---
abstract: 'We investigate an impact of the axial-vector interaction on spatial modulation of quark matter. A magnetic field coupled with baryon density leads to a topological axial current, so that the effect of the axial-vector interaction is crucially enhanced then. Using the Sakai-Sugimoto model we have found that, contrary to a naïve expectation, the spatially modulated phase is less favored for a stronger magnetic field, which is realized by the presence of topological current.'
author:
- Kenji Fukushima
- 'Pablo A. Morales'
title: Spatial modulation and topological current in holographic QCD matter
---

#### Introduction

The phase diagram of hot and dense matter out of quarks and gluons has not been clarified satisfactorily based on the first-principle theory of the strong interaction, i.e. quantum chromodynamics (QCD). The most severe obstacle is the notorious sign problem of the Dirac determinant at finite quark density $\rho$ or chemical potential $\mu$, which prevents us from the direct application of the Monte-Carlo simulation in the region with $\mu\gtrsim
T$ [@Fukushima:2010bq].

Instead of the lattice simulation, one could have deduced possible phase structures using chiral effective models, see e.g. [@Herbst:2013ail] for a recent work. It is conjectured from model studies that the chiral phase transition might be of first order at high density, so that a second-order critical point called the QCD critical point [@Asakawa:1989bq] could appear on the phase diagram, the discovery of which is one of the major goals of the beam-energy scan program in heavy-ion collision experiments [@Kumar:2012fb]. The model setup, however, suffers from uncontrolled uncertainties and the QCD critical point is a model-dependent prediction. It is well understood by now that the vector-type interaction $\sim (\bar{\psi}\gamma_\mu\psi)^2$, which gives rise to the density-density interaction $\sim \rho^2$ even in the mean-field level, crucially affects the liquid-gas phase transition of dense quark matter [@Kitazawa:2002bc; @Fukushima:2012mz] (see also [@Contrera:2012wj]). Moreover, nowadays, spatially inhomogeneous states are becoming a more and more realistic candidate that may supersede the conventional first-order phase boundary [@Deryagin:1992rw], which is rather robust against the vector interaction [@Fukushima:2012mz; @Carignano:2010ac].

The simplest Ansatz to introduce spatial modulation is the chiral spiral or the dual chiral-density wave, $$\langle\bar{\psi}\psi\rangle = \Delta\cos({\boldsymbol{k}}\cdot{\boldsymbol{x}})\;,\quad
 \langle\bar{\psi}\gamma_5\tau^3\psi\rangle
  = \Delta\sin({\boldsymbol{k}}\cdot{\boldsymbol{x}})\;,$$ which is reminiscent of the $p$-wave $\pi^0$ condensate in symmetric nuclear matter. Recalling the history of the pion condensation [@pion], one may well consider that a spin-isospin short-range interaction could significantly diminish the reality of chiral spirals; it was indeed the case for the pion condensation that is disfavored by the so-called Landau-Migdal parameters $g'$ associated with short-range effective interaction in Fermi liquid theory (see also [@Tatsumi:2003fa] for some arguments in favor of the pion condensation). In the relativistic language, thus, it is conceivable that the axial-vector interaction $\sim (\bar{\psi}\gamma_5 \gamma_\mu \boldsymbol{\tau}\psi)^2$ may be influential on spatial modulation of quark matter, though the vector interaction is not. This is an important question but, to the best of our knowledge, there is no theoretical investigation on this issue. The difficulty lies in the fact that the axial-vector has no mean-field contribution unlike the density in the vector channel, and therefore one should go beyond the mean-field approximation. So far, the renormalization-group improvement has been successful for the homogeneous states only [@Herbst:2013ail].

This situation would be drastically changed if we turn an external magnetic field $B$ on. Such a system of dense quark matter at strong magnetic field has been intensely investigated. It was pointed out first in the Sakai-Sugimoto model [@Sakai:2004cn] which is a holographic dual of large-${N_{\text{c}}}$ QCD that $B$ lowers the critical $\mu$ [@Preis:2010cq]. This observation turns out to be generic in chiral models [@Preis:2012fh] and is often referred to as the inverse magnetic catalysis in contrast to the enhancement of chiral symmetry breaking at zero density [@Gusynin:1994re]. In this way, clarification of the QCD phase diagram along larger-$B$ direction is an intriguing subject and many studies have been devoted to it [@Gatto:2012sp].

