# Role of slow, out-of-equilibrium modes on the dynamic structure factor near
the QCD critical point

Golam Sarwar<EMAIL_ADDRESS>Theory Division, Physical Research
Laboratory, Navrangpura, Ahmedabad 380 009, India  Md Hasanujjaman
<EMAIL_ADDRESS>Department of Physics, Darjeeling Government College,
Darjeeling- 734101, India  Jan-e Alam<EMAIL_ADDRESS>Variable Energy
Cyclotron Centre, 1/AF Bidhan Nagar, Kolkata- 700064, India  Homi Bhabha
National Institute, Training School Complex, Mumbai - 400085, India

###### Abstract

The role of slow out of equilibrium modes (OEM), introduced to extend the
validity of hydrodynamics near the QCD critical point on the power spectrum of
dynamical density fluctuations has been studied. We have used the equation of
motion of slow modes for the situation when the extensive nature of
thermodynamics is not altered due to the introduction of OEM. We find that the
extensivity condition puts an extra constraint on the coupling of OEM with the
four divergence of velocity. The dynamic structure factor
($\mathscr{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$) in presence of the OEM
shows four Lorentzian peaks asymmetrically positioned about
$\omega(\text{frequency})=0$, whereas the dynamic structure factor without the
presence of any OEM shows three well-known Lorentzian peaks. The width of the
peaks are reduced in presence of OEM. We find that the asymmetric peaks
originate due to the coupling of the out-of-equilibrium modes with the
hydrodynamic modes. It is also shown that the OEM has negligible effects on
$\mathscr{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ if first order
hydrodynamics (relativistic Navier-Stokes) is used. The introduction of OEM
reduces the width of the Rayleigh peak indicating the reduction in the decay
rate of the fluctuation which leads to slowing down, a well-known
characteristics of the critical end point (CEP).

##  I Introduction

One of the outstanding issue in Relativistic High Energy Collision Experiments
(RHIC-E) is to detect the Critical End Point (CEP) located at some critical
baryonic chemical potential ($\mu_{c}$) and critical temperature ($T_{c}$) in
the QCD (Quantum Chromodynamics) phase diagram. The existence of the CEP in
QCD was suggested in Refs. Halasz et al. (1998); Barducci et al. (1994);
Barducci et al. (1990a); Berges and Rajagopal (1999); Kiriyama et al. (2000)
based on the effective field theoretic models and by lattice QCD simulations
Fodor and Katz (2002, 2004). The experimental search for the CEP has been
taken up through the beam energy scan (BES) program at the Relativistic Heavy
Ion Collider (RHIC) Aggarwal et al. (2010). The search will continue in future
experiments at the Facility for Anti-proton and Ion Research (FAIR) and
Nuclotron- based Ion Collider fAcility (NICA). The exact location of the CEP
is not known from the first principle because of the difficulties associated
with the sign problem of the spin 1/2 Dirac particle (quark) in lattice QCD
calculations Gavai (2015). Some of the QCD based effective models such as NJL
and PNJL predict the location of the CEP Masayuki and Koichi (1989); Barducci
et al. (1990b); Barducci et al. (1994); Kiriyama et al. (2000) but the results
are dependent on the parameters of the models. However, the prediction of the
location of the CEP is not the only problem. Even if the location of the CEP
is predicted accurately its experimental detection is extremely challenging
because the measured quantities are obtained by integration over the space-
time evolution history of the fireball produced in relativistic heavy ion
collisions, that is, the experimental results are superposition of all
temperatures and densities through which the system passes, not only from the
single point ($\mu_{c}$, $T_{c}$) of the phase diagram. The space time
evolution of the fireball are studied by solving relativistic hydrodynamic
equations controlling the conservation of the density of conserved quantities.
The hydrodynamic modeling has been used to study the evolution of the fireball
created in relativistic nuclear collisions with the inclusion of the CEP Du et
al. (2021); Bleicher and Herold (2012); Aguiar et al. (2007); Nonaka and
Asakawa (2005). However, the hydrodynamics breaks down near the CEP Stephanov
and Yin (2018) due to long range correlations and enhanced fluctuations
Asakawa et al. (2000); Jeon and Koch (2003); Koch (2008). The CEP causes the
system to linger its relaxation to equilibrium, termed as the ‘critical
slowing down’ Berdnikov and Rajagopal (2000); Stanley (1987); Ma (1976). It
has been shown in Ref. Stephanov and Yin (2018) that the validity of the
hydrodynamics can be extended near CEP by introducing a scalar non-
hydrodynamic field.

The hydrodynamics is used to describe the slowly evolving modes of the
macroscopic system while the faster non-hydrodynamic modes are set by the
collision dynamics at the microscopic level. The hydrodynamics can be applied
to systems in local equilibrium. The time required to achieve local
equilibrium is much shorter than the time required to attain global
equilibrium and this separation of time scale permits the application of
hydrodynamics. At the CEP the correlation length diverges leading to the
divergence of the time scale for local equilibration which leads to the break
down of the hydrodynamics. In other words, if the system encounters CEP then
the correlation length ($\xi$) diverges and the relaxation time which evolves
as $\sim\xi^{3}$, also diverges leading to critical slowing down.
Consequently, the system stay away from local thermal equilibrium and the
hydrodynamics becomes inapplicable. In such situation, a new variable, $\phi$
representing non-hydrodynamic mode is included in the definition of entropy
along with other hydrodynamical variables to extend the validity of the
hydrodynamics in the vicinity of the CEP Stephanov and Yin (2018). The slow
out-of-equilibrium modes (OEM) are not treated separately Stephanov and Yin
(2018); Rajagopal et al. (2020) as they are coupled with hydrodynamic modes.
In this regards, it is very important to understand the role of $\phi$ on the
critical point dynamics. In the present work the effect of $\phi$ on the
spectral structure will be studied using relativistic causal dissipative
hydrodynamics proposed by Israel and Stewart (IS) Israel (1976). The
dispersion relations for the hydrodynamic modes are governed by the set of
hydrodynamic equations. The extensivity of thermodynamics restricts the
coupling of slow modes with gradients of other hydrodynamic fields.

