# Pion dynamics in a soft-wall AdS-QCD model

Xuanmin Cao1<EMAIL_ADDRESS>Matteo Baggioli2,3<EMAIL_ADDRESS>Hui Liu1
<EMAIL_ADDRESS>Danning Li1<EMAIL_ADDRESS>1Department of physics and
Siyuan Laboratory, Jinan University, Guangzhou 510632, China.  2Wilczek
Quantum Center, School of Physics and Astronomy, Shanghai Jiao Tong
University, Shanghai 200240, China  3Shanghai Research Center for Quantum
Sciences, Shanghai 201315, China

###### Abstract

Pseudo-Goldstone modes appear in many physical systems and display robust
universal features. First, their mass $m$ obeys the so-called Gell-Mann-Oakes-
Renner (GMOR) relation $f^{2}\,m^{2}=H\,\bar{\sigma}$, with $f$ the Goldstone
stiffness, $H$ the explicit breaking scale and $\bar{\sigma}$ the spontaneous
condensate. More recently, it has been shown that their damping $\Omega$ is
constrained to follow the relation $\Omega=m^{2}D_{\varphi}$, where
$D_{\varphi}$ is the Goldstone diffusivity in the purely spontaneous phase.
Pions are the most paradigmatic example of pseudo-Goldstone modes and they are
related to chiral symmetry breaking in QCD. In this work, we consider a
bottom-up soft-wall AdS-QCD model with broken
${\rm{SU}}(2)_{L}\times{\rm{SU}}(2)_{R}$ symmetry and we study the nature of
the associated pseudo-Goldstone modes – the pions. In particular, we perform a
detailed investigation of their dispersion relation in presence of
dissipation, of the role of the explicit breaking induced by the quark masses
and of the dynamics near the critical point. Taking advantage of the
microscopic information provided by the holographic model, we give
quantitative predictions for all the coefficients appearing in the effective
description. In particular, we estimate the finite temperature behavior of the
kinetic parameter $\mathfrak{r^{2}}$ defined as the ration between the
Goldstone diffusivity $D_{\varphi}$ and the pion attenuation constant $D_{A}$.
Interestingly, we observe important deviations from the value
$\mathfrak{r^{2}}=3/4$ computed in chiral perturbation theory in the limit of
zero temperature.

###### Contents

  1. I Introduction
  2. II The damping of pseudo-Goldstone modes
    1. II.1 The case of Pions
  3. III The holographic model
    1. III.1 The Pseudo-Goldstone modes in the soft-wall model
  4. IV Results
    1. IV.1 The chiral condensate and phase transition
    2. IV.2 Pions in the chiral limit
    3. IV.3 Pions as pseudo-Goldstone modes
    4. IV.4 The relaxation rate of Pions
  5. V Outlook
  6. A Comparison between the holographic result for $G_{\varphi\varphi}^{R}$ and the thermal chiral effective field theory predictions

##  I Introduction

Nambu-Goldstone modes (NGMs) are massless excitations related to the
spontaneous symmetry breaking (SSB) of global symmetries [1]. Their appearance
and their properties follow from the Goldstone theorem and its generalizations
[2, 3, 4, 5]. Because of their protected massless nature, NGMs have a strong
impact on the infrared (IR) low energy dynamics and they constitute the
fundamental building blocks for low-energy effective field theory and
hydrodynamics [6, 7, 8, 9].

To provide a more concrete manifestation of these statements, let us consider
two paradigmatic examples. The first one is that of superfluids, phases of
matter which spontaneously break a global ${\rm{U}}(1)$ symmetry [10]. The
low-energy properties of superfluids are described in terms of the
${\rm{U}}(1)$ Goldstone mode [8] which is ultimately responsible for the
propagation of second sound [11] and their infinite DC conductivity as
well.111See [12] for a modern derivation of this statement using higher-form
mixed anomalies. A second, and perhaps even more familiar, example is that of
solids. Solids are physical systems which spontaneously break translational
invariance with their atoms sit at preferred positions, a.k.a. “long-range
order”. Solids exhibit Goldstone modes usually called phonons which correspond
to what you hear knocking your desk.222This identification is slightly
imprecise. For example, transverse sound in solids does not correspond
directly to transverse phonons but rather to a collective mixture of
transverse phonons and transverse momentum fluctuations [7]. As a matter of
fact, Goldstones and the SSB of translations are the ultimate reason for the
rigidity of your desk.
Despite symmetries and NGMs are an old subject, there are still several open
questions in particular when dissipative/open systems or out-of equilibrium
setups are considered, e.g. [13, 14, 15], or non “standard” symmetries are
spontaneously broken, e.g. [16].

In the past decades, Holography, or gauge-gravity duality, has emerged as a
useful complementary tool to study physical systems where standard
perturbative methods fail, with many applications to condensed matter physics
[17, 18], QCD [19], etc. SSB has historically been one of the first
applications of this duality in the form of the so-called holographic
superconductor model [20].333Because of historical reasons, we keep using this
jargon well aware that the model does not describe the physics of
superconductors since the boundary ${\rm{U}}(1)$ symmetry remains global, at
least using standard boundary conditions. The holographic model mimics the
condensation of a charged scalar operator which induces the spontaneous
symmetry breaking of a global ${\rm{U}}(1)$ symmetry – a superfluid phase. The
low-energy dynamics of the dual field theory is in perfect agreement with
relativistic superfluid hydrodynamics [21] and displays, as expected, the
presence of a well-defined Goldstone mode in the spectrum. Similarly, in more
recent years, large effort has been devoted to holographic phases which break
spontaneously translations [7]. In a large class of models, the presence of
propagating Goldstone bosons (phonons)444The same argument of footnote 2
applies here. has been verified, e.g. [22, 23], and their dynamics matched to
relativistic viscoelasticity theory [24, 25]. Interestingly, even more exotic
type of Goldstone modes have been discussed in holography such as type II/B
NGMs [26, 27], out-of-equilibrium NGMs [28] and NGMs of spontaneously broken
higher-form symmetries [29, 30]. The graviton itself has been recently
identified as a Goldstone boson of an exotic biform symmetry [31].

Interestingly, the dynamics of pNGMs displays a certain degree of universality
which was first realized in the context of chiral perturbation theory for
pions [32]. In particular, it was early recognized that their mass $m_{\pi}$
satisfies the so-called Gell-Mann-Oakes-Renner (GMOR) relation which, at zero
temperature, is given by:

$f_{\pi}^{2}m_{\pi}^{2}\,=\,\left(m_{u}+m_{d}\right)\,\bar{\sigma}$ (1)

in terms of the up and down quark masses $m_{u},m_{d}$ which explicitly break
chiral symmetry. $\bar{\sigma}\equiv\langle\bar{\psi}\psi\rangle$ is the
chiral condensate which signals the spontaneous breaking of chiral symmetry
and $f_{\pi}$ is the pion decay constant which, as we will see in detail
later, plays the role of the Goldstone susceptibility. The mass $m_{\pi}$ is
usually denoted as screening mass in contrapposition to the pole mass.555As we
will explicitly see, at finite temperature, the two definitions of mass do not
coincide. From a physical perspective, the GMOR relation, Eq. (1), is
tantamount to recognizing that the mass squared of the pNGMs is linearly
proportional to the strength of explicit breaking (the quark masses) and the
amount of SSB (the chiral condensate). More in general, we do expect the mass
of pNGMs to vanish by setting either the explicit breaking scale or the
spontaneous one to zero. As we will see, this property is a generic feature of
pseudo-Goldstone modes which goes beyond the specific case of pions. The
validity of the GMOR relation, Eq.(1), has been verified directly in
holographic QCD models both at zero temperature [33] and finite temperature
[34]. Moreover, a similar expression has been discussed at length in the
context of holographic phonons [35, 36, 37] and also in holographic models
with pseudo-spontaneous breaking of a global ${\rm{U}}(1)$ symmetry [38, 39].
More formal analyses can be found in [40, 41].

By direct computation in a large class of holographic models with broken
translations [36, 42, 35, 37, 43, 44, 45], it was only recently realized that
in finite temperature dissipative systems also the damping, e.g. relaxation
rate or linewidth, of pNGMs follow a universal relation in terms of other
physical parameters of the system. In particular, at small frequency (energy),
the dispersion relation of the pNGM is expected to have a universal form of
the type:

$\omega=\pm\,\omega_{0}-\frac{i}{2}\,\Omega\,+\,\dots$ (2)

where $\omega_{0}=v\,k_{0}$ is the pole mass and $\Omega$ is the already
mentioned relaxation rate (at zero wave-vector).666In order to avoid
confusion, let us clarify our jargon. Given a dispersion relation $\omega(k)$,
its imaginary part, $-\mathrm{Im}[\omega(k)]$, represents the inverse of its
relaxation time, $\tau^{-1}$. The latter is also indicated as “relaxation
rate”, “damping” or “linewidth”. $v$ is the speed of the propagating sound
mode in the purely spontaneous phase777For simplicity, we consider only type I
Goldstone modes. Same arguments can be made for type II NMGs [46]. and $k_{0}$
is the screening mass, which was indicated as $m_{\pi}$ for the pions above.
As we will describe in detail in the next Section, the damping $\Omega$ in the
limit of soft-explicit breaking is universally given by:

$\Omega=D_{\varphi}k_{0}^{2}\,,$ (3)

where $D_{\varphi}$ is the Goldstone diffusivity (see more details later). The
universal relation, Eq. (3), has been verified numerically and via
perturbative semi-analytical methods in a large class of holographic models
for translations [36, 42, 35, 37, 43, 44, 45] and also in holographic models
with global $U(1)$ symmetry [38, 39]. Moreover, it has been derived in
effective field theories for quasicrystals and phason modes in [47, 48], it
has been proven within pions hydrodynamics using positivity of entropy
production in [49], and more generally in [50], and it has finally been
demonstrated using the locality requirements of the hydrodynamics framework in
[46].

This universal relation might have important physical consequences for several
phenomena in nature, such as the $T$-linear resistivity of strange metals [46,
7]. An even more natural framework where this relation might be at play is
QCD, in particular around the chiral critical point. In the limit of two
flavors of massless quarks, the chiral phase transition in QCD is in the same
universality class as the classical four components Heisenberg antiferromagnet
[51] and it can be effectively described by a non-abelian version of
superfluid hydrodynamics reflecting the SU$(2)_{L}\times$ SU$(2)_{R}\simeq
O(4)$ symmetry [52]. The pseudo-Goldstone mode nature of the pions might
produce strong effects and their mass $m_{\pi}$ can modify the hydrodynamics
description in the chiral limit with potentially measurable consequences. In
particular, in the presence of a finite pion mass, the hydrodynamic theory is
ordinary hydrodynamics at long distances, and superfluid-like at short
distances [49]. Recently, a more systematic study of this problem has been
initiated in [53] and an initial exploration using numerical simulations has
appeared in [54]. In this series of works, the universal relation in Eq. (3)
has been derived using positivity of entropy production but its validity has
not been confirmed yet in QCD simulations or experiments.