There are also some theoretical works focused on inhomogeneous states of dense quark matter at finite $B$: In the strong-$B$ limit quarks are dimensionally reduced into a (1+1)-dimensional system, so that the ground-state structure should be a chiral spiral, i.e. chiral magnetic spiral [@Basar:2010zd]. It is also possible that another spiral can develop due to the presence of $B$ [@Ferrer:2012zq]. In view of such results, it should be a natural expectation that a stronger $B$ may ease a barrier to form spirals.

Here, in this work, we would address one important physical effect that has been overlooked in these preceding works. That is, the inevitable generation of the topological current, $${\boldsymbol{j}}_A = {N_{\text{c}}}\sum_f \frac{q_f^2 \mu}{2\pi^2}{\boldsymbol{B}}\;,
\label{eq:cse}$$ having the origin in quantum anomaly [@Metlitski:2005pr], should be incorporated. ${N_{\text{c}}}$ is the number of color, $f$ runs over flavor degrees of freedom, and $q_f$ is the electric charge of flavor $f$. Interestingly, if ${\boldsymbol{j}}_A\neq0$ at finite $\mu$ and $B$, the axial-vector interaction has a mean-field contribution ${\boldsymbol{j}}_A^2$ in the same way as $\rho^2$ emerging from the vector interaction, which could have played a role similar to the Landau-Migdal interaction and thus disfavored spirals contrary to the naïve expectation. Although there are countless works to study such chiral magnetic and separation effects as in Eq. , nobody has ever considered its impact on the phase structure at finite $\mu$ and $B$.

For the purpose to address these issues, the Sakai-Sugimoto model suits the best. We could use conventional methods, but then it is difficult to quantify the axial-vector interaction. There is no such ambiguity in the holographic approach. Besides, the holographic technique for the phase diagram research has been successfully advanced recently and the instability toward spatially modulated phase has been discovered [@Ooguri:2010xs]. In the presence of chiral chemical potential, also, similar instability leading to a spiral has been identified in the Sakai-Sugimoto model [@BallonBayona:2012wx].

#### Holographic Description

The gauge/gravity (or generally bulk/boundary) correspondence states that the full quantum generating functional of 4-dimensional field theory is equivalent to the on-shell action of the gravity theory with corresponding source at the ultraviolet (UV) boundary. Thus, ${N_{\text{c}}}$ D4 branes compactified along the $x_4$-direction represent the gluonic degrees of freedom [@Witten:1998zw] and ${N_{\text{f}}}$ ${\rm D8}$-$\overline{\rm D8}$ branes realize the spontaneous breaking of ${\rm U}({N_{\text{f}}})_{\rm L}\times{\rm U}({N_{\text{f}}})_{\rm R}$ chiral symmetry in QCD [@Sakai:2004cn]. In the same way as in the first paper of [@Ooguri:2010xs] we focus on the situation where ${\rm D8}$ and $\overline{\rm D8}$ are separate above the deconfinement transition. There, the induced-metric on the flavor branes is, $$ds^2 = u^{3/2}\bigl[ f(u) d\tau^2 + d{\boldsymbol{x}}^2 \bigl]
  + \Bigl[ u^{3/2} x_4'(u)^2 + \frac{1}{u^{3/2} f(u)} \Bigr] du^2 \;,
\label{eq:metric}$$ where $f(u)=1-u_T^3/u^3$. We note that all variables are made dimensionless by the AdS radius. The horizon at $u=u_T$ defines the Hawking temperature, which is translated to the physical temperature as $T = 3u_T^{1/2}/(4\pi)$. In the chiral symmetric phase D8 and $\overline{\rm D8}$ are simply straight, so that ${x_4'(u)}=0$ is chosen.

