LOQIT Sapere Aude grant DFF \#4005-00370).
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[^1]: contributed equally to this work
[^2]: contributed equally to this work
[^3]: We note that single-photon emission is preserved if one integrates both the signal from the X and CX lines.[@Sallen2009]
|
---
author:
- Suhani Vora
- Reza Mahjourian
- Soeren Pirk
- Anelia Angelova
bibliography:
- 'ms.bib'
title: Future Semantic Segmentation Using 3D Structure
---
|
---
abstract: 'The market economy deals with many interacting agents such as buyers and sellers who are autonomous intelligent agents pursuing their own interests. One such multi-agent system (MAS) that plays an important role in auctions is the combinatorial auctioning system (CAS). We use this framework to define our concept of fairness in terms of what we call as “basic fairness” and “extended fairness.” The assumptions of quasilinear preferences and dominant strategies are taken into consideration while explaining fairness. We give an algorithm to ensure fairness in a CAS using a Generalized Vickrey Auction (GVA). We use an algorithm of Sandholm to achieve optimality. Basic and extended fairness are then analyzed according to the dominant strategy solution concept.'
author:
- |
Sumanth Sudeendra sumanth.s@iiitb.net\
International Institute of Information Technology - Bangalore,\
Bangalore, India.\
Megha Saini msaini1@iit.edu\
Illinois Institute of Technology,\
Chicago, USA.\
Shrisha Rao srao@iiitb.ac.in\
International Institute of Information Technology - Bangalore,\
Bangalore, India.\
title: Fairness in Combinatorial Auctions
---
**Keywords:** fairness, optimality, multi-agent systems, combinatorial auctions, mechanism design
Introduction {#intro}
============
The term “auction” refers to a mechanism of allocating single or multiple resources to one or more agents (or bidders) [@23]. In recent years, computer scientists, rather than just economists, are interested in auctions. The increase in computing power and improved algorithms have paved the way for combinatorial auctions. Here multiple items are for sale by the auctioneer and bidders can bid for a bundle of items (also called packages). In a multi-agent system (MAS), we consider these bidders and the auctioneer as autonomous agents who act in a self-interested manner in their dealings with one another. Similarly, even in MAS dealing with resource allocation other than by auction, there are self-interested autonomous agents . We study a framework where optimality is a desirable property but fairness is a required property. An excellent example of such a framework is a combinatorial auctioning system (CAS) where the two most important issues pertaining to resource allocation are optimality and fairness. A CAS is a kind of MAS whereby the bidders can express preferences over combinations of items [@15; @13].
We assume in this paper that an agent’s valuation of an item does not change based on other agents’ private information (i.e., some evidence which affects the valuation of an agent), that utilities are quasilinear (i.e., utility is linear in terms of money), and that there are no externalities (i.e., an agent that does not win an item neither cares which other agent wins it, nor worries about how much other agents pay for it) [@23]. This is realistic, as seen for example in relation to the Nigerian Communications Commission auction described below, where, “The decision to charge bidders what they bid was accepted by bidders and observers as fair and transparent despite the difference in some of the payments for identical licenses.” [@29 p. 30] In such scenarios, each agent holds different preferences over the various possible allocations and hence concepts like individual rationality, fairness, optimality, efficiency, etc., are important .
We introduce the concept of fairness in the auction mechanism. Although the notion of fairness is of course well known in general, it does not seem to have been clearly defined with respect to auction processes in particular. We propose two types of fairness, namely *basic fairness* and *extended fairness*. We explain basic fairness using the concept of equitable distribution along with the respective preferences. Extended fairness is explained such that envy-freeness prevails in the allocation and the entire resource is allocated to the winning bidder. We introduce a *fairness table* consisting of fair values as perceived by bidders and auctioneer; this is sealed at the start of the bidding process. We give emphasis to fairness, unlike the classical approach where revenue maximization is the only goal required in auctions. To achieve fairness, the proposed algorithm explains a novel payment scheme which is applied at the end of the bidding process, where we determine the final amount payable to the auctioneer by the winning bidder. We ensure that this process is considered to be fair by both bidders and the auctioneer by means of extended fairness. We handle the special case of a tie in the bidding process using equitable distribution, and ensure that basic fairness is achieved. The mathematical formulations of fairness concepts in combinatorial auctions are explained, and a detailed analysis is presented to highlight some of the properties exhibited by our payment scheme.
In our mechanism, there are self-interested bidders and an auctioneer, who express their perceptions of the fair value of the resources through a data structure called the fairness table. Here an auctioneer acts as a facilitator to ensure that an item achieves its fair value. We consider optimality as the desired property and fairness as the required property. We illustrate this using a combinatorial auction framework, in which multiple items are simultaneously up for sale and a bidder can bid for any bundle of items. The optimal allocation of resources is discussed using an algorithm of Sandholm where we obtain the winning bidders. The incentives to the winning bidders are provided through the Generalized Vickrey Auction (GVA) and Algorithm \[Algorithm1\]. We apply the fairness concept using the fairness table and Algorithm \[Algorithm1\].
The auctioning of the electromagnetic spectrum is one of the well-known applications of combinatorial auctions. The first-ever combinatorial auctioning of the radio spectrum was held in Nigeria in 2002 [@29]. A single-round sealed-bid combinatorial auction (not the Simultaneous Multiple Round Auctions (SMRA) used in other countries) was conducted for regional fixed wireless access (FWA) licenses; the decision for this format was made as SMRA was impractical in Nigeria due to its insufficient communication infrastructure, and as the NCC did not wish to have a lengthy auction. Some 67 out of the 80 licenses available were allocated, with successful bids amounting to 3.78 billion naira (38 million US dollars). Here, the complimentarity and substitutability of licenses were the important factors for choosing a combinatorial auction. The cost of the allocation process was an important factor, and the Nigerian Communications Commission (NCC) did not want to discourage smaller bidders, with its primary goal being efficiency and transparency.
We find in the case of the combinatorial auction conducted by NCC the following problem: “The final choice of auction design rested heavily on information revealed about the regional structure of demand from initial applications. It was therefore critical that the application process created incentives for bidders to reveal such information.” [@29 p. 24] This problem is taken care of in our approach as we introduce the fairness table which is to be populated at the beginning of the auctioning process. This table allows us to see the regional structure of demands through the fair values assigned to the resources. We provide higher rewards for bidders who truthfully give their fair values, by Theorems \[Thm1\] and \[Thm2\].
It was also observed by the NCC that bidders defaulted on their winning bids in a significant number of cases, though not enough to undermine the overall process [@29]. We can decrease the number of bidders defaulting provided we satisfy Theorem \[Thm2\]. This makes it less rewarding for bidders to bid way beyond their capacity, which in turn decreses the possibility of winning bidders defaulting.
Since the importance here is on transparent and fair allocation, we can apply our method to ensure fairness in combinatorial auctions. We start with introducing some of the related work in Section \[related\]. Next, we explain different notions of fairness with formal definitions in Section \[definitions\]. This is followed by our study of CAS in Section \[fairCAS\]; Algorithm \[Algorithm1\] in Section \[algo\] is used to extend the payment scheme to achieve fairness in CAS with an example. Section \[analysis\] gives a detailed analysis of fairness using mechanism design under quasilinear settings [@22]. We conclude with Section \[conclusion\], which offers some conclusions about our efforts, and some suggestions for further work along these lines.
Related Work {#related}
============
In this section, we review different definitions of fairness as they have been proposed in the multi-agent literature. The problem of fair allocation is being resolved in various MAS by using different procedures, depending upon the technique of allocation of goods and the nature of goods. Its welfare implications in different systems were explored by Rabin [@16]. Brams and Taylor give an analysis of procedures for allocating divisible and indivisible items and for resolving disputes among the self-interested agents [@3]. One of the procedures described by them is the “divide and choose” method of allocation of divisible goods among two agents to ensure the fair allocation of goods which also exhibits the property of “envy-freeness,” a property first introduced by Foley [@10]. Lucas’ method of markers, and Knaster’s method of sealed bids are described for MAS comprising more than two players and for the division of indivisible items. The adjusted-winner (AW) procedure is also defined by Brams [@2] for envy-freeness and equitability in two-agent systems. Various other procedures like moving-knife procedures for cake cutting are defined for the MAS comprising three or more agents [@2; @1].
The auction mechanism proposed by Biggart [@24] provides an economic sociology perspective. There, fairness can mean different things for bidders and auctioneer. The auctioneer may consider a process fair which in fact only gives him the maximum revenue, whereas the bidders may consider a process fair which only gives the auctioneer the least return on all items. The most important consideration overall is to sustain the community’s faith in the fairness of the process. This does not mean that buyers and sellers cannot press their advantage, but they are allowed to do so only insofar as the community as a whole considers their actions appropriate and acceptable.
A concept of verifiable fairness in Internet auctions has been proposed by Liao and Hwang [@25]. This was to promote trust in Internet auctions. The scheme proposed provides evidence regarding policies implemented so that the confidence of bidders increases and they consider it to be fair. Most of these auctions see transparency in the auctioning process and rules as the basis for ensuring fairness in the system, but clarity regarding fairness still remains wanting.
The Nash bargaining concept is used by many economists. In Nash bargaining, there is no particular winner against a bargain. If the amount requested is within the total amount available at the owner then they get their share, but if the demand is more then they get nothing. In our case, this is not the case as we have a winner in all circumstances even if the auctioneer is facing a loss. Also the extended fairness concept is not present in Nash bargaining to acquire a desired product whereas in our case bidders pay a price to achieve extended fairness.
The game-theoretic concept of Shapley value [@27] describes the fairest allocation of collectively-gained profits between several collaborative agents. This is one approach used in coalitional games. Though this deals with fair allocation, it is restricted to a mechanism where the actors contribute in a coalition. The profits obtained are allocated in a fair manner. The Shapley value is different from our approach, as we do not take into account prior understanding or coalition among the bidding agents in our discussion.
Fairness as a collective measure has been considered by Moulin [@26], who proposes aggregate or collective welfare which is measured in terms of an objective standard or index that assumes equivalence between this measure and a particular mix of economic and non-economic goods which gives happiness to a varying set of individual utility functions. This tries to capture social welfare and commonwealth to be incorporated into every individuals’ happiness equations. Though debatable, it provides an excellent introduction to the concept of fairness.
A Distributed Combinatorial Auctioning System (DCAS) consisting of auctioneers and bidders who communicate by message passing has been proposed [@30]. Their work uses a fair division algorithm that is based on DCAS concept and model. It also discusses how basic and extended fairness implementations may be achieved in distributed resource allocation.
The fair package assignment model proposed by Lahaie and Parkes [@36] is defined on items having pure complements or super additive valuations. This model does not address combinatorial package assignments which involve both complements and substitutes in general. Their model provides fairness to a “core” which contains a set of all distributions which are considered competitive—no fairness is posited for other distributions. Hence the bidders whose distributions lie outside the core do not get the benefits of fair assessment. In the case of multiple-round combinatorial auctions, for example, bidders whose bids are not in the core during earlier rounds are not in contention in later ones. This scheme seems unfair in a fundamental way, as it effectively discriminates against bidders who cannot make it into the core. In our model only truthfulness in bidding is considered, and no bidders are distinguished based on whether their bids lie inside or outside a putative core.
However, the term *fairness* is defined differently in various MAS with regard to the resource allocation. In some MAS, it can be defined as equitable distribution of resources such that each recipient believes that he receives his fair share. Thus, no agent wants somebody else’s share more than its own share; such division is therefore also known as envy-free division of resources [@2]. Thus fair allocation is achieved if it is efficient, envy-free and equitable [@3].
Definitions of Fairness {#definitions}
=======================
Our additional notions of fairness in various MAS are basic fairness and extended fairness. This section defines the various notions about fairness in combinatorial auctions in a MAS.
Terminology {#term}
-----------
Let our CAS be a MAS which is defined by the following entities:
1. The total number of resources is represented by $m$ and the total number of bidders by $n$.
2. The set $R= \{r_0, r_1, r_2, \ldots, r_{m-1}\}$ is a set of $m$ resources $r_i$, and $2^R$ denotes the power set of $R$.
3. The set $B= \{b_0, b_1, b_2, \ldots, b_{n-1}\}$ is a set of $n$ bidders $b_j$.
4. $a$ is the auctioneer who initially owns all the resources.
5. A package $S$ is some subset of the set of resources, i.e., $S
\subseteq 2^{R}$.
6. $\mathbb{R}$ is the set of real numbers.
For instance, consider a CAS that comprises three bidders $b_0$, $b_1$, $b_2$, an auctioneer denoted as $a$, and three resources $r_0$, $r_1$ ,$r_2$. Each bidder is privileged to bid upon any combination of these resources. We denote the combinations or subsets of these resources as {$r_0$ }, {$r_1$}, {$r_2$ }, $\{r_0 , r_1\}$, $\{r_0 , r_2\}$, $\{r_1 , r_2\}$, $\{r_0 , r_1 , r_2\}$.
\[ex3\]
A package for a bidder winning the subsets {$r_0$} and {$r_1$} is defined as $\{\{r_0\}, \{r_1\}\}$.
We also consider the concept of weight while assigning the fair value. Here, weight is not the physical weight but is used as a multiplicative factor for describing the desirability of the package by the bidder. If a higher weight is assigned to a package, then it will result in a higher fair value. This expresses the well-known fact that a bidder is likely to assign a higher fair value to a resource that is desired or needed than to one that is not, even when the two resources have the same intrinsic value (e.g., a starving man is likely to assign a far higher value to a meal than to any other commodity of equivalent intrinsic worth).
\[def3\]
Let us define some important terms used in our later discussion, as follows:
1. The *initial value* of an item is defined as $\Omega: B \times R
\rightarrow \mathbb{R}$, where $\Omega(b_i,r_j)$ is the initial amount attached by bidder $b_i \in B$ to a resource $r_j \in R$.
2. The *weight* a package is defined as $\Theta: B \times 2^{R}
\rightarrow \mathbb{R}$, where $\Theta(b_i,S)$ is the weight for bidder $b_i \in B$ of package $S$.
3. The *fair value* for a resource is defined as $\Pi: B
\times R \rightarrow \mathbb{R}$, where $\Pi(b_i,r_j) =
\Theta(b_i,{r_j}) \times \Omega(b_i,r_j)$ for a bidder $b_i \in B$ on resource $r_j \in R$.
The fair value for a package is defined as $\Pi: B \times 2^{R}
\rightarrow \mathbb{R}$, where $\Pi(b_i,S)$ is the value obtained as $\sum \Pi(b_i,r_j), \forall r_j \in S $.
4. The *bid value* of a package is defined as $\Upsilon: B
\times 2^{R} \rightarrow \mathbb{R}$, where $\Upsilon(b_i,S)$ is the amount that the bidder $b_i \in B$ is willing to give in exchange for the package $S$.
5. The *utility value* of a package is defined as $\Gamma: B
\times 2^{R} \rightarrow \mathbb{R}$, where $\Gamma(b_i,S) =
\Upsilon(b_i,S) - \Pi(b_i,S)$.
6. The *package cost* is defined as $\Psi: B \times 2^{R}
\rightarrow \mathbb{R}$, where $\Psi(b_i,S)$ gives the final winning amount for bidder $b_i$ on package $S$ after the bidding has ended.
Assume that the auctioneer and each bidder all have fair values for each of the individual resources (say, in dollars) as shown in Table \[tab:fairtable\]. Every bidding process will have a base value initially assigned to an item from where the bidding proceeds. The fair values by a bidder and an auctioneer for each resource represent their measures of its actual value, and depend on their weights and their initial values (Definition \[def3\]). Thus, a bidder is willing to consider a resource at his fair value. Similarly, the auctioneer is willing to sell a resource at *his* fair value. However, bid value may be higher or lower than fair value and hence result in higher or lower utility values (Definition \[def3\]) depending on the need of the resource. Fair value for a combination of resources in the fairness table can be calculated as the sum of the fair value for each of the resources in that combination (fair values are considered additive where as the bid values are combinatorial in nature and not additive).
[c c c c]{}\
(r)[1-4]{} & Resource $r_0$ & Resource $r_1$ & Resource $r_2$\
Bidder $b_0$ &5 &8 &8\
Bidder $b_1$ &10 &2 &8\
Bidder $b_2$ &10 &5 &10\
Auctioneer $a$ &8 &10 &15\
From Table \[tab:fairtable\], we can see that bidder $b_0$ values resource $r_0$ at \$5, $r_1$ at \$8 and $r_2$ at \$8. This means that bidder $b_0$ is willing to pay \$5 for $r_0$, \$8 for $r_1$, and \$8 also for $r_2$; $b_0$ believes that no loss is incurred by the auctioneer in this trade. The fair value for the subset $\{r_0 , r_2
\}$ for the bidder $b_0$ is calculated as the sum of his the fair values for $r_0$ and $r_2$, i.e., \$5 + \$8 = \$13. Similarly, the fair value for a package is the sum of the fair values of the comprising sets (Definition \[def3\]), i.e., for a package {$r_0$}, $\{r_1 , r_2\}$, the fair value is the sum of the fair values of {$r_0$} and $\{r_1 , r_2\}$.
A bidder participates in the bidding process by quoting his bid for the packages. Let the bids raised by the bidders for the individual resource and different combination of resources be as given in Table $2$. It can be seen that the bids raised by each of the bidder for different sets of resources may or may not be equal to the fair value of the respective set of resources. This is because the combinations may be complimentary or substitutes.
A bidder is considered to make bid zero for any sets of resources he does not wish to procure.
[c c c c c c c c]{}\
(r)[1-8]{} & $r_0$ &$r_1$ &$r_2$ &{$r_0$ , $r_1$} &{$r_0$ , $r_2$} &{$r_1$ , $r_2$} &{$r_0$ , $r_1$ , $r_2$}\
Bidder $b_0$ &0 &10 &5 &10 &20 &15 &50\
Bidder $b_1$ &10 &5 &10 &30 &0 &0 &50\
Bidder $b_2$ &10 &0 &15 &20 &30 &15 &30\
With this terminology we proceed to explain fairness in subsequent sections.
Basic Fairness {#Basic}
--------------
In many MAS, there occurs a need of allocating the resources in an equitable manner, i.e., each agent gets an equitable share of the resources. Such allocations leave the agents with a feeling that they have received a fair share [@3]. For example, if we consider a method that would leave two agents feeling as if they had received $60$% of the good then we would call it equitable. If one felt to be favored and had received $80$% while the other agent believed to have received $60$% then it would not be equitable [@3]. This is quite difficult to access and tends to quite subjective in many cases. We give a mechanism where this applies only in case of a tie, hence we consider a divisible resource which does not lose its value upon division and divide it equitably among bidders in proportion to their assigned weights. Each agent has a set of allocations he deems fair. An allocation is then is said to achieve basic fairness in resource allocation if all agents deem it fair.
Each bidder $b_i$ wants to maximize his chances of procuring the resource and individual utility given by $\Gamma(b_i,S)$ represents the satisfaction of obtaining the resource. The most simple approach is that the satisfaction of a bidder does not depend on other bidders’ satisfactions. The representation below considers that a package $S$ is divisible and can be divided equitably among $n$ bidders in proportion to their utility values.
The resource can be divided equitably in the ratio: $\Gamma(b_i,S) / \sum_{i=1}^{n}\{\Gamma(b_i,S)\}$, where weights are set freely by agents.
\[def1\]
If each bidder $b_i$ has a utility for a package $S$ given by $\Gamma(b_i,S)$, and the package $S$ can be divided equitably among $n$ bidders in the ratio $\Gamma(b_i,S) / \sum_{i=1}^{n}\{\Gamma(b_i,S)\}$, then basic fairness is said to be achieved.
\[ex1\]
Consider there to be three bidders for the divisible package $S$. The bidders’ bid values, fair values and utility values are shown in Table \[tab:basicfair\].
[c c c c]{}\
(r)[1-4]{} & Bidder $b_0$ & Bidder $b_1$ & Bidder $b_2$\
Bid Value &$24$ &$16$ &$20$\
Fair Value &$18$ &$12$ &$16$\
Utility Value &$6$ &$4$ &$4$\
The calculations of ratios are done as shown below.
For bidder $b_0$, $6$/$14$ = $0.43$.
For bidder $b_1$, $4$/$14$ = $0.285$.
For bidder $b_2$, $4$/$14$ = $0.285$.
If the winning amount is \$$100$ then it is divided in the ratio $0.43 : 0.285 : 0.285$ to achieve basic fairness, i.e., bidder $b_0$ has to pay \$$43$, bidder $b_1$ has to pay \$$28.50$, bidder $b_2$ has to pay \$$28.50$.
This method of equitable allocation ensures that all agents deem the allocation to be fair. Therefore, we say that every agent believes that the set of resources is divided fairly among all the agents. This concept of fairness is termed as basic fairness.
This kind of fairness is required in applications wherein fairness is the key issue, rather than the individual satisfactions of the self-interested agents. In such applications, it becomes necessary to divide a package in an equitable fashion so that every agent believes that it is receiving its fair share from the set of resources. Hence, we see that every agent enjoys material equality and this ensures basic fairness among them.
Extended Fairness {#extended}
-----------------
In order to ensure egalitarian social welfare , basic fairness is alone not sufficient. We also need to address envy-freeness [@2]. Envy-free allocations result in each agent being at least as happy with its share of the goods as it would be with any of the other agents shares despite the difference in some payments for identical goods [@3; @29]. Here, we need to ensure that the allocation is perceived by all agents to be a fair allocation.
In a MAS, every agent assigns a fair value to each resource that determines its estimate of the value of the resource in quantitative terms. The fair value attached to each resource can be expressed in monetary terms in most MAS. Here, the agent believes that the allocated resource is fair if he receives the entire allocation and the value is according to his fair estimate.
However, it is important to mention that the fair value attached to each resource by an agent does not necessarily reflect the bid value of the resource. An agent may hold a higher or lower bid value for a resource irrespective of the fair value attached to the resource. Rather, the fair value attached to a resource is an estimate of the actual value of the resource in the system as perceived by an agent in quantitative terms. It means that an agent is always willing to trade a resource at its fair value.
Let there be $k$ bidders who bid for a package $S$. Let each bidder $b_i$ and auctioneer $a$ (who is here only as a facilitator for achieving the items’ fair value) give their fair values in the fairness table (as in the example in Table \[tab:fairtable\]), which is open for all to see at the *end* of the bidding process.
\[def4\]
Let us define some of the terms used in our discussion.
1. $C$ is defined as the winning amount after the bidding process for a package $S$.
2. $\xi: {a} \times 2^{R} \rightarrow \mathbb{R}$ defines the fair value of the auctioneer $a$ for a package $S$ denoted by $\xi(S)$.
3. $\Pi(b_i,S)$ is defined as the fair value of the bidder $b_i$ for a package $S$.
4. The *profit* denoted by $\Phi$ is defined as the net amount $C$ above the fair value of the auctioneer $\xi(S)$ given by the bidder $b_i$ for the package $S$ and is calculated as difference $C-\xi(S)$.
5. The function $distribute$ is defined as the amount $x: B \times 2^{R}
\rightarrow \mathbb{R}$ to be given back to the losing bidders $b_i$ who bid for the winning package $S$.
6. The value $reward$ is defined as $reward: B \times 2^{R}
\rightarrow \mathbb{R}$, where $reward = \Phi - x$.
Now let us define extended fairness in resource allocation.
\[def2\] An allocation is said to satisfy extended fairness, if when a winning bidder $b_i$ is allocated a package $S$: (i) if $\Upsilon(b_i,S) > \xi(S)$, then a losing bidder $b_j$ is rewarded $\Phi \times \left(\frac{\Pi(b_j,S)-\xi(S)}{\xi(S)}\right)$; and (ii) if $\Upsilon(b_i,S) \leq \xi(S)$, then no one gets a reward.
Consider the following scenarios:
1. The auctioneer makes a profit more than his fair value assigned initially for that package $S$. He distributes the profit among the losing bidders in proportion to their fair values for that package $S$ as follows:
Let $C$ be the winning bid which is greater than the fair value of the auctioneer, i.e., $\xi(S)$. Therefore, $\Phi = C-\xi(S)$ by Definition \[def4\]. The profit to be distributed for each losing bidder $i$ is calculated by:
$$distribute \left(\Phi \times \left(\frac{\Pi(b_i,S)-\xi(S)}{\xi(S)}\right)\right)$$
Now, the incentive for the winning bidder is $reward = \Phi -
distribute\left(\Phi \times \left(\frac{\Pi(b_i,S)-\xi(S)}{\xi(S)}\right)\right)$. Thus, the auctioneer has obtained his fair value and hence considers this allocation as fair. All the bidders get amounts according to their fair values, which makes them envy free.
2. The auctioneer gets a winning bid $C$ which is exactly the same as the fair value $\xi(S)$ associated with the package $S$. Now the profit is zero. Therefore, the auctioneer has obtained his fair value and hence considers this allocation as fair. All the bidders, though did not get any reward consider this allocation as envy-free as auctioneer too did not make any profits more than his own fair value.
3. The auctioneer gets a winning bid $C$ less than the fair value $\xi(S)$ attached by him for the package $S$. In this case, we try to minimize his loss as follows:
If the fair value given by bidder $\Pi(b_i,S) \geq \xi(S)$, then bidder $b_i$ pays $\xi(S)$. Thus, the auctioneer has no loss as he gets his fair value and the bidder too is envy free since he considers that paying his fair value as fair. The other bidders are still envy free since the amount paid by the winning bidder is more than he actually won in the bidding process.
If the fair value given by bidder $\Pi(b_i,S) < \xi(S)$ and $\Pi(b_i,S) \leq C$, then the payment does not change and he pays $C$, else he pays $\Pi(b_i,S)$. Only in this case auctioneer fails to get his fair value and the bidder does not get the distributed profit amount. The allocation is still envy-free for all the bidders but not for the auctioneer. This can be avoided if both bidder and auctioneer remain truthful in their fair values.
\[ex2\]
An example of such a system can be explained with a scenario of auctioning of a painting. The contending bidders express their fair values through their sealed bids that is submitted to the auctioneer, i.e., each contending bidder believes that his quotation fulfills the value expected by the auctioneer and he is a competitive contender for the painting. We assume here that all bids are truthful. An unbiased auctioneer selects the bid which is the maximum for revenue maximization and the painting is allotted to him. Here the auctioneer would distribute the profits among losing bidders when he gets back his fair value. This takes care of envy-freeness. Hence, the allocation is perceived to be fair by the winning bidder and by all other bidders as it is allocated to the most deserving among all the bidders. The auctioneer also perceives this to be fair since he will obtain the fair value for the resource. Thus all participants perceive the allocation to an agent to be fair irrespective of the fair values attached by them. Therefore, extended fairness is said to be achieved.
To make these notions of fairness mathematically precise, we need a framework where fairness is a required property in resource allocation. However, we also see that resource allocation deals with another key issue of optimality in various MAS. The best example of resource allocation framework where both optimality and fairness are the key issues is Combinatorial Auctioning Systems (CAS).
Fairness in Combinatorial Auctioning Systems (CAS) {#fairCAS}
==================================================
Combinatorial Auctioning Systems are a kind of MAS which comprise an auctioneer and a number of self-interested bidders. The auctioneer aims at allocating the available resources among the bidders who, in turn, bid for sets of resources to procure them in order to satisfy their needs. The bidders aim at procuring the resources at minimum value during the bidding process, while the auctioneer aims at maximizing the revenue generated by the allocation of these resources. Thus, CAS refers to a scenario where the bidders bid for the set of resources and the auctioneer allocates the same to the highest-bidding agent in order to maximize the revenue. Hence, we see that optimality is one of the key issues in CAS.
An algorithm of Sandholm is used here to attain optimal allocation of resources. Sandholm proposes various methods for winner determination in combinatorial auctions [@18]. The search methodology can be used to obtain optimal allocation of resources. We can represent the Table \[tab:bidtable\] as a Bid tree using an algorithm of Sandholm [@18]. We can also carry out some preprocessing steps to make the steps faster without compromising the optimality [@13; @18]. Thus we can determine the winning bidders.
However, besides optimality, another key issue desired by some auctioning systems is fairness. To incorporate this significant property in this resource allocation procedure, we propose an algorithm which uses the concept of extended fairness for each agent with basic fairness in case of a tie and determines the final payment made by the winning bidders.
The algorithm that we describe is based upon a CAS that uses an algorithm of Sandholm for achieving optimality, and an incentive compatible mechanism called Generalized Vickrey Auction (GVA) along with Algorithm \[Algorithm1\] as the pricing mechanism that determines the payments to be given by the winning bidders.
The Generalized Vickrey Auction (GVA) has a payoff structure that is designed in a manner such that each winning agent gets a discount on its actual bid. This discount is called a Vickrey Discount, and is defined by [@13] as the extent by which the total revenue to the seller is increased due to the presence of that winning bidder, i.e., the marginal contribution of the winning bidder to the total revenue. The GVA framework requires significant transfer payments from bidders to auctioneer hence a redistribution mechanism is required to reduce the cost of implementation [@37; @38]. Hence, after we obtain winning bidders from the algorithm of Sandholm, the GVA mechanism can be applied to get Package Cost (Definition \[def3\]) and Algorithm \[Algorithm1\] can be used for redistribution of payments back to the bidders to achieve fair allocation. We give mathematical formulations to show that both kinds of fairness can be achieved in CAS.
Notion of Fairness in Combinatorial Auctions
--------------------------------------------
1. Each bidder and the auctioneer define its fair values in the fairness table (Table \[tab:fairtable\]) before the start of the bidding process. It is a sealed matrix and is unsealed at the end of bidding process.
2. An allocation tree is constructed at the end of the bidding process to determine the optimum allocation and the winning bidders [@18]. Information about all the bidders in a tie is not discarded using some pre-defined criteria.
3. Calculate the package cost $\Psi(b_i,S_j)$ (Definition \[def3\]) denoted by $P_{i,j}$ which is the final winning bid amount for bidder $b_i$ on package $S_j$ is obtained after applying GVA scheme.
4. The fair value of the package won by each bidder is calculated, and the value is denoted as $Q_{i,j}$ for the bidder $b_i$ who wins the package $S_j$.
5. The fair value of each package is calculated using the fairness table of the auctioneer and is denoted as $Q_{a,j}$ for a package $S_j$.
6. The values of $Q_{a,j}$ and $P_{i,j}$ are compared to determine the final payment by the bidder which is considered fair.
Now, we propose an algorithm which satisfies extended fairness in all cases, except in case of a tie.
Notations Used in Algorithm \[Algorithm1\] {#notations}
------------------------------------------
- $b_i$ is an arbitrary bidder $i$ who belongs to the set of bidders $B$.
- $S_j$ is the winning package with complimentaries and substitutes included, which is a subset of set $R$.
- In general, the Fairness Table for bidder $b_i$ and Auctioneer $a$ is defined as shown in Table \[tab:generalfairtable\].
[c c c c c c]{}\
(r)[1-6]{} & Resource $r_0$ & Resource $r_1$ & Resource $r_2$ & …& Resource $r_{m-1}$\
Bidder $b_0$ & $\Pi(b_0,r_0)$ & $\Pi(b_0,r_1)$ & $\Pi(b_0,r_2)$ & …& $\Pi(b_0,r_{m-1})$\
Bidder $b_1$ & $\Pi(b_1,r_0)$ & $\Pi(b_1,r_1)$ & $\Pi(b_1,r_2)$ & …& $\Pi(b_1,r_{m-1})$\
. & . & . & . & …& .\
. & . & . & . & …& .\
. & . & . & . & …& .\
Bidder $b_{n-1}$ & $\Pi(b_{n-1},r_0)$ & $\Pi(b_{n-1},r_1)$ & $\Pi(b_{n-1},r_2)$ & …& $\Pi(b_{n-1},r_{m-1})$\
Auctioneer $a$ & $\xi(r_0)$ & $\xi(r_1)$ & $\xi(r_2)$ & …& $\xi(r_{m-1})$\
- The fair value function by a bidder $b_i$ for a resource $r_j$ is given by $\Pi (b_i, r_j) = d$, where d $\in \mathbb{N}$.
- $Q_{i,j}$ is the fair value of resource $r_j$ by bidder $b_i$ where $r_j\in$ R and $b_i \in$ B.
- $Q_{a,j}$ is the fair value of resource $r_j$ by auctioneer $a$ where $r_j\in$ R. Here we consider only a single auctioneer.
- The package Cost $P_{i,j}$ (Definition \[def3\]) for bidder $b_i$ obtained from the GVA scheme is represented as $\Psi
(b_i,S_j)$.(The package cost is a function of bid values on the bundles of resources)
- The pay function by a bidder $b_i$ is represented as $pay(c)$ is the final payment to be made to the auctioneer by the bidder $b_i$ where $c$ is the bid amount.
- $\Phi$ (Definition \[def4\]) is the net amount above the fair value distributed by the auctioneer $a$ to the bidders for a package $S_j$.
- $distribute$ is a function which calculates the amount to be given back to the bidders who bid for the winning package $S_j$ (Definition \[def4\]).
- $loss$ is the net amount below the fair value given by the auctioneer to the package $S_j$.
Flow of Algorithm \[Algorithm1\] {#flow}
--------------------------------
In Algorithm \[Algorithm1\], we calculate the package cost and the fair values of bidder and auctioneer given in the lines 1–3. These are calculated in the beginning and are represented as $P_{i,j}$, $Q_{i,j}$ and $Q_{a,j}$ respectively.
In lines 4–9, we have the first [**if**]{} condition where the package cost $P_{i,j}$ is greater than the fair value assigned by the auctioneer for the package $Q_{a,j}$. If this evaluates to <span style="font-variant:small-caps;">TRUE</span>, then the bidder pays the amount but the net profit calculated is distributed among all the bidders who bid for that package proportional to their bids.\
\
In lines 10–12, we have the second [**if**]{} condition where the package cost $P_{i,j}$ is equal to the fair value assigned by the auctioneer for the package $Q_{a,j}$. If this evaluates to <span style="font-variant:small-caps;">TRUE</span>, since there is no profit the bidder still pays and there is no amount distributed to the winning package bidders.\
\
In lines 13–32, we have the third ‘if’ condition where the package cost $P_{i,j}$ is less than the fair value assigned by the auctioneer for the package $Q_{a,j}$. If this evaluates to <span style="font-variant:small-caps;">TRUE</span>, there is a loss for the auctioneer so we try to minimize the loss by checking the additional cases as follows.\
\
First at line $15$, if the fair value of bidder $Q_{i,j}$ is greater than fair value of auctioneer $Q_{a,j}$ evaluates to <span style="font-variant:small-caps;">TRUE</span>, then the bidder will have to pay only $Q_{a,j}$. This prevents loss for auctioneer and also the bidder deems it as fair.\
\
Secondly at line $19$, if the fair value of bidder $Q_{i,j}$ is equal to the fair value of auctioneer $Q_{a,j}$ evaluates to <span style="font-variant:small-caps;">TRUE</span>, then the bidder will have to pay only $Q_{a,j}$ as in the previous condition. Similar to the previous condition this prevents loss for auctioneer and also the bidder deems it as fair.\
\
Finally at line $23$, if the fair value of bidder $Q_{i,j}$ is less than fair value of auctioneer $Q_{a,j}$ evaluates to <span style="font-variant:small-caps;">TRUE</span>, then we have to see the additional two conditions as follows.\
\
If the fair value of bidder $Q_{i,j}$ is less than or equal to package cost of bidder $P_{i,j}$ then the bidders’ final payment remains the same, i.e., $P_{i,j}$.\
\
If the fair value of bidder $Q_{i,j}$ is greater than the package cost of bidder $P_{i,j}$ then the bidders’ final payment is $Q_{i,j}$. These are presented in Algorithm \[Algorithm1\].
Algorithm to Incorporate Extended Fairness {#algo}
------------------------------------------
\
[$Q_{i,j} \leftarrow \Pi(b_i,S_j)$]{}\
[$Q_{a,j} \leftarrow \xi(S_j)$]{}
Handling a Case of a Tie—Incorporating Basic Fairness
-----------------------------------------------------
Unlike traditional algorithms, we do not discard the bids in the case of a tie on the basis of some pre-decided criterion. We consider these cases in our algorithm to provide basic fairness to the bidders. In case of a tie, we shall measure the utility value of the resource to each bidder in the tie. The utility value of a resource to a bidder is the quantified measure of satisfaction or happiness derived by the procurement of the resource (Definition \[def3\]).
The bidders maximize this utility value to quantify the importance and their need for the resource. Thus, the higher the utility value, the greater is the need for the package. In such a case, fairness can be achieved if the package $S$ is divided among all the bidders in a proportional manner, i.e., in accordance to the utility value attached to the package by each bidder.
\[ex5\]
Let us consider the same example to explain the concept of basic fairness in our system. From Table \[tab:bidtable\], we observe that the optimum allocation attained through allocation tree comprises the package [$r_0$, $r_1$ , $r_2$ ]{} as it generates the maximum revenue of \$$50$. However, we see that this bid is submitted by the two bidders, $b_0$ and $b_1$.
Thus, we calculate the fair value of the package $S_j$ = {$r_0$, $r_1$ , $r_2$} for the bidder $b_0$ and $b_1$, i.e., $\Pi(b_0,
S_j) = 5+8+8 = \$21$ and $\Pi(b_1, S_j) = 10+2+8 = \$20$. Thus, the utility value of the package $S$ for the bidder $b_0$ and $b_1$ is as follows:<span style="font-variant:small-caps;"></span>
For bidder $b_0$ , $\Gamma(b_0, S_j) = 50 - 21 = \$29$, and
For bidder $b_1$ , $\Gamma(b_1, S_j) = 50 - 20 = \$30$.
Hence, the package $S$ is divided among bidders $b_0$ and $b_1$, in the ratio of $29:30$. In other words, bidder $b_0$ gets $49.15$% and bidder $b_1$ gets $50.85$% of the package $S_j$.
The payment made by the bidders is also done in the similar proportional manner similar to Example \[ex1\].
The bidders $b_0$ and $b_1$ make their respective payments in the ratio of $29:30$ to make up a total of \$$50$ for the auctioneer, i.e., bidder $b_0$ pays \$$24.65$ and bidder $b_1$ pays \$$25.35$ to the auctioneer for their respective shares.
Hence, we see that extended fairness as well as basic fairness are achieved in a CAS using our approach. We take into account the fair estimates of the auctioneer and the bidders for each resource to ensure that fairness is achieved to auctioneer as well as the bidders. A detailed analysis of our mechanism is in the following section.
Analysis
========
Using the solution concept of dominant strategies and mechanism design with quasilinear preferences, we can analyze the following.
We say that the agents’ preferences are quasilinear when they satisfy the conditions given below: first we are in a setting where the mechanism can choose to charge or reward an agent an arbitrary amount. Second, and more restrictive, is that an agent’s utility of a choice cannot depend on the money that he has, i.e., his value is the same whether he is rich or poor. Finally, the agents care only about the choice selected and their own payments, i.e., they are not concerned about monetary payments made or received by other agents.
Fairness
--------
We say that extended fairness is achieved when a bidder procures a resource for an amount that is equal to his estimate of fair value of that resource. In such a case, the bidder believes that the resource was procured by it at a fair amount irrespective of other bidders estimate of fair value of that resource. This is according to the last condition of quasilinear preference. Thus, the allocation is believed to be extendedly fair as per the estimates of the winning bidder.
We also see that basic fairness is achieved in our system when there is more than one bidder who has raised equal bid for the same set of resources. In such a case, we divide the set of resources among all the bidders so as to ensure fairness to all the bidders in a tie. However, this division of resources set is done in a proportional manner. We intend to divide the resource such that the bidder holding highest utility value to it should get the biggest share. To ensure this, we calculate the utility value (i.e., $\Gamma(b_i, S_j) =
\Upsilon(b_i,S_j) - \Pi(b_i,S_j))$ of the set of resources to each bidder and divide the set in the ratio of these values among the respective bidders. Thus, we see that each bidder procures his basic share of the set of resources in accordance to the basic importance attached by the bidder to the set of resources. Due to the achievement of fairness through our payment scheme, the bidders are expected to show willingness to participate in the auctions.
Higher Rewards {#rewards}
--------------
Here we show that a bidder is encouraged to bid higher as he gets rewards proportional to his bids which are fair.
\[Thm1\]
Given a CAS, a bidder has an incentive to bid higher using the extended fairness algorithm \[Algorithm1\] as he gets higher rewards which are fair provided:
1. the bid value $P_{i,j}$ is always greater than or equal to fair value $Q_{a,j}$ on package $S_j$.
2. $Q_{i,j}$ of winning bidder is always greater than or equal to $Q_{i,j}$ of the losing bidder.
Assume any two bidders $b_x, b_y \in B$ who bid for package $S_j$. Assume values: $P_{x,j},Q_{x,j}$ for bidder $b_x$. $P_{y,j},Q_{y,j}$ for bidder $b_y$. $Q_{a,j}$ for auctioneer $a$.
The first condition is $P_{x,j} > Q_{a,j}$ and $Q_{x,j} >
Q_{y,j}$ hence we have a profit $\Phi_x$ which is distributed in the ratio $\Phi_x\times \left(\frac{Q_{x,j} -
Q_{a,j}}{Q_{a,j}}\right)$ to $b_x$ and $\Phi_x\times\left(\frac{Q_{y,j} - Q_{a,j}}{Q_{a,j}}\right)$ to $b_y$. This gives proportional as well as fair incentives to $b_x$ and $b_y$. This also gives a higher reward to $b_x$ since $Q_{x,j}
> Q_{y,j}$.
For notational convinience, let $k$ represent $\frac{Q_{x,j} - Q_{a,j}}{Q_{a,j}}$ and $l$ represent $\frac{Q_{y,j} -
Q_{a,j}}{Q_{a,j}}$.
The second condition is $P_{x,j} > Q{a,j}$ and $Q_{x,j} >
Q_{y,j}$ hence we have a profit $\Phi_y$ which is distributed as $\Phi_y \times k$ and $\Phi_y \times l$ where $k, l$ are constant for $b_x$ and $b_y$. When $\Phi_y > \Phi_x$ we see a greater amount of reward is given to higher bidding and hence bidders are encouraged to bid more.
\[Thm2\]
Given a CAS, a bidder gets higher rewards using the extended fairness algorithm \[Algorithm1\] if his fair value $Q_{i,j}$ satisfies the condition $Q_{a,j}> Q_{i,j} > 2\times Q_{a,j}$ for a package $S_j$.
Assume a bidder $b_i \in B$ who bids for package $S_j$ and wins it. Assume values: $P_{i,j},Q_{i,j}$ for bidder $b_i$. $Q_{a,j}$ for auctioneer $a$.
The first condition is when the winning bidder $b_i$ has given a a low fair value for a package $S_j$ intentionally, i.e., $Q_{i,j} <
Q_{a,j}$. Now, his ratio is calculated as $k=\frac{Q_{i,j} -
Q_{a,j}}{Q_{a,j}}$ which is negative. He has to distribute an extra amount of the same proportion, i.e., $distribute\left(\Phi\times\left(2 \times \frac{Q_{i,j} -
Q_{a,j}}{Q_{a,j}}\right)\right)$. Hence, $reward=\Phi -
distribute \left(\Phi\times \left(2 \times \frac{Q_{i,j} -
Q_{a,j}}{Q_{a,j}}\right)\right)$. Therefore, bidder has to pay $P_{i,j} - reward$.
The second condition is when the winning bidder $b_i$ has given a very high fair value for a package $S_j$, i.e., $Q_{i,j} > 2 \times
Q_{a,j}$. Now, his ratio is calculated as $k=\frac{Q_{i,j} -
Q_{a,j}}{Q_{a,j}}$ which is greater than 1. He has to distribute $\Phi\times\left(\frac{Q_{i,j} - Q_{a,j}}{Q_{a,j}}\right)$ which is greater than $\Phi$. Therefore, the extra amount to be distributed would be added to $P_{i,j}$ and hence ends up paying higher amount without any rewards.
The third condition is when the winning bidder $b_i$ has given a true fair value for a package $S_j$, i.e., $Q_{a,j}>Q_{i,j} > 2
\times Q_{a,j}$. Now, his ratio is calculated as $k=\frac{Q_{i,j}
- Q_{a,j}}{Q_{a,j}}$ which is a proper fraction. Hence, $reward=\Phi - distribute \left(\Phi\times \left(\frac{Q_{i,j} -
Q_{a,j}}{Q_{a,j}}\right)\right)$. Therefore, bidder $b_i$ has to pay $P_{i,j} - reward$ from definition \[def4\].
Clearly, we can see that the maximum reward is possible only in the third condition, where the fair value is neither too low nor very high. Thus, Algorithm \[Algorithm1\] provides higher rewards if fair value for a package $S_j$ satisfies the condition $Q_{a,j}>
Q_{i,j} > 2\times Q_{a,j}$.
Other Issues {#otherissues}
------------
Now let us discuss some issues considering the quasilinear mechanism.
### Truthfulness {#truth}
Consider the following definition by [@22].
\[truthdef\]
A quasilinear mechanism is *truthful* if it is direct and $\forall i$, bidder $b_i$’s equilibrium strategy is $\Upsilon(b_i,S_j) = \Pi(b_i,S_j).$
(Note that [@22] uses $v_i$ for what we denote by $\Pi(b_i,S_j)$ and $\hat{v_i}$ for what we denote by $\Upsilon(b_i,S_j)$.)
\[truththm\] In our mechanism, the bidder $b_i$’s equilibrium strategy is $\Upsilon(b_i,S_j) = \Pi(b_i,S_j)$ so it is truthful.
Assume a bidder $b_i \in B$ who bids for package $S_j$ provides his fair value $\Pi(b_i,S_j)$ in the fairness table. Let us denote the strategy choosen by $b_i$. i.e., $\Upsilon(b_i,S_j) = \Pi(b_i,S_j)$ to be $d_i$ which is a dominant strategy as per our assumption (Line 10 in Algorithm \[Algorithm1\] and Theorem \[Thm2\]).
Assume that the bidder $b_i$ would be better off declaring a fair value $\Pi(b_i,S_j)^\prime$ instead of $\Pi(b_i,S_j)$ to our mechanism. This implies that $b_i$ has chosen a different strategy $d_i^\prime$ instead of $d_i$ which is not in equilibrium, contradicting our assumption that $d_i$ is the dominant strategy for $b_i$.
This means that here the only action available to an agent is to reveal his private information. Any solution to a mechanism design problem can be converted into one in which agents always reveal their true preferences, if the new mechanism “lies for the agents” in just the way they would have chosen to lie to the original mechanism. Thus the new mechanism is dominant-strategy truthful [@22].
In our algorithm the bidder or auctioneer benefit only when they give their fair value truthfully as in cases where $P_{i,j} > Q_{a,j}$ and $P_{i,j} = Q_{a,j}$, where he gets the incentives as profits are distributed. But if the fair value is not truthful then he risks going to the case $P_{i,j} < Q_{a,j}$ and $Q_{i,j} < Q_{a,j}$ where naturally he is denied of any benefits. Thus if he lies to the mechanism to gain profits he would not succeed as he would have chosen a strategy which leads to loss.
### Efficiency
Consider the efficiency with respect to the package won by the bidder $b_i$ denoted by $S_j$. We define $S_j^\prime$ as a subset of resources which are not won by the bidder $b_i$.
Consider the following definition by [@22].
\[effdef\]
A quasilinear mechanism is *strictly Pareto efficient*, or just *efficient*, if in equilibrium it selects a choice $S_j$ such that $\sum_i \Pi(S_j) \geq \sum_i \Pi(S_j^\prime)$
(Note that [@22] uses $x$ for what we denote by $S_j$.)
\[effthm\] In our mechanism, the bidder $b_i$’s equilibrium strategy is to select choice $S_j$ such that $\sum_i \Pi(S_j) \geq \sum_i \Pi(S_j^\prime)$ so it is efficient.
Assume a bidder $b_i \in B$ has chosen a dominant strategy $d_i$ selects a choice $S_j$ such that $\sum_{b_i} \Pi(b_i,S_j) < \sum_{b_i}
\Pi(b_i,S_j^\prime)$. This implies that sum of fair values of items in selected package $S_j$, is not more efficient than the sum of items not in package $S_j$. Thus there is another strategy $d_i^\prime$ which selects a choice $S_j^\prime$ which is more efficient than $S_j$(Theorem \[Thm1\]). Hence, $d_i$ was not in equilibrium as $d_i^\prime$ is the dominant strategy. This is a contradiction to our assumption that $d_i$ was the dominant strategy.
An efficient mechanism selects the choice which maximizes the sum of the agents’ utilities, disregarding the the monetary payments they are required to pay. This can be shown in our algorithm concept where the choice is made on the agents’ fair values which helps in maximizing its profits. Thus, the efficiency is defined in terms of the true fair values and not the declared value in the bid table (Table \[tab:bidtable\]).
### Incentive Compatibility {#incentive}
The combinatorial auction can be made incentive compatible using the Generalized Vickrey Auction (GVA) and Algorithm \[Algorithm1\]. The payment using GVA can be explained by assuming that all agents follow their dominant strategies and declare their values truthfully. Each agent is made to pay his social cost; the aggregate impact that his participation has on other agents utilities [@22].
The payment mechanism described in our system is incentive compatible, i.e., they fare best when they reveal their private information truthfully in certain cases. As shown in Theorem \[Thm1\] and Theorem \[Thm2\] the bidders following dominant strategies in Algorithm \[Algorithm1\] is bound to get higher incentives.
Thus the GVA and Algorithm \[Algorithm1\] enables our mechanism to be incentive compatible.
### Optimality {#optimal}
Optimality is a significant property that is desired in a CAS. We ensure this property by the use of an algorithm of Sandholm [@18] in our system. It is used to obtain the optimum allocation of resources so as to maximize the revenue generated for the auctioneer. Thus, the output obtained is the most optimal output and there is no other allocation that generates more revenues than the current allocation.
Conclusion
==========
We have shown that fairness can be incorporated in CAS from our methodology. Extended fairness as well as basic fairness can be attained through our payment mechanism. Optimal allocation is obtained through an algorithm of Sandholm, and the other significant properties like allocative efficiency and incentive compatibility are also achieved. This is an improvement because in the existing world of multi-agent systems, there do not seem to be many studies that attempt to incorporate optimality as well as fairness. The present paper addresses this lack in a specific multi-agent system, namely, the CAS.
The Nigerian Communications Commission (NCC) faced problems in giving incentives to bidders who divulge their preferences and bidders were not keen on divulging it since it may lead to more adverse competition. Our algorithm for extended fairness takes care of this problem as bidders receive more incentives with higher bids. Since the preferences given by the bidders in the fairness table is confidential and sealed, they need not worry about their preferences being disclosed to competitors.
The framework described can also be extended in several ways: first is to de-centralize the suggested algorithm, to avoid use of a single dedicated auctioneer. Especially in distributed computing environments, it would be best for there to be a method to implement the suggested algorithm (or something close to it) without requiring an agent to act as a dedicated auctioneer [@30].
A second important extension would be to find applications for the work. Some applications that suggest themselves include distribution of land (a matter of great concern for governments and people the world over) in a fair manner. In land auctions where a tie occurs, no pre-defined or idiosyncratic method need be used to break the tie; rather, the allocation can be done fairly in the manner suggested.
A third important extension is to experiment with the grid computing framework . The applicability of fairness scheme in grid computing while allocating resources and its impact on the expected revenue would be an interesting application area.
Fairness is also an important and pressing concern in the computing sciences and information technology, particularly, in distributed computing [@12]. It is therefore also of interest to see how our method for achieving fairness could be applied in such contexts.
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|
---
abstract: 'Light curves of the contact systems visible in the direction of Baade’s Window have been analyzed using the first coefficients of the Fourier representation. The results confirm that the geometric contact between components is usually weak. Systems showing significant differences in the depths of eclipses are very rare in the volume-limited sample to 3 kpc: only 2 among 98 contact systems show the difference larger than 0.065 mag; for most systems the difference is $< 0.04$ mag. If this relative frequency of 1/50 is representative, then one among 12,500 – 15,000 Main Sequence F–K spectral-type stars is either a semi-detached or poor-thermal-contact system. Below the orbital period of 0.37 day, no systems with appreciable differences in the eclipse depths have been discovered. Since large depth differences are expected to be associated with the “broken-contact” phase of the Thermal Relaxation Oscillations, this phase must be very short for orbital periods above 0.37 day and possibly entirely absent for shorter periods. In the full sample, which is dominated by intrinsically bright, distant, long-period systems, larger eclipse-depth differences are more common with about 9% of binaries showing this effect. Sizes of these differences correlate with the sense of light-curve asymmetries (differing heights of maxima) for systems with orbital periods longer than 0.4 day suggesting an admixture of semi-detached systems with accretion hot spots on cooler components. The light-curve amplitudes in the full sample as well as in its volume-limited sub-sample are surprisingly small and strongly suggest a mass-ratio distribution steeply rising toward more dissimilar components. Since the sky-field sample is dominated by contact binaries with large amplitudes, it is suggested that a large fraction of low mass-ratio systems remains to be discovered among bright stars in the sky. For a mass-ratio distribution emphasizing low values, an approach based on the statistics of the inner contact angles of totally-eclipsing systems may offer a better means of determining this distribution than the statistics of the variability amplitudes.'
author:
- |
Slavek M. Rucinski\
e-mail: [*rucinski@astro.utoronto.ca*]{}
title: |
Eclipsing Binaries in the OGLE Variable Star Catalog. II.\
Light Curves of the W UMa-type Systems in Baade’s Window
---
INTRODUCTION
============
The first paper of this series (Rucinski 1997 = R97) demonstrated the potential of W UMa-type systems as tracers of the Galactic structure and population content. This paper should be consulted for several details on the OGLE eclipsing-binary sample and on its analysis. R97 used almost exclusively the time-independent data, such as the orbital periods, maximum brightness magnitudes and colors, and upper limits to the interstellar reddening and extinction. The present paper will concentrate on properties of contact binaries which are accessible through analysis of the light curves.
The only information from light curves used in R97 was in the preparation of an automatically selected sample of contact systems observed with relatively small errors and fulfilling shape criteria for contact configurations. The algorithm, based on Fourier decomposition of light curves for all 933 eclipsing systems in nine OGLE fields in Baade’s Window (BW), defined a much smaller sample of 388 contact binaries (the “restricted-” or R-sample) than the original visual classification of the OGLE project which contained 604 systems (the O-sample). For both samples, the additional constraint was that the period should be shorter than one day and that the $V-I$ colors should be available. The current paper will present results mostly for the R-sample.
For consistency with R97, the same data from the OGLE Catalog, Parts I – III are used in this paper (Udalski et al. 1994, 1995a, 1995b). The new data for BW9 – BW11 (Udalski et al. 1996) have not been included here also because the extinction/reddening data are not available for these fields.
It has been shown in R97 that the distances of the W UMa systems in the BW are distributed rather evenly in space all the way to the Bulge. Determinations of distances required assumption of the length scale of the interstellar absorption layer along the line of sight, $d_0$. Two extreme assumptions on $d_0$ were considered: (1) extinction extending uniformly to the Bulge, $d_0 = 8$ kpc, and (2) extinction truncated at $d_0 = 2$ kpc, with the latter assumption giving a more consistent picture. The choice is however not critical for the results in the present paper and enters only through a definition of the local sample, judged to be complete to distance of 3 kpc. When applied to the R-sample, this choice will be designated R$_2$, in consistence with R97.
The properties of the contact binaries which were extensively discussed in R97 and will not be repeated here, although of relevance for the present paper, were the following: Most of the W UMa-type systems belong to the population of old Turn-Off-Point stars. Their periods and colors are confined to relatively narrow ranges of $0.25 < P <
0.7$ days and $0.4 < (V-I)_0 < 1.4$. The [*apparent*]{} density of contact binaries is about $(7 - 10) \times 10^{-5}$ systems per pc$^3$ and the [*apparent*]{} frequency, relative to nearby dwarfs of similar colors is one contact system per about 250 – 300 Main Sequence stars. A correction for undetected systems with low orbital inclination would increase the above numbers by a factor of about 2 times. Judging from these results, which were derived from the volume-limited sample, most of the contact binaries belong to the Old Disk population, but with a possibility of an admixture of Halo and Thick Disk systems.
The present paper is organized as follows: Sections \[fill\] – \[asym\] contain discussions of various properties of the contact binaries which can be analyzed using Fourier coefficients of the light-curve decomposition. Section \[fill\] discusses the degree-of-contact (sometimes also called “over-contact”, although such an expression seems to be awkward). Section \[ptc\] discusses the statistics of occurrence of the poor thermal contact in contact binaries. Section \[asym\] addresses the poorly explored matter of light-curve asymmetries. Sections \[ampl\] and \[total\] address the matter of the mass-ratio distribution determinations from the amplitude distribution and from the distribution of angles of totality for totally eclipsing systems. Section \[sum\] gives a summary of the paper.
DEGREE OF CONTACT {#fill}
=================
As was discussed in R97, the two even cosine terms of the Fourier decomposition of light curves, $a_2$ and $a_4$, can be used to separate contact binaries from detached binaries (see Figure 4 in R97). This separation was used in the definition of the R-sample, which was selected on the basis of (1) a good overall fit of the cosine representations to the light curves, with the mean standard error of the fit better than 0.04 mag (i.e. about 2-sigma error for most systems), and (2) a shape criterion based on the $a_2$ and $a_4$ coefficients. This criterion can be directly linked to to the degree-of-contact parameter used in Rucinski (1993 = R93). As was shown in R93, a crude estimate of the degree of contact can be made by interpolation in the $a_4 - a_2$ plane. Figure 1 shows the O-sample and R-sample data, compared with the theoretical predictions of R93. The three domains marked in the figure are defined as envelopes for all combinations of mass-ratios ($0.05 \le q \le 1$) and inclinations ($30 \le i \le 90$ degrees) considered in R93, for three cases of the degree of contact: when the stars just fill the inner critical envelope ($f=0$, the steepest rising and the narrowest domain), when the stars fill the outermost common envelope ($f=1$, the largest and lowest domain in the figure) and for one intermediate case ($f=0.5$). The upper edge of the marginal contact domain is basically the same as the cut-off line for the Fourier filter: $a_4 = a_2\,(0.125-a_2)$. The degree of contact $f$ is defined in terms of the potential, as in R93.
In addition to the R-sample, the O-sample is included in Figure 1 to show how many systems have been lost in moving from the original OGLE classification to the restrictive R-sample. A large fraction of rejected systems had poorly defined light curves, but some very interesting objects, very close to contact, did not pass the Fourier filter. One example of particular importance is the shortest-known period Main Sequence binary \#3.038 (P = 0.198 day) which consists of two very similar, strongly distorted, but detached, M-type dwarfs (a full light-curve synthesis solution by Maceroni & Rucinski is in preparation). Since it was decided to concentrate on genuine contact systems, the R-sample will be used from this point on. However, one should realize the limitations of the selection, which was based on the Fourier coefficients calculated in R93 for one effective temperature and one spectral band ($V$-filter). Since the contact-binary variability is dominated by geometrical changes and weakly depends on the atmospheric properties, it can be argued that these limitations should not strongly affect any [*relations*]{} between the coefficients (as opposed to their numerical values). However, this approach cannot really replace full synthesis solutions of the light curves, especially for derivation of the values of $f$.
As we can see in Figure 1, most contact systems occupy a band corresponding to moderate degrees of contact, of about $0 < f < 0.5$. This is in agreement with the previous results (Lucy 1973; Rucinski 1973; Rucinski 1985, Fig.3.1.9). However, this method of estimating $f$ can be used only in a qualitative sense. In addition, it entirely loses sensitivity for small amplitudes and nothing can be said about cases with $|a_2| < 0.1$. In their majority, these will be small mass-ratio systems, as with the decrease of $q$, sizes of components become progressively more different leading to shallower eclipses.
Figure 1 contains interesting contribution to the matter of the most common values of the mass-ratio. The broken lines join loci of $a_2$ and $a_4$ combinations which can be reached for edge-on orbits (inclination $i=90^\circ$) for fixed values of the mass ratio, 0.1, 0.3, 0.5 and 1.0. The concentration of observational points close to the origin, with strong fall-off toward the upper right in Figure 1, strongly suggests that large (close to unity) values of the mass-ratio are very infrequent. The mass-ratio distribution is apparently skewed with a preference for small values. We will return to the important matter of the mass-ratio distribution in Sections \[ampl\] and \[total\].
POOR THERMAL CONTACT AND SEMI-DETACHED SYSTEMS {#ptc}
==============================================
Only the two even cosine coefficients, $a_2$ and $a_4$ of the Fourier decomposition of the light curves have been used so far. Now we will consider the first odd term, $a_1$. Figure 2 shows the two first cosine coefficients, $a_1$ and $a_2$, for the full sample of 933 eclipsing systems in the Catalog, divided into groups according to the original OGLE classification. This figure corresponds to Figure 4 in R97 which gave the $a_2$ – $a_4$ dependence for the whole OGLE sample. The filled circles are EW-type systems according to the OGLE classification, i.e. the contact binaries. Open circles mark EB systems. In the OGLE Catalog, all EB systems have periods longer than one day[^1]. Crosses mark all remaining systems with classes E, EA and E?. These systems frequently have eclipses of unequal depths, so that values of $a_1$ for them may differ substantially from zero. Indeed, as in Figure 4 in R97, we see a clean separation between the contact systems, occupying a band within $-0.02 < a_1 < 0$ (or difference in the eclipse depths less than 0.04 mag), and other binaries which sometimes show strongly negative values of $a_1$. For contact binaries with good energy exchange between components, the $a_1$ term is expected to be very small (R93), reflecting almost identical depths of eclipses, in accordance with almost constant effective temperature over the whole contact configuration. The equality of the effective temperatures is a defining feature of the contact binaries of the W UMa-type; it was initially one of the most difficult properties to explain and led to development of the successful contact model by Lucy(1968).
A closer look at Figure 2 reveals that a certain fraction of systems classified by OGLE as EW or contact binaries also appear to have negative values of $a_1$. These are the systems of interest in this section. There are two reasons why systems appearing as contact binaries may show differences in eclipse depths: Some of them may be very close, semi-detached binaries (SD), and some may be in contact, but with the energy exchange constricted or diminished for some reason. We will call the latter poor-thermal-contact (PTC) systems. There exists a third reason why systems in good geometrical contact may show deviations from the Lucy model. This is the so-called “W-type syndrome” related to slightly higher surface brightnesses of the less-massive components. It seems to be limited to cool systems and may be related to their chromospheric activity. There have been several attempts to explain this light-curve and a small temperature (about 5%) excess of the less-massive component seems to be preferred (for an exhaustive discussion of the effect for the prototype case of W UMa and for the literature, see Linnell 1991a, 1991b, 1991c). The W-type syndrome is too small to be addressed in this paper and actually must be hiding in the spread of points within $-0.02 < a_1 < 0$ in Figure 2. We are interested here in much larger effects produced by the SD or PTC causes.
It is very difficult to distinguish observationally between contact systems with inhibited energy exchange and semi-detached systems with components close to the inner critical Roche lobe. Systems which show large differences in depths of eclipses, yet appear to be in contact, are sometimes called the B-type systems for their $\beta$ Lyrae-type light curves. A relation between short-period semi-detached systems and contact systems is expected to exist on the basis of the theoretically predicted Thermal Relaxation Oscillations (Lucy 1976; Flannery 1976; for the most recent review, see Eggleton 1996). In the simplest version, the TRO cycles should consist of two main phases, the good geometrical and thermal contact state and the semi-detached state. In reality, switching between these two may involve the PTC state. Relative durations of the contact and semi-detached branches of the TRO cycles are expected to scale as some larger-than-one power of the mass-ratio: with the contact stage lasting long and the semi-detached stage quite brief. But nothing is known how long the PTC state could last. Observations do not give us a clear picture, mostly because of the poor statistics, spotty coverage of stellar parameter space and difficulties with separation of the SD and PTC cases.
After the work of Lucy & Wilson (1979), which explicitly addressed the question of systems in the “broken-contact” phase, several short-period eclipsing systems with unequal depths of the minima have been studied by Kałużny and Hilditch with their respective collaborators (Kałużny 1983, 1986a, 1986b, 1986c; Hilditch et al. 1984, 1988; Bell et al. 1990). These and other results have been discussed in terms of the temperature difference between the components in the compilation of Lipari & Sistero (1988). These investigations did not give an answer about the evolutionary state of such systems. Some of them seem to be genuine PTC contact systems with components of unequal temperatures, some may be in the semi-detached state (with either the more- or less-massive components filling their critical lobes), mimicking contact systems. Usually, a full set of photometric and spectroscopic data is needed in individual cases. One property however is clear: Irrespectively what produces the large differences in depths of eclipses, such systems are very rare, even in spite of certain advantage in discovery relative to normal contact systems. We see them only among systems having orbital period longer than a certain threshold value: the SD or PTC binaries do not occur among very cool, short-period systems. The border line is currently at about 0.4 days, defined by the shortest-period PTC system known at this time, W Crv with $P = 0.388$ day (Odell 1996).
The OGLE data give us a first chance to look into statistics of the occurrence of unequally deep eclipses. However, we should note an important limitation of results based solely on such photometric data: Unless we see total eclipses, we do not know which star, more or less massive, is eclipsed at each minimum. In view of the surface-brightness deviations from the contact model, this is a serious complication. Here, we follow the conventional way of counting phases, from the deeper minimum, i.e. from the eclipse of the hotter star. A more meaningful convention, normally used in light-curve synthesis programs, would be with phases counted from the eclipse of the more-massive star.
Figure 3 shows the same data as in Figure 2, but for the R-sample. Its sub-sample with the distances smaller than 3 kpc (the reddening model with $d_0 = 2$ kpc, designated as R$_2$, see the Introduction), will be from now on called the “local sample”. As was discussed in R97, this distance delineates a volume-limited sample of contact systems in the OGLE data. We can immediately see that contact systems with large differences in depths of minima are all distant, hence intrinsically bright. Among the 98 R-sample systems to 3 kpc, only 2 have $a_1 < -0.03$ (difference in depth larger than 0.065 mag), so that the phenomenon of unequal eclipse depths is very rare and affects only some 2% of the systems. For the full R-sample extending all the way to the Bulge, with the same magnitude-difference threshold, we have 36 among 388 or 9% of all systems. The two systems in the local R-sample with unequal depths of minima are \#3.012 and \#6.005. Their orbital periods are 0.370 and 0.698 days. The first system sets a new short-period limit of the occurrence of the phenomenon, replacing W Crv with 0.388 day. The light curve for \#3.012 is shown in Figure 4. The system \#6.005 (discussed in the next section, its light curve is in Figure 11) is the only nearby systems with large eclipse-depth difference and relatively long orbital period.
The results on $a_1$ for the R$_2$ sample are shown as an orbital-period dependence in Figure 5 and as the intrinsic-color dependence in Figure 6. As was discussed in R97, the range of the observed intrinsic colors for most systems is relatively narrow, due to their concentration in the Turn-Off Point region for an old stellar population. The distribution is additionally compressed on the red side due to progressive elimination with distance of faint, red systems from the full, magnitude-limited R-sample. As we already said, the local systems, with a relatively wider range of colors, show small values of $a_1$. This applies also to the reddest binaries which we will discuss now in a small detour from the main subject.
The three reddest systems in Figure 6, with the observed colors $V-I > 2$ and the intrinsic colors $(V-I)_0 > 1.5$ belong to the local sample with distances of 600 – 1400 parsecs. These small distances may be erroneous if the colors are red due to some observational problems rather than to genuinely low effective temperatures, as very red colors lead to low intrinsic luminosities in our absolute-magnitude calibration. The red colors are really unusual when compared with the current short-period, red-color limit for the contact binaries determined by CC Com with $P=0.2207$ day, $B-V=1.24$, $V-I=1.39$ and the spectral type about $K5$ (Rucinski 1976; Bradstreet 1985). All three systems, \#3.053, \#7.112 and \#8.072, are quite unremarkable as far as their light curves are concerned. Only \#8.072 has a short orbital period of 0.284, in some accord with the color, but even in this case the color is well beyond what has been observed before for contact systems: $V-I = 2.04$ and $(V-I)_0 = 1.77$, implying $(B-V)_0 \simeq 1.4$. \#3.053 is unusual in that it has a moderately long period of 0.466 day in combination with a very red color: $V-I = 2.59$ and $(V-I)_0 = 2.31$. The corresponding $(B-V)_0 \simeq 1.5$ implies the spectral type as late as M1 or M2. The light curves of \#3.053 and \#8.072 have moderately large amplitudes of $\Delta I = 0.45$ and 0.54 so that blending of images with other, very red stars would be difficult to postulate. Such blending may be the explanation for the low amplitude of \#7.112 ($\Delta I = 0.17$), which is unusually red ($V-I=2.58$) for its orbital period of 0.590 day. All three systems require further observations. We note that the possible error of 0.1 in the OGLE mean colors for red stars cannot be an explanation here, as the stars are simply too red.
LIGHT-CURVE ASYMMETRIES {#asym}
=======================
Light-curve asymmetries are fairly common in contact binaries. The difference in heights of minima is sometimes called the O’Connell effect (for earlier references, see Linnell 1982 and Milone et al. 1987). There is no generally accepted interpretation of such asymmetries, but the most obvious explanations would be in terms of stellar spots or some streaming motions deflected by Coriolis forces. The starspot explanation seems to work well in cool systems which are expected to be very active. Little is known about causes of persistent asymmetries for systems of spectral types earlier than F-type since large magnetic spots would be difficult to imagine to exist on these stars. Sometimes, the asymmetry is so large that it must be caused by some streaming/accretion phenomena. Of particular importance here is the short-period ($P=0.301$ day), late-type (K3V) detached system V361 Lyr which shows a huge asymmetry (Kałużny 1990, 1991; Gray et al. 1995) apparently due to an accretion process currently taking place between components[^2]. The maximum after the deeper eclipse is higher in this case, indicating in-fall on the cooler component as the most likely direction of the gas streaming.
No statistics are currently available as to how large are the asymmetries and how often do they occur. Selection of sky-field objects for individual observations is obviously highly biased. A large sample, such as the OGLE sample, which can be subdivided into magnitude- and volume-limited ones, is of great usefulness here. Before presenting the results, the same warning as in Section \[ptc\]: Our photometric sample suffers from the ambiguity in the origin of phases by half of the orbital period. If the asymmetry phenomena are driven by the Coriolis force, they are expected to show signatures possibly related to the relative masses of components eclipsed at each minimum. We do not have this information, but instead we have information about the relative effective temperatures of components. These might correlate with the masses, but this is not obvious that this must necessarily be the case.
The asymmetries have been analyzed in the simplest possible way by inclusion of one sine term in the Fourier decomposition. No major differences exist in the results for the O- and R-samples, so that only the latter sample will be discussed here. Figure 7 contains the sine coefficients, $b_1$, plotted versus the largest cosine coefficient $a_2$. The sign of $b_1$ indicates which maximum is higher after the deeper eclipse. No obvious correlation between both coefficients seems to be present. However, the next Figure 8, with the dependence of $b_1$ on the orbital period, brings up an interesting property: While for the local, mostly short-period systems we see about equal numbers of positive and negative values of $b_1$, for systems with periods longer than about 0.4 day, the positive asymmetries dominate. This effect is not very strong, but is definitely present, as is shown in the histogram of the $b_1$ coefficients for 302 systems of the R-sample with $P > 0.4$ day in Figure 9. The distribution is only mildly asymmetric: the mean is slightly positive, $+0.0013$, and the skewness is not significant ($0.33 \pm 0.22$). However, if we treat the both branches as independent distributions, the $\chi^2$ test gives probability of only 0.0047 that the differences in numbers in 8 bins of $|b_1|$ could be due to random fluctuations. If we eliminate the two bins close to the origin ($-0.002$ to $+0.002$) which are most susceptible to random fluctuations, then the probability that the two branches are different due to a statistical fluctuation decreases to as little as 0.0026. Thus, the difference in the two sides of the distribution is significant at the level of 0.995 to 0.997. We see no way how the asymmetry in the distribution of $b_1$ could be generated by ways of data reductions. It must be real. Apparently, for long-period systems, incidence of the elevated maximum after the deeper eclipse is higher than the opposite case.
The most obvious candidate for the cause of the asymmetry in the distribution of $b_1$ would be the Coriolis force acting on gas streams between components. The sense of the deviations induced by the Coriolis force is such that – if we see hot spots due to accretion phenomena (as seems to be the case for V361 Lyr) – the visibility of spots [*before the secondary minima*]{} implies that the spots are located on the cooler components. This would be exactly what should happen for binaries in the semi-detached state with more massive, hotter components losing mass for their less massive companions. At this point, without spectroscopic data, this is a hypothesis rather than a definite statement.
If some of the light-curve asymmetries are caused by mass-transfer phenomena in semi-detached systems of similar type, then we could expect a correlation between the sense and size of the asymmetry and the difference in the depths of eclipses. Such a correlation cannot be very tight as it would reflect the percentage admixture of the semi-detached systems among all systems showing the light curve asymmetries. The scatter plot for $a-1$ and $b_1$ is shown in Figure 10 where some correlation between these quantities is indeed visible. The formal linear (Pearson) correlation coefficient $-0.29 \pm 0.08$ indicates presence of a correlation with a relatively high significance (probability that the observed value resulted from a correlation of independent distributions is at the level 0.001). The three systems with the largest deviations in Figure 10 ($a_1 \le -0.05$ and $b_1 \ge 0.01$), and thus the best candidates to be the semi-detached systems with mass-transfer are the following: \#0.111, \#3.094 and \#8.133. Their periods are relatively long, 0.639, 0.912 and 0.509, which coupled with their blue colors gives the distances 4.5, 7.9 and 5.3 kpc (under the R$_2$ assumption). System \#3.094 was suggested in R97 to be in the Galactic Bulge.
A typical light curve for a system showing a large asymmetry is shown in Figure 11. T he system is \#6.005 with the orbital period of 0.698 day which was already mentioned in the previous section as one of the two nearby systems with a large difference in the depth of the eclipses. Thus, this system directly shows the correlation between $a_1$ and $b_1$ and may be taken as a prototype of the class.
The dependence of the $b_1$ coefficient on the intrinsic color is shown in Figure 12. This plot seems to indicate that for very cool systems, negative $b_1$ may be more common, but because of the small number of such systems, the significance to this indication is low.
AMPLITUDE DISTRIBUTION {#ampl}
======================
Amplitudes of light variations of contact binaries contain an important information on the mass-ratio ($q$) distribution: Large amplitudes can be observed only for large mass-ratios, close to unity; for small mass-ratios, only small amplitudes can be observed. This simple relation is illustrated in Figure 13 for a few theoretical distributions calculated on the basis of R93 for mass-ratios within narrow ranges of $\Delta q = 0.1$ (every second interval in $q$). We see not only the obvious dependence between the largest possible amplitudes and the mean value of $q$, but also the effects of total eclipses producing similar amplitudes for large fractions of systems within $q$-dependent ranges of inclinations close to $i=90^\circ$. The distributions in Figure 13 cannot be obviously observed as weighting by the mass-ratio distribution $Q(q)$ occurs. Exactly this weighting opens up a possibility of observational determination of $Q(q)$ which would be of great importance for our understanding of formation and evolution of contact binary stars.
Application of the line of reasoning described above is not easy, even for such a large sample as the one being analyzed now. First of all, for an inverse problem of determination of $Q(q)$ from the amplitude distribution, $A(a)$, very good statistics for [*each bin*]{} of $A$ is needed. This is the main obstacle why our attempts at determination of $Q(q)$ through a solution of the integral equation $A(a) = \int Q(q) K(q,\,a)\,da$ have been so far unsuccessful and will not be discussed here. The second problem is that the magnitude-limited sample is biased by the presence of distant, bright systems. As was shown in Section \[ptc\], these are systems with the highest frequency of occurrence of unequally deep eclipses. For such systems, primary minima are deeper and secondary minima are shallower than for good thermal contact models leading to a corrupted statistics of the amplitudes (although use of the $a_2$ coefficient could possibly help here). If we eliminate those bright and distant systems, and utilize only the systems of the local sample (which seems to give a fair representation of most typical systems), the sample becomes too small for definite applications. Finally, the distribution of the third geometrical parameter, the degree-of-contact, is not known but it does have some influence on the amplitude distribution, judging by the theoretical results, as in Figure 14.
Figure 14 gives the observed distributions $A(a)$, where $a = \Delta I$ in the OGLE data, as well as some additional theoretical predictions. The latter are shown by lines for the flat mass-ratio distribution $Q(q)$ and for two cases of the degree-of-contact, $f=0$ and $f=0.5$ (upper panel), as well as for for a strongly falling distribution, $Q(q) = 1 - q$ with $f=0$ (lower panel). These theoretical distributions show some sensitivity to the assumptions on the shape of $Q(q)$ and on the value of $f$, but look very different from the observed distributions. The full R-sample (upper panel) and the local R$_2$ sample to 3 kpc (lower panel) are shown as histograms. They indicate dominance of low-amplitude systems. Large amplitude systems are almost non-existent in the OGLE sample, which agrees very well with the old open-cluster data (Kałużny & Rucinski 1993; Rucinski & Kałużny 1994). The most obvious explanation for dominance of small amplitudes would be by a $Q(q)$ distribution which steeply increases for small values of $q$.
The OGLE sample gives us a very different picture than for the whole sky. In the sky field, we observe several bright contact systems which have large variability amplitudes, and some of them indeed have spectroscopically-determined mass-ratios very close to unity. The extreme cases are SW Lac with $q = 0.73 \pm 0.01$ (Hrivnak 1992), OO Aql with $q=0.84 \pm 0.02$ (Hrivnak 1989) and VZ Psc with $q=0.92 \pm 0.03$ (Hrivnak & Milone 1989). Thus, although $q=1$ apparently never happens, some mass-ratios can be quite close to unity. The large values of $q$ for the field systems are consistent with the amplitude distribution for the whole sky which peaks at $a \simeq 0.55$ and extends to amplitudes as large as one magnitude (Rucinski & Kałużny 1994; shown as a dotted histogram in the upper panel of Figure 14). Judging by the OGLE data, these are atypical cases, emphasized by the ease with which large-amplitude systems are detected in the sky surveys. More typical contact binaries show small variation amplitudes which are related to their small mass-ratios. Such systems remain to be discovered among bright stars of the sky field.
MASS-RATIO DISTRIBUTION FROM TOTALLY ECLIPSING SYSTEM {#total}
=====================================================
Since inversion of the amplitude distribution for derivation of $Q(q)$ presents several problems, one can consider a different approach: Mochnacki & Doughty (1972a, 1972b) described a method of element determinations utilizing the angles of internal eclipse contact for totally eclipsing systems. This method can obviously be applied only to a sub-sample of all systems, but has an advantage of being very weakly sensitive to the degree-of-contact: Basically, for contact configurations, only two geometrical parameters, $q$ and $i$, determine the the angle of the inner eclipse contact, $\phi = \phi (q,\,i)$.
The method of determination of the mass-ratio distribution $Q(q)$ would be quite simple: The distributions $Q(q)$ and of the orbital inclinations $I(i) = \sin i$ must be statistically independent. Thus, from the distribution of the inner-contact angles, $\Phi(\phi)$, one could determine $Q(q)$ by solving the integral equation: $$\Phi(\phi) = \int_{q(0^\circ,\phi)}^{q(90^\circ,\phi)}
Q(q) \, \sin[i(\phi,q)]\>
\vert \partial i(\phi,q)/\partial \phi \vert \> d q$$ At present, we have not been able to apply this approach because the sample of totally eclipsing systems among the OGLE systems was too small. Also, because of the relatively large observational errors of about 0.02 mag, it was difficult to set up an automatic-selection process of finding totally-eclipsing systems. However, the approach holds a great potential and, in the future, should be used on large samples of well observed systems. Since, as we have shown in Section \[ampl\], the mass-ratio distribution must be skewed to small values of $q$, a relatively large fraction of contact binaries should show total eclipses.
SUMMARY {#sum}
=======
The paper contains analysis of light curves of the W UMa-type binaries in the OGLE Catalog of Periodic Variable Stars for fields BWC – BW9. It is a continuation of R97, but concentrates on properties of the contact systems, rather than on their usefulness for galactic-structure studies. The important result of R97 that the contact binaries belong mostly to the Turn-Off Point population of old stars is not amplified here; the stress is on structural properties of the systems, as they can be gauged using simple methods of light curve characterization. Since the observations have moderate accuracy of about 0.02 mag, the low-order Fourier decomposition was judged to provide an adequate tool for such a characterization.
Rough estimates of the degree-of-contact, obtained using the $a_2$ and $a_4$ cosine coefficients confirm that the most frequent values are concentrated with $0 < f < 0.5$, suggesting weak contact. The quality of this determination is low and it is somewhat qualitative. A much more interesting results came from the analysis of the differences in depths of eclipses which was based on the first cosine coefficient, $a_1$. In the volume-limited “local” sample to 3 kpc, among 98 systems, only 2 show appreciable depth differences, so that good thermal and geometrical contact is a norm rather than an exception. One of these systems, \#3.012, sets a new short-period limit of 0.370 day for the occurrence of unequal eclipses. For the full magnitude-limited sample, which is favorably biased toward the intrinsically bright, distant systems, incidence of the large depth differences is more common with some 9% showing differences in depths of minima larger than 0.065 mag ($a_1 < -0.03$). Some of these may be in good geometrical contact and with the inhibited energy exchange, the poor-thermal-contact systems (PTC), but some may be actually semi-detached (SD) systems, very close to the contact configuration. The present data, in only one spectral band and without spectroscopic support, do not permit to distinguish between these possibilities in individual cases.
Analysis of the light curve maximum-light asymmetries – measured by the first sine term $b_1$ – which weakly (but significantly) correlate with the differences in the depths of the eclipses for periods longer than about 0.4 day, suggests that the semi-detached state with mass-exchange is common among systems with unequally deep eclipses. The asymmetries would be then caused by mass-exchange streams impinging outer layers of cooler components. Since the asymmetries may be also caused by photospheric spots, the statistics of asymmetries cannot be used to determine the number of SD systems mimicking contact binaries. The second system in the local sample which shows a large difference of the eclipse depths, \#6.005, is a perfect illustration of the correlation between the difference in the depth of eclipses and the sense of the light curve asymmetry. The overall rarity of systems with large eclipse-depth differences suggests that the admixture of semi-detached or poor-thermal-contact systems to the totality of contact systems is very small. These are however the intrinsically brightest systems which are visible to large distances so that they are preferentially represented in magnitude-limited samples. In the future, it would be highly desirable to obtain light curves in a few spectral bands, say in the $U$, $B$, $V$ and $I$ filters. This would permit decoupling of the effects of the accretion streams and temperature differences between components from the geometric issues of contact versus semi-detached configuration.
Presence of only two SD/PTC systems among 98 contact binaries of the local sample can explain why we do not see such systems among bright stars. Using the apparent frequency of contact binaries found in R97 of 1/250 – 1/300, we can estimate the expected frequency of SD/PTC systems among Main Sequence dwarfs to be about 1/12,500 – 1/15,000; since this estimate is based on 2 cases, it carries a Poissonian uncertainty of a factor of about 1.4. A complete sample of bright stars to $V
\simeq 7 - 8$ is accessible from the Hipparcos Input Catalogue (CD-ROM Version, Turon et al. 1994). By counting luminosity-class stars IV and V in the HIP, within the color range where contact binaries occur (R97), $0.4 < B-V < 1.2$, to the successively deeper limiting magnitudes $V =5$, 6, 7, and 8, one obtains 128, 497, 1620 and 4561 stars. Thus, no SD/PTC systems are expected to the brightness levels at which the Hipparcos sample starts showing selection effects. However, by going one magnitude deeper, we could expect some 3–4 time more stars, so that we should be able to detect one or two SD/PTC system. This is well confirmed by the actual numbers as the brightest among such systems, AG Vir and FT Lup, appear at $V=8.5$ and 9.2 (Kałużny 1986b, 1986c). As an aside, we mention here that the sample of bright contact binaries, when compared with the Hipparcos numbers, fully confirms the apparent frequency of 1/250 – 1/300: To the same $V$-magnitude limits as above, the variable-star lists contain 3, 3, 6, and 14 contact systems.
The distribution of the variability amplitudes suggests that the mass-ratio distribution $Q(q)$ increases toward small values of $q$. Since the increase appears for $\Delta I < 0.6$, but is modified by discovery selection already below $\Delta I < 0.3$, the present distribution is just too narrow to attempt a determination of the mass-ratio distribution $Q(q)$. A distribution emphasizing low values of $q$ should lead to frequent occurrence of totally eclipsing systems. Such systems may allow an entirely independent determination of $Q(q)$ based on the angles of the inner eclipse contacts. We note that the sky-field is surprisingly devoid of low-amplitude contact systems; we suspect that they have been simply overlooked in non-systematic surveys, a result which has led to an over-representation of large-amplitude contact systems.
Thanks are due to Dr. Bohdan Paczyński and Dr. Janusz Kałużny for useful comments and suggestions and to the OGLE team for making their data available through the computer networks.
Research grant from the Natural Sciences and Engineering Council of Canada is acknowledged here.
Note: The data on the W UMa-type stars in the OGLE survey of fields BWC to BW8 which were used in R97 and in this paper are available from the author or from http://www.astro.utoronto/$\sim$rucinski/rucinski.html
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[^1]: Many of the EB systems classified by OGLE would formally fulfill our Fourier filter for inclusion in the R-sample, but we considered only systems with $P < 1$ day.
[^2]: To the author’s knowledge, no radial-velocity of this important system has been performed so that all inferences about it are still based solely on photometric data.
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abstract: 'Following a polynomial approach, many robust fixed-order controller design problems can be formulated as optimization problems whose set of feasible solutions is modelled by parametrized polynomial matrix inequalities (PMI). These feasibility sets are typically nonconvex. Given a parametrized PMI set, we provide a hierarchy of linear matrix inequality (LMI) problems whose optimal solutions generate inner approximations modelled by a single polynomial superlevel set. Those inner approximations converge in a well-defined analytic sense to the nonconvex original feasible set, with asymptotically vanishing conservatism. One may also impose the hierarchy of inner approximations to be nested or convex. In the latter case they do not converge any more to the feasible set, but they can be used in a convex optimization framework at the price of some conservatism. Finally, we show that the specific geometry of nonconvex polynomial stability regions can be exploited to improve convergence of the hierarchy of inner approximations.'
author:
- 'Didier Henrion$^{1,2,3}$, Jean-Bernard Lasserre$^{1,2,4}$'
title: '**Inner approximations for polynomial matrix inequalities and robust stability regions**'
---
: polynomial matrix inequality, linear matrix inequality, robust optimization, robust fixed-order controller design, moments, positive polynomials.
Introduction
============
Linear system stability can be formulated semialgebraically in the space of coefficients of the characteristic polynomial. The region of stability is generally [*nonconvex*]{} in this space, and this is a major obstacle when solving fixed-order and/or robust controller design problems. Using the Hermite stability criterion, these problems can be formulated as parametrized polynomial matrix inequalities (PMIs) where parameters account for uncertainties and the decision variables are controller coefficients. Recent results on real algebraic geometry and generalized problems of moments can be used to build up a hierarchy of convex linear matrix inequality (LMI) [*outer*]{} approximations of the region of stability, with asymptotic convergence to its convex hull, see e.g. [@hl04] for a software implementation and examples, and see [@hl06] for an application to PMI problems arising from static output feedback design.
If outer approximations of nonconvex semialgebraic sets can be readily constructed with these LMI relaxations, [*inner*]{} approximations are much harder to obtain. However, for controller design purposes, inner approximations are essential since they correspond to sufficient conditions and hence guarantees of stability or robust stability. In the robust systems control literature, convex inner approximations of the stability region have been proposed in the form of polytopes [@n06], ellipsoids [@hpas03] or more general LMI regions [@hsk03; @hkl06] derived from polynomial positivity conditions. Interval analysis can also be used in this context, see e.g. [@wj94].
In this paper we provide a numerical scheme for approximating from inside the feasible set $\P \subset \R^n$ of a parametrized PMI $P(x,u)\succeq0$ (for some matrix polynomial $P$), that is, the set of points $x$ such that $P(x,u)\succeq0$ for [*all*]{} values of the parameter $u$ in some specified domain $\U\subset\R^p$ (assumed to be a basic compact semialgebraic set[^1]). This includes as a special case the approximation of the stability region (and the robust stability region) of linear systems. The particular case where $P(x,u)$ is affine in $x$ covers parametrized LMIs with many applications in robust control, as surveyed e.g. in [@s06].
Given a compact set $\B \subset \R^n$ containing $\P$, this numerical scheme consists of building up a sequence of inner approximations $\G_d\subset\P\subset\B$, $d\in\N$, which fulfils two essential conditions:
1. The approximation converges in a [*well-defined analytic*]{} sense ;
2. Each set $\G_d$ is defined in a [*simple*]{} manner, as a superlevel set of a single polynomial. In our mind, this feature is essential for a successful implementation in practical applications.
More precisely, we provide a hierarchy of inner approximations $(\G_d)$ of $\P$, where each $\G_d = \{x\in\B:g_d(x)\geq0\}$ is a basic semi-algebraic set for some polynomial $g_d$ of degree $d$. The vector of coefficients of the polynomial $g_d$ is an optimal solution of an LMI problem. When $d$ increases, the convergence of $(\G_d)$ to $\P$ is very strong. Indeed, the Lebesgue volume of $\G_d$ converges to the Lebesgue volume of $\P$. In fact, on any (a priori fixed) compact set $\B$, the sequence $(g_d)$ converges for the $L_1$-norm on $\B$ to the function $x\mapsto \lm(x)=\min_{u\in \U} \lm(x,u)$ where $\lm(x,u)$ is the minimum eigenvalue of the matrix-polynomial $P(x,u)$ associated with the PMI. Consequently, $g_d\to \lm$ in (Lebesgue) measure on $\B$, and $g_{d_k}\to\lm$ almost everywhere and almost uniformly on $\B$, for a subsequence $(g_{d_k})$. In addition, if one defines the piecewise polynomial $\bar{g}_d:=\max_{k\leq d}g_k$, then $\bar{g}_d\to\lm$ almost everywhere, almost uniformly and in (Lebesgue) measure on $\B$.
In addition, we can easily enforce that the inner approximations $(\G_d)$ are nested and/or convex. Of course, for the latter convex approximations, convergence to $\P$ is lost if $\P$ is not convex. However, on the other hand, having a convex inner approximation of $\P$ may reveal to be very useful, e.g., for optimization purposes.
On the practical and computational sides, the quality of the approximation of $\P$ depends heavily on the chosen set $\B\supset \P$ on which to make the approximation of the function $\lm$. The smaller $\B$, the better the approximation. In particular, it is worth emphasizing that when the set $\P$ to approximate is the stability or robust stability region of a linear system, then its particular geometry can be exploited to construct a tight bounding set $\B$. Therefore, a good approximation of $\P$ is obtained significantly faster than with an arbitrary set $\B$ containing $\P$.
Finally, let us insist that the main goal of the paper is to show that it is possible to provide a tight and explicit inner approximation with no quantifier, of nonconvex feasible sets described with quantifiers. Then this new feasible set can be used for optimization purposes and we are facing two cases:
- the convex case: if $f$ and $-g$ are convex polynomials, $\B = \{x \in \R^n : \|x\|_{\infty} \leq1\}$ and $\G = \{x \in \B : g(x) \geq 0\}$ then the optimization problem $\min_x f(x) \,\mathrm{s.t.}\, x \in \G$ is polynomially solvable. Indeed, functions $f(x)$, $g(x)$, $\|x\|_{\infty}$ are polynomially computable, of polynomial growth, and the feasible set is polynomially bounded. Then polynomial solvability of the problem follows from [@bn01 Theorem 5.3.1].
- the nonconvex case: if $-g$ is not convex then notice that firstly we still have an optimization problem with no quantifier, a nontrivial improvement. Secondly we are now faced with an polynomial optimization problem with a single polynomial constraint and possibly bound constraints $x \in \B$. One may then apply the hierarchy of convex LMI relaxations described in [@l09 Chapter 5]. Of course, in general, polynomial optimization is NP-hard. However, if the size of the problem is relatively small and the degree of $g$ is small, practice seems to reveal that the problem is solved exactly with few relaxations in many cases, see [@l09 §5.3.3]. In addition, if some structured sparsity in the data is present then one may even solve problems of potentially large size by using an appropriate sparse version of these LMI relaxations as described in [@wkkm06], see also [@l09 §4.6].
The outline of the paper is as follows. In Section \[problem\] we formally state the problem to be solved. In Section \[lmi\] we describe our hierarchy of inner approximations. In Section \[control\], we show that the specific geometry of the stability region can be exploited, as illustrated on several standard problems of robust control. The final section collects technical results and the proofs.
Problem statement {#problem}
=================
Let $\R[x]$ denote the ring or real polynomials in the variables $x=(x_1,\ldots,x_n)$, and let $\R[x]_d$ be the vector space of real polynomials of degree at most $d$. Similarly, let $\Sigma[x]\subset\R[x]$ denote the convex cone of real polynomials that are sums of squares (SOS) of polynomials, and $\Sigma[x]_d\subset\Sigma[x]$ its subcone of SOS polynomials of degree at most $2d$. Denote by $\s^m$ the space of $m\times m$ real symmetric matrices. For a given matrix $A\in\s^m$, the notation $A \succeq 0$ means that $A$ is positive semidefinite, i.e., all its eigenvalues are real and nonnegative.
Let $P:\R[x,u]\to\s^m$ be a matrix polynomial, i.e. a matrix whose entries are scalar multivariate polynomials of the vector indeterminates $x$ and $u$. Then $$\label{setp}
{\mathbf P} := \{x \in {\mathbb R}^n \: :\: \forall u \in {\mathbf U}, \: P(x,u) \succeq 0\}$$ defines a parametrized polynomial matrix inequality (PMI) set, where $x \in {\mathbb R}^n$ is a vector of decision variables, $u \in {\mathbb R}^p$ is a vector of uncertain parameters belonging to a compact semialgebraic set $$\label{setu}
{\mathbf U} := \{u \in {\mathbb R}^p \: :\: a_i(u) \geq 0, \: i=1,\ldots,n_a\}$$ described by given polynomials $a_i(u) \in {\mathbb R}[u]$, and $P(x,u)$ is a given symmetric polynomial matrix of size $m$. As $\U$ is compact, without loss of generality we assume that for some $i=i^*$, $a_{i^*}(u) = R^2-u^Tu$, where $R$ is sufficiently large.
We also assume that $\mathbf P$ is bounded and that we are given a compact set $\mathbf B \supset \mathbf P$ with explicitly known moments $y=(y_\alpha)$, $\alpha\in\N^n$, of the Lebesgue measure on $\B$, i.e. $$\label{momb}
y_{\alpha} := \int_{\mathbf B} x^{\alpha} dx$$ where $x^{\alpha} := \prod_{i=1}^n x^{\alpha_i}_i$. Typical choices for $\mathbf B$ are a box or a ball. To fix ideas, let $$\B\,:=\,\{x\in\R^n\::\: b_j(x)\geq 0,\:j=1,\ldots,n_b\}$$ for some polynomials $b_j\in\R[x]$. Again, with no loss of generality, we may and will assume that for some $j=j^*$, $b_{j^*}(x) = R^2-x^Tx$, where $R$ is sufficiently large. Finally, denote by $\vol\,\A$ the Lebesgue volume of any Borel set $\A\subset\B$.
We are now ready to state our polynomial inner approximation problem.
[**(Inner Approximations)**]{}\[inner\] Given set $\mathbf P$, build up a sequence of basic closed semialgebraic sets ${\mathbf G}_d=\{x\in\B\,:\,g_d(x)\geq0\}$, for some $g_d\in\R[x]$, such that $${\mathbf G}_d \subseteq \mathbf P, \quad d=1,2,\ldots
\quad\mbox{and}\quad \lim_{d \rightarrow \infty} \vol\,{\mathbf G}_d\,=\,\vol\,\P.$$
In addition, we may want the sequence of inner approximations to satisfy additional nesting or convexity conditions.
[**(Nested Inner Approximations)**]{}\[nest\] Solve Problem \[inner\] with the additional constraint $$\G_{d}\, \subseteq \,\G_{d+1}\,\subseteq\,\P, \quad d=1,2,\ldots$$
[**(Convex Inner Approximations)**]{}\[convex\] Given set $\mathbf P$, build up a sequence of nested basic closed convex semialgebraic sets ${\mathbf G}_d=\{x\in\B\,:\,g_d(x)\geq0\}$, for some $g_d\in\R[x]$, such that $$\G_{d}\, \subseteq \,\G_{d+1}\,\subseteq\,\P, \quad d=1,2,\ldots$$
A hierarchy of semialgebraic inner approximations {#lmi}
=================================================
Given a polynomial matrix $P(x,u)$ which defines the set $\mathbf P$ in (\[setp\]), polynomials $a_i\in\R[u]$ which define the uncertain set $\mathbf U$ in (\[setu\]), let ${\mathbf V} = \{v \in {\mathbb R}^m \: :\: v^Tv = 1\}$ denote the Euclidean unit sphere of $\R^m$ and let $\lm:\B\to\R$ be the function: $$\label{funp}
x\mapsto \lm(x) = \min_{u \in \U} \min_{v \in \V} v^T P(x,u) v$$ as the robust minimum eigenvalue function of $P(x,u)$. Function $\lm$ is continuous but not necessarily differentiable. It allows to define set $\mathbf P$ alternatively as the superlevel set $${\mathbf P} = \{x \in {\mathbb R}^n \: :\: \lm(x) \geq 0\}.$$
Primal SOS SDP problems
-----------------------
Let $a_0\in\R[u]$ be the constant polynomial $1$. Let $2d_0\geq \max(2+{\rm deg}\,P,\max_i {\rm deg} a_i, \max_j {\rm deg} b_j)$, and consider the hierarchy of convex optimization problems indexed by the parameter $d\in\N$, $d\geq d_0$: $$\label{sdp}
\begin{array}{rl}
\rho_d\,=\,\displaystyle\int_\B\lm(x)\,dx\,-\,\min_{g,r,s,t} & \displaystyle\int_\B g(x)\,dx\\[1em]
\mathrm{s.t.} & v^T P(x,u) v - g(x) \,=\, r(x,u,v) (1-v^T v) \\
& +\displaystyle\sum_{i=0}^{n_a} s_i(x,u,v) a_i(u)+\displaystyle\sum_{j=1}^{n_b} t_j(x,u,v) b_j(x) \quad\forall (x,u,v)\\
\end{array}$$ where decision variables are coefficients of polynomials $g\in\R[x]_{2d}$, $r\in\R[x,u,v]_{2d_r}$ and coefficients of SOS polynomials $s_i\in\Sigma[x,u,v]_{d_{s_i}}$, $i=0,1,\ldots,n_a$, and $t_j\in\Sigma[x,u,v]_{d_{t_j}}$, $j=1,\ldots,n_b$. Note in particular that the degrees of the polynomials should be such that $d_r \geq d-1$, $d_{s_i} \geq d-\lceil ({\rm deg} \,a_i)/2\rceil$ and $d_{t_j} \geq d-\lceil ({\rm deg} \,b_j)/2\rceil$. Since higher degree terms may cancel, the degrees can be chosen strictly greater than these lower bounds. However, in the experiments described later on in the paper, we systematically chose the lowest possible degrees.
For each $d\in\N$ fixed, the associated optimization problem (\[sdp\]) is a semidefinite programming (SDP) problem. Indeed, stating that the two polynomials in both sides of the equation in (\[sdp\]) are identical translates into linear equalities between the coefficients of polynomials $g, r, (s_i), (t_j)$ and stating that some of them are SOS translates into semidefiniteness of appropriate symmetric matrices. For more details, the interested reader is referred to e.g. [@l09 Chapter 2].
Dual moment SDP problems
------------------------
To define the dual to SDP problem (\[sdp\]) we must introduce some notations.
With a sequence $y=(y_\alpha)$, $\alpha\in\N^n$, let $L_y:\R[x]\to\R$ be the linear functional $$f\quad (=\sum_{\alpha}f_{\alpha}\,x^\alpha)\quad\mapsto\quad
L_y(f)\,=\,\sum_{\alpha}f_{\alpha}\,y_{\alpha},\quad f\in\R[x].$$ With $d\in\N$, the moment matrix of order $d$ associated with $y$ is the real symmetric matrix $M_d(y)$ with rows and columns indexed in $\N^n_d$, and defined by $$\label{moment}
M_d(y)(\alpha,\beta)\,:=\,L_y(x^{\alpha+\beta})\,=\,y_{\alpha+\beta},\qquad\forall\alpha,\beta\in\N^n_d.$$ A sequence $y=(y_\alpha)$ has a representing measure if there exists a finite Borel measure $\mu$ on $\R^n$, such that $y_\alpha=\int x^\alpha d\mu$ for every $\alpha\in\N^n$.
With $y$ as above and $h\in\R[x]$, the localizing matrix of order $d$ associated with $y$ and $h$ is the real symmetric matrix $M_d(h\,y)$ with rows and columns indexed by $\N^n_d$, and whose entry $(\alpha,\beta)$ is given by $$\label{local}
M_d(y)(h\,y)(\alpha,\beta)\,:=\,L_y(h(x)\,x^{\alpha+\beta})\,=\,
\sum_{\gamma}h_\gamma \,y_{\alpha+\beta+\gamma},\quad\forall\alpha,\beta\in\N^n_d.$$
With these notations, the dual to SDP problem (\[sdp\]) is given by: $$\label{sdp*}
\begin{array}{rl}
\rho^*_d=\displaystyle\int_\B\lm(x)dx\,-\,\min_y &L_y(v^TP(x,u)v)\\
\mbox{s.t.}&M_d(y)\succeq0,\:M_{d-1}((1-v^Tv)\,y)=0\\
&M_{d-d_{a_i}}(a_i\,y)\succeq0,\quad i=0,1,\ldots,n_a\\
&M_{d-d_{b_j}}(b_j\,y)\succeq0,\quad j=1,\ldots,n_b\\
&L_y(x^\alpha)\,=\,\int_\B x^\alpha\,dx,\quad\forall\alpha\in\N^n_{2d}
\end{array}$$ where $y\in\N^{n+p+m}_{2d}$.
Convergence
-----------
Before stating our main results, let us recall some standard notions of functional analysis. Let $g : \B \to \R$ be a function of $x$, and let $(g_d)$ denote a sequence of functions of $x$ indexed by $d \in {\mathbb N}$. Lebesgue space $L_1(\B)$ is the Banach space of integrable functions on $\B$ equipped with the norm $$\Vert g\Vert_1=\int_\B \vert g\vert dx.$$ Regarding sequence $(g_d)$, we use the following notions of convergence in $\B$ when $d\to\infty$:
- $g_d\to g$ in $L_1$ norm means $\displaystyle\lim_{d\to\infty} \Vert g-g_d\Vert_1 = 0$;
- $g_d\to g$ in Lebesgue measure means that for every $\varepsilon>0$, $$\lim_{d\to\infty}\mathrm{vol}\{x : |g(x)-g_d(x)|\geq\varepsilon\}=0;$$
- $g_d\to g$ almost everywhere means that $\lim_{d\to\infty}g_d(x)=g(x)$ pointwise except possibly for $x \in \A \subset \B$ with $\mathrm{vol}\,\A = 0$;
- $g_d\to g$ almost uniformly means that for every given $\varepsilon>0$, there is a set $\A \subset \B$ such that $\mathrm{vol}\,\A < \varepsilon$ and $g_d\to g$ uniformly on $\B\setminus\A$;
- finally, with the notation $g_d\uparrow g$ we mean that $g_d\to g$ while satisfying $g_d(x)\leq g_{d+1}(x)$ for all $d$.
For more details on these related notions of convergence, see [@ash §2.5].
\[lemma1\] For every $d\geq d_0$, SDP problem (\[sdp\]) has an optimal solution $g_d\in\R[x]_{2d}$ and $$\label{lem1-1}
\rho_d\,=\,\int_\B(\lm(x)-g_d(x))\,dx\,=\,\Vert \lm-g_d\Vert_1.$$
A detailed proof of Lemma \[lemma1\] can be found in §\[proof-lemma1\]. In particular we prove that there is no duality gap between SOS SDP problem (\[sdp\]) and moment SDP problem (\[sdp\*\]), i.e. $\rho_d = \rho^*_d$.
For every $d\geq d_0$, let $\bar{g}_d:\B\to\R$ be the piecewise polynomial $$\label{piecewise}
x\mapsto \bar{g}_d(x):=\max_{d_0\leq k\leq d}\,g_k(x).$$ We are now in position to prove our main result.
\[thmain\] Let $g_d\in\R[x]_{2d}$ be an optimal solution of SDP problem (\[sdp\]) and consider the associated sequence $(g_d)\subset L_1(\B)$ for $d\geq d_0$. Then:
[(a)]{} $g_d\to\lm$ in $L_1$ norm and in Lebesgue measure;
[(b)]{} $\bar{g}_d\uparrow \lm$ almost everywhere, almost uniformly and in Lebesgue measure.
A detailed proof of Theorem \[thmain\] can be found in §\[proof-thmain\]. It relies on the Stone-Weierstrass theorem, Putinar’s Positivstellensatz, Lebesgue’s dominated convergence theorem and Egorov’s theorem.
Polynomial and piecewise polynomial inner approximations
--------------------------------------------------------
\[coro1\] For every $d\geq d_0$, let $g_d\in\R[x]_{2d}$ be an optimal solution of SDP problem (\[sdp\]), let $\bar{g}_d$ be the piecewise polynomial defined in (\[piecewise\]), and let $$\label{coro1-0}
\G_d \,:=\, \{x \in \B \: :\: g_d(x) \geq 0\},\quad
\bar{\G}_d\,:=\, \{x \in \B \: :\: \bar{g}_d(x) \geq 0\}.$$ Then $$\begin{aligned}
\label{coro1-1}
\G_d\subset\P\quad\forall\,d\geq d_0&\mbox{and}& \lim_{d\to\infty} \vol(\P\setminus \G_d) = 0.\\
\label{coro1-2}
\bar{\G}_{d_0}\subseteq\cdots\subseteq\bar{\G}_d\subseteq\cdots\subset\P
&\mbox{and}& \lim_{d\to\infty} \vol(\P\setminus \bar{\G}_d) = 0.\end{aligned}$$ That is, sequence $(\G_d)$ solves Problem \[inner\] and sequence $(\bar{\G}_d)$ solves Problem \[nest\] if piecewise polynomials are allowed.
A proof can be found in §\[proof-coro1\].
Nested polynomial inner approximations
--------------------------------------
We now consider Problem \[nest\] where $g_d$ is constrained to be a polynomial instead of a piecewise polynomial. We need to slightly modify SDP problem (\[sdp\]). Suppose that at step $d-1$ in the hierarchy we have already obtained an optimal solution $g_{d-1}\in\R[x]_{2d-2}$, such that $g_{d-1}\geq g_{d_0}$ on $\B$, for all $d_0\leq d-1$. At step $d$ we now solve SDP problem (\[sdp\]) with the additional constraint $$\label{add1}
g(x)-g_{d-1}(x)\,=\,c_0(x)+\displaystyle\sum_{j=1}^{n_b} c_j(x)b_j(x),\quad\forall x$$ with unknown SOS polynomials $c_0\in\Sigma[x]_d$ and $c_j\in\Sigma[x]_{d-d_{b_j}}$.
\[coro2\] Let $g_d\in\R[x]_{2d}$ be an optimal solution of SDP problem (\[sdp\]) with the additional constraint (\[add1\]) and let $\G_d$ be as in (\[coro1-0\]) for $d\geq d_0$. Then the sequence $(\G_d)$ solves Problem \[nest\].
For a proof see §\[proof-coro2\].
Convex nested polynomial inner approximations
---------------------------------------------
Finally, for $g\in\R[x]_{2d}$, denote by $\nabla^2 g(x)$ the Hessian matrix of $g$ at $x$, and consider SDP problem (\[sdp\]) with the additional constraint $$\label{add2}
v^T \nabla^2 g(x) v = c_0(x,v) + \sum_{j=1}^{n_b} c_j(x,v)b_j(x)+c_{n_b+1}(x,v) (1-v^T v),$$ for some SOS polynomials $c_0\in\Sigma[x,v]_d$, $c_j\in\Sigma[x,v]_{d-d_{b_j}}$ and $c_{n_b+1}\in\Sigma[x,v]_{d-1}$.
Let $g\in\R[x]_{2d}$ be an optimal solution of SDP problem (\[sdp\]) with the additional constraint (\[add2\]) and let $\G_d$ be as in (\[coro1-0\]) for $d\geq d_0$. Then the sequence $(\G_d)$ solves Problem \[convex\].
The proof follows along the same lines as the proof of Corollary \[coro2\].
Example
-------
Consider the nonconvex planar PMI set $${\mathbf P} = \{x \in {\mathbb R}^2 \: :\:
P(x) = \left[\begin{array}{cc} 1-16x_1x_2 & x_1 \\
x_1 & 1-x_1^2-x_2^2 \end{array}\right] \succeq 0\}$$ which is Example II-E in [@hl06] scaled to fit within the unit box $${\mathbf B} = \{x \in {\mathbb R}^2 \: :\: \|x\|_{\infty} \leq 1\}$$ whose moments (\[momb\]) are readily given by $$y_{\alpha} = \frac{4}{(\alpha_1+1)(\alpha_2+1)}.$$
![Degree two (left) and four (right) inner approximations (light gray) of PMI set (dark gray) embedded in unit box (dashed).\[figqmi24\]](qmi2 "fig:"){width="49.00000%"} ![Degree two (left) and four (right) inner approximations (light gray) of PMI set (dark gray) embedded in unit box (dashed).\[figqmi24\]](qmi4 "fig:"){width="49.00000%"}
On Figure \[figqmi24\] we represent the degree two and degree four solutions to SDP problem (\[sdp\]), modelled by YALMIP 3 and solved by SeDuMi 1.3 under a Matlab environment. We see in particular that the degree four polynomial superlevel set ${\mathbf G}_2$ is somewhat smaller than expected. This is due to the fact that the objective function in problem (\[sdp\]) is the integral of $g(x)$ over the whole box $\mathbf B$, not only over PMI set $\mathbf P$. There is a significant role played by the components of the integral on complement set ${\mathbf B}\backslash{\mathbf P}$, and this deteriorates the inner approximation.
This issue can be addressed partly by embedding $\mathbf P$ in a tighter set $\mathbf B$, for example here the unit disk $${\mathbf B} = \{x \in {\mathbb R}^2 \: :\: \|x\|_2 \leq 1\}$$ whose moments (\[momb\]) are given by $$y_{\alpha} = \frac{\Gamma(\frac{\alpha_1+1}{2})\Gamma(\frac{\alpha_2+1}{2})}
{\Gamma(2+\frac{\alpha_1+\alpha_2}{2})}$$ where $\Gamma$ is the gamma function such that $\Gamma(k)=(k-1)!$ for integer $k$. See [@lz01 Theorem 3.1] for the general expression[^2] of moments of the unit disk in ${\mathbb R}^n$.
![Degree two (left) and four (right) inner approximations (light gray) of PMI set (dark gray) embedded in unit disk (dashed).\[figqmidisk24\]](qmidisk2 "fig:"){width="49.00000%"} ![Degree two (left) and four (right) inner approximations (light gray) of PMI set (dark gray) embedded in unit disk (dashed).\[figqmidisk24\]](qmidisk4 "fig:"){width="49.00000%"}
![Degree six (left) and eight (right) inner approximations (light gray) of PMI set (dark gray) embedded in unit disk (dashed).\[figqmidisk68\]](qmidisk6 "fig:"){width="49.00000%"} ![Degree six (left) and eight (right) inner approximations (light gray) of PMI set (dark gray) embedded in unit disk (dashed).\[figqmidisk68\]](qmidisk8 "fig:"){width="49.00000%"}
On Figure \[figqmidisk24\] we represent the degree two and degree four solutions to SDP problem (\[sdp\]). Comparing with Figure \[figqmi24\], we see that the approximations embedded in the unit disk are much tighter than the approximations embedded in the unit box. Finally, on Figure \[figqmidisk68\] we represent the tighter degree six and degree eight inner approximations within the unit disk.
Geometry of control problems {#control}
============================
As explained in the introduction, inner approximations of the stability regions are essential for fixed-order controller design. The PMI regions arising from parametric stability conditions have a specific geometry that can be exploited to improve the convergence of the hierarchy of inner approximations. In this section, we first recall Hermite’s PMI formulation of (discrete-time) stability conditions. Then we recall that the PMI stability region is the image of a unit box through a multi-affine mapping, which allows to derive explicit expressions for the moments of the full-dimensional stability region, as well as tight polytopic outer approximations of low-dimensional affine sections of the stability region. Numerical examples illustrate these techniques for fixed-order nominal and robustly stabilizing controller design.
Hermite’s PMI
-------------
Derived by the French mathematician Charles Hermite in 1854, the Hermite matrix criterion is a symmetric version of the Routh-Hurwitz criterion for assessing stability of a polynomial. Originally it was derived for locating the roots of a polynomial in the open upper half of the complex plane, but with a fractional transform it can be readily transposed to the open unit disk and discrete-time stability. The criterion says that a polynomial $x(z)=z^n+x_1z^{n-1}+\cdots+x_{n-1}z+x_n$ has all its roots in the open unit disk if and only if its Hermite matrix $P(x)=T^T_1(x)T_1(x)-T^T_2(x)T_2(x)$ is positive definite, where $$T_1(x) = \left[\begin{array}{cccc}
1 & x_1 & x_2 \\
0 & 1 & x_1 \\
0 & 0 & 1 \\
& & & \ddots
\end{array}\right] \quad
T_2(x) = \left[\begin{array}{cccc}
x_n & x_{n-1} & x_{n-2} \\
0 & x_n & x_{n-1} \\
0 & 0 & x_n \\
& & & \ddots
\end{array}\right]$$ are $n$-by-$n$ upper-right triangular Toeplitz matrices, see e.g. the entrywise formulas of [@barnett Theorem 3.13] or the construction explained in [@hpas03]. The Hermite matrix is $n$-by-$n$, symmetric and quadratic in coefficients $x=(x_1,x_2,\ldots,x_n)$, so that the interior of the PMI set $$\P = \{x \in \R^n \: :\: P(x) \succeq 0\}$$ is the parametric stability domain which is bounded, connected but nonconvex for $n\geq 3$. Optimal controller design amounts to optimizing over semialgebraic set $\P$.
Multiaffine mapping of the unit box {#multiaffine}
-----------------------------------
As explained e.g. in [@n06] or [@sd11 §3.5] and references therein, stability domain $\P$ can also be constructed as the image of the unit box (in the space of so-called reflection coefficients) through a multiaffine mapping. More explicitly $\P = f(\K)$ where $\K = \{k \in \R^n \: :\: \|k\|_{\infty} \leq 1\}$ and multiaffine mapping $f : \R^n \rightarrow \R^n$ is defined by $$\begin{array}{rcl}
f(k) & = &
\left[\begin{array}{ccccc}0&1&0&0&0\\0&0&1&0&0\\0&0&0&1&0\end{array}\right]
\left[\begin{array}{cccc}1&0&0&k_3\\0&1&k_3&0\\0&k_3&1&0\\k_3&0&0&1\\0&0&0&0\end{array}\right]
\left[\begin{array}{ccc}1&0&k_2\\0&1+k_2&0\\k_2&0&1\\0&0&0\end{array}\right]
\left[\begin{array}{cc}1&k_1\\k_1&1\\0&0\\\end{array}\right]
\left[\begin{array}{c}1\\0\end{array}\right] \\
& = &
\left[\begin{array}{c}k_2k_3+k_1(1+k_2)\\k_2+k_1k_3(1+k_2)\\k_3\end{array}\right]
\end{array}$$ in the case $n=3$. The general expression of $f$ for other values of $n$ is not given here for space reasons, but it follows readily from the construction outlined above.
Using this mapping we can obtain moments (\[momb\]) of $\B=\P$ analytically: $$\label{mom3}
y_{\alpha} = \int_{\mathbf P} x^{\alpha} dx = \int_{\mathbf K}
(k_2k_3+k_1(1+k_2))^{\alpha_1}
(k_2+k_1k_3(1+k_2))^{\alpha_2}
k_3^{\alpha_3}
\det \nabla f(k) dk$$ where $\det \nabla f(k) = (1+k_2)(1-k_3^2)$ is the determinant of the Jacobian of $f$, in the case $n=3$. For space reasons we do not give here the explicit value of $y_{\alpha}$ as a function of $\alpha$, but it can be obtained by integration by parts.
Finally, let us mention a well-known geometric property of $\P$: its convex hull is a polytope whose vertices correspond to the $n+1$ polynomials with roots equal to $-1$ or $+1$. For example, when $n=3$, we have $$\label{conv3}
\mathrm{conv}\,\P = \mathrm{conv}\{(-3,3,-1),\,(-1,-1,1),\,(1,-1,-1),\,(3,3,1)\}.$$
Third degree stability region
-----------------------------
![Two views of a degree two inner approximation (red) of nonconvex third-degree stability region (gray).\[fighermite3d2\]](hermite3d2a "fig:"){width="49.00000%"} ![Two views of a degree two inner approximation (red) of nonconvex third-degree stability region (gray).\[fighermite3d2\]](hermite3d2b "fig:"){width="49.00000%"}
![Two views of a degree four inner approximation (red) of nonconvex third-degree stability region (gray).\[fighermite3d4\]](hermite3d4a "fig:"){width="49.00000%"} ![Two views of a degree four inner approximation (red) of nonconvex third-degree stability region (gray).\[fighermite3d4\]](hermite3d4b "fig:"){width="49.00000%"}
![Two views of a degree six inner approximation (red) of nonconvex third-degree stability region (gray).\[fighermite3d6\]](hermite3d6a "fig:"){width="49.00000%"} ![Two views of a degree six inner approximation (red) of nonconvex third-degree stability region (gray).\[fighermite3d6\]](hermite3d6b "fig:"){width="49.00000%"}
Consider the problem of approximating from the inside the nonconvex stability region $\P$ of a discrete-time third degree polynomial $z \mapsto z^3+x_1z^2+x_2z+x_3$. An ellipsoidal inner approximation was proposed in [@hpas03]. The Hermite polynomial matrix defining $\P$ as in (\[setp\]) is given by $$P(x) = \left[\begin{array}{ccc}
1 - x_3^2 & x_1 - x_2x_3 & x_2 - x_1x_3 \\
x_1 - x_2x_3 & 1+x_1^2-x_2^2-x_3^2 & x_1-x_2x_3 \\
x_2 - x_1x_3 & x_1-x_2x_3 & 1-x_3^2
\end{array}\right].$$ The boundary of $\P$ consists of two triangles and a hyperbolic paraboloid. The convex hull of $\P$ is the simplex described in (\[conv3\]). We have analytic expressions (\[mom3\]) for the moments (\[momb\]) of $\B=\P$.
On Figures \[fighermite3d2\], \[fighermite3d4\] and \[fighermite3d6\] we respectively represent the degree two, four and six inner approximations of $\P$, scaled within the unit box for visualization purposes. We observe that the degree six approximation is very tight, thanks to the availability of the moments of the Lebesgue measure on $\P$.
Fixed-order controller design {#design}
-----------------------------
Consider the linear discrete-time system with characteristic polynomial $z \mapsto z^4-(2x_1+x_2)z^3+2x_1z+x_2$ depending affinely on two real design parameters $x_1$ and $x_2$. It follows from Hermite’s stability criterion that this polynomial has its roots in the open unit disk if and only if $$\small
P(x) = \left[\begin{array}{@{\;}c@{\;}c@{\;}c@{\;}c@{\;}}
1-x_2^2 & -2x_1-x_2-2x_1x_2 & 0 & 2x_1+2x_1x_2+x_2^2 \\
-2x_1-x_2-2x_1x_2 & 1+4x_1x_2 & -2x_1-x_2-2x_1x_2 & 0 \\
0 & -2x_1-x_2-2x_1x_2 & 1+4x_1x_2 & -2x_1-x_2-2x_1x_2 \\
2x_1+2x_1x_2+x_2^2 & 0 & -2x_1-x_2-2x_1x_2 & 1-x_2^2
\end{array}\right]$$ is positive definite. As recalled in (\[multiaffine\]), the convex hull of the four-dimensional stability domain of a degree four polynomial is the simplex with vertices $(-4,6,-4,1)$, $(-2,0,2,-1)$, $(0,-2,0,1)$, $(2,0,-2,-1)$, $(4,6,4,1)$ corresponding to the five polynomials with zeros equal to $-1$ or $+1$. Using elementary linear algebra, we find out that the image of this simplex through the affine mapping $(-(2x_1+x_2),0,2x_1,x_2)$ parametrized by $x \in \R^2$ is the two-dimensional simplex $$\B = \mathrm{conv}\{(-\frac{1}{4},1),\,(\frac{7}{8},-\frac{1}{2}),\,
(-\frac{5}{8},-\frac{1}{2})\}.$$ The (closure of the) stability region $\P = \{x \in \R^2 \: :\: P(x) \succeq 0\}$ is therefore included in $\B$, whose moments (\[momb\]) are readily obtained e.g. by the explicit formulas of [@la01].
![Degree two (left) and four (right) inner approximations (light gray) of PMI stability region (dark gray) embedded in simplex (dashed).\[fighermite4a\]](hermite4d2 "fig:"){width="49.00000%"} ![Degree two (left) and four (right) inner approximations (light gray) of PMI stability region (dark gray) embedded in simplex (dashed).\[fighermite4a\]](hermite4d4 "fig:"){width="49.00000%"}
![Degree six (left) and eight (right) inner approximations (light gray) of PMI stability region (dark gray) embedded in simplex (dashed).\[fighermite4b\]](hermite4d6 "fig:"){width="49.00000%"} ![Degree six (left) and eight (right) inner approximations (light gray) of PMI stability region (dark gray) embedded in simplex (dashed).\[fighermite4b\]](hermite4d8 "fig:"){width="49.00000%"}
On Figures \[fighermite4a\] and \[fighermite4b\] we represent the degree two, four, six and eight inner approximations to $\P$, corresponding to stability regions for the linear system. We observe that the approximations become tight rather quickly. This is due to the fact that $\B$ is a good outer approximation of $\P$ with known moments. Tighter outer approximations $\B$ would result in tighter inner approximations of $\P$, but then the moments of $\B$ can be hard to compute, see [@hls09].
![Degree two (left) and four (right) inner approximations (light gray) of robust PMI stability region (dark gray) embedded in simplex (dashed).\[fighermite4robusta\]](hermite4d2robust "fig:"){width="49.00000%"} ![Degree two (left) and four (right) inner approximations (light gray) of robust PMI stability region (dark gray) embedded in simplex (dashed).\[fighermite4robusta\]](hermite4d4robust "fig:"){width="49.00000%"}
Robust controller design
------------------------
Now consider the uncertain polynomial $z \mapsto x_2+u+2x_1z-(2x_1+x_2)z^3+z^4$ with $u \in {\mathbf U} = \{u \in {\mathbb R} \: :\:
u^2 \leq \frac{1}{16}\}$ with uncertain Hermite matrix $P(x,u)$ and the corresponding parametrized PMI stability region $\P$ in (\[setp\]). Let us use the same bounding set $\B$ as in §\[design\].
On Figure \[fighermite4robusta\] we represent the degree two and degree four inner approximations to $\mathbf P$, corresponding to robust stability regions for the linear system. Comparing with Figure \[fighermite4a\] we see that the approximations are smaller, and in particular they do not touch the stability boundary to cope with the robustness requirements.
Conclusion
==========
We have constructed a hierarchy of inner approximations of feasible sets defined by parametrized or uncertain polynomial matrix inequalities (PMI). Each inner approximation is computed by solving a convex linear matrix inequality (LMI) problem. The hierarchy converges in a well-defined analytic sense, so that conservatism of the approximation is guaranteed to vanish asymptotically. In addition, the inner approximations are simple polynomial or piecewise-polynomial superlevel sets, so that optimization over these sets is significantly simpler than optimization over the original parametrized PMI set. In particular, we remove the possibly complicated dependence of the problem data on the uncertain parameters.
One may also impose the hierarchy of inner approximations to be nested. Finally, one may also impose the inner approximations to be convex. In this latter case they do not converge any more to the feasible set but, on the other hand, optimization over the parametrized PMI set can be reformulated as a convex polynomial optimization problem (of course at the price of some conservatism). Ideally, beyond convexity, we may also want the inner convex approximation to be semidefinite representable (as an explicit affine projection of an affine section of the SDP cone), and deriving such a representation may be an interesting research direction.
The tradeoff to be found is between tightness of the inner approximation and degree of the defining polynomials. In the context of robust control design, a satisfactory inner approximation can be possibly computed off-line, and then used afterwards on-line in a feedback control setup.
Our methodology is valid for general parametrized PMI problems. However, in the case of parametrized PMI problems coming from fixed-order robust controller design problems, geometric insight can be exploited to improve convergence of the hierarchy. The key information is the knowledge of the moments of the Lebesgue measure on a compact set which tightly contains the parametrized PMI set we want to approximate from the inside. In turns out that for robust control problems this knowledge is available easily, as illustrated in the paper by several examples.
The main limitation of the approach lies in the ability of solving primal moment and dual polynomial sum-of-squares LMI problems. State-of-the-art general-purpose semidefinite programming solvers can currently address problems of relatively moderate dimensions, but problem structure and data sparsity can be exploited for larger problems.
Acknowledgements {#acknowledgements .unnumbered}
================
The first author acknowledges support by project No. 103/10/0628 of the Grant Agency of the Czech Republic. We are grateful to Luca Zaccarian for pointing out a mistake in a previous version of this paper.
Appendix {#technical}
========
Proof of Lemma \[lemma1\] {#proof-lemma1}
-------------------------
The dual of polynomial SOS SDP problem (\[sdp\]) is moment SDP problem (\[sdp\*\]). Slater’s condition cannot hold for (\[sdp\*\]) because $\V$ has empty interior in $\R^m$. However it turns out that Slater’s condition holds for an equivalent version of SDP problem (\[sdp\*\]), i.e., the latter has a strictly feasible solution $\hat{y}$. Indeed, let $J\subset\R[v]$ be the ideal generated by the polynomial $v\mapsto \theta(v):=1-v^Tv$ so that the real variety $V_\R(J):=\{v\in\R^m: \theta(v)=0\}$ associated with $J$ is just the unit sphere $\V$. It turns out that the real radical[^3] of $J$ is $J$ itself, that is, $I(V_\R(J))=J$ (where for $S\subset\R^m$, $I(S)$ denotes the vanishing ideal). And after embedding $J$ in $\R[x,u,v]$, we still have $I(V_\R(J))=J$.
Let $H:=\{(\alpha,\beta,\gamma)\in\N^n\times \N^p\times \N^m:\gamma_m\leq 1\}$, and let $H_d:=\{(\alpha,\beta,\gamma)\in H: \sum_i\alpha_i+\sum_j\beta_j+\sum_\ell \gamma_\ell\leq d\}$. The monomials $(x^\alpha u^\beta v^\gamma)$, $(\alpha,\beta,\gamma)\in H$, form a basis of the quotient space $\R[x,u,v]/J$. Moreover, for every $(\alpha,\beta,\gamma)\in \N^{n+p+m}_d$, $$x^{\alpha} u^{\beta} v^{\gamma}\,=\,\sum_{(a,b,c)\in H_{d}}p_{abc}\,x^au^{b} v^{c}+ \underbrace{h(x,u,v)}_{\in\R[x,u,v]_{d-2}}\,(1-v^Tv),$$ for some real coefficients $(p_{abc})$, and some $h\in\R[x,u,v]_{d-2}$. This is because every time one sees a monomial $x^a u^b v^c$ with $c_m\geq2$, one uses $v_m^2=1-\sum_{j\neq m}v_j^2$ to reduce this monomial modulo $\theta=(1-v^Tv)$. For instance $$\begin{aligned}
x^au^bv_1^{c_1}\cdots v_{m-1}^{c_{m-1}}v_m^3&=&x^au^bv_1^{c_1}\cdots v_{m-1}^{c_{m-1}}v_m\times
\underbrace{v_m^2}_{=-\theta+(1-\sum_{j\neq m}v_j^2)}\\
&=&x^au^bv_1^{c_1}\cdots v_{m-1}^{c_{m-1}}v_m
-\sum_{j\neq m}x^au^bv_1^{c_1}\cdots v_j^{c_j+2}\cdots v_{m-1}^{c_{m-1}}v_m\\
&&-\underbrace{x^au^bv_1^{c_1}\cdots v_{m-1}^{c_{m-1}}v_m}_{\in\R[x,u,v]_{d-2}}\,\theta(v),\end{aligned}$$ etc. Therefore, for every $(\alpha,\beta,\gamma),(\alpha',\beta',\gamma')
\in H_d$, $$\label{substitute}
x^{\alpha+\alpha'} u^{\beta+\beta'} v^{\gamma+\gamma'}\,=\,\sum_{(a,b,c)\in H_{2d}}
p_{abc}\, x^au^{b} v^{c}+ \underbrace{h(x,u,v)}_{\in\R[x,u,v]_{2d-2}}\,(1-v^Tv),$$ for some real coefficients $(p_{abc})$, and some $h\in\R[x,u,v]_{2d-2}$.
So because of the constraints $M_{d-1}((1-v^Tv)\,y)=0$, the semidefinite program (\[sdp\*\]) is equivalent to the semidefinite program: $$\label{sdp2*}
\begin{array}{rl}
\rho^*_d=\displaystyle\int_\B\lm(x)dx\,-\,\min_y &L_y(v^TP(x,u)v)\\
\mbox{s.t.}&\hat{M}_d(y)\succeq0,\:M_{d-1}((1-v^Tv)\,y)=0\\
&\hat{M}_{d-d_{a_i}}(a_i\,y)\succeq0,\quad i=0,1,\ldots,n_a\\
&\hat{M}_{d-d_{b_j}}(b_j\,y)\succeq0,\quad j=1,\ldots,n_b\\
&L_y(x^\alpha)\,=\,\int_\B x^\alpha\,dx,\quad\forall\alpha\in\N^n_{2d}
\end{array}$$ where the smaller moment matrix $\hat{M}_d(y)$ is the submatrix of $M_d(y)$ obtained by looking only at rows and columns indexed in the monomial basis $(x^\alpha y^\beta v^\gamma)$, $(\alpha,\beta,\gamma)\in H_d$, instead of $\N^{n+p+m}_d$. Similarly, the smaller localizing matrix $\hat{M}_{d-d_{a_i}}(a_i\,y)$ is the submatrix of $M_{d-d_{a_i}}(a_i\,y)$ obtained by looking only at rows and columns indexed in the monomial basis $(x^\alpha y^\beta v^\gamma)$, $(\alpha,\beta,\gamma)\in H_{d-d_{a_i}}$, instead of $\N^{n+p+m}_{d-d_{a_i}}$; and similarly for $\hat{M}_{d-d_{b_j}}(b_j\,y)$.
Indeed, in view of (\[substitute\]) and using $M_d((1-v^Tv)\,y)=0$, every column of $M_d(y)$ associated with $(\alpha,\beta,\gamma)\in\N^{n+p+m}$ is a linear combination of columns associated with $(\alpha',\beta',\gamma')\in H_d$. And similary for $M_{d-d_{a_i}}(a_i\,y)$ and $M_{d-d_{b_j}}(b_j\,y)$. Hence, $M_d(y)\succeq0\Leftrightarrow \hat{M}_d(y)\succeq0$, and $$M_{d-d_{a_i}}(a_i\,y)\succeq0\Leftrightarrow \hat{M}_{d-d_{a_i}}(a_i\,y)\succeq0;\quad
M_{d-d_{b_j}}(b_j\,y)\succeq0\Leftrightarrow \hat{M}_{d-d_{b_j}}(b_j\,y)\succeq0,$$ for all $i=1,\ldots,n_a$, $j=1,\ldots,n_b$.
Next, let $\hat{y}$ be the sequence of moments of the (product) measure $\mu$ uniformly distributed on $\B\times\U\times\V$, and scaled so that for all $(\alpha,\beta,\gamma)\in\N^{n+p+m}_{2d}$ $$\hat{y}_{\alpha\beta\gamma}\,=\,\int_{\B\times\U\times\V}x^\alpha\,u^\beta v^\gamma\,d\mu(x,u,v)\,=\,\frac{1}{{\rm vol}\,\U\times \V}
\int_\B\int_\U\int_\V x^\alpha\,u^\beta v^\gamma\,\underbrace{dx\,du\,d\lambda(v)}_{d\mu(x,u,v)}$$ (with $\lambda$ the rotation invariant measure on $\V$). Therefore, for every $\alpha\in\N^n_{2d}$, $$\hat{y}_{\alpha 00}\,=\,L_y(x^\alpha)\,=\,\int_{\B\times\U\times\V}x^\alpha\,d\mu(x,u,v)\,=\,\int_\B x^\alpha dx.$$ Moreover, $M_{d-1}((1-v^Tv)\,y)=0$ for every $d$ and importantly, $\hat{M}_d(\hat{y}) \succ 0$, $\hat{M}_{d-d_{a_i}}(a_i\,\hat{y})\succ0$ and $\hat{M}_{d-d_{b_j}}(b_j\,\hat{y})\succ0$. To see why, suppose for instance that $h^T\hat{M}_d(\hat{y})h=0$ for some vector $h\neq0$. This means that for some non trivial polynomial $h\in\R[x,u,v]/J$ of degree $d$, $$h^T\hat{M}_d(\hat{y})h\,=\,\int_{\B\times\U\times\V}h^2\,d\mu\,=\,0,$$ that is, $h(x,u,v)=0$ for $\mu$-almost all $(x,u,v)\in\B\times\U\times\V$, and so $h(x,u,v)=0$ for all $(x,u,v)\in\B\times\U\times\V$ because $h$ is continuous. But as $\B\times\U$ has nonempty interior in $\R^n\times\R^p$, then necessarily $h\in I(V_\R(J))\,(=J)$ – see Lemma \[lem-ideal\] in section \[ideal\] – which contradicts $0\neq h\in\R[x,u,v]/J$. Therefore $\hat{y}$ is a strictly feasible solution of (\[sdp2\*\]) and so Slater’s condition holds for (\[sdp2\*\]).
Denote by $\hat{\Sigma}[x,u,v]_d$ the space of polynomials of degree at most $2d$, that are SOS of polynomials in $\R[x,u,v]/J$. As $\hat{y}$ is a strictly feasible solution of the semidefinite program (\[sdp2\*\]), by a standard result of convex optimization, there is no duality gap between (\[sdp2\*\]) and its dual $$\label{sdp2}
\begin{array}{rl}
\rho'_d\,=\,\displaystyle\int_\B\lm(x)\,dx\,-\,\min_{g,r,s,t} & \displaystyle\int_\B g(x)\,dx\\[1em]
\mathrm{s.t.} & v^T P(x,u) v - g(x) \,=\, r(x,u,v) (1-v^T v) \\
& +\displaystyle\sum_{i=0}^{n_a} s_i(x,u,v) a_i(u)+\displaystyle\sum_{j=1}^{n_b} t_j(x,u,v) b_j(x) \quad\forall (x,u,v)\\
\end{array}$$ where now the decision variables are coefficients of polynomials $g\in\R[x]_{2d}$, $r\in\R[x,u,v]_{2d_r}$, and coefficients of SOS polynomials $s_i\in\hat{\Sigma}[x,u,v]_{d_{a_i}}$, $i=0,1,\ldots,n_a$, and $t_j\in\hat{\Sigma}[x,u,v]_{d_{b_j}}$, $j=1,\ldots,n_b$. That is, $\rho'_d=\rho_d^*$ and so $\rho_d=\rho^*_d$ because $\rho'_d\leq\rho_d\leq\rho^*_d$. If $\rho'_d<\infty$ then (\[sdp2\]) is guaranteed to have an optimal solution $(g^*,r^*,s^*,t^*)$. But observe that such an optimal solution $(g^*,r^*,s^*,t^*)$ is also feasible in (\[sdp\]), and so having value $\rho'_d=\rho_d=\rho^*_d$, $(g^*,r^*,s^*,t^*)$ is also an optimal solution of (\[sdp\]).
It remains to prove that $\rho_d$ is bounded. For any feasible solution $y$ of (\[sdp\]), $y_0\leq {\rm vol}\,\B$, and $$\label{bounds}
L_y(x_i^{2d})\,\leq\,R^{2d}y_0^d\,;\quad L_y(u_j^{2d})\,\leq\,R^{2d}y_0^d\,;\quad L_y(v_k^{2d})\,\leq\,y_0^d,$$ for all $i=1,\ldots,n$, $j=1,\ldots,p$, $k=1,\ldots,m$. This follows from $M_{d-d_{a_{i^*}}}(a_{i^*}\,y)\succeq0$, $M_{d-d_{b_{j^*}}}(b_{j^*}\,y)\succeq0$ and $M_{d-1}((1-v^Tv)\,y)=0$, where $a_{i^*}(x)\,=\,R^2-x^Tx$ and $b_{j^*}(x)\,=\,R^2-u^Tu$; see the comments after (\[setu\]) and (\[momb\]). Then by [@lasnetzer Lemma 4.3], one obtains $\vert y_\alpha\vert\leq R^{2d}({\rm vol}\,\B)^d$, for all $\alpha\in\N^n_{2d}$, which shows that the feasible set of (\[sdp\*\]) is compact. Hence (\[sdp\*\]) has an optimal solution and $\rho_d$ is finite; therefore its dual (\[sdp\]) also has an optimal solution, the desired result.
Proof of Theorem \[thmain\] {#proof-thmain}
---------------------------
\(a) Let $\K:=\B\times\U\times\V\subset\R^{n+p+m}$ and consider the infinite-dimensional optimization problem $$\label{momp}
\begin{array}{rcll}
\rho & = & \displaystyle\min_{\mu\in M(\K)} & \displaystyle\int_\K v^TP(x,u)v\, d\mu(x,u,v) \\
& & \mathrm{s.t.} & \displaystyle\int_\K x^\alpha d\mu\,=\,\int_\B x^\alpha\,dx,\quad\alpha\in\N^n
\end{array}$$ where $M(\K)$ is the space of finite Borel measures on $\K$. Problem (\[momp\]) has an optimal solution $\mu^*\in M(\K)$. Indeed, $\rho\geq\int_\B\lm(x) dx$ because for every $(x,u,v)\in\K$, $v^TP(x,u)v\geq \lm(x)$; and so for every feasible solution $\mu\in M(\K)$, $$\int_\K v^TP(x,u,v)v\,d\mu(x,u,v)\,\geq\,\int_\K\lm(x)\,d\mu(x,u,v)\,=\,\int_\B\lm(x)\,dx$$ because $\int_\K x^\alpha d\mu=\int_\B x^\alpha dx$ for all $\alpha\in\N$ and hence the marginal of $\mu$ on $\R^n$ is the Lebesgue measure on $\B$. On the other hand, observe that for every $x\in\B$, $\lm(x)=v_x^TP(x,u_x)v_x$ for some $(u_x,v_x)\in \U\times\V$. Therefore, let $\mu^*\in M(\K)$ be the Borel measure concentrated on $(x,u_x,v_x)$ for all $x\in\B$, i.e. $$\mu^*(\B'\times\U'\times\V')\,:=\,\int_{\B'\cap\U'}1_{\U'\times\V'}(u_x,v_x)\,dx,\qquad\forall
(\B',\U',\V')\in B(\B)\times B(\U)\times B(\V)$$ where $x\mapsto 1_\B(x)$ denotes the indicator function of set $\B$ and $B(\B)$ denotes the Borel $\sigma$-algebra of subsets of $\B$. Then $\mu^*$ is feasible for problem (\[momp\]) with value $$\int_\K v^TP(x,u)v\,d\mu^*(x,u,v)=\int_\B\lm(x)\,dx$$ which proves that $\rho=\int_\B\lm(x)\,dx$.
Next, $\lm$ being continuous on compact set $\B$, by the Stone-Weierstrass theorem [@ash §A7.5], for every $\varepsilon>0$ there exists a polynomial $h_\varepsilon\in\R[x]$ such that $$\sup_{x\in \B}\vert \lm(x)-h_\varepsilon(x)\vert<\frac{\varepsilon}{2}.$$ Hence the polynomial $p_\varepsilon:=h_\varepsilon -\varepsilon$ satisfies $\lm-p_\varepsilon>0$ on $\B$ and so $v^TP(x,u)v-p_\varepsilon >0$ on $\B\times\U\times\V$. By Putinar’s Positivstellensatz, see e.g [@l09 Section 2.5], there exists SOS polynomials $r_\varepsilon\in\R[x,u,v]$, and $s_{i\varepsilon},t_{j\varepsilon} \in\Sigma[x,u,v]$ such that equation (\[sdp\]) is satisfied. Hence for $d$ sufficiently large, say $d\geq d_\varepsilon$, $(p_\varepsilon,r_\varepsilon,s_{i\varepsilon},t_{j\varepsilon})$ is a feasible solution of (\[sdp\]) with associated value $$\int_\B(\lm(x)-p_\varepsilon(x))\,dx\,\leq\,\frac{3\varepsilon}{2}\int_\B dx.$$ Hence $0\leq\rho_d\leq \frac{3\varepsilon}{2}\int_B dx$ whenever $d\geq d_\varepsilon$ where $\rho_d$ is defined in (\[lem1-1\]). As $\varepsilon>0$ was arbitrary, we obtain the desired result $$\lim_{d\to\infty}\rho_d=0.$$ Observe that since $g_d\leq\lm$ for all $d$, $$\rho_d=\int_\B(\lm(x)-g_d(x))\,dx=\int_\B\vert \lm(x)-g_d(x)\vert\,dx$$ so that the convergence $\rho_d\to0$ is just the convergence $g_d\to\lm$ for the $L_1$ norm on $\B$. Finally the convergence $g_d\to\lm$ in Lebesgue measure on $\B$ follows from [@ash Theorem 2.5.1].
\(b) For each $x\in\B$, fixed and arbitrary, the sequence $(\bar{g}_d)$ is monotone nondecreasing and bounded above by $\lm$. Therefore there exists $g^*:\B\to\R$ such that for every $x\in\B$, $\bar{g}_d(x)\uparrow g^*(x)\leq \lm(x)$ as $d\to\infty$. Since $\bar{g}_d\geq\bar{g}_0$ and $\int_\B \bar{g}_0dx>-\infty$, by Lebesgue’s Dominated Convergence Theorem [@ash §1.6.9] $$\int_\B g^*(x)dx\,=\,\lim_{d\to\infty}\int_\B \bar{g}_d(x)dx\,=\,\int_\B\lm(x)dx,$$ and so from $g^*(x)\leq\lm(x)$ we deduce that $g^*(x)=\lm(x)$ for almost all $x\in\B$. Combining the latter with $\bar{g}_d\uparrow g^*$, we obtain that $\bar{g}_d\to \lm$ almost everywhere in $\B$. But then since the Lebesgue measure is finite on $\B$, by Egorov’s theorem [@ash Theorem 2.5.5], $\bar{g}_d\to \lm$ almost uniformly in $\B$. Finally, convergence in Lebesgue measure on $\B$ also follows from [@ash Theorem 2.5.2].
Proof of Corollary \[coro1\] {#proof-coro1}
----------------------------
By Theorem \[thmain\], $\lim_{d\to\infty}\Vert \lm-g_d\Vert_1=0$. Therefore, by [@ash Theorem 2.5.1] the sequence $(g_d)$ converges to $\lm$ in Lebesgue measure, i.e. for every $\varepsilon>0$, $$\label{aux}
\lim_{d\to\infty}
\vol\{x\,:\, \vert\lm(x)-g_d(x)\vert\,\geq\,\varepsilon\}=0.$$ Let $\varepsilon>0$ be fixed, arbitrary, and let $\P_\varepsilon:=\{x\in\B\,:\,\lm(x)\geq\varepsilon\}$, so that $\lim_{\varepsilon\to0}\vol\,\P_\varepsilon = \vol\,\P$. By (\[aux\]), $\lim_{d\to\infty}\vol(\P_\varepsilon\cap\{x\in\B\,:\,g_d(x)<0\})=0$. Next, for all $d\in\N$, $$\vol\,\P_\varepsilon\,=\,\vol(\P_\varepsilon\cap\{x\in\B\,:\,g_d(x)<0\})
+\vol(\P_\varepsilon\cap\{x\in\B\,:\,g_d(x)\geq0\}).$$ Therefore, taking the limit as $d\to\infty$ yields $$\begin{aligned}
\vol\,\P_\varepsilon&=&\underbrace{\lim_{d\to\infty}\vol(\P_\varepsilon\cap\{x\in\B\,:\,g_d(x)<0\})}_{=0\mbox{ by (\ref{aux})}}
+\lim_{d\to\infty}\vol(\P_\varepsilon\cap\underbrace{\{x\in\B\,:\,g_d(x)\geq0\}}_{=\G_d})\\
&=&\lim_{d\to\infty}\vol(\P_\varepsilon\cap\G_d)\,\leq\,\lim_{d\to\infty}\vol\,\G_d.\end{aligned}$$ As $\varepsilon>0$ was arbitrary and $\G_d\subset\P$, we obtain the desired result (\[coro1-1\]). The proof of (\[coro1-2\]) is similar.
Proof of Corollary \[coro2\] {#proof-coro2}
----------------------------
Let $0<\varepsilon<\frac{1}{3}$ be fixed, arbitrary. As in the proof of Theorem \[thmain\], for every $k\in\N$ there exists a polynomial $h_k\in\R[x]$ such that $\sup_{x\in \B}\vert \lm(x)-h_k(x)\vert<\varepsilon^k$. Hence for all $x\in\B$ and all $k\geq1$, $$\lm(x)-3\varepsilon^k< h_k(x)-2\varepsilon^k<\lm(x)-\varepsilon^k < \lm(x)-3\varepsilon^{k+1}<h_{k+1}(x)-2\varepsilon^{k+1}
<\lm(x)-\varepsilon^{k+1}$$ and so the polynomial $x\mapsto p_k(x):=h_k(x) -2\varepsilon^k$ satisfies $p_{k+1}(x) >p_{k}(x)$ and $\lm(x)>p_k(x)$ for all $x\in\B$. Again, by Putinar’s Positivstellensatz, see e.g [@l09 Section 2.5], $p_k$ is feasible for (\[sdp\]) with the additional constraint (\[add1\]), provided that $d$ is sufficiently large, and with associated value $$\int_\B\vert \lm(x)-p_k(x)\vert dx\,=\,\int_\B (\lm(x)-p_k(x))dx\,<3\varepsilon^k\int_\B dx\quad\to0\quad\mbox{as }k\to\infty.$$
An auxiliary result for the proof of Lemma \[lemma1\] {#ideal}
-----------------------------------------------------
Remember that $J\subset \R[v]$ is the ideal generated by $1-v^Tv$ and the real radical $I(V_\R(J))$ of $J$ is $J$ itself. And when $J$ is embedded in $\R[x,u,v]$ (with same name of simplicity) we also have $I(V_\R(J))=J$.
\[lem-ideal\] If $f\in\R[x,u,v]_d$ is such that $f(x,u,v)=0$ for all $(x,u,v)\in\B\times\U\times\V$ then $f\in J$.
Write $$f(x,u,v)\,=\,\sum_{\alpha\in\N^m_d}g_\alpha(x,u)\,v^\alpha,$$ for some polynomials $(g_\alpha)\subset\R[x,u]_d$, $\alpha\in\N^m_d$. Next, let $(x_0,u_0)\in\B\times\U$ be fixed, so that $v\mapsto f(x_0,u_0,v)=0$ for all $v\in\V$. Therefore, as a polynomial of $\R[v]$, it vanishes on $\V=V_\R(J)$ and as $I(V_\R(J))=J$, $v\mapsto f(x_0,u_0,v)\in J$, that is, $$\label{reduction}
f(x_0,u_0,v)\,=\,\sum_{\alpha\in\N^m_d}g_\alpha(x_0,u_0)\,v^\alpha\,=\,(1-v^Tv)\,\theta^{x_0,v_0}(v),$$ for some polynomial $v\mapsto\theta^{x_0,v_0}(v)\in\R[v]_{d}$. The coefficients ($\theta_\alpha^{x_0,u_0})$ of the polynomial $\theta^{x_0,u_0}(v)=\sum_\alpha\theta_\alpha^{x_0,u_0}v^\alpha$ are linear in the coefficients $(g_\beta(x_0,u_0))$, $\beta\in\N^m_d$, of $f$. Indeed one may reduce each monomial $v^\alpha$ using $v_m^{2}=1-\sum_{i\neq m}v_i^2$, until there is no monomial $v_m^\beta$ with $\beta>1$. For instance, $$v_1^{\alpha_1}\cdots v_{m-1}^{\alpha_{m-1}}v_m^{2}=v_1^{\alpha_1}\cdots v_{m-1}^{\alpha_{m-1}}(v^Tv-1)+v_1^{\alpha_1}\cdots v_{m-1}^{\alpha_{m-1}}-\sum_{j\neq m}v_1^{\alpha_1}\cdots v_j^{\alpha_j+2}\cdots v_{m-1}^{\alpha_{m-1}},$$ and $$v_1^{\alpha_1}\cdots v_{m-1}^{\alpha_{m-1}}v_m^{3}=v_1^{\alpha_1}\cdots v_{m-1}^{\alpha_{m-1}}v_m(v^Tv-1)+v_1^{\alpha_1}\cdots v_{m-1}^{\alpha_{m-1}}v_m-\sum_{j\neq m}v_1^{\alpha_1}\cdots v_j^{\alpha_j+2}\cdots v_{m-1}^{\alpha_{m-1}}v_m,$$ etc., to finally obtain $$v^\alpha=p_\alpha(v)(1-v^Tv)+r_\alpha,\qquad \forall \alpha\in \N^m_d,$$ for some $p_\alpha\in\R[v]_{\vert\alpha\vert-2}$ and $r_\alpha\in \R[v]/J$. Therefore, summing up over all $\alpha\in\N^m_d$ yields: $$\begin{aligned}
\nonumber
f(x_0,u_0,v)&=&\sum_{\alpha\in\N^m_d}g_\alpha(x_0,u_0)\,v^\alpha\\
\nonumber
&=&(v^Tv-1)\underbrace{\sum_{\alpha\in\N^m_d}g_\alpha(x_0,u_0)\,p_\alpha (v)}_{h(x_0,u_0,v)}
+\underbrace{\sum_{\alpha\in\N^m_d}g_\alpha(x_0,u_0)\,r_\alpha(v)}_{\mbox{$=0$ as $v\mapsto f(x_0,u_0,v)\in J$}}\\
\label{reduction3}
&=&(v^Tv-1)\,h(x_0,u_0,v),\quad\forall v\in\R^m,\end{aligned}$$ for some $h\in\R[x,u,v]$. But since (\[reduction3\]) holds for every $(x,u)\in\B\times\U$, we obtain $$f(x,u,v)\,=\,(v^Tv-1)\,h(x_0,u_0,v),\quad\forall (x,u,v)\in\B\times\U\times\R^m,$$ and as $\B\times\U\times\R^m$ has nonempty interior, $$f(x,u,v)\,=\,(v^Tv-1)\,h(x_0,u_0,v),\quad\forall (x,u,v)\in\R^n\times\R^p\times\R^m,$$ i.e., $f=(v^Tv-1)h$, which proves the desired result that $f\in J$.
[XX]{}
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[^1]: A basic semialgebraic set is a set defined by intersecting a finite number of polynomial superlevel sets.
[^2]: Note however that there is an incorrect factor $2^{-n}$ in the right handside of equation (3.3) in [@lz01].
[^3]: $\V$ is Zariski dense in $V_{\mathbb{C}}(J)\,(=\{v\in \mathbb{C}^n:\theta(v)=0\})$ so that $I(V_\R(J))=I(V_\mathbb{C}(J))$. But $\theta$ being irreducible, $J$ is a prime ideal and so $I(V_\mathbb{C}(J))=J$.
|
---
author:
- Catherine Pfaff
bibliography:
- 'BirecurrencyCondition.bib'
title: 'Ideal Whitehead Graphs in $Out(F_r)$ I: Some Unachieved Graphs'
---
Introduction
============
For a compact surface $S$, the *mapping class group $\mathcal{MCG}(S)$* is the group of isotopy classes of homeomorphisms $h \colon S\to S$. A generic (see, for example, [@m11]) mapping class is *pseudo-Anosov*, i.e. has a representative leaving invariant a pair of transverse measured singular minimal foliations. From the foliation comes a singularity index list. Masur and Smillie determined precisely which singularity index lists, permitted by the Poincare-Hopf index formula, arise from pseudo-Anosovs [@ms93]. The search for an analogous theorem in the setting of an outer automorphism group of a free group is still open.
We let $Out(F_r)$ denote the outer automorphism group of the free group of rank r. Analogous to pseudo-Anosov mapping classes are fully irreducible outer automorphisms, i.e. those such that no power leaves invariant the conjugacy class of a proper free factor. In fact, some fully irreducible outer automorphisms, called *geometrics*, are induced by pseudo-Anosovs. The index lists of geometrics are understood through the Masur-Smillie index theorem.
In [@gjll], Gaboriau, Jaeger, Levitt, and Lustig defined singularity indices for fully irreducible outer automorphisms. Additionally, they proved an $Out(F_r)$-analogue to the Poincare-Hopf index equality, namely the index sum inequality $i(\phi) \geq 1-r$ for a fully irreducible $\phi \in Out(F_r)$.
Having an inequality, instead of just an equality, makes the search for an analogue to the Masur-Smillie theorem richly more complicated. Toward this goal, Handel and Mosher asked in [@hm11]:
[\[Q:Q1\]]{} Which index types, satisfying $i(\phi) > 1-r$, are achieved by nongeometric fully irreducible $\phi \in Out(F_r)$?
There are several results on related questions. For example, [@jl09] gives examples of automorphisms with the maximal number of fixed points on $\partial F_r$, as dictated by a related inequality in [@gjll]. However, our work focuses on an $Out(F_r)$-version of the Masur-Smillie theorem. Hence, in this paper, in [@p12c], and in [@p12d] we restrict attention to fully irreducibles and the [@gjll] index inequality.
Beyond the existence of an inequality, instead of just an equality, “ideal Whitehead graphs” give yet another layer of complexity for fully irreducibles. An ideal Whitehead graph describes the structure of singular leaves, in analogue to the boundary curves of principle regions in Nielsen theory [@n86]. In the surface case, ideal Whitehead graphs are all circles. However, the ideal Whitehead graph $\mathcal{IW}(\phi)$ for a fully irreducible $\phi \in Out(F_r)$ (see [@hm11] or Definition \[D:whiteheadgraphs\] below) gives a strictly finer outer automorphism invariant than just the corresponding index list. Indeed, each connected component $C_i$ of $\mathcal{IW}(\phi)$ contributes the index $1-\frac{k_i}{2}$ to the list, where $C_i$ has $k_i$ vertices. One can see many complicated ideal Whitehead graph examples, including complete graphs in every rank (in [@p12c]) and in the eighteen of the twenty-one connected, five-vertex graphs achieved by fully irreducibles in rank-three ([@p12d]). The deeper, more appropriate question is thus:
[\[Q:Q2\]]{} Which isomorphism types of graphs occur as the ideal Whitehead graph $\mathcal{IW}(\phi)$ of a fully irreducible outer automorphism $\phi$?
[@p12d] will give a complete answer to Question \[Q:Q2\] in rank 3 for the single-element index list $(-\frac{3}{2})$. In Theorem \[T:MainTheorem\] of this paper we provide examples in each rank of connected (2r-1)-vertex graphs that are not the ideal Whitehead graph $\mathcal{IW}(\phi)$ for any fully irreducible $\phi \in Out(F_r)$, i.e. that are *unachieved* in rank r:
*For each $r \geq 3$, let $\mathcal{G}_r$ be the graph consisting of $2r-2$ edges adjoined at a single vertex.* \
A.
: *For no fully irreducible $\phi \in Out(F_r)$ is $\mathcal{IW}(\phi) \cong \mathcal{G}_r$.*\
B.
: *The following connected graphs are not the ideal Whitehead graph $\mathcal{IW}(\phi)$ for any fully irreducible $\phi \in Out(F_3)$:*\
{width="2.6in"}
Nongeometric fully irreducible outer automorphisms are either “ageometric” or “parageometric,” as defined by Lustig. Ageometric outer automorphisms are our focus, since the index sum for a parageometric, as is true for a geometric, satisfies the Poincare-Hopf equality [@gjll]. Parageometrics have been studied in papers including [@hm07]. In [@bf94], Bestvina and Feighn prove the [@gjll] index inequality is strict for ageometrics.
For a fully irreducible $\phi \in Out(F_r)$, to have the index list $(\frac{3}{2}-r)$, $\phi$ must be ageometric with a connected, (2r-1)-vertex ideal Whitehead graph $\mathcal{IW}(\phi)$. We chose to focus on the single-element index list $(\frac{3}{2}-r)$ because it is the closest to that achieved by geometrics, without being achieved by a geometric. We denote the set of connected (2r-1)-vertex, simplicial graphs by $\mathcal{PI}_{(r; (\frac{3}{2}-r))}$.
One often studies outer automorphisms via geometric representatives. Let $R_r$ be the $r$-petaled rose, with its fundamental group identified with $F_r$. For a finite graph $\Gamma$ with no valence-one vertices, a homotopy equivalence $R_r \to \Gamma$ is called a *marking*. Such a graph $\Gamma$, together with its marking $R_r \to \Gamma$, is called a *marked graph*. Each $\phi \in Out(F_r)$ can be represented by a homotopy equivalence $g\colon \Gamma \to \Gamma$ of a marked graph ($\phi= g_{*}\colon \pi_1(\Gamma) \to \pi_1(\Gamma)$). Thurston defined such a homotopy equivalence to be a *train track map* when $g^k$ is locally injective on edge interiors for each $k>0$. When $g$ induces $\phi \in Out(F_r)$ and sends vertices to vertices, one says $g$ is a *train track (tt) representative* for $\phi$ [@bh92].
To prove Theorem \[T:MainTheorem\]A, we give a necessary *Birecurrency Condition* (Proposition \[P:BC\]) on “lamination train track structures.” For a train track representative $g \colon \Gamma \to \Gamma$ on a marked rose, we define a *lamination train track (ltt) Structure* *$G(g)$* obtainable from $\Gamma$ by replacing the vertex $v$ with the “local Whitehead graph” $\mathcal{LW}(g; v)$. The local Whitehead graph encodes how lamination leaves enter and exit $v$. In our circumstance, $\mathcal{IW}(\phi)$ will be a subgraph of $\mathcal{LW}(g; v)$, hence of $G(g)$.
The lamination train track structure $G(g)$ is given a smooth structure so that leaves of the expanding lamination are realized as locally smoothly embedded lines. It is called *birecurrent* if it has a locally smoothly embedded line crossing each edge infinitely many times as $\bold{R}\to \infty$ and as $\bold{R}\to -\infty$.
**(Birecurrency Condition)** *The lamination train track structure for each train track representative of each fully irreducible outer automorphism $\phi \in Out(F_r)$ is birecurrent.*
Combinatorial proofs (not included here) of Theorem \[T:MainTheorem\]A exist. However, we include a proof using the Birecurrency Condition to highlight what we have observed to be a significant obstacle to achievability, namely the birecurrency of ltt structures. The Birecurrency Condition is also used in our proof of Theorem \[T:MainTheorem\]B. We use it in [@p12c], where we prove the achievability of the complete graph in each rank. Finally, the condition is used in [@p12d] to prove precisely which of the twenty-one connected, simplicial, five-vertex graphs are $\mathcal{IW}(\phi)$ for fully irreducible $\phi \in Out(F_3)$.
In Proposition \[P:IdealDecomposition\] we show that each $\phi$, such that $\mathcal{IW}(\phi) \in \mathcal{PI}_{(r;(\frac{3}{2}-r))}$, has a power $\phi^R$ with a rotationless representative whose Stallings fold decomposition (see Subsection \[SS:StallingsFoldDecompositions\]) consists entirely of proper full folds of roses (see Subsection \[SS:Folds\]). The representatives of Proposition \[P:IdealDecomposition\] are called “ideally decomposable.” We define in Section \[Ch:AMDiagrams\] automata, ideal “decomposition ($\mathcal{ID}$) diagrams” with ltt structures as nodes. Every ideally decomposed representative is realized by a loop in an $\mathcal{ID}$ diagram. To prove Theorem \[T:MainTheorem\]B we show ideally decomposed representatives cannot exist by showing that the $\mathcal{ID}$ diagrams do not have the correct kind of loops.
We again use the ideally decomposed representatives and $\mathcal{ID}$ diagrams in [@p12c] and [@p12d] to construct ideally decomposed representatives with particular ideal Whitehead graphs.
To determine the edges of the $\mathcal{ID}$ diagrams, we prove in Section \[Ch:AMProperties\] a list of “Admissible Map (AM) properties” held by ideal decompositions. In Section \[Ch:Peels\] we use the AM properties to determine the two geometric moves one applies to ltt structures in defining edges of the $\mathcal{ID}$ diagrams. The geometric moves turn out to have useful properties expanded upon in [@p12c] and [@p12d].
Acknowledgements {#acknowledgements .unnumbered}
----------------
The author would like to thank Lee Mosher for his truly invaluable conversations and Martin Lustig for his interest in her work. She also extends her gratitude to Bard College at Simon’s Rock and the CRM for their hospitality.
Preliminary definitions and notation
====================================
[\[Ch:PrelimDfns\]]{}
We continue with the introduction’s notation. ***Further we assume throughout this document that all representatives $g$ of $\phi \in Out(F_r)$ are train tracks (tts).***
We let $\mathcal{FI}_r$ denoted the subset of $Out(F_r)$ consisting of all fully irreducible elements.
**2.1. Directions and turns**
In general we use the definitions from [@bh92] and [@bfh00] when discussing train tracks. We give further definitions and notation here. $g: \Gamma \to \Gamma$ will represent $\phi \in Out(F_r)$.
$\mathcal{E}^+(\Gamma)= \{E_1, \dots, E_{n}\}= \{e_1, e_1, \dots, e_{2n-1}, e_{2n} \}$ will be the edge set of $\Gamma$ with some prescribed orientation. For $E \in \mathcal{E}^+(\Gamma)$, $\overline{E}$ will be $E$ oppositely oriented. $\mathcal{E}(\Gamma)$:$=\{E_1, \overline{E_1}, \dots, E_n, \overline{E_n} \}$. If the indexing $\{E_1, \dots, E_{n}\}$ of the edges (thus the indexing $\{e_1, e_1, \dots, e_{2n-1}, e_{2n} \}$) is prescribed, we call $\Gamma$ an *edge-indexed* graph. Edge-indexed graphs differing by an index-preserving homeomorphism will be considered equivalent.
$\mathcal{V}(\Gamma)$ will denote the vertex set of $\Gamma$ ($\mathcal{V}$, when $\Gamma$ is clear) and $\mathcal{D}(\Gamma)$ will denote $\underset{v \in \mathcal{V}(\Gamma)}{\cup} \mathcal{D}(v)$, where $\mathcal{D}(v)$ is the set of directions (germs of initial edge segments) at $v$.
For each $e \in \mathcal{E}(\Gamma)$, $D_0(e)$ will denote the initial direction of $e$ and $D_0 \gamma := D_0(e_1)$ for each path $\gamma=e_1 \dots e_k$ in $\Gamma$. *$Dg$* will denote the direction map induced by $g$. We call $d \in \mathcal{D}(\Gamma)$ *periodic* if $Dg^k(d)=d$ for some $k>0$ and *fixed* if $k=1$.
$Per(x)$ will consist of the periodic directions at an $x \in \Gamma$ and $Fix(x)$ of those fixed. $Fix(g)$ will denote the fixed point set for $g$.
$\mathcal{T}(v)$ will denote the set of turns (unordered pairs of directions) at a $v \in \mathcal{V}(\Gamma)$ and $D^tg$ the induced map of turns. For a path $\gamma=e_1e_2 \dots e_{k-1}e_k$ in $\Gamma$, we say $\gamma$ *contains (or crosses over)* the turn $\{\overline{e_i}, e_{i+1}\}$ for each $1 \leq i < k$. Sometimes we abusively write $\{\overline{e_i}, e_j\}$ for $\{D_0(\overline{e_i}), D_0(e_j)\}$. Recall that a turn is called *illegal* for $g$ if $Dg^k(d_i)=Dg^k(d_j)$ for some $k$ ($d_i$ and $d_j$ are in the same *gate*).
**2.2. Periodic Nielsen paths and ageometric outer automorphisms**
Recall [@bf94] that a *periodic Nielsen path (pNp)* is a nontrivial path $\rho$ between $x,y \in Fix(g)$ such that, for some $k$, $g^k(\rho) \simeq \rho$ rel endpoints (*Nielsen path (Np)* if $k=1$). In later sections we use [@gjll] that a $\phi \in \mathcal{FI}_r$ is ageometric if and only if some $\phi^k$ has a representative with no pNps (closed or otherwise). $\mathcal{AFI}_r$ will denote the subset of $\mathcal{FI}_r$ consisting precisely of its ageometric elements.
**2.3. Local Whitehead graphs, local stable Whitehead graphs, and ideal Whitehead graphs**[\[S:IWGs\]]{}
Please note that the ideal Whitehead graphs, local Whitehead graphs, and stable Whitehead graphs used here (defined in [@hm11]) differ from other Whitehead graphs in the literature. We clarify a difference. In general, Whitehead graphs record turns taken by immersions of 1-manifolds into graphs. In our case, the 1-manifold is a set of lines, the attracting lamination. In much of the literature the 1-manifolds are circuits representing conjugacy classes of free group elements. For example, for the Whitehead graphs of [@cv86], edge images are viewed as cyclic words. This is not true for ours.
The following can be found in [@hm11], though it is not their original source, and versions here are specialized. See [@p12a] for more extensive explanations of the definitions and their invariance. ***For this subsection $g: \Gamma \to \Gamma$ will be a pNp-free train track.***
[\[D:whiteheadgraphs\]]{} Let $\Gamma$ be a connected marked graph, $v \in \Gamma$, and $g: \Gamma \to \Gamma$ a representative of $\phi \in Out(F_r)$. The *local Whitehead graph* for $g$ at $v$ (denoted *$\mathcal{LW}(g; v)$*) has:
\(1) a vertex for each direction $d \in \mathcal{D}(v)$ and
\(2) edges connecting vertices for $d_1, d_2 \in \mathcal{D}(v)$ where $\{d_1, d_2 \}$ is taken by some $g^k(e)$, with $e \in \mathcal{E}(\Gamma)$.
The *local Stable Whitehead graph* $\mathcal{SW}(g; v)$ is the subgraph obtained by restricting precisely to vertices with labels in $Per(v)$. For a rose $\Gamma$ with vertex $v$, we denote the single local stable Whitehead graph $\mathcal{SW}(g; v)$ by $\mathcal{SW}(g)$ and the single local Whitehead graph $\mathcal{LW}(g; v)$ by $\mathcal{LW}(g)$.
For a pNp-free $g$, the *ideal Whitehead graph of $\phi$*, *$\mathcal{IW}(\phi)$*, is isomorphic to $\underset{\text{singularities v} \in \Gamma}{\bigsqcup} \mathcal{SW}(g;v)$, where a *singularity* for $g$ in $\Gamma$ is a vertex with at least three periodic directions. In particular, when $\Gamma$ is a rose, $\mathcal{IW}(\phi) \cong \mathcal{SW}(g)$.
[\[Ex:whiteheadgraphs\]]{} Let $g: \Gamma \to \Gamma$, where $\Gamma$ is a rose and $g$ is the train track such that the following describes the edge-path images of its edges: $$g =
\begin{cases}
a \mapsto abacbaba\bar{c}abacbaba \\
b \mapsto ba\bar{c} \\
c \mapsto c\bar{a}\bar{b}\bar{a}\bar{b}\bar{a}\bar{b}\bar{c}\bar{a}\bar{b}\bar{a}c
\end{cases}.$$
The vertices for $\mathcal{LW}(g)$ are $\{a, \bar a, b, \bar b, c, \bar c \}$ and the vertices of $\mathcal{SW}(g)$ are $\{a, \bar a, b, c, \bar c \}$: The periodic (actually fixed) directions for $g$ are $\{a, \bar a, b, c, \bar c \}$. $\bar b$ is not periodic since $Dg(\bar b)=c$, which is a fixed direction, meaning that $Dg^k(\bar b)=c$ for all $k \geq 1$, and thus $Dg^k(\bar{b})$ does NOT equal $\bar{b}$ for any $k \geq 1$.
The turns taken by the $g^k(E)$, for $E \in \mathcal{E}(\Gamma)$, are $\{a,\bar{b}\}$, $\{\bar{a},\bar{c}\}$, $\{b,\bar{a}\}$, $\{b,\bar{c}\}$, $\{c,\bar{a}\}$, and $\{a, c\}$. Since $\{a,\bar{b}\}$ contains the nonperiodic direction $\bar{b}$, this turn does not give an edge in $\mathcal{SW}(g)$, though does give an edge in $\mathcal{LW}(g)$. All other turns listed give edges in both $\mathcal{SW}(g)$ and $\mathcal{LW}(g)$.
$\mathcal{LW}(g)$ and $\mathcal{SW}(g)$ respectively look like (reasons for colors become clear in Subsection 2.4): \
{width="2in"} \[fig:IdealWhiteheadGraphs\]\
**2.4. Lamination train track structures**[\[SS:Realltts\]]{}
We define here “lamination train track (ltt) structures.” Bestvina, Feighn, and Handel discussed in their papers slightly different train track structures. However, those we define contain as smooth paths lamination (see [@bfh00]) leaf realizations. This makes them useful for deeming unachieved particular ideal Whitehead graphs and for constructing representatives (see [@p12c] and [@p12d]). ***Again, $g: \Gamma \to \Gamma$ will be a pNp-free train track on a marked rose with vertex $v$.***
The *colored local Whitehead graph $\mathcal{CW}(g)$ at $v$*, is $\mathcal{LW}(g)$, but with the subgraph $\mathcal{SW}(g)$ colored purple and $\mathcal{LW}(g)- \mathcal{SW}(g)$ colored red (nonperiodic direction vertices are red).
Let $\Gamma_N=\Gamma-N(v)$ where $N(v)$ is a contractible neighborhood of $v$. For each $E_i \in \mathcal{E}^+$, add vertices $d_i$ and $\overline{d_i}$ at the corresponding boundary points of the partial edge $E_i-(N(v) \cap E_i)$. A *lamination train track (ltt) Structure* *$G(g)$* for $g$ is formed from $\Gamma_N \bigsqcup \mathcal{CW}(g)$ by identifying the vertex $d_i$ in $\Gamma_N$ with the vertex $d_i$ in $\mathcal{CW}(g)$. Vertices for nonperiodic directions are red, edges of $\Gamma_N$ black, and all periodic vertices purple.
An ltt structure $G(g)$ is given a *smooth structure* via a partition of the edges at each vertex into two sets: $\mathcal{E}_b$ (containing the black edges of $G(g)$) and $\mathcal{E}_c$ (containing the colored edges of $G(g)$). A *smooth path* we will mean a path alternating between colored and black edges.
An edge connecting a vertex pair $\{d_i, d_j \}$ will be denoted \[$d_i, d_j$\], with interior ($d_i, d_j$). Additionally, $[e_i]$ will denote the black edge \[$d_i, \overline{d_i}$\] for $e_i \in \mathcal{E}(\Gamma)$.
For a smooth (possibly infinite) path $\gamma$ in $G(g)$, the *path (or line) in $\Gamma$ corresponding to $\gamma$* is $\dots e_{-j}e_{-j+1} \dots e_{-1}e_0e_1 \dots e_j \dots$, with $\gamma= \dots [d_{-j}, \overline{d_{-j}}][\overline{d_{-j}}, d_{-j+1}] \dots [d_0, \overline{d_0}][\overline{d_0}, d_1] \dots [d_j, \overline{d_j}] \dots,$ where each $d_i=D_0(e_i)$, each $[d_i, \overline{d_i}]$ is the black edge $[e_i]$, and each $[d_i, \overline{d_{i+1}}]$ is a colored edge. We denote such a path $\gamma= [\dots, d_{-j}, \overline{d_{-j}}, d_{-j+1}, \dots, \overline{d_{-1}}, d_0, \overline{d_0}, d_1, \dots, d_j, \overline{d_j} \dots].$
[\[Ex:G(g)\]]{} Let $g$ be as in Example \[Ex:whiteheadgraphs\]. The vertex $\bar b$ in $G(g)$ is red. All others are purple. $G(g)$ has a purple edge for each edge in $\mathcal{SW}(g)$ and a single red edge for the turn $\{a,\bar{b}\}$ (represented by an edge in $\mathcal{LW}(g)$, but not in $\mathcal{SW}(g)$). $\mathcal{CW}(g)$ is $\mathcal{LW}(g)$ with the coloring of Example \[Ex:whiteheadgraphs\]. And $G(g)$ is obtained from $\mathcal{CW}(g)$ by adding black edges connecting the vertex pairs $\{a,\bar{a}\}$, $\{b,\bar{b}\}$, and $\{c,\bar{c}\}$ (corresponding precisely to the edges $a, b,$ and $c$ of $\Gamma$). \
{width="1in"} \[fig:lttExample\]\
Once can check that each $g(e)$ is realized by a smooth path in $G(g)$.
If $\Gamma$ had more than one vertex, one could define $G(g)$ by creating a colored graph $\mathcal{CW}(g;v)$ for each vertex, removing an open neighborhood of each vertex when forming $\Gamma_N$, and then continuing with the identifications as above in $\Gamma_N \bigsqcup (\cup \mathcal{CW}(g;v))$.
Ideal decompositions
====================
[\[Ch:IdealDecompositions\]]{}
In this section we prove (Proposition \[P:IdealDecomposition\]): if $\mathcal{G} \in \mathcal{PI}_{(r;(\frac{3}{2}-r))}$ is $\mathcal{IW}(\phi)$ for a $\phi \in \mathcal{AFI}_r$, then $\phi$ has a rotationless power with a representative satisfying several nice properties, including that its Stallings fold decomposition consists entirely of proper full folds of roses. We call such a decomposition an *ideal decomposition*. Proving an ideal decomposition cannot exist will suffice to deem a $\mathcal{G}$ unachieved.
We remind the reader of definitions of folds and a Stallings fold decomposition before introducing ideal decompositions, as our Proposition \[P:IdealDecomposition\] proof relies heavily upon them.
Folds
-----
[\[SS:Folds\]]{}
Stallings introduced “folds” in [@s83] and Bestvina and Handel use several versions in their train track algorithm of [@bh92].
Let $g: \Gamma \to \Gamma$ be a homotopy equivalence of marked graphs. Suppose $g(e_1)=g(e_2)$ as edge paths, where $e_1, e_2 \in \mathcal{E}(\Gamma)$ emanate from a common vertex $v \in \mathcal{V} (\Gamma)$. One can obtain a graph $\Gamma_1$ by identifying $e_1$ and $e_2$ in such a way that $g:\Gamma \to \Gamma$ projects to $g_1: \Gamma_1 \to \Gamma_1$ under the quotient map induced by the identification of $e_1$ and $e_2$. $g_1$ is also a homotopy equivalence and one says $g_1$ and $\Gamma_1$ are obtained from $g$ by an *elementary fold* of $e_1$ and $e_2$.
To generalize one requires $e_1' \subset e_1$ and $e_2' \subset e_2$ only be maximal, initial, nontrivial subsegments of edges emanating from a common vertex such that $g(e_1')=g(e_2')$ as edge paths and such that the terminal endpoints of $e_1$ and $e_2$ are in $g^{-1}(\mathcal{V}(\Gamma))$. Possibly redefining $\Gamma$ to have vertices at the endpoints of $e_1'$ and $e_2'$, one can fold $e_1'$ and $e_2'$ as $e_1$ and $e_2$ were folded above. We say $g_1\colon\Gamma_1 \to \Gamma_1$ is obtained by \
- a *partial fold* of $e_1$ and $e_2$: if both $e_1'$ and $e_2'$ are proper subedges;\
- a *proper full fold* of $e_1$ and $e_2$: if only one of $e_1'$ and $e_2'$ is a proper subedge (the other a full edge);\
- an *improper full fold* of $e_1$ and $e_2$: if $e_1'$ and $e_2'$ are both full edges.
Stallings fold decompositions
-----------------------------
[\[SS:StallingsFoldDecompositions\]]{}
Stallings [@s83] also showed a tight homotopy equivalence of graphs is a composition of elementary folds and a final homeomorphism. We call such a decomposition a *Stallings fold decomposition*.
A description of a Stallings Fold Decomposition can be found in [@s89], where Skora described a Stallings fold decomposition for a $g\colon \Gamma \to \Gamma'$ as a sequence of folds performed continuously. Consider a lift $\tilde{g}\colon \tilde{\Gamma} \to \tilde{\Gamma}'$, where here $\tilde{\Gamma}'$ is given the path metric. Foliate $\tilde{\Gamma}$ x $\tilde{\Gamma}'$ with the leaves $\tilde{\Gamma}$ x $\{x'\}$ for $x' \in \Gamma'$. Define $N_t(\tilde{g})=\{(x,x') \in \tilde{\Gamma}$ x $\tilde{\Gamma}'$ $\vert$ $d(\tilde{g}(x),x') \leq t \}$. For each $t$, by restricting the foliation to $N_t$ and collapsing all leaf components, one obtains a tree $\Gamma_t$. Quotienting by the $F_r$-action, one sees the sequence of folds performed on the graphs below over time.
Alternatively, at an illegal turn for $g\colon \Gamma \to \Gamma$, fold maximal initial segments having the same image in $\tilde{\Gamma}'$ to obtain a map $g^1: \Gamma_1 \to \Gamma'$ of the quotient graph $\Gamma_1$. Repeat for $g^1$. If some $g^k$ has no illegal turn, it will be a homeomorphism and the fold sequence is complete. Using this description, we can assume only the final element of the decomposition is a homeomorphism. Thus, a Stallings fold decomposition of $g:\Gamma \to \Gamma$ can be written $\Gamma_0 \xrightarrow{g_1} \Gamma_1 \xrightarrow{g_2} \cdots \xrightarrow{g_{n-1}} \Gamma_{n-1} \xrightarrow{g_n} \Gamma_n$ where each $g_k$, with $1 \leq k \leq n-1$, is a fold and $g_n$ is a homeomorphism.
Ideal Decompositions
--------------------
[\[SS:Folds\]]{}
In this subsection we prove Proposition \[P:IdealDecomposition\]. For the proof, we need [@hm11]: For $\phi \in \mathcal{AFI}_r$ such that $\mathcal{IW}(\phi) \in \mathcal{PI}_{(r;(\frac{3}{2}-r))}$, $\phi$ is *rotationless* if and only if the vertices of $\mathcal{IW}(\phi)$ are fixed by the action of $\phi$. We also need that a representative $g$ of $\phi \in Out(F_r)$ is rotationless if and only if $\phi$ is rotationless. Finally, we need the following lemmas.
[\[L:pNpFreePreserved\]]{} Let $g \colon \Gamma \to \Gamma$ be a pNp-free tt representative of $\phi \in \mathcal{FI}_r$ and $\Gamma = \Gamma_0 \xrightarrow{g_1} \Gamma_1 \xrightarrow{g_2} \cdots \xrightarrow{g_{n-1}} \Gamma_{n-1} \xrightarrow{g_n} \Gamma_n = \Gamma$ a decomposition of $g$ into homotopy equivalences of marked graphs with no valence-one vertices. Then the composition $h \colon \Gamma_k \xrightarrow{g_{k+1}} \Gamma_{k+1} \xrightarrow{g_{k+2}} \cdots \xrightarrow{g_{k-1}} \Gamma_{k-1} \xrightarrow{g_k} \Gamma_k$ is also a pNp-free tt representative of $\phi$ (in particular, $\mathcal{IW}(h) \cong \mathcal{IW}(g)$).
Suppose $h$ had a pNp $\rho$ and $h^p(\rho) \simeq \rho$ rel endpoints. Let $\rho_1=g_n \circ \cdots \circ g_{k+1}(\rho)$. If $\rho_1$ were trivial, $h^p(\rho)=(g_k \circ \cdots \circ g_1 \circ g^{p-1})(g_n \circ \cdots \circ g_{k+1}(\rho))=(g_k \circ \cdots \circ g_1 \circ g^{p-1})(\rho_1)$ would be trivial, contradicting $\rho$ being a pNp. So assume $\rho_1$ is not trivial.
$g^p(\rho_1)=g^p((g_k \circ \cdots \circ g_1)(\rho))=(g_n \circ \cdots \circ g_{k+1}) \circ h^p(\rho)$. Now, $h^p(\rho) \simeq \rho$ rel endpoints and so $(g_n \circ \cdots \circ g_{k+1}) \circ h^p(\rho) \simeq (g_n \circ \cdots \circ g_{k+1})(\rho)$ rel endpoints. So $g^p(\rho_1)=g^p((g_k \circ \cdots \circ g_1)(\rho))=(g_n \circ \cdots \circ g_{k+1}) \circ h^p(\rho)$ is homotopic to $(g_n \circ \cdots \circ g_{k+1})(\rho)=\rho_1$ rel endpoints. This makes $\rho_1$ a pNp for $g$, contradicting that $g$ is pNp-free. Thus, $h$ is pNp-free.
Let $\pi\colon R_r \to \Gamma$ mark $\Gamma_1$. Since $g_1$ is a homotopy equivalence, $g_1 \circ \pi$ gives a marking on $\Gamma$. So $g$ and $h$ differ by a change of marking and thus represent the same outer automorphism $\phi$.
Finally, we show $h$ is a train track. For contradiction’s sake suppose $h(e)$ crossed an illegal turn $\{d_1, d_2 \}$. Since each $g_j$ is necessarily surjective, some $(g_k \circ \cdots \circ g_1)(e_i)$ would traverse $e$. So $(g_k \circ \cdots \circ g_1)(e_i)$ would cross $\{d_1, d_2 \}$. And $g^2(e_i)=(g_n \circ \cdots \circ g_{k+1}) \circ h \circ (g_k \circ \cdots \circ g_1)(e_i)$ would cross $\{D(g_n \circ \cdots \circ g_{k+1})(d_1), D(g_n \circ \cdots \circ g_{k+1})(d_2) \}$, which would either be illegal or degenerate (since $\{d_1, d_2 \}$ is an illegal turn). This would contradict that $g$ is a tt. So $h$ is a tt.
[\[L:GateCollapsing\]]{} Let $g: \Gamma \to \Gamma$ be a pNp-free tt representative of $\phi \in \mathcal{FI}_r$ with $2r-1$ fixed directions and Stallings fold decomposition $\Gamma_0 \xrightarrow{g_1} \Gamma_1 \xrightarrow{g_2} \cdots \xrightarrow{g_{n-1}} \Gamma_{n-1} \xrightarrow{g_n} \Gamma_n$. Let $g^i$ be such that $g=g^i \circ g_i \circ \cdots \circ g_1$. Let $d_{(1,1)}, \dots, d_{(1,2r-1)}$ be the fixed directions for $Dg$ and let $d_{j,k}=D(g_j \circ \cdots \circ g_1)(d_{1,k})$ for each $1 \leq j \leq n$ and $1 \leq k \leq 2r-1$. Then $D(g^i)$ is injective on $\{d_{(i,1)}, \dots, d_{(i,2r-1)}\}$.
Let $d_{(1,1)}, \dots, d_{(1,2r-1)}$ be the fixed directions for $Df$. If $D(g^i)$ identified any of $d_{(i,1)}, \dots, d_{(i,2r-1)}$, then $Df$ would have fewer than 2r-1 directions in its image.
[\[P:IdealDecomposition\]]{} Let $\phi \in Out(F_r)$ be an ageometric, fully irreducible outer automorphism whose ideal Whitehead graph $\mathcal{IW}(\phi)$ is a connected, (2r-1)-vertex graph. Then there exists a train track representative of a power $\psi=\phi^R$ of $\phi$ that is: \
1. on the rose,\
2. rotationless,\
3. pNp-free, and\
4. decomposable as a sequence of proper full folds of roses.
In fact, it decomposes as $\Gamma = \Gamma_0 \xrightarrow{g_1} \Gamma_1 \xrightarrow{g_2} \cdots \xrightarrow{g_{n-1}}
\Gamma_{n-1} \xrightarrow{g_n} \Gamma_n = \Gamma$, where: (I) the index set $\{1, \dots, n \}$ is viewed as the set $\mathbf {Z}$/$n \mathbf {Z}$ with its natural cyclic ordering; (II) each $\Gamma_k$ is an edge-indexed rose with an indexing $\{e_{(k,1)}, e_{(k,2)}, \dots, e_{(k,2r-1)}, e_{(k,2r)}\}$ where: \
Since $\phi \in \mathcal{AFI}_r$, there exists a pNp-free tt representative $g$ of a power of $\phi$. Let $h=g^k: \Gamma \to \Gamma$ be rotationless. Then $h$ is also a pNp-free tt representative of some $\phi^R$ and $h$ (and all powers of $h$) satisfy (2)-(3). Since $h$ has no pNps (meaning $\mathcal{IW}(\phi^R) \cong \underset{\text{singularities v} \in \Gamma}{\bigsqcup} \mathcal{SW}(h;v)$ and, if $\Gamma$ is the rose, $\mathcal{SW}(h) \cong \mathcal{IW}(\phi^R)$ ), since $h$ fixes all its periodic directions, and since $\mathcal{IW}(\phi)$ (hence $\mathcal{IW}(\phi^R)$) is in $\mathcal{PI}_{(r;(\frac{3}{2}-r))}$, $\Gamma$ must have a vertex with $2r-1$ fixed directions. Thus, $\Gamma$ must be one of: \
{width="3.3in"} \[fig:HigherRankGraphChoices\]\
If $\Gamma=A_1$, $h$ satisfies (3). We show, in this case, we also have the decomposition for (4). However, first we show $\Gamma$ cannot be $A_2$ or $A_3$ by ruling out all possibilities for folds in $h$’s Stallings decomposition.
If $\Gamma=A_2$, $v$ has to be the vertex with 2r-1 fixed directions. $h$ has an illegal turn unless it it is a homeomorphism, contradicting irreducibility. Note $w$ could not be mapped to $v$ in a way not forcing an illegal turn at $w$, as this would force either an illegal turn at $v$ (if $t$ were wrapped around some $b_i$) or we would have backtracking on $t$. Because all 2r-1 directions at $v$ are fixed by $h$, if $h$ had an illegal turn, it would have to occur at $w$ (no two fixed directions can share a gate).
The turns at $w$ are $\{a, \bar{a}\}$, $\{a, t\}$, and $\{\bar{a}, t\}$. By symmetry we only need to rule out illegal turns at $\{a, \bar{a}\}$ and $\{a, t\}$.
First, suppose $\{a, \bar{a}\}$ were illegal and the first fold in the Stallings decomposition. Fold $\{a, \bar{a}\}$ maximally to obtain $(A_2)_1$. Completely collapsing $a$ would change the homotopy type of $A_2$. \
{width="3.5in"} \[fig:RosewithStemPetalFold\]\
Let $h_1: (A_2)_1 \to (A_2)_1$ be the induced map of \[BH92\]. Since the fold of $\{a, \bar{a}\}$ was maximal, $\{a_1, \overline{a_1}\}$ must be legal. Since $h$ was a train track, $\{t_1, a_1\}$ and $\{t_1, \overline{a_1}\}$ would also be legal. But then $h_1$ would fix all directions at both vertices of $\Gamma_1$ (since it still would need to fix all directions at $v$). This would make $h_1$ a homeomorphism, again contradicting irreducibility. So $\{a, \bar{a}\}$ could not have been the first turn folded. We are left to rule out $\{a, t\}$.
Suppose the first turn folded in the Stallings decomposition were $\{a, t\}$. Fold $\{a, t\}$ maximally to obtain $(A_2)'_1$. Let $h_1'\colon (A_2)'_1 \to (A_2)'_1$ be the induced map of \[BH92\]. Either A. all of $t$ was folded with a full power of $a$; B. all of $t$ was folded with a partial power of $a$; or C. part of $t$ was folded with either a full or partial power of $a$.
If (A) or (B) held, $(A_2)_1'$ would be a rose and $h_1'$ would give a representative on the rose, returning us to the case of $A_1$. So we just need to analyze (C).
Consider first (C), i.e. suppose that part of $t$ is folded with either a full or partial power of $a$: \
{width="3.8in"} \[fig:RoseWithStemChoices2\]\
If $h=h^1 \circ g_1$, where $g_1$ is the single fold performed thus far, then $h^1$ could not identify any directions at $w'$: identifying $a_2$ and $t_2$ would lead to $h$ back-tracking on $t$; identifying $t_2$ and $\bar{a}$ would lead to $h$ back-tracking on $a$; and $h^1$ could not identify $t_2$ and $\overline{a_3}$ because the fold was maximal. But then all directions of $(A_2)_{1}'$ would be fixed by $h^1$, making $h^1$ a homeomorphism and the decomposition complete. However, this would make $h$ consist of the single fold $g_1$ and a homeomorphism, contradicting $h$’s irreducibility. Thus, all cases where $\Gamma = A_2$ are either impossible or yield the representative on the rose for (1).
Now assume $\Gamma=A_3$. $v$ must have $2r-1$ fixed directions. As with $A_2$, since $h$ must fix all directions at $v$, if $h$ had an illegal turn (which it still has to) it would be at $w$. Without losing generality assume $\{b, d\}$ is an illegal turn and that the first Stallings fold maximally folds $\{b, d\}$. Folding all of $b$ and $d$ would change the homotopy type. So assume (again without generality loss) either: \
- all of $b$ is folded with part of $d$ or\
- only proper initial segments of $b$ and $d$ are folded with each other.\
If all of $b$ is folded with part of $d$, we get a pNp-free tt on the rose. So suppose only proper initial segments of $b$ and $d$ are identified. Let $h_1\colon (A_3)_1 \to (A_3)_1$ be the [@bh92] induced map. \
{width="4.8in"} \
\[fig:TheFold\]\
The new vertex $w'$ has 3 distinct gates: $\{b', d'\}$ is legal since the fold was maximal and $\{b', \bar{e}\}$ and $\{d', \bar{e}\}$ must be legal or $h$ would have back-tracked on $b$ or $d$, respectively. This leaves that the entire decomposition is a single fold and a homeomorphism, again contradicting $h$’s irreducibility.
We have ruled out $A_3$ and proved for (1) that we have a pNp-free representative on the rose of some $\psi=\phi^R$. We now prove (4).
Let $h$ be the pNp-free tt representative of $\phi^R$ on the rose and $\Gamma_0 \xrightarrow{g_1} \Gamma_1 \xrightarrow{g_2} \cdots \xrightarrow{g_{n-1}} \Gamma_{n-1} \xrightarrow{g_n} \Gamma_n$ the Stallings decomposition. Each $g_i$ is either an elementary fold or locally injective (thus a homeomorphism). We can assume $g_n$ is the only homeomorphism. Let $h^i=g_n \circ \dots \circ g_{i+1}$. Since $h$ has precisely $2r-1$ gates, $h$ has precisely one illegal turn. We first determine what $g_1$ could be. $g_1$ cannot be a homeomorphism or $h=g_1$, making $h$ reducible. So $g_1$ must maximally fold the illegal turn. Suppose the fold is a proper full fold. (If it is not, see the analysis below of cases of improper or partial folds.) \
{width="3.8in"} \
\
By Lemma \[L:GateCollapsing\], $h^1$ can only have one turn $\{d_1, d_2\}$ where $Dh^1(\{d_1, d_2\})$ is degenerate (we call such a turn an *order-1 illegal turn* for $h^1$). If it has no order-1 illegal turn, $h^1$ is a homeomorphism and the decomposition is determined. So suppose $h^1$ has an order-1 illegal turn (with more than one, $h$ could not have 2r-1 distinct gates). The next Stallings fold must maximally fold this turn. With similar logic, we can continue as such until either $h$ is obtained, in which case the desired decomposition is found, or until the next fold is not a proper full fold. The next fold cannot be an improper full fold or the homotopy type would change. Suppose after the last proper full fold we have: \
{width="1.2in"} \[fig:AfterLastPFF\]\
Without losing generality, suppose the illegal turn is $\{a_j, \overline{a_j}\}$. Maximally folding $\{a_j, \overline{a_j}\}$ yields $A_2$, as above. This cannot be the final fold in the decomposition since $A_1$ is not homeomorphic to $A_2$. By Lemma \[L:pNpFreePreserved\], the illegal turn must be at $w$. The fold of Figure 3 cannot be performed, as our fold was maximal. If the fold of Figure \[fig:HigherRankRoseProperFullFold\] were performed, there would be backtracking on $a$.
Now suppose, without loss of generality, that the first Stallings fold that is not a proper full fold is a partial fold of $b'$ and $c'$, as in the following figure. \
{width="3in"} \
\[fig:ImproperFold\]\
As in the case of $\Gamma=A_3$ above, the next fold has to be at $w$ or the next generator would be a homeomorphism, contradicting that the image of $h$ is a rose, while $A_3$ is not a rose. Since the previous fold was maximal, the next fold cannot be of $\{b'', c''\}$. Also, $\{b'', \bar{d}\}$ and $\{c'', \bar{d}\}$ cannot be illegal turns or $h$ would have had edge backtracking. Thus, $h_i$ was not possible in the first place, meaning that all folds in the Stallings decomposition must be proper full folds between roses, proving (4).
Since all Stallings folds are proper full folds of roses, for each $1 \leq k \leq n-1$, one can index $\mathcal{E}_k = \mathcal{E}(\Gamma_k)$ as $\{E_{(k,1)},\overline{E_{(k,1)}}, E_{(k,2)}, \overline{E_{(k,2)}}, \dots, E_{(k,r)}, \overline{E_{(k,r)}} \} = \{e_{(k,1)}, e_{(k,2)}, \dots, e_{(k,2r-1)}, e_{(k,2r)}\}$ so that (a) $g_k\colon e_{k-1,j_k} \mapsto e_{k,i_k} e_{k,j_k}$ where $e_{k-1,j_k} \in \mathcal{E}_{k-1}$, $e_{k,i_k}, e_{k,j_k} \in \mathcal{E}_k$ and (b) $g_k(e_{k-1,i})=e_{k,i}$ for all $e_{k-1,i} \neq e_{k-1,j_k}^{\pm 1}$. Suppose we similarly index the directions $D(e_{k,i}) = d_{k,i}$.
Let $g_n=h'$ be the Stallings decomposition’s homeomorphism and suppose its edge index permutation were nontrivial. Some power $p$ of the permutation would be trivial. Replace $h$ by $h^p$, rewriting $h^p$’s decomposition as follows. Let $\sigma$ be the permutation defined by $h'(e_{n-1,i})= e_{n-1,\sigma(i)}$ for each $i$. For $n \leq k \leq 2n-p$, define $g_k$ by $g_k: e_{k-1,\sigma^{-s+1}(j_t)} \mapsto e_{k,\sigma^{-s+1}(i_t)} e_{k,\sigma^{-s+1}(j_t)}$ where $k=sp+t$ and $0 \leq t \leq p$. Adjust the corresponding proper full folds accordingly. This decomposition still gives $h^p$, but now the homeomorphism’s edge index permutation is trivial, making it unnecessary for the decomposition.
Representatives with a decomposition satisfying (I)-(II) of Proposition \[P:IdealDecomposition\] will be called and *ideally decomposable ($\mathcal{ID}$)* representative with an *ideal decomposition*.
[\[N:IdealDecompositions\]]{} **(Ideal Decompositions)** We will consider the notation of the proposition standard for an ideal decomposition. Additionally,
1. We denote $e_{k-1,j_k}$ by $e^{pu}_{k-1}$, denote $e_{k,j_k}$ by $e^u_k$, denote $e_{k,i_k}$ by $e^a_k$, and denote $e_{k-1,i_{k-1}}$ by $e^{pa}_{k-1}$.
2. $\mathcal{D}_k$ will denote the set of directions corresponding to $\mathcal{E}_k$.
3. $f_k:= g_k \circ \cdots \circ g_1 \circ g_n \circ \cdots \circ g_{k+1}: \Gamma_k \to \Gamma_k$.
4. $$g_{k,i}:=
\begin{cases}
g_k \circ \cdots \circ g_i\colon \Gamma_{i-1} \to \Gamma_k \text{ if $k>i$}\text{ and } \\
g_k \circ \cdots \circ g_1 \circ g_n \circ \cdots \circ g_i \text{ if $k<i$}
\end{cases}.$$
5. $d^u_k$ will denote $D_0(e^u_k)$, sometimes called the ***u**nachieved direction* for $g_k$, as it is not in $Im(Dg_k)$.
6. $d^a_k$ will denote $D_0(e^a_k)$, sometimes called the *twice-**a**chieved direction* for $g_k$, as it is the image of both $d^{pu}_{k-1}$ ($=D_0(e_{k-1,j_k})$) and $d^{pa}_{k-1}$ ($=D_0(e_{k-1,i_k})$) under $Dg_k$. $d^{pu}_{k-1}$ will sometimes be called the ***p**re-unachieved direction* for $g_k$ and $d^{pa}_{k-1}$ the ***p**re-twice-achieved direction* for $g_k$.
7. $G_k$ will denote the ltt structure $G(f_k)$
8. $G_{k,l}$ will denote the subgraph of $G_l$ containing \
9. For any $k,l$, we have a direction map $Dg_{k,l}$ and an induced map of turns $Dg_{k,l}^t$. The *induced map of ltt Structures* $Dg_{k,l}^T: G_{l-1} \mapsto G_k$ (which we show below exists) is such that \
[\[Ex:InducedMap\]]{} We describe an induced map of rose-based ltt structures for $g_2: x \mapsto xz$. \
{width="2.7in"}
10. $\mathcal{C}(G_k)$ will denote the subgraph of $G_k$, coming from $\mathcal{LW}(f_k)$ and containing all colored (red and purple) edges of $G_k$.
11. Sometimes we use $\mathcal{PI}(G_k)$ to denote the purple subgraph of $G_k$ coming from $\mathcal{SW}(f_k)$.
12. $Dg_{k,l}^C$ will denote the restriction (which we show below exists) to $\mathcal{C}(G_{l-1})$ of $Dg_{k,l}^T$.
13. If we additionally require $\phi \in \mathcal{AFI}_r$ and $\mathcal{IW}(\phi) \in \mathcal{PI}_{(r;(\frac{3}{2}-r))}$, then we will say $g$ *has $(r;(\frac{3}{2}-r))$ potential*. (By saying $g$ *has $(r;(\frac{3}{2}-r))$ potential*, it will be implicit that, not only is $\phi \in \mathcal{AFI}_r$, but $\phi$ is ideally decomposed, or at least $\mathcal{ID}$).
For typographical clarity, we sometimes put parantheses around subscripts. We refer to $E_{k,i}$ as $E_i$, and $\Gamma_k$ as $\Gamma$, for all $k$ when $k$ is clear.
Birecurrency Condition
======================
[\[S:bc\]]{}
Proposition \[P:BC\] of this section gives a necessary condition for an ideal Whitehead graph to be achieved. We use it to prove Theorem \[T:MainTheorem\]a, and implicitly throughout this paper and [@p12d].
Definitions of lines and the attracting lamination for a $\phi \in Out(F_r)$ will be as in [@bfh00]. A complete summary of relevant definitions can be found in [@p12a]. We use [@bfh00] that a $\phi \in \mathcal{FI}_r$ has a unique attracting lamination (we denote by $\Lambda_{\phi}$) and that attracting laminations contain birecurrent leaves.
Note that there is both notational and terminology variance in the name assigned to an attracting lamination. It is called a *stable lamination* in [@bfh97] and is sometimes also referred to in the literature as an *expanding lamination*. In [@bfh97] and [@bfh00], it is denoted $\Lambda^+_{\phi}$, or just $\Lambda^+$, while the authors of [@hm11] used the notation $\Lambda_-$, more consistent with dynamical systems terminology.
A *train track (tt) graph* is a finite graph $G$ satisfying: \
tt1:
: $G$ has no valence-1 vertices;\
tt2:
: each edge of $G$ has 2 distinct vertices (single edges are never loops); and\
tt3:
: the edge set of $G$ is partitioned into two subsets, $\mathcal{E}_b$ (the “black” edges) and $\mathcal{E}_c$ (the “colored” edges), such that each vertex is incident to at least one $E_b \in \mathcal{E}_b$ and at least one $E_c \in \mathcal{E}_c$.\
tt graphs are *equivalent* that are isomorphic as graphs via an isomorphism preserving the edge partition. And a path in a tt graph is *smooth* that alternates between edges in $\mathcal{E}_b$ and edges in $\mathcal{E}_c$.
The ltt structure $G(g)$ for a pNp-free representative $g$ on the rose is a train track graph where the black edges are in $\mathcal{E}_b$ and $\mathcal{E}_c$ is the edge set of $\mathcal{C}(G(g))$.
A smooth tt graph is *birecurrent* if it has a locally smoothly embedded line crossing each edge infinitely many times as $\bold{R}\to \infty$ and as $\bold{R}\to -\infty$.
[\[P:BC\]]{}**(Birecurrency Condition)** The lamination train track structure for each train track representative of each fully irreducible outer automorphism is birecurrent.
Our proof requires the following lemmas relating $\mathcal{LW}(g)$ and realization of leaves of $\Lambda_{\phi}$. The proofs use lamination facts from [@bfh97] and [@hm11].
[\[L:LeafTurns\]]{} Let $g: \Gamma \to \Gamma$, with $(r;(\frac{3}{2}-r))$ potential, represent $\phi \in Out(F_r)$. The only possible turns taken by the realization in $\Gamma$ of a leaf of $\Lambda_{\phi}$ are those giving edges in $\mathcal{LW}(g)$. Conversely, each turn represented by an edge of $\mathcal{LW}(g)$ is a turn taken by some (hence all) leaves of $\Lambda_{\phi}$ (as realized in $\Gamma$).
First note that, since $g$ is irreducible, each $E_i \in \mathcal{E}(\Gamma)$ has an interior fixed point. Thus, for each $E_i \in \mathcal{E}(\Gamma)$, there is a periodic leaf of $\Lambda_{\phi}$ obtained by iterating a neighborhood of a fixed point of $E_i$.
Consider any turn $\{d_1, d_2\}$ taken by the realization in $\Gamma$ of a leaf of $\Lambda_{\phi}$. Since periodic leaves are dense in the lamination, each periodic leaf of the lamination contains a subpath taking the turn. In particular, the leaf obtained by iterating a neighborhood of a fixed point of $e$ for any $e \in \mathcal{E}(\Gamma)$ takes the turn, so $\overline{e_1} e_2$ (where $D_0(e_1)=d_1$ or $D_0(e_2)=d_2$) is contained in some $g^k(e)$, for each $e \in \mathcal{E}(\Gamma)$. So $\{d_1, d_2\}$ is represented by an edge in $\mathcal{LW}(g)$, concluding the forward direction.
If $[d_1, d_2]$ is an edge of $\mathcal{LW}(g)$ then, for some $i$ and $k$, $\overline{e_1} e_2$ is a subpath of $g^k(E_i)$. Again, each $E_i \in \mathcal{E}(\Gamma)$ has an interior fixed point and hence $\Lambda_{\phi}$ has a periodic leaf obtained by iterating a neighborhood of $E_i$’s fixed point. $g^k(E_i)$ is a subpath of this periodic leaf and (by periodic leaf density) of every leaf of $\Lambda_{\phi}$. Since the leaves contain $g^k(E_i)$ as a subpath, they contain $\overline{e_1} e_2$ as a subpath, so $\{d_1, d_2\}$.
[\[L:SmoothPathsforLeaves\]]{} Let $g\colon\Gamma \to \Gamma$ represent $\phi \in \mathcal{AFI}_r$. Then $G(g)$ contains a smooth path corresponding to the realization in $\Gamma$ of each leaf of $\Lambda_{\phi}$.
Consider the realization $\lambda$ of a leaf of $\Lambda_{\phi}$ and any single subpath $\sigma=e_1 e_2 e_3$ in $\lambda$. If it exists, the representation in $G(g)$ of $\sigma$ would be the path $[d_1, \overline{d_1}, d_2, \overline{d_2}, d_3, \overline{d_3}]$. Lemma \[L:LeafTurns\] tells us $[\overline{d}_1, d_2]$ and $[\overline{d_2}, d_3]$ are edges of $\mathcal{LW}(g)$, hence are in $\mathcal{C}(G(g))$. The path representing $\sigma$ in $G(g)$ thus exists and alternates between colored and black edges. Analyzing overlapping subpaths to verifies smoothness.
We show that the path $\gamma$ corresponding to the realization $\lambda$ of a leaf of $\Lambda_{\phi}$ is a locally smoothly embedded line in $G(g)$ traversing each edge of $G(g)$ infinitely many times as $\bold{R}\to \infty$ and as $\bold{R}\to -\infty$. By Lemma \[L:LeafTurns\], for any colored edge \[$d_i, d_j$\] in $G(g)$, $\lambda$ must contain either $\overline{e_i} e_j$ or $\overline{e_2} e_1$ as a subpath. Fully irreducible outer automorphism lamination leaf birecurrency implies $\gamma$ must traverse the subpath $\overline{e_i} e_j$ or $\overline{e_j} e_i$ infinitely many times as $\bold{R}\to \infty$ and as $\bold{R}\to -\infty$. By Lemma \[L:SmoothPathsforLeaves\], this concludes the proof for a colored edge. Consider a black edge $[d_l, \overline{d_l}]=[e_l]$. Each vertex is shared with a colored edge. Let \[$d_l, \overline{d_m}$\] be such an edge. As shown above, $\overline{e_l} e_m$ or $\overline{e_m} e_l$ occur in a realization $\lambda$ infinitely many times as $\bold{R}\to \infty$ and as $\bold{R}\to -\infty$. So $\lambda$ traverses $e_l$ infinitely many times as $\bold{R}\to \infty$ and as $\bold{R}\to -\infty$. Thus $\gamma$ traverses $[e_l]$ infinitely many times as $\bold{R}\to \infty$ and as $\bold{R}\to -\infty$.
Admissible map properties
=========================
[\[Ch:AMProperties\]]{}
We prove that the ideal decomposition of a $(r;(\frac{3}{2}-r))$ potential representative satisfies “Admissible Map Properties” listed in Proposition \[P:am\]. In Section \[Ch:Peels\] we use the properties to show there are only two possible (fold/peel) relationship types between adjacent ltt structures in an ideal decomposition. Using this, in Section \[Ch:AMDiagrams\], we define the “ideal decomposition diagram” for $\mathcal{G} \in \mathcal{PI}_{(r;(\frac{3}{2}-r))}$.
The statement of Proposition \[P:am\] comes at the start of this section, while its proof comes after a sequence of technical lemmas used in the proof.
***$g:\Gamma \to \Gamma$ will represent $\phi\in Out(F_r)$, have $(r;(\frac{3}{2}-r))$ potential, and be ideally decomposed as: $\Gamma = \Gamma_0 \xrightarrow{g_1} \Gamma_1 \xrightarrow{g_2} \cdots \xrightarrow{g_{n-1}}\Gamma_{n-1} \xrightarrow{g_n} \Gamma_n = \Gamma$. We use the standard \[N:IdealDecompositions\] notation.***
[\[P:am\]]{} $g$ satisfies each of the following.
AM Property I:
: Each $G_j$ is birecurrent.\
AM Property II:
: For each $G_j$, the illegal turn $T_j$ for the generator $g_{j+1}$ exiting $G_j$ contains the unachieved direction $d^u_j$ for the generator $g_j$ entering $G_j$, i.e. either $d^u_j=d^{pa}_j$ or $d^u_j=d^{pu}_j$.\
AM Property III:
: In each $G_j$, the vertex labeled $d^u_j$ and edge $[t^R_j]=[d^u_j, \overline{d^a_j}]$ are both red.\
AM Property IV:
: If $[d_{(j,i)}, d_{(j,l)}]$ is in $C(G_j)$, then $D^Cg_{m,j+1}$(\[$d_{(j,i)}, d_{(j,l)}$\]) is a purple edge in $G_m$, for each $m \neq j$.\
AM Property V:
: For each $j$, $[t^R_j]=[d^u_j, \overline{d^a_j}]$ is the unique edge containing $d^u_j$.\
AM Property VI:
: Each $g_j$ is defined by $g_j:e^{pu}_{j-1} \mapsto e^a_j e^u_j$ (where $D_0(e^u_j)=d^u_j$, $D_0(\overline{e^a_j})=\overline{d^a_j}$, $e^u_j=e_{j,m}$, and $e^{pu}_{j-1}=e_{j-1,m}$).\
AM Property VII:
: $Dg_{l,j+1}$ induces an isomorphism from $SW(f_j)$ onto $SW(f_l)$ for all $j \neq l$.\
AM Property VIII:
: For each $1 \leq j \leq r$: \
- there exists a $k$ such that either $e^u_k=E_{k,j}$ or $e^u_k= \overline{E_{k,j}}$ and\
- there exists a $k$ such that either $e^a_k=E_{k,j}$ or $e^a_k= \overline{E_{k,j}}$.
The proof of Proposition \[P:am\] will come at the end of this subsection.
An edge path $\gamma=e_1 \dots e_k$ in $\Gamma$ has *cancellation* if $\overline{e_i} =e_{i+1}$ for some $1 \leq i \leq k-1$. We say $g$ has *no cancellation on edges* if for no $l>0$ and edge $e \in \mathcal{E}(\Gamma)$ does $g^l(e)$ have cancellation.
[\[L:PreLemma\]]{} For this lemma we index the generators in the decomposition of all powers $g^p$ of $g$ so that $g^p=g_{pn} \circ g_{pn-1} \circ \dots \circ g_{(p-1)n} \circ \dots \circ g_{(p-2)n} \circ \dots \circ g_{n+1} \circ g_n \circ \dots \circ g_1$ ($g_{mn+i}=g_i$, but we want to use the indices to keep track of a generator’s place in the decomposition of $g^p$). With this notation, $g_{k,l}$ will mean $g_k \circ \dots \circ g_l$. Then: 1. for each $e \in \mathcal{E}(\Gamma_{l-1}$), no $g_{k,l}(e)$ has cancellation; 2. for each $0 \leq l \leq k$ and $E_{l-1,i} \in \mathcal{E}^+(\Gamma_{l-1})$, the edge $E_{k,i}$ is in the path $g_{k,l} (E_{l-1,i})$; and 3. if $e^u_k=e_{k,j}$, then the turn $\{\overline{d^a_k}, d^u_k \}$ is in the edge path $g_{k,l}(e_{l-1,j})$, for all $0 \leq l \leq k$.
Let $s$ be minimal so that some $g_{s,t} (e_{t-1,j})$ has cancellation. Before continuing with our proof of (1), we first proceed by induction on $k-l$ to show that (2) holds for $k<s$. For the base case observe that $g_{l+1}(e_{l,j})=e_{l+1,j}$ for all $e_{l+1,j} \neq (e^{pu}_l)^{\pm 1}$. Thus, if $e_{l,j} \neq e^{pu}_l$ and $e_{l,j} \neq \overline{e^{pu}_l}$ then $g_{l+1}(e_{l,j})$ is precisely the path $e_{l+1,j}$ and so we are only left for the base case to consider when $e_{l,j} = (e^{pu}_l)^{\pm 1}$. If $e_{l,j} = e^{pu}_l$, then $g_{l+1}(e_{l,j})=e^a_{l+1} e_{l+1,j}$ and so the edge path $g_{l+1}(e_{l,j})$ contains $e_{l+1,j}$, as desired. If $e_{l,j} =\overline{e^{pu}_l}$, then $g_{l+1}(e_{l,j})=e_{l+1,j} \overline{e^a_{l+1}}$ and so the edge path $g_{l+1}(e_{l,j})$ also contains $e_{l+1,j}$ in this case. Having considered all possibilities, the base case is proved.
For the inductive step, we assume $g_{k-1,l+1} (e_{l,j})$ contains $e_{k-1,j}$ and show $e_{k,j}$ is in the path $g_{k,l+1}(e_{l,j})$. Let $g_{k-1,l+1}(e_{l,j})= e_{i_1}\dots e_{i_{q-1}} e_{k-1,j} e_{i_{q+1}}\dots e_{i_r}$ for some edges $e_i \in \mathcal{E}_{k-1}$. As in the base case, for all $e_{k-1,j} \neq (e^u_k)^{\pm 1}$, $g_k(e_{k-1,j})$ is precisely the path $e_{k,j}$. Thus (since $g_k$ is an automorphism and since there is no cancellation in $g_ {j_1,j_2}(e_{j_1,j_2})$ for $1 \leq j_1 \leq j_2 \leq k$), $g_{k,l+1} (e_{l,j})=\gamma_1 \dots \gamma_{q-1} (e_{k,j}) \gamma_{q+1} \dots \gamma_m$ where each $\gamma_{i_j}= g_l({e_{i_j}})$ and where no $\{\overline{\gamma_i}, \gamma_{i+1} \}$, $\{\overline{e_{k,j}}, \gamma_{q+1} \}$, or $\{\overline{\gamma_{q-1}}, e_{k,j} \}$ is an illegal turn. So each $e_{k,j}$ is in $g_{k,l+1} (e_{l,j})$. We are only left to consider for the inductive step the cases $e_{k-1,j}= e^{pu}_k$ and $e_{k-1,j} =\overline {e^{pu}_k}$.
If $e_{k-1,j} = e^{pu}_k$, then $g_k(e_{k-1,j})=e^a_k e_{k,j}$, and so $g_{k,l+1}(e_{l,j}) = \gamma_1 \dots \gamma_{q-1} e^a_k e_{k,j} \gamma_{q+1} \dots \gamma_m$ (where no $\{\overline{\gamma_i}, \gamma_{i+1} \}$, $\{\overline{e_{k,j}}, \gamma_{q+1} \}$, or $\{\overline{\gamma_{q-1}}, e^a_k \}$ is an illegal turn), which contains $e_{k,j}$, as desired. If instead $e_{k-1,j} =\overline {e^{pu}_k}$, then $g_k(e_{k-1,j})=e_{k,j} \overline{e^a_k}$ and so $g_{k,l+1}(e_{l,j})=\gamma_1 \dots \gamma_{q-1} e_{k,j} \overline{e^a_k} \gamma_{q+1} \dots\gamma_m$, which also contains $e_{k,j}$. Having considered all possibilities, the inductive step is now also proven and the proof is complete for (2) in the case of $k<s$.
We finish the proof of (1). $s$ is still minimal. So $g_{s,t}(e_{t-1,j})$ has cancellation for some $e_{t-1,j} \in \mathcal{E}_j$. Suppose $g_{s,t}(e_{t-1,j})$ has cancellation. For $1 \leq j \leq m$, let $\alpha_j \in \mathcal{E}_{s-1}$ be such that $g_{s-1,t}(e_{t-1,j})= \alpha_1 \cdots \alpha_m$. By $s$’s minimality, either $g_s(\alpha_i)$ has cancellation for some $1 \leq i \leq m$ or $Dg_s(\overline{\alpha_i})=Dg_s(\alpha_{i+1})$ for some $1 \leq i < m$. Since each $g_s$ is a generator, no $g_s(\alpha_i)$ has cancellation. So, for some $i$, $Dg_s(\overline{\alpha_i})= Dg_s(\alpha_{i+1})$. As we have proved (1) for all $k<s$, we know $g_{t-1,1}(e_{0,j})$ contains $e_{t-1,j}$. So $g_{s,1}(e_{0,j})= g_{s,t}(g_{t-1,1}(e_{0,j}))$ contains cancellation, implying $g^p(e_{0,j})= g_{pn,s+1}(g_{s,1}(e_{0,j}))= g_{s,t}(\dots e_{t-1,j} \dots)$ for some $p$ (with $pn>s+1$) contains cancellation, contradicting that $g$ is a train track.
We now prove (3). Let $e^u_k=e_{k,l}$. By (2) we know that the edge path $g_{k-1,l}(e_{l-1,j})$ contains $e_{k-1,j}$. Let $e_1, \dots e_m \in \mathcal{E}_{k-1}$ be such that $g_{k-1,l}(e_{l-1,j})=e_1 \dots e_{q-1} e_{k-1,j} e_{q+1} \dots e_m$. Then $g_{k,l}(e_{l-1,j})=\gamma_1 \dots \gamma_{q-1} e^a_k e^u_k \gamma_{q+1} \dots \gamma_r$ where $\gamma_j=g_k(e_j)$ for all $j$. Thus $g_{k,l}(e^{pu}_{k-1})$ contains $\{ \overline{d^a_k}, d^u_k \}$, as desired.
[\[L:fk\]]{} (Properties of $f_k= g_k \circ g_{k-1} \circ \cdots \circ g_{k+2} \circ g_{k+1}\colon\Gamma_k\to\Gamma_k$) \
a.
: Each $f_k$ represents the same $\phi$. In particular, if $g$ has $(r;(\frac{3}{2}-r))$ potential, then so does each $f_k$.\
b.
: Each $f_k$ is rotationless. In particular, all periodic directions are fixed.\
c.
: Each $f_k$ has 2r-1 gates (and thus periodic directions).\
d.
: For each $k$, $d^u_k \notin \mathcal{IM}(Df_k)$. Thus, $d^u_k$ is the unique nonperiodic (in fact nonfixed) direction for $Df_k$.\
e.
: If $\Gamma = \Gamma_0 \xrightarrow{g_1} \Gamma_1 \xrightarrow{g_2} \cdots \xrightarrow{g_{n-1}}\Gamma_{n-1} \xrightarrow{g_n} \Gamma_n = \Gamma$ is an ideal decomposition of $g$, then $\Gamma_k \xrightarrow{g_{k+1}} \Gamma_{k+1} \xrightarrow{g_{k+2}} \cdots \xrightarrow{g_{k-1}} \Gamma_{k-1} \xrightarrow{g_k} \Gamma_k$ is an ideal decomposition of $f_k$.
Lemma \[L:pNpFreePreserved\] implies (a). Each $f_k$ is rotationless, as it represents a rotationless $\phi$. This gives (b). We prove (c). The number of gates is the number of periodic directions, which here (by (b)) is the number of fixed directions. $f_k$ is on the rose, so has a single local Stable Whitehead graph. Lemma \[L:pNpFreePreserved\] implies $f_k$, as $g$, has no pNps. So $\mathcal{SW}(f_k) \cong \mathcal{IW}(\phi)$, which has 2r-1 vertices. So $f_k$ has 2r-1 periodic directions, thus gates. We prove (d). By (b) and (c), $Df_k$ has 2r-1 fixed directions. Since $d^u_k \notin \mathcal{IM}(Dg_k)$, it cannot be in $\mathcal{IM}(Df_k)$, so is the unique nonfixed direction. We prove (e). Ideal decomposition properties (I)-(IIb) hold for $f_k$’s decomposition, as they hold for $g$’s decomposition and the decompositions have the same $\Gamma_i$ and $g_i$ (renumbered). (IIc) holds for $f_k$’s decomposition by (d).
***We add to the notation already established: $t^R_k=\{\overline{d^a_k},d^u_k\}$, $e^R_k=[t^R_k]$, and $T_k=\{d^{pa}_k, d^{pu}_k \}$.***
[\[L:IllegalTurn\]]{} The following hold for each $T_k=\{d^{pa}_k, d^{pu}_k \}$. \
a.
: $T_k$ is an illegal turn for $g_{k+1}$ and, thus, also for $f_k$.\
b.
: For each $k$, $T_k$ contains $d^{u}_{k}$.\
Recall that $T_k=\{d^{pa}_k, d^{pu}_k \}$. Since $D^tg_{k+1}(\{d^{pa}_k, d^{pu}_k \})=\{Dg_{k+1}(d^{pa}_k), Dg_k(d^{pu}_k) \}= \{d^a_{k+1}, d^a_{k+1} \}$, $D^tf_k(\{d^{pa}_k, d^{pu}_k\})=D^t(g_{k,k+2} \circ g_{k+1}) (\{d^{pa}_k, d^{pu}_k\})= D^t(g_{k,k+2}) (D^tg_{k+1} (\{d^{pa}_k, d^{pu}_k\}))= D^tg_{k,k+2} (\{d^a_{k+1}, d^a_{k+1}\})$ $=\{D^tg_{k,k+2}(d^a_{k+1}), D^tg_{k,k+2}(d^a_{k+1})\}$, which is degenerate. So $T_k$ is an illegal turn for $f_k$, proving (a).
For (b) suppose $g$ has $2r-1$ periodic directions and, for contradiction’s sake, the illegal turn $T_k$ does not contain $d^u_k=d_{k,i}$. Let $d^u_{k+1}=d_{k+1,s}$ and $d^a_{k+1}=d_{k+1,t}$. Then $Dg_k(d_{k-1,s})=d_{k,s}$ and $Dg_k(d_{k-1,t})=d_{k,t}$, so $D^t(g_{k+1} \circ g_k)(\{d_{(k-1,s)}, d_{(k-1,t)} \})= \{D(g_{k+1} \circ g_k)(d_{(k-1,s)}), D(g_{k+1} \circ g_k)(d_{(k-1,t)})\}= \{ Dg_{k+1}(d_{k,s}= d^{pu}_k), Dg_{k+1}(d_{k,t}= d^{pa}_k)\} = \{ d^a_{k+1}, d^a_{k+1} \}$. So $d_{k-1,s}$ and $d_{k-1,t}$ share a gate. But $d_{k-1,i}$ already shares a gate with another element and we already established that $d_{k-1,i} \neq d_{k-1,s}$ and $d_{k-1,i} \neq d_{k-1,t}$. So $f_{k-1}$ has at most $2r-2$ gates. Since each $f_k$ has the same number of gates, this implies $g$ has at most $2r-2$ gates, giving a contradiction. (b) is proved.
[\[C:UnachievedDirection\]]{} **(of Lemma \[L:IllegalTurn\])** For each $1 \leq k \leq n$, \
a.
: $t^R_k= \{\overline{d^a_k}, d^u_k\}$, must contain either $d^{pu}_k$ or $d^{pa}_k$ and\
b.
: The vertex labeled $d^u_k$ in $G_k$ is red and $[t^R_k]= [\overline{d^a_k}, d^u_k]$ is a red edge in $G_k$.
We start with (a). Lemma \[L:IllegalTurn\] implies each $T_k$ contains $d^u_k$. At the same time, we know $t^R_k= \{\overline{d^a_k}, d^u_k\}$, implying $t^R_k$ contains $d^u_k$, thus either $d^{pa}_k$ or $d^{pu}_k$. We now prove (b). By Lemma \[L:fk\]d, $d^u_k$ is not a periodic direction for $Df_k$, so is not a vertex of $\mathcal{SW}(f_k)$. Thus, $d^u_k$ labels a red vertex in $G_k$. To show $[t^R_k]$ is in $\mathcal{LW}(f_k)$ it suffices to show $t^R_k$ is in $f_k(e^u_k)$. Let $e^u_k=e_{k,l}$. By Lemma \[L:PreLemma\], the path $g_{k-1,k+1} (e^u_k=e_{k,l})$ contains $e_{k-1,l}$. Let $e_j \in \mathcal{E}_{l-1}$ be such that $g_{k-1,k+1}(e^u_k)= e_1 \dots e_{q-1} e_{k-1,l} e_{q+1} \dots e_m$. Then $f_k(e^u_k ) =g_{k,k+1} (e^u_k )= \gamma_1 \dots \gamma_{q-1} e^a_k e^u_k \gamma_{q+1}\dots \gamma_m$ where $\gamma_j =g_k(e_{i_j})$ for all $j$. So $f_k(e^u_k)$ contains $\{ \bar d^a_k, d^u_k \}$ and $\mathcal{LW}(f_k)$ contains \[$t^R_k$\]. Since $[\overline{d^a_k}, d^u_k]$ contains the red vertex $d^u_k$, it is red in $G_k$.
[\[L:EdgeImage\]]{} If $[d_{(l,i)},d_{(l,j)}]$ is in $\mathcal{C}(G_l)$, then $[D^tg_{k,l+1}(\{d_{(l,i)},d_{(l,j)}\})]$ is a purple edge in $G_k$.
It suffices to show two things: (1) $D^tg_{k,l+1}(\{d_{(l,i)},d_{(l,j)}\})$ is a turn in some edge path $f_l^p(e_{l,m})$ with $p \geq 1$ and (2) $Dg_{k,l+1}(d_{l,i})$ and $Dg_{k,l+1}(d_{l,j})$ are periodic directions for $f_l$. We use induction. Start with (1). For the base case assume $[d_{(k-1,i)}, d_{(k-1,j)}]$ is in $\mathcal{C}(G_{k-1})$, so $f_{k-1}^p(e_{k-1,t})= s_1 \dots \overline{e_{(k-1,i)}} e_{(k-1,j)} \dots s_m$ for some $e_{(k-1,t)},s_1, \dots s_m \in \mathcal{E}_{k-1}$ and $p \geq 1$. By Lemma \[L:PreLemma\], $e_{k-1,t}$ is in the path $g_{k-1} \circ \cdots \circ g_1 \circ g_n \circ \cdots \circ g_{k+1} (e_{k,t})$. Thus, since $f_{k-1}^p(e_{k-1,t})=s_1 \dots \overline{e_{(k-1,i)}} e_{(k-1,j)} \dots s_m$ and no $g_{i,j}(e_{j-1,t})$ can have cancellation, $s_1 \dots \overline{e_{(k-1,i)}} e_{(k-1,j)} \dots s_m$ is a subpath of $f_{k-1}^p \circ g_{k-1} \circ \cdots \circ g_1 \circ g_n \circ \cdots \circ g_{k+1} (e_{k,t})$. Apply $g_k$ to $f_{k-1}^p \circ g_{k-1} \circ \cdots \circ g_1 \circ g_n \circ \cdots \circ g_k (e_{k-1,t})$ to get $f_k^{p+1}(e_{k,t})$.
Suppose $Dg_k(e_{k-1,i})=e_{k,i}$ and $Dg_k(e_{k-1,j})=e_{k,j}$. Then $g_k(\dots\overline{e_{k-1,i}} e_{k-1,j} \dots) = \dots \overline{e_{(k,i)}} e_{(k,j)} \dots$, with possibly different edges before and after $\overline{e_{k,i}}$ and $e_{k,j}$ than before and after $\overline{e_{k-1,i}}$ and $e_{k-1,j}$. Thus, here, $f^{p+1}_k( \dots \overline{e_{(k-1,i)}} e_{(k-1,j)} \dots)$ contains $\{d_{(k,i)},d_{(k,j)}\}$, which here is $D^tg_k(\{d_{(k-1,i)}, d_{(k-1,j)}\})$. So \[$D^tg_k(\{d_{(k-1,i)}, d_{(k-1,j)}\})$\] is an edge in $G_k$.
Suppose $g_k\colon e_{k-1,j} \mapsto e_{k,l} e_{k,j}$. Then $g_k(\dots \overline{e_{k-1,i}} e_{k-1,j}\dots)= \dots \overline{e_{k,i}} e_{k,l}e_{k,j} \dots$, (again with possibly different edges before and after $\overline{e_{k,i}}$ and $e_{k,j}$). So $g_k(\dots \overline{e_{(k-1,i)}} e_{(k-1,j)} \dots)$ contains $\{\overline{d_{(k,l)}},d_{(k,j)}\}$, which here is $D^tg_k(\{d_{(k-1,i)},d_{(k-1,j)}\})$, so \[$D^tg_k(\{d_{(k-1,i)},d_{(k-1,j)}\})$\] again is in $G_k$.
Finally, suppose $g_k: e_{k-1,j} \mapsto e_{k,j}e_{k,l}$ defined $g_k$. Unless $\overline{e_{k-1,i}} = e_{(k-1,j)}$, we have $g_k(\dots \overline{e_{(k-1,i)}} e_{(k-1,j)} \dots)=\dots \overline{e_{(k,i)}} e_{(k,j)} e_{(k,l)} \dots$, containing $\{d_{(k,i)},d_{(k,j)}\}= D^tg_k(\{d_{(k-1,i)},d_{(k-1,j)}\})$. So \[$D^tg_k(\{d_{(k-1,i)},d_{(k-1,j)}\})$\] is an edge in $G_k$ here also.
If $\overline{e_{k-1,i}} = e_{k-1,j}$, we are in a reflection of the previous case. The other cases ($g_k: \overline{e_{k-1,i}} \mapsto \overline{e_{k,i}} e_{k,l}$ and $g_k: \overline{e_{k-1,i}} \mapsto e_{k,l} \overline{e_{k,i}}$) follow similarly by symmetry. The base case for (1) is complete.
We prove the base case of (2). Since $[D^tg_k(\{d_{(k-1,i)},d_{(k-1,j)}\})]= [Dg_k(d_{(k-1,i)}),Dg_k(d_{(k-1,j)})]$, both vertex labels of $[D^tg_k(\{d_{(k-1,i)},d_{(k-1,j)}\})]$ are in $\mathcal{IM}(Dg_k)$. By Lemma \[L:fk\]d, this means both vertices are periodic. So $[D^tg_k(\{d_{(k-1,i)},d_{(k-1,j)}\})]$ is in $\mathcal{PI}(G_k)$. The base case is proved. Suppose inductively $[d_{(l,i)},d_{(l,j)}]$ is an edge in $\mathcal{C}(G_l)$ and $[D^tg_{k-1,l+1}(\{d_{(l,i)},d_{(l,j)}\})]$ is an edge in $\mathcal{PI}(G_{k-1})$. The base case implies $[D^tg_k(D^tg_{k-1,l+1}(\{d_{(l,i)},d_{(l,j)}\})]$ is an edge in $\mathcal{PI}(G_k)$. But $D^tg_k(D^tg_{k-1,l+1}(\{d_{(l,i)},d_{(l,j)}\}))=$ $D^tg_{k,l+1}(\{d_{(l,i)},d_{(l,j)}\})$. The lemma is proved.
[\[L:RedEdgeImage\]]{} (Properties of $t^R_k$ and $e^R_k$). For each $1 \leq l,k \leq n$ \
a.
: $[D^tg_{l,k}(\{\overline{d^a_{k-1}}, d^u_{k-1} \})]$ is a purple edge in $G_l$.\
b.
: $[\overline{d^a_k}, d^u_k]$ is not in $D^Cg_k(G_{k-1})$.
By Lemma \[L:EdgeImage\], it suffices to show for (a) that $[\overline{d^a_{k-1}}, d^u_{k-1}]$ is a colored edge of $G_{k-1}$. This was shown in Corollary \[C:UnachievedDirection\]b. By Lemma \[L:EdgeImage\], each edge in $\mathcal{C}(G_{k-1})$ is mapped to a purple edge in $G_k$. On the other hand, $[\overline{d^a_k}, d^u_k]$ is a red edge in $G_k$. Thus, $[\overline{d^a_k}, d^u_k]$ is not in $D^Cg_k(G_{k-1})$ and (b) is proved.
Each $G_k$ has a unique red edge ($e^R_k=[t^R_k]=[\overline{d^a_k},d^u_k]$):
[\[L:1Edge\]]{} $\mathcal{C}(G_k)$ can have at most 1 edge segment connecting the nonperiodic direction red vertex $d^u_k$ to the set of purple periodic direction vertices.
First note that the nonperiodic direction $d^u_k$ labels the red vertex in $G_k$. If $g_k(e_{k-1,i})= e_{k,i}e_{k,j}$, then the red vertex in $G_k$ is $\overline{d_{k,i}}$ (where $d_{k,i}=D_0(e_{k,i})$ and $d_{k,j}=D_0(e_{k,j})$). The vertex $\overline{d_{k,i}}$ will be adjoined to the vertex for $d_{k,j}$ and only $d_{k,j}$: each occurrence of $e_{k-1,i}$ in the image under $g_{k-1,1}$ of any edge has been replaced by $e_{k,i}e_{k,j}$ and every occurrence of $\overline{e_{k,i}}$ has been replaced by $\overline{e_{k,i}} \overline{e_{k,j}}$, ie, there are no copies of $e_{k,j}$ without $e_{k,i}$ following them and no copies of $\overline{e_{k,i}}$ without $\overline{e_{k,j}}$ preceding them.
The red edge and vertex of $G_k$ determine $g_k$:
[\[L:NG\]]{} Suppose that the unique red edge in $G_k$ is $[t^R_k]= [d_{(k,j)}, \overline{d_{(k,i)}}]$ and that the vertex representing $d_{k,j}$ is red. Then $g_k(e_{k-1,j}) = e_{k,i} e_{k,j}$ and $g_k(e_{k-1,t})=e_{k,t}$ for $e_{k-1,t} \neq (e_{k-1,j})^{\pm 1}$, where $D_0(e_{s,t})=d_{s,t}$ and $D_0(\overline{e_{s,t}}) = \overline{d_{s,t}}$ for all $s$, $t$.
By the ideal decomposition definition, $g_k$ is defined by $g_k: e_{k-1,j} \mapsto e_{k,i} e_{k,j}$. Corollary \[C:UnachievedDirection\] implies $D_0(e_{k,j})=d^u_k$, i.e. the direction associated to the red vertex of $G_k$. So the second index of $d^u_k$ uniquely determines the index $j$, so $e_{k-1,j}= e^{pu}_{k-1}$ and $e_{k,i}=e^a_k$. Additionally, Corollary \[C:UnachievedDirection\]’s proof implies $[\overline{d_{(k,i)}}, d_{(k,j)}]$ is $G_k$’s red edge. So $e_{k,i}=e^a_k$. And $g_k$ must be $g_k : e^{pu}_{k-1} \mapsto e^a_k e^u_k$, i.e, $e_{k-1,j} \mapsto e_{k,i} e_{k,j}$.
[\[L:PurpleEdgeImages\]]{} (Induced maps of ltt structures) \
a.
: $D^Cf_k$ maps $\mathcal{PI}(G_k)$ isomorphically onto itself via a label-preserving isomorphism.\
b.
: The set of purple edges of $G_{k-1}$ is mapped by $D^Cg_k$ injectively into the set of purple edges of $G_k$.\
c.
: For each $0 \leq l,k \leq n$, $Dg_{l, k+1}$ induces an isomorphism from $\mathcal{SW}(f_k)$ onto $\mathcal{SW}(f_l)$.\
We prove (a). Lemma \[L:EdgeImage\] implies that $D^Cf_k$ maps $\mathcal{PI}(G_k)$ into itself. However, $Df_k$ fixes all directions labeling vertices of $\mathcal{SW}(f_k)=\mathcal{PI}(G_k)$. Thus, $D^Cf_k$ restricted to $\mathcal{PI}(G_k)$, is a label-preserving graph isomorphism onto its image.
We prove (b). Since $d^a_k$ is the only direction with more than one $Dg_k$ preimage, and these two preimages are $d^{pa}_{k-1}$ and $d^{pu}_{k-1}$, the $[d_{(k,i)}, d^a_k]$ are the only edges in $G_k$ with more than one $D^Cg_k$ preimage. The two preimages are the edges $[d_{(k-1,i)}, d^{pa}_{k-1}]$ and $[d_{(k-1,i)}, d^{pu}_{k-1}]$ in $G_{k-1}$. However, by Lemma \[L:IllegalTurn\], either $e^u_{k-1}= e^{pu}_{k-1}$ or $e^u_{k-1}=e^{pa}_{k-1}$. So one of the preimages of $d^a_k$ is actually $d^u_{k-1}$, i.e. one of the preimage edges is actually $[d_{(k-1,i)}, d^u_{k-1}]$. Since $[t^R_{k-1}]$ is the only edge of $C(G_{k-1})$ containing $d^u_{k-1}$, one of the preimages of $[d_{(k,i)}, d^a_k]$ must be $[t^R_{k-1}]$, leaving only one possible purple preimage.
We prove (c). By (b), the set of $G_k$’s purple edges is mapped injectively by $D^Cg_{l, k+1}$ into the set of $G_l$’s purple edges. Likewise, the set of $G_l$’s purple edges is mapped injectively by $D^Cg_{k, l+1}$ into $G_k$. (a) implies $D^Cf_k=(D^Cg_{k, l+1}) \circ (D^Cg_{l, k+1})$ and $D^Cf_l=(D^Cg_{l, k+1}) \circ (D^Cg_{k, l+1})$ are bijections. So, the map $D^Cg_{l, k+1}$ induces on the set of $G_k$’s purple edges is a bijection. It is only left to show that two purple edges share a vertex in $G_k$ if and only if their $D^Cg_{l, k+1}$ images share a vertex in $G_l$.
If $[x, d_1]$ and $[x, d_2]$ are in $\mathcal{PI}(G_k)$, $D^Cg_{l, k+1}([x, d_1])=[Dg_{l, k+1}(x), Dg_{l, k+1}(d_1)]$ and $D^tg_{l, k+1}([x, d_2])= [Dg_{l, k+1}(x), Dg_{l, k+1}(d_2)]$ share $Dg_{l, k+1}(x)$. On the other hand, if $[w, d_3]$ and $[w, d_4]$ in $\mathcal{PI}(G_l)$ share $w$, then $[D^tg_{k, l+1}(\{w, d_3 \})]=[Dg_{k, l+1}(w), Dg_{k, l+1}(d_3)]$ and $[D^tg_{k, l+1}(\{w, d_4 \})]=[Dg_{k, l+1}(w), Dg_{k, l+1}(d_4)]$ share $Dg_{k, l+1}(w)$. Since $D^Cf_l$ is an isomorphism on $\mathcal{PI}(G_l)$, $D^Cg_{l, k+1}$ and $D^Cg_{k, l+1}$ act as inverses. So the preimages of $[w, d_3]$ and $[w, d_4]$ under $D^Cg_{l, k+1}$ share a vertex in $G_l$.
Lemmas \[L:irred\] gives properties stemming from irreducibility (though not proving irreducibility):
[\[L:irred\]]{} For each $1 \leq j \leq r$ \
a.
: there exists a $k$ such that either $e^u_k=E_{k,j}$ or $e^u_k= \overline{E_{k,j}}$ and\
b.
: there exists a $k$ such that either $e^a_k=E_{k,j}$ or $e^a_k= \overline{E_{k,j}}$.\
We start with (a). For contradiction’s sake suppose there is some $j$ so that $e^u_k \neq E_{k,j}^{\pm 1}$ for all $k$. We inductively show $g(E_{0,j})=E_{0,j}$, implying $g$’s reducibility. Induction will be on the $k$ in $g_{k-1,1}$.
For the base case, we need $g_1(E_{0,j})=E_{1,j}$ if $e_1^u \neq E_{1,j}^{\pm 1}$. $g_1$ is defined by $e^{pu}_0 \mapsto e^a_1 e^u_1$. Since $e_1^u \neq E_{1,j}$ and $\overline{e_1^u} \neq \overline{E_{(1,j)}}$, we know $e^{pu}_0 \neq E_{(0,j)}^{\pm 1}$. Thus, $g_1(E_{0,j})=E_{(1,j)}$, as desired. Now inductively suppose $g_{k-1,1}(E_{0,j})=E_{k-1,j}$ and $e_k^u \neq E_{k,j}^{\pm 1}$. Then $e^{pu}_{k-1} \neq E_{k-1,j}^{\pm 1}$. Thus, since $e^{pu}_{k-1} \mapsto e^a_k e^u_k$ defines $g_k$, we know $g_k(E_{k-1,j})=E_{k,j}$. So $g_{k, 1}(E_{0,j})=g_k (g_{k-1,1}(E_{0,j}))=g_k (E_{k-1,j})= E_{k,j}$. Inductively, this proves $g(E_{0,j})=E_{0,j}$, we have our contradiction, and (b) is proved.
We now prove (b). For contradiction’s sake, suppose that, for some $1 \leq j \leq r$, $e^a_k \neq E_{k,j}$ and $e^a_k \neq \overline{E_{k,j}}$ for each $k$. The goal will be to inductively show that, for each $E_{0,i}$ with $E_{0,i} \neq E_{0,j}$ and $E_{0,i} \neq \overline{E_{(0,j)}}$, $g(E_{0,i})$ does not contain $E_{0,j}$ and does not contain $\overline{E_{0,j}}$ (contradicting irreducibility).
We prove the base case. $g_1$ is defined by $e^{pu}_0 \mapsto e^a_1 e^u_1$. First suppose $E_{0,j}=(e^{pu}_0)^{\pm 1}$. Then $e^{pu}_0 \neq E_{0,i}^{\pm 1}$ (since $E_{0,i} \neq E_{0,j}^{\pm 1}$). So $g_1(E_{0,i})= E_{1,i}$, which does not contain $E_{1,j}^{\pm 1}$. Now suppose that $E_{0,j} \neq e^{pu}_0$ and $E_{0,j} \neq \overline{e^{pu}_0}$. Then $e^a_1 e^u_1$ does not contain $E_{1,j}$ or $\overline{E_{1,j}}$ (since $e^a_k \neq (E_{k,j})^{\pm 1}$ by assumption). So $E_{1,j}^{\pm 1}$ are not in the image of $E_{0,i}$ if $E_{0,i} = e^{pu}_0$ (since the image of $E_{0,i}$ is then $e^a_1 e^u_1$) and are not in the image of $\overline {E_{0,i}}$ (since the image is $\overline{e^u_1} \overline{e^a_1}$) and are not in the image $E_{0,i}$ if $E_{0,i} \neq (e^{pu}_0)^{\pm 1}$ (since the image is $E_{1,i}$ and $E_{1,i} \neq E_{1,j}^{\pm 1}$). The base case is proved.
Inductively suppose $g_{k-1,1}(E_{0,i})$ does not contain $E_{k-1,j}^{\pm 1}$. Similar analysis as above shows $g_k(E_{k-1,i})$ does not contain $E_{k,j}^{\pm 1}$ for any $E_{k,i} \neq E_{k,j}^{\pm 1}$. Since $g_{k-1,1}(E_{k-1,i})$ does not contain $E_{k-1,j}^{\pm 1}$, $g_{k-1, 1}(E_{0,i})= e_1 \dots e_m$ with each $e_i \neq E_{k-1,j}^{\pm 1}$. Thus, no $g_k(e_i)$ contains $E_{k,j}^{\pm 1}$. So $g_{k, 1}(E_{0,i})= g_k(g_{k-1,1}(E_{0,i}))= g_k(e_1) \dots g_k(e_m)$ does not contain $E_{k,j}^{\pm 1}$. This completes the inductive step, thus (b).
Lemma \[L:irred\] is necessary, but not sufficient, for $g$ to be irreducible. For example, the composition of $a \mapsto ab$, $b \mapsto ba$, $c \mapsto cd$, and $d \mapsto dc$ satisfies Lemma \[L:irred\], but is reducible.
AM property I follows from Proposition \[P:BC\] and Lemma \[L:fk\]; AM property II from Lemma \[L:IllegalTurn\]; AM property III from Corollary \[C:UnachievedDirection\]; AM property IV from Lemma \[L:EdgeImage\]; AM property V from Lemma \[L:1Edge\] and Corollary \[C:UnachievedDirection\]; AM property VI from Lemma \[L:NG\]; AM property VII from Lemma \[L:PurpleEdgeImages\]; and AM property VIII from Lemma \[L:irred\].
Lamination train track (ltt) structures
=======================================
[\[Ch:ltt\]]{}
In Subection \[SS:Realltts\] we defined ltt structures for ideally decomposed representatives with $(r;(\frac{3}{2}-r))$ potential. Both for defining $\mathcal{ID}$ diagrams and for applying the Birecurrency Condition, we need abstract definitions of ltt structures motivated by the AM properties of Section \[Ch:AMProperties\].
Abstract lamination train track structures
------------------------------------------
(See Example \[Ex:G(g)\]) A *lamination train track (ltt) structure $G$* is a pair-labeled colored train track graph (black edges will be included, but not considered colored) satisfying: \
ltt1:
: Vertices are either purple or red.\
ltt2:
: Edges are of 3 types ($\mathcal{E}_b$ comprises the black edges and $\mathcal{E}_c$ comprises the red and purple edges):\
\
ltt3:
: No pair of vertices is connected by two distinct colored edges.\
The purple subgraph of $G$ will be called the *potential ideal Whitehead graph associated to $G$*, denoted *$\mathcal{PI}(G)$*. For a finite graph $\mathcal{G} \cong \mathcal{PI}(G)$, we say $G$ *is an ltt Structure for $\mathcal{G}$*.
An *$(r;(\frac{3}{2}-r))$ ltt structure* is an ltt structure $G$ for a $\mathcal{G} \in \mathcal{PI}_{(r;(\frac{3}{2}-r))}$ such that: \
ltt(\*)4:
: $G$ has precisely 2r-1 purple vertices, a unique red vertex, and a unique red edge.\
ltt structures are *equivalent* that differ by an ornamentation-preserving (label and color preserving), homeomorphism.
[\[N:ltt\]]{} **(ltt Structures)** For an ltt Structure $G$:
1. An edge connecting a vertex pair $\{d_i, d_j \}$ will be denoted \[$d_i, d_j$\], with interior ($d_i, d_j$). (While the notation \[$d_i, d_j$\] may be ambiguous when there is more than one edge connecting the vertex pair $\{d_i, d_j \}$, we will be clear in such cases as to which edge we refer to.)
2. $[e_i]$ will denote \[$d_i, \overline{d_i}$\]
3. Red vertices and edges will be called *nonperiodic*.
4. Purple vertices and edges will be called *periodic*.
5. *$\mathcal{C}(G)$* will denote the colored subgraph of $G$, called the *colored subgraph associated to* (or *of*) $G$.
6. $G$ will be called *admissible* if it is birecurrent.
For an $(r;(\frac{3}{2}-r))$ ltt structure $G$ for $\mathcal{G}$, additionally:
1. $d^u$ will label the unique red vertex and be called the *unachieved direction*.
2. $e^R=[t^R]$, will denote the unique red edge and $\overline{d^a}$ its purple vertex’s label. So $t^R= \{d^u, \overline{d^a} \}$ and $e^R= [d^u, \overline{d^a}]$.
3. $\overline{d^a}$ is contained in a unique black edge, which we call the *twice-achieved edge*.
4. $d^a$ will label the other twice-achieved edge vertex and be called the *twice-achieved direction*.
5. If $G$ has a subscript, the subscript carries over to all relevant notation. For example, in $G_k$, $d^u_k$ will label the red vertex and $e^R_k$ the red edge.
A 2r-element set of the form $\{x_1, \overline{x_1}, \dots, x_r, \overline{x_r} \}$, with elements paired into *edge pairs* $\{x_i, \overline{x_i}\}$, will be called a *rank*-$r$ *edge pair labeling set*. It will then be standard to say $\overline{\overline{x_i}}=x_i$. A graph with vertices labeled by an edge pair labeling set will be called a *pair-labeled* graph. If an indexing is prescribed, it will be called an *indexed pair-labeled* graph.
[\[D:Indexedltt\*\]]{} For an ltt structure to be considered *indexed pair-labeled*, we require:
1. It is index pair-labeled (of rank $r$) as a graph.\
2. The vertices of the black edges are indexed by edge pairs.\
Index pair-labeled ltt structures are *equivalent* that are equivalent as ltt structures via an equivalence preserves the indexing of the vertex labeling set.
By index pair-labeling (with rank $r$) an $(r;(\frac{3}{2}-r))$ ltt structure $G$ and edge-indexing the edges of an $r$-petaled rose $\Gamma$, one creates an identification of the vertices in $G$ with $\mathcal{D}(v)$, where $v$ is the vertex of $\Gamma$. With this identification, we say $G$ is *based* at $\Gamma$. In such a case it will be standard to use the notation $\{d_1, d_2, \dots, d_{2r-1}, d_{2r} \}$ for the vertex labels (instead of $\{x_1, x_2, \dots, x_{2r-1}, x_{2r} \}$). Additionally, $[e_i]$ will denote $[D_0(e_i), D_0(\overline{e_i})] = [d_i, \overline{d_i}]$ for each edge $e_i \in \mathcal{E}(\Gamma)$.
A $\mathcal{G} \in \mathcal{PI}_{(r;(\frac{3}{2}-r))}$ will be called *(index) pair-labeled* if its vertices are labeled by a $2r-1$ element subset of the rank $r$ (indexed) edge pair labeling set.
Maps of lamination train track structures
-----------------------------------------
[\[SS:BasedlttStructureMaps\]]{}
***Let $G$ and $G'$ be rank-$r$ indexed pair-labeled $(r;(\frac{3}{2}-r))$ ltt structures, with bases $\Gamma$ and $\Gamma'$, and $g:\Gamma \to \Gamma'$ a tight homotopy equivalence taking edges to nondegenerate edge-paths.***
Recall that $Dg$ induces a map of turns $D^tg: \{a,b\} \mapsto \{Dg(a), Dg(b)\}$. $Dg$ additionally induces a map on the corresponding edges of $\mathcal{C}(G)$ and $\mathcal{C}(G')$ if the appropriate edges exist in $\mathcal{C}(G')$:
When the map sending
1. the vertex labeled $d$ in $G$ to that labeled by $Dg(d)$ in $G'$ and
2. the edge \[$d_i, d_j$\] in $\mathcal{C}(G)$ to the edge \[$Dg(d_i), Dg(d_j)$\] in $\mathcal{C}(G')$ also satisfies that
3. each $\mathcal{PI}(G)$ is mapped isomorphically onto $\mathcal{PI}(G')$,
we call it the *map of colored subgraphs induced by $g$* and denote it $D^C(g): C(G) \to C(G')$.
When it exists, the map $D^T(g): G \to G'$ *induced by $g$* is the extension of $D^C(g): C(G) \to C(G')$ taking the interior of the black edge of $G$ corresponding to the edge $E \in \mathcal{E}(\Gamma)$ to the interior of the smooth path in $G'$ corresponding to $g(E)$.
ltt structures are ltt structures
---------------------------------
[\[Ch:lttMeansltt\]]{}
By showing that the ltt structures of Subsection \[SS:Realltts\] are indeed abstract ltt structures, we can create a finite list of ltt structures for a particular $\mathcal{G} \in \mathcal{PI}_{(r;(\frac{3}{2}-r))}$ to apply the birecurrency condition to.
[\[L:PF\]]{} Let $g:\Gamma \to \Gamma$ be a representative of $\phi \in Out(F_r)$, with $(r;(\frac{3}{2}-r))$ potential, such that $\mathcal{IW}(g) \cong \mathcal{G}$. Then $G(g)$ is an $(r;(\frac{3}{2}-r))$ ltt structure with base graph $\Gamma$. Furthermore, $\mathcal{PI}(G(g)) \cong \mathcal{G}$.
This is more or less just direct applications of the lemmas above. [@p12a] gives a detailed proof of a more general lemma.
Generating triples
------------------
[\[SS:GeneratingTriples\]]{}
Since we deal with representatives decomposed into Nielsen generators, we use an abstract notion of an “indexed generating triple.”
[\[D:Triple\]]{} A *triple* $(g_k, G_{k-1}, G_k)$ will be an ordered set of three objects where $g_k: \Gamma_{k-1} \to \Gamma_k$ is a proper full fold of roses and, for $i=k-1,k$, $G_i$ is an ltt structure with base $\Gamma_i$.
[\[D:GeneratingTriple\]]{} A *generating triple* is a triple $(g_k, G_{k-1}, G_k)$ where \
(gtI)
: $g_k: \Gamma_{k-1} \to \Gamma_k$ is a proper full fold of edge-indexed roses defined by
- $g_k(e_{k-1,j_k})= e_{k,i_k} e_{k,j_k}$ where $d^a_k=D_0(e_{k,i_k})$, $d^u_k=D_0(e_{k,j_k})$, and $e_{k,i_k} \neq (e_{k,j_k})^{\pm 1}$ and\
- $g_k(e_{k-1,t})= e_{k,t}$ for all $e_{k-1,t} \neq (e_{k,j_k})^{\pm 1}$;
(gtII)
: $G_i$ is an indexed pair-labeled $(r;(\frac{3}{2}-r))$ ltt structure with base $\Gamma_i$ for $i=k-1,k$; and
(gtIII)
: The induced map of based ltt structures $D^T(g_k): G_{k-1} \to G_k$ exists and, in particular, restricts to an isomorphism from $\mathcal{PI}(G_{k-1})$ to $\mathcal{PI}(G_k)$.
[\[N:GeneratingTriples\]]{} **(Generating Triples)** For a generating triple $(g_k, G_{k-1}, G_k)$:
1. We call $G_{k-1}$ the *source ltt structure* and $G_k$ the *destination ltt structure*.\
2. [\[N:IngoingGeneratorTerminology\]]{} $g_k$ will be called the *(ingoing) generator* and will sometimes be written $g_k: e^{pu}_{k-1} \mapsto e^a_k e^u_k$ (“p” is for “pre”). Thus, $d_{k-1,j_k}$ will sometimes be written $d^{pu}_{k-1}$.\
3. $e^{pa}_{k-1}$ denotes $e_{k-1,i_k}$ (again “p” is for “pre”).\
4. If $G_k$ and $G_{k-1}$ are indexed pair-labeled $(r;(\frac{3}{2}-r))$ ltt structures for $\mathcal{G}$, then $(g_k, G_{k-1}, G_k)$ will be a generating triple *for $\mathcal{G}$*.
While $d^u_i$ is determined by the red vertex of $G_i$ (and does not rely on other information in the triple), $d^{pu}_{k-1}$ and $d^{pa}_{k-1}$ actually rely on gtI, and cannot be determined by knowing only $G_{k-1}$.
[\[E:GeneratingTripleEx\]]{} The triple $(g_2, G_1, G_2)$ of Example \[Ex:InducedMap\] is an example of a generating triple where $x$ denotes both $E_{(1,1)}$ and $E_{(2,1)}$, $y$ denotes both $E_{(1,2)}$ and $E_{(2,2)}$, and $z$ denotes both $E_{(1,3)}$ and $E_{(2,3)}$.
[\[D:GeneratorExtendsTolttStructures\]]{} Suppose $(g_i, G_{i-1}, G_i)$ and $(g_i', G_{i-1}', G_i)'$ are generating triples. Let $g_i^T: G_{i-1} \to G_i$ be induced by $g_i: \Gamma_{i-1} \to \Gamma_i$ and $g_i^T: G_{i-1}' \to G_i'$ by $g_i: \Gamma_{i-1}' \to \Gamma_i'$. We say $(g_i, G_{i-1}, G_i)$ and $(g_i', G_{i-1}', G_i')$ are *equivalent* if there exist indexed pair-labeled graph equivalences $H_{i-1}: \Gamma_{i-1} \to \Gamma_{i-1}'$ and $H_i: \Gamma_i \to \Gamma_i'$ such that:
1. for $k=i,i-1$, $H_i: \Gamma_i \to \Gamma_i'$ induces indexed pair-labeled ltt structure equivalence of $G_i$ and $G_i'$
2. and $H_i \circ g_i = g_i' \circ H_{i-1}$.
Peels, extensions, and switches
===============================
[\[Ch:Peels\]]{}
Suppose $\mathcal{G} \in \mathcal{PI}_{(r;(\frac{3}{2}-r))}$. By Section \[Ch:IdealDecompositions\], if there is a $\phi \in \mathcal{AFI}_r$ with $IW(\phi) \cong \mathcal{G}$, then there is an ideally decomposed $(r;(\frac{3}{2}-r))$-potential representative $g$ of a power of $\phi$. By Section \[Ch:AMProperties\], such a representative would satisfy the AM properties. Thus, if we can show that a representative satisfying the properties does not exist, we have shown there is no $\phi \in \mathcal{AFI}_r$ with $IW(\phi) \cong \mathcal{G}$ (we use this fact in Section \[Ch:UnachievableGraphs\]). In this section we show what triples $(g_k, G_{k-1}, G_k)$ satisfying the AM properties must look like. We prove in Proposition \[P:ExtensionsSwitches\] that, if the structure $G_k$ and a purple edge $[d, d^a_k]$ in $G_k$ are set, then there is only one $g_k$ possibility and at most two $G_{k-1}$ possibilities (one generating triple possibility will be called a “switch” and the other an “extension”). Extensions and switches are used here only to define ideal decomposition diagrams but have interesting properties used (and proved) in [@p12c] and [@p12d].
Peels
-----
[\[S:Peels\]]{}
As a warm-up, we describe a geometric method for visualizing “switches” and “extensions” as moves, “peels,” transforming an ltt structure $G_i$ into an ltt structure $G_{i-1}$.
Each peel of an ltt structure $G_i$ involves three directed edges of $G_i$: \
- The *First Edge of the Peel* (*New Red Edge* in $G_i$): the red edge from $d^u_i$ to $\overline{d^a_i}$.\
- The *Second Edge of the Peel* (*Twice-Achieved Edge* in $G_i$): the black edge from $\overline{d^a_i}$ to $d^a_i$.\
- The *Third Edge of the Peel* (*Determining Edge* for the peel): a purple edge from $d^a_i$ to $d$. (In $G_{i-1}$, this vertex $d$ will be the red edge’s attaching vertex, labeled $\overline{d^a_{i-1}}$).\
{width="2.8in"}
For each determining edge choice $[d^a_i, d]$ in $G_i$, there is one “peel switch” (Figure 8) and one “peel extension” (Figure 7). When $G_i$ has only a single purple edge at $d^a_i$, the switch and extension differ by a color switch of two edges and two vertices. We start by explaining this case. After, we explain the preliminary step necessary for any switch where more than one purple edge in $G_i$ contains $d^a_i$.
We describe how, when $G_i$ has only a single purple edge at $d^a_i$, the two peels determined by $[d^a_i, d]$ transform $G_i$ into $G_{i-1}$. While keeping $d$ fixed, starting at vertex $\overline{d^a_i}$, peel off black edge $[\overline{d^a_i}, d^a_i]$ and the third edge $[d^a_i, d]$, leaving copies of $[\overline{d^a_i}, d^a_i]$ and $[d^a_i, d]$ and creating a new edge $[d^u_i, d]$ from the concatenation of the peel’s first, second, and third edges (Figure 7 or 8).
In a *peel extension*: $[d^u_i, \overline{d^a_i}]$ disappears into the concatenation and does not exist in $G_{i-1}$, the copy of $[\overline{d^a_i}, d^a_i]$ left behind stays black in $G_{i-1}$, the copy of $[d^a_i, d]$ left behind stays purple in $G_{i-1}$, the edge $[d^u_i, d]$ formed from the concatenation is red in $G_{i-1}$, and nothing else changes from $G_i$ to $G_{i-1}$ (if one ignores the first indices of the vertex labels). The triple $(g_i, G_{i-1}, G_i)$, with $g_i$ as in AM property VI, will be called the *extension determined by $[d^a_i, d]$*. \
[\[fig:PeelExtension\]]{} {width="2.6in"}\
In a *peel switch* (where $[d^a_i, d]$ was the only purple edge in $G_i$ containing $d^a_i$): Again $[d^u_i, \overline{d^a_i}]$ has disappeared into the concatenation and the copy of $[\overline{d^a_i}, d^a_i]$ left behind stays black in $G_{i-1}$. But now the edge $[d^u_i, d]$ formed from the concatenation is purple in $G_{i-1}$, the copy of $[d^a_i, d]$ left behind and vertex $d^a_i$ are both red in $G_{i-1}$ (so that $d^a_i$ is now actually $d^u_{i-1}$), and vertex $d^u_i$ is purple in $G_{i-1}$. The triple $(g_i, G_{i-1}, G_i)$, with $g_i$ as in AM property VI, will be called the *switch determined by $[d^a_i, d]$*. \
[\[fig:PeelSwitch\]]{} {width="2.4in"}\
Preliminary step for a switch where purple edges other than the determining edge $[d^a_i, d]$ contain vertex $d^a_i$ in $G_i$: For each purple edge $[d^a_i, d']$ in $G_i$ where $d \neq d'$, form a purple concatenated edge $[d', d^u_i]$ in $G_{i-1}$ by concatenating $[d', d^a_i]$ with a copy of $[d^a_i, \overline{d^a_i}, d^u_i]$, created by splitting open, as in Figure 9, $[d^a_i, \overline{d^a_i}]$ from $d^a_i$ to $\overline{d^a_i}$ and $[\overline{d^a_i}, d^u_i]$ from $\overline{d^a_i}$ to $d^u_i$. \
{width="4.1in"} \[fig:PeelSwitch2\]\
To check the peel switch was performed correctly, one can: remove $G_i$’s red edge, lift vertex $d^a_i$ (with purple edges containing it dangling from one’s fingers), and drop vertex $d^a_i$ in the spot of vertex $d^u_i$, while leaving behind a copy of $[d^a_i, d]$ to become the new red edge of $G_{i-1}$ (with $d^{pa}_{i-1}$ as the red vertex).
Extensions and switches
-----------------------
***Throughout this section $G_k$ will be an indexed pair-labeled $(r;(\frac{3}{2}-r))$ ltt structure for a $\mathcal{G} \in \mathcal{PI}_{(r;(\frac{3}{2}-r))}$ with rose base graph $\Gamma_k$. We use the standard notation.***
We define extensions and switches “entering” an indexed pair-labeled admissible $(r;(\frac{3}{2}-r))$ ltt structure $G_k$ for $\mathcal{G}$. However, we first prove that determining edges exist.
There exists a purple edge with vertex $d^a_k$, so that it may be written $[d^a_k, d_{k,l}]$.
If $d^a_k$ were red, the $e^R_k$ would be $[d^a_k, \overline{d^a_k}]$, violating that $\mathcal{G} \in \mathcal{PI}_{(r;(\frac{3}{2}-r))}$. $d^a_k$ must be contained in an edge $[d^a_k, d_{k,l}]$ or $\mathcal{G}$ would not have 2r-1 vertices. If $d_{k,l}$ were red, i.e. $d_{k,l}=d^u_k$, then both $[d^u_k, \overline{d^a_k}]$ and $[d^u_k, d^a_k]$ would be red, violating \[ltt(\*)4\]. So $[d^a_k, d_{k,l}]$ must be purple.
[\[D:Extension\]]{} (See Figure \[fig:ExtensionDiagram\]) For a purple edge $[d^a_k, d_{k,l}]$ in $G_k$, the *extension determined by* $[d^a_k, d_{k,l}]$, is the generating triple $(g_k, G_{k-1}, G_k)$ for $\mathcal{G}$ satisfying: \
(extI):
: The restriction of $D^T(g_k)$ to $\mathcal{PI}(G_{k-1})$ is defined by sending, for each $j$, the vertex labeled $d_{k-1,j}$ to the vertex labeled $d_{k,j}$ and extending linearly over edges.\
(extII):
: $d^u_{k-1}= d^{pu}_{k-1}$, i.e. $d^{pu}_{k-1}= d_{k-1,j_k}$ labels the single red vertex in $G_{k-1}$.\
(extIII):
: $\overline{d^a_{k-1}}= d_{k-1,l}$.
(extIII) implies that the single red edge $e^{R}_{k-1}= [d^u_{k-1}, \overline{d^{a}_{k-1}}]$ of $G_{k-1}$ can be written, among other ways, as $[d^{pu}_{k-1}, d_{(k-1,l)}]$.
Explained in Section \[S:Peels\], but with this section’s notation, an extension transforms ltt structures as: \
![[]{data-label="fig:ExtensionDiagram"}](ExtensionDiagramNew.eps "fig:"){width="4.3in"} \
[\[L:ExtensionUniqueness\]]{} Given an edge $[d^a_k, d_{k,l}]$ in $\mathcal{PI}(G_k)$, the extension $(g_k, G_{k-1}, G_k)$ determined by $[d^a_k, d_{k,l}]$ is unique. \
I.
: $G_{k-1}$ can be obtained from $G_k$ by the following steps: \
II.
: The fold is such that the corresponding homotopy equivalence maps the oriented $e_{k-1,j_k} \in \mathcal{E}_{k-1}$ over the path $e_{k,i_k} e_{k,j_k}$ in $\Gamma_k$ and then each oriented $e_{k-1,t} \in \mathcal{E}_{k-1}$ with $e_{k-1,t} \neq e_{k-1,j_k}^{\pm 1}$ over $e_{k,t}$.
The proof is an unraveling of definitions. A full presentation can be found in [@p12a].
[\[D:Switch\]]{} (See Figure \[fig:SwitchDiagram\]) The *switch* determined by a purple edge $[d^a_k, d_{(k,l)}]$ in $G_k$ is the generating triple $(g_k, G_{k-1}, G_k)$ for $\mathcal{G}$ satisfying: \
(swI):
: $D^T(g_k)$ restricts to an isomorphism from $\mathcal{PI}(G_{k-1})$ to $\mathcal{PI}(G_k)$ defined by $$\mathcal{PI}(G_{k-1}) \xrightarrow{d^{pu}_{k-1} \mapsto d^a_k=d_{k, i_k}} \mathcal{PI}(G_k)$$ ($d_{k-1,t} \mapsto d_{k,t}$ for $d_{k-1,t} \neq d^{pu}_{k-1}$) and extended linearly over edges.\
(swII):
: $d^{pa}_{k-1} = d^u_{k-1}$.\
(swIII):
: $\overline{d^a_{k-1}} = d_{k-1,l}$.
(swII) implies that the red edge $e^R_{k-1} = [d^u_{k-1}, d^a_{k-1}]$ of $G_{k-1}$ can be written $[d^{pa}_{k-1}, \overline{d^a_{k-1}}]$, among other ways. (swIII) implies that $e^R_{k-1}$ can be written $[d_{(k-1,i_k)}, d_{(k-1,l)}]$.
Explained in Section \[S:Peels\], but with this section’s notation, a switch transforms ltt structures as follows: \
![[]{data-label="fig:SwitchDiagram"}](SwitchDiagramNew.eps "fig:"){width="4.3in"} \
[\[L:SwitchUniqueness\]]{} Given an edge $[d^a_k, d_{k,l}]$ in $\mathcal{PI}(G_k)$, the switch $(g_k, G_{k-1}, G_k)$ determined by $[d^a_k, d_{k,l}]$ is unique. \
I.
: $G_{k-1}$ can be obtained from $G_k$ by the following steps: \
II.
: The fold is such that the corresponding homotopy equivalence maps the oriented $e_{k-1,j_k} \in \mathcal{E}_{k-1}$ over the path $e_{k,i_k} e_{k,j_k}$ in $\Gamma_k$ and then each oriented $e_{k-1,t} \in \mathcal{E}_{k-1}$ with $e_{k-1,t} \neq e_{k-1,j_k}^{\pm 1}$ over $e_{k,t}$.
The proof is an unraveling of definitions. A full presentation can be found in [@p12a].
Recall (Proposition \[P:am\]) that each triple in an ideal decomposition satisfies the AM properties. Thus, to construct a diagram realizing any ideally decomposed $(r;(\frac{3}{2}-r))$-potential representative with ideal Whitehead graph $\mathcal{G}$, we want edges of the diagram to correspond to triples satisfying the AM properties. Proposition \[P:ExtensionsSwitches\] tells us each such a triple is either an admissible switch or admissible extension.
[\[P:ExtensionsSwitches\]]{} Suppose $(g_k, G_{k-1}, G_k)$ is a triple for $\mathcal{G}$ such that: 1. $\mathcal{G} \in \mathcal{PI}_{(r;(\frac{3}{2}-r))}$ and 2. $G_i$ is an indexed pair-labeled $(r;(\frac{3}{2}-r))$ ltt structure for $\mathcal{G}$ with base graph $\Gamma_i$, for $i=k,k-1$. Then $(g_k, G_{k-1}, G_k)$ satisfies AM properties I-VII if and only if it is either an admissible switch or an admissible extension. In particular, in the circumstance where $d^u_{k-1}= d^{pa}_{k-1}$, the triple is a switch and, in the circumstance where $d^u_{k-1}= d^{pu}_{k-1}$, the triple is an extension.
For the forward direction, assume $(g_k, G_{k-1}, G_k)$ satisfies AM properties I-VII and (1)-(2) in the proposition statement. We show the triple is either a switch or an extension (AM property I give birecurrency). Assumption (1) in the proposition statement implies (gtII).
By AM property VI, $g_k$ is defined by $g_k(e^{pu}_{k-1})=e^a_k e^u_k$ and $g_k(e_{k-1,i})=e_{k,i}$ for $e_{k-1,i} \neq (e^{pu}_{k-1})^{\pm 1}$, $D_0(e^u_k)= d^u_k$, $D_0(\overline{e^a_k})= \overline{d^a_k}$, and $e^{pu}_{k-1}= e_{(k-1,j)}$, where $e^u_k=e_{k,j}$. We have (gtI).
By AM property VII, $Dg_k$ induces on isomorphism from $SW(G_{k-1})$ to $SW(G_k)$. Since the only direction whose second index is not fixed by $Dg_k$ is $d^{pu}_{k-1}$, the only vertex label of $SW(G_{k-1})$ not determined by this isomorphism is the preimage of $d^a_k$ (which AM property IV dictates to be either $d^{pu}_{k-1}$ or $d^{pa}_{k-1}$). When the preimage is $d^{pa}_{k-1}$, this gives (extI). When the preimage is $d^{pu}_{k-1}$, this gives (swI). For the isomorphism to extend linearly over edges, we need that images of edges in $G_{k-1}$ are edges in $G_k$, i.e. $[Dg_k(d_{k-1,i}), Dg_k(d_{k-1,j})]$ is an edge in $G_k$ for each edge $[d_{(k-1,i)}, d_{(k-1,j)}]$ in $G_{k-1}$. This follows from AM property IV. We have (gtIII).
AM property II gives either $d^u_{k-1}= d^{pa}_{k-1}$ or $d^u_{k-1}=d^{pu}_{k-1}$. In the switch case, the above arguments imply $d^{pu}_{k-1}$ labels a purple vertex. So $d^u_{k-1}=d^{pa}_{k-1}$ (since AM property III tells us $d^u_{k-1}$ is red). This gives (swII) once one appropriately coordinates notation. In the extension case, the above arguments give instead that $d^{pa}_{k-1}$ labels a purple vertex, meaning $d^u_{k-1}=d^{pu}_{k-1}$ (again since AM property III tells us $d^u_{k-1}$ is red). This gives us (extII). We are left with (extIII) and (swIII). What we need is that $[d^a_k, d_{k,l}]$ is a purple edge in $G_k$ where $\overline{d^a_{k-1}}= d_{k-1,l}$.
By AM property V, $G_{k-1}$ has a single red edge $[t^R_{k-1}] = [\overline{d^a_{k-1}}, d^u_{k-1}]$. By AM property IV, $D^Cg_k([t^R_{k-1}])$ is in $\mathcal{PI}(G_k)$. First consider what we established is the switch case, i.e. assume $d^u_{k-1}=d^{pa}_{k-1}$. The goal is to determine $[t^R_{k-1}]= [d_{(k-1,i_k)}, d_{(k-1,l)}]$, where $d^a_k=d_{k,i_k}$ ($d_{k-1,i_k}=d^{pa}_{k-1}$) and $[d^a_k, d_{k,l}]$ is in $\mathcal{PI}(G_k)$ (making $(g_k, G_{k-1}, G_k)$ the switch determined by $[d^a_k, d_{k,l}]$). Since $d^u_{k-1}=d^{pa}_{k-1}$, we know $[t^R_{k-1}]= [\overline{d^a_{k-1}}, d^u_{k-1} ]=[\overline{d^a_{k-1}}, d^{pa}_{k-1}]$. We know $\overline{d^a_{k-1}} \neq d^{pa}_{k-1}$ (since (tt2) implies $\overline{d^a_{k-1}} \neq d^u_{k-1}$, which equals $d^{pa}_{k-1}$). Thus, AM property VI says $D^Cg_k([t^R_{k-1}]) = D^Cg_k([\overline{d^a_{k-1}}, d^{pa}_{k-1}]) = [d_{(k,l)}, d^{a}_{k}]$ where $\overline{d^a_{k-1}}=e_{k-1,l}$. So $[d_{(k,l)}, d^{a}_{k}]$ is in $\mathcal{PI}(G_k)$. We thus have (swIII). Now consider what we established is the extension case, i.e. assume $d^u_{k-1}= d^{pu}_{k-1}$. We need $[t^R_{k-1}]=[d_{(k-1,j_k)}, d_{(k-1,l)}]$, where $d^u_{k-1}=d_{k-1,j_k}$ and $[d^a_k, d_{k,l}]$ is in $\mathcal{PI}(G_k)$ (making $(g_k, G_{k-1}, G_k)$ the extension determined by $[d^a_k, d_{k,l}]$). Since $d^u_{k-1}= d^{pu}_{k-1}$, we know $[t^R_{k-1}]= [\overline{d^a_{k-1}}, d^u_{k-1} ]= [\overline{d^a_{k-1}}, d^{pu}_{k-1}]$. We know $\overline{d^a_{k-1}} \neq d^{pu}_{k-1}$ (since (tt2) implies $\overline{d^a_{k-1}} \neq d^u_{k-1}$, which equals $d^{pu}_{k-1}$). Thus, by AM property VI, $D^Cg_k([t^R_{k-1}]) = D^Cg_k([\overline{d^a_{k-1}}, d^{pu}_{k-1}]) = [d_{(k,l)}, d^{a}_{k}]$, where $\overline{d^a_{k-1}}= e_{k-1,l}$. We have (extIII) and the forward direction.
For the converse, assume $(g_k, G_{k-1}, G_k)$ is either an admissible switch or extension. Since we required extensions and switches be admissible, $G_{k-1}$ and $G_k$ are birecurrent. We have AM property I.
The first and second parts of AM property II are equivalent and the second part holds by (extII) for an extension and (swII) for a switch. For AM property III note that there is only a single red vertex (labeled $d^u_k$) in $G_k$ and is only a single red vertex (labeled $d^u_{k-1}$) in $G_{k-1}$ because of the requirement in (gtII) that $G_k$ and $G_{k-1}$ are $(r;(\frac{3}{2}-r))$ ltt structures (see the standard notation for why this is notationally consistent with the AM properties). What is left of AM property III is that the edge $[t^R_k]= [d^u_k, \overline{d^a_k}]$ in $G_k$ and the edge $[t^R_{k-1}]= [d^u_{k-1}, \overline{d^a_{k-1}}]$ in $G_{k-1}$ are both red. This follows from (gtI) combined with (extII) for an extension and (swII) for a switch.
(gtIII) implies AM property IV. For AM property V, note: AM property III implies $e^R_k$ is a red edge containing the red vertex $d^u_k$. (ltt(\*)4) implies the uniqueness of both the red edge and direction.
Since AM property VI follows from (gtI), combined with (extII) for an extension and (swII) for a switch, and AM property VII follows from (gtIII), we have proved the converse.
In light of Proposition \[P:ExtensionsSwitches\], an *admissible map* will mean a triple for a $\mathcal{G} \in \mathcal{PI}_{(r;(\frac{3}{2}-r))}$ that is an admissible switch or admissible extension or (equivalently) satisfies AM properties I-VII.
Ideal decomposition ($\mathcal{ID}$) diagrams
=============================================
[\[Ch:AMDiagrams\]]{}
***Throughout this section $\mathcal{G} \in \mathcal{PI}_{(r;(\frac{3}{2}-r))}$.*** We define the “ideal decomposition ($\mathcal{ID}$) diagram” for $\mathcal{G}$, as well as prove that representatives with $(r;(\frac{3}{2}-r))$ potential are realized as loops in these diagrams. We use $\mathcal{ID}$ diagrams to prove Theorem \[T:MainTheorem\]B and to construct examples in [@p12d].
A *preliminary ideal decomposition diagram for $\mathcal{G}$* is the directed graph where \
1. the nodes correspond to equivalence classes of admissible indexed pair-labeled $(r;(\frac{3}{2}-r))$ ltt structures for $\mathcal{G}$ and\
2. for each equivalence class of an admissible generator triple ($g_i$, $G_{i-1}$, $G_i$) for $\mathcal{G}$, there exists a directed edge $E(g_i, G_{i-1}, G_i)$ from the node \[$G_{i-1}$\] to the node \[$G_i$\].
The disjoint union of the maximal strongly connected subgraphs of the preliminary ideal decomposition diagram for $\mathcal{G}$ will be called the *ideal decomposition ($\mathcal{ID}$) diagram for $\mathcal{G}$* (or *$\mathcal{ID}(\mathcal{G})$*).
[@p12a] gives a procedure for constructing $\mathcal{ID}$ diagrams (there called “AM Diagrams”).
We say an ideal decomposition $\Gamma_0 \xrightarrow{g_1} \Gamma_1 \xrightarrow{g_2} \cdots \xrightarrow{g_{k-1}}\Gamma_{k-1} \xrightarrow{g_k} \Gamma_k$ of a tt $g$ with indexed $(r;(\frac{3}{2}-r))$ ltt structures $G_0 \to G_1 \to \cdots \to G_{k-1} \to G_k$ for $\mathcal{G}$ is *realized* by $E(g_1, G_{0}, G_1) * \dots * E(g_k, G_{k-1}, G_k)$ in $\mathcal{ID}(\mathcal{G})$ if the oriented path $E(g_1, G_{0}, G_1) * \dots * E(g_k, G_{k-1}, G_k)$ in $\mathcal{ID}(\mathcal{G})$ from \[$G_0$\] to \[$G_k$\], traversing the $E(g_i, G_{i-1}, G_i)$ in order of increasing $i$ (from $E(g_1, G_{0}, G_1)$ to $E(g_k, G_{k-1}, G_k)$), exists.
[\[P:ReferenceLoop\]]{} If $g=g_{k} \circ \cdots \circ g_1$, with ltt structures $G_0 \to G_1 \to \cdots \to G_{k-1} \to G_k$, is an ideally decomposed representative of $\phi \in Out(F_r)$, with $(r;(\frac{3}{2}-r))$ potential, such that $\mathcal{IW}(\phi)=\mathcal{G}$, then $E(g_1, G_{0}, G_1) * \dots * E(g_k, G_{k-1}, G_k)$ exists in $\mathcal{ID}(\mathcal{G})$ and forms an oriented loop.
This follows from Proposition \[P:ExtensionsSwitches\] and Proposition \[P:am\].
[\[C:ReferenceLoop\]]{} **(of Proposition \[P:ReferenceLoop\])** If no loop in $\mathcal{ID}(\mathcal{G})$ gives a potentially-$(r;(\frac{3}{2}-r))$ representative of a $\phi \in Out(F_r)$ with $\mathcal{IW}(\phi) = \mathcal{G}$, such a $\phi$ does not exist. In particular, any of the following $\mathcal{ID}(\mathcal{G})$ properties would prove such a representative does not exist: \
1. For at least one edge pair $\{d_i, \overline{d_i}\}$, where $e_i \in \mathcal{E}(\Gamma)$, no red vertex in $\mathcal{ID}(\mathcal{G})$ is labeled by $d_i^{\pm 1}$.
2. The representative corresponding to each loop in $\mathcal{ID}(\mathcal{G})$ has a pNp.
As a result of Corollary \[C:ReferenceLoop\](1) we define:
**Irreducibility Potential Test:** Check whether, in each connected component of $\mathcal{ID}(\mathcal{G})$, for each edge vertex pair $\{d_i, \overline{d_i}\}$, there is a node $N$ in the component such that either $d_i$ or $\overline{d_i}$ labels the red vertex in the structure $N$. If it holds for no component, $\mathcal{G}$ is unachieved.
Let $\{x_1, \overline{x_1}, \dots, x_{2r}, \overline{x_{2r}}\}$ be a rank-r edge pair labeling set. We call a permutation of the indices $1 \leq i \leq 2r$ combined with a permutation of the elements of each pair $\{x_i, \overline{x_i}\}$ an *Edge Pair (EP) Permutation*. Edge-indexed graphs will be considered *Edge Pair Permutation (EPP) isomorphic* if there is an EP permutation making the labelings identical (this still holds even if only a subset of $\{x_1, \overline{x_1}, \dots, x_{2r}, \overline{x_{2r}}\}$ is used to label the vertices, as with a graph in $\mathcal{PI}_{(r;(\frac{3}{2}-r))}$).
When checking for irreducibility, it is only necessary to look at one EPP isomorphism class of each component (where two components are in the same class if one can be obtained from the other by applying the same EPP isomorphism to each triple in the component).
Several unachieved ideal Whitehead graphs
=========================================
[\[Ch:UnachievableGraphs\]]{}
[\[T:MainTheorem\]]{} For each $r \geq 3$, let $\mathcal{G}_r$ be the graph consisting of $2r-2$ edges adjoined at a single vertex. \
A.
: For no fully irreducible $\phi \in Out(F_r)$ is $\mathcal{IW}(\phi) \cong \mathcal{G}_r$.\
B.
: The following connected graphs are not the ideal Whitehead graph $\mathcal{IW}(\phi)$ for any fully irreducible $\phi \in Out(F_3)$:\
{width="2.6in"}
We first prove (A). By Proposition \[P:BC\], it suffices to show that no admissible $(r;(\frac{3}{2}-r))$ ltt structure for $\mathcal{G}$ is birecurrent. Up to EPP-isomorphism, there are two such ltt structures to consider, neither birecurrent): \
{width="2in"} \[fig:NotBirecurrent\]\
These are the only structures worth considering as follows: Call the valence-($2r-2$) vertex $v_1$. Either (1) some valence-1 vertex is labeled by $\overline{v_1}$ or (2) the set of valence-$1$ vertices $\{x_1, \overline{x_1}, \dots, x_{r-1}, \overline{x_{r-1}}\}$ consists of $r-1$ edge-pairs. Suppose (2) holds. The red edge cannot be attached in such a way that it is labeled with an edge-pair or is a loop and attaching it to any other vertex yields an EPP-isomorphic ltt structure to that on the left. Suppose (1) holds. Let $x_i$ label the red vertex. The valence-$1$ vertex labels will be $\{\overline{v_1}, x_2, \overline{x_2}, \dots, x_{i-1}, \overline{x_{i-1}}, \overline{x_i}, x_{i+1}, \overline{x_{i+1}}, \overline{x_i} \dots, x_{r}, \overline{x_{r}}\}$. The red edge cannot be attached at $\overline{x_i}$. So either it will be attached at $v_1$, $\overline{v_1}$, or some $x_j$ with $x_j \neq x_i^{\pm 1}$. Unless it is attached at $\overline{v_1}$, $\overline{v_1}$ is a valence-$1$ vertex of \[$v_1, \overline{v_1}$\] in the local Whitehead graph, making $[v_1, \overline{v_1}]$ an edge only traversable once by a smooth line. If the red edge is attached at $\overline{v_1}$, we have the structure on the right.
We prove (B). The left graph is covered by A. The following is a representative of the EPP isomorphism class of the only significant component of $\mathcal{ID}(\mathcal{G})$ where $\mathcal{G}$ is the right-most structure: \
{width="3.7in"} \[fig:IllustrativeAMDiagram\]\
Since $\mathcal{ID}(\mathcal{G})$ contains only red vertices labeled $z$ and $\bar{x}$ (leaving out $\{y, \overline{y}\}$), unless some other component contains all 3 edge vertex pairs ($\{x, \overline{x}\}$, $\{y, \overline{y}\}$, and $\{z, \overline{z}\}$), the middle graph would be unachieved. Since no other component does contain all 3 edge vertex pairs as vertex labels (all components are EPP-isomorphic), the middle graph is indeed unachieved.
Again, for the right-hand, the $\mathcal{ID}$ Diagram lacks irreducibility potential. A component of the $\mathcal{ID}$ diagram is given below (all components are EPP-isomorphic). The only edge pairs labeling red vertices of this component are $\{x, \overline{x} \}$ and $\{z, \overline{z}\}$: \
{width="4in"} \[fig:IllustrativeAMDExample\]
|
---
abstract: 'In characteristic $p=0$ or $p>5$, we show that a K3 surface with an order $60$ automorphism is unique up to isomorphism. As a consequence, we characterize the supersingular K3 surface with Artin invariant 1 in characteristic $p\equiv 11$ (mod 12) by a cyclic symmetry of order 60.'
address: 'School of Mathematics, Korea Institute for Advanced Study, Seoul 130-722, Korea '
author:
- JongHae Keum
date: 'May 2012, September 2013'
title: K3 surfaces with an order $60$ automorphism and a characterization of supersingular K3 surfaces with Artin invariant 1
---
Ø ß
[^1]
Let $X$ be a K3 surface over an algebraically closed field $k$ of characteristic $p\ge 0$. An automorphism $g$ of $X$ is called *symplectic* if it preserves a regular 2-form $\omega_X$, and *purely non-symplectic* if no power of $g$ is symplectic except the identity.
Over $k={\mathbb{C}}$, Xiao [@Xiao] and Machida and Oguiso [@MO] proved that a positive integer $N$ is the order of a purely non-symplectic automorphism of a complex K3 surface if and only if $\phi(N)\le 20$ and $N\neq 60$, where $\phi$ is the Euler function. On the other hand, there is a K3 surface with an automorphism of order 60 ([@K] Example 3.2): $$\label{formula}
X_{60}:= (y^2+x^3+t_0t_1^{11}-t_0^{11}t_1 = 0)\subset
\mathbb{P}(4,6,1,1),$$ $$\label{form2}
g_{60}(t_0,t_1,x,y)=(t_0,\zeta_{60}^6t_1,\zeta_{60}^2x,\zeta_{60}^3y)$$ where $\zeta_{60}\in k$ is a primitive 60th root of unity. The K3 surface $X_{60}$ is defined over the integers and both the surface and the automorphism have a good reduction mod $p$ unless $p=2$, 3, 5.
For an automorphism $g$ of finite order of a K3 surface $X$, we write $${\rm ord}(g)=m.n$$ if $g$ is of order $mn$ and the natural homomorphism $$\langle g\rangle\to {\rm GL}(H^0(X,
\Omega^2_X))$$ has kernel of order $m$ and image of order $n$. We call $n$ the *non-symplectic order* of $g$.
The main result of the paper is the following.
\[main\] Let $k$ be an algebraically closed field of characteristic $p=0$ or $p>5$. Let $X$ be a K3 surface defined over $k$ with an automorphism $g$ of order $60$. Then
1. ${\rm ord}(g)=5.12$;
2. the pair $(X, \langle g\rangle)$ is isomorphic to the pair $(X_{60}, \langle g_{60}\rangle)$, i.e. there is an isomorphism $f:X\to X_{60}$ such that $f \langle g\rangle f^{-1}=\langle g_{60}\rangle$.
The non-existence of a complex K3 surface with a purely non-symplectic automorphism of order 60 was proved by Machida and Oguiso [@MO]. Their proof does not extend to the positive characteristic case, as it uses the holomorphic Lefschetz formula and the notion of transcendental lattice, both not available in positive characteristic.
Theorem \[main\] and Main Theorem of [@K] determine completely the list of all non-symplectic orders in characteristic $p>0$:
\[cor\] In any fixed characteristic $p>0$, a positive integer $N$ is the non-symplectic order of an automorphism of a K3 surface if and only if $p\nmid N$, $N\neq 60$ and $\phi(N)\le 20$.
It is well known that the Fermat quartic surface $$x_0^4+x_1^4+x_2^4+x_3^4 = 0$$ is a supersingular K3 surface with Artin invariant 1, if the characteristic $p\equiv 3$ (mod 4). This can be seen by using the algorithm for determining the Artin invariant of a weighted Delsarte surface whose minimal resolution is a K3 surface ([@Shioda], [@Goto]). The same algorithm shows that in characteristic $p\equiv 11$ (mod 12) the surface $X_{60}$ is a supersingular K3 surface with Artin invariant 1, hence is isomorphic to the Fermat quartic surface, since a supersingular K3 surface with Artin invariant 1 is unique up to isomorphism ([@Ogus], [@Ogus2]).
In characteristic $p\equiv 11$ $({\rm mod}\,\, 12)$, the Fermat quartic surface is the only K3 surface with an order $60$ automorphism.
Over $k={\mathbb{C}}$, Oguiso [@Og2] proved that the Fermat quartic surface is the only K3 surface with a faithful action of a nilpotent group of order $512=2^9$. Over $k={\mathbb{C}}$, the surface $X_{60}$ is not isomorphic to the Fermat quartic surface, as the former admits a purely non-symplectic automorphism of order $12$, while the latter has Picard number 20, hence by Nikulin [@Nik] does not admit a purely non-symplectic automorphism of order $n$ with $\phi(n)>2$.
In characteristic $p=11$ the Fermat quartic surface also admits a cyclic action of order 66 (Example 7.5 [@K]) and a symplectic action of the simple groups $M_{22}$, $M_{11}$ and $L_2(11)$, where $M_{r}$ is one of the Mathieu groups [@DK3].
Throughout this paper, whenever we work with $l$-adic cohomology we assume $l$ is any prime different from the characteristic.
[**Notation**]{}
- ${\rm NS}(X)$ : the Néron-Severi group of a variety $X$;
- $X^g={\rm Fix}(g)$ : the fixed locus of an automorphism $g$ of $X$;
- $e(g):=e({\rm Fix}(g))$, the Euler characteristic of ${\rm Fix}(g)$ for $g$ tame;
- ${\textup{Tr}}(g^*|H^*(X)):=\sum_{j=0}^{2\dim X} (-1)^j{\textup{Tr}}(g^*|H^j_{\rm et}(X,{{\mathbb{Q}}}_l))$.
For an automorphism $g$ of a K3 surface $X$,
<!-- -->
- ${\rm ord}(g)=m.n$ : $g$ is of order $mn$ and the representation of the group $\langle g^*\rangle$ on $H^0(X, \Omega^2_X)$ has kernel of order $m$;
- $[g^*]=[\lambda_1, \ldots, \lambda_{22}]$ : the list of the eigenvalues of $g^*|H^2_{\rm et}(X,{{\mathbb{Q}}}_l)$.
- $\zeta_a$ : a primitive $a$-th root of unity in $\overline{{\mathbb{Q}}_l}$;
- $[\zeta_a:\phi(a)]\subset [g^*]$ : all primitive $a$-th roots of unity appear in $[g^*]$ where $\phi(a)$ indicates the number of them.
- $[\lambda.r]\subset [g^*]$ : $\lambda$ repeats $r$ times in $[g^*]$.
- $[(\zeta_a:\phi(a)).r]\subset [g^*]$ : the list $\zeta_a:\phi(a)$ repeats $r$ times in $[g^*]$.
Preliminaries
=============
We first recall the following basic result.
\[integral\]$($3.7.3 [@Illusie]$)$ Let $g$ be an automorphism of a projective variety $X$ over an algebraically closed field $k$ of characteristic $p> 0$. Let $l$ be a prime $\neq p$. Then the following hold true.
1. The characteristic polynomial of $g^*|H_{\rm et}^j(X,{\mathbb{Q}}_l)$ has integer coefficients for each $j$. In particular, if for some positive integer $m$ a primitive $m$-th root of unity appears with multiplicity $r$ as an eigenvalue of $g^*|H_{\rm et}^j(X,{\mathbb{Q}}_l)$, then so does each of its conjugates.
2. The characteristic polynomial of $g^*$ does not depend on the choice of cohomology, $l$-adic or crystalline.
\[integral’\] Let $g$ be an automorphism of a projective variety $X$ over an algebraically closed field $k$ of characteristic $p> 0$. Let $l$ be a prime $\neq p$. Then the following hold true.
1. If $g$ is of finite order, then $g$ has an invariant ample divisor, and $1$ is an eigenvalue of $g^*|H_{\rm et}^2(X,{\mathbb{Q}}_l)$.
2. If $X$ is a K3 surface, $g$ is tame and $g^*|H^0(X,\Omega_X^2)$ has $\zeta_n\in k$ as an eigenvalue, then $g^*|H_{\rm et}^2(X,{\mathbb{Q}}_l)$ has $\zeta_n\in \overline{{\mathbb{Q}}_l}$ as an eigenvalue.
\(1) For any ample divisor $D$ the sum $\sum g^{i}(D)$ is $g$-invariant. A $g^*$-invariant ample line bundle gives a $g^*$-invariant vector in the 2nd crystalline cohomology $H_{\rm crys}^2(X/W)$ under the Chern class map $$c_1: {\textup{Pic}}(X)\to H_{\rm crys}^2(X/W).$$ It follows that $1$ is an eigenvalue of $g^*|H_{\rm crys}^2(X/W)$. Here $W=W(k)$ is the ring of Witt vectors. Now apply Proposition \[integral\](2).
\(2) The quotient module $$H_{{\text{crys}}}^2(X/W)/pH_{{\text{crys}}}^2(X/W)$$ is a finite dimensional $k$-vector space isomorphic to the algebraic de Rham cohomology $H_{{\text{DR}}}^2(X)$. See [@Illusie] for the crystalline cohomology. It is known that the Hodge to de Rham spectral sequence $$E_1^{t,s}:=H^s(X, \Omega_X^t)\Rightarrow H_{{\text{DR}}}^*(X)$$ degenerates at $E_1$, giving the Hodge filtration on $H_{{\text{DR}}}^2(X)$ and the following canonical exact sequences: $$0\to F^1\to F^0=H_{{\text{DR}}}^2(X)\to H^2(X, \mathcal{O}_X)\to 0$$ $$0\to F^2=H^0(X,\Omega_X^2)\to F^1\to H^1(X,\Omega_X^1)\to 0.$$ In particular $g^*|H_{{\text{DR}}}^2(X)$ has $\zeta_n\in k$ as an eigenvalue. The corresponding eigenvalue of $g^*|H_{{\text{crys}}}^2(X/W)$ must be an $np^r$-th root of unity for some $r$, since $n$ is not divisible by $p$. Then $g^{p^r*}|H_{{\text{crys}}}^2(X/W)$ has an $n$-th root of unity as an eigenvalue. Since $g$ is tame, so does $g^{*}|H_{{\text{crys}}}^2(X/W)$.
Recall that for a nonsingular projective variety $Z$ in characteristic $p>0$, there is an exact sequence of ${\mathbb{Q}}_l$-vector spaces $$\label{trans}
0\to {\text{NS}}(Z)\otimes {\mathbb{Q}}_l \to H_{\rm et}^2(Z,{\mathbb{Q}}_l) \to
T_l^2(Z)\to 0$$ where $T_l^2(Z) = T_l(\textup{Br}(Z))$ in the standard notation in the theory of étale cohomology (see [@Shioda2]). The Brauer group ${\text{Br}}(Z)$ is known to be a birational invariant.
\[diminv\] Let $Z$ be a nonsingular projective variety in characteristic $p>0$. Let $g$ be an automorphism of $Z$ of finite order. Assume $l\ne p$. Then the following assertions are true.
1. Both traces of $g^*$ on ${\rm NS}(Z)$ and on $T_l^2(Z)$ are integers.
2. ${\textup{rank}}~{\rm NS}(Z)^g={\textup{rank}}~{\rm NS}(Z/\langle g \rangle).$
3. $\dim H^2_{\rm et}(Z,{{\mathbb{Q}}}_l)^g={\textup{rank}}~{\rm NS}(Z)^g+\dim T_l^2(Z)^g$.
4. If the minimal resolution $Y$ of $Z/\langle g \rangle$ has $T_l^2(Y)=0$, then $$\dim H^2_{\rm et}(Z,{{\mathbb{Q}}}_l)^g={\textup{rank}}~{\rm NS}(Z)^g.$$
The condition of $(4)$ is satisfied if $Z/\langle g\rangle$ is rational or is birational to an Enriques surface.