The fluctuations in condensed matter system has been extensively studied.
Specifically, the power spectrum of the correlation of density fluctuations or
the spectral structure
($\mathscr{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$) has been studied both
experimentally and theoretically and shown to be crucially dependent on
transport coefficients Stanley (1987); Ma (2018); E. Reichl . The thermally
excited fluctuations are governed by the the transport coefficients of the
medium according to the Onsager’s hypothesis Onsager (1931). The light and
neutron scatterings experiments have been performed to investigate the
properties of fluctuations experimentally in condensed matter physics. The
spectrum of scattered light contains separately identifiable peaks of
Lorentzian distribution, called Rayleigh (R) peak Rayleigh (1881) at angular
frequency, $\omega=0$, and Brillouin (B) peaks located symmetrically in the
opposite side of R-peak, experimentally detected by Fleury and Boon Fleury and
Boon (1969). The R-line arises from the entropy or temperature fluctuations at
constant pressure, whereas B-lines arise from the pressure fluctuations at
constant entropy. The width of the R-line is connected to $\kappa/(\rho
C_{P})$ Stanley (1987), where $\kappa$ is the thermal conductivity, $C_{P}$ is
the isobaric specific heat of the fluid with mass density, $\rho$. Therefore,
if the order of divergence for $\kappa$ is weaker than $C_{P}$, then a narrow
R-line appears. Since the width represents the decay rate of the fluctuation a
narrow width will indicate the slower decay of fluctuation. Two Brillouin-
peaks are positioned at $\pm c_{s}k$ with respect to the frequency of the
incident light, where $c_{s}$ is the speed of sound and $k$ is the wave
vector. The finite width of the B-line provides information about the values
of transport coefficients such as shear, and bulk viscosities. The ratio of
intensities of R-line to B-line is:
$I_{R}/2I_{B}=C_{P}/C_{V}-1=K_{T}/K_{S}-1$, called the Landau-Placzek ratio
where $K_{T}$ and $K_{S}$ are isothermal and isothermal and adiabatic
compressibilities respectively. Therefore, the study of the structure factor
is very useful to understand the behaviour of the fluid near the CEP as some
of the transport coefficients and response functions change drastically. Near
the CEP the B-peaks tend to vanish giving rise to severe modification in
$\mathscr{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$. In QCD no external
probe exists to measure such drastic change in
$\mathscr{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$. However, it is a well-
known feature of the critical phenomena that the behaviour of a large class of
systems near CEP are independent of the dynamical details. Since, the CEP in
QCD belongs to same universality class, $\mathscr{O}(4)$ Halasz et al. (1998);
Berges and Rajagopal (1999), as that of liquid-gas critical point, the role of
such slow modes can be tested in liquid-gas system, and the knowledge can be
very helpful to guide the theoretical modeling relevant to the CEP of QCD. In
this context the qualitative and quantitative effects of the slow modes on
various physical quantities should be studied. In the present work the effects
of these modes on the spectral structure of the fluctuations near the CEP
Minami and Kunihiro (2009, 2010); Hasanujjaman et al. (2021) has been
investigated. More specifically, we will study role of the slow modes on the
Rayleigh and Brillouin peaks of
$\mathscr{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$. Thermal fluctuations
can cause entropy production leading to fluctuation in multiplicity. However,
their ensemble average is unaffected Nagai et al. (2016). Thermal fluctuations
may not affect lower flow harmonics but affect their correlation. Recently,
CMS collaborations have measured higher flow harmonics which again carry a
strong signature of thermal fluctuations Chatrchyan et al. (2013) and hence
that of CEP.

The paper is organized as follows. In the next section the expression for
$\mathscr{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ will be derived by
including the variable, $\phi$ representing the non-hydrodynamic mode. In
Sec.III the results are presented and Sec.IV is devoted to summary and
discussion.

##  II Hydrodynamics with out-of-equilibrium modes

The evolution of fluid near the CEP within the scope of fluid dynamics can be
extended by introducing an extra slow degree of freedom representing soft mode
Stephanov and Yin (2018). Here we use the IS hydrodynamical model Israel
(1976) in Eckart frame Eckart (1940) to study the effects of this slow degrees
of freedom. The signature metric for the Minkowski space time is taken as,
$g^{\mu\nu}=(-1,\,1,\,1\,,1)$. The fluid velocity field, $u^{\mu}$ is
normalized as $u^{\mu}u_{\mu}=-1$. The energy-momentum and charge conservation
equations are:

$\partial_{\mu}T^{\mu\nu}=0$ (1)

and

$\partial_{\mu}J^{\mu}=0.$ (2)

where,

$T^{\mu\nu}=\epsilon
u^{\mu}u^{\nu}+(P+\Pi)\Delta^{\mu\nu}+h^{\mu}u^{\nu}+u^{\mu}h^{\nu}+\pi^{\mu\nu},$
(3)

$J^{\mu}=nu^{\mu}+n^{\mu}\,,$ (4)

where $q^{\mu}=h^{\mu}-n^{\mu}(\epsilon+P)/n$ is the heat flux, $h^{\mu}$ is
the vector dissipation or dissipative energy flow, $u^{\mu}$ is the flow
velocity, with $u_{\mu}h^{\mu}=u_{\mu}\pi^{\mu\nu}=0$, $n$ is conserved charge
number density (net baryon density for RHIC-E), $n^{\mu}$ is dissipative
current density, $\epsilon$ is energy density, $P$ is the pressure, $\Pi$ is
the scalar dissipation or the bulk-stress, $\pi^{\mu\nu}$ is the tensor
dissipation or shear stress tensor and
$\Delta^{\mu\nu}=g^{\mu\nu}+u^{\mu}u^{\nu}$ is the projection operator, such
that $\Delta^{\mu\nu}u_{\nu}=0$. In Eckart frame, $n^{\mu}=0$ and
$q^{\mu}=h^{\mu}$. The relaxation equation for additional scalar soft mode
$\phi$ is introduced as Stephanov and Yin (2018):

$D\phi=-F_{\phi}+A_{\phi}\theta\,,$ (5)

where, $\theta=\partial_{\mu}u^{\mu}$ and $D=u^{\mu}\partial_{\mu}$. Forms of
$F_{\phi}$ and $A_{\phi}$ can be obtained by imposing second law of
thermodynamics which states:

$\displaystyle\partial_{\mu}s^{\mu}\geq 0\,,$ (6)

where, $s^{\mu}$ is the entropy four current and is defined as

$\displaystyle s^{\mu}=su^{\mu}+\Delta s^{\mu}\,.$ (7)

In hydro+ formalism, the partial equilibrium entropy gets separate
contribution from the slow modes represented by, $\phi$. The change in entropy
density due to the introduction the scalar field, $\phi$ is given by,

$ds_{+}=\beta_{+}d\epsilon-\alpha_{+}dn-\pi d\phi\,,$ (8)

where, $\pi$ is the ‘energy cost’ for addition of $\phi$ modes in the system
can be called the corresponding chemical potential. Here
$\alpha_{+}=\mu_{+}/T$, where, $T$ is the temperature $\mu_{+}$ is the
chemical potential.