Then, the DBI action in the flavor sector can be expressed with the metric from Eq.  and the ${\rm U}(1)$ field strength tensor $F_{\alpha\beta}$ which is split into $B$ in the $z$-direction (under simplification that all ${N_{\text{f}}}$ flavors have the same electric charge), the background ${\bar{a}}_0$ and ${\bar{a}}_z$ corresponding to $\mu$ and $j_A^z$, and spatially inhomogeneous fluctuations $f_{\alpha\beta}$. The 5-dimensional effective action reads, $$\begin{split}
 S_{\text{D8}}^{\text{DBI}} &= \mathcal{N}\int d\tau\,d^3 x\,du\,
  u^{1/4}\sqrt{-\det(g_{\alpha\beta}+F_{\alpha\beta})} \\
  &= {\mathcal{N}}\int d\tau\, d^3 x\, du\, u^{5/2}
  \sqrt{{\mathcal{A}}\cdot{\mathcal{B}}}\;(1+{\mathcal{X}})
 \end{split}$$ with an overall (irrelevant) constant ${\mathcal{N}}$ and $${\mathcal{A}}= 1 - {\bar{a}}_0'(u)^2 + f(u) {\bar{a}}_z'(u)^2 \;, \quad
 {\mathcal{B}}= 1 + B^2 u^{-3} \;,
\label{eq:A_B}$$ and the fluctuation part ${\mathcal{X}}$ up to the quadratic order with respect to $f_{xy}=\partial_x a_y-\partial_y a_x$, and $f_{yz}$, $f_{zx}$, $f_{ux}$, $f_{uy}$, $f_{uz}$ with similar definitions.

Hence, together with the Chern-Simons action, $S^{\text{CS}}=(\mathcal{N}/8)\int d\tau\,d^3 x\,du\,
\epsilon^{\mu_1\mu_2\mu_3\mu_4\mu_5}A_{\mu_1}F_{\mu_2\mu_3}F_{\mu_4\mu_5}$, we can define variables conjugate to ${\bar{a}}_0'$ and ${\bar{a}}_z'$ using the full action $S=S_{\text{D8}}^{\text{DBI}}+S^{\text{CS}}$, as $$\begin{aligned}
 \rho &= -\frac{\delta S}{\delta {\bar{a}}_0'(u)} = u^{5/2} {\bar{a}}_0'(u)
 \sqrt{\frac{{\mathcal{B}}}{{\mathcal{A}}}} - 3 B {\bar{a}}_z(u) \;,
\label{eq:rho}\\
 b &= \frac{\delta S}{\delta {\bar{a}}_z'(u)} = u^{5/2} f(u) {\bar{a}}_z'(u)
  \sqrt{\frac{{\mathcal{B}}}{{\mathcal{A}}}} - 3 B {\bar{a}}_0(u) \;.
\label{eq:b}\end{aligned}$$ Because $S$ is dependent on not ${\bar{a}}_0$ and ${\bar{a}}_z$ but ${\bar{a}}_0'$ and ${\bar{a}}_z'$ only, $\rho$ and $b$ fixed from the equations of motion are $u$-independent. We find $b=0$ by evaluating it at $u=u_T$, and from the boundary condition ${\bar{a}}_0(\infty)=\mu$, we can get the asymptotic forms as $${\bar{a}}_z(u) \simeq -2\mu B u^{-3/2} \;,\quad
 {\bar{a}}_0(u) \simeq \mu-\frac{9}{8}\rho u^{-3/2} \;,$$ near the UV boundary ($u\sim\infty$). This asymptotic behavior of ${\bar{a}}_z(u)$ represents the topological vector and axial-vector currents  [@Preis:2010cq; @Preis:2012fh; @BallonBayona:2012wx; @Yee:2009vw]. In our numerical calculations we fully solve Eqs.  and for a given density $\rho$ to obtain the whole profile of ${\bar{a}}_0(u)$ and ${\bar{a}}_z(u)$.