The following is well known, see for example Deligne-Lusztig (Theorem 3.2 [@DL]).
\[trace\]$($Lefschetz fixed point formula$)$ Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic $p> 0$ and let $g$ be a tame automorphism of $X$. Then $X^g={\rm Fix}(g)$ is smooth and $$e(g):=e(X^g)={\textup{Tr}}(g^*|
H^*(X)).$$
A tame symplectic automorphism $h$ of a K3 surface has finitely many fixed points, the number of fixed points $f(h)$ depends only on the order of $h$ and the list of possible pairs $({\rm ord}(h),
f(h))$ is the same as in the complex case (Theorem 3.3 and Proposition 4.1 [@DK2]): $$({\rm ord}(h),
f(h))=(2,8),\,\,(3,6),\,\,(4,4),\,\,(5,4),\,\,(6,2),\,\,(7,3),\,\,(8,2).$$ Thus by the Lefschetz fixed point formula (Proposition \[trace\]), we obtain the following.
\[Lefschetz\] Let $h$ be a tame symplectic automorphism of a K3 surface $X$. Then $h^*|H_{\rm et}^2(X,{\mathbb{Q}}_l)$ has eigenvalues $$\begin{array}{lll} {\rm ord}(h)=2&:&[h^*]=[1,\, 1.13,\, -1.8]\\
{\rm ord}(h)=3&:&[h^*]=[1,\, 1.9,\, (\zeta_3:2).6]\\
{\rm ord}(h)=4&:&[h^*]=[1,\, 1.7,\, (\zeta_4:2).4,\, -1.6]\\
{\rm ord}(h)=5&:&[h^*]=[1,\, 1.5,\, (\zeta_5:4).4]\\
{\rm ord}(h)=6&:&[h^*]=[1,\, 1.5,\, (\zeta_3:2).4,\, (\zeta_6:2).2,\, -1.4]\\
{\rm ord}(h)=7&:&[h^*]=[1,\, 1.3,\, (\zeta_7:6).3]\\
{\rm ord}(h)=8&:&[h^*]=[1,\,
1.3,\,(\zeta_8:4).2,\,(\zeta_4:2).3,\, -1.4]\end{array}$$ where the first eigenvalue corresponds to an invariant ample divisor.
We need the following information on a special involution of a K3 surface.
\[nsym2\] Let $X$ be a K3 surface in characteristic $p\neq 2$. Assume that $h$ is an automorphism of order $2$ with $\dim
H^2_{\rm et}(X,{{\mathbb{Q}}}_l)^h =2$. Then $h$ is non-symplectic and has an $h$-invariant elliptic fibration $\psi:X\to {\bf P}^1$, $$X/\langle h\rangle\cong {\bf F}_e$$ a rational ruled surface, and $X^h$ is either a curve of genus $9$ which is a $4$-section of $\psi$ or the union of a section and a curve of genus $10$ which is a $3$-section. In the first case $e=0, 1$ or $2$, and in the second $e=4$. Each singular fibre of $\psi$ is of type $I_1$ $($nodal$)$, $I_2$, $II$ $($cuspidal$)$ or $III$, and is intersected by $X^h$ at the node and two smooth points if of type $I_1$, at the two singular points if of type $I_2$, at the cusp with multiplicity $3$ and a smooth point if of type $II$, at the singular point tangentially to both components if of type $III$. If $X^h$ contains a section, then each singular fibre is of type $I_1$ or $II$.
Since $\dim H^2_{\rm et}(X,{{\mathbb{Q}}}_l)^h=2$, the eigenvalues of $h^*|H^2_{\rm et}(X,{{\mathbb{Q}}}_l)$ must be $$[h^*]=[1.2,\,-1.20], \,\,{\rm so}\,\,\,{\textup{Tr}}(h^*|H^*(X))=-16.$$ By Lemma \[Lefschetz\], $h$ is non-symplectic, thus $X^h$ is a disjoint union of smooth curves and the quotient $X/\langle h\rangle$ is a nonsingular rational surface. By Proposition \[diminv\], $X/\langle h\rangle$ has Picard number 2, hence is isomorphic to a rational ruled surface ${\bf F}_e$. Note that $e(X^h)=-16$, so $X^h$ is non-empty and has at most 2 components. Thus $X^h$ is either a curve $C_9$ of genus 9 or the union of two curves $C_0$ and $C_{10}$ of genus 0 and 10, respectively. In the first case, the image $C_9'\subset {\bf F}_e$ of $C_9$ satisfies $C_9'^2=32$ and $C_9'K=-16$, hence $C_9'\equiv 4S_0+(4+2e)F$, where $S_0$ is the section with $S_0^2=-e$, and $F$ a fibre of ${\bf F}_e$. Since $S_0C_9'\ge 0$, we have $e\le 2$. In the second case, the image $C_0'$ of $C_0$ has $C_0'^2=-4$, hence $C_0'=S_0$ and $e=4$, then it is easy to see that $C_{10}'\equiv 3(S_0+4F)$.
In characteristic $p\neq 3$ the pull-back of the ruling on ${\bf F}_e$ gives an $h$-invariant elliptic fibration $\psi:X\to {\bf P}^1$. Each singular fibre has at most 2 components since it is the pull-back of a fibre of ${\bf F}_e$.
In characteristic $p=3$ we have to show that the pull-back is not a quasi-elliptic fibration. Suppose it is. The closure of the cusps of irreducible fibres is a smooth rational curve and must be fixed pointwise by $h$, then the genus 10 curve must be a section of the quasi-elliptic fibration, impossible.
The following easy lemmas also will be used frequently.
\[fix\] Let $S$ be a set and ${\rm Aut}(S)$ be the group of bijections of $S$. For any $g\in {\rm Aut}(S)$ and positive integers $a$ and $b$,
1. ${\rm Fix}(g)\subset {\rm Fix}(g^a)$;
2. ${\rm Fix}(g^a)\cap {\rm Fix}(g^b)={\rm Fix}(g^d)$ where $d=\gcd (a, b)$;
3. ${\rm Fix}(g)= {\rm Fix}(g^a)$ if ${\rm ord}(g)$ is finite and prime to $a$.
\[sum\] Let $R(n)$ be the sum of all primitive $n$-th root of unity in $\overline{{\mathbb{Q}}}$ or in $\overline{{\mathbb{Q}}_l}$. Then $$R(n)=\left\{\begin{array}{ccl} 0&{\rm if}& n\,{\rm has\,\, a\,\, square\,\, factor},\\
(-1)^t&{\rm if}& n\,{\rm is\,\, a\,\, product\,\, of}\,\,t\,\,{\rm distinct\,\, primes}.\\
\end{array} \right.$$
The following lemma will play a key role in our proof.
\[60\] Let $g$ be an automorphism of order $60$ of a K3 surface in characteristic $p\neq 2$, $3$, $5$. If $$[g^{*}]=[1,\,\zeta_{60}:16,\,\zeta_{12}:4,\,\pm 1],$$ then
1. there is a $g^{}$-invariant elliptic fibration $\psi:X\to {\bf P}^1$ with $12$ cuspidal fibres, say $F_{\infty}$, $F_0$, $F_{t_1}, \ldots, F_{t_{10}}$;
2. ${\rm Fix}(g^{30})$ consists of a section $R$ of $\psi$ and a curve $C_{10}$ of genus $10$ which is a $3$-section passing through each cusp with multiplicity $3$;
3. the action of $g$ on the base ${\bf P}^1$ is of order $10$, fixing $2$ points, say $\infty$ and $0$, and makes the $10$ points $t_1, \ldots, t_{10}$ to form a single orbit;
4. ${\rm Fix}(g^{10})=R\cup\,\{{\rm the\,
cusps\, of\, the\, 12\, cuspidal\, fibres}\};$
5. ${\rm Fix}(g^{12})={\rm Fix}(g)$ and it consists of the $4$ points, $$R\cap F_{\infty},\,\,R\cap F_{0},\,\,C_{10}\cap F_{\infty},\,\,C_{10}\cap F_{0};$$
6. $[g^{*}]=[1,\,\zeta_{60}:16,\,\zeta_{12}:4,\, 1].$
Note that $[g^{30*}]=[1,\,-1.16,\,-1.4,\, 1],$ and $$[g^{10*}]=[1,\,(\zeta_{6}:2).8,\,(\zeta_{6}:2).2,\, 1],\quad e(g^{10})=14.$$ Thus, we can apply Lemma \[nsym2\] to $h=g^{30}$. Since $${\rm Fix}(g^{d})\subset {\rm Fix}(g^{30})$$ for any $d$ dividing 30, we see that ${\rm Fix}(g^{10})$ consists of 14 points if ${\rm Fix}(g^{30})$ is irreducible. If ${\rm Fix}(g^{30})$ is a curve $C_{9}$ of genus $9$, then $g^{10}$ acts on $C_9$ with 14 fixed points, too many for an order 3 automorphism. Thus ${\rm Fix}(g^{30})$ consists of a section $R$ of a $g^{30}$-invariant elliptic fibration $$\psi:X\to {\bf P}^1$$ and a curve $C_{10}$ of genus $10$ which is a $3$-section. We know that $$X/\langle g^{30} \rangle\cong {\bf F}_4$$ a rational ruled surface. Every automorphism of ${\bf F}_4$, hence the one induced by $g$, preserves the unique ruling, so $g$ preserves the elliptic fibration. Let $a$ and $b$ be the number of singular fibres of type $I_1$ and $II$ respectively. Then $$a+2b=e(X)=24, \quad 12\le a+b\le 24.$$ Note that $g^{30}$ acts trivially on the base ${\bf P}^1$. Neither $g^5$ nor $g^6$ acts trivially on ${\bf P}^1$. Otherwise, ${\rm Fix}(g^{5})$ or ${\rm Fix}(g^{6})$ must contain the section $R$, the nodes of the nodal fibres and the cusps of the cuspidal fibres, too many, as we compute $e(g^{5})=3\pm 1$ and $e(g^{6})=4$. Our automorphism $g$ acts on the set of the base points of the $a+b$ singular fibres. An orbit of this action has length 1, 2, 3, 5, 6, 10 or 15, i.e. a divisor of 30. If an orbit has length 3, 5 or 6, then $g^5$ or $g^6$ fixes all points in the orbit, hence acts trivially on the base ${\bf P}^1$. Thus no orbit has length 3, 5, 6. If an orbit has length 15, then $a\ge 16$ and $g^2$ fixes more than two points on the base ${\bf P}^1$. We have proved that every orbit has length 1, 2, or 10. Then $g^{10}$ fixes all base points of the singular fibres. Thus it acts trivially on the base ${\bf P}^1$ and ${\rm Fix}(g^{10})$ contains $R$ and the nodes and the cusps of the singular fibres. Since $ e(g^{10})=14$, we infer that $a=0$ and $b=12$. Then the action of $g$ on the 12 base points of the cuspidal fibres has an orbit of length 10; otherwise $g^2$ would act trivially on the base. If the remaining two points, say $\infty$ and 0, are interchanged by $g$, then $g$ fixes 2 points on the base ${\bf P}^1$ away from the 12 points, then $g^2$ fixes 4 points on the base, so acts trivially on the base. Thus $g$ fixes $\infty$ and 0. This proves (1), (2) and (3).
The statement (4) follows from (3) and the fact that ${\rm Fix}(g^{10})$ has Euler number 14 and is contained in $R\cup C_{10}$.
By (3) ${\rm Fix}(g)$ consists of the $4$ points, hence $e(g)={\textup{Tr}}(g^*|H^*(X))=4$. Again, by (3) ${\rm Fix}(g^{12})$ is a subset of $F_{\infty}\cup F_{0}$. Since $e(g^{12})=4$, ${\rm Fix}(g^{12})$ cannot contain any point other than the 4 points of ${\rm Fix}(g)$. This proves (5) and (6).
Proof: the Tame Case
====================
Throughout this section, we assume that the characteristic $p>0$, $p\neq 2$, 3, 5 and $g$ is an automorphism of order $60$ of a K3 surface. We first prove that $g$ cannot be purely non-symplectic.
\[1.60\] ${\rm ord}(g)\neq 1.60$.
Suppose that ${\rm ord}(g)=1.60$. Then by Proposition \[integral’\] the action of $g^*$ on $H_{\rm et}^2(X,{\mathbb{Q}}_l)$, $l\neq{\rm char}(k)$, has $\zeta_{60}\in \overline{{\mathbb{Q}}_l}$ as an eigenvalue and $$[g^{*}]=[1,\, \zeta_{60}:16,\, \eta_1,\ldots, \eta_5]$$ where $[\eta_1,\ldots, \eta_5]$ is a combination of $\zeta_{12}:4$, $\zeta_{10}:4$, $\zeta_{5}:4$, $\zeta_{6}:2$, $\zeta_{4}:2$, $\zeta_{3}:2$, $\pm 1$, and the first eigenvalue corresponds to a $g$-invariant ample divisor.
Claim 1: $[\eta_1,\ldots, \eta_5]\neq [\zeta_{10}:4,\,\pm 1]$, $[\zeta_{5}:4,\,\pm 1]$.\
Suppose that $[\eta_1,\ldots, \eta_5]=[\zeta_{10}:4,\,\pm 1]$ or $[\zeta_{5}:4,\,\pm 1]$. Then Lefschetz fixed point formula gives $$e(g^{30})={\textup{Tr}}(g^{30*}|H^*(X))=-8$$ and ${\rm Fix}(g^{30})$ consists of $d$ smooth rational curves and a curve $C_{d+5}$ of genus $d+5$. We have $0\le d\le 5$, since each fixed curve gives an invariant vector in $\dim H_{\rm et}^2(X,{\mathbb{Q}}_l)$. Note that $e(g^2)={\textup{Tr}}(g^{2*}|H^*(X))=1$. Since ${\rm Fix}(g^2)\subset{\rm Fix}(g^{30})$, we infer that ${\rm Fix}(g^2)$ consists of a point. Note that $C_{d+5}\nsubseteq
{\rm Fix}(g^{10})$, since $e(g^{10})=16> e(g^{30})$. If $d=1, 2$ or $4$, then $g$ acts on the $d$ smooth rational curves and $g^2$ preserves at least one of them, hence fixes at least 2 points. If $d=3$, then $g$ must rotate the $3$ smooth rational curves and $g^{10}$ acts on the curve $C_8$ with $16$ fixed points, which is impossible. If $d=0$, then $g^{10}$ gives an order 3 automorphism of the curve $C_5$ with $16$ fixed points, impossible. If $d=5$, then $g$ must rotate the $5$ smooth rational curves and $g^{5}$ preserves each of them, hence $e(g^{5})\ge 10$. But ${\textup{Tr}}(g^{5*}|H^*(X))\le 8$, contradicting the Lefschetz fixed point formula.
Claim 2: $[\eta_1,\ldots, \eta_5]\neq [\zeta_{6}:2,\,\pm 1,\,\pm
1,\,\pm 1]$, $[\zeta_{3}:2,\,\pm 1,\,\pm 1,\,\pm 1]$.\
Suppose that $[\eta_1,\ldots, \eta_5]=[\zeta_{6}:2,\,\pm 1,\,\pm 1,\,\pm 1]$ or $[\zeta_{3}:2,\,\pm 1,\,\pm 1,\,\pm 1]$. This case can be handled similarly. We see that $e(g^{30})=-8$ and ${\rm
Fix}(g^{30})$ consists of $d$ smooth rational curves and a curve $C_{d+5}$ of genus $d+5$, $0\le d\le 5$. We also see that $e(g^2)=3$ and ${\rm Fix}(g^2)$ consists of either 3 points or a point and a ${\mathbb{P}}^1$. Note that $C_{d+5}\nsubseteq {\rm
Fix}(g^{10})$, since $e(g^{10})=13> e(g^{30})$. If $d=0$ or 1, then $g^{10}$ gives an order 3 automorphism of the curve $C_{d+5}$ with at least $11$ fixed points, which is impossible. If $d=2$, then $g^2$ preserves 2 smooth rational curves, hence fixes at least 4 points. If $d=3$, then $g$ must rotate the $3$ smooth rational curves and $g^{10}$ acts on the curve $C_8$ with $13$ fixed points, impossible. If $d=4$, then $g^{3}$ preserves each of them, hence $e(g^{3})\ge 8$ or $e(g^{3})=8+e(C_9)=-8$, which is possible only if $[g^{*}]=[1,\,\zeta_{60}:16,\,\zeta_{3}:2,\,1,\,1,\,1]]$. Then $e(g)=5> e(g^{2})$, but ${\rm Fix}(g)$ and ${\rm Fix}(g^{2})$ consist of isolated points and some ${\mathbb{P}}^1$’s. If $d=5$, then $g$ must rotate the $5$ smooth rational curves and $g^{5}$ preserves each of them, hence $e(g^{5})\ge 10$. But ${\textup{Tr}}(g^{5*}|H^*(X))\le 7$, contradicting the Lefschetz formula.
Claim 3: $[\eta_1,\ldots, \eta_5]\neq [(\zeta_{6}:2).2,\,\pm 1]$, $[(\zeta_{3}:2).2,\,\pm 1]$, $[\zeta_{6}:2,\,\zeta_{3}:2,\,\pm
1]$.\
Suppose that $[\eta_1,\ldots, \eta_5]=[(\zeta_{6}:2).2,\,\pm
1]$, $[(\zeta_{3}:2).2,\,\pm 1]$ or $[\zeta_{6}:2,\,\zeta_{3}:2,\,\pm 1]$. Note that $e(g^{30})=-8$ and ${\rm Fix}(g^{30})$ consists of $d$ smooth rational curves and a curve $C_{d+5}$ of genus $d+5$, $0\le d\le
5$. We see that $e(g^2)=0$. Since ${\rm
Fix}(g^{2})\subseteq {\rm Fix}(g^{30})$, ${\rm
Fix}(g^2)=\emptyset$, thus ${\rm Fix}(g)=\emptyset$ and $[g^{*}]=[1,\,\zeta_{60}:16,\,(\zeta_{3}:2).2,\,-1]$. Note that $C_{d+5}\nsubseteq {\rm Fix}(g^{10})$, since $e(g^{10})=10> e(g^{30})$. If $d=0$, then $g^{10}$ gives an order 3 automorphism of the curve $C_{5}$ with $10$ fixed points, which is impossible. If $d=1, 2$ or 4, then $g^2$ preserves at least one smooth rational curve, hence fixes at least 2 points. If $d=3$, then $g$ must rotate the $3$ smooth rational curves, hence $g^{15}$ acts freely on the curve $C_8$, since $e(g^{15})=6$. But no genus 8 curve admits a free involution. If $d=5$, then $g$ must rotate the $5$ smooth rational curves and $g^{5}$ preserves each of them, hence $e(g^{5})\ge 10$. But ${\textup{Tr}}(g^{5*}|H^*(X))=0$.
Claim 4: $[\eta_1,\ldots, \eta_5]\neq
[\zeta_{4}:2,\,\zeta_{6}:2,\,\pm 1]$, $[\zeta_{4}:2,\,\zeta_{3}:2,\,\pm 1]$.\
Suppose that $[\eta_1,\ldots, \eta_5]=[\zeta_{4}:2,\,\zeta_{6}:2,\,\pm 1]$ or $[\zeta_{4}:2,\,\zeta_{3}:2,\,\pm 1]$. In this case, $e(g^{30})=-12$ and ${\rm Fix}(g^{30})$ consists of $d$ smooth rational curves and a curve $C_{d+7}$ of genus $d+7$, $0\le d\le
3$. We compute $$e(g^2)={\textup{Tr}}(g^{2*}|H^*(X))=-1> e(g^{30}),$$ hence $C_{d+7}\nsubseteq {\rm Fix}(g^{2})$. But then $e(g^2)\ge 0$.
Claim 5: $[\eta_1,\ldots, \eta_5]\neq [(\zeta_{4}:2).2,\,\pm 1]$.\
Suppose that $[\eta_1,\ldots, \eta_5]=[(\zeta_{4}:2).2,\,\pm 1]$. In this case, $e(g^{30})=-16$ and ${\rm Fix}(g^{30})$ consists of $d$ smooth rational curves and a curve $C_{d+9}$ of genus $d+9$, $0\le d\le 1$. Since $e(g^2)=-2>
e(g^{30}),$ $C_{d+9}\nsubseteq {\rm Fix}(g^{2})$, but then $e(g^2)\ge 0$.
Claim 6: $[\eta_1,\ldots, \eta_5]\neq [\zeta_{4}:2,\,\pm 1,\,\pm
1,\,\pm 1]$.\
Suppose that $[\eta_1,\ldots,
\eta_5]=[\zeta_{4}:2,\,\pm 1,\,\pm 1,\,\pm 1]$. In this case, $e(g^{30})=-12$ and ${\rm Fix}(g^{30})$ consists of $d$ smooth rational curves and a curve $C_{d+7}$ of genus $d+7$, $0\le
d\le 3$. We compute $$e(g^2)={\textup{Tr}}(g^{2*}|H^*(X))=2> e(g^{30}),$$ hence $C_{d+7}\nsubseteq {\rm Fix}(g^{2})$ and ${\rm
Fix}(g^{2})$ consists of either 2 points or a ${\mathbb{P}}^1$, since ${\rm Fix}(g^{2})\subset{\rm Fix}(g^{30})$. Since ${\rm
Fix}(g)\subset{\rm Fix}(g^{2})$, we infer that $$e(g)=2\,\,\,{\rm or}\,\,\, 0.$$ By computing $[g^{15*}]$ and $[g^{10*}]$, we see that $$e(g)=e(g^{15})\,\,\,{\rm and}\,\,\, e(g^{10})=12.$$ If $d=0$, then $g^{10}$ gives an order 3 automorphism of the curve $C_{7}$ with $12$ fixed points, impossible. If $d=2$, then $g^2$ preserves both smooth rational curves, hence $e(g^2)\ge 4$. If $d=3$, then $g^2$ cannot preserve two of the three smooth rational curves, hence $g$ must rotate the three, then $g^{15}$ preserves each of the three, hence $e(g^{15})\ge 6$. If $d=1$, then $g^{15}$ acts freely on the curve $C_8$. But no genus 8 curve admits a free involution.
Claim 7: $[\eta_1,\ldots, \eta_5]\neq [\pm 1,\,\pm 1,\,\pm 1,\,\pm
1,\,\pm 1]$.\
Suppose that $[\eta_1,\ldots, \eta_5]=[\pm 1,\,\pm
1,\,\pm 1,\,\pm 1,\,\pm 1]$. In this case,\
$e(g^{30})=-8$ and ${\rm Fix}(g^{30})$ consists of $d$ smooth rational curves and a curve $C_{d+5}$ of genus $d+5$, $0\le d\le
5$. We also compute $$e(g^2)=6,\,\,\,e(g^{15})=e(g), \,\,\,e(g^{10})=16.$$ Since $e(g^2)>e(g^{30})$, we see that $C_{d+5}\nsubseteq {\rm Fix}(g^{2})$ and $$e(g^{15})=e(g)\le e(g^2)=6.$$ If $d\le
2$, then $g^{10}$ gives an order 3 automorphism of the curve $C_{d+5}$ with $16-2d$ fixed points, which is impossible. Assume $d\ge 4$. If $g^{15}$ preserves at least 4 of the $d$ smooth rational curves, then $e(g^{15})\ge 8>6$. If $g^{15}$ preserves at most 2 of the $d$ smooth rational curves, then $g^{2}$ preserves at least 4, hence $e(g^{2})\ge 8>6$. If $g^{15}$ preserves exactly 3 of the $d$ smooth rational curves, then $d=5$ and $g^{15}$ acts freely as an involution on the curve $C_{10}$, a contradiction. Assume $d=3$. If $g$ rotates the $3$ smooth rational curves or fixes each of them, then $g^{15}$ fixes each of them, hence acts freely on the curve $C_8$, a contradiction. If $g$ fixes exactly one of the $3$ smooth rational curves, then $g^2$ fixes each of them, hence acts freely on the curve $C_8$, then $g$ acts freely on the curve $C_8$ and $e(g)=2$, then $g^{15}$ has $e(g^{15})=2$, hence acts freely on the curve $C_8$. This proves the claim.
We may assume that $[g^{*}]=[1,\,\zeta_{60}:16,\,\zeta_{12}:4,\,\pm 1].$ Then by Lemma \[60\] $$[g^{*}]=[1,\,\zeta_{60}:16,\,\zeta_{12}:4,\, 1].$$ Consider the order 5 automorphism $g^{12}$. It is non-symplectic and the quotient $$X':=X/\langle g^{12}\rangle$$ is a singular rational surface with $K_{X'}$ numerically trivial. Furthermore, by Proposition \[diminv\] Picard number $\rho(X')=6$ .
Claim 8: $X'=X/\langle g^{12} \rangle$ has four singular points, one of type $\frac{1}{5}(3,3)$ and three of type $\frac{1}{5}(2,4)$.\
To prove the claim, note first that ${\rm Fix}(g^{12})$ consists of the 4 points from Lemma \[60\], 2 points of $R$ and 2 points of $C_{10}$. Since $$g^{12*}\omega_X=\zeta_5\omega_X\,\,\, {\rm for\,\, some}\,\, \zeta_5\in k,$$ there are two types of local action of $g^{12}$ at a fixed point, $\frac{1}{5}(3,3)$ and $\frac{1}{5}(2,4)$. Let $a$ and $b$ be the number of points respectively of the two types. Then $$a+b=4.$$ Let $\varepsilon: Y\to X'$ be a minimal resolution. Then $$K_Y=\varepsilon^*K_{X'}-\sum D_p$$ where $D_p$ is an effective ${\mathbb{Q}}$-divisor supported on the exceptional set of the singular point $p\in X'$. Here $``="$ means numerical equivalence. Thus $$K_Y^2=\sum D_p^2=-\sum K_YD_p.$$ See, e.g., Lemma 3.6 [@HK1] for the formulas of $D_p$ and $K_YD_p$, which are valid not only in the complex case but also for tame quotient singular points in positive characteristic. We compute $$K_Y^2= 10-\rho(Y)= 10-\{\rho(X')+a+2b\}=4-a-2b.$$ On the other hand, $K_YD_p=\frac{9}{5}$ if $p$ is of type $\frac{1}{5}(3,3)$, and $K_YD_p=\frac{2}{5}$ if $p$ is of type $\frac{1}{5}(2,4)$, thus $$K_Y^2= -\frac{9}{5}a-\frac{2}{5}b.$$ Solving the system, we get $a=1$ and $b=3$. This proves the claim.
Now by Claim 8, we compute that $$K_Y=-\frac{3A}{5}-\sum_{i=1}^3 \frac{A_{1i}+2A_{2i}}{5}$$ where $A$ and $A_{ji}$ are exceptional curves with $A^2=-5$, $A_{1i}^2=-2$, $A_{2i}^2=-3$, $A_{1i}.A_{2i}=1$. If the 2 points of $R$ are of type $\frac{1}{5}(2,4)$, then the proper transform $R'$ of the image of $R$ in $X'$ has intersection number with $K_Y$, $$K_Y.R'=-\frac{1}{5}-\frac{1}{5}, \,\,-\frac{1}{5}-\frac{2}{5}\,\,{\rm or}\,\,-\frac{2}{5}-\frac{2}{5},$$ none is an integer. If the 2 points of $C_{10}$ are of type $\frac{1}{5}(2,4)$, then the proper transform $C_{10}'$ of the image of $C_{10}$ in $X'$ has intersection number with $K_Y$ which cannot be an integer, a contradiction.
\[2.30\] ${\rm ord}(g)\neq 2.30$, $3.20$, $4.15$, $6.10$.
These cases are much simpler than the previous one, and are contained in [@K], Lemma 4.5 and 4.7.
\[5.12\] If ${\rm ord}(g)=5.12$, then $$[g^{*}]=[1,\,\zeta_{12}:4,\,1,\,\zeta_{60}:16].$$
Since $g^{12}$ is symplectic of order 5, $$[g^{12*}]=[1,\,1.5,\, (\zeta_5:4).4]$$ and for any positive integer $a$ dividing 12, ${\rm Fix}(g^{a})\subset {\rm Fix}(g^{12})$ and $$0\le e(g^{a})\le e(g^{12})=4.$$ By Proposition \[integral’\], $\zeta_{12}\in [g^{*}]$. Thus we infer that $$[g^{*}]=[1,\,\zeta_{12}:4,\,\pm 1,\, \eta_1,\ldots, \eta_{16}]$$ where $[\eta_1,\ldots, \eta_{16}]$ is a combination of $\zeta_{5}:4$, $\zeta_{10}:4$, $\zeta_{15}:8$, $\zeta_{20}:8$, $\zeta_{30}:8$, $\zeta_{60}:16$ and the first eigenvalue corresponds to a $g$-invariant ample divisor.
Assume that $[\eta_1,\ldots, \eta_{16}]$ contains $[\zeta_{15}:8]$ or $[\zeta_{30}:8]$. Then $$[g^{2*}]=[1,\, (\zeta_6:2).2,\,1,\,\zeta_{15}:8,\,\tau_1,\ldots, \tau_8]$$ where $[\tau_1,\ldots, \tau_8]$ is a combination of $\zeta_{5}:4$, $\zeta_{10}:4$, $\zeta_{15}:8$, hence $\sum\tau_j\ge -2$ and $e(g^2)={\textup{Tr}}(g^{2*}|H^*(X))=7+\sum\tau_j\ge 5$, contradicting $e(g^{2})\le 4.$
Assume that $[\eta_1,\ldots, \eta_{16}]$ contains $[\zeta_{20}:8]$. In this case, $$[g^{2*}]=[1,\, (\zeta_6:2).2, \, 1,\,(\zeta_{10}:4).2,\,\tau_1,\ldots,
\tau_8]$$ where $[\tau_1,\ldots, \tau_8]$ is a combination of $\zeta_{5}:4$, $\zeta_{10}:4$, $\zeta_{15}:8$. Since $\sum\tau_j\ge -2$, $e(g^2)={\textup{Tr}}(g^{2*}|H^*(X))=8+\sum\tau_j\ge 6$, contradicting $e(g^{2})\le 4.$
Assume that $[\eta_1,\ldots, \eta_{16}]$ is a combination of $\zeta_{5}:4$, $\zeta_{10}:4$. Then $$[g^{6*}]=[1,\, -1.4, \, 1,\,(\zeta_{5}:4).4]$$ and $e(g^6)={\textup{Tr}}(g^{6*}|H^*(X))=-4,$ contradicting $e(g^{6})\ge 0$.\
Therefore $[\eta_1,\ldots, \eta_{16}]=[\zeta_{60}:16]$. Now Lemma \[60\] applies.
[**Proof of Theorem \[main\].**]{}
\(1) follows from Lemmas \[1.60\] and \[2.30\].
\(2) We know ${\rm ord}(g)=5.12$. By Lemma \[5.12\] we can apply Lemma \[60\], and will use the elliptic structure and the notation there. Let $$y^2+x^3+A(t_0,t_1)x+B(t_0,t_1) = 0$$ be the Weierstrass equation of the $g$-invariant elliptic pencil, where $A$ (resp. $B$) is a binary form of degree $8$ (resp. $12$). By Lemma \[60\], $g$ leaves invariant the section $R$ and the action of $g$ on the base of the fibration $\psi:X\to {\bf P}^1$ is of order 10. After a linear change of the coordinates $(t_0,t_1)$ we may assume that $g$ acts on the base by $$g:(t_0,t_1)\mapsto (t_0,\zeta_{60}^6t_1).$$ We know that $g$ preserves two cuspidal fibres $F_0$, $F_{\infty}$ and makes the remaining 10 cuspidal fibres to form one orbit. Thus the discriminant polynomial $$\Delta =
-4A^3-27B^2=ct_0^2t_1^2(t_1^{10}-t_0^{10})^2$$ for some constant $c\in k$, as it must have two double roots (corresponding to the fibres $F_0$, $F_{\infty}$) and one orbit of double roots. We know that the zeros of $A$ correspond to either cuspidal fibres or nonsingular fibres with “complex multiplication” automorphism of order 6. Since this set is invariant with respect to the order 10 action of $g$ on the base, we see that the only possibility is $A
= 0$. Then the above Weierstrass equation can be written in the form $$y^2+x^3+at_0t_1(t_1^{10}-t_0^{10}) = 0$$ for some constant $a$. A suitable linear change of variables makes $a=1$ without changing the action of $g$ on the base. Thus $$X \cong X_{60}$$ as an elliptic surface. We may assume that $$g^*\Big(\frac{dx\wedge dt}{y}\Big)=\zeta_{60}^5\frac{dx\wedge dt}{y}$$ for some primitive 12th root of unity $\zeta_{60}^5$. Here a choice of such a root of unity is equivalent to a choice of a generator of the cyclic group $\langle g\rangle$. Since $g^{10}$ is of order 6 and acts trivially on the base, it is a complex multiplication of order 6 on a general fibre, so $$g^{10}(x, y, t_0, t_1)=(\zeta_6^2x, \zeta_6^3y, t_0, t_1).$$ Note that $${\rm Fix}(g)=\{{\rm the\,
two\,cusps\, of}\,F_0\,{\rm and}\, F_{\infty}\}\cup (R\cap
F_0)\cup (R\cap F_{\infty}).$$ Analysing the local action of $g$ at the fixed point $(x, y, t_0, t_1)=(0,0,1,0)$, the cusp of $F_0$, we infer that $$g(x, y, t_0, t_1)=(\zeta_{60}^2x, \zeta_{60}^3y, t_0,
\zeta_{60}^6t_1).$$ Here we first determine the linear terms, then see that the higher degree terms must vanish. This completes the proof of Theorem \[main\] in the positive characteristic case.
Proof: the Complex Case
=======================
Throughout this section, $X$ is a complex K3 surface.
A non-projective K3 surface cannot admit a non-symplectic automorphism of finite order (see [@Ueno], [@Nik]), and its automorphisms of finite order are symplectic, hence of order $\le 8$. Thus we may assume that $X$ is projective. The proofs of Lemma \[nsym2\], \[60\], \[1.60\], \[2.30\] and \[5.12\] go word for word, once the $l$-adic cohomology $H^2_{\rm et}(X,{\mathbb{Q}}_l)$ is replaced by the integral singular cohomology $H^2(X,{\mathbb{Z}})$, and Proposition \[trace\] by the usual topological Lefschetz fixed point formula. “Proof of Theorem \[main\]" also goes word for word.
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[^1]: Research supported by National Research Foundation of Korea (NRF grant).
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---
abstract: 'We study the workload processes of two restricted M/G/1 queueing systems: in Model 1 any service requirement that would exceed a certain capacity threshold is truncated; in Model 2 new arrivals do not enter the system if they have to wait more than a fixed threshold time in line. For Model 1 we obtain several results concerning the rate of convergence to equilibrium. In particular we derive uniform bounds for geometric ergodicity with respect to certain subclasses. However, we prove that for the class of all Model 1 workload processes there is no uniform bound. For Model 2 we prove that geometric ergodicity follows from the finiteness of the moment-generating function of the service time distribution and derive bounds for the convergence rates in special cases. The proofs use the coupling method.'
address:
- 'University of Warwick, Department of Statistics, Coventry, CV4 7AL, United Kingdom'
- 'Institute of Mathematics, University of Osnabrück, Albrechtstr. 28A, 49076 Osnabrück, Germany'
- 'Institute of Mathematics, University of Osnabrück, Albrechtstr. 28A, 49076 Osnabrück, Germany'
author:
- Martin Kolb
- Wolfgang Stadje
- Achim Wübker
title: 'The Rate of Convergence to Stationarity for M/G/1 Models with Admission Controls via Coupling'
---
Introduction
============
In this paper we consider the long-run behavior of the workload processes $V_t$ of the two most important $M/G/1$ queueing systems with admission restrictions. We are interested in the rate of convergence toward the equilibrium (stationary) distribution $\pi$ and measure this rate in terms of the total variation distance, which is defined as $$\label{main_quantity}
d(x,t)=||\mathbb{P}_x \bigl(V_t \in\cdot\bigr)-\pi||_{TV}=\sup_{A\in\mathcal{B}}|\mathbb{P}_x
\bigl(V_t\in A\bigr)-\pi(A)|,$$ where of course $\mathbb{P}_{x}\bigl(V_t\in A\bigr)= \mathbb{P}\bigl(V_t\in A \mid
V_0=x\bigr)$ and $\mathcal{B}$ is the Borel $\sigma$-field in $\mathbb{R}_+$. The main purpose of this paper is the investigation of $d(x,t)$ as $t\rightarrow\infty$ for two $M/G/1$-type models. Let $T_{n}$ be the arrival time of the $n$th customer at the queue and $T_{0}=0$. The inter-arrival times $I_i=T_i-T_{i-1},i\in\mathbb{N}$, are assumed to be i.i.d. and exponential with mean $1/\lambda$. Let $S_n$ be the service requirement of the $n$th customer; $(S_n)_{n\in\mathbb{N}}$ is assumed to be an i.i.d. sequence with common distribution $G$.\
[**Model I: truncated service at the capacity limit**]{}. The workload process $V^{1,x}_t$ of this $M/G/1$ queue in a system with capacity 1 is formally defined by $$\label{M/G/1}
V^{1,x}_t=\left\{\begin{array}{cc}
x & t=0\\
\max[V^{1,x}_{T_{n-1}}-(t-T_{n-1}),0],& T_{n-1}\le t<T_n,n\ge 1\\
V^{1,x}_{T_n -}+ \min [S_n + V^{1,x}_{T_n -},1], &t=T_n,n\ge 1
\end{array}
\right.$$ This model, which has been referred to as the “*truncated service policy*” in the literature (see e.g. [@PeStaZa]), can be described as follows: whenever the total workload would increase beyond the capacity threshold, it is reduced such that this threshold is exactly reached but not exceeded. Note that under this rule every customer is admitted to the system.\
[**Model 2: bounded waiting time policy**]{}. In the second model new arrivals whose waiting time in line would exceed some constant are not admitted to enter the system. According to this policy, admission is interrupted as long as the workload process stays above the threshold, say 1. The workload process is thus given by $$\label{defiV2}
V^{2,x}_t=\left\{\begin{array}{cc}
x & t=0\\
\max[V^{2,x}_{T_{n-1}}-(t-T_{n-1}),0],& T_{n-1}\le t<T_n,n\ge 1,\\
V^{2,x}_{T_n -}+S_{n}\mathbf{1}_{\{V^{2,x}_{T_n-}<1\}},&t=T_n,n\ge1.
\end{array}
\right.$$ Note that the distribution of $V^2_{x_t}$ has support $[0,\infty)$ if $G$ has unbounded support.\
A comprehensive account of Model 1 for interarrival and service time distributions with rational Laplace-Stieltjes transforms (LSTs) was already given by Cohen in his monograph [@Co] (Ch. III.5). His method is based on Pollaczek’s classical contour integral equation which, in the case of rational LSTs, leads to explicit, albeit very complicated formulas. In [@PeStaZa] the busy period distributions in the $M/G/1$ and in the $G/M/1$ case are derived directly in terms of certain transforms of the underlying distributions. Early papers on the waiting times in Model 2 are [@Da; @C1; @LT; @GS; @Ho]. In the more general context of queues with state-dependent arrival and service rates some aspects of restricted $M/G/1$ queues were investigated in [@GK]. For other related models (e.g. partial refusal of overload work) see [@BPSZ].
Investigations concerning the rate of convergence to equilibrium for queueing systems have a long history, see e.g. [@KaMc2; @Ca2; @Ch; @StPa; @VaZe1; @VaZePa; @GaGo]. Much of this work is based on the spectral representation for birth and death processes due to Karlin and McGregor [@KaMc1], whose application requires exponentially distributed service times, so that this technique works well for $M/M/1$, $M/M/n$ and $M/M/\infty$, but is not applicable to $M/G/1$-type queues.
Our approach is based on the [*coupling method*]{}, which turns out to be flexible enough for dealing with general service distributions. In [@Th1; @Th2; @LuMeTw] coupling has been used to estimate convergence rates to equilibrium for standard $M/G/1$ queues without boundary modifications, but our construction is different. To the best of our knowledge, convergence rates for the processes $V_t^{1,x}$ and $V_t^{2,x}$ defined above have not yet been derived.
The paper is organized as follows. In Section 2 we analyze the asymptotic behavior of $V_t^{1,x}$ for $t\rightarrow \infty$. We determine the density $\tilde{\pi}$ of the invariant distribution $\pi$ and give a new formula for the distribution function of $\pi$. (Another expression was derived in [@Co] and [@Da] by different methods.) Then the general coupling method and the associated coupling inequality that will be used in this paper is presented. We show uniform ergodicity with respect to the arrival rate and to $G\in \mathcal{G}_{\rho,p}=\{G\in\mathcal{G}:G[\rho,\infty)\ge p\}$) ($\rho, p >0 $ fixed) and also with respect to all service time distributions for fixed $\lambda>0$. However, uniformity fails to hold over all $\lambda$ and $G$. At the end of Section 2 we discuss two examples. Section 3 is devoted to Model 2. We derive the invariant density, prove that geometric ergodicity follows from the finiteness of the moment-generating function of the service time distribution, and derive a bound for the convergence rate in the case of bounded service times.
Analysis of Model 1 {#section_1}
===================
The invariant distribution
--------------------------
The Markov process $V_t^{1,x}$ is geometrically ergodic and therefore has an uniquely determined invariant distribution $\pi$ satisfying $$d(x,t)=||\mathbb{P}\bigl(V_t^{1,x}\in \cdot\bigr)-\pi||_{TV}\le C_{x,\alpha} \exp(-\alpha t), \ \ \ t\ge 0,\ x\in[0,1]$$ for certain constants $\alpha >0$ and $C_{x,\alpha}>0$. To see this, let $\tilde{T}_i$ be the time of the $i$th arrival of $V_t^{1,x}$ to $1$. Clearly $(V_{\tilde{T}_i+t}^{1,x})_{t\ge 0}$ has the same distribution for all $i\in\mathbb{N}$, i.e., $1$ is a regenerative point. It follows from the general theory of regenerative processes (see e.g. [@As], Ch. 6) that if $$\label{y1} Y_1=\tilde{T}_2-\tilde{T}_1$$ is spread-out and $\mathbb{E}\bigl(Y_1\bigr)<\infty$, then the Markov process $V_t^{1,x}$ is geometrically ergodic with uniquely determined invariant distribution $\pi$. In our case the spread-out condition as well as the finiteness of the expectation of $Y_1$ are clearly satisfied. Of course, $\pi$ is also the asymptotic distribution of $V^{1,x}_t$ as $t\to \infty$ (see e.g. [@MeTw], [@Nu]).
The invariant measure can be immediately written down in the form $$\label{PiwithY}
\pi(A)=\frac{1}{\mathbb{E}_{1}\bigl(Y_1\bigr)}\int_{0}^{\infty}
\mathbb{P}\bigl(V_t^{1,1}\in A \mid Y_1>t\bigr)\mathbb{P}_{1}\bigl(Y_1>t\bigr)dt.$$ Eq. expresses $\pi$ in terms of the transient distributions of $V_t^{1,1}$; it is not very useful for explicit computations (except possibly for simulations). A formula expressing $\pi$ in terms of the system primitives $\lambda$ and $G$ is also well-known (see [@Co] and [@Da]): we have for the invariant distribution function $$\label{Pi}
\pi(x)=\frac{\sum_{n=0}^{\infty}\int_{0}^{x}\dfrac{e^{\lambda(x-u)}[-\lambda(x-u)]^n}{n!}
dG_{n}(u)}{\sum_{n=0}^{\infty}\int_{0}^{1}\dfrac{e^{\lambda(x-u)}[-\lambda(x-u)]^n}{n!}dG_{n}(u)},\,\,\,0\le x\le 1,$$ where $G_{n}$ is the $n$fold convolution of $G$ with itself and $\pi(x)$ is an abbreviation for $\pi[0,x]$.\
A quick and neat direct approach leading to the [*density*]{} $\bar{\pi}$ of $\pi$ on $(0,1]$, and then via integration also to a new explicit formula for $\pi (x)$, is as follows. By the standard level crossing technique (see e.g. [@brill]), $\bar{\pi} (x) $ is equal to the downcrossing rate of level $x$, which in turn is equal to the upcrossing rate of $x$. An upcrossing of $x$ occurs if for some $y\in [0,x)$ a customer with a service requirement of size larger than $x-y$ arrives and the current workload is equal to $y$. Hence, setting $\bar{G}(x)=1-G(x)$, $$\label{flow_out}
\bar{\pi} (x)=\int_{0}^{x}\mathbb{P}\bigl(S_1>x-y\bigr)\lambda\pi(dy)=\lambda\pi(0)\bar{G}(x)+\lambda
(\bar{G}\ast\bar{\pi})(x).$$ Iteration yields, for every $n\in \mathbb{N}$, $$\begin{aligned}
\label{tildepi}
\bar{\pi}(x)&=&\lambda\pi(0)\bar{G}(x)+\lambda\bar{G}\ast(\lambda\pi(0)\bar{G}+\lambda\bar{G}\ast\bar{\pi})(x)\nonumber\\
&=&\ldots
=\pi(0)\sum_{i=1}^{n}\lambda^{i}\bar{G}^{\ast i}(x) + \lambda^n(\bar{G}^{\ast n}\ast\bar{\pi})(x). \end{aligned}$$ Since the left-hand side of is finite and all terms are nonnegative it follows that $\sum_{i=1}^{\infty}\lambda^{i}\bar{G}^{\ast i}(x) <\infty$ and, consequently, $\lim_{n\to \infty} \lambda^n(\bar{G}^{\ast n}\ast\bar{\pi})(x)=0$. We thus obtain $$\begin{aligned}
\label{tildepi1}
\bar{\pi}(x)&=& \pi(0)\sum_{i=1}^{\infty}\lambda^{i}\bar{G}^{\ast i}(x). \end{aligned}$$ $\pi(0)$ can be computed by taking the integral on both sides: $$1-\pi(0)= \pi(0)\sum_{i=1}^{\infty}\lambda^{i}\int_{0}^1\bar{G}^{\ast i}(x)dx.$$ This yields $$\label{pi_0}
\pi(0)=\frac{1}{1+\sum_{i=1}^{\infty}\lambda^{i}\int_{0}^1\bar{G}^{\ast i}(x)dx}.$$ We have proved
\[invariant\_density\] The density $\bar{\pi}$ of the invariant distribution $\pi$ for $x\in (0,1]$ is given by $$\label{invariant_density1}
\bar{\pi}(x)=\frac{1}{1+\sum_{i=1}^{\infty}\lambda^{i}\int_{0}^1\bar{G}^{\ast i}(x)dx}\sum_{i=1}^{\infty}\lambda^{i}\bar{G}^{\ast i}(x)
%\label{invariant_distribution}$$ and we have $$\pi(x)=\frac{1+\sum_{i=1}^{\infty}\lambda^{i}\int_{0}^{x}\bar{G}^{\ast i}(y)dy}{1+\sum_{i=1}^{\infty}\lambda^{i}\int_{0}^1\bar{G}^{\ast i}(x)dx}.$$
The rate of convergence to equilibrium and the coupling inequality {#speed}
------------------------------------------------------------------
We now prove that the process $V_t^{1,x}$ is uniformly geometrically ergodic, i.e., there exist constants $\alpha>0$ and $C=C_{\alpha}\in\mathbb{R}_{+}$ such that $$\label{uniform_ergodicity}
d(t):=\sup_{x\in[0,1]}d(x,t)=\sup_{x\in[0,1]}||\mathbb{P}\bigl(V_t^{1,x}
\in \cdot\bigr)-\pi||_{TV}\le C_{\alpha} \exp(-\alpha t).$$ In this case, $$\rho:=\limsup_{t\rightarrow\infty}d(t)^{\frac{1}{t}}\le e^{-\alpha}<1$$ and the quantity $1-\rho$ is called the [*spectral gap*]{}. We say that $V_t$ satisfies the spectral gap property (SGP) if $\rho < 1$. Different characterizations of uniform and geometric ergodicity can be found in the monograph [@MeTw]. For birth and death processes, sufficient conditions for geometric ergodicity are established in [@Va1] and a characterization of uniform ergodicity in terms of the birth and death rates can be found in [@Ma] and the references therein.
Let us describe the coupling method that will be used here. It can be easily established that $$\label{dt}
\sup_{x\in[0,1]}||\mathbb{P}\bigl(V_t^{1,x}\in
\cdot\bigr)-\pi||_{TV}\le \sup_{x,y\in[0,1]}||\mathbb{P}\bigl(V_t^{1,x}
\in \cdot\bigr)-\mathbb{P}\bigl(V_t^{1,y}\in \cdot\bigr)||_{TV}=:\bar{d}(t).$$ From the triangle inequality it follows that $$\label{barbound}
\frac{\bar{d}(t)}{2}\le d(t)\le \bar{d}(t),$$ and thus the asymptotics of $d(t)$ can be obtained from by those of $\bar{d}(t)$. There are two main reasons for considering $\bar{d}(t)$ instead of $d(t)$. First, it is known that $\bar{d}(t)$ is sub-multiplicative (see [@LePeWi]) in the sense that $$\label{subm}
\bar{d}(t+s)\le \bar{d}(t)\bar{d}(s).$$ Second, $\bar{d}(t)$ can be studied without any knowledge of $\pi$, although it determines the convergence rate of $\mathbb{P}\bigl(V_t^{1,\cdot}\in \cdot\bigr)$ to $\pi$. The value of $\bar{d}(t)$ can be bounded by using the following standard coupling inequality: We have $$\begin{aligned}
||\mathbb{P}\bigl(V_{t}^{1,x}\in\cdot\bigr)-\mathbb{P}\bigl(V_{t}^{1,y}\in\cdot\bigr)||_{TV}&=&\sup_{A\in\mathcal{B}}|\mathbb{P}\bigl(V_t^{1,x}\in A\bigr)-\mathbb{P}\bigl(V_t^{1,y}\in A\bigr)|\nonumber\\
&\le&\sup_{A\in\mathcal{B}}\mathbb{P}\bigl(\{V_t^{1,x}\in A\}\cap \{V_t^{1,y}\notin A\}\bigr)\nonumber\\
&\le&\mathbb{P}\bigl(V_t^{1,x}\not=V_t^{1,y}\bigr)\nonumber\\
&=&\mathbb{P}\bigl(T^{x,y}>t\bigr),\end{aligned}$$ where $T^{x,y}=\inf\{t\ge 0:V_t^{1,x}=V_t^{1,y}\}$. This yields $$\label{coupling_inequalityI}
\bar{d}(t)\le\sup_{x,y}\mathbb{P}\bigl(T^{x,y}>t\bigr).$$ The strength of the above coupling inequality depends of course heavily on the choice of the coupling. In the following we will consider processes $V_t^{1,x}$ and $V_t^{1,y}$ that are based on the same sequences $(T_i)_{i\in\mathbb{N}}$ and $(S_i)_{i\in\mathbb{N}}$ of arrival times and service requirements. We immediately see that that $V_t^{1,x}\le V_t^{1,y}$ whenever $x<y$; hence $V_t^{1,x}$ is a stochastically ordered Markov process in the sense of [@LuMeTw]. This has the advantage that the coupling time $T^{x,y}$ can be related to certain hitting times as has been done for example by Lund and Tweedie [@LuTw; @LuMeTw; @RoTw; @ScTw]. These papers have been written in the setting of an unbounded state space, where uniform ergodicity mostly fails to be true, and focus on improving bounds that had been previously obtained by the Lyapunov function approach. Moreover, while in [@LuTw; @LuMeTw; @RoTw; @ScTw] the tails of the coupling time are bounded from above by the tails of the hitting times of the “minimal element” of the state space, in our setting a simultaneous consideration of hitting the minimal or the maximal element leads to the desired bounds.
Let us introduce the first times when the process that starts in $x$ hits $0$ or $1$, respectively: $$U_{0}^{x}:=\inf\{t\ge0:V_t^{1,x}=0\},\,\,\,\,U_{1}^{x}:=\inf\{t\ge0:V_t^{1,x}=1\}.
\nonumber$$ The following Lemma turns out to be very useful.
\[main\] $$\begin{aligned}
\mathbb{P}\bigl(T^{x,y}>t\bigr)&\le&\mathbb{P}\bigl(U_{0}^{1}\wedge U_{1}^{0}>t\bigr)\label{coupling_ineq}\\
&\le& \min(\mathbb{P}\bigl(U_{0}^{1}>t\bigr),\mathbb{P}\bigl( U_{1}^{0}>t\bigr))\label{coupling_ineq_II}.\end{aligned}$$
Since the coupling preserves the order, i.e., $$V_t^{1,x}\le V_t^{1,y}\,\,\forall x\le y,
\nonumber$$ it follows that $V_t^{1,y}=0$ implies $V_t^{1,x}=0$ and $V_t^{1,x}=1$ implies $V_t^{1,y}=1$.
How much is lost when working with and as upper bounds for the tails of $\mathbb{P}\bigl(T^{x,y}>t\bigr)$? In Example 1 below an application of results in the exact rate of convergence to equilibrium, while yields rates that are far from being optimal.
We start with establishing uniform ergodicity for $V_t^{1,x}$.
\[first\_cor\] For all $t\ge 1$, $$\label{un}
\sup_{x\in[0,1]}||\mathbb{P}(V_t^{1,x}\in\cdot)-\pi||_{TV}\le(1-e^{-\lambda})^{t}.$$
Using the above coupling we obtain by applying Lemma \[main\] that $$\mathbb{P}\bigl(T^{x,y}>1\bigr)\le
\mathbb{P}\bigl(T_{1}<1\bigr)=1-\mathbb{P}\bigl(T_{1}\ge1\bigr)=1-\mathbb{P}\bigl(U_{0}^1\le1\bigr)=1-e^{-\lambda}.$$ Hence, yields $$\bar{d}(1)\le 1-e^{-\lambda}.$$ Thus, by , $$\label{simple}
\bar{d}(t)\le\bar{d}(1)^{t}=(1-e^{-\lambda})^{t}$$ Now the assertion follows from and .
The bound in becomes poor for large $\lambda$. However, in this case the process reaches level 1 quickly so that one might expect that can be used to show, for fixed $G$, uniform ergodicity with respect to $\lambda$. The following result shows that an even stronger statement holds.
\[t:second\] For every $\beta> 0$ and $p>0$ the process $V_t^{1,x}$ has the SGP uniformly on $G\in\mathcal{G}_{\beta,p}=\{G\in\mathcal{G}:G(\beta,1]\ge p\}$ and uniformly in $\lambda$, i.e., for every $G\in\mathcal{G}_{\beta,p}$ and $\lambda >0$ the corresponding spectral gap $\rho = \rho (G,\lambda)$ satisfies $$\label{rho}
\rho (G,\lambda) \le 1-e^{-\lambda_0}<1,$$ where $\lambda_0=\lambda_0(p,\beta)$ is the unique solution $\lambda \in (1/p\beta ,\infty)$ of $$\label{unique}
1-e^{-\lambda}=(\lambda \,p)^{\frac{1}{\beta}}e^{1-\lambda\,p}.$$
As in Proposition 1 we can easily derive that for all $\lambda >0$ we have $$\label{not_optimal}
\mathbb{P}\bigl(U^1_0>t)\le \mathbb{P}\bigl(U^1_0>1)^t\le(1-e^{-\lambda})^t$$ However, the right-hand side of tends to zero as $\lambda\rightarrow\infty$ and hence does not yield any uniformity. Consider an arbitrary $G\in \mathcal{G}_{\beta,p}$ and define the process $\hat{V}_t^{1,x}$ as $V_t^{1,x}$ with the difference that
- All jumps of size $<\beta$ are not recognized
- All jumps of size $\ge\beta$ are reduced to size $\beta$.
Observe that the arrival times of the jumps of size $\beta$ form a Poisson process with intensity $\lambda p$ and that, obviously, $\hat{V}_t^{1,x}\le V_t^{1,x}$ for all $t\in\mathbb{R}_{+}$. Now let $\hat{U}_1^x$ be defined as $U_1^x$ but referring to $\hat{V}_t^{1,x}$ instead of $V_t^{1,x}$ in its definition. Then we have $\hat{U}_1^x>U_1^x$ and therefore $$\begin{aligned}
\label{probably_optimal}
\mathbb{P}\bigl(U_{1}^x>t\bigr)&\le&\mathbb{P}\bigl(U_{1}^0>t\bigr)\le\mathbb{P}\bigl(\hat{U}_1^0>t\bigr)\nonumber\\
&\le& \mathbb{P}\bigl(\mbox{ less than }\lceil\frac{1+t}{\beta}\rceil
\mbox{ jumps of size at least }\beta \mbox{ occur up to time }t\bigr)\nonumber\\
&=&1- \sum_{i=\lceil\frac{1+t}{\beta}\rceil}^{\infty}e^{-\lambda p t}\frac{(\lambda p t)^{i}}{i!}=\sum_{i=0}^{{\lceil\frac{1+t}{\beta}\rceil}-1}e^{-\lambda p t}\frac{(\lambda p t)^{i}}{i!}. \end{aligned}$$ Now Lemma \[main\] yields $$\label{to_be_optimized}
\mathbb{P}\bigl(T^{x,y}>t\bigr)\le
\min\bigl[(1-e^{-\lambda})^t,\sum_{i=0}^{{\lceil\frac{1+t}{\beta}\rceil}-1}e^{-\lambda p t}\frac{(\lambda p t)^{i}}{i!}\bigr]$$ and hence for $\rho=\rho (G,\lambda)$ $$\begin{aligned}
\label{to_be_optimized_II}
\rho (G,\lambda )
&=&\limsup_{t\rightarrow\infty}d(t)^{\frac{1}{t}}
\le\limsup_{t\rightarrow\infty}\mathbb{P}\bigl(T^{x,y}>t\bigr)^{\frac{1}{t}}\nonumber\\
&\le& \min\bigg( 1-e^{-\lambda},\limsup_{t\rightarrow\infty}
\Big(\sum_{i=0}^{{\lceil\frac{1+t}{\beta}\rceil}-1}e^{-\lambda p t}\frac{(\lambda p t)^{i}}{i!}\Big)^{\frac{1}{t}}\bigg)
\nonumber\\
&=& \min \Big(1-e^{-\lambda},\mathbf{1}_{\{\lambda p \le 1\}}+\mathbf{1}_{\{\lambda p > 1\}}(\lambda \,p)^{\frac{1}{\beta}}
e^{1-\lambda\,p}\Big). \end{aligned}$$ Let us consider the right-hand side of : While $\lambda \mapsto 1-e^{-\lambda}$, $\lambda \in (0,\infty)$, is strictly increasing from 0 to 1, the function $\lambda \mapsto
\mathbf{1}_{\{\lambda p \le 1\}}+\mathbf{1}_{\{\lambda p > 1\}}(\lambda \,p)^{\frac{1}{\beta}}e^{1-\lambda\,p}$ equals $1$ for $\lambda p\le 1$, is strictly increasing for $1< \lambda\,p\le\frac{1}{\beta}$ to a value larger than 1 and strictly decreasing to 0 for $\lambda\,p>\frac{1}{\beta}$. This implies that there exists a unique $\lambda_0 \in (1/p\beta ,\infty)$ for which holds true, and this $\lambda_0$ satisfies .
[**Remarks**]{}. 1. Observe that yields a lower bound for the spectral gap $1-\rho$ for every given triple $\lambda,p,\beta$.\
2. Since $\lambda_0>\frac{1}{p\,\lambda}$, the above lower bound for the spectral gap converges to 0 for fixed $p>0$ and $\beta\rightarrow 0$. Below we deal with the question whether geometric ergodicity holds uniformly on the set of [*all*]{} service time distributions.\
3. As another approach to compute an upper bound, one could try the following: $$\begin{aligned}
\label{not_easy_to_handle}
&&\mathbb{P}\bigl(U_{1}^{0}<t\bigr)\le\mathbb{P}\bigl(\sup_{s\le t}V_t^{1,0}<1\bigr)\nonumber\\
&\le&\sum_{i=0}^{\infty}\mathbb{P}\bigl(\sup_{s\le t}V_t^{1,0}<1|J_{t}=i\bigr)\mathbb{P}\bigl(J_{t}=i\bigr)\nonumber\\
&=&\sum_{i=0}^{\infty}
e^{-\lambda t} \lambda^{i}\int_{[0,1]^{i}}\mathbf{1}_{\{x_1<x_2<\ldots<x_i\le t\}}\nonumber\\
&&\mathbb{E}\bigl[[\,\,[\ldots[[S_1-(x_2-x_1)]^{+}+S_2-(x_3-x_2)]^{+}+\ldots]
\nonumber\\ && \hspace{5cm}
+S_i-(t-x_i)]^{+}\bigr]dx_1\ldots dx_i.
%\nonumber\\\end{aligned}$$ However, the calculation of the integral in seems to be difficult.
Some special cases {#examples}
------------------
Let us consider two examples in which Lemma \[main\] can be used directly. The first example exhibits a surprising behavior.
[**Example 1.**]{} Assume that the service time distribution $G$ has its support in $[1,\infty)$. Consequently, whenever a customer enters the system both processes $V^{1,x}_t$ and $V^{1,y}_t$ merge immediately and then remain together forever. On the other hand, if no customer enters the system during the first unit of time, both processes arrive at state $0$ independently of the initial values $x$ and $y$. Consequently, $$\begin{aligned}
\mathbb{P}\bigl(T^{x,y}>t\bigr)=\mathbb{P}\bigl(T^{x,y}>t,T_1\le t\bigr)+\mathbb{P}\bigl(T^{x,y}>t,T_1> t\bigr)
\le e^{-\lambda t}\mathbf{1}_{[0,1)}(t).\end{aligned}$$ In particular we have $\bar{d}(t)=0$ for $t\ge 1$ and hence $d(t)=0$ for $t\ge 1$ by . The fast speed of convergence is quite surprising, since it means that the process is already in equilibrium after one unit of time regardless of its initial value. This result shows the power of the simple coupling inequality .\
What is the distribution $\pi$ of $V^{1,x}_1\,\,$? Since $$\bar{G}(x)=
\left\{
\begin{array}{c c}
1,&\,\,x\in[0,1)\\
0,&x \ge 1
\end{array}
\right.$$ a straightforward calculation shows that $$\label{sf1}
\sum_{i=1}^{\infty}\lambda^{i}\bar{G}^{\ast i}(x)=\lambda x-\lambda +\lambda e^{\lambda x}$$ and hence $$\label{sf2}
\sum_{i=1}^{\infty}\lambda^{i}\int_0^1 \bar{G}^{\ast i}(x)dx=e^{\lambda}-\frac{\lambda}{2}-1.$$ Now insert and in . This yields $$\bar{\pi}(x)=\frac{\lambda x -\lambda +\lambda e^{\lambda x}}{e^{\lambda}-\frac{\lambda}{2}}.$$ Adding the atom at 0 it is readily seen that the distribution function $\pi (x)$ is given by $$\pi(x)=\frac{e^{\lambda x}-\lambda x+\frac{\lambda}{2}x^2}{e^{\lambda}-\frac{\lambda}{2}}.$$
[**Example 2**]{}. Assume that $p=\mathbb{P}\bigl(G\ge 1\bigr)>0$. Then we have $$\label{ef}
\sup_{x\in[0,1]}||\mathbb{P}(V_t^{1,x}\in\cdot)-\pi||_{TV}\le e^{-\lambda p t}.$$ To see this, we use use the same coupling as before. Whenever a jump of size larger than one occurs, both processes glue together regardless of their initial values. The arrival times of the jumps of size larger than one is a Poisson process with intensity $\lambda p$. Hence follows from $$\mathbb{P}\bigl(T^{x,y}>t\bigr)\le\mathbb{P}\bigl(T_1>t\bigr)\le e^{-p\lambda t}.$$ On the other hand, we have that $$\begin{aligned}
\mathbb{P}\bigl(T^{x,y}>1\bigr)&\le& \mathbb{P}\bigl(\text{ for $t\in[0,1]$
the process has at least one jump of size $<1$}\bigr)\nonumber\\
&=&1-e^{-\lambda (1-p)}\end{aligned}$$ and hence $$\label{ef_2}
\sup_{x\in[0,1]}||\mathbb{P}(V_t^{1,x}\in\cdot)-\pi||_{TV}\le (1-e^{-\lambda (1-p)})^t.$$ Now and together yield the following lower bound for spectral gap: $$1-\rho\ge \min(1-e^{-\lambda p },e^{-\lambda (1-p)}).$$ It follows immediately that $e^{-\lambda_0 (1-p)}$ is a lower bound which is uniform in $\lambda$, where $\lambda_0$ is the unique solution of $e^{-\lambda (1-p)}=1-e^{-\lambda p}$.
[The SGP does not hold uniformly]{}\[general\] Let $\mathcal G$ be the set of all distributions on $(0,\infty)$. For general service distribution $G
\in \mathcal{G}$, it is not easy to analyze the time when the processes $V_t^{1,x}$ and $V_t^{1,y}$ merge.
We show now that there is no universal bound for the spectral gap valid for all $\lambda$ and all $G$. We will see in the proof of this result that the spectral gap converges to zero when taking the point mass at $\epsilon$ as service distribution, choosing $\lambda=\lambda_{\epsilon}\rightarrow\infty$ in a balanced way and letting $\epsilon\rightarrow 0$.
\[second\_prop\] $$\inf_{G\in \mathcal{G},\lambda>0}(1-\rho (G,\lambda))=0.$$
Let $\epsilon>0$ and take $G=\delta_{\epsilon}$, the point mass at $\epsilon$. Then $$\mathbb{P}\bigl(T_1>\epsilon\bigr)=e^{-\lambda\epsilon}.
\nonumber$$ Moreover, let $$R_{0}^{(\epsilon,\lambda)}=0,\,R_1^{(\epsilon,\lambda)}=
\min\{\epsilon,T_1\},\,R_{i+1}^{(\epsilon,\lambda)}
=\min\{R_i^{(\epsilon,\lambda)}+\epsilon,\min \{T_j:T_j\ge R_i,j\in\mathbb{N}\}\},
\nonumber$$ where as before the $T_j$ denote the arrival times of the process. We have, for $x\in(\epsilon,1-\epsilon)$, $$\begin{aligned}
\mathbb{P}_{x}\bigl(V_{R_1^{(\epsilon,\lambda)}}\in\cdot\bigr)&=&
\mathbb{P}_{x}\bigl(V_{R_1^{(\epsilon,\lambda)}}
\in\cdot \mid T_1\ge \epsilon\bigr)\mathbb{P}_{x}\bigl(
T_1\ge \epsilon\bigr)
\nonumber\\
&& \hspace{3cm} +\int_0^{\epsilon}\mathbb{P}_{x}\bigl(
V_{R_1^{(\epsilon,\lambda)}}\in\cdot \mid
T_1= s\bigr)\mathbb{P}_{x}\bigl(T_1\in ds \bigr)\nonumber\\
&=&e^{-\lambda\epsilon}\delta_{x-\epsilon}+\int_0^{\epsilon}\delta_{x+\epsilon -s}(\cdot)\lambda e^{-\lambda s}ds.
\nonumber \end{aligned}$$ This implies that $$\mathbb{P}_{x}\bigl(V_{R_1^{(\epsilon,\lambda)}}
\in\cdot\bigr)\rightarrow \delta_{x-\epsilon},\,\,\,\lambda\rightarrow 0\mbox{ and }
\mathbb{P}_{x}\bigl(V_{R_1^{(\epsilon,\lambda)}}\in\cdot\bigr)\rightarrow \delta_{x+\epsilon},\,\,\,\lambda\rightarrow \infty,
\nonumber$$ where the convergence is with respect to the weak topology. In particular, $$\label{expectation}
\mathbb{E}_{x}\bigl(V_{R_1^{(\epsilon,\lambda)}}\bigr)\rightarrow {x-\epsilon},\,\,\,\lambda\rightarrow 0 \mbox{ and }
\mathbb{E}_{x}\bigl(V_{R_1^{(\epsilon,\lambda)}}\bigr)\rightarrow x+\epsilon,\,\,\,\lambda\rightarrow \infty.$$ Observe that $\mathbb{E}_{x}\bigl(V_{R_1^{(\epsilon,\lambda)}}\bigr)$ depends continuously on $\lambda$. Hence by the intermediate value theorem there exists a $\tilde{\lambda}$ such that $\mathbb{E}_{x}\bigl(V_{R_1^{(\epsilon,\tilde{\lambda})}}\bigr)=x$. We can write $$\label{sum}
V_{R_n^{(\epsilon,\tilde{\lambda})}}=V_{R_1^{(\epsilon,\tilde{\lambda})}}
+\sum_{i=1}^{n-1}\bigl(V_{R_{i+1}^{(\epsilon,\tilde {\lambda})}}-V_{R_{i}^{(\epsilon,\tilde{\lambda})}}\bigr).$$ Since the inter-arrival times are exponentially distributed, it follows that for fixed $n$ and sufficiently small $\epsilon\le\tilde{\epsilon}(n,x)$ the sum in is a sum of i.i.d. random variables with expectation zero. Here, $\tilde{\epsilon}(n,x)$ must be chosen such that the process started at $x$ cannot reach the boundary up to time $R_n$. Now let $\tilde{V}_{R_n^{(\epsilon,\tilde{\lambda})}}$ be the boundary-free version of $V_{R_n^{(\epsilon,\tilde{\lambda})}}$, i.e., let $\tilde{V}_{R_n^{(\epsilon,\tilde{\lambda})}}$ be defined analogously to $V_{R_n^{(\epsilon,\tilde{\lambda})}}$, where in the definition of $V^{1}$ we have to replace $\bar{S}_n$ by $S_n$. Moreover, let $$M_n=M_n^{(\epsilon)}=\sum_{i=1}^{n-1}\bigl(\tilde{V}_{R_{i+1}^{(\epsilon,\tilde {\lambda})}}-\tilde{V}_{R_{i}^{(\epsilon,\tilde{\lambda})}}).$$ Observe that $M_n$ is a martingale with respect to the filtration $\sigma(M_1,M_2,\ldots,M_n),n\in\Nset$. Let $N_t:=\max\{i\in\mathbb{N}:R_i\le t\}$ and $h$ be a function such that $h(\epsilon)\rightarrow 0$ for $\epsilon\rightarrow 0$, but $h(\epsilon)/\epsilon^{\alpha}\rightarrow \infty$ for all $\alpha>0$. Then if $x$ satisfies $x\ge\frac{3}{4}+\epsilon$ and $0< \epsilon<\frac{1}{4}$ we obtain $$\begin{aligned}
\mathbb{P}_x\bigl(V_t^{(\epsilon,\tilde{\lambda})}<\frac{1}{2}\bigr)&=&\mathbb{P}_{x}\bigl(V_{R_1^{(\epsilon,\tilde{\lambda})}}+\sum_{i=1}^{N_t-1}\bigl(V_{R_{i+1}^{(\epsilon,\tilde{\lambda})}}-V_{R_i^{(\epsilon,\tilde{\lambda})}}\bigr)+V_t-V_{R_{N_t}}<\frac{1}{2}\bigr)\nonumber\\
&\le&\mathbb{P}_{x}\bigl(\sum_{i=1}^{N_t-1}\bigl(V_{R_{i+1}^{(\epsilon,\tilde{\lambda})}}-V_{R_i^{(\epsilon,\tilde{\lambda})}}\bigr)<\frac{1}{2}-x+\epsilon\bigr)\nonumber\\
&\le&\mathbb{P}_{x}\bigl(\sum_{i=1}^{N_t}\bigl(V_{R_{i+1}^{(\epsilon,\tilde{\lambda})}}-V_{R_i^{(\epsilon,\tilde{\lambda})}}\bigr)<-\frac{1}{4}\bigr)\nonumber\\
&\le&\mathbb{P}_{x}\bigl(\sum_{i=1}^{N_t}(V_{R_{i+1}^{(\epsilon,\tilde{\lambda})}}-V_{R_{i}^{(\epsilon,\tilde{\lambda})}})<-\frac{1}{4},N_t\le \frac{h(\epsilon)}{\epsilon^2}\bigr)+\mathbb{P}_{x}\bigl(N_t> \frac{h(\epsilon)}{\epsilon^2}\bigr)\nonumber\\
&\le&\mathbb{P}_{x}\bigl(\sum_{i=1}^{N_t\wedge \frac{h(\epsilon)}{\epsilon^2}}(V_{R_{i+1}^{(\epsilon,\tilde{\lambda})}}
-V_{R_{i}^{(\epsilon,\tilde{\lambda})}})<-\frac{1}{4}\bigr)+\mathbb{P}_{x}\bigl(N_t> \frac{h(\epsilon)}{\epsilon^2}\bigr)
\nonumber\\
&\le&\mathbb{P}_{x}\bigl(|\sum_{i=1}^{N_t\wedge \frac{h(\epsilon)}{\epsilon^2}}(V_{R_{i+1}^{(\epsilon,\tilde{\lambda})}}
-V_{R_{i}^{(\epsilon,\tilde{\lambda})}})|\ge\frac{1}{4}\bigr)+\mathbb{P}_{x}\bigl(N_t> \frac{h(\epsilon)}{\epsilon^2}\bigr).
\nonumber\end{aligned}$$ Hence, $$\begin{aligned}
\mathbb{P}_x\bigl(V_t^{(\epsilon,\tilde{\lambda})}
&\le&\mathbb{P}_{x}\bigl(\sup_{j\le\frac{h(\epsilon)}{\epsilon^2}}\|\sum_{i=1}^{j}(V_{R_{i+1}^{(\epsilon,\tilde{\lambda})}}-V_{R_{i}^{(\epsilon,\tilde{\lambda})}})\|\ge\frac{1}{4}\bigr)+\mathbb{P}_{x}\bigl(N_t> \frac{h(\epsilon)}{\epsilon^2}\bigr)\nonumber\\
&\le&\mathbb{P}_{x}\bigl(\sup_{j\le\frac{h(\epsilon)}{\epsilon^2}}\|\sum_{i=1}^{j}(V_{R_{i+1}^{(\epsilon,\tilde{\lambda})}}-V_{R_{i}^{(\epsilon,\tilde{\lambda})}})\|\ge 1-x\bigr)+\mathbb{P}_{x}\bigl(N_t> \frac{h(\epsilon)}{\epsilon^2}\bigr)\nonumber\\
&=&\mathbb{P}_{x}\bigl(\sup_{j\le\frac{h(\epsilon)}{\epsilon^2}}\|\sum_{i=1}^{j}(\tilde{V}_{R_{i+1}^{(\epsilon,\tilde{\lambda})}}-\tilde{V}_{R_{i}^{(\epsilon,\tilde{\lambda})}})\|\ge 1-x\bigr)+\mathbb{P}_{x}\bigl(N_t> \frac{h(\epsilon)}{\epsilon^2}\bigr)\nonumber\\
&\le& \frac{1}{(1-x)^2} \mathbb{E}\bigl(\|\sum_{i=1}^{\lfloor\frac{h(\epsilon)}{\epsilon^2}\rfloor}(\tilde{V}_{R_{i+1}^{(\epsilon,\tilde{\lambda})}}-\tilde{V}_{R_{i}^{(\epsilon,\tilde{\lambda})}})\|^2\bigr)+\mathbb{P}_{x}\bigl(N_t> \frac{h(\epsilon)}{\epsilon^2}\bigr)\label{doob}\\
&=& \frac{1}{(1-x)^2}\lfloor\frac{h(\epsilon)}{\epsilon^2}\rfloor \mathbb{E}\bigl((\tilde{V}_{R_{2}^{(\epsilon,\tilde{\lambda})}}-\tilde{V}_{R_{1}^{(\epsilon,\tilde{\lambda})}})^2\bigr)+\mathbb{P}_{x}\bigl(N_t> \frac{h(\epsilon)}{\epsilon^2}\bigr)\nonumber\\
&\le&\frac{1}{(1-x)^2}h(\epsilon)+\mathbb{P}_{x}\bigl(N_t> \frac{h(\epsilon)}{\epsilon^2}\bigr).\label{last}\end{aligned}$$ In we have used Doob’s maximal inequality for martingales. Next note that $$\begin{aligned}
\mathbb{P}_{x}\bigl(N_t> \frac{h(\epsilon)}{\epsilon^2}\bigr)&\le&\mathbb{P}\bigl(R_1+\sum_{i=1}^{\lfloor\frac{h(\epsilon)}{\epsilon^2}\rfloor-1}(R_{i+1}-R_{i})<t\bigr)\nonumber\\
&\le&\mathbb{P}\bigl(\sum_{i=1}^{\lfloor\frac{h(\epsilon)}{\epsilon^2}\rfloor-1}(R_{i+1}-R_{i})<t+\epsilon\bigr)\nonumber\\
&\le&\mathbb{P}\bigl(\frac{1}{\sigma_{\epsilon}\sqrt{\lfloor\frac{h(\epsilon)}{\epsilon^2}\rfloor}}
\sum_{i=1}^{\lfloor\frac{h(\epsilon)}{\epsilon^2}\rfloor-1}\bigl((R_{i+1}-R_{i})
-\mathbb{E}\bigl(R_{2}-R_{1}\bigr)\bigr)\label{minus_infinity}
\\ && \hspace{4cm}
<\frac{t+\epsilon-\lfloor\frac{h(\epsilon)}{\epsilon^2}\rfloor\mathbb{E}
\bigl(R_{2}-R_{1}\bigr)}{\sigma_{\epsilon}\sqrt{\lfloor\frac{h(\epsilon)}{\epsilon^2}\rfloor}}\bigr), \nonumber \end{aligned}$$ where $\sigma_{\epsilon}^2=\mbox{Var}(R_2-R_1)$. From the standard central limit theorem it follows that $$\frac{1}{\sigma_{\epsilon}\sqrt{\frac{h(\epsilon)}{\epsilon^2}}}\sum_{i=1}^{\lfloor\frac{h(\epsilon)}{\epsilon^2}\rfloor-1}\bigl((R_{i+1}-R_{i})-\mathbb{E}\bigl(R_{2}-R_{1}\bigr)\bigr)\rightarrow N(0,1)$$ in distribution. On the other hand, the right-hand side in converges to $-\infty$, and hence we have $$\mathbb{P}_{x}\bigl(N_t> \frac{h(\epsilon)}{\epsilon^2}\bigr)\rightarrow 0\mbox{ for }\epsilon\rightarrow 0.$$ This together with implies that $$\label{right_concentration}
\mathbb{P}_{x}\bigl(V_t^{(\epsilon,\tilde{\lambda})}<\frac{1}{2}\bigr)\rightarrow 0\mbox{ for }\epsilon\rightarrow 0.$$ Now we can carry out a similar calculation for $y\le \frac{1}{4}-\epsilon$ ($0<\epsilon<\frac{1}{4}$), yielding $$\label{left_concentration}
\mathbb{P}_{y}\bigl(V_t^{(\epsilon,\tilde{\lambda})}>\frac{1}{2}\bigr)\rightarrow 0\mbox{ for }\epsilon\rightarrow 0.$$ Let $\bar{d}_{\epsilon}(t)=\sup_{x,y}\|\mathbb{P}_{x}\bigl(V_t^{(\epsilon,\tilde{\lambda})}
\in\cdot\bigr)-\mathbb{P}_{y}\bigl(V_t^{(\epsilon,\tilde{\lambda})}\in\cdot\bigr)\|_{TV}$. Then it follows from and $$\bar{d_{\epsilon}}(t)\rightarrow 1,\,\,\,\epsilon \rightarrow 0\mbox{ for all }t>0,$$ from which the result follows.
[Results for Model 2]{} In this section we present the basic analysis of Model 2. It is shown that $\mathbb{E}(S_1)<\infty$ implies that the process $V_t^{2,\cdot}$ has an invariant distribution $\pi$ and determine an explicit formula for $\pi$. A condition ensuring geometric ergodicity is given and an estimate for the rate of convergence in the case of bounded jumps is derived.
[The invariant distribution]{}
The process $V^{2,x}_t$ has an invariant distribution if $\mathbb{E}\bigl(S_1\bigr)<\infty$. In this case the invariant density $\tilde{\pi}$ on $(0,\infty)$ is given by $$\tilde{\pi}(x)=\left\{
\begin{array}{c c}
\dfrac{\sum_{i=1}^{\infty}\lambda^{i}\bar{G}^{\ast i}(x)}{1+\lambda\mathbb{E}
\bigl(S_1\bigr)\bigl(1+\sum_{i=1}^{\infty}\lambda^{i}\int_{0}^{b}\bar{G}^{\ast i}(y)dy\bigr)},&x\in(0,1]\\
\dfrac{\lambda\bar{G}(x)+\lambda\sum_{i=1}^{\infty}\lambda^{i}\int_{0}^{b}\bar{G}(x-y)
\bar{G}^{\ast i}(y) dy}{1+\lambda\mathbb{E}\bigl(S_1\bigr)\bigl(1+\sum_{i=1}^{\infty}\lambda^{i}\int_{0}^{b}
\bar{G}^{\ast i}(y)dy\bigr)}, &x\in (1,\infty)
\end{array}\right.$$
The condition $\mathbb{E}\bigl(S_1\bigr)<\infty$ ensures that the expected time between two consecutive visits of $V^{2,x}_t$ at level 1 is finite so that the limit theorem for regenerative processes can be applied. Setting the invariant density $\tilde{\pi}(x)$ equal to the upcrossing rate of level $x\in(0,1]$ we get $$\begin{aligned}
\tilde{\pi}(x)&=&\lambda\int_0^x \bar{G}(x-y)\pi(dy)=
\lambda\pi(0)\bar{G}(x)+\lambda\int_{0}^x\bar{G}(x-y)\tilde{\pi}(y)dy\nonumber\\
&=&\lambda\pi(0)\bar{G}(x)+\lambda\bar{G}\ast\tilde{\pi}(x)\end{aligned}$$ As in the proof of Theorem 1 this yields for $x\in(0,1]$ $$\label{eq1}
\tilde{\pi}(x)
=\lambda\pi(0)\bar{G}(x)+\lambda\bar{G}\ast\tilde{\pi}(x)=\pi(0)\sum_{i=1}^{\infty}\lambda^{i}\bar{G}^{\ast i}(x).$$ For $x\in(1,\infty)$ the same arguments as above show that $$\label{eq2}
\tilde{\pi}(x)=\lambda\pi(0)\bar{G}(x)+\lambda\int_{0}^{1}\bar{G}(x-y)\tilde{\pi}(y)dy.$$ If we define $\tilde{\tilde{\pi}}(x)=\tilde{\pi}(x)\mathbf{1}_{(0,1]}(x)$, we obtain from and that for all $x\in(0,\infty)$ we have $$\label{eq3}
\tilde{\pi}(x)=\lambda\pi(0)\bar{G}(x)+\lambda
\int_{0}^{x}\bar{G}(x-y)\tilde{\tilde{\pi}}(y)dy=\lambda\pi(0)\bar{G}(x)+\lambda (\bar{G}\ast\tilde{\tilde{\pi}})(x).$$ Taking the integral in , an application of Fubini’s theorem and leads to $$\begin{aligned}
1-\pi(0)&=&\lambda\pi(0)\mathbb{E}(S_1)+\lambda \mathbb{E}(S_1)\pi(1)=\lambda\pi(1)\mathbb{E}(S_1)\\
&=&\pi(0)\lambda\mathbb{E}\bigl(S_1\bigr)\bigl(1+\sum_{i=1}^{\infty}\lambda^{i}\int_{0}^{b}\bar{G}^{\ast i}(y)dy\bigr),\end{aligned}$$ which yields $$\label{pi0}
\pi(0)=\frac{1}{1+\lambda\mathbb{E}\bigl(S_1\bigr)\bigl(1+\sum_{i=1}^{\infty}\lambda^{i}\int_{0}^{1}\bar{G}^{\ast i}(y)dy\bigr)}.$$ The claim follows now from , and .
[A sufficient condition for geometric ergodicity]{} For jump distributions with unbounded support $V^{2,x}_t$ is in general not geometrically ergodic. The next theorem gives a sufficient condition.
The process $V_t^{2,x}$ is geometrically ergodic if $$\mathbb{E}\bigl(r^{S_1}\bigr)< \infty \ \mbox{ for some } r>1.$$
Let $$\tau_{C}^x=\inf_{t}\{t>0 :V_t^{2,\cdot}\in C\}.$$ The proof is based on Theorem 15.0.1 in [@MeTw] which, translated to our setting, essentially states the following: If there exists a petite set $C\in\mathcal{B}(\mathbb{R}_+)$ (for a definition of the term ‘petite’ we refer to [@MeTw]) and $r>1$ such that $$\sup_{x\in C}\mathbb{E}_x\bigl(r^{\tau_C^x}\bigr)<\infty,$$ then $V_t^{2,x}$ is geometrically ergodic. Now we can choose $C=\{0\}$ and the claim follows.
[Jump distributions with compact support]{} In this subsection we assume that $G$ has compact support. Let $b$ be minimal such that $$\mbox{supp}(G)\subset[0,b].$$ By definition of the process $V^{2,x}_t$ it follows that $$V^{2,x}_t\subset [0,b+1].$$ In order to estimate $d(t)$, let us bound $\bar{d}(t)$ for this example by using once again , where $T^{x,y}$ is defined here is as before in the sense that in the former definition of $T^{x,y}$ one simply has to replace $V_t^{1,x}$ by $V_t^{2,x}$ and $V_t^{1,y}$ by $V_t^{2,y}$.\
Let $0=x_0<x_1,\ldots<x_{N(\epsilon)-1}=b+1$ be a decomposition of the interval $[0,b+1]$ such that $x_{i+1}-x_{i}\le\epsilon$ for $i\in\{0,\ldots,N(\epsilon)-1\}$ $$\begin{aligned}
\bar{d}(t)&=&\sup_{x,y\in[0,b+1]}||\mathbb{P}\bigl(V_t^{1,x}\in \cdot\bigr)-\mathbb{P}\bigl(V_t^{1,y}\in \cdot\bigr)\|_{TV}\nonumber\\
&\le&\sum_{i=0}^{N(\epsilon)-1}\sup_{x,y\in[x_i,x_{i+1})}||\mathbb{P}\bigl(V_t^{1,x}\in \cdot\bigr)-\mathbb{P}\bigl(V_t^{1,y}\in \cdot\bigr)\|_{TV}\nonumber\\
&\le& \frac{b+1}{\epsilon}\bigl(1-e^{-\lambda(1+\epsilon)}\bigr)^{\lfloor\frac{t}{b+1}\rfloor}.\end{aligned}$$ This implies $$\limsup_{t\rightarrow\infty}-\frac{1}{t}\log\bar{d}(t)\ge -\frac{1}{b+1}\log\bigl(1-e^{-\lambda(1+\epsilon)}\bigr)\,\,\,\,\forall\epsilon>0,$$ which immediately yields that $$\label{deterministic}
\limsup_{t\rightarrow\infty}-\frac{1}{t}\log d(t)\ge -\frac{1}{b+1}\log\bigl(1-e^{-\lambda}\bigr).$$ Therefore, $$\lim_{t\rightarrow\infty} e^{\alpha t} d(t)=0$$ for every $\alpha < \frac{1}{b+1}|\log\bigl(1-e^{-\lambda}\bigr)|$.
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abstract: |
We compute symmetry algebras of a system of two equations $y^{(k)}(x)=z^{(l)}(x)=0$, where $2\le k\le l$. It appears that there are many ways to convert such system of ODEs to an exterior differential system. They lead to different series of finite-dimensional symmetry algebras. For example, for $(k,l)=(2,3)$ we get two non-isomorphic symmetry algebras of the same dimension. We explore how these symmetry algebras are related to both Sternberg prolongation of $G$-structures and Tanaka prolongation of graded nilpotent Lie algebras.
Surprisingly, the case $(k,l)=(2,3)$ provides an example of a linear subalgebra ${\mathfrak a}$ in ${\mathfrak{gl}}(5,{\mathbb R})$ such that the Sternberg prolongations of ${\mathfrak a}$ and ${\mathfrak a}^{t}$ are both of the same dimension, but are non-isomorphic.
We also discuss the non-linear case and the link with flag structures on smooth manifolds.
address:
- 'Belarussian State University, Nezavisimosti Ave. 4, Minsk 220050, Belarus; E-mail: doubrov@islc.org'
- 'Department of Mathematics, Texas A$\&$M University, College Station, TX 77843-3368, USA; E-mail: zelenko@math.tamu.edu'
author:
- Boris Doubrov
- Igor Zelenko
title: Symmetries of trivial systems of ODEs of mixed order
---
Introduction
============
The goal of this paper is to show that symmetry computations for systems of ODEs of mixed order exhibit new phenomena not visible in case of systems of ODEs of uniform order. It is sufficient to consider a system of two trivial ODEs of different order to demonstrate these phenomena: $$\label{odekl}
y^{(k)}(x)=0, \quad z^{(l)}(x)=0,$$ where $y(x)$ and $z(x)$ are two unknown functions. We shall always assume that $2\le k \le l$, as under these conditions the symmetry algebra of this system becomes finite-dimensional.
The first phenomenon is that mixed order systems admit different reformulations in terms of exterior differential systems, which lead to different symmetry algebras. And unlike inclusions between Lie algebras of point, contact and internal symmetries, there is no inclusion between symmetry algebras coming from different EDS. For example, we show that in the simplest non-trivial case of $(k,l)=(2,3)$ there are two different EDS’s. Their symmetry algebras are both 15-dimensional, but one of them is isomorphic to ${\mathfrak{gl}}(3,{\mathbb R}){\rightthreetimes}S^2({\mathbb R}^3)$, while another is isomorphic to ${\mathfrak{csp}}(4,{\mathbb R}){\rightthreetimes}{\mathbb R}^4$. For other pairs of $2\le k < l$ these symmetry algebras even have different dimensions.
The second phenomenon is that the same case $(k,l)=(2,3)$, interpreted in terms of $G$-structures, leads to an explicit example of a linear subalgebra ${\mathfrak a}$ in ${\mathfrak{gl}}(5,{\mathbb R})$ such that the Sternberg prolongations of ${\mathfrak a}$ and ${\mathfrak a}^{t}$ are both of the same dimension, but are non-equivalent to each other. In fact, we show that both symmetry algebras can be obtained as total prolongations of certain $G$-structures related to the orbits of $SL(2,{\mathbb R})$-action on Grassmann varieties ${\operatorname{Gr}}_2({\mathbb R}^{k+l})$ and ${\operatorname{Gr}}_{k+l-2}({\mathbb R}^{k+l})$.
Finally, the third phenomenon is related to the use of Tanaka theory of graded nilpotent Lie algebras and their prolongations for computing the symmetry algebras. It appears that in one of the EDS reformulations one needs to consider graded nilpotent Lie algebras ${\mathfrak{m}}=\sum_{i<i}{\mathfrak{m}}_i$, which are not generated by ${\mathfrak{m}}_{-1}$. It appears that Tanaka theory produces the expected result in this case as well, if we slightly modify the notion of Tanaka prolongation.
The paper is organized as follows. In Section \[sec:eds\] we show how systems of mixed order can be turned into exterior differential systems in two different ways. In Section \[sec:sym\] we compute symmetry algebras for each of these exterior differential systems and show that we get non-isomorphic symmetry algebras. In Sections \[sec:tan\] and \[sec:g\] we show how these symmetry algebras appear naturally as Tanaka prolongation of certain graded nilpotent Lie algebras and as Sternberg prolongation of subalgebras in ${\mathfrak{gl}}(k+l,{\mathbb R})$. In Sections \[sec:nlin\] we discuss the case of non-linear systems of mixed order. In Section \[sec:flag\] we link the geometry of non-linear systems of mixed order with so-called flag structures on smooth manifolds. Finally, in Section \[sec:shifts\] we describe other ways to associate exterior differential systems with equations of mixed order.
Two EDS interpretations {#sec:eds}
=======================
The EDS of first kind
---------------------
Let $J^l({\mathbb R},{\mathbb R}^2)$ be the $l$-th jet space of maps from ${\mathbb R}$ to ${\mathbb R}^2$ with the coordinate system: $$(x,y,y_1,\dots,y_l,z,z_1,\dots,z_l).$$ The system can be prolonged the the system of equations: $$y^{(k)}=y^{(k+1)}=\dots=y^{(l)}=0,\quad z^{(l)}=0,$$ which can be considered as a submanifold ${\mathcal{E}}\subset J^l({\mathbb R},{\mathbb R}^2)$ defined by equations: $$y_{k}=y_{k+1}=\dots=y_{l}=0,\quad z_{l}=0.$$ The remaining set of jet space coordinates $(x,y,z,y_1,\dots,y_{k-1},z_1,\dots,z_{l-1})$ forms a coordinate system on ${\mathcal{E}}$ itself. The system ${\mathcal{E}}$ defines also a one-dimensional vector bundle on ${\mathcal{E}}$ tangent to lifts of all solutions of : $$\label{distE}
E = \left\langle {\frac{{\partial}}{{\partial}x}} + y_1{\frac{{\partial}}{{\partial}y}} + \dots + y_{k-1}{\frac{{\partial}}{{\partial}y_{k-2}}}+ z_1{\frac{{\partial}}{{\partial}z}} +
\dots + z_{l-1}{\frac{{\partial}}{{\partial}y_{l-2}}}\right\rangle.$$ Recall that the *contact system* $C$ on $J^l({\mathbb R},{\mathbb R}^2)$ is defined by contact differential forms $dy_i-y_{i+1}dx$, $dz_i-z_{i+1}dx$, $i=0,\dots,l-1$, where $y_0=y$ and $z_0=z$. Note that the distribution $E$ can be defined as the intersection of the contact system $C$ with $T{\mathcal{E}}$, or, in other words, by the restriction of all contact forms on ${\mathcal{E}}$.
Due to the Lie theorem this contact system canonically defines a sequence of integrable distributions complementary to $E$: $$V_i = \langle dx, dy_r, dz_r \mid r = 0,\dots, i-1 \rangle^{\perp},$$ where $i=1, \dots, l$. These distributions can also be defined as tangent spaces to fibers of the canonical projections $\pi_{l,i-1}\colon J^l({\mathbb R},{\mathbb R}^2)\to J^{i-1}({\mathbb R},{\mathbb R}^2)$. In particular, all symmetries of the contact system $C$ are exactly the prolongations of vector fields from $J^0({\mathbb R},{\mathbb R}^2)={\mathbb R}^3$. Such vector fields are called $\emph{point vector fields}$.
Symmetries (or, rather, *infinitesimal symmetries*) of ${\mathcal{E}}$ are defined as point vector fields, which are at the same time tangent to the equation submanifold ${\mathcal{E}}$. Let $X$ be a restriction of such symmetry to ${\mathcal{E}}$ itself. Then it preserves both the vector distribution $D$ given by and all the intersections $V_i\cap T{\mathcal{E}}$. Denote these intersections by $F_i$. In particular, we have: $$F_1 = \left \langle {\frac{{\partial}}{{\partial}y_i}}, i=1,\dots,k-1; {\frac{{\partial}}{{\partial}z_j}},j=1,\dots,l-1\right\rangle.$$ Simple computation shows that $E$ and $F_1$ alone allow to recover all other $F_i$, $i>1$, via: $$F_{i+1} = \{ Y \in F_i \mid [Y, E]\subset F_i \}.$$ Note that if $l>k$, then the smallest of $F_i$ is $$F_{l-1} = \left\langle {\frac{{\partial}}{{\partial}z_{l-1}}} \right\rangle.$$ On the contrary, the pair $F_{l-1}$ and $E$ does not recover the complete sequence of $F_i$, as the inclusion $[E,F_{k}]\subset F_{k-1}$ is strict, and the distribution $E+F_{k}+[E,F_{k}]$ is completely integrable.
This motivates the following
The *exterior differential system of first kind* associated with the system of ODEs is given by a pair of vector distributions $E$ and $F_1$ on the equation manifold ${\mathcal{E}}\subset J^l({\mathbb R},{\mathbb R}^2)$.
*Symmetries of first kind* are the vector fields $S$ on ${\mathcal{E}}$ that preserve both $E$ and $F_1$, that is $[S,E]\subset E$ and $[S,F_1]\subset F_1$.
The EDS of second kind
----------------------
Another way of defining the exterior differential system by is to start from the mixed jet space $J^{k,l}({\mathbb R},{\mathbb R}^2)$ with the local coordinate system $(x,y,y_1,\dots,y_k,z,z_1,\dots,z_l)$ and define the equation submanifold ${\mathcal{E}}$ by equations $$y_k=0, z_l=0.$$ Then, as above, the coordinates $(x,y,z,y_1,\dots,y_{k-1},z_1,\dots,z_{l-1})$ form a coordinate system on ${\mathcal{E}}$. Similar to the jet space $J^l({\mathbb R},{\mathbb R}^2)$ we define the contact system $C$ on $J^{k,l}({\mathbb R},{\mathbb R}^2)$ by the collection of contact forms $$\begin{aligned}
& dy_i - y_{i+1}dx, i= 0,\dots,k-1;
& dz_j - z_{j+1}dx, j= 0,\dots,l-1.\end{aligned}$$ The restrictions of these form to the equation manifold ${\mathcal{E}}\subset J^{k,l}({\mathbb R},{\mathbb R}^2)$ define the same one-dimensional vector distribution $E$ given by . So, up to now everything looks very similar to the above.
Let $\pi\colon {\mathcal{E}}\to J^{k-1,l-1}({\mathbb R},{\mathbb R}^2)$ be the projection of ${\mathcal{E}}$ to the lower order jet space. The pull-back of the contact distribution from $J^{k-1,l-1}({\mathbb R},{\mathbb R}^2)$ to ${\mathcal{E}}$ is a 3-dimensional vector distribution $D$ on ${\mathcal{E}}$ defined by 1-forms: $$\begin{aligned}
& dy_i - y_{i+1}dx, i= 0,\dots,k-2;\\
& dz_j - z_{j+1}dx, j= 0,\dots,l-2.\end{aligned}$$ or, in terms of vector fields by: $$\begin{aligned}
X &= {\frac{{\partial}}{{\partial}x}} + y_1{\frac{{\partial}}{{\partial}y}} + \dots + y_{k-1}{\frac{{\partial}}{{\partial}y_{k-2}}}+ z_1{\frac{{\partial}}{{\partial}z}} +
\dots + z_{l-1}{\frac{{\partial}}{{\partial}z_{l-2}}},\\
Y &= {\frac{{\partial}}{{\partial}y_{k-1}}},\\
Z &= {\frac{{\partial}}{{\partial}z_{l-1}}}.\end{aligned}$$
The *exterior differential system of second kind* associated with the system of ODEs is given by a pair of vector distributions $E \subset D$ on the equation manifold ${\mathcal{E}}\subset J^{k,l}({\mathbb R},{\mathbb R}^2)$.
*Symmetries of second kind* are the vector fields $S$ on ${\mathcal{E}}$ that preserve both $D$ and $E$, that is $[S,D]\subset D$ and $[S,E]\subset E$.
As in the case of differential systems of first kind, we can define a sequence of complementary completely integrable distributions in the following way. Let $F_{l-1}$ be the 2-dimensional completely integrable distribution generated by vector fields $Y$ and $Z$. As this is the only 2-dimensional completely integrable subdistribution of $D$, it is preserved by all symmetries of $D$. As $D=E\oplus F_{l-1}$, we see that the symmetries of second kind are exactly the vector fields preserving both $E$ and $F_{l-1}$.
Taking iterative brackets, we can further define: $$F_{i-1} = F_i + [E,F_i]\quad\text{for all } i \le l-1.$$ These distributions are all complementary to $E$ and completely integrable.
Symmetry computation {#sec:sym}
====================
All symmetries of first kind are prolongations of the vector fields from $J^0({\mathbb R},{\mathbb R}^2)={\mathbb R}^3$. They are described as follows.
\[propI\] The Lie algebra ${\mathfrak g}^{I}_{k,l}$ of symmetries of first kind of equation is spanned over ${\mathbb R}$ by the following vector fields:
If $k=2$, $l>2$: $$\begin{aligned}
& {\frac{{\partial}}{{\partial}x}}, {\frac{{\partial}}{{\partial}y}}, \label{a1}\\
& x{\frac{{\partial}}{{\partial}x}}, x{\frac{{\partial}}{{\partial}y}}, y{\frac{{\partial}}{{\partial}x}}, y{\frac{{\partial}}{{\partial}y}},z{\frac{{\partial}}{{\partial}z}}, \label{a2}\\
& x^2{\frac{{\partial}}{{\partial}x}}+xy{\frac{{\partial}}{{\partial}y}}+(l-1)xz{\frac{{\partial}}{{\partial}z}}, xy{\frac{{\partial}}{{\partial}x}} + y^2{\frac{{\partial}}{{\partial}y}} +(l-1)yz{\frac{{\partial}}{{\partial}z}},\label{a3}\\
& x^iy^j{\frac{{\partial}}{{\partial}z}}, 0\le i+j\le l-1.\label{a4}\end{aligned}$$ Algebraically ${\mathfrak g}^{I}_{2,l}$ is isomorphic to ${\mathfrak{gl}}(3,{\mathbb R}){\rightthreetimes}S^{l-1}({\mathbb R}^3)$, where the subalgebra ${\mathfrak{gl}}(3,{\mathbb R})$ is spanned by – and the commutative ideal $S^{l-1}({\mathbb R}^3)$ is spanned by .
If $2<k<l$: $$\begin{aligned}
& {\frac{{\partial}}{{\partial}x}}, x{\frac{{\partial}}{{\partial}x}}, x^2{\frac{{\partial}}{{\partial}x}}+(k-1)xy{\frac{{\partial}}{{\partial}y}}+(l-1)xz{\frac{{\partial}}{{\partial}z}}, y{\frac{{\partial}}{{\partial}y}}, z{\frac{{\partial}}{{\partial}z}},\label{b1}\\
& {\frac{{\partial}}{{\partial}y}}, x{\frac{{\partial}}{{\partial}y}}, \dots, x^{k-1}{\frac{{\partial}}{{\partial}y}},\label{b2}\\
& x^iy^j{\frac{{\partial}}{{\partial}z}}, 0\le i+(k-1)j\le l-1.\label{b3}\end{aligned}$$ Algebraically ${\mathfrak g}^{I}_{k,l}$ is isomorphic to $({\mathbb R}\times{\mathfrak{gl}}(2,{\mathbb R}){\rightthreetimes}V_k){\rightthreetimes}\left(\sum_{i=0}^{[l/k]} V_{l-ki}\right)$, where $V_r$ is an $r$-dimensional irreducible representation of ${\mathfrak{gl}}(2,{\mathbb R})$. The subalgebra ${\mathbb R}\times{\mathfrak{gl}}(2,{\mathbb R})$ is spanned by , the subalgebra $V_k$ is spanned by , and the commutative ideal $\sum_{i=0}^{[l/k]} V_{l-ki}$ is spanned by .
Take an arbitrary vector field $A{\frac{{\partial}}{{\partial}x}} + B{\frac{{\partial}}{{\partial}y}}+C{\frac{{\partial}}{{\partial}z}}$ on ${\mathbb R}^3$, prolong it to $J^l({\mathbb R},{\mathbb R}^2)$ and check that it preserves the equation. This results in a system of PDEs on the functions $A,B,C$, which are easy to solve. This results in the above Lie algebras of point symmetries.
To describe the symmetries of the second kind we need the following technical result, which is also of its own interest. Let $D$ be the operator of total derivative on $J^{r+1}({\mathbb R},{\mathbb R})$ with the standard coordinate system $(x,z_0,\dots,z_{r+1})$: $$D = {\frac{{\partial}}{{\partial}x}} + z_1{\frac{{\partial}}{{\partial}z_0}} + \dots + z_r {\frac{{\partial}}{{\partial}z_{r-1}}}+\dots +z_{r+1} {\frac{{\partial}}{{\partial}z_r}}.$$ Denote by $g_{i,j}$ the following function on $J^{r+1}({\mathbb R},{\mathbb R})$: $$g_{i,j} = \frac{x^{i+j}}{(i+j)!} D^i(z_0/x^j), \quad i,j\ge 0.$$ Let $g_{i,j}^{(s)}=\frac{\partial^{s} g_{i,j}}{\partial x^s}$ for any $s\ge 0$. In particular, we see that $g_{i,j}^{(0)}=g_{i,j}$ and $g_{i,j}^{(s)}=0$ for $s>i$.
It is easy to see that each $g_{i,j}^{(s)}$ is a polynomial in $x,z_{i-s},z_{i-s+1},\dots,z_i$, linear in $z_{i-s},z_{i-s+1},\dots,z_i$ and having a constant coefficient at $z_{i-s}$. In particular, for fixed $i,j$ all functions $g_{i,j}^{(s)}$, $s=0,\dots,i$ are algebraically independent (even over ${\mathbb R}[x]$).
\[lem:g\]
1. The following identity holds: $$\label{eq:ij}
g_{i,j} - (x/i)g^{(1)}_{i,j} + j g_{i-1,j+1} = 0,\quad\text{for all } i,j>0.$$
2. The space of solutions for the system of linear equations: $$\label{eq:d2}
\begin{aligned}
D^2 ( f ) &= 0;\\
\frac{\partial f}{\partial z_{r+1}} &= 0,
\end{aligned}$$ is $r+3$-dimensional and is spanned by the functions $1$, $x$ and $g_{r,2}^{(s)}$, $s=0,\dots,r$.
The space of solutions for the following system of linear equations: $$\label{eq:dk}
\begin{aligned}
D^{p+1} ( f ) &= 0, \quad p\ge 1;\\
\frac{\partial f}{\partial z_{r+1}} &= 0,
\end{aligned}$$ is spanned by $W^p=\{ f_1\dots f_{p} \mid f_i\in W\}$, where $W$ is the above $(r+3)$-dimensional space of solutions of .
The space of solutions for the following system of linear equations: $$\label{eq:dkl}
\begin{aligned}
D^{p+1} ( f ) &= 0, \quad p\ge 1;\\
\frac{\partial f}{\partial z_{r+1-q}} & =\dots =\frac{\partial f}{\partial z_{r+1}} = 0,\quad q\ge 0;
\end{aligned}$$ is spanned by $x^i g_{r-q,q+2}^{(s_1)} \dots g_{r-q,q+2}^{(s_j)}$, where $i+(q+1)j\le p$.
\(a) Easily follows from the the fact that $D$ commutes with ${\frac{{\partial}}{{\partial}x}}$ and from the identity: $$D^i(z_0/x^j) = D^i(x\cdot z_0/x^{j+1}) = x D^i(z_0/x^{j+1}) + i D^{i-1}(z_0/x^{j+1}).$$
\(b) It is clear that $1$ and $x$ are solutions of system . Further, we have: $$\begin{gathered}
D^2(g_{r,2}) =\tfrac{x^r}{r!} D^r(z_0/x^2) + 2\tfrac{x^{r+1}}{(r+1)!}D^{r+1}(z_0/x_2) + \tfrac{x^{r+2}}{(r+2)!}D^{r+2}(z_0/x^2) \\
= \tfrac{x^r}{r!} \sum_{i=0}^r \tbinom{r}{i} D^{r-i}(x^{-2})D^i(z_0) + 2\tfrac{x^{r+1}}{(r+1)!} \sum_{i=0}^{r+1} \tbinom{r+1}{i} D^{r+1-i}(x^{-2})D^i(z_0) + \\ + \tfrac{x^{r+2}}{(r+2)!} \sum_{i=0}^{r+1} \tbinom{r+2}{i} D^{r+2-i}(x^{-2})D^i(z_0)\\
= \tfrac{x^r}{r!} \sum_{i=0}^{r+1} z_{i}\left(\tbinom{r}{i} D^{r-i}(x^{-2})+\tfrac{2x}{r+1}\tbinom{r+1}{i}D^{r+1-i}(x^{-2})+\tfrac{x^2}{(r+1)(r+2)}\tbinom{r+2}{i}D^{r+2-i}(x^{-2})\right)\\
= \tfrac{x^r}{i!(r-i)!}\sum_{i=0}^{r+1} z_i \left(D^{r-i}(x^{-2})+\tfrac{2x}{r+1-i}D^{r+1-i}(x^{-2})+\tfrac{x^2}{(r+1-i)(r+2-i)}D^{r+2-i}(x^{-2})\right)\\
= \tfrac{x^r}{i!(r-i)!}\sum_{i=0}^{r+1} (-1)^{r-i}z_i \left(\tfrac{(r+1-i)!}{x^{r-i+2}}-\tfrac{2x}{r+1-i}\tfrac{(r+2-i)!}{x^{r+3-i}}+\tfrac{x^2}{(r+1-i)(r+2-i)}\tfrac{(r+3-i)!}{x^{r+4-i}}\right)\\
= \sum_{i=0}^{r+1} (-1)^{r-i}\frac{z_i x^{i-2}(r+1-i)!}{i!(r-i)!} \left(1 - \frac{2(r+2-i)}{r+1-i} + \frac{r+3-i}{r+1-i}\right) = 0.\end{gathered}$$ As operators $D$ and ${\frac{{\partial}}{{\partial}x}}$ commute, we see that all functions $g_{r,2}^{(k)}$, $k=0,\dots,r$, indeed satisfy system .
Denote by $Z_i$, $i=0,\dots,r+1$, the differential operator ${\frac{{\partial}}{{\partial}z_{i}}}$. Note that $[Z_i, D] = Z_{i-1}$, or what is the same $D Z_i = Z_i D - Z_{i-1}$ for all $i=1,\dots,r+1$. Taking $i=r+1$, we get that $$(Z_{r+1} D - Z_r)f = 0$$ for any solution $f$ of system . Multiplying this identity by $D$ from the left and using bracket relations between $D$ and $Z_i$, we get $$D(Z_{r+1} D - Z_r)f = (Z_{r+1}D^2-2Z_r D + Z_{r-1})f= (-2Z_r D + Z_{r-1})f = 0.$$ Proceeding in the similar way we get that: $$(Z_{i+1} D - (r+1-i)Z_i)f=0\quad\text{for all }i=0,\dots,r.$$ and also $Z_0Df=DZ_0f=0$. It follows that $Z_iZ_j(f)=0$ for all $i,j=0,\dots,r$ and, thus, $f$ is linear with respect to $z_0,\dots,z_r$. Then easy computation shows that the solution space of system for has dimension $r+3$.
\(c) It is easy to see that all elements of $W^p$ satisfy system for arbitrary $p\ge 2$. The proof that the solution space of system for any $p\ge 2$ coincides with $W^p$ follows from the dimension count similar to the case $p=1$.
\(d) Follows by induction by $q$ from (a) and (c).
The symmetries of second kind are not always prolongations of vector fields on ${\mathbb R}^3$, but they can still be given as prolongations of vector fields from mixed jet space $J^{0,l-k}({\mathbb R},{\mathbb R}^2)$ with the standard coordinate system $(x,y,z,z_1,\dots,z_{l-k})$. The vector fields from $J^{0,0}({\mathbb R},{\mathbb R}^2)={\mathbb R}^3$ can be naturally prolonged to $J^{0,l-k}({\mathbb R},{\mathbb R}^2)$ using the standard prolongation formulas for variables $z_i$ alone.
\[propII\] The Lie algebra ${\mathfrak g}^{II}$ of symmetries of second kind is spanned over ${\mathbb R}$ by the following vector fields:
If $k=2,l=3$: $$\begin{aligned}
& (x^2z_1/2 -zx){\frac{{\partial}}{{\partial}x}}+(xyz_1/2-yz){\frac{{\partial}}{{\partial}y}}+(x^2z_1^2/4-z^2){\frac{{\partial}}{{\partial}z}}+(xz_1^2/2-zz_1){\frac{{\partial}}{{\partial}z_1}},\\
& 2(xz_1-z){\frac{{\partial}}{{\partial}x}}+yz_1{\frac{{\partial}}{{\partial}y}}+xz_1^2{\frac{{\partial}}{{\partial}z}}+z_1^2{\frac{{\partial}}{{\partial}z_1}},\quad z_1{\frac{{\partial}}{{\partial}x}}+z_1^2/2{\frac{{\partial}}{{\partial}z}},\\
& x^2 {\frac{{\partial}}{{\partial}x}} +xy{\frac{{\partial}}{{\partial}y}}+2xz{\frac{{\partial}}{{\partial}z}}+2z{\frac{{\partial}}{{\partial}z_1}},\\
& x{\frac{{\partial}}{{\partial}x}}-z_1{\frac{{\partial}}{{\partial}z_1}},\quad z{\frac{{\partial}}{{\partial}z}}+z_1{\frac{{\partial}}{{\partial}z_1}},\\
& x^2 {\frac{{\partial}}{{\partial}z}}+2x{\frac{{\partial}}{{\partial}z_1}},\quad x{\frac{{\partial}}{{\partial}z}}+{\frac{{\partial}}{{\partial}z_1}},\\
& {\frac{{\partial}}{{\partial}x}},{\frac{{\partial}}{{\partial}z}},\quad y{\frac{{\partial}}{{\partial}y}},\\
& \quad (xz_1-2z){\frac{{\partial}}{{\partial}y}}, x{\frac{{\partial}}{{\partial}y}}, z_1{\frac{{\partial}}{{\partial}y}}, {\frac{{\partial}}{{\partial}y}}.\end{aligned}$$ Algebraically ${\mathfrak g}^{II}_{2,3}$ is isomorphic to ${\mathfrak{csp}}(4,{\mathbb R}){\rightthreetimes}{\mathbb R}^4$, where the subalgebra ${\mathfrak{csp}}(4,{\mathbb R})$ is spanned by vector fields in the first 6 lines and the commutative ideal ${\mathbb R}^4$ is spanned by the vector fields in the last line.
If $2\le k<l$ and $(k,l)\ne (2,3)$: $$\begin{aligned}
& {\frac{{\partial}}{{\partial}x}}, x{\frac{{\partial}}{{\partial}x}}, x^2{\frac{{\partial}}{{\partial}x}}+(k-1)xy{\frac{{\partial}}{{\partial}y}}+(l-1)xz{\frac{{\partial}}{{\partial}z}}, y{\frac{{\partial}}{{\partial}y}}, z{\frac{{\partial}}{{\partial}z}},\label{d1}\\
& {\frac{{\partial}}{{\partial}z}}, x{\frac{{\partial}}{{\partial}z}},\dots,x^{l-1}{\frac{{\partial}}{{\partial}z}},\label{d2}\\
& {\frac{{\partial}}{{\partial}y}}, x{\frac{{\partial}}{{\partial}y}},\dots,x^{k-1}{\frac{{\partial}}{{\partial}y}},\label{d3}\\
& g_{l-k,k}^{(s)}{\frac{{\partial}}{{\partial}y}}, 0\le s \le l-k, \label{d4}\end{aligned}$$ Here vector fields and are assumed to be prolonged to $J^{0,l-k}({\mathbb R},{\mathbb R}^2)$.
Algebraically ${\mathfrak g}^{II}_{k,l}$ is isomorphic to $({\mathbb R}\times {\mathfrak{gl}}(2,{\mathbb R}){\rightthreetimes}W){\rightthreetimes}{V_k\oplus V_l}$, where ${\mathbb R}\times {\mathfrak{gl}}(2,{\mathbb R})$ is spanned by , $V_l$ is spanned by , $V_k$ is spanned by , and $W$ is an $(l-k+1)$-dimensional space spanned by , which can be identified with an irreducible submodule of ${\operatorname{Hom}}(V_l, V_k)\subset {\mathfrak{gl}}(V_k\oplus V_l)$.
Any symmetry of second kind is a prolongation of the vector field $$X = A{\frac{{\partial}}{{\partial}x}}+B{\frac{{\partial}}{{\partial}y}}+C_0{\frac{{\partial}}{{\partial}z}}+C_1{\frac{{\partial}}{{\partial}z_1}}+\dots+C_{l-k}{\frac{{\partial}}{{\partial}z_{l-k}}}$$ from the mixed order jet space $J^{0,l-k}({\mathbb R},{\mathbb R}^2)$. As it preserves contact forms $dz_i-z_{i+1}d{x}$, $i=0,\dots,l-k-1$, it is easy to see that it projects to a contact vector field on $J^{l-k}({\mathbb R},{\mathbb R})$: $$\overline X = A {\frac{{\partial}}{{\partial}x}} + C_0{\frac{{\partial}}{{\partial}z_0}} + \dots + C_{l-k}{\frac{{\partial}}{{\partial}z_{l-k}}}.$$ Moreover, simple computation shows that $\overline X$ is contact symmetry of the equation $z_l=0$, while the kernel of the projection $X\mapsto \overline X$ consists of vector fields: $$\big(cy + f(x,z_0,\dots,z_{l-k})\big){\frac{{\partial}}{{\partial}y}},$$ where the function $f$ satisfies system for $r=l-2$, $p=k-1$ and $q=k-2$.
Direct computation shows that all contact symmetries of the equation $z_l=0$ can be extended to the symmetries of the EDS of the second kind. In particular, in case $l=3$ the contact symmetry algebra is isomorphic to $\mathfrak{sp}(4,{\mathbb R})$, and it is embedded as a subalgebra into ${\mathfrak g}^{II}$.
Further, item (d) of Lemma \[lem:g\] implies that $f$ is a linear combination of products $x^i g^{(s_1)}_{l-k,k}\dots g^{(s_j)}_{l-k,k}$, where $i+(k-1)j\le k-1$. Thus, either $j=0$, $i=0,\dots,k-1$ or $j=1$, $i=0$. This completes the proof of the theorem.
Symmetry algebras via Tanaka prolongation {#sec:tan}
=========================================
In this section we show how symmetry algebras of first and second kind can be obtained as Tanaka prolongations of certain graded nilpotent Lie algebras.
Let us fix the following frame on the equation ${\mathcal{E}}\subset J^{k,l}({\mathbb R},{\mathbb R}^2)$: $$\begin{aligned}
Y_i &= {\frac{{\partial}}{{\partial}y_i}},\quad i = 0,\dots, k-1;\\
Z_j &= {\frac{{\partial}}{{\partial}z_j}},\quad j = 0,\dots, l-1;\\
D &= {\frac{{\partial}}{{\partial}x}} + y_1{\frac{{\partial}}{{\partial}y_0}} + \dots + y_{k-1}{\frac{{\partial}}{{\partial}y_{k-2}}} + z_1{\frac{{\partial}}{{\partial}z_0}} + \dots + z_{l-1}{\frac{{\partial}}{{\partial}z_{l-2}}}.\end{aligned}$$ It is easy to see that these vector fields for a basis of a nilpotent Lie algebra ${\mathfrak{n}}$ with the only non-zero Lie brackets being: $$[Y_i, D] = Y_{i-1}, \ [Z_j, D] = Z_{j-1}, \quad i,j\ge 1.$$ Let us introduce two different gradings on ${\mathfrak{n}}$. Both gradings are concentrated in negative degree and have $\deg D = -1$. The first grading is defined by $\deg Z_i = \deg Y_i = i-l$. In particular, $\deg Z_{l-1} = -1$ and $\deg Y_{k-1} = -1 + (k-l)$. The second grading is defined by $\deg Y_{i} = i-k, \deg Z_{j} = j-l$. In particular, $\deg Y_{k-1} = \deg Z_{l-1} = -1$. To distinguish these two cases we shall denote the Lie algebra ${\mathfrak{n}}$ equipped with a first grading as ${\mathfrak{n}}^{I}$ and with a second grading as ${\mathfrak{n}}^{II}$.
Let us recall the notion of Tanaka prolongation of graded nilpotent Lie algebras. Let ${\mathfrak{m}}$ be an arbitrary negatively graded nilpotent Lie algebra of depth $\mu$, that is ${\mathfrak{m}}=\sum_{i=1}^\mu {\mathfrak{m}}_{-i}$. We recall that ${\mathfrak{m}}$ is called *fundamental*, if ${\mathfrak{m}}$ is generated by ${\mathfrak{m}}_{-1}$.
For the above graded nilpotent Lie algebras we have: $$\begin{aligned}
{\mathfrak{n}}^{I}_{-1} &= \langle D, Z_{l-1} \rangle\quad \text{for } k < l;\\
{\mathfrak{n}}^{I}_{-1} &= \langle D, Z_{l-1}, Y_{k-1} \rangle\quad \text{for } k=l;\\
{\mathfrak{n}}^{II}_{-1} &= \langle D, Z_{l-1}, Y_{k-1} \rangle\quad \text{for all } k\le l.\end{aligned}$$ Thus, we see that ${\mathfrak{n}}^{I}$ is **not fundamental** in case of $k<l$.
The *universal (Tanaka) prolongation of ${\mathfrak{m}}$* is defined as a largest graded Lie algebra ${\mathfrak g}({\mathfrak{m}})$ satisfying the following two conditions:
1. ${\mathfrak g}_{i}({\mathfrak{m}}) = {\mathfrak{m}}_{i}$ for all $i<0$;
2. for any $X\in {\mathfrak g}_i({\mathfrak{m}})$, $i\ge 0$, the equality $[X,{\mathfrak g}_{-}({\mathfrak{m}})]=0$ implies $X=0$.
If ${\mathfrak{m}}$ is fundamental, then the second condition can be replaced by:
1. for any $X\in {\mathfrak g}_i({\mathfrak{m}})$, $i\ge 0$, the equality $[X,{\mathfrak g}_{-1}({\mathfrak{m}})]=0$ implies $X=0$.
The universal Tanaka prolongation has a natural geometric sense in terms of symmetries of left-invariant EDS’s on Lie groups. Namely, let $M$ be a Lie group with the Lie algebra ${\mathfrak{m}}$. Define the sequence $T^{-k}M$ of left-invariant vector distributions on $M$ by the condition: $$T^{-k}_eM = \sum_{i=1}^k {\mathfrak{m}}_{-i}.$$ In extreme cases we have $T^0M=0$ and $T^{-\mu}M=TM$. If ${\mathfrak{m}}$ is fundamental, then the complete sequence is defined by $T^{-1}M$.
We call the flag $\{ T^{-k} M\}$ *the standard flag of type ${\mathfrak{m}}$*. An infinitesimal symmetry of flag $\{ T^{-k} M\}$ is a vector field $X$ on $M$ such that $[X, T^{-i}M]\subset T^{-i}M$ for all $i=1,\dots,\mu$. Denote by $\bar g({\mathfrak{m}})$ the Lie algebra of all germs of infinitesimal symmetries at the identity of $M$. It is easy to show that this Lie algebra is well-defined. In general, it can be infinite-dimensional. But as it contains all germs of right-invariant vector fields on $M$, it is transitive at the identity $e$ (i.e., the values of all elements from $\bar {\mathfrak g}({\mathfrak{m}})$ at $e$ span all tangent space $T_eM$).
This Lie algebra $\bar{\mathfrak g}({\mathfrak{m}})$ can be equipped with a natural decreasing filtration by setting: $$\bar{\mathfrak g}^{-i}({\mathfrak{m}}) = \{ X\in \bar{\mathfrak g}({\mathfrak{m}}) \mid X_e \subset T^{-i}_eM \},\quad\text{for all } i\ge 0.$$ and extending it in the positive direction as follows: $$\bar{\mathfrak g}^{i}({\mathfrak{m}}) = \{ X\in \bar{\mathfrak g}({\mathfrak{m}}) \mid [X, \bar{\mathfrak g}^{-j}({\mathfrak{m}})]\subset \bar{\mathfrak g}^{i-j}({\mathfrak{m}}) \forall j>0\},\quad \text{for } i > 0.$$ In particular, $\bar{\mathfrak g}^{-\mu}({\mathfrak{m}})=\bar {\mathfrak g}^{-\mu}({\mathfrak{m}})$ and $\bar{\mathfrak g}^{0}({\mathfrak{m}})$ is a subalgebra of all germs of infinitesimal symmetries that vanish at $e$.
Finally, we define ${\mathfrak g}({\mathfrak{m}})$ as a graded Lie algebra associated with the filtered Lie algebra $\bar{\mathfrak g}({\mathfrak{m}})$: $${\mathfrak g}_i({\mathfrak{m}}) = \bar{\mathfrak g}^{i}({\mathfrak{m}})/\bar{\mathfrak g}^{i+1}({\mathfrak{m}}),\quad\text{for all } i\in\mathbb{Z}.$$
The fundamental result of N. Tanaka and K. Yamaguchi says:
\[thm:yama\] The graded Lie algebra ${\mathfrak g}({\mathfrak{m}})$ coincides with a universal Tanaka prolongation of the graded nilpotent Lie algebra ${\mathfrak{m}}$. If, moreover, ${\mathfrak g}({\mathfrak{m}})$ (or $\bar {\mathfrak g}({\mathfrak{m}})$) is finite-dimensional, then $\bar {\mathfrak g}({\mathfrak{m}})$ is isomorphic to ${\mathfrak g}({\mathfrak{m}})$ as filtered Lie algebras.
We note that Tanaka and Yamaguchi prove this result only in the case when ${\mathfrak{m}}$ is fundamental, i.e. is generated by ${\mathfrak{m}}_{-1}$. But his prove works only without modification in the case of arbitrary graded nilpotent Lie algebras assuming we define the Tanaka prolongation as above.
This result can be generalized to the case when we additionally put extra linear restrictions at degree $0$. Note that ${\mathfrak g}_0({\mathfrak{m}})$ is exactly the Lie algebra ${\operatorname{Der}}_0({\mathfrak{m}})$ of all degree preserving derivations of ${\mathfrak{m}}$. Let ${\mathfrak g}_0$ be an arbitrary subalgebra in ${\operatorname{Der}}_0({\mathfrak{m}})$. Then we can define the universal prolongation of the pair $({\mathfrak{m}},{\mathfrak g}_0)$ as the largest graded Lie algebra ${\mathfrak g}({\mathfrak{m}},{\mathfrak g}_0)$ satisfying the following conditions:
1. ${\mathfrak g}_{i}({\mathfrak{m}},{\mathfrak g}_0) = {\mathfrak{m}}_{i}$ for all $i<0$ and ${\mathfrak g}_{0}({\mathfrak{m}},{\mathfrak g}_0) = {\mathfrak g}_0$;
2. for any $X\in {\mathfrak g}_i({\mathfrak{m}},{\mathfrak g}_0)$, $i\ge 0$, the equality $[X,{\mathfrak g}_{-}({\mathfrak{m}})]=0$ implies $X=0$.
Now suppose for simplicity that ${\mathfrak g}_0$ is defined as a subalgebra of ${\operatorname{Der}}_0({\mathfrak{m}})$ that stabilizes a family of graded subspaces $\{E_i\}_{i\in I}$ in ${\mathfrak{m}}$. We shall write this as ${\mathfrak g}_0={\operatorname{Stab}}(\{E_i\}_{i\in I})$. Then the Lie algebra ${\mathfrak g}({\mathfrak{m}},{\mathfrak g}_0)$ can also be interpreted in terms of symmetries of a family of left-invariant vector distributions on a Lie group ${\mathfrak{m}}$ with Lie algebra ${\mathfrak{m}}$. Namely, subspaces $E_i$, $i\in I$, extend to left-invariant vector distributions on $M$, and we define a filtered Lie algebra $\bar {\mathfrak g}({\mathfrak{m}},{\mathfrak g}_0)$ as symmetries of these distributions together with the distributions $T^{-i}M$ defined above. The proof of Theorem \[thm:yama\] stays valid in this generalized case as well: the graded Lie algebra ${\mathfrak g}({\mathfrak{m}},{\mathfrak g}_0)$ coincides with a universal Tanaka prolongation of the pair $({\mathfrak{m}},{\mathfrak g}_0)$.
We can now formulate the following result.
The Lie algebras ${\mathfrak g}^{I}_{k,l}$ and ${\mathfrak g}^{II}_{k,l}$ from Section \[sec:sym\] coincide with Tanaka prolongations of the pairs $({\mathfrak{n}}^{I}_{k,l},{\operatorname{Stab}}(E))$ and $({\mathfrak{n}}^{II}_{k,l},{\operatorname{Stab}}(E))$ where $E=\langle D \rangle$.
Directly follows from the simple observation that both EDS’s of first and second kind associated with the system coincide with left-invariant distributions defined by Lie algebras ${\mathfrak{n}}^{I}_{k,l}$ (first kind) ${\mathfrak{n}}^{II}_{k,l}$ (second kind) and the complementary left-invariant distribution corresponding to the subspace $E$.
Consider the case $(k,l)=(2,3)$. According to the above theorem, the universal Tanaka prolongation is isomorphic to the Lie algebra ${\mathfrak g}^{I}_{2,3}$. Let us describe the corresponding grading of ${\mathfrak g}^{I}_{2,3}$. We have: $$\begin{aligned}
&\deg -3: && \langle {\frac{{\partial}}{{\partial}y}}, {\frac{{\partial}}{{\partial}z}}\rangle;\\
&\deg -2: && \langle x{\frac{{\partial}}{{\partial}y}}, x{\frac{{\partial}}{{\partial}z}}\rangle;\\
&\deg -1: && \langle x^2{\frac{{\partial}}{{\partial}z}}, {\frac{{\partial}}{{\partial}x}}\rangle;\\
&\deg 0: && \langle x{\frac{{\partial}}{{\partial}x}}, y{\frac{{\partial}}{{\partial}y}}, z{\frac{{\partial}}{{\partial}z}}, y{\frac{{\partial}}{{\partial}z}}\rangle;\\
&\deg 1: && \langle x^2{\frac{{\partial}}{{\partial}x}}+xy{\frac{{\partial}}{{\partial}y}}+2xz{\frac{{\partial}}{{\partial}z}}, xy{\frac{{\partial}}{{\partial}z}} \rangle;\\
&\deg 2: && \langle y{\frac{{\partial}}{{\partial}x}} \rangle;\\
&\deg 3: && \langle xy{\frac{{\partial}}{{\partial}x}}+y^2{\frac{{\partial}}{{\partial}y}}+2yz{\frac{{\partial}}{{\partial}z}}, y^2{\frac{{\partial}}{{\partial}z}} \rangle.\end{aligned}$$
More generally, for arbitrary $2\le k < l$ the grading of ${\mathfrak g}^{I}_{k,l}$ can be determined by setting $\deg x = 1$, $\deg y=\deg z = l$ in formulas of Proposition \[propI\].
The grading of ${\mathfrak g}^{II}_{2,3}$ can be determined by setting $\deg x = 1$, $\deg y=\deg z_1=2$, $\deg z = 3$ in Proposition \[propII\]: $$\begin{aligned}
&\deg -3: && \langle {\frac{{\partial}}{{\partial}z}}\rangle;\\
&\deg -2: && \langle {\frac{{\partial}}{{\partial}y}}, x{\frac{{\partial}}{{\partial}z}}+{\frac{{\partial}}{{\partial}z_1}}\rangle;\\
&\deg -1: && \langle {\frac{{\partial}}{{\partial}x}}, x{\frac{{\partial}}{{\partial}y}}, x^2{\frac{{\partial}}{{\partial}z}}+2x{\frac{{\partial}}{{\partial}z_1}}\rangle;\\
&\deg 0: && \langle x{\frac{{\partial}}{{\partial}x}}-z_1{\frac{{\partial}}{{\partial}z_1}}, z{\frac{{\partial}}{{\partial}z}}+z_1{\frac{{\partial}}{{\partial}z_1}}, y{\frac{{\partial}}{{\partial}y}}, z_1{\frac{{\partial}}{{\partial}y}}\rangle;\\
&\deg 1: && \langle x^2 {\frac{{\partial}}{{\partial}x}} +xy{\frac{{\partial}}{{\partial}y}}+2xz{\frac{{\partial}}{{\partial}z}}+2z{\frac{{\partial}}{{\partial}z_1}}, z_1{\frac{{\partial}}{{\partial}x}}+z_1^2/2{\frac{{\partial}}{{\partial}z}}, (xz_1-2z){\frac{{\partial}}{{\partial}y}} \rangle;\\
&\deg 2: && \langle 2(xz_1-z){\frac{{\partial}}{{\partial}x}}+yz_1{\frac{{\partial}}{{\partial}y}}+xz_1^2{\frac{{\partial}}{{\partial}z}}+z_1^2{\frac{{\partial}}{{\partial}z_1}} \rangle;\\
&\deg 3: && \langle (x^2z_1/2 -zx){\frac{{\partial}}{{\partial}x}}+(xyz_1/2-yz){\frac{{\partial}}{{\partial}y}}+(x^2z_1^2/4-z^2){\frac{{\partial}}{{\partial}z}}+(xz_1^2/2-zz_1){\frac{{\partial}}{{\partial}z_1}}.\end{aligned}$$
Similarly, for all other $(k,l)$ the grading of ${\mathfrak g}^{II}_{k,l}$ is defined by $\deg x = 1$, $\deg y=k$, $\deg z_i = l-i$, $i=0,\dots,l-k$.
Symmetry algebras via Sternberg prolongation {#sec:g}
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In this section we show how symmetry algebras ${\mathfrak g}^{I}_{k,l}$ and ${\mathfrak g}^{II}_{k,l}$ can be obtained as Sternberg prolongations of certain subalgebras in ${\mathfrak{gl}}(k+l,{\mathbb R})$. These subalgebras are exactly the symmetry algebras of homogeneous rational curves in ${\operatorname{Gr}}(k+l-2,k+l)$ and ${\operatorname{Gr}}(2,k+l)$ in the cases of symmetries of first and second kind respectively.
Let us recall the notion of Sternberg prolongation of a linear Lie algebra. Let $V$ be an arbitrary finite-dimensional vector space and let ${\mathfrak a}\subset {\mathfrak{gl}}(V)$ be a linear Lie algebra. Denote also by $D(V)$ the Lie algebra of polynomial vector fields on $V$: $$D(V) = \sum_{i=0}^{\infty} S^i(V^*)\otimes V.$$ It is a graded Lie algebra, where $D_k(V)=S^{k+1}(V^*)\otimes V$, so that $D_{-1}(V)=V$ and $D_0(V)$ is identified with ${\mathfrak{gl}}(V)$. By Sternberg prolongation of ${\mathfrak a}\subset {\mathfrak{gl}}(V)$ we understand a graded subalgebra ${\mathfrak g}$ of $D(V)$ that can be defined via three equivalent ways:
1. ${\mathfrak g}$ is a largest graded subalgebra of $D(V)$ such that ${\mathfrak g}_{-1}=D_{-1}(V)=V$ and ${\mathfrak g}_0={\mathfrak a}$;
2. ${\mathfrak g}_{-1}=D_{-1}(V)$, ${\mathfrak g}_0={\mathfrak a}$ and ${\mathfrak g}_{i+1}=\{u\in D_{i+1}(V)\mid [u,{\mathfrak g}_{-1}]\subset {\mathfrak g}_i$ for all $i\ge 0$;
3. ${\mathfrak g}_{i} = S^{i+1}(V^*)\otimes V \cap S^{i-1}(V^*)\otimes {\mathfrak a}$ for all $i\ge -1$.
Let $V$ be now the $(k+l)$-dimensional vector space with the basis $\{ e_0, e_1,\dots, e_{k-1}, f_0, f_1, \dots, f_{l-1} \}$. Define the nilpotent linear operator $X\colon V\to V$ by: $$\begin{aligned}
\label{Xdef}
X(e_i)&=e_{i-1}, \quad i=1,\dots,k-1;\quad X(e_0)=0;\\
X(f_i)&=f_{i-1}, \quad i=1,\dots,l-1;\quad X(f_0)=0 \nonumber.\end{aligned}$$ Denote by $\exp(tX)$ the corresponding one-parameter group in $GL(V)$.
We shall consider two different gradings of $V$:
1. the grading of first kind: $\deg e_i = \deg f_i = i-l$;
2. the grading of second kind: $\deg e_i = i-k$, $\deg f_j = j-l$.
Note that for both gradings the operator $X$ has degree $-1$. But they induce different gradings on the Lie algebra ${\mathfrak{gl}}(V)$. And for both gradings all elements of $V$ are concentrated in negative degree.
Define a curve $\gamma^{I}$ in ${\operatorname{Gr}}(k+l-2,k+l)$ as the closure of the orbit of the one-parameter subgroup $\exp(tX)$ through the codimension two subspace $V_1=\langle e_1, \dots, e_{k-1}, f_1,\dots, f_{l-1}\rangle$. Similarly, define a curve $\gamma^{II}$ in ${\operatorname{Gr}}(2,k+l)$ as a closure of the orbit of $\exp(tX)$ through the two-dimensional subspace $V_2=\langle e_{k-1}, f_{l-1}\rangle$. The curve $\gamma^{II}$ is well-known in projective geometry as a rational normal scroll $S_{k,l}$ (see [@harris]), while $\gamma^{I}$ is its dual curve. Let $A^{I}_{k,l}, A^{II}_{k,l}\subset GL(V)$ be the symmetry groups of curves $\gamma^{I}$ and $\gamma^{II}$ respectively. Denote by ${\mathfrak a}^{I}_{k,l}$ and ${\mathfrak a}^{II}_{k,l}$ the corresponding subalgebras in ${\mathfrak{gl}}(V)$.
These symmetry algebras can be easily computed in a purely algebraic way as follows:
\[flagprolong\] The subalgebras ${\mathfrak a}^{I}_{k,l}, {\mathfrak a}^{II}_{k,l}$ are the largest graded subalgebras of ${\mathfrak{gl}}(V)$ whose negative part is one-dimensional and is generated by $X$. They can be constructed inductively as: $$\begin{aligned}
{\mathfrak a}_{-1} &= \langle X \rangle; \\
{\mathfrak a}_{i} & = \{ u\in {\mathfrak{gl}}_{i}(V) \mid [u, X]\subset {\mathfrak a}_{i-1} \},\quad\text{for }i\ge 0,\end{aligned}$$ where ${\mathfrak a}$ is either ${\mathfrak a}^{I}_{k,l}$ or ${\mathfrak a}^{II}_{k,l}$ and $V$ is equipped with the grading of first or second kind respectively.
In fact, symmetry algebras of rational normal scrolls are well-known. If $k<l$, then $${\mathfrak a}^{II}_{k,l} = \left\{\left.
\begin{pmatrix} \rho_k(A)+ e_1 E_{k} & B \\
0 & \rho_l(A) + e_2 E_{l} \end{pmatrix}
\,\right|\, \begin{matrix} A\in {\mathfrak{sl}}(2,{\mathbb R}), e_1,e_2\in{\mathbb R}, \\ B \subset V(l-k) \end{matrix}
\right\},$$ where $\rho_r\colon {\mathfrak{sl}}(2,{\mathbb R})\to {\mathfrak{gl}}(r,{\mathbb R})$ is an irreducible $r$-dimensional representation of ${\mathfrak{sl}}(2,{\mathbb R})$, and $V(l-k)$ is an irreducible component of dimension $l-k+1$ in the decomposition of the tensor product of $\rho_k$ and $\rho_l$.
For example, in the simplest case of $(k,l)=(2,3)$ we have: $${\mathfrak a}^{II}_{2,3}=
\left\{
\begin{pmatrix}
a+e_1 & c & p & q & 0 \\
b & -a+e_1 & 0 & p & q \\
0 & 0 & 2a+e_2 & 2c & 0\\
0 & 0 & b & e_2 & c \\
0 & 0 & 0 & 2b & -2a + e_2
\end{pmatrix}\right\}.$$
If $k=l$, then $${\mathfrak a}^{II}_{k,k} = \left\{\left.
\begin{pmatrix} \rho_k(A)+c_{11} E_{k} & c_{12} E_{k} \\
c_{21} E_{k} & \rho_k(A) + c_{22} E_{k} \end{pmatrix}
\,\right|\, \begin{matrix} A\in {\mathfrak{sl}}(2,{\mathbb R}), \\ c_{11},c_{12},c_{21},c_{22}\in{\mathbb R}\end{matrix}
\right\}.$$
As $\gamma^{II}$ is dual to $\gamma^{I}$, the subalgebra ${\mathfrak a}^{I}_{k,l}$ is conjugate to the transposed of ${\mathfrak a}^{II}_{k,l}$.
Now we can formulate the main result of the paper.
Let ${\mathfrak a}^{I}_{k,l},{\mathfrak a}^{II}_{k,l}\subset {\mathfrak{gl}}(V)$ be symmetry algebras of the curves $\gamma_1$ and $\gamma_2$ respectively. The Sternberg prolongations of ${\mathfrak a}^{I}_{k,l},{\mathfrak a}^{II}_{k,l}$ are finite dimensional and coincide with Lie algebras ${\mathfrak g}^{I}_{k,l}$ and ${\mathfrak g}^{II}_{k,l}$.
The proof of this theorem is a direct corollary of the following technical result.
Let ${\mathfrak{m}}$ be a graded nilpotent Lie algebra, $V$ a graded commutative ideal stable with respect to ${\operatorname{Der}}_0({\mathfrak{m}})$, and let $E\subset {\mathfrak{m}}$ be a commutative subalgebra in ${\mathfrak{m}}_{-1}$ such that ${\mathfrak{m}}=E\oplus V$.
Assume that the action of $E$ on $V$ is faithful and denote by ${\operatorname{ad}}E$ the corresponding subalgebra in ${\mathfrak{gl}}(V)$, concentrated in degree $-1$. Define ${\mathfrak a}$ as a largest graded subalgebra of ${\mathfrak{gl}}(V)$ such that ${\mathfrak a}_{-}={\operatorname{ad}}E$. Then the Tanaka prolongation of the pair $({\mathfrak{m}}, {\operatorname{Stab}}(E))$ coincides with the Sternberg prolongation of the subalgebra ${\mathfrak a}\subset {\mathfrak{gl}}(V)$.
Let ${\mathfrak g}$ be the Sternberg prolongation of the subalgebra ${\mathfrak a}\subset{\mathfrak{gl}}(V)$. We equip it with grading inherited from the grading of $V\subset{\mathfrak{m}}$. Let us show that ${\mathfrak g}$ satisfies conditions (1)-(2) or the Tanaka prolongation of the pair $({\mathfrak{m}},{\operatorname{Stab}}(E))$ and hence is naturally embedded into ${\mathfrak g}({\mathfrak{m}},{\operatorname{Stab}}(E))$.
It is clear that both $V$ and ${\operatorname{ad}}E \subset {\mathfrak a}$ are negatively graded and lie in ${\mathfrak g}$. Suppose $u$ is a homogeneous element of ${\mathfrak g}$ of negative degree. If it lies in $S^1(V^*)\otimes V = {\mathfrak{gl}}(V)$, then by construction of ${\mathfrak a}$ it is contained in ${\operatorname{ad}}E$. If $u\in S^k(V^*)\otimes V$ with $k\ge 2$, then we can find homogeneous elements $v_1,\dots,v_{k-1}\in V$ such that $\bar u=[\dots [u,v_1],\dots, v_{k-1}]$ is a non-zero element of ${\mathfrak a}$. As all $v_1,\dots,v_{k-1}\in V$ are negatively graded and $\deg \bar u\ge -1$, we see that $\deg u \ge 0$. Thus, negative part of ${\mathfrak g}$ coincides with $({\operatorname{ad}}E) \oplus V$ and is naturally isomorphic to ${\mathfrak{m}}$.
Let $u\in{\mathfrak g}$ be an arbitrary element of degree $0$. Let us show that it lies in ${\mathfrak a}_0$ and thus preserves ${\operatorname{ad}}E$. Another option would be that $u\in S^2(V^*)\otimes V$ and $[u,v]\in{\mathfrak a}$ for all $v\in V$. As $[u,v]$ is necessarily negatively graded, this would imply that $[u,v]\in{\operatorname{ad}}E$. Thus, ${\operatorname{ad}}u$ would define a non-zero degree preserving derivation of ${\mathfrak{m}}$ that takes $V$ to $E$. But this contradicts the assumption that $V$ is stable with respect to ${\operatorname{Der}}_0({\mathfrak{m}})$. This proves that ${\mathfrak g}_0$ stabilizes $E$.
Now let $u\in{\mathfrak g}$ be an arbitrary non-zero element of non-negative degree. Then by the property of Sternberg prolongation we have $[u,V]\ne 0$. This completes the proof that ${\mathfrak g}$ is naturally embedded into the Tanaka prolongation of $({\mathfrak{m}},{\operatorname{Stab}}(E))$.
Let us now prove that ${\mathfrak g}({\mathfrak{m}},{\operatorname{Stab}}(E))$ is embedded into the Sternberg prolongation of the subalgebra ${\mathfrak a}\subset {\mathfrak{gl}}(V)$. As $V$ naturally lies in ${\mathfrak g}({\mathfrak{m}},{\operatorname{Stab}}(E))$ as a commutative subalgebra, we just need to prove that there is a complementary subalgebra ${\mathfrak g}_0$ to $V$ such that each non-zero element in it has a non-zero bracket with $V$. We define ${\mathfrak g}_0$ as $E+\sum_{i\ge 0} {\mathfrak g}_i({\mathfrak{m}},{\operatorname{Stab}}(E))$. Indeed, by definition of ${\mathfrak g}({\mathfrak{m}},{\operatorname{Stab}}(E))$ we have $[E,{\mathfrak g}_0({\mathfrak{m}},{\operatorname{Stab}}(E))]\subset E$ and $[E,{\mathfrak g}_i({\mathfrak{m}},{\operatorname{Stab}}(E))]\subset {\mathfrak g}_{i-1}({\mathfrak{m}},{\operatorname{Stab}}(E))$ for $i>0$. Next, let $u$ be an arbitrary non-zero element inside ${\mathfrak g}_0$. Suppose $[u,V]=0$. This means that $u$ lies in the centralizer $Z(V)$ of $V$ in ${\mathfrak g}({\mathfrak{m}},{\operatorname{Stab}}(E))$. But as $[E,V]\subset V$, the centralizer $Z(V)$ is also stable with respect to the adjoint action of $E$. Hence, by the property (2) of Tanaka prolongation taking sufficiently many brackets of $u$ with elements from $E$, we get a non-zero element $\bar u$ from $Z_0(V)=Z(V)\cap {\mathfrak g}_0({\mathfrak{m}},{\operatorname{Stab}}(E))$. Moreover, this element acts trivially on $V$, but non-trivially on $E$. Thus, there is an element $e\in E$ such that $[\bar u, e]\ne 0$. But then for any element $v\in V$ we have: $$[[\bar u, e], v] = [\bar u, [e, v]] - [e, [\bar u, v]] =0$$ This contradicts to the assumption that the action of $E$ on $V$ is faithful.
According to the above theorem the Sternberg prolongation of the Lie algebra ${\mathfrak a}^{I}_{2,3}$ is equal to ${\mathfrak g}^{I}_{2,3}$ and is isomorphic to ${\mathfrak{gl}}(3,{\mathbb R}){\rightthreetimes}S^2({\mathbb R}^3)$. Let us describe the corresponding grading of ${\mathfrak g}^{I}_{2,3}$: $$\begin{aligned}
&\deg -1: && \langle {\frac{{\partial}}{{\partial}y}}, {\frac{{\partial}}{{\partial}z}}, x{\frac{{\partial}}{{\partial}y}}, x{\frac{{\partial}}{{\partial}z}}, x^2{\frac{{\partial}}{{\partial}z}}\rangle;\\
&\deg 0: && \langle {\frac{{\partial}}{{\partial}x}}, x{\frac{{\partial}}{{\partial}x}}, x^2{\frac{{\partial}}{{\partial}x}}+xy{\frac{{\partial}}{{\partial}y}}+2xz{\frac{{\partial}}{{\partial}z}}, y{\frac{{\partial}}{{\partial}y}}, z{\frac{{\partial}}{{\partial}z}}, y{\frac{{\partial}}{{\partial}z}}, xy{\frac{{\partial}}{{\partial}z}}\rangle;\\
&\deg 1: && \langle y{\frac{{\partial}}{{\partial}x}}, y^2{\frac{{\partial}}{{\partial}z}}, xy{\frac{{\partial}}{{\partial}x}}+y^2{\frac{{\partial}}{{\partial}y}}+2yz{\frac{{\partial}}{{\partial}z}} \rangle.\\\end{aligned}$$ As expected, the degree $-1$ component is a commutative subalgebra.
In general, in case of arbitrary $2\le k < l$ the grading of ${\mathfrak g}^I_{k,l}$ according to Sternberg prolongation can be determined by setting $\deg x = 0$, $\deg y=\deg z = 1$ in formulas of Proposition \[propI\]. Note that the number of non-zero prolongations of ${\mathfrak a}^{I}_{k_l}$ can be arbitrarily high. For example, if $k=2$, then the $(l-2)$-nd prolongation is still non-zero, while $(l-1)$-st one already vanishes.
The grading of ${\mathfrak g}^{II}_{2,3}$ viewed as Sternberg prolongation of ${\mathfrak a}^{II}_{2,3}$ can be determined by setting $\deg x = 0$, $\deg y=\deg z = \deg z_1=1$ in Proposition \[propII\].
For $(k,l)\ne (2,3)$ the grading of ${\mathfrak g}^{II}_{k,l}$ is defined by $\deg x = 0$, $\deg y=\deg z = 1$.
Non-linear mixed order equations {#sec:nlin}
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Non-linear mixed-order equations can be treated via the notion of $G$-structures on filtered manifolds introduced by N. Tanaka. Consider a system: $$\label{nlin:kl}
\begin{aligned}
y^{(k)} &= f(x,y,y',\dots,y^{(k-1)},z,z',\dots,z^{(l-1)}),\\
z^{(l)} &= g(x,y,y',\dots,y^{(k-1)},z,z',\dots,z^{(l-1)}),
\end{aligned}$$ where as above $2\le k<l$. It can be viewed as a codimension 2 submanifold ${\mathcal{E}}$ of the mixed jet space $J^{k,l}({\mathbb R},{\mathbb R}^2)$, which is transversal to the fibers of the projection $\pi\colon J^{k,l}({\mathbb R},{\mathbb R}^2) \to J^{k-1,l-1}({\mathbb R},{\mathbb R}^2)$.
The canonical contact system on $J^{k,l}({\mathbb R},{\mathbb R}^2)$ restricted to the equation manifold ${\mathcal{E}}$ is given by the following 1-forms: $$\begin{aligned}
& dy_i - y_{i+1}dx, i=0,\dots k-2, \quad dz_j - z_{j+1}dx, j=0,\dots,l-2;\\
& dy_{k-1} - f(x,y_0,y_1,\dots,y_{k-1},z_0,z_1,\dots,z_{l-1}) dx;\\
& dz_{l-1} - g(x,y_0,y_1,\dots,y_{k-1},z_0,z_1,\dots,z_{l-1}) dx.\\\end{aligned}$$ It defines a one-dimensional vector distribution $E$, whose integral curves are lifts of solutions of the given system to the jet space $J^{k,l}({\mathbb R},{\mathbb R}^2)$.
As in the case of two different notions of symmetries, there are also two ways to introduce another vector distribution (or even a flag of distributions) complementary to $E$. Namely, the first way is to define a complementary foliation $F$ as the kernel of the projection $\pi_1\colon {\mathcal{E}}\to J^{0,0}({\mathbb R},{\mathbb R})={\mathbb R}^3$. The second way is to define $F$ as the kernel of the projection $\pi_2\colon {\mathcal{E}}\to J^{k-2,l-2}({\mathbb R},{\mathbb R}^2)$.
Let us first consider the second case in more detail. As in case of trivial ODEs, let $F$ be the 2-dimensional completely integrable distribution tangent to the fibers of the projection $\pi_2\colon {\mathcal{E}}\to J^{k-2,l-2}({\mathbb R},{\mathbb R}^2)$. Next, we define $D=E\oplus F$, which can also be defined as a pull-back of the standard contact system on $J^{k-1,l-1}({\mathbb R},{\mathbb R}^2)$. In particular, the weak derived flag of $D$ defines the filtration of the tangent bundle $T{\mathcal{E}}$ and turns ${\mathcal{E}}$ into a filtered manifold of type ${\mathfrak{n}}^{II}_{k,l}$. The decomposition $D=E\oplus F$ is known as so-called *pseudo-product structure* and introduced and first studied by N. Tanaka [@tan3]. As in case of trivial ODEs, we shall call this pseudo-product structure *the EDS of second kind associated with a non-linear system of ODEs of mixed order*.
Its algebraic prolongation is already computed in Section \[sec:tan\] and is equal to ${\mathfrak g}^{II}_{k,l}$. In particular, using the results of Tanaka [@tan1] (see also Zelenko [@zel:tan]) we immediately arrive at the following result.
\[propkind2\] For any EDS of second kind associated with system there exists a natural frame bundle $P\to {\mathcal{E}}$ and an absolute parallelism structure $\omega\colon TP \to {\mathfrak g}^{II}_{k,l}$.
Let us now consider the EDS interpretation of that contains the fibers of the projection $\pi_1\colon {\mathcal{E}}\to J^{0,0}({\mathbb R},{\mathbb R})={\mathbb R}^3$ as a part of its data. As in case of trivial systems, define $F_1$ as a distribution tangent to the fibers of the projection $\pi_1$: $$\label{F1def}
F_1 = \left \langle {\frac{{\partial}}{{\partial}y_i}}, i=1,\dots,k-1; {\frac{{\partial}}{{\partial}z_j}},j=1,\dots,l-1\right\rangle.$$ Further, define a sequence of distributions: $$\label{Fidef}
F_{i+1} = \{ Y \in F_i \mid [Y, E]\subset F_i \}.$$ We call a pair of distributions $(E,F_1)$ *the EDS of first kind associated with a non-linear pair of ODEs of mixed order*. However, in general, the dimensions of these distributions in the non-trivial case may differ from the dimensions of these distributions for the trivial system of equations. To see this explicitly, consider the first non-trivial case of $(k,l)=(2,4)$ when this can be observed explicitly. In this case the vector distribution $E$ is spanned by: $$X = {\frac{{\partial}}{{\partial}x}} + y_1{\frac{{\partial}}{{\partial}y_0}} + f{\frac{{\partial}}{{\partial}y_2}} + z_1{\frac{{\partial}}{{\partial}z_0}} + z_2{\frac{{\partial}}{{\partial}z_1}} + z_3{\frac{{\partial}}{{\partial}z_2}} + g{\frac{{\partial}}{{\partial}z_3}},$$ and $F_1$ is spanned by vector fields ${\frac{{\partial}}{{\partial}y_1}}$ and ${\frac{{\partial}}{{\partial}z_i}}$, $i=1,2,3$. Simple computation shows that, as expected, $F_2$ is spanned by ${\frac{{\partial}}{{\partial}z_i}}$, $i=2,3$. However, for $F_3$ we have already the branching: $$\begin{aligned}
F_3 &= 0,\qquad&\text{if } \frac{\partial f}{\partial z_3} \ne 0;\\
F_3 &= \left\langle {\frac{{\partial}}{{\partial}z_3}} \right\rangle, &\text{otherwise}.\end{aligned}$$
This example can be easily extended to arbitrary $k<l$. A straightforward generalization of Tanaka prolongation procedure [@tan3; @zel:tan] to filtered structures with constant non-fundamental symbol gives:
\[propkind1\] Let ${\mathcal{E}}$ be a non-linear pair of ODEs such that $\frac{\partial f}{\partial z_i} = 0$, $i=k+1,\dots,l-1$. Then there exists a frame bundle $P\to {\mathcal{E}}$ and an absolute parallelism structure $\omega\colon TP \to {\mathfrak g}^{I}_{k,l}$ naturally associated with the EDS of first kind for system .
Flag structures {#sec:flag}
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Non-linear mixed order equations can be viewed as a particular case of another class of geometric structures. Let $M$ be an arbitrary smooth manifold of dimension $n$. Let $\alpha=(\alpha_1,\dots,\alpha_r)$ be any increasing sequence of integers $1\le \alpha_1 < \dots < \alpha_r < n$. Denote by $F_{\alpha}(T_pM)$, $p\in M$, the flag variety of subspaces in $T_pM$ of dimensions $\alpha_1,\dots,\alpha_r$ and by $F_{\alpha}(TM)$ or simply by $F_\alpha(M)$ the bundle of flag varieties at all points $p\in M$.
Let $V$ be a vector space of dimension $n$ and let $F_\alpha(V)$ be the flag variety of subspaces in $V$. To stick with the notational agreements of the Tanaka theory we number the subspaces in flags from $F_{\alpha}(V)$ by negative integers, i.e. an element of $F_{\alpha}(V)$ is a decreasing by inclusion tuple $\{V_{-i}\}_{i=1}^r$ of subspaces of $V$ such that $\dim V_{-i}=\alpha_i$. The flag variety $F_\alpha(V)$ is naturally equipped with a transitive action of the Lie group $GL(V)$. The bundle $F_\alpha(M)$ can also be viewed as bundle associated to the principal $GL(V)$-bundle $\mathcal{F}(M)$ of all frames in $M$.
For any curve $W(t)\subset V$ of subspaces of fixed dimension $m$ (i.e., a curve in a Grassmann variety ${\operatorname{Gr}}_m(V)$) we define $W'(t)$ as follows. Choose a family of smooth curves $v_1(t),\dots,v_m(t)$ in $V$ such that $$W(t) = \langle v_1(t),\dots,v_m(t) \rangle,\quad t\in {\mathbb R}.$$ Then we define: $$W'(t) = W(t) + \langle v'_1(t),\dots,v'_m(t) \rangle,\quad t\in {\mathbb R}.$$ It is easy to check that $W'(t)$ does not depend on the choice if the vectors $v_i(t)$. Moreover if $w(t)$ is a smooth curve in $V$ such that $w(t)\in W(t)$ then for any $t_0$ the image of the vector $w'(t_0)$ to the factor space $V/W(t_0)$ depends on the vector $w(t_0)$ only and not on the curve $w(t)$, i.e an element of ${\operatorname{Hom}}\bigl(W(t_0), V/W_(t_0)\bigr)$ is assigned to the tangent vector to the curve $t\mapsto W(t)$ at $t_0$.
Let now $\Gamma\subset F_\alpha(V)$ be an unparametrized curve. Fixing any local parameter $t$ on $\Gamma$, we can explicitly write it as: $$\label{flagcurve}
0 \subset V_{r}(t) \subset V_{r-1}(t)\subset\dots \subset V_{1}(t)\subset V_0 \subset V.$$ We say that the curve $\Gamma$ is *integral* or *compatible with respect to differentiation*, if it $V_i'(t)\subset V_{i-1}(t)$ for all $i=1,\dots,r)$. Similar to the previous paragraph, the tangent line to the curve $\Gamma$ at the point corresponding to a parameter $t$ can be associated with a line generated by a degree $-1$ endomorphisms $X_t$ of the graded space $\displaystyle{\bigoplus_{i=0}^{r} V_i(t)/V_{i+1}(t)}$, where $V_{r+1}(t):=0$. Identifying all graded spaces $\displaystyle{\bigoplus_{i=0}^{r} V_i(t)/V_{i+1}(t)}$ with one grading of the space $V$ we can consider the line $\langle X_t\rangle$ of degree $-1$ endomorphisms of $V$, defined up to the conjugations by linear isomorphisms preserving the grading. This line (or more precisely its equivalence class with respect to the above conjugation) is called the *flag symbol of the curve at the point $t$*. The curve of flags is said to be of constant type $\langle X\rangle$ if its flag symbols at any point belong to the same equivalence class of the line $\langle X\rangle$ (with respect to the conjugations by linear isomorphisms preserving the grading). Note that absolutely the same constructions can be done if the indices in are shifted somehow.
A *flag structure on a manifold $M$* is a smooth one-dimensional subbundle $\mathcal C$ of the flag bundle $F_\alpha(TM)$ such that $C_p\subset F_\alpha(T_pM)$ is an integral curve for all $p\in M$. We say that a flag structure has a constant flag symbol $\langle X\rangle$ if all its fibers are curves of flags with constant flag symbol $\langle X\rangle$.
Flag structures, and even in more general setting, were studied in our recent preprint [@quasi]. Assume that, like in Proposition \[flagprolong\], ${\mathfrak a}(\langle X\rangle)$ is the largest graded subalgebras of ${\mathfrak{gl}}(V)$ whose negative part is one-dimensional and is generated by $X$. The algebra ${\mathfrak a}(\langle X\rangle)$ is called the *universal prolongation of the flag symbol $\langle X\rangle$ in ${\mathfrak{gl}}(V)$.* Let ${\mathfrak g}(\langle X\rangle)$ be the Sternberg prolongation of the algebra $\langle X\rangle$, as described in section 5. The following Theorem is a particular case of Theorem 2.4 in [@quasi]:
\[flagthm\] For any flag structure on a manifold $M$ with constant flag symbol $\langle X\rangle$ there exists a natural frame bundle $P\to {\mathcal{E}}$ and an absolute parallelism structure $\omega\colon TP \to {\mathfrak g}(\langle X\rangle)$.
Now we show how to assign a natural flag structure with constant symbol to the EDS, both of first and second kind, associated with a systems of ODEs on the space of solutions of this system. Let ${\operatorname{Fol}}(\mathcal E)$ be the foliation of the equation submanifold $\mathcal E$ the (prolonged) solutions of the system or, equivalently, by the integral curves of the rank $1$ distribution $E$. Then Propositions \[propkind2\] and \[propkind1\] can be seen as particular cases of Theorem \[flagthm\].
Indeed, let us pass to the quotient manifold of $\mathcal E$ by the foliation ${\operatorname{Fol}}(\mathcal E)$, i.e. to the space of solutions of . Locally we can assume that there exists a quotient manifold $$\text{Sol}(\mathcal E)=\mathcal E/{\operatorname{Fol}}(E),$$ whose points are leaves of ${\operatorname{Fol}}(E)$ or, equivalently, (prolonged) solutions of . Let $\Phi\colon \mathcal E\to {\operatorname{Fol}}(E)$ be the canonical projection to the quotient manifold.
Fix a leaf $\gamma$ of ${\operatorname{Fol}}(E)$. For the EDS of the second kind, as in section 6, let $F$ be the 2-dimensional completely integrable distribution tangent to the fibers of the projection $\pi_2\colon {\mathcal{E}}\to J^{k-2,l-2}({\mathbb R},{\mathbb R}^2)$: $$\label{Ji}
V_{-1}(x):=\Phi_*\bigl(F(x)\bigr), \quad x\in \gamma.$$ Then the curve $x\mapsto V_{-1}(x), x\in\gamma$ is a curve in the Grassmannian of planes in $T_\gamma {\operatorname{Fol}}(E)$, Taking differentiation, one can generate from this curve the following curve of flags in $T_\gamma {\operatorname{Fol}}(E)$: $$C^{II}_\gamma=\{x\mapsto \{0 \subset V_{l-1}(x) \subset V_{l-2}\dots \subset V_{0}(x)= T_\gamma {\operatorname{Fol}}(E)\}: x\in\gamma\},$$ where the spaces $V_{i}(x)$ are defined inductively: $V_{i-1}(x):=V_{i}'(x)$. Note that each curve $C^{II}_\gamma$ is a curve with constant flag symbol, generated by $X$ as in with the grading of the second kind (see the paragraph after the formula ).
Then the bundle $C^{II}\rightarrow {\operatorname{Fol}}(E)$ is the flag structure on ${\operatorname{Fol}}(E)$ with the constant flag symbol generated by this $X$ and the problem of equivalence of the EDSs of second kind is the same as the problem of equivalence of the corresponding flag structures $C^{II}$. Proposition \[propkind2\] is the consequence of Proposition \[flagprolong\] and Theorem \[flagthm\].
In the same way for the EDS of the first kind, if $F_i$ with $1\leq i\leq l-1$ are the distributions defined by relations - and $F_0=T {\operatorname{Fol}}(E)$, then let $$C^{I}_\gamma=\{x\mapsto\{\Phi_*\bigl(F_i(x)\bigr)\}_{i=0}^{l-1}: x\in\gamma\},$$ Note that by the definition the curve of flags $C^{I}_\gamma$ it is compatible with respect to the differentiation and for EDS of the first kind associated with ODE such that$\frac{\partial f}{\partial z_i} = 0$, $i=k+1,\dots,l-1$ the curve $C^{I}_\gamma$ is a curve with constant flag symbol, generated by $X$ as in with the grading of the first kindm(see the paragraph after the formula ). Then for such EDS the bundle $C^{I}\rightarrow {\operatorname{Fol}}(E)$ is the flag structure on ${\operatorname{Fol}}(E)$ with the constant flag symbol generated by this $X$ and the problem of equivalence of the EDSs of first kind is the same as the problem of equivalence of the corresponding flag structures $C^{I}$. Proposition \[propkind1\] is the consequence of Proposition \[flagprolong\] and Theorem \[flagthm\].
Other kinds of EDS associated with an ODE system of mixed order {#sec:shifts}
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As shown above, the two different EDS interpretations of a given trivial or general system of two ODEs of mixed order come from fixing different jet space projections.
Namely, in case of an EDS of first kind we assume that the projection $J^{k,l}({\mathbb R},{\mathbb R}^2)\to J^{0,0}({\mathbb R},{\mathbb R}^2)$ is preserved. Hence, all symmetries of the first kind are just point transformations that preserve equation ${\mathcal{E}}\subset J^{k,l}({\mathbb R},{\mathbb R}^2)$. Moreover, due to the classical Lie theorem any transformation preserving the contact system on $J^{r,r}({\mathbb R},{\mathbb R}^2)$, $r\ge 0$, also preserves the projection $J^{r,r}({\mathbb R},{\mathbb R}^2)\to J^{0,0}({\mathbb R},{\mathbb R}^2)$. Therefore, defining an EDS of first kind we could as well assume that any of the projections $J^{k,l}({\mathbb R},{\mathbb R}^2)\to J^{r,r}({\mathbb R},{\mathbb R}^2)$, $r=0,\dots,k$, is preserved.
In the same manner, an EDS of second kind is defined by assuming that the projection $J^{k,l}({\mathbb R},{\mathbb R}^2)\to J^{k-1,l-1}({\mathbb R},{\mathbb R}^2)$ is preserved. This automatically implies that any of the projections $J^{k,l}({\mathbb R},{\mathbb R}^2)\to J^{k-r,l-r}({\mathbb R},{\mathbb R}^2)$, $r=0,\dots,k$ is also preserved.
But we can define other kinds of EDS assuming that we preserve any of the projections $J^{k,l}({\mathbb R},{\mathbb R}^2)\to J^{k-r,l-s}({\mathbb R},{\mathbb R}^2)$ for arbitrary $r=1,\dots,k$, $s=1,\dots,l$. Similar to Lie theorem, we can prove that this automatically implies that the projections $J^{k,l}({\mathbb R},{\mathbb R}^2)\to J^{k-r-1,l-s-1}({\mathbb R},{\mathbb R}^2)$ and $J^{k,l}({\mathbb R},{\mathbb R}^2)\to J^{k-r+1,l-s+1}({\mathbb R},{\mathbb R}^2)$ are also preserved, assuming that all indexes are non-negative. Thus, only the difference $\delta=(k-r)-(l-s)$ is important.
In more detail, we have:
An *EDS of shift $\delta$* associated with a trivial system of equations $y^{(k)}=z^{(l)}=0$, $k\le l$ is defined as a pair of vector distributions on the equation ${\mathcal{E}}={y_k=z_l=0}\subset J^{k,l}$: $$\begin{aligned}
E &= \langle {\frac{{\partial}}{{\partial}x}}+y_1{\frac{{\partial}}{{\partial}y_0}} + \dots y_{k-1} {\frac{{\partial}}{{\partial}y_{k-2}}} + z_1{\frac{{\partial}}{{\partial}z_0}} + \dots z_{l-1} {\frac{{\partial}}{{\partial}y_{l-2}}} \rangle;\\
F &= \langle {\frac{{\partial}}{{\partial}y_1}}, \dots, {\frac{{\partial}}{{\partial}y_{k-1}}}, {\frac{{\partial}}{{\partial}z_{\delta+1}}}, \dots, {\frac{{\partial}}{{\partial}z_{l-1}}}\rangle.\end{aligned}$$ if $\delta \ge 0$, and: $$\begin{aligned}
E &= \langle {\frac{{\partial}}{{\partial}x}}+y_1{\frac{{\partial}}{{\partial}y_0}} + \dots y_{k-1} {\frac{{\partial}}{{\partial}y_{k-2}}} + z_1{\frac{{\partial}}{{\partial}z_0}} + \dots z_{l-1} {\frac{{\partial}}{{\partial}y_{l-2}}} \rangle;\\
F &= \langle {\frac{{\partial}}{{\partial}y_{-\delta+1}}}, \dots, {\frac{{\partial}}{{\partial}y_{k-1}}}, {\frac{{\partial}}{{\partial}z_1}}, \dots, {\frac{{\partial}}{{\partial}z_{l-1}}}\rangle.\end{aligned}$$ if $\delta < 0$.
Note that this definition includes also the case of $k=l$, where, as we shall see below, different values of the shift lead to different symmetry algebras. Although the definition makes sense for arbitrary values of $s$, we shall be mainly interested in non-trivial cases, when $\delta$ is in the range from $-k+2$ to $l-2$.
As above, the vector distribution $E$ is tangent to the solutions of the system, while the distribution $F$ is tangent to the fibers of the projection $\pi\colon {\mathcal{E}}\to J^{0,\delta}$. In particular, we see that the EDS of first kind corresponds to the shift $\delta=0$, while the EDS of second kind corresponds to the shift $\delta=l-k$.
There is a graphical way to encode an arbitrary EDS of shift $s$ using a skew Young tableau (i.e., the tableau does not need to be aligned to the left). Namely, the tableau consists of two rows with $l$ and $k$ boxes respectively, the rows are right aligned for $\delta=0$, the row with $l$ cells is shifted by extra $\delta$ boxes to the right if $\delta$ is positive or to $-\delta$ cells to the left otherwise. The restriction $-k+2 \le \delta \le l-2$ means that the rows overlap at least in two cells.
Such graphical notation provides an easy way to describe the grading on the vector space $V$ and degree $-1$ operator $X\in {\mathfrak{gl}}(V)$ as defined in Section \[sec:sym\]. Namely, each box in the tableau corresponds to a basis element in $V$. The grading is defined in such way that it decreases by $1$ from left to right starting from $-1$ with basis elements in the same column having the same degree. And the operator $X$ maps each basis element to another basis element corresponding to the box on the right, or to $0$, if there is no right neighbor. The corresponding graded nilpotent Lie algebra ${\mathfrak{m}}$ is defined as ${\mathbb R}X\oplus V$. Below we list a number of examples of such tableaux.
Tableaux and preserved jet space projections for systems of mixed order $(2,3)$: $$\begin{aligned}
\delta = 1: &\quad\young(\ \ \ ,\ \ )\quad J^{2,3}({\mathbb R},{\mathbb R}^2)\to J^{1,2}({\mathbb R},{\mathbb R}^2) \to J^{0,1}({\mathbb R},{\mathbb R}^2);\\[2mm]
\delta = 0: &\quad\young(\ \ \ ,:\ \ )\quad J^{2,3}({\mathbb R},{\mathbb R}^2)\to J^{1,1}({\mathbb R},{\mathbb R}^2) \to J^{0,0}({\mathbb R},{\mathbb R}^2).\end{aligned}$$
Tableaux and preserved jet space projections for systems of mixed order $(2,4)$: $$\begin{aligned}
\delta = 2: &\quad \young(\ \ \ \ ,\ \ )\quad J^{2,4}({\mathbb R},{\mathbb R}^2)\to J^{1,3}({\mathbb R},{\mathbb R}^2) \to J^{0,2}({\mathbb R},{\mathbb R}^2);\\[2mm]
\delta = 1: &\quad \young(\ \ \ \ ,:\ \ )\quad J^{2,4}({\mathbb R},{\mathbb R}^2)\to J^{1,2}({\mathbb R},{\mathbb R}^2) \to J^{0,1}({\mathbb R},{\mathbb R}^2);\\[2mm]
\delta = 0: &\quad \young(\ \ \ \ ,::\ \ )\quad J^{2,4}({\mathbb R},{\mathbb R}^2)\to J^{1,1}({\mathbb R},{\mathbb R}^2) \to J^{0,0}({\mathbb R},{\mathbb R}^2).\end{aligned}$$
Tableaux and preserved jet space projections for systems of mixed order $(3,4)$: $$\begin{aligned}
\delta = 2: &\quad \young(:\ \ \ \ ,\ \ \ )\quad J^{3,4}({\mathbb R},{\mathbb R}^2)\to J^{1,3}({\mathbb R},{\mathbb R}^2) \to J^{0,2}({\mathbb R},{\mathbb R}^2);\\[2mm]
\delta = 1: &\quad \young(\ \ \ \ ,\ \ \ )\quad J^{3,4}({\mathbb R},{\mathbb R}^2)\to J^{2,3}({\mathbb R},{\mathbb R}^2) \to J^{1,2}({\mathbb R},{\mathbb R}^2)\to J^{0,1}({\mathbb R},{\mathbb R}^2);\\[2mm]
\delta = 0: &\quad \young(\ \ \ \ ,:\ \ \ )\quad J^{3,4}({\mathbb R},{\mathbb R}^2)\to J^{2,2}({\mathbb R},{\mathbb R}^2) \to J^{1,1}({\mathbb R},{\mathbb R}^2)\to J^{0,0}({\mathbb R},{\mathbb R}^2);\\[2mm]
\delta = -1: &\quad \young(\ \ \ \ ,::\ \ \ )\quad J^{3,4}({\mathbb R},{\mathbb R}^2)\to J^{2,1}({\mathbb R},{\mathbb R}^2) \to J^{1,0}({\mathbb R},{\mathbb R}^2).\end{aligned}$$
In fact, such kind tableaux can be used to encode different EDS associated with mixed order systems of ODEs with an arbitrary number of equations. As this topic lies outside of the scope of the current paper, we just show one particular example corresponding to an EDS associated with a system of order $(2,3,4)$: $$\young(\ \ \ \ ,::\ \ \ ,::\ \ )$$
Using the results from Section \[sec:sym\], we can easily compute the symmetry algebra of an EDS of shift $\delta$ associated with a system . All other results including the relationship between Spenser and Tanaka prolongations and the existence of natural frame bundles in case of non-linear systems are also easily generalized to the EDSs of arbitrary shift $\delta$.
The symmetry algebra of an EDS of shift $\delta$ associated with a system $y^{(k)}=z^{(l)}=0$ is described as follows.
1\. If $\delta$ satisfies $0<\delta <l-k$ (i.e., when one of the rows in the corresponding skew Young tableau lies strictly within the other row), then the symmetry algebra is generated by prolongations of vector fields: $$\begin{aligned}
&{\frac{{\partial}}{{\partial}x}}, x{\frac{{\partial}}{{\partial}x}}, x^2{\frac{{\partial}}{{\partial}x}} + (k-1)xy_0{\frac{{\partial}}{{\partial}y_0}} + (l-1)xz_0{\frac{{\partial}}{{\partial}z_0}},\\
& y_0{\frac{{\partial}}{{\partial}y_0}}, z_0{\frac{{\partial}}{{\partial}z_0}},\\
& x^i {\frac{{\partial}}{{\partial}y_0}},x^j{\frac{{\partial}}{{\partial}z_0}}\quad i=0,\dots,k-1;\ j=0\dots,l-1.\end{aligned}$$ The cases $\delta = 0$ and $\delta = l-k$ are covered by Propositions \[propI\] and \[propII\] respectively.
2\. If $3=k\le l$ and $\delta=-1$, the symmetry algebra is generated by prolongations of the following vector fields:
$$\begin{aligned}
& x(2y_0-xy_1){\frac{{\partial}}{{\partial}x}}+(2y_0^2-\frac{1}{2}x^2y_1^2){\frac{{\partial}}{{\partial}y_0}}+y_1(2y_0-xy_1){\frac{{\partial}}{{\partial}y_1}} + (l-1)z_0(2y_0-xy_1){\frac{{\partial}}{{\partial}z_0}},\\
& 2(y_0-xy_1){\frac{{\partial}}{{\partial}x}}-xy_1^2{\frac{{\partial}}{{\partial}y_0}}-y_1^2{\frac{{\partial}}{{\partial}y_1}}-(l-1)z_0y_1{\frac{{\partial}}{{\partial}z_0}},\\
& x^2{\frac{{\partial}}{{\partial}x}}+2xy_0{\frac{{\partial}}{{\partial}y_0}}+2y_0{\frac{{\partial}}{{\partial}y_1}}+(l-1)xz_0{\frac{{\partial}}{{\partial}z_0}},\\
&x{\frac{{\partial}}{{\partial}x}}-y_1{\frac{{\partial}}{{\partial}y_1}},\ y_0{\frac{{\partial}}{{\partial}y_0}} +y_1{\frac{{\partial}}{{\partial}y_1}},\ z_0{\frac{{\partial}}{{\partial}z_0}},\\
& y_1{\frac{{\partial}}{{\partial}x}} + \frac{1}{2}y_1^2{\frac{{\partial}}{{\partial}y_0}},\ {\frac{{\partial}}{{\partial}x}},\ {\frac{{\partial}}{{\partial}y_0}},x{\frac{{\partial}}{{\partial}y_0}}+{\frac{{\partial}}{{\partial}y_1}},x^2{\frac{{\partial}}{{\partial}y_0}}+2x{\frac{{\partial}}{{\partial}y_1}},\\
&x^i(xy_1-2y_0)^{j_0} y_1^{j_1}{\frac{{\partial}}{{\partial}z_0}},\quad i+j_0+j_1\le l-1.\end{aligned}$$
This symmetry algebra is isomorphic to $\mathfrak{csp}(4,{\mathbb R}){\rightthreetimes}S^{l-1}({\mathbb R}^4)$.
4\. If $3\le k\le l$ and $\delta > l-k$, then the symmetry algebra is generated by the prolongations of the following vector fields: $$\begin{aligned}
&{\frac{{\partial}}{{\partial}x}}, x{\frac{{\partial}}{{\partial}x}}, x^2{\frac{{\partial}}{{\partial}x}} + (k-1)xy_0{\frac{{\partial}}{{\partial}y_0}} + (l-1)xz_0{\frac{{\partial}}{{\partial}z_0}},\\
& y_0{\frac{{\partial}}{{\partial}y_0}}, z_0{\frac{{\partial}}{{\partial}z_0}}, x^i {\frac{{\partial}}{{\partial}z_0}},\quad i=0,\dots,l-1;\\
& x^i g_{\delta,l-\delta}^{(s_1)} \dots g_{\delta,l-\delta}^{(s_j)}{\frac{{\partial}}{{\partial}y_0}}, \quad j\ge 0, i+ (l-\delta-1) j\le k-1,\end{aligned}$$ where the functions $g_{\delta,l-\delta}^{(s)}$ are defined in Section \[sec:sym\].
5\. The symmetry algebra in the case $4 \le k\le l$ and $\delta<0$ is obtained from the previous item by exchanging $y$ and $z$, $k$ and $l$ and replacing $\delta$ by $-\delta$.
The proof is similar to the proof of Proposition \[propII\] and is reduced to solving equations for $p=k-1$ and $q=l-\delta-2$.
This result can be considered as a direct generalization of Propositions \[propI\] and \[propII\]. The results of Sections \[sec:tan\] andref[sec:g]{} also generalize to the EDS systems of arbitrary shift. In particular, all symmetry algebras from the above proposition can be obtained as Tanaka prolongations of the corresponding (possibly, non-fundamental) graded nilpotent Lie algebras as well as Sternberg prolongations of certain subalgebras in ${\mathfrak{gl}}(k+l,{\mathbb R})$.
The existence of natural frame bundles in case of non-linear systems and the relation to the geometry flag structures are also easily generalized. See [@quasi] for more detail. In particular, the equivalence problem for EDS of arbitrary shift $\delta$ coincides with $(k-l-\delta)$-equivalence of non-linear systems of ODEs of mixed order $(k,l)$ introduced in [@quasi Section 3].
[99]{} Boris Doubrov, Boris Komrakov, *Classification of homogeneous submanifolds in homogeneous spaces*, Lobachevskij Journal of Mathematics, **3** (1999), 19–38.
Boris Doubrov, *Projective reparametrization of homogeneous curves*, Archivum Mathematicum, **41** (2005), 129–133.
Boris Doubrov, Igor Zelenko, *On geometry of curves of flags of constant type*, Central European Journal of Mathematics, **10** (2012), 1836–1871.
Boris Doubrov, Igor Zelenko, *Prolongation of quasi-principal frame bundles and geometry of flag structures on manifolds*, preprint, arXiv:1210.7334 \[math.DG\], 47 pages.
Joe Harris, *Algebraic geometry: a first course*, Springer–Verlag, NY, 1997.
S. Sternberg, *Lectures on differential geometry*, Englewood Cliffs, Printice–Hall, 1964.
V.W. Guillemin, S. Sternberg, *An algebraic model of transitive differential geometry* Bull. Amer. Math. Soc., **70** (1964), 16–47.
Noboru Tanaka, *On differential systems, graded Lie algebras and pseudo-groups*, J. Math. Kyoto. Univ., **10** (1970), 1–82.
Noboru Tanaka, *On the equivalence problems associated with simple graded Lie algebras*, Hokkaido Math. J., **6** (1979), 23–84.
Noboru Tanaka, *On affine symmetric spaces and the automorphism groups of product manifolds*, Hokkaido Math. J., **14** (1985), 277–351.
Keizo Yamaguchi, *Differential systems associated with simple graded Lie algebras*, Adv. Studies in Pure Math., **22** (1993), 413–494.
Igor Zelenko, *On Tanaka’s prolongation procedure for filtered structures of constant type*, SIGMA, **5** (2009), 094, 21p.
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****
**Djurdje Cvijović**
\
**E-Mail: djurdje@vinca.rs**\
> **Abstract.** A new explicit closed-form formula for the multivariate $(n, k)$th partial Bell polynomial $B_{n,k} (x_1, x_2, \ldots, x_{n - k + 1})$ is deduced. The formula involves multiple summations and makes it possible, for the first time, to easily evaluate $B_{n,k}$ directly for given values of $n$ and $k$ ($n\geq k, k =2, 3,\ldots$). Also, a new addition formula (with respect to $k$) is found for the polynomials $B_{n,k}$ and it is shown that they admit a new recurrence relation. Several special cases and consequences are pointed out, and some examples are also given.
**2010 *Mathematics Subject Classification.*** $\;$ Primary 11B83, 11B75, 11B99; Secondary 11B73, 11B37. ***Key Words and Phrases.***
Partial Bell polynomial; Recurrence relation; Stirling number of the second kind.
Introduction
============
For $n$ and $k$ non-negative integers, the (exponential) $(n, k)$th partial Bell polynomial in the variables $x_1, x_2,\ldots,x_{n - k + 1}$ denoted by $B_{n,k} \equiv B_{n,k}(x_1, x_2, \ldots, x_{n - k + 1})$ may be defined by the formal power series expansion see, for instance, [@Comtet pp. 133, Eq. (3a’)] $$\frac{1}{k!} \left(\sum_{m \,= 1}^{\infty} x_m \,\frac{t^m}{m!}\right)^k = \sum_{n\,= k}^{\infty} B_{n,k}(x_1, x_2, \ldots, x_{n - k + 1})\,\frac{t^n}{n!}\qquad(k\geq0),$$ or, what amounts to the same, by the explicit formula [@Hazewinkel p. 96] $$B_{n,k} = \sum \frac{n!}{\ell_1!\, \ell_2! \ldots \ell_{n-k+1}!} \left(\frac{x_1}{1!}\right)^{\ell_1} \left(\frac{x_2}{2!}\right)^{\ell_2}\ldots \left(\frac{x_{n-k+1}}{(n-k+1)!}\right)^{\ell_{n-k+1}},$$ where (multiple) summation is extended over all partitions of a positive integer number $n$ into exactly $k$ parts (summands), [*i.e.*]{}, over all solutions in non-negative integers $\ell_{\alpha},$ $1\leq\alpha\leq n-k+1,$ of a system of the two simultaneous equations $$\ell_1 + 2 \,\ell_2 + \cdots + (n-k+1) \,\ell_{n-k+1} = n$$ and $$\ell_1 + \ell_2 + \cdots + \ell_{n-k+1} = k.$$
For fixed $n$ and $k$, $B_{n,k}$ has positive integral coefficients and is a homogenous and isobaric polynomial in its $(n-k+1)$ variables $x_1, x_2, \ldots, x_{n-k+1}$ of total degree $k$ and total weight $n$, [*i.e.*]{}, it is a linear combination of monomials $x_1^{\ell_1} x_2^{\ell_2}\ldots x_{n-k+1}^{\ell_{n-k+1}}$ whose partial degrees and weights are constantly given by $\ell_1 + \ell_2 + \ldots + \ell_{n-k+1} = k$ and $\ell_1 + 2 \ell_2 + \ldots + ({n-k+1})\ell_{n-k+1} = n$. For some examples of these polynomials see Section 3.
The partial Bell polynomials are quite general polynomials, they have a number of applications and more details about them can be found in Bell [@Bell], Comtet [@Comtet pp. 133–137], Hazewinkel [@Hazewinkel pp. 95–98], Charalambides [@Charalambides pp. 412–417] and Aldrovandi [@Aldrovandi pp. 151–182]. However, the following formulae for $B_{n,k}$ $$B_{n,k}= \frac{1}{x_1}\cdot\frac{1}{n-k} \sum_{\alpha\,=1}^{n-k} \binom{n}{\alpha}\left[(k+1)-\frac{n+1}{\alpha+1}\right] x_{\alpha+1} B_{n-\alpha,k},$$ $$B_{n, k_1 + k_2} = \frac{k_1!\, k_2!}{(k_1 + k_2)!} \sum_{\alpha\,=0}^n \binom{n}{\alpha} B_{\alpha, k_1} B_{n - \alpha, k_2}$$ and $$\begin{aligned}
B_{n, k + 1} = & \frac{1}{(k+1)!} \underbrace{\sum_{\alpha_1\,= k}^{n-1} \, \sum_{\alpha_2\,= k-1}^{\alpha_1-1} \cdots \sum_{\alpha_k\, = 1}^{\alpha_{k-1}-1} }_{k }
\overbrace{\binom{n}{\alpha_1} \binom{\alpha_1}{\alpha_2} \cdots \binom{\alpha_{k-1}}{\alpha_k}}^{k}\nonumber
\\
&\cdot x_{n-\alpha_1} x_{\alpha_1 -\alpha_2} \cdots x_{\alpha_{k-1}-\alpha_k} x_{\alpha_k} \qquad(n\geq k+1, k\,=1, 2, \ldots)\end{aligned}$$ appear not to have been noticed in any work on the subject which we have seen. In this note it is aimed to provide short proofs of these results, show some immediate consequences of them and provide some application examples (see also Section 3).
Proof of the main results
=========================
We begin by showing that the identity (1.3) follows without difficulty from the definition of partial Bell polynomials $B_{n,k}$ by means of the generating relation (1.1), given that the next auxiliary result for powers of series is used. Consider $$\left(\sum_{n\,= 1}^{\infty} f_{n} x^n\right)^k = \sum_{n\,= k}^{\infty} g_{n} (k)\, x^n.$$ For a fixed positive integer $k,$ we have that: $$g_{k}(k) = f_{1}^k, \tag{2.2a}$$ $$g_n (k) = \frac{1}{(n-k) f_1 } \sum_{\alpha\,=1}^{n-k} \Big[(\alpha+1) (k+1)-(n+1)\Big] f_{\alpha+1} \,g_{n-\alpha}(k)\tag{2.2b}$$ $$(n\geq k+1).$$
Indeed, by comparing (2.1) with the definition of $B_{n,k}$ in (1.1) and upon setting $g_n(k)= k! B_{n,k}/n!$ and $f_n= x_{n}/n!,$ we arrive at the proposed formula (1.3) by utilizing (2.2b).
Note that (2.2b) may be found in the literature (see [@Gould]) but it is not as widely known (and even less used) as it should be. It is exactly for this reason that we derive it starting from the following more general (and equally little known) recurrence relation involving the series coefficients $f_n$ and $g_n (k)$ in $\left(\sum_{n\,= 0}^{\infty} f_{n} x^n\right)^k = \sum_{n\,= 0}^{\infty} g_{n} (k)\, x^n.$ $$\sum_{\alpha\,= 0}^n \big[\alpha (k + 1) - n\big] f_{\alpha} \, g_{n-\alpha}(k) = 0 \qquad(n\geq0).\tag{2.3}$$
First, upon taking logarithms of each side of the equation $g(x) = \big[f(x)\big]^k$ and then differentiating both sides of the result with respect to $x$, we obtain $f (x) g'(x) = k \,f'(x) g (x).$ Next, insert the power series expansions of the various functions in this equation and multiply both sides by $x$, to get $$\sum_{m\,= 0}^{\infty} f_{m} x^m \cdot \sum_{m\,= 0}^{\infty} m \,g_{m}(k) \,x^m = k \sum_{m\,= 0}^{\infty} m\,f_{m} x^m\cdot \sum_{m\,= 0}^{\infty} g_{m}(k)\, x^m. \tag{2.4}$$
Now, recall that if $\sum\nolimits_{m\,=0}^{\infty} a_m$ and $\sum\nolimits_{m\,= 0}^{\infty} b_m$ are two series, then their Cauchy product is the series $\sum\nolimits_{n\,= 0}^{\infty} c_n$ where $c_n = \sum\nolimits_{k\,= 0}^{n} a_k b_{n-k}$. This is to say that in the particular case at hand, by equating the coefficients of a given power of $x$, say $x^n,$ on both sides of (2.4), we have $\sum\nolimits_{\alpha\,= 0}^{n} (n -\alpha) f_{\alpha}g_{n-\alpha}(k) = k\, \sum\nolimits_{\alpha\,= 0}^{n} \alpha f_{\alpha} g_{n-\alpha}(k),$ which eventually gives (2.3). The recurrence relation (2.3) is clearly valid for an arbitrary real or complex number $k$ and it can be used to compute successively as many of the unknown $g_{m}(k)$ values as desired, in order $g_{0}(k), g_{1}(k),g_{2}(k),\ldots$, if $g_{0}(k)$ is known. The special case of (2.3) solved for $g_n(k),$ for $k$ a positive integer and $f_0 \neq 0,$ appears in various editions of the standard reference book by Gradshteyn and Ryzhik (see, for instance, [@Gradshteyn p. 17, Entry 0.314])
Finally, if we suppose $f_0 = 0$ and $f_1\neq0$ then, from $\left(\sum_{n\,= 0}^{\infty} f_{n} x^n\right)^k = \sum_{n\,= 0}^{\infty} g_{n}(k)\, x^n,$ where $k$ is a positive integer, it is obvious that the coefficient $g_{n}(k),$ $n = 0, 1, \ldots, k,$ is only nonzero when $n = k,$ $g_{k}(k)$ then equals $f_{1}^k$ (see (2.2a)), while $\left(\sum_{n\,= 0}^{\infty} f_{n} x^n\right)^k = \sum_{n\,= 0}^{\infty} g_{n}(k)\, x^n$ reduces to (2.1). Therefore, since $g_{0}(k)= g_{1}(k)= \ldots = g_{k-1}(k)=0$ and $f_0 = 0,$ the recurrence relation (2.3) becomes $$\sum_{\alpha\,= 1}^{n - k} \big[\alpha (k + 1) - n\big] f_{\alpha} \, g_{n-\alpha}(k) = 0 \qquad(n\geq k),$$ so that, upon replacing $n$ by $n+1$, putting $\alpha + 1$ for $\alpha$ and solving for $g_n(k)$, we have that the coefficients $g_n(k)$, $n\geq k +1,$ are given by (2.2b) above.
In order to prove (1.4) we shall again resort to the generating relation for $B_{n,k}$ (1.1). Let us by $[t^n] \phi(t)$ denote the coefficient of $t^n$ in the power series of an arbitrary $\phi(t)$. Put $f(t)= \sum_{m \,= 1}^{\infty} x_m \,\frac{t^m}{m!}$, then by (1.1), we have $$k_1! B_{n,k_1} = n! \,[t^n] f(t)^{k_1} \qquad(n\geq k_1)$$ and $$k_2! B_{n,k_2} = n! \,[t^n] f(t)^{k_2}\qquad(n\geq k_2),$$ thus $$\begin{aligned}
(k_1 + k_2)! B_{n,k_1 + k_2} & = n! \,[t^n] f(t)^{k_1 + k_2} = n! \,[t^n] \Big( f(t)^{k_1} \cdot f(t)^{k_2}\Big)\nonumber
\\
& \hskip-20mm = n! \,\sum_{\alpha\,= 0}^n [t^{\alpha}] f(t)^{k_1}\cdot [t^{n- \alpha}] f(t)^{k_2} = n!\,\sum_{\alpha\,=0}^n \frac{k_1! B_{\alpha,k_1}}{\alpha!}\frac{k_2! B_{n-\alpha,k_2}}{(n-\alpha)!},\tag{2.5}\end{aligned}$$ $$(n\geq k_1 + k_2)$$ since $[t^n] \Big(\phi(t) \psi(t)\Big) = \sum_{\alpha\,= 0}^n [t^{\alpha}] \phi(t)\cdot [t^{n-\alpha}] \psi(t)$ (the Cauchy product of two series). We conclude the proof by noting that the required expression (1.4) follows by rewriting (2.5).
Lastly, we shall prove the closed-form formula (1.5) by making use of (1.4). It suffices to show that the addition formula for $B_{n,k}$ (1.4) may be used to deduce the following: $$B_{n,2} = \frac{1}{2!} \sum_{\alpha\,= 1}^{n-1} \binom{n}{\alpha} x_{n - \alpha}\, x_{\alpha} \qquad(n\geq 2),\tag{2.6}$$ $$B_{n,3} = \frac{1}{3!} \sum_{\alpha\,= 2}^{n-1} \sum_{\beta\,= 1}^{\alpha -1} \binom{n}{\alpha} \binom{\alpha}{\beta} x_{n - \alpha} \, x_{\alpha - \beta} \, x_{\beta}\qquad(n\geq 3)\tag{2.7}$$ and $$B_{n,4} = \frac{1}{4!} \sum_{\alpha\,= 3}^{n-1} \sum_{\beta\,= 2}^{\alpha -1} \sum_{\gamma\,= 1}^{\beta-1}\binom{n}{\alpha} \binom{\alpha}{\beta} \binom{\beta}{\gamma} x_{n - \alpha} \, x_{\alpha - \beta} \, x_{\beta - \gamma}\, x_{\gamma}\qquad(n\geq 4).\tag{2.8}$$
By bearing in mind that $B_{n,1} = x_{n}$ (this is a simple consequence of the definition $B_{n,k}$ in (1.1)) and upon noticing that $x_{0}= 0$ (again, see (1.1)), the expression for $B_{n,2}$ given in (2.6) follows by (1.4) with $k_1 = 1$ and $k_2 =1$. Further, this result for $B_{n,2}$ together with (1.4), where $k_1 = 2$ and $k_2 =1,$ leads to (2.7). It is clear that by repeating this procedure recursively we may obtain $B_{n,4},$ and so on.
Further results and concluding remarks
======================================
We remark that the explicit closed-form formula for $B_{n,k}(x_1, x_2, \ldots, x_{n - k + 1})$ given by (1.5) is particularly useful. Namely, it is hard to work with the formula (1.2) which explicitly defines $B_{n,k}$ due to complicated multiple summations, and, for instance, it is virtually impossible by its use to write down a polynomial for given values of $n$ and $k$. However, formula (1.5), although also involves multiple summations, makes this possible. In other words, it is now possible to directly evaluate $B_{n,k}$ for given $n$ and $k$ ($n\geq k, k =2, 3,\ldots$) by utilizing (1.5) instead of computing it recursively by making use of some recurrence relations (see, for instance, (1.3)). It is noteworthy to mention that the practical evaluation is greatly facilitated by wide availability of various symbolic algebra programs. In order to demonstrate an application of this result, we list several of the polynomials $B_{n,k}$ determined by the formula (1.5), where all the computations were carried out by using Mathematica 6.0 (Wolfram Research)
$$\begin{aligned}
&B_{8,7} = 28 x_1^6 x_2,\qquad B_{9,7} = 378 x_1^5 x_2^2 + 84 x_1^6 x_3,
\\
& B_{10,7} = 3150 x_1^4 x_2^3 + 2520 x_1^5 x_2 x_3 + 210 x_1^6 x_4,
\\
& B_{11,7} = 17325 x_1^3 x_2^4 + 34650 x_1^4 x_2^2 x_3 + 4620 x_1^5 x_3^2 +
6930 x_1^5 x_2 x_4 + 462 x_1^6 x_5,
\\
& B_{12,7} = 62370 x_1^2 x_2^5 + 277200 x_1^3 x_2^3 x_3 +
138600 x_1^4 x_2 x_3^2 + 103950 x_1^4 x_2^2 x_4
\\
& \quad \quad \,\,+
27720 x_1^5 x_3 x_4 + 16632 x_1^5 x_2 x_5 + 924 x_1^6 x_6,
\\
& B_{13,7} = 135135 x_1 x_2^6 + 1351350 x_1^2 x_2^4 x_3 +
1801800 x_1^3 x_2^2 x_3^2 + 200200 x_1^4 x_3^3
\\
& \quad \quad \,\,+ 900900 x_1^3 x_2^3 x_4 + 900900 x_1^4 x_2 x_3 x_4 +
45045 x_1^5 x_4^2 + 270270 x_1^4 x_2^2 x_5
\\
& \quad \quad \,\, +72072 x_1^5 x_3 x_5 + 36036 x_1^5 x_2 x_6 + 1716 x_1^6 x_7.\end{aligned}$$
It should be noted that our results for $B_{8,7}$ $B_{9,7}$ and $B_{10,7}$ are in full agreement with those recorded in the work (for instance) of Charalambides [@Charalambides p. 417].
One further illustration of an application of (1.5) is the following (presumably) new explicit formula $$\begin{aligned}
S(n,k+1) = & \frac{1}{(k+1)!} \underbrace{\sum_{\alpha_1\,= k}^{n-1} \, \sum_{\alpha_2\,= k-1}^{\alpha_1-1} \cdots \sum_{\alpha_k\, = 1}^{\alpha_{k-1}-1} }_{k }
\overbrace{\binom{n}{\alpha_1} \binom{\alpha_1}{\alpha_2} \cdots \binom{\alpha_{k-1}}{\alpha_k}}^{k}\end{aligned}$$ $$(n\geq k+1, k\,=1, 2, \ldots)$$ for the Stirling numbers of the second kind $S(n,k)$ defined by means of (see [@Comtet Chapter 5]) $$S(n,k)=\frac{1}{k!} \sum_{\alpha\,=0}^k (-1)^{k-\alpha} \binom{k}{\alpha}\alpha^n,$$ which is an immediate consequence of the relationship $S(n,k)= B_{n,k}(1,\ldots,1)$ [@Comtet p. 135, Eq. (3g)]. Moreover, for given $k$, it is easy to sum the multiple sum (3.1) by repeated use of the familiar result $(1+x)^n = \sum_{k\,=0}^n \binom{n}{k} x^k$, so that we have: $$\begin{aligned}
&S(n,2)= \frac{1}{2} \Big(2^n -2\Big) = 2^{n-1}-1,
\\
&S(n,3)= \frac{1}{6} \Big(3^n - 3\cdot 2^n+3\Big),
\\
&S(n,4)= \frac{1}{24} \Big(4^n - 4\cdot 3^n +3\cdot 2^{n+1}-4\Big),
\\
&S(n,5)= \frac{1}{120} \Big(5^n - 5 \cdot4^n + 10\cdot 3^n - 10\cdot 2^{n } +5 \Big),
\\
&S(n,6)= \frac{1}{720} \Big(6^n -6\cdot 5^n + 15\cdot 4^n -20\cdot 3^n + 15\cdot 2^n -6 \Big),\end{aligned}$$ and these expressions agree fully with those which are obtained by using the defining relation (3.2).
Acknowledgements {#acknowledgements .unnumbered}
================
[The author acknowledges financial support from Ministry of Science of the Republic of Serbia under Research Projects 144004 and 142025.]{}
[20]{}
L. Comtet, [*Advanced Combinatorics: The Art of Finite and Infinite Expansions*]{}, D. Reidel Publishing Co., Dordrecht, 1974.
M. Hazewinkel (Ed.), [*Encyclopedia of Mathematics, Supplement I*]{}, Kluwer Academic Publishers, Dordrehct, 1997.
E.T. Bell, Exponential polynomials, [*Ann. Math.*]{} [**35**]{} (1934), 258–277.
C.A. Charalambides, [*Enumerative Combinatorics*]{}, Chapman and Hall/CRC, Boca Raton, 2002.
R. Aldrovandi, , World Scientific, Singapore, 2001.
H.W. Gould, Coefficient identities for powers of Taylor and Dirichlet series, [*Amer. Math. Monthly*]{} [**81**]{} (1974), 3–14.
I.S. Gradshteyn and I.M. Ryzhik, [*Table of Integrals, Series, and Products*]{}, Seventh Edition, Academic Press, 2007.
|
---
abstract: 'We present 17 transit light curves of the ultra-short period planetary system WASP-103, a strong candidate for the detection of tidally-induced orbital decay. We use these to establish a high-precision reference epoch for transit timing studies. The time of the reference transit midpoint is now measured to an accuracy of 4.8s, versus 67.4s in the discovery paper, aiding future searches for orbital decay. With the help of published spectroscopic measurements and theoretical stellar models, we determine the physical properties of the system to high precision and present a detailed error budget for these calculations. The planet [[has a Roche lobe filling factor of 0.58]{}]{}, leading to a significant asphericity; we correct its measured mass and mean density for this phenomenon. A high-resolution [*Lucky Imaging*]{} observation shows no evidence for faint stars close enough to contaminate the point spread function of WASP-103. Our data were obtained in the Bessell $RI$ and the SDSS $griz$ passbands and yield a larger planet radius at bluer optical wavelengths, to a confidence level of $7.3\sigma$. Interpreting this as an effect of Rayleigh scattering in the planetary atmosphere leads to a measurement of the planetary mass which is too small by a factor of five, implying that Rayleigh scattering is not the main cause of the variation of radius with wavelength.'
title: 'High-precision photometry by telescope defocussing. VII. The ultra-short period planet WASP-103 [^1]'
---
stars: planetary systems — stars: fundamental parameters — stars: individual: WASP-103
Introduction {#sec:intro}
============
An important factor governing the tidal evolution of planetary systems is the stellar tidal quality factor $Q_\star$ [e.g. @GoldreichSoter66icar], which represents the efficiency of tidal dissipation in the star. Its value is necessary for predicting the timescales of orbital circularisation, axial alignment and rotational synchronisation of binary star and planet systems. Short-period giant planets suffer orbital decay due to tidal effects, and most will ultimately be devoured by their host star rather than reach an equilibrium state [@Levrard++09apj; @Jackson++09apj]. The magnitude of $Q_\star$ therefore influences the orbital period distribution of populations of extrasolar planets.
Unfortunately, $Q_\star$ is not well constrained by current observations. Its value is often taken to be $10^6$ [@OgilvieLin07apj] but there exist divergent results in the literature. A value of $10^{5.5}$ was found to be a good match to a sample of known extrasolar planets by @Jackson++08apj2, but theoretical work by @PenevSasselov11apj constrained $Q_\star$ to lie between $10^8$ and $10^{9.5}$ and an observational study by @Penev+12apj found $Q_\star > 10^7$ to 99% confidence. Inferences from the properties of binary star systems are often used but are not relevant to this issue: $Q_\star$ is not a fundamental property of a star but depends on the nature of the tidal perturbation [@Goldreich63mn; @Ogilvie14xxx]. $Q_\star$ should, however, be observationally accessible through the study of transiting extrasolar planets (TEPs).
@Birkby+14mn assessed the known population of TEPs for their potential for the direct determination of the strength of tidal interations. The mechanism considered was the detection of tidally-induced orbital decay, which manifests itself as a decreasing orbital period. These authors found that WASP-18 [@Hellier+09nat; @Me+09apj] is the most promising system, due to its short orbital period (0.94d) and large planet mass (10.4[$\,{\rm M}_{\rm Jup}$]{}), followed by WASP-103 [@Gillon+14aa hereafter G14], then WASP-19 [@Hebb+10apj; @Mancini+13mn].
Adopting the canonical value of $Q_\star = 10^6$, @Birkby+14mn calculated that orbital decay would cause a shift in transit times – over a time interval of 10yr – of 350s for WASP-18, 100s for WASP-103 and 60s for WASP-19. Detection of this effect clearly requires observations over many years coupled with a precise ephemeris against which to measure deviations from strict periodicity. High-quality transit timing data are already available for WASP-18 [@Maxted+13mn] and WASP-19 [@Tregloan++13mn; @Abe+13aa; @Lendl+13aa; @Mancini+13mn], but not for WASP-103.
WASP-103 was discovered by G14 and comprises a TEP of mass 1.5[$\,{\rm M}_{\rm Jup}$]{} and radius 1.6[$\,{\rm R}_{\rm Jup}$]{} in a very short-period orbit (0.92d) around an F8V star of mass 1.2[$\,{\rm M}_\odot$]{} and radius 1.4[$\,{\rm R}_\odot$]{}. G14 obtained observations of five transits, two with the Swiss Euler telescope and three with TRAPPIST, both at ESO La Silla. The Euler data each cover only half a transit, whereas the TRAPPIST data have a lower photometric precision and suffer from 180 field rotations during the transits due to the nature of the telescope mount. The properties of the system could therefore be measured to only modest precision; in particular the ephemeris zeropoint is known to a precision of only 64s. In this work we present 17 high-quality transit light curves which we use to determine a precise orbital ephemeris for WASP-103, as well as to improve measurements of its physical properties.
Observations and data reduction {#sec:obs}
===============================
---------------- ------------ ------------ ---------- --------------- --------------- ---------------- -------- ---------------------------- -------- ------------ ---------------- ---------
Instrument Date of Start time End time $N_{\rm obs}$ $T_{\rm exp}$ $T_{\rm dead}$ Filter Airmass Moon Aperture $N_{\rm poly}$ Scatter
first obs (UT) (UT) (s) (s) illum. radii (px) (mmag)
DFOSC 2014 04 20 05:08 09:45 134 100–105 18 $R$ 1.54 $\to$ 1.24 $\to$ 1.54 0.725 14 25 45 1 0.675
DFOSC 2014 05 02 05:45 10:13 113 110–130 19 $I$ 1.28 $\to$ 1.24 $\to$ 2.23 0.100 14 22 50 1 0.815
DFOSC 2014 06 09 04:08 08:22 130 100 16 $R$ 1.24 $\to$ 3.08 0.888 16 27 50 1 1.031
DFOSC 2014 06 23 01:44 06:19 195 50–120 16 $R$ 1.35 $\to$ 1.24 $\to$ 1.86 0.168 14 22 40 1 1.329
DFOSC 2014 06 24 00:50 04:25 112 100 16 $R$ 1.54 $\to$ 1.24 $\to$ 1.31 0.103 19 25 40 1 0.647
DFOSC 2014 07 06 01:28 05:20 118 100 18 $R$ 1.28 $\to$ 1.24 $\to$ 1.80 0.564 16 26 50 1 0.653
DFOSC 2014 07 18 01:19 05:45 139 90–110 16 $R$ 1.25 $\to$ 3.00 0.603 17 25 60 1 0.716
DFOSC 2014 07 18 23:04 04:20 181 60–110 16 $R$ 1.52 $\to$ 1.24 $\to$ 1.73 0.502 16 24 55 2 0.585
\[3pt\] GROND 2014 07 06 00:23 05:27 122 100–120 40 $g$ 1.45 $\to$ 1.24 $\to$ 1.87 0.564 24 65 85 2 1.251
GROND 2014 07 06 00:23 05:27 119 100–120 40 $r$ 1.45 $\to$ 1.24 $\to$ 1.87 0.564 24 65 85 2 0.707
GROND 2014 07 06 00:23 05:27 125 100–120 40 $i$ 1.45 $\to$ 1.24 $\to$ 1.87 0.564 24 65 85 2 0.843
GROND 2014 07 06 00:23 05:27 121 100–120 40 $z$ 1.45 $\to$ 1.24 $\to$ 1.87 0.564 24 65 85 2 1.106
GROND 2014 07 18 22:55 03:59 125 98–108 41 $g$ 1.64 $\to$ 1.24 $\to$ 1.93 0.502 30 50 85 2 0.882
GROND 2014 07 18 22:55 04:43 143 98–108 41 $r$ 1.64 $\to$ 1.24 $\to$ 1.93 0.502 25 45 70 2 0.915
GROND 2014 07 18 22:55 04:39 142 98–108 41 $i$ 1.64 $\to$ 1.24 $\to$ 1.88 0.502 28 56 83 2 0.656
GROND 2014 07 18 22:55 04:43 144 98–108 41 $z$ 1.64 $\to$ 1.24 $\to$ 1.93 0.502 30 50 80 2 0.948
\[3pt\] CASLEO 2014 08 12 23:22 03:10 129 90–120 4 $R$ 1.29 $\to$ 2.12 0.920 20 30 60 4 1.552
---------------- ------------ ------------ ---------- --------------- --------------- ---------------- -------- ---------------------------- -------- ------------ ---------------- ---------
![\[fig:lc:dk\] DFOSC light curves presented in this work, in the order they are given in Table\[tab:obslog\]. Times are given relative to the midpoint of each transit, and the filter used is indicated. Dark blue and dark red filled circles represent observations through the Bessell $R$ and $I$ filters, respectively.](plotLCall-dk.eps){width="\columnwidth"}
![\[fig:lc:grond\] GROND light curves presented in this work, in the order they are given in Table\[tab:obslog\]. Times are given relative to the midpoint of each transit, and the filter used is indicated. $g$-band data are shown in light blue, $r$-band in green, $i$-band in orange and $z$-band in light red.](plotLCall-grond.eps){width="\columnwidth"}
![\[fig:lc:casleo\] The CASLEO light curve of WASP-103. Times are given relative to the midpoint of the transit.](plotLCall-casleo.eps){width="\columnwidth"}
DFOSC observations {#sec:obs:dfosc}
------------------
Eight transits were obtained using the DFOSC (Danish Faint Object Spectrograph and Camera) instrument on the 1.54m Danish Telescope at ESO La Silla, Chile, in the context of the MiNDSTEp microlensing program [@Dominik+10an]. DFOSC has a field of view of 13.7[$^\prime$]{}$\times$13.7[$^\prime$]{} at a plate scale of 0.39[$^{\prime\prime}$]{}pixel$^{-1}$. We windowed down the CCD to cover WASP-103 itself and seven good comparison stars, in order to shorten the dead time between exposures.
The instrument was defocussed to lower the noise level of the observations, in line with our usual strategy [see @Me+09mn; @Me+14mn]. The telescope was autoguided to limit pointing drifts to less than five pixels over each observing sequence. Seven of the transits were obtained through a Bessell $R$ filter, but one was taken through a Bessell $I$ filter by accident. An observing log is given in Table\[tab:obslog\] and the light curves are plotted individually in Fig.\[fig:lc:dk\].
GROND observations {#sec:obs:grond}
------------------
We observed two transits of WASP-103 using the GROND instrument [@Greiner+08pasp] mounted on the MPG 2.2m telescope at La Silla, Chile. Both transits were also observed with DFOSC. GROND was used to obtain light curves simultaneously in passbands which approximate SDSS $g$, $r$, $i$ and $z$. The small field of view of this instrument (5.4$^{\prime}$$\times$5.4$^{\prime}$ at a plate scale of 0.158$^{\prime\prime}$pixel$^{-1}$) meant that few comparison stars were available and the best of these was several times fainter than WASP-103 itself. The scatter in the GROND light curves is therefore worse than generally achieved [e.g. @Nikolov+13aa; @Mancini+14aa; @Mancini+14aa2], but the data are certainly still useful. The telescope was defocussed and autoguided for both sets of observations. Further details are given in the observing log (Table\[tab:obslog\]) and the light curves are plotted individually in Fig.\[fig:lc:grond\].
CASLEO observations
-------------------
We observed one transit of WASP-103 (Fig.\[fig:lc:casleo\]) using the 2.15m Jorge Sahade telescope located at the Complejo Astronómico El Leoncito in San Juan, Argentina[^2]. We used the focal reducer and Roper Scientific CCD, yielding an unvignetted field of view of 9$^{\prime}$ radius at a plate scale of 0.45$^{\prime\prime}$pixel$^{-1}$. The CCD was operated without binning or windowing due to its short readout time. The observing conditions were excellent. The images were slightly defocussed to a FWHM of 3$^{\prime\prime}$, and were obtained through a Johnson-Cousins Schuler $R$ filter.
Data reduction {#sec:obs:defot}
--------------
instrument Filter
------------ -------- --------- -------- ---- --------- --- ---------
DFOSC $R$ 2456767 719670 0 0008211 0 0006953
DFOSC $I$ 2456779 746022 0 0004141 0 0008471
DFOSC $R$ 2456817 679055 0 0002629 0 0009941
DFOSC $R$ 2456831 578063 -0 0040473 0 0035872
DFOSC $R$ 2456832 540736 0 0018135 0 0006501
DFOSC $R$ 2456844 566748 0 0000224 0 0006405
DFOSC $R$ 2456856 559865 -0 0004751 0 0006952
DFOSC $R$ 2456857 466519 -0 0000459 0 0005428
GROND $g$ 2456844 521804 0 0008698 0 0013610
GROND $r$ 2456844 521804 0 0003845 0 0007863
GROND $i$ 2456844 521804 0 0005408 0 0009272
GROND $z$ 2456844 521804 -0 0013755 0 0012409
GROND $g$ 2456857 459724 -0 0005583 0 0024887
GROND $r$ 2456857 461385 0 0000852 0 0009575
GROND $i$ 2456857 461385 -0 0016371 0 0015169
GROND $z$ 2456857 459724 0 0012460 0 0015680
CASLEO $R$ 2456882 47535 0 00114 0 00121
: \[tab:lcdata\] Sample of the data presented in this work (the first datapoint of each light curve). The full dataset will be made available at the CDS.
The DFOSC and GROND data were reduced using the [defot]{} code [@Me+09mn] with the improvements discussed by @Me+14mn. Master bias, dome flat fields and sky flat fields were constructed but not applied, as they were found not to improve the quality of the resulting light curves [see @Me+14mn]. Aperture photometry was performed using the [idl[^3]/astrolib[^4]]{} implementation of [daophot]{} [@Stetson87pasp]. Image motion was tracked by cross-correlating individual images with a reference image.
We obtained photometry on the instrumental system using software apertures of a range of sizes, and retained those which gave light curves with the smallest scatter (Table\[tab:obslog\]). We found that the choice of aperture size does influence the scatter in the final light curve, but does not have a significant effect on the transit shape.
The instrumental magnitudes were then transformed to differential-magnitude light curves normalised to zero magnitude outside transit. The normalisation was enforced with first- or second-order polynomials (see Table\[tab:obslog\]) fitted to the out-of-transit data. The differential magnitudes are relative to a weighted ensemble of typically five (DFOSC) or two to four (GROND) comparison stars. The comparison star weights and polynomial coefficients were simultaneously optimised to minimise the scatter in the out-of-transit data.
The CASLEO data were reduced using standard aperture photometry methods, with the IRAF tasks [ccdproc]{} and [apphot]{}. We found that it was necessary to flat-field the data in order to obtain a good light curve. The final light curve was obtained by dividing the flux of WASP-103 by the average flux of three comparison stars. An aperture radius of three times the FWHM was used, as it minimised the scatter in the data.
Finally, the timestamps for the datapoints were converted to the BJD(TDB) timescale [@Eastman++10pasp]. We performed manual time checks for several images and have verified that the FITS file timestamps are on the UTC system to within a few seconds. The reduced data are given in Table\[tab:lcdata\] and will be lodged with the CDS[^5].
High-resolution imaging {#sec:obs:li}
-----------------------
![\[fig:li\] High-resolution Lucky Image of the field around WASP-103. The upper panel has a linear flux scale for context and the lower panel has a logarithmic flux scale to enhance the visibility of any faint stars. Each image covers $8{\ensuremath{^{\prime\prime}}}\times 8{\ensuremath{^{\prime\prime}}}$ centred on WASP-103. A bar of length $1{\ensuremath{^{\prime\prime}}}$ is superimposed in the bottom-right of each image. The image is a sum of the best 2% of the original images.](wasp103map01lin.eps "fig:"){width="\columnwidth"} ![\[fig:li\] High-resolution Lucky Image of the field around WASP-103. The upper panel has a linear flux scale for context and the lower panel has a logarithmic flux scale to enhance the visibility of any faint stars. Each image covers $8{\ensuremath{^{\prime\prime}}}\times 8{\ensuremath{^{\prime\prime}}}$ centred on WASP-103. A bar of length $1{\ensuremath{^{\prime\prime}}}$ is superimposed in the bottom-right of each image. The image is a sum of the best 2% of the original images.](wasp103map01log.eps "fig:"){width="\columnwidth"}
Several images were taken of WASP-103 with DFOSC in sharp focus, in order to test for the presence of faint nearby stars whose photons might bias our results [@Daemgen+09aa]. The closest star we found on any image is 42 pixels south-east of WASP-103, and 5.3mag fainter in the $R$-band. It is thus too faint and far away to contaminate the inner aperture of our target star.
We proceeded to obtain a high-resolution image of WASP-103 using the Lucky Imager (LI) mounted on the Danish telescope [see @Skottfelt+13aa]. The LI uses an Andor 512$\times$512 pixel electron-multiplying CCD, with a pixel scale of 0.09[$^{\prime\prime}$]{}pixel$^{-1}$ giving a field of view of $45{\ensuremath{^{\prime\prime}}}\times45{\ensuremath{^{\prime\prime}}}$. The data were reduced using a dedicated pipeline and the 2% of images with the smallest point spread function (PSF) were stacked together to yield combined images whose PSF is smaller than the seeing limit. A long-pass filter was used, resulting in a response which approximates that of SDSS $i$$+$$z$. An overall exposure time of 415s corresponds to an effective exposure time of 8.3s for the best 2% of the images. The FWHM of the PSF is 5.9pixels (0.53[$^{\prime\prime}$]{}) in both dimensions. The LI image (Fig.\[fig:li\]) shows no evidence for a point source closer than that found in our DFOSC images.
Transit timing analysis {#sec:porb}
=======================
{width="\textwidth"}
--------------- --------- ---------- ------------ ------------------------
Time of min. Error Cycle Residual Reference
(BJD/TDB) (d) number (d)
2456459.59957 0.00079 $-$407.0 0.00019 G14
2456767.80578 0.00017 $-$74.0 $-$0.00029 This work (DFOSC $R$)
2456779.83870 0.00022 $-$61.0 0.00054 This work (DFOSC $I$)
2456817.78572 0.00027 $-$20.0 0.00019 This work (DFOSC $R$)
2456831.66843 0.00039 $-$5.0 $-$0.00029 This work (DFOSC $R$)
2456832.59401 0.00024 $-$4.0 $-$0.00025 This work (DFOSC $R$)
2456844.62641 0.00019 9.0 0.00005 This work (DFOSC $R$)
2456844.62642 0.00034 9.0 0.00006 This work (GROND $g$)
2456844.62633 0.00019 9.0 $-$0.00003 This work (GROND $r$)
2456844.62647 0.00023 9.0 0.00011 This work (GROND $i$)
2456844.62678 0.00030 9.0 0.00042 This work (GROND $z$)
2456856.65838 0.00018 22.0 $-$0.00007 This work (DFOSC $R$)
2456857.58383 0.00014 23.0 $-$0.00016 This work (DFOSC $R$)
2456857.58390 0.00022 23.0 $-$0.00009 This work (GROND $g$)
2456857.58398 0.00017 23.0 0.00001 This work (GROND $r$)
2456857.58400 0.00025 23.0 0.00001 This work (GROND $i$)
2456857.58421 0.00025 23.0 0.00022 This work (GROND $z$)
2456882.57473 0.00050 50.0 0.00100 This work (CASLEO $R$)
--------------- --------- ---------- ------------ ------------------------
: \[tab:minima\] Times of minimum light and their residuals versus the ephemeris derived in this work.
We first modelled each light curve individually using the [jktebop]{} code (see below) in order to determine [[the]{}]{} times of mid-transit. In this process the errorbars for each dataset were also scaled to give a reduced $\chi^2$ of $\chi^2_\nu = 1.0$ versus the fitted model. This step is needed because the uncertainties from the [aper]{} algorithm are often moderately underestimated.
We then fitted the times of mid-transit with a straight line versus cycle number to determine a new linear orbital ephemeris. We included the ephemeris zeropoint from G14, which is also on the BJD(TDB) timescale and was obtained by them from a joint fit to all their data. Table\[tab:minima\] gives all transit times plus their residual versus the fitted ephemeris. We chose the reference epoch to be that which gives the lowest uncertainty in the time zeropoint, as this minimises the covariance between the reference time of minimum and the orbital period. The resulting ephemeris is $$T_0 = {\rm BJD(TDB)} \,\, 2\,456\,836.296445 (55) \, + \, 0.9255456 (13) \times E$$ where $E$ gives the cycle count versus the reference epoch and the bracketed quantities indicate the uncertainty in the final digit of the preceding number.
The $\chi^2_\nu$ of the fit is excellent at 1.055. The timestamps from DFOSC and GROND are obtained from different atomic clocks, so are unrelated to each other. The good agreement between them is therefore evidence that both are correct.
Fig.\[fig:minima\] shows the residuals of the times of mid-transit versus the linear ephemeris we have determined. The precision in the measurement of the midpoint of the reference transit has improved from 64.7s (G14) to 4.8s, meaning that we have established a high-quality set of timing data against which orbital decay could be measured in future.
Light curve analysis {#sec:lc}
====================
![\[fig:lcfit\] Phased light curves of WASP-103 compared to the [jktebop]{} best fits. The residuals of the fits are plotted at the base of the figure, offset from unity. Labels give the source and passband for each dataset. The polynomial baseline functions have been removed from the data before plotting.](plotLCfit.eps){width="\columnwidth"}
Source
----------------- ---------------- ---------------- ---------------- ---------------- -------------- ------------- ---------------- ---------------- ----------------- -----------------
DFOSC $R$-band 0.3703 0.0055 0.1129 0.0009 88.1 2.2 0.3328 0.0048 0.03755 0.00074
DFOSC $I$-band 0.3766 0.0146 0.1118 0.0013 84.8 4.2 0.3387 0.0128 0.03788 0.00175
GROND $g$-band 0.3734 0.0140 0.1183 0.0022 86.3 3.9 0.3339 0.0123 0.03949 0.00201
GROND $r$-band 0.3753 0.0102 0.1150 0.0011 85.2 3.2 0.3366 0.0087 0.03870 0.00124
GROND $i$-band 0.3667 0.0132 0.1091 0.0017 87.1 3.6 0.3307 0.0116 0.03606 0.00129
GROND $z$-band 0.3661 0.0111 0.1106 0.0016 89.9 2.4 0.3297 0.0099 0.03645 0.00120
CASLEO $R$-band 0.3665 0.0203 0.1117 0.0055 89.6 4.5 0.3296 0.0167 0.03683 0.00316
Final results [**0.3712**]{} [**0.0040**]{} [**0.1127**]{} [**0.0009**]{} [**87.3**]{} [**1.2**]{} [**0.3335**]{} [**0.0035**]{} [**0.03754**]{} [**0.00049**]{}
G14 86.3 2.7
We analysed our light curves using the [jktebop]{}[^6] code [@Me++04mn] and the [*Homogeneous Studies*]{} methodology [@Me12mn and references therein]. The light curves were divided up according to their passband (Bessell $R$ and $I$ for DFOSC and SDSS $griz$ for GROND) and each set was modelled together.
The model was parameterised by the fractional radii of the star and the planet ($r_{\rm A}$ and $r_{\rm b}$), which are the ratios between the true radii and the semimajor axis ($r_{\rm A,b} = \frac{R_{\rm A,b}}{a}$). The parameters of the fit were the sum and ratio of the fractional radii ($r_{\rm A} + r_{\rm b}$ and $k = \frac{r_{\rm b}}{r_{\rm A}}$), the orbital inclination ($i$), limb darkening coefficients, and a reference time of mid-transit. We assumed an orbital eccentricity of zero (G14) and the orbital period found in Section\[sec:porb\]. We also fitted for the coefficients of polynomial functions of differential magnitude versus time [@Me+14mn]. One polynomial was used for each transit light curve, of the order given in Table\[tab:obslog\].
Limb darkening was incorporated using each of five laws [see @Me08mn], with the linear coefficients either fixed at theoretically predicted values[^7] or included as fitted parameters. We did not calculate fits for both limb darkening coefficients in the four two-coefficient laws as they are very strongly correlated [@Me++07aa; @Carter+08apj]. The nonlinear coefficients were instead perturbed by $\pm$0.1 on a flat distribution during the error analysis simulations, in order to account for imperfections in the theoretical values of the coefficients.
Error estimates for the fitted parameters were obtained in three ways. Two sets were obtained using residual-permutation and Monte Carlo simulations [@Me08mn] and the larger of the two was retained for each fitted parameter. We also ran solutions using the five different limb darkening laws, and increased the errorbar for each parameter to account for any disagreement between these five solutions. Tables of results for each light curve can be found in the Appendix and the best fits can be inspected in Fig.\[fig:lcfit\].
Results
-------
For all light curves we found that the best solutions were obtained when the linear limb darkening coefficient was fitted and the nonlinear coefficient was fixed but perturbed. We found that there is a significant correlation between $i$ and $k$ for all light curves, which hinders the precision to which we can measure the photometric parameters. The best fit for the CASLEO and the GROND $z$-band data is a central transit ($i \approx 90^\circ$), but this does not have a significant effect on the value of $k$ measured from these data.
Table\[tab:lcfit\] holds the measured parameters from each light curve. The final value for each parameter is the weighted mean of the values from the different light curves. We find a good agreement for all parameters except for $k$, which is in line with previous experience (see @Me12mn and references therein). The [$\chi_\nu^{\,2}$]{} of the individual values of $k$ versus the weighted mean is 3.1, and the errorbar for the final value of $k$ in Table\[tab:lcfit\] has been multiplied by $\sqrt{3.1}$ to force a [$\chi_\nu^{\,2}$]{} of unity. Our results agree with, but are significantly more precise than, those found by G14.
Physical properties {#sec:absdim}
===================
[l l l r@[$\pm$]{}c@[$\pm$]{}l r@[$\pm$]{}l]{} Quantity & Symbol & Unit & &\
Stellar mass & $M_{\rm A}$ & [$\,{\rm M}_\odot$]{}& 1.204 & 0.089 & 0.019 &\
Stellar radius & $R_{\rm A}$ & [$\,{\rm R}_\odot$]{}& 1.419 & 0.039 & 0.008 &\
Stellar surface gravity & $\log g_{\rm A}$ & cgs & 4.215 & 0.014 & 0.002 &\
Stellar density & $\rho_{\rm A}$ & [$\,\rho_\odot$]{}& &\
Planet mass & $M_{\rm b}$ & [$\,{\rm M}_{\rm Jup}$]{}& 1.47 & 0.11 & 0.02 & 1.490 & 0.088\
Planet radius$^\star$ & $R_{\rm b}$ & [$\,{\rm R}_{\rm Jup}$]{}& 1.554 & 0.044 & 0.008 &\
Planet surface gravity & $g_{\rm b}$ & [ms$^{-2}$]{}& & 15.7 & 1.4\
Planet density$^\star$ & $\rho_{\rm b}$ & [$\,\rho_{\rm Jup}$]{}& 0.367 & 0.027 & 0.002 &\
Equilibrium temperature & [$T_{\rm eq}^{\,\prime}$]{} & K & &\
Safronov number & [$\Theta$]{} & & 0.0311 & 0.0019 & 0.0002 &\
Orbital semimajor axis & $a$ & au & 0.01978 & 0.00049 & 0.00010 & 0.01985 & 0.00021\
Age & $\tau$ & Gyr & &\
\
Planet radius & & [$\,{\rm R}_{\rm Jup}$]{}&\
Planet density & & [$\,\rho_{\rm Jup}$]{}&\
We have measured the physical properties of the WASP-103 system using the results from Section\[sec:lc\], five grids of predictions from theoretical models of stellar evolution [@Claret04aa; @Demarque+04apjs; @Pietrinferni+04apj; @Vandenberg++06apjs; @Dotter+08apjs], and the spectroscopic properties of the host star. Theoretical models are needed to provide an additional constraint on the stellar properties as the system properties cannot be obtained from only measured quantities. The spectroscopic properties were obtained by G14 and comprise effective temperature (${\ensuremath{T_{\rm eff}}}= 6110 \pm 160$K), metallicity (${\ensuremath{\left[\frac{\rm Fe}{\rm H}\right]}}= 0.06 \pm 0.13$) and velocity amplitude ($K_{\rm A} = 271 \pm 15$[ms$^{-1}$]{}). The adopted set of physical constants is given in [@Me11mn].
We first estimated the velocity amplitude of the [*planet*]{}, $K_{\rm b}$, and used this along with the measured $r_{\rm A}$, $r_{\rm b}$, $i$ and $K_{\rm A}$ to determine the physical properties of the system. $K_{\rm b}$ was then iteratively refined to find the best match between the measured $r_{\rm A}$ and the calculated $\frac{R_{\rm A}}{a}$, and the observed [$T_{\rm eff}$]{} and that predicted by a theoretical model for the obtained stellar mass, radius and [$\left[\frac{\rm Fe}{\rm H}\right]$]{}. This was done for a grid of ages from the zero-age main sequence to beyond the terminal-age main sequence for the star, in 0.01Gyr increments, and the overall best $K_{\rm b}$ was adopted. The statistical errors in the input quantities were propagated to the output quantities by a perturbation approach.
We ran the above analysis for each of the five grids of theoretical stellar models, yielding five different estimates of each output quantity. These were transformed into a single final result for each parameter by taking the unweighted mean of the five estimates and their statistical errors, plus an accompanying systematic error which gives the largest difference between the mean and individual values. The final results of this process are a set of physical properties for the WASP-103 system, each with a statistical error and a systematic error. The stellar density, planetary surface gravity and planetary equilibrium temperatures can be calculated without resorting to theoretical predictions [@SeagerMallen03apj; @Me++07mn; @Me10mn], so do not have an associated systematic error.
Results
-------
--------------------- ------------- ------- ------------- ------------- ------------------- -----------------------------------------
Output
parameter $K_{\rm A}$ $i$ $r_{\rm A}$ $r_{\rm b}$ [$T_{\rm eff}$]{} [$\left[\frac{\rm Fe}{\rm H}\right]$]{}
Age 0.012 0.035 0.873 0.471
$a$ 0.030 0.020 0.797 0.601
\[2pt\] $M_{\rm A}$ 0.029 0.020 0.796 0.602
$R_{\rm A}$ 0.027 0.469 0.703 0.530
$\log g_{\rm A}$ 0.019 0.706 0.563 0.424
$\rho_{\rm A}$ 0.002 1.000 0.001
\[2pt\] $M_{\rm b}$ 0.809 0.014 0.012 0.466 0.352
$R_{\rm b}$ 0.025 0.017 0.544 0.668 0.504
$g_{\rm b}$ 0.901 0.016 0.434
$\rho_{\rm b}$ 0.772 0.014 0.006 0.564 0.232 0.174
--------------------- ------------- ------- ------------- ------------- ------------------- -----------------------------------------
: \[tab:err\] Detailed error budget for the calculation of the system properties of WASP-103 from the photometric and spectroscopic parameters, and the Y$^2$ stellar models. Each number in the table is the fractional contribution to the final uncertainty of an output parameter from the errorbar of an input parameter. The final uncertainty for each output parameter is the quadrature sum of the individual contributions from each input parameter.
Our final results are given in Table\[tab:model\] and have been added to TEPCat[^8]. We find a good agreement between the five different model sets (TableA8). Some of the measured quantities, in particular the stellar and planetary mass, are still relatively uncertain. To investigate this we calculated a complete error budget for each output parameter, and show the results of this analysis in Table\[tab:err\] when using the Y$^2$ theoretical stellar models [@Demarque+04apjs]. The error budgets for the other four model sets are similar.
The uncertainties in the physical properties of the planet are dominated by that in $K_{\rm A}$, followed by that in $r_{\rm b}$. The uncertainties in the stellar properties are dominated by those in [$T_{\rm eff}$]{} and [$\left[\frac{\rm Fe}{\rm H}\right]$]{}, followed by $r_{\rm A}$. Improvements in our understanding of the WASP-103 system would most easily be achieved by obtaining new spectra from which additional radial velocity measurements and improved [$T_{\rm eff}$]{} and [$\left[\frac{\rm Fe}{\rm H}\right]$]{} measurements could be obtained.
To illustrate the progress possible from further spectroscopic analysis, we reran the analysis but with smaller errorbars of $\pm$50K in [$T_{\rm eff}$]{} and $\pm$0.05dex in [$\left[\frac{\rm Fe}{\rm H}\right]$]{}. The precision in $M_{\rm A}$ changes from 0.091[$\,{\rm M}_\odot$]{} to 0.041[$\,{\rm M}_\odot$]{}. Similar improvements are seen for $a$, and smaller improvements for $R_{\rm A}$, $R_{\rm b}$ and $\rho_{\rm b}$. Augmenting this situation by adopting an errorbar of $\pm$5[ms$^{-1}$]{} in $K_{\rm A}$ changes the precision in $M_{\rm b}$ from 0.11[$\,{\rm M}_{\rm Jup}$]{} to 0.047[$\,{\rm M}_{\rm Jup}$]{} and yields further improvements for $R_{\rm b}$ and $\rho_{\rm b}$.
Comparison with G14
-------------------
Table\[tab:model\] also shows the parameter values found by G14, which are in good agreement with our results. Some of the errorbars, however, are smaller than those in the current work, despite the fact that G14 had much less observational data at their disposal. A possible reason for this discrepancy is the additional constraint used to obtain a determinate model for the system. We used each of five sets of theoretical model predictions, whilst G14 adopted a calibration of $M_{\rm A}$ as a function of $\rho_{\rm A}$, [$T_{\rm eff}$]{} and [$\left[\frac{\rm Fe}{\rm H}\right]$]{} based on semi-empirical results from the analysis of low-mass detached eclipsing binary (dEB) systems [@Torres++08apj; @Enoch+10aa; @Me09mn; @Me11mn]. The dEB calibration suffers from an astrophysical scatter of the calibrating objects which is much greater than that of the precision to which the calibration function can be fitted [see @Me11mn]. G14 accounted for the uncertainty in the calibration by perturbing the measured properties of the calibrators during their Markov [[chain]{}]{} Monte Carlo analysis [@Gillon+13aa]. They therefore accounted for the observational uncertainties in the measured properties of the calibrators, but [[neglected]{}]{} the astrophysical scatter.
There is supporting evidence for this interpretation of why our errorbars for some measurements are significantly larger than those found by G14. Our own implementation of the dEB calibration [@Me11mn] explicitly includes the astrophysical scatter and yields $M_{\rm A} = 1.29 \pm 0.11$[$\,{\rm M}_\odot$]{}, where the greatest contribution to the uncertainty is the scatter of the calibrators around the calibration function. [[G14 themselves found a value of $M_{\rm A} = 1.18 \pm 0.10$[$\,{\rm M}_\odot$]{} from an alternative approach (comparable to our main method) of using the CLÉS theoretical models [@Scuflaire+08apss] as their additional constraint. This is much less precise than their default value of $M_{\rm A} = {\ensuremath{1.220^{+0.039}_{-0.036}}}$[$\,{\rm M}_\odot$]{} from the dEB calibration.]{}]{} M. Gillon (private communication) confirms our interpretation of the situation.
Correction for asphericity
--------------------------
Symbol Description
--------------------------------------------- ----------------------------- ------- -------
$R_{\rm sub}$ ([$\,{\rm R}_{\rm Jup}$]{}) Radius at substellar point 1.721 0.075
$R_{\rm back}$ ([$\,{\rm R}_{\rm Jup}$]{}) Radius at antistellar point 1.710 0.072
$R_{\rm side}$ ([$\,{\rm R}_{\rm Jup}$]{}) Radius at sides 1.571 0.047
$R_{\rm pole}$ ([$\,{\rm R}_{\rm Jup}$]{}) Radius at poles 1.537 0.043
$R_{\rm cross}$ ([$\,{\rm R}_{\rm Jup}$]{}) Cross-sectional radius 1.554 0.045
$R_{\rm mean}$ ([$\,{\rm R}_{\rm Jup}$]{}) Mean radius 1.603 0.052
\[3pt\] $f_{\rm RL}$ Roche lobe filling factor 0.584 0.033
\[3pt\] $R_{\rm sub}/R_{\rm side}$ 1.095 0.017
$R_{\rm sub}/R_{\rm pole}$ 1.120 0.020
$R_{\rm side}/R_{\rm pole}$ 1.022 0.003
$R_{\rm back}/R_{\rm sub}$ 0.994 0.002
$R_{\rm mean}/R_{\rm cross}^{\ 3}$ [*density correction*]{} 1.096 0.015
: \[tab:sphere\] Specification of the shape of WASP-103b obtained using Roche geometry.
[$\,{\rm R}_{\rm Jup}$]{}, the equatorial radius of Jupiter, is adopted to be 71492km.
As pointed out by @Li+10nat for the case of WASP-12b, some close-in extrasolar planets may have significant departures from spherical shape. @Budaj11aj calculated the Roche shapes of all transiting planets known at that time, as well as light curves and spectra taking into account the non-spherical shape. He found that WASP-19b and WASP-12b had the most significant tidal distortion of all known planets. The Roche model assumes that the object is rotating synchronously with the orbital period, there is a negligible orbital eccentricity, and that masses can be treated as point masses. The Roche shape has a characteristic pronounced expansion of the object towards the sub-stellar point, and a slightly less pronounced expansion towards the anti-stellar point. The radii on the side of the object are smaller, and the radii at the rotation poles are the smallest. @Leconte++11aa developed a model of tidally distorted planets which takes into account the tidally distorted mass distribution within the object assuming an ellipsoidal shape. @Burton+14apj studied the consequences of the Roche shape on the measured densities of exoplanets.
The Roche shape of a planet is determined by the semi-major axis, mass ratio, and a value of the surface potential. Assuming the parameters found above ($a = 4.25 \pm 0.11$[$\,{\rm R}_\odot$]{}, $M_{\rm A}/M_{\rm b} = 854 \pm 4$ and $R_{\rm b} = 1.554 \pm 0.45$[$\,{\rm R}_{\rm Jup}$]{}) one can estimate the tidally distorted Roche potential, i.e. the shape of the planet which would have the same cross-section during the transit as the one inferred from the observations under the assumption of a spherical planet. The shape of WASP-103b is described by the parameters $R_{\rm sub}$, $R_{\rm back}$, $R_{\rm side}$ and $R_{\rm pole}$ (see @Budaj11aj for more details). The descriptions and values of these are given in Table\[tab:sphere\]. The uncertainties in Table\[tab:sphere\] are the quadrature addition of those due to each input parameter; they are dominated by the uncertainty in the radius of the planet.
The cross-sectional radius, $R_{\rm cross} = \sqrt{R_{\rm side}R_{\rm pole}}$, is the radius of the circle with the same cross-section as the Roche surface during the transit. $R_{\rm cross}$ is the quantity measured from transit light curves using spherical-approximation codes such as [jktebop]{}. $R_{\rm mean}$ is the radius of a sphere with the same volume as that enclosed by the Roche surface.
Table\[tab:sphere\] also gives ratios between $R_{\rm sub}$, $R_{\rm back}$, $R_{\rm side}$ and $R_{\rm pole}$. Moderate changes in the planetary radius lead to very small changes in the ratios. In the case that future analyses yield a revised planetary radius, these ratios can therefore be used to rescale the values of $R_{\rm sub}$, $R_{\rm back}$, $R_{\rm side}$ and $R_{\rm pole}$ appropriately. In particular, the quantity $R_{\rm mean}/R_{\rm cross}^{\ 3}$ is the correction which must be applied to the density measured in the spherical approximation to convert it to the density obtained using Roche geometry.
WASP-103b [[has a Roche lobe filling factor ($f_{\rm RL}$) of 0.58, where $f_{\rm RL}$ is defined to be the radius of the planet at the substellar point relative to the radius of the L1 point. The planet]{}]{} is therefore well away from Roche-lobe overflow but is significantly distorted. The above analysis provides corrections to the properties measured in the spherical approximation. The planetary radius increases by 2.2% to $R_{\rm b} = 1.603 \pm 0.052$[$\,{\rm R}_{\rm Jup}$]{}, and its density falls by 9.6% to $\rho_{\rm b} = 0.335 \pm 0.025$[$\,\rho_{\rm Jup}$]{}. These revised values include the uncertainty in the correction for asphericity and are included in Table\[tab:model\]. These departures from sphericity mean WASP-103b is one of the three most distorted planets known, alongside WASP-19b and WASP-12b.
Variation of radius with wavelength {#sec:rayleigh}
===================================
![\[fig:rvary\] Measured planetary radius ($R_{\rm b}$) as a function of the central wavelength of the passbands used for different light curves. The datapoints show the $R_{\rm b}$ measured from each light curve. The vertical errorbars show the relative uncertainty in $R_{\rm b}$ (i.e. neglecting the common sources of error) and the horizontal errorbars indicate the FWHM of the passband. The datapoints are colour-coded consistently with Figs. \[fig:lc:dk\] and \[fig:lc:grond\] and the passbands are labelled at the top of the figure. The [[dotted]{}]{} grey line to the right of the figure shows the measured value of $R_{\rm b}$ from Table\[tab:model\], which includes all sources of uncertainty. The [[solid]{}]{} grey line to the left of the figure shows how big ten pressure scale heights is.](plotvarywave.eps){width="\columnwidth"}
---------- ----------------- ------- --------- ---------
Passband Central FWHM
wavelength (nm) (nm)
$g$ [[477.0]{}]{} 137.9 0.03911 0.00029
$r$ [[623.1]{}]{} 138.2 0.03806 0.00022
$R$ [[658.9]{}]{} 164.7 0.03770 0.00010
$i$ [[762.5]{}]{} 153.5 0.03643 0.00026
$I$ [[820.0]{}]{} 140.0 0.03703 0.00022
$z$ [[913.4]{}]{} 137.0 0.03698 0.00033
---------- ----------------- ------- --------- ---------
: \[tab:rb\] Values of $r_{\rm b}$ for each of the light curves as plotted in Fig.\[fig:rvary\]. Note that the errorbars in this table exclude all common sources of uncertainty in $r_{\rm b}$ so should only be used to compare different values of $r_{\rm b}(\lambda)$.
If a planet has an extended atmosphere, then a variation of opacity with wavelength will cause a variation of the measured planetary radius with wavelength. The light curve solutions (Table\[tab:lcfit\]) show a dependence between the measured value of $k$ and the central wavelength of the passband used, in that larger $k$ values occur at bluer wavelengths. This implies a larger planetary radius in the blue, which might be due to Rayleigh scattering from a high-altitude atmospheric haze [e.g. @Pont+08mn; @Sing+11mn; @Pont+13mn].
We followed the approach of @Me+12mn2 to tease out this signal from our light curves. We modelled each dataset with the parameters $r_{\rm A}$ and $i$ fixed at the final values in Table\[tab:lcfit\], but still fitting for $T_0$, $r_{\rm b}$, the linear limb darkening coefficient and the polynomial coefficients. [[We did not consider solutions with both limb darkening coefficients fixed, as they had a significantly poorer fit, or with both fitted, as this resulted in unphysical values of the coefficients for most of the light curves.]{}]{} This [[process]{}]{} yielded a value of $r_{\rm b}$ for each light curve with all common sources of [[uncertainty]{}]{} removed from the errorbars (Table\[tab:rb\]), which we then converted to $R_{\rm b}$ using the semimajor axis from Table\[tab:model\]. We discluded the CASLEO data from this analysis due to the low precision of the $r_{\rm b}$ it gave.
Fig.\[fig:rvary\] shows the $R_{\rm b}$ values found from the individual light curves as a function of wavelength. The value of $R_{\rm b}$ from Table\[tab:model\] is indicated for context. We calculated the atmospheric pressure scale height, $H$, of WASP-103b using this formula [e.g. @DepaterLissauer01book]: $$H = \frac{ k_{\rm B} {\ensuremath{T_{\rm eq}^{\,\prime}}}}{ \mu g_{\rm b} }$$ where $k_{\rm B}$ is Botzmann’s constant and $\mu$ is the mean molecular weight in the atmosphere. We adopted $\mu = 2.3$ following @DewitSeager13sci, and the other parameters were taken from Table\[tab:model\]. This yielded $H = 597$km $= 0.00834$[$\,{\rm R}_{\rm Jup}$]{}. The relative errors on our individual $R_{\rm b}$ values are therefore in the region of few pressure scale heights, and the total variation we find between the $g$ and $i$ bands is $13.3H$. For comparison, @Sing+11mn [their fig.14] found a variation of $6H$ between 330nm and 1$\mu$m in transmission spectra of HD189733b. Similar or larger effects have been noted in transmission photometry of HAT-P-5 [@Me+12mn2], GJ3470 [@Nascimbeni+13aa2] and Qatar-2 [@Mancini+14mn].
Is this variation with wavelength plausible? To examine this we turned to the MassSpec concept proposed by @DewitSeager13sci. The atmospheric scale height depends on surface gravity and thus the planet mass: $$M_{\rm b} = \frac{ k_{\rm B} {\ensuremath{T_{\rm eq}^{\,\prime}}}R_{\rm b}^2 }{ \mu G H }$$ where $G$ is the gravitational constant. The variation of the measured radius with wavelength due to Rayleigh scattering depends on the atmospheric scale height under the assumption of a power law relation between the wavelength and cross-section of the scattering species. Rayleigh scattering corresponds to a power law coefficient of $\alpha = -4$ [@Lecavelier+08aa] where $$\alpha H = \frac{{\rm d}R_{\rm b}(\lambda)}{{\rm d}\ln\lambda}$$ which yields the equation $$M_{\rm b} = - \frac{ \alpha k_{\rm B} {\ensuremath{T_{\rm eq}^{\,\prime}}}[ R_{\rm b}(\lambda) ]^2 }{ \mu G \frac{{\rm d}R_{\rm b}(\lambda)}{{\rm d}\ln\lambda} }$$
We applied MassSpec to our $R_{\rm b}(\lambda)$ values for WASP-103b. The slope of $R_{\rm b}$ versus $\ln\lambda$ is detected to a significance of $7.3\sigma$ and corresponds to a planet mass of $0.31 \pm 0.05$[$\,{\rm M}_{\rm Jup}$]{}. The slope is robustly detected, but gives a planet mass much lower than the mass of $M_{\rm b} = 1.49 \pm 0.11$[$\,{\rm M}_{\rm Jup}$]{} found in Section\[sec:absdim\]. The gradient of the slope is greater than it should be under the scenario outlined above. We can equalise the two mass measurements by adopting a stronger power law with a coefficient of $\alpha = 19.0 \pm 1.5$, which is extremely large. We conclude that our data are not consistent with Rayleigh scattering so are either affected by additional physical processes or are returning spurious results.
The presence of unocculted starspots on the visible disc of the star could cause a trend in the measured planetary radius similar to what we see for WASP-103. Unocculted spots cause an overestimate of the ratio of the radii [e.g. @Czesla+09aa; @Ballerini+12aa; @Oshagh+13aa], and are cooler than the surrounding photosphere so have a greater effect in the blue. They therefore bias planetary radius measurements to higher values, and do so more strongly at bluer wavelengths. Occulted plage can cause an analagous effect [@Oshagh+14aa], but we are aware of only circumstantial evidence for plage of the necessary brightness and extent in planet host stars.
The presence of starspots has not been observed on WASP-103A, which at ${\ensuremath{T_{\rm eff}}}= 6110$K is too hot to suffer major spot activity. G14 found no evidence for spot-induced rotational modulation down to a limiting amplitude of 3mmag, and our light curves show no features attributable to occultations of a starspot by the planet. This is therefore an unlikely explanation for the strong correlation between $R_{\rm b}$ and wavelength. Further investigation of this effect requires data with a greater spectral coverage and/or resolution.
Summary and conclusions {#sec:summary}
=======================
The recently-discovered planetary system WASP-103 is well suited to detailed analysis due to its short orbital period and the brightness of the host star. These analyses include the investigation of tidal effects, the determination of high-precision physical properties, and the investigation of the atmospheric properties of the planet. We have obtained 17 new transit light curves which we use to further our understanding in all three areas.
The extremely short orbital period of the WASP-103 system makes it a strong candidate for the detection of tidally-induced orbital decay [@Birkby+14mn]. [[Detecting this effect could yield a measurement of the tidal quality factor for the host star, which is vital for assessing the strength of tidal effects such as orbital circularisation, and for predicting the ultimate fate of hot Jupiters.]{}]{} The prime limitation in [[attempts to observe this effect]{}]{} is that the strength of the signal, and therefore the length of the observational program needed to detect it, is unknown. Our high-precision light curves improve the measurement of the time of midpoint of a transit at the reference epoch from 67.4s (G14) to 4.8s (this work). There are currently no indications of a change in orbital period, but these effects are expected to take of order a decade to become apparent. Our work establishes a high-precision [[transit timing]{}]{} at the reference epoch against which future observations can be measured.
We modelled our light curves with the [jktebop]{} code following the [*Homogeneous Studies*]{} methodology in order to measure high-precision photometric parameters of the system. These were combined with published spectroscopic results and with five sets of theoretical stellar models in order to determine the physical properties of the system to high precision and with robust error estimates. We present an error budget which shows that more precise measurements of the [$T_{\rm eff}$]{} and [$\left[\frac{\rm Fe}{\rm H}\right]$]{} of the host star would be an effective way of further improving these results. A high-resolution [*Lucky Imaging*]{} observation shows no evidence for the presence of faint stars at small (but non-zero) angular separations from WASP-103, which might have contaminated the flux from the system and thus caused us to underestimate the radius of the planet.
The short orbital period of the planet means it is extremely close to its host star: its orbital separation of 0.01978au corresponds to only $3.0R_{\rm A}$. This distorts the planet from a spherical shape, and causes an underestimate of its radius when light curves are modelled in the spherical approximation. We determined the planetary shape using Roche geometry [@Budaj11aj] and utilized these results to correct its measured radius and mean density for the effects of asphericity.
Our light curves were taken in six passbands spanning much of the optical wavelength region. There is a trend towards finding a larger planetary radius at bluer wavelengths, at a statistical significance of $7.3\sigma$. We used the MassSpec concept [@DewitSeager13sci] to convert this into a measurement of the planetary mass under the assumption that the slope is caused by Rayleigh scattering. The resulting mass is too small by a factor of five, implying that Rayleigh scattering is not the main culprit for the observed variation of radius with wavelength.
We recommend that further work on the WASP-103 system includes a detailed spectral analysis for the host star, transit depth measurements in the optical and infrared with a higher spectral resolution than achieved here, and occultation depth measurements to determine the thermal emission of the planet and thus constrain its atmospheric energy budget. Long-term monitoring of its times of transit is also necessary in order to detect the predicted orbital decay due to tidal effects. [[Finally, the system is a good candidate for observing the Rossiter-McLaughlin effect, due to the substantial rotational velocity of the star ($v \sin i = 10.6 \pm 0.9$[kms$^{-1}$]{}; G14).]{}]{}
Acknowledgements {#acknowledgements .unnumbered}
================
The operation of the Danish 1.54m telescope is financed by a grant to UGJ from the Danish Natural Science Research Council (FNU). This paper incorporates observations collected using the Gamma Ray Burst Optical and Near-Infrared Detector (GROND) instrument at the MPG 2.2m telescope located at ESO La Silla, Chile, program 093.A-9007(A). GROND was built by the high-energy group of MPE in collaboration with the LSW Tautenburg and ESO, and is operated as a PI-instrument at the MPG 2.2m telescope. We thank Mike Gillon for helpful discussions. The reduced light curves presented in this work will be made available at the CDS ([http://vizier.u-strasbg.fr/]{}) and at [http://www.astro.keele.ac.uk/$\sim$jkt/]{}. JSouthworth acknowledges financial support from STFC in the form of an Advanced Fellowship. JB acknowledges funding by the Australian Research Council Discovery Project Grant DP120101792. Funding for the Stellar Astrophysics Centre is provided by The Danish National Research Foundation (Grant agreement no. DNRF106). TH is supported by a Sapere Aude Starting Grant from The Danish Council for Independent Research. This publication was supported by grant NPRP X-019-1-006 from Qatar National Research Fund (a member of Qatar Foundation). TCH is supported by the Korea Astronomy & Space Science Institute travel grant \#2014-1-400-06. CS received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 268421. OW (FNRS research fellow) and JSurdej acknowledge support from the Communauté française de Belgique - Actions de recherche concertées - Académie Wallonie-Europe. The following internet-based resources were used in research for this paper: the ESO Digitized Sky Survey; the NASA Astrophysics Data System; the SIMBAD database and VizieR catalogue access tool operated at CDS, Strasbourg, France; and the ar$\chi$iv scientific paper preprint service operated by Cornell University.
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[^1]: Based on data collected by MiNDSTEp with the Danish 1.54m telescope, and data collected with GROND on the MPG 2.2m telescope, both located at ESO La Silla.
[^2]: Visiting Astronomer, Complejo Astronómico El Leoncito operated under agreement between the Consejo Nacional de Investigaciones Científicas y Técnicas de la República Argentina and the National Universities of La Plata, Córdoba and San Juan
[^3]: The acronym [idl]{} stands for Interactive Data Language and is a trademark of ITT Visual Information Solutions. For further details see: [http://www.ittvis.com/ProductServices/IDL.aspx]{}.
[^4]: The [astrolib]{} subroutine library is distributed by NASA. For further details see: [http://idlastro.gsfc.nasa.gov/]{}.
[^5]: [http://vizier.u-strasbg.fr/]{}
[^6]: [jktebop]{} is written in [fortran77]{} and the source code is available at [http://www.astro.keele.ac.uk/jkt/codes/jktebop.html]{}
[^7]: Theoretical limb darkening coefficients were obtained by bilinear interpolation in [$T_{\rm eff}$]{} and [$\log g$]{} using the [jktld]{} code available from: [http://www.astro.keele.ac.uk/jkt/codes/jktld.html]{}
[^8]: TEPCat is The Transiting Extrasolar Planet Catalogue [@Me11mn] at: [http://www.astro.keele.ac.uk/jkt/tepcat/]{}
|
Introduction
============
Recently, Markovic and Shapiro [@MS], motivated by some observational suggestions of a positive cosmological constant [@Lambdaobs], have reexamined the effect of this constant on the evolution of a homogeneous dust ball embedded in vacuum. This paper extends their analysis so as to include the inhomogeneous and degenerate cases. The qualitative behavior of the boundary histories are shown by way of effective potential and Penrose-Carter diagrams. The case $\Lambda < 0$ is included as it provides for an interesting contrast. The well known case $\Lambda = 0$ is not included.
Dust
====
The study of spherically symmetric distributions of matter without pressure in the general theory of relativity has a long history. It is fair to say that the dynamics of this “Lemaître - Tolman - Bondi" solution are well understood, even with a non-vanishing cosmological constant [@omer]. Whereas the discovery of “shell-focusing" singularities in dust added a new dimension to the dynamics [@ES], these singularities are now well studied [@lake1] and are not considered here.
We review the dynamics to set the notation. First, recall that the flow lines of all dust distributions are *geodesic*. As a consequence, with spherical symmetry we can choose synchronous comoving coordinates $(\textsf{r},\theta,\phi,\tau)$ so that the line element associated with the dust takes the form $$\label{dust}
ds^2 = e^{\alpha(\textsf{r},\tau)}{d\textsf{r}^2} + R(\textsf{r},\tau)^2(d\theta^2 + \sin^2 \theta d\phi^2)-d\tau^2.$$ As long as $R^{'} \not= 0 ~(^{'}\equiv \frac{\partial}{\partial \textsf{r}})$ [@ruban] we obtain $$\label{alpha}
e^{\alpha(\textsf{r},\tau)} = \frac{{R^{'}}^{2}}{1+2 E(\textsf{r})}.$$ A further integration gives one more independent function of $\textsf{r}$ $$\label{M}
{{R^{*}}}^{2} - 2 E(\textsf{r})-\frac{\Lambda R^2}{3} = \frac{2 M(\textsf{r})}{R}$$ where $^{*}\equiv \frac{\partial}{\partial \tau}$. The energy density follows as $$\label{rho}
4 \pi \rho(\textsf{r},\tau) = \frac{M^{'}}{R^{2}R^{'}}.$$ Many explicit forms of $R(\textsf{r},\tau)$ are known, but these are not of interest here.
Vacuum
======
The $\Lambda$ generalization of the Schwarzschild vacuum is well known. In terms of familiar curvature coordinates $(r,\theta,\phi,t)$ the line element is given by $$\label{vacuum}
ds^2 = \frac{dr^2}{f(r)} + r^2(d\theta^2 + \sin^2 \theta d\phi^2)-f(r)dt^2,$$ where $$\label{f}
f(r) = 1-\frac{2 m}{r}-\frac{\Lambda r^2}{3}.$$ The associated generalization of the Birkhoff theorem is well known [@bonnor]. It is interesting to note that the $\Lambda$ generalization of the Israel theorem [@israel] is not known. Geodesically complete forms of the metric (\[vacuum\]) along with Penrose - Carter diagrams are now well know [@lake2].
The coordinates $(r,\theta,\phi,t)$ are adapted to two Killing vectors and so geodesics of the metric (\[vacuum\]) have two constants of motion. The orbits are stably planar and we choose the plane to be $\theta = \pi/2$. The momentum conjugate to $\phi$ is the orbital angular momentum *l*, $$\label{l}
r^2\dot{\phi} = \textit{l},$$ and the momentum conjugate to $t$ is the energy $\gamma$, $$\label{energy}
f(r)\dot{t} = \gamma.$$
For timelike geodesics we can take $^. = \frac{d}{d\lambda}$ where $\lambda$ is the proper time. In what follows we are interested in radial motion so that $\textit{l} = 0$. $\gamma$, however, plays a central role. The timelike geodesic equations reduce to $$\label{geodesics}
\gamma^2 - \dot{r}^2 = -f(r).$$ We can write $P(r) \equiv -f(r)$ and treat $P$ as the effective potential of elementary mechanics.
Junction
========
The junction of dust and vacuum in spherical symmetry by way of the Darmois - Israel conditions is well understood [@ML]. To summarize, the continuity of the first fundamental form associated with the boundary ($\Sigma$) ensures that the continuity of $\theta$ and $\phi$ in metrics (\[dust\]) and (\[vacuum\]) is allowed and that the history of the boundary is given by $$\label{history}
R(\textsf{r}_{\Sigma},t) = r_{\Sigma}.$$ The continuity of the second fundamental form guarantees that the flow lines of the boundary particles are simultaneously geodesic of both enveloping 4-geometries. The junction conditions demand that $$\label{msurf}
M(\textsf{r}_{\Sigma}) = m,$$ and that for $R^{'} \not= 0$ $$\label{gammab}
E(\textsf{r}_{\Sigma}) = \frac{\gamma^2-1}{2}.$$ The case $R^{'} = 0$ gives $\gamma = 0$.
Discussion
==========
The qualitative history of the geodesics of (\[vacuum\]), and via (\[gammab\]) therefore of the dust boundary $\Sigma$, can be obtained from a sketch of $P$ (and in particular the requirement that $\gamma^2 \geq P$). These are shown in Figure 1. (The roots $(r_0,r_2,r_3)$ are given explicitly in [@lake2].) Note that for $\Lambda < 0$ *all* orbits are closed, in contrast to $\Lambda \geq 0$. The case $\gamma = 0$ is unique in the sense that $\Sigma$ traverses the bifurcation of the Killing horizons (in the non-degenerate cases).
The Penrose - Carter diagrams are shown in Figure 2. The possible histories of $\Sigma$ are shown. The dust can be matched to the left or to the right. The degenerate case $3 m = 1/\sqrt{\Lambda}$ requires a special coordinate construction [@lake3]. Note that here the case $\gamma = 0$ is associated with unstable equilibrium at the points of internal infinity.
Acknowledgments {#acknowledgments .unnumbered}
---------------
This work was supported by a grant from the Natural Sciences and Engineering Research Council of Canada. I would like to thank Sean Hayward for pointing out the work by Nakao and José Lemos for reminding me of his work on Oppenheimer-Snyder collapse.
D. Markovic and S. L. Shapiro, Phys. Rev. D, [**61**]{} (084029) (gr-qc/9912066). This problem has also been considered by K. Nakao, G.R.G. [**24**]{}, 1069 (1992). See also A. Ilha, J. P. S. Lemos, Phys. Rev. D, [**55**]{}, 1788 (1997) and A. Ilha, A. Kleber, J. P. S. Lemos, J. Math. Phys., [**40**]{} 3509 (1999). See, for example, A. G. Riess [*et al.*]{}, Astron. J. [**116**]{}, 1009 (1998), S. Perlmutter [*et al.*]{}, Astrophys. J. [**517**]{}, 565 (1999), I. Zehavi and A. Dekel, Nature [**401**]{}, 252 (1999). G. C. Omer, Proc. Nat. Ac. Sci. [**53**]{}, 1 (1965). D. M. Eardley and L. Smarr, Phys. Rev. D [**19**]{}, 2239 (1979). See, for example, K. Lake, Phys. Rev. Lett. [**68**]{}, 3129 (1992). Following Ruban (V. A. Ruban, JETP [**56**]{}, 1914 (1969)), the case $R^{'} = 0$ can be examined as a limit of the subclass $R^{'} = \Gamma(\textsf{r})e^{\alpha/2}$. In this case the junction conditions impose the restriction $\Gamma(\textsf{r}_{\Sigma}) = \gamma$. See, for example, W. B. Bonnor in *Recent Development in General Relativity* (Pergamon, New York, 1962). W. Israel, Phys. Rev. [**164**]{}, 1776 (1967). For an early independent construction see K. Lake and R. C. Roeder, Phys. Rev. D [**15**]{} 3513, (1977). See, for example, P. Musgrave and K. Lake, Class. Quant. Grav. **13** 1885 (1996) (gr-qc/9510052). K. Lake, Phys. Rev. D [**20**]{} 370 (1979).
|
---
abstract: 'We consider the Higgs boson anomalous FCNC interactions with $u$, $c$, $d$, $s$ and $b$ quarks using the effective field theory framework. Constraints on anomalous couplings are derived from experimental results on Higgs boson production with subsequent decay into $b \bar{b}$ pair at LHC with $\sqrt{s} = 13 $ TeV. Upper limits on the branching fractions of $H \to b\bar{s}$ and $H \to b\bar{d}$ are set by performing a realistic detector simulation and accurately reproducing analysis selections of the CMS Higgs boson measurement in the four-lepton final state at $\sqrt{s} = 13$ TeV. The searches are projected into operation conditions of HL-LHC. Sensitivity at FCC-hh to anomalous FCNC interactions is studied based on Higgs boson production with $H \rightarrow \gamma \gamma$ decay channel. It is shown that at FCC-hh machine one can expect to set the upper limits of the order of $10^{-2}$ at $95\%$ CL for $\mathcal{B}(H \rightarrow b\bar{s})$ and $\mathcal{B}(H \rightarrow b\bar{d})$.'
address:
- 'NRC “Kurchatov Institute” - IHEP, Protvino, Moscow Region, Russia'
- 'Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, Russia'
-
author:
- 'M. Ilyushin'
- 'P. Mandrik'
- 'S. Slabospitsky'
bibliography:
- 'HiggsCouplings.bib'
title: Constraints on the Higgs boson anomalous FCNC interactions with light quarks
---
FCNC ,Higgs ,Flavor violation ,LHC ,HL-LHC ,FCC-hh ,EFT ,Anomalous interactions ,BSM ,HEP
Introduction {#section_intro}
============
The discovery of Higgs boson by the Large Hadron Collider (LHC) [@Aad:2012tfa; @Chatrchyan:2012xdj] experiments has opened up new area of direct searches for physics Beyond Standard Model (BSM). One of the possible anomalous interaction is the Higgs-mediated flavour-changing neutral currents (FCNC). These processes are forbidden in Standard Model (SM) at tree level and are strongly suppressed in loop corrections by the Glashow-Iliopoulos-Maiani mechanism [@PhysRevD.2.1285].
The Higgs mediated FCNC in top-quark sector is actively investigated at LHC [@Aad:2015pja; @Aaboud:2017mfd; @Aaboud:2018pob; @Khachatryan:2016atv; @Sirunyan:2017uae] by searching for $t\bar{t}$ production with one top quark decay through a FCNC channel and other follow the dominant SM decay $t \rightarrow bW$. The results of the searches are summarized in Table \[table:top\_fcnc\_results\].
\[htbp\]
**Detector** $\mathcal{B}(t \rightarrow u H)$ $\mathcal{B}(t \rightarrow c H)$ Ref.
------------------------------- ---------------------------------- ---------------------------------- ---------------------
ATLAS, 13 TeV, 36.1 fb$^{-1}$ $1.9 \times 10^{-3}$ $1.6 \times 10^{-3}$ [@Aaboud:2018pob]
CMS, 13 TeV, 35.9 fb$^{-1}$ $4.7 \times 10^{-3}$ $4.7 \times 10^{-3}$ [@Sirunyan:2017uae]
: The current experimental upper limits on FCNC decays of top-quark at 95% CL.[]{data-label="table:top_fcnc_results"}
The FCNC couplings of the Higgs to the rest SM quarks can affect various low-energy precision measurements. The strongest indirect bounds on FCNC quark-quark-Higgs couplings came from measurement of $B_{d,s} - \bar{B}_{d,s}$, $K^0 - \bar{K}^0$ and $D^0 - \bar{D}^0$ oscillations [@Harnik:2012pb]. The corresponding constraints on FCNC couplings translated into upper limits on branching fractions of the FCNC decays of Higgs boson to $u,d,s,c,b$ quarks are summarized in the Table \[table:light\_fcnc\_results\]. Due to huge QCD background the experiments at LHC are less sensitive to searching for FCNC decays of the Higgs boson. On the other hand the direct probes of such processes could complement the indirect limits. In addition in possible BSM scenarios the branching ratio of $H \rightarrow q q'$ can be enhanced with keeping other low-energy flavour observables approximately at their SM values [@Crivellin:2017upt; @Altmannshofer:2019ogm]. Therefor, the searches for FCNC Higgs boson interactions are very important and could be considered as a complementary probe of new physics.
At the moment there is no any experimental evidence of the FCNC process. Future research and increase of the experimental sensitivity are related to the proposed energy-frontier colliders [@Mandrik:2018gud; @Barducci:2017ioq; @Mandrik:2018yhe; @Arroyo-Urena:2019qhl] such as High Luminosity LHC (HL-LHC) [@Apollinari:2116337] and Future Circular Collider (FCC-hh) project, defined by the target of 100 TeV proton-proton collisions with a total integrated luminosity of 30 ab$^{-1}$ [@Benedikt:2651300; @Mangano:2651294].
\[htbp\]
**Observable** **Constraint**
---------------------- ------------------------------------------------------------------
$D^0$ oscillations $\mathcal{B}(H \rightarrow u \bar{c}) \lesssim 2 \times 10^{-5}$
$B^0_d$ oscillations $\mathcal{B}(H \rightarrow d \bar{b}) \lesssim 8 \times 10^{-5}$
$K^0$ oscillations $\mathcal{B}(H \rightarrow d \bar{s}) \lesssim 2 \times 10^{-6}$
$B^0_s$ oscillations $\mathcal{B}(H \rightarrow s \bar{b}) \lesssim 7 \times 10^{-3}$
: The upper limits on FCNC decays of Higgs boson to the light quarks at 95% CL from experiments with mesons (see [@Harnik:2012pb] for details). []{data-label="table:light_fcnc_results"}
In this article we invested the contribution of FCNC interactions to the single Higgs boson production (fig. \[feyman\_fcnc\_0\], left) and Higgs boson production in association with a light quark (fig. \[feyman\_fcnc\_0\], center and right). The limits on Higgs boson FCNC interactions based on recent LHC data are obtained and the searches are projected into operation conditions of HL-LHC [@Apollinari:2116337] and FCC-hh projects. The cross section ratio for the different processes are presented in table 1.
[0.32]{} ![Example of diagrams for Higgs boson production (left) and Higgs boson associated production with quark (center and right) mediated by FCNC couplings.[]{data-label="feyman_fcnc_0"}](fey_Hcoupl_0.pdf "fig:"){height="3cm"}
[0.32]{} ![Example of diagrams for Higgs boson production (left) and Higgs boson associated production with quark (center and right) mediated by FCNC couplings.[]{data-label="feyman_fcnc_0"}](fey_qg_to_Hq_s_channel.pdf "fig:"){height="3cm"}
[0.32]{} ![Example of diagrams for Higgs boson production (left) and Higgs boson associated production with quark (center and right) mediated by FCNC couplings.[]{data-label="feyman_fcnc_0"}](fey_qg_to_Hq.pdf "fig:"){height="3cm"}
The constraints from the current Higgs production cross-sections {#constraints}
================================================================
The flavor-violating couplings may arise from different sources [@Agashe:2013hma]. In this article we use the effective field theory approach (EFT) [@Weinberg:1978kz; @Buchmuller:1985jz; @Arzt:1994gp] for describing the effects of BSM physics in Higgs interactions. The effective Lagrangian (up to dimension-six gauge-invariant effective operators) has the form as follows [@AguilarSaavedra:2004wm; @AguilarSaavedra:2009mx]: $$\label{eq_lagrangian}
\mathcal{L}_{BSM} = -\frac{ 1 }{ \sqrt{2} } \bar{q}
(\kappa_{qq'H}^L P_L + \kappa_{qq'H}^R P_R) q' H$$ where $P_{L,R} = \frac{1}{2}(1 \pm \gamma^5)$, $q, q' \in (u,c,t)$ or $q, q' \in (d,s,b)$. The couplings $\kappa_{qq'H}^L$ and $\kappa_{qq'H}^R$ are complex in general.
Note, that in our analysis these couplings are appeared in the combination $$\begin{aligned}
|\kappa^L_{qq'}|^2 + |\kappa^R_{qq'}|^2 =
(\hbox{Re}\,\kappa^L_{qq'})^2 + (\hbox{Im}\,\kappa^L_{qq'})^2 +
(\hbox{Re}\,\kappa^R_{qq'})^2 + (\hbox{Im}\,\kappa^R_{qq'})^2\end{aligned}$$ Thus, in what follows we set $$\begin{aligned}
\left.
\begin{array}{l}
\kappa \equiv |\kappa^L_{qq'}| = |\kappa^R_{qq'}|
\\
\lambda \equiv |\hbox{Re}\,\kappa^L_{qq'}| = |\hbox{Im}\,\kappa^L_{qq'}| =
|\hbox{Re}\,\kappa^R_{qq'}| = |\hbox{Im}\,\kappa^R_{qq'}| \\
\to \kappa = \sqrt{2} \lambda
\end{array}
\right\}
\label{coupling1}\end{aligned}$$ The Higgs decays width resulted from (\[eq\_lagrangian\]) equals: $$\begin{aligned}
\Gamma(H \to q \bar{q}') =
\frac{3 ( |\kappa^L_{qq'}|^2 + |\kappa^R_{qq'}|^2) M_H}{32 \pi}
= \frac{3 |\kappa|^2 M_H}{16 \pi}
= |\lambda|^2 \times 14.92 \;\;\hbox{GeV}
\label{wid1}\end{aligned}$$ The very rough estimates of the coupling $\kappa_{qq'}$ could be obtained from the Higgs production in the $pp$-collisions at LHC [@Aaboud:2018zhk; @Sirunyan:2018kst]: $$\begin{aligned}
pp \, \to \, H \, X, \;\; pp \, \to \, H \; W/Z X, \quad H \to b \bar{b}
\label{reaction0}\end{aligned}$$ We use the experimental results from ATLAS and CMS collaborations: $$\begin{aligned}
\mu_b = \displaystyle \frac{\sigma^{exp}(p p \to \, H \, X) }
{\sigma^{theor}(p p \to \, H \, X) }
\label{rat1}\end{aligned}$$
------------------------------------------------------------------------------------------
$p p \to \, H \; W/Z \, X$ $p p \to \, H \, X$
------- ---------------------------------- ------------------------- ---------------------
ATLAS $ \mu_b = 0.98^{+ 0.22}_{-0.21}$ $\mu_b = 1.01 \pm 0.20$ [@Aaboud:2018zhk]
CMS $\mu_b = 1.01 \pm 0.22$ $\mu_b = [@Sirunyan:2018kst]
1.04 \pm 0.20$
------------------------------------------------------------------------------------------
and for estimates we set $$\begin{aligned}
0.8 \leq \mu_b \leq 1.2 \label{rat2}\end{aligned}$$ In order to get the constraints on anomalous constants $\kappa_{qq'}$ we consider the ratio: $$\begin{aligned}
&& \tilde{\mu}_b =
\frac{\sigma(pp \to H)_{SM+FCNC} \, {\mathcal{B}}_{det}(H \to b \bar{b})_{SM+FCNC} }
{\sigma(pp \to H)_{SM} \, {\mathcal{B}}_{det}(H \to b \bar{b})_{SM }}
\label{ratio2}\end{aligned}$$ where $(...)_{SM}$ and $(...)_{SM+FCNC}$ stands for SM and SM+FCNC contributions to Higgs production and decays. The value ${\mathcal{B}}_{det}$ equals branching fractions of the Higgs decays into quark-antiquark pair times the $B$-tagging and $B$ miss-tagging efficiencies (from ATLAS paper [@Aaboud:2018zhk]) $$\begin{aligned}
\varepsilon_b = 70\%, \;
\varepsilon_c = 12\%, \;
\varepsilon_{q} = 0.3\%, \; q = d,u,s \label{btag}
\end{aligned}$$ So, for SM and SM+FCNC scenarios we have: $$\begin{aligned}
{\mathcal{B}}_{det}(H \to b \bar{b})_{SM } &=& {\mathcal{B}}_{sm}(H\to b \bar{b})
\varepsilon_b^2 \\
{\mathcal{B}}_{det}(H \to b \bar{b})_{SM+FCNC} & =&
B_{fcnc}(H\to b \bar{b}) \varepsilon_b^2 +
B_{fcnc}(H \to q_1 \bar{q}_2) \varepsilon_{q_1} \varepsilon_{q_2}
\end{aligned}$$ We use the [<span style="font-variant:small-caps;">MG5\_</span>]{}a[<span style="font-variant:small-caps;">MC@NLO</span>]{} 2.5.2 [@Alwall:2014hca] package (see section \[Event\_generation\]) for estimation of the Higgs anomalous production cross-sections at $\sqrt{s} = 13$ TeV: $$\begin{aligned}
\sigma_{sm} \approx 50\, \hbox{pb} &&
\sigma(b \bar{s} + \bar{b} s)_{fcnc} =
|\lambda|^2 \times 18000 \;\; \hbox{pb} \\
\sigma(b \bar{d} + \bar{b} d)_{anom} =
|\lambda|^2 \times 45600 \; \hbox{pb}
&& \sigma(c \bar{u} + \bar{c} u)_{anom} = |\lambda|^2 \times 82000 \;
\hbox{pb}\end{aligned}$$ Then, from requirement on $\tilde{\mu}_b$ from (\[rat2\]) we get the constraints on the anomalous couplings $\kappa_{q q'}$. To avoid ambiguities due to different normalizations of the couplings in the Lagrangian, the branching ratios of the corresponding FCNC processes are also used for presentation of the results.
$q q'$ $\kappa$ $\lambda$ $\Gamma(q \bar{q}')$ MeV $\mathcal{B}(q \bar{q}')$
-------- ----------------- ----------- -------------------------- ---------------------------
$bs$ $ \leq 0.0085$ $0.006$ 0.54 10%
$bd$ $ \leq 0.0089$ $ 0.0063$ 0.60 11%
$cu$ $ \leq 0.0096$ $0.0068$ 0.69 13%
: The upper limits on the anomalous couplings, the Higgs boson decay widths (in MeV) and branching fractions. []{data-label="fcnc_xsec_table_22"}
Certainly, these constraints are much worse as indirect constraints, given if the Table \[table:light\_fcnc\_results\]. However, these constraints are first ones resulted from direct searches of the Higgs FCNC interactions with the light quarks.
Event generation {#Event_generation}
================
The estimation based on (\[ratio2\]) does not take into account the differences in kinematics of the SM and FCNC Higgs boson production processes. In order to accurately incorporate detector effects and reconstruction efficiencies for the next sections we are performing Monte-Carlo (MC) simulation of related processes. We use the Lagrangian (\[eq\_lagrangian\]) for the signal simulation. The Lagrangian (\[eq\_lagrangian\]) is implemented in FeynRules [@Alloul:2013bka] based on [@Amorim:2009mx] and the model is interfaced with generators using the UFO module [@Degrande:2011ua]. The events are generated using the [<span style="font-variant:small-caps;">MG5\_</span>]{}a[<span style="font-variant:small-caps;">MC@NLO</span>]{} 2.5.2 [@Alwall:2014hca] package, with subsequent showering and hadronization in [<span style="font-variant:small-caps;">Pythia</span>]{} 8.230 [@Sjostrand:2014zea]. The [<span style="font-variant:small-caps;">NNPDF3.0</span>]{} [@Ball:2014uwa] PDF sets are used. The detector simulation has been performed with the fast simulation tool [<span style="font-variant:small-caps;">Delphes</span>]{} 3.4.2 [@deFavereau2014] using the corresponding detectors parameterization cards. No additional pileup interactions are added to the simulation. The cross-sections for Higgs boson productions associated with zero or one jet and mediated by FCNC couplings in proton-proton collisions for different centre-of-mass energy are given in the Table \[fcnc\_xsec\_table\_1\]. Note, these values are evaluated for Higss production with 0 or 1 jet using the MLM matching scheme [@Alwall:2007fs]. Therefore, they are greater then those used in previous section.
------------ --------------------------------------- --------------------------------------- --------------------------------------- ---------------------------------------
subprocess
13 TeV 14 TeV 27 TeV 100 TeV
$ucH$ $ 9.08 \times 10^{4}\lambda_{ucH}^2 $ $ 9.85 \times 10^{4}\lambda_{ucH}^2 $ $ 2.01 \times 10^{5}\lambda_{ucH}^2 $ $ 7.3 \times 10^{5}\lambda_{ucH}^2 $
$dsH$ $ 8.25 \times 10^{4}\lambda_{dsH}^2 $ $ 9.02 \times 10^{4}\lambda_{dsH}^2 $ $ 1.91 \times 10^{5}\lambda_{dsH}^2 $ $ 7.23 \times 10^{5}\lambda_{dsH}^2 $
$dbH$ $ 4.81 \times 10^{4}\lambda_{dbH}^2 $ $ 5.32 \times 10^{4}\lambda_{dbH}^2 $ $ 1.18 \times 10^{5}\lambda_{dbH}^2 $ $ 4.77 \times 10^{5}\lambda_{dbH}^2 $
$sbH$ $ 2.32 \times 10^{4}\lambda_{sbH}^2 $ $ 2.61 \times 10^{4}\lambda_{sbH}^2 $ $ 6.67 \times 10^{4}\lambda_{sbH}^2 $ $ 3.27 \times 10^{5}\lambda_{sbH}^2 $
------------ --------------------------------------- --------------------------------------- --------------------------------------- ---------------------------------------
: The cross-sections of Higgs boson + 0, 1 jet productions mediated by FCNC couplings in proton-proton collisions for different centre-of-mass energies. []{data-label="fcnc_xsec_table_1"}
Constrain from Higgs boson measurement in the four-lepton final state at $\sqrt{s} = 13$ TeV {#section_ZZ}
============================================================================================
In this section, we obtain upper the limit on the $\mathcal{B}(H \rightarrow b\bar{s})$ and $\mathcal{B}(H \rightarrow b\bar{d})$ branching fractions using constraints on Higgs boson measurement in the four-lepton final state at $\sqrt{s} = 13$ TeV from CMS experiment at LHC [@Sirunyan:2017exp] from the reaction as follows: $$\begin{aligned}
p \, p \, \to \, H X, \;\; \to \, H \, j \, X, \quad
H \to Z Z, \;\; Z \to \ell^+, \ell^-, \; \ell = \mu, e
\label{reacHZZ}\end{aligned}$$ In order to accurately incorporate the effects of the analyses efficiency different for the SM and FCNC Higgs boson production we reproduce the events selections from [@Sirunyan:2017exp].
The four-lepton candidates build $ZZ$ pairs. One $Z$ candidate is defined as pairs of two opposite charge and matching flavour leptons $(e^+ e^-, \mu^+ \mu^-)$ that satisfy $12<m_{ll}<120$ GeV. Electrons are reconstructed within the geometrical acceptance defined by pseudorapidity $|\eta^e|<2.5$ and for transverse momentum $p_{T}^{e}>7$ GeV. Muons are reconstructed within the geometrical acceptance $|\eta^\mu|<2.4$ and $p_T^\mu>5$ GeV. All leptons within $ZZ$ pairs must be separated in angular space by at least $\Delta R(l_i,l_j)>0.02$. Two of the four selected leptons should have $p_{T,i} > 20$ GeV and $p_{T,j}>10$ GeV.
The $Z$ candidate with reconstructed mass $m_{ll}$ closest to the nominal $Z$ boson mass is denoted as $Z_1$, and the second one is denoted as $Z_2$. The $Z_1$ invariant mass must be larger than 40 GeV. In the $4\mu$ and $4e$ sub-channels the $ZZ$ event with reconstructed mass $m_{Z2}\ge12$ GeV and $m_{Z1}$ closest to the nominal $Z$ boson mass. All four opposite-charge lepton pairs that can be built with the four leptons (irrespective of flavor) are required to satisfy $m_{l_i^+l_j^-}>4$ GeV. Finally, the four-lepton invariant mass should be of the Higgs boson in a $118<m_{4l}<130$ GeV.
The comparison of selection efficiencies for FCNC Higgs boson production processes are presented in Table \[CMS\_eff\]. The simulation of the SM Higgs boson production with Delphes show good agreement with reference Geant4 results taken from [@Sirunyan:2017exp]. The selection efficiency is different for different FCNC Higgs boson productions processes due to the presence of the valence $d$ quark in $bdH$ vertex (as compared to $b s \to H$ production).
Higgs production $4e$ $2e2\mu$ $4\mu$ total total ($m_{4l}$ cut)
------------------ ------ ---------- -------- ------- ----------------------
SM (Geant4) 5.1% 12.9% 10.2% 28.3% 24.9%
SM (Delphes) 4.9% 13.1% 9.3% 27.2% 25.6%
FCNC ($dbH$) 3.6% 9.5% 6.5% 19.5% 17.8%
FCNC ($sbH$) 4.9% 12.8% 9% 26.7% 24.5%
: The comparison of selection efficiency for FCNC and SM Higgs boson productions for different $ZZ$ decay channels before the cut on invariant mass reconstructed Higgs boson $m_{4l} \in [118,130]$ GeV and after the cut. The reference Geant4 results are taken from [@Sirunyan:2017exp]. []{data-label="CMS_eff"}
Statistical analyses is performed based on the number of selected events (after the cut on $118 < m_{4l} < 130$ GeV) where the expected number of signal FCNC events is from our modeling and the observed and expected number of background events are taken from the CMS experimental results [@Sirunyan:2017exp]. For the signal processes lepton energy resolution (20%), lepton energy scale (0.3%), lepton identification (9% on the overall event yield) and luminosity (2.6%) uncertainties are taken into account. The uncertainty from the renormalization and factorization scale is determined by varying these scales between 0.5 and 2 times their nominal value while keeping their ratio between 0.5 and 2 [@deFlorian:2016spz]. PDF uncertainty is determined by taking the root mean square of the variation when using different replicas of the default PDF set [@Butterworth:2015oua]. Contributions of the systematic uncertainties to selection efficiency of the FCNC Higgs boson production are summarized in the Table \[CMS\_sys\_contribution\]. The total uncertainties on the number of selected signal and background (extracted from [@Sirunyan:2017exp]) events are incorporated into statistical model as a nuisances neglecting the correlations.
Process $\mathcal{B}(H \rightarrow b\bar{s})$ $\mathcal{B}(H \rightarrow b\bar{d})$
-------------------------- --------------------------------------- --------------------------------------- -- --
Lepton energy resolution $< \pm 0.2\%$ $< \pm 0.2\%$
Lepton energy scale $< \pm 0.5\%$ $< \pm 0.5\%$
Lepton identification $\pm 9\%$ $\pm 9\%$
Luminosity $\pm 2.6\%$ $\pm 2.6\%$
QCD scale $-19.6\%$ $+18.1$ $-17\%$ $+15.2\%$
PDF $\pm 8\%$ $\pm 3.4$
Total $-23.1\%$ $+21.9\%$ $-19.7\%$ $+18.2\%$
: Summary of contribution of the systematic uncertainties to the selection efficiency of the FCNC Higgs boson production. []{data-label="CMS_sys_contribution"}
Bayesian inference is used to derive the posterior probability based on the following likelihood function: $$\begin{aligned}
\mathcal{L} = \mathcal{G}\Big( N_{obs}| N_{back} + (N_{SM} + N_{FCNC}) \cdot \frac{\mathcal{B}_{FCNC + SM}}{\mathcal{B}_{SM}}, \sqrt{N_{obs}} \Big) \times \\
\times \mathcal{G}\Big( N_{back} | N_{back}^{exp}, \sigma_{N_{back}^{exp}} \Big) \times \\
\times \mathcal{G}\Big( N_{FCNC} | N_{FCNC}^{exp}(\lambda), \sigma_{N_{FCNC}^{exp}(\lambda)} \Big)\end{aligned}$$ where the $\mathcal{G}$ - Gaussian function, $N_{back}^{exp},N_{SM}^{exp},N_{FCNC}^{exp}$ - the expected from the MC simulation number of background, SM and FCNC Higgs boson production events respectively, $\sigma_{N_{...}^{exp}}$ - its uncertainty, $\mathcal{B}_{FCNC + SM}$ - branching of $H \rightarrow 4\ell$ ($\ell = e, \mu$) in the presence of FCNC.
The 95% C.L. expected exclusion limits on the anomalous couplings and the branching fractions are given in Table \[FCC\_limits\].
Sensitivity at HL-LHC
=====================
The reconstruction efficiency estimated in section \[section\_ZZ\] can be used to project the FCNC searches into HL-LHC conditions, defined by total integrated luminosity of 3 ab$^{-1}$ and collision energy of 14 TeV, respectively. For the rescaling the crossections of SM Higgs boson productions are taken from [@Cepeda:2019klc]. The rescaling factors for crossections of $qq \rightarrow ZZ$ and $gg \rightarrow ZZ$ background processes are taken from [@CMS-PAS-FTR-18-014]. The rescaling factros for crossections of “$Z+X$” background processes is estimated using the corresponding crossections from [<span style="font-variant:small-caps;">MG5\_</span>]{}a[<span style="font-variant:small-caps;">MC@NLO</span>]{} 2.5.2 [@Alwall:2014hca] simulation of dominated $Z + jets$ process. The cross section ratio for the different processes are summurised in table \[table:xsec\_rescale\_table\]. Statistical analyses from section \[section\_ZZ\] is reproduced for the new conditions. The dominated systematic uncertainties on the simulation originating from theoretical sources are scaled by 50$\%$ following the treatment of systematic uncertainties in [@Cepeda:2019klc]. In this considered scenario the theoretical uncertainties are expected to improve over time due to developments in the calculations, techniques and orders considered. The 95% C.L. expected exclusion limits on the anomalous couplings and the branching fractions are given in Table \[FCC\_limits\].
\[htbp\]
**Process** $\sigma_{\text{14 TeV}}/\sigma_{\text{13 TeV}}$
--------------------- -------------------------------------------------
$qq \rightarrow ZZ$ 1.17
$gg \rightarrow ZZ$ 1.13
“$Z+X$” 1.11
SM Higgs 1.13
FCNC Higgs ($dbH$) 1.10
FCNC Higgs ($sbH$) 1.13
: Cross section ratios $\sigma_{\text{14 TeV}}/\sigma_{\text{13 TeV}}$ for FCNC and background processes. []{data-label="table:xsec_rescale_table"}
Sensitivity at FCC-hh
=====================
In this section the sensitivity to single Higgs boson production through FCNC in $bdH$ and $bsH$ subprocesses is explored for the FCC-hh experimental conditions following the [@L.Borgonovi:2642471] SM study. The $H \rightarrow \gamma \gamma$ decay channel is used in this analysis. The SM single Higgs production is considered as background in additional to QCD di-photon productions including the huge tree level $qq \rightarrow \gamma \gamma$ component, generated up to two merged extra-jets, and a smaller loop-induced component, $gg \rightarrow \gamma \gamma$, generated up to one additional merged jet. A conservative K-factor of 2 is applied to both QCD contributions. The signal and background process generation and detector simulation are described in \[Event\_generation\] chapter.
The photons with $p_T > 25$ GeV, $|\eta| < 4$ and relative isolation $< 0.15$ are used in the following analyses. Jets are reconstructed using anti$-kT$ algorithm with distance parameter $R=0.4$ and required to have $p_T > 30$ GeV, $|\eta| < 3$. The events are selected using the following baseline criteria:
1. at least 2 selected photons and at least one of them with $p_T > 30$ GeV;
2. mass of the Higgs boson candidate reconstructed from the two photons with the highest $p_T$ should be $|m_H - 125| < 5$ GeV.
Distributions of the kinematic variables obtained after baseline selections are presented at Fig. \[fig:fcc\_plots\], Fig. \[fig:fcc\_plots\_2\] and Fig. \[fig:fcc\_plots\_3\].
![ Distributions of the kinematic variables obtained after basic selections: $\Delta R$ between two selected photons with the highest $p_T$ (top-left), $p_T$ of the Higgs boson candidate (top-right), $p_T$ of the leading jet (bottom-left), $p_T$ of the leading b-tagged jet (bottom-right). The signal processes have arbitrary normalization for the illustration purpose. []{data-label="fig:fcc_plots"}](B_dR_photons_xfactor_1_tmva_fcc.pdf "fig:"){width="0.48\columnwidth"} ![ Distributions of the kinematic variables obtained after basic selections: $\Delta R$ between two selected photons with the highest $p_T$ (top-left), $p_T$ of the Higgs boson candidate (top-right), $p_T$ of the leading jet (bottom-left), $p_T$ of the leading b-tagged jet (bottom-right). The signal processes have arbitrary normalization for the illustration purpose. []{data-label="fig:fcc_plots"}](B_H_pt_xfactor_1_tmva_fcc.pdf "fig:"){width="0.48\columnwidth"}\
![ Distributions of the kinematic variables obtained after basic selections: $\Delta R$ between two selected photons with the highest $p_T$ (top-left), $p_T$ of the Higgs boson candidate (top-right), $p_T$ of the leading jet (bottom-left), $p_T$ of the leading b-tagged jet (bottom-right). The signal processes have arbitrary normalization for the illustration purpose. []{data-label="fig:fcc_plots"}](B_lj_pt_xfactor_1_tmva_fcc.pdf "fig:"){width="0.48\columnwidth"} ![ Distributions of the kinematic variables obtained after basic selections: $\Delta R$ between two selected photons with the highest $p_T$ (top-left), $p_T$ of the Higgs boson candidate (top-right), $p_T$ of the leading jet (bottom-left), $p_T$ of the leading b-tagged jet (bottom-right). The signal processes have arbitrary normalization for the illustration purpose. []{data-label="fig:fcc_plots"}](B_lbj_pt_xfactor_1_tmva_fcc.pdf "fig:"){width="0.48\columnwidth"}\
![ Distributions of the kinematic variables obtained after basic selections: leading photons $p_{T}^{\gamma_{1}}$ (top-left), second photons $p_{T}^{\gamma_{2}}$ (top-right), $\Delta R$ between Higgs boson candidate and leading jet (bottom-left), $\Delta R$ between Higgs boson candidate and leading b-tagged jet (bottom-right). The signal processes have arbitrary normalization for the illustration purpose. []{data-label="fig:fcc_plots_2"}](B_photon_1v_pt_xfactor_1_tmva_fcc.pdf "fig:"){width="0.48\columnwidth"} ![ Distributions of the kinematic variables obtained after basic selections: leading photons $p_{T}^{\gamma_{1}}$ (top-left), second photons $p_{T}^{\gamma_{2}}$ (top-right), $\Delta R$ between Higgs boson candidate and leading jet (bottom-left), $\Delta R$ between Higgs boson candidate and leading b-tagged jet (bottom-right). The signal processes have arbitrary normalization for the illustration purpose. []{data-label="fig:fcc_plots_2"}](B_photon_2v_pt_xfactor_1_tmva_fcc.pdf "fig:"){width="0.48\columnwidth"}\
![ Distributions of the kinematic variables obtained after basic selections: leading photons $p_{T}^{\gamma_{1}}$ (top-left), second photons $p_{T}^{\gamma_{2}}$ (top-right), $\Delta R$ between Higgs boson candidate and leading jet (bottom-left), $\Delta R$ between Higgs boson candidate and leading b-tagged jet (bottom-right). The signal processes have arbitrary normalization for the illustration purpose. []{data-label="fig:fcc_plots_2"}](B_dR_H_lj_xfactor_1_tmva_fcc.pdf "fig:"){width="0.48\columnwidth"} ![ Distributions of the kinematic variables obtained after basic selections: leading photons $p_{T}^{\gamma_{1}}$ (top-left), second photons $p_{T}^{\gamma_{2}}$ (top-right), $\Delta R$ between Higgs boson candidate and leading jet (bottom-left), $\Delta R$ between Higgs boson candidate and leading b-tagged jet (bottom-right). The signal processes have arbitrary normalization for the illustration purpose. []{data-label="fig:fcc_plots_2"}](B_dR_H_lbj_xfactor_1_tmva_fcc.pdf "fig:"){width="0.48\columnwidth"}\
![ Distributions of the kinematic variables obtained after basic selections: $\Delta\varphi$ between reconstructed Higgs boson candidate and leading b-tagged jet (top-left), disbalance in energy of photons from Higgs decay (top-right, see text for the description) and mass of the reconstructed Higgs boson candidate (bottom-left, without cut on mass), $\eta$ of the reconstructed Higgs boson candidate (bottom-right). The signal processes have arbitrary normalization for the illustration purpose. []{data-label="fig:fcc_plots_3"}](B_dPhi_H_lbj_xfactor_1_tmva_fcc.pdf "fig:"){width="0.48\columnwidth"} ![ Distributions of the kinematic variables obtained after basic selections: $\Delta\varphi$ between reconstructed Higgs boson candidate and leading b-tagged jet (top-left), disbalance in energy of photons from Higgs decay (top-right, see text for the description) and mass of the reconstructed Higgs boson candidate (bottom-left, without cut on mass), $\eta$ of the reconstructed Higgs boson candidate (bottom-right). The signal processes have arbitrary normalization for the illustration purpose. []{data-label="fig:fcc_plots_3"}](B_ED_photons_xfactor_1_tmva_fcc.pdf "fig:"){width="0.48\columnwidth"}\
![ Distributions of the kinematic variables obtained after basic selections: $\Delta\varphi$ between reconstructed Higgs boson candidate and leading b-tagged jet (top-left), disbalance in energy of photons from Higgs decay (top-right, see text for the description) and mass of the reconstructed Higgs boson candidate (bottom-left, without cut on mass), $\eta$ of the reconstructed Higgs boson candidate (bottom-right). The signal processes have arbitrary normalization for the illustration purpose. []{data-label="fig:fcc_plots_3"}](B_H_m_xfactor_tmva_fcc.pdf "fig:"){width="0.48\columnwidth"} ![ Distributions of the kinematic variables obtained after basic selections: $\Delta\varphi$ between reconstructed Higgs boson candidate and leading b-tagged jet (top-left), disbalance in energy of photons from Higgs decay (top-right, see text for the description) and mass of the reconstructed Higgs boson candidate (bottom-left, without cut on mass), $\eta$ of the reconstructed Higgs boson candidate (bottom-right). The signal processes have arbitrary normalization for the illustration purpose. []{data-label="fig:fcc_plots_3"}](B_H_eta_xfactor_1_tmva_fcc.pdf "fig:"){width="0.48\columnwidth"}
A Boosted Decision Tree (BDT) constructed in the TMVA framework [@Hocker:2007ht] is used to separate the signal signature from the background contributions. 10% of events selected for training and the remainder are used in the statistical analysis of the BDT discriminants with the CombinedLimit package. The following input variables are used for training:
1. Higgs boson candidate $M_H$, $p_{T}^{H}$ and $\eta_{H}$;
2. leading jet (LJ) $p_{T}^{LJ}$ and $\eta_{LJ}$;
3. leading b-tagged jet (LBJ) $p_{T}^{LBJ}$ and $\eta_{LBJ}$;
4. leading photons $p_{T}^{\gamma_{1}}$, $\eta^{\gamma_{1}}$ and second photons $p_{T}^{\gamma_{2}}$, $\eta^{\gamma_{2}}$;
5. Number of jets $N_{jets}$ and number of b-tagged jets $N_{b-jets}$;
6. $\Delta R(\gamma,\gamma)$ between leading and second photon;
7. $\Delta R(H,LBJ)$ between Higgs boson candidate and leading jet;
8. $\Delta R(H, LJ)$ between Higgs boson candidate and leading b-tagged jet.