The role of non-hydrodynamic mode on the extensivity condition was ignored in
earlier calculations. In the present work it is found that, this condition is
important for determining the form of $A_{\phi}$. From here onward, we drop
the subscript “$+$” and write the following relations in presence of $\phi$
as:

$\displaystyle dP$ $\displaystyle=$ $\displaystyle sdT+nd\mu+\phi d\pi\,,$ (9)
$\displaystyle sT$ $\displaystyle=$ $\displaystyle\epsilon+P-\mu n-\pi\phi\,.$
(10)

The equation for the soft mode is already of relaxation type in its form,
therefore, the coupling of slowly evolving soft modes at first order is
adequate to maintain causality. The coupling with the soft mode at first order
can be achieved as follows:

$\partial_{\mu}s^{\mu}=(F_{\phi}-b\partial_{\mu}q^{\mu})\pi-q^{\mu}\partial_{\mu}(\beta+b\pi)-\beta(\partial_{\mu}u_{\nu})\Delta
T^{\mu\nu}+\partial_{\mu}(\Delta s^{\mu}+\beta q^{\mu}+b\pi
q^{\mu})+S_{n}\theta\,,$ (11)

where $b$ represents the coupling strength and

$S_{n}=s-\beta(\epsilon+P)+\mu\beta n+\pi A_{\phi}\,.$ (12)

In the Eckart frame the soft mode couples with the heat flux.
$\partial_{\mu}s^{\mu}\geq 0$ can be satisfied, with $S_{n}=0$ and
redefinition of $\pi$, $q^{\mu}$, $\Delta T^{\mu\nu}$ and $s^{\mu}$,
therefore, we have

$\displaystyle\Delta s^{\mu}=-\beta q^{\mu}-b\pi q^{\mu}\,,$ (13)

and

$\displaystyle q^{\mu}$ $\displaystyle=$ $\displaystyle-\kappa
T[Du^{\mu}-\frac{1}{\beta}\Delta^{\mu\nu}\partial_{\nu}(\beta+b\pi)]\,,$ (14)
$\displaystyle F_{\phi}$ $\displaystyle=$
$\displaystyle\gamma\pi-b\partial_{\mu}[\kappa
TDu^{\mu}-\frac{\kappa}{\beta}\Delta^{\mu\nu}\partial_{\nu}(\beta+b\pi)]\,,$
(15)

where the proportionality constants $\kappa\geq 0$, $\gamma\geq 0$ are
respectively thermal conductivity and relaxation rate of slow modes.
Comparison of Eqs.(10) and (12) gives $A_{\phi}=\phi$ for $S_{n}=0$.
Therefore, the modified equations for fluxes in the second order IS theory
with the coupling to soft modes read as follows:

$\displaystyle\Pi=-\frac{1}{3}\zeta\Big{[}\partial_{\mu}u^{\mu}+\beta_{0}D\Pi-\tilde{\alpha}_{0}\partial_{\mu}q^{\mu}\Big{]}\,,$
$\displaystyle q^{\mu}=-\kappa
T\Delta^{\mu\nu}\Big{[}\beta\partial_{\nu}(T+b\pi)+Du_{\nu}+\beta_{1}Dq_{\nu}-\tilde{\alpha}_{0}\partial_{\nu}\Pi-\tilde{\alpha}_{1}\partial_{\lambda}\pi^{\lambda}_{\nu}\Big{]}\,,$
$\displaystyle\pi^{\mu\nu}=-2\eta\Big{[}\Delta^{\mu\nu\rho\lambda}\partial_{\rho}u_{\lambda}+\beta_{2}D\pi^{\mu\nu}-\tilde{\alpha}_{1}\Delta^{\mu\nu\rho\lambda}\partial_{\rho}q_{\lambda}\Big{]}\,,$
(16)

with constants of proportionality $\eta\geq 0$, $\zeta\geq 0$ where $\eta$ and
$\zeta$ are the shear and bulk viscous coefficients respectively. The
quantities, $\tilde{\alpha_{0}}$ and $\tilde{\alpha_{1}}$ are coupling
coefficients, $\beta_{0},\tilde{\beta_{1}},\beta_{2}$ are relaxation
coefficients. The relations of $\beta_{0},\tilde{\beta_{1}}$ and $\beta_{2}$
to the relaxation time scales are given by Muronga (2004, 2002):

$\tau_{\Pi}=\zeta\beta_{0},\,\,\,\,\tau_{q}=\kappa
T\beta_{1},\,\,\,\,\tau_{\pi}=2\eta\beta_{2}\,,$ (17)

The coupling coefficients, which couple to heat flux, and bulk pressure
$(l_{q\Pi},l_{\Pi q})$, the heat flux, and shear tensor $(l_{q\pi},l_{\pi
q})$, are related to the relaxation lengths by the following relations,

$l_{\Pi q}=\zeta\alpha_{0},\,\,\,\,l_{q\Pi}=\kappa
T\alpha_{0},\,\,\,\,l_{q\pi}=\kappa T\alpha_{1},\,\,\,\,l_{\pi
q}=2\eta\alpha_{1}~{}.$ (18)

The expressions for relaxation and coupling coefficients are taken from Ref.
Hasanujjaman et al. (2021).

###  II.1 Linearized equations and the dynamic structure factor

Next we linearize the hydrodynamic equations for small deviations from
equilibrium field quantities. We assume
$\mathscr{Q}=\mathscr{Q}_{0}+\mathscr{\delta Q}$, where $\mathscr{Q}$,
$\mathscr{Q}_{0}$ and $\delta\mathscr{Q}$ represents hydrodynamic variables,
its average value and the fluctuation respectively. $\mathscr{Q}_{0}=0$ is its
average value for dissipative degrees of freedom and
$u_{0}^{\mu}=(-1,\,0,\,0,\,0)$ and $\delta
u^{\mu}=(0,\,\delta\mathchoice{\mbox{\boldmath$u$}}{\mbox{\boldmath$u$}}{\mbox{\boldmath$\scriptstyle
u$}}{\mbox{\boldmath$\scriptscriptstyle u$}})$.