From the concrete form of ${\mathcal{X}}$ we can get the equations of motion with respect to fluctuations $a_i$ ($i=x,y,z$) as $$\begin{aligned}
 & u^{-1/2}\!\sqrt{\frac{{\mathcal{A}}}{{\mathcal{B}}}} \Bigl(
  \frac{\partial_y f_{yx}}{{\mathcal{B}}} \!+\! {\mathcal{C}}\partial_z f_{zx} \Bigr)
  + \partial_u\! \Bigl[ \frac{u^{5/2} f(u) f_{ux}}{\sqrt{{\mathcal{A}}\cdot {\mathcal{B}}}} \Bigr] \notag\\
 &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\:\:\:
  + 3 {\bar{a}}_0' f_{yz} = 0 \;,
\label{eq:eom1}\\
 & u^{-1/2}\!\sqrt{\frac{{\mathcal{A}}}{{\mathcal{B}}}} \Bigl(
  \frac{\partial_x f_{xy}}{{\mathcal{B}}} \!+\! {\mathcal{C}}\partial_z f_{zy} \Bigr)
  + \partial_u\! \Bigl[ \frac{u^{5/2} f(u) f_{uy}}{\sqrt{{\mathcal{A}}\cdot {\mathcal{B}}}} \Bigr] \notag\\
 &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\:
  + 3 {\bar{a}}_0' f_{zx} = 0 \;,
\label{eq:eom2}\\
 & u^{-1/2}\!\sqrt{\frac{{\mathcal{A}}}{{\mathcal{B}}}}\, {\mathcal{C}}\bigl(
  \partial_x f_{xz} + \partial_y f_{yz} \bigr) + \partial_u
  \Bigl[ u^{5/2}\!\sqrt{\frac{{\mathcal{B}}}{{\mathcal{A}}}}\,{\mathcal{C}}f(u) f_{uz}
  \Bigr] \notag\\
 &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\:
  + 3 {\bar{a}}_0' f_{xy} = 0 \;,
\label{eq:eom3}\end{aligned}$$ where ${\mathcal{C}}= 1 - f(u) {\bar{a}}_z'(u)^2 / {\mathcal{A}}$.

![Smallest eigenvalue of the matrix ${\mathcal{M}}$ as a function of $k_x$ (perpendicular to $B$) and $k_z$ (parallel to $B$) at $\rho=3.72u_T^{5/2}$ at $B=0$ (surface in the middle) and at $B=u_T^{3/2}$ with ${\bar{a}}_z$ (surface in the top) and without ${\bar{a}}_z$ (surface in the bottom).[]{data-label="fig:critical"}](critical){width="0.95\columnwidth"}

#### Numerical Results

A finite $B$ breaks rotational symmetry and we cannot find the eigenmodes as done in [@Ooguri:2010xs]. Let us here explain how to proceed to numerical analyses. Our goal is to locate the critical $\rho$ or $\mu$ (denoted by $\mu_c$ hereafter) at which Eqs. , , and have normalizable solutions with some momenta $k_x$, $k_y$, $k_z$ in Fourier space. In fact the normalizability condition or the boundary conditions $a_i(\infty)\to0$ dictate how the energy dispersion relations behave. Since we drop time dependence, our solutions describe the dispersion relation at zero energy. If a zero-energy excitation is realized with non-zero momenta, a homogeneous state should become unstable.

To solve three differential equations for $a_i$ from $u=u_T$ to $u=\infty$, we need to specify the initial condition for $a_i'(u_T)$. These are uniquely taken if we require the solutions to be non-singular at $u=u_T$; since $f(u)$ vanishes at $u=u_T$, only the term with $\partial_u$ acting on $f(u)$ remains non-zero unless $a_i''(u_T)$ is singular. Then, we can easily express $a_i'(u_T)$ using $a_i(u_T)$. For example, we can deduce $a_x'(u_T)$ from Eq.  as $$\begin{aligned}
 a_x'(u_T) &= \frac{{\mathcal{A}}}{3u_T^2}\Bigl[\frac{k_y^2 a_x
  -k_x k_y a_y}{{\mathcal{B}}} + {\mathcal{C}}\bigl( k_z^2 a_x - k_z k_x a_z\bigr)
  \Bigr] \notag\\
 &\qquad\qquad
  -i \sqrt{{\mathcal{A}}\cdot{\mathcal{B}}}\, {\bar{a}}_0' u_T^{-3/2}(k_y a_z - k_z a_y) \;,\end{aligned}$$ as well as $a_y'(u_T)$ and $a_z'(u_T)$ similarly.