![ Expected exclusion limits at 95% C.L. on the FCNC $H \rightarrow b\bar{s}$ and $H \rightarrow b\bar{d}$ branching fractions (left) and FCNC couplings (right) as a function of integrated luminosity. []{data-label="fig:fcc_plots_lumi"}](lumi_H_bsH.pdf "fig:"){width="0.48\columnwidth"} ![ Expected exclusion limits at 95% C.L. on the FCNC $H \rightarrow b\bar{s}$ and $H \rightarrow b\bar{d}$ branching fractions (left) and FCNC couplings (right) as a function of integrated luminosity. []{data-label="fig:fcc_plots_lumi"}](lumi_H_kappa_bsH.pdf "fig:"){width="0.48\columnwidth"}
For each background a 20% normalisation uncertainty is assumed and incorporated in statistical model as nuisance parameter. The asymptotic frequentist formulae [@Cowan:2010js] is used to obtain an expected upper limit on signal cross section based on an Asimov data set of background-only model. The 95% C.L. expected exclusion limits on the branching fractions are given in Table \[FCC\_limits\]. Figure \[fig:fcc\_plots\_lumi\] shows the expected exclusion limits at 95% C.L. on the FCNC $H \rightarrow b\bar{s}$ and $H \rightarrow b\bar{d}$ branching fractions and FCNC couplings as a function of integrated luminosity.
if( TMath::Abs( dPhi\_H\_lbj ) < 2.00 ) weight\_sel\_3 = 0; //if( TMath::Abs( dPhi\_H\_lj ) > 1.75 ) weight\_sel\_3 = 0; if( TMath::Abs( dPhi\_photons ) < 2.15 ) weight\_sel\_3 = 0; if( ED\_photons > 0.5 ) weight\_sel\_3 = 0; if( H\_pt > 50 ) weight\_sel\_3 = 0; if( lbj\_pt > 100 ) weight\_sel\_3 = 0; if( lj\_pt > 100 ) weight\_sel\_3 = 0;
Experiment $\mathcal{B}(H \rightarrow b\bar{s})$ $\mathcal{B}(H \rightarrow b\bar{d})$
----------------------------------------------------------------------- --------------------------------------- ---------------------------------------
Meson oscillations [@Harnik:2012pb] $7 \times 10^{-3}$ $8 \times 10^{-5}$
CMS LHC $H \rightarrow ZZ \rightarrow 4\ell$ (35.9 fb$^{-1}$, 13 TeV) $27 \times 10^{-2}$ $32 \times 10^{-2}$
HL-LHC $H \rightarrow ZZ \rightarrow 4\ell$ (3 ab$^{-1}$, 14 TeV) $5.8 \times 10^{-2}$ $6.0 \times 10^{-2}$
FCC-hh $H\rightarrow \gamma\gamma$ (30 ab$^{-1}$, 100 TeV) $1.5 \times 10^{-2}$ $1.1 \times 10^{-2}$
Experiment $\lambda_{sbH}$ $\lambda_{dbH}$
Meson oscillations [@Harnik:2012pb] $1.9 \times 10^{-3}$ $2.1 \times 10^{-4}$
CMS LHC $H \rightarrow ZZ \rightarrow 4\ell$ (35.9 fb$^{-1}$, 13 TeV) $13 \times 10^{-3}$ $14 \times 10^{-3}$
HL-LHC $H \rightarrow ZZ \rightarrow 4\ell$ (3 ab$^{-1}$, 14 TeV) $4.0 \times 10^{-3}$ $4.1 \times 10^{-3}$
FCC-hh $H\rightarrow \gamma\gamma$ (30 ab$^{-1}$, 100 TeV) $2.0 \times 10^{-3}$ $1.8 \times 10^{-3}$
: The 95% C.L. expected exclusion limits at FCC-hh on the branching fractions of Higgs FCNC decays and flavor-violating couplings in comparison with present experimental limits. []{data-label="FCC_limits"}
Conclusions
===========
In this work, we demonstrate that the contribution of flavour violation interaction to the production of the Higgs boson in high energy proton-proton collisions can be used for the direct search. The realistic detector simulation and accurately reproducing analysis selections of the CMS Higgs boson measurement in the four-lepton final state at $\sqrt{s} = 13$ TeV allow to set upper limits on the branching fractions of $H \to b\bar{s}$ and $H \to b\bar{d}$ and project the searches into HL-LHC conditions. We also examine the sensitivity at FCC-hh based on Higgs boson production with $H \rightarrow \gamma \gamma$ decay channel. Expected upper limits of the order of $10^{-2}$ at $95\%$ CL for $\mathcal{B}(H \rightarrow b\bar{s})$ and $\mathcal{B}(H \rightarrow b\bar{d})$ are competitive with the indirect limits from meson oscillations experiments. The outcome of our study is summarised in Table \[FCC\_limits\]. Further improvements are possible through the combination of results of different Higgs boson decay and interaction searhes such as pair Higgs boson production.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to H. Gray, F. Moortgat and M. Selvaggi for permission to use the MC samples with background processes used in FCC-hh sensitivity study. We also would like to thank V.F.Kachanov, A.M.Zaitesv for useful discussions.
|
---
abstract: 'A new proof for the completeness of the coherent states $D(\alpha )\mid f>$ for the Heisenberg Weyl group and the groups $SU(2)$ and $SU(1,1)$ is presented. Generalizations of these results and their consequences are disussed.'
address:
- |
Physical Research Laboratory\
Navrangpura\
Ahmedabad - 580 009 (INDIA)
- |
School of Physics\
University of Hyderabad\
Hyderabad - 500 046 (INDIA)
author:
- 'G.S. Agarwal'
- 'S. Chaturvedi'
title: 'A non group theoretic proof of completeness of arbitrary coherent states $D(\alpha)\mid f>$'
---
[**Introduction**]{}
Resolution of the identity operator in terms of the eigenstates of suitable operators proves to be an important calculational tool in quantum mechanics. One comes across numerous instances where quantum mechanical calculations are greatly simplified by a judicious use of the resolution of the identity in terms of the eigenstates of appropriate operators. Among the various resolutions of the identity, the one which has played a key role in quantum optics is that in terms of the coherent states $\mid\alpha>$ \[1-3\], the eigenstates of the annihilation operator $${1\over\pi} \int d^2\alpha \mid\alpha><\alpha\mid = \bi\,\,\,\,,$$ where $$\mid\alpha> = D(\alpha)\mid0>~~;~~D(\alpha)=\exp(\alpha\ad-\alpha^*a)~~;~~
[a,\ad] = \bi\,\,\,.$$
The coherent states $\mid\alpha>$ together with (1) have not only led to new calculational techniques but also led to new conceptual developments such as the notion of quasi probability distributions.
The proof of (1) found in most text books on quantum optics and quantum mechanics proceeds by expanding $\mid\alpha>$ in terms of Fock states and carrying out the $\alpha$-integration and by using the completeness of Fock states. In recent times states like $D(\alpha )\mid n>$, the displaced number states \[4-6\], have been used in quantum optics and it is known that these also form a complete set for each $n$ \[5\]. In fact, from a group theoretic point of view \[7,8\] one has a more general result $${1\over\pi}\int d^2\alpha D(\alpha)\mid f><f\mid D^\dagger(\alpha) =
\bi\,\,\,\,\,,$$ where $\mid f>$, referred to as the fiducial state, is any fixed normalizable state. (In (3) it has been assumed that $\mid f>$ is normalized to unity.) The states $$\mid \alpha; f> = D(\alpha)\mid f>\,\,\,\,,$$ are referred to as generalized coherent states. (To avoid confusion with other notions of generlized coherent states, we would, hereafter, refer to them as $f$-coherent states.) The choice $\mid f>=\mid n>$ in (3), for instance, leads to the resolution of the identity in terms of the displaced number states. The group theoretical proof of (3), using Schur’s Lemma, is based on the following observations
- $D(\beta)$ provide an irreducible representation (upto a phase) of the Heisenberg Weyl group.
- the operator $$X_1(f) \equiv {1\over\pi}\int d^2\alpha D(\alpha)\mid f> <f\mid D^\dagger
(\alpha) \,\,\,\,,$$ commutes with the $D(\beta)$’s and hence, by Schur’s Lemma, is proportional to the identity operator $$X_1(f) = c(f) \bi\,\,\,\,\,,$$
- the constant $c(f)$ can be calculated by taking the matrix element of $X_{1}(f)$ between any normalizable state. (For consistency, $c(f)$ should be $<\infty$ which, for coherent states for certain groups leads to restrictions on the fiducial states.) For the Heisenberg-Weyl group, it is easy to show that for any fiducial state $\mid f>$; $<f\mid f>=1$, $c(f)=1$ and hence one has (3). By expanding $\mid f>$ in terms of Fock states (3) may equivalently be written as $${1\over\pi}\int d^2\alpha\mid \alpha; n><\alpha;m\mid =\bi \delta_{nm}~~~;~~~
\mid \alpha;n> \equiv D(\alpha)\mid n>\,\,\,\,.$$
The considerations given above apply to other groups like $SU(2)$ and $SU(1,1)$ as well \[7,8\]. For the case of $SU(2)$ $$[S_+, S_-] = 2S_z~~~ ; ~~~[S_z, S_\pm] = \pm S_\pm \,\,\,\,,$$ one has $$X_2(m) \equiv {2S+1\over4\pi} \int {d^2\zeta\over(1+\mid \zeta\mid ^2)^2}
\,\,\mid \zeta;m><\zeta;m\mid = \bi\,\,\,\,,$$ where $$\mid \zeta;m> \equiv D(\xi)\mid S,m>~~; ~~ D(\xi) = \exp(\xi S_+-\xi^*S_-)
\,\,\,\,,$$ and $\mid S,m>$ are eigenstates of $S^2$ and $S_z$. The variables $\zeta$ and $\xi$ are related to each other as follows $$\xi = {\theta\over2} e^{-i\phi} ~~~ ; ~~~ \zeta = \tan{\theta\over2}
e^{-i\phi} \,\,\,\,\,,$$ and the integration in (9) is over the entire $\zeta$-plane.
Similarly, for $SU(1,1)$ $$[K_-, K_+] = 2K_z ~~ ; ~~ [K_z, K_\pm] = \pm K_\pm ~~~ ,$$ realized via $$K_+ = {1\over2} a^{\dagger2} ~~ ; ~~K_-={1\over2}a^2 ~~ ; ~~
K_z={1\over2}(\ad a + {1\over2}) \,\,\,\,,$$ one has $$X_3(n) \equiv {1\over2\pi} \int {d^2\zeta\over(1-\mid \zeta\mid ^2)^2}
\mid \zeta;2n+1><\zeta;2n+1\mid = \bi_{odd} \,\,\,\,,$$ where $$\mid \zeta;2n+1>\equiv D(\xi)\mid 2n+1>~;~D(\xi)=\exp(\xi K_+-\xi^*K_-)~;~
K_z\mid 2n+1>=(n+{3\over4})\mid 2n+1> \,\,\,\,,$$ and $\zeta$ and $\xi$ are related to each other as follows $$\xi=\mid \xi\mid e^{-i\phi}~~~ ;~~~ \zeta=\tanh\mid \xi\mid e^{-i\phi}~~~.$$ The operator $\bi_{odd}$ in (14) denotes the unit operator in the odd sector of the Fock space. $$\bi_{odd} \equiv \sum_{k=0}^\infty \mid 2k+1><2k+1\mid \,\,\,\,,$$ and the integration in (14) is over the unit disc centered at the origin in the complex $\zeta$-plane.
[**New proof of completeness of $f$-coherent states**]{}
We first consider (3). To prove (3) in a rather elegant way we make use of the following results:
- resolution of the identity (1) in terms of coherent states.
- the fact that an operator is uniquely determined by its diagonal elements \[9\].
$$<\beta\mid G\mid \beta> = 1 ~\mbox{for all} ~ \beta ~~\mbox{if and only if} ~~
G = \bi\,\,\,\,.$$
Now consider the operator $X_1(f)$ $$X_1(f) \equiv {1\over\pi} \int d^2\alpha D(\alpha)
\mid f><f\mid D^\dagger(\alpha)\,\,\,\,.$$ Consider the diagonal elements of $X_1(f)$ $$\begin{aligned}
<\beta\mid X_1(f)\mid \beta> & = & {1\over\pi}\int d^2\alpha
<\beta\mid D(\alpha)\mid f><f\mid D^\dagger(\alpha)\mid \beta>\,\,\,\,,\nonumber\\
& = &
{1\over\pi}\int d^2\alpha
<0\mid D^\dagger(\beta)D(\alpha)\mid f><f\mid D^\dagger(\alpha)D(\beta)\mid 0>\,\,\,\,,\end{aligned}$$ which on using the algebraic property of the displacement operator $D(\alpha)$ $$D^\dagger(\beta)D(\alpha) = D(\alpha-\beta) \exp[(\beta^*\alpha -
\beta\alpha^*)/2]\,\,\,\,,$$ reduces to $$<\beta\mid X_1(f)\mid \beta> = {1\over\pi} \int
d^2\alpha\mid <0\mid D^\dagger(\beta-\alpha)\mid f>\mid ^2\,\,\,\,.$$ On rewriting the integrand (22) in terms of coherent states and changing the variable of integration (22) becomes $$\begin{aligned}
<\beta\mid X_1(f)\mid \beta> & = & {1\over\pi} \int
d^2\alpha\mid <\beta-\alpha\mid f>\mid ^2\nonumber\\
& = & {1\over\pi} \int d^2\alpha<f\mid \alpha><\alpha\mid f>\nonumber\\
& = & <f\mid {1\over\pi}\int d^2\alpha\mid \alpha><\alpha\mid f> = 1 \,\,\,\,.\end{aligned}$$ Thus the diagonal coherent elements of $X_1(f)$ for all values of $\beta$ are equal to unity and therefore using the property (18) we conclude that $$X_1(f) = \bi\,\,\,\,.$$ This constitutes a direct proof of the completeness of the $f$-coherent states of the Heisenberg-Weyl group.
Next we consider the $SU(2)$ case. In this the analogues of (i) and (ii) above are
\(i) completeness of the atomic coherent states $\mid \zeta;-S>$ \[10\] $${2S+1\over4\pi} \int{d^2\zeta\over(1+\mid \zeta\mid ^2)^2}
\,\,\mid \zeta;-S><\zeta;-S\mid = \bi \,\,\,\,,$$ (ii) $<\zeta;-S\mid G\mid \zeta;-S> = 1$ for all $\zeta$ if an only if $G=\bi$. (26)
We consider the diagonal matrix elements of $X_2(m)$ defined in (9) between the atomic coherent states $\mid \zeta^\prime;-S>$. We follow the same procedure as above and use the following algebraic properties.
$$D(\xi_1)D(\xi_2) = D(\xi_3)\exp[i\Phi(\xi_1,\xi_2)S_z]\,\,\,\,,$$
where $$\Phi(\xi_1,\xi_2) = {1\over i} \ln\left[{1-\zeta_1\zeta_2^*\over
1-\zeta_1^*\zeta_2}\right]\,\,\,\,,$$ and $$\zeta_3 = {\zeta_1+\zeta_2\over1-\zeta_1^*\zeta_2} \,\,\,\,.$$ Further, under the change of variables from $\zeta_2$ to $\zeta_3$ the measure of integration in (9) is invariant $${d^2\zeta_2\over(1+\mid \zeta_2\mid ^2)^2} =
{d^2\zeta_3\over(1+\mid \zeta_3\mid ^2)^2} \,\,\,\,.$$ Using these relations we obtain $$<\zeta^\prime;-S\mid X_2\mid \zeta^\prime;-S>={2S+1\over4\pi} \int
{d^2\zeta^{\prime\prime}\over(1+\mid \zeta^{\prime\prime}\mid ^2)^2}
<S,N\mid \zeta^{\prime\prime};-S><\zeta^{\prime\prime};-S\mid S,N> \,\,\,,$$ which, on using the completeness of the atomic coherent states yields $$<\zeta^\prime;-S\mid X_2(m)\mid \zeta^\prime;-S> = 1
~~\mbox{for all}~~
\zeta^\prime \,\,\,\,,$$ and hence $X_2(m)=\bi$. It is important to note that the fiducial state in this case must be an eigenstate of $S_z$ otherwise the phase factor which arises from the use of (27) will not cancel.
Similarly, in the $SU(1,1)$ case, we use the following algebraic properties. $$D(\xi_1)D(\xi_2) = D(\xi_3)\exp[i\Phi(\xi_1,\xi_2)K_z]\,\,\,\,,$$ where $$\begin{aligned}
\Phi(\xi_1,\xi_2) & = & {1\over i} \ln\left[{1+\zeta_1\zeta_2\over
1+\zeta_1^*\zeta_2}\right] \,\,\,\,,\\
\zeta_3 & = & {\zeta_1+\zeta_2\over1+\zeta_1^*\zeta_2} \,\,\,\,.\end{aligned}$$ The measure of integration is invariant under the change of variables from $\zeta_2$ to $\zeta_3$ $${d^2\zeta_2\over(1-\mid \zeta_2\mid ^2)^2} =
{d^2\zeta_3\over(1-\mid \zeta_3\mid ^2)^2} \,\,\,\,.$$ On using the completeness of $\mid \zeta;1>$, one can show that $$<\zeta;1\mid X_3(n)\mid \zeta;1> = 1~~ \mbox{for all} ~~\zeta\,\,\,\,,$$ and hence $X_3(n)=\bi$.
[**Outlook:**]{}
We have thus shown that $$\int d\mu(\zeta) D(\zeta)\mid f><f\mid D^\dagger(\zeta) = \bi \,\,\,,$$ for the $f$-coherent states for the three groups considered above. The relation (38) is amenable to further generalisations. In the case of Heisenberg- Weyl group, by expanding the state $\mid f>$ in (38) in terms of the number states $\mid n>$ one obtains $$\int d\mu(\zeta) D(\zeta)\mid m><n\mid D^\dagger(\zeta) = \bi
\delta_{mn} \,\,\,\,,$$ and hence $$\int d\mu(\zeta) D(\zeta)\mid f_1><f_2\mid D^\dagger(\zeta) = \bi
<f_1\mid f_2> \,\,\,\,.$$ In view of (39), one has $$\int d\mu(\zeta) D(\zeta)\rho_o D^\dagger(\zeta) = \bi \,\,\,\,,$$ where $\rho_o$ is an arbitrary density matrix. For $SU(2)$ and $SU(1,1)$, (38) implies (41) with $\rho_o$ subject to the conditions $$[\rho_o, S_z] = 0 ~~\mbox{and} ~~[\rho_o, K_z] = 0\,\,\,\,,$$ respectively. It may be noted that, in the context of Heisenberg-Weyl group, resolutions of the identity of the type (41) have been derived by Vourdas and Bishop \[11\] for two specific choices of $\rho_o$. The fact that, for the Heisenberg-Weyl group (41) is valid for an arbitrary $\rho_o$ does not seem to be generally appreciated.
The results given above enable us to derive interesting identities involving orthogonal polynomials. For example the following integral[^1] involving the Jacobi polynomials $P_n^{(\alpha,\beta)}(x)$ \[12\] $${1\over2} \left[{\Gamma(n+1)~ \Gamma(p+3/2)\over\Gamma(p+1)~
\Gamma(n+3/2)}\right] ~ \int_o^1 {dx\over(1-x)^{1/2}} ~ x^{p-n}
\left[ P_n^{p-n,1/2)} (1-2x)\right]^2 = 1 \,\,\,\,,$$ can be derived from (38) by applying it to the $SU(1,1)$ case and using the relations[^2] $$\begin{aligned}
<2m+1\mid D(\xi)\mid 2n+1> & = & e^{-i(m-n)\phi} \left[{\Gamma(n+1)
\Gamma(m+3/2)\over\Gamma(m+1) ~ \Gamma(n+3/2)}\right]^{1/2}
(\mid \zeta\mid )^{m-n}(1-\mid \zeta\mid ^2)^{3/4}\nonumber\\
\nonumber\\
&&~~~~~~~~~~~~~~~~ P_n^{(m-n,1/2)}(1-2\mid \zeta\mid ^2) ~~\mbox{for} ~~
m\ge n\,\,\,\,,\\
\nonumber\\
& = & e^{-i(n-m)\phi} \left[{\Gamma(m+1) ~
\Gamma(n+3/2)\over\Gamma(n+1) ~ \Gamma(m+3/2)}\right]^{1/2}
(-\mid \zeta\mid )^{n-m}(1-\mid \zeta\mid ^2)^{3/4}\nonumber\\
\nonumber\\
&&~~~~~~~~~~~~~~~~ P_m^{(n-m,1/2)}(1-2\mid \zeta\mid ^2) ~~\mbox{for} ~~
m\le n\,\,\,\,,\end{aligned}$$ In conclusion, we also note the possibility of using relations like (1) to construct new classes of quasi-probability distributions. Thus, for instance, for any density operator $\rho$, one can define a generalised Q-function as follows $$Q(\zeta) = Tr[\rho D(\zeta)\rho_o D^\dagger(\zeta)]$$ We hope to discuss this in detail elsewhere.
[00]{}
J R Klauder, Ann. Phys. [**11**]{} (1960) 123.
R.J. Glauber, Phys. Rev. Lett. [**10**]{} (1963) 84.
E.C.G. Sudarshan, Phys. Rev. Lett. [**10**]{} (1963) 277.
M. Boiteux and A. Levelut, J. Phys. [**A6**]{} (1973) 589.
S.M. Roy and V. Singh, Phys. Rev. [**D25**]{} (1982) 3413.
F.A.M.de Oliveira, M.S. Kim, P.L. Knight and V. Buzek, Phys. Rev. [**A41**]{} (1990) 2645.
A Perelomov, [*Generalized Coherent States and Their Applications*]{}, (Springer, Berlin, 1986).
J.R. Klauder in [*Coherent States*]{} J.R. Klauder and Bo-Sture Skagerstam (World Scientific, 1985).
C.L. Mehta and E.C.G. Sudarshan, Phys. Rev. [**138**]{} (1965) 274.
F.T. Arecchi, E. Courtens, R. Gilmore and H. Thomas, Phys. Rev. [**A6**]{} (1972) 2211.
A. Vourdas and R.F. Bishop, Phys. Rev. [**A50**]{} (1994) 3331 ; ibid [**A51**]{} (1995) 2353.
I.S. Gradshteyn and I.M. Ryzhik [*Tables of Integrals, Series and Products*]{} (Academic Press, N.Y., 1965); M.A. Abramowitz and I.A. Stegun, [*Handbook of Mathematical Functions*]{} (Dover, N.Y., 1970).
G. Racah, Phys. Rev. [**61**]{} (1942) 186; ibid [**62**]{} (1942) 432.
[^1]: A direct proof of (43) appears to be difficult. We have succeeded in proving it using Racah identities \[13\].
[^2]: Expressions for these matrix elements in terms of associated Legendre functions may be found in \[7\].
|
[**Holomorphic bundles\
on diagonal Hopf manifolds** ]{}\
Misha Verbitsky[^1]\
[verbit@maths.gla.ac.uk, verbit@mccme.ru]{}
[**Abstract**]{}\
Let $A\in GL(n, {{\Bbb C}})$ be a diagonal linear operator, with all eigenvalues satisfying $|\alpha_i|<1$, and $M = ({{\Bbb C}}^n\backslash 0)/\langle A\rangle$ the corresponding Hopf manifold. We show that any stable holomorphic bundle on $M$ can be lifted to a $\tilde G_F$-equivariant coherent sheaf on ${{\Bbb C}}^n$, where $\tilde G_F \cong ({{\Bbb C}}^*)^l$ is a commutative Lie group acting on ${{\Bbb C}}^n$ and containing $A$. This is used to show that all stable bundles on $M$ are filtrable, that is, admit a filtration by a sequence $F_i$ of coherent sheaves, with all subquotients $F_i/F_{i-1}$ of rank 1.
[ ]{}
Introduction {#_Intro_Section_}
============
In this paper we study the Hopf manifolds of form $M = ({{\Bbb C}}^n\backslash 0)/\langle A\rangle$, where $A\in GL(n, {{\Bbb C}})$ is a linear operator with all eigenvalues satisfying $|\alpha_i| <1$ (such an operator is called a linear contraction). Deforming $A$ to an operator $\lambda \cdot Id$, $0<|\lambda|<1$, we find that $M$ is diffeomorphic to $S^{2n-1}\times ({{\Bbb R}}/{{\Bbb Z}}) \cong S^{2n-1}\times S^1$. The odd Betti numbers of $M$ are odd, hence $M$ is not Kähler. This is the first example of non-Kähler manifold known in algebraic geometry.
When $A$ is diagonal and has form $A = \tau \cdot Id$, $M$ is elliptically fibered over ${{\Bbb C}}P^{n-1}$, with all fibers isomorphic to an elliptic curve $C_\tau = {{\Bbb C}}^*/\langle \tau\rangle$. In this (so-called “classical”) case, the algebraic dimension is maximal possible. For arbitrary $A$, the algebraic dimension of $M$ can reach any value from 0 to $n-1$.
Algebraic geometry of Hopf manifolds, especially Hopf surfaces, is well studied ([@_Kato1_], [@_Kato2_], [@_Brinzanescu_Moraru:FM_], [@_Brinzanescu_Moraru:stable_], [@_Moraru:Hopf_]). For $\dim M =2$, one has a good understanding of the geometry of holomorphic vector bundles on $M$ ([@_Moraru:Hopf2_]). A typical stable vector bundle in this situation is non-filtrable, and actually contains no proper holomorphic subsheaves.
For $\dim_{{\Bbb C}}M >2$, geometry of holomorphic vector bundles is drastically different. In [@_Verbitsky:Sta_Elli_], it was shown that any bundle (and any coherent sheaf) on a classical Hopf manifold $$({{\Bbb C}}^n\backslash 0)/\langle \lambda \cdot Id\rangle,
\ \ n>2, \ \ 0< |\lambda|<1$$ is filtrable. In the present paper, we generalize this theorem to an arbitrary diagonal Hopf manifold.
[ ]{}\[\_main\_filtra\_Theorem\_\] Let $A\in GL(n, {{\Bbb C}})$ be a diagonal linear operator, with all eigenvalues satisfying $|\alpha_i|<1$, and $M = ({{\Bbb C}}^n\backslash 0)/\langle A\rangle$ the corresponding Hopf manifold. Then any coherent sheaf $F\in {\operatorname{Coh}}(M)$ is [**filtrable**]{}, that is, admits a filtration $$0 = F_0 \subset F_1 \subset ... \subset F_m = F$$ with ${\operatorname{rk}}F_i /F_{i-1} {\leqslant}1$.
[**Proof:**]{} Using induction, we can always assume that any sheaf $F'$ with ${\operatorname{rk}}F' < {\operatorname{rk}}F$ is filtrable. Then $F$ is filtrable unless $F$ has no proper coherent shubsheaf. In the lattter case, $F$ is stable. Therefore, \[\_main\_filtra\_Theorem\_\] is implied by the following theorem, which is proven in Section \[\_equi\_shea\_Section\_\] by the means of gauge theory.
[ ]{}\[\_stable\_filtra\_main\_Theorem\_\] Let $A\in GL(n, {{\Bbb C}})$ be a diagonal linear operator, with all eigenvalues satisfying $|\alpha_i|<1$, and $M = ({{\Bbb C}}^n\backslash 0)/\langle A\rangle$ the corresponding Hopf manifold. We choose a locally conformally Kähler Hermitian structure on $M$ as in Subsection \[\_LCK\_Hopf\_Subsection\_\]. Let $F$ be a holomorphic bundle (or a reflexive coherent sheaf) which is stable with respect to this Hermitian structure.[^2] Then $F$ is filtrable.
[**Proof:**]{} See \[\_equi\_shea\_and\_shea\_on\_Hopf\_Remark\_\].
The proof of \[\_stable\_filtra\_main\_Theorem\_\] goes as follows. Using the Kobayashi-Hitchin correspondence on complex Hermitian manifolds (Section \[\_stable\_Kobaya\_Section\_\]), we show that any stable bundle on a diagonal Hopf manifold is equivariant with respect to a certain holomorphic flow (\[\_all\_stable\_equi\_Corollary\_\]). Taking a completion of this flow in $GL(n, {{\Bbb C}})$, we obtain an abelian Lie group, which is isomorphic to $({{\Bbb C}}^*)^l$ (\[\_tilde\_G\_C\^\*\^l\_Proposition\_\]). This allows us to treat stable holomorphic bundles (or reflexive sheaves) on $M$ as objects in a category $({{\Bbb C}}^*)^l$-equivariant coherent sheaves on ${{\Bbb C}}^n\backslash 0$ (\[\_equi\_shea\_and\_shea\_on\_Hopf\_Remark\_\]). Then we show that all objects in this category are filtrable (\[\_Coh\_G\_F\_filtra\_Theorem\_\]).
Diagonal Hopf manifolds in Vaisman geometry
===========================================
An introduction to Vaisman geometry
-----------------------------------
[ ]{}Let $M$ be a complex manifold, $\dim_{{\Bbb C}}M >1$, and $\tilde M$ its covering. Assume that $\tilde M$ is equipped with a Kähler form $\omega_K$, in such a way that the deck transform of $\tilde M/M$ acts on $(\tilde M, \omega_K)$ by homotheties. The form $\omega_K$ defines on $M$ a conformal class by $[\omega_K]$. The pair ($M,[\omega_K]$) is called [**locally conformally Kähler (LCK)**]{}. A Hermitian form $\omega_H$ on $M$ is called an LCK-form if it belongs to the conformal class $[\omega_K]$.
[ ]{}Consider an LCK-manifold $M$ with an LCK-form $\omega_H$. A pullback of $\omega_H$ to $\tilde M$ is written as $f\omega_K$, where $f$ is a function and $\omega_K$ is the Kähler form on $\tilde M$. Therefore, $d \omega_H = \omega_H \wedge \theta$, where $\theta= \frac {d f} f$ is a 1-form on $M$. Clearly, $\theta$ is defined uniquely. Since $\theta=d\log f$, $\theta$ is also closed. This form is called [**the Lee form**]{} of $(M, \omega_H)$.
[ ]{}For a general Hermitian complex manifold $(M, \omega_H)$, the Lee form is defined as ${d^c}^*\omega_H$, where $d^c =I \circ d
\circ I^{-1}$ is the twisted de Rham differential, and ${d^c}^*$ its Hermitian adjoint. It is not difficult to check that this definition is compatible with the one we used above.
[ ]{}\[\_Gauduchin\_defi\_Definition\_\] Let $(M, \omega_H)$ be a Hermitian complex manifold, $\dim_{{\Bbb C}}M=n$. Then $\omega_H$ is called [**a Gauduchon metric**]{} if $d^*{d^c}^*\omega_H=0$, or, equivalently, $d d^c(\omega_H^{n-1}) =0$.
[ ]{} In [@_Gauduchon_1984_], P. Gauduchon proved that such a metric on $M$ exists and is unique, up to a constant multiplier, in any conformal class, provided that the manifold $M$ is compact.
[ ]{}On a compact LCK-manifold, this result translates into an existence of a unique metric with a harmonic Lee form $\theta$. Indeed, ${d^c}^*\omega_H=\theta$ is always closed, hence the Gauduchon condition $d^*{d^c}^*\omega_H=0$ is equivalent to $d^*\theta=0$.
Further on, we shall always fix a choice of a Hermitian metric on an LCK-manifold by choosing a Gauduchon metric.
[ ]{}Let $M$ be an LCK-manifold equipped with a Gauduchon metric $\omega_H$, $\theta$ its Lee form and $\nabla$ the Levi-Civita connection associated with $\omega_H$. Assume that $\theta$ is parallel: $\nabla\theta=0$. Then $M$ is called [**a Vaisman manifold.**]{}
[ ]{}\[\_Kamishima\_Ornea\_Remark\_\] According to Kamishima-Ornea ([@_Kamishima_Ornea_]), a compact LCK-manifold $M$ is Vaisman if and only if it admits a holomorphic vector field acting on $M$ conformally, in such a way that its lifting to $\tilde M$ is not an isometry of $(\tilde M, \omega_K)$.
[ ]{} It is easy to see ([@_Dragomir_Ornea_]) that the condition $\nabla \theta=0$ implies that the dual to $\theta$ vector field $\theta^\sharp$ (called [**the Lee field**]{}) is a holomorphic isometry of $M$ and acts on $\tilde M$ by non-isometric conformal automorphisms. This gives the “only if” part of Kamishima-Ornea theorem.
For further results, details and calculations in Vaisman geometry, the reader is referred to [@_Dragomir_Ornea_], [@_Gauduchon_Ornea_], [@_OV:Structure_], [@_OV:Immersion_], [@_OV:Potential_].
Further on, we shall use the following lemma, which is proven in [@_Verbitsky:LCHK_] (see also [@_OV:Immersion_]).
[ ]{}\[\_omega\_0\_in\_Vaisman\_Lemma\_\] Let $M$ be a Vaisman manifold, $\theta^\sharp$ it Lee field, and $\Sigma$ the complex holomorphic foliation generated by $\theta^\sharp$. Denote by $\omega_0:= d^c \theta$ the real (1,1)-form obtained as a $d^c = I \circ d \circ I^{-1}$-differential of the Lee field $\theta$. Then $\omega_0{\geqslant}0$, and the null direction of $\omega_0$ is precisely $\Sigma$.
[ ]{}\[\_weight\_bu\_Remark\_\] Let $L_{{{\Bbb R}}}$ be a real flat line bundle on $M$ with the same automorphy factors as the Kähler form $\omega_K$ (in conformal geometry, it is known as [**the weight bundle**]{}). Any non-degenerate positive section of $L_{{{\Bbb R}}}$ corresponds uniquely to a metric on $M$ conformally equivalent to $\omega_K$, and the converse is also true. The Gauduchon metric gives a rise to a section $\mu_G$ of $L_{{{\Bbb R}}}$. Consider $L:= L_{{{\Bbb R}}}\otimes_{{{\Bbb R}}} {{\Bbb C}}$ as a holomorphic Hermitian line bundle, with a holomorphic structure induced from the flat connection on $L=L_{{{\Bbb R}}}\otimes_{{{\Bbb R}}} {{\Bbb C}}$, and Hermitian structure defined by $|\mu_G|=const$. Denote by $\nabla_C$ the corresponding Chern connection. Then $\omega_0$ is the curvature of $\nabla_C$ ([@_Verbitsky:LCHK_], [@_OV:Immersion_]).
LCK structure on diagonal Hopf manifolds {#_LCK_Hopf_Subsection_}
----------------------------------------
The main examples of LCK and Vaisman geometries are provided by the theory of Hopf manifolds.
[ ]{}Let $A\in GL(n)$ be a linear transform, acting on ${{\Bbb C}}^n$ with all eigenvalues satisfying $|\alpha_i|<1$. Denote by $\langle A \rangle\subset GL(n, {{\Bbb C}})$ the cyclic group generated by $A$. The quotient $({{\Bbb C}}^n\backslash 0)/\langle A \rangle$ is called [**a linear Hopf manifold**]{}. If $A$ is diagonalizable, $({{\Bbb C}}^n\backslash 0)/\langle A \rangle$ is called [**a diagonal Hopf manifold**]{}.
[ ]{}If one takes an arbitrary holomorphic contraction $A$ instead of a linear contraction, one obtains the general definition of a Hopf manifold (see e.g. [@_Kato1_], [@_Kato2_] for details).
[ ]{}Izu Vaisman, who introduced the subject and studied the Vaisman manifolds at great length (see [@_Vaisman:Dedicata_], [@_Vaisman:Torino_]), called them the generalized Hopf manifolds. This name is not suitable because many Hopf manifolds are not Vaisman. For linear Hopf manifolds, $({{\Bbb C}}^n\backslash 0)/\langle A\rangle$ is Vaisman if and only if $A$ is diagonalizable (see [@_OV:Potential_]).
Let $A\in GL(n, {{\Bbb C}})$ be a diagonal linear transform: $$\begin{bmatrix} \alpha_1 & 0 & \dots & 0\\
0 & \alpha_2 & \dots & 0\\
\vdots & \vdots & \ddots & \vdots\\
0 & 0 & \dots & \alpha_n,
\end{bmatrix}, \hfill \hfill |\alpha_i| <1$$ Consider the Kähler metric $\omega_K:= -\1{\partial}{\overline}{\partial}{\varphi}$ on ${{\Bbb C}}^n \backslash 0$, defined using the Kähler potential ${\varphi}:\; {{\Bbb C}}^n \backslash 0{{\:\longrightarrow\:}}{{\Bbb R}}$. The ${\varphi}$ is defined via the formula $$\label{_potential_phi_formula_Equation_}
{\varphi}(t_1, ... , t_n) = \sum |t_i|^{\beta_i},$$ where $\beta_i:= \log_{|\alpha_i|^{-1}} C$ are positive real numbers which satisfy $|\alpha_i|^{-\beta_i}= C$ for some fixed real constant $C>1$, chosen in such a way that all $\beta_i$ satisfy $|\beta_i| {\geqslant}2$, and $t_i$ are complex coordinates. By construction, $A^*{\varphi}= C^{-1} {\varphi}$. Indeed, $$A^*{\varphi}(t_1, ..., t_n) = {\varphi}(A(t_1, ..., t_n))=
\sum |\alpha_i t_i|^{\beta_i}= C^{-1}{\varphi}(t_1, ..., t_n).$$ Therefore, $\omega_K:= -\1{\partial}{\overline}{\partial}{\varphi}$ is a Kähler form which satisfies $A^* \omega_K =C^{-1}\omega_K$. This implies that the diagonal Hopf manifold $({{\Bbb C}}^n\backslash 0)/\langle A \rangle$ is LCK. To see that it is Vaisman, we notice that the holomorphic vector field $\log A$ acts on $({{\Bbb C}}^n\backslash 0, \omega_K)$ conformally and apply the Kamishima-Ornea theorem (\[\_Kamishima\_Ornea\_Remark\_\]).
We proceed with computing the Lee field for the Gauduchon metric on $({{\Bbb C}}^n\backslash 0)/\langle A \rangle$, equipped with a conformal structure defined by the Kähler form described above.
Consider the action of the complex Lie group $V(t)= e^{{{\Bbb C}}v}$ generated by the holomorphic vector field $v:= \sum_i -t_i\log |\alpha_i| \frac{d}{dt_i}$. By construction, $V(\lambda)$ is a linear operator which can be written as $\sum e^{|\alpha_i|\lambda} t_i$. For $\lambda$ real, this operator multiplies ${\varphi}$ by a constant $C^\lambda$ (this is proven in the same was as one proves that $A({\varphi})=C^{-1}{\varphi}$), and for $\lambda$ purely imaginary, $V(\lambda)$ preserves ${\varphi}$ (this is clear). Therefore, $v^c:= I(v)$ acts on $({{\Bbb C}}^n\backslash 0, \omega_K)$ by holomorphic isometries.
The corresponding moment map $\mu:\; {{\Bbb C}}^n\backslash 0{{\:\longrightarrow\:}}{{\Bbb R}}$ is given by $d\mu = \omega_K(v^c, \cdot)$. The latter differential form is written as $$\label{_omega_K(v^c)_Equation_}
(d d^c {\varphi}){\hspace{2pt}\raisebox{1pt}{\text{$\lrcorner$}}\hspace{2pt}}v^c =
{\operatorname{Lie}}_{v^c} d^c {\varphi}- d(d^c{\varphi}{\hspace{2pt}\raisebox{1pt}{\text{$\lrcorner$}}\hspace{2pt}}v^c).$$ The first term of the right hand side of vanishes because $v^c$ acts on $({{\Bbb C}}^n\backslash 0)$ preserving ${\varphi}$ and a complex structure. This gives $$\omega_K(v^c, \cdot)=(d d^c {\varphi}){\hspace{2pt}\raisebox{1pt}{\text{$\lrcorner$}}\hspace{2pt}}v^c = -d(d^c{\varphi}{\hspace{2pt}\raisebox{1pt}{\text{$\lrcorner$}}\hspace{2pt}}v^c)
= d(d{\varphi}{\hspace{2pt}\raisebox{1pt}{\text{$\lrcorner$}}\hspace{2pt}}v)= \log C \cdot d{\varphi}$$ (the last equation holds because $d{\varphi}{\hspace{2pt}\raisebox{1pt}{\text{$\lrcorner$}}\hspace{2pt}}v= {\operatorname{Lie}}_v {\varphi}= \log C\cdot {\varphi}$). We obtained that $\log(C){\varphi}$ is the moment map for $V(t)$ acting on $({{\Bbb C}}^n\backslash 0, \omega_K)$.
We obtained the following claim, which is well known in many similar situations.
[ ]{}\[\_moment\_map\_Claim\_\] Let $A\subset GL(n)$ be a diagonal contraction of ${{\Bbb C}}^n$, with all eigenvalues $\alpha_i$ satisfying $|\alpha_i| <1$. Consider a Kähler metric $\omega_K:= -\1 {\partial}{\overline}{\partial}{\varphi}$ on ${{\Bbb C}}^n\backslash 0$, where the Kähler potential ${\varphi}$ is defined by the formula , and let $V(t)= e^{{{\Bbb C}}v}$ be the holomorphic flow generated by $v:=\sum_i -t_i\log |\alpha_i| \frac{d}{dt_i}$. Let $v^c:= I(v)$ be the complex adjoint of $v$. Then $e^{{{\Bbb R}}v^c}\subset V(t)$, preserves the Kähler structure on ${{\Bbb C}}^n\backslash 0$, and the corresponding moment map is $\log (C){\varphi}$: $$d(\log (C) {\varphi}) = \omega_k(v^c, \cdot).$$
Consider the Hermitian form $\omega_H = \frac{\omega_K}{\varphi}$ on $M = ({{\Bbb C}}^n\backslash 0)/\langle A \rangle$. The corresponding Lee form $\theta$ is obtained via $$d\omega_H = - \frac{\omega_K}{{\varphi}^2}= - \omega_H \wedge \log d{\varphi},$$ hence $\theta= \frac{d{\varphi}}{{\varphi}}$. The dual under $\omega_H$ vector field (Lee field) is given by $\theta^{\sharp}= v$, where $v=\log (C)\sum_i -t_i\log |\alpha_i| \frac{d}{dt_i}$. This is clear because $v$ is dual to $\log (C)d{\varphi}$ with respect to $\omega_K$ as \[\_moment\_map\_Claim\_\] implies, and $\omega_H = \frac{\omega_K}{\varphi}$.
This gives the following Proposition.
[ ]{}\[\_Lee\_field\_on\_Hopf\_Proposition\_\] In assumptions of \[\_moment\_map\_Claim\_\], consider the Hermitian form $\omega_H = \frac{\omega_K}{\varphi}$ on $M = ({{\Bbb C}}^n\backslash 0)/\langle A \rangle$. Then the corresponding Lee field is given as $$\label{_theta_sharp_via_alpha_i_Equation_}
\theta^{\sharp}=\log (C)\sum_i -t_i\log |\alpha_i| \frac{d}{dt_i}.$$ Moreover, $\omega_H$ is Gauduchon.
[**Proof:**]{} The equation is proven above. To see that $\omega_H$ is Gauduchon, it suffices to see that $|\theta^\sharp|_{\omega_H}$ is constant. Indeed, from the definition of $d^*$ it follows easily that $$d^*\theta = \nabla_{\theta^\sharp}\theta^\sharp.$$ However, $\theta^\sharp$ is Killing, because ${\operatorname{Lie}}_{\theta^\sharp} {\varphi}= C {\varphi}$, ${\operatorname{Lie}}_{\theta^\sharp} \omega_K = C \omega_K$, and therefore $${\operatorname{Lie}}_{\theta^\sharp} \omega_H = C \omega_H - C \omega_H=0.$$ By another definition of Killing fields, this means that $$(\nabla_X\theta^\sharp, Y)_{\omega_H} =
-(\nabla_Y\theta^\sharp, X)_{\omega_H}$$ for all vector fields $X, Y$. Taking $Y = \theta^\sharp$, and applying ${\operatorname{Lie}}_X(\theta^\sharp, \theta^\sharp)_{\omega_H}=0$, we obtain $$0 = (\nabla_X\theta^\sharp, \theta^\sharp)_{\omega_H}=
-(\nabla_{\theta^\sharp}\theta^\sharp, X)_{\omega_H}.$$ As $X$ is arbitrary, this implies $\nabla_{\theta^\sharp}\theta^\sharp=0$. Therefore, \[\_Lee\_field\_on\_Hopf\_Proposition\_\] is implied by the equation $\omega_H(\theta^\sharp, {\overline}\theta^\sharp)=const$, or, equivalently, $$\label{_omega_K_theta_sharp_const_phi_Equation_}
\omega_K(\theta^\sharp, {\overline}\theta^\sharp)=const\cdot {\varphi}.$$ Writing $\omega_K$ as $$\omega_K = -\1 {\partial}{\overline}{\partial}{\varphi}=
\sum_i dt_i \wedge d {\overline}t_i |t_i|^{\beta_i-2} \frac{\beta_i^2}{4},$$ and using $\theta^\sharp = \log (C)\sum_i -t_i\log |\alpha_i| \frac{d}{dt_i}$, we obtain $$\label{omega_k_of_theta_explic_Equation_}
\omega_K(\theta^\sharp, {\overline}\theta^\sharp) =
\log (C)^2\sum_i (\log |\alpha_i|)^2 |t_i|^{\beta_i}\frac{\beta_i^2}{4}.$$ By definition, $e^{-\log |\alpha_i| \beta_i}=C$, in other words, $\beta_i = -\frac{\log C}{\log \alpha_i}$. Plugging this into , we obtain $$\omega_K(\theta^\sharp, {\overline}\theta^\sharp)=
\sum_i |t_i|^{\beta_i}\frac{(\log C)^4}{4} = \frac {(\log C)^4} 4 {\varphi}.$$ This proves \[\_Lee\_field\_on\_Hopf\_Proposition\_\].
Stable bundles on Hermitian manifolds {#_stable_Kobaya_Section_}
=====================================
Gauduchon metrics and stability
-------------------------------
[ ]{}Let $M$ be a compact complex Hermitian manifold. Choose a Gauduchon metric in the same conformal class.[^3] Consider a torsion-free coherent sheaf $F$ on $M$. Denote by $\det F$ its determinant bundle. Pick a Hermitian metric $\nu$ on $\det F$, and let $\Theta$ be the curvature of the associated Chern connection. We define the degree of $F$ as follows: $$\deg F := \int_M \Theta \wedge \omega^{\dim_{{\Bbb C}}M-1},$$ where $\omega\in \Lambda^{1,1}(M)$ is the Hermitian form of the Gauduchon metric. This notion is independent from the choice of the Hermitian structure $\nu$ in $F$. Indeed, if $\nu' = e^\psi \nu$, $\psi \in C^\infty(M)$, then the associated curvature form is written as $\Theta' = \Theta + {\partial}{\overline}{\partial}\psi$, and $$\int_M {\partial}{\overline}{\partial}\psi\wedge \omega^{\dim_{{\Bbb C}}M-1}=0$$ because $\omega$ is Gauduchon.
If $F$ is a Hermitian vector bundle, $\Theta_F$ its curvature, and the metric $\nu$ is induced from $F$, then $\Theta= {\operatorname{Tr}}_F\Theta_F$. In Kähler case this allows one to relate the degree of a bundle with the first Chern class. However, in non-Kähler case, the degree is not a topological invariant — it depends fundamentally on the holomorphic geometry of $F$. Moreover, the degree is not discrete, as in the Kähler situation, but takes values in continuum.
Further on, we shall see that one can in some cases construct a holomorphic structure of any given degree $\lambda\in {{\Bbb R}}$ on a fixed $C^\infty$-bundle. In our examples, such holomorphic structures are constructed on a topologically trivial line bundle over a Vaisman manifold (\[\_arbi\_degree\_Remark\_\]).
[ ]{}Let $F$ be a non-zero torsion-free coherent sheaf on $M$. Then ${\operatorname{slope}}(F)$ is defined as $${\operatorname{slope}}(F) := \frac{\deg F}{{\operatorname{rk}}F}.$$ The sheaf $F$ is called\
-------------------- ---------------------------------------------------------------------------------------------------------------
[**stable**]{} if for all subsheaves $F'\subset F$, we have ${\operatorname{slope}}(F')< {\operatorname{slope}}(F)$
[**semistable**]{} if for all subsheaves $F'\subset F$, we have ${\operatorname{slope}}(F'){\leqslant}{\operatorname{slope}}(F)$
[**polystable**]{} if $F$ can be represented as a direct sum of stable
coherent sheaves with the same slope.
-------------------- ---------------------------------------------------------------------------------------------------------------
[ ]{}This definition is stability is “good” as most standard properties of stable and semistable bundles hold in this situation as well. In particular, all line bundles are stable; all stable sheaves are simple; the Jordan-Hölder and Harder-Narasimhan filtrations are well defined and behave in the same way as they do in the usual Kähler situation ([@_Lubke_Teleman:Book_], [@_Bruasse:Harder_Nara_]).
However, not all bundles are [**filtrable**]{}, that is, are obtained as successive extensions by coherent sheaves of rank 1. There are non-filtrable holomorphic vector bundles on most non-algebraic K3 surfaces.
Kobayashi-Hitchin correspondence {#_Koba_Hi_Subsection_}
--------------------------------
The statement of Kobayashi-Hitchin correspondence (Donaldson-Uhlenbeck-Yau theorem) is translated to the Hermitian situation verbatim, following Li and Yau ([@_Li_Yau_]).
[ ]{}Let $B$ be a holomorphic Hermitian vector bundle on a Hermitian manifold $M$, and $\Theta\in \Lambda^{1,1}(M)\otimes {\operatorname{End}}(B)$ the curvature of its Chern connection $\nabla$. Consider the operator $\Lambda:\; \Lambda^{1,1}(M)\otimes {\operatorname{End}}(B){{\:\longrightarrow\:}}{\operatorname{End}}(B)$ which is a Hermitian adjoint to $b {{\:\longrightarrow\:}}\omega\otimes b$, $\omega$ being the Hermitian form on $M$. The connection $\nabla$ is called [**Hermitian-Einstein**]{} (or [**Yang-Mills**]{}) if $\Lambda \Theta = {{\it const}}\cdot {\operatorname{Id}}_B$.
[ ]{} (Kobayashi-Hitchin correspondence) Let $B$ be a holomorphic vector bundle on a compact complex manifold equipped with a Gauduchon metric. Then $B$ admits a Hermitian-Einstein connection $\nabla$ if and only if $B$ is polystable. Moreover, the Hermitian-Einstein connection is unique.
[**Proof:**]{} See [@_Li_Yau_], [@_Lubke_Teleman:Book_], [@_Lubke_Teleman:Universal_].
Stable bundles on Vaisman manifolds
===================================
Existence of the positive exact (1,1)-form $\omega_0$, defined in \[\_omega\_0\_in\_Vaisman\_Lemma\_\], brings many consequences for algebraic geometry of the Vaisman manifolds (see e.g. [@_Verbitsky:LCHK_] and [@_OV:Immersion_]). One of these is the structure theorem for Hermitian-Einstein bundles of degree 0.
The following result was stated and proven as Theorem 4.3, [@_Verbitsky:Sta_Elli_] for positive principal elliptic fibrations, which admit a similar structure. These manifolds are not always Vaisman (e.g. Calabi-Eckmann manifolds are not Vaisman). However, the proof of this theorem can be repeated almost verbatim in the Vaisman situation.
[ ]{}\[\_Hermi\_Einste\_curva\_equi\_Theorem\_\] Let $M$ be a compact Vaisman manifold, $\dim_{{\Bbb C}}M >2$, and $B$ a stable bundle of degree 0 on $M$. Denote by $\Sigma$ the 1-dimensional complex holomorphic foliation generated by the Lee field $\theta^\sharp$. Then $\Theta(v, \cdot)=0$ for any $v\in \Sigma$. In particular, $B$ is equivariant with respect to the complex Lie group $V(t)$ generated by $\theta^\sharp$, and this equivariant structure is compatible with the connection.
[**Proof:**]{} Consider the map $$\Lambda:\; \Lambda^{1,1}(M, {\operatorname{End}}(B)) {{\:\longrightarrow\:}}{\operatorname{End}}(B)$$ defined in Subsection \[\_Koba\_Hi\_Subsection\_\]. By definition, $\Theta$ is [**primitive**]{}, that is, satisfies $\Lambda\Theta=0$. Then \[\_Hermi\_Einste\_curva\_equi\_Theorem\_\] is implied by the following proposition.
[ ]{}\[\_primi\_form\_equi\_Proposition\_\] Let $M$ be a compact Vaisman manifold, $\dim_{{\Bbb C}}M >2$, $B$ a Hermitian bundle with connection, and $\Theta \in \Lambda^{1,1}(M, {{\Bbb R}}) \otimes_{{\Bbb R}}{{\frak}u}(B)$ a closed skew-Hermitian real (1,1)-form. Assume that $\Theta$ is primitive, that is, $\Lambda\Theta=0$. Then $\Theta(v, \cdot) =0$ for any $v\in \Sigma$.
[**Proof:**]{} Rescaling the metric, we normalize the Lee form $\theta$ so that $|\theta|=1$. Let $\theta$, $\theta_1, ... , \theta_{n-1}$ be an orthonormal basis in $\Lambda^{1,0}(M)$, with $\theta\in \Sigma$, $\theta_i\in \Sigma^\bot$. Consider the form $\omega_0$ (\[\_omega\_0\_in\_Vaisman\_Lemma\_\]). This form is exact, positive, and has $n-1$ strictly positive eigenvalues. Using the basis described above, we can write $$\label{_omega,_omega_0_explicit_Equation_}
\omega_H= -\1
\left(\theta\wedge{\overline}\theta +\sum_{i}\theta_i \wedge {\overline}\theta_i \right),
\ \ \ \omega_0 = -\1 \left (\sum_{i}\theta_i \wedge {\overline}\theta_i\right )$$ where $\omega_H$ is the Hermitian form of $M$ (see [@_Verbitsky:LCHK_], Proposition 6.1).
In this basis, we can write $\Theta$ as $$\begin{aligned}
\label{_Theta_basis_Equation_}
\Theta &= \sum_{i\neq j}(\theta_i \wedge {\overline}\theta_j
+ {\overline}\theta_i \wedge {\overline}\theta_j) \otimes b_{ij} +
\sum_{i}(\theta_i \wedge {\overline}\theta_i) \otimes a_i \\
& + \sum_{i}(\theta \wedge {\overline}\theta_i
+ {\overline}\theta \wedge {\overline}\theta_i) \otimes b_{i}
+ \theta\wedge{\overline}\theta\otimes a,\end{aligned}$$ with $b_{ij}$, $b_i$, $a_i$, $a\in {{\frak}u}(B)$ being skew-Hermitian endomorphisms of $B$.
Let $\Xi:= {\operatorname{Tr}}(\Theta\wedge \Theta)$. This is a closed (2,2)-form on $M$. Then implies $$(\1)^n\Xi \wedge \omega_0^{n-2} = {\operatorname{Tr}}\left(-\sum b_i^2 +a \left(\sum a_i\right)
\right)$$ On the other hand, $ \sum a_i + a = \Lambda\Theta=0$, hence $$(\1)^n\Xi \wedge \omega_0^{n-2} = {\operatorname{Tr}}\left(-\sum b_i^2 -a^2 \right).$$ Since $u {{\:\longrightarrow\:}}{\operatorname{Tr}}(-u^2)$ is a positive definite form on ${{\frak}u}(B)$, the integral $$\label{_integral_Xi_Equation_}
\int_M (\1)^n\Xi \wedge \omega_0^{n-2}$$ is non-negative, and positive unless $b_i$ and $a$ both vanish everywhere. Using $n>2$, we find that vanishes, because $\omega_0$ is exact and $\Xi$ is closed. Therefore, $b_i$ and $a$ are identically zero, which is exactly the claim of \[\_primi\_form\_equi\_Proposition\_\]. We proved \[\_Hermi\_Einste\_curva\_equi\_Theorem\_\].
[ ]{}\[\_arbi\_degree\_Remark\_\] The results of \[\_Hermi\_Einste\_curva\_equi\_Theorem\_\] can be applied to arbitrary stable bundle on $M$ using the following trick. Consider the line bundle $L$ (\[\_weight\_bu\_Remark\_\]). Write the Chern connection on $L$ as $$\nabla_C = \nabla_{triv} -\1\theta^c,$$ where $\theta^c= I(\theta)$ is the complex conjugate of $\theta$ (see [@_Verbitsky:LCHK_], (6.11)), and $\nabla_{triv}$ is a trivial connection associated to the trivialization of $L$ constructed in \[\_weight\_bu\_Remark\_\]. Since $d\theta^c = \omega_0$, $L$ has a degree $\delta:= \int \omega_0 \wedge \omega_H^{n-1}$ which is clearly positive (see ). Given $\lambda\in {{\Bbb R}}$, denote by $L_\lambda$ a holomorphic Hermitian bundle with the connection $\nabla_{triv} -\1\frac{\lambda}{\delta}\theta^c$. Then $L_\lambda$ has degree $\lambda$. We obtain that a Vaisman manifold admits a line bundle $L_\lambda$ of arbitrary degree $\lambda$. Moreover, $L_\lambda$ is by construction $V(t)$-equivariant (the form $\theta^c$ is $V(t)$-invariant, as $V(t)$ acts on $M$ preserving the metric and the holomorphic structure). This brings the following corollary.
[ ]{}\[\_all\_stable\_equi\_Corollary\_\] Let $M$ be a compact Vaisman manifold, and $B$ a stable bundle. Consider a complex holomorphic flow $V(t)= e^{t\theta^\sharp}$ generated by the Lee field $\theta^\sharp$. Then $B$ admits a natural $V(t)$-equivariant structure.
[**Proof:**]{} Tensoring $B$ by $L_{\lambda}$ for appropriate choice of $\lambda\in {{\Bbb R}}$, we obtain a stable bundle of degree 0. Then \[\_Hermi\_Einste\_curva\_equi\_Theorem\_\] implies \[\_all\_stable\_equi\_Corollary\_\].
Stable bundles on Hopf manifolds and coherent sheaves on ${{\Bbb C}}^n$
=======================================================================
Admissible Hermitian structures on reflexive sheaves
----------------------------------------------------
[ ]{}\[\_refle\_Definition\_\] Let $X$ be a complex manifold, and $F$ a coherent sheaf on $X$. Consider the sheaf $F^*:= {\cal H}om_{{{\cal O}}_X}(F, {{\cal O}}_X)$. There is a natural functorial map $\rho_F:\; F {{\:\longrightarrow\:}}F^{**}$. The sheaf $F^{**}$ is called [**a reflexive hull**]{}, or [ **reflexization**]{}, of $F$. The sheaf $F$ is called [**reflexive**]{} if the map $\rho_F:\; F {{\:\longrightarrow\:}}F^{**}$ is an isomorphism.
[ ]{}For all coherent sheaves $F$, the map $\rho_{F^*}:\; F^* {{\:\longrightarrow\:}}F^{***}$ is an isomorphism ([@_OSS_], Ch. II, the proof of Lemma 1.1.12). Therefore, a reflexive hull of a sheaf is always reflexive.
Reflexive hull can be obtained by restricting to an open subset and taking the pushforward.
[ ]{}\[\_refle\_pushfor\_Lemma\_\] Let $X$ be a complex manifold, $F$ a coherent sheaf on $X$, $Z$ a closed analytic subvariety, ${\operatorname{codim}}Z{\geqslant}2$, and $j:\; (X\backslash Z) \hookrightarrow X$ the natural embedding. Assume that the pullback $j^* F$ is reflexive on $(X\backslash Z)$. Then the pushforward $j_* j^* F$ is also reflexive.
[**Proof:**]{} This is [@_OSS_], Ch. II, Lemma 1.1.12.
[ ]{}\[\_refle\_pushfor\_Remark\_\] From \[\_refle\_pushfor\_Lemma\_\], it is apparent that one could obtain a reflexization of a non-singular in codimension 1 coherent sheaf $F$ by taking $j_* j^* F$, where $j:\; (X\backslash Z) \hookrightarrow X$ the natural open embedding, and $Z$ the singular locus of $F$.
Using the results of [@_Bando_Siu_], we are able to apply the Kobayashi-Hitchin correspondence to reflexive sheaves.
[ ]{}\[\_admissi\_Definition\_\] [@_Bando_Siu_] Let $F$ be a coherent sheaf on $M$ and $\nabla$ a Hermitian connection on $F$ defined outside of its singularities. Denote by $\Theta$ the curvature of $\nabla$. Then $\nabla$ is called [**admissible**]{} if the following holds
(i)
: $\Lambda \Theta\in {\operatorname{End}}(F)$ is uniformly bounded
(ii)
: $|\Theta|^2$ is integrable on $M$.
[ ]{}\[\_B\_S\_exie\_admissi\_Theorem\_\] [@_Bando_Siu_] Any torsion-free coherent sheaf admits an admissible connection. An admissible connection can be extended over the place where $F$ is smooth. Moreover, if a bundle $B$ on $M\backslash Z$, ${\operatorname{codim}}_{{\Bbb C}}Z{\geqslant}2$ is equipped with an admissible connection, then $B$ can be extended to a coherent sheaf on $M$.
A version of Donaldson-Uhlenbeck-Yau theorem exists for coherent sheaves (\[\_UY\_for\_shea\_Theorem\_\]); given a torsion-free coherent sheaf $F$, $F$ admits an admissible Hermitian-Einstein connection $\nabla$ if and only if $F$ is polystable.
[ ]{}\[\_UY\_for\_shea\_Theorem\_\] Let $M$ be a compact Kähler manifold, and $F$ a coherent sheaf without torsion. Then $F$ admits an admissible Hermitian-Einstein metric is and only if $F$ is polystable. Moreover, if $F$ is stable, then this metric is unique, up to a constant multiplier.
[**Proof:**]{} [@_Bando_Siu_], Theorem 3.
This proof can be adapted for Hermitian complex manifolds with Gauduchon metric.
Hermitian-Einstein bundles on Hopf manifolds and admissibility
--------------------------------------------------------------
[ ]{}\[\_from\_stable\_to\_refle\_Theorem\_\] Let $M= ({{\Bbb C}}^n\backslash 0)/\langle A\rangle$ be a diagonal Hopf manifold, $n{\geqslant}3$, and $B$ a stable holomorphic bundle on $M$ of degree 0. Denote by $\tilde B$ the pullback of $B$ to ${{\Bbb C}}^n \backslash 0$. Then $\tilde B$ can be extended to a reflexive coherent sheaf $F$ on ${{\Bbb C}}^n$. Moreover, $F$ is $V(t)$-equivariant, where $V(t)$, is the complex holomorphic flow on ${{\Bbb C}}^n$ generated by the Lee field $\theta^\sharp=\log (C)\sum_i -t_i\log |\alpha_i| \frac{d}{dt_i}$.
[**Proof:**]{} Consider a Hermitian-Einstein metric on $B$, and lift it to $\tilde B$. Denote by $\tilde \Theta$ the curvature of $\tilde B$. To extend $\tilde B$ to ${{\Bbb C}}^n$, we apply the Bando-Siu theorem (\[\_B\_S\_exie\_admissi\_Theorem\_\]). We need to show that $\tilde B$ is admissible, in the sense of \[\_admissi\_Definition\_\]. The Kähler metric $\omega_K$ on ${{\Bbb C}}^n$ is conformally equivalent to that lifted from $M$, hence $\Lambda \tilde \Theta=0$ (this condition means that $\tilde \Theta$ is orthogonal to the Hermitian form pointwise, and therefore it is conformally invariant). To prove that $\tilde B$ is admissible, it remains to show that $\tilde \Theta$ is square-integrable. The function $|\tilde \Theta|^2$ can be expressed, using the Hodge-Riemann relations, as follows.
[ ]{}\[\_tilde\_Theta\_Lemma\_\] Let $B_1$ be a Hermitian bundle on a Hermitian almost complex manifold $M_1$, of dimension $n$, and $$\nu \in \Lambda^{1,1}(M_1, {\frak}{su}(B_1))$$ a ${\frak}{su}(B_1)$-valued (1,1)-form satisfying $\Lambda(\nu)=0$. Then $$\label{_square_form_via_H_E_Equation_}
|\nu|^2 = -\1\frac{n-1}{2n} {\operatorname{Tr}}(\Lambda^2 (\nu \wedge \nu)),$$ where $$\Lambda:\; \Lambda^{p,q}(M_1, {\frak}{su}(B_1))
{{\:\longrightarrow\:}}\Lambda^{p-1,q-1}(M_1, {\frak}{su}(B_1))$$ is the standard Hodge operator on differential forms.
[**Proof:**]{} An elementary calculation, and essentally the same as one which proves the Hodge-Riemann bilinear relations (see e.g. [@_Bando_Siu_]).
[ ]{} The equation can be stated as $$\label{_square_form_volume_via_H_E_Equation_}
|\nu|^2{\operatorname{Vol}}(M_1) =
-\1\frac{n-1}{2n\cdot 2^n\cdot n!} {\operatorname{Tr}}(\nu \wedge \nu)\wedge \omega_1^{n-2},$$ where $\omega$ is the Hermitian form on $M_1$, and ${\operatorname{Vol}}(M_1)$ the Riemannian volume. This is clear from the definition of $\Lambda$ and the relation ${\operatorname{Vol}}(M_1) = \frac1 {2^n n!} \omega_1^n$.
Using , we obtain that $L^2$-integrability of $\tilde \Theta$ is equivalent to integrability of the form $$\label{_square_curv_Equation_}
{\operatorname{Tr}}(\tilde \Theta\wedge \tilde \Theta)\wedge \omega_K^{n-2}.$$
The form $\tilde \Theta$ is by construction $A$-invariant, and $\omega_K$ satisfies $A^*(\omega_K) = c \omega_K$ because $M$ is LCK. Therefore, the form is homogeneous with respect to the action of $A$: $$\label{_curv_volu_homoge_Equation_}
A^* \left({\operatorname{Tr}}(\tilde \Theta\wedge \tilde \Theta)\wedge \omega_K^{n-2}\right)=
c^{n-2}{\operatorname{Tr}}(\tilde \Theta\wedge \tilde \Theta)\wedge \omega_K^{n-2},
c<1.$$ Denote by $D$ the fundamental domain for $\langle A\rangle$, $$D:= \{ x\in {{\Bbb C}}^n\backslash 0\ | \ 1{\leqslant}\rho(x) <C\}$$ Then ${{\Bbb C}}^n \backslash 0 = \cap_{i\in {{\Bbb Z}}} A^i(D)$. To check that $\tilde \Theta$ is $L^2$-integrable in a neighbourhood of 0, we need to show that the series $$\sum_{i=0}^{\infty} \int_{A^i(D)} |\tilde \Theta|^2 {\operatorname{Vol}}=
-\1\frac{n-1}{2n\cdot 2^n \cdot n!}\sum_{i=0}^{\infty} \int_{A^i(D)}
{\operatorname{Tr}}(\tilde \Theta\wedge \tilde \Theta)\wedge \omega_K^{n-2}$$ converges. However, by homogeneity, the latter integral is power series, and implies that it converges whenever $n>2$. We have shown that $\tilde B$ is admissible. Now, Bando-Siu theorem (\[\_B\_S\_exie\_admissi\_Theorem\_\]) implies the first assertion of \[\_from\_stable\_to\_refle\_Theorem\_\]. The second assertion is implied immediately by \[\_refle\_pushfor\_Lemma\_\]. Indeed, let $({{\Bbb C}}^n \backslash 0)\stackrel j \hookrightarrow {{\Bbb C}}^n$ be the standard embedding. Then $F = j_* \tilde B$ (\[\_refle\_pushfor\_Lemma\_\]). By \[\_all\_stable\_equi\_Corollary\_\], $\tilde B$ is $V(t)$-equivariant. Then $j_* \tilde B$ is also $V(t)$-equivariant.
[ ]{}\[\_exte\_equiva\_theorem\_to\_refle\_Remark\_\] Using the Bando-Siu version of Donaldson-Uhlenbeck-Yau theorem, we can extend \[\_from\_stable\_to\_refle\_Theorem\_\] verbatim to reflexive coherent sheaves.
Equivariant sheaves on ${{\Bbb C}}^n$ {#_equi_shea_Section_}
=====================================
Extending $V(t)$-equivariance to $({{\Bbb C}}^*)^l$-equivariance
----------------------------------------------------------------
Let $M= ({{\Bbb C}}^n\backslash 0)/\langle A\rangle$ be a diagonal Hopf manifold, and $V(t)= e^{{{\Bbb C}}\theta^\sharp}$, $t\in {{\Bbb C}}$ the holomorphic flow generated by the Lee field $\theta^\sharp$ as above. Then $V(t)$ acts on $M$ by holomorphic isometries ([@_Kamishima_Ornea_]). Consider the closure $G$ of $V(t)$, $t\in {{\Bbb C}}$, within the group ${\operatorname{Iso}}(M)$ of isometries of $M$. Denote by $\tilde G$ the the lifting of $G$ to ${\operatorname{Aut}}(\tilde M)$ ([@_OV:Structure_], [@_OV:Immersion_]). By construction, $\tilde G$ is the smallest closed Lie subgroup of $GL(n, {{\Bbb C}})$ containing $V(t)$ and $A$. It is easy to check that $\tilde G$ is a reductive complex commutative Lie group. A similar result is true for all Vaisman manifolds.
[ ]{}\[\_tilde\_G\_C\^\*\^l\_Proposition\_\] For any Vaisman manifold $M$, let $\theta^\sharp$ be its Lee field, $G$ the closure of the corresponding complex holomorphic flow within ${\operatorname{Iso}}(M)$, and $\tilde G$ its lift to ${\operatorname{Aut}}(\tilde M)$. Then $\tilde G\cong ({{\Bbb C}}^*)^k$, and the deck transform map $\gamma\in {\operatorname{Aut}}(\tilde M, M)$ lies in $\tilde G$.
[**Proof:**]{} This is [@_OV:Immersion_], Proposition 4.3.
[ ]{}\[\_tilde\_G\_F\_equiv\_shea\_Theorem\_\] In assumptions of \[\_from\_stable\_to\_refle\_Theorem\_\], consider the action of the group $V(t)$ on $\Gamma({{\Bbb C}}^n, F)$. Consider the adic topology on ${{\cal O}}_{{{\Bbb C}}^n}$ and $\Gamma({{\Bbb C}}^n, F)$, with $\lim f_i {{\:\longrightarrow\:}}0$ as $[f_i]_0{{\:\longrightarrow\:}}\infty$, where $[f_i]_0$ denotes the order of zeroes of $f_i$ in $0\in {{\Bbb C}}^n$. Clearly, $V(t)$ is continuous in adic topology. Let $\tilde G_F$ be the closure of $V(t)$-action on $\Gamma({{\Bbb C}}^n, F) \times {{\cal O}}_{{{\Bbb C}}^n}$ in adic topology. Then
(i)
: The natural map $\tilde G_F \stackrel \rho {{\:\longrightarrow\:}}GL(F/{\frak}m F)\times GL({\frak}m/{\frak}m^2)$ is injective, where ${\frak}m$ is the maximal ideal of $0$ in ${{\cal O}}_{{{\Bbb C}}^n}$.
(ii)
: $\tilde G_F$ is a closure of $V(t)$ under the natural map $V(t){{\:\longrightarrow\:}}GL(F/{\frak}m F)\times GL({\frak}m/{\frak}m^2)$.
(iii)
: Consider the natural projection $\tilde G_F\stackrel \pi {{\:\longrightarrow\:}}\tilde G$ induced by $$GL(F/{\frak}m F)\times GL({\frak}m/{\frak}m^2){{\:\longrightarrow\:}}GL({\frak}m/{\frak}m^2).$$ Then $\pi$ satisfies $g(af) = \pi(g)(a)g(f)$, for any $f\in \Gamma({{\Bbb C}}^n, F)$, $a \in {{\cal O}}_{{{\Bbb C}}^n}$, $g\in \tilde G_F$. This gives a $\tilde G_F$-equivariant structure on $F$.
(iv)
: The group $\tilde G_F$ is isomorphic to $({{\Bbb C}}^*)^l$.
[**Proof:**]{} \[\_tilde\_G\_F\_equiv\_shea\_Theorem\_\] (i) is clear from Nakayama’s lemma. \[\_tilde\_G\_F\_equiv\_shea\_Theorem\_\] (ii) is immediately implied by \[\_tilde\_G\_F\_equiv\_shea\_Theorem\_\] (i). \[\_tilde\_G\_F\_equiv\_shea\_Theorem\_\] (iii) follows from \[\_tilde\_G\_F\_equiv\_shea\_Theorem\_\] (ii) and $V(t)$-equivariance of $F$.
To prove \[\_tilde\_G\_F\_equiv\_shea\_Theorem\_\] (iv), we use \[\_tilde\_G\_F\_equiv\_shea\_Theorem\_\] (ii), and notice that $\tilde G_F$ is commutative as a closure of a 1-parametric group within a Lie group $GL(F/{{\frak}m}F)\times GL({\frak}m/{\frak}m^2)$. To show that $\tilde G_F\cong ({{\Bbb C}}^*)^l$, we need to prove that it is reductive, that is, to show that $V(t)$ acts diagonally on $(F/{\frak}m F)\times({\frak}m/{\frak}m^2)$.
The group $V(t)$ acts on $M$ holomorphically and conformally. Since the Hermitian-Einstein metric on $B$ is unique, up to a constant multiplier, the group $V(t)$ acts on $B$ also conformally. Then, $V(t)$ acts conformally on the Hermitian space $\Gamma(B_{{{\Bbb C}}^n}, F)$ of holomorphic sections of $F$ on an open ball $B_{{{\Bbb C}}^n}\subset{{\Bbb C}}^n$ and on $\Gamma(B_{{{\Bbb C}}^n}, {{\cal O}}_{{{\Bbb C}}^n})$. Since orthogonal matrices in finite dimension are diagonalizable, $V(t)$ acts diagonally on any finite-dimensional subspace in $\Gamma(B_{{{\Bbb C}}^n}, F)\times \Gamma(B_{{{\Bbb C}}^n}, {{\cal O}}_{{{\Bbb C}}^n})$ preserved by $V(t)$. Using the same classical Poincare-Dulac argument as used in the proof of Theorem 3.3 in [@_OV:Potential_], we find that $\Gamma(B_{{{\Bbb C}}^n}, F)\times \Gamma(B_{{{\Bbb C}}^n}, {{\cal O}}_{{{\Bbb C}}^n})$ contains a dense (in appropriate, e.g. ${\frak}m$-adic topology) subspace which is generated by finite-dimensional $V(t)$-invariant subspaces. Then $V(t)$-action on the space $\Gamma(B_{{{\Bbb C}}^n}, F)\times \Gamma(B_{{{\Bbb C}}^n}, {{\cal O}}_{{{\Bbb C}}^n})$ is diagonal in a dense subspace. Therefore, this action is diagonal on its quotient $(F/{\frak}m F)\times({\frak}m/{\frak}m^2)$. We proved \[\_tilde\_G\_F\_equiv\_shea\_Theorem\_\] (iv).
[ ]{}\[\_gene\_equi\_to\_shea\_Remark\_\] Using the Bando-Siu version of Donaldson-Uhlenbeck-Yau theorem (see \[\_exte\_equiva\_theorem\_to\_refle\_Remark\_\]), we can extend \[\_tilde\_G\_F\_equiv\_shea\_Theorem\_\] verbatim to reflexive coherent sheaves.
[ ]{}\[\_equi\_shea\_and\_shea\_on\_Hopf\_Remark\_\] Denote by ${{\Bbb C}}^n_*$ the complex manifold ${{\Bbb C}}^n\backslash 0$. Given a $\tilde G_F$-equivariant coherent sheaf on ${{\Bbb C}}^n_*$, we can obtain a coherent sheaf on ${{\Bbb C}}^n_*/\langle A\rangle$. Indeed, coherent sheaves on ${{\Bbb C}}^n_* /\langle A\rangle$ are the same as $\langle A\rangle$-equivariant sheaves on ${{\Bbb C}}^n_*$, and $\langle A\rangle$ lies in $\tilde G$ as \[\_tilde\_G\_C\^\*\^l\_Proposition\_\] implies. Therefore, to prove the filtrability of a stable bundle $B$ on $M = ({{\Bbb C}}^n_*) /\langle A\rangle,$ it suffices to show that the corresponding $\tilde G_F$-equivariant coherent sheaf $F$ is filtrable on ${{\Bbb C}}^n_*$ in the category ${\operatorname{Coh}}_{\tilde G_F}({{\Bbb C}}^n_*)$ of $\tilde G_F$-equivariant coherent sheaves. Then, the following theorem proves \[\_stable\_filtra\_main\_Theorem\_\].
[ ]{}\[\_Coh\_G\_F\_filtra\_Theorem\_\] Let $\tilde G_F\cong ({{\Bbb C}}^*)^l$ be a commutative Lie group, acting on ${{\Bbb C}}^n_*$ via a homomorphism $\tilde G_F\stackrel \pi {{\:\longrightarrow\:}}GL({{\Bbb C}}, n)$, and ${\operatorname{Coh}}_{\tilde G_F}({{\Bbb C}}^n_*)$ be the category of $\tilde G_F$-equivariant coherent sheaves on ${{\Bbb C}}^n_*$. Assume that $\pi(\tilde G_F)$ contains an endomorphism with all eigenvalues $<1$. Then all objects of ${\operatorname{Coh}}_{\tilde G_F}({{\Bbb C}}^n_*)$ are filtrable by $\tilde G_F$-equivariant coherent sheaves of rank at most 1.
We prove \[\_Coh\_G\_F\_filtra\_Theorem\_\] in Subsection \[\_C\^\*\^l\_equiv\_Subsection\_\].
$({{\Bbb C}}^*)^l$-equivariant coherent sheaves on ${{\Bbb C}}^n\backslash 0$ {#_C^*^l_equiv_Subsection_}
-----------------------------------------------------------------------------
We work in assumptions of \[\_Coh\_G\_F\_filtra\_Theorem\_\].
[ ]{}\[\_R\_gene\_by\_fini\_G\_F\_inv\_Lemma\_\] Let $R\in {\operatorname{Coh}}_{\tilde G_F}({{\Bbb C}}^n_*)$ be a $\tilde G_F$-equivariant coherent sheaf over ${{\Bbb C}}^n_*:= {{\Bbb C}}^n\backslash 0$. Then $R$ is generated over ${{\cal O}}_{{{\Bbb C}}^n_*}$ by a finite-dimensional $\tilde G_F$-invariant space $V\subset \Gamma(R, {{\Bbb C}}^n_*)$.
[**Proof:**]{} The images of ${{\Bbb C}}^*$ are dense in $\tilde G_F\cong ({{\Bbb C}}^*)^l$. Therefore, there exists an embedding ${{\Bbb C}}^*\stackrel \mu \hookrightarrow \tilde G_F$ acting on ${{\Bbb C}}^n$ with all eigenvalues different from 1. This action can be written as $$t {{\:\longrightarrow\:}}\begin{bmatrix} t^{k_1} & 0 & \dots & 0\\
0 & t^{k_2} & \dots & 0\\
\vdots & \vdots & \ddots & \vdots\\
0 & 0 & \dots & t^{k_n}
\end{bmatrix}$$ where all $k_i$ are integers different from 0. Clearly, $\mu$ acts on ${{\Bbb C}}^n_*$ freely in generic point, and the quotient ${{\Bbb C}}^n_* /\mu ({{\Bbb C}}^*)$ is well defined. This quotient is known as a [**weighted projective space**]{}, denoted by ${{\Bbb C}}P^{n-1}(k_1, k_2, ... k_n)$, and it is a projective orbifold. To give a $\mu$-equivariant coherent sheaf on ${{\Bbb C}}^n_*$ is by definition the same as to give a coherent sheaf on the orbifold ${{\Bbb C}}P^{n-1}(k_1, k_2, ... k_n)$. Let $R_0$ be the sheaf on ${{\Bbb C}}P^{n-1}(k_1, k_2, ... k_n)$ corresponding to $R$, considered as a $\mu$-equivariant sheaf on ${{\Bbb C}}^n_*$. The sections of $R_0\otimes {{\cal O}}(i)$ correspond to the sections of $R$ on which $\mu({{\Bbb C}}^*)$ acts with the weight $i$. We obtain a sequence of finite-dimensional subspaces $$\Gamma(R_0\otimes {{\cal O}}(i))\subset \Gamma(R).$$ Since ${{\cal O}}(1)$ is ample, the sheaf $R_0\otimes {{\cal O}}(N)$ is globally generated for $N$ sufficiently big (here we use the Kodaira-Nakano theorem for orbifolds, [@_Baily_]). Then $\oplus_{i{\leqslant}N} \Gamma(R_0\otimes {{\cal O}}(i))$ will generate $\Gamma(R)$ over ${{\cal O}}_{{{\Bbb C}}^n_*}$. Since $\tilde G_F$ commutes with $\mu({{\Bbb C}}^*)$, the space $\Gamma(R_0\otimes {{\cal O}}(i))\subset \Gamma(R)$ is $\tilde G_F$-invariant. This proves \[\_R\_gene\_by\_fini\_G\_F\_inv\_Lemma\_\].
Now we can prove the filtrability of arbitrary $R\in {\operatorname{Coh}}_{\tilde G_F}({{\Bbb C}}^n_*)$. By \[\_R\_gene\_by\_fini\_G\_F\_inv\_Lemma\_\], for any $R\in {\operatorname{Coh}}_{\tilde G_F}({{\Bbb C}}^n_*)$, there exists a surjective $\tilde G_F$-equivariant map $R_1 {{\:\longrightarrow\:}}R{{\:\longrightarrow\:}}0$, where $R_1 = {{\cal O}}_{{{\Bbb C}}^n_*}\otimes_{{\Bbb C}}W$, and $W$ is a finite-dimensional representation of $\tilde G_F$. Since $\tilde G_F$ is commutative, $W= \oplus W_i$, where $W_i$ are $\tilde G_F$-invariant 1-dimensional subspaces of $W$. This gives an epimorphism $$\oplus ({{\cal O}}_{{{\Bbb C}}^n_*}\otimes W_i) {{\:\longrightarrow\:}}R$$ where all the summands ${{\cal O}}_{{{\Bbb C}}^n_*}\otimes W_i$ are $\tilde G_F$-equivariant line bundles. Then, $R$ is clearly filtrable within ${\operatorname{Coh}}_{\tilde G_F}({{\Bbb C}}^n_*)$. This proves \[\_Coh\_G\_F\_filtra\_Theorem\_\]. \[\_stable\_filtra\_main\_Theorem\_\] is also proven.
[**Acknowledgements:**]{} I am grateful to Ruxandra Moraru who posed the problem, and Dmitry Novikov for a private lecture on normal forms and the Poincare-Dulac theorem.
[XXX]{}
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[Misha Verbitsky\
University of Glasgow, Department of Mathematics,\
15 University Gardens, Glasgow G12 8QW, Scotland.]{}\
\
[Institute of Theoretical and Experimental Physics\
B. Cheremushkinskaya, 25, Moscow, 117259, Russia ]{}\
\
verbit@maths.gla.ac.uk, verbit@mccme.ru
[^1]: Misha Verbitsky is an EPSRC advanced fellow supported by CRDF grant RM1-2354-MO02 and EPSRC grant GR/R77773/01
[^2]: For a definition of stability on Hermitian manifolds, see Section \[\_stable\_Kobaya\_Section\_\].
[^3]: A Hermitian metric on a complex manifold of dimension $n$ is called [**Gauduchon**]{} if ${\partial}{\overline}{\partial}(\omega^{n-1})=0$, where $\omega$ is its Hermitian form (\[\_Gauduchin\_defi\_Definition\_\]). On a compact manifold, a Gauduchon metric exists in any conformal class, and is unique up to a constant multiplier, see [@_Gauduchon_1984_].
|
---
abstract: 'We calculate the one-loop perturbative correction to the coefficient of the term in non-abelian gauge theory in the presence of Higgs fields, with a variety of symmetry-breaking structures. In the case of a residual $U(1)$ symmetry, radiative corrections do not change the coefficient of the term. In the case of an unbroken non-abelian subgroup, the coefficient of the relevant term (suitably normalized) attains an integral correction, as required for consistency of the quantum theory. Interestingly, this coefficient arises purely from the unbroken non-abelian sector in question; the orthogonal sector makes no contribution. This implies that the coefficient of the term is a discontinuous function over the phase diagram of the theory.'
address: 'Laboratoire de Physique Nucléaire, Université de Montréal C.P. 6128, succ. centreville, Montréal, Québec, Canada, H3C 3J7'
author:
- 'Avinash Khare,[^1] R. B. MacKenzie, P. K. Panigrahi[^2] and M. B. Paranjape'
title: 'Spontaneous Symmetry Breaking and the Renormalization of the Chern-Simons Term'
---
\#1, \#2, \#3, 1\#4\#5\#6[ [*\#1 *]{}[**\#2**]{}, \#3 (1\#4\#5\#6)]{}
Yang Mills theories in 2+1 dimensions have attracted much attention in recent years. This is primarily because of the possibility of adding a new topological term to the action, the (CS) term [@1; @desjactem], which has had diverse applications from condensed matter physics to pure mathematics. The possibility of particles obeying arbitrary statistics, anyons [@any], can be elegantly formulated by using a CS term[@csterm]. Anyons are known to play an important role in the fractional quantum Hall effect[@fqhe], and provide a mechanism for superconductivity[@anysup]. The limit where the action is the pure CS term[@purecs] results in a topological field theory[@topol]. The only observables of this theory are Wilson loops, whose expectation values give rise to knot invariants. Considering such a theory with a non-compact gauge group, for example the 2+1 dimensional Poincare group, gives a consistent quantum theory of 2+1 dimensional gravity[@2dg]. Three-dimensional field theories are the high temperature limits of corresponding 3+1 dimensional field theories[@hit]. Thus parity violating theories in 3+1 dimensions, like the Standard Model, will in general contain the CS term in their effective actions in the high temperature limit. Therefore it is interesting to study 2+1 dimensional Yang Mills theory with the CS term.
For a non-abelian gauge theory with gauge group ${\cal G}$, the CS action, although invariant under small gauge transformations, changes by an integer multiple of $8\pi^2\mu/g^2$ under a large gauge transformation, where $\mu$ is the coefficient of the CS term and $g$ is the gauge coupling constant. This leads to the celebrated quantization condition[@desjactem] q=n,n=0,1,2,\[one\] in order to maintain the invariance of $e^{iS}$.
An important question to address is whether this quantization condition is respected by quantum corrections. This issue was considered by Pisarski and Rao[@pisrao], for the case of a pure gauge theory with dynamics governed by the usual Yang Mills term and the CS term. They found that the quantization condition is indeed preserved to one loop; however, the integer on the R.H.S. of (\[one\]) is shifted by $N$ for ${\cal
G}=SU(N)$. This calculation has been extended to two loops by Giavirini, [@2loop], who found no further correction in the limit of pure CS interaction, confirming the expectation that there are no corrections beyond one loop in that theory[@pisrao].
Subsequently, the question of quantization was considered for the case of a completely spontaneously broken gauge theory by Khlebnikov and Shaposhnikov[@khlsha]. They found that at the one loop level $q$ is multiplicatively renormalized by a complicated function of the three mass scales in the problem ($\mu$, the symmetry breaking mass scale and the physical Higgs mass scale), and violates (\[one\]).
At first sight, this result indicates loss of gauge invariance and inconsistency of the theory. However, the situation is not as catastrophic as it first appears: the underlying theory can continue to maintain gauge invariance, as manifested for example in the effective action. The effective action would be a completely gauge invariant functional of external gauge and scalar fields, but would cease to appear this way when the scalar field is evaluated at its vacuum expectation value. Indeed, it was observed in [@khlsha] that in the presence of spontaneous symmetry breaking, other terms exist in the effective action which reduce to the CS term when $\phi\rightarrow \langle\phi\rangle=v$, but which are nonetheless [*invariant*]{} under large gauge transformations. If this is the case, the calculation in [@khlsha] is not, in fact, a calculation of the coeffecient of the CS term alone. Rather, it is the sum of the coeffecients of the CS term and the other terms which reduce to it. This leaves the possibility, which should be verified, that the non-quantized result obtained in [@khlsha] is the sum of a quantized CS coeffecient and non-quantized (yet perfectly acceptable) contributions from the other terms.
In a slightly different context (namely, the spontaneously-broken abelian case, where quantization of the coefficient of the CS term is not required for consistency), a complicated radiatively induced correction to (apparently) the CS term[@abssb] was found to be due to other terms in the effective action which reduce to the CS term in the symmetry-breaking phase, exactly as described above: the coefficient of the CS term itself is in fact unchanged in that model[@khamacpar].
In this work, we consider the case of partial breaking of a non-abelian gauge symmetry. For an abelian unbroken subgroup, this is a proving ground for the Coleman-Hill theorem[@colhil]. The theorem asserts that if there are no massless states in the theory other than the photon, and if there is manifest Lorentz invariance, then the renormalization of the CS term is zero except for the one-loop contribution of fermions. They specifically exclude the case of unbroken non-abelian gauge theories, where each generator forms an abelian subgroup, since there exists no quantization which respects the conditions of the theorem. The case of a partially broken non-abelian gauge theory is more interesting. With symmetry breaking to just $U(1)$, all the remaining gauge bosons attain explicit masses; hence, the Coleman-Hill theorem implies no renormalization of the CS term. On the other hand, for a non-abelian unbroken subgroup ($SU(M)$, where $M<N$, say), the Coleman-Hill theorem does not apply. We must still have, at the very least, a quantized coefficient of the CS term for the gluons of the unbroken subgroup to have a consistent theory[@desjactem]. Although there is no formal proof, there is a suggestion in [@pisrao] that there might be an equivalent non-abelian version of the Coleman-Hill theorem which states that under suitable general conditions the renormalization of the CS term in $SU(N)$ theories is by $q\to q+N$, as found in [@pisrao]. If this is indeed the case, one might expect that the renormalization of the CS term for an unbroken, non-abelian sector to be that which corresponds to the unbroken subgroup only ($q\to q+M$ for unbroken $SU(M)$). This is because one can construct a Lorentz-invariant gauge where there are no massless particles outside the unbroken sector, and the conjectured non-abelian version of the Coleman-Hill theorem (assuming its assumptions are those of the abelian version) would imply that no contribution to $q$ will arise from the broken sector.
In the following, we find that exactly this scenario takes place, to one loop.
We begin by studying the gauge group $SU(3)$ spontaneously broken to an $SU(2)$ subgroup. The latter being nonabelian, we must demand the quantization of $\qs$, the value of $q$ for the unbroken $SU(2)$ subgroup, including radiative corrections. (It is important to observe that no terms exist in the effective action which would reduce under spontaneous symmetry breaking to the unbroken CS term.) We find, confirming the work of Chen, [@dunne] and correcting the calculation of the original version of this paper, that $\qs\to\qrs = \qs+2$.
The coefficient of the CS term remaining quantized in accordance with (\[one\]), these results do not uncover any inconsistency in the quantum version of the theory of the type discussed in [@desjactem]. Nonetheless, we find the results rather perplexing: the coefficient of the CS term is apparently a discontinuous function over the phase diagram of the theory. Specifically, when the pattern of symmetry breaking changes, the coefficient of the CS term jumps by a discrete amount. Although such behaviour has been seen before, for example in the case of massive fermions where the contribution to the CS term is proportional to the sign of the fermion mass, we do not expect this to arise from scalars.
This behaviour seems to be the rule rather than the exception. Indeed, we have generalized the above to include several different patterns of symmetry breaking, and we find in all cases results consistent with the nonabelian generalization of the Coleman-Hill theorem conjectured above: when $SU(N)$ is broken to a subgroup which contains in general $SU(M)$ and $U(1)$ factors, the renormalization of the CS term for unbroken $U(1)$ subgroups is zero, while that for unbroken $SU(M)$ subgroups is by $\delta q=M$. The calculation in all cases separates into contributions from within the subgroup under consideration and from the orthogonal sector of the theory; the former gives the simple result just stated while the latter gives a vanishing contribution. As for the renormalization of $q$ for the broken sector of the theory, one can, as outlined above, construct terms in the effective action which reduce to the CS term when the scalar field is replaced by its expectation value, and thus the type of calculation undertaken here is insufficient to determine its renormalization.
We begin with the case of $SU(3)$ spontaneously broken to $SU(2)$ via a triplet of Higgs in the fundamental representation. The corresponding Lagrangian is given by
[l]{} =-[14]{}F\_[a]{}F\_a\^-[2]{}\_( A\_a\^\^A\_a\^+[13]{}gf\_[abc]{}A\_a\^A\_b\^A\_c\^)\
+( D\^)\^\_A( D\_)\_A+m\^2(\^\_A \_A)-(\^\_A\_A)\^2, \[two\]
where
[l]{} F\_[a]{}=\_A\_[a]{}-\_A\_[a]{}+gf\_[abc]{}A\_[b]{} A\_[c]{}\
( D\_)\_A=\_\_A-ig[\_[AB]{}\^a2]{}A\_[a]{}\_B \[three\]
and $\lambda^a_{AB}$ are the Gell-Mann matrices with $f_{abc}$ being the structure constants. The choice of the Higgs potential implies a non-zero vacuum expectation value for $\phi$. We write $\phi = \phi^\prime +\langle\phi\rangle_0$ with \_0=, v=. \[four\] The gauge fixing and ghost terms are given by
[l]{} \^=-[12]{}(\_A\_a\^-ig( \_0\^\^-\^ [\^a2]{}\_0))\^2\
+\_|\_a\^\_a -igf\_[abc]{}\_|\_a A\^\_b\_c. \[five\]
The vertices are standard for a spontaneously broken non-abelian gauge theory; however, the CS term introduces an extra (parity odd) three-gluon vertex. The gluon propagator is, however, more involved. It is colour diagonal; for colour indices in the unbroken sector, $(a=1,2,3)$, in Landau gauge $(\xi =0)$, it is given by i\_=-i[(g\_-[k\_k\_k\^2]{})-i\_[k\^k\^2]{}k\^2-\^2]{}, \[six\] while for the broken generators we have [@pisrao; @paukha] i\_=-i[(g\_-[k\_k\_k\^2]{})(k\^2-m\_W\^2)-i\_k\^(k\^2-m\_+\^2)(k\^2-m\_-\^2)]{}. \[seven\] Here m\_= \[eight\] and $m_W$ is the contribution to the gluon mass from the Higgs mechanism. We have $m_W=vg/2\equiv m_D$ for the iso-doublet massive vectors $(a=4,5,6,7)$ and $m_W=vg/\sqrt 3\equiv m_S$ for the iso-singlet massive vector $(a=8)$.
Following Pisarski and Rao [@pisrao], we calculate $\qr$ according to =[4g\^2]{}Z\_mZ\^2 \[nine\] to one loop in Landau gauge, for the unbroken $SU(2)$ subgroup. Here $Z_m$ and $\tilde Z$ are, respectively, the renormalization constants for the odd part of the gluon self-energy and the ghost self-energy. We note that $\qr$ for the broken generators will be different than that for the unbroken $SU(2)$ subgroup since the physical Higgs contributes to $Z_m$ in this case. We do not present the details of the calculation since it is amply described in [@dunne]. For $\qrs$, the renormalized $q$ for the unbroken $SU(2)$ subgroup, we must find $Z_m$ and $\tilde Z$ for that sector. Beyond those calculated in [@pisrao], there are additional contributions, $\delta Z_m$, to the gluon self-energy coming from the massive iso-doublet vector bosons circulating in the gluon loop and from the loop containing unphysical scalars. The ghost self-energy is also augmented by an additional contribution $\delta \tilde Z$, from the loop containing massive iso-doublet vector bosons. The massive iso-singlet vector boson actually does not contribute at this order. We find
[l]{} Z\_m=g\^2( -2[(p\^2+m\_D\^2)\^2(p\^2+m\^2\_+)\^2(p\^2+m\^2\_-)\^2]{} +[163]{}[p\^2(p\^2+m\_D\^2)(p\^2+m\^2\_+)\^2(p\^2+m\^2\_-)\^2]{}.\
.-[23]{}[\^2(p\^2+m\_D\^2)(p\^2+m\^2\_+)\^2(p\^2+m\^2\_-)\^2]{} +2[\^2p\^2(p\^2+m\^2\_+)\^2(p\^2+m\^2\_-)\^2]{} +2[m\_D\^23p\^2(p\^2+m\^2\_+)(p\^2+m\^2\_-)]{}), \[ten\]
and Z=-[23]{}g\^2. \[eleven\] To these, we add the Pisarski-Rao contributions (coming from the unbroken $SU(2)$ sector), yielding: Z\_m=1+[7g\^26m]{}+Z\_m, Z=1-[g\^23m]{}+Z, \[elevena\] which, in (\[nine\]), yields =q+2+q(Z\_m+2Z). \[elevenb\] The remaining integrals are straightforward; one finds that $\delta Z_m+2\delta\tilde Z=0$, and =q+2, \[elevenc\] in agreement with [@dunne].
Some comments are in order. First, $\qrs$ is quantized, as it must be for consistency of the theory.[^3] Second, there is nonetheless some peculiar behaviour exhibited. On the one hand, our final result (\[elevenc\]) is completely independent of the expectation value of the scalar field, while on the other hand if we were to redo the entire calculation in the symmetric phase (in the absence of spontaneous symmetry breaking), $\qs$ would attain a renormalization [*exactly as in the pure gauge theory with the full gauge group*]{} $SU(3)$: one would find $\qrs=q+3$.[^4] The limit of symmetry restoration in the above calculation must be examined with care: we must consider how the limit affects the integrals (\[ten\]) and (\[eleven\]) rather than merely studying the final result (\[elevenc\]). In fact, the problem can be traced to the third and fifth terms in the integrand of (\[ten\]). For instance, the fifth term is simply not present in the symmetric phase since there is no gluon-gluon-scalar vertex (the vertex is proportional to $v$, whose presence in the fifth term is contained in $m_D$), while the symmetric limit ($v\to0$) of the integral of this term is nonzero. This ambiguity is due to an infrared problem which appears in the integrand as $v\to0$. In this limit, $m_-\to0$, and the integral is linearly divergent. That term’s contribution to $\delta Z_m$ is of the order $g^2 {m_D}^2/m_-$. Since $m_D\propto v$ while as $v\to0$ $m_-\sim v^2$ (as can be seen from (\[eight\])), the contribution of that term is finite and nonzero as $v\to0$, in disagreement with the zero result one would have obtained obtained in the symmetric theory. Similar considerations apply to the third term, while the other terms in (\[ten\]) and (\[eleven\]) are well-behaved in the symmetric limit. Thus, we conclude that evaluating the integral and taking the limit of no symmetry breaking do not commute. Non-commutativity of limits has also been observed in perturbative calculations in CS theories in several other situations [@khlsha; @abssb; @lebtho].
The calculations outlined above can be easily modified to handle other cases. We have studied the following patterns of symmetry breaking: $SU(2)\to U(1)$ via a real triplet which attains an expectation value $\langle\phi_a\rangle=v\delta_{a,3}$; $SU(3)\to SU(2)\times U(1)$ via an adjoint which attains an expectation value $\langle\Phi\rangle=v T_8\sim diag(1,1,-2)$; $SU(3)\to U(1)\times U(1)$ via an adjoint which attains an expectation value $\langle\Phi\rangle=v T_3\sim diag(1,-1,0)$. In all cases, the Feynman rules are found in a straightforward way, and the diagrams which contribute to $Z_m$ and to $\tilde Z$ for an unbroken gluon are identical to those as calculated above. Differences arise only in the values of coupling constants, masses, and group theoretical factors. The calculation of the $Z$s naturally separates into contributions from the unbroken group and possible contributions from the orthogonal (broken) sector. In the case of an unbroken group which is a direct product, $\delta
q$ can be computed for each subgroup of the direct product. Straightforward group theoretical factors imply that the subgroups decouple: each subgroup of the unbroken group only contributes to its own $\delta q$. As for the contribution from the broken sector, in all cases it was found that $\delta Z_m+2\delta\tilde Z=0$, and the net result was that $\delta q$ is that value one would have calculated from the pure gauge theory of the unbroken subgroup under consideration. Thus, in all cases the correction to $\qu$ was zero, in keeping with expectations based on the Coleman-Hill theorem [@colhil]. Furthermore, in the second case, the unbroken $SU(2)$ correction is as above: $\delta
\qs=2$. The generalization is clear: the $q$ of any residual $U(1)$ symmetry receives no radiative correction, while that of any residual non-abelian group receives a radiative correction which is as if the orthogonal sector of the theory was not there.
We thank A.S. Goldhaber, M. Leblanc, G. W. Semenoff and V. P. Spiridonov for useful discussions. We are particularly indebted to G. Dunne for discussions of his results which enabled us to locate a couple of errors in the original version of this work. A. Khare thanks the Laboratoire de Physique Nucléaire for the kind invitation and hospitality during his visit. This work supported in part by NSERC of Canada and FCAR du Québec.
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[^1]: permanent address: Institute of Physics, Sachivalaya Marg, Bhubaneswar, 751005, India.
[^2]: permanent address: Department of Physics, University of Hyderabad, Hyderabad, 500134, India.
[^3]: It is perhaps worth reiterating that computational errors in the original version of this paper led us to a different conclusion, namely, that $\qrs$ is not quantized.
[^4]: Note that, even though the symmetry is not broken to $SU(2)$ here, we are free to calculate the radiative correction to $q$ for the gluons of an $SU(2)$ subgroup.
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abstract: 'The search for di-Higgs final states is typically limited at the LHC to the dominant gluon fusion channels, with weak boson fusion only assuming a spectator role. In this work, we demonstrate that when it comes to searches for resonant structures that arise from iso-singlet mixing in the Higgs sector, the weak boson fusion sideline can indeed contribute to winning the discovery game. Extending existing experimental resonance searches by including both contributions is therefore crucial.'
author:
- Rahool Kumar Barman
- Christoph Englert
- Dorival Gonçalves
- Michael Spannowsky
bibliography:
- 'references.bib'
title: 'Di-Higgs resonance searches in weak boson fusion'
---
Introduction {#sec:intro}
============
The search for new physics beyond the Standard Model (SM) is a key pillar of the Large Hadron Collider (LHC) physics programme. As significant deviations from the SM expectation have remained elusive after the Higgs boson’s discovery so far, the nature of the electroweak scale is still fundamentally unknown. A particularly relevant process in this context is the production of multiple Higgs bosons. Firstly, multi-Higgs production directly probes aspects of spontaneous symmetry breaking that cannot be accessed with weak boson or heavy quark physics. Secondly, the inclusive production cross section of Higgs pairs of around 30 fb [@Dawson:1998py; @Borowka:2016ehy; @deFlorian:2016spz; @Grazzini:2018bsd; @DiMicco:2019ngk; @Baglio:2020ini] is about three orders of magnitude smaller than single Higgs production, thus highlighting the statistical difficulty that experimental investigations face in this area.
Multi-Higgs production is phenomenologically limited to Higgs pairs [@Plehn:2005nk], at least in the near future [@Papaefstathiou:2015paa; @Chiesa:2020awd], and as with single Higgs production, gluon fusion (GF) contributes to the bulk of the production cross section. While Higgs production via weak boson fusion (WBF) with its distinct phenomenological properties [@Cahn:1983ip; @Rainwater:1998kj; @Rainwater:1999sd; @Plehn:1999xi] and large cross section plays an important role in the investigation of the Higgs boson’s properties, di-Higgs production from weak boson fusion will be statistically limited at the LHC [@Dolan:2015zja; @Bishara:2016kjn; @Arganda:2018ftn]. WBF-type analysis are further hampered by the importance of the top threshold for gluon fusion production [@Dolan:2013rja] and the necessity to relax central jet vetos to retain a reasonable WBF signal count through central $h\to b\bar b$ decays. Experimental analyses typically mitigate the non-applicability of central jet vetos in the WBF selection by considering stringent invariant jet pair masses, see e.g. Ref. [@Aad:2020kub]. While such a selection serves to purify signal samples towards the WBF component, forward jets will also arise from gluon fusion samples [@DelDuca:2001eu; @DelDuca:2003ba; @delDuca:2007nwt; @DelDuca:2006hk] when biased towards valence quark-flavoured initial state processes and the question of the size of the potential, model-dependent GF component remains.
Resonant phenomena in weak boson fusion are less studied from a phenomenological perspective than their GF counterparts. A bias towards GF-like production is understandable as two-Higgs-doublet extensions of the SM in particular as prototypes of supersymmetric theories lead to gauge-phobic scalars, and WBF production of exotic states, e.g. the additional CP-odd scalar proceeds dominantly through GF. However, the observation of resonances in WBF would have exciting theoretical implications. Introducing a new resonant BSM scalar in the WBF modes rests on non-alignment [@Grzadkowski:2018ohf], CP violation [@Fontes:2017zfn], a significant non-doublet component of the electroweak vacuum (e.g. [@Georgi:1985nv; @Gunion:1989ci; @Hartling:2014aga]), or combinations of these.
Electroweak symmetry breaking from triplets faces theoretical reservation related to the fine-tuning of the rho parameter [@Gunion:1990dt].[^1] Phenomenologically, (tree-level) custodial triplet extensions lead to a range of additional exotic final states, most notably a doubly-charged Higgs that is predominantly produced through weak boson fusion as part of a fermiophobic custodial quintet [@Godfrey:2010qb; @Cheung:2002gd; @Englert:2013wga; @Zaro:2015ika; @Degrande:2015xnm]. Electrically uncharged components of the custodial triplet will not decay promptly to the 125 GeV state if the latter is identified as a doublet like state, again due to custodial isospin. CP violation is typically a small effect in actual scans [@Fontes:2017zfn] such that a competitive production through WBF is typically suppressed.
The possibility of non-alignment (i.e. the physical 125 GeV Higgs boson not being fully aligned with fluctuations around the electroweak vacuum) remains as an a priori relevant parameter space for WBF to be relevant. The mixing of isospin singlet states is present in any Higgs sector extension, but most transparently analysed in the so-called Higgs portal scenario [@Binoth:1996au]. This model also fully correlates the exotic Higgs production with observed $m_h\simeq 125$ GeV Higgs boson phenomenology, which turns any sensitivity projection for heavy Higgs states into a conservative estimate as new, non-singlet fields will loosen the tight correlations of the singlet extensions.
The relevance of WBF production is further highlighted in singlet scenarios by the fact that for SM Higgs-like states with masses ${\cal{O}}$(TeV), GF and WBF productions become comparable [@Dittmaier:2011ti]. This strongly indicates that if such a state is realised in nature, both GF and WBF play a priori equally important role in the discovery of new physics. As there is accidental destructive interference of $pp\to H\to t\bar t$ with QCD continuum top pair production [@Gaemers:1984sj; @Dicus:1994bm; @Frederix:2007gi; @Carena:2016npr; @Hespel:2016qaf; @BuarqueFranzosi:2017qlm; @Englert:2019rga] which particularly affects the sensitivity in the singlet extension scenario [@Basler:2019nas], gaining sensitivity in the $H\to hh$ decays is not only necessary, but also possibly the only phenomenological robust avenue to successfully detect such scenarios. Depending on the Higgs potential these channels might be favoured over the decays into massive electroweak gauge bosons, which are additional relevant channels.
In this work we perform a detailed investigation of WBF production of exotic Higgs bosons $pp\to H jj $ arising from iso-singlet mixing, in particular in their decay $H\to hh$. We include the gluon fusion component keeping the full $m_t$ dependence and highlight the interplay of both production modes and their relevance to hone the discovery potential at the LHC. In particular, we show that gluon fusion remains phenomenologically relevant and should therefore be reflected as an appropriate signal contribution in any analysis that seeks to inform further theoretical investigations.
We organise this paper as follows: In Sec. \[sec:model\], we provide a short summary of the key phenomenological aspects of the singlet extension scenario, which acts as the vehicle of this work. We stress that our findings readily generalise to more complex scenarios. Sec. \[sec:model\] is devoted to the WBF di-Higgs resonance analysis. We conclude in Sec. \[sec:conc\].
The model {#sec:model}
=========
We consider the extension of the SM with Higgs doublet $\Phi_s$ by an additional singlet $\Phi_h$ under the SM gauge group $$V= \mu_s^2 |\Phi_s|^2 + \lambda_s |\Phi_s|^4 + \mu_h^2 |\Phi_h|^2 + \lambda_h |\Phi_h|^4 + \eta |\Phi_s|^2 |\Phi_h|^2 \,.$$ Expanding around the vacuum expectation values of the respective fields $$v_i^2={1\over \lambda_i} \left(-\mu_i^2-{\eta \over 2} v_{j\neq i}^2\right),\quad {i,j=s,h}$$ via $\Phi_i=(v_i+H_i)/\sqrt{2}$ leads to a mixing of Lagrangian eigenstates in the mass basis $$\begin{split}
h &= \phantom{-}\cos\theta\, H_s + \sin\theta \,H_h \\
H &= -\sin\theta \, H_s + \cos\theta \, H_h\,.
\end{split}$$ We will implicitly identify $h$ with the observed, lighter $m_h\simeq 125~\text{GeV}$ boson aligned with the SM expectation, i.e. we will be particularly interested in the region ${\cos\theta\lesssim 1}$. The masses are given by $$m^2_{h,H} = (\lambda_sv_s^2 +\lambda_h v_h^2) \mp\sqrt{ (\lambda_s v_s^2 - \lambda_h v_h^2)^2 +\eta^2 v_s^2 v_h^2 }\,,$$ and $$\tan 2\theta = {\eta\,v_s v_h\over \lambda_s v_s^2 - \lambda_h v_h^2}$$ while $v_s\simeq 246$ GeV from electroweak symmetry breaking in the SM.
We assume no additional decay channels which means that signal strengths of the SM-like Higgs are modified $\mu=\cos^2\theta$. $H$ boson production cross sections as a function of $m_H$ can be obtained from the SM ones [@Dittmaier:2011ti] by rescaling with $\sin^2\theta$; branching ratios are unmodified for $m_H< 2m_h$. We are particularly interested in the region $m_H\geq 2m_h$ where cascade decays $H\to hh$ are open. In this case the heavy Higgs partner receives an leading order additional contribution to its decay width $$\Gamma(H\to hh) = {c_{Hhh}^2 \over 32 m_H \pi} \sqrt{ 1 - {4 m_h^2 \over m_H^2}}$$
with $$c_{Hhh} = 3 \sin 2\theta \left( \lambda_sv_s \cos \theta - \lambda_hv_h \sin\theta \right)- \tan 2\theta \\
\left( \lambda_s v_s^2- \lambda_h^2 v_h^2 \right) \left[ ( 1-3\cos^2\theta) {\sin\theta \over v_h} -
( 1-3\sin^2\theta) {\cos\theta \over v_s} \right]\,.$$
The potential measurement of $\Gamma(H\to hh)$, together with the masses $m_{h,H}$ and SM signal strength and weak boson masses allows us to fully reconstruct the singlet-extended Higgs potential. A range of precision computations from a QCD and electroweak point of view have become available recently [@Chen:2014ask; @Bojarski:2015kra; @Dawson:2017jja; @Lopez-Val:2014jva; @Falkowski:2015iwa] with strongest constraints typically arising from the $W$ mass measurement [@Lopez-Val:2014jva; @Robens:2015gla].
Analysis {#sec:ana}
========
We derive the LHC sensitivity to di-Higgs resonances in the Vector Boson Fusion (VBF) channel $pp \rightarrow Hjj$, with $H\rightarrow h h \rightarrow 4b$. The signal is characterized by four bottom tagged jets in association with two light flavor jets. The leading backgrounds for this process are $pp\rightarrow 4b + 2j$, $2b + 4j$, and $t\bar{t}b\bar{b}$.
We generate the WBF and QCD $pp\to (H\to hh) j j$ signal samples with [Vbfnlo]{} [@Arnold:2008rz], which we have modified to include the $H\to hh$ decay. The backgrounds are generated with [MadGraph5aMC@NLO]{} [@Alwall:2014hca]. All samples are generated at leading order with center of mass energy of ${\sqrt{s}=13}$ TeV. Parton shower, hadronization, and underlying event effects are accounted for with [Pythia8]{} [@Sjostrand:2007gs]. Jets are defined through the anti-k$_T$ algorithm with ${R=0.4}$, $p_{Tj}>30$ GeV, and $|\eta_j|<4.5$ via [FastJet]{} [@Cacciari:2011ma]. We assume $70\%$ $b$-tagging efficiency and 1% mistag rate.
We start our analysis demanding at least six jets in the final state, where four of those are $b$-tagged. We impose a minimum threshold for the invariant mass for the four $b$-jets of $m_{4b}>350$ GeV and veto leptons with ${p_{T\ell}>12}$ GeV and $|\eta_{\ell}|<2.5$. The two light-flavor jets with highest rapidity, $j_{1,2}$, satisfy the VBF topology falling in different hemispheres of the detector ${\eta_{j1} \times \eta_{j2}<0}$, with large rapidity separation ${|\eta_{j1}- \eta_{j2}|>4.2}$, and sizable invariant mass ${m_{jj}>1}$ TeV.
![Normalized distribution of $\eta_{j_{3}}^{\star}=|\eta_{j3}-(\eta_{j1}+\eta_{j2})/2|$ for the dominant $4b$ background (blue) and the WBF signal events $M_{H} = 0.5~{\rm TeV}$ (red) and 1 TeV (black) after imposing the basic selection cuts and the VBF selections: ${\eta_{j1} \times \eta_{j2}<0}$, ${|\eta_{j1}- \eta_{j2}|>4.2}$ and ${m_{jj}>1}$ TeV.[]{data-label="fig:etaj3"}](etaj3_sep)
[ C[3.1cm]{} | C[2.6cm]{} C[2.6cm]{} C[2.6cm]{} ]{} Process & Basic selections & VBF topology & Double Higgs reconstruction\
$4b$ & 250 & 47 & 1.2\
$2b2j$ & $4.9 \times 10^{-1}$ & $1.0 \times 10^{-1}$ & -\
$t\bar{t}b\bar{b}$ & 90 & 3.7 & $3.0 \times 10^{-3}$\
WBF ${m_{H} = 500~\text{GeV}}$ & $2.6 \times 10^{-1}$ & $1.3 \times 10^{-1}$ & $5.0 \times 10^{-2}$\
GF ${m_{H} = 500~\text{GeV}}$ & $2.2 \times 10^{-1}$ & $7.1 \times 10^{-2}$ & $2.8 \times 10^{-2}$\
WBF ${m_{H} = 1~\text{TeV}}$ & $9.4 \times 10^{-2}$ & $5.4 \times 10^{-2}$ & $3.2 \times 10^{-2}$\
GF ${m_{H} = 1~\text{TeV}}$ & $2.2 \times 10^{-2}$ & $8.3 \times 10^{-3}$ & $4.7\times 10^{-3}$\
While the WBF signal displays suppressed extra jet emissions in the central region of the detector, the bulk of the QCD background radiation is centered around this regime [@Derrick:1987uy; @Bjorken:1992er; @Barger:1991ar; @Barger:1994zq]. In Fig. \[fig:etaj3\], we illustrate this property displaying two mass scenarios for the WBF signal samples, $m_H=0.5$ TeV and 1 TeV. The more massive is the signal resonance, the further forward the tagging jets hit the detector. This phenomenological pattern is related to gauge boson scattering $VV\rightarrow hh$ around the heavy Higgs pole, where the longitudinal and transverse scattering amplitudes scale as $\mathcal{A}_{LL}/\mathcal{A}_{TT}\sim m_{H}^2/m_V^2$ for $m_H\gg m_V$ [@Dawson:1984gx; @Figy:2007kv; @Goncalves:2017gzy]. We explore this feature to further suppress the backgrounds imposing that the rapidity for the third jet $\eta_{j3}$ satisfies the relation $$\left| \eta_{j3}-\frac{\eta_{j1}+\eta_{j2}}{2}\right|>2.5\,.$$ After establishing the VBF topology, the next step of the analysis focuses on the Higgs bosons reconstruction. This is performed by identifying among the four $b$-jets the pair whose invariant mass $m_{h1}$ is closest to the Higgs mass, $m_h=125$ GeV. The remaining $b$-jet pair defines the second Higgs boson candidate $h_2$. In the two dimensional space defined by the masses of the Higgs boson candidates $(m_{h1},m_{h2})$, the signal region is defined to be within the circular region $$\sqrt{\left(\frac{m_{h1}-125~\text{GeV}}{20~\text{GeV}}\right)^2+\left(\frac{m_{h2}-125~\text{GeV}}{20~\text{GeV}}\right)^2}<1\,.$$ To further improve the $m_{4b}$ mass resolution, each Higgs boson candidate’s four-momentum is scaled by the correction factor $m_h/m_{h1(2)}$. This improves the signal $m_{4b}$ resolution from $20$ to $40\%$, depending on the heavy Higgs mass hypothesis, and presents sub-leading effects to the background $m_{4b}$ distribution [@Sirunyan:2018zkk].
![Stacked $m_{4b}$ distribution for the signal and background events after the complete cut-flow analysis shown in Table \[tab:cutflow\]. The VBF signal hypotheses are also shown in the non-stacked format with the WBF (solid line) and GF (dashed line) components independently displayed. We assume $\text{BR}(H\rightarrow hh)=1$ and $\sin\theta=0.3$ with the LHC running at $\sqrt{s}=13~\text{TeV}$ and integrated luminosity $\mathcal{L}=3~\text{ab}^{-1}$.[]{data-label="fig:mhh"}](mhh)
Since very few multi-jet background events pass the cut-flow analysis with large $m_{4b}$, we follow a similar statistical procedure performed by the ATLAS collaboration in their $pp\rightarrow H\rightarrow hh\rightarrow 4b$ study [@Aaboud:2018knk]. Namely, the statistical precision for the $m_{4b}$ distribution at high energies is improved by fitting the background distribution at low invariant masses $m_{4b}<1$ TeV with the functional form $$F(m_{4b})=a\frac{s}{m_{4b}^2}\left(1-\frac{m_{4b}}{\sqrt{s}}\right)^{b-c \log\frac{m_{4b}}{\sqrt{s} }} \,,$$ where $a,~b,$ and $c$ are real free parameters and $\sqrt{s}$ the LHC center of mass energy. This also emulates a data-driven approach that is typically the method of choice when backgrounds are only poorly understood from a systematic and theoretical perspective, see e.g. [@Aaboud:2018urx; @Aad:2020kop]. As we are looking for a resonance on top of a steeply falling background such a method provides a particularly motivated approach to reduce uncertainties.
In Fig. \[fig:mhh\], we illustrate the invariant mass distribution $m_{4b}$ for the signal and background components after the full cut-flow analysis shown in Table \[tab:cutflow\]. While the WBF signal component displays dominant contributions to the event rate, the VBF GF signal can result into non-negligible additions to the event count. It should be noted that the larger the signal mass $m_H$ is, the larger the relative WBF component becomes.
![95% CL limit on the Higgs-singlet mixing as a function of the heavy Higgs boson mass $m_H$. We show both the VBF $pp\rightarrow H jj \rightarrow 4bjj$ (red solid) and GF $pp\rightarrow H\rightarrow 4b$ (black) limits. To estimate the importance of the VBF GF signal component to the VBF analysis, we also show the bound considering only the WBF signal component (red dashed). We assume the heavy Higgs boson branching ratio to di-Higgs $\text{BR}(H\rightarrow hh) =1/4$ and the LHC at 13 TeV with integrated luminosity ${\mathcal{L}=3~\text{ab}^{-1}}$.[]{data-label="fig:limit"}](sintheta_limit.pdf){width="8cm"}
To estimate the HL-LHC sensitivity to the resonant VBF $hh$ signal, we calculate a binned log-likelihood analysis based on the $m_{4b}$ distribution using the CL$_s$ method [@Read:2002hq]. We assume the integrated luminosity ${\mathcal{L}=3~\text{ab}^{-1}}$. In Fig. \[fig:limit\], we present the 95% CL sensitivity to the heavy Higgs-singlet mixing $\sin\theta$ as a function of the Heavy Higgs boson mass $m_H$. Motivated by the Goldstone boson equivalence theorem for $m_H\gg m_W$, we assume the heavy Higgs branching ratio to di-Higgs ${\text{BR}}(H\rightarrow hh) =1/4$. To illustrate the importance of the VBF GF signal component, we separately show the signal sensitivity accounting for the full VBF sample and only for its WBF component. We observe that the VBF GF results in non-negligible contributions for the low mass regime $500~\text{GeV}<m_H<900~\text{GeV}$.
0.6cm
To compare our new VBF di-Higgs resonance search with the existing limits, we use the CMS $pp\rightarrow H\rightarrow hh \rightarrow 4b$ study [@Sirunyan:2018zkk]. CMS derives the 95% CL limit on the heavy Higgs cross section $\sigma(pp\rightarrow H \rightarrow hh \rightarrow 4b)$ as a function of its mass $m_H$. We translate this bound in terms of the mixing $\sin\theta$ in Fig. \[fig:limit\], using the heavy Higgs production cross section at NNLO+NNLL QCD, including top and bottom quark mass effects up to NLO [@Dittmaier:2011ti; @Heinemeyer:2013tqa; @deFlorian:2016spz]. The CMS limit on the heavy Higgs cross section was scaled to the HL-LHC integrated luminosity, $\mathcal{L}=3~\text{ab}^{-1}$. The discontinuity on the CMS limit at $m_H\sim 580$ GeV arises from the two distinct strategies separating low and high mass resonances.
We observe that the double Higgs resonant search in the VBF mode can significantly contribute to the heavy Higgs resonant analyses. The increase in the ratio $\sigma_\text{VBF}/\sigma_\text{GF}$ for larger $m_H$ leads to comparable sensitivities between the VBF and GF channels for $m_H\sim 900$ GeV. Whereas the VBF search displays stronger limits at high $m_H$ regime, it can also contribute to further constrain the low mass scenarios ${500~\text{GeV}<m_H<900~\text{GeV}}$ via a combination between the GF and VBF analyses.
In order to understand the relevance of the GF and VBF limits on the singlet extension scenario discussed in Sec. \[sec:model\], we interpret the constraints in the aforesaid model. We scan over the singlet model parameter space for $|\lambda_i|\leq 4\pi$ and include the $W$ mass constraint from Ref. [@Lopez-Val:2014jva; @Robens:2015gla] as it typically imposes the strongest constraint on the model’s parameter space. The results are shown in Fig. \[fig:scan\]. The constraints from gluon fusion $gg\to hh$ are displayed in blue points while those of $pp\to hh j j$ are given in orange squares. We see that the vector boson fusion provides significant sensitivity for higher masses where the gluon fusion projection becomes insensitive.
While there is a region where gluon fusion and VBF overlap and can be used to further hone the LHC sensitivity to this scenario through a statistical combination, we also see regions in branching ratio $H\to hh$ where VBF provides genuine, new sensitivity that cannot be accessed with the gluon fusion analysis. This region is characterised by 125 GeV Higgs boson signal strength modifiers of $\lesssim 4\%$. Given the HL-LHC projections of Ref. [@deBlas:2019rxi], this suggest that the resonance search in the WBF channel can also explore the model’s parameter space beyond the precision that can be obtained from 125 GeV signal studies.
QCD contributions to $pp\to hh j j$ are not the dominant contribution in this mass region (it is a sizable contribution for the theoretical interpretation of the results of Ref. [@Aad:2020kub]), it nonetheless is sizable and should be included in investigations possibly as separate signal contribution to enable a consistent theoretical interpretation.
Summary and Conclusions {#sec:conc}
=======================
Weak boson fusion through its distinct phenomenological properties provides a unique opportunity for new physics searches. In scenarios with isospin singlet mixing decays of a heavy Higgs partner into 125 GeV Higgs bosons can be preferred while more obvious decays into top quarks suffer from interference distortion [@Basler:2019nas], and decays into massive weak bosons might be less dominant. Given that the weak boson fusion production cross section becomes comparable to gluon fusion cross section for SM-like production at around 1 TeV, the WBF production at small mixing angles becomes a phenomenologically relevant channel. In this paper we have investigated the WBF production of heavy Higgs partners with subsequent decay $H\to h h$. We show that this channel, which has been somewhat overlooked in the past, provides additional relevant new physics potential. In parallel, we show that the gluon fusion component to the vector boson fusion channel remains sizeable and should be included in experimental analysis to enable a consistent theoretical interpretation of reported results.
[*[Acknowledgements]{}*]{} — We thank Stephen Brown and Peter Galler for helpful conversations.
CE is supported by the UK Science and Technology Facilities Council (STFC), under grant ST/P000746/1. CE also acknowledges support through the IPPP associate scheme. DG was supported by the US Department of Energy under grant number [DE-SC 0016013]{}. MS is supported by the STFC under grant ST/P001246/1.
[^1]: It remains as a possibility of strong electroweak symmetry breaking in realistic UV constructions [@Ferretti:2014qta; @Golterman:2015zwa].
|
---
abstract: 'Granular mixtures rapidly segregate radially by size when tumbled in a partially filled horizontal drum. The smaller component moves toward the axis of rotation and forms a buried core, which then splits into axial bands. Models have generally assumed that the axial segregation is opposed by diffusion. Using narrow pulses of the smaller component as initial conditions, we have characterized axial transport in the core. We find that the axial advance of the segregated core is well described by a self-similar concentration profile whose width scales as $t^\alpha$, with $\alpha \sim 0.3 < 1/2$. Thus, the process is subdiffusive rather than diffusive as previously assumed. We find that $\alpha$ is nearly independent of the grain type and drum rotation rate within the smoothly streaming regime. We compare our results to two one-dimensional PDE models which contain self-similarity and subdiffusion; a linear fractional diffusion model and the nonlinear porous medium equation.'
author:
- 'Zeina S. Khan and Stephen W. Morris'
title: Subdiffusive axial transport of granular materials in a long drum mixer
---
An interesting property of dry granular materials is their tendency to separate by size under a wide variety of flow conditions [@Duranspg; @ristow]. Granular segregation is widely found in nature, and plagues industrial processes as well. Probably the best controlled and most widely studied example is segregation along the axis of a partially filled, horizontal “drum mixer" [@Oyama; @shinbrot; @KHprl; @KHpre; @KHspg; @KCprl; @KCpre; @ottino_prl; @our_EPL; @chris; @ristownak; @nakagawa; @shattuck]. After hundreds of drum rotations, an initially mixed binary distribution of different-sized grains sorts itself into almost periodic bands along the axis of the drum. These bands are threaded by a radial core of the smaller grains which develops prior to axial band formation [@KHprl; @KHpre; @ristownak; @nakagawa; @shattuck; @our_EPL]. The radial core typically forms after just a few drum rotations. Accounting for this rich dynamical behaviour has been the goal of cellular automata models [@automata], molecular-dynamics simulations [@rapaport] and several continuum theories [@Savage; @Ziketal; @Levitan; @Levinechaos; @ATprl; @ATVpre; @elperin]. The axial bands must somehow be sustained against being mixed away by the random motion of the grains. Continuum models have generally assumed that the random motions mimic normal diffusion, and therefore that normal Laplacian gradient terms determine the short-wavelength cutoff of the axial band pattern. In this Letter, we experimentally challenge this common assumption. We find, surprisingly, that the axial transport of the radially segregated core along the drum is much slower than diffusion, [*i.e.*]{} that it is subdiffusive. It is nevertheless described by a self-similar profile which scales approximately as $t^{1/3}$. We also find that the self-diffusion of the larger particles is subdiffusive. These results have strong implications for models of axial segregation, and possibly for other theories of granular mixing.
Early theoretical models regarded axial band formation as the result of a diffusion process with a negative diffusion coefficient [@Savage; @Ziketal; @Levinechaos]. These models ignore the radially segregated core, and they cannot account for the oscillatory transient that precedes axial band formation in some mixtures [@KCprl; @KCpre; @ottino_prl]. This oscillatory travelling wave state apparently demands that the basic dynamics be at least second order in time. A later model due to Aranson [*et. al*]{} [@ATprl; @ATVpre] reproduced both axial segregation as well as the oscillatory transient, while still ignoring the core. We have recently shown experimentally that this model is also inadequate[@our_EPL], and that the core dynamics are themselves oscillatory. Another model due to Elperin [*et. al*]{} [@elperin] regards axial segregation as resulting from a radial core instability leading to a spatially periodic thickening of the core. This model, unfortunately, also cannot account for the travelling wave state. In all cases, these models explain the short wavelength cutoff of the axial band pattern as the result of the supposed axial diffusion of the smaller grains. We show experimentally below that axial transport is not well-described by normal diffusion. This is true of either the smaller grains in a binary mixture or of the larger grains in a self-mixing process. This falsifies a common, basic assumption of segregation models.
A few studies have investigated the axial transport of grains experimentally [@old_diff; @ristownak; @nakagawa; @shattuck], but none have systematically investigated the effects of varying grain type and drum rotation rate. Here we report experiments which characterize the axial transport of radially segregated grains using several different grain types and drum rotation rates, starting with a narrow pulse initial condition.
The drum mixer used in all experiments consisted of a horizontal Pyrex tube, 600 mm long with an inner diameter of 28.5 mm, rotated about its long axis at a constant rotation rate of 0.31 $\textrm{rev}/\textrm{s}$ or 0.62 $\textrm{rev}/\textrm{s}$. The larger grains were either cubic white table salt or transparent glass spheres and had a size range of 300-420 $\mu$m. The smaller grains were either irregularly shaped black hobby sand or bronze spheres, with a size range of 177-212 $\mu$m. The filled volume fraction of the drum was 28 %. In order to reproducably fill the drum, the grains were loaded into a long U-shaped channel, which was inserted lengthwise into the drum and rotated to deposit its contents.
This procedure ensures a uniform filling fraction of the drum. To obtain reproducible, quantitative dynamical information, we used a pulse initial condition. The pulse was made by placing thin spacers in the U-shaped channel 1.5 mm apart. The 1.5 mm space was filled with the smaller grains, and the remaining space was filled with the larger grains.
After a few drum revolutions, the pulse of small grains forms a subsurface radial core and cannot be observed using standard surface lighting and video imaging techniques. Instead, we used a bulk visualisation technique developed by Khan [*et al*]{} [@our_EPL]. The large grains are transluscent and the small grains are opaque. When a bright light source is placed behind the rotating drum, one can observe a shadow of the radial core on the front face of the granular sample. This shadow is a two-dimensional projection of the radial core. A computer controlled high speed camera was used to observe the radial core shadow. Five images per drum revolution were obtained and averaged to determine the evolution of the radial core. Figure \[Ffig1\]a shows a typical image of the radial core shadow. Using edge detection, the radial core
height $h(x,t)$ was measured as shown in figure \[Ffig1\]b, and expressed as a fraction of the full height of the material in the drum. If we assume that any cross section of the three dimensional structure of the radial core is an ellipsoid, the square of the diameter of the radial core $h^2$ is proportional to the volume of small grains contained in the radial core at each axial position $x$. Figure \[Ffig1\]c shows
----------------- ----------------- ------------------------------ -----------------
Large grains Small grains Rotation rate $\alpha$
300-420 $\mu m$ 177-212 $\mu m$ $ (\textrm{rev}/\textrm{s})$
salt sand 0.31 0.38 $\pm$ 0.03
salt sand 0.62 0.37 $\pm$ 0.03
glass bronze 0.31 0.31 $\pm$ 0.04
glass bronze 0.62 0.29 $\pm$ 0.01
glass sand 0.31 0.35 $\pm$ 0.03
glass $-$ 0.31 0.34 $\pm$ 0.04
salt $-$ 0.31 0.29 $\pm$ 0.01
----------------- ----------------- ------------------------------ -----------------
: \[Ttable1\]Collapse parameters for the self-similar spreading of radial cores in various grain types and rotation frequencies.
the time evolution of the $x$ integral of $h$, which increases with time. Figure \[Ffig1\]d shows that the $x$ integral of $h^2$ is constant in time, as it should be for a conserved quantity. This validates our assumption about the core shape, and demonstrates that $h^2$ can be used as a local concentration measure. The error in measurement of $h^2$ corresponding to an error in $h$ of $\pm$ 2 pixels.
For a normal diffusive process, the width of a narrow pulse initial condition grows as $t^{1/2}$. In our experiment, the pulse of small grains do not mix into the larger ones, but instead the pulse sinks below the surface of the larger grains forming a radial core, which then spreads axially. We can nevertheless ask if this axial spreading is analogous to normal diffusion, as is assumed in models[@Savage; @Ziketal; @Levinechaos; @ATprl; @ATVpre; @elperin]. Figure \[Ffig2\]a shows the radial core concentration profile at different times, for a mixture of small sand grains and large salt grains. Plotting the full-width at half-maximum of the concentration profile against time, we determined the power-law dependence of the radial core width with time, as shown in figure \[Ffig2\]b. From this, we determined that the width scales as $t^{\alpha}$, where $\alpha < 1/2$. This analysis, however, only determines the power-law time dependence of one arbitrarily chosen dimension of a pulse (here, the half-maximum width) and not the whole pulse shape. For a symmetric initial condition, data collapse can test the scaling of the entire pulse. Figure \[Ffig2\]c shows collapsed data corresponding to the concentration profiles in \[Ffig2\]a, where the axial length scale was transformed as $x \rightarrow {x}{t^{-\alpha}}$ and the axial concentration of small grains $C(x,t)$ was transformed as $C \rightarrow {C}{t^{\alpha}}$. The pulse width increases at the same rate as the pulse amplitude decreases, thus the spreading process is self-similar. This implies that the integrated concentration is constant and that no grains are lost from the core. The average collapse parameter for large salt grains and small sand grains with a drum rotation rate of 0.62 $\textrm{rev}/\textrm{s}$ is $\alpha = 0.37 \pm 0.03$, averaged over 10 runs. Similar experiments were repeated for different combinations of grains at two drum rotation rates. The results are shown in table 1 \[Ttable1\]. We conclude that cores of small grains spread axially as $t^{\alpha}$ where $\alpha \sim 1/3 < 1/2$, independent of grain type and drum rotation rate within the smoothly streaming regime.
It is interesting to compare the spreading of radially segregated cores of small grains with the non-segregating self-diffusion of the large grains alone. To observe this experimentally, some of the large grains were dyed black. These dyed grains were loaded into a drum full of otherwise identical white grains with a 1.5 mm wide pulse as the initial condition. The space-time evolution was observed using standard surface-lighting and imaging techniques [@KCprl; @KCpre]. Figure \[Ffig3\]a shows the concentration profile of dyed salt particles at various times. This data was collapsed in a similar way as discussed previously, as shown in figure \[Ffig3\]b. Again, we find a collapse parameter $\alpha < 1/2$. For runs using salt grains, $\alpha = 29 \pm 0.01 $ and for runs using glass spheres, $\alpha = 34 \pm 0.04 $, each averaged over 5 runs. Thus, we conclude that the self-diffusion of grains in the rotating drum is also subdiffusive, even when no segregation is involved. We discuss some differences between these two cases below.
In addition to examining the temporal scaling of the pulse, we can also measure in detail the functional shape of the scaling solution. Here is it possible to distinguish between different subdiffusive processes. We have investigated two candidate models for radial core spreading; the fractional diffusion equation (FDE) and the porous medium equation (PME). The fractional diffusion equation is $$\frac{\partial^{\gamma}}{\partial t^{\gamma}} C(x,t) = D\frac{\partial^2}{\partial x^2}C(x,t),$$ where $\gamma = 2\alpha$ denotes the order of a fractional time derivative[@wyss; @metzler]. Solutions of this linear equation have the property that the width of a narrow pulse initial condition grows as $t^{\alpha}$, where $\alpha \le 1/2$. If $\alpha = 1/2$, the solution reduces to normal Fick diffusion. This FDE model is often used to describe processes which occur in spaces where there are temporal or spatial constraints, such as the flow of tracers through porous media [@henry]. The FDE has an analytic series solution in terms of Fox’s H-Functions [@wyss; @metzler], which forms the self-similar scaling solution. We also examined the porous medium equation (PME), $$\frac{\partial}{\partial t}C(x,t) =\tilde D\frac{\partial^2}{\partial x^2}(C(x,t)^2).$$ This nonlinear model describes the spreading of a compact mound, and has the property that for a narrow pulse initial condition, the width grows as $t^{1/3}$, and the scaling solution has a parabolic profile[@barenblatt].
We fit radial core concentration data collapsed with $\alpha$ as a free parameter to the series solution of the FDE, and data collapsed with $\alpha = 1/3$ to a numerical solution of the PME, as shown in figure \[Ffig4\]a and b respectively. We find that while both solutions model the collapsed concentration profiles reasonably well within experimental error, the PME has a smaller systematic discrepancy, since the profiles are better described as parabolic. The FDE solution has exponential wings and inflection points that are not obvious in the data. We note, however, that our projection visualization technique may simply be too insensitive to detect these tails.
We also fit the non-segregating self-diffusion of the large grains to both models and find that the FDE gives a qualitatively better fit because in this case, the concentration profiles have tails within experimental resolution, while the parabolic PME solution does not. Examples of these fits to collapsed concentration profiles of mixing salt grains are shown in figures \[Ffig4\]c-d. In all cases, however, fits to ordinary Fick diffusion with $\alpha = 1/2$ are very poor.
In conclusion, our results show that the axial transport of grains in a rotating tube is a subdiffusive process. This is true of both small particles comprising a segregated radial core as well as for surface mixing of larger grains. In all cases, we find temporally self-similar concentration profiles that scale approximately as $t^{1/3}$. These conclusions suggest that spontaneous axial segregation patterns in such tubes are more weakly damped, in the sense that they are sustained against slower mixing processes, than has been previously supposed. The goal of our future work is to elucidate the connection between axial band formation and the axial transport of grains, which is still unclear.
We wish to thank Wayne Tokaruk, Mary Pugh and Frank Van Bussel. This work was supported by the Natural Science and Engineering Research Council of Canada.
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|
---
author:
- Vahid Mokhtari
- Luís Seabra Lopes
- 'Armando J. Pinho'
- Roman Manevich
bibliography:
- 'ref.bib'
title: |
Learning Task Knowledge and its Scope of Applicability\
in Experience-Based Planning Domains
---
=1
|
---
author:
- The ATLAS Collaboration
bibliography:
- 'sct2011.bib'
title: Operation and performance of the ATLAS semiconductor tracker
---
=1
Introduction
============
The SCT detector
================
Operation
=========
Offline reconstruction and simulation
=====================================
Monitoring and data quality assessment
======================================
Performance
===========
Detector occupancy
------------------
Noise
-----
Alignment stability {#sec:fsires}
-------------------
Intrinsic hit efficiency
------------------------
Lorentz angle
-------------
Energy loss and particle identification
---------------------------------------
Measurement of $\delta$-ray production
--------------------------------------
Radiation effects
=================
Conclusions
===========
The operation and performance of the ATLAS semiconductor tracker during 2009–2013 are described in this paper. During this period, more than 99% of detector modules were operational, and more than 99% of data collected by the ATLAS experiment had good SCT data quality. The frequency-scannning interferometry system showed the position of the detector to be stable at the micron level over long periods of time. Measurements of the increase in leakage currents with time are consistent with the radiation-damage predictions. The differences between data and simulation are typically less than 30%. This level of agreement exceeds expectations, and provides confidence in the fluence predictions. The verification of the simulations will be repeated at higher beam energies in future. Single event upsets have been identified and measured in the barrel module data, and a strategy for their mitigation implemented.
The detector occupancy was found to vary linearly with the number of interactions per bunch crossing, up to the maximum of 70 interactions per crossing, where it is less than 2% in the innermost barrel layer (which has highest occupancy). The intrinsic hit efficiency of the detector was measured to be (99.74$\pm$0.04)%, and the noise occupancies of almost all chips remained below the design requirement of $5 \times 10^{-4}$.
Measured values of the Lorentz angle are compatible with model predictions within at most twice the estimated uncertainties on those predictions. The measured values for sensors with $<$100$>$ crystal orientation are approximately $1^{\circ}$ lower than for those with $<$111$>$ crystal orientation, contrary to the expectation that a higher expected charge-carrier mobility in the sensors with $<$100$>$ crystal orientation should result in a higher value of the Lorentz angle.
Despite the binary readout, some particle identification from energy-loss measurements is possible: the discriminating power arises from the number of time bins above threshold and cluster widths. The position of the proton energy-loss peak was found to be stable at the 5–10% level during 2010–2012. The position of this peak may become a useful tool for monitoring radiation damage in future. The production of $\delta$-rays in the silicon sensors was measured, and is found to be in good agreement with expectations.
|
---
abstract: 'This note answers a question raised in [@FALC]: Is it consistent that for an arbitrary tall summable ideal $\I_g$ there exists an $\I_g$-ultrafilter which is not rapid? We show that assuming Martin’s Axiom for $\sigma$-centered posets such ultrafilters exist for every tall summable ideal $\I_g$.'
---
[Rapid ultrafilters and summable ideals]{}\
<span style="font-variant:small-caps;">Jana Flašková</span> [^1]\
[Department of Mathematics, University of West Bohemia]{}
Introduction
============
This note follows up the author’s paper “$\I$-ultrafilters and summable ideals" [@FALC] in which the connections between rapid ultrafilters and $\I_g$-ultrafilters have been studied. We will use the same notation and recall the most important definitions and facts in this introduction.
An ultrafilter $\U$ is called a [*rapid ultrafilter*]{} if the enumeration functions of sets in $\U$ form a dominating family in $({}^{\omega}\omega, \leq^{\ast})$, where the enumeration function of a set $A$ is the unique strictly increasing function $e_A$ from $\omega$ onto $A$. An ultrafilter $\U$ is called a [*$Q$-point*]{} if for every partition $\{Q_n: n \in \omega\}$ of $\omega$ into finite sets there is $A \in \U$ such that $|A \cap Q_n| \leq 1$ for every $n \in \omega$. Clearly, every $Q$-point is a rapid ultrafilter, but the converse is not true (see e.g. [@Mi]).
For a function $g: \omega \rightarrow (0,+\infty)$ such that $\sum\limits_{n \in \omega} g(n) = +\infty$ the family $$\I_g = \{A \subseteq \omega: \sum_{a\in A} g(a) < +\infty\}$$ is an ideal on $\omega$, which we call the [*summable ideal determined by function*]{} $g$. A summable ideal is tall if and only if $\lim\limits_{n \rightarrow \infty} g(n) = 0$.
The following description of rapid ultrafilters can be found in [@V]:
\[MainRapid\] For an ultrafilter $\U \in \omega^{\ast}$ the following are equivalent:
1. $\U$ is rapid
2. $\U \cap \I_g \neq \emptyset$ for every tall summable ideal $\I_g$
The definition of an $\I$-ultrafilter was given by Baumgartner in [@B]: Let $\I$ be a family of subsets of a set $X$ such that $\I$ contains all singletons and is closed under subsets. Given an ultrafilter $\U$ on $\omega$, we say that ${\U}$ is an [*$\I$-ultrafilter*]{} if for every function $F:\omega \rightarrow X$ there exists $A \in \U$ such that $F[A] \in \I$.
We say that an ultrafilter $\U$ is a [*hereditarily rapid ultrafilter*]{} if it is a rapid ultrafilter such that for every $\V \leq_{RK} \U$ the ultrafilter $\V$ is again a rapid ultrafilter. Since every hereditarily rapid ultrafilter is obviously a rapid ultrafilter, the existence of hereditarily rapid ultrafilters is not provable in ZFC because Miller proved in [@Mi] that there are no rapid ultrafilters in Laver model. On the other hand, every selective ultrafilter is hereditarily rapid, thus the existence of hereditarily rapid ultrafilters is consistent with ZFC.
The following characterization of hereditarily rapid ultrafilters follows from the definition, Theorem \[MainRapid\] and from the fact that the class of $\I$-ultrafilters is downwards closed with respect to the Rudin-Keisler order on ultrafilters.
\[herrapid\] For an ultrafilter $\U \in \omega^{\ast}$ the following are equivalent:
1. $\U$ is hereditarily rapid
2. $\U$ is an $\I$-ultrafilter for every tall summable ideal $\I$
It was proved in [@FALC] that $Q$-points (and consequently rapid ultrafilters) need not be $\I_g$-ultrafilters in a strong sense.
\[QnotI\] $(MA_{\small \rm ctble})$ There is a $Q$-point which is not an $\I_g$-ultrafilter for any summable ideal $\I_g$.
\[rapidnotI\] $(MA_{\rm ctble})$ For an arbitrary summable ideal $\I_g$ there exists a rapid ultrafilter which is not an $\I_g$-ultrafilter.
We also showed in [@FALC] that $\I_g$-ultrafilters need not be $Q$-points by the following counterpart of Theorem \[QnotI\].
\[InotQ\] $(MA_{\rm ctble})$ There exists $\U \in \omega^{\ast}$ such that $\U$ is an $\I_g$-ultrafilter for every tall summable ideal $\I_g$ and $\U$ is not a $Q$-point.
However, we did not prove a counterpart for Corollary \[rapidnotI\]. Asssuming Martin‘s axiom for countable posets an $\I_{1/n}$-ultrafilter which is not rapid was constructed in [@FTh], but the question remained open for an arbitrary tall summable ideal $\I_g$.
The aim of this note is to provide a construction of an $\I_g$-ultrafilter which is not rapid for an arbitrary tall summable ideal $\I_g$.
Some properties of the summable ideals
======================================
Let us first recall the definition of Katětov order $\leq_K$ for ideals on $\omega$: For $\I$ and $\J$ ideals on $\omega$ we write $\I \leq_K \J$ if there is a function $f: \omega \rightarrow \omega$ such that $f^{-1}[A] \in \J$ for all $A \in \I$.
The structure of the summable ideals ordered by Katětov order was investigated by Meza [@M-A]. We are particularly interested how the comparability of two ideals in Katětov order reflects to the inclusion of the corresponding classes of $\I$-ultrafilters.
Obviously, if $\I \leq_K \J$ then every $\I$-ultrafilter is a $\J$-ultrafilter. This implication cannot be reversed in general. However, in Theorem \[IgnotIh\] we prove that assuming Martin’s Axiom for $\sigma$-centered posets the converse is also true whenever $\I$ and $\J$ are tall summable ideals.
From now on all summable ideals will be tall and determined by a decreasing function $g$ (notice that every tall summable ideal can be mapped to such an ideal by a permutation). These ideals are invariant with respect to the translation which is formulated more precisely in the next lemma. For the sake of simplicity of its formulation let us fix the following notation: If $A$ is a subset of $\omega$ enumerated increasingly as $A=\{a_n: n \in \omega\}$ then $A+1 = \{a_n+1: n \in \omega\}$.
\[translates\] Assume $\I_g$ is a tall summable ideal determined by a decreasing function $g$, $A$ is a subset of $\omega$ and $B \subseteq A$. Then
1. $A \in \I_g$ if and only if $A+1 \in \I_g$
2. $A \in \I_g$ if and only if $B+1 \cup (A\setminus B) \in \I_g$
1\. Since the function $g$ is decreasing, $g(a_n) \geq g(a_n+1) \geq g(a_{n+1})$. Thus for every $A \subseteq \omega$ the following inequalities hold $$\sum_{a\in A} g(a) = \sum_{n \in \omega} g(a_n) \geq \sum_{n \in \omega} g(a_n+1) = \sum_{a\in A+1} g(a)$$ and $$\sum_{a\in A} g(a)= \sum_{n \in \omega} g(a_n) \leq g(0) + \sum_{n \in \omega} g(a_n+1)=g(0) + \sum_{a\in A+1} g(a)$$ which implies that $A \in \I_g$ if and only if $A+1 \in \I_g$.
2\. follows directly from 1.: $A \in \I_g$ if and only if both $B \in \I_g$ and $A \setminus B \in \I_g$. This is by 1. equivalent to $B+1 \in \I_g$ and $A \setminus B \in \I_g$ which holds if and only if $B+1 \cup A \setminus B \in \I_g$.
\[choosingA\] Assume $f \in \omega^{\omega}$, $\I_g$ and $\I_h$ are tall summable ideals with $\I_g \not\leq_K \I_h$. If $H$ is an infinite subset of $\omega$ such that $H
\not\in \I_h$ and $f[H] \not\in \I_g$ then there exists $A \subseteq f[H]$ such that $A \in \I_g$ and $f^{-1}[A] \cap H \not\in \I_h$.
Let us denote by $\EE$ the set of all even numbers and $\OO$ the set of all odd numbers.
Define $\tilde{f}: \omega \rightarrow \omega$ by $$\tilde{f}(n) = \left\{\begin{array}{ll}
f(n) & \hbox{if } n \in H \cap f^{-1}[\EE] \hbox{ or } n \in (\omega \setminus H) \cap f^{-1}[\OO] \\
f(n)+1 & \hbox{if } n \in H \cap f^{-1}[\OO] \hbox{ or } n \in (\omega \setminus H) \cap f^{-1}[\EE].
\end{array} \right.$$ Notice that the sets $\tilde{f}[H]$ and $\tilde{f}[\omega \setminus H]$ are disjoint because $\tilde{f}[H] \subseteq \EE$ and $\tilde{f}[\omega \setminus H] \subseteq \OO$. Since $f[H] \not\in \I_g$ and $\tilde{f}[H] = (f[H] \cap \EE) \cup (f[H] \setminus \EE)+1$, we have $\tilde{f}[H] \not\in \I_g$ by Lemma \[translates\].
It follows from $\I_g \not\leq_K \I_h$ that $\I_g \restriction \tilde{f}[H] \not\leq_K \I_h$, so there exists a set $\tilde{A} \in \I_g \restriction \tilde{f}[H]$ such that $\tilde{f}^{-1}[\tilde{A}] \not\in \I_h$. Put $A = f[\tilde{f}^{-1}[\tilde{A}]]$. It remains to verify that $A$ has all the required properties:
$\bullet$ $A \subseteq f[H]$ because $\tilde{f}^{-1}[\tilde{A}] \subseteq H$.
$\bullet$ $A \in \I_g$ by Lemma \[translates\] because $\tilde{A} \in \I_g$ and $\tilde{A} = (A \cap \EE) \cup (A \setminus \EE)+1$
$\bullet$ $f^{-1}[A] \cap H \supseteq \tilde{f}^{-1}[\tilde{A}] \cap H = \tilde{f}^{-1}[\tilde{A}]
\not\in \I_h$.
\[oneset\] ($MA_{\sigma-\hbox{\small\rm centered}}$) Assume $\I_h$ is a tall summable ideal and $\F$ is a filter base with $|\F| < \ccc$ such that $\F \cap \I_h = \emptyset$. Then there exists a set $H \subseteq \omega$ such that $H \not\in \I_h$ and $H \setminus F$ is finite for every $F \in \F$.
Define a poset $$\PP = \{\langle K, \D \rangle : K \in [\omega]^{<\omega}, \D \in [\F]^{<\omega}\}$$ with partial order given by $\langle K, \D \rangle \leq_{\PP} \langle L, \E \rangle$ iff $K \supseteq L$, $\min K \setminus L > \max L$, $K \setminus L \subseteq \bigcap \E$ and $\D \supseteq \E$. It is not difficult to see that $(\PP, \leq_{\PP})$ is a $\sigma$-centered poset.
Now for every $m \in \omega$ define $B_m = \{\langle K, \D \rangle \in \PP: \sum_{k \in K} h(k) \geq m\}$ and for every $F \in \F$ put $B_F =\{\langle K, \D \rangle \in \PP: F \in \D\}$.
[*Claim. $B_m$ and $B_F$ are dense in $\PP$ for every $m \in \omega$ and for every $F \in \F$.*]{}
Consider arbitrary $\langle L, \E \rangle \in \PP$. Since $\bigcap \E \not \in \I_h$ and $\sum_{k \in \bigcap \E} h(k) = +\infty$, there exists $L' \subseteq \bigcap \E$ with $\min L' > \max L$ such that $\sum_{k \in L'} h(k) \geq m$. Put $K = L \cup L'$ and notice that $\langle K, \E \rangle \leq_{\PP} \langle L, \E \rangle$ and $\langle K, \E \rangle \in B_m$. For the second part put $\D= \E \cup \{F\}$ and observe that $\langle L, \D \rangle \leq_{\PP} \langle L, \E \rangle$ and $\langle L, \D \rangle \in B_F$. $\Box$
According to the assumption $MA_{\sigma-\hbox{\small centered}}$ there exists a generic filter $\G$ on $\PP$. Define $G = \bigcup \{K \in [\omega]^{<\omega}: (\exists \D \in [\F]^{<\omega}) \langle K, \D \rangle \in \G\}$.
\(1) $G \not\in \I_h$
For every $m \in \omega$ and every $K \in \G \cap B_{m}$ we have $G \supset K$ and $\sum_{k \in K} h(k) \geq m$. Thus $\sum_{k \in G} h(k) = +\infty$ and $G \not\in \I_h$.
\(2) $(\forall F \in \F)$ $G \subseteq^{\ast} F$
For every $F \in \F$ there exists $\langle K_F, \D_F \rangle \in \G \cap B_F$. Because $\G$ is a filter for every $\langle K, \D \rangle \in \G$ there exists $\langle L_F, \E_F \rangle \in \G$ such that $\langle L_F, \E_F \rangle \leq_{\PP} \langle K, \D \rangle$ and $\langle L_F, \E_F \rangle \leq_{\PP} \langle K_F, \D_F \rangle$. It follows that $K \setminus K_F \subseteq L_F \setminus K_F \subseteq \bigcap \D_F \subseteq F$. Thus $G \setminus K_F \subseteq F$ and $G \subseteq^{\ast} F$.
Main result
===========
We will use the fact that rapid ultrafilters are precisely those ultrafilters which have nonempty interesection with every tall summable ideal. Thus in order to construct an $\I_g$-ultrafilter which is not rapid, we want to construct an $\I_g$-ultrafilter which has an empty intersection with another summable ideal $\I_h$.
\[Istep\] ($MA_{\sigma-\hbox{\small centered}}$) Assume $\I_g$ and $\I_h$ are two tall summable ideals such that $\I_g \not \leq_K \I_h$. Assume $\F$ is a filter base with $|\F| < \ccc$ such that $\F \cap \I_h = \emptyset$ and a function $f \in \omega^{\omega}$ is given. Then there exists $G \subseteq \omega$ such that $f[G] \in \I_g$ and $G \cap F
\not\in \I_h$ for every $F \in \F$.
We may apply Lemma \[oneset\] on $\F$ and $\I_h$. So there is a $H \not\in \I_h$ such that $|H \setminus F| < \omega$ for every $F \in \F$.
If $f[H] \in \I_g$ then put $G = H$.
If $f[H] \not\in \I_g$ we may apply Lemma \[choosingA\] which provides $A
\subseteq f[H]$ such that $A \in \I_g$ and $f^{-1}[A] \cap H \not\in \I_h$. Put $G = f^{-1}[A]$.
$\bullet$ $f[G] = A \in \I_g$
$\bullet$ Since $G \cap H \not\in \I_h$ and $(G\cap H) \setminus F$ is finite for every $F \in \F$ it follows that $G \cap F \not\in \I_h$ for every $F \in \F$.
\[IgnotIh\] ($MA_{\sigma-\hbox{\small centered}}$) For arbitrary tall summable ideals $\I_g$ and $\I_h$ such that $\I_g \not
\leq_K \I_h$ there is an $\I_g$-ultrafilter $\U$ with $\U \cap \I_h = \emptyset$.
Enumerate all functions in ${}^{\omega}\omega$ as $\{f_{\alpha}:\alpha <
\ccc\}$. By transfinite induction on $\alpha < \ccc$ we construct filter bases $\F_{\alpha}$ such that the following conditions are satisfied:
\(i) $\F_0$ is the Fréchet filter
\(ii) $\F_{\alpha} \supseteq \F_{\beta}$ whenever $\alpha \geq \beta$
\(iii) $\F_{\gamma} = \bigcup_{\alpha < \gamma} \F_{\alpha}$ for $\gamma$ limit
\(iv) $(\forall \alpha)$ $|\F_{\alpha}| \leq |\alpha + 1| \cdot \omega$
\(v) $(\forall \alpha)$ $\F_{\alpha} \cap \I_h = \emptyset$
\(vi) $(\forall \alpha)$ $(\exists F \in \F_{\alpha+1})$ $f_{\alpha}[F] \in \I_g$
Conditions (i)–(iii) allow us to start the induction and keep it going. Moreover (iii) ensures that (iv)–(vi) are satisfied at limit stages of the construction, so it is necessary to verify conditions (iv)–(vi) only at non-limit steps.
Induction step: Suppose we already know $\F_{\alpha}$.
Due to (iv) and (v) we may apply Lemma \[Istep\] to $f_{\alpha}$ and $\F_{\alpha}$. Let $\F_{\alpha+1}$ be the filter base generated by $\F_{\alpha}$ and $G$. The filter base $\F_{\alpha+1}$ satisfies (iv)–(vi).
Finally, let $\F = \bigcup_{\alpha < \ccc} \F_{\alpha}$. Because of condition (vi) every ultrafilter which extends $\F$ is an $\I_g$-ultrafilter. Because of condition (v) $\F$ has empty intersection with $\I_h$ and thus can be extended to an ultrafilter $\U$ with $\U \cap \I_h = \emptyset$.
\[noMinimals\] For every tall summable ideal $\I_g$ there is a tall summable ideal $\I_h$ such that $\I_g \not\leq_K \I_h$.
Since $\I_g$ is a tall summable ideal we may fix a partition of $\omega$ into finite consecutive intervals $I_n$, $n \in \omega$ such that
\(i) $I_0 \neq \emptyset$
\(ii) $|I_{n+1}| \geq n |\bigcup_{j \leq n} I_j|$ for every $n \in \omega$
\(iii) for every $n > 0$ if $m \in I_n$ then $g(m) < \frac{1}{2^n}$
Now define $h: \omega \rightarrow (0,\infty)$ by $$h(m) = \left\{\begin{array}{ll}
1 & \hbox{for } m \in I_0 \\
\frac{1}{n} & \hbox{for } m \in I_n \hbox{ with } n \geq 1
\end{array} \right.$$ It remains to verify that $\I_g \not\leq_K \I_h$. We will show that for every $f: \omega \rightarrow \omega$ there exists $A \in \I_g$ such that $f^{-1}[A] \not\in \I_h$.
Consider $f: \omega \rightarrow \omega$ arbitrary. For every $n \in \omega$ define $$B_n = \{m \in I_n: f(m) < \min I_n\} \quad \qquad C_n = \{m \in I_n: f(m) \geq \min I_n\}$$
[*Case I. $A_0 = \{n \in \omega: |B_n| \geq |C_n|\}$ is infinite*]{}
Since $B_n \cup C_n = I_n$ we have $|B_n| \geq \frac{1}{2} |I_n| \geq \frac{n}{2} |\bigcup_{j < n} I_j| = \frac{n}{2} (\min I_n - 1)$. Thus for every $n \in A_0$ there exists $m_n \in f[B_n]$ such that $|f^{-1}(m_n) \cap B_n| \geq \frac{n}{2}$.
If $A = \{m_n: n \in A_0\}$ is finite then, of course $A \in \I_g$. Otherwise there exists an infinite set $A \subseteq \{m_n: n \in A_0\}$ such that $A \in \I_g$ because $\I_g$ is a tall ideal. In both cases $\tilde{A}_0 =\{n \in A_0: m_n \in A\}$ is infinite and $f^{-1}[A] \not\in \I_h$ because $$\sum_{a \in f^{-1}[A]} h(a) \geq \sum_{n \in \tilde{A}_0} \sum_{a \in f^{-1}[A] \cap I_n} h(a) \geq \sum_{n \in \tilde{A}_0} |f^{-1}(m_n) \cap I_n| \cdot \frac{1}{n} \geq \sum_{n \in \tilde{A}_0} \frac{1}{2} = \infty$$
[*Case II. $A_0 = \{n \in \omega: |B_n| \geq |C_n|\}$ is finite*]{}
According to the assumption there is $n_0 \in \omega$ such that $|B_n| < |C_n|$ for every $n \geq n_0$. Pick $m_n \in C_n$ for every $n \geq n_0$. Put $M = \{m_n: n \geq n_0\}$ and $A = f[M]$. Since $m_n \in C_n$ one has $f(m_n) \geq \min I_n$ and therefore $g(f(m_n)) \leq \frac{1}{2^n}$. It is easy to see that $A \in \I_g$ because $$\sum_{a \in A} g(a) \leq \sum_{n \geq n_0} g(f(m_n)) \leq \sum_{n \geq n_0} \frac{1}{2^n} = \frac{1}{2^{n_0-1}}.$$ It remains to verify that $f^{-1}[A] \not\in \I_h$. To see this notice that $$\sum _{a \in f^{-1}[A]} h(a) \geq \sum_{a \in M} h(a) = \sum_{n \geq n_0} h(m_n) = \sum_{n \geq n_0} \frac{1}{n} = \infty.$$
\[Ignotrapid\] ($MA_{\sigma-\hbox{\small centered}}$) For an arbitrary tall summable ideal $\I_g$ there is an $\I_g$-ultrafilter which is not rapid.
This is an immediate consequence of Theorem \[IgnotIh\], Proposition \[noMinimals\] and the characterization of rapid ultrafilters in Theorem \[MainRapid\].
One possible generalization and its limits
==========================================
Once we have proved Theorem \[Ignotrapid\], which so to speak reverses Corollary \[rapidnotI\], we may ask whether it is possible that an ultrafilter is an $\I_g$-ultrafilter for “many" tall summable ideals simultaneously and still not a rapid ultrafilter. Certainly, “many" cannot mean all tall summable ideals, because of Theorem \[herrapid\]. We will show that in fact $\ddd$ many may be too much, but less than $\bbb$ is not.
\[atmostD\] There exists a family $\mathcal{D}$ of tall summable ideals such that $|\mathcal{D}| = \ddd$ and an ultrafilter $\U \in \omega^{\ast}$ is rapid if and only if it has a nonempty intersection with every tall summable ideal in $\mathcal{D}$.
Let us first construct the family $\mathcal{D}$: Assume $\mathcal{F} \subseteq {}^{\omega}\omega$ is a dominating family and $|\mathcal{F}| = \ddd$. Without loss of generality we may assume that all functions in $\mathcal{F}$ are strictly increasing and $f(j+1) \geq f(j)+j+1$ for every $j \in \omega$. For every $f \in \mathcal{F}$ define $g_f: \omega \rightarrow (0,+\infty)$ by $$g_f(m)=\left\{
\begin{array}{l @{\quad} l}
1 & \hbox{if } m < f(0) \\
\frac{1}{j+1} & \hbox{if } m \in [f(j),f(j+1))
\end{array}
\right.$$ Let $\mathcal{D}=\{\I_{g_f}: f \in \mathcal{F}\}$.
Now, one implication is clear since every rapid ultrafilter has a nonempty intersection with all tall summable ideals, in particular it has a nonempty intersection with every ideal from $\mathcal{D}$.
It remains to verify that if an ultrafilter has nonempty intersection with every ideal in $\mathcal{D}$, then it has nonempty intersection with all tall summable ideals and therefore is rapid. To this end, assume $\I_g$ is an arbitrary tall summable ideal. One can define a strictly increasing function $f_g$ such that for every $j \in \omega$:
- $f_g(j+1) \geq f_g(j)+j+1$
- if $m \geq f_g(j)$ then $g(m) \leq \frac{1}{2^j}$
Remember that family $\mathcal{F}$ was dominating. Hence there exists $f \in
\mathcal{F}$ and $k_0 \in \omega$ such that $f(k) \geq f_g(k)$ for every $k
\geq k_0$. For a every $n \geq f(k_0)$ there exists a unique $j \geq k_0$ such that $n \in [f(j),f(j+1))$. Since $n \geq f(j) \geq f_g(j)$ we get $g(n) \leq
\frac{1}{2^j} \leq \frac{1}{j+1} = g_f(n)$. From $g \leq^{\ast} g_f$ follows that $\I_{g_f} \subseteq \I_g$. Thus every ultrafilter $\U \in \omega^{\ast}$ which has a nonempty intersection with all ideals from $\mathcal{D}$ has a nonempty intersection with $\I_g$ and since $\I_g$ was arbitrary, $\U$ is a rapid ultrafilter,
\[Bcentr\] If $\mathcal{D}$ is a family of tall summable ideals and $|\mathcal{D}| < \bbb$ then there exists a tall summable ideal $\I_g$ such that $\I_g \subseteq \I_h$ for every $\I_h \in \mathcal{D}$.
For every $\I_h \in \mathcal{D}$ define a strictly increasing function $f_h \in
{}^{\omega}\omega$ such that whenever $m \geq f_h(j)$ then $h(m) \leq
\frac{1}{2^j}$.
According to the assumptions, the family of functions $\mathcal{F}=\{f_h: \I_h
\in \mathcal{D}\}$ is bounded, so there exists $f \in {}^{\omega}\omega$ such that $f_h \leq^{\ast} f$ for every $f_h \in \mathcal{F}$. We may assume that $f$ is strictly increasing. Define $g: \omega \rightarrow (0,+\infty)$ by $$g(m)=\left\{
\begin{array}{l @{\quad} l}
1 & \hbox{if } m < f(0) \\
\frac{1}{j+1} & \hbox{if } m \in [f(j),f(j+1))
\end{array}
\right.$$
For a given function $f_h \in \mathcal{F}$ there exists $k_h \in \omega$ such that $f_h(k) \leq f(k)$ for every $k \geq k_h$. For every $n \geq f(k_h)$ there is exactly one $j \geq k_h$ such that $n \in [f(j),f(j+1))$. Since $n
\geq f(j) \geq f_h(j)$ we get $h(n) \leq \frac{1}{2^j} \leq \frac{1}{j+1} =
g(n)$. From $h \leq^{\ast} g$ follows that $\I_g \subseteq \I_h$.
($MA_{\sigma-\hbox{\small centered}}$) If $\mathcal{D}$ is a family of tall summable ideals and $|\mathcal{D}| < \ccc$ then there exists an ultrafilter $\U \in \omega^{\ast}$ such that $\U$ is an $\I$-ultrafilter for every $\I \in \mathcal{D}$, but $\U$ is not a rapid ultrafilter.
Combine Theorem \[Ignotrapid\] and Proposition \[Bcentr\] and the fact that $\bbb = \ccc$ under $MA_{\sigma-\hbox{\small centered}}$.
Open questions
==============
Let $\mathcal{D}$ be a family of tall summable ideals.
What is the minimal size of the family $\mathcal{D}$ such that rapid ultrafilters can be characterized as those ultrafilters on the natural numbers which have a nonempty intersection with all ideals in the family $\mathcal{D}$?
Due to Proposition \[atmostD\] the size of such a family is at most $\ddd$. But is $\ddd$ really the minimum?
[9]{}
J. Baumgartner, Ultrafilters on $\omega$, [*J. Symbolic Logic*]{} [**60**]{}, no. 2, 624–639, 1995.
J. Flašková, Ultrafilters and small sets, [*Ph.D. thesis,*]{} Charles University, Prague 2006.
J. Flašková, $\I$-ultrafilters and summable ideals, in: [*Proceedings of the 10th Asian Logic Conference*]{} (Kobe 2008), 113 – 123, World Scientific, Singapore, 2010.
D. Meza Alcántara, Ideals and filters on countable sets. [*Ph.D. thesis.*]{} UNAM México, 2009.
A. W. Miller, There are no $Q$-points in Laver’s model for the Borel conjecture, [*Proc. Amer. Math. Soc.*]{} [**78**]{}, no. 1, 103–106, 1980.
P. Vojtáš, On $\omega^{\ast}$ and absolutely divergent series, [*Topology Proceedings*]{} [**19**]{}, 335 – 348, 1994.
[^1]: Work done patially during a visit to the Institut Mittag-Leffler (Djursholm, Sweden) and partially supported from the European Science Foundation in the realm of the activity entitled ’New Frontiers of Infinity: Mathematical, Philosophical and Computational Prospects’.
|
---
abstract: 'The supplementary materials describe the evolution of the non-linear transport in non-local geometry, from a series of resonances corresponding to escape and creation of edge channels (presented in the main article) towards a zero differential resistance state when the voltage drop is measured on length scales much larger than the mean free path. The magnetic field ($B \rightarrow -B$) and DC current ($I \rightarrow -I$) symmetry properties of the reported zero-differential state strongly supports its edge transport origin. Finally we provide a more detailed derivation for the equations of the continuum theory.'
author:
- |
A.D. Chepelianskii$^{(a,b)}$, J. Laidet$^{(c)}$, I. Farrer$^{(a)}$, D.A. Ritchie$^{(a)}$, K. Kono$^{(b)}$, H. Bouchiat$^{(c)}$\
(a) Cavendish Laboratory, University of Cambridge, J J Thomson Avenue, Cambridge CB3 OHE, UK\
(b) Low Temperature Physics Laboratory, RIKEN, Wako, Saitama 351-0198, Japan\
(c) LPS, Univ. Paris-Sud, CNRS, UMR 8502, F-91405, Orsay, France\
title: 'Quantized escape and formation of edge channels at high Landau levels : supplementary materials'
---
I. Non local differential resistance with distant voltage probes
================================================================
![Dependence of the non local differential resistance $d V_{nl;F}/d I$ on magnetic field and DC current amplitude, this quantity was measured in a geometry where the separation between voltage probes was $D_x \simeq 500\;{\rm \mu m}$ on the $\mu = 10^7\;{\rm cm^2/Vs}$ sample from the main text. Temperature was $T = 1.2\;{\rm K}$. []{data-label="Supp1"}](megadvEKdiLMVxfig){width="8.5cm"}
We have measured the non local differential resistance (NLDR) $d V_{nl;F}/d I$ in a geometry where the voltage probes were separated by a distance $D_x = 500\;{\rm \mu m}$ larger than the mean free path $\ell_e = 100\;{\rm \mu m}$ in the sample. The experiment was performed on the same sample as in the main text but with a different arrangement of voltage and current probes, the current sources were located $500\;{\rm \mu m}$ away from the voltage probes. In the linear response regime the dependence of $R_{nl;F} = d V_{nl;F}/d I(I = 0)$ on the magnetic field, was very similar to the data shown on Fig. 2 (from main article). The quantity $R_{nl:F}$ was finite for positive magnetic fields and almost vanished for $B < 0$. The dependence of $d V_{nl;F}/d I$ on the magnetic field $B$ and on the DC current amplitude $I$ is represented on Fig. \[Supp1\] for $B > 0$. The oscillating features as function of the DC current $I$ are not resolved contrarily to measurements where $D_x$ was smaller than the mean free path (see data on Fig. 3 from main article). Negative values of $d V_{nl;F}/d I$ at negative current $I$ are still observed in this geometry.
II. Formation of zero differential resistance states in a macroscopic sample
============================================================================
![Dependence of NLDR $d V_{nl;L}/d I$ on magnetic field and DC current amplitude for the $\mu = 3 \times 10^6\;{\rm cm^2/Vs}$ mobility sample. NLDR was measured in the geometry sketched in the top panel, temperature was $T = 0.3\;{\rm K}$.[]{data-label="Supp2"}](figNonLocLow){width="8.5cm"}
We have also studied NLDR in a macroscopic geometry with geometrical parameters larger than the mean free path. The sample was made in a lower mobility 2DEG, with mobility $\mu = 3 \times 10^6\;{\rm cm^2/Vs}$ and a carrier density of $n_e = 3.2\;\times 10^{11} {\rm cm}^{-2}$. The geometry of the measurement is sketched in in Fig. \[Supp2\]. This figures summarizes our results on the NLDR in this sample, for positive magnetic fields for which the non local resistance is non-vanishing.
The strong asymmetry between positive and negative currents is also observed in this lower mobility 2DEG, however the characteristic magnetic field where the asymmetry appears is around a factor three stronger as compared to the $\mu = 10^7\;{\rm cm^2/Vs}$ sample, this difference is consistent with the ratio between the mobilities of the two samples. As in Fig. \[Supp1\], the separation between the voltage probes was larger than the mean free path $\ell_e = 30\;{\rm \mu m}$ and the oscillations as a function of the DC current cannot be resolved. However, in the present experiment NLDR is almost zero in a large region of negative currents which contrasts with previous data where NLDR could be negative for $I < 0$ (see Fig. 2 from the main article and Fig. \[Supp1\]).
In order to highlight the presence of a zero differential resistance state (ZDRS), we have calculated the dependence of $V_{nl;L}$ on current by integrating the experimental differential resistance data. The results obtained after this procedure are represented on Fig. \[Supp3\] which shows that the voltage $V_{nl;L}$ exhibits a plateau at negative $I$ where it is almost independent on current in a wide range of magnetic fields while for positive currents the voltage dependence is almost ohmic. The inset in Fig. \[Supp3\], shows the dependence of the voltage on the magnetic field for several values of current inside the ZDRS plateau. These results confirm that the voltage saturates to a constant value independent on current in this regime, the value of the saturation voltage grows almost linearly with magnetic field with weak oscillations that are probably related to the Shubnikov-de Haas oscillations in the longitudinal resistance.
![Non local voltage/current characteristics $V_{nl;L}(I)$ for the $\mu = 3 \times 10^6\;{\rm cm^2/Vs}$ mobility sample at several magnetic fields (for resemblance with data from ZDRS experiments in local geometries, we have shown $-V_{nl;L}$ as function of $-I$ in this figure). The inset shows the voltage as a function of magnetic field for several currents inside the plateau regime. Temperature was $T = 0.3\;{\rm K}$.[]{data-label="Supp3"}](figVofHhighfield){width="9cm"}
The observed zero-differential state possesses the symmetry of an edge effect. It appears only for the sign of magnetic field which ensures guiding towards the voltage probe electrodes from the distant current sources, and for a specific sign of the DC current that creates a voltage drop along the edge tending to stabilize propagation along edges. Therefore it seems likely that an edge transport related mechanism is leading to the formation of ZDRS in this case. On the higher mobility sample where the dimension of the voltage probes were smaller than the mean free path, negative values of NLDR were observed (see Fig. 3 from main article and Fig. \[Supp1\]), this suggests that ZDRS is formed due to the clamping of the potential on large length scales by the voltage probe electrodes. On the contrary, if the electrodes are not invasive the potential exhibits sharp variations whenever the energy of the electrons propagating along the edge is changed by an amount close to $\hbar \omega_c$ (see main text). These voltage oscillations are probably indicative of a spatially modulated charge density distribution, and could explain the observation of oscillating/negative differential resistances in our experiments. It would be highly interesting to understand the role played by the edge mediated ZDRS mechanism in ZDRS experiments realized in the conventional longitudinal resistance measurement geometry. However due to the absence of a reliable theoretical framework to describe the edge effects reported in this article, it is not possible to estimate the amplitude of their contribution in the measurement of longitudinal resistance.
III. Continuum theory
=====================
In this section we provide a mode detailed derivation of formulas from continuum theory that we used in the main article.
We start our calculations from the potential created by a point source of current I located at z = 0 in a semi-infinite two dimensional electron gas. It is convenient to represent points in the 2DEG as complex numbers z = x + iy where (x, y) are the point Cartesian coordinates, and the half plane fills the space $y > 0$. In this case we find the potential $V_p(z) = R_p(z) I$ with: $$\begin{aligned}
R_p(z) &= \frac{\rho_{xx}}{\pi} \left( \log |z| + \alpha \arg z \right) \end{aligned}$$ where we have introduced the Hall angle $\alpha = \frac{\rho_{xx}}{\rho_{xy}}$.
A stripe geometry described by $z = x + i y$ with $y \in (0, W)$ can be mapped onto this half plane using the conformal mapping $z′ = \exp\left(\frac{\pi z}{W}\right)$. This allows to find the potential $V_-(z, x_0) = R_-(z,x_0) I$ created by a point source located on the bottom edge of the stripe at $z = x_0$ ($x_0$ real): $$\begin{aligned}
V_-(z, x_0) = R_p( \exp(\frac{\pi z}{W}) \exp(\frac{-\pi x_0}{W}) - 1) I.\end{aligned}$$
The potential $V_+(z, x_0) = R_+(z,x_0) I$ created by a source on the top edge of the stripe at $z = x_0 + i W$ reads: $$\begin{aligned}
V_+(z, x_0) = R_p( \exp(\frac{\pi z}{W}) \exp(\frac{-\pi x_0}{W}) + 1) I.\end{aligned}$$
Subtracting these two expressions we find the potential $V = V_+(z, 0) − V_−(z, 0)$ created by a current between point-like sources and drains located opposite to each other along the channel (respectively at $z = i W$ and $z = 0$). For the particular case of the potential generated along the top edge $y = i W$ , far from the sources $|x| \gg W$, we find the following expression: $$\begin{aligned}
V(x) = \frac{2}{\pi} I \rho_{xx} \exp\left( \frac{-\pi |x|}{W} \right) - \rho_{xy} I \eta(-x)\end{aligned}$$
where $\eta(x)$ is the Heaviside function. This gives the expression for the nonlocal resistance given in the main text: $$\begin{aligned}
R_{nl} = \frac{2 \rho_{xx} D_x}{W} \exp\left( -\frac{\pi L}{W} \right)\end{aligned}$$
where $D_x$ is the spacing between the voltage probes and $L$ is their distance from the source along the channel (for simplicity we have assumed $D_x \ll W$).
|
---
author:
- 'T. Maiolino, P. Laurent, L. Titarchuk, M. Orlandini, F. Frontera.'
bibliography:
- 'bibliografiaarxiv.bib'
title: 'Red-skewed K$\alpha$ iron lines in GX 13+1.'
---
Introduction {#sec:introduction}
============
The strong Fe $K_\alpha$ emission line (in the $\sim6.4-7.0$ keV X-ray energy band) has been observed as broadened, asymmetric, and red-skewed in a number of sources, which include extragalactic and Galactic accreting X-ray compact objects: Seyfert-type I active galactic nuclei (AGNs) [e.g., @Tanaka1995b; @Nandra1997; @Fabian2000; @Reynolds2003; @Miller2007; @Hagino2016], black hole (BH) low-mass X-ray binaries (LMXBs) [e.g., @Fabian1989; @Miller2002b; @Miller2007; @Reynolds2003], and neutron star (NS) LMXBs [e.g., @Bhattacharyya2007; @Cackett2008; @Pandel2008; @Shaposhnikov2009; @Reis2009; @Cackett2010; @Ludlam2017; @LudlamXTE2017].
For accreting NSs and BHs, the standard interpretation for the Fe line asymmetry invokes general relativity (GR) effects through the strong gravitational field that is present close to the compact object, where the emission line is assumed to be produced. Although the relativistic interpretation is commonly accepted to explain the asymmetry of the observed iron line profiles, there are some considerations before this prevailing scenario can be accepted:
1. [At the time of writing, the relativistic line interpretation has some difficulties to explain all the characteristics that are observed in these sources (e.g., X-ray variability, see section \[sec:NSline\]).]{}
2. [Fe line profiles with similar characteristics (i.e., broadened and red-skewed) were observed in the X-ray spectra of accreting white dwarfs (WDs) [@Hellier2004; @Vrielmann2005; @Titarchuk2009 Maiolino et al. in preparation]. These systems require an alternative explanation for the line profile because GR effects do not play a role around WDs.]{}
3. [A line model based on relativistic effects is not unique in describing the broad and asymmetric Fe K emission line profiles [e.g., @Hagino2016]]{}.
An alternative model, called the wind line model (<span style="font-variant:small-caps;">windline</span>, hereafter; presented by @Laurent2007), is able to explain the red-skewed Fe emission line profile observed in NS and BH LMXBs [@Titarchuk2009; @Shaposhnikov2009], and in systems containing WD as the accreting compact object (i.e., cataclysmic variables) because the line model does not require GR effects [@Titarchuk2009]. In the <span style="font-variant:small-caps;">windline</span> framework, the asymmetric and red-skewed line profile is produced by repeated down-scattering of the line photons by electrons in a diverging outflow (wind). The line is generated in a partially ionized and thin inner shell of the outflow by irradiation of hard X-ray photons coming from the central source [for more details about the model, see @Laurent2007 and discussion therein].
Motivated by the potential of the <span style="font-variant:small-caps;">windline</span> model in describing red-skewed iron line profiles in all three types of accreting X-ray powered sources, we applied this model on the residual excess in the Fe K energy range. This excess is observed in the spectra of all types of Galactic accreting compact objects: NSs, BHs, and WDs. In this first paper we present the investigation of the iron emission line profile in the spectrum of the LMXB GX 13+1, which hosts a neutron star as a compact object.
GX 13+1 is a perfect target for testing the <span style="font-variant:small-caps;">windline</span> model because strong red-skewed Fe emission lines are reported in its spectra [e.g., @Trigo2012], with simultaneous observations of blueshifted Fe absorption lines. This indicates an outflow around the source [@Boirin2005], (see section \[sec:gx13p1\]).
This paper is structured as follows: in the section \[sec:NSline\] we present a concise background on broad and red-skewed iron lines in accreting NSs and their relevance. In section \[sec:windXSPEC\] we describe the parameters of the <span style="font-variant:small-caps;">windline</span> and in \[sec:gx13p1\] the characteristics of the source GX 13+1. In section \[sec:data\] we show the XMM-Newton observations that we analyzed and the data reduction. In section \[sec:datanalysis\] we describe the data analysis and compare the <span style="font-variant:small-caps;">windline</span> and the relativistic <span style="font-variant:small-caps;">diskline</span> fits. We discuss our results and conclude in section \[sec:dis\].
Broad and red-skewed iron lines in accreting neutron stars {#sec:NSline}
----------------------------------------------------------
Fluorescent Fe K emission line profiles were first observed as broadened, asymmetric, and red-skewed in Galactic and extragalactic accreting BHs; in LMXBs and Seyfert-type I galaxies [e.g., @Fabian1989; @Tanaka1995b; @Miller2007 and references therein]. Because the Fe K band was successfully modeled by a relativistic line model, this led to the conclusion that the breadth and asymmetry of the line in BH sources is due to effects of Doppler-broadening, transverse Doppler shift, relativistic beaming, and gravitational redshifts produced in the innermost part of the accretion disk, where GR effects are playing a role.
The subsequent discovery of red-skewed Fe emission line in the NS LMXB Serpens X–1 [@Bhattacharyya2007; @Cackett2008], followed by the observation in a few more NS LMXB sources, and the satisfactory fits of the line profiles using the relativistic line model extended the interpretation of the asymmetric line profiles in terms of GR effects on accreting NSs.
In the framework of the relativistic models, the fluorescent Fe emission line is produced by the reflection of hard photons (described by a power-law spectrum, coming from the inner part of the source) by the iron (ions) on the surface of a cold ($<$ 1 keV) accretion disk. The relativistic scenario is even now commonly accepted as an explanation for red-skewed Fe K emission lines in NS (and BH) sources. The line in NS LMXBs is usually modeled assuming a Schwarzschild potential around the accreting compact object (<span style="font-variant:small-caps;">diskline</span> model [@Fabian1989]). However, models considering a Kerr metric, created primarily for describing the emission line profiles coming from the accretion disks around rotating BHs, have also been used to fit to the iron emission lines in accreting NSs, see, for example, the <span style="font-variant:small-caps;">laor</span> [@Laor1991], <span style="font-variant:small-caps;">relconv</span> [@Dauser2010], and <span style="font-variant:small-caps;">relxill</span> [@Garzia2014] models. The main motivation for applying the relativistic scenario to NS sources is the fact that the inner disk radii predicted by the relativistic model have been found to agree well with the radii implied by kilohertz quasi-periodic oscillation (kHz QPO) frequency in 4U 1820-30 and GX394+2 sources as a Keplerian frequency at those radii (supporting the inner disk origin for kHz QPOs) [@Cackett2008].
However, @Shaposhnikov2009 and @Titarchuk2009, using *Suzaku* and *XMM-Newton* data, examined the red-skewed line profile observed in the spectra of two NS LMXB sources, Cyg X–2 and Serpens X-1, in the framework of the <span style="font-variant:small-caps;">windline</span> model. They found that this nonrelativistic model is able to reproduce the red-skewed line profile with a fit quality (on the basis of $\chi^2$-statistic) similar to that obtained by relativistic reflection models. Although they were unable to conclusively rule out one of the models, they pointed out that the <span style="font-variant:small-caps;">windline</span> model appears to give a more adequate explanation because it does not require the accretion disk inner edge to advance close to the NS surface. In addition, the <span style="font-variant:small-caps;">windline</span> model explains a timing source property that the <span style="font-variant:small-caps;">diskline</span> model has difficulty to account for. They verified a lack of erratic fast variability when the Fe line is present in the Cyg X–2 spectrum (which was also observed in a BH LMXB source, GX 339$-$4; [@Titarchuk2009]). This result weakens the red-skewed line connection with kHz QPOs and strengthens its connection to outflow phenomena because the suppression of fast variability can be explained by smearing in a strong outflow (wind) [see Figure 8, Sections 3.1 and 3.2 in @Titarchuk2009].
Furthermore, @Lyu2014 used *Suzaku*, *XMM-Newton*, and *RXTE* observations of the NS 4U 1636-53 to study the correlation between the inner disk radius (obtained from the Fe line relativistic fits) and the flux. They observed that the line did not change significantly with flux states of the source and concluded that the line is broadened by mechanisms other than just relativistic broadening. This behavior is not only observed in NS LMXBs. For example, observations of MCG–6-30-15, NGC 4051, and MCG–5-23-16 [@Reynolds1997; @Reeves2006; @Marinucci2014] show that the broad and red-skewed Fe lines in AGNs show little variability despite the large changes in the continuum flux. The iron line and the reflected emission flux do not respond to the X-ray continuum level of variability on short timescales [@Reeves2006], which means that the fast changes that are almost certain to occur near the compact object are not observed in the line. This point is difficult to explain by the reflection and GR interpretation of the iron line formation, and has been suggested to be caused by strong gravitational light bending [see @Miniutti2004; @Chiang2011].
In addition, @Mizumoto2018 demonstrated that the short lag time of the reprocessed Fe-K line energy band observed in AGNs can also be explained by scattering of photons coming from the X-ray central source by an outflow or disk-wind (with velocity of $\sim$ 0.1c, placed at large distances, $\sim$ 100 R$_g$, from the compact object). Their simulations resulted in features in the lag-energy plot and characteristics similar to those observed in 1H0707-495 and NGC 4151. They showed that the observation of short lag times does not necessarily indicate that the line is produced very close to the event horizon of the BH, as expected by the relativistic scenario [see @Mizumoto2018 and references therein].
As previously mentioned, the prevailing interpretation that has been used to model the red-skewed Fe line profiles in accreting NS (and also BHs) with reflection and GR effects is not entirely successful, and there is the alternative <span style="font-variant:small-caps;">windline</span> model that does not need relativistic effects, but successfully describes the observed line profiles in these sources. The importance to make this comparative study between the models is not just to either confirm or reject the relativistic framework in NS LMXBs, but also to confirm or reject the implications to the NS physics derived by the models. By interpreting the broadening and red-skew of the Fe K emission line as due to GR effects, the relativistic models allow us to derive directly from the spectral fit parameters such as disk inclination, spin of the compact object, and inner radius of the accretion disk. The inner radius has been used to set an upper limit on the magnetic field strength and on the radius of the neutron stars, which in turn is used to study constraints on the NS equation of state [@Ludlam2017]. Although the relativistic models allow this powerful derivation, it is important to study the asymmetry of the Fe K line with other models that provide an alternative as well as a more consistent physical scenario of the NS LMXBs spectral emission.
<span style="font-variant:small-caps;">windline</span> model into XSPEC {#sec:windXSPEC}
-----------------------------------------------------------------------
The main obstacle to analyzing the spectral data with the <span style="font-variant:small-caps;">windline</span> model is that the analytical solution is not available [@Shaposhnikov2009]. Therefore, the <span style="font-variant:small-caps;">windline</span> model was inserted within XSPEC with a number of FITS tables containing the results of the wind line model implemented using the Monte Carlo approach [@Laurent2007]. Each FITS table is constructed for a fixed emission line energy, running a grid of values for the free parameters. The free parameters of the model are as follows:
- the optical depth of the wind/outflow ($\tau_w$);
- the temperature of the electrons in the wind/outflow ($kT_{ew}$ in keV);
- the velocity ($\beta$) of the wind/outflow in unity of $c$ (speed of light);
- the redshift of the source; **
- the normalization (which is the number of photons in the whole spectrum, in units of photon/keV).
In this model, the line profiles strongly depend on the optical depth $\tau_w$ and on the velocity $\beta$ (v/c) of the outflow (wind). The shape of the red wing, below the broad peak, follows a power law with an index that is a strong function of $\tau_w$ and $\beta$. If the temperature of the electrons in the wind is low, kT$_{ew} < 1$ keV, the temperature does not change the line profile much, and most of the effect is due to the wind velocity, that is, the velocity effect is strong enough to change the red wing of the line at the observed level.
Source GX 13+1 {#sec:gx13p1}
--------------
The source GX 13+1 is classified as a LMXB with a NS as a compact object. In LMXBs the secondary star (which is an evolved late-type K5 III giant star in GX 13+1) fills the Roche lobe and transfers mass onto the compact object through the inner Lagrangian point (L1), feeding the primary star through an accreting disk [@Lewin2006].
This source is located at a distance of $7 \pm 1$ kpc and is also classified as a type I X-ray burster [@Fleischman1985; @Matsuba1995]. Its bright persistent X-ray emission shows characteristics of both atoll and Z LMXBs [see, e.g., @Trigo2012; @Fridriksson2015 and references therein]. GX 13+1 is likely an X-ray dipping source. Dipping sources are binary systems with shallow X-ray eclipses, so called dippers. These eclipses are likely caused by the periodic obscuration of the central X-ray source by a structure called bulge (or hot spot), or by outflows in the outer disk [@Lewin1995; @pintore2014]. The bulge is created by the collision between the accretion flow coming from the secondary star with the outer edge of the accretion disk, and it contains optically thick material. Therefore, the observation of dippers is associated with LMXB sources with very high inclination. The strong energy-dependent obscuration in GX 13+1, the absence of eclipses, and the spectral type of the secondary star indicate an inclination of 60-80$^{\circ}$ [@Trigo2012].
The source GX 13+1 is known as one of the two NS LMXBs showing winds [@Trigo2012; @Ueda2004]; the other source is IGR J17480-2446 [@Miller2011]. A thermal driving mechanism of the wind is expected [see discussion in @Trigo2012 and references therein].
Spectral characteristics of GX 13+1 {#spectral-characteristics-of-gx-131 .unnumbered}
-----------------------------------
The continuum emission of GX 13+1 has previously been modeled with a multicolor blackbody component plus either a blackbody or a nonthermal component (such as Comptonization, power law, or cutoff power law) without significantly worsening the quality of the fit [see @Trigo2012; @pintore2014 and references therein].
Several narrow absorption lines, characteristics of dipping sources, are observed in the energy spectrum and are associated with a warm absorber (highly photoionized plasma) around the source, driven by outflows from the outer regions of the accretion disk [@pintore2014; @Trigo2012]. The absorption lines indicate bulk outflow velocities of $\sim$ 400 km s$^{-1}$ [see @pintore2014 and references therein].
Narrow resonant absorption lines near 7 keV were found with an ASCA observation of GX 13+1 [@Ueda2001]. Afterward, K$_{\alpha}$ and K$_{\beta}$ lines of He- and H-like Fe ions, the K$_{\alpha}$ line of H-like Ca (Ca XX) ion, and a deep Fe XXV absorption edge (at 8.83 keV) were observed with XMM-Newton [@Sidoli2002]. @Trigo2012 additionally observed a Fe XXVI absorption edge (at 9.28 keV). The depth of the absorption features in the XMM-Newton 2008 observations changed significantly on timescale of a few days, and more subtle variations are seen on shorter timescales of a few hours [@Sidoli2002]. Because the narrow absorption lines are observed throughout the orbital cycle, the absorption plasma is likely cylindrically distributed around the compact object [see @pintore2014 and references therein].
Another spectral feature that is superposed on the continuum and the narrow absorption lines is the broad emission line in the K-shell Fe XVIII-XXVI energy range ($\sim$ 6.4 - 7.0 keV). This emission line is the main feature we study here. It is usually modeled, as well as in all NS sources, by relativistic line models (see section \[sec:NSline\]).
XMM Newton observations {#sec:data}
=======================
We analyzed two public XMM-Newton EPIC-pn observations of GX13+1. The first observation, Obs. ID 0505480101 (hereafter called Obs. 1), has 13.8 ks EPIC-pn exposure time, starting on 2008 March 09 at 18:24:01 and ending at 22:48:30 UTC. The second observation, Obs. ID 0505480201 (hereafter called Obs. 2) has 13.7 ks EPIC-pn exposure time, starting on 2008 March 11 at 23:48:02 and ending at 03:15:09 UTC of the next day (see Table \[tabObs.\] for a log of the XMM-Newton observations used in this analysis). Both observations were taken with the camera operating in timing mode.
All the five public XMM-Newton EPIC-pn observations analyzed by @Trigo2012 were previously considered for this study. However, a pre-analysis of the spectra led to the selection of the two observations used in this paper. Because all observations are affected by pile-up, we extracted two spectra for each observation, A and B, respectively. A different RAWX range was excised for each during the data processing for pile-up correction: one column was excised for A spectra from the bright central region, and either two or three RAWX columns were excised for B spectra. We fit the A and B spectra with the following total model: <span style="font-variant:small-caps;">tbabs\*edge$_1$\*edge$_2$\*(diskbb+bbodyrad+gaussians)</span>, which corresponds to *Model 1* in @Trigo2012 (*Model 1*; hereafter). The Gaussian components in this case correspond to negative Gaussians. We used this total model to check for the iron emission line because the absorption lines can mimic the emission line. To select the observations, we used the following criterion: observations in which the line barely appears, that is, with a residual fluctuation < $2.5-3.0~\sigma$, when one or two columns are excised, were discarded. It is important to state that when the pile-up correction was performed excising five columns, as performed by @Trigo2012, the emission line disappears in all observations. Therefore, we extracted two columns in the selected two observations (see section \[sec:datared\]) to find a balance between pile-up correction and the presence of the line. The continuum is not strongly affected by the difference in the RAWX extraction in these two observations (i.e., by excising two or five columns).
The pile-up correction makes the spectra softer. Consequently, if the line is weak in the observation, it decreases the evidence of its presence. In addition, pile-up correction can deteriorate the statistics of the counts in the hard part of the spectrum, which contributes to a poorer constraining of the continuum model. This in turn affects the emission line presence perception and its shape definition.
-------- ---------------- --------------------------- -------------------------- ------
Obs. observation ID Exposure Start Time (UTC) Exposure End Time (UTC) Exp.
year month day hh:mm:ss year month day hh:mm:ss (ks)
Obs. 1 0505480101 2008 March 09 18:24:01 2008 March 09 22:48:30 13.8
Obs. 2 0505480201 2008 March 11 23:48:02 2008 March 12 03:15:09 10.3
-------- ---------------- --------------------------- -------------------------- ------
Data reduction {#sec:datared}
--------------
All light curves and spectra were extracted through the Science Analysis Software (SAS) version 14.0.0. Following the recommendation for the pn camera on the timing mode of observation, we selected events in the 0.6-10 keV energy range to avoid the increased noise, and only single and double events were taken into account for the spectrum extraction (pattern in \[1:4\]). In order to provide the most conservative screening criteria, we used the FLAG=0 in the selection expression for the standard filters, which excludes border pixels (columns with offset), for which the pattern type and the total energy is known with significantly lower precision (see XMM-SOC CAL-TN-0018, and The XMM-Newton ABC Guide).
The calibration of the energy scale in EPIC-pn in timing mode requires a complex chain of corrections, which are important to yield a more accurate energy reconstruction in the iron line (6 - 7 keV) energy range (see XMM-SOC-CAL-TN-0083). In order to achieve the best calibration, we applied a) the X-ray loading (XRL) correction through the parameters runepreject=yes and withxrlcorrection=yes; b) the special gain correction, which is default in SASv14.0 (and can be applied through the parameter withgaintiming=yes); and c) the energy scale rate-dependent PHA (RDPHA) correction, which was introduced in SASv13.0, and taken as default in SASv14.0 for Epic-pn timing mode (XMM-SOC-CAL-TN-0018). This correction is expected to yield a better energy scale accuracy than the rate-dependent charge transfer inefficiency (RDCTI) correction for the EPIC-pn timing mode, respectively. @pintore2014 tested both rate dependence corrections in GX 13+1 in EPIC-pn timing mode. They showed that the RDPHA calibration provides more consistent centroid line energies and more physical fit parameters (such as the emission Gaussian line centroid energy and source inclination parameters obtained through the diskline model best fit). Using this correction, they found the energy of the iron mission line at 6.6 keV, and stated that it was consistent with the energies found by @Trigo2012 and @Dai2014. In the RDPHA correction, the energy scale is calibrated by fitting the peaks in derivative PHA spectra corresponding to the Si ($\sim$1.7 keV) and Au ($\sim$ 2.3keV) edges of the instrumental responses, where the gradient of the effective area is largest (XMM-SOC-CAL-TN-0306).
We extracted source+background spectrum and light curves of both observations (Obs. 1 and 2) from 10 < RAWX < 60 and the background from 3 < RAWX < 9. The SAS task *epatplot* was used as a diagnostic tool for pile-up in the pn-camera. It showed that both observations are affected by pile-up. In order to mitigate its effect, we excised the brightest central columns RAWX 38-39 during the spectral extraction.
The ancillary and response matrices were generated trough the SAS task *arfgen* (in the appropriated way to account for the area excised due to pile-up) and *rmfgen*. The EPIC-pn spectra were rebinned in order to have at least 25 counts in each background-subtracted channel and in order to avoid oversampling the intrinsic energy resolution by a factor larger than 3.
The light curves were produced trough the SAS task *epiclccorr*, which corrects the light curve for various effects affecting the detection efficiency (such as vignetting, bad pixels, PSF variation and quantum efficiency), as well as for variations affecting the stability of the detection within the exposure (such as dead time and GTIs). Because all these effects can affect source and background light curves in a different manner, the background subtraction was made accordingly through this SAS task.
Light curves {#sec:LCs}
------------
Figure \[fig:LC\] shows the EPIC-pn light curves extracted from Obs. 1 and 2 of GX 13+1. We present the light curves in the 0.7 - 10.0 keV, 0.7 - 4.0 keV, and 4.0 -10.0 keV energy ranges, with a bin size of 100.0 s, and also the hardness ratio between the photon counts in the last two energy bands. Both light curves show high count rate variability.
Although we see the same shape of the light curves as in @Trigo2012, who analyzed the same EPIC-pn observations of GX13+1, we obtained a slightly higher count rate. We tried to find the origin of the discrepancy on the count rate (of $\sim$50 count/s). In order to do this, we first extracted the light curve in the same way as @Trigo2012, that is, considering the same extraction region (private communication). We obtained the same count rate as before, which means that the difference is not due to the different extraction region used. We then extracted the event list in the same way as @Trigo2012, that is, we applied the RDCTI calibration correction, and extracted the light curve without applying background subtraction, to see whether somewhat unexpectedly, the origin could be in these points. As expected, we obtained the same count rate as before and thus failed to find the origin of the discrepancy. The only possibility we see lies in the different SAS versions that were used in each analysis. We used SAS version 14.0.0 and @Trigo2012 used SAS version 10.0.0. Consequently, we applied the XRL correction inserted in SAS 13.5/14.0.0. @Trigo2012 calculated the “residual” of the offset map by subtracting the offset map of a nearby observation taken with closed filter from the offset map, and applied the XRL correction excising the inner region, which they found to be affected by XRL (which is the same region as was excised for pile-up correction). The differences in the XRL correction procedures might have contributed for the discrepancy in the count rate. Furthermore, the RDCTI calibration correction has been changed since the release of SASv10.0.0; the RDCTI correction applied by us (for the comparison) is not the same as was applied by @Trigo2012.
In Obs. 1 we observed a maximum fractional variation in the hardness ratio of $\sim$ 7.3% with a mean value of 0.688(1). A similar variation was identified in Obs. 2, where we observed a maximum fractional variation of $\sim$ 5.2% with a mean value of 0.76(7). We confirm what was found by [@Trigo2012]: the count rate variability is not associated with significant change in the hardness ratio. Because we did not find any significant change in the hardness ratio, we used the total EPIC-pn exposure time in the spectral extraction for both observations.
\[fig:LC101\] \[fig:LC201\]
Spectral variability {#sec:spec}
--------------------
Although the variation in hardness ratio is not significant, @Trigo2012 pointed out that Obs. 1 shows a significant spectral change when the spectrum is divided into two intervals (“high and low”). However, we did not obtain this result. We checked the possible spectral variation by dividing the spectra into two parts: the first part using the good time interval (GTI) for times shorter than $\sim$ 7000 s or longer than $\sim 1.25 \times 10^4$ s, and the second part was extracted using the GTI between these two values. The spectra do not show any spectral variability.
Conservatively, we also extracted two spectra from Obs. 2: the first using the GTI in which the detected count rate was lower than 900 count $s^{-1}$ , and the second using a count rate higher than this amount. We did not find spectral variation in either observation (only the normalization parameter was different), therefore we present the spectra using the total EPIC-pn exposure time.
Data analysis {#sec:datanalysis}
=============
In our spectral analysis we used the 2.5-10.0 keV energy range. We excluded the soft energy range in which weak absorption features at $\sim$ 1.88 keV and $\sim$ 2.3 keV were observed in previous analyses and were referred to as being most likely caused by residual calibration uncertainties. We observed the feature at $\sim$ 1.88 keV in both observations (which could be Si XIII, or systematic residuals at the Si edge owing to a deficient calibration), but we did not observe the feature at $\sim$ 2.2-2.3 keV (which could be residuals of calibration around the instrumental edge of Au-M at 2.3 keV) [@Trigo2012; @pintore2014]. The absence of the weak absorption line at $\sim$2.3 keV in our analysis could be due to improvement in the calibration in SAS version 14.0.0. Obs. 1 and 2 in this paper correspond to observations 4 and 6 in @Trigo2012.
We fit the spectral continuum with a blackbody component (<span style="font-variant:small-caps;">bbodyrad</span> in XSPEC) to model the thermal emission from the neutron star surface, plus a Comptonization component (<span style="font-variant:small-caps;">compTT</span> in XSPEC), which represents the emission attributed to a corona. The continuum of GX 13+1 is modified by significant total photon absorption by the material present in the line of sight of the observer. We used only one component (<span style="font-variant:small-caps;">tbabs</span> in XSPEC) to indicate both the Galactic absorption due to neutral hydrogen column and the variable absorbing material close to the source. The <span style="font-variant:small-caps;">tbabs</span> parameter NH was therefore let free in all fits.
Both observations show a broad and red-skewed emission line in the Fe K$_{\alpha}$ energy band (see Figure \[fig:emline\]). In previous analyses [@pintore2014; @Trigo2012], these features were fit either using a simple broad Gaussian or the relativistic <span style="font-variant:small-caps;">diskline</span> model. In this work, we fit the broad red-skewed line profile using both the nonrelativistic <span style="font-variant:small-caps;">windline</span> and the <span style="font-variant:small-caps;">diskline</span> model, and we compared the results obtained in each fit. The spectral analyses were performed using the XSPEC astrophysical spectral package version 12.09.0j [@Arnaud1996].
In the same energy band in which the Fe emission line is apparent, we observed three narrow absorption lines in Obs. 1 and one in Obs. 2. These features are characteristics of dipping sources and were modeled with absorbed Gaussians, the <span style="font-variant:small-caps;">lgabs</span> model, which differs from the <span style="font-variant:small-caps;">gabs</span> model by the fact that it is not an exponentially multiplicative model (suitable for describing cyclotron resonance features), but a multiplicative model of the form $\propto (1-\text{Gaussian})$ [@Soong1990]. Because <span style="font-variant:small-caps;">lgabs</span> is a multiplicative model, it is not possible to obtain the EW of these lines through XSPEC. The addition of edges at 8.83 keV (Fe XXV) and 9.28 keV (Fe XXVI) did not improve the quality of the fit significantly. The best-fit parameters are shown in Table \[tab:parameters\].
In Obs. 1, the three narrow absorption lines are identified as coming from highly ionized iron: K$_{\alpha}$ Fe He-like line (Fe XXV), K$_{\alpha}$ H-like line (Fe XXVI), and K$_{\beta}$ Fe He-like line. In Obs. 2, only the K$_{\alpha}$ Fe He-like absorption line is present. The addition of a narrow absorbed K$_{\beta}$ Fe H-like line did not significantly improve the fit.
Fitting the broad iron emission line {#fitting-the-broad-iron-emission-line .unnumbered}
------------------------------------
When we fit the Fe emission line with a simple Gaussian in Obs. 1, we obtained the line centroid energy $E$ equal to $6.872^{+0.030}_{-0.021}$ keV, $\sigma$ equal to $0.59^{+0.03}_{-0.05}$ keV, and the equivalent width (EW) equal to $226^{+142}_{-222}$ eV; in Obs. 2, we obtained the line centroid energy equal to $6.58^{+0.05}_{-0.05}$ keV, $\sigma$ equal to $0.64^{+0.04}_{-0.04}$ keV, and EW equal to $118^{+60}_{-87}$ eV. The centroid energies are different between the two observations. The EW of the lines are poorly constrained in both fits, and the broadening of the lines agrees within the errors between the observations at 90% confidence level. This constancy of the line width is in agreement with what was reported by @Cackett2013.
Comparing our results with @Trigo2012, we obtained a higher Gaussian centroid energy in Obs. 1 (see Table \[tab:compDT12\]). However, when we fit the spectra of Obs. 1 and 2 with the same total model used by @Trigo2012 (see Table \[tab:compDT12\], rows 1 and 3), we obtained that all parameters, except for the $\sigma$ in Obs. 1, agree at 90% confidence level with the values found by @Trigo2012. The $\sigma$ in Obs. 1 appears slightly narrower in our analysis.
Because @Trigo2012 applied RDCTI correction in their analysis, the RDPHA correction applied by us could have led to higher centroid energies, larger broadness, and higher intensity of the emission line [see @pintore2014]. However, we did not observe these differences in the line when we fit the data with the same total *Model 1* used by @Trigo2012 (see Table \[tab:compDT12\], rows 1 and 3). Therefore, the different centroid energy found by us in Obs. 1 (see Table \[tab:compDT12\], row 4, column 3) is likely due to the differences in the total model used to fit the continuum and the absorbed narrow lines. However, in Obs. 2 we did not observe a shift in the line centroid energy with respect to the fit performed by @Trigo2012.
\[tab:compDT12\]
[lccccccccc]{} & & & & &\
& & & & &\
\
Line model &Total model &E (keV) & $\sigma$ (keV) & EW (eV) & & E (keV) & $\sigma$ (keV) & EW (eV)&Reference\
Gaussian & Model 1&$6.53^{+0.11}_{-0.09}$ &$0.72^{+0.14}_{-0.16}$ &$185^{+62}_{-62}$ & &$6.63^{+0.13}_{-0.13}$ &$0.74^{+0.19}_{-0.19}$ &$108^{+27}_{-27}$ &[@Trigo2012 Table 3]\
Gaussian & Model 2 &$6.56^{+0.10}_{-0.07}$ &$0.88^{+0.12^{(h)}}_{-0.07}$ &$299^{+77}_{-77}$ & &$6.71^{+0.12}_{-0.16}$ &$0.77^{+0.23^{(h)}}_{-0.15}$ &$91^{+48}_{-23}$ &[@Trigo2012 Table 4]\
Gaussian &Model 1 &$6.67^{+0.06}_{-0.03}$ &$0.50^{+0.03}_{-0.10}$ &$128^{+18}_{-18}$ & &$6.57^{+0.06}_{-0.07}$ &$0.50^{+0.05}_{-0.07}$ &$78^{+18}_{-17}$ &The present paper\
Gaussian &Model 3&$6.872^{+0.030}_{-0.021}$ &$0.59^{+0.03}_{-0.05}$ &$226^{+142}_{-222}$ & &$6.58^{+0.05}_{-0.05}$ &$0.64^{+0.04}_{-0.04}$ &$118^{+60}_{-87}$ &The present paper\
Diskline &Model 4&$6.674_{-0.002}^{+0.040}$ &$-$ &$256_{-73}^{+128}$ &$$ &$6.26_{-0.04}^{+0.05}$ &$-$ &$119_{-25}^{+21}$ &The present paper\
Windline &Model 5&$[6.6]$ &$-$ &$195_{-47}^{+26}$ &$$ &$[6.6]$ &$-$ &$130_{-24}^{+25}$ &The present paper\
<span style="font-variant:small-caps;">windline</span> fit {#sec:windlinefit}
----------------------------------------------------------
When we fit the <span style="font-variant:small-caps;">windline</span> model to the residual excess in the iron line energy range, we obtained the best fit-parameters shown in Table \[tab:parameters\]; columns 4 and 5 show the best-fit parameters for Obs. 1 and Obs. 2, respectively. All fits were performed in order to leave the maximum number of free parameters.
Figure \[fig:spec101\] shows the best spectral fit in the 2.5 to 10 keV energy range for Obs. 1 (*left*) and Obs. 2 (*right* ) when the <span style="font-variant:small-caps;">windline</span> model was used to fit the residual excess in the iron line energy range. Figure \[fig:spec201\] shows the unfolded spectrum in the energy range in which both the broad emission line and the narrow absorption features appear simultaneously.
The best fit was found in both observations with a 6.6 keV emission line, which corresponds to a $K_{\alpha}$ transition of the Fe XXI-XXIII ions. This line appears with an EW equal to $195_{-47}^{+26}$ in Obs. 1, and $130_{-24}^{+25}$ in Obs. 2 (see Table \[tab:compDT12\], \[tab:parameters\]). We obtained the highest value of the hydrogen column in Obs. 1, which is in agreement with @Trigo2012, although different absorption column models were used by them (see Table 4 therein, where observations No. 4 and 6 correspond to Obs. 1 and 2).
In Obs. 1, the temperature of the electrons kT$_{ew}$ and the velocity of the outflow were not constrained by the fit. To obtain consistent physical parameters, we therefore fixed these two parameters (kT$_{ew}$ and $\beta$) with values similar to those found by the best fit in Obs. 2 (see Table \[tab:parameters\]).
In Obs. 1 and 2 we obtained $\tau_w > 1$, $\tau > \tau_{w}$, temperature of the electrons in the wind kT$_{ew}$ $\sim$ 0.6 keV, and $\beta~\sim~10^{-2}$. The velocity of the outflow is equal to $2.25 \times 10^4$ km s$^{-1}$ in Obs. 1 and ${2.01^{+0.18}_{-0.21} \times 10^4}$ km s$^{-1}$ in Obs. 2.
These outflow velocities are $\text{about ten}$ times higher than the blueshifted absorption feature velocities found in previous analyses: @Trigo2012 found blueshifts in GX13+1 between $\sim$ 2100 and 3700 km s$^{-1}$; @Ueda2004 indicated, based on Chandra HETGS, a bulk outflow plasma velocity of $\sim$ 400 km s$^{-1}$; and @Allen2016 likewise used Chandra HETG observations and found an outflow plasma velocity $>~500$ km s$^{-1}$.
<span style="font-variant:small-caps;">diskline</span> fit
----------------------------------------------------------
When we fit the <span style="font-variant:small-caps;">diskline</span> model to the residual excess in the iron line energy range, we obtained the best-fit parameters shown in Table \[tab:parameters\]. All fits were performed in order to leave the maximum number of free parameters. Column 6 shows the best-fit parameters for Obs. 1 (Fit A, hereafter); columns 7 and 8 show the best-fit parameters for Obs. 2 when the energy of the iron emission line was not constrained (Fit B, hereafter), and when the energy line was constrained to the energy range of 6.4 to 6.97 keV (Fit C, hereafter). Figure \[fig:DKlines\] shows the unfolded spectrum (of Fits A and C) in the energy range in which both the broad emission line and the narrow absorption features appear simultaneously.
We found an inclination of the source ($i$) equal to $60_{-6}^{+4\circ}$ in Obs. 1, $75^{\circ}$ pegged at the hard limit and $61_{-5}^{+5\circ}$ in Obs. 2, in Fits B and C, respectively. All inclination values found are in the $60 - 80^{\circ}$ inclination range expected for GX 13+1 [@Trigo2012]. We checked the effect of the inclination on the best-fit parameters in both observations. For inclination values within the expected range the best-fit parameters do not change significantly.
The line emissivity as a function of the accretion disk radius ($r$) is an unknown function. For simplicity, the <span style="font-variant:small-caps;">diskline</span> model assumes that the line emissivity varies as $r^{B_{10}}$, and for the most part, set $B_{10}$ is equal to $-2$ [see @Fabian1989]. This emissivity index is expected where $r$ is approximately a few outer radii ($r_c$) of the corona. The best fit led to a power-law index $B_{10}$ of $\sim$ $-2.3$ for both observations: $B_{10}$ is equal to $-2.38^{+0.06}_{-0.06}$ in Obs. 1 and equal to $-2.25^{+0.20}_{-0.15}$ and $-2.40^{+0.12}_{-0.10}$ in Obs. 2 in Fits B and C, respectively.
In both observations the inner disk radius ($R_{in}$) of the reflected line component is equal to $\sim$ 10 gravitational radii ($R_g$): $R_{in}$ is equal to $9.6^{+0.9}_{-0.8}$ $R_g$ in Obs. 1 and equal to $14^{+4}_{-7}$ and $14^{+7}_{-4}$ in Obs. 2, in the Fits B and C, respectively.
The outer radius ($R_{out}$) in Obs. 1 was fixed to 1000 $R_g$, because the thawing of the parameters led to unphysical results: either $R_{out}$ becomes smaller than $R_{in}$, or $R_{out}$ assumes the value of $\sim$ 10.000 $R_g$. On the other hand, the $R_{out}$ parameter in Obs. 2 was left free because it appears better constrained by the fit.
We found an emission line energy of $6.674^{+0.040}_{-0.002}$ keV in Obs. 1, which corresponds to a K$_{\alpha}$ He-like Fe emission line. On the other hand, in Obs. 2 Fit B we found an anomalous emission line energy of $6.26^{+0.05}_{-0.04}$ keV, which represents a less physical energy of an iron line. The closest iron line energy is expected to be found at 6.4 keV, which corresponds to a K$_{\alpha}$ emission line from a neutral iron atom. In Fit C we therefore allowed the line energy parameter to vary only in a physical energy range, and in this case, the line appeared pegged at the lower limit. The line appeared with an EW equal to $256_{-73}^{+128}$ eV in Obs. 1, $119_{-25}^{+21}$ eV in Obs. 2 Fit B, and not well constrained, equal to $111_{-107}^{+18}$ eV, in Obs. 2 Fit C (see Table \[tab:compDT12\], \[tab:parameters\]).
Freezing the $R_{out}$ parameter in Obs. 2 Fit B to the same value found in Obs. 1 (i.e., to 1000 $R_g$) leads to a physical emission line energy value of $\sim$ 6.6 keV, but the inclination of the source assumes the value of $\sim$ 30$^{\circ}$, which is not consistent with the dips observed in this source. In this fitting we observed an increment of the $\chi^2$-red.\
We also determined how different line emissivity indices affect the best-fit parameters in our analysis. We list our results below.
- Forcing $B_{10}$ equal to zero, which is expected for $r$ < $r_c$, led to less physical values of the source inclination ($i$ > $90^{\circ}$) and an emission line energy of $\sim$ 6.8 keV in Obs. 1. In Obs. 2 (Fit B), the emission energy line shifts to a physical value of $\sim$ 6.7 keV, but all other parameters become unconstrained and assume unphysical values, that is, $R_{out}$ becomes smaller than $R_{in}$, and the source is found to be viewed face-on ($i$ $\sim$ $3^{\circ}$). All these fits led to an increment of the $\chi^2$-red. **\
- Forcing $B_{10}$ equal to -3, which is expected for $r$ beyond $r_c$, where the coronal radiation intercepts the disk only obliquely, led $R_{out}$ to assume unphysical value if it was not frozen, and the emission line energy shifts to $\sim$ 6.6 keV in Obs. 1. In Obs. 2 we did not observe significant changes in the best-fit parameters. All these fits led to an increment of the $\chi^2$-red.
<span style="font-variant:small-caps;">windline</span> versus <span style="font-variant:small-caps;">diskline</span> fit
------------------------------------------------------------------------------------------------------------------------
In both observations the best fit performed with the <span style="font-variant:small-caps;">windline</span> model was found for an emission line energy at 6.6 keV. Despite the evolution of the continuum between the two observations, the iron emission line remains remarkably constant. In contrast, when the excess in the energy range of the Fe K emission line was fit with the <span style="font-variant:small-caps;">diskline</span> model, different line energies were found: we found a $6.674^{+0.040}_{-0.002}$ keV emission line energy in Obs. 1, a $6.26^{+0.05}_{-0.04}$ keV emission line in Obs. 2 Fit B, and a 6.4 keV emission line pegged at the lower limit in Obs. 2 Fit C, when the line energy was constrained to a physical energy range (see Table \[tab:parameters\]). In Obs. 2, the line energy changes considerably in relation to the <span style="font-variant:small-caps;">windline</span> model, and it is low to be emitted by an iron atom in Fit B. The best fit in this case is not able to constrain the energy line parameter to a range of physical values, probably because of the large line broadening.
The reconstructed shape of the line also changes considerably between the two models. The line profile described by the <span style="font-variant:small-caps;">diskline</span> model (see Figure \[fig:DKlines\]) appears double-peaked as a result of the Doppler effects on the accretion disk. The high disk inclination of the source produces a broader line in which the two peaks are very well separated. Aberration, time dilation, and blueshift by the fraction of the disk that is relevant together with the disk inclination make the blue horn brighter than the red horn [@Fabian1989]. In contrast to the relativistic line profiles, the <span style="font-variant:small-caps;">windline</span> line profiles (see Figure \[fig:spec201\]) show a broad single-peaked line, whose peak corresponds to the direct component of the line.
All fits with the <span style="font-variant:small-caps;">diskline</span> model determined the values of the source inclination in the range expected for GX 13+1, and in Obs. 1 and Obs. 2 Fit C the inclination parameter remains constant at 90% confidence level. The <span style="font-variant:small-caps;">windline</span> model shows no inclination effect on the line profile, the flow is spherically symmetric, with radial velocities, which means that there is no preferred angle between the flow stream and the line of sight, that is, observers from any direction see the same modified line, produced by a mean Doppler shift over all flow directions.
We obtained that not only the energy of the emission line was affected when different line models were used. It also slightly affected the continuum and the centroid energy of the narrow He-like Fe K$_{\alpha}$ absorption line (see Table \[tab:parameters\] and the discussion below).
Of the continuum parameters, in Obs. 1 only the <span style="font-variant:small-caps;">bbodyrad</span> temperature remains in agreement at the 90% confidence level, when different line models are used. In Obs. 2, however, only the value of the hydrogen column remains in agreement at the 90% confidence level (see Table \[tab:parameters\]).
The EWs of the broad iron emission line in Obs. 1 and Obs. 2 are poorly constrained, which is partially due to the presence of the absorption lines that intercept the broad emission line, as previously noted by @Trigo2012. The width of the narrow absorption lines are not constrained by the fits either. Because of the error bars, we therefore cannot infer any change in the EW of the emission line when different line models are used in an observation.
@Cackett2013 [see Figure 3 therein], using the results of @Trigo2012 [see Table 4 therein], found no statistically significant correlation between the warm absorber column density and Fe emission line EW in GX 13+1. They obtained that a change in the measured emission line EW with increasing N$_H$ depends on the continuum parameters. That may explain the different values of EW among Obs. 1 and 2 at the 90% confidence level in the fits performed with the <span style="font-variant:small-caps;">diskline</span> model. This may explain the different values of EW among Obs. 1 and 2 at the 90% confidence level in the fits performed with the <span style="font-variant:small-caps;">diskline</span> model.
### Narrow and absorbed Gaussians {#narrow-and-absorbed-gaussians .unnumbered}
When we fit the spectra with the total model used by @Trigo2012, that is, using Gaussians with negative normalizations to fit the absorbed lines, we obtained that in Obs. 1 the EW ($30 \pm 4$ eV) of the Fe He-like absorption line at $\sim$ 6.7 keV is slightly greater than the value found by @Trigo2012 ($20 \pm 5$ eV). For all other absorbed lines in Obs.1 and for the line in Obs.2, the EWs agree at the 90% confidence level with @Trigo2012 [see Table 3 therein]. The values of the centroid energies of the narrow and absorbed Gaussians found in the two observation for the <span style="font-variant:small-caps;">windline</span> and <span style="font-variant:small-caps;">diskline</span> fits are compatible with the values found by @Trigo2012 and @pintore2014 [for the RDPHA correction, although a different observation was used by them] at the 90% confidence level.
However, comparing the line energies obtained in the fits performed with the <span style="font-variant:small-caps;">diskline</span> model with the fits performed with the <span style="font-variant:small-caps;">windline</span> model, we observed that in Obs. 1 the He-like Fe K$_{\alpha}$ line appears to be blueshifted when the <span style="font-variant:small-caps;">diskline</span> model is used to fit to the broad iron emission line, whereas in the <span style="font-variant:small-caps;">windline</span> fit, it was not possible to distinguish between a blueshifted, redshifted, or not shifted (affected) line because of the error bars of this parameter. The other two narrow lines, H-like Fe K$_{\alpha}$ and He-like Fe K$_{\beta}$, showed the same ambiguity in the line shift in both observations for the <span style="font-variant:small-caps;">windline</span> and <span style="font-variant:small-caps;">diskline</span> fits.
When the narrow line indeed appeared to be shifted, the velocity of the warm absorber producing the shift in the line energy is around ten times lower than the velocity of the more central <span style="font-variant:small-caps;">outflow (wind)</span> where the iron emission line is formed in the <span style="font-variant:small-caps;">windline</span> framework. For all other cases, in which the ambiguity in the line shift is found, when any shift is present, the radial velocity of the outflow is around 100 times lower than the central outflow velocity.
Run-test and statistical assessment of the goodness-of-fit
----------------------------------------------------------
Table \[tab:NSruntest\] shows (see columns 6, 7 and 8) that all the different line models are statistically equivalent on the basis of the $\chi^2$ goodness-of-fit test. For example, in Obs. 1 the F-test gives a probability of chance improvement of 16$\%$, when instead of the <span style="font-variant:small-caps;">windline</span> model the <span style="font-variant:small-caps;">diskline</span> model is used to describe the data. In Obs. 2 the F-test gives a probability of chance improvement of 48$\%$ and 41$\%$ when Fits B and C are compared with the <span style="font-variant:small-caps;">widnline</span> fit, respectively. In order to distinguish this ambiguity between the models, we used a different approach to this problem.
Because our main goal is to determine whether the iron line profile is intrinsically asymmetric (i.e., the skewness coming from physical grounds) and to distinguish between the relativistic and nonrelativistic cases, we need a statistical test that takes the ** of the feature into account. To this aim, we used the run-test, also known as the Wald-Wolfowitz test [@Barlow1989; @Eadie1971], which inspects the residuals of the fit. If the model perfectly describes the data, the positive ($+$) and negative ($-$) residuals are expected to be randomly distributed around zero, and the number of runs, that is, the number of sequences of consecutive $+$ or $-$ residuals, is expected to be large [see, e.g., @Redman2009; @Orlandini2012 for an application of the run-test in an astrophysical context].
The run-test gives the cumulative probability of obtaining by chance the number of observed runs (run-test probability, hereafter). For a good fit, that is, for a random distribution of the residuals, the run-test probability will be high. If, on the other hand, the fit does not describe the shape of the feature, the data will not be randomly distributed with respect to the fitting model. In this case, the run-test probability will be lower, meaning that there is a residual underlying trend.
It is important to mention that the run-test and the $\chi^2$-statistic are independent. The $\chi^2$-statistic does not depend on the ordering of the bins or on the signs of the residuals in each bin, while the run-test accounts for both the ordering and the signs, providing effectively additional information on the model goodness [@Eadie1971].
Table \[tab:NSruntest\] also shows the run-test probabilities in the 6.0 to 6.7 keV emission line energy range for the Gaussian (column 3), the <span style="font-variant:small-caps;">windline</span> (column 4), and the <span style="font-variant:small-caps;">diskline</span> (column 5) fits. For the Gaussian fit we obtained a run-test probability of $12.6\%$ in Obs. 1, and $55.2\%$ in Obs. 2. In the case of the <span style="font-variant:small-caps;">diskline</span> fit, we found a run-test probability of $14.4\%$ in Obs. 1, $43.0\%$ in Obs. 2 when the line energy was not constrained in the fit, and $32.4\%$ when the line energy was constrained in the 6.4 - 6.97 keV energy range. When we fitted the data with the <span style="font-variant:small-caps;">windline</span> model, this probability becomes $20.0\%$ in Obs. 1 and $56.9\%$ in Obs. 2. For all line models (<span style="font-variant:small-caps;">gaussian</span>, <span style="font-variant:small-caps;">diskline,</span> and <span style="font-variant:small-caps;">windline</span>) the hypothesis that the residuals are randomly distributed is accepted for a test at $5\%$ significance level.
\[fig:SP101\] \[fig:SP101uf\]
\[fig:SP201\] \[fig:SP201uf\]
\[fig:DK101uf\] \[fig:DK201uf\]
\[tab:parameters\]
Component Parameter Unit Obs.1 Obs.2 Obs.1 Obs.2 Obs.2
------------- ---------------- --------------------------- --------------------------- ----------------------------------- --------------------------- --------------------------- ---------------------------
Tbabs nH 10$^{22}$ atoms cm$^{-2}$ $4.90_{-0.15}^{+0.15}$ 2.9$_{-0.3}^{+0.6}$ $6.06_{-0.06}^{+0.40}$ 2.48$_{-0.27}^{+0.17}$ $2.48_{-0.39}^{+0.12}$
bbodyrad $kT_{bb}$ keV $1.221_{-0.008}^{+0.003}$ 1.29$_{-0.03}^{+0.03}$ $1.220_{-0.003}^{+0.003}$ 1.505$_{-0.050}^{+0.023}$ $1.503_{-0.040}^{+0.025}$
CompTT $kT_0$ keV $0.132_{-0.022}^{+0.018}$ 0.51$_{-0.06}^{+0.03}$ $0.188_{-0.008}^{+0.060}$ 0.644$_{-0.019}^{+0.025}$ $0.65_{-0.01}^{+0.04}$
$kT_e$ keV $2.015_{-0.009}^{+0.007}$ 2.46$_{-0.17}^{+0.60}$ $2.503_{-0.003}^{+0.018}$ 3.84$_{-0.17}^{+0.22}$ $3.84_{-0.23}^{+0.10}$
$\tau$ $3.80_{-0.15}^{+0.40}$ 5.7$_{-1.0}^{+0.6}$ $2.54_{-0.21}^{+0.08}$ 3.01$_{-0.20}^{+0.40}$ $3.00_{-0.16}^{+0.40}$
Windline $\tau_w$ $2.55_{-0.21}^{+0.08}$ 2.57$_{-0.18}^{+0.15}$
$kT_{ew}$ keV \[0.7\] 0.6$_{-0.4}^{+0.7}$
$\beta$ \[7.5$\times 10^{-2}$\] 6.7$_{-0.7}^{+0.6}\times 10^{-2}$
Diskline $E_{L}$ keV $6.674_{-0.002}^{+0.040}$ 6.26$_{-0.04}^{+0.05}$ $6.4$
$\beta_{10}$ $-2.38_{-0.06}^{+0.06}$ -2.25$_{-0.15}^{+0.20}$ $-2.40_{-0.10}^{+0.12}$
$R_{in}$ $R_G$ $10_{-2}^{+3}$ 14$_{-7}^{+4}$ $14_{-4}^{+7}$
$R_{out}$ $R_G$ $[1000]$ 413$_{-111}^{+212}$ $956_{-319}^{+510}$
$i_{out}$ deg $60_{-6}^{+4}$ 75 $61_{-5}^{+5}$
EW eV 195$_{-47}^{+26}$ 130$_{-24}^{+25}$ $256_{-73}^{+128}$ $119_{-25}^{+21}$ $111_{-107}^{+18}$
Gaus$_1$ $E_{L}$ keV 6.701$_{-0.014}^{+0.008}$ $6.733_{-0.013}^{+0.003}$
$\sigma$ keV 3.9$\times 10^{-3}$ 2.8$\times 10^{-3}$
Gaus$_2$ $E_{L}$ keV 7.005$_{-0.015}^{+0.015}$ 7.005$_{-0.014}^{+0.014}$ $7.005_{-0.015}^{+0.002}$ 7.005$_{-0.015}^{+0.014}$ $7.004_{-0.014}^{+0.010}$
$\sigma$ keV 1.1$\times 10^{-4}$ 1.4$\times10^{-3}$ 1.1$\times 10^{-4}$ $6.7\times10^{-4}$ $1.1\times10^{-4}$
Gaus$_3$ $E_{L}$ keV 7.877$_{-0.018}^{+0.015}$ $7.860_{-0.015}^{+0.015}$
$\sigma$ keV 2.7$\times 10^{-3}$ $1.6\times10^{-4}$
Fit quality $\chi^2$/d.o.f 1784.81/1481 1572.35/1485 1690.81/1478 1573.37/1483 1588.06/1483
$\chi^2_{red}$ 1.21 1.06 1.14 1.06 1.07
------ ---------- ---------- ---------- ---------- -- --------------------- --------------------- --------------------- --
Obs. Energy
(keV) Gaussian windline diskline Gaussian windline diskline
1 6.0–6.67 12.6 20.0 14.4 1.16 (1714.58/1480) 1.21 (1784.81/1481) 1.14 (1690.81/1478)
2 6.0–6.67 55.2 56.9 43.0 1.06 (1578.99/1486) 1.06 (1572.35/1485) 1.06 (1573.37/1483)
2 6.0–6.67 32.4 1.07 (1588.06/1483)
------ ---------- ---------- ---------- ---------- -- --------------------- --------------------- --------------------- --
Discussion and conclusions {#sec:dis}
==========================
The fluorescent iron line is produced by hard X-ray irradiation of a cold gas. However, where and how the asymmetry of the line is created differs between the two asymmetric Fe emission line models.
Because of the high disk inclination and the assumption that the line is created in the inner part of the accretion disk, the line profile appears to be double-peaked when it is fit with the <span style="font-variant:small-caps;">diskline</span> model. On the other hand, the line profile appears to be single-peaked when it is fit with the <span style="font-variant:small-caps;">windline</span> model because the red wing is produced by the indirect component of the line photons, which interact multiple times with the electrons in the outflow before escaping to the observer. Nonetheless, the different line model frameworks are both capable in terms of $\chi^2$-statistic to fit the broad Fe K emission line profiles.
The Compton bump in the $\sim$ 10 - 40 keV energy range is expected to be observed in the reflection scenario, and therefore in the relativistic line scenario [@Fabian2000]. However, the bump is expected in the <span style="font-variant:small-caps;">windline</span> framework as well. @Laurent2007 used Monte Carlo simulations of the continuum spectrum emerging from a pure scattering wind, and their analytical description showed that when the incident spectrum is described by a hard power law ($\Gamma < 1$) and the wind has physical parameters such that $\beta \lesssim 0.3$ and $\tau$ of a few, a photon accumulation bump at energies around $\gtrsim ~10$ keV and a softening of the spectrum at higher energies are observed. Because the number of photons is conserved, the down-scattered high-energy photons, which are removed from the high-energy part of the incident spectrum, are detected at lower energy. They pointed out that as the optical depth increases, more prominent bumps are formed in the outflow. In general, the shape of the emerging continuum depends on the mean number of scatterings suffered by the photons in the outflowing plasma (which is a combined effect of the wind optical depth $\tau$ and the wind velocity), and on the incident spectrum shape.
The <span style="font-variant:small-caps;">windline</span> model allows determining the possible outflow physical parameters. For both observations, we found the
- optical depth of the wind ($\tau_w) > 1$;
- temperature of the electrons in the wind (kT$_{ew})$ of $\sim$ 0.6 keV, which is the possible outflow temperature [@Laming2004]; **
- outflow velocity ($\beta$) of $\sim~10^{-2}c$ (see sec. \[sec:windlinefit\]).
The outflow velocities determined by the <span style="font-variant:small-caps;">windline</span> model $\text{are}$ ten times higher than the velocity determined by the blueshift velocities of the absorption features found in previous analyses. The different velocities can still be explained considering that the emission and absorption lines are produced in different regions: the emission line might be produced in a more internal region of the source, at the bottom of the wind or outflow (in an inner shell) where the velocity is high; on the other hand, the resonance absorption lines are produced in a more distant region from the compact object by the interaction of photons with highly ionized species present in a cylindrical absorbing plasma around the source, driven by outflows from the outer regions of the accretion disk. Therefore, the presence of the outflow and its velocity variation along the cloud radius may explain the simultaneous observation of narrow absorption and broad emission lines in sources with high inclination, such as GX13+1. The absence of absorption lines in the internal outflow, responsible for red-skewing the line, can be explained by relatively high temperature of the outflow, which is on the order 0.3-0.6 keV, see the calculations of the temperature structure of the outflow in @Laming2004.
The wind can be launched from 3 to 10$^4$ R$_g$. In the <span style="font-variant:small-caps;">windline</span> model, the wind is considered as a spherical shell of internal radius of 100 R$_g$ and external radius of 120 R$_g$, so that the wind starts at 100 R$_g$. These values have no great influence on the emitted spectrum as long as the inner radius of the wind is far from the central compact object, where outflowing plasma is not fully ionized and strong gravitational effects are not expected.
The critical mass outflow rate M is given by $4 \pi r^2 \rho(r) m_{p} \beta c = 1.7 \beta \tau 10^{21}$ g/s = $2.6 \beta \tau 10^{-5}$ M$_{\odot}$ yr$^{-1}$ (where m$_{p}$ is the mass of the proton) [see @TS2007 appendix E]. For the fit values we found, $\tau \approx 2$ and $\beta \approx 0.01$, we find M = 3 $\times$ 10$^{19}$ g/s, which is more than ten times what was previously found [see @Trigo2012 and references therein]. However, this is consistent with the velocity difference between the inner and outer parts of the flow, outer parts where absorption lines take place. The given mass outflow rate is an upper limit because it was calculated considering that the wind is spherical throughout the system. A lower mass outflow rate may be emitted by a wind that partially covers the system. The mass outflow rate was self-consistently calculated by @Laming2004 and reproduced by @Laurent2007 using Monte Carlo simulations.
When the different asymmetric line models were fit to the same observation, slight differences in the continuum were observed, and the EW remained the same at the $90\%$ confidence level. The fits with the two different line models to the complex spectrum of the dipping source GX13+1 also introduced differences in the energy of the fluorescent Fe emission line and in the energy of the K$_\alpha$ Fe He-like (Fe XXV) absorption line.
The best fit with the <span style="font-variant:small-caps;">diskline</span> model in Obs. 2 gives an emission line energy equal to 6.26$^{+0.05}_{-0.04}$ keV, which is lower than the energy expected from photons coming from a neutral or ionized iron atom. It is, for example, 2.8$\sigma$ from the K$_{\alpha}$ energy line emitted by a neutral iron, at 6.4 keV. We checked if the total continuum model could affect the emission line energy in this observation, but fitting the continuum and the absorbed Gaussians as in *Model 1* , that is, with <span style="font-variant:small-caps;">tbabs\*edge$_1$\*edge$_2$\*</span>(<span style="font-variant:small-caps;">diskbb+bbodyrad+diskline+gaus$_1$</span>) model in XSPEC, did not lead to a significant change in the emission line energy (in this case, the energy of the line is equal to $6.28^{+0.04}_{-0.10}$). To obtain a physically consistent fit, we constrained the line energy parameter to the expected energy range. In this case, although the energy line appears pegged at the lower limit, the geometric parameters of the disk are found in good agreement with the fit in Obs. 1.
In the fits using the <span style="font-variant:small-caps;">windline</span> model, the outflow parameters, and consequently the properties of the inner portion of the outflow, are approximately constant in the two observations. The emission line generated in the outflow remains remarkably constant although the underlying continuum evolves. However, as in the <span style="font-variant:small-caps;">diskline</span> fit to Obs. 2, in Obs. 1 two <span style="font-variant:small-caps;">widnline</span> parameters ($kT_{ew}$ and $\beta$) had to be frozen to obtain a physically consistent fit. The constant properties of the inner outflow may be explained by the intrinsic variability of the accretion flow.
The ambiguity in terms of $\chi^2$-statistic test between the emission line models was expected and confirmed in the spectral fits presented in this paper. The $\chi^2$-statistic was not sufficient to lead to an unambiguous statement on the line profiles because it squares the differences between data and model and consequently looses the information about the form of the residuals in the emission line energy range. Therefore, we used the statistical run-test to take the shape of the residuals into account. We tried to break the degeneracy between the relativistic and nonrelativistic line models in the GX 13+1 spectra because it is found in the literature that this source presents clear evidence of both disk-wind (outflow) and strong, broad, and skewed iron emission lines. However, this source has a complex spectrum, with narrow absorption iron lines in the same energy range as the broad emission line. The absorbed iron lines complicate a distinction between the two line models because the run-test is not performed in the entire energy range of the emission line.
For the two observations analyzed in this paper, we were not able to reject the hypothesis that the residuals are randomly distributed at the $5\%$ significance level, and we cannot rule out one of the line models. We obtained a higher cumulative probability of observing by chance the number of runs around the fitting line for the <span style="font-variant:small-caps;">windline</span> spectral fit. However, when we modeled the continuum using the total *Model 1*, we obtained the opposite, that is, a higher cumulative probability for the <span style="font-variant:small-caps;">diskline</span> model. Therefore, we conclude that for the observations analyzed in this paper the broad emission Fe line profiles can be described as a signature of a wind or outflow and also by GR effects in terms of $\chi^2$-statistic and run-test.
The statistical run-test may allow a better assessment of the goodness of the nonrelativistic (<span style="font-variant:small-caps;">windline</span>) versus the relativistic (<span style="font-variant:small-caps;">diskline</span>) line profiles. To break the degeneracy between the relativistic and nonrelativistic broad iron lines, a study considering several other NS LMXB sources, containing strong and broad iron emission lines in their spectra could be performed. It could be performed in a perfect scenario, considering mainly sources with a more straightforward total spectrum, and using observations that are not affected by pile-up.
In the <span style="font-variant:small-caps;">windline</span> model, the large amount of mass outflowing with high velocity implies that the Earth observer should see broad and redshifted iron lines formed in the outflow. The velocity of the outflows in NS LMXBs given by the model could, for example, be compared with velocities found by P Cygni profiles in optical, UV, and X-ray because these lines indicate outflows or disk-wind outflows [e.g., @Brandt2000; @Schulz2002]. If the <span style="font-variant:small-caps;">windline</span> model is found to be more appropriate to fit broad and skewed iron lines in NS LMXBs, the observation of such profiles will be of utmost importance for understanding outflowing in these systems; this is also an important tool for studying the inflowing-outflowing connection, as stated by @Trigo2012.
We emphasize that in addition to the moderate spectral resolution of the X-ray detectors that are available today, a careful analysis of the Fe K emission line can bring important information for understanding the physics behind the asymmetric Fe fluorescent emission lines and their implication on the neutron star physics. The next generation of X-ray observatories loaded with instruments with high spectral resolution will certainly improve the understanding of the asymmetric Fe line formation in accreting compact objects.
We conclude this paper by also pointing out the importance of a timing variability study to constrain the physical process that leads to the broad and red-skewed iron line profiles. It can provide important and additional information for breaking the ambiguity between the relativistic and nonrelativistic line models.
Acknowledgments {#acknowledgments .unnumbered}
===============
T. Maiolino acknowledges the financial support given by the Erasmus Mundus Joint Doctorate Program by Grants Number 2013-1471 from the agency EACEA of the European Commission, the CNES/INTEGRAL, and the CNR/INAF-IASF Bologna. T. Maiolino would also like to thank Clément Stahl and Lorella Angelini for their valuable comments that contributed to this manuscript. We would like to thank Maria Díaz Trigo for sharing details of previous XMM-Newton Epic pn data reprocessing. Finally, we thank the anonymous referee for their critical comments that considerably improved the content of the paper.
|
---
author:
- |
\
Low Temperature Laboratory, Helsinki University of Technology, P.O.Box 2200, FIN-02015 HUT, Finland\
L.D. Landau Institute for Theoretical Physics, 119334 Moscow, Russia\
E-mail:
title: 'Fermi-point scenario for emergent gravity'
---
Natural values of physical quantities
=====================================
In emergent physics the natural value of a physical quantity means that this value naturally emerges in the effective low energy theory without fine tuning. Both in particle physics and condensed matter the natural value of a quantity depends on whether this quantity is determined by macroscopic or microscopic physics, see Table \[NaturalValues\]:
$$\matrix{
\begin{array}{lccc}
{\rm physical~quantity} &~~{\rm natural~value} ~~ &~~{\rm dimensional~analysis} ~~&~~{\rm observation} \cr
\cr
\hline
\hline
\cr
{\rm Newton ~constant} &E_{\rm P}^{-2} &E_{\rm P}^{-2} &E_{\rm P}^{-2} \cr
{\rm running ~coupling ~constant} & 1 &1 &\sim 1 \cr
{\rm mass~of~Higgs~boson} &E_{\rm P} & E_{\rm P} &\approx 0 \cr
\cr
\hline
\cr
{\rm temperature~of~Universe} &0 &E_{\rm P} &\approx 0 \cr
{\rm cosmological ~constant~\& ~vacuum~pressure} &0 &E_{\rm P}^4 &\approx 0 \cr
{\rm volume~of~Universe} &\infty &E_{\rm P}^{-3} &\infty \cr
\cr
\hline
\cr
{\rm mass~of~elementary~particle} &E_{\rm P} ~~{\rm or}~~0 & E_{\rm P} &\approx 0 \cr
\cr
\hline
\cr
\end{array}
}
\label{NaturalValues}$$
The first column in the Table \[NaturalValues\] contains the natural values of the physical quantities. In the second column the estimates of these quantities are shown, which follow from dimensional analysis assuming that the role of the fundamental energy scale is played by Planck energy $E_{\rm P} $. In the third column the observational values are given; here we neglect the magnitudes which are much smaller than the Planck scale values. For example, the observed masses of elementary particles, the upper limit for the mass of Higgs boson, the observed value of the cosmological constant, and the highest temperature in the Universe are many orders of magnitude smaller than their Planck scale values, and thus are considered as almost identical zero.
Most of the quantities are determined by microscopic physics and are expressed in terms of the corresponding microscopic scale, which is the Planck scale $E_{\rm P}$ in our Universe or atomic scale in condensed matter systems. An example is the Newton constant $G=a_G E_{\rm P}^{-2}$. For emergent gravity, the dimensionless prefactor $a_G$ depends on the vacuum content and is of order unity in units $\hbar=c=1$ (in the Fermi-point scenario which we discuss here, $\hbar$ is the fundamental constant, while the parameter $c$ – the maximum attainable speed of the low-energy particles – is determined by the microscopic physics). In principle, the parameter $a_G$ can be zero, but this requires fine-tuning between different scalar, vector and spinor fields in the vacuum. That is why the natural value of $G$ is $E_{\rm P}^{-2}$. The natural value of the mass of the Higgs boson is the Planck energy, $M_{\rm Higgs}=a_H E_{\rm P}$. However, from the condensed matter systems we know that the prefactor $a_H$ depends much on the complexity of the system, and can be exponentially reduced. This is what happens to the transition temperature $T_c$ in superconductors and Fermi superfluids, which is exponentially suppressed almost in all systems except for the high-$T_c$ cuprates. The running coupling constants $\alpha_n$ also falls into this category, since they depend on the ultraviolet cut-off together with the infra-red cut-off $E_{\rm IR}$: $\alpha_n^{-1}\sim \ln (E_{\rm P}/E_{\rm IR})$ .
Temperature, pressure, and volume belong to the category determined by macroscopic physics – thermodynamics. These thermodynamic quantities do not depend on the micro-physics or on momentum-space topology; they only depend on the environment. In the absence of forces from the environment, the pressure and temperature of any system relax to zero. The same should hold for the temperature of the Universe and for the vacuum pressure. The vacuum pressure is, with a minus sign, the cosmological constant, $\Lambda=\epsilon_{\rm vac}=-p_{\rm vac}$. Whatever is the vacuum content, and independently of the history of phase transitions in the quantum vacuum, the cosmological constant must relax to zero or to the small value which compensates the other partial contributions to the total pressure of the system: it is the total pressure of the system that must be zero in equilibrium.
Masses of elementary particles fall into a special category. The naive estimation tells us that these masses should be on the order of the Planck energy scale: $M_{\rm expected} \sim E_{\rm P}\sim 10^{19}$ GeV. This highly contradicts observations: the observed masses of known particles are many orders of magnitude smaller, being below the electroweak energy scale $M_{\rm real}<E_{\rm ew}\sim 1$ TeV. This represents the main hierarchy problem. In the “natural” Universe, where all masses are of order $E_{\rm P}$, all fermionic degrees of freedom are completely frozen out because of the Bolzmann factor $e^{-M/T}$, which is about $e^{-E_{\rm P} /E_{\rm ew}} \sim e^{-10^{16}}$ already at the temperature corresponding to the highest energy reached in accelerators. There is no fermionic matter in such a Universe.
![Characteristic energy scale in the vacuum of the “natural Universe” is the Planck energy $E_P$. Compared to that energy, the high-energy physics and cosmology operate at extremely ultra-low temperatures.[]{data-label="MainProblem"}](ULTPhysicsPaper.eps){width="1.0\linewidth"}
That we survive in our Universe is not the result of the anthropic principle (the latter chooses the Universes which are fine-tuned for life but have an extremely low probability). On the contrary, this simply indicates that our Universe is also natural, and its vacuum is generic though it belongs to a universality class which is different from the Universes with massive particles. Indeed, the momentum space topology suggests that, both in relativistic quantum field theories and in fermionic condensed matter, there are several universality classes of quantum vacua (ground states) [@FrogNielBook; @Book; @Horava]. One of them contains vacua with trivial topology, whose fermionic excitations are massive (gapped) fermions. The natural mass of these fermions is on the order of $E_{\rm P}$.
The other classes contain gapless vacua. Their fermionic excitations live either near Fermi surface (as in metals), or near a Fermi point (as in superfluid $^3$He-A), or near some other topologically stable manifold of zeroes in the energy spectrum. The gaplessness of these fermions is protected by topology, and thus is not sensitive to the details of the microscopic (trans-Planckian) physics. Irrespective of the deformation of the parameters of the microscopic theory, the natural value of the gap in the energy spectrum of these fermions remains strictly zero.
Emergent gravity in vacua with Fermi points
===========================================
For our Universe, which obeys Lorentz invariance, only those vacua are important that are either Lorentz invariant, or acquire Lorentz invariance as an effective symmetry emerging at low energy. This excludes the vacua with Fermi surface and leaves the class of vacua with a Fermi point of chiral type (the hedgehog in momentum space, see Fig. \[EmergentPhysics\]), in which fermionic excitations behave as left-handed or right-handed Weyl fermions [@FrogNielBook; @Book], and the class of vacua with the nodal point obeying $Z_2$ topology, where fermionic excitations behave as massless Majorana neutrinos [@Horava].
![Relativistic quantum fields and gravity emerging near a Fermi point – topologically protected hedgehog in momentum space. Spin of a right-handed fermion is directed along spines of the hedgehog[]{data-label="EmergentPhysics"}](EmergentPhysics.eps){width="1.0\linewidth"}
The advantage of the vacua with Fermi points is that practically all the main physical laws (except for quantum mechanics) can be considered as effective laws, which naturally emerge at low energy. This is the consequence of the so-called Atiyah-Bott-Shapiro construction (see Ref. [@Horava]), which leads to the following general form of expansion for the Hamiltonian of fermionic quasiparticles near the Fermi point: $$H=e_i^k\Gamma^i(p_k-p_k^0)+~{\rm higher~order~terms}~.
\label{Atiyah-Bott-Shapiro}$$ Here the $\Gamma^i$ are Dirac matrices; the expansion parameters (the vector $p_k^0$ indicating the position of the Fermi point in momentum space and the matrix $e_i^k$) depend on the space and time coordinates and thus are dynamic fields. This expansion demonstrates that close to the Fermi point, the low-energy electrons behave as relativistic Weyl fermions. The vector field $p_k^0$ plays the role of the effective $U(1)$ gauge field acting on these fermions. If $p_k^0$ is the matrix field, it gives rise to effective non-Abelian gauge fields. The matrix field $e_i^k$ acts on the quasiparticles as a vierbein field, and thus describes dynamical gravity. As a result, close to the Fermi point, matter fields (all ingredients of Standard Model: chiral fermions and Abelian and non-Abelian gauge fields) emerge together with geometry, relativistic spin, Dirac matrices, and physical laws: Lorentz and gauge invariance, equivalence principle, etc.
The existence of the Fermi point in the vacuum of our Universe is an experimental fact, since all our elementary particles, quarks and leptons, are chiral Weyl fermions. It is still not excluded that some of neutrinos are Majorana fermions, but this only changes the topological characteristic of the Fermi point. Does that mean gravity in our Universe is not fundamental? At the moment there is no experimental evidence that the fundamental theory must be abandoned. Moreover, the existence of a Fermi point can be a property of the fundamental theory too, in this case the second order and all higher order terms in Eq.(\[Atiyah-Bott-Shapiro\]) are absent due to Lorentz symmetry.
What about emergent gravity? Can gravity be an effective low-energy phenomenon? Usually the induced Sakharov type gravity is abandoned using the argument that the induced cosmological constant is proportional to $E_{\rm P}^4$. From the renormalization group point of view, this is the dominant term in the gravitational action, and its effect is that there is no distance scale in the universe longer than $E_{\rm P}^{-1}$. So one has no classical regime of gravity at all, and only a Planck scale Universe. However, in emergent scenarios this argument against effective gravity does not work: it is clear from the thermodynamic arguments that for any effective theory of gravity the natural value of $\Lambda$ is zero. This result does not depend on the microscopic structure of the vacuum from which gravity emerges, and is actually the final result of the renormalization dictated by macroscopic physics (more on that see in Re. [@New]).
Emergent gravity vs fundamental gravity
=======================================
![Equations for the metric field $g_{\mu\nu}$ emerging near the Fermi point depend on hierarchy of ultraviolet cut-off’s: Planck energy scale $E_{\rm P}$ vs Lorentz violating scale $E_{\rm Lorentz }$. []{data-label="GravityAndHierarchy"}](GravityAndHierarchy.eps){width="1.0\linewidth"}
The Fermi-point scenario gives a particular mechanism of emergent gravity. This mechanism has many consequences, and some of them can be used to falsify this scenario. So, let us assume that gravity is the low-energy effective theory emerging from the Atiyah-Bott-Shapiro construction. What are the consequences?
If gravity emerges from the Fermi point scenario, then:
\(1) Gravity emerges together with matter (Fig. \[EmergentPhysics\]). This means that the so-called “quantum gravity” must be the unified theory of the underlying quantum vacuum, where the gravitational degrees of freedom cannot be separated from all other microscopic degrees of freedom, which give rise to the matter fields (fermions and gauge fields).
\(2) Fermionic matter, which emerges together with gravity, consists of Weyl fermions. This agrees with the fermionic content of our Universe, where the elementary particles are left-handed and right-handed quarks and leptons.
\(3) Gravity cannot be quantized. Gravity is the result of an up-down procedure: it is the low-energy macroscopic classical output of the high energy microscopic quantum vacuum. The inverse down-up procedure from classical to quantum gravity is highly restricted. The first steps in quantization are allowed: it is possible to quantize gravitational waves to obtain their quanta – gravitons; it is possible to obtain some quantum corrections to Einstein equation; to extend classical gravity to the semiclassical and stochastic [@Hu] levels, etc. But one cannot cannot obtain “quantum gravity” by full quantization of Einstein equations.
\(4) Effective gravity may essentially differ from fundamental gravity even in principle. Since in effective gravity general covariance is lost at high energy, metrics which for the low-energy observers appear equivalent, since they can be transformed into each other by mathematical coordinate transformation, need not be equivalent physically. As a result, in emergent gravity some metrics, which are natural in general relativity, are simply forbidden. For example, emergent gravity is not able to incorporate the geodesically-complete Einstein Universe with spatial section $S^3$ [@KlinkhamerVolovikCoexisting]. It, therefore, appears that the original static $S^3$ Einstein Universe [@Einstein] can exist only within the context of fundamental general relativity. Some coordinate transformations in GR are not allowed in emergent gravity: these are either singular transformations of the original coordinates, or transformations which remove some parts of spacetime (or add the extra parts). The non-equivalence of different metrics is especially important in the presence of an event horizon. For example, in emergent gravity the Painlevé-Gullstrand metric is more appropriate for the description of a black hole, than the Schwarzschild metric, which is (coordinate) singular at the horizon.
\(5) The Universe is naturally flat. In fundamental general relativity, the isotropic and homogeneous Universe means the space with constant curvature. In emergent gravity with an effective metric, the isotropic and homogeneous Universe corresponds to flat space. In general relativity the flatness of the Universe requires either fine tuning or inflationary scenario in which the curvature term is exponentially suppressed if the exponential inflation of the Universe irons out curved space to make it extraordinarily flat. The observed flatness of our Universe is in favor of emergent gravity.
![Momentum-space topology is the main source of massless elementary particles. But it must be accompanied by discrete symmetries between Fermi points (see Fig. \[TwoScenarios\]). [*bottom right*]{}: from “Knots in art” by Piotr Pieranski. []{data-label="SymVsTopology.eps"}](SymVsTopologyS.eps){width="1.0\linewidth"}
\(6) The cosmological constant is naturally small or zero. In general relativity, the cosmological constant is an arbitrary constant, and thus its smallness requires fine-tuning. Thus observations are in favor of emergent gravity. The unsolved problem is: what is the physical mechanism of relaxation of $\Lambda$ towards zero? (See Ref. [@Barcelo] and references therein.) The present small value of $\Lambda$ indicates that the Universe is so close to equilibrium that the current relaxation rate is very slow.
\(7) In the Fermi point scenario space-time is naturally 4-dimensional. This is a fundamental property of the Fermi-point topology, which as distinct from the string theory does not require the higher-dimensional space-times.
![([*top*]{}) In Standard Model the Fermi points with positive $N_3=+1$ and negative $N_3=-1$ topological charges are at the same point ${\bf p}=0$. It is the discrete symmetry between the Fermi points which prevents their mutual annihilation. When this symmetry is violated or spontaneously broken, there are two topologically different scenarios: ([*bottom left*]{}) either Fermi point annihilate each other and Dirac mass is formed; ([*bottom right*]{}) or Fermi points split [@Splitting]. It is possible that actually the splitting exists at the microscopic level, but in our low energy corner we cannot observe it because of the emergent gauge symmetry: in some cases splitting can be removed by gauge transformation.[]{data-label="TwoScenarios"}](HiggsVsSplitting.eps){width="0.7\linewidth"}
\(8) The underlying physics must contain discrete symmetries (Fig. \[SymVsTopology.eps\]). Their role is extremely important. The main role is to prohibit the cancellation of Fermi points with opposite topological charges (see Fig. \[TwoScenarios\]). As a side effect, in the low-energy corner discrete symmetries are transformed into gauge symmetries, and give rise to gauge fields. They also reduce the number of massless gauge bosons. To justify the Fermi point scenario, one should find the discrete symmetry which leads in the low energy corner to one of the GUT or Pati-Salam models.
![From history of hedgehogs, or three elements of modern physics: (i) quantum mechanics (or quantum field theory); (ii) Grand Unification based on the phenomenon of broken symmetry at low energy (GUT symmetry is restored when the Planck energy scale is approached from below); and (iii) anti-GUT based on the opposite phenomenon – GUT symmetry gradually emerges when the Planck energy scale is approached from above. A hedgehog-like topological defect in momentum space – the Fermi point – gives rise to symmetry emergent at Planck-GUT scales. In turn, symmetry breaking occurring at lower energy, gives rise to topological defects in real space (e.g., a hedgehog-like object) and life (e.g., a real hedgehog).[]{data-label="TriKita"}](TriKita.eps){width="1.0\linewidth"}
\(9) Lorentz symmetry must persist well above the Planck energy. The requirement for two well separated energy scales follows from the high precision of physical laws in our Universe [@Bjorken2001], and it represents the most crucial test of the emergent scenario. In the case when the Lorentz violating scale $E_{\rm Lorentz }< E_{\rm P}$, the metric field does not obey Einstein equations; instead it is governed by the hydrodynamic type equations (see Fig. \[GravityAndHierarchy\]). The Einstein equations emerge in the limit $E_{\rm Lorentz }\gg E_{\rm P}$, and their accuracy is determined by the small parameter $E_{\rm P}^2/E_{\rm Lorentz }^2\ll 1$. For example, in the Frolov-Fursaev version of Sakharov induced gravity [@FrolovFursaev1998], the ultraviolet cut-off is much larger than the Planck energy, and Einstein equations are reproduced. The observed bounds on the violation of Lorentz symmetry can be obtained from ultra-high-energy cosmic rays. For example, according to conservative estimations the relative value of the Lorentz violating terms in the Maxwell equations is below $10^{-18}$ [@KlinkhamerRisse]. This suggests that $E_{\rm Lorentz } > 10^9E_{\rm P}$, which is in favor of emergent scenario.
All this implies that physics continues far beyond the Planck scale, and this opens new possibilities for construction of microscopic theories. Since in the Fermi point scenario bosons are composite objects, the ultraviolet cut-off is different for fermions and bosons [@KlinkhamerVolovikMerging] (a similar situation occurs in condensed matter [@Chubukov]). The smaller (composite) scale can be associated with $E_{\rm P}$, while the “atomic” structure of the quantum vacuum will be only revealed at much higher Lorentz-violating scale $E_{\rm Lorentz}$. The opposite situation, when fermionic degrees of freedom emerge in the underlying bosonic quantum vacuum, is possible (see [@Kitaev; @Wen] and especially Ref. [@YueYu], where the Fermi points emerge in some model of spins on a three-dimensional lattice), but at the moment the generic mechanism for the emergence of gravity and matter in bosonic vacuum has not yet been found.
\(10) Finally, what about quantum mechanics? Actually both schemes for the classification of quantum vacua: by symmetry (GUT scheme) and by topology in momentum space (anti-GUT scheme) are based on quantum mechanics (Fig. \[TriKita\]). While general relativity is assumed to be as fundamental as quantum mechanics, emergent gravity with its emergent metric of the effective low-energy space-time is a secondary phenomenon. It is the byproduct of quantum field theory or of many-body quantum mechanics. As a result, in the Fermi-point scenario there are no principle contradictions between quantum mechanics and gravity. Due to the same reason, emergent gravity cannot be responsible for the issues related to foundations of quantum mechanics, and in particular for the collapse of the wave function. Also, item (9) implies that if quantum mechanics is not fundamental, the scale at which it emerges is far beyond the Planck scale.
I thank Frans Klinkhamer for fruitful discussions. This work has been supported in part by the European Science Foundation network programme “Quantum Geometry and Quantum Gravity” and by the Russian Foundation for Fundamental Research.
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---
abstract: 'We present flavour-symmetric results for the couplings of quark-antiquark systems to meson-meson channels in the harmonic-oscillator expansion. We tabulate their values for all possible open and closed decay channels of pseudo-scalar, vector and scalar mesons. We compare the predictions of a model that employs these flavour-symmetric couplings, both with the results of a model which uses explicitly flavour-dependent couplings, and with experiment.'
author:
- |
Eef van Beveren\
[*Departamento de Física, Universidade de Coimbra*]{}\
[*P-3000 Coimbra, Portugal*]{}\
[eef@malaposta.fis.uc.pt]{}\
\[.3cm\]
- |
George Rupp\
[*Centro de Física das Interacções Fundamentais*]{}\
[*Instituto Superior Técnico, Edifício Ciência*]{}\
[*P-1096 Lisboa Codex, Portugal*]{}\
[george@ajax.ist.utl.pt]{}\
\[.3cm\] [PACS number(s): 14.40.Cs, 12.39.Pn, 13.75.Lb]{}\
\[.3cm\] [hep-ph/9806248]{}
title: Flavour symmetry of mesonic decay couplings
---
24.cm 17.3cm -1.7cm -1.8cm 1.5cm 1.5cm 1.2em
.7cm
Introduction
============
Particle interactions are described by point-particle vertices in fundamental theories. Quarks, the basic particles for strong interactions, are point objects, to our best knowledge, hence assumed to interact via point-particle vertices in the existing theories: through a quark-gluon vertex in Quantum Chromodynamics (QCD) [@Fritzsch], through a four-quark vertex in the Nambu-Jona-Lasinio model (NJL) [@Nambu].
QCD exhibits good agreement with experiment, qualitatively for low and medium energies, and moreover quantitatively at high energies [@QCD], whereas NJL shows good agreement with experiment only for energies below 1 GeV [@NJL]. So in the energy interval crucial to meson physics, i.e., ranging from the two-pion threshold to energies as high as the states in the bottomonium system, no fundamental theory possesses a satisfactory descriptive power: not QCD, because the relevant momentum transfers are too low and thus the effective color coupling constant is too large for a perturbative approach, not NJL, because the energies are too high. Consequently, for a quantitative description of the spectra and scattering of mesons and baryons, neither of the two theories has sufficient predictive power for the time being. Therefore, the use of quark models is still opportune in this domain of hadronic physics.
Now, ideally a quark model should be derived from QCD, but this is rather utopian as yet. As a matter of fact, not even a direct relation between QCD and confinement has been established and so confinement usually has to be imposed on the valence quarks of the model [@confinement]. Different models follow distinct strategies to achieve this and the manner in which confinement is approached distinguishes models among each other. Moreover, each model has its own very specific purpose, often not mentioned in too much detail by the authors, which makes it difficult to compare models. For instance, there exist heavy-quark potential models made to measure in order to reproduce, with great accuracy, the radial and angular spectra of charmonium and bottomonium, as well as the electromagnetic properties of these systems. But if the same potentials are used in the light quark sector, the results are normally quite bad, especially for radial excitations, and even possible relativistic corrections are insufficient to cure the discrepancies. Conversely, sophisticated relativistic models for the light mesons usually fail to reproduce the correct radial spacings in the charmonium and bottomonium spectra.
On the other hand, most quark models treat hadrons as manifestly stable bound states of quarks, simply ignoring the fact that most hadrons are resonances, some of them even extremely broad, so as to make their very existence questionable. The standard justification is the conjecture that the effect of strong decay will be to produce predominantly imaginary mass shifts, thus allowing to first fit the real parts of the spectra and then to treat the hadronic widths *a posteriori, *with perturbative methods. However, we know from fundamental principles in scattering theory that real shifts are generally of the same order as or even larger than the imaginary ones. Moreover, hadronic loops, i.e., *virtual *decay channels, give rise to *attractive *forces, so that the shifts due to, in principle, all closed decay channels, must be added up so as to produce a negative mass shift. So not even the true bound states can be treated as pure quark states. The usual excuse is the unsupported assertion that the effect of closed channels will be negligible, except near threshold.******
However, the Nijmegen unitarised meson model (NUMM), devised to simultaneously describe meson spectra and meson-meson scattering, from the light pseudoscalars and vectors [@radial], via the usually awkward scalars [@scalar], all the way up to the $b\bar{b}$ sector [@charm], showed that both premises are indeed wrong: real shifts are generally comparable with or larger than the imaginary ones, and the damping of closed channels is insufficient to make their influence on the ground states of the spectra negligible. On the contrary, due to the nodal structure of the radial wave functions and the mentioned additivity property, the shifts — real and negative – of ground states are usually largest [@influence]. Furthermore, no drastic enhancement takes place near threshold, so that these states cannot be singled out [@threshold].
Having come to the conclusion that, for a truly quantitative description of mesonic spectra, one must include the coupling to meson-meson channels, the crucial questions to be raised are how to calculate the involved coupling constants and which two-meson channels to take along. Here, one should step back and have another look at the QCD Lagrangian. Realising that, at least qualitatively, there should be no obvious disagreement between QCD and whatever meson model to be used, we are led to respect manifest flavour blindness. This will impose stringent conditions on how couplings can be computed and how to select classes of decay channels, since obviously one cannot take into account an infinite number.
Models which describe the scattering of mesons (and/or baryons) often imitate the fundamental theories in the sense that interactions take place via effective point-particle vertices. However, meson (and baryons) are composite systems, built out of strongly interacting valence quarks, glue, and a quark-antiquark sea. So it seems obvious that, when mesons (and baryons) are considered point objects, some information must get lost. In this paper we will demonstrate that this can indeed be the case and how it manifests itself in the flavour non-independence of the such-described strong interactions, when not dealt with carefully.
Point interactions are a powerful tool in constructing theories that not only consider relativistic kinematics, but also take into account the property of particle creation and annihilation. However, in applying point interactions to composite systems, one should include all hidden degrees of freedom. Flavour is just one such degree of freedom. Angular momentum and spin are others which should be properly included. At present, it is opportune to model the internal degrees of freedom and next to integrate them out for the determination of effective point couplings. A consistent way of doing so, which moreover preserves flavour independence, is described below.
The organisation of this paper is as follows. In section \[flavour\], we discuss the general philosophy behind a simple model for flavour symmetry. This model is then exposed in section \[flavmod\]. The intensities of the three-meson vertices for meson decay into meson-meson pairs are given in section \[3mesonvertex\]. Results are discussed in section \[results\]. The consequences of flavour (in)dependence are studied for two different, though similar, models in section \[comparison\]. Some essential formulae are collected in the appendices \[rearrangement\] and \[scalarmixing\].
Flavour symmetry {#flavour}
================
Since strong interactions are independent of flavour, the probability to create a quark-antiquark pair out of the vacuum cannot depend on the flavour of the quark and the antiquark. However, it obviously depends on the masses involved. But if for a moment we assume that the flavour masses, or at least the effective quark masses in the relevant energy interval of the three lowest flavours, [*up*]{}, [*down*]{} and [*strange*]{}, are equal, then the corresponding probabilities of pair creation should be equal. Let us apply this principle to the strong coupling of a meson to a pair of mesons. Here, we assume that the related strong decay processes are triggered by the creation of a flavourless quark-antiquark pair.
In order to set the picture, we consider a simple model in which the initial meson is described by a confined quark-antiquark system of any flavour, given by
$$a\bar{b}
\;\;\; ,
\label{qqbar}$$
where $a$ and $b$ represent any of the three flavours under discussion, and the final pair of decay products is described by a system of two freely moving mesons, which represent any of the three combinations
$$(a\bar{u})\; +\; (u\bar{b})\;\; ,\;\;
(a\bar{d})\; +\; (d\bar{b})\;\; ,\;\;
(a\bar{s})\; +\; (s\bar{b})\;\; .
\label{mesonmeson}$$
Also, let us, for a moment, assume that no further quantum numbers are involved. Then, flavour symmetry demands that the probabilities are equal for the $a\bar{b}$ system to decay into any of the three channels of formula (\[mesonmeson\]). In particular, when under full flavour symmetry the six mesons represented in formula (\[mesonmeson\]) have all the same mass, then the experimental results for the decay of system (\[qqbar\]) into any of the three channels of formula (\[mesonmeson\]) should be indistinguishable.
Let the three decay coupling constants of the process under consideration be represented by, respectively,
$$g(a,b;u)\;\; ,\;\;
g(a,b;d)\;\; ,\;\;
g(a,b;s)\;\; .
\label{koppeling}$$
Then, assuming full flavour symmetry, we have the identities
$$\left\{ g(a,b;u)\right\}^{2}\; =\;
\left\{ g(a,b;d)\right\}^{2}\; =\;
\left\{ g(a,b;s)\right\}^{2}
\;\;\; ,
\label{flasym1}$$
and, moreover, for the total decay intensity $\Gamma (a,b)$ the relation
$$\Gamma (a,b)\; =\; A\;\left[\left\{ g(a,b;u)\right\}^{2}\; +\;
\left\{ g(a,b;d)\right\}^{2}\; +\;
\left\{ g(a,b;s)\right\}^{2}\right]
\;\;\; ,
\label{flasym2}$$
where the proportionality factor $A$ is also completely flavour symmetric, which means constant here. Furthermore, one has that, under flavour symmetry, $\Gamma (a,b)$ must be independent of the flavours $a$ and $b$.
Unfortunately, quarks are fermions and mesons are spatially extended systems, and hence spin and spatial quantum numbers do play an important rôle in the decay of a meson into a pair of mesons. Nevertheless, it remains possible to construct coupling constants which have the property that the total decay probability is independent of the flavour of the decaying meson in the limit of equal masses, as we will see below.
Modelling full flavour symmetry {#flavmod}
===============================
When a normalised wave function $\psi$ is expanded on a complete orthonormal basis $\phi_{n}$, for $n=0,1,2,\dots$, according to
$$\psi\; =\;\sum_{n=0}^{\infty}\; c_{n}\phi_{n}
\;\;\; ,
\label{expand}$$
one has for the expansion coefficients $c_{n}$ the property
$$\sum_{n=0}^{\infty}\; {\left| c_{n}\right|}^{2}\; =\; 1
\;\;\; .
\label{complete}$$
It is exactly property (\[complete\]) that leads to flavour symmetry.
Let us consider a system of two quarks and two antiquarks, like any of the three combinations of formula (\[mesonmeson\]). One complete basis for the Hilbert space of such a system can be constructed by taking products of the internal wave function of the $a\bar{b}$ system, specifying thereby its internal spatial and flavour quantum numbers, the internal wave function of the $q\bar{q}$ (either $u\bar{u}$, $d\bar{d}$ or $s\bar{s}$) system, and the relative wave function of the two subsystems. Another complete basis for this Hilbert space consists of products of the internal wave function of the $a\bar{q}$ system, the internal wave function of the $q\bar{b}$ system, and the relative wave function of those two subsystems. Any wave function describing one of the two-quark-two-antiquark systems (\[mesonmeson\]) can be expanded in either of the above-defined bases.
Such an expansion takes a particularly manageable form if the four partons are supposed to move in a harmonic oscillator potential with universal frequency. In that case the spatial quantum numbers are linearly related to the total energy of the system, which gives rise to finite bases at each energy level and hence to finite expansions. The flavour-symmetry condition (\[complete\]) then becomes a finite sum, which makes it easy for verification. Furthermore, the restriction to harmonic oscillators is not a real limitation, since any other basis can always be expanded in the corresponding harmonic oscillator basis, $\{ n\}$, according to
$$\left\langle M_{1}M_{2}\left| V\right| M\right\rangle\; =\;
\sum_{\{ n,n'\}}
\left\langle M_{1}M_{2}\left| n'\right.\right\rangle
\left\langle n' \left| V\right| n\right\rangle
\left\langle n\left| M\right.\right\rangle
\;\;\; .
\label{anypot}$$
Here, $V$ represents the interaction Hamiltonian which describes the transitions between the quark-antiquark system $a\bar{b}$ and the two-meson channels. We assume that the spatial, or momentum-dependent, part of the matrix elements $\left\langle n' \left| V\right| n\right\rangle$ is flavour independent and that the flavour-dependent parts are constants.
The expansion of a particular many-particle wave function into a specific basis for well-defined subsystems, or [*recoupling*]{}, has been studied a great deal in the past. The related coefficients for the harmonic oscillator basis are known as [*Moshinsky brackets*]{}. Moshinsky brackets are well-known coefficients of recoupling in Nuclear Physics; see Ref. [@Talmi] for their definition. The group-theoretical implications of parton recoupling in the harmonic-oscillator approximation have been studied exhaustively in Ref. [@Barg60]. Their application to meson decay has for the first time been formulated in Ref. [@Ribe82]. A full generalisation for the spatial part of the recoupling constants, which includes all possible quantum numbers for any number of (bosonic) partons, can be found in Ref. [@rekpplng]. The inclusion of fermionic and flavour degrees of freedom, which leads to an analytic expression for the coupling constants of any meson to any of its two-meson real or virtual decay channels, is given in Ref. [@kpplng], for the case that the new valence pair is created with $^{3}P_{0}$ quantum numbers. A Fortran source program is available on request.
Coupling constants for three-meson vertices {#3mesonvertex}
===========================================
Within the above-outlined formalism, we assume that mesons can be classified by the quantum numbers of their valence constituent quark-antiquark distributions, i.e.,
$${{\textstyle \mbox{\rm meson}}}\left( j,M,\ell ,s,n,{\cal M}\right)
\;\;\; .
\label{meson}$$
The quantum numbers $j$ and $M$ in formula (\[meson\]) represent, respectively, the spin of the meson and its $z$-component. Alternatively, $j$ represents the total angular momentum of the relative motion in the quark+antiquark system which describes the meson. The quantum numbers $\ell$, $s$ and $n$ stand, respectively, for the orbital angular momentum, the spin, and the radial excitation of the constituents of the meson. Finally, $\cal M$ represents the $3\times 3$ flavour matrix which indicates the valencies of the quark and the antiquark.
Here, we study the decay intensities for the following processes:
$${{\textstyle \mbox{\rm meson}}}\left( J,J_{z},\ell ,s,n,{\cal M}_{C}\right)
\;\longrightarrow\;
{{\textstyle \mbox{\rm meson}}}\left( j_{1},M_{1},\ell_{1},s_{1},n_{1},{\cal M}_{A}\right) +
{{\textstyle \mbox{\rm meson}}}\left( j_{2},M_{2},\ell_{2},s_{2},n_{2},{\cal M}_{B}\right)
\; .
\label{MMdecay}$$
This is not a completely satisfactory notation, since the spin $z$-components $M_{1}$ and $M_{2}$ of the decay products in formula (\[MMdecay\]) are not supposed to be observable and, moreover, the quantum numbers which characterise the relative motion of the decay products are not specified in formula (\[MMdecay\]), despite being equally important. Let us indicate the orbital quantum numbers of the two-meson system by means of an index, $r$, and hence denote the orbital angular momentum of the two mesons by $\ell_{r}$, the total spin by $s_{r}$, and the radial excitation of the relative motion in the two-meson system by $n_{r}$. The total angular momentum of the two-meson system and its $z$-component are, due to angular-momentum conservation, given by $J$ and $J_{z}$, respectively.
The decay probability for the process (\[MMdecay\]) is then, following the formalism developed in Ref. [@kpplng], given by the following matrix element:
$$\begin{aligned}
& & \;\;\;\;\;{\mbox{$\left\langle J,J_{z},j_{1},\ell_{1},s_{1},n_{1},
j_{2},\ell_{2},s_{2},n_{2},\ell_{r},s_{r},n_{r},A,B\left| J,J_{z},\ell ,s,n,C\right.\right\rangle$}}\; =
\label{decint} \\ [.3cm] & &
=\;{{\textstyle \mbox{\rm Tr}}}\left\{\;
{\cal M}_{A}{\cal M}_{B}{{\cal M}_{C}}^{T}
\;{\mbox{$\left\langle J,J_{z},j_{1},\ell_{1},s_{1},n_{1},
j_{2},\ell_{2},s_{2},n_{2},\ell_{r},s_{r},n_{r}\left| J,J_{z},\ell ,s,n,{\mbox{\boldmath $\alpha$}}_{ABC}\right.\right\rangle$}}
\right.\; +
\nonumber \\ [.3cm] & &
+\left.\;
{\cal M}_{B}{\cal M}_{A}{{\cal M}_{C}}^{T}
\;{\mbox{$\left\langle J,J_{z},j_{1},\ell_{1},s_{1},n_{1},
j_{2},\ell_{2},s_{2},n_{2},\ell_{r},s_{r},n_{r}\left| J,J_{z},\ell ,s,n,{\mbox{\boldmath $\alpha$}}_{BAC}\right.\right\rangle$}}\;
\right\}\;\;\; .
\nonumber\end{aligned}$$
The spatial parts in each of the two terms of the transition element (\[decint\]) are denoted by
$${\mbox{$\left\langle J,J_{z},j_{1},\ell_{1},s_{1},n_{1},
j_{2},\ell_{2},s_{2},n_{2},\ell_{r},s_{r},n_{r}\left| J,J_{z},\ell ,s,n,{\mbox{\boldmath $\alpha$}}\right.\right\rangle$}}
\;\;\; ,
\label{spatial}$$
and are defined and explained in Appendix \[rearrangement\].
Since for many purposes it is sufficient to have flavour-symmetric coupling constants for the corresponding strong decay channels of pseudoscalar, vector, and scalar mesons, we tabulate the probabilities for the three-meson vertices of those decay processes; for a pseudoscalar meson in Table \[pseudoscalar\], for a vector meson in Table \[vector\], and for a scalar meson in Table \[scalar\]. In order to maintain the tables as condensed as possible, we represent mesons by symbols and by their quantum numbers. Since we assume that isospin is indeed a perfect symmetry, we may represent all members of an isomultiplet by the same symbol, for which we just have chosen the letters and numbers $t$, $d$, $8$ and $1$, according to the identification given in Table \[partid\]. Now let us just analyse one horizontal line of one of the three tables, to make sure that the reader understands what the numbers represent. Let us take the fourth line of Table \[pseudoscalar\]. In the first column we find four zeroes, representing the internal spatial quantum numbers $j$, $\ell$, $s$, and $n$ of the first decay product, $M_{1}$, which hence characterises a meson out of the lowest-lying ($n=0$) pseudoscalar nonet. In the second column we find similarly that the second decay product, $M_{2}$, represents a meson out of the lowest-lying vector nonet. In the third column we find the quantum numbers for the relative motion of $M_{1}$ and $M_{2}$, i.e., $P$-wave ($\ell_{r}=1$) with total spin one ($s_{r}=1$) in the lowest radial excitation ($n_{r}=0$). Since the table refers to the real or virtual decays of the lowest-lying pseudoscalar meson nonet ($J\ell sn=0000$, indicated in the top of the table), the next four columns refer to its isotriplet member, which is the pion. We then find that the pion couples with a strength $\sqrt{1/6}$ to the $tt$ (isotriplet-isotriplet) channel, which, following Table \[partid\] and the above-discussed particle assignments to $M_{1}$ and $M_{2}$, i.e., pseudoscalar and vector respectively, represents in this case the $\pi\rho$ channel. Following a similar reasoning, we find that the pion couples with a strength $\sqrt{1/12}$ to $KK^{\ast}$. The total coupling of a pion to pseudoscalar-vector channels is given in the column under $T$ by $\sqrt{1/4}$, which is the square root of the quadratic sum of the two previous couplings, i.e., $\sqrt{1/6+1/12}$.
The next set of coupling constants refer to the real or virtual (actually only virtual) decays of a kaon. We find $\sqrt{1/8}$ to $td$, which represents both of the possibilities pseudoscalar (isotriplet) + vector (isodoublet), i.e., $\pi K^{\ast}$, and pseudoscalar (isodoublet) + vector (isotriplet), i.e., $K\rho$, each with half of the intensity that is given in the table, and therefore one has for the kaon the coupling constants $\sqrt{1/16}$ to $\pi K^{\ast}$ and $\sqrt{1/16}$ to $K\rho$. Next, we find in the table that the kaon couples with $\sqrt{1/8}$ to $d8$, which represents both of the possibilities, pseudoscalar (isodoublet) + vector ($SU_{3}$-octet isoscalar), i.e., $K$ + some admixture of the $\omega$ and $\phi$ mesons, and pseudoscalar ($SU_{3}$-octet isoscalar) + vector (isodoublet), i.e., some admixture of $\eta$’s + $K^{\ast}$, each with half of the intensity that is given in the table, and hence one extracts for the kaon the coupling constants $\sqrt{1/16}$ to $K+(\omega ,\phi )$ and $\sqrt{1/16}$ to $(\eta ,\eta ')+K^{\ast}$. The kaon does not couple to the $d1$ channels in pseudoscalar + vector, which represent the channels with one isodoublet and one $SU_{3}$-singlet. The total coupling for the kaon to its pseudoscalar + vector decay channels sums up to $\sqrt{1/4}$, as one verifies in the column under $T$. The next two sets of coupling constants refer similarly to the decay modes of the isoscalar, either $SU_{3}$-octet or $SU_{3}$-singlet, partners of the pseudoscalar nonet. Mixings can be done by hand as examplified in Appendix \[scalarmixing\].
Notice that for the $SU_{3}$-octet members one has flavour symmetry for each horizontal line in the tables. This does not go through for the $SU_{3}$-singlet partners, with the exception of scalar meson decay (Table \[scalar\]), where all horizontal lines have the same total coupling in each subsection of the table. However, all columns under $T$ sum up to 1, representing full flavour symmetry once all possible decay channels have been accounted for.
Results
=======
The pole structures of the scattering matrices for $P$-wave meson-meson scattering, or equivalently, the radial spectra of heavy and light pseudoscalar and vector mesons, were studied in the, largely non-relativistic, coupled-channel NUMM, published in Ref. [@radial] hereafter referred to as B83, in which the authors parametrised confinement by a universal frequency, the same for all flavours, including charm and bottom. The universal frequency and a flavour-independent overall coupling constant, representing the probability for the creation of a $^{3}P_{0}$ light quark-antiquark pair, were sufficient to obtain theoretical predictions for phase shifts and scattering cross sections, or equivalently, for central resonance positions and widths, which were in reasonable agreement with the data. All relative couplings were exactly taken as given in Tables \[pseudoscalar\] and \[vector\], though some of these had been derived in a more empiric way, and then extended in order to also include the heavy-quark systems. This extension is quite trivial and will not be discussed here. The only flavour non-invariance came from the quark masses and the two-meson thresholds, all other ingredients were the same for all flavours. Of course, many of the decay channels were omitted, assuming their thresholds to be high enough in energy, so as not to have too much importance for the details of the scattering processes at much lower energies. But this is only a practical ingredient, not to be confused with flavour breaking.
In Ref. [@EMtrans], the electromagnetic transitions in the charmonium and bottomonium systems were studied, using the quark and meson distributions from B83, with good results, indicating that not only the pole structures of the scattering matrices, but also the related wave functions stood the confrontation with experiment.
In Ref. [@scalar], hereafter referred to as B86, the pole structure of the scattering matrix was inspected for $S$-wave meson-meson scattering. Since the model was the same as for Ref. [@radial], using exactly the same universal frequency, flavour-independent overall coupling constant, and quark masses, the calculated phase shifts and scattering cross sections could be considered genuine theoretical predictions. The agreement with the data was unexpected, especially because it had not been the objective of the model, neither was the model constructed towards fitting the $S$-wave scattering data. All relative couplings were exactly taken as given in Table \[scalar\]. Also here, only those two-meson channels were taken into account which contain members of the lowest-lying pseudoscalar and vector nonets. That such a procedure does not break flavour invariance may be explicitly verified by checking the first, third, and last line in Table \[scalar\].
The observed flavour independence of strong interactions is a very important ingredient for low-energy hadron physics and woven into the NUMM; first, by the universal frequency, which makes the ratio of the kinetic term and the potential term of the model flavour independent and hence also the level splittings; second, by the intensities of the three-meson vertices for the coupling to the various decay channels.
Comparison of two models {#comparison}
========================
As stated in the introduction, it is not easy to compare meson models, but here we will pay attention to the comparison of the NUMM with a model [@Toern95], hereafter referred to as T95, which is tailor-made for scalar mesons or, in other words, for $S$-wave meson-meson scattering. The latter model, a revised version of the Helsinki unitarised quark model (HUQM), was confronted with experiment in an analysis published in Ref. [@Toern96], hereafter referred to as TR96. Based on the good agreement of its theoretical predictions with the available experimental phase shifts, one may be inclined to accept all further conclusions presented in the same publication, such as the existence and location of resonances. However, the authors failed to find the complex-energy pole corresponding to the established $f_0$(1500) resonance. Furthermore, they also did not find a light $K_0^*$, i.e., the old $\kappa$, somewhere between 700 and 1100 MeV. Although the latter resonance is not (yet) established experimentally, it has recently received renewed phenomenological and theoretical support [@Ish97; @Sch98; @Rij98] (see also Ref. [@comment]). Moreover, its absence in Nature would imply a breaking of the conventional nonet pattern for mesons. On the other hand, if a light $K_0^*$ is confirmed, then there exist unmistakable experimental candidates for *two *complete scalar nonets, as predicted by B86. So it is intriguing to figure out why model T95/TR96, which is very similar in its philosophy and also in observing a resonance doubling, at least for some states, does not reproduce this resonance.**
In Table \[tabspsps\], we collect the intensities for strong scalar-meson decay into a pair of pseudoscalar mesons, under the assumption that pions, kaons, and eta’s have equal masses, as given by models B86 and T95/TR96. For the purpose of comparison, we have multiplied the values given in B86 by a constant factor. The resulting values can also be read from the first line of Table \[scalar\] when renormalised (i.e., multiplied by a factor 24), and when isoscalar mixing has been dealt with as outlined in Appendix \[scalarmixing\]. Now notice that, by using formula (\[flasym2\]), the total decay intensities stemming from Ref. [@scalar] become equal to $A$ for all scalar mesons, as demanded by flavour symmetry, i.e.,
$$\Gamma\left( a_{0}\right)\; =\;
\Gamma\left( \kappa\right)\; =\;
\Gamma\left( f_{0}\; ,n\bar{n}\right)\; =\;
\Gamma\left( f_{0}\; ,s\bar{s}\right)\; =\; A
\;\;\; .
\label{flasym3}$$
For model T95/TR96, the comparable intensities are derived from a point-particle approach to the three-meson vertex, which results in coupling constants given by
$${{\textstyle \mbox{\rm Tr}}}\left({\cal M}_{A}{\cal M}_{B}{\cal M}_{C}\right)
\;\;\; ,
\label{simple}$$
where $A$, $B$ and $C$ stand for the three mesons involved at the vertex $C\rightarrow AB$, and ${\cal M}_{X}$ is the $3\times 3$ flavour matrix for meson $X$. Flavour symmetry as from Eq. (\[simple\]), and using formula (\[flasym2\]), yields in this case the flavour-dependent result
$$\Gamma\left( a_{0}\right)\; =\;
\Gamma\left(\kappa\right)\; =\;
{\frac{\textstyle 3}{\textstyle 5}}\Gamma\left( f_{0}\; n\bar{n}\right)\; =\;
{\frac{\textstyle 3}{\textstyle 4}}\Gamma\left( f_{0}\; s\bar{s}\right)\; =\; A
\;\;\; ,
\label{flasym4}$$
in contrast with the results shown in Eq. (\[flasym3\]).
It might be surprising that such a seemingly flavour-symmetric vertex leads to flavour [*non*]{}-independence when applied to strong meson decay. However, if one does not take into account the internal structure of the various quark-antiquark systems involved, then normalisations, so essential to wave functions or distributions, are swept under the rug, and thus the above vertex intensity (\[simple\]) leads to flavour-dependent results. It appears to be for this reason that the authors of TR96, which base their calculations on the coupling constants from T95, do not observe any resonance doubling for the isodoublet and one of the two isoscalars, and therefore miss the $K^{\ast}_{0}$(700-1100) and $f_0$(1500) poles needed to complete two scalar meson nonets.
The normalisation factors that are relevant to formula (\[flasym3\]) are given in formulae (\[decABC\]) and (\[normtab\]) of Appendix \[rearrangement\]. In Appendix \[scalarmixing\], we show how they lead exactly to the factors ${\mbox{$\frac{3}{5}$}}$ and ${\mbox{$\frac{3}{4}$}}$ which are necessary to compensate the flavour-dependence of formula (\[flasym4\]).
Rearrangement coefficients {#rearrangement}
==========================
The spatial parts of the matrix elements (\[decint\]) are, following the formalism developed in Ref. [@kpplng], in the approximation of equal flavour masses just given by Clebsch-Gordonary and some overlap integrals, amounting to
$$\begin{aligned}
& & \;\;\;\;\;\left.\begin{array}{l}
{\mbox{$\left\langle J,J_{z},j_{1},\ell_{1},s_{1},n_{1},
j_{2},\ell_{2},s_{2},n_{2},\ell_{r},s_{r},n_{r}\left| J,J_{z},\ell ,s,n,{\mbox{\boldmath $\alpha$}}_{ABC}\right.\right\rangle$}}\\ [.5cm]
{\mbox{$\left\langle J,J_{z},j_{1},\ell_{1},s_{1},n_{1},
j_{2},\ell_{2},s_{2},n_{2},\ell_{r},s_{r},n_{r}\left| J,J_{z},\ell ,s,n,{\mbox{\boldmath $\alpha$}}_{BAC}\right.\right\rangle$}}\end{array}\right\}\; =
\nonumber \\ [.5cm] & &
=\;{\frac{\textstyle 1}{\textstyle \sqrt{1+
{\mbox{$\left\langle C\left| SU(3)_{{\textstyle \mbox{\rm \scriptsize flavour}}}{{\textstyle \mbox{\rm -singlet}}}\right.\right\rangle$}}
\delta (^{2s+1}\ell_{J},^{3}P_{0})\delta_{n0}}}}\;\;
\sum_{{\textstyle \{\mu\} ,\{ m\} ,\{ M\}}}
\label{decABC} \\ [.3cm] & &
{\mbox{$\left(\begin{array}{rrr}
s_{r} & \ell_{r} & J \\ [.1cm] M_{r} & m_{r} & J_{z} \end{array}\right)$}}
{\mbox{$\left(\begin{array}{rrr}
j_{1} & j_{2} & s_{r} \\ [.1cm] M_{1} & M_{2} & M_{r} \end{array}\right)$}}
{\mbox{$\left(\begin{array}{rrr}
\ell_{1} & s_{1} & j_{1} \\ [.1cm] m_{1} & \mu_{1} & M_{1} \end{array}\right)$}}
{\mbox{$\left(\begin{array}{rrr}
\ell_{2} & s_{2} & j_{2} \\ [.1cm] m_{2} & \mu_{2} & M_{2} \end{array}\right)$}}\;\times
\nonumber \\ [.3cm] & & \times\;
{\mbox{$\left(\begin{array}{rrr}
\ell & s & J \\ [.1cm] m_{\ell} & \mu_{s} & J_{z} \end{array}\right)$}}
{\mbox{$\left(\begin{array}{rrr}
1 & 1 & 0 \\ [.1cm] m & -m & 0 \end{array}\right)$}}
{\mbox{$\left(\begin{array}{rrr}
{\mbox{$\frac{1}{2}$}} & {\mbox{$\frac{1}{2}$}} & s_{1} \\ [.1cm] \mu_{a} & \mu_{b} & \mu_{1} \end{array}\right)$}}
{\mbox{$\left(\begin{array}{rrr}
{\mbox{$\frac{1}{2}$}} & {\mbox{$\frac{1}{2}$}} & s_{2} \\ [.1cm] \mu_{c} & \mu_{d} & \mu_{2} \end{array}\right)$}}\;\times
\nonumber \\ [.3cm] & & \times\;
\left\{\begin{array}{l}
{\mbox{$\left(\begin{array}{rrr}
{\mbox{$\frac{1}{2}$}} & {\mbox{$\frac{1}{2}$}} & s \\ [.1cm] \mu_{a} & \mu_{d} & \mu_{s} \end{array}\right)$}}
{\mbox{$\left(\begin{array}{rrr}
{\mbox{$\frac{1}{2}$}} & {\mbox{$\frac{1}{2}$}} & 1 \\ [.1cm] \mu_{c} & \mu_{b} & -m \end{array}\right)$}}
\left(\begin{array}{ccccccc}
n & \ell & m_{\ell} & & n_{1} & \ell_{1} & m_{1}\\ [.1cm]
0 & 1 & m & & n_{2} & \ell_{2} & m_{2}\\ [.1cm]
0 & 0 & 0 & & n_{r} & \ell_{r} & m_{r}
\end{array}\right)_{{\textstyle {\mbox{\boldmath $\alpha$}}_{ABC}}}\\ [.6cm]
{\mbox{$\left(\begin{array}{rrr}
{\mbox{$\frac{1}{2}$}} & {\mbox{$\frac{1}{2}$}} & s \\ [.1cm] \mu_{c} & \mu_{b} & \mu_{s} \end{array}\right)$}}
{\mbox{$\left(\begin{array}{rrr}
{\mbox{$\frac{1}{2}$}} & {\mbox{$\frac{1}{2}$}} & 1 \\ [.1cm] \mu_{a} & \mu_{d} & -m \end{array}\right)$}}
\left(\begin{array}{ccccccc}
n & \ell & m_{\ell} & & n_{1} & \ell_{1} & m_{1}\\ [.1cm]
0 & 1 & m & & n_{2} & \ell_{2} & m_{2}\\ [.1cm]
0 & 0 & 0 & & n_{r} & \ell_{r} & m_{r}
\end{array}\right)_{{\textstyle {\mbox{\boldmath $\alpha$}}_{BAC}}}\end{array}\right.
\nonumber\end{aligned}$$
where the sum is over all $\mu$’s, $m$’s, and $M$’s that appear in the formula, where
$$\begin{aligned}
{\mbox{$\left\langle C\left| SU(3)_{{\textstyle \mbox{\rm \scriptsize flavour}}}{{\textstyle \mbox{\rm -singlet}}}\right.\right\rangle$}}\; = & &
\nonumber \\ [.3cm]
0 & {{\textstyle \mbox{\rm for}}} & {\mbox{$\left| C\right\rangle$}}\;\;\;{{\textstyle \mbox{\rm orthogonal to the }}}
SU(3)_{{\textstyle \mbox{\rm \scriptsize flavour}}}{{\textstyle \mbox{\rm -singlet state}}}
\;\;\; ,\nonumber \\ [.3cm]
{\mbox{$\frac{1}{3}$}} & {{\textstyle \mbox{\rm for}}} & {\mbox{$\left| C\right\rangle$}}\; =\;
{\mbox{$\left| u\bar{u}\right\rangle$}}\; ,\;\;\;
{\mbox{$\left| d\bar{d}\right\rangle$}}\; ,\;\;{{\textstyle \mbox{\rm or}}}\;\;\;
{\mbox{$\left| s\bar{s}\right\rangle$}}
\;\;\; ,\nonumber \\ [.3cm]
{\mbox{$\frac{2}{3}$}} & {{\textstyle \mbox{\rm for}}} & {\mbox{$\left| C\right\rangle$}}\; =\;\sqrt{{\mbox{$\frac{1}{2}$}}}\left\{
{\mbox{$\left| u\bar{u}\right\rangle$}}+{\mbox{$\left| d\bar{d}\right\rangle$}}\right\}
\;\;\; ,\label{normtab} \\ [.3cm]
1 & {{\textstyle \mbox{\rm for}}} & {\mbox{$\left| C\right\rangle$}}\; =\;\sqrt{{\mbox{$\frac{1}{3}$}}}\left\{
{\mbox{$\left| u\bar{u}\right\rangle$}}+{\mbox{$\left| d\bar{d}\right\rangle$}}+{\mbox{$\left| s\bar{s}\right\rangle$}}\right\}
\nonumber\end{aligned}$$
and where
$$\delta (^{2s+1}\ell_{J},^{3}P_{0})\; =\;
\delta_{J0}\delta_{\ell 1}\delta_{s1}
\;\;\; .