In the linearized domain the equations of motion read as:

$\displaystyle 0$ $\displaystyle=$
$\displaystyle-\frac{\partial\delta\epsilon}{\partial
t}-(\epsilon_{0}+P_{0})+\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}\cdot\delta\mathchoice{\mbox{\boldmath$u$}}{\mbox{\boldmath$u$}}{\mbox{\boldmath$\scriptstyle
u$}}{\mbox{\boldmath$\scriptscriptstyle
u$}}-\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}\cdot\delta\mathchoice{\mbox{\boldmath$q$}}{\mbox{\boldmath$q$}}{\mbox{\boldmath$\scriptstyle
q$}}{\mbox{\boldmath$\scriptscriptstyle q$}}\,,$ (19a) $\displaystyle 0$
$\displaystyle=$ $\displaystyle-(\epsilon_{0}+P_{0})\frac{\partial}{\partial
t}\delta u^{i}-\partial^{i}(\delta P+\delta\Pi)+\frac{\partial}{\partial
t}\delta q^{i}-\partial_{j}\Pi^{ij}\,,$ (19b) $\displaystyle 0$
$\displaystyle=$ $\displaystyle-\frac{\partial}{\partial t}\delta
n-n_{0}\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}\cdot\delta\mathchoice{\mbox{\boldmath$u$}}{\mbox{\boldmath$u$}}{\mbox{\boldmath$\scriptstyle
u$}}{\mbox{\boldmath$\scriptscriptstyle u$}}\,,$ (19c) $\displaystyle 0$
$\displaystyle=$
$\displaystyle\delta\Pi+\frac{1}{3}\zeta[\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}\cdot\delta\mathchoice{\mbox{\boldmath$u$}}{\mbox{\boldmath$u$}}{\mbox{\boldmath$\scriptstyle
u$}}{\mbox{\boldmath$\scriptscriptstyle u$}}+\beta_{0}\frac{\partial}{\partial
t}\delta\Pi-\tilde{\alpha_{0}}\mathchoice{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}\cdot\delta\mathchoice{\mbox{\boldmath$q$}}{\mbox{\boldmath$q$}}{\mbox{\boldmath$\scriptstyle
q$}}{\mbox{\boldmath$\scriptscriptstyle q$}}]\,,$ (19d) $\displaystyle 0$
$\displaystyle=$ $\displaystyle\delta q^{i}+\kappa T_{0}\nabla^{i}\delta
T+\kappa T_{0}\frac{\partial}{\partial t}\delta u^{i}+\kappa
T_{0}\beta_{1}\frac{\partial}{\partial t}\delta q^{i}-\kappa
T_{0}\tilde{\alpha}_{0}\nabla^{i}\delta\Pi-\kappa
T_{0}\tilde{\alpha}_{1}\nabla_{j}\pi^{ij}\,,$ (19e) $\displaystyle 0$
$\displaystyle=$
$\displaystyle\delta\pi^{ij}+2\eta\delta^{ijlm}(\partial_{l}\delta
u_{m}-\tilde{\alpha}_{1}\partial_{l}\delta
q_{m})+2\eta\beta_{2}\frac{\partial}{\partial t}\delta\pi^{ij}\,,$ (19f)
$\displaystyle 0$ $\displaystyle=$ $\displaystyle-\frac{\partial}{\partial
t}\delta\phi-(\gamma+T_{0}^{2}\frac{K_{q\pi}^{2}}{\kappa}\nabla^{2})C_{\phi\pi}\delta\phi-\big{[}\gamma+(T_{0}^{2}\frac{K_{q\pi}^{2}}{\kappa}-\frac{K_{q\pi}}{C_{T\pi}})\nabla^{2}\big{]}C_{T\pi}\delta
T$ (19g)
$\displaystyle-(\gamma+T_{0}^{2}\frac{K_{q\pi}}{\kappa}\nabla^{2})C_{\pi
n}\delta n-T_{0}K_{q\pi}\frac{\partial}{\partial t}(\nabla_{i}\delta
u^{i})+\tilde{\phi}\nabla_{i}\delta u^{i},$

where, $K_{q\pi}=b\kappa$ and

$\displaystyle C_{A\pi}=\frac{\partial\pi}{\partial
A},\,\,\,\,\text{with}\,\,\,A\equiv(T,n,\phi)\,.$ (20)

By applying Fourier-Laplace transformation i.e,
$\displaystyle{\lim_{x\to\infty}}\int
d^{3}x\int_{0}^{\infty}\exp[{(z-\epsilon)t-i\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}}\cdot\mathchoice{\mbox{\boldmath$x$}}{\mbox{\boldmath$x$}}{\mbox{\boldmath$\scriptstyle
x$}}{\mbox{\boldmath$\scriptscriptstyle x$}}}]$ from the left on the above
equation and performing the integration, then, replacing $z$ by $-i\omega$ we
get a set of linear equations in $\omega-k$ space given in the Appendix A.

The set of equations given in Eqs. (29)-(39) can be written in matrix form as

$\displaystyle\mathds{M}\delta\mathscr{Q}=\mathscr{A}\,,$ (21)

where, $\mathds{M}$, is an $11\times 11$ matrix, arranged row wise of
coefficients of $\delta n,\delta T,\delta u_{||},\delta
u_{\perp},\delta\Pi,\delta q_{||}$,
$\delta
q_{\perp},\delta\pi_{||\,||},\delta\pi_{||\,\perp},\delta\pi_{\perp\,\perp}$,
and $\delta\phi$ respectively The set of linear equations can be solved as

$\delta\mathscr{Q}=\mathds{M}^{-1}\mathscr{A}\,,$ (22)

In the present work we are interested in evaluating the two point correlation
of density fluctuation. The solution of the set of equations represented by
Eq. (22) leads to the following expression for density fluctuation ($\delta
n$),

$\displaystyle\delta
n(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},\omega)=\mathds{M}^{-1}_{11}\big{[}-\big{(}\frac{\partial{\epsilon}}{\partial
n}\big{)}\delta
n(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},0)-\big{(}\frac{\partial{\epsilon}}{\partial T}\big{)}\delta
T(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},0)-\big{(}\frac{\partial{\epsilon}}{\partial\phi}\big{)}\delta\phi(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},0)\Big{]}-\mathds{M}^{-1}_{14}\big{[}-\delta
n(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)\Big{]}.$ (23)

Now we define the correlation of density fluctuations, as
$\mathscr{S^{\prime}}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ by the following
expression:

$\displaystyle\mathscr{S^{\prime}}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)=\Big{<}\delta
n(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)\delta
n(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)\Big{>}\,.$ (24)

Since, the correlation between two independent thermodynamic variables, say,
$\mathscr{Q}_{i}$ and $\mathscr{Q}_{j}$ vanishes i.e