Now, we are ready for solving Eqs. , , and numerically, and the final values $a_i(\infty)$ are then given as functions of the initial values $a_i(u_T)$, which can be expressed, thanks to the linearity, as follows; $$\begin{pmatrix} a_x(\infty) \\ a_y(\infty) \\ a_z(\infty)
 \end{pmatrix}
 = {\mathcal{M}}\begin{pmatrix} a_x(u_T) \\ a_y(u_T) \\ a_z(u_T)
  \end{pmatrix} \;,$$ where ${\mathcal{M}}$ is a $3\times3$ matrix, having three eigenvalues. If an eigenvalue turns out to be vanishing at some momenta, the initial condition set with the corresponding eigenvector leads to the desired boundary conditions, $a_x(\infty)=a_y(\infty)=a_z(\infty)=0$.

Figure \[fig:critical\] shows the smallest eigenvalue of ${\mathcal{M}}$ as a function of $k_x$ and $k_z$ (we can set $k_y=0$ without loss of generality). We can get rid of $u_T$-dependence by rescaling $\rho$, $\mu$, $B$, and $k_i$. We find that $\rho=3.72u_T^{5/2}$ is the critical value for $B=0$ at which the smallest eigenvalue touches zero at $|\boldsymbol{k}|=2.3u_T^{1/2}$ (which confirms [@Ooguri:2010xs]). When we increase $B$, the smallest eigenvalue is pushed up as depicted by the upper surface in Fig. \[fig:critical\], and thus the critical density should get larger. This means that a larger $B$ disfavors the spatially modulated phase. Though it is not visually clear from Fig. \[fig:critical\], the eigenvalue is slightly tilted in the presence of $B$ and the minimum of the eigenvalues is located on $k_x\neq0$ and $k_z=0$.

In terms of the chemical potential the relation between $\mu_c$ and $B$ is more complicated. As seen by the solid curve in Fig. \[fig:changeB\] $\mu_c$ rather goes down with increasing $B$ as long as the magnetic field is small enough, $B/u_T^{3/2}\lesssim 1$, even though the critical $\rho$ monotonically grows up. This is simply because the phase space is enhanced by $B$; if $B$ is raised up for a fixed $\mu$, the corresponding density $\rho$ becomes larger.

![Critical chemical potential $\mu_c$ as a function of $B$. The upper solid curve represents the result with ${\bar{a}}_z$ taken into account and the lower dashed line represents the result without ${\bar{a}}_z$.[]{data-label="fig:changeB"}](changeB){width="0.85\columnwidth"}

#### Discussions

It could have been more intuitively understandable if $B$ favored more modulation in view of the chiral magnetic spirals at $B\to\infty$. Here, in order to think of the effect of the topological current , let us drop ${\bar{a}}_z(u)$ off from the calculation. Of course, ${\bar{a}}_z(u)=0$ is *not* a solution of the equation of motion, but this artificial manipulation in the present holographic treatment can mimic the common approximation to neglect ${\boldsymbol{j}}_A$ in most non-holographic calculations.

In this case without ${\bar{a}}_z$ we find that the smallest eigenvalue is significantly pushed down by $B$ as seen in the bottom surface in Fig. \[fig:critical\]. This indicates that the critical density is lowered by $B$ which makes a sharp contrast to the case with ${\bar{a}}_z$ (and thus ${\boldsymbol{j}}_A$). Needless to say, the critical chemical potential $\mu_c$ also exhibits an opposite behavior to the previous case with ${\bar{a}}_z$, which is evident from the dashed curve in Fig. \[fig:changeB\].