\label{delta3P0}$$
The central part of formula (\[decABC\]) is constituted by the rearrangement coefficients, which can be given by the following diagramatic representation
$$\left(\begin{array}{ccccccc}
n & \ell & m_{\ell} & & n_{1} & \ell_{1} & m_{1}\\ [.1cm]
0 & 1 & m & & n_{2} & \ell_{2} & m_{2}\\ [.1cm]
0 & 0 & 0 & & n_{r} & \ell_{r} & m_{r}
\end{array}\right)_{{\textstyle {\mbox{\boldmath $\alpha$}}}}\; =\;\;\;\;
\begin{picture}(240,80)(0,60)
\linethickness{2pt}
\multiput(0,0)(0,60){3}{\line(1,0){60}}
\multiput(180,0)(0,60){3}{\line(1,0){60}}
\thinlines
\put(60,0){\line(1,0){50}}
\put(120,0){\makebox(0,0){$\alpha_{33}$}}
\put(130,0){\line(1,0){50}}
\put(60,0){\line(2,1){35}}
\put(100,20){\makebox(0,0){$\alpha_{23}$}}
\put(105,22.5){\line(2,1){75}}
\put(60,0){\line(1,1){15}}
\put(80,20){\makebox(0,0){$\alpha_{13}$}}
\put(85,25){\line(1,1){95}}
\put(60,60){\line(2,-1){71}}
\put(140,20){\makebox(0,0){$\alpha_{32}$}}
\put(149,15.5){\line(2,-1){31}}
\put(60,60){\line(1,0){20}}
\put(90,60){\makebox(0,0){$\alpha_{22}$}}
\put(100,60){\line(1,0){80}}
\put(60,60){\line(2,1){75}}
\put(140,100){\makebox(0,0){$\alpha_{12}$}}
\put(145,102.5){\line(2,1){35}}
\put(60,120){\line(1,0){50}}
\put(120,120){\makebox(0,0){$\alpha_{11}$}}
\put(130,120){\line(1,0){50}}
\put(60,120){\line(2,-1){31}}
\put(100,100){\makebox(0,0){$\alpha_{21}$}}
\put(109,95.5){\line(2,-1){71}}
\put(60,120){\line(1,-1){15}}
\put(80,100){\makebox(0,0){$\alpha_{31}$}}
\put(85,95){\line(1,-1){95}}
\put(30,5){\makebox(0,0)[cb]{$0,0,0$}}
\put(30,65){\makebox(0,0)[cb]{$0,1,m$}}
\put(30,125){\makebox(0,0)[cb]{$n,\ell ,m_{\ell}$}}
\put(210,5){\makebox(0,0)[cb]{$n_{r},\ell_{r},m_{r}$}}
\put(210,65){\makebox(0,0)[cb]{$n_{2},\ell_{2},m_{2}$}}
\put(210,125){\makebox(0,0)[cb]{$n_{1},\ell_{1},m_{1}$}}
\end{picture}
\label{reardiagram}$$
The upper-left external line carries the relevant quantum numbers of the initial meson in formula (\[MMdecay\]), the middle-left external line the quantum numbers of the $^{3}P_{0}$ $q\bar{q}$-pair, and the lower-left external line the quantum numbers of the relative motion of the two quark-antiquark systems, which, to lowest order, is supposed to be in its ground state. The external lines on the right-hand side of the diagram carry the quantum numbers of the decay products of formula (\[MMdecay\]) and their relative motion.
As explained in Ref. [@rekpplng], each of the internal lines ${ij}$ of diagram (\[reardiagram\]) carries the set of quantum numbers $\left\{ n_{ij},\ell_{ij},m_{ij}\right\}$, over all possibilities of which must be summed, thereby respecting partial quantum-number conservation at each vertex, i.e.,
$$\begin{aligned}
& & \;\;\;\;\;
\left(\begin{array}{ccccccc}
n & \ell & m_{\ell} & & n_{1} & \ell_{1} & m_{1}\\ [.1cm]
0 & 1 & m & & n_{2} & \ell_{2} & m_{2}\\ [.1cm]
0 & 0 & 0 & & n_{r} & \ell_{r} & m_{r}
\end{array}\right)_{{\textstyle {\mbox{\boldmath $\alpha$}}}}
\; =\;
(-1)^{{\textstyle n+n_{1}+n_{2}+n_{r}}}\;\left({\frac{\textstyle \pi}{\textstyle 4}}\right)^{3}\;
\sqrt{\left( n!\; n_{1}!\; n_{2}!\; n_{r}!\right)}
\nonumber \\ [.3cm] & &
\sqrt{\left({\frac{\textstyle
\Gamma\left( 2n+\ell+{\mbox{$\frac{3}{2}$}}\right)
\Gamma\left({\mbox{$\frac{5}{2}$}}\right)
\Gamma\left({\mbox{$\frac{3}{2}$}}\right)
\Gamma\left( 2n_{1}+\ell_{1}+{\mbox{$\frac{3}{2}$}}\right)
\Gamma\left( 2n_{2}+\ell_{2}+{\mbox{$\frac{3}{2}$}}\right)
\Gamma\left( 2n_{r}+\ell_{r}+{\mbox{$\frac{3}{2}$}}\right)}{\textstyle \left( 2\ell +1\right)
\left( 3\right)
\left( 2\ell_{1}+1\right)
\left( 2\ell_{2}+1\right)
\left( 2\ell_{r}+1\right)}}\right)}
\nonumber \\ [.3cm] & &
\sum_{{\textstyle \left\{ n_{ij},\ell_{ij},m_{ij}\right\}}}\;\;
\prod_{{\textstyle i,j}}\;\; \left(\alpha_{ij}\right)^{{\textstyle 2n_{ij}+\ell_{ij}}}
{\frac{\textstyle \left( 2\ell_{ij}+1\right)}{\textstyle n_{ij}!\;\Gamma\left( n_{ij}+\ell_{ij}+{\mbox{$\frac{3}{2}$}}\right)}}
\nonumber \\ [.3cm] & &
\delta\left( 2\left[ n_{11}+n_{21}+n_{31}\right] +
\ell_{11}+\ell_{21}+\ell_{31}\; ,\; 2n+\ell\right)
\nonumber \\ [.3cm] & &
\delta\left( 2\left[ n_{12}+n_{22}+n_{32}\right] +
\ell_{12}+\ell_{22}+\ell_{32}\; ,\; 1\right)
\nonumber \\ [.3cm] & &
\delta\left( 2\left[ n_{13}+n_{23}+n_{33}\right] +
\ell_{13}+\ell_{23}+\ell_{33}\; ,\; 0\right)
\nonumber \\ [.3cm] & &
\delta\left( 2\left[ n_{11}+n_{12}+n_{13}\right] +
\ell_{11}+\ell_{12}+\ell_{13}\; ,\; 2n_{1}+\ell_{1}\right)
\nonumber \\ [.3cm] & &
\delta\left( 2\left[ n_{21}+n_{22}+n_{23}\right] +
\ell_{21}+\ell_{22}+\ell_{23}\; ,\; 2n_{2}+\ell_{2}\right)
\nonumber \\ [.3cm] & &
\delta\left( 2\left[ n_{31}+n_{32}+n_{33}\right] +
\ell_{31}+\ell_{32}+\ell_{33}\; ,\; 2n_{r}+\ell_{r}\right)
\nonumber \\ [.3cm] & &
\left(\begin{array}{ccc|c}
\ell_{11} & \ell_{21} & \ell_{31} & \ell\\ [.1cm]
m_{11} & m_{21} & m_{31} & m_{\ell}\end{array}\right)
\left(\begin{array}{ccc|c}
\ell_{12} & \ell_{22} & \ell_{32} & 1\\ [.1cm]
m_{12} & m_{22} & m_{32} & m\end{array}\right)
\left(\begin{array}{ccc|c}
\ell_{13} & \ell_{23} & \ell_{33} & 0\\ [.1cm]
m_{13} & m_{23} & m_{33} & 0\end{array}\right)
\nonumber \\ [.3cm] & &
\left(\begin{array}{ccc|c}
\ell_{11} & \ell_{12} & \ell_{13} & \ell_{1}\\ [.1cm]
m_{11} & m_{12} & m_{13} & m_{1}\end{array}\right)
\left(\begin{array}{ccc|c}
\ell_{21} & \ell_{22} & \ell_{23} & \ell_{2}\\ [.1cm]
m_{21} & m_{22} & m_{23} & m_{2}\end{array}\right)
\left(\begin{array}{ccc|c}
\ell_{31} & \ell_{32} & \ell_{33} & \ell_{r}\\ [.1cm]
m_{31} & m_{32} & m_{33} & m_{r}\end{array}\right)\; ,
\label{rearformula}\end{aligned}$$
where the angular-momenta recoupling coefficients are defined by
$$\begin{aligned}
& & \;\;\;\;\;
\left(\begin{array}{ccc|c}
\ell_{1} & \ell_{2} & \ell_{3} & \ell\\ [.1cm]
m_{1} & m_{2} & m_{3} & m\end{array}\right)
\; =
\nonumber \\ [.3cm] & = &
\sum_{L,M}\;
{\mbox{$\left(\begin{array}{rrr}
\ell_{1} & \ell_{2} & L \\ [.1cm] m_{1} & m_{2} & M \end{array}\right)$}}
{\mbox{$\left(\begin{array}{rrr}
L & \ell_{3} & \ell \\ [.1cm] M & m_{3} & m \end{array}\right)$}}
{\mbox{$\left(\begin{array}{rrr}
\ell_{1} & \ell_{2} & L \\ [.1cm] 0 & 0 & 0 \end{array}\right)$}}
{\mbox{$\left(\begin{array}{rrr}
L & \ell_{3} & \ell \\ [.1cm] 0 & 0 & 0 \end{array}\right)$}}\; ,\end{aligned}$$
and where the -matrices, for the case of equal constituent flavour masses, are given by
$${\mbox{\boldmath $\alpha$}}_{ABC}\; =\;
\left(\begin{array}{ccc}
{\mbox{$\frac{1}{2}$}} & {\mbox{$\frac{1}{2}$}} & -\sqrt{{\mbox{$\frac{1}{2}$}}}\\ [.1cm]
{\mbox{$\frac{1}{2}$}} & {\mbox{$\frac{1}{2}$}} & \sqrt{{\mbox{$\frac{1}{2}$}}}\\ [.1cm]
-\sqrt{{\mbox{$\frac{1}{2}$}}} & \sqrt{{\mbox{$\frac{1}{2}$}}} & 0\end{array}\right)
\;\;\;{{\textstyle \mbox{\rm and}}}\;\;\;
{\mbox{\boldmath $\alpha$}}_{BAC}\; =\;
\left(\begin{array}{ccc}
{\mbox{$\frac{1}{2}$}} & {\mbox{$\frac{1}{2}$}} & \sqrt{{\mbox{$\frac{1}{2}$}}}\\ [.1cm]
{\mbox{$\frac{1}{2}$}} & {\mbox{$\frac{1}{2}$}} & -\sqrt{{\mbox{$\frac{1}{2}$}}}\\ [.1cm]
\sqrt{{\mbox{$\frac{1}{2}$}}} & -\sqrt{{\mbox{$\frac{1}{2}$}}} & 0\end{array}\right)\; .
\label{alphamat}$$
The allowed values for the quantum numbers of the internal lines of diagram (\[reardiagram\]), given by $n_{ij}$ and $\ell_{ij}$ in formula (\[rearformula\]), are non-negative integers and hence, because of partial quantum number conservation at each vertex of the diagram, which is moreover expressed by the Kronecker delta’s in formula (\[rearformula\]), we find
$$n_{12}\; =\;
n_{22}\; =\;
n_{32}\; =\;
n_{13}\; =\;
n_{23}\; =\;
n_{33}\; =\;
\ell_{13}\; =\;
\ell_{23}\; =\;
\ell_{33}\; =\; 0\;\;\; ,$$
which, also substituting the -matrices (\[alphamat\]), simplifies the expression for the rearrangement coefficients to
$$\begin{aligned}
& & \;\;\;\;\;
\left(\begin{array}{ccccccc}
n & \ell & m_{\ell} & & n_{1} & \ell_{1} & m_{1}\\ [.1cm]
0 & 1 & m & & n_{2} & \ell_{2} & m_{2}\\ [.1cm]
0 & 0 & 0 & & n_{r} & \ell_{r} & m_{r}
\end{array}\right)_{{\textstyle
\left\{\begin{array}{c}{\mbox{\boldmath $\alpha$}}_{ABC}\\{\mbox{\boldmath $\alpha$}}_{BAC}\end{array}
\right\} }} =
(-1)^{{\textstyle n+n_{1}+n_{2}+n_{r}}}\left({\frac{\textstyle \pi}{\textstyle 4}}\right)^{3}
\sqrt{\left( n!\; n_{1}!\; n_{2}!\; n_{r}!\right)}
\nonumber \\ [.3cm] & &
\sqrt{\left({\frac{\textstyle
\Gamma\left( 2n+\ell+{\mbox{$\frac{3}{2}$}}\right)
\Gamma\left({\mbox{$\frac{5}{2}$}}\right)
\Gamma\left({\mbox{$\frac{3}{2}$}}\right)
\Gamma\left( 2n_{1}+\ell_{1}+{\mbox{$\frac{3}{2}$}}\right)
\Gamma\left( 2n_{2}+\ell_{2}+{\mbox{$\frac{3}{2}$}}\right)
\Gamma\left( 2n_{r}+\ell_{r}+{\mbox{$\frac{3}{2}$}}\right)}{\textstyle \left( 2\ell +1\right)
\left( 3\right)
\left( 2\ell_{1}+1\right)
\left( 2\ell_{2}+1\right)
\left( 2\ell_{r}+1\right)}}\right)}
\nonumber \\ [.3cm] & &
\left({\mbox{$\frac{1}{2}$}}\right)^{{\textstyle 2n+\ell +1-n_{r}-{\mbox{$\frac{1}{2}$}}\ell_{r}}}\;\;
\sum_{{\textstyle \left\{ n_{ij},\ell_{ij},m_{ij}\right\}}}\;\;
\left\{\begin{array}{c} (-1)^{{\textstyle \ell_{31}}}\\[.5cm]
(-1)^{{\textstyle \ell_{32}}}\end{array}\right\}
\nonumber \\ [.3cm] & &
{\frac{\textstyle 1}{\textstyle n_{11}!\;n_{21}!\;n_{31}!}}\;
{\frac{\textstyle
\left( 2\ell_{11}+1\right)
\left( 2\ell_{12}+1\right)
}{\textstyle
\Gamma\left( n_{11}+\ell_{11}+{\mbox{$\frac{3}{2}$}}\right)
\Gamma\left(\ell_{12}+{\mbox{$\frac{3}{2}$}}\right)
\Gamma\left({\mbox{$\frac{3}{2}$}}\right)
}}
\nonumber \\ [.3cm] & &
{\frac{\textstyle
\left( 2\ell_{21}+1\right)
\left( 2\ell_{22}+1\right)
}{\textstyle
\Gamma\left( n_{21}+\ell_{21}+{\mbox{$\frac{3}{2}$}}\right)
\Gamma\left(\ell_{22}+{\mbox{$\frac{3}{2}$}}\right)
\Gamma\left({\mbox{$\frac{3}{2}$}}\right)
}}\;
{\frac{\textstyle
\left( 2\ell_{31}+1\right)
\left( 2\ell_{32}+1\right)
}{\textstyle
\Gamma\left( n_{31}+\ell_{31}+{\mbox{$\frac{3}{2}$}}\right)
\Gamma\left(\ell_{32}+{\mbox{$\frac{3}{2}$}}\right)
\Gamma\left({\mbox{$\frac{3}{2}$}}\right)
}}
\nonumber \\ [.3cm] & &
\delta\left( 2\left[ n_{11}+n_{21}+n_{31}\right] +
\ell_{11}+\ell_{21}+\ell_{31}\; ,\; 2n+\ell\right)\;
\delta\left( \ell_{12}+\ell_{22}+\ell_{32}\; ,\; 1\right)
\nonumber \\ [.3cm] & &
\delta\left( 2n_{11} +\ell_{11}+\ell_{12}\; ,\; 2n_{1}+\ell_{1}\right)\;
\delta\left( 2n_{21} +\ell_{21}+\ell_{22}\; ,\; 2n_{2}+\ell_{2}\right)
\nonumber \\ [.3cm] & &
\delta\left( 2n_{31} +
\ell_{31}+\ell_{32}\; ,\; 2n_{r}+\ell_{r}\right)
\nonumber \\ [.3cm] & &
\left(\begin{array}{ccc|c}
\ell_{11} & \ell_{21} & \ell_{31} & \ell\\ [.1cm]
m_{11} & m_{21} & m_{31} & m_{\ell}\end{array}\right)
{\mbox{$\left(\begin{array}{rrr}
\ell_{11} & \ell_{12} & \ell_{1} \\ [.1cm] m_{11} & m_{12} & m_{1} \end{array}\right)$}}
{\mbox{$\left(\begin{array}{rrr}
\ell_{11} & \ell_{12} & \ell_{1} \\ [.1cm] 0 & 0 & 0 \end{array}\right)$}}
\nonumber \\ [.3cm] & &
{\mbox{$\left(\begin{array}{rrr}
\ell_{21} & \ell_{22} & \ell_{2} \\ [.1cm] m_{21} & m_{22} & m_{2} \end{array}\right)$}}
{\mbox{$\left(\begin{array}{rrr}
\ell_{21} & \ell_{22} & \ell_{2} \\ [.1cm] 0 & 0 & 0 \end{array}\right)$}}
{\mbox{$\left(\begin{array}{rrr}
\ell_{31} & \ell_{32} & \ell_{r} \\ [.1cm] m_{31} & m_{32} & m_{r} \end{array}\right)$}}
{\mbox{$\left(\begin{array}{rrr}
\ell_{31} & \ell_{32} & \ell_{r} \\ [.1cm] 0 & 0 & 0 \end{array}\right)$}}
\; .
\label{rrrngmnt}\end{aligned}$$
Notice that the Kronecker deltas in formula (\[rrrngmnt\]) amount to
$$2n+\ell +1\; =\;
2\left( n_{1}+n_{2}+n_{r}\right) +\ell_{1}+\ell_{2}+\ell_{r}\;\;\; ,
\label{finstat}$$
which is precisely the important relation that limits the number of possible decay channels.
Moreover, for an initial pseudoscalar or vector meson out of the lowest-lying flavour nonets, one has in formula (\[decint\]) for the $q\bar{q}$ quantum numbers $n$ and $\ell$ that
$$n\; =\;\ell\; =\; 0\;\;\; .$$
Consequently, through the use of the Kronecker deltas in Eq. (\[rrrngmnt\]), we find for the quantum numbers of the internal lines of diagram (\[reardiagram\]) that
$$n_{11}\; =\; n_{21}\; =\; n_{31}\; =\;
\ell_{11}\; =\; \ell_{21}\; =\; \ell_{31}\; =\; 0
\;\;\; ,$$
and moreover
$$n_{1}\; =\; n_{2}\; =\; n_{r}\; =\; 0
\;\;\;{{\textstyle \mbox{\rm and}}}\;\;\;
\ell_{1}\; +\; \ell_{2}\; +\; \ell_{r}\; =\; 1
\;\;\; .
\label{psvecrel}$$
Relations (\[psvecrel\]) can be checked against the first three colums of Tables (\[pseudoscalar\]) and (\[vector\]), where for all possible channels the radial excitations $n_{1}$, $n_{2}$, or $n_{r}$ vanish, and, moreover, the sums of $\ell_{1}$, $\ell_{2}$, and $\ell_{r}$ equal 1. This is a consequence of formula (\[finstat\]) and drastically limits the number of possible quantum numbers and hence decay channels.
For the relevant rearrangement coefficients we find, using formula (\[rrrngmnt\]), in this case
$$\left(\begin{array}{ccccccc}
0 & 0 & 0 & & 0 & \ell_{1} & m_{1}\\ [.1cm]
0 & 1 & m & & 0 & \ell_{2} & m_{2}\\ [.1cm]
0 & 0 & 0 & & 0 & \ell_{r} & m_{r}
\end{array}\right)_{{\textstyle
\left\{\begin{array}{c}{\mbox{\boldmath $\alpha$}}_{ABC}\\{\mbox{\boldmath $\alpha$}}_{BAC}\end{array}
\right\} }}\; =\;
\left({\mbox{$\frac{1}{2}$}}\right)^{{\textstyle 1-{\mbox{$\frac{1}{2}$}}\ell_{r}}}\;\;
\left\{\begin{array}{c} +1\\[.5cm] (-1)^{{\textstyle \ell_{r}}}\end{array}\right\}\;
\delta\left( \ell_{1}+\ell_{2}+\ell_{r}\; ,\; 1\right)\; .
\label{rearps+v}$$
For the decay of a meson out of the lowest-lying scalar nonet, one has
$$n\; =\; 0\;\;\;{{\textstyle \mbox{\rm and}}}\;\;\;\ell\; =\; s\; =\; 1\;\;\; ,$$
and hence, by the use of formula (\[rrrngmnt\]), one obtains for the relevant rearrangement coefficients in this case
$$\begin{aligned}
& & \;\;\;\;\;
\left(\begin{array}{ccccccc}
0 & 1 & m_{\ell} & & n_{1} & \ell_{1} & m_{1}\\ [.1cm]
0 & 1 & m & & n_{2} & \ell_{2} & m_{2}\\ [.1cm]
0 & 0 & 0 & & n_{r} & \ell_{r} & m_{r}
\end{array}\right)_{{\textstyle
\left\{\begin{array}{c}{\mbox{\boldmath $\alpha$}}_{ABC}\\{\mbox{\boldmath $\alpha$}}_{BAC}\end{array}
\right\} }}\; =
\nonumber \\ [.3cm] & = &
(-1)^{{\textstyle n_{1}+n_{2}+n_{r}}}\; 8\;
\sqrt{\left( n_{1}!\; n_{2}!\; n_{r}!\right)}
\nonumber \\ [.3cm] & &
\sqrt{\left({\frac{\textstyle
\Gamma\left( 2n_{1}+\ell_{1}+{\mbox{$\frac{3}{2}$}}\right)
\Gamma\left( 2n_{2}+\ell_{2}+{\mbox{$\frac{3}{2}$}}\right)
\Gamma\left( 2n_{r}+\ell_{r}+{\mbox{$\frac{3}{2}$}}\right)}{\textstyle 2\pi^{3/2}\;
\left( 2\ell_{1}+1\right)
\left( 2\ell_{2}+1\right)
\left( 2\ell_{r}+1\right)}}\right)}
\nonumber \\ [.3cm] & &
\left({\mbox{$\frac{1}{2}$}}\right)^{{\textstyle 2n_{1}+2n_{2}+n_{r}+\ell_{1}+\ell_{2}
+{\mbox{$\frac{1}{2}$}}\ell_{r}}}\;\;
\sum_{{\textstyle \left\{ n_{ij},\ell_{ij},m_{ij}\right\}}}\;\;
\left\{\begin{array}{c} (-1)^{{\textstyle \ell_{31}}}\\[.5cm]
(-1)^{{\textstyle \ell_{32}}}\end{array}\right\}
\nonumber \\ [.3cm] & &
\delta\left(\ell_{11}+\ell_{21}+\ell_{31}\; ,\; 1\right)\;
\delta\left(\ell_{12}+\ell_{22}+\ell_{32}\; ,\; 1\right)
\nonumber \\ [.3cm] & &
\delta\left(\ell_{11}+\ell_{12}\; ,\; 2n_{1}+\ell_{1}\right)\;
\delta\left(\ell_{21}+\ell_{22}\; ,\; 2n_{2}+\ell_{2}\right)
\delta\left(\ell_{31}+\ell_{32}\; ,\; 2n_{r}+\ell_{r}\right)
\nonumber \\ [.3cm] & &
{\mbox{$\left(\begin{array}{rrr}
\ell_{11} & \ell_{12} & \ell_{1} \\ [.1cm] m_{11} & m_{12} & m_{1} \end{array}\right)$}}
{\mbox{$\left(\begin{array}{rrr}
\ell_{11} & \ell_{12} & \ell_{1} \\ [.1cm] 0 & 0 & 0 \end{array}\right)$}}
\nonumber \\ [.3cm] & &
{\mbox{$\left(\begin{array}{rrr}
\ell_{21} & \ell_{22} & \ell_{2} \\ [.1cm] m_{21} & m_{22} & m_{2} \end{array}\right)$}}
{\mbox{$\left(\begin{array}{rrr}
\ell_{21} & \ell_{22} & \ell_{2} \\ [.1cm] 0 & 0 & 0 \end{array}\right)$}}
{\mbox{$\left(\begin{array}{rrr}
\ell_{31} & \ell_{32} & \ell_{r} \\ [.1cm] m_{31} & m_{32} & m_{r} \end{array}\right)$}}
{\mbox{$\left(\begin{array}{rrr}
\ell_{31} & \ell_{32} & \ell_{r} \\ [.1cm] 0 & 0 & 0 \end{array}\right)$}}
\; .
\label{rearscalar}\end{aligned}$$
Mixing for scalar mesons {#scalarmixing}
========================
Since we assume $^{3}P_{0}$ quantum numbers for the creation of a $q\bar{q}$ pair out of the vacuum, the mixing of the isoscalar flavour-nonet members for lowest-lying scalar meson decay is not completely trivial. So we will outline here some of the necessary ingredients.
In order to simplify the discussion, let us separate the normalisation factor and the summations in formula (\[decABC\]). Moreover, the two Kronecker deltas under the square root in formula (\[decABC\]) do not vanish for the lowest-lying scalar mesons. Therefore, let us denote
$$\begin{aligned}
& & \;\;\;\;\;\left.\begin{array}{l}
{\mbox{$\left\langle 0,0,j_{1},\ell_{1},s_{1},n_{1},
j_{2},\ell_{2},s_{2},n_{2},\ell_{r},s_{r},n_{r}\left| 0,0,1,1,0,{\mbox{\boldmath $\alpha$}}_{ABC}\right.\right\rangle$}}\\ [.5cm]
{\mbox{$\left\langle 0,0,j_{1},\ell_{1},s_{1},n_{1},
j_{2},\ell_{2},s_{2},n_{2},\ell_{r},s_{r},n_{r}\left| 0,0,1,1,0,{\mbox{\boldmath $\alpha$}}_{BAC}\right.\right\rangle$}}\end{array}\right\}\; =
\label{decsimp} \\ [.5cm] & &
=\;{\frac{\textstyle 1}{\textstyle \sqrt{1+
{\mbox{$\left\langle C\left| SU(3)_{{\textstyle \mbox{\rm \scriptsize flavour}}}{{\textstyle \mbox{\rm -singlet}}}\right.\right\rangle$}}}}}\;\;
\times\;\left\{\begin{array}{l} {\mbox{$\left\langle ABC\right\rangle$}} \\ [.3cm] {\mbox{$\left\langle BAC\right\rangle$}}
\end{array}\right.
\;\;\;\; ,
\nonumber\end{aligned}$$
where ${\mbox{$\left\langle ABC\right\rangle$}}$ stands for the upper summation in Eq. (\[decABC\]) and ${\mbox{$\left\langle BAC\right\rangle$}}$ for the lower.
As one notices from formulae (\[decABC\]), (\[reardiagram\]), (\[rearformula\]) or just only from formula (\[rearscalar\]), the transition coefficients ${\mbox{$\left\langle ABC\right\rangle$}}$ and ${\mbox{$\left\langle BAC\right\rangle$}}$ do not , for full $SU(3)$ flavour symmetry, i.e., for equal up, down and strange quark masses, depend on the flavour contents of the three mesons $A$, $B$, and $C$ involved in the transition process (\[MMdecay\]), but just on the orbital and intrinsic spin quantum numbers of the system, which circumstance is also expressed by the notation of formula (\[spatial\]). In fact, for the case of equal quark masses, those transition coefficients are equal for each different set of spatial quantum numbers, up to a sign. This sign is positive for all possible couplings in the case of the lowest-lying scalar mesons. Consequently, since for Table \[tabspsps\] only the transitions between scalar mesons and pairs of pseudoscalar mesons are relevant, and for mixing in general only the restriction to a specific set of spatial quantum numbers has to be considered, we may put here
$${\mbox{$\left\langle ABC\right\rangle$}}\; =\;{\mbox{$\left\langle BAC\right\rangle$}}
\;\;\; .
\label{ABCequal}$$
For the transitions of the flavour-octet members to pairs of mesons, one has, according to formula (\[normtab\]), a unity normalisation factor. Let us study then the matrix elements for a representant, $u\bar{d}$, of the isotriplets, which is denoted by $t$ in Table \[scalar\], coupled to an isoscalar, $\phi$, and an isotriplet, for which we also take as a representant the $u\bar{d}$ state, and which, moreover, is also denoted by $t$ in Table \[scalar\], i.e.,
$${\mbox{$\left\langle \left( u\bar{d}\right)\phi\left| u\bar{d}\right.\right\rangle$}}
\;\;\; .
\label{isotriplet}$$
If $\phi$ represents the flavour-octet-member isoscalar $\phi_{8}$, we find
$${\mbox{$\left\langle \left( u\bar{d}\right)\phi_{8}\left| u\bar{d}\right.\right\rangle$}}\; =\;
{\mbox{$\left\langle \left( u\bar{d}\right)\sqrt{{\mbox{$\frac{1}{6}$}}}
\left( u\bar{u}+d\bar{d}-2s\bar{s}\right)\left| u\bar{d}\right.\right\rangle$}}
\;\;\; .$$
Obviously, the matrix element for the $s\bar{s}$ contribution to $\phi_{8}$ vanishes, hence
$${\mbox{$\left\langle \left( u\bar{d}\right)\phi_{8}\left| u\bar{d}\right.\right\rangle$}}\; =\;
\sqrt{{\mbox{$\frac{1}{6}$}}}\left\{
{\mbox{$\left\langle \left( u\bar{d}\right)\left( u\bar{u}\right)\left| u\bar{d}\right.\right\rangle$}}\; +\;
{\mbox{$\left\langle \left( u\bar{d}\right)\left( d\bar{d}\right)\left| u\bar{d}\right.\right\rangle$}}
\right\}
\;\;\; ,$$
which, also using Eq. (\[ABCequal\]), gives
$${\mbox{$\left\langle \left( u\bar{d}\right)\phi_{8}\left| u\bar{d}\right.\right\rangle$}}\; =\;
\sqrt{{\mbox{$\frac{1}{6}$}}}\left\{{\mbox{$\left\langle BAC\right\rangle$}}+{\mbox{$\left\langle ABC\right\rangle$}}\right\}
\; =\;\sqrt{{\mbox{$\frac{2}{3}$}}}{\mbox{$\left\langle ABC\right\rangle$}}
\;\;\; .
\label{isot8}$$
Following a similar reasoning when $\phi$ in Eq. (\[isotriplet\]) represents the flavour-singlet isoscalar $\phi_{1}$, we find for its matrix elements the result
$${\mbox{$\left\langle \left( u\bar{d}\right)\phi_{1}\left| u\bar{d}\right.\right\rangle$}}\; =\;
{\mbox{$\left\langle \left( u\bar{d}\right)
\sqrt{{\mbox{$\frac{1}{3}$}}}\left( u\bar{u}+d\bar{d}+s\bar{s}\right)\left| u\bar{d}\right.\right\rangle$}}\; =\;
\sqrt{{\mbox{$\frac{4}{3}$}}}{\mbox{$\left\langle ABC\right\rangle$}}
\;\;\; .
\label{isot1}$$
In Table \[scalar\], for the quadratic matrix elements under $t8$ and $t1$ in the sector [*isotriplets*]{}, one may verify the factor 2 that follows from formulae (\[isot8\]) and (\[isot1\]),
For the ideally-mixed isoscalars, $\phi_{n}$ (non-strange) and $\phi_{s}$ (strange), defined by
$$\phi_{n}\; =\;
\sqrt{{\mbox{$\frac{1}{2}$}}}\left( u\bar{u}+d\bar{d}\right)\; =\;
\sqrt{{\mbox{$\frac{2}{3}$}}}\phi_{1}+
\sqrt{{\mbox{$\frac{1}{3}$}}}\phi_{8}
\;\;\;{{\textstyle \mbox{\rm and}}}\;\;\;
\phi_{s}\; =\; s\bar{s}\; =\;
\sqrt{{\mbox{$\frac{1}{3}$}}}\phi_{1}-
\sqrt{{\mbox{$\frac{2}{3}$}}}\phi_{8}
\;\;\; ,
\label{idealmix}$$
one finds the matrix elements
$$\begin{aligned}
{\mbox{$\left\langle \left( u\bar{d}\right)\phi_{n}\left| u\bar{d}\right.\right\rangle$}} & = &
\sqrt{2}{\mbox{$\left\langle ABC\right\rangle$}}
\; =\;\sqrt{3}{\mbox{$\left\langle \left( u\bar{d}\right)\phi_{8}\left| u\bar{d}\right.\right\rangle$}}
\; =\;\sqrt{{\mbox{$\frac{3}{2}$}}}{\mbox{$\left\langle \left( u\bar{d}\right)\phi_{1}\left| u\bar{d}\right.\right\rangle$}}
\;\;\;{{\textstyle \mbox{\rm and}}}
\nonumber \\ [.3cm]
{\mbox{$\left\langle \left( u\bar{d}\right)\phi_{s}\left| u\bar{d}\right.\right\rangle$}}
& = & 0\;\;\; .
\label{isotns}\end{aligned}$$
Besides the multiplicative factor of 24 which is discussed in Section \[comparison\], formula (\[isotns\]) establishes the relation between the values given in the first line of Table \[scalar\] and the values given in Table \[tabspsps\] for the following matrix elements:
$${\mbox{$\left\langle \pi\eta_{n}\left| a_{0}\right.\right\rangle$}}\; =\;
\sqrt{24}\sqrt{3}{\mbox{$\left\langle t8\left| t\right.\right\rangle$}}\; =\;
\sqrt{{\mbox{$\frac{2}{3}$}}}
\;\;\;{{\textstyle \mbox{\rm and}}}\;\;\;
{\mbox{$\left\langle \pi\eta_{s}\left| a_{0}\right.\right\rangle$}}\; =\; 0
\;\;\; .
\label{ppea0}$$
For the coupling of an isodoublet lowest-lying scalar meson to the isodoublet-isoscalar pair, we may also select representants. Let us consider the matrix element
$${\mbox{$\left\langle \left( u\bar{s}\right)\phi\left| u\bar{s}\right.\right\rangle$}}
\;\;\; .$$
In this case, the matrix element for the $d\bar{d}$ contribution vanishes. Consequently, for the isoscalar flavour singlets and octets, we end up with
$${\mbox{$\left\langle \left( u\bar{s}\right)\phi_{1}\left| u\bar{s}\right.\right\rangle$}}\; =\;
\sqrt{{\mbox{$\frac{4}{3}$}}}{\mbox{$\left\langle ABC\right\rangle$}}
\;\;\;{{\textstyle \mbox{\rm and}}}\;\;\;
{\mbox{$\left\langle \left( u\bar{s}\right)\phi_{8}\left| u\bar{s}\right.\right\rangle$}}\; =\;
-\sqrt{{\mbox{$\frac{1}{6}$}}}{\mbox{$\left\langle ABC\right\rangle$}}
\;\;\; ,
\label{isod18}$$
which explains the factor 8 in Table \[scalar\] for the quadratic matrix elements under $d8$ and $d1$, in the sector under [*isodoublets*]{}.
From Eq. (\[isod18\]), we obtain for the ideally mixed combinations (\[idealmix\]) the relations
$$\begin{aligned}
{\mbox{$\left\langle \left( u\bar{s}\right)\phi_{n}\left| u\bar{s}\right.\right\rangle$}} & = &
\sqrt{{\mbox{$\frac{1}{2}$}}}{\mbox{$\left\langle BAC\right\rangle$}}\; =\;
\sqrt{{\mbox{$\frac{3}{8}$}}}{\mbox{$\left\langle \left( u\bar{s}\right)\phi_{1}\left| u\bar{s}\right.\right\rangle$}}
\;\;\;{{\textstyle \mbox{\rm and}}}
\\ [.3cm]
{\mbox{$\left\langle \left( u\bar{s}\right)\phi_{s}\left| u\bar{s}\right.\right\rangle$}} & = &
{\mbox{$\left\langle ABC\right\rangle$}}\; =\;
\sqrt{{\mbox{$\frac{3}{4}$}}}{\mbox{$\left\langle \left( u\bar{s}\right)\phi_{1}\left| u\bar{s}\right.\right\rangle$}}
\;\;\; ,\end{aligned}$$
which, by the use of the first line of Table \[scalar\] and when, moreover, multiplied by the factor $\sqrt{24}$, gives the matrix elements of $\kappa\rightarrow K\eta_{n}$ and $\kappa\rightarrow K\eta_{s}$, i.e.,
$${\mbox{$\left\langle K\eta_{n}\left| \kappa\right.\right\rangle$}}\; =\;
\sqrt{24}\sqrt{{\mbox{$\frac{3}{8}$}}}
{\mbox{$\left\langle d1\left| d\right.\right\rangle$}}\; =\;
\sqrt{{\mbox{$\frac{1}{6}$}}}
\;\;\;{{\textstyle \mbox{\rm and}}}\;\;\;
{\mbox{$\left\langle K\eta_{s}\left| \kappa\right.\right\rangle$}}\; =\;
\sqrt{24}\sqrt{{\mbox{$\frac{3}{4}$}}}
{\mbox{$\left\langle d1\left| d\right.\right\rangle$}}\; =\;
\sqrt{{\mbox{$\frac{1}{3}$}}}
\;\;\; ,$$
the quadratic sum ($=\displaystyle{\mbox{$\frac{1}{2}$}}$) of which is found for model B86 in Table \[tabspsps\].
Now, according to formula (\[normtab\]), for the couplings of lowest-lying scalar isoscalars to meson pairs, the normalisation may be not unity. Let us begin with the coupling to a pair of isotriplets, e.g.
$${\mbox{$\left\langle \left( u\bar{d}\right)\left( d\bar{u}\right)\left| \phi\right.\right\rangle$}}
\;\;\; .$$
When $\phi$ represents a flavour-singlet isoscalar, then, also using formulae (\[normtab\]) and (\[ABCequal\]), one finds
$${\mbox{$\left\langle \left( u\bar{d}\right)\left( d\bar{u}\right)\left| \phi_{1}\right.\right\rangle$}}=
{\mbox{$\left\langle \left( u\bar{d}\right)\left( d\bar{u}\right)\left| \sqrt{{\mbox{$\frac{1}{3}$}}}\left( u\bar{u}+d\bar{d}+s\bar{s}\right)\right.\right\rangle$}}=
\sqrt{{\mbox{$\frac{1}{3}$}}}
\left\{{\frac{\textstyle {\mbox{$\left\langle ABC\right\rangle$}}}{\textstyle \sqrt{2}}}+{\frac{\textstyle {\mbox{$\left\langle BAC\right\rangle$}}}{\textstyle \sqrt{2}}}\right\}
=\sqrt{{\mbox{$\frac{2}{3}$}}}{\mbox{$\left\langle ABC\right\rangle$}}
\;\;\; ,$$
whereas, when $\phi$ represents a flavour-octet isoscalar, it follows that
$${\mbox{$\left\langle \left( u\bar{d}\right)\left( d\bar{u}\right)\left| \phi_{8}\right.\right\rangle$}}\; =\;
\sqrt{{\mbox{$\frac{1}{6}$}}}
\left\{{\mbox{$\left\langle ABC\right\rangle$}}+{\mbox{$\left\langle BAC\right\rangle$}}\right\}\; =\;
\sqrt{{\mbox{$\frac{2}{3}$}}}{\mbox{$\left\langle ABC\right\rangle$}}
\;\;\; .$$
Indeed, in Table \[scalar\] the quadratic matrix elements under $tt$ in the sector for the flavour-octet isoscalars are the same as in the sector for the flavour-singlet isoscalars.
For the ideally mixed combination (\[idealmix\]), also applying formulae (\[normtab\]) and (\[ABCequal\]), one obtains
$${\mbox{$\left\langle \left( u\bar{d}\right)\left( d\bar{u}\right)\left| \phi_{n}\right.\right\rangle$}}\; =\;
\sqrt{{\mbox{$\frac{1}{2}$}}}
\left\{{\frac{\textstyle {\mbox{$\left\langle ABC\right\rangle$}}}{\textstyle \sqrt{{\mbox{$\frac{5}{3}$}}}}}+
{\frac{\textstyle {\mbox{$\left\langle BAC\right\rangle$}}}{\textstyle \sqrt{{\mbox{$\frac{5}{3}$}}}}}\right\}
\; =\;
\sqrt{{\mbox{$\frac{6}{5}$}}}{\mbox{$\left\langle ABC\right\rangle$}}
\; =\;
\sqrt{{\mbox{$\frac{9}{5}$}}}
{\mbox{$\left\langle \left( u\bar{d}\right)\left( d\bar{u}\right)\left| \phi_{1}\right.\right\rangle$}}
\;\;\; .
\label{isoscttn}$$
When, moreover, multiplied by the factor $\sqrt{24}$, formula (\[isoscttn\]) establishes the relation between the first line of Table \[scalar\] and the matrix element of $\eta_{n}\rightarrow\pi\pi$, according to
$${\mbox{$\left\langle \pi\pi\left| \eta_{n}\right.\right\rangle$}}\; =\;
\sqrt{24}\sqrt{{\mbox{$\frac{9}{5}$}}}
{\mbox{$\left\langle tt\left| \phi_{1}\right.\right\rangle$}}\; =\;
\sqrt{{\mbox{$\frac{3}{5}$}}}
\;\;\; ,$$
as is found for model B86 in Table \[tabspsps\].
Next, let us also study the coupling of isoscalars to a pair of isodoublets, e.g.
$${\mbox{$\left\langle \left( u\bar{s}\right)\left( s\bar{u}\right)\left| \phi\right.\right\rangle$}}
\;\;\; .$$
When $\phi$ represents a flavour-singlet isoscalar, then, again through the use of formulae (\[normtab\]) and (\[ABCequal\]), we have
$${\mbox{$\left\langle \left( u\bar{s}\right)\left( s\bar{u}\right)\left| \phi_{1}\right.\right\rangle$}}\; =\;
\sqrt{{\mbox{$\frac{1}{3}$}}}
\left\{{\frac{\textstyle {\mbox{$\left\langle ABC\right\rangle$}}}{\textstyle \sqrt{2}}}+{\frac{\textstyle {\mbox{$\left\langle BAC\right\rangle$}}}{\textstyle \sqrt{2}}}\right\}
\; =\;
\sqrt{{\mbox{$\frac{2}{3}$}}}{\mbox{$\left\langle ABC\right\rangle$}}
\;\;\; .$$
Similarly, for a flavour-octet isoscalar we find
$${\mbox{$\left\langle \left( u\bar{s}\right)\left( s\bar{u}\right)\left| \phi_{8}\right.\right\rangle$}}\; =\;
\sqrt{{\mbox{$\frac{1}{6}$}}}
\left\{{\mbox{$\left\langle ABC\right\rangle$}}-2{\mbox{$\left\langle BAC\right\rangle$}}\right\}
\; =\; -\sqrt{{\mbox{$\frac{1}{6}$}}}{\mbox{$\left\langle ABC\right\rangle$}}
\;\;\; ,$$
which result explains the factor ${\mbox{$\frac{1}{4}$}}$ in Table \[scalar\] between the quadratic matrix elements under $dd$ in the sectors for the flavour-octet isoscalars and for the flavour-singlet isoscalars.
For the ideally mixed isoscalars defined in formula (\[idealmix\]), using once again formula (\[normtab\]), we obtain
$$\begin{aligned}
{\mbox{$\left\langle \left( u\bar{s}\right)\left( s\bar{u}\right)\left| \phi_{n}\right.\right\rangle$}} & = &
\sqrt{{\mbox{$\frac{1}{2}$}}}{\frac{\textstyle {\mbox{$\left\langle ABC\right\rangle$}}}{\textstyle \sqrt{{\mbox{$\frac{5}{3}$}}}}}
\; =\;\sqrt{{\mbox{$\frac{3}{10}$}}}{\mbox{$\left\langle ABC\right\rangle$}}\; =\;
\sqrt{{\mbox{$\frac{9}{20}$}}}
{\mbox{$\left\langle \left( u\bar{s}\right)\left( s\bar{u}\right)\left| \phi_{1}\right.\right\rangle$}}
\;\;\;{{\textstyle \mbox{\rm and}}} \\ [.3cm]
{\mbox{$\left\langle \left( u\bar{s}\right)\left( s\bar{u}\right)\left| \phi_{s}\right.\right\rangle$}} & = &
\sqrt{{\mbox{$\frac{1}{2}$}}}{\frac{\textstyle {\mbox{$\left\langle BAC\right\rangle$}}}{\textstyle \sqrt{{\mbox{$\frac{4}{3}$}}}}}
\; =\;\sqrt{{\mbox{$\frac{3}{4}$}}}{\mbox{$\left\langle ABC\right\rangle$}}\; =\;
\sqrt{{\mbox{$\frac{9}{8}$}}}
{\mbox{$\left\langle \left( u\bar{s}\right)\left( s\bar{u}\right)\left| \phi_{1}\right.\right\rangle$}}
\;\;\; ,\end{aligned}$$
which nicely explains the values for the matrix elements of $\left(\eta_{n}/\eta_{s}\right)\rightarrow K\bar{K}$, i.e.,
$${\mbox{$\left\langle K\bar{K}\left| \eta_{n}\right.\right\rangle$}}\; =\;
\sqrt{24}\sqrt{{\mbox{$\frac{9}{20}$}}}
{\mbox{$\left\langle dd\left| \phi_{1}\right.\right\rangle$}}\; =\;
\sqrt{{\mbox{$\frac{1}{5}$}}}
\;\;\;{{\textstyle \mbox{\rm and}}}\;\;\;
{\mbox{$\left\langle K\bar{K}\left| \eta_{s}\right.\right\rangle$}}\; =\;
\sqrt{24}\sqrt{{\mbox{$\frac{9}{8}$}}}
{\mbox{$\left\langle dd\left| \phi_{1}\right.\right\rangle$}}\; =\;
\sqrt{{\mbox{$\frac{1}{2}$}}}
\;\;\; ,$$
given for model B86 in Table \[tabspsps\].
Similar straightforward calculations lead to the matrix elements ${\mbox{$\left\langle \eta_{n}\eta_{n}\left| \eta_{n}\right.\right\rangle$}}$ and ${\mbox{$\left\langle \eta_{s}\eta_{s}\left| \eta_{s}\right.\right\rangle$}}$.
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-------------- ----------------------------------
symbol multiplet
$t$ isotriplets
\[.3cm\] $d$ isodoublets
\[.3cm\] $8$ isoscalar $SU_{3}$-octet members
\[.3cm\] $1$ $SU_{3}$-singlets
-------------- ----------------------------------
: Particle identification used in this paper.[]{data-label="partid"}
------------------------------ --------------------- -------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- -------------------------------
\[.3cm\] $M_{1}$ $M_{2}$ rel.
\[.3cm\] $\!\! j\ell sn\!\!$ $\!\! j\ell sn\!\!$ $\!\! \ell sn\!\!$ $\!\! tt\!\!$ $\!\! dd\!\!$ $\!\! t8\!\!$ $\!\! t1\!\!$ $\!\! T\!\!$ $\!\! td\!\!$ $\!\! d8\!\!$ $\!\! d1\!\!$ $\!\! T\!\!$ $\!\! tt\!\!$ $\!\! dd\!\!$ $\!\! 88\!\!$ $\!\! 11\!\!$ $\!\! T\!\!$ $\!\! tt\!\!$ $\!\! dd\!\!$ $\!\! 88\!\!$ $\!\! 11\!\!$ $\!\! T\!\!$
\[.3cm\] $\!\! 0000\!\!$ $\!\! 0110\!\!$ $\!\! 000\!\!$ - $\!\!\frac{ 1}{ 24}\!\!$ $\!\!\frac{ 1}{ 36}\!\!$ $\!\!\frac{ 1}{ 18}\!\!$ $\!\!\frac{ 1}{ 8}\!\!$ $\!\!\frac{ 1}{ 16}\!\!$ $\!\!\frac{ 1}{ 144}\!\!$ $\!\!\frac{ 1}{ 18}\!\!$ $\!\!\frac{ 1}{ 8}\!\!$ $\!\!\frac{ 1}{ 24}\!\!$ $\!\!\frac{ 1}{ 72}\!\!$ $\!\!\frac{ 1}{ 72}\!\!$ $\!\!\frac{ 1}{ 18}\!\!$ $\!\!\frac{ 1}{ 8}\!\!$ $\!\!\frac{ 1}{ 12}\!\!$ $\!\!\frac{ 1}{ 9}\!\!$ $\!\!\frac{ 1}{ 36}\!\!$ $\!\!\frac{ 1}{ 36}\!\!$ $\!\!\frac{ 1}{ 4}\!\!$
\[.3cm\] $\!\! 1010\!\!$ $\!\! 1100\!\!$ $\!\! 000\!\!$ - $\!\!\frac{ 1}{ 24}\!\!$ $\!\!\frac{ 1}{ 36}\!\!$ $\!\!\frac{ 1}{ 18}\!\!$ $\!\!\frac{ 1}{ 8}\!\!$ $\!\!\frac{ 1}{ 16}\!\!$ $\!\!\frac{ 1}{ 144}\!\!$ $\!\!\frac{ 1}{ 18}\!\!$ $\!\!\frac{ 1}{ 8}\!\!$ $\!\!\frac{ 1}{ 24}\!\!$ $\!\!\frac{ 1}{ 72}\!\!$ $\!\!\frac{ 1}{ 72}\!\!$ $\!\!\frac{ 1}{ 18}\!\!$ $\!\!\frac{ 1}{ 8}\!\!$ $\!\!\frac{ 1}{ 12}\!\!$ $\!\!\frac{ 1}{ 9}\!\!$ $\!\!\frac{ 1}{ 36}\!\!$ $\!\!\frac{ 1}{ 36}\!\!$ $\!\!\frac{ 1}{ 4}\!\!$
\[.3cm\] $\!\! 1010\!\!$ $\!\! 1110\!\!$ $\!\! 000\!\!$ $\!\!\frac{ 1}{ 6}\!\!$ $\!\!\frac{ 1}{ 12}\!\!$ - - $\!\!\frac{ 1}{ 4}\!\!$ $\!\!\frac{ 1}{ 8}\!\!$ $\!\!\frac{ 1}{ 8}\!\!$ - $\!\!\frac{ 1}{ 4}\!\!$ - $\!\!\frac{ 1}{ 4}\!\!$ - - $\!\!\frac{ 1}{ 4}\!\!$ - - - - -
\[.3cm\] $\!\! 0000\!\!$ $\!\! 1010\!\!$ $\!\! 110\!\!$ $\!\!\frac{ 1}{ 6}\!\!$ $\!\!\frac{ 1}{ 12}\!\!$ - - $\!\!\frac{ 1}{ 4}\!\!$ $\!\!\frac{ 1}{ 8}\!\!$ $\!\!\frac{ 1}{ 8}\!\!$ - $\!\!\frac{ 1}{ 4}\!\!$ - $\!\!\frac{ 1}{ 4}\!\!$ - - $\!\!\frac{ 1}{ 4}\!\!$ - - - - -
\[.3cm\] $\!\! 1010\!\!$ $\!\! 1010\!\!$ $\!\! 110\!\!$ - $\!\!\frac{ 1}{ 12}\!\!$ $\!\!\frac{ 1}{ 18}\!\!$ $\!\!\frac{ 1}{ 9}\!\!$ $\!\!\frac{ 1}{ 4}\!\!$ $\!\!\frac{ 1}{ 8}\!\!$ $\!\!\frac{ 1}{ 72}\!\!$ $\!\!\frac{ 1}{ 9}\!\!$ $\!\!\frac{ 1}{ 4}\!\!$ $\!\!\frac{ 1}{ 12}\!\!$ $\!\!\frac{ 1}{ 36}\!\!$ $\!\!\frac{ 1}{ 36}\!\!$ $\!\!\frac{ 1}{ 9}\!\!$ $\!\!\frac{ 1}{ 4}\!\!$ $\!\!\frac{ 1}{ 6}\!\!$ $\!\!\frac{ 2}{ 9}\!\!$ $\!\!\frac{ 1}{ 18}\!\!$ $\!\!\frac{ 1}{ 18}\!\!$ $\!\!\frac{ 1}{ 2}\!\!$
------------------------------ --------------------- -------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- -------------------------------
: Coupling constants for the decay processes of pseudoscalar mesons into meson pairs. The interpretation of the content of the table is explained in the text.[]{data-label="pseudoscalar"}
------------------------------ --------------------- -------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- -------------------------------
\[.3cm\] $M_{1}$ $M_{2}$ rel.
\[.3cm\] $\!\! j\ell sn\!\!$ $\!\! j\ell sn\!\!$ $\!\! \ell sn\!\!$ $\!\! tt\!\!$ $\!\! dd\!\!$ $\!\! t8\!\!$ $\!\! t1\!\!$ $\!\! T\!\!$ $\!\! td\!\!$ $\!\! d8\!\!$ $\!\! d1\!\!$ $\!\! T\!\!$ $\!\! tt\!\!$ $\!\! dd\!\!$ $\!\! 88\!\!$ $\!\! 11\!\!$ $\!\! T\!\!$ $\!\! tt\!\!$ $\!\! dd\!\!$ $\!\! 88\!\!$ $\!\! 11\!\!$ $\!\! T\!\!$
\[.3cm\] $\!\! 0000\!\!$ $\!\! 1100\!\!$ $\!\! 010\!\!$ - $\!\!\frac{ 1}{ 72}\!\!$ $\!\!\frac{ 1}{ 108}\!\!$ $\!\!\frac{ 1}{ 54}\!\!$ $\!\!\frac{ 1}{ 24}\!\!$ $\!\!\frac{ 1}{ 48}\!\!$ $\!\!\frac{ 1}{ 432}\!\!$ $\!\!\frac{ 1}{ 54}\!\!$ $\!\!\frac{ 1}{ 24}\!\!$ $\!\!\frac{ 1}{ 72}\!\!$ $\!\!\frac{ 1}{ 216}\!\!$ $\!\!\frac{ 1}{ 216}\!\!$ $\!\!\frac{ 1}{ 54}\!\!$ $\!\!\frac{ 1}{ 24}\!\!$ $\!\!\frac{ 1}{ 36}\!\!$ $\!\!\frac{ 1}{ 27}\!\!$ $\!\!\frac{ 1}{ 108}\!\!$ $\!\!\frac{ 1}{ 108}\!\!$ $\!\!\frac{ 1}{ 12}\!\!$
\[.3cm\] $\!\! 0000\!\!$ $\!\! 1110\!\!$ $\!\! 010\!\!$ $\!\!\frac{ 1}{ 18}\!\!$ $\!\!\frac{ 1}{ 36}\!\!$ - - $\!\!\frac{ 1}{ 12}\!\!$ $\!\!\frac{ 1}{ 24}\!\!$ $\!\!\frac{ 1}{ 24}\!\!$ - $\!\!\frac{ 1}{ 12}\!\!$ - $\!\!\frac{ 1}{ 12}\!\!$ - - $\!\!\frac{ 1}{ 12}\!\!$ - - - - -
\[.3cm\] $\!\! 1010\!\!$ $\!\! 1100\!\!$ $\!\! 010\!\!$ $\!\!\frac{ 1}{ 18}\!\!$ $\!\!\frac{ 1}{ 36}\!\!$ - - $\!\!\frac{ 1}{ 12}\!\!$ $\!\!\frac{ 1}{ 24}\!\!$ $\!\!\frac{ 1}{ 24}\!\!$ - $\!\!\frac{ 1}{ 12}\!\!$ - $\!\!\frac{ 1}{ 12}\!\!$ - - $\!\!\frac{ 1}{ 12}\!\!$ - - - - -
\[.3cm\] $\!\! 1010\!\!$ $\!\! 1110\!\!$ $\!\! 010\!\!$ - $\!\!\frac{ 1}{ 18}\!\!$ $\!\!\frac{ 1}{ 27}\!\!$ $\!\!\frac{ 2}{ 27}\!\!$ $\!\!\frac{ 1}{ 6}\!\!$ $\!\!\frac{ 1}{ 12}\!\!$ $\!\!\frac{ 1}{ 108}\!\!$ $\!\!\frac{ 2}{ 27}\!\!$ $\!\!\frac{ 1}{ 6}\!\!$ $\!\!\frac{ 1}{ 18}\!\!$ $\!\!\frac{ 1}{ 54}\!\!$ $\!\!\frac{ 1}{ 54}\!\!$ $\!\!\frac{ 2}{ 27}\!\!$ $\!\!\frac{ 1}{ 6}\!\!$ $\!\!\frac{ 1}{ 9}\!\!$ $\!\!\frac{ 4}{ 27}\!\!$ $\!\!\frac{ 1}{ 27}\!\!$ $\!\!\frac{ 1}{ 27}\!\!$ $\!\!\frac{ 1}{ 3}\!\!$
\[.3cm\] $\!\! 0110\!\!$ $\!\! 1010\!\!$ $\!\! 010\!\!$ - $\!\!\frac{ 1}{ 24}\!\!$ $\!\!\frac{ 1}{ 36}\!\!$ $\!\!\frac{ 1}{ 18}\!\!$ $\!\!\frac{ 1}{ 8}\!\!$ $\!\!\frac{ 1}{ 16}\!\!$ $\!\!\frac{ 1}{ 144}\!\!$ $\!\!\frac{ 1}{ 18}\!\!$ $\!\!\frac{ 1}{ 8}\!\!$ $\!\!\frac{ 1}{ 24}\!\!$ $\!\!\frac{ 1}{ 72}\!\!$ $\!\!\frac{ 1}{ 72}\!\!$ $\!\!\frac{ 1}{ 18}\!\!$ $\!\!\frac{ 1}{ 8}\!\!$ $\!\!\frac{ 1}{ 12}\!\!$ $\!\!\frac{ 1}{ 9}\!\!$ $\!\!\frac{ 1}{ 36}\!\!$ $\!\!\frac{ 1}{ 36}\!\!$ $\!\!\frac{ 1}{ 4}\!\!$
\[.3cm\] $\!\! 0000\!\!$ $\!\! 0000\!\!$ $\!\! 100\!\!$ $\!\!\frac{ 1}{ 36}\!\!$ $\!\!\frac{ 1}{ 72}\!\!$ - - $\!\!\frac{ 1}{ 24}\!\!$ $\!\!\frac{ 1}{ 48}\!\!$ $\!\!\frac{ 1}{ 48}\!\!$ - $\!\!\frac{ 1}{ 24}\!\!$ - $\!\!\frac{ 1}{ 24}\!\!$ - - $\!\!\frac{ 1}{ 24}\!\!$ - - - - -
\[.3cm\] $\!\! 1010\!\!$ $\!\! 1010\!\!$ $\!\! 100\!\!$ $\!\!\frac{ 1}{ 108}\!\!$ $\!\!\frac{ 1}{ 216}\!\!$ - - $\!\!\frac{ 1}{ 72}\!\!$ $\!\!\frac{ 1}{ 144}\!\!$ $\!\!\frac{ 1}{ 144}\!\!$ - $\!\!\frac{ 1}{ 72}\!\!$ - $\!\!\frac{ 1}{ 72}\!\!$ - - $\!\!\frac{ 1}{ 72}\!\!$ - - - - -
\[.3cm\] $\!\! 0000\!\!$ $\!\! 1010\!\!$ $\!\! 110\!\!$ - $\!\!\frac{ 1}{ 18}\!\!$ $\!\!\frac{ 1}{ 27}\!\!$ $\!\!\frac{ 2}{ 27}\!\!$ $\!\!\frac{ 1}{ 6}\!\!$ $\!\!\frac{ 1}{ 12}\!\!$ $\!\!\frac{ 1}{ 108}\!\!$ $\!\!\frac{ 2}{ 27}\!\!$ $\!\!\frac{ 1}{ 6}\!\!$ $\!\!\frac{ 1}{ 18}\!\!$ $\!\!\frac{ 1}{ 54}\!\!$ $\!\!\frac{ 1}{ 54}\!\!$ $\!\!\frac{ 2}{ 27}\!\!$ $\!\!\frac{ 1}{ 6}\!\!$ $\!\!\frac{ 1}{ 9}\!\!$ $\!\!\frac{ 4}{ 27}\!\!$ $\!\!\frac{ 1}{ 27}\!\!$ $\!\!\frac{ 1}{ 27}\!\!$ $\!\!\frac{ 1}{ 3}\!\!$
\[.3cm\] $\!\! 1010\!\!$ $\!\! 1010\!\!$ $\!\! 120\!\!$ $\!\!\frac{ 5}{ 27}\!\!$ $\!\!\frac{ 5}{ 54}\!\!$ - - $\!\!\frac{ 5}{ 18}\!\!$ $\!\!\frac{ 5}{ 36}\!\!$ $\!\!\frac{ 5}{ 36}\!\!$ - $\!\!\frac{ 5}{ 18}\!\!$ - $\!\!\frac{ 5}{ 18}\!\!$ - - $\!\!\frac{ 5}{ 18}\!\!$ - - - - -
------------------------------ --------------------- -------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- -------------------------------
: Coupling constants for the decay processes of vector mesons into meson pairs. The interpretation of the content of the table is explained in the text.[]{data-label="vector"}
------------------------------ --------------------- -------------------- --------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- -------------------------------
\[.3cm\] $M_{1}$ $M_{2}$ rel.
\[.3cm\] $\!\! j\ell sn\!\!$ $\!\! j\ell sn\!\!$ $\!\! \ell sn\!\!$ $\!\! tt\!\!$ $\!\! dd\!\!$ $\!\! t8\!\!$ $\!\! t1\!\!$ $\!\! T\!\!$ $\!\! td\!\!$ $\!\! d8\!\!$ $\!\! d1\!\!$ $\!\! T\!\!$ $\!\! tt\!\!$ $\!\! dd\!\!$ $\!\! 88\!\!$ $\!\! 11\!\!$ $\!\! T\!\!$ $\!\! tt\!\!$ $\!\! dd\!\!$ $\!\! 88\!\!$ $\!\! 11\!\!$ $\!\! T\!\!$
\[.3cm\] $\!\! 0000\!\!$ $\!\! 0000\!\!$ $\!\! 001\!\!$ - $\!\!\frac{ 1}{ 72}\!\!$ $\!\!\frac{ 1}{ 108}\!\!$ $\!\!\frac{ 1}{ 54}\!\!$ $\!\!\frac{ 1}{ 24}\!\!$ $\!\!\frac{ 1}{ 48}\!\!$ $\!\!\frac{ 1}{ 432}\!\!$ $\!\!\frac{ 1}{ 54}\!\!$ $\!\!\frac{ 1}{ 24}\!\!$ $\!\!\frac{ 1}{ 72}\!\!$ $\!\!\frac{ 1}{ 216}\!\!$ $\!\!\frac{ 1}{ 216}\!\!$ $\!\!\frac{ 1}{ 54}\!\!$ $\!\!\frac{ 1}{ 24}\!\!$ $\!\!\frac{ 1}{ 72}\!\!$ $\!\!\frac{ 1}{ 54}\!\!$ $\!\!\frac{ 1}{ 216}\!\!$ $\!\!\frac{ 1}{ 216}\!\!$ $\!\!\frac{ 1}{ 24}\!\!$
\[.3cm\] $\!\! 0000\!\!$ $\!\! 0001\!\!$ $\!\! 000\!\!$ - $\!\!\frac{ 1}{ 144}\!\!$ $\!\!\frac{ 1}{ 216}\!\!$ $\!\!\frac{ 1}{ 108}\!\!$ $\!\!\frac{ 1}{ 48}\!\!$ $\!\!\frac{ 1}{ 96}\!\!$ $\!\!\frac{ 1}{ 864}\!\!$ $\!\!\frac{ 1}{ 108}\!\!$ $\!\!\frac{ 1}{ 48}\!\!$ $\!\!\frac{ 1}{ 144}\!\!$ $\!\!\frac{ 1}{ 432}\!\!$ $\!\!\frac{ 1}{ 432}\!\!$ $\!\!\frac{ 1}{ 108}\!\!$ $\!\!\frac{ 1}{ 48}\!\!$ $\!\!\frac{ 1}{ 144}\!\!$ $\!\!\frac{ 1}{ 108}\!\!$ $\!\!\frac{ 1}{ 432}\!\!$ $\!\!\frac{ 1}{ 432}\!\!$ $\!\!\frac{ 1}{ 48}\!\!$
\[.3cm\] $\!\! 1010\!\!$ $\!\! 1010\!\!$ $\!\! 001\!\!$ - $\!\!\frac{ 1}{ 216}\!\!$ $\!\!\frac{ 1}{ 324}\!\!$ $\!\!\frac{ 1}{ 162}\!\!$ $\!\!\frac{ 1}{ 72}\!\!$ $\!\!\frac{ 1}{ 144}\!\!$ $\!\!\frac{ 1}{ 1296}\!\!$ $\!\!\frac{ 1}{ 162}\!\!$ $\!\!\frac{ 1}{ 72}\!\!$ $\!\!\frac{ 1}{ 216}\!\!$ $\!\!\frac{ 1}{ 648}\!\!$ $\!\!\frac{ 1}{ 648}\!\!$ $\!\!\frac{ 1}{ 162}\!\!$ $\!\!\frac{ 1}{ 72}\!\!$ $\!\!\frac{ 1}{ 216}\!\!$ $\!\!\frac{ 1}{ 162}\!\!$ $\!\!\frac{ 1}{ 648}\!\!$ $\!\!\frac{ 1}{ 648}\!\!$ $\!\!\frac{ 1}{ 72}\!\!$
\[.3cm\] $\!\! 1010\!\!$ $\!\! 1011\!\!$ $\!\! 000\!\!$ - $\!\!\frac{ 1}{ 432}\!\!$ $\!\!\frac{ 1}{ 648}\!\!$ $\!\!\frac{ 1}{ 324}\!\!$ $\!\!\frac{ 1}{ 144}\!\!$ $\!\!\frac{ 1}{ 288}\!\!$ $\!\!\frac{ 1}{ 2592}\!\!$ $\!\!\frac{ 1}{ 324}\!\!$ $\!\!\frac{ 1}{ 144}\!\!$ $\!\!\frac{ 1}{ 432}\!\!$ $\!\!\frac{ 1}{ 1296}\!\!$ $\!\!\frac{ 1}{ 1296}\!\!$ $\!\!\frac{ 1}{ 324}\!\!$ $\!\!\frac{ 1}{ 144}\!\!$ $\!\!\frac{ 1}{ 432}\!\!$ $\!\!\frac{ 1}{ 324}\!\!$ $\!\!\frac{ 1}{ 1296}\!\!$ $\!\!\frac{ 1}{ 1296}\!\!$ $\!\!\frac{ 1}{ 144}\!\!$
\[.3cm\] $\!\! 1010\!\!$ $\!\! 1210\!\!$ $\!\! 000\!\!$ - $\!\!\frac{ 5}{ 108}\!\!$ $\!\!\frac{ 5}{ 162}\!\!$ $\!\!\frac{ 5}{ 81}\!\!$ $\!\!\frac{ 5}{ 36}\!\!$ $\!\!\frac{ 5}{ 72}\!\!$ $\!\!\frac{ 5}{ 648}\!\!$ $\!\!\frac{ 5}{ 81}\!\!$ $\!\!\frac{ 5}{ 36}\!\!$ $\!\!\frac{ 5}{ 108}\!\!$ $\!\!\frac{ 5}{ 324}\!\!$ $\!\!\frac{ 5}{ 324}\!\!$ $\!\!\frac{ 5}{ 81}\!\!$ $\!\!\frac{ 5}{ 36}\!\!$ $\!\!\frac{ 5}{ 108}\!\!$ $\!\!\frac{ 5}{ 81}\!\!$ $\!\!\frac{ 5}{ 324}\!\!$ $\!\!\frac{ 5}{ 324}\!\!$ $\!\!\frac{ 5}{ 36}\!\!$
\[.3cm\] $\!\! 1100\!\!$ $\!\! 1100\!\!$ $\!\! 000\!\!$ - $\!\!\frac{ 1}{ 144}\!\!$ $\!\!\frac{ 1}{ 216}\!\!$ $\!\!\frac{ 1}{ 108}\!\!$ $\!\!\frac{ 1}{ 48}\!\!$ $\!\!\frac{ 1}{ 96}\!\!$ $\!\!\frac{ 1}{ 864}\!\!$ $\!\!\frac{ 1}{ 108}\!\!$ $\!\!\frac{ 1}{ 48}\!\!$ $\!\!\frac{ 1}{ 144}\!\!$ $\!\!\frac{ 1}{ 432}\!\!$ $\!\!\frac{ 1}{ 432}\!\!$ $\!\!\frac{ 1}{ 108}\!\!$ $\!\!\frac{ 1}{ 48}\!\!$ $\!\!\frac{ 1}{ 144}\!\!$ $\!\!\frac{ 1}{ 108}\!\!$ $\!\!\frac{ 1}{ 432}\!\!$ $\!\!\frac{ 1}{ 432}\!\!$ $\!\!\frac{ 1}{ 48}\!\!$
\[.3cm\] $\!\! 0110\!\!$ $\!\! 0110\!\!$ $\!\! 000\!\!$ - $\!\!\frac{ 1}{ 48}\!\!$ $\!\!\frac{ 1}{ 72}\!\!$ $\!\!\frac{ 1}{ 36}\!\!$ $\!\!\frac{ 1}{ 16}\!\!$ $\!\!\frac{ 1}{ 32}\!\!$ $\!\!\frac{ 1}{ 288}\!\!$ $\!\!\frac{ 1}{ 36}\!\!$ $\!\!\frac{ 1}{ 16}\!\!$ $\!\!\frac{ 1}{ 48}\!\!$ $\!\!\frac{ 1}{ 144}\!\!$ $\!\!\frac{ 1}{ 144}\!\!$ $\!\!\frac{ 1}{ 36}\!\!$ $\!\!\frac{ 1}{ 16}\!\!$ $\!\!\frac{ 1}{ 48}\!\!$ $\!\!\frac{ 1}{ 36}\!\!$ $\!\!\frac{ 1}{ 144}\!\!$ $\!\!\frac{ 1}{ 144}\!\!$ $\!\!\frac{ 1}{ 16}\!\!$
\[.3cm\] $\!\! 1110\!\!$ $\!\! 1110\!\!$ $\!\! 000\!\!$ - $\!\!\frac{ 1}{ 36}\!\!$ $\!\!\frac{ 1}{ 54}\!\!$ $\!\!\frac{ 1}{ 27}\!\!$ $\!\!\frac{ 1}{ 12}\!\!$ $\!\!\frac{ 1}{ 24}\!\!$ $\!\!\frac{ 1}{ 216}\!\!$ $\!\!\frac{ 1}{ 27}\!\!$ $\!\!\frac{ 1}{ 12}\!\!$ $\!\!\frac{ 1}{ 36}\!\!$ $\!\!\frac{ 1}{ 108}\!\!$ $\!\!\frac{ 1}{ 108}\!\!$ $\!\!\frac{ 1}{ 27}\!\!$ $\!\!\frac{ 1}{ 12}\!\!$ $\!\!\frac{ 1}{ 36}\!\!$ $\!\!\frac{ 1}{ 27}\!\!$ $\!\!\frac{ 1}{ 108}\!\!$ $\!\!\frac{ 1}{ 108}\!\!$ $\!\!\frac{ 1}{ 12}\!\!$
\[.3cm\] $\!\! 0000\!\!$ $\!\! 1110\!\!$ $\!\! 110\!\!$ - $\!\!\frac{ 1}{ 18}\!\!$ $\!\!\frac{ 1}{ 27}\!\!$ $\!\!\frac{ 2}{ 27}\!\!$ $\!\!\frac{ 1}{ 6}\!\!$ $\!\!\frac{ 1}{ 12}\!\!$ $\!\!\frac{ 1}{ 108}\!\!$ $\!\!\frac{ 2}{ 27}\!\!$ $\!\!\frac{ 1}{ 6}\!\!$ $\!\!\frac{ 1}{ 18}\!\!$ $\!\!\frac{ 1}{ 54}\!\!$ $\!\!\frac{ 1}{ 54}\!\!$ $\!\!\frac{ 2}{ 27}\!\!$ $\!\!\frac{ 1}{ 6}\!\!$ $\!\!\frac{ 1}{ 18}\!\!$ $\!\!\frac{ 2}{ 27}\!\!$ $\!\!\frac{ 1}{ 54}\!\!$ $\!\!\frac{ 1}{ 54}\!\!$ $\!\!\frac{ 1}{ 6}\!\!$
\[.3cm\] $\!\! 1010\!\!$ $\!\! 1100\!\!$ $\!\! 110\!\!$ - $\!\!\frac{ 1}{ 18}\!\!$ $\!\!\frac{ 1}{ 27}\!\!$ $\!\!\frac{ 2}{ 27}\!\!$ $\!\!\frac{ 1}{ 6}\!\!$ $\!\!\frac{ 1}{ 12}\!\!$ $\!\!\frac{ 1}{ 108}\!\!$ $\!\!\frac{ 2}{ 27}\!\!$ $\!\!\frac{ 1}{ 6}\!\!$ $\!\!\frac{ 1}{ 18}\!\!$ $\!\!\frac{ 1}{ 54}\!\!$ $\!\!\frac{ 1}{ 54}\!\!$ $\!\!\frac{ 2}{ 27}\!\!$ $\!\!\frac{ 1}{ 6}\!\!$ $\!\!\frac{ 1}{ 18}\!\!$ $\!\!\frac{ 2}{ 27}\!\!$ $\!\!\frac{ 1}{ 54}\!\!$ $\!\!\frac{ 1}{ 54}\!\!$ $\!\!\frac{ 1}{ 6}\!\!$
\[.3cm\] $\!\! 1010\!\!$ $\!\! 1010\!\!$ $\!\! 220\!\!$ - $\!\!\frac{ 5}{ 54}\!\!$ $\!\!\frac{ 5}{ 81}\!\!$ $\!\!\frac{ 10}{ 81}\!\!$ $\!\!\frac{ 5}{ 18}\!\!$ $\!\!\frac{ 5}{ 36}\!\!$ $\!\!\frac{ 5}{ 324}\!\!$ $\!\!\frac{ 10}{ 81}\!\!$ $\!\!\frac{ 5}{ 18}\!\!$ $\!\!\frac{ 5}{ 54}\!\!$ $\!\!\frac{ 5}{ 162}\!\!$ $\!\!\frac{ 5}{ 162}\!\!$ $\!\!\frac{ 10}{ 81}\!\!$ $\!\!\frac{ 5}{ 18}\!\!$ $\!\!\frac{ 5}{ 54}\!\!$ $\!\!\frac{ 10}{ 81}\!\!$ $\!\!\frac{ 5}{ 162}\!\!$ $\!\!\frac{ 5}{ 162}\!\!$ $\!\!\frac{ 5}{ 18}\!\!$
------------------------------ --------------------- -------------------- --------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------- -------------------------------
: Coupling constants for the decay processes of scalar mesons into meson pairs. The interpretation of the content of the table is explained in the text.[]{data-label="scalar"}
---------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------
initial meson
\[.3cm\]
$a_{0}$ or $\delta$ ${\frac{\textstyle 1}{\textstyle 3}}$ ($K\bar{K}$) + ${\frac{\textstyle 2}{\textstyle 3}}$ ($\pi\eta_n$) ${\frac{\textstyle 1}{\textstyle 3}}$ ($K\bar{K}$) and ${\frac{\textstyle 2}{\textstyle 3}}$ (sum $\pi\eta$’s)
\[.3cm\] $K^{\ast}_{0}$ or $\kappa$ ${\frac{\textstyle 1}{\textstyle 2}}$ ($K\pi$) + ${\frac{\textstyle 1}{\textstyle 2}}$ ($K\eta_n$+$K\eta_s$) ${\frac{\textstyle 1}{\textstyle 2}}$ ($K\pi$) and ${\frac{\textstyle 1}{\textstyle 2}}$ (sum $K\eta$’s)
\[.3cm\] $f_{0}$ or $\epsilon /S$ $n\bar{n}$ ${\frac{\textstyle 3}{\textstyle 5}}$ ($\pi\pi$) + ${\frac{\textstyle 1}{\textstyle 5}}$ ($K\bar{K}$) + ${\frac{\textstyle 1}{\textstyle 5}}$ ($\eta_n\eta_n$) 1 ($\pi\pi$), ${\frac{\textstyle 1}{\textstyle 3}}$ ($K\bar{K}$) and ${\frac{\textstyle 1}{\textstyle 3}}$ (sum $\eta\eta$’s)
\[.3cm\] $f_{0}$ or $\epsilon /S$ $s\bar{s}$ ${\frac{\textstyle 1}{\textstyle 2}}$ ($K\bar{K}$) + ${\frac{\textstyle 1}{\textstyle 2}}$ ($\eta_s\eta_s$) ${\frac{\textstyle 2}{\textstyle 3}}$ ($K\bar{K}$) and ${\frac{\textstyle 2}{\textstyle 3}}$ (sum $\eta\eta$’s)
---------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------
: Quadratic coupling constants for the decay process of a scalar meson into a pair of pseudoscalar mesons.[]{data-label="tabspsps"}
|
1 [kb20@yandex.ru]{} 2 [sergey\_sushkov@mail.ru]{}
Introduction
============
It is well known that the existence of traversable Lorentzian as solutions to the equations of general relativity requires “exotic matter”, i.e., matter that violates the null energy condition [@thorne; @hoh-vis]. In particular, for configurations with a minimally coupled scalar field as a source, solutions are only possible if the scalar field is phantom, i.e., has a wrong sign of kinetic energy [@br73; @vac1; @SusZha]. In alternative theories of gravity, such as scalar-tensor, multidimensional and curvature-nonlinear theories, also turn out to be possible only if some of the degrees of freedom are of phantom nature [@br73; @bstar07] (see also the review [@lobo] and references therein).
Meanwhile, macroscopic phantom matter has not yet been observed, which puts to doubt the very possibility of obtaining realistic even in a remote future and even by a highly advanced civilization.
In this paper, we would like to discuss an interesting opportunity of obtaining configurations in general relativity with a kind of matter which possesses phantom properties only in a restricted region of space, somewhere close to the throat, whereas far away from it all standard energy conditions are observed. As an example of such matter, we consider configurations of a minimally coupled scalar field with the Lagrangian \[L\_s\] L\_s = -h() g\_\_- V(), where $h(\phi)$ and $V(\phi)$ are arbitrary functions. If $h(\phi)$ has a variable sign, it cannot be absorbed by re-definition of $\phi$ in its whole range. A case of interest is that $h >0$ (that is, the scalar field is canonical, with positive kinetic energy) in a weak field region and $h < 0$ (the scalar field is of phantom, or ghost nature) in some restricted region where a throat can be expected. In this sense it can be said that the ghost is trapped. A possible transition between $h > 0$ and $h < 0$ in cosmology was considered in [@rubin].
The paper is organized as follows. In Section 2 we present the basic equations and show why the above strategy cannot be realized for massless fields ($V(\phi)\equiv 0$). In Section 3 we describe some general properties of the system and obtain explicit examples of “trapped-ghost” solutions using the inverse-problem method, and Section 4 is a conclusion.
Scalar fields with a variable kinetic term
==========================================
The general static, spherically symmetric metric can be written as \[ds\] ds\^2 = -\^[2(u)]{}dt\^2 + \^[2(u)]{}du\^2 + \^[2(u)]{} d\^2. where $u$ is an arbitrary radial coordinate and $d\Omega^2 = (d\theta^2 +
\sin^2\theta d\varphi^2)$ is the linear element on a unit sphere. A scalar field $\phi(u)$ with the Lagrangian (\[L\_s\]) in a space-time with the metric (\[ds\]) has the stress-energy tensor (SET) T= h(u) \^[-2]{} ’(u)\^2 (1, -1, 1, 1) + V(u). \[SET\] (the prime denotes $d/du$). The kinetic energy density is positive if $h(\phi) >0$ and negative if $h(\phi) < 0$, so solutions sought for must be obtained with $h >0$ at large values of the spherical radius $r(u)
= \e^{\beta}$ and $h < 0$ at smaller radii $r$. One can show, however, that this goal cannot be achieved for a massless field ($V(\phi) \equiv 0$).
Indeed, in the massless case, the SET (\[SET\]) has the same structure as for a usual massless scalar field with $h = \pm 1$. Therefore the metric has the same form as in this simple case and should be reduced to the Fisher metric [@Fis] if $h > 0$ and to the corresponding solution for a phantom scalar, first found by Bergmann and Leipnik [@BerLei] (it is sometimes called “anti-Fisher”) in case $h(\phi) < 0$. Let us reproduce this solution for our scalar (\[L\_s\]) in the simplest joint form, following [@br73].
Two combinations of the Einstein equations \[EE\] R- R = - T for the metric (\[ds\]) and the SET (\[SET\]) with $V\equiv 0$ read $R^0_0 =0$ and $R^0_0 + R^2_2 =0$. Choosing the harmonic radial coordinate $u$, such that $\alpha(u) = 2\beta(u) + \gamma(u)$, we easily solve these equations. Indeed, the first of them reads simply $\gamma'' =0$, while the second one is written as $\beta'' + \gamma'' = \e^{2(\beta+\gamma)}$. Solving them, we have = - mu, \[s\] \^[--]{} = s(k,u) := where $k$ and $m$ are integration constants; two more integration constants have been suppressed by choosing the zero point of $u$ and the scale along the time axis. As a result, the metric has the form [@br73] \[ds1\] ds\^2 = -\^[-2mu]{} dt\^2 + (note that spatial infinity here corresponds to $u=0$ and $m$ has the meaning of the Schwarzschild mass). Moreover, with these metric functions, the ${1\choose 1}$ component of the Einstein equations (\[EE\]) leads to \[int\] k\^2 k = m\^2 + h() ’\^2. It means that $h(\phi) \phi'^2 = \const$, that is, $h(\phi)$ cannot change its sign within a particular solution which is characterized by certain values of the constants $m$ and $k$. The situation remains the same if, instead of a single scalar field, there is a nonlinear sigma model with multiple scalar fields $\phi^a$ and the Lagrangian \[L-sigma\] L\_= -h\_[ab]{} g\_\^a\_\^b, where $h_{ab}$ are functions of $\phi^a$: the metric then has the same form (\[ds1\]), and a relation similar to (\[int\]) reads [@sigma] $$k^2 \sign k = m^2 + \half h_{ab} \phi^a{}' \phi^b{}'.$$ Therefore the quantity that determines the canonical or phantom nature of the scalars, $h_{ab} \phi^a{}' \phi^b{}'$, is constant. If the matrix $h_{ab}$ is not positive- or negative-definite, some solutions due to (\[L-sigma\]) can be while others correspond to a canonical scalar and have a Fisher central singularity [@sigma], but there are no solutions of trapped-ghost character.
Returning to our system with the Lagrangian (\[L\_s\]), we can assert that trapped-ghost can only exist with a nonzero potential $V(\phi)$. To find such configurations, it is helpful to use the so-called quasiglobal gauge $\alpha + \gamma=0$, so that the metric (\[ds\]) takes the form ds\^2 = - A(u) dt\^2 + + r\^2(u)d\^2, \[ds2\] where $A(u)$ is called the redshift function and $r(u)$ the area function. Then the Einstein-scalar equations can be written as (A r\^2 h’)’ - Ar\^2 h’’ r\^2 dV/d, \[phi\] (A’r\^2)’ - 2r\^2 V; \[00\] 2 r”/r - h()[’]{}\^2 ; \[01\] A (r\^2)” - r\^2 A” 2, \[02\] \[11\] -1 + A’ rr’ + Ar’\^2 r\^2 (h A ’\^2 -V), where the prime again denotes $d/du$. (\[phi\]) follows from (\[00\])–(\[02\]), which, given the potential $V(\phi)$ and the kinetic function $h(\phi)$, form a determined set of equations for the unknowns $r(u)$, $A(u)$, $\phi(u)$. (\[11\]) (the ${1\choose 1}$ component of the Einstein equations), free from second-order derivatives, is a first integral of (\[phi\])–(\[02\]) and can be obtained from (\[00\])–(\[02\]) by excluding second-order derivatives. Moreover, (\[02\]) can be integrated giving B’(u) (A/r\^2)’ = 2(3m - u)/r\^4, \[B’\] where $B(u) \equiv A/r^2$ and $m$ is an integration constant equal to the Schwarzschild mass if the metric (\[ds\]) is as $u\to \infty$ ($r \approx u$, $A = 1 - 2m/u + o(1/u)$). If there is a flat asymptotic as $u\to -\infty$, the Schwarzschild mass there is equal to $-m$ ($r \approx |u|$, $A = 1 + 2m/|u| + o(1/u)$.
Thus in any solution with two flat asymptotics we inevitably have masses of opposite signs, just as is the case in the well-known special solution — the anti-Fisher [@ellis; @cold08; @SusZha] whose metric in the gauge (\[ds2\]) reads $$ds^2 = -\e^{-2mz} dt^2 + \e^{2mz} [du^2 + (k^2 + u^2) d\Omega^2],$$ with $k < 0$ and $z = |k|^{-1} \cot^{-1} (u/|k|)$ \[the constants $m$ and $k$ have the same meaning as in (\[s\])–(\[int\])\].
It is also clear that $m = 0$ in all symmetric solutions to (\[phi\])–(\[11\]), such that $r(u)$ and $A(u)$ are even functions. Indeed, in this case $B'(u)$ is odd, hence $m=0$ in (\[B’\]).
Models with a trapped ghost
===========================
If one specifies the functions $V(\phi)$ and $h(\phi)$ in the Lagrangian (\[L\_s\]), it is, in general, hard to solve the above equations. Alternatively, to find examples of solutions possessing some particular properties, one may employ the inverse problem method, choosing some of the functions $r(u)$, $A(u)$ or $\phi(u)$ and then reconstructing the form of $V(\phi)$ and/or $h(\phi)$. We will do so, choosing a function $r(u)$ that describes a profile. Then $A(u)$ is found from (\[B’\]) and $V(u)$ from (\[00\]). The function $\phi(u)$ is found from (\[01\]) provided $h(\phi)$ is known; however, using the scalar field parametrization freedom, we can, vice versa, choose a monotonic function $\phi(u)$ (which will yield an unambiguous function $V(\phi)$) and find $h(u)$ from (\[01\]).
Let us discuss what kind of function $r(u)$ is required for our purpose.
1. The throat ($u=0$ without loss of generality) is a minimum of $r(u)$, that is, r(0) = a, r’(0) =0, r”(0) > 0 with $a=\const >0$ (these requirements are sometimes called the flare-out conditions). So $r(u)$ must have such a minimum.
2. In a trapped-ghost wormhole, by definition, the kinetic coupling function $h(\phi)$ is negative near the throat and positive far from it. According to (\[01\]), this means that $r''$ is positive at small $|u|$ and negative at sufficiently large $|u|$.
3. If the wormhole is asymptotically flat at large $|u|$, we should have $$r(u) \approx |u|\qquad {\rm as}\qquad u\to \pm \infty.$$
A simple example of the function $r(u)$ satisfying the requirements 1–3 is (see Fig. \[figr\]): \[r\] r(u) = a , = > 2. where $x = u/a$, and $a$ is the (arbitrary) throat radius.
![Plots of $r(u)/a$ given by [(\[r\])]{} with $\lambda=3; 5; 10$ (solid, dashed, and dotted lines, respectively). \[figr\]](r.eps){width="8cm"}
Now we can integrate [(\[B’\])]{}. Assuming $m=0$, we find (see Fig.\[figA\]) \[A\] A(u) = .
![Plots of $A(u)$ given by [(\[A\])]{} with $\lambda=3; 5; 10$ (solid, dashed, and dotted lines, respectively). \[figA\]](A.eps){width="8cm"}
Substituting the expressions [(\[r\])]{} and [(\[A\])]{} into [(\[00\])]{}, we obtain the potential $V$ as a function of $u$ or $x = u/a$: \[V\] V(u) = . One can notice that $V(u) < 0$ at large $|u|$. The negative sign of the potential in a certain range of $u$ is not a shortcoming of this particular model but a direct consequence of the field equations. Indeed, as follows from (\[B’\]), we have $A'r^2 {\mathop {\ \longrightarrow\ }\limits }_{u\to\pm\infty} 2m$ at both flat aymptotics. Consequently, due to (\[00\]), $$\int_{-\infty}^{+\infty} r^2 V(u) du = 0,$$ so that if $V(u)\not \equiv 0$, it has an alternate sign.
To construct $V$ as an unambiguous function of $\phi$ and to find $h(\phi)$, it makes sense to choose a monotonic function $\phi(u)$. It is convenient to assume \[phi\_2\] (u) = , \_0 = , and $\phi$ has a finite range: $\phi \in (- \phi_0, \phi_0)$, which is common to kink configurations. Thus we have $x = u/a =
\lambda\tan(\pi\phi/2\phi_0) $, whose substitution into [(\[V\])]{} gives an expression for $V(\phi)$ defined in this finite range. The function $V(\phi)$ can be extended to the whole real axis, $\phi \in \R$, by supposing $V(\phi)\equiv 0$ at $|\phi| \geq \phi_0$. Plots of the extended potential $V(\phi)$ are shown in Fig.\[figV\].
![Plots of $V(\phi)$ given by [(\[V\])]{} with $\lambda=3; 5; 10$ (solid, dashed, and dotted lines, respectively). \[figV\]](V.eps){width="8cm"}
The expression for $h(\phi)$ is found from [(\[01\])]{} as follows: \[h2\] h() = , where $x = \lambda\tan(\pi\phi/2\phi_0)$. The function $h(\phi)$ given by [(\[h2\])]{} is also defined in the interval $(-\phi_0,\phi_0)$ and can be extended to $\R$ by supposing $h(\phi)\equiv 1$ at $|\phi|\geq \phi_0$. The extended kinetic coupling function $h(\phi)$ is plotted in Fig.\[figh\].
![Plots of $h(\phi)$ given by [(\[h2\])]{} with $\lambda=3; 5; 10$ (solid, dashed, and dotted lines, respectively). \[figh\]](h2.eps){width="8cm"}
It is well known that the null energy condition (NEC) holds for a canonical scalar field and is violated for a phantom one. In our case, it happens for $h(\phi) > 0$ and $h(\phi) < 0$, respectively. Let us illustrate this using our solution (\[r\]), (\[A\]), (\[phi\_2\]) as an example. The NEC reads $-T_{\mu\nu}k^\mu k^\nu\ge 0$, where $k^\mu$ is an arbitrary null vector. Due to the Einstein equations [(\[EE\])]{}, it can be equivalently written as $G_{\mu\nu}k^{\mu}k^{\nu}\ge 0$. Taking the radial null vector $k^\mu=(A^{-1/2},A^{1/2},0,0)$ in the metric [(\[ds2\])]{} and denoting $\Xi: = G_{\mu\nu}k^{\mu}k^{\nu}$, we find (u) = -2 = , \[Xi\] i.e., it is a multiple of $h(\phi)$ with a positive factor. The plot of $h(\phi)$ thus completely characterizes NEC violation taking place near the throat.
Conclusion
==========
We have shown that a minimally coupled scalar field may change its nature from canonical to ghost in a smooth way without creating any space-time singularities. This feature, in particular, allows for construction of models (trapped-ghost ) where the ghost is present in some restricted region around the throat (of arbitrary size) whereas in the weak-field region far from it the scalar has usual canonical properties. One can speculate that if such ghosts do exist in Nature, they are all confined to strong-field regions (“all genies are sitting in their bottles”), but just one of them, having been released, has occupied the whole Universe and plays the part of dark energy (if dark energy is really phantom, which is more or less likely but not certain).
We have also found some general properties of models in the Einstein-scalar field system under consideration:
(i)
: trapped-ghost are only possible with nonzero potentials $V(\phi)$;
(ii)
: in all with two flat asymptoyics, $V(\phi)$ has an alternate sign (unless $V \equiv 0$);
(iii)
: in any with two flat asymptoyics, if the Schwarzschild mass equals $m$ at one of them, it equals $-m$ at the other. Hence mirror symmetry with respect to the throat (i.e., the metric functions are even in the radial coordinate $u$) implies $m = 0$.
Acknowledgments {#acknowledgments .unnumbered}
---------------
This work was supported in part by the Russian Foundation for Basic Research grants No. 08-02-91307, 08-02-00325, and 09-02-00677a.
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|
---
abstract: 'In this work we use a recently developed nonintegrability theorem of Morales and Ramis to prove that the Friedmann Robertson Walker cosmological model with a conformally coupled massive scalar field is nonintegrable.'
author:
- 'L. A. A. Coelho'
- 'J. E. F. Skea'
- 'T. J. Stuchi'
title: 'Friedmann Robertson Walker models with Conformally Coupled Massive Scalar Fields are Non-integrable'
---
Introduction
============
In recent years the search for nonintegrablility criteria for Hamiltonian systems in the complex domain has acquired more relevance [@mora]-[@yosh]. Such techniques are potentially of particular importance in cosmology because of controversies over both integrability, and the existence of chaos in cosmological models [@tere]-[@bianchi]. Part of the problem is that certain methods used to traditionally measure chaos in non-relativistic systems, such as the Lyapunov exponents, are no longer valid in General Relativity where there is no absolute time coordinate.
In this work we use a recently developed theorem by Morales and Ramis [@mora] which establishes a relation between two differents concepts of integrability: the complete integrability of complex analytical Hamiltonian systems (given by Liouville’s theorem) and the integrability of homogeneous linear ordinary differential equations (LODEs) in terms of Liouvillian functions in the complex plane. A Liouvillian function is a function which can be written as a combination of elementary functions, algebraic functions (solutions of polynomial equations), their indefinite integrals or exponentials of these integrals. Since we are working in the complex domain this definition includes elementary functions such as logarithms, trigonometric functions and their inverses.
The model we study in this paper is a Friedman-Robertson-Walker (FRW) cosmological model with a conformally coupled scalar field, $\phi$, of mass $m$. We use the conformal form of the metric $${\rm d}s^2=a^2(\eta)\left[{\rm d}\eta^2-\frac{1}{1-kr^2}\,{\rm d}r^2-r^2\,{\rm d}\Omega^2\right],
\label{line}$$ with $\eta$ the conformal time, $a(\eta)$ the scale factor and $k=0,\pm 1$ the curvature.
The dynamics of this model has been discussed and studied before using numerical methods [@tere]-[@hamils] but, as far as we are aware, no completely rigorous conclusion has been reached about its integrability. In [@hami2] the integrability of a generalisation of the model studied here is considered using Painlevé analysis via the ARS algorithm [@ARS1; @ARS2; @ARS3]. Though there is a strong connection between integrability and the Painlevé property, and the latter has been remarkably successful in indicating possibly integral cases, it is worth noting that the lack of the Painlevé property is not a rigorous obstruction to integrability [@Grammaticos]. Additionally, the ARS algorithm is not a foolproof method for determining whether a system possesses the Painlevé property, and its application can lead to false conclusions [@mix1; @mix2; @mix3], particularly when applied to determining the non-integrability of a dynamical system. Finally, certain expressions in [@hami2] are undefined for our model, requiring a separate analysis.
The Morales-Ramis Theorem (MRT) which we use in our study rigorously provides necessary conditions for the integrability of a Hamiltonian system and so sufficient conditions for non-integrability. The theorem can be used to reduce the question of integrability to one of the existence of Liouvillian solutions of a homogeneous second-order linear ODE. This problem can in turn be solved using Kovacic’s algorithm. Though complex to write down, the algorithm is, as we shall see, straightforward to apply to the problem considered here. To use the MRT for our cosmological model we first require a Hamiltonian which generates the field equations. In this case it is known [@tere]-[@hamils] that a suitable Hamiltonian is
$$H=\frac{1}{2}\left[(p_\phi^2+k\phi^2)-(p_a^2+ka^2)+m^2a^2\phi^2\right]=0,
\label{Hamil}$$
where $p_a$ and $p_\phi$ are the momenta conjugate to $a$ and $\phi$ respectively.
In the next section we give the main results of the Morales-Ramis theorem. Since the various versions of Kovacic’s algorithm in the literature [@mora; @bianchi; @kova; @duval] have slight differences in presentation and conventions, we include the version of the algorithm as used by us. We then show how the algorithm quickly determines that the Hamiltonian is nonintegrable for $k\neq 0$. Finally for the case $k= 0$ the analysis based on the invariant planes $a=p_a=0$ and $\phi=p_\phi=0$ is inconclusive. However, because the potential is homogeneous in this case, there exist particular nonsingular solutions which do not lie in these planes which can be used as a basis for the analysis. Fortunately the case of homogeneous potentials has been exhaustively studied by Yoshida [@yosh] and Morales-Ramis [@mora] and so we can simply apply those results.
The Morales-Ramis Theorem
=========================
The Morales-Ramis theorem is a nonitegrability criterion: it gives a necessary condition for a Hamiltonian system to be integrable and therefore a sufficient condition for nonintegrability. The theorem is based on the analysis of the variational equations (in particular the normal variational equation, or NVE) for the perturbations of a non-equilibrum particular solution. The basic idea is that if the flow of the Hamiltonian system has a regular behaviour (is integrable), then the linearized flow along a particular integral curve given by the NVE must also be regular (integrable). Conversely if the linearized flow is nonintegrable the system as a whole will be nonintegrable.
A Hamiltonian system, $X_H$, of dimension $n$ is called integrable if there exist $n$ independent constants of the motion in involution. By considering the differential Galois group of the NVE, the theorem of Morales-Ramis links this concept of integrability to an apparently different concept of integrability – the existence of Liouvillian solutions of the NVE of $X_H$. The theorem may be stated as
If there are $n$ first integrals of $X_H$ that are independent and in involution, then the identity component of the Galois group of the NVE is abelian.
It is known that [@kapla] for an ODE to admit a Liouvillian solution, the identity component of its Galois group must be soluble. Hence, if the solutions are not Liouvillian, the identity component of the Galois group is not soluble and, therefore, non-Abelian.
Our strategy will therefore be:
[**1:**]{} Select a particular solution (in our case an invariant plane).\
[**2:**]{} Write the variational equations and the NVE.\
[**3:**]{} Check if the solutions of the NVE are Liouvillian functions.
To decide the third step, we use Kovacic’s algorithm [@kova] which we now turn to describe.
Kovacic’s algorithm
===================
Kovacic’s algorithm provides a procedure for computing the Liouvillian solutions of a homogeneous linear second order differential equation. If the algorithm terminates negatively, we can conclude that no such solutions exist.
Let ${\mathbb{C}}(x)$ be the field of rational complex functions (ratios of polynomials in $x$ with complex coefficients). It is well-known that by using the change of dependent variable $$y=\xi\,\exp\left({1\over 2}\int b\,{\rm d}x\right)
\label{trans1}$$ the second order homogeneous LODE $$y''+b(x)\,y'+c(x)\,y = 0$$ can be transformed to the so-called reduced invariant form $$\xi''-g\xi=0,
\label{kova1}$$ where $$g(x)=\frac{1}{2}{b}'(x)+\frac{1}{4}b(x)^2-c(x).
\label{gdef}$$ Note that, if $b(x)$ and $c(x)~\in{\mathbb{C}}(x)$ then $g(x)~\in{\mathbb{C}}(x)$.
Moreover, using a further change of variables $v=\xi'/\xi$, equation (\[kova1\]) is transformed into the Riccati equation $$v'+v^2=g.
\label{kova2}$$
Now equation (\[kova1\]) is integrable, if and only if equation (\[kova2\]) has an algebraic solution, that is $v$ solves a polynomial equation $f(v)=0$, where the degree of $f$ (the minimal polynomial) in $v$ belongs to the set $L=\left\{1,2,4,6,12\right\}$.
Kovacic’s algorithm can be divided into three main steps: the first step is the determination of the subset of $L$ relevant for the LODE under consideration; the two other steps are devoted respectively to determining the existence of the minimal polynomial, and its construction. If the algorithm does not terminate successfully (ie, equation (\[kova2\]) has no algebraic solution) then equation (\[kova1\]) has no solution in terms of Liouvillian functions.
In the version used of the algorithm we essentially follow [@mora; @bianchi; @duval; @luis]. Let $$g=g(x)=\frac{s(x)}{t(x)},
\label{gx}$$ with $s(x),t(x)$ relatively prime polynomials, and $t(x)$ monic. Define the function $h$ on the set $L_{max}=\{1,2,4,6,12\}$ by $h(1)=1$, $h(2)=4$, $h(4)=h(6)=h(12)=12$.
[**Step 1**]{} (determination of possible orders of the minimal polynomial)
If $t(x)=1$ then set $m=0$, else factorize $t(x)$ into monic relatively prime polynomials $$t(x)=t_1(x)\,t^2_2(x)\ldots t^m_m(x),$$ where $t_i$ have no multiple roots and $t_m\neq1$.
Then
[**1.1**]{} Let $\Gamma'$ be the set of roots of $t(x)$ (i.e.,the singular points in the finite complex plane) and let $\Gamma=\Gamma'\cup
\infty $ be the set of singular points.
Then the order of a singular point $c\in \Gamma'$ is, as usual, $o(c)=i$ if $c$ is a root of multiplicity $i$ of $t_i$. The order at infinity is defined by $o(\infty)=\mbox{max}(0,4+\mbox{deg}(s)-\mbox{deg}(t))$. We call $m^+=\mbox{max}(m,o(\infty))$.
For $0\leqslant i\leqslant m^+$, denote by $\Gamma_i=\left\{c\in\Gamma\mid o(c)=i\right\}$ the subset of all elements of order $i$. [**1.2**]{} If $m^+\geq 2$ then we write $\gamma_2=\mbox{card}(\Gamma_2)$, else $\gamma_2=0$. Then we compute
$\displaystyle{\gamma=\gamma_2+\mbox{card}\left(
\bigcup_{\stackrel{\mbox{\scriptsize$3\leq k\leq m^+$}}
{k~\mbox{odd}}
} \Gamma_k\right)}.$
[**1.3**]{} For the singular points of order one or two, $c\in \Gamma_2
\cup \Gamma_1$, we compute the principal parts of $g$:
$\displaystyle{g_c=\alpha_{c}(x-c)^{-2}+\beta_{c}(x-c)^{-1}+O(1),}$
if $c\in \Gamma'$, and
$\displaystyle{g_{\infty}=\alpha_{\infty}x^{-2}+\beta_{\infty}x^{-3}+O(x^{-4}),}$
for the point at infinity.
[**1.4**]{} We define the subset $L'$ (of all possible values for the degree of minimal polynomial) as $\{1\}\subset L'$ if $\gamma=\gamma_2$, $\{2\}\subset L'$ if $\gamma \geq 2$ and $\{4,6,12\}\subset L'$ if $m^+
\leq 2$.
[**1.5**]{} We have the three following mutually exclusive cases:
1.5.1 If $m^+ > 2$, then $L=L'$.
1.5.2 Define $\Delta_c = \sqrt{1+4\alpha_c}$. If $m^+ \leq 2$ and $\forall c\in \Gamma_1\cup\Gamma_2$, $\Delta_c\in {\mathbb Q}$, then $L=L'$.
1.5.3 If cases (1.5.1) and (1.5.2) do not hold, then $L=L'-\{4,6,12\}$.
[**1.6**]{} If $L={\emptyset}$, then equation (\[kova1\]) is non-integrable with Galois group $SL(2,{\mathbb{C}})$, else one writes $n$ for the minimum value in $L$.
For the second and third steps of the algorithm we consider a fixed value of $n$.
[**Step 2**]{}
[**2.1**]{} If $\infty$ has order $0$ we write the set
$\displaystyle{E_{\infty}=\left\{0,\frac{h(n)}{n},2\frac{h(n)}{n},3\frac{h(n)}{n},\ldots,n\frac{h(n)}{n}\right\}.}$
[**2.2**]{} If $c$ has order 1, then $E_{c}=\{h(n)\}$.
[**2.3**]{} If $n=1$, for each $c$ of order 2 we define
$\displaystyle{E_{c}=\left\{\frac{1}{2}(1+\Delta_c),\frac{1}{2}(1-\Delta_c)\right\}}$
[**2.4**]{} If $n\geq 2$, for each $c$ of order 2, we define
$$E_{c}={\mathbb Z}\cap
\left\{\frac{h(n)}{2}(1-\Delta_c)+\frac{h(n)}{n}k\Delta_c:
k=0,1,\ldots ,n\right\}.$$
[**2.5**]{} If $n=1$, for each singular point of even order $2\nu$, with $\nu >1$, we compute the numbers $\alpha_c$ and $\beta_c$ defined (up to a sign) by the following conditions:
> 2.5.1 If $c\in \Gamma'$,
$\displaystyle{
g_c=\left\{{\alpha_{c}\over (x-c)^{\nu}}
+\sum_{i=2}^{\nu-1}{\mu_{i,c}\over (x-c)^{i}}\right\}^2
+{\beta_c\over\left(x-c\right)^{\nu+1}}+
%O\left(\left(x-c\right)^{-\nu}\right)
O\left(x-c\right)^{-\nu}
}$
and we write
$\displaystyle{\sqrt{g_c}:=\alpha_c\left(x-c\right)^{-\nu}+\sum_{i=2}^{\nu-1}{\mu_{i,c}\left(x-c\right)^{-i}}.}$
> 2.5.2 If $c=\infty $,
$\displaystyle{g_{\infty}=\left\{\alpha_{\infty}x^{\nu-2}+\sum_{i=0}^{\nu-3}{\mu_{i,\infty}x^{i}}\right\}^2-\beta_{\infty}x^{\nu-3}+O\left(x^{\nu-4}\right),}$
and we write
$\displaystyle{\sqrt{g_{\infty}}:=\alpha_{\infty}x^{\nu-2}+\sum_{i=0}^{\nu-3}{\mu_{i,\infty}x^{i}}.}$
Then for each $c$ as above, we compute
$\displaystyle{E_c=\left\{\frac{1}{2}\left(\nu+\epsilon
\frac{\beta_c}{\alpha_c}\right):\epsilon =\pm 1\right\},}$
and the sign function on $E_c$ is defined by
$\displaystyle{\mbox{sign}\left(\frac{1}{2}\left(\nu+\epsilon
\frac{\beta_c}{\alpha_c}\right)\right)=\epsilon,}$
being $+1$ if $\beta_c=0$.
[**2.6**]{} If $n=2$, for each $c$ of order $\nu$, with $\nu \geq 3$, we write $E_c=\{\nu\}$.
[**Step 3**]{}
[**3.1**]{} For $n$ fixed, we try to obtain elements ${\bf e}=(e_c)_{c\in
\Gamma}$ in the Cartesian product $\prod_{c\in \Gamma}^{}{}E_c$, such that:
[(i) $\displaystyle{d({\bf e}):= n-\frac{n}{h(n)}\sum_{c\in
\Gamma}^{}{}e_c}$ is a non-negative integer,]{}
\(ii) If $n=2$ or $n=6$ then [**e**]{} has an even number of elements which are odd integers
\(iii) when $n=4$, then [**e**]{} has at least two elements not divisible by 3, and the sum of all elements not divisible by 3 is divisible by 3.
If no such set ${\bf e}$ is obtained, we select the next value in $L$ and repeat Step 2, else $n$ is the maximum value in $L$ and the Galois group is $SL(2,{\mathbb{C}})$ (and equation (\[kova1\]) is non-integrable).
[**3.2**]{} For each family ${\bf e}$ as above, we try to obtain a rational function $Q$ and a polynomial $P$, such that
(i)
$\displaystyle{Q=\frac{n}{h(n)}\sum_{c\in
\Gamma'}^{}{}\frac{e_c}{x-c}+\delta_{n1}\sum_{c\in \bigcup_{\nu
>1}^{}{}\Gamma_{2\nu}}^{}{}\mbox{sign}(e_c)\sqrt{g_c},}$
where $\delta_{n1}$ is the Kronecker delta.
\(ii) $P$ is a polynomial of degree $d({\bf e})$ and its coefficients are found as a solution of the (in general, overdetermined) system of equations
$\displaystyle{P_{-1}=0,}$
$\displaystyle{P_{i-1}=-(P_i)'-QP_i-(n-i)(i+1)gP_{i+1},~~n\geq
i\geq 0,}$
$\displaystyle{P_n=-P.}$
If a pair $(P,Q)$ as above is found, then equation (\[kova1\]) is integrable and the Riccati equation (\[kova2\]) has an algebraic solution $v$ given by any root $v$ of the equation
$\displaystyle{f(v)=\sum_{i=0}^{n}{\frac{P_i}{(n-i)!}v^i=0.}}$
If no pair as above is found we take the next value in $L$ and we go to Step 2. If $n$ is the greatest value in $L$ then the Galois group of (\[kova1\]) is $SL(2,{\bf{C}})$ and the ODE is non-integrable.
Application and Result
======================
The case $k\neq 0$
------------------
We apply the theorem of Morales-Ramis to (\[Hamil\]). We choose as our set of non-equilibrium particular solutions the invariant plane $p_a=a=0$. The NVEs relative to this plane are $$\frac{{\rm d}^2\delta a}{{\rm d}t^2}=\left(-k+m^2\phi^2\right)\delta a.
\label{kova4}$$
Changing the independent variable to $\phi$ and renaming $\delta a=y$, we obtain the equation
$$\frac{{\rm d}^2 y}{{\rm d}\phi^2}+\frac{1}{\phi}\frac{{\rm d}y}{{\rm d}\phi}
+\left(\frac{m^2}{k} -\frac{1}{\phi^2}\right)y=0.
\label{eq45}$$
This equation is a second-order, linear and homogeneous ODE with coefficients which are rational functions of $\phi$, and we can therefore apply Kovacic’s algorithm to determine any Liouvillian solutions.
Using (\[trans1\]) and (\[gdef\]) we transform (\[eq45\]) into the reduced invariant form (\[kova1\])
$$\xi''= \left(\frac{3k-4m^2\phi^2}{4k\phi^2}\right)\,\xi.
\label{kova7}$$
Equation (\[eq45\]) has no Liouvillian solutions when $k\neq 0$ and $m\neq 0$.
[*Proof*]{}: by application of Kovacic’s algorithm to equation (\[kova7\]).
[**Step 1**]{}
$g(\phi)$ has one finite pole, at $\phi=0$, of order 2 and the pole at infinity, of order 4 (since, by assumption, we are treating the massive case, $m\neq 0$). This implies that $m^+=4$ and $\gamma=\gamma_2=1$. Since the pole at $\phi=0$ belongs to $\Gamma_2$, we calculate the Laurent series (when $k\neq0$) as
$$g_0=\frac{3}{4}\phi^{-2}-\frac{m^2}{k}$$
Hence $\displaystyle{\alpha_0=\frac{3}{4}}$ and $\beta_0=0$. Thus we have $L=\{1\}$.
[**Step 2**]{}
Because $L=\{1\}$ the unique value for $n$ is $n=1$. Through the items $2.3$ and $2.5$ we calculate the sets $E_c$. From $2.3$ we have that $\displaystyle{E_0=\left\{\frac{3}{2},-\frac{1}{2}\right\}}$. In item $2.5.2$ we need to expand $g$ around $\phi=\infty$. Doing this we obtain
$$g_\infty=\frac{3}{4}\phi^{-2}-\frac{m^2}{k}\Longrightarrow E_\infty=\{1\}.$$
Summarizing,
$$E_0=\left\{\frac{3}{2},-\frac{1}{2}\right\} ~~~~~~\mbox{and}~~~~~~E_\infty=\{1\} .$$
[**Step 3**]{}
In this step we need to calculate $\prod_{c\in \Gamma} E_c$, using the sets determined in the previous step. We obtain the set of sets given by
$$\prod_{c\in \Gamma}^{}{}E_c= \left\{ \left\{\frac{3}{2},1\right\},\left\{-\frac{1}{2},1\right\} \right\}.$$
From 3.1(i) we calculate the values of $d(\bf{e})$ as $\displaystyle{d=-\frac{3}{2}}$ and $\displaystyle{d=\frac{1}{2}}$ respectively. Since neither of theses values satisfies 3.1(i) and there are no other values of $n$ in $L$, the Galois group of (\[kova7\]) is $SL(2,{\mathbb{C}})$, equation (\[kova7\]) is nonintegrable in terms of Liouvillian functions, and therefore the system represented by the Hamiltonian (\[Hamil\]) is also nonintegrable when $k\neq0$. This completes the proof.
The case $k=0$
--------------
When $k=0$ the only first integral of the Hamiltonian system (\[Hamil\]) is the Hamiltonian, and the system is therefore nonintegrable.
[*Proof:*]{} In order to prove this lemma we observe that when $k=0$ the Hamiltonian (\[Hamil\]) has a homogeneous potential. Hamiltonians with homogeneous potentials have been exhaustively studied using the MRT and particular results obtained which we now outline.
Let
$$H={1\over 2}\sum_{i=1}^n p_i^2 + V(q_1,\ldots,q_n)
\label{hamhomog}$$
where $A$ is a constant and $V$ is a homogeneous potential, i.e. $V(A\overrightarrow{Q})=A^{g}\,V(\overrightarrow{Q})$ with $g$ being the degree of the potential. To put our Hamiltonian in the form (\[hamhomog\]) we perform the canonical transformation, $x=ia$ and $P_x=-iP_a$, after which
$$H=\frac{1}{2}\left[p_x^2+p_\phi^2-m^2x^2\phi^2\right]
\label{hamhom}$$
and $g=4$. The MRT for homogeneous potentials is given by [@mora; @yosh].
Let $V(q_1, \ldots, q_n)$ be a homogeneous potential function of integer degree $g$, $c$ a solution of the equation $c=\overrightarrow{V}'(c)$, and $\lambda_i$ (the Yoshida coefficients) the eigenvalues of the matriz $V''(c)$. One of these eigenvalues is trivial, in that it corresponds to the tangential variational equation, and has value $g-1$.
If a Hamiltonian system of the form (\[hamhomog\]) is completely integrable (with holomorphic or meromorphic first integrals) then each pair $(g,\lambda_i)$ belongs to one of the following list (where we do not consider the trivial case $g=0$) $$\begin{array}{ll}
(1) & (g,p+p(p-1)g/2) \\
(2) & (2,\mbox{arbitrary complex number}) \\
(3) & (-2,\mbox{arbitrary complex number}) \\
(4) & (-5,\frac{49}{40}-\frac{1}{40}(\frac{10}{3}+10p)^2) \\
(5) & (-5,\frac{49}{40}-\frac{1}{40}(4+10p)^2) \\
(6) & (-4,\frac{9}{8}-\frac{1}{8}(\frac{4}{3}+4p)^2) \\
(7) & (-3,\frac{25}{24}-\frac{1}{24}(2+6p)^2) \\
(8) & (-3,\frac{25}{24}-\frac{1}{24}(\frac{3}{2}+6p)^2) \\
(9) & (-3,\frac{25}{24}-\frac{1}{24}(\frac{6}{5}+6p)^2) \\
(10) & (-3,\frac{25}{24}-\frac{1}{24}(\frac{12}{5}+6p)^2) \\
(11) & (3,-\frac{1}{24}+\frac{1}{24}(2+6p)^2) \\
(12) & (3,-\frac{1}{24}+\frac{1}{24}(\frac{3}{2}+6p)^2) \\
(13) & (3,-\frac{1}{24}+\frac{1}{24}(\frac{6}{5}+6p)^2) \\
(14) & (3,-\frac{1}{24}+\frac{1}{24}(\frac{12}{5}+6p)^2) \\
(15) & (4,-\frac{1}{8}+\frac{1}{8}(\frac{4}{3}+4p)^2) \\
(16) & (5,-\frac{9}{40}+\frac{1}{40}(\frac{10}{3}+10p)^2) \\
(17) & (5,-\frac{9}{40}+\frac{1}{40}(4+10p)^2) \\
(18) & (g,\frac{1}{2}(\frac{g-1}{g}+p(p+1)g)) \\
\end{array}$$
where $p$ is an arbitrary integer.
For the system represented by (\[hamhom\]) the only non-trivial Yoshida coefficient is $\lambda=-1$. For $g=4$ the only possibilities for satisfying Theorem 3 are (1), (15) and (18). For all these cases there are no integer values of $p$ which solve $\lambda=-1$, and we conclude that the system represented by (\[hamhom\]) is nonintegrable. This completes the proof of lemma 2.
We can now enunciate the following theorem
The Friedmann Robertson Walker model with a conformally coupled massive scalar field represented by the Hamiltonian (\[Hamil\]) is not completely integrable.
[*Proof:*]{} By lemma 1 there are no Liouvillian solutions of the NVE for the plane $a=p_a=0$. This implies that the identity component of its Galois group is not soluble and therefore non-Abelian. Using theorem 1 we have that the only first integral of the Hamiltonian system is the Hamiltonian itself, and that the system is not completely integrable in this case. By lemma 2 the Hamiltonian system is also not completely integrable when $k=0$. Therefore the Hamiltonian system represented by (\[Hamil\]) is nonintegrable for all values of $k$.
Conclusion
==========
From our analysis we have shown rigorously using analytic methods that FRW universes with a conformally coupled massive scalar field are nonintegrable. This is compatible with results from numerical analysis based on Poincaré sections [@tere] which indicate that the behaviour of the system is mathematically chaotic.
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---
abstract: 'This note is devoted to the theory of projective limits of finite-dimensional Lie groups, as developed in the recent monograph \[Hofmann, K.H. and S.A. Morris, “The Lie Theory of Connected Pro-Lie Groups,” EMS Publ. House, 2007\]. We replace the original, highly non-trivial proof of the One-Parameter Subgroup Lifting Lemma given in the monograph by a shorter and more elementary argument. Furthermore, we shorten (and correct) the proof of the so-called Pro-Lie Group Theorem, which asserts that pro-Lie groups and projective limits of Lie groups coincide.'
---
[**Simplified Proofs for the Pro-Lie Group Theorem\
and the One-Parameter Subgroup Lifting Lemma**]{}\
[**Helge Glöckner**]{}
By a famous theorem of Yamabe [@Yam], every identity neighbourhood of a connected (or almost connected) locally compact group $G$ contains a closed normal subgroup $N$ such that $G/N$ is a Lie group, and thus is a so-called pro-Lie group. Therefore locally compact pro-Lie groups form a large class of locally compact groups, which has been studied by many authors (see, e.g., [@Iwa], [@Las], [@MaZ] as well as [@HMS] and the references therein). Although a small number of papers broached on the topic of non-locally compact pro-Lie groups (like [@Hof] and [@Glo]), a profound structure theory of such groups was only begun recently in [@HMo] and then fully worked out in the monograph [@HaM]. The novel results accomplished in [@HaM] make it clear that the study of general pro-Lie groups is fruitful also for the theory of locally compact groups.\
We recall from [@HaM]: For $G$ a Hausdorff topological group, ${{\mathcal N}}(G)$ denotes the set of all closed normal subgroups $N$ of $G$ such that $G/N$ is a (finite-dimensional) Lie group. If $G$ is complete and ${{\mathcal N}}(G)$ is a filter basis which converges to $1$, then $G$ is called a *pro-Lie group*. It is easy to see that every pro-Lie group is, in particular, a projective limit of Lie groups. Various results which are known in the locally compact case become much more complicated to prove for non-locally compact pro-Lie groups. For example, it is not too hard to see that every locally compact group which is a projective limit of Lie groups is a pro-Lie group (see [@App] for an elementary argument; the appeal to the solution of Hilbert’s fifth problem in the earlier proof in [@HMS] is unnecessary). Also, it has been known for a long time [@HWY] that one-parameter subgroups can be lifted over quotient morphisms $q\colon G\to H$ between locally compact groups, i.e., for each continuous homomorphism $X\colon {{\mathbb R}}\to H$ there exists a continuous homomorphism $Y\colon {{\mathbb R}}\to G$ such that $X=q\circ Y$. The original proofs for analogues of the preceding two results for general pro-Lie groups as given in [@HMo] and [@HaM] (called the “Pro-Lie Group Theorem” and “One-Parameter Subgroup Lifting Lemma” there) were quite long and complicated. Later, A.A. George Michael gave a short alternative proof of the Pro-Lie Group Theorem, which however was not self-contained but depended on a non-elementary result from outside, the Gleason–Palais Theorem: *If $G$ is a locally arcwise connected topological group in which the compact metrizable subsets are of bounded dimension, then $G$ is a Lie group* [@GaP Theorem 7.2].\
The goal of this note is to record two short and simple arguments, which together with some 10 pages of external reading[^1] provide elementary and essentially self-contained proofs for both the Pro-Lie Group Theorem and the One-Parameter Subgroup Lifting Lemma (up to well-known facts). In this way, the proof of the latter shrinks from over 3 pages to 8 lines, and the proof of the former by 6 pages. Moreover, the author noticed that the proof of the Pro-Lie Group Theorem in [@HaM] (and [@HMo]) depends on an incorrect assertion,[^2] making it the more important to have a correct elementary proof available.\
Let us now re-state and prove the theorem and lemma in contention. Notations from [@HaM] will be used without explanation.
Every projective limit of Lie groups is a pro-Lie group.
Let $G$ be a projective limit of a projective system $((G_j)_{j\in J},(f_{jk})_{j\leq k})$ of Lie groups $G_j$ and morphisms $f_{jk}\colon G_k\to G_j$. By [@HaM Proposition 3.27], $G$ will be a pro-Lie group if we can show that $G/\ker(f_j)$ is a Lie group for each limit map $f_j\colon G\to G_j$. Let $H_j$ be the analytic subgroup of $G_j$ with Lie algebra ${{\mathcal L}}(f_j)({{\mathcal L}}(G))$ (equipped with its Lie group topology). By [@HaM Lemmas 3.23 and 3.24], $f_j$ restricts and corestricts to a quotient morphism $\phi_j\colon G_0\to H_j$. Given $g\in G$, write $I_g^G\colon G\to G$, $I_g^G(h):=ghg^{-1}$. Since $\phi_j\circ I_g^G|_{G_0}=I_{f_j(g)}^{G_j}\circ \phi_j$, we see that $I_{f_j(g)}^{G_j}(H_j)\subseteq H_j$ and $I_{f_j(g)}^{G_j}|_{H_j}\colon H_j\to H_j$ is continuous. Hence $Q_j:=f_j(G)$ can be made a Lie group with $H_j$ as an open subgroup. Then the corestriction $q_j\colon G\to Q_j$ of $f_j$ to $Q_j$ is a surjective homomorphism, which is open since so is $f_j|_{G_0}^{H_j}=\phi_j$. If we can show that $q_j$ is continuous, then $q_j$ will be a quotient morphism and thus $G/\ker(f_j)\cong Q_j$ a Lie group. However, by [@HaM Lemma 3.21], there exists some $k\in I$ such that $k\geq j$ and $f_{jk}((G_k)_0){\subseteq}H_j$. Also, it is shown in the proof of [@HaM Lemma 3.24] that the map ${\overline}{f}_{jk}\colon (G_k)_0\to H_j$, $x{\mapsto}f_{jk}(x)$ is continuous. Since $U:=f_k^{-1}((G_k)_0)$ is a neighbourhood in $G$ and $q_j|_U\!=\!{\overline}{f}_{jk}\!\circ\! f_k|_U^{(G_k)_0}$ is continuous, the homomorphism $q_j$ is continuous.
Let $G$ and $H$ be pro-Lie groups and $f\colon G\to H$ be a quotient morphism of topological groups. Then every one-parameter subgroup $X$ of $H$ lifts to one of $G$, i.e., there exists a one-parameter subgroup $Y\colon {{\mathbb R}}\to G$ such that $X=f \circ Y$.
We adapt an argument from [@HaM p.193]. By Lemmas 4.16, 4.17 and 4.18 in [@HaM], we may assume that $H={{\mathbb R}}$ and have to show that $f$ is a retraction. If $f$ was not a retraction, then we would have ${{\mathcal L}}(f)({{\mathcal L}}(G))=\{0\}$ and hence $f(G_0)=\{1\}$, using that $\exp_G({{\mathcal L}}(G))$ generates a dense subgroup of $G_0$ (by Lemma 3.24 and the proof of Lemma 3.22 in [@HaM]), and $f\circ \exp_G=\exp_H\circ \, {{\mathcal L}}(f)=1$. Hence $f$ factors to a quotient morphism $G/G_0\to {{\mathbb R}}$. Since $G/G_0$ is proto-discrete by [@HaM Lemma 3.31], it would follow that also its quotient ${{\mathbb R}}$ is proto-discrete (see [@HaM Proposition 3.30(b)]) and hence discrete (as ${{\mathbb R}}$ has no small subgroups). We have reached a contradiction.
We mention that the Pro-Lie Group Theorem has no analogue for projective limits of Banach-Lie groups. In fact, consider a Fréchet space $E$ which is not a Banach space but admits a continuous norm $\|.\|$ (e.g., $E=C^\infty([0,1],{{\mathbb R}})$). Then $E$ is a projective limit of Banach spaces. The $\|.\|$-unit ball $U$ is a $0$-neighbourhood in $E$ which does not contain any non-trivial subgroup of $E$. If there existed a quotient morphism $q\colon E\to G$ to a Banach-Lie group $G$ with kernel in $U$, then we would have $\ker(q)=\{0\}$. Hence $q$ would be an isomorphism, entailing that the Banach-Lie group $G$ is abelian and simply connected and therefore isomorphic to the additive group of a Banach space. Since $E$ is not a Banach space, we have reached a contradiction.
[99]{} George Michael, A.A., *On inverse limits of finite-dimensional Lie groups*, J. Lie Theory [**16**]{} (2006), 221–224. Gleason A. and R. Palais, *On a class of transformation groups*, Amer. J. Math. [**79**]{} (1957), 631–648. Glöckner, H., *Approximation by $p$-adic Lie groups*, Glasgow Math. J. [**44**]{} (2002), 231–239. Glöckner, H., *Real and $p$-adic Lie algebra functors on the category of topological groups*, Pac. J. Math. [**203**]{} (2002), 321–368. Hofmann, K.H., *Category-theoretical methods in topological algebra*, in: E. Binz and H. Herrlich (Eds.), “Categorical Topology,” Springer-Verlag, 1976. Hofmann, K.H. and S.A. Morris, *Projective limits of finite-dimensional Lie groups*, Proc. London Math. Soc. (2003), 647–676. Hofmann, K.H. and S.A. Morris, “The Structure of Connected Pro-Lie Groups,” EMS Tracts in Math. [**2**]{}, Europ. Math. Soc. Publ. House, Zurich, 2007. Hofmann, K.H., S.A. Morris and M. Stroppel, *Locally compact groups, residual Lie groups, and varieties generated by Lie groups*, Topology Appl. [**71**]{} (1996), 63–91. Hofmann, K.H., T.S. Wu and J.S. Yang, *Equidimensional immersions of locally compact groups*, Math. Proc. Camb. Philos. Soc. (1989), 253–261. Iwasawa, K., *On some types of topological groups*, Ann. of Math. [**50**]{} (1949), 507–558. Lashof, R.K., *Lie algebras of locally compact groups*, Pac. J. Math. [**7**]{} (1957), 1145–1162. Montgomery, D. and L. Zippin, “Topological Transformation Groups,” Interscience, New York, 1955. Yamabe, H., *On the conjecture of Iwasawa and Gleason*, Ann. of Math. [**58**]{} (1953), 48–54.
[, TU Darmstadt, FB Mathematik AG AGF, Schlossgartenstr.7, 64289 Darmstadt,\
Germany. E-Mail: [gloeckner@mathematik.tu-darmstadt.de]{}]{}
[^1]: Lemmas 3.20–3.24, Propositions 3.27 and 3.30, Lemma 3.31 and Lemmas 4.16–4.18 in [@HaM].
[^2]: Parts (iii) and (iv) of the “Closed Subgroup Theorem” [@HaM Theorem 1.34] are false, as the example $G={\mathbb R}$, $H={\mathbb Z}$, ${\mathcal N}=
\{\{0\},\sqrt{2} \, {\mathbb Z}\}$ shows. This invalidates the proof of part (iii) of the “First Fundamental Lemma” [@HaM Lemma 3.29], which is used in [@HaM] to prove the Pro-Lie Group Theorem (the proof of Lemma 3.29 (iv) also seems to be defective, because elements $M\in {{\mathcal M}}$ are of the form $M=\ker(f_j)\cap G_0$, rather than $M=\ker(f_j)$).
|
---
abstract: 'Non-Gaussian entangled states of light have been found to improve the success of quantum teleportation. Earlier works in the literature focussed mainly on two-mode non-Gaussian states generated by de-Gaussification of two-mode squeezed vacuum states. In the current work, we study quantum teleportation with a class of non-Gaussian entangled resource states that are generated at the output of a passive beam splitter (BS) with different input single mode non-Gaussian states. In particular, we consider input states that are generated under successive application of squeezing and photon addition/subtraction operations in various orders. We focus on identifying what attributes of the resource states are necessary or sufficient for quantum teleportation (QT). To this end we first evaluate two attributes considered in the literature, viz. squeezed vacuum affinity (SVA) and EPR correlation. While SVA is not non-zero for all two-mode resource states, EPR correlation is neither necessary nor sufficient of QT. We consider yet another attribute, viz. two-mode quadrature squeezing as defined by Simon *et. al.* \[Phys. Rev. A **49**, 1567 (1994)\]. Our numerical results on the de-Gaussified two-mode squeezed vacuum state as well as the BS generated non-Gaussian states lead us to the conclusion that two-mode quadrature squeezing is a [*necessary condition*]{} for QT, in general. We further demonstrate the plausibility of this conclusion by giving an analytical proof that two-mode quadrature squeezing is a necessary condition for QT in the case of symmetric two-mode Gaussian resource states.'
author:
- Soumyakanti Bose
- 'M. Sanjay Kumar'
title: 'Quantum Teleportation with a Class of Non-Gaussian Entangled Resources'
---
I. Introduction {#i.-introduction .unnumbered}
===============
Quantum teleportation (QT) [@book_qit] is one of the most important information processing tasks that serves as a building block to several other protocols of quantum information technology [@qt_tp]. It was first proposed by Bennett *et. al.* [@TP_B] for qubits that could be realized in several systems such as atomic spin, polarization of light *etc* [@book_qit]. Later, an experimentally realizable extension of the protocol to quantum optical systems was proposed by Braunstein and Kimble [@TP_BK]. Quantum teleportation with entangled optical resources, implementing Braunstein-Kimble (BK) protocol, has also been experimentally realized [@tp_exp1; @tp_exp2; @tp_exp3; @tp_exp4; @tp_exp5].
The most commonly used Gaussian entangled quantum optical resource in teleportation is the two-mode squeezed vacuum state (TMSV) which could be generated in parametric down conversion [@OPDC]. However, certain de-Gaussification processes such as photon addition and subtraction alongwith their coherent superposition, quantum catalysis *etc.* have been found to improve the amount of entanglement as well as the success of teleportation compared to TMSV. [@ent_grangier; @tp_illuminati; @tp_yang; @tp_lee; @ent_benlloch; @tp_wang; @tp_agarwal; @tp_zubairy].
Dell’Anno *et. al.* [@tp_illuminati] showed that optimized teleportation could be achieved by tuning entanglement, non-Gaussianity (NG) and squeezed vacuum affinity of the entangled resource state. Later developments [@tp_yang; @tp_lee; @tp_wang; @tp_agarwal] have pointed to the possibility that Einstein-Podolosky-Rosen (EPR) correlation of the resource states could be a sufficient condition for QT. However, Lee *et. al.* [@tp_lee] and Wang *et. al.* [@tp_wang] have argued that EPR correlation is not always necessary for QT - for example, the symmetrically photon added TMSV yields QT even without EPR correlation. Further, Hu *et. al.* [@tp_zubairy] have addressed the question of whether there could be other aspects of the resource states, besides EPR correlation, that are crucial for QT. In this respect they considered the Hillery-Zubairy (HZ) correlation. However, they concluded that EPR correlation is a better witness of QT than HZ correlation, i.e., there exists resource states that yield QT that are not HZ correlated but are EPR correlated. Clearly then, the question of what may be the *necessary and sufficient condition* (s) for quantum teleportation is very much open.
It may be noted that all the non-Gaussian entangled states, in earlier works [@ent_grangier; @tp_illuminati; @tp_yang; @tp_lee; @ent_benlloch; @tp_wang; @tp_agarwal; @tp_zubairy], were generated by de-Gaussifying the TMSV. Another way to generate non-Gaussian entangled states is by using a passive BS with single mode nonclassical non-Gaussian states at one of the input ports. The BS output states are guaranteed to be entangled in view of the result on the necessary and sufficient condition for BS output entanglement [@nc_bsent]. In our previous work [@bose_kumar], we have studied various aspects of BS output entanglement with a class of input single mode non-Gaussian states, viz. the states that are generated under multiple nonclassicality inducing operations (MNIO) [@qs_mnc]. In the present work, we explore such states in the context of quantum teleportation.
The non-Gaussian resource states we consider here are generated under BS action with specific input states. These input states are generated under successive application of various nonclassicality (NC)-inducing operations, viz., photon addition/subtraction and quadrature squeezing on the single mode vacuum. The specific input states are the photon added squeezed vacuum state (PAS), the photon subtracted squeezed vacuum state (PSS) and squeezed number state (SNS). The analysis in this paper hinges on our numerical results on the dependence of the teleportation fidelity on the squeeze parameter ($r$) for various values of the photon addition/subtraction number ($m$) in the case of specific input states. We analyze our numerical results on teleportation in the light of various properties of resource states that have been considered in the literature to be crucial for QT, in particular, EPR correlation [@tp_yang; @tp_lee; @tp_wang; @tp_agarwal; @tp_zubairy] and squeezed vacuum affinity [@tp_illuminati].
It was believed [@tp_yang; @tp_agarwal; @tp_zubairy] that EPR correlation could be a necessary/sufficient condition for QT. However, as argued in [@tp_lee; @tp_wang], EPR correlation is not always necessary for QT - a counterexample being that of the TMSV with photon added symmetrically in both modes that yields QT even without EPR correlation. Furthermore for a large subset of states that we have considered in this paper, we have found that EPR correlation is not even sufficient for QT. This result of ours in conjunction with the results of [@tp_lee; @tp_wang] indicates that EPR correlation is *neither necessary nor sufficient* for QT. Although it has not been stated explicitly by Dell’Anno *et. al.* [@tp_illuminati], it is implicit in their work that squeezed vacuum affinity is a necessary ingredient for QT. It so happens that for the states that they have considered, SVA is always nonzero. Resource states for which SVA is non-zero zero may in principle yield QT. In fact, some of the resource states that we have considered in this paper do have this property. However, we would like to emphasize here SVA is not non-zero, in general. It is easily seen that for any bipartite state other than that having the form $\langle n1,n1|\rho|m1,m1\rangle=\delta_{n1,n2}~\delta_{m1,m2}~ \langle n1,n2|\rho|m1,m2\rangle$, SVA vanishes. In fact, in the case of most of the states we have considered, SVA becomes trivially zero. Hence, it is clear that SVA can’t be regarded as an essential ingredient for QT.
In view of the above discussion, the question arises as to what is the property of the resource states, besides entanglement, that contributes to QT when the resource states are not EPR correlated and SVA too is not non-zero. In this paper we find that such a property is, in fact, the two-mode quadrature squeezing of the resource states as defined by Simon *et. al.* [@qs_simon]. Our numerical results on the class of non-Gaussian resource states studied in this paper indicate that two-mode quadrature squeezing is, indeed, a *necessary condition* for QT, in the sense that in all cases where the resource state is not two-mode quadrature squeezed the fidelity of teleportation is $<1/2$, i.e., there is no QT. However, two-mode quadrature squeezing is not a sufficient condition.
The paper is organized as follows. In [**Sec. II**]{} we present our numerical results on the teleportation of a coherent state with BS generated entangled non-Gaussian resource states. These states are obtained with various single mode non-Gaussian and nonclassical at one of the input ports while the other left with vacuum. In [**Sec III**]{} we presents a detailed analysis of the entanglement, NG and SVA of the BS entangled states with a view to understand teleportation. In [**Sec. IV**]{} we discuss EPR correlation of the resource states in light of the results on teleportation. In [**Sec. V**]{} we analyze our results in terms of two-mode quadrature squeezing character of the BS entangled resources. Here, we point out that two-mode quadrature squeezing is indeed a necessary condition for QT. [**Sec VI**]{} contains summary of the work.
II. Teleportation of a Coherent State using the BS Generated non-Gaussian Entangled Resources {#ii.-teleportation-of-a-coherent-state-using-the-bs-generated-non-gaussian-entangled-resources .unnumbered}
=============================================================================================
In this section, we under take a qualitative study of QT with BS generated resource states. For simplicity we consider nonclassical non-Gaussian single mode state at one of the input ports of the BS with vacuum at the other port. The specific input states that we consider are the photon added squeezed vacuum state (PAS), the photon subtracted squeezed vacuum state (PSS) and the squeezed number state (SNS). These input states are mathematically described as,
\[psi\_single\] $$\begin{aligned}
&|\psi_{\rm{pas}}\rangle =\frac{1}{\sqrt{\rm{N^{m}_{pas}}}} a^{\dagger m}S(r)|0\rangle \\
&|\psi_{\rm{pss}}\rangle =\frac{1}{\sqrt{\rm{N^{m}_{pss}}}} a^{m}S(r)|0\rangle \\
&|\psi_{\rm{sns}}\rangle =S(r)|m\rangle,\end{aligned}$$
where, $S(r)=\exp[\frac{r}{2}(a^{\dagger 2}-a^{2})]$ is the single mode squeezing operator and the quantities $\rm{N^{m}_{pas}}$ and $\rm{N^{m}_{pss}}$ are defined by the relations $\rm{N^{m}_{pas}}=m!\mu^{m}P_{m}(\mu)$, $\rm{N^{m}_{pss}}=m!\nu^{2m}\sum_{k=0}^{m} \frac{m!}{(m-k)! k!}$ $(\frac{-\mu}{2\nu})^{k}\frac{H_{k}^{2}(0)}{k!}$, $\mu=\cosh r$ and $\nu=\sinh r$. Here $P_{n}(x)$ and $H_{n}(x)$ are respectively $n^{\rm{th}}$ order Legendre and Hermite polynomials.
A passive $50:50$ BS is described by the following transformation matrix between the input and the output mode operators, $$\begin{pmatrix}
A \\
B
\end{pmatrix}=
\begin{pmatrix}
1/\sqrt{2} & 1/\sqrt{2}\\
-1/\sqrt{2} & 1/\sqrt{2}
\end{pmatrix}
\begin{pmatrix}
a \\
b
\end{pmatrix} .
\label{bs_trans}$$ where $\lbrace A,B\rbrace$ and $\lbrace a,b\rbrace$ are the output and input mode operators respectively. Since the input states are nonclassical, it is guaranteed that the corresponding BS output states will be entangled [@nc_bsent]. It is well-known that entanglement is necessary for QT. Next we analyze QT with these BS entangled resource states.
The teleportation protocol we consider is the standard Braunstein-Kimble (BK) [@TP_BK] protocol. The performance/success of the teleportation is measured in terms of the fidelity of teleportation ($F$), defined as the overlap between the unknown input state and the output state (the retrieved state), $F=Tr[\rho_{\rm{in}}\rho_{\rm_{out}}]$. The evaluation $F$ becomes particularly simple in the characteristic function (CF) description [@TPF_CF]. The CF of an $n$ mode quantum optical state $\rho$ is defined as $\chi_{\rho}(\lbrace \lambda_{i} \rbrace)=Tr[\rho D(\lbrace \lambda_{i} \rbrace)]$ where $D(\lbrace \lambda_{i} \rbrace)=\Pi_{i=1}^{n}\exp [\lambda_{i} a^{\dagger}_{i}-\lambda^{*}_{i}a_{i}]$; $a_{i}$ being the $i^{\rm{th}}$ mode operator. For any two-mode state $\rho_{\rm{AB}}$ as a resource, the fidelity of teleportation of an unknown input state $\rho_{\rm{in}}$ can be expressed as [@TPF_CF], $$F=\int \frac{d^{2}\lambda}{\pi}~ \chi_{\rm{in}}(-\lambda)~\chi_{\rm{in}}(\lambda)~\chi_{\rm{AB}}(\lambda,\lambda^{*}),
\label{def_telfid}$$ where, $\chi_{\rm{in}}(\lambda)$ and $\chi_{\rm{AB}}(\lambda,\lambda^{*})$ are the CFs of $\rho_{\rm{in}}$ and $\rho_{\rm{AB}}$ respectively. For simplicity we consider a coherent state as the unknown input state and BS generated entangled states as resource. In this case, Eqn. (\[def\_telfid\]) simplifies to, $$F=\int \frac{d^{2}\lambda}{\pi}~ e^{-\lambda^{2}}~\chi_{\rm{BS}}^{\rm{out}}(\lambda,\lambda^{*}) ,
\label{telfid_coh}$$ where, $\chi_{\rm{BS}}^{\rm{out}}(\lambda,\lambda^{*})$ corresponds to the characteristic function of the BS output state. Henceforth, we shall use Eq. \[telfid\_coh\] while discussing $F$. The maximum fidelity of teleportation of a coherent state attainable by a separable state in the BK protocol is $1/2$ [@TPF_CS]. Hence, $F> \frac{1}{2}$ indicates QT.
![(Color Online) Plot of $F$ vs $r$ for $m=0$ (black solid line), $1$ (yellow dashed line), $2$ (green dotted line), $3$ (blue dashed dotted line) and $4$ (red dashed double dotted line) with BS output states generated from the input states [**(a)**]{} PAS, [**(b)**]{} PSS and [**(c)**]{} SNS. Violet long dashed line corresponds to the maximum limit for the “classical” teleportation, i.e, $1/2$. \[fig\_tf\]](TF_BSO.eps)
In Fig \[fig\_tf\] we plot the dependence of $F$ on the squeeze parameter $r$ and the number of photon addition/subtraction $m$ in the case of non-Gaussian BS output entangled states generated from single mode input states \[Eq. (\[psi\_single\]a), (\[psi\_single\]b) and (\[psi\_single\]c)\]. As is evident from Fig. (\[fig\_tf\]), in the case of all three input states, the teleportation fidelity $F$ exhibits a rather complex, in particular non-monotonic, dependence on the state parameters $r$ and $m$.
The rest of the paper is devoted to understanding the various ramifications of the principle numerical results in Fig. \[fig\_tf\]. In the next few sections we shall assess the role of various attributes of the resource states, viz. entanglement, non-Gaussianity (NG), squeezed vacuum affinity (SVA) and EPR correlation on teleportation fidelity in respect of the results in Fig. \[fig\_tf\].
III. Attributes of the Resource States I: Entanglement, NG and SVA {#iii.-attributes-of-the-resource-states-i-entanglement-ng-and-sva .unnumbered}
==================================================================
In this section we essentially extend the analysis of Dell’Anno *et. al.* [@tp_illuminati] to BS generated resource states. It is pertinent to recall here the observation of Dell’Anno *et. al.* [@tp_illuminati], in the context of resource states generated by certain de-Gaussifications of the TMSV, that in order to achieve optimal teleportation, one has to tune values of entanglement, NG and SVA of the resource states. The purpose of this section is to verify if this observation of Dell’Anno *et. al.* is borne out in the case of BS generated resource states with $|\psi_{\rm{pas}}\rangle$, $|\psi_{\rm{pss}}\rangle$ and $|\psi_{\rm{sns}}\rangle$ at the input.
III-A. Entanglement and Teleportation Fidelity {#iii-a.-entanglement-and-teleportation-fidelity .unnumbered}
----------------------------------------------
We denote the BS generated entanglement with input $\vert\psi\rangle$ by $E_{\rm{BS}}^{|\psi\rangle}$. In Fig \[fig\_ent\], we plot the dependence of BS entanglement for different input states. The specific dependence of $E_{\rm{BS}}^{|\psi_{\rm{pas}}\rangle}$ \[Fig. \[fig\_ent\](a)\] and $E_{\rm{BS}}^{|\psi_{\rm{sns}}\rangle}$ \[Fig. \[fig\_ent\](c)\] on $r$ and $m$ have already been discussed in detail in [@bose_kumar]. Here, we reproduce the figures for $E_{\rm{BS}}^{|\psi_{\rm{pas}}\rangle}$ and $E_{\rm{BS}}^{|\psi_{\rm{sns}}\rangle}$ from our previous work [@bose_kumar] for the sake of future discussion. However, the results for $E_{\rm{BS}}^{|\psi_{\rm{pss}}\rangle}$ are new and we discuss them in some detail.
In the case of $E_{\rm{BS}}^{|\psi_{\rm{pss}}\rangle}$ \[Fig. \[fig\_ent\](b)\], we find that for small $r$ ($\leq 0.40$), odd photon subtracted states \[$m=1,3$\] are more entangled than the even photon subtracted states \[$m=2,4$\]. However, with increase in $r$, $E_{\rm{BS}}^{|\psi_{\rm{pss}}\rangle}$ for even photon subtracted states becomes higher than that for odd photon subtracted states. In general, $E_{\rm{BS}}^{|\psi_{\rm{pss}}\rangle}$, for all values of $m$, increases monotonically with increase in $r$.
As it is quite explicit from Fig. \[fig\_tf\] and Fig. \[fig\_ent\], the dependence of the fidelity of teleportation on input parameters $r$ and $m$ for different input states is very different from that of the respective BS output entanglement. In the cases of both PAS, PSS and SNS as input, BS output entanglement, for all non-zero values of $m$ and $r$, is always greater than that for the input Gaussian single mode squeezed vacuum state ($m=0$). However, in the case of teleporation, we observe that $F$ for all input states, except for the case of even PSS in the small $r$ ($\lesssim 0.60$) limit, is always smaller compared to the case of input squeezed vacuum state for all non-zero values of $m$ and $r$.
In the case of even PSS input, in the small $r$ ($\lesssim 0.30$) region, all input odd PSSs yields more entanglement at the output of BS than the input even PSSs. However, $F$ for all even PSSs at BS input is greater than all input odd PSSs. These results indicate the well-known fact that, although entanglement is necessary for QT, increase in entanglement does not always ensure increase in fidelity of teleportation.
![(Color Online) Dependence of $E_{\rm{BS}}$ on $r$ for $m=0$ (black solid line), $1$ (yellow dashed line), $2$ (green dotted line), $3$ (blue dashed dotted line) and $4$ (red dashed double dotted line) for the input states [**(a)**]{} PAS, [**(b)**]{} PSS and [**(c)**]{} SNS. \[fig\_ent\]](ENT_BSO.eps)
III-B. NG and Teleportation Fidelity {#iii-b.-ng-and-teleportation-fidelity .unnumbered}
------------------------------------
In this subsection we study how teleportation fidelity depends on the NG of the BS generated resource states. There have been several proposals for quantification of the non-Gaussian character of any state in terms of Hilbert-Schmidt distance [@ngm_hsd], relative entropy [@ngm_re], Wehrl entropy [@ngm_we] *etc*. In the present paper, we consider the Wehrl entropy based measure of NG.
For a quantum state of light, described by the density operator $\rho$, its non-Gaussianity is defined as, $$\delta(\rho)=H_{\rm{w}}(\rho^{\rm{G}})-H_{\rm{w}}(\rho),
\label{def_ng}$$ where $H_{\rm{w}}(\rho)$ \[$=-\int\frac{d^{2}z}{\pi}Q_{\rho}(z)\log Q_{\rho}(z)$\] is the Wehrl entropy of $\rho$ defined in terms of the Husimi-Kano $Q_{\rho}(z)$ \[$=\langle z|\rho|z\rangle$\] distribution. Here $\rho^{\rm{G}}$ is the Gaussian counterpart of $\rho$, the state formed with the first and the second moments equal to those of $\rho$ itself.
It is further shown by Ivan *et. al.* [@ngm_we] that, in the case of product state input at any passive linear system like BS, NG of the output state becomes equal to the sum of NG of the input states, i.e., $$\delta(\rho_{\rm{out}})=\delta(\mathscr{U}_{\rm{BS}}(\rho_{a}\otimes \rho_{b})\mathscr{U}_{\rm{BS}}^{\dagger})=\delta(\rho_{a}) + \delta(\rho_{a}),
\label{NG_BsOut}$$ where, $\mathscr{U}_{\rm{BS}}$ is the unitary operation corresponding to the evolution of the input state ($\rho_{a}\otimes \rho_{b}$) through BS. In the current work we have considered the cases where one of the input ports BS is fed with single mode non-Gaussian states $\rho_{\rm{in}}$ while the other port is left with vacuum. Since, vacuum ($|0\rangle$) is a Gaussian state with $\delta(|0\rangle)=0$, Eq. (\[NG\_BsOut\]) immediately implies that the NG of the BS generate resource states ($\rho_{\rm{out}}$) we have considered here is same as the NG of the corresponding input state $\rho_{\rm{in}}$.
![(Color Online) Plot of $\delta$ of the BS output states vs $r$ for $m=1$ (black solid line), $2$ (yellow dashed line), $3$ (green dotted line) and $4$ (blue dashed dotted line) for the input states [**(a)**]{} PAS, [**(b)**]{} PSS and [**(c)**]{} SNS. \[fig\_ng\]](NG_BSO.eps)
In Fig. \[fig\_ng\] we plot $\delta$ with $r$ for different $m$, for the BS output states generated from different input states. By $\delta^{|\psi\rangle}$ we denote the NG of the BS output state generated from the input state $|\psi\rangle$. It is clear from Fig. \[fig\_tf\] and Fig. \[fig\_ng\] that the fidelity of teleportation ($F$) does not depend monotonically on the NG ($\delta$) of the resource states. In the case of input PAS, $\delta$ increases monotonically with increase in both $m$ and $r$, while corresponding $F$ shows a non-monotonic dependence. In the case of input PSS, while the odd $m$ states are more non-Gaussian than the even $m$ states at low $r$ ($\lesssim 0.30$) limit, $F$ for input even PSSs is always higher than that for input odd PSSs. Besides, in the case of input SNS, $\delta$ shows a non-monotonic dependence on $r$ for higher values of $m$ while corresponding $F$ is a monotonically increasing function of $r$ for all values of $m$.
III-C. SVA and Teleportation Fidelity {#iii-c.-sva-and-teleportation-fidelity .unnumbered}
-------------------------------------
Dell’Anno *et. al.* [@tp_illuminati] identified yet another attribute called squeezed vacuum affinity ($\eta$) that entangled quantum optical resources must possess to achieve QT. For any bipartite entangled state $\rho_{\rm{AB}}$, $\eta$ is defined as its maximal overlap with the TMSV ($|\xi(s)\rangle$), $$\eta=\max_{s}|\langle \xi(s)|\rho|\xi(s)\rangle|^{2} .
\label{def_sva}$$
First, We have analyzed the case of even photon added/subtracted states ($m=0,2,4$) at input of the BS for which the output states have nonzero $\eta$. In Fig. \[fig\_sva\], we have shown the dependence of $\eta$ of the BS generated resource states for different input states. As evident, in the case of all input states, $\eta$ for the BS output resource states decrease with increase in $r$ for different values of $m$. The maximum SVA is obtained for $r=0$ and $m=0$ that corresponds to the vacuum state ($|0\rangle$).
However, we have noticed that $\eta$ becomes trivially zero in the case of all input states with odd photon addition/subtraction. This could be explained in the following way. The state TMSV has a symmetric expansion in number state basis $|\xi(s)\rangle=\frac{1}{\mu_{s}}\sum_{k}\tau_{s}^{k}|k,k\rangle$, where $\mu_{s}=\cosh s$ and $\tau_{s}=\tanh s$. Let’s now consider a bipartite state $\rho=\sum_{\substack{m,n\\ k,l}}~C_{m,n}^{k,l}$ $|m,n\rangle\langle k,l|$. The overlap between $|\xi(s)\rangle$ and $\rho$ is given by, $$\rm{overlap}=\langle \xi(s)|\rho|\xi(s)\rangle= \frac{1}{\mu_{s}}\sum_{\substack{m,n\\ k,l}}~C_{m,n}^{k,l}~\tau_{s}^{m+k}~\delta_{m,n}~\delta_{k,l}.
\label{overlap}$$
Evidently, in the case of a bipartite state $\rho$ for which the diagonal elements for all $m$ and $k$ vanish (e.g., $C_{m,m}^{k,k}=0$), SVA is identically zero. Note that a passive BS simply redistributes the photons in the input modes among the output modes. As a consequence, for all odd number ($m=2p+1$, $p$ is any positive integer) of photon added/subtracted states at input, BS output state have diagonal elements identically equal to zero, i.e, $C_{m,m}^{k,k}=0$ leading to $\eta=0$.
![$\eta$ for BS output states with input [**(a)**]{} PAS, [**(b)**]{} PSS and [**(c)**]{} SNS for even $m$. We consider $m=0$ (black solid line), $2$ (yellow dashed line) and $4$ (green dotted line). \[fig\_sva\]](SVA.eps)
It is quite clear from Fig. \[fig\_ent\], \[fig\_ng\] and \[fig\_sva\] for entanglement, NG and SVA respectively, that these attributes do not behave quite the same way as the teleprotation fidelity \[Fig. \[fig\_tf\]\], as far as their dependence on $r$ and $m$ is concerned. In other words, $F$ depends non-monotonically on each of these attributes. One can’t achieve a larger value of $F$ merely by increasing any one of these attributes. Thus, our results in the case of those BS generated resource states for which SVA is non-zero are consistent with those of Dell’Anno *et. al.* in the case of de-Gaussified two-mode squeezed vacuum states.
IV. Attributes of the Resource States II: EPR Correlation {#iv.-attributes-of-the-resource-states-ii-epr-correlation .unnumbered}
=========================================================
In recent years, besides entanglement, Einstein-Podolsky-Rosen (EPR) correlation [@epr_corr] of the two-mode resource states have been found to be an important ingredient in achieving QT [@tp_yang; @tp_agarwal; @tp_zubairy]. However, Lee *et. al.* [@tp_lee] and Wang *et. al.* [@tp_wang] have have pointed to examples of states that yield QT even without EPR correlation. In this section, we study this attribute in the case of BS generated non-Gaussian entangled resource states.
In the seminal paper on completeness of quantum mechanics [@EPR], Einstein, Podolsky and Rosen proposed an ideal bipartite state which is a common eigenstate of the relative position and total momentum of the subsystems. In the case of any two-mode quantum optical state one can define an EPR correlation parameter known as EPR uncertainty $\Delta_{\rm{EPR}}$ [@epr_corr] as $$\begin{aligned}
\Delta_{\rm{EPR}}&=\langle(\Delta(X_{\rm{A}}-X_{\rm{B}}))^{2}\rangle+\langle(\Delta(P_{\rm{A}}+P_{\rm{B}}))^{2}\rangle \nonumber \\
\begin{split}
&=2\big([1+\langle A^{\dagger}A\rangle+\langle B^{\dagger}B\rangle-\langle A^{\dagger}B^{\dagger}\rangle-\langle AB\rangle] \nonumber
\end{split}\\
&~~~~~~~~~~~~~~~~~~~~~~~~~[\langle A^{\dagger}\rangle-\langle B\rangle][\langle A\rangle-\langle B^{\dagger}\rangle]\big), \label{def_epr}\end{aligned}$$ where, the quadrature operators $\lbrace X_{\rm{A}},P_{\rm{A}},X_{\rm{B}},P_{\rm{B}}\rbrace$ are defined as $X_{\rm{A}}=\frac{1}{\sqrt{2}}(A+A^{\dagger})$, $P_{\rm{A}}=\frac{1}{i\sqrt{2}}(A-A^{\dagger})$, $X_{\rm{B}}=\frac{1}{\sqrt{2}}(B+B^{\dagger})$ and $P_{\rm{B}}=\frac{1}{i\sqrt{2}}(B-B^{\dagger})$. EPR uncertainty ($\Delta_{\rm{EPR}}$) being zero indicates perfect correlation between the modes. The correlated state considered by Einstein *et. al.* which is known as the EPR state [@epr_st], could be realized in terms of TMSV in the limit of infinite squeezing strength ($r\rightarrow \infty$). In the case of two-mode states with $\Delta_{\rm{EPR}}>0$, smaller the value of $\Delta_{\rm{EPR}}$ more correlated the modes are. Further, as shown by Duan *et. al.* [@epr_corr] $\Delta_{\rm{EPR}}<2$ indicates that the two-mode state is entangled.
In this section, we evaluate EPR correlation for the BS generated entangled resources for the different input non-Gaussian states we have considered in this paper. Using the transformation matrix for a $50$:$50$ BS \[Eq. (\[bs\_trans\])\], $\Delta_{\rm{EPR}}$ \[Eq. (\[def\_epr\])\] for the BS generated resource states can be expressed in terms of the input mode operators as, $$\begin{aligned}
\Delta_{\rm{EPR}}&= 2\big( 1+\langle a^{\dagger}a\rangle + \langle b^{\dagger}b\rangle - \langle a^{\dagger}\rangle\langle a\rangle - \langle b^{\dagger}\rangle\langle b\rangle \big) - \nonumber \\
&~~~~~~~~~~~~~~~~~~ \big( \langle a^{\dagger 2}\rangle + \langle a^{2}\rangle -\langle a^{\dagger}\rangle^{2} - \langle a\rangle^{2} \big) - \nonumber \\
&~~~~~~~~~~~~~~~~~~ \big( \langle b^{\dagger 2}\rangle + \langle b^{2}\rangle -\langle b^{\dagger}\rangle^{2} - \langle b\rangle^{2} \big) . \label{def_epr_bs}\end{aligned}$$
We have considered single mode nonclassical states at one of the input ports (say mode $a$) while other port (mode $b$) is left in the vacuum state. This leads to $\langle b\rangle=\langle b^{\dagger}\rangle=\langle b^{2}\rangle=\langle b^{\dagger 2}\rangle=\langle b^{\dagger}b\rangle=0$. Besides, for the input nonclassical states we have considered, $\langle a\rangle=\langle a^{\dagger}\rangle=0$ and $\langle a^{2}\rangle=\langle a^{\dagger 2}\rangle$. With these results, EPR uncertainty for the BS output states \[Eq. \[def\_epr\_bs\]\] reduces to, $$\Delta_{\rm{EPR}}=2\big( 1 + \langle a^{\dagger}a\rangle - \langle a^{2}\rangle \big) \label{def_epr_bs_in}.$$
We denote the $\Delta_{\rm{EPR}}$ in the case of input state $|\psi\rangle$ as $\Delta_{\rm{EPR}}^{|\psi\rangle}$. Using the expression of Eq. \[def\_epr\_bs\_in\], we find the analytic forms of the $\Delta_{\rm{EPR}}$, for input PAS, PSS and SNS as,
\[epr\_psi\_input\] $$\begin{aligned}
\Delta_{\rm{EPR}}^{|\psi_{\rm{pas}}\rangle}&=2\Big[ \frac{N^{m+1}_{\rm{pas}}}{N^{m}_{\rm{pas}}} + ~\frac{\mu^{2m}(m+2)!}{N^{m}_{\rm{pas}}}~\Big(\frac{\mu\nu}{2}\Big) \sum_{k=0}^{m} \begin{pmatrix}
m\\
k
\end{pmatrix} \nonumber\\
&~~~~~~~~~~~~~~~~~~~~~ \Big(\frac{-\nu}{2\mu}\Big)^{k}~\frac{H_{k}(0)H_{k+2}(0)}{(k+2)!} \Big], \\
\Delta_{\rm{EPR}}^{|\psi_{\rm{pss}}\rangle}&=2\Big[1+ \frac{N^{m+1}_{\rm{pss}}}{N^{m}_{\rm{pss}}} + ~\frac{\nu^{2m}(m+2)!}{N^{m}_{\rm{pss}}}~\Big(\frac{\mu\nu}{2}\Big)~ \sum_{k=0}^{m} \begin{pmatrix}
m\\
k
\end{pmatrix} \nonumber \\
&~~~~~~~~~~~~~~~~~~~~~~ \Big(\frac{-\mu}{2\nu}\Big)^{k}~ \frac{H_{k}(0)H_{k+2}(0)}{(k+2)!} \Big], \\
\Delta_{\rm{EPR}}^{|\psi_{\rm{sns}}\rangle}&=2\Big[1+ m(\mu-\nu)^{2}-\nu(\mu-\nu) \Big],\end{aligned}$$
where, $\mu=\cosh r$, $\nu=\sinh r$ and $H_{n}(x)$ is the $n^{\rm{th}}$ order Hermite polynomial. The expression $\begin{pmatrix}
m\\
k
\end{pmatrix}$ is the binomial coefficient and the normalization constants $N^{m}_{\rm{pas}}$ and $N^{m}_{\rm{pss}}$ are defined in Eq. (\[psi\_single\]).
In Fig. \[fig\_epr\] we have plotted $\Delta_{\rm{EPR}}$ as a function of $r$ for various values of $m$ for the BS output states, generated from the input single mode states. It is evident from a comparison of Fig. \[fig\_tf\] for the teleportation fidelity and Fig. \[fig\_epr\] for the EPR correlation in the case of BS generated resource states with either of the $|\psi_{\rm{pas}}\rangle$, $|\psi_{\rm{pss}}\rangle$ and $|\psi_{\rm{sns}}\rangle$, that there exists particular region of $r$ where resource states are EPR correlated ($\Delta_{\rm{EPR}}<2$) yet they don’t yield QT ($F>1/2$).
![(Color Online) Dependence of $\Delta_{\rm{EPR}}$ on $r$ for different $m=0$ (black solid line), $1$ (yellow dashed line), $2$ (green dotted line), $3$ (blue dashed dotted line) and $4$ (red dashed double dotted line) for input [**(a)**]{} PAS, [**(b)**]{} PSS and [**(c)**]{} SNS. The long dashed violet line correspoonds to $\Delta_{\rm{EPR}}=2.0$. \[fig\_epr\]](EPR_BSO.eps)
This leads to the conclusion that EPR correlation is not sufficient for QT. Further, as shown by Lee *et. al.* [@tp_lee] and Wang *et. al.* [@tp_wang] in the case of non-Gaussian entangled states, QT can be achieved even when the resource state is not EPR correlated, i.e. $\Delta_{\rm{EPR}}>2$. Thus, in view of our results together with the results of [@tp_lee; @tp_wang], we conclude that EPR correlation is *neither necessary nor sufficient* for QT.
Let us summarize our analysis of the various attributes of the resource states so far. The results of Sec. II-C lead us to conclude that SVA, as it is not non-zero in general and in particular in the case of BS generated resource states, it cannot be regarded. Further, when it is non-zero it is not even sufficient. Further, the results of Sec. IV make it clear that EPR correlation is neither necessary nor sufficient. In the backdrop of these results the question of what other attributes of the resource states, beside entanglement, play an essential role in QT remains open.
We propose yet another attribute of resource states that has not been considered in the literature in the context of QT, namely the $U(2)$-invariant two-mode quadrature squeezing as defined by Simon *et. al.* [@qs_simon]. In the next section we examine the role of this attribute in the context of QT.
V. Attributes of the Resource States III: Two-mode Quadrature Squeezing {#v.-attributes-of-the-resource-states-iii-two-mode-quadrature-squeezing .unnumbered}
=======================================================================
We recall here the definition of the $U(2)$-invariant two-mode quadrature squeezing as defined by Simon *et. al.* [@qs_simon]. Let’s consider an two-mode quantum state of light $\rho$ with mode annihilation operators $a_{k}$ \[$k=1,2$\] satisfying the commutation relations, $$\begin{aligned}
[a_{k},a^{\dagger}_{l}]&=\delta_{k,l}~~\rm{and} \nonumber \\
[a_{k},a_{l}]&=[a_{k}^{\dagger},a_{l}^{\dagger}]=0 .\end{aligned}$$
In terms of the quadrature components, namely $x_{k}=\frac{1}{\sqrt{2}}(a_{k}+a^{\dagger}_{k})$ and $p_{k}=\frac{1}{i\sqrt{2}}(a_{k}-a^{\dagger}_{k})$ ($k=1,2$), one can define a column vector as $\overrightarrow{R}=(x_{1},p_{1},x_{2},p_{2})^{T}$, where “$T$” stands for transposition. The variance matrix of a two-mode state $\rho$ can be written in a compact form as $V_{k,l}=\frac{1}{2}~\rm{Tr}[\rho\lbrace \Delta R_{k},\Delta R_{l}\rbrace]$, where $\Delta R_{k}=R_{k}-\rm{Tr}[\rho R_{k}]$. The state $\rho$ is said to quadrature squeezed if $$\lambda_{\rm{min}}<\frac{1}{2},
\label{def_sq}$$ where, $\lambda_{\rm{min}}$ is the least eigenvalue of its variance matrix $V$ [@qs_simon]. Accordingly, the degree of quadrature squeezing is defined as, $$f_{\rm{sq}}=\frac{1}{\sqrt{2\lambda_{\rm{min}}}} ,
\label{def_degsq}$$ and in line with Eq. \[def\_sq\], the state is said to be quadrature squeezed if $f_{\rm{sq}}>1$. Henceforth, throughout the rest of the paper, any discussion on two-mode quadrature squeezing will correspond to the $U(2)$-invariant squeezing as described by Eq. (\[def\_sq\]) and Eq. (\[def\_degsq\]) accordingly.
In the following, we shall compute two-mode quadrature squeezing for two different class of entangled resources, namely, [**(a)**]{}: states obtained by symmetrically single photon addition/subtraction on TMSV and two-mode squeezed number states considered by Dell’Anno [*et. al.*]{} [@tp_illuminati] and [**(b)**]{}: BS generated entangled states considered in this paper.
V-A. Two-mode quadrature squeezing for states considered by Dell’Anno [*et. al.*]{} {#v-a.-two-mode-quadrature-squeezing-for-states-considered-by-dellanno-et.-al. .unnumbered}
-----------------------------------------------------------------------------------
Let’s denote the states considered in [@tp_illuminati] by,
\[psi\_illuminati\] $$\begin{aligned}
|\psi_{\rm{TMSV}}\rangle&=S_{a,b}(r)|0,0\rangle, \\
|\psi_{\rm{tmpa}}\rangle&=\frac{1}{N_{+}}a^{\dagger}b^{\dagger}S_{a,b}(r)|0,0\rangle, \\
|\psi_{\rm{tmps}}\rangle&=\frac{1}{N_{-}}abS_{a,b}(r)|0,0\rangle, \\
|\psi_{\rm{tmsn}}\rangle&=S_{a,b}(r)|1,1\rangle,\end{aligned}$$
where, $S_{a,b}(r)=e^{r(a^{\dagger}b^{\dagger}-ab)}$, $N_{+}$ and $N_{-}$ are the normalization constants.
We obtain analytic expressions for the $\lambda_{\rm{min}}$ for the states \[Eq. (\[psi\_illuminati\]a), (\[psi\_illuminati\]b), (\[psi\_illuminati\]c) and (\[psi\_illuminati\]d)\] as, $$\begin{aligned}
\lambda_{\rm{min}}\big( |\psi_{\rm{TMSV}}\rangle \big)&=\frac{1}{2}-\nu(\mu-\nu), \nonumber \\
\lambda_{\rm{min}}\big( |\psi_{\rm{tmpa}}\rangle \big)&=\frac{1}{2}+(1-\tau)(1-3\tau+\tau^{2}-\tau^{3}) , \nonumber \\
\lambda_{\rm{min}}\big( |\psi_{\rm{tmps}}\rangle \big)&=\frac{1}{2}-2\tau(1-\tau)(1-\tau+\tau^{2}) , \nonumber \\
\lambda_{\rm{min}}\big( |\psi_{\rm{tmsn}}\rangle \big)&=\frac{1}{2}+(\mu-2\nu)(\mu-\nu),\end{aligned}$$ where, $\mu=\cosh r$, $\nu=\sinh r$ and $\tau=\tanh r$. The degree of squeezing for the states, then, is calculated using Eq. (\[def\_degsq\]). In Fig. \[fig\_tmsq\_illu\], we plot the dependence of the degree of two-mode quadrature squeezing ($f_{\rm{sq}}^{\rm{tm}}$), for states given in Eq. (\[psi\_illuminati\]a), (\[psi\_illuminati\]b) and (\[psi\_illuminati\]c), upon squeezing strength $r$. We also plot, in the same figure, $f_{\rm{sq}}^{\rm{tm}}$ for TMSV as reference.
![(Color Online) Plot of $f_{\rm{sq}}^{\rm{tm}}$ vs $r$ for TMSV (black solid line), $|\psi_{\rm{pa}}\rangle$ (yellow dashed line), $|\psi_{\rm{ps}}\rangle$ (green dotted line) and $|\psi_{\rm{sn}}\rangle$ (blue dashed dotted line). \[fig\_tmsq\_illu\]](TMSQ_ILLU.eps)
The degree of squeezing ($f_{\rm{sq}}^{\rm{tm}}$) for TMSV is found to be always greater than unity for all non-zero values of $r$. With increase in $r$, $f_{\rm{sq}}^{\rm{tm}}$ increases monotonically. In the case of $|\psi_{\rm{pa}}\rangle$, we notice that the state shows two-mode squeezing ($f_{\rm{sq}}^{\rm{tm}}>1.0$) beyond $r\sim 0.30$. However, it leads to quantum teleportation for higher $r$ values [@tp_illuminati]. In the case of $|\psi_{\rm{tmps}}\rangle$, we observe the presence of two-mode squeezing for all values of $r$ which falls in line with the curve for corresponding teleportation fidelity [@tp_illuminati]. It is worth noting that for a small squeeze parameter ($r\lesssim 0.65$), photon subtracted TMSV is more two-mode quadrature squeezed than the TMSV; however, for higher $r$ ($\gtrsim 0.70$) TMSV becomes more squeezed. In comparison with the specific curve for fidelity of teleportation [@tp_illuminati], two-mode quadrature squeezing appears to be necessary for QT; however, not sufficient. In the case of $|\psi_{\rm{tmsn}}\rangle$ we find the dependence of $f_{\rm{sq}}^{\rm{tm}}$ on $r$ very similar to the case of $|\psi_{\rm{tmpa}}\rangle$.
V-B. Two-mode quadrature squeezing for the BS generated entangled states with specific input states {#v-b.-two-mode-quadrature-squeezing-for-the-bs-generated-entangled-states-with-specific-input-states .unnumbered}
---------------------------------------------------------------------------------------------------
Using the relation between variance matrices of the input and output state of a BS, it is easy to show (Appendix A) that the $\lambda_{\rm{min}}$ for the BS output states is given by, $\lambda_{\rm{min}}=\min [1/2,\Delta Q]$, where, $\Delta Q$ is the value of the uncertainty of the squeezed quadrature of the input state. We denote the the degree of squeezing for the BS output states as $f_{\rm{sq}}^{\rm{bs}}$.
In Fig. \[fig\_tmsq\_bs\] we show the dependence of $f_{\rm{sq}}^{\rm{bs}}$ on $r$ for the BS output two-mode states generated from input PAS, PSS and SNS. In the case of input PAS \[Fig. \[fig\_tmsq\_bs\](a)\], $f_{\rm{sq}}^{\rm{bs}}$, for all $m\geq 1$, becomes greater than unity beyond a moderate squeezing strength ($r\gtrsim 0.60$). However, these states yield QT ($F>1/2$) for higher $r$. In comparison to the results on $F$ \[Fig. \[fig\_tf\](a)\], it explains the absence of QT below $r\sim 0.60$.
In the case of input PSS, all the even photon subtracted states \[$m=2$, $4$\] as well as no photon subtracted state \[$m=0$\] possess two-mode quadrature squeezing ($f_{\rm{sq}}^{\rm{bs}}>1.0$) \[Fig. \[fig\_tmsq\_bs\](b)\] for all values of $r$. However, all the odd photon subtracted states attain $f_{\rm{sq}}^{\rm{bs}}>1.0$ for higher values of $r$. In comparison to the corresponding results on $F$ \[Fig. \[fig\_tf\](b)\], it is clear that the states, we consider here, yield quantum teleportation provided they possess two-mode quadrature squeezing.
In the case of input SNS, we observe that $f_{\rm{sq}}^{\rm{bs}}$ \[Fig. \[fig\_tmsq\_bs\](c)\] for $m\neq 0$ becomes greater than unity for high values of $r$. The threshold value of $r$ for two-mode squeezing ($f_{\rm{sq}}^{\rm{bs}}>1.0$) increases with the increase in $m$. In the case of corresponding results on $F$ \[Fig. \[fig\_tf\](c)\] also, we notice that for $m\neq 0$ states quantum teleportation ($F>1/2$) is attained for higher values of $r$.
It is noteworthy that all the BS output resource states that we have considered attain two-mode quadrature squeezing, depending upon the value of $m$, beyond a certain value of squeeze parameter $r$. This could be explained in the following manner. Using the relation between the variance of matrix of the state at input of the BS and that of the output state, it can be easily shown (Appendix) that the output state is quadrature squeezed if and only if the input single mode is quadrature squeezed. Since, the input single mode states become quadrature squeezed ($f_{\rm{sq}}>1$) beyond a moderate value of squeeze parameter $r$, depending upon the value of $m$, the same is reflected in the quadrature squeezing of the output states.
![(Color Online) Plot of $f_{\rm{sq}}^{\rm{bs}}$ vs $r$ for different $m=0$ (black solid line), $1$ (yellow dashed line), $2$ (green dotted line), $3$ (blue dashed dotted line) and $4$ (red dashed double dotted line) for the BS output states with single mode [**(a)**]{} PAS, [**(b)**]{} PSS and [**(c)**]{} SNS at input. \[fig\_tmsq\_bs\]](TMSQ_BSO.eps)
A close examination of the numerical results on $f_{\rm{sq}}^{\rm{tm}}$ for the states considered by Dell’Anno [*et. al.*]{} [@tp_illuminati] as well as the BS generated states that we have considered in this work, indicates that [*two-mode quadrature squeezing is necessary for QT*]{}. However, two-mode quadrature squeezing is not a sufficient condition.
In this connection it is instructive to examine if two-mode quadrature squeezing is necessary for QT in the case of Gaussian resource states. In fact, it turns out (as we show in the next subsection) that in the case of symmetric Gaussian states two-mode quadrature squeezing is indeed necessary for QT.
V-C. Quantum Teleportation with symmetric Gaussian resource states {#v-c.-quantum-teleportation-with-symmetric-gaussian-resource-states .unnumbered}
------------------------------------------------------------------
Let’s consider a symmetric Gaussian state with two-mode variance matrix $V$ of the following specific form, $$V=\begin{bmatrix}
\eta& 0& c& 0\\
0& \eta& 0& -c\\
c& 0& \eta& 0\\
0& -c& 0& \eta
\end{bmatrix} .
\label{vm_gen}$$
The necessary condition on $V$ (set by the uncertainty relation) to be a bona fide quantum variance matrix is that its symplectic eigenvalues ($\kappa_{i}$, $i=1,2$) (elements in the Williamson’s diagonal form) must be no less than $1/2$, i.e. $\kappa_{i}\geq 1/2$. These, symplectic eigenvalues are obtained as the ordinary eigenvalues of $|iV\Omega|$, where, $$\Omega=\begin{bmatrix}
J& 0\\
0& J
\end{bmatrix}~;~J=\begin{bmatrix}
0& 1\\
-1& 0
\end{bmatrix} .
\label{def_symplectic_metric}$$
The condition $\kappa_{i}\geq 1/2$, for the variance matrix $V$ \[Eq. \[vm\_gen\]\] leads to, $$\sqrt{(\eta+c)(\eta-c)}\geq 1/2
\label{cond_vm_bf} .$$
According to the condition of two-mode quadrature squeezing as defined by Simon [*et. al.*]{} [@qs_simon], the variance matrix $V$ is said to be quadrature squeezed if its “[*least eigenvalue*]{}” becomes less than $1/2$. For the variance matrix $V$ given in Eq. \[vm\_gen\], its eigenvalues are $l=\eta\pm c$. Evidently, the condition of two-mode quadrature squeezing for $V$ yields $$l_{\rm{min}}=\eta-c<1/2 .
\label{cond_vm_qs}$$
Let’s now look at the teleportation of the coherent state with the Gaussian resource states. For any Gaussian state with variance matrix $V=\begin{bmatrix}
A& C\\
C^{T}& B
\end{bmatrix}$, where, $A$, $C$ and $B$ are $2\times 2$ matrices, the fidelity of teleportation of a coherent state (Eq. \[def\_telfid\]) becomes [@Pirandola_LasPhys], $$F=\frac{1}{\sqrt{\det[\mathscr{M}]}},
\label{def_tf_vm}$$ where, $\mathscr{M}=A-\lbrace \sigma_{z},C\rbrace+\sigma_{z}B\sigma_{z}+I$. $\sigma_{z}$ is the Pauli spin matrix, $\sigma_{z}=\begin{bmatrix}
1& 0\\
0& -1\\
\end{bmatrix}$.
For the symmetric Gaussian states with variance matrix given in Eq. \[vm\_gen\], we have $B=A=diag(\eta,\eta)$ and $C=C^{\rm{T}}=diag(c,-c)$. This leads to $\mathscr{M}=diag(1+2\overline{\eta-c},1+2\overline{\eta-c})$ with $\det [\mathscr{M}]=(1+2\overline{\eta -c})^{2}$. Now the condition of QT, i.e., $F>1/2$, leads to, $$\sqrt{\det[\mathscr{M}]}\leq 2 \Rightarrow \eta-c \leq 1/2 .
\label{cond_vm_qt}$$
Evidently, the condition for quantum teleportation (Eq. \[cond\_vm\_qt\]) and and the condition for quadrature squeezing (Eq. \[cond\_vm\_qs\]), are identical. This implies that quadrature squeezing is a necessary condition for QT with symmetric Gaussian resouce states. Further it also implies that, in this case, it is also sufficient. However note that we have considered teleportation of a coherent state via Braunstein-Kimble protocol. We do not expect that two-mode quadrature squeezing would be sufficient in the case of teleportation of a general single-mode state and with general Gaussian resource states.
In view of the result for the symmetric Gaussian states obtained above, that two-mode quadrature squeezing is necessary for QT, it is plausible that in the case of non-Gaussian entangled resource states as well [*two-mode quadrature squeezing is necessary for QT*]{}.
VI. Conclusion {#vi.-conclusion .unnumbered}
==============
In summary, we have studied QT with a class of non-Gaussian resource states. These resource states are generated by a passive BS with specific single mode non-Gaussian states at one of the input ports, viz., the photon added squeezed vacuum state, the photon subtracted squeezed vacuum state and squeezed number state. In contrast, the non-Gaussian resource states studied in the literature in the context of QT normally are those generated from various de-Gaussifications of the TMSV. The analysis in this paper hinges on our numerical results on the dependence of the teleportation fidelity on the squeeze parameter ($r$) for various values of the photon addition/subtraction ($m$), in the case of different resource states.
Firstly, we have extended the analysis of Dell’Anno *et. al.* [@tp_illuminati] to the BS generated non-Gaussian resource states and studied in detail the dependence of QT on entanglement, NG and SVA. While Dell’Anno *et. al.* used the Hilbert-Schmidt distance based NG measure, we instead have used the Wehrl entropy based measure. Consistent with the results of Dell’Anno *et. al.*, we have found that the teleportation fidelity doesn’t depend monotonically on either of these properties but one has to tune the values of these to achieve optimal QT fidelity.
Our next focus has been to identify what all attributes of the resource states, apart from entanglement, are necessary and/or sufficient for QT. To this end, we have studied SVA and EPR correlation which have been considered in the literature as being critical for QT. However, we have found that SVA is, in general, not non-zero for all resource states. In particular, it turns out to be zero in most of the cases for the class of states that we have considered. On the other hand, while the fact that EPR correlation is not necessary for QT has been known in the literature [@tp_wang; @tp_lee], numerical results on our class of states indicate that it is not also sufficient.
We have proposed that two-mode $U(2)$-invariant squeezing [@qs_simon] is an appropriate attribute to consider in this context. Our numerical results on both the class of BS generated non-Gaussian resource states as well as other de-Gaussified TMSV lead us to the conclusion that $U(2)$-invariant squeezing is, in fact, a *necessary condition* that all resource states must satisfy. To argue that this is a plausible conclusion we have given an analytical proof, in the case of symmetric Gaussian states, that $U(2)$-invariant two-mode quadrature squeezing is indeed *necessary* for QT. It turns out that in the special case of QT of a coherent state via Braunstein-Kimble protocol, and with symmetric Gaussian resouce states, two-mode quadrature squeezing is also sufficient. It would be nice to give an analytical proof that two-mode quadrature squeezing is necessary for all Gaussian entangled resource states. We shall return to this question elsewhere.
Appendix: Least Eigenvalue of the Variance Matrix of BS Output States Generated from Single Mode Input States {#appendix-least-eigenvalue-of-the-variance-matrix-of-bs-output-states-generated-from-single-mode-input-states .unnumbered}
=============================================================================================================
Here, we discuss the least eigenvalue of the two-mode variance matrix of the BS output states generated from a single mode nonclassical state at one of the input ports while the other port is left with vacuum. Let’s consider the column vectors $R_{\rm{in}}$ and $R_{\rm{out}}$ for the input and output quadrature operators as $$R_{\rm{in}}=\begin{bmatrix}
x_{a} \\
p_{a} \\
x_{b} \\
p_{b} \\
\end{bmatrix},~~
R_{\rm{out}}=\begin{bmatrix}
x_{A} \\
p_{A} \\
x_{B} \\
p_{B} \\
\end{bmatrix} .$$ The quadrature operators $x_{i}$, $p_{i}$ corresponding to annihilation and creation operators $a_{i}, a^{\dagger}_{i}$ are defined as $x_{i}=\frac{1}{\sqrt{2}}(a_{i}+a^{\dagger}_{i})$ and and $p_{i}=\frac{1}{i\sqrt{2}}(a_{i}-a^{\dagger}_{i})$.
Using the transformation matrix between input and output mode operators Eq. (\[bs\_trans\]) for a $50:50$ BS, it is easy to show that $R_{\rm{out}}$ is related to $R_{\rm{in}}$ by the transformation, $R_{\rm{out}}=S R_{\rm{in}}$, i.e., $$\begin{bmatrix}
x_{A} \\
p_{A} \\
x_{B} \\
p_{B} \\
\end{bmatrix}=\begin{bmatrix}
\frac{1}{\sqrt{2}}& 0& \frac{1}{\sqrt{2}}& 0\\
0& \frac{1}{\sqrt{2}}& 0& \frac{1}{\sqrt{2}}\\
-\frac{1}{\sqrt{2}}& 0& \frac{1}{\sqrt{2}}& 0\\
0& -\frac{1}{\sqrt{2}}& 0& \frac{1}{\sqrt{2}}
\end{bmatrix}
\begin{bmatrix}
x_{a} \\
p_{a} \\
x_{b} \\
p_{b} \\
\end{bmatrix} .
\label{tran_bs_coor}$$ It is well known that, under the linear transformation $S$, the input variance matrix $V_{\rm{in}}$ transforms as $SV_{\rm{in}}S^{T}$. In this paper we consider the class of states for which the input variance matrix $V_{\rm{in}}$ is given by, $$V_{\rm{in}}=\begin{bmatrix}
\eta_{a}& 0& 0& 0\\
0& \zeta_{a}& 0& 0\\
0& 0& \frac{1}{2}& 0\\
0& 0& 0& \frac{1}{2}
\end{bmatrix} .$$
Using the transformation $S$ given in Eq. (\[tran\_bs\_coor\]) we get the output variance matrix as, $$V_{\rm{out}}=SV_{\rm{in}}S^{T}=\begin{bmatrix}
\frac{\eta_{a}+1/2}{2}& 0& \frac{-\eta_{a}+1/2}{2}& 0\\
0& \frac{\zeta_{a}+1/2}{2}& 0& \frac{-\zeta_{a}+1/2}{2}\\
\frac{-\eta_{a}+1/2}{2}& 0& \frac{\eta_{a}+1/2}{2}& 0\\
0& \frac{-\zeta_{a}+1/2}{2}& 0& \frac{\zeta_{a}+1/2}{2}
\end{bmatrix} .$$
It can be easily shown that the least eigenvalue of $V_{\rm{out}}$ is given by $\lambda_{\rm{min}}=\min[1/2,\eta_{a},\zeta_{a}]$. To be specific, let’s assume $\eta_{a}\geq\zeta_{a}$. In this case the minimum eigenvalue will be given by $\lambda_{\rm{min}}=\min[1/2,\zeta_{a}]$.
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|
---
abstract: 'The Ultra Compact Dwarf (UCD) galaxies recently discovered in the Fornax and Virgo clusters exhibit structural similarity to the dense nuclei of nucleated dEs indicating that the progenitor galaxy and its halo have been entirely tidally disrupted. Using high resolution $N$-body simulations with up to ten million particles we investigate the evolution and tidal stripping of substructure halos orbiting within a host potential. We find that complete disruption of satellite halos modeled following the NFW density profile occurs only for very low values of concentration in disagreement with the theoretical predictions of CDM models. This discrepancy is further exacerbated when we include the effect of baryons since disk formation increases the central density.'
author:
- 'Stelios Kazantzidis, Ben Moore & Lucio Mayer'
title: 'Galaxies and Overmerging: What Does it Take to Destroy a Satellite Galaxy?'
---
\#1[[*\#1*]{}]{} \#1[[*\#1*]{}]{} = @scaling[.95]{} \#1[@scaling[\#1]{}]{} \#1[=@scaling]{} \#1
1.25in .125in .25in
Introduction
============
High resolution numerical simulations and sophisticated semi-analytic modeling have significantly improved our understanding of the properties and the evolution of cold dark matter (CDM) substructure. It has been demonstrated (Moore [[et al. ]{}]{}1998, Colpi, Mayer, & Governato 1999; Moore [[et al. ]{}]{}1999; Klypin [[et al. ]{}]{}1999; Taffoni [[et al. ]{}]{}2003) that tidal distruption is very inefficient for these low mass subhalos. Satellites represent earlier generations of the merging hierarchy and are typically denser and more concentrated than their more massive hosts. While gravitational tides serve to unbind mass associated with these subhalos, it is still not clear whether or not complete disruption of substructure halos can take place in a CDM potential.
Direct evidence of tidal disruption processes operating effectively within galaxy clusters has recently been provided by the discovery of a new population of subluminous and extremely compact objects in the Fornax (Drinkwater [[et al. ]{}]{}2000; Phillipps [[et al. ]{}]{}2001) and Virgo (Drinkwater, private communication) clusters. These Ultra Compact Dwarf (UCD) galaxies are dynamically distinct systems with intrinsic sizes $\lsim 100$ pc, and properties, including velocity dispersions, absolute magnitudes and mass-to-light (M/L) ratios, considerably higher than any normal globular cluster (Drinkwater [[et al. ]{}]{}2003). There is accumulating evidence that the UCDs constitute the remnant nuclei of dwarf galaxies whose extended stellar component and DM halo have been both entirely disrupted by gravitational interactions within their host cluster. Their derived M/L ratios range from 2 to 4 and are consistent with those of stellar populations suggesting that these systems contain no dark matter. Other factors consistent with the interpretation that the UCDs are the products of tidal disruption processes include their strong structural similarity to the dense nuclei of nucleated dwarf ellipticals (dEs) and the lack of extended stellar envelopes around them in photographic images (Phillipps [[et al. ]{}]{}2001).
Our goal is not only to investigate how probable complete disruption of substructure halos at the typical distances of known UCDs (within about 30% of the cluster virial radius) is, but also to place constraints on the structure of DM halos and examine whether the existence of the UCDs is consistent with the theoretical predictions of CDM models.
Numerical Simulations
=====================
We study the evolution of dwarf satellites comparable in mass to typical cluster nucleated dEs in the external potential of the Fornax cluster. The NFW density profile (Navarro, Frenk, & White 1996) is used for both the live satellites and the spherically symmetric static host potential. The latter represents a cluster halo with virial mass $M_{\rm prim}=0.5 \times10^{14} \,h^{-1} {{\rm M_\odot}}$ and $c_{\rm prim}=8.5$ (hereafter, $h=0.5$). The satellite’s virial mass is $M_{\rm sat}=2 \times10^{10} \,h^{-1} {{\rm M_\odot}}$. Our simulations neglect the effects of dynamical friction and the response of the primary to the presence of the satellite. Considering the vast difference in the mass and size of the two main systems we do not expect our results to be affected by this choice. In this investigation all satellite models are Monte Carlo realizations of the exact phase-space distribution function under the assumptions of spherical symmetry and isotropic velocity dispersion tensors, $f=f(E)$. Kazantzidis, Magorrian, & Moore (2003) have explicitly demonstrated that the choice of initial conditions is vital for studies like the present. Out of equilibrium initial conditions may artificially accelerate the mass loss of the model satellites by decreasing their central density and changing the character of their orbital anisotropy.
The current positions of the UCDs which give an indication of the apocenter of their orbits coupled with theoretical studies of halo orbital properties, will be used to constrain the orbital parameters of the satellites. In particular, we shall adopt an apocenter radius equal to $r_{\rm apo}=1.77\ R_{\rm s}$, where $R_{\rm s}$ is the scale radius of the host halo and $(r_{\rm apo}/r_{\rm peri})=\,$(5:1) close to the median ratio of apocentric to pericentric radii found in cosmological simulations (Ghigna [[et al. ]{}]{}1998). The pericenter of the orbit is $50$ kpc in all the simulations presented here. We evolve our models using PKDGRAV (Stadel 2001) for 10 Gyr. In all our runs, the total energy was conserved to better than 0.1%. In Figure 1 (left panel) we perform a quantitative comparison of the evolution of the bound satellite mass for three different mass resolutions. We use the group finder SKID (Stadel 2001) to identify the remaining bound mass. We define complete disruption of a satellite system when we do not find any gravitationally bound structure at a scale of $2 \varepsilon_s$ or larger, where $\varepsilon_s$ is the gravitational softening for our runs. The satellite halo has a $c_{\rm sat}=5$ and is simulated with $N=10^5$ (filled squares), $N=10^6$ (open circles), and $N=10^7$ (filled circles) particles. The evolution of the bound mass is plotted up to the point where complete disruption of the satellite system occurs. The halo resolved with just $10^5$ particles fully disrupts after just two orbits, but this is clearly a resolution effect since the same halos resolved with more particles survive significantly longer. Convergence is achieved when we adopt a mass resolution of more than $10^6$ particles per halo.
In order to avoid an unecessary computational cost, we adopt a mass resolution of $N=10^6$ in the following runs. We set the gravitational softening to $\epsilon=50$ pc, hence, our force resolution being equal to $2 \varepsilon$ corresponds roughly to the upper limit inferred for the UCD size. At this force resolution only completely disrupted systems might be suitable UCD progenitors. In the right panel of Figure 1 we demonstrate that the satellite with $c_{\rm sat}=9$ defines the upper limit of the concentration parameter for disruption at the timescales of interest. Note that the thick horizontal solid line denotes the region below which the halo may resemble that of a UCD galaxy. Satellites with lower values of concentration disrupt earlier and therefore could be associated with UCD progenitors. These values are significantly lower than those measured in cosmological simulations for halos in these mass scales (Bullock [[et al. ]{}]{}2001).
The Effect of Baryons on Satellite Survival
===========================================
The theoretical predictions of CDM models for the concentration values are only for precollapse halos. A real galaxy would always have an “effective concentration”, $c_{\rm eff}$, higher than that of a pure DM halo system. The baryons steepen the inner density profile by both adding mass to the center and causing the halo to adiabatically contract responding to their infall, increasing considerably the resilience of satellites to tidal stripping. In Figure 2 we present the rotation curves of two disk models constructed using the semi-analytical modeling of Mo, Mao, & White (1998) together with the rotation curve of a pure DM halo with concentration equal to the upper limit of disruption ($c=9$) for the adopted standard orbit. Both models have typical values of the disk mass fraction $m_{\rm d}=0.05$ and halo spin parameter $\lambda=0.05$. In the left panel is illustrated that a typical disk greatly increases the effective concentration of a pure DM halo making the disruption of a satellite galaxy significantly more problematic. In order to achieve an effective concentration of $c_{\rm eff} \le 9$ we need to start from $c_{\rm halo} < 3$ for the same typical disk parameters (right panel). This value is more than 4 $\sigma$ lower than the theoretical predictions for the mass range of our satellites.
We are unable to explain the origin of the UCDs within the CDM models. We shall explore further the dependence of satellite disruption on their orbital properties, central density slopes, and host halo structure to address this issue in more detail (Kazantzidis, Mayer & Moore, in preparation).
Bullock, J. S., [[et al. ]{}]{}2001, , 321, 559 Colpi, M., Mayer, L., & Governato, F. 1999, , 525, 720 Drinkwater, M. J., Jones, J. B., Gregg, M. D., & Phillipps, S. 2000, PASA, 17, 227 Drinkwater, M. J., [[et al. ]{}]{}2003, Nature, 423, 519 Ghigna, S., [[et al. ]{}]{}1998, , 300, 146 Kazantzidis, S., Magorrian, J., & Moore, B. 2003, ApJ submitted Klypin, A., Kravtsov, A. V., Valenzuela, O., & Prada, F. 1999, , 522, 82 Mo, H. J., Mao, S., & White, D. M. 1998, , 295, 319 Moore, B., Governato, F., Quinn, T., Stadel, J., & Lake, G. 1998, , 499, L5 Moore, B., [[et al. ]{}]{}1999, , 524, L19 Stadel, J. 2001, PhD thesis, U.Washington. Taffoni, G., Mayer, L., Colpi, M., & Governato, F. 2003, , 341, 434 Phillipps, S., Drinkwater, M. J., Gregg, M. D., & Jones, J. B. 2001, , 560, 201
|
---
abstract: 'We investigate the formation of star clusters in an unbound GMC, where the supporting kinetic energy is twice as large as the cloud’s self-gravity. This cloud manages to form a series of star clusters and disperse, all within roughly 2 crossing times (10 Myr), supporting recent claims that star formation is a rapid process. Simple assumptions about the nature of the star formation occurring in the clusters allows us to place an estimate for the star formation efficiency at about 5 to 10%, consistent with observations. We also propose that unbound clouds can act as a mechanism for forming OB associations. The clusters that form in the cloud behave as OB subgroups. These clusters are naturally expanding from one another due to unbound nature of the flows that create them. The properties of the cloud we present here are are consistent with those of classic OB associations.'
author:
- |
Paul C. Clark$^1$ [^1] Ian A. Bonnell$^1$, Hans Zinnecker$^2$ & Matthew R. Bate$^3$\
$^1$ School of Physics & Astronomy, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS.\
$^2$ Astrophysikalisches Institut Potsdam, An der Sternwarte 16, D-14482 Potsdam, Germany\
$^3$ School of Physics, University of Exeter, Stocker Road, Exeter, EX4 4QL
bibliography:
- 'GMC.bib'
title: |
Star formation in unbound giant molecular clouds:\
the origin of OB associations?
---
Molecular clouds, turbulence, IMF
Introduction {#intro}
============
Practically all star formation is thought to occur in clusters that are embedded in giant molecular clouds (GMCs) [@Ladas2003]. This suggests that a clustering environment, where multiple objects compete for a common gas reservoir [@Zinnecker1982; @Larson1992], plays an important role in early stages of protostellar evolution, such as dictating the form of the stellar initial mass function (IMF) (@Bonnelletal1997; @Bonnelletal2001b). Furthermore, @Elmegreen2000 has collected observational evidence suggesting that star formation is a rapid process, occurring on roughly the crossing time of the region at a variety of scales. Not only does he propose that the star formation in a typical GMC occurs within 4 Myr (approximately the crossing time for standard GMC) but that the cloud’s dispersal occurs within a few crossing times, or $\la$ 10 Myr.
The combined implication of these observations is that star formation occurs quickly and in groups and that the sites of star formation disperse quickly. Our proposal in this paper is that this is possible if GMCs are dynamically unbound objects, with the internal turbulent energy greater than that of the cloud’s self-gravity. This follows from the work of @Semadenietal1995 who showed that transient (unbound) GMC sized objects can be formed from flows in the ISM. We also find that unbound GMCs may provide a natural mechanism for the creation of OB associations, a notion first suggested by @Ambart1958.
In the rest of this first section we discuss the ideas behind rapid star formation and the dynamical state of GMCs. We also include in this section a discussion of OB associations. In section [\[setup\]]{} we describe the details of the simulation and section [\[evol\]]{} follows the general evolution of the GMC. In section [\[SFE\]]{} we give estimates of the star formation efficiency in the GMC based on some simple assumptions. In section [\[OB\]]{} we highlight the similarities between the simulation and the general structure in an OB associations. A summary of the paper’s main conclusions can be found in section [\[finish\]]{}.
GMC Lifetimes and Rapid Star Formation {#GMCS}
--------------------------------------
Until the last decade or so GMCs were generally believed to be long-lived structures, with some estimates of ages reaching as high as $10^{8}$ Myr [@Solomonetal1979; @Scovilleetal1979; @ScovilleHersh1979]. It was generally believed that the chemistry of turning atomic species into molecules would require millions of years before an object like a GMC would be detectable via its CO abundance [@Jura1975]. One also had the problem that the CO mass in the galaxy, coupled with estimates of the star formation rate, suggested that GMCs had to live for tens of millions of years if the star formation efficiency was to remain at the observed level of a few percent [@ZuckermanEvans1974; @ZuckermanPalmer1974].
Recent observations of embedded clusters tend to suggest that the whole process of star formation, including GMC formation and dispersal, occurs on roughly the crossing time for the region [@Elmegreen2000]. Not only do most molecular clouds in the solar neighbourhood contain signs of star formation in the form of clusters, but the age determination of these clusters suggests they are very young, typically less than 10Myr [@Hartmann2000]. This suggests that star formation occurs quickly in GMCs after their formation. The fact that clusters with ages greater than 5 Myr are seldom associated with molecular gas, suggests that clouds disperse quickly [@Leisawizetal1989].
In the original cloud lifetime proposition, it was assumed that all of the CO observed in the galaxy was associated with molecular hydrogen involved in star formation. We now realise that the vast majority of the gas that comprises a GMC is never involved in the star formation process. In fact the star formation efficiency in GMCs is only a few percent. The reason behind this lies with the fact that little of the cloud is actually dense and bound enough to turn into stars in the cloud’s lifetime [@Padoan1995; @Hartmann1998; @Zinnecker2002]. Also, if GMCs are short-lived features then there is little time for the more tenuous parts of the cloud to get involved in the star formation via accretion.
The old GMC model also required the cloud to be supported and in virial equilibrium, since that would permit them to remain as coherent structures for as long as was necessary. This support pressure had to be in the form of non-thermal kinetic energy, such as turbulence [@Larson1981], since the thermal energy component of these clouds is typically very small. To counteract the gravity on the large scales however requires motions which are supersonic, and it is known that these quickly damp in shocks (@Maclowetal1998; @Stoneetal1998), even in the presence of magnetic fields. Thus the bound GMC model requires some method of continually driving the turbulence on the large scale. These driving mechanisms are not necessary in the short cloud lifetime model, and there also is no need to assume that the clouds are in virial equilibrium. GMCs can therefore exist in a variety of dynamical states.
@Heyeretal2001 have examined the stability of molecular clouds in the outer galaxy and come to the conclusion that most clouds are indeed globally unbound by their internal motions. They also point out the difficulty in producing mass estimates (which a great number of papers on the subject of GMCs pass over) and note that even mass estimates determined via CO measurements (both $^{12}$CO and $^{13}$CO) assumes at some stage the cloud is bound. Although they do find the clouds approach dynamical stability at large masses ($>10^{5}$), there is still considerable scatter in the data, suggesting that there is no typical dynamic state for GMCs.
@Pringleetal2001 have shown that it might also be possible to build GMCs by accumulating very low density hydrogen gas, already in a molecular state. Their study came in response to the ideas presented by @Elmegreen2000, in an attempt to provide a new mechanism for GMC formation that can occur quickly. They point out that it is quite possible that a large fraction of the interstellar medium may be in molecular form, but either simply too low a density to be detectable by current methods or too far away from illuminating sources. The GMCs are then formed from large scale shocks, from spiral arm passage or feedback from high mass stars, such as winds and supernovae. This cloud formation can occur within a few million years. @Pringleetal2001 also point out that GMCs are probably not in virial equilibrium, and note that their wind-swept appearance suggests that they are anything but.
The simulation that we present here draws on the above studies for motivation. We assume that large scale flows are able to create an unbound GMC in a few millions years. Instead of being contained by external forces (e.g. @Heyeretal2001), we assume that the cloud is free to expand into the ISM. Thus the flows that created the cloud are assumed to have been used up in its formation. Since the cloud is assumed to be short lived and not quasi-static, there is no need for the internal turbulent energy, which will dissipate on the crossing time, to be replenished [@Paredesetal1999; @Elmegreen2000].
The Origin of OB Associations {#originOB}
-----------------------------
OB associations are historically identified simply as extended groups of OB stars, having diameters of tens of parsecs [@Ambart1955]. Furthermore they are rather more diffuse than open clusters, with the mass density of OB type stars at $\sim 0.1$ pc$^{-3}$ [@Blaauw1964; @Ambart1955; @Garmany1994; @Ladas2003]. It was found that these associations contain considerable substructure which are referred to as ‘OB subgoups’ [@Blaauw1964]. These subgroups are unbound from one another as was deduced from their expansion about the centre of the region [@Blaauw1952]. Some regions or ‘subgroups’ are shown to be associated with molecular gas. In general these regions are not coeval but can exhibit a spread of ages between the subgroup population as large as 10 Myr [@Blaauw1964]. The fact that OB associations are very young, with some of the subgroups possessing ages of the order of a millions years, suggests that unbound nature of the subgroups from one another is primordial.
The relationship between OB associations and other types of clusters found in the galactic disc, such as open clusters and embedded clusters, is still rather unclear. The OB associations do however have a classic theory regarding their formation. @ElmegreenLada1977 proposed that OB associations form via triggering, prompted by the ionised regions produced by previous generations of OB stars. In this manner, the star formation is self propagating, with one generations of OB stars triggering the formation of the next. Since the shocked layer in which the new group of OB stars forms is moving away from the older OB stars, at a few kms$^{-1}$, the new group is unbound from its parent group. The region then naturally has the dynamics of the observed OB groups. Motivation came from observations of stars forming at the boundaries of molecular clouds and HII, such as NGC7538, M17 and M8 (@Habingetal1972; @Ladaetal1976).
The issue is complicated however when one considers the detailed stellar population of OB associations [@Garmany1994; @Brown2001]. In the self propagating model, OB type stars form in the shocked layers where conditions are naturally more suited to forming high mass stars. Low mass stars form spontaneously in the rest of the cloud. Thus the model assumes a two step formation process whereby low mass stars and high mass stars are formed by different mechanisms and in physically separated locations. The IMF of the OB associations however do not exhibit this feature and generally possess the standard field star IMF, at least within the Salpeter range (e.g. Sco OB2, @deGeus1992; @PreibischZinnecker1999). Since up to nearly 90% of star formation is thought to occur in embedded clusters, with a field star IMF and primordial mass segregation (for a discussion see @Ladas2003), it may be that the formation of OB associations has more in common with standard clustered star formation.
@BBV2003 and @BVB2004 have modelled cluster formation in a turbulently supported cloud. They modelled a 1000molecular cloud that was initially supported against collapse by a turbulent velocity field. It was found that the dissipation of the large scale supersonic flows produced a number of distinct subclusters. Each subcluster contains at the core a massive star. The subclusters were mass segregated and each had a protostellar population consistent with that of the observed field star IMF, both of which are the result of competitive accretion. Since the cloud was initially bound, even more so after the dissipation of the turbulent energy, the whole system of subclusters are themselves bound to one another. They quickly merge within roughly 0.5 Myr (roughly twice the free-fall time for the original cloud). If this merging process was to occur on large scales, such as a whole GMC, one would never be able to form OB associations. The massive stars at the centres of the subclusters would find themselves in one large cluster.
Our proposal in this paper is that OB associations are just a series of clusters that form in [*[unbound]{}*]{} GMCs. The expanding cloud produces a series of clusters that are unbound from one another due to the fact that the flows that form them are also unbound from one another. The clusters, which become OB subgroups, then simply expand away from their mutual centre of mass along with the gas from the cloud, rather than merge into a single cluster. Thus only one star formation mechanism is at play here: clustered formation. The OB association therefore will have the universally observed IMF.
Details of the GMC Simulation {#setup}
=============================
The fluid was modelled using the Lagrangian particle method of smoothed particle hydrodynamics, or SPH [@Lucy1977; @GingoldMonaghan1977]. The smoothing lengths are variable in both time and space, with the constraint that there must be roughly constant number of neighbours for each particle, which is chosen to be roughly 50 (with a fluctuation from 30 to 70 neighbours). We use the standard artificial viscosity suggested by @GingoldMonaghan1983 with $\alpha = 1$ and $\beta = 2$. Gravitational forces are calculated using dipole and quadrupole moments obtained via a tree structure [@Benzetal1990], which is also used to construct particle neighbour lists. The code has been parallelised by Bate using OpenMP and the simulation presented here was performed on the UK Astrophysical Fluids Facility (UKAFF).
Our simulation starts with a uniform density sphere of molecular hydrogen of radius 20pc with a mass of $1 \times 10^{5}$ . The gas is isothermal and has a temperature of 10K. These numbers (mass, size and temperature) are typical of those reported for GMCs in the solar neighbourhood [@Blitz1991]. We model the gas with 500,000 SPH particles and are thus able to accurately follow the formation of self-gravitating regions down to a mass of 20[@BateBurkert1997; @Whitworth1998]. The free fall time associated with this cloud, the time taken for the unsupported gas to collapse under gravity to a central point, is roughly 4.7 Myr. The cloud has an initial Jeans mass of 30.4. We do not include any feedback processes, such as stellar winds and jets or the effects of massive stars such as ionisation fronts and supernovae.
To model the turbulence, we support the cloud with a Gaussian random velocity field with a power spectrum of $P(k) \propto k^{-4}$ which is consistent with a velocity field with a Larson-type relation of $\sigma \propto L^{0.5}$ where $\sigma$ is the velocity dispersion and $L$ is the length scale of the region [@MyersGammie1999]. At the beginning of the calculation the ratio of gravitational to kinetic energy is 0.5 ($\mathrm{E_{kin} = 2 E_{grav}}$). We stress that the turbulent kinetic energy is able to decay freely in this simulation since we include no driving mechanism. The timescale for the energy decay is the crossing time [@Maclowetal1998; @Stoneetal1998] which for the initial velocity field is $t_{cr} = $ 4.2 Myr, slightly less than the free fall time.
The SPH code includes the modification by @Bateetal1995 which replaces dense, self-gravitating, regions of the gas with point masses, or ‘sink particles’. These sinks allow the code to model the dynamical evolution of accreting protostars, without integration time steps becoming prohibitively small. We set the sink particles to form at a density of 1000 times the initial density, with a subsequent accretion radius of 0.17pc. When a particle finds itself at the centre of a dense, bound and collapsing region it is turned into a sink particle and its 50 to 100 neighbours are accreted onto it. With the resolution used for this simulation the sink particles start with a mass of at least 15before further accretion. Therefore we cannot think of these point mass objects as ‘protostars’, as was the case in @Bateetal2003, but instead assume that they represent ‘proto-clusters’. To prevent the ‘sink particles’ behaving as point masses in gravitational interactions, we smooth the sink-sink gravitational forces to a distance of $r_{min}$ = 0.2 pc in the form $\mathrm{F_{ij} = -Gm_{i}m_{j}/(r_{ij} + r_{min})^{2}}$ between particles $i$ and $j$.
In the analysis that follows, we discuss the properties of star formation centres, or ‘’. These can either comprise of a single protocluster (or sink particle) or a coherent group of protoclusters. To identify where more than one sink particle is involved, we make use of the mass segregation that occurs naturally when the protoclusters interact in self gravitating groups (see for example @BVB2004). First we sort all the protoclusters by mass. We then take the most massive protocluster and tag it and its fellow protocluster neighbours within 0.5pc to be a members of ‘1’. We then go down the mass sorted list of protoclusters until we arrive at the next most massive protocluster that has not been associated with 1. It becomes tagged as being a member of ‘2’. All the protoclusters within 0.5pc of this protocluster are now tagged as being members of 2, unless they are already members of 1. This process continues down the list of protoclusters until all have been assigned membership to a . There is thus a resolution of 0.5pc which distinguishes one from another. We find that the choice of 0.5 pc used in attributing memberships does not significantly affect the population, since the protoclusters formed in the simulation are either well separated (and thus in isolation) or exist in dense groups. The radius of the is given by the radius of the furthest protocluster from the centre of mass. If there is only one protocluster then the radius is simply the accretion radius, which is 0.17pc.
One problem with trying to model turbulence in a numerical simulation of this type is that it is not always possible to resolve the velocity structure at all scales. Turbulence is assumed to be hierarchical, following a Larson-type relation of $\sigma \propto L^{\alpha}$ [@Larson1981]. In SPH, while a particle can have a kinetic energy based on its velocity, it can have no internal velocity structure. As a result, the kinetic energy below a certain mass scale ( actually the mass of an SPH particle and its neighbours) is not included in the calculation. We therefore stress that our simulation is lacking the kinetic energy that should be present at scales of less than 20. The details of how individual stars form are thus not available from this simulation, and we must restrict ourselves to the large scale properties of star formation and the formation of clusters.
General Evolution {#evol}
=================
Figure [\[piccies\]]{} shows column density images from different points during the simulation and allows us to see clearly the evolution of the gas and regions of star formation. We see from the figure that the structure of the cloud changes remarkably quickly. It starts as a churning network of gaseous filaments and within 10Myr (when the simulation was terminated) evolves into an ensemble of distinct clusters, by which time the gas has lost much of its early character. The fact that an unbound GMC can form stars and star clusters reinforces the predictions made in @ClarkBonnell2004.
The point at which the first bound objects condense out of the unbound flows occurs at roughly 2.4 Myr. This is roughly half the crossing time for the region (although some authors use $t_{cr} = R/V$ instead of $t_{cr} = 2R/V$ as is used here). This time is consistent with the kinetic energy dissipation rate [@Maclowetal1998; @Stoneetal1998] and the formation of a turbulently dominated density structure [@Padoanetal2001].
Rather than simply discuss the individual protoclusters that form (the sink particles) it makes sense here to discuss the bound groups of these protoclusters as well, which we will simple refer to here as ‘star formation centres’ or . The formation of the actually occurs very rapidly. The mass of the 16 most massive of these centres is shown as a function of time in figure [\[bigclusters\]]{}. Within 5Myr (or 2.5 Myr after the onset of star formation) the 15 most massive all have masses greater than about 100 , and are beginning to get to a size where there is good possibility of them forming massive stars (this will be discussed in section [\[SFE\]]{}).
A desirable feature of an initially unbound GMC is that cloud dispersal and star formation are occurring simultaneously. This removes the necessity for feedback mechanisms to disperse the cloud, or at the very least makes their task much easier. The timescale for star formation is thus comparable to the timescale for the cloud’s dispersal. The dynamics of an unbound cloud is therefore naturally in keeping with the recent observations that star formation and cloud dispersal occur in a few crossing times. There is also the added bonus that star formation efficiencies will be kept low, since most of the gas around a protocluster clump will be unbound to it and moving away. This prevents the material getting involved in the accretion once a starts to form.
Figure [\[rhodist\]]{} shows the density distribution of the gas at three points in the simulation. The vertical dot-dashed line marks the original density of the cloud. Just before the first protocluster forms at 2.4 Myr we see that the most common density is roughly $7 \times 10^{-22}$ gcm$^{-3}$ (the solid line curve), an order of magnitude higher than at the start of the simulation. Note however that very little material at this point is as dense as $7 \times 10^{-21}$ gcm$^{-3}$, showing that the turbulence does not allow much material to get up to typical star forming densities [@Falgaroneetal1991; @Padoan1995; @Zinnecker2002].
After 7 Myr the peak in the distribution falls back to roughly the starting density, however there is much more spread in the distribution. This spread is controlled by two mechanisms. The high density tail increases as the grow by accretion and the subsequent rise in the potential energy. This causes yet more material to fall into the star forming regions. The low density tail increases since the cloud is freely expanding. By 13 Myr, only $\sim 3 t_{cr}$, we see that the majority of the gas has fallen to very low densities. By this point it is unlikely that observations of such a cloud would reveal much in the way of molecular gas and would instead only be visible as HI. The cloud can now be assumed to be ‘dispersed’. Even if the GMC fails to be a site of massive star formation, the dispersal would still occur on a timescale consistent with Elmegreen’s (2000) observations.
Note also from figure [\[piccies\]]{} that the cloud contains cavities and dense regions of star formation. These are created in the simulation purely by the turbulence. This type of structure in star forming clouds is often attributed to the effects of high mass stellar feedback, such as winds and supernovae, and is thought to be the trigger for star formation in the region (e.g. @ElmegreenLada1977) Instead, we realise that turbulence can mimick these effects. Furthermore the cavities in the simulation would be easily ionised by any high mass stars that form in the [@Daleetal2004]. We would then have a series of separated by a region of HII gas, just as is found in the classic picture of triggered star formation.
The Formation of Stars and Expected Efficiency {#SFE}
==============================================
In this section we use some simple assumptions about the star formation that occurs in the to determine the numbers of high mass stars and the star formation efficiency that one might expect from the simulation. It is still beyond the capabilities of current computational resources to model the details of how individual stars form in a body of gas as large as a GMC. In the simulation presented here we cannot model any gas dynamics below the 20scale. We can however give the reader a feel for the star formation that is present by using the results of previous simulations, along with some assumptions about the star formation efficiency and the form of the IMF.
It has been shown from numerical simulations that star formation occurs on roughly the local crossing time for the turbulence when the region is dynamically bound [@Bateetal2003; @Klessen2001]. On the small scales such as those represented by our protoclusters, 0.1pc, the crossing time is of the order $10^{5}$ years. We can therefore assume that all of our protoclusters form stars and that the star formation in our protoclusters takes place quickly, rapidly enough to be regarded here as instantaneous compared to the evolution of the whole GMC.
The simulation presented also has no method of incorporating feedback into the GMC model. As is shown in figure [\[bigclusters\]]{} in the right hand plot, the mass accreted into the gradually increases as the simulation progresses. At the point where the simulation is terminated, 30% of the GMC has been accreted by the protoclusters. It is unlikely that this value is representative of how much mass would actually be involved in the star formation by this time, since feedback mechanisms such as ionisation, winds and supernovae would seriously alter the amount of gas that would be available for accretion into the . What is needed is an estimate of when one would expect the star formation process to be halted by feedback mechanisms. This requires some knowledge of the star formation taking place within the .
We have already pointed out that the protoclusters in the simulation group into large . From now on in the paper we will use the details of these regions, rather than the individual protoclusters, to assess the nature of the star formation in the GMC. Table [\[clusterinfo\]]{} gives the details of the after 9 Myr. The masses quoted for the in table [\[clusterinfo\]]{} includes all particles (SPH and protoclusters) that fall within the radius of the region. The gas particle component is however quite small, generally less than 20%.
Although the star formation efficiency of GMCs is thought to be in the range of 1 to 10%, at the cluster level it is thought to be about 20 to 50% depending on the region (for a discussion we point the reader to @Ladas2003 and @Kroupa2001). In this paper we assume that the star formation efficiency in our is 50%, but will include a discussion about the case in which 100% of the mass is turned into stars. The assumed efficiency here is high but this is deliberate since it actually assumes as little as possible about the effect that the feedback mechanisms from the young stars are having on the accretion processes in the . We will also assume that the IMF of the stellar population in the follow a two step power law form, $dN
\propto m^{-\alpha} dm$, with $\alpha$ = 1.5 for $0.08 < m/M_{\odot} \le 0.5$ and $\alpha$ = 2.35 [@Salpeter1955] for $0.5 < m/M_{\odot} \le 100 $. This IMF, in conjunction with our assumption that 50% of the mass of the is turned into stars, allows us to estimate the stellar population produced by the simulation.
As already mentioned in the previous section, figure [\[bigclusters\]]{} shows in the left hand plot how the mass of the 15 largest evolves with time. The horizontal lines mark the point at which high mass stars can form. From our IMF model, 15% of the mass should be contained in stars with masses greater that 10. Thus a 10star will be present provided that there is 10/0.15 =67in the stellar population. Applying our assumed star formation efficiency of 50%, the must therefore have a mass of 134if they are to harbour a 10star. The horizontal long-dashed line in the figure denotes the point at which the achieve this mass. Doing the same for 25stars, which should comprise 7.7% of the stellar mass in our chosen IMF, we find that the need to contain $25/(0.077
\times 0.5)$ = 650if they are to contain a 25star. This is represented by the horizontal short-dashed line in the figure.
From our simple assumptions about the small scale efficiency and the form of the IMF, we can estimate at what point in the simulation the star formation process will be disrupted by feedback mechanisms. From figure [\[bigclusters\]]{}, we can estimate that the formation of 10stars would occur at about 0.8(or at 4 Myr). A star of mass 25would form after 1.1(or 5 Myr). Since the mass of the is increasing fairly rapidly at this point, stars with even higher masses would be expected to be present shortly after this, within 0.5 Myr or so. It would thus appear that the GMC is able to get enough mass into the for them to be able to form a full stellar population within about 1 Myr. This is consistent with the observations of the small age spread in the stellar population of the Orion cluster [@Hillenbrandetal2001].
Very rapidly after the first stars form we see that 10objects will be present. This means that shortly after their formation, are going to contain ionising sources. Such stars are commonly suggested to be responsible for controlling the star formation efficiency by expelling the gas from the cluster in which they form (such as our ), thus preventing the protostellar population from accreting or preventing new stars from forming. However @Daleetal2004 have noted that the ionisation from these stars does not appear to significantly affect the accretion rate in the clusters. The clumpy/fractal nature of the gas at the centre of the cluster where the OB type stars are situated acts to shield vast regions of the cluster from ionisation. Rather than pushing through the dense material, the ionising photons just find the path with the least resistance out of the cluster. This is low density gas which would not normally be associated with protostellar accretion in the first place. Similarly, the gas structure may also prevent the winds from OB stars expelling gas from the cluster. It has been suggested that winds are able to escape via the fractal holes, without imparting much momentum to the dense regions [@Henning1989].
It is therefore not clear if ionisation or winds will be able to expel the gas from cluster, thus halting the star formation process. One mechanism that certainly will produce the desired effect is a supernova explosion. In fact it has been estimated that these events will not only remove the gas from a cluster, but also be able to disperse the natal GMC. Thus a high mass star’s death will definitely mark the end of the star formation period in our cloud. Stars with masses greater than 25have very short main-sequence lifetimes, of about 3-5 Myr, and we see from the figure that they form at about 5 Myr after GMC formation. If we assume that a supernova event will occur at about 4 Myr after the formation of the very high mass stars, then we estimate the first supernova event to occur at about 9 Myr or when the OB stars are 4 Myr old.
Assuming the supernova event will halt the star formation, we can now get an estimate of the star formation efficiency in the GMC. The vertical dashed line in figure [\[bigclusters\]]{} denotes the point at which we might see the first SN event. At this time, 0.1 to 0.2 of the GMC’s mass is contained in the . However we have assumed up until now that the efficiency in the is not 100% but 50%, therefore our estimate of the star formation efficiency in the GMC is roughly 5 to 10%. This is easily comparable to the expected efficiencies in GMCs by Elmegreen’s (2000) rapid cloud formation/dispersal model.
The above analysis relied on a lot of assumptions about the nature of the star formation in the . In particular, it is guilty of invoking a star formation efficiency in the in order to determine the star formation efficiency of the cloud: one could argue that this is not entirely self-consistent. Here we redo the above analysis but without the efficiency assumption. If all the mass in the is used in forming a stellar population, then the mass required by a before a 25star forms is 25/0.077 = 325. The first to achieve this mass does so at about 4 Myr, only 1 Myr less than our previous estimate. Thus we can predict the supernova to occur at 8 Myr. At this point in the simulation there is about 7 - 8% of the cloud incorporated in the . Thus the expected star formation efficiency is still less than 10%, suggesting that our analysis is not heavily dependent on the assumptions.
----- ---------------------- --------------- ------ -------------------
No. $\mathrm{M_{\sfcp}}$ $\Delta$V R $\mathrm{t_{cr}}$
() (km s$^{-1}$) pc Myr
1 1763 3.27 0.71 0.42
2 761 2.68 0.45 0.33
3 706 3.24 0.29 0.17
4 624 2.24 0.53 0.46
5 455 2.37 0.35 0.29
6 362 1.77 0.50 0.55
7 338 1.77 0.46 0.51
8 329 2.88 0.17 0.12
9 305 1.68 0.46 0.54
10 212 2.31 0.17 0.14
11 174 1.37 0.39 0.56
12 151 1.95 0.17 0.17
13 150 1.95 0.17 0.17
14 144 1.91 0.17 0.17
15 141 1.63 0.23 0.27
16 140 1.39 0.31 0.44
----- ---------------------- --------------- ------ -------------------
: \[clusterinfo\] The table gives the properties of all the star formation centres () that would be expected to contain massive stars by t = 9 Myr (see section [\[SFE\]]{} for a discussion of this). $\Delta$V is the internal velocity dispersion and assumes the region is in virial equilibrium such that $\mathrm{\Delta V = (GM_{\sfcp}/R)^{1/2}}$. The crossing time is then calculated from $\mathrm{t_{cr} = 2R/\Delta V}$. Note that $\mathrm{M_{\sfcp}}$ is the total (gas + stars) mass enclosed within R.
Dynamical Evolution of the Clusters and their relation to OB associations {#OB}
=========================================================================
As already discussed, the simulation produces a series of star formation centres (). Of these regions, 16 of them (see table [\[clusterinfo\]]{}) are massive enough to contain a star of greater than 10 by the time at which we estimate a SN explosion will destroy the GMC. From figure [\[piccies\]]{} we see that these are expanding as a group away from one another (in fact this is true of the GMC structure in general). Furthermore, the distance between is roughly 10pc after about 13Myr. Thus the group of clusters have the appearance of an OB association, with the individual being OB subgroups that are expanding about some common point. In this section we examine the properties of the and compare them to the observations of OB associations.
In figure [\[OBcluster\]]{}, the left hand panel shows the positions of the that are large enough to contain stars greater than 10at t = 9Myr, assuming a star formation efficiency of 50% and the IMF presented in section [\[SFE\]]{}. Their positions are plotted at the time we estimate the SN event to start expelling gas from the cloud. We now assume that the SN event removes the gas from the GMC quickly enough such that the motions of objects have no time to adjust to the change in potential. We assume that this is true both at the scale of the motions and at the smaller scale of the stellar motions inside the . The right hand panel shows the positions of the after a further 4 Myr, i.e. at t = 13 Myr, assuming that they continue on the path they had before the gas expulsion. The circles denote the size of the at t = 13 Myr, which have been evaluated from $\rm{r = r_{SN} +
\Delta V \times 4}$ Myr, where $\rm{\Delta V}$ is the region’s internal velocity dispersion and $\rm{r_{SN}}$ is the radius of the at the point when the SN event occurs (i.e. at 9 Myr). This assumes that the star forming regions will disperse at roughly their internal velocity dispersion once the gas has been expelled.
Figure [\[OBcluster\]]{} clearly shows that the are expanding away from one another. By determining the average radius of the from their common centre of mass, both at t = 9 Myr and at t = 13 Myr, we can get an estimate of their expansion velocity from their dynamical centre. At t = 9 Myr the average distance of the from their centre of mass is 18.4pc, while after 13 Myr it is 25pc. This corresponds to an expansion velocity (that is a 3-dimensional velocity) for the of 1.5kms$^{-1}$. We compare this, for example, to the observed expansion of the OB subgroups in Per OB which is roughly 2kms$^{-1}$ [@Fredrick1956; @Blaauw1964]. We also see that if the themselves are able to expand after the gas expulsion, they become an extended distribution of stars after only 13 Mys, as is shown by the circles in the figure.
The stellar mass density is another important feature of OB associations. Generally, OB associations have a density of OB stars of roughly 0.1pc$^{-3}$ (see the introduction for references). In this simulation at t = 9 Myr, the density of OB type stars is 0.16pc$^{-3}$, and the mass density of stars $\ge$ 25is 0.1pc$^{-3}$. This is calculated by taking the total mass in the of stars of greater than the required mass type and dividing by the volume of region containing all the with these types of stars. After the system has had time to evolve for 4 Myr, the densities are 0.06 and 0.04pc$^{-3}$ for the OB type stars and those with masses $\ge$ 25respectively. Note these figures are based on the having a star formation efficiency of 50% and containing the IMF of stellar objects that was presented in section [\[SFE\]]{}.
We can compare this to the density of high mass stars in the at the point of the SN explosion. If we take the largest , with mass 1763and radius of 0.7pc, and assume again that 50% of this is contained in stars. Then the total mass in stars of mass greater than 10is $1763 \times 0.15 / 2
=$ 132. The density of massive stars is then $132/0.7^{3} = 384$ pc$^{-3}$.
The turbulent flows are thus able to create a series of star forming regions that have roughly the same properties as those found in OB associations. Since the regions (the ) are formed within large flows, the stars that form will have roughly the same motion as the gas stream that formed them, potentially explaining why @Blaauw1991 finds that the gas surrounding OB subgroups is generally moving with the group.
Observations of some OB associations also indicate distinct age spreads between their subgroups. This generally takes the form of an age progression from one side of the association to the next. The ages are generally derived from where the very high mass stars turn off the main sequence, which is a much more reliable method than using pre-main-sequence (PMS) tracks of low mass objects. Also the high mass end is normally the only part of the mass spectrum that is well established in OB associations. This age spread has been the motivation behind the triggered sequential star formation model developed by @ElmegreenLada1977, which is in turn motivated by the observations of @Blaauw1964. However, we note here that the Orion OB association exhibits no discernible age progression in the subgroups [@Brownetal1999].
Does our simulation show a convincing age spread between the subgroups/? In figure [\[ages\]]{} we plot the age of the protoclusters (the sink particles that group together to form the ) and their position from their common centre of mass. All the points are plotted at t = 13 Myr, with the positions being the y-direction in the simulation, since the are more spaced out in this direction. The ages are determined in two ways. The crosses denote the ages determined by when the protocluster first forms. The filled hexagons are determined by the time when the protoclusters reach a mass of 134, the point at which a 10star can form. Note that in the previous sections we determined when 10stars could form based on the mass contained in an , rather than its constituent protoclusters and gas. We are forced to use the individual protoclusters here since the are not coherent objects throughout the entire evolution of the simulation.
We see clearly from figure [\[ages\]]{} that while a large range of ages exist at any particular distance from the centre of mass, no trend is present in the ages with distance. Thus our simulation predicts that the OB association would be essentially coeval. However this is just a symptom of our idealised initial conditions. The initial uniform density sphere, with multiple Jeans masses, allows the entire cloud to proceed directly to star formation, via the dissipation of kinetic energy. Since the turbulence is the same throughout the cloud, star formation occurs simultaneously in quite separate locations. If on the other hand our GMC needs to accumulated in a large scale shock, as suggested by @Pringleetal2001, then there would naturally be an age spread as the layer in which the GMC forms starts to grow. The most important point in this picture is that the whole region would not be at the same density, but instead would have to evolve to star forming densities as the GMC accumulates.
Conclusions {#finish}
===========
The simulation presented in this paper highlights that GMCs need not be regarded as objects in virial equilibrium, or even bound, for them to be sites of star formation. Globally unbound GMCs can form stellar clusters very quickly, on roughly their crossing time. Furthermore, the unbound state of the cloud ensures that whole region is also dispersing while it is forming stars. They are thus naturally transient features. This evolutionary picture of a cloud forming, producing a stellar population, and then dispersing has been shown by Elmegreen (2000) to be apparent in a number of independent observations.
Using some simple assumptions about the form of the star formation in the star formation centres () of our simulation, we have provided an estimate of the star formation efficiency in the GMC. At the point one would expect the first supernova events, we find that the star formation efficiency is about 5 - 10%. This assumes that the environment in the simulation yields an efficiency of 50%. Removing this assumption about the , and letting the SN event be the only control over the efficiency, we find that the cloud has a global star formation efficiency of 7-8% (for our assumed IMF).
We argue that unbound GMCs may provide a simple mechanism for forming OB associations, a concept that was touched upon by @Ambart1955 [@Ambart1958]. OB stars form at the centre of a population of . These , which condense out of the unbound flows in the GMC are naturally expanding away from one another, as the positive energy disperses the cloud’s gas. Not only does the mechanism explain the OB association dynamics but it also explains the observed substructure, generally referred to as OB subgroups. Since the OB association is just a series of independently formed clusters, one would also expect the association to have the field star IMF.
Acknowledgments {#acknowledgments .unnumbered}
===============
The computations reported here were performed using the UK Astrophysical Fluids Facility (UKAFF). We also acknoweledge the assistance of the EC-network grant EC-RTN1-1999-00436. The authors would like to dedicate this paper to Adriaan Blaauw, on the occasion of his 90th birthday.
[^1]: E-mail: pcc@st-and.ac.uk
|
---
abstract: 'This work studies the joint problem of power and trajectory optimization in an unmanned aerial vehicle (UAV)-enabled mobile relaying system. In the considered system, in order to provide convenient and sustainable energy supply to the UAV relay, we consider the deployment of a power beacon (PB) which can wirelessly charge the UAV and it is realized by a properly designed laser charging system. To this end, we propose an efficiency (the weighted sum of the energy efficiency during information transmission and wireless power transmission efficiency) maximization problem by optimizing the source/UAV/PB transmit powers along with the UAV’s trajectory. This optimization problem is also subject to practical mobility constraints, as well as the *information-causality constraint* and *energy-causality constraint* at the UAV. Different from the commonly used alternating optimization (AO) algorithm, two joint design algorithms, namely: the concave-convex procedure (CCCP) and penalty dual decomposition (PDD)-based algorithms, are presented to address the resulting non-convex problem, which features complex objective function with multiple-ratio terms and coupling constraints. These two very different algorithms are both able to achieve a stationary solution of the original efficiency maximization problem. Simulation results validate the effectiveness of the proposed algorithms.'
author:
- 'Ming-Min Zhao, Qingjiang Shi, and Min-Jian Zhao [^1]'
bibliography:
- 'references.bib'
title: 'Efficiency Maximization for UAV-Enabled Mobile Relaying Systems with Laser Charging'
---
Mobile relaying, trajectory and power optimization, UAV communication, wireless power transfer.
Introduction
============
Thanks to the continuous cost reduction and device miniaturization in unmanned aerial vehicles (UAVs), wireless communications equipped and enabled by UAVs have attracted a lot of attentions recently, such as relaying, data gathering, secure transmission and information dissemination, etc [@Zeng2016Mag; @Xiao2016Mag; @Zeng2016; @mei2018cellular; @wu2019fundamental; @Zhang2019; @Zhang2019Cellular; @zeng2019accessing; @Mei2019NOMAUAV; @Cui2019UAV; @You2019]. In order to provide wireless data service for devices without infrastructure coverage due to, e.g., severe blocking by urban or mountainous terrain, communications infrastructure failure caused by natural disasters, etc., UAV-enabled wireless communication exhibits great potential in providing throughput/reliability improvement and coverage extension. Among the various applications enabled by UAVs, the use of UAVs as relay nodes for achieving high-speed and reliable wireless communications between two or more distant users whose direct communication links are blocked or corrupted, is expected to play an important role in future communication systems [@Zeng2016Mag; @Zeng2016].
Related Works and Motivation
----------------------------
UAV relays can be generally categorized into two types, i.e., statistic relaying and mobile relaying. The researches on statistic UAV relaying usually aim to find the best UAV position that maximizes the performance of the wireless network, along with the corresponding resource allocation strategy [@Chen2017ICC; @Chen2018CL; @Esrafilian2018asilomar; @Fan2018CL; @Xue2018access; @Chen2018TWC; @Li2019TMC; @li2019uav]. Specifically, in [@Chen2017ICC], an algorithm was proposed to find the optimal position of the UAV based on the fine-grained line-of-sight (LoS) information. The work [@Chen2018CL] investigated the optimum placement of UAV, where the total power loss, the overall outage and bit error rate were derived as reliability measures. The work [@Esrafilian2018asilomar] studied the optimal placement problem of a UAV relay without the need of any prior knowledge on the user locations and the underlying wireless channel pathloss parameters. In [@Fan2018CL], a system of multiple communication pairs with one UAV relay was considered, the node placement and resource allocation was jointly optimized. In [@Xue2018access], joint 3D location and power optimization was investigated. Placement of multiple UAVs was considered in [@Chen2018TWC], where the cases that multiple UAVs form either a single multi-hop link or multiple dual-hop links were analyzed. The work [@Li2019TMC] proposed to use UAVs as floating relaying nodes in order to resolve the problem of undesirable channel conditions of indoor users. The work [@li2019uav] considered a UAV-enabled two-way relaying system, where the joint optimization of UAV positioning and transmit powers was studied.
Compared to the statistic relaying scheme, the deployment of UAVs which serve as mobile relaying nodes is a more cost-effective solution to extend the wireless communication range and offer more reliable connectivities. Generally, two distinct advantages can be achieved by UAV-enabled mobile relaying systems: 1) enhanced performance brought up by the dynamic adjustment of relay locations to better coordinate with the environment; 2) the high mobility of UAVs enables the system to provide more flexible and responsive serves. As a result, the exploitation and exploration of UAV-enabled mobile relaying for more efficient physical layer designs have received a lot of attention recently [@Anazawa2015Globecom; @Zeng2016; @Jiang2018access; @Zhang2018access; @Zhang2017ICC; @Zhang2018CL]. In particular, the work [@Anazawa2015Globecom] proposed to use a mobile relay to carry data for several isolated communities and a genetic algorithm was designed where the trajectories of the mobile relay were represented by chromosomes that evolve to approximate the optimal solution. In [@Zeng2016], the throughput maximization problem in a decode-and-forward (DF) mobile relaying system was studied by jointly optimizing the source/relay transmit powers and the relay trajectory. An alternating optimization (AO)-based algorithm was proposed to optimize the power allocation and relay trajectory in a sequential manner. The work [@Zhang2018access] extended that of [@Zeng2016] to the multi-hop scenario, where a single multi-hop link was considered. The works [@Jiang2018access] and [@Zhang2018CL] investigated the use of amplify-and-forward (AF) relay strategy. In [@Zhang2017ICC], the spectrum efficiency and energy efficiency were optimized by assuming that the circular trajectory and time-division duplexing (TDD) were adopted. Furthermore, UAV-enabled mobile relaying can also be utilized to facilitate secure transmissions [@Wang2017WCL; @Wang2018access; @Li2018globecom; @xiao2018secrecy; @Cheng2019TCOM], full-duplex communications [@Wang2018JSAC] and wireless power transfer (WPT) [@Xie2019IOT], etc.
Despite the various benefits brought about by UAV-enabled mobile relaying, the UAV’s operations are usually restricted by many energy-consuming factors, such as the propulsion power to support its mobility, communication with the ground devices, etc. Therefore, many of the advantages of UAV-enabled wireless communication systems would be untouchable if the UAV’s battery capacity is limited and no additional power supply is available. Recently, laser power is becoming a viable solution to prolong the flight time of UAVs [@ZhangDLC2018; @Ouyang2018ICCworkshops]. Compared to other WPT techniques enabled by wind, sunlight, or radio frequency (RF) signals, the laser-beamed power supply is more stable and it can deliver much larger energy amounts. It is regarded as an important technique for emergency responses, military operations, and also to accelerate the pace of implementing 5G-oriented UAV networks [@Huo2019]. Moreover, the field tests conducted in [@Nugent2010] have validated the feasibility of laser-powered UAVs. Therefore, in order to provide convenient and sustainable energy supply to the UAV, we consider the employment of a laser power beacon (PB), which is able to send laser beams to charge the UAV in flight. As a result, in the considered mobile relaying system, we need to take the *energy-causality constraint* at the UAV relay into consideration, i.e., the total energy consumption of the UAV relay at the current time slot cannot exceed its remaining battery storage, in order to maintain its sustainable operations.
Our Contributions
-----------------
To this end, we propose an efficiency maximization problem, where the energy efficiency during information transmission and the laser power transmission efficiency are both taken into consideration by adding an adjustable weighting factor between them. In the considered problem, the UAV’s trajectory and the transmit powers of the source, UAV and laser PB are jointly optimized under the mobility constraints, information-causality and energy-causality constraints at the UAV. This joint design problem is very challenging due to the facts that the objective function is in a multiple-ratio form, the constraints are highly non-convex and the optimization variables are tightly coupled both in the objective and constraints. By taking advantage of the problem structure, we propose two algorithms which can both converge to the set of stationary solutions. The first algorithm, i.e., the concave-convex procedure (CCCP)-based algorithm, is designed by carefully introducing auxiliary variables and approximating the underlying non-convex components in the considered problem by convex ones. To derive the second algorithm, we employ the penalty dual decomposition (PDD) framework [@ShiPDD2017] and demonstrate that the optimization variables as well as the introduced auxiliary variables can be decoupled into several separate blocks. Then, the joint design problem can be addressed by iterating over a sequence of simple and efficient updates in each block of variables. These two algorithms exhibit similar performance in simulations, but they are essentially very different and each of them offers different advantages, i.e., the CCCP-based algorithm is able to converge within fewer iterations, while the PDD-based algorithm is more implementation-friendly.
The main contributions of this work can be summarized as follows:
1\) A general optimization framework for joint power allocation and trajectory design in a UAV-enabled mobile relaying system with laser charging is proposed. In particular, the weighted sum of the information transmission efficiency and power transmission efficiency is proposed as the objective function, the source/UAV/PB transmit powers and the relay trajectory are jointly optimized under the mobility, information-causality and energy-causality constraints.
2\) Despite the highly non-convexity of the considered problem and the intrinsic coupling in the optimization variables, two joint design algorithms, i.e., the CCCP and PDD-based algorithms, are proposed which are both guaranteed to converge to the set of stationary solutions.
3\) In order to validate the effectiveness of the proposed algorithms, computer simulations are conducted and the performance of the AO-based algorithm is also investigated for comparison. We demonstrate that the proposed joint design algorithms are able to outperform the commonly used AO-based algorithm. Furthermore, the impacts of different laser wavelengths and weather conditions are shown, as well as the tradeoff between the information/power transmission efficiencies.
Organization of the Paper and Notations
---------------------------------------
The rest of the paper is organized as follows. In Section \[sec\_system\_model\], we present the considered UAV-enabled mobile relaying system model and the corresponding problem formulation. In Section \[sec\_CCCP\] and \[sec\_PDD\], the proposed CCCP and PDD-based algorithms are developed, respectively, along with their complexity analysis. In Section \[sec\_simulations\], simulations are conducted to characterize the performance of the proposed algorithms and Section \[sec\_conclusion\] concludes the paper.
*Notations:* Scalars, vectors and matrices are respectively denoted by lower case, boldface lower case and boldface upper case letters. For a matrix $\mathbf{X}$, $\mathbf{X}^T$ and $\mathbf{X}^H$ denote its transpose and conjugate transpose, respectively. $\mathbf{a} \boldsymbol{\cdot} \mathbf{b}$ represents the dot product between the vectors $\mathbf{a}$ and $\mathbf{b}$. $\|\cdot\|$ denotes the Euclidean norm of a complex vector, $\Pi_{[a,b]}$ represents the projection operator onto the interval $[a,b]$ and $\odot$ denotes the Hadamard product. The set difference is defined as $\mathcal{A}\backslash \mathcal{B} \triangleq \{x| x\in\mathcal{A},x\notin \mathcal{B}\}$.
System Model and the Relay Problem {#sec_system_model}
==================================
In this work, we consider a UAV-enabled mobile relaying system which contains a source node, a destination node, a UAV and a laser PB, as shown in Fig. \[systemmodel\]. We assume that the direct link between the source and the destination is sufficiently weak and hence can be ignored due to e.g., severe blockage, and the UAV serves as a mobile relay node to assist their communications [@Zeng2016]. Furthermore, we assume that the UAV is wireless-powered by a PB which is realized by a properly designed laser charging system [@ZhangDLC2018].
We consider a Cartesian coordinate system without loss of generality, where the source, the destination and the PB are located at $\mathbf{q}_S\triangleq(0, 0, 0)$, $\mathbf{q}_D \triangleq (x_D, y_D, 0)$ and $\mathbf{q}_P \triangleq(x_{PB},y_{PB},0)$ respectively. For simplicity, we assume that the UAV is flying at a fixed altitude $H$ and $H$ could be chosen to be the minimum altitude that is required for terrain or building avoidance without frequent aircraft ascending or descending.[^2] Moreover, we focus on the UAV’s operation during flight and ignore its take-off and landing phases. We discretize the time interval $T$ into $N$ equally spaced time slots, i.e., $T = N\delta_t$ , where $\delta_t$ denotes the elemental slot length, which is chosen to be sufficiently small. Thus, the trajectory of the UAV $(x(t), y(t),H)$ over $T$ can be approximated by the $N$-length sequences $(\mathbf{q}_n \triangleq(x_n, y_n,H))_{n=1}^N$, where $(x_n, y_n)$ denotes the UAV’s $x-y$ coordinate at slot $n \in \mathcal{N} \triangleq\{1,\cdots,N\}$. Let $\mathbf{q}_I \triangleq(x_I, y_I, H)$ and $\mathbf{q}_F \triangleq(x_F, y_F , H)$ denote the initial and final locations of the UAV relay, which are given depend on various factors [@Zeng2016]. Furthermore, let $v_{\textrm{max}}$ denote the maximum UAV speed, then we assume $v_{\textrm{max}} \geq \|\mathbf{q}_F - \mathbf{q}_I\|/T$ is always satisfied such that there exists at least one feasible trajectory. With regards to the mobility constraints of the UAV [@JeongTVT2018], we have[^3]
\[mobility\_cons1\] $$\begin{aligned}
& \mathbf{q}_1 = \mathbf{q}_I, \; \mathbf{q}_N = \mathbf{q}_F,\label{mobility_cons1_sub2}\\
& \|\mathbf{v}_n\| \triangleq {\|\mathbf{q}_{n+1} - \mathbf{q}_n\|}/{\delta_t} \leq v_{\textrm{max}},\; \forall n\in \mathcal{N}\backslash \{N\}.
\end{aligned}$$
Information Transmission Model
------------------------------
We assume that LoS links dominate the wireless channels from the source to the UAV and that from the UAV to the destination, and the Doppler effect due to the mobility of the UAV can be perfectly compensated [@Zeng2016]. Therefore, at slot $n$, the channel power from the source to the UAV follows the free-space path loss model, which can be expressed as $\bar h_n^{sr} = \beta_0 (d_n^{sr})^{-2} = {\beta_0}/{\|\mathbf{q}_n-\mathbf{q}_S\|^2},\; n \in \mathcal{N}$, where $\beta_0$ denotes the channel power at the reference distance $d_0 = 1$ meter (m), whose value depends on the carrier frequency, antenna gain, etc., and $d_n^{sr} = \|\mathbf{q}_n- \mathbf{q}_S\|$ is the link distance between the source and the UAV at slot $n$. Similarly, the channel power from the UAV to the destination at slot $n$ can be expressed as $\bar h_n^{rd} = {\beta_0}/{\|\mathbf{q}_n-\mathbf{q}_D\|^2}$.
Let $p_n^s$ and $p_n^r$ denote the transmit powers of the source and the UAV at slot $n$, then the maximum transmission rate from the source to the UAV and from the UAV to the destination in bits/second/Hz (bps/Hz) at slot $n$ can be expressed as $R_n^s = \log_2\left(1 + {p_n^s\bar h_n^{sr}}/{\sigma^2}\right) = \log_2\left(1 + {p_n^s\gamma_0}/{\|\mathbf{q}_n-\mathbf{q}_S\|^2}\right)$ and $R_n^r = \log_2\left(1 + {p_n^r\bar h_n^{rd}}/{\sigma^2}\right) = \log_2\left(1 + {p_n^r\gamma_0}/{\|\mathbf{q}_n-\mathbf{q}_D\|^2}\right)$, where $\sigma^2$ is the noise power and $\gamma_0 = \beta_0/\sigma^2$ denotes the reference signal-to-noise ratio (SNR).
Wireless Power Transmission Model
---------------------------------
In this work, we model the PB as a laser charging system which was proposed in [@ZhangDLC2018], where the optical components are divided into two separate parts, the transmitter and the receiver, respectively. Consequently, the received power $P_n^r$ of the UAV at slot $n$ can be expressed as $P_n^r = \eta_{el} \eta_n^{lt} \eta_{le} P_n^s$, where $\eta_{el}$, $\eta_n^{lt}$ and $\eta_{le}$ denote the electricity-to-laser conversion efficiency, the laser transmission efficiency and the laser-to-electricity conversion efficiency, respectively [@ZhangDLC2018], $P_n^s$ represents the transmit power of the PB at slot $n$. Furthermore, $\eta_n^{lt}$ can be modeled as $\eta_n^{lt} = e^{-\alpha d_n^{rp}}$ [@LiuSemiconductor2005], where $\alpha$ denotes the laser attenuation coefficient and $d_n^{rp}$ is the distance between the UAV and the PB at slot $n$. $\alpha$ can be further depicted as $\alpha = \frac{\varepsilon}{\kappa}\left({\lambda}/{\chi}\right)^{-\varrho}$, where $\varepsilon$ and $\chi$ are two constants, $\kappa$, $\lambda$ and $\varrho$ denote the visibility, wavelength and size distribution of the scattering particles, respectively.
Employing the approximation method in [@ZhangDLC2018], we can alternatively model the received power $P_n^r$ as follows: $$\small
P_n^r = \left\{ {\begin{array}{*{20}{l}}
{a_1 a_2 \eta_n^{lt} P_n^s + a_2 b_1 \eta_n^{lt} + b_2,\;P_n^s \geq P_{\textrm{min}}^s},\\
{0,\;0 \leq P_n^s < P_{\textrm{min}}^s },
\end{array}} \right.$$ where $P_{\textrm{min}}^s$ denotes the minimum supply power that is required to activate the corresponding circuits of the laser transceiver, and the involved parameters are listed in Table \[tab:laser\_parameter\]. Note that $P_n^r$ is a non-convex function with respect to the UAV’s trajectory $\mathbf{q}_n$.
-- -- -- -- --
-- -- -- -- --
: Laser Power Transmission Parameters
\[tab:laser\_parameter\]
Energy Consumption Model
------------------------
Note that the energy consumption of the UAV is dominated by the propulsion power for maintaining the UAV aloft and supporting its mobility, which is usually much higher than the communication power consumption (e.g., hundreds of watts versus a few watts or even mW) [@Zeng2016Mag]. As a result, we consider the model in [@JeongTVT2018] and [@Xue2014] to characterize the energy consumption of the UAV due to flying, which postulates the flying energy at each slot $n$ to depend only on the velocity vector $\mathbf{v}_n$ as $$\label{EC_flying} \small
E_n^F(\mathbf{v}_n) = \omega \|\mathbf{v}_n\|^2,$$ where $\omega = 0.5M \delta_t$ and $M$ is the UAV’s mass, including its payload.[^4]
Problem Formulation
-------------------
In this work, we aim to maximize the information transmission efficiency of the UAV-enabled relay system and the laser power transmission efficiency simultaneously subject to the information/energy-causality constraints, the power budget constraints and the UAV’s mobility constraints . Specifically, the information-causality constraints mean that the UAV can only forward the data that has already been received from the source at each slot $n$ and by assuming that the processing delay at the UAV is one slot, we have $$\label{information-causality} \small
\sum\limits_{n=2}^{m} {R_n^r} \leq \sum\limits_{n=1}^{m-1} {R_n^s},\; m\in \mathcal{N}\backslash \{1\}.$$ It is obvious that the source should not transmit at the last slot $N$ and thus we can see that $R_N^s = R_1^r = 0$ should be satisfied (and hence $p_N^s = p_1^r = 0$) without loss of optimality. For simplicity, we assume that the UAV is equipped with a data buffer with sufficiently large storage size. Similarly, in order to guarantee that the UAV can safely reach the final location with enough battery level in case of emergence and to avoid overcharging, the following energy-causality constraint should also be satisfied: $$\label{energy-causality} \small
\theta \leq \mathcal{E} - \sum\limits_{n=1}^m E_n^F(\mathbf{v}_n) + \sum\limits_{n=1}^m P_n^r \delta_t \leq \mathcal{E},\; m\in \mathcal{N},$$ where $\mathcal{E}$ represents the UAV’s energy budget (i.e., the maximum energy storage capacity of the UAV’s battery if we assume that the UAV is fully charged before taking off) and $\theta$ is a predefined threshold which characterizes the minimum energy storage during the flight.
Furthermore, the energy efficiency of the UAV during information transmission can be expressed as $$\label{f_EE_function}
f_{\textrm{EE}} (\{\mathbf{q}_n,p_n^s,p_n^r\})\triangleq {\sum\limits_{n = 2}^{N} {R_{n}^r}} \Big{/} \left(\upsilon^s\sum\limits_{n = 1}^{N-1} p_n^s + \upsilon^r\sum\limits_{n = 2}^{N} p_n^r + NP_{\textrm{on}}\right),$$ where $\upsilon^s \geq 1$ and $\upsilon^r \geq 1$ are the power inefficiencies of the amplifiers in the source and the UAV, respectively, $P_{\textrm{on}}$ denotes the constant link on-power induced mainly by signal processing (it will be elaborated in Section \[sec\_simulations\]). The laser power transmission efficiency is given by $$\small
f_{\textrm{PE}} (\{\mathbf{q}_n,P_n^s\}) \triangleq {\sum\limits_{n=1}^N P_n^r}\Big{/}\left(\sum\limits_{n=1}^N P_n^s\right).$$ Therefore, the considered optimization problem can be formulated as
\[multi\_ratio\_problem\] $$\begin{aligned}
&\mathop {\max }\limits_{\{ \mathbf{q}_n,\;p_n^s,\;p_n^r,\;P_n^s\} } f_{\textrm{EE}} (\{\mathbf{q}_n,p_n^s,p_n^r\})+ \gamma f_{\textrm{PE}} (\{\mathbf{q}_n,P_n^s\}) \label{objective_function_original_problem}\\
&\textrm{s.t.}\; 0 \leq p^s_n \leq p_{\textrm{max}}^s,\; n\in \mathcal{N}\backslash\{N\},\; 0 \leq p_{n}^r \leq p_{\textrm{max}}^r,\;n\in \mathcal{N}\backslash\{1\}, \label{power_cons_ori_UAV} \\
& P_{\textrm{min}}^s \leq P_n^s \leq P_{\textrm{max}}^s,\;n\in \mathcal{N}, \label{power_cons_ori_PB}\\
& \sum\limits_{n=2}^{N} {R_n^r} \geq R_{\textrm{sum}},\label{mini_rate}\\
& \eqref{mobility_cons1},\;\eqref{information-causality}\; \textrm{and}\; \eqref{energy-causality}, \notag\end{aligned}$$
where $\gamma$ denotes a weighting factor that accounts for the priority of $f_{\textrm{PE}} (\cdot)$ over $ f_{\textrm{EE}} (\cdot)$; and denote the transmit power constraints of the UAV and the PB, respectively; $R_{\textrm{sum}}$ represents the minimum sum-rate that should be achieved during the flight. Note that in order to maximize $f_{\textrm{EE}} (\cdot)$, the UAV should fly close to the source and destination, however for the maximization of $f_{\textrm{PE}} (\cdot)$, the UAV should be close to the PB instead. Since the source, the destination and the PB are not co-located in general, these two efficiencies are usually conflict with each other and there exists a tradeoff between them. Throughout this paper, we assume that the flight duration $T$ is sufficiently long such that the UAV must harvest energy from the PB otherwise its battery would be drained out.[^5]
*Problem is highly non-convex, which involves multiple fractional terms in the objective function and the optimization variables are coupled in the constraints. It cannot be directly solved by standard convex optimization techniques. Moreover, neither the Dinkelbach’s transformation [@Dinkelbach1967] nor the fractional programming technique [@Shen2018] can be directly applied to solve this problem, since the former cannot deal with objective functions with multiple-ratio terms and the latter is not designed to handle coupling constraints. A feasible approach for problem is the AO-based algorithm, which alternating between power optimization and trajectory optimization, however, no optimality (e.g., to stationary solutions) can be theoretically declared for such an algorithm as has been shown in [@Zeng2016; @Zhang2018; @Jiang2018], etc. To tackle this difficulty, in this work, we propose two algorithms to address problem with different design techniques and both of them are guaranteed to achieve stationary solutions of problem .*
*In this work, we assume that the energy supplys of the communication and propulsion systems of the UAV are independent for emergency purposes, e.g., sending localization signals when the UAV does not have enough power to maintain aloft, etc. As a result, in , the denominator does not contain the propulsion power . Besides, the PB’s location will affect the overall performance, however, it is regarded as a fixed infrastructure in this work and its location is considered to be a predefined parameter that cannot be optimized. The case that propulsion power dominates the denominator of and the placement of the PB are left for future work. Moreover, the efficiencies of the communication and propulsion systems are formulated and optimized as two separate terms, i.e., $f_{\textrm{EE}} (\{\mathbf{q}_n,p_n^s,p_n^r\})$ and $f_{\textrm{PE}} (\{\mathbf{q}_n,P_n^s\})$. Otherwise, the objective function would become ${\sum\limits_{n = 2}^{N} {R_{n}^r}}\big{/}\Big({\sum\limits_{n=1}^N\omega \|\mathbf{v}_n\|^2 - \sum\limits_{n=1}^N P_n^r}\Big)$ and due to the fact that $P_n^s$ is not considered in this case, the laser power transmission efficiency would be ignored.*
The Proposed CCCP-based Algorithm {#sec_CCCP}
=================================
In this section, in order to make problem more tractable, we propose to first transform it into an equivalent form by properly introducing auxiliary variables; we then present a CCCP-based algorithm to address the resulting problem. The proposed algorithm is motivated by the observation that by some skillful mathematical manipulations, the objective function with multiple-ratio terms , the pivotal coupling constraint and can be expressed as difference of convex (DC) functions. Thus, we can use the CCCP technique [@CCCP2009] to iteratively solve problem , where in each iteration only a convex subproblem is needed to be solved.
Problem Transformation {#problem_transformation_CCCP}
----------------------
We first introduce auxiliary variables $s_n^r$ and $s_n^s$, which satisfy
\[rate\_auxi\] $$\begin{aligned}
{p_n^r\gamma_0}/({H^2+\|\mathbf{q}_n - \mathbf{q}_D\|^2}) \geq s_n^r,\;\forall n\in \mathcal{N}\backslash \{1\},\\
{p_n^s\gamma_0}/({H^2+\|\mathbf{q}_n - \mathbf{q}_S\|^2}) \geq s_n^s,\;\forall n\in \mathcal{N}\backslash \{N\}.\end{aligned}$$
It can be seen that constraints must be satisfied with equality at optimality. If either of these two inequalities are satisfied with strict inequality, we can always decrease $p_n^r$ or $p_n^s$, such that a higher objective value can be achieved without violating any constraints. As a result, problem can be transformed into
\[CCCP2\_problem\_equi\] $$\begin{aligned}
&\mathop {\max }\limits_{\{ \mathbf{q}_n,p_n^s,p_n^r,P_n^s,s_n^r,s_n^s\} } \bar{f}_{\textrm{EE}} (\{\mathbf{q}_n,p_n^s,p_n^r,s_n^r\})+ \gamma f_{\textrm{PE}} (\{\mathbf{q}_n,P_n^s\}) \\
&\textrm{s.t.}\; \sum\limits_{n=2}^{m} \log_2\left(1 +s_n^r\right) \leq \sum\limits_{n=1}^{m-1} \log_2\left(1 +s_n^s\right),\; m \in \mathcal{N}\backslash\{1\}, \label{info_casu_1}\\
& \sum\limits_{n=2}^{N} \log_2\left(1 +s_n^r\right) \geq R_{\textrm{sum}},\label{rate2}\\
&\eqref{mobility_cons1},\;\eqref{energy-causality},\; \eqref{power_cons_ori_UAV},\; \eqref{power_cons_ori_PB} \;\textrm{and}\;\eqref{rate_auxi},\end{aligned}$$
where $$\small
\bar{f}_{\textrm{EE}} (\{\mathbf{q}_n,p_n^s,p_n^r,s_n^r\})\triangleq {\sum\limits_{n = 2}^{N} \log_2\left(1 +s_n^r\right) }\Big{/} \left(\upsilon^s\sum\limits_{n = 1}^{N-1} p_n^s + \upsilon^r\sum\limits_{n = 2}^{N} p_n^r + N P_{\textrm{on}}\right),$$ and we can see that problem is equivalent to .
Then, we proceed to handle the objective function which is in a multiple-ratio form and the main idea is also to introduce some auxiliary variables. Specifically, for the information transmission efficiency part, i.e., $\bar{f}_{\textrm{EE}} (\{\mathbf{q}_n,p_n^s,p_n^r,s_n^r\})$, we resort to the employment of auxiliary variables $\tilde R$, $\tilde p$ and $E_i$, which satisfy
\[E\_i\_constraints\] $$\begin{aligned}
& \sum\limits_{n = 2}^{N} \log_2\left(1 +s_n^r\right) \geq \tilde R,\; \upsilon^s\sum\limits_{n = 1}^{N-1} p_n^s + \upsilon^r\sum\limits_{n = 2}^{N} p_n^r + N P_{\textrm{on}} \leq \tilde p,\label{IE_cons2}\\
& \tilde R \geq \tilde p E_i. \label{IE_cons3}\end{aligned}$$
With the help of these variables, we can observe that $\bar{f}_{\textrm{EE}} (\{\mathbf{q}_n,p_n^s,p_n^r,s_n^r\})$ can be replaced by a simple scalar variable $E_i$ and three additional inequality constraints in . It can be shown that this transformation incurs no loss of optimality by a similar argument as for constraints . The power transmission efficiency part, i.e., $f_{\textrm{PE}} (\{\mathbf{q}_n,P_n^s\})$, can also be transformed into its equivalent form in a similar vein. To be specific, introduce auxiliary variables $\{t_n\}$ and $\{\hat t_n\}$ which satisfy $$\label{pow_1} \small
e^{-\alpha \sqrt{H^2 + \|\mathbf{q}_n - \mathbf{q}_P\|^2}} \geq t_n, (\textrm{usually}\; t_n <1)$$ $$\label{pow_2} \small
t_n P_n^s \geq \hat t_n,$$ respectively, then $P_n^r$ can be rewritten as $P_n^r = a_1 a_2 \hat{t}_n + a_2 b_1 t_n + b_2$. As a result, $f_{\textrm{PE}} (\{\mathbf{q}_n,P_n^s\})$ can be equivalently expressed as $E_e$, with the help of the following constraints:
\[E\_e\_constraints\] $$\begin{aligned}
&\sum\limits_{n=1}^N a_1 a_2 \hat t_n + a_2 b_1 t_n + b_2 \geq \tilde{t},\;\sum\limits_{n=1}^N P_n^s \leq \tilde P,\label{EE_cons2}\\
&\tilde t \geq \tilde P E_{e},\end{aligned}$$
where $\tilde{t}$, $\tilde P$ and $E_e$ are the introduced auxiliary variables. Therefore, we can see that the original objective function , which is very difficult to handle, can now be equivalently transformed into the weighted sum of two scalar variables, i.e., $E_i +\gamma E_e$. However, as a cost for this simple representation, we have to deal with the additional constraints , , and , which will be detailed in the next subsection.
Next, we focus on constraints , which are also difficult to address due to the fact that $\frac{x}{y^2} \geq z$ is non-convex. To tackle this difficulty, we resort to the help of two auxiliary variables $d_n^D$ and $d_n^S$, which measure the upper bounds of the squared distances from the UAV to the source and destination. Accordingly, constraints can be decomposed into
\[equi\_1\] $$\begin{aligned}
&H^2+\|\mathbf{q}_n - \mathbf{q}_D\|^2 \leq d_n^D, \label{rate_1}\\
& s_n^r d_n^D - p_n^r\gamma_0 \leq 0, \label{non_convex1}\\
&H^2+\|\mathbf{q}_n - \mathbf{q}_S\|^2 \leq d_n^S, \label{rate_2}\\
& s_n^s d_n^S - p_n^s\gamma_0 \leq 0.\label{non_convex2}\end{aligned}$$
Note that constraint must be satisfied with equality at optimality, otherwise we can always decrease $d_n^D$, increase $s_n^r$ and $\tilde{R}$, and then properly adjust $E_i$ to increase the objective function. A similar argument also holds for constraint , therefore we omit the details for brevity.
To summarize, we conclude that problem can be equivalently transformed into the following problem:
\[equivalent\_problem\] $$\begin{aligned}
& \mathop {\max }\limits_{\bm{\mathcal{X}}} \; E_i+\gamma E_e\\
& \textrm{s.t.} \; \theta \leq \mathcal{E} - \sum\limits_{n=1}^m E_n^F(\mathbf{v}_n) + \sum\limits_{n=1}^m (a_1 a_2 \hat t_n + a_2 b_1 t_n + b_2) \delta_t \leq \mathcal{E},\; m\in \mathcal{N}, \label{energy_cons}\\
& \eqref{mobility_cons1},\; \eqref{power_cons_ori_UAV},\;\eqref{power_cons_ori_PB},\;\eqref{info_casu_1},\;\eqref{rate2},\;\eqref{E_i_constraints}-\eqref{equi_1}, \notag \end{aligned}$$
where $\bm{\mathcal{X}} \triangleq \{\mathbf{q}_n,p_n^s,p_n^r,P_n^s,s_n^r,s_n^s,\tilde{R},\tilde{p},E_i,E_e,\tilde{P},\tilde{t},t_n,\hat{t}_n, d_n^S,d_n^D\}$. Although problem is now in a much simpler form than that of , it is still highly non-convex and difficult to address. In the following, we present the design methodology to iteratively solve problem by the concept of CCCP.
Algorithm Design
----------------
Non-convex constraints are generally difficult to handle, e.g., , , , , , , and etc. Among them, constraints and are more difficult since the logarithm and exponential functions are involved. In the following, we show that these constraints can be expressed in DC forms by proper transformations and then by employing the CCCP concept, problem can be iteratively solved to stationary solutions. Unless otherwise stated, we use subscript $l$ to indicate the variables obtained in the $l$-th iteration.
Firstly, let us focus on constraints and . Since the $\log_2(\cdot)$ function is concave, can be readily viewed as a DC function. By approximating the convex function $-\sum\limits_{n=2}^{m} \log_2\left(1 +s_n^r\right) $ in the $l$-th iteration by its first order Taylor expansion around the current point $\{s_{n,l}^r\}$, we can obtain $$\label{info_caus2} \small
-\sum\limits_{n=1}^{m-1} \log_2\left(1 +s_n^s\right) + \sum\limits_{n=2}^{m} \Big(\log_2(1+s_{n,l}^r) + \frac{1}{(1+s_{n,l}^r)\ln(2)}(s_n^r - s_{n,l}^r)\Big) \leq 0,\;m\in\mathcal{N}\backslash\{1\}.\\$$ As for constraint , the following equivalent form can be obtained: $\sqrt{H^2 + \|\mathbf{q}_n - \mathbf{q}_P\|^2} \leq -({\ln t_n})/{\alpha} ,\; n \in \mathcal{N}$, and since the $\ln(\cdot)$ function is also concave and the left hand side is a second order cone (SOC) which is convex, this equivalent inequality is also in DC form and can be approximated by the following convex constraint: $$\label{SOCP_1} \small
\sqrt{H^2 + \|{\mathbf{q}}_{n} - {\mathbf{q}}_P \|^2} + ({\ln t_{n,l}})/{\alpha} + (t_n-t_{n,l})/(\alpha t_{n,l}) \leq 0,\; n \in \mathcal{N}.\\$$
Secondly, we consider constraints , , , and . It can be observed that these constraints are all in the form of $xy-z\leq 0$ or $xy-z\geq 0$, which can be further expressed as $\frac{1}{2}(x+y)^2-\frac{1}{2}x^2 -\frac{1}{2}y^2-z \leq 0$ or $z - \frac{1}{2}(x+y)^2+\frac{1}{2}x^2 +\frac{1}{2}y^2 \leq 0$. They are also DC functions, and by using the CCCP concept, they can be approximated by convex function without any difficulty. The detailed expressions of these approximations will be given below.
Finally, it can be easily seen that can be decomposed into one convex constraint and one DC constraint, which can be handled in a similar way. Therefore, in the $l$-th iteration of the proposed CCCP-based algorithm, we have the following convex problem:
\[CCCP2\_inner\] $$\begin{aligned}
&\mathop {\max }\limits_{\{ \bm{\mathcal{X}}\} }\; E_i+\gamma E_e\\
&\textrm{s.t.}\; \eqref{mobility_cons1},\; \eqref{power_cons_ori_UAV},\;\eqref{power_cons_ori_PB},\;\eqref{rate2},\;\eqref{IE_cons2},\;\eqref{EE_cons2},\;\eqref{rate_1},\;\eqref{rate_2},\;\eqref{info_caus2},\;\eqref{SOCP_1},\notag\\
& (s_n^s+d_n^S )^2 + (s_{n,l}^s)^2 +(d_{n,l}^S)^2 - 2s_{n,l}^s s_n^s - 2d_{n,l}^S d_{n}^S - 2p_n^s\gamma_0 \leq 0,\\
& (s_n^r+d_n^D )^2 + (s_{n,l}^r)^2 +(d_{n,l}^D)^2 - 2 s_{n,l}^r s_n^r - 2d_{n,l}^D d_{n}^D - 2p_n^r\gamma_0 \leq 0,\\
&( \tilde p+ E_i)^2 + \tilde p_l^2 + E_{i,l}^2 - 2\tilde p_l \tilde p - 2E_{i,l} E_{i} - 2\tilde R \leq 0,\\
& (\tilde P +E_{e})^2 + {\tilde P_l}^2 + E_{e,l}^2 -2\tilde P_l\tilde P - 2E_{e,l} E_{e} - 2\tilde t \leq 0,\\
& 2\hat t_n + t_n^2 + (P_n^s)^2 +(t_{n,l} +P_{n,l}^s)^2 - 2(t_{n,l} +P_{n,l}^s)(t_n+P_{n}^s) \leq 0,\\
& \theta - \mathcal{E} - \sum\limits_{n=1}^m (a_1 a_2 \hat t_n + a_2 b_1 t_n + b_2) + \sum\limits_{n=1}^m \kappa \|\mathbf{v}_n\|^2 \leq 0,\\
& \sum\limits_{n=1}^m \kappa\left(\frac{-\|{\mathbf{q}}_{n+1,l} - {\mathbf{q}}_{n,l}\|^2 + 2({\mathbf{q}}_{n+1,l} - {\mathbf{q}}_{n,l})^T({\mathbf{q}}_{n+1} - {\mathbf{q}}_{n})}{\delta_t^2}\right) - \sum\limits_{n=1}^m (a_1 a_2 \hat t_n + a_2 b_1 t_n + b_2) \delta_t \geq 0,
\end{aligned}$$
which is a second-order cone program (SOCP) and it can be solved by some off-the-shelf solvers, such as CVX [@cvx]. The proposed CCCP-based algorithm to solve problem is summarized in Algorithm \[CCCP\_algorithm\] and we have the following proposition regarding its convergence property:
\[prop1\] Every limit point of the sequence generated by Algorithm \[CCCP\_algorithm\] is a stationary solution of problem .
Please refer to reference [@CCCP2009] for the detailed proof.
Initialize with a feasible solution $\bm{\mathcal{X}}_0$ and set $l = 0$. Solve problem with fixed $\bm{\mathcal{X}}_l$ and assign the solution to $\bm{\mathcal{X}}_{l+1}$. Update the iteration index: $l = l + 1$.
Furthermore, the computational complexity of Algorithm \[CCCP\_algorithm\] is dominated by solving problem $L$ times, where $L$ denotes the total iteration number. Since problem involves $7N+1$ linear constraints, $5N-1$ SOCs with dimension $3$, $3N-2$ SOCs with dimension $4$ and the number of variables $n$ is on the order of $\mathcal{O}(11N)$, we can see that the complexity of Algorithm 1 is on the order of $\mathcal{O}(11NL\sqrt{23N-5} (198N^2+96N-37))$ according to the basic elements of complexity analysis as used in [@Wang2014]. Therefore, by letting $N \rightarrow \infty$, the worst-case asymptotic complexity of Algorithm \[CCCP\_algorithm\] can be evaluated as $\mathcal{O}(LN^{3.5})$.
The Proposed PDD-based Algorithm {#sec_PDD}
================================
In the previous section, we proposed the CCCP-based algorithm (i.e., Algorithm \[CCCP\_algorithm\]), where in each iteration, an SOCP problem is required to be solved. The main idea is to replace the complex objective function and constraints with simpler ones and possibly with some linear approximations, thus employing convex solvers is inevitable. However, since the intrinsic structure of problem may not be fully exploited, off-the-shelf software solvers might be inefficient in many scenarios. In this section, we take an alternative by embracing the PDD framework and present a PDD-based algorithm. Specifically, we first transform problem into an equivalent form by introducing auxiliary variables and some additional equality constraints. Different from Algorithm \[CCCP\_algorithm\], in this case, our aim is to make this problem fully decomposable, i.e., to relief the coupling of the constraints. Then, instead of directly handling the equivalent problem with many constraints, we focus on its augmented Lagrangian (AL) problem, where the equality constrains are augmented onto the objective function with certain dual variables and a penalty parameter. As a result, we obtain a twin-loop PDD-based algorithm, where the inner loop seeks to (approximately) solve the AL problem using a block minimization technique, while the outer loop updates the dual variables and the penalty parameter. Especially, we show that each subproblem can be solved either in closed-form or by the bisection method.
Problem Transformation {#problem-transformation}
----------------------
Firstly, we introduce the following variable substitutions:
$$\begin{aligned}
& H^2+\|\mathbf{q}_n - \mathbf{q}_D\|^2 = d_n^D, \; H^2+\|\mathbf{q}_n - \mathbf{q}_S\|^2 = d_n^S,\; H^2 + \|\mathbf{q}_n - \mathbf{q}_P\|^2 = \left(\frac{\ln t_n}{\alpha}\right)^2 ,\label{traj3}\\
& p_n^r\gamma_0 = s_n^r d_n^D, \; p_n^s\gamma_0 = s_n^s d_n^S,\label{traj5}\\
& t_n P_n^s = \hat{t}_n,\label{traj6}\end{aligned}$$
where the purposes of $s_n^r$, $s_n^s$, $d_n^D$, $d_n^S$, $t_n$ and $\hat{t}_n$ are similar to those in Section \[problem\_transformation\_CCCP\], only in this case, we prefer to directly introduce equality constraints such that the PDD framework can be naturally blended in.
Next, since the trajectory variables $\{\mathbf{q}_n\}$ are coupled in the velocity vectors and appear multiple times in , in order to break these couplings, we further introduce four redundancy copies, i.e., $\dot{\mathbf{q}}_n = \mathbf{q}_n$, $\bar{\mathbf{q}}_n = \mathbf{q}_n$, $\hat{\mathbf{q}}_n = \mathbf{q}_n$, $\tilde{\mathbf{q}}_n = \bar{\mathbf{q}}_n$. Let $\bar{v}_{n} = {\|\tilde{\mathbf{q}}_{n+1} - \mathbf{q}_n\|^2}/{\delta_t^2}$ and $ \tilde{v}_{m+1} = \sum\limits_{n=1}^m \bar{v}_n$ represent the squared velocity at slot $n$ and the sum of squared velocity from slot $1$ to $m$ and introduce $\tilde{v}_m = \dot{v}_m$, $\dot{v}_m = \breve{v}_m$ (due to the same reason with that of $\{\mathbf{q}_n\}$). Then, it can be seen that ${\|\tilde{\mathbf{q}}_{n+1} - \mathbf{q}_n\|^2}/{\delta_t^2} = \breve{v}_{n+1} - \tilde{v}_n$ holds.
Finally, in order to decompose the information-causality and energy-causality constraints, the following auxiliary variables are employed:
\[PDD\_cons\_1\] $$\begin{aligned}
& \log_2\left(1 + s_n^r\right) = \bar{s}_n^r, \; \log_2\left(1 + s_n^s\right) = \bar{s}_n^s,\label{PDD_cons1_2}\\
& \sum\limits_{n=2}^{m} \bar{s}_n^r - \sum\limits_{n=1}^{m-1} \bar{s}_n^s = \tilde{s}_m,\label{PDD_cons1_3}\\
& \ln(t_n)/\alpha = t_n^L,\label{PDD_cons1_4}\\
& \breve{t}_{n}=a_1a_2\hat{t}_n+a_2b_1 t_n,\label{PDD_cons1_5}\\
& - \sum\limits_{n=1}^m \kappa \bar{v}_n + \Big(\sum\limits_{i=1}^m \breve{t}_i + m b_2\Big)\delta_t = e_m,\label{PDD_cons1_6}\end{aligned}$$
where the main motivation is to make these coupling constraints separable among each other and among different time slots. Therefore, we have the following optimization problem:
\[PDD\_problem5\] $$\begin{aligned}
&\mathop {\max }\limits_{\bm{\mathcal{Y}} }\; \hat{f}_{\textrm{EE}} (\{p_n^s,p_n^r,\bar{s}_n^r\})+ \gamma \hat{f}_{\textrm{PE}} (\{t_n,\hat{t}_n,P_n^s\})\\
&\textrm{s.t.}\; \tilde{s}_m \leq 0,\; m \in \mathcal{N}\backslash\{1\}, \label{PDD_problem_cons1} \\
& \sum\limits_{n=2}^{N} \bar{s}_n^r \geq R_{\textrm{sum}},\label{PDD_problem_cons2}\\
& \mathcal{E} \geq \mathcal{E} +e_m \geq \theta,\; m \in \mathcal{N},\label{PDD_problem_cons3}\\
& H^2+\|\dot{\mathbf{q}}_n - \mathbf{q}_D\|^2 = d_n^D,\;
H^2+\|\bar{\mathbf{q}}_n - \mathbf{q}_S\|^2 = d_n^S,\;
H^2 + \|\hat{\mathbf{q}}_n - \mathbf{q}_P\|^2 = \left(t_n^L\right)^2,\;n \in \mathcal{N} \label{PDD_problem_cons6}\\
& {\|\tilde{\mathbf{q}}_{n+1} - \mathbf{q}_{n}\|^2}/{\delta_t^2} =\breve{v}_{n+1} - \tilde{v}_n,\;n \in \mathcal{N}, \label{PDD_problem_cons7}\\
& \breve{v}_{n+1} - \tilde{v}_n = \bar{v}_n,\; \breve{v}_{n} = \dot{v}_n,\;\tilde{v}_{n} = \dot{v}_n,\; n \in \mathcal{N},\label{v_variables}\\
& \bar{v}_n\leq v_{\textrm{max}}^2,\label{PDD_problem_cons8} \\
& \dot{\mathbf{q}}_n = \mathbf{q}_n,\;
\bar{\mathbf{q}}_n = \mathbf{q}_n,\;
\hat{\mathbf{q}}_n = \mathbf{q}_n,\;
\tilde{\mathbf{q}}_n = \bar{\mathbf{q}}_n,\; n \in \mathcal{N}, \label{q_variables}\\
& \eqref{mobility_cons1_sub2},\; \eqref{power_cons_ori_UAV},\;\eqref{power_cons_ori_PB},\;\eqref{traj5},\;\eqref{traj6},\;\eqref{PDD_cons_1}, \notag
\end{aligned}$$
where $\bm{\mathcal{Y}} \triangleq \{ \mathbf{q}_n,p_n^s,p_n^r,P_n^s,d_n^S,d_n^D,t_n,\hat{t}_n,\breve{t}_n,t_n^L,s_n^s,s_n^r,\bar{s}_n^s,\bar{s}_n^r,\dot{\mathbf{q}}_n, \bar{\mathbf{q}}_n,\hat{\mathbf{q}}_n,\tilde{\mathbf{q}}_n,\bar{v}_n,\breve{v}_n,\tilde{v}_n,\dot{v}_n,\tilde{s}_m, e_m\} $, $$\small
\hat{f}_{\textrm{EE}} (\{p_n^s,p_n^r,\bar{s}_n^r\}) \triangleq {\sum\limits_{n = 2}^{N} \bar{s}_n^r}\Big{/}\left(\upsilon^s\sum\limits_{n = 1}^{N-1} p_n^s + \upsilon^r\sum\limits_{n = 2}^{N} p_n^r + N P_{\textrm{on}}\right),$$ $$\small
\hat{f}_{\textrm{PE}} (\{t_n,\hat{t}_n,P_n^s\}) \triangleq \left(\sum\limits_{n=1}^N a_1a_2\hat{t}_n+a_2b_1 t_n+b_2\right)\Big{/}{\sum\limits_{n=1}^N P_n^s}.$$ Note that problem and are equivalent, since to this end, we are basically introducing equality constraints. The roles and necessities of these additional variables and constraints would be clear in the next subsection.
Algorithm Design
----------------
In this subsection, our aim is to solve problem by proposing an efficient PDD-based algorithm. We first formulate the AL problem of as follows:
\[AL\] $$\begin{aligned}
&\mathop {\max }\limits_{\bm{\mathcal{Y}} } \; \hat{f}_{\textrm{EE}} (\{p_n^s,p_n^r,\bar{s}_n^r\})+ \gamma \hat{f}_{\textrm{PE}} (\{t_n,\hat{t}_n,P_n^s\}) - f_{\textrm{AL}}(\bm{\mathcal{Y}}, \bm{\Lambda})\\
&\textrm{s.t.}\; \eqref{mobility_cons1_sub2},\; \eqref{power_cons_ori_UAV},\;\eqref{power_cons_ori_PB},\;\eqref{PDD_problem_cons1}-\eqref{PDD_problem_cons8},
\end{aligned}$$
where $f_{\textrm{AL}}(\bm{\mathcal{Y}}, \bm{\Lambda})$ represents the AL part which is obtained by augmenting the equality constraints with certain dual variables and penalty functions.[^6] $\bm{\Lambda}$ denotes the collection of all dual variables, which is listed in Table \[tab:dual\_variables\] with their corresponding equality constraints.
Constraints and and
---------------- ---------------------- -------------------- ----------------------- ------------------------ ------------------ --------------------------------------- --------------------------------------------------------------------------------------------
Dual Variables $ \mu_n^D, \mu_n^S$ $\xi_n^S, \xi_n^L$ $\zeta_n^r,\zeta_n^s$ $\zeta_m^i, \zeta_m^e$ $\tilde{\eta}_n$ $\bar{\tau}_n, \tau_n,\tilde{\tau}_n$ $\dot{\bm{\lambda}}_n, \bar{\bm{\lambda}}_n, \hat{\bm{\lambda}}_n, \tilde{\bm{\lambda}}_n$
: A List of Introduced Dual Variables
\[tab:dual\_variables\]
Next, we propose to divide the optimization variables $\bm{\mathcal{Y}}$ into the following groups: $\{s_n^s,s_n^r\}$, $\{\bar{s}_n^s,\bar{s}_n^r\}$, $\{p_n^s,p_n^r\}$, $\{\tilde{s}_m,e_m\}$, $\{\bar{\mathbf{q}}_n, d_n^S, \hat{\mathbf{q}}_n,t_n^L,\dot{\mathbf{q}}_n,d_n^D\}$, $\{\mathbf{q}_{n},\tilde{\mathbf{q}}_{n+1},\breve{v}_{n+1}, \tilde{v}_n\}$, $\{\hat{t}_n, t_n, \breve{t}_n\}$ and $\{\dot{v}_n, \bar{v}_n,$ $ P_n^s\}$, and iteratively solve problem by employing the block successive upper-bound minimization (BSUM) method [@Hong2016].[^7]
***1) Block $\{s_n^s,s_n^r\}$***: we have the following problem: $$\label{subproblem_1} \small
\begin{array}{l}
\mathop {\min }\limits_{\{ s_n^s,\; s_n^r\} } \;
\sum\limits_{n\in\mathcal{N}}(p_n^r\gamma_0 - s_n^r d_n^D + \rho \mu_n^D)^2
+ \sum\limits_{n\in\mathcal{N}}(p_n^s\gamma_0 - s_n^s d_n^S + \rho \mu_n^S)^2 \\
+ \sum\limits_{n\in\mathcal{N}\backslash\{1\}}\left(\log_2\left(1 + s_n^r\right) - \bar{s}_n^r+ \rho \zeta_n^r\right)^2
+ \sum\limits_{n\in\mathcal{N}\backslash\{N\}}\left(\log_2\left(1 + s_n^s\right) - \bar{s}_n^s+ \rho \zeta_n^s\right)^2 \\
\textrm{s.t.} \; s_n^r \geq 0,\;s_n^s \geq 0,\;{s}_1^r = 0,\; {s}_N^s = 0.
\end{array}$$ It can be observed that the optimization of $s_n^r$ and $s_n^s$ is separable and their updates for different time slot $n$ can be proceeded in parallel. Since the objective of problem is non-convex, one may need to employ the fixed-point method to directly solve it. In this work, we take an alternative by minimizing an approximate function of the objective and the solution can be obtained in closed-form, the details are relegated to Appendix \[appendix\_subproblem1\].
***2) Block $\{\bar{s}_n^s,\bar{s}_n^r\}$***: the corresponding optimization problem can be expressed as $$\label{subproblem_2} \small
\begin{array}{l}
\mathop {\min }\limits_{\{ \bar{s}_n^s,\; \bar{s}_n^r\} } \;-\hat{f}_{\textrm{EE}} (\{p_n^s,p_n^r,\bar{s}_n^r\})
+ \frac{1}{2\rho}\sum\limits_{n\in\mathcal{N}\backslash\{1\}}\left(\log_2\left(1 + s_n^r\right) - \bar{s}_n^r+ \rho \zeta_n^r\right)^2\\
+ \frac{1}{2\rho}\sum\limits_{n\in\mathcal{N}\backslash\{N\}}\left(\log_2\left(1 + s_n^s\right) - \bar{s}_n^s+ \rho \zeta_n^s\right)^2
+ \frac{1}{2\rho}\sum\limits_{m\in\mathcal{N}\backslash\{1\}}\left(\sum\limits_{n=2}^{m} \bar{s}_n^r - \sum\limits_{n=1}^{m-1} \bar{s}_n^s - \tilde{s}_m+ \rho \zeta_m^i\right)^2 \\
\textrm{s.t.}\; \eqref{PDD_problem_cons2},\; \bar{s}_1^r = 0,\; \bar{s}_N^s = 0,
\end{array}$$ which is convex. It can be easily verified that problem satisfies the Slater’s condition [@ConvexOptimization], therefore strong duality holds for and it can be globally solved by resorting to its Lagrangian dual problem. Specifically, a closed-form solution can be derived and the details are demonstrated in Appendix \[appendix\_subproblem2\].
***3) Block $\{p_n^s,p_n^r\}$***: we have the following problem: $$\label{subproblem_3} \small
\begin{array}{l}
\mathop {\min }\limits_{\{ p_n^s,\;p_n^r\} } -\hat{f}_{\textrm{EE}} (\{p_n^s,p_n^r,\bar{s}_n^r\})
+ \frac{1}{2\rho}\sum\limits_{n\in\mathcal{N}}\left((p_n^r\gamma_0 - s_n^r d_n^D + \rho \mu_n^D)^2
+ (p_n^s\gamma_0 - s_n^s d_n^S + \rho \mu_n^S)^2\right)\\
\textrm{s.t.}\; \eqref{power_cons_ori_UAV}.
\end{array}$$ Since $\hat{f}_{\textrm{EE}} (\{p_n^s,p_n^r,\bar{s}_n^r\})$ is convex with respect to $ p_n^s$ and $p_n^r$, the objective function of problem is in DC form. Thus, by employing the BSUM method, the updates of these variables can also be conducted in closed-form, which is detailed in Appendix \[appendix\_subproblem3\].
***4) Block $\{\tilde{s}_m,e_m\}$***: this subproblem can be written as $$\label{subproblem_4} \small
\begin{array}{l}
\mathop {\min }\limits_{\{\tilde{s}_m, e_m\} }
\sum\limits_{m\in\mathcal{N}\backslash\{1\}}\left(\sum\limits_{n=2}^{m} \bar{s}_n^r - \sum\limits_{n=1}^{m-1} \bar{s}_n^s - \tilde{s}_m+ \rho \zeta_m^i\right)^2\\
+\sum\limits_{m\in\mathcal{N}}\left(- \kappa \tilde{v}_{m+1} +\Big( \sum\limits_{i=1}^m \breve{t}_i + m b_2\Big)\delta_t - e_m+ \rho \zeta_m^e\right)^2\\
\textrm{s.t.}\; \eqref{PDD_problem_cons1},\;\eqref{PDD_problem_cons3}.
\end{array}$$ Due to the convexity of problem , It can be readily seen that its optimal solution can be obtained by $ \tilde{s}_m = \Pi_{(-\infty,0]} \Big(\sum\limits_{n=2}^{m} \bar{s}_n^r - \sum\limits_{n=1}^{m-1} \bar{s}_n^s + \rho \zeta_m^i\Big)$ and $e_m = \Pi_{[\theta-\mathcal{E},0]}\Big(- \kappa \tilde{v}_{m+1} + \big(\sum\limits_{i=1}^m \breve{t}_i + m b_2\big) \delta_t+ \rho \zeta_m^e\Big)$.
***5) Block $\{\bar{\mathbf{q}}_{n}, d_n^S, \hat{\mathbf{q}}_{n}, t_n^L, \dot{\mathbf{q}}_n,d_n^D\}$***: in this case, we can observe that the variables $\{\hat{\mathbf{q}}_{n}, t_n^L\}$, $\{\bar{\mathbf{q}}_{n}, d_n^S\}$ and $\{\dot{\mathbf{q}}_n,d_n^D\}$ are already decoupled both in the objective function and the constraints, and the optimization for each slot $n$ can be proceed in parallel. Moreover, the optimization problems of these three sub-blocks exhibit a similar structure, i.e., they are all quadratically constrained quadratic programs with only one constraint (QCQP-1). Therefore, these three subproblems can be globally solved to their optimal solutions. Consider the optimization of $\{\bar{\mathbf{q}}_{n}, d_n^S\}$, we have the following problem: $$\label{subproblem_5} \small
\begin{array}{l}
\mathop {\min }\limits_{\{ \bar{\mathbf{q}}_n,d_n^S\} } \sum\limits_{n\in\mathcal{N}}(\bar{\mathbf{q}}_n - \mathbf{q}_n + \rho \bar{\bm{\lambda}}_n)^2+
\sum\limits_{n\in\mathcal{N}}(\tilde{\mathbf{q}}_n - \bar{\mathbf{q}}_n + \rho \tilde{\bm{\lambda}}_n)^2+ \sum\limits_{n\in\mathcal{N}}(p_n^s\gamma_0 - s_n^s d_n^S + \rho \mu_n^S)^2\\
\textrm{s.t.}\; H^2+\|\bar{\mathbf{q}}_n - \mathbf{q}_S\|^2 = d_n^S,
\end{array}$$ whose optimal solution and the corresponding derivation are detailed in Appendix \[appendix\_subproblem5\]. The optimization of the other two sub-blocks can be similarly addressed, and thus they are omitted here for brevity.
***6) Block $\{\mathbf{q}_{n},\tilde{\mathbf{q}}_{n+1},\breve{v}_{n+1}, \tilde{v}_n\}$***: the following optimization problem can be obtained: $$\label{subproblem_6} \small
\begin{array}{l}
\mathop {\min }\limits_{\{\mathbf{q}_{n},\tilde{\mathbf{q}}_{n+1}, \breve{v}_{n+1}, \tilde{v}_n\} } \; \sum\limits_{n\in\mathcal{N}} \left(\|\dot{\mathbf{q}}_{n} - \mathbf{q}_{n} + \rho \bar{\bm{\lambda}}_{n}\|^2 + \|\bar{\mathbf{q}}_{n} - \mathbf{q}_{n} + \rho \bar{\bm{\lambda}}_{n}\|^2
+ \|\hat{\mathbf{q}}_{n} - \mathbf{q}_{n} + \rho \hat{\bm{\lambda}}_{n}\|^2 \right) \\
+ \sum\limits_{n\in\mathcal{N}-1}\|\tilde{\mathbf{q}}_{n+1} - \bar{\mathbf{q}}_{n+1} + \rho \tilde{\bm{\lambda}}_{n+1}\|^2 + \sum\limits_{n\in\mathcal{N}}\left(- \kappa \tilde{v}_{n+1} + \Big(\sum\limits_{i=1}^n \breve{t}_i +n b_2\Big)\delta_t - e_n+ \rho \zeta_n^e\right)^2\\
+ \sum\limits_{n\in\mathcal{N}-1} \left( \breve{v}_{n+1} - \dot{v}_{n+1} +\rho \tau_{n+1} \right)^2 + \sum\limits_{n\in\mathcal{N}} \left( \left( \tilde{v}_{n} - \dot{v}_{n} +\rho \tilde{\tau}_n \right)^2+\left( \breve{v}_{n+1} - \tilde{v}_{n} -\bar{v}_n+\rho \bar{\tau}_n \right)^2 \right)
\\
\textrm{s.t.}\; \frac{\|\tilde{\mathbf{q}}_{n+1} - \mathbf{q}_{n}\|^2}{\delta_t^2} =\breve{v}_{n+1} - \tilde{v}_n,\;\forall n,\\
\end{array}$$ which is also a QCQP-1 problem when restricting to one particular $n$. Therefore, the method proposed in Appendix \[appendix\_subproblem5\] can be easily modified to solve problem . However, in this block, three special cases need to be considered: 1) when $n=1$, we set $\mathbf{q}_1 = \mathbf{q}_I$ and the other variables can be obtained by solving the resulting problem; 2) when $n=N-1$, $\tilde{\mathbf{q}}_{n+1} = \mathbf{q}_F$ should be satisfied; 3) when $n=N$, we set $\mathbf{q}_N = \mathbf{q}_F$ and the optimization problem of $\tilde{v}_N$ can be expressed as $$\label{subproblem_6_1} \small
\begin{array}{l}
\mathop {\min }\limits_{\tilde{v}_N }
\left( \breve{v}_{N} - \tilde{v}_{N} +\rho \tau_N \right)^2 + \left(- \kappa \tilde{v}_{N} + (\tilde{t}_N + (N-1)b_2)\delta_t - e_{N-1}+ \rho \zeta_{N-1}^e\right)^2\\
+ \left(- \kappa \tilde{v}_{N} +( \tilde{t}_{N+1}+ N b_2)\delta_t - e_{N}+ \rho \zeta_{N}^e\right)^2.
\end{array}$$ which is an unconstrained quadratic program (QP) and can be easily solved.
***7) Block $\{\hat{t}_n, t_n, \breve{t}_n\}$***: in this block, since $\hat{t}_n$, $t_n$ and $\breve{t}_n$ are coupled in the objective function of problem , we propose to optimize them using the one-iteration block coordinate descent (BCD) method and some proper approximations are employed when necessary. Specifically, for $t_n$, we have the following non-convex problem: $$\label{block7_1} \small
\begin{array}{l}
\mathop {\min }\limits_{\{ t_n\} }\; - \gamma \frac{a_2 b_1 t_n }{\sum\limits_{n=1}^N P_n^s} + \frac{1}{2\rho}\sum\limits_{n\in\mathcal{N}}(t_n P_n^s - \hat{t}_n + \rho \xi_n^S)^2\\
+ \frac{1}{2\rho}\sum\limits_{n\in\mathcal{N}} \left( \ln(t_n) - \alpha t_n^L +\rho \xi_n^L \right)^2
+ \frac{1}{2\rho}\sum\limits_{n \in\mathcal{N}} (\breve{t}_{n}-(a_1a_2\hat{t}_n+a_2b_1 t_n)+\rho\tilde{\eta}_n)^2.
\end{array}$$ Since $e^{-\alpha \sqrt{H^2 + \|\mathbf{q}_n - \mathbf{q}_P\|^2}} = t_n$, we can infer that $e^{-\alpha \sqrt{d_{\textrm{max}}}} \leq t_n \leq e^{-\alpha \sqrt{H^2}}$ must be satisfied, where $d_{\textrm{max}}$ denotes the maximum squared distance between the UAV and the PB, which can be obtained by $d_{\textrm{max}} = \max(\|\mathbf{q}_S-\mathbf{q}_P\|^2, \|\mathbf{q}_D-\mathbf{q}_P,\|\mathbf{q}_I-\mathbf{q}_P\|^2,\|\mathbf{q}_F-\mathbf{q}_P\|^2)$. Consequently, according to a similar derivation as in ***Block 1***, problem can be approximated by $$\label{subproblem_t} \small
\min\limits_{t_n}\; a t_n^2+b t_n,$$ where $a = \frac{1}{2\rho} (P_n^s)^2+ \frac{1}{2\rho} (a_2 b_1)^2 + \frac{\phi}{2\rho}$, $ b = -\gamma\frac{ a_2 b_1}{\sum\limits_{n=1}^N P_n^s} + \frac{1}{\rho} P_n^s(\rho \xi_n^S-\hat{t}_n)+ \frac{1}{\rho}\left(\ln(\tilde{t}_n) - \alpha t_n^L +\rho \xi_n^L\right)\frac{1}{\tilde{t}_n}
+ \frac{1}{\rho} (a_1a_2\hat{t}_n- \breve{t}_{n}- \rho \tilde{\eta}_n)a_2b_1 - \frac{ \phi}{\rho} \tilde{t}_n $, $\tilde{t}_n$ denotes the value of $t_n$ in the previous iteration and $\phi = \frac{1- (\ln(e^{-\alpha \sqrt{d_{\textrm{max}}}}) - \alpha t_n^L+\rho \xi_n^L)}{(e^{-\alpha \sqrt{d_{\textrm{max}}}})^2}$. The optimal solution to problem is $t_n = -0.5b/a$. As for the optimization of $\hat{t}_n$ and $\breve{t}_n$, we only need to solve two unconstrained QPs, which can be done without much difficulty. ***8) Block $\{\dot{v}_n, \bar{v}_n, P_n^s\}$***: in this case, the variables $\dot{v}_n$, $\bar{v}_n$ and $P_n^s$ are mutually separable and independent of each other. To be specific, the subproblems with respect to $\dot{v}$ and $\bar{v}_n$ can be expressed as $$\small
\mathop {\min }\limits_{\{ \dot{v}_n\} } \; \sum\limits_{n\in\mathcal{N}} \left( \breve{v}_{n} - \dot{v}_{n} +\rho \tau_n \right)^2 + \sum\limits_{n\in\mathcal{N}} \left( \tilde{v}_{n} - \dot{v}_{n} +\rho \tilde{\tau}_n \right)^2,$$ $$\label{block8_1} \small
\begin{array}{l}
\min\limits_{\bar{v}_n} \; \sum\limits_{n\in\mathcal{N}} \left( \breve{v}_{n+1} - \tilde{v}_{n} -\bar{v}_n+\rho \bar{\tau}_n \right)^2\\
\textrm{s.t.}\; \bar{v}_{n}\leq v_{\textrm{max}}^2,\; \forall n.
\end{array}$$ Their optimal solutions can be obtained by $\dot{v}_n = (\breve{v}_{n}+\rho \tau_n+\tilde{v}_{n} +\rho \tilde{\tau}_n)/2$ and $\bar{v}_n=\Pi_{[0, v_{\textrm{max}}^2]}(\breve{v}_{n+1} - \tilde{v}_{n} +\rho \bar{\tau}_n)$, respectively. Finally, the optimization problem of $P_n^S$ can be written as $$\label{block8_2} \small
\begin{array}{l}
\mathop {\min }\limits_{\{ P_n^s\} } \; - \gamma \hat{f}_{\textrm{PE}} (\{t_n,\hat{t}_n,P_n^s\})+ \frac{1}{2\rho}\sum\limits_{n\in\mathcal{N}}(t_n P_n^s - \hat{t}_n + \rho \xi_n^S)^2\\
\textrm{s.t.}\; P_{\textrm{min}}^s \leq P_n^s \leq P_{\textrm{max}}^s,\forall n.