$\displaystyle\Big{<}\delta\mathscr{Q}_{i}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},\omega)\delta\mathscr{Q}_{j}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)\Big{>}=0,\,\,\,\,i\neq j\,.$
(25)

Therefore, $\mathscr{S}^{{}^{\prime}}_{nn}$ can be written as

$\displaystyle\mathscr{S}^{{}^{\prime}}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},\omega)=-\Big{[}\big{(}\frac{\partial{\epsilon}}{\partial
n}\big{)}\mathds{M}^{-1}_{11}-\mathds{M}^{-1}_{14}\Big{]}\Big{<}\delta
n(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)\delta
n(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)\Big{>}\,,$ (26)

The final expression for the
$\mathscr{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ can be obtained as:

$\displaystyle\mathscr{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},\omega)=\frac{\mathscr{S}^{{}^{\prime}}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)}{\Big{<}\delta
n(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)\delta
n(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},0)\Big{>}}=-\Big{[}\big{(}\frac{\partial{\epsilon}}{\partial
n}\big{)}\mathds{M}^{-1}_{11}-\mathds{M}^{-1}_{14}\Big{]}\,.$ (27)

This is the spectral correlation in density fluctuation with the inclusion of
the extra degree of freedom, $\phi$.

##  III Results and discussion

Figure 1: (Color online) (a) Variation of
$\mathscr{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ with $\omega$ for
$k=0.1\,fm^{-1}$ and $\eta/s=\zeta/s=\kappa T/s=1/4\pi$, when the system is
away from CEP ($r=0.2$). The peaks are appearing as very narrow due to the
scale chosen along x-axis, change in x-axis scale makes the width visible (see
the inset for the R-peak). (b) The system is close to the CEP ($r=0.01$). The
results are obtained with the EoS containing the CEP and the transport
coefficients, thermodynamic response function and the relaxation coefficients
are estimated from the scaling behaviour.

The aim of this work is to find out the spectral structure of density
fluctuation within the scope of relativistic causal hydrodynamics with its
validity extended near the CEP by introducing a scalar field as discussed
above. The effect of the CEP is taken into consideration through an Equation
of State (EoS), containing the critical point. The EoS is constructed from the
universality hypothesis which suggests that the CEP of the QCD belongs to same
universality class as the 3D Ising model. We do not repeat the discussion on
the construction of the EoS here but refer to the appropriate literature
Parotto et al. (2020); Nonaka and Asakawa (2005); Hasanujjaman et al. (2020,
2021) for details. The behaviour of various transport coefficients and
response functions play crucial roles to determine the structure factor in the
presence of the CEP. The scaling behaviour of various transport coefficients
and thermodynamic functions near the CEP have been taken Kapusta and Torres-
Rincon (2012); Guida and Zinn-Justin (1997); Rajagopal and Wilczek (1993) into
account by using the following relations:

$\displaystyle\kappa_{T}=\kappa_{T}^{0}|r|^{-\gamma^{\prime}},C_{V}=C_{0}|r|^{-\alpha},C_{P}=\frac{\kappa_{0}T_{0}}{n_{0}}\Big{(}\frac{\partial
P}{\partial T}\Big{)}^{2}_{n}|r|^{-\gamma^{\prime}},$ $\displaystyle
c^{2}_{s}=\frac{T_{0}}{n_{0}h_{0}C_{0}}\Big{(}\frac{\partial P}{\partial
T}\Big{)}^{2}_{n}|r|^{\alpha},\alpha_{p}=\kappa_{0}\Big{(}\frac{\partial
P}{\partial T}\Big{)}_{n}|r|^{-\gamma^{\prime}}$
$\displaystyle\eta=\eta_{0}|r|^{1+a_{\kappa}/2-\gamma^{\prime}},\zeta=\zeta_{0}|r|^{-\alpha_{\zeta}},\kappa=\kappa_{0}|r|^{-a_{\kappa}}\,,$
(28)

where $r=(T-T_{c})/T_{c}$ is the reduced temperature and
$\alpha,\,\gamma^{\prime},\,a_{\zeta},\,a_{\kappa}$ are the critical
exponents. Here $\alpha=0.11$ and $\gamma^{\prime}=1.2$. We take the values
$a_{\zeta}=\nu d-\alpha=1.78$ (here, $d=3,\nu=0.63$), and $a_{\kappa}=0.63$.
The partial derivative, $\left(\frac{\partial P}{\partial T}\right)_{n}$ has
been evaluated by using the EoS for consistency. The critical behaviours of
the second-order coupling and relaxation coefficients of the IS hydrodynamics
have also been taken into account (see Hasanujjaman et al. (2021) for
details).

Figure 2: (Color online) Variation of
$\mathscr{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ with $\omega$ and $k$
for $r=0.2$, i.e when the system is away from CEP. Figure 3: (Color online)
Variation of
$\mathscr{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ with $\omega$ for
$k=0.1\,fm^{-1}$ and $r=0.2$, i.e the system is away from CEP for
$K_{q\pi}=0.$

In Fig.1, the variation of
$\mathscr{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ with $\omega$ is shown
for $k=0.1$ fm-1, when the system is away from the CEP (left panel). In
absence of $\phi$, the
$\mathscr{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ has three peaks. These
are the R-peak located at $\omega=0$ and the B-peaks located symmetrically on
either sides of the R-peak. The
$\mathscr{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ admits four peaks in the
presence of $\phi$ which is introduced to extend the validity of
hydrodynamics. It is interesting to note that there is no elastic peak
(R-peak) at $\omega=0$. Two away side peaks are identified as the Brillouin
peaks (B-peaks). The Stokes component (left side) and the anti-Stokes
component (right side) are located asymmetrically on either side of the origin
with unequal magnitudes. The other peak is due to the coupling of $\phi$ with
the heat flux. This profile of
$\mathscr{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ is different from the
one evaluated earlier Hasanujjaman et al. (2021), in absence of $\phi$, where
the Stokes and anti-Stokes components of equal magnitude are symmetrically
located about $\omega=0$. The asymmetry in the B-peaks may arise due to the
local inhomogeneity present in the system. The B-peaks arise from propagating
sound modes associated with pressure fluctuations at constant entropy. In
condensed matter physics, the asymmetry of the B-peaks are understood from the
fact that two sound modes with different $\omega$ values, $-c_{s}k$ and
$+c_{s}k$ originate from different temperature zone’s Rayleigh (1916); Zarate
and Sengers (2006). The asymmetry present in Rayleigh component is exactly
opposite to the Brillouin components. In the vicinity of the CEP (right panel)
the
$\mathscr{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ shows two peaks as the
B-peaks disappear due to the absorption of sound at the CEP.