Holography has been playing a complementary role also in this direction.
Chiral symmetry breaking and the effects of finite quark masses have been
studied in many instances [55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66,
67]. More recently, the properties of pions and their dynamics in finite
temperature holographic QCD models have been discussed [68, 34, 69, 70], as
well as in the NJL model [71]. Nevertheless, a complete description is still
lacking. In this work, we provide a step forward in this program by continuing
the study of pions in finite temperature holographic QCD models. Among the
various tasks, we will test directly the validity of the universal relation in
Eq. (3) in a more realistic QCD model, generalizing the results obtained in
the ${\rm{U}}(1)$ holographic toy models [39].

The manuscript is organized as follows. In Section II we review the basic
properties of pseudo-Goldstone modes at finite temperature; in Section II.1 we
focus on the case of pions and summarize the main concepts therein; in Section
III we present the holographic QCD model; in Section IV we describe in detail
all our results and finally in Section V we provide an outlook and some open
questions for the future. Appendix A provides further details about the
methods and the numerical techniques.

##  II The damping of pseudo-Goldstone modes

In this Section, we provide more details about the low-energy description of
pNGMs and a more formal definition of the universal relation in Eq.(3). For a
simple treatment in the case of a global ${\rm{U}}(1)$ symmetry we refer to
[39], for a more complete discussion and formal derivation see [46, 50].
Let us start by considering a system with a spontaneously broken symmetry and
a type I Goldstone mode with dispersion relation

$\omega=\pm v\,k-\frac{i}{2}\,D_{A}\,k^{2}+\dots$ (4)

with $v$ the velocity and $D_{A}$ the leading quadratic coefficient of
attenuation constant. The latter has not to be confused with the wave-vector
dependent damping $\Gamma_{k}$ used later which is defined as:

$\Gamma_{k}\equiv-\mathrm{Im}\left[\omega(k)\right]\,.$ (5)

Eq.(4) can be for example identified with the dispersion relation of second
sound in a superfluid or that of shear sound in a standard solid. $D_{A}$
arises because of the dissipative finite temperature effects and it is usually
not discussed in the zero temperature field theory treatments.

At zero wave-vector $k=0$, and finite frequency $\omega$, the Goldstone888The
Goldstone mode $\varphi$ has always to be identified with the phase of the
complex order parameter $\psi=A\exp(i\varphi)$ where $A$ is the amplitude. The
fluctuations of the amplitude mode $\delta A$ are known to be gapped and they
are therefore usually not included in the low-energy effective description.
For simplicity, they will not be considered in this work either. retarded
Green’s function is given by:

$G_{\varphi\varphi}(\omega,k=0)=\frac{1}{\chi_{Q}\,\omega^{2}}-\frac{i}{\omega}\,\Xi+\dots$
(6)

where $\chi_{Q}$ is the susceptibility of the conserved quantity $Q$
associated to the spontaneously broken symmetry. The ellipsis indicates
higher-order corrections in the frequency $\omega$ which are neglected. For
example, in the case of a spontaneously broken $U(1)$ (superfluid), we have
$\chi_{Q}\equiv\chi_{\rho\rho}$ which is the charge density susceptibility; in
the case of spontaneously broken translations (solid),
$\chi_{Q}\equiv\chi_{PP}$ is the momentum susceptibility. The $\Xi$ parameter
appearing in Eq.(6) is formally defined via the following Kubo’s formula

$\Xi=\lim_{\omega\rightarrow
0}\,\omega\,\mathrm{Im}G_{\varphi\varphi}(\omega,k=0)$ (7)

and it is related to the Goldstone diffusivity $D_{\varphi}$ as:

$D_{\varphi}=\frac{\Xi}{\chi_{\varphi\varphi}}$ (8)

with $\chi_{\varphi\varphi}$ the Goldstone susceptibility. Let us give once
more some concrete examples. For a superfluid state, $\Xi$ is usually denoted
at $\zeta_{3}$ [21] and $\chi_{\varphi\varphi}=\mu/\rho_{s}$ with $\mu$ the
chemical potential and $\rho_{s}$ the superfluid density. In a solid,
$\chi_{\varphi\varphi}=1/G$ for transverse phonons and $1/(K+G)$ for
longitudinal ones, with $G,K$ respectively the bulk and shear moduli [7].
Importantly, the Goldstone diffusivity is not the attenuation constant $D_{A}$
in Eq.(4) but only part of it. More precisely $D_{A}=D_{Q}+D_{\varphi}$ with
$D_{Q}$ the diffusion constant of the conserved charge associated to the
spontaneously broken symmetry. Also, $D_{Q}=\sigma_{Q}/\chi_{Q}$ with
$\sigma_{Q}$ the “conductivity” associated to the conserved current $J_{Q}$
and given by:

$\sigma_{Q}\equiv-\lim_{\omega\rightarrow
0}\frac{1}{\omega}\mathrm{Im}G_{J^{Q}J^{Q}}(\omega,k=0)\,.$ (9)

For a superfluid $\sigma_{Q}$ is the electric conductivity and $J_{Q}$ the
electric current; for translations $\sigma_{Q}$ is the viscosity and
$J_{Q}\equiv T_{ij}$ the shear stress. Finally, let us remind the Reader that
the velocity in Eq.(4) satisfies the following identity:

$v^{2}=\frac{1}{\chi_{Q}\,\chi_{\varphi\varphi}}\,.$ (10)

Let us now introduce a small external source breaking explicitly the already
spontaneously broken symmetry. In the regime of small explicit breaking, the
associated current $J_{Q}$ is an “almost” conserved quantity and the
corresponding conserved charge $Q$ slowly relaxes in time with a rate
proportional to the amount of explicit breaking itself. In this limit, the
dispersion in Eq.(4) gets modified into:

$\omega=\pm\,\omega_{0}-\frac{i}{2}\,\Omega\,+\dots$ (11)

where, for simplicity, the wave-vector dependent terms are not displayed (see
[39, 46, 50] for a treatment of those corrections).
This implies that the original Goldstone mode acquires both a finite mass
$\omega_{0}$ and a finite relaxation rate $\Omega$ which is now independent of
the wave-vector $k$ (cfr. Eq.(4)). In the language of hydrodynamics, the
damping $\Omega$ is known as “phase-relaxation rate” and it is appears in the
Josephson constraint as a relaxation term

$\left(\partial_{t}+\Omega\right)\varphi=\dots$ (12)

where the ellipsis indicates terms which are not relevant to the present
discussion and therefore omitted. Before continuing, let us pause for a moment
on the meaning of the parameter $\Omega$. Even in absence of explicit
breaking, a phase-relaxation mechanism ($\Omega\neq 0$) is known to appear in
presence of topological defects, e.g. vortices in superfluids [72] or
dislocations/disclinations in solids [73]. In our scenario, the origin of the
damping term $\Omega$ is different. It is not connected to the existence of
any defects and it disappears in absence of explicit breaking. In other words,
this damping term is an effect of the pseudo-spontaneous breaking of the
global symmetry.

Back to our discussion, the pNGM mass $\omega_{0}$ is proportional to both the
explicit and spontaneous breaking scales and obeys a GMOR-type relation. It
appears in the static Goldstone correlator:

$G_{\varphi\varphi}(\omega=0,k)\,=\,-\frac{\chi_{\varphi\varphi}}{k^{2}+k_{0}^{2}}$
(13)

where

$k_{0}\equiv\frac{\omega_{0}}{v}$ (14)

is the screening mass. The latter can be therefore obtained independently from
the static correlator in Eq.(13) and from the pNGM dispersion at zero wave-
vector, Eq.(11).
Finally, as already discussed above, the damping $\Omega$ in the limit of soft
explicit breaking satisfies the following relation:

$\Omega=k_{0}^{2}D_{\varphi}$ (15)

which can be thought as the dissipative counterpart of the more known GMOR
relation. Notice that, following Eq.(11), the damping can be directly
extracted from the pNGM dispersion using:

$\Omega\equiv-2\lim_{k\rightarrow 0}\,\mathrm{Im}\left[\omega(k)\right]\,.$
(16)

###  II.1 The case of Pions

In order to connect more explicitly the language of [46, 50] with the QCD
framework recently discussed in [49, 53, 54], in this short section, we
provide more details about the case of pions.
Neglecting the dynamics of the heavier quarks, QCD can be described by a
${\rm{U}}(1)\times{\rm{SU}}(2)_{L}\times{\rm{SU}}(2)_{R}$ symmetry which is
spontaneously broken below the chiral critical point to
${\rm{U}}(1)\times{\rm{SU}}(2)_{V}$ by the presence of a finite chiral
condensate $\bar{\sigma}\equiv\langle\bar{\psi}\psi\rangle$. In the limit of
zero quark masses $m_{q}=0$, i.e., the chiral limit, the low-energy
description needs to take into account the presence of massless Goldstone
degrees of freedom and it takes the form of a non-abelian superfluid
hydrodynamics introduced in [52] and then formally generalized in [74]. In
presence of finite quark masses $m_{q}\neq 0$, then chiral symmetry is not an
exact symmetry from the start. This effect must be taken into account in the
hydrodynamics description as done in [75, 76, 77] and more recently in [49,
53].