\end{array}$$ Let $\mathbf{x} = [P_1^s,\cdots,P_N^s]^T$, it can be seen that $ \gamma \hat{f}_{\textrm{PE}} (\{t_n,\hat{t}_n,P_n^s\})$ is jointly concave over the variables in $\mathbf{x}$, therefore the objective function of is a DC function with respect to $\mathbf{x}$. The detailed procedure to solve problem is relegated to Appendix \[appendix\_subproblem8\].
Besides, the dual variables can be updated according to ${\bm{\lambda}} = {\bm{\lambda}} + \frac{1}{\rho}(\mathbf{x}-\mathbf{y})$, where $\mathbf{x}=\mathbf{y}$ denotes a toy example of the equality constraint in Table \[tab:dual\_variables\] and ${\bm{\lambda}}$ denotes the corresponding dual variable. To summarize, the proposed PDD-based algorithm is shown in Algorithm \[PDD\_algorithm\]. As for its convergence property, we have the following proposition:
\[prop2\] Every limit point of the sequence generated by Algorithm \[PDD\_algorithm\] is a stationary solution of problem .
Please refer to reference [@ShiPDD2017] for the detailed proof.
Furthermore, we can observe that the complexity of Algorithm \[PDD\_algorithm\] is dominated by solving problem $L^o L^i$ times, where $L^o$ and $L^i$ denote the required numbers of outer and inner iterations. Therefore, the complexity of Algorithm \[PDD\_algorithm\] is on the order of $\mathcal{O}(L^o L^i (2N-2)^3)$.
Initialize $\bm{\mathcal{Y}}_0$ and $\rho_0$, choose $q<1$. Set the outer iteration number $l^o = 0$. Set the inner iteration number $l^i = 0$. Update the variables in $\bm{\mathcal{Y}}$ by successively optimizing them in **Blocks 1-8**. $l^i \leftarrow l^i +1 $. Update the dual variables and set $\rho \leftarrow q \rho $. $l^o \leftarrow l^o +1 $.
Simulation Results {#sec_simulations}
==================
In this section, we provide simulation results to validate the effectiveness of our proposed algorithms and mobile relaying design. In the considered system, the location of the destination is set to $\mathbf{q}_D = (x_D=1000\textrm{m},y_D=0,0)$, i.e., the source and the destination is separated by $1000$m. The nominal system configuration is defined by the following choice of parameters: $\gamma_0= 80$dB, $v_{\textrm{max}}=15$m/s, $H = 100$m, $P_{\textrm{min}}^s = 10$W, $P_{\textrm{max}}^s = 100$W, $p_{\textrm{max}}^s = p_{\textrm{max}}^r=20$dBm, $R_{\textrm{sum}} = 100$bps/Hz, $\upsilon^s = \upsilon^r = 5$, $M=9.7$kg, $T=120$s, $\delta_t=4$s, $\mathcal{E} = 10^5$J and $\theta = 10^3$J, where the UAV-related parameters are set according to [@DJIUAV]. Unless otherwise stated, the parameters correspond to the 810nm laser and the clear air weather condition in Table \[tab:laser\_parameter\] are used throughout this paper. The constant power consumption $P_{\textrm{on}}$ is set as follows [@Cui2004; @Shi2016EE]: $$\small
P_{\textrm{on}} = 2(P_{\textrm{DAC}} + P_{\textrm{mix}} + P_{\textrm{filt}}) +3 P_{\textrm{syn}} + 2(P_{\textrm{LNA}} + P_{\textrm{mix}}+P_{\textrm{IFA}} + P_{\textrm{filr}}+P_{\textrm{ADC}}),$$ where $P_{\textrm{DAC}}$, $P_{\textrm{mix}}$, $P_{\textrm{filt}}$, $P_{\textrm{syn}}$, $P_{\textrm{LNA}}$, $P_{\textrm{IFA}}$, $P_{\textrm{filr}}$ and $P_{\textrm{ADC}}$ denote the power consumption of the digital to analog converter (DAC), the mixer, the active filters at the transmitter side, the frequency synthesizer, the low-noise amplifier (LNA), the intermediate frequency amplifier (IFA), the active filters at the receiver side, and the analog to digital converter (ADC), respectively. For the detailed values of these parameters, please refer to [@Cui2004] and [@Shi2016EE]. For comparison, we also provide the performance of the AO-based algorithm [@Zeng2016], where the optimization variables are divided into two groups, i.e., 1) the transmit powers of the source and the UAV; 2) the transmit power of the PB and the trajectory of the UAV. These two groups of variables are alternatively optimized with the other fixed. Note that in the AO-based algorithm, the concept of CCCP is also needed to solve the optimization problem of the second group, therefore it is also a double-loop algorithm. In our simulations, a maximum of $100$ iterations are employed to optimize the variables in the second group.
### Convergence property
We first investigate the convergence behaviors of the proposed algorithms, i.e., Algorithm \[CCCP\_algorithm\] (the CCCP-based algorithm) and Algorithm \[PDD\_algorithm\] (the PDD-based algorithm), with different values of $\gamma$, and the results are shown in Fig. \[convergence\_CCCP\_AO\] and \[convergence\_PDD\]. It can be observed from Fig. \[convergence\_CCCP\_AO\] that the CCCP-based algorithm is monotonic convergent, i.e., the obtained objective value in the current iteration is always larger than or equal to that obtained in the preceding iteration, and Algorithm \[CCCP\_algorithm\] needs a few hundreds of iterations to obtain steady performance. An appealing property of this algorithm is that the solution obtained in each iteration is always feasible, thus even if it is terminated before convergence, the resulting solution is still applicable. In Fig. \[convergence\_PDD\], we demonstrate the convergence behavior of Algorithm \[PDD\_algorithm\] in terms of the objective value and the constraint violation.[^8] As can be seen, Algorithm \[PDD\_algorithm\] converges within $700$ iterations, although this number is larger than that required by Algorithm \[CCCP\_algorithm\], this does not necessarily mean that Algorithm \[PDD\_algorithm\] is more complex. On the contrary, Algorithm \[PDD\_algorithm\] is much more simple and implementation-friendly since in each block, the variables for different time slots can be updated in parallel, which makes distributed computing possible. Furthermore, each updating step can either be completed in closed-form or by the bisection method, and this attractive characteristic of Algorithm \[PDD\_algorithm\] avoids the usage of software solvers (usually treated as black-boxes).
Then, in Table \[tab:performance\_comparison\], we list the steady state performance achieved by the considered algorithms, where we assume that the UAV’s initial and final $x-y$ coordinates are predetermined to $(x_I, y_I) = (0, 500\textrm{m})$ and $(x_F , y_F) = (1000\textrm{m}, 500\textrm{m})$ and the location of the PB is set to $(x_{PB},y_{PB}) = (500\textrm{m},800\textrm{m})$. It is observed that when $\gamma=1$, the weighted efficiencies obtained by the considered algorithms are close to each other and the proposed Algorithm \[CCCP\_algorithm\] achieves the best performance. When $\gamma$ is larger, i.e., $\gamma=100$ or $1000$, Algorithms \[CCCP\_algorithm\] and \[PDD\_algorithm\] achieve superior performance gains over the AO-based algorithm, i.e., in these cases the performance of the AO-based algorithm is not competitive anymore. Meanwhile, Algorithms \[CCCP\_algorithm\] and \[PDD\_algorithm\] can achieve a similar performance. As the AO-based algorithm is recognized as the must commonly used algorithm (also it can be viewed as the state-of-the-art) for joint power and trajectory optimization in UAV-enabled mobile relaying systems, our results suggest that the proposed CCCP and PDD-based algorithms are more powerful when handling difficult objective functions and constraints and better performance can be achieved. Moreover, generally, the AO-based algorithm has no theoretical guarantee on the quality of the converged solution, while for our proposed Algorithms \[CCCP\_algorithm\] and \[PDD\_algorithm\], stationary solutions can be assured according to Propositions \[prop1\] and \[prop2\], respectively.
$\gamma=1$ $\gamma=100$ $\gamma=1000$
------------------------------- ------------ -------------- --------------- -- -- -- -- -- -- -- --
AO 5.64 20.38 168.75
Algorithm \[CCCP\_algorithm\] 5.66 25.03 220.34
Algorithm \[PDD\_algorithm\] 4.94 25.17 223.97
: Steady state performance comparison
\[tab:performance\_comparison\]
### Impacts of $\gamma$
In Fig. \[fig\_traj\] and Fig. \[fig\_PS\_ps\_pr\], we illustrate the trajectories and transmit powers obtained by the considered algorithms, where the same simulation parameters as that in Table \[tab:performance\_comparison\] are used. As can be seen, when $f_{\textrm{PE}}(\cdot)$ is of relatively low priority (i.e., when $\gamma=1$), the trajectories obtained by the considered three algorithms are similar to each other, i.e., the UAV tends to first fly towards the source to have a better receive SNR (or equivalently receive data rate). Then, it flies close to the destination to deliver the received information from the source to the destination. For the AO-based algorithm and Algorithm \[CCCP\_algorithm\], the source and the PB are prone to transmit with a larger power when they are near the UAV, and the UAV’s transmit power is also in positive proportion to its distance to the destination. However, Algorithm \[PDD\_algorithm\] gets stuck in an unfavorable stationary point in this case. When the priority of $f_{\textrm{PE}}(\cdot)$ gets higher (i.e., when $\gamma=100$ or $1000$), it can be observed that the optimized trajectories are trying to get close to the PB such that the power transmission efficiency would be larger. It is also interesting to see that the trajectories obtained by the considered three algorithms are totally different when $\gamma=100$, this is mainly due to the different design methodologies when deriving the considered algorithms. Furthermore, it is observed that the UAV flies with a faster speed in some less rewarding locations, e.g., when $\gamma=1$, the UAV maintains a slower speed when it is close to the source and the destination since in this case, $f_{\textrm{EE}}(\cdot)$ plays a more important role than $f_{\textrm{PE}}(\cdot)$. When $\gamma=1000$, the UAV slows down when it is near the PB such that a higher power transmission efficiency can be achieved.
In Fig. \[traj\_evo\], we show the UAV’s trajectories obtained by the considered algorithms with different numbers of iterations when $\gamma=100$. For Algorithm \[CCCP\_algorithm\], we can see that due to the characteristic of the CCCP method, the solution obtained in the current iteration is heavily dependent on that of the previous iteration. In other words, since we approximate the original non-convex feasible set in around the previous solution by a convex subset, the trajectory obtained in the current iteration tends to improve the previous one and thus it is expected that these two trajectory would not be two far away from each other. For the AO-based algorithm, it can be observed that it converges very rapidly, i.e., the trajectory achieved in the first iteration is already very close to the converged one. Also, we can infer that the converged solution would be sensitive to the initialization. For Algorithm \[PDD\_algorithm\], it is observed that the trajectories in different iterations are not that related as those in the AO-based algorithm and Algorithm \[CCCP\_algorithm\]. This is mainly due to the fact that in the initial few iterations, the parameter $\rho$ is set to be large (a larger $\rho$ means less penalty), therefore the obtained solutions are not always feasible to problem . As a result, Algorithm \[PDD\_algorithm\] might be able to explore in a larger region of the variable space and search for solutions which can achieve a potentially larger objective value (but not necessarily feasible). Note that this distinguishing property of Algorithm \[PDD\_algorithm\] is very different from those of the AO-based algorithm and Algorithm \[CCCP\_algorithm\], who usually find a solution in certain subsets of the original feasible set. With the increasing of the iteration number and the decreasing of the parameter $\rho$, the equality constraints are forced to be satisfied and thus a feasible solution (also a stationary solution according to Proposition \[prop2\]) can be found.
### Impacts of the total flight time $T$
In Fig. \[figure\_Loiter\], the UAV’s trajectories obtained by Algorithm \[CCCP\_algorithm\] with different values of $T$ are plotted, where $\gamma$ is set to 20. It can be observed that when $T$ is sufficiently large (e.g., $T=160$s or $T=240$s), the UAV would keep a very low speed near the source (or the destination) for a certain period before it moves towards the destination (or the final location). Therefore, a possible loiter phase is implicitly included in the proposed formulation and this phase is observable when $T$ is large enough. Moreover, we can see that the longer the total flight time $T$, the closer the UAV flies to the source and the destination.
### Impacts of the laser wavelength and the weather condition
Finally, in Fig. \[figure\_weather\_wavelength\], we show the UAV’s trajectories obtained by Algorithm \[CCCP\_algorithm\] with different laser wavelengths and weather conditions, where $\gamma$ is fixed to $100$. From Fig. \[figure\_weather\_wavelength\] (a) (the weather condition is set to be clear air), we can observe that the trajectory obtained when $\lambda=1550$nm is more prone to be close to the PB. This is because the power transmission efficiency of the $1550$nm laser is lower than that of the $810$nm laser when the distance between the PB and the UAV is less than about $5$km [@ZhangDLC2018], thus the UAV should fly towards the PB for a higher power transmission efficiency when the $1550$nm laser is used. A similar observation can also be made from Fig. \[figure\_weather\_wavelength\] (b) (the $810$nm laser is used), i.e., when the weather condition is worse (in our case, fog is worse than haze and haze is worse than clear air), the UAV should be more close to the PB for a high power transmission efficiency.
Conclusion {#sec_conclusion}
==========
This paper proposed a new UAV-enabled mobile relaying system, where a laser PB is employed to wirelessly charge the energy-constrained UAV relay. We aimed to maximize the information/power transmission efficiency of the system by jointly optimizing the transmit powers and the UAV’s trajectory. Two efficient algorithms, i.e., the CCCP and PDD-based algorithms, were proposed to address the resulting problem, which is highly non-convex and challenging to solve. Numerical results were presented to validate the effectiveness of the proposed algorithms. We have demonstrated that the proposed algorithms outperform the conventional AO-based algorithm, especially when the weighting factor $\gamma$ is large. It was also shown that there is a tradeoff between maximizing the information transmission efficiency and the power transmission efficiency, and the UAV’s trajectory is highly related to the laser wavelength and the weather condition.
Solution to Problem {#appendix_subproblem1}
--------------------
Let us first focus on the optimization of $s_n^r$. Since the objective of problem is not concave and also does not exhibit a DC structure, simple linear approximation does not work in this case. As a result, we consider the following quadratic upper bound of the objective: $$\small
u_n(s_n^r, \hat{s}_n^r) = (p_n^r\gamma_0 - s_n^r d_n^D + \rho \mu_n^D)^2 + 2(\log_2(1+\hat{s}_n^r)-\bar{s}_n^r+\rho\zeta_n^r)\frac{1}{\ln2(1+\hat{s}_n^r)} s_n^r + \frac{1}{2}\phi(s_n^r-\hat{s}_n^r)^2,$$ where $\hat{s}_n^r$ denotes the value of the variable $s_n^r$ in the previous iteration, $\phi$ is a scalar which should satisfy $\phi-\bar{\phi} \geq 0$ [@Hong2016] and $\bar{\phi}$ denotes the second-order derivative of $\left(\log_2\left(1 + s_n^r\right) - \bar{s}_n^r+ \rho \zeta_n^r\right)^2$. Since $s_n^r$ is bounded by $0 \leq s_n^r \leq \frac{p_r^{\textrm{max}}\gamma_0}{H^2}$, we can set the value of $\phi$ to $$\small
\phi=\max\left(\frac{2}{(\ln2)^2(1+\hat{s}_n^r)^2} -2(\log_2(1+\hat{s}_n^r)-\bar{s}_n^r+\rho\zeta_n^r)\frac{1}{\ln2 (1+\hat{s}_n^r)^2}\right)\\
= \frac{2+2\ln 2(\bar{s}_n^r-\rho\zeta_n^r)}{(\ln2)^2}.$$
With the aforementioned approximation, we have the following problem: $$\small
\begin{array}{l}
\min \limits_{s_n^r} \; u_n(s_n^r, \hat{s}_n^r)\\
\textrm{s.t.} \; s_n^r \geq 0,\; s_1^r = 0,
\end{array}$$ whose optimal solution can be obtained by a simple projection operation, i.e., $$\small
s_n^r = \Pi_{[0,+\infty)} \left(\frac{\phi \hat{s}_n^r+2d_n^Dp_n^r \gamma_0+2\rho d_n^D\mu_n^D - \frac{2}{\ln 2(1+\hat{s}_n^r)} (\log_2(1+\hat{s}_n^r)-\bar{s}_n^r+\rho \zeta_n^r) }{2(d_n^D)^2+\phi}\right),\;n\in \mathcal{N}\backslash\{1\},$$ and $s_1^r = 0$. The optimization of $s_n^s$ can be similarly tackled without difficulty.
Optimal Solution to Problem {#appendix_subproblem2}
----------------------------
We first introduce a Lagrange multiplier $\lambda$ to the first constraint of problem and define the following partial Lagrangian: $\mathcal{L}(\{\bar{s}_n^s\},\{\bar{s}_n^r\},\lambda) \triangleq f_{\eqref{subproblem_2}}(\bar{s}_n^s, \bar{s}_n^r)+ \lambda\Big(R_{\textrm{sum}} - \sum\limits_{n=2}^{N} \bar{s}_n^r \Big)$, where $f_{\eqref{subproblem_2}}(\bar{s}_n^s, \bar{s}_n^r)$ denotes the objective function of problem . Then, the dual function, denoted by $d(\lambda)$, can be written as $$\label{dual_function} \small
\begin{array}{l}
d(\lambda) \triangleq \min\limits_{\{\bar{s}_n^s\},\{\bar{s}_n^r\}} \mathcal{L}(\{\bar{s}_n^s\},\{\bar{s}_n^r\},\lambda).
\end{array}$$ We need to find a nonnegative $\lambda$ to minimize the dual function $d(\lambda) $, i.e., solving the dual problem: $\max\limits_{\lambda \geq 0} \; d(\lambda)$.
In order to express problem in a more compact form, we introduce the following notations: $$\small
\begin{array}{l}
\mathbf{x} = [\bar{s}_1^s,\cdots,\bar{s}_{N-1}^s,\bar{s}_2^r,\cdots,\bar{s}_N^r]^T,\;
\mathbf{a}_1 = [\mathbf{0}_{N-1},-\mathbf{1}_{N-1}]^T,\\
\mathbf{a}_m = [-\mathbf{1}_{(m-1)\times1},\mathbf{0}_{(N-m)\times 1},\mathbf{1}_{(m-1)\times1},\mathbf{0}_{(N-m)\times 1} ]^T,\;m\in\mathcal{N}\backslash\{1\},\;
a_m = \tilde{s}_m - \rho \zeta_m^i,\\
b_n^r = \log_2\left(1 + s_n^r\right) + \rho \zeta_n^r,\;
b_n^s = \log_2\left(1 + s_n^s\right) + \rho \zeta_n^s,\;
\mathbf{b}_n^r = [\mathbf{0}_{(N-1)\times 1},\mathbf{e}_{n-1}]^T,\;
\mathbf{b}_n^s = [\mathbf{e}_{n}, \mathbf{0}_{(N-1)\times 1}]^T,\\
\mathbf{B} = \frac{1}{2\rho} \sum\limits_{n\in\mathcal{N}\backslash\{1\}}\mathbf{b}_n^{r}\mathbf{b}_n^{rT} + \frac{1}{2\rho}\sum\limits_{n\in\mathcal{N}\backslash\{N\}} \mathbf{b}_n^{s} \mathbf{b}_n^{sT} + \frac{1}{2\rho}\sum\limits_{m\in\mathcal{N}\backslash\{1\}} \mathbf{a}_m \mathbf{a}_m^T ,\\
\mathbf{b} = -\frac{1}{\rho} \sum\limits_{n\in\mathcal{N}\backslash\{1\}} b_n^r \mathbf{b}_n^r - \frac{1}{\rho}\sum\limits_{n\in\mathcal{N}\backslash\{N\}} b_n^s \mathbf{b}_n^s- \frac{1}{\rho}\sum\limits_{m\in\mathcal{N}\backslash\{1\}} a_m \mathbf{a}_m+\frac{\mathbf{a}_1 }{\upsilon^s\sum\limits_{n = 1}^{N-1} p_n^s + \upsilon^r\sum\limits_{n = 2}^{N} p_n^r + N P_{\textrm{on}}} ,\\
\end{array}$$ where $\mathbf{e}_n$ denotes a vector with a single non-zero component (equals to $1$) located at $n$, as a result, problem can be equivalently formulated as $$\small
\min\limits_{\{\bar{s}_n^r,\;\bar{s}_n^s\}} \;\mathbf{x}^T \mathbf{B} \mathbf{x} + \mathbf{x}^T (\mathbf{b}+\lambda\mathbf{a}_1) + \lambda R_{\textrm{sum}},$$ By resorting to the first order optimality condition, we have $\mathbf{x}^* = -\frac{1}{2}\mathbf{B}^{-1}(\mathbf{b}+\lambda^*\mathbf{a}_1)$ and $\mathbf{a}_1^T \mathbf{x}^* + R_{\textrm{sum}} = 0$, where $\mathbf{x}^*$ and $\lambda^*$ denote the optimal primal and dual variables. Therefore, the optimal dual variable $\lambda$ can be obtained by $\lambda^* = \frac{2R_{\textrm{sum}} - \mathbf{a}_1^T \mathbf{B}^{-1}\mathbf{b}}{\mathbf{a}_1^T \mathbf{B}^{-1} \mathbf{a}_1}$, and the optimal solution of problem can be expressed as $$\small
\mathbf{x}^* = \left\{ \begin{array}{l}
-\frac{1}{2}\mathbf{B}^{-1}\mathbf{b},\; \textrm{if} \; \frac{1}{2}\mathbf{a}_1^T \mathbf{B}^{-1} \mathbf{b} \geq R_{\textrm{sum}},\\
-\frac{1}{2}\mathbf{B}^{-1}(\mathbf{b}+\lambda^*\mathbf{a}_1),\;\textrm{otherwise}.
\end{array}
\right.$$
Solution to Problem {#appendix_subproblem3}
--------------------
According to the BSUM method, we consider the following problem, which is obtained by replacing the objective of by its linear approximation, $$\label{subproblem_3_1} \small
\begin{array}{l}
\mathop {\min }\limits_{\{ p_n^s,p_n^r\} } -\left( \sum\limits_{n = 1}^{N-1} g_n(\hat{p}_n^r,\hat{p}_n^s)(p_n^r - \hat{p}_n^r) + \sum\limits_{n = 2}^{N} g_n(\hat{p}_n^r,\hat{p}_n^s)(p_n^s - \hat{p}_n^s)\right)\\
+ \frac{1}{2\rho}\sum\limits_{n\in\mathcal{N}}(p_n^r\gamma_0 - s_n^r d_n^D + \rho \mu_n^D)^2
+ \frac{1}{2\rho}\sum\limits_{n\in\mathcal{N}}(p_n^s\gamma_0 - s_n^s d_n^S + \rho \mu_n^S)^2\\
\textrm{s.t.}\; \eqref{power_cons_ori_UAV}, \end{array}$$ where $g_n(\hat{p}_n^r,\hat{p}_n^s) = {-\sum\limits_{n = 2}^{N} \bar{s}_n^r }\Big{/}{\left(\upsilon^s\sum\limits_{n = 1}^{N-1} \hat{p}_n^s + \upsilon^r\sum\limits_{n = 2}^{N} \hat{p}_n^r + N P_{\textrm{on}}\right)^2}$, $\hat{p}_n^r$ and $\hat{p}_n^s$ denote the values of ${p}_n^r$ and ${p}_n^s$ in the previous iteration. As can be seen, problem is separable among different $n$ and its optimal solution can be obtained by a simple projection operation, i.e. $p_n^r = \Pi_{[0,p_{\textrm{max}}^r]}\left(\frac{g_n(\hat{p}_n^r,\hat{p}_n^s)\rho}{\gamma_0^2} + \frac{s_n^r d_n^D - \rho \mu_n^D}{\gamma_0}\right)$ and $
p_n^s = \Pi_{[0,p_{\textrm{max}}^s]}\left(\frac{g_n(\hat{p}_n^r,\hat{p}_n^s)\rho}{\gamma_0^2} + \frac{s_n^s d_n^S - \rho \mu_n^S}{\gamma_0}\right)$.
Optimal Solution to Problem {#appendix_subproblem5}
----------------------------
In order to express problem in a standard form, we introduce the following notations: $\mathbf{x} = [\bar{\mathbf{q}}_n^T, d_n^S]^T$, $\mathbf{a} = [(- \mathbf{q}_n + \rho \bar{\bm{\lambda}}_n)^T,0]^T$, $\tilde{\mathbf{a}} = [(- \tilde{\mathbf{q}}_n - \rho \tilde{\bm{\lambda}}_n)^T,0]^T$, $\bar{\mathbf{a}} = [0,0,s_n^s]^T$, $\tilde{\mathbf{A}} = \mathbf{I} - \mathbf{e}_3\mathbf{e}_3^T$, $c = -p_n^s\gamma_0 - \rho \mu_n^S$, $\mathbf{A} = 2\tilde{\mathbf{A}}+ \bar{\mathbf{a}}\bar{\mathbf{a}}^T$, $\mathbf{b} = 2\tilde{\mathbf{A}}^T\mathbf{a} + 2\tilde{\mathbf{A}}^T\tilde{\mathbf{a}} +2\bar{\mathbf{a}}c$, $\mathbf{c} = [\mathbf{q}_S^T,0]^T$, $\mathbf{d} = [0,0,1]^T$, $\tilde{\mathbf{c}} = -2\tilde{\mathbf{A}}^T\mathbf{c} - \mathbf{d}$ and $d = \mathbf{c}^T\mathbf{c} + H^2 $. Consequently, problem can be equivalently written as $$\label{QCQP_1_3D} \small
\begin{array}{l}
\min\limits_{\mathbf{x}}\;\mathbf{x}^T \mathbf{A} \mathbf{x} + \mathbf{x}^T \mathbf{b}\\
\textrm{s.t.}\;\mathbf{x}^T\tilde{\mathbf{A}}\mathbf{x} + \mathbf{x}^T \tilde{\mathbf{c}} + d = 0.
\end{array}$$ The Lagrangian of problem can be expressed as $\mathcal{L} = \mathbf{x}^T \mathbf{A} \mathbf{x} + \mathbf{x}^T \mathbf{b} + \lambda(\mathbf{x}^T\tilde{\mathbf{A}}\mathbf{x} + \mathbf{x}^T \tilde{\mathbf{c}} + d)$, where $\lambda$ denotes the Lagrangian multiplier. According to the first-order optimality condition, we have $\mathbf{x} = (2 \mathbf{A} + 2\lambda \tilde{\mathbf{A}})^{-1}(-\mathbf{b}-\lambda \tilde{\mathbf{c}} )$. Since $2 \mathbf{A} + 2\lambda \tilde{\mathbf{A}} \succeq \mathbf{0}$ should be satisfied in order to make problem feasible, thus $\lambda \geq \max(-1,-(s_n^s)^2)$ holds. Then, the optimal dual variable $\lambda^*$ can be found by resorting to the bisection method or the Newton method and then the optimal solution of problem can be obtained.
Solution to Problem {#appendix_subproblem8}
--------------------
By applying first-order approximation to $ \gamma \hat{f}_{\textrm{PE}} (\{t_n,\hat{t}_n,P_n^s\})$, we can obtain the following convex approximation of problem : $$\label{subproblem_8} \small
\begin{array}{l}
\min \limits_{\mathbf{x}}\; \mathbf{x}^T \mathbf{A}\mathbf{x} + \mathbf{x}^T \mathbf{b}\\
\textrm{s.t.} \; P_{\textrm{min}}^s \mathbf{1} \leq \mathbf{x} \leq P_{\textrm{max}}^s \mathbf{1},
\end{array}$$ where $\mathbf{A} = \frac{1}{2\rho}\sum\limits_{n\in \mathcal{N}}t_n^2 \mathbf{e}_n \mathbf{e}_n^T $, $\mathbf{b} = \frac{1}{\rho} \sum \limits_{n \in \mathcal{N}}t_na_n \mathbf{e}_n+ \sum\limits_{n\in \mathcal{N}} \frac{\gamma \sum\limits_{n=1}^N (a_1 a_2 \hat{t}_n + a_2 b_1 t_n + b_2) }{\left(\sum\limits_{n=1}^N \tilde{P}_n^s\right)^2}\mathbf{e}_n$, $a_n = - \hat{t}_n + \rho \xi_n^S$ and $\tilde{P}_n^s$ denotes the value of ${P}_n^s$ in the previous iteration. It can be observed that with this approximation, problem is fully decomposed for different $n$, due to the fact that $\mathbf{A}$ is a diagonal matrix. Then, the optimal solution of problem can be expressed as $\mathbf{x} = \Pi_{[P_{\textrm{min}}^s,P_{\textrm{max}}^s]}\left((-\mathbf{b}/2)\odot(1/\textrm{diag}(\mathbf{A}))\right)$.
[^1]: M. M. Zhao and M. J. Zhao are with the College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310027, China (e-mail: {zmmblack, mjzhao}@zju.edu.cn).
Q. Shi is with the School of Software Engineering, Tongji University, Shanghai 200092, China (e-mail: shiqj@tongji.edu.cn).
[^2]: Note that the proposed algorithms can be extended to the case where the UAV’s altitude $H$ is also a design variable without much difficulty.
[^3]: For a fixed-wing UAV, the mobility constraints should further include $\|\mathbf{v}_n \| \geq v_{\textrm{min}}$ and $\arccos\Big(\frac{(\mathbf{q}_{n+1}-\mathbf{q}_n) \boldsymbol{\cdot} (\mathbf{q}_{n}-\mathbf{q}_{n-1}) }{\|\mathbf{q}_{n+1}-\mathbf{q}_n\| \|\mathbf{q}_{n}-\mathbf{q}_{n-1}\|}\Big) \leq \varpi_{\textrm{max}}$, where $v_{\textrm{min}}$ denotes the stall speed and $\varpi_{\textrm{max}}$ represents the maximum angular turn rate in $\textrm{rad}/s$. However, in order to better focus on laser charging, we only consider constraint when dealing with the UAV’s mobility. Further investigation into more sophisticated UAV controls is left for future work.
[^4]: There are more practical models which assume that the energy $E_n^F$ also depends on the acceleration vector $\mathbf{a}_n$ [@Leishman2006; @Zeng2017]. Furthermore, for rotary-wing aircrafts, there would be energy consumption when the UAV is in hover state [@Dorling2017UAV]. However, in order to illustrate the merits of the proposed algorithms and to simplify derivations, we focus on model in this work.
[^5]: Note that if the UAV has enough energy during the whole flight, the considered problem would reduce to the conventional UAV-enabled relay system, a similar problem has been considered in [@Zeng2016] and it is out of the scope of this paper.
[^6]: For example, consider an equality constraint $\mathbf{x}=\mathbf{y}$, the corresponding AL part can be expressed as $f_{\textrm{AL}}(\mathbf{x},\mathbf{y}, \bm{\lambda}) = \frac{1}{2\rho} \|\mathbf{x}-\mathbf{y}+\rho \bm{\lambda}\|^2$, where $\bm{\lambda}$ denotes the dual variable and $\rho$ is the penalty parameter. In this work, since the exact expression of $f_{\textrm{AL}}(\bm{\mathcal{Y}}, \bm{\Lambda})$ is kind of tedious, we omit it for brevity but its components will be presented in the following.
[^7]: For simplicity, some of the notations are reused in these blocks and we note that the definitions of these notations are only valid in the current block.
[^8]: The constraint violation is defined as the maximum absolute value of all the equality constraints listed in Table \[tab:dual\_variables\].
|
---
author:
- Alon Levy
bibliography:
- 'semistable\_reduction\_v2.bib'
title: 'The Semistable Reduction Problem for the Space of Morphisms on $\mathbb{P}^{n}$'
---
Introduction and the Statement of the Problem
=============================================
The moduli spaces of dynamical systems on $\mathbb{P}^{n}$ are the spaces of morphisms, and more in general rational maps, defined by polynomials of degree $d$; the case of interest is $d > 1$, in which case those rational maps are not automorphisms. For each $n$ and $d$, we write each rational map $\varphi$ as $(\varphi_{0}:\ldots:\varphi_{n})$, so that the space is parametrized by the monomials of each $\varphi_{i}$ and is naturally isomorphic to a large projective space, $\mathbb{P}^{N}$. By an elementary computation, $N = (n+1){n+d \choose d} - 1$. As we will not consider more than one of these moduli spaces at a time, there is no ambiguity in writing just $N$, without explicit dependence on $n$ and $d$.
Within the space of rational maps, the space of morphisms is an affine open subvariety, denoted ${\operatorname{Hom}}_{d}^{n}$. The group ${\operatorname{PGL}}(n+1)$ acts on $\mathbb{P}^{N}$ by conjugation, corresponding to coordinate change, i.e. $A$ maps $\varphi$ to $A\varphi A^{-1}$; this action preserves ${\operatorname{Hom}}_{d}^{n}$, since the property of being a morphism is independent of coordinate change.
We study the quotient of the action using geometric invariant theory [@GIT]. To do this, we need to replace ${\operatorname{PGL}}(n+1)$ with ${\operatorname{SL}}(n+1)$, which projects onto ${\operatorname{PGL}}(n+1)$ finite-to-one. Geometric invariant theory defines stable and semistable loci for the ${\operatorname{SL}}(n+1)$-action. To take the quotient, we need to remove the unstable locus, defined as the complement of the semistable locus. The quotient of ${\operatorname{Hom}}_{d}^{n}$ by ${\operatorname{SL}}(n+1)$ is denoted $\mathrm{M}_{d}^{n}$, and parametrizes morphisms on $\mathbb{P}^{n}$ up to coordinate change. The stable and semistable loci for the action of ${\operatorname{SL}}(n+1)$ on $\mathbb{P}^{N}$ are denoted by ${\operatorname{Hom}}_{d}^{n,
s}$ and ${\operatorname{Hom}}_{d}^{n, ss}$, and their quotients are denoted by $\mathrm{M}_{d}^{n, s}$ and $\mathrm{M}_{d}^{n, ss}$.
It is a fact that every regular map is in the stable locus. More precisely, we have the following prior results, due to [@Sil96], [@PST], and [@Lev]:
\[containment\]${\operatorname{Hom}}_{d}^{n, s}$ and ${\operatorname{Hom}}_{d}^{n, ss}$ are open subvarieties of $\mathbb{P}^{N}$ such that ${\operatorname{Hom}}_{d}^{n} \subsetneq {\operatorname{Hom}}_{d}^{n, s}
\subseteq {\operatorname{Hom}}_{d}^{n, ss} \subsetneq \mathbb{P}^{N}$. The middle containment is an equality if and only if $n = 1$ and $d$ is even.
\[finite\]The stabilizer group in ${\operatorname{PGL}}(n+1)$ of each element of ${\operatorname{Hom}}_{d}^{n}$ is finite and bounded in terms of $d$ and $n$.
$\mathrm{M}_{d}^{n, ss}$ is a proper variety, as it is the quotient of the largest semistable subspace of $\mathbb{P}^{N}$ for the action of ${\operatorname{SL}}(n+1)$. We make the following simplifying,
\[ss\]A rational map $\varphi \in \mathbb{P}^{N}$ is called semistable if it is in the semistable space ${\operatorname{Hom}}_{d}^{n, ss}$.
The semistable reduction theorem states the following, answering in the affirmative a conjecture for $\mathbb{P}^{1}$ in [@STW]:
\[SSR1\]If $C$ is a complete curve with $K(C)$ its function field, and if $\varphi_{K(C)}$ is a semistable rational map on $\mathbb{P}^{n}_{K(C)}$, then there exists a curve $D$ mapping finite-to-one onto $C$ with a $\mathbb{P}^{n}$-bundle $\mathbf{P}(\mathcal{E})$ on $D$ with a self-map $\Phi$ such that,
1. The restriction of $\Phi$ to the fiber of each $x \in D$, $\varphi_{x}$, is a semistable rational self-map.
2. $\Phi$ is a semistable map over $K(D)$, and is equivalent to $\varphi_{K(D)}$ under coordinate change.
This can be seen by using an alternative formulation. Semistable reduction can be thought of as extending a rational map defined over a field $K$ to a rational map defined over a discrete valuation ring $R$ whose fraction field is $K$, in a way that is not too degenerate. The reason a discrete valuation ring suffices is that once we know we can extend to a discrete valuation ring, we can extend to some larger integral domain.
We thus obtain the following equivalent formulation of semistable reduction:
\[SSR2\]Let $G$ be a geometrically reductive group acting on a projective variety $X$ whose stable and semistable spaces are $X^{s}$ and $X^{ss}$ respectively. Let $R$ be a discrete valuation ring with fraction field $K$, and let $x_{K} \in X^{s}_{K}$. Then for some finite extension $K'$ of $K$, with $R'$ the integral closure of $R$ in $K'$, $x_{K}$ has an integral model over $R'$ with semistable reduction modulo the maximal ideal. In other words, we can find some $A \in
G(\overline{K})$ such that $A\cdot x_{K}$ has semistable reduction. If $x_{K} \in X^{ss}_{K}$, then the same result is true, except that $x_{R'}$ could be an integral model for some $x'_{K'}$ mapping to the same point of $X^{ss}//G$ such that $x'_{K'} \notin G\cdot x_{K}$.
We follow the method used in [@Zha]. Let $C$ be the Zariski closure of $x_{K}$ in $X^{ss}_{R}//G$, and reduce it modulo the maximal ideal to obtain $x_{k}$, where $k$ is the residue field of $R$. Observe that $C$ is a one-dimensional subscheme of $X^{ss}_{\overline{k}}//G$ which is isomorphic to ${\operatorname{Spec}}R$, and is as a result connected. Since $G$ is connected, the preimage $\pi^{-1}(C)$ is also connected: when $x_{K}$ is stable it follows from the fact that $\pi^{-1}(C)$ is the Zariski closure of $G\cdot x_{K}$ in $X^{ss}$, and even when it is not, $\pi^{-1}(C)$ is the union of connected orbits whose closures intersect. Since further $\pi^{-1}(C)$ surjects onto $C$, we can find an integral one-dimensional subscheme mapping surjectively to $C$. This subscheme necessarily maps finite-to-one onto $C$ by dimension counting, so it is isomorphic to some finite extension ring $R'$, giving us $K'$ as its fraction field.
Theorem \[SSR2\] can also be proven in a much more explicit way, producing for each $\varphi_{K} \in {\operatorname{Hom}}_{d}^{n, ss}$ a sequence of $A$’s conjugating it to a model with semistable reduction.
This leads to the natural question of which vector bundle classes can occur for each $C$, and more generally for each choice of $n$ and $d$. One interesting subquestion is whether, for every $C$, we can choose the bundle to be trivial. Equivalently, it asks whether for each $C$ we can find a proper $D
\subseteq {\operatorname{Hom}}_{d}^{n, ss}$ that maps finite-to-one onto $C$. For most curves upstairs, the answer should be positive, by simple dimension counting: as demonstrated in [@Sil96] and [@Lev], the complement of ${\operatorname{Hom}}_{d}^{n, ss}$ has high codimension, equal to about half of $N$. However, it turns out that the answer is sometimes negative, and in fact, for every $n$ and $d$ we can find a $C$ with only nontrivial bundle classes. More precisely:
\[bad1\]For every $n$ and $d$, there exists a curve with no trivial bundle class satisfying semistable reduction.
\[bad1.1\]An equivalent formulation for Theorem \[bad1\] is that for every $n$ and $d$ we can find a curve $C \subseteq \mathrm{M}_{d}^{n, ss}$ such that there does not exist a curve $D \subseteq {\operatorname{Hom}}_{d}^{n, ss}$ mapping onto $C$ under $\pi$.
Although most curves in ${\operatorname{Hom}}_{d}^{n, ss}$ can be completed, it does not imply we can find a nontrivial bundle on an open dense set of the Chow variety of $\mathrm{M}_{d}^{n, ss}$. In fact, as we will see in section \[goodcase\], there exist components of the Chow variety of $\mathrm{M}_{d}^{n, ss}$ where, at least generically, a nontrivial bundle is required.
Our study of bundle classes will now split into two cases. In the case of curves satisfying semistable reduction with a trivial bundle, the reformulation of Remark \[bad1.1\], in its positive form, means that we can study $D$ directly as a curve in $\mathbb{P}^{N}$. We can bound the degree of the map from $D$ to $C$ in terms of the stabilizer groups that occur on $D$. More precisely:
\[GIT\]Let $X$ be a projective variety over an algebraically closed field with an action by a geometrically reductive linear algebraic group $G$. Using the terminology of geometric invariant theory, let $D$ be a complete curve in the stable space $X^{s}$ whose quotient by $G$ is a complete curve $C$; say the map from $D$ to $C$ has degree $m$. Suppose the stabilizer is generically finite, of size $h$, and either $D$ or $C$ is normal. Then there exists a finite subgroup $S_{D} \subseteq G$, of order equal to $mh$, such that for all $x \in D$ and $g \in G$, $gx \in D$ iff $g \in S_{D}$.
\[GITc\]With the same notation and conditions as in Proposition \[GIT\], the map from $D$ to $C$ is ramified precisely at points $x \in D$ where the stabilizer group is larger than $h$, and intersects $S_{D}$ in a larger subgroup than in the generic case.
If the genus of $C$ is $0$, then the only way the the map from $D$ to $C$ could have high degree is if it ramifies over many points; therefore, Corollary \[GITc\] forces the degree to be small, at least as long as $C$ is contained in the stable locus.
In the case of curves that only satisfy semistable reduction with a nontrivial bundle, we do not have a description purely in terms of coordinates. Instead, we will study which bundle classes can be attached to every curve $C$. The question of which bundles occur is an invariant of $C$; therefore, it is essentially an invariant that we can use to study the scheme ${\operatorname{Hom}}(C, \mathrm{M}_{d}^{n, ss})$. In the sequel, we will study the scheme using the bundle class set and height invariants.
For the study of which nontrivial bundle classes can occur, first observe that fixing a $D$ for which a bundle exists, we can apply the reformulation of Theorem \[SSR2\] to obtain a unique extension of $\varphi$ locally. This can be done at every point, so it is true globally, so we have,
\[uni\]Using the notation of Theorem \[SSR1\], the bundle class $\mathbf{P}(\mathcal{E})$ depends only on $D$ and its trivialization $U_{i}$, $U_{i} \hookrightarrow {\operatorname{Hom}}_{d}^{n, ss}$.
Note that the bundle class does not necessarily depend only on $D$, regarded as an abstract curve with a map to $C$. The reason is that a point of $D$ may not be stable, which means it may correspond to one of several different orbits, whose closures intersect. However, there are only finitely many orbits corresponding to each point, so the bundle class depends on $D$ up to a finite amount; if $C$ happens to be contained in the stable locus, then it depends only on $D$.
Thus we can study which bundle classes occur for a given $C$. We will content ourselves with rational curves, for which there is a relatively easy description of all projective bundles. Recall that every vector bundle over $\mathbb{P}^{1}$ splits as a direct sum of line bundles, and that the bundle $\bigoplus_{i}\mathcal{O}(m_{i})$ is projectively equivalent to $\bigoplus_{i}\mathcal{O}(l + m_{i})$ for all $l \in \mathbb{Z}$. In other words, a $\mathbb{P}^{n}$-bundle over $\mathbb{P}^{1}$ can be written as $\mathcal{O} \oplus \mathcal{O}(m_{1}) \oplus \ldots \oplus \mathcal{O}(m_{n})$; if the $m_{i}$’s are in non-decreasing order, then the expression uniquely determines the bundle’s class. We will show that,
\[multi\]There exists a curve $C$ for which multiple non-isomorphic bundle classes can occur. In fact, suppose $C$ is isomorphic to $\mathbb{P}^{1}$, and there exists $U \subseteq {\operatorname{Hom}}_{d}^{n, ss}$ mapping finite-to-one into $C$ such that $U$ is a projective curve minus a point. Then there are always infinitely many possible classes: if the class of $U$ is thought of as splitting as $\mathbf{P}(\mathcal{E}) = \mathcal{O} \oplus \mathcal{O}(m_{1}) \oplus \ldots \oplus
\mathcal{O}(m_{n})$, where $m_{i} \in \mathbb{N}$, then for every integer $l$ the class $\mathcal{O} \oplus \mathcal{O}(lm_{1}) \oplus \ldots \oplus
\mathcal{O}(lm_{n})$ also occurs.
Proposition \[multi\] frustrated our initial attempt to obtain an easy classification of bundles based on curves. However, it raises multiple interesting questions instead. First, the construction uses a rational $D$ mapping finite-to-one onto $C$, and going to higher $m$ involves raising the degree of the map $D \to
C$. It may turn out that bounding the degree bounds the bundle class; we conjecture that if we fix the degree of the map then we obtain only finitely many bundle classes. Furthermore, in analogy with the consequences of Corollary \[GITc\], we should conversely be able to bound the degree of the map in terms of $C$ and the bundle class, at least for rational $C$.
Second, it is nontrivial to find the minimal $m_{i}$’s for which a bundle splitting as $\mathcal{O} \oplus \mathcal{O}(m_{1}) \oplus \ldots \oplus
\mathcal{O}(m_{n})$ would satisfy semistable reduction; the case of $n = 1$ could be stated particularly simply, as the question would be about the minimal $m$ for which $\mathcal{O} \oplus \mathcal{O}(m)$ occurs.
In section \[GITrecap\], we recap the basics of geometric invariant theory, which we will use in the proof of Theorem \[bad1\]. In sections \[exbad\] and \[pfbad\] we will illustrate Theorem \[bad1\]: in section \[exbad\] we will give some examples and compute the bundle classes that occur, proving Proposition \[multi\] on the way, while in section \[pfbad\] we will prove the theorem. In section \[goodcase\] we will focus on the trivial bundle case, proving Proposition \[GIT\] and defining the height function, which will impose constraints on which curves admit a trivial bundle; this will allow us to obtain a large family of curves $C$ in $\mathrm{M}_{2}^{ss}$ with no trivial bundle.
A Description of The Stable and Semistable Spaces {#GITrecap}
=================================================
Recall from geometric invariant theory that a when a geometrically reductive linear algebraic group $G$ has a linear action on a projectivized vector space $\mathbb{P}(V)$, we have,
\[sss\]A point $x \in V$ is called semistable (resp. stable) if any of the following equivalent conditions hold:
1. There exists a $G$-invariant homogeneous section $s$ such that $s(x) \neq 0$ (resp. same condition, and the action of $G$ on $x$ is closed).
2. The closure of $G\cdot x$ does not contain $0$ (resp. $G\cdot x$ is closed).
3. Every one-parameter subgroup $T$ acts on $x$ with both nonnegative and nonpositive weights (resp. negative and positive weights).
The last condition in the definition is equivalent to having nonpositive (resp. negative) weights. This is because if we can find a subgroup acting with only negative weights, then we can take its inverse and obtain only positive weights.
Observe that for every nonzero scalar $k$, $x$ is stable (resp. semistable) iff $kx$ is. So the same definitions of stability and semistability hold for points of $\mathbb{P}(V)$. The definitions also descend to every $G$-invariant projective variety $X \subseteq \mathbb{P}(V)$; in fact, in [@GIT] they are defined for $X$ in terms of a $G$-equivariant line bundle $L$. When $L$ is ample, as in the case of the space under discussion in this paper, this reduces to the above definition.
The importance of stability is captured in the following prior results:
The space of all stable points, $X^{s}$, and the space of all semistable points, $X^{ss}$, are both open and $G$-invariant.
There exists a quotient $Y = X^{ss}//G$, called a good categorical quotient, with a natural map $\pi:X \to Y$, satisfying the following properties:
1. $\pi$ is a $G$-equivariant map, where $G$ acts on $Y$ trivially.
2. Every $G$-equivariant map $X \to Z$, where $G$ acts on $Z$ trivially, factors through $\pi$.
3. $\pi$ is an open submersion.
4. $\pi(x_{1}) = \pi(x_{2})$ iff the closures of $G\cdot x_{1}$ and $G\cdot x_{2}$ intersect.
5. For every open $U \subseteq Y$, $\mathcal{O}_{U} = \mathcal{O}(\pi^{-1}(U))^{G}$.
In addition, $Y$ is proper.
There exists a quotient $Z = X^{s}//G$, called a good geometric quotient, with a natural map $\pi:X \to Z$ satisfying all enumerated conditions of a good categorial quotient, as well as the following:
1. $\pi(x_{1}) = \pi(x_{2})$ iff $G\cdot x_{1} = G\cdot x_{2}$.
2. $Z$ is naturally an open subset of $X^{ss}//G$.
On $X^{s}$, the dimension of the stabilizer group ${\operatorname{Stab}}_{G}(x)$ is constant.
Returning to our case of self-maps of $\mathbb{P}^{n}$, we write the stable and semistable spaces for the conjugation action as ${\operatorname{Hom}}_{d}^{n, s}$ and ${\operatorname{Hom}}_{d}^{n, ss}$. This involves a fair amount of abuse of notation, since those two spaces are open subvarieties of $\mathbb{P}^{N}$ and in fact properly contain ${\operatorname{Hom}}_{d}^{n}$, which consists only of regular maps.
In [@Lev] we proved the fact that ${\operatorname{Hom}}_{d}^{n} \subsetneq {\operatorname{Hom}}_{d}^{n, s}$ by describing ${\operatorname{Hom}}_{d}^{n, s}$ and ${\operatorname{Hom}}_{d}^{n, ss}$ more or less explicitly. We will recap the results, which are very technical but help us answer the question of when we can obtain a trivial bundle class in the semistable reduction problem and when we cannot.
We use the Hilbert-Mumford criterion, which is the last condition in Definition \[sss\]. In more explicit terms, the criterion for semistability (resp. stability) states that for every one-parameter subgroup $T \leq {\operatorname{SL}}(n+1)$, the action of $T$ on $\varphi$ can be diagonalized with eigenvalues $t^{a_{I}}$ and at least one $a_{I}$ is nonpositive (resp. negative). Now, assume by conjugation that this one-parameter subgroup is in fact diagonal, with diagonal entries $t^{a_{0}}, \ldots, t^{a_{n}}$, and that $a_{0} \geq \ldots \geq a_{n}$; we may also assume that the $a_{i}$’s are coprime, as dividing throughout by a common factor would not change the underlying group. Note also that $a_{0} + \ldots + a_{n} = 0$. Our task is made easy by the fact that our standard coordinates for $\mathbb{A}^{N+1}$ are the monomials, on which $T$ already acts diagonally. Throughout this analysis, we fix $\mathbf{a} = (a_{0}, \ldots, a_{n})$, and similarly for $\mathbf{x}$ and $\mathbf{d}$.
Now, $T$ acts on the $x_{0}^{d_{0}}\ldots x_{n}^{d_{n}}$ monomial of the $i$th polynomial, $\varphi_{i}$, with weight $a_{i} - \mathbf{a}\cdot\mathbf{d}.$ A map $\varphi \in \mathbb{P}^{N}$ is unstable (resp. not stable) iff, after conjugation, there exists a choice of $a_{i}$’s such that whenever the $\mathbf{x^{d}}$-coefficient of $\varphi_{i}$ satisfies $\mathbf{a}\cdot\mathbf{d} \leq a_{i}$ (resp. $<$), it is equal to zero.
While in principle there are infinitely many possible $T$’s, parametrized by a hyperplane in $\mathbb{P}^{n}(\mathbb{Q})$, in practice there are up to conjugation only finitely many. This is because each diagonal $T$ imposes conditions of the form “the $\mathbf{x^{d}}$-coefficient of $\varphi_{i}$ is zero,” and there are only finitely many such conditions. Thus the stable and semistable spaces are indeed open in $\mathbb{P}^{N}$.
The conjugation conditions we have chosen for $T$ are such that the conditions they impose for $\varphi$ to be unstable (or merely not stable) are the most stringent on $\varphi_{n}$ and least stringent on $\varphi_{0}$, and are the most stringent on monomials with high $x_{0}$-degrees and least stringent on monomials with high $x_{n}$-degrees.
If $n = 1$, we have a simpler description, due to Silverman [@Sil96]:
$\varphi \in \mathbb{P}^{N}$ is unstable (resp. not stable) iff it is equivalent under coordinate change to a map $(a_{0}x^{d} + \ldots +
a_{d}y^{d})/(b_{0}x^{d} + \ldots + b_{d}y^{d})$, such that:
1. $a_{i} = 0$ for all $i \leq (d-1)/2$ (resp. $<$).
2. $b_{i} = 0$ for all $i \leq (d+1)/2$ (resp. $<$).
The description for $n = 1$ can be thought of as giving a dynamical criterion for stability and semistability. A point $\varphi \in \mathbb{P}^{N}$ is unstable if there exists a point $x \in \mathbb{P}^{1}$ where $\varphi$ has a bad point of degree more than $(d+1)/2$, or $\varphi$ has a bad point of degree more than $(d-1)/2$ where it in addition has a fixed point. Following Rahul Pandharipande’s unpublished reinterpretation of [@Sil96], we define “bad point” as a vertical component of the graph $\Gamma_{\varphi} \subseteq \mathbb{P}^{1} \times \mathbb{P}^{1}$, and “fixed point” as a fixed point of the unique non-vertical component of $\Gamma_{\varphi}$. When $n = 1, d = 2$, this condition reduces to having a fixed point at a bad point, or alternatively a repeated bad point.
The conditions for higher $n$ are not as geometric. However, if we interpret fixed points liberally enough, there are still strong parallels with the $n =
1$ case. One can show that the unstable space for $n = 2$ and $d = 2$ consists of two irreducible components, which roughly generalize the $n = 1, d = 2$ condition of having a fixed point at a bad point; in this case, one needs to define a limit of the value of $\varphi(x)$ as $x$ approaches the bad point, though this limit can be defined purely in terms of degrees of polynomials, without needing to resort to a specific metric on the base field.
Examples of Nontrivial Bundles {#exbad}
==============================
The space ${\operatorname{Rat}}_{2} = {\operatorname{Hom}}_{2}^{1}$ and its quotient $\mathrm{M}_{2}$ have been analyzed with more success than the larger spaces, yielding the following prior structure result [@D1C] [@Sil96]:
\[Sil\]$\mathrm{M}_{2} = \mathbb{A}^{2}$; $\mathrm{M}_{2}^{s} = \mathrm{M}_{2}^{ss} = \mathbb{P}^{2}$. The first two elementary symmetric polynomials in the multipliers of the fixed points realize both isomorphisms.
Recall that within $\mathbb{P}^{N} = \mathbb{P}^{5}$, a map $(a_{0}x^{2} + a_{1}xy + a_{2}y^{2})/(b_{0}x^{2} + b_{1}xy + b_{2}y^{2})$ is unstable iff it is in the closure of the ${\operatorname{PGL}}(2)$-orbit of the subvariety $a_{0} = b_{0} = b_{1} = 0$. In other words, it is unstable iff there the map is degenerate and has a double bad point, or a fixed point at a bad point.
\[poly\]A map on $\mathbb{P}^{1}$ is a polynomial iff there exists a totally invariant fixed point. Taking such a point to infinity turns the map into a polynomial in the ordinary sense. In ${\operatorname{Rat}}_{d}$, or generally in $\mathbb{P}^{N} = \mathbb{P}^{2d+1}$, a map is polynomial iff it is in the closure of the ${\operatorname{PGL}}(2)$-orbit of the subvariety defined by zeros in all coefficients in the denominator except the $y^{d}$-coefficient.
A totally invariant fixed point is not necessarily a totally fixed point. A totally invariant fixed point is one that is totally ramified. A totally fixed point is the root of the fixed point polynomial when it is unique, i.e. when the polynomial is a power of a linear term. In fact by an easy computation, a map has a totally invariant, totally fixed point $x$ iff it is degenerate linear with a multiplicity-$d-1$ bad point at $x$, in which case it is necessarily unstable.
The polynomial maps define a curve in $\mathrm{M}_{2}^{ss}$; we will show,
\[Example\]The polynomial curve in $\mathrm{M}_{2}^{ss}$ only satisfies semistable reduction with nontrivial bundles.
First, note that in $\mathbb{P}^{5}$, the polynomial maps are those that can be conjugated to the form $(a_{0}x^{2} + a_{1}xy +
a_{2}y^{2})/b_{2}y^{2}$, in which case the totally invariant fixed point is $\infty = (1:0)$. We will call the polynomial map locus $X$. If $a_{0} = 0$ then the map is unstable; we will show that every curve in $X$ contains a map for which $a_{0} = 0$. Clearly, the set of all maps with a given totally invariant fixed point is isomorphic to $\mathbb{P}^{3}$, and the unstable locus within it is isomorphic to $\mathbb{P}^{2}$ as a linear subvariety, so for there to be any hope of a trivial bundle, a curve in $X$ cannot lie entirely over one totally invariant point.
Now, the fixed point equation for a map of the form $f/g$ is $fy - gx$; the homogeneous roots of this equation are the fixed points, with the correct multiplicities. For our purposes, when the totally invariant point is $\infty$, the fixed point equation is $a_{0}x^{2}y + (a_{1} - b_{2})xy^{2} + a_{2}y^{3}$. We get that $a_{0} = 0$ iff the totally invariant point is a repeated root of the fixed point equation.
There exists a map from $X$ to $\mathbb{P}^{1} \times \mathbb{P}^{2}$, mapping $\varphi$ to its totally invariant point in $\mathbb{P}^{1}$, and to the two elementary symmetric polynomials in the two other fixed points in $\mathbb{P}^{2}$. Write $(x:y)$ for the image in $\mathbb{P}^{1}$ and $(a:b:c)$ for the image in $\mathbb{P}^{2}$. Now $(x:y)$ is a repeated root if $ax^{2} + bxy + cy^{2} = 0$. The equation defines an ample divisor, so every curve in $\mathbb{P}^{1} \times
\mathbb{P}^{2}$ will meet it. Finally, a curve in $X$ maps either to a single point in $\mathbb{P}^{1} \times \mathbb{P}^{2}$, in which case it must contain points with $a_{0} = 0$ as above, or to a curve, in which case it intersects the divisor $ax^{2} + bxy + cy^{2} = 0$. In both cases, the curve contains unstable points. Thus there is no global semistable curve $D$ in ${\operatorname{Hom}}_{d}^{n, ss}$ mapping down to $C$.
Note that in the above proof, maps conjugate to $x^{2}$ have two totally invariant points, so a priori the map from $X$ to $\mathbb{P}^{1} \times
\mathbb{P}^{2}$ is not well-defined at them. However, for any curve $D$ in $X$, there is a well-defined completion of this map, whose value at $x^{2}$ on the $\mathbb{P}^{1}$ factor is one of the two totally invariant points. Thus this complication does not invalidate the above proof.
Let us now compute the vector bundle classes that do occur for the polynomial curve. We work with the description $x^{2} + c$, which yields an affine curve that maps one-to-one into $C$, missing only the point at infinity, which is conjugate to $\frac{x^{2} - x}{0}$. To hit the point at infinity, we choose the alternative parametrization $cx^{2} - cx + 1$, which, when $c = \infty$, corresponds to the unique (up to conjugation) semistable degenerate constant map. For any $c$, this map is conjugate to $x^{2} - cx + c$ and thence $x^{2} + c/2 - c^{2}/4$, using the transition function $[c, -1/2; 0, 1]$. Thus the bundle splits as $\mathcal{O} \oplus \mathcal{O}(1)$.
This bundle depends on the choice of $D$. In fact, if we choose another parametrization for $D$, for example $c^{2}x^{2} - c^{2}x + 1$, then the transition function $[c^{2}, -1/2; 0, 1]$, which leads to the bundle $\mathcal{O} \oplus \mathcal{O}(2)$. This is not equivalent to $\mathcal{O} \oplus \mathcal{O}(1)$. This then leads to the question of which classes of bundles can occur over each $C$. In the example we have just done, the answer is every nontrivial class: for every positive integer $m$, we can use $c^{m}x^{2} - c^{m}x + 1$ as a parametrization, leading to $\mathcal{O} \oplus \mathcal{O}(m)$, which exhausts all nontrivial projective bundle classes.
Recall the result of Proposition \[multi\]:
Suppose $C$ is isomorphic to $\mathbb{P}^{1}$, and there exists $U \subseteq {\operatorname{Hom}}_{d}^{n, ss}$ mapping finite-to-one into $C$ such that $U$ is a projective curve minus a point. Then there are always infinitely many possible classes: if the class of $U$ is thought of as splitting as $\mathbf{P}(\mathcal{E}) =
\mathcal{O} \oplus \mathcal{O}(m_{1}) \oplus \ldots \oplus \mathcal{O}(m_{n})$, where $m_{i} \in \mathbb{N}$, then for every integer $l$ the class $\mathcal{O}
\oplus \mathcal{O}(lm_{1}) \oplus \ldots \oplus \mathcal{O}(lm_{n})$ also occurs.
Imitating the analysis of the polynomial curve above, we can parametrize $C$ by one variable, say $c$, and choose coordinates such that the sole bad point in the closure of $U$ corresponds to $c = \infty$. Now, we can by assumption find a piece $U'$ above the infinite point with a transition function determining the vector bundle $\mathcal{O} \oplus \mathcal{O}(m_{1}) \oplus \ldots \oplus \mathcal{O}(m_{n})$. Now let $V$ be the composition of $U'$ with the map $c \mapsto
c^{l}$. Then $U$ and $V$ determine a vector bundle satisfying semistable reduction, of class $\mathcal{O} \oplus \mathcal{O}(lm_{1}) \oplus \ldots \oplus
\mathcal{O}(lm_{n})$, as required.
The example in Theorem \[Example\], of polynomial maps, is equivalent to a multiplier condition. When $d = 2$, a map is polynomial iff it has a superattracting fixed point, i.e. one whose multiplier is zero; see the description in the first chapter of [@ADS]. One can imitate the proof that semistable reduction does not hold for a more general curve, defined by the condition that there exists a fixed point of multiplier $t \neq 1$. In that case, the condition $b_{1} = 0$ is replaced by $b_{1} = ta_{0}$, and the point is a repeated root of the fixed point equation iff $a_{0} = b_{1}$, in which case we clearly have $a_{0}
= b_{1} = 0$ and the point is unstable.
When the multiplier is $1$, the fixed point in question is automatically a repeated root, with $b_{1} = a_{0}$. The condition that the point is the only fixed point corresponds to $b_{2} = a_{1}$, which by itself does not imply that the map fails to be a morphism, let alone that it is unstable.
Instead, the condition that gives us $b_{1} = a_{0} = 0$ is the condition that the fixed point is totally invariant. Specifically, the fixed point’s two preimages are itself and one more point; when the fixed point is $\infty$, the extra point is $-b_{2}/b_{1}$. Now we can map $X$ to $\mathbb{P}^{1} \times
\mathbb{P}^{1}$ where the first coordinate is the fixed point and the second is its preimage. This map is well-defined on all of $X$ because only one point can be a double root of a cubic. Now the diagonal is ample in $\mathbb{P}^{1} \times \mathbb{P}^{1}$, so the only way a curve $D$ can avoid it is by mapping to a single point; but in that case, $D$ lies in a fixed variety isomorphic to $\mathbb{P}^{3}$ where the unstable locus is $\mathbb{P}^{2}$, so it will intersect the unstable locus.
The fact that any condition of the form “there exists a fixed point of multiplier $t$” induces a curve for which semistable reduction requires a nontrivial bundle means that there is no hope of enlarging the semistable space in a way that ensures we always have a trivial bundle. We really do need to think of semistable reduction as encompassing nontrivial bundle classes as well as trivial ones.
Specifically: it is trivial to show that the closure of the polynomial locus in ${\operatorname{Rat}}_{2}$ includes all the unstable points (fix $\infty$ to be the totally invariant point and let $a_{0}$ go to zero). At least some of those unstable points will also arise as closures of other multiplier-$t$ conditions. However, different multiplier-$t$ conditions limit to different points in $\mathrm{M}_{2}^{ss}\setminus\mathrm{M}_{2}$.
The General Case {#pfbad}
================
So far we have talked about nontrivial classes in $\mathrm{M}_{2}$. But we have a stronger result, restating Theorem \[bad1\]:
\[bad2\]For all $n$ and $d$, over any base field, there exists a curve with no trivial bundle class satisfying semistable reduction.
In all cases, we will focus on **polynomial maps**, which we will define to be maps that are ${\operatorname{PGL}}(n+1)$-conjugate to maps for which the last polynomial $q_{n}$ has zero coefficients in every monomial except possibly $x_{n}^{d}$.
\[poly1\]The set of polynomial maps, defined above, is closed in $\overline{{\operatorname{Hom}}_{d}^{n}} = \mathbb{P}^{N}$.
Clearly, the set of polynomial maps with respect to a particular hyperplane – for example, $x_{n} = 0$ – is closed. Now, for each hyperplane $a_{0}x_{0} + \ldots + a_{n}x_{n} = 0$, we can check by conjugation to see that the condition that the map is polynomial corresponds to the condition that $a_{0}q_{0} + \ldots + a_{n}q_{n} = c(a_{0}x_{0} + \ldots + a_{n}x_{n})^{d}$, where $c$ may be zero. As $\mathbb{P}^{n}$ is proper, it suffices to show that the condition “$\varphi$ is polynomial with respect to $a_{0}x_{0} + \ldots + a_{n}x_{n} = 0$” is closed in $\left(\mathbb{P}^{n}\right)^{*} \times \mathbb{P}^{N}$.
Now, we may construct a rational function $f$ from $\left(\mathbb{P}^{n}\right)^{*} \times \mathbb{P}^{N}$ to ${\operatorname{Sym}}^{d}(\mathbb{P}^{n}) \times
{\operatorname{Sym}}^{d}(\mathbb{P}^{n})$ by $((a_{0}x_{0} + \ldots + a_{n}x_{n}), \varphi) \mapsto ((a_{0}x_{0} + \ldots + a_{n}x_{n})^{d}, a_{0}q_{0} + \ldots + a_{n}q_{n})$. The map $\varphi$ is polynomial with respect to $a_{0}x_{0} + \ldots + a_{n}x_{n} = 0$ iff $f$ is ill-defined at $((a_{0}x_{0} + \ldots + a_{n}x_{n}), \varphi)$ or $f((a_{0}x_{0} + \ldots + a_{n}x_{n}), \varphi) \in \Delta$, the diagonal subvariety. The ill-defined locus of $f$ is closed, and the preimage of $\Delta$ is closed in the well-defined locus.
In fact, the condition of $\varphi$ being polynomial with respect to any number of distinct hyperplanes in general position – in other words, the condition that $\varphi$ is conjugate to a map for which $q_{i} = c_{i}x_{i}^{d}$ for all $i > 0$ (or $i > 1$, etc.) – is more or less closed as well. It is not closed, but a sufficiently good condition is closed. Namely:
\[flag\]For each $1 \leq i \leq n$, consider the ${\operatorname{PGL}}(n+1)$-orbit of the space of maps in which, for each $j \geq i$, $q_{j}$ has zero coefficients in every monomial containing any term $x_{k}$ with $k < j$. This orbit is closed in $\mathbb{P}^{N}$.
Observe that the above-defined space of maps consists of maps that are polynomial with respect to $x_{n} = 0$, such that the induced map on the totally invariant hyperplane $x_{n} = 0$ is polynomial with respect to $x_{n-1} = 0$, and so on until we reach the induced map on the totally invariant subspace $x_{i+1} =
\ldots = x_{n} = 0$.
Now we use descending induction. Lemma \[poly1\] is the base case, when $i = n$. Now suppose it is true down to $i$. Then for $i - 1$, the condition of having no nonzero $x_{k}$ term in $q_{i-1}$ with $k < i - 1$ is equivalent to the condition that the induced map on the totally invariant subspace $x_{i} = x_{i+1}
= \ldots = x_{n} = 0$ is polynomial; this condition is closed in the space of all maps that are polynomial down to $x_{i}$, which we assume closed by the induction hypothesis.
We call maps of the form in the above lemma **polynomial with respect to $B$**, where $B$ is the Borel subgroup preserving the ordered basis of conditions. In the case above, $B$ is the upper triangular matrices.
We need one final result to make computations easier:
\[last\]Let $X$ be a curve of polynomial maps, all with respect to a Borel subgroup $B$, and let $\varphi$ be a semistable map in $\overline{{\operatorname{PGL}}(n+1)\cdot X}$. Then $\varphi \in \overline{B\cdot X}$.
Let $C$ be the closure of the image of $X$ in $\mathrm{M}_{d}^{n, ss}$. By semistable reduction, there exists some affine curve $Y \ni \varphi$ mapping finite-to-one to $C$, i.e. dominantly. We need to find some open $Z \subseteq Y$ containing $\varphi$ and some $f: Z \to {\operatorname{PGL}}(n+1)$ such that $f(\varphi)$ is the identity matrix, and $Z' = \{(f(z)\cdot z)\}$ consists of maps which are polynomial with respect to $B$. Such map necessarily exists: we have a map $h$ from $Y$ to the flag variety of $\mathbb{P}^{n}$ sending each $y$ to the subgroup with respect to which it is polynomial (possibly involving some choice if generically $y$ is polynomial with respect to more than one flag), which then lifts to $G$, possibly after deleting finitely many points. Generically, a point of $X$ maps to a point of $C$ that is in the image of $Z$; therefore, picking the correct points in $X$, we get that $\varphi \in \overline{B\cdot X}$.
With the above lemmas, let us now prove the theorem with $n = 1$, which is slightly easier than the higher-$n$ case, where the more complicated Lemma \[flag\]. We will use the family $x^{d} + c$, where $c \in \mathbb{A}^{1}$. In projective notation, this is $\frac{a_{0}x^{d} + a_{d}y^{d}}{b_{d}y^{d}}$, which is a one-dimensional family modulo conjugation. We have,
Let $V$ be the closure of the ${\operatorname{PGL}}(2)$-orbit of the family $\frac{a_{0}x^{d} + a_{d}y^{d}}{b_{d}y^{d}}$ in $\mathbb{P}^{N}$. Then:
1. In characteristic $0$ or $p \nmid d$, every $\varphi \in V$ is actually in the ${\operatorname{PGL}}(2)$-orbit of the family, or else it is a degenerate linear map, conjugate to $\frac{a_{d-1}xy^{d-1} + a_{d}y^{d}}{b_{d}y^{d}}$.
2. In characteristic $p \mid d$, with $p^{m} \mid\mid d$ and $p^{m} \neq d$, every $\varphi \in V$ is in the ${\operatorname{PGL}}(2)$-orbit of the family or is a degenerate map conjugate to $\frac{a_{d-p^{m}}xy^{d-p^{m}} + a_{d}y^{d}}{b_{d}y^{d}}$.
3. In characteristic $p$ with $d = p^{m}$, set $V$ to be the closure of the orbit of the family $\frac{a_{0}x^{d} + a_{d-1}xy^{d-1}}{b_{d}y^{d}}$; then every $\varphi \in V$ is actually in the orbit of the family, or else it is a degenerate linear map, conjugate to $\frac{a_{d-1}xy^{d-1} + a_{d}y^{d}}{b_{d}y^{d}}$, and furthermore $a_{d-1} = b_{d}$.
Observe that the first two cases are really the same: case $2$ is reduced to case $1$ viewed as a degree-$\frac{d}{p^{m}}$ map in $(x^{p^{m}}:y^{p^{m}})$. So it suffices to prove case $1$ to prove $2$; we will start with the family $\frac{a_{0}x^{d} + a_{d}y^{d}}{b_{d}y^{d}}$ and see what algebraic equations its orbit satisfies. As polynomials are closed in $\overline{{\operatorname{Rat}}_{d}}$, every point in the closure of the orbit is a polynomial. We may further assume it is polynomial with respect to $y = 0$; therefore, by Lemma \[last\], it suffices to look at the action of upper triangular matrices. Further, the condition of being within the family $\frac{a_{0}x^{d} + a_{d}y^{d}}{b_{d}y^{d}}$ is stabilized by diagonal matrices; therefore, it suffices to look at the action of matrices of the form $[1, t; 0, 1]$.
Now, the conjugation action of $[1, t; 0, 1]$ fixes $b_{d}y^{d}$ and maps $a_{0}x^{d} + a_{d}y^{d}$ to $a_{0}(x - ty)^{d} + (a_{d} + tb_{d})y^{d}$. Clearly, there is no hope of obtaining any condition on $b_{d}$ or $a_{d}$. Now, the conditions on the terms $a_{0}, \ldots, a_{d-1}$ are that for some $t$, they fit into the pattern $a_{0}(x^{d} - dtx^{d-1}y + \ldots \pm dt^{d-1}xy^{d-1})$, i.e. $a_{i} = (-t)^{i}{d \choose i}a_{0}$. To remove the dependence on $t$, note that when $i
+ j = k + l$, we have ${d \choose i}{d \choose j}a_{i}a_{j} = {d \choose k}{d \choose l}a_{k}a_{l}$, as long as $i, j, k, l < d$.
Let us now look at what those conditions imply. Setting $j = i, k = i-1, l = i+1$, we get conditions of the form ${d \choose i}^{2}a_{i}^{2} = {d \choose
i-1}{d \choose i+1}a_{i-1}a_{i+1}$, whenever $i + 1 < d$. If $a_{0} \neq 0$, then the value of $a_{1}$ uniquely determines the value of $a_{2}$ by the condition with $i = 1$; the value of $a_{2}$ uniquely determines $a_{3}$ by the condition with $i = 2$; and so on, until we uniquely determine $a_{d-1}$. In this case, choosing $t = -\frac{a_{1}}{da_{0}}$ will conjugate this map back to the family $\frac{a_{0}x^{d} + a_{d}y^{d}}{b_{d}y^{d}}$. If $a_{0} = 0$, then the equation with $i = 1$ will imply that $a_{1} = 0$; then the equation with $i = 2$ will imply that $a_{2} = 0$; and so on, until we set $a_{d-2} = 0$. We cannot ensure $a_{d-1} =
0$ because $a_{d-1}$ always appears in those equations multiplied by a different $a_{i}$, instead of squared. Hence we could get a degenerate-linear map.
In case $3$, we again look at the action of matrices of the form $[1, t; 0, 1]$. Such matrices map $\frac{a_{0}x^{d} + a_{d-1}xy^{d-1}}{b_{d}y^{d}}$ to $\frac{a_{0}x^{d} + a_{d-1}xy^{d-1} + (-a_{0}t^{d} - a_{d-1}t + b_{d}t)y^{d}}{b_{d}y^{d}}$. Now the only way a map of the form $\frac{a_{0}x^{d} + a_{d-1}xy^{d-1} +
a_{d}y^{d}}{b_{d}y^{d}}$ could degenerate is if the image of the polynomial map $t \mapsto -a_{0}t^{d} - a_{d-1}t + b_{d}t$ misses $a_{d}$, which could only happen if the polynomial were constant, i.e. $a_{0} = 0$ and $a_{d-1} = b_{d}$, giving us a degenerate-linear map.
The importance of the lemma is that in all degenerate cases, the map is necessarily unstable, since $d-1$ (or, in case $2$, $d-p^{m}$) is always at least as large as $d/2$.
We can now prove the theorem for $n = 1$. So if we can always find a $D \subseteq {\operatorname{Hom}}_{d}^{n, ss}$ that works globally, we can find one over a family in which every map is conjugate to $\frac{a_{0}x^{d} + a_{d}y^{d}}{b_{d}y^{d}}$, or, in characteristic $p$ with $d = p^{m}$, $\frac{a_{0}x^{d} +
a_{d-1}xy^{d-1}}{b_{d}y^{d}}$. It suffices to show that there exists a map with $a_{0} = 0$. For this, we use the fixed point polynomial, which is well-defined on this family. If the polynomial is fixed, then all maps in the family may be simultaneously conjugated to the form $\frac{a_{0}x^{d} + a_{d}y^{d}}{b_{d}y^{d}}$ (or $\frac{a_{0}x^{d} + a_{d-1}xy^{d-1}}{b_{d}y^{d}}$), and then one map must have $a_{0} = 0$. If the polynomial varies, then some map will have the point at infinity colliding with another fixed point. This will force the map to be ill-defined at infinity; recall that totally invariant points are simple roots of the fixed point polynomial, unless they are bad. This will force $a_{0}$ to be zero, again.
For higher $n$, the proof is similar. The lemma we need is similar to the lemma we use above, but is somewhat more complicated:
Let $V$ be the closure of the ${\operatorname{PGL}}(n+1)$-orbit of the family $(c_{0}x_{0}^{d} + bx_{1}^{d}:q_{1}:\ldots:q_{n})$, where $q_{i}$ is $x_{j}$-free for all $j < i$.
1. If the characteristic does not divide $d$, then every $\varphi \in V$ is actually in the ${\operatorname{PGL}}(n+1)$-orbit of the family, or else it is a degenerate map, whose only possible nonzero coefficients in $q_{0}$ are those without an $x_{0}$ term and those of the form $x_{0}p_{0}$ where there is no nonzero $x_{0}$-term in $p_{0}$.
2. If the characteristic $p$ satisfies $p \mid d$, with $d \neq p^{m} \mid\mid d$ then the same statement as in case $1$ holds as long as each $q_{i}$ is in terms of $x_{j}^{p^{m}}$, but with $x_{0}p_{0}$ replaced by $x_{0}^{p^{m}}p_{0}$.
3. If the characteristic $p$ satisfies $d = p^{m}$ then, changing the family to $(c_{0}x_{0}^{d} + bx_{0}x_{1}^{d-1}:q_{1}:\ldots:q_{n})$, with $q_{i}$ in terms of $x_{j}^{d}$ as in case $2$, the same statement as in case $1$ holds.
As in the one-dimensional case, case $2$ is reducible to case $1$ with $d$ replaced with $\frac{d}{p^{m}}$ and $x_{i}$ with $x_{i}^{m}$. By Lemma \[last\], we only need to conjugate by upper triangular matrices. Further, we only need to conjugate by just matrices of the family $E$, with first row $(1,
t_{1}, \ldots, t_{n})$ and other rows the same as the identity matrix. This is because we can control the diagonal elements because the condition of being in the family is diagonal matrix-invariant, and we can control the rest by projecting any curve $Z$ of unipotent upper triangular matrices onto $E$.
Set $a_{\mathbf{d}}$ to be the $\mathbf{x^{d}}$-coefficient in $q_{0}$. For all vectors $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$, $\mathbf{l}$ with $\mathbf{i} + \mathbf{j} = \mathbf{k} + \mathbf{l}$, we have ${d \choose \mathbf{i}}{d \choose \mathbf{j}}a_{\mathbf{i}}a_{\mathbf{j}} = {d \choose \mathbf{k}}{d
\choose \mathbf{l}}a_{\mathbf{k}}a_{\mathbf{l}}$, as long as none of $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$, or $\mathbf{l}$ is in the span of $\mathbf{e}_{i}$ for $i > 0$. Note that $i$ and $\mathbf{i}$ are two separate quantities, one an index of coordinates and one an index of monomials.
As in the one-dimensional case, we may set $\mathbf{j} = \mathbf{i}$ and $\mathbf{k} = \mathbf{i} - \mathbf{e}_{0} + \mathbf{e}_{i}$. If $c_{0} = a_{(d, 0,
\ldots, 0)} \neq 0$, then by the same argument as before, the values of the $x_{0}^{d-1}x_{i}$-coefficients determine all the rest, and we can conjugate the map back to the desired form. And if $c_{0} = 0$, then the value of every coefficient that can occur as $\mathbf{i}$ in the above construct is zero; the only coefficients that cannot are those with no $x_{0}$ component and those with a linear $x_{0}$ component.
In case $3$, we restrict to matrices of the same form as in case $1$, and observe that those matrices only generate extra $x_{i}^{d}$ and $x_{i}x_{1}^{d-1}$ in $q_{0}$. The statement is vacuous if $c_{0} = 0$, so assume $c_{0} \neq 0$. For $i = 1$, this is identical to the one-dimensional case, so if $c_{0} \neq 0$ then we can find an appropriate $t_{1}$. For higher $i$, if $b \neq 0$ then we can extract $t_{i}$ from the $x_{i}x_{1}^{d-1}$ coefficient, which will necessarily work for the $x_{i}^{d}$ coefficient as well, making the map conjugate to the family; if $b = 0$, then the same equations as for $i = 1$ hold for higher $i$, and we can again find $t_{i}$’s conjugating the map to the family.
While we could also control the terms involving a linear (or $p$-power) $x_{0}$ coefficient in the above construction, it is not necessary for our purposes.
To finish the proof of the theorem, first note that in the closure of the family above, any map for which $c_{0} = 0$ is unstable. Indeed, the one-parameter subgroup of ${\operatorname{PGL}}(n+1)$ with diagonal coefficients $t_{0} = n, t_{i} = -1$ for $i > 0$, shows instability. Recall that a map is unstable with respect to such a family if $t_{i} > t_{0}d_{0} + \ldots + t_{n}d_{n}$ whenever the $x_{0}^{d_{0}}\ldots x_{n}^{d_{n}}$-coefficient of $q_{i}$ is nonzero. With the above one-parameter subgroup, we have $t_{0}d_{0} + \ldots + t_{n}d_{n} = -d < -1$ for the only nonzero monomials in $q_{i}$ with $i > 0$; in $q_{0}$, the maximal value of $t_{0}d_{0} + \ldots + t_{n}d_{n}$ is $t_{0} + t_{i}(d-1) = n - (d-1) < n$.
Now we need to show only that for some map in the family, $c_{0}$ will indeed be zero. So suppose on the contrary that $c_{0}$ is never zero. Then all maps are, after conjugation, in the family $(c_{0}x_{0}^{d} + bx_{1}^{d}:q_{1}\ldots:q_{n})$, where the linear subvariety $q_{i} = q_{i+1} = \ldots = q_{n}$ is totally invariant. Now look at the action on the line $x_{2} = \ldots = x_{n} = 0$. Every morphism will induce a morphism on this line, so there will be three fixed points on it, counting multiplicity. We now imitate the proof in the one-dimensional case: the totally invariant fixed point on this line, $(1:0:\ldots:0)$, will collide with another fixed point, so the map will be ill-defined at it. This means that $(1:0:\ldots:0)$ is a bad point, which cannot happen unless $c_{0} = 0$.
Trivially, the above theorem for curves shows the same for higher-dimensional families in $\mathrm{M}_{d}^{n, ss}$. An interesting question could be to generalize semistable reduction to higher-dimensional families, for which we may get projective vector bundles just like in the case of curves. Trivially, if we have two proper subvarieties of $\mathrm{M}_{d}^{n, ss}$, $V_{1} \subseteq V_{2}$, and a bundle class occurs for $V_{2}$, then its restriction to $V_{1}$ occurs for $V_{1}$. In particular, if we have the trivial class over $V_{2}$ then we also have it over $V_{1}$, as well as any other subvariety of $V_{2}$. This leads to the following question: if the trivial class occurs for every proper closed subvariety of $V_{2}$, does it necessarily occur for $V_{2}$? What if we weaken the condition and only require the trivial class to occur for subvarieties that cover $V_{2}$?
The Trivial Bundle Case {#goodcase}
=======================
For most curves $C \subseteq \mathrm{M}_{d}^{n, ss}$, there occurs a trivial bundle. Since the complement of ${\operatorname{Hom}}_{d}^{n, ss}$ in $\mathbb{P}^{N}$ has high codimension, this is true by simple dimension counting. Therefore, it is useful to analyze those curves separately, as we have more tools to work with. Specifically, we can use more machinery from geometric invariant theory. We will start by proving Proposition \[GIT\], restated below:
\[GIT2\]Let $X$ be a projective variety over an algebraically closed field with an action by a geometrically reductive linear algebraic group $G$. Using the terminology of geometric invariant theory, let $D$ be a complete curve in the stable space $X^{s}$ whose quotient by $G$ is a complete curve $C$; say the map from $D$ to $C$ has degree $m$. Suppose the stabilizer is generically finite, of size $h$, and either $D$ or $C$ is normal. Then there exists a finite subgroup $S_{D} \subseteq G$, of order equal to $mh$, such that for all $x \in D$ and $g \in G$, $gx \in D$ iff $g \in S_{D}$.
For $x \in D$, we define $S_{D}(x) = \{g \in G: gx \in D\}$. This is a map of sets from an open dense subset of $D$ to ${\operatorname{Sym}}^{mh}(G)$, and is regular on an open dense subset. We have:
\[sym\]The map from ${\operatorname{Sym}}^{mh}(G) \times X^{s}$ to ${\operatorname{Sym}}^{mh}(X^{s}) \times X^{s}$ defined by sending each $(\{g_{1}, \ldots, g_{mh}\}, x)$ to $(\{g_{1}\cdot x, \ldots, g_{mh}\cdot x\}, x)$ is proper.
By standard geometric invariant theory, the map from $G \times X^{s}$ to $X^{s} \times X^{s}$, $(g, x) \mapsto (g\cdot x, x)$, is proper. Therefore the map from $G^{mh} \times (X^{s})^{mh}$ to $(X^{s})^{mh} \times (X^{s})^{mh}$ defined by $(g_{i}, x_{i}) \mapsto (g_{i}\cdot x_{i}, x_{i})$ is also proper, as the product of proper maps. Now closed immersions are proper, so the map remains proper if we restrict it to $G^{mh} \times X^{s}$ where we embed $X^{s}$ into $(X^{s})^{mh}$ diagonally; the image of this map lands in $(X^{s})^{mh} \times X^{s}$. Finally, we quotient out by the symmetric group $S_{k}$, obtaining:
$$\xymatrix@R+2em@C+2em{
G^{mh} \times X^{s} \ar[r] \ar[d]_{\pi} & (X^{s})^{mh} \times X^{s} \ar[d]^{\pi} \\
{\operatorname{Sym}}^{mh}(G) \times X^{s} \ar[r] & {\operatorname{Sym}}^{mh}(X^{s}) \times X^{s}
}$$
The map on the bottom is already separated and finite-type; we will show it is universally closed. Extend it by some arbitrary scheme $Y$. If $V \subseteq
{\operatorname{Sym}}^{mh}(G) \times X^{s} \times Y$ is closed, then so is $\pi^{-1}(V) \subseteq G^{mh} \times X^{s} \times Y$. The map on top is universally closed, so its image is closed in $(X^{s})^{mh} \times X^{s} \times Y$. But the map on the right is proper, so the image of $V$ is also closed in ${\operatorname{Sym}}^{mh}(X^{s}) \times X^{s} \times
Y$.
Now, the rational map $f_{D}(x) = S_{D}(x)\cdot x \in {\operatorname{Sym}}^{mh}(D)$ can be extended to a morphism on all of $D$, since both $D$ and ${\operatorname{Sym}}^{mh}(D)$ are proper. This is trivial if $D$ is normal; if it is not normal, but $C$ is normal, then observe that the map factors through $C$ since it is constant on orbits, and then analytically extend it through $C$. But now $(f_{D}(x), x)$ embeds into ${\operatorname{Sym}}^{mh}(X^{s}) \times X^{s}$ as a proper curve. The preimage in ${\operatorname{Sym}}^{mh}(G) \times
X^{s}$ of this curve is also proper; for each $(f_{D}(x), x)$, it is a finite set of points of the form $(S, x)$ satisfying $S\cdot x = f_{D}(x)$, including $(S_{D}(x), x)$. Projecting onto the ${\operatorname{Sym}}^{mh}(G)$ factor, we still get a proper set, which means it must be a finite set of points, as ${\operatorname{Sym}}^{mh}(G)$ is affine. One of these points will be $S_{D}$, which is then necessarily finite.
Finally, if $g, h \in S_{D}$ and $x \in D$ then $g\cdot h\cdot x \in g \cdot D = D$; therefore $S_{D}$ is a group.
The proposition essentially says that the cover $D \to C$ is necessarily Galois. The generic stabilizer is necessarily a group $H$, normal in $S_{D}$.
With the same notation and conditions as in Proposition \[GIT2\], the map from $D$ to $C$ ramifies precisely at points $x \in D$ such that ${\operatorname{Stab}}(x)$ intersects $S_{D}$ in a strictly larger group than $H$. Furthermore, the ramification degree is exactly $[{\operatorname{Stab}}(x)\cap S_{D}:H]$.
For high $n$ or $d$, the stabilized locus of ${\operatorname{Hom}}_{d}^{n}$ is of high codimension. Furthermore, most curves in ${\operatorname{Hom}}_{d}^{n, ss}$ lie in ${\operatorname{Hom}}_{d}^{n,
s}$. Therefore, generically not only is $H$ trivial, but also there are no points on $D$ with nontrivial stabilizer. Thus for most $C$ and $D$, the map $D \to C$ must be unramified. Thus, when $C$ is rational, generically the degree is $1$.
It’s based on this observation that we conjecture the bounds for the nontrivial bundle case in both directions – that is, that if we fix $C$ and the bundle class $\mathbf{P}(\mathcal{E})$, then the degree of the map $\pi: D \to C$ is bounded.
Using the structure result on $\mathrm{M}_{2}^{ss} = \mathbb{P}^{2}$, we can prove much more:
If $C$ is a generic line in $\mathrm{M}_{2}^{ss}$, then it requires a nontrivial bundle.
Generically, $C$ is not the line consisting of the resultant locus, $\mathrm{M}_{2}^{ss}\setminus\mathrm{M}_{2}$. So it intersects this line at exactly one point. Furthermore, since the resultant ${\operatorname{Res}}_{2}$ is an ${\operatorname{SL}}(2)$-invariant section, we have $D.{\operatorname{Res}}_{2} = m\cdot C.{\operatorname{Res}}_{2}$; we abuse notation and use ${\operatorname{Res}}_{d}^{n}$ to refer to the resultant divisor both upstairs and downstairs. Since the degree of the resultant upstairs is $(n+1)d^{n} = 4$ [@Jou], we obtain $4\cdot D.\mathcal{O}(1) = m$. In other words, $m \geq 4$.
However, using Proposition \[GIT2\], we will show $m \leq 2$ generically. The generic stabilizer is trivial, and the stabilized locus is a cuspidal cubic in $\mathbb{P}^{2}$, on which the stabilizer is isomorphic to $\mathbb{Z}/2\mathbb{Z}$, except at the cusp, where it is $S_{3}$. The generic line $C$ will intersect this cuspidal curve at three points, none of which is the cusp. Therefore, $h = 1$, and there are at most three points of ramification, with ramification degree $2$. By Riemann-Hurwitz, the maximum $m$ is $2$, contradicting $m \geq 4$.
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abstract: |
Using SU(3) chiral perturbation theory we calculate the density-dependent complex-valued spin-orbit coupling strength $U_{\Sigma ls}(k_f)+ i\, W_{\Sigma
ls}(k_f)$ of a $\Sigma$ hyperon in the nuclear medium. The leading long-range $\Sigma N$ interaction arises from iterated one-pion exchange with a $\Lambda$ or a $\Sigma$ hyperon in the intermediate state. We find from this unique long-range dynamics a sizeable “wrong-sign” spin-orbit coupling strength of $U_{\Sigma ls}(k_{f0}) \simeq -20$MeVfm$^2$ at normal nuclear matter density $\rho_0 = 0.16\,$fm$^{-3}$. The strong $\Sigma N\to \Lambda N$ conversion process contributes at the same time an imaginary part of $W_{
\Sigma ls}(k_{f0}) \simeq -12$MeVfm$^2$. When combined with estimates of the short-range contribution the total $\Sigma$-nuclear spin-orbit coupling becomes rather weak.
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25.1cm 17.cm -2.cm -0.6 cm -0.6 cm \#1\#2
\#2cm
N. Kaiser\
PACS: 13.75.Ev, 21.65.+f, 21.80.+a, 24.10.Cn\
Hypernuclear physics has a long and well-documented history [@dover; @chrien; @dovgal]. One primary goal in this field is to determine from the experimental data the nuclear mean-field potentials relevant for the hyperon single-particle motion. For the $\Lambda$ hyperon the situation is by now rather clear and the following quantitative features have emerged. The attractive nuclear mean-field potential for a $\Lambda$ hyperon is about half as strong as the one for nucleons in nuclei: $U_\Lambda \simeq -28\,$MeV [@2]. With this value of the potential depth the empirical single-particle energies of a $\Lambda$ bound in hypernuclei are well described over a wide range in mass number. On the other hand, the $\Lambda$-nucleus spin-orbit interaction is found to be extraordinarily weak. For example, recent precision measurements [@ajimura] of $E1$-transitions from $p$- to $s$-shell orbitals in $^{13}_\Lambda C$ give a $p_{3/2}-p_{1/2}$ spin-orbit splitting of only $(152\pm 65)\,$keV to be compared with a value of about $6$MeV in ordinary $p$-shell nuclei.
In case of the $\Sigma$ hyperon recent developments have lead to a revision concerning the sign and magnitude of its nuclear mean-field potential [@galneu]. Whereas an earlier analysis of the shifts and widths of x-ray transitions in $\Sigma^-$ atoms came up with an attractive (real) $\Sigma$-nucleus optical potential of about $-27\,$MeV [@dover], there is currently good experimental and phenomenological evidence for a substantial $\Sigma$-nucleus repulsion. A reanalysis of the $\Sigma^-$ atom data in Ref.[@batty] including the then available precise measurements of W and Pb atoms and employing phenomenological density-dependent fits has lead to a $\Sigma$-nucleus potential with a strongly repulsive core (of height $\sim 95\,$MeV) and a shallow attractive tail outside the nucleus. The inclusive $(\pi^-,K^+)$ spectra on medium-to-heavy nuclear targets measured at KEK [@noumi; @saha] give more direct evidence for a strongly repulsive $\Sigma$-nucleus potential. In the framework of the distorted wave impulse approximation, a best fit of the measured $(\pi^-,K^+)$ inclusive spectra on Si, Ni, In and Bi targets is obtained with a $\Sigma$-nucleus repulsion of about $90\,$MeV. However, the detailed description of the $\Sigma^-$ production mechanism plays an important role for the extracted value of the $\Sigma$-nucleus repulsion. Within a semiclassical distorted wave model [@kohno], which avoids the factorization approximation by an averaged differential cross section, the KEK data can also be well reproduced with a complex $\Sigma$-nucleus potential of strength $(30- 20\,i)$MeV. Concerning the $\Sigma$-nucleus spin-orbit coupling there exist so far no experimental hints for it. Most theoretical models [@pirner; @bouyssy] predict the $\Sigma$-nucleus spin-orbit coupling to be strong (i.e. comparable to the one of nucleons). The basic argument for a strong spin-orbit coupling is provided by the large and positive value of the tensor-to-vector coupling ratio of the $\omega$ meson to the $\Sigma$ hyperon assuming vector meson dominance and the non-relativistic quark model with SU(6) spin-flavor symmetry. The G-matrix calculations by the Kyoto-Niigata group [@fuji] using the hyperon-nucleon interaction as derived from their SU(6) quark model predict a $\Sigma$-nucleus spin-orbit coupling which is about half as strong as the one of nucleons. However, due to the presence of the strong $\Sigma N\to \Lambda N$ conversion process in the nuclear medium one expects the $\Sigma$-nucleus spin-orbit coupling strength to have also an imaginary part. This possibility has generally been ignored in quark and one-boson exchange models.
Recently, we have applied chiral effective field theory to calculate the hyperon mean-fields in nuclear matter [@lambdapot]. In this approach the small $\Lambda$-nuclear spin-orbit interaction finds a novel explanation in terms of an almost complete cancellation between short-range contributions (estimated from the known nucleonic spin-orbit coupling strength) and long-range terms generated by iterated one-pion exchange with intermediate $\Sigma$ hyperons. The exceptionally small $\Sigma\Lambda$ mass splitting of $M_\Sigma -M_\Lambda =77.5\,$MeV influences hereby prominently the effect coming from the second order $1\pi$-exchange tensor interaction. Furthermore, it has been shown in Ref.[@jorge] that the proposed cancellation mechanism does not get disturbed by the inclusion of analogous two-pion exchange processes involving decuplet baryons ($\Delta(1232)$ and $\Sigma^*(1385)$) in the intermediate state with considerably larger mass splittings. The density-dependent complex $\Sigma$-nuclear mean-field $U_\Sigma(k_f)+ i\,
W_\Sigma(k_f)$ has also been calculated in the same framework in Ref.[@sigmapot]. It has been found that genuine long-range[^1] contributions from iterated one-pion exchange with intermediate $\Lambda$ and $\Sigma$ hyperons sum up to a moderately repulsive (real) single-particle potential of $U_\Sigma(k_{f0}) \simeq 59\,$MeV at normal nuclear matter density $\rho_0 = 0.16\,$fm$^{-3}$. The $\Sigma N\to \Lambda N$ conversion process induced by one-pion exchange generates at the same time an imaginary single-particle potential of $W_\Sigma(k_{f0}) \simeq -21.5$MeV. This value is in fair agreement with empirical determinations [@batty] and quark model predictions [@kohno2]. The purpose of the present short paper is to calculate in the same chiral effective field theory framework the density-dependent complex-valued $\Sigma$-nuclear spin-orbit coupling strength. As for the $\Lambda$ hyperon [@lambdapot] we do find a sizeable “wrong-sign” spin-orbit coupling from the second-order one-pion exchange tensor interaction. When combined with estimates of the short-range contribution (employing QCD sum rule predictions) the total $\Sigma$-nuclear spin-orbit coupling becomes rather weak.
Let us begin with some basic considerations. The pertinent quantity to extract the $\Sigma$-nuclear spin-orbit coupling is the spin-dependent part of the self-energy of a $\Sigma$ hyperon interacting with weakly inhomogeneous isospin-symmetric (spin-saturated) nuclear matter. Let the $\Sigma$ hyperon scatter from initial momentum $\vec p- \vec q/2$ to final momentum $\vec p+
\vec q/2$. The spin-orbit part of the self-energy is then: $$\Sigma_{\rm spin} = {i \over 2} \,\vec \sigma \cdot (\vec q
\times \vec p\,) \, \Big[ U_{\Sigma ls}(k_f)+i\, W_{\Sigma ls}(k_f)\Big] \,,$$ where the density-dependent spin-orbit coupling strength $U_{\Sigma ls}(k_f)
+i\,W_{\Sigma ls}(k_f)$ is taken in the limit of homogeneous nuclear matter (characterized by its Fermi momentum $k_f$) and zero external $\Sigma$-momenta: $\vec p =\vec q =0$. The more familiar spin-orbit Hamiltonian follows from Eq.(1) by multiplication with a density form factor and Fourier transformation $\int d^3 q \exp(i \vec q \cdot \vec r\,)$. For orientation, consider first the $\omega$ meson exchange between the $\Sigma$ hyperon and the nucleons. The non-relativistic expansion of the vector (and tensor) coupling vertex between Dirac spinors of the $\Sigma$ hyperon gives rise to a spin-orbit term proportional to $i\,\vec \sigma \cdot (\vec q
\times \vec p\,)/4M_\Sigma^2$. Next one takes the limit of homogeneous nuclear matter (i.e. $\vec q=0$), performs the remaining integral over the nuclear Fermi sphere and arrives at the familiar result: $$U_{\Sigma ls}(k_f)^{(\omega)} = {g_{\omega \Sigma}(1+2
\kappa_{\omega \Sigma}) g_{\omega N} \over 2M_\Sigma^2 m_\omega^2} \, \rho \,,$$ linear in density $\rho= 2k_f^3/3\pi^2$. Here, $\kappa_{\omega\Sigma}$ denotes the tensor-to-vector coupling ratio of the $\omega$ meson to the $\Sigma$ hyperon.
The crucial observation is now that the (left) iterated one-pion exchange diagram in Fig.1 generates also a (sizeable) spin-orbit coupling term. The prefactor ${i\over2}\vec\sigma \times \vec q$ is immediately identified by rewriting the product of $\pi\Sigma B$-interaction vertices $\vec\sigma\cdot(
\vec l-\vec q/2)\,\vec \sigma \cdot (\vec l + \vec q/2) = {i\over 2}(\vec
\sigma \times \vec q\,)\cdot (-2 \vec l\,) +\dots $ at the open baryon line. For all remaining parts of the diagram one can then take the limit of homogeneous nuclear matter (i.e. $\vec q=0$). The other essential factor $\vec p$ comes from the energy denominator $-\Delta^2+\vec l\cdot(\vec l-\vec
p_1+\vec p\,)$. The $\Sigma\Lambda$ mass splitting is rewritten here in terms of the small scale parameter $\Delta = \sqrt{M_B (M_\Sigma -M_\Lambda)} \simeq
285\,$MeV with $M_B=(2M_N+M_\Lambda+M_\Sigma)/4\simeq 1047\,$MeV a mean baryon mass. It serves the purpose to average out small differences in the kinetic energies of the various baryons involved. Keeping only the term linear in the external momentum $\vec p$ one finds from the left diagram in Fig.1 with a $\Lambda$ hyperon in the intermediate state the following contribution to the $\Sigma$-nuclear spin-orbit coupling strength: $$\begin{aligned}
U_{\Sigma ls}(k_f)^{(2\pi\Lambda)} + i\, W_{\Sigma ls}(k_f)^{
(2\pi\Lambda)} &=& - {2D^2 g_A^2 \over 9f_\pi^4}\! \int\limits_{|\vec p_1| <
k_f} \!\!\!{d^3 p_1 d^3 l\over (2\pi)^6} { M_B\, {\vec l}\,^4 \over (m_\pi^2
+{\vec l}\,^2 )^2\, [ -\Delta^2 -i0+{\vec l}\,^2 -\vec l \cdot \vec p_1]^2 }
\nonumber \\ &=& {2\over 3} {\partial \over \partial \Delta^2} \Big[U_\Sigma(
k_f)^{(2\pi\Lambda)}+i\, W_\Sigma(k_f)^{(2\pi\Lambda)} \Big] \,. \end{aligned}$$ Here, $D=0.84$ and $F=0.46$ [@lambdapot] denote the SU(3) axial vector coupling constants together with $g_A = D+F =1.3$ the nucleon axial vector coupling constant. $f_\pi = 92.4\,$MeV is the pion decay constant and $m_\pi=138\,$MeV the average pion mass. Note that the loop integral in Eq.(3) is convergent as its stands. Most useful is actually the representation of the spin-orbit coupling strength as a derivative of the $\Sigma$-nuclear potential $U_\Sigma(k_f)+i\, W_\Sigma(k_f)$ with respect to the (mass splitting) parameter $\Delta^2$. Using the analytical expressions in Ref.[@sigmapot] to evaluate this derivative we find for the real and imaginary part: $$U_{\Sigma ls}(k_f)^{(2\pi\Lambda)} = {D^2 g_A^2 M_B m_\pi^2
\over 72 \pi^3 f_\pi^4} \bigg\{ (4+2\delta) \arctan{ \sqrt{u}\over 1+\delta}
- {3u+( 1+\delta)(4+2\delta)\over u +(1+\delta)^2} \sqrt{u} \bigg\} \,,$$ $$\begin{aligned}
W_{\Sigma ls}(k_f)^{(2\pi\Lambda)} &=& {D^2 g_A^2 M_B m_\pi^2
\over 72 \pi^3 f_\pi^4} \Bigg\{- {u+(1+\delta)(2+\delta)\over u +(1+\delta)^2}
\sqrt{u(4\delta+u)}\nonumber \\ && +(4+2\delta) \ln{u+2+2\delta+\sqrt{u(4
\delta+u)} \over 2[ u +(1+\delta)^2]^{1/2}} \Bigg\} \,, \end{aligned}$$ with the abbreviations $u= k_f^2/m_\pi^2$ and $\delta = \Delta^2/m_\pi^2$. The right diagram in Fig.1 with two medium insertions represents the Pauli blocking correction. In comparison to the expression in Eq.(3) the sign is reverse and the momentum transfer $\vec l$ gets replaced by $\vec l = \vec p_1
- \vec p_2$ with $\vec p_2$ to be integrated over a Fermi sphere of radius $k_f$, i.e. $|\vec p_2| < k_f$. In case of the real part one is left with a double-integral of the form: $$\begin{aligned}
U_{\Sigma ls}(k_f)^{(2\pi\Lambda)}_{\rm Pauli} &=&{D^2g_A^2M_B
m_\pi^2\over 36\pi^4 f_\pi^4} -\!\!\!\!\!\!\int_0^u \!\!dx \int_0^u \!\!dy \,
{1 \over (2\delta+1+x-y)^2 } \, \Bigg\{ {(2\delta+x-y)^2 \sqrt{xy}\over 2(
\delta-y)^2-2x y} \nonumber\\ &&+ {2 \sqrt{xy} \over (1+x+y)^2-4x y} +{2\delta
+x-y \over 2\delta+1+x-y} \ln{|\delta - y -\sqrt{xy} | (1+x+y - 2\sqrt{xy})
\over |\delta - y +\sqrt{xy}| (1+x+y + 2\sqrt{xy})}\Bigg\} \,, \nonumber \\\end{aligned}$$ where the first term in brackets has to be treated as a principal value integral. In practice this is done by solving the $\int_0^u\! dx$-integral analytically and converting the occurring logarithms into logarithms of absolute values. The Pauli blocking correction to the imaginary part $W_{\Sigma ls}(k_f)$ can even be written in closed analytical form: $$\begin{aligned}
W_{\Sigma ls}(k_f)^{(2\pi\Lambda)}_{\rm Pauli}& = &{D^2g_A^2
M_B m_\pi^2\over 72\pi^3 f_\pi^4}\, \theta(\sqrt{2}k_f-\Delta) \Bigg\{
{u\over 2}- \delta -1 +{1 \over 1+2 \delta}+ { u \delta \over u+\delta^2}
\nonumber\\ &&+ {u (1-\delta) \over 2u+2(1+\delta)^2}+ {u+(1+\delta)(2+\delta)
\over 2u +2(1 +\delta)^2} \sqrt{u(4 \delta+u)} +2\ln(2+4 \delta) \nonumber\\
&& + \delta \ln(2+2 \delta^2 u^{-1}) -(2+\delta)
\ln\Big[u+2+2\delta+\sqrt{u(4\delta+u)}\,\Big] \Bigg\} \,. \end{aligned}$$ Interestingly, there is a threshold condition $k_f>\Delta/\sqrt{2}$ for Pauli blocking to become active in the imaginary part. The threshold opens at about one half of nuclear matter saturation density $\rho_{\rm th}= 0.072\,{\rm fm
}^{-3} =0.45\rho_0$.
The additional contributions from the iterated one-pion exchange diagrams with a $\Sigma$ hyperon in the intermediate state are obtained by substituting axial vector coupling constants, $D^2\to 6 F^2$, and dropping the $\Sigma
\Lambda$ mass splitting, $\delta \to 0$. The explicit expressions for these contributions to the complex $\Sigma$-nuclear spin-orbit coupling strength read: $$U_{\Sigma ls}(k_f)^{(2\pi\Sigma)} = {F^2 g_A^2 M_B m_\pi^2
\over 12 \pi^3 f_\pi^4} \bigg\{ 4 \arctan \sqrt{u} - {4+3u \over 1+u} \sqrt{u}
\bigg\} \,,$$ $$W_{\Sigma ls}(k_f)^{(2\pi\Sigma)} = -W_{\Sigma ls}(k_f)^{
(2\pi\Sigma)}_{\rm Pauli} = {F^2 g_A^2 M_B m_\pi^2 \over 12 \pi^3 f_\pi^4}
\bigg\{2\ln(1+u)- {2u+u^2 \over 1+u} \bigg\} \,,$$ $$\begin{aligned}
U_{\Sigma ls}(k_f)^{(2\pi\Sigma)}_{\rm Pauli} &=& {F^2 g_A^2
M_B m_\pi^2 \over 12 \pi^4 f_\pi^4}\Bigg\{ 6\sqrt{u} \arctan(2\sqrt{u})-2u
-{2\sqrt{u} \over \sqrt{1+u}}\ln(\sqrt{u}+\sqrt{1+u}) \nonumber\\ &&
-{3\over 2} \ln(1+4u)+\int_0^u \!\!dx \,{1+2u-2x \over (1+u-x)^2}\ln{(\sqrt{u}-
\sqrt{x})(1+u+x +2 \sqrt{ux}) \over (\sqrt{u}+\sqrt{x})(1+u+x -2 \sqrt{ux}) }
\Bigg\} \,, \nonumber\\ \end{aligned}$$ where now almost all integrals could be solved for the Pauli blocking correction.
Summing up all calculated two-loop terms written in Eqs.(4-10) we show in Fig.2 the resulting complex $\Sigma$-nuclear spin-orbit coupling strength $U_{\Sigma ls}(k_f)+i\,W_{\Sigma ls}(k_f)$ as a function of the nucleon density in the region $0\leq \rho\leq 0.2\,$fm$^{-3}$ (corresponding to Fermi momenta $k_f \leq 283\,$MeV). It is expected that higher-loop contributions related to pion-absorption on two nucleons, in-medium nucleon and pion self-energy corrections etc. are small in this low-density region. The upper curve for the imaginary part $W_{\Sigma ls}(k_f)$ clearly displays the onset of the Pauli blocking effect at the threshold density $\rho_{\rm th}= 0.072\,{\rm fm}^{-3}$. It may come as a surprise that Pauli blocking increases the magnitude of the negative imaginary part. But going back to the original expression Eq.(3) one sees that the squared energy denominator introduces as a weight function for imaginary part the derivative of a delta-function. Therefore the usual argument of phase space reduction by Pauli blocking becomes insufficient even for a qualitative estimate. At normal nuclear matter density $\rho_0 = 0.16\,$fm$^{-3}$ (corresponding to a Fermi momentum of $k_{f0} = 263\,$MeV) one finds for the total imaginary part $W_{\Sigma ls}(k_{f0})=(-6.83-4.89)\,$MeVfm$^2 =-11.7$MeVfm$^2$, where the second entry stems from Pauli blocking. The physics behind this imaginary spin-orbit coupling strength is, of course, the $\Sigma N \to \Lambda N$ conversion process induced by $1\pi$-exchange. One can also see from Fig.2 that the cusp effect in the imaginary part $W_{\Sigma ls}(k_f)$ causes some non-smooth behavior of the real part $U_{\Sigma ls}(k_f)$. The almost linear decrease with density gets interrupted at the threshold density $\rho_{\rm
th}= 0.072\,{\rm fm}^{-3}$. At saturation density one finds a “wrong-sign” $\Sigma$-nuclear spin-orbit coupling strength of $U_{\Sigma ls}(k_{f0}) = [(
-1.83-2.32)+(-18.21+2.43)]\,$MeVfm$^2=-19.9$MeVfm$^2$, where the individual entries correspond to respective terms written in Eqs.(4,6,8,10), in that order. It is somewhat larger than the “wrong-sign” spin-orbit coupling of a $\Lambda$ hyperon, $U_{\Lambda ls}(k_{f0}) =-15\,$MeVfm$^2$ [@lambdapot]. This is our major result: The second order $1\pi$-exchange tensor interaction generates sizeable “wrong-sign” spin-orbit couplings for the $\Lambda$ and the $\Sigma$ hyperon together. The negative sign in case of the $\Sigma$ hyperon is however less obvious, because the relevant loop integrals are derivatives of six-dimensional principal value integrals (see Eq.(3)). As an aside we note that in the chiral limit ($m_\pi=0)$ the $\Sigma$-nuclear spin-orbit coupling strength changes to $U_{\Sigma ls}(k_{f0})+i\,W_{\Sigma
ls}(k_{f0}) =(-25.0-13.0\, i)\,$MeVfm$^2$, with the real part coming now entirely from the Pauli blocking corrections.
It is expected that the additional $2\pi$-exchange effects of Ref.[@jorge] including decuplet baryons in the intermediate state do not change the present results in a significant way. First, the additional mass splittings in the energy denominators are so high that no new contribution to the imaginary part $W_{\Sigma ls}(k_f)$ is generated for $\rho\leq \rho_0$. Secondly, the approximate cancellation between the contributions from $\Delta(1232)$ and $\Sigma^*(1385)$ intermediate states works for $\Lambda$ and $\Sigma$ hyperons together, since it is based on different signs of spin-sums [@jorge].
The short-range part of the $\Sigma$-nuclear spin-orbit interaction results from a variety of processes, one of them being the $\omega$-exchange piece presented in Eq.(2). Following Ref.[@lambdapot], we relate the short-distance spin-orbit coupling of the $\Sigma$ hyperon to the one of the nucleon as follows: $$U_{\Sigma ls}(k_f)^{(\rm sh)}= C_{ls}{M_N^2 \over
M_\Sigma^2} \, U_{N ls}(k_f)^{(\rm sh)}\,.$$ The factor $(M_N/M_\Sigma)^2 = 0.62$ results from the replacement of the nucleon by a $\Sigma$ hyperon in these relativistic spin-orbit terms. The coefficient $C_{ls}$ parameterizes the ratio of the relevant coupling constants. The expectation from the naive quark model would be $C_{ls}=2/3$. On the other hand, QCD sum rule calculations of $\Sigma$ hyperons in nuclear matter [@jin] indicate that the Lorentz scalar and vector mean fields of a $\Sigma$ hyperon are similar to the corresponding ones of a nucleon, i.e. $C_{ls} \simeq 1$. In case of the Lorentz scalar mean field, the QCD sum rule calculations are subject to uncertainties due to poorly known contributions from four-quark condensates. Ref.[@jin] concludes that due to a significant SU(3) symmetry breaking in nuclear matter the short-range spin-orbit term of a $\Sigma$ hyperon may be comparable to the one of a nucleon. For the further discussion we take for the short-range nucleonic spin-orbit coupling strength $U_{N ls}(k_f)^{(\rm sh)}= 3 \rho W_0/2
= 30 \,{\rm MeVfm}^2 \rho/\rho_0$ with $W_0 =124\,{\rm MeVfm}^5$ the spin-orbit parameter in the Skyrme phenomenology [@sly]. Employing $C_{ls}
\simeq 1$, as indicated by the sum rule calculations, one estimates the short-range $\Sigma$-nuclear spin-orbit coupling strength to $U_{\Sigma
ls}(k_{f0})^{(\rm sh)} \simeq 18.6\,$MeVfm$^2$. This would lead to an almost complete cancellation of the long-range component generated by iterated one-pion exchange, resulting in a rather weak $\Sigma$-nuclear spin-orbit coupling (admittedly with large uncertainties). Finally, we note that the long-range and short-range pieces are distinguished by markedly different dependences on the pion mass $m_\pi$ (or light quark mass $m_q \sim m_\pi^2$) and the density $\rho=2k_f^3/3\pi^2$. Therefore, there seems to be no double counting problem when adding long-range and short-range components.
In summary, we have calculated in this work the $\Sigma$-nuclear spin-orbit coupling generated by iterated one-pion exchange with a $\Lambda$ or a $\Sigma$ hyperon in the intermediate state. We find from this unique long-range dynamics a sizeable “wrong-sign” spin-orbit coupling strength of $U_{\Sigma ls}(k_{f0}) \simeq -20$MeVfm$^2$. When combined with estimates of the short-range component a weak $\Sigma$-nuclear spin-orbit coupling will result in total. Unfortunately, the prospects for an experimental check of this feature are poor. The recently established repulsive nature of the $\Sigma$-nucleus optical potential [@galneu] precludes a rich spectroscopy of heavy $\Sigma$-hypernuclei which could reveal spin-orbit splittings.
Acknowledgments: I thank A. Gal and W. Weise for suggesting this work and for informative discussions.
[99]{} C.B. Dover, D.J. Millener, and A. Gal, [*Phys. Reports*]{} [**184**]{}, 1 (1989), and refs. therein. R.E. Chrien and C.B. Dover, [*Annu. Rev. Nucl. Part. Sci.*]{} [**39**]{}, 113 (1989), and refs. therein. C.B. Dover and A. Gal, [*Prog. Part. Nucl. Phys.*]{} [ **12**]{}, 171 (1984). D.J. Millener, C.B. Dover, and A. Gal, [*Phys. Rev.*]{} [**C38**]{}, 2700 (1988). S. Ajimura et al., [*Phys. Rev. Lett.*]{} [**86**]{}, 4255 (2001). A. Gal, [*Prog. Theor. Phys. Suppl.*]{} [**156**]{}, 1 (2004). C.J. Batty, E. Friedman, and A. Gal, [*Phys. Reports*]{} [**287**]{}, 385 (1997), and refs. therein. H. Noumi et al., [*Phys. Rev. Lett.*]{} [**89**]{}, 072301 (2002); [**90**]{}, 049902 (E) (2003). P.K. Saha et al., [*Phys. Rev.*]{} [**C70**]{}, 044613 (2004). M. Kohno et al., [*Prog. Theor. Phys.*]{} [**112**]{}, 895 (2004); [*Phys. Rev.*]{} [**C74**]{}, 064613 (2006). H.J. Pirner, [*Phys. Lett.*]{} [**B85**]{}, 190 (1979). A. Bouyssy, [*Nucl. Phys.*]{} [**A381**]{}, 445 (1982). Y. Fujiwara, Y. Suzuki, and C. Nakamoto, [*Prog. Part. Nucl. Phys.*]{} [**58**]{}, 439 (2007). N. Kaiser and W. Weise, *Phys. Rev.* **C71**, 015203 (2005). J. Martin Camalich and M.J. Vicente Vacas, *Phys. Rev.* **C75**, 035207 (2007). N. Kaiser, [*Phys. Rev.*]{} **C71**, 068201 (2005). M. Kohno et al., [*Nucl. Phys.*]{} [**A674**]{}, 229 (2000). X. Jin and M. Nielsen, [*Phys. Rev.*]{} [**C51**]{}, 347 (1995), and refs. therein. E. Chabanat et al., [*Nucl. Phys.*]{} [**A635**]{}, 231 (1998).
[^1]: Genuine long-range means that (unique) part of the pion-loop which depends exclusively on small scales ($k_f, m_\pi, \Delta$), but not any high-momentum cutoff. In case of the $\Sigma$-nuclear mean field $U_\Sigma(k_f)$ it seems that the net short-range contribution is small [@sigmapot]. For the $\Lambda$ single-particle potential $U_\Lambda(k_f)$ an attractive short-range contribution [@lambdapot] is however necessary in order to reproduce the empirical potential depth of $-28\,$MeV. A deeper understanding of this feature is presently missing.
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abstract: |
The HCN 1–0 hyperfine lines have been observed toward 24 young stellar objects (YSOs) of class 0 and I. The hyperfine lines are well separated in most cases and show such rich structures as asymmetric double peaks and strong wings. We examined how their line shapes and velocity shifts vary along with their relative optical depths and compared them with those of CS 2–1, H$_2$CO 2$_{12}$–1$_{11}$, and HCO$^+$ 4–3 & 3–2 transitions previously observed by Mardones et al. (1997) and Gregersen et al. (1997).
It is found that all these molecular species do not always exhibit the same sense of line asymmetry and the correlation of velocity shift is better between HCN and CS than between HCN and H$_2$CO. The most opaque transition of HCN $F$=2–1 has about the same velocity shift as that of CS despite of the larger beam size of this study, which suggests that HCN $F$=2–1 line may be more sensitive to the internal motion of YSOs than CS line. Systematic changes of the velocity shift are noted for many sources, as one goes from $F$=0–1 to 2–1. The monotonic decrease of velocity (blue shift) is apparently more frequent.
A detailed model of radiative transfer allowing line overlap of HCN is employed to L483 which shows convincing signatures of infall on a scale of $\sim 0.1$ pc. It appears that the observed line is not compatible with the standard Shu (1977) model, but is fitted with augmentations of density and infall velocity, by factors of 6 and 0.5, respectively, and with an inclusion of a diffuse, static, turbulent, and geometrically thick envelope.
The distribution of hyperfine line intensity ratios for these YSOs does not accord with the LTE condition and is essentially the same as ones previously noted in cold dark clouds or small translucent cores. Though this anomaly may be explained in terms of radiative transfer effect in the cores which are either static or under systematic motion, some of them seem to invoke an existence of scattering envelope. It is confirmed that HCN is detected more selectively in class 0 and I sources than in starless cores or class II objects, which implies that the core embedding YSO(s) form a dense ($\sim 10^6$cm$^{-3}$) envelope with a significant HCN abundance in a narrow time span of their evolution ([@afo98]).
author:
- 'Y.-S. Park, Jongsoo Kim, and Y. C. Minh'
title: 'A Survey of the HCN $J$=1–0 Hyperfine Lines towards Class 0 and I Sources'
---
Introduction
============
Since the pioneering works of Zhou and his colleagues ([@zho93]; [@zho94]; [@zho95]), observational signatures of infall motion in the early phase of star formation have been discovered towards many young stellar objects (YSOs). They are mainly based on the spectral features that an opaque molecular line has a self-absorption with the blue peak stronger than the red one (hereafter the blue asymmetry), while an optically thin line of single peak is located between the two peaks of the opaque one. In an attempt to search for infalling YSOs, surveys of class 0 and I sources have been undertaken using the transitions of, e.g., CS, H$_2$CO, and HCO$^+$. However, only a few sources like B335 and IRAS 16293–2422 seem to provide compelling evidence of collapse motion ([@zho93]; [@zho95]; [@nar98]). Line profiles of some sources exhibit different asymmetry from one molecular species to another as well as from transition to transition in a molecule ([@mar97]; [@leh97]; [@gre97]). It is likely that the infall motion is not so simple or monotonic as one expects. Not only outflow motion but also aspherical geometry make the problem more complex. Thus, previous studies have always come to a conservative conclusion that class 0 objects seem to undergo infall phase in a statistical sense only and it is unclear for class I objects ([@mar97]; [@gre97]).
In order to have a comprehensive view on the internal motion of YSOs, one needs to make observations with various molecules and transitions using both single dish and interferometer. As a complement to the previous single dish observations by Mardones et al. (1997) and by Gregersen et al. (1997), we have carried out an HCN $J$=1–0 survey of class 0 and I YSOs. We can probe denser regions with HCN $J$=1–0 lines than with CS $J$=2–1 one, the most popular density tracer. The $J$=1–0 transition of HCN has three hyperfine lines ($F$=1–0, 1–1, and 2–1) whose optical depths are scaled to $1:3:5$ in the LTE condition. The hyperfine lines enable us to investigate how the line shape changes with their optical depths. Moreover, molecular abundance and spatial resolutions are the same for the three hyperfine components. This is an advantage over using other combinations of molecules or transitions with different beam sizes or abundances. Recently, it is known that the HCN emission peak of each Bok globule is coincident with the position of embedded source within 6$''$ and the detection rate of HCN lines is higher in class I and probably class 0 sources than in starless cores and class II sources ([@afo98]). Thus HCN may be a good tracer of class 0 and I objects.
The main goal of this study is to provide another set of line profiles of YSOs. Based on the data set, we investigate the tendency of line asymmetry among three hyperfine components, compare the asymmetry with that of other molecular lines, and probe the internal structure and kinematics of the cores. Probing with various molecular species may also be helpful in investigating molecular chemistry in the protostellar cores ([@raw96]).
After presenting observation and data reduction procedure (section 2), we describes observational results in section 3. A simple radiative transfer model of L483 is detailed in section 4. Discussions on the relevant molecular chemistry and hyperfine line ratios of HCN are given in section 5. Finally, we summarize results in section 6.
Observations
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The sources listed in Table 1 have been selected based on the data set in Mardones et al. (1997) and Gregersen et al. (1997). We detected HCN hyperfine lines toward 22 of 24 with an rms level of a few tens of mK.
Observations were carried out with a radome-enclosed 14 meter telescope in Taeduk Radio Astronomy Observatory, Korea, during April and May 1997. We used an SIS receiver and an autocorrelation spectrometer with 20 KHz (0.068 km s$^{-1}$ at HCN $F$=2–1 frequency) resolution. System temperatures were typically $400-600$ K (SSB) and pointing was good to an accuracy of 10$''$ in both directions of azimuth and elevation. The spectra were Hanning-smoothed, calibrated by a standard chopper wheel method, and presented in an antenna temperature ($T_A^*$) scale.
The rest frequency of $F$=2–1 transition is 88.631847 GHz, and the separations of $F$=1–1 and $F$=0–1 lines with respect to $F$=2–1 are 4.84 and $-7.07$ km s$^{-1}$, respectively ([@lov92]). Measured FWHM beam size and beam efficiency are 61$''$ and 40%, respectively, at this frequency. We used frequency switching mode with $\Delta f = 6$ MHz and obtained the spectra with the S/N ratios better than 15.
Results
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Line profiles
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Observed line profiles are presented in Fig. 1 (a-d), where we see spectral features like self-absorption, asymmetry, and broad wings. The hyperfine lines are well separated except for a few cases. Line asymmetries of many sources seem to grow as the optical depth increases. All three hyperfine lines appear to be optically thin for L146, Serp SMM3, L673A, and L1152, whereas they all are optically thick for NGC1333-4A, L483, and B335. The line wings of many sources are generally weaker than those in other studies ([@mar97]; [@gre97]), which may result from a larger telescope beam of this study.
Individual sources
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We outline the characteristics of HCN line profiles of the sources and, when available, compare them with those of CS 2–1, H$_2$CO 2$_{12}$–1$_{11}$, and N$_2$H$^+$ 1–0 in Mardones et al. (1997) and with HCO$^+$ 4–3 and 3–2 in Gregersen et al. (1997). Mainly sources with dissimilarities in line shapes among molecular species are discussed. It should be noted that the two surveys are made with FWHM beam size of $\stackrel{<}{_{\sim}}20''$, while ours are with 61$''$.
L1448-IRS3: $F$=2–1 line looks similar to CS line, whereas $F$=0–1 line to N$_2$H$^+$ line. HCO$^+$ 3–2 line is significantly shifted to the red.
L1448mm: All HCN hyperfine components show the blue asymmetry like CS spectrum. On the other hand, all transitions of H$_2$CO and HCO$^+$ have the red asymmetry.
NGC1333-2: All three lines differ from those of CS and H$_2$CO in their shapes. Instead, they are similar to N$_2$H$^+$ line, which seems to be also optically thick.
L43: It shows how the absorption dip grows as the optical depth increases. The line shape of $F$=2–1 transition is quite similar to that of CS. $F$=0–1 line is relatively strong compared to the other two hyperfine lines.
L146: The shape of line profiles are closer to that of H$_2$CO. Lines are gradually shifted to the red, as the transition becomes optically thick. As in L43, $F$=0–1 line is brighter than $F$=1–1 line. $F$=2–1 line is quite narrow.
L483: This is a good example showing how the line shape varies with the optical depth. It seems to be consistent with a Shu (1977) type infall motion; the red shoulder in the least opaque $F$=0–1 transition changes into the absorption dip in the most opaque $F$=2–1 one. $F$=2–1 lines are more asymmetric than those of CS and H$_2$CO. However, the asymmetry is reversed to the red in the 20$''$ beam observation of HCO$^+$. It is interesting to note that $F$=0–1 line is the brightest among three components. The depression of $F$=2–1 line can not be explained by self-absorption only (see section 4).
S68N: $F$=2–1 line has a deep self-absorption with a wider blue component. The general shape of HCN resembles that of CS, though the blue wing of HCN is weaker. By contrast, H$_2$CO has a clear blue asymmetry.
FIRS1: This is a typical object demonstrating diverse asymmetries of different molecular transitions. $F$=2–1 component of HCN shows a prominent blue asymmetry like CS, H$_2$CO line does the red one, and HCO$^+$ lines are nearly symmetric.
SMM4: This shows a progressive blue shift of HCN lines and deviation from symmetry with an increasing optical depth. However, the shift may not result from infall, but from outflow motion, judged from long tail to the blue. The opaque lines are similar in shape to both CS and H$_2$CO lines. HCO$^+$ lines with the central absorption dips have the same asymmetry, but do not have any long tail to the blue.
SMM3: All three HCN components are fitted well with Gaussian. Other transitions of CS, H$_2$CO, and HCO$^+$ are also symmetric. Relative intensities are quite close to the values of optically thin limit under the LTE condition.
B335: The asymmetry of HCN lines seems to change from the blue to the red as we move from $F$=0–1 line to $F$=2–1 one, although the S/N ratio is not so high. B335 core has been well known as a prototype of YSOs with convincing evidence of core collapse, and thus well studied ([@zho93], 1994; [@zho95]; [@cho95]). However, our observation contradicts most existing observations (cf. [@kam85]), except CCS observations ([@vel95]). DC 303.8-14.2 is another example showing a reversal of asymmetry among HCN hyperfine lines ([@leh97]). Most observations suggestive of collapse motion have been made with a beam of $\stackrel{<}{_{\sim}} 20''$, while observations of this study and Velusamy et al. (1995) are with $\sim 1'$ beam. Line profiles obtained with larger beams may be contaminated by outflow motion. However, we can not rule out a possibility that the outer part of B335 is slowly expanding ([@leh97]).
L1157: $F$=2–1 and 1–1 lines have long tails to the blue with single peaks, while $F$=0–1 line is almost symmetric. Both CS and H$_2$CO lines show the blue asymmetry as well. On the contrary, HCO$^+$ lines have rather strange spectral features, the blue asymmetry in $J$=3–2 and the red one in $J$=4–3 transition.
L1251B: Shoulders in the red of $F$=2–1 and 1–1 transitions coincide with that of CS. H$_2$CO has the blue asymmetry like HCN and CS, but its self-absorption is much deeper than those of HCN and CS.
In summary, for a large fraction of sources, the asymmetries are the same in all the transitions of HCN, CS, H$_2$CO, and HCO$^+$. However, for several sources, it depends on the molecular species and, particularly for B335 and L1157, it varies from transition to transition of a molecule. It should be pointed out that the asymmetry of HCN is relatively more similar to that of CS than to that of H$_2$CO, which will be quantified in next subsection.
Correlations
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For more detailed investigation of line profiles, one needs to quantify the line velocity associated with internal motion. We use a parameter defined by Mardones et al. (1997). At first, we measure $V_{\rm G}$($i$–$j$), the Gaussian peak velocity in the $F$=$i$–$j$ transition of HCN after fitting the three hyperfine lines with three Gaussians. Fitting procedure is straightforward for the lines of simple shape. For lines having another weak peak or a shoulder, we fit the profile after masking such features. If intensities of two peaks differ by less than $2 \sigma$, where $\sigma$ is the rms noise of spectrum, the whole velocity span is taken into account. The resulting $V_{\rm G}$($i$–$j$) and its standard deviation of each hyperfine component are listed in Table 1. The standard deviation is usually less than one channel width of the spectrometer. Since some hyperfine transitions of S68N, Serp FIRS1, and Serp SMM5 are blended each other, their $V_{\rm G}$($i$–$j$)s are not included in the table. The measure of velocity shift or degree of asymmetry $\delta V$ is then defined as $$\delta V = (V_{\rm G} - V_{\rm thin})/\Delta V_{\rm thin}.$$ We use the line velocity and FWHM of N$_2$H$^+$ for $V_{\rm thin}$ and $\Delta V_{\rm thin}$, respectively, in Mardones et al. (1997).
We plot relations between $\delta V$(HCN) and $\delta V$(CS), and between $\delta V$(HCN) and $\delta V$(H$_2$CO) in Fig. 2. The velocity shifts of CS and H$_2$CO molecules are also from Mardones et al. (1997). It is found that $\delta V$(0–1) correlates well with $\delta V$(CS). And we do not find any significant differences in their relations between class 0 and I. The close relation itself implies that HCN $F$=0–1 transition probes as similar kinematics as CS molecule does. It would be possible only if the distribution of HCN molecule is similar to that of CS inside the core. However, the slope of 0.47 suggests that $F$=0–1 transition is moderately optically thick; if the transition is as thin as that of N$_2$H$^+$, the slope should be around zero. It would not be so optically thick, since their line shapes are usually simple and show a single peak. The slope significantly less than unity may also be due to a difference in the beam size of two surveys; our larger telescope beam, which covers more volume of static envelope, may lessen the slope.
As going toward the $F$=2–1 transition, we notice the worse correlation as well as the gradual increase of the slope. The increase of the slope up to around unity implys that the most opaque line of HCN is as sensitive to the internal motion of the core as the CS, again suggesting similar spatial distributions of both HCN and CS molecules. General similarity in the line shapes between HCN $F$=2–1 and CS $J$=2–1, as shown in the previous subsection, supports this argument. An increasing scatter can be interpreted in terms of the opacity of the transition; since the opaque line is formed where the optical depth is about one, it does not reflect a global property, but does a local one. Thus small differences in excitation conditions and chemistry between HCN and CS would result in large differences in their line profiles.
In the lower panel of Fig. 2, we plot the relations between $\delta V$(HCN) and $\delta V$(H$_2$CO). The correlation between them is not so good as the case of $\delta V$(HCN) and $\delta V$(CS); we can hardly find a linear relationship between $\delta V$(0–1) and $\delta V$(H$_2$CO). Their distributions mimic the relation between $\delta V$(CS) and $\delta V$(H$_2$CO) illustrated in Fig 3, which is drawn from data set of Mardones et al. (1997). However, if we confine ourselves to the sources of $\delta V$(HCN)$<0$ and $\delta V$(H$_2$CO)$<0$, then we can see as similar trend as in the upper panel – the concomitant increase of the slope to unity with the optical depth.
Good correlations of $\delta V$s between HCN and CS indicate a similar distribution of both molecular species in the core. It is intriguing that, in spite of the larger beam size of our HCN observation, the slope is roughly unity for both pairs of $\delta V$(HCN)–$\delta V$(CS) and $\delta V$(HCN)–$\delta V$(H$_2$CO) for the most opaque transition of HCN. It appears that HCN is even more sensitive to the internal motion of the core. The critical density of HCN $J$=1–0, $2.5\times 10^6$ cm$^{-3}$, is larger than those of CS and H$_2$CO, $5.7\times 10^5$ cm$^{-3}$ and $1.1\times 10^6$ cm$^{-3}$, respectively. Thus HCN molecule traces deeper and denser regions of YSOs than CS or H$_2$CO. If HCN lines are observed with as similar beam size as that of CS or H$_2$CO ($\sim 20''$), we may be able to see more clearly the internal motion of the cores.
Fig. 4 quantifies the $\delta V$(HCN) of individual sources as a function of relative optical depth. If there is any systematic internal motion, we may expect gradual increase or decrease of $\delta V$ ([@mye95]; [@zha98]). It was not so easy to find such a relation from combinations of different molecules or transitions often sparsely distributed in frequency space, since the chemistry and beam size are different from each other. Most YSOs exhibit a monotonic increase or decrease of $\delta V$ with respect to the optical depth. Seven sources (L1448mm, NGC1333-4A, L483, Serp SMM4, L1157, L1172, and L1251B) suggest infall motion, while three sources (L43, L146, and B335) do outward motion. A few YSOs (L1448-IRS3, serp SMM3, and 18331-0035) seem to be static. A few large blobs or clumps may also give rise to such a systematic velocity shift. This possibility is, however, ruled out by the smoothness of optically thin N$_2$H$^+$ lines which are almost symmetric.
Radiative Transfer Model of L483
================================
Line profiles of L483 in Fig. 1 demonstrate clearly how the absorption dip develops under inward velocity field as the optical depth increases. Line intensity ratios among hyperfine transitions significantly deviate from the ‘standard hyperfine ratios’ (see section 5). Thus, synthesizing them with radiative transfer code will be useful in understanding the infall motion as well as structure of L483. We do not try to reproduce detailed features of the observed line profiles, but focus on quantitative comparisons between observed and synthesized lines.
The radiative transfer of HCN is complicated due to line overlap caused by hyperfine splitting of energy levels. In the case of cold cores like L483, the hyperfine lines do not overlap each other in $J$=1–0 transition, but they do in the transitions of higher $J$s, which affects the excitation condition of $J$=1–0 transition. The problem of the line overlap has been successfully treated by the Monte Carlo method ([@gon93]; [@lap89]) as well as by a conventional one ([@tur97]). We will use the former scheme and impose the condition of infall motion. Levels up to $J$=4 are taken into account, which include 13 individual energy levels and 21 radiative transitions among them. The Einstein $A$ coefficients and line frequencies are provided by Gonzáles-Alfonso (1998) and by Turner (1998), and the collisional rate coefficients by Monteiro & Stutzki (1986). With those molecular constants, we made a model of one dimensional radiative transfer, and confirmed its performance, by reproducing results of Gonzáles-Alfonso & Cernicharo (1993).
The Shu model ([@shu77]) could be a start point for the distributions of gas density and motion inside the core. Based on the observation of high density tracing molecules, HC$_3$N and NH$_3$ ([@ful93]), we fix the size of the core as $1.\!\!'72$ or $R=0.10$ pc at a distance of 200 pc. The core is divided into 30 concentric shells with radii running as $r_i \propto i^{0.7} (i=0,...,30)$. Since temperature is found to be 12 K near the [*IRAS*]{} source and fall to 9 K in the outer region ([@ful93]; [@par91]), $T_k$ is assumed to be constant at 10 K in the first set of calculations. The sound speed is then 0.2 km s$^{-1}$. An e-folding non-thermal turbulence of $v_{\rm turb}=0.4$ km s$^{-1}$ is found from Myers et al. (1995), and the abundance of HCN relative to H$_2$, $5\times 10^{-9}$ is from Turner et al. (1997) and from Gonzáles-Alfonso & Cernicharo (1993). Model calculations are then carried out and resulting line profiles are convolved with the telescope beam. In order to compare the synthetic line profile with the observed one in brightness temperature unit, we divided the latter one by 0.5, a compromise between the main beam efficiency (0.4) and the forward beam coupling efficiency (0.7).
The Shu model is completely described by an infall radius, $r_{\rm inf}$ or an elapsed time after the onset of infall. Fig. 5 shows synthesized line profiles for $r_{\rm inf} = 0, 0.4 R$, and $0.8 R$, respectively. From the synthesized lines (top in Fig. 5), we find that i) the asymmetry is negligible in three hyperfine lines, ii) the synthesized lines are weaker than the observed ones, and iii) the line intensity ratios are maintained as $I(F=0-1):I(F=1-1):I(F=2-1) = 1:(1.3-2):(2-3)$. Then we applied the kinetic temperature varying as $T_k(r)=10 (R/r)^{0.4}$ K, where the $r^{-0.4}$ dependence is based on the distribution of dust temperature ([@mye95]; [@wan95]). The sound speed was accordingly changed to 0.25 km s$^{-1}$. In fact, the Shu model assumes ‘isothermal’ cloud, and thus the model with kinetic temperature decreasing outward is far from consistency. However, we need to assume ‘constant’ sound speed in order to use his simple expression of density and velocity field. With this new temperature distribution, the synthetic lines (middle in Fig. 5) become stronger and more asymmetric. However, the $F$=0–1 line is still weaker than the observed one, and the hyperfine ratios differ from observation. It appears that the inside-out collapse model can not produce the observational features well. One viable option is to increase the density of Shu model core and to introduce an extended diffuse envelope, as proposed by Wang et al. (1995) and by Gonzáles-Alfonso & Cernicharo (1993). The role of the diffuse envelope is to attenuate the optically thicker line more. A large turbulence in the envelope is required for rather uniform attenuation across the line. If the envelope is sufficiently opaque, the line core of $F$=2–1 component may be formed in the envelope. The general weakening of lines is then compensated by an augmentation of density in the core. A successful fit after several trial and errors is shown in the bottom of Fig. 5. The resulting model is that a core with $r_{\rm inf}=R=0.1$ pc and $v_{\rm turb}=0.3$ km s$^{-1}$ is embedded in a static envelope with $R=0.3$ pc, $n({\rm H}_2)=5\times 10^3$ cm$^{-3}$, $v_{\rm turb}=1$ km s$^{-1}$, and $T_k=10$ K. In the core, the density increases by a factor of 6 and the infall velocity decreases by a factor of 2 with respect to those of the Shu model, respectively. The line center optical depths of $F$=0–1, 1–1, and 2–1 transitions towards the center of core are 3, 8, and 11, respectively, which are not beam averaged. The difference in peak brightness temperature is, of course, due to different excitation temperatures of the transitions. The red shift of absorption dip shown in Fig. 1b justifies rather large infall radius; without the inward motion of outer layer, the absorption dip will be located at the velocity of optically thin line. The size and density of envelope may change in such a way that its optical depth is kept constant. Different combinations, however, give rise to deeper or shallower absorption dips due to different excitation conditions of HCN in the envelope. Thus, we come to a model different from that of Shu (1977), as invoked already by Wang et al. (1995). In fact, higher density and slower infall speed are characteristic features of a magnetically supported core model, where the contraction occurs quasi-statically and the density of envelope increases as the core evolves ([@cio94], 1995).
Obviously, the real structure of L483 seems to be more complex than our model. It is shown that a near IR image exhibits an elongated structure in the East-West direction and the axis of CO molecular outflow is also aligned in this direction ([@par91]; [@lad91]; [@ful95]). Gregersen et al. (1997) noticed that HCO$^+$ $J$=3–2 and 4–3 lines show the red asymmetry, contrary to ours. However, the possibility of outward motion suggested by HCO$^+$ near the center is ruled out by the recent VLA observation of NH$_3$ indicating inward motion down to $\sim 0.\!\!'2$ scale ([@ful99]). Reasons for different asymmetry of HCO$^+$ may be attributable to differences in molecular chemistry and excitation conditions (see section 5).
Discussions
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The similarity of line shapes between CS and HCN and the dissimilarity between HCO$^+$ and HCN were already mentioned in section 3. We further examine the line asymmetries of 9 class 0 objects which have been observed in all transitions of four molecular species (CS, H$_2$CO, HCN, and HCO$^+$). As summarized in Table 2, only one source each shows the reversal of asymmetry between CS and HCN, and between HCO$^+$ and H$_2$CO. On the other hand, three sources change their asymmetries between HCN and HCO$^+$. Thus, though the number of samples is not large, it seems that the line asymmetry is similar within each pair of CS–HCN and HCO$^+$–H$_2$CO, but different between two pairs. It is interesting to note that CS and probably HCN seem to prefer the blue asymmetry compared with HCO$^+$.
Why do the two pairs show different asymmetry? Let us first consider if the critical densities of four molecules are grouped in the same way as their line shapes. A sequence of transitions is, however, CS 2–1, H$_2$CO 2$_{12}$–1$_{11}$, HCN 1–0, and HCO$^+$ 3–2 & 4–3 in an order of increasing critical density (The critical densities of HCO$^+$ 3–2 and 4–3 are $0.5\times 10^7$ and $1.3\times 10^7$ cm $^{-3}$, respectively; [@mon85]), suggesting that excitation condition is of little importance. Then molecular chemistry such as depletion in the innermost region and enhancement in the outflow may play important roles in the line formation. It is known that HCN as well as CS may freeze out onto grains in cold dense region ([@ber95]; [@bla92]; [@mcm94]), whereas HCO$^+$ will not, since it has a small dipole moment and there is no chemical reaction route to consume it in this region ([@raw96]; [@van98]). However, there seems to be little observational evidence of significantly depleted CS or HCN. If so, the blue asymmetry would be more frequent in HCO$^+$ than in CS or HCN, provided that motion is more like inside-out collapse, which is usually confined to the central region of core. However, this is not the case, as shown in Table 2.
Outflows prevalent in these sources may make certain molecules more abundant. L1157 is one of the examples which shows dramatic enhancements of various kinds of molecules ([@bac97]), where four molecules of our concern, CS, H$_2$CO, HCN, and HCO$^+$ are all enhanced. In the interferometric observations of 9 class 0 objects, HCO$^+$ and HCN show so different sensitivity to the envelope and outflow from source to source that one could not find any general tendency ([@cho99]). Recent BIMA observation of L483 suggests that HCO$^+$ traces outflow, while HCN does thick disk or envelope ([@par99]). The molecular chemistry is very complicated in this way, if both outflow and infalling envelope coexist; even in the case that the outflow occupies a small fraction of volume, the line shape will be significantly affected, if the abundance increases drastically (e.g., $>100$) in the flow region. The chemistry may also be related with the evolution of YSOs. Observations with high spatial resolution and more elaborated time-dependent chemistry models are required for further understanding of line formation and molecular chemistry in YSOs ([@mun95]).
We noted in section 3 that hyperfine line intensities of Serp SMM4 and SMM3 are scaled to the relative LTE optical depth. It is also interesting to note that, despite of their strong self-absorptions, the intensity ratios of three sources, S68N, Serp FIRS1, and SMM5, are also close to $1:3:5$, an optically thin limit in the LTE condition. In Fig. 6, excluding these three sources, we plot the hyperfine line ratios, $R_{02}$ and $R_{12}$, defined by, $$\begin{aligned}
R_{02} = {T_{\rm max}(F=0-1) \over T_{\rm max}(F=2-1)}, \ \ \ \ \
R_{12} = {T_{\rm max}(F=1-1) \over T_{\rm max}(F=2-1)}. \end{aligned}$$ It is found that $R_{12}$ lies between 0.4 and 0.7, except one source, while $R_{02}$ spans from 0.2 to 1.0. Similar distributions have been known for quiescent dark clouds ([@har89]; [@gon93]) and for small translucent cores ([@tur97]), though their physical conditions are quite different each other. The parameter space with $R_{12}\stackrel{<}{_{\sim}}1$ and $R_{02}\stackrel{<}{_{\sim}}0.8$ can be explained in terms of radiative transfer effect on the clouds/cores which are either static or under systematic motion ([@gon93]; [@tur97]). However, the region of $R_{12}\stackrel{>}{_{\sim}}1$ or $R_{02}\stackrel{>}{_{\sim}}0.8$, where the $F$=2–1 component is significantly suppressed with respect to the $F$=0–1 and $F$=1–1 components, seems to invoke the existence of diffuse envelope, as we have shown in section 4.
Recently, Afonso et al. (1998) carried out an HCN $J$=1–0 survey towards YSOs in Bok globules. They found that HCN is detected with a higher probability in class I and probably class 0 objects than in starless cores and class II sources. Because there is only one class 0 source in their survey, such a preference for class 0 was uncertain. Our study implies the high detection rate of almost unity for both class 0 and I sources. Thus there seems to be a unique phase of class 0 and I in the evolution of YSOs when a dense envelope of $\sim 1'$ ($0.05-0.1$ pc at the distances of $150-300$ pc) in size is formed.
Summary
=======
We have carried out a survey of HCN $J$=1–0 hyperfine lines for 24 objects identified as class 0 and I with a spectral resolution of 0.068 km s$^{-1}$. 22 sources are detected with around 30 mK rms level in $T_A^*$ unit.
It is found that three hyperfine components show a variety of spectral features such as deep self-absorption, asymmetry, and broad wings which are more prominent in the optically thicker lines. Moreover, for a large fraction of sources, HCN hyperfine lines show a progressive shift to the blue, as the optical depth increases. Only a few sources show a gradual shift to the red, which implies that an inward motion is predominant in the core embedding YSOs. When compared with previous CS and H$_2$CO surveys, the velocity shifts of HCN correlate better with those of CS than with those of H$_2$CO. Little difference in the correlation is noted between class 0 and I.
L483 is confirmed as a candidate infalling source on $\sim 0.1$ pc scale, from a growing degree of asymmetry and self-absorption with an increasing optical depth. We synthesized its hyperfine lines, by solving radiative transfer in a collapsing core model with the Monte Carlo method. The synthetic lines based on the Shu (1977) model do not fit the observed ones. We reproduced the observed ones successfully with the modification of the Shu model, an overall increase of gas density by a factor of 6 and the decrease of infall velocity by a factor of 2. Furthermore, in order to explain the line intensity ratios, a diffuse, static, and geometrically thick envelope surrounding the modified Shu core is essential.
Authors are grateful to Dr. C.W. Lee and Dr. E. Gonzáles-Alfonso for helpful discussions. This study was supported by Korea Astronomy Observatory through KAO grant 97-5400-000.
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[lrrccrrr]{} L1448-IRS3 & 3 22 31.5 & 30 34 49 & 0 & & 4.53$\pm0.02$ & 4.52$\pm0.02$ & 4.54$\pm0.01$ L1448mm & 3 22 34.4 & 30 33 35 & 0 & & 4.76$\pm0.03$ & 4.84$\pm0.02$ & 4.82$\pm0.02$ NGC1333-2 & 3 25 49.9 & 31 04 16 & 0 & & 7.65$\pm0.04$ & 7.59$\pm0.03$ & 7.50$\pm0.02$ NGC1333-4A & 3 26 04.8 & 31 03 13 & 0 & & 7.04$\pm0.04$ & 6.76$\pm0.02$ & 6.70$\pm0.01$ L43 & 16 31 37.7 & -15 40 52 & I & & 0.58$\pm0.01$ & 0.54$\pm0.02$ & 0.71$\pm0.01$ L146 & 16 54 27.2 & -16 04 48 & I & & 5.25$\pm0.01$ & 5.24$\pm0.02$ & 5.34$\pm0.01$ L483 & 18 14 50.6 & -04 40 49 & 0 & & 5.30$\pm0.01$ & 5.11$\pm0.03$ & 5.05$\pm0.02$ S68N & 18 27 15.2 & 01 14 57 & 0 & & & & Serp FIRS1 & 18 27 17.4 & 01 13 16 & 0 & & & & Serp SMM5 & 18 27 18.9 & 01 14 36 & 0 & & & & Serp SMM4 & 18 27 24.3 & 01 11 11 & 0 & & 7.81$\pm0.03$ & 7.62$\pm0.02$ & 7.71$\pm0.01$ Serp SMM3 & 18 27 27.3 & 01 11 55 & 0 & & 7.80$\pm0.03$ & 7.70$\pm0.01$ & 7.68$\pm0.01$ Serp SMM2 & 18 27 28.0 & 01 10 45 & 0 & & 7.45$\pm0.03$ & 7.30$\pm0.02$ & 7.28$\pm0.01$ 18331-0035 & 18 33 07.6 & -00 35 48 & 0 & &10.86$\pm0.03$ & 10.81$\pm0.02$ & 10.89$\pm0.01$ L723 & 19 15 41.3 & 19 06 47 & 0 & & & & L673A & 19 18 04.6 & 11 14 12 & 0 & & 7.02$\pm0.02$ & 6.94$\pm0.01$ & 6.99$\pm0.01$ B335 & 19 34 35.7 & 07 27 20 & 0 & & 8.38$\pm0.02$ & 8.39$\pm0.02$ & 8.60$\pm0.01$ IRAS20050 & 20 05 02.5 & 27 20 09 & 0 & & & & L1152 & 20 35 19.4 & 67 42 30 & I & & 2.67$\pm0.01$ & 2.65$\pm0.02$ & 2.62$\pm0.01$ L1157 & 20 38 39.6 & 67 51 33 & 0 & & 2.60$\pm0.02$ & 2.38$\pm0.02$ & 2.31$\pm0.01$ L1172 & 21 01 44.2 & 67 42 24 & I & & 2.84$\pm0.02$ & 2.60$\pm0.02$ & 2.57$\pm0.02$ L1251A & 22 34 22.0 & 75 01 32 & I & &-4.90$\pm0.04$ & -4.98$\pm0.04$ & -5.15$\pm0.02$ L1251B & 22 37 40.8 & 74 55 50 & I & &-3.94$\pm0.03$ & -4.06$\pm0.03$ & -4.13$\pm0.01$ L1262 & 23 23 48.7 & 74 01 08 & I & & 4.12$\pm0.02$ & 4.08$\pm0.03$ & 4.25$\pm0.02$
[lccccc]{} L1448-IRS3 & R & R & & R & N L1448mm & B & R & R & R & B NGC1333-4A & B & B & B & B & B L483 & B & B & R & R & B Serp FIRS1 & B & R & N & N & B Serp SMM4 & B & B & B & B & B Serp SMM3 & R & N & N & N & N B335 & B & B & B & B & R L1157 & B & B & R & B & B
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abstract: 'Quantum non-demolition (QND) measurement of collective variables by off-resonant optical probing has the ability to create entanglement and squeezing in atomic ensembles. Until now, this technique has been applied to real or effective spin one-half systems. We show theoretically that the build-up of Raman coherence prevents the naive application of this technique to larger spin atoms, but that dynamical decoupling can be used to recover the ideal QND behavior. We experimentally demonstrate dynamical decoupling by using a two-polarization probing technique. The decoupled QND measurement achieves a sensitivity 5.7(6) dB better than the spin projection noise.'
author:
- 'M. Koschorreck'
- 'M. Napolitano'
- 'B. Dubost'
- 'M. W. Mitchell'
bibliography:
- 'TPP.bib'
title: 'QND Measurement of Large-Spin Ensembles by Dynamical Decoupling'
---
Quantum non-demolition measurement plays a central role in quantum networking and quantum metrology for its ability to simultaneously detect and generate non-classical quantum states. The original proposal by Braginsky [@Braginsky1974UFNv114p41] in the context of gravitational wave detection has been generalized to the optical [@Poizat1994APv19p265; @Holland1990PRAv42p2995], atomic [@Kuzmich1998ELv42p481] and nano-mechanical [@Ruskov2005PRBv71p235407] domains. In the atomic domain, QND by dispersive optical probing of spins or pseudo-spins has been demonstrated using ensembles of cold atoms on a clock transition [@Windpassinger2009MSTv20p55301; @Schleier-Smith2010PRLv104p73604], and with polarization variables [@Takano2009PRLv102p33601; @Koschorreck2010PRLv104p93602], but thus far only with real or effective spin-1/2 systems.
QND measurement of larger spin systems offers a metrological advantage, e.g., in magnetometry [@Geremia2005PRLv94p203002], and may be essential for the detection of different quantum phases of degenerate atomic gases that intrinsically rely on large-spin systems [@Eckert2007NPv4p50; @Eckert2007PRLv98p100404; @Roscilde2009NJPv11p55041]. Dispersive interactions with large-spin atoms are complicated by the presence of non-QND-type terms in the effective Hamiltonian describing the interaction [@Geremia2006PRAv73p42112; @Madsen2004PRAv70p52324; @Echaniz2005JOBv7p548]. As we show, and contrary to what has often been assumed [@Kuzmich2000PRLv85p1594; @Eckert2007NPv4p50; @Eckert2007PRLv98p100404; @Roscilde2009NJPv11p55041], these terms spoil the QND performance, even in the large-detuning limit. The non-QND terms introduce noise into the measured variable, or equivalently decoherence into the atomic state. The problem is serious for both large and small ensembles, so that naive application of dispersive probing fails for several of the above-cited proposals.
We approach this problem using the methods of dynamical decoupling [@Viola1998PRAv58p2733; @Viola1999PRLv82p2417; @Facchi2005PRAv71p22302], which allow us to effectively cancel the non-QND terms in the Hamiltonian while retaining the QND term. To our knowledge, this is the first application of this method to quantum non-demolition measurements. Dynamical decoupling has been extensively applied in magnetic resonance [@Morton2008Nv455p1085; @Biercuk2009Nv458p996], used to suppress collisional decoherence in a thermal vapor [@Search2000PRLv85p2272], to extend coherence times in solids [@Taylor2008NPv4p810], in Rydberg atoms [@Minns2006PRLv97p], and with photon polarization [@Damodarakurup2009PRLv103p40502]. Other approaches include application of a static perturbation [@Smith2004PRLv93p163602; @Fraval2004PRLv92p].
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We consider an ensemble of spin-$f$ atoms interacting with a pulse of near-resonant polarized light. As described in references [@Geremia2006PRAv73p42112; @Madsen2004PRAv70p52324; @Echaniz2005JOBv7p548], the light and atoms interact by the effective Hamiltonian $\hat{H}_{\rm eff}$ $$\tau {\hat{H}}_{\mathrm{eff}} = G_1 \Sz \Jz+G_2 ( \Sx \Jx + \Sy \Jy)\,\,, \label{eq:H_full}$$ where $\tau$ is the duration of the pulse and $G_{1,2}$ are coupling constants that depend on the atomic absorption cross section, the beam geometry, the detuning from resonance $\Delta$, and the hyperfine structure of the atom [@Kubasik2009PRAv79p43815]. The atomic variables $\hat{{{\bf J}}}$ (described below) are collective spin and alignment operators. The light is described by the Stokes operators $\hat{{{\bf S}}}$ defined as $\hat{S}_i \equiv \frac{1}{2}(\hat{a}_+^\dagger,\hat{a}_-^\dagger) \sigma_i (\hat{a}_+,\hat{a}_-)^T$, where the $\sigma_i$ are the Pauli matrices and $\hat{a}_\pm$ are annihilation operators for the temporal mode of the pulse and circular plus/minus polarization. Bold subscripts, e.g., $\mathbf{x}$, are used to label non-spatial directions for atomic and light variables. The $G_1$ term describes a QND interaction, while the $G_2$ describes a more complicated coupling. In the dispersive, i.e. far-detuned, regime, $G_1$ and $G_2$ scale as $\Delta^{-1}$ and $\Delta^{-2}$, respectively. It has sometimes been assumed that the $G_2$ terms can be neglected for sufficiently large $\Delta$, leaving an approximate QND interaction. As we show below, this scaling argument fails, and the $G_2$ terms remain important. We note an important symmetry: $\hat{H}_{\rm eff}$ commutes with $\Sz + \Jz$, and is thus invariant under simultaneous rotation of $\hat{{{\bf J}}}$ and $\hat{{{\bf S}}}$ about the $z$ axis.
The atomic collective variables are $\hat{J}_k \equiv \sum_{i}^{N_{A}} \hat{j}^{(i)}_k$ where the superscript indicates the $i$-th atom and $\jx \equiv (\hat{f}_x^2 - \hat{f}_y^2 )/2$, $\jy \equiv (\hat{f}_x \hat{f}_y + \hat{f}_y \hat{f}_x)/2$, $\jz \equiv \hat{f}_z/2$ and ${\hat{j}_{[\mathbf{x,y}]}}\equiv -i [\jx,\jy] = \hat{f}_z (\hat{f}^2 - \hat{f}_z^2 -1/2 ) $. These obey commutation relations $[\jz,\jx] = i \jy$, $ [\jy,\jz] = i \jx$, $ [\jx,\jy] = i {\hat{j}_{[\mathbf{x,y}]}}$. For $f=1/2$, $\jx,\jy$ and ${\hat{j}_{[\mathbf{x,y}]}}$ vanish identically while for $f=1$, ${\hat{j}_{[\mathbf{x,y}]}}=\jz$ so that $\jx,\jy,$ and $\jz$ describe a pseudo-spin $\hat{\bf j}$.
In the QND scenario, an initial coherent polarization state with $\dexpect{\hat{{\bf S}}} = (N_L/2,0,0)$ is passed through the ensemble and experiences a rotation due to the $G_1$ term such that the component $\Sy$ (the ‘meter’ variable) indicates the value of $\Jz$ (the ‘system’ variable). We assume that $\Jx = N_A/2$. For a weak pulse, i.e., for $\dexpect{\hat{{\bf S}}} $ sufficiently small, we have the $\tau$-linear input-output relations $\hat{A}{^{({\rm out})} }= \hat{A}{^{({\rm in})} }- i \tau [\hat{A}{^{({\rm in})} },\hat{H}_{\rm eff}]$. Of specific interest are
$$\begin{aligned}
\Jz{^{({\rm out})} }&=& \Jz{^{({\rm in})} }{}+ G_2 \Sx \Jy{^{({\rm in})} }-{G_2 \Sy{^{({\rm in})} }\Jx} \,\,,\label{JzInOut} \\
\Jy{^{({\rm out})} }&=& \Jy{^{({\rm in})} }- G_1 \Sz{^{({\rm in})} }\Jx - G_2 \Sx {\hat{J}_{[\mathbf{x,y}]}}{^{({\rm in})} }\,\,, \label{JyInOut} \\
\Sy{^{({\rm out})} }&=& \Sy{^{({\rm in})} }+ G_1 \Sx \Jz{^{({\rm in})} }-{G_2 \Sz{^{({\rm in})} }\Jy} \,\,,\label{SyInOut}\end{aligned}$$
which describe the change in the system variable, its conjugate, and the meter variable. In the case of $f=1/2$, the $G_2$ terms vanish identically and we have a pure QND measurement: information about $\Jz$ enters $\Sy$ and there is a back-action on $\Jy$, but not on $\Jz$. The input noise $\var{\Sy{^{({\rm in})} }} = \mSx/2$ limits the performance of the measurement, and corresponds to a spin sensitivity of $\delta \Jz^2 = (2 G_1^2 \Sx)^{-1}$. For comparison, the projection noise of an ${\mathbf{x}}$-polarized spin state is $\var{\Jz} = \Jx/2$, so that projection noise sensitivity is achieved for $\Sx = (G_1^2 \Jx)^{-1} \equiv {S_{\rm SNR}}$.
This ideal QND regime does not occur naturally except for $f=1/2$. In the interesting regime $\Sx \approx {S_{\rm SNR}}$, we find that ${G_2 \Sx \Jy} \approx \Jy(G_{2}/G_{1}^2)/\Jx$ is independent of $\Delta$, and cannot be neglected based on detuning. To get an order of magnitude, we note that for large detuning, $G_1 \approx {\sigma_0 \Gamma}/{4 A \Delta}$, $G_2 \approx G_1 \Delta_{\rm HFS}/\Delta$ where $\sigma_0$ is the on-resonance scattering cross-section, $A$ is the effective area of the beam, and $\Gamma$ and $\Delta_{\rm HFS}$ are the natural linewidth and hyperfine splitting, respectively, of the excited states. In terms of the on-resonance optical depth $d_0 \equiv \sigma_0 N_A / A$, we find $G_2/G_1^2 \mJx \approx 8 \Delta_{\rm HFS}/d_0 \Gamma$. In a typical experiment with rubidium on the $D_2$ line, $\Delta_{\rm HFS}/\Gamma \sim 30$ and $d_0 \sim 50$ [@Kubasik2009PRAv79p43815], so the contribution of this term is important.
In contrast, the last term in Eq. (\[JyInOut\]) and (\[SyInOut\]), respectively, contribute variances $\left<G_2^2 \Sy^2 \Jx^2\right>$ and $\left<G_2^2 \Sz^2 \Jy^2\right>$ which scale as $\Delta^{-2}$. We will henceforth drop these terms.
The system variable $\Jz$ is coupled to a degree of freedom, $\Jy$, which is neither system nor meter in the QND measurement. This coupling introduces noise into the system variable, and decoherence into the state of the ensemble. To remove the decoherence associated with this coupling $G_2 \Sx \Jy$, we adopt the strategy of “bang-bang” dynamical decoupling [@Viola1998PRAv58p2733; @Viola1999PRLv82p2417; @Facchi2005PRAv71p22302]. In this method, a unitary ${\hat{U}_{b}}$ and its inverse ${\hat{U}_{b}}^\dagger$ are alternately and periodically applied to the system $p$ times during the evolution, so that the total evolution is $[{\hat{U}_{b}}^\dagger {\hat{U}}_H(t/2p) {\hat{U}_{b}}{\hat{U}}_H(t/2p)]^p$ where ${\hat{U}}_H(t)$ describes unitary evolution under $\hat{H}$ for a time $t$. With this evolution, those system variables that are unchanged by ${\hat{U}_{b}}$ continue to evolve under $\hat{H}$, while others are rapidly switched from one value to another, preventing coherent evolution. For large $p$, the system evolves under a modified Hamiltonian $\hat{H}' = \hat{P} \hat{H}$, where $\hat{P}$ projects onto the commutant (i.e., the set of operators which commute with) of $\{ {\hat{U}_{b}},{\hat{U}_{b}}^\dagger \}$ [@Facchi2005PRAv71p22302].
To eliminate $G_2 (\Sx \Jx + \Sy \Jy)$, while keeping $G_1 \Sz \Jz$ we choose a ${\hat{U}_{b}}$ which commutes with $\Jz$, but not with $\Jx$ or $\Jy$, namely a $\pi$ rotation about $\Jz$, ${\hat{U}_{b}}= \exp[i \pi \Jz]$. This leaves $\Jz$ unchanged, but inverts $\Jx$ and $\Jy$. By the symmetry of $\hat{H}_{\rm eff}$, this is equivalent to inverting $\Sx$ and $\Sy$, which suggests a practical implementation: probe with pulses of alternating $\Sx$, and define a ‘meter’ variable taking into account the inversion of $\Sy$.
We consider sequential interaction of the ensemble with a pair of pulses, with $\Sx^{(1)} = -\Sx^{(2)}=N_L/4p$. We define also the new ‘meter’ variable $S_y^{(\rm diff)} \equiv \Sy^{(1)} -\Sy^{(2)}$. We describe the atomic variables before, between, and after the two pulses with superscripts $(\rm in),(mid),(out)$, respectively. We apply Equations (\[JzInOut\]-\[SyInOut\]) to find: $$\begin{aligned}
\Jz{^{({\rm mid})} }&=& \Jz{^{({\rm in})} }{}+ G_2 \Sx{^{({1})} }\Jy{^{({\rm in})} }\label{JzInOutFP} \\
\Jy{^{({\rm mid})} }&=& \Jy{^{({\rm in})} }- G_1 \Sz{^{(1,{\rm in})} }\Jx - G_2 \Sx{^{({1})} }{\hat{J}_{[\mathbf{x,y}]}}{^{({\rm in})} }\label{JzInOutFP} \\
\Sy{^{(1,{\rm out})} }&=& \Sy{^{(1,{\rm in})} }+ G_1 \Sx{^{({1})} }\Jz{^{({\rm in})} }\label{SyInOutFP}\end{aligned}$$ and $$\begin{aligned}
\Jz{^{({\rm out})} }&=& \Jz{^{({\rm in})} }\label{JzInOutTP} \\ \Sy^{(\rm diff,out)} &=& \Sy^{(\rm diff,in)} + 2G_1 \Sx{^{({1})} }\Jz{^{({\rm in})} }\label{SyInOutTP}\end{aligned}$$ plus terms in $G_1 G_2 \Sx \Sz \Jx $, $G_2^2 \Sx^2 {\hat{J}_{[\mathbf{x,y}]}}$ and $G_1 G_2 \Sx^2 \Jy$ which become negligible in the limit of large $p$. The ideal QND form is recovered by the dynamical decoupling.
The presence of the $G_2$ term can be detected by noise scaling properties. While in the ideal QND of Equations (\[JzInOutTP\]),(\[SyInOutTP\]) the variance of the system variable is $\propto \Jx$ giving a variance for the meter variable linear in $\Jx$, for the imperfect QND of Equations (\[JzInOut\]) to (\[SyInOut\]) this is not the case: from Equation (\[JzInOutFP\]), we see that $\Jy$ acquires a back-action variance $\propto \Jx^2$, which then is fed into the system variable by the $G_2$ term. This additional $\Jx^2$ noise is also reflected in the meter variable, and provides a measurable indication of $G_2$.
We use the two-polarization decoupling technique to perform QND measurement on an ensemble of $\sim10^{6}$ laser cooled $^{87}$Rb atoms in the $F=1$ ground state. In the atomic ensemble system, described in detail in reference [@Kubasik2009PRAv79p43815], $\mu$s pulses interact with an elongated atomic cloud and are detected by a shot-noise-limited polarimeter. The experiment achieves projection noise limited sensitivity, as calibrated against a thermal spin state [@Koschorreck2010PRLv104p93602].
 Experimental sequence for projection noise measurement. The CSS is prepared once and its magnitude $\dexpect{ \Jx}$ is measured. This serves as a measure of the spin polarization prior to the QND probing. We prepare the CSS a second time and assume it has the same spin polarization as in the first preparation. The state is probed with a train of pulses of alternating polarization. Measuring the spin polarization after the QND measurement tells us the amount of depolarization introduced in the QND probing. The QND probing scatters a non-negligible fraction of atoms into $F=2$, which are removed from the trap with resonant light in order to reduce the number of atoms in the trap. The whole cycle is repeated 10 times during one trap loading. ](TPP_scheme2){width="1\columnwidth"}
The experimental sequence is shown schematically in Fig. \[fig:TPP scheme\]. In each measurement cycle the atom number $N_A$ is first measured by a dispersive atom-number measurement (DANM) [@Koschorreck2010PRLv104p93602]. A $\Jx$-polarized coherent spin state (CSS) is then prepared and probed with pulses of alternating polarization to find the QND signal $\Sy \equiv \sum_i \hat{s}_{\mathbf{y},i}{^{({\rm out})} }(-1)^{i+1}$. Immediately after, $\dexpect{\Jx}$ is measured to quantify depolarization of the sample and any atoms having made transitions to the $F=2$ manifold are removed from the trap, reducing $N_A$ for the next cycle and allowing a range of $N_A$ to be probed on a single loading. This sequence of state preparation and probing is repeated ten times for each loading of the trap. The trap is loaded 350 times to acquire statistics.
The optical dipole trap, formed by a weakly-focused ($52\,\mu$m) beam of a Yb:YAG laser at $1030\,$nm with $6\,$W of optical power, is loaded from a conventional two stage magneto-optical trap (MOT) during $4\,$s. Sub-Doppler cooling produces atom temperatures down to $25\,\mu$K as measured in the dipole trap [@Kubasik2009PRAv79p43815]. In the DANM, we prepare a $\Jx$-polarized CSS, i.e., all atoms in a coherent superposition of hyperfine states $\left|\uparrow/\downarrow\right\rangle \equiv \left|F=1,m_F = \pm1\right\rangle $, by optically pumping with vertically-polarized light tuned to the transition $F=1\rightarrow F'=1$, while also applying repumping on the $F=2\rightarrow F'=2$ transition and a weak magnetic field along $x$ to prevent spin precession. The atoms arrive to this dark state after scattering fewer than two photons on average. To measure $\dexpect{\Jx}$, we send ten circularly-polarized probe pulses, i.e., with $\dexpect{\Sz} = N_L/2$, tuned $190\,$MHz to the red of the transition $F=1\rightarrow F'=0$. Each pulse, of $1\,\mu$s duration, contains $2.6\times10^{6}$ photons and produces a signal $\dexpect{\Sy} \propto G_{2}\dexpect{ \Sz}\dexpect{ \Jx}$. The coherent state for the QND measurement is prepared in the same way, but in zero magnetic field.
To measure $ \Jz$, i.e., one half the population difference between $\left|\uparrow\right\rangle $ and $\left|\downarrow\right\rangle $, we send probe pulses of either vertical $\msx = n_L/2$ or horizontal $\msx = -n_L/2$ polarization through atomic sample and record their polarization rotation as $\hat{s}_{\mathbf{y},i}^{\rm (out)}$. The number of individual probe pulses is $2p$ and the total number of probe photons $N_{L}=2pn_{L}$.
 Variance of polarimeter signal as a function of atom number, comparing naive probing, i.e., a single input polarization, to “bang-bang” dynamically-decoupled probing of different orders $p$. Grey curves indicate simulation results for: naive probing (solid), and decoupled probing with $p=1$ (widely dashed), $p=2$ (dashed), and $p=5$ (dotted). The black solid line shows the expected projection noise for $p\rightarrow \infty$, or the ideal QND interaction $G2 = 0$. All curves are calculated using the independently measured interaction strength $G_1 = 1.27(5)\times 10^{?7}$ and have no free parameters. Red squares are measured data using dynamical decoupling with $p=5$. Blue circles are measured data with naive probing. Technical noise from laboratory fields dominates the naive probing results, and pushes them above the theoretical curve, while technical noise is suppressed in the dynamically-decoupled probing. ](TPP_data3.pdf){width="1\columnwidth"}
In Fig. \[fig:Exp and Sim\] we plot the measured noise versus atom number, which confirms the linear scaling characteristic of the QND measurement. The black squares indicate the variance $\var{\Sy}$ normalized to the optical polarization noise, measured in the absence of atoms. Independent measurements confirm the polarimetry is shot-noise limited in this regime. The black solid line is the expected projection noise scaling $4\var{ \Sy}/N_L=1+G_{1}^{2}N_{L}\var{ \Jz}$, calculated from the independently measured interaction strength $G_{1}$ and number of probe photons $N_{L}=8\times10^{8}$. The QND measurement achieves projection-noise limited sensitivity, i.e., the measurement noise is $5.7(6)\,$dB below the projection noise.
Also shown are results of covariance matrix calculations, following the techniques of reference [@Koschorreck2009JPBv42p9], including loss and photon scattering. The scenarios considered include the naive QND measurement, i.e., with a single polarization, and the “bang-bang” or two-polarization QND measurement, with $p=1,2,5$. These show a rapid decrease in the quadratic component with increasing $p$. This confirms the removal of $G_2$ due to the dynamical decoupling. Also included in these simulations is the term $ \Sy \Jy$ which introduces noise into $ \Jz$ proportional to $G_{2}^{2}\var{ \Sy}\dexpect{ \Jx}^{2}$. For our experimental parameters this term leads to an increase of $\var{ \Jz}$ of less then $2\,$% and as noted above could be reduced with increased detuning.
The dynamical decoupling also suppresses technical noise which would otherwise enter into $ \Jz$ through the interaction $G_{2}( \Sx \Jx+ \Sy \Jy)$. An imperfect preparation of the atomic and and/or light state, e.g., $\dexpect{ \Jy}\neq0$ or $\dexpect{ \Sy}\neq0$, would otherwise be transferred into $ \Jz$.
Using dynamical decoupling techniques, we have demonstrated optical quantum non-demolition measurement of a large-spin system. We first identify an often-overlooked impediment to this goal: the tensorial polarizability causes decoherence of the measured variable, and prevents (naive) QND measurement of small ensembles. We then identify an appropriate dynamical decoupling strategy to cancel the tensorial components of the effective Hamiltonian, and implement the strategy with an ensemble of $\sim 10^6$ cold $^{87}$Rb atoms and two-polarization probing. The dynamically-decoupled QND measurement achieves a sensitivity $5.7(6)$ dB better than the projection noise level. The technique will enable the use of large-spin ensembles in quantum metrology and quantum networking, and permit the QND measurement of exotic phases of large-spin condensed atomic gases.
We gratefully acknowledge fruitful discussions with Ivan H. Deutsch and Robert Sewell. This work was funded by the Spanish Ministry of Science and Innovation under the ILUMA project (Ref. FIS2008-01051) and the Consolider-Ingenio 2010 Project QOIT.
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abstract: 'In this paper, we present an approach based on reinforcement learning for eye tracking data manipulation. It is based on two opposing agents, where one tries to classify the data correctly and the second agent looks for patterns in the data, which get manipulated to hide specific information. We show that our approach is successfully applicable to preserve the privacy of a subject. In addition, our approach allows to evaluate the importance of temporal, as well as spatial, information of eye tracking data for specific classification goals. In general, this approach can also be used for stimuli manipulation, making it interesting for gaze guidance. For this purpose, this work provides the theoretical basis, which is why we have also integrated a section on how to apply this method for gaze guidance.'
author:
- Wolfgang Fuhl
bibliography:
- 'aaatemplate.bib'
title: Reinforcement learning for the manipulation of eye tracking data
---
<ccs2012> <concept> <concept\_id>10010147.10010257.10010258.10010261.10010275</concept\_id> <concept\_desc>Computing methodologies Multi-agent reinforcement learning</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010147.10010257.10010293.10010294</concept\_id> <concept\_desc>Computing methodologies Neural networks</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010147.10010178.10010224</concept\_id> <concept\_desc>Computing methodologies Computer vision</concept\_desc> <concept\_significance>300</concept\_significance> </concept> </ccs2012>
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Introduction
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Related Work
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Method
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Evaluation
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Applicability to gaze guidance
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Conclusion
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abstract: 'Incorporating information about the target distribution in proposal mechanisms generally increases the efficiency of Markov chain Monte Carlo algorithms, comparatively to those based on naive random walks. Hamiltonian Monte Carlo is a successful example of fixed-dimensional algorithms incorporating gradient information. In trans-dimensional algorithms, [@green2003trans] recommended to generate the parameter proposals during model switches from normal distributions with informative means and covariance matrices. These proposal distributions can be viewed as approximating the limiting parameter distributions, where the limit is with regard to the sample size. Models are typically proposed naively. In this paper, we build on the approach of [@zanella2019informed] for discrete spaces to incorporate information about neighbouring models. More specifically, we rely on approximations to posterior model probabilities that are asymptotically exact, as the sample size increases. We prove that, as expected, the samplers combining this approach with that of [@green2003trans] behave like those able to generate from both the model distribution and parameter distributions in the large sample regime. We also prove that the proposed strategy is optimal when the posterior model probabilities concentrate. We review generic methods improving parameter proposals when the sample size is not large enough. We show how we can leverage these methods to improve model proposals as well. The methodology is applied to a real-data example. Detailed guidelines to fully automate the methodology implementation are provided. The code is available online.[^1]'
author:
- 'Philippe Gagnon $^{1}$'
bibliography:
- 'reference.bib'
title: A step further towards automatic and efficient reversible jump algorithms
---
$^{1}$Department of Statistics, University of Oxford, United Kingdom.
Keywords: Bayesian statistics; large sample asymptotics; Markov chain Monte Carlo methods; model selection; trans-dimensional Markov chains; variable selection; weak convergence.
Introduction {#sec_intro}
============
Reversible jump algorithms
--------------------------
Reversible jump (RJ, [@green1995reversible]) algorithms are Markov chain Monte Carlo (MCMC) methods that one uses to sample from a target distribution $\pi(\, \cdot \mid \mathbf{D}_n)$ defined on a union of sets $\bigcup_{k \in \mathcal{K}} \{k\} \times \operatorname{\mathbb{R}}^{d_k}$, $\mathcal{K}$ being some countable set and $d_k$ positive integers. This distribution corresponds in Bayesian statistics to a joint posterior of a model indicator $K\in \mathcal{K}$ and the parameters of Model $K$, $\mathbf{X}_K\in\operatorname{\mathbb{R}}^{d_K}$, $\mathbf{D}_n$ representing a data sample of size $n$. Such a posterior distribution allows to jointly infer about $(K, \mathbf{X}_K)$, or in other words, simultaneously achieve model selection and parameter estimation. In the following, we assume for simplicity that the parameters of all models are continuous random variables. Again for simplicity, we will abuse notation by also using $\pi(\, \cdot \mid \mathbf{D}_n)$ to denote the joint posterior density with respect to a product of the counting and Lebesgue measures.
At each iteration of a RJ algorithm, a proposal is first made for the model to explore next, which can be represented by a proposal of the form $k\mapsto k'$ ($k'$ may be equal to $k$), where $k'$ is generated from a probability mass function (PMF) $g(k,\cdot\,)$, $(k, \mathbf{x}_k)$ being the current state of the Markov chain. A proposal is next made the parameters of Model $k'$. This is usually achieved through two steps:
1. generate $\mathbf{u}_{k\mapsto k'}\sim q_{k\mapsto k'}$ (this vector can be viewed as auxiliary variables that are used, for instance, to propose values for the parameters of Model $k'$), where $q_{k\mapsto k'}$ is a probability density function (PDF),
2. apply the function $\mathcal{D}_{k\mapsto k'}$ to $(\mathbf{x}_k,\mathbf{u}_{k\mapsto k'})$, $\mathcal{D}_{k\mapsto k'}(\mathbf{x}_k,\mathbf{u}_{k\mapsto k'})=:(\mathbf{y}_{k'},\mathbf{u}_{k'\mapsto k})$, where the vector $\mathbf{y}_{k'}$ represents the proposal for the parameters of Model $k'$ (equal to $\mathbf{u}_{k\mapsto k'}$ in our example in Step 1), and $\mathcal{D}_{k\mapsto k'}$ is a diffeomorphism.
Finally, the whole proposal is accepted, i.e. the next state of the chain is $(k',\mathbf{y}_{k'})$, with the following probability (assuming that the current state has positive density under the target): $$\begin{aligned}
\label{eqn_acc_prob_RJ}
\alpha_{\text{RJ}}((k,\mathbf{x}_{k}),(k',\mathbf{y}_{k'})):=1\wedge \frac{g(k',k) \, \pi(k',\mathbf{y}_{k'}\mid \mathbf{D}_n) \, q_{k'\mapsto k}(\mathbf{u}_{k'\mapsto k})}{g(k,k') \, \pi(k,\mathbf{x}_{k}\mid \mathbf{D}_n) \, q_{k\mapsto k'}(\mathbf{u}_{k\mapsto k'}) \, |J_{\mathcal{D}_{k\mapsto k'}}(\mathbf{x}_{k}, \mathbf{u}_{k\mapsto k'})|^{-1}},\end{aligned}$$ where $|J_{\mathcal{D}_{k\mapsto k'}}(\mathbf{x}_{k}, \mathbf{u}_{k\mapsto k'})|$ is the Jacobian of the function $\mathcal{D}_{k\mapsto k'}$. If the proposal is rejected, the chain remains at the same state $(k,\mathbf{x}_{k})$ for another time interval.
Looping over the steps described above produces Markov chains that are reversible with respect to the target distribution. If in addition the chains are irreducible and aperiodic, they are then ergodic (see [@tierney1994markov]), which guarantees that the Law of Large Numbers holds.
Problem and perspective of analysis
-----------------------------------
Implementing RJ is well know for being a difficult task considering the large number of functions that need to be specified and the often lack of intuition about how one should achieve their specifications. Significant amount of work has been carried out to address the specification of the functions $\mathcal{D}_{k \mapsto k'}$ and $q_{k \mapsto k'}$ when no prior information about the targets can be exploited or a more automatic perspective is adopted (see, e.g., [@green2003trans] and [@brooks2003efficient]). The approaches of these authors are arguably the most popular. Their objective is the following: given $\mathbf{x}_k \sim \pi(\, \cdot \mid k,\mathbf{D}_n)$, we want to identify $q_{k\mapsto k'}, q_{k'\mapsto k}$ and $\mathcal{D}_{k\mapsto k'}$ such that applying the transformation $\mathcal{D}_{k\mapsto k'}$ to $(\mathbf{x}_k, \mathbf{u}_{k\mapsto k'}) \sim \pi(\, \cdot \mid k,\mathbf{D}_n)\otimes q_{k\mapsto k'}$ leads to $(\mathbf{y}_{k'},\mathbf{u}_{k'\mapsto k}) \sim \pi(\, \cdot \mid k',\mathbf{D}_n)\otimes q_{k'\mapsto k}$ (at least approximatively). We essentially look for a way to generate from the conditional distributions $\pi(\, \cdot \mid k',\mathbf{D}_n)$, in this constrained framework. This in turn aims at increasing the acceptance probability $\alpha_{\text{RJ}}$ defined in towards $$\begin{aligned}
\label{acc_prob_marginal}
\alpha_{\text{marginal}}(k, k'):=1\wedge \frac{g(k',k) \, \pi(k'\mid \mathbf{D}_n)}{g(k,k') \, \pi(k \mid \mathbf{D}_n)},\end{aligned}$$ which corresponds to the acceptance probability in a marginal sampler targeting the PMF $\pi(k \mid \mathbf{D}_n)$. The approach of [@green2003trans], for instance, proceeds as if the conditional distributions $\pi(\, \cdot \mid k,\mathbf{D}_n)$ were normal.
Notwithstanding the merit of this objective, it is to be noticed that even when the goal is achieved “half” of the work for maximising $\alpha_{\text{RJ}}$ is done as poor models may often be proposed if $g$ is not well designed (see ). Note that for model switches, ordering two proposal distributions $g$ which allow reaching the same neighbouring values $k'$ through their associated acceptance probabilities $\alpha_{\text{RJ}}$ represents a first step towards proving that one is better than the other (see [@peskun1973optimum] and [@tierney1998note] for detailed explanations about why higher acceptance probabilities are often better in that case).
$\begin{array}{ccc}
\hspace{-3mm}\includegraphics[width = 0.33\textwidth, trim = {0 0 20mm 0}, clip]{Fig1_a.pdf} & \hspace{-4mm}\includegraphics[width = 0.33\textwidth, trim = {0 0 20mm 0}, clip]{Fig1_b.pdf} & \hspace{-4mm}\includegraphics[width = 0.33\textwidth, trim = {0 0 20mm 0}, clip]{Fig1_c.pdf} \cr
\hspace{-4mm}\textbf{(a) Posterior probabilities} & \hspace{-4mm} \textbf{(b) Uninformed uniform sampler} & \hspace{-4mm} \textbf{(c) Informed sampler}
\end{array}$
The specification of $g$ has been overlooked; this PMF is indeed typically set to a uniform as in (b). The first objective of this paper is to incorporate information about neighbouring models in its design so that fully informed RJ are available. We thus focus on transitions involving model switches, i.e. proposals $k \mapsto k'$ with $k' \neq k$. For proposals with $k' = k$, also called *parameter updates*, we consider in our analysis that all algorithms proceed in the same manner. In our numerical examples, we employ Hamiltonian Monte Carlo (HMC, see, e.g., [@neal2011mcmc]), which is a well known efficient informed. Its implementation is now fully automated in, for instance, the R package RStan ([@RStan]).
The first obstacle to achieving our first objective is that we typically do not have direct access to model information, because it involves integrals over the parameter space. Drawing inspiration from the approach [@green2003trans] that can be viewed as approximating the limit of $\pi(\, \cdot \mid k,\mathbf{D}_n)$ (under regularity conditions), we propose to use approximations to $\pi(k \mid \mathbf{D}_n)$ whose accuracy increases with $n$. To study the efficiency of the proposed approach, we study the limiting behaviour of RJ relying on it and the approach of [@green2003trans]. We in particular analyse the case where the posterior model probabilities as well as the posterior parameter densities concentrate as $n$ increases. In this situation, the parameter space continues to be explored, but at different scales given that the parameters are continuous parameters. In contrast, fewer models are visited during an algorithm run as more of them have negligible mass. We mathematically represent this limiting situation which represents an approximation to what one encounters in practice, and prove that the proposed approach is optimal in the limit.
Given that the sample may not be sufficiently large for the approximations to be accurate, two existing generic methods improving the parameter proposal mechanisms are presented. It is realised that they are useful for improving the model proposal mechanisms as well. In particular, we show that as the precision parameters of these methods increase without bounds the sampler converges towards an ideal one that is able to generate from $\pi(\, \cdot \mid k, \mathbf{D}_n)$ and that has access to $\pi(k \mid \mathbf{D}_n)$, for fixed $n$.
The second objective of this paper is to make clear how each function required for implementation should be specified, allowing a fully automated implementation procedure. This procedure can be executed if the log conditional densities $\log \pi(\, \cdot \mid k, \mathbf{D}_n)$ have well defined mode and second derivatives. It can be executed even if the model space is large or infinite, as long as the model probabilities concentrate on a reasonable number of models, which is expected in practice.
Organisation of the paper
-------------------------
In , we discuss the specification of the function $g$; more specifically in the case where the model space is relatively small in , and in the complementary case in . We present in the methods bridging the gap when the large sample regime is not attained. In particular, we review two generic methods allowing to generate parameter proposals from distributions arbitrarily close to $\pi(\, \cdot \mid k, \mathbf{D}_n)$ in , and in we propose a novel approach building on these methods for achieving the same objective, but for model proposals. In , the implementation procedure is detailed. The methodology is evaluated on a robust variable selection application to real data in . The paper finishes in with retrospective comments and possible directions for future research.
Design of the function $g$ {#sec_design_g}
==========================
The design of the function $g$ starts with the definition of neighbourhoods around all models which specify the support of $g(k,\cdot\,)$ for all $k$. It is typically possible to achieve this in a natural way in model selection. For instance in mixture modelling, $k$ represents the number of components and the neighbourhood around $k$, denoted by $\mathcal{N}(k)$, may be defined as the models that have plus or minus $0, 1, 2, \ldots, c$ components, where $c$ is positive integer. More precisely, $\mathcal{N}(k):=\{k':|k'-k|\in\{0, 1,2,\ldots,c\}\}$. It is also possible to define natural neighbourhoods when there is no such “ordering” between the models. For instance in variable selection, we use $k$ as a label. Model $k_0$ may represent the model with covariates 1, 2, 3 and 6 (the covariates are also labelled, as in ), and neighbouring models can be defined as the models obtained by adding or removing one variable to the current model.
In practice, $g$ is commonly set to the uniform distribution over $\mathcal{N}(k)$: $g(k,k'):=1/|\mathcal{N}(k)|$ for $k'\in \mathcal{N}(k)$, where $|\mathcal{N}(k)|$ represents the cardinality of $\mathcal{N}(k)$. Our goal is to extract information from the neighbourhood and include it in the PMF $g(k, \cdot \,)$ to skew the latter towards high probability models. We focus in this paper on the case where there exists no natural ordering between the models, as [@gagnon2019NRJ] recently proposed non-reversible trans-dimensional samplers reaching high efficiency in the situation where a natural ordering exists.
In the related regular discrete sampling context, i.e. Metropolis–Hastings (MH, [@metropolis1953equation] and [@hastings1970monte]) algorithms used to target PMF, [@zanella2019informed] recently proposed a solution. The author analysed high-dimensional regimes and recommended to construct what he called *locally balanced* PMF of the form $$g(\mathbf{x},\mathbf{y})\propto h\left(\frac{\pi(\mathbf{y}\mid \mathbf{D}_n)}{\pi(\mathbf{x}\mid \mathbf{D}_n)}\right){\mathds{1}}(\mathbf{y}\in \mathcal{N}(\mathbf{x})),$$ where $\mathbf{x}$ and $\mathbf{y}$ belong to a discrete domain, $h$ is a continuous function respecting the condition $h(x)=x \, h(1/x)$ for all positive $x$ (the square root satisfies this condition for instance), and ${\mathds{1}}(\, \cdot \,)$ is the indicator function. Incorporating information about the neighbourhood, these proposals lead to faster mixing. A Peskun ordering (see [@peskun1973optimum] and [@tierney1998note]) is proved in some specific high-dimensional situations, allowing to establish optimality of the strategy. In these high-dimensional regimes, the sizes of the neighbourhoods are seen to be extremely small comparatively to that of the domain. The author also explained that the natural choice $g(\mathbf{x},\mathbf{y})\propto \pi(\mathbf{y}\mid \mathbf{D}) \, {\mathds{1}}(\mathbf{y}\in \mathcal{N}(\mathbf{x}))$ (called *globally balanced* in that paper) makes sense when the sizes of the neighbourhoods are comparable to that of the domain.
In this paper, we extend the strategy of [@zanella2019informed] to the RJ framework in a natural fashion. The peculiarity of this framework is that we typically do not have access to the marginal posterior probabilities $\pi(k \mid \mathbf{D}_n)$. This is why we have to use approximations. The Laplace approximation to $\pi(k \mid \mathbf{D}_n)$ is a natural choice. It is indeed consistent (see, e.g., [@davison1986approximate]) when the conditional density $\pi(\, \cdot \mid k, \mathbf{D}_n)$ has a well defined mode. It requires finding this mode and computing the second derivatives of the log of this density. We use this approximation in the numerical examples. We now consider two cases for the specific design of the function $g$.
Case 1: the neighbourhoods are equal to the model domain {#sec_Case1}
--------------------------------------------------------
In some situations, the size of $\mathcal{K}$ is small and it is feasible to switch from any model to any other one, meaning that we may want to set $\mathcal{N}(k):=\mathcal{K}$ for all $k$. Using the globally balanced proposal in this case — i.e. $g(k, k') := \widehat{\pi}(k' \mid \mathbf{D}_n)$, where $\widehat{\pi}(k' \mid \mathbf{D}_n)$ is an approximation to $\pi(k' \mid \mathbf{D}_n)$ — makes intuitively a lot of sense as it corresponds to independent sampling for $K$ in the limit, as the approximations to the posterior parameter distributions and posterior model distribution get better and better. This represents our recommandation. Following the analysis of [@zanella2019informed] in a discrete sampling context, this recommendation is expected to be also valid when the neighbourhoods are not exactly equal to the model domain, but of comparable sizes to it.
In the rest of the section, we prove that indeed as $n \longrightarrow \infty$ the sampler behaves like an ideal one that generates from the conditional densities $\pi(\, \cdot \mid k, \mathbf{D}_n)$ and posterior model PMF, corresponding to regular Monte Carlo sampling. We also evaluate the efficiency of the proposed approach by comparing the limiting sampler to others using different $g$.
We first consider the following assumption on the posterior model probabilities and their approximations.
\[ass1\_Case1\] Each pair of random variables $(\widehat{\pi}(k\mid \mathbf{D}_n), \pi(k\mid \mathbf{D}_n))$ (where the randomness comes from $\mathbf{D}_n$) is such that $|\widehat{\pi}(k\mid \mathbf{D}_n) - \pi(k\mid \mathbf{D}_n)|$ and $|\pi(k\mid \mathbf{D}_n) - \bar{\pi}(k)|$ converge in probability towards 0 as $n \longrightarrow \infty$, $\bar{\pi}(k)$ thus being the liming value of $\pi(k\mid \mathbf{D}_n)$.
The independent sampling mentioned above will in fact only happen if the normal approximations to the parameter posteriors make sense. We consider in the following analysis that it is the case, at least in the limit. In other words, we consider that we have a Bernstein-von Mises convergence for the conditional distributions $\pi(\, \cdot \mid k,\mathbf{D}_n)$ (see, e.g., [@van2000asymptotic]).
\[ass2\_Case1\] For all $k$, there exist a mean vector $\boldsymbol\mu_k$ and a covariance matrix $\boldsymbol\Sigma_k$ such that $\text{TV}(\pi(\, \cdot \mid k,\mathbf{D}_n), \mathcal{N}(\widehat{\boldsymbol\mu}_k, \boldsymbol\Sigma_k / n))$ converges in probability towards 0 as $n \longrightarrow \infty$, where $\text{TV}$ denotes the total variation and $\widehat{\boldsymbol\mu}_k$ is an estimator of $\boldsymbol\mu_k$.
Use $\{(K,\mathbf{X}_K)_{\text{ideal}}(m): m\in\operatorname{\mathbb{N}}\}$ to denote the Markov chain associated to the ideal RJ that targets a distribution that is such that the marginal probabilities on $K$ are given by $\bar{\pi}(k)$ and the conditional distribution of the parameters given $K$ is normal with mean and variance given by $\widehat{\boldsymbol\mu}_K$ and $\boldsymbol\Sigma_K / n$, respectively. This ideal RJ has access to $\bar{\pi}(k)$ and therefore sets its model proposal distribution, denoted by $g_{\text{ideal}}$, to $g_{\text{ideal}}(k, k') := \bar{\pi}(k')$ for all $k$. Its functions used for parameter proposals, denoted by $\mathcal{D}_{k \mapsto k'}^{\text{ideal}}$ and $q_{k \mapsto k'}^{\text{ideal}}$, can be set such that the acceptance probability is exactly equal to 1 for any model switches. The obvious way is to set $q_{k \mapsto k'}^{\text{ideal}}:=\mathcal{N}(\widehat{\boldsymbol\mu}_{k'}, \boldsymbol\Sigma_{k'} / n)$ and $\mathcal{D}_{k \mapsto k'}^{\text{ideal}}$ such that $\mathbf{y}_{k'} := \mathbf{u}_{k \mapsto k'}$. Another way generates less random variables and uses linear transformations. If for instance $d_{k'}>d_k$, one can generate $\mathbf{u}_{k\mapsto k'}\sim \mathcal{N}(\mathbf{0}, \mathbf{I}_{d_{k'}-d_k})$, and set $\mathbf{y}_{k'} := \widehat{\boldsymbol\mu}_{k'} + \mathbf{V}_{k'}\boldsymbol\Lambda_{k'}^{1/2}\mathbf{z}_{k \mapsto k'}$, where $\mathbf{I}_{d_{k'}-d_k}$ is the identity matrix of size ${d_{k'}-d_k}$ and $\mathbf{z}_{k \mapsto k'}^T:=((\boldsymbol\Lambda_{k}^{-1/2}\mathbf{V}_{k}^T (\mathbf{x}_k - \widehat{\boldsymbol\mu}_k))^T, \mathbf{u}_{k\mapsto k'}^T)$, $\mathbf{V}_{k}$ and $\boldsymbol\Lambda_{k}$ being the matrices containing the eigenvectors and eigenvalues of $\boldsymbol\Sigma_k/n$, respectively. We consider that the ideal RJ sets $\mathcal{D}_{k \mapsto k'}^{\text{ideal}}$ and $q_{k \mapsto k'}^{\text{ideal}}$ in either of these manners. Use $\{(K,\mathbf{Z}_K)_{\text{ideal}}(m): m\in\operatorname{\mathbb{N}}\}$ to denote the standardised version of $\{(K,\mathbf{X}_K)_{\text{ideal}}(m): m\in\operatorname{\mathbb{N}}\}$, where $(K,\mathbf{Z}_K)_{\text{ideal}}(m):= (K, \sqrt{n} (\mathbf{X}_K - \widehat{\boldsymbol\mu}_K))_{\text{ideal}}(m)$ for all $m$. We denote the stationary distribution of this Markov chain by $\bar{\pi}$ which is such that $\bar{\pi}(\, \cdot\mid k):=\mathcal{N}(0, \boldsymbol\Sigma_k)$.
Now use $\{(K,\mathbf{X}_K)_n(m): m\in\operatorname{\mathbb{N}}\}$ to denote the Markov chain associated to the RJ that targets $\pi(\, \cdot \mid \mathbf{D}_n)$, with conditional distributions that are typically non-Gaussian (for fixed $n$). This RJ is not able to generate from the posterior model probabilities and the conditional distributions of the parameters and thus uses the approximations instead, namely $g(k, k'):=\widehat{\pi}(k'\mid \mathbf{D}_n)$ and $q_{k \mapsto k'}:=\mathcal{N}\left(\widehat{\boldsymbol\mu}_{k'}, \widehat{\boldsymbol\Sigma}_{k'} / n\right)$ (when analysing for instance the convergence towards the sampler using $q_{k \mapsto k'}^{\text{ideal}}:=\mathcal{N}(\widehat{\boldsymbol\mu}_{k'}, \boldsymbol\Sigma_{k'} / n)$), where $\widehat{\boldsymbol\Sigma}_k$ is an estimator of $\boldsymbol\Sigma_k$. But, the regular RJ uses the same functions $\mathcal{D}_{k \mapsto k'}$ as its ideal counterpart. Use $\{(K,\mathbf{Z}_K)_{n}(m): m\in\operatorname{\mathbb{N}}\}$ to denote the standardised version of $\{(K,\mathbf{X}_K)_{n}(m): m\in\operatorname{\mathbb{N}}\}$, where $(K,\mathbf{Z}_K)_{n}(m):= (K, \sqrt{n} (\mathbf{X}_K - \widehat{\boldsymbol\mu}_{K}))_{n}(m)$ for all $m$.
Before presenting our first weak convergence result, we require the estimators $\widehat{\boldsymbol\mu}_{k'}$ and $\widehat{\boldsymbol\Sigma}_{k'}$ to be consistent. This desired property is generally satisfied when $\widehat{\boldsymbol\mu}_{k'}$ is the maximum a posteriori probability (MAP) estimate and $\widehat{\boldsymbol\Sigma}_{k'}$ is the inverse of the second derivative matrix of $\log \pi(\, \cdot \mid k', \mathbf{D}_n)$ evaluated at $\widehat{\boldsymbol\mu}_{k'}$, which are used in our numerical examples. Note that when the prior is non-informative and proportional to 1, $\widehat{\boldsymbol\mu}_{k'}$ and $\widehat{\boldsymbol\Sigma}_{k'}$ correspond to the maximum likelihood estimate and the inverse of the observed information matrix, respectively.
\[ass3\_Case1\] For all $k$, the random variables $\widehat{\boldsymbol\mu}_k$ and $\widehat{\boldsymbol\Sigma}_k$ (where the randomness comes from $\mathbf{D}_n$) converge in probability towards $\boldsymbol\mu_k$ and $\boldsymbol\Sigma_k$, respectively.
We now present our first weak convergence result.
\[thm\_conv\_Case1\]
Under Assumptions \[ass1\_Case1\] to \[ass3\_Case1\] and assuming that $(K,\mathbf{X}_K)_n(0)\sim \pi(\, \cdot \mid \mathbf{D}_n)$ and $(K,\mathbf{Z}_K)_{\text{ideal}}(0)\sim \bar{\pi}$, we have that $$\{(K,\mathbf{Z}_K)_n(m): m\in\operatorname{\mathbb{N}}\}\Longrightarrow \{(K,\mathbf{Z}_K)_{\text{ideal}}(m): m\in\operatorname{\mathbb{N}}\} \quad \text{in probability as $n\longrightarrow \infty$},$$ where “$\Longrightarrow$” is used to denote weak convergence.
See .
This result tells us that the implementable RJ (the one using the approximations) asymptotically behaves like the ideal RJ that has access to the posterior model probabilities and for which the posterior parameter distributions are normals. In particular, the acceptance probabilities in the implementable algorithm are exactly (and asymptotically) equal to one.
To (approximately) evaluate the efficiency of our recommendation for $g$, we thus rely on the comparison between the ideal RJ presented above with another RJ targeting $\bar{\pi}$ as well, but using another model proposal distribution $\tilde{g}$. The latter RJ also sets the functions for the parameter proposals like the ideal RJ, which implies that the acceptance probabilities are as in the marginal sampler for $K$ given in with $\tilde{g}$ instead of $g$ (and $\bar{\pi}(k)$ instead of $\pi(k\mid\mathbf{D}_n)$).
The ideal RJ samples $K$ as in regular Monte Carlo, which is commonly considered as the ideal sampling framework. Nevertheless, it is natural to ask whether it is possible to establish a Peskun ordering? We answer this question by focusing on the marginal behaviour of $K$ through the iterations, which is our main concern regarding the design of $g$. It is interesting to realise that the stochastic process associated with $K$ for the ideal RJ is a reversible Markov chain with a transition kernel given by $$P_{\text{ideal}}(k, k'):=g_{\text{ideal}}(k,k'):=\bar{\pi}(k').$$ The stochastic process associated with $K$ for the other RJ using $\tilde{g}$ is also a reversible Markov chain. The difference is that the transition kernel is given by $$\tilde{P}(k, k'):=\tilde{g}(k, k')\left(1\wedge \frac{\bar{\pi}(k')}{\bar{\pi}(k)}\frac{\tilde{g}(k', k)}{\tilde{g}(k, k')} \right) + \delta_{k'}(k)\sum_{i\neq k} \tilde{g}(k, i)\left(1 - 1\wedge \frac{\bar{\pi}(i)}{\bar{\pi}(k)}\frac{\tilde{g}(i, k)}{\tilde{g}(k, i)} \right).$$ To answer the question above, we have to show that $P_{\text{ideal}}(k, k')\geq \tilde{P}(k, k')$ for all $k, k'$ such that $k'\neq k$. This is however not true in general even if the ideal RJ proceeds as regular Monte Carlo. For instance, consider the case where there are more than 2 models and $\tilde{g}(k,k')=c \, \bar{\pi}(k')$ and $\tilde{g}(k',k)=c \, \bar{\pi}(k)$ for one specific pair $(k,k')$, $c \geq 1$ being a constant. In this case, $\tilde{P}(k, k') = c \, \bar{\pi}(k') \geq \bar{\pi}(k') = P_{\text{ideal}}(k, k')$. Note that $\tilde{P}$ does not dominate $P_{\text{ideal}}$ either.
If $\tilde{g}(k, \cdot \,)$ is a uniform on $\mathcal{K}$, it is possible to show that the condition becomes: $$\begin{aligned}
\label{eq_condition_unif}
\max\{\bar{\pi}(k'), \bar{\pi}(k)\}-\frac{1}{|\mathcal{K}|}\geq 0,\end{aligned}$$ for all $k,k'$ such that $k'\neq k$. This condition is satisfied when for instance $|\mathcal{K}|=2$ and $\bar{\pi}(1)\neq\bar{\pi}(2)$.
In the situation where the marginal posterior of $K$ concentrates, in the sense that $\bar{\pi}(k^*)= 1$ for some value $k^*$ (see for instance [@johnson2012bayesian] in linear regression), the mixing of the stochastic process associated with $K$ becomes less of an issue. Nevertheless, our recommendation for the PMF $g$ seems intuitively appropriate, because this PMF (asymptotically) only proposes to update the parameters of Model $k^*$ if the chain is currently at $k^*$. In the current setting where $\mathcal{N}(k):=\mathcal{K}$ for all $k$, this model is reached in (asymptotically) one step.
In contrast, when $\tilde{g}(k, \cdot \,)$ is a uniform on $\mathcal{K}$ and the current model is Model $k^*$, the sampler may spend a lot of time trying to switch to the other models, with an acceptance probability of (asymptotically) $0=1\wedge \bar{\pi}(k')/\bar{\pi}(k^*)$ when $k'\neq k^*$. Note that the acceptance probability for $k'\neq k^*$ is 0 for any choice of proposal PMF $\tilde{g}$. The acceptance probabilities associated with moves from Model $k^*$ to Model $k'$ (with $k'\neq k^*$) do not actually exist for the ideal RJ, because no other value than $k^*$ is proposed. In fact, the ideal sampler with $g_{\text{ideal}}$ dominates any other sampler with $\tilde{g}$ but the same functions $q_{k \mapsto k'}^{\text{ideal}}$ and $\mathcal{D}_{k \mapsto k'}^{\text{ideal}}$ and same parameter update scheme, as assumed above. To prove this, we analyse this time the transition kernel of the whole Markov chain evaluated at any set $\{k^*\}\times A_{k^*}$ to which the present state $(k^*,\mathbf{z}_{k^*})$ is subtracted. With $g_{\text{ideal}}$, it is $$\begin{aligned}
&P_{\text{ideal}}^{\text{complete}}((k^*,\mathbf{z}_{k^*}),\{k^*\}\times A_{k^*}\setminus \{(k^*,\mathbf{z}_{k^*})\} ) \cr
&\hspace{40mm} :=\bar{\pi}(k^*)\, {\mathbb{P}}(\mathbf{y}_{k^*} \in A_{k^*} \text{ is proposed and accepted}\mid k^*\mapsto k^* \text{ is proposed}) \cr
&\hspace{40mm} \hspace{1mm}= {\mathbb{P}}(\mathbf{y}_{k^*} \in A_{k^*}\text{ is proposed and accepted}\mid k^*\mapsto k^* \text{ is proposed}),\end{aligned}$$ where the latter probability corresponds to the probability of accepting a parameter update. With $\tilde{g}$, it is $$\begin{aligned}
&\tilde{P}^{\text{complete}}((k^*,\mathbf{z}_{k^*}),\{k^*\}\times A_{k^*}\setminus \{(k^*,\mathbf{z}_{k^*})\} )\cr
&\hspace{40mm} := \tilde{g}(k^*, k^*)\, {\mathbb{P}}(\mathbf{y}_{k^*} \in A_{k^*} \text{ is proposed and accepted}\mid k^*\mapsto k^* \text{ is proposed}) \cr
&\hspace{40mm} \hspace{1mm}\leq {\mathbb{P}}(\mathbf{y}_{k^*} \in A_{k^*}\text{ is proposed and accepted}\mid k^*\mapsto k^* \text{ is proposed}).\end{aligned}$$ This allows to conclude that $P_{\text{ideal}}^{\text{complete}}$ dominates $\tilde{P}^{\text{complete}}$ in terms of asymptotic variance of ergodic averages.
Case 2: the neighbourhoods are smaller than the model domain {#sec_Case2}
------------------------------------------------------------
The situation of interest for RJ users typically corresponds to that where the size of $\mathcal{K}$ is large, which points towards setting neighbourhoods $\mathcal{N}(k)$ with smaller sizes. The high-dimensional regime analysed by [@zanella2019informed] is represented by a limiting case where this difference in size is seen to grow without bounds. As mentioned at the beginning of , this author suggests to use *locally-balanced* proposals in this situation. This follows from several observations that are summarised in this section. These observations are relevant to our context, and we thus recommend to set $$\begin{aligned}
\label{eqn_g_case2}
g(k, k') := h\left(\frac{\widehat{\pi}(k' \mid \mathbf{D}_n)}{\widehat{\pi}(k \mid \mathbf{D}_n)}\right) \bigg/ c_k, \quad k' \in \mathcal{N}(k),\end{aligned}$$ where $c_k$ is the normalising constant of $g(k, \cdot \,)$. [@zanella2019informed] analyses two choices for the function $h$: $h(x):=\sqrt{x}$ and $h(x):=x/(1+x)$. The choice $h(x):=x/(1+x)$ is called the *Barker proposal* by the author because of the connection with [@barker1965monte]’s acceptance probability choice: $$\frac{\widehat{\pi}(k' \mid \mathbf{D}_n) \, \big/ \, \widehat{\pi}(k \mid \mathbf{D}_n)}{1 + \widehat{\pi}(k' \mid \mathbf{D}_n) \, \big/ \, \widehat{\pi}(k \mid \mathbf{D}_n)}=\frac{\widehat{\pi}(k' \mid \mathbf{D}_n)}{\widehat{\pi}(k' \mid \mathbf{D}_n) + \widehat{\pi}(k \mid \mathbf{D}_n)}.$$ The analysis of [@zanella2019informed] suggests that this latter choice is superior. In our numerical analyses, both choices lead to similar performances. Putting these analysis results together points towards a recommendation of setting $h(x):=x/(1+x)$.
In the rest of the section, we study as in the limiting behaviour of the sampler. A difference is that, in this case, the limiting ideal sampler proposes models using $g_{\text{ideal}}(k, k'):=h(\bar{\pi}(k')/\bar{\pi}(k))/c_k^{\text{ideal}}$ for $k'\in \mathcal{N}(k)$, where $c_k^{\text{ideal}}$ is the normalising constant of $g_{\text{ideal}}(k, \cdot \,)$. We prove that when the marginal posterior of $K$ concentrates, the sampler is optimal whenever $h$ is such that $h(0)=0$ and $h(1)>0$ (which is the case for $h(x):=\sqrt{x}$ and $h(x):=x/(1+x)$).
Use as in $\{(K,\mathbf{X}_K)_{\text{ideal}}(m): m\in\operatorname{\mathbb{N}}\}$ to denote the Markov chain associated to the ideal RJ that targets a distribution that is such that the marginal probabilities on $K$ are given by $\bar{\pi}(k)$ and the conditional distribution of the parameters given $K$ is normal with mean and variance given by $\widehat{\boldsymbol\mu}_K$ and $\boldsymbol\Sigma_K / n$, respectively. This ideal RJ sets $g_{\text{ideal}}(k, k'):=h(\bar{\pi}(k')/\bar{\pi}(k))/c_k^{\text{ideal}}$ for $k'\in \mathcal{N}(k)$. The functions $\mathcal{D}_{k \mapsto k'}^{\text{ideal}}$ and $q_{k \mapsto k'}^{\text{ideal}}$ are set as in . Due to the form of $g_{\text{ideal}}$ the acceptance probabilities are given by $$\begin{aligned}
\label{eqn_acc_ideal}
\alpha_{\text{ideal}}(k, k'):=1 \wedge \frac{\bar{\pi}(k')}{\bar{\pi}(k)} \frac{h\left(\frac{\bar{\pi}(k)}{\bar{\pi}(k')}\right)}{h\left(\frac{\bar{\pi}(k')}{\bar{\pi}(k)}\right)} \frac{c_{k}^{\text{ideal}}}{c_{k'}^{\text{ideal}}},\end{aligned}$$ which are, even if they do not depend on the parameters and their proposals, in general not equal to 1. The functions $h$ such that $h(x)=x \, h(1/x)$ all have in common that their use leads to acceptance probabilities of the following form: $$\alpha_{\text{ideal}}(k, k')=1 \wedge \frac{c_{k}^{\text{ideal}}}{c_{k'}^{\text{ideal}}}.$$
Use $\{(K,\mathbf{Z}_K)_{\text{ideal}}(m): m\in\operatorname{\mathbb{N}}\}$ to denote the standardised version of $\{(K,\mathbf{X}_K)_{\text{ideal}}(m): m\in\operatorname{\mathbb{N}}\}$, where $(K,\mathbf{Z}_K)_{\text{ideal}}(m):= (K, \sqrt{n} (\mathbf{X}_K - \widehat{\boldsymbol\mu}_K))_{\text{ideal}}(m)$ for all $m$. Use again $\{(K,\mathbf{X}_K)_n(m): m\in\operatorname{\mathbb{N}}\}$ to denote the Markov chain associated to the RJ that targets $\pi(\, \cdot\mid \mathbf{D}_n)$. This RJ sets $g$ as in , and the functions $q_{k \mapsto k'}$ and $\mathcal{D}_{k \mapsto k'}$ as in . Use $\{(K,\mathbf{Z}_K)_{n}(m): m\in\operatorname{\mathbb{N}}\}$ to denote the standardised version of $\{(K,\mathbf{X}_K)_{n}(m): m\in\operatorname{\mathbb{N}}\}$, where $(K,\mathbf{Z}_K)_{n}(m):= (K, \sqrt{n} (\mathbf{X}_K - \widehat{\boldsymbol\mu}_{K}))_{n}(m)$ for all $m$.
We now present our second weak convergence result in which we assume that $\mathcal{K}$ is finite. It is possible to extend the result of to the case where $\mathcal{K}$ is countably infinite under more technical versions of Assumptions \[ass1\_Case1\] to \[ass3\_Case1\].
\[thm\_conv\_Case2\]
Under Assumptions \[ass1\_Case1\] to \[ass3\_Case1\] and assuming that $|\mathcal{K}| < \infty$, $(K,\mathbf{X}_K)_n(0)\sim \pi(\, \cdot \mid \mathbf{D}_n)$ and $(K,\mathbf{Z}_K)_{\text{ideal}}(0)\sim \bar{\pi}$, we have that $$\{(K,\mathbf{Z}_K)_n(m): m\in\operatorname{\mathbb{N}}\}\Longrightarrow \{(K,\mathbf{Z}_K)_{\text{ideal}}(m): m\in\operatorname{\mathbb{N}}\} \quad \text{in probability as $n\longrightarrow \infty$}.$$
Analogous to that of after realising that $|g(k,k') - g_{\text{ideal}}(k, k')|$ converges in probability towards 0 as $n\longrightarrow \infty$, for all $k,k'$ (see in ).
As , our second weak convergence result tells us that the implementable RJ asymptotically behaves like the ideal RJ. To again evaluate the efficiency of $g$, we rely on the comparison of the ideal RJ with another RJ targeting $\bar{\pi}$ as well, but using another proposal distribution $\tilde{g}$ for the model switches.
We first consider that the marginal posterior of $K$ concentrates. We observe that $$g_{\text{ideal}}(k^*, k'):= \frac{h\left(\frac{\bar{\pi}(k')}{\bar{\pi}(k^*)}\right)}{h\left(\frac{\bar{\pi}(k^*)}{\bar{\pi}(k^*)}\right) + \sum_{\{l:l\in \mathcal{N}(k^*), l\neq k^*\}} h\left(\frac{\bar{\pi}(l)}{\bar{\pi}(k^*)}\right)}=\delta_{k^*}(k'),$$ for any $h$ such that $h(0)=0$ and $h(1)>0$. Using the same arguments as in the end of shows that the ideal RJ is optimal in that case.
But beyond knowing how the chain performs after reaching the mode, it is interesting to understand how it gets there. Consider that $n$ is finite, but sufficiently large. The probability $\pi(k^*\mid \mathbf{D}_n)$ is thus close to 1, and $\pi(k\mid\mathbf{D}_n)$ is close to 0 for all $k\neq k^*$ (and the estimates $\widehat{\pi}(k\mid\mathbf{D}_n)$ are close to $\pi(k\mid\mathbf{D}_n)$). Consider without loss of generality that all models have strictly positive posterior probabilities. Finally, consider that the initial state $k(0)\neq k^*$. All paths eventually lead to $k^*$ if the chain is irreducible. There is a subset of $\mathcal{K}$, that we denote by $\mathcal{K}^*\subset\mathcal{K}$, that is formed of all the models connected to $k^*$. Once a chain reaches that set, it goes to $k^*$ next with high probability. We are thus interested more specifically by the situation where $k(0)\notin \mathcal{K}^* \cup \{k^*\}$ and the behaviour of the chain while it explores $\mathcal{K}\setminus \mathcal{K}^* \cup \{k^*\}$.
[@zanella2019informed] provides conditions under which the ratios of normalising constants $c_k/c_{k'}\longrightarrow 1$, but this time, as $|\mathcal{K}|$ increases. This convergence surely does not hold for $k\in \mathcal{K}\setminus \mathcal{K}^* \cup \{k^*\}$ and $k'\in \mathcal{K}^* \cup \{k^*\}$ given that $c_k/c_{k'}\approx 0$. However, the conditions are realistic for the case where $k, k' \in \mathcal{K}\setminus \mathcal{K}^* \cup \{k^*\}$. In other words, as $|\mathcal{K}|$ increases, the relative mass of the neighbourhoods of $k, k' \in \mathcal{K}\setminus \mathcal{K}^* \cup \{k^*\}$ become similar. This in turn implies that the acceptance probabilities associated to these moves $k \mapsto k'$ are 1 in the limit. As shown in , yielding acceptance probabilities of 1 is not enough for a proposal distribution $g$ to be optimal. [@zanella2019informed] however proves that informed PMF like $g$ produce Markov chains with better mixing properties than *uninformed* (uniform) samplers when the probabilities vary within neighbourhoods (as shown by ). Therefore, if for instance at each step along some paths that lead from $k(0)$ to $k^*$ the models have progressively higher probabilities, and in particular they have higher probabilities than their neighbours, $g$ is expected to effectively make the chains follow these paths.
Note that when the marginal posterior of $K$ does not concentrate, the analysis presented in the last paragraph about the ratios $c_k/c_{k'}$ and the mixing properties of $g$ holds, but this time, on the whole domain $\mathcal{K}$.
Improving the approximations {#sec_improve_approx}
============================
In practice, the sample size may not be large enough for the approximations to be accurate. Fortunately, there exist methods that allow to compensate for functions $q_{k\mapsto k'}, q_{k'\mapsto k}$ and $\mathcal{D}_{k\mapsto k'}$ that are not sufficiently well designed. In this sense, using locations and variances of $(\widehat{\boldsymbol\mu}_k, \widehat{\boldsymbol\Sigma}_k / n)$ and $(\widehat{\boldsymbol\mu}_k', \widehat{\boldsymbol\Sigma}_{k'} / n)$ in normal approximations represent a first step towards ending up with random variables distributed as $\pi(\,\cdot\mid k',\mathbf{D}_n)\otimes q_{k'\mapsto k}$, starting from random variables distributed as $\pi(\,\cdot\mid k,\mathbf{D}_n)\otimes q_{k\mapsto k'}$. The methods presented in , which are those of [@karagiannis2013annealed] and [@andrieu2018utility], allow to bridge the gap. They turn out to be useful for improving the approximations forming the model proposal distribution $g$ as well, as explained in .
Improving the parameter proposal distributions {#sec_improve_parameters}
----------------------------------------------
### RJ incorporating the method of [@karagiannis2013annealed] {#sec_andrieu_2013}
For finite $n$, the shapes of the posteriors under Models $k$ and $k'$ may be quite different from each other, in addition to being different from bell curves. This explains why jumping (“in one step”) from the former to the latter may be difficult. [@karagiannis2013annealed] introduce a sequence of artificial and intermediate models that form a bridge between Models $k$ and $k'$, allowing to take several tinier steps instead (in the sense that the intermediate models are closer to each other). A path is followed along that bridge via inhomogeneous Markov kernels. The artificial models take the form of annealing intermediate distributions: for $t=0,\ldots,T$, define $$\begin{aligned}
\label{eqn_def_rho}
\rho_{k\mapsto k'}^{(t)}(\mathbf{x}_k^{(t)},\mathbf{u}_{k\mapsto k'}^{(t)})&\propto \left[\pi(k,\mathbf{x}_k^{(t)}\mid\mathbf{D}_n)\, q_{k\mapsto k'}(\mathbf{u}_{k\mapsto k'}^{(t)}) \, |J_{\mathcal{D}_{k\mapsto k'}}(\mathbf{x}_{k}^{(t)}, \mathbf{u}_{k\mapsto k'}^{(t)})|^{-1}\right]^{1-\gamma_t} \left[\pi(k',\mathbf{y}_{k'}^{(t)}\mid\mathbf{D}_n) \, q_{k'\mapsto k}(\mathbf{u}_{k'\mapsto k}^{(t)})\right]^{\gamma_t}, \cr
\rho_{k'\mapsto k}^{(t)}(\mathbf{y}_{k'}^{(t)},\mathbf{u}_{k'\mapsto k}^{(t)})&\propto \left[\pi(k,\mathbf{x}_k^{(t)}\mid\mathbf{D})\, q_{k\mapsto k'}(\mathbf{u}_{k\mapsto k'}^{(t)}) \, |J_{\mathcal{D}_{k\mapsto k'}}(\mathbf{x}_{k}^{(t)}, \mathbf{u}_{k\mapsto k'}^{(t)})|^{-1}\right]^{1-\gamma_{T-t}} \left[\pi(k',\mathbf{y}_{k'}^{(t)}\mid\mathbf{D}_n) \, q_{k'\mapsto k}(\mathbf{u}_{k'\mapsto k}^{(t)})\right]^{\gamma_{T-t}},\end{aligned}$$ where $T$ is a positive integer, $\gamma_0:=0, \gamma_T:=1$ and $\gamma_t\in[0,1]$ for $t\in\{1,\ldots,T-1\}$. In our numerical examples, we set $\gamma_t:=t/T$, as done in [@karagiannis2013annealed].
We notice that when switching from Model $k$ to Model $k'$, we start with distributions $\rho_{k\mapsto k'}^{(t)}$ close to $(\pi(k,\,\cdot\,\mid\mathbf{D}_n)\otimes q_{k\mapsto k'}) \, |J_{\mathcal{D}_{k\mapsto k'}}|^{-1}$ to finish, after a transition phase, with distributions close to $\pi(k',\,\cdot\,\mid\mathbf{D}_n)\otimes q_{k'\mapsto k}$. We wrote $\rho_{k\mapsto k'}^{(t)}$ as a function of $(\mathbf{x}_k^{(t)},\mathbf{u}_{k\mapsto k'}^{(t)})$ to emphasise that the starting point is $(\mathbf{x}_k^{(0)},\mathbf{u}_{k\mapsto k'}^{(0)})$. It is in fact also a function of $(\mathbf{y}_{k'}^{(t)}, \mathbf{u}_{k'\mapsto k}^{(t)})$, but recall that these are functions of $(\mathbf{x}_k^{(t)},\mathbf{u}_{k\mapsto k'}^{(t)})$: $(\mathbf{y}_{k'}^{(t)},\mathbf{u}_{k'\mapsto k}^{(t)}) := \mathcal{D}_{k\mapsto k'}(\mathbf{x}_k^{(t)},\mathbf{u}_{k\mapsto k'}^{(t)})$.
The annealing distributions above are called *geometric annealing distributions* in [@karagiannis2013annealed]. Another choice of distributions is presented in that paper. We present only geometric annealing distributions here because they seem to be the most practical.
It is generally impossible to sample from $\rho_{k\mapsto k'}^{(t)}$ which is why Markov kernels $K_{k \mapsto k'}^{(t)}$ that are reversible with respect to $\rho_{k\mapsto k'}^{(t)}$ are used to generate the path from Model $k$ to Model $k'$. We now present in the RJ incorporating the method of [@karagiannis2013annealed]. In Step 2.(b), the path can be generated through $(\mathbf{y}_{k'}^{(t)},\mathbf{u}_{k'\mapsto k}^{(t)})$ instead. It is simply a question of which choice is the most practical. Note that corresponds to regular RJ when $T=1$; no path is generated in Step 2.(b).
1. Generate $k'\sim g(k, \cdot \,)$.
2. If $k' = k$, attempt a parameter update.
3. If $k' \neq k$, attempt a model switch from Model $k$ to Model $k'$. Generate $\mathbf{u}_{k \mapsto k'}^{(0)} \sim q_{k \mapsto k'}$ and $u_a\sim\mathcal{U}(0, 1)$, and set $\mathbf{x}_{k}^{(0)}:=\mathbf{x}_{k}$. Generate a path $(\mathbf{x}_{k}^{(1)}, \mathbf{u}_{k \mapsto k'}^{(1)}),\ldots,(\mathbf{x}_{k}^{(T-1)}, \mathbf{u}_{k \mapsto k'}^{(T-1)})$, where $(\mathbf{x}_{k}^{(t)}, \mathbf{u}_{k \mapsto k'}^{(t)})\sim K_{k \mapsto k'}^{(t)}((\mathbf{x}_{k}^{(t-1)}, \mathbf{u}_{k \mapsto k'}^{(t-1)}), \cdot \,)$. If $$\begin{aligned}
\label{eqn_acc_ratio_andrieu_2013}
u_a \leq \alpha_{\text{RJ2}}((k,\mathbf{x}_{k}^{(0)}),(k',\mathbf{y}_{k'}^{(T-1)}))&:=1 \wedge \frac{g(k',k)}{g(k,k')}\prod_{t=0}^{T-1}\frac{\rho_{k\mapsto k'}^{(t+1)}(\mathbf{x}_k^{(t)},\mathbf{u}_{k\mapsto k'}^{(t)})}{\rho_{k\mapsto k'}^{(t)}(\mathbf{x}_k^{(t)},\mathbf{u}_{k\mapsto k'}^{(t)})},
\end{aligned}$$ set the next state of the chain to $(k',\mathbf{y}_{k'}^{(T-1)})$. Otherwise, set it to $(k, \mathbf{x}_k)$.
4. Go to Step 1.
The authors explain that the product in , that we denote by $$\begin{aligned}
\label{eqn_ratio_A2013}
r_{\text{RJ2}}((k,\mathbf{x}_{k}^{(0)}),(k',\mathbf{y}_{k'}^{(T-1)})):=\prod_{t=0}^{T-1}\frac{\rho_{k\mapsto k'}^{(t+1)}(\mathbf{x}_k^{(t)},\mathbf{u}_{k\mapsto k'}^{(t)})}{\rho_{k\mapsto k'}^{(t)}(\mathbf{x}_k^{(t)},\mathbf{u}_{k\mapsto k'}^{(t)})},\end{aligned}$$ represent a consistent estimator of $\pi(k'\mid \mathbf{D}_n)/\pi(k\mid \mathbf{D}_n)$ as $T\longrightarrow\infty$. This implies that $\alpha_{\text{RJ2}}\longrightarrow \alpha_{\text{marginal}}$ defined in as $T\longrightarrow\infty$. In fact, it is proved in [@gagnon2019NRJ] that under regularity conditions the Markov chain associated with converges weakly to that of the RJ which is able to sample from $\pi(\,\cdot\,\mid k, \mathbf{D}_n)$ for all $k$ with acceptance probabilities $\alpha_{\text{marginal}}$, as $T\longrightarrow\infty$ for fixed $n$. In other words, increasing $T$ yields proposals with distributions closer and closer to $\pi(\,\cdot\,\mid k', \mathbf{D}_n)$, even when the latter is not normal. Note that the weak convergence in that case is not in probability because the target is considered non-random (contrarily to the framework in which Theorems \[thm\_conv\_Case1\] and \[thm\_conv\_Case2\] are stated).
[@karagiannis2013annealed] prove that under two conditions is valid, in the sense that the target distribution is an invariant distribution. These conditions are the following.
Symmetry condition:
: For $t=1,\ldots,T-1$ the pairs of transition kernels $K_{k \mapsto k'}^{(t)}(\,\cdot \,, \cdot \,)$ and $K_{k' \mapsto k}^{(T-t)}(\,\cdot \,, \cdot \,)$ satisfy $$\begin{aligned}
\label{eqn_symmetry}
K_{k \mapsto k'}^{(t)}((\mathbf{x}_{k}, \mathbf{u}_{k \mapsto k'}), \cdot \,) = K_{k' \mapsto k}^{(T-t)}((\mathbf{x}_{k}, \mathbf{u}_{k \mapsto k'}), \cdot \,) \quad \text{for any } (\mathbf{x}_{k}, \mathbf{u}_{k \mapsto k'}).
\end{aligned}$$
Reversibility condition:
: For $t=1,\ldots,T-1$, and for any $(\mathbf{x}_k,\mathbf{u}_{k\mapsto k'})$ and $(\mathbf{x}_{k}', \mathbf{u}_{k \mapsto k'}')$, $$\begin{aligned}
\label{eqn_reversibility}
\rho_{k\mapsto k'}^{(t)}(\mathbf{x}_k,\mathbf{u}_{k\mapsto k'})K_{k \mapsto k'}^{(t)}((\mathbf{x}_{k}, \mathbf{u}_{k \mapsto k'}), (\mathbf{x}_{k}', \mathbf{u}_{k \mapsto k'}'))=\rho_{k\mapsto k'}^{(t)}(\mathbf{x}_k',\mathbf{u}_{k\mapsto k'}')K_{k \mapsto k'}^{(t)}((\mathbf{x}_{k}', \mathbf{u}_{k \mapsto k'}'), (\mathbf{x}_{k}, \mathbf{u}_{k \mapsto k'})).
\end{aligned}$$
As mentioned in [@karagiannis2013annealed], is verified if for all $t$, $K_{k \mapsto k'}^{(t)}(\,\cdot \,, \cdot \,)$ and $K_{k' \mapsto k}^{(T-t)}(\,\cdot \,, \cdot \,)$ are MH kernels sharing the same proposal distributions. We recommend to use MALA (Metropolis adjusted Langevin, [@roberts1998optimal]) proposals whenever this is possible; see [@karagiannis2013annealed] for other examples. We present in a procedure to automatically tune the scaling parameter of the MALA within this context.
The other additional input that needs to be specified is $T$. In fact, to run the algorithm we need to specify a value for each couple $(k, k')$; we thus define $T_{k, k'}$ to be the value for the couple $(k ,k')$. Typically, they are all set to the same value $T$ to simplify the problem, as done in [@karagiannis2013annealed]. This may be sub-optimal when the model space is large. In this paper, we instead use a value $T_{k, k'}$ specific to each couple $(k, k')$, and we achieve this in a way that scales well with the number of models. We explain in this section how to specify $T_{k, k'}$ for given $(k, k')$, and present in how we proceed for the collection $\{T_{k, k'}\}$.
In [@karagiannis2013annealed], it is explained that one should expect by gradually increasing $T_{k, k'}$ to observe at the beginning a steady increase in the quality of the approximations translating into an increase of the acceptance probabilities towards $\alpha_{\text{marginal}}$ defined in , until the samplers are close enough to the limiting RJ; after this point the increase is less marked (see (a)). The strategy is to find the approximate location of this point and to choose a suitable smaller value for $T_{k, k'}$. This may be done in two steps. Firstly, identify the value of $T_{k, k'}$ for which the increase is most marked (which is at $T_{k, k'} = 2$ using the slope of the polynomial regression in (b)). Secondly, determine where the rate starts to decrease (which is around $T_{k, k'} = 10$ in (b)), implying a diminishing return, and presumably, that the asymptotic regime is reached. We recommend to set $T_{k, k'}$ to the closest value to the middle of the interval (i.e. $T_{k, k'} := (2 + 10) / 2 = 6$ in the example) so that there is still work to do for the method presented in the next section. Note that for the data and models on which Figures \[fig\_1\] and \[fig\_2\] are based, the normal approximations to the parameter distributions are good as the acceptance probabilities are close to the limiting value even for small values of $T_{k, k'}$ (notice the y-axis scale in (a)).
$\begin{array}{cc}
\hspace{-2.5mm}\includegraphics[width = 0.5\textwidth]{Fig2_a.pdf} & \hspace{-2.5mm}\includegraphics[width = 0.5\textwidth]{Fig2_b.pdf} \cr
\hspace{-0mm}\textbf{(a)} & \hspace{-0mm} \textbf{(b)}
\end{array}$
The potential benefit associated with the additional feature in (compared with vanilla RJ) certainly comes at a computational cost. As shown in [@karagiannis2013annealed], this cost may be offset by a large enough increase in effective sample size (ESS) resulting in a net increase in ESS per unit time.
### RJ additionally incorporating the method of [@andrieu2018utility] {#sec_andrieu_2018}
As mentioned in the last section, $r_{\text{RJ2}}$ (see ) can be seen as an estimator of $\pi(k'\mid\mathbf{D}_n)/\pi(k\mid\mathbf{D}_n)$. It seems a good idea to independently produce in parallel $N$ paths ending with $N$ proposals, that we denote by $\mathbf{y}_{k'}^{(T-1, 1)},\ldots,\mathbf{y}_{k'}^{(T-1, N)}$, and therefore $N$ estimates $r_{\text{RJ2}}((k,\mathbf{x}_{k}^{(0)}),(k',\mathbf{y}_{k'}^{(T-1,1)})),\ldots, r_{\text{RJ2}}((k,\mathbf{x}_{k}^{(0)}),(k',\mathbf{y}_{k'}^{(T-1,N)}))$ to average the latter for obtaining a better estimate of $\pi(k'\mid\mathbf{D}_n)/\pi(k\mid\mathbf{D}_n)$ (we simplify the notation by omitting the subscript $k, k'$ in $T_{k, k'}$). Denote this average (with simplified notation) by $$\bar{r}(k, k'):=\frac{1}{N} \sum_{j=1}^N r_{\text{RJ2}}((k,\mathbf{x}_{k}^{(0)}),(k',\mathbf{y}_{k'}^{(T-1,j)})).$$ Applying this method naively does however not lead to valid algorithms. The approach of [@andrieu2018utility] exploits this averaging idea while leading to valid RJ. In fact, these authors present a general method that can be used in a broad range of sampling situations (not only when using RJ).
We now present in the RJ additionally incorporating the method of [@andrieu2018utility].
1. Generate $k'\sim g(k, \cdot \,)$.
2. If $k' = k$, attempt a parameter update.
3. If $k'\neq k$, attempt a model switch from Model $k$ to Model $k'$. Generate $u_a, u_c\sim \mathcal{U}(0, 1)$. If $u_c\leq 1/2$ go to Step 2.(b-i), otherwise go to Step 2.(b-ii).
4. Generate $N$ proposals $\mathbf{y}_{k'}^{(T-1, 1)},\ldots,\mathbf{y}_{k'}^{(T-1, N)}$ as in Step 2.(b) of . Generate $j^*$ from a PMF such that ${\mathbb{P}}(J^*=j) \propto r_{\text{RJ2}}((k,\mathbf{x}_{k}), (k', \mathbf{y}_{k'}^{(T-1,j)}))$. If $$\begin{aligned}
u_a &\leq \frac{g(k',k)}{g(k,k')} \, \bar{r}(k, k'),
\end{aligned}$$ set the next state of the chain to $(k',\mathbf{y}_{k'}^{(T-1,j^*)})$. Otherwise, set it to $(k, \mathbf{x}_k)$.
5. Generate one forward path as in Step 2.(b) of . Denote the endpoint by $\mathbf{y}_{k'}^{(T-1, 1)}$. From $\mathbf{y}_{k'}^{(T-1, 1)}$, generate $N - 1$ reverse paths again as in Step 2.(b) of , yielding $N - 1$ proposals for the parameters of Model $k$. If $$\begin{aligned}
u_a &\leq \frac{g(k',k)}{g(k,k')} \, \bar{r}(k', k)^{-1},
\end{aligned}$$ set the next state of the chain to $(k', \mathbf{y}_{k'}^{(T-1,1)})$. Otherwise, set it to $(k, \mathbf{x}_k)$.
6. Go to Step 1.
No additional assumptions to those presented in are required to guarantee that is valid. [@andrieu2018utility] prove that increasing $N$ decreases the asymptotic variance of the Monte Carlo estimates produced by RJ incorporating their approach. It is expected that increasing $N$ (as increasing $T$ in the last section) leads to a steady increase in the quality of the approximations until the samplers are close enough to the limiting RJ. Therefore the same strategy as in the last section to find the approximate location of the threshold may be applied. We recommend in this case to set $N$ to the value for which the rate starts to decrease (see ). In this paper, we in fact use a value that we denote by $N_{k, k'}$ specific to each couple $(k, k')$. We present in how we proceed for specifying the collection of values $\{N_{k, k'}\}$.
An advantage of the approach presented in this section is that the additional computational cost (over ) is negligible considering that one can generate the $N_{k, k'}$ proposals $\mathbf{y}_{k'}^{(T-1, 1)},\ldots,\mathbf{y}_{k'}^{(T-1, N_{k, k'})}$ and compute the $N_{k, k'}$ estimates $r_{\text{RJ2}}((k,\mathbf{x}_{k}^{(0)}),(k',\mathbf{y}_{k'}^{(T-1,1)})), \ldots, r_{\text{RJ2}}((k,\mathbf{x}_{k}^{(0)}),(k',\mathbf{y}_{k'}^{(T-1,N_{k, k'})}))$ in parallel, requiring essentially the same amount of time as generating one proposal and computing one estimate.
Improving the model proposal distribution {#sec_improve_models}
-----------------------------------------
We have seen in that $\bar{r}(k, k')$ and the ratios $r_{\text{RJ2}}$ forming it are estimators of $\pi(k'\mid\mathbf{D}_n)/\pi(k\mid\mathbf{D}_n)$. They can thus be used to enhance the approximation $\widehat{\pi}(k'\mid\mathbf{D}_n)/\widehat{\pi}(k\mid\mathbf{D}_n)$ in $g(k,k')$, in the case where the neighbourhoods are smaller than the model domain (). We focus on improving the approximations in this case rather than in the case where the neighbourhoods are equal to the domain, as in the latter $\widehat{\pi}(k\mid\mathbf{D}_n)$ may be adjusted after trial runs given that the size of $\mathcal{K}$ is typically small.
If we want to enhance the PMF $g(k, \cdot \,)$, we need to improve $\widehat{\pi}(l\mid\mathbf{D}_n) / \widehat{\pi}(k\mid\mathbf{D}_n)$ for all $l\in \mathcal{N}(k)$ as these are all involved in the construction of the PMF. Also, once the proposal for the model to explore next $k'$ is generated, we need to do the same for $g(k', \cdot \,)$ given that this PMF comes into play in the computation of the acceptance probabilities (see, e.g., ). We thus need parameter proposals $\mathbf{y}_{l}^{(T-1,1)}, \ldots, \mathbf{y}_{l}^{(T-1,N)}$ for all Models $l\in \mathcal{N}(k)$, and also for all models belonging to $\mathcal{N}(k')$, which will be denoted by $\mathbf{z}_{s}^{(T-1,1)}, \ldots, \mathbf{z}_{s}^{(T-1,N)}$, $s\in \mathcal{N}(k')$ (we simplify the notation by omitting the subscript $k, k'$ in $T_{k, k'}$ and $N_{k, k'}$). The ratios $r_{\text{RJ2}}$ are next computed.
There are several ways to combine these ratios with $\widehat{\pi}(l\mid\mathbf{D}_n) / \widehat{\pi}(k\mid\mathbf{D}_n)$ (or $\widehat{\pi}(s\mid\mathbf{D}_n) / \widehat{\pi}(k'\mid\mathbf{D}_n)$) to improve the estimation of $\pi(l\mid\mathbf{D}_n) / \pi(k\mid\mathbf{D}_n)$ (or $\pi(s\mid\mathbf{D}_n) / \pi(k'\mid\mathbf{D}_n)$). We define the improved version of the PMF $g$ as follows to reflect this flexibility: $$\begin{aligned}
\label{eqn_g_imp}
g_{\text{imp.}}(k, l, \mathbf{x}_{k}^{(0)}, \mathbf{y}_{\bullet}^{(0:T-1, \bullet)}, \mathbf{u}_{\bullet \mapsto k}^{(0:T-1, \bullet)}) := h\left(\frac{\tilde{\pi}(l\mid \mathbf{D}_n)}{\tilde{\pi}(k\mid \mathbf{D}_n)}\right) \bigg/ c_k^{\text{imp.}},\end{aligned}$$ where $$\frac{\tilde{\pi}(l\mid \mathbf{D}_n)}{\tilde{\pi}(k\mid \mathbf{D}_n)} := \varrho\left(\frac{\widehat{\pi}(l\mid\mathbf{D}_n)}{\widehat{\pi}(k\mid\mathbf{D}_n)}, r_{\text{RJ2}}((k,\mathbf{x}_{k}^{(0)}),(l,\mathbf{y}_{l}^{(T-1,1)})), \ldots, r_{\text{RJ2}}((k,\mathbf{x}_{k}^{(0)}),(l,\mathbf{y}_{l}^{(T-1,N)}))\right),$$ $\mathbf{y}_{\bullet}^{(0:T-1, \bullet)}$ is the vector containing $\mathbf{y}_{l}^{(0, j)}, \ldots, \mathbf{y}_{l}^{(T-1, j)}$ for all $j\in\{1,\ldots,N\}$ and $l\in \mathcal{N}(k)$, $ \mathbf{u}_{\bullet \mapsto k}^{(0:T-1, \bullet)}$ is the vector containing $\mathbf{u}_{l \mapsto k}^{(0, j)}, \ldots, \mathbf{u}_{l \mapsto k}^{(T-1, j)}$ for all $j\in\{1,\ldots,N\}$ and $l\in \mathcal{N}(k)$, and $c_k^{\text{imp.}}$ is the normalising constant, $\varrho$ being a function aiming at putting together the information whose choice is discussed below. Note that $\tilde{\pi}(l\mid \mathbf{D}_n) / \tilde{\pi}(k\mid \mathbf{D}_n)$ is in fact an estimator of $\pi(l\mid \mathbf{D}_n) / \pi(k\mid \mathbf{D}_n)$ and a function of $\mathbf{x}_{k}^{(0)},\mathbf{y}_{l}^{(T-1,1)},\ldots, \mathbf{y}_{l}^{(T-1,N)}$ additionally to $k$ and $l$; we used this notation to simplify and make the connection with $\widehat{\pi}(l\mid\mathbf{D}_n) / \widehat{\pi}(k\mid\mathbf{D}_n)$.
includes the idea of improving $\widehat{\pi}(l\mid\mathbf{D}_n) / \widehat{\pi}(k\mid\mathbf{D}_n)$ using ratios $ r_{\text{RJ2}}$ in a valid way (as indicated by below). It is noticed that the computations for the two main steps (Steps 2.(i) and 2.(ii)) can be performed in parallel. The computation time is thus roughly doubled compared to that for Steps 2.(b-i) and 2.(b-ii) in . The main drawback of is that it requires to perform the computations for $g_{\text{imp.}}(k', \cdot \,)$ even when $k'=k$. This is because $g_{\text{imp.}}(k, k', \mathbf{x}_{k}^{(0)}, \mathbf{y}_{\bullet}^{(0:T-1, \bullet)}, \mathbf{u}_{\bullet \mapsto k}^{(0:T-1, \bullet)})$ is different from $g_{\text{imp.}}(k', k, \mathbf{y}_{k'}^{(T-1,j^*)}, \mathbf{z}_{\bullet}^{(0:T-1, \bullet)}, \mathbf{u}_{\bullet \mapsto k'}^{(0:T-1, \bullet)})$ even when $k' = k$.
1. Generate $u_a, u_c\sim \mathcal{U}(0, 1)$. If $u_c\leq 1/2$ go to Step 2.(i), otherwise go to Step 2.(ii).
2. For all $l\in \mathcal{N}(k)$, generate $\mathbf{y}_{l}^{(T-1,1)}, \ldots, \mathbf{y}_{l}^{(T-1,N)}$ as in Step 2.(b-i) in and compute $g_{\text{imp.}}(k, \cdot \,)$ (see ). Generate $k'\sim g_{\text{imp.}}(k, \cdot \,)$ and $j^*$ from a PMF such that ${\mathbb{P}}(J^*=j) \propto r_{\text{RJ2}}((k,\mathbf{x}_{k}^{(0)}),(k',\mathbf{y}_{k'}^{(T-1,j)}))$, and compute $\bar{r}(k, k')$. Now, for all $s\in \mathcal{N}(k')\setminus \{k\}$, generate $N$ endpoints $\mathbf{z}_{s}^{(T-1,1)}, \ldots, \mathbf{z}_{s}^{(T-1,N)}$ as in Step 2.(b-ii) in from $\mathbf{y}_{k'}^{(T-1,j^*)}$. Compute $g_{\text{imp.}}(k', \cdot \,)$ using the same estimates as in the first part for approximating $\pi(k \mid \mathbf{D}_n) / \pi(k' \mid \mathbf{D}_n)$. If $$\begin{aligned}
u_a &\leq \frac{g_{\text{imp.}}(k', k, \mathbf{y}_{k'}^{(T-1,j^*)}, \mathbf{z}_{\bullet}^{(0:T-1, \bullet)}, \mathbf{u}_{\bullet \mapsto k'}^{(0:T-1, \bullet)})}{g_{\text{imp.}}(k, k', \mathbf{x}_{k}^{(0)}, \mathbf{y}_{\bullet}^{(0:T-1, \bullet)}, \mathbf{u}_{\bullet \mapsto k}^{(0:T-1, \bullet)})} \, \bar{r}(k, k'),
\end{aligned}$$ set the next state of the chain to $(k',\mathbf{y}_{k'}^{(T-1,j^*)})$. Otherwise, set it to $(k, \mathbf{x}_k)$.
3. For all $l\in \mathcal{N}(k)$, generate $\mathbf{y}_{l}^{(T-1,1)}, \ldots, \mathbf{y}_{l}^{(T-1,N)}$ as in Step 2.(b-ii) in and compute $g_{\text{imp.}}(k, \cdot \,)$. Generate $k'\sim g_{\text{imp.}}(k, \cdot \,)$ and compute $\bar{r}(k', k)^{-1}$. Now, for all $s\in \mathcal{N}(k')\setminus \{k\}$, generate $N$ endpoints $\mathbf{z}_{s}^{(T-1,1)}, \ldots, \mathbf{z}_{s}^{(T-1,N)}$ as in Step 2.(b-i) in from $\mathbf{y}_{k'}^{(T-1, 1)}$. Compute $g_{\text{imp.}}(k', \cdot \,)$ using the same estimates as in the first part for approximating $\pi(k \mid \mathbf{D}_n) / \pi(k' \mid \mathbf{D}_n)$. If $$\begin{aligned}
u_a &\leq \frac{g_{\text{imp.}}(k', k, \mathbf{y}_{k'}^{(T-1, 1)}, \mathbf{z}_{\bullet}^{(0:T-1, \bullet)}, \mathbf{u}_{\bullet \mapsto k'}^{(0:T-1, \bullet)})}{g_{\text{imp.}}(k, k', \mathbf{x}_{k}^{(0)}, \mathbf{y}_{\bullet}^{(0:T-1, \bullet)}, \mathbf{u}_{\bullet \mapsto k}^{(0:T-1, \bullet)})} \, \bar{r}(k', k)^{-1},
\end{aligned}$$ set the next state of the chain to $(k',\mathbf{y}_{k'}^{(T-1, 1)})$. Otherwise, set it to $(k, \mathbf{x}_k)$.
4. Go to Step 1.
\[prop\_inv\_algo\_model\_imp\] Under the two assumptions presented in , -, is valid.
See .
It is natural to set $\tilde{\pi}(l\mid \mathbf{D}_n) / \tilde{\pi}(k\mid \mathbf{D}_n)$ to 1 when $l = k$. This implies that we in fact do not need to generate proposals for Model $k$ in the first parts of Steps 2.(i) and 2.(ii). If $k' \neq k$, they do not need to be generated at all. Also, in the second parts of Steps 2.(i) and 2.(ii), it is not required to generate proposals for $s = k'$ for the same reason.
The function $\varrho$ in specifies the way the information is combined. It may be set for instance to the simple average: $$\begin{aligned}
\label{eqn_est_ratio}
\frac{\tilde{\pi}(l\mid \mathbf{D}_n)}{\tilde{\pi}(k\mid \mathbf{D}_n)} := \frac{1}{N + 1} \left( \frac{\widehat{\pi}(l\mid\mathbf{D}_n)}{\widehat{\pi}(k\mid\mathbf{D}_n)} + \sum_{j=1}^N r_{\text{RJ2}}((k,\mathbf{x}_{k}^{(0)}),(l,\mathbf{y}_{l}^{(T-1,j)}))\right).\end{aligned}$$ One may alternatively take the average of $\widehat{\pi}(l\mid\mathbf{D}_n) / \widehat{\pi}(k\mid\mathbf{D}_n)$ and $\bar{r}(k, l)$: $$\frac{\tilde{\pi}(l\mid \mathbf{D}_n)}{\tilde{\pi}(k\mid \mathbf{D}_n)} := \frac{1}{2} \left( \frac{\widehat{\pi}(l\mid\mathbf{D}_n)}{\widehat{\pi}(k\mid\mathbf{D}_n)} + \bar{r}(k, l)\right) := \frac{1}{2} \left( \frac{\widehat{\pi}(l\mid\mathbf{D}_n)}{\widehat{\pi}(k\mid\mathbf{D}_n)} + \frac{1}{N} \sum_{j=1}^N r_{\text{RJ2}}((k,\mathbf{x}_{k}^{(0)}),(l,\mathbf{y}_{l}^{(T-1,j)}))\right).$$ These reflect a choice of putting more or less weight on $\widehat{\pi}(l\mid\mathbf{D}_n) / \widehat{\pi}(k\mid\mathbf{D}_n)$. We know that if $T$ and $N$ are large enough then $\bar{r}(k, l)$ is close to $\pi(l\mid\mathbf{D}_n) / \pi(k\mid\mathbf{D}_n)$, which may not be the case for $\widehat{\pi}(l\mid\mathbf{D}_n) / \widehat{\pi}(k\mid\mathbf{D}_n)$ when $n$ is not sufficiently large. The latter ratio may thus act as outlying/conflicting information against which these averages above are not robust. A robust approach consists in setting $\varrho$ to be the median of $\widehat{\pi}(l\mid\mathbf{D}_n) / \widehat{\pi}(k\mid\mathbf{D}_n), r_{\text{RJ2}}((k,\mathbf{x}_{k}^{(0)}),(l,\mathbf{y}_{l}^{(T-1,1)})), \ldots, r_{\text{RJ2}}((k,\mathbf{x}_{k}^{(0)}),(l,\mathbf{y}_{l}^{(T-1,N)}))$. We recommend this approach and use it in our numerical examples.
Furthermore, as $T,N\longrightarrow \infty$, $\tilde{\pi}(l\mid \mathbf{D}_n)/\tilde{\pi}(k\mid \mathbf{D}_n)\longrightarrow \pi(l\mid\mathbf{D}_n) / \pi(k\mid\mathbf{D}_n)$ when the median or is used (recall the properties of $r_{\text{RJ2}}$ and $\bar{r}$ mentioned in ), for fixed $n$. Therefore, if the function $h$ is such that $h(x) = x \, h(1/x)$ for $x>0$, then the acceptance probabilities in converge towards $1 \wedge \bar{c}_k / \bar{c}_{k'}$, where $\bar{c}_k$ and $\bar{c}_{k'}$ are the limiting normalising constants with $$\bar{c}_k := \sum_{l \in \mathcal{N}(k)} h\left(\frac{\pi(l\mid\mathbf{D}_n)}{\pi(k\mid\mathbf{D}_n)}\right).$$ In fact, the same technique as in the proof of Theorem 1 in [@gagnon2019NRJ] allows to prove that the Markov chain associated with converges weakly for fixed $n$ to that of an ideal RJ which has access to the posterior probabilities $\pi(k\mid \mathbf{D}_n)$ and is able to sample from the conditional distributions $\pi(\, \cdot\mid k,\mathbf{D}_n)$ (and for which the acceptance probabilities are $1 \wedge \bar{c}_k / \bar{c}_{k'}$), with its good mixing properties as discussed in .
Implementation {#sec_implementation}
==============
Several authors (see, e.g., [@green2003trans]) mentioned that informed RJ samplers may be problematic when it is require to gather information for each model before running them, because this is infeasible for large (or infinite) model spaces. We explain in this section that, for the samplers presented so far, the information gathering can be done on the fly as the chains reach new models. This strategy is often more efficient and can in fact make the implementation of informed RJ samplers possible, even if the model space is large or infinite, provided that the posterior probabilities concentrate on a reasonable number of models (in the sense that the number of different models visited during algorithm runs is on average reasonable). When the probabilities concentrate on few models, this implementation strategy is expected to be highly effective as the information required for model switches and parameter updates will essentially be gathered in practice only for these few models.
To start running , for instance, several inputs may seem to be required, like $\widehat{\pi}(k\mid \mathbf{D}_n)$, $\widehat{\boldsymbol\mu}_k$, and $\widehat{\boldsymbol\Sigma}_k$ for all $k\in\mathcal{K}$. The estimates $\widehat{\boldsymbol\mu}_k$ are typically maximisers of likelihood functions or posterior densities and $\widehat{\pi}(k\mid \mathbf{D}_n)$ and $\widehat{\boldsymbol\Sigma}_k$ are based on them. It is thus actually unnecessary to compute all of them beforehand; during a run the computations may be done on the fly as the chain reaches new models, and the estimates may be stored to be reused next time the models are visited. The reason why is because these maximisers are independent of the chain path; they are the same whether they are computed before or at the same time the algorithm is running. This is the key idea. The current state may even be used to identify starting points for the optimisers as the output is *in theory* independent.
The same principle may be applied for identifying suitable values for $T_{k, k'}$ and $N_{k, k'}$. One may generate several parameter proposals for Model $k'$ from $\widehat{\boldsymbol\mu}_k$, this for several values for $T_{k, k'}$ to find a suitable one according to the strategy presented in . There is no need to generate parameter proposals for Model $k$ from $\widehat{\boldsymbol\mu}_{k'}$ as the process is reversible. If MALA is used to generate the paths, its step size $\epsilon_{k, k'}$ is tuned at the same time. We recommend to apply the following procedure (assuming that a grid $\{T_{k, k'}^{\min}, \ldots, T_{k, k'}^{\max}\}$ has been prespecified for the values to try) that can be executed using parallel computing.
1. 2. Tune the value of $\epsilon_{k, k'}$ so that the acceptance rate is around 0.55. Denote by $\epsilon_{k, k'}^\text{start}$ an identified value.
3. Generate a grid around $\epsilon_{k, k'}^\text{start}$: $\{\epsilon_{k, k'}^1, \ldots, \epsilon_{k, k'}^{j_0} := \epsilon_{k, k'}^\text{start}, \ldots, \epsilon_{k, k'}^{L}\}$, where $L$ is a positive integer.
4. For each $\epsilon_{k, k'}^j$, generate several parameter proposals for Model $k'$ from $\widehat{\boldsymbol\mu}_k$. For each of these proposals, evaluate the total squared distance $\text{TSD} := \sum_{t = 1}^{T_{k, k'} - 1} \|(\mathbf{x}_k^{(t)}, \mathbf{u}_{k \mapsto k'}^{(t)}) - (\mathbf{x}_k^{(t - 1)}, \mathbf{u}_{k \mapsto k'}^{(t - 1)})\|_2^2$, and compute the acceptance probability according to with $g(k, \cdot \,)$ and $g(k', \cdot \,)$ set as in .
5. Identify the value $\epsilon_{k, k'}^*$ associated to the largest average $\text{TSD}$ and estimate the probability of accepting a proposal using the data collected at the previous step to identify a suitable value for $T_{k, k'}$.
Once this is done, the same strategy (except the $\epsilon_{k, k'}$ part) may be applied to identify a suitable value for $N_{k,k'}$ using $\epsilon_{k, k'}^*$ and the selected value for $T_{k, k'}$, as explained in . Note that, instead of starting all the paths from $\widehat{\boldsymbol\mu}_k$ in Step 3, one may use different starting points obtained by sequentially applying parameter update steps with $\widehat{\boldsymbol\mu}_k$ as starting value to diversify the sample and robustify the selected values for $T_{k, k'}$ and $N_{k,k'}$. Again, the idea is to store and reuse these values (in this case, of $\epsilon_{k, k'}^*$, $T_{k, k'}$ and $N_{k,k'}$).
If HMC is used to update the parameters, the step sizes and trajectory lengths can also be tuned on the fly. In our numerical examples, we use the step sizes identified by RStan (with the option *static HMC*) and tune the trajectory lengths by trying several values on a grid. The merit of each trajectory length is evaluated via its associated ESS. Also, if HMC is used, the momentum needs to be refreshed. Theoretically, we may consider that a momentum refreshment is performed every odd iteration, and that the algorithms proceed as in Algorithms \[algo\_RJ\_andrieu\_2013\], \[algo\_RJ\_andrieu\_2018\] or \[algo\_RJ\_imp\_model\] for instance every even iteration. Also, we need (in theory) to add or withdraw momentum variables when switching models. In practice, we do not have to proceed in this way. Given that momentum variables are only required when updating the parameters, we may generate them only when it is known that a parameter update is proposed (i.e. $k' = k$).
Application: variable selection in wholly robust linear regression {#sec_application}
==================================================================
A new technique emerged to gain robustness against outliers in parametric modelling: replace the traditional distribution assumption (which is a normal assumption in the problems studied) by a super heavy-tailed distribution assumption (see [@desgagne2015robustness], [@gagnon2017PCR], [@gagnon2018regression], and [@DesGag2019]). The rationale is that this latter assumption is more adapted to the eventual presence of outliers by giving higher probabilities to extreme values. The proof of effectiveness of the approach resides in the following: the posterior distribution converges towards that based on the nonoutliers only (i.e. excluding the outliers) as the outliers move further and further away from the bulk of the data. This theoretical result corresponds to a concept in Bayesian statistics called *whole robustness*. As explained in these papers cited above, the models have built-in robustness that resolve conflicts due to contradictory information in a sensitive way. It takes full consideration of nonoutliers and excludes observations that are undoubtedly outlying; in between these two extremes, it balances and bounds the impact of possible outliers, reflecting the uncertainty about the nature of these observations.
In [@gagnon2018regression], the convergence is proved within the most general linear regression framework, encompassing analysis of variance and covariance (ANOVA and ANCOVA), and variable selection. In this section, we apply the methodology presented in the previous sections to sample from a joint posterior distribution of robust linear regressions and their parameters. The data analysed are the same prostate cancer data as in . RJ is required comparatively to the case where the error distribution is assumed to be a Student ([@1984west431]). Using a heavy-tailed distribution like the Student only allows for partial robustness ([@andrade2011bayesian]), which may lead to regression coefficients with inflated variances, and ultimately contaminated model selection.
The super heavy-tailed distribution used is called the *log-Pareto-tailed normal* (LPTN). Its density matches the normal on the central part, while having log-Pareto tails. The model with the LPTN is thus expected to behave similarly to the traditional one in the absence of outliers (that latter model is known for being the benchmark in terms of efficiency in that situation). Not only that is the case in absence of outliers, but the limiting LPTN posterior distribution (as the distance between the outliers and the bulk of the data approaches infinity) is also similar to the normal posterior, but that based on the nonoutliers only. Given that the robust approach naturally gives rise to an outlier detection method, we can thus identify a “common” data set and compare the MCMC outputs to the values that we are able to explicitly compute for the normal models. That allows ensuring that there is no problem with the computer code. Note that all the details for the normal and robust models can be found in the supplementary material (). It is also proved in the supplementary material that a simple modification to a uniform prior on $K$ prevents the Jeffreys-Lindley paradox from arising when the usual non-informative priors are used for the parameters.
The performances of the different algorithms are summarised in and . The results are based on 1,000 runs of 100,000 iterations for each algorithm, with burn-ins of 10,000. The model switching acceptance rate and model visit rate are related. The former is simply the (average) acceptance rate, but computed considering only the iterations in which model switches are proposed; the latter is the (average) number of model switches in one run, reported per iteration. For both these measures, we count the number of accepted model switches, and this number is divided by either the number of proposed model switches or total number of iterations. The error reduction is the relative decrease in total variation between the empirical and true marginal posterior distributions of $K$, with respect to the naive RJ.
The model acceptance rate is close to 1 when the parameter proposals are approximately distributed as $\pi(\, \cdot \mid k, \mathbf{D}_n)$ and $g(k',k) / g(k,k') \approx \pi(k \mid \mathbf{D}_n) / \pi(k' \mid \mathbf{D}_n)$ (see, for instance, $\alpha_{\text{RJ}}$ in ). As mentioned in , getting a model acceptance rate closer to 1 is in our framework a first step towards optimality. This may indeed lead to larger off-diagonal elements in the model switch transition matrix, which is better in the sense of [@peskun1973optimum]. A higher model visit rate reflects larger off-diagonal elements. In this example, we observe that both measures are positively correlated with the error reduction. In particular, we notice that using informed proposal distributions $g$ significantly enhances the algorithms. In RJ with $h$ as in (but without the techniques included in Algorithms \[algo\_RJ\_andrieu\_2013\], \[algo\_RJ\_andrieu\_2018\] and \[algo\_RJ\_imp\_model\]), the designs of both the parameter proposal distributions and model proposal distribution are based on approximations whose accuracy increases as $n \longrightarrow \infty$. Algorithms \[algo\_RJ\_andrieu\_2013\] and \[algo\_RJ\_andrieu\_2018\] allow to bridge the gap with regard to the parameter proposal distributions while enhance the model proposals. The results show that, even if the asymptotic regime is not attained, a sample size of $n = 97$ is relatively large for such a robust linear regression problem with a total of nine covariates. Finally, we note that the robust linear regression analysis indicates that there are no outliers (at least no severe ones).
Discussion {#sec_discussion}
==========
In this paper, we showed that using an informed model proposal distribution contributes to the global efficiency of RJ algorithms. In particular, informed proposals are crucial when the model probabilities and parameters densities vary significantly within neighbourhoods. They vary significantly when the target concentrates as $n \longrightarrow \infty$. But we noticed in our numerical example that they do, even when this large sample regime is not reached. The proposed RJ show major improvement as the chains spend less iterations at the same state, comparatively to naive samplers which often try to reach low probability models and thus suffer from high-rejection rates. In particular, improving the approximations for both the model proposals and parameter proposals successfully reaches a model switching acceptance rate of $0.85$ in our numerical example, which is close to the rate of $0.91$ for the limiting RJ (as $T_{k, k'}, N_{k, k'} \longrightarrow \infty$) accepting model proposals with the same rate as a marginal sampler for $K$ having access to $\pi(k \mid \mathbf{D}_n)$.
Yet, the proposed samplers are reversible which allows them to return to recently visited models often. The next step in this line of research of trans-dimensional samplers for non-nested model selection is to propose sampling schemes which do not suffer from this diffusive behaviour, but instead induce persistent movement in the model indicator.
Proofs {#sec_proofs}
======
To prove this result, we use Theorem 2 of [@schmon2018large]. We thus have to verify the following three conditions.
1. $(K,\mathbf{Z}_K)_n(0)\Longrightarrow (K,\mathbf{Z}_K)_{\text{ideal}}(0)$ in probability as $n\longrightarrow \infty$.
Denote the distribution of $(K,\mathbf{Z}_K)_n(0)$ by $\pi_{K, \mathbf{Z}_K}(\, \cdot \mid \mathbf{D}_n)$. Given that $\mathcal{K}$ is finite, it suffices to verify that $$\left|\pi(k\mid \mathbf{D}_n)\,{\mathbb{P}}(\mathbf{Z}_{k,n} \in A) - \bar{\pi}(k)\,{\mathbb{P}}(\mathbf{Z}_{k,\text{ideal}} \in A) \right|\longrightarrow 0 \quad \text{in probability},$$ for any $k$ and measurable set $A$, where ${\mathbb{P}}(\mathbf{Z}_{k,n} \in A)$ and ${\mathbb{P}}(\mathbf{Z}_{k,\text{ideal}} \in A)$ are computed using the conditional distributions given that $K=k$. Using the triangle inequality, we have that $$\begin{aligned}
& \left|\pi(k\mid \mathbf{D}_n)\,{\mathbb{P}}(\mathbf{Z}_{k,n} \in A) - \bar{\pi}(k)\,{\mathbb{P}}(\mathbf{Z}_{k,\text{ideal}} \in A) \right| \cr
&\quad \leq \left|\pi(k\mid \mathbf{D}_n)\,{\mathbb{P}}(\mathbf{Z}_{k,n} \in A) - \bar{\pi}(k)\,{\mathbb{P}}(\mathbf{Z}_{k,n} \in A) \right| + \left| \bar{\pi}(k)\,{\mathbb{P}}(\mathbf{Z}_{k,n} \in A) - \bar{\pi}(k)\,{\mathbb{P}}(\mathbf{Z}_{k,\text{ideal}} \in A) \right|.\end{aligned}$$ We now show that both absolute values converge towards 0 in probability which will allow to conclude by Slutsky’s theorem and monotonicity of probabilities. We first have that $$\left|\pi(k\mid \mathbf{D}_n)\,{\mathbb{P}}(\mathbf{Z}_{k,n} \in A) - \bar{\pi}(k)\,{\mathbb{P}}(\mathbf{Z}_{k,n} \in A) \right|\leq \left|\pi(k\mid \mathbf{D}_n) - \bar{\pi}(k)\right|\longrightarrow 0 \quad \text{in probability},$$ by and the fact that ${\mathbb{P}}(\mathbf{Z}_{k,n} \in A)\leq 1$. Using now that $\bar{\pi}(k)\leq 1$, we have that $$\begin{aligned}
\left| \bar{\pi}(k)\,{\mathbb{P}}(\mathbf{Z}_{k,n} \in A) - \bar{\pi}(k)\,{\mathbb{P}}(\mathbf{Z}_{k,\text{ideal}} \in A)\right|&\leq \left| {\mathbb{P}}(\mathbf{Z}_{k,n} \in A) - {\mathbb{P}}(\mathbf{Z}_{k,\text{ideal}} \in A)\right| \cr
&=\left| {\mathbb{P}}(\mathbf{X}_{k,n} \in A_n) - {\mathbb{P}}(\mathbf{X}_{k,\text{ideal}} \in A_n)\right| \cr
&\leq \int \left|\pi(\mathbf{x}_{k}\mid k, \mathbf{D}_n) - \varphi(\mathbf{x}_{k}; \widehat{\boldsymbol\mu}_k, \boldsymbol\Sigma_k/n)\right| \, d\mathbf{x}_{k} \longrightarrow 0,\end{aligned}$$ in probability, by , where $A_n$ is the set $A$ after applying the inverse transformation to retrieve the original random variables, and $\varphi(\mathbf{x}_{k}; \widehat{\boldsymbol\mu}_k, \boldsymbol\Sigma_k/n)$ is the density of a normal with mean and variance of $\widehat{\boldsymbol\mu}_k$ and $\boldsymbol\Sigma_k/n$, respectively, evaluated at $\mathbf{x}_{k}$. Note that in the last inequality, we used that $A_n\subseteq \operatorname{\mathbb{R}}^{d_k}$.
2. Use $P_n$ and $P_{\text{ideal}}$ to denote the transition kernels of $\{(K,\mathbf{Z}_K)_n(m): m\in\operatorname{\mathbb{N}}\}$ and $\{(K,\mathbf{Z}_K)_{\text{ideal}}(m): m\in\operatorname{\mathbb{N}}\}$, respectively. These are such that $$\sum_k \int \left|P_n \phi(k, \mathbf{z}_k) - P_{\text{ideal}} \phi(k, \mathbf{z}_k)\right| \, \pi_{K, \mathbf{Z}_K}(k, \mathbf{z}_k \mid \mathbf{D}_n) \, d\mathbf{z}_k \longrightarrow 0 \quad \text{in probability},$$ as $n\longrightarrow\infty$ for all $\phi\in \text{BL}$, where $\text{BL}$ denotes the set of bounded Lipschitz functions.
We have that $$\begin{aligned}
P_{\text{ideal}}((k, \mathbf{z}_k), (k', \mathbf{y}_{k'})) := \bar{\pi}(k') \, \varphi(\mathbf{y}_{k'}; \mathbf{0}, \boldsymbol\Sigma_{k'}),\end{aligned}$$ where we considered for simplicity that the ideal (nonstandardised) RJ is such that $q_{k \mapsto k'}:=\mathcal{N}(\widehat{\boldsymbol\mu}_{k'}, \boldsymbol\Sigma_{k'} / n)$ and $\mathcal{D}_{k \mapsto k'}$ such that $\mathbf{y}_{k'} := \mathbf{u}_{k \mapsto k'}$. The proof is similar for the other cases.
By definition, $$\begin{aligned}
P_{\text{ideal}} \phi(k, \mathbf{z}_k)&:= \sum_{k'}\int \phi(k', \mathbf{y}_{k'}) \, P_{\text{ideal}}((k, \mathbf{z}_k), (k', \mathbf{y}_{k'})) \, d\mathbf{y}_{k'} \cr
&\hspace{1mm}=\sum_{k'}\int \phi(k', \mathbf{y}_{k'}) \, \bar{\pi}(k') \, \varphi(\mathbf{y}_{k'}; \mathbf{0}, \boldsymbol\Sigma_{k'}) \, d\mathbf{y}_{k'},\end{aligned}$$ which is constant with respect to $(k, \mathbf{z}_k)$.
We also have that $$\begin{aligned}
&P_n((k, \mathbf{z}_k), (k', \mathbf{y}_{k'})) := \widehat{\pi}(k'\mid\mathbf{D}_n) \, \varphi(\mathbf{y}_{k'}; \mathbf{0}, \widehat{\boldsymbol\Sigma}_{k'}) \, \alpha((k, \mathbf{z}_k), (k', \mathbf{y}_{k'})) \cr
&\quad + \delta_{(k, \mathbf{z}_k)}(k', \mathbf{y}_{k'}) \sum_{l}\int \left(1 - \alpha((k, \mathbf{z}_k), (l, \mathbf{u}_{k\mapsto k'}))\right) \widehat{\pi}(l\mid\mathbf{D}_n) \, \varphi(\mathbf{u}_{k\mapsto k'}; \mathbf{0}, \widehat{\boldsymbol\Sigma}_{k'}) \, d\mathbf{u}_{k\mapsto k'},\end{aligned}$$ where in this case $$\alpha((k, \mathbf{z}_k), (k', \mathbf{y}_{k'})) = 1 \wedge \frac{\widehat{\pi}(k\mid\mathbf{D}_n) \, \pi_{K, \mathbf{Z}_K}(k',\mathbf{y}_{k'} \mid \mathbf{D}_n) \, \varphi(\mathbf{z}_{k}; \mathbf{0}, \widehat{\boldsymbol\Sigma}_{k})}{\widehat{\pi}(k'\mid\mathbf{D}_n) \, \pi_{K, \mathbf{Z}_K}(k,\mathbf{z}_{k} \mid \mathbf{D}_n) \, \varphi(\mathbf{y}_{k'}; \mathbf{0}, \widehat{\boldsymbol\Sigma}_{k'})}.$$ Therefore, $$\begin{aligned}
P_n \phi(k, \mathbf{z}_k)&:= \sum_{k'}\int \phi(k', \mathbf{y}_{k'}) \, P_n((k, \mathbf{z}_k), (k', \mathbf{y}_{k'})) \, d\mathbf{y}_{k'} \cr
&= \sum_{k'}\int \phi(k', \mathbf{y}_{k'}) \, \widehat{\pi}(k'\mid\mathbf{D}_n) \, \varphi(\mathbf{y}_{k'}; \mathbf{0}, \widehat{\boldsymbol\Sigma}_{k'}) \, \alpha((k, \mathbf{z}_k), (k', \mathbf{y}_{k'})) \, d\mathbf{y}_{k'} \cr
& \qquad + \phi(k, \mathbf{z}_{k}) \sum_{l}\int \left(1 - \alpha((k, \mathbf{z}_k), (l, \mathbf{u}_{k\mapsto k'}))\right) \, \widehat{\pi}(l\mid\mathbf{D}_n) \, \varphi(\mathbf{u}_{k\mapsto k'}; \mathbf{0}, \widehat{\boldsymbol\Sigma}_{k'}) \, d\mathbf{u}_{k\mapsto k'}.\end{aligned}$$ Consequently, $$\begin{aligned}
\label{eqn_proof1_1}
&\sum_k \int \left|P_n \phi(k, \mathbf{z}_k) - P_{\text{ideal}} \phi(k, \mathbf{z}_k)\right| \, \pi_{K, \mathbf{Z}_K}(k, \mathbf{z}_k \mid \mathbf{D}_n) \, d\mathbf{z}_k \cr
&\quad\leq \sum_k \int \left|\sum_{k'}\int \phi(k', \mathbf{y}_{k'})\left(\widehat{\pi}(k'\mid\mathbf{D}_n) \, \varphi(\mathbf{y}_{k'}; \mathbf{0}, \widehat{\boldsymbol\Sigma}_{k'}) \, \alpha((k, \mathbf{z}_k), (k', \mathbf{y}_{k'})) \right.\right. \cr
&\hspace{70mm} \left.\left.- \bar{\pi}(k') \, \varphi(\mathbf{y}_{k'}; \mathbf{0}, \boldsymbol\Sigma_{k'})\right)d\mathbf{y}_{k'}\right| \pi_{K, \mathbf{Z}_K}(k, \mathbf{z}_k \mid \mathbf{D}_n) \, d\mathbf{z}_k \cr
&\qquad + \sum_k \int \left|\phi(k, \mathbf{z}_{k}) \sum_{l}\int \left(1 - \alpha((k, \mathbf{z}_k), (l, \mathbf{u}_{k\mapsto k'}))\right) \widehat{\pi}(l\mid\mathbf{D}_n) \, \varphi(\mathbf{u}_{k\mapsto k'}; \mathbf{0}, \widehat{\boldsymbol\Sigma}_{k'}) \, d\mathbf{u}_{k\mapsto k'}\right| \cr
&\hspace{110mm} \times \pi_{K, \mathbf{Z}_K}(k, \mathbf{z}_k \mid \mathbf{D}_n) \, d\mathbf{z}_k,\end{aligned}$$ using the triangle inequality.
We now show that both terms on the right-hand side (RHS) in converge towards 0 in probability which will allow to conclude by Slutsky’s theorem and monotonicity of probabilities. Firstly, $$\begin{aligned}
\label{eqn_proof1_2}
&\sum_k \int \left|\sum_{k'}\int \phi(k', \mathbf{y}_{k'})\left(\widehat{\pi}(k'\mid\mathbf{D}_n) \, \varphi(\mathbf{y}_{k'}; \mathbf{0}, \widehat{\boldsymbol\Sigma}_{k'}) \, \alpha((k, \mathbf{z}_k), (k', \mathbf{y}_{k'})) \right.\right. \cr
&\hspace{70mm} \left.\left.- \bar{\pi}(k') \, \varphi(\mathbf{y}_{k'}; \mathbf{0}, \boldsymbol\Sigma_{k'})\right)d\mathbf{y}_{k'}\right| \pi_{K, \mathbf{Z}_K}(k, \mathbf{z}_k \mid \mathbf{D}_n) \, d\mathbf{z}_k \cr
&\quad\leq M \sum_{k, k'} \int \left| \widehat{\pi}(k'\mid\mathbf{D}_n) \, \varphi(\mathbf{y}_{k'}; \mathbf{0}, \widehat{\boldsymbol\Sigma}_{k'}) \, \alpha((k, \mathbf{z}_k), (k', \mathbf{y}_{k'})) - \bar{\pi}(k') \, \varphi(\mathbf{y}_{k'}; \mathbf{0}, \boldsymbol\Sigma_{k'}) \right| \cr
& \hspace{110mm} \times \pi_{K, \mathbf{Z}_K}(k, \mathbf{z}_k \mid \mathbf{D}_n) \, d\mathbf{y}_{k'} \, d\mathbf{z}_k \cr
&\quad\leq M \sum_{k, k'} \int \left| \widehat{\pi}(k'\mid\mathbf{D}_n) \, \varphi(\mathbf{y}_{k'}; \mathbf{0}, \widehat{\boldsymbol\Sigma}_{k'}) \, \alpha((k, \mathbf{z}_k), (k', \mathbf{y}_{k'})) - \widehat{\pi}(k'\mid\mathbf{D}_n) \, \varphi(\mathbf{y}_{k'}; \mathbf{0}, \widehat{\boldsymbol\Sigma}_{k'}) \right| \cr
& \hspace{110mm} \times \pi_{K, \mathbf{Z}_K}(k, \mathbf{z}_k \mid \mathbf{D}_n) \, d\mathbf{y}_{k'} \, d\mathbf{z}_k \cr
&\qquad + M \sum_{k, k'} \int \left|\widehat{\pi}(k'\mid\mathbf{D}_n) \, \varphi(\mathbf{y}_{k'}; \mathbf{0}, \widehat{\boldsymbol\Sigma}_{k'}) - \bar{\pi}(k') \, \varphi(\mathbf{y}_{k'}; \mathbf{0}, \boldsymbol\Sigma_{k'}) \right| \, \pi_{K, \mathbf{Z}_K}(k, \mathbf{z}_k \mid \mathbf{D}_n) \, d\mathbf{y}_{k'} \, d\mathbf{z}_k\end{aligned}$$ using Jensen’s inequality and the fact that there exists a positive constant $M$ such that $|\phi|\leq M$ in the first inequality, and the triangle inequality in the second one. Again, we show that each of the last two terms converges in probability towards 0. We start by the second term: $$\begin{aligned}
&\sum_{k, k'} \int \left|\widehat{\pi}(k'\mid\mathbf{D}_n) \, \varphi(\mathbf{y}_{k'}; \mathbf{0}, \widehat{\boldsymbol\Sigma}_{k'}) - \bar{\pi}(k') \, \varphi(\mathbf{y}_{k'}; \mathbf{0}, \boldsymbol\Sigma_{k'}) \right| \, \pi_{K, \mathbf{Z}_K}(k, \mathbf{z}_k \mid \mathbf{D}_n) \, d\mathbf{y}_{k'} \, d\mathbf{z}_k \cr
&\quad \leq \sum_{k'} \int \left|\widehat{\pi}(k'\mid\mathbf{D}_n) \, \varphi(\mathbf{y}_{k'}; \mathbf{0}, \widehat{\boldsymbol\Sigma}_{k'}) - \widehat{\pi}(k'\mid\mathbf{D}_n) \, \varphi(\mathbf{y}_{k'}; \mathbf{0}, \boldsymbol\Sigma_{k'}) \right| \, d\mathbf{y}_{k'} \cr
&\qquad + \sum_{k'} \int \left| \widehat{\pi}(k'\mid\mathbf{D}_n) \, \varphi(\mathbf{y}_{k'}; \mathbf{0}, \boldsymbol\Sigma_{k'}) - \bar{\pi}(k') \, \varphi(\mathbf{y}_{k'}; \mathbf{0}, \boldsymbol\Sigma_{k'}) \right| \, d\mathbf{y}_{k'} \cr
&\quad = \sum_{k'} \widehat{\pi}(k'\mid\mathbf{D}_n) \int \left| \varphi(\mathbf{y}_{k'}; \mathbf{0}, \widehat{\boldsymbol\Sigma}_{k'}) - \varphi(\mathbf{y}_{k'}; \mathbf{0}, \boldsymbol\Sigma_{k'}) \right| \, d\mathbf{y}_{k'} + \sum_{k'} \left|\widehat{\pi}(k'\mid\mathbf{D}_n) - \bar{\pi}(k')\right|.\end{aligned}$$ The second term is seen to converges towards 0 in probability by and Slutsky’s theorem. For the first term, we extract a subsequence $\{n_j: j\in\operatorname{\mathbb{N}}\}$ such that $\widehat{\boldsymbol\Sigma}_{k'}^{n_j}\longrightarrow \boldsymbol\Sigma_{k'}$ almost surely. This implies that for all $\mathbf{y}_{k'}$, $f_{n_j}(\mathbf{y}_{k'}):=\varphi(\mathbf{y}_{k'}; \mathbf{0}, \widehat{\boldsymbol\Sigma}_{k'}^{n_j})\longrightarrow f(\mathbf{y}_{k'}):=\varphi(\mathbf{y}_{k'}; \mathbf{0}, \boldsymbol\Sigma_{k'})$ almost surely, which in turn implies that $\int|f_{n_j} - f|\longrightarrow 0$ (Scheffé’s lemma) almost surely. That allows to show that the first term converges towards 0 in probability.
We now return to the first term at the RHS of the last inequality in . It is equal to (up to the constant $M$) $$\sum_{k, k'} \int \left| \alpha((k, \mathbf{z}_k), (k', \mathbf{y}_{k'})) - 1 \right| \pi_{K, \mathbf{Z}_K}(k, \mathbf{z}_k \mid \mathbf{D}_n) \, \widehat{\pi}(k'\mid\mathbf{D}_n) \, \varphi(\mathbf{y}_{k'}; \mathbf{0}, \widehat{\boldsymbol\Sigma}_{k'}) \, d\mathbf{y}_{k'} \, d\mathbf{z}_k.$$ Define the set $A$ such that on this set $\alpha\leq 1$. On $A^c$, the integral is exactly 0. On $A$, it is equal to $$\begin{aligned}
&\sum_{k, k'} \int_A \left| \pi_{K, \mathbf{Z}_K}(k', \mathbf{y}_{k'} \mid \mathbf{D}_n) \, \widehat{\pi}(k\mid\mathbf{D}_n) \, \varphi(\mathbf{z}_{k}; \mathbf{0}, \widehat{\boldsymbol\Sigma}_{k}) - \pi_{K, \mathbf{Z}_K}(k, \mathbf{z}_k \mid \mathbf{D}_n) \, \widehat{\pi}(k'\mid\mathbf{D}_n) \, \varphi(\mathbf{y}_{k'}; \mathbf{0}, \widehat{\boldsymbol\Sigma}_{k'}) \right| \, d\mathbf{y}_{k'} \, d\mathbf{z}_k \cr
&\leq \sum_{k, k'} \int \left| \pi_{K, \mathbf{Z}_K}(k', \mathbf{y}_{k'} \mid \mathbf{D}_n) \, \widehat{\pi}(k\mid\mathbf{D}_n) \, \varphi(\mathbf{z}_{k}; \mathbf{0}, \widehat{\boldsymbol\Sigma}_{k}) - \pi_{K, \mathbf{Z}_K}(k', \mathbf{y}_{k'} \mid \mathbf{D}_n) \, \pi_{K, \mathbf{Z}_K}(k, \mathbf{z}_{k} \mid \mathbf{D}_n) \right| \, d\mathbf{y}_{k'} \, d\mathbf{z}_k \cr
&+ \sum_{k, k'} \int \left| \pi_{K, \mathbf{Z}_K}(k', \mathbf{y}_{k'} \mid \mathbf{D}_n) \, \pi_{K, \mathbf{Z}_K}(k, \mathbf{z}_{k} \mid \mathbf{D}_n) - \pi_{K, \mathbf{Z}_K}(k, \mathbf{z}_k \mid \mathbf{D}_n) \, \widehat{\pi}(k'\mid\mathbf{D}_n) \, \varphi(\mathbf{y}_{k'}; \mathbf{0}, \widehat{\boldsymbol\Sigma}_{k'})\right| \, d\mathbf{y}_{k'} \, d\mathbf{z}_k,\end{aligned}$$ using the definition of $\alpha$ and next the triangle inequality and that $A\subseteq \operatorname{\mathbb{R}}^{d_{k'}}\times \operatorname{\mathbb{R}}^{d_k}$. We show that the first term converges towards 0 in probability. The proof is similar for the second one. We have that $$\begin{aligned}
&\sum_{k, k'} \int \left| \pi_{K, \mathbf{Z}_K}(k', \mathbf{y}_{k'} \mid \mathbf{D}_n) \, \widehat{\pi}(k\mid\mathbf{D}_n) \, \varphi(\mathbf{z}_{k}; \mathbf{0}, \widehat{\boldsymbol\Sigma}_{k}) - \pi_{K, \mathbf{Z}_K}(k', \mathbf{y}_{k'} \mid \mathbf{D}_n) \, \pi_{K, \mathbf{Z}_K}(k, \mathbf{z}_{k} \mid \mathbf{D}_n) \right| \, d\mathbf{y}_{k'} \, d\mathbf{z}_k \cr
& \quad =\sum_k \int \left| \widehat{\pi}(k\mid\mathbf{D}_n) \, \varphi(\mathbf{z}_{k}; \mathbf{0}, \widehat{\boldsymbol\Sigma}_{k}) - \pi_{K, \mathbf{Z}_K}(k, \mathbf{z}_{k} \mid \mathbf{D}_n) \right| \, d\mathbf{z}_k \cr
& \quad \leq \sum_k \int \left| \widehat{\pi}(k\mid\mathbf{D}_n) \, \varphi(\mathbf{z}_{k}; \mathbf{0}, \widehat{\boldsymbol\Sigma}_{k}) - \pi(k\mid\mathbf{D}_n) \, \varphi(\mathbf{z}_{k}; \mathbf{0}, \widehat{\boldsymbol\Sigma}_{k}) \right| \, d\mathbf{z}_k \cr
&\qquad + \sum_k \int \left| \pi(k\mid\mathbf{D}_n) \, \varphi(\mathbf{z}_{k}; \mathbf{0}, \widehat{\boldsymbol\Sigma}_{k}) - \pi_{K, \mathbf{Z}_K}(k, \mathbf{z}_{k} \mid \mathbf{D}_n) \right| \, d\mathbf{z}_k,\end{aligned}$$ using again the triangle inequality. The first term is equal to $\sum_k|\widehat{\pi}(k\mid\mathbf{D}_n) - \pi(k\mid\mathbf{D}_n)|$, which converges in probability towards 0 by and Slutsky’s theorem. For the second term we first use that $\int|\varphi(\mathbf{z}_{k}; \mathbf{0}, \widehat{\boldsymbol\Sigma}_{k})-\varphi(\mathbf{z}_{k}; \mathbf{0}, \boldsymbol\Sigma_{k})|\, d\mathbf{z}_{k}$ converges in probability towards 0 as explained previously. Therefore, we deal with a a sum of integrals $\int |\varphi(\mathbf{x}_{k}; \widehat{\boldsymbol\mu}_k, \boldsymbol\Sigma_{k}/n) - \pi(\mathbf{x}_{k} \mid k, \mathbf{D}_n) | \, d\mathbf{x}_{k}$ after a change of variable. This is seen to converge towards 0 in probability by .
The second term on the RHS in converges towards 0 in probability following the same arguments. The second condition is thus verified.
3. The transition kernel $P_{\text{ideal}}$ is such that $P_{\text{ideal}} \phi(k, \mathbf{z}_k)$ is continuous in $(k, \mathbf{z}_k)$ for any $\phi\in\mathcal{C}_b$ (the set of continuous bounded functions).
In our case, it has been seen that $P_{\text{ideal}} \phi(k, \mathbf{z}_k)$ is constant with respect to $(k, \mathbf{z}_k)$. This concludes the proof.
\[lemma\_conv\_g\]
Under , $|g(k,k') - g_{\text{ideal}}(k, k')|$ converges in probability towards 0 as $n\longrightarrow \infty$, for $g(k,k')$ and $g_{\text{ideal}}(k, k')$ defined in and for all $k,k'$.
We consider two cases.
1. $\bar{\pi}(k)>0$. In this case, using Slutsky’s theorem, it suffices to show that $$\begin{aligned}
\label{eqn1_lemma1}
\left|\frac{\widehat{\pi}(k'\mid \mathbf{D}_n)}{\widehat{\pi}(k\mid \mathbf{D}_n)} - \frac{\bar{\pi}(k')}{\bar{\pi}(k)} \right| \longrightarrow 0,
\end{aligned}$$ in probability for any $k,k'$ as $h$ is continuous and $c_k, c_k^{\text{ideal}}$ are finite sums of $h$ applied to ratios like those in . holds as a result of and Slutsky’s theorem.
2. $\bar{\pi}(k)=0$. Consider that $\widehat{\pi}(k\mid \mathbf{D}_n)>0$ for all $k$, for finite $n$. This is usually the case in practice. We simply define $g_{\text{ideal}}(k, k')$ as the limit (in probability) of $$g(k,k'):= \frac{h\left(\frac{\widehat{\pi}(k'\mid \mathbf{D}_n)}{\widehat{\pi}(k\mid \mathbf{D}_n)}\right)}{\sum_{l \in \mathcal{N}(k)} h\left(\frac{\widehat{\pi}(l\mid \mathbf{D}_n)}{\widehat{\pi}(k\mid \mathbf{D}_n)}\right)}.$$
We prove the result for the case $T_{k, k'} := 1$ (without annealing intermediate distributions), to simplify; the general case is proved similarly. We prove that the probability to reach the state $\{k'\}\times \{\mathbf{y}_{k'}\in A_{k'}\}$, from $\{k\}\times \{\mathbf{x}_{k}\in A_{k}\}$, is equal to the probability of the reverse move. We denote by $P$ the Markov kernel. We thus prove that $$\begin{aligned}
&\int_{\{\mathbf{x}_{k}\in A_{k}\}} \pi(k, \mathbf{x}_k \mid \mathbf{D}_n)\int_{\{\mathbf{y}_{k'}\in A_{k'}\}} P((k, \mathbf{x}_{k}), (k', \mathbf{y}_{k'})) \, d\mathbf{y}_{k'} \, d\mathbf{x}_{k} \cr
&\qquad = \int_{\{\mathbf{y}_{k'}\in A_{k'}\}} \pi(k', \mathbf{y}_{k'} \mid \mathbf{D}_n)\int_{\{\mathbf{x}_{k}\in A_{k}\}} P((k', \mathbf{y}_{k'}), (k, \mathbf{x}_{k})) \, d\mathbf{x}_{k} \, d\mathbf{y}_{k'}.
\end{aligned}$$ Note that we abused notation by denoting the measures associated with the kernel $d\mathbf{y}_{k'}$ or $d\mathbf{x}_{k}$ because a group of vectors $\mathbf{u}_{l\mapsto s}^{(j)}$ are used in the transition and they are not of the same dimension as $\mathbf{y}_{k'}$ and $\mathbf{x}_{k}$. The vector $\mathbf{u}_{l\mapsto s}^{(j)} := \mathbf{u}_{l\mapsto s}^{(0, j)}$ is used here to denote the $j$-th auxiliary vector that makes the $j$-th proposal $\mathbf{y}_{s}^{(j)} := \mathbf{y}_{s}^{(0, j)}$, $j\in\{1,\ldots,N\}$.
We now introduce notation to improve readability. We define three joint densities that are used to enhance the approximations when Step 2.(i) is applied to generate the proposal: $$\begin{aligned}
\bar{q}_{k \mapsto \mathcal{N}(k)\setminus\{k'\}} (\bar{\mathbf{u}}_{k \mapsto \mathcal{N}(k)\setminus\{k'\}}) &:= \prod_{l \in \mathcal{N}(k) \setminus\{k'\}} \prod_{j=1}^N q_{k\mapsto l}(\mathbf{u}_{k\mapsto l}^{(j)}), \quad \bar{q}_{k \mapsto k'}(\bar{\mathbf{u}}_{k \mapsto k'}) := \prod_{j=1}^N q_{k\mapsto k'}(\mathbf{u}_{k\mapsto k'}^{(j)}), \cr
\bar{\bar{q}}_{k' \mapsto \mathcal{N}(k')\setminus\{k\}}(\bar{\bar{\mathbf{u}}}_{k' \mapsto \mathcal{N}(k')\setminus\{k\}}) &:= \prod_{l\in \mathcal{N}(k') \setminus \{k\}} q_{k'\mapsto l}(\mathbf{u}_{k'\mapsto l}^{(j^*)}) \prod_{j=1 (j\neq j^*)}^N q_{l\mapsto k'}(\mathbf{u}_{l\mapsto k'}^{(j)}).
\end{aligned}$$ The densities $\bar{q}_{k \mapsto \mathcal{N}(k)\setminus\{k'\}}$ and $\bar{q}_{k \mapsto k'}$ together represent the joint density of the random variables generated in the first part of Step 2.(i). The density $\bar{\bar{q}}_{k' \mapsto \mathcal{N}(k')\setminus\{k\}}$ represents the joint density of the random variables generated in the second part of Step 2.(i).
We now define three joint densities that are used to enhance the approximations when Step 2.(ii) is applied to generate the proposal: $$\begin{aligned}
\tilde{q}_{k \mapsto \mathcal{N}(k) \setminus \{k'\}} (\tilde{\mathbf{u}}_{k \mapsto \mathcal{N}(k) \setminus \{k'\}}) &:= \prod_{l \in \mathcal{N}(k) \setminus \{k'\}} q_{k\mapsto l}(\mathbf{u}_{k\mapsto l}^{(j^*)}) \prod_{j=1 (j\neq j^*)}^N q_{l\mapsto k}(\mathbf{u}_{l\mapsto k}^{(j)}), \cr
\tilde{q}_{k \mapsto k'} (\tilde{\mathbf{u}}_{k \mapsto k'}) &:= q_{k\mapsto k'}(\mathbf{u}_{k\mapsto k'}^{(j^*)}) \prod_{j=1 (j\neq j^*)}^N q_{k' \mapsto k}(\mathbf{u}_{k' \mapsto k}^{(j)}), \cr
\tilde{\tilde{q}}_{k' \mapsto \mathcal{N}(k') \setminus \{k\}}(\tilde{\tilde{\mathbf{u}}}_{k' \mapsto \mathcal{N}(k') \setminus \{k\}})&:= \prod_{l \in N(k') \setminus \{k\}} \prod_{j=1}^N q_{k'\mapsto l}(\mathbf{u}_{k'\mapsto l}^{(j)}).
\end{aligned}$$ The densities $\tilde{q}_{k \mapsto \mathcal{N}(k) \setminus \{k'\}}$ and $ \tilde{q}_{k \mapsto k'} $ together represent the joint density of the random variables generated in the first part of Step 2.(ii). The density $\tilde{\tilde{q}}_{k' \mapsto \mathcal{N}(k') \setminus \{k\}}$ represents the joint density of the random variables generated in the second part of Step 2.(ii).
We have that $$\begin{aligned}
P((k, \mathbf{x}_{k}), (k', \mathbf{y}_{k'}))&:= \frac{1}{2} \, \bar{q}_{k \mapsto \mathcal{N}(k)\setminus\{k'\}} (\bar{\mathbf{u}}_{k \mapsto \mathcal{N}(k)\setminus\{k'\}}) \, \bar{q}_{k \mapsto k'}(\bar{\mathbf{u}}_{k \mapsto k'}) \, g_{\text{imp.}}(k, k', \mathbf{x}_k, \bar{\mathbf{u}}_{k \mapsto k'}, \bar{\mathbf{u}}_{k \mapsto \mathcal{N}(k)\setminus\{k'\}}) \cr
&\hspace{-30mm} \times \frac{r_{\text{RJ}}((k,\mathbf{x}_{k}),(k',\mathbf{y}_{k'}))}{N \bar{r}(k, k', \mathbf{x}_{k}, \bar{\mathbf{u}}_{k \mapsto k'})} \, \bar{\bar{q}}_{k' \mapsto \mathcal{N}(k')\setminus\{k\}}(\bar{\bar{\mathbf{u}}}_{k' \mapsto \mathcal{N}(k')\setminus\{k\}}) \left(1 \wedge \frac{g_{\text{imp.}}(k', k, \mathbf{x}_k, \bar{\mathbf{u}}_{k \mapsto k'}, \bar{\bar{\mathbf{u}}}_{k' \mapsto \mathcal{N}(k')\setminus\{k\}})}{g_{\text{imp.}}(k, k', \mathbf{x}_k, \bar{\mathbf{u}}_{k \mapsto k'}, \bar{\mathbf{u}}_{k \mapsto \mathcal{N}(k)\setminus\{k'\}})} \, \bar{r}(k, k', \mathbf{x}_{k}, \bar{\mathbf{u}}_{k \mapsto k'}) \right) \cr
& + \frac{1}{2} \, \tilde{q}_{k \mapsto \mathcal{N}(k) \setminus \{k'\}} (\tilde{\mathbf{u}}_{k \mapsto \mathcal{N}(k) \setminus \{k'\}}) \, \tilde{q}_{k \mapsto k'} (\tilde{\mathbf{u}}_{k \mapsto k'}) \, g_{\text{imp.}}(k, k', \mathbf{x}_k, \tilde{\mathbf{u}}_{k \mapsto k'}, \tilde{\mathbf{u}}_{k \mapsto \mathcal{N}(k) \setminus \{k'\}}) \, \frac{1}{N} \cr
&\times \tilde{\tilde{q}}_{k' \mapsto \mathcal{N}(k') \setminus \{k\}}(\tilde{\tilde{\mathbf{u}}}_{k' \mapsto \mathcal{N}(k') \setminus \{k\}}) \left(1 \wedge \frac{g_{\text{imp.}}(k', k, \mathbf{x}_k, \tilde{\mathbf{u}}_{k \mapsto k'}, \tilde{\tilde{\mathbf{u}}}_{k' \mapsto \mathcal{N}(k') \setminus \{k\}})}{g_{\text{imp.}}(k, k', \mathbf{x}_k, \tilde{\mathbf{u}}_{k \mapsto k'}, \tilde{\mathbf{u}}_{k \mapsto \mathcal{N}(k) \setminus \{k'\}})} \, \bar{r}(k', k, \mathbf{x}_k, \tilde{\mathbf{u}}_{k \mapsto k'})^{-1} \right) \cr
& \quad + \delta_{(k', \mathbf{y}_{k'})}(k, \mathbf{x}_{k}) \, {\mathbb{P}}(\text{rejection}\mid (k, \mathbf{x}_{k})),
\end{aligned}$$ where ${\mathbb{P}}(\text{rejection}\mid (k, \mathbf{x}_{k}))$ is the rejection probability given that the current state is $(k, \mathbf{x}_{k})$. Note that we considered that in Step 2.(ii) we set uniformly at random the index of the proposal. This is however in practice not important (which is why in we set it to be 1) because of the form of the acceptance ratio. Note also that we use the notation $\bar{r}(k, k', \mathbf{x}_{k}, \bar{\mathbf{u}}_{k \mapsto k'})$ to be clear about which variables is involved.