In Fig. 2 the structure factor in $\omega-k$ plane has been plotted when the
system is away from CEP. The structure of
$\mathscr{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ with non-zero $\phi$ is
quite different from the case when $\phi=0$. In case of $\phi=0$ there is a
$R$-peak and two $B-$ peaks at $(\omega=0)$ and $(\omega=pmc_{s}k)$
respectively with diminishing peak values as $k$ increases. However, with
$\phi\neq 0$ peaks appear at non-zero $k$ values too due to the coupling of
$\phi$ with other hydrodynamic fields. The symmetric structure of
$\mathscr{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ with R and B peaks can
be reproduced when the coupling of $\pi$ (the chemical potential corresponding
to the variable $\phi$) with flux $q$ is set to zero (Fig. 3).

Figure 4: (Color online) Variation of
$\mathscr{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ with $\omega$ for
$k=0.1\,fm^{-1}$ and $r=0.2$, i.e the system is away from CEP but with higher
value of $(\frac{\partial P}{\partial\phi})$. The asymmetry in the B-peaks is
clearly visible.

The asymmetry in
$\mathscr{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ with respect to $\omega$
increases with increase in $\big{(}\partial P/\partial\phi\big{)}$. This is
distinctly visible in Fig. 4 in comparison to results displayed in Fig. 1. The
locations of the four peaks in $\omega$ are obtained from the dispersion
relation provided in the Appendix B.

Figure 5: (Color online) Variation of
$\mathscr{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ with $\omega$ for
$k=0.1\,fm^{-1}$ and $r=0.2$, i.e the system is away from CEP with increased
$\phi$ modes in the system which increases the thermal fluctuation.

Fig. 5 displays the structure factor with increased $\phi$ mode. The height of
the peaks get enhanced significantly due to the enhancement in $\phi$ value.

Figure 6: (Color online) Variation of
$\mathscr{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ with $\omega$ for
$k=0.1\,fm^{-1}$ and $r=0.2$. The results for second order and first order
hydrodynamics are compared here.

In Fig. 6 the structure factor for second-order hydrodynamics has been
compared with first-order hydrodynamics (relativistic Navier-Stokes).
Interestingly the structure factor for first-order hydrodynamics admits a
R-peak at the origin and two symmetric B-peaks located on the opposite sides
of the R-peak. The effects of $\phi$ seems to be inconsequential in the first-
order theory because of the vanishing of various coupling and relaxation
coefficients.

Figure 7: (Color online) Variation of
$\mathscr{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ with $\omega$ for
$k=0.1\,fm^{-1}$ and $r=0.2$ with (dotted line) and without (solid) $\phi$
mode, when the longitudinal modes are only considered.

The $\omega$ dependence of
$\mathscr{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ derived from the
longitudinal dispersion relation with and without $\phi$ has been depicted in
Fig. 7. We find that the width of the B-peaks are small due to the
introduction of the $\phi$ field (blue dashed line) and closer to the R-peak
compared to the case when $\phi=0$. The rate of decay of the thermal
fluctuation reflected through the width of the R-peak becomes smaller due to
the introduction of the field $\phi$, that is, the decay of the fluctuation
becomes slower in the presence of $\phi$. The comparison of this result with
that displayed in Fig. 1 indicate that the extra peak appeared in Fig. 1 is
due to the coupling of the transverse modes with $\phi$. It is interesting to
note that the presence of slow modes can cause the extra peak in the dynamic
structure factor in the presence of the transverse modes in the causal theory
of hydrodynamics.

##  IV Summary and discussions

It has been shown that the validity of second-order hydrodynamics can be
extended near the CEP by introducing an extra degree of freedom Stephanov and
Yin (2018). Here we have derived the equation of slow modes for the situation
when the extensive nature of thermodynamics is not altered due to introduction
of slow modes. We find that the extensivity condition puts extra constraint on
the coupling of scalar slow modes to the velocity four divergence. The role of
this extra out-of-equilibrium mode on the spectral properties of dynamical
density fluctuations near the QCD critical point has been investigated. The
dynamical spectral structure
($\mathscr{S}_{nn}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$) in presence of the out-
of equilibrium modes admits four Lorentzian peaks. Whereas the dynamic
structure factor without out-of-equilibrium modes admits three Lorentzians
peak structure. We find that the asymmetric peaks originate due to coupling of
the out-of-equilibrium modes with the hydrodynamic modes. It is also shown
that the first-order hydrodynamics is unaffected by the extra variable
introduced to broaden the scope of hydrodynamics. The presence of slow modes
can cause the extra peak in the dynamic structure factor due to the presence
of the transverse modes in the causal theory of hydrodynamics. These effects
of slow modes on the dynamic structure factor may help in the experimental
investigation of role of slow modes near the $\mathscr{O}(4)$ critical points
other than that is expected in heavy ion collisions, such as condensed matter
systems. Consequently, by virtue of universality class, the acquired knowledge
on the role of slow modes from experiments on the critical point may be useful
in estimating the effect of the QCD CEP through modeling the hydrodynamic
evolution of system formed in heavy ion collisions near the QCD critical
point. The field representing the OEM, $\phi$ plays a crucial role. It reduces
the width of the distribution representing the thermal fluctuation which
increases the decay time of the fluctuation.

##  V Acknowledgement

GS and MH acknowledge Guruprasad Kadam for fruitful discussions. GS also
thanks Hiranmaya Mishra for support during part of this work. MH would also
like to thank Department of Higher Education, Govt. of West Bengal, India.

##  Appendix A The set linear equations derived for the perturbations in
$\omega-k$ space

The set of equations for the perturbations in hydrodynamic and non-
hydrodynamic ($\phi$) fields in $\omega-k$ space are given in the followings.
Here
$\delta\epsilon(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ is in the $\omega$-space
and
$\delta\tilde{\epsilon}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},t=0)$ is defined in $t$-space.
Similar notation has been used for other fields.