For simplicity, we will focus on QCD in the limit of two flavors. In this
approximation, the low-energy description is based on a four-component vector
field:

$\phi_{a}=\left(\sigma,\vec{\pi}\right)$ (17)

where $\sigma$ is the meson and $\vec{\pi}$ the pions degrees of freedom. The
vacuum expectation value of this vector is given by:

$\langle\phi_{a}\rangle=\left(\bar{\sigma},0\right)$ (18)

where $\bar{\sigma}$ is the chiral condensate breaking chiral symmetry. The
pions degrees of freedom can be described using the phase fluctuations
$\varphi=\pi/\bar{\sigma}$ which will be indicated in the rest of this work as
the Goldstone degrees of freedom.
Under precise assumptions, the hydrodynamic equations describing this system
are then given by (see for example [78]):

$\displaystyle\partial_{t}\varphi=\,-\mu_{5}+D_{\varphi}\left(\nabla^{2}-m_{scr}^{2}\right)\varphi\,,$
(19)
$\displaystyle\partial_{t}\mu_{5}=v^{2}\left(-\nabla^{2}+m_{scr}^{2}\right)\varphi+D_{5}\nabla^{2}\mu_{5}\,,$
(20)
$\displaystyle\partial_{t}\delta\sigma=D_{\varphi}\left(\nabla^{2}-m_{\sigma}^{2}\right)\delta\sigma\,.$
(21)

Here $\varphi$ indicates the Goldstone degrees of freedom, $\mu_{5}$ the axial
chemical potential and $\delta\sigma$ the meson fluctuations. Additionally,
$m_{scr}$ is the pion screening mass and $m_{\sigma}$ the sigma meson mass.
The Goldstone degrees of freedom couple to the axial charge fluctuations where
$\mu_{5},D_{5}$ are the axial chemical potential and diffusion constant. On
the contrary, at least in this approximation, the $\sigma$ mode, which plays
the role of the Higgs/amplitude mode, appears decoupled. This decoupling
approximation might not always be valid and the effects of the amplitude mode
could become important close to the critical point (see for example [79, 80]).
We will return on this point in the outlook; for the moment, we will ignore
the effects of the amplitude mode in our description.
In Fourier space, the dispersion relation of the pNGMs is the solution of the
following equation:

$-\omega^{2}+\omega_{k}^{2}-i\,\omega\,\Gamma_{k}\,=\,0$ (22)

with:

$\omega_{k}^{2}=v^{2}\left(k^{2}+m_{scr}^{2}\right)+\dots$ (23)

where $v^{2}=f_{s}^{2}/f_{t}^{2}$ give the pion velocity and $m_{scr}$ its
screening mass. $f_{s}$ is usually called pion decay constant while
$f_{t}^{2}$ corresponds to the axial charge susceptibility. The two coincide
at zero temperature. Finally, one can define the pole mass:

$m_{p}^{2}=v^{2}m_{scr}^{2}$ (24)

which was indicated in the previous section as $\omega_{0}$. The pion decay
constant corresponds to what was indicated as Goldstone susceptibility in the
previous section, $f_{s}^{2}=1/\chi_{\varphi\varphi}$, and the axial
susceptibility with $f_{t}^{2}=\chi_{Q}$. This treatment is in complete
analogy with spin wave fluctuations in Heisenberg antiferromagnets where
$f_{s}$ corresponds to the so-called stiffness and $f_{t}^{2}$ to the magnetic
susceptibility in a direction perpendicular to magnetization [81].

Additionally, the damping is given, up to quadratic order, by999Originally
[77], the damping was assumed to be:
$\Gamma_{k}=\kappa_{1}m_{scr}^{2}+\left(D_{5}+\kappa_{2}\right)k^{2}\,$ (25)
with $\kappa_{1}\neq\kappa_{2}$. In Eq.(26), following the more recent results
in [53], we have fixed $\kappa_{1}=\kappa_{2}=D_{\varphi}$. As discussed in
the main text, this is equivalent to assume the universal relation (3) for the
pseudo-Goldstone modes damping .:

$\Gamma_{k}=D_{\varphi}m_{scr}^{2}+\left(D_{5}+D_{\varphi}\right)k^{2}\,.$
(26)

Here, $D_{5}$ is the axial charge diffusion constant corresponding to what was
indicates as $D_{Q}$ in the previous general section. Moreover, the product
$D_{\varphi}m_{scr}^{2}$ coincides with the damping $\Omega$ implying the
universal damping relation discussed in the previous section. Likewise, from
Eq.(25), we can identify the coefficient $D_{A}$ used above as
$D_{A}=D_{5}+D_{\varphi}$.
Finally, one can define a useful phenomenological parameter:

$\mathfrak{r}^{2}\equiv\frac{D_{\varphi}}{D_{\varphi}+D_{5}}=\frac{D_{\varphi}}{D_{A}}$
(27)

which parametrizes the ratio between the Goldstone diffusivity and the pion
attenuation constant. In chiral perturbation theory, the latter is found to be
equal to $3/4$ in the limit of zero temperature [82].

##  III The holographic model

In the holographic framework, the hard-wall [83] and soft-wall [84] AdS/QCD
models provide an ideal starting point to study the spontaneous symmetry
breaking of chiral symmetry, by promoting the 4D global symmetry
${\rm{SU}}(N_{f})_{L}\times{\rm{SU}}(N_{f})_{R}$ of QCD to a gauge symmetry in
5D. The chiral symmetry is spontaneously broken to ${\rm{SU}}(N_{f})_{V}$ when
the bulk scalar field in the model gets a non-vanishing expectation value,
which could be self-consistently derived from its bulk equation of motion. As
a result, the pions appear as the Goldstone modes of the spontaneous chiral
symmetry breaking. Another advantage of these two models is that the explicit
breaking could also be easily considered, when the quark masses are introduced
as non-vanishing sources for the corresponding operators. Thus, in such
holographic QCD models, one has all the ingredients to investigate the nature
of the pseudo-Goldstone modes.

Compared to the hard-wall model, the soft-wall model introduces an IR scale
through the dilaton field profile

$\Phi(z)=\mu_{g}^{2}z^{2},$ (28)

which breaks the conformal symmetry of the dual field theory. Here $\mu_{g}$
is a parameter with mass dimension one related to the dynamics of confinement
and to the $\Lambda_{QCD}$ scale. This parameter provides the possibility to
describe the Regge behavior, i.e. the observation that the mass square of the
highly excited states $m_{n}^{2}$ satisfies $m_{n}^{2}\propto n$, with $n$ the
excitation quantum number. As a phenomenological model, we will keep the
dilaton field as a background field without dynamics, just like the original
soft-wall model. Also, we will consider only the lightest two flavors, i.e. u,
d quarks. This implies $N_{f}=2$. 101010If you are interested in more flavors
case, you can read the $N_{f}=4$ case in [85]

The action of the soft-wall model is easily constructed from symmetries and it
reads

$\displaystyle S=\int
d^{5}x\sqrt{g}\,e^{-\Phi}\,{\rm{Tr}}\left[|DX|^{2}-V(|X|)-\frac{1}{g_{5}^{2}}\left(F_{L}^{2}+F_{R}^{2}\right)\right],$

with $g$ the determinant of the metric $g_{MN}$, $g_{5}$ the gauge coupling.
$X$ is a bulk matrix-valued scalar field dual to the 4D operator
$\bar{\psi}^{\alpha}\psi^{\beta}$ (with $\alpha$ and $\beta$ flavor indices).
$F_{MN,L/R}$ is the left/right hand gauge field strengths. The latter are
defined via the gauge potentials $A_{L,R}=A_{L,R}^{a}\tau^{a}$ ($\tau^{a}$ the
generator of ${\rm{SU}}(N_{f})$ group) using:

$F_{MN,L/R}=\partial_{M}A_{N,L/R}-\partial_{N}A_{M,L/R}-i[A_{M,L/R},A_{N,L/R}]$
(30)

where $D_{M}X=\partial_{M}X-iA_{LM}X+iXA_{RM}$ is the covariant derivative of
scalar field with $M,N=\\{0,1,2,3,4\\}$. Finally, $V(|X|)$ is the scalar
potential which will be specified in the following. By mapping the holographic
result for the two-point function of the vector current to the 4D field theory
calculation, one could fix $g_{5}^{2}=12\pi^{2}/N_{c}$ [86]. In this work, we
will focus on the realistic case with $N_{c}=3$.

In the original soft-wall model, the background metric is taken to simply be
that of five-dimensional Anti-de Sitter spacetime,

$ds^{2}=\frac{1}{z^{2}}\left(dt^{2}-dx^{i}dx^{i}-dz^{2}\right),$ (31)

with $i=1,2,3$ representing the spatial dimensions and z the fifth radial
dimension. Moreover, only the mass term is kept in the scalar potential which
reads $V(|X|)=m_{5}^{2}\,|X|^{2}$. In order to have an operator with conformal
dimension $\Delta=3$, we fix $m_{5}^{2}=-3$. For convenience, the AdS radius
is also fixed to be $L=1$. It is interesting to notice that this simple
construction accurately describes the spectrum of light vector mesons,
including the highly excited states [84].

To study the thermal properties of hadrons, finite temperature effects could
be introduced by considering black hole solutions. The temperature $T$ is then
identified with the Hawking temperature of the black hole. A simple choice is
to consider the AdS-Schwarzchild black hole whose metric reads

$ds^{2}=\frac{1}{z^{2}}\left(f(z)dt^{2}-dx^{i}dx^{i}-\frac{1}{f(z)}dz^{2}\right),$
(32)

with

$\displaystyle f(z)=1-\frac{z^{4}}{z_{h}^{4}}\,.$ (33)

Here, $z_{h}$ is the horizon radius of the black hole, which is related to the
temperature $T$ by $z_{h}=1/(\pi T)$.

As emphasized in Refs.[64, 65, 66], the above model with
$\Phi(z)=\mu_{g}^{2}z^{2},m_{5}^{2}=-3$, is not able to describe spontaneous
chiral symmetry breaking. On the contrary, it gives a vanishing chiral
condensate in the chiral limit (where all the quarks are massless). To
introduce spontaneous chiral symmetry breaking, one has to modify either the
dilaton profile [64, 65, 66] or the $5$D mass of the bulk scalar field $X$
[87]. According to the study in Ref.[66], the key point of these modifications
is to guarantee that the mass squared of the scalar meson vanishes at a
critical temperature $T_{c}$. Above such a critical value, the scalar mass
squared becomes negative and a second order phase transition occurs in the
chiral limit (for more details, we refer to Ref. [66]). Additionally, a
correct description of the meson spectra and of the low temperature chiral
condensate requires higher order corrections in the scalar potential. A
quartic term is the simplest correction that can be added. Taking all these
considerations into account, we will use the following extension of the
original soft-wall model potential,

$V(|X|)=m_{5}^{2}(z)\,|X|^{2}+\lambda\,|X|^{4},$ (34)

where the $z$ dependent 5D mass $m_{5}^{2}(z)$ takes the following form

$m_{5}^{2}=-3-\mu_{c}^{2}z^{2}.$ (35)

Here, two additional parameters, $\lambda$ and $\mu_{c}$, have been
introduced. Since the dilaton will not be modified and its profile will remain
as in Eq.(28), the negative corrections in $m_{5}^{2}(z)$ could also be
thought as originating from the interaction between $\phi$ and $X$ (e.g., a
bulk term of the form $\phi|X|^{2}$). However, here we will treat it as a
simple phenomenological model without caring about the microscopic origin of
this term.