In the holographic approach, generally, it is hard to carve distinct physical effects out from the final results, and we did not spell out the axial-vector interaction $\sim (\bar{\psi}\gamma_5 \gamma_\mu \boldsymbol{\tau}\psi)^2$. Nevertheless, our finding based on the comparison with and without ${\bar{a}}_z$ is suggestive enough to demonstrate the importance of the axial-vector interaction along the same direction as the Landau-Migdal interaction disfavoring the $p$-wave pion condensation. It is an intriguing future problem to implement the axial-vector interaction in conventional chiral models such as the (Polyakov-loop coupled) Nambu–Jona-Lasinio model and the quark-meson model to confirm our finding and elucidate more microscopic dynamics. In fact, in these chiral models, ${\boldsymbol{j}}_A$ should be treated as a mean-field variable and ${\boldsymbol{j}}_A$ is “renormalized” then [@Fukushima:2010zza]. Similar corrections on the topological current are reported also with explicit QED calculations [@Gorbar:2013upa].

![Phase boundaries of the onset of the spatially modulated phase at $B=0$ (solid curve), $B=0.5$ (dashed curve), and $B=1.5$ (dotted curve) in the unit of not $u_T$ but the AdS radius. For reference the phase boundary for the homogeneous chiral transition [@Bergman:2007wp] is also shown.[]{data-label="fig:boundary"}](boundary){width="0.85\columnwidth"}

#### Summary

We calculated the critical density and the critical chemical potential $\mu_c$ for spatial modulation at finite $B$. We found that the spatial modulation is disfavored for a larger $B$, which becomes manifest on the phase diagram as summarized in Fig. \[fig:boundary\]. When $B=0$, we can find $\mu_c\simeq 1.59u_T=27.9T^2$ that draws a solid curve in Fig. \[fig:boundary\] (as seen in [@Ooguri:2010xs]). This phase boundary is shifted toward larger $\mu$ with increasing $B$, so that a stronger $B$ causes shrinkage of the region with spatial inhomogeneity on the phase diagram. The effect of $B$ appears tamed at higher $T$, which can be explained from Eq.  in which $B^2/u_T^3$ becomes negligible for high $T$ and thus large $u_T$. By comparing the results with and without the background ${\bar{a}}_z(u)$, we conclude that the disfavor of spatially modulated phase at finite $B$ is attributed to the topological currents and presumably the axial-vector interaction strengthened by ${\boldsymbol{j}}_A$.

We are now making progress to explore the whole structure of the holographic QCD phase diagram at finite $T$, $\mu$, and $B$ including the effect of spontaneous chiral-symmetry breaking and baryon density source that both make $x_4(u)$ take a non-trivial shape. This will be reported elsewhere.

We thank S. Nakamura, I. Shovkovy, and W. Weise for useful discussions. K. F. was supported by JSPS KAKENHI Grant \# 24740169.

[99]{}

For reviews, see; S. Muroya, A. Nakamura, C. Nonaka and T. Takaishi, Prog. Theor. Phys.  [**110**]{}, 615 (2003); K. Fukushima and T. Hatsuda, Rept. Prog. Phys.  [**74**]{}, 014001 (2011). T. K. Herbst, J. M. Pawlowski and B. -J. Schaefer, arXiv:1302.1426 \[hep-ph\].

M. Asakawa and K. Yazaki, Nucl. Phys. A [**504**]{}, 668 (1989); A. Barducci, R. Casalbuoni, S. De Curtis, R. Gatto and G. Pettini, Phys. Lett.  B [**231**]{}, 463 (1989); A. Barducci, R. Casalbuoni, G. Pettini and R. Gatto, Phys. Rev.  D [**49**]{}, 426 (1994).

L. Kumar \[STAR Collaboration\], arXiv:1211.1350 \[nucl-ex\]; J. T. Mitchell \[PHENIX Collaboration\], arXiv:1211.6139 \[nucl-ex\].