$\displaystyle
i\omega\delta\epsilon(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)+ik\delta
q_{||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},\omega)+ik(\epsilon_{0}+P_{0})\delta
u_{||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},\omega)=-\delta\tilde{\epsilon}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},t=0),$ (29)

$\displaystyle i\omega(\epsilon_{0}+P_{0})\delta
u_{||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)-ik\delta
P(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},\omega)-ik\delta\Pi(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ $\displaystyle+$
$\displaystyle i\omega\delta
q_{||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},\omega)-ik\delta\pi_{||\,||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ (30) $\displaystyle=$
$\displaystyle(\epsilon_{0}+P_{0})\tilde{\delta
u}_{||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},t=0)-\delta
q_{||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega),$
$\displaystyle-i\omega(\epsilon_{0}+P_{0})\delta
u_{\perp}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)-i\omega\delta
q_{\perp}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},\omega)-ik\delta\pi_{\perp\,||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},\omega)=(\epsilon_{0}+P_{0})\delta
u_{\perp}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)-\tilde{\delta
q}_{\perp}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},t=0),$ (31) $\displaystyle
i\omega\delta
n(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)+ikn_{0}\delta
u_{||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)=-\tilde{\delta
n}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},t=0),$ (32)
$\displaystyle(1+i\omega\frac{1}{3}\zeta\beta_{0})\delta\Pi(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)+ik\frac{1}{3}\zeta\delta
u_{||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},\omega)-ik\frac{1}{3}\zeta\tilde{\alpha}_{0}\delta
q_{||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},\omega)=-\frac{1}{3}\zeta\beta_{0}\tilde{\delta\Pi}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},t=0),$ (33)
$\displaystyle(1+i\omega\kappa T_{0}\beta_{1})\delta
q_{||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)+ik\kappa T_{0}\delta
u_{||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)+ik\big{[}\kappa-
C_{T\pi}K_{q\pi}T_{0}^{2}\big{]}\delta
T(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},\omega)-ikT_{0}^{2}K_{q\pi}C_{\phi\pi}\delta\phi(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ $\displaystyle-ik\kappa
T_{0}\tilde{\alpha}_{0}\delta\Pi(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},\omega)-ikT_{0}^{2}K_{q\pi}C_{n\pi}\delta
n(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)-ik\kappa
T_{0}\tilde{\alpha}_{1}\delta\pi_{||\,||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ $\displaystyle\hskip
245.71754pt=-\kappa T_{0}\tilde{\delta
u}_{||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},t=0)-\beta_{1}\kappa
T_{0}\tilde{\delta
q}_{||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},t=0),$ (34)
$\displaystyle(1+i\omega\beta_{1}kT_{0})\delta
q_{\perp}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)+i\kappa
T_{0}(\omega\delta
u_{\perp}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},\omega)-k\tilde{\alpha}_{1}\delta\pi_{\perp||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)\big{)}=-\kappa
T_{0}\big{[}\delta
u_{\perp}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},0)-\beta_{1}\tilde{\delta
q}_{\perp}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},t=0)\big{]}$ (35)
$\displaystyle(1+2i\omega\eta\beta_{2})\delta\pi_{||\,||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)+ik\frac{4}{3}\eta\delta
u_{||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)-\frac{4}{3}\delta
q_{||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},\omega)=-2\eta\beta_{2}\tilde{\delta\pi}_{||\,||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},t=0),$ (36)
$\displaystyle(1+2i\omega\eta\beta_{2})\delta\pi_{\perp\,\perp}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)-ik\frac{4}{3}\eta\delta
u_{||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},\omega)+ik\frac{4}{3}\eta\tilde{\alpha}_{1}\delta
q_{||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},\omega)=-2\eta\beta_{2}\tilde{\delta\pi}_{\perp\,\perp}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},t=0),$ (37)
$\displaystyle(1+2i\omega\eta\beta_{2})\delta\pi_{\perp\,\perp}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)+ik\eta\delta
u_{\perp}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)-ik\delta
q_{\perp}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle
k$}},\omega)=-2\eta\beta_{2}\tilde{\delta\pi}_{\perp\,||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},t=0),$ (38)
$\displaystyle(i\omega+C_{\phi\pi}(\gamma-
T_{0}^{2}\frac{K_{q\pi}}{\kappa}k^{2}))\delta\phi(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)+\\{\gamma-
K_{q\pi}(T_{0}^{2}\frac{K_{q\pi}}{\kappa}-\frac{1}{C_{T\pi}})k^{2}\\}C_{T\pi}\delta
T(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ $\displaystyle+(\gamma-
T_{0}^{2}\frac{K_{q\pi}^{2}}{\kappa}k^{2})C_{n\pi}\delta
n(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)-ik(\tilde{\phi}-i\omega
T_{0}K_{q\pi})\delta
u_{||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},\omega)$ $\displaystyle\hskip
144.54pt=-(1+ik\frac{K_{q\pi}^{2}}{\kappa}T_{0}C_{\phi\pi})\tilde{\delta\phi}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},t=0)-ikT_{0}K_{q\pi}\tilde{\delta
u}_{||}(\mathchoice{\mbox{\boldmath$k$}}{\mbox{\boldmath$k$}}{\mbox{\boldmath$\scriptstyle
k$}}{\mbox{\boldmath$\scriptscriptstyle k$}},t=0)\,,$ (39)

where, subscripts ”$||$” and ”$\perp$” stand for projection along and
perpendicular to $k$ respectively. After expressing $\delta\epsilon$ and
$\delta p$ as:

$\displaystyle\delta\epsilon$ $\displaystyle=$
$\displaystyle\big{(}\frac{\partial\epsilon}{\partial n}\big{)}\delta
n+\big{(}\frac{\partial\epsilon}{\partial T}\big{)}\delta
T+\big{(}\frac{\partial\epsilon}{\partial\phi}\big{)}\delta\phi,$ (40)
$\displaystyle\delta P$ $\displaystyle=$ $\displaystyle\big{(}\frac{\partial
P}{\partial n}\big{)}\delta n+\big{(}\frac{\partial P}{\partial
T}\big{)}\delta T+\big{(}\frac{\partial P}{\partial\phi}\big{)}\delta\phi\,.$
(41)

##  Appendix B Speed of sound and the roots of $\omega(k)$

In this appendix we provide the expressions for the speed of sound and the
roots of $\omega(k)$ with the inclusion of out-of-equilibrium mode $\phi$. The
speed of sound ($c_{s}$) is given by,