So far, we have three free parameters, $\mu_{c},\mu_{g},\lambda$, in the
model. By comparing the model predictions with the experimental data for the
light mesons, the best fitting of these parameters could be obtained as [87]

$\displaystyle\mu_{g}=440\,\rm{MeV},\ \ \ \ \mu_{c}=1450\,\rm{MeV},\ \ \ \ \
\lambda=80\,.$ (36)

This choice reveals the presence of massless pions in the spectrum in the
chiral limit, confirming their Goldstone nature within this model [34, 69].
Also, when taking the physical and finite values for the quark masses, it
gives a pion mass of around $140$ MeV and a reasonable value of chiral
condensate, about $0.015\rm{GeV}^{3}=(246MeV)^{3}$. In this work, we would
like to extend the study of this model to finite temperature and to focus on
the dissipative properties of the (pseudo-) Goldstone modes, i.e. the pions.

###  III.1 The Pseudo-Goldstone modes in the soft-wall model

As mentioned above, the 5D matrix-valued scalar $X^{\alpha\beta}$ is dual to
the 4D operator $\bar{\psi}^{\alpha}\psi^{\beta}$. The spontaneous breaking of
chiral symmetry is due to the non-zero expectation value for the operator
$\bar{\psi}\psi$. Thus, in this case, we do expect a non-zero background field
$X$. On the contrary, the meson states are treated as perturbations above the
vacuum. Therefore, following Ref.[84], we take a simple ansatz for $X$ given
by

$X=\frac{\Sigma}{2}\,e^{i2\varphi}\,.$ (37)

Here, $\Sigma$ describes the chiral condensate and it is considered as a
background field, while $\varphi\equiv\varphi^{a}\tau^{a}$ describes the
perturbations of the phase, which is dual to the pseudo-Goldstone mode (i.e.
the pion meson). For convenience, other kinds of perturbations which are
irrelevant for our analysis will be neglected.

Inserting this ansatz into the action in Eq. (III), one can easily derive the
equation of motion (EOM) for $\Sigma$ as

$\Sigma^{\prime\prime}+\left(3A^{\prime}+\frac{f^{\prime}}{f}-\Phi^{\prime}\right)\Sigma^{\prime}-\frac{e^{2A}}{f}\left(m_{5}^{2}+\frac{\lambda\Sigma^{2}}{2}\right)\Sigma=0,$
(38)

where ′ indicates the derivative with respect to $z$. From this equation, one
can obtain the asymptotic solution near the UV boundary ($z=0$) as

$\displaystyle\Sigma(z)$ $\displaystyle=$ $\displaystyle m_{q}\zeta
z+\frac{1}{4}m_{q}\zeta[(m_{q}\zeta)^{2}\lambda+4\mu_{g}^{2}-2\mu_{c}^{2}]z^{2}\ln(z)+$
(39)
$\displaystyle\frac{\bar{\sigma}}{\zeta}z^{3}+\mathcal{O}\left(z^{3}\right).$

From the dual field theory perspective, the two integration constants $m_{q}$
and $\bar{\sigma}$ correspond to the quark mass and the chiral condensate
respectively, i.e.

$m_{q}=m_{u}=m_{d},\qquad\bar{\sigma}=\langle\bar{\psi}\psi\rangle.$ (40)

The normalization constant $\zeta=\sqrt{N_{c}}/(2\pi)$ is introduced, in order
to match the holographic results of two-point function of the scalar operator
to the 4D field theory calculation (for more details, see Ref. [86]).

At the horizon $z=z_{h}$, $\Sigma(z)$ has an asymptotic solution given by

$\Sigma(z)=c_{0}+\frac{c_{0}(c_{0}^{2}\lambda-2z_{h}^{2}\mu_{c}^{2}-6)}{8z_{h}}(z_{h}-z)+\mathcal{O}[(z_{h}-z)^{2}],$
(41)

with $c_{0}$ an arbitrary integration constant. Taking concrete values for the
temperature $T$ (or equivalently the horizon radius $z_{h}$) and the quark
mass $m_{q}$, and imposing regularity conditions for $\Sigma$ at the horizon,
one can numerically solve Eq. (38) and extract the chiral condensate
$\bar{\sigma}$. For more details about the numerical algorithm, we refer the
Reader to Ref. [69]. Following this scheme, the chiral condensate is self-
consistently extracted dynamically, instead of being externally fixed as in
the hard-wall model [83]. In the next sections, we will apply these techniques
and show the corresponding numerical results for the condensate and other
quantities.

After constructing the background solution with finite chiral condensate, we
are interested in the dynamics of the fluctuations on top of it. We take the
four dimensional wave-vector to be $q=(\omega,0,0,k)$. Using these notations,
the fluctuations of the axial-vector field divide into the transverse ones,
$a_{1}$ and $a_{2}$, and the longitudinal ones, $a_{0}$ and $a_{3}$ in the
$a_{z}=0$ gauge. In the 4D momentum space, we can derive the linearised EOMs
for coupled fields $a_{0}$, $a_{3}$ and $\varphi$,

$\displaystyle{a_{0}}^{\prime\prime}+\left(A^{\prime}-\Phi^{\prime}\right){a_{0}}^{\prime}-\left(\frac{k^{2}+e^{2A}g_{5}^{2}\Sigma^{2}}{f}\right)a_{0}-$
$\displaystyle\left(\frac{\omega ka_{0}+i\omega
e^{2A}g_{5}^{2}\Sigma^{2}}{f}\right)\varphi=0\,,$ (42a)
$\displaystyle{a_{3}}^{\prime\prime}+\left(A^{\prime}+\frac{f^{\prime}}{f}-\Phi^{\prime}\right){a_{3}}^{\prime}+\left(\frac{\omega^{2}-e^{2A}g_{5}^{2}\Sigma^{2}f}{f^{2}}\right)a_{3}+$
$\displaystyle\left(\frac{\omega
ka_{3}+ike^{2A}g_{5}^{2}\Sigma^{2}f}{f^{2}}\right)\varphi=0\,,$ (42b) and
$\displaystyle\varphi^{\prime\prime}+(3A^{\prime}+\frac{f^{\prime}}{f}-\Phi^{\prime}+\frac{2\Sigma^{\prime}}{\Sigma})\varphi^{\prime}+$
$\displaystyle\left(\frac{\omega^{2}-k^{2}f}{f^{2}}\right)\varphi-i\left(\frac{\omega
a_{0}+kfa_{3}}{f^{2}}\right)=0.$ (42c)

By using Eqs. (42)(a), (b) and (c), the equation for the bulk field $\varphi$
can be reduced to a non-dynamical first-order differential equation which
reads:

$\displaystyle ikfa_{3}^{\prime}+i\omega
a_{0}^{\prime}-e^{2A}\Sigma^{2}g_{5}^{2}f{\varphi}^{\prime}=0.$ (43)

To determine the two point Green’s functions, we consider the on-shell action
for the longitudinal axial-vector and pseudo-scalar channels which is given by

$\displaystyle S_{\rm{on}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2g_{5}^{2}}\int
dq^{4}\bigg{\\{}e^{A(z)-\Phi(z)}\big{[}{a_{0}}(-q,z){a_{0}}^{\prime}(q,z)-$
(44)
$\displaystyle{a_{3}}(-q,z)f(z){a_{3}}^{\prime}(q,z)\big{]}+e^{3A-\Phi}g_{5}^{2}f(z)\varphi(-q,z)$
$\displaystyle\Sigma(z)^{2}\varphi^{\prime}(q,z)\bigg{\\}}_{z=\epsilon},$

with $\epsilon$ an infinitesimally small constant representing our UV cutoff.

When the quark mass is finite, i.e. $m_{q}\neq 0$, the $\rm{SU}(2)$ symmetry
is explicitly broken and the second order chiral phase transition turns to be
a crossover. The leading order of the asymptotically solution of $\Sigma(z)$
is $m_{q}\zeta z$. As a result, at the boundary $z=0$, the asymptotic
solutions to Eqs. (42) behave as

$\displaystyle a_{0}(z)$ $\displaystyle=$ $\displaystyle
a_{t0}+a_{tl}z^{2}\ln(z)+a_{t2}z^{2}+\mathcal{O}(z^{3}),$ (45a) $\displaystyle
a_{3}(z)$ $\displaystyle=$ $\displaystyle
a_{x0}+a_{xl}z^{2}\ln(z)+a_{x2}z^{2}+\mathcal{O}(z^{3}),$ (45b)
$\displaystyle\varphi(z)$ $\displaystyle=$
$\displaystyle\varphi_{0}+\varphi_{l}z^{2}\ln(z)+\varphi_{2}z^{2}+\mathcal{O}(z^{3}),$
(45c)

with

$\displaystyle a_{tl}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left[\omega(a_{t0}k+a_{x0}\omega)-(m_{q}\zeta)^{2}g_{5}^{2}(a_{x0}-ik\varphi_{0})\right],$
$\displaystyle a_{xl}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[k(a_{t0}k+a_{x0}\omega)+(m_{q}\zeta)^{2}g_{5}^{2}(a_{t0}+i\omega\varphi_{0})\right],$
$\displaystyle\varphi_{l}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[i(a_{t0}\omega+a_{x0}k)-(\omega^{2}-k^{2})\varphi_{0}],$

in which $a_{t0}$, $a_{x0}$, $a_{t2}$, $a_{x2}$, $\varphi_{0}$ and
$\varphi_{2}$ are integration constants. Since the second order differential
equation for $\varphi$ can be reduced to a first order equation, one can
express $\varphi_{2}$ in terms of the other integration constants

$\varphi_{2}=\frac{i(a_{t2}\omega+a_{x2}k)}{g_{5}^{2}(m_{q}\zeta)^{2}}\,.$
(46)

Considering these asymptotic solutions, the on-shell action reduces to a
boundary term,

$\displaystyle S_{\rm{on}}$ $\displaystyle=$ $\displaystyle-\int
dq^{4}\left\\{\frac{a_{t0}(-q)a_{t2}(q)}{g_{5}^{2}}-\frac{a_{x0}(-q)a_{x2}}{g_{5}^{2}}+\right.$
(47)
$\displaystyle\left.{\varphi_{0}(-q)\varphi_{2}(q)(m_{q}\zeta)^{2}}\right\\}$

where higher-order corrections are not shown. According to the holographic
dictionary, we take $a_{t0}(q)$ and $a_{x0}(q)$ as the 4D field theory sources
for the axial vector current $A_{\mu}$ and $2\varphi_{0}m_{q}\zeta$ as the
source for the 4D pion operator $\pi$, respectively. Consequently, the
subleading terms in the expansions $\frac{a_{t2}}{g_{5}^{2}}$,
$\frac{a_{x2}}{g_{5}^{2}}$ and $(1/2)\varphi_{2}m_{q}\zeta$ represent the
expectation values for the corresponding field theory operators.

Following the standard prescriptions, we derive the retarded Green’s functions
by taking the second derivative of the on-shell action with respect to the
corresponding sources. From Eq. (52), the correlator for the dimensionless
pion field (i.e. the phase) is then given by 111111To verify the validity of
this procedure, in appendix A, we compare our results with those in Refs. [77,
76].