M. Kitazawa, T. Koide, T. Kunihiro and Y. Nemoto, Prog. Theor. Phys.  [**108**]{}, 929 (2002); C. Sasaki, B. Friman and K. Redlich, Phys. Rev. D [**75**]{}, 074013 (2007); K. Fukushima, Phys. Rev. D [**78**]{}, 114019 (2008). K. Fukushima, Phys. Rev. D [**86**]{}, 054002 (2012). G. A. Contrera, A. G. Grunfeld and D. B. Blaschke, arXiv:1207.4890 \[hep-ph\].

D. V. Deryagin, D. Y. .Grigoriev and V. A. Rubakov, Int. J. Mod. Phys. A [**7**]{}, 659 (1992); E. Shuster and D. T. Son, Nucl. Phys. B [**573**]{}, 434 (2000); E. Nakano and T. Tatsumi, Phys. Rev. D [**71**]{}, 114006 (2005); D. Nickel, Phys. Rev. D [**80**]{}, 074025 (2009); T. Kojo, Y. Hidaka, L. McLerran and R. D. Pisarski, Nucl. Phys. A [**843**]{}, 37 (2010). S. Carignano, D. Nickel and M. Buballa, Phys. Rev. D [**82**]{}, 054009 (2010). For a comprehensive collection of reviews, see; R. Tamagaki [*et al.*]{}, in [*Various Phases in High-Density Nuclear Matter and Neutron Stars*]{} (Prog. Theor. Phys. Suppl. [**112**]{} (1993) edited by T. Kunihiro, T. Muto, T. Takatsuka, and T. Tatsumi).

T. Tatsumi, nucl-th/0302009.

T. Sakai and S. Sugimoto, Prog. Theor. Phys.  [**113**]{}, 843 (2005). F. Preis, A. Rebhan and A. Schmitt, JHEP [**1103**]{}, 033 (2011). F. Preis, A. Rebhan and A. Schmitt, Lect. Notes Phys.  [**871**]{}, 51 (2013). V. P. Gusynin, V. A. Miransky and I. A. Shovkovy, Phys. Rev. Lett.  [**73**]{}, 3499 (1994); Phys. Rev. D [**52**]{}, 4718 (1995). For recent reviews, see; R. Gatto and M. Ruggieri, Lect. Notes Phys.  [**871**]{}, 87 (2013); E. S. Fraga, Lect. Notes Phys.  [**871**]{}, 121 (2013). G. Basar, G. V. Dunne and D. E. Kharzeev, Phys. Rev. Lett.  [**104**]{}, 232301 (2010). E. J. Ferrer, V. de la Incera and A. Sanchez, Acta Phys. Polon. Supp.  [**5**]{}, 679 (2012). M. A. Metlitski and A. R. Zhitnitsky, Phys. Rev. D [**72**]{}, 045011 (2005); K. Fukushima, D. E. Kharzeev and H. J. Warringa, Phys. Rev. D [**78**]{}, 074033 (2008); A. Vilenkin, Phys. Rev. D [**22**]{}, 3080 (1980); M. Giovannini and M. E. Shaposhnikov, Phys. Rev. D [**57**]{}, 2186 (1998). H. Ooguri and C. -S. Park, Phys. Rev. Lett.  [**106**]{}, 061601 (2011); W. -y. Chuang, S. -H. Dai, S. Kawamoto, F. -L. Lin and C. -P. Yeh, Phys. Rev. D [**83**]{}, 106003 (2011). A. Ballon-Bayona, K. Peeters and M. Zamaklar, JHEP [**1211**]{}, 164 (2012). E. Witten, Adv. Theor. Math. Phys.  [**2**]{}, 505 (1998). H. -U. Yee, JHEP [**0911**]{}, 085 (2009); A. Rebhan, A. Schmitt and S. A. Stricker, JHEP [**1001**]{}, 026 (2010). K. Fukushima and M. Ruggieri, Phys. Rev. D [**82**]{}, 054001 (2010). E. V. Gorbar, V. A. Miransky, I. A. Shovkovy and X. Wang, arXiv:1304.4606 \[hep-ph\].

O. Bergman, G. Lifschytz and M. Lippert, JHEP [**0711**]{}, 056 (2007).