The probability of reaching the state $\{k'\}\times \{\mathbf{y}_{k'}\in A_{k'}\}$, from $\{k\}\times \{\mathbf{x}_{k}\in A_{k}\}$, is thus given by $$\begin{aligned}
\label{eqn1_proof_prop}
&\int_{\{\mathbf{x}_{k}\in A_{k}\}} \pi(k, \mathbf{x}_k \mid \mathbf{D}_n)\int_{\{\mathbf{y}_{k'}\in A_{k'}\}}\frac{1}{2} \, \bar{q}_{k \mapsto \mathcal{N}(k)\setminus\{k'\}} (\bar{\mathbf{u}}_{k \mapsto \mathcal{N}(k)\setminus\{k'\}}) \, \bar{q}_{k \mapsto k'}(\bar{\mathbf{u}}_{k \mapsto k'}) \, g_{\text{imp.}}(k, k', \mathbf{x}_k, \bar{\mathbf{u}}_{k \mapsto k'}, \bar{\mathbf{u}}_{k \mapsto \mathcal{N}(k)\setminus\{k'\}}) \cr
& \times \frac{r_{\text{RJ}}((k,\mathbf{x}_{k}),(k',\mathbf{y}_{k'}))}{N \bar{r}(k, k', \mathbf{x}_{k}, \bar{\mathbf{u}}_{k \mapsto k'})} \, \bar{\bar{q}}_{k' \mapsto \mathcal{N}(k')\setminus\{k\}}(\bar{\bar{\mathbf{u}}}_{k' \mapsto \mathcal{N}(k')\setminus\{k\}}) \cr
&\times \left(1 \wedge \frac{g_{\text{imp.}}(k', k, \mathbf{x}_k, \bar{\mathbf{u}}_{k \mapsto k'}, \bar{\bar{\mathbf{u}}}_{k' \mapsto \mathcal{N}(k')\setminus\{k\}})}{g_{\text{imp.}}(k, k', \mathbf{x}_k, \bar{\mathbf{u}}_{k \mapsto k'}, \bar{\mathbf{u}}_{k \mapsto \mathcal{N}(k)\setminus\{k'\}})} \, \bar{r}(k, k', \mathbf{x}_{k}, \bar{\mathbf{u}}_{k \mapsto k'}) \right) \, d\bar{\bar{\mathbf{u}}}_{k' \mapsto \mathcal{N}(k')\setminus\{k\}} \, d\bar{\mathbf{u}}_{k \mapsto \mathcal{N}(k)\setminus\{k'\}} \, d\bar{\mathbf{u}}_{k \mapsto k'} \, d\mathbf{x}_{k} \cr
&+\int_{\{\mathbf{x}_{k}\in A_{k}\}} \pi(k, \mathbf{x}_k \mid \mathbf{D}_n)\int_{\{\mathbf{y}_{k'}\in A_{k'}\}}\frac{1}{2} \, \tilde{q}_{k \mapsto \mathcal{N}(k) \setminus \{k'\}} (\tilde{\mathbf{u}}_{k \mapsto \mathcal{N}(k) \setminus \{k'\}}) \, \tilde{q}_{k \mapsto k'} (\tilde{\mathbf{u}}_{k \mapsto k'}) \, g_{\text{imp.}}(k, k', \mathbf{x}_k, \tilde{\mathbf{u}}_{k \mapsto k'}, \tilde{\mathbf{u}}_{k \mapsto \mathcal{N}(k) \setminus \{k'\}}) \cr
& \times \frac{1}{N} \, \tilde{\tilde{q}}_{k' \mapsto \mathcal{N}(k') \setminus \{k\}}(\tilde{\tilde{\mathbf{u}}}_{k' \mapsto \mathcal{N}(k') \setminus \{k\}}) \cr
&\times \left(1 \wedge \frac{g_{\text{imp.}}(k', k, \mathbf{x}_k, \tilde{\mathbf{u}}_{k \mapsto k'}, \tilde{\tilde{\mathbf{u}}}_{k' \mapsto \mathcal{N}(k') \setminus \{k\}})}{g_{\text{imp.}}(k, k', \mathbf{x}_k, \tilde{\mathbf{u}}_{k \mapsto k'}, \tilde{\mathbf{u}}_{k \mapsto \mathcal{N}(k) \setminus \{k'\}})} \, \bar{r}(k', k, \mathbf{x}_k, \tilde{\mathbf{u}}_{k \mapsto k'})^{-1} \right) \, d\tilde{\tilde{\mathbf{u}}}_{k' \mapsto \mathcal{N}(k') \setminus \{k\}} \, d\tilde{\mathbf{u}}_{k \mapsto \mathcal{N}(k) \setminus \{k'\}} \, d\tilde{\mathbf{u}}_{k \mapsto k'} \, d\mathbf{x}_{k} \cr
&+\int_{\{\mathbf{x}_{k}\in A_{k}\}} \pi(k, \mathbf{x}_k \mid \mathbf{D}_n)\int_{\{\mathbf{y}_{k'}\in A_{k'}\}} \delta_{(k', \mathbf{y}_{k'})}(k, \mathbf{x}_{k}) \, {\mathbb{P}}(\text{rejection}\mid (k, \mathbf{x}_{k})) \, d\mathbf{y}_{k'} \, d\mathbf{x}_{k}.
\end{aligned}$$ We now prove that the first part can be rewritten as that corresponding to Step 2.(ii) for the reverse move; the second part corresponds instead to Step 2.(i), and the last term to the probability of rejecting from $(k',\mathbf{y}_{k'})$.
The first part can be rewritten as $$\begin{aligned}
&\int_{\{\mathbf{x}_{k}\in A_{k}\} \times \{\mathbf{y}_{k'}\in A_{k'}\}} \pi(k', \mathbf{y}_{k'} \mid \mathbf{D}_n) \, |J_{\mathcal{D}_{k\mapsto k'}}(\mathbf{x}_{k}, \mathbf{u}_{k\mapsto k'}^{(j^*)})| \, \frac{1}{2} \, \tilde{q}_{k' \mapsto \mathcal{N}(k') \setminus \{k\}} (\tilde{\mathbf{u}}_{k' \mapsto \mathcal{N}(k') \setminus \{k\}}) \, \tilde{q}_{k' \mapsto k} (\tilde{\mathbf{u}}_{k' \mapsto k}) \cr
&\times g_{\text{imp.}}(k', k, \mathbf{x}_k, \bar{\mathbf{u}}_{k \mapsto k'}, \bar{\bar{\mathbf{u}}}_{k' \mapsto \mathcal{N}(k')\setminus\{k\}}) \frac{1}{N} \, \tilde{\tilde{q}}_{k \mapsto \mathcal{N}(k) \setminus \{k'\}}(\tilde{\tilde{\mathbf{u}}}_{k \mapsto \mathcal{N}(k) \setminus \{k'\}}) \cr
& \times \left(1 \wedge \frac{g_{\text{imp.}}(k, k', \mathbf{x}_k, \bar{\mathbf{u}}_{k \mapsto k'}, \bar{\mathbf{u}}_{k \mapsto \mathcal{N}(k)\setminus\{k'\}})}{g_{\text{imp.}}(k', k, \mathbf{x}_k, \bar{\mathbf{u}}_{k \mapsto k'}, \bar{\bar{\mathbf{u}}}_{k' \mapsto \mathcal{N}(k')\setminus\{k\}})} \, \bar{r}(k, k', \mathbf{x}_{k}, \bar{\mathbf{u}}_{k \mapsto k'})^{-1} \right) \, d\bar{\bar{\mathbf{u}}}_{k' \mapsto \mathcal{N}(k')\setminus\{k\}} \, d\bar{\mathbf{u}}_{k \mapsto \mathcal{N}(k)\setminus\{k'\}} \, d\bar{\mathbf{u}}_{k \mapsto k'} \, d\mathbf{x}_{k},
\end{aligned}$$ given that $$\begin{aligned}
r_{\text{RJ}}((k,\mathbf{x}_{k}),(k',\mathbf{y}_{k'})) &:= \frac{\pi(k', \mathbf{y}_{k'} \mid \mathbf{D}_n) \, q_{k'\mapsto k}(\mathbf{u}_{k'\mapsto k}^{(j^*)})}{\pi(k, \mathbf{x}_k \mid \mathbf{D}_n) \, q_{k\mapsto k'}(\mathbf{u}_{k\mapsto k'}^{(j^*)}) \, |J_{\mathcal{D}_{k\mapsto k'}}(\mathbf{x}_{k}, \mathbf{u}_{k\mapsto k'}^{(j^*)})|^{-1}}, \cr
\bar{q}_{k \mapsto \mathcal{N}(k)\setminus\{k'\}} (\bar{\mathbf{u}}_{k \mapsto \mathcal{N}(k)\setminus\{k'\}}) &\hspace{1mm}= \tilde{\tilde{q}}_{k \mapsto \mathcal{N}(k) \setminus \{k'\}}(\tilde{\tilde{\mathbf{u}}}_{k \mapsto \mathcal{N}(k) \setminus \{k'\}}) \cr
\bar{q}_{k \mapsto k'}(\bar{\mathbf{u}}_{k \mapsto k'}) \, \frac{q_{k'\mapsto k}(\mathbf{u}_{k'\mapsto k}^{(j^*)})}{q_{k\mapsto k'}(\mathbf{u}_{k\mapsto k'}^{(j^*)})} &\hspace{1mm}= \tilde{q}_{k' \mapsto k} (\tilde{\mathbf{u}}_{k' \mapsto k}) \cr
\bar{\bar{q}}_{k' \mapsto \mathcal{N}(k')\setminus\{k\}}(\bar{\bar{\mathbf{u}}}_{k' \mapsto \mathcal{N}(k')\setminus\{k\}}) &\hspace{1mm}= \tilde{q}_{k' \mapsto \mathcal{N}(k') \setminus \{k\}} (\tilde{\mathbf{u}}_{k' \mapsto \mathcal{N}(k') \setminus \{k\}}).
\end{aligned}$$ Therefore, the first term can be rewritten as $$\begin{aligned}
&\int_{\{\mathbf{y}_{k'}\in A_{k'}\}} \pi(k', \mathbf{y}_{k'} \mid \mathbf{D}_n)\int_{\{\mathbf{x}_{k}\in A_{k}\}} \, \frac{1}{2} \, \tilde{q}_{k' \mapsto \mathcal{N}(k') \setminus \{k\}} (\tilde{\mathbf{u}}_{k' \mapsto \mathcal{N}(k') \setminus \{k\}}) \, \tilde{q}_{k' \mapsto k} (\tilde{\mathbf{u}}_{k' \mapsto k}) \cr
&\times g_{\text{imp.}}(k', k, \mathbf{y}_{k'}, \tilde{\mathbf{u}}_{k' \mapsto k}, \tilde{\mathbf{u}}_{k' \mapsto \mathcal{N}(k') \setminus \{k\}}) \frac{1}{N} \, \tilde{\tilde{q}}_{k \mapsto \mathcal{N}(k) \setminus \{k'\}}(\tilde{\tilde{\mathbf{u}}}_{k \mapsto \mathcal{N}(k) \setminus \{k'\}}) \cr
& \times \left(1 \wedge \frac{g_{\text{imp.}}(k, k', \mathbf{y}_{k'}, \tilde{\mathbf{u}}_{k' \mapsto k}, \tilde{\tilde{\mathbf{u}}}_{k \mapsto \mathcal{N}(k) \setminus \{k'\}})}{g_{\text{imp.}}(k', k, \mathbf{y}_{k'}, \tilde{\mathbf{u}}_{k' \mapsto k}, \tilde{\mathbf{u}}_{k' \mapsto \mathcal{N}(k') \setminus \{k\}})} \, \bar{r}(k, k', \mathbf{y}_{k'}, \tilde{\mathbf{u}}_{k' \mapsto k})^{-1} \right) \, d\tilde{\tilde{\mathbf{u}}}_{k \mapsto \mathcal{N}(k) \setminus \{k'\}} \, d\tilde{\mathbf{u}}_{k' \mapsto \mathcal{N}(k') \setminus \{k\}} \, d\tilde{\mathbf{u}}_{k' \mapsto k} \, d\mathbf{y}_{k'} ,
\end{aligned}$$ after the change of variable $\mathcal{D}_{k\mapsto k'}(\mathbf{x}_{k}, \mathbf{u}_{k\mapsto k'}^{(j^*)})=(\mathbf{y}_{k'}, \mathbf{u}_{k'\mapsto k}^{(j^*)})$.
The analysis of the second part in uses the same arguments. Finally, the third part in can be rewritten as $$\int_{\{\mathbf{y}_{k'}\in A_{k'}\}} \pi(k', \mathbf{y}_{k'} \mid \mathbf{D}_n) \int_{\{\mathbf{x}_{k}\in A_{k}\}} \delta_{(k, \mathbf{x}_{k})}(k', \mathbf{y}_{k'}) \, {\mathbb{P}}(\text{rejection}\mid (k', \mathbf{y}_{k'}) \, d\mathbf{x}_{k} \, d\mathbf{y}_{k'},$$ which concludes the proof.
Supplementary material {#sec_supp_mat}
======================
We present in all the details to compute estimates for the normal linear regression model. These are followed in by the required quantities to implement the MCMC algorithms for the robust linear model. In , we also prove that the noninformative prior used does lead to a consistent variable selection procedure.
Normal linear regression {#sec_normal_reg}
------------------------
We present in this section a result giving the precise form of the joint posterior density for the normal linear regression model. But, beforehand, we need to introduce notation. We define $\gamma_1, \ldots, \gamma_n \in \operatorname{\mathbb{R}}$ to be $n$ data points from the dependent variable. We denote the full design matrix containing $n$ observations from all covariates by $\mathbf{C} \in \operatorname{\mathbb{R}}^{n \times p}$, where $p$ is a positive integer. For simplicity, we refer to the first column of $\mathbf{C}$ as the *first* covariate even if, as usual, it is a column of $1$’s. The design matrix associated with Model $k$ whose columns form a subset of $\mathbf{C}$ is denoted by $\mathbf{C}_k$, with lines denoted by $\mathbf{c}_{i,k}^T$. We use $d_k$ to denote the number of covariates in Model $k$; we therefore slightly abuse notation given that the number of parameters for Model $k$ is $d_k+1$ (one regression coefficient per covariate plus the scale parameter of the error term).
As typically done in linear regression, we assume that the covariates are fixed and known; the random quantities are $\gamma_1,\ldots,\gamma_n$ and the parameters. The former are random through random errors $\epsilon_{1,K}, \ldots, \epsilon_{n, K} \in \operatorname{\mathbb{R}}$ and models as follows: $$\gamma_i = \mathbf{c}_{i, K}^T \, \boldsymbol\beta_K + \epsilon_{i, K}, \quad i = 1, \ldots, n, \quad K \in \mathcal{K},$$ where $\boldsymbol\beta_K$ is the random vector containing the regression coefficients of Model $K$. We finally assume that $\epsilon_{1,K}, \ldots, \epsilon_{n, K}$ and $\boldsymbol\beta_K$ are $n+1$ conditionally independent random variables given $(K, \sigma_K)$, with $\sigma_K>0$ being the scale parameter of the errors of Model $K$. The conditional density of $\epsilon_{i, K}$ is given by $$\epsilon_{i, K} \mid K, \sigma_K, \boldsymbol\beta_K \stackrel{d}{=} \epsilon_{i, K} \mid K, \sigma_K \stackrel{d}{\sim} (1 / \sigma_K) f(\epsilon_{i, K} / \sigma_K), \quad i = 1, \ldots, n.$$
The precise form of the posterior density of $K, \boldsymbol\beta_K, \sigma_K$ given $\boldsymbol\gamma_n := (\gamma_1, \ldots, \gamma_n)^T$ is given in .
\[prop\_posterior\_normal\] If $f := \mathcal{N}(0, 1)$ and $\pi(\boldsymbol\beta_k, \sigma_k \mid k) \propto 1 / \sigma_k$, then $$\begin{aligned}
\label{eqn_post_k_reg}
\pi(k \mid \boldsymbol\gamma_n) \propto \pi(k) \, \frac{\Gamma((n - d_k) / 2) \, \pi^{d_k / 2}}{\|\boldsymbol\gamma_n - \widehat{\boldsymbol\gamma}_k\|_2^{n - d_k} \, |\mathbf{C}_k^T \mathbf{C}_k|^{1/2}},
\end{aligned}$$ $$\begin{aligned}
\pi(\sigma_k \mid k, \boldsymbol\gamma_n) = \frac{2^{1-\frac{n-d_k}{2}} \|\boldsymbol\gamma_n - \widehat{\boldsymbol\gamma}_k\|_2^{n - d_k}}{\Gamma((n-d_k)/2) \, \sigma_k^{n-d_k+1}} \, \exp\left\{-\frac{1}{2\sigma_k^2}\, \|\boldsymbol\gamma_n - \widehat{\boldsymbol\gamma}_k\|_2^2\right\},
\end{aligned}$$ and $$\boldsymbol\beta_K \mid K, \sigma_K, \boldsymbol\gamma_n \sim \mathcal{N}((\mathbf{C}_K^T \mathbf{C}_K)^{-1} \mathbf{C}_K^T \boldsymbol\gamma_n, \sigma_K^2 (\mathbf{C}_K^T \mathbf{C}_K)^{-1}),$$ where $\widehat{\boldsymbol\gamma}_k := \mathbf{C}_k (\mathbf{C}_k^T \mathbf{C}_k)^{-1} \mathbf{C}_k^T \boldsymbol\gamma_n$ and $\| \cdot \|_2$ is the Euclidian norm. Note that the normalisation constant of $\pi(k \mid \boldsymbol\gamma_n)$ is the sum over $k$ of the expression on the RHS in .
Note that $\sigma_K^2 \mid K, \boldsymbol\gamma_n$ has an inverse-gamma distribution with shape and rate parameters given by $(n - d_k) / 2$ and $\|\boldsymbol\gamma_n - \widehat{\boldsymbol\gamma}_k\|_2^2 / 2$, respectively.
The proof relies essentially on straightforward calculations. We have $$\begin{aligned}
\pi(k, \boldsymbol\beta_k, \sigma_k \mid \boldsymbol\gamma_n) &\propto \pi(k) \, \frac{1}{\sigma_k} \, \prod_{i=1}^n \frac{1}{\sqrt{2 \pi} \, \sigma_k} \exp\left(- \frac{1}{2 \sigma_k} \, (\gamma_i - \mathbf{c}_{i, k}^T \, \boldsymbol\beta_k)^2\right) \cr
&\propto \pi(k) \, \frac{1}{\sigma_k^{n + 1}} \, \exp\left(- \frac{1}{2 \sigma_k} \sum_{i=1}^n (\gamma_i - \mathbf{c}_{i, k}^T \, \boldsymbol\beta_k)^2\right).
\end{aligned}$$ In [@gagnon2018supp], it is proved that $$\begin{aligned}
\sum_{i=1}^n (\gamma_i - \mathbf{c}_{i, k}^T \, \boldsymbol\beta_k)^2 = (\boldsymbol\beta_k - \widehat{\boldsymbol\beta}_k)^T \mathbf{C}_K^T \mathbf{C}_K (\boldsymbol\beta_k - \widehat{\boldsymbol\beta}_k) + \|\boldsymbol\gamma_n - \widehat{\boldsymbol\gamma}_k\|_2^2,
\end{aligned}$$ where $\widehat{\boldsymbol\beta}_k := (\mathbf{C}_k^T \mathbf{C}_k)^{-1} \mathbf{C}_k^T \boldsymbol\gamma_n$. Multiplying and dividing by the appropriate terms yields $$\begin{aligned}
\pi(k, \boldsymbol\beta_k, \sigma_k \mid \boldsymbol\gamma_n) &\propto \pi(k) \, \frac{\Gamma((n - d_k) / 2) \, \pi^{d_k / 2}}{\|\boldsymbol\gamma_n - \widehat{\boldsymbol\gamma}_k\|_2^{n - d_k} \, |\mathbf{C}_k^T \mathbf{C}_k|^{1/2}} \cr
&\qquad \times \frac{2^{1-\frac{n-d_k}{2}} \|\boldsymbol\gamma_n - \widehat{\boldsymbol\gamma}_k\|_2^{n - d_k}}{\Gamma((n-d_k)/2) \, \sigma_k^{n-d_k+1}} \, \exp\left\{-\frac{1}{2\sigma_k^2}\, \|\boldsymbol\gamma_n - \widehat{\boldsymbol\gamma}_k\|_2^2\right\} \cr
&\qquad \times \frac{ |\mathbf{C}_k^T \mathbf{C}_k|^{1/2}}{(2 \pi)^{d_k / 2} \sigma_k^{d_k}} \exp\left(-\frac{1}{2\sigma_k^2} (\boldsymbol\beta_k - \widehat{\boldsymbol\beta}_k)^T \mathbf{C}_K^T \mathbf{C}_K (\boldsymbol\beta_k - \widehat{\boldsymbol\beta}_k)\right),
\end{aligned}$$ which concludes the proof.
Relying on improper priors such as $\pi(\boldsymbol\beta_k, \sigma_k \mid k) = c_k / \sigma_k$ may lead to inconsistencies in model selection (see, e.g., [@casella2009consistency]). When this problem happens, the phenomenon is referred to as the Jeffreys-Lindley paradox ([@lindley1957paradox] and [@jeffreys1967prob]) in the literature. This paradox arises, for instance, when one select different constants $c_k$ in different models so as to yield desired conclusions. We now show that the Jeffreys-Lindley paradox does not arise in the normal linear regression framework described above. It is thus expected to not arise either under the robust LPTN distribution, given the similarity of the latter with the normal except in the tails leading to similar posteriors (as explained in ).
Consider two distinct models: Models $j$ and $s$. The ratio of the posterior probabilities of these two models is given by (see ) $$\begin{aligned}
\label{eqn_post_ratio}
\frac{\pi(j \mid \boldsymbol\gamma_n)}{\pi(s \mid \boldsymbol\gamma_n)} &= \frac{\Gamma((n - d_s) / 2 - (d_j - d_s) / 2)}{\Gamma((n - d_s) / 2) ((n - d_s) / 2)^{-(d_j - d_s) / 2}} \, n^{-(d_j - d_s) / 2} \left(\frac{\|\boldsymbol\gamma_n - \widehat{\boldsymbol\gamma}_s\|_2^2 / n}{\|\boldsymbol\gamma_n - \widehat{\boldsymbol\gamma}_j\|_2^2 / n}\right)^{n / 2} \cr
&\qquad \times \frac{\pi^{d_j / 2}}{\pi^{d_s / 2}} \frac{\left(\|\boldsymbol\gamma_n - \widehat{\boldsymbol\gamma}_j\|_2^2 / n \right)^{d_j / 2}}{\left(\|\boldsymbol\gamma_n - \widehat{\boldsymbol\gamma}_s\|_2^2 / n \right)^{d_s / 2}} \frac{((n - d_s) / 2)^{-(d_j - d_s) / 2}}{n^{-(d_j - d_s) / 2}} \cr
&\qquad \times \frac{\pi(j)}{\pi(s)} \frac{|\mathbf{C}_s^T \mathbf{C}_s|^{1/2}}{ |\mathbf{C}_j^T \mathbf{C}_j|^{1/2}} \, n^{(d_j - d_s) / 2}.\end{aligned}$$ The difference between the Bayesian information criterion (BIC, [@schwarz1978estimating]) of Models $j$ and $s$ is given by $$\begin{aligned}
\text{BIC}_j - \text{BIC}_s &= n\log\left(\|\boldsymbol\gamma_n - \widehat{\boldsymbol\gamma}_j\|_2^2 / n\right) + (d_j + 1)\log n \cr
&\qquad - n\log\left(\|\boldsymbol\gamma_n - \widehat{\boldsymbol\gamma}_s\|_2^2 / n\right) - (d_s + 1)\log n \cr
&= n \log\left(\frac{\|\boldsymbol\gamma_n - \widehat{\boldsymbol\gamma}_j\|_2^2 / n}{\|\boldsymbol\gamma_n - \widehat{\boldsymbol\gamma}_s\|_2^2 / n}\right) + (d_j - d_s)\log n.\end{aligned}$$ Given that the first ratio on the RHS of converges to 1 as $n \longrightarrow \infty$, we have that $\exp\{-(\text{BIC}_j - \text{BIC}_s) / 2\}$ asymptotically behaves like the first term on the RHS of . The terms $\left(\|\boldsymbol\gamma_n - \widehat{\boldsymbol\gamma}_j\|_2^2 / n \right)^{d_j / 2}$ on the RHS in converge towards a constant (in $n$). All terms on the second row on the RHS in are thus asymptotically constant. Therefore, if the prior on $K$ is set to $\pi(k) \propto |\mathbf{C}_k^T \mathbf{C}_k|^{1/2} / n^{d_k / 2}$, the product in the third row on the RHS in is equal to 1. Consequently, $\pi(j \mid \boldsymbol\gamma_n) / \pi(s \mid \boldsymbol\gamma_n) \longrightarrow \infty$ whenever $\exp\{-(\text{BIC}_j - \text{BIC}_s) / 2\}\longrightarrow \infty$, and $\pi(j \mid \boldsymbol\gamma_n) / \pi(s \mid \boldsymbol\gamma_n) \longrightarrow 0$ whenever $\exp\{-(\text{BIC}_j - \text{BIC}_s) / 2\}\longrightarrow 0$. In other words, the Bayesian variable selection procedure associated with the normal linear regression framework described above is consistent (in the same sense as [@casella2009consistency]) whenever BIC is consistent, which is the case under regularity conditions (see, e.g., [@chib2016bayes]). If the “true” model is among the models considered, then its posterior probability converges to 1 as $n$ increases. We set the prior accordingly in the numerical examples.
When the covariates are orthonormal, $|\mathbf{C}_k^T \mathbf{C}_k|^{1 / 2} = |n \mathbf{I}_{d_k}|^{1 / 2} = n^{d_k / 2}$ (if the standardisation has been performed using a standard deviation in which the divisor is $n$). The prior on $K$ can thus be seen as a relative adjustment of the volume spanned by the columns of $\mathbf{C}_K^T \mathbf{C}_K$.
We work on the log scale for the scale parameters so that all the parameters take values on the real line, and presumably, are closer to having normal distributions. We thus define $\eta_k := \log \sigma_k$. The associated conditional distribution is given by $$\pi(\eta_k \mid k, \boldsymbol\gamma_n) := \frac{2^{1-\frac{n-d_k}{2}} \|\boldsymbol\gamma_n - \widehat{\boldsymbol\gamma}_k\|_2^{n - d_k}}{\Gamma((n-d_k)/2) \, {\mathrm{e}}^{(n - d_k) \eta_k}} \, \exp\left\{-\frac{1}{2 {\mathrm{e}}^{2 \eta_k}}\, \|\boldsymbol\gamma_n - \widehat{\boldsymbol\gamma}_k\|_2^2\right\}.$$ To implement the algorithms, we need to identify maximisers of the conditional posterior densities. This is achieved easily using and the conditional density of $\eta_k$: $$\widehat{\boldsymbol\mu}_k := (\widehat{\boldsymbol\beta}_k, \widehat{\eta}_k),$$ where $$\widehat{\eta}_k := \log \sqrt{\frac{1}{n - d_k} \, \|\boldsymbol\gamma_n - \widehat{\boldsymbol\gamma}_k\|_2^2}.$$ We also need to identify the Fisher information matrix: $$\mathcal{I}(\boldsymbol\beta_k, \eta_k) := \left(\begin{array}{cc}
\mathbf{C}_k^T \mathbf{C}_k / {\mathrm{e}}^{2 \eta_k} & \mathbf{0} \cr
\mathbf{0} & 2n
\end{array}
\right),$$ which implies that $$\begin{aligned}
\label{eqn_inverse_fisher}
\mathcal{I}^{-1}(\boldsymbol\beta_k, \eta_k) := \left(\begin{array}{cc}
{\mathrm{e}}^{2 \eta_k}(\mathbf{C}_k^T \mathbf{C}_k)^{-1} & \mathbf{0} \cr
\mathbf{0} & 1 / 2n
\end{array}
\right).\end{aligned}$$ We thus set $q_{k \mapsto k'} := \mathcal{N}( (\widehat{\boldsymbol\beta}_{k'}, \widehat{\eta}_{k'}), \mathcal{I}^{-1}(\widehat{\boldsymbol\beta}_{k'}, \widehat{\eta}_{k'}))$ and $\mathcal{D}_{k \mapsto k'}$ such that $\mathbf{y}_{k'} := \mathbf{u}_{k \mapsto k'}$ in the RJ.
To use the annealing distributions in the algorithms, we work with the log densities; therefore we simply multiply $\log \pi(\, \cdot \mid k, \boldsymbol\gamma_n)$ by $1 - t / T$ and $\log \pi(\, \cdot \mid k', \boldsymbol\gamma_n)$ by $t / T$ to obtain $\log \rho_{k \mapsto k'}^{(t)}$. To use MALA proposals, we however need to compute the gradient of $\log \rho_{k \mapsto k'}^{(t)}$. We now do that (the proportional sign “$\propto$” is with respect to everything that are not the parameters and their proposals): $$\begin{aligned}
\pi(\mathbf{x}_k^{(t)} \mid k, \mathbf{D}_n)^{1 - t / T} q_{k' \mapsto k}(\mathbf{x}_k^{(t)})^{t / T} &= \left[\frac{|\mathbf{C}_k^T \mathbf{C}_k|^{1/2}}{(2\pi)^{d_k/2} {\mathrm{e}}^{d_k \eta_k}} \exp\left(-\frac{1}{2{\mathrm{e}}^{2\eta_k}}(\boldsymbol\beta_k - \widehat{\boldsymbol\beta}_k)^T (\mathbf{C}_k^T \mathbf{C}_k)(\boldsymbol\beta_k - \widehat{\boldsymbol\beta}_k) \right)\right]^{1 - t/T} \cr
&\qquad \times \left[ \frac{2^{1-\frac{n-d_k}{2}} \|\boldsymbol\gamma_n - \widehat{\boldsymbol\gamma}_k\|_2^{n - d_k}}{\Gamma((n-d_k)/2) \, {\mathrm{e}}^{(n - d_k) \eta_k}} \, \exp\left(-\frac{1}{2 {\mathrm{e}}^{2 \eta_k}}\, \|\boldsymbol\gamma_n - \widehat{\boldsymbol\gamma}_k\|_2^2\right)\right]^{1 - t/T} \cr
&\qquad\times \left[\frac{|\mathbf{C}_k^T \mathbf{C}_k|^{1/2}}{(2\pi)^{d_k/2} {\mathrm{e}}^{d_k \widehat{\eta}_k}} \exp\left(-\frac{1}{2{\mathrm{e}}^{2\widehat{\eta}_k}}(\boldsymbol\beta_k - \widehat{\boldsymbol\beta}_k)^T (\mathbf{C}_k^T \mathbf{C}_k)(\boldsymbol\beta_k - \widehat{\boldsymbol\beta}_k) \right)\right]^{t/T} \cr
& \qquad\times \left[\frac{1}{\sqrt{2\pi(1/(2n))}} \exp\left(-\frac{1}{2(1/(2n))}(\eta_k - \widehat{\eta}_k)^2\right)\right]^{t/T} \cr
&\hspace{-20mm}\propto \frac{1}{{\mathrm{e}}^{d_k((1 - t/T) \eta_k + (t/T) \widehat{\eta}_k)}} \exp\left(-\frac{1}{2}(\boldsymbol\beta_k - \widehat{\boldsymbol\beta}_k)^T [(1 - t/T){\mathrm{e}}^{-2\eta_k} + (t/T) {\mathrm{e}}^{-2\widehat{\eta}_k}](\mathbf{C}_k^T \mathbf{C}_k)(\boldsymbol\beta_k - \widehat{\boldsymbol\beta}_k) \right) \cr
&\qquad \times \frac{1}{{\mathrm{e}}^{(n-d_k)(1 - t/T) \eta_k }} \exp\left(-\frac{(1- t/T)}{2{\mathrm{e}}^{2\eta_k}} \|\boldsymbol\gamma_n - \widehat{\boldsymbol\gamma}_k\|_2^2 \right) \cr
&\qquad \times \exp\left(- n(t / T)(\eta_k - \widehat{\eta}_k)^2\right),\end{aligned}$$ where we omitted the superscript “(t)” for the variables to simplify. Therefore, $$\begin{aligned}
\frac{\partial}{\partial \boldsymbol\beta_k} \log \pi(\mathbf{x}_k^{(t)} \mid k, \mathbf{D}_n)^{1 - t / T} q_{k' \mapsto k}(\mathbf{x}_k^{(t)})^{t / T} = -[(1 - t/T){\mathrm{e}}^{-2\eta_k} + (t/T) {\mathrm{e}}^{-2\widehat{\eta}_k}](\mathbf{C}_k^T \mathbf{C}_k)(\boldsymbol\beta_k - \widehat{\boldsymbol\beta}_k),\end{aligned}$$ and $$\begin{aligned}
\frac{\partial}{\partial \eta_k} \log \pi(\mathbf{x}_k^{(t)} \mid k, \mathbf{D}_n)^{1 - t / T} q_{k' \mapsto k}(\mathbf{x}_k^{(t)})^{t / T} &= -(1 - t/T) d_k + (1 - t/T) {\mathrm{e}}^{-2 \eta_k} (\boldsymbol\beta_k - \widehat{\boldsymbol\beta}_k)^T (\mathbf{C}_k^T \mathbf{C}_k)(\boldsymbol\beta_k - \widehat{\boldsymbol\beta}_k) \cr
&\qquad -(1 - t/T)(n - d_k) + (1 - t/T) {\mathrm{e}}^{-2 \eta_k} \|\boldsymbol\gamma_n - \widehat{\boldsymbol\gamma}_k\|_2^2 \cr
&\qquad - 2 n (t/T) (\eta_k - \widehat{\eta}_k).\end{aligned}$$
Robust linear regression {#sec_robust_reg}
------------------------
The density of the LPTN with parameter $\rho\in (2\Phi(1) - 1, 1) \approx (0.6827, 1)$ is given by $$\label{eqn_log_pareto_pcr}
f(x):=\left\{
\begin{array}{lcc}
\varphi(x) & \text{ if } & {\left|x\right|}\leq \tau, \\
\varphi(\tau)\,\frac{\tau}{|x|}\left(\frac{\log \tau}{\log |x|}\right)^{\lambda+1} & \text{ if } & {\left|x\right|}>\tau, \\
\end{array}
\right.$$ where $x\in\operatorname{\mathbb{R}}$, and $\varphi(\,\cdot\,)$ and $\Phi(\,\cdot\,)$ are the PDF and cumulative distribution function (CDF) of a standard normal. The terms $\tau>1$ and $\lambda>0$ are functions of $\rho$ and satisfy $$\begin{aligned}
& \tau:=\Phi^{-1}((1+\rho)/2) := \{\tau : {\mathbb{P}}(-\tau \leq Z \leq \tau)= \rho \,\text{ for }\, Z\, \sim \, \mathcal{N}(0,1)\}, \\
& \lambda:=2(1-\rho)^{-1}\varphi(\tau) \, \tau \log(\tau), \nonumber
\end{aligned}$$ with $\Phi^{-1}(\,\cdot\,)$ being the inverse CDF of a standard normal. Setting $\rho$ to $0.95$ has proved to be suitable for practical purposes (see [@gagnon2018regression]). Accordingly, this is the value that is used in our numerical analyses.
The joint posterior density is: $$\begin{aligned}
\pi(k, \boldsymbol\beta_k, \sigma_k \mid \boldsymbol\gamma_n) \propto \frac{|\mathbf{C}_k^T \mathbf{C}_k|^{1/2}}{n^{d_k / 2}} \frac{1}{\sigma_k} \prod_{i = 1}^n \frac{1}{\sigma_k} f\left(\frac{\gamma_i - \mathbf{c}_{i, k}^T \boldsymbol\beta_k}{\sigma_k}\right).
\end{aligned}$$
With the change of variable $\eta_k := \log \sigma_k$, we have $$\begin{aligned}
\pi(k, \boldsymbol\beta_k, \eta_k \mid \boldsymbol\gamma_n) \propto \frac{|\mathbf{C}_k^T \mathbf{C}_k|^{1/2}}{n^{d_k / 2}} \frac{1}{{\mathrm{e}}^{\eta_k n}} \prod_{i = 1}^n f\left(\frac{\gamma_i - \mathbf{c}_{i, k}^T \boldsymbol\beta_k}{{\mathrm{e}}^{\eta_k}}\right).
\end{aligned}$$
We need log conditionals and their gradients for optimisers and HMC: $$\log \pi(\boldsymbol\beta_k, \eta_k \mid k, \boldsymbol\gamma_n) \propto - n \eta_k + \sum_{i = 1}^n \log f\left(\frac{\gamma_i - \mathbf{c}_{i, k}^T \boldsymbol\beta_k}{{\mathrm{e}}^{\eta_k}}\right),$$ where the proportional sign has to be understood in the original scale, and $$\begin{aligned}
\frac{\partial}{\partial \boldsymbol\beta_k} \log \pi(\boldsymbol\beta_k, \eta_k \mid k, \boldsymbol\gamma_n) &= \sum_{i = 1}^n {\mathrm{e}}^{-2 \eta_k} (\gamma_i - \mathbf{c}_{i, k}^T \boldsymbol\beta_k) \mathbf{c}_{i, k} {\mathds{1}}\left((\gamma_i - \mathbf{c}_{i, k}^T \boldsymbol\beta_k) / {\mathrm{e}}^{\eta_k} \leq \tau \right) \cr
& \hspace{-20mm} + \left[ \frac{\text{sgn}(\gamma_i - \mathbf{c}_{i, k}^T \boldsymbol\beta_k) \mathbf{c}_{i, k}}{|\gamma_i - \mathbf{c}_{i, k}^T \boldsymbol\beta_k|} + (\lambda + 1) \frac{\text{sgn}(\gamma_i - \mathbf{c}_{i, k}^T \boldsymbol\beta_k) \mathbf{c}_{i, k}}{|\gamma_i - \mathbf{c}_{i, k}^T \boldsymbol\beta_k| \log\left((\gamma_i - \mathbf{c}_{i, k}^T \boldsymbol\beta_k) / {\mathrm{e}}^{\eta_k}\right)}\right] {\mathds{1}}\left((\gamma_i - \mathbf{c}_{i, k}^T \boldsymbol\beta_k) / {\mathrm{e}}^{\eta_k} > \tau \right),\end{aligned}$$ and $$\begin{aligned}
\frac{\partial}{\partial \eta_k} \log \pi(\boldsymbol\beta_k, \eta_k \mid k, \boldsymbol\gamma_n) &= -n + \sum_{i = 1}^n {\mathrm{e}}^{-2 \eta_k} (\gamma_i - \mathbf{c}_{i, k}^T \boldsymbol\beta_k)^2 {\mathds{1}}\left((\gamma_i - \mathbf{c}_{i, k}^T \boldsymbol\beta_k) / {\mathrm{e}}^{\eta_k} \leq \tau \right) \cr
& \qquad + \left[1 + (\lambda + 1) \frac{1}{ \log\left((\gamma_i - \mathbf{c}_{i, k}^T \boldsymbol\beta_k) / {\mathrm{e}}^{\eta_k}\right)}\right] {\mathds{1}}\left((\gamma_i - \mathbf{c}_{i, k}^T \boldsymbol\beta_k) / {\mathrm{e}}^{\eta_k} > \tau \right),\end{aligned}$$
As approximations to the posterior model probabilities, we use Laplace approximations to $$\begin{aligned}
\int \pi(k, \boldsymbol\beta_k, \eta_k \mid \boldsymbol\gamma_n) \, d\boldsymbol\beta_k \, d\eta_k \propto \frac{|\mathbf{C}_k^T \mathbf{C}_k|^{1/2}}{n^{d_k / 2}} \int \frac{1}{{\mathrm{e}}^{\eta_k n}} \prod_{i = 1}^n f\left(\frac{\gamma_i - \mathbf{c}_{i, k}^T \boldsymbol\beta_k}{{\mathrm{e}}^{\eta_k}}\right) \, d\boldsymbol\beta_k \, d\eta_k,\end{aligned}$$ which yield $$\begin{aligned}
\frac{|\mathbf{C}_k^T \mathbf{C}_k|^{1/2}}{n^{d_k / 2}} (2\pi)^{(d_k + 1) / 2} \pi(\widehat{\boldsymbol\beta}_k, \widehat{\eta}_k \mid k, \boldsymbol\gamma_n)|\mathcal{I}(\widehat{\boldsymbol\beta}_k, \widehat{\eta}_k)|^{1/2} = \frac{1}{n^{(d_k + 1) / 2}\sqrt{2}} (2\pi)^{(d_k + 1) / 2}\frac{1}{{\mathrm{e}}^{\widehat{\eta}_k (n - d_k)}} \prod_{i = 1}^n f\left(\frac{\gamma_i - \mathbf{c}_{i, k}^T \widehat{\boldsymbol\beta}_k}{{\mathrm{e}}^{\widehat{\eta}_k}}\right).\end{aligned}$$ Note that we use the same Fisher information matrix as the normal regression to simplify. Note also that when $f := \mathcal{N}(0, 1)$, $$\begin{aligned}
\frac{\widehat{\pi}(j \mid \boldsymbol\gamma_n)}{\widehat{\pi}(s \mid \boldsymbol\gamma_n)} &= n^{-(d_j - d_s) / 2} \left(\frac{\|\boldsymbol\gamma_n - \widehat{\boldsymbol\gamma}_s\|_2^2 / (n - d_s)}{\|\boldsymbol\gamma_n - \widehat{\boldsymbol\gamma}_j\|_2^2 / (n - d_j)}\right)^{n / 2} \cr
&\qquad \times \frac{(2 \pi)^{d_j / 2}}{(2 \pi)^{d_s / 2}} \frac{\left(\|\boldsymbol\gamma_n - \widehat{\boldsymbol\gamma}_j\|_2^2 / (n - d_j) \right)^{d_j / 2}}{\left(\|\boldsymbol\gamma_n - \widehat{\boldsymbol\gamma}_s\|_2^2 / (n - d_s) \right)^{d_s / 2}} \frac{{\mathrm{e}}^{d_j / 2}}{{\mathrm{e}}^{d_s / 2}},\end{aligned}$$ which behaves asymptotically like . This confirms that the estimators are consistent under the normality assumption. We expect the same to hold for the LPTN regression.
As for the normal regression, to use the annealing distributions in the algorithms, we work with the log densities; therefore we simply multiply $\log \pi(\, \cdot \mid k, \boldsymbol\gamma_n)$ by $1 - t / T$ and $\log \pi(\, \cdot \mid k', \boldsymbol\gamma_n)$ by $t / T$ to obtain $\log \rho_{k \mapsto k'}^{(t)}$. To use MALA proposals, we however need to compute the gradient of $\log \rho_{k \mapsto k'}^{(t)}$. We now do that (the proportional sign “$\propto$” is with respect to everything that are not the parameters and their proposal): $$\begin{aligned}
\pi(\mathbf{x}_k^{(t)} \mid k, \mathbf{D}_n)^{1 - t / T} q_{k' \mapsto k}(\mathbf{x}_k^{(t)})^{t / T} &= \left[ \pi(\boldsymbol\beta_k, \eta_k \mid k, \boldsymbol\gamma_n)\right]^{1 - t/T} \cr
&\qquad\times \left[\frac{|\mathbf{C}_k^T \mathbf{C}_k|^{1/2}}{(2\pi)^{d_k/2} {\mathrm{e}}^{d_k \widehat{\eta}_k}} \exp\left(-\frac{1}{2{\mathrm{e}}^{2\widehat{\eta}_k}}(\boldsymbol\beta_k - \widehat{\boldsymbol\beta}_k)^T (\mathbf{C}_k^T \mathbf{C}_k)(\boldsymbol\beta_k - \widehat{\boldsymbol\beta}_k) \right)\right]^{t/T} \cr
& \qquad\times \left[\frac{1}{\sqrt{2\pi(1/(2n))}} \exp\left(-\frac{1}{2(1/(2n))}(\eta_k - \widehat{\eta}_k)^2\right)\right]^{t/T} \cr
&\propto \left[ \pi(\boldsymbol\beta_k, \eta_k \mid k, \boldsymbol\gamma_n)\right]^{1 - t/T} \cr
&\qquad \times \frac{1}{{\mathrm{e}}^{d_k (t/T) \widehat{\eta}_k}} \exp\left(-\frac{(t / T)}{2 {\mathrm{e}}^{2\widehat{\eta}_k}}(\boldsymbol\beta_k - \widehat{\boldsymbol\beta}_k)^T (\mathbf{C}_k^T \mathbf{C}_k)(\boldsymbol\beta_k - \widehat{\boldsymbol\beta}_k) \right) \cr
&\qquad \times \exp\left(- n(t / T)(\eta_k - \widehat{\eta}_k)^2\right),\end{aligned}$$ where we omitted the superscript “(t)” for the variables to simplify. Therefore, $$\begin{aligned}
\frac{\partial}{\partial \boldsymbol\beta_k} \log \pi(\mathbf{x}_k^{(t)} \mid k, \mathbf{D}_n)^{1 - t / T} q_{k' \mapsto k}(\mathbf{x}_k^{(t)})^{t / T} = (1 - t/T) \frac{\partial}{\partial \boldsymbol\beta_k} \log \pi(\boldsymbol\beta_k, \eta_k \mid k, \boldsymbol\gamma_n)
- (t/T) {\mathrm{e}}^{-2\widehat{\eta}_k}(\mathbf{C}_k^T \mathbf{C}_k)(\boldsymbol\beta_k - \widehat{\boldsymbol\beta}_k),\end{aligned}$$ and $$\begin{aligned}
\frac{\partial}{\partial \eta_k} \log \pi(\mathbf{x}_k^{(t)} \mid k, \mathbf{D}_n)^{1 - t / T} q_{k' \mapsto k}(\mathbf{x}_k^{(t)})^{t / T} &= (1 - t/T) \frac{\partial}{\partial \eta_k} \log \pi(\boldsymbol\beta_k, \eta_k \mid k, \boldsymbol\gamma_n) - 2 n (t/T) (\eta_k - \widehat{\eta}_k).\end{aligned}$$
[^1]: See the ArXiv page of this paper.
|
---
abstract: 'H1743–322 is one of the few black hole candidates (BHCs) in low-mass X-ray binaries that shows mHz quasi-periodic oscillations (QPOs) that are not associated with the more common type A, B and C oscillations seen in the X-ray light curves of typical BHCs systems. To better understand the physical origin of the mHz oscillations, we carried out a phase-resolved spectroscopic study of two RXTE observations of this source. As previously reported, the averaged energy spectra of H1743–322 shows a strong iron line at $\sim6.5$ keV. Here we found evidence that the line flux appears to be modulated at twice the frequency of the mHz QPO. This line flux modulation is very similar to the one previously found for the type-C QPO in this source. We interpret the possibly periodic line flux modulation with this mHz QPO in terms of Lense-Thirring precession of the inner flow, and discuss the possible connection with the modulation of the line properties with the type-C QPO frequency.'
author:
- |
Zheng Cheng$^1$[^1], Mariano Méndez$^1$, Diego Altamirano$^2$, Aru Beri$^2$, Yanan Wang$^1$\
$^1$Kapteyn Astronomical Institute, University of Groningen, Postbus 800, 9700 AV Groningen, The Netherlands\
$^2$School of Physics and Astronomy, University of Southampton, Southampton, Hampshire SO17 1BJ, UK\
bibliography:
- 'paper\_7.bib'
title: 'Phase-resolved spectral analysis of the 11 millihertz quasi-periodic oscillation in the black-hole candidate H1743–322'
---
accretion, accretion discs – black hole physics – relativistic processes – X-rays: individual: H1743–322
Introduction {#intro}
============
Low-mass X-ray binaries (LMXB) consist of a compact object, a neutron star or black hole, and a low-mass companion star. As the companion fills its Roche lobe, matter is transferred to the compact star through an accretion disc [@frank2002]. These systems show occasional outbursts, during which the X-ray luminosity increases by several orders of magnitude. These outbursts last from a few days up to months, and display distinct temporal and spectral features [e.g., @tanaka1995; @homan2005b; @remillard2006].
Quasi periodic oscillations (QPOs) have been found in accreting neutron-stars and black-holes systems, spanning a wide range of centroid frequencies, from milli-Hertz to $\sim1300$ Hertz [@klis2006book; @remillard2006]. In particular, black-hole X-ray binaries (BHXRB) generally show three types of low-frequency (LF) QPOs, called types A, B, and C [@wijnands1999; @casella2005], and high-frequency (HF) QPOs in a few cases [@strohmayer2001; @homan2005; @belloni2012]. The short variability time scale of HFQPOs in BHXRBs suggests that these QPOs are produced in the innermost region of the accretion disc [@strohmayer2001; @kluzniak2001]. The origin of the different types of LFQPOs is, however, not fully understood yet. Type-C QPOs are by far the most common type of LFQPOs among the three groups; type-C QPOs have been observed in hard states and transitions to/from the soft states with frequencies ranging from tens of mHz to $\sim$30 Hz, and always accompanied by a strong flat-top red noise component [@motta2015; @casella2005]. Type-B QPOs, usually observed in the intermediate states, are strong and are accompanied by a weak red noise component [@belloni2005]. There are only a handful of detections of type-A QPOs; they are detected when the source is in the soft state as weak and broad peaks, and are still poorly understood [@motta2016]. The different types of LFQPOs are believed to originate from either intrinsic luminosity variations or to geometric variability of the emission area. The former could, for instance, be due to fluctuations of the mass accretion rate [@tagger1999; @cabanac2010], shocks in the accretion flow [@chakrabarti1993] or intrinsic variability of the jet [@giannios2004]; the latter may result from Lense-Thirring precession of the accretion inner flow [@stella1998; @ingram2009], a precessing region related to the base of the jet [@fender2009; @millerjones2012] or the effect of a warp in the disc [@fragile2008].
[@schnittman2006] proposed a precessing ring model for the hot gas in the innermost regions of the accretion disc; in this context, the fractional rms amplitude of the type-C QPOs should be higher for high inclination system. [@heil2015], [@motta2015], [@eijnden2017] recently reported the dependence of the QPO properties upon the source inclination using a large sample. For instance, [@motta2015] found that the type-C QPOs show higher rms amplitude for high inclination systems, while type B QPOs are stronger for low inclination ones. This is consistent with the idea of the Lense-Thirring precession origin of type-C QPO. The behaviour of type-B QPOs, on the other hand, supports the hypothesis of them being related to the jet in these sources.
Phase-resolved spectroscopy can provide more details about the origin of LFQPOs. By combining the energy spectra and timing information, we can study changes of the energy spectra of a source in the QPO time scale. [@ingram2015] detected a modulation of the equivalent width of the iron line with the phase of the type-C QPO in GRS 1915+105 which, they argued, supports the geometric origin of this QPO. [@stevens2016] studied the variation of the blackbody and power-law components in the phase-resolved spectra of the type-B QPO in GX 339-4; they interpreted their findings as evidence that the type-B QPO originates at the base of the jet in this source. [@ingram2016; @ingram2017] found a modulation of the centroid energy of the iron emission line with the phase of the type-C QPO in the BHXRB H1743–322, which they regarded as strong evidence that the type-C QPO is produced by Lense-Thirring precession of the inner flow near the central compact object (see also @miller2005).
H1743–322 was discovered with the Ariel V All Sky Monitor by [@kaluzienski1977]. [@white1984] classified the source as a black-hole candidate (BHC) on the basis of its X-ray spectral characteristics. In March 2003, ESA’s INTEGRAL satellite detected a bright and variable source, named IGR J17464-3213, in the same region in the sky [@revnivtsev2003]. Using the Rossi X-Ray Timing Explorer (*RXTE*), [@markwardt2003] confirmed that this new transient was actually H1743–322. A radio counterpart was found with the Very Large Array (VLA) during the 2003 outburst by [@rupen2003a], and relativistic jets were observed from this source by [@corbel2005] using the Australia Telescope Compact Array (ATCA). Due to its X-ray dipping behaviour [@homan2005; @miller2006], the system is believed to have a relative high inclination with respect to the line of sight ($\sim60^\circ$-$70^\circ$), and it is located at a distance of $8.5\pm0.8$ kpc [@steiner2012].
[@diego2012] reported the discovery of QPOs at $\sim11$ mHz in two *RXTE* and one *Chandra* observation of H1743–322. Those observations also displayed the more common type-C QPOs typically seen in this source as well as in other BHCs systems [@motta2015]. It is interesting to try and understand why this QPO is unique, and the mechanism driving the phenomenon. In this paper we reanalyse the *RXTE* data and carry out phase resolved spectroscopy on the period of the 11 mHz QPO. We describe the details of the observations and the data reduction and analysis in §\[data\], we show the results in §\[results\], and we discuss our findings in §\[discussion\].
Data reduction {#data}
==============
The BHC H1743–322 was observed with the Proportional Counter Array (PCA, @jahoda2006) on board the Rossi X-ray Timing Explorer (*RXTE*, @bradt1993) a total of 558 times between 2003 March and 2011 June. An unusual QPO at $\sim$11 mHz was detected by [@diego2012] twice, in the rising phase of the 2010 outburst (using *RXTE* and *Chandra*) and the 2011 outburst (using *RXTE*). This QPO has only been detected in these three observations (*RXTE* obsID 95368-01-01-00 and 96425-01-01-00; *Chandra* ID number 401083) in this source so far.
For our analysis we used the *RXTE* PCA data of obsID 95368-01-01-00 (2010, orbits 1 and 2, GoodXenon mode) and obsID 96425- 01-01-00 (2011 orbit 3, Event mode). We created Good Time Intervals (GTI) using the following filtering criteria: elevation angle larger than $10^\circ$, target offset less than 0.02$^\circ$, and two Proportional Counter Units (PCUs) on, which was the maximum number of PCUs that were active during these observations: PCU 2 and 4 were on in orbit 1 and orbit 3, and PCU 1 and 2 were on in orbit 2. We extracted data from all three layers of those PCUs. The GoodXenon mode uses two Event Analyzers (EAs) simultaneously, such that the events are split between the two EA’s in two separate files. Therefore, we first combined the event files and then applied the GTIs to carry out the analysis. We extracted 1-second resolution source light curves in the 3–25 keV energy range, which we call the full-band light curves in this paper, from which we subtracted the background light curve constructed using 16 seconds resolution Standard 2 mode data after applying the dead time correction. As there is a gap ($\sim$3000 seconds) between the two orbits in this observation, to study the variability, we computed the Lomb-Scargle periodograms [@lomb1976; @scargle1982; @press2007] in the analysis for each of the orbits separately. For the second observation taken with a 64-channel Event mode, the background was subtracted using the Standard 2 mode data after rebinning them to the same number of channels as the Event mode data.
We modelled the spectra in the 3–25 keV range using [XSPEC]{} version of 12.9.1a. The interstellar absorption was considered by including the component [tbabs]{} using the solar abundance table of [@wilms2000] with the photo-electric absorption cross-sections from [@balucinska1992] and the He cross-section by [@yan1998]. We used the component [diskbb]{} [@mitsuda1984; @makishima1986] to model the soft emission from the accretion disc. To account for the high-energy photons produced by inverse Compton scattering in a corona of hot electrons, we added a thermal Comptonisation component, [nthcomp]{} [@zdziarski1996; @zycki1999]. Compared with a power-law, the [nthcomp]{} component gives a better description of the continuum in the bandpass where the seed photons come from; the source of seed photons in our case is the accretion disc, therefore we assumed that the seed photons have a disc-blackbody shape and linked the seed-photon temperature in [nthcomp]{}, $kT_{\rm seed}$, to the temperature at the inner disc radius, $kT_{\rm dbb}$, in [diskbb]{}. As our spectra only extend up to 25 keV, and the data showed no cut off up to that energy, we fixed the electron temperature, $kT_{\rm e}$, in [nthcomp]{} to 1000 keV. The redshift of this component was fixed to 0 through out the analysis. We also included a Gaussian component to fit emission of an iron line that was apparent in the residuals at $\sim6.4$ keV.
The *Chandra* observation was taken in continuous-clocking (CC) mode, in which the background spectrum cannot be accurately extracted; the properties of the line, on the other hand, are very sensitive to the shape of the continuum above $\sim7-8$ keV, where the background emission is comparable to the source. Because of this, and the impossibility of obtaining an accurate background spectrum, we did not use the *Chandra* data further, as the line parameters cannot be retrieved from this observation. To check whether the *Chandra* data were generally consistent with the *RXTE* observation, we subtracted a rough model of the background to the data and fitted the spectrum with the same model we used for the *RXTE* spectra. Unlike in the case of the *RXTE* spectra, the parameters of the Gaussian component cannot be well constrained by the *Chandra* data. We therefore fixed the centroid energy and the width of the Gaussian component to the value we found in the *RXTE* observations, and fitted only the normalisation of the line. We confirmed that the 95% upper limit of the flux of the line in the *Chandra* data is consistent with the value we measured in the *RXTE* observations.
Results
=======
Average Spectra
---------------
Model Parameters Orbit 1 Orbit 2 Orbit 3
---------------------------------------------------- ------------------------ ------------------------ ------------------------
$N_{\rm H}$ (cm$^{-2}$)
$kT_{\rm dbb}$ (keV) 1.35$\pm$0.07 1.35$\pm$0.06 1.35$\pm$0.07
$N_{\rm dbb}$ 4.90$_{-0.48}^{+0.67}$ 4.95$_{-0.39}^{+0.53}$ 3.91$_{-0.41}^{+0.54}$
$\Gamma$ 1.60$\pm$0.01 1.59$\pm$0.01 1.53$\pm$0.01
$kT_{\rm e}$ (keV)
$N_{\rm nth}$ 0.13$\pm$0.01 0.13$\pm$0.01 0.11$\pm$0.01
$E_{\rm line}$ (keV) 6.53$\pm$0.09 6.39$\pm$0.05 6.49$\pm$0.08
$\sigma$ (keV) 0.48$\pm$0.16 0.39$\pm$0.14 0.39$\pm$0.13
$N_{\rm line}$ ($\times10^{-3}$) 1.32$\pm$0.31 1.35$\pm$0.25 1.19$\pm$0.22
$F_{\rm X}$ ($10^{-9}$ erg cm$^{-2}$ s$^{-1}$) 2.80$\pm$0.01 2.87$\pm$0.01 2.64$\pm$0.01
$F_{\rm dbb}$ ($10^{-9}$ erg cm$^{-2}$ s$^{-1}$) 0.14$\pm$0.02 0.14$\pm$0.02 0.11$\pm$0.02
$F_{\rm nth}$ ($10^{-9}$ erg cm$^{-2}$ s$^{-1}$) 2.64$\pm$0.02 2.72$\pm$0.02 2.51$\pm$0.02
$F_{\rm line}$ ($10^{-11}$ erg cm$^{-2}$ s$^{-1}$) 1.38$\pm$0.32 1.39$\pm$0.25 1.24$\pm$0.23
$\chi^2$/d.o.f.
We initially fitted the average spectra for the three orbits simultaneously using only the thermal Comptonisation model plus a soft disc component. Because the PCA instrument is not sensitive below 3 keV, the hydrogen column density could not be well constrained using our data. Different values of $N_{\rm H}$ have been previously reported for this source: $N_{\rm H}= 1.35 \times10^{22} \rm{cm}^{-2}$ [@ingram2016], $N_{\rm H}= 2.01 \times10^{22} \rm{cm}^{-2}$ [@stiele2016] and $N_{\rm H}= 2.2 \times10^{22} \rm{cm}^{-2}$ [@mcclintock2009]. [@prat2009] used 1.8 $\times10^{22} \rm{cm}^{-2}$, which they determined from *Swift* and *XMM-Newton* observations. [@chaty2015] agreed with this value, as they suggested that a relative low absorption would give spectral indices more compatible with thermal emission in the soft state. We therefore fixed $N_{\rm H}=1.8 \times10^{22} \rm{cm}^{-2}$, but since our analysis focused on the spectra above 3 keV, especially the energy range of the iron emission line, our results are not affected by the choice of the value of $N_{\rm H}$.
We could not get an acceptable fit using the model [tbabs\*(diskbb+nthcomp)]{} : the best fit yielded a $\chi^{2}$ of 418.8 for 121 degrees of freedom; the residuals indicated the need of an extra component at around 6.5 keV, consistent with emission from the iron K-$\alpha$ line, possibly coming from reflection off the accretion disc. We therefore added a Gaussian component to our model, allowing the centroid energy to vary between 6 and 7 keV; on the other hand, the width and the normalisation were left free and allowed to vary in each orbit. The fit with the Gaussian line yielded a $\chi^{2}$ of 152.4 for 113 degrees of freedom. An F-test yields a null-hypothesis probability of $1.4\times10^{-21}$, indicating that the Gaussian component is significantly required by the fit. The best-fitting parameters are shown in Table \[t1\].
Phase resolved spectra
----------------------
![ Lomb-Scargle periodograms of the three orbits of H1743–322 for three energy bands. The inset plot in each panel shows a zoom of the frequency axis from 8 to 15 mHz. The red, green and blue lines indicate the Lomb-Scargle periodograms in the 3-5 keV, 5-9 keV and 9–25 keV bands, respectively. The centroid frequency of the full-band mHz QPOs are 11.3 mHz, 11.4 mHz and 11.1 mHz for the three orbits, respectively. The type-C QPO and its second harmonic are also apparent in this figure. The fundamental frequency of the full-band type-C QPO in the three orbits is 0.92 Hz, 0.92 Hz and 0.43 Hz, respectively. Neither the frequency of the mHz QPOs or the type-C QPOs show any energy dependence. []{data-label="p1"}](p1_5.pdf){width="7.9cm"}
We computed the Lomb-Scargle periodogram of the light curves of the three orbits in three different energy bands, 3–5 keV, 5–9 keV and 9–25 keV; we found that, in each orbit, the frequency of the mHz QPO is consistent with being the same in all three energy bands. We show the Lomb-Scargle periodogram of the three orbits in Figure \[p1\]. Both the $\sim11$ mHz QPOs and the type-C QPOs (with its harmonic) can be seen in the figure. The false-alarm probability for mHz QPOs in these three orbits, taking into account the number of trials, is $1.4\times10^{-24}$, $1.8\times10^{-81}$, and $2.9\times10^{-21}$, respectively. The frequency of the mHz QPOs is quite stable; the centroid frequency of the mHz QPOs for the three orbits measured in the full-band periodogram are at $11.3\pm0.2$, $11.4\pm0.2$ and $11.1\pm0.2$ mHz, respectively. The frequency of the type-C QPO, however, is a factor of $\sim2$ lower in orbit 3 compared with the first two orbits. The frequency of the type-C QPO for the three orbits are $0.92\pm0.03$ Hz, $0.92\pm0.04$ Hz and $0.43\pm0.02$ Hz, respectively.
![ Deviations from the average spectra of H1743–322 in orbit 1 for three different QPO phase intervals. The blue circles, red squares and green triangles indicate the deviations from the average spectrum at, respectively, $\phi = 0.0$–$0.1$, corresponding to the minimum of the light curve folded at the QPO frequency, $\phi=0.5$–$0.6$, corresponding to the maximum, and $\phi=0.9$–$1.0$, corresponding to the mean value of the light curve. []{data-label="p2"}](p2_6.pdf){width="6cm"}
Based on the QPO frequency, we folded the background-subtracted light curves and created the average oscillation waveforms for the three orbits separately. As the observation is in the rising phase of the outburst, there is a significant increase of the count rate between the two orbits in the first observation. Therefore, instead of combining the spectra of these two orbits together, we analysed the spectra for each of the orbits separately.
We divided the QPO cycle into ten phase bins with the same duration for each orbit, where we defined the phase $\phi=0.0-0.1$ at the first minimum of the folded light curve (see below). We then computed the GTIs for each of the phase bins and extracted the phase resolved spectra using the [ftool]{} command [seextrct]{}.
For each orbit separately, we used the model [tbabs\*(diskbb+nthcomp+gaussian)]{} to fit the ten phase-resolved spectra simultaneously. We first tried to let the Gaussian line centroid energy and normalisation free to change between phases, however, the 1-$\sigma$ errors of the best-fitting energy of the line were $\sim0.2-0.5$ keV, which are too large to detect the expected variability, c.f. $\sim0.3$ keV as found by [@ingram2016], given also that the energy resolution of the PCA at 6 keV is $\sim1$ keV. For the rest of the analysis we decided to link the centroid energy and width of the line to be the same in all phases and let the normalisation free to change.
In Figure \[p2\] we plot the spectra of three selected QPO phases in the first orbit, phase 0.0–0.1, 0.5–0.6 and 0.9–1.0 of the QPO cycle, respectively, from which we subtracted the average spectra. From this Figure it is apparent that the spectra show more variability in the soft than in the hard part. The fitting results also support this idea: the contribution of the soft component ([diskbb]{}) changes from 2% of the total flux between 3–25 keV in phase 0.0–0.1 to 9% of the total flux in phase 0.6–0.7, whereas the photon index of the [nthcomp]{} component in the different phases is roughly consistent with the value obtained from fitting the average spectra. The coupling of the [nthcomp]{} and [diskbb]{} components cause the parameters to be less well constrained, resulting in large errors of the fitting parameters. Because of this, we decided to link the photon index and the normalisation of [nthcomp]{} component across phases.
{width="16cm"}
In Figure \[p3\], we show the fitting parameters as a function of the QPO phase for the two orbits of observation 95368-01-01-00. The change of the disc temperature follows the modulation of the count rate in both orbits, whereas the [diskbb]{} normalisation changes in the opposite way. A clear modulation of the Gaussian normalisation is also apparent in the first orbit; this modulation has a period of half the period of the QPO, with peaks at phase 0.1–0.2 and 0.6–0.7 of the QPO cycle. The fits yield a $\chi^{2}$ of 526.0 for 486 degrees of freedom, with the best-fitting centroid energy and width of the line being 6.55$\pm$0.07 keV, 0.49$\pm$0.15 keV, respectively. If we fit the phase-resolved spectra by linking all the Gaussian normalisations to the same value, the best fit yields a $\chi^{2}$ 552.4 for 495 degrees of freedom. The F-test yields a significance of 2.9$\sigma$ for this variability.
Even though the source shows more variability in the soft part of the spectrum, the soft component contributes less than $\sim5$% of the total flux in the time averaged spectrum, and the hard component dominates the spectrum during the full QPO cycle. For this reason, we also fitted the data linking the temperature and normalisation of the [diskbb]{} component between different phases, letting the parameters of the [nthcomp]{} component free. In this case the fits yield a $\chi^{2}$ of 523.2 for 486 degrees of freedom, and the best-fitting centroid energy and width of the line are 6.56$\pm$0.07 keV, 0.45$\pm$0.15 keV, respectively. The photon index is consistent with the value we got from the time averaged spectra within errors, and the normalisation of the Gaussian line shows exactly the same behaviour as in the case when we linked the parameters of the hard component. The F-test yields a significance of 1.9$\sigma$ for the variability of the line in this case. The results of these fits are also shown in Figure \[p3\].
We get consistent results, but with larger errors bars, if we let both the photon index and normalisation of [nthcomp]{} free; the best fit yields $\chi^{2}=$ 501.9 for 468 degrees of freedom for orbit 1, and $\chi^{2}=$ 480.2 for 498 degrees of freedom for orbit 2. In this case an F-test indicates that the significance level of variability of the Gaussian normalisation in orbit 1 is 1.6$\sigma$. The Gaussian normalisation does not show a similar modulation in the second and third orbit.
In order to confirm that the modulation is not affected by the choice of the model, we first excluded the energy range between 5–8 keV of the phase-resolved spectra, and fitted the spectra with a model without the iron emission line, [tbabs\*(diskbb+nthcomp)]{}. We got the best-fitting parameters by letting the temperature and normalisation of [diskbb]{}, and the photon index and normalisation of [nthcomp]{} free to change across phases. We then fitted the full energy range (3–25 keV) spectra with a model that included an extra Gaussian component. All the parameters of the continuum components were fixed at the best-fitting values we found when we excluded the 5–8 keV range. The centroid energy and the width of the Gaussian component were linked to be the same, while the normalisation of the Gaussian component was free to change, as a function of phase. The Gaussian normalisation shows the same modulation, and an F-test yields a significance of 3.9$\sigma$ for this variability. Additionally, to confirm such modulation in a model-independent way, we extracted two light curves, one between 5.5–7.5 keV (the mean count rate is 86.5$\pm$0.2 counts/s) to include the energy range of the line, and the second one in the 7.5–12.0 keV energy band (the mean count rate is 84.9$\pm$0.2 counts/s). We subtracted one light curve from the other to remove the contribution of the continuum, and then computed the Lomb-Scargle periodogram of the subtracted light curve; the result is shown in Figure \[p6\]. There is a peak in the Lomb-Scargle periodogram at 0.0227 Hz, which is at twice the frequency of the 11.3 mHz QPO shown in Figure \[p1\]. This is consistent with the modulation of the Gaussian normalisation found in the first orbit. The single trial false-alarm probability of this modulation is 5$\times$$10^{-5}$. This shows that the flux modulation of the iron emission line is model-independent and highly significant.
To explore whether the modulation of the Gaussian normalisation is present only during parts of orbits 2 and 3, and whether the modulation changed during orbit 1, we computed a dynamical power spectrum (DPS) of the three orbits. For this we calculated the Lomb-Scargle periodograms in a sliding window of 1200 seconds, shifted by 100 seconds each time to produce periodograms for a series of overlapping time intervals for the three orbits separately. The DPS are shown in Figure \[p4\]. The colour indicates the Lomb-Scargle power at the corresponding frequency, while the white dashed line indicates the frequency that we used to create the phase-resolved spectra in Figure \[p3\]. The QPO frequency in all orbits is relatively stable; variations are less than 10%, yet in the second orbit the QPO frequency shows slightly more variability compared to the first and the third orbits.
We created phase-resolved spectra for each sub-datasets within a sliding data window of $\sim$1200 seconds, each time fitting the spectra with the same model, focusing on the fitting result of the Gaussian normalisation only. The width of the window ensured that we included 14 full QPO cycles. We found the same modulation of the Gaussian normalisation in all the sub-datasets of the first orbit; in the second orbit the modulation is present at the beginning of the observation (Figure \[p5\]A) which shows a variability of 1.7$\sigma$ significant, except that the phase of the modulation was shifted with respect to the phases of the first orbit, and appears to be absent in the rest of the observations (Figure \[p5\]). We show the DPS for the second orbit in the left panel of Figure \[p5\], and in the right panels we show the results of the Gaussian normalisation for the three selected sub-datasets (marked with red triangles). Sub-panel A shows a modulation similar to what we have found in the first orbit, with the two peaks appearing at phase 0.3 and 0.9 of the QPO cycle. This modulation corresponds to the time interval marked with the red line in the left panel of the plot. Point B indicates the time of the second orbit at which the QPO period shows the largest deviation from the mean value of the full dataset. At point C, the period of the QPO returns back to the mean value.
![Dynamic power spectra of H1743–322 for the three *RXTE* orbits. The colour indicates the Lomb-Scargle power at the corresponding frequency in a sliding ‘data window’ of 1200 seconds width. The horizontal axis marks the start time of the ‘data window’ which was shifted 100 seconds each time. The white dashed line indicates the frequency of the peak Lomb-Scargle power for the full dataset of each orbit.[]{data-label="p4"}](p5_4.pdf){width="8cm"}
Discussion
==========
The black-hole candidate H1743–322 showed a unique QPO at $\sim11$ mHz during its 2010 and 2011 outbursts, so far only detected in two *RXTE* and one *Chandra* observations of this source. Here we analysed the two *RXTE* observations that show this mHz QPO. The frequency of the mHz QPO is the same in these two outbursts, whereas the frequency of a type-C QPO in these observations varies by a factor of two. We carried out phase-resolved spectroscopy on the period of the mHz QPO using the *RXTE* observations, and found a modulation of the Gaussian line normalisation in the first orbit of the *RXTE* data, with the line normalisations showing two maxima and two minima in one QPO cycle. This modulation is not affected by the continuum model. Similar, yet weaker modulation, is present at the beginning of the data of the second orbit, but this modulation disappeared as the observation progressed. The line flux (proportional to the Gaussian normalisation, since the line energy and width are linked across phases) modulation is very similar to what [@ingram2016] found for the type-C QPO. We interpret the line flux modulation of this mHz QPO in terms of Lense-Thirring precession of the inner flow and discuss the possible connection with a similar modulation of the line parameters with the period of the type-C QPO in the following sections.
Possible interpretation of the modulation of the line flux
----------------------------------------------------------
Our data show a modulation of the line flux with the 11 mHz QPO in the first orbit, reminiscent to the modulation of the Gaussian line centroid energy and flux with twice the frequency of the type-C QPO in this source [@ingram2016; @ingram2017]. In that case the frequency of the type-C QPO was $\sim0.20$ to $\sim0.25$ Hz, about 20 times higher than the frequency of the mHz QPO we analysed here. [@ingram2016] combined five *XMM-Newton* observations and found a modulation of the iron line centroid energy with a significance of 3.7$\sigma$, whereas the modulation of the line flux was only 1.3$\sigma$ significant in the other observations. For another observation, which they called anomalous (see @ingram2016), the line flux shows a large amplitude modulation in phase with the modulation of the line energy, but anticorrelated with the modulation of the line energy seen in the other data sets. [@ingram2016] explained the red and blue-shift of the iron line as due to Lense-Thirring precession of the hot inner flow around the BH. The disc is truncated at some radius larger than the innermost stable circular orbit (ISCO), forming a large scale-height hot inner flow in this region, which is misaligned with respect to the accretion disc [@done2007]. Different azimuths of the disc will be illuminated by both the front and back part of the precessing inner flow, resulting in oscillations of the line energy with two maxima and two minima in one QPO cycle when the approaching and the receding sides of the disc are illuminated; the maximum of the line flux occurs when the illuminated part of the disc faces towards us.
Our data show a modulation of the line flux that changes by a factor of $\sim4$ in the first orbit, which is similar to the case of the anomalous data set in [@ingram2016] where it was interpreted as due to the change of the ionisation state of the disc. According to [@ingram2016], as the misalignment between the disc and inner flow caused by the precession motion of the inner flow changes, the disc ionisation state will also change within the QPO cycle, due to changes of the number of hard continuum photons that are intercepted by the disc; this in turn leads to a modulation of the line flux in phase with the modulation of the line energy. Unfortunately, the limited *RXTE* energy resolution makes it impossible to resolve a possible energy shift of the emission line, so it is unclear whether or how the line energy changes with the line flux in our observation.
A weaker modulation likely present at the beginning of the second [*R*XTE]{} orbit (Figure \[p5\]A), that disappeared as the observation progressed further, suggests that the process that drives the modulation is unstable. This could be due to an unstable inner flow geometry, e.g., if the misalignment angle between the disc and the inner flow decreases, the disc ionisation would also decrease, leading to a lack of line flux modulation. We also noticed that, in the data of the second orbit, the line flux modulation has a phase shift compared to the modulation in the first orbit. Fits to these two modulations with a sine function yields a phase difference of 0.2 of the oscillation period, suggesting a possible change of the inner flow geometry between these two orbits. For a 10$M_{\odot}$ BH, the structure, for instance two hot spots in the inner flow, that produce this $\sim11$ mHz QPO would move $\sim1000$ km within $\sim2400$ seconds interval between the first two orbits of [*R*XTE]{} observation.
Line flux modulation for different types of QPOs
------------------------------------------------
Source QPO frequency rms(%) X-ray state$^{a}$ inclination note
-------------------- ------------------------ ------------------- ------------------- -------------------------------- ------
H1743–322 11 mHz $\sim3$ (rising) LHS $\sim60^{\circ}-70^{\circ}$ 1
V404 Cyg 18 mHz $18\pm2$(*Swift*) (rising) LHS $67^{\circ}$$_{-1}^{+3}$ 2
73 mHz $27\pm3$ (rising) LHS
136 mHz $8\pm2$ SIMS
IC 10 X-1 6.3 mHz $11\pm3$ - $\ge63^{\circ}$ 3
Swift J1357.2–0933 5.9 mHz $12\pm3$ LHS $\gtrsim70^{\circ}$ 4
XTE J1118+480 69-159 mHz $\sim4-10$ LHS $68^{\circ}-79^{\circ}$ 5
LMC X-1 $\sim$27 mHz $\sim1-2$ HSS $36.38^{\circ}\pm1.92^{\circ}$ 6
Cygnus X-3 8.5 mHz & 30 mHz $\lesssim3$ HSS likely low inclination 7
9 mHz, 21 mHz & 31 mHz $\lesssim2$ HSS
MAXI J1820+070 30-42 mHz - LHS/HIMS - 8
$^{a}$LHS refers to Low-Hard state, HSS refers to High-Soft state, SIMS refers to Soft-Intermediate state, HIMS refers to High-Intermediate state.
Notes: 1. Type-C QPOs have been detected simultaneously with the $\sim11$ mHz QPOs in the same observations during the 2010 and 2011 outbursts at 0.919 Hz (rms 12%) and 0.424 Hz (rms 11%), respectively [@diego2012]; 2. A LFQPO at 1.03 Hz (rms 46%) has been detected simultaneously with the 73 mHz QPO in the [*C*handra]{} observation. The fractional rms amplitude is strongly energy-dependent, there is strong radio activity coincident with the X-ray flaring [@huppenkothen2017; @mooley2015]; 3. The fractional rms amplitude of the QPO is roughly energy-independent in the $0.3-1.5$ keV energy range [@pasham2013; @laycock2015]; 4. The source shows optical dips at frequencies similar to the frequency of the mHz QPOs [@padilla2014]; 5. Similar QPOs feature were observed in X-rays, EUV and optical [@wood2000; @khargharia2013; @revnivtsev2000; @haswell2000]; 6. The presence of the mHz QPOs and the broad iron line in the X-ray emission appears to depend on the presence of a strong power-law component [@alam2014; @orosz2009]; 7. The presence of mHz QPOs is associated to major radio flaring events [@koljonen2011; @vilhu2013]; 8. [@mereminskiy2018].
\[t2\]
[@ingram2016; @ingram2017] found that the centroid energy and flux of the iron line oscillate at twice the frequency of the type-C QPO, which at that time was $\sim$0.20–0.25 Hz; here we found that the line flux is modulated at twice the frequency of the QPO at $\sim$11 mHz. It is then quite interesting to know how the iron line flux can be modulated by different types of QPOs. Type-C QPOs were also detected in our observations; [@diego2012] reported the type-C QPO and its second harmonic at 0.9 Hz and 1.8 Hz in the 2010 observation, and at 0.4 Hz and 0.8 Hz in the 2011 observation (see also Figure \[p1\]). The mHz QPO shows a similar frequency in both observations, however, the frequency of the type-C QPO in these two observations differs by a factor of $\sim$2; because of this [@diego2012] concluded that the mHz and type-C QPO may be produced by different mechanisms.
Assuming that the modulation of the iron line energy/normalization found by [@ingram2016] is always present (including at the time of the observations analyzed in this paper), our results imply that the line flux modulation remains the same even if the QPO frequency changes by more than an order of magnitude. Under the interpretation of Lense-Thirring precession [@ingram2016], our results could be explained if the mHz QPO is due to a precessing torus with a radius that is $\sim3-5$ times larger than the radius of the precessing inner flow predicted by the frequency of the type-C QPOs.
The possibility of the existence of this two-components inner flow is interesting, yet it still needs to be tested. [@qu2010] and [@yan2012] found that the centroid frequency of the LF QPOs in GRS 1915+105 decreases with energy when the QPO frequency is below $\sim2$ Hz, but increases with energy when the QPO is above $\sim2$ Hz. A similar behaviour has also been reported for XTE J1550–564 [@li2013a] and H1743–322 [@li2013b]: in both sources the frequency increases with energy when the QPO frequency is higher than a certain value (this turn-over frequency is different for different sources). However, GRS 1915+105 is the only source that shows decreases of QPO frequency with energy, which is still unexplained by current models. To explain these results, [@eijnden2016a] proposed a geometric toy model of differential precession [@ingram2009] by considering the inner flow as two separate components: an outer and an inner half, with the latter always producing a QPO at higher frequency than the former. For higher full-band QPO frequencies, the spectrum of the inner half is harder than that of the outer half, whereas the outer half is harder when the full-band QPO frequencies are lower than the turn-over frequency. This spectral evolution could either be due to the ionisation state changes of the reflected component which dominated the outer part, or to the presence of an extra cooling process very close to the black hole.
This is consistent with what we have observed here, as the phase-resolved spectroscopy of H1743–322 shows that the variability of the spectra among different phases are dominated by the soft photons for this mHz QPO, which are likely coming from the outer half of the inner flow. [@ingram2016] showed that in H1743–322 the power-law index changes from $\sim1.26$ to $\sim1.29$ during the $\sim0.2$ Hz QPO cycles, which is significantly harder than the values found in the average spectra of our observations for the mHz QPO, $1.60\pm0.01$ for orbit 1, $1.59\pm0.01$ for orbit 2, and $1.53\pm0.01$ for orbit 3. These results indicate that the $\sim11$ mHz QPOs may originate in the outer (cooler) region of the precessing inner flow, while the type-C QPOs originate in the hotter inner half of that flow.
The detailed structure of an inner flow that can produce these two independent LF QPOs simultaneously needs to be investigated. For the case of GRS 1915+105, [@qu2010] found that the largest variation of the QPO frequency with energy occurs when the QPO is at $\sim6$ Hz, where it changes from $\sim5.9$ Hz in the 1.94–5.12 keV band to $\sim6.7$ Hz in the 18.09–38.44 keV band. If $R_{\rm in}$ is the distance from the BH to the inner half precessing inner flow, for GRS 1915+105 the outer half could extend up to a radius of 1.04$R_{\rm in}$. For the case of H1743–322, the mass of the BH is thought to be $\sim$10 M$_{\odot}$ with a spin parameter of 0.2 [@steiner2012]. If the type-C QPO is due to Lense-Thirring precession, the inner half of the inner flow would have a radius of 10 $r_{\rm g}$ during the 2010 outburst, and 13 $r_{\rm g}$ during the 2011 outburst; the frequency of the mHz QPO is stable in these two outbursts, and the outer half of the inner flow would have a radius of 46 $r_{\rm g}$. Compared to GRS 1915+105, this requires a large truncation radius to form such an extended structure. [@diego2012] found that the frequency of this mHz QPO changes by less than $\sim$1.5 mHz during the 60 ks *Chandra* observation in the 2010 outburst, whereas [@ingram2016] found that the type-C QPO frequency changed from $\sim0.2$ Hz to $\sim0.25$ Hz during the *XMM-Newton* observation in the 2014 outburst. Those facts indicate that, if the mHz QPO and the type-C QPO may both originate from the precessing inner flow (at different radii), the structure of these two components should be relatively independent, and the extreme environment of the inner region near the BH must be much more unstable and complicated than the outer region.
Besides the $\sim11$ mHz QPOs in H1743–322 [@diego2012], several other sources have shown QPOs with frequency in the mHz range (see Table \[t2\]). Unlike the typical LFQPO in BHXRBs, these mHz QPOs appear at frequencies that are lower than most of the type-A ($\sim$ 8 Hz), type-B ($\sim$5-6 Hz) and type-C ($\sim$0.1-15 Hz) QPOs. Except for LMC X-1 and Cygnus X-3, the mHz QPOs in these systems have relatively large fractional rms amplitude and appear during the Low-Hard State (LHS), which distinguishes them from the so-called ‘heartbeat’ QPOs found in GRS 1915+105 [@belloni2000] and IGR J17091–3624 [@altamirano2011] in the mHz range.
Most of the sources listed in Table \[t2\] are believed to have high orbital inclinations, which appears to connect these mHz QPOs to the so-called 1-Hz QPO observed in dipping neutron star X-ray binaries. The frequency of the 1-Hz QPO has been observed to vary between 0.4 and 3.0 Hz [@homan1999; @homan2012]; these QPOs have been explained as obscuration by a radiation-driven warping of the accretion disc [@jonker1999; @pringle1996], or to a precessing inner accretion disc [@homan2012]. From the list of sources in Table \[t2\], Swift J1357.2–0933 and XTE J1118+480 have shown both optical dips at a similar frequency as the mHz QPOs [@padilla2014; @wood2000]; this raises the possibility that the structure obscuring the optical emission may also cause the mHz QPO. However, in Swift J1357.2–0933 the optical dips were detected in several occasions at a frequency that decreased as the X-ray luminosity decreased, whereas the X-ray mHz QPOs were only detected in one observation and were significantly absent in the other cases in which optical dips were observed [@padilla2014]. Therefore, there is no direct evidence that these two phenomena are physical related. Given the presence of the mHz QPOs associated with radio flaring events in V404 Cyg and Cygnus X-3 [@huppenkothen2017; @koljonen2011], an alternative explanation is that these QPOs are produced at the base of a precessing jet [@markoff2005; @kalamkar2016].
LMC X-1 and Cygnus X-3 share more common features, but appear to be part of a different group compared with the other six sources with mHz QPOs listed in Table \[t2\]: in these two sources the mHz QPOs have relative low fractional rms amplitudes ($\lesssim$ 3%), the QPOs appear in the High-Soft State (HSS), and the two systems are believed to have low orbital inclinations. Those facts suggest that the mHz QPOs in low-inclination systems would either have a different origin compared with mHz QPOs in high-inclination systems or, just like the type-C QPOs, the mHz QPOs may also have a geometric origin which leads to the inclination dependent properties: the oscillations are stronger for high-inclination than for low-inclination sources [@motta2015; @eijnden2017]. If this is true, then it should not be a surprise that the line flux modulation with the mHz QPO frequency is similar to that of the type-C QPOs. However, H1743–322 is exceptional in this respect, as the strong hard dips found in the light curves in this source suggest a high system inclination ($\sim$60$^{\circ}$-70$^{\circ}$, @homan2005) but with a relative low rms amplitude ($\sim$ 3%, @diego2012). [@steiner2012] measured the jets inclination of H1743–322 as $75^{\circ}\pm3^{\circ}$, indicating that there is possibly a very large misalignment between the BH equatorial plane and the accretion disc compared with other sources, and this may be the reason that makes the geometry of the inner flow in H1743–322 different from that in the other high inclination systems.
We also noticed that in the observations of H1743–322 and V404 Cyg that show two distinct LFQPOs simultaneously at, respectively, 10–70 mHz and 0.5–1 Hz, the QPOs at 0.5–1 Hz have a significantly higher rms amplitude than the QPOs at mHz frequencies (see Table \[t2\] Notes 1 and 2; there is not enough information about the LFQPOs in Cygnus X-3 yet to draw a similar conclusion). It remains to be seen whether this trend holds in all cases.
Since the list in Table \[t2\] only includes eight sources with mHz QPO detection, while there is not much information yet available for the newly discovered BHC MAXI J1820+070, the reliability of the geometric origin of these mHz QPOs is still unclear. As the other interpretations still have difficulties in explaining the line flux modulation of the mHz QPOs, the Lense-Thirring precession of the inner flow at large radii is more likely to be the cause of the mHz QPOs in H1743–322. More observations of these LFQPOs are required to reveal the possible relations between the different types QPOs, to help us better understanding their physical origins and, perhaps, the geometry of the accreting flow close to the BH.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research has made use of data and software provided by the High Energy Astrophysics Science Archive Research Center (HEASARC), which is a service of the Astrophysics Science Division at NASA/GSFC and the High Energy Astrophysics Division of the Smithsonian Astrophysical Observatory. ZC thanks Jamie M.C. Court for the helpful discussion about the algorithm in creating GTIs of each phase. DA acknowledges support from the Royal Society. A.B acknowledges support from the Royal Society and SERB (Science & Engineering Research Board, India) through Newton-Bhabha Fund.
[^1]: Email: zheng@astro.rug.nl
|
---
abstract: 'The Cramér-Rao product of the Fisher information $F[\rho]$ and the variance $\langle \mathbf{x}^2\rangle\equiv\int \mathbf{x}^2\rho(\mathbf{x})d\mathbf{x}$ of a probability density $\rho(\mathbf{x})$, defined on a domain $\Delta \subset \mathbb{R}^D$, is found to have a minimum value reached by the density associated with the ground state of the harmonic oscillator in $\Delta$, when $\Delta$ is an unbounded domain. If $\Delta$ is bounded, the minimum value of the Fisher information is achieved by the ground state of the quantum box described itself by this domain.'
author:
-
bibliography:
- 'cramer.bib'
title: 'Generalized Cramér-Rao relations for non-relativistic quantum systems'
---
Introduction
============
The Cramér-Rao inequality belongs to a natural family of information-theoretic inequalities [@cover_thomas; @johnson_04; @bercher.is09] which play a relevant role in a great variety of scientific and technological fields ranging from probability theory [@cramer_46; @renyi_70], communication theory [@dembo.itit91], signal processing [@kay_93] and approximation theory [@dehesa.jcam06] to quantum physics of $D$-dimensional systems with a finite number of particles [@romera.pra94; @dehesa.jpa07; @angulo_10]. Recently the general Cramér-Rao inequality [@cover_thomas; @dembo.itit91] $$F\langle \mathbf{x}^2\rangle \ge D^2,
\label{eq.cramerrao}$$ valid for all ground and excited states of any $D-$dimensional quantum system, has been substantially improved [@dehesa.jpa07] in the case of centrally-symmetric potentials as $$F\langle \mathbf{x}^2\rangle \ge 4\left(1-\frac{2|m|}{2l+D-2}\right)\left(l+\frac{D}{2}\right)^2$$ where the integer numbers $(l,m)$ denote the hyperquantum orbital and magnetic quantum numbers having the values $0 \le l \le n-1$ and $-l \le m \le + l,$ the integer $n$ being the hyperquantum principal number. It is worth pointing out that the lower bound to the Cramér-Rao product is equal to $D^2$ when $l=0$, i.e. for states $s$.
Moreover, the Cramér-Rao inequality has been shown to be closely connected to the Heisenberg uncertainty inequality by various authors [@dembo.itit91]. In particular, it is fulfilled that $$F\langle \mathbf{x}^2\rangle \ge 4\left(1-\frac{2|m|}{2l+D-2}\right)\langle r^2\rangle\langle p^2\rangle$$ is valid for central potentials [@dehesa.jpa07]. Since the Cramér-Rao inequality (\[eq.cramerrao\]) is saturated by Gaussian probability densities, it can be interpreted as a measure of the amount of non-normality of the quantum-mechanical probability density which describes the involved quantum state of the physical system under consideration [@dembo.itit91].
This work, which has been motivated by the recent findings in [@bercher.is09], provides the best Cramér-Rao lower bound for general $D$-dimensional systems on unbounded domains in Section \[sec2\]. It is shown that this lower bound is reached for the probability density characterizing the oscillator ground state, and that its value is controlled by the corresponding ground state energy. Next, in Section \[sec3\], we show that the minimal value of the Fisher information of general $D$-dimensional systems defined on bounded domains is achieved for the ground state of the quantum box described by this region.
Unbounded $D$-dimensional domains {#sec2}
=================================
Notations and problem
---------------------
Let us consider an unbounded domain $\Delta$ in $\mathbb{R}^D$. The problem consists in finding the normalized probability density $\rho(\mathbf{x})=u^2(\mathbf{x})$, with $\mathbf{x}\in\Delta$, that minimizes the Fisher information with the constraints that the variance $\langle \mathbf{x}^2\rangle$ has a given value, and that $u(\mathbf{x})=0$ $\forall \mathbf{x}\in \mathcal{D}(\Delta)$, the frontier of $\Delta$. The Fisher information will be denoted as $F[\rho]$ or simply $F$ when there is no ambiguity about the considered distribution; it is defined as
$$F[\rho] = 4\int_\Delta |\boldsymbol\nabla u(\mathbf{x})|^2d\mathbf{x}.$$
The Lagrangian of this problem is $$\begin{aligned}
\mathcal{L}=4\int_\Delta |\boldsymbol\nabla u(\mathbf{x})|^2d\mathbf{x}
+\alpha\left[\int_\Delta u^2(\mathbf{x})d\mathbf{x}-1\right] \\
+\beta\left[\int_\Delta \mathbf{x}^2 u^2(\mathbf{x})d\mathbf{x}-\langle\mathbf{x}^2\rangle\right],\end{aligned}$$ where $d\mathbf{x}=\prod_{i=1}^{D}dx_{i}$, $\mathbf{x}^2=\sum_{i=1}^{D}x_{i}^2$, and $$|\boldsymbol\nabla u(\mathbf{x})|^2=\sum_{i=1}^D\left(\frac{\partial u}{\partial x_i}\right)^2.$$ The associated Euler-Lagrange equation states that $$\sum_{i=1}^D\frac{\partial}{\partial x_i}\frac{\partial \mathcal{L}}{\partial\left(\frac{\partial u}{\partial x_i}\right)}-\frac{\partial \mathcal{L}}{\partial u}=0,$$ which yields the differential equation $$8\boldsymbol\nabla^2u(\mathbf{x})-2\alpha u(\mathbf{x})-2\beta \mathbf{x}^2u(\mathbf{x})=0.$$ This equation, together with the boundary condition $u(\mathbf{x})=0$, $\forall \mathbf{x}\in \mathcal{D}(\Delta)$, coincides with the Schrödinger equation $$-\frac12\boldsymbol\nabla^2u(\mathbf{x})+V(\mathbf{x})u(\mathbf{x})=-\frac{\alpha}{8}u(\mathbf{x})$$ with the potential $$\begin{aligned}
\label{potentialunbounded}
V(\mathbf{x})=\left\{
\begin{array}{ll}
\frac{\beta}{8}\mathbf{x}^2 & \text{if } \mathbf{x}\in\Delta,\\
0 & \text{if }\mathbf{x}\notin\Delta.
\end{array}
\right.\end{aligned}$$ Then, the Lagrange parameter $\beta$ must be strictly positive in order to obtain integrable solutions.
The set of densities with a given variance $\langle\mathbf{x}^2\rangle$ is a convex set, and since the Fisher information is a convex functional, its minimum, if it exists, is unique. However, it is not possible to find a minimal value of the Fisher information regardless of the value of $\langle \mathbf{x}^2\rangle$: the reason is that, due to the nature of the harmonic potential, the factor $\beta^\frac14$ acts only as a scale factor for the resulting probability density function. Then, since the Fisher information scales as $\beta^{-1/2}$, it is possible to reach arbitrarily low values of the Fisher information by modifying this scale factor. Nevertheless, since the product $F\langle\mathbf{x}^2\rangle$ is scale invariant, it is independent of $\beta$, and a minimum should exist. This minimum is characterized by the following theorem.
In the case of an unbounded domain $\Delta$, the minimum value of the Cramér-Rao product $F\langle\mathbf{x}^2\rangle$ verifies $$E^2=\frac{\beta}{16}F\langle\mathbf{x}^2\rangle$$ where $E$ is the energy of the ground state of the quantum system with potential defined by (\[potentialunbounded\]). It is reached by the probability density associated to the ground state of this system.
We apply the virial theorem, that ensures the existence of this minimum and points to it: for the potential (\[potentialunbounded\]), the virial theorem establishes that the kinetic energy $\langle T\rangle=\langle V\rangle$. Then, the total energy $E=\langle T\rangle+\langle V\rangle=2\langle T\rangle=2\langle V\rangle$, and $E^2=4\langle T\rangle\langle V\rangle$. As $\langle T\rangle=F/8$ for real wave functions, and since $\langle V\rangle=\frac{\beta}{8}\langle \mathbf{x}^2\rangle$, we have that $$%\label{FVunbounded}
E^2=\frac{\beta}{16}F\langle\mathbf{x}^2\rangle.$$ Then, the minimum value of the product $F\langle\mathbf{x}^2\rangle$ is reached for the minimum value of the energy $E$, i.e. by the ground state of this quantum system.
Example 1
---------
Let us consider the case of the positive half plane $$\Delta=\{(x,y)\in\mathbb{R}^2, x>0\}.$$
The Schrödinger equation reads $$\begin{gathered}
-\frac12 \left(\frac{\partial^2 u(x,y)}{\partial x^2}+\frac{\partial^2 u(x,y)}{\partial y^2}\right)
+\frac{\beta}{8}(x^2+y^2)u(x,y)\\
=-\frac{\alpha}{8}u(x,y),\end{gathered}$$ with the constraint $u(0,y)=0 \,\, \forall y \in \mathbb{R}$.
The solutions are of the form $$\begin{aligned}
u_{n_1,n_2}(x,y)=\sqrt{\frac{2^{-2n_1-n_2-1}\sqrt{\beta}}{\pi (2n_1+1)! n_2!}} e^{-\frac{\sqrt{\beta}}{4}(x^2+y^2)} \\
\times H_{2n_1+1}\left(\frac{\beta^\frac14}{\sqrt{2}}x\right) H_{n_2}\left(\frac{\beta^\frac14}{\sqrt{2}}y\right) \end{aligned}$$ where $H_{n}(x)$ is the Hermite polynomial of degree $n$. The corresponding energy levels are $$E_{n_1,n_2}=-\frac{\alpha}{8}=\frac{\sqrt{\beta}}{2}(2n_1+n_2+2).$$
The density $\rho_{n_1,n_2}$ is defined as $$\rho_{n_1,n_2}(x,y)=u_{n_1,n_2}^2(x,y);$$ its variance is $$\langle x^2+y^2\rangle_{n_1,n_2}=\frac{2}{\sqrt{\beta}}(2n_1+n_2+2),$$ and its Fisher information $$F[\rho_{n_1,n_2}]=2\sqrt{\beta}(2n_1+n_2+2).$$
Notice that, as $\beta$ is a scale factor, the product of these two quantities does not depend on $\beta$ and equals $$F[\rho_{n_1,n_2}]\langle x^2+y^2\rangle_{n_1,n_2}=4(2n_1+n_2+2)^2.$$ As predicted by the virial theorem, its minimum value is obtained for the ground state described by the density $$\rho_{0,0}(x,y)=\frac{\beta}{\pi} x^{2} e^{-\frac{\sqrt{\beta}}{2}(x^2+y^2)}$$ and is equal to $$F[\rho_{0,0}]\langle x^2+y^2\rangle_{0,0}=16,$$
Bounded $D$-dimensional domains {#sec3}
===============================
Statement of the problem
------------------------
In the case of a bounded domain $\Delta$, the sign of the Lagrange multiplier $\beta$ in the equation $$-\frac12\boldsymbol\nabla^2u(\mathbf{x})+\frac{\beta}{8}\mathbf{x}^2u(\mathbf{x})=-\frac{\alpha}{8}u(\mathbf{x})
\label{eq.schrodinger}$$ cannot be fixed as in the case of unbounded domains.
As the values of the $D$-dimensional variable $\mathbf{x}$ are bounded in $\Delta$, its variance $\langle \mathbf{x}^2\rangle$ is also bounded. Then, there must exist a value of the constraint $\langle \mathbf{x}^2\rangle_{*}$ for which the minimal Fisher information $F_*$ is achieved. This value $F_*$ would be the minimal value of the Fisher information among all the densities defined in $\Delta$.
We need the two following propositions to find this minimal value and the density that achieves it.
Let $$g_\epsilon(\mathbf{x})=\epsilon u^2(\mathbf{x})+(1-\epsilon) v^2(\mathbf{x})$$ with $0\le \epsilon <1$. Then a first order expansion of the Fisher information of $g_\epsilon$ is $$F[g_\epsilon]=F[v^2]-\beta\epsilon (\langle\mathbf{x}^2\rangle_u-\langle\mathbf{x}^2\rangle_v)+o(\epsilon^2).$$
The Fisher information of $g_\epsilon$ is $$\begin{aligned}
&F[g_\epsilon]=\int_\Delta \frac{|\boldsymbol\nabla g_\epsilon(\mathbf{x})|^2}{g_\epsilon(\mathbf{x})} d\mathbf{x}\\
&=4\int_\Delta \frac{v^2(\mathbf{x})|\boldsymbol\nabla v(\mathbf{x})|^2}{g_\epsilon(\mathbf{x})} d\mathbf{x}\\
&+8\epsilon \int_\Delta \frac{u(\mathbf{x})v(\mathbf{x})\boldsymbol\nabla u(\mathbf{x})\cdot\boldsymbol\nabla v(\mathbf{x})-v^2(\mathbf{x})|\boldsymbol\nabla v(\mathbf{x})|^2}{g_\epsilon(\mathbf{x})} d\mathbf{x}\\
&+4\epsilon^2 \int_\Delta \frac{u^2(\mathbf{x})|\boldsymbol\nabla u(\mathbf{x})|^2+v^2(\mathbf{x})|\boldsymbol\nabla v(\mathbf{x})|^2}{g_\epsilon(\mathbf{x})} d\mathbf{x}\\
&+4\epsilon^2 \int_\Delta \frac{-2u(\mathbf{x})v(\mathbf{x})\boldsymbol\nabla u(\mathbf{x})\cdot\boldsymbol\nabla v(\mathbf{x})}{g_\epsilon(\mathbf{x})} d\mathbf{x}\\
%&=&4\int_\Delta \frac{v^2(\mathbf{x})|\mathbf{\nabla}v(\mathbf{x})|^2+2\epsilon v(\mathbf{x}) (u(\mathbf{x})\mathbf{\nabla}v(\mathbf{x})\cdot\mathbf{\nabla}u(\mathbf{x})-v(\mathbf{x})|\mathbf{\nabla}v(\mathbf{x})|^2)}{\epsilon u^2(\mathbf{x})+(1-\epsilon) v^2(\mathbf{x})} d\mathbf{x}\\
%&& + 4\epsilon^2 \int_\Delta \frac{u^2(\mathbf{x})|\mathbf{\nabla}u(\mathbf{x})|^2+v^2(\mathbf{x})|\mathbf{\nabla}v(\mathbf{x})|^2-2u(\mathbf{x})v(\mathbf{x})\mathbf{\nabla}u(\mathbf{x})\cdot\mathbf{\nabla}v(\mathbf{x})}{\epsilon u^2(\mathbf{x})+(1-\epsilon) v^2(\mathbf{x})} d\mathbf{x}\end{aligned}$$ Denoting the integrand in the two first integrals by $G(u,v)$, we perform a Taylor expansion in terms of $\epsilon$ around $\epsilon=0$, yielding the expression $$G(u,v)=G_0(u,v)+\epsilon G_1(u,v)+o(\epsilon^2),$$ where $$G_0(u,v)=|\boldsymbol\nabla v(\mathbf{x})|^2,$$ and $$G_1(u,v)=\boldsymbol\nabla v(\mathbf{x})\cdot\boldsymbol\nabla \left(\frac{u^2(\mathbf{x})-v^2(\mathbf{x})}{v(\mathbf{x})}\right).$$
Then, $$F[g_\epsilon]=F[v^2]+4\epsilon \int_\Delta \boldsymbol\nabla v(\mathbf{x})\cdot\boldsymbol\nabla \left(\frac{u^2(\mathbf{x})-v^2(\mathbf{x})}{v(\mathbf{x})}\right) d\mathbf{x}+o(\epsilon^2).$$ An integration by parts yields: $$\begin{aligned}
\int_\Delta\boldsymbol\nabla v(\mathbf{x})\cdot\boldsymbol\nabla \left(\frac{u^2(\mathbf{x})-v^2(\mathbf{x})}{v(\mathbf{x})}\right) d\mathbf{x}\\
=\int_{\mathcal{D}(\Delta)} \frac{u^2(\mathbf{x})-v^2(\mathbf{x})}{v(\mathbf{x})} \boldsymbol\nabla v(\mathbf{x})\cdot \mathbf{\nu} d\gamma_\Delta\\
-\int_\Delta \frac{u^2(\mathbf{x})-v^2(\mathbf{x})}{v(\mathbf{x})} \boldsymbol\nabla^2v(\mathbf{x})d\mathbf{x}\end{aligned}$$ where $d\gamma_\Delta$ is the infinitesimal element of the border $\mathcal{D}(\Delta)$ of $\Delta$ and $\mathbf{\nu}$ is a vector perpendicular to it.
As $u(\mathbf{x})=0$ and $v(\mathbf{x})=0$ for $x\in \mathcal{D}(\Delta)$, this result yields $$F[g_\epsilon]=F[v^2]-4\epsilon\int_\Delta \frac{u^2(\mathbf{x})-v^2(\mathbf{x})}{v(\mathbf{x})} \boldsymbol\nabla^2v(\mathbf{x})d\mathbf{x}.$$ Now we use the Schrödinger equation $$-\frac12\boldsymbol\nabla^2v(\mathbf{x})+\frac{\beta}{8}\mathbf{x}^2 v(\mathbf{x})=-\frac{\alpha}{8}v(\mathbf{x}),$$ so that we obtain the result $$F[g_\epsilon]=F[v^2]-\beta\epsilon(\langle \mathbf{x}^2\rangle_u-\langle\mathbf{x}^2\rangle_v)+o(\epsilon^2).$$
Let us denote by $\langle \mathbf{x}^2\rangle_*$ the value of the variance that, after solving the variational problem, corresponds to the minimal value of the Fisher information $F_*$. Then the Lagrange multiplier $\beta$ of the problem has the same sign as $\langle \mathbf{x}^2\rangle_{*} - \langle \mathbf{x}^2\rangle$, that is
- if $\langle \mathbf{x}^2\rangle<\langle \mathbf{x}^2\rangle_*$ then $\beta>0$,
- if $\langle \mathbf{x}^2\rangle>\langle \mathbf{x}^2\rangle_*$ then $\beta<0$,
- if $\langle \mathbf{x}^2\rangle=\langle \mathbf{x}^2\rangle_*$ then $\beta=0$.
Let $u^2(\mathbf{x})$ and $v^2(\mathbf{x})$ be two distributions with respective Fisher informations $F[u^2]$ and $F[v^2]$. According to the previous proposition, if $g_\epsilon(\mathbf{x})=\epsilon u^2(\mathbf{x})+(1-\epsilon) v^2(\mathbf{x})$. Then, $$F[g_\epsilon]=F[v^2]-\beta\epsilon (\langle\mathbf{x}^2\rangle_u-\langle\mathbf{x}^2\rangle_v)+o(\epsilon^2).
\label{eq.fgepsilon}$$ The convexity of the Fisher information gives $$F[g_\epsilon]<\epsilon F[u^2]+(1-\epsilon)F[v^2]$$ Let us now take $\langle x^2\rangle_u=\langle x^2\rangle_*$. Then, $F[u^2]=F_*$ is the minimum Fisher information, so we have the majoration $$\epsilon F[u^2]+(1-\epsilon)F[v^2]\le F[v^2].$$ Therefore, $F[g_\epsilon]<F[v^2]$, and, as a consequence, from (\[eq.fgepsilon\]) and the previous inequality, we obtain $$\beta\epsilon(\langle\mathbf{x}^2\rangle_u-\langle\mathbf{x}^2\rangle_v)>0,$$ which gives the first and second cases in the proposition. The third case follows by continuity.
For $\beta=0$, the Schrödinger equation (\[eq.schrodinger\]) becomes $$\label{schroedinger}
-\frac12\boldsymbol\nabla^2u(\mathbf{x})=-\frac{\alpha}{8}u(\mathbf{x})$$ for $\mathbf{x}\in\Delta$, with $u(\mathbf{x})=0$ $\forall \mathbf{x}\in \mathcal{D}(\Delta)$, that is the equation of the $D$-dimensional infinite potential well defined in $\Delta$.
We can now characterize the minimum Fisher information in the case of a bounded domain.
The minimum Fisher information in a bounded domain of $\mathbb{R}^D$ is reached by the probability density associated to the ground state of the quantum system whose potential is the infinite well in this domain.
Notice that if $u(\mathbf{x})$ is a solution of equation (\[schroedinger\]), then $|u(\mathbf{x})|$ is also a solution. Since the solution of the problem of finding the minimum value of $F$ for a given value of $\langle \mathbf{x}^2\rangle$ must be unique, we conclude that $u(\mathbf{x})=|u(\mathbf{x})|$, so $u(\mathbf{x})>0$ for $\mathbf{x}\in\Delta$. But the only eigenstate of the previous Schrödinger equation that does not have any zero is the ground state. Thus, the density that minimizes the Fisher information in a bounded domain is that associated to the ground state of the corresponding infinite well defined on that domain.
Example 2
---------
Let us consider the case of the rectangular domain $$\Delta=\{(x,y)\in\mathbb{R}^2,0<x<2,-1<y<2\}.$$
The Schrödinger equation reads $$-\frac12 \left(\frac{\partial^2 u(x,y)}{\partial x^2}+\frac{\partial^2 u(x,y)}{\partial y^2}\right)=-\frac{\alpha}{8}u(x,y),$$ with the constraint $u(x,y)=0$ if $(x,y)\notin\Delta$.
The solutions are of the form $$u_{n_{1},n_{2}}(x,y)=\sqrt{\frac32}\sin\left(\frac{n_1\pi}{2}x\right)\sin\left(\frac{n_2\pi}{3}(y+1)\right)$$ with energy levels $$E_{n_1,n_2}=\frac{\pi^2}{8}\left(n_1^2+\frac49 n_2^2\right).$$
The associated density is $$\rho_{n_1,n_2}(x,y)=u_{n_{1},n_{2}}^{2}(x,y).$$ The Fisher information of this density is $$F[\rho_{n_1,n_2}]=\pi ^2\left( n_1^2+\frac49 n_2^2\right).$$ (Notice that $E_{n_1,n_2}=F[\rho_{n_1,n_2}]/8$ as $\langle V\rangle=0$).
The ground state, corresponding to $n_1=n_2=1,$ reads $$\rho_{1,1}(x,y)=\frac32 \sin^2\left(\frac{\pi}{2}x\right)\sin^2\left(\frac{\pi}{3}(y+1)\right),$$ and is the only eigenstate without zeros. Its Fisher information $$F[\rho_{1,1}]=\frac{13\pi^2}{9}$$ is the minimum Fisher information that can be achieved for any density defined in the domain $\Delta$.
The expectation value of $\langle \mathbf{x}^2\rangle$ has the value $$\langle x^2+y^2\rangle_{n_1,n_2} = \frac73-\frac{1}{2\pi^2}\left(\frac{4}{n_1^2}+\frac{9}{n_2^2}\right).$$ Then, $$\langle x^2+y^2\rangle_{1,1}F[\rho_{1,1}]=\left( \frac73-\frac{13}{2\pi^2}\right)\frac{13\pi^2}{9} \approx 23.875$$ is found to be the minimal value of the Cramér-Rao product.
Conclusions
===========
In summary, we have studied the probability densities yielding the minimum value of the Cramer-Rao product in general $D$-dimensional unbounded domains and the densities minimizing the Fisher information itself in bounded domains $\Delta\in {\cal R}^D$. The present developments constitute a substantial generalization of previous results obtained by Becher and Vignat for the one dimensional case [@bercher.is09]. In the case of unbounded domains the density optimising the Cramer-Rao product turns out to be the one corresponding to the ground state of the ($D$-dimensional) harmonic oscillator in the domain. On the other hand, the optimal density minimising the Fisher information in a bounded domain $\Delta$ is given by the ground state of a quantum free particle confined within a rigid box described by the boundary of $\Delta$. This may have applications for the study of important quantum systems such as quantum billiards.
It is intriguing that the above mentioned optima are achieved by the ground states of two systems that are among the most basic (and most important) in quantum physics. This constitutes new evidence pointing towards the fundamental role played by Fisher information both in quantum mechanics and information theory, and particularly in the interface between these two fields.
Acknowledgements {#acknowledgements .unnumbered}
================
This work has been partially supported by the MICINN grant FIS2008-2380 and the grants FQM-2245 and FQM-4643 of the Junta de Andalucía. JSD, ARP and PSM belong to the research group FQM-207. C. V. thanks Pr. Dehesa for his kind invitation to Granada in November 2009 that allowed to initiate this work.
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abstract: 'The highest-mass stars have the lowest frequency in the stellar IMF, and they are also the most easily observed stars. Thus, the counting statistics for OB stars provide important tests for the fundamental nature and quantitative parameters of the IMF. We first examine some local statistics for the stellar upper-mass limit itself. Then, we examine the parameter space and statistics for extremely sparse clusters that contain OB stars, in the SMC. We find that thus far, these locally observed counting statistics are consistent with a constant stellar upper-mass limit. The sparse OB star clusters easily fall within the parameter space of Monte Carlo simulations of cluster populations. If the observed objects are representative of their cluster birth masses, their existence implies that the maximum stellar mass is largely independent of the parent cluster mass.'
author:
- 'M. S. Oey$^1$, J. B. Lamb$^1$, J. K. Werk$^1$, and C. J. Clarke$^2$'
bibliography:
- 'Oey\_MS.bib'
title: OB Stars in Stochastic Regimes
---
Introduction
============
Since massive stars are exceedingly rare, any local, empirical determination of their statistical properties is necessarily a stochastic problem. This applies to determining the slope of the initial mass function (IMF) and stellar upper-mass limit. While the properties of massive stars are inferred from models for both stellar atmospheres and stellar evolution, we can begin to set constraints on the statistical properties of the population once we believe we can make consistent, systematic estimates of the stellar properties. However, it is important to bear in mind the systematic uncertainties inherent in these determinations (Massey, these Proceedings). Here, I will examine a few stochastic analyses of local observations of the OB star population.
UP: The Stellar Upper-Mass Limit
================================
If the IMF truly behaves as a probability density function, which is indeed the way that we ordinarily assume that it does, then in principle we can collect data from many different clusters and combine them to increase the size of our sample to beat down stochastic noise. We applied this technique to evaluate the stellar upper-mass limit [@OeyClarke2005], using data from local OB associations. @Massey1995 obtained uniform, high-quality spectroscopic classifications of the upper IMF in a number of OB associations in the Milky Way and Magellanic Clouds. To evaluate the upper-mass limit, we are interested only in clusters having ages $\lesssim 3$ Myr, which are young enough so that we are observing these systems before the most massive stars have expired as supernovae (SNe). Eight objects studied by Massey et al. (1995) qualify: IC 1805, Berkeley 86, NGC 7380, NGC 1893, NGC 2244, Tr 14/16, LH 10, and LH 117/118. These clusters contain, respectively, 24, 10, 11, 19, 12, 82, 65, and 40 stars having masses $\geq 10$ M$_\odot$, for a total of 263 stars. In addition to this sample, Massey & Hunter (1998) used the same techniques to obtain the census of the upper IMF in the R136a super star cluster in the 30 Doradus complex in the LMC, using [ *HST*]{}/STIS spectroscopy. Their strong lower limit for the number of stars having masses $\geq 10$ M$_\odot$ is 650. Combining the samples of ordinary OB associations and R136a yields a grand total of 913 stars.
The maximum stellar mass in the entire sample is around 120 – 150 M$_\odot$. We can compare this with the average expected maximum mass for an IMF with a @Salpeter1955 power-law slope $\gamma= 2.35$, for an assumed upper-mass limit $m_{\rm up}$ of the parent distribution, and a given number of stars $N_*$ in the ensemble. As before, $N_*$ includes only stars of mass $m\geq10$ M$_\odot$. The expected maximum mass $\langle
m_{\rm max}\rangle$ is given by: $$\label{eqmmax}
\langle m_{\rm max}\rangle = m_{\rm up} -
\int_{0}^{m_{\rm up}} \Biggl[\int_{0}^M \phi(m)\ dm\Biggr]^N \ dM \quad .$$ Figure \[werkfig\] shows the average expected $\langle m_{\rm
max}\rangle$ as a function of $m_{\rm up}$ for three different values of $N_*$ (dotted lines). For a true upper-mass limit $m_{\rm up} = 1000$ M$_\odot$, an ensemble having $N_*=250$ has an expected $\langle m_{\rm max}\rangle \sim 450$ M$_\odot$; an ensemble having $N_*=1000$ has $\langle m_{\rm max}\rangle \sim 650$ M$_\odot$. These values of $N_*$ are similar to those for our observed combined samples of Milky way and LMC associations, excluding and including R136a. We note that the values of $\langle m_{\rm max}\rangle$ do have some dependence on the assumed form of the parent IMF. The dotted lines in Figure \[werkfig\] show the results from equation \[eqmmax\] for a simple Salpeter power-law truncated at $m_{\rm up}$. However, if we adopt a “softer” truncation of the form, $$\label{eqaltIMF}
n(m)\ dm \propto \Biggl[\Biggl(\frac{m}{m_{\rm
up}}\Biggr)^{-\gamma}-1\Biggr]\ dm \quad ,$$ then instead equation \[eqmmax\] yields the solid lines in Figure \[werkfig\].
In any case, as far as we know, no stars approaching these expected $m_{\rm max}$ values on the order of a few hundred M$_\odot$ are known to be observed, and the example calculations assume a true $m_{\rm up}$ of merely 1000 M$_\odot$, much less infinity. Indeed, inverting the argument, the observed values of $m_{\rm max}\sim 150$ M$_\odot$ imply that the parent $m_{\rm up}$ has a similar value. @Elmegreen2000 used this reasoning to estimate that if $m_{\rm up}=\infty$, then somewhere in the Milky Way, there should be a star having $m_{\rm max}$ = 10,000 M$_\odot$. No star remotely approaching this mass is suggested to have been seen.
However, given that the preceding is based on only 9 clusters, we may worry that, because of stochasticity, these may not be representative of the massive star population. We can quantify the degree to which we might be this unlucky by evaluating the probabilities $p(m_{\rm max})$ of obtaining the observed $m_{\rm max}$ seen in each cluster, for an assumed $m_{\rm up}$. Under normal circumstances, we should be uniformly lucky and unlucky, and so these $p(m_{\rm max})$ should represent a uniform distribution when $m_{\rm up}$ corresponds to its actual value in the parent distribution. Figure \[f\_pmmax\] shows the histograms of $p(m_{\rm max})$ for assumed $m_{\rm up} = 10^4$, 200, 150, and 120 M$_\odot$. The probabilities $P$ that these histograms are drawn from uniform distributions for these respective cases are $P < 0.002,\ < 0.02,\ 0.12,$ and $< 0.47$. Thus we confirm that, only when $m_{\rm up}$ has values similar to the observed $m_{\rm max}$, do we see significance in $P$, demonstrating that $m_{\rm up}$ in this particular sample is indeed around 150 M$_\odot$ at the significance implied by $P$. We note that @Aban2006 point out that the statistical estimator for the maximum value of a truncated, inverse power-law distribution is indeed the maximum value in the dataset, corresponding to the observed $m_{\rm max}$.
It is especially remarkable that this apparent value of $m_{\rm up} \sim 150$ M$_\odot$ is seen over a great range in star-forming conditions. It holds for both ordinary OB associations in both the Milky Way, a massive, spiral galaxy; and the LMC, a lower-mass, irregular galaxy with presumably a lower-pressure ISM. At the same time, the same upper-mass limit applies in the R136a super star cluster, an extreme environment in the LMC. And @Figer2005 finds the same upper-mass limit in the Arches cluster near the Galactic Center, which is another extreme, yet very different, environment. We caution that these results are based on the inferred stellar masses for these systems, taken at face value, and assuming that the highest-mass stars have not yet expired.
@Koen2006 used the same dataset for R136a [@MasseyHunter1998] to evaluate $m_{\rm up}$ and the IMF slope $\gamma$ from the cumulative stellar mass function. The cumulative distribution more clearly reveals the existence of a truncated $m_{\rm up}$. Koen simultaneously fit $m_{\rm up}$ and $\gamma$ for both sets of masses provided by Massey & Hunter, based on two different calibrations for spectral type to stellar effective temperature. The fit was carried out with both a least-squares method and a maximum likelihood method. Across all four cases, the results again are largely consistent with a Salpeter slope and $m_{\rm up}\sim 150$ M$_\odot$. Fixing $m_{\rm up}=\infty$ results in a very steep slope, $\gamma\sim 4$ to 5, strongly inconsistent with the Salpeter value. An infinite $m_{\rm up}$ is eliminated with a probability of $<0.0005$ and $<0.002$ for the least-squares and maximum-likelihood methods, respectively, consistent with our results (Oey & Clarke 2005).
Sparse OB Star Groups
=====================
Given the apparent robustness of the upper-mass limit in a variety of local environments, we seek to examine other extreme circumstances. An especially interesting case is field OB stars. While a substantial fraction of these are likely to be runaway stars ejected from clusters, many may well be members of small, low-mass clusters. @Oey2004 demonstrated that individual SMC field OB stars, as defined by a friends-of-friends algorithm, do fall smoothly on a continuous power-law distribution in the number of stars per cluster $N_*$, which is akin to the cluster mass function. This supports the scenario that most field OB stars are members of low-mass clusters.
We obtained F555W and F814W SNAP observations of eight, apparently isolated OB stars from among these SMC field stars, using the [*HST*]{} ACS camera [@Lamb2010]. Applying both a stellar density analysis and a friends-of-friends algorithm, we confirmed that three of these objects have stellar density enhancements, corresponding to sparse stellar groups. An additional object registered a positive signal for companions using the friends-of-friends algorithm, but not the stellar density analysis. The remaining four stars appear isolated to within the detection limit of F814W = 22 mag. Radial velocity observations subsequently revealed two of these stars to be runaway stars. This leaves two stars remaining as candidates for [*in-situ*]{} isolated, field OB stars.
We constructed Monte-Carlo simulations of cluster populations to examine the parameter space occupied by our observed, sparse OB star groups. We first adopted a cluster mass function (MF) given by a simple power-law distribution: $$\label{cmf}
N(M_{\rm cl})\ dM_{\rm cl} \propto M_{\rm cl}^{-2}\ dM_{\rm cl} \quad ,$$ where $N(M_{\rm cl})$ is the number of clusters in the mass range $M_{\rm cl}$ to $M_{\rm cl} + dM_{\rm cl}$. Each cluster drawn from this distribution is then populated with a stellar IMF drawn from a @Kroupa2001 IMF, which has the form, $$% \[
n(m)\ dm \left\{
\begin{array}{ll}
m^{-1.3}\ dm\ , \quad 0.08 {\rm M_\odot}\leq m < 0.5\ {\rm M_\odot} \\
m^{-2.35}\ dm\ , \quad 0.5 {\rm M_\odot}\leq m < 150\ {\rm M_\odot}
\end{array}
\right.
% \]$$ We adopt a default model having a power-law index of –2 for the cluster MF (equation \[cmf\]), and lower-mass limit of $M_{\rm cl,lo} =
20$ M$_\odot$. This model best reproduces the frequency of single-O star clusters in the SMC and Milky Way (see Lamb et al. 2010).
Figure \[mrat\]$a$ shows the mass ratio distribution of the second-highest to highest mass stars $m_{\rm max,2}/m_{\rm max}$ vs $m_{\rm max}$ in these simulations. Only clusters having a single OB star, defined in this figure as having mass $m\ge 18$ M$_\odot$ are shown. Our observed objects are overplotted with the black squares, showing upper limits for the apparently isolated objects. We see that while our observed objects appear to fall in a densely-populated region of the parameter space, the simulations peter out at the lowest values of $m_{\rm max,2}/m_{\rm max}$, since the probability of drawing an extremely low mass ratio is tiny. All of our data fall within the lowest 20th percentile in $m_{\rm max,2}/m_{\rm max}$, and these frequencies also result when our clusters are simulated based on cluster membership number $N_*$ instead of a MF (Figure \[mrat\]$b$). This model analogously uses the form $N_*^{-2}$, the Kroupa IMF, and a lower limit of $N_{\rm *,lo} = 40$ stars, which is equivalent to $M_{\rm cl,lo}$ used above. That our objects all fall in this low-frequency regime is partly due to our selection of apparently isolated OB stars as the [ *HST*]{} targets. We note that the simulations reproduce this regime fairly easily.
Figure \[mmaxmcl\] shows the relation between the $m_{\rm max}$ and $M_{\rm cl}$ for the simulations based on the cluster mass function (equation \[cmf\]), now showing all clusters having at least one OB star. Since there is no relation imposed between the $m_{\rm max}$ and $M_{\rm cl}$, the simulations occupy a large parameter space. The solid lines show the 10th, 25th, 50th, 75th, and 90th percentiles of $m_{\rm max}$ for a given $M_{\rm cl}$, and the dashed line shows the mean, equivalent to equation \[eqmmax\]. Our observed objects are shown with the black squares as before, with $M_{\rm cl}$ computed by assuming that a fully populated IMF exists below the detection threshold of $\sim 1.5\ {\rm M_\odot}$. Open diamonds show the sample of observed clusters compiled by @Weidner2010.
We see that for almost all of our observed objects, 90% of the simulated clusters have lower $m_{\rm max}$ for a given $M_{\rm cl}$, whereas the sample from Weidner et al. (2010) is largely at the opposite extreme. Our data, taken at face value, are in substantial conflict with the premise of a well-defined $m_{\rm max}$ – $M_{\rm cl}$ relation, proposed by @WeidnerKroupa2005. The observed data show a scatter spanning two orders of magnitude in $M_{\rm cl}$ for the $m_{\rm max}$ in our sparse OB groups. While it is essential to understand the consequences of cluster dynamical evolution, this range in values, which is also consistent with the unconstrained simulations, strongly suggests that a $m_{\rm max}$ – $M_{\rm cl}$ relation is weak at best, as also found by @MaschbergerClarke2008. @Testi1999 discovered sparse groups around Galactic Herbig Ae/Be stars, which are newly formed objects, and thus their parent systems are unlikely to be strongly depleted by dynamical evaporation. Our observations extend such findings to OB stars.
The existence of a relation between $m_{\rm max}$ and $M_{\rm cl}$ is widely debated. A clear relation would strongly impact the integrated galaxy IMF (IGIMF; Weidner & Kroupa 2005), affecting interpretations of stellar populations, star-formation histories, and inferred galaxy evolution. Moreover, competitive accretion theories for star formation predict the existence of such a relation: $m_{\rm max}\propto M_{\rm cl}^{2/3}$ [@Bonnell2004]. It is therefore of great interest to further investigate whether sparse OB star groups are representative of their birth conditions.
Summary
=======
With samples of uniformly derived OB star masses, we can begin to quantitatively evaluate the properties of the stellar upper IMF, based on a stochastic understanding of its nature. We demonstrated the existence of a stellar upper-mass limit $m_{\rm up}\sim 150$ M$_\odot$ for a Salpeter slope, using data for the massive star census in a sample of ordinary Milky Way and LMC OB associations, and the super star cluster R136a in 30 Doradus (Oey & Clarke 2005). Based the probability of observing the highest-mass stars in their respective clusters, we confirmed the existence of a strong deficit above this value of $m_{\rm up}$. Koen (2006) examined the cumulative distribution function of the stellar masses, and by jointly fitting $m_{\rm up}$ together with a power-law distribution for R136a, he confirmed a value for $m_{\rm up} \sim 150$ M$_\odot$ and a Salpeter-like slope as the best-fit values for these parameter estimations. The variety of environments in which this value of the upper-mass limit applies is remarkable: ordinary OB associations, the R136a super star cluster, and the Galactic Center environment.
To further examine the robustness of this upper-mass limit, we searched for low-mass companions around eight SMC field OB stars with [*HST*]{}/ACS. We confirmed the existence of sparse groups associated with 3 – 4 of these field massive stars, while 2 targets are runaway OB stars, and 2 – 3 remain candidates for isolated OB stars that formed [*in situ*]{} (Lamb et al. 2010). We generate Monte Carlo simulations of cluster populations based on simple power-law sampling for both cluster and stellar masses, and we find that the cluster lower-mass limit is most consistent with the frequency of SMC and Galactic field O stars at $M_{\rm cl,lo} \sim 20$ M$_\odot$ or $N_{\rm *,lo}\sim 40$ for a Kroupa IMF. Our sparse OB groups all fall in the lowest 20th percentile in $m_{\rm max,2}$/$m_{\rm max}$, regardless of whether the clusters are populated by $M_{\rm cl}$ or by $N_*$.
Our sparse OB groups generally fall in the highest 10th percentile of $m_{\rm max}$ for a given $M_{\rm cl}$, at an opposite extreme from the sample of objects compiled by Weidner et al. (2010), which fall in the locus of the most massive objects, at a given $m_{\rm max}$, in our simulations. Taken at face value, our results therefore contradict the existence of a well-defined relation between $m_{\rm max}$ and $M_{\rm cl}$. If our sample is not predominantly the result of dynamical evaporation, then this finding may pose difficulties for the predicted steepening of the IGIMF based on a suggested $m_{\rm max}$ – $M_{\rm cl}$ relation (e.g., Weidner & Kroupa 2005) and competitive accretion theories for star formation (Bonnell et al. 2004), which also rely on the existence of such a relation.
This work was supported by NASA HST-GO-10629.01 and NSF grant AST-0907758.
|
---
abstract: 'Robotic calibration allows for the fusion of data from multiple sensors such as odometers, cameras, etc., by providing appropriate relationships between the corresponding reference frames. For wheeled robots equipped with camera/lidar along with wheel encoders, calibration entails learning the motion model of the sensor or the robot in terms of the data from the encoders and generally carried out before performing tasks such as simultaneous localization and mapping (SLAM). This work puts forward a novel Gaussian Process-based non-parametric approach for calibrating wheeled robots with arbitrary or unknown drive configurations. The procedure is more general as it learns the entire sensor/robot motion model in terms of odometry measurements. Different from existing non-parametric approaches, our method relies on measurements from the onboard sensors and hence does not require the ground truth information from external motion capture systems. Alternatively, we propose a computationally efficient approach that relies on the linear approximation of the sensor motion model. Finally, we perform experiments to calibrate robots with un-modelled effects to demonstrate the accuracy, usefulness, and flexibility of the proposed approach.'
author:
- 'Mohan Krishna Nutalapati, Lavish Arora, Anway Bose, Ketan Rajawat, and Rajesh M Hegde [^1]'
bibliography:
- 'refer.bib'
title: '**Model Free Calibration of Wheeled Robots Using Gaussian Process** '
---
INTRODUCTION
============
[R]{}[obotic]{} calibration is an essential first step necessary for carrying out various sophisticated tasks such as simultaneous localization and mapping (SLAM) [@SLAM1; @SLAM2], object detection and tracking [@objectdetec], and autonomous navigation [@Autonav]. For most wheeled robot configurations equipped with wheel encoders and exteroceptive sensors like camera/lidar, the calibration process entails learning a mathematical model that can be used to fuse odometry and sensor data. In the case when the motion model of the robot is unavailable, due to some unmodelled effects calibration involves learning the relationships that describe the sensor motion in terms of the odometry measurements. Precise calibration is imperative since calibration errors are often systematic and tend to accumulate over time [@new]. Conversely, an accurately specified odometric model complements the exteroceptive sensor, e.g. to correct for measurement distortions if any [@LOAM], and continues to provide motion information even in featureless or geometrically degenerate environments [@degraded].
[Traditional approaches]{} [@extend; @intrinsic3] for calibration of wheeled robots focus on learning a parametric motion model of the robot/sensor. A common issue among these approaches was the need for external measurement setup such as calibrated video cameras or motion capture systems. On the other hand [@censi; @tricycle] overcome this issue by performing simultaneous calibration of odometry and sensor parameters using measurements from the sensor. More generic calibration routines for arbitrary robot configurations were presented in [@agv25; @SLAMC] where solution to calibration parameters is found along with robot state variables. All these techniques essentially learn the parameters associated with the motion model of the robot/sensor to generate accurate odometry. However, the performance degrades due to uncertainty arising from interactions with the ground and hence lead to bad odometry estimates [@new]. Moreover, modeling such uncertainties that arise due to non-systematic errors is a very challenging and difficult task. To this end, non-parametric methods [@blimp; @new] employ tools from Gaussian process (GP) estimation learn the residuals between the parametric model and ground truth measurements from external motion capture systems. A common assumption that all these methods make is the residual function being zero mean, which holds true only when the kinematic model of the robot is accurately known. However, for robots with misaligned wheel axis or other unknown offsets that may arise due to unsupervised assembly [@nomodel], excessive wear-and-tear etc., zero mean assumption is not valid rendering the approach suboptimal.
[.25]{} ![Deformed Turtlebot3 mecanum drive robot used for experimental evaluations. (a) Unaligned wheel axis deformation, (b) Tilted wheel deformation.[]{data-label="meca"}](mec_def_1_ "fig:"){width=".90\linewidth"}
[.25]{} ![Deformed Turtlebot3 mecanum drive robot used for experimental evaluations. (a) Unaligned wheel axis deformation, (b) Tilted wheel deformation.[]{data-label="meca"}](mec_def_2 "fig:"){width=".90\linewidth"}
This work puts forth a more general framework that subsumes existing non-parametric approaches, while also applicable to scenarios where the motion model of the robot/sensor is distorted or not known. Different from the existing non-parametric approaches, the proposed method learns the whole sensor/robot motion model. To this end, the key contributions of the work are
- Formulation of a Gaussian-process regression framework that captures the arbitrary or unknown motion model of the sensor/robot. The entire calibration routine can be carried out using measurements from onboard sensors capable of sensing ego-motion
- A computationally efficient approach for near-optimal and model-free calibration
The rest of the paper is organized as follows. Sec. II details the system setup and the problem formulation. The proposed algorithm is described in Sec. III. Detailed experimental evaluations are carried out to validate the performance of the proposed method, and the results are discussed in Sec. IV. Finally, Sec. V concludes the paper. The notation used in the paper is summarized in Table \[notation\].
Problem Formulation {#PF}
===================
------------------------------------------------------------------------------------------------------------------------
Parameters
------------------------------------------------------------------------------------------------------------------------
$\begin{matrix} \p = (\underbrace{\ell_x,\ell_y,\ell_\theta}_\l,\r) & \text{parameters to be estimated} \end{matrix}$
$\begin{matrix} \l & \text{position of extrinsic sensor w.r.t robot frame} \end{matrix}$
$\begin{matrix} \r & \text{robot instrinsic parameters} \end{matrix}$
Measurements
$\begin{matrix}
\mathcal{U} & \text{raw data log of odometry sensor} & \\
\mathcal{V} & \text{Measurememts from exteroceptive sensor} & \\
\q(t)\ \ = &[q_x(t)\ q_y(t)\ q_\theta(t)]^T\ \text{Pose of robot at any time }\emph{t} \\
\hat{\s}_{jk} & \text{sensor displacement estimate for time interval}\ \emph{$[t_j,t_k)$}
\end{matrix}$
More Symbols
$\begin{matrix}
\oplus &\text{Roto-translation operator} & \\
\circleddash &\text{inverse of} \oplus \text{operator} & \\
& \begin{bmatrix}
a_x\\
a_y\\
a_\theta
\end{bmatrix} \oplus \begin{bmatrix}
b_x\\
b_y\\
b_\theta
\end{bmatrix} \overset{\Delta}{=} \begin{bmatrix}
a_x + b_x \cos a_\theta - b_y \sin a_\theta\\
a_y + b_x \sin a_\theta + b_y \cos a_\theta \\
a_\theta + b_\theta
\end{bmatrix} & \\ \\
& \circleddash \begin{bmatrix}
a_x\\
a_y\\
a_\theta
\end{bmatrix} \overset{\Delta}{=} \begin{bmatrix}
-a_x \cos a_\theta - a_y \sin a_\theta\\
a_x \sin a_\theta - a_y \cos a_\theta\\
-a_\theta
\end{bmatrix}
\end{matrix}$
------------------------------------------------------------------------------------------------------------------------
: Nomenclature used in the paper[]{data-label="notation"}
In this section we first start with introducing preliminary notations (see Table \[notation\]) used through out the paper. Consider a general robot with an arbitrary drive configuration, equipped with $m$ rotary encoders on its wheels and/or joints and an exteroceptive sensor such as a lidar or a camera. The exteroceptive sensor can sense the environment and generate scans or images $\V=\{\Z(t)\}_{t\in \mathcal{T}}$ that can be used to estimate its ego motion. Here, $\mathcal{T}:=\{t_1, t_2, \ldots, t_n\}$ denotes the set of discrete time instants at which the measurements are made. The rotary encoders output raw odometry data in the form of a sequence of wheels angular velocities $\U = \{\db(t)\}_{t\in \mathcal{T}}$. Given two time instants $t_j$ and $t_k$ such that $\Delta t_{jk}:=t_k-t_j > 0$ is sufficiently small, it is generally assumed that $\db(t):=\db_{jk}$ for all $t_j \leq t < t_k$. Traditionally, the odometry data is pre-processed to yield relative translation motion and orientation information, and is subsequently fused with the ego motion estimates from exteroceptive sensors. This pre-processing step necessitates the use of the motion model $\f_r$ of the robot that acts upon the odometry data $\db_{jk}$ to yield the relative pose of the robot $\q_{jk} := \circleddash \q_j \oplus \q_k =\f_r(\db_{jk})$ for the time interval $\Delta t_{jk}$. Here $\q_j := (q_j^x,q_j^y,q_j^\theta)^\intercal$ denote the position of the robot at time $t = t_j$. Note that if the exteroceptive sensor is mounted exactly on the robot frame of reference, the sensor motion model denoted by $\f$ is the same as the robot motion model $\f_r$. In general however, if the pose of the exteroceptive sensor with respect to the robot is denoted by $\l$, the sensor motion model is given by $\f(\db_{jk}) = \circleddash \l \oplus \f_r(\db_{jk}) \oplus \l$, where generally $\l$ is unknown.
[.48]{} ![Block diagram describing the system setup and data flow []{data-label="meth"}](methodflow4 "fig:"){width="0.99\linewidth"}
Having the preliminary notations at hand, we now describe the system setup displayed in Fig. \[meth\]. The goal of the calibration phase is to estimate the function $\f$, given $\mathcal{U}$ and $\mathcal{V}$. The estimated motion model, denoted by $\hat{\f}$, is subsequently used in the operational phase to augment or even complement the motion estimates provided by the exteroceptive sensor. For instance, accurate odometry can be used to correct distortions in the sensor measurements [@LOAM]. Note here that the parametric form of the function $\f$ exists when the robot motion model $\f_r$ is well defined. For instance two-wheel differential drive robot [@censi] , four wheel mecanum drive [@mecanum] etc. In other words, $\f(\bigcdot) = \g(\bigcdot\ ;\ \p)$ where $\g$ is a known function and $\p$ is the set of unknown parameters, such as the dimensions of the wheel, sensor position w.r.t robot frame of reference etc. State-of-the-art techniques like [@censi; @tricycle; @trad1] learns $\f$ under this assumption. A significantly more challenging scenario occurs when the form of $\f$ is not known, e.g. due to excess wear-and-tear, or is difficult to handle, e.g. due to non-differentiability. For such cases, the parametric approaches [@censi; @tricycle; @trad1] are no longer feasible and the unknown function $\f$ is generally infinite-dimensional. Towards this end, a low-complexity approach is proposed (see Sec. \[linear\_model\]), wherein a simple but generic (e.g. linear) model for $\f$ is postulated. A more general and fully non-parametric Gaussian process framework is also put forth that is capable of handling more complex scenarios and estimate a broader class of motion models $\f$. It is remarked that in this case, unless the exteroceptive sensor is mounted on the robot axis, additional information may be required to also estimate the robot motion model $\f_r$.
Model Free Calibration using GP {#GP}
===============================
When no information about the kinematic model of the robot is available, it becomes necessary to estimate $\f$ directly. As in Sec. \[PF\], let $\mathcal{T}$ be the set of time instants at which measurements are made. For certain time interval $[t_j,t_k)$ for which $\Delta t_{jk}$ is not too large, let the exteroceptive sensor generates motion estimates $\hat{\s}_{jk}$. Given data of the form $\D:=(\db_{jk}, \hat{\s}_{jk})_{(j,k)\in\E}$, where $\E$ represents set of indices of all chosen key measurement pairs of the sensor and $n:=|\E|$, the goal is to learn the function $\f:\Rn^m\rightarrow \Rn^3$ that adheres to the model $$\begin{aligned}
\hat{\s}_{jk} = \f(\db_{jk}) + \boldsymbol{\varepsilon}_{jk}\end{aligned}$$ for all $(j,k)\in\E$, where $\boldsymbol{\varepsilon}_{jk} \sim \mathcal{N}(\boldsymbol{\mu},\boldsymbol{\Sigma}_{jk}) \in \Rn^{3}$ models the noise in the measurements, and $\db_{jk}$ now represents the number of wheel ticks recorded in the time interval $\Delta t_{jk}$. Here we assume that the noise covariance $\Sig_{jk}$ is known before hand. Given an estimated $\hat{\f}$ of the sensor motion model, a new odometry measurements $\db_e$ for an edge $e$ (pair of times at which measurements are made) can be used to directly yield sensor pose changes $\s_e = \hat{\f}(\db_e)$. As remarked earlier, it may be possible to obtain the robot pose change $\q_e$ from $\s_e$ if the sensor pose $\l$ is known a priori. Since the functional variable $\f$ is infinite dimensional in general, it is necessary to postulate a finite dimensional model that is computationally tractable. Towards solving the functional estimation problem, we detail two methods, that are very different in terms of computational complexity and usage flexibility.
[.33]{} {width="0.90\linewidth"}
[.33]{} {width="0.90\linewidth"}
[.33]{} {width="0.90\linewidth"}
\
[.33]{} {width="0.90\linewidth"}
[.33]{} {width="0.90\linewidth"}
[.33]{} {width="0.90\linewidth"}
Calibration via Gaussian process regression (CGP)
-------------------------------------------------
The GP regression approach assumes that the measurement is Gaussian distributed and that the function $\f$ is a Gaussian process, whose mean and variance functions depend on the data. Specifically, we have that $$\begin{aligned}
\hat{\s}_{jk} \sim \mathcal{N}(\f(\db_{jk}), \Sig_{jk})\end{aligned}$$ or equivalently, $\boldsymbol{\varepsilon}_{jk} \sim \mathcal{N}(\textbf{0},\Sig_{jk})$. Given inputs $\{\db_{jk}\}$, let $\fb$ denote the $\{3n \times 1\}$ vector that collects $\{\f(\db_{jk})\}$ for $\{(j,k)\in\E \}$. Defining $\Sig \in \Rn^{3n \times 3n}$ as the block diagonal matrix with entries $ \Sig_{jk}$ and $\hat{\s} \in \Rn^{3n}$ as the vector that collects all the measurements $\{\hat{\s}_{jk}\}_{(j,k)\in\E}$. Having this we can equivalently write the joint likelihood as $$\begin{aligned}
p(\hat{\s}|\fb) = \mathcal{N}(\hat{\s}|\fb, \Sig)\end{aligned}$$
Unlike the parametric model based approaches [@censi], we impose a Gaussian process prior on $\f$ directly. Equivalently, we have that $$\begin{aligned}
p(\fb) = \mathcal{N}(\fb|\mathbf{\bar{\mub}}, \K)\end{aligned}$$ where $\bar{\mub} \in \Rn^{3n}$ is the mean vector with stacked entries of $\mub(\db_{jk}) \in \Rn^3$, and $\K \in \Rn^{3n \times 3n}$ is the covariance matrix with a block of entries $[\K_{i,i'}] = \boldsymbol{\kappa}(\db_{jk},\db_{j'k'})$ for $(j,k)$ and $(j',k') \in \E$ and $i,i' \in \{1,\cdots, n\}$. The choice of the mean function $\mub:\Rn^m \rightarrow \Rn^3 $ and the kernel function $\boldsymbol{\kappa} :\Rn^{m} \times \Rn^m \rightarrow \Rn^{3\times3}$ is generally important and application specific. Popular choices include the linear, squared exponential, polynomial, Laplace, and Gaussian, among others. With a Gaussian prior and noise model, the posterior distribution of $\f$ given $\D$ is also Gaussian. For a new odometry measurement $\db_e$ with noise variance $\Sig_e$, let $\k_e \in \Rn^{3n \times 3}$ be the vector that collects $\{\boldsymbol{\kappa}(\db_e, \db_{jk})\}_{(j,k)\in\E}$. Then the distribution of $\hat{\f}(\db_e)$ for given $\hat{\s}$ is $$\begin{aligned}
p(\hat{\f}(\db_e)|\hat{\s}) = \mathcal{N}(\hat{\f}(\db_e)\ | \ \hat{\mub}_e, \hat{\Sig}_e)\end{aligned}$$ where $\hat{\mub}_e = \k_e^T(\K+\Sig)^{-1}(\hat{\s}-\bar{\mub}) + \mub(\db_e) $ and the covariance $\hat{\Sig}_e = \boldsymbol{\kappa}(\db_e,\db_e) - \k_e^T(\K+\Sig)^{-1}\k_e$. Note that in general, the choice of the mean and kernel functions is important and specific to the type of robot in use. In the present case, we use the linear mean function $$\mub(\textbf{x}) = \textbf{C} \textbf{x}$$ where $\textbf{x} \in \Rn^m $ is the vector of wheel ticks recorded in a time interval and $\textbf{C} \in \Rn^{3\times m}$ is the associated hyper-parameter of the mean function. Recall that $m$ represents total number of wheels equipped with wheel encoders. Intuitively, the implication of this choice of linear mean function is that the relative position of the robot varies linearly with the wheel ticks recorded in the corresponding time interval. Such a relationship generally holds for arbitrary drive configurations if the time interval is sufficiently small. A widely used kernel function is the radial basis function as follows $$\label{covse}
[\boldsymbol{\kappa}_{rbf}(\textbf{x},\textbf{x}')]_{i,i'} = \sigma_{i,i'}^2 \exp\left(-\frac{1}{2}(\textbf{x}-\textbf{x}')^T \textbf{B}_{i,i'}^{-1}(\textbf{x}-\textbf{x}')\right)$$ where $\textbf{x},\textbf{x}'$ $\in \mathbb{R}^m$ are the data inputs with hyper-parameters $\Xi = [\sigma_{i,i'},\textbf{B}_{i,i'}]$, here $i,i' = 1,2,3$. It will be shown in section \[expres\] that for the two-wheel differential drive robot in use here, the squared exponential kernel with the linear mean function yielded better results than others. On the other hand for four-wheel Mecanum drive in use here, the inner product kernel, which amounts to a linear transformation of the feature space, $$[\boldsymbol{\kappa}_{lin}(\textbf{x},\textbf{x}')]_{i,i'} = \langle\ \textbf{x},\textbf{x}'\rangle$$ performed better. We remark here that for our experiments we have assumed $\boldsymbol{\kappa}(\textbf{x},\textbf{x}')$ is a diagonal matrix with diagonal entries $\{[\boldsymbol{\kappa}(\textbf{x},\textbf{x}')]_{i,i}\}$. In general, the choice of the mean and kernel functions and that of the associated hyper-parameters is made a priori. For our experiments we infer the hyper-parameters by optimizing the corresponding log marginal likelihood. However, they may also be determined during the calibration phase via cross-validation.
Collect measurements from sensors. **Training Phase :** Run sensor displacement algorithm for each selected interval, to get the estimates $\{\hat{\s}_{jk}\}$ with the corresponding wheel ticks $\db_{jk}$ and stack them. Now pre-compute the following quantities : $(\boldsymbol{K}+\boldsymbol{\Sigma})^{-1}(\hat{\s}-\bar{\mub}) $ and $(\boldsymbol{K}+\boldsymbol{\Sigma})^{-1}$ **Testing Phase :** For every test input $\db_e$, evaluate the following, $\hat{\mub}_{e} = \boldsymbol{k}_e(\boldsymbol{K}+\boldsymbol{\Sigma})^{-1}(\hat{\s}-\bar{\mub}) + \mub(\db_e)$ $\hat{\boldsymbol{\Sigma}}_e = \boldsymbol{\kappa}(\db_e,\db_e) - \k_e^T(\K+\Sig)^{-1}\k_e $ Report $\hat{\s}_e $, where $p(\hat{\s}_{e}) = \mathcal{N}(\hat{\s}_e|\hat{\mub}_{e},\hat{\boldsymbol{\Sigma}}_{e})$
Approximate linear motion model {#linear_model}
-------------------------------
As an alternative to the general and flexible CGP approach that is applicable to any robot, we also put forth a computationally simple approach that relies on a linear approximation of $\f$. Specifically, if $\Delta t_{jk}$ is sufficiently small, so are elements of $\db_{jk}$. Therefore, it follows from the first order Taylor’s series expansion, that $\f$ is approximately linear. This assertion if further verified empirically for the two-wheel differential drive. As evident from Fig. \[fig2\], for $\Delta t_{jk}$ sufficiently small, the elements of $\db_{jk}$ are concentrated around zero and the surface fitting them is indeed approximately linear. Motivated by the observation in Fig. \[fig2\], we let $\f(\db_{jk}) = \W\db_{jk}$, where $\W \in \Rn^{3 \times m}$ is the unknown weight matrix. The following robust linear regression problem can subsequently be solved to yield the weights: $$\begin{aligned}
\label{rlr}
\widehat{\W} = \arg\min_{\W} \sum_{(j,k)\in\E}\sum_{i\in\{x,y,\theta\}} \rho_c\left(\frac{\hat{\s}_{jk}^i - [\W\db_{jk}]_i}{\sigma_{jk}^i}\right) \end{aligned}$$ where $\rho_c$ is the Huber loss function [@huber]. Here, is a convex optimization problem and can be solved efficiently with complexity $\O(n^3)$. It is remarked that the entries of $\W$ do not have any physical significance and cannot generally be related to the intrinsic or extrinsic robot parameters, especially after wheel deformation. Note that while making predictions the complexity of the linear model is $\mathcal{O}(m)$ where as for CGP it is $\mathcal{O}(n^2)$.
Experimental Evaluations
========================
This section details the experiments carried out to test the proposed CGP algorithm. We begin with the performance metrics used for validating the accuracy of the estimated model followed by details regarding the experimental setup and results.
Performance Metrics
-------------------
In the absence of wheel slippages, it is remarked that the accuracy of estimated model is quantified by the closeness of the robot/sensor trajectory estimate obtained from odometry to the ground truth trajectory. Since ground truth data was not available for the experiments, we instead used a SLAM algorithm to localize the sensor and build a map of the environment. While SLAM output would itself be not as accurate as compared to the ground truth, of which some of them [@slam] do not require odometry measurements and consequently serves as a benchmark for all calibration algorithms. Specifically, the *google cartographer* algorithm, which leverages a robust scan to sub-map joining routine, is used for generating the trajectory and the map [@slam] of the environments. It is remarked that in the absence of extrinsic calibration parameters, SLAM outputs only the sensor trajectory (and not the robot trajectory), which is subsequently used for comparisons.
Various sensor trajectory estimates are compared on the basis of Relative Pose Error (RPE) and the Absolute Trajectory Error (ATE) motivated from [@benchmark]. The RPE measures the local accuracy of the trajectory, and is indicative of the drift in the estimated trajectory as compared to the ground truth. At any time $t_k \in \mathcal{T}$ , let the odometry and SLAM pose estimates be denoted by $\hat{\x}_k$ and $\x_k$, respectively. Then, relative pose change between times $t_k$ and $t_{k+1}$ estimated via odometry and SLAM are given by $\circleddash\ \hat{\x}_k \oplus \hat{\x}_{k+1}$ and $\circleddash\ \x_k \oplus \x_{k+1}$, respectively. Defining $\e^r_k := \circleddash\ (\circleddash\ \hat{\x}_k \oplus \hat{\x}_{k+1})\oplus (\circleddash\ \x_k \oplus \x_{k+1})$, the RPE is defined as the root mean square of the translational components of $\{\e^r_k\}_{k=1}^{n-1}$, i.e., $$\text{RPE} := \left ( \frac{1}{n-1} \sum\limits_{k=1}^{n-1} \left \|\text{trans}(\e^{r}_k) \right \|^2 \right )^{1/2}$$ where trans$(\textbf{e}_k)$ refers to the translational components of $\textbf{e}_k$. In contrast, the ATE measures the global (in)consistency of the estimated trajectory and is indicative of the absolute distance between the poses estimated by odometry and SLAM at any time $t_k$. Defining the absolute pose error at time $t_k$ as $\e^a_k:=\circleddash\ \hat{\x}_k \oplus \x_k$, the ATE is evaluated as the root mean square of the pose errors for all times $t_k\in \mathcal{T}$, i.e., $$\text{ATE} := \left ( \frac{1}{n} \sum\limits_{k=1}^{n} \left \|\text{trans}(\e^a_k) \right \|^2 \right )^{1/2}.$$ Next, we detail the experimental setup used to test model estimates from different forms of Gaussian process (GP).
[.25]{}
[.25]{}
[.33]{}
[.33]{}
[.33]{}
\[traj\_all\]
[|c|c|c|c|]{} **Robot** & **Configuration** & & **Test Data**\
*FireBird VI* & **F1** & WSN Lab &
------------
Tomography
Lab
------------
: List of experimental configurations with labels and locations[]{data-label="config"}
\
& **T1** & &\
& **T2** & &\
------------- --------------- ------------- --------------
**Mean fn** **Kernel fn** **ATE (m)** **RPE (mm)**
Zero RBF 6.273 9.634
Linear RBF **0.592** **9.367**
Zero Linear [0.687]{} [9.367]{}
Zero RBF + Linear 0.716 9.34
Linear RBF + Linear 0.732 9.343
**0.687** **9.367**
1.546 9.361
------------- --------------- ------------- --------------
: ATE and RPE for Configurations **F1**[]{data-label="2wheel"}
Experimental Setup
------------------
### Robots
We have used a two-wheel differential drive *FireBird VI* robot (see Fig. \[fireb\]) having a particular set of intrinsic parameters [@firebird_web] and a four-wheel mecanum drive *Turtlebot3* robot (see Fig. \[meca\]). The *Fire Bird VI* is primarily a research robot with diameter 280 mm, weight of 12 kilograms, and maximum translational velocity of 1.28 m/s. All *FireBird* encoders publish data at the rate of 10 Hz with a resolution of 3840 ticks per revolution. Similarly *Turtlebot3 mecanum* is also a research robot from the Robotis group with all wheels diameter of 60 mm. It weighs 1.8 kilograms and maximum translational velocity is 0.26m/s. The dynamixels used publish data at 10 Hz with an approximate resolution of 4096 ticks per revolution. For the purposes of the experiment, we made use of an on board computer with i5 processor, 8GB RAM, running ROS kinetic for processing the data from lidar and wheel encoders, performing SLAM for validation, and running the calibration algorithms.
### Lidar Sensor
RPLidar A2 is a low cost $360^\circ$, $2D$ laser scanner with a detection range of 6 meters, a distance resolution less than 0.5 m and an adjustable operating frequency of 5 to 15 Hz. This scanner was mounted on the both the robots with the frequency of 10 Hz resulting in an angular resolution of 0.9$^\circ$.
### Scan Matching
We used point-to-line ICP (PLICP) variant [@P2LICP] in order to estimate the sensor displacements $\hat{\s}_{jk}$. It is remarked that all ICP-like methods also output the corresponding covariance value in closed-form [@covariance] that can be used by the CGP algorithm.
### Data Processing
For the purposes of the experiments, we ensure that scans are collected at times spaced $T$ seconds apart. The choice of $T$ is not trivial. For instance, choosing a small $T$ often makes the algorithm too sensitive to un-modeled effects arising due to synchronization of sensors, robot’s dynamics. It is lucrative to choose far scan pairs as more information is capured about the parameters however both the scan matching output as well as the motion model become inaccurate when $T$ is large. For the experiments, we chose the largest value of $T$ that yielded a reliable scan matching output in the form of sensor motion, resulting in $T=0.6$ second for *Turtlebot3 mecanum* and $T = 0.3$ seconds for the *Firebird VI* robot. These values are chosen based on maximum wheel speeds such that slippages are minimized during experimentation. Note that since the odometry readings are acquired at a rate, higher than the scans, temporally closest odometry reading is associated to a given scan. With the chosen $T$ the robots would move a maximum displacement of 15cm in x and y and 8$^\circ$ in yaw, under such conditions PLICP achieves 99.51 $\%$ accuracy [@P2LICP].
### Deformed Robot Configurations
In order to demonstrate the non-availability of the robot model, one of the wheels of the *Firebird VI* robot is deformed with a thick tape (see Fig. \[fireb\](b)), this configuration is referred as **F1**. Care was taken to ensure that the deformation was not too large, so as to avoid wobbling of the robot and the scan plane of the Lidar. In the case of *Turtlebot3* robot two different configurations (**T1** and **T2**) are constructed (see Fig. \[meca\]), by changing the position of the wheels from the regular configuration. We will see further that the amount of deformation in tilted wheel configuration **T2** (as in Fig. \[meca\](b)) is more as opposed to unaligned wheels configuration **T1** (as in Fig. \[meca\](a)). Next, experiments comprising of training and testing phases, are carried out using these deformed robots for all the specified configurations (see Table \[config\]). While training data is used to learn the motion model of the robot/sensor, the test data is used to evaluate the accuracy of the learned model. It is remarked that the collected test data involves short and long trajectories with varied robot motions. Each experiment is labeled for reference, with details provided as shown in Table \[config\]. For example, configuration **T1** refers to the experiment done using *Turtlebot3* robot, where both training and test data are collected in ACES library.
Experimental Results {#expres}
--------------------
We first perform offline calibration of *FireBird VI* robot with configuration **F1** using the proposed CGP algorithm along with the model based CMLE [@censi] algorithm. Note that in the case of CGP algorithm various kernel and mean functions are trained to determine which of them captures the sensor motion model accurately. Note that we have also trained on composite kernel functions like $\kappa_{rbf} + \kappa_{lin} (\textbf{x},\textbf{x}')$. After the model is learned, predictions are made on the test data. The predicted trajectories are then compared with SLAM trajectory as reference. Error metrics for these trajectories are generated and displayed in Table \[2wheel\]. It is observed that CGP with squared exponential kernel function with linear mean function outperforms other trained models, also CMLE. We remark here that although CMLE predicts the radius of the left wheel to be slightly more than that of the right wheel, the predictions are worse due to non applicability of the parametric model as the wheel looses its notion of circularity. Observe that CGP with linear kernel function is comparable to the best case. The predicted trajectories generated for CMLE [@censi], CGP with linear kernel and SLAM [@slam] are displayed in Fig. \[traj\_all\](a). It is evident that the proposed CGP with linear kernel predicts test trajectory close to SLAM.
Similar procedure is carried out in the case of Turtlebot3 robot with configurations **T1** and **T2**. Note that since the analysis of CMLE [@censi] is restricted to two wheel differential drive robots, we use parametric motion model of four wheel mecanum drive robot [@mecanum] with manufacturer specified parameters for robot intrinsics and nominal hand measured parameters for lidar extrinsics to perform predictions on test data. Table \[erro\_tur\] displays error metrics evaluated for parametric and various non-parametric models. Observe that the proposed CGP algorithm with linear kernel function outperforms other learned models. The corresponding test trajectories for configurations **T1** and **T2** are displayed in Fig. \[traj\_all\](b) and Fig. \[traj\_all\](c) respectively.
Interestingly it can be observed from Table.\[2wheel\] and Table \[erro\_tur\] that the linear model approximation is sufficient to explain the motion model with the set deformities in all configurations. Here we notice that learning a linear approximation of $\f$ is sufficient to accurately predict robot odometry, this is in lines with our discussion in Sec. \[linear\_model\].
Conclusion
==========
We develop a novel odometry and sensor calibration framework applicable to wheeled mobile robots operating in planar environments. The key idea is to utilize the ego-motion estimates from the exteroceptive sensor to estimate the motion model of the sensor/robot. The proposed framework is general as it applies to robots whose motion model is not known. We advocate a non-parametric Gaussian process regression-based approach that directly learns the relationship between the wheel odometry and the sensor motion. The method does not require ground-truth measurements from an external setup, and henceforth the calibration routine can be carried out without interrupting the robot operation. A computationally efficient method that relies on a linear approximation of the sensor motion model is shown to perform on par with the proposed calibration via Gaussian process (CGP) algorithm. Experiments are performed on robots with un-modelled deformations and is shown to outperform existing parametric approaches. Moreover, all the MATLAB codes are made available online[^2]. The method being general is applied to wheeled robots operating in planar environments but does not make any assumptions regarding the same. As part of the future work, it would be interesting to test the performance in non-planar settings.
[^1]: [The authors are with the Department of Electrical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India, (e-mail: $\left\{\text{nmohank, lavi, anwayb, ketan, rhegde} \right\}$@iitk.ac.in).]{}
[^2]: http://www.tinyurl.com/GPcalibration
|
---
author:
- |
for the VERITAS collaboration[^1]\
DESY, Platanenallee 6, 15738 Zeuthen, Germany\
E-mail:
bibliography:
- 'ofdb\_lsI.bib'
title: 'VERITAS observations of exceptionally bright TeV flares from [LS I +61$^{\circ}$ 303]{}'
---
Introduction
============
The class of TeV-emitting high-mass X-ray binaries (HMXBs) consists of only a few sources: LS 5039 [@2005Sci...309..746A], PSR B1259-63 , [LS I +61$^{\circ}$ 303]{} [@Albert2006], HESS J0632+057 [@2009ApJ...698L..94A], and the newest member of the class 1FGL J1018.6-5856 . Only the compact object of PSR B1259-63 has been firmly identified as a pulsar, while the natures of the compact objects in the other systems have not yet been unambiguously determined. Consequently, the fundamental setup that produces the TeV emission along with its characteristic variability on the timescale of one orbital period remains unknown.
[LS I +61$^{\circ}$ 303]{} is the only known TeV binary in the Northern Hemisphere that has a sufficiently short orbital period to allow for regular study with TeV instruments. Located at a distance of $\sim2$kpc [@1991AJ....101.2126F], the system comprises a B0 Ve star and a compact object [@HandC1981; @Casares2005]. The observed emission is variable and modulated with a period of $P \approx 26.5$ days, believed to be associated with the orbital structure of the binary system . Radial velocity measurements show the orbit to be elliptical $e = 0.537\pm0.034$, with periastron occurring around phase $\phi=0.275$ and apastron at $\phi=0.775$ [@Aragona2009]. The periastron distance between the star and the compact object is estimated at $2.84 \times 10^{10}$m (0.19AU) and the apastron distance at $9.57 \times 10^{10}$m (0.64AU) . The inclination of the system is expected to lie in the range $10^\circ$–$60^\circ$ , but the lack of exact knowledge of this parameter leads to considerable uncertainty of the other orbital parameters.
In this work, the results of the VERITAS campaign on [LS I +61$^{\circ}$ 303]{} in late 2014 are presented. During this time, VERITAS observed historically bright flares from the binary around apastron, with the source exhibiting flux levels a factor of 2–3 times higher than ever previously observed.
Observations
============
VERITAS is an imaging atmospheric-Cherenkov telescope (IACT) array located at the Fred Lawrence Whipple Observatory (FLWO) in southern Arizona (1.3km a.s.l., 31$^{\circ}$40’N, 110$^{\circ}$57’W). It consists of four 12m diameter Davies-Cotton design optical telescopes, each with a reflector composed of 345 tessellated hexagonal mirror facets that focus light onto a camera at the focal plane with a field of view of $3.5^\circ$. It is sensitive to photons with energies from 85[GeV]{} to $>30$[TeV]{} with an energy resolution of 15–25%, an angular resolution of $<0.1^\circ$ at 1[TeV]{}, and the ability to detect a 1% Crab Nebula source in approximately 25 hours[^2] [@parkicrc]. For a full description of the hardware components and analysis methods utilized by VERITAS, see [@VERITAS; @KiedaVTSUpgrade; @VERITASLSIDetection].
In the 2014 season, VERITAS observations of [LS I +61$^{\circ}$ 303]{} were taken from October 16 (MJD 56946) to December 12 (MJD 57003) for a total of 24.7 hours of quality selected livetime. These observations covered three separate orbital periods, sampling the orbital phase regions of $\phi = 0.5-0.2$ (see Figure \[f:fluxphase\] and Table \[t:fluxphase\]). Over the entire set of observations, a total of 449 excess events above an energy threshold of 300[GeV]{} were detected above background, equivalent to a statistical significance of 21 standard deviations above background ($21\sigma$, calculated using Equation 17 of [@LiMa]).
![Light curve of [LS I +61$^{\circ}$ 303]{} during the 2014 observation season in nightly bins. The data for the first orbit (October) are shown with orange circles, the data for the second orbit are shown with purple diamonds, and the data for the third orbit are shown with blue squares. Flux upper limits at the 99% confidence level (using the unbounded approach of [@Rolke]) are shown for points with $<3\,\sigma$ significance and are represented by arrows. []{data-label="f:fluxphase"}](./figs/fluxvphase_300.pdf){width="80.00000%"}
Date observed \[MJD\] Orbital phase Flux($>300$[GeV]{}) \[$\times 10^{-11}$ cm$^{-2}$ s$^{-1}$\]
----------------------- --------------- --------------------------------------------------------------
56946 0.52 $<$0.15
56947 0.55 2.32 $\pm$ 0.54
56948 0.60 3.20 $\pm$ 0.34
56950 0.67 $<$0.47
56951 0.71 $<$0.64
56952 0.75 $<$0.55
56954 0.82 $<$0.28
56956 0.90 $<$0.86
56958 0.97 $<$0.38
56960 0.05 $<$0.46
56962 0.12 $<$0.96
56973 0.54 $<$0.53
56974 0.58 1.30 $\pm$ 0.34
56975 0.61 2.02 $\pm$ 0.22
56976 0.65 1.55 $\pm$ 0.25
56977 0.69 $<$0.94
56979 0.76 $<$0.85
56981 0.84 $<$0.53
56982 0.88 $<$1.02
56983 0.92 $<$0.58
56985 0.99 $<$0.82
56987 0.06 $<$0.69
56989 0.14 $<$1.13
57001 0.59 $<$0.66
57002 0.63 0.72 $\pm$ 0.12
57003 0.67 1.06 $\pm$ 0.20
: VERITAS observations of [LS I +61$^{\circ}$ 303]{} in 2014[]{data-label="t:fluxphase"}
During the first orbit observed (in October), the source presented the largest of its flares (referred to as “F1” from here on), beginning on 2014 October 17 (MJD 56947, $\phi = 0.55$) with emission reaching a peak of $(31.9 \pm 3.4_{\mathrm{stat}}) \times10^{-12}$[ cm$^{-2}$ s$^{-1}$]{} on October 18 (MJD 56948). This is the largest flux ever detected from this source. Observations following the peak were limited by poor weather conditions and only recommenced on October 20 (MJD 56950), by which time the flux from the source had already decreased. During the second orbital passage in November, VERITAS detected another period of elevated flux (“F2”) from the source at similar orbital phases ($\phi = 0.5-0.6$) with peak emission of $(20.2 \pm 2.2_{\mathrm{stat}}) \times10^{-12}$[ cm$^{-2}$ s$^{-1}$]{} on November 14 (MJD 56975).
The rise and fall times of the flares were determined by fitting Equation 7 of @2010ApJ...722..520A to the light curve of each orbit and fixing $t_0$ to the MJD of the observed peak. The fit to the first orbit is not very good, with a fit probability of $3 \times 10^{-7}$. Nevertheless, the rise and fall times of F1 are found to be $0.39 \pm 0.07$ days and $0.37 \pm 0.23$ days, respectively. The second orbit is well fit by this function, with a fit probability of $1 \times 10^{-1}$, possibly due to the better data sampling throughout this flare. The rise and fall times of F2 are thus found to be $0.65 \pm 0.13$ days and $0.83 \pm 0.12$ days, respectively. A piecewise defined exponential function was also fit to F1 and F2, but resulted in a very poor fit probability for both flares. Variability on a nightly timescale was tested using the method described in [@2013ApJ...779...88A]. Similar to their findings for this source, a hint of nightly variability at a significance level of $\sim3\,\sigma$ post trials was found in F1.
Follow-up observations conducted by VERITAS during the next month (2014 December 10–12) covered the orbital phases of $\phi=0.59-0.67$ and detected the source at a lower flux level, reaching only $(7.2 \pm 1.2_{\mathrm{stat}}) \times10^{-12}$[ cm$^{-2}$ s$^{-1}$]{} around the orbital phase at which the flares were detected in the previous orbits. The observations during this month appear to exclude the type of peaked flaring behavior seen at the same phase range in the previous two orbital cycles, perhaps indicating some orbit-to-orbit variations in the system.
The average differential energy spectrum from all observations of [LS I +61$^{\circ}$ 303]{} during the 2014 observing season was found to be well fit with a power law of the form $$\frac{dN}{dE} = N_0 \left( \frac{E}{1\mbox{{\,TeV}{}}} \right)^{\Gamma},$$ in which $N_0$ is the normalization at the pivot energy of 1[TeV]{}, and $\Gamma$ is the spectral index. The measured parameters are consistent with past observations. Differential energy spectra were also extracted from F1 (October 17–18) and F2 (November 13–15) and show a similar spectral shape, albeit with a higher normalization constant. The parameters from the spectral fits are given in Table \[t:specfits\]. An uncertainty on the energy scale of 15–25% results in a systematic uncertainty of $\sim50\%$ on the flux normalization and $\sim40\%$ on the integral flux, assuming a spectral index of -2.5. The systematic uncertainty on the spectral index is estimated at $\sim 0.3$, accounting for uncertainties on the collection efficiency, sky brightness, analysis cuts and simulation model. All spectra are shown in Figure \[spec\] along with previous spectral measurements for comparison.
Observations Normalization \[$\times 10^{-12}$ cm$^{-2}$ s$^{-1}$ TeV$^{-1}$\] Spectral index
---------------- ------------------------------------------------------------------- ---------------------------------------------------------
All (average) $1.7 \pm 0.7_{\mathrm{stat}} \pm 0.9_{\mathrm{sys}}$ $-2.35 \pm 0.32_{\mathrm{stat}} \pm 0.3_{\mathrm{sys}}$
F1 (Oct 17–18) $8.6 \pm 1.0_{\mathrm{stat}} \pm 4.3_{\mathrm{sys}}$ $-2.24 \pm 0.12_{\mathrm{stat}} \pm 0.3_{\mathrm{sys}}$
F2 (Nov 13–15) $4.8 \pm 0.4_{\mathrm{stat}} \pm 2.4_{\mathrm{sys}}$ $-2.36 \pm 0.12_{\mathrm{stat}} \pm 0.3_{\mathrm{sys}}$
: Spectral parameters of the power law fits to the observations of [LS I +61$^{\circ}$ 303]{} in the energy range 0.3–20[TeV]{}.[]{data-label="t:specfits"}
After the detection of F2 , VERITAS released an ATel [@2015VTSATEL] notifying the astronomical community of the historic flux levels and triggering observations by multiwavelength partners, as well as additional observations with the MAGIC TeV experiment. The data from this campaign are under analysis and will be presented in an upcoming publication. Initial results of TeV, GeV and X-ray observations during this campaign are presented in [@Karicrc].
Discussion and Conclusion
=========================
The nature of the compact object in [LS I +61$^{\circ}$ 303]{} is not firmly established, so proposed models of the system cover a range of possibilities. In general, the models fall into one of two main categories: a microquasar ($\mu$Q) scenario or a pulsar binary (PB) scenario. The majority of both model types employ a leptonic origin of the observed non-thermal emission.
![Average and flare differential energy spectra of [LS I +61$^{\circ}$ 303]{} from the VERITAS 2014 observations, shown in comparison with the average spectra from [@VERITASLSIDetection] and [@Aleksic].[]{data-label="spec"}](figs/all_spectra_coloured.pdf){width="65.00000%"}
In the $\mu$Q scenario, non-thermal particle acceleration processes occur in the jet of an accreting compact object. Soft photons from the optical star and its accretion disk, and also from the synchrotron radiation in the jet can be upscattered by the ultrarelativistic electrons in the jet [@1538-4357-650-2-L123; @Bednarek01102006]. Support for this type of model comes from the detection of an elongation of the source on the scale of tens of milliarcseconds in radio wavelengths [@Massi2001; @Massi2004], interpreted as a Doppler-boosted jet with precession .
The PB scenario utilizes the presence of a shocked wind in which particle acceleration is the result of the interaction between the stellar and the pulsar winds. For example, the author of [@Dubus2006] proposes the compact object to be a young pulsar with a spin-down rate of the order of $10^{36}$ erg s$^{-1}$ that generates a pulsar wind composed of principally monoenergetic electrons/positrons. The pulsar wind expands and shocks as it interacts with the stellar wind from the optical star. This shock has a “bow” or “comet” shape with a tail extending away from the optical star. The stellar photons can be upscattered to TeV energies by the particles accelerated at the shock. Support for this type of model comes from the detection of cometary-shaped radio emission, pointed away from the high-mass star [@Dhawan2006].
A general PB scenario is presented by [@Paredes-Fortuny2014] describing an inhomogeneous stellar wind in which the Be star disc is disrupted and fragments, and the resulting clumps of the disc fall into the shock region, pushing it closer to the pulsar. The reduction in size of the pulsar wind termination shock could allow for increased acceleration efficiency on the timescale of a few hours, depending on the size and density of the disc fragments. Such a scenario could qualitatively account for the exceptionally bright TeV flares and orbit-to-orbit variations seen in [LS I +61$^{\circ}$ 303]{}.
The detection of pulsed emission from [LS I +61$^{\circ}$ 303]{} would unambiguously identify the source as a pulsar binary. While no pulsations have been detected to date, it is also possible that the dense stellar environment of the source could hinder such a detection. Regardless, further observations of [LS I +61$^{\circ}$ 303]{} with TeV instruments are necessary to fully understand the varying TeV emission from the source.
[^1]: veritas.sao.arizona.edu
[^2]: <http://veritas.sao.arizona.edu>
|
---
author:
- Sebastian Unsleber
- Sebastian Maier
- 'Dara P. S. McCutcheon'
- 'Yu-Ming He'
- Michael Dambach
- Manuel Gschrey
- Niels Gregersen
- Jesper Mørk
- Stephan Reitzenstein
- Sven Höfling
- Christian Schneider
- Martin Kamp
title: 'Observation of resonance fluorescence and the Mollow-triplet from a coherently driven site-controlled quantum dot'
---
Introduction
============
Semiconductor self-assembled quantum dots (QDs) are prime candidates for solid state quantum emitters [@Michler2000; @Santori02]. Many groundbreaking experiments have shown this great potential, e.g. in quantum key distribution experiments [@Waks2002; @Heindel2012], the demonstration of spin–photon entanglement [@DeGreve2012; @Gao2012], and the emission of highly indistinguishable single photons [@He2013; @Wei2014a]. Furthermore, quantum dots offer advantages compared to alternative platforms, e.g. cold atoms or trapped ions, such as their ability to be electrically contacted [@Heindel2010; @Ellis2008; @Yuan2002], and the possibility to be implemented into photonic architectures or networks [@Hoang12; @Yao09; @Joens2015].
Thus far, most relevant demonstrations were carried out based on randomly positioned QDs. Scalable schemes for the implementation of solid state quantum bits or quantum emitters, however, require position control over the QDs. This has triggered extensive research activities to realize site-controlled QD (SCQD) arrays [@Ishikawa2000; @Schmidt2007; @Schneider2009]. While the demonstration of single photon emission [@Baier2004; @Schneider2009a], emission of indistinguishable [@Joens2013] and polarization entangled photons [@Juska2013] in this system has been demonstrated, the implementation of resonant fluorescence in this system remains elusive. Resonant coupling of a laser to the quantum emitter, however, is key towards the emission of single photons with high indistinguishability and to coherently control the state of the excitonic qubit. In this work, we demonstrate for the first time the resonant excitation of a SCQD. We observe the characteristic Mollow triplet [@Ates2009a; @Flagg2009; @Vamivakas2009] in the resonance fluorescence spectra under continuous wave excitation conditions, and demonstrate the coherent control of the excited state SCQD population via pulsed resonant excitation. Furthermore, we analyze the light matter coupling for varying temperatures and driving strengths, which allows us to characterize the strength of exciton–phonon interactions and assess the lateral exciton extension in the quantum dot.
Sample structure and setup {#sec:Sample structure and setup}
==========================
Our sample consists of stacked site-controlled InAs quantum dots which are embedded in a single-sided planar cavity to improve the light extraction, as shown in Fig. \[Fig1\](a). After the growth of the bottom distributed Bragg reflector (DBR) with $30$ quarter-wavelength thick AlAs/GaAs mirror pairs and a $85$ nm thick GaAs layer via molecular beam epitaxy, arrays of nanoholes with a $2~\mu$m pitch are defined on the wafer by means of electron beam lithography and wet-chemical etching [@Schneider2009]. The structure design in principle allows to yield photon extaction efficiencis on the order of $\approx10-12~\%$ [@Royo2001] for the microscope objective with $NA=0.42$ we used. We note that this number can be further increased by utilising micropillar cavities (based on devices with top DBR [@Gazzano2013; @Heindel2010; @Schneider2009a], or by shaping the confinement in broadband approaches via dielectric lenses [@Maier2014; @Gschrey2015; @Sapienza2015]. The pre-patterned sample is deoxidized with thermally activated hydrogen before the second epitaxial growth step is performed. The nanoholes are then capped by $10$ nm GaAs and a first InAs quantum dot layer (seeding layer) at a substrate temperature of $540~^{\circ}$C to maximize the migration length and to ensure that nucleation preferentially takes place in the nanoholes. To improve the optical quality of the quantum dots, a second layer of InAs quantum dots is separated by a $35$ nm thick GaAs separation layer, while the positioning of the SCQDs is maintained by the vertical strain field (Fig. \[Fig1\](b)). In Fig. \[Fig1\](c) a scanning electron microscopy (SEM) image shows SCQDs with a $2~\mu$m period on an uncapped reference sample, recorded under an angle of $70^{\circ}$ to enhance the imaging contrast. The emission wavelength range of the buried QDs in our device is shifted towards $900$ nm by performing an in-situ annealing step, before a $131$ nm thick GaAs layer completes the structure. A titanium grid on the surface of the sample with a pitch of $12~\mu$m and $300$ nm width (Fig. \[Fig1\](a)) is defined by means of e-beam lithography. This serves as a coordinate system for orientation, with each grid square containing a regular array of 36 SCQDs.
![(a) Schematic drawing of the sample structure showing the lower DBR and the $\lambda$-thick spacer with embedded SCQDs. (b) Detailed layer sequence with two quantum dot layers and the $35$ nm thick GaAs separation layer in between. (c) SEM image of SCQDs on an uncapped calibration sample. (d) $\mu$PL-map of a $14\times 14~\mu\text{m}^2$ area. The bright spots are SCQDs which are located on a 2 $\mu$m grid which was predefined by nanoholes. The blue grid serves as a guide to the eye, the red arrow indicates the SCQD which is subject to the in depth study.[]{data-label="Fig1"}](Fig1.pdf){width="48.00000%"}
The sample is mounted on the cold-finger of a He flow cryostat and is excited by a continuous wave laser at 532 nm. Fig. \[Fig1\](d) shows a $14\times 14~\mu$m$^2$ large array scan of the photoluminescence (PL)-signal of the sample. The bright spots between the titanium frames can be attributed to the PL signal of SCQDs which are located on the expected positions of the array. SEM images of the uncapped sample shown in Fig. \[Fig1\](c), indicate that SCQDs are present at all expected positions. As such, we attribute dark spots on the grid to a large variation in quantum efficiencies [@Albert2010]. The SCQD indicated with a red arrow in Fig. \[Fig1\](d) is the basis of detailed subsequent $\mu$PL investigations. The QD is excited either by a fiber coupled continuous wave diode laser or a pulsed Ti:Sapphire laser with a repetition rate of $f=82~$MHz and a pulse length of $\Delta t\approx1.2~$ps. In order to suppress the resonant laser, we use a cross polarization configuration where the excitation laser and the detected SCQD signal have perpendicular polarizations. The emitted photons are then analyzed via a double monochromator with two $1200~\mathrm{lines}/\mathrm{mm}$ gratings and a nitrogen cooled Si-CCD ($\Delta E_{\mathrm{Res}}\approx 20~\mu$eV).
Theory {#sec:Theory}
======
For the interpretation of our experimental results, we employ basic concepts of the theory of a resonantly driven quantum dot coupled to longitudinal acoustic (LA) phonons. In Ref. [@McCutcheon2013] it was shown that provided $\Omega<k_{B} T<\omega_{c}$, with $\Omega$ the Rabi frequency, $k_B$ the Boltzmann constant, $T$ the sample temperature, and $\omega_{c}$ the cut-off frequency of the phonon environment, the effect of phonons on the excitonic dynamics can be accurately captured by a renormalization of the bare Rabi frequency and the introduction of a phonon-induced pure-dephasing rate. We write the optical Bloch equations describing the excitonic degrees of freedom as $$\begin{aligned}
\dot{\alpha}_x&=-\Gamma_2\,\alpha_x,\label{alphaxdot}\\
\dot{\alpha}_y&=-\Gamma_2\,\alpha_y - \Omega_r \alpha_z,\label{alphaydot}\\
\dot{\alpha}_z&=-\Gamma_1\,\alpha_z + \Omega_r \alpha_z-\Gamma_1,\label{alphazdot}\end{aligned}$$ where $\alpha_i=\left\langle\sigma_i\right\rangle=\mathrm{Tr}(\rho \sigma_{i})$ for $i=x,y,z$ and $\rho$ the exciton density operator, and we work in a basis where $\sigma_{x}=\left|e\right\rangle\left\langle g\right|+\left|g\right\rangle\left\langle e\right|$, $\sigma_{y}=-i\left|e\right\rangle\left\langle g\right|+i\left|g\right\rangle\left\langle e\right|$, $\sigma_{z}=\left|e\right\rangle\left\langle e\right|-\left|g\right\rangle\left\langle g\right|$ with $\left|e\right\rangle$ and $\left|g\right\rangle$ the single exciton and ground states respectively. The total dephasing rate is given by $\Gamma_{2}=\frac{1}{2}\Gamma_{1}+\gamma_{PD}+\gamma_{0}$ with $\Gamma_{1}=1/T_{1}$ spontaneous emission rate, $\gamma_{0}$ captures dephasing not attributed to phonons, and $\gamma_{PD}$ is the phonon induced dephasing rate, which we define below. The effective Rabi frequency is given by (for $\hbar=1$) $$\Omega_{r}=\mu E_{0} R~(=\kappa\sqrt{P}),
\label{Omegar}$$ with $\mu$ the dipole moment of the emitter, $E_{0}$ the electric field strength of the laser light, $R$ a phonon induced renormalization factor, $\kappa$ representing the product $\mu R$ and $P$ the laser power.
Exciton–phonon coupling is characterized by the spectral density, which for coupling to LA phonons takes the form $J(\omega)=\alpha \omega^{3} \exp[-(\omega/\omega_{c})^{2}]$, with $\alpha$ capturing the overall strength of the interaction [@Ramsay2010a; @Ramsay2010]. In terms of the spectral density, the phonon-induced renormalisation factor is given by [@McCutcheon2010; @McCutcheon2013; @Wei2014] $$R=\exp\bigg[-\frac{1}{2}\int_0^{\infty}\mathrm{d}\omega \frac{J(\omega)}{\omega^2}\coth(\omega/2k_B T)\bigg].
\label{Bav}$$ In the weak exciton–phonon coupling limit, the phonon induced pure-dephasing rate is given by $$\gamma_{PD}=(\pi/2)J(\Omega_r)\coth(\Omega_r/2k_B T),
\label{gammaPDFull}$$ which becomes $$\gamma_{PD}=\pi \alpha k_B T \Omega_r^2~(=\chi \Omega_r^2),
\label{gammaPD}$$ provided $\Omega_r\ll k_B T,\omega_c$ and $\chi$ being the product $\pi \alpha k_B T $. We note that this last condition is typically met in continuous wave measurements due to the relatively small Rabi frequencies. For pulsed measurements, however, instantaneous Rabi frequencies can reach levels comparable to $\omega_c,k_B T \sim 1~\mathrm{meV}$ (at $T=10~\mathrm{K}$). In the following, we therefore use the simpler expression given in Eq. ([\[gammaPD\]]{}) for the continuous wave measurements, but the full expression in Eq. ([\[gammaPDFull\]]{}) when investigating pulsed excitation conditions.
The incoherent component of the resonance fluorescence spectrum is given by $S(\omega)\propto \mathrm{Re}[\int_0^{\infty}\mathrm{d}\tau (g^{(1)}(\tau)-g^{(1)}(\infty))\exp[-i\omega\tau]]$, where $g^{(1)}(\tau)=\lim_{t\to\infty}\left\langle \sigma^{\dagger}(t+\tau)\sigma(t)\right\rangle$ with $\sigma=\left|g\right\rangle\left\langle e\right|$ is the steady-state first order field correlation function. From Eqs. ([\[alphaxdot\]]{}) to ([\[alphazdot\]]{}) calculation of the field correlation function and fluorescence spectrum proceeds by invoking the quantum regression theorem [@carmichael; @mccutcheon2015]. It is found that above saturation ($\Omega_{r}\gg \Gamma_1$), the fluorescence spectrum consists of three Lorentzian peaks, one centred at the laser driving frequency, and two more positioned on either side at a distance of $\Omega_{r}$. In this regime the full width at half maximum (FWHM) of the sidebands is given by [@Wei2014] $$\Delta \omega = \frac{3}{2}\Gamma_1+\gamma_{PD}+\gamma_0.
\label{SidebandWidths}$$ From these expressions we see that we expect the sideband widths to increase linearly with $\Omega_{r}^{2}$, with an intersect (as $\Omega_{r} \to 0$) given by $\smash{(3/2)\Gamma_1+\gamma_0}$. Moreover, since $\smash{E_0\sim\sqrt{P}}$, for constant $\mu$ one expects the gradient $(\mathrm{d}\Omega_r/\mathrm{d}\sqrt{P})\sim \mu R$ to decrease with temperature since $R$ decreases, as was experimentally observed in Ref. [@Wei2014].
Experimental Results and Analysis {#sec:Experimental Results and Analysis}
=================================
![(a) Second order autocorrelation function for pulsed non-resonant excitation and a sample temperature of $T=6.8$ K. We extract a $g^{(2)}(0)$-value of $g^{(2)}(0)=0.39\pm0.02$. (b) Resonance fluorescence (RF) spectrum of a SCQD for a pump power of $P=746~$nW at $T=5~$K. The triplet peak structure is a signature of coherent coupling between the quantum dot exciton and the laser field.[]{data-label="Fig2"}](Fig2.pdf){width="48.00000%"}
First, we study the emission properties of the SCQD under pulsed nonresonant excitation. To do so, we couple the spectrally filtered microphotoluminescence emission into a fiber-based Hanbury Brown and Twiss setup in order to measure the second order autocorrelation function. A histogram of coincidence events measured as a function of detection time delay is shown in Fig. \[Fig2\](a). The suppressed peak around $\tau\approx0$ ns is a clear signature for single photon emission from our SCQD. We extract a $g^{(2)}(0)$-value by dividing the area of the central peak by the average area of the surrounding peaks, which yields a value of $g^{(2)}(0)=0.39\pm0.02$ (corrected according to [@Michler2002] by spectral background emission $g^{(2)}(0)=0.22\pm0.02$). Remaining two-photon detection events arise from refilling of the SCQD due to the non-resonant excitation. In addtition, fitting the sidepeaks with a two-sided exponential function allows us to extract the lifetime of the exciton and we find $T_1=(561\pm68)$ ps. Compared to standard In(Ga)As QDs grown under comparable conditions [@Johansen2008], the $T_1$-time is slightly reduced. We believe this is a consequence of the presence of non-radiative recombination channels which play a significant role for QDs grown on patterned substrates [@Albert2010].
Next, we study the SCQD’s emission properties under strict resonant excitation. Therefore, we drive the single exciton transition on resonance with a continuous wave laser, and analyze the spectrum of emitted radiation. Fig. \[Fig2\](b) shows a representative spectrum of resonance fluorescence from the investigated SCQD corresponding to a pump power of $P=746~$nW and at a temperature of $T=5~$K. Due to imperfections of the sample surface the laser is not fully suppressed, resulting in a ratio of the sidepeak area to the central peak of approximately $1:12$. We fit the spectrum to a sum of two Lorenztians for the sidepeaks and a Gaussian for the central peak, which is resolution limited. From the fit, for this pump power and temperature we obtain sidepeak widths of $\Delta\omega_F=(31\pm7)~\mu$eV and $\Delta\omega_T=(43\pm9)~\mu$eV and a Rabi splitting of $\Omega_r=72~\mu$eV.
For increasing pump power but at fixed temperature, the sidepeak splitting increases as predicted by theory. Fig. \[Fig3\](a) shows this splitting as a function of the square root of the laser power. The splitting follows a clear linear dependence, as is expected from Eq. ([\[Omegar\]]{}) and recalling that the field amplitude $E_0\propto\sqrt{P}$ [@Vamivakas2009]. From the fits, we obtain a slope of $\kappa=\pm(5.04\pm0.01)~\mu\text{eV}/\sqrt{\mu\text{W}}$, which is slightly smaller than it is reported for self-organized QDs ([@Wei2014]:$\kappa=7.6~\mu\text{eV}/\sqrt{\mu\text{W}}$). Fig. [\[Fig3\]]{}(b) shows the variation of the sideband linewidths as a function of the square of the Rabi frequency. We observe a linear behaviour with gradients of $\chi_F=(2.0\pm0.6)\cdot10^{-4}~\mu\text{eV}^{-1}$ for the F-line and $\chi_T=(6.3\pm0.7)\cdot10^{-4}~\mu\text{eV}^{-1}$ for the T-line. The linear behaviour is consistent with Eq. ([\[gammaPD\]]{}), suggesting excitation induced dephasing caused by coupling to LA phonons in our sample. Additionally, from Eq. ([\[SidebandWidths\]]{}) we see the intersects with the y-axis give non-phonon induced dephasing rates of $\gamma_{0}\approx 30~\mu\mathrm{eV}$. We note that for resonant excitation, our theory predicts F- and T-sidebands of equal width. We attribute the systematic asymmetry to a slight detuning of the QD level from the excitation laser, which becomes more pronounced at the high pump powers we use in order to maximize the Rabi splitting. Such asymmetries have been predicted in [@Ulhaq2013] for systems with relatively high dephasing rates, and arise as a result of the exciton-phonon coupling interaction producing behaviour which departs from a simple pure-dephasing model. The larger broadening of the T-line seen in our measurements suggests our laser is blue shifted from the QD transition. A systematic investigation of these off-resonance effects and their underlying physical mechanisms has yet to be performed, and this is a topic which we plan to explore fully in future work.
![(a) Rabi splitting as a function of the square root of the pump power at $T=5~$K. A clear linear increase is observed with effective dipole moment of $\kappa=\pm(5.04\pm0.01)~\frac{\mu\text{eV}}{\sqrt{\mu\text{W}}}$. (b) Linewidth of the Mollow sidepeaks as a function of the Rabi frequency. A linear trend is consistent with Eq. ([\[gammaPD\]]{}) suggesting dephasing caused by coupling to LA phonons.[]{data-label="Fig3"}](Fig3.pdf){width="48.00000%"}
To further characterize our system, we performed measurements of the sidepeak splitting and linewidth for temperatures up to $35~$K. In Fig. \[Fig4\](a) we plot $\chi$ (see Eq. \[gammaPD\]) as a function of the sample temperature . $\chi$ is the parameter which captures the power broadening of the mollow sidepeaks. We observe a monotonous increase of $\chi$ with the sample temperature, and by fitting to Eq. \[gammaPD\] we can extract $\alpha_F=(0.077\pm0.015)~\mathrm{ps}^2$ and $\alpha_T=(0.113\pm0.012)~\mathrm{ps}^2$ which are consistent with other experiments carried out on standard In(Ga)As QDs [@Wei2014; @Ramsay2010; @Ramsay2010a]. The solid lines in Fig. \[Fig4\](a) show $\gamma_{PD}/\Omega_r^2=\pi\alpha k_B T$ using these extracted values. We attribute the slight scattering of the datapoints to a slight detuning caused by the high driving strength we used in the experiment. As mentioned above, the power dephasing coefficient is a sensitive function to the laser detuning, which we cannot exclude to slightly change during the experiment.
![(a) Change in the Mollow sideband widths with squared Rabi frequency as a function of temperature. The lines show a fitting to the exciton–phonon coupling model. (b) Gradient of the renormalised Rabi splitting with increasing pump power versus the temperature. We observe a notable increase for temperatures exceeding $15$ K, which is attributed to a localization effect of the excitonic wave function for low temperatures.[]{data-label="Fig4"}](Fig4.pdf){width="48.00000%"}
Fig. [\[Fig4\]]{}(b) shows the renormalized dipole moment $\kappa=\mu ~R$ (see equ. \[Omegar\]) as the temperature is increased. We first obtain a decrease in the effective dipole moment for temperatures up to $15$ K, followed by a clear increase for higher temperatures. From Eq. ([\[Omegar\]]{}), we read $\mathrm{d}\Omega_r/\mathrm{d}\sqrt{P}\propto\mu ~R$. Since the phonon renormalisation parameter $R$ decreases with rising temperature, one also expects a decrease of $\mathrm{d}\Omega_r/\mathrm{d}\sqrt{P}$ with temperature. This is what we observe for T<15 K. Above 15 K, the notable increase suggests an increase in the dipole moment $\mu$ with temperature. We assume that the Indium concentration in our QD is non-uniform, which leads to shallow potential minima within QD that trap the exciton at very low temperatures. Once a characteristic activation energy is thermally overcome, the exciton wavefunction spreads over the entire QD, leading to an increase of the dipole moment $\mu$, and thus to a stronger coupling to the resonant laser field. Similar observations were found via magneto optical studies on self-assembled QDs [@Musial2014].
For potential applications of our SCQDs, it is desirable that the excited state population can be coherently controlled, which we now demonstrate using pulsed resonant excitation. Fig. \[Fig5\](a) shows the emission spectrum of the same SCQD under pulsed resonant excitation (f$_{Rep}=82$ MHz, $\Delta t\approx1.2$ ps). A broad remaining laser background is observed due to the temporally narrow excitation pulses. To extract the quantum dot emission intensity, we fit the QD emission in each single spectrum with a Gaussian function and extract the integrated intensity from the fit results.
![(a) Quantum dot spectrum under pulsed resonant excitation and for a sample temperature of $T=5.8$ K. The fit is a double Gaussian function for the dot emission and the remaining laser background. (b) Integrated intensity of the quantum dot emission as a function of the square root of the pump power. The intensity is obtained from fitting the single spectra. The solid line is a fit to theory including coupling to phonons.[]{data-label="Fig5"}](Fig5.pdf){width="48.00000%"}
The results are shown in Fig. \[Fig5\](b), where a damped sinusoidal behaviour is observed. This is consistent with our exciton–phonon coupling model, which (according to Eq. ([\[gammaPD\]]{})) predicts an increase in dephasing for larger pulse strengths, and has been observed elsewhere for conventional, self assembled QDs [@Ramsay2010a; @Ramsay2010; @McCutcheon2010]. In fact, we can use Eqs. ([\[alphaxdot\]]{}) to ([\[alphazdot\]]{}) to fit the experimentally measured intensities as a function of pulse area. The integrated emission intensity is proportional to the final excited state population, which we can simulate using the optical Bloch equations. To do so, we make the replacement $\Omega_r\to R(\Theta/2\tau\sqrt{\pi}) \exp[-(t/2\tau)^2]$, with $\tau=\Delta t/4\sqrt{\ln 2}$, $\Theta$ the pulse area, and we set $\Gamma_1=0$ since we are interested in timescales of the order $\Delta t \ll T_1$. We then numerically solve Eqs. ([\[alphaxdot\]]{}) to ([\[alphazdot\]]{}) using the full phonon-induced pure-dephasing rate in Eq. ([\[gammaPDFull\]]{}) for each excitation power, allowing for a scaling factor between $\sqrt{P_{\mathrm{laser}}}$ and $\Theta$, fixing $\gamma_0=30~\mu\mathrm{eV}$ from our previous fits, and allowing the exciton–phonon coupling parameter $\alpha$ and the cut-off frequency $\omega_c$ to vary.
The result of this fitting procedure is shown by the solid line in Fig. \[Fig5\](b), and the exciton–phonon coupling parameters we extract are $\alpha=(0.054\pm0.016)~\mathrm{ps}^2$ and $\omega_c=(4.34\pm1.21)~\mathrm{ps}^{-1}$. We see that our model is able to capture the experimental data well, though the exciton–phonon coupling strength $\alpha$ we extract differs slightly from those found for continuous wave driving. There are a number of possible reasons for this discrepancy. Firstly, we note that similarly good fits can be found in the pulsed excitation case by fixing $\alpha$ to a value closer to that found for continuous driving, and allowing instead the background dephasing rate $\gamma_0$ to vary. As such, in the case of pulsed excitation it is not straightforward to distinguish phonon-induced dephasing from other sources. We also note that while the fundamental exciton–phonon coupling parameters $\alpha$ and $\omega_c$ should not depend on the excitation scheme, the background rate $\gamma_0$ may well differ in the two cases, particularly for temporally short pulses which have spectral components which are not resonant with the exciton transition energy [@Wei2014a]. Finally, we note that in replacing the Rabi frequency $\Omega_r$ in Eq. ([\[gammaPDFull\]]{}) with a time dependent quantity requires making a Markovian approximation, which assumes the phonon environment relaxes on a timescale shorter than the excitonic dynamics [@Ramsay2010; @McCutcheon2010]. For the short temporal pulses used here this approximation may not be strictly valid, and we note that it will be interesting to explore potential non-Markovian effects in these systems in future works [@mccutcheon2015].
Summary {#sec:Summary}
=======
We have shown for the first time the coherent coupling of a resonant laser field to the excitonic state of a SCQD. We observed the characteristic Mollow triplet in the spectral domain and an increase in the Rabi splitting with increasing temperature, which we attribute to a localization of the SCQD wave function for low temperatures due to a gradient in the Indium concentration inside the SCQD. Furthermore, we analyzed in detail the power and temperature dependent dephasing channels in our system. Finally, we demonstrate the coupling of the QD exciton to a pulsed resonant laser field by measuring the dependency of the SCQD emission on the pulse area of the excitation laser. The observed Rabi oscillations feature a distinct damping, which points towards the presence of exciton–phonon coupling in our system. The possibility to generate resonance fluorescence photons from positioned quantum dots opens new experimental opportunities for realizing scalable solid state quantum emitters. It is a crucial step towards the deterministic generation of single photons with unity indistinguishability [@he2013demand], and resonant techniques are key for obtaining optical coherent control over single spins in QDs [@Press2008]. The implementation of such schemes, based on a fully scalable quantum emitter architecture (such as provided by site controlled quantum dots), remains one of the big challenges in solid state quantum emitter research. As a next step towards this architecture, an implementation of site-controlled quantum dots within microcavities is desirable. Due to the control over the nucleation spot, one can expect an almost perfect coupling of the SCQD to the fundamental optical mode of such a device which gives rise to increased extraction efficiencies and high degrees of indistinguishability, which is key for realizing an efficient source of indistinguishable photons.
Funding Information {#funding-information .unnumbered}
===================
State of Bavaria and the German Ministry of Education and Research (BMBF) within the projects Q.com-H and the Chist-era project SSQN. German Research Foundation via the SFB 787 “Semiconductor Nanophotonics: Materials, Models, Devices”. D.P.S.M. acknowledges support from project SIQUTE (contract EXL02) of the European Metrology Research Programme (EMRP). The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union. Y. H. acknowledges support from the Sino-German (CSC-DAAD) Postdoc Scholarship Program. J. M. acknowledges support from Villum Fonden via the NATEC Centre.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors thank M. Emmerling for sample preparation.
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abstract: 'Let $M$ be an analytic manifold modelled on an ultrametric Banach space over a complete ultrametric field ${{\mathbb K}}$. Let $f\colon M \to M$ be an analytic diffeomorphism and $p$ be a fixed point of $f$. We discuss invariant manifolds around $p$, like stable manifolds, centre-stable manifolds and centre manifolds, with an emphasis on results specific to the case that $M$ has finite dimension. The results have applications in the theory of Lie groups over totally disconnected local fields.'
---
[**Invariant manifolds for finite-dimensional non-archimedean dynamical systems**]{}\
[**Helge Glöckner**]{}
[: 37D10 (Primary) 46S10, 26E30 (Secondary)\
[*Key words*]{}: Dynamical system, fixed point, invariant manifold, stable manifold, centre manifold, centre-stable manifold, ultrametric field, local field, non-archimedean analysis, analytic map, Lie group, contractive automorphism, contraction group, scale function]{}
[**Introduction and statement of main results**]{}
Guided by the classical theory of invariant manifolds for time-discrete smooth dynamical systems over the real ground field (cf. [@HaK], [@HPS], [@Ir1], [@Wel]), invariant manifolds have recently also been constructed for time-discrete analytic dynamical systems over a complete ultrametric field $({{\mathbb K}},|.|)$ [@EXP]. The invariant manifolds are useful in the theory of Lie groups over local fields, where they allow results to be extended to ground fields of positive characteristic, which previously where available only in characteristic $0$ (i.e., for $p$-adic Lie groups). To enable these Lie theoretic applications, the general theory from [@EXP] is not sufficient, and additional, more specific results concerning ultrametric invariant manifolds are needed. The goal of this article is to provide such complementary results, including simplifications of the theory from [@EXP] for finite-dimensional dynamical systems, which make it applicable in the situations at hand.\
As in the real case, hyperbolicity assumptions are essential for a discussion of invariant manifolds. Roughly speaking, a continuous linear self-map $\alpha\colon E\to E$ of an ultrametric Banach space $E$ over ${{\mathbb K}}$ is called hyperbolic if $E$ admits a decomposition $E=E_{{\footnotesize\rm s}}\oplus E_{{\footnotesize\rm u}}$ into a stable subspace $E_{{\footnotesize\rm s}}$ on which $\alpha$ is contractive and an unstable subspace $E_{{\footnotesize\rm u}}$ on which $\alpha$ is expansive. More precisely, $\alpha$ is called *hyperbolic* if it is $1$-hyperbolic in the following sense [@EXP]:\
[**Definition.**]{} The continuous linear map $\alpha\colon E\to E$ is said to be *$a$-hyperbolic* for $a\in \; ]0,\infty[$ if there exist $\alpha$-invariant vector subspaces $E_{a,{{\footnotesize\rm s}}}$ and $E_{a,{{\footnotesize\rm u}}}$ of $E$ such that $E=E_{a,{{\footnotesize\rm s}}} \oplus E_{a,{{\footnotesize\rm u}}}$, and an ultrametric norm $\|.\|$ on $E$ defining its topology, with properties (a)–(c):
- $\|x+y\|=\max\{\|x\|,\|y\|\}$ for all $x\in E_{a, {{\footnotesize\rm s}}}$ and $y\in E_{a,{{\footnotesize\rm u}}}$;
- $\alpha_2:=\alpha|_{E_{a,{{\footnotesize\rm u}}}}$ is invertible;
- $\|\alpha_1\|< a$ and $\frac{1}{\|\alpha_2^{-1}\|}>a$ holds for the operator norms with respect to $\|.\|$, where $\alpha_1:=\alpha|_{E_{a,{{\footnotesize\rm s}}}}$ (and $\frac{1}{0}:=\infty$).
Then $E_{a,{{\footnotesize\rm s}}}$ is uniquely determined and if $\alpha$ is invertible or $E$ finite-dimensional, then also $E_{a,{{\footnotesize\rm u}}}$ is unique (see [@EXP Remark 6.4] and Remark \[centunique\] below). If $a=1$, we also write $E_{{\footnotesize\rm s}}:=E_{1,{{\footnotesize\rm s}}}$ and $E_{{\footnotesize\rm u}}:=E_{1,{{\footnotesize\rm u}}}$.\
Similarly, $E$ may have an $a$-centre-stable subspace $E_{a,{{\footnotesize\rm cs}}}$ such that $$E=E_{a,{{\footnotesize\rm cs}}}\oplus E_{a,{{\footnotesize\rm u}}},$$ or an $a$-centre subspace $E_{a,{{\footnotesize\rm c}}}$ such that $$E=E_{a,{{\footnotesize\rm s}}}\oplus E_{a,{{\footnotesize\rm c}}}\oplus E_{a,{{\footnotesize\rm u}}};$$ see Definitions \[defcsub\] and \[defacentre\] for details. We omit the subscript “$a$” if $a=1$.\
It is useful to fix a notation for the set of absolute values of eigenvalues, in the finite-dimensional case.\
[**Definition.**]{} Let $\alpha\colon E\to E$ be a linear self-map of a finite-dimensional vector space $E$ over a complete ultrametric field $({{\mathbb K}},|.|)$. We use the same symbol, $|.|$, for the unique extension of $|.|$ to an absolute value on an algebraic closure ${\overline}{{{\mathbb K}}}$ of ${{\mathbb K}}$ (see [@Sch Theorem 16.1]). We write $R(\alpha){\subseteq}[0,\infty[$ for the set of all $|\lambda|$ such that $\lambda\in {\overline}{{{\mathbb K}}}$ is an eigenvalue of $\alpha{\otimes}_{{\mathbb K}}\operatorname{id}_{{\overline}{{{\mathbb K}}}}$.\
The above definition of hyperbolicity is a good basis for theorems, but may be difficult to verify directly. Fortunately, in the finite-dimensional case, an easier (and more concrete) description of hyperbolicity can be obtained. Also, the existence of centre subspaces and centre-stable subspaces is automatic:\
[**Theorem A.**]{} *Let $\alpha\colon E\to E$ be a linear self-map of a finite-dimensional vector space $E$ over a complete ultrametric field ${{\mathbb K}}$. Then $E$ admits an $a$-centre-stable subspace and an $a$-centre subspace, for each $a\in\;]0,\infty[$. Moreover, $\alpha$ is $a$-hyperbolic if and only if $a\not\in R(\alpha)$.*\
Let $M$ be an analytic manifold modelled on an ultrametric Banach space $E$ over ${{\mathbb K}}$ (as in [@Bo1]). An analytic diffeomorphism $\kappa\colon U\to V$ from an open set $U{\subseteq}M$ onto an open set $V{\subseteq}E$ is called a *chart* for $M$. An analytic map $f\colon N\to M$ between analytic manifolds is called an *immersion* if, for each $x\in N$, the tangent map $T_x(f)\colon T_x(N)\to T_{f(x)}(M)$ is a homeomorphism onto its image $\operatorname{im}T_x(f)$, and $\operatorname{im}T_x(f)$ is complemented in $T_{f(x)}(M)$ as a topological vector space. If $M$ and $N$ have finite dimension, this simply means that $T_x(f)$ is injective for each $x\in N$. An analytic manifold $N$ is called an *immersed submanifold* of $M$ if $N{\subseteq}M$ as a set and the inclusion map $\iota\colon
N\to M$ is an immersion. For $x\in N$, we identify $T_x(N)$ with the vector subspace $\operatorname{im}T_x(\iota)$ of $T_x(M)$.\
As before, let $M$ be an analytic manifold modelled on an ultrametric Banach spaces $E$ over a complete ultrametric field $({{\mathbb K}},|.|)$. Let $f\colon M\to M$ be an analytic diffeomorphism, and $p\in M$ be a fixed point of $f$.\
[**Definition.**]{} Given $a\in \;]0,1]$, we define $W_a^{{\footnotesize\rm s}}(f,p){\subseteq}M$, the *$a$-stable set* around $p$ with respect to $f$, as the set of all $x\in M$ such that $$\label{dewaseq}
\mbox{$f^n(x)\to p\;$ as $\;n\to\infty\;$
and $\; a^{-n}\|\kappa(f^n(x))\|\to 0\, $,}$$ for some (and hence every) chart $\kappa\colon U\to V{\subseteq}E$ of $M$ with $p\in U$ such that $\kappa(p)=0$, and some (and hence every) ultrametric norm $\|.\|$ on $E$ defining its topology.[^1]\
It is clear from the definition that $W_a^{{\footnotesize\rm s}}:=W_a^{{\footnotesize\rm s}}(f,p)$ is stable under $f$, i.e., $f(W_a^{{\footnotesize\rm s}})=W_a^{{\footnotesize\rm s}}$. If the tangent map $T_p(f)\colon T_p(M)\to T_p(M)$ is $a$-hyperbolic (which can be checked using Theorem A), then $W_a^{{\footnotesize\rm s}}$ is an analytic manifold, the *$a$-stable manifold* around $p$ with respect to $f$:\
[**Ultrametric Stable Manifold Theorem**]{} (cf. [@EXP Theorem 1.3]). *Let $M$ be an analytic manifold modelled on an ultrametric Banach space over a complete ultrametric field ${{\mathbb K}}$. Let $f\colon M\to M$ be an analytic diffeomorphism, $p\in M$ be a point fixed by $f$, and $a\in \;]0,1]$. If the tangent map $\alpha:=T_p(f)\colon T_p(M)\to T_p(M)$ is $a$-hyperbolic* (*which is satisfied if $M$ is finite-dimensional and $a\not\in R(\alpha)$*), *then there exists a unique analytic manifold structure on $W_a^{{\footnotesize\rm s}}:=W_a^{{\footnotesize\rm s}}(f,p)$ such that* (a)–(c) *hold*:
- *$W_a^{{\footnotesize\rm s}}$ is an immersed submanifold of $M$*;
- *$W_a^{{\footnotesize\rm s}}$ is tangent to the $a$-stable subspace $T_p(M)_{a, {{\footnotesize\rm s}}}$* (*with respect to $T_p(f)$*), *i.e., $T_p(W_a^{{\footnotesize\rm s}})= T_p(M)_{a,{{\footnotesize\rm s}}}$*;
- *$f$ restricts to an analytic diffeomorphism $W_a^{{\footnotesize\rm s}}\to W_a^{{\footnotesize\rm s}}$.*
*Moreover, each neighbourhood of $p$ in $W_a^{{\footnotesize\rm s}}$ contains an open neighbourhood $\Omega$ of $p$ in $W_a^{{\footnotesize\rm s}}$ which is a submanifold of $M$, is $f$-invariant $($i.e., $f(\Omega){\subseteq}\Omega)$, and such that $W_a^{{\footnotesize\rm s}}=\bigcup_{n=0}^\infty f^{-n}(\Omega)$.*\
If $T_p(f)$ is hyperbolic, then $W_1^{{\footnotesize\rm s}}$ is simply called the *stable manifold* around $p$, and denoted $W^{{\footnotesize\rm s}}$.\
Now consider the following local situation:\
Let $M$ be an analytic manifold modelled on an ultrametric Banach space over a complete ultrametric field ${{\mathbb K}}$. Let $M_0{\subseteq}M$ be open, $f\colon M_0 \to M$ be an analytic mapping, $p\in M_0$ be a fixed point of $f$, and $a\in \;]0,1]$. The following four definitions are taken from [@EXP].\
[**Definition.**]{} If $T_p(M)$ has an $a$-centre-stable subspace $T_p(M)_{a,{{\footnotesize\rm cs}}}$ with respect to $T_p(f)$, we call an immersed submanifold $N{\subseteq}M_0$ an *$a$-centre-stable manifold* around $p$ with respect to $f$ if (a)–(d) are satisfied:
- $p\in N$;
- $N$ is tangent to $T_p(M)_{a,{{\footnotesize\rm cs}}}$ at $p$, i.e., $T_p(N)=T_p(M)_{a,{{\footnotesize\rm cs}}}$;
- $f(N){\subseteq}N$; and
- $f|_N\colon N\to N$ is analytic.
If $a=1$, we simply speak of a *centre-stable manifold*.\
[**Definition.**]{} If $T_p(f)$ is an automorphism and $T_p(M)$ has a centre subspace $T_p(M)_{{\footnotesize\rm c}}$ with respect to $T_p(f)$, we say that an immersed submanifold $N{\subseteq}M_0$ is a *centre manifold* around $p$ with respect to $f$ if (a), (c) and (d) from the preceding definition hold as well as
- $N$ is tangent to $T_p(M)_{{\footnotesize\rm c}}$ at $p$, i.e., $T_p(N)=T_p(M)_{{\footnotesize\rm c}}$.
[**Definition.**]{} In the situation above, assume that $T_p(f)$ is $a$-hyperbolic. An immersed submanifold $N{\subseteq}M_0$ is called a *local $a$-stable manifold* around $p$ with respect to $f$ if (a), (c) and (d) just stated are satisfied as well as
- $N$ is tangent at $p$ to the $a$-stable subspace $T_p(M)_{a,{{\footnotesize\rm s}}}$ with respect to $T_p(f)$, i.e., $T_p(N)=T_p(M)_{a,{{\footnotesize\rm s}}}$.
If $a=1$, we simply speak of a *local stable manifold*.\
[**Definition.**]{} In the situation above, assume that $T_p(f)$ is $a$-hyperbolic. An immersed submanifold $N{\subseteq}M_0$ is called a *local $a$-unstable manifold* around $p$ with respect to $f$ if
- $p\in N$;
- $N$ is tangent at $p$ to the $a$-unstable subspace $T_p(M)_{a,{{\footnotesize\rm u}}}$ with respect to $T_p(f)$, i.e., $T_p(N)=T_p(M)_{a,{{\footnotesize\rm u}}}$;
- There exists an open neighbourhood $U$ of $p$ in $N$ such that $f(U){\subseteq}N$ and $f|_U\colon U\to N$ is analytic.
Combining Theorem A with [@EXP Theorems 1.9, 1.10, 6.6 and 8.3] (which contain further information), we obtain in the finite-dimensional case:\
[**Local Invariant Manifold Theorem.**]{} *Let $M$ be a finite-dimensionalanalytic manifold over a complete ultrametric field ${{\mathbb K}}$, $M_0{\subseteq}M$ be an opensubset, $f\colon M_0\to M$ be an analytic map and $p\in M_0$ a point fixed by $f$.If $a\in \;]0,1]$, then* (a)–(c) *hold*:
- *There exists an $a$-centre-stable manifold $N$ around $p$ with respect to $f$, such that $N$ is a submanifold of $M$*;
- *If $\alpha:=T_p(f)$ is an automorphism, then there exists an $a$-centre manifold $N$ around $p$ with respect to $f$ which is a submanifold of $M$, such that $f(N)=N$*;
- *If $a\not\in R(\alpha)$, then there exists a local $a$-stable manifold $N$ around $p$ with respect to $f$, which is a submanifold of $M$*.
*For $a\geq 1$, we have*:
- *If $a\not\in R(\alpha)$, then there exists a local $a$-unstable manifold $N$ around $p$ with respect to $f$, which is a submanifold of $M$*.
*In all of* (a)–(d), *the germ of $N$ at $p$* (*as an analytic manifold*) *is uniquely determined. Moreover, there is a basis of open neighbourhoods $N'$ of $p$ in $N$ such that $N'$ has the property of $N$ described in* (a)–(d), *respectively.*\
If $\alpha:=T_p(f)\colon T_p(M)\to T_p(M)$ is an automorphism in the preceding situation, then properties of the spectrum of $\alpha$ and properties of the fixed point $p$ of $f$ can be related. The next theorem collects results of this type from Propositions \[sin\], \[typeR\] and \[unicon\]. We say that a fixed point $p\in M_0$ of $f\colon M_0\to M$ is *uniformly attractive* if each neighbourhood of $p$ in $M_0$ contains a neighbourhood $Q$ of $p$ in $M_0$ such that $f(Q){\subseteq}Q$ and $$\lim_{n\to\infty}f^n(x)=p\quad\mbox{for all $\,x\in Q$}$$ (cf. Definition \[typesfp\]).\
[**Theorem B.**]{} *Let $M$ be a finite-dimensional analytic manifold over a complete ultrametric field ${{\mathbb K}}$, $M_0{\subseteq}M$ be an open subset, $f\colon M_0\to M$ be an analytic map and $p\in M_0$ a fixed point of $f$ such that $\alpha:=T_p(f)$ is an automorphism. Then* (a)–(c) *hold*:
- *$R(\alpha){\subseteq}\;]0,1]$ if and only if each neighbourhood $P$ of $p$ in $M_0$ contains a neighbourhood $Q$ of $p$ such that $f(Q){\subseteq}Q$*;
- *$R(\alpha){\subseteq}\{1\}$ if and only if each each neighbourhood $P$ of $p$ in $M_0$ contains a neighbourhood $Q$ of $p$ such that $f(Q)=Q$*;
- *$R(\alpha){\subseteq}\;]0,1[$ if and only if $p$ is a uniformly attractive fixed point of $f$.*
In the $1$-dimensional case, fixed (and periodic) points were already classified into attractive, repelling and indifferent ones in [@Khr]. Results concerning attractive and repelling fixed points, as well as Siegel disks were also obtained in [@Aga], which amount to the sufficiency (but not the necessity) of the spectral condition in (b) and (c) of Theorem B.\
It is useful to have conditions ensuring that the (global) stable manifold $W^{{\footnotesize\rm s}}$ is not only an immersed submanifold, but a submanifold. In view of Theorem A, our Proposition \[Bangd\] below subsumes the following:\
[**Theorem C.**]{} *Let $M$ be a finite-dimensional analytic manifold over a complete ultrametric field. Let $p\in M$ be a fixed point of an analytic diffeomorphism $f\colon M\to M$, and $\alpha:=T_p(f)$. If $R(\alpha){\subseteq}\,]0,1]$, then $W_a^{{\footnotesize\rm s}}(f,p)$ is a submanifold of $M$, for each $a\in \;]0,1]$ such that $T_p(f)$ is $a$-hyperbolic.*\
If $\beta \colon G\to G$ is an automorphism of a finite-dimensional analytic Lie group $G$ over a complete ultrametric field, then the neutral element $e\in G$ is a fixed point for $\beta$, but we cannot expect in general that $T_e(\beta)$ is hyperbolic. Nonetheless, it is always possible to turn the stable set $$U_\beta:=W^{{{\footnotesize\rm s}}}(\beta,e):=\{x\in G\colon \lim_{n\to\infty}\beta^n(x)=e\}$$ (the so-called *contraction group*) into a manifold:\
[**Theorem D.**]{} *If $\beta\colon G\to G$ is an automorphism of a finite-dimensional analytic Lie group $G$ over a complete ultrametric field, then there is a unique immersed submanifold structure on $U_\beta =W^{{\footnotesize\rm s}}(\beta,e)$ such that conditions* (a)–(c) *of the Ultrametric Stable Manifold Theorem* (*with $\beta$ in place of $f$*) *are satisfied. This immersed submanifold structure makes $U_\beta$ an immersed Lie subgroup of $G$.*\
To explain the motivation for the current article, and to show the utility of its results, we now briefly describe three Lie-theoretic applications which are only available through the use of invariant manifolds.\
[**Applications in Lie theory.**]{} Let $G$ be an analytic finite-dimensional Lie group over a local field ${{\mathbb K}}$ and $\beta\colon G\to G$ be an analytic automorphism. The *Levi factor* of $\beta$ is the subgroup $$M_\beta:=\{x\in G\colon \mbox{$\beta^{{\mathbb Z}}(x)$ is relatively compact in $G$} \},$$ where $\beta^{{\mathbb Z}}(x):=\{\beta^n(x)\colon n\in {{\mathbb Z}}\}$ (see [@BaW]). Using invariant manifolds, one can prove the following results in arbitrary characteristic (the $p$-adic case of which is due to J.S.P. Wang [@Wan]):
- *The group $U_\beta$ is always nilpotent* (see [@MaZ Theorem B]).
- *If $U_\beta$ is closed, then $U_\beta$, $U_{\beta^{-1}}$ and $M_\beta$ are Lie subgroups of $G$. Moreover, $U_\beta M_\beta U_{\beta^{-1}}$ is an open subset of $G$ and the “product map”* $$\pi\colon U_\beta\times M_\beta \times U_{\beta^{-1}}\to
U_\beta M_\beta U_{\beta^{-1}}\,,\quad (x,y,z){\mapsto}xyz$$ *is an analytic diffeomorphism* (see [@SPO]).
In fact, the $a_j$-stable manifolds $G_j:=W_{a_j}^{{\footnotesize\rm s}}(\beta,e)$ provide a central series $\{1\}=G_1{\subseteq}G_2{\subseteq}\cdots {\subseteq}G_n=G$ of Lie subgroups of $G$, for suitable real numbers $0<a_1<\cdots < a_n<1$ (see [@MaZ]). And to get (b), one heavily uses the (stable) manifold structures on $U_\beta=W^{{\footnotesize\rm s}}(\beta,e)$ and $U_{\beta^{-1}}=W^{{\footnotesize\rm s}}(\beta^{-1},e)$ discussed here, and the fact that $M_\beta$ contains a centre manifold for $\beta$ around $e$ (see [@SPO]; the result was also announced with a sketch of proof in [@SUR Theorem 9.1]).
- Using (b) as a tool, it is also possible to calculate the “scale” $s(\beta)$ (introduced in [@Wi1], [@Wi2])[^2] if $U_\beta$ is closed, in terms of the eigenvalues of the tangent map $L(\beta):=T_e(\beta)$ (see [@SPO]; cf. [@SUR Theorem 9.3] for a more detailed announcement with a sketch of proof). Previously, this was only possible in the $p$-adic case (see [@SCA]; cf. also [@BaW] for the scale of inner automorphisms of reductive algebraic groups).
*Structure of the article.* We first provide notation, basic facts and further definitions of invariant vector subspaces in a preparatory section (Section 1). Sections 2–6 are devoted to the proofs of Theorems A–D, and related results.
Preliminaries and notation {#secprepa}
==========================
In this section, we fix notation and recall some basic facts. We also define (and briefly discuss) centre subspaces and centre-stable subspaces.\
In this article, ${{\mathbb N}}:=\{1,2,\ldots\}$ and ${{\mathbb N}}_0:={{\mathbb N}}\cup\{0\}$. We write ${{\mathbb Z}}$ for the integers and ${{\mathbb R}}$ for the field of real numbers. If $f\colon M\to M$ and $n\in {{\mathbb N}}$, we write $f^n:=f\circ\, \cdots\, \circ f$ for the $n$-fold composition, and $f^0:=\operatorname{id}_M$. If $f$ is invertible, we define $f^{-n}:=(f^{-1})^n$.\
Recall that an *ultrametric field* is a field ${{\mathbb K}}$, together with an absolute value $|.|\colon {{\mathbb K}}\to [0,\infty[$ which satisfies the ultrametric inequality. We shall always assume that the metric $d\colon {{\mathbb K}}\times {{\mathbb K}}\to [0,\infty[$, $d(x,y):=|x-y|$, defines a non-discrete topology on ${{\mathbb K}}$. If the metric space $({{\mathbb K}},d)$ is complete, then the ultrametric field $({{\mathbb K}},d)$ is called *complete*. A totally disconnected, locally compact, non-discrete topological field is called a *local field*. Any such admits an ultrametric absolute value making it a complete ultrametric field [@Wei]. See, e.g., [@Sch] for background concerning complete ultrametric fields.\
An *ultrametric Banach space* over an ultrametric field ${{\mathbb K}}$ is a complete normed space $(E,\|.\|)$ over ${{\mathbb K}}$ whose norm $\|.\| \colon E\to [0,\infty[$ satisfies the *ultrametric inequality*, $\|x+y\|\leq \max\{\|x\|, \|y\|\}$ for all $x,y\in E$ (cf. [@Roo]). The ultrametric inequality entails that $$\label{domi}
\|x+ y\|= \|x\|\quad\mbox{for all $x,y\in E$ such that $\|y\|<\|x\|$.}$$ Given $x\in E$ and $r\in \;]0,\infty]$, we set $B^E_r(x):=\{y\in E\colon \|y-x\|<r\}$.\
If $A\colon E\to F$ is a continuous linear map between ultrametric Banach spaces $(E,\|.\|_E)$ and $(F,\|.\|_F)$, we write $\|A\|:=\sup\{\|Ax\|_F/\|x\|_E\colon 0\not= x \in E\}$ for its operator norm. The following observation is immediate.
\[expafac\] If $(E,\|.\|)$ is an ultrametric Banach space over ${{\mathbb K}}$ and $A\colon E\to E$ an invertible continuous linear map, then $\frac{1}{\|A^{-1}\|}$ can be interpreted as an expansion factor, in the sense that $\|A y\|\geq \frac{1}{\|A^{-1}\|}\|y\|$ for all $y\in E$ (as in the familiar case of real Banach spaces).
We refer to [@Bo1] for the concept of an analytic map $f\colon U\to F$, where $(E,\|.\|_E)$ and $(F,\|.\|_F)$ are ultrametric Banach spaces and $U$ is an open subset of $E$; compare [@Ser] if $E$ and $F$ have finite dimension. Thus, in the terminology of Non-Archimedean Geometry, the mappings we consider are *locally* analyticmaps. If $f$ is as before and $x\in U$, we write $f'(x)\colon E\to F$ for the total differential of $f$ at $x$. We shall use that $f$ is *strictly* differentiable at $x$ (see [@Bo1]):
\[remstrict\] If $f\colon E\supseteq U\to F$ is analytic and $x\in U$, write $$\label{raute}
f(y)=f(x)+f'(x).(y-x)+R(y)\quad\mbox{for $\, y\in U$.}$$ Then $R|_{B^E_{\varepsilon}(x)}$ is Lipschitz for small ${\varepsilon}>0$ in the sense that $$\operatorname{Lip}(R|_{B^E_{\varepsilon}(x)}):=\sup\left\{\frac{\|R(z)-R(y)\|_F}{\|z-y\|_E}
\colon y\not= z\in B^E_{\varepsilon}(x)\right\}<\infty,$$ and $$\lim_{{\varepsilon}\to0}\operatorname{Lip}(R|_{B^E_{\varepsilon}(x)})=0.$$ If $E=F$ and $f'(x)$ is an automorphism, then $$\operatorname{Lip}(R|_{B^E_{\varepsilon}(x)})<\frac{1}{\|f'(x)^{-1}\|}$$ for ${\varepsilon}>0$ small enough. Hence, by (\[domi\]) and (\[raute\]), for all $y,z\in B^E_{\varepsilon}(x)$ we have $$\label{flower}
\|f(z)-f(y)\|=\|f'(x)(z-y)+R(z)-R(y)\|=\|f'(x).(z-y)\|.$$
An *analytic manifold* modelled on an ultrametric Banach space $E$ over a complete ultrametric field ${{\mathbb K}}$ is defined as usual (as a Hausdorff topological space $M$, together with a (maximal) set ${{\cal A}}$ of homeomorphisms (“charts”) $\phi\colon U_\phi\to V_\phi$ from open subsets of $M$ onto open subsets of $E$, such that $M=\bigcup_{\phi\in {{\cal A}}}U_\phi$ and the mappings $\phi\circ \psi^{-1}$ are analytic for all $\phi,\psi\in {{\cal A}}$). Also the tangent space $T_pM$ of $M$ at $p\in M$, the tangent bundle $TM$, analytic maps $f\colon M\to N$ between analytic manifolds, and the tangent maps $T_pf\colon T_pM\to T_{f(p)}N$ as well as $Tf\colon TM\to TN$ can be defined as usual (cf. [@Bo1]). If $f\colon M\to V$ is an analytic map to an open subset $V$ of an ultrametric Banach space $F$, then we identify $TV$ with $V\times F$ in the natural way and let $df\colon TM\to F$ be the second component of the map $Tf\colon M\to V\times F$. An *analytic Lie group* $G$ over ${{\mathbb K}}$ is a group, equipped with an analytic manifold structure modelled on an ultrametric Banach space over ${{\mathbb K}}$, such that the group inversion and group multiplication are analytic (cf. [@Bo2]). As usual, we write $L(G):=T_e(G)$ and $L(\beta):=T_e(\beta)$, if $\beta\colon G\to H$ is an analytic homomorphism between analytic Lie groups. Let $M$ be an analytic manifold modelled on an ultrametric Banach space $E$. A subset $N{\subseteq}M$ is called a *submanifold* of $M$ if there exists a complemented vector subspace $F$ of the modelling space of $M$ such that each point $p\in N$ is contained in the domain $U$ of some chart $\phi\colon U\to V$ of $M$ such that $\phi(N\cap U)=F\cap V$. By contrast, an analytic manifold $N$ is called an *immersed submanifold* of $M$ if $N{\subseteq}M$ as a set and the inclusion map $\iota \colon N\to M$ is an immersion. Subgroups of Lie groups with analogous properties are called *Lie subgroups* and *immersed Lie subgroups*, respectively. If we call a mapping $f$ an analytic diffeomorphism between two manifolds (or an analytic automorphism of a Lie group), then also the inverse map $f^{-1}$ is assumed analytic.\
Let us now complete the definitions of invariant vector subspaces from the Introduction. In the remainder of this section, let $E$ be an ultrametric Banach space over ${{\mathbb K}}$. Let $\alpha\colon E\to E$ be a continuous linear map, and $a\in \;]0,\infty[$.
\[centunique\] We mention that the spaces $E_{a,{{\footnotesize\rm s}}}$ and $E_{a,{{\footnotesize\rm u}}}$ in the definition of $a$-hyperbolicity stated in the Introduction are uniquely determined, in the case of an endomorphism $\alpha\colon E\to E$ of a finite-dimensional ${{\mathbb K}}$-vector space $E$. See [@EXP Remark 6.4] for the assertion if $\alpha$ is an automorphism. In the general case, the argument in the cited remark still provides uniqueness of $E_{a,{{\footnotesize\rm s}}}$. Let us write $E^+:=\bigcap_{k\in {{\mathbb N}}}\alpha^k(E)$ for the Fitting one component of $E$ (see, e.g., [@HaN Lemma 5.3.11]). Then $\alpha$ restricts to an automorphism $\beta$ of $E^+$. Now $E^+=(E_{a,{{\footnotesize\rm s}}})^+ \oplus E_{a,{{\footnotesize\rm u}}}$ is a decomposition for the $a$-hyperbolic automorphism $\beta$ and thus also $E_{a,{{\footnotesize\rm u}}}$ is unique.
\[defcsub\] An $\alpha$-invariant vector subspace $E_{a,{{\footnotesize\rm cs}}} {\subseteq}E$ is called an *$a$-centre-stable subspace* with respect to $\alpha$ if there exists an $\alpha$-invariant vector subspace $E_{a,{{\footnotesize\rm u}}}$ of $E$ such that $E=E_{a,{{\footnotesize\rm cs}}} \oplus E_{a,{{\footnotesize\rm u}}}$ and $\alpha_2:= \alpha|_{E_{a,{{\footnotesize\rm u}}}}\colon E_{a,{{\footnotesize\rm u}}}\to E_{a,{{\footnotesize\rm u}}}$ is invertible, and there exists an ultrametric norm $\|.\|$ on $E$ defining its topology, with the following properties:
- $\|x+y\|=\max\{\|x\|,\|y\|\}$ for all $x\in E_{a,{{\footnotesize\rm cs}}}$, $y\in E_{a,{{\footnotesize\rm u}}}$; and
- $\|\alpha_1\|\leq a$ and $\frac{1}{\|\alpha_2^{-1}\|}>a$ holds for the operator norms with respect to $\|.\|$, where $\alpha_1:=\alpha|_{E_{a,{{\footnotesize\rm cs}}}}$.
Then $E_{a,{{\footnotesize\rm cs}}}$ is uniquely determined and if $\alpha$ is invertible, then $E_{a,{{\footnotesize\rm u}}}$ is unique (see [@EXP Remark 3.3]). Arguing as in Remark \[centunique\], we see that $E_{a,{{\footnotesize\rm u}}}$ is also unique if $E$ is finite-dimensional.
\[defacentre\] We say that an $\alpha$-invariant vector subspace $E_{a,{{\footnotesize\rm c}}} {\subseteq}E$ is an *$a$-centre subspace* with respect to $\alpha$ if there exist $\alpha$-invariant vector subspaces $E_{a,{{\footnotesize\rm s}}}$ and $E_{a,{{\footnotesize\rm u}}}$ of $E$ such that $E=E_{a,{{\footnotesize\rm s}}}\oplus E_{a,{{\footnotesize\rm c}}} \oplus E_{a,{{\footnotesize\rm u}}}$, and an ultrametric norm $\|.\|$ on $E$ defining its topology, with the following properties:
- $\|x+y+z \|=\max\{\|x\|,\|y\|,\|z\|\}$ for all $x\in E_{a,{{\footnotesize\rm s}}}$, $y\in E_{a,{{\footnotesize\rm c}}}$ and $z\in E_{a,{{\footnotesize\rm u}}}$;
- $\|\alpha(x)\|=a\|x\|$ for all $x\in E_{a,{{\footnotesize\rm c}}}$;
- $\alpha_3:=\alpha|_{E_{a,{{\footnotesize\rm u}}}}$ is invertible;[^3] and
- $\|\alpha_1\|<a$ and $\frac{1}{\|\alpha_3^{-1}\|}>a$ hold for the operator norms with respect to $\|.\|$, where $\alpha_1:=\alpha|_{E_{a,s}}$.
If $\alpha$ is an automorphism, then $E_{a,{{\footnotesize\rm s}}}$, $E_{a,{{\footnotesize\rm c}}}$ and $E_{a,{{\footnotesize\rm u}}}$ are uniquely determined (see [@EXP Remark 4.3]). If $E$ is finite-dimensional, then $E_{a,{{\footnotesize\rm s}}}$ is unique by its description in [@EXP Remark 4.3], and hence also $E_{a,{{\footnotesize\rm c}}}$ and $E_{a,{{\footnotesize\rm u}}}$ are unique by the argument from Remark \[centunique\]. $E_{a,{{\footnotesize\rm s}}}$ and $E_{a,{{\footnotesize\rm u}}}$ are called the *$a$-stable* and *$a$-unstable* subspaces of $E$ with respect to $\alpha$, respectively. If $a=1$, we simply speak of stable, centre and unstable subspaces, and write $E_{{\footnotesize\rm s}}$, $E_{{\footnotesize\rm c}}$ and $E_{{{\footnotesize\rm u}}}$ instead of $E_{1,{{\footnotesize\rm s}}}$, $E_{1,{{\footnotesize\rm c}}}$ and $E_{1,{{\footnotesize\rm u}}}$.
Spectral interpretation of hyperbolicity {#secfin}
========================================
In this section, we consider the special case where $\alpha$ is an automorphism of a *finite-dimensional* vector space over a complete ultrametric field $({{\mathbb K}},|.|)$. We shall interpret $a$-hyperbolicity as the absence of eigenvalues of absolute value $a$ (in an algebraic closure of ${{\mathbb K}}$). Moreover, we shall see that an $a$-centre subspace and an $a$-centre-stable subspace always exist.
\[findiset\] Let $({{\mathbb K}},|.|)$ be a complete ultrametric field, $E$ be a finite-dimensional ${{\mathbb K}}$-vector space, and $\alpha\colon E\to E$ be a linear map. We define ${\overline}{{{\mathbb K}}}$, the extension $|.|$ and $R(\alpha)$ as in the Introduction, using the ${\overline}{{{\mathbb K}}}$-linear self-map $\alpha_{{\overline}{{{\mathbb K}}}}:=\alpha\otimes \operatorname{id}_{{\overline}{{{\mathbb K}}}}$ of the ${\overline}{{{\mathbb K}}}$-vector space $E_{{\overline}{{{\mathbb K}}}}:=E{\otimes}_{{\mathbb K}}{\overline}{{{\mathbb K}}}$ obtained from $E$ by extension of scalars. For each $\lambda\in {\overline}{{{\mathbb K}}}$, we let $$(E_{{\overline}{{{\mathbb K}}}})_{(\lambda)}\, :=\, \{ x\in E_{{\overline}{{{\mathbb K}}}}\!:
(\alpha_{{\overline}{{{\mathbb K}}}}-\lambda)^dx=0\}$$ be the generalized eigenspace of $\alpha_{{\overline}{{{\mathbb K}}}}$ in $E_{{\overline}{{{\mathbb K}}}}$ corresponding to $\lambda$ (where $d$ is the dimension of the ${{\mathbb K}}$-vector space $E$). Given $\rho\in [0,\infty[$, we define $$\label{dfspacerho}
(E_{{\overline}{{{\mathbb K}}}})_\rho\; :=\;
\bigoplus_{|\lambda|=\rho} (E_{{\overline}{{{\mathbb K}}}})_{(\lambda)}\,{\subseteq}\, E_{{\overline}{{{\mathbb K}}}} \, ,\vspace{-.7mm}$$ where the sum is taken over all $\lambda\in {\overline}{{{\mathbb K}}}$ such that $|\lambda|=\rho$. As usual, we identify $E$ with $E{\otimes}1{\subseteq}E_{{\overline}{{{\mathbb K}}}}$.
The following fact (cf. (1.0) on p.81 in [@Mar Chapter II]) is important:[^4]
For each $\rho\in R(\alpha)$, the vector subspace $(E_{{\overline}{{{\mathbb K}}}})_\rho$ of $E_{{\overline}{{{\mathbb K}}}}$ is defined over ${{\mathbb K}}$, i.e., $(E_{{\overline}{{{\mathbb K}}}})_\rho= (E_\rho)_{{\overline}{{{\mathbb K}}}}$ with $E_\rho:=(E_{{\overline}{{{\mathbb K}}}})_\rho\cap E$. Thus $$\label{isdsum}
E\; =\; \bigoplus_{\rho\in R(\alpha)} E_\rho\,,$$ and each $E_\rho$ is an $\alpha$-invariant vector subspace of $E$.[$\Box$]{}
It is essential for us that certain well-behaved norms exist on $E$ (as in \[findiset\]).
\[defnadpt\] A norm $\|.\|$ on $E$ is *adapted to $\alpha$* if the following holds:
- $\|.\|$ is ultrametric;
- $\big\|\sum_{\rho\in R(\alpha)} x_\rho\big\|=\max\{\|x_\rho\|
\colon \rho\in R(\alpha)\}$ for each $(x_\rho)_{\rho\in R(\alpha)}
\in \prod_{\rho\in R(\alpha)} E_\rho$; and
- $\|\alpha (x)\|=\rho\|x\|$ for each $0\not= \rho\in R(\alpha)$ and $x\in E_\rho$.
\[propadapt\] Let $E$ be a finite-dimensional vector space over a complete ultrametric field $({{\mathbb K}},|.|)$ and $\alpha\colon E\to E$ be a linear map. Let ${\varepsilon}>0$ and $E_0:=\{x\in E\colon (\exists n\in {{\mathbb N}})\;\alpha^n(x)=0\}$. Then $E$ admits a norm $\|.\|$ adapted to $\alpha$, such that $\alpha|_{E_0}$ has operator norm $<{\varepsilon}$ with respect to $\|.\|$.
The proof uses the following lemma:
\[normrho\] For each $\rho\in R(\alpha)\setminus\{0\}$, there exists an ultrametric norm $\|.\|_\rho$ on $E_\rho$ such that $\|\alpha(x)\|_\rho=\rho\|x\|_\rho$ for each $x\in E_\rho$.[$\Box$]{}
If $\alpha$ is an automorphism, then the assertion holds by [@SUR Lemma 4.4]. The general case follows if we replace $\alpha$ by the map $\alpha|_{E_\rho}\colon E_\rho\to E_\rho$, which is an automorphism as $\ker(\alpha){\subseteq}E_0$ and thus $E_\rho\cap \ker(\alpha)=\{0\}$.
The next lemma takes care of the case $\rho=0$.
\[rhozero\] Let $E$ be a finite-dimensional vector space over a complete ultrametric field $({{\mathbb K}},|.|)$ and $\alpha\colon E\to E$ be a nilpotent linear map. Let ${\varepsilon}>0$. Then there exists an ultrametric norm $\|.\|$ on $E$ with respect to which $\alpha$ has operator norm $<{\varepsilon}$.
Assume first that there exists a basis $v_1,\ldots, v_m$ of $E$ with respect to which $\alpha$ has Jordan normal form with a single Jordan block, i.e., $\alpha(v_1)=0$ and $\alpha(v_k)=v_{k-1}$ for $k\in \{2,\ldots, m\}$. The case $E=\{0\}$ being trivial, we may assume that $m\geq 1$. Choose $\lambda\in {{\mathbb K}}$ such that $0<|\lambda|<{\varepsilon}$ and define $w_k:=\lambda^k v_k$ for $k\in \{1,\ldots, m\}$. Then $\alpha(w_k)=\lambda^k v_{k-1}=\lambda w_{k-1}$ for $k\in\{2,\ldots, m\}$ and $\alpha(w_1)=0$, entailing that $\alpha$ has operator norm $<{\varepsilon}$ with respect to the maximum norm $\|.\|$ on $E$ with respect to the basis $w_1,\ldots, w_m$, $$\left\|\sum_{k=1}^m t_kw_k\right\|\; :=\;
\max\{|t_k|\colon k=1,\ldots, m\}\quad \mbox{for $\, t_1,\ldots, t_m\in {{\mathbb K}}$.}$$ In the general case, we write $E$ as a direct sum $\bigoplus_{j=1}^nE_j$ of $\alpha$-invariant vector subspaces $E_j{\subseteq}E$ such that the Jordan decomposition of $\alpha|_{E_j}$ has a single Jordan block. For each $j$, there exists an ultrametric norm $\|.\|_j$ on $E_j$ with respect to which $\alpha|_{E_j}$ has operator norm $<{\varepsilon}$, by the above special case. Then $\alpha$ has operator norm $<{\varepsilon}$ with respect to the ultrametric norm $\|.\|$ on $E$ given by $\|v_1+\cdots +v_n\|:=\max\{\|v_j\|_j\colon j=1,\ldots, n\}$ for $v_j\in E_j$.
[**Proof of Proposition \[propadapt\].**]{} For each $\rho\in R(\alpha)\setminus\{0\}$, we choose a norm $\|.\|_\rho$ on $E_\rho$ as described in Lemma \[normrho\]. Lemma \[rhozero\] provides an ultrametric norm $\|.\|_0$ on $E_0$, with respect to which $\alpha|_{E_0}$ has operator norm $<{\varepsilon}$. Then $$\Big\| \sum_{\rho\in R(\alpha)} x_\rho\Big\| \; :=
\; \max\, \big\{ \,\|x_\rho\|_\rho\colon \rho\in R(\alpha)\,\big\}\quad
\mbox{for $\,(x_\rho)_{\rho\in R(\alpha)} \in \prod_{\rho\in R(\alpha)}
E_\rho$}$$ defines a norm $\|.\|\colon E\to[0,\infty[$ which, by construction, is adapted to $\alpha$ and with respect to which $\alpha|_{E_0}$ has operator norm $<{\varepsilon}$.[$\Box$]{} We are now ready to prove Theorem A from the Introduction.\
[**Proof of Theorem A.**]{} By Proposition \[propadapt\], there exists an ultrametric norm $\|.\|{\widetilde}{\;}$ on $E$ which is adapted to $\alpha$, and with respect to which $\alpha|_{E_0}$ has operator norm $<a$.
*Centre-stable subspaces.* The conditions from Definition \[defcsub\] are satisfied with $\|.\|:=\|.\|{\widetilde}{\;}$ and $$\label{sodecacs}
E_{a,{{\footnotesize\rm cs}}}:=\bigoplus_{\rho\leq a} E_\rho\quad\mbox{ and }\quad
E_{a,{{\footnotesize\rm u}}}:= \bigoplus_{\rho >a} E_\rho\,.$$ *Centre subspaces.* The conditions of Definition \[defacentre\] are satisfied with $\|.\|:=\|.\|{\widetilde}{\;}$ and $$\label{sodecac}
E_{a,{{\footnotesize\rm s}}}:=\bigoplus_{\rho< a} E_\rho,\quad
E_{a,{{\footnotesize\rm c}}}:=E_a,\quad
\quad\mbox{ and }\quad
E_{a,{{\footnotesize\rm u}}}:= \bigoplus_{\rho >a} E_\rho\,.$$ *Hyperbolicity.* If $a\not\in R(\alpha)$, then the conditions from the definition of $a$-hyperbolicity (stated in the Introduction) are satisfied with $\|.\|:=\|.\|{\widetilde}{\;}$, $$\label{soahyp}
E_{a,{{\footnotesize\rm s}}}:=\bigoplus_{\rho< a} E_\rho \quad
\quad\mbox{ and }\quad
E_{a,{{\footnotesize\rm u}}}:= \bigoplus_{\rho >a} E_\rho\,.$$ If $a\in R(\alpha)$, then $\alpha$ cannot be $a$-hyperbolic. In fact, if $\alpha$ was $a$-hyperbolic, we obtain a norm $\|.\|$ and a splitting $E=E_{a,{{\footnotesize\rm s}}}\oplus E_{a,{{\footnotesize\rm u}}}$ as in the cited definition. Define $\alpha_1:=\alpha|_{E_{a,{{\footnotesize\rm s}}}}$ and $\alpha_2:=\alpha|_{E_{a,{{\footnotesize\rm u}}}}$. Because the norms $\|.\|$ and $\|.\|{\widetilde}{\;}$ are equivalent, there exists $C>0$ such that $C^{-1}\|.\|\leq \|.\|{\widetilde}{\;}\leq C\|.\|$. Let $0\not=v\in E_a$. Write $v=x+y$ with $x\in E_{a,{{\footnotesize\rm s}}}$ and $y\in E_{a,{{\footnotesize\rm u}}}$. If $y\not=0$, then $$\|v\|{\widetilde}{\;}=a^{-n}\|\alpha^n(v)\|{\widetilde}{\;}
\geq
a^{-n} C^{-1}\|\alpha^n(v)\|
\geq C^{-1} \left(\frac{1}{a \|\alpha_2^{-1}\|}\right)^n \|y\|$$ for all $n\in {{\mathbb N}}$, which is absurd because $\frac{1}{a\|\alpha_2^{-1}\|}>1$. Hence $y=0$ and thus $x=v\not=0$. But then $$\|v\|{\widetilde}{\;}=a^{-n}\|\alpha^n(v)\|{\widetilde}{\;}
\leq a^{-n} C \|\alpha^n(v)\|
\leq C \left(\frac{\|\alpha_1\|}{a}\right)^n\|v\|
\quad
\mbox{for all $\, n\in {{\mathbb N}}$.}$$ Since $\frac{\|\alpha_1\|}{a}<1$, this is absurd. Thus $\alpha$ cannot be $a$-hyperbolic.[$\Box$]{}
Behaviour close to a fixed point {#seccon}
================================
We now relate the behaviour of a dynamical system $(M,f)$ around a fixed point $p$ and properties of the linear map $T_p(f)$.
\[situnow\] Let $M$ be an analytic manifold modelled on an ultrametric Banach space over a complete ultrametric field $({{\mathbb K}},|.|)$. Let $f\colon M_0\to M$ be an analytic mapping on an open subset $M_0{\subseteq}M$ and $p\in M_0$ be a fixed point of $f$, such that $T_p(f)\colon T_p(M)\to T_p(M)$ is an automorphism.
\[sin\] In [\[situnow\]]{}, the following conditions are equivalent:
- $T_p(M)$ admits a centre-stable subspace with respect to $T_p(f)$, and each neighbourhood $P$ of $p$ in $M_0$ contains a neighbourhood $Q$ of $p$ such that $f(Q){\subseteq}Q$.
- There exists a norm $\|.\|$ on $T_p(M)$ defining its topology, such that $\|T_p(f)\|\leq 1$ holds for the corresponding operator norm.
If, moreover, $M$ is a finite-dimensional manifold, then [(a)]{} and [(b)]{} are also equivalent to the following condition:
- Each eigenvalue $\lambda$ of $T_p(f)\otimes_{{\mathbb K}}\operatorname{id}_{{\overline}{{{\mathbb K}}}}$ in an algebraic closure ${\overline}{{{\mathbb K}}}$ of ${{\mathbb K}}$ has absolute value $|\lambda|\leq 1$.
\(b) means that $E:=T_p(M)$ coincides with its centre-stable subspace with respect to $\alpha:=T_p(f)$. If $E$ is finite-dimensional, this property is equivalent to $R(\alpha){\subseteq}\;]0,1]$ and hence to (c), by (\[sodecacs\]) (using that $E_{{{\footnotesize\rm cs}}}$ is unique). If (b) holds, then (a) follows with [@EXP Theorem 1,9(c)].[^5]
(a)${\Rightarrow}$(b): If (a) holds, then $E$ admits a decomposition $E=E_{1,{{\footnotesize\rm cs}}}\oplus E_{1,{{\footnotesize\rm u}}}$ and a norm $\|.\|$, as described in Definition \[defcsub\] (with $a=1$). After shrinking $M_0$, we may assume that $M_1:=f(M_0)$ is open in $M$ and $f \colon M_0\to M_1$ is a diffeomorphism (by the Inverse Function Theorem).
If $E_{1,{{\footnotesize\rm u}}}\not=\{0\}$, we let $P{\subseteq}M_0\cap M_1$ be an open neighbourhood of $p$ such that $f(P){\subseteq}P$, and consider the map $g:=f^{-1}\colon M_1\to M$. Then $E_{1,{{\footnotesize\rm u}}}$ is the stable subspace of $E$ with respect to $T_p(g)=\alpha^{-1}$. Pick $b\in \;] \|\alpha^{-1}|_{E_{1,{{\footnotesize\rm u}}}}\| ,1[$. Then $\alpha^{-1}$ is $b$-hyperbolic, and $$E_{b,{{\footnotesize\rm s}}}=E_{1,{{\footnotesize\rm u}}} \quad \mbox{ as well as }\quad E_{b,{{\footnotesize\rm u}}}=E_{1,{{\footnotesize\rm cs}}}$$ (with respect to the automorphisms $\alpha^{-1}$ and $\alpha$ on the left and right of the equality signs, respectively). By [@EXP Theorem 6.6] (applied to $g|_P\colon P\to M$), there exists a local $b$-stable manifold $N{\subseteq}P$ with respect to $g$, such that $g^n(x)\to p$ as $n\to\infty$, for all $x\in N$. Since $N$ is tangent to $E_{1,{{\footnotesize\rm u}}}\not=\{0\}$, we have $N\not=\{p\}$ and thus find a point $x\in N\setminus\{p\}$. By hypothesis (a), there is an open $p$-neighbourhood $Q{\subseteq}P\setminus \{x\}$ with $f(Q){\subseteq}Q$. Since $g^n(x)\to p$, there exists $m\in {{\mathbb N}}$ with $y:=g^m(x)\in Q$. Then $x=f^m(y)\in f^m(Q){\subseteq}Q$, contradicting the choice of $Q$. Hence $E_{1,u}=\{0\}$ (and thus (b) holds).
\[typeR\] In [\[situnow\]]{}, the following conditions are equivalent:
- $T_p(M)$ admits a centre subspace with respect to $T_p(f)$, and eachneighbourhood $P$ of $p$ in $M_0$ contains a neighbourhood $Q$ of $p$ such that $f(Q)=Q$.
- There exists a norm $\|.\|$ on $T_p(M)$ defining its topology, which makes $T_p(f)$ an isometry.
If, moreover, $M$ is a finite-dimensional manifold, then [(a)]{} and [(b)]{} are also equivalent to the following condition:
- Each eigenvalue $\lambda$ of $T_p(f)\otimes_{{\mathbb K}}\operatorname{id}_{{\overline}{{{\mathbb K}}}}$ in an algebraic closure ${\overline}{{{\mathbb K}}}$ of ${{\mathbb K}}$ has absolute value $|\lambda|=1$.
\(b) means that $E:=T_p(M)$ coincides with its centre subspace with respect to $\alpha:=T_p(f)$. If $E$ is finite-dimensional, this property is equivalent to $R(\alpha){\subseteq}\{1\}$ and hence to (c), by (\[sodecac\]) (using the uniqueness of $E_{{{\footnotesize\rm c}}}$). If (b) holds, then (a) follows with [@EXP Theorem 1.10(c)].[^6]
(a)${\Rightarrow}$(b): After shrinking $M_0$, we may assume that $M_1:=f(M_0)$ is open in $M$ and $f \colon M_0\to M_1$ is a diffeomorphism. If (a) holds, then there is a decomposition $E=E_{1,{{\footnotesize\rm s}}}\oplus E_{1,{{\footnotesize\rm c}}}\oplus E_{1,{{\footnotesize\rm u}}}$ and a norm $\|.\|$, as in Definition \[defacentre\] (with $a=1$). By “(a)${\Rightarrow}$(b)” in Proposition \[sin\], we have $E_{1,{{\footnotesize\rm u}}}=\{0\}$. Applying Proposition \[sin\] to $g:=f^{-1}\colon M_1\to M$, we see that also $E_{1,{{\footnotesize\rm s}}}=\{0\}$ (because this is the unstable subspace of $T_p(M)$ with respect to $T_p(g)=\alpha^{-1}$). Thus $E=E_{1,{{\footnotesize\rm c}}}$, establishing (b).
The proofs show that $Q$ can always be chosen as an *open* subset of $M_0$, in part (a) of Proposition \[sin\] and \[typeR\].
\[typesfp\] In the situation of [\[situnow\]]{}, we use the following terminology:
- $p$ is said to be an *attractive* fixed point of $f$ if $p$ has a neighbourhood $P{\subseteq}M_0$ such that $f^n(x)$ is defined for all $x\in P$ and $n\in {{\mathbb N}}$, and $\lim_{n\to\infty}f^n(x)=p$ for all $x\in P$.
- We say that $p$ is *uniformly attractive* if it is attractive and, moreover, every neighbourhood of $p$ in $M_0$ contains a neighbourhood $Q$ of $p$ such that $f(Q){\subseteq}Q$.
\[unicon\] In [\[situnow\]]{}, the following conditions are equivalent:
- $T_p(M)$ admits a centre subspace with respect to $T_p(f)$, and $p$ isuniformly attractive;
- There exists a norm $\|.\|$ on $T_p(M)$ defining its topology, such that $\|T_p(f)\|<1$ holds for the corresponding operator norm.
If, moreover, $M$ is a finite-dimensional manifold, then [(a)]{} and [(b)]{} are also equivalent to the following condition:
- Each eigenvalue $\lambda$ of $T_p(f)\otimes_{{\mathbb K}}\operatorname{id}_{{\overline}{{{\mathbb K}}}}$ in an algebraic closure ${\overline}{{{\mathbb K}}}$ of ${{\mathbb K}}$ has absolute value $|\lambda| < 1$.
\(b) means that $E:=T_p(M)$ coincides with its stable subspace with respect to $\alpha:=T_p(f)$. If $E$ is finite-dimensional, this property is equivalent to $R(\alpha){\subseteq}\;]0,1[$ and hence to (c), by (\[sodecac\]) (using the uniqueness of $E_{{{\footnotesize\rm s}}}$). If (a) holds, then also (b), as shall be verified in Remark \[rmweaker\].
If (b) holds and $P{\subseteq}M_0$ is an open neighbourhood of $p$, then [@EXP Theorem 6.6][^7] (applied to $f|_P$ instead of $f$) provides a local stable manifold $N{\subseteq}P$ such that $\lim_{n\to\infty}f^n(x)=p$ for all $x\in N$. Because $T_p(N)=E=T_p(M)$, it follows that $N$ is open in $M$. Since, moreover, $f(N){\subseteq}N$ by definition of $N$, we have verifed that $p$ is uniformly attractive.
\[rmweaker\] If $p$ is merely attractive (but possibly not uniformly) and $E:=T_p(M)$ admits a centre subspace with respect to $T_p(f)$, we can still conclude that $E_{1,{{\footnotesize\rm c}}}=\{0\}$.\
\[After shrinking $M_0$, we may assume that $f$ is injective. Let $P{\subseteq}M_0$ be as in Definition \[typesfp\](a). If $E_{1,{{\footnotesize\rm c}}}\not=\{0\}$, we let $Q{\subseteq}P$ be a centre manifold with respect to $f$, such that $f(Q)=Q$ (see [@EXP Theorem 1.10(c)]). Since $E_{1,{{\footnotesize\rm c}}}\not=\{0\}$, we must have $Q\not=\{p\}$, enabling us to pick $x_0\in Q\setminus\{p\}$. Using [@EXP Theorem 1.10(c)] again, we find a centre manifold $S{\subseteq}Q\setminus\{x_0\}$ with respect to $f$, such that $f(S)=S$. Since $f$ is injective, it follows that $f(Q\setminus S)=Q\setminus S$ and thus $f^n(x_0)\in Q\setminus S$ for all $n\in {{\mathbb N}}_0$. As $Q$ is a neighbourhood of $p$, we infer $f^n(x_0)\not\to p$ as $n\to\infty$. Since $x_0\in P$, this contradicts the choice of $P$.\]
When [$W_a^{{\footnotesize\rm s}}(f,p)$]{} is not only immersed {#notonly}
===============================================================
In general, $W_a^{{\footnotesize\rm s}}$ is only an *immersed* submanifold of $M$, not a submanifold (cf.[@SUR §7.1] for an easy example). We now describe a criterion (needed in [@MaZ]) which prevents such pathologies.
\[Bangd\] Let $M$ be an analytic manifold modelled on an ultrametric Banach space over a complete ultrametric field. Let $p\in M$ be a fixed point of an analytic diffeomorphism $f\colon M\to M$, such that $E:=T_p(M)$ admits a centre-stable subspace with respect to $T_p(f)$, and $E_{1,{{\footnotesize\rm u}}}=\{0\}$. Then $W_a^{{\footnotesize\rm s}}(f,p)$ is a submanifold of $M$, for each $a\in \;]0,1]$ such that $T_p(f)$ is $a$-hyperbolic.
Let $W_a^{{\footnotesize\rm s}}:=W_a^{{\footnotesize\rm s}}(f,p)$ and $\Omega {\subseteq}W_a^{{\footnotesize\rm s}}$ be as in the Ultrametric Stable Manifold Theorem from the Introduction. Since $f$ restricts to a diffeomorphism of $W_a^{{\footnotesize\rm s}}$, the image $f(\Omega)$ is relatively open in $\Omega$. Hence, there exists an open $p$-neighbourhood $Q{\subseteq}M$ such that $\Omega\cap Q{\subseteq}f(\Omega)$. By “(b)${\Rightarrow}$(a)” in Proposition \[sin\], we may assume that $f(Q){\subseteq}Q$, after replacing $Q$ with a smaller neighbourhood of $p$ if necessary. We claim that $$\label{goclaim}
W_a^{{\footnotesize\rm s}}\cap Q
=\Omega\cap Q\,.$$ If this is true, then $W_a^{{\footnotesize\rm s}}\cap Q$ is a submanifold of $M$, and hence also $$f^{-n}(W_a^{{\footnotesize\rm s}}\cap Q)= f^{-n}(W_a^{{\footnotesize\rm s}})\cap f^{-n}(Q)=W_a^{{\footnotesize\rm s}}\cap f^{-n}(Q)$$ is a submanifold of $M$ (as $f^{-n}\colon M\to M$ is a diffeomorphism). Since $\bigcup_{n\in {{\mathbb N}}_0}f^{-n}(Q)$ is an open subset of $M$ which contains $W_a^{{\footnotesize\rm s}}$ (exploiting that $f^n(x)\in Q$ for large $n$, for each $x\in W_a^{{\footnotesize\rm s}}$), we deduce that $W_a^{{\footnotesize\rm s}}$ is a submanifold of $M$ (and the submanifold structure coincides with the given immersed submanifold structure on $W^{{\footnotesize\rm s}}_a$, as both structures coincide on each of the sets $f^{-n}(W^{{\footnotesize\rm s}}_a\cap Q)$, $n\in{{\mathbb N}}_0$, which form an open cover for $W^{{\footnotesize\rm s}}_a$).\
To prove (\[goclaim\]), suppose that $x\in W_a^{{\footnotesize\rm s}}\cap Q$ but $x\not\in \Omega\cap Q$ (and hence $x\not\in \Omega$). Since $f(Q){\subseteq}Q$, we then have $$f^n(x)\in Q\quad\mbox{for all $\,n\in {{\mathbb N}}_0$.}$$ By definition of $\Omega$, there exists $n\in {{\mathbb N}}_0$ such that $f^n(x)\in \Omega$. We choose $n$ minimal and note that $n\geq 1$ as $x\not\in \Omega$ by hypothesis. Then $f^n(x)\in \Omega\cap Q{\subseteq}f(\Omega)$ and hence $f^{n-1}(x)=f^{-1}(f^n(x)) \in f^{-1}(f(\Omega))=\Omega$, contradicting the minimality of $n$. Hence $x$ cannot exist and thus $W_a^{{\footnotesize\rm s}}\cap Q{\subseteq}\Omega\cap Q$. The converse inclusion, $\Omega\cap Q{\subseteq}W_a^{{\footnotesize\rm s}}\cap Q$, being trivial, (\[goclaim\]) is proved.
Dependence of [$a$]{}-stable manifolds on [$a>0$]{} {#adepend}
===================================================
We collect further results in the finite-dimensional case required in Section 6 and [@MaZ]. In particular, we study the dependence of $a$-stable manifolds on the parameter $a$.
\[adepprop\] Let $M$ be an analytic manifold modelled on a finite-dimensional vector space over a complete ultrametric field $({{\mathbb K}},|.|)$. Let $p\in M$ be a fixed point of an analytic diffeomorphism $f\colon M\to M$. Abbreviate $\alpha:=T_p(f)$ and define $R(\alpha)$ as in the Introduction. Then the following holds:
- If $R(\alpha){\subseteq}\;]0,1]$, then $W_a^{{\footnotesize\rm s}}(f,p)$ is a submanifold of $M$, for each $a\in \;]0,1]\setminus R(\alpha)$.
- If $0<a<b\leq 1$ and $[a,b]\cap R(\alpha)=\emptyset$, then $W_a^{{\footnotesize\rm s}}(f,p)=W_b^{{\footnotesize\rm s}}(f,p)$.
- If $a\in \;]0,1]$ and $\,]0,a]\cap R(\alpha)=\emptyset$, then $W_a^{{\footnotesize\rm s}}(f,p)=\{p\}$.
\(a) follows from Proposition \[Bangd\] (using (\[sodecac\]) and Theorem A).
\(b) Define $E:=T_p(M)$. Let $\|.\|$ be a norm on $E$ adapted to $\alpha:=T_p(f)$, and $R(\alpha)$ as well as the subspaces $E_\rho{\subseteq}E$ for $\rho>0$ be as in \[findiset\]. Choose a chart $\kappa\colon P\to U{\subseteq}E$ of $M$ around $p$ such that $\kappa(p)=0$. Let $Q{\subseteq}P$ be an open neighbourhood of $p$ such that $f(Q){\subseteq}P$; after shrinking $Q$, we may assume that $\kappa(Q)=B_r^E(0)$ for some $r>0$. Then $g:=\kappa\circ f|_Q\circ\kappa^{-1}|_{B_r^E(0)}\colon B^E_r(0)\to E$ expresses $f|_Q$ in the local chart $\kappa$. By hypothesis on $a$ and $b$, we have $$X :=\bigoplus_{\rho<a}E_\rho=\bigoplus_{\rho<b}E_\rho
\quad\mbox{ and }\quad
Y :=\bigoplus_{\rho>a}E_\rho=\bigoplus_{\rho>b}E_\rho\,.$$ Hence $E_{a,{{\footnotesize\rm s}}}=E_{b,{{\footnotesize\rm s}}}=X$ and $E_{a,{{\footnotesize\rm u}}}=E_{b,{{\footnotesize\rm u}}}=Y$, by (\[soahyp\]). Now let $\Omega_a$ and $\Omega_b$ be an $\Omega$ as in the Ultrametric Stable Manifold Theorem, applied with $a$ and $b$, respectively. By [@EXP Theorem 6.2(f)] and the proof of Theorem 1.3 in [@EXP], we may assume that $\Omega_a=\kappa^{-1}(\Gamma_a)$ and $\Omega_b=\kappa^{-1}(\Gamma_b)$, where $$\begin{aligned}
\Gamma_a & =& \{ z\in B_r^E(0)\colon \mbox{($\forall n\in {{\mathbb N}}_0$) $g^n(z)$
is defined and $\|g^n(z)\|\leq a^nr$}\}\; \mbox{and}\notag\\
\Gamma_b & =& \{ z\in B_t^E(0)\colon \mbox{($\forall n\in {{\mathbb N}}_0$) $g^n(z)$
is defined and $\|g^n(z)\|\leq b^nt$}\}\label{graess}\end{aligned}$$ for certain $r,t>0$. Moreover, by [@EXP Theorem 6.2(e)], we may assume that $r=t$, after replacing both $r$ and $t$ by $\min\{r,t\}$. Then $\Gamma_a{\subseteq}\Gamma_b$ by (\[graess\]), and hence $\Gamma_a=\Gamma_b$ (since both sets are graphs of functions on the same domain, by the cited theorem). Thus $\Omega_a=\Omega_b$, entailing that $W_a^{{\footnotesize\rm s}}(f,p)=W_b^{{\footnotesize\rm s}}(f,p)$ as a set and also as an immersed submanifold of $M$ (cf. proof of [@EXP Theorem 1.3]).
\(c) By (\[soahyp\]), we have $E_{a,{{\footnotesize\rm s}}}=\bigoplus_{\rho<a}E_\rho=\{0\}$, whence $\Omega=\kappa^{-1}(\Gamma)=\{p\}$ in [@EXP Theorem 1.3] and its proof. Thus $W_a^{{\footnotesize\rm s}}(f,p)=\bigcup_{n\in {{\mathbb N}}_0}f^{-n}(\Omega)=\{p\}$.
Results for automorphisms of Lie groups {#seclie}
=======================================
Throughout this section, $G$ is an analytic Lie group modelled on an ultrametric Banach space over a complete ultrametric field $({{\mathbb K}},|.|)$, and $\beta\colon G\to G$ an analytic automorphism. Then the neutral element $e\in G$ is a fixed point of $\beta$, and hence our general theory applies. We now compile some additional conclusions which are specific to automorphisms. Like results of the previous sections, these are needed for the farther-reaching Lie-theoretic applications described in the introduction.\
We begin with a corollary to Proposition \[unicon\]. An automorphism $\beta\colon G\to G$ is called *contractive* if $\lim_{n\to\infty}\beta^n(x)=e$ for each $x\in G$.
If $G$ is finite-dimensional and $\beta\colon G\to G$ a contractiveautomorphism, then every eigenvalue $\lambda$ of $ L(\beta) {\otimes}_{{\mathbb K}}\operatorname{id}_{{\overline}{{{\mathbb K}}}}$ in an algebraicclosure ${\overline}{{{\mathbb K}}}$ has absolute value $|\lambda|<1$.
$G$ is complete by [@FOR Proposition 2.1(a)], and metrizable. Since every identity neighbourhood $P$ in $G$ contains an open subgroup $U$ of $G$ (see, e.g., [@FOR Proposition 2.1(a)]), Lemma 1(a) in [@Sie] provides a $\beta$-invariant open subgroup $Q:=U_{(0)}{\subseteq}U{\subseteq}P$ of $G$. Hence $e$ is a uniformly contractive fixed point of $\beta$, and thus “(a)${\Rightarrow}$(c)” in Proposition \[unicon\] applies.
\[proprev\] If $a\in \;]0,1]$ and $L(\beta)$ is $a$-hyperbolic, the following holds:
- The $a$-stable manifold $W_a^{{\footnotesize\rm s}}(\beta,e)$ is an immersed Lie subgroup of $G$.
- If, moreover, $L(G)$ admits a centre subspace with respect to $L(\beta)$ and $L(G)_{1,{{\footnotesize\rm u}}}=\{0\}$, then $W_a^{{\footnotesize\rm s}}(\beta,e)$ is a Lie subgroup of $G$.
\(a) The proof of [@MaZ Proposition 4.6] applies without changes.[^8]
\(b) is a special case of Proposition \[Bangd\].
If $G$ is finite-dimensional, then the extra hypotheses in Proposition \[proprev\](b) mean that $R(L(\beta)){\subseteq}\;]0,1]$ (see Theorem A and (\[sodecac\])).\
In the following situation, hyperbolicity is not needed to make $W^{{\footnotesize\rm s}}$ a manifold.
\[notmostgen\] If $\beta\colon G\to G$ is an automorphism and $L(G)$ admits a centre subspace with respect to $L(\beta)\colon L(G)\to L(G)$, then the following holds:
- There exist a local stable manifold $V_1$ and a centre manifold $V_0$ around $e$ with respect to $\beta$, and a local stable manifold $V_{-1}$ around $e$ with respect to $\beta^{-1}$, such that $V_1V_0V_{-1}$ is open in $G$ and the product map $$\label{thepromp}
\pi \colon V_1\times V_0\times V_{-1} \to V_1V_0V_{-1}\,\quad (x,y,z){\mapsto}xyz$$ is an analytic diffeomorphism.
- There is a unique immersed submanifold structure on $W^{{\footnotesize\rm s}}(\beta,e)$ such that conditions [(a)–(c)]{} of the Ultrametric Stable Manifold Theorem $($from the Introduction$)$ are satisfied. This immersed submanifold structure makes $W^{{\footnotesize\rm s}}(\beta,e)$ an immersed Lie subgroup of $G$, and also the final assertion of the cited theorem holds. Moreover, $W^{{\footnotesize\rm s}}(\beta,e)=W_a^{{\footnotesize\rm s}}(\beta,e)$ for some $a\in \;]0,1[$ such that $L(\beta)$ is $a$-hyperbolic.
\(a) Set $E:=L(G)$ and let $E=E_1\oplus E_0\oplus E_{-1}$ be the decomposition into a stable subspace $E_1$, centre subspace $E_0$ and unstable subspace $E_{-1}$ with respect to $L(\beta)$, and $\|.\|$ be an ultrametric norm as in Definition \[defacentre\]. There is $a\in \;]0,1[$ such that $\|L(\beta)|_{E_1}\|<a$ and $\frac{1}{\|L(\beta)^{-1}|_{E_{-1}}\|}>\frac{1}{a}$. Then $L(\beta)$ is $a$-hyperbolic with $a$-stable subspace $E_1$ and $a$-unstable subspace $E_0\oplus E_{-1}$ (and the norm $\|.\|$ as before). Also $L(\beta)^{-1}$ is $a$-hyperbolic, with $a$-stable subspace $E_{-1}$ and $a$-unstable subspace $E_0\oplus E_1$ (and the norm $\|.\|$ as before). We let $V_1$ be a local $a$-stable manifold around $e$ with respect to $\beta$ and $V_{-1}$ be a local $a$-stable manifold around $e$ with respect to $\beta^{-1}$ (see [@EXP Theorem 6.6(a)]); by [@EXP Theorem 6.6(c)], we may assume that $V_1{\subseteq}W_a^{{\footnotesize\rm s}}(\beta,e)$. Also, we let $V_0$ be a centre manifold around $p$ with respect to $\beta$ (see [@EXP Theorem 1.10(a)]). Then $T_e(V_1)=E_1$, $T_e(V_0)=E_0$ and $T_e(V_{-1})=E_{-1}$, whence $$L(G)\, =\, T_e(V_1)\oplus T_e(V_0) \oplus T_e(V_{-1})\, .$$ Thus, after shrinking $V_1$, $V_0$ and $V_{-1}$ (which is possible by [@EXP Theorems 6.6(c) and 1.10(c)]), we may assume that $P:=V_1V_0V_{-1}$ is open in $G$ and the product map (\[thepromp\]) is an analytic diffeomorphism (by the Inverse Function Theorem [@Bo1]).
\(b) Shrinking $V_1$, $V_0$ and $V_{-1}$ further if necessary, we may assume that there are $r>0$ and charts $\kappa_j\colon V_j\to B^{E_j}_r(0)$ with $\kappa_j(e)=0$ and $d\kappa_j=\operatorname{id}$ for $j\in \{-1,0,1\}$. There is $s\in \;]0,r]$ such that $\beta(\kappa^{-1}_j(B^{E_j}_s(0))){\subseteq}V_j$ for all $j\in \{-1,0,1\}$. Let $g_j:=\kappa_j\circ \beta \circ \kappa_j^{-1}|_{B^{E_j}_s(0)}$. Shrinking $s$, we achieve that $$\begin{aligned}
\|g_0(x)\|\, &=& \,\;\|x\|\hspace*{1.08mm} \quad
\; \mbox{for each $x\in B^{E_0}_s(0)$,}\label{useelswh2}\\
\|g_1(x)\|\, & < & \; a \|x\| \quad
\, \hspace*{.1mm}\mbox{for each $x\in B^{E_1}_s(0)$, and } \label{useelswh3}\\
\|g_{-1}(x)\| & > & a^{-1} \|x\| \;\; \mbox{for each $x\in B^{E_{-1}}_s(0)$} \label{useelswh4}\end{aligned}$$ (using (\[flower\])). Then $$\kappa:=(\kappa_1\times \kappa_0\times \kappa_{-1})\circ \pi^{-1}\colon
P\to B^E_r(0)$$ is a chart of $G$ around $e$. We set $g:=g_1\times g_0 \times g_{-1} \colon B^E_s(0)\to
B^E_r(0)$ (where $B^E_s(0)=B^{E_1}_s(0)\times B^{E_0}_s(0) \times B^{E_{-1}}_s(0)$). Abbreviate $Q:=\kappa^{-1}(B^E_s(0))$. Then $$\label{locconj}
\beta|_Q=\kappa^{-1}\circ g \circ\kappa|_Q\,.$$ If $z\in W^{{\footnotesize\rm s}}(\beta,e)$, there is $n_0\in {{\mathbb N}}_0$ such that $\beta^n(z)\in Q$ for all $n\geq n_0$, and $$\label{wllwspr}
\|\kappa(\beta^n(z))\|\to 0\quad\mbox{as $\,n\to\infty$.}$$ After replacing $z$ with $\beta^{n_0}(z)$, we may assume that $n_0=0$. Now $x=(x_1, x_0, x_{-1}):=\kappa(z)$ is an element of $B_s^E(0)$ such that $g^n(x)=\kappa(\beta^n(z))\in B^E_s(0)$ for all $n\in {{\mathbb N}}_0$ (cf. (\[locconj\])). Also $$\label{zwstp}
\lim_{n\to\infty}\|g^n(x)\|=0\,,$$ by (\[wllwspr\]). Since $\|g^n(x)\|=\max\{\|g_1^n(x_1)\|, \| g_0^n(x_0)\|,\|g_{-1}^n(x_{-1})\|\}$ for all $n\in {{\mathbb N}}_0$, using (\[useelswh2\]) and (\[useelswh4\]) we obtain a contradiction to (\[zwstp\]) unless $x_0=0$ and $x_{-1}=0$. Thus $x=x_1\in E_1$ and thus $z=\kappa_1^{-1}(x_1)\in V_1{\subseteq}W_a^{{\footnotesize\rm s}}(\beta,e)$,entailing that $W^{{\footnotesize\rm s}}(\beta,e){\subseteq}W_a^{{\footnotesize\rm s}}(\beta,e)$. The converse inclusion being trivial, we deduce that $W^{{\footnotesize\rm s}}(\beta,e)
=W_a^{{\footnotesize\rm s}}(\beta,e)$. We give $W^{{\footnotesize\rm s}}(\beta,e)$ the manifold structure of $W_a^{{\footnotesize\rm s}}(\beta,e)$. It then is tangent to $E_{a,{{\footnotesize\rm s}}}=E_1$ at $e$. Hence $W^{{\footnotesize\rm s}}(\beta,e)$ satisfies conditions (a)–(c) of the Ultrametric Stable Manifold Theorem and also the final assertion of the theorem. To obtain the uniqueness of the immersed submanifold structure subject to these conditions, note that for any such structure on $W^{{\footnotesize\rm s}}$, each neighbourhood of $e$ in $W^{{\footnotesize\rm s}}$ contains an open $\beta$-invariant neighbourhood of $e$ (as this only requires (\[domi\]) and \[remstrict\]). Now one shows as in the proof of [@EXP Theorem 6.6(b)] that the germ of the latter coincides with the germ wealready have, and this entails as in the proof of the uniqueness part of[@EXP Theorem 1.3] that the new manifold structure on $W^{{\footnotesize\rm s}}$ coincides with the one we already had (further explanations are omitted, because the assertion is not central). All other assertions follow from Proposition \[proprev\].
[**Proof of Theorem D.**]{} We now prove Theorem D. The proof will provide additional information: *$W^{{\footnotesize\rm s}}(\beta,e)=W_a^{{\footnotesize\rm s}}(\beta,e)$ for each $a\in \;]0,1[$ such that $[a,1[\,\cap R(L(\beta))=\emptyset$ and $\;]1,\frac{1}{a}]\cap R(L(\beta))=\emptyset$*.\
If we choose $\|.\|$ as a norm adapted to $L(\beta)$ (as in Definition \[defnadpt\]) in the proof of Proposition \[notmostgen\], then $E_1$, $E_0$ and $E_{-1}$ are the direct sum of all $L(G)_\rho$ with $\rho\in R(L(\beta))$, such that $\rho\in \;]0,1[$ (resp., $\rho=1$, resp., $\rho\in \;]1,\infty[$), by (\[sodecac\]). If $a$ is as described at the beginning of the proof, then $\|L(\beta)\|<a$ and $\|L(\beta)^{-1}\|<a$ (as is clear from (b) and (c) in Definition \[defnadpt\]). Therefore the proof of Proposition \[notmostgen\] applies with this choice of $a$.[$\Box$]{}
[99]{} Aguayo, J., M. Saavedra, M. Vallas, *Attracting and repelling points of analytic dynamical systems of several variables in a non-archimedean formulation*, Theoret. and Math. Phys. [**140**]{} (2004), 1175–1181. Baumgartner, U. and G.A. Willis, *Contraction groups and scales of automorphisms of totally disconnected, locally compact groups*, Israel J. Math. [**142**]{} (2004), 221–248. Bourbaki, N., “Variétés différentielles et analytiques. Fascicule de résultats,” Hermann, Paris, 1967. Bourbaki, N., “Lie Groups and Lie Algebras” (Chapters 1–3), Springer, Berlin 1989. Glöckner, H., [*Scale functions on $p$-adic Lie groups*]{}, Manuscr. Math. [**97**]{} (1998), 205–215. Glöckner, H., *Every smooth $p$-adic Lie group admits a compatible analytic structure*, Forum Math. [**18**]{} (2006), 45–84. Glöckner, H., *Contractible Lie groups over local fields*, Math. Z. 260 (2008), 889–904. Glöckner, H., *Lectures on Lie groups over local fields*, preprint, arXiv:0804.2234v4. Glöckner, H., *Invariant manifolds for analytic dynamical systems over ultrametric fields*, Expo. Math. [**31**]{} (2013), 116–150. Glöckner, H., [*Automorphisms of Lie groups over local fields of positive characteristic*]{}, in preparation. Hasselblatt, B. and A. Katok, “Handbook of Dynamical Systems,” Volume 1A, Elsevier, 2002. Hilgert, J. and K.-H. Neeb, “Structure and Geometry of Lie Groups,” Springer, 2012. Hirsch, M.W., C.C. Pugh and M. Shub, “Invariant Manifolds,” Springer, 1977. Irwin, M.C., [*On the stable manifold theorem*]{}, Bull. London Math. Soc. [**2**]{} (1970), 196–198. Khrennikov, A. and M. Nilsson, “$p$-Adic Deterministic and Random Dynamical Systems,” Kluwer, 2004. Margulis, G.A., “Discrete Subgroups of Semisimple Lie Groups,” Springer, 1991. Schikhof, W.H., “Ultrametric Calculus,” Cambridge University Press, 1984. Serre, J.-P., “Lie Algebras and Lie Groups,” Springer, Berlin, 1992. Siebert, E., *Semisimple convolution semigroups and the topology of contraction groups*, pp.325–343 in: H. Heyer (ed.), “Probability Measures on Groups IX” (Oberwolfach 1988), Springer, Berlin, 1989. van Rooij, A.C.M., ‘Non-Archimedean Functional Analysis,’’ Marcel Dekker, 1978. Wang, J.S.P., [*The Mautner phenomenon for $p$-adic Lie groups*]{}, Math. Z. [**185**]{} (1984), 403–412. Weil, A., “Basic Number Theory,” Springer, 1967. Wells, J.C., [*Invariant manifolds of non-linear operators*]{}, Pacific J. Math. [**62**]{}(1976), 285–293. Willis, G.A., [*The structure of totally disconnected, locally compact groups*]{}, Math. Ann. [**300**]{} (1994), 341–363. Willis, G.A, [*Further properties of the scale function on a totally disconnected group*]{}, J. Algebra [**237**]{}(2001), 142–164.
Helge Glöckner, Universität Paderborn, Institut für Mathematik, Warburger Str. 100,\
33098 Paderborn, Germany. E-Mail: glocknermath.upb.de
[^1]: See [@EXP Remark 6.5] for the independence of the choice of $\kappa$ and $\|.\|$.
[^2]: The scale can be defined as the minimum index $s(\beta):=\min_V [V:V\cap \beta^{-1}(V)]$, for $V$ ranging through the set of all compact, open subgroups of $G$.
[^3]: This hypothesis can be omitted (as it then follows from the others) if $E$ has finite dimension (since $\ker\alpha{\subseteq}E_{a,{{\footnotesize\rm s}}}$) or $\alpha$ is an automorphism.
[^4]: In [@Mar p.81], ${{\mathbb K}}$ is a local field, but the proof works also for complete ultrametric fields.
[^5]: If $E$ is finite-dimensional, this corresponds to the conclusions concerning centre-stable manifolds in the Local Invariant Manifold Theorem stated above.
[^6]: If $E$ is finite-dimensional, see also the conclusions concerning centre manifolds in the Local Invariant Manifold Theorem above.
[^7]: If $E$ is finite-dimensional, see also the conclusions concerning local stable manifolds in the Local Invariant Manifold Theorem above.
[^8]: In $\diamondsuit$, read “$\,\leq a^n\,$” as “$\,< a^n r$.”
|
---
abstract: 'In this paper we study models and coordination policies for intermodal , wherein a fleet of self-driving vehicles provides on-demand mobility jointly with public transit. Specifically, we first present a network flow model for intermodal , where we capture the coupling between and public transit and the goal is to maximize social welfare. Second, leveraging such a model, we design a pricing and tolling scheme that allows to achieve the social optimum under the assumption of a perfect market with selfish agents. Finally, we present a real-world case study for New York City. Our results show that the coordination between fleets and public transit can yield significant benefits compared to an system operating in isolation.'
author:
- 'Mauro Salazar$^{1,2}$, Federico Rossi$^2$, Maximilian Schiffer$^{2,3}$, Christopher H. Onder$^1$, and Marco Pavone$^2$ [^1][^2][^3]'
bibliography:
- '../../../bib/main.bib'
- '../../../bib/ASL\_papers.bib'
title: |
**On the Interaction between Autonomous Mobility-on-Demand\
and Public Transportation Systems**
---
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Dr. Daniele Vigo and Dr. Guido Gentile for the fruitful discussions, and Dr. Ilse New for her assistance with the proofreading and useful advice. The first author would like to thank Dr. Lino Guzzella for his support. This research was supported by the National Science Foundation under CAREER Award CMMI-1454737 and the Toyota Research Institute (TRI). This article solely reflects the opinions and conclusions of its authors and not NSF, TRI, or any other entity. This paper is dedicated to Stella.
[^1]: $^1$Institute for Dynamic Systems and Control ETH Zürich, Zurich (ZH), Switzerland [maurosalazar@idsc.mavt.ethz.ch]{}
[^2]: $^2$Autonomous Systems Lab, Stanford University, Stanford (CA), United States [{frossi2,pavone}@stanford.edu]{}
[^3]: $^3$TUM School of Management, Technical University of Munich, 80333 Munich, Germany [maximilian.schiffer@tum.de]{}
|
---
abstract: |
We explore the correlation between morphological types and mid-infrared (MIR) properties of an optically flux-limited sample of 154 galaxies from the Forth Data Release (DR4) of Sloan Digital Sky Survey (SDSS), cross-correlated with [*Spitzer*]{} SWIRE ([*Spitzer*]{} Wide-Area InfraRed Extragalactic Survey) fields of ELAIS-N1, ELAIS-N2 and Lockman Hole. Aperture photometry is performed on the SDSS and [*Spitzer*]{} images to obtain optical and MIR properties. The morphological classifications are given based on both visual inspection and bulge-disk decomposition on SDSS $g-$ and $r-$band images. The average bulge-to-total ratio ($B/T$) is a smooth function over different morphological types. Both the $8\mu
m$(dust) and $24\mu m$(dust) luminosities and their relative luminosity ratios to $3.6\mu m$ (MIR dust-to-star ratios) present obvious correlations with both the Hubble $T$ type and $B/T$. The early-type galaxies notably differ from the late-types in the MIR properties, especially in the MIR dust-to-star ratios. It is suggested that the MIR dust-to-star ratio of either $\nu
L_{\nu}[8\mu m(dust)]/\nu L_{\nu}[3.6\mu m]$ or $\nu L_{\nu}[24\mu
m(dust)]/\nu L_{\nu}[3.6\mu m]$ is an effective way to separate the early-type galaxies from the late-type ones. Based on the tight correlation between the stellar mass and the $3.6\mu m$ luminosity, we have derived a formula to calculate the stellar mass from the latter. We have also investigated the MIR properties of both edge-on galaxies and barred galaxies in our sample. Since they present similar MIR properties to the other sample galaxies, they do not influence the MIR properties obtained for the entire sample.
author:
- 'Hai-Ning Li, Hong Wu, Chen Cao, Yi-Nan Zhu'
title: |
Morphological Dependence of MIR Properties\
of SDSS Galaxies in the [*Spitzer*]{} SWIRE Survey
---
INTRODUCTION {#section:introduction}
============
Ever since Hubble’s famous paper outlined his classification system [@Hubble1926; @Hubble1936], morphological classification in conjunction with physical measurement has become an important tool in extragalactic astrophysics. A number of quantitative classifiers have been developed or extended over the years to probe the structure of galaxies. There are parametric classifiers like radial multi-Gaussian deconvolution [@Bendinelli1991], bulge-disk decomposition [@Byun1995], etc., and the nonparametric ones such as the C-A system [@Abraham1994; @Abraham1996], artificial neural nets trained from visual classification sets [@Odewahn1996], Gini Coefficient [@Lotz2004] and so on. As one of the main quantitative criteria and a function of the Hubble classification, the bulge-disk decomposition has now been widely used as an effective method to examine galaxy structures and morphological properties, [e.g., @deJong1996; @Baggett1998; @Tasca2005; @Allen2006 etc].
The variations of galaxy physical properties with morphology and environment are crucial in our understanding of the evolution of galaxies [@Kennicutt1998; @Brinchmann2004]. Numbers of properties such as the integrated birthrate variation [@Sandage1986], the optical and infrared photometric properties [@Boselli2001; @Shimasaku2001; @Popescu2002a; @Davoodi2006], the star formation properties of galaxies in clusters [@Yuan2005], the circumnuclear H$\alpha$ luminosity and bar structures [@Shi2006], etc, which have been investigated present regularity or correlation along different morphological types. Through substantial former investigations, it is known that as a result of different stellar populations and the amount of dust and gas for the environment of star-forming, the early-type galaxies (ellipticals and lenticulars) exhibit rather different properties compared with the late-type ones (spirals, irregulars and so on).
As we know, averagely one third of the total luminosity from normal galaxies is absorbed and re-radiated by dust [@Mathis1990; @Popescu2002b], and even higher fraction from galaxies with the most active star-forming activity [e.g. luminous infrared galaxies (LIGs), @Sanders1996; @Wang2006]. Both the Infrared Space Observatory [ ISO - @Kessler1996] and [*Spitzer*]{} Space Telescope [@Werner2004] continued to explore the importance of dust. Studies of dusty starburst galaxies [@Poggianti2000; @Flores2004] have shown that, most of the activities (e.g., star-formation and/or AGN emission) in these galaxies are hidden by dust, and the bolometric luminosities of the active systems are mostly emitted in the infrared. This suggests that the infrared emission is a sensitive tracer of the young stellar population and star formation rates (SFRs), and suffers weaker extinction. The MIR dust emissions mainly consist of polycyclic aromatic hydrocarbons (PAHs) emission and the continuum emission feature of warm dust component. Both of these two components have been investigated as reliable measures of the SFRs of galaxies as a whole [@Peeters2004; @Wu2005a]. Whereby studies in the MIR properties will certainly improve our understanding of the galaxies which have been well studied in the optical bands, and give an insight into the details of their star formation histories.
This work tries to explore the relationships between the morphology and the MIR properties, for a flux-limited sample of normal galaxies which were selected from the galaxies of SDSS-DR4 [@Adelman-McCarthy2006] cross-correlated with the [*Spitzer*]{} SWIRE [@Lonsdale2003]. Considering that the [*Spitzer*]{} IRAC (Fazio et al. 2004) $8\mu m$ band covers the strongest PAH feature ($7.7\mu m$), and the MIPS (Rieke et al. 2004) $24\mu m$ band covers the continuum emission of very small grains (VSGs) free of PAH features, we tried to employ the IRAC bands and MIPS $24\mu m$ band to investigate the MIR dust properties of entire galaxies. We performed elliptical aperture photometric analysis in both the optical and MIR bands. The $8\mu m$ dust and $24\mu m$ dust luminosity and their dust-to-star ratios are used to quantitatively investigate the MIR properties of our sample galaxies. Two ways to classify the morphologies of the sample galaxies were adopted: the $T$ type of the revised Hubble classification system [@deVaucouleurs1976] by visual inspection, and the quantitative parameter of $B/T$ by the bulge-disk decomposition.
In §2, we describe the infrared and optical data, the sample construction, and the elliptical aperture photometry. The morphological classifications are presented in §3. In §4, we analyze the optical and MIR colors, and give the statistical results of the MIR dust properties against galaxy morphological types and the corresponding discussions. The conclusions are presented in §5. Throughout this paper, we assume a Hubble constant $H_{0} = 70
{\rm km s^{-1} Mpc^{-1}}$ and $\Omega_M=0.3$, $\Omega_\Lambda=0.7$ in calculating the distance and the luminosity.
DATA AND DATA REDUCTION {#sec:data.process}
=======================
The Sample {#subsec:sample}
----------
We used the IRAC $3.6, 4.5, 5.8, 8.0\mu m$ and the MIPS $24\mu m$ images from the northern SWIRE fields of Lockman Hole, ELAIS-N1, and ELAIS-N2. The BCD (Basic Calibrated Data) images of the four IRAC bands were obtained from [*Spitzer*]{} Science Center, which include flat-field corrections, dark subtraction, linearity and flux calibrations [@Fazio2004]. The IRAC images (in all four bands) were mosaiced from the BCD images after pointing refinement, distortion correction and cosmic-ray removal with the final pixel scale of $0.6\arcsec$ as described by @Huang2004 and @Wu2005a; likewise the MIPS $24\mu m$ images were mosaiced in the similar way with the final pixel scale of $1.225\arcsec$ [@Cao2007; @Wen2007]. Sources detected by SExtractor [@Bertin1996] in IRAC four bands and MIPS $24\mu m$ were matched with the Two Micron All Sky Survey (2MASS) sources to achieve astrometric uncertainties of around $0.1\arcsec$.
The SDSS data provide full coverage of the SWIRE fields of Lockman and ELAIS-N2 but cover only one third of ELAIS-N1 field. The $ugriz$ corrected frames of spectroscopically observed galaxies were taken from the SDSS-DR4. The pixel scale of SDSS images is $0.4\arcsec$ [@Stoughton2002]. The SDSS-DR4 $ugriz$ spectrophotometric catalogue was cross-correlated with SWIRE MIR catalogue measured by SExtractor [@Bertin1996] by a radius of $2\arcsec$. The total survey area of the three northern SWIRE fields is $\sim$24 deg$^{2}$, of which the overlap with SDSS is $\sim$15 deg$^{2}$. To obtain the reliable morphological classification [@Fukugita2004], 163 bright galaxies with Petrosian magnitude $r\leqslant15.9$ were selected. Only four SDSS galaxies were not matched with SWIRE MIR sources by 2$\arcsec$, since the SDSS fiber observations mistakenly pointed to the off-nucleus regions rather than the nuclear regions of galaxies. This provides a preliminary sample of 159 galaxies.
We excluded further five galaxies in our magnitude limited sample of 159 galaxies, because they failed in either the sky-background subtraction or the bulge-disk decomposition, or were severely contaminated by nearby bright stars. This led to a SDSS $r-$band flux-limited final sample of 154 galaxies. 142 of this sample have been imaged by IRAC four bands and 137 have been imaged by MIPS 24$\mu$m band. Finally, a sub-sample of 125 galaxies has images in all five MIR bands, and is used for further statistical discussion in §\[subsec:statistics\]. All of theses objects are local, with redshift less than 0.13. In our sample, six galaxies were with the absolute B magnitude fainter than -18, and thus classified as dwarf galaxies [@Mateo1998]. Here, the B magnitude can be obtained from the SDSS $g-$ and $r-$ magnitudes [@Smith2002]. The distributions of SDSS $r-$band Petrosian magnitudes, the redshift, and B-band absolute magnitudes for the 154 sample galaxies as well as those for the 125 sub-sample galaxies are plotted in Figure \[fig:properties\]. All these distributions show that the sub-sample can well represent the flux-limited sample.
Among the 154 sample galaxies, 133 with emission line detections could be classified with the traditional $\log$([/H$\beta$]{})-$\log$([/H$\alpha$]{}) diagnostic diagram [@Baldwin1981; @Veilleux1987]. The emission line fluxes were derived from the catalogue of @Tremonti2004. The criteria given by @Kewley2001 was adopted to distinguish the potential star-forming galaxies from AGNs, as is shown in Figure \[fig:BPT\].
Photometry {#subsec:photometry}
----------
To obtain the accurate photometry, sky background fitting and subtraction were done [@Zheng1999; @Wu2002; @Wu2005b] on both the SDSS corrected frames and the [*Spitzer*]{} images. All objects detected by SExtractor were masked to generate a background-only image, and the fitting sky-background was then subtracted. Photometry was performed on the background-subtracted images by IRAF task ELLIPSE [@Jedrzejewski1987]. To embrace almost all the flux of these extended sources in the different bands, an elliptical isophote with the B-band surface brightness of 26 mag $arcsec^{-2}$ was adopted as the photometric aperture, based on the SDSS $g-$band images. With such elliptical apertures, the total fluxes were measured in all the wavelengths including the SDSS $ugriz$, the IRAC four bands and the MIPS $24\mu m$ band (see sample apertures marked in Figure \[fig:sample\]) with IRAF task POLYPHOT. Note that the point source functions (PSFs) of MIPS $24\mu m$ are rather extended, therefore further aperture corrections were performed on this band. For objects with the equivalent radius of the elliptical apertures smaller than $15\arcsec$, aperture corrections were applied to calibrate the integrated flux to an equivalent radius of $15\arcsec$. The photometric accuracies of these different bands are quite small, less than 0.03 mag on average. Flux calibration accuracies of the IRAC four bands [@Fazio2004] and MIPS $24\mu
m$ band [@Rieke2004] are less than $10\%$. The final errors include both the above errors.