$\displaystyle G_{\varphi\varphi}(q)$ $\displaystyle=$
$\displaystyle\frac{\delta
S_{\rm{on}}}{2m_{q}^{2}\bar{\sigma}^{2}\delta\varphi_{0}(-q)\delta\varphi_{0}(q)}$
(48) $\displaystyle=$
$\displaystyle\frac{1}{(2m_{q}\bar{\sigma})^{2}}\frac{2\varphi_{2}(q)(m_{q}\zeta)^{2}}{\varphi_{0}(q)}$
$\displaystyle=$
$\displaystyle\frac{1}{(2m_{q}\bar{\sigma})^{2}}\frac{2i[a_{t2}(q)\omega+a_{x2}(q)k]}{g_{5}^{2}\varphi_{0}(q)}.$

In order to compute the retarded correlator, the appropriate boundary
conditions at the horizon must be imposed [88]. For that purpose, we consider
“incoming” boundary conditions at the horizon, $z=z_{h}$. We then obtain the
solutions close to the horizon which are given by

$\displaystyle a_{0}(z)=(z_{h}-z)^{-\frac{i\omega}{4\pi
T}}\bigg{\\{}-\frac{4(a_{b0}kz_{h}^{2}+4ic_{0}^{2}\pi^{2}\varphi_{b0})}{(4i+z_{h}\omega)z_{h}^{2}}(z_{h}-z)$
$\displaystyle\ \ \ \ \ \
+\mathcal{O}[(z_{h}-z)^{2}]\bigg{\\}}-i\varphi_{b1}\omega,$ (49a)
$\displaystyle a_{3}(z)=(z_{h}-z)^{-\frac{i\omega}{4\pi
T}}\bigg{\\{}a_{b0}+\bigg{\\{}\frac{4\pi^{2}c_{0}^{2}\varphi_{b0}k}{4i+\omega
z_{h}}+a_{b0}z_{h}\omega$ $\displaystyle\ \ \ \ \ \
\bigg{[}\frac{z_{h}^{2}k^{2}}{4i+\omega
z_{h}}+\frac{4c_{0}^{2}\pi^{2}}{z_{h}\omega}+\frac{i(8z_{h}^{2}\mu_{g}^{2}+3iz_{h}\omega-2)}{4}\bigg{]}\bigg{\\}}\bigg{/}$
$\displaystyle\ \ \ \ \ \ \bigg{[}2z_{h}(2-i\omega
z_{h})\bigg{]}(z_{h}-z)+\mathcal{O}[(z_{h}-z)^{2}]\bigg{\\}}+i\varphi_{b1}k,$
(49b) $\displaystyle\varphi(z)=(z_{h}-z)^{-\frac{i\omega}{4\pi
T}}\bigg{\\{}\varphi_{b0}+\bigg{\\{}\frac{2(c_{0}^{2}\pi^{2}\varphi_{b0}\omega-
a_{b0}z_{h}k)}{(2i+z_{h}\omega)(4-iz_{h}\omega)}+$ $\displaystyle\ \ \
\varphi_{b0}\bigg{[}\frac{4iz_{h}k^{2}-c_{0}^{2}\lambda\omega+z_{h}(2z_{h}\mu_{c}^{2}-8z_{h}\mu_{g}^{2}-3i\omega)\omega}{8(2i+z_{h}\omega)}\bigg{]}\bigg{\\}}$
$\displaystyle\ \ \ \ \ \
(z_{h}-z)+\mathcal{O}[(z_{h}-z)^{2}]\bigg{\\}}+\varphi_{b1},$ (49c)

where $a_{b0}$, $\varphi_{b0}$ and $\varphi_{b1}$ are integration constants.
Combing the boundary and horizon expansions, Eqs. (45) and (49), one can
numerically find unique solutions for bulk EOMs in Eqs. (42).

From the holographic model, one can extract further information about the
dynamics of pions. First, the pion decay constant is defined as
$f_{\pi}q_{\mu}=\langle 0|A_{\mu}|\pi(q)\rangle$ at zero temperature, in which
$|0\rangle$ is the vacuum state and $|\pi\rangle$ the one-pion state. For the
zero temperature case, there are some studies in the holographic AdS/QCD
models [83, 68]. At finite temperature, Lorentz symmetry is broken. Therefore,
the pion decay constant at finite temperature splits into a time component
$f_{t}$ and a space one $f_{s}$. Following the analysis in Ref. [83], the
generalized pion decay constants are given by

$\displaystyle f_{t}^{2}=-\frac{2a_{t2}(0)}{a_{t0}(0)g_{5}^{2}},\ \ \ \
f_{s}^{2}=-\frac{2a_{x2}(0)}{a_{x0}(0)g_{5}^{2}}.$ (50)

By looking at the solution around the boundary, one can find that in Ref. [83]
the term ($F[q,z]/m_{q}$) appears. This term is clearly divergent in the
chiral limit, $m_{q}$ approaching zero, and therefore the results of Ref. [83]
must be revisited for zero quark mass.

In order to do that, we observe that, in the chiral limit, the expansions of
the solutions at the horizon remain unchanged. Since the leading term of the
expansion for $\Sigma$ becomes $(\bar{\sigma}/\zeta)z^{3}$ setting $m_{q}=0$,
the asymptotic solution of $\varphi$ turns to be

$\displaystyle\varphi(z)$ $\displaystyle=$ $\displaystyle
z^{-2}\left\\{\bar{\varphi}_{0}+\frac{1}{2}(q^{2}-2\mu_{g}^{2}+\mu_{c}^{2})\bar{\varphi}_{0}z^{2}\ln(z)+\right.$
(51) $\displaystyle\left.\bar{\varphi}_{2}z^{2}+\mathcal{O}(z^{3})\right\\},$

with
$\bar{\varphi}_{0}=-\frac{i(a_{t2}\omega+a_{x2}k)}{g_{5}^{2}(\bar{\sigma}/\zeta)^{2}}.$
Thus, the on-shell action becomes

$\displaystyle S^{\prime}_{\rm{on}}$ $\displaystyle=$ $\displaystyle-\int
dq^{4}\bigg{\\{}\frac{a_{t0}(-q)a_{t2}(q)}{g_{5}^{2}}-\frac{a_{x0}(-q)a_{x2}}{g_{5}^{2}}-$
(52)
$\displaystyle{\bar{\varphi}_{0}(-q)\bar{\varphi}_{2}(q)(\bar{\sigma}/\zeta)^{2}}\bigg{\\}}.$

According to the holographic dictionary, the quantity
$2\bar{\varphi}_{0}(q)\bar{\sigma}/\zeta$ has to be identified as the source
for pion operator and $\bar{\sigma}/(2/\zeta)\bar{\varphi}_{2}(q)$ as its
expectation value. Following the same steps as before, the correlator for the
phase field $\varphi$ is given by121212In appendix A, we have verified that
our holographic results in the chiral limit are consistent with the thermal
chiral effective field theory (EFT) analysis of Ref. [77, 76].

$\displaystyle G_{\varphi\varphi}(q)$ $\displaystyle=$
$\displaystyle\frac{\delta
S^{\prime}_{\rm{on}}}{4(\bar{\sigma}/\zeta)^{4}\delta\bar{\varphi}_{0}(-q)\delta\bar{\varphi}_{0}(q)}$
(53) $\displaystyle=$
$\displaystyle-\frac{1}{(\bar{\sigma}/\zeta)^{2}}\frac{\bar{\varphi}_{2}(q)}{2\bar{\varphi}_{0}(q)}$
$\displaystyle=$
$\displaystyle\frac{\bar{\varphi}_{2}(q)g_{5}^{2}}{2i[a_{t2}(q)\omega+a_{x2}(q)k]}.$

##  IV Results

Based on the above discussion, we are now in the position to explore all the
features of our holographic model. In this section, we will present the main
results of our work.

###  IV.1 The chiral condensate and phase transition

Figure 1: The chiral condensate
$\bar{\sigma}\equiv\langle\bar{\psi}\psi\rangle$ as a function of the $T_{c}$
normalized temperature $T/T_{c}$, for different values of the current quark
mass $m_{q}$. The black solid line represents the result in the chiral limit,
which implies a second order phase transition at $T_{c}=163\rm{MeV}$. The red,
blue, purple, orange dashed and cyan dotted lines represent the results for
$m_{q}=1,3.22,10,20,40\rm{MeV}$ respectively, and show a smooth crossover.

The Goldstone modes are tightly related with the spontaneous chiral symmetry
breaking, whose order parameter is given by the chiral condensate
$\bar{\sigma}\equiv\langle\bar{\psi}\psi\rangle$. In Fig.1, we show the
behavior of the chiral condensate $\bar{\sigma}$ in function of the reduced
temperature $T/T_{c}$ for different values of the quark mass $m_{q}$. The
result in the chiral limit, $m_{q}=0$, corresponds to the solid black line
therein. In that limit, the chiral condensate $\bar{\sigma}$ is finite at low
temperature and reaches a constant value $\approx 0.015\rm{GeV}^{3}$. At the
critical temperature, $T_{c}\approx 0.163\rm{MeV}$, the chiral condensate
vanishes following a square root law, $\propto\sqrt{T_{c}-T}$, typical of mean
field second order phase transitions. Above the critical temperature, we have
the chiral symmetry restored phase with zero condensate, usually also labelled
as the normal phase.

Then, we assume a finite and increasing quark mass,
$m_{q}=1,3.22,10,20,40\,\rm{MeV}$131313Here, when $m_{q}=3.22\,\rm{MeV}$ (blue
line), the pion mass in the vacuum is around the physical value
$139.5\,\rm{MeV}$. Therefore, we will consider this value as the “physical
quark mass”., and re-compute the chiral condensate in those cases. The results
are shown by the red, blue, purple, orange, cyan lines in Fig.1 respectively.
There, we can see that when the quark mass becomes finite, the chiral
condensate at low temperature becomes larger than its value in the chiral
limit. This might be due to the contribution from the explicit symmetry
breaking. Moreover, the condensate $\bar{\sigma}$ does not vanish at any
temperature. A finite tail appears in the high temperature region, which could
be considered as the pure contribution from the quark masses. Near $T_{c}$, we
could see that the transition from the broken phase to the symmetry restored
phase is no longer sharp. Instead, it becomes a continuous crossover. This is
reasonable, since the finite quark mass breaks the exact symmetry explicitly
rendering the notion of spontaneous symmetry breaking imprecise.141414Notice
how that is not necessarily always the case. See for example the discussions
in [39, 89].

###  IV.2 Pions in the chiral limit

\begin{overpic}[width=346.89731pt]{ReGphiphi.pdf} \put(85.0,63.0){\bf{(a)}}
\end{overpic}\begin{overpic}[width=346.89731pt]{ImGphiphi.pdf}
\put(85.0,62.0){\bf{(b)}}
\end{overpic}\begin{overpic}[width=325.215pt]{chiQ.pdf}
\put(90.0,50.0){\bf{(c)}}
\end{overpic}\begin{overpic}[width=325.215pt]{Xi.pdf}
\put(90.0,50.0){\bf{(d)}} \end{overpic}

Figure 2: The low frequency behavior of the Goldstone correlator. (a)
$\omega^{2}{\rm{Re}[G_{\varphi\varphi}]}$ and (b)
$-\omega{\rm{Im}}[G_{\varphi\varphi}]$ as a function of $\omega$ at $k=0$. The
thermodynamic susceptibilities extracted from the low-frequency limit of the
dynamical Goldstone correlator, Eq.(6). (c) The dimensionless axial charge
susceptibility $1/f_{t}^{2}$ and (d) the Goldstone diffusivity $\Xi$ as a
function of the reduced temperature $T/T_{c}$.

After extracting the information about the chiral phase transition from the
order parameter, we continue to investigate the dynamical and thermal
properties of the low-energy (quasi-)particles. In this section, we will focus
on the case with massless quarks. We start by studying the properties of the
Goldstone modes, the pions.

From Eq. (8), the pion diffusivity can be obtained from the ratio of the
dissipative coefficient $\Xi$ and the Goldstone susceptibility
$\chi_{\varphi\varphi}$. These two quantities can be extracted from the
temporal correlations in Eq. (13).

We take $m_{q}=0$ and compute the dynamical correlation function of the
Goldstone modes $G_{\varphi\varphi}(\omega,k=0)$ numerically. The results are
shown in panels (a) and (b) of Fig.2, where we have taken
$T=70,90,110,130,150,160\rm{MeV}$ (cyan, orange, purple, blue, red, black
lines respectively). In order to show the scaling behavior near $\omega=0$, we
have rescaled the real part and the imaginary part of $G_{\varphi\varphi}$
with additional factors of $\omega^{2}$ and $\omega$ respectively. From the
figures, one can easily seen that both the rescaled real and imaginary parts
are almost constant when $\omega$ is small. Thus, we have

$\mathrm{Re}G_{\varphi\varphi}(\omega,k=0)\propto\omega^{-2}\,,\qquad\mathrm{Im}G_{\varphi\varphi}(\omega,k=0)\propto\omega^{-1}\,,$
(54)

which is exactly of the same form as Eq.(6).

Then, $f_{t}^{2}$ and $\Xi$ could be obtained numerically from the values of
the reduced real and imaginary parts in the limit of small frequency. The
numerical results are given in Fig. 2(c) and (d) respectively. From Fig.2(c),
we could see that, at low temperature, $1/f_{t}^{2}(T=20\rm{MeV})=1/0.0048\ \
(\rm{GeV})^{-2}$, which is very close to the square of the zero temperature
value, $1/f_{t}^{2}\approx 1/0.0049\ {\rm{(GeV)}}^{-2}$ [87]. Then, it
decreases with increasing the temperature and reaches a finite value around
$1/f_{t}^{2}(T=163\rm{MeV})=1/0.0090\ (\rm{GeV})^{-2}$ near $T_{c}$. This
implies that $f_{t}$ is finite at $T_{c}$, which is consistent with the
analysis in Ref. [77, 76]. From Fig.2(d), $\Xi$ vanishes at low temperature,
as expected for any dissipative coefficient. Along with the increasing of
temperature, $\Xi$ increases and becomes of order $10^{4}(\rm{GeV^{-3})}$ near
$T_{c}$.

\begin{overpic}[width=368.57964pt]{dcphidq.pdf} \end{overpic} Figure 3: The
reduced temperature dependence of the dimensionless diffusivities
$D_{\varphi}$ and $D_{Q}$. The inset is a zoom for $T/T_{c}$ in the range
$[0.75,1]$.

\begin{overpic}[width=346.89731pt]{soundvelocity.pdf}
\put(85.0,57.0){\bf{(a)}}
\end{overpic}\begin{overpic}[width=368.57964pt]{gamma.pdf}
\put(85.0,55.0){\bf{(b)}} \end{overpic}

Figure 4: (a) The pion velocity $v_{0}^{2}$ in the chiral limit in function of
the reduced temperature $T/T_{c}$. The horizontal gray dashed line is as a
guide to the eye. (b) The attenuation constant $D_{A}$ in function of the
reduced temperature $T/T_{c}$. The solid line are the values extracted using
the quantities appearing in the various correlators. The symbols are the
values extracted by fitting the numerical dispersion relations in Fig.5.

\begin{overpic}[width=346.89731pt]{dschirallimitre.pdf}
\put(65.0,50.0){\bf{(a)}}
\end{overpic}\begin{overpic}[width=368.57964pt]{dschirallimitim.pdf}
\put(65.0,50.0){\bf{(b)}} \end{overpic}

Figure 5: (a) The real and (b) imaginary part of the dispersion relation in
the chiral limit. The gray dashed line guides the eyes towards the light-like
dispersion, ${\rm{Re}}[\omega]=k$.
\begin{overpic}[width=368.57964pt]{tausquared.pdf} \end{overpic} Figure 6: The
value of the $\mathfrak{r}^{2}$ parameter defined in Eq.(27). The symbols are
the fitting results from the dispersion relations shown in Fig.5. The dashed
line indicates the chiral perturbation theory value $\mathfrak{r}^{2}=3/4$
[82].

After obtaining numerically the susceptibility $f_{t}$ and the dissipative
coefficient $\Xi$, it is straightforward to get the Goldstone diffusivity
$D_{\varphi}$ from their ratio. In Fig. 3, we show the Goldstone diffusivity
$D_{\varphi}$ together with the charge diffusivity $D_{Q}$ extracted from the
ratio between the conductivity $\sigma_{Q}$ in Eq. (9) and the corresponding
susceptibility $f_{t}$. The two diffusivities display a similar behavior as a
function of the reduced temperature $T/T_{c}$. More precisely, they vanish at
low temperature and they approach a constant value close to the critical point
$T\approx T_{c}$, {$D_{\varphi}(T=163\rm{MeV})=0.7269(\rm{Gev}^{-1})$ ,
$D_{Q}(T=163\rm{MeV})=0.6815(\rm{Gev}^{-1})$}. 151515In general, the sound
attenuation constant $D_{A}$ is expected to scale as
$D_{A}\propto(1-T/T_{c})^{\nu(d-4)/2}$ [77] where $\nu$ is the critical
exponent determining the class of universality of the critical point. The
holographic model we used in this paper belongs to the mean field universality
class, i.e. ${\nu(d-4)/2}=0$. Therefore we consistently find that
$D_{A}\rightarrow\text{const.}$ for $T\rightarrow T_{c}$.

We then move to the discussion of the pion velocity and attenuation constant,
$v_{0}^{2},D_{A}$ (where the subfix $0$ indicates the chiral limit). According
to the definition of the sound velocity in Eq. (10), we extract the results
and show them in Fig.4(a). The pion velocity approaches the speed of light at
low temperature and, as expected, it goes to zero at the critical temperature
as:

$v_{0}^{2}\propto(T_{c}-T)\,.$ (55)

The attenuation constant, $D_{A}=D_{\varphi}+D_{Q}$, is shown in the panel (b)
of Fig.4. It goes from a constant value at the critical point $T_{c}$ to zero
in the limit of $T\rightarrow 0$.

To confirm our computations, we numerically obtain the dispersion relation of
the pions in the chiral limit. The results are shown in Fig.5 for three
different temperatures. We find that the dispersion relation is consistently
described by Eq.(4) in the small wave-vector limit. By fitting the dispersion
relations, we can then find the value of the attenuation constant $D_{A}$ and
the sound speed $v_{0}$. The results are shown with colored symbols in Fig.4
and are in perfect agreement with the predictions from the correlators (solid
lines therein).

Finally, in Fig.6, we show the $\mathfrak{r}$ parameter defined in Eq.(27). It
is predicted to be equal to $3/4$ in the limit of zero temperature using
chiral perturbation theory (see Ref. [82]. In our holographic model, we find
that $\mathfrak{r}^{2}$ approaches $1$ in the zero temperature limit. It
monotonously decreases by increasing temperature and reaches a constant value
$\approx 0.5$ in the limit $T\rightarrow T_{c}$. We will comment about this
outcome in the conclusions.

###  IV.3 Pions as pseudo-Goldstone modes

\begin{overpic}[width=346.89731pt]{GphiphikT.pdf} \put(80.0,62.0){\bf{(a)}}
\end{overpic}\begin{overpic}[width=346.89731pt]{Gphiphikmq.pdf}
\put(80.0,62.0){\bf{(b)}}
\end{overpic}\begin{overpic}[width=346.89731pt]{fpimq.pdf}
\put(80.0,55.0){\bf{(c)}}
\end{overpic}\begin{overpic}[width=346.89731pt]{mscr.pdf}
\put(80.0,55.0){\bf{(d)}} \end{overpic}

Figure 7: The combination
$-(k^{2}+m_{\rm{scr}}^{2})(m_{\rm{scr}}^{2}/k^{2})G_{\varphi\varphi}(k)$: (a)
at different temperatures and fixed $m_{q}=3.22$ MeV and (b) at different
quark masses and $T=100$ MeV. (c) The pion decay constant. (d) The temperature
dependence of screening masses for different quark masses. Figure 8: The
validity of the GMOR relation at finite quark masses. The horizon gray dashed
line as a guide to the eye.

So far, we have focused on the chiral phase transition and the dynamics of the
pions in the chiral limit where the masses of the quarks are assumed to be
zero. However, in real world, chiral symmetry appears to be softly broken
since the quarks are not massless. We therefore extend here the discussion in
presence of a small quark mass.