The Galactic extinction in each SDSS filter from SDSS-DR4 catalogue was adopted, and then intrinsic extinction was derived from the Balmer decrement [@Calzetti2001]. The photometric $K$-correction was calculated using the method of @Blanton2003 [[Kcorrect V4-1-4]{}]. No extinction correction has been performed on photometric results in the MIR bands since extinction effect is rather negligible in the infrared compared with optical wavelength. Due to the fact that our sample are all low redshift galaxies and, as of this work, there are no reliable $K$-correction for these MIR spectral ranges available yet, we applied no $K$-correction to the MIR photometry. All the measurements were converted to AB magnitudes [@Oke1983].
Although PAH and VSG emissions dominate $8\mu m$ and $24\mu m$ bands for the majority of our sample, there is still a stellar continuum in these bands, especially for the early-type galaxies. Herewith a subtraction of the stellar contribution using the $3.6\mu m$ luminosity was adopted, with a scale factor of 0.232 for the $8\mu
m$ band and 0.032 for the $24\mu m$ band [@Helou2004]. Hereafter, we denoted the 8$\mu m$(dust) and 24$\mu m$(dust) representing the dust emissions after subtracting the stellar contribution [@Wu2005a].
MORPHOLOGICAL CLASSIFICATION {#sec:mor.class}
============================
We conducted the morphological classification by two methods: visual inspection and bulge-disk decomposition with GIM2D [Galaxy IMage 2D: @Simard1998; @Simard2002].
GIM2D Fitting {#subsec:gim2d.fit}
-------------
GIM2D is a two-dimensional photometric decomposition fitting algorithm which fits each image to a superposition of an bulge component with a Sérsic profile, and a disk component with an exponential profile [@Simard1998; @Simard2002]. GIM2D was employed to obtain the structural parameters of galaxies in our sample. The bulge component of the model is a profile of Sérsic form [@Sersic1968]: $$\Sigma(r) = \Sigma_e\cdot exp\{-b[(r/r_e)^{1/n}-1]\}
\label{eqn:Sersic.profile}$$ where $\Sigma(r)$ is the surface brightness at a radius $r$ and $\Sigma_e$ is the characteristic value (i.e. the effective surface brightness), defined as the brightness at the effective radius $r_e$. Parameter $b$ is related to the Sérsic index $n$ and chosen equal to $1.9992n-0.3271$ so that $r_e$ remains the projected radius enclosing half of the light in this component [@Ciotti1991].
The disk component is an exponential profile of the form: $$\Sigma(r) = \Sigma_0\cdot exp(-r/r_d)
\label{eqn:Exponential.profile}$$ where $\Sigma_0$ is the central surface brightness, and $r_d$ is the disk scale length.
Decomposition was performed based on this model, with a Gaussian PSF. A total of twelve parameters were adjusted in fitting the galaxy image and retrieved as output from our decomposition: the total luminosity $L$, the bulge fraction $B/T$, the bulge effective radius $r_e$, the bulge ellipticity $e$, the bulge position angle $\phi_b$, the disk scale length $r_d$, the disk inclination angle $i$, the disk position angle $\phi_d$, the centroiding offsets $dx$ and $dy$, the S$\acute{e}$rsic index $n$, and the residual sky background level $db$. The $B/T$ which is defined as the fraction of the total flux in the bulge component has been extracted as a quantitative indicator of morphology. $B/T=1$ corresponds to a pure bulge, while $B/T=0$ to a pure disk. With all the objects detected by SExtractor flagged except the galaxy of interest, the $g-$band mask images were adopted not only in $g-$band but also in $r-$band, throughout the decomposition fitting with GIM2D. Figure \[fig:sample\] shows example (science, mask, model, and residual) images of different morphological types in GIM2D fitting. The returned $\chi ^2$ values of $g-$ and $r-$band fitting have the mean values of 1.14 and 1.18 with deviations of 0.26 and 0.33 respectively, representing rather convincing fitting results.
To check the results obtained, we compared the parameters $B/T$ of the 154 galaxies estimated from $g-$ and $r-$band in Figure \[fig:gim2d\]. The $B/T$ values obtained from both bands agree well with small amount of deviation. Among the three most deviated sources, SDSS J161222.61+525827.9 is an early-type galaxy with highly centralized surface brightness distribution in bluer band. SDSS J151723.30+593517.0 is a peculiar galaxy and UGC5888 is an irregular. We adopted $B/T$ values derived from $r-$band images throughout the following investigation since the decomposition results in the two SDSS bands hold quite good agreement.
Visual Classification {#subsec:visual.class}
---------------------
The morphological types of revised Hubble sequence [@deVaucouleurs1964] of our sample galaxies were classified by visual inspection based on features like bulge ratios, the presence of spiral arms and/or bars, signs of interaction, multiple nuclei etc. All galaxies in our sample were classified into six morphological classes: $T=0$(E or S0), 1(Sa), 3(Sb), 5(Sc), 7(Sd), and 9(Irr). Notice that we assigned an additional class 10 corresponding to galaxies with peculiar morphology possibly related to galactic interactions, mergers, etc. We performed the visual classification in both SDSS $g-$ and $r-$band images. The classifications done independently by the four of us, agree in over 90% of the sample. The classifications were verified for 36 of those galaxies whose morphological types were found in the NASA/IPAC Extragalactic Database (NED) [^1] and were found to be accurate to $\Delta T=\pm1$.
Considering the possible effects that inclinations and bar structures may cause, we further divided sample galaxies into 3 types: barred galaxies, edge-on galaxies, and the rest defined as general galaxies for further consideration. Note that when defining edge-on galaxies, we adopted a standard of GIM2D fitted disk inclination angle $i > 70$. Table \[tab:mor.class\] shows the numbers and corresponding fractions of different morphological types in our sample. In Table \[tab:mor.spitzer\], we present the numbers of galaxies observed by different MIR bands in each morphological type. It can be drawn from Table 2 that the morphological fractions E/S0: S(Sa-Sd): Irr of optical-$8\mu m$ and optical-$24\mu m$ samples are 0.47: 0.51: 0.014 and 0.44: 0.54: 0.016 respectively. Both are consistant with E(E/S0-S0): S(S0a-Sdm): Im of 0.40: 0.57: 0.014 obtained from an optically selected sample of SDSS galaxies [@Fukugita2007].
Comparison {#subsec: gim2d.compare}
----------
A comparison between Hubble $T$ type and the bulge-to-total ratio $B/T$ has been carried out, as is shown in Figure \[fig:T.Bratio\]. Except for the peculiar galaxies which exhibit a diversity of $B/T$, there is a smooth inclination in $B/T$ along Hubble sequence from $T=9$ to $T=0$, i.e., for normal galaxies, except the peculiar, the late-types exhibit lower $B/T$ than the early-type ones. Hence $B/T$ does act as a reliable measure of morphological types for normal galaxies in our sample. In order to statistically examine the properties of our sample, galaxies have been divided into two morphological types: the early-type (E/S0) with T=0, and the late-type (Sa, Sb, Sc, Sd, and Irr) with T from 1 to 9. Consequently, according to @Simien1986, such a classification roughly corresponds to the division of $B/T=0.4$. 128 out of the 144 normal sample objects coincide with their morphological types classified with $T$. The divisions are shown in Figure \[fig:T.Bratio\]. Since as a whole, divisions with the two methods agree with each other, we have adopted T type to divide the sample galaxies into either the early-type or the late-type in the following statistics.
RESULT AND DISCUSSION {#sec:result}
=====================
Color-Color Diagram
-------------------
The optical-MIR color-color diagrams are shown in the left panels of Figure \[fig:color\], and there are anti-correlations between optical and MIR colors, which is consistent with the result of @Hogg2005. Such a trend can be explained rather naturally: the optically blue color is related to the recent star formation, indicating the existence of notable amount of dust, and the MIR dust emissions can be shown in both $8\mu m$ and $24\mu m$ bands. On the other hand, optically red galaxies are always too old to contain plenty of dust, and hence present weak MIR dust emissions.
The right panels of Figure \[fig:color\] show the relationship between MIR colors. It can be detected that \[3.6\]-\[4.5\] colors in our sample are quite blue, with an average of around -0.52, indicating that there is no extremely active galactic nuclei (QSO etc.) in our sample, because both the emissions of $3.6\mu m$ and $4.5\mu m$ of normal galaxies are dominated by decreasing stellar continuum of the old stellar population. Yet these two bands of very active galaxies like quasars are dominated by the power law spectra of quasars and consequently should exhibit redder \[3.6\]-\[4.5\] colors. Both colors of \[3.6\]-\[8\] and \[3.6\]-\[24\] can roughly characterize the relative strengths of the MIR dust emissions of PAHs and VSGs. The stronger the dust emissions, the redder the colors. Therefore, it is not surprised to find that most of the peculiar galaxies of our sample at the redder part of these panels because they contain large amount of dust for their violent star formation [@Sanders1996].
Estimation of the Stellar Mass {#subsec:I1.mass}
------------------------------
The luminosity of $3.6\mu m$ band is often treated as a tracer of stellar component [@Wu2005a; @Davoodi2006 etc] as well as a test of validity of mass determination [@Hancock2007]. Furthermore @Hancock2007 has compared the mass of clumps in Arp 82 derived from [*R*]{} band fluxes and broadband colors against luminosities of $3.6\mu m$, and has found strong correlation between them. Hereby the relationship between the stellar mass and the $3.6\mu m$ luminosity of our sample galaxies is examined. Based on the optical photometries as described in § \[subsec:photometry\], we calculated the stellar mass of our sample, following @Bell2003: $$\log(\frac{M_{\star}}{M_{\odot}}) = -0.4\times(M_{r,AB}-4.67) + [a_{r}+b_{r}\times (g-r)_{AB}+0.15]
\label{eqn:mass.Bell}$$ where $M_{r,AB}$ is the $r-$band absolute magnitude, $(g-r)_{AB}$ is the rest-frame color in the AB magnitude system. The coefficients $a_r$ and $b_r$ come from Table 7 of @Bell2003. A @Salpeter1955 stellar mass initial function (IMF) has been adopted with $\alpha = 2.35$ and $0.1M_{\odot} < M < 100M_{\odot}$. The distribution of the stellar mass for our sample galaxies is presented in Figure \[fig:properties\], in a range between $10^{9}
M_{\odot}$ and $10^{12} M_{\odot}$, with the average mass around $10^{11} M_{\odot}$, as intermediate mass galaxies.
A tight correlation of the stellar mass against the luminosity of $3.6\mu m$ is detected for our sample, as is shown in Figure \[fig:I1.mass\]. Our result is consistent with that found by @Hancock2007 but with smaller scatters, thus it confirms the capability of the $3.6\mu m$ luminosity as a measure of the stellar mass of galaxies. Based on our 145 galaxies and 24 clumps in Arp 82 [@Hancock2007], we fit the relation between the stellar mass and the $3.6\mu m$ luminosity, and obtain a nearly linear correlation: $$\log(\frac{M_{\star}}{M_{\odot}}) = (1.34\pm0.09) + \\
(1.00\pm0.01)\times \log(\frac{\nu L_{\nu}[3.6\mu m]}{L_{\odot}})
\label{eqn:I1.mass}$$
As is pointed out by @Charlot1996 and @Madau1998, the mass-to-infrared light ratio is relatively insensitive to the star formation history, and remains very close to unity, independent of either galaxy colors or Hubble types. This relation is probably due to the fact that in the near-infrared, older stellar populations may dominate both the galaxy luminosities and the stellar masses. Therefore such a correlation provides a proxy way to estimate the stellar mass of galaxies directly from the integrated $3.6\mu m$ luminosity.
MIR Properties and Morphology {#subsec:MIR.mor}
-----------------------------
Figure \[fig:mor.lum\] shows the $8\mu m$ and $24\mu m$ dust luminosities as the function of different morphological types in our sample. For normal galaxies, except the dwarfs, along either the Hubble $T$ types or the $B/T$ ratios, there exist obvious declinations of the MIR luminosities from the late-type to the early-type galaxies, especially showing a steep change around the division of the late-type and the early-type galaxies. All the peculiar galaxies show relatively higher MIR luminosities, independent of their $B/T$ ratios. The phenomenon could be attributed to the fact that early-type galaxies are dominated by older population and are deficient of dust, resulting in lower MIR dust luminosities, while the late-type contain larger amount of young stars and more dust thus present stronger MIR emissions. Peculiar galaxies which are undergoing strong star forming activities contain great amount of dust and thereby show averagely high MIR luminosities. As @Wu2005a has pointed out, both the $8\mu m$ and $24\mu m$ dust luminosities can be used as measures of SFRs of entire galaxies; therefore, the correlations between the MIR dust luminosities and morphological types reflect a consequent relationship between the galactic SFRs and morphological types. This also confirms the previous results of @Sandage1986, @Kennicutt1998, etc. As for the six dwarf galaxies, because they all have low mass of around $10^9 M_{\odot}$ (from the previous mass determination), so they contain less dust and thus show lower MIR dust emissions [@Hogg2005].
The MIR dust-to-star ratios are also plotted against different morphological types, in Figure \[fig:mor.ratio\]. The prominent correlations are also detected between the MIR dust-to-star ratios and both the Hubble $T$ types and the $B/T$ ratios. Furthermore, such correlations seem to be more obvious than those between the MIR luminosities and morphological types. Since both the $8\mu m$ and $24\mu m$ dust luminosities possess correlations with SFRs for normal galaxies, and the $3.6\mu m$ luminosity is a reliable tracer of stellar component, such ratios can be treated as the dust-to-star ratios. The correlations between the MIR dust-to-star ratios and the morphological types could represent the distribution of SFRs per unit stellar mass [@Wen2007] over different morphologies. Contrary to the behavior in Figure \[fig:mor.lum\], the dwarf galaxies are roughly consistent with the other late-type galaxies within the error bars, but still present a little lower dust-to-star ratios. This could be explained with the fact that the gravitation potentials of these low mass galaxies may not be strong enough to retain as much dust and gas against the radiation fields as those of the normal-mass galaxies, and hereby all the dwarf galaxies show slightly lower MIR dust-to-star ratios. Almost all the peculiar galaxies present higher MIR dust-to-star ratios.
The ratio of the $8\mu m$ dust luminosity to the $24\mu m$ dust luminosity is compared with different morphological types in Figure \[fig:mor.dust\]. Considering the $8\mu m$ dust emissions mainly represent emission of PAHs heated by B type stars while $24\mu m$ dust emissions stand for emissions from hot dust mostly heated by O type stars [@Peeters2004], the ratio can be treated as the estimation of ratio between these two components. In general, $8\mu m$(dust)-to-$24\mu m$(dust) ratios remain constant on average along the Hubble sequence, while the larger scatter in distribution of the early-type probably arises from existence of nuclear AGNs which destroy PAH emissions presumably due to photodestruction of the PAH molecules by EUV/X-ray photons [@Genzel1998; @Siebenmorgen2004] or the outer diffuse PAH emissions heated by the older stars [@Sauvage2005]. This can also be seen in later discussion in Figure \[fig:SF.AGN\] where AGNs present lower $8\mu m$(dust) luminosities. It should be noted that the two dwarf irregular galaxies exhibit lower $8\mu
m$(dust)-to-$24\mu m$(dust) ratios than the most of the late-type do, possibly due to their low metallicities [^2] [@Engelbracht2005].
The above result indicates that for the normal galaxies except the dwarfs, the ratio of these two dust components does not vary much against morphology. Therefore the $8\mu m$(dust) luminosities are as capable to trace the galaxy SFRs as the $24\mu m$(dust) luminosities as @Wu2005a has pointed out.
Statistics of MIR Properties {#subsec:statistics}
----------------------------
In order to compare the statistical MIR properties of different types of sample galaxies, we performed all the following statistics on the sub-sample of 125 galaxies which have photometric information for all the optical and MIR bands. Since from Figure \[fig:properties\] and K-S test results this sub-sample does not exhibit notable differences from the 154-galaxy sample, we suggest that the following statistical discussions are representative of all the sample galaxies.
Distributions of the MIR luminosities and dust-to-star ratios for both the early-type and the late-type galaxies in the sub-sample are presented in Figure \[fig:early.late\]. It is clear that, in general, the late-type galaxies exhibit quite different MIR properties to early-type ones, presenting a distinct and statistically higher MIR luminosities and dust-to-star ratios. The Gaussian fitting is performed on each distribution, and the lines crossing intersection points of the two Gaussian distributions are presented as the division of the two morphological types. In Part A of Table \[tab:type\], the median values and the mean values together with scatters of distributions of the early-type and the late-type galaxies are listed. Probabilities that the distributions of these two types can match are all smaller than $8.8\times
10^{-5}$, indicating that they present quite distinct properties. Therefore, both the MIR luminosities and dust-to-star ratios can be used to separate the early-type galaxies from the late-type ones. Table \[tab:type.division\] displays specific values of divisions and the reliability of such classifications. We define the [reliability]{} of classification as the fraction of galaxies from the subsample that are selected by Hubble $T$ types. For example, out of the 117 normal galaxies which have images in all IRAC bands, the T type criterion selects 64 late-type galaxies and 53 early-types. 47 of the 64 late-type galaxies have $\log\nu L_{\nu}(8\mu
m)/L_{\odot}\geqslant8.91$ and thus are consistently classified as late-type, while 47 of the 53 early-types have $\log\nu L_{\nu}(8\mu
m)/L_{\odot}\leqslant8.91$ and thus are classified as early-type. Therefore the $\log\nu L_{\nu}(8\mu m)/L_{\odot}=8.91$ maintains reliability of 73% for selecting late-type galaxies and 89% for early-types. Correspondingly, $\log\nu L_{\nu}(24\mu
m)/L_{\odot}=8.30$ gives a reliability of 70% for the late-type and 77% for the early-type; $\log\nu L_{\nu}(8\mu m)/\nu L_{\nu}(3.6\mu
m)=-1.15$ can give a 88% for the late-type and 83% for the early-type; and $\log\nu L_{\nu}(24\mu m)/\nu L_{\nu}(3.6\mu
m)=-1.45$ can give a 84% for the late-type and 85% for the early-type; Hereby the MIR dust luminosities especially the MIR dust-to-star ratios can be effective tools to divide the early-type galaxies from the late-type ones.
The comparisons between MIR properties of star-forming galaxies and AGNs (see §\[subsec:sample\]) are shown in Figure \[fig:SF.AGN\]. Generally, star-forming galaxies present stronger MIR emissions than those possess AGN activities. From Part B of Table \[tab:type\], the star-forming galaxies and AGNs show the probability of matching each other in $8\mu m$ and $24\mu m$ dust luminosites and dust-to-star ratios with less than $2.1\times
10^{-3}$, indicating quite different MIR properties, although not so distinct as these between the early-type and late-type ones.
Some of the sample galaxies are edge-on galaxies and some are barred galaxies. Will the galaxies in edge-on view or with bars present different MIR properties? We compare the distributions of the MIR dust luminosities and the MIR dust-to-star ratios between edge-on galaxies, barred galaxies and normal galaxies in Figure \[fig:edge.on\] and Figure \[fig:bar\]. The statistical results are listed in Part C and D of Table \[tab:type\], with the probabilities in matching each by few to tenth percent, indicating rather resembling distributions. Thus, neither galaxies in edge-on view nor galaxies with bars present statistical differences from the other normal galaxies in both MIR properties. Furthermore, although based on limited data points, one can still find out in Figure \[fig:edge.on\] that the edge-on galaxies can also be classified into early-type and late-type ones with the division criteria described in Table \[tab:type.division\]. Therefore, the inclination of galaxies and the existence of bars do not affect our previous results.
We have also checked the distributions of total sample of 154 galaxies in the above parameters and type comparisons, and yielded quite similar results, therefore the selection of this sub-sample does not affect the reliability of our statistical results.
SUMMARY {#section:summary}
=======
We investigated the correlations between the morphological types and the MIR properties of a local optically flux-limited sample of galaxies selected from the spectroscopic catalogue of galaxies in SDSS-DR4, cross-correlated with the Lockman Hole, ELAIS-N1 and ELAIS-N2 of [*Spitzer*]{} SWIRE survey. Aperture photometry has been performed on all these galaxies in all optical and MIR bands. Morphological classifications have been performed by both visual inspection and the bulge-disk decomposition with GIM2D. Our major results are as follows:
[(1) The presented analysis clearly shows that the bulge-to-total ratio $B/T$ obtained by the bulge-disk decomposition is proved a qualified quantitative measure of the Hubble $T$ types. Galaxies with earlier morphological types possess larger bulge ratios while later type ones have more dominant disk structures. ]{}
[(2) The $3.6\mu m$ luminosity presents a tight correlation with the stellar mass, and this provides us a new tool to estimate the stellar mass. The empirical formula to calculate the stellar mass by the $3.6\mu m$ luminosity is given in Equation \[eqn:I1.mass\]. ]{}
[(3) Except for the dwarf galaxies and a few peculiar objects, the MIR dust luminosities of $8\mu m$ and $24\mu m$ exhibit correlations with either Hubble $T$ type or the bulge-to-total ratio $B/T$. Such correlations are much more obvious if we used the MIR dust-to-star ratios (either $\nu L_{\nu}[8\mu m (dust)]/\nu
L_{\nu}[3.6\mu m]$ or $\nu L_{\nu}[24\mu m (dust)]/\nu
L_{\nu}[3.6\mu m]$) instead of the MIR luminosities. ]{}
[(4) The MIR dust luminosity ratios of $\nu L_{\nu}[8\mu m
(dust)]/\nu L_{\nu}[24\mu m(dust)]$ turn to be roughly constant against the morphological types, especially for the late-type galaxies. Therefore, on average, the $8\mu m$(dust) luminosity can as reliably measure the global SFRs of normal galaxies as the $24\mu
m$(dust) luminosity can, regardless of the morphological types. ]{}
[(5) Distributions of the MIR dust luminosities and the MIR dust-to-star ratios of both the early-type and the late-type galaxies are very different. The late-type galaxies present higher MIR dust luminosities and MIR dust-to-star ratios than the early-type galaxies. The MIR dust luminosities of $8\mu m$ and $24\mu m$ and especially the MIR dust-to-star ratios of $\nu
L_{\nu}[8\mu m (dust)]/\nu L_{\nu}[3.6\mu m]$ and $\nu L_{\nu}[24\mu
m (dust)]/\nu L_{\nu}[3.6\mu m]$ can provide an effective tool to distinguish the late-type galaxies from the early-types. ]{}
[(6) The star-forming galaxies and AGNs also present different statistical MIR properties, with the former showing higher MIR luminosities and MIR dust-to-star ratios. The statistical results show that either galaxies in edge-on view or galaxies with bars do not present quite different MIR properties from other sample galaxies. ]{}
The authors would like to appreciate the anonymous referee for helpful comments and suggestions, and we are grateful for the help of D.B. Sanders, S. Mao, Z.-L. Zhou, X.-Y. Xia, Z.-G. Deng, Z. Wang, J.-S. Huang, C.-N. Hao, F.-S. Liu and J.-L. Wang. This project was supported by the National Science Foundation of China through grants 10273012, 10333060 and 10473013.
This work is based in part on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under NASA contract 1407.
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, Cambridge University, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.
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[lccccccc|c]{} General & 53 & 13 & 13 & 11 & 1 & 2 & 93 & 7\
Barred & 6 & 6 & 8 & 7 & 1 & 0 & 28 & 2\
Edge-on & 7 & 6 & 5 & 4 & 1 & 0 & 23 & 1\
Total & 66 & 25 & 26 & 22 & 3 & 2 & 144 & 10\
Fraction% & 42.9 & 16.2 & 16.9 & 14.3 & 1.9 & 1.3 & 93.5 & 6.5\
[l|cccccccc]{} 3.6$\mu$m & 61 & 22 & 26 & 21 & 3 & 2 & 10 & 145\
4.5$\mu$m & 66 & 24 & 25 & 21 & 2 & 2 & 10 & 150\
5.8$\mu$m & 61 & 22 & 26 & 21 & 3 & 2 & 10 & 145\
8.0$\mu$m & 66 & 24 & 25 & 21 & 2 & 2 & 10 & 150\
24$\mu$m & 57 & 22 & 23 & 22 & 3 & 2 & 8 & 137\
all bands & 53 & 19 & 20 & 21 & 2 & 2 & 8 & 125\
[lcccccccc]{} $<Part A>$ & & & & & & & &\
$\log \nu L_{\nu}(8\mu m)/L_{\odot}$ & 8.30 & 9.30 & 8.28(0.09) & 9.13(0.10) & 53 & 64 & 0.72 & $2.9\times10^{-6}$\
$\log \nu L_{\nu}(24\mu m)/L_{\odot}$ & 7.76 & 8.67 & 7.77(0.09) & 8.55(0.10) & 53 & 64 & 0.63 & $8.8\times10^{-5}$\
$\log \nu L_{\nu}(8\mu m)/\nu L_{\nu}(3.6\mu m)$ & -1.55 &-0.22 & -1.49(0.07)&-0.33(0.07)&53&64&0.83& $4.7\times10^{-8}$\
$\log \nu L_{\nu}(24\mu m)/\nu L_{\nu}(3.6\mu m)$&-2.06&-0.81&-2.00(0.08)&-0.92(0.07)&53&64&0.77&$6.9\times10^{-7}$\
$<Part B>$ & & & & & & & &\
$\log \nu L_{\nu}(8\mu m)/L_{\odot}$ & 9.31 & 8.78 & 8.99(0.14) & 8.72(0.10) & 51 & 46 & 0.38 & $1.8\times10^{-3}$\
$\log \nu L_{\nu}(24\mu m)/L_{\odot}$ & 8.75 & 8.27 & 8.49(0.13) & 8.16(0.10) & 51 & 46 & 0.38 & $2.1\times10^{-3}$\
$\log \nu L_{\nu}(8\mu m)/\nu L_{\nu}(3.6\mu m)$ & -0.13 & -1.16 & -0.37(0.10) & -1.04(0.09) & 51 & 46 & 0.38 & $1.8\times10^{-3}$\
$\log \nu L_{\nu}(24\mu m)/\nu L_{\nu}(3.6\mu m)$ & -0.70 & -1.65 & -0.87(0.09)& -1.60(0.10) & 51 & 46 & 0.38 & $2.1\times10^{-3}$\
$<Part C>$ & & & & & & & &\
$\log \nu L_{\nu}(8\mu m)/L_{\odot}$ & 8.14 & 8.81 & 8.52(0.22) & 8.80(0.08) & 21 & 96 & 0.33 & 0.05\
$\log \nu L_{\nu}(24\mu m)/L_{\odot}$ & 7.82 & 8.31 & 7.96(0.22) & 8.21(0.09) & 21 & 96 & 0.31 & 0.08\
$\log \nu L_{\nu}(8\mu m)/\nu L_{\nu}(3.6\mu m)$ & -1.26 & -0.75 & -0.94(0.19) & -0.83(0.08) & 21 & 96 & 0.25 & 0.25\
$\log \nu L_{\nu}(24\mu m)/\nu L_{\nu}(3.6\mu m)$ & -1.83 & -1.37 & -1.51(0.18) & -1.42(0.09) & 21 & 96 & 0.22 & 0.40\
$<Part D>$ & & & & & & & &\
$\log \nu L_{\nu}(8\mu m)/L_{\odot}$ & 9.30 & 9.44 & 9.12(0.16) & 9.26(0.15) & 19 & 31 & 0.24 & 0.51\
$\log \nu L_{\nu}(24\mu m)/L_{\odot}$ & 8.66 & 8.78 & 8.54(0.16) & 8.67(0.14) & 19 & 31 & 0.17 & 0.86\
$\log \nu L_{\nu}(8\mu m)/\nu L_{\nu}(3.6\mu m)$ & -0.15 & -0.22 & -0.26(0.12) & -0.27(0.09) & 19 & 31 & 0.16 & 0.91\
$\log \nu L_{\nu}(24\mu m)/\nu L_{\nu}(3.6\mu m)$ & -0.73 & -0.81 & -0.84(0.14) & -0.86(0.08) & 19 & 31 & 0.21 & 0.68\
[lccc]{} $\log \nu L_{\nu}(8\mu m)/L_{\odot}$ & 8.91 & 73% & 89%\
$\log \nu L_{\nu}(24\mu m)/L_{\odot}$ & 8.30 & 70% & 77%\
$\log \nu L_{\nu}(8\mu m)/\nu L_{\nu}(3.6\mu m)$ & -1.15 & 88% & 83%\
$\log \nu L_{\nu}(24\mu m)/\nu L_{\nu}(3.6\mu m)$ & -1.45 & 84% & 85%\
[^1]: See http://nedwww.ipac.caltech.edu/index.html
[^2]: with metallicity $12+\log(O/H)=8.55$ and $8.35$ respectively, about $3/4$ and $1/2$ the solar value [@Asplund2004].
|
8.5in -.5in 6.25in -.25in \#1\#2[3.6pt]{} \#1[[$^{\ref{#1}}$]{}]{}
UMD-PP-96-17\
hep-ph/9508365\
August 1995\
[**$CP$ violation in multi-Higgs** ]{}
[**supersymmetric models** ]{}
Manuel Masip$^{(a,b)}$ and Andrija Rašin$^{(a)}$
*$^{(a)}$Department of Physics*
*University of Maryland*
*College Park, MD 20742, U.S.A.*
*$^{(b)}$Departamento de Física Teórica y del Cosmos*
*Universidad de Granada*
*18071 Granada, Spain*
**Abstract**
> We consider supersymmetric extensions of the standard model with two pairs of Higgs doublets. We study the possibility of spontaneous $CP$ violation in these scenarios and present a model where the origin of $CP$ violation is soft, with all the complex phases in the Lagrangian derived from complex masses and vacuum expectation values (VEVs) of the Higgs fields. The main ingredient of the model is an approximate global symmetry, which determines the order of magnitude of Yukawa couplings and scalar VEVs. We assume that the terms violating this symmetry are suppressed by powers of the small parameter $\epsilon_{PQ}=O(m_b/m_t)$. The tree-level flavor changing interactions are small due to a combination of this global symmetry and a flavor symmetry, but they can be the dominant source of $CP$ violation. All $CP$-violating effects occur at order $\epsilon_{PQ}^2$ as the result of exchange of [*almost*]{}-decoupled extra Higgs bosons and/or through the usual mechanisms with an [*almost*]{}-real CKM matrix. On dimensional grounds, the model gives $\epsilon_K\approx \epsilon_{PQ}^2$ and predicts for the neutron electric dipole moment (and possibly also for $\epsilon'_K$) a suppression of order $\epsilon_{PQ}^2$ with respect to the values obtained in standard and minimal supersymmetric scenarios. The predicted $CP$ asymmetries in $B$ decays are generically too small to be seen in the near future. The mass of the lightest neutral scalar, the strong $CP$ problem, and possible contributions to the $Z$ decay into $b$ quarks (the $R_b$ puzzle) are also briefly addressed in the framework of this model.
Introduction
============
Although the standard model is today in impressive agreement with all particle physics data, its scalar sector has not been proven yet. A scalar sector defined by elementary fields seems to contradict the possibility of two very different mass scales (namely, the electroweak and the grand unification or Planck scales). Supersymmetry (SUSY) [@nill81] would offer an explanation for the stability at the quantum level of the different scales of the theory, provided that it is broken only by soft terms below the TeV region. A lot of attention has been paid to the minimal SUSY extension of the standard model (MSSM), which presents appealing features such as a consistent grand unification of the gauge couplings or a candidate for the cold dark matter of the universe. Experimentally, the MSSM has been so far [*flexible*]{} enough to avoid conflict with any measurement, but its most compelling prediction, the presence of a light neutral Higgs (lighter than the $Z$ boson at the tree level), is still missing.
The MSSM would also offer distinctive predictions for $CP$ violating processes. Arbitrary complex phases $\psi$ in soft gaugino masses and scalar trilinears would give fermion electric dipole moments (EDMs) well above their present experimental limits. This implies generically $\psi \leq 10^{-2}$ [@deru90], a somewhat unnaturally small number. The MSSM would also predict unsupressed flavor changing neutral currents (FCNCs) unless there is some degree of degeneracy between squark masses (something which occurs for supergravity Lagrangians with canonical kinetic terms) and correlation between the Cabibbo-Kobayashi-Maskawa (CKM) matrix and its equivalent in the squark sector. In the usual SUSY scenario [@bigi90] $CP$ violation in $K$ and $B$ physics depends essentially on only one phase (the CKM phase $\phi$), whereas the set of small phases in soft terms (uncorrelated to the family structure) may have experimental relevance only in fermion EDMs. This scenario, however, holds only for highly degenerated squark masses. In general, taking the experimental limit ${ {\Delta m^2_{\tilde{q}} }
\over { m^2_{\tilde{q}} } } \leq {1 \over 30}$ [@elli82] from $K-\bar{K}$ mixing one obtains that acceptable complex phases in gaugino masses may have an impact on the $K$ system. For complex gluino masses this was shown in [@lang84], and a model with small phases has been recently proposed in [@babu94a]. It was also shown [@poma93] that due to large top-quark effects acceptable complex phases in chargino mass terms may also contribute to $CP$ asymetries in the $K$ system.
A different approach to the origin of $CP$ violation which is specially appealing in SUSY models is the idea of spontaneous $CP$ violation (SCPV) [@tlee73]. All the phases in the Lagrangian (initially $CP$-conserving) would have their origin in a small number of complex vacuum expectation values (VEVs) of scalar fields. Moreover, the sizes of these phases could be correlated by approximate symmetries suppressing some couplings in the Lagrangian [@jliu87]. Unfortunately, in the minimal Higgs sector of the MSSM there is no room for SCPV [@maek92; @poma92], and [*hard*]{} $CP$ violation is required. The possibility of SCPV has been also studied in SUSY extensions with singlet fields [@poma93; @babu94b], where it can be obtained but seems to require certain ammount of fine tuning.
The SUSY models with more than two Higgs doublets are an obvious extension of the MSSM [@flor83]. They are minimal in the sense that no new species are introduced, but just repeated. From the model building point of view there is no compelling reason to disregard them, and they could appear naturally in models with fermion-Higgs unification (like $E_6$) [@gurs76] or left-right symmetric scenarios, where two bidoublets are required in order to obtain realistic fermion masses and mixings. Four Higgs doublet (4HD) models require an intermediate scale to be consistent with grand unification, but even this could be more in line with recent data on $\alpha_s(M_Z)$ than the desert scenario [@brah95]. Since more than one Higgs doublet couples to quarks of a given charge, a possible concern in this type of models is the presence of FCNC at the tree level. The experimental limits, however, can be easily avoided just by invoking the action of an approximate flavor symmetry (see next section). On the other hand, a nonminimal scalar sector opens the possibility of SCPV and, in general, widens the parameter space relevant in low-energy precision measurements (this could be convenient, for example, if the anomalous value of $R_b$ persists).
In this paper we present a 4HD SUSY model which seems to contain satisfactory answers to many phenomenological questions. $CP$ violation appears softly, in complex Higgs masses and VEVs. The main ingredient of the model is a Peccei-Quinn like approximate symmetry which determines the order of magnitude of Yukawa couplings and scalar VEVs. We define this symmetry in such a way that the additional pair of doublets has small VEVs with order one complex phases and is weakly coupled to all matter fields. As a consequence, the ratio $m_b/m_t$, $CP$-violating effects in $K$ physics, and the neutron EDM will appear suppressed by powers of the small parameter $\epsilon_{PQ}$ that parametrizes the violation of this symmetry. The $CP$ asymmetries in $B$ decays are predicted to be typically two or three orders of magnitude smaller than in CKM scenarios (the CKM matrix in the model is essentially real), a signal that can be used to discriminate this 4HD model with respect to the MSSM or the standard model.
The plan of the paper is as follows. In Section 2 we write the generic Lagrangian for 4HD SUSY models and review previous results on spontaneous $CP$ violation. We show that a realistic scenario for soft $CP$ violation requires complex Higgs masses in the initial effective model. In Section 3 we define our model and minimize the Higgs potential. We show that the order of magnitude of Yukawa couplings, complex scalar VEVs, and the CKM complex phase are correlated by the approximate global symmetry. In Section 4 we explore the implications of the model on $K$ and $B$ physics as well as on the neutron EDM. In Section 5 we discuss the resulting spectrum of scalar fields (in particular, the mass of the lightest neutral mode) and other possible phenomenological impacts of the 4HD model. Section 6 is devoted to conclusions. Details about the minimization of the scalar potential can be found in the Appendix.
Complex VEVs in four Higgs doublet models
===========================================
The most general superpotential with four higgs doublets is given by W & = & Q ( [**h**]{}\_1 H\_1 + [**h**]{}\_3 H\_3) D\^c + Q ( [**h**]{}\_2 H\_2 + [**h**]{}\_4 H\_4) U\^c + L([**h**]{}\_1\^e H\_1 + [**h**]{}\_3\^e H\_3) E\^c\
& + & \_[12]{} H\_1 H\_2 + \_[32]{} H\_3 H\_2 + \_[14]{} H\_1 H\_4 + \_[34]{} H\_3 H\_4 , \[eq:superpot\] where $Q$ stands for quark doublets, $D^c$ for down quark singlets, $U^c$ for up quark singlets, $L$ for lepton doublets, $E^c$ for charged lepton singlets, and ${\bf h}_i$ are the Yukawa matrices (family indices are omitted). The Higgs doublets $H_1,\;H_3$ and $H_2,\;H_4$ have hypercharges $-1$ and $+1$, respectively.
Including soft SUSY breaking terms the effective potential for the Higgs fields is V & = & m\_1\^2 H\_1\^H\_1+ m\_2\^2 H\_2\^H\_2 + m\_3\^2 H\_3\^H\_3 + m\_4\^2 H\_4\^H\_4 +\
& + & (m\^2\_[12]{} H\_1 H\_2 + h.c.) + (m\^2\_[32]{} H\_3 H\_2 + h.c.) +\
& + & (m\^2\_[14]{} H\_1 H\_4 + h.c.) + (m\^2\_[34]{} H\_3 H\_4 + h.c.) +\
& + & (m\^2\_[13]{} H\_1\^H\_3 + h.c.) + (m\^2\_[24]{} H\_2\^H\_4 + h.c.) + V\^[4HD]{}\_D + V, \[eq:pot\] where $V^{4HD}_D$ contains the D-terms and $\Delta V$ the radiative corrections. For the neutral components $\phi_i$ of the doublets one has V\^[4HD]{}\_D & = & (g\^2+g’\^2) \[ \_1\^\_1 +\_3\^\_3 -\_2\^\_2 - \_4\^\_4 \]\^2. The radiative contributions $\Delta V$ are generated by SUSY breaking effects. In our arguments it will suffice to consider the terms derived from large top (and possibly bottom) quark Yukawa interactions [@elli91]: V & = & \_[q]{} { m\^4\_ \[ ln([m\^2\_ Q\^2]{}) - [3 2]{} \] - m\^4\_q \[ ln([m\^2\_q Q\^2]{}) - [3 2]{} \] }, \[eq:radiative\] where $q=t,b$, $m_t^2 = | h_{2t} \phi_2 + h_{4t} \phi_4|^2$, $m_b^2 = | h_{1b} \phi_1 + h_{3b} \phi_3|^2$, and $m_{\tilde{q}}^2 = m_{s}^2 + m_q^2$.
Our first comment about the viability of 4HD models should make reference to the size of FCNCs via Yukawa interactions. If the Yukawa matrices in (\[eq:superpot\]) are uncorrelated, there is no reason to expect that the unitary transformations defining mass eigenstates also diagonalize (in flavor space) the couplings to the extra Higgs doublets. This would introduce unsuppressed FCNCs at tree level. The observed pattern of quark masses and mixings, however, strongly suggests the possibility of an approximate flavor symmetry as the origin of the hierarchies required in the Yukawa matrices. In the simplest scenarios [@frog79; @anta92] the effect of such a symmetry would be to generate fermion matrices with off-diagonal elements of order $O(\sqrt{m_im_j}/v)$, where $m_i$ is the mass of the $i$th quark and $v$ is the weak scale. If the extra Higgs doublets are a replica (with respect to this flavor symmetry) of the first doublet, they will introduce Yukawa matrices with the same approximate structure. In that case the smallness of Yukawa couplings is enough to keep all FCNC within the experimental limits. In particular, for extra Higgs masses around 1 TeV the tree-level contributions to $K-\bar{K}$ and $B-\bar{B}$ mixings would be of the same order as the standard contributions [@anta92; @hall93]. In our 4HD model we will assume this type of approximate flavor symmetry at work.
Our main motivation to study 4HD models concerns the origin of $CP$ violation. In these models explicit $CP$ violation seems even more inconvenient than in the MSSM, due to new processes mediated by the Yukawa interactions described above (the approximate flavor symmetry would not explain, for example, the small value of $\epsilon_K$ [@hall93]). The possibility of SCPV in 4HD models has been addressed in a recent paper [@masi95]. There we assume that all the parameters in the Lagrangian are real and the $CP$-violating phases appear via VEVs $v_i e^{i\delta_i}$ of the Higgs fields. We showed that at tree level ([*i.e.*]{}, $\Delta V=0$) the minimum equations for the phases can be solved in terms of a simple geometrical object but the remaining conditions are then uncompatible. Namely, after a redefinition of masses and fields that cancels the terms $m^2_{13}$ and $m^2_{24}$, the four tree-level minimum conditions for the VEVs $v_i$ with nonzero phases read [@masi95] v\_1 & = & v\^2\_1 \[ m\_1\^2 -[[m\^2\_[12]{} m\^2\_[34]{} - m\^2\_[14]{} m\^2\_[32]{}]{}]{} [1h([**v**]{})]{} + g([**v**]{}) \] = 0\
v\_2 & = & v\^2\_2 \[ m\_2\^2 - m\^2\_[12]{} m\^2\_[32]{} h([**v**]{}) - g([**v**]{}) \] = 0\
v\_3 & = & v\^2\_3 \[ m\_3\^2 +[[m\^2\_[12]{} m\^2\_[34]{} - m\^2\_[14]{} m\^2\_[32]{}]{}]{} [1 h([**v**]{})]{} + g([**v**]{}) \] = 0\
v\_4 & = & v\^2\_4 \[ m\_4\^2 + m\^2\_[14]{} m\^2\_[34]{} h([**v**]{}) - g([**v**]{}) \] = 0 , \[eq:tritvevtwo\] where $g({\bf v}) = {1 \over 8} (g^2 + g'^2)
[ v_1^2 + v_3^2 - v_2^2 - v_4^2 ]$, and h([**v**]{}) = . Since the four equations above depend on only two combinations of VEVs, $g({\bf v})$ and $h({\bf v})$, there will be no solution ([*i.e.*]{}, phases different from 0 or $\pi$) unless a fine tuned value of the mass parameters is imposed. Moreover, if this fine tuning were used it would imply the presence of two massless scalar fields.
The effect of the radiative corrections is twofold: they relax the ammount of fine tuning required in the equations above, and they generate masses for the two massless modes. For these two effects to be sizeable we need (see $\Delta V$ in Eq. \[eq:radiative\]) large squark masses and at least two large Yukawa couplings. These could be the two top quark couplings $h_{2t}$ and $h_{4t}$ or one top ($h_{2t}$) plus one bottom ($h_{1b}$) coupling. However, for $m_{s} \leq 5$ TeV and Yukawas smaller than $\approx 1.2$ (as required to avoid Landau poles before the Plank scale) we find that the two light scalar fields have masses smaller than $\approx 30$ GeV. In consequence we conclude that in 4HD models with all the parameters real the presence of nontrivial complex phases in the Higgs VEVs implies two scalar fields apparently too light. (A more detailed examination of the parameter space might show, however, that this possibility is not entirely excluded by current limits on the masses of the scalar fields.) The situation here is then similar to the MSSM (where the allowed mass of the light scalar field is already excluded [@poma92]) or the singlet model (which relay on radiative effects to give mass to a mode with negative tree-level mass [@babu94b]).
In 4HD scenarios there is, however, still another possibility which seems consistent with the idea of SCPV. It requires that the four Higgs mass parameters $\mu$ in the superpotential (or, equivalently, the six parameters $m^2_{ij}$ in Eq. \[eq:pot\]) are allowed to be complex. This could be justified since the Higgses are the only superfields which are not protected of mass contributions by the gauge symmetry. They could acquire their masses in an intermediate scale, via complex VEVs of singlet fields with no sizeable effect on the rest of the low-energy effective Lagrangian. We will not assume complex phases on all soft SUSY-breaking terms, since in principle these singlets do not couple to gauginos or squarks (we will neglect the possibility of further phases or new contributions to SUSY-breaking parameters due to the presence of nonsinglet heavy fields [@dann85]). The hypothesis of complex Higgs masses, consistent with a [*soft*]{} origin of $CP$ violation, is not possible in the MSSM or the singlet model, since there (unlike here) all the Higgs masses can be made real by field redefinitions.
In the next sections we study the implications of a 4HD model where all the parameters in the initial Lagrangian are real except for the Higgs mass parameters.
Definition of the model
=========================
The approximate flavor symmetry described in the previous section suppresses all FCNC amplitudes to acceptable limits. However, multi-Higgs models face a potential problem also with $CP$ violation: if the Yukawa couplings are complex with phases of order one, $CP$-violating signals in $K$ physics would be too large. In particular, $\epsilon_K$ would be typically two or three orders of magnitude larger than observed [@hall93]. Thus it seems that a general model of many Higgs doublets requires another ingredient in addition to the flavor symmetry. Its effect should be either a suppression of the Yukawa couplings of the new doublets, or to make the complex phases small. The first approach is typical in models with natural flavor conservation (NFC) [@glas77], whereas a natural suppression of the phases has been obtained in the superweak model with SCPV proposed in [@jliu87]. In our scenario these two effects will be achieved by the action of a global symmetry.
We will assume that the effective Lagrangian of the model obeys an approximate Peccei-Quinn like symmetry with the following assignment of charges [@rasi95]: Q(H\_3) = +1Q(H\_4) = -1Q(D\^c) = +1. All other superfields have zero charge[^1]. The symmetry is approximate in the sense that couplings of operators violating the symmetry are suppressed by powers of a small parameter $\epsilon_{PQ}$. In this section we discuss the impact of this global symmetry first on the Higgs scalar part of the potential and then on the Yukawa sector.
The assignment of charges tells us that in the scalar potential $m_{14}^2$, $m_{32}^2$, $m_{13}^2$, $m_{24}^4$ are suppressed by $\epsilon_{PQ}$. $m_{12}^2$ and $m_{34}^2$ remain unsuppressed (of the order of the SUSY-breaking scale $m_s\leq 1$ TeV). For easy reading we will write the suppression factors explicitly; for example $m^2_{32}$ becomes $\epsilon_{PQ} m^2_{32}$, where $m^2_{32}=O(m^2_s)$. The tree-level scalar potential (we neglect radiative corrections in the following) involving only neutral Higgs fields is then given by V & = & (
[cc]{} \_1\^& \_3\^\
) (
[cc]{} m\_1\^2 & \_[PQ]{} m\_[13]{}\^2\
\_[PQ]{} m\_[13]{}\^[2\*]{} & m\_[3]{}\^2\
) (
[c]{} \_1\
\_3\
)\
& + & (
[cc]{} \_2\^& \_4\^\
) (
[cc]{} m\_2\^2 & \_[PQ]{} m\_[24]{}\^2\
\_[PQ]{} m\_[24]{}\^[2\*]{} & m\_[4]{}\^2\
) (
[c]{} \_2\
\_4\
)\
& + & \[ (
[cc]{} \_1 & \_3\
) (
[cc]{} m\_[12]{}\^2 & \_[PQ]{} m\_[14]{}\^2\
\_[PQ]{} m\_[32]{}\^2 & m\_[34]{}\^2\
) (
[c]{} \_2\
\_4\
) + h.c. \]\
& + & [1 8]{} (g\^2+g’\^2) \[ (
[cc]{} \_1\^& \_3\^\
) (
[c]{} \_1\
\_3\
) - (
[cc]{} \_2\^& \_4\^\
) (
[c]{} \_2\
\_4\
) \]\^2 . \[eq:potapprox\]
As explained in the previous section, we assume that the mass parameters $m^2_{ij}$ are complex. The first two mass matrices above can be diagonalized through two unitary transformations of order $\epsilon_{PQ}$ of the scalar fields: (
[c]{} ’\_1\
’\_3\
) = [**U**]{}\_1 (
[c]{} \_1\
\_3\
); (
[c]{} ’\_2\
’\_4\
) = [**U**]{}\_2 (
[c]{} \_2\
\_4\
). \[eq:unit\] The quartic term in the potential will not change its form and can be obtained just by replacing unprimed by primed fields. The relative size of the four complex masses in the third mass matrix above will stay the same ([*i.e.*]{}, the off-diagonal elements are still suppressed by $\epsilon_{PQ}$). A phase transformation of the fields $\phi_i$ can be used to remove three of the four phases, leaving only one phase $\alpha$ in the scalar potential. Dropping the prime to specify transformed quantities, the mass terms read V\_m & = & (
[cc]{} \_1\^& \_3\^\
) (
[cc]{} m\_1\^2 & 0\
0 & m\_[3]{}\^2\
) (
[c]{} \_1\
\_3\
) + (
[cc]{} \_2\^& \_4\^\
) (
[cc]{} m\_2\^2 & 0\
0 & m\_[4]{}\^2\
) (
[c]{} \_2\
\_4\
)\
& + & \[ (
[cc]{} \_1 & \_3\
) (
[cc]{} m\_[12]{}\^2 & \_[PQ]{} e\^[i]{} m\_[14]{}\^2\
\_[PQ]{} m\_[32]{}\^2 & m\_[34]{}\^2\
) (
[c]{} \_2\
\_4\
) + h.c. \] . where the masses are now real and the phase $\alpha$ of $m^2_{14}$ has been written explicitly. We assume $\alpha$ to be of order one, since there is no symmetry reason for it to be suppressed. It is easy to see how the Higgs field redefinitions above change the mass parameters $\mu_{ij}$ in the superpotential $W$ (these parameters are relevant since they will appear in scalar trilinears). We obtain $\mu_{12}$ and $\mu_{34}$ real up to order $\epsilon_{PQ}^2$ whereas $\mu_{14}$ and $\mu_{23}$ will be mass coefficients with arbitrary complex phases but suppreessed by a power of $\epsilon_{PQ}$.
We now go to the minimization of the Higgs potential. In particular, we want to find what are the relative size and the phases of the scalar VEVs suggested by the approximate symmetry. We write = [1 ]{} v\_1 ; <\_3> = [1 ]{} v\_3 e\^[i\_3]{} , and = [1 ]{} v\_2 e\^[i\_2]{} ; <\_4> = [1 ]{} v\_4 e\^[i\_4]{} , where a global hypercharge transformation has been used to rotate away the phase of $<\phi_1>$. A detailed discussion of the minimum equations can be found in the Appendix. The results are the following. For nonzero values of the phase $\alpha$ the minimum is allways complex. The suppression in terms of $\epsilon_{PQ}$ of the mass parameters determines the order of magnitude of the VEVs and phases: v\_1 , v\_2 & = & O(v)\
v\_3 , v\_4 & = & O(\_[PQ]{} v)\
\_2 & = & O(\^2\_[PQ]{})\
\_3 , \_4 & = & O(1) \[eq:structure\] where $v$ denotes the weak scale. A remarkable feature of the model is that one can understand its structure in terms of an expansion in $\epsilon_{PQ}$ from the model with just two doublets. In the limit $\epsilon_{PQ}=0$ the sectors $(H_1,H_2)$ and $(H_3,H_4)$ decouple; the minimum gives then equations for $v_1$ and $v_2$ identical to the VEVs in the MSSM, whereas $v_3=v_4=0$. The phase $\delta_2$ is then zero, while the phases $\delta_3$ and $\delta_4$ are irrelevant. Turning on a small value of $\epsilon_{PQ}$ gives (proportionally) VEVs to the extra pair of scalars. Simultaneously it allows a nonzero value of $\alpha$, which translates into unsuppressed complex phases in the $(H_3,H_4)$ sector and a phase of order $\epsilon^2_{PQ}$ in the $(H_1,H_2)$ [*standard*]{} sector. The mixings between the two sectors are small: either in a basis of scalar mass eigenstates or in a basis where $v'_3=v'_4=0$ (useful when discussing FCNCs via Yukawas), both are obtained from the original basis just by unitary transformations of order $\epsilon_{PQ}$.
We now turn to the Yukawa sector of the theory. The charge assignments dictates that the matrix ${\bf h}_2$ is unsuppressed, ${\bf h}_1$ and ${\bf h}_4$ are suppressed by a factor of $\epsilon_{PQ}$, while ${\bf h}_3$ is suppressed by $\epsilon_{PQ}^2$. Making this suppression explicit the Yukawa sector for the quark fields reads \_Y & = & Q ( \_[PQ]{} [**h**]{}\_1 H\_1 + \_[PQ]{}\^2 [**h**]{}\_3 H\_3) D\^c + Q ( [**h**]{}\_2 H\_2 + \_[PQ]{} [**h**]{}\_4 H\_4) U\^c , \[eq:superpotsup\] where now ${\bf h}_i$ ($i=1,...,4$) just carry the suppression from the flavor symmetry. In our model the Yukawa couplings of the initial Lagrangian are real. However, the field redefinitions performed to leave only one phase $\alpha$ in the Higgs potential will also redefine the Yukawa couplings and introduce complex phases. As we show, these phases translate into a CKM phase and complex FCNC couplings which are naturally suppressed by the approximate global symmetry.
We first performed the unitary transformations in Eq. (\[eq:unit\]), which redefine the fields $H_i$ by (complex) factors of order $\epsilon_{PQ}$. They imply a redefinition of the Yukawa matrices which introduces phases of order $\epsilon_{PQ}^2$ in ${\bf h}_1$ and ${\bf h}_2$ and of order one in ${\bf h}_3$ and ${\bf h}_4$. Then we performed the (order one) phase redefinitions of the Higgs doublets that make all mass parameters real except for $m^2_{14}$. This translates into overall phases of order one multiplying the Yukawa matrices ${\bf h}_i$. However, we can still redefine the quark fields and absorb the phases which multiply ${\bf h}_1$ and ${\bf h}_2$ (the leading Yukawa couplings). The net result is that the Lagrangian in (\[eq:superpotsup\]) expressed in terms of the Higgs fields used to minimize the potential has real (up to order $\epsilon_{PQ}^2$) couplings in ${\bf h}_1$ and ${\bf h}_2$ and arbitrary complex phases in ${\bf h}_3$ and ${\bf h}_4$.
After spontaneous symmetry breaking, the structure of VEVs in (\[eq:structure\]) suggests (note that $v_1$ amd $v_2$ are not suppressed by powers of $\epsilon_{PQ}$) $\tan \beta\equiv \sqrt{v_2^2+v_4^2\over
v_1^2+v_3^2}=O(1)$ and $\epsilon_{PQ} = O (m_b / m_t)$. Thus, the approximate global symmetry is used to accommodate the small ratio $m_b/m_t$, while the hierarchy between generations of the same charge is left to the flavor symmetry (the flavor symmetry would be exact in the limit with only the third generation being massive). The complex Yukawas ${\bf h}_3$ and ${\bf h}_4$ and the corresponding VEVs ($v_3$ and $v_4$) are both suppressed by a power of $\epsilon_{PQ}$, whereas the leading Yukawas and VEVs are real up to order $\epsilon_{PQ}^2$. We then obtain quark mass matrices where all the entries have complex phases of order $\epsilon_{PQ}^2$. In consequence, the complex phase in the CKM matrix is also of order $\epsilon_{PQ}^2$.
It is also easy to see what is the pattern of FCNC and $CP$ violation via scalar exchange predicted by the model. It will be convenient to define a basis where only two of the four Higgs fields develop VEV and then only the second pair of scalars mediates FCNC processes. Again, this involves a unitary transformation of order $\epsilon_{PQ}$, which leaves almost decoupled the extra pair of Higgses (essentially ($H_3,H_4$)). In addition, the mixings in the scalar mass matrices between the two sectors are also suppressed. The overall suppression by a power of $\epsilon_{PQ}$, when added to the one with origin in the flavor symmetry, renders these tree-level FCNCs smaller than CKM (box) diagrams typically by a factor of $\epsilon^2_{PQ}$. In particular, the $K-\bar{K}$ and $B-\bar{B}$ mixings are dominated here by the standard contributions, like the $W$-exchange box diagram in Fig. \[fig:wexchange\]. This fact will distinguish our scenario from typical multi-Higgs models with soft $CP$ violation where the tree-level superweak interactions are the main source of flavor changing processes [@jliu87]. On the other hand, the sector ($H_3,H_4$) involves arbitrary phases in Yukawa couplings and scalar VEVs. Although suppressed by a factor of $\epsilon^2_{PQ}$, these couplings can be the dominant source of $CP$ violation through diagrames like the one shown in Fig. \[fig:treelevelfc\]. In particular, as shown in the next section, they are the main source of complex phases in $K$ physics and compete with box contributions in $B$ physics.
We need as well that FCNC contributions via SUSY particles (wino and gluino box diagrams) are within the experimental limits, which in general requires certain degree of squark-quark alignment and squark degeneracy. In fact, the squark-quark alignment could appear here as a natural consequence of the flavor symmetry[@ynir93]. Since we assumed no complex phases in soft-SUSY parameters others than Higgs masses, their $CP$ violating effects will follow the same pattern described above. We explore these and other phenomenological implications of the model in the next sections.
$CP$ violation in $K$ and $B$ physics and the neutron EDM
===========================================================
[*$K$ physics.*]{} As our first example we look at the $K$ system. In this scenario the FCNC processes via Yukawa interactions (see previous section) are highly suppressed, and the dominant contribution to Re $\Delta M_{12}$ comes from the box diagram in Fig. \[fig:wexchange\]. The leading imaginary contribution to $\Delta M_{12}$, however, will come from the neutral Higgs exchange in Fig. \[fig:treelevelfc\]. The flavor-changing Yukawa couplings are complex (with phases of order 1) and generically suppressed by the Peccei-Quinn and the flavor symmetries (for example, in Fig. \[fig:treelevelfc\] the couplings are of order $\epsilon_{PQ}
\sqrt{m_d m_s} / v$). When the mass of the exchanged scalar is around 1 TeV, this (complex) diagram is roughly suppressed by $\epsilon^2_{PQ}$ with respect to the box diagram in Fig. \[fig:wexchange\]. Since the CKM matrix and then the box contributions are approximately real, the leading contribution to Im $(\Delta M_{12})$ comes from the Higgs exchange in Fig. \[fig:treelevelfc\]. On dimensional grounds, the $CP$ violating parameter $\epsilon_K$ (see [@chau83] for definitions and notation) in the $K$ system and $\epsilon_{PQ}$ are related: |\_[K]{}| \_[PQ]{}\^2. The parameter $\epsilon_{PQ}$ sets the overall strength of Yukawa couplings of the Higgs doublets and suggests the order of magnitude of all the scalar VEVs. In particular (see section 3), one expects $\tan\beta= O(1)$ and $\epsilon_{PQ}=O(m_b/m_t)$. Then the relation above establishes $\epsilon_K\approx 10^{-3}$, as experimentally required.
Other sizeable contributions to $\epsilon_K$ may come from SUSY box diagrams with chargino or gluino exchange. Both of them require large SUSY contributions to $\Delta M_{12}$ (of the same order as the standard box diagram). Chargino box diagrams would then give [@poma93] contributions of order $\epsilon_K\approx
10^{-1}\epsilon_{PQ}^2$, whereas gluino boxes [@babu94a] could be as large as $\epsilon_K\approx \epsilon_{PQ}^2$ ([*i.e.*]{}, of the same order as the dominant tree-level scalar exchange). The factors $\epsilon_{PQ}^2$ above derive from the suppresion in Yukawa couplings or extra scalar VEVs. Large gluino box contributions, however, also require large left-right squark mixing $\delta_{LR}\equiv m^2_{LR}/m^2_s\approx 10^{-3}$ (a naive estimate would give $\delta_{LR}\approx
{A\sqrt{m_dm_s}\over m_s^2}\approx 10^{-4}$). In addition, the tree-level contributions to $\epsilon_K$ can be easily enhanced [@rasi95] assuming Higgs masses ligheter than 1 TeV, so the clear tendency in our model is that this type of nonstandard Higgs exchange provides the dominant contribution to $\epsilon_K$.
In contrast, the expected value for $\epsilon'_K$ differs in principle from the standard model prediction. An estimate of $\epsilon'_K$ can be obtained from the phase $t_0\equiv$ Im$A_0$/Re$A_0$, where $A_i$ is the decay amplitude of a $K^0$ into two pions of isospin $i$ (see [@chau83] for notation). In particular one has the experimental constraint t\_0 |[A\_0A\_2]{}| |’\_K| 10\^[-4]{}. The dominant contribution to Re$A_0$ arises from the standard penguin diagram: \_p \_c +H.c., where = (|[s]{}\_L\_T\^a d\_L) (|[q]{}\_R\_T\^a q\_R) and $T^a$ are the 3-dimensional generators of $SU(3)$. In our scenario, however, the imaginary part of this penguin diagram is suppressed by the smallness of the phase (of order $\epsilon_{PQ}^2\approx 10^{-3}$) in the CKM matrix. Since the standard model prediction for $t_0$ is of order $s_{13}s_{23}/s_{12}\approx 10^{-3}$, we obtain a first contribution of order $10^{-6}$. Other contributions to $t_0$ may come from penguin diagrams with chargino and stop (Figure \[fig:charginoep\]) and tree-level diagrams with charged scalars. The first contributions have been studied in [@poma93] in the context of SUSY models with SCPV. It is found that they are typically of order $t_0\approx 10^{-3}\sin\delta$, being $\delta$ the complex phase in the VEV of $H_2$ (the scalar field giving mass to the top quark). In our model the Higgs fields $H$ and $\tilde{H}$ in Fig. \[fig:charginoep\] can correspond to $H_2$ or $H_4$. In the first case the phase $\delta$ is of order $\epsilon_{PQ}^2$, and in the second case the same degree of suppression comes from the small VEV and the small Yukawa coupling of $H_4$. The contributions via exchange of charged Higgses have been analyzed in [@jliu87] in the context of two-Higgs doublets models. In our model either the Yukawas are [*almost*]{} real (for the doublets with dominant couplings, as it happens in [@jliu87]), or the Yukawas themselves are suppressed. Hence, from the three types of diagrams we obtain contributions (taking $\epsilon^2_{PQ}\approx 10^{-3}$) of order $t_0\approx 10^{-6}$ or ${\epsilon'_K\over \epsilon_K}\approx 10^{-5} \;$ (for $|A_2/A_0|\approx 1/22$).
Potentially larger contributions to $\epsilon'_K$ are expected from gluino-mediated penguin diagrams (Figure \[fig:gluinoep\]). Although gluino masses in our model are real, there will be complex phases of order $\epsilon_{PQ}^2$ (see discussion of chargino penguin above) in left-right squark mixings. This type of contributions to $\epsilon'_K$ have been studied in detail in [@babu94a]. There it is found that for complex phases in gluino masses of order $10^{-3}$ they could result in values as large as ${\epsilon'_K \over \epsilon_K}
\approx 10^{-3}$. An analogous result has been recently obtained in [@poma95], with contributions from squark mixings of CKM type and small $CP$-violating phases (which are natural in our scenario). These values are only obtained, however, when gluino box diagrams saturate the value of $\epsilon_K$. Since we have assumed that the FCNCs are here dominated by standard box diagrams, we expect a value for $\epsilon'_K$ typically smaller. Modulo hadronic matrix element uncertainties, we estimate 10\^[-4]{} - 10\^[-5]{}, with the possibility to consistently increase this value via gluino penguin contributions.
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[*$B$ physics.*]{} We consider now $CP$ violation in $B$ physics [@ynir92]. Although today the only observed $CP$ violation is in the $K$ system, the standard model predicts clear signals in $B$ decays that should be observed in the near future. These $CP$ asymmetries are generally parametrized in terms of the complex phases $\lambda_{iq}$, which in turn depend on the product of phases in three amplitudes: the direct $b$ decay, the $B-\bar{B}$ mixing, and (possibly) the $K-\bar{K}$ mixing. In CKM scenarios the phases $\lambda_{iq}$ are constrained by unitarity and have a simple geometrical interpretation (these predictions are not expected to change much in minimal SUSY models). However, in our 4HD scenario the three amplitudes above have complex phases of order $\epsilon^2_{PQ}$ (see below), and the predictions change to the extent that no $CP$ asymmetries will be observed at the projected $B$ factory at SLAC.
To see why this is so, we will first consider the decay amplitude of a $b$ quark into lighter flavors. The main contribution corresponds to a tree-level diagram with $W$ exchange. Since it will be proportional to elements of the CKM matrix, its imaginary component will be suppressed by a factor of $\epsilon_{PQ}^2$. The decay via charged Higgs are suppressed by the same factor due to the relative smallness of their Yukawas and the smallness of the mixing (in the scalar mass matrix) between the standard and the extra Higgs sectors. In non-SUSY models with NFC the second effect can give significant contributions (proportional to $m_t$) in $B$ decays and in flavor-changing processes [@jliu87].
The main contribution in this model to $B-\bar{B}$ mixing $\Delta M_{B\overline B}$ comes from the standard box diagrams, and is proportional to CKM elements. The tree-level diagrams with exchange of neutral scalar give contributions which are suppressed by the flavor and the Peccei-Quinn symmetries, with a relative factor of $\epsilon_{PQ}^2$ with respect to the box diagrams. Both types of diagrams introduce imaginary components of order $\epsilon_{PQ}^2$ with respect to the main real component, and the contribution to the complex phase $\lambda_{iq}$ from $B-\bar{B}$ mixing is negligible (of that order). As discussed above, the same conclusion applies to the contribution from $K-\bar{K}$ mixing.
In consequence, in this scenario one expects that all $CP$ asymmetries in $B$ decays negligibly small (of order $\epsilon^2_{PQ} \approx 10^{-3}$). This type of prediction is shared, for example, by non-SUSY multi-Higgs doublet models [@jliu87; @hall93] or SUSY models with real Yukawa matrices [@poma93; @babu94a]. In CKM scenarios the situation is essentially different. There the smallness of $CP$ violation in the $K$ system is atributed to the smallness of the CKM elements involving the third family of quarks, whereas $CP$-violating asymmetries in the $B$ system are large: the $B-\bar{B}$ mixing and the $b$ decays are proportional to elements of the CKM matrix with arbitrary complex phases. The absence of $CP$ asymmetries at the SLAC $B$ factory would point to a non-CKM origin of $CP$ violation, and many-Higgs doublet model (SUSY or non-SUSY) would appear as a natural candidate.
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[*Neutron EDM:*]{} As in usual SUSY scenarios, the prediction of our model for the neutron EDM $d_n$ is much larger than in the non-SUSY standard model. In the MSSM the explicit phases $\psi$ in SUSY-breaking gaugino mases and scalar trilinears introduce contributions which roughly require a suppression of 2 or 3 orders of magnitude: $d_n \approx 10^{-25} ({ \psi \over {7\times10^{-3}}})\;e$ cm [@deru90]. These diagrams are also present in our scenario, but their contribution has a natural suppression of order $\epsilon^2_{PQ}$ respect to the MSSM. The origin of this factor is (again!) either the smallnes of the complex phase $\delta_2$ (of order $\epsilon_{PQ}^2$) in the two standard Higgs doublets, or the combined relative smallness of VEVs and Yukawa couplings (both suppressed by a factor of $\epsilon_{PQ}$) of the two extra doublets.
To illustrate this fact, let us consider the contribution from the the one-loop chargino-squark diagram (Figure \[fig:charginonedm\]). When $H$ corresponds to $H_1$, then the complex phase in the Yukawa coupling is suppressed. When $H$ is $H_3$, then the VEV and the Yukawa coupling are of order $\epsilon_{PQ}$. In consequence, in this 4HD one expects d\_n 10\^[-23]{} \_[PQ]{}\^2e 10\^[-26]{} e, a value which is close to the present experimental limit $|d_n|<1.2\times10^{-25}e$ cm [@smit90].
Here we also comment on the lepton sector. We still have the freedom to assign a global symmetry charge to $E^c$ (or even $L$). For simplicity let us consider $E^c = +1$, which would be consistent with $m_\tau=O(m_b)$. In this sector all FCNC processes via nonstandard scalars will be completely negligible (the size of Yukawas suggested by the flavor symmetry would be enough to control all these processes). The pattern of $CP$ violation will be analogous to the one discussed in the quark sector, with the relevant complex phases suppressed by a factor of $\epsilon^2_{PQ}$. The leading contribution to the electron EDM comes from a diagram similar to the one shown in Fig. \[fig:charginonedm\]. If all the superparticles have comparable masses it is expected that $d_e \approx 10^{-2} d_n$ [@bern91], so that in our model the electron EDM is not far from the present experimental limits.
Other phenomenological implications
=====================================
As shown by Flores and Sher in [@flor83], the presence of a light neutral scalar field (with a tree-level mass smaller than $M_Z$) is a prediction shared by all SUSY models with Higgs doublets only, regardless of the number of doublets. Since in the limit $\epsilon_{PQ}\rightarrow 0$ the scalar sector of our 4HD model essentially coincides with the MSSM, we expect small corrections to the standard predictions.
To see how these corrections arise we will first consider the model with $\alpha=0$ and, consequently, with all the VEVs real. The approximate symmetry dictates that $v\approx v_1\approx v_2$ and $\epsilon_{PQ} v\approx v_3\approx v_4$. We can perform two rotations of order $\epsilon_{PQ}$ of the Higgs fields (one in the space $\phi_1-\phi_3$ and another in $\phi_2-\phi_4$) in such a way that $v_3=v_4=0$. It is then straightforward to find the mass $4\times 4$ matrix $M^2_h$ M\^2\_h=(
[cc]{} M\^2\_0 & M\^2\_1\
M\^[2T]{}\_1 & M\^2\_2\
) \[eq:mass\] for the $CP$-even scalar fields $h_i$. The $2\times 2$ matrix $M^2_0$ corresponding to $h_1-h_2$ is identical to the one obtained in the MSSM, with an eigenvalue $m^2_h$ smaller than $M^2_Z$ and another $m^2_H\approx m^2_{s}$. The submatrix $M^2_2$ in the $h_3-h_4$ sector has two eigenvalues of order $m^2_s$. The Peccei-Quinn symmetry forces all the elements in $M^2_1=O(\epsilon_{PQ} m^2_s)$ and, through mixing, tends to lower the lightest eigenvalue in $M^2_h$ by terms of order $\epsilon^2_{PQ} m^2_s$. For nonzero values of $\alpha$ the scalar VEVs will be allways complex (see Section 3), introducing mixing betweeen CP-odd and $CP$-even states. Due to the approximate symmetry, however, the mixings of the lightest scalar field with $CP$-odd states are small and introduce corrections of the same order. In consequence, we conclude that these corrections do not change significantly the tree-level bound $m_h<M_Z$ (for $\epsilon^2_{PQ}=10^{-3}$ and $m_s=500$ GeV this bound is lowered by less than 2 GeV). However, we expect radiative top quark effects to be much more important.
We note that the spontaneous breaking of the (approximate) global symmetry do not introduce light fields. The reason for this is that in the limit of exact symmetry ($\epsilon_{PQ}=0$) the only VEVs breaking the symmetry ($v_3$ and $v_4$) go to zero too: there are no light pseudo-goldstone states because the size of the spontaneous and the explicit symmetry breaking terms is of the same order.
It is also easy to see that this model accommodates the small ratio $m_b/m_t$ without need of fine tuning to avoid too light charginos [@nels93] (in the MSSM, a small mass ratio $m_b/m_t$ based on a large value of $\tan\beta$ implies such fine-tuning problem). The chargino mass matrix is here (
[ccc]{} \_[12]{} & \_[PQ]{} \_[14]{} & [[g v]{} ]{}\
\_[PQ]{}\_[32]{} &\_[34]{}&[[g \_[PQ]{}v]{}]{}\
[[g v]{} ]{}& [[g \_[PQ]{} v]{}]{} & M\
), where we used the VEVs in (\[eq:structure\]) and denoted the gaugino mass by $M$. This structure has no light eigenvalues.
Another possible implication of 4HD models concerns the value of $R_b$. Within the standard model, the partial width of the $Z$ boson to $b\overline b$ seems to be very sensitive to top-quark radiative correction. For the top observed in CDF the predicted value is well below (a three-$\sigma$ deviation) the present experimental limits [@pdg94]. In minimal SUSY scenarios the main correction results from the balance between $Z$ vertices with Higgs-top and their SUSY partners, and the anomalous value of $R_b$ can be alleviated for light charginos and/or light stop scalars [@sola95]. In 4HD SUSY models the situation is similar (especially in our scenario, due to the global approximate symmetry assumed in the Yukawa sector), with more freedom than in the MSSM to adjust the corrections. Note, for example, that large bottom Yukawa couplings do not imply necessarily a large value of $\tan\beta$ $(\equiv \sqrt{v_2^2+v_4^2\over v_1^2+v_3^2})$.
Our last comment concerns the strong $CP$ problem. In the model under consideration there are (tree-level) contributions to $\theta$ of order $\epsilon_{PQ}^2\approx 10^{-3}$, a value much bigger than the present experimental limit ($\theta<10^{-9}$). It seems possible, however, that the intermediate scale used to break $CP$ would also define a realistic axion scenario. Some of the ingredients (a Peccei-Quinn symmetry or singlet VEVs breaking the global symmetries) are already present in the model. Of course, for this scenario to work other requirements (on the dimension of the operators breaking the anomalous Peccei-Quinn symmetry, on the ratio of the scales involved,...) are also needed.
Conclusions
===========
The origin of $CP$ nonconservation in SUSY models provides a good reason to explore nonminimal extensions. In the usual MSSM scenario $CP$-violating phases occur in two different sectors: in Yukawa couplings, where they would be responsible for $CP$ violation in $K$ and $B$ physics, and in SUSY-breaking terms (gaugino masses and scalar trilinears), where they would induce too large fermion EDMs unless suppressed by two or three orders of magnitude.
We have presented here an extension of the MSSM with four Higgs doublets where the complex phases appear spontaneously, induced by explicit phases in Higgs masses. An approximate Peccei-Quinn symmetry [*almost*]{} decouples the pair of extra Higgs fields, but their small couplings (also suppressed by the flavor symmetry) turn out to be responsible for all $CP$-violating phenomena. In particular, tree-level FCNC diagrams are irrelevant in Re$\Delta M_{12}$ but responsible for $\epsilon_K$. The resulting CKM matrix of the model has a negligible complex phase of order $\epsilon_{PQ}^2\approx 10^{-3}$. This suppression appears in all $CP$ signals either from small phases in the dominant scalar sector or from small ratios of VEVs and Yukawa couplings in the extra sector.
On dimensional grounds, the parameter $\epsilon_{PQ}$ specifying the violation of the Peccei-Quinn symmetry sets:
$\bullet$ the ratio $m_b/m_t \approx \epsilon_{PQ}$, and the relative suppression of the Yukawa couplings of the extra Higgses (${\bf h}_3/{\bf h}_1\approx{\bf h}_4/{\bf h}_2\approx \epsilon_{PQ}$);
$\bullet$ the parameter $\epsilon_K \approx \epsilon_{PQ}^2$ and the ratio $\epsilon'_K/\epsilon_K \le \epsilon_{PQ}^2$ (with a preferred value ($10^{-1}-10^{-2})\epsilon_{PQ}^2$);
$\bullet$ the neutron EDM $\approx 10^{-23}\epsilon_{PQ}^2 e$ cm, being $10^{-23}$ an estimate for typical SUSY-breaking parameters;
$\bullet$ and the $CP$ violating asymmetries $\lambda_{iq}
\approx \epsilon_{PQ}^2$ involved in $B$ physics.
In consequence, a neutron EDM close to its present experimental limit, negligible $CP$-violating effects on $B$ physics, and a small value of the $\epsilon'_K$ parameter could be regarded as typical predictions of the model. In addition, we have estimated the effects of the extra sector on the mass of the lightest neutral scalar and commented on other aspects of the model ($R_b$ and the strong $\theta$ parameter).
We think that 4HD models constitute an interesting possibility in SUSY extensions which, however, seems almost absent in the literature. We have defined a scenario where $CP$ violation is brought under control in a consistent way (due to the action of an approximate symmetry), in contrast to SUSY models where the complex phases are assumed small without explanation. Although we have analyzed a particular model, we think that it contains essential ingredients which may be shared by any satisfactory multi-Higgs SUSY model. In a generic multi-Higgs model hard (CKM-like) $CP$ violation seems to imply too large imaginary FCNCs mediated by the extra Higgs fields. This fact strongly suggests a soft origin of $CP$ violation. Then the problem of containing simultaneously FCNC and too large $CP$ violation forces these models to have, for example, unobservable $CP$ asymmetries in $B$ decays, a prediction that will be tested in the near future.
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[**Acknowledgments**]{}
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We thank D. Chang, L. Hall, R. Mohapatra and A. Pomarol for helpful suggestions and comments. The work of M. M. has been partially supported by a grant from the Junta de Andalucía (Spain). The work of A. R. was supported by the NSF grant No. PHY9421385.
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[**Appendix: Solutions to minimum equations**]{}
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The vacuum expectation value of the scalar potential is & = &[1 2]{} m\_1\^2 v\_1\^2 + [1 2]{} m\_2\^2 v\_2\^2 + [1 2]{} m\_3\^2 v\_3\^2 + [1 2]{} m\_4\^2 v\_4\^2 + m\_[12]{}\^2 v\_1 v\_2 \_2 +\
& + & \_[PQ]{} m\_[14]{}\^2 v\_1 v\_4 (\_4 + ) + \_[PQ]{} m\_[32]{}\^2 v\_3 v\_2 (\_3+\_2) + m\_[34]{}\^2 v\_3 v\_4 (\_3+\_4) +\
& + & [1 32]{} (g\^2 + g’\^2) \[ v\_1\^2 + v\_3\^2 - v\_2\^2 - v\_4\^2 \]\^2 . \[eq:vevpot1\] The conditions at the minimum are v\_1 & = & m\_1\^2 v\^2\_1 + m\_[12]{}\^2 v\_1 v\_2 \_2 + \_[PQ]{} m\_[14]{}\^2 v\_1 v\_4 (\_4+) + v\^2\_1 g([**v**]{}) = 0 ,\
v\_2 & = & m\_2\^2 v\^2\_2 + m\_[12]{}\^2 v\_1 v\_2 \_2 + \_[PQ]{} m\_[32]{}\^2 v\_3 v\_2 (\_3 + \_2) - v\^2\_2 g([**v**]{}) = 0 ,\
v\_3 & = & m\_3\^2 v\^2\_3 + \_[PQ]{} m\_[32]{}\^2 v\_3 v\_2 (\_3 + \_2) + m\_[34]{}\^2 v\_3 v\_4 (\_3 + \_4) + v\^2\_3 g([**v**]{}) = 0 ,\
v\_4 & = & m\_4\^2 v\^2\_4 + m\_[34]{}\^2 v\_3 v\_4 (\_3 + \_4) + \_[PQ]{} m\_[14]{}\^2 v\_1 v\_4 (\_4 + ) - v\^2\_4 g([**v**]{}) = 0 \[eq:minvevspq\] where $g({\bf v}) = {1 \over 8} (g^2 + g'^2)
[ v_1^2 + v_3^2 - v_2^2 - v_4^2 ]$, and - & = & m\_[12]{}\^2 v\_1 v\_2 \_2 + \_[PQ]{} m\_[32]{}\^2 v\_3 v\_2 (\_3 + \_2) = 0 ,\
- & = & \_[PQ]{} m\_[32]{}\^2 v\_3 v\_2 (\_3 + \_2) + m\_[34]{}\^2 v\_3 v\_4 (\_3 + \_4) = 0 ,\
- & = & m\_[34]{}\^2 v\_3 v\_4 (\_3 + \_4) + \_[PQ]{} m\_[14]{}\^2 v\_1 v\_4 (\_4 + )= 0 . \[eq:minphasespq\]
First, we can estimate the relative sizes of VEVs. The solutions to (\[eq:minvevspq\]) are consistent with either one of the pairs of VEVs $(v_1,v_2)$ or $(v_3,v_4)$ being suppressed by order $\epsilon_{PQ}$ with respect to the weak scale. Depending on the sizes of the unsupressed parameters $m^2_1,m^2_3,m^2_{13}$ and $m^2_2,m^2_4,m^2_{24}$ the absolute minimum will prefer one of the above choices. This can easily be seen from the following. Imagine for a moment that $\epsilon_{PQ}=0$. The equations in (\[eq:minvevspq\]) reduce to two pairs of equations, with first pair depending on the ratio ${v_1 \over v_2}$ and $g({\bf v})$, and the second on ${v_3 \over v_4}$ and the same function $g({\bf v})$. Thus, one pair of VEVs is forced to be zero. Turning back on a small $\epsilon_{PQ}$, the terms in the Lagrangian suppressed by $\epsilon_{PQ}$ can make the previously trivial pair nonzero, but suppressed. The only question left is which pair of the VEVs is small, and this will depend on the choice of the unsuppressed mass parameters in the potential. We will assume that the parameters are such that $(v_1,v_2)$ are unsuppressed (and of the order weak scale), while $(v_3,v_4)$ are of the order $\epsilon_{PQ}$ times the weak scale. This assumption does not involve fine tuning but only “halves" the available parameter space. For the phases $\delta_i$, it was shown in [@masi95] that in the limit $\alpha=0$ there is no nontrivial solution ([*i.e.*]{}, $\delta_i$ different from 0 or $\pi$). It is easy to see, however, that for $\alpha\neq 0$ the equations (\[eq:minvevspq\]) do [*not*]{} have this trivial solution, and complex phases are guaranteed. In particular, for $\alpha=O(1)$ from the two first equations in (\[eq:minphasespq\]) it follows that $\delta_2$ is of order $\epsilon_{PQ}^2$ (modulo $\pi$, depending on the sign of $m^2_{12}$) and $\delta_3$ and $\delta_4$ are unsuppressed.
In summary, the structure of values of VEVs and their phases is v\_1 , v\_2 & = & O(v)\
v\_3 , v\_4 & = & O(\_[PQ]{} v)\
\_2 & = & O(\^2\_[PQ]{})\
\_3 , \_4 & = & O(1) \[eq:structure1\] where $v$ denotes the weak scale.
We now explore this structure in more detail (at first order in $\epsilon_{PQ}$). The first equation in (\[eq:minphasespq\]) gives $\delta_2$ to be of order $\epsilon^2_{PQ}$ up to[^2] a factor of $\pi$. Such $\delta_2$ does not contribute to leading order (its contributions are O($\epsilon^2_{PQ}$)) to the minimum of the scalar potential (\[eq:vevpot1\]), and can be neglected in the rest of equations. The minimum equations (\[eq:minvevspq\]) to leading order are v\_1 & = & m\_1\^2 v\^2\_1 + m\_[12]{}\^2 v\_1 v\_2 + v\^2\_1 g\_o([**v**]{}) = 0 ,\
v\_2 & = & m\_2\^2 v\^2\_2 + m\_[12]{}\^2 v\_1 v\_2 - v\^2\_2 g\_o([**v**]{}) = 0 ,\
v\_3 & = & m\_3\^2 v\^2\_3 - \_[PQ]{} m\_[32]{}\^2 v\_3 v\_2 \_3 + m\_[34]{}\^2 v\_3 v\_4 (\_3 + \_4) + v\^2\_3 g\_o([**v**]{}) = 0 ,\
v\_4 & = & m\_4\^2 v\^2\_4 + m\_[34]{}\^2 v\_3 v\_4 (\_3 + \_4) + \_[PQ]{} m\_[14]{}\^2 v\_1 v\_4 (\_4 + ) - v\^2\_4 g\_o([**v**]{}) = 0, \[eq:minvevspqsm\] where $g_o({\bf v}) = {1 \over 8} (g^2 + g'^2)
[ v_1^2 - v_2^2 ]$, and - & = & - \_[PQ]{} m\_[32]{}\^2 v\_3 v\_2 \_3 + m\_[34]{}\^2 v\_3 v\_4 (\_3 + \_4) = 0 ,\
- & = & m\_[34]{}\^2 v\_3 v\_4 (\_3 + \_4) + \_[PQ]{} m\_[14]{}\^2 v\_1 v\_4 (\_4 + )= 0 . \[eq:minphasespqsm\] The first two equations in (\[eq:minvevspqsm\]) are just the equations of the MSSM, and they fix $v_1$ and $v_2$ in the usual way. Thus, we expect both $v_1$ and $v_2$ to be of the order of weak scale and $\tan\beta=O(1)$ ([*i.e.*]{} no supression by $\epsilon_{PQ}$, and no fine tuning producing large $\tan \beta$).
The goal now is to find the phases $\delta_3$ and $\delta_4$ in terms of quantities $c = \epsilon_{PQ} m^2_{32} v_3 v_2$, $f= m^2_{34} v_3 v_4$ and $y = - \epsilon_{PQ} m^2 v_1 v_4$ and the angle $\alpha$ in order to eliminate them in the third and fourth equation of VEVs in (\[eq:minvevspqsm\]). We note that the three quantities $c$, $f$, and $y$ are of order $O(\epsilon^2_{PQ})$, and so we expect $\delta_3$ and $\delta_4$ unsupressed. In order to find $\delta_3$ and $\delta_4$ we use a geometrical interpretation similar to the one devised in [@masi95]. It is possible to see that the two equations (\[eq:minphasespqsm\]) define one of the objects shown in Figure \[fig:trinontriv\] (which one it is will depend whether $1/c$, $1/$f and $1/y$ can form a triangle or not). The quantities $p$ and $q$ there are not independent, and can be expressed in terms of $c$, $f$, $y$ and $\alpha$. The difference between the two types of solutions can be understood in the limit $\alpha \to 0$, where only the trivial solutions $\delta_3 = 0$ and $\delta_4=\pi$ exist. For $\alpha=0$ the object in Fig. \[fig:trinontriv\](b) implies nonzero $\delta_3$ and $\delta_4$, a type of solution which requires fine tuning between mass parameters once it is substituted in the equations for $v_3$ and $v_4$. In consequence, for small values of $\alpha$ only the solutions of the type in Fig. \[fig:trinontriv\](a) appear. When $\alpha$ is nonzero the fine tuning is lifted, and both types of objects define possible solutions to the minimum equations.
We performed numerical solutions to the above equations when $\alpha \neq 0$ and large (order 1) and we found that the minima satisfy the structure given in (\[eq:structure1\]). For simplicity and to illustrate the discussion above we will give the equations for $v_3$ and $v_4$ at first order in $\alpha$.
[**Fig. \[fig:trinontriv\](a):**]{} When $\alpha \to 0$ we see that $\delta_3 \to 0$ and $\delta_4 \to \pi$, while $1/p \to 1/c + 1/f$ and $1/q \to 1/y - 1/f$. From the figure we first find cosines of the relevant angles ($\delta_3+\delta_4$, $\delta_3$ and $\delta_4+\alpha$) to leading order in $\alpha$. Then we are in the position to find $v_3$ and $v_4$ by substituting these expressions into the two last equations of (\[eq:minvevspqsm\]) v\_3 & = & m\_3\^2 v\^2\_3 - c \[1 - [\^2 2]{} ([[ 1 ]{} ]{})\^2\] - f \[1 - [\^2 2]{} ([[ 1 ]{} ]{})\^2\] + v\^2\_3 g([**v**]{}) = 0 ,\
v\_4 & = & m\_4\^2 v\^2\_4 - f \[1 - [\^2 2]{} ([[ 1 f]{} ]{})\^2\] + y \[1 - [\^2 2]{} ([ [ 1 y]{} ]{})\^2\] - v\^2\_4 g([**v**]{}) = 0 where, again, $c = \epsilon_{PQ} m^2_{32} v_3 v_2$, $f= m^2_{34} v_3 v_4$ and $y = - \epsilon_{PQ} m^2 v_1 v_4$. These equations, although still complicated, can be solved in $v_3$ and $v_4$, (remember that $v_1$ and $v_2$ are already fixed). Then we can go back and find $\delta_3$ and $\delta_4$, thus completing the search for the first case.
[**Fig. \[fig:trinontriv\](b):**]{} In this case $1/p \to 1/y$ and $1/q \to 1/c$, while $\delta_3$ and $\delta_4$ tend to go to angles in the triangle with sides $1/c$, $1/f$ and $1/y$ (we denote this (order O(1)) asymptotic angles as $\delta^o_3$ and $\delta^o_4$). We can again find the relevant angles to leading order in $\alpha$ and then find $v_3$ and $v_4$ by substituting these expressions into the two last equations of (\[eq:minvevspqsm\]) v\_3 & = & v\^2\_3 \[ m\^2\_3 + + g\^0([**v**]{}) \] + 2 f = 0 ,\
v\_4 & = & v\^2\_4 \[ m\^2\_4 + - g\^0([**v**]{}) \] + 2 f = 0 ,\
Remembering that $v_1$ and $v_2$ are already fixed in terms of $m_1^2$, $m^2_2$ and $m^2_{12}$, we see clearly the fine tuning for vanishing $\alpha$[@masi95]: the terms in square brackets would be forced to vanish, implying two relations between mass parameters. However, once we include $\alpha \neq 0$ these degeneracies get lifted.
[50]{} For a review on SUSY see H.P. Nilles, Phys. Rep. [**110**]{} (1981) 1. A. de Rújula, M.B. Gavela, P. Pène, and F.J. Vegas, Phys. Lett. [**B245**]{} (1990) 640. See for example I.I. Bigi and F. Gabbiani, in [*Proceedings of the BNL Summer Study on $CP$ violation*]{} Brookhaven National Laboratory, 1990, edited by S. Dawson and A. Soni. J. Ellis, D.V. Nanopoulos, Phys. Lett. [**B110**]{} (1982) 44. P. Langacker and B. Sathiapalan, Phys. Lett. [**B144**]{} (1984) 401. K.S. Babu and S.M. Barr, Phys. Rev. Lett. [**72**]{} (1994) 2831. A. Pomarol, Phys. Rev. [**D47**]{} (1993) 273. T.D. Lee, Phys. Rev. [**D8**]{} (1973) 1226. J. Liu and L. Wolfenstein, Nucl. Phys. [**B289**]{} (1987) 1. N. Maekawa, Phys. Lett. [**B282**]{} (1992) 387. A. Pomarol, Phys. Lett. [**B287**]{} (1992) 331. K.S. Babu and S.M. Barr, Phys. Rev. [**D49**]{} (1994) 2156. For previous work on four Higgs doublet SUSY models see R.A. Flores and M. Sher, Ann. Phys. [**148**]{} (1983) 95; K. Griest and M. Sher, Phys. Rev. [**D42**]{} (1990) 3834; H. Haber and Y. Nir, Nucl. Phys. [**B335**]{} (1990) 363. F. Gursey, P. Sikivie, and P. Ramond, Phys. Lett. [**B60**]{} (1976) 177; A string model with four Higgs doublets has been proposed in F. del Aguila, M. Masip, and L. da Mota, Nucl. Phys. [**B440**]{} (1995) 3. B. Brachmachari and R.N. Mohapatra, University of Maryland preprint No. UMD-PP-95-138 (hep-ph/9505347). J. Ellis, G. Ridolfi and F. Zwirner, Phys. Lett. [**B257**]{} (1991) 83. C.D. Froggatt and H.B. Nielsen, Nucl. Phys. [**B147**]{} (1979) 277; T.P. Cheng and M. Sher, Phys. Rev. [**D35**]{} (1987) 3484. A. Antaramian, L.J. Hall and A. Rašin, Phys. Rev. Lett. [**69**]{} (1992) 1871. L. Hall and S. Weinberg, Phys. Rev. [**D48**]{} (1993) 979. M. Masip and A. Rašin, University of Maryland preprint No. UMD-PP-95-143 (hep-ph/9506471), to appear in Phys. Rev. [**D**]{}. A. Dannenberg, L. Hall and L. Randall, Nucl. Phys. [**B271**]{} (1985) 575. S. Glashow and S. Weinberg, Phys. Rev. [**D15**]{} (1977) 1958. A. Rašin, talk given at the XXXth Moriond conference on Electroweak Interactions and Unified Theories, Les Arcs, 1995, University of Maryland preprint No. UMD-PP-95-142 (hep-ph/9507239). Y. Nir and N. Seiberg, Phys. Lett. [**B309**]{} (1993) 337. L.L. Chau, Phys. Rep. [**95**]{} (1983) 1. A. Pomarol and D. Tommasini, CERN preprint No. CERN-TH/95-207 (hep-ph/9507462). For a review and further references see Y. Nir and H. Quinn, Ann. Rev. Nucl. Part. Sci. [**42**]{} (1992) 211. I.S. Altarev [*et al.*]{}, Phys. Lett. [**B276**]{} (1992) 242; K.F. Smith [*et al.*]{}, Phys. Lett. [**B234**]{} (1990) 191. W. Bernreuther and M. Suzuki, Rev. Mod. Phys. [**63**]{} (1991) 313. A.E. Nelson and L. Randall, Phys. Lett. [**B316**]{} (1993) 516. , Phys. Rev. [**D50**]{} (1994) 1173. See for example D. García, R.A. Jiménez, and J. Solà, Phys. Lett. [**B347**]{} (1995) 2737.
5.75in =1.00mm
(33.66,182.34) (30.00,100.00)[(1,0)[90.00]{}]{} (30.00,139.00)[(1,0)[90.00]{}]{} (36.00,100.00)[(1,0)[10]{}]{} (65.00,100.00)[(1,0)[10]{}]{} (96.00,100.00)[(1,0)[10]{}]{} (53.00,139.00)[(-1,0)[10]{}]{} (82.00,139.00)[(-1,0)[10]{}]{} (113.00,139.00)[(-1,0)[10]{}]{} (46.00,105.00)[(0,0)\[rc\][$d$]{}]{} (78.00,105.00)[(0,0)\[rc\][$u,c,t$]{}]{} (106.00,105.00)[(0,0)\[rc\][$s$]{}]{} (46.00,144.00)[(0,0)\[rc\][$s$]{}]{} (78.00,144.00)[(0,0)\[rc\][$u,c,t$]{}]{} (106.00,144.00)[(0,0)\[rc\][$d$]{}]{} (60,101.5)[(3,3)\[r\]]{} (60,104.5)[(3,3)\[l\]]{} (60,107.5)[(3,3)\[r\]]{} (60,110.5)[(3,3)\[l\]]{} (60,113.5)[(3,3)\[r\]]{} (60,116.5)[(3,3)\[l\]]{} (60,119.5)[(3,3)\[r\]]{} (60,122.5)[(3,3)\[l\]]{} (60,125.5)[(3,3)\[r\]]{} (60,128.5)[(3,3)\[l\]]{} (60,131.5)[(3,3)\[r\]]{} (60,134.5)[(3,3)\[l\]]{} (60,137.5)[(3,3)\[r\]]{} (90,101.5)[(3,3)\[r\]]{} (90,104.5)[(3,3)\[l\]]{} (90,107.5)[(3,3)\[r\]]{} (90,110.5)[(3,3)\[l\]]{} (90,113.5)[(3,3)\[r\]]{} (90,116.5)[(3,3)\[l\]]{} (90,119.5)[(3,3)\[r\]]{} (90,122.5)[(3,3)\[l\]]{} (90,125.5)[(3,3)\[r\]]{} (90,128.5)[(3,3)\[l\]]{} (90,131.5)[(3,3)\[r\]]{} (90,134.5)[(3,3)\[l\]]{} (90,137.5)[(3,3)\[r\]]{} (70.00,121.00)[(0,0)\[rc\][W]{}]{} (100.00,121.00)[(0,0)\[rc\][W]{}]{}
5.75in =1.00mm
(33.66,182.34) (30.00,100.00)[(1,1)[20.00]{}]{} (30.00,140.00)[(1,-1)[20.00]{}]{} (100.00,120.00)[(1,1)[20.00]{}]{} (100.00,120.00)[(1,-1)[20.00]{}]{} (50.00,120.00)(5,0)[10]{}[(1,0)[4.00]{}]{} (45.00,105.00)[(0,0)\[rc\][$d$]{}]{} (47.00,135.00)[(0,0)\[rc\][$s^c$]{}]{} (105.00,105.00)[(0,0)\[rc\][$s$]{}]{} (107.00,135.00)[(0,0)\[rc\][$d^c$]{}]{} (78.00,125.00)[(0,0)\[rc\][$H$]{}]{} (80.00,120.00)[(-1,0)[5]{}]{} (30.00,100.00)[(1,1)[10]{}]{} (30.00,140.00)[(1,-1)[10]{}]{} (100.00,120.00)[(1,1)[10]{}]{} (100.00,120.00)[(1,-1)[10]{}]{}
5.75in =1.00mm
(33.66,182.34) (30.00,120.00)[(1,0)[100.00]{}]{} (60.00,120.00)(5,5)[4]{}[(1,1)[4.00]{}]{} (100.00,120.00)(-5,5)[4]{}[(-1,1)[4.00]{}]{} (40.00,123.00)[(0,0)\[rc\][$s$]{}]{} (120.00,123.00)[(0,0)\[rc\][$d$]{}]{} (74.00,124.00)[(0,0)\[rc\][$\tilde{W}$]{}]{} (90.00,124.00)[(0,0)\[rc\][$\tilde{H}$]{}]{} (85.00,160.00)[(0,0)\[rc\][$g$]{}]{} (72.00,135.00)[(0,0)\[rc\][$<H>$]{}]{} (64.00,128.00)[(0,0)\[rc\][$\tilde{t}$]{}]{} (97.00,134.00)[(0,0)\[rc\][$\tilde{t}^c$]{}]{} (35.00,120.00)[(1,0)[10]{}]{} (105.00,120.00)[(1,0)[10]{}]{} (80.00,120.00)[(-1,0)[10]{}]{} (96.00,120.00)[(-1,0)[10]{}]{} (81.80,120.00)[(0,0)\[rc\][$\times$]{}]{} (75.50,134.00)[(0,0)\[rc\][$+$]{}]{} (86.80,115.00)[(0,0)\[rc\][$<H>$]{}]{} (65,125)[(1,1)[5]{}]{} (95,125)[(-1,1)[5]{}]{}
(80,141.5)[(3,3)\[r\]]{} (80,144.5)[(3,3)\[l\]]{} (80,147.5)[(3,3)\[r\]]{} (80,150.5)[(3,3)\[l\]]{} (80,153.5)[(3,3)\[r\]]{} (80,156.5)[(3,3)\[l\]]{} (80,159.5)[(3,3)\[r\]]{}
5.75in =1.00mm
(33.66,182.34) (30.00,120.00)[(1,0)[100.00]{}]{} (60.00,120.00)(5,5)[4]{}[(1,1)[4.00]{}]{} (100.00,120.00)(-5,5)[4]{}[(-1,1)[4.00]{}]{} (40.00,123.00)[(0,0)\[rc\][$s$]{}]{} (123.00,123.00)[(0,0)\[rc\][$d^c$]{}]{} (74.00,124.00)[(0,0)\[rc\][$\tilde{g}$]{}]{} (90.00,124.00)[(0,0)\[rc\][$\tilde{g}$]{}]{} (85.00,160.00)[(0,0)\[rc\][$g$]{}]{} (72.00,135.00)[(0,0)\[rc\][$<H>$]{}]{} (64.00,128.00)[(0,0)\[rc\][$\tilde{q}$]{}]{} (97.00,133.00)[(0,0)\[rc\][$\tilde{q}^c$]{}]{} (35.00,120.00)[(1,0)[10]{}]{} (125.00,120.00)[(-1,0)[10]{}]{} (77.00,120.00)[(-1,0)[10]{}]{} (83.00,120.00)[(1,0)[10]{}]{} (81.80,120.00)[(0,0)\[rc\][$\times$]{}]{} (75.50,134.00)[(0,0)\[rc\][$+$]{}]{} (80,141.5)[(3,3)\[r\]]{} (80,144.5)[(3,3)\[l\]]{} (80,147.5)[(3,3)\[r\]]{} (80,150.5)[(3,3)\[l\]]{} (80,153.5)[(3,3)\[r\]]{} (80,156.5)[(3,3)\[l\]]{} (80,159.5)[(3,3)\[r\]]{} (65,125)[(1,1)[5]{}]{} (95,125)[(-1,1)[5]{}]{}
5.75in =1.00mm
(33.66,182.34) (30.00,120.00)[(1,0)[100.00]{}]{} (60.00,120.00)(5,5)[4]{}[(1,1)[4.00]{}]{} (100.00,120.00)(-5,5)[4]{}[(-1,1)[4.00]{}]{} (40.00,123.00)[(0,0)\[rc\][$d$]{}]{} (123.00,123.00)[(0,0)\[rc\][$d^c$]{}]{} (74.00,124.00)[(0,0)\[rc\][$\tilde{W}$]{}]{} (90.00,124.00)[(0,0)\[rc\][$\tilde{H}$]{}]{} (85.00,160.00)[(0,0)\[rc\][$\gamma$]{}]{} (86.00,115.00)[(0,0)\[rc\][$<H>$]{}]{} (68.00,135.00)[(0,0)\[rc\][$\tilde{q}$]{}]{} (98.00,135.00)[(0,0)\[rc\][$\tilde{q}$]{}]{} (81.80,120.00)[(0,0)\[rc\][$\times$]{}]{} (80,141.5)[(3,3)\[r\]]{} (80,144.5)[(3,3)\[l\]]{} (80,147.5)[(3,3)\[r\]]{} (80,150.5)[(3,3)\[l\]]{} (80,153.5)[(3,3)\[r\]]{} (80,156.5)[(3,3)\[l\]]{} (80,159.5)[(3,3)\[r\]]{} (65,125)[(1,1)[5]{}]{} (87,133)[(1,-1)[5]{}]{} (35.00,120.00)[(1,0)[10]{}]{} (125.00,120.00)[(-1,0)[10]{}]{} (76.00,120.00)[(-1,0)[10]{}]{} (83.00,120.00)[(1,0)[10]{}]{}
5.75in =1.00mm
(33.66,182.34) (15.00,185.00)[(0,0)\[rc\][(a)]{}]{} (10.00,150.00)[(1,0)[133.00]{}]{} (10.00,150.00)[(3,1)[114.00]{}]{} (70.00,170.00)[(3,-2)[30.00]{}]{} (124.00,188.00)[(1,-2)[19.00]{}]{} (106.00,146.00)(6,-4)[10]{}[(3,-2)[4.00]{}]{} (145.00,146.00)(4,-8)[5]{}[(1,-2)[2.00]{}]{} (138.00,153.00)[(0,0)\[rc\][$\delta_3$]{}]{} (73.00,165.00)[(0,0)\[rc\][$\delta_4+\alpha$]{}]{} (47.00,153.00)[(0,0)\[rc\][$\pi-(\delta_3+\delta_4)$]{}]{} (125.00,181.00)[(0,0)\[rc\][$\delta_4$]{}]{} (155.00,120.00)[(0,0)\[rc\][$\alpha$]{}]{} (87.00,165.00)[(0,0)\[rc\][$1/f$]{}]{} (140.00,170.00)[(0,0)\[rc\][$1/f$]{}]{} (60.00,145.00)[(0,0)\[rc\][$1/y$]{}]{} (90.00,140.00)[(0,0)\[rc\][$1/p$]{}]{} (42.00,164.00)[(0,0)\[rc\][$1/q$]{}]{} (79.00,180.00)[(0,0)\[rc\][$1/c$]{}]{} (10.00,146.00)[(0,0)\[rc\][A]{}]{} (100.00,146.00)[(0,0)\[rc\][D]{}]{} (72.00,173.00)[(0,0)\[rc\][E]{}]{} (150.00,150.00)[(0,0)\[rc\][B]{}]{} (126.00,192.00)[(0,0)\[rc\][C]{}]{} (10.00,30.00)[(1,0)[145.00]{}]{} (90.00,30.00)[(3,4)[48.00]{}]{} (10.00,30.00)[(2,1)[128.00]{}]{} (155.00,30.00)[(-2,1)[72.50]{}]{} (89.00,33.00)[(0,0)\[rc\][$\delta_3$]{}]{} (125.00,80.00)[(0,0)\[rc\][$\delta_4$]{}]{} (110.00,48.00)[(0,0)\[rc\][$\alpha$]{}]{} (45.00,33.00)[(0,0)\[rc\][$\pi-(\delta_3+\delta_4)$]{}]{} (88.00,61.00)[(0,0)\[rc\][$\delta_4+\alpha$]{}]{} (15.00,90.00)[(0,0)\[rc\][(b)]{}]{} (122.00,62.00)[(0,0)\[rc\][$1/f$]{}]{} (130.00,50.00)[(0,0)\[rc\][$1/f$]{}]{} (50.00,26.00)[(0,0)\[rc\][$1/p$]{}]{} (85.00,20.00)[(0,0)\[rc\][$1/y$]{}]{} (80.00,75.00)[(0,0)\[rc\][$1/c$]{}]{} (55.00,55.00)[(0,0)\[rc\][$1/q$]{}]{} (10.00,26.00)[(0,0)\[rc\][A]{}]{} (90.00,26.00)[(0,0)\[rc\][B]{}]{} (143.00,95.00)[(0,0)\[rc\][C]{}]{} (160.00,26.00)[(0,0)\[rc\][D]{}]{} (82.00,69.00)[(0,0)\[rc\][E]{}]{}
[^1]: There should also be charge assignments in the lepton sector, [*e.g.*]{} $Q(E^c) = +1$. We will comment on this possibility when discussing the electron EDM.
[^2]: In the following we choose $m^2_{12}$ positive without loss of generality and thus $\delta_2 \sim \pi + O(\epsilon^2_{PQ})$.
|
---
author:
- Avraham Gal
title: 'Structure and Width of the d$^\ast$(2380) Dibaryon'
---
Walter Greiner: recollections {#sec:intro}
=============================
This contribution is dedicated to the memory of Walter Greiner whose wide-ranging interests included exotic phases of matter. My first physics encounter with Walter was in Fall 1983 in a joint physics symposium hosted by him, see Fig. \[fig:greiner\].
Greiner’s wide-ranging interests included also superheavy elements, so it was quite natural for him to ask my good colleague Eli Friedman, a leading figure in exotic atoms, whether extrapolating pionic atoms to superheavy elements would shed light on a then-speculated pion condensation phase. Subsequently in 1984 Eli spent one month in Frankfurt at Greiner’s invitation, concluding together with Gerhard Soff [@FS85] that the strong-interaction $\pi^-_{1s}$ repulsive energy shift known from light pionic atoms persists also in superheavy elements, as shown in Fig. \[fig:FS85\]-left, thereby ruling out pion condensation for large $Z$. However, quite surprisingly, they also found that 1s & 2p $\pi^-$ atomic states in normal heavy elements up to $Z\approx 100$ have abnormally small widths of less than 1 MeV owing to the repulsive $\pi$-nucleus strong interaction within the nuclear volume. Hence ‘deeply bound’ states (DBS) in pionic atoms are experimentally resolvable, although they cannot be populated radiatively as in light pionic atoms because the absorption width in the higher 3d state exceeds its radiative width by almost two orders of magnitude, as shown in Fig. \[fig:FS85\]-right.
Friedman and Soff’s 1985 prediction of DBS was repeated three years later by Toki and Yamazaki [@TY88], who apparently were not aware of it, and verified experimentally in 1996 at GSI in a (d,$\,^3$He) reaction on $^{208}
$Pb [@Yam96]. Subsequent experiments on Pb and Sn isotopes have yielded accurate data on several other pionic-atom DBS [@yama12], showing clear evidence in support of Weise’s 1990 conjecture of partial chiral symmetry restoration in the nuclear medium due to a renormalized isovector $s$-wave $\pi N$ interaction through the decrease of the pion-decay constant $f_{\pi}
$ [@weise90]. However, the few DBS established so far are still short of providing on their own the precision reached by comprehensive fits to [*all*]{} (of order 100) pionic atom data, dominantly in higher atomic orbits, in substantiating this conjecture; for a recent review on the state of the art in pionic atoms see Ref. [@FG14].
![First prediction of deeply bound pionic atom states [@FS85]. The left panel shows binding energies and widths in 1s deeply bound pionic states [@FS85], and the right panel shows the width saturation in circular states of pionic atoms of Pb [@FG07].[]{data-label="fig:FS85"}](fg07fig4.eps "fig:"){width=".48\linewidth"} ![First prediction of deeply bound pionic atom states [@FS85]. The left panel shows binding energies and widths in 1s deeply bound pionic states [@FS85], and the right panel shows the width saturation in circular states of pionic atoms of Pb [@FG07].[]{data-label="fig:FS85"}](Pbwidths2.eps "fig:"){width=".48\linewidth" height="5.5cm"}
My own encounters with Walter Greiner and several of his colleagues in Frankfurt during several Humboldt-Prize periods in the 1990s focused on developing the concept of Strange Hadronic Matter [@SHM93; @SHM94; @SHM00] and also on studying Kaon Condensation [@KC94]. I recall fondly that period. Here I highlight another exotic phase of matter: non-strange Pion Assisted Dibaryons, reviewed by me recently in Ref. [@gal16].
Pion assisted $N\Delta$ and $\Delta\Delta$ dibaryons
====================================================
The Dyson-Xuong 1964 prediction
-------------------------------
Non-strange $s$-wave dibaryon resonances ${\cal D}_{IS}$ with isospin $I$ and spin $S$ were predicted by Dyson and Xuong in 1964 [@DX64] as early as SU(6) symmetry proved successful, placing the nucleon $N(939)$ and its $P_{33}$ $\pi N$ resonance $\Delta(1232)$ in the same ${\bf 56}$ multiplet. These authors chose the ${\bf 490}$ lowest-dimension SU(6) multiplet in the $\bf{56\times 56}$ direct product containing the flavor-SU(3) $\overline{\bf
10}$ and ${\bf 27}$ multiplets in which the deuteron, ${\cal D}_{01}$, and $NN$ virtual state, ${\cal D}_{10}$, are classified. Four more non-strange dibaryons emerged in this scheme, with masses listed in Table \[tab:dyson\] in terms of constants $A$ and $B$. Identifying $A$ with the $NN$ threshold mass 1878 MeV, the value $B\approx 47$ MeV was derived by assigning ${\cal D}_{12}$ to the $pp\leftrightarrow \pi^+ d$ coupled-channel resonance behavior noted then at 2160 MeV, near the $N\Delta$ threshold (2.171 MeV). This led in particular to a predicted mass $M=2350$ MeV for the $\Delta\Delta$ dibaryon candidate ${\cal D}_{03}$ assigned at present to the recently established d$^\ast$(2380) resonance [@clement17].
${\cal D}_{IS}$ ${\cal D}_{01}$ ${\cal D}_{10}$ ${\cal D}_{12}$ ${\cal D}_{21}$ ${\cal D}_{03}$ ${\cal D}_{30}$
-------------------- -- --------------------- -- ----------------- -- ----------------- -- ----------------- -- --------------------- -- -----------------
$BB'$ $NN$ $NN$ $N\Delta$ $N\Delta$ $\Delta\Delta$ $\Delta\Delta$
SU(3)$_{\rm f}$ $\overline{\bf 10}$ ${\bf 27}$ ${\bf 27}$ ${\bf 35}$ $\overline{\bf 10}$ ${\bf 28}$
$M({\cal D}_{IS})$ $A$ $A$ $A+6B$ $A+6B$ $A+10B$ $A+10B$
: Predicted masses of non-strange $L=0$ dibaryons ${\cal D}_{IS}$ with isospin $I$ and spin $S$, using the Dyson-Xuong SU(6)$\to$SU(4) mass formula $M=A+B[I(I+1)+S(S+1)-2]$ [@DX64].
\[tab:dyson\]
In retrospect, the choice of the ${\bf 490}$ lowest-dimension SU(6) multiplet, with Young tableau denoted \[3,3,0\], is not accidental. This \[3,3,0\] is the one adjoint to \[2,2,2\] for color-SU(3) singlet six-quark (6q) state, thereby ensuring a totally antisymmetric color-flavor-spin-space 6q wavefunction, assuming a totally symmetric $L=0$ orbital component. For non-strange dibaryons, flavor-SU(3) reduces to isospin-SU(2), whence flavor-spin SU(6) reduces to isospin-spin SU(4) in which the \[3,3,0\] representation corresponds to a ${\bf 50}$ dimensional representation consisting of precisely the $I,S$ values of the dibaryon candidates listed in Table \[tab:dyson\], as also noted recently in Ref. [@PPL15]. Since the ${\bf 27}$ and $\overline{
\bf 10}$ flavor-SU(3) multiplets accommodate $NN$ $s$-wave states that are close to binding ($^1S_0$) or weakly bound ($^3S_1$), we focus here on the ${\cal D}_{12}$ and ${\cal D}_{03}$ dibaryon candidates assigned to these flavor-SU(3) multiplets.
Pion assisted dibaryons
-----------------------
The pion plays a major role as a virtual particle in binding or almost binding $NN$ $s$-wave states. The pion as a real particle interacts strongly with nucleons, giving rise to the $\pi N$ $P_{33}$ $p$-wave $\Delta$(1232) resonance. Can it also assist in binding two nucleons into $s$-wave $N\Delta$ states? And once we have such $N\Delta$ states, can the pion assist in binding them into $s$-wave $\Delta\Delta$ states? This is the idea behind the concept developed by Garcilazo and me of pion assisted dibaryons [@GG13; @GG14], or more generally meson assisted dibaryons to go beyond the non-strange sector, see Ref. [@gal16] for review.
As discussed in the next subsection, describing $N\Delta$ systems in terms of a stable nucleon ($N$) and a two-body $\pi N$ resonance ($\Delta$) leads to a well defined $\pi NN$ three-body model in which $IJ=12$ and $21$ resonances identified with the ${\cal D}_{12}$ and ${\cal D}_{21}$ dibaryons of Table \[tab:dyson\] are generated. This relationship between $N\Delta$ and $\pi NN$ may be generalized into relationship between a two-body $B\Delta$ system and a three-body $\pi NB$ system, where the baryon $B$ stands for $N,
\Delta, Y$ (hyperon) etc. In order to stay within a three-body formulation one needs to assume that the baryon $B$ is stable. For $B=N$, this formulation relates the $N\Delta$ system to the three-body $\pi NN$ system. For $B=\Delta$, once properly formulated, it relates the $\Delta\Delta$ system to the three-body $\pi N\Delta$ system, suggesting to seek $\Delta\Delta$ dibaryon resonances by solving $\pi N\Delta$ Faddeev equations, with a stable $\Delta$. The decay width of the $\Delta$ resonance is considered then at the penultimate stage of the calculation. In terms of two-body isobars we have then a coupled-channel problem $B\Delta
\leftrightarrow\pi D$, where $D$ stands generically for appropriate dibaryon isobars: (i) ${\cal D}_{01}$ and ${\cal D}_{10}$, which are the $NN$ isobars identified with the deuteron and virtual state respectively, for $B=N$, and (ii) ${\cal D}_{12}$ and ${\cal D}_{21}$ for $B=\Delta$.
![Diagrammatic representation of the integral equation for the $B\Delta$ $T$ matrix, derived by using separable pairwise interactions in $\pi NB$ Faddeev equations [@GG14] and solved numerically to calculate $B\Delta$ dibaryon resonance poles for $B=N,\,\Delta$.[]{data-label="fig:piNB"}](BDel.eps){width="80.00000%"}
Within this model, and using separable pairwise interactions, the coupled-channel $B\Delta -\pi D$ eigenvalue problem reduces to a single integral equation for the $B\Delta$ $T$ matrix shown diagrammatically in Fig. \[fig:piNB\], where starting with a $B\Delta$ configuration the $\Delta$-resonance isobar decays into $\pi N$, followed by $NB\to NB$ scattering through the $D$-isobar with a spectator pion, and ultimately by means of the inverse decay $\pi N\to\Delta$ back into the $B\Delta$ configuration. We note that the interaction between the $\pi$ meson and $B$ is neglected for $B=\Delta$, for lack of known $\pi\Delta$ isobar resonances in the relevant energy range.
$N\Delta$ dibaryons
-------------------
The ${\cal D}_{12}$ dibaryon of Table \[tab:dyson\] shows up clearly in the Argand diagram of the $NN$ $^1D_2$ partial wave which is coupled above the $NN\pi$ threshold to the $I=1$ $s$-wave $N\Delta$ channel. Values of ${\cal D}_{12}$ and ${\cal D}_{21}$ pole positions $W=M-{\rm i}\Gamma/2$ from our hadronic-model three-body $\pi NN$ Faddeev calculations [@GG13; @GG14] described in the previous subsection are listed in Table \[tab:NDel\] together with results of phenomenological studies that include (i) early $NN$ phase shift analyses [@arndt87] and (ii) $pp \leftrightarrow np\pi^+$ coupled-channels analyses [@hosh92]. The ${\cal D}_{12}$ mass and width values calculated in the Faddeev hadronic model version using $r_{\Delta}
\approx\,1.3$ fm are remarkably close to the phenomenologically derived ones, whereas the mass evaluated in the version using $r_{\Delta}\approx\,0.9$ fm agrees with that assumed in the Dyson-Xuong pioneering discussion [@DX64].
----------------- -- ------------------ -- ------------------ -- ---------------------- -- -------------------------- -- ------------------
$N\Delta$
${\cal D}_{IS}$ Ref. [@arndt87] Ref. [@hosh92] $r_{ $r_{\Delta}\approx\,0.9$ Ref. [@ueda82]
\Delta}\approx\,1.3$
${\cal D}_{12}$ 2148$-{\rm i}$63 2144$-{\rm i}$55 2147$-{\rm i}$60 2159$-{\rm i}$70 2116$-{\rm i}$61
${\cal D}_{21}$ – – 2165$-{\rm i}$64 2169$-{\rm i}$69 –
----------------- -- ------------------ -- ------------------ -- ---------------------- -- -------------------------- -- ------------------
: ${\cal D}_{12}$ and ${\cal D}_{21}$ $N\Delta$ dibaryon $S$-matrix pole positions $W=M-{\rm i}\frac{\Gamma}{2}$ (in MeV), obtained by solving the $N\Delta$ $T$-matrix integral equation of Fig. \[fig:piNB\] [@GG14], are listed for two choices of the $\pi N$ $P_{33}$ form factor specified by a radius parameter $r_{\Delta}$ (in fm) together with two phenomenological values. The last column lists the results of a nonrelativistic meson-exchange Faddeev calculation.
\[tab:NDel\]
Recent $pp\to pp\pi^+\pi^-$ production data [@wasa18] locate the ${\cal D}_{21}$ dibaryon resonance almost degenerate with the ${\cal D}_{12}$. Our $\pi NN$ Faddeev calculations produce it about 10-20 MeV higher than the ${\cal D}_{12}$, see Table \[tab:NDel\]. The widths of these near-threshold $N\Delta$ dibaryons are, naturally, close to that of the $\Delta$ resonance. We note that only $^3S_1$ $NN$ enters the calculation of the ${\cal D}_{12}$ resonance, while for the ${\cal D}_{21}$ resonance calculation only $^1S_0$ $NN$ enters, both with maximal strength. Obviously, with the $^1S_0$ interaction the weaker of the two, one expects indeed that the ${\cal D}_{21}$ resonance lies above the ${\cal D}_{12}$ resonance. Moreover, these two dibaryon resonances differ also in their flavor-SU(3) classification, see Table \[tab:dyson\], which is likely to push up the ${\cal D}_{21}$ further away from the ${\cal D}_{12}$. Finally, the $N\Delta$ $s$-wave states with $IJ=$ $11$ and $22$ are found not to resonate in the $\pi NN$ Faddeev calculations [@GG14].
$\Delta\Delta$ dibaryons
------------------------
![$d^{\ast}$(2380) dibaryon resonance signatures in recent WASA-at-COSY Collaboration experiments. Left: from the peak observed in the $pn\to d\pi^0\pi^0$ reaction [@wasa11]. Right: from the Argand diagram of the $^3D_3$ partial wave in $pn$ scattering [@wasa14].[]{data-label="fig:WASA"}](tot_xs_pn.eps "fig:"){width="48.00000%" height="5cm"} ![$d^{\ast}$(2380) dibaryon resonance signatures in recent WASA-at-COSY Collaboration experiments. Left: from the peak observed in the $pn\to d\pi^0\pi^0$ reaction [@wasa11]. Right: from the Argand diagram of the $^3D_3$ partial wave in $pn$ scattering [@wasa14].[]{data-label="fig:WASA"}](arg_3D3_a.eps "fig:"){width="48.00000%" height="5cm"}
The ${\cal D}_{03}$ dibaryon of Table \[tab:dyson\] shows up in the $^3D_3$ nucleon-nucleon partial wave above the $NN\pi\pi$ threshold owing to the coupling between the $I=0$ $^3D_3$ $NN$ channel and the $I=0$ $^7S_3$ $\Delta
\Delta$ channel, i.e. the coupling between the two-body $NN$ channel and the four-body $NN\pi\pi$ channel. Indeed its best demonstration is by the relatively narrow peak about 80 MeV above the $\pi^0\pi^0$ production threshold and 80 MeV below the $\Delta\Delta$ threshold, with $\Gamma_{d^{
\ast}}\approx 70$ MeV, observed in $pn\to d\pi^0\pi^0$ by the WASA-at-COSY Collaboration [@wasa11] and shown in Fig. \[fig:WASA\]-left. The $I=0$ isospin assignment follows from the isospin balance in $pn \to
d\pi^0\pi^0$, and the $J^P=3^+$ spin-parity assignment follows from the measured deuteron angular distribution. The $d^{\ast}$(2380) was also observed in $pn\to d\pi^+\pi^-$ [@wasa13a], with cross section consistent with that measured in $pn\to d\pi^0\pi^0$, and studied in several $pn\to NN\pi\pi$ reactions [@wasa13b; @wasa13c; @hades15]. Recent measurements of $pn$ scattering and analyzing power [@wasa14] have led to the $pn$ $^3D_3$ partial-wave Argand diagram shown in Fig. \[fig:WASA\]-right, supporting the ${\cal D}_{03}$ dibaryon resonance interpretation.
${\cal D}_{IS}$ ($BB'$) [@DX64] [@MAS80] [@OY80] [@cvetic80] [@MT83] [@Gold89] [@beijing99] [@sal02] exp./phen.
---------------------------------- --------- ---------- --------- ------------- --------- ------------- -------------- ---------- ---------------
${\cal D}_{03}$ ($\Delta\Delta$) 2.35 2.36 2.46 2.42 2.38 $\leq$2.26 2.40 2.46 2.38
${\cal D}_{12}$ ($N\Delta$) 2.16 2.36 – – 2.36 – – 2.17 $\approx$2.15
: ${\cal D}_{03}$ mass (in GeV) predicted in several quark-based calculations prior to 2008. Wherever calculated, the mass of ${\cal D}_{12}$ is also listed.
\[tab:QM\]
The history and state of the art of the ${\cal D}_{03}$ dibaryon, now denoted d$^\ast$(2380), were reviewed recently by Clement [@clement17]. In particular, its mass was predicted in several quark-based calculations, as listed in Table \[tab:QM\] in the columns following the symmetry-based value predicted first by Dyson and Xuong [@DX64]. Also listed are ${\cal D}_{12}$ mass values, wherever available from such calculations. Remarkably, none of these quark-based predictions managed to reproduce the empirical mass values listed in the last column for [*both*]{} ${\cal D}_{12}$ and ${\cal D}_{03}$. More recent quark-based calculations, following the 2008 first announcement of observing the ${\cal D}_{03}$ [@clement08], are discussed below.
Values of ${\cal D}_{03}$ and ${\cal D}_{30}$ pole positions $W=M-{\rm i}
\Gamma/2$ from our hadronic-model three-body $\pi N\Delta$ Faddeev calculations [@GG13; @GG14] are listed in Table \[tab:DelDel\]. The ${\cal D}_{03}$ mass and width values calculated in the Faddeev hadronic model version using $r_{\Delta}\approx\,1.3$ fm are remarkably close to the experimentally reported ones, whereas the mass evaluated in the model version using $r_{\Delta}\approx\,0.9$ fm agrees, perhaps fortuitously so, with that derived in the Dyson-Xuong pioneering discussion [@DX64]. For smaller values of $r_{\Delta}$ one needs to introduce explicit vector-meson and/or quark-gluon degrees of freedom which are outside the scope of the present model. In contrast, the calculated widths $\Gamma$ are determined primarily by the phase space available for decay, displaying little sensitivity to the radius $r_{\Delta}$ of the $\pi N$ $P_{33}$ form factor.
----------------- -- -------------------------- -- -------------------------- -- ------------------ -- ------------------
$\Delta\Delta$
${\cal D}_{IS}$ $r_{\Delta}\approx\,1.3$ $r_{\Delta}\approx\,0.9$ Ref. [@wang14] Ref. [@dong16]
${\cal D}_{03}$ 2383$-{\rm i}$41 2343$-{\rm i}$24 2393$-{\rm i}$75 2380$-{\rm i}$36
${\cal D}_{30}$ 2411$-{\rm i}$41 2370$-{\rm i}$22 2440$-{\rm i} –
$100
----------------- -- -------------------------- -- -------------------------- -- ------------------ -- ------------------
: ${\cal D}_{03}$ and ${\cal D}_{30}$ $\Delta\Delta$ dibaryon $S$-matrix pole position $W=M-{\rm i}\frac{\Gamma}{2}$ (in MeV), obtained in our hadronic model by solving the $\Delta\Delta$ $T$-matrix integral equation of Fig. \[fig:piNB\], are listed for two choices of the $\pi N$ $P_{33}$ form factor specified by a radius parameter $r_{\Delta}$ (in fm). The last two columns list results of post 2008 quark-based RGM calculations with hidden-color $\Delta_8\Delta_8$ components.
\[tab:DelDel\]
The ${\cal D}_{30}$ dibaryon resonance is found in our $\pi N\Delta$ Faddeev calculations to lie about 30 MeV above the ${\cal D}_{03}$. These two states are degenerate in the limit of equal $D={\cal D}_{12}$ and $D={\cal D}_{21}$ isobar propagators in Fig. \[fig:piNB\]. Since ${\cal D}_{12}$ was found to lie lower than ${\cal D}_{21}$, we expect also ${\cal D}_{03}$ to lie lower than ${\cal D}_{30}$ as satisfied in our Faddeev calculations. Moreover, here too the difference in their flavor-SU(3) classification will push the ${\cal D}_{30}$ further apart from the ${\cal D}_{03}$. The ${\cal D}_{30}$ has not been observed and only upper limits for its production in $pp\to
pp\pi^+\pi^+\pi^-\pi^-$ are available [@wasa16].
Finally, we briefly discuss the ${\cal D}_{03}$ mass and width values, listed in the last two columns of Table \[tab:DelDel\], from two recent quark-based resonating-group-method (RGM) calculations [@wang14; @dong16] that add $\Delta_{\bf 8}\Delta_{\bf 8}$ hidden-color (CC) components to a $\Delta_{\bf
1}\Delta_{\bf 1}$ cluster. Interestingly, the authors of Ref. [@wang14] have just questioned the applicability of admixing CC components in dibaryon calculations [@wang17]. The two listed calculations generate mass values that are close to the mass of the d$^{\ast}$(2380). The calculated widths, however, differ a lot from each other: one calculation generates a width of 150 MeV [@wang14], exceeding substantially the reported value $\Gamma_{d^{
\ast}(2380)}$=80$\pm$10 MeV [@wasa14], the other one generates a width of 72 MeV [@dong16], thereby reproducing the d$^{\ast}$(2380) width. While the introduction of CC components has moderate effect on the resulting mass and width in the chiral version of the first calculation, lowering the mass by 20 MeV and the width by 25 MeV, it leads to substantial reduction of the width in the second (also chiral) calculation from 133 MeV to 72 MeV. The reason is that the dominant CC $\Delta_{\bf 8}\Delta_{\bf 8}$ components, with $68\%$ weight [@dong16], cannot decay through single-fermion transitions $\Delta_{\bf 8}\to N_{\bf 1}\pi_{\bf 1}$ to asymptotically free color-singlet hadrons. However, as argued in the next section, these quark-based width calculations miss important kinematical ingredients that make the width of a single compact $\Delta_{\bf 1}\Delta_{\bf 1}$ cluster considerably smaller than $\Gamma_{d^{\ast}(2380)}$. The introduction of substantial $\Delta_{
\bf 8}\Delta_{\bf 8}$ components only aggravates the disagreement.
The width of d$^\ast$(2380), small or large? {#sec:width}
============================================
The width derived for the $d^{\ast}$(2380) dibaryon resonance by the WASA-at-COSY Collaboration and the SAID Data-Analysis-Center is $\Gamma_{d^{
\ast}(2380)}$=80$\pm$10 MeV [@wasa14]. It is much smaller than 230 MeV, twice the width $\Gamma_{\Delta}\approx 115$ MeV [@SP07; @anisovich12] of a single free-space $\Delta$, expected naively for a $\Delta\Delta$ quasibound configuration. However, considering the reduced phase space, $M_{\Delta}=1232
\Rightarrow E_{\Delta}=1232-B_{\Delta\Delta}/2$ MeV in a bound-$\Delta$ decay, where $B_{\Delta\Delta}=2\times 1232-2380=84$ MeV is the $\Delta\Delta$ binding energy, the free-space $\Delta$ width gets reduced to 81 MeV using the in-medium single-$\Delta$ width $\Gamma_{\Delta\to N\pi}$ expression obtained from the empirical $\Delta$-decay momentum dependence $$\Gamma_{\Delta\to N\pi}(q_{\Delta\to N\pi})=\gamma\,
\frac{q^3_{\Delta\to N\pi}}{q_0^2+q^2_{\Delta\to N\pi}},
\label{eq:gamma}$$ with $\gamma=0.74$ and $q_0=159$ MeV [@BCS17]. Yet, this simple estimate is incomplete since neither of the two $\Delta$s is at rest in a deeply bound $\Delta\Delta$ state. To take account of the $\Delta\Delta$ momentum distribution, we evaluate the bound-$\Delta$ decay width ${\overline{\Gamma}}_{\Delta\to N\pi}$ by averaging $\Gamma_{\Delta\to
N\pi}(\sqrt{s_{\Delta}})$ over the $\Delta\Delta$ bound-state momentum-space distribution, $${\overline{\Gamma}}_{\Delta\to N\pi}\equiv\langle \Psi^{\ast}(p_{\Delta\Delta})
|\Gamma_{\Delta\to N\pi}(\sqrt{s_{\Delta}})|\Psi(p_{\Delta\Delta})\rangle
\approx \Gamma_{\Delta\to N\pi}(\sqrt{{\overline{s}}_{\Delta}}),
\label{eq:av}$$ where $\Psi(p_{\Delta\Delta})$ is the $\Delta\Delta$ momentum-space wavefunction and the dependence of $\Gamma_{\Delta\to N\pi}$ on $q_{\Delta\to
N\pi}$ for on-mass-shell nucleons and pions was replaced by dependence on $\sqrt{s_{\Delta}}$. The averaged bound-$\Delta$ invariant energy squared ${\overline{s}}_{\Delta}$ is defined by $${\overline{s}}_{\Delta}=(1232-B_{\Delta\Delta}/2)^2-P_{\Delta\Delta}^2 ,
\label{eq:s}$$ in terms of a $\Delta\Delta$ bound-state r.m.s. momentum $P_{\Delta\Delta}
\equiv{\langle p_{\Delta\Delta}^2\rangle}^{1/2}$.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$R_{\Delta\Delta}$ (fm) $\sqrt{{\overline{s}}_{\Delta}}$ (MeV) ${\overline{q}}_{\Delta\to N\pi}$ (MeV) ${\overline{\Gamma}}_{\Delta\to ${\overline{\Gamma}}_{\Delta\Delta\to NN\pi\pi}$ (MeV)
N\pi}$ (MeV)
--------------------------- -------------------------------------------- ---------------------------------------------- ----------------------------------- ----------------------------------------------------------
0.6 1083 38.3 1.6 2.6
0.7 1112 96.6 19.3 32.1
0.8 1131 122.0 33.5 55.8
1.0 1153 147.7 50.6 84.4
1.5 1174 170.4 67.4 112.3
2.0 1181 177.9 73.2 122.0
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: Values of $\sqrt{{\overline{s}}_{\Delta}}\,$, of the corresponding decay-pion momentum ${\overline{q}}_{\Delta\to N\pi}$ and of ${\overline{
\Gamma}}_{\Delta\to N\pi}$ (\[eq:av\]), listed as a function of $R_{
\Delta\Delta}$ using $P_{\Delta\Delta}R_{\Delta\Delta}=\frac{3}{2}$ in Eq. (\[eq:s\]). The last column lists deduced values of ${\overline{
\Gamma}}_{\Delta\Delta\to NN\pi\pi}$, approximating it by $\frac{5}{3}{
\overline{\Gamma}}_{\Delta\to N\pi}$ (see text).
\[tab:width\]
In Table \[tab:width\], taken from my recent work [@gal17], we list values of $\sqrt{{\overline{s}}_{\Delta}}$ and the associated in-medium decay-pion momentum ${\overline{q}}_{\Delta\to N\pi}$ for several representative values of the r.m.s. radius $R_{\Delta\Delta}\equiv {\langle
r_{\Delta\Delta}^2\rangle}^{1/2}$ of the bound $\Delta\Delta$ wavefunction, using the equality sign in the uncertainty relationship $P_{\Delta\Delta}R_{
\Delta\Delta}\geq 3/2$. Listed also are values of the in-medium single-$\Delta$ width ${\overline{\Gamma}}_{\Delta\to N\pi}$ obtained from Eq. (\[eq:av\]). It is implicitly assumed here that the empirical momentum dependence (\[eq:gamma\]) provides a good approximation also for off-mass-shell $\Delta$s. Finally, The last column of the table lists values of ${\overline{\Gamma}}_{\Delta\Delta\to NN\pi\pi}$ obtained by multiplying ${\overline{\Gamma}}_{\Delta\to N\pi}$ by two, for the two $\Delta$s, while applying to one of them the isospin projection factor 2/3 introduced in the Gal-Garcilazo hadronic model [@GG13; @GG14] to obey the quantum statistics requirements in the leading final $NN\pi\pi$ decay channels. The large spread of ${\overline{\Gamma}}_{\Delta\Delta\to NN\pi\pi}$ width values exhibited in the table, all of which are much smaller than the 162 MeV obtained by ignoring in Eq. (\[eq:s\]) the bound-state momentum distribution, demonstrates the importance of momentum-dependent contributions. It is seen that a compact d$^{\ast}$(2380) with r.m.s. radius $R_{\Delta\Delta}$ less than 0.8 fm is incompatible with the experimental value $\Gamma_{d^{\ast}(2380)}$=80$\pm
$10 MeV from WASA-at-COSY and SAID even upon adding a non-pionic partial width $\Gamma_{\Delta\Delta\to NN}\sim 10$ MeV [@wasa14].
![d$^{\ast}$(2380) $\Delta\Delta$ wavefunction with r.m.s. radius $R_{\Delta\Delta}=0.76$ fm (Left) and deuteron wavefunction with r.m.s. radius $R_d\approx 2$ fm (Right) from recent quark-based RGM calculations [@dong16; @huang15]. Figure adapted from Ref. [@huang15].[]{data-label="fig:wf"}](d-star-ext-1.eps "fig:"){width="48.00000%"} ![d$^{\ast}$(2380) $\Delta\Delta$ wavefunction with r.m.s. radius $R_{\Delta\Delta}=0.76$ fm (Left) and deuteron wavefunction with r.m.s. radius $R_d\approx 2$ fm (Right) from recent quark-based RGM calculations [@dong16; @huang15]. Figure adapted from Ref. [@huang15].[]{data-label="fig:wf"}](Deuteron-ext-1.eps "fig:"){width="48.00000%"}
Fig. \[fig:wf\] shows d$^{\ast}$(2380) and d(1876) wavefunctions from quark-based RGM calculations [@dong16]. The d$^{\ast}$(2380) appears quite squeezed compared to the diffuse deuteron. Its size, $R_{\Delta\Delta}
$=0.76 fm, leads to unacceptably small upper limit of about 47 MeV for the d$^{\ast}$(2380) pionic width. This drastic effect of momentum dependence is missing in quark-based width calculations dealing with pionic decay modes of $\Delta_{\bf 1}\Delta_{\bf 1}$ components, e.g. Ref. [@dong16]. Practitioners of quark-based models ought therefore to ask “what makes $\Gamma_{d^{\ast}(2380)}$ so much larger than the width calculated for a compact $\Delta\Delta$ dibaryon?" rather than “what makes $\Gamma_{d^{\ast}
(2380)}$ so much smaller than twice a free-space $\Delta$ width?"
The preceding discussion of $\Gamma_{d^{\ast}(2380)}$ suggests that the quark-based model’s finding of a tightly bound $\Delta\Delta$ $s$-wave configuration is in conflict with the observed width. Fortunately, our hadronic-model calculations [@GG13; @GG14] offer resolution of this insufficiency by coupling to the tightly bound and compact $\Delta\Delta$ component of the d$^{\ast}$(2380) dibaryon’s wavefunction a $\pi N\Delta$ resonating component dominated asymptotically by a $p$-wave pion attached loosely to the near-threshold $N\Delta$ dibaryon ${\cal D}_{12}$ with size about 1.5–2 fm. Formally, one can recouple spins and isospins in this $\pi{\cal D}_{12}$ system, so as to assume an extended $\Delta\Delta$-like object. This explains why the preceding discussion of $\Gamma_{d^{\ast}\to
NN\pi\pi}$ in terms of a $\Delta\Delta$ constituent model required a size larger than provided by quark-based RGM calculations [@dong16] to reconcile with the reported value of $\Gamma_{d^{\ast}(2380)}$. We recall that the width calculated in our diffuse-structure $\pi N\Delta$ model [@GG13; @GG14], as listed in Table \[tab:DelDel\], is in good agreement with the observed width of the d$^{\ast}$(2380) dibaryon resonance.
![Invariant mass distributions in ELPH experiment [@ELPH17] $\gamma d \to d \pi^0 \pi^0$ at $\sqrt{s}=2.39$ GeV.[]{data-label="fig:ELPH"}](ELPH.eps){width="90.00000%"}
Support for the role of the $\pi{\cal D}_{12}$ configuration in the decay of the d$^{\ast}$(2380) dibaryon resonance is provided by a recent ELPH $\gamma d
\to d \pi^0 \pi^0$ experiment [@ELPH17] looking for the d$^{\ast}$(2380). The cross section data agree with a relativistic Breit-Wigner resonance shape with mass of 2370 MeV and width of 68 MeV, but the statistical significance of the fit is low, particularly since most of the data are from the energy region above the d$^{\ast}$(2380). Invariant mass distributions from this experiment at $\sqrt{s}=2.39$ GeV, recorded in Fig. \[fig:ELPH\], are more illuminating. The $\pi\pi$ mass distribution shown in (a) suggests a two-bump structure, fitted in solid red. The lower bump around 300 MeV is perhaps a manifestation of the ABC effect [@ABC60], already observed in $pn\to d\pi^0\pi^0$ by WASA-at-COSY [@wasa11; @BCS17] and interpreted in Ref. [@gal17] as due to a tightly bound $\Delta\Delta$ decay with reduced $\Delta \to N \pi$ phase space. The upper bump in (a) is consistent then with the d$^{\ast}(2380)\to
\pi {\cal D}_{12}$ decay mode, in agreement with the $\pi d$ mass distribution shown in (b) that peaks slightly below the ${\cal D}_{12}$(2150) mass.
![The $pn\to d\pi^0\pi^0$ WASA-at-COSY $M_{d\pi}$ invariant-mass distribution [@wasa11] and, in solid lines, as calculated [@PK16] for two input parametrizations of ${\cal D}_{12}(2150)$. The dot-dashed line gives the $\pi{\cal D}_{12}(2150)$ contribution to the two-body decay of the d$^{\ast}$(2380) dibaryon, and the dashed line gives a $\sigma$-meson emission contribution. Figure adapted from Ref. [@PK16].[]{data-label="fig:plato"}](platonova.eps){width="60.00000%"}
Theoretical support for the relevance of the ${\cal D}_{12}(2150)$ $N\Delta$ dibaryon to the physics of the d$^{\ast}$(2380) resonance is demonstrated in Fig. \[fig:plato\] [@PK16] by showing a $d\pi$ invariant-mass distribution peaking near the $N\Delta$ threshold as deduced from the $pn\to
d\pi^0\pi^0$ reaction by which the d$^{\ast}$(2380) was found [@wasa11]. However, the peak is shifted to about 20 MeV below the mass of the ${\cal D}_{
12}(2150)$ and the width is smaller by about 40 MeV than the ${\cal D}_{12}
(2150)$ width, agreeing perhaps fortuitously with $\Gamma_{d^{\ast}(2380)}$. Both of these features, the peak downward shift and the smaller width, can be explained by the asymmetry between the two emitted $\pi^0$ mesons, only one of which arises from the $\Delta\to N\pi$ decay within the ${\cal D}_{12}(2150)
$ [@PK16] (I am indebted to Heinz Clement for confirming to me this explanation).
decay channel $d\pi^0\pi^0$ $d\pi^+\pi^-$ $pn\pi^0\pi^0$ $pn\pi^+\pi^-$ $pp\pi^-\pi^0$ $nn\pi^+\pi^0$ $NN\pi$ $NN$ total
--------------- --------------- --------------- ---------------- ---------------- ---------------- ---------------- --------- ---------- -------
BR(th.) 11.2 20.4 11.6 25.8 4.7 4.7 8.3 13.3 100
BR(exp.) 14$\pm$1 23$\pm$2 12$\pm$2 30$\pm$5 6$\pm$1 6$\pm$1 $\leq$9 12$\pm$3 103
: d$^{\ast}$(2380) decay width branching ratios (BR in percents) from Ref. [@gal17] for theory and from Refs. [@BCS15; @wasa17] for experiment.
\[tab:BR\]
Recalling the $\Delta\Delta$ – $\pi{\cal D}_{12}$ coupled channel nature of the d$^{\ast}$(2380) in our hadronic model [@GG13; @GG14], one may describe satisfactorily the d$^{\ast}$(2380) total and partial decay widths in terms of an incoherent mixture of these relatively short-ranged ($\Delta\Delta$) and long-ranged ($\pi{\cal D}_{12}$) channels. This is demonstrated in Table \[tab:BR\] where the $NN\pi\pi$ calculated partial widths, totaling $\approx$60 MeV, are assigned a weight $\frac{5}{7}$ from $\Delta\Delta$ and a weight $\frac{2}{7}$ from $\pi{\cal D}_{12}$. This choice, ensuring that the partial decay width $\Gamma_{d^{\ast}\to NN\pi}$ does not exceed the upper limit of BR$\leq$9% determined recently from [*not*]{} observing the single-pion decay branch [@wasa17], is by no means unique and the weights chosen here may be varied to some extent. For more details, see Ref. [@gal17].
Conclusion {#sec:concl}
==========
Substantiated by systematic production and decay studies in recent WASA-at-COSY experiments [@clement17], the d$^{\ast}$(2380) is the most spectacular dibaryon candidate at present. Following its early prediction in 1964 by Dyson and Xuong [@DX64], it has been assigned in most theoretical works to a $\Delta\Delta$ quasibound state. Given the small width $\Gamma_{d^{
\ast}(2380)}=80\pm 10$ MeV [@wasa14] with respect to twice the width of a free-space $\Delta$, $\Gamma_{\Delta}\approx 115$ MeV, its location far from thresholds makes it easier to discard a possible underlying threshold effect. However, as argued in this review following Ref. [@gal17], the observed small width is much larger than what two [*deeply bound*]{} $\Delta$ baryons can yield upon decay. The d$^{\ast}$(2380) therefore cannot be described in terms of a single compact $\Delta\Delta$ state as quark-based calculations derive it [@wang14; @dong16]. A complementary quasi two-body component is provided within a $\pi N\Delta$ three-body hadronic model [@GG13; @GG14] by the $\pi{\cal D}_{12}$ channel, in which the d$^{\ast}$(2380) resonates. The ${\cal D}_{12}$ dibaryon stands here for the $I(J^P)=1(2^+)$ $N\Delta$ near-threshold system that might possess a quasibound state $S$-matrix pole. It is a loose system of size 1.5–2 fm, as opposed to a compact $\Delta\Delta$ component of size 0.5–1 fm. It was also pointed out here, following Ref. [@gal17], how the ABC low-mass enhancement in the $\pi^0\pi^0$ invariant mass distribution of the $pn\to d\pi^0\pi^0$ fusion reaction at $\sqrt{s}=2.38$ GeV might be associated with a compact $\Delta\Delta$ component. The $\pi{\cal D}_{12}$ channel, in contrast, is responsible to the higher-mass structure of the $\pi^0\pi^0$ distribution and, furthermore, it gives rise to a non-negligible $d^{\ast}\to NN\pi$ single-pion decay branch, considerably higher than that obtained for a quark-based purely $\Delta\Delta$ configuration [@dong17], but consistently with the upper limit of $\leq
$9% determined recently by the WASA-at-COSY Collaboration [@wasa17]. A precise measurement of this decay width and BR will provide a valuable constraint on the $\pi{\cal D}_{12}$–$\Delta\Delta$ mixing parameter.
We end with a brief discussion of possible 6q admixtures in the essentially hadronic wavefunction of the d$^{\ast}$(2380) dibaryon resonance. For this we refer to the recent 6q non-strange dibaryon variational calculation in Ref. [@PPL15] which depending on the assumed confinement potential generates a $^3S_1$ 6q dibaryon about 550 to 700 MeV above the deuteron, and a $^7S_3$ 6q dibaryon about 230 to 350 MeV above the d$^{\ast}$(2380). Taking a typical 20 MeV potential matrix element from deuteron structure calculations and 600 MeV for the energy separation between the deuteron and the $^3S_1$ 6q dibaryon, one finds admixture amplitude of order 0.03 and hence 6q admixture probability of order 0.001 which is compatible with that discussed recently by Miller [@miller14]. Using the same 20 MeV potential matrix element for the $\Delta\Delta$ dibaryon candidate and 300 MeV for the energy separation between the d$^{\ast}$(2380) and the $^7S_3$ 6q dibaryon, one finds twice as large admixture amplitude and hence four times larger 6q admixture probability in the d$^{\ast}$(2380), altogether smaller than 1%. These order-of-magnitude estimates demonstrate that long-range hadronic and short-range quark degrees of freedom hardly mix also for $\Delta\Delta$ configurations, and that the d$^{\ast}$(2380) is extremely far from a pure 6q configuration. This conclusion is at odds with the conjecture made recently by Bashkanov, Brodsky and Clement [@BBC13] that 6q CC components dominate the wavefunctions of the $\Delta\Delta$ dibaryon candidates ${\cal D}_{03}$, identified with the observed d$^{\ast}$(2380), and ${\cal D}_{30}$. Unfortunately, most of the quark-based calculations discussed in the present work combine quark-model input with hadronic-exchange model input in a loose way which discards their predictive power.
I’m indebted to the organizers of the Frontiers of Science symposium in memory of Walter Greiner, held at FIAS, Frankfurt, June 2017, particularly to Horst Stöcker, for inviting me to participate in this special event and for supporting my trip. Special thanks are due to Humberto Garcilazo, together with whom the concept of pion assisted dibaryons was conceived, and also due to Heinz Clement for many stimulating exchanges on the physics of dibaryons and Jerry Miller for instructive discussions on 6q contributions to dibaryons.
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abstract: 'This is the second in our series of papers concerning some reversed Hardy–Littlewood–Sobolev inequalities. In the present work, we establish the following sharp reversed Hardy–Littlewood–Sobolev inequality on the half space $\operatorname{\mathbf{R}}_+^n$ for any nonnegative functions $f\in L^p(\partial \operatorname{\mathbf{R}}_+^n)$, $g\in L^r(\operatorname{\mathbf{R}}_+^n)$, and $p,r\in (0,1)$, $\lambda > 0$ such that $(1-1/n)1/p + 1/r -(\lambda-1) /n =2$. Some estimates for ${\mathscr C}_{n,p,r}$ as well as the existence of extrema functions for this inequality are also considered. New ideas are also introduced in this paper.'
address:
- |
Department of Mathematics\
College of Science, Viêt Nam National University\
Hà Nôi, Viêt Nam.
- |
School of Mathematical Sciences\
Tel Aviv University\
Tel Aviv 69978, Israel.
- |
Institut de Mathématiques de Toulouse\
Université Paul Sabatier\
31062 Toulouse cédex 09, France.
author:
- Quôc Anh Ngô
- Van Hoang Nguyen
date: ' **at** '
title: 'Sharp reversed Hardy–Littlewood–Sobolev inequality on the half space $\operatorname{\mathbf{R}}_+^n$'
---
Introduction
============
This is the second in our series of papers concerning some reversed Hardy–Littlewood–Sobolev (HLS) inequalities which tell us how to bound $\int \int f(x) |x-y|^\lambda g(y) dxdy$ in terms of $\|f\|_p \|g\|_r$ for suitable numbers $p$ and $r$. In the literature, the classical HLS inequality named after Hardy–Littlewood [@hl1928; @hl1930] and Sobolev [@sobolev1938] concerns the following estimate $$\label{eq:HLSineq}
\int_{\operatorname{\mathbf{R}}^n} \int_{\operatorname{\mathbf{R}}^n} \frac{f(x) g(y)}{|x-y|^{-\lambda}} dx dy \leqslant {\mathscr N}_{n,\lambda,p} \Big( \int_{\operatorname{\mathbf{R}}^n} |f|^p dx \Big)^{1/p} \Big( \int_{\operatorname{\mathbf{R}}^n} |g|^r dx \Big)^{1/r}$$ for some constant ${\mathscr N}_{n,\lambda,p} > 0$ and for all $f \in L^p(\operatorname{\mathbf{R}}^n)$ and $g \in L^r(\operatorname{\mathbf{R}}^n)$ where the positive constants $p$ and $r$ are related via the following $1/p+1/r-\lambda/n=2$ for some $\lambda <0$ is crucial. As the impact of the HLS inequality on quantitative theories of solutions of (partial) differential equations is now clear, the HLS inequality as well as its variants have captured much attention by many mathematicians.
In one way or another, the HLS inequality and other famous inequalities are related to each other; see [@beckner1993]. To see this more precise, it is quite a surprise to remark that the HLS inequality and the Sobolev inequality are indeed dual for certain families of exponents. First, we let $\lambda = n-2s$ in and rewrite the right hand side of with the fact that ${2^{ - 2s}}{\pi ^{ - n/2}} \Gamma (n/2-s)/\Gamma (s)$ is simply the Green function of the operator $(-\Delta)^s$ in $\operatorname{\mathbf{R}}^n$ for each $s \in (0,n/2)$ to get $$\label{eqHLS->Sobolev}
\int_{\operatorname{\mathbf{R}}^n} f (-\Delta)^{-s}(f) dx \leqslant {\mathscr S}_{n,s} \Big( \int_{\operatorname{\mathbf{R}}^n} |f|^{2n/(n+2s)} dx \Big)^{1+2s/n}.$$ Hence, the sharp HLS inequality can imply the sharp Sobolev inequality. Further seminal works also reveal that the sharp HLS inequality can also imply the Moser–Trudinger–Onofri inequality and the logarithmic HLS inequality [@beckner1993], as well as the Gross logarithmic Sobolev inequality [@gross1975]. Clearly, all these inequalities have many important applications in analysis and geometry, as well as in quantum field theory.
Although the HLS inequality (not in the sharp form) was proved earlier, it took quite a long time to find the sharp version with the precise sharp constant of ; see [@l1983]. Although Lieb was able to prove the existence of optimizers for for any $p$ and $r$, neither the sharp constant nor the precise form of optimizers are known except in the diagonal case $p=r$. In this special case, $p=r=2n/(2n+\lambda)$, the sharp constant ${\mathscr N}_{n,\lambda,p}$ is $${\mathscr N}_{n,\lambda,p} ={\mathscr N}_{n,\lambda} = {\pi ^{\lambda /2}}\frac{{\Gamma (n/2 - \lambda /2)}}{{\Gamma (n - \lambda /2)}}{\left( {\frac{{\Gamma (n)}}{{\Gamma (n/2)}}} \right)^{1-\lambda /n}} .$$ As we have already mentioned before, in the last two decades, the classical HLS inequality has captured much attention by many mathematicians. Some remarkable extensions and generalizations have already been drawn, for example, one has HLS inequalities on Heisenberg groups, on compact Riemannian manifolds, and on weighted forms; see [@fl2012a; @hanzhu; @st1958] for details.
Among extensions and generalizations in the literature, let us mention the following two results. The first result concerns a so-called reversed HLS inequality which recently proved by Dou and Zhu in [@dz2014] for the case of the whole space $\operatorname{\mathbf{R}}^n$. By using an extension of the classical Marcinkiewicz interpolation theorem applying to certain singular integral operators, Dou and Zhu established a reversed HLS inequality in the whole space $\operatorname{\mathbf{R}}^n$ which turns out to be useful when studying some curvature equations with negative critical Sobolev exponents; see [@zhu2014]. Their result can be stated as follows.
\[thmDouZhuReversed\] Let $p,r\in (0,1)$ and $\lambda > 0$ such that $1/p + 1/r -\lambda /n =2$. Then there exists a best constant ${\mathscr C}_{n,p,r}>0$ such that for any nonnegative functions $f\in L^p(\operatorname{\mathbf{R}}^n)$ and $g\in L^r(\operatorname{\mathbf{R}}^n)$, we have $$\label{eq:RHLS}
\int_{\operatorname{\mathbf{R}}^n} \int_{\operatorname{\mathbf{R}}^n} f(x) |x-y|^\lambda g(y) dx dy \geqslant {\mathscr C}_{n,p,r} \Big( \int_{\operatorname{\mathbf{R}}^n} |f|^p dx \Big)^{1/p} \Big( \int_{\operatorname{\mathbf{R}}^n} |g|^r dx \Big)^{1/r} .$$
In the first paper of our series [@NgoNguyen2015], we provided an alternative way to reprove Theorem \[thmDouZhuReversed\] which is simpler and more direct that the method used in [@dz2014]. We also calculated the sharp constant ${\mathscr C}_{n,\lambda}$ in the diagonal case $p=r=2n/(2n+\lambda)$. It it quite interesting to note that the sharp constant ${\mathscr C}_{n,\lambda}$ also takes the same form as ${\mathscr N}_{n,\lambda}$. (However, the value of ${\mathscr C}_{n,\lambda}$ and ${\mathscr N}_{n,\lambda}$ are different since we require $\lambda<0$ in ${\mathscr N}_{n,\lambda}$ and $\lambda >0$ in ${\mathscr C}_{n,\lambda}$.)
The second result that we wish to address is a so-called HLS inequality on the upper half space $\operatorname{\mathbf{R}}_+^n$ which can be seen as an extension of which was established by Dou and Zhu in [@dz2013].
In order to state this result, some notation and conventions are needed. First, given $n \geqslant 3$, by $\operatorname{\mathbf{R}}_+^n$, we mean the Euclidean half space given by $$\operatorname{\mathbf{R}}_+^n = \{y = (y',y_n) \in \operatorname{\mathbf{R}}^n : y' \in \operatorname{\mathbf{R}}^{n-1}, y_n > 0\}$$ and by $\partial \operatorname{\mathbf{R}}_+^n$, we mean the boundary of $\operatorname{\mathbf{R}}_+^n$; hence we can identify $\partial \operatorname{\mathbf{R}}_+^n = \operatorname{\mathbf{R}}^{n-1}$. Upon using the above notations and for the sake of simplicity, for each $y \in \operatorname{\mathbf{R}}_+^n$ and $x \in \partial \operatorname{\mathbf{R}}_+^n$, we can write $$|x-y| = \sqrt{|y'-x|^2 + y_n^2}$$ with, of course, $y=(y',y_n)$. Conventionally and for the sake of clarity, throughout the present work, by $x$ we usually mean a point in $\partial \operatorname{\mathbf{R}}_+^n$ while by $y$ we mean a point in $\operatorname{\mathbf{R}}_+^n$.
We are now in a position to state the HLS inequality on the upper half space $\operatorname{\mathbf{R}}_+^n$ in [@dz2013].
\[thmDouZhu\] For any $n\geqslant 2$, $\lambda \in (1-n, 0)$ and $p,r>1$ satisfying $(n-1)/(np) +1/r - \lambda /n =2-1/n$, there exists a best constant ${\mathscr N}_{n,\alpha,p}^+ > 0$ depending only on $n,\alpha$ and $p$ such that for any nonnegative functions $f\in L^p(\partial \operatorname{\mathbf{R}}_+^n)$ and $g\in L^r(\operatorname{\mathbf{R}}_+^n)$, it holds $$\label{eq:DZreverseHLS}
\int_{\partial \operatorname{\mathbf{R}}_+^n} \int_{\operatorname{\mathbf{R}}_+^n} \frac{f(x) g(y)}{|x-y|^{-\lambda}} dy dx \leqslant {\mathscr N}_{n,\alpha,p}^+ \Big( \int_{\partial\operatorname{\mathbf{R}}_+^n} |f|^p dx \Big)^{1/p} \Big( \int_{\operatorname{\mathbf{R}}_+^n} |g|^r dy \Big)^{1/r} .$$
Motivated by Theorems \[thmDouZhuReversed\] and \[thmDouZhu\], the aim of the present paper is to propose a reversed version of the classical HLS inequality on the upper half space $\operatorname{\mathbf{R}}_+^n$. The following theorem is our main result of the paper.
\[thmMAIN\] For any $n\geqslant 2$, $\lambda > 0$ and $p,r\in (0,1)$ satisfying $$\label{eq:condHLS}
\frac{n-1}n \frac1p + \frac1r -\frac{\lambda}n =2-\frac 1n,$$ there exists a best constant ${\mathscr C}_{n,\alpha,p}^+ > 0$ depending only on $n,\alpha$ and $p$ such that for any nonnegative functions $f\in L^p(\partial \operatorname{\mathbf{R}}_+^n)$ and $g\in L^r(\operatorname{\mathbf{R}}_+^n)$, there holds $$\label{eq:reverseHLS}
\int_{\operatorname{\mathbf{R}}_+^n} \int_{\partial \operatorname{\mathbf{R}}_+^n} f(x) |x-y|^\lambda g(y) dx dy \geqslant {\mathscr C}_{n,\alpha,p}^+ \Big( \int_{\partial\operatorname{\mathbf{R}}_+^n} |f|^p dx \Big)^{1/p} \Big( \int_{\operatorname{\mathbf{R}}_+^n} |g|^r dy \Big)^{1/r}.$$
To prove Theorem \[thmMAIN\], we adopt the standard approach, based on the layer cake representation, for the classical HLS inequality on$\operatorname{\mathbf{R}}^n$; see [@liebloss2001 Section 4.3]. Once we can establish Theorem \[thmMAIN\], it is natural to ask whether the extremal functions for the reversed HLS inequality actually exist. To this purpose, inspired by [@dz2013], let us first introduce an “Laplacian-type extension" operator $E_\lambda$ for any function $f$ on $\partial \operatorname{\mathbf{R}}_+^n$ to a function on $\operatorname{\mathbf{R}}_+^n$ as follows $$(E_\lambda f)(y) = \int_{\partial \operatorname{\mathbf{R}}_+^n} |x-y|^\lambda f(x) dx$$ for $y\in \operatorname{\mathbf{R}}_+^n$. (Note that $E_\lambda f$ does not agree with $f$ on $\partial \operatorname{\mathbf{R}}_+^n$ since $(E_\lambda f)(y',0) = \int_{\partial \operatorname{\mathbf{R}}_+^n} |x-y'|^\lambda f(x) dx$.) Then the reversed HLS inequality is equivalent to the following inequality $$\label{eq:anotherform}
\Big( \int_{ \operatorname{\mathbf{R}}_+^n} |E_\lambda f|^q dy \Big)^{1/q} \geqslant {\mathscr C}_{n,\alpha,p}^+ \Big( \int_{\partial \operatorname{\mathbf{R}}_+^n} |f|^p dx \Big)^{1/p}$$ for any non-negative function $f\in L^p(\partial \operatorname{\mathbf{R}}_+^n)$ with $q <0$ satisfies $$\label{eq:cond1}
\frac1q = \frac{n-1}n\, \left(\frac1p -1\right ) - \frac\lambda n$$ which can be computed in terms of $r$ as follows: $1/q=1-1/r$. As a convention, for any $\vartheta<1$ any any function $\phi : \Omega \to \operatorname{\mathbf{R}}$, by the notation $\phi \in L^\vartheta (\Omega)$, we mean $\int_\Omega |\phi|^\vartheta < +\infty$ although this integral is no longer a norm for $L^\vartheta (\Omega)$ with $\vartheta<1$ since the triangle inequality fails to hold. Being a linear topological space, it is well-known that $L^\vartheta (\Omega)$ with $\vartheta \in (0,1)$ has trivial dual.
Similar, one can consider the “restriction" operator $R_\lambda $ which maps any function $g$ on $\operatorname{\mathbf{R}}_+^n$ to a function on $\partial \operatorname{\mathbf{R}}_+^n$ as the following $$(R_\lambda g)(x) = \int_{\operatorname{\mathbf{R}}_+^n} |y-x|^\lambda g(y) dy$$ for $x\in \partial \operatorname{\mathbf{R}}_+^n$. Note that the operators $I_\lambda$ and $R_\lambda$ are dual in the sense that for any functions $f$ on $\partial \operatorname{\mathbf{R}}_+^n$ and $g$ on $\operatorname{\mathbf{R}}_+^n$, the following identity $$\int_{\operatorname{\mathbf{R}}_+^n} (I_\lambda f)(y) g(y) dy = \int_{\partial \operatorname{\mathbf{R}}_+^n} f(x) (R_\lambda g)(x) dx$$ holds, thanks to the Tonelli theorem. Once we introduce $R_\lambda $, we can easily see that the reversed HLS inequality is equivalent to the following inequality $$\label{eq:anotherform1}
\Big( \int_{\partial \operatorname{\mathbf{R}}_+^n} |R_\lambda g|^q dx \Big)^{1/q} \geqslant {\mathscr C}_{n,\alpha,p}^+ \Big( \int_{ \operatorname{\mathbf{R}}_+^n} |g|^r dx \Big)^{1/r}$$ for any non-negative function $g\in L^r( \operatorname{\mathbf{R}}_+^n)$ with $q<0$ satisfies $$\label{eq:cond2}
\frac{1}{q} = \frac{n}{{n - 1}}\left( {\frac{1}{r} - 1} \right) - \frac{\lambda }{n - 1}$$ which can be computed in terms of $p$ as follows: $1/q=1-1/p$.
In view of , to study the existence of extremal functions for , it is equivalent to studying the following minimizing problem $$\label{eq:variationalprob}
{\mathscr C}_{n,\alpha,p}^+ :=\inf_f \Big\{ \|E_\lambda f\|_{L^q(\operatorname{\mathbf{R}}_+^n)} : f\geqslant 0, \|f\|_{L^p(\partial \operatorname{\mathbf{R}}_+^n)} =1\Big\}.$$ It is elementary to verify that extremal functions for are those solving the minimizing problem . In the following result, we prove that such a extremal function for indeed exists.
\[Existence\] There exists some function $f\in L^p(\partial \operatorname{\mathbf{R}}_+^n)$ such that $f \geqslant 0$, $\|f\|_{L^p(\partial \operatorname{\mathbf{R}}_+^n)} =1$ and $\|E_\lambda f\|_{L^q(\operatorname{\mathbf{R}}_+^n)} = {\mathscr C}_{n,\alpha,p}^+$. Moreover, if $f$ is a minimizer of then there exist a non-negative, strictly decreasing function $h$ on $[0, +\infty)$ and some $x_0\in \partial \operatorname{\mathbf{R}}_+^n$ such that $f(x) = h(|x+x_0|)$ a.e. $x\in \partial \operatorname{\mathbf{R}}_+^n$.
To prove the existence of extremal functions for , we borrow Talenti’s proof of the sharp Sobolev inequality by considering within the set of symmetric decreasing rearrangements. In view of , if we denote by $f^\star$ the symmetric decreasing rearrangement of some function $f \in L^p(\partial \operatorname{\mathbf{R}}_+^n)$, then it is easy to see that $\|f^\star\|_{L^p(\partial \operatorname{\mathbf{R}}_+^n)} = \|f\|_{L^p(\partial \operatorname{\mathbf{R}}_+^n)} $. Therefore, it suffices to compare $\|E_\lambda f\|_{L^q(\operatorname{\mathbf{R}}_+^n)}$ and $\|E_\lambda f^\star\|_{L^q(\operatorname{\mathbf{R}}_+^n)} $. The key ingredient in our analysis is to characterize $E_\lambda f^\star$ by showing that it depends on two parameters: one is the distance from the boundary and the other is some radial variable on this boundary; see Lemma \[lemmaexistence\]. Using the characterization of $E_\lambda f^\star$, we successfully obtain the existence result.
We now turn our attention to the extremal function found in Theorem \[Existence\] above. In order to discuss further, let us first denote the following functional $$F_\lambda (f,g) = \int_{\operatorname{\mathbf{R}}_+^n} \int_{\partial\operatorname{\mathbf{R}}_+^n} f(x) |x-y|^\lambda g(y) dx dy$$ for any nonnegative functions $f\in L^p(\partial \operatorname{\mathbf{R}}_+^n)$ and $g\in L^r(\operatorname{\mathbf{R}}_+^n)$. Then, in order to study the existence of extremal functions, it is necessary to minimize the functional $F_\lambda$ along with the following two constraints $\int_{\partial\operatorname{\mathbf{R}}_+^n} |f(x)|^p dx =1$ and $\int_{\operatorname{\mathbf{R}}_+^n} |g(y)|^r dy =1$. Upon a simple calculation, with respect to the function $f$, the first variation of the functional $F_\lambda$ is nothing but $$D_f (F_\lambda) (f,g) (h) = \int_{\partial\operatorname{\mathbf{R}}_+^n} \Big( \int_{\operatorname{\mathbf{R}}_+^n} |x-y|^\lambda g(y) dy \Big) h(x) dx$$ while the first variation of the constraint $\int_{\partial\operatorname{\mathbf{R}}_+^n} |f(x)|^p dx =1$ is as follows $$p \int_{\partial\operatorname{\mathbf{R}}_+^n} |f(x)|^{p-2} f(x) h(x) dx.$$ Therefore, by the Lagrange multiplier theorem, there exists some constant $\alpha$ such that $$\int_{\partial\operatorname{\mathbf{R}}_+^n} \Big( \int_{\operatorname{\mathbf{R}}_+^n} |x-y|^\lambda g(y) dy \Big) h(x) dx = \alpha \int_{\partial\operatorname{\mathbf{R}}_+^n} |f(x)|^{p-2} f(x) h(x) dx$$ holds for all $h$ defined in $\partial \operatorname{\mathbf{R}}_+^n$. From this we know that $f$ and $g$ must satisfy the following equation $$\alpha |f(x)|^{p-2} f(x) = \int_{\operatorname{\mathbf{R}}_+^n} |x-y|^\lambda g(y) dy.$$ Interchanging the role of $f$ and $g$, we also know that $f$ and $g$ must fulfill the following $$\beta |g(y)|^{r-2} g(y) = \int_{\partial \operatorname{\mathbf{R}}_+^n} |x-y|^\lambda f(x) dx$$ for some new constant $\beta$. The balance condition guarantees that $\alpha = \beta = 1/F_\lambda (f,g)$. Hence, up to a constant multiple and simply using the following changes $u =f^{p-1}$ and $v = g^{r-1}$, the two relations above lead us to studying the following integral system $$\label{eq:systemDouZhu}
\left\{
\begin{split}
u(x) &= \int_{\operatorname{\mathbf{R}}_+^n} |x-y|^\lambda v(y)^{1/(r-1)} dy,\\
v(y) &= \int_{\partial \operatorname{\mathbf{R}}_+^n} |x-y|^\lambda u(x)^{1/(p-1)} dx.
\end{split}
\right.$$ Note that the exponents $1/(r-1)$ and $1/(p-1)$ in are all negative. Concerning to the integral system , in the case when $1-n < \lambda <0$ and when $r=2n/(2n+\lambda)$ and $p=2(n-1)/(2n+\lambda-2)$, all non-negative integrable solutions of was already classified in [@dz2013 Section 3] using an integral form of the well-known method of moving spheres. In the literature, the method of moving spheres was first introduced by Li and Zhu in [@lz1995], see also [@l2004; @xu2005], which is a variant of the well-known method of moving planes introduced by Aleksandrov [@a1958], see also [@s1971; @gnn1979; @cgs1989; @cl1991; @clo2005; @clo2006].
The main result in [@dz2013 Section 3] is to show that, up to translations and dilations, any non-negative, measurable solution $(u,v)$ of must be the following form $$\left\{
\begin{split}
u(x) =& a_1{\left( {|x - \overline x{|^2} + b^2} \right)^{(\alpha - n)/2}}, \\
v(x,0) = &a_2 {\left( {|x - \overline x{|^2} + b^2} \right)^{(\alpha - n)/2}},
\end{split}
\right.$$ where $a_1, a_2, b>0$ and $x, \overline x \in \partial \operatorname{\mathbf{R}}_+^n$. Motivated by the above classification by Dou and Zhu, in the last part of our present paper, we also classify solutions of integral systems of the form where $\lambda>0$. To be precise, we are interested in classification of nonnegative, measurable functions of the following general system $$\label{eqIntegralSystem}
\left\{
\begin{split}
u(x) &= \int_{\operatorname{\mathbf{R}}_+^n} |x-y|^\lambda v(y)^{-\kappa} dy,\\
v(y) &= \int_{\partial\operatorname{\mathbf{R}}_+^n} |x-y|^\lambda u(x)^{-\theta} dx,
\end{split}
\right.$$ with $\lambda, \kappa, \theta>0$. To achieve that goal, we first establish the following necessary condition.
\[lemNECESSARY\] For $n \geqslant 3$, $\lambda>0$, $ \kappa>0$ and $\theta>0$, then a necessary condition for $\kappa$ and $\theta$ in order for to admit a $C^1$ solution $(u,v)$ defined in $\partial\operatorname{\mathbf{R}}_+^n \times \operatorname{\mathbf{R}}_+^n$ is $$\label{eq:NecessaryCond}
\frac{{n - 1}}{n}\frac{1}{{\theta - 1}} + \frac{1}{{\kappa - 1}} = \frac{\lambda }{n}.$$
The condition usually refers to the critical condition for . Then, we provide the following classification result for solutions of in the case $\kappa = \theta + 2/\lambda$.
\[thmCLASSIFICATION\] Given $n \geqslant 3$, suppose that $\lambda>0$, $\kappa>0$ and $\theta>0$ satisfy $\kappa = \theta + 2/\lambda$. Let $(u,v)$ be a pair of nonnegative Lebesgue measurable functions defined in $\partial\operatorname{\mathbf{R}}_+^n \times \operatorname{\mathbf{R}}_+^n$ satisfying . Then $\kappa = 1 + 2n/\lambda$ and hence $\theta = 1 + (2n - 2)/\lambda$ and for some constants $a, b>0$ and some point $\overline x \in \partial\operatorname{\mathbf{R}}_+^n$, $u$ and $v$ take the following form $$u(x)=v(x,0) = a( |x-\overline x|^2 + b^2)^{\lambda/2}$$ for $x \in \partial \operatorname{\mathbf{R}}_+^n$.
To prove Proposition \[thmCLASSIFICATION\], we make use of the method of moving spheres introduced in [@lz1995]; see also [@lz2003]. For the classical HLS inequality, it is deserved to note that Frank and Lieb [@fl2010] successfully used reflection positivity via inversions in spheres to replace the moving spheres argument. It is likelihood that the inversion positivity could be used in this new scenario and we hope we could treat this issue elsewhere.
As an immediate consequence of Proposition \[thmCLASSIFICATION\], we can explicitly compute the sharp constant in the reversed HLS inequality for the special case when $\lambda =2$. Note that in this special case, there hold $p = (n-1)/n$ and $r = n/(n+1)$.
\[explicit\] Let $n\geqslant 2$, then $$\label{eq:explicit}
{\mathscr C}_{n, 1-1/n, n/(n+1)}^+ =\frac{2^{-1+1/n}}\pi \left(\frac{\Gamma(n)}{\Gamma(n/2)}\right)^{1/n} \left(\frac{\Gamma(n-1)}{\Gamma((n-1)/2)}\right)^{1/(n-1)}.$$
Again, we note that Corollary \[explicit\] only applies to the case $\lambda = 2$. For $0 < \lambda \ne 2$, it is not easy to obtain a precise value for the sharp constant ${\mathscr C}_{n,\alpha,p}^+$. This is because we do not know much information of $v$ out of the boundary $\partial \operatorname{\mathbf{R}}_+^n$. For general case, it has just come to our attention that the usage of the Gegenbauer polynomials could be useful and we will address this issue in future work; see [@fl2010; @fl2011; @fl2012a].
Finally, we study the limiting case of when $\lambda =0$ which will be called the log-HLS inequality on half space; see [@CarlenLoss92].
\[log-HLSonhalfspace\] Let $n\geqslant 2$. There exists a constant $C_n$ such that for any positive functions $f\in L^1(\partial\operatorname{\mathbf{R}}_+^n)$ and $g\in L^1(\operatorname{\mathbf{R}}_+^n)$ such that $\int_{\partial\operatorname{\mathbf{R}}_+^n} f(x) dx = \int_{\operatorname{\mathbf{R}}_+^n} g(y) dy =1$, and $$\int_{\partial\operatorname{\mathbf{R}}_+^n} f(x)\ln(1 +|x|^2) dx < +\infty,\quad\text{and}\quad \int_{\operatorname{\mathbf{R}}_+^n} g(y) \ln(1 + |y|^2) dy < +\infty,$$ then there holds $$\begin{aligned}
\label{eq:log-HLS}
-\int_{\operatorname{\mathbf{R}}_+^n}& \int_{\partial\operatorname{\mathbf{R}}_+^n} f(x) \ln(|x-y|) g(y) dx dy\notag\\
&\leqslant \frac1{2(n-1)}\int_{\partial\operatorname{\mathbf{R}}_+^n} f(x) \ln f(x) dx + \frac1{2n}\int_{\operatorname{\mathbf{R}}_+^n} g(y) \ln g(y) dy - C_n.\end{aligned}$$ The constant $C_n$ is given by $$\begin{aligned}
\label{bestconstantlogHLS}
C_n = &-\frac1{2(n-1)} \ln |\mathbb S^{n-1}| + \frac1{2(n-1)|\mathbb S^{n-1}|}\int_{\partial\operatorname{\mathbf{R}}_+^n}f_0(x) \ln f_0(x) dx\notag\\
& - \frac1{2n} \ln\Big(\int_{\operatorname{\mathbf{R}}_+^n} \exp \Big(-\frac{2n}{|\mathbb S^{n-1}|}\int_{\partial\operatorname{\mathbf{R}}_+^n} \ln(|x-y|) f_0(x)dx \Big) dy \Big),\end{aligned}$$ with $$f_0(x) = \left(\frac2{1+|x|^2}\right)^{-n+1},\quad x \in \partial \operatorname{\mathbf{R}}_+^n.$$ Moreover, the inequality is sharp, and equality occurs if $f = f_0/|\mathbb S^{n-1}|$ and $$g(x) = c_n\exp\Big(-\frac{2n}{|\mathbb S^{n-1}|} \int_{\partial\operatorname{\mathbf{R}}_+^n} f_0(x) \ln |x-y| dx\Big),$$ where the constant $c_n$ is chosen in such a way that $\int_{\operatorname{\mathbf{R}}^n_+} g(y) dy =1$.
In the forthcoming article [@NgoNguyen2015Heisenberg], we shall study the reversed HLS inequality on the Heisenberg group $\mathbb H^n$.
Proof of reversed HLS inequality in $\operatorname{\mathbf{R}}_+^n$: Proof of Theorem \[thmMAIN\]
=================================================================================================
In this section, we prove the reversed HLS inequality . By homogeneity, we can normalize $f$ and $g$ in such a way that $\|f\|_{L^p(\partial \operatorname{\mathbf{R}}_+^n)} = \|g\|_{L^r(\operatorname{\mathbf{R}}_+^n)} =1$. For each point $y\in \operatorname{\mathbf{R}}^n$, let us denote $$B_c(x) = \{z\in \operatorname{\mathbf{R}}^n\, :\, |z-y| \leqslant c\}.$$ In the special case when $y = 0$, we simply denote $B_c(0)$ by $B_c$; hence $B_c = \{y\in \operatorname{\mathbf{R}}^n\, :\, |y| \leqslant c\}$. For $a,b> 0$, we also denote $$u(a) = \mathscr L^{n-1}(\{x \in \partial \operatorname{\mathbf{R}}_+^n \, :\, f(x) > a\})$$ and $$v(b) =\mathscr L^n(\{y \in \operatorname{\mathbf{R}}_+^n \,:\, g(y) > b\}),$$ where $\mathscr L^k$ stands for the $k$-dimensional Lebesgue measure with positive integers $k$. Then by the layer cake representation [@liebloss2001 Theorem 1.13] and our normalization it follows that $$p\int_0^\infty u(a) a^{p-1} da = \|f\|_{L^p(\partial \operatorname{\mathbf{R}}_+^n)}^p =1$$ and $$r\int_0^\infty v(b) b^{r-1} db = \|g\|_{L^r(\operatorname{\mathbf{R}}_+^n)}^r =1.$$ Next we denote $\lambda =\alpha -n >0$ and $$I(f,g) = \int_{\operatorname{\mathbf{R}}_+^n} \int_{\partial \operatorname{\mathbf{R}}_+^n} f(x) |x-y|^{\alpha-n} g(y) dx dy .$$ As the first step, we establish a rough form for by showing that there is some constant $C>0$ depending only on $n,p, \lambda$ such that $I(f,g) \geqslant C$. To this purpose, by applying the layer cake representation again, we obtain $$f(x) = \int_0^\infty \chi_{\{f> a\}}(x) da,\quad g(y) = \int_0^\infty \chi_{\{g> b\}}(y) db,$$ and $$|x-y|^{\lambda} = \lambda \int_0^\infty c^{\lambda-1} \chi_{\R^n\setminus B_c}(x-y) dc.$$ From this and the Fubini theorem, it follows that $$\label{eq:layercake}
I(f,g) = \lambda \int_0^\infty\int_0^\infty\int_0^\infty c^{\lambda-1} I(a,b,c) da db dc,$$ where $$I(a,b,c) = \int_{\operatorname{\mathbf{R}}_+^n} \int_{\partial \operatorname{\mathbf{R}}_+^n} \chi_{\{f > a\}}(x) \chi_{\R^n\setminus B_c}(x-y) \chi_{\{g> b\}}(y) dx dy.$$
**Step 1**. Our first step to prove is to claim the following: There holds $$\label{eq:Step1}
I(a,b,c) \geqslant \frac{u(a)\, v(b)}2$$ for any $c$ satisfying $$\label{eq:claim}
c\leqslant \max\Big\{\Big(\frac{u(a)}{2\omega_{n-1}}\Big)^{1/(n-1)},\Big(\frac{v(b)}{\omega_n}\Big)^{1/n}\Big\}$$ where $\omega_k$ denotes the volume of unit ball in the $k$-dimensional space $\operatorname{\mathbf{R}}^k$, that is, $\omega_k = \text{vol}(\mathbb B^k)$. Indeed, there are two possible cases regarding to the right hand side of .
. Suppose that $( u(a)/(2\omega_{n-1}) )^{1/(n-1)} \leqslant (v(b)/\omega_n)^{1/n}$. Then by the Fubini theorem, we can estimate $I(a,b,c)$ as follows $$\begin{aligned}
I(a,b,c)& = \int_{\partial \operatorname{\mathbf{R}}_+^n} \chi_{\{f> a\}}(x) \mathscr L^n \big(\{g> b\} \cap \{y\in \operatorname{\mathbf{R}}_+^n\,:\, |x-y| > c\} \big) dx\\
& = \int_{\partial \operatorname{\mathbf{R}}_+^n} \chi_{\{f> a\}}(x) \Big(v(b) - \mathscr L^n \big(\{g> b\} \cap \{y\in \operatorname{\mathbf{R}}_+^n\,:\, |x-y| \leqslant c\} \big) \Big) dx\\
&\geqslant \int_{\partial \operatorname{\mathbf{R}}_+^n} \chi_{\{f> a\}}(x)\Big(v(b) -\frac{\omega_n\, c^n}2 \Big) dx\\
&\geqslant \frac{u(a) \, v(b)}2,\end{aligned}$$ which implies .
. Otherwise, we suppose that $( v(b)/\omega_n)^{1/n} \leqslant (u(a)/(2\omega_{n-1}))^{1/(n-1)}$. In this scenario, by repeating the same arguments as above, we can also bound $I(a,b,c)$ from below as $I(a,b,c) \geqslant u(a)v(b)/2$. Hence we conclude that holds.
**Step 2**. Using , and the nonnegativity of function $I(a,b,c)$, we get $$\label{eq:step1}
\begin{split}
I(f,g) &\geqslant \int_0^\infty\int_0^\infty \Big(\lambda \int_0^{\max\left\{ ( u(a) / (2\omega_{n-1}) )^{1/(n-1)}, ( v(b)/\omega_n )^{1/n}\right\}}c^{\lambda-1}I(a,b,c) dc\Big)da\, db \\
&\geqslant \int_0^\infty\int_0^\infty \frac{u(a)\, v(b)}2\, \Big(\max\Big\{\left(\frac{u(a)}{2\omega_{n-1}}\right)^{1/(n-1)},\left(\frac{v(b)}{\omega_n}\Big)^{1/n}\right\}\Big)^\lambda\, da\, db \\
&\geqslant C \int_0^\infty\int_0^\infty u(a)\, v(b) \, \max\left\{u(a)^{\lambda/(n-1)}, v(b)^{\lambda /n}\right\}\, da\, db,
\end{split}$$ where $C> 0$ depends only on $n$. In the sequel, we use $C$ to denote a positive constant which depends only on $n,p, \lambda$ (or equivalently, on $n,p,\alpha$) and whose value can be changed from line to line.
Denote $\beta =np/((n-1)r)$. We split the integral $\int_0^\infty$ evaluated with respect to the variable $b$ into two integrals as follows $\int_0^\infty = \int_0^{a^\beta} + \int_{a^\beta}^\infty$. Thus, the integrals in can be estimated from below as the the following $$\label{eq:step2}
\begin{split}
\int_0^\infty\int_0^\infty & u(a)\, v(b) \, \max\left\{u(a)^{ \lambda/(n-1)}, v(b)^{\lambda /n}\right\}\, da\, db, \\
\geqslant &\int_0^\infty u(a) \int_0^{a^\beta } v(b)^{1+\lambda /n} db da + \int_0^\infty u(a)^{1+ \lambda/(n-1)} \int_{a^\beta }^\infty v(b) db da, \\
= &\int_0^\infty u(a) \int_0^{a^\beta } v(b)^{1+ \lambda /n} db da + \int_0^\infty v(b) \int_0^{b^{1/\beta }} u(a)^{1+ \lambda /(n-1)} da db\\
=&I+II.
\end{split}$$
**Step 3**. We continue estimating the two integrals $I$ and $II$ in . First, by using the reversed Hölder inequality, we get $$\label{eq:firstterm}
\begin{split}
\int_0^{a^\beta } v(b)^{1+ \lambda /n} db& = \int_0^{a^\beta } v(b)^{1+ \lambda /n} b^{(r-1)(1+ \lambda /n)}b^{-(r-1)(1+ \lambda /n)} db \\
&\geqslant \Big(\int_0^{a^\beta } v(b) b^{r-1} db\Big)^{1+\lambda /n} \Big(\int_0^{a^\beta } b^{(r-1)(1+ \lambda /n)\lambda /n} db\Big)^{-\lambda /n} \\
&= C a^{p-1} \Big(\int_0^{a^\beta } v(b) b^{r-1} db\Big)^{1+\lambda /n}.
\end{split}$$ Similarly, we also get $$\label{eq:secondterm}
\int_0^{b^{1/\beta }} u(a)^{1+\lambda /(n-1)} da \geqslant C b^{r-1} \Big(\int_0^{b^{1/\beta }} u(a) a^{p-1} da \Big)^{1+\lambda /(n-1)}.$$ Upon using our normalization $1 =r\int_0^\infty v(b) b^{r-1} dr$ and the fact that $1 + \lambda /n < 1 +\lambda /(n-1)$, we deduce $$\label{eq:compare}
\Big(\int_0^{a^\beta } v(b) b^{r-1} db\Big)^{1+ \lambda /n} \geqslant C \Big(\int_0^{a^\beta } v(b) b^{r-1} db\Big)^{1+ \lambda /(n-1)}$$ for any $a>0$. Setting $\gamma = 1 +\lambda /(n-1)$ and plugging , , and into , we eventually arrive at $$\begin{split}
\int_0^\infty\int_0^\infty u(a)\, v(b) \, \max &\left\{u(a)^{\lambda /(n-1)}, v(b)^{\lambda /n}\right\}\, da\, db \\
\geqslant &C \int_0^\infty u(a) a^{p-1}\Big(\int_0^{a^\beta } v(b) b^{r-1} db\Big)^{\gamma } da \\
&+ C\int_0^\infty v(b) b^{r-1} \Big(\int_0^{b^{1/\beta }} u(a) a^{p-1} da \Big)^{\gamma } db.
\end{split}$$ Using the relation $r\int_0^\infty v(b) b^{r-1} dr = p\int_0^\infty u(a) a^{p-1} da =1$ and the convexity of the function $\Phi(t) = t^\gamma $ we obtain, thanks to the Jensen inequality, the following $$\label{eq:step3}
\begin{split}
\int_0^\infty\int_0^\infty u(a)\, v(b) \, \max &\left\{u(a)^{\lambda /(n-1)}, v(b)^{\lambda /n}\right\}\, da\, db\\
\geqslant C &\Big(\int_0^\infty u(a) a^{p-1}\int_0^{a^\beta } v(b) b^{r-1} db da\Big)^\gamma\\
&+C\Big(\int_0^\infty v(b) b^{r-1} \int_0^{b^{1/\beta }} u(a) a^{p-1} da db\Big)^\gamma .
\end{split}$$ Also by the convexity of the function $\Phi(t) = t^\gamma $, we have the following elementary inequality $A^\gamma + B^\gamma \geqslant 2^{1-\gamma } (A+B)^\gamma$ for all $A,B>0$. By applying this elementary inequality to and again using the Fubini theorem, we conclude that $$\begin{aligned}
\label{eq:finish}
\int_0^\infty &\int_0^\infty u(a)\, v(b) \, \max\left\{u(a)^{\lambda /(n-1)}, v(b)^{\lambda /n}\right\}\, da\, db,\notag\\
&\geqslant C\Big(\int_0^\infty u(a) a^{p-1}\int_0^{a^\beta } v(b) b^{r-1} db da + \int_0^\infty v(b) b^{r-1} \int_0^{b^{1/\beta }} u(a) a^{p-1} da db\Big)^\gamma \notag\\
&= C\Big(\int_0^\infty u(a) a^{p-1}\int_0^{a^\beta } v(b) b^{r-1} db da + \int_0^\infty u(a) a^{p-1}\int_{a^\beta }^\infty v(b) b^{r-1} db da\Big)^\gamma \notag\\
& = C \big(\frac1{pr}\big)^\gamma .\end{aligned}$$ Combining and gives the estimate $I(f,g) \geqslant C$ for some constant $C>0$. From this, the sharp reversed Hardy-Littlewood-Sobolev inequality on the upper half space follows where the sharp constant ${\mathscr C}_{n,\alpha,p}^+$ is characterized by .
Existence of extremal functions for reversed HLS inequality: Proof of Theorem \[Existence\]
===========================================================================================
Recall that $p,r \in (0,1)$ and $\lambda >0$ satisfy $(n-1)/np + 1/r - \lambda /n = 2 - 1/n$. As in Introduction and for the sake of simplicity, we still denote $q = r/(r-1) < 0$. Clearly, $1/p -\lambda /n = 1 + 1/q$. Let $f$ be a function on $\partial\operatorname{\mathbf{R}}^n_+$ which vanishes at infinity, its symmetric decreasing rearrangement is denoted by $f^\star$; see [@liebloss2001] or [@Bur] for definition. It is well-known that if $f\in L^p(\partial\operatorname{\mathbf{R}}^n_+)$ with $p > 0$, then $f^\star\in L^p(\partial\operatorname{\mathbf{R}}^n_+)$ and $\|f\|_{L^p(\partial\operatorname{\mathbf{R}}^n_+)} = \|f^\star\|_{L^p(\partial\operatorname{\mathbf{R}}^n_+)}$.
We start the proof of Theorem \[Existence\] by the following simple lemma which says more about the interaction between $f$ and $f^\star$.
\[lemmaexistence\] We have the following claims:
- There exists a positive function $F$ on $\operatorname{\mathbf{R}}_+^2$ which is strictly increasing in each variable (when the other is fixed) such that $$(E_\lambda f^\star) (x',x_n) = F(|x'|,x_n)$$ for any $(x',x_n) \in \operatorname{\mathbf{R}}^n_+$.
- For any non-negative function $f\in L^p(\partial\operatorname{\mathbf{R}}^n_+)$, there holds $$\label{eq:decreasenorm}
\int_{\operatorname{\mathbf{R}}^n_+} |E_\lambda f|^q dy \leqslant \int_{\operatorname{\mathbf{R}}^n_+} |E_\lambda f^\star |^q dy ,$$ where $q = r/(r-1) < 0$ with the equality occurs if and only if $f^\star$ is a strictly decreasing and there exists $x_0\in \partial\operatorname{\mathbf{R}}^n_+$ such that $$f(x) = f^\star(x +x_0)$$ a.e. $x\in \partial\operatorname{\mathbf{R}}^n_+$.
The Lemma is immediately derived from the definition of the extension $E_\lambda$ and Lemma $1$ in [@NgoNguyen2015].
We are now in a position to prove Theorem \[Existence\]. The radial symmetry and strictly decreasing of the minimizers for immediately follow from Lemma \[lemmaexistence\]. We only have to prove the existence of a minimizer for this problem. For the sake of clarity, we divide our proof into several steps.
**Step 1**. *Selecting a suitable minimizing sequence for .*
We start our proof by letting $\{f_j\}_j$ be a minimizing sequence for the problem , so is the sequence $\{f_j^\star \}_j$. Hence, without loss of generality, we can assume at the very beginning that $\{f_j\}_j$ is nonnegative radially symmetric and non-increasing minimizing sequence.
By abusing notations, we shall write $f_j(x)$ by $f_j(|x|)$ or even by $f_j (r)$ where $r=|x|$. Under this convention, by the normalization $\|f_j\|_{L^p(\partial \operatorname{\mathbf{R}}_+^n)} =1$, we have $$\begin{split}
1 = & (n-1)\omega_{n-1} \int_0^\infty f_j(r)^p r^{n-2} dr \geqslant \omega_{n-1} f_j(R)^p R^{n-1}
\end{split}$$ for any $R > 0$. From this, we obtain the following estimate $0\leqslant f_j(r) \leqslant C r^{-(n-1)/p}$ for any $r > 0$ and for some constant $C$ independent of $j$.
In order to go further, we need the following lemma whose proof mimics that of [@dz2014 Lemma 3.2]; see also [@l1983 Lemma 2.4].
\[farawayzero\] Suppose that $f\in L^p(\operatorname{\mathbf{R}}^n)$ is non-negative, radially symmetric, and $$f(|y|) \leqslant \epsilon |y|^{-(n-1)/p}$$ for all $y\in \partial\operatorname{\mathbf{R}}_+^n$. Then for any $p_1\in (0, p)$, there exists a constant $C_1 > 0$, independent of $f$ and $\epsilon$ such that $$\label{eq:farawayzero}
\int_{\operatorname{\mathbf{R}}_+^n} |E_\lambda f|^q dy \leqslant C_1 \epsilon^{q(1 - p)/p_1} \Big( \int_{\partial\operatorname{\mathbf{R}}_+^n} | f|^p dx \Big)^{q/p_1}$$ where $q = r/(r-1) < 0$.
Define $F: \operatorname{\mathbf{R}}\to \operatorname{\mathbf{R}}$ by setting $$F(t) = e^{(n-1)t/p} f(e^t).$$ Then we can easily see that $$\label{eq:relationfF}
\int_{\partial\operatorname{\mathbf{R}}^n_+} |f|^p dx = (n-1)\omega_{n-1} \int_{-\infty} ^{+\infty} |F|^p dt$$ and that $$\label{eq:relationfF2}
\|F\|_{L^\infty(\operatorname{\mathbf{R}})} \leqslant \epsilon,$$ where, as before, $\omega_k = 2\pi^{k/2} / k \Gamma(k/2)$ is the volume of the $k$-dimensional unit ball $\mathbb B^k$. Writing $y =(y',y_n) \in \operatorname{\mathbf{R}}^n_+$, it is easy to see that $E_\lambda f$ is radially symmetric in $y'$. Now we define $H: \operatorname{\mathbf{R}}\times\operatorname{\mathbf{R}}_+\to \operatorname{\mathbf{R}}$ by letting $$H(t,y_n) = e^{nt/q} (E_\lambda f)(e^t, e^t y_n).$$ By a simple change of variables, we then obtain $$\label{eq:relationEf}
(n-1)\omega_{n-1} \int_{\operatorname{\mathbf{R}}_+^2} |H(t,y_n)|^q dtdy_n = \int_{\operatorname{\mathbf{R}}_+^n} |E_\lambda f|^q dy .$$ Thanks to and recall $1/q = 1-1/r$ from the beginning of this section, we know that $n/q + \lambda /2 = -(n-1) (1-1/ p) - \lambda /2$. If for each real number $s \geqslant 0$ we use $\overrightarrow s$ to denote some vector sitting in $\partial \operatorname{\mathbf{R}}_+^n$ with length $s$, then we clearly have $$\begin{split}
H(t,y_n) = & e^{nt/q} \int_{\partial \operatorname{\mathbf{R}}_+^n} \Big| | \overrightarrow{e^t}-x|^2+e^{2t}y_n^2 \Big|^{\lambda/2} f(x) dx \\
= & e^{nt/q + t\lambda/2} \int_{\partial \operatorname{\mathbf{R}}_+^n} \big| e^t (1+y_n^2) + e^{-t} |x|^2 -2 \overrightarrow{1} \cdot x \big|^{\lambda/2} f(x) dx \\
= & e^{(n/q+ \lambda/2)t} \int_{-\infty} ^{+\infty} \int_{\partial B^{n-2}(0, e^s)} \big| e^{t-s} (1+y_n^2) + e^{-(t-s)}-2 \overrightarrow{e^{-s}} \cdot x \big|^{\lambda/2} \times \\
&\qquad \qquad \qquad \qquad \qquad \qquad \times e^{s(1+\lambda /2)} f(e^s) d\sigma ds \\
= & e^{(n/q+ \lambda/2)t} \int_{-\infty} ^{+\infty} \int_{\mathbb S^{n-2}} \big| e^{t-s} (1+y_n^2) + e^{-(t-s)}-2 \overrightarrow{1} \cdot \xi \big|^{\lambda/2} \times \\
&\qquad \qquad \qquad \qquad \qquad \qquad \times e^{s(n-1+\lambda /2)} f(e^s) d\xi ds .
\end{split}$$ Hence, thanks to $n/q + \lambda + n - 1 = (n - 1)/p$, we can readily obtain an explicit form of $H$ as follows $$H(t,y_n) = \int_{-\infty} ^{+\infty} L(t-s,y_n) F(s) ds,$$ where $L(s,y_n) = e^{(n/q + \lambda/2)s} Z(s,y_n)$ with $$Z(s,y_n) =
\begin{cases}
\displaystyle \int_{\mathbb S^{n-2}} \big( e^s(1+y_n^2) + e^{-s} - 2 \overrightarrow{1} \cdot \xi \big)^{\lambda/2} d\xi & \text{ if } n\geqslant 3,\\
\big( \big( e^s(1+y_n^2) + e^{-s} -2\big)^{\lambda/2} + \big( e^s(1+y_n^2) + e^{-s} +2\big)^{\lambda/2} \big) & \text{ if } n=2.
\end{cases}$$ Clearly, there exists a constant $c > 0$ such that $L(t,y_n) \geqslant c$ for all $(t,y_n) \in \operatorname{\mathbf{R}}_+^2$ and $$L(t,y_n) \sim (e^t(1+y_n^2) +e^{-t})^{\lambda/2}$$ as $t^2 +y_n^2 \to +\infty$. Since $n/q + \lambda = (n-1) (1/p - 1) > 0$, we know that $\lambda q + 1 <0$. From this we conclude, for any $s < 0$, that $$\label{eq:Step1-BoundForAllNegativeS}
\int_{\operatorname{\mathbf{R}}} \left(\int_0^\infty L^q(t,y_n) dy_n\right)^{s/q} dt < + \infty.$$ For any $p_1 \in (0, p)$, we choose $s_1$ such that $1/p_1 + 1/s_1 = 1 + 1/q$. Since $p_1 \in (0,1)$ and $q<0$, we clearly have $s_1 < 0$. In addition, it follows from $s_1/p_1 < s_1$ that $q/s_1 > 1$. From these facts and by the reversed Young inequality [@dz2014 Lemma 2.2], for any $y_n > 0$ we have $$\int_{\operatorname{\mathbf{R}}} |H(t, y_n)|^q dt \leqslant \Big( \int_{\operatorname{\mathbf{R}}} |L(t, y_n)|^{s_1} dt \Big)^{q/s_1} \Big(\int_{\operatorname{\mathbf{R}}} |F|^{p_1} dt\Big)^{q/p_1}.$$ which, by integrating both sides with respect to $y_n$ over $[0, +\infty)$, implies $$\label{eq:rYoungineq}
\int_{\operatorname{\mathbf{R}}^2_+} |H(t, y_n)|^q dt dy_n \leqslant \Big(\int_{\operatorname{\mathbf{R}}} |F|^{p_1} dt\Big)^{q/p_1} \int_0^\infty\Big(\int_{\operatorname{\mathbf{R}}} L(t,y_n)^{s_1} dt\Big)^{q/s_1} dy_n.$$ To estimate the right hand side of , on one hand, we observe by the Minkowski inequality that $$\label{eq:Minkowskiineq}
\Big(\int_0^\infty\Big(\int_{\operatorname{\mathbf{R}}} L^{s_1}(t,y_n) dt\Big)^{q/s_1} dy_n\Big)^{s_1/q} \leqslant \int_{\operatorname{\mathbf{R}}} \left(\int_0^\infty L^q(t,y_n) dy_n\right)^{s_1/q} dt.$$ Thanks to $q/s_1 > 0$, we conclude from and that $$\int_0^\infty\Big(\int_{\operatorname{\mathbf{R}}} L^{s_1}(t,y_n) dt\Big)^{q/s_1} dy_n < +\infty,$$ which then helps us to conclude from that $$\label{eq:Step1-OneHand}
\int_{\operatorname{\mathbf{R}}^2_+} |H(t, y_n)|^q dt dy_n \leqslant C \Big(\int_{\operatorname{\mathbf{R}}} |F|^{p_1} dt\Big)^{q/p_1} .$$ On the other hand, if we write $F ^{p_1} = F ^p F ^{p_1-p} \geqslant F ^p \|F\|_{L^\infty(\operatorname{\mathbf{R}})} ^{p_1-p}$, then thanks to and we get $$\label{eq:Step1-OtherHand}
\int_{\operatorname{\mathbf{R}}} |F|^{p_1} dt \geqslant C \epsilon^{p_1-p} \int_{\partial\operatorname{\mathbf{R}}^n_+} |f|^p dx .$$ Simply plugging into gives $$\label{eq:Step1-BothHands}
\int_{\operatorname{\mathbf{R}}^2_+} |H(t, y_n)|^q dt dy_n \leqslant C\epsilon^{q(1-p)/p_1} \Big( \int_{\partial\operatorname{\mathbf{R}}^n_+} |f|^p dx \Big)^{q/p_1} .$$ Thus, combining and gives as claimed.
**Step 2**. *Existence of a potential minimizer $f_0$ for .*
For each $j$ we set $$a_j = \sup_{r > 0} r^{(n-1)/p} f_j(r)$$ which obviously belongs to $[0,C]$. Thanks to the normalization $\|f_j\|_{L^p(\partial\operatorname{\mathbf{R}}^n_+)} = 1$ and the fact $\|E_\lambda f_j\|_{L^q(\operatorname{\mathbf{R}}_+^n)} \to {\mathscr C}_{n,p,\lambda}^+ < +\infty$, we obtain from Lemma \[farawayzero\] the following estimate $a_j \geqslant 2c_0$ for some $c_0 >0$. For each $j$, we choose $\lambda_j > 0$ in such a way that $\lambda_j^{(n-1)/p} f_j(\lambda_j) > c_0$. Then we set $$g_j(x) = \lambda_j^{n/p} f_j(\lambda_j x).$$ From this, it is routine to check that $\{g_j\}_j$ is also a minimizing sequence for problem , and $g_j(1) > c_0$ for any $j$ by our choice for $\lambda_j$. Consequently, by replacing the sequence $\{f_j\}_j$ by the new sequence $\{g_j\}$, if necessary, we can further assume that our sequence $\{f_j\}_j$ obeys $f_j(1) > c_0$ for any $j$.
Similar to Lieb’s argument in [@l1983], which is based on the Helly theorem, by passing to a subsequence, we have $f_j\to f_0$ a.e. in $\partial\operatorname{\mathbf{R}}_+^n$. It is now evident that $f_0$ is non-negative radially symmetric, non-increasing and is in $L^p(\partial\operatorname{\mathbf{R}}_+^n)$. The rest of our arguments is to show that $f_0$ is indeed the desired minimizer for .
By Lemma \[lemmaexistence\], we know that $(E_\lambda f_j)(y',y_n)$ is radially symmetric and strictly decreasing in $y'$ and strictly increasing in $y_n$ for any $j$. Moreover, for all $y \in \operatorname{\mathbf{R}}_+^n$, there holds $$\label{eq:lowerbound}
(E_\lambda f_j) (y) \geqslant c_0\int_{|x| \leqslant 1} |x-y|^\lambda dx \geqslant C_2 (1 + |y|^\lambda)=: g(y)$$ for some new constant $C_2$ independent of $j$.
**Step 3**. *The function $f_0$ is indeed a minimizer for*
For each $y\in \operatorname{\mathbf{R}}_+^n$, set $$k(y) = \liminf_{j\to\infty} (E_\lambda f_j) (y) .$$ By , we have $k(y) \geqslant g(y)$ for any $y\in \operatorname{\mathbf{R}}^n_+$. It is easy to see that $$g(y)^q - k(y)^q = \liminf_{j\to\infty}(g(y)^q - (E_\lambda f_j) (y)^q).$$ Again by and the Fatou lemma, we have $$\begin{split}
\int_{\operatorname{\mathbf{R}}_+^n} (g(y)^q -k(y)^q) dy \leqslant & \liminf_{j\to\infty} \int_{\operatorname{\mathbf{R}}_+^n} (g(y)^q - (E_\lambda f_j) (y)^q) dy \\
= &\int_{\operatorname{\mathbf{R}}_+^n} g(y)^q dy - \big( {\mathscr C}_{n,p,\lambda}^+ \big) ^q.
\end{split}$$ Therefore $$\label{eqIntegralK=C}
\int_{\operatorname{\mathbf{R}}_+^n} g(y)^q dy \geqslant \int_{\operatorname{\mathbf{R}}_+^n} k(y)^q dy \geqslant \big( {\mathscr C}_{n,p,\lambda}^+ \big) ^q.$$ These inequalities imply that the set $\{y\in \operatorname{\mathbf{R}}_+^n\, :\, 0 < k(y) < +\infty\}$ has positive measure. Hence we can take a point $y_1\in \operatorname{\mathbf{R}}_+^n$ and extract a subsequence of $E_\lambda f_j$, still denoted by $E_\lambda f_j$, such that $$\lim_{j\to +\infty} (E_\lambda f_j) (y_1) = a_1 \in (0, +\infty).$$ Repeating the above arguments and extracting a subsequence of $E_\lambda f_j$ if necessary, we can choose a point $y_2 \in \operatorname{\mathbf{R}}_+^n$ such that $y_2\not = y_1$ and that $$\lim_{j\to +\infty} (E_\lambda f_j) (y_2) = a_2 \in (0, +\infty).$$ Then there exists some constant $C_5 > 0$ such that $(E_\lambda f_j) (y_i) \leqslant C_5$ for $i=1,2$ and for all $j \geqslant 1$. Using the simple inequality $|a+b|^\lambda \leqslant \max\{1,2^{\lambda-1}\} (|a|^\lambda + |b|^\lambda)$ for any $a,b\in\operatorname{\mathbf{R}}^n$, we have $$\begin{split}
|y_1-y_2|^\lambda \int_{\partial\operatorname{\mathbf{R}}_+^n}f_j(x)dx \leqslant &\max\{1,2^{\lambda-1}\} \int_{\partial\operatorname{\mathbf{R}}_+^n} |y_1-x|^\lambda f_j(x) dx \\
&+ \max\{1,2^{\lambda-1}\} \int_{\partial\operatorname{\mathbf{R}}_+^n} |y_2-x|^\lambda f_j(x) dx \\
=&\max\{1,2^{\lambda-1}\} \big( (E_\lambda f_j) (y_1) + (E_\lambda f_j) (y_2) \big)\\
\leqslant &2\max\{1,2^{\lambda-1}\} C_5.
\end{split}$$ Thus, there exists another constant $C_6> 0$ such that $\int_{\partial\operatorname{\mathbf{R}}_+^n}f_j(x)dx \leqslant C_6$ for all $j\geqslant 1$. On one hand, for any $R > 2|y_1|$, there holds $|y_1-x| \geqslant |x|/3$ for any $x$ in the region $\{3R/4 \leqslant |x|\leqslant R\}$. Therefore, by a simple variable change, we can estimate $$\begin{split}
C_5 \geqslant &\int_{\{3R/4\leqslant |y|\leqslant R\}} |y_1-x|^\lambda f_j(x) dx \\
\geqslant & 3^{-\lambda} f_j(R) R^{n-1+\lambda} \int_{\{3/4\leqslant |x|\leqslant 1\}} |x|^\lambda dx.
\end{split}$$ Note that in the preceding estimate, we have used the fact that $f_j$ is radial symmetric and non-increasing. Hence, there exists some $C_7 > 0$ such that $f_j(r) \leqslant C_7 r^{-n-\lambda+1}$ for any $r > 2 |y_1|$ and for all $j\geqslant 1$.
Making use of the above estimate $f_j(r) \leqslant C_7 r^{-n-\lambda+1}$, for any $r > 2|y_1|$, we further have $$\label{eq:outsideball}
\begin{split}
\int_{\{|x| > R\}} f_j(x)^p dx \leqslant & C_7^p \int_{\{|x| > R\}}|x|^{-p(n+\lambda-1)}dx \\
=& -\frac{(n-1)\omega_{n-1} q}{np} C_7^pR^{np/q}.
\end{split}$$ Thanks to $\int_{\operatorname{\mathbf{R}}^n}f_j(x)dx \leqslant C_6$, we also have $$\label{eq:fjgeqR}
\int_{\{f_j > R\}} f_j(x)^p dx \leqslant R^{p-1} \int_{\partial\operatorname{\mathbf{R}}_+^n} f_j(x) dx \leqslant C_6 R^{p-1}.$$ In view of and , for arbitrary $\epsilon >0$, we can select $R > 2|y_1|$ sufficiently large in such a way that $$\int_{\{|x| > R\}} f_j(x)^p dx < \frac\epsilon 2$$ and that $$\int_{\{f_j > R\}} f_j(x)^p dx < \frac\epsilon 2.$$ We now set $g_j(x) = \min\{f_j(x),R\}$ for each $j\geqslant 1$. Since $\int_{\partial\operatorname{\mathbf{R}}_+^n} f_j(x)^p dx =1$, we have $$\begin{split}
\int_{\{|x|\leqslant R\}}g_j(x)^p dx& \geqslant \int_{\{|x|\leqslant R\}\cap \{f_j \leqslant R\}} f_j(x)^p dx\\
&=1 - \int_{\{|x|\leqslant R\}\cap \{f_j > R\}} f_j(x)^p dx -\int_{\{|x| > R\}} f_j(x)^p dx \geqslant 1-\epsilon.
\end{split}$$ For each $R$ fixed, the dominated convergence theorem guarantees that $$\lim_{j\to\infty} \int_{\{|x|\leqslant R\}}g_j(x)^p dx = \int_{\{|x|\leqslant R\}}\left(\min\{f_0(x),R\}\right)^p dx.$$ Therefore, by sending $R \to +\infty$, we arrive at $$\int_{\partial\operatorname{\mathbf{R}}_+^n}f_0(x)^p dx \geqslant 1-\epsilon,$$ for any $\epsilon >0$. From this we can conclude $\int_{\partial\operatorname{\mathbf{R}}_+^n}f_0(x)^p dx \geqslant 1$. On the other hand, by the Fatou lemma, we have $\int_{\partial\operatorname{\mathbf{R}}_+^n}f_0(x)^p dx \leqslant 1$. Therefore, we have just proved $\|f_0\|_{L^p(\partial\operatorname{\mathbf{R}}_+^n)} =1$.
In the last part of the step, to realize that $f_0$ is indeed a minimizer for , we apply the Fatou lemma once again to get $$k(x) = \liminf_{j\to\infty} E_\lambda f_j(x) \geqslant E_\lambda f_0(x),$$ for a.e. $x$ in $\operatorname{\mathbf{R}}_+^n$. Hence, combining the preceding estimate and gives $${\mathscr C}_{n,p,\lambda}^+ = {\mathscr C}_{n,p,\lambda}^+ \|f_0\|_{L^p(\partial\operatorname{\mathbf{R}}_+^n)} \leqslant \|E_\lambda f_0\|_{L^q(\operatorname{\mathbf{R}}_+^n)} \leqslant \Big(\int_{\operatorname{\mathbf{R}}_+^n} k(y)^{q} dy \Big)^{1/q} \leqslant {\mathscr C}_{n,p,\lambda}^+.$$ This shows that $f_0$ is a minimizer for ; hence finishing the proof.
Basic properties of (\[eqIntegralSystem\]) and Proof of Lemma \[lemNECESSARY\]
==============================================================================
In this section, we first establish some basic properties of the system which shall be needed to prove Lemma \[lemNECESSARY\] and Proposition \[thmCLASSIFICATION\].
Preliminaries
-------------
In this subsection, we setup some preliminaries necessarily for our analysis. Here and in what follows, by $\lesssim$ and $\gtrsim$ we mean inequalities up to $p$, $q$, and dimensional constants. For the sake of simplicity, we denote $B_{\partial \operatorname{\mathbf{R}}_+^n} (x,R) = \{ \xi \in \partial \operatorname{\mathbf{R}}_+^n : |\xi - x| \leqslant R\}$ and $B_{ \operatorname{\mathbf{R}}_+^n} (x,R) = \{ \eta \in \operatorname{\mathbf{R}}_+^n : |\eta - x| \leqslant R\}$. We also denote $$\Sigma_{x,R}^{n-1} = \operatorname{\mathbf{R}}_+^n \backslash \overline{B_{\operatorname{\mathbf{R}}_+^n} (x, R)} , \quad \Sigma_{x,R}^n= \partial \operatorname{\mathbf{R}}_+^n \backslash \overline{B_{\partial \operatorname{\mathbf{R}}_+^n} (x,R)} .$$ We now establish the most important part of this section known as a prior estimates for solutions of as stated in Lemma \[lem-Growth\] below.
\[lem-Growth\] For $n \geqslant 1$ and $\lambda,\kappa,\theta>0$, let $(u,v)$ be a pair of non-negative Lebesgue measurable functions in $\partial \operatorname{\mathbf{R}}_+^n \times \operatorname{\mathbf{R}}_+^n$ satisfying . Then there hold $$\label{eqGrowth1}
\int_{\partial \operatorname{\mathbf{R}}_+^n} {(1 + |x|^\lambda )u(x)^{ - \theta}dx} < +\infty , \quad \int_{\operatorname{\mathbf{R}}_+^n} {(1 + |y|^\lambda )v(y)^{ - \kappa}dy} < +\infty ,$$ and $$\label{eqGrowth2}
\begin{split}
\mathop {\lim }\limits_{|x| \to +\infty } \frac{u(x)}{|x|^\lambda} =& {\int_{\operatorname{\mathbf{R}}_+^n} {v(y)^{ - \kappa}dy} } , \quad \mathop {\lim }\limits_{|y| \to +\infty } \frac{{v(y)}}{{|y|^\lambda }} = {\int_{\partial\operatorname{\mathbf{R}}_+^n} {u(x)^{ - \theta }dx} } ,
\end{split}$$ and $u$ and $v$ are bounded from below in the following sense $$\label{eqGrowth3}
\frac{1 + |x|^\lambda }{C} \leqslant u(x) \leqslant C(1 + |x|^\lambda )$$ for all $x \in \partial\operatorname{\mathbf{R}}_+^n$ and $$\label{eqGrowth4}
\frac{1 + |y|^\lambda }{C} \leqslant v(y) \leqslant C(1 + |y|^\lambda )$$ for all $y \in \operatorname{\mathbf{R}}_+^n$ for some constant $C \geqslant 1$.
To prove our lemma, we first observe from that both $u$ and $v$ are strictly positive everywhere in their domains and are finite within a set of positive measure. Hence there exist some $R>1$ sufficiently large and some Lebesgue measurable set $E \subset \overline{\operatorname{\mathbf{R}}_+^n}$ such that $$\label{eqSetE}
E \subset \{ z : u(z) < R, v(z) < R\} \cap B_{\overline{\operatorname{\mathbf{R}}_+^n}} (0,R)$$ with $\mathscr L^n (E) \geqslant 1/R$. Using this, we can easily bound $v$ from below as follows $$\begin{split}
v(y) \geqslant \int_E {|x - y|^\lambda u(x)^{ - q}dx} \geqslant & \frac{1}{R^q}\int_E {|x - y|^\lambda dx} =\frac{1}{R^q}\int_{E+y} {|x|^\lambda dx}
\end{split}$$ for any $y \in \operatorname{\mathbf{R}}_+^n$. Now we choose $\varepsilon > 0$ small enough and then fix it in such a way that $\operatorname{\textrm{vol}}(B_{\operatorname{\mathbf{R}}_+^n} (0,\varepsilon)) < \mathscr L^n (E) /2$. Then we can estimate $$\begin{split}
\int_{E+y} {|x|^\lambda dx} & \geqslant \int_{E+y \backslash B_{\operatorname{\mathbf{R}}_+^n} (0,\varepsilon)} {|x|^\lambda dx} \\
& \geqslant \varepsilon^\lambda \int_{E+y \backslash B_{\operatorname{\mathbf{R}}_+^n} (0,\varepsilon)} dy \\
& = \varepsilon^\lambda \big( \mathscr L^n (E+y) -\operatorname{\textrm{vol}}(B_{\operatorname{\mathbf{R}}_+^n} (0,\varepsilon)) \big).
\end{split}$$ From this, it is clear that $v$ is bounded from below by some positive constant. The same reason applied to $u$ shows that there exists some constant $C_0>0$ such that $$\label{eqUVBoundedFromBelow}
u(x), v(y) > C_0$$ for all $x \in \partial\operatorname{\mathbf{R}}^n$ and $y \in \operatorname{\mathbf{R}}^n$.
**Proof of** . To prove this, we first consider $|x| \geqslant 2R$ where $R$ is defined through . Note that for every $y \in E \subset B_{\overline{\operatorname{\mathbf{R}}_+^n}}(0,R)$, there holds $|x - y| \geqslant |x| - |y| \geqslant |x|/2$, thanks to $|x| \geqslant 2R$. Using this we can estimate $$v(y) \geqslant \frac{1}{R^q}\int_E {|x - y|^\lambda dx} \geqslant \frac{\text{vol}(E)}{(2R)^\lambda} |y|^\lambda$$ for any $|y| \geqslant 2R$. A similar argument also shows $u(x) \geqslant \text{vol}(E) (2R)^{-\lambda}|x|^\lambda $ in the region $\{x : |x| \geqslant 2R\}$. Hence, it is easy to select a large constant $C>1$ in such a way that holds in the region $\{ |x| \geqslant 2R\}$. Thanks to , we can further decrease $C$, if necessary, to obtain the estimate in the ball $\{x \in \overline{\operatorname{\mathbf{R}}_+^n}: |x| \leqslant 2R\}$; hence the proof of follows.
**Proof of** . We only need to estimate $v$ since $u$ can be estimated similarly. To this purpose, we first show that $u^{-q} \in L^1(\partial \operatorname{\mathbf{R}}_+^n)$. Clearly for some $\overline x$ satisfying $1 \leqslant |\overline x| \leqslant 2$, there holds $$\int_{\partial \operatorname{\mathbf{R}}_+^n} {|\overline x - x|^\lambda u{(x)^{ - \theta}}dx} = v(\overline x ) \in (0, + \infty ).$$ Observer that for any $x \in \partial \operatorname{\mathbf{R}}_+^n \backslash B_{\partial \operatorname{\mathbf{R}}_+^n} (0,4)$, there holds $|\overline x - x | \geqslant |x| -|\overline x | > 1$; hence $$\int_{\partial \operatorname{\mathbf{R}}_+^n \backslash B_{\partial \operatorname{\mathbf{R}}_+^n} (0,4) } u(x)^{ - \theta}dx < \int_{\operatorname{\mathbf{R}}^n} {|\overline x - x|^\lambda u{(x)^{ - \theta }}dx} < +\infty.$$ In the small ball $B_{\partial \operatorname{\mathbf{R}}_+^n} (0,4)$, thanks to , it is obvious to verify that $$\int_{B_{\partial \operatorname{\mathbf{R}}_+^n} (0,4) } u(x)^{ - \theta}dx \lesssim \int_{B_{\partial \operatorname{\mathbf{R}}_+^n} (0,4)} (1+|x|^\lambda))^{-\theta} dx < +\infty.$$ Thus, we have just shown that $u^{-\theta} \in L^1(\partial \operatorname{\mathbf{R}}_+^n)$. In view of , it suffices to prove that $$\label{eqGrowth1-suffice}
\int_{\partial \operatorname{\mathbf{R}}_+^n} |x|^\lambda u(x)^{ - \theta}dx < +\infty.$$ To see this, we again observe that $|x| \leqslant 2|\overline x - x| $ for all $x \in \partial \operatorname{\mathbf{R}}_+^n \backslash B_{\partial \operatorname{\mathbf{R}}_+^n} (0,4) $. Therefore, $$\int_{\partial \operatorname{\mathbf{R}}_+^n \backslash B_{\partial \operatorname{\mathbf{R}}_+^n}(0,4) } |x|^\lambda u(x)^{ - q}dx \lesssim \int_{\partial \operatorname{\mathbf{R}}_+^n \backslash B_{\partial \operatorname{\mathbf{R}}_+^n} (0,4)} {|\overline x - x|^\lambda u{(x)^{ -\theta}}dx} < +\infty.$$ In the small ball $B_{\partial \operatorname{\mathbf{R}}_+^n} (0,4)$, it is obvious to see that $$\int_{ B_{\partial \operatorname{\mathbf{R}}_+^n} (0,4) } |x|^\lambda u(x)^{ - \theta }dx \lesssim \int_{B_{\partial \operatorname{\mathbf{R}}_+^n} (0,4)}u(x)^{-\theta} dx < +\infty,$$ thanks to $u^{- \theta} \in L^1(\partial \operatorname{\mathbf{R}}_+^n)$. From this, follows, so does .
**Proof of** . We only consider the limit $|y|^{-\lambda} v(y)$ as $|y| \to +\infty$ since the limit $|x|^{-\lambda} u(x)$ can be proved similarly. Indeed, using , we first obtain $$\label{eqProofGrowth2-1}
\begin{split}
\mathop {\lim }\limits_{|y| \to +\infty } \frac{ v(y) }{|y|^\lambda } =& \mathop {\lim }\limits_{|y| \to +\infty } {\int_{\partial \operatorname{\mathbf{R}}_+^n} {\frac{{|x - y|^\lambda }}{{|y|^\lambda}} u{(x)^{ - \theta}}dx} }.
\end{split}$$ Observe that as $|y| \to +\infty$, $( |x - y|/|y| )^\lambda u(x)^{ - \theta} \to u(x)^{ - \theta}$ almost everywhere $y$ in $\operatorname{\mathbf{R}}_+^n$. Hence we can apply the Lebesgue dominated convergence theorem to pass to the limit to conclude provided we can show that $|x - y|^\lambda |x|^{ - \theta}u(x)^{ - q}$ is bounded by some integrable function. To this end, we observe that $|x-y|^\lambda \lesssim |x|^\lambda +|y|^\lambda $; hence, if $|x| > 1$ then $${\left( {\frac{|x - y|}{|y|}} \right)^\lambda}u(x)^{ - \theta } \lesssim (1+|x|^\lambda)u(x)^{ - \theta}.$$ Our proof now follows by observing $ (1 + |x|^\lambda )u(x)^{ -\theta} \in L^1(\partial \operatorname{\mathbf{R}}_+^n)$ by .
**Proof of** . To see this, we first observe from that there exists some large number $k>1/R$ such that $$\frac{u(x)}{|x|^\lambda} < 1 + \int_{\operatorname{\mathbf{R}}_+^n} v(y)^{-\kappa} dy$$ in $\partial \operatorname{\mathbf{R}}_+^n \backslash B_{\partial \operatorname{\mathbf{R}}_+^n} (0,kR)$. In the ball $B_{\partial \operatorname{\mathbf{R}}_+^n} (0,kR)$, it is easy to estimate $|x-y|^\lambda \lesssim |x|^\lambda + |y|^\lambda$ which helps us to conclude that $$u(x) \lesssim (kR)^\lambda \int_{\operatorname{\mathbf{R}}_+^n} (1+|y|^\lambda) v(y)^{-\kappa} dy$$ in the ball $B_{\partial \operatorname{\mathbf{R}}_+^n} (0,kR)$. From this and our estimate for $u$ outside $B_{\partial \operatorname{\mathbf{R}}_+^n}(0,kR)$, we obtain the desired estimate in $\partial \operatorname{\mathbf{R}}_+^n$. Our estimate for $v$ follows similarly; hence we obtain as claimed.
Inspired by , we prove the following simple observation.
\[lem-IntegralU=IntegralV\] There holds $u \in L^{1-\theta} (\partial\operatorname{\mathbf{R}}_+^n)$ and $v \in L^{1-\kappa} (\operatorname{\mathbf{R}}_+^n)$. Moreover, $$\int_{\partial\operatorname{\mathbf{R}}_+^n} u^{1-\theta} (x) dx = \int_{\operatorname{\mathbf{R}}_+^n} \int_{\partial\operatorname{\mathbf{R}}_+^n} v(y)^{-\kappa} |x-y|^{\lambda } u^{-\theta}(x) dx dy = \int_{\operatorname{\mathbf{R}}_+^n} v^{1-\kappa} (y) dy .$$
To see $u \in L^{1-\theta} (\partial\operatorname{\mathbf{R}}_+^n)$, we observe from and that $$\int_{\partial\operatorname{\mathbf{R}}_+^n} u^{1-\theta} (x) dx \lesssim \int_{\partial \operatorname{\mathbf{R}}_+^n} {(1 + |x|^\lambda )u(x)^{ - \theta}dx} < +\infty.$$ A similar argument shows $v \in L^{1-\kappa} (\operatorname{\mathbf{R}}_+^n)$ as claimed. The way to obtain the desired relation is elementary since $$u^{1-\theta} (x) = u^{-\theta}(x) \int_{\operatorname{\mathbf{R}}_+^n} v(y)^{-\kappa} |x-y|^{\lambda } dy$$ and $$v^{1-\kappa} (y) = v(y)^{-\kappa} \int_{\partial\operatorname{\mathbf{R}}_+^n} |x-y|^{\lambda } u^{-\theta}(x) dx .$$ Integrating both sides over suitable domains gives the desired result.
In the following result, we prove a regularity result similar to [@l2004 Lemma 5.2] obtained by Li.
\[lem-Regularity\] For $n \geqslant 1$ and $\lambda,\kappa,\theta>0$, let $(u,v)$ be a pair of non-negative Lebesgue measurable functions in $\partial \operatorname{\mathbf{R}}_+^n \times \operatorname{\mathbf{R}}_+^n$ satisfying . Then $u$ and $v$ are smooth.
Our proof is similar to that of [@l2004 Lemma 5.2]. Let $R>0$ be arbitrary, first we decompose $u$ and $v$ into the following way $$\begin{split}
u(x) =& \big(u_R^1 + u_R^2 \big) (x) = \Big(\int_{|y| \leqslant 2R} + \int_{|y| > 2R} \Big)|x - y|^\lambda v{(y)^{ - \kappa}}dy, \\
v(y) =& \big(v_R^1 + v_R^2 \big) (y) = \Big(\int_{|x| \leqslant 2R} + \int_{|x| > 2R} \Big)|x - y|^\lambda u(x)^{ - \theta}dx. \\
\end{split}$$ Thanks to , we immediately see that we can continuously differentiate $u_R^2$ and $v_R^2$ under the integral sign for any $x \in \partial \operatorname{\mathbf{R}}_+^n$ satisfying $|x|<R$. Consequently, $u_R^2 \in C^\infty (B_{\partial \operatorname{\mathbf{R}}_+^n}(0,R))$ and $v_R^2 \in C^\infty (B_{\operatorname{\mathbf{R}}_+^n}(0,R))$.
In view of and , we know that $u^{-\theta} \in L^\infty (B_{\partial \operatorname{\mathbf{R}}_+^n} (0, 2R))$. This and the following elementary inequality $\big| |x-y|^\lambda - |z-y|^\lambda \big| \lesssim |x-z|^{\min\{\lambda, 1\}}$ for all $x, z \in B(0, R)$ and all $y \in B(0, 2R)$ conclude that $v_R^1$ is at least Hölder continuous in $B_{\operatorname{\mathbf{R}}_+^n}(0,R)$. Similar reasons tell us that $u_R^1$ is also at least Hölder continuous in $B_{\partial \operatorname{\mathbf{R}}_+^n}(0,R)$. Hence, we have just proved that $u$ and $v$ are at least Hölder continuous in $B(0,R)$, so are at least Hölder continuous in $\partial \operatorname{\mathbf{R}}_+^n$ and $\operatorname{\mathbf{R}}_+^n$ respectively since $R>0$ is arbitrary.
Standard bootstrap argument shows $u \in C^\infty (\partial \operatorname{\mathbf{R}}_+^n)$ and at the same time $v \in C^\infty (\operatorname{\mathbf{R}}_+^n)$ follows the same lines.
Proof of Lemma \[lemNECESSARY\]
-------------------------------
To prove this lemma, we borrow the idea in [@lei2015]. As the first step in the proof, we make use of the integrability of $u$ and $v$ in $L^{1-\theta} (\partial\operatorname{\mathbf{R}}_+^n)$ and in $L^{1-\kappa} (\operatorname{\mathbf{R}}_+^n)$ respectively to derive and below. For this purpose, we let $\zeta : [0,+\infty) \to [0,1]$ be a smooth cut-off function such that $$\zeta (t)=
\begin{cases}
0 &\text{ if } t \geqslant 2,\\
1 &\text{ if } 0 \leqslant t \leqslant1,\\
\end{cases}$$ and $\zeta' \in [0,1]$. Then for some fixed $N>0$, by integration by parts, we obtain $$\label{eq-p5-eq3}
\begin{split}
\int_{B_{\partial\operatorname{\mathbf{R}}_+^n}(0,N)} \zeta \Big( \frac {|x|}R\Big) \langle \nabla u^{1-\theta} (x), x \rangle dx = & - (n-1) \int_{B_{\partial\operatorname{\mathbf{R}}_+^n}(0,N)} \zeta \Big( \frac {|x|}R\Big) u^{1-\theta} (x) dx \\
&- \int_{B_{\partial\operatorname{\mathbf{R}}_+^n}(0,N)} \Big\langle \nabla \zeta \Big( \frac {|x|}R\Big) , x \Big\rangle u^{1-\theta} (x) dx \\
& + \int_{\partial B_{\partial\operatorname{\mathbf{R}}_+^n} (0,N)} \zeta \Big( \frac {|x|}R\Big) u^{1-\theta} (x) \Big\langle x, \frac x{|x|}\Big\rangle d\sigma.
\end{split}$$ For fixed $R$, by taking $N> 2R$, we deduce from the definition of $\zeta$ that the last term on the right hand side of vanishes. $III =0$. Therefore, taking the limit as $N \to +\infty$ gives $$\label{eq-p5-eq3'}
\begin{split}
\int_{ \partial\operatorname{\mathbf{R}}_+^n} \zeta \Big( \frac {|x|}R\Big) \langle \nabla u^{1-\theta} (x), x \rangle dx = & - (n-1) \int_{ \partial\operatorname{\mathbf{R}}_+^n } \zeta \Big( \frac {|x|}R\Big) u^{1-\theta} (x) dx \\
&- \int_{ \partial\operatorname{\mathbf{R}}_+^n } \Big\langle \nabla \zeta \Big( \frac {|x|}R\Big) , x \Big\rangle u^{1-\theta} (x) dx \\
=& I + II.
\end{split}$$ To estimate $II$, we note by a standard computation that $$\Big|\Big\langle \nabla \zeta \Big( \frac {|x|}R\Big) , x \Big\rangle\Big| \leqslant \frac {2|x|}R$$ holds. From this one can conclude that $II \to 0$ as $R \to +\infty$ since $$\begin{split}
\int_{\partial\operatorname{\mathbf{R}}_+^n} \Big\langle \nabla \zeta \Big( \frac {|x|}R\Big) , x \Big\rangle u^{1-\theta} (x) dx = & \int_{\partial\operatorname{\mathbf{R}}_+^n \cap \{R \leqslant |x| \leqslant 2R\}} \Big\langle \nabla \zeta \Big( \frac {|x|}R\Big) , x \Big\rangle u^{1-\theta} (x) dx\\
\leqslant & 4 \int_{\partial\operatorname{\mathbf{R}}_+^n \cap \{R \leqslant |x| \leqslant 2R\}} u^{1-\theta} dx.
\end{split}$$ Thus, by the dominated convergence theorem, we obtain $$\label{eq-p5-eq4}
\begin{split}
\lim_{R \to +\infty} \int_{\partial\operatorname{\mathbf{R}}_+^n} \zeta& \Big( \frac {|x|}R\Big) \langle \nabla u^{1-\theta} (x), x \rangle dx = - (n-1) \int_{\partial\operatorname{\mathbf{R}}_+^n} u^{1-\theta} (x) dx.
\end{split}$$ As a consequence of , we obtain $$\label{eq-p5-eqKeyU}
\int_{\partial\operatorname{\mathbf{R}}_+^n} \langle \nabla u^{1-\theta} (x), x \rangle dx = - (n-1) \int_{\partial\operatorname{\mathbf{R}}_+^n} u^{1-\theta} (x) dx.$$ A similar argument shows $$\label{eq-p5-eqKeyV}
\int_{ \operatorname{\mathbf{R}}_+^n} \langle \nabla v^{1-\kappa} (y), y \rangle dy = - n \int_{ \operatorname{\mathbf{R}}_+^n} v^{1-\kappa} (y) dy.$$ Notice that by the Fubini theorem, there holds $$\label{eq-p5-eqKeyIdentity}
\begin{split}
\frac{\kappa}{\kappa-1} \int_{\operatorname{\mathbf{R}}_+^n} \langle \nabla v^{1-\kappa} (y), y \rangle dy =& \int_{\operatorname{\mathbf{R}}_+^n} \langle \nabla v^{-\kappa} (y), y \rangle v(y) dy \\
=& \int_{\operatorname{\mathbf{R}}_+^n} \langle \nabla v^{-\kappa} (y), y \rangle \int_{\partial \operatorname{\mathbf{R}}_+^n} |x-y|^\lambda u(x)^{-\theta} dx dy\\
=& \int_{\partial \operatorname{\mathbf{R}}_+^n} u(x)^{-\theta} \int_{\operatorname{\mathbf{R}}_+^n} |x-y|^\lambda \langle \nabla v^{-\kappa} (y), y \rangle dy dx.
\end{split}$$ For $\mu>0$, we set $y=\mu z$. A simple variable change tells us that $$\begin{split}
u(\mu x) =& \int_{ \operatorname{\mathbf{R}}_+^n} |\mu x-y|^\lambda v(y)^{-\kappa} dy = \mu^{n+\lambda} \int_{ \operatorname{\mathbf{R}}_+^n} |x-z|^\lambda v(\mu z)^{-\kappa} dz.
\end{split}$$ Differentiating with respect to $\mu$ gives $$\begin{split}
\langle x, (\nabla u) (\mu x) \rangle =& (n+\lambda) \mu^{n+\lambda-1} \int_{ \operatorname{\mathbf{R}}_+^n} |x-z|^\lambda v(\mu z)^{-\kappa} dz\\
&+ \mu^{n+\lambda } \int_{ \operatorname{\mathbf{R}}_+^n} |x-z|^\lambda \langle z, (\nabla v^{-\kappa})(\mu z) \rangle dz,
\end{split}$$ which implies, after setting $\mu =1$ and using , the following $$\label{eq-p5-eqGradientUIdentity}
\begin{split}
\langle x, \nabla u ( x) \rangle =& (n+\lambda) u(x) + \int_{ \operatorname{\mathbf{R}}_+^n} |x-z|^\lambda \langle z, (\nabla v^{-\kappa})( z) \rangle dz.
\end{split}$$ Hence, by multiplying both sides of by $u^{-\theta}$ and making use of , we have just shown that $$\label{eq-p5-eqKeyIdentity2}
\begin{split}
\int_{\partial\operatorname{\mathbf{R}}_+^n} \langle \nabla u (x), x \rangle u^{ -\theta} (x) dx =& (n+\lambda) \int_{\partial\operatorname{\mathbf{R}}_+^n} u^{1-\theta} (x) dx + \frac{\kappa}{\kappa-1} \int_{\operatorname{\mathbf{R}}_+^n} \langle \nabla v^{1-\kappa} (y), y \rangle dy.
\end{split}$$ Thanks to and , it follows from that $$- \frac{n-1}{1-\theta} \int_{\partial\operatorname{\mathbf{R}}_+^n} u^{1-\theta} (x) dx =(n+\lambda) \int_{\partial\operatorname{\mathbf{R}}_+^n} u^{1-\theta} (x) dx - \frac{\kappa n}{\kappa-1} \int_{\operatorname{\mathbf{R}}_+^n} v^{1-\kappa} (y) dy.$$ Thus, we have just proved that $$\Big( n+\lambda - \frac{n-1}{ 1-\theta } \Big) \int_{\partial\operatorname{\mathbf{R}}_+^n} u^{1-\theta} (x) dx = \frac{\kappa n}{\kappa-1} \int_{\operatorname{\mathbf{R}}_+^n} v^{1-\kappa} (y) dy.$$ Thanks to Lemma \[lem-IntegralU=IntegralV\], we conclude that $(1 - 1/n)/(\theta - 1) + 1/(\kappa - 1) = \lambda /n$ as claimed.
It is worth noticing that the necessary condition is exactly the same as the condition in the reversed HLS inequality if one replaces $\kappa$ and $\theta$ by $1/(1-r)$ and $1/(1-p)$ respectively.
\[lem->0>0\] For $n \geqslant 1$, $\lambda>0$, $ \kappa>0$ and $\theta>0$ satisfying $\theta = \kappa - 2/\lambda$, there holds $$2n - \kappa \lambda + \lambda \geqslant 0, \quad 2n -2 - \theta \lambda + \lambda \geqslant 0.$$
Suppose that $\kappa \geqslant 1 + 2n/\lambda$ and hence $\theta \geqslant 1 + (2n - 2)/\lambda$, we make use of Lemma \[lemNECESSARY\] to conclude that $\lambda>0$ and $ \kappa>0$ fulfill , that is, $(n - 1)/(n(\theta - 1)) +1/(\kappa - 1) = \lambda /n$. Resolving this equation with the condition $\theta = \kappa - 2/\lambda$ gives $\kappa = 1 + 2n/\lambda$ and $\theta = 1 + (2n - 2)/\lambda$. From this we obtain the equalities since $2n - \kappa \lambda + \lambda = 0$ and $2n -2 - \theta \lambda + \lambda = 0$.
Otherwise, there holds $\kappa < 1 + 2n/\lambda$ and hence $\theta < 1 + (2n - 2)/\lambda$. Form this, it is immediate to see that $2n - \kappa \lambda + \lambda > 0$ and $2n -2 - \theta \lambda + \lambda > 0$.
A classification of solutions of (\[eqIntegralSystem\]): Proof of Proposition \[thmCLASSIFICATION\]
===================================================================================================
Recall that $\kappa = 1 + 2n/\lambda$ and that $\theta = 1 + (2n - 2)/\lambda$, thanks to and our hypothesis $\kappa =\lambda + 2/\theta$.
The method of moving spheres for systems
----------------------------------------
Let $w$ be a positive function on $\overline{\operatorname{\mathbf{R}}_+^n}$ where we denote $\overline{\operatorname{\mathbf{R}}_+^n} = \operatorname{\mathbf{R}}_+^n \cup \partial \operatorname{\mathbf{R}}_+^n$. For $x \in \partial \operatorname{\mathbf{R}}_+^n$ and $\nu>0$ we define $$\label{eqFunctionChange}
w_{x,\nu}( \xi ) = \big( |\xi-x| / \nu \big)^\lambda w( \xi^{x,\nu} )$$ for all $\xi \in \overline{\operatorname{\mathbf{R}}_+^n}$ where $\xi^{x,\nu}$ is the Kelvin transformation of $\xi$ with respect to the ball $B_{ \operatorname{\mathbf{R}}_+^n} (x,\nu) \subset \operatorname{\mathbf{R}}_+^n$, given as follows $$\label{eqVariableChange}
{\xi^{x,\nu}} = x + \nu^2\frac{\xi - x}{{| \xi - x |}^2}.$$ It is important to note that in the whole moving spheres arguments in this section, only spheres centered on the boundary hyperplane $\partial \operatorname{\mathbf{R}}_+^n$ can be used. This provides a reason why we cannot capture further information for $v$ out of $\partial \operatorname{\mathbf{R}}_+^n$ as indicated in Proposition \[thmCLASSIFICATION\]. Clearly, upon the change of variable $y=z^{x,\nu}$ with $z\in \overline{\operatorname{\mathbf{R}}_+^n}$, we then have $$\label{eqJacobian}
dy=
\begin{cases}
\left( \nu /|z-x| \right)^{2n} dz &\text{ if } z \in \operatorname{\mathbf{R}}_+^n\\
\left( \nu /|z-x| \right)^{2n-2} dz &\text{ if } z \in \partial\operatorname{\mathbf{R}}_+^n.
\end{cases}$$
\[lem0\] For any solutions $(u,v)$ of , we have $${u_{x,\nu}}(\xi ) = \int_{\operatorname{\mathbf{R}}_+^n} | \xi - z|^\lambda v_{x,\nu} (z)^{ - \kappa} dz$$ for any $\xi \in \partial\operatorname{\mathbf{R}}_+^n$ and $${v_{x,\nu}}(\eta ) = \int_{\partial\operatorname{\mathbf{R}}_+^n} | \eta - z|^\lambda u_{x,\nu} (z)^{ - \theta} dz$$ for any $\eta \in \operatorname{\mathbf{R}}_+^n$.
Using our system , we obtain $$\begin{split}
{u_{x,\nu}}(\xi ) =& {\left( {\frac {|\xi - x|}{\nu}} \right)^\lambda}u({\xi ^{x,\nu}}) ={\left( {\frac {|\xi - x|}{\nu}} \right)^\lambda}\int_{\operatorname{\mathbf{R}}_+^n} | {\xi ^{x,\nu}} - y|^\lambda v{(y)^{ - \kappa}}dy\\
=& \int_{\operatorname{\mathbf{R}}_+^n} | \xi - z|^\lambda {\left( {\frac{\lambda } {|z - x|} } \right)^{2n - \kappa \lambda + \lambda}}{v_{x,\nu}}{(z)^{ - \kappa}}dz.
\end{split}$$ From this we obtain the desired formula for $u$, thanks to $2n - \kappa \lambda + \lambda =0$. The formula for $v$ follows the same line as above with a little difference since we need to integrate over $\partial\operatorname{\mathbf{R}}_+^n$.
Next, we estimate $u_{x,\nu}(\xi ) - u(\xi )$ and $v_{x,\nu}(\eta ) - v (\eta )$. Since the computation is elementary and well-known in other contexts, we omit its details.
\[lem1\] For any solutions $(u,v)$ of any $\lambda >0$ and any $x \in \partial\operatorname{\mathbf{R}}_+^n$, we have $${u_{x,\nu}}(\xi ) - u(\xi ) = \int_{\Sigma_{x,\nu}^n} k(x,\nu ;\xi ,z) \big[v(z)^{-\kappa} - {v_{x,\nu}}{(z)^{ - \kappa}} \big] dz,$$ for any $\xi \in \partial\operatorname{\mathbf{R}}_+^n$ and $${v_{x,\nu}}(\eta ) - v (\eta ) = \int_{\Sigma_{x,\nu}^{n-1}} k(x,\nu ;\eta ,z) \big[ u(z)^{-\theta} - {u_{x,\nu}}{(z)^{ - \theta}} \big] dz,$$ for any $\eta \in \operatorname{\mathbf{R}}_+^n$ where $$k(x,\nu ;\zeta ,z) = \left( {\frac {|\zeta - x|}{\nu}} \right)^\lambda |{\zeta ^{x,\nu}} - z|^\lambda - |\xi - z|^\lambda .$$ Moreover, $k(x,\nu; \zeta, z)>0$ for any $|\zeta - x| > \lambda>0$ and $|z - x| > \lambda>0$.
In the following lemma, we prove that the method of moving spheres can get started starting from a very small radius.
\[lemStartMS\] For each $x \in \partial \operatorname{\mathbf{R}}_+^n$, there exists some $\nu_0(x)>0$ such that for any $\nu \in (0, \nu_0(x))$ $${u_{x,\nu}}(\xi) \geqslant u(\xi)$$ for any point $\xi \in \Sigma_{x,\nu}^{n-1}$ and $${v_{x,\nu}}(\eta) \geqslant v(\eta)$$ for any point $\eta \in \Sigma_{x,\nu}^n$.
Since $u$ is a positive $C^1$-function in $\partial\operatorname{\mathbf{R}}_+^n$ and $\lambda>0$, there exists some $r_0>0$ small enough such that $$\nabla _\xi \big( |\xi-x|^{-\lambda/2} u(\xi) \big) \cdot (\xi-x) < 0$$ for all $\xi \in \partial\operatorname{\mathbf{R}}_+^n$ with $0<|\xi-x|<r_0$. Consequently, we can estimate $$\begin{split}
u_{x,\nu} (\xi) =& {\left( {\frac{|\xi-x|}{\nu}} \right)^\lambda}u (\xi^{x,\nu}) = |\xi-x|^{\lambda/2} |\xi^{x,\nu}-x|^{-\lambda/2} u (\xi^{x,\nu})> u(\xi)
\end{split}$$ for all $\xi \in \partial\operatorname{\mathbf{R}}_+^n$ with $0<\lambda < |\xi-x| < r_0$. Note that in the previous estimate, we made use of the fact that if $|\xi-x| > \lambda$ then $|\xi^{x,\nu} -x| < \lambda$. Note that for small $\nu_0 \in (0, r_0)$ and for each $0<\lambda<\nu_0$, we have $$u_{x, \lambda }(\xi) \geqslant {\left( {\frac{|\xi-x|}{\nu}} \right)^\lambda}\mathop {\inf }\limits_{B(x, r_0)} u \geqslant u(\xi)$$ for all $|\xi-x| \geqslant r_0$. Hence, we have just shown that $u_{x, \lambda }(\xi) \geqslant u(\xi)$ for all point $\xi \in \partial\operatorname{\mathbf{R}}_+^n$ and any $\lambda$ such that $|\xi-x| \geqslant \lambda$ with $0 < \nu < \nu_0$. A similar argument also shows that $v_{x, \lambda }(\eta) \geqslant v(\eta)$ for all point $\eta \in \operatorname{\mathbf{R}}_+^n$ and any $\lambda$ such that $|\eta-x| \geqslant \lambda$ with $0 < \lambda< \lambda_1$ for some $\lambda_1 \in (0, r_1)$. Simply setting $\nu_0 (x) = \min\{ \nu_0, \lambda_1\}$ we obtain the desired result.
For each $x \in \partial \operatorname{\mathbf{R}}_+^n$ we define $$\overline \nu (x) = \sup \left\{ {\mu > 0:{u_{x,\nu}}(\xi) \geqslant u(\xi), {v_{x,\nu}}(\eta) \geqslant v(\eta), \forall 0 < \nu < \mu , \xi \in \Sigma_{x,\nu}^{n-1} , \eta \in \Sigma_{x,\nu}^{n} } \right\}.$$ In view of Lemma \[lemStartMS\] above, we get $0 < \overline \nu (x) \leqslant + \infty$. In the next few lemmas, we show that whenever $\nu (x)$ is finite for some point $x$, we can write down precisely the form of $(u,v)$.
\[lem3\] If $\overline \nu (x_0) <\infty$ for some point $x_0 \in \partial\operatorname{\mathbf{R}}_+^n$ then $$u_{x_0,\overline \nu (x_0)} \equiv u, \quad {v_{{x_0},\overline \nu ({x_0})}} \equiv v$$ in $\partial \operatorname{\mathbf{R}}_+^n$ and $\operatorname{\mathbf{R}}_+^n$, respectively. In addition, we obtain $q = 1 + 2n/p$.
By the definition of $\overline \nu (x_0)$, we know that $$\label{eqProof0}
{u_{x_0,\overline \nu (x_0)}}(\xi) \geqslant u(\xi), \quad {v_{x_0,\overline \nu (x_0)}}(\eta) \geqslant v(\eta)$$ for any $\xi \in \Sigma_{x_0,\overline \nu (x_0)}^{n-1}$ and $\eta \in \Sigma_{x_0,\overline \nu (x_0)}^n$. In view of Lemma \[lem1\], we obtain $$\label{eqProof1}
\begin{split}
{u_{x_0,\overline \nu (x_0)}}(\xi) - u(\xi) = \int_{ \Sigma_{x_0,\overline \nu (x_0)}^n } & k( x_0,\overline \nu (x_0);\xi,z ) \big[v(z)^{-\kappa} - {v_{x_0,\overline \nu (x_0)}} (z)^{ -\kappa} \big] dz,
\end{split}$$ and $$\label{eqProof2}
\begin{split}
{v_{x_0,\overline \nu (x_0)}}(\eta) - v(\eta) = \int_{ \Sigma_{x_0,\overline \nu (x_0)}^{n-1} } & k( x_0,\overline \nu (x_0);\eta,z ) \big[u(z)^{-\theta} - {u_{x_0,\overline \nu (x_0)}}{(z)^{ - \theta}}\big] dz.
\end{split}$$ Keep in mind that $2n - \kappa \lambda + \lambda \geqslant 0$ and $2n-2 -\theta \lambda + \lambda \geqslant 0$ by Lemma \[lem->0>0\]; hence there are two possible cases:
**Case 1**. Either ${u_{x_0,\overline \nu (x_0)}}(\xi) = u(\xi)$ for any $\xi \in \Sigma_{x_0,\overline \nu (x_0)}^{n-1}$ or ${v_{x_0,\overline \nu (x_0)}}(\eta) = v(\eta)$ for any $\eta \in \Sigma_{x_0,\overline \nu (x_0)}^n$. Without loss of generality, we assume that the formal case occurs. Using and the positivity of the kernel $k$, we get that $2n - \kappa \lambda + \lambda=0$ and that ${v_{x_0,\overline \nu (x_0)}}(\eta) = v(\eta)$ for any $\eta \in \Sigma_{x_0,\overline \nu (x_0)}^n$. Hence by we conclude that ${u_{x_0,\overline \nu (x_0)}}(\xi) = u(\xi)$ in the whole $\partial\operatorname{\mathbf{R}}_+^n$. A similar argument also shows that $2n-2 -\theta \lambda + \lambda=0$ and that ${v_{x_0,\overline \nu (x_0)}}(\eta) = v(\eta)$ in $\operatorname{\mathbf{R}}_+^n$ and we are done.
**Case 2**. Or ${u_{x_0,\overline \nu (x_0)}}(\xi) > u(\xi)$ for any $\xi \in \Sigma_{x_0,\overline \nu (x_0)}^{n-1}$ and ${v_{x_0,\overline \nu (x_0)}}(\eta) > v(\eta)$ for any $\eta \in \Sigma_{x_0,\overline \nu (x_0)}^n$. In this case, we derive a contradiction by showing that we can slightly move spheres a little bit over $\overline \nu (x_0)$ which then violates the definition of $\overline \nu (x_0)$.
In order to achieve that goal, first we can estimate $$\label{eqProof3}
\begin{split}
{u_{x_0,\overline \nu (x_0)}}(\xi) &- u(\xi) \geqslant \int_{\Sigma_{x_0,\overline \nu (x_0)}^n} k( x_0,\overline \nu (x_0);\xi, z )\Big [v(z)^{-\kappa} - v_{x_0,\overline \nu (x_0)} (z)^{-\kappa}\Big] dz,
\end{split}$$ thanks to the positivity of the kernel $k$.
**Estimate of $u_{x_0, \nu} - u$ outside $B_{\partial\operatorname{\mathbf{R}}_+^n}(x_0,\overline \nu (x_0) + 1)$.** First, by the Fatou lemma and , we obtain $$\begin{split}
\mathop {\lim \inf }\limits_{|\xi| \to +\infty }& \big( |\xi|^{-\lambda} ({u_{x_0,\overline \nu (x_0)}} - u)(\xi) \big) \\
\geqslant & \mathop {\lim \inf }\limits_{|\xi| \to +\infty } \int_{\Sigma_{x_0,\overline \nu (x_0)}^n} |\xi|^{-\lambda} k( x_0,\overline \nu (x_0);\xi,z ) \Big[v(z)^{-\kappa} - v_{x_0,\overline \nu (x_0)} (z)^{-\kappa}\Big]dz \\
\geqslant& \int_{\Sigma_{x_0,\overline \nu (x_0)}^n} \big( {{\big( |z| / \overline \nu (x_0) \big) ^\lambda} - 1} \big) \big[v(z)^{-\kappa} - v_{x_0,\overline \nu (x_0)} (z)^{-\kappa} \big]dz > 0.
\end{split}$$ As a consequence, outside a large ball, we would have $({u_{x_0,\overline \nu (x_0)}} - u)(\xi) \gtrsim |\xi|^\lambda$ while in that ball and outside of $B(x_0, \overline \nu (x_0) + 1)$ we would also have $(u_{x_0,\overline \nu (x_0)} - u)(\xi) \gtrsim |\xi|^\lambda$ thanks to the smoothness of $u_{x_0,\overline \nu (x_0)} - u$ and our assumption $u_{x_0,\overline \nu (x_0)}(\xi) > u(\xi)$. Therefore, there exists some $\varepsilon_1 >0$ such that $$({u_{x_0,\overline \nu (x_0)}} - u)(\xi) \geqslant {\varepsilon _1}|\xi|^\lambda$$ for all $|\xi-x_0| \geqslant \overline \nu (x_0) + 1$. Recall that ${u_{x_0,\overline \nu (x_0)}}(\xi) = (|x_0 - \xi|/\lambda)^\lambda u({\xi^{x_0,\overline \nu (x_0)}})$; hence there exists some $\varepsilon_2 \in (0, \varepsilon_1)$ such that $$\label{eqProof4}
\begin{split}
(u_{x_0, \nu} - u)(\xi) =& ({u_{x_0,\overline \nu (x_0)}} - u)(\xi) + ({u_{x_0, \nu}} - {u_{x_0,\overline \nu (x_0)}})(\xi) \\
\geqslant& {\varepsilon _1}|\xi|^\lambda + ({u_{x_0, \nu}} - {u_{x_0,\overline \nu (x_0)}})(\xi) \geqslant \frac{{{\varepsilon _1}}}{2}|\xi|^\lambda
\end{split}$$ for all $|\xi-x_0| \geqslant \overline \nu (x_0) + 1$ and all $\lambda \in (\overline \nu (x_0),\overline \nu (x_0) + {\varepsilon _2})$. Repeating the above arguments shows that is also valid for $v_{x_0, \nu} - v$, that is $$\label{eqProof4-ForV}
\begin{split}
(v_{x_0, \nu} - v)(\eta) \geqslant \frac{{{\varepsilon _1}}}{2}|\eta|^\lambda
\end{split}$$ for a possibly new constant $\varepsilon_1>0$.
**Estimate of $u_{x_0, \nu} - u$ inside $B_{\partial\operatorname{\mathbf{R}}_+^n}(x_0,\overline \nu (x_0) + 1)$.** Now for $\varepsilon \in (0,{\varepsilon _2})$ to be determined later and for $\lambda \in (\overline \nu (x_0),\overline \nu (x_0) + \varepsilon ) \subset (\overline \nu (x_0),\overline \nu (x_0) + {\varepsilon _2})$ and for $\lambda \leqslant |\xi-x_0| \leqslant \overline \nu (x_0) + 1$, from , we estimate $$\begin{split}
({u_{x_0, \nu}} - u)(\xi) \geqslant& \int_{\Sigma_{x_0,\overline \nu (x_0)}^n} k( x_0, \nu;\xi,z)[v(z)^{-\kappa} - {v_{x_0, \nu}}{(z)^{ - \kappa}}]dz \\
\geqslant &\int_{\overline \nu (x_0) + 1 \geqslant |z - x_0| \geqslant \lambda } k( x_0, \nu; \xi , z)[v(z)^{-\kappa} - {v_{x_0, \nu}}{(z)^{ - \kappa}}]dz \\
& + \int_{\overline \nu (x_0) + 3 \geqslant |z - x_0| \geqslant \overline \nu (x_0)+ 2} k( x_0, \nu; \xi , z)[v(z)^{-\kappa} - {v_{x_0, \nu}}{(z)^{ - \kappa}}]dz \\
\geqslant &\int_{\overline \nu (x_0) + 1 \geqslant |z - x_0| \geqslant \lambda } k( x_0, \nu; \xi , z)[v_{x_0,\overline \nu (x_0)} (z)^{-\kappa} - {v_{x_0, \nu}}{(z)^{ - \kappa}}]dz \\
&+ \int_{\overline \nu (x_0) + 3 \geqslant |z - x_0| \geqslant \overline \nu (x_0)+ 2} k( x_0, \nu; \xi , z)[v(z)^{-\kappa} - {v_{x_0, \nu}}{(z)^{ - \kappa}}]dz \\
=& I + II.
\end{split}$$ As we shall see later, there holds $I+II \geqslant 0$ provided $\varepsilon >0$ is small enough. To see this, we estimate $I$ and $II$ term by term.
**Estimate of $II$**. Thanks to , there exists $\delta_1>0$ such that $\big( v^{-\kappa} - v_{x_0,\nu}^{-\kappa} \big)(z) \geqslant \delta_1$ for any $\overline \nu (x_0) +2 \leqslant |z-x_0| \leqslant \overline \nu (x_0) +3$. By the definition of $k$ given in Lemma \[lem1\] we note that $$k(x_0,\nu; \xi ,z)=k(0, \nu; \xi -x_0,z -x_0)$$ and that $$\nabla _\xi k(0, \nu ;\xi ,z) \cdot y \big|_{|\xi| = \nu } = p |\xi -z| ^{\lambda-2} \big(|z|^2 - |\xi|^2\big) > 0$$ for all $\overline \nu (x_0) +2 \leqslant |z| \leqslant \overline \nu (x_0) +3$. Hence, there exists some constant $\delta_2>0$ independent of $\varepsilon$ such that $$k(0, \nu ;\xi,z) \geqslant {\delta _2}(|\xi| - \nu )$$ for all $\overline \nu (x_0) \leqslant \nu \leqslant |\xi| \leqslant \overline \nu (x_0) + 1$ and all $\overline \nu (x_0) + 2 \leqslant |z| \leqslant \overline \nu (x_0) + 3$. Simply replacing $y$ by $y-x_0$ and $z$ by $z-z_0$ and making use of the rule $k(x_0,\nu; \xi,z)=k(0, \nu; \xi-x_0,z -x_0)$, we obtain with the same constant $\delta_2>0$ as above the following estimate $$k( x_0, \nu; \xi , z) \geqslant \delta_2 (|\xi-x_0| - \nu )$$ for all $\overline \nu (x_0) \leqslant \nu \leqslant |y-x_0| \leqslant \overline \nu (x_0) + 1$ and all $\overline \nu (x_0) + 2 \leqslant |z-x_0| \leqslant \overline \nu (x_0) + 3$. Thus, we have just proved that $$\label{eqEstimateII}
II \geqslant \delta_1 \delta_2 (|\xi-x_0|-\nu) \int_{\overline \nu (x_0) + 3 \geqslant |z - x_0| \geqslant \overline \nu (x_0) + 2} dz.$$
**Estimate of $I$**. To estimate $I$, we first observe that $| v_{x_0, \nu } ^{ - \kappa} - {v^{ - \kappa}}|(z) \lesssim \nu - \overline \nu (x_0) \lesssim \varepsilon$ for all $z$ satisfying $\overline \nu (x_0) \leqslant \nu \leqslant |z-x_0| \leqslant \overline \nu (x_0) + 1$ and all $\overline \nu (x_0) \leqslant \nu \leqslant \overline \nu (x_0) + \varepsilon$ and that $$\begin{split}
\int_{\lambda \leqslant |z-x_0| \leqslant \overline \nu (x_0)+ 1} & {k( x_0, \nu; \xi , z)dz} \\
= & \int_{\lambda \leqslant |z| \leqslant \overline \nu(x_0) + 1} {k(0, \nu ; \xi-x_0,z)dz}\\
\leqslant & \int_{\lambda \leqslant |z| \leqslant \overline \nu (x_0)+ 1} \Big| \big( |\xi-x_0| /\lambda \big)^\lambda-1 \Big| { |(\xi-x_0)^{0,\lambda } - z|^\lambda dz} \\
& + \int_{\lambda \leqslant |z| \leqslant \overline \nu(x_0) + 1} { \big(|(\xi-x_0)^{0,\lambda } - z|^\lambda - |(\xi-x_0) - z|^\lambda \big)dz} \\
\leqslant& C(|\xi-x_0| - \lambda ) + C|{(\xi-x_0)^{0,\lambda }} - (\xi-x_0)| \\
\leqslant & C(|\xi-x_0| - \lambda ).
\end{split}$$ where $C>0$ is constant independent of $\varepsilon$. Thus, we obtain $$\label{eqEstimateI}
I \geqslant -C\varepsilon \int_{\overline \nu (x_0) + 1 \geqslant |z - x_0| \geqslant \lambda } k( x_0, \nu; \xi , z) dz.$$ By combining and , for some small $\varepsilon>0$ we have $$\begin{split}
({u_{x_0, \nu }} - u)(\xi) \geqslant& \Big( \delta_1 \delta_2 \int_{\overline \nu (x_0) + 3 \geqslant |z - x_0| \geqslant \overline \nu (x_0) + 2} dz -C\varepsilon \Big) (|\xi-x_0|-\nu) \geqslant 0
\end{split}$$ for $\overline \nu (x_0) \leqslant \nu \leqslant \overline \nu (x_0) + \varepsilon$ and $\nu \leqslant |y-x_0| \leqslant \overline \nu (x_0) + 1$.
**Estimates of $u_{x_0, \nu} - u$ in $B_{\partial\operatorname{\mathbf{R}}_+^n}(x_0,\overline \nu (x_0) + 1)$ and $v_{x_0, \nu} - v$ in $B_{\operatorname{\mathbf{R}}_+^n}(x_0,\overline \nu (x_0) + 1)$.** Combining the preceding estimate for $u_{x_0, \nu} - u$ inside the ball $B(x_0, \overline \nu (x_0) + 1)$ and above gives $$\begin{split}
({u_{x_0, \nu}} - u)(\xi) \geqslant 0
\end{split}$$ for $\overline \nu (x_0) \leqslant \nu \leqslant \overline \nu (x_0) + \varepsilon$ and $\nu \leqslant |y-x_0|$. Again by repeating the whole procedure above for the difference $v_{x_0,\nu} -v$, we can conclude that $$(v_{x_0,\nu} -v)(y) \geqslant 0$$ for $\overline \nu (x_0) \leqslant \nu \leqslant \overline \nu (x_0) + \varepsilon$ and $\nu \leqslant |y-x_0|$ where $\varepsilon$ could be smaller if necessary; thus giving us a contradiction to the definition of $\overline\nu (x_0)$.
In the last lemma of the current section, we prove that whenever $\overline \nu (x_0) <\infty$ for some point $x_0 \in \partial \operatorname{\mathbf{R}}_+^n$, there must hold $\overline \nu (x) <\infty$ for any point $x \in \partial \operatorname{\mathbf{R}}_+^n$.
\[lemLamdaVanishAtEveryPoint\] If $\overline \nu (x_0) <\infty$ for some point $x_0 \in \partial \operatorname{\mathbf{R}}_+^n$ then $\overline \nu (x) <\infty$ for any point $x \in \partial \operatorname{\mathbf{R}}_+^n$; hence $${u_{x,\overline \nu (x)}} \equiv u, \quad {v_{x,\overline \nu (x)}} \equiv v$$ for all $x \in \partial \operatorname{\mathbf{R}}_+^n$.
Suppose that there exists some $x_0 \in \partial \operatorname{\mathbf{R}}_+^n$ such that $\overline\nu (x_0)<\infty$, by Lemma \[lem3\] and for $\xi \in \partial\operatorname{\mathbf{R}}_+^n$ with $|\xi|$ sufficiently large, we have $$\begin{aligned}
{| \xi |^{-\lambda} }u(y) &= | \xi |^{-\lambda} u_{x_0,\overline \nu (x_0)} (\xi)\\
& = {| \xi |^{-\lambda} }{\left( {\frac{{\overline \nu ({x_0})}}{{\left| {\xi - {x_0}} \right|}}} \right)^{-\lambda} } u\Big( {{x_0} + \lambda {{({x_0})}^2}\frac{\xi - x_0}{{{{\left| {\xi - x_0} \right|}^2}}}} \Big) \\
&= \overline \nu {({x_0})^{-\lambda}}{\left( {\frac{{\left| \xi- x_0 \right|}}{| \xi |}} \right)^\lambda } u\Big( {{x_0} + \lambda {{({x_0})}^2}\frac{{\xi - x_0}}{{{{\left| {\xi - x_0} \right|}^2}}}} \Big) \end{aligned}$$ which implies $$\label{eq14}
\mathop {\lim }\limits_{| \xi | \to +\infty } {| \xi |^{-\lambda} }u(\xi) = \overline \nu {({x_0})^{-\lambda}}u({x_0}).$$ Repeating the above argument then gives $$\label{eq15}
\mathop {\lim }\limits_{| \eta | \to +\infty } {| \eta |^{-\lambda} } v(\eta) = \overline \nu {({x_0})^{-\lambda}} v(x_0).$$ Let $x \in \partial \operatorname{\mathbf{R}}_+^n$ be arbitrary, by the definition of $\overline\nu (x)$ we get $$u_{x,\nu} (\xi) \geqslant u(\xi), \quad v_{x,\nu} (\eta) \geqslant v(\eta)$$ for all $0 < \nu < \overline \nu (x)$ and all $\xi \in \Sigma_{x, \nu }^{n-1}$ and $\eta \in \Sigma_{x, \nu }^n$. Then by a direct computation and thanks to , one can easily see that $$\label{eq16}
\begin{split}
\mathop {\liminf }\limits_{| \xi | \to +\infty } {| \xi |^{-\lambda}}u(\xi) &\leqslant \mathop {\lim \inf }\limits_{| \xi | \to +\infty } {| \xi |^{-\lambda}}{u_{x,\nu}}(\xi) \\
&= \mathop {\liminf }\limits_{| \xi | \to +\infty } {| \xi |^{-\lambda} }{\left( {\frac{\nu}{{\left| {\xi - x} \right|}}} \right)^{-\lambda} }u\Big( {x + \nu^2 \frac{\xi-x}{{{{\left| {\xi - x } \right|}^2}}}} \Big) \\
&= \nu ^{-\lambda} u(x)
\end{split}$$ for all $0 < \lambda < \overline \nu (x)$. Combining and gives $ \overline \nu (x_0)^{-\lambda} u( x_0 ) \leqslant \nu ^{-\lambda} u(x)$ for all $0 < \lambda < \overline \nu (x)$. From this, we conclude $\overline \nu (x) < +\infty$ for all $x \in {\operatorname{\mathbf{R}}^n}$ as claimed.
Proof of Proposition \[thmCLASSIFICATION\]
------------------------------------------
To conclude Proposition \[thmCLASSIFICATION\], we make use of the following three lemmas from [@lz2003 Appendix B]. The first lemma concerns functions $f$ satisfying the inequality $f_{x,\nu}(\xi) \leqslant f(\xi)$, which can be the case in view of Lemma \[lemStartMS\].
\[lemKey1\] For $\nu \in \operatorname{\mathbf{R}}$ and $f$ a function defined on $\partial\operatorname{\mathbf{R}}_+^n$ and valued in $[-\infty, +\infty]$ satisfying $$\Big( \frac{\nu }{ | \xi - x | } \Big)^\lambda f\Big( x + \nu^2\frac{\xi - x}{ | \xi - x |^2} \Big) \leqslant f(\xi)$$ for all $x, \xi \in \partial\operatorname{\mathbf{R}}_+^n$ satisfying $| x - \xi | > \nu > 0$. Then $f$ is constant or is identical to infinity.
In the second lemma, if the inequality in Lemma \[lemKey1\] becomes equality, then we can characterize the function $f$ completely; see also Lemma \[lem3\].
\[lemKey2\] For $\nu \in \operatorname{\mathbf{R}}$ and $f$ a continuous function in $\partial\operatorname{\mathbf{R}}_+^n$. Suppose that for every $x\in \partial\operatorname{\mathbf{R}}_+^n$, there exists $\lambda(x)>0$ such that $$\Big( {\frac{{\nu (x)}}{ | \xi - x | }} \Big)^\lambda f\Big( {x + \nu (x)^2\frac{\xi - x}{{| \xi - x |}^2}} \Big) = f(\xi),$$ for all $\xi \in \partial\operatorname{\mathbf{R}}_+^n \backslash \{ x\} $. Then $$f(x) = \pm a{\big( d + | x - \overline x |^2 \big)^{-\lambda /2}}$$ for some $a \geqslant 0$, $d>0$ and $\overline x \in \partial \operatorname{\mathbf{R}}_+^n$.
It is worth noting that Lemma \[lemKey2\] also holds for a larger class consisting measures, known as the characterization of inversion invariant measures; see [@fl2010 Theorem 1.4]. The last lemma is in the same fashion of Lemma \[lemKey2\] above for functions defined on $\overline{\operatorname{\mathbf{R}}_+^n}$.
\[lemKey3\] For $\nu \in \operatorname{\mathbf{R}}$ and $f$ a function defined on $\overline{\operatorname{\mathbf{R}}_+^n}$ and valued in $[-\infty, +\infty]$ satisfying $$\Big( {\frac{\nu }{ | y - x | }} \Big)^\lambda f\Big( {x + \nu^2\frac{{y - x}}{{| y - x |}^2}} \Big) \leqslant f(y)$$ for all $x \in \partial \operatorname{\mathbf{R}}_+^n$ and $y \in \operatorname{\mathbf{R}}_+^n$ satisfying $| x - y | > \nu > 0$. Then $f$ restricted to $\partial \operatorname{\mathbf{R}}_+^n$ is constant or is identical to infinity. In other words, $f$ depends only on the last coordinate.
With all ingredients above, we are now in a position to prove Proposition \[thmCLASSIFICATION\]. First, there are two possible cases:
**Case 1.** If $\overline\nu (x)=\infty$, then for any $x \in \partial \operatorname{\mathbf{R}}_+^n$ we know that $u_{x,\nu}(\xi) \geqslant u(\xi)$ for all $\lambda > 0$ and for any $x \in \partial \operatorname{\mathbf{R}}_+^n$ and $\xi \in \partial \operatorname{\mathbf{R}}_+^n$ satisfying $|\xi-x| \geqslant \lambda$. By Lemma \[lemKey1\], $u$ must be constant, say $u_0$. Similarly, Lemma \[lemKey3\] tells us that $v$ depends only on the last coordinate in the sense that $$v(y) = v(0, y_n) = \int_{\partial \operatorname{\mathbf{R}}_+^n} \big| |x|^2 + y_n^2 \big|^\lambda (u_0)^{-\theta} dx .$$ Upon a change of variables, we deduce that $$v(y) = v(0, y_n) = C y_n^{2\lambda + n-1} \int_0^{+\infty} \big| \rho^2 + 1 \big|^\lambda \rho^{n-2} d\rho .$$ From this we conclude that $v(0, y_n) = +\infty$ provided $y_n\ne 0$ since $\lambda>0$ and $n-2 \geqslant 0$. Thus, $(u,v)$ does not solve .
**Case 2.** Otherwise, there exists some $x_0 \in \partial \operatorname{\mathbf{R}}_+^n$ such that $\overline\nu (x_0)<\infty$. Then by Lemma \[lemLamdaVanishAtEveryPoint\], we deduce that $\overline\nu (x)<\infty$ for any point $x \in \partial \operatorname{\mathbf{R}}_+^n$. We are now in a position to apply Lemma \[lemKey2\] to conclude that $u$ is of the form $$\label{eqForm4U}
u(x) = a ( b ^2 + | {x - \overline x } | ^2)^{\lambda/2}$$ for some $a , d >0$ and some point $\overline x \in \partial \operatorname{\mathbf{R}}_+^n$. Using the form of $u$ and the equation satisfied by $v$ in , we get $$v(y) = a \int_{\partial \operatorname{\mathbf{R}}_+^n} \frac{|x-y|^\lambda}{ ( b ^2 + | x - \overline x | ^2)^{n-1+ \lambda/2} } dx$$ for any $y \in \operatorname{\mathbf{R}}_+^n$. If we restrict $y$ to $\partial \operatorname{\mathbf{R}}_+^n$, we clearly get $$u(y|_{\partial \operatorname{\mathbf{R}}_+^n}) = a\int_{\partial \operatorname{\mathbf{R}}_+^n} \frac{|x-y|_{\partial \operatorname{\mathbf{R}}_+^n}|^\lambda}{ ( b^2 + | {x - \overline x } | ^2)^{n-1+ \lambda/2} } dx.$$ From this, we obtain $v(y|_{\partial \operatorname{\mathbf{R}}_+^n}) = u(y|_{\partial \operatorname{\mathbf{R}}_+^n})$. Hence, $v$ is of the following form $$\label{eqForm4V}
v(x,0)= a ( b^2 + | x - \overline x | ^2)^{\lambda/2}.$$ Our proof of Proposition \[thmCLASSIFICATION\] is now complete.
Finally, we conclude this section by giving the proof of Corollary \[explicit\]. Recall that under the current situation we have $\lambda =2$. By Proposition \[thmCLASSIFICATION\] and up to a constant multiplication, translation, and dilation, we know that the extremal function $f$ has the following form $$f(x) = (1 +|x|^2)^{-n},\quad x\in \partial \operatorname{\mathbf{R}}_+^n,$$ and thanks to our system $$g(y) = \alpha \Big(\int_{\partial\operatorname{\mathbf{R}}_+^n} |x-y|^2 f(x) dx\Big)^{-n-1},\quad y \in \operatorname{\mathbf{R}}_+^n,$$ for some constant $\alpha >0$. An easy computation shows that $$\label{eq:comp}
\int_{\partial\operatorname{\mathbf{R}}_+^n} (1+|x|^2)^{-n} dx = \int_{\partial\operatorname{\mathbf{R}}_+^n} (1+|x|^2)^{-n} |x|^2 dx = \frac{\pi^{(n-1)/2} \Gamma((n+1)/2)}{\Gamma(n)}.$$ Therefore we conclude that $$g(y) = \beta (1+|y|^2)^{-n-1}$$ for some $\beta > 0$. Denote $h(y) = (1+|y|^2)^{-n-1}$, it is evident that $$\label{eq:a0}
{\mathscr C}_{n,1-1/n, n/(n+1)}^+ = \frac{\int_{\operatorname{\mathbf{R}}_+^n}\int_{\partial\operatorname{\mathbf{R}}_+^n} f(x) |x-y|^2 h(y) dxdy}{\|f\|_{L^{(n-1)/n}(\partial\operatorname{\mathbf{R}}_+^n)} \|h\|_{L^{n/(n+1)}(\operatorname{\mathbf{R}}_+^n)}}.$$ We now estimate term by term. First, in view of , we have $$\label{eq:a1}
\int_{\operatorname{\mathbf{R}}_+^n}\int_{\partial\operatorname{\mathbf{R}}_+^n} f(x) |x-y|^2 h(y) dxdy = \frac{\pi^{(n-1)/2} \Gamma((n+1)/2)}{\Gamma(n)}\int_{\operatorname{\mathbf{R}}_+^n} (1+|y|^2)^{-n} dy.$$ Regarding to $\|f\|_{L^{(n-1)/n}(\partial\operatorname{\mathbf{R}}_+^n)}$ and $\|h\|_{L^{n/(n+1)}(\operatorname{\mathbf{R}}_+^n)}$, it is easy to check that $$\label{eq:a2}
\int_{\partial\operatorname{\mathbf{R}}_+^n} (1+|x|^2)^{-n+1} dx = \pi^{(n-1)/2} \frac{\Gamma((n-1)/2)}{\Gamma(n-1)},$$ and that $$\label{eq:a3}
\int_{\operatorname{\mathbf{R}}_+^n} (1+|y|^2)^{-n} dy = \frac{\pi^{n/2}}2 \frac{\Gamma(n/2)}{\Gamma(n)}.$$ Plugging , and into , we obtain as claimed.
A log-HLS inequality on half space: Proof of Theorem \[log-HLSonhalfspace\]
===========================================================================
Throughout this section, we choose $p = 2(n-1)/(2(n-1) + \lambda)$ and $r = 2n/(2n +\lambda)$. The benefit of this particular choice for $p$ and $r$ is that there exists an optimizer pair $(f,g)$ with $f(x)=u(x)^{1/(p-1)}=(1+|x|^2)^{1-n - \lambda/2}$ on $\partial \operatorname{\mathbf{R}}_+^n$ and $g(y',0)=v(y',0)^{1/(r-1)}=(1+|y'|^2)^{-n-\lambda/2}$, up to dilations and translations. Consider the stereographic projection $\mathcal{S}: \operatorname{\mathbf{R}}^n \to \mathbb S^n$ given by $$\mathcal{S}(x) = \Big(\frac{2x_1}{1+|x|^2},\frac{2x_2}{1+|x|^2},\ldots, \frac{1-|x|^2}{1+|x|^2}, \frac{2x_n}{1+|x|^2}\Big).$$ It is easy to see that $\mathcal{S}$ transforms $\operatorname{\mathbf{R}}_+^n$ into $\mathbb S^n_+ = \{(\xi_1,\cdots,\xi_{n+1})\in \mathbb S^n\, :\, \xi_{n+1} > 0\}$ and transforms $\partial\operatorname{\mathbf{R}}_+^n$ into $\mathbb S^n_0 = \{(\xi_1,\cdots,\xi_{n+1})\in \mathbb S^n\, :\, \xi_{n+1} = 0\}$ which can be identified with $\mathbb S^{n-1}$. It is well-known that the Jacobian of $\mathcal{S}$ at any $x\in \operatorname{\mathbf{R}}^n$ is $$J_{\mathcal{S}}(x) = \big(2/(1+|x|^2) \big)^{n}$$ and the Jacobian of $\mathcal{S}|_{\partial\operatorname{\mathbf{R}}_+^n}$ at any $y\in \partial\operatorname{\mathbf{R}}_+^n$ is nothing but $$J_{\mathcal{S}|_{\partial\operatorname{\mathbf{R}}_+^n}}(y) = \big(2/(1+|y|^2)\big)^{n-1}.$$ Let $f,g$ be nonnegative functions on $\partial\operatorname{\mathbf{R}}_+^n$ and $\operatorname{\mathbf{R}}_+^n$, respectively. We let $F:\mathbb S^n_0 \to \operatorname{\mathbf{R}}$ and $G: \mathbb S^n_+ \to \operatorname{\mathbf{R}}$ be functions given by the following $$f(y) = F(\mathcal{S}|_{\partial\operatorname{\mathbf{R}}_+^n}(y)) J_{\mathcal{S}|_{\partial\operatorname{\mathbf{R}}_+^n}}(y)^{1/p},\quad y\in \partial\operatorname{\mathbf{R}}_+^n,$$ and $$g(x) = G(\mathcal{S}(x)) J_{\mathcal{S}}(x)^{1/r},\quad x\in \operatorname{\mathbf{R}}_+^n.$$ We then can readily check that $$\label{eq:changevariable}
\int_{\partial\operatorname{\mathbf{R}}_+^n} f(y)^p dy = \int_{\mathbb S^n_0} F(\eta)^p d\eta,\quad \int_{\operatorname{\mathbf{R}}_+^n} g(x)^r dx = \int_{\mathbb S^n_+} G(\xi)^r d\xi,$$ where $d\xi$ and $d\eta$ denote the Lebesgue measure on $\mathbb S^n$ and on $\mathbb S_0^n$, respectively. (Note that the Lebesgue measure on $\mathbb S_0^n$ is also the Lebesgue measure on the sphere $\mathbb S^{n-1}$.) Recall that for $x,y\in \operatorname{\mathbf{R}}^n$, there holds $$\label{eq:crucialequality}
|\mathcal{S}(x) - \mathcal{S}(y)|^2 = \frac2{1+|y|^2} |x-y|^2 \frac{2}{1+|x|^2}.$$ From , we easily imply that $$\label{eq:transferonsphere}
\int_{\operatorname{\mathbf{R}}^n_+} \int_{\partial\operatorname{\mathbf{R}}_+^n} g(x) |x-y|^\lambda f(y) dx dy = \int_{\mathbb S^n_+}\int_{\mathbb S_0^n} G(\xi) |\xi-\eta|^\lambda F(\eta) d\xi d\eta.$$ Combining , , and gives us a spherical forms of the reversed HLS on the half space as follows $$\label{eq:sphericalform}
\int_{\mathbb S^n_+}\int_{\mathbb S_0^n} G(\xi) |\xi-\eta|^\lambda F(\eta) d\xi d\eta \geqslant C(n,\lambda) \|F\|_{L^p(\mathbb S^n_0)} \|G\|_{L^r(\mathbb S^n_+)},$$ for any nonnegative functions $F\in L^p(\mathbb S^n_0)$ and $G\in L^r(\mathbb S^n_+)$ with the sharp constant $$C(n,\lambda) = \|E_\lambda f_0\|_{L^q(\operatorname{\mathbf{R}}_+^n)} \|f_0\|_{L^p(\partial\operatorname{\mathbf{R}}_+^n)}^{-1}$$ and the optimizer function $$f_0(y) = \big(1+|y|^2\big)^{1-n-\lambda/2}.$$ Similar to the classical HLS inequality on $\operatorname{\mathbf{R}}^n$ which can be lifted to the sphere $\mathbb S^n$ for which the competing symmetries argument can be used. In view of the spherical form of , it seems that a competing symmetries argument could work in this scenario. Recall that $q = -2n/\lambda$, from this it is easy to check that $$\label{eq:constantonsphere}
C(n,\lambda) = |\mathbb S^{n-1}|^{-\lambda/2(n-1)} \Big(\int_{\mathbb S^n_+}\Big(\int_{\mathbb S^n_0} |\xi -\eta|^\lambda d\sigma(\eta)\Big)^{-2n/\lambda} d\xi\Big)^{-\lambda/2n},$$ where $d\sigma$ is the normalization of the Lebesgue measure on $\mathbb S^n_0$ in the sense that $\sigma(\mathbb S^n_0) = 1$.
The next Proposition gives us the behavior of the constant $C(n,\lambda)$ when $\lambda$ tends to zero.
\[constantnearzero\] Let $n\geqslant 2$, there holds $$\label{eq:nearzero}
C(n,\lambda) = 1-\frac\lambda{2(n-1)} \ln |\mathbb S^{n-1}| -\frac\lambda{2n} \ln\Big(\int_{\mathbb S^n_+} e^{-2n \int_{\mathbb S^n_0} \ln(|\xi-\eta|) d\sigma(\eta)} d\xi\Big) + o(\lambda)$$ with the error $o(\lambda)/\lambda \to 0$ as $\lambda\to 0$.
We first observe that there exists a constant $C > 0$ which is independent of $\xi\in \mathbb S^n_+$ such that $$\int_{\mathbb S^n_0} \big(\ln |\xi -\eta| \big)^2 d\sigma(\eta) \leqslant C$$ for all $\xi \in \mathbb S^n_+$. Hence $$\int_{\mathbb S^n_0} |\xi-\eta|^\lambda d\sigma(\eta) = 1 + \lambda \int_{\mathbb S^n_0} \ln |\xi-\eta| d\sigma(\eta) + o(\lambda)$$ uniformly in $\xi \in \mathbb S^n_+$ when $\lambda\to 0$. For the sake of simplicity, let us denote $$H(\xi) = \int_{\mathbb S^n_0} \ln |\xi-\eta| d\sigma(\eta)$$ for each $\xi \in \mathbb S^n_+$. Since $H$ is bounded, we obtain $$\int_{\mathbb S^n_0} |\xi-\eta|^\lambda d\sigma(\eta) = (1 + \lambda H(\xi))(1 + o(\lambda))$$ uniformly in $\xi \in \mathbb S^n_+$ when $\lambda\to 0$. Therefore, we have $$\Big(\int_{\mathbb S^n_+}\Big(\int_{\mathbb S^n_0} |\xi -\eta|^\lambda d\sigma(\eta)\Big)^{-2n/\lambda} d\xi\Big)^{-\lambda/2n} = (1+ o(\lambda)) \Big(\int_{\mathbb S^n_+}\left(1+\lambda H(\xi)\right)^{-2n/\lambda} d\xi\Big)^{-\lambda/2n}.$$ Thanks to the boundedness of $H$, we then have $$\big(1+ \lambda H(\xi) \big)^{-2n/\lambda} = e^{-2n H(\xi)} (1 + o(1))$$ uniformly in in $\xi \in S^n_+$ when $\lambda\to 0$. Hence $$\Big(\int_{\mathbb S^n_+}\big(1+\lambda H(\xi)\big)^{-2n/\lambda} d\xi\Big)^{-\lambda/2n} = (1+o(\lambda)) \Big(\int_{\mathbb S^n_+} e^{-2n H(\xi)} d\xi\Big)^{-\lambda/2n}.$$ Finally, we have $$\label{eq:nearzero1}
\Big(\int_{\mathbb S^n_+}\Big(\int_{\mathbb S^n_0} |\xi -\eta|^\lambda d\sigma(\eta)\Big)^{-2n/\lambda} d\xi\Big)^{-\lambda/2n} = (1 +o(\lambda)) \Big(\int_{\mathbb S^n_+} e^{-2n H(\xi)} d\xi\Big)^{-\lambda/2n}.$$ The expansion now follows from and .
We are now in a position to prove Theorem \[log-HLSonhalfspace\].
It is clear that $p,r\to 1$ when $\lambda \to 0$, and by Proposition \[constantnearzero\] we have $C(n,\lambda)\to 1$ when $\lambda\to 0$. Our assumptions on $f$ and $g$ ensure that $$\lim_{\lambda\to 0} \frac 1\lambda \Big( \int_{\operatorname{\mathbf{R}}_+^n}\int_{\partial\operatorname{\mathbf{R}}_+^n} g(x) |x-y|^\lambda f(y) dx dy - 1 \Big) = \int_{\operatorname{\mathbf{R}}_+^n}\int_{\partial\operatorname{\mathbf{R}}_+^n} g(x) \ln |x-y| f(y) dx dy.$$ From this, we obtain $$\begin{aligned}
\lim_{\lambda\to 0} &\frac{C(n,\lambda)\|f\|_{L^p(\partial\operatorname{\mathbf{R}}_+^n)}\|g\|_{L^r(\operatorname{\mathbf{R}}_+^n)}-1}\lambda\\
& =-\int_{\partial\operatorname{\mathbf{R}}_+^n} f(y) \ln f(y) dy - \int_{\operatorname{\mathbf{R}}_+^n} g(x) \ln g(x) dx \\
&\quad -\frac1{2(n-1)}\ln |\mathbb S^{n-1}| -\frac1{2n} \ln\Big(\int_{\mathbb S^n_+} e^{-2n \int_{\mathbb S^n_0} \ln |\xi-\eta| d\sigma(\eta)} d\xi\Big).\end{aligned}$$ It is easy to see that $$C_n = -\frac1{2(n-1)}\ln |\mathbb S^{n-1}| -\frac1{2n} \ln\Big(\int_{\mathbb S^n_+} e^{-2n \int_{\mathbb S^n_0} \ln |\xi-\eta| d\sigma(\eta)} d\xi\Big).$$ The proof of Theorem \[log-HLSonhalfspace\] is then finished.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors want to thank Professor Meijun Zhu for stimulating discussion on the references [@dz2013; @dz2014; @hanzhu] as well as for sending us his preprint [@zhu2015]. Both authors are very grateful to the three anonymous referees for careful reading of our manuscript and for critical comments and suggestions which substantially improved the exposition of the article. Especially, we are indebted to one of the three referees for bringing the usage of the Gegenbauer polynomials as well as the spherical reflection positivity to out attention. V.H. Nguyen is supported by a grant from the European Research Council (grant number $305629$).
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---
abstract: 'Nanoporous supercapacitors play an important role in modern energy storage systems, and their modeling is essential to predict and optimize the charging behaviour. Two classes of models have been developed that consist of finite and infinitely long pores. Here, we show that although both types of models predict qualitatively consistent results, there are important differences emerging due to the finite pore length. In particular, we find that the ion density inside a finite pore is not constant but increases linearly from the pore entrance to the pore end, where the ions form a strongly layered structure. This hinders a direct quantitative comparison between the two models. In addition, we show that although the ion density between the electrodes changes appreciably with the applied potential, this change has a minor effect on charging. Our simulations also reveal a complex charging behaviour, which is adsorption-driven at high voltages, but it is dominated either by co-ion desorption or by adsorption of both types of ions at low voltages, depending on the ion concentration.'
author:
- Konrad Breitsprecher
- Manuel Abele
- Svyatoslav Kondrat
- Christian Holm
title: The effect of finite pore length on ion structure and charging
---
Introduction
============
Electrical double-layer capacitors, or supercapacitors, are an important player on the market of energy storage devices. Supercapacitors store energy by electrosorption of counter-charge into the porous electrodes and provide high power densities and cyclability, but the stored energies are relatively low [@miller:sci:08; @simon_gogotsi:acr:13]. To increase the energy storage, electrodes with *subnanometer* pores are used, which show anomalously high capacitances (per surface area) [@pinero:carbon:06; @gogotsi:sci:06; @gogotsi:08] and hence high stored energies [@kondrat:ees:12]. This anomalous increase of capacitance is due to the emergence of a *superionic state* [@kondrat:jpcm:11; @kondrat:pccp:11], [[[*i.e.*]{}]{}]{}, screening of the electrostatic interactions in narrow conducting pores [@rochester:cpc:13; @schmickler:ec:14; @mohammadzadeh_schmickler:ea:2015:NanotubeScreening]. In this work we shall focus on electrodes with such narrow pores only.
Modelling plays a crucial role in understanding and predicting the properties of supercapacitors, such as capacitance, energy storage and charging times. There have been many models and methods developed, but they can be conventionally spit into two classes. One class consists of models that literary ‘mimic’ a supercapacitor, [[[*i.e.*]{}]{}]{}, they consider an ionic liquid confined between two electrodes with porous structures, either with slit-shaped [@borodin:10; @feng:jpcl:11; @wu:qiao:jpcl:12; @vatamanu:jpcl:energystorage:13; @vatamanu:acsnano:15] or cylindrical [@borodin:10; @shim:10] pores, or even featuring a complex pore network [@merlet:natmat:12; @burt_salanne:jpcl:16:ElectrolyteEffect]. However, since (typically molecular dynamics) simulations of such models are computationally demanding, they are necessarily ‘scaled-down’ as compared to their experimental counterparts. For instance, the pore length in a typical simulation is tens of nanometers at best, while in the experimental systems the pores can be micrometer long, as follows from the carbon particle sizes (see, [[[*e.g.*]{}]{}]{}, Refs. [@li:07:ShortPores; @dyatkin_gogotsi:jps:16], and Ref. ); likewise, the size of the region between the electrodes is of the order of nanometers in simulations, but it is hundreds of micrometers or millimeters in real supercapacitors.
On the other hand, there is a class of simplified models that consider an ionic liquid confined to a *single* pore, either slit-shaped [@kondrat:jpcm:11; @kondrat:pccp:11; @kiyohara:jpcc:07; @asaka:jcp:10; @kiyohara:jcp:11; @jiang:nanolett:11; @jiang:nanoscale:14; @dudka:jpcm:16] or cylindrical [@kornyshev:fd:14; @lee:prl:14; @schmickler:ea:2015:harmonicOscillator; @rochester:jpcc:16]. These models are likely to describe more closely the long pores of real porous electrodes , but their deficiency is that the effects related to the pore closing and opening are ignored and the charging dynamics are not straightforward to study. Often such simplified models can be treated analytically [@kondrat:jpcm:11; @kornyshev:fd:14; @lee:prl:14; @schmickler:ea:2015:harmonicOscillator; @rochester:jpcc:16; @dudka:jpcm:16], but also Monte Carlo (MC) simulations [@kiyohara:jpcc:07; @asaka:jcp:10; @kiyohara:jcp:11; @kondrat:pccp:11; @rochester:jpcc:16] and classical density functional theories [@jiang:nanolett:11; @jiang:nanoscale:14; @wu_li:arpc:07:DFT] have been applied.
The main goal of this work is to connect these two types of models and to study the effects of finite and infinite pore lengths on the ion structure and charging. To this end, we take a variation of the model of a single slit nanopore of infinite length developed in Refs. [@kondrat:jpcm:11; @kondrat:pccp:11]. We shall also present a *new model* for supercapacitors, which consists of two electrodes with slit nanopores; in contrast to other similar models [@feng:jpcl:11; @wu:qiao:jpcl:12; @vatamanu:jpcl:energystorage:13; @vatamanu:acsnano:15], our model considers pores that are open on one side only ([[[*cf.*]{}]{}]{}[Fig. \[fig:model\]]{}a), [[[*i.e.*]{}]{}]{}, we take into account the pore closings explicitly. We shall first show how the superionic state emerges in this model ([Section \[sec:mm\]]{}; in this section we also describe the details of both models and the methods used to study them). Then, we shall discuss how the parameters of the two models can be connected ([Section \[sec:0V\]]{}) and how to adjust the simulations and the analysis to make a meaningful comparison ([Section \[sec:md\]]{}). The results obtained by these two models are compared and discussed in [Section \[sec:mcmd:cmp\]]{}. In [Section \[sec:pore\_walls\]]{} we discuss how the ion structure and charging behaviour depend on the pore wall-ion interactions. We summarize and conclude in [Section \[sec:summary\]]{}.
Models and Methods {#sec:mm}
==================
![Supercapacitor models in Monte Carlo (MC) and molecular dynamics (MD) simulations. (a) In MD simulations an ionic liquid is confined between two electrodes with a slit pore of *finite* length each. The electrodes are constructed from the carbon atoms, and the electrostatic potential $2U$ is applied between the electrodes. (b) In Monte Carlo simulations the ions are confined in a single metallic slit pore, *infinitely* extended in the lateral directions. The electrostatic potential $U$ is applied to the pore walls with respect to the bulk ([[[*i.e.*]{}]{}]{}, unconfined) electrolyte. (c) Force $f$ between two ions confined in a metallic slit pore of width ${w}=9.37$Å. The open squares denote the force calculated by the [ICC$^*$]{}method in the middle of a finite pore; this method is used for constant-potential MD simulations. The solid line shows the force obtained directly from the interaction potential for an *infinite* pore, [Eq. (\[eq:mc:u2\])]{}, [[[*i.e.*]{}]{}]{}, $f_{\alpha\beta} = -d v_{\alpha\beta}/dR$ (note that $f_{++} = f_{--} = - f_{+-} \equiv f$). The ion diameters are $\sigma = \sigma_\pm = 5$Å and their centers are located on the symmetry plane of the slit, [[[*i.e.*]{}]{}]{}, $z_1 = z_2 = 0$ in [Eq. (\[eq:mc:u2\])]{}. The Coulomb force (dash line) is shown for comparison. (d) Force $f_\mathrm{self}$ acting on a single ion inside a slit metallic pore due to image forces. The open squares show $f_\mathrm{self}$ calculated in the middle of a finite pore by using the [ICC$^*$]{}method, which is compared with the force for the infinitely long pore, $f_\mathrm{self} = -d{E_\mathrm{self}}/dz$ obtained from the interaction potential (\[eq:mc:u1\]) used in the MC simulations (solid line). \[fig:model\] ](force_nolog_snapshots_2){width="90.00000%"}
We compare two models of nanoporous supercapacitors. In one model we consider a supercapacitor that consists of two electrodes separated by a distance ${H}$, with each electrode featuring a slit nanopore of the specified width ${w}$ and *finite* length ${l}$ ([Fig. \[fig:model\]]{}a). This model will be used in MD simulations as described below; we shall call it the MD model.
In addition, we consider a model consisting of a single nanopore, [[[*i.e.*]{}]{}]{}, an ionic liquid confined between two parallel metal plates, which are *infinitely* extended in the lateral directions ([Fig. \[fig:model\]]{}b); this model will be treated by MC simulations and will be called the MC model henceforth. We apply a potential $U$ at the plates of the MC model (see [Section \[sec:methods:mc\]]{}), which by symmetry corresponds to the applied potential $2U$ between the two electrodes of the MD model.
In both models and in all simulations, we shall use the same pore width ${w}=9.37$Å. We note that the pore width accessible to the ions is smaller, as will be discussed below.
Ionic liquid
------------
In both MD and MC simulations, we have taken charged soft spheres to model an ionic liquid. The ion-ion soft potential was the repulsive-only Week-Chandler-Anderson (WCA) potential [@wca:jpc:71], $\phi_\mathrm{WCA}$. We recall that $\phi_\mathrm{WCA}$ is the standard 12-6 Lennard-Jones potential cut at the ion-ion separation $R_\mathrm{min} = 2^{1/6} \sigma$, where $\sigma$ is the ion diameter, and shifted such that $\phi_\mathrm{WCA}=0$ at $R=R_\mathrm{min}$; this is to ensure that the corresponding force is a continuous function of $R$.
The following parameters have been used in all simulations: Interaction parameter $\epsilon = 1$kJ/mol and the ion diameter $\sigma = \sigma_\pm = 5$Å. In the solvent-free case, these parameters give the pressure $1$atm for the ion volume fraction $0.134$ (molar concentration $0.6$M) and at temperature $T=400$K.
Pore walls {#sec:mm:pore_walls}
----------
In the MD model the electrodes have been constructed from carbon atoms, which, however, we modeled as WCA particles with the following parameters: $\sigma_c = 3.37$Å and $\epsilon_{c}=1$kJ/mol. The hexagonal structure of the pore walls was obtained by subdividing the surface into equilateral triangles and placing the atoms in their centers. A side length of $2.5$Å provides an atom-atom bond length of $1.44$Å, similar as in graphene. The pore entrance was curved with a radius of $4$Å, and the pore closing was curved with a radius of $2$Å.