We start by considering the behavior of the pion correlator in the static
limit, $\omega=0$. In panels (a) and (b) of Fig. 7, we plot the combination
$-(k^{2}+m_{\rm{scr}}^{2})(m_{\rm{scr}}^{2}/k^{2})G_{\varphi\varphi}(\omega=0,k)$
as a function of the momentum $k$ at different temperatures and different
quark masses. At low momentum, all the curves approach a constant value which
is well described by the leading order approximation of the static Green’s
function, Eq. (13). From such a limit we can define the pion decay constant as
$f_{s}^{2}=1/\chi_{\varphi\varphi}$. As evident in Eq. (50), we can calculate
the finite pion decay constant using the solution of the axial-vector field
equation at the singularity pole $q=0$. As shown in Fig. 7(c), the pion decay
constant decreases with temperature and goes to zero near $T_{c}$ in the limit
of zero quark mass, $m_{q}=0$. In the chiral limit, the pion decay constant
behaves as $f_{\pi}^{2}\propto|T-T_{c}|$ near the critical point.161616Our
numerical fit gives a power scaling $|T-T_{c}|^{1.2}$. The deviation from the
expected linear power is just a result of numerical accuracy and of a too
small number of points close to $T_{c}$. Away from the chiral limit, the pion
decay constant does not goes anymore to zero at the bare critical temperature
$T_{c}$ and it has a finite tail reminiscent of the behavior of the chiral
condensate in Fig.1. Following the same analogy, the value of the pion decay
constant grows with the quark mass $m_{q}$. At last, in Fig. 7(d) 171717The
data for the pion screening mass in the chiral limit is obtained from Ref.
[34]., we plot the pion screening mass as functions of temperature for
different quark masses. The screening mass is related to the inverse of the
correlation length ($\zeta$) and increases monotonically with temperature. In
the chiral limit $m_{q}=0$, the screening mass equals to zero in the chiral
broken phase and behaves as $m_{scr}^{-1}=\zeta\propto|T-T_{c}|^{-\nu}$ with
$\nu=0.5$ in the chiral restored phase [34]. This is yet another manifestation
of the mean-field nature of the critical point in the chiral limit.

\begin{overpic}[width=130.08731pt]{ret80.pdf} \put(80.0,10.0){\bf{(a)}}
\end{overpic}\begin{overpic}[width=130.08731pt]{ret100.pdf}
\put(80.0,10.0){\bf{(b)}}
\end{overpic}\begin{overpic}[width=130.08731pt]{ret130.pdf}
\put(80.0,10.0){\bf{(c)}}
\end{overpic}\begin{overpic}[width=130.08731pt]{imt80.pdf}
\put(80.0,10.0){\bf{(d)}}
\end{overpic}\begin{overpic}[width=130.08731pt]{imt100.pdf}
\put(80.0,10.0){\bf{(e)}}
\end{overpic}\begin{overpic}[width=130.08731pt]{imt130.pdf}
\put(80.0,10.0){\bf{(f)}} \end{overpic}

Figure 9: Real part (${\rm{Re}}[\omega]$) and imaginary part
($-{\rm{Im}}[\omega]$) of the dispersion relation of the pions at (a) and (d)
$T=80$ MeV; (b) and (e) $T=100$ MeV; (c) and (f) $T=130$ MeV. The gray dashed
lines guide the eyes towards the light-like dispersion, ${\rm{Re}}[\omega]=k$.

Since we already have the pion decay constant, the sigma condensate and the
screening mass, we can immediately verify the validity of the famous GMOR
relation

$f_{s}^{2}m_{\rm{scr}}^{2}=2m_{q}\bar{\sigma}$ (56)

valid in the limit of finite but small quark masses. We plot the ratio
$2m_{q}\bar{\sigma}/(f_{s}^{2}m_{\rm{scr}}^{2})$ as a function of $T/T_{c}$ in
Fig. 8. In the limit of small quark masses, the ratio is very close to $1$
apart from a very small region close to the critical temperature. Close to
$T_{c}$, the quark condensate becomes very small and we are not anymore in the
limit of pseudo-spontaneous breaking of chiral symmetry since the explicit
breaking term, the quark mass, becomes larger than the spontaneous scale, the
condensate. Our analysis indicates that the GMOR relation works well in the
chiral symmetry broken phase when the quark mass is very small. By increasing
the value of the quark mass, which parametrizes the explicit breaking scale,
the GMOR ceases to be accurate. Nevertheless, we interestingly find that the
GMOR value still acts as a lower bound for the pion mass:

$m_{\rm{scr}}^{2}f_{s}^{2}\geq 2m_{q}\bar{\sigma}\,.$ (57)

The larger the quark masses, the more the pion mass deviates from such a lower
bound. It would be interesting to verify whether this outcome is a general
property or it is specific to our holographic model. We are not aware of any
discussion of this sort in the literature.

We are now ready to look at the dispersion relation of the pseudo-goldstone
modes, the pions away from the chiral limit. In Fig.9 we show the dispersion
relation as a function of the quark mass for different values of temperature.
At small wave-vector, the dispersion relation of the pions is well
approximated by Eq. (22) where the renormalized energy $\omega$ is given in
Eq.(23) and the $k-$dependent damping $\Gamma_{k}$ by the expression in
Eq.(25). A lot of information can be extracted from the dispersion relation.
First, the real part of the frequency at zero wave-vector gives the pole mass,
i.e. $m_{\rm{p}}={\rm{Re}}[\omega(k=0)]$. We plot the pole mass as a function
of $T/T_{c}$ for different $m_{q}$ in Fig. 10(a). The pole mass increases with
the quark mass as expected. Using the values for the screening mass in Fig.
7(d), we can calculate the sound velocity $v$ as described in Eq. (24). We
plot the ratio between $v^{2}$ and the square of sound velocity in the chiral
limit $v_{0}^{2}$ as a function of $T/T_{c}$ in Fig. 10(b). At low
temperature, the sound velocity approaches its value in the chiral limit. More
precisely, we see that the ratio is approximately one up to $T/T_{c}\sim 0.6$.
Approaching the critical point, the ratio grows rapidly and it increases with
the value of the quark mass. As shown in the inset, the velocity always
approaches the speed of light in the zero temperature limit and decreases with
the increasing temperature. In order to confirm that the pion velocity
extracted from Eq.(24) is correct, we present also an alternative derivation.
In particular, we directly fit the pion dispersion relation with the formula

$\mathrm{Re}[\omega(k)]=\sqrt{m_{p}^{2}+v^{2}k^{2}}\,.$ (58)

Part of the data for the dispersion relations are shown in Fig.9. In panel (b)
of Fig.10, we indicate the results from the fitting with open symbols for
three different temperatures and different values of the quark mass. We find
perfect agreement between the velocity extracted with Eq.(24) and that
obtained by fitting the real part of the dispersion relation with Eq.(58).

Always from the dispersion relation, the imaginary part corresponds to the
damping $\Gamma_{k}$ (or thermal width) as defined in Eq. (26). Here, we are
particularly interested in its value at $k=0$ which corresponds to the phase
relaxation rate $\Omega$ as defined in Eq. (16). The numerical results for
$\Omega$ are shown in Fig. 11. Panel (a) of Fig. 11 displays the phase
relaxation rate as a function of $T/T_{c}$. The inset shows the details in the
low temperature regime. Panel (b) shows the phase relaxation rate as a
function of the quark mass $m_{q}$. Away from the critical point, for $T\ll
T_{c}$, the phase relaxation rate is linear in the quark mass:

$\Omega\propto m_{q}\,.$ (59)

On the contrary, close to the critical point, $T\approx T_{c}$, it satisfies a
scaling relation:

$\Omega\propto m_{q}^{\nu z/(\beta\delta)}$ (60)

with the critical exponents $\beta=1/2$ and $\delta=3$. This behavior is
consistent with the analysis of the critical point in Ref. [90]. This
observation is also compatible with the nonlinear scalings found for the GMOR
relation and for the phase relaxation rate in certain holographic models with
broken translations [89, 37, 91].

\begin{overpic}[width=359.90538pt]{mpole.pdf} \put(85.0,58.0){\bf{(a)}}
\end{overpic}\begin{overpic}[width=346.89731pt]{soundvelocitymq.pdf}
\put(85.0,56.0){\bf{(b)}} \end{overpic}

Figure 10: (a) Temperature dependence of the pole mass $m_{\rm{p}}$ with
different quark masses. (b) The square of the sound velocity $v$ as a function
of the reduced temperature $T/T_{c}$. The open symbols are the values of the
sound speed obtained by fitting the dispersion relations in Fig. 9 with the
formula ${\rm{Re}}[\omega]=\sqrt{m_{p}^{2}+v^{2}k^{2}}$. The inset figure
shows the ratio between the velocity at finite quark mass $v$ and the bare one
in chiral limit $v_{0}$.

\begin{overpic}[width=346.89731pt]{bigomega.pdf} \put(80.0,50.0){\bf{(a)}}
\end{overpic}\begin{overpic}[width=346.89731pt]{bigomegamq.pdf}
\put(77.0,50.0){\bf{(b)}} \end{overpic}

Figure 11: (a) The phase relaxation rate $\Omega$ as a function of the reduced
temperature $T/T_{c}$ for different quark masses $m_{q}$. The inset is a semi-
log plot of the same data. (b) The same quantity as a function of $m_{q}$ for
different $T$. The dashed lines are fitting results to a power-law scaling of
the form $\Omega\propto m_{q}^{b}$. We have $b\approx 0.62$ at $T=163$ MeV and
$b\approx 0.98$ at $T=116$ MeV.

###  IV.4 The relaxation rate of Pions

Finally, we are in the position to verify the universal damping relation
$\Omega=D_{\varphi}m_{\rm{scr}}^{2}$ which is expected to be valid for any
pseudo-Goldstone modes. In Fig. 12, we plot the dimensionless ratio
$\Omega/(m_{\rm{scr}}^{2}D_{\varphi})$ as a function of reduced temperature
$T/T_{c}$ for different values of the quark mass. Whenever the explicit
symmetry breaking parameter $m_{q}$ is small, this ratio is equal to $1$ in
almost all the range of temperatures apart from a very small region close to
the critical point. Around the critical point, once more, the condensate
becomes small and therefore the pseudo-spontaneous limit ceases to be valid.
Moreover, if the breaking parameter becomes large, the universal relation is
gradually breaking (as expected) at any temperature. Still, for reasonable
values of the mass, the universal relation still holds approximately well at
intermediate temperatures. Finally, let us notice how this relation appears to
be violated in both directions. This means, that, differently from the GMOR
relation and the pion mass, there is no lower or upper bound for the phase
relaxation rate $\Omega$ away from the chiral limit.

##  V Outlook

In summary, we have studied in detail the low-energy dynamics of pseudo-
Goldstone modes (pions) in a holographic soft-wall QCD model with
${\rm{SU}}(2)_{L}\times{\rm{SU}}(2)_{R}$ broken symmetry at finite
temperature. By matching to the hydrodynamics framework [53], we have
extracted the dissipative transport coefficients and the dispersion relation
of the pNGMs. The holographic model serves as a “microscopic” description of
the pions dynamics which could play a role complementary to the kinetic theory
[82] and simulations [54] approaches. As a direct test of the results in [53],
and more generally in [46, 50], we have numerically verified the universal
relation for the damping of pseudo-Goldstone modes, Eq.(3), which was
previously verified only in toy models with $\rm{U}(1)$ symmetry [39] or for
the case of translations [36].