In the MC model we neglect the pore wall structure and consider flat soft walls instead. To this end we propose the 10-4 Lennard-Jones (LJ) interaction potential $$\begin{aligned}
\label{eq:phi104}
\phi_{\mathrm{wall-ion}}^{\mathrm{MC}}(z) = 2 \pi \epsilon_{\mathrm{wall-ion}}\sigma_{\mathrm{wall-ion}}^2 \rho_{\mathrm{wall}}\left[ \frac{2}{5}\left(\frac{\sigma_{\mathrm{wall-ion}}}{z-z_0}\right)^{10} -
\left(\frac{\sigma_{\mathrm{wall-ion}}}{z-z_0}\right)^4 \right]\end{aligned}$$ where $\epsilon_{\mathrm{wall-ion}}$ and $\sigma_{\mathrm{wall-ion}}$ are the wall-ion energy and diameter parameters, $\rho_{\mathrm{wall}}$ is the *two-dimensional* number density of carbon atoms, and $z_0$ is the location of the wall. This potential is obtained by integrating the LJ inter-particle interaction potential over a *surface* of LJ particles, where the surface is infinitely extended in the $(x,y)$ directions.
In order to match the MC and MD models, we have fitted the interaction potential (\[eq:phi104\]) to the *averaged* potential that an ion experiences when approaching an atomistic wall. We emphasize that (i) this fit is difficult to do accurately in the whole range of the wall-ion distances, and (ii) the atomistic wall-ion potential is not homogeneous in the lateral directions ([Supplementary Fig. \[si:fig:wall-ion-wca\]]{}). We will discuss this in [Section \[sec:pore\_walls\]]{}, where we also consider the case of hard pore walls with the accessible pore width ${w_\mathrm{acc}}=6$Å for comparison.
Grand canonical Monte Carlo simulations {#sec:methods:mc}
---------------------------------------
In the MC model a slit pore is *infinitely* extended in the $(x,y)$ directions ([Fig. \[fig:model\]]{}b), which is modeled by applying periodic boundary conditions in these directions. The electrostatic potential $U$ was applied to the pore walls, which amounts to setting the electro-chemical potential to $\mu_\pm = \pm eU + \delta E_\pm$, where $e$ is the elementary charge and $\delta E_\pm$ is an energy of transfer of a $\pm$ ion from the bulk of a supercapacitor into the pore (assumed equal for anions and cations). Note that $\delta E$ does *not* include the change in the ion’s self-energy ([[[*cf.*]{}]{}]{}[Eq. (\[eq:mc:u1\])]{}), and that its typical values lie between $-5k_BT$ and $45k_BT$ [@lee:prx:16].
The electrostatic interaction energy between two ions confined in a metal slit pore is [@kondrat:jpcm:11] $$\begin{aligned}
\label{eq:mc:u2}
v_{\alpha\beta}(z_1, z_2, R) =
\frac{4q_\alpha q_\beta }{\varepsilon {w}}
\sum_{n=1}^{\infty} K_0(\pi n R /{w})
\sin\boldsymbol{(}\pi n(z_1+1/2)/{w}\boldsymbol{)}
\sin\boldsymbol{(}\pi n(z_2+1/2)/{w}\boldsymbol{)}\end{aligned}$$ where $q_{\alpha}$ and $q_\beta$ are the ion charges ($=\pm e$ in this work), $R$ is the lateral distance between the ions, $z_1, z_2 \in [-{w}/2, {w}/2]$ are their positions across the pore, and $\varepsilon$ is the dielectric constant (taken $\varepsilon = 4$ in this work).
An ion confined in a narrow conducting nanopore experiences an image-force attraction to the pore walls. For a slit metallic pore, infinitely extended in the lateral directions, this interaction energy can be calculated analytically [@kondrat:jpcm:11] $$\begin{aligned}
\label{eq:mc:u1}
{E_\mathrm{self}}^{(\alpha)} (z) = - \frac{q_\alpha^2 }{\varepsilon{w}} \int_0^\infty\left[\frac{1}{2}
- \frac{\sinh\boldsymbol(k(1/2-z/{w})\boldsymbol)\sinh\boldsymbol(k(1/2+z/{w})\boldsymbol)}
{\sinh(k)}\right]dk,\end{aligned}$$ where $z$ is the position across the pore. Note that ${E_\mathrm{self}}^{(\alpha)}$ does not depend on the ion charge but only on its valency (taken $1$ in this work); thus, we shall omit index $\alpha$ in ${E_\mathrm{self}}^{(\alpha)}$.
The interaction potentials (\[eq:mc:u2\]) and (\[eq:mc:u1\]) constitute the superionic state. They were implemented in towhee simulation package [@towhee; @kondrat:pccp:11] and grand canonical Monte Carlo simulations were performed using the Widom insertion-deletion move [@widomMC:1963], translational move, and the molecular-type swap move [@kondrat:pccp:11]. We performed $5\times10^6$ equilibration runs and up to $10^7$ production runs at temperature $T=400$K.
Molecular dynamics simulations
------------------------------
MD simulations have been performed using simulation package [@espresso; @limbach:cpc:06:Espresso; @arnold_holm:espresso:2013] with the velocity-Verlet algorithm for integration and a Langevin thermostat (at temperature $T=400 K$ and damping constant $\xi=10$ps$^{-1}$) to model a NVT ensemble.
For constant potential simulations, we used the [ICC$^*$]{}algorithm [@arnold_holm:cpc:02:MMM2D; @kesselheim10a; @arnold_holm:entropy:13:ICC] in combination with the 3D-periodic electrostatic solver [P$^3$M]{}[@ballenegger08a]. In the [ICC$^*$]{}method, the induced [ICC$^*$]{}charges are defined on a discretized closed surface, and their values are determined iteratively each simulation step. Since the standard [P$^3$M]{}solver does not take into account the applied potential between the electrodes, we have additionally superimposed the corresponding external electrostatic potential, which has been pre-calculated numerically by solving the Laplace equation with the appropriate boundary conditions; this has been done iteratively on a equidistant lattice using a seven-point-stencil relaxation algorithm [@press07a].
To test how the superionic state emerges within the [ICC$^*$]{}approach, we have calculated the force between two ions in the pore middle, and compared it with the force obtained from [Eq. (\[eq:mc:u2\])]{} as $f_{\alpha\beta}= - dv_{\alpha\beta}/dR$ (note that $f_{++} = f_{--}=-f_{+-} \equiv f$). [Figure \[fig:model\]]{}c demonstrates an excellent agreement between the two methods. We have also compared the force due to image forces acting between an ion and the pore walls obtained by the [ICC$^*$]{}approach and from [Eq. (\[eq:mc:u1\])]{} as $f_\mathrm{self}=-{E_\mathrm{self}}/dz$; again the agreement is very good ([Fig. \[fig:model\]]{}d). We note however that the corresponding *potential* acquires an additional contribution due to periodicity (see [Supplementary Fig. \[si:fig:Eself\]]{}), which can be corrected by considering larger systems. However, this shift in ${E_\mathrm{self}}$ does not influence the ion-pore walls *forces*, as [Fig. \[fig:model\]]{}d demonstrates, and therefore the results of the MD simulations ([[[*i.e.*]{}]{}]{}, it only shifts the energy level, but the energy differences remain the same).
Non-polarized nanopores {#sec:0V}
=======================
![Ion packing fraction in the pore at zero voltage from molecular dynamics (MD) and Monte Carlo (MC)simulations. (a) Ion packing fraction as a function of the total concentration ${c_\mathrm{IL}}$ of an ionic liquid (IL) from MD simulations. (b) Ion packing fraction in the pore from MC simulations as a function of $\delta E$, an energy of transfer of an ion from the bulk electrolyte into the pore. Large symbols (rhombus, square, up and down triangles and circle) show the systems taken for a comparison of MC and MD models ([[[*cf.*]{}]{}]{}[Figs. \[fig:mcmd:eta\_pore\]]{} and \[fig:mcmd:cap\]). \[fig:mcmd:eta\] ](eta_wca_sw104){width="70.00000%"}
In order to facilitate a comparison between the two models ([Fig. \[fig:model\]]{}a-b), we have matched the in-pore ion packing fractions at zero potential, $U=0$. We calculated this packing fraction as ${\eta_{\mathrm{in-pore}}}= \pi \sigma^3/(6\Omega)$, where $\sigma=\sigma_\pm=5$Å is the ion diameter and $\Omega = S {w_\mathrm{acc}}$ is the volume of a pore; here $S$ is the lateral area and ${w_\mathrm{acc}}$ the accessible pore width. In the MD model, unless otherwise specified, only the middle parts of the pores were taken into account when calculating ${\eta_{\mathrm{in-pore}}}$, [[[*i.e.*]{}]{}]{}, the entrance and the closing of the pores were excluded (see [Section \[sec:md:ion\_struct\]]{}). Since ${w_\mathrm{acc}}$ is not known *a priori* and is expected to vary with the applied potential, for definiteness we took ${w_\mathrm{acc}}= {w}- \sigma_c$, where $\sigma_c = 3.37$Å is the diameter of the carbon atom. The pore width ${w}=9.37$Å gives ${w_\mathrm{acc}}=6$Å. Note that this is the accessible pore width for a system with hard pore walls.
In the MC model, the pore occupation is controlled by the ion transfer energy $\delta E_\pm = \delta E$. [Figure \[fig:mcmd:eta\]]{}a shows that the in-pore packing fraction decreases as $\delta E$ increases, and the pore becomes more ionophobic [@kondrat:nh:16; @lee:prx:16].
In the MD model, the in-pore ion packing fraction can be controlled by changing the total concentration of ions in a supercapacitor, ${c_\mathrm{IL}}$. Physically this can be realized by varying the pressure in the case of neat ionic liquids or by varying the salt concentration in the case of electrolyte solutions. [Figure \[fig:mcmd:eta\]]{}b demonstrates that the pore becomes less populated as ${c_\mathrm{IL}}$ decreases. However, at extremely low concentrations our MD simulations predict the formation of ionic liquid clusters in the bulk electrolyte ([[[*i.e.*]{}]{}]{}, between the electrodes), which influence the charging behaviour; we have therefore decided not to consider such cases in this work.
After having matched the pore occupancies at no applied potential, we perform voltage-dependent MC and MD simulations for the systems shown by symbols in [Fig. \[fig:mcmd:eta\]]{}.
Charging of nanopores of finite length (MD model) {#sec:md}
=================================================
There are two important parameters in the MD model that are not present in our MC system, [[[*viz.*]{}]{}]{}the pore length ${l}$ and the size ${H}$ of the bulk of a supercapacitor. In order to understand better their potential impact on charging, we first discuss how they influence the structure of an ionic liquid, in comparison to the MC model of a single infinitely-long nanopore.
Ion structure {#sec:md:ion_struct}
-------------
![Ion density profiles $\rho_\pm$ from MD simulations. The green line shows $\rho_+=\rho_-$ at no voltage, and the blue and red lines show $\rho_+$ and $\rho_-$ at applied potential $U=3V$. The grey areas denote the location of the pores. Without the appropriate correction steps, the co-ions get trapped in the pores, and the density of ions in the bulk electrolyte between the electrodes depends on the applied voltage. The total concentration of an ionic liquid in the supercapacitor is ${c_\mathrm{IL}}\approx 1.1$M. \[fig:md:rhoy\] ](rho_y){width="45.00000%"}
[Figure \[fig:md:rhoy\]]{} shows the average ion density profiles $\rho_\pm (y)$ between the two electrodes for zero and non-zero applied potentials. We can make the following observations:
1. The average ion density in the bulk electrolyte depends on the applied voltage. This implies that the chemical potential of the bulk ionic liquid changes with voltage, while it is taken constant in the MC model.
2. Some *co-ions* become trapped near the pore closings on the time scales of our MD simulations. This means that the system has not reached equilibrium. Note that this is unlikely to happen in the MC model as we perform grand canonical simulations.
3. The ions exhibit a clear layering near the pore closings and openings, while they seem to form a nearly homogeneous structure in the middle of a pore. However, for non-zero potentials the counter-ion density *is not constant along the pore* and increases from the pore entrance to the pore end. Clearly, in the MC model the average ion densities are position independent.
We shall now discuss how to correct the MD simulations to be able to approach more closely the single-pore supercapacitor models. We will see, however, that although points (1) and (2) can be corrected relatively easily, point (3) is more subtle and makes it difficult to compare the MC and MD models *quantitatively*.
Bulk density calibration {#sec:md:bulk}
------------------------
![Calibration of bulk density in MD simulations. (a) Total packing fraction ${\eta_{\mathrm{bulk}}}$ of the ions in the *bulk* electrolytes between the electrodes (see [Fig. \[fig:model\]]{}a); ${\eta_{\mathrm{bulk}}}$ can change significantly with the applied potential. To keep ${\eta_{\mathrm{bulk}}}$ constant (dash-dot line for ${c_\mathrm{IL}}\approx 1.1$M) we calibrate the bulk density during equilibration runs by inserting ion pairs into the system (see the main text). (b) Total packing fraction ${\eta_{\mathrm{in-pore}}}$ of the ions in the pore with and without density calibration for ${c_\mathrm{IL}}=1.1$M; for the remaining two concentrations from panel (a) see [Supplementary Fig. \[si:fig:md:bulk\_calibration\]]{}. Such a calibration influences ${\eta_{\mathrm{in-pore}}}$, albeit weakly, but its effect on the accumulated charge is negligible, however (see the inset). \[fig:md:calibration\] ](eta){width="80.00000%"}
Clearly, also in experimental systems and commercially fabricated supercapacitors, the ion density in the bulk ([[[*i.e.*]{}]{}]{}, between the supercapacitor electrodes) can vary with the applied voltage. However, the volume of a bulk region in these systems is typically large, as compared to the total pore volume, and this change is expected to be small. In MD simulations, the size of a bulk region (${H}$) is often comparable to the pore size [@feng:jpcl:11; @wu:qiao:jpcl:12; @vatamanu:jpcl:energystorage:13; @vatamanu:acsnano:15], and its effect on the bulk density can therefore be significant ([Fig. \[fig:md:calibration\]]{}a).
One way to deal with this problem is to consider systems with sufficiently large ${H}$, which would, however, increase the computational costs accordingly. We have therefore taken a different route. We chose to *calibrate* the total number of ions in a system during equilibration runs each time the voltage is changed. This was done by inserting ion pairs into the system (or removing them from the system when necessary) until the bulk density $\rho_{\mathrm{bulk}}(U\ne 0)$ equilibrates to $\rho_{\mathrm{bulk}}(U=0)$. After the calibration we run production runs as usual.
Surprisingly at first glance, we have found that although the total ion density in the pore is slightly altered by calibration, it has practically no effect on the charge storage ([Fig. \[fig:md:calibration\]]{}b). This is likely because a change $\Delta \mu_\pm$ in the chemical potential due to the change in the ion density is small compared to what the system gains from the applied potential, [[[*i.e.*]{}]{}]{}, $\Delta \mu_\pm \ll eU$. We could not accurately estimate $\Delta \mu_\pm$ for our system, but we expect it to be of the order of few $k_BT$’s [@kato:jpcb:08:ILChemPot]. This is supported by an observation that the change in the transfer energy of about $3k_BT$ (in the MC model) corresponds to the change in the total ion concentration from $1$M to more than $2$M (see [Fig. \[fig:mcmd:eta\]]{}). For comparison, $eU \approx 30 k_BT$ at $T=400$K and at an applied potential of $1$V.
Avoiding ion trapping
---------------------
![Co-ion trapping in MD simulations. (a) Charge density heat-map for a step-voltage charging when a voltage of $2U=6$V is abruptly applied between the electrodes. Some co-ions are trapped near the pore closings even after $40$ns of simulation time. (b) Co-ion trapping can be avoided by charging the system slowly with a linear voltage ramp $U(t) = k t$, where $k=0.25$V/ns. There are no co-ions trapped in the pores after $10$ns of simulation time. These plots show the simulation results averaged over the last $8$ns of a $40$ns simulation. \[fig:md:trapping\] ](heatmaps_combined){width="80.00000%"}
[Figures \[fig:md:rhoy\]]{} and \[fig:md:trapping\]a show that at high voltages the co-ions in the pore become ‘trapped’ near the pore closing on the time scales of our MD simulations ($\sim 100$ns). Such a co-ion trapping leads to a decreased charge storage and sluggish dynamics [@pak_hwang:jpcc:16:IonTrapping]. To avoid ion trapping in this work we have used a linear *voltage ramp* $U(t) = k t$ to charge our system, instead of a typically used step-voltage charging as in [Fig. \[fig:md:trapping\]]{}a. [Figure \[fig:md:trapping\]]{}b demonstrates that this strategy allows us to avoid co-ion trapping on computationally accessible time scales.
The analysis of this approach will be presented in detail elsewhere.
Effect of finite pore length, and pore entrance and closing {#sec:md:pore_length}
-----------------------------------------------------------
![Effect of pore length on ion structure and charging. (a) Counter-ion densities from MD simulations for different pore lengths ${l}$ at applied potential $U=3$V (there are no coions in the pore at this voltage). The horizontal orange line shows the corresponding MC result. The MD results have been obtained using bulk calibration and ramp-voltage charging with the rate $k=0.25$V/ns. The grey areas highlight the location of the pores, and the vertical dash lines show the pore entrance and closings. Only half of a supercapacitor is shown. (b) Accumulated charge per surface area as a function of pore length. The charge has been calculated for the whole pore and when only the middle part of a pore is taken into account ($2$nm from the pore entrance and $3.5$nm from the pore end). The total concentration of an ionic liquid in the supercapacitor is ${c_\mathrm{IL}}\approx 1.6$M and the transfer energy in MC simulations is $\delta E = 21.75k_BT$. These parameters have been chosen such that the MC and MD models give the same ion densities at no applied voltage (see [Fig. \[fig:mcmd:eta\]]{}). \[fig:mcmd:rhoy\] ](rho_y_mcmd_v2){width="90.00000%"}
We have pointed out in [Section \[sec:md:ion\_struct\]]{} that the pore entrance and closing influence the in-pore structure of ions, particularly strongly at non-zero voltages ([Fig. \[fig:md:rhoy\]]{}). [Figure \[fig:mcmd:rhoy\]]{}a shows that this behaviour remains true also for long pores (${l}=20$nm), [[[*i.e.*]{}]{}]{}, the counter-ion density is systematically higher at the pore end and correspondingly lower where it begins; we note that these ion density profiles persist for long simulation runs (up to 200ns) and are likely the equilibrium profiles. Such a behaviour is understandable because image forces are weaker at the entrance, where the conducting pore walls end; in contrast, near the pore end the additional (closing) surface amplifies the image-force effects and hence attracts more ions. As a result, the counter-ion density along the pore changes approximately linearly between these two points.
Interestingly, the MC simulations predict the counter-ion densities that are closer to those at the pore end, rather than at the entrance (at the same applied potentials and for the same densities at zero potential). The exact reason for this is not clear to us, but it is tempting to speculate that it is actually the pore entrance that reduces the ion density, and in this way affects the ion concentration in the whole pore and hence its charging behaviour. On the other hand, it is also possible that the atomistic pore-wall structure in the MD model induces a weak ordering of an ionic liquid, and thus leads to an additional entropic cost for dense ion packing (recall that the pore walls in the MC model are flat). This implies that in the MD model a higher voltage is needed to induce the same counter-ion density as in the MC model, and this is indeed what we observe ([[[*cf.*]{}]{}]{}[Fig. \[fig:mcmd:eta\_pore\]]{}). It is worth noting that similar effects have been reported for ionic liquids at flat (non-porous) electrodes [@breitsprecher:jpcc:15:ElectrodModels].
This inhomogeneity of the ion distribution manifests itself in the accumulated charge ($Q_\mathrm{whole-pore}$), which depends sensitively on the pore length and shows a non-monotonic behaviour (squares in [Fig. \[fig:mcmd:rhoy\]]{}b). Interestingly, $Q_\mathrm{whole-pore}$ decreases as $l$ increases (for long pores), which is because the ion density at the pore entrance becomes less affected by the pore closing, attending a lower value ([Fig. \[fig:mcmd:rhoy\]]{}a). Since the effects due to the pore closing and opening weaken with increasing $l$, $Q_\mathrm{whole-pore}$ approaches $Q_\mathrm{pore-middle}$ as $l\to\infty$, where $Q_\mathrm{pore-middle}$ takes into account only the middle part of a pore. However, both $Q_\mathrm{pore-middle}$ and $Q_\mathrm{whole-pore}$ are smaller than $Q_\mathrm{MC}$ obtained within the MC model featuring an infinitely long pore. As discussed, this is likely due to the effect that the pore entrance has on the in-pore ion density ([Fig. \[fig:mcmd:rhoy\]]{}a).
Thus, [Fig. \[fig:mcmd:rhoy\]]{} demonstrates that *there is no well-defined bulk region inside the pore*, where the average ion density would be constant along the pore. This behaviour affects the charging behaviour and hinders a direct *quantitative* comparison between the MC and MD models. We therefore restrict further discussions mainly to qualitative comparisons.
Comparison of the results for finite and infinite pores {#sec:mcmd:cmp}
=======================================================
We are now in position to compare the simulation results of the MD and MC models with finite and infinitely long nanopores, respectively. In the MD simulations, we have taken a voltage ramp of $k=0.25$V/ns whenever necessary (at voltages $\lesssim 2$V we observe no trapping for the pore length considered); the size of a bulk region was ${H}=80$Å. The pore length was ${l}=80$Å, but only the middle region of size $25$Å was used to analyse the results (this is to exclude the contribution from strong ionic-liquid layering at the pore ends and entrances).
Since in MD simulations we had to adjust the [ICC$^*$]{} charges each simulation step to keep a constant potential on the electrode surfaces, these *constant-potential* simulations are computationally demanding, and we have performed them only for a limited number of voltages. As a result, since the differential capacitance and the charging parameter $X_D$ (see below) require numerical differentiations with respect to voltage, we have calculated them with lower resolutions. This can be contrasted with our MC simulations, in which the metallic nature of the electrodes is taken into account via the interaction potentials (\[eq:mc:u2\]) and (\[eq:mc:u1\]). This allows to reduce the computational costs significantly (note that such analytical solutions exist only for a few simple geometries [@rochester:13]).
Pore filling and charging mechanisms
------------------------------------
![ Total packing fraction ${\eta_{\mathrm{in-pore}}}$ of ions in a pore and charging parameter $X_D$ from (a) MD and (b-c) MC simulations. Transfer energies ($\delta E$) and the ionic liquid concentrations (${c_\mathrm{IL}}$) have been matched such as to have approximately the same ${\eta_{\mathrm{in-pore}}}$ at zero voltage. MC and MD models predict similar behaviours of ${\eta_{\mathrm{in-pore}}}$ at low and intermediate voltages. Small discrepancies appear only at higher potentials likely due to the differences in the pore wall structures in the MD and MC models. (c) Charging parameter $X_D$, [Eq. (\[eq:XD\])]{}, obtained from MC simulations shows the regions where charging is dominated by adsorption ($X_D>0$) and desorption ($X_D< 0$). $X_D=0$ corresponds to swapping of co-ions for counterions. The same symbols and line codes are used in (c) as in (b). The comparison of $X_D$ obtained within the MC and MD models is shown in [Supplementary Fig. \[si:fig:mcmd:XD\]]{}. \[fig:mcmd:eta\_pore\] ](eta_sw104_XD){width="42.00000%"}
[Figure \[fig:mcmd:eta\_pore\]]{} shows the total ion packing fraction ${\eta_{\mathrm{in-pore}}}$ as a function of the applied potential, and demonstrates that pore filling proceeds similarly in the MC and MD models. Interestingly, in all cases of *strongly ionophilic* pores, [[[*i.e.*]{}]{}]{}, the pores with a substantial amount of an ionic liquid at no voltage, ${\eta_{\mathrm{in-pore}}}$ first decreases for increasing voltage, and starts to increase only when there are no co-ions left in the pore. In other words, at low voltages charging is dominated by *co-ion desorption*, while it is the *counter-ion adsorption* that drives charging at higher applied potentials [@kondrat:jpcm:11; @wu:qiao:jpcl:12; @vatamanu:jpcl:energystorage:13; @kondrat:nh:16]. This is in agreement with the recent observation [@kondrat:nh:16] showing that desorption (and swapping) are thermodynamically preferable over adsorption in most cases, except of a narrow window of parameters in which desorption and swapping are infeasible due to the lack of co-ions.
To characterize charging mechanisms in more detail, we introduce a charging parameter, similar to the parameter $X$ of @forse:jacs:16:chmec [@forse:jacs:16:chmec], $$\begin{aligned}
\label{eq:XD}
X_D(U) = \frac{e}{C(U)}\frac{d N}{d U},\end{aligned}$$ where $e$ is the elementary charge, $C(U)= dQ/dU$ the differential capacitance, $Q$ denotes the accumulated charge and $N$ the total number of ions. $X_D$ expresses how charging is related to pore filling or de-filling, and thus describes which charging mechanism takes place. If charging is driven solely by swapping of coions for counter-ions, then the total ion density does not change, $N=\mathrm{const}$, and hence $X_D = 0$. For pure electrosorption we have $edN/dU = dQ/dU$ and thus $X_D = 1$, while for desorption $dQ/dU = - edN/dU$ and so $X_D = -1$. The parameter $X$ of @forse:jacs:16:chmec is related to $X_D$ in a similar fashion as the integral capacitance is related to the differential capacitance, [[[*i.e.*]{}]{}]{}, $$\begin{aligned}
\label{eq:XI}
X(U) = \frac{1}{Q} \int_{0}^{U} X_D (u) C (u)du,\end{aligned}$$ which can be seen as a voltage-averaged $X_D$ with the weight $C(u)$, where $Q = \int_{0}^{U} C(u) du$ is a normalization constant.
The charging parameter $X_D$ obtained from MC simulations is presented in [Fig. \[fig:mcmd:eta\_pore\]]{}c. It shows that at high voltages charging is solely due to counter-ion adsorption, [[[*i.e.*]{}]{}]{}, $X_D \approx 1$, but at low voltages it can be either co-ion desorption or counter-ion adsorption, depending on the transfer energy $\delta E$. Interestingly, for high values of $\delta E$, [[[*i.e.*]{}]{}]{}, when the pore is nearly empty at no applied potential, the parameter $X_D$ is significantly greater unity, which means that *both* counter and co-ions are adsorbed into the pore at low voltages. This is likely because the additional ‘ion pairs’ screen the interactions between the counter-ions, reducing the thermodynamic cost of adsorption (note that at low densities the entropic cost of ion insertion is low).
Charging and differential capacitance
-------------------------------------
![Charging from MC and MD simulations. (a) Accumulated charge $Q$ as a function of applied potential $U$ from MD simulations for a few values of the total ionic liquid concentration ${c_\mathrm{IL}}$ in a supercapacitor. (b) $Q$ from MC simulations for a few values of the transfer energy $\delta E$. We have adjusted $\delta E$ and ${c_\mathrm{IL}}$ to give approximately the same in-pore ion packing fractions at zero voltage (see [Fig. \[fig:mcmd:eta\]]{}). For a detailed comparison of the $Q(V)$ curves see [Fig. \[fig:mcmd:walls\]]{}b and [Supplementary Fig. \[si:fig:mcmd:charge\]]{}. (c) Differential capacitance $C$ as a function of voltage from MC simulations. The comparison of the capacitances obtained within the MC and MD models is shown in [Supplementary Fig. \[si:fig:mcmd:cap\]]{}. \[fig:mcmd:cap\] ](charge_cap_sw104){width="42.00000%"}
Figure \[fig:mcmd:cap\] compares the accumulate charge $Q(U)$ from MC and MD simulations for pores with different occupancies at zero voltage, and demonstrates that also the charging process proceeds similarly in the MC and MD models. Interestingly, $Q(U)$ is practically independent of the transfer energies $\delta E$ (MC simulations) and ionic liquid concentrations ${c_\mathrm{IL}}$ (MD simulations). This is because the electrostatic contribution ($\pm eU$) to the total electrochemical potential dominates the contribution due to $\delta E$ and ${c_\mathrm{IL}}$, respectively (see [Section \[sec:md:bulk\]]{}).
Fine details of the charging process are captured by the differential capacitance $C=dQ/dU$. Although $Q(U)$ does not seem to vary significantly with $\delta E$ ([Fig. \[fig:mcmd:cap\]]{}b), $C(U)$ shows nevertheless a complex behaviour, particularly for densely populated ionophilic pores. For such pores, the capacitance exhibits a first maximum corresponding to the co-ion/counter-ion swapping and a second maximum associated with the co-ion desorption, before it finally decreases as the pore becomes more and more occupied by counter-ions at high voltages. For weakly ionophobic pores there is only one maximum in $C(U)$ at low pore occupancies, while at high potentials the charging proceeds similarly for all pores. For strongly ionophobic pores, the charging curves are shifted to higher voltages ([Supplementary Fig. \[si:fig:mc:capen\]]{}). However, we have not been able to obtain such ionophobic pores in the MD model by varying the ionic liquid concentration ${c_\mathrm{IL}}$; thus, we shall not discuss this case further in this work.
Although charging in the MC and MD models show the same qualitative behaviour, there are some quantitative differences ([Fig. \[fig:mcmd:cap\]]{}a-b, and [Supplementary Fig. \[si:fig:mcmd:charge\]]{} and \[si:fig:mcmd:cap\]), as discussed in [Section \[sec:md:pore\_length\]]{}.
Ion structure {#ion-structure}
-------------
![ Ion packing in narrow pores from (a-c) MC and (d-f) MD simulations. The transfer energy in (a-c) is $\delta E_\pm = 21.75k_BT$. In (d-f) the molar concentration of an ionic liquid is ${c_\mathrm{IL}}\approx 1.635$M. The remaining parameters are the same as in [Figs. \[fig:mcmd:eta\_pore\]]{} and \[fig:mcmd:cap\]. \[fig:mcmd:rhoz\] ](rhoz_mcmd_sw104){width="75.00000%"}
We have also looked at the ion structure across the pore. This is shown in [Fig. \[fig:mcmd:rhoz\]]{}, where we compare the density profiles obtained from MC and MD simulations for different voltages. The agreement is very good; small discrepancies are because we could not match exactly the ion densities at zero voltage.
At zero voltage the ion density has a maximum at the middle of the pore. This might seem surprising at first glance, since the image-force wall-ion attraction exhibits a *maximum* at the pore center ([Fig. \[fig:model\]]{}d). However, for ultranarrow pores considered in this work, this is altered by the wall-ion repulsive van-der Waals interactions, which produce a minimum rather than a maximum in the total wall-ion interaction potential ([Supplementary Fig. \[si:fig:wall-ion\]]{}). Nevertheless, at high applied potentials, the counter-ions prefer to locate themselves at the pore walls. This is because the electrostatic energy ($\sim eU$) dominates the unfavourable van der Waals interactions between the walls and the ions, while the increased ion density pushes the counter-ions closer to the walls.
Effect of pore walls {#sec:pore_walls}
====================
![ Effect of pore walls. (a) Ion density profiles across the pore in the case of the hard and soft pore walls at zero applied voltage. For the profiles at non-zero voltages see [Supplementary Fig. \[si:fig:mc:rhoz\]]{}. (b) Accumulated charge as a function of applied voltage for pores with hard and soft walls obtained from MC simulations. The case of carbon walls (MD simulations) is shown for comparison. The ion transfer energies are $(\delta E)_\mathrm{hard} = 21.5k_BT$ and $(\delta E)_\mathrm{soft} = 21.75 k_BT$ for the hard and soft pore walls, respectively; their values have been chosen such that the ion packing fraction ${\eta_{\mathrm{in-pore}}}\approx 0.3$ in the non-polarised pores. \[fig:mcmd:walls\] ](sw104_vs_hw){width=".95\textwidth"}
In addition to the soft pore walls, which interact with the ions via [Eq. (\[eq:phi104\])]{}, we have considered the model of *hard* walls, which are widely used in the literature [@kondrat:pccp:11; @kiyohara:jpcc:07; @asaka:jcp:10; @kiyohara:jcp:11; @jiang:nanolett:11; @jiang:nanoscale:14]. [Figure \[fig:mcmd:walls\]]{} shows that the hard walls strongly influence the ionic liquid structure inside a pore, but their effect on the charge storage is moderate. For low voltages, the accumulated charge in both systems practically coincides, and the only significant differences arise at high applied potentials, where the pore with hard walls saturates while the soft-wall pore can accommodate more charge.
We have also considered the soft walls with the standard 9-3 Lennard-Jones interaction potential between the walls and the ions. This interaction potential is more difficult to fit to the average interaction potential between the ions and the atomistic wall of the MD model, which is not very surprising. We have found that the in-pore ion structure depends sensitively on the fitting parameters (results not shown). However, as in the case of the hard walls, it has no significant effect on the charging behaviour.
We can conclude that although fine details of the non-electrostatic wall-ion interactions are important for the ion structure, their impact on charging is minor, at least at low and intermediates voltages.
Conclusions {#sec:summary}
===========
We have studied charge storage in supercapacitors with slit narrow pores using two models. In one model, treated by MD simulations, a supercapacitor consisted of two electrodes, each with a slit pore of a *finite* length. In the second model, we focused on a single pore, *infinitely* extended in the lateral directions; this model was studied grand-canonically by MC simulations ([Fig. \[fig:model\]]{}). Our main conclusion is that although these two models are *qualitatively* consistent with each other ([Figs. \[fig:mcmd:eta\_pore\]]{}, \[fig:mcmd:cap\] and \[fig:mcmd:rhoz\]), there are some important differences due to the finite pore length. In particular, the pore entrances and closings seem to have a vivid effect on the ion structure inside a pore. At high concentrations and/or high applied potentials, the ion density is not constant along the pore but varies roughly linearly between the pore entrance and the pore end, where it exhibit a strongly oscillatory structure ([Fig. \[fig:mcmd:rhoy\]]{}). This impedes a direct *quantitative* comparison of the two models.
We have also shown that:
- In the MD model with finite pores, the ion density $\rho_{\mathrm{bulk}}$ between the electrodes of a supercapacitor can vary appreciably with the applied voltage. This can be corrected by calibrating $\rho_{\mathrm{bulk}}$ during equilibration runs, to keep it constant, as in the single-pore MC model. However, this change in the bulk density seems to have a minor effect on the charging behaviour ([Fig. \[fig:md:calibration\]]{}). This result means that it is safe to consider relatively small electrode-electrode separations in supercapacitor models and refrain from computationally expansive bulk calibrations. Note that the studies of the charging dynamics would not be straightforward if the bulk calibration were necessary.
- At intermediate and high voltages (and for long pores), the co-ions become trapped in the pores on our typical simulation time sales ($\sim 100 $ns), producing non-equilibrium states. We have overcome this difficulty by using a voltage-ramp charging, instead of an abrupt step-voltage charging typically used in simulations ([Fig. \[fig:md:trapping\]]{}).
As co-ion trapping is expected to occur in experimental systems as well, we have studied this problem in more detail. In particular, we worked out a method to accelerate charging in systems with co-ion trapping and determined optimal charge/discharge regimes; the results of this study will be presented in a separate article.
- At high voltages, charging proceeds exclusively via counter-ion adsorption, while at low voltages the charging process is dominated by either co-ion desorption or counter-ion adsorption, depending on the ion transfer energy or the total ion concentration ([Fig. \[fig:mcmd:eta\_pore\]]{}). Remarkably, at high transfer energies, implying low ion concentrations, both counter and co-ions are adsorbed into the pore at low voltages (the charging parameter $X_D > 1$, see [Eq. (\[eq:XD\])]{} and [Fig. \[fig:mcmd:eta\_pore\]]{}c).
- Interestingly, the accumulated charge seems to be only weekly dependent on the total ions density in a supercapacitor ([Fig. \[fig:mcmd:cap\]]{}). This observation provides an additional degree of freedom for optimizing the charging dynamics by varying the ion concentration without significantly compromising the energy density (note that the fine details of the charging process are resolved by the differential capacitance, which does depend on the total ion concentration/ion transfer energy, [Fig. \[fig:mcmd:cap\]]{}c).
- Even though hard and soft pore walls lead to significant differences in the in-pore ion structure, they show practically the same charging behaviour ([Fig. \[fig:mcmd:walls\]]{}). This is because the applied potential ‘overrules’ all fine details of the non-electrostatic wall-ion interactions and the resulting ionic liquid structure.
In the context of the recent studies on ionophobicity of pores, it is instructive to emphasize that the pore occupancy at zero voltage, which determines the pore ionophobicity, can be varied by changing the total ion concentration (${c_\mathrm{IL}}$) in a supercapacitor. However, by changing ${c_\mathrm{IL}}$ alone we could not achieve a state corresponding to strongly ionophobic pores, which provide high stored energies [@kondrat:nh:16; @lee:prx:16; @lian_wu:jpcm:16:Ionophobic] and fast charging [@kondrat:jpcc:13; @kondrat:nm:14; @lee:nanotech:14]. Thus, another method must be proposed to control effectively the ionophobicity of pores.
Finally, we have considered only a few aspects of modeling supercapacitors and restricted our attention to charged soft spheres as a model for ionic liquids. Our results on the pore walls suggest that non-electrostatic interactions and image forces can have a profound impact on the in-pore ion structure. It will thus be interesting and fruitful to understand the effects due to the differences in ionic liquid models [@breitsprecher14a:CG; @breitsprecher14b:CG], and whether such simple and computationally inexpensive models can capture the charging behaviour correctly.
Supplementary Material
======================
Supplementary figures show the ion-wall repulsive interaction potential ([Fig. \[si:fig:wall-ion-wca\]]{}); the ion’s self-energy across the pore ([Fig. \[si:fig:Eself\]]{}); the total wall-ion interaction potential ([Fig. \[si:fig:wall-ion\]]{}); the effect of calibration on the ion density and accumulated charge ([Fig. \[si:fig:md:bulk\_calibration\]]{}); the comparison of the charging parameter $X_D$ (see [Eq. (\[eq:XD\])]{}) obtained within the MD and MC models ([Fig. \[si:fig:mcmd:XD\]]{}); the accumulated charge within the MD and MC models ([Fig. \[si:fig:mcmd:charge\]]{}); the differential capacitance within the MD and MC models ([Fig. \[si:fig:mcmd:cap\]]{}); the differential capacitance and stored energy from MC simulations ([Fig. \[si:fig:mc:capen\]]{}); and the ion density profiles across the pore for pores with soft and hard walls from MC simulations ([Fig. \[si:fig:mc:rhoz\]]{}).
C.H. and K.B. acknowledge the Deutsche Forschungsgemeinschaft (DFG) through the cluster of excellence “Simulation Technology” and the SFB 716 for financial support. We are also grateful for the computing resources on the Cray XC40 (Hazel Hen) from the HLRS in Stuttgart. This project has received funding from the European Unions Horizon 2020 research and innovation programme under the Marie Sk[ł]{}odowska-Curie grant agreement No 734276 (S.K. contribution).
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---
abstract: 'We present new radio continuum observations of NGC 253 from the Murchison Widefield Array at frequencies between 76 and 227 MHz. We model the broadband radio spectral energy distribution for the total flux density of NGC 253 between 76 MHz and 11 GHz. The spectrum is best described as a sum of central starburst and extended emission. The central component, corresponding to the inner 500 pc of the starburst region of the galaxy, is best modelled as an internally free-free absorbed synchrotron plasma, with a turnover frequency around 230 MHz. The extended emission component of the NGC 253 spectrum is best described as a synchrotron emission flattening at low radio frequencies. We find that 34% of the extended emission (outside the central starburst region) at 1 GHz becomes [ partially]{} absorbed at low radio frequencies. Most of this flattening occurs in the western region of the SE halo, and may be indicative of synchrotron self-absorption of shock re-accelerated electrons or an intrinsic low-energy cut off of the electron distribution. Furthermore, we detect the large-scale synchrotron radio halo of NGC 253 in our radio images. At 154–231 MHz the halo displays the well known X-shaped/horn-like structure, and extends out to $\sim 8$ kpc in $z$-direction (from major axis).'
author:
- 'A. D. Kapińska$^{1,2}$, L. Staveley-Smith$^{1,2}$, R. Crocker$^{3}$, G. R. Meurer$^1$, S. Bhandari$^{4,2}$, N. Hurley-Walker$^{5}$, A.R. Offringa$^{6}$, D.J. Hanish, N. Seymour$^5$, R. D. Ekers$^7$, M. E. Bell$^{7}$, J.R. Callingham$^{7,8,2}$, K. S. Dwarakanath$^{9}$, B.-Q. For$^{1}$, B. M. Gaensler$^{10,8,2}$, P. J. Hancock$^{5,2}$, L. Hindson$^{11,12}$, M. Johnston-Hollitt$^{12}$, E. Lenc$^{8,2}$, B. McKinley$^{13}$, J. Morgan$^{5}$, P. Procopio$^{13,2}$, R. B. Wayth$^{5,2}$, C. Wu$^1$, Q. Zheng$^{12}$, N. Barry$^{14}$, A. P. Beardsley$^{14}$, J. D. Bowman$^{15}$, F. Briggs$^3$, P. Carroll$^{15}$, J. S. Dillon$^{16}$, A. Ewall-Wice$^{16}$, L. Feng$^{16}$, L. J. Greenhill$^{17}$, B. J. Hazelton$^{14}$, J. N. Hewitt$^{16}$, D. J. Jacobs$^{15}$, H.-S. Kim$^{13,2}$, P. Kittiwisit$^{15}$, J. Line$^{13,2}$, A. Loeb$^{17}$, D. A. Mitchell$^{3,2}$, M. F. Morales$^{14}$, A. R. Neben$^{16}$, S. Paul$^{9}$, B. Pindor$^{13,2}$, J. C. Pober$^{18}$, J. Riding$^{13,2}$, S. K. Sethi$^{9}$, N. Udaya Shankar$^{9}$, R. Subrahmanyan$^{9,2}$, I. S. Sullivan$^{14}$, M. Tegmark$^{16}$, N. Thyagarajan$^{15}$, S. J. Tingay$^{5,19,2}$, C. M. Trott$^{5}$, R. L. Webster$^{13,2}$, S. B. Wyithe$^{13,2}$, R. J. Cappallo$^{20}$, A. A. Deshpande$^{9}$, D. L. Kaplan$^{21}$, C. J. Lonsdale$^{20}$, S. R. McWhirter$^{20}$, E. Morgan$^{16}$, D. Oberoi$^{22}$, S. M. Ord$^{5,7,2}$, T. Prabu$^{9}$, K. S. Srivani$^{9}$, A. Williams$^{5}$, C. L. Williams$^{16}$'
title: Spectral energy distribution and radio halo of NGC 253 at low radio frequencies
---
INTRODUCTION {#sec:intro}
============
Observing at low radio frequencies ($\lesssim0.5$ GHz) is of a particular value; low energy and old plasma can be revealed, tracing and constraining physical conditions in galaxies. In star forming galaxies the low surface brightness plasma forms e.g. extended halos associated with winds and large scale magnetic fields, or diffuse emission from galactic disks. Furthermore, measurements at low radio frequencies can help to distinguish for instance between thermal and non-thermal plasma, and their absorbing mechanisms, responsible for the level of observed radio emission. It is expected that the Square Kilometre Array (SKA) will unravel a large star-forming galaxy population [e.g. @2015aska.confE..70B; @2015aska.confE..68J], but before we can embark on a large scale study of star-forming and starburst galaxies and their evolution with continuum radio surveys, we need to understand the origin of the complex radio spectral energy distributions and morphologies of these galaxies. Nearby objects are ideal laboratories for this task.
NGC 253 is the dominant galaxy in the nearby Sculptor Group, at a distance of 3.94 Mpc from the Local Group [@2003AA...404...93K] and velocity $cz=240$ km s$^{-1}$. It is an almost edge-on SBc type galaxy [@1991rc3..book.....D] observed at an inclination of $78.5^{\circ}$ [@1980ApJ...239...54P] and is considered a prototype of nuclear starburst galaxies [@1980ApJ...238...24R]. Its estimated stellar mass is $\sim4\times10^{10}$ M$_{\odot}$, with a prominent stellar halo of $2.5\times10^{9}$ M$_{\odot}$ extending up to 30 kpc above the disk [@2011ApJ...736...24B]. As one of the closest and most prominent galaxies, NGC 253 has been extensively studied in all wavelengths, including broadband radio continuum, polarization and [ H[i]{}]{} observations .
Radio emission from starburst galaxies originates from two principal components: the non-thermal synchrotron emission from relativistic electrons spiralling in the interstellar magnetic field, and the thermal emission from electrons colliding with ions in the ionized interstellar medium (ISM) around hot stars. The sources of the non-thermal emission are predominantly cosmic rays accelerated by supernova remnants (SNR) that in NGC 253 ultimately create a prominent synchrotron radio halo [@1992ApJ...399L..59C]. [ Studies of the NGC 253 magnetic field suggest the disk wind model and large-scale dynamo action to be shaping the vertical structure of the field, which in turn enhances the cosmic ray transport through a collimation of strong, starburst driven superwind]{} .
The starburst region of NGC 253 is violently active; the supernova rate of the inner 300 pc of the galaxy is estimated to be between $0.14$ and $2.4$ yr$^{-1}$, and the star formation rate is $\sim5$ M$_{\odot}$ yr$^{-1}$ [@2006AJ....132.1333L; @2014AJ....147....5R; @2015MNRAS.450L..80B; @2015MNRAS.450.3935L]. It has been suggested that up to half of the radio sources in the central starburst region are dominated by thermal emission: i.e. H[ii]{} regions characterized by a flat radio spectral index[^1] $\alpha \simeq 0.1$ and including at least one large supercluster of stars [@1997ApJ...488..621U; @1999ApJ...518..183K]. Outside the central starburst region the radio emission at GHz frequencies is dominated by steep spectrum diffuse emission and SNRs, but several strong thermal sources are detected [@2000AJ....120..278U]. Based on integrated radio continuum spectra, estimated 10% of the NGC 253 flux density at 1 GHz to be of thermal origin, increasing to 35% at 10 GHz.
At low radio frequencies both of these principal components become pronounced. Synchrotron emission has steep spectrum, becoming dominant at sub-GHz frequencies due to the population of old, low energy electrons. However, such emission may be also subject to self-absorption in the case of compact objects. Thermal emission also becomes increasingly more absorbed with decreasing frequency. The free-free absorption in the central starburst of NGC 253 has previously been measured .
Here, we present extensive low radio frequency ($<230$ MHz) imaging of NGC 253 obtained with the Murchison Widefield Array [MWA; @2013PASA...30...31B; @2013PASA...30....7T]. Our images are some of the deepest yet at these frequencies, and at low angular resolution they are especially sensitive to large-scale diffuse structure, allowing us to investigate the extent and frequency dependence of the radio halo. The paper is structured as follows. Our radio data and methods, including assumed models of radio spectra and model fitting, are described in §\[sec:data\] and §\[sec:methods\] respectively. Results are presented in §\[sec:results\]. The synchrotron radio halo of NGC 253 is discussed in §\[sec:halo\]. We discuss low frequency radio emission from NGC 253, its radio spectral energy distribution and radio spectral maps in §\[sec:spectral-props\]. Conclusions are given in §\[sec:conclude\].
Observations and data reduction {#sec:data}
===============================
We use radio continuum data from the Galactic and Extragalactic All-Sky MWA Survey [GLEAM; @2015PASA...32...25W] and the MWA Epoch of Reionization experiment [MWA/EoR; @2013PASA...30...31B; @2016ApJ...819....8P]. The GLEAM survey provides unprecedented spectral coverage between 72 and 231 MHz, while the MWA/EoR image at 169 MHz is [ almost twice as deep as the most sensitive GLEAM image at 200 MHz (rms noise 4.1 mJy beam$^{-1}$ and 7.3 mJy beam$^{-1}$ respectively)]{}. In addition, the data have been observed and processed independently, providing a verification of our flux density calibration.
The GaLactic and Extragalactic All-Sky MWA Survey (GLEAM)
---------------------------------------------------------
The GLEAM survey observed the entire radio sky south of declination $+30^{\circ}$ at an angular resolution of approximately 1.7 arcmin (227 MHz) to 5 arcmin (76 MHz). At 154 MHz the GLEAM survey is sensitive to structures up to $10$ deg in angular scale, and has an instantaneous field of view of $25\times25$ deg$^{2}$. The observations were made in a meridian drift scan mode covering frequencies between 72 and 231 MHz with bandwidths of 7.68 MHz grouped in five 30.72 MHz-wide bands. These bands, centred on 87.7, 118.4, 154.2, 185.0 and 215.7 MHz (hereafter 88, 118, 154, 185 and 216 MHz), were observed sequentially as 112 sec snapshots; each frequency was observed every 10 min. During a night typically 8–10 h in hour angle were observed. Frequencies between 134 and 137 MHz were avoided due to satellite interference. For more details on the survey parameters and strategy see [@2015PASA...32...25W].
Here we use GLEAM data from the first year of observing [Data Release 1 from 2013 August – 2014 June; @2016MNRAS.GLEAM.subm]. The sky area covering NGC 253 was observed on 2013 August 10 and 2013 November 25. The full data reduction process is described in detail in [@2016MNRAS.GLEAM.subm]; here we summarize only the main calibration and imaging steps.
The correlated data were first pre-processed with the [cotter]{} pipeline which performs flagging of data affected by radio frequency interference (RFI) and averaging of the data to 1s time and 40 kHz frequency resolution [@2010MNRAS.405..155O; @2015PASA...32....8O]. Standard calibration (phase and amplitude bandpass calibration) was done with [CASA]{} [Common Astronomy Software Applications package; @2007ASPC..376..127M]. Imaging and self-calibration were then performed using [WSClean]{} imager [@2014MNRAS.444..606O] that corrects for wide field $w$-term effects. Images of a 7.68 MHz bandwidth at 20 frequencies continuously distributed between 72 and 231 MHz (avoiding 134–137 MHz) and using a robust weighting $r=-1.0$ [@1995AAS...18711202B] were then created. Deconvolution has been performed at this stage, and details are provided in [@2016MNRAS.GLEAM.subm].
The primary beam correction of our GLEAM observations was done with the [@2015RaSc...50...52S] model down to the 10% level of the beam response. An additional calibration stage was necessary to correct for residual declination dependence of the flux density scale in the final mosaics arising from the limited accuracy of the adopted primary beam model. This was done by comparing flux density measurements of all unresolved sources extracted from GLEAM images above $8\sigma$ rms noise level to their radio spectra as predicted by three catalogues: VLA Low-Frequency Sky Survey redux [VLSSr; @2014MNRAS.440..327L], MRC and NRAO VLA Sky Survey [NVSS; @1998AJ....115.1693C]. The absolute flux density scale of the GLEAM images is accurate to 8%, which is included in the quoted uncertainties of the measurements [for details see @2016MNRAS.GLEAM.subm].
Images for each of five central frequencies centered on frequencies of 88, 118, 154, 185 and 216 MHz and of bandwidth 30.72 MHz were made. The two highest frequency images are further combined to create a ‘deep’ image at 200 MHz with a 61.4 MHz bandwidth. We also use the 7.68-MHz images for construction of the high resolution radio spectrum of NGC 253. The final synthesised beam sizes, rms and background noise levels in the deep 200 MHz image are $2.22\times2.12$ arcmin$^2$, ${\rm PA}=-78^\circ$, 11 mJy beam$^{-1}$ and 7.3 mJy beam$^{-1}$ respectively, and [ their range between the lowest and highest GLEAM frequencies is listed in Table \[tab:noisebm\]]{}.
------- ------------------ ----------- ------------------- ------------------- -- --
$\nu$ rms noise background
(MHz) bmaj$\times$bmin PA (mJy beam$^{-1}$) noise
(arcmin$^2$) (deg) (mJy beam$^{-1}$)
76 $5.03\times4.72$ $-18.8$ $107$ $-44$
227 $1.73\times1.67$ $-26.0$ $12.8$ $-3.3$
------- ------------------ ----------- ------------------- ------------------- -- --
: Range of angular resolution and noise values of the GLEAM data. []{data-label="tab:noisebm"}
![The 330 MHz image of NGC 253 from with overlaid contours from the TGSS ADR1 survey (white) and the MWA/EoR image (red). The TGSS contours start at $4\sigma$ local rms noise level ($\sigma=11.7$ mJy beam$^{-1}$) and increase as $\sigma2^i$ for $i>0$. The MWA/EoR0 contour marks the $4\sigma$ radio intensity at 169 MHz (16.4 mJy beam$^{-1}$). The sizes of the synthesised beams at 169 MHz (red), 330 MHz (green) and 150 MHz (black) are drawn in the top right corner. Background sources, not associated with the intrinsic emission of NGC 253, are labelled with numbers (see §\[sec:bkg-srcs\]). The color scale is in units of Jy beam$^{-1}$, and the pixel size is $5\times5$ arcsec$^2$.\
[]{data-label="rys:tgss-carilli"}](carilli-tgss-eor.eps){width="86mm"}
\[rys:carilli-halo\] \[rys:bkg-sources\]
MWA Epoch of Reionization (EoR) data
------------------------------------
The observed MWA/EoR field that contains the Sculptor Group (EoR0 field) is centered on $\text{RA} = 0^{\rm h}$, $\text{Dec} = -27^{\circ}$, and was observed for a total of 30 hours between August and October 2013 in a combination of a tracking and drifting modes. In this hybrid mode the telescope tracks a set of discrete pointing centers through which the field of interest is drifting. The observations cover frequencies between $138.9-197.7$ MHz observed as two bands (low and high) each with an instantaneous bandwidth of $\Delta\nu=30.72$ MHz.
The correlated MWA/EoR0 data were pre-processed with the [cotter]{} pipeline [@2015PASA...32....8O] and averaged to 4s time and 40 kHz frequency resolution. Calibration of the data was performed as a direction-independent self-calibration using the [mitchcal]{} tool [@2008ISTSP...2..707M] and was based on a bootstrapped sky model. The initial sky model was generated from the MWA Commissioning Survey [@2014PASA...31...45H], the MRC catalogue and the Sydney University Molonglo Sky Survey [SUMSS; @2003MNRAS.342.1117M]. Imaging was performed with the [WSClean]{} software that corrects for the non-zero $w$-term effects. During the imaging process 2,500 sources were peeled, and the images were created with a uniform weighting. The primary beam, and so the flux density scale, was corrected by applying the [@2015RaSc...50...52S] model. As shown by [@2014PASA...31...45H], this model is accurate to 10%, hence we add this error in quadrature to the quoted uncertainties of our measurements.
The final image used in this paper is centred at 169.6 MHz (thereafter 169 MHz) with a total bandwidth of $\Delta\nu=58.8$ MHz, synthesised beam size $2.3\times2.3$ arcmin$^2$ and rms noise $4.1$ mJy beam$^{-1}$. The calibration and imaging process of the EoR0 data is presented and discussed in detail in [@2016MNRAS.458.1057O].
Other low frequency radio surveys
---------------------------------
There are additional two all-sky low frequency radio surveys that include NGC 253: the 74 MHz VLSSr [@2014MNRAS.440..327L] and the 150 MHz Tata Institute of Fundamental Research (TIFR) Giant Metrewave Radio Telescope (GMRT) Sky Survey [ (TGSS) Alternative Data Release 1]{} [ADR1; @2016arXiv160304368I].
The TGSS survey observed the whole radio sky north of declination $-53^{\circ}$ at a frequency 150 MHz (bandwidth $\Delta\nu=16.7$ MHz) at an angular resolution $25\times 25/\text{cos}(\delta-19^{\circ})$ arcsec$^2$ at declinations south of $+19^{\circ}$. The instantenous field of view of the survey at half power at 150 MHz is $3.1\times3.1$ deg$^2$, with sensitivity to structures up to 68 arcmin in angular scale [@2016arXiv160304368I]. Since the absolute flux density calibration of the TGSS ADR1 may be uncertain up to 50% in some sky regions[^2], we independently verified the calibration in the area of NGC 253. We selected unresolved sources with flux density $>1$ Jy from the $5\times5$ deg$^2$ mosaic that included NGC 253 (R03\_D17). We compared the TGSS ADR1 flux densities of these sources with the predicted values based on the spectral modeling in which we used the VLSSr, GLEAM (deep 200 MHz), MRC and NVSS surveys. We found that the TGSS mosaic required scaling by a factor 1.02 in flux density, and the absolute flux density calibration was accurate to 7%; we further added this error in quadrature to the quoted uncertainties of our measurements.
The VLSS survey [@2007AJ....134.1245C] observed the radio sky north of $-30^{\circ}$ at a frequency 74 MHz. Here we use the recent re-reduction of the survey data, the VLSSr [@2014MNRAS.440..327L]. VLSSr images have an angular resolution of $75\times75$ arcsec$^2$ and a theoretical sensitivity to structures of 13–37 arcmin in angular scale.
We find that neither TGSS nor VLSSr are sensitive to the extended emission of NGC 253 (Figure \[rys:tgss-carilli\]). For this reason we use the TGSS data for the flux density measurement of the central starburst region, and both TGSS and VLSSr for measurements of the flux densities of background sources only.
{width="180mm"}
Methods {#sec:methods}
=======
Flux density measurements {#sec:flux-dens}
-------------------------
Measurements of the total flux density of NGC 253 were performed with [CASA]{} task [imstat]{} that provides a summed flux density within a specified regions of the image corrected for the synthesised beam. We masked all pixels below $2.6\sigma$ local rms noise level [ [@2012MNRAS.425..979H]]{}. For point sources the flux density was measured with the [AIPS]{} task [jmfit]{}; for each unresolved source we fit for two components, a Gaussian and a zero-level with a slope. The absolute flux density scale is set to the [@1977AA....61...99B] scale.
Models of radio spectra {#sec:models-spectra}
-----------------------
Radio sources often show simple spectra that can be approximated by a power-law. However, at low radio frequencies (a few hundred MHz and below) radio spectra are consistently more curved until a turnover frequency below which the spectra become inverted. The spectral turnover is typically caused by either synchrotron self-absorption and/or thermal free-free absorption [@2015ApJ...809..168C and references therein]. If there is no evidence for a spectral turnover in the radio spectra we construct here [ (see §\[sec:modselct\] for model selection method)]{}, we proceed with fitting a polynomial. The curved radio spectra are then modeled with an $n$th-order polynomial, that in the logarithmic scale takes a general form of $$\begin{aligned}
\text{log}(S_{\nu}) & = \sum^n_{i=0} A_i \text{log}^i\left(\frac{\nu}{\nu_0}\right) \\
& = A_0 + A_1 \text{log}\left(\frac{\nu}{\nu_0}\right) + ... + A_n \text{log}^n\left(\frac{\nu}{\nu_0}\right),
\end{aligned}
\label{eqn:poly}$$ where $A_0$ is an offset parameter (equivalent to log($S_0$) in the simple power-law case), $A_1$ is the spectral index $-\alpha$, and $A_n$ are curvature parameters ($c_n$). In the linear space the model takes the following form $$S_\nu = \prod^n_{i=0} 10^{A_i\text{log}^i(\nu/\nu_0)},$$ which we use in our modeling to preserve Gaussian noise characteristics of the measurements.
Where the data suggest or show a spectral turnover, the following models are tested: synchrotron self-absorption, free-free absorption or a combination of these and power-law components.
### Synchrotron self-absorption (SSA)
At low radio frequencies the intensity of the synchrotron radiation may become sufficiently high (optically thick regime) for re-absorption, termed synchrotron self-absorption, to take over. The process may be important, or even dominant, for compact sources [@1990SvAL...16..339S; @1998ApJ...499..810C]. We model the synchrotron radio spectra that may turnover due to self-absorption at low radio frequencies as [e.g. @2003AJ....126..723T] $$S_\nu = S_{\tau=1} \left( \frac{\nu}{\nu_{\tau=1}} \right)^{-\alpha} \left ( \frac{1 - e^{-\tau(\nu)}}{\tau(\nu)}\right ),
\label{eqn:SSA}$$ $$\tau(\nu) = (\nu/\nu_{\tau=1})^{-(\alpha+2.5)},$$ where $\nu_{\tau=1}$ is a frequency at which the optical depth ($\tau$) reaches unity.
----------- ----------------------- ----------------- -------------- ---------------- ------------
Frequency [Angular]{} References
$[$MHz$]$ [resolution]{}
(arcsec$^2$)
74 $263 \pm76^{\dagger}$ $75\times75$ a,
150 $212 \pm19$ $15.6^{\rm up}$ $162\pm17$ $36\times24$ b,
200 $233 \pm23$ $138\times126$ c
330 $190\pm10$ $72\times72$ d
610 $84 \pm15$ $114\times24$ e
843 $97 \pm10$ $47\times43$ f
1465 $63.0\pm2.5$ $66\times38$ g
1465 $53.0\pm3.0$ $17.2\pm1.0$ $26\pm2$ $30\times30$ d
1490 $5.6 \pm0.5$ $53.0\pm0.5$ $19.0\pm0.5$ h
4850 $19.0\pm1.0$ $6.6\pm0.3$ $7.9\pm0.4$ $30\times30$ d
8350 $9.5\pm0.5$ $84\times84$ d
----------- ----------------------- ----------------- -------------- ---------------- ------------
$\dagger$ Tentative detection ($2.5\sigma$). $^{\rm up}$ Upper limit, equal $3\times$ local rms noise level. Our measurement based on images from the quoted survey. [**References.**]{} (a) [@2014MNRAS.440..327L], (b) [@2016arXiv160304368I], (c) This publication, (d) , (e) , (f) [@1983PASAu...5..235R], (g) , (h) [@1987ApJS...65..485C].
------------ ------------ ----------------- ---------------- ----------------- ----- ----------
Background polynomial $S_{1 \rm GHz}$ $\alpha$ $c_1$ dof $\chi^2$
source order (mJy)
1 2 $80.7 \pm1.9$ $0.79 \pm0.02$ $-0.25 \pm0.04$ 7 19.2
2 2 $21.9 \pm1.0$ $0.32 \pm0.07$ $-0.63 \pm0.11$ 1 3.5
3 1 $33.1 \pm1.4$ $0.88 \pm0.03$ – 1 2.8
------------ ------------ ----------------- ---------------- ----------------- ----- ----------
----------- ---------------- ---------------------------------------------- ------------------------------- ------------
Frequency [ Angular]{} [ Largest ]{} References
$[$MHz$]$ [ resolution]{} [ angular scale]{}
(arcmin$^2$) (deg)
76 $23.9\pm2.0$ $5.1\times4.7$ 29 a
80 $23.7\pm4.0$ $3.7\times3.7$ no info b,c
84 $22.6\pm1.9$ $5.1\times4.7$ 27 a
92 $20.9\pm1.7$ $5.1\times4.7$ 24 a
99 $20.0\pm1.6$ $5.1\times4.7$ 23 a
107 $20.3\pm1.7$ $5.1\times4.7$ 21 a
115 $20.0\pm1.6$ $5.1\times4.7$ 19 a
122 $18.8\pm1.6$ $5.1\times4.7$ 18 a
130 $17.8\pm1.5$ $5.1\times4.7$ 17 a
143 $17.9\pm1.5$ $5.1\times4.7$ 16 a
151 $16.6\pm1.4$ $5.1\times4.7$ 15 a
158 $16.1\pm1.3$ $5.1\times4.7$ 14 a
166 $15.7\pm1.3$ $5.1\times4.7$ 13 a
169 $15.0\pm1.2$ $5.1\times4.7$ 13 a
174 $16.4\pm1.3$ $5.1\times4.7$ 13 a
181 $15.7\pm1.3$ $5.1\times4.7$ 12 a
189 $15.3\pm1.2$ $5.1\times4.7$ 12 a
197 $15.4\pm1.2$ $5.1\times4.7$ 11 a
204 $15.8\pm1.3$ $5.1\times4.7$ 11 a
212 $15.1\pm1.2$ $5.1\times4.7$ 11 a
220 $14.7\pm1.2$ $5.1\times4.7$ 10 a
227 $15.0\pm1.2$ $5.1\times4.7$ 10 a
330 $16.5 \pm1.9$ $1.2\times 1.2$ 1.2 d, e
408 $15.7 \pm1.9$ $2.9\times2.86$ no info f
468 $15.1 \pm1.5$ $2.1\times2.1$, $5.2\times5.2 ^\diamondsuit$ extr. single dish$^\clubsuit$ c, g
610 $9.4 \pm0.6$ $1.9\times0.4$ no info h
843 $9.0 \pm0.9$ $0.72\times0.78$ no info (th: 1.1) i
960 $8.0 \pm0.12$ $20.2\times20.2$ single dish c, g
1100 $6.7 \pm0.08$ $4.2\times4.2$ no info, dense core j
1200 $6.68 \pm0.10$ $3.4\times3.4$ no info, dense core j
1300 $6.22 \pm0.07$ $3.2\times3.2$ no info, dense core j
1400 $5.89 \pm0.16$ $2.9\times2.9$ no info, dense core j
1410 $6.12 \pm0.12$ $15.5\times15.5$ single dish c, g
1430 $5.7 \pm0.5$ $0.91\times0.83$ no info (th: 0.8) h
1465 $5.9 \pm0.1$ $1.1\times0.63$ extr. single dish$^\clubsuit$ k
1465 $6.3 \pm1.1$ $0.5\times0.5$ 0.25 d, e
1490 $5.6 \pm0.5$ $0.9\times0.9$ 0.27 m
2650 $3.85 \pm0.12$ $8.3\times8.3$ single dish c, g
2695 $4.26 \pm0.14$ $4.9\times4.9$ single dish n
2700 $3.49 \pm0.12$ $8.0\times8.0$ single dish c, g
4850 $2.93 \pm0.13$ $2.7\times2.7$ single dish n
4850 $2.71 \pm0.14$ $0.5\times0.5$ single dish e
4850 $2.69 \pm0.10$ $4.2\times4.2$ single dish p
5009 $2.50 \pm0.23$ $4.0\times4.0$ single dish c, g
5009 $2.12 \pm0.09$ $4.0\times4.0$ single dish c, r
8350 $1.66 \pm0.08$ $1.4\times1.4$ single dish e
8700 $2.06 \pm0.12$ $1.5\times1.5$ single dish n
10550 $1.98 \pm0.18$ $1.2\times1.2$ single dish s
10700 $1.95 \pm0.15$ $1.2\times1.2$ single dish t
----------- ---------------- ---------------------------------------------- ------------------------------- ------------
$\diamondsuit$ Conflicting details given in the reference. $\clubsuit$ Corrected for zero-spacing missing flux density with extrapolation. [**References.**]{} (a) This publication, (b) [@1973AuJPA..27....1S], (c) [@1981AAS...45..367K], (d) [@1992ApJ...399L..59C], (e) , (f) [@1971MNRAS.152..403C], (g) [@1975AuJPA..38....1W], (h) , (i) [@1983PASAu...5..235R], (j) [@2010ApJ...710.1462W], (k) , (m) [@1987ApJS...65..485C], (n) [@1979AA....77...25B], (p) [@1994ApJS...90..179G], (r) [@1976AuJPA..39....1W], (s) , (t) [@1983AA...127..177K].
----------- ----------------------- ---------------- ------------
Frequency [Angular]{} References
$[$MHz$]$ [resolution]{}
(arcsec$^2$)
150 $2.16\pm0.15$ $36\times24$ a,
330 $2.67 \pm0.16$ $33\times21$ b
610 $2.3 \pm0.2$ $114\times24$ c
1413 $2.33 \pm0.14\dagger$ $3\times1.8$ d
1450 $2.07 \pm0.04$ $33\times21$ b
1465 $2.04 \pm0.10$ $30\times30$ e
1660 $1.96 \pm0.04$ $33\times21$ b
4520 $1.36 \pm0.04$ $33\times21$ b
4850 $1.27 \pm0.06$ $30\times30$ e
4890 $1.30 \pm0.04$ $33\times21$ b
6700 $1.13 \pm0.04$ $37\times37$ f
7000 $1.04 \pm0.04$ $35\times35$ f
8090 $0.93 \pm0.03$ $33\times21$ b
8350 $0.98 \pm0.05$ $84\times84$ e
8470 $0.89 \pm0.03$ $33\times21$ b
----------- ----------------------- ---------------- ------------
: Flux density measurements of NGC 253 central starburst region. All measurements are in the same absolute flux density scale of [@1977AA....61...99B]. []{data-label="tab:spectra-core"}
$\dagger$ Integrated. Our measurement based on images from the quoted survey. [**References.**]{} (a) [@2016arXiv160304368I], (b) , (c) , (d) [@1982ApJ...252..102C], (e) , (f) [@2010ApJ...710.1462W].
{width="180mm"}
{width="103mm"} {width="75mm"}
### Free-free absorption (FFA)
The self-absorbed bremstrahlung (i.e. free-free absorbed) radio spectrum can be expressed as [e.g. @2002MNRAS.334..912M] $$S_\nu = S_{\tau=1} \left (\frac{\nu}{\nu_{\tau=1}}\right)^{2} (1 - e^{-\tau_{\rm ff}(\nu)}),$$ where the opacity coefficient is given by $$\tau_{\rm ff}(\nu) = (\nu/\nu_{\tau=1})^{-2.1}.
\label{eqn:tau-FFA}$$
As discussed in §\[sec:intro\] radio emission from NGC 253 is a mixture of synchrotron emitting cosmic rays from SNR and thermal emission H[ii]{} regions. The free-free absorption is expected to start dominating at low radio frequencies, where the intensity of the electrons in the ionized gas becomes high (optically thick regime). For NGC 253 it is a natural assumption that the thermal plasma co-exists with the synchrotron emitting electrons, hence the radio spectrum can be modeled as a synchrotron power-law with an internal free-free absorbing screen [SFA; @2003AJ....126..723T], $$S_\nu = S_{0} \left(\frac{\nu}{\nu_0} \right)^{-\alpha} \left ( \frac{1 - e^{-\tau_{\rm ff}(\nu)}}{\tau_{\rm ff}(\nu)}\right ).
\label{eqn:SFA}$$
Weighted non-linear least squares fitting {#sec:correlated-noise}
-----------------------------------------
All measurements in this paper are considered independent of each other (in the GLEAM survey valid for flux densities $\gtrsim5$ Jy; see [@2016MNRAS.GLEAM.subm], thus a simple form of $\chi^2$ statistic is used for the fitting of the radio spectra, which at the same time is the goodness-of-fit of the fitted model ($M_i$), $$\chi^2 = \sum_{i}^n \left( \frac{\text{M}_i - \text{data}_i}{\text{error}_i} \right)^2
\label{eqn:chi2}$$ for $i=1,..,n$ data points. In minimization of Eqn. \[eqn:chi2\] we use the Levenberg-Marquardt algorithm [@1944Levenberg; @1963Marquardt] implemented in the Python[^3] module [lmfit]{} [@2014PythonLMFIT].
Model selection {#sec:modselct}
---------------
For the formal model selection we use the Bayesian inference method. We follow the prescription outlined in [@2015ApJ...809..168C], with the log-likelihood function (the probability of observing the data given model parameters $\theta$) in the form of
$$\text{ln}\mathcal{L}(\theta) = -\frac{1}{2}\sum_{i}^{n}\left[\frac{(\text{M}_i - \text{data}_i)^2}{\text{error}_i^2} + \text{ln}(2\pi\, \text{error}_i^2)\right].$$
Under the hypothesis that the models being compared ($M_1$, $M_2$) are equally likely, the model selection can be performed based solely on the Bayesian evidence ($Z$), where
$$\Delta \text{ln}(Z) = \text{ln}(Z_2) - \text{ln}(Z_1)$$
and $$Z_{1,2} = \int \int ... \int \mathcal{L}(\theta) \Pi(\theta) d(\theta).$$ The dimensionality of the integration depends on the number of model parameters. If $\Delta\text{ln}(Z)\geqslant3$ model $M_2$ is strongly favoured over $M_1$. If $1<\Delta\text{ln}(Z)<3$ model $M_2$ is only moderately favoured over $M_1$, and if $\Delta\text{ln}(Z)<1$ the preference of one model over the other is inconclusive. For more discussion on the theoretical background of the method used see [@2015ApJ...809..168C]. We use the [MultiNest]{} tool for our calculations of the Bayesian evidence.
[crrrrrrrr]{} Region & & $S_{\rm 169 MHz}$ & $S_{\rm 1.4GHz}^{\rm 169 res}$ & & $S_{\rm 200 MHz}$ & $S_{\rm 1.4GHz}^{\rm 200 res}$\
& & & & & &\
1 & $0.54 \pm0.06$ & $242\pm26$ & $75\pm10$ & $0.57\pm0.06$ & $233\pm25$ & $75\pm10$\
2 & $0.63 \pm0.04$ & $586\pm59$ & $149\pm13$ & $0.68\pm0.03$ & $581\pm51$ & $149\pm13$\
3 & $0.57 \pm0.02$ & $883\pm89$ & $257\pm11$ & $0.62\pm0.02$ & $884\pm73$ & $257\pm11$\
4 & $0.67 \pm0.04$ & $415\pm42$ & $97\pm9$ & $0.72\pm0.04$ & $412\pm36$ & $97\pm9$\
5 & $0.31 \pm<0.01$ & $6558\pm656$& $3341\pm14$ & $0.34\pm<0.01$ & $6589\pm528$& $3341\pm14$\
6 & $0.60 \pm0.07$ & $224\pm24$ & $61\pm10$ & $0.70\pm0.07$ & $246\pm25$ & $61\pm10$\
7 & $0.58 \pm0.03$ & $488\pm49$ & $139\pm10$ & $0.63\pm0.03$ & $487\pm42$ & $139\pm10$\
8 & $0.50 \pm0.02$ & $616\pm62$ & $211\pm9$ & $0.51\pm0.02$ & $578\pm49$ & $211\pm9$\
9 & $0.53 \pm0.06$ & $216\pm23$ & $69\pm9$ & $0.51 \pm0.06$ & $192\pm21$ & $69\pm9$\
Results {#sec:results}
=======
In what follows we refer to the ‘halo’ as the radio emission beyond the boundary of the optical disk of the galaxy. Given the low angular resolution of the MWA observations we distinguish only between the galaxy disk and the extended synchrotron halo.
Background sources {#sec:bkg-srcs}
------------------
There are three discrete radio sources located within the extended emission of NGC 253; the sources are marked in Figure \[rys:bkg-sources\], and their positions are based on the NVSS and measurements.
The discrete radio source no. 1 is located at RA(J2000)=$00^h47^m59^s.10$, Dec(J2000)=$-25^\circ18'22''.45$ at 200 MHz, and is most likely a background AGN [@1992ApJ...399L..59C]. The radio source is detected in the GLEAM images, but at low frequencies becomes increasingly confused with the NGC 253 halo emission. We measure the flux density of the source only in the deep GLEAM image (Table \[tab:spectra\]). The discrete radio source no. 2 is located at RA(J2000)=$00^h47^m12^s.01$, Dec(J2000)=$-25^\circ17'43''.9$ and is most likely a faint background AGN . This source is heavily embeded in the NGC 253 extended emission in the MWA images. The discrete radio source no. 3 is located at RA(J2000)=$00^h47^m44^s.91$, Dec(J2000)=$-25^\circ13'38''.4$. This source is embeded in the extended emission in our MWA images, but is clearly detected in the TGSS ADR1 image (Figure \[rys:bkg-sources\]). The flux density measurements of the background sources are listed in Table \[tab:spectra\], and the spectral modeling results are given in Table \[tab:bkgmodels\]. We subtract the estimated flux density contribution of these sources from the total flux density measurements of NGC 253. In addition, we model the background source no. 1 as a point source with a peak flux density $240$ mJy at 169 MHz and $223$ mJy at 200 MHz, and for pictorial purposes we subtract it directly from the radio image plane. The resulting radio contours are overlaid on H$\alpha$ and X-ray images in Figure \[rys:multi-images\] and discussed in §\[sec:halo\].
NGC 253 {#sec:ngc253}
-------
Radio images of NGC 253 at six chosen radio frequencies are presented in Figure \[rys:radio-images\]. At 200 MHz, [ the deepest image from the presented here GLEAM observations,]{} the size of NGC 253 is 1310 arcsec (major axis) and 535 arcsec (minor axis) measured at a $\text{PA}=52^\circ$, with a total radio luminosity density $2.4(\pm0.1)\times10^{22}$ W Hz$^{-1}$. [ At 169 MHz (MWA/EoR0 image), the size increases by 3–6 percent, to 1440 arcsec (major axis) and 615 arcsec (minor axis), which may be a combination of the intrinsic increase in size and the uncertainty of the measurement]{}.
### Total radio emission
Radio continuum spectra of NGC 253 between 76 MHz and 10.7 GHz are plotted in Figure \[rys:radio-spectra\] and the flux density measurements are tabulated in Table \[tab:spectra\]. Background radio sources (§\[sec:bkg-srcs\]) located within the diffuse emission of NGC 253 were subtracted from the total flux density measurements. In the construction of the radio spectrum we used archival data provided the measurements were of angular resolution comparable to GLEAM or were sensitive to low brightness emission on angular scales of at least 0.5 deg, the total flux density was integrated over the diffuse emission of NGC 253 and not fitted by Gaussian components, and the absolute flux density scale and the uncertainties of the measurements were quoted. We do not use measurements at angular resolution of $>20$ arcmin because of confusion of NGC 253 with nearby sources.
We find the best fitting model to be a 2nd-order polynomial with $S_0=7.30\pm0.04$ Jy, $\alpha=0.56\pm0.01$ and a curvature $c_1=-0.12\pm0.01$ at a reference frequency of 1 GHz ($\chi^2=140$, with degrees of freedom: dof $=45$; Figure \[rys:radio-spectra\]), which is significantly preferred to a simple power-law ($\Delta \text{ln}(Z)=100.5\pm0.3$).
### Central starburst region {#sec:nucleus}
The angular resolution of the MWA data is too low to resolve the central starburst region of NGC 253; the highest angular resolution achieved is 102 arcsec at 227 MHz (GLEAM) and 138 arcsec at 169 MHz (EoR), which is over three times the size of the NGC 253 starburst region . We construct radio spectra of the central starburst region using data from the literature and new measurements from the TGSS ADR1 survey (Figure \[rys:radio-spectra\], Table \[tab:spectra\]). We limit the measurements to those that are at an angular resolution comparable to the size of the central starburst region (approximately 20–30 arcsec).
We find the best fitting model to be a 2nd order polynomial with $S_0=2.28\pm0.02$ Jy, $\alpha=0.20\pm0.01$ and a curvature $c_1=-0.24\pm0.01$ at a reference frequency of 1 GHz ($\chi^2=12.8$, dof $=13$; Figure \[rys:radio-spectra\]). We further attempt to model the spectral turnover, and we find SFA (Eqn. \[eqn:SFA\]) to be the best fitting model with $S_{\tau=1}=4.43\pm0.14$ Jy, $\alpha=0.43\pm0.01$ and $\nu_{\tau=1}=238\pm15$ MHz ($\chi^2=42.9$, dof $=13$). Based on the Bayesian evidence the SFA model (synchrotron plasma absorbed by an internal free-free absorbing screen) is preferred to the pure synchrotron self-absorption model, SSA ($\Delta \text{ln}(Z)=8.3\pm0.3$).
{width="182mm"}
{width="78mm"} {width="78mm"}
### Spectral index maps {#sec:spindx-maps}
Using the total flux density images at 200 MHz (GLEAM), 169 MHz (MWA/EoR) and 1.46 GHz [@1992ApJ...399L..59C], we created spectral index distribution maps and their corresponding uncertainty and signal-to-noise ratio maps (Figure \[rys:sp-idx\]). [ To create the spectral index maps we convolved the 1.46 GHz image to the resolution of the GLEAM image (for the 200 MHz–1.46 GHz spectral index map), and the MWA/EoR image (169 MHz–1.46 GHz spectral index map).]{} The spectral index maps suggest an apparent variation in $\alpha$ across the NGC 253 disc and halo. The central regions of the galaxy are dominated by a flat component ($\alpha=0.31-0.34$), coinciding with the central starburst. The gradual steepening along the minor axis seen by is seen only on the northern side of the galaxy in our maps. Further, the region extending SW from the central starburst region seems to be flatter ($\alpha \lesssim 0.53$) than in the other parts of the galaxy outside the central starburst ($\alpha \sim 0.60-0.65$).
To verify the significance of the spectral index variation across the galaxy, we used the T-T method [@1962MNRAS.124..297T]. The method allows one to estimate a spectral index within defined regions of a source between two frequencies. We define nine regions within NGC 253 (Figure \[rys:sp-idx\]). Due to the low angular resolution of our observations and to avoid oversampling, only one data point is associated with each region. The results are shown in Table \[tab:TTplots\] and Figure \[rys:TTplots\].
We find that between 200 MHz and 1.465 GHz the apparently flat regions (Region 8 and 9) are statistically different from the other regions within NGC 253 apart from Region 1 (Figure \[rys:TTplots\], Table \[tab:TTplots\]). This spectral flattening is not present in the radio spectral index distribution map between 330 MHz and 1.46 GHz of , even though the same high frequency map is used. This clearly indicates that the flattening occurs at $<300$ MHz. There is also a slight flattening of the spectral index in the NE region perpendicular to the major axis (Region 1), although we find this flattening to be statistically different only from Region 2 (eastern NW halo), 4 (radio spur) and 5 (including central starburst) in the $\alpha^{\rm 200 MHz}_{\rm 1.4 GHz}$ map. All regions further flatten at 169 MHz, reducing the differences between spectral indices of the regions.
Discussion
==========
Low-frequency synchrotron radio halo {#sec:halo}
------------------------------------
A large-scale radio halo in NGC 253 was discovered and confirmed by [@1992ApJ...399L..59C] and extensively studied by , and . This synchrotron halo is most pronounced at low radio frequencies, with the estimated scale heights of $1.7\pm0.1$ kpc at 1.4 GHz and $2.5\pm0.2$ kpc at 330 MHz . Both the deep 200 MHz GLEAM image and the MWA/EoR image at 169 MHz reveal the extended synchrotron halo, which is at least as extensive as one detected in the [@1992ApJ...399L..59C] 330 MHz map (Figure \[rys:carilli-halo\]).
### Maximum vertical extent
We measured the observed, projected maximum vertical extent of the NGC 253 disk and halo at 169 MHz as a function of the distance from the nuclear region along the major axis (Figure \[rys:scaleheights\], Table \[tab:scaleheights\]). The extent is measured perpendicular to the major axis (at $\text{PA} =-38^\circ$, [*z*]{}-direction) in steps of 132 arcsec (2.4 kpc) separately for the North (filled circles) and South (empty circles) side of the disk and halo as divided by the major axis ($\text{PA} = 52^\circ$). We find the projected radio halo to extend up to 4.75 kpc above the optical [$B$ band including 90 per cent total light, @1989spce.book.....L] and 6.3 kpc above the infrared [total $K_{\rm s}$ band, @2003AJ....125..525J] edge of the galaxy, reaching up to a total 7.9 kpc in $z$-direction. This is consistent with previous radio measurements at higher radio frequencies , as well as broadband X-ray observations of the galaxy’s extended extraplanar emission . attributed decrease of scale height they measured and modeled to the increased synchrotron losses in the central regions, where the magnetic field is highest.
### Halo morphology
The shape of the radio halo in our MWA radio images (Figure \[rys:multi-images\]) resembles the ‘horn-like’ or ‘X-shaped’ structure seen at GHz radio frequencies , in H[i]{} , X-rays , H$\alpha$ (G. Meurer, priv.comm.; Figure \[rys:multi-images\]), UV [@2005ApJ...619L..99H] and far-IR [@2009ApJ...698L.125K].
The radio halo was investigated in detail by who, through modeling of the large-scale magnetic field, attributed its origin to disk wind, confirming previous suggestions . This is also in line with the H$\alpha$ and optical analyses of the inner starburst-driven superwind [@2011MNRAS.414.3719W]. In our MWA maps both the NE and NW halo regions are pronounced. The extended soft X-ray emission ($<1$ keV) of the halo, detected in the north-western direction from the NGC 253 disk, is interpreted as bubbles of hot low density gas . postulates that the large-scale magnetic field of the halo follows the walls of these bubbles, where it may be compressed, producing the X-shaped synchrotron radio halo as well as heating up pre-existing cold gas to X-ray energies. The northern halo can also be easily seen, in projection, in our Figure \[rys:multi-images\] where we overlay MWA/EoR intensity contours on the soft X-ray emission [*XMM-Newton*]{} image.
------ ------- ----- ----- ----- -----
-658 -11.8 113 2.0 100 1.8
-526 -9.5 213 3.8 211 3.8
-395 -7.1 262 4.7 327 5.9
-263 -4.7 396 7.1 440 7.9
-132 -2.4 416 7.5 367 6.6
0 0.0 311 5.6 291 5.2
132 2.4 270 4.9 291 5.2
263 4.7 297 5.5 323 5.8
395 7.1 279 5.0 260 4.7
526 9.5 201 3.6 171 3.1
658 11.8 56 1.0 100 1.8
------ ------- ----- ----- ----- -----
: Projected maximum vertical extent of the NGC 253 disk and halo at 169 MHz ($h$) measured at a distance $x$ from the nuclear region along the major axis, where 0 is centred on the nucleus of the galaxy. The NE direction along the major axis is negative and SW is positive. The extent is measured perpendicular to the major axis (PA$ =-38^\circ$) separately for the North and South side of the disk and halo. The uncertainties on the measurements are 21.6 arcsec (equivalent to 0.4 kpc). See §\[sec:halo\] for discussion. []{data-label="tab:scaleheights"}
![ Projected maximum vertical extent of the NGC 253 disk and halo at 169 MHz ($h$) as a function of the distance from the nuclear region along the major axis ($x$), where 0 is centred on the nucleus of the galaxy. The NE direction along the major axis is negative and SW is positive. The extent is are measured perpendicular to the major axis (at $\text{PA} =-38^\circ$) at a step of 132 arcsec (2.4 kpc) separately for the North (filled circles) and South (empty circles) side of the disk and halo as divided by the major axis ($\text{PA} = 52^\circ$). The synthesised beam size of the radio image, $138\times138$ arcsec$^{2}$, is not included in the uncertainties and is drawn as solid horizontal line. Measured sizes of NGC 253 at optical $B$ band including 90 per cent of total light (dash-dotted line) and total infrared $K_{\rm s}$ band (dashed line) are drawn for reference. Plotted values are tabulated in Table \[tab:scaleheights\].[]{data-label="rys:scaleheights"}](NGC253-scaleheight.eps){width="88mm"}
The SE region of the extended halo, the ‘spur’ [@1992ApJ...399L..59C], is contaminated by a background source. We modeled the background source as unresolved (at MWA angular resolution) and subtracted it from the 169 MHz EoR and deep 200 MHz GLEAM images as described in §\[sec:bkg-srcs\]. The residual emission, which we consider intrinsic to the ‘spur’ is shown in Figure \[rys:multi-images\] overplotted on the H$\alpha$ and X-ray images. Although slightly offset , the feature broadly coincides with the extended outflows at both frequencies as clearly seen in our figure. The spur has been previously interpreted as originating from the active star formation region in the NE end of the bar [@1988AJ.....95.1057W; @1992ApJ...399L..59C]. We confirm the association of the radio ‘spur’ with an extended H$\alpha$ outflow, which can be clearly seen in our Figure \[rys:multi-images\] [G. Meurer, priv.comm.; @2003PASP..115..928K]. In our radio spectral map between 200 MHz and 1.46 GHz the spur displays the steepest spectral index within the galaxy (Figure \[rys:sp-idx\]), gradually steepening from $\alpha\sim0.7$ to $\alpha\sim0.9$ outwards from the galaxy disk (though it still contains the background source no.1 with $\alpha=0.57$), which is indicative of ageing unabsorbed synchrotron plasma.
Spectral properties of NGC 253 {#sec:spectral-props}
------------------------------
### Broadband spectrum of total radio emission {#sec:nonthermal-emission}
The broadband spectrum of the total radio emission from NGC 253 is steep, although flattening at MHz radio frequencies (Figure \[rys:radio-spectra\]). The total radio emission originates from SNRs, H[ii]{} regions (predominantly central starburst region; e.g. @1997ApJ...488..621U [-@1997ApJ...488..621U], @2000AJ....120..278U [-@2000AJ....120..278U], but cf. @1988AJ.....95.1057W [-@1988AJ.....95.1057W], @1996AJ....112.1429H [-@1996AJ....112.1429H]) and electrons (cosmic rays) freely spiralling in the large scale magnetic field . The steep spectrum is understood to be of a synchrotron origin. The flattening of the spectrum, however, may be due to a number of reasons, including some degree of absorption of the synchrotron emission (Eqn. \[eqn:SSA\]) and a low energy cut-off of the electron population, where in general it is assumed the electron energy ($E$) spectrum can be described as a power-law $N(E)\propto E^{-p}$ with the index $p$ related to the radio spectral index as $\alpha=(p-1)/2$. The low radio frequency flattening of the spectra of starburst galaxies is not unusual and it has been observed previously [e.g. @2015AJ....149...32M].
{width="170mm"} {width="87mm"} {width="82.5mm"}
We attempt to separate the contribution of the extended and central starburst emission to the total flux densities to see if the [ spectral]{} flattening can be attributed mostly to the absorption occurring in the central starburst region. To do this we simultaneously fitted the total flux density and the central starburst region spectra, assuming an underlying 2- or 3-component spectrum composed of the central starburst region (component C) and extended emission (component E, composed of one or two components). As an example of our fitting method, in the case where the component E is modeled as a power-law and component C as a free-free absorbed synchrotron emission, the model equation ($S_{\nu}^{\rm mod})$ takes the following form
$$S_{\nu}^{\rm mod} = ( S_\nu^{\rm E} + S_\nu^{\rm C} )^{\rm tot} + ( S_\nu^{\rm C} )^{\rm cor},$$
where $$S_\nu^{\rm E} = S_0^{\rm ext} \left( \frac{\nu}{\nu_0}\right ) ^{-\alpha^{\rm ext}},$$ $$S_\nu^{\rm C} = S_0^{\rm cor} \left( \frac{\nu}{\nu_0}\right ) ^{-\alpha^{\rm cor}} \left (\frac{1-e^{-\tau_{\rm ff}(\nu)}}{\tau_{\rm ff}(\nu)} \right ).$$ $\tau_{\rm ff}(\nu)$ is given by Eqn. \[eqn:tau-FFA\] and the indices indicate: $\rm tot$ – total, $\rm cor$ – core, $\rm C$ – component C, $\rm E$ – component E, $\rm ext$ – extended. This model equation is then compared to our observed data ($S_{\nu}^{\rm obs}$), where $S_{\nu}^{\rm obs} = S_{\nu}^{\rm obs, tot} + S_{\nu}^{\rm obs, cor}$. [ In the 2-component model, we fit the component E with either simple power-law or a curved spectrum (2nd degree polynomial; Eqn. \[eqn:poly\]). In the 3-component model, we fit the component E with a combination of power-law and either a curved spectrum, synchrotron self-absorbed (Eqn. \[eqn:SSA\]) or synchrotron free-free absorbed component (Eqn. \[eqn:SFA\]). The component C is modeled with either 2nd-degree polynomial, self-absorbed synchrotron or synchrotron power-law emission with a free-free absorbing screen. ]{}
We find the best fitting model to be the 3-component model, with (1) the component E modeled as a combination of a simple power-law and 2nd order polynomial, with a total flux density $S_0 = 5.08\pm0.50$ Jy and $\alpha = 0.71\pm0.01$ at reference frequency 1 GHz, and 34% of $S_0$ becoming absorbed at low frequencies as described by 2nd order polynomial with $c_1=-0.76\pm0.17$, and (2) the component C modeled as a synchrotron plasma with an internal free-free absorbing screen, with $S_{\tau=1} = 4.34\pm0.11$ Jy, $\nu_{\tau=1} = 231\pm14$ MHz, $\alpha_{\rm SSA} =0.41\pm0.01$. The 3-component model ($\chi^2 = 173$, dof $=58$) is favoured over any 2-component model, even if the component E is modeled as a 2nd order polynomial ($\Delta\text{ln}(Z)>4.3\pm0.3$).
Preference of the 3-component model indicates that flattening of the component E spectrum at the lower frequencies is non-negligible. In principle, this flattening could be attributed to synchrotron self-absorption caused by shock re-acceleration of the halo/disk plasma, an external free-free absorbing screen, or an intrinsic low-energy cut-off of the electron distribution. Thermal free-free absorption can be [ largely excluded based on the limited]{} evidence for high thermal content in the NGC 253 halo, [ especially in the SW region (see §\[sec:spectral-maps\] for details)]{}. Although our Bayesian inference tests indicate that the flattening caused by the synchrotron self-absorption is moderately favoured over the low-energy cut-off in the electron distribution which could be inferred from the 2nd order polynomial fit ($\Delta\text{ln}(Z)=2.5\pm0.3$), we find that the former fit is associated with very high uncertainties. We also find that any model invoking multiple internal components that we tested is strongly favoured over an external free-free absorbing screen. Our results do not change in the absence of the data points between 300 and 600 MHz that may seem unusually high, which further strengthens our result that the radio emitting plasma in the disk and halo of NGC 253 is composed of at least two spectral components that behave differently. This result is also in line with our findings on the radio spectral index variation across the galaxy (discussed further in the next section).
Furthermore, another important result of our spectral modeling is that the central starburst region is best modeled by the SFA model. Although, in principle, the 2nd order polynomial is statistically favoured, the SFA model is more realistic. Curved radio spectra can be explained by low energy cut-off of electron population, SSA, SFA or FFA models. As we have shown the SSA model is statistically ruled out (§\[sec:nucleus\]). Given the overwhelming evidence of significant thermal component in the central starburst [e.g. Figure \[rys:multi-images\]; @1997ApJ...488..621U; @1999ApJ...518..183K; @2011ApJ...739L..24K] coexisting with synchrotron plasma, the synchrotron free-free absorption model is more likely than the low energy cut-off in electron population. The plasma becomes optically thick around frequency $230$ MHz. Given this result, and under a simplified assumption that a uniform optical depth holds across the region, we estimate a typical emission measure [@1961RvMP...33..525O; @1978ppim.book.....S] of the absorbing gas towards the central starburst region to be very high, of the order $4\times10^5$ pc cm$^{-6}$ .
### Radio spectral index distribution maps {#sec:spectral-maps}
We now consider the origin of the $\alpha$ variation across the NGC 253 disk and halo. The southern flattening occurs beyond the SW spiral arm, in the halo region. In Figure \[rys:lucero\] we overplot the $\alpha^{\rm 200 MHz}_{\rm 1.4 GHz}$ and extraplanar H[i]{} contours on the [*XMM-Newton*]{} soft X-ray image of NGC 253. The diffuse X-ray emission indicates ionized hot gas. As pointed out by [@2015MNRAS.450.3935L], the neutral cold H[i]{} gas seems to surround the X-ray emitting regions. The radio spectral index spatial variations seem to follow the distribution of the X-ray emission, with the $\alpha$ steepening occurring in the regions of intense soft X-ray emission (radio spur and NW halo) and the flattening around the voids of diffuse X-ray plasma (western SE halo, eastern NW halo). This distribution seems to also match the extraplanar H[i]{} emission, especially in the western SE halo region. [ In H$\alpha$ we detect faint diffuse emission in the NE halo and the southern ‘spur’ (Figure \[rys:multi-images\]), with the line fluxes measured down to $3\times 10^{-18}$ erg s$^{-2}$ s$^{-1}$ arcsec$^{-2}$. In these regions the spectral index seems to steepen (Figure \[rys:lucero\]B). There is, however, almost no H$\alpha$ emission present in the western SE halo, while most of the flattening of the component E in the modeling of §\[sec:nonthermal-emission\] is due to this region (Figure \[rys:lucero\]C, regions 8 and 9 in Figure \[rys:sp-idx\]).]{}
It has previously been suggested that the halo gas originates from both galactic ‘fountains’ from the star-forming disk and a galactic superwind . This strong superwind may be pushing and collimating the neutral cold gas in the halo [@2015MNRAS.450.3935L]. In the case of strong collimation shocks one may observe flattening of radio spectra due to synchrotron self-absorption in transverse shocks. Our results seem to favour such a scenario. It is also worth noting that the spectrum flattening of the SW halo corresponds to an extended loop, or arch, seen in optical images . However, if the SW halo is predominantly diffuse, and of low density, the flattening may be rather due to an intrinsic low-energy cut off of the electron distribution.
Another important note is that, based on our broadband radio spectrum modeling, the SW halo region cannot be fully responsible for the total spectrum flattening. Radio emission that becomes absorbed at lower frequencies constitutes more than 30% of the total extended radio emission at 1 GHz ($1.73\pm0.36$ Jy), while the SW region is only 0.77 Jy at that frequency, which means that the flattening must be also occurring, although in a smaller degree, in other regions across the disk and halo.
Although the SW flattening of the radio spectral index is most likely of a synchrotron origin, an external free-free absorbing screen was also previously suggested. The foreground absorption model was favoured by [@2008AA...489.1029B] based on their X-ray data modeling and apparent differences of radio and X-ray halo morphologies. As proved later , deep continuum and polarization radio observations at both GHz and MHz frequencies reveal the horn-like structure of the radio halo, which directly resembles the X-ray diffuse emission. Based on the equipartition assumptions, find the magnetic field within the halo to be very high, $7-12 \mu$G, reaching as much as $160\pm20 \mu$G in the central regions and $46\pm10 \mu$G in the starburst outflow. The magnetic field in the central regions is strong enough for synchrotron emission to contribute a few per cent to the total X-ray emission [@2013ApJ...762...29L]. As discussed in the previous section, we also find that an external free-free absorbing screen is not a statistically preferred model. These new findings support models in which the total X-ray emission may indeed come from a combination of thermal and synchrotron plasma rather than multi-temperature pure thermal plasma with an externally caused absorption [cf. @2008AA...489.1029B].
Conclusions {#sec:conclude}
===========
We present deep, low-frequency radio continuum images and flux density measurements of a nearby, archetypal starburst galaxy, NGC 253. Our data are part of the Galactic and Extragalactic All-Sky MWA Survey and the MWA EoR observations. The images span frequencies between 76 and 231 MHz at angular resolution of 1.7 – 5 arcmin and rms noise levels of 4 – 75 mJy (depending on frequency), and present the deepest measurements of NGC 253 at these low radio frequencies yet.
Our main findings are summarized as follows.
1. We detect a large-scale synchrotron radio halo that at 154–231 MHz displays the X-shaped/horn-like structure seen at GHz radio frequencies, and is broadly consistent with other multiwavelength observations of NGC 253.
2. The projected maximum vertical extent of the synchrotron emission at 169 MHz extends up to 7.5 kpc NW (7.9 kpc SE) from the major axis of NGC 253, consistent with large-scale soft X-ray emission (extending 9 kpc NW) and X-ray outflow (6.3 kpc SE).\
3. The radio spectrum of the central starburst region of NGC 253 is significantly curved at low radio frequencies, with the spectral turnover occurring around 230–240 MHz, which is for the first time statistically constrained.
4. The radio spectral index maps show significant spectral variations in the structure of NGC 253 between 200 MHz and 1.465 GHz. In particular, we isolate a region of statistically significant spectral flattening to the western side of the SE halo. However, as the SW region is rather faint at 1.46 GHz it cannot be fully responsible for the total spectrum flattening, which indicates that the flattening must be also occurring, likely in a smaller degree, in other regions across the disk and halo.
5. The broadband spectrum of integrated total radio emission of NGC 253 is best described as a sum of central starburst and extended emission, where the central starburst component is best modeled as an internally free-free absorbed synchrotron plasma, and the extended emission as synchrotron emission flattening at low radio frequencies. We also find that an external free-free absorbing screen is not a statistically preferred model when compared to models including multiple internal components.
6. We find that the extended emission of NGC 253 is best modeled by a combination of two synchrotron components, one of which becomes significantly absorbed at low radio frequencies. The flattening occurs at frequencies below $300$ MHz, and may be attributed to synchrotron self-absorption of shock re-accelerated electrons or an intrinsic low-energy cut off of the electron distribution.
Acknowledgments {#acknowledgments .unnumbered}
===============
ADK thanks P. A. Curran for valuable discussions on data modeling and constant encouragement in achieving the goals. The authors thank the anonymous referee for careful reading of the manuscript and suggestions that improved this paper. The authors thank W. Pietsch and D. Lucero for providing, respectively, X-ray and H[i]{} fits images of NGC 253, and O.I. Wong and X. Sun for helpful comments. The authors thank V. Heesen for 1.465 GHz image of NGC 253 and for helpful discussions. SB acknowledges funding for the ICRAR Summer Scholarship.
This research was conducted under financial support of the Australian Research Council Centre of Excellence for All-sky Astrophysics (CAASTRO), through project number CE110001020. This scientific work makes use of the Murchison Radio-astronomy Observatory, operated by CSIRO. We acknowledge the Wajarri Yamatji people as the traditional owners of the Observatory site. Support for the operation of the MWA is provided by the Australian Government (NCRIS), under a contract to Curtin University administered by Astronomy Australia Limited. We acknowledge the Pawsey Supercomputing Centre which is supported by the Western Australian and Australian Governments. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This publication uses the following radio data reduction software: the Multichanel Image Reconstruction, Image Analysis and Display software [; @1995ASPC...77..433S], the Common Astronomy Software Applications package [; @2007ASPC..376..127M] and the Astronomical Image Processing System [AIPS]{}. [AIPS]{} is produced and maintained by the National Radio Astronomy Observatory, a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.
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\[lastpage\]
[^1]: Radio spectral index $\alpha$ is defined such that the flux density $S_\nu$ at a frequency $\nu$ is $S_\nu \propto \nu^{-\alpha}$.
[^2]: http://tgssadr.strw.leidenuniv.nl/doku.php
[^3]: http://www.python.org
|
---
abstract: 'Pion propagation in a hadronic fluid with a non-homogeneous relativistic flow is studied in terms of the linear sigma model. The wave equation turns out to be equivalent to the equation of motion for a massless scalar field propagating in a curved spacetime geometry. The metric tensor depends locally on the soft pion dispersion relation and the four-velocity of the fluid. For a relativistic flow in curved spacetime the apparent and trapping horizons may be defined in the same way as in general relativity. An expression for the analog surface gravity is derived.'
author:
- |
Neven Bilić and Dijana Tolić\
Rudjer Bošković Institute,\
P.O. Box 180, 10001 Zagreb, Croatia\
E-mail: bilic@thphys.irb.hr, dijana.tolic@irb.hr
title: Trapped surfaces in a hadronic fluid
---
Introduction
============
In current understanding the matter created in heavy-ion collisions behaves as a nearly perfect expanding fluid [@heinz] under extreme conditions of very high density and temperature. This hydrodynamic behavior was observed at Brookhaven’s Relativistic Heavy Ion Collider (RHIC) and recently confirmed by the ALICE collaboration in Pb-Pb collisions at the LHC [@floris; @cern]. In high energy collisions the produced particles are predominantly pions. The agreement between the pion production results reported in [@floris] and the theoretical hydrodynamical model predictions [@shen] is truly remarkable. A realistic hydrodynamic model may be constructed [@dumitru-kolb] in which a transverse expansion is superimposed on a longitudinal boost invariant expansion [@bjorken]. It is often stated by particle physicists that heavy ion collisions create mini Big Bangs[^1] – events in which matter is created under extreme conditions of high density and high temperature resembling the conditions in the early universe a fraction of a second after the Big Bang. The expansion of hadronic matter that takes place immediately after a heavy ion collision has certain similarity with the cosmological expansion. However, the analogy is rather superficial since in a cosmological expansion of spacetime after the Big Bang the gravity plays the essential role, whereas high energy collisions and subsequent expansion does not involve gravity at all. Although the Minkowski spacetime with expanding hadronic matter can be mapped into an expanding spacetime, the resulting spacetime is still flat. However, we will demonstrate here that in high energy collisions a much closer analogy with cosmology may be drawn owing to the effective analog gravity with essentially curved geometry. Various aspects of analog gravity (for a review and extensive list of references see [@barcelo]) have been studied in acoustics [@visser] optics [@philbin], superfluidity [@jacobson], black hole accretion [@moncrief; @abraham], and hadronic fluid near the QCD chiral phase transition [@tolic]. In this paper we study in detail the framework of analog gravity provided by a hadronic fluid at nonzero temperature for the whole range of temperatures below the chiral phase transition. We show that the analog cosmological spacetime corresponds to a contracting FRW universe with a nontrivial apparent horizon.
Strongly interacting matter is described at the fundamental level by a nonabelian gauge theory called quantum chromodynamics (QCD). At large distances or small momenta, the QCD exhibits the phenomena of quark confinement and chiral symmetry breaking. At low energies, the QCD vacuum is characterized by a nonvanishing expectation value [@shifman]: $\langle \bar\psi\psi\rangle \approx$ (235 MeV)$^3$, the so called quark condensate, which describes the density of quark-antiquark pairs found in the QCD vacuum and its nonvanishing value is the manifestation of chiral symmetry breaking. The phenomenological importance of the chiral transition and possible experimental signatures have been discussed by Harris and Müller [@harris].
The chiral symmetry breaking and restoration at finite temperature may be conveniently studied using the linear sigma model [@bilic; @bilic1] originally proposed as a model for strong nuclear interactions [@gell]. Today, the linear sigma model serves as an effective model for the low-energy (low-temperature) phase of QCD. The basic model involves four scalar fields (three pions and a sigma meson) and two-flavor constituent quarks. In the chirally symmetric phase at temperatures above the chiral transition point the mesons are massive with equal masses and quarks are massless. In the chirally broken phase the pions are massless, whereas the quarks and sigma meson acquire a nonzero mass proportional to the chiral condensate. At temperatures below the chiral phase transition point the pions, although being massless, propagate slower than light [@pisarski2; @son1; @son2] with a velocity approaching zero at the critical temperature. Hence, it is very likely that there exists a region where the flow velocity exceeds the pion velocity and the analog trapped region may form. In our previous paper [@tolic] we have demonstrated that a region containing analog trapped surfaces forms near the chiral phase transition. The purpose of this paper is to study general conditions for the formation of a trapped region with the inner boundary as a marginally trapped surface which we refer to as the [*analog apparent horizon*]{}. Our approach is based on the linear sigma model combined with a boost invariant Bjorken type spherical expansion. A similar model has been previously studied in the context of disoriented chiral condensate [@lampert].
The remainder of the paper is organized as follows. In Sec. \[chiral\] we describe the properties and the dynamics of the chiral fluid at finite temperature. The analog geometry of the expanding chiral fluid is studied in Sec. \[analog\] in which we derive the condition for the analog apparent horizon and study the analog Hawking effect. In the concluding section, Sec. \[conclusion\], we summarize our results and discuss physical consequences. Finally, in Appendix \[trapped\] we outline basic notions related to trapped surfaces in black hole physics and cosmology.
Chiral fluid {#chiral}
============
In this section we focus on physics of hadrons at finite temperature and study the properties and the dynamics of an expanding chiral fluid. We base our study on a linear sigma model with no fermions which we describe in Sec. \[linear\]. In Sec. \[velocity\] we calculate the effective velocity of pions propagating in a chiral medium. We model the fluid expansion on a boost invariant spherical expansion of the Bjorken type which we describe in Sec. \[bjorken\]
Linear sigma model {#linear}
------------------
Consider a linear sigma model at finite temperature in a general curved spacetime background. For our purpose it is sufficient to study the model with no constituent fermions. The thermal bath provides a medium which may have an inhomogeneous velocity field. The dynamics of mesons in such a medium is described by an effective chirally symmetric Lagrangian of the form [@bilic2] $$\label{eq1}
{\cal{L}} =
\frac{1}{2}(a\, g^{\mu\nu}
+b\, u^{\mu}u^{\nu})\partial_{\mu} \varphi
\partial_{\nu} \varphi
- \frac{m_0^2}{2}
\varphi^2
- \frac{\lambda}{4}
(\varphi^2)^2 ,$$ where $u_{\mu}$ is the velocity of the fluid, and $g_{\mu\nu}$ is the background metric. The mesons $\varphi\equiv (\sigma ,$ [$\pi$]{}) constitute the $(\frac{1}{2},\frac{1}{2})$ representation of the chiral SU(2)$\times$SU(2). The parameters $a$ and $b$ depend on the local temperature $T$ and on the parameters of the model $m_0$ and $\lambda$ and may be calculated in perturbation theory. At zero temperature the medium is absent in which case $a=1$ and $b=0$.
If $m_0^{2} < 0$ the chiral symmetry will be spontaneously broken. At the classical level, the $\sigma$ and $\pi$ fields develop nonvanishing expectation values such that at zero temperature $$\label{eq2}
\langle \sigma \rangle^{2} + \langle \mbox{\boldmath$\pi$} \rangle^{2}=
- \frac{m_0^{2}}{\lambda} \equiv f_{\pi}^{2} .$$ It is convenient to choose here $$\label{eq3}
\langle \pi_{i} \rangle = 0, \;\;\;\;\; \; \langle \sigma \rangle =
f_{\pi} .$$ At nonzero temperature the expectation value $\langle \sigma \rangle$ is temperature dependent and vanishes at the chiral transition point. Redefining the fields $$\label{eq9}
\varphi \rightarrow
\varphi +\varphi'(x) =
(\sigma,\mbox{\boldmath$\pi$})+
(\sigma'(x),\mbox{\boldmath$\pi$}'(x)) ,$$ where [$\pi'$]{} and $\sigma'$ are quantum fluctuations around the constant values [$\pi$]{} = 0 and $\sigma
=\langle \sigma \rangle$ respectively, we obtain the effective Lagrangian in which the chiral symmetry is explicitly broken: $$\label{eq5}
{\cal{L}'} =
\frac{1}{2}(a\, g^{\mu\nu}
+b\, u^{\mu}u^{\nu})\partial_{\mu} \varphi'
\partial_{\nu} \varphi'
- \frac{m_{\sigma}^{2}}{2} \sigma'^{2}
- \frac{m_{\pi}^{2}}{2}
\mbox{\boldmath$\pi$}'^{2}
-g \sigma'\varphi'^2
- \frac{\lambda}{4}
(\varphi'^2)^2 .$$ The fields $\sigma'$ and $\mbox{\boldmath$\pi$}'$ correspond to the physical sigma meson and pions, respectively. The effective masses and the trilinear coupling $g$ are functions of $\sigma$ defined as $$\begin{aligned}
\label{eq11}
m_{\sigma}^{2} & = &
m_0^{2} + 3 \lambda \sigma^2 ,
\nonumber \\
m_{\pi}^{2} & = & m_0^{2}+\lambda \sigma^{2} ,
\\
g & = & \lambda \sigma .
\nonumber\end{aligned}$$ For temperatures below the chiral transition point the meson masses are given by $$m_{\pi}^2 = 0\, ; \;\;\;\;\;\;
m_{\sigma}^2 = 2\lambda \sigma^{2} ,
\label{eq43}$$ in agreement with the Goldstone theorem. The temperature dependence of the chiral condensate $\sigma$ is obtained by minimizing the thermodynamical potential $\Omega=-(T/V) \ln Z$ with respect to $\sigma$ at fixed inverse temperature $\beta$. At one loop order, the extremum condition reads [@bilic1] $$\sigma^{2}=
f_{\pi}^{2}
- 3 \int \frac{d^3p}{(2\pi)^3}
\: \frac{1}{\omega_{\sigma}} \: n_{B} (\omega_{\sigma})
- 3 \int \frac{d^3p}{(2\pi)^3}
\: \frac{1}{\omega_{\pi}} \: n_{B} (\omega_{\pi}) \, ,
\label{eq032}$$ where $$\label{eq28}
\omega_{\pi}
=|\mbox{\boldmath $p$}|
\, ; \;\;\;\;\;\;
\omega_{\sigma}
=(\mbox{\boldmath $p$}^2+m_{\sigma}^{2})^{1/2}$$ are the energies of the $\pi$ and $\sigma$ particles respectively and $$\label{eq30}
n_{B}(\omega) = \frac{1}{e^{\beta \omega} - 1}$$ is the Bose-Einstein distribution function. Eq. (\[eq032\]) has been derived from the zero-order thermodynamical potential with meson masses at one loop order [@bilic1]. This approximation corresponds to the leading order in $1/N$ expansion, where $N$ is the number of scalar fields [@meyer]. In our case, $N=4$. The right-hand side of (\[eq032\]) depends on $\sigma$ through the mass $m_{\sigma}$ given by (\[eq43\]). The behavior of $\sigma$ near the critical temperature should be analyzed with special care. A straightforward solution to (\[eq032\]) as a function of temperature exhibits a weak first-order phase transition [@bilic1; @rod]. However, Pisarski and Wilczek have shown on general grounds that the phase transition in SU(2)$\times$SU(2) chiral models should be of second order [@pis]. Hence, it is generally believed that a first-order transition is an artifact of the one loop approximation. Two loop calculations [@baacke] make an improvement and confirm the general analysis of [@pis]. It is possible to mimic the second order phase transition even with (\[eq032\]) by making the $\sigma$-meson mass temperature independent all the way up to the critical temperature and equal to its zero-temperature mean field value given by $$m_{\sigma}^2 = 2\lambda f_\pi^2 ,
\label{eq31}$$ instead of (\[eq43\]). We fix the coupling $\lambda$ from the values of $m_\sigma$ and $f_\pi$ for which we take $m_\sigma=1$ GeV and $f_\pi = 92.4$ MeV as a phenomenological input. In Fig. \[fig1\] we plot the solutions to (\[eq032\]) for both temperature dependent and temperature independent $m_\sigma$ exhibitting apparent first and second order phase transitions, respectively. In the rest of our paper we employ the solution that corresponds to the second order phase transition. For our choice of parameters we find numerically $T_{\rm c}=182.822$ MeV.
The propagation of pions is governed by the equation of motion $$\frac{1}{\sqrt{-g}}
\partial_{\mu}
\left[
{\sqrt{-g}}\,
( a\, g^{\mu\nu}+ b\,
u^{\mu}
u^{\nu}) \partial_{\nu}\mbox{\boldmath{$\pi$}}\right]
+V(\sigma,
\mbox{\boldmath{$\pi$}})
\mbox{\boldmath{$\pi$}}=0,
\label{eq013}$$ where $$V(\sigma,
\mbox{\boldmath{$\pi$}})=
m_\pi^2 + g\sigma+ \lambda (\sigma^2+\mbox{\boldmath{$\pi$}}^2)
\label{eq213}$$ is the interaction potential. In the comoving reference frame in flat spacetime, equation (\[eq013\]) reduces to the wave equation $$(\partial_t^2 -
c_{\pi}^2
\Delta +\frac{c_{\pi}^2}{a}V)
\mbox{\boldmath{$\pi$}}=0 ,
\label{eq014}$$ where the quantity $c_{\pi}$ defined by $$c_{\pi}^2=\left(1+\frac{b}{a}\right)^{-1} ,
\label{eq015}$$ is the pion velocity. As we shall demonstrate in the next section. The constants $a$ and $b$ may be derived from the finite-temperature perturbation expansion of the pion self energy.
Pion velocity {#velocity}
-------------
At temperatures below the chiral transition point the pions are massless. However, the velocity of massless particles in a medium is not necessarily equal to the velocity of light - in the chiral fluid pions usually propagate slower than light[^2]. The pion velocity in a sigma model at finite temperature has been calculated at one loop level by Pisarski and Tytgat in the low temperature approximation [@pisarski2] and by Son and Stephanov for temperatures close the chiral transition point [@son1; @son2]. It has been found that the pion velocity vanishes as one approaches the critical temperature. Here we summarize the calculation of the parameters $a$ and $b$ in the entire range of temperatures in the chiral symmetry broken phase [@bilic2].
The pion velocity may be derived from the self energy $\Sigma(q,T)$ in the limit when the external momentum $q$ approaches 0. For a flat background geometry $g_{\mu\nu}=\eta_{\mu\nu}$, the inverse pion propagator $\Delta^{-1}$ derived directly from the effective Lagrangian (\[eq5\]) as $$\Delta^{-1}=a q^{\mu}q_{\mu}
+b(q^{\mu}u_{\mu})^2 -m_{\pi}^2 ,
\label{eq200}$$ may in the limit $q\rightarrow 0$ be expressed in the form $$Z_{\pi}\Delta^{-1}=
q^{\mu}q_{\mu}-
\frac{1}{2!}
q^{\mu}q^{\nu}\left[\frac{\partial}{\partial q^{\mu}}
\frac{\partial}{\partial q^{\nu}}
(\Sigma(q,T)
- \Sigma(q,0)) \right]_{q=0}
+\dots ,
\label{eq201}$$ where the ellipsis denotes the terms of higher order in $q^{\mu}$. The $q^{\mu}$ independent term of the self energy absorbs in the renormalized pion mass, equal to zero in the chiral symmetry broken phase. The subtracted $T=0$ term has been absorbed in the wave function renormalization factor $Z_{\pi}$. By comparing this equation with Eq. (\[eq200\]) written in the comoving frame as $$\Delta^{-1}=(a+b)q_0^2
-a \mbox{\boldmath $q$}^2-m_\pi^2,
\label{eq202}$$ we can express the parameters $a$ and $b$, and hence the pion velocity, in terms of second derivatives of $\Sigma(q,T)$ evaluated at $q^{\mu}=0$. At one loop level the only diagram that gives a nontrivial q-dependence of $\Sigma$ is the bubble diagram. Subtracting the $T=0$ term one finds [@son2] $$\begin{aligned}
\Sigma(q)
\!&\! \equiv \!&\!
\Sigma(q,T)
- \Sigma(q,0)
= -4{g}^2 \int\!
\frac{d^3p}{(2\pi)^3}
\frac{1}{2\omega_{\pi}
2\omega_{\sigma,q}}
\nonumber\\
\!&\!\!&\!
\left\{ [n_B(\omega_{\pi})+
n_B(\omega_{\sigma,q})]
\left(\frac{1}{\omega_{\sigma,q}+
\omega_{\pi}}
+ \frac{1}{\omega_{\sigma,q}+
\omega_{\pi}+q_0}\right)\right. \nonumber\\
\!&\!\!&\!
+ \left. [n_B(\omega_{\pi})-
n_B(\omega_{\sigma,q})]\left(
\frac{1}{\omega_{\sigma,q}-
\omega_{\pi}} +
\frac{1}{\omega_{\sigma,q}-
\omega_{\pi}+ q_0}\right)\right\} ,
\label{eq203}\end{aligned}$$ where $\omega_{\sigma,q}=
[(\mbox{\boldmath $p$}-
\mbox{\boldmath $q$})^2
+m_\sigma^2]^{1/2}$. Here we take $m_\sigma$ to be a function of $\sigma$ through Eq. (\[eq43\]). A straightforward evaluation of the second derivatives of $\Sigma(q)$ at $q_{\mu}=0$ yields $$%\begin{eqnarray}
a =
% \!&\! = \!&\!
1+ \frac{16 {g}^2}{m_{\sigma}^4} \int\!
\frac{d^3p}{(2\pi)^3}
\left[ \frac{n_B(\omega_{\pi})}{4\omega_{\pi}}+
\frac{n_B(\omega_{\sigma})
}{4\omega_{\sigma}}
%\right. \nonumber\\
% \!&\!\!&\!
% \left.
- \frac{1}{3}
\frac{\omega_{\pi}^2}{m_{\sigma}^2}
\left(
\frac{n_B(\omega_{\pi})}{\omega_{\pi}} -
\frac{n_B(\omega_{\sigma})
}{\omega_{\sigma}}
\right)\right] ,
\label{eq204}$$ $$%\begin{eqnarray}
b =
% \!&\! = \!&\!
\frac{16{g}^2}{m_{\sigma}^4} \int\!
\frac{d^3p}{(2\pi)^3}
\left[
\frac{\omega_{\pi} n_B(\omega_{\pi})
}{m_{\sigma}^2}-
\frac{\omega_{\sigma}
n_B(\omega_{\sigma})
}{m_{\sigma}^2}
%\right. \nonumber\\
% \!&\!\!&\!
% \left.
+ \frac{1}{3}
\frac{\omega_{\pi}^2}{m_{\sigma}^2}
\left(
\frac{n_B(\omega_{\pi})}{\omega_{\pi}} -
\frac{n_B(\omega_{\sigma})
}{\omega_{\sigma}}
\right)\right] .
\label{eq205}$$ The pion velocity $c_{\pi}$ as given by (\[eq015\]) depends on temperature explicitly through the thermal distribution function $n_B$ and implicitly through the chiral condensate $\sigma$ given by Eq. (\[eq032\]).
In Fig. \[fig1a\] we plot $c_{\pi}$ as a function of temperature corresponding to two solutions depicted in the left panel.
Spherical Bjorken expansion {#bjorken}
---------------------------
In order to explore the analogy between the chiral-fluid and cosmological expansions we consider a boost invariant spherically symmetric Bjorken type expansion [@bjorken] in Minkowski background spacetime. In radial coordinates $x^\mu=(t,r,\vartheta,\varphi)$ the fluid velocity is given by $$u^\mu=(\gamma,\gamma v, 0,0)= (t/\tau, r/\tau,0,0),
% \hspace{.5in} u_\mu=(\gamma,-\gamma v, 0,0) ,
\label{eq144}$$ where $v=r/t $ is the radial three-velocity and $\tau=\sqrt{t^2-r^2}$ is the [*proper time*]{}. Using the so called [*radial rapidity*]{} $$y=\frac{1}{2} \ln \frac{t+r}{t-r} ,
\label{eq145}$$ the velocity is expressed as $$u^\mu=(\cosh y,\sinh y,0, 0),
% \hspace{.5in} u_\mu=( \cosh y, -\sinh y,0,0)
\label{eq146}$$ and hence, the radial three-velocity is $$v=\tanh y.
\label{eq246}$$ It is convenient to change $(t,r,\vartheta,\varphi)$ to new coordinates $(\tau,y,\vartheta,\varphi)$ via the transformation $$\begin{aligned}
& &t=\tau \cosh y ,
\nonumber \\
& & r=\tau \sinh y .
\label{eq147}\end{aligned}$$ In these coordinates the background Minkowski metric takes the form $$g_{\mu\nu}
=
\left(\begin{array}{cccc}
1 & & & \\
& -\tau^2 & & \\
& & -\tau^2\sinh^2\! y & \\
& & & -\tau^2\sinh^2\! y \sin^2 \vartheta
\end{array} \right).
\label{eq218}$$ and the velocity componets become $u^\mu=(1,0,0,0)$. Hence, the new coordinate frame is comoving. The metric (\[eq218\]) corresponds to the Milne cosmological model – a homogeneous, isotropic, expanding universe with the cosmological scale $a=\tau$ and negative spatial curvature.
The functional dependence of $T$ on $\tau$ follows from the energy-momentum conservation. For a perfect relativistic fluid the energy-momentum tensor is given by $$T_{\mu\nu}=(p+\rho) u_{\mu}u_{\nu}-p g_{\mu\nu} ,
\label{eq001}$$ where $p$ and $\rho$ denote respectively the pressure and the energy density of the fluid. From the energy-momentum conservation $${T^{\mu\nu}}_{;\nu}=0
\label{eq102}$$ applied to (\[eq001\]) we find $$u^\mu \rho_{,\mu}+(p+\rho){u^\mu}_{;\mu}=0,
\label{eq003}$$ where the subscript $;\mu$ denotes the covariant differentiation associated with the background metric. Since our fluid is dominated by massles pions at nonzero temperature, it is a reasonable approximation to assume the equation of state $p=\rho/3$ of an ideal gas of massless bosons. Then, Eq. (\[eq003\]) in comoving coordinates reads $$\frac{\partial\rho}{\partial\tau} + \frac{4\rho}{\tau} =0
\label{eq148}$$ with the solution $$\rho=\rho_0 \left(\frac{\tau_0}{\tau}\right)^4.
\label{eq006}$$ This expression combined with the density of the pion gas [@landau] $$\rho=\frac{\pi^2}{10}T^4,
\label{eq106}$$ implies the temperature profile $$T=T_0 \frac{\tau_0}{\tau}.
\label{eq007}$$ The constants $T_0$ and $\tau_0$ may be fixed from the phenomenology of high energy collisions. For example, if we choose $T_0=1{\rm GeV}$, then a typical value of $\rho=1 {\rm GeV/fm^3}$ at $\tau\approx 5 \,{\rm fm}$ [@kolb-russkikh] is obtained with $\tau_0 = 1.5\, {\rm fm}$. In our case, with these values the interesting range of temperatures $T$ between 100 and 200 MeV corresponds to $\tau$ between 15 and 30 fm. In the following we work with $T_0=1{\rm GeV}$ and keep $\tau_0$ unspecified so that physical quantities of dimension of time or length are expressed in units of $\tau_0$.
Analog cosmology {#analog}
================
In this section we turn to study the analog metric and formation and properties of the apparent horizon in an expanding chiral fluid. To this end we outline the formalism in the first subsection and derive a condition for the apparent horizon for a general hyperbolic spacetime. In Sec. \[horizon\] we derive the analog metric for the expanding chiral fluid and study the properties of the analog apparent horizon. Then, in Sec. \[surface\] we exploit the Kodama-Hayward definition of surface gravity to derive the Hawking temperature as a function of the parameters of the chiral fluid, in particular, as a function of the local fluid temperature.
Radial null geodesics {#radial}
---------------------
To study the apparent horizon in an expanding chiral fluid we need to examine the behavior of radial null geodesics of the analog metric which we shall derive in Sec. \[horizon\]. With hindsight, we first consider a space time of the form $$ds^2= \beta(\tau)^2 d\tau^2 -\alpha(\tau)^2 (dy^2 + \sinh^2\! y \, d\Omega^2),
\label{eq008}$$ where $\beta$ and $\alpha$ are arbitrary functions of $\tau$. The metric tensor is $$G_{\mu\nu}
=
\left(\begin{array}{cccc}
\beta^2 & & & \\
& -\alpha^2 & & \\
& & -\alpha^2\sinh^2\! y & \\
& & & -\alpha^2\sinh^2\! y \sin^2 \vartheta
\end{array} \right).
\label{eq243}$$ This metric represents the class of hyperbolic ($k=-1$) FRW spacetimes including the flat spacetime example (\[eq218\]). We denote by $l_+^\mu$ and $l_-^\mu$ the vectors tangent to outgoing and ingoing affinely parameterized radial null geodesics normal to a sphericall two-dimensional surface $S$. The tangent vectors are null with respect to the metric (\[eq243\]), i.e., $$G_{\mu\nu}l_+^\mu l_+^\nu =G_{\mu\nu}l_-^\mu l_-^\nu= 0 .
\label{eq143}$$ Using the geodesic equation $$l^\mu \nabla_\mu{l^\nu}=0,
\label{eq009}$$ where the symbol $\nabla_\mu$ denotes a covariant derivative associated with the metric (\[eq243\]), one easily finds the tangent null vectors corresponding to four types of radial null geodesics, $$l_\pm^\mu =q_\pm \alpha^{-1}\left( \beta^{-1}, \pm\alpha^{-1},0,0\right),
\label{eq010}$$ tangent to future directed and $$l_\pm^\mu =\tilde{q}_\pm \alpha^{-1}\left( - \beta^{-1}, \pm\alpha^{-1},0,0\right),
\label{eq011}$$ to past directed null geodesics , where $q_+$, $q_-$, $\tilde{q}_+$, and $\tilde{q}_-$ are arbitrary positive constants. The corresponding affine parameters $\lambda_+$ and $\lambda_-$ for the outgoing and ingoing null geodesics, respectively, are found to satisfy $$\frac{d\lambda_\pm}{d\tau}= \frac{1}{q_\pm}\alpha\beta
\label{eq012}$$ for future directed and $$\frac{d\lambda_\pm}{d\tau}=- \frac{1}{\tilde{q}_\pm}\alpha\beta ,
\label{eq019}$$ for past directed null geodesics. For simplicity, from now on we set $q_+ =q_-=\tilde{q}_+=\tilde{q}_-=1$.
The null vectors $l_+^\mu$ and $l_-^\mu$ point towards increasing and decreasing $y$, respectively. Hence, we adopt the usual convention and refer to $l_+^\mu$ ($l_-^\mu$) and the corresponding null geodesics as outgoing (ingoing) although increasing (decreasing) $y$ does not necessarily imply increasing (decreasing) of the radial coordinate $r$. As we move along a geodesic the changes of the coordinates $\tau$ and $y$ are subject to the condition $ds=0$ of radial null geodesics, i.e., $$d\tau=\pm \frac{\alpha}{\beta} dy
\label{eq017}$$ along the geodesic. Here the signs determine whether $y$ is increasing or decreasing as we move along the geodesic. For example, for future directed null geodesics, it follows from (\[eq012\]) and (\[eq017\]) that an outgoing geodesic is directed along increasing $y$, i.e., $y$ increases with $\lambda_+$, whereas an ingoing geodesic is directed along decreasing $y$, i.e., $y$ decreases with $\lambda_-$.
The key roles in the study of trapped surfaces are played by the expansion parameters $\varepsilon_+$ and $\varepsilon_-$ $$\varepsilon_\pm=\nabla_\mu l_\pm^\mu
\label{eq244}$$ of outgoing and ingoing null geodesics, respectively. Particularly important are the values of $\varepsilon_+$ and $\varepsilon_-$ and their Lie derivatives $$\frac{d\varepsilon_+}{d\lambda_-} \equiv
l^\mu_-\partial_\mu \varepsilon_+ ;
\hspace{1cm}
\frac{d\varepsilon_-}{d\lambda_+} \equiv
l^\mu_+\partial_\mu \varepsilon_-$$ in the neighborhood of a marginally trapped surface. As we shall shortly demonstrate, the relevant marginally trapped surface in the expanding chiral fluid is future inner marginally trapped. According to our convention described in Appendix \[trapped\], a two-dimensional surface $H$ is said to be [*future inner marginally trapped*]{} if the future directed null expansions on $H$ satisfy the conditions: $\varepsilon_+|_H=0$, $l_-^\mu\partial_\mu\varepsilon_+|_H>0$ and $\varepsilon_-|_H<0$. The future inner marginally trapped surface is the [**inner**]{} boundary of a future trapped region consisting of trapped surfaces with negative ingoing and outgoing null expansions. From now on we refer to this surface as the [*apparent horizon*]{}.
From (\[eq010\]) and (\[eq011\]) we find $$%\varepsilon_\pm =\frac{2 q_\pm}{\alpha^2}\left(\frac{\dot{\alpha}}{\beta}\pm \frac{1}{v} \right)
\varepsilon_\pm =\frac{2}{\alpha^2}\left(\frac{\dot{\alpha}}{\beta}\pm \frac{1}{\tanh y} \right)
\label{eq110}$$ for future directed and $$%\varepsilon_\pm =\frac{2 \tilde{q}_\pm}{\alpha^2}\left(-\frac{\dot{\alpha}}{\beta}\pm \frac{1}{v} \right)
\varepsilon_\pm =\frac{2}{\alpha^2}\left(-\frac{\dot{\alpha}}{\beta}\pm \frac{1}{\tanh y} \right)
\label{eq111}$$ for past directed radial null geodesics, where the overdot denotes a partial derivative with respect to $\tau$. The respective Lie derivatives are given by $$\frac{d\varepsilon_\pm}{d\lambda_\mp} \equiv
l^\mu_\mp\partial_\mu \varepsilon_\pm
=
%\frac{2q_+ q_-}{\alpha^2\beta^2}\left[\frac{\ddot{\alpha}}{\alpha}
\frac{2}{\alpha^2\beta^2}\left[\frac{\ddot{\alpha}}{\alpha}
-\frac{\dot{\alpha}\dot{\beta}}{\alpha\beta}
-\left( 1-\frac{1}{\tanh^2\! y} \right)\frac{\beta^2}{\alpha^2}\right]
%-\frac{2q_\mp\dot{\alpha}}{\alpha^2\beta}\varepsilon_\pm,
-\frac{2\dot{\alpha}}{\alpha^2\beta}\varepsilon_\pm,
\label{eq112}$$ for future directed and $$\frac{d\varepsilon_\pm}{d\lambda_\mp} \equiv
l^\mu_\mp\partial_\mu \varepsilon_\pm =
% \frac{2\tilde{q}_+\tilde{q}_-}{\alpha^2\beta^2}\left[\frac{\ddot{\alpha}}{\alpha}
\frac{2}{\alpha^2\beta^2}\left[\frac{\ddot{\alpha}}{\alpha}
-\frac{\dot{\alpha}\dot{\beta}}{\alpha\beta}
+\left( 1-\frac{1}{\tanh^2\! y} \right)\frac{\beta^2}{\alpha^2}\right]
%+\frac{2\tilde{q}_\mp\dot{\alpha}}{\alpha^2\beta}\varepsilon_\pm,
+\frac{2\dot{\alpha}}{\alpha^2\beta}\varepsilon_\pm,
\label{eq113}$$ for past directed radial null geodesics.
For a spherically symmetric spacetime, the condition that one of the null expansions vanishes on the apparent horizon $H$ is equivalent to the condition that the vector $n_\mu$, normal to the surface of spherical symmetry, is null on $H$. In other words, the condition $$\nabla_\mu l^\mu|_H=0,
\label{eq115}$$ where $l^\mu$ denotes either $l^\mu_+$ or $l^\mu_+$, is equivalent to the condition $$G^{\mu\nu}n_\mu n_\nu|_H=0.
\label{eq116}$$ This may be seen as follows. For the metric (\[eq243\]) the normal $n_\mu$ is given by $$n_\mu=\partial_\mu (\alpha \sinh y).
\label{eq114}$$ The expansion $\varepsilon_+$ (or $\varepsilon_-$) defined in (\[eq244\]) may be written as $$\nabla_\mu l^\mu=\frac{1}{\sqrt{-G}}\partial_\mu (\sqrt{-G} l^\mu)
=\frac{1}{\sqrt{-h}}\partial_\mu (\sqrt{-h} l^\mu) + \frac{2}{\alpha\sinh y}l^\mu n_\mu
\label{eq120}$$ where $h$ denotes the determinant of the metric $$h_{\alpha\beta}
=
\left(\begin{array}{cc}
\beta^2 & 0 \\
0 & -\alpha^2 \\
\end{array} \right),
\label{eq122}$$ of the two-dimensional space normal to the surface of spherical symmetry. It may be shown that the first term on the right hand side of (\[eq120\]) vanishes identically by the geodesic equation. Hence, the expansion $\nabla_\mu l^\mu$ vanishes on $H$ if and only if $$l^\beta n_\beta|_H=0.
\label{eq123}$$ Suppose one of the expansions vanishes on $H$, i.e., Eq. (\[eq123\]) holds for either $l^\mu_+$ or $l^\mu_-$. Since $l^\mu$ is null and both $l^\mu$ and $n^\mu$ are normal to $H$ and hence tangent to the two-dimensional space $(\tau, y)$ with the metric (\[eq122\]), Eq. (\[eq123\]) implies $h_{\alpha\beta}n^\alpha n^\beta|_H=0$. Hence, $\nabla_\mu l^\mu|_H=0$ implies $G^{\mu\nu}n_\mu n_\nu|_H=0$.
To prove the reverse it is sufficient to show that $l^\beta_+ n_\beta\neq 0$ and $l^\beta_- n_\beta\neq 0$ implies $h_{\alpha\beta}n^\alpha n^\beta \neq 0$, which may be easily shown for a general two-dimensional metric in diagonal gauge. Then, the following statement holds: the vanishing of $h_{\alpha\beta}n^\alpha n^\beta$ on $H$ implies either $l^\beta_+ n_\beta|_H=0$ or $l^\beta_- n_\beta|_H=0$. This together with (\[eq120\]), implies either $\varepsilon_+|_H=0$ or $\varepsilon_-|_H=0$.
Either from (\[eq115\]) or from (\[eq116\]) one finds the condition for the apparent horizon $$\frac{\dot{\alpha}}{\beta}\pm \frac{1}{\tanh y} =0
\label{eq117}$$
Analog horizons {#horizon}
---------------
Next we derive the analog metric and define the analog Hubble and the apparent horizons. Equation (\[eq013\]) may be written in the form [@moncrief; @bilic3; @visser2] $$\frac{1}{\sqrt{-G}}\,
\partial_{\mu}
(\sqrt{-G}\,
G^{\mu\nu})
\partial_{\nu}
\mbox{\boldmath{$\pi$}}
+\frac{c_{\pi}^2}{a}V(\sigma,
\mbox{\boldmath{$\pi$}})
\mbox{\boldmath{$\pi$}}=0 ,
\label{eq028}$$ with the analog metric tensor, its inverse, and its determinant given by $$G_{\mu\nu} =\frac{a}{c_{\pi}}
[g_{\mu\nu}-(1-c_{\pi}^2)u_{\mu}u_{\nu}] ,
\label{eq022}$$ $$G^{\mu\nu} =
\frac{c_{\pi}}{a}
\left[g^{\mu\nu}-(1-\frac{1}{c_{\pi}^2})u^{\mu}u^{\nu}
\right],
\label{eq029}$$ $$G = \frac{a^4}{c_{\pi}^2}g .
\label{eq030}$$ Hence, the pion field propagates in a (3+1)-dimensional effective geometry described by the metric $G_{\mu\nu}$.
In the comoving coordinate frame defined by the coordinate transformation (\[eq147\]) the velocity is $u^\mu=(1,0,0,0)$ and, as a consequence, the analog metric (\[eq022\]) is diagonal $$G_{\mu\nu}
=\frac{a}{c_\pi}
\left(\begin{array}{cccc}
c_\pi^2 & & & \\
& -\tau^2 & & \\
& & -\tau^2\sinh^2\! y & \\
& & & -\tau^2\sinh^2\! y \sin^2 \vartheta
\end{array} \right).
\label{eq018}$$ Here, the parameters $a$ and $c_\pi$ are functions of the temperature $T$ which in turn is a function of $\tau$. In the following we assume that these functions are positive. The metric (\[eq018\]) is precisely of the form (\[eq243\]) with $$\beta(\tau)=\sqrt{ac_\pi} ; \hspace{0.5in} \alpha(\tau)=\tau\sqrt{\frac{a}{c_\pi}} .
\label{eq108}$$ The physical range of $\tau$ is fixed by Eq. (\[eq007\]) since the available temperature ranges between $T=0$ and $T=T_{\rm c}$. Hence, the proper time range is $\tau_{\rm c}\leq \tau < \infty$ where the critical value $\tau_{\rm c}$ is related to the critical temperature as $\tau_{\rm c}/\tau_0= T_0/T_{\rm c}$. The metric is singular at $\tau=\tau_{\rm c}$.
In Fig. \[fig2\] we plot the expansions $\varepsilon_+$ and $\varepsilon_-$ of outgoing and ingoing radial null geodesics, respectively, as functions of $r$ for an arbitrarily chosen fixed time $t=6 t_0$ and, similarly, as functions of $y$ for a fixed $\tau=5.77 \tau_0$. In the lower two panels we plot the derivative of the outgoing null expansion $\varepsilon_+$ along the ingoing null geodesic. The outgoing null expansion decreases with increasing $r$ from positive to negative values and vanishes at the point $r=r_H$, whereas the ingoing null expansion remains negative. At this point the derivative of $\varepsilon_+$ with respect to $\lambda_-$ is positive. According to the standard convention described in Appendix \[trapped\] the region $\{r>r_H, t=6\tau_0\}$ is future trapped and the location $r_H$ marks its inner boundary. Thus, the sphere at $r_H$ is future inner marginally trapped.
Spacetime diagrams corresponding to the metric (\[eq018\]) are presented in Fig. \[fig3\] showing future directed radial null geodesics. The origin in the plots in both panels corresponds to the critical value $\tau_{\rm c}$ at which $c_\pi$ vanishes. Numerically, with the chosen $T_0=1$ GeV we have $\tau_{\rm c}/\tau_0=5.47$. The geodesic lines are constructed using $$y= \pm \int_{\tau_{\rm c}}^\tau d\tau' c_\pi(\tau')/\tau' +{\rm const}
\label{eq216}$$ which follows from (\[eq017\]) with (\[eq108\]). As mentioned in Sec. \[radial\], increasing (decreasing) $y$ does not necessarily imply increasing (decreasing) of the radial coordinate $r$. With the help of the coordinate transformation (\[eq147\]) the shift $dy$ along a geodesic may be expressed in terms of $dr$ $$dy=
\frac{c_\pi}{( c_\pi\pm v )\tau\cosh y } dr .
\label{eq016}$$ where we have used (\[eq246\]) and (\[eq108\]). We note that if $v<c_\pi$, increasing(decreasing) $y$ correspond to increasing (decreasing) $r$ for both signs, whereas if $v>c_\pi$ increasing $y$ corresponds to increasing $r$ for an outgoing and decreasing $r$ for an ingoing geodesic.
Using (\[eq216\]) we introduce null coordinates $$w= \frac{1}{2}\left(y+\int_{\tau_{\rm c}}^\tau d\tau' \beta(\tau')/\alpha(\tau')\right);
\hspace{1cm}
u= \frac{1}{2}\left(-y+\int_{\tau_{\rm c}}^\tau d\tau' \beta(\tau')/\alpha(\tau')\right),
\label{eq316}$$ ranging in the intervals $[0,+\infty)$ and $(-\infty,+\infty)$, respectively, with a condition $0\leq w\pm u < \infty$. In these coordinates the metric (\[eq008\]) becomes $$ds^2= \alpha^2\left( 4 du dw - \sinh^2\! (w-u) \, d\Omega^2\right).
\label{eq208}$$ The singularity at $\tau=\tau_{\rm c}$ is mapped onto the entire $u+w=0$ line.
Next, we compactify the spacetime using the coordinate transformation $$W=\tanh w;
\hspace{1cm}
U= \tanh u.
\label{eq317}$$ The coordinates $W$ and $U$ range in the intervals $[0,1]$ and $[-1,1]$, respectively, with a condition $0\leq W\pm U \leq 2$. Furthermore, the rotation $$T=\frac{1}{\sqrt{2}}(W+U);
\hspace{1cm}
R= \frac{1}{\sqrt{2}}(W-U) ,
\label{eq319}$$ brings the metric to a conformally flat form $$ds^2= \frac{2\alpha^2}{(1-U^2)(1-W^2)}\left[ dT^2 -dR^2 - R^2 \, d\Omega^2\right],
\label{eq309}$$ where both coordinates $R$ and $T$ range in the interval $[0,\sqrt{2}]$ with a condition $R+T \leq \sqrt{2}$. The conformal diagram representing our analog spacetime is depicted in Fig. \[fig3a\]. The singularity at $\tau=\tau_{\rm c}$ is mapped onto the segment $[0,\sqrt{2}]$ on the horizontal axis. The coordinate transformation $$t'=\int \beta d\tau
\label{eq138}$$ brings the metric (\[eq018\]) to the standard form of an open $k=-1$ FRW spacetime metric with the cosmological time $t'$. The time coordinate $t'$ is related to the original time $t$ via $\tau$ and the transformation (\[eq147\]). The analog cosmological scale is $a(t')=\alpha(\tau(t'))/r_0$, where the costant $r_0$ is related to the spatial Gaussian curvature $K=-1/r_0^2$. The proper distance is $d_{\rm p}=\alpha y$ and the analog Hubble constant $${\cal H}=\frac{\dot{\alpha}}{\alpha\beta},
\label{eq238}$$ where the overdot denotes a partial derivative with respect to $\tau$. Then, we define the [*analog Hubble horizon*]{} as a two-dimensional spherical surface at which the magnitude of the analog recession velocity $$v_{\rm rec}= {\cal H}d_{\rm p} = y\frac{\dot{\alpha}}{\beta}
\label{eq038}$$ equals the velocity of light. Hence, the condition $$%y= \frac{\sqrt{a c_\pi}}{\left|\partial_\tau \left(\tau\sqrt{a/c_\pi}\right)\right|}
y =\frac{\beta}{\left|\dot{\alpha}\right|}
\label{eq318}$$ defines the location of the analog Hubble horizon. Note that the radial fluid velocity $v$ in (\[eq246\]) and the analog recession velocity (\[eq038\]) are quite distinct quantities – in an expanding fluid $v$ is allways positive and less than the velocity of light $c=1$ whereas $v_{\rm rec}$ may be positive or negative depending on the sign of $\dot{\alpha}$ and its magnitude may be arbitrary large.
A two-dimensional spherical surface on which the radial velocity $v$ equals the velocity of pions $c_\pi$ defines another horizon, which we refer to as the [*naive Hubble horizon*]{}. This horizon is obviously distinct from the analog Hubble horizon defined above. The evolution of the naive and the analog Hubble horizons with $\tau$ are depicted in Fig. \[fig3\].
Next we introduce the concept of analog marginally trapped surface or [*analog apparent horizon*]{} following closely the general considerations of Sec. \[radial\] and Appendix \[trapped\]. Formation of an analog apparent horizon in an expanding hadronic fluid is similar to the formation of a black hole in a gravitational collapse although the role of an outer trapped surface is exchanged with that of an inner trapped surface. Unlike a black hole in general relativity, the formation of which is indicated by the existence of an **outer** marginally trapped surface, the formation of an analog black (or white) hole in an expanding fluid is indicated by the existence of a future or past **inner** marginally trapped surface.
Equation (\[eq117\]) with (\[eq108\]) defines a hypersurface which we refer to as the [*analog trapping horizon*]{}. Any solution to Eq. (\[eq117\]) , e.g., in terms of $r$ for fixed $t$, gives the location of the analog apparent horizon $r_H$. For example, the radius $r_H=1.53\tau_0$ computed using (\[eq117\]) for fixed $t=6t_0$ is the point of vanishing outgoing null expansion which marks the location of the apparent horizon in the top left panel of Fig. \[fig2\]. From (\[eq110\]) it follows that the region of spacetime for which $$\tanh y \geq |\beta/\dot{\alpha}|
\label{eq158}$$ is trapped. It is future trapped if $\dot{\alpha} < 0$ and past trapped if $\dot{\alpha} > 0$. The condition (\[eq158\]) can be met only if $|\beta/\dot{\alpha}| \leq 1$ which holds for $\tau$ between $\tau_{\rm c}$ and $\tau_{\rm max}$. At $\tau=\tau_{\rm max}$ we have $|\beta/\dot{\alpha}|=1$ so the endpoint of the trapping horizon in Fig. \[fig3\] is at $\tau=\tau_{\rm max}$, $\tanh y=1$.
We find that $\dot{\alpha}$ is negative for $\tau$ in the entire range $\tau_{\rm c} \leq \tau \leq \tau_{\rm max}$ and, according to (\[eq238\]), the analog Hubble constant is always negative. Hence, our analog cosmological model is a contracting FRW spacetime with a negative spatial curvature. The shaded area left of the bold line in Fig. \[fig3\] represents the time evolution of the future trapped region. Note that the analog Hubble horizon is always behind the apparent horizon whereas the naive Hubble horizon may be located ahead of or behind the apparent horizon depending on the magnitude of $\dot\alpha$. The naive Hubble and apparent horizons coincide if $a$ and $c_\pi$ are $\tau$ independent constants.
The apparent horizon is generally not a Killing horizon and normally does not coincide with the event horizon (one exception is de Sitter spacetime). Moreover, the apparent horizon exists in all FRW universes [@ellis], whereas the event horizon does not exist in eternally expanding FRW universes with the equation of state $w>-1/3$ (see, e.g., [@davis]). For the metric (\[eq018\]), the event horizon is defined by $$y= \int_\tau^\infty d\tau' \frac{c_\pi (\tau')}{\tau'}.
\label{eq207}$$ In our chiral fluid model the integral on the righthand side diverges at the upper limit because $c_\pi\rightarrow 1$ as $\tau\rightarrow \infty$ and hence, the analog event horizon does not exist. In contrast, as we have demonstrated, the analog apparent horizon does exist.
Analog Hawking effect {#surface}
---------------------
One immediate effect related to the apparent horizon is the Hawking radiation. Unfortunately, in a non-stationary spacetime, the surface gravity associated to the apparent horizon is not uniquely defined [@nielsen2]. Several ideas have been put forward how to generalize the definition of surface gravity for the case when the apparent horizon does not coincide with the event horizon [@fodor; @mukohyama-booth; @hayward; @hayward2]. In this paper we use the prescription of [@hayward2] which we have adapted to analog gravity in our previous paper [@tolic]. This prescription involves the so called Kodama vector $K^\mu$ [@kodama] which generalizes the concept of the time translation Killing vector to non-stationary spacetimes. The analog surface gravity $\kappa$ is defined by $$\kappa =\frac{1}{2} \frac{1}{\sqrt{-h}} \partial_\alpha ( \sqrt{-h}h^{\alpha\beta}k n_\beta),
\label{eq228}$$ where the quantities on the righthand side should be evaluated on the trapping horizon. The metric $h_{\alpha\beta}$ of the two-dimensional space normal to the surface of spherical symmetry and the vector $n_\alpha$ normal to that surface are given by (\[eq122\]) and (\[eq114\]), respectively.
The definition (\[eq228\]) differs from the original expression for the dynamical surface gravity [@hayward2; @hayward3] by a normalization factor $k$ which we have introduced in order to meet the requirement that $K^\mu$ should coincide with the time translation Killing vector $\xi^\mu$ for a stationary geometry. For the metric (\[eq018\]) with (\[eq108\]) we have found [@tolic] $$k= \beta \left(\cosh^2y-\sinh^2y \frac{\tau \dot{\alpha}}{\alpha}\right).
\label{eq324}$$ Then, the definition (\[eq228\]) yields $$\kappa =
\frac{v}{2 \beta\gamma(\alpha -\tau \dot{\alpha}v^2)^2}
\left[\alpha(\dot{\alpha}^2+\alpha\ddot{\alpha}-\beta^2)
+2\beta^2 \left(\tau\dot{\alpha}-\alpha\right)v
+ (\alpha \dot{\alpha}^2-2\tau \dot{\alpha}^3+\beta^2 \tau \dot{\alpha}) v^2
\right]
\label{eq229}$$ evaluated on the trapping horizon. The above expression may be somewhat simplified by making use of the horizon condition (\[eq117\]). We find $$\kappa =
\frac{c_\pi}{2\tau }
\frac{1 + 2c_\pi v (1-v)- (2+c_\pi) v^3}{\gamma v(1+c_\pi v )^2}
+\frac{\ddot{\alpha}}{2\beta}\frac{v}{\gamma (1+c_\pi v)^2}
\label{eq231}$$ evaluated on the trapping horizon.
It is worthwhile analysing the limitting case of (\[eq229\]) when the quantities $a$, and $c_\pi$ are constants. Then $\dot{\alpha}=\alpha/\tau$, $\ddot{\alpha}=0$ and the apparent horizon is fixed by the condition $v = c_\pi$. At any chosen time $t=\tau (1-c_\pi^2)^{-1/2}$ the horizon is located at $r_H=c_\pi t$ and the expression for $\kappa$ reduces to $$\kappa =\frac{1}{2t}=\frac{\sqrt{1-c_\pi^2}}{2\tau}
\label{eq230}$$ Hence, the analog surface gravity is finite for any physical value of $c_\pi$ and is maximal when $c_\pi=0$. However, with $c_\pi=0$ the horizon degenerates to a point located at the origin $r=0$.
In the left panel of Fig. \[fig4\] we plot $\kappa$ as a function of $\tau$ as given by (\[eq231\]). The corresponding temperature defined as $$T_H=\frac{\kappa}{
2\pi}
\label{eq044}$$ represents the analog Hawking temperature of thermal pions emitted at the apparent horizon as measured by an observer near infinity. Since the background geometry is flat, this temperature equals the locally measured Hawking temperature at the horizon. Thus, equation (\[eq044\]) with (\[eq229\]) corresponds to the flat spacetime Unruh effect.
As we move along the trapping horizon the radius of the apparent horizon increases and the Hawking temperature decreases rapidly with $\tau$. Hence, there is a correlation between $T_H$ and the local fluid temperature $T$ which is related to $\tau$ by (\[eq007\]). In the right panel of Fig. \[fig4\] we show the Hawking temperature $T_H$ as a function of the fluid temperature $T$ at the apparent horizon.
In our previous paper [@tolic] we have shown that the surface gravity diverges as $$\kappa = (\eta+1/2)(\tau-\tau_{\rm c})^{-1}
\label{eq048}$$ at the singular point, where $\eta$ is a constant related to the scaling of the quantity $\sqrt{a/c_\pi}$ $$\sqrt{\frac{a}{c_\pi}}\propto (T_{\rm c}-T)^{-\eta}
\label{eq045}$$ in the neighborhood of the critical point. The constant $\eta$ may be roughly estimated as follows. The estimate of the function $\Sigma (q)$ defined in (\[eq203\]) in the neighborhood of $q^\mu=0$ for small $\sigma$ yields [@son2] $$\Sigma(0, \mbox{\boldmath $q$}^2) \sim
\frac{T}{\sigma} \mbox{\boldmath $q$}^2; \hspace{1cm} \Sigma(q_0, 0) \sim \frac{T^2}{\sigma^2} q_0^2
\label{eq248}$$ By comparing this with (\[eq202\]) we deduce the behavior of the quantities $a$ and $b$ for small $\sigma$ $$a \sim
\frac{T}{\sigma} ; \hspace{1cm} a+b \sim \frac{T^2}{\sigma^2}.
\label{eq249}$$ Then, from (\[eq015\]) the pion velocity goes to zero approximately as $c_\pi \propto \sigma^{1/2}$ whereas the ratio $a/c_\pi$ diverges as $a/c_\pi \propto \sigma^{-3/2}$. From Eq. (\[eq032\]) we find $\sigma \propto (T_{\rm c}-T)^{1/2}$ near the critical point which yields $\eta=3/8$.
Numerically, by fitting $\sqrt{a/c_\pi}$ in the close neighborhood of $T_{\rm c}$ to the function (\[eq045\]) with the critical temperature $T_{\rm c}=182.822$ MeV obtained numerically from (\[eq032\]), we find $\eta=0.253$. A more refined analysis based on scaling and universality arguments of Son and Stephanov [@son1] yields $\eta=0.1975$ [@tolic].
Summary and discussion {#conclusion}
======================
We have demonstrated that, owing to the analog gravity effects in high energy collisions, a close analogy may be drawn between the evolution of a hadronic fluid and the spacetime expansion. Using the formalism of relativistic acoustic geometry we have analyzed the expanding chiral fluid in the regime of broken chiral symmetry. The expansion which takes place after the collision is modelled by spherically symmetric Bjorken type expansion. The propagation of massless pions in the chiral fluid provides a geometric analog of expanding spacetime equivalent to an open ($k=-1$) FRW cosmology. The geometry depends on the parameters $a$ and $b$ of the effective Lagrangian defined in Sec.\[chiral\]. The elements of the analog metric tensor are functions of the spacetime coordinates via the temperature dependence of $a$ and the pion velocity $c_\pi$. The pions propagate slower than light with a velocity close to zero in the neighborhood of the critical point of the chiral phase transition.
A trapped region forms for radial velocities of the fluid beyond the value defined by Eq. (\[eq117\]). This value defines a hypersurface shown in Fig. \[fig3\] which we refer to as the analog trapping horizon, at which the outgoing radial null expansion vanishes. Our trapping horizon is foliated by future inner marginally trapped surfaces and is equivalent to the trapping horizon in contracting FRW spacetime, i.e., in dynamical spacetime with a negative Hubble constant. The shaded area in Fig. \[fig3\] represents the time evolution of the future trapped region, with the future inner marginally trapped surface (or the future apparent horizons) as its inner boundary. This marginally trapped surface may be regarded as an “outer” white hole: the ingoing pions (future directed ingoing null geodesics) freely cross the apparent horizon whereas the outgoing cannot penetrate the apparent horizon. This is opposite to an expanding FRW universe where the inner marginally trapped surface acts as a black hole: the future directed ingoing null geodesics cannot escape the apparent horizon whereas the outgoing null geodesics freely cross the apparent horizon.
We have studied the Hawking effect associated with the analog apparent horizon using the Kodama-Hayward definition of surface gravity adapted to the analog gravity geometry. The Hawking temperature correlates with the local temperature of the fluid at the apparent horizon and diverges at the critical point. In contrast to the usual general relativistic Hawking effect, where the Hawking temperature is tiny compared with the temperature of the background, the analog horizon temperature is of the order or even larger than the local temperature of the fluid.
The analog Hawking radiation of pions should not be confused with the Hawking-Unruh radiation of hadrons of Castorina et al. [@castorina]. The latter is a usual Unruh effect due to the acceleration of quark-antiquark pairs produced in particle collisions, whereas the former is an analog thermal radiation due to effective geometry of the chiral fluid.
A spherically symmetric Bjorken expansion model considered here may be phenomenologically viable as a model of hadron production in $e^+e^-$ but it is certainly not the best model for description of high energy heavy ion collisions. It would be desirable to apply our formalism to a more realistic hydrodynamic model that involves a transverse expansion superimposed on a longitudinal boost invariant expansion. In this case the calculations become rather involved as the formalism for general nonspherical spacetimes is not yet fully diveloped. This work is in progress.
In conclusion, we believe that the study of analog gravity in high energy collision may in general improve our understanding of both particle physics phenomenology and dynamical general relativistic systems.
Acknowledgments {#acknowledgments .unnumbered}
---------------
This work was supported by the Ministry of Science, Education and Sport of the Republic of Croatia under Contract No. 098-0982930-2864 and partially supported by the ICTP-SEENET-MTP grant PRJ-09 “Strings and Cosmology“ in the frame of the SEENET-MTP Network.
Trapped surfaces in general relativity {#trapped}
======================================
Following Hayward [@hayward] we summarize here the relevant definitions related to trapped surfaces.
- [**Trapped surface**]{}. Let $\Sigma$ denote a spacelike hypersurface, e.g., a hypersurface defined by $t=\rm const$. A two-dimensional surface $S\subset \Sigma$ with spherical topology is called a [*trapped surface*]{} on $\Sigma$ if the families of ingoing and outgoing null geodesics normal to the surface are both converging or both diverging. More precisely, the null expansions $\varepsilon_+=l^\mu_{+;\mu}$ and $\varepsilon_-=l^\mu_{-;\mu}$ on a trapped surface $S$ should satisfy $\varepsilon_+\varepsilon_- >0$. One distinguishes between a [*past trapped surface*]{} for which both $\varepsilon_+$ and $\varepsilon_-$ are positive, and a [*future trapped surface*]{} for which both $\varepsilon_+$ and $\varepsilon_-$ are negative.
- [**Trapped region**]{}. A set of future trapped surfaces (or closed trapped surfaces [@ellis; @hawking]) on $\Sigma$ is referred to as a [*future trapped region*]{}. Similarly, a set of past trapped surfaces on $\Sigma$ is called a [*past trapped region*]{}.
- [**Marginally trapped surface**]{}. A two-dimensional surface $H$ is said to be [*marginally trapped*]{} if one of the null expansions vanishes on $H$, i.e., if either $\varepsilon_+|_H=0$ or $\varepsilon_-|_H=0$. A marginally trapped surface is also referred to as an [*apparent horizon*]{} although, strictly speaking, the original definition of Ellis and Hawking [@hawking] involves in addition the assumption of asymptotic flatness.
- [**Outer marginally trapped surface**]{}. A surface $H$ is said to be future (past) outer marginally trapped if on $H$ the future (past) directed outgoing null expansion vanishes, its derivative along the ingoing null geodesic is negative, and the ingoing null expansion is negative, i.e., if $\varepsilon_+|_H=0$, $l_-^\mu\partial_\mu\varepsilon_+|_H <0$, and $\varepsilon_-|_H<0$. Equivalently, the future (past) outer marginally trapped surface may be defined as the surface on which the past (future) directed ingoing null expansion vanishes, its derivative along the outgoing null geodesic is negative, and the outgoing null expansion is positive, i.e., if $\varepsilon_-|_H=0$, $l_+^\mu\partial_\mu\varepsilon_-|_H <0$, and $\varepsilon_+|_H>0$.
- [**Inner marginally trapped surface**]{}. A surface $H$ is said to be future (past) inner marginally trapped if on $H$ the future (past) directed outgoing null expansion vanishes, its derivative along the ingoing null geodesic is positive, and the ingoing null expansion is negative, i.e., if $\varepsilon_+|_H=0$, $l_-^\mu\partial_\mu\varepsilon_+|_H>0$, and $\varepsilon_-|_H<0$. Equivalently, the future (past) inner marginally trapped surface may be defined as the surface on which the past (future) directed ingoing null expansion vanishes, its derivative along the outgoing null geodesic is positive, and the outgoing null expansion is positive, i.e., ($\varepsilon_-|_H=0$, $l_+^\mu\partial_\mu\varepsilon_-|_H >0$, and $\varepsilon_+|_H>0$).
- [**Trapping horizon**]{}. A hypersurface foliated by inner or outer marginally trapped surfaces is referred to as an inner or outer [*trapping horizon*]{}, respectively.
According to this classification we distinguish four physically relevant classes:
1. [*Future **outer** marginally trapped surface*]{} is the outer boundary of a future trapped region and is typical of a black hole.
2. [*Past **outer** marginally trapped surface*]{} is the outer boundary of a past trapped region and is typical of a white hole.
3. [*Future **inner** marginally trapped surface*]{} is the inner boundary of a future trapped region and represents an “outer” white hole. This situation is physically relevant in the cosmological context for a shrinking universe, i.e., for an FRW spacetime with $\dot{a} < 0$.
4. [*Past **inner** marginally trapped surface*]{} is the inner boundary of a past trapped region and represents an “outer” black hole.This situation is physically relevant in the context of an expanding FRW universe [@ellis; @faraoni].
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[^1]: A few recent quotations from the press: “What we’re doing is reproducing the conditions that existed at the very early universe, a few millionths of a second after the Big Bang," [@tuts]; “The Large Hadron Collider has successfully created a “mini-Big Bang" by smashing together lead ions instead of protons.” [@bbc]; “The collisions generated mini Big Bangs and the highest temperatures and densities ever achieved in an experiment” [@evans1].
[^2]: If the chiral fermions are present pions become superluminal in certain range of temperature and baryon chemical potential [@bilic2].
|
---
author:
- Yuan Liu
- Jin Zhang
- Shuang Nan Zhang
date: '...'
title: 'Intranight optical variability of radio-quiet BL Lacertae objects'
---
[Intranight variation (or microvariation) is a common phenomenon of radio-loud BL Lac objects. However, it is not clear whether the recently found radio-quiet BL Lac objects have the same properties. The occurrence rate of intranight variation is helpful in distinguishing the mechanism of the continuum of radio-quiet BL Lac objects.]{} [We conducted a photometric monitoring of 8 radio-quiet BL Lac objects by the Xinglong 2.16m and Lijiang 2.4m telescopes. The differential light curves are calculated between each target and two comparison stars. To quantify the variation, the significance of variation is examined by a scaled $F$-test.]{} [No significant variation is found in the 11 sessions of light curves of 8 radio-quiet BL Lac objects (one galactic source is excluded). The lack of microvariation in radio-quiet BL Lac objects is consistent with the detection rate of microvariation in normal radio-quiet AGNs, but much lower than for radio-loud AGNs. This result indicates that the continua of the radio-quiet BL Lac objects are not dominated by jets that will induce frequent microvariations. ]{}
Introduction
============
Active galactic nuclei (AGNs) are characterized by their broad band continua, strong emission lines, and fast variability. However, a handful of abnormal AGNs have recently been discovered in the SDSS data, i.e., weak line quasars and radio-quiet BL Lac objects (Diamond-Stanic et al. 2009; Plotkin et al. 2010). The UV/optical emission lines are absent in their UV/optical spectroscopies, though the shape and luminosity of their continua are comparable to the normal AGNs. The fraction of these special AGNs is small ($\sim$1/1000) in the SDSS DR 7 sample; however, it could be an important stage in the evolving sequence of AGNs (Hryniewicz et al. 2010; Liu & Zhang 2011). In the early stage of an active cycle of AGNs, the radiative feedback can expel the gas from broad line regions and result in weak or even in the disappearance of broad emission lines (Liu & Zhang 2011). Other models, such as of a cold accretion disk, an extremely high accretion rate, shielding gas, or abnormal BL Lac objects, have also been proposed (Shemmer et al. 2010; Plotkin et al. 2010; Laor & Davis 2011; Wu et al. 2012). The origin of such weak-line AGNs is still not clear. Both the thermal (accretion disk) or non-thermal (jet) component may explain the weak line feature.
The optical continua of classical BL Lac objects are dominated by the synchrotron emission from the relativistic jets, therefore high polarization and fast variability are important characteristics of classical BL Lac objects. Heidt & Nilsson (2011) found the polarization degrees of the radio-quiet BL Lac candidates are low. Plotkin et al. (2010) investigated the long-term variability of radio-quiet BL Lac objects using the data from SDSS stripe 82 and find that the variation amplitude of radio-quiet BL Lac objects is smaller than that of radio-loud BL Lac objects. However, owing to the small size of their sample, this is not very conclusive.
The variation in short time scale (intranight) is another characteristic of classical BL Lac objects. During a very short period, e.g. several hours, the flux of classical BL Lac object can change by several tenths of a magnitude (Wagner & Witzel 1995; Heidt & Wagner 1996; Bai et al. 1998; de Diego et al. 1998). However, it is still unclear whether the intranight variation is frequent in the radio-quiet BL Lac objets, which will be helpful in distinguishing the origin of their continua. Gopal-Krishna et al. (2013, hereafter GJC2013) and Chand et al. (2014, hereafter CKG2014) claim that they have detected a considerable fraction of intranight variation in a sample of weak-line AGNs (duty cycle$\sim$5%), and this fraction can be higher if the signal-to-noise ratio of the light curve is further increased. However, as we discuss in this paper, some galactic sources may contaminate their sample.
In this paper, we report the result of our monitoring campaign of radio-quiet BL Lac objects. Observations and data reduction are described in Section 2, and then the significance of the intranight variation is shown in Section 3. Section 4 discusses the implication of our results and presents the conclusions.
Object (SDSS) RA DEC $R$ Redshift Date Telescope Filter Duration (h) N
--------------------- --------- -------- ------- ---------- ----------- ----------- -------- -------------- ----
J081250.80+522530.8 123.212 52.425 17.85 1.152 2011.2.12 L $R$ 4.4 27
2012.1.27 L $R$ 5.7 32
J085025.60+342750.9 132.607 34.464 18.51 1.389 2011.2.13 L $R$ 3.3 20
J090107.64+384658.8 135.282 38.783 17.87 unknown 2012.1.29 L $R$ 6.9 32
J094533.99+100950.1 146.392 10.164 17.45 1.662 2011.2.10 L $V$ 6.1 25
2012.1.28 L $R$ 6.0 34
J125219.48+264053.9 193.081 26.682 17.51 1.292 2011.4.23 X $R$ 6.3 38
2013.4.14 X $R$ 8.4 47
J132809.59+545452.8 202.040 54.915 17.59 2.096 2011.4.24 X $R$ 4.6 48
J134601.29+585820.1 206.505 58.972 17.46 1.667 2011.4.25 X $R$ 4.1 18
J142943.64+385932.2 217.432 38.992 17.26 0.930 2013.4.13 X $R$ 6.2 33
\
Observations and data reduction
===============================
The amplitude of intranight variation is normally several tenths of a magnitude, so the desired error of our observed magnitude is $\lesssim$0.05 mag with the exposure time not longer than 10 min. The corresponding magnitude threshold is $R\sim18.5$ for the 2 m class telescopes we used. Seven radio-quiet BL Lac objects were selected from Plotkin et al. (2010), and SDSS J094533.99+100950.1 was selected from Hryniewicz et al. (2010). The above selection criteria are similar to those in GJC2013 and CKG2014. Actually, five sources in our sample are shared with GJC2013 and CKG2014 who based their selection primarily on classification by Plotkin et al. (2010) as a ‘high-confidence BL Lac candidate’. We additionally included in our sample some low-confidence BL Lac candidates. The sources are classified as low-confidence only because the continuum near the emission line is hard to define, and the equivalent width of emission lines will be larger or smaller than 5 ${\AA}$ depending on the continuum assumptions, which is mainly due to the noisy spectra around some emission lines. We therefore think there should be no systematic difference between high- and low-confidence sources and will investigate the variation property of subsamples in future works.
The observations were carried out by BFOSC (BAO Faint Object Spectrograph and Camera) on the Xinglong (China) 2.16 m telescope and YFOSC (Yunnan Faint Object Spectrograph and Camera ) on the Lijiang (China) 2.4 m telescope. All observations were performed in Johnson R band, except for SDSS J094533.99+100950.1 in Johnson V band. The exposure time was 300 s or 600 s depending on the weather conditions. The detailed information about the sample and observations is shown in Table 1. In total, there are 11 sessions of light curves of these eight sources.
Object (SDSS) Date RA (J2000) DEC (J2000) $r$ (SDSS) $g-r$ (SDSS)
--------------------- ----------- ------------- ------------- ------------ --------------
J081250.80+522530.8 2011.2.12 08 12 50.80 +52 25 30.8 18.05 0.3
Star 1 08 12 51.29 +52 26 46.4 17.28 1.4
Star 2 08 12 49.52 +52 26 26.2 17.89 1.3
J081250.80+522530.8 2012.1.27 08 12 50.80 +52 25 30.8 18.05 0.3
Star 1 08 12 51.29 +52 26 46.4 17.28 1.4
Star 2 08 12 49.52 +52 26 26.2 17.89 1.3
J085025.60+342750.9 2011.2.13 08 50 25.60 +34 27 50.9 18.66 0.4
Star 1 08 50 26.96 +34 26 35.9 19.22 1.0
Star 2 08 50 17.77 +34 26 50.5 18.53 0.7
J090107.64+384658.8 2012.1.29 09 01 07.64 +38 46 58.8 18.12 0.1
Star 1 09 01 06.48 +38 47 08.7 18.14 1.4
Star 2 09 01 05.15 +38 48 24.5 17.36 1.3
J094533.99+100950.1 2011.2.10 09 45 33.99 +10 09 50.1 17.66 0.4
Star 1 09 45 27.96 +10 08 47.8 16.89 0.4
Star 2 09 45 37.93 +10 08 08.9 18.01 0.7
J094533.99+100950.1 2012.1.28 09 45 33.99 +10 09 50.1 17.66 0.4
Star 1 09 45 27.96 +10 08 47.8 16.89 0.4
Star 2 09 45 37.93 +10 08 08.9 18.01 0.7
J125219.48+264053.9 2011.4.23 12 52 19.48 +26 40 53.9 17.70 0.2
Star 1 12 52 27.12 +26 38 49.7 17.44 1.1
Star 2 12 52 14.26 +26 39 11.5 17.15 1.3
J125219.48+264053.9 2013.4.14 12 52 19.48 +26 40 53.9 17.70 0.2
Star 1 12 52 23.02 +26 38 42.9 15.82 0.6
Star 2 12 52 23.82 +26 41 42.6 16.43 0.3
J132809.59+545452.8 2011.4.24 13 28 09.59 +54 54 52.8 17.84 0.1
Star 1 13 27 58.21 +54 54 00.2 17.54 0.8
Star 2 13 28 22.83 +54 55 54.7 18.14 0.5
J134601.29+585820.1 2011.4.25 13 46 01.29 +58 58 20.1 17.74 0.2
Star 1 13 46 06.60 +58 58 08.2 18.01 1.5
Star 2 13 45 55.76 +58 57 34.8 18.46 1.3
J142943.64+385932.2 2013.4.13 14 29 43.64 +38 59 32.2 17.55 0.0
Star 1 14 29 39.99 +39 02 19.6 17.21 0.9
Star 2 14 29 30.47 +39 00 08.7 16.33 0.6
The photometric data were reduced with the standard routines in the Image Reduction and Analysis Facility (IRAF) software. The bias frames were extracted from no fewer than ten frames, and the flat frames did not have fewer than five frames in one band for one night of observation. The dark of the CCD is negligible (compared with the readout noise and the flat fluctuation) and therefore not considered. The flat frames for the same band were combined by average, and then the normalized flat frame was generated; the normalized bias frame was generated by median combination. Then the source frames were corrected by the normalized bias frame and flat frame.
With the corrected source images, we used the package APPHOT to perform aperture photometry. The values of *enclosed*, *moffat*, and *direct* for the comparison stars and the target source were used to estimate the mean full width at half maximum (FWHM). The apertures of the photometry for individual frame were carried with $2.5\sim3$ times of FWHM. If the value of FWHM significantly changed during one night, we took different values of FWHM even for the same source.
Results
=======
To detect the underlying variation of the target, we first calculated the differential light curves (DLCs) between the target and comparison stars. Two nearby comparison stars (noted as Star 1 and Star 2 hereafter) with similar magnitudes to the target were selected and the DLCs of AGN$-$Star 1, AGN$-$Star 2, and Star 1$-$Star 2 are shown in Figure 1. The position and $g-r$ color of targets and comparison stars are shown in Table 2. Since the color difference between target and star pairs is smaller than 1.5, the variation in air mass during the observation has little effect on DLCs (Carini et al. 1992; Stalin et al. 2004). We also tried different companion stars, and the final significance of intranight variation is quite robust. Some exposures with bad weather were excluded from the DLCs, which led to some gaps in the DLCs.
{width="45.00000%"} {width="45.00000%"} {width="45.00000%"} {width="45.00000%"} {width="45.00000%"} {width="45.00000%"}
{width="45.00000%"} {width="45.00000%"} {width="45.00000%"} {width="45.00000%"} {width="45.00000%"}
[[**Fig. \[fig:1\].**]{} *continued*]{}
To quantify the significance of the variation of light curves, we performed a scaled $F$-test, which is more powerful and reliable than the traditional $C$-test (de Diego 2010). The scaled $F$ value (Howell et al. 1988) is defined as $${F} = \frac{{s_{{\rm{AGN - Star~1}}}^2}}{{{\Gamma ^2}s_{{\rm{Star~1 - Star~2}}}^2}}, \eqno{(1)}$$ where $s_x^2 = \frac{1}{N-1}\sum\limits_{i = 1}^N {{{({X_i} - \bar
X)}^2}} $, and $x$ can stand for AGN-Star 1 or Star 1-Star 2.
The definition of $\Gamma ^2$ is $${\Gamma ^2} = {\left(
{\frac{{{N_{{\rm{Star~2}}}}}}{{{N_{{\rm{AGN}}}}}}} \right)^2}\left[
{\frac{{N_{{\rm{Star~1}}}^2(N_{{\rm{AGN}}}^{} + P) +
N_{{\rm{AGN}}}^2(N_{{\rm{Star~1}}}^{} +
P)}}{{N_{{\rm{Star~2}}}^2(N_{{\rm{Star~1}}}^{} + P) +
N_{{\rm{Star~1}}}^2(N_{{\rm{Star~2}}}^{} + P)}}} \right], \eqno{(2)}$$ which is the scaled factor to account for the different accuracies between the photometries of the target and comparison stars (Howell et al. 1988). The variables $N_{\rm{AGN}}$, $N_{\rm{Star~1}}$, and $N_{\rm{Star~2}}$ are the total counts (sky-subtracted) of target, Star 1, and Star 2, respectively. The variable $P$ is defined as $P=n_p(N_S+N_r^2)$, where $n_p$ is the number of pixels in the applied measuring aperture, the variable $N_S$ is the sky photons per pixel, and $N_r$ is the readout noise ($e^-$/pixel). The value of $\Gamma ^2$ can be calculated frame-by-frame. However, the variation in $\Gamma ^2$ of our observations during one night is no more than 10% owing to the small variation of our targets. Therefore, we have taken the median value of $\Gamma ^2$ for the exposures in one night. Our final result is not sensitive to this choice.
The significance of the variation is determined by the $F$ distribution with $N_{\rm{AGN-Star~1}}-1$ and $N_{\rm{Star~1-Star~2}}-1$ degrees of freedom, where $N_{\rm{AGN-Star~1}}$ and $N_{\rm{Star~1-Star~2}}$ are the number of observations in the AGN-Star 1 and Star 1-Star 2 DLCs, respectively.
The results of the significance are listed in Table 3. Since we exchanged the position of Star 1 and Star 2 in equations (1) and (2), there are two values of significance for ${\rm{(AGN - Star~1)/(Star~1 - Star~2)}}$ and ${\rm{(AGN - Star~2)/(Star~2 - Star~1)}}$.
As indicated by the results of $F$-test, we only detect a significant variation ($\sim$3$\sigma$ level) in SDSS J090107.64+384658.8. However, due to the large proper motion of this source (62$\pm$11 mas/yr from Monet et al. 2003), its extragalactic nature is doubtable. Therefore, we would like to exclude it from the final sample of radio-quiet BL Lac objects. As a result, there is no significant variation detected in our observations of radio-quiet BL Lac objects.
Object (SDSS) Date $\Gamma_{1}$ $\Gamma_{2}$ $F_{1}$ $F_{2}$ Significance$_{1}$ Significance$_{2}$
--------------------- ----------- -------------- -------------- --------- --------- -------------------- --------------------
J081250.80+522530.8 2011.2.12 1.16 1.66 0.70 1.34 18.4% 76.9%
J081250.80+522530.8 2012.1.27 1.50 1.85 0.51 0.72 3.44% 18.7%
J085025.60+342750.9 2011.2.13 1.09 0.59 1.20 1.41 64.9% 77.0%
J090107.64+384658.8 2012.1.29 1.84 1.34 1.75 2.68 93.8% 99.6%
J094533.99+100950.1 2011.2.10 0.45 1.25 1.15 0.93 63.5% 43.0%
J094533.99+100950.1 2012.1.28 0.51 1.26 0.74 1.16 20.1% 66.2%
J125219.48+264053.9 2011.4.23 1.80 1.58 0.39 0.36 0.25% 0.13%
J125219.48+264053.9 2013.4.14 6.76 7.15 0.49 0.46 0.91% 0.50%
J132809.59+545452.8 2011.4.24 0.79 1.29 1.05 0.91 56.6% 37.3%
J134601.29+585820.1 2011.4.25 0.60 0.96 0.67 1.18 20.9% 63.1%
J142943.64+385932.2 2013.4.13 2.05 1.40 1.51 2.23 87.4% 98.7%
and ${\rm{(AGN -
Star~2)/(Star~2 - Star~1)}}$, respectively.
Discussions and conclusions
===========================
Radio-loud AGNs and blazars can exhibit microvariation with a large amplitude up to $\sim$100%. However, some microvariation events are also observed in radio-quiet AGNs with high significance (Stalin et al. 2004; Gupta & Joshi 2005). The mechanism of the microvariation in radio-loud AGNs is believed to be the fluctuation caused by the shocks in jets. However, the instability or flares in the accretion disk can also induce microvariation even for the radio-quiet AGNs (Mangalam & Wiita 1993). A weak blazar component in radio-quiet AGNs is an alternative to microvariation (Czerny et al. 2008). Though the occurrence of microvariations is not a smoking gun of jets, the fraction and amplitude of the microvariations in radio-quiet and radio-loud ones are quite different.
Gupta & Joshi (2005) compiled the microvariations of different classes of AGNs and found the detection fractions of microvariation in radio-quiet and radio-loud (non-blazars) AGNs are $\sim$10% and $\sim$35-40%, respectively. For blazars, the fractions are $\sim$60-65% and $\sim$80-85% for the observations that are less than and more than 6 h, respectively. In addition, they also claim that the amplitude of the microvariation of radio-loud ones is larger than that of radio-quiet ones.
Carini et al. (2007) established a sample of 117 radio-quiet AGNs that have been investigated for microvariations and found a detection rate of microvariations for the entire sample of 21.4%. If the criteria for ‘radio-quiet’ are strengthened to $R<1$ ($R$ is the ratio of the radio \[5 GHz\] flux to optical \[4400 ${\AA}$\] flux), the detection rate of microvariations is only 15.9%.
Goyal et al. (2013) analyzed 262 sessions of light curves of 77 AGNs from their uniform AGN monitoring data and found the duty cycles of intranight variation of radio-quiet quasars, radio-intermediate quasars, lobe-dominated quasars, low optical-polarization core-dominated quasars, high optical-polarization core-dominated quasars, and TeV blazars are 10%, 18%, 5%, 17%, 43%, and 45%, respectively.
No significant microvariation is detected in our final sample, in ten sessions of light curves. The 1$\sigma$ upper limit of the fraction of microvariation is 15% using the method of Cameron (2011)[^1]. In deriving this upper limit, we treated the sources equally. The weights of sources should not be the same owing to different exposure times, signal-to-noise ratios, and observation numbers; however, this potential minor correction will not change our final conclusion. This low fraction in our sample of radio-quiet BL Lac objects is consistent with that of the radio-quiet AGNs but much lower than for the radio-loud ones and blazars. This indicates that the continuum of radio-quiet BL Lac objects is not dominated by the jet component. Actually, the SED of radio-quiet BL Lac objects is similar to the normal radio-quiet AGNs (Lane et al. 2011), which further supports the accretion disk origin of the continuum. Accurate black hole mass measurements can determine the accretion state of radio-quiet BL Lac objects and further distinguish the different models related to the accretion disk origin of their continua, which will be explored in our future works.
GJC2013 and CKG2014 detected significant variations (confidence level $>$99% for two comparison stars) of SDSS J090843.25+285229.8 and SDSS J121929.45+471522.8 in their 29 light curves. Based on these two events, they derived a duty cycle $\sim$5% of intranight variation from their sample on weak-line AGNs. However, two sources in their sample (SDSS J090107.64+384658.8 and SDSS J121929.45+471522.8) are likely to be galactic sources owing to the large proper motion. The proper motions of SDSS J090107.64+384658.8 and SDSS J121929.45+471522.8 from USNO-B are 62$\pm$11 mas/yr and 112$\pm$4 mas/yr, respectively (Monet et al. 2003). These two bright sources ($V$$\sim$18) are well above the completeness limit $V$=21 of the USNO-B catalog (Monet et al. 2003). The positional error of one epoch is $\sim$200 mas. Thus, the proper motion of $\sim$100 mas/yr can be accurately detected with the epoch difference of $\sim$40 years. The systematic error of the proper motion is comparable to the statistical error. Munn et al. (2004) and Roeser et al. (2010) have further calibrated the USNO-B catalog with SDSS and 2MASS astrometry, and the resulting proper motions of these two sources are consistent with those from the USNO-B catalog. No radio or X-ray counterpart is found near their positions. Actually, Rebassa-Mansergas et al. (2010) lists SDSS J121929.45+471522.8 as a DC white dwarf in their catalog. SDSS J090107.64+384658.8 is classified as an uncertain DC white dwarf by Eisenstein et al. (2006) and as an uncertain DC+M binary system by Kleinman et al. (2013). The variation observed is likely due to the oscillation or accretion of the white dwarfs (Winget & Kepler 2008; Fontaine & Brassard 2008).
The DC white dwarfs are generally difficult to positively identify due to their featureless spectra and the discrepancy indeed exists in different catalogs. However, we think these possible galactic sources should be excluded from the sample of AGNs to be safe. We detected the intranight variation in SDSS J090107.64+384658.8 but excluded it from the sample. After the possible white dwarfs are removed from the sample of GJC2013 and CKG2014, only one significant variation (SDSS J090843.25+285229.8) is detected in the remaining 22 light curves, which is also the only significant event in all 32 light curves (including 10 additional ones of this paper) of weak-line AGNs up to now. This low occurrence rate is consistent with the rate for the normal radio-quiet AGNs.
Our present sample of radio-quiet BL Lac objects is still too small to constrain the duty cycle of the microvariation well. We will enlarge our sample, especially for the monitoring time longer than six hours, and improve the accuracy of photometry to $\sim$ 0.01 mag to detect smaller variations.
Observations in more bands will help to investigate the color-behavior and further constrain the mechanism of the continuum of the radio-quiet BL Lac objects.
The authors thank the referee for useful comments that improved the paper. This work is supported by 973 Program of China under grant 2014CB845802, by the National Natural Science Foundation of China under grant Nos. 11103019, 11133002, and 11103022, and 11373036, and by the Strategic Priority Research Program “The Emergence of Cosmological Structures" of the Chinese Academy of Sciences, Grant No. XDB09000000. We acknowledge the support of the staff of the Xinglong 2.16m and the Lijiang 2.4m telescope. Funding for the Lijiang 2.4m telescope has been provided by CAS and the People’s Government of Yunnan Province. This work was partially supported by the Open Project Program of the Key Laboratory of Optical Astronomy, NAOC, CAS.
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[^1]: Given the sample size $n$ and observed success counts $k$, the upper limit $p_u$ is defined by $\int_{{p_u}}^1 {\frac{{(a + b - 1)!}}{{(a - 1)!(b - 1)!}}{p^{a - 1}}{q^{b - 1}}dp} = (1 - c)/2$, where $a=k+1$, $b=n-k+1$, $q=1-p$, and $c$ is the confidence level.
|
---
abstract: 'Deep Convolutional Neural Networks (CNNs) have demonstrated excellent performance in image classification, but still show room for improvement in object-detection tasks with many categories, in particular for cluttered scenes and occlusion. Modern detection algorithms like Regions with CNNs [@girshick14] rely on Selective Search [@uijlings13] to propose regions which with high probability represent *objects*, where in turn CNNs are deployed for classification. Selective Search represents a family of sophisticated algorithms that are engineered with multiple segmentation, appearance and saliency cues, typically coming with a significant run-time overhead. Furthermore, [@hosang14] have shown that most methods suffer from low reproducibility due to unstable superpixels, even for slight image perturbations. Although CNNs are subsequently used for classification in top-performing object-detection pipelines, current proposal methods are agnostic to how these models parse objects and their rich learned representations. As a result they may propose regions which may not resemble high-level objects or totally miss some of them. To overcome these drawbacks we propose a boosting approach which directly takes advantage of hierarchical CNN features for detecting regions of interest fast. We demonstrate its performance on ImageNet $2013$ detection benchmark and compare it with state-of-the-art methods.'
bibliography:
- 'boost\_conv\_feats.bib'
nocite: '[@*]'
---
Introduction {#intro}
============
Visual object detection is at the heart of many applications in science and engineering, ranging from microscopic scales in medicine to macroscopic scale in space exploration. Historically these problems are mostly formulated as classification problems in a sliding-window framework. To this end a classifier is trained to differentiate one or more object classes from a background class by sliding a window over the whole image with a given stride and on multiple scales and aspect ratios predicting the class label for every single window. In practice, this demands the classification of more than a million windows for common images, resulting in a trade-off between run-time performance and classification accuracy for real-time applications.
Recent years have seen a paradigm shift in object-detection systems replacing the sliding window step with object-proposal algorithms prior to classification [@girshick14; @szegedy14scalable; @cinbis13; @wang13; @he14]. These algorithms propose regions in an image which are predicted to contain the objects of interest with high likelihood. Such an approach not only reduces the number of regions which have to be classified from a million to a few thousands but also allows for spanning across a larger range of scales and aspect ratios for regions of interest. Depending on the execution time of the proposal method and the classifier, the reduction of regions for classification can lead to a significantly faster run time and allow the use of more sophisticated classifiers.
Frameworks with region proposal methods and subsequent classification which are based on Convolutional Neural Networks [@girshick14] achieve state-of-the-art performance on the ImageNet [@deng09] detection. A popular pipeline consists of three main steps: (i) several regions that are likely to be objects are generated by a proposal algorithm; (ii) deep CNNs with multi-way Softmax or Support Vector Machines (SVMs) on the top classify these regions in order to detect all possible classes and instances per image; (iii) finally, an optional regression step further refines the location of the detected objects.
A large body of work of region proposal algorithms exists [@hosang14]. The majority of them are engineered to merge different segmentation, saliency and appearance cues, and some hierarchical scheme, with parameters tuned to specific datasets. Here we propose a data-driven region-proposal method which is based on features extracted directly from lower convolutional layers of a CNN. We use a fast binary boosting framework [@appel13] to predict the objectness of regions, and finally deploy a regressor which uses features from the upper convolutional layer after pooling to refine the localization. The proposed framework achieves the top recall rate on ImageNet $2013$ detection for Intersection over Union (IoU) localization between $50-65\%$. Finally, we apply our proposal algorithm instead of Selective Search in the baseline Regions-with-CNN pipeline [@girshick14] resulting in $8\%$ improvement over the state-of-the-art with a single prediction model while achieving considerably faster test time.
In the two following subsections we briefly review region proposal algorithms and present the motivation behind our work. In Section \[algo\] we present the details of our algorithm and in Section \[experiments\] we show the experimental study and results. In Section \[imagenet\], we place our algorithm in line with the Regions-with-CNN framework and benchmark its performance on the ImageNet $2013$ detection challenge. Finally, in Section \[discussion\] we state our conclusions and point out directions for future research.
Prior work
----------
Most currently leading frameworks use various segmentation methods and engineered cues in hierarchical algorithms to merge smaller areas (e.g., superpixels) to larger ones. During this hierarchical process boxes which likely are objects are proposed. In turn CNNs are applied as region descriptors, whose receptive fields are rectangular. This relaxes the need to perform accurate segmentation. Instead a rectangle around regions that are most probably objects is sufficient.
We briefly review some representative methods which are evaluated in detail in [@hosang14].
*Selective Search* [@uijlings13], which is currently the most popular algorithm, involves no learning. Its features and score functions are carefully engineered on Pascal VOC and ILSVRC so that low-level superpixels [@felzenszwalb04] are gradually merged to represent high-level objects in a greedy fashion. It achieves very high localization accuracy due to the initial over-segmentation at a time overhead. *RandomizedPrim’s* [@manen13] is similar to Selective Search in terms of features and the process of merging superpixels. However, the weights of the merging function are learned and the whole merging process is randomized.
Then there is a family of algorithms which invest significant time in a good high-level segmentation. *Constrained Parametric Min-Cuts (CPMC)* [@carreira12] generates a set of overlapping segments. Each proposal segment is the solution of a binary segmentation problem. Up to $10,000$ segments are generated per image, which are subsequently ranked by objectness using a trained regressor. [@rantalankila14], similar in principle to [@uijlings13] and [@carreira12], merges a large pool of features in a hierarchical way starting from superpixels. It generates several segments via seeds like CPMC does.
[@endres10; @endres14] combine a large set of cues and deploy a hierarchical segmentation scheme. Additionally, they learn a regressor to estimate boundaries between surfaces with different orientations. They use graph cuts with different seeds and parameters to generate diverse segments similar to CPMC. *Multiscale Combinatorial Grouping (MCG)* [@arbelaez14] combines efficient normalized cuts and CPMC [@carreira12] and achieves competitive results within a reasonable time budget.
In the literature several methods which engineer objectness have been proposed in the past. *Objectness* [@alexe12] was one of the first to be published, although its performance is inferior to most modern algorithms [@hosang14]. Objectness estimates a score based on a combination of multiple cues such as saliency, colour contrast, edge density, location and size statistics, and the overlap of proposed regions with superpixels. [@rahtu11] improves Objectness by proposing new cues and combine them more effectively.
Moreover, fast algorithms with approximate detection have been recently introduced. *Binarized Normed Gradients for Objectness (BING)* [@cheng14] is a simple linear classifier over edge features and is used in a sliding window manner. In stark contrast to most other methods, BING takes on average only 0.2s per image on CPU. *EdgeBoxes* [@zitnick14] is similar in spirit to BING. A scoring function is evaluated in a sliding window manner, with object boundary estimates and features which are obtained via structured decision forests.
Given that engineering an algorithm for specific data is not always a desired strategy, there are recent region-proposal algorithms which are data-driven. Data-driven Objectness [@kang14] is a practical method, where the likelihood of an image corresponding to a scene object is estimated through comparisons with large collections of example object patches. This method can prove very effective when the notion of object is not well defined through topology and appearance, such as daily activities.
A contemporary work which follows the data-driven path is Scalable, High-quality object detection [@szegedy14scalable]. After they revisited their Multibox algorithm [@erhan14], they are able to integrate region proposals and classification in one step end-to-end. By deploying an ensemble of models with robust loss function and their newly introduced *contextual features*, they achieve state-of-the-art performance on the detection task.
Motivation
----------
Recent top-performing approaches on ILSVRC detection are based on hand-crafted methods for region proposals, such as Selective Search [@uijlings13]. However, although these methods are tuned on this benchmark, they miss several objects. For example, Selective Search proposes on average $2,403$ regions per image in [@girshick14] with $91.6\%$ recall of ground truth objects (for $0.5$ IoU threshold). In that case, even with *oracle* classification and subsequent localization, more than $8\%$ of object instances will not be detected. This leaves significant room for improvement in future algorithms.
Furthermore, algorithms which build on superpixels can be unstable even for slight image perturbations resulting in low generalizations [@hosang14] on different data.
Object proposal algorithms which are based on low-level cues are agnostic on how a learned network perceives the class of objects in the space of natural images. A CNN which is trained in a supervised manner to recognize $1000$ object categories on ILSVRC, has learned a rich set of representations to identify local parts and hierarchically merge them toward certain class instances at the top layers. As opposed to Segmentation-based methods which merge segments based on simple binary criteria like existence of boundaries and color or not, CNN features ideally span the manifold of natural images which is a very small subspace inside a high dimensional feature space.
In practice, Selective Search is not scale-invariant. Nevertheless, it is engineered to work well on ILSVRC and Pascal VOC data with careful parameter tuning. To this end, Regions with CNN [@girshick14] resizes all images to a width of $500$ pixels to serve its purpose. However, a data-driven algorithm which is not constrained to build on superpixels bypasses this step.
Regions with CNN uses a linear regressor after classifying the proposed regions to better localize the bounding boxes around the object. For this purpose they deploy *pool-5* CNN features. This techinque can be applied to proposals in the first place. Of course, class-specific regressors are applicable only after classification, but neverthess generic object regressors can also enhance region localization.
All in all, the motivation of this work is to address the conceptual discontinuity between the object proposal method and the subsequent classification. Using hand-crafted scores in the proposal stage and applying a convolutional neural network for classification results not only typically in slow run-time but also in the aforementioned instabilities. The key idea is to utilize the convolutional responses of a network whose weights are learned to recognize different object classes also for proposals. Finally, we formulate the problem with a boosting framework to guarantee fast execution time.
{width="0.9\linewidth"}
Algorithm {#algo}
=========
[**Detector:**]{} We are deploying a binary boosting framework to learn a classifier with desired output $y_i \in \{ -1, 1 \}$ for an image patch $i, i \in \{ 1, \ldots, N \}$, where $1$ stands for *object* and $-1$ for *background*. The input samples $x_i$ are feature vectors which describe an image patch $i$. The features $x_i$ are a selected subset of convolutional responses $conv_j^{k_j}$ from a Proposal CNN (cf. Fig. \[figure1\]), where $j$ pertains to convolutional layer $j \in \{ 1, \ldots, L \}$ and $k_j$ spans the number of feature maps for this layer (e.g., *alexNet* [@krizhevsky12] uses $L=5$ and $k_1 \in \{ 1, \ldots, 96 \}$). Our Proposal CNN is the *VGGs* model from [@chatfield14], whose first-layer convolutional responses are $110 \times 110$ pixels, and therefore provides double resolution compared to alexNet.
Aggregate-channel features from [@dollar09] are used, where deep convolutional responses serve the role of channels, while we deploy a modified version of the fast setting provided by [@appel13]. Thus, efficient AdaBoost is used to train and combine $2,048$ depth-two trees over the $d \times d \times F$ candidate features (channel pixel lookups), where $d$ is the baseline classifier’s size and $F$ is the number of convolutional responses, i.e., the patch descriptors (e.g., VGGs architecture has $F=96$ kernels in the first layer). The convolutional responses from all positive and negative patches which are extracted from the training set are rescaled to a fixed $d \times d$ size (e.g, $d=25$) before they serve as input to the classifier. In practice, classifiers with various $d$ can be trained to capture different resolutions of these representations. On testing all classifiers are applied to the raw image and their detections are aggregated and non-maximally suppressed jointly.
[**Hierarchical features:**]{} In order to evaluate *objectness* in different patches, we train the classifier with several positive and negative samples, which are extracted from Pascal 2007 VOC [@everingham10]. Positive samples are the ones that correspond to the ground truth annotated objects, while negatives are defined as rectangular samples randomly extracted from the training set at different scales and aspect ratios, which have less than $0.3$ Intersection over Union overlap with the positives. For our experiments we considered patches sampled from the validation sets of VOC 2007 and ImageNet 2013 detection datasets, since both of them are exhaustively annotated[^1]. Naturally there is a margin for improvement with more sophisticated sampling, given that the VOC and ILSVRC categories do not include all possible object classes that can appear in an image.
In order to properly crop the objects from the convolutional responses along the hierarchy, the image level annotations have to be mapped to the corresponding regions from the intermediate representations effectively. Therefore, the supports of pooling and convolutional kernels determine a band within the rectangular box which has to be cropped out, so that information outside the object’s area can be safely ignored. In practice after two pooling stages the area that should be cropped without including too much background information becomes very narrow. In our experiments we consider filter responses from the first two layers. We have found that using only kernels from the first convolutional layer, before any spatial pooling is applied, gives the best performance.
Most first-layer kernels resemble anisotropic Gaussians and color blobs (cf. Fig. \[figure1\]). As a matter of fact our method relates to BING and EdgeBoxes, which use edge features, and methods that use scores which account for color similarity (e.g., Selective Search). By inspecting the convolutional responses of the first layer, we observe that some kernels are able to capture the texture of certain objects and thus *disentangle* them from the background and the other objects (such as the first and third kernels at Fig. \[figure2\]). This family of features provides quick detection, as the time-consuming high-level segmentation is avoided. However, it has the drawback that local information from lower layers cannot naturally compose non-rigid high-level objects. Nevertheless, a subsequent regression step which leverages features from the upper convolutional layers can help with these cases. This is described at a following paragraph.
![An image from Pascal VOC [@everingham10] and its convolutional responses with representative first-layer filters. In order to classify object candidates a binary boosting framework is trained with positive (green) and negative (red) samples which are extracted from CNN’s lower layers.[]{data-label="figure2"}](im19_pos_neg-eps-converted-to.pdf){width="0.8\linewidth"}
![An image from Pascal VOC [@everingham10] and its convolutional responses with representative first-layer filters. In order to classify object candidates a binary boosting framework is trained with positive (green) and negative (red) samples which are extracted from CNN’s lower layers.[]{data-label="figure2"}](filter10-eps-converted-to.pdf){width="0.8cm"}
![An image from Pascal VOC [@everingham10] and its convolutional responses with representative first-layer filters. In order to classify object candidates a binary boosting framework is trained with positive (green) and negative (red) samples which are extracted from CNN’s lower layers.[]{data-label="figure2"}](filter60-eps-converted-to.pdf){width="0.8cm"}
![An image from Pascal VOC [@everingham10] and its convolutional responses with representative first-layer filters. In order to classify object candidates a binary boosting framework is trained with positive (green) and negative (red) samples which are extracted from CNN’s lower layers.[]{data-label="figure2"}](filter90-eps-converted-to.pdf){width="0.8cm"}
![An image from Pascal VOC [@everingham10] and its convolutional responses with representative first-layer filters. In order to classify object candidates a binary boosting framework is trained with positive (green) and negative (red) samples which are extracted from CNN’s lower layers.[]{data-label="figure2"}](filter20-eps-converted-to.pdf){width="0.8cm"}
![An image from Pascal VOC [@everingham10] and its convolutional responses with representative first-layer filters. In order to classify object candidates a binary boosting framework is trained with positive (green) and negative (red) samples which are extracted from CNN’s lower layers.[]{data-label="figure2"}](im19_map10_pos_neg-eps-converted-to.pdf){width="2.6cm"}
![An image from Pascal VOC [@everingham10] and its convolutional responses with representative first-layer filters. In order to classify object candidates a binary boosting framework is trained with positive (green) and negative (red) samples which are extracted from CNN’s lower layers.[]{data-label="figure2"}](im19_map60_pos_neg-eps-converted-to.pdf){width="2.6cm"}
![An image from Pascal VOC [@everingham10] and its convolutional responses with representative first-layer filters. In order to classify object candidates a binary boosting framework is trained with positive (green) and negative (red) samples which are extracted from CNN’s lower layers.[]{data-label="figure2"}](im19_map90_pos_neg-eps-converted-to.pdf){width="2.6cm"}
![An image from Pascal VOC [@everingham10] and its convolutional responses with representative first-layer filters. In order to classify object candidates a binary boosting framework is trained with positive (green) and negative (red) samples which are extracted from CNN’s lower layers.[]{data-label="figure2"}](im19_map20_pos_neg-eps-converted-to.pdf){width="2.6cm"}
[**Testing:**]{} In order to detect objects in previously unseen images we apply the learned classifier in a sliding window manner densely in $S$ different scales and $R$ aspect ratios. We typically sample $S=12$ scales and $R=3$ aspect ratios. Non-maximum suppression (NMS) is used to reject detections with more than $U$ Intersection over Union overlap for every (scale, aspect ratio) combination. Finally, after detections from all scales and aspect ratios are aggregated, another joint NMS with $V$ IoU is applied. We experimented with different parameters and we use $U=63\%$ and $V=90\%$ in our reported results in Fig. \[figure3\].
[**Bounding-Box Regression:**]{} After extracting the proposals per image, a subsequent regression step can be deployed to refine their localization. As proposed by [@girshick14], a linear regressor is used with regularization constant $\lambda = 1,000$. For training we use all ground truth annotations $G^i$ and our best detection $P^i$ per ground truth for all training images $i, i \in \{ 1, \ldots, N \}$ from Pascal VOC 2007. The best detection is defined as the one with the highest overlap with the ground truth. We throw away pairs with less than $70\%$ IoU overlap. The goal of the regressor is to learn how to shift the locations of $P$ towards $G$ given the description of detected bounding box $\phi$. The transformations are modeled as linear functions of $pool_5$ features, which are obtained by forward propagating the $P$ regions through the Proposal CNN.
Experiments
===========
In order to test the efficacy and performance of our algorithm we performed experiments on data from the ImageNet $2013$ detection challenge[^2] [@deng09]. We follow the approach proposed in the review paper of [@hosang14] and we report the obtained performance vs. localization accuracy (Fig. \[figure3\]) and number of candidates per image (Fig. \[figure4\]). Specifically, we calculate the recall of ground truth objects for various localization thresholds using the Intersection-over-Union (IoU) criterion, which is the standard metric on Pascal VOC. In Fig. \[figure3\] we demonsrate our performance compared to state-of-the-art methods and three baselines, as they have been evaluated by [@hosang14]. Each algorithm is allowed to propose up to $10,000$ regions per image on average. The methods are sorted based on the Area-Under-the-Curve (AUC) metric, while in parentheses is the average number of proposed regions per image.
{width="20cm"}
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Recall ($\%$) for various IoU thresholds $IoU \geq 0.5$ $IoU \geq 0.65$ $IoU \geq 0.8$ Testing time (s)
------------------------------------------ ---------------- ----------------- ------------------------------------------------------------------------------ ------------------
Selective Search [@uijlings13] 94.6 89.0 **72.2 & 10\
Randomized Prim [@manen13] & 92.3 & 82.0 & 61.2 & 1\
MCG [@arbelaez14] & 91.8 & 81.0 & 60.6 & 30\
Edge Boxes [@zitnick14] & 93.1 & 86.6 & 49.7 & 0.3\
Boosting Convolutional Features & **98.1 & **89.4 & 38.7 & 2\
Endres $2010$ [@endres10] & 81.1 & 67.7 & 46.4 & 100\
BING [@cheng14] & 95.5 & 43.0 & 7.2 & 0.2\
Boosting Conv Features and Selective Search & **97.7 & **91.9 & **75.3 & 12\
Gaussian & 85.3 & 72.5 & 51.3 & 0\
Sliding window & 90.8 & 56.9 & 14.1 & 0\
Superpixels & 51.3 & 26.7 & 10.0 & 1\
************
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[**Recall vs. localization:**]{} Our method belongs to the family of algorithms with fast and approximate object detection, such as BING and EdgeBoxes. These algorithms provide higher recall rate but poorer localization as opposed to methods that use high-level segmentation cues like Selective Search. The latter ones are considerably slower but more accurate in localizing the objects due to boundary information. In Table \[table1\] we provide the recall rate for varying localization accuracy via IoU criterion. Our method provides the highest recall until around $65\%$ IoU overlap. We also provide in Fig. \[figure3\] and Table \[table1\] the gain in performance when we jointly use Selective Search and our method while still constraining the number of proposals to be less than $10,000$. In that case the benefit is mutual, as Selective Search provides better localization, while our algorithm higher recall, i.e, higher retrieval rate of ground truth objects for localization accuracy less than $65\%$ IoU. A small subset of images have been blacklisted in the evaluation process per ILSVRC policy.
[**Time analysis:**]{} The complexity on testing is linear to each of the number of deployed classifiers, scales, aspect ratios, and image size, when the other parameters are held constant. In Table \[table1\] an estimation of average time on testing is shown at the last column for our framework and others, as the latter ones have been evaluated by [@hosang14]. Our algorithm is quite efficient in both training and testing. More specifically, extracting convolutional responses for the validation image set of ImageNet $2013$ detection benchmark takes only a few minutes with Caffe [@jia14] on a modern machine (e.g., testing one image with a single $K40$ GPU takes $2ms$) mainly because of time needed to save the features. However, this can be done offline for popular datasets. Training a modified version of boosting framework [@appel13] which is now part of Dollar’s Matlab toolbox [@dollar13] on Pascal VOC 2007 data (train-val and test, i.e., 9,963 images with $24,640$ annotated objects) with a high-end multi-core CPU takes about three hours. This consists of training on all positives and $20k$ negatives, and additionally three rounds of bootstrapping, when at each round $20k$ more negatives are extracted among classifier’s false positives. The training time increases for larger values of baseline detector’s size, such as $d=40$. But this still does not affect the testing time.
When testing the learned classifier is applied densely in sliding window fashion on $20,121$ validation images from ImageNet detection. All possible windows in $S=12$ different scales and $R=3$ different aspect ratios are evaluated by the classifier, which outputs for each window its confidence to be an object. Greedy non-maximal suppression is performed, where bounding boxes are processed in order of decreasing confidence, and once a box is suppressed it can no longer suppress other boxes. Separate and joint NMS are deployed with $U=63 \%$ and $V=90 \%$ IoU thresholds, correspondingly. Testing on a multi-core CPU takes about $2s$ per image.
In Fig. \[figure4\] we show performance comparisons in terms of at least $50\%$ IoU recall for different number of proposed regions. Our scheme is the most effective when at least $1,000$ regions are proposed. For a smaller number of proposals the performance degrades fast. This is mainly because of significant non-maximal suppression that is applied to reduce the number of proposals, while starting from a large number of scales and aspect ratios. In practice a more sophisticated design for less proposals can improve the recall rate further especially in $[100-1,000]$ region. For small number of candidates an alternative strategy could be to deploy regression to cluster neighboring regions in unique detection, instead of using non-maximum suppression.
![Proposals quality on ImageNet 2013 validation set in terms of detected objects with at least 50% IoU for various average number of candidates per image. Compared to all other methods from [@hosang14], our method is the most effective in terms of ground truth object retrieval when at least $1,000$ regions are proposed and accurate localization is not a major concern.[]{data-label="figure4"}](recall_vs_candidates-eps-converted-to.pdf){width="9cm"}
ImageNet detection challenge {#imagenet}
============================
In order to investigate in practice how the recall-localization trade-off affects the effectiveness of the object candidates on a detection task, we put the challenge to the test and evaluate the overall performance on ImageNet 2013 benchmark. We introduce our algorithm into Regions-with-CNN detection framework [@girshick14] by replacing Selective Search [@uijlings13] and proposing our regions instead at the first step.
In Table \[table2\] we show the mean and median average precision on a subset of validation set. We use the $\{ val1, val2 \}$ split as was performed by [@girshick14]. We use their pretrained CNN and SVM models as category CNN, which are trained on $ \{val_1, train_{1k} \}$, i.e., 9,887 validation images and 1,000 ground truth positives per class from c lassification set. The deployed Proposal CNN is the *VGGs* model from [@chatfield14], which is pretrained on ILSVRC2012 classification dataset. Given that Selective Search is not scale-invariant, all images are rescaled to have $900$ pixels width while preserving the aspect ratio. Thus, Selective Search proposes on average $5,826$ regions per image. In [@girshick14] all images are rescaled to have $500$ pixels width, which yields $29.7$ and $29.2$ mean and median AP for $2,403$ regions on average. For our method we used the model that we demonstrate in Figs. \[figure3\] and \[figure4\], which generates $9,927$ proposals on average.
Average Precision (AP) Mean AP Median AP
------------------------ --------- ----------- --
Selective Search 31.5 30.2
Boosting Deep Feats 34.0 32.5
: Mean and median average precision on ImageNet 2013 detection task. We deploy the state-of-the-art Regions with Convolutional Neural Networks (R-CNN). Comparison when regions are proposed by Selective Search and our method, correspondingly. There is no post-processing regression step.[]{data-label="table2"}
Our improvement is a product of two factors; first, higher recall of ground truth objects within roughly $50-70\%$ IoU threshold and, second, a larger number of proposals. Coarse localization is corrected to some extent from subsequent steps of R-CNN due to the robustness of convolutional neural network in terms of object location and partial occlusion. Further improvement is expected if class-specific regression is introduced at the top of prediction, so that the detected bounding boxes are better located around the objects, which could prove to be especially helpful in our algorithm given that we leverage no boundary/segmentation cues. Finally, an ensemble of models along with more sophisticated architectures (e.g., GoogLeNet [@szegedy14deep], MSRA PReLU-nets [@he15], very-deep nets [@simonyan14], etc.) could further improve our results.
Discussion
==========
Features learned in convolutional neural networks have been proven to be very discriminative in recognizing objects in natural images by mapping them on small manifolds of a very higher-dimensional feature space. The boosting classifier learns to map image data points to the union of all these low-dimensional manifolds. We hypothesize that it is still a relatively small subspace, which preserves the notion of object and includes most instances that can be found in popular visual datasets such as Pascal VOC and ILSVRC.
In this paper we propose a framework which is able to benefit from the hierarchical and data-adaptive features extracted from a convolutional neural network, as well as from the quick training and test time of state-of-the-art boosting algorithms. There are many directions to explore this idea further: in this paper the feature responses extracted from different layers have equal weight during training, but exploiting the hierarchy in a top-down fashion might result in faster and more accurate predictions. Additionally, a data-driven regression mechanism which captures gradient information could improve the localization of regions proposed by our framework.
Our framework can also be applied to other areas, such as medical imaging, text detection and planetary science. Hand-engineering proposal detectors is quite challenging, as we need to come up with new similarity metrics, saliency and segmentation cues. However, instead of designing score functions and features from scratch for each new domain, learned deep convolutional features can be used, after a network has been trained on a sufficiently large and representative sample of related data. Finally, non-linear tree-based classifiers like boosting or random forests can provide a framework for fast inference, while avoiding the overhead of complete propagation in deep neural networks and at the same time being flexible to opt for a subset of actionable features for the task.
[^1]: This means that all object instances belonging to $C$ classes are fully annotated ($C=20$ for Pascal, and $C=200$ for ImageNet data), which prevents us from extracting supposedly negative samples which actually correspond to objects.
[^2]: As opposed to the training set for both the boosting framework and the regressor, which is Pascal $2007$ VOC.
|
---
abstract: 'Magnetic structures are investigated by means of neutron diffraction to shine light on the intricate details which are believed key to understanding the magnetoelectric effect in LiCoPO$_4$. At zero field, a spontaneous spin canting of $\varphi = 7(1)^{\circ}$ is found. The spins tilt away from the easy $b$-axis towards $c$. Symmetry considerations lead to the magnetic point group $m''_z$ which is consistent with the previously observed magnetoelectric tensor form and weak ferromagnetic moment along $b$. For magnetic fields applied along $a$, the induced ferromagnetic moment couples via the Dzyaloshinskii-Moriya interaction to yield an additional field-induced spin canting. An upper limit to the size of the interaction is estimated from the canting angle.'
author:
- Ellen Fogh
- Oksana Zaharko
- Jürg Schefer
- Christof Niedermayer
- 'Sonja Holm-Dahlin'
- Michael Korning Sørensen
- Andreas Bott Kristensen
- Niels Hessel Andersen
- David Vaknin
- Niels Bech Christensen
- 'Rasmus Toft-Petersen'
title: 'Dzyaloshinskii-Moriya interaction and the magnetic ground state in magnetoelectric LiCoPO$_4$'
---
Introduction
============
In a number of insulators, an external electric or magnetic field can induce a finite magnetization or electric polarization respectively. This so-called magnetoelectric (ME) effect was first theoretically predicted [@landau_lifshitz; @dzyaloshinskii1959] and shortly thereafter experimentally observed in Cr$_2$O$_3$ [@astrov1960; @astrov1961]. Since then, a collection of materials displaying the ME effect has been identified but the underlying microscopic mechanisms are not yet fully understood.
The Dzyaloshinskii-Moriya (DM) interaction has proved a key ingredient in explaning the induced or spontaneous electric polarization in a number of compounds such as $R$MnO$_3$ ($R$ = Gd, Tb, Dy) [@sergienko2006], Ni$_3$V$_2$O$_8$[@kenzelmann2006] and CuFeO$_2$[@kimura2006]. In these systems, the non-collinear incommensurate order of the magnetic moments results in a displacement of the oxygen ions situated in between neighboring moments and a net displacement of charge is generated[@kimura2007]. Non-collinear order may appear as a consequence of competing interactions, so-called spin frustration. Such systems are associated with large ME effects[@kimura2007; @spaldin2008].
The lithium orthophosphate family (space group *Pnma*), Li$M$PO$_4$ ($M$ = Co, Ni, Mn, Fe), is in many ways an excellent model system for studying the ME effect. All family members exhibit commensurate near-collinear antiferromagnetic order as well as the ME effect in their low-temperature and low-field ground state. In recent studies, additional ME phases were found at elevated magnetic fields applied along the respective easy axes in LiNiPO$_4$ [@toftpetersen2017] and LiCoPO$_4$ [@khrustalyov2016]. In both materials, these high-field ME phases are also accompanied by commensurate antiferromagnetic order [@fogh2017; @toftpetersen2017].
The magnetically induced linear ME coupling is described as $P_i = \alpha_{ij} H_j$, where $P_i$ is the electric polarization, $H_j$ the external magnetic field and $\alpha_{ij}$ are the ME tensor elements with $i,j = \lbrace a,b,c \rbrace$. Allowed tensor elements are dictated by the magnetic symmetry of the system. For collinear (anti)ferromagnets one may think of tensor elements for which the magnetic field is *perpendicular* to the spin orientation, $\alpha_{\perp}$, and those for which the field is *parallel* to the spins, $\alpha_{\parallel}$. Magnitudes and temperature dependencies for $\alpha_{\perp}$ and $\alpha_{\parallel}$ have been computed from first principles for ME compounds such as Cr$_2$O$_3$ [@iniguez2008; @mostovoy2010; @malashevich2012; @mu2014], and LiFePO$_4$ [@scaramucci2012]. In these studies it is possible to separate effects due to ion displacements within the unit cell (lattice contribution) and effects due to electronic motion around ’clamped’ ions (electronic contribution). In both cases one distinguishes between spin and orbital effects.
The *ab initio* calculations show that $\alpha_{\perp}$ is generally dominated by the spin-lattice contribution and the ME coupling is relativistic in origin, e.g. via the DM interaction. The predicted temperature dependence of $\alpha_{\perp}$ follows that of the order parameter [@malashevich2012; @scaramucci2012]. This corresponds well with observations in the lithium orthophosphate family except for a slight variation in the curve for LiNiPO$_4$ \[see Fig. \[fig:ME\]\].
The behavior of $\alpha_{\parallel}$ is altogether more tricky and *ab initio* calculations indicate that orbital contributions may play an important role [@malashevich2012; @scaramucci2012]. When disregarding orbital contributions, the computed temperature dependence of $\alpha_{\parallel}$ displays a maximum below the transition temperature and then goes to zero for $T \rightarrow 0$ [@mostovoy2010]. The comparison of the measured and predicted temperature dependencies of $\alpha_{\parallel}$ for LiMnPO$_4$ is excellent \[see Fig. \[fig:ME\]\]. However, for the remaining family members $\alpha_{\parallel} \neq 0$ for $T \rightarrow 0$ and the prediction is clearly lacking. It is even worse in the case of Cr$_2$O$_3$ (not shown) where $\alpha_{\parallel}$ changes sign as a function of temperature [@rado1961]. The orbital moment is almost entirely quenched for LiMnPO$_4$ but not for LiFePO$_4$, LiCoPO$_4$ and LiNiPO$_4$. Hence, the discrepancy between the predicted and measured values of $\alpha_{\parallel}$ for $T \rightarrow 0$ for the latter three compounds may be related to the orbital moment. Moreover, the maximum magnitude of the observed ME tensor elements also appears linked to the orbital moment with $\Delta g/g = 0$ and $|\alpha_{\mathrm{max}}| = 0.8\,\mathrm{ps/m}$ for LiMnPO$_4$ and $\Delta g/g = 0.3$ and $|\alpha_{\mathrm{max}}| = 30\,\mathrm{ps/m}$ for LiCoPO$_4$. However, recent first-principle calculations on LiFePO$_4$ taking into account orbital contributions (both lattice and electronic parts) still fail to encapsulate the low-temperature behavior of $\alpha_{\parallel}$ [@scaramucci2012].
In this paper, we focus on LiCoPO$_4$ which has, by far, the strongest ME effect in the lithium orthophosphate family [@mercier; @wieglhofer]. Although intensively studied, there is as of yet no satisfactory theory for the underlying microscopic mechanism. LiCoPO$_4$ has lattice parameters $a = 10.20\,\mathrm{Å}$, $b = 5.92\,\mathrm{Å}$ and $c = 4.70\,\mathrm{Å}$ [@newnham1965] and the four magnetic Co$^{2+}$ ions ($S = \frac{3}{2}$) of the crystallographic unit cell form an almost face-centered structure with the positions ${\bf r}_1 = (1/4+\varepsilon, 1/4, 1-\delta)$, ${\bf r}_2 = (3/4+\varepsilon, 1/4, 1/2+\delta)$, ${\bf r}_3 = (3/4-\varepsilon, 3/4, \delta)$ and ${\bf r}_4 = (1/4-\varepsilon, 3/4, 1/2-\delta)$ and with the displacements $\varepsilon = 0.0286$ and $\delta = 0.0207$ [@kubel1994]. The zero-field commensurate antiferromagnetic structure of LiCoPO$_4$ has spins along $b$ (easy axis) and the four magnetic ions in a $C = (\uparrow \uparrow \downarrow \downarrow)$ arrangement [@santoro1966]. Here $\uparrow$/$\downarrow$ denotes spin up/down for ions on site number $1-4$. The transition temperature is $T_N = 21.6\,\mathrm{K}$ [@szewczyk2011; @vaknin2002] and the saturation field is $\sim28\,\mathrm{T}$ with saturated moment $3.6\,\mathrm{\mu_B}$/ion [@kharchenko2010]. A number of studies establish that the magnetic point group of the zero-field magnetic structure is $2'_x$ rather than $mmm'$ as previously believed [@santoro1966]. This is based on the observation of a weak ferromagnetic moment [@rivera1994; @kharchenko2003], symmetry of the susceptibility tensor of optical second harmonic generation [@zimmermann2009; @vanAken2008] and the discovery of a toroidal moment [@vaknin2002; @ederer2007; @spaldin2008; @vanAken2007; @zimmermann2014]. The magnetic phase diagram of LiCoPO$_4$ was previously characterized up to $25.9\,\mathrm{T}$ applied along $b$ by magnetization measurements, neutron diffraction and electric polarization measurements [@kharchenko2010; @fogh2017; @khrustalyov2017]. At $11.9\,\mathrm{T}$, the commensurate low-field structure gives way to an elliptic spin cycloid propagating along $b$ with a period of thrice the crystallographic unit cell. The magnetic moments are in the $(b,c)$-plane with the major axis along $b$. In the field interval $20.5-21.0\,\mathrm{T}$, the magnetic unit cell size remains but the spins re-orient. Above $21.0\,\mathrm{T}$, there is a re-entrance of commensurate magnetic order accompanied by the ME effect.
In this work we investigate the possible role of the spin-orbit coupling for explaining the ME effect in LiCoPO$_4$ as well as its sister compounds. In order to do so we look into the details of the zero-field magnetic structure of LiCoPO$_4$ and study the effects of a magnetic field applied along $a$ by means of neutron diffraction and magnetometry. A spontaneous canting of spins away from the $b$-axis towards $c$ is revealed. The resulting structure has magnetic point group $m'_z$ and we discuss the implications related to the ME tensor form and with regards to previous studies. In order to investigate the DM interaction in LiCoPO$_4$ we perform a neutron diffraction experiment with magnetic fields applied along $a$, i.e. perpendicular to the easy axis. The induced ferromagnetic moment couples via the DM interaction to yield a field-induced spin canting. We estimate the size of the DM interaction and discuss how this interaction may play a part as generator of the ME effect in LiCoPO$_4$.
Experimental details
====================
Vibrating sample magnetization measurements were performed with a standard CRYOGENIC cryogen free measurement system. Magnetic fields of $0-16\,\mathrm{T}$ were applied along $a$ for temperatures in the interval $2-300\,\mathrm{K}$.
The zero-field magnetic structure was determined at the TriCS diffractometer at the Paul Scherrer Institute (PSI) employing an Euler cradle, a closed-cycle He refrigerator, open collimation and a Ge(311) monochromator with wavelength $\lambda = 1.18\,\mathrm{Å}$. No $\lambda/2$ contamination of the beam is possible due to the diamond structure of Ge. 193 inequivalent peaks were collected at $30\,\mathrm{K}$ and $5\,\mathrm{K}$.
Canting components of the zero-field structure could not be unambigiously determined at TriCS due to extinction effects and the large absorption cross section of Co. Instead, these components were investigated at the triple-axis spectrometer RITA-II at the PSI where a low background is obtained by energy discrimination. The instrument was operated in elastic mode with incoming and outgoing wavelength $\lambda = 4\,\mathrm{Å}$. A PG(002) monochromator and 80’ collimation between monochromator and sample were used and a liquid nitrogen cooled Be filter after the sample ensured removal of $\lambda/2$ neutrons. A cryomagnet supplied vertical magnetic fields up to $12.2\,\mathrm{T}$ along $a$ and $b$ for samples oriented with scattering planes $(0,K,L)$ and $(H,0,L)$ respectively.
A high quality LiCoPO$_4$ single crystal measuring $2\times2\times5\,\mathrm{mm^3}$ ($\sim20\,\mathrm{mg}$) was used for magnetization measurements with magnetic fields applied along $a$ and for neutron diffraction experiments in zero field and with magnetic fields applied along $b$. A second sample with dimensions $3\times4\times4\,\mathrm{mm^3}$ ($\sim40\,\mathrm{mg}$) was used for the neutron diffraction experiment performed with fields applied along $a$.
Results & discussion
====================
The atomic and magnetic structures of LiCoPO$_4$ were determined by combining data from the TriCS and RITA-II experiments. Based on the *Pnma* space group and 241 Bragg peaks, atomic displacements of $\varepsilon = 0.028$ and $\delta = 0.020$ were refined in Fullprof [@rodriguez-carvajal1993] ($R_F = 11.9$%) in fair agreement with literature[@kubel1994]. The zero-field magnetic structure was determined from 130 Bragg peaks and is mainly of $C_y$ symmetry ($R_F = 17.2$%), a result conforming with earlier findings[@santoro1966; @vaknin2002]. The refined magnetic moment is $3.54(5)\,\mathrm{\mu_B}$, consistent with previous magnetization measurements [@kharchenko2010]. Note that the Li occupancy was refined to $1.03(5)$ and hence the sample is stoichiometric within the precision of the experiment. Refinement results with the Li occupancy fixed to unity are listed in Table \[tab:fullprof\].
Atom Site $x$ $y$ $z$ $R_y$
------ ------ ----------- ----------- ----------- ---------
Li 4a 0 0 0 –
Co 4c 0.278(2) 0.25 0.980(3) 3.54(5)
P 4c 0.0945(8) 0.25 0.419(2) –
O1 4c 0.0986(7) 0.25 0.743(2) –
O2 4c 0.4545(7) 0.25 0.203(1) –
O3 8d 0.1669(5) 0.0463(7) 0.2826(9) –
: Atomic positions for LiCoPO$_4$ obtained from Fullprof refinement ($R_F = 11.9$%) using 241 Bragg peaks collected at TriCS at $(30\,\mathrm{K}, 0\,\mathrm{T})$ and using the *Pnma* space group. The Debye-Waller factor was fixed to $B_{\mathrm{iso}} = 0.20$. The magnetic moment in units of $\mu_B$ as refined using a $C_y$ symmetry component is given in the rightmost column. This results from refinement ($R_F = 17.2$%) using 130 commensurate magnetic peaks collected at $(2\,\mathrm{K},0\,\mathrm{T})$. The lattice parameters used in the refinements were $a = 10.20\,\mathrm{Å}$, $b = 5.92\,\mathrm{Å}$ and $c = 4.70\,\mathrm{Å}$ as given in Ref..[]{data-label="tab:fullprof"}
Other magnetic structures including a minor spin rotation towards $c$ ($C_z$) or a spin canting towards $c$ ($A_z$) were proposed but these refinements were not sufficiently different to distinguish them from the one regarding only a $C_y$ component. Extinction effects and the large neutron absorption cross section of cobalt result in significantly different intensities for equivalent Bragg peaks and hence, the TriCS data only enabled identification of the major symmetry component, $C_y$.
Minor spin components in zero field and for magnetic fields applied along $b$ and $a$ were investigated at RITA-II by measuring a few key Bragg peaks: $(3,0,1)$, $(0,1,0)$, $(1,0,0)$, $(0,2,1)$, $(0,1,2)$ and $(0,0,1)$. Of these only $(0,1,0)$ has zero magnetic intensity. The calculated magnetic structure factors for the four basis vectors, $|S_R({\bf Q})|^2$, $R = \left\lbrace A, G, C, F \right\rbrace$, and spin polarization factors, $|P_i({\bf Q})|^2$, $i = \left\lbrace x,y,z \right\rbrace$, for these peaks are listed in Table \[tab:structurefactors\]. The magnetic neutron intensity may then be expressed as: $$I({\bf Q}) \propto S^2 \, f({\bf Q})^2 \, \sum_{R} |S_R({\bf Q})|^2 \sum_{i} |P_i({\bf Q})|^2,
\label{eq:intensity}$$ where $f({\bf Q})$ is the magnetic form factor and $S$ is the thermal average of the magnetic moment. The following analysis is based on a process of eliminating possible structures and is not a full structure refinement.
----------- --------------------------------------------- --------------------------------------------- --------------------------------------------- ----------------------------------------- ------ ------ ------
$A$ $G$ $C$ $F$ $x$ $y$ $z$
$(\uparrow \downarrow \downarrow \uparrow)$ $(\uparrow \downarrow \uparrow \downarrow)$ $(\uparrow \uparrow \downarrow \downarrow)$ $(\uparrow \uparrow \uparrow \uparrow)$ $a$ $b$ $c$
$(3,0,1)$ 0.07 0.22 11.73 3.98 0.34 1 0.66
$(0,1,0)$ 0 0 16 0 1 0 1
$(1,0,0)$ 15.51 0.49 0 0 0 1 1
$(0,2,1)$ 0 15.71 0.29 0 1 0.28 0.72
$(0,1,2)$ 0 1.14 14.86 0 1 0.86 0.14
$(0,0,1)$ 0 15.71 0.29 0 1 1 0
----------- --------------------------------------------- --------------------------------------------- --------------------------------------------- ----------------------------------------- ------ ------ ------
: Absolute squares of structure and polarization factors for the magnetic basis vectors reflected by the key Bragg peaks used to establish the magnetic structure of LiCoPO$_4$. The factors are normalized to unit spin lengths. Note that the crystallographic directions $a$, $b$ and $c$ may be used interchangeably with $x$, $y$ and $z$ respectively.[]{data-label="tab:structurefactors"}
Spontaneous spin canting at zero field
--------------------------------------
In addition to the major $C_y$ spin component, a smaller symmetry component was identified by observation of magnetic intensity at the $(1,0,0)$ position. This peak mainly represents magnetic structures of $A$ symmetry with spins polarized along either $b$ or $c$. It is approximately one order of magnitude weaker than $(3,0,1)$ \[compare Figs. \[fig:rita\](a) and \[fig:rita\](b)\] which may be assumed to represent the major spin component when regarding the following argument: both $(3,0,1)$ and $(0,1,0)$ appear if a $C$ component is present but the two peaks represent different spin polarizations. $(3,0,1)$ is present for any spin orientation whereas $(0,1,0)$ is only present for components along $a$ or $c$. Since $(0,1,0)$ has no magnetic intensity \[see Fig. \[fig:rita\](c)\] we can exclude those two spin directions entirely. Hence, the $(3,0,1)$ magnetic Bragg peak may be assumed to solely represent a $C_y$ spin arrangement.
Next, the basis vector corresponding to the $(1,0,0)$ Bragg peak is identified. The thermal average of the spin is most often maximized at low temperatures. Since an $A$ type component with spins along $b$ would produce spins of varying lengths, it is therefore reasonable to assume that the observed magnetic intensity at $(1,0,0)$ is instead due to a spin component along $c$. The result is a canting of the spins in the $(b,c)$-plane and the canting angle, $\varphi$, is estimated by comparing the intensity of $(1,0,0)$ with that of $(3,0,1)$. Following the above arguments, it is assumed that $(3,0,1)$ represents only a $C_y$ symmetry component and $(1,0,0)$ represents only an $A_z$ component such that the measured intensities may be written as in Eq. $$\begin{aligned}
I_{(1,0,0)} & \propto \left|S_A^{(1,0,0)}\right|^2 \left|P_z^{(1,0,0)}\right|^2 f^2_{(1,0,0)},\\
I_{(3,0,1)} & \propto \left|S_C^{(3,0,1)}\right|^2 \left|P_b^{(3,0,1)}\right|^2 f^2_{(3,0,1)},\end{aligned}$$ The spontaneous canting angle is then calculated from the corrected intensities, $I^{\mathrm{corr}}_{(1,0,0)}$ and $I^{\mathrm{corr}}_{(3,0,1)}$, as $\tan \varphi = \sqrt{I^{\mathrm{corr}}_{(1,0,0)}/I^{\mathrm{corr}}_{(3,0,1)}}$. The usual Lorentz factor for two-axis diffractometers, $\sin (2 \theta)$, is also taken into account and although not entirely correct for the triple-axis setup [@pynn1975], the correction is estimated to introduce an error of maximum 10% for the two implicated Bragg peaks. The calculated angle is shown in Fig. \[fig:rita\](e) where both data at $0\,\mathrm{T}$ and $10\,\mathrm{T}$ along $b$ are shown. The canting angle is temperature independent below the transition temperature and it is also independent of the applied magnetic field. The magnetic structure is thus locked in with a spontaneous canting angle of $\varphi = 7(1)^{\circ}$ as estimated from a weighed mean of all data points in Fig. \[fig:rita\](e). The resulting zero-field structure is illustrated in Fig. \[fig:zero\](a). Note that the $(3,0,1)$ Bragg peak is relatively strong compared to $(1,0,0)$ and is therefore, to a larger extent, subject to extinction effects. Consequently, the calculated angle may be overestimated.
$\Gamma_1$ $\Gamma_2$ $\Gamma_3$ $\Gamma_4$ $\Gamma_5$ $\Gamma_6$ $\Gamma_7$ $\Gamma_8$
------------ ------------ ------------ ------------ ------------ ------------ ------------ ------------
$Pnma$ $Pnm'a'$ $Pn'ma'$ $Pn'm'a$ $Pn'm'a'$ $Pn'ma$ $Pnm'a$ $Pnma'$
$F_x$ $G_x$ $C_x$ $A_x$
$G_y$ $F_y$ $A_y$ $C_y$
$G_z$ $F_z$ $A_z$ $C_z$
: Irreducible representations, magnetic space groups and corresponding basis vectors for *Pnma*.[]{data-label="tab:irreps"}
Both $(3,0,1)$ and $(1,0,0)$ appear at the same transition temperature – see Fig. \[fig:rita\](d) – and therefore reflect the same order parameter. Indeed, a power law with collectively fitted transition temperature, $T_N=21.55(2)\,\mathrm{K}$, and critical exponent, $\beta = 0.34(1)$, describe the recorded data well. However, note that the $C$ type structure polarized along $b$ and the $A$ type structure polarized along $b$ or $c$ are not contained within the same irreducible representation of the lithium orthophosphates, see Table \[tab:irreps\].
The Bragg peaks $(0,2,1)$, $(0,1,2)$ and $(0,0,1)$ also have magnetic intensity at $0\,\mathrm{T}$. These peaks are all present for a $C_y$ structure but may also represent a $G$ type component polarized along either $a$ or $b$, see Table \[tab:structurefactors\]. A $G_y$ component is unlikely due to maximized moments at low temperatures and is not compatible with the observed ME effect, toroidal moment and weak ferromagnetism. Furthermore, $G_x$ is paired with $F_z$ in the irreducible representations, see Table \[tab:irreps\], and $F_z$ is not present [@kharchenko2003]. Therefore, the magnetic intensity at the $(0,2,1)$, $(0,1,2)$ and $(0,0,1)$ positions at $0\,\mathrm{T}$ may be subscribed to the major $C_y$ spin component.
It is commented that the determined zero-field structure does not fully agree with earlier findings. A $C_z$ type rotation of the spins away from the $b$-axis was reported in Ref. based on the observation of the $(0,1,0)$ magnetic peak. However, as seen in Fig. \[fig:rita\](c), we observe zero magnetic intensity at the $(0,1,0)$ position. A maximum of the rotation angle of $0.7(3)^{\circ}$ is estimated from the error on the measured zero intensity. This is contrasted by the $4.6^{\circ}$ reported in Ref. . One possible explanation for the discrepancy might be found in slightly different levels of Li in different samples. Previously, changes in atomic bond lengths and magnetic properties of Li$_z$CoPO$_4$ with $z = 0.2,0.7$ as compared to the stoichiometric compound, LiCoPO$_4$, were reported [@ehrenberg2009]. It is conceivable that small variations in Li contents between samples may bring about small differences in the exact magnetic structure. As already mentioned, our sample was found to have a Li occupancy of $1.03(5)$.
It has been repeatedly suggested [@vaknin2002; @vanAken2008; @zimmermann2009] that the zero-field structure of LiCoPO$_4$ has lower symmetry than the originally proposed magnetic point group $mmm'$ [@santoro1966]. The observed $4.6^{\circ}$ rotation of spins restricts symmetry to $2_x'/m_x$ which is further reduced to $2'_x$ when requiring a weak ferromagnetic component along $b$. Indeed, optical second harmonic generation measurements advocate that the point group symmetry is $2'_x$ [@zimmermann2009]. This point group allows for a toroidal moment [@schmid2008] and the linear ME effect with tensor elements $\alpha_{ab}, \alpha_{ba} \neq 0$ [@rivera2009], consistent with measurements [@rivera1994]. In addition, $2'_x$ allows the tensor elements $\alpha_{ac}, \alpha_{ca} \neq 0$ which are not measurably different from zero [@rivera1994] but the spin rotation angle introduces only a small deviation from $mmm'$. Furthermore, as the point group merely yields the *allowed* ME tensor elements they are not *necessarily* active.
Thus neutron diffraction [@vaknin2002], SQUID [@kharchenko2001] and optical second harmonic generation measurements [@zimmermann2009; @vanAken2008] all paint a picture of LiCoPO$_4$ having magnetic point group $2'_x$ in its zero-field state. In contrast, our observation of a spontaneous spin canting rather than a rotation leads to the magnetic point group $2_z/m'_z$. This point group also allows for a toroidal moment and the ME tensor elements $\alpha_{aa},\alpha_{ab},\alpha_{ba},\alpha_{bb},\alpha_{cc} \neq 0$ where only the off-diagonal elements are measurably different from zero. Again, we note that the canting angle only presents a small deviation from $mmm'$. $2_z/m'_z$ does not support a ferromagnetic moment along $b$ rendering it inconsistent with observations. However, removing the twofold axis enables a ferromagnetic moment in the $(a,b)$-plane. Thus, the magnetic point group $m_z'$ is consistent with our neutron diffraction data and a weak ferromagnetic moment along $b$. Note, however, that it is not consistent with the observed optical second harmonic generation signal [@zimmermann2009; @vanAken2008].
Interestingly, $m'_z$ is also consistent with the previous neutron diffraction study when using a different – but still correct – interpretation of the presented data. The rotation of the spins towards $c$ was established based on observation of the $(0,1,0)$ magnetic Bragg peak. However, this rotation might equally well be towards $a$. Assuming such a rotation results in magnetic point group $2_z/m'_z$ which again needs relaxing to $m'_z$ to allow for a ferromagnetic moment along $b$. In addition, the $C_x$ component belongs to the same irreducible representation as the $A_z$ component \[see Table \[tab:irreps\]\] and as is deducted in the next section; the two components combined yield a favorable energy term via the DM interaction. Therefore, our observations may in fact be consistent with the previous studies and the magnetic point group of the zero-field structure of LiCoPO$_4$ is $m'_z$.
Field-induced spin canting for H$\parallel$a
--------------------------------------------
For magnetic fields applied along $a$, LiCoPO$_4$ is linearly magnetized with the field as seen in the magnetization data in Fig. \[fig:a-axis\](a). A ferromagnetic contribution to the spin structure is induced with $S^a = \alpha H$ and fitted slope $\alpha = 0.0395(1)\,\mu_B/\mathrm{T}$. Furthermore, yet another antiferromagnetic component exists in addition to the established main structure of $C_y$ symmetry and the minor $A_z$ component. This extra component is manifested by an increase in the intensity of the $(0,2,1)$ magnetic Bragg peak as a function of applied field, see Fig. \[fig:a-axis\](b). The magnetic origin of the $(0,2,1)$ intensity is confirmed by its temperature dependence which follows a Curie-Weiss law squared, see Fig. \[fig:a-axis\](c).
The $(0,2,1)$ peak represents mainly spin arrangements of symmetry $G$ and to a smaller extent structures of symmetry $C$, *cf.* Table \[tab:structurefactors\]. All spin orientations are possible and more information is therefore needed in order to pin down which magnetic structure the additional intensity of $(0,2,1)$ signifies. Again, the argument follows a process of elimination using two other magnetic Bragg peaks: $(0,1,2)$ and $(0,0,1)$.
The $(0,1,2)$ peak is present for any $C$ spin structures. This peak has no additional field-induced intensity \[see Fig. \[fig:a-axis\](b)\] and consequently any additional $C$ spin elements are ruled out. Finally, $(0,0,1)$ represents $G$ symmetry with spins polarized along $a$ or $b$. Again, this peak shows no change upon applying a magnetic field along $a$ \[see Fig. \[fig:a-axis\](b)\] and these magnetic structure types may too be rejected. The only remaining possible magnetic structure as a contributor to the $(0,2,1)$ field-induced intensity is then $G_z$. This component comes as an addition to the already established major $C_y$ component and the smaller $A_z$ component. An asymmetry is introduced in the canting angles such that spins $(1,2)$ and $(3,4)$ form pairs with canting angles $\phi+\theta$ and $\phi-\theta$ respectively. Here $\theta \equiv \theta(H)$ is the field-induced canting angle. The resulting magnetic structure for magnetic fields applied along $a$ is shown in Fig. \[fig:zero\](b).
The size of $\theta$ is now estimated. As previously argued, it may be assumed that at $0\,\mathrm{T}$, $(0,2,1)$ only reflects the $C_y$ structure. Any additional intensity upon applying a field then originates from the $G_z$ component: $$I_{(0,2,1)}(H) - I_{(0,2,1)}(0\,\mathrm{T}) \propto \left|S_{G}^{(0,2,1)}\right|^2 \left|P_z^{(0,2,1)}\right|^2.$$ This is to be compared to the intensity of $(0,2,1)$ at $0\,\mathrm{T}$: $$I_{(0,2,1)}(0\,\mathrm{T}) \propto \left|S_{C}^{(0,2,1)}\right|^2 \left|P_y^{(0,2,1)}\right|^2.$$ Since only one peak is involved in the determination of the field-induced canting angle there is no need to correct for the magnetic form factor or Lorentz factor and any extinction or absorption effects may be neglected. The field-induced canting angle is then calculated as $\tan \theta = \sqrt{ \frac{I^{\mathrm{corr}}_{(0,2,1)}(H) - I^{\mathrm{corr}}_{(0,2,1)}(0\,\mathrm{T}) }{I^{\mathrm{corr}}_{(0,2,1)}(0\,\mathrm{T})}}$ and is to a good approximation linear as a function of applied field along $a$: $\theta = \beta H$ with fitted slope $\beta = 0.012(1)\,\mathrm{rad/T}$ \[see Fig. \[fig:a-axis\](a)\]. The field-induced canting angle as deduced from the magnetization, $\sin \theta = M/M_S$, is also shown in Fig. \[fig:a-axis\](a) and substantiates the link between $F_x$ and $G_z$. Furthermore, since the neutron intensity is proportional to the ordered magnetic moment squared, a linear coupling between the ferromagnetic moment and canted moment would result in a quadratic increase in the neutron intensity of $(0,2,1)$ as a function of applied field. This is indeed the case as shown in Fig. \[fig:a-axis\](b). Here the solid line is a fit to a quadratic dependence, $I \propto H^2$. The measured intensity is clearly well described by the fit. Additionally, the symmetry elements $G_z$ and $F_x$ belong to the same irreducible representation, see Table \[tab:irreps\].
Dzyaloshinskii-Moriya interaction
---------------------------------
An estimate of the size of the DM interaction in LiCoPO$_4$ may be obtained from the field-induced spin canting. A similar calculation was previously performed for the sister compound LiNiPO$_4$ and the analysis in Ref. is directly applicable here. Symmetry arguments lead to the only allowed DM coefficients ${\bf D}_{14} = (0,D_{14}^b,0) = - {\bf D}_{23}$ and ${\bf D}_{12} = (0,D_{12}^b,0) = {\bf D}_{34}$. These yield terms in the Hamiltonian of the form: $$\begin{aligned}
\mathcal{H}_{\mathrm{DM}}^1 & = {\bf D}_{14} \cdot \left( {\bf S}_1 \times {\bf S}_4 \right) - {\bf D}_{14} \cdot \left( {\bf S}_2 \times {\bf S}_3 \right)\\
& = D_{14}^b \left( S_1^c S_4^a - S_1^a S_4^c - S_2^c S_3^a + S_2^a S_3^c \right) \quad \mathrm{and}\\
\mathcal{H}_{\mathrm{DM}}^2 & = {\bf D}_{12} \cdot \left( {\bf S}_1 \times {\bf S}_2 \right) + {\bf D}_{12} \cdot \left( {\bf S}_3 \times {\bf S}_4 \right)\\
& = D_{12}^b \left( S_1^c S_2^a - S_1^a S_2^c + S_3^c S_4^a - S_3^a S_4^c \right).\end{aligned}$$ The spin component along $a$ is finite for $H \parallel a$ and assumed equal at all sites, i.e. $S_1^a = S_2^a = S_3^a = S_4^a = S^a > 0$. In this case, both terms favor a $G_z$ type order and this is exactly what we observe. The ferromagnetic moment along $a$ therefore induces – via the DM interaction – an antiferromagnetic spin component of symmetry $G_z$.
The field-induced $G_z$ component leaves the nearest neighbor spin pairs $(1,4)$ and $(2,3)$ antiparallel and hence no energy change is to be expected from the term $\mathcal{H}^1_{\mathrm{DM}}$ nor from the nearest neighbor exchange term. On the other hand, the term $\mathcal{H}^2_{\mathrm{DM}}$ does indeed yield a finite energy contribution for a $G_z$ component. The strength of the DM interaction may be estimated by balancing the different energy contributions for spins deviating from the easy axis, $b$: $$\begin{aligned}
\left.
\begin{matrix}
\mathcal{H}_{\mathrm{DM}} = 4 D^b_{12} S^a S \sin \theta \\
\mathcal{H}_{\mathrm{ani}} = 4 \mathfrak{D}^c S^2 \sin^2 \theta
\end{matrix}
\right\rbrace \Rightarrow \frac{D^b_{12}}{\mathfrak{D}^c} = \frac{-S \sin \theta}{S^a} \approx -S\frac{\theta}{S^a},\end{aligned}$$ where $\mathfrak{D}^c$ is the single-ion anisotropy constant for spin components along $c$, $S=3.6\,\mathrm{\mu_B}$ the saturated moment, $\sin \theta \approx \theta$ holds for small canting angles, $\theta = \beta H$ and $S^a = \alpha H$. With the fitted coefficients $\beta = 0.012(1)\,\mathrm{rad/T}$ and $\alpha = 0.0395(1)\,\mathrm{\mu_B/T}$ the ratio becomes $D^b_{12} / \mathfrak{D}^c \approx -1.1$. Note that this is an upper bound for the size of the DM interaction as the above simple calculation neglects any competing exchange interactions which may also influence the spin canting.
Thus, the DM interaction in LiCoPO$_4$ may be as large as the single-ion anisotropy along $c$. The full spin Hamiltonian of LiCoPO$_4$ has not been determined yet, but limited inelastic neutron scattering data shows an almost dispersionless spin excitation along the $(0,K,0)$ direction and a single-ion anisotropy constant of $\mathfrak{D}^c \approx 0.7\,\mathrm{meV}$ is suggested [@vaknin2002; @tian2008; @Note1]. This is a very strong DM interaction and its possible role as a generator for the ME effect in LiCoPO$_4$ is discussed in the following.
Magnetostrictive mechanisms successfully explain the ME effect in LiNiPO$_4$ [@jensen2009_2; @toftpetersen2017] and LiFePO$_4$ [@toftpetersen2015] based on magnetic field-induced changes in the exchange and DM interactions respectively. A similar model would be expected to describe the effect in LiCoPO$_4$. However, so far a satisfactory model has eluded all our efforts – both when considering magnetic field-induced changes in the exchange and DM interactions individually and combined. Such microscopic models inherently result in a ME coefficient, $\alpha_{\parallel}$, proportional to $\chi_{\parallel} \langle S \rangle^2$, i.e. the magnetic susceptibility and the order parameter. The susceptibility drops at low temperatures in a collinear antiferromagnet and the order parameter levels out after the initial increase at the transition. Hence the temperature dependence of $\alpha_{\parallel}$ has a maximum below the transition as seen in LiMnPO$_4$, LiNiPO$_4$ and LiFePO$_4$ \[re-visit Fig. \[fig:ME\]\]. However, for LiCoPO$_4$ $\alpha_{\parallel}$ does not display such maximum as a function of temperature. In fact, its temperature profile resembles that of the order parameter and the curves are similar for $\alpha_{\parallel}$ and $\alpha_{\perp}$.
As discussed in the introduction, it was previously proposed that the spin-orbit coupling is a central element in fully understanding the ME effect in the lithium orthophosphates. However, *ab initio* calculations considering both spin and orbital momentum on equal footing still fail to correctly predict the size of $\alpha_{\parallel}$ for $T \rightarrow 0$ in LiFePO$_4$ [@scaramucci2012]. The spin-orbit coupling is expected to be larger in the sister compound, LiCoPO$_4$ and similar first-principle computations may be expected to produce larger ME coefficients. To our best knowledge such calculations have yet to be performed. Nevertheless, our neutron diffraction data show that there is indeed a large DM interaction in LiCoPO$_4$ which in turn relates to the spin-orbit coupling. Therefore, it remains that the spin-orbit coupling plays an important role in generating the ME effect in LiCoPO$_4$ – and most likely in the entire family of compounds. This emphasizes the need for more theoretical work and improved *ab initio* calculations in order to elucidate the missing mechanism(s) governing the linear ME effect in LiCoPO$_4$ and even better; explain the link between the spin-orbit coupling and the ME effect in the lithium orthophosphates in general. Moreover, spin excitation measurements would enable modeling of the spin Hamiltonian of LiCoPO$_4$ and thereby provide a better understanding of the magnetic interactions in the system.
Conclusions
===========
Intricate details of the zero-field magnetic structure of LiCoPO$_4$ were investigated in hope of illuminating the microscopic mechanism behind the large magnetoelectric effect in LiCoPO$_4$. The Co$^{2+}$ ions mainly order in a commensurate antiferromagnetic structure of $C_y$ symmetry. Additionally, we discover a spontaneous spin canting of $\varphi = 7(1)^{\circ}$ originating in an $A_z$ spin component. The resulting zero-field magnetic structure belongs to the magnetic point group $m'_z$, consistent with previously reported experimental results.
For magnetic fields applied along $a$, a second minor spin component of symmetry $G_z$ is induced. The canting angle increases to a good approximation linearly with the applied field and is shown to be induced via the Dzyaloshinskii-Moriya interaction by the ferromagnetic moment along $a$. The upper limit for the size of the Dzyaloshinskii-Moriya interaction was estimated to be approximately equal to that of the single-ion anisotropy constant along $c$. This shows that the spin-orbit coupling is strong in LiCoPO$_4$ and we discuss how it may be linked to its large magnetoelectric effect.
Acknowledgements {#acknowledgements .unnumbered}
================
Work was supported by the Danish Agency for Science and Higher Education under DANSCATT. Neutron diffraction experiments were performed at the Swiss spallation neutron source SINQ, Paul Scherrer Institute, Villigen, Switzerland. Ames Laboratory is operated by the U.S. Department of Energy by Iowa State University under Contract No. DE-AC02-07CH11358.
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abstract: 'We report 345 GHz continuum observations of the host galaxies of gamma-ray bursts (GRBs) 021004 and 080607 at $z>2$ using the Atacama Large Millimeter/Submillimeter Array (ALMA) in Cycle 0. Of the two bursts, GRB021004 is one of the few GRBs that originates in a Lyman limit host, while GRB080607 is classified as a “dark burst” and its host galaxy is a candidate of dusty star forming galaxy at $z\sim3$. With an order of magnitude improvement in the sensitivities of the new imaging searches, we detect the host galaxy of GRB080607 with a flux of $S_{345} = 0.31\pm 0.09$ mJy and a corresponding infrared luminosity of $L_{\rm IR}=(2.4$–$4.5)\times10^{11}~L_\sun$. However, the host galaxy of GRB021004 remains undetected and the ALMA observations allow us to place a 3-$\sigma$ upper limit of $L_{\rm IR}<3.1\times10^{11}~L_\sun$ for the host galaxy. The continuum imaging observations show that the two galaxies are not ultraluminous infrared galaxies but are at the faintest end of the dusty galaxy population that gives rise to the submillimeter extragalactic background light. The derived star formation rates of the two GRB host galaxies are less than 100 $M_\sun$ yr$^{-1}$, which are broadly consistent with optical measurements. The result suggests that the large extinction ($A_V\sim3$) in the afterglow of GRB080607 is confined along its particularly dusty sightline, and not representative of the global properties of the host galaxy.'
author:
- 'Wei-Hao Wang, Hsiao-Wen Chen, and Kui-Yun Huang'
title: ALMA Submillimeter Continuum Imaging of the Host Galaxies of GRB021004 and GRB080607
---
Introduction
============
Long-duration gamma-ray bursts (GRBs) are believed to originate in the death of massive stars (see [@woosley06] for a recent review), and are thus expected to trace star formation in galaxies [e.g., @wijers98; @totani99]. Because of the extreme luminosity of the prompt emission and the afterglow, GRBs are a powerful probe of star formation in early times [e.g., @tanvir09; @salvaterra09]. In order to establish the link between GRBs and the cosmic star formation, it is important to understand the properties of the GRB host galaxies [@hjorth12]. A critical measurement is the star formation rates (SFRs) of the host galaxies.
There exist various SFR indicators for high-redshift galaxies in different spectral windows. One key concern is the effect of dust extinction. Even with arguably good extinction corrections in optical data, highly obscured components may still exist and would only appear at the far-infrared and radio wavelengths. The presence of such components would indicate a significant spatial variation in dust content in which case a global extinction correction would not apply. For GRB host galaxies, systematic surveys were carried out to observe continuum emission in the radio [@michalowski12] and submillimeter [@berger03; @tanvir04; @priddey06] frequency ranges for constraining dust enshrouded SFR. Because of the synchrotron spectral slope, radio observations are only effective in detecting GRB host galaxies at $z\lesssim1$ [e.g., @michalowski12]. Dust continuum emission in the submillimeter has a spectral slope that can nearly cancel the effect of luminosity distance from $z\sim1$ to $z\sim10$, making the submillimeter wavelengths an effective window for detecting faint galaxies at high redshifts [@blain93]. However, the 850 $\mu$m survey of 21 GRB host galaxies at $z<3.5$ by @tanvir04 using the James Clerk Maxwell Telescope (JCMT) only uncovered three host galaxies at $>3\sigma$ confidence levels, and all three hosts are at $z<1.5$. Submillimeter single-dish telescopes are confusion limited at roughly a few mJy at 850 $\mu$m, and therefore can only detect galaxies with infrared luminosities of $L_{\rm IR}$ (8 to 1000 $\mu$m) $> 10^{12.5}~L_\sun$, or SFR $\gtrsim1000~M_\sun$ yr$^{-1}$. This SFR limit is much larger than the typical SFR of GRB host galaxies measured in the optical [@christensen04; @savaglio09]. Deeper submillimeter measurements are thus required to better constrain their infrared luminosity and the underlying SFR.
A particularly interesting class of GRB is “dark GRBs” [@djorgovsky01; @jakobsson05], defined by their faint optical afterglow, relative to the bright X-ray emission. A definition for dark bursts is those with the optical-to-X-ray spectral index of $\beta_{\rm OX} < 0.5$ [@jakobsson04], which is physically motivated based on theoretical predictions of the synchrotron model. Approximately 30%–50% of long-duration GRBs have suppressed optical fluxes relative to their X-ray emission [@melandri08; @cenko09; @melandri12]. The weaker optical emission can be caused by either intergalactic medium absorption at $z>6$ [@kawai06; @greiner09; @tanvir09; @salvaterra09] or dust extinction in their host galaxies [@perley09]. In the latter case, dark GRBs may serve as a tracer of dust enshrouded star formation across cosmic time. However, uncertainty remains regarding whether the observed dust obscuration is representative of the global properties of the host galaxies or merely local to the progenitor site. This uncertainty can be addressed by comparing the rest-frame infrared luminosities between dark GRB host galaxies and the rest of the host galaxy population.
In this *letter*, we present initial results from a pilot study of GRB host galaxies in the submillimeter frequency range using the Atacama Large Millimeter/Submillimeter Array (ALMA). In Cycle 0, we observed the host galaxies of GRB021004 ($z=2.330$) and GRB080607 ($z=3.036$) at 345 GHz. The GRB fields were selected to have early-time afterglow spectra available for constraining the ISM absorption properties of the host galaxies [e.g., @fynbo05; @prochaska09; @sheffer09]. They are among the best studied events and they represent two extremes in the integrated total ISM column density alone the afterglow line of sight. GRB021004 is one of a few GRBs arising in a Lyman limit absorber with $N_{\rm H I} = 10^{19.5\pm0.5}$ cm$^{-2}$, and the host of GRB080607 is a damped Ly$\alpha$ absorber with an unprecedentedly high gas density of $N_{\rm H I} = 10^{22.70\pm0.15}$ cm$^{-2}$. GRB080607 is a dark burst with highly extinguished afterglow ($A_V\sim3.2$, [@prochaska09; @perley11]), and the extinction suggests that the host galaxy may have detectable submillimeter dust emission. The host galaxies of the two GRBs are found to have optical SFRs of 10–40 $M_\sun$ yr$^{-1}$ [@jakobsson05; @castro10; @chen11a], consistent with normal star forming galaxies at $z>2$. Here we use the ALMA results to estimate the infrared luminosities of the two host galaxies, and to examine whether there exist highly obscured star-forming regions that are not revealed by optical observations. We describe our ALMA observations and data reduction in Section \[obs\], and the results and estimate the infrared luminosities and the SFRs of the GRB host galaxies in Section \[result\]. The implication of our observations is discussed in Section \[discussion\]. We adopt cosmological parameters of $H_0=71$ km s$^{-1}$ Mpc$^{-1}$, $\Omega_M=0.27$, and $\Omega_\Lambda=0.73$, and we convert the previous results to this set of cosmology.
Observation and Data Reduction {#obs}
==============================
Observations of the continuum emission at 345 GHz from the host galaxies of GRB021004 and GRB080607 ware obtained using the ALMA 12-m array. Four spectral windows were tuned to center at 338, 340, 350, and 352 GHz, each with a 2 GHz bandwidth. Bandpass calibrators and flux calibrators were observed prior to the observations of the science targets. Bright quasars near the GRB fields were observed every $\sim11$ minutes for phase and amplitude calibrations. For each science target, a total of 0.7–0.8 hr of on-target integration was collected. Table \[tab1\] summarizes the basic observing parameters and the various calibrators.
We received the data from the Joint ALMA Observatory (JAO) a few weeks after the observations. The delivered data were already bandpass, flux, and gain (phase and amplitude) calibrated by JAO, and reference images were also provided. All the above calibration and imaging were carried out using Common Astronomy Software Applications [CASA, @mcmullin07]. We further inspected the JAO calibration in CASA, and Fourier-transformed the complex visibility to make our own images. To obtain the highest S/N, we gave all visibility data equal weights regardless of their density distribution in the $uv$ plane (i.e., “natural weighting”). The resulting synthesized beams and sensitivities are summarized in Table \[tab1\]. We do not detect the host galaxy of GRB021004, and thus the imaging remains in the “dirty” stage. We detect the host galaxy of GRB080607, and therefore “CLEANed” the sidelobes of the detected object in CASA.
[lcc]{} Observing Date & Oct 22, 2011 & Nov 16, 2011\
& Nov 5, 2011 & Jan 12, 2012\
Number of Antennas & 17 & 20\
On-Target Integration & 42.6 min & 48.2 min\
Bandpass Calibrator & 3c454.3 & 3c273\
Flux Calibrator & Callisto & Titan\
Gain Calibrators & B0007+016 & J1239+075\
& J0010+109 & B1236+077\
Sensitivity (1 $\sigma$) & 0.113 mJy & 0.098 mJy\
Synthesized Beam & $1\farcs55 \times 1\farcs25$, $-20\arcdeg$ & $1\farcs56 \times 0\farcs87$, $3\arcdeg$\
(major $\times$ minor, PA)
Results {#result}
=======
GRB021004 {#sec_GRB021004}
---------
We do not detect the host galaxy of GRB021004 (Figure \[fig1\]). The 345 GHz point-source flux measured at the location of the host galaxy is $0.17\pm0.11$ mJy. @tanvir04 measured an 850 $\mu$m flux of $0.77 \pm 1.25$ mJy using SCUBA on JCMT. @smith05 improved the previous SCUBA result slightly to $-1.4\pm1.0$ mJy. Our measurement is $10\times$ deeper than these previous observations but the host galaxy remains undetected.
Given the redshift of $z=2.330$ and adopting the 3-$\sigma$ upper limit of 0.33 mJy, we can estimate the upper limits of its rest-frame infrared luminosity and SFR. To do so with single-band photometry, we need to assume a dust temperature, and this can be done using the infrared spectral energy distribution (SED) library of @chary01 [hereafer CE01]. The SEDs in CE01 are luminosity dependent (based on a locally calibrated luminosity—dust temperature relation) and do not allow for scaling of the SEDs. The library contains a broad range of infrared luminosity, from $2\times10^8$ to $4\times10^{13}~L_\sun$, and each template has its unique dust temperature. We thus redshift the CE01 SEDs to $z=2.330$ and look for those with observed 345 GHz fluxes below our upper limit. Of the 105 templates provided by CE01, 65 have 345 GHz fluxes lower than 0.33 mJy (Figure \[fig2\]), with corresponding infrared luminosities between $2.6\times10^8$ and $3.1\times10^{11}$ $L_\sun$. An ultraluminous infrared galaxy (ULIRG, $L_{\rm IR}>10^{12}~L_\sun$) is clearly ruled out for the host of GRB021004, which can be at most a modest infrared luminous galaxy of $L_{\rm IR}\sim3\times10^{11}$ $L_\sun$. Combining available optical photometric measurements (open squares in Figure \[fig2\]) with the ALMA upper limit further reveals a blue SED that is inconsistent with any of the CE01 templates, suggesting that the host of GRB021004 contains primarily young stars with little dust [e.g., @chen09] and that the infrared luminosity may be substantially lower than the observed limit. If we adopt the SFR conversion of star-forming galaxies, SFR ($M_\sun$ yr$^{-1}$) $= 1.7\times10^{-10}$ $L_{\rm IR}/L_\sun$ [@kennicutt98], then the 3-$\sigma$ upper limit of the SFR of the host galaxy is 53 $M_\sun$ yr$^{-1}$. These results are summarized in Table \[tab2\].
@castro10 obtained an optical spectrum of the early-time afterglow of GRB021004. The authors estimated an unobscured SFR of $\sim40$ $M_\sun$ yr$^{-1}$ based on the observed H$\alpha$ flux. @jakobsson05 obtained a Ly$\alpha$ spectrum of the host galaxy and estimated an SFR of $\sim11$ $M_\sun$ yr$^{-1}$, but this is uncertain because of the complex radiative transfer of Ly$\alpha$. Both results are within our 3-$\sigma$ upper limit. The ALMA imaging observation confirms the low SFR of the host galaxy, and also shows that more sensitive, multi-wavelength ALMA submillimeter imaging is needed to constrain the infrared luminosity and dust SED of this object.
GRB080607 {#sec_GRB080607}
---------
A significant 345 GHz flux is detected at the location of GRB080607. Figure \[fig1\] shows our ALMA image of the field around GRB080607, and its 345 GHz flux contours overlaid on an *HST* WFC3 F160W image. The rms noise is measured to be 0.094 mJy beam$^{-1}$ within the primary beam. There exists a peak of 345 GHz emission at the location of the host galaxy, with a peak flux of 0.27 mJy beam$^{-1}$. We measure a flux of 0.31 mJy by fitting the emission with a point-source model. On the other hand, the contours in Figure \[fig1\] suggest that the emission is elongated, and the elongation is similar to that observed in the *HST* image. We therefore also fit the emission with an extended 2-D Gaussian, and obtain a slightly higher integrated flux of 0.32 mJy. However, the fitted Gaussian is still consistent with a point source, which is not surprising given the low S/N. In the subsequent analyses, we adopt the more conservative measurement of 0.31 mJy with a statistical significance of 3.3 $\sigma$. Approximately $3\farcs5$ south of the GRB host galaxy, a marginal ($\sim3\sigma$) submillimeter emission is also detected at an optically bright absorber at $z=1.3399$ (H.-W. Chen et al. 2012, in preparation). We do not give further consideration to this object in this paper, but we note that this absorber could also be a faint submillimeter source.
The key question here is whether we can consider the $\sim3.3\sigma$ emission as a detection of the GRB host galaxy. First, the fitted 345 GHz peak in both the point-source and the Gaussian cases has an offset of $0\farcs08$ from the centroid of the optical emission. This offset is negligible given the S/N and the synthesized beam size of $1\farcs56\times0\farcs87$. Thus the confidence of the detection is enhanced by its coincident position with the GRB host galaxy.
Second, we consider the probability for this peak to be spurious. Figure \[fig3\] shows the histogram of pixel brightness in the primary beam. The distribution can be well fitted with a Gaussian with $\sigma=0.096$ mJy beam$^{-1}$ (solid curve), consistent with our measured noise of 0.094 mJy beam$^{-1}$ (dotted curve). Following the Gaussian distribution function, the probability of finding a $>3.3\sigma$ noise spike is $5\times10^{-4}$. However, there are hints of a non-Gaussian noise. The histogram in Figure \[fig3\] suggests an excess of positive pixels (at $>0.2$ mJy). In the image (Figure \[fig1\]), additional to the GRB host galaxy, there is a second $>3.3\sigma$ spike to the north-east of the GRB host, which has no known optical counterpart. Within the ALMA primary beam (FWHM = $17\farcs4$), there are approximately 220 independent resolution elements. This additional $>3.3\sigma$ spike suggests a probability of $1/220 = 5\times10^{-3}$, which is $10\times$ higher than the Gaussian probability. This is an upper limit, since we cannot rule out this spike as a real submillimeter source. Thus the probability of finding a $>3.3\sigma$ spike at the location of our target is between $5\times10^{-4}$ (assuming a Gaussian noise) and $5\times10^{-3}$ (assuming that the second 3.3 $\sigma$ spike is due to noise). Both these values are sufficiently small. Therefore, the fact that the observed emission coincides with the position of the GRB host substantially increases the confidence level of the detection of the host. In this paper, we consider this a detection, but we also point out that it will be worthwhile to confirm this with ALMA in future larger GRB host galaxy surveys.
With the above measured flux and the redshift of $z=3.036$, we then estimate the infrared luminosity of the host galaxy of GRB080607, using the CE01 library. We redshift the CE01 templates to $z=3.036$ and find seven of the 105 templates fall in the observed range of $0.313\pm0.094$ mJy (Figure \[fig2\]), with $L_{\rm IR} = 2.4\times10^{11}$ to $4.5\times10^{11}$ $L_\sun$. Including available optical and near-infrared photometric measurements, the bottom panel of Figure \[fig2\] further shows that the SED is well represented by known dusty templates of local galaxies across the full spectral range. The agreement strongly supports the conclusion that the host galaxy of GRB080607 is similar to a luminous infrared galaxy (LIRG, $L_{\rm}>10^{11}~L_\sun$) in the local universe. The inferred SFR is between 41 and 77 $M_\sun$ yr$^{-1}$. The above results are summarized in Table \[tab2\].
@chen11a presented SED fitting for the host of GRB080607 at $\lambda_{\rm rest} \sim 0.4$–4 $\mu$m. Adopting a Milky-Way type dust extinction law, they found $A_V=1.24$ and an extinction corrected SFR of $\sim16$–24 $M_\sun$ yr$^{-1}$, roughly $3\times$ lower than the submillimeter SFR. Given known uncertainties in both the optical extinction correction and the infrared SED of galaxies and possible variation in the distribution of dust content, the factor of three difference between the optical and submillimeter SFRs only suggests a modest amount of dust enshrouded star formation.
[lcccc]{} GRB021004 & 2.330 & $<0.33$ & $<3.1$ & $<53$\
GRB080607 & 3.036 & $0.31\pm0.09$ & 2.4–4.5 & 41–77
Discussion
==========
With the pilot ALMA imaging program of two GRB host galaxies at $z>2$, we attempt to constrain the far-infrared properties of GRB host galaxies. In our sample, GRB021004 has a very bright afterglow, but the host galaxy does not appear to show unusual dust content. On the other hand, GRB080607 is a dark GRB with large extinction along the line of sight. Their host galaxies have measured 345 GHz fluxes of $0.17\pm0.11$ and $0.31\pm0.09$ mJy, respectively. Statistically, we cannot rule out the possibility that the two host galaxies have comparable submillimeter fluxes. Despite that we had already pushed the sensitivities to roughly an order of magnitude deeper than previous measurements, deeper ALMA observations (and a larger sample) are clearly needed to tell the difference between the host galaxies of typical and dark GRBs.
On the other hand, the ALMA sensitivity limit is deep enough to probe beyond the ULIRG regime. @chen10 suggest that the high-redshift infrared luminous galaxy population contributes to the GRB host galaxy population. The ALMA sensitivity thus allows us to examine whether the two host galaxies are similar to typical dusty galaxies selected by submillimeter telescopes (submillimeter galaxies, hereafter SMGs). First, it is established that bright 345 GHz selected SMGs primarily reside in the redshift range of $z=1.5$–3.5 [@barger00; @chapman03; @chapman05; @wardlow11]. The host galaxies of GRB021004 and GRB080607 have redshifts in the range of these bright SMGs. The integrated source counts indicate that bright SMGs of $S_{345}>2$ mJy contribute to $\sim30\%$ of the extragalactic background light in this wavelength range [e.g., @coppin06]. Faint-end counts derived from lensing cluster surveys indicate that approximately 50% of the background arises from fainter sources in the flux range of $S_{345}=0.5$–2 mJy [@cowie02; @knudsen08; @chen11b]. Our ALMA detection and tight upper limit on the host galaxies of GRB021004 and GRB080607 thus put them at the still fainter end of the 345 GHz population.
We further compare the two GRB host galaxies with normal star forming galaxies at high redshift. There exists a correlation between SFR and stellar mass of star forming galaxies at high redshift [e.g., @daddi07; @pannella09; @karim11; @rodighiero11]. This correlation is often referred to as the star formation “main sequence” of galaxies. Galaxies at the main sequence are suggested to be disks that undergo quasi-steady star formation, and outliers are suggested to be starbursts with star formation boosted by gas-rich mergers [@daddi10; @genzel10]. For $z\sim2$, @daddi07 found SFR = 200 $M_{11}^{0.9}$ ($M_\sun$ yr$^{-1}$) for main-sequence galaxies, where $M_{11}$ is stellar mass in units of $10^{11} M_\sun$. The host galaxy of GRB021004 has estimated stellar masses of $1.6\times10^{10} M_\sun$ [@savaglio09] and $2.6\times10^{9} M_\sun$ [@chen09]. With our ALMA SFR upper limit ($3\sigma$) of 53 $M_\sun$ yr$^{-1}$, its SFR/$M_{11}^{0.9}$ has values of $<280$ or $<1400$, depending on the adopted stellar mass. The former is consistent with main-sequence galaxies, while the latter is close to a starburst. However, both values are 3-$\sigma$ upper limits. The host galaxy of GRB080607 has a better constrained stellar mass of 1–3 $\times10^{10} M_\sun$ [@chen11a]. If we adopt our ALMA SFR of 41–77 $M_\sun$ yr$^{-1}$, then it has SFR/$M_{11}^{0.9}=100$–600. This exercise shows that both GRB host galaxies are consistent with being main-sequence star-forming galaxies.
Finally, the afterglow spectrum of GRB080607 shows a fairly large dust extinction of $A_V=3.2$, and unprecedentedly high gas densities of $N_{\rm H I} = 10^{22.70\pm0.15}$ cm$^{-2}$ and $N_{\rm CO} = 10^{16.5\pm0.3}$ cm$^{-2}$, with a warm CO excitation temperature of $T_{\rm ex}^{\rm CO} > 100$ K [@prochaska09]. However, Prochaska et al. also suggest that the intervening molecular cloud is not the birth place of the GRB. The extinction in the afterglow is significantly larger than that for the host galaxy ($A_V=1.2$, [@chen11a]). All the above, together with our ALMA result of a relatively normal SFR, indicates that GRB080607 is not in a rare dusty galaxy, but the sightline happens to pass through a molecular cloud in its host galaxy.
We thank the referee for the useful comments. This paper makes use of the ALMA data: ADS/JAO.ALMA\#2011.0.00101.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan), in cooperation with the Republic of Chile. The JAO is operated by ESO, AUI/NRAO and NAOJ. This work is supported by the NSC grant 99-2112-M-001-012-MY3 (W.H.W.).
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|
---
author:
- 'Jose Luis Blanco-Claraco[^1]'
title: 'OLAE-ICP: Robust and fast alignment of geometric features with the optimal linear attitude estimator[^2]'
---
Introduction
============
The registration problem at hand consists of finding the transformation that best fits a set of noisy observations of geometric primitives taken from the frames of references $\mathbf{x}_A \in SE(3)$ and $\mathbf{x}_B \in SE(3)$. That is, we are interested in finding the relative pose $\mathbf{x}_B \ominus \mathbf{x}_A$ that minimizes a particular cost function.
Fundamentally different problems arise next depending on whether the *correspondences* between the two sets are known or unknown. If they are known, an *optimal transformation* algorithm, which usually has a closed-form solution, must be employed; otherwise, the usual approach is to apply a version of the Iterative Closest Point (ICP) ([@besl1992method; @pomerleau2015review]) to iterate between establishing correspondences and finding the corresponding optimal transformation (using one of the algorithms of the former group) until convergence is achieved.
Existing ICP algorithms have been reviewed extensively in the literature, for example, in [@pomerleau2015review]. Most proposed versions deal only with point-to-point correspondences, or propose point-to-plane correspondences, as in [@segal2009generalized]. Closed-form solution for plane to plane pairings was already proposed in the seminar work [@faugeras1986representation], although line to line correspondences lacked a closed-form solution and a nonlinear iterative solution was proposed in that case.
The present approach aims at integrating planes and lines into *existing* optimal attitude solvers, an idea that seems not to be proposed in related works. Note that a fundamental limitation of our proposal in its present form is that planes and lines only contribute information about the relative *orientation* between two frames of references, thus at least one point to point pairing is required to completely solve the full SE(3) relative pose.
The goal of this work is, therefore, to: (i) provide a way to integrate different geometric primitives into the existing Horn’s and OLAE solvers, and (ii) perform an experimental evaluation of their performance.
The rest of this paper is organized as follows. Section \[sect:background\] reviews the foundations of the related methods. Then, the proposed method, together with a throughout discussion of how to reliably implement OLAE is provided in Section \[sect:proposed\]. Experimental results are then exposed in the next section and we finally provide some discussion and conclusions in Section \[sec:conclusion\].
Background {#sect:background}
==========
The algorithm in [@horn1987closed] to find the optimal transformation between pairs of points, relative to their corresponding point cloud centroids, is very well-known in the robotics and computer vision communities, so it will be not further described here.
On the other hand, we have a large body of research focused on finding the optimal relative orientation, i.e. a SO(3) transformation, between a set of *unit vector observations*. Note that this is in contrast to *relative point observations*, which may have arbitrary norms, leading to a similar, but different problem. The vector observation problem arises naturally in spacecraft and satellite localization, hence it has been of the maximum interest to the aerospace community since the original Wahba’s 1965 formulation (see [@whaba1965least]):
$$\label{eq:wahba}
\mathbf{R}^\star = \operatorname*{\arg\!\min}_{\mathbf{R} \in SO(3)} \sum_{i=1}^N w_i || \mathbf{v}^a_i -\mathbf{R} \mathbf{v}^b_i ||^2$$
where $\mathbf{R}^\star$ is the sought optimal rotation matrix of $b$ with respect to $a$, $\mathbf{v}^a_i$ and $\mathbf{v}^b_i$ are the unit vector observations, taken in frames of reference $a$ and $b$, respectively, and $w_i$ are the relative scalar weights for each observation. If weights are identified as the inverse of each observation variance, the problem becomes a maximum likelihood estimator.
For a survey of solutions to Eq.(\[eq:wahba\]) that have been proposed in the literature over the years, please refer to [@markley1999estimate; @markley2000quaternion].
The present work focuses on a the Optimal Linear Attitude Estimator (OLAE), as introduced in [@mortari2007olae]. OLAE allows the global estimation (noniterative, without any initial guess) of the optimal relative attitude between a set of unit vector observations. In a sense, it is *similar* to Wahba’s problem (becoming equivalent only in the limit case of zero noise), but with a different target cost function and using a Caley-Gibbs-Rodrigues rotation vector parameterization ([@bauchau2003vectorial]). This formulation leads to the formation of a linear system, with a strictly quadratic cost function:
$$\label{eq:olae.org}
\mathbf{M}_m \mathbf{g} = -\mathbf{z}$$
with $\mathbf{g}$ the sought Gibbs vector. Refer to Eq.(25) in [@mortari2007olae] for the expression of $\mathbf{M}_m$; the expression for $\mathbf{z}$ will be described in §\[sect:olae\].
Recently, [@lourakis2018efficient] proposed to use OLAE as an alternative to Horn’s method for determining the orientation inside the general loop of ICP, but that work did not include any way to also handle planes or lines in the matching process.
Proposed approach {#sect:proposed}
=================
In the following we provide details on how to build an ICP system, capable of handling point, line, and plane correspondences. At the core of the ICP algorithm we are free to choose any optimal transformation algorithm, hence we will describe three of them first: the well-known Horn’s optimal quaternion solution in §\[sect:horn\], the OLAE in §\[sect:olae\], and an iterative, Gauss-Newton method in §\[sect:gauss-newton\] as baseline.
Unifying primitives for Horn’s optimal solution {#sect:horn}
-----------------------------------------------
Let $A=\{\mathbf{a}_i\}_{i=1}^{N_a}$ and $B=\{\mathbf{b}_i\}_{i=1}^{N_b}$ be the sets of geometric primitives as observed from frames $\mathbf{x}_A$ and $\mathbf{x}_B$, respectively. At this point we will assume that correspondences have been already being established between the two sets, hence the $i-th$ element of $A$ is believed to correspond to the $i-th$ element of $B$, so we let $N=N_a=N_b$ be the total number of features for simplicity of notation.
Horn’s method accepts as input two sets of points, hence it is not directly suitable to handle other kind of geometric primitives. However, a key observation is that [@horn1987closed] actually estimates the transformation in three steps: first the rotation, then the scale, and finally the translation. Rotation is decoupled by means of redefining points in *local coordinates* with respect to their corresponding point cloud *centroid*. Therefore, the core of the Horn’s method actually takes as input two sets of *vectors*, $A_H=\{\mathbf{v}^a_i\}_{i=1}^{N}$ and $B_H=\{\mathbf{v}^b_i\}_{i=1}^{N}$, which are always assumed to represent *local coordinates* of points in a point cloud.
However, nothing prevents us to include additional vectors into these sets, with any other geometric meaning. In particular, we propose to build the sets of vectors $A_H$ and $B_H$ from the set of geometric primitives $A$ and $B$ as follows:
1. [If the $i$-th pair $(\mathbf{a}_i,\mathbf{b}_i)$ corresponds to a pair of points, we follow the standard procedure ([@horn1987closed]) and define: $$\begin{aligned}
\mathbf{v}^a_i &=& \mathbf{a}_i - \bar{\mathbf{c}}_a \\
\mathbf{v}^b_i &=& \mathbf{b}_i - \bar{\mathbf{c}}_b
\end{aligned}$$ where $\bar{\mathbf{c}}_{a,b}$ are the weighted centroids of all point features in $A$ and $B$. Due to the importance of the centroid in this method, we propose to remove those pairs that can be clearly classified as outliers, based on a scale mismatch detector. Elaborating on this later idea, please note that, if all pairings were inliers, and under the assumption of zero-mean additive random noise, it should be fulfilled that $E[|\mathbf{v}^a_i|]-E[|\mathbf{v}^b_i|]=0, \forall i$, with $E[\cdot]$ the mathematical expectation. Equivalently, we could write this condition as $E[|\mathbf{v}^a_i|]/E[|\mathbf{v}^b_i|]=1$, leading to the following test for early classification of pairings as outliers: $$\begin{aligned}
\label{eq:outlier.test}
\frac{\max(|\mathbf{v}^a_i|,|\mathbf{v}^b_i|)}{\min(|\mathbf{v}^a_i|,|\mathbf{v}^b_i|)} - 1 < s_t
\end{aligned}$$ with $s_t$ being the scale outlier detection threshold. Reasonable values for this threshold are in the range $(0.1,1.0)$. Smaller values tend to discard good pairings, while larger ones will only filter the most severe and obvious outliers. If the noise distribution of points *and* outliers is known, its value can be precisely determined to fulfill with a predetermined level of confidence. A remark on how to make the detection of outliers more robust is given in the discussion of future works in §\[sec:conclusion\]. ]{}
2. [If the $i$-th pair $(\mathbf{a}_i,\mathbf{b}_i)$ corresponds to a pair of lines, we define $\mathbf{v}^a_i$ and $\mathbf{v}^b_i$ as the *unit* director vectors of the corresponding lines. To solve the direction ambiguity of the director vector (i.e. any point and both $\mathbf{v}$ and $-\mathbf{v}$ define the same line), a criterion must be taken regarding the relative orientation of lines when observed from the sensor. For example, we could arbitrarily impose that director vectors should make an angle smaller than $\pi$ radians with respect to the unit vector from the sensor to the closest point of the line with respect to the sensor. ]{}
3. [If the $i$-th pair $(\mathbf{a}_i,\mathbf{b}_i)$ corresponds to a pair of planes, we define $\mathbf{v}^a_i$ and $\mathbf{v}^b_i$ as the *unit* normal vectors of the corresponding planes. Again, ambiguity should be addressed to enforce that the same plane, when observed from two different view points, has a vector that points in the same direction. A natural choice in this case is to select the outwards-pointing direction of measured surfaces. ]{}
Once we have all the vectors stacked into the lists $A_H$ and $B_H$, including their pairwise relative weights $w_i$, we follow the standard method described in [@horn1987closed] to recover the optimal SO(3) transformation, then use the point cloud centroids (excluding outliers) to solve for optimal translation.
Unifying primitives for OLAE {#sect:olae}
----------------------------
Just like in the former section, we start with the input sets $A=\{\mathbf{a}_i\}_{i=1}^{N}$ and $B=\{\mathbf{b}_i\}_{i=1}^{N}$. Conversely to Horn’s method, vector-based attitude estimator as OLAE or any other solution to the Wahba’s problem, accepts two sets of *unit vectors*, $A_O$ and $B_O$, as input. Again, we propose to include other entities, like points, by converting them appropriately. We can build the sets of unit vectors $A_O=\{\mathbf{v}^a_i\}_{i=1}^{N}$ and $B_O=\{\mathbf{v}^b_i\}_{i=1}^{N}$ from the set of geometric primitives $A$ and $B$ as follows:
1. [For pairs of points $(\mathbf{a}_i,\mathbf{b}_i)$, we start following the same procedure than described for the Horn’s method above, then we *normalize* the local coordinates of the points with respect to the centroids $\bar{\mathbf{c}}_{a,b}$, that is: $$\begin{aligned}
\label{eq:olae.vec.norm}
\bar{\mathbf{v}}^a_i &=& \mathbf{a}_i - \bar{\mathbf{c}}_a \quad \quad \mathbf{v}^a_i = \frac{\bar{\mathbf{v}}^a_i}{|\bar{\mathbf{v}}^a_i|} \\
\bar{\mathbf{v}}^b_i &=& \mathbf{b}_i - \bar{\mathbf{c}}_b \quad \quad \mathbf{v}^b_i = \frac{\bar{\mathbf{v}}^b_i}{|\bar{\mathbf{v}}^b_i|}
\end{aligned}$$ Note that this simple transformation enables attitude estimation algorithms to also cope with point correspondences. Outliers can be also detected in this case using Eq. (\[eq:outlier.test\]) without modifications. ]{}
2. [For line or plane correspondences, the unit director and normal vectors, respectively, already are unit vectors (the natural input to OLAE), so no further actions are required. ]{}
Once we have all observations stacked into the lists of unit vectors $A_O$ and $B_O$, and we are given a vector of relative weights $w_i$, we can find the $3 \times 3$ attitude profile matrix $\mathbf{B}$ and the $\mathbf{z}$ vector in Eq. (\[eq:olae.org\]) as:
$$\begin{aligned}
\mathbf{B} &=& \sum_{i=1}^N w_i \mathbf{v}^b_i (\mathbf{v}^a_i)^\top \\
\mathbf{v} &=& - \sum_{i=1}^N w_i \mathbf{v}^b_i \times \mathbf{v}^a_i\end{aligned}$$
from which can be approximate $\mathbf{M}_m$ in Eq. (\[eq:olae.org\]) as $\mathbf{M}_m \approx \mathbf{M}_w$ (which is more convenient for reasons that will be clear when dealing with the sequential rotation method), and where $\mathbf{M}_w$ is (see [@mortari2007olae]):
$$\label{eq:olae.Mw}
\mathbf{M}_w = \begin{bmatrix}
S_{11} - p & S_{12} & S_{13} \\
S_{12} & S_{22} - p & S_{23} \\
S_{13} & S_{23} & S_{33} -p
\end{bmatrix}$$
where it has been used:
$$\begin{aligned}
\mathbf{S} &=& \mathbf{B} + \mathbf{B}^\top \\
p&=& tr(B)+1 \\
m&=& tr(B)-1\end{aligned}$$
leading to the linear system of equations:
$$\begin{aligned}
\label{eq:olae.final.Mw}
\mathbf{M}_w &=& \mathbf{g} \mathbf{z}\end{aligned}$$
from which the optimal rotation can be solved for as the Gibbs vector $\mathbf{g} = (g_x, g_y, g_z)$. In order to recover a quaternion $(q_r, q_x,q_y,q_z)$ from $\mathbf{g}$, we can use:
$$\begin{aligned}
q_r &=& \frac{1}{\sqrt{1+g_x^2+g_y^2+g_z^2}} \\
q_x &=& q_r g_x \\
q_y &=& q_r g_y \\
q_z &=& q_r g_z\end{aligned}$$
The only edge case that remains to be dealt with is the singularity of the Gibbs vector representation when representing a rotation of 180 degrees. In fact, the accuracy of the solution of the linear system in Eq. (\[eq:olae.final.Mw\]) may in theory be compromised when working near the singularity. In practice, for reasonable noise levels in the input data, the solution is robust even with rotations of 179 degrees.
Nevertheless, in order to work near the optimal conditions of OLAE, we propose to evaluate *four* different solutions: one for the unmodified linear system in Eq. (\[eq:olae.final.Mw\]), and three for systems that have undergone a *rotation* around each one of the axes (x,y,z). The approximation $\mathbf{M}_m \approx \mathbf{M}_w$ reveals useful here, since the rotated $\mathbf{M}_w$ matrices can be evaluated in closed form from $\mathbf{B}$, avoiding the need to rotate all the input vectors and going through the summation again. This method, originally described in [@shuster1981attitude], can be found in Eqs.(35)-(37) in [@mortari2007olae]
We propose to evaluate the determinant of the matrix of coefficients of the linear system (i.e. $\mathbf{M}_w$ for the unrotated case, $\mathbf{M}_w^{x,y,z}$ for the rotated systems) and use the system with the largest absolute value, since it will ensure that the system has a full rank of 3.
Gauss-Newton iterative solver {#sect:gauss-newton}
-----------------------------
In order to validate the implementation of the multi-primitive optimal transformation algorithms based on the Horn’s method and OLAE, we also implemented a baseline method based on a non-linear, iterative solver. It support the same kind of pairings than the aforementioned methods, plus additional ones, e.g. point-to-plane. The interested reader is referred to the source code for details on the cost functions. Closed-form Jacobians have been found for each kind of pairing by applying the chain rule to the corresponding target function and making use of the expressions in [@blanco2010tutorial] to solve for SE(3) on-manifold increments at each iteration.
Complete ICP algorithm {#sect:complete.icp}
----------------------
Once the optimal transformation estimators have been defined, we can build a complete multi-primitive alignment algorithm by iterating between selecting closest correspondences between the two maps, and finding the optimal transformation that raises from the selected pairings. We use the nearest neighbor criterion for pairing points, using KD-trees for efficiency. Planes are paired by looking for the nearest centroids in the reference map whose normal vector makes an angle below a certain threshold in the map to be aligned. A robust kernel is optionally applied by multiplying the relative weights of each pair with a robust loss function, i.e. $w_i' = w_i f(\mathbf{v}^a_i,\mathbf{v}^b_i)$. The interested reader is referred to the source code for further details.
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Experimental results {#sect:experiments}
====================
Next we describe different numerical simulations aimed at evaluating the algorithms proposed in former sections. The implementation used to obtain these results has been released as open source for the sake of reproducibility.
Sensitivity to noise
--------------------
To benchmark the performance of the implemented methods against noise, sets of points and plane centroids are randomly drawn following an uniformly distributed in the cube (0,0,0)-(50,50,50), and then transformed following a random SE(3) pose which is then estimated from the three methods described in the text above. Points in the transformed map are corrupted with additive Gaussian noise. Plane normals are also rotated following random SO(3) rotations whose rotation angle also follows a Gaussian distribution. Errors are evaluated in two parts: rotation error with respect to the ground truth is measured as the norm of the matrix logarithm of the rotation error, while translation is measured using Euclidean distances. All the experiments have been repeated 1000 times to obtain significant statistics.
Figure \[fig:results.pt100.pl0\] shows the statistical results for 100 paired points, while Figure \[fig:results.pt1.pl100\] shows the results for one point and 100 plane pairs, and finally Figure \[fig:results.pt100.pl100\] illustrates the case for 100 points and 100 planes. No outliers are introduced in this test set.
Absolute outlier rejection {#sect:results.outlier}
--------------------------
Next we evaluate the outlier rejection method based on the scale mismatch threshold, as described by Eq. (\[eq:outlier.test\]). The outlier detector has been integrated into both, OLAE and Horn’s solution, and the error achieved after removing outliers (those that were successfully detected) is shown in Figure \[fig:results.outlier.rejection\]. For comparison, we added the result of the Gauss-Newton and the original Horn’s method, both of them without any outlier filter. Here we used the threshold $s_t=0.2$.
\
Robust loss cost
----------------
When a guess for the solution is available, e.g. in the final refining stages of ICP, when the solution is near convergence, we can enable the additional robust loss weight factor mentioned in §\[sect:complete.icp\] to further mitigate the effect of outliers. Figure \[fig:results.robust.loss\] shows a comparison of OLAE with the robust loss function and the classic Horn’s solution (without loss function).
![Error for different outlier ratios with and without the robust loss function to modify pair weights in the proposed ICP system. Note that this feature is only available when a reasonable initial estimate of the solution is known in advance.[]{data-label="fig:results.robust.loss"}](experiments/results_olae_vs_horn.png){width="0.8\columnwidth"}
Complete ICP system
-------------------
In order to test the complete ICP system described in §\[sect:complete.icp\], we used 3D pointcloud models publicly available in the Stanford’s dataset ([@curless1996volumetric]). In particular, in this section we used the *Bunny* dataset, downsampled to 1000 points, as a reference pointcloud. Then, a transformed pointcloud is generated by translating and rotating the model using a random SE(3) pose with translation in X,Y, and Z, uniformly drawn in the range $[-0.25 b, 0.25 b]$, with $b$ the maximum length of the model bounding box, and with random rotations built from values of yaw, pitch, and roll drawn from a uniform distribution in the range $[-20 deg, +20 deg]$. Note that, while the optimal transformation methods (OLAE, Horn’s) are able to find a global optimal transformation without iterating, the *association* stage required in ICP limits the convergence volume of the state space; that explains the relatively small translations and rotations used in this test. The whole process is repeated 10 times, and we measured the SO(3) rmse, the translation rmse, and CPU time, of the final estimation of the ICP algorithm when using each of the three different algorithms discussed above at its core while solving for optimal transformations. The statistical results can be seen in Figure \[fig:results.icp\].
Discussion and conclusions {#sec:conclusion}
==========================
We have presented a methodology to allow point, line, and plane features to be integrated into Horn’s method and into OLAE, which originally only supported point and vector observations, respectively.
It is worth mentioning that some information is lost due to the vector normalization stage in Eq. (\[eq:olae.vec.norm\]) when using OLAE (but not in Horn’s method), and this has an impact in the solution accuracy: when most correspondences are points, attitude-base methods (i.e. OLAE) performed slightly worse than Horn’s method, which in turn considers the full scale of relative point coordinates. From the statistical results, it can be concluded that OLAE performs identical to Horn’s method when most features are vector-like (i.e. planes or lines). OLAE is faster than Horn’s method only when working with a reduced number of correspondences (roughly less than 10 primitive pairings), thus it should be the preferred choice only for applications that require the fast evaluation of small sets, e.g. inside a RANSAC loop.
Future works include methods to avoid relying on a centroid. This has revealed as a weak point of the Horn’s classic solution, which may be mitigated by using relative vectors for different graph topologies, as recently proposed in [@yang2019polynomial].
[15]{} \[1\][\#1]{} \[1\][`#1`]{} urlstyle \[1\][doi: \#1]{}
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F Landis Markley and Daniele Mortari. How to estimate attitude from vector observations. In *AAS/AIAA Astrodynamics Specialist Conference*, 1999.
F Landis Markley and Daniele Mortari. Quaternion attitude estimation using vector observations. *Journal of the Astronautical Sciences*, 480 (2):0 359–380, 2000.
Daniele Mortari, F Landis Markley, and Puneet Singla. Optimal linear attitude estimator. *Journal of Guidance, Control, and Dynamics*, 300 (6):0 1619–1627, 2007.
Olivier A Bauchau and Lorenzo Trainelli. The vectorial parameterization of rotation. *Nonlinear dynamics*, 320 (1):0 71–92, 2003.
Manolis Lourakis and George Terzakis. Efficient absolute orientation revisited. In *IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)*, pages 5813–5818. IEEE, 2018. URL <https://ieeexplore.ieee.org/abstract/document/8594296/>.
M.D. Shuster and S.D. Oh. Three-axis attitude determination from vector observations. *Journal of Guidance, Control, and Dynamics*, 40 (1):0 70–77, 1981.
Jose-Luis Blanco. A tutorial on se(3) transformation parameterizations and on-manifold optimization. *University of Malaga, Tech. Rep*, 3, 2010.
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[^1]: Department of Engineering, University of Almeria (04120), Spain, `jlblanco@ual.es`
[^2]: This technical report extends the preliminary description of the method mentioned in ‘A Modular Optimization Framework for Localization and Mapping’ on the 2019 ‘Robotics: Science and Systems (RSS 2019)’ conference.
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---
abstract: 'Magnetic particle imaging (MPI) is a tracer-based technique for medical imaging where the tracer consists of ironoxide nanoparticles. The key idea is to measure the particle response to a temporally changing external magnetic field to compute the spatial concentration of the tracer inside the object. A decent mathematical model demands for a data-driven computation of the system function which does not only describe the measurement geometry but also encodes the interaction of the particles with the external magnetic field. The physical model of this interaction is given by the Landau-Lifshitz-Gilbert (LLG) equation. The determination of the system function can be seen as an inverse problem of its own which can be interpreted as a calibration problem for MPI. In this contribution the calibration problem is formulated as an inverse parameter identification problem for the LLG equation. We give a detailed analysis of the direct as well as the inverse problem in an all-at-once as well as in a reduced setting. The analytical results yield a deeper understanding of inverse problems connected to the LLG equation and provide a starting point for the development of robust numerical solution methods in MPI.'
author:
- 'Barbara Kaltenbacher[^1]'
- 'Tram Thi Ngoc Nguyen[^2]'
- 'Anne Wald[^3]'
- 'Thomas Schuster[^4]'
bibliography:
- 'bibliography\_llg.bib'
title: 'Parameter identification for the Landau-Lifshitz-Gilbert equation in Magnetic Particle Imaging'
---
magnetic particle imaging, time-dependent inverse problems, parameter identification, Landau-Lifshitz-Gilbert equation, all-at-once formulation
**Acknowledgements:** The work of Anne Wald and Thomas Schuster was partly funded by Hermann und Dr. Charlotte Deutsch-Stiftung and by the German Federal Ministry of Education and Research (Bundesministerium für Bildung und Forschung, BMBF) under 05M16TSA.
[^1]: Alpen-Adria-Universität Klagenfurt, Austria ([barbara.kaltenbacher@aau.at]{})
[^2]: Alpen-Adria-Universität Klagenfurt, Austria ([tram.nguyen@aau.at]{})
[^3]: Department of Mathematics, Saarland University, PO Box 15 11 50, 66123 Saarbrücken, Germany ([anne.wald@num.uni-sb.de]{})
[^4]: Department of Mathematics, Saarland University, PO Box 15 11 50, 66123 Saarbrücken, Germany ([thomas.schuster@num.uni-sb.de]{})
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-CONF-[03]{}/012\
hep-ex/[0000]{}\
July 2003\
**Evidence for the Rare Decay $B\rightarrow \bf{J/\psi \eta K}$**
The Collaboration\
\
**Abstract**
abstract.tex
Presented at the International Europhysics Conference On High-Energy Physics (HEP 2003), 7/17—7/23/2003, Aachen, Germany
[*Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309*]{}\
------------------------------------------------------------------------
Work supported in part by Department of Energy contract DE-AC03-76SF00515.
authors\_eps2003.tex
base.tex
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