There are several direct questions which need to be addressed. As a concrete
example, following [49, 53], it would be interesting to compute the effects of
a finite pion mass on the QCD transport properties such as the shear viscosity
or the axial conductivity and investigate further the scaling behaviors of the
transport coefficients close to the critical point. A different question
regards the value of the parameter $\mathfrak{r}^{2}$ and the observed
deviations from the chiral perturbation theory value $3/4$ [82] together with
their meaning and universality (e.g. in different holographic QCD models). Is
it an effect of strong coupling? Is it an effect of large-N? Further
investigation in this direction is needed.

Additionally, in our analysis the dynamics and role of the amplitude mode (the
$\sigma$ meson) has been completely neglected. Close to the critical
temperature, the effects of the amplitude mode could be dramatic (see for
example [92]).181818This could be simply understood from the fact that the
amplitude mode becomes massless at $T_{c}$. Moreover, whenever the symmetry is
also broken explicitly a clear separation between the massless Goldstone modes
and the amplitude mode is missing even away from the critical point. This
could produce even stronger effects. It would be interesting to enlarge the
hydrodynamic framework by considering the amplitude mode and use holography as
a tool in this direction as initiated in [79, 80].

Finally, it would be fruitful to extend our analysis to the time dependent
dynamics and analyze the thermalization properties of this system near the
critical point as done in the chiral limit in [90].

We leave some of these questions for the near future.

Figure 12: The dimensionless ratio $\Omega/({m_{\rm{scr}}^{2}}D_{\varphi})$ as
a function of the dimensionless temperature $T/T_{c}$ four different fixed
quark masses from $m_{q}=0.1$ MeV to $m_{q}=10$ MeV.

### Acknowledgments

We thank S. Grieninger, A. Donos, V. Ziogas, A. Soloviev and B. Goutéraux for
fruitful discussions on the topic of this paper. X.C. is supported by the
National Natural Science Foundation of China under Grant Nos. 12275108 and the
Fundamental Research Funds for the Central Universities under grant No.
21622324. M.B. acknowledges the support of the Shanghai Municipal Science and
Technology Major Project (Grant No.2019SHZDZX01) and the sponsorship from the
Yangyang Development Fund. M.B. would like to thank IFT Madrid, NORDITA and
GIST for the warm hospitality during the completion of this work and
acknowledges the support of the NORDITA distinguished visitor program and GIST
visitor program. H.L. is supported by the National Natural Science Foundation
of China under Grant No. 11405074. D.L. is supported by the National Natural
Science Foundation of China under Grant Nos. 12275108, 12235016, 11805084, and
the Guangdong Pearl River Talents Plan under Grant No. 2017GC010480.

##  Appendix A Comparison between the holographic result for
$G_{\varphi\varphi}^{R}$ and the thermal chiral effective field theory
predictions

In this work, we focused on the dissipative properties of soft pions, which
are intrinsically embodied in the pion correlators. Thus, it is very important
to verify the validity of the procedure to extract the correlators derived
within the holographic soft-wall model. In this Appendix, we compare the
results for the holographic correlators to the correlators obtained in thermal
chiral effective field theory in Refs. [77, 76].

\begin{overpic}[width=134.42113pt]{shootingr2.pdf}
\end{overpic}\begin{overpic}[width=134.42113pt]{shootingr1.pdf}
\end{overpic}\begin{overpic}[width=134.42113pt]{shootingr3.pdf}
\end{overpic}\begin{overpic}[width=134.42113pt]{shootingi2.pdf}
\end{overpic}\begin{overpic}[width=134.42113pt]{shootingi1.pdf}
\end{overpic}\begin{overpic}[width=134.42113pt]{shootingi3.pdf} \end{overpic}

Figure 13: The real and imaginary parts of numerical solutions of $a_{0}(z)$,
$a_{3}(z)$ and $\varphi(z)$ for our benchmark example discussed in Appendix A.
The colored red and blue dashed lines are the solutions from the IR shooting.
The black dot-dashed and gray dashed lines those obtained from the UV
shooting.

\begin{overpic}[width=134.42113pt]{acorrelatormq1.pdf}
\put(85.0,55.0){\bf{(a)}}
\end{overpic}\begin{overpic}[width=134.42113pt]{acorrelatormq3.pdf}
\put(85.0,55.0){\bf{(b)}}
\end{overpic}\begin{overpic}[width=134.42113pt]{acorrelatormq0.pdf}
\put(85.0,55.0){\bf{(c)}} \end{overpic}

Figure 14: The comparison of $G_{\varphi\varphi}(0,k)$ at $T=130{\rm{MeV}}$
between the holographic numerical results and the results from thermal chiral
effective field theory obtained in Refs. [77, 76]. (a) $m_{q}=1{\rm{MeV}}$;(b)
$m_{q}=3.22{\rm{MeV}}$; (c) chiral limit, $m_{q}=0$. The vertical gray dashed
lines mark the momentum positions $k^{2}=m_{\rm{scr}}^{2}$.

In section III.1, we have introduced the correlator of the pion operator in
the soft-wall AdS/QCD model. To obtain the correlator, we calculate the three
coupled second-order differential equations in Eqs (42). At the AdS boundary,
we have three solutions (45)- (51) with six undetermined integration
constants, $a_{t0}$, $a_{t2}$, $a_{x0}$, $a_{x2}$, $\varphi_{0}$ (or
$\bar{\varphi}_{1}$) and $\varphi_{2}$ (or
$\bar{\varphi}_{2}$).191919Considering the reduced functions in Eq. (43) would
cancel out one of the integration constants. Near the horizon, we have
asymptotic solutions (49) with three undetermined integration constants
($a_{b0}$, $\varphi_{b0}$, $\varphi_{b1}$). Besides, the frequency $\omega$
and wave-vector $k$ appear also in the coupled equations.

Since these are linear equations, one can set one of the integration constants
to a fixed value. For convenience, we fix $\varphi_{b0}=1$. We solve the
equations using the so-called “double shooting method”, in which the equations
are integrated shooting both from the horizon $z=z_{h}$ and the UV boundary
$z=0$ and then matched at an intermediate radial position, $0<z_{0}<z_{h}$.

As a concrete example of this procedure, we consider the pion correlator in
Eq. (53) at $T=90$ MeV, $k=1$ MeV and $m_{q}=1$ MeV. Since we are not
interested in other correlators, we set the source for the gauge field
operator $A_{0}$ and $A_{3}$ to be zero, i.e. $a_{t0}=a_{x0}=0$. To obtain the
pole of the correlator, one needs to search for the value of the frequency
$\omega$ which satisfies the zero source boundary condition at the UV
boundary, $\varphi_{0}=0$. To continue, we have five undetermined integration
constants, $a_{t2}$, $a_{x2}$, $\varphi_{2}$, $a_{b0}$ and $\varphi_{b1}$, one
free parameter, the frequency $\omega$, and six matching conditions. The
latter are given by the continuity of the bulk functions and their first
derivatives at the matching point $z_{0}$,

$\displaystyle a_{0}^{b}(z_{0})=a^{h}_{0}(z_{0})\,,\quad
a_{3}^{b}(z_{0})=a^{h}_{3}(z_{0})\,,$
$\displaystyle\varphi_{0}^{b}(z_{0})=\varphi^{h}_{0}(z_{0})\,,\quad\partial_{z}a_{0}^{b}(z_{0})=\partial_{z}a^{h}_{0}(z_{0})\,,$
$\displaystyle\partial_{z}a_{3}^{b}(z_{0})=\partial_{z}a^{h}_{3}(z_{0})\,,\quad\partial_{z}\varphi_{0}^{b}(z_{0})=\partial_{z}\varphi^{h}_{0}(z_{0})$
(61)

in which the indices $b$ and $h$ label respectively the numerical solutions
solved with boundary conditions from the UV, $z=\epsilon$, or with horizon
conditions from the horizon $z=(1-\epsilon)z_{h}$. In our numerics, we take
$\epsilon=10^{-6}$. Besides, we numerically obtain the chiral condensate
$\bar{\sigma}=0.0149$ $\rm{GeV}^{3}$ and $c_{0}=0.756149$ GeV through the EOM
for $\Sigma(z)$ in Eq. (38) using a similar method. We plot the numerical
results form the various bulk functions $A_{0}(z)$, $A_{3}(z)$ and
$\varphi(z)$ in Fig. 13. The two sets of solutions for the functions
$A_{0}(z)$, $A_{3}(z)$, $\varphi(z)$ and their first derivatives are
continuous at the matching point $z_{0}$. In this case, the choice of
parameters which satisfies all the matching conditions is given by:

$\displaystyle a_{t2}=0.527998+0.035872i\ \rm{GeV}^{2}\,,$ $\displaystyle
a_{x2}=-0.006417-0.000432{\rm{GeV}^{2}}\,,$
$\displaystyle\omega=0.074525-0.000023i\ {\rm{GeV}}\,,$
$\displaystyle\varphi_{2}=-886.85+13114.6\ {\rm{GeV}^{2}}\,,$ $\displaystyle
a_{b0}=0.010709+0.000357i\ {\rm{GeV}}\,,$
$\displaystyle\varphi_{b1}=-5.600+67.91i\ {\rm{GeV}^{2}}\,.$

If we are not only interested in the poles of the correlator but in its full
structure, the procedure is slightly different. We now keep the value of the
frequency $\omega$ as an external parameter and solve on the contrary the
matching conditions in terms of the integration constant $\varphi_{0}$.

To verify the validity of the correlator in Eq. (53), we compare
$G_{\varphi\varphi}(q)$ at $T=130$ MeV as a benchmark example. The expected
form for such a correlator is given by [77, 76]

$G_{\varphi\varphi}(0,k)=-\frac{1}{f_{s}^{2}(k^{2}+m_{\rm{scr}}^{2})}\,.$ (62)

In Fig. 14(a) and (b), we choose $m_{q}=1$ MeV and $m_{q}=3.22$ MeV and we
plot the static correlator $G_{\varphi\varphi}(0,k)$ as a function of the
wave-vector $k$. We expand the correlator around $k^{2}=m_{\rm{scr}}^{2}$ and
only retain the leading term $(m_{\rm{scr}}^{2}/k^{2})G_{\varphi\varphi}$. The
dashed lines are the results from Eq. (62) with the pion decay constants
obtained through Eq. (50), $f_{s}^{2}=0.003772$ and $0.003892\
{\rm{GeV}^{2}}$, respectively. In Fig. 14(c), we follow the same procedure in
the chiral limit, $m_{q}=0$. We compare once more the holographic results with
those from Eq. (62) in which the pion decay constant $f_{s}^{2}=0.003718\
{\rm{GeV}^{2}}$ is obtained using Eq. (50). When the quark mass is small, we
find that the holographic results are in perfect agreement with the
predictions from thermal chiral effective field theory confirming the validity
of our computations.

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