$\displaystyle c_{s}^{2}$ $\displaystyle=$ $\displaystyle\Big{(}\frac{\partial
p}{\partial\epsilon}\Big{)}_{s/n}$ (42) $\displaystyle=$
$\displaystyle\frac{sdT+nd\mu+\phi d\pi}{Tds+\mu dn+\pi d\phi}$
$\displaystyle=$ $\displaystyle\mathscr{A}/\mathscr{B}\,,$

where, $\mathscr{A}$ and $\mathscr{B}$ are given by

$\displaystyle\mathscr{A}={s+n\mathscr{F}+\phi\mathscr{G}}\,,$ (43)
$\displaystyle\mathscr{B}$ $\displaystyle=$ $\displaystyle T(\frac{\partial
s}{\partial T})_{\mu,\pi}+T\mathscr{F}(\frac{\partial
s}{\partial\mu})_{T,\pi}+T\mathscr{G}(\frac{\partial
s}{\partial\pi})_{T,\mu}+\mu(\frac{\partial n}{\partial
T})_{\mu,\pi}+\mu\mathscr{F}(\frac{\partial n}{\partial\mu})_{T,\pi}$ (44)
$\displaystyle+$ $\displaystyle\mu\mathscr{G}(\frac{\partial
n}{\partial\pi})_{T,\mu}+T(\frac{\partial s}{\partial
T})_{\mu,\pi}+T\mathscr{F}(\frac{\partial
s}{\partial\mu})_{T,\pi}+T\mathscr{G}(\frac{\partial
s}{\partial\pi})_{T,\mu}\,.$

where,

$\displaystyle\mathscr{F}$ $\displaystyle=$
$\displaystyle\frac{\Big{[}(\frac{\partial s}{\partial
T})-\frac{s}{n}(\frac{\partial n}{\partial
T})\Big{]}+(\frac{\partial\pi}{\partial T})\Big{[}(\frac{\partial
s}{\partial\pi})-\frac{s}{n}(\frac{\partial
n}{\partial\pi})\Big{]}}{\frac{s}{n}(\frac{\partial
n}{\partial\mu})_{T}-(\frac{\partial s}{\partial\mu})_{T}}\,,$ (45)
$\displaystyle\mathscr{G}$ $\displaystyle=$
$\displaystyle\frac{\Big{[}(\frac{\partial s}{\partial
T})-\frac{s}{n}(\frac{\partial n}{\partial
T})\Big{]}+(\frac{\partial\mu}{\partial T})\Big{[}(\frac{\partial
s}{\partial\mu})-\frac{s}{n}(\frac{\partial
n}{\partial\mu})\Big{]}}{\frac{s}{n}(\frac{\partial
n}{\partial\mu})_{T}-(\frac{\partial s}{\partial\mu})_{T}}\,.$ (46)

The four roots of $\omega$ indicating the location of peaks in the structure
factor obtained from the dispersion relation are given below.

$\displaystyle\omega_{1}=\frac{-\sqrt{-(\epsilon_{0}+P_{0})^{2}}+i\epsilon_{0}+iP_{0}}{2T_{0}\chi}+\frac{\eta
k^{2}\sqrt{-(\epsilon_{0}+P_{0}){}^{2}}}{(\epsilon_{0}+P_{0}){}^{2}}+\mathscr{O}(k^{3})\,,$
(47)
$\displaystyle\omega_{2}=\frac{\sqrt{-(\epsilon_{0}+P_{0}){}^{2}}+i\epsilon_{0}+iP_{0}}{2T_{0}\chi}-\frac{\eta
k^{2}\sqrt{-(\epsilon_{0}+P_{0}){}^{2}}}{(\epsilon_{0}+P_{0}){}^{2}}+\mathscr{O}(k^{3})\,,$
(48) $\displaystyle\omega_{3}$ $\displaystyle=$
$\displaystyle\frac{i\gamma(C_{\phi\pi}\epsilon_{T}-C_{{T\pi}}\epsilon_{\phi})}{3\epsilon_{T}}+\frac{ik^{2}}{9\epsilon_{T}(\epsilon_{0}+P_{0})}\Big{[}3T_{0}^{2}\bar{\phi}C_{\phi\pi}\epsilon_{T}\kappa_{{q\pi}}-3T_{0}^{2}\bar{\phi}C_{{T\pi}}\epsilon_{\phi}\kappa_{{q\pi}}+3\chi\bar{\phi}\epsilon_{\phi}$
(49) $\displaystyle-$ $\displaystyle
3T_{0}^{2}C_{{n\pi}}\epsilon_{T}n_{0}\kappa_{{q\pi}}+3T_{0}^{2}C_{{T\pi}}\epsilon_{n}n_{0}\kappa_{{q\pi}}-3\chi\epsilon_{n}n_{0}-3\epsilon_{\phi}P_{0}\kappa_{{q\pi}}-3\epsilon_{0}\epsilon_{\phi}\kappa_{{q\pi}}-3T_{0}\epsilon_{\phi}P_{T}\kappa_{{q\pi}}$
$\displaystyle+$ $\displaystyle
3T_{0}\epsilon_{T}P_{\phi}\kappa_{{q\pi}}+\zeta\epsilon_{T}+4\eta\epsilon_{T}+3T_{0}\chi
P_{T}\Big{]}+\mathscr{O}(k^{3})\,,$ $\displaystyle\omega_{4}$ $\displaystyle=$
$\displaystyle\frac{i\gamma(C_{\phi\pi}\epsilon_{T}-C_{{T\pi}}e_{\phi})}{3\epsilon_{T}}+k\sqrt{\frac{\bar{\phi}\epsilon_{\phi}P_{T}-\bar{\phi}\epsilon_{T}P_{\phi}+\epsilon_{T}n_{0}P_{n}-\epsilon_{n}n_{0}P_{T}+\epsilon_{0}P_{T}+P_{0}P_{T}}{\epsilon_{T}(\epsilon_{0}+P_{0})}}$
(50) $\displaystyle+$
$\displaystyle\frac{ik^{2}}{{9\epsilon_{T}(\epsilon_{0}+P_{0})}}\Big{[}(3T_{0}^{2}\bar{\phi}C_{\phi\pi}\epsilon_{T}\kappa_{{q\pi}}-3T_{0}^{2}\bar{\phi}C_{{T\pi}}\epsilon_{\phi}\kappa_{{q\pi}}+3\chi\bar{\phi}\epsilon_{\phi}-3T_{0}^{2}C_{{n\pi}}\epsilon_{T}n_{0}\kappa_{{q\pi}}$
$\displaystyle+$ $\displaystyle
3T_{0}^{2}C_{{T\pi}}\epsilon_{n}n_{0}\kappa_{{q\pi}}-3\chi\epsilon_{n}n_{0}-3\epsilon_{\phi}P_{0}\kappa_{{q\pi}}-3\epsilon_{0}\epsilon_{\phi}\kappa_{{q\pi}}-3T_{0}\epsilon_{\phi}P_{T}\kappa_{{q\pi}}+3T_{0}\epsilon_{T}P_{\phi}\kappa_{{q\pi}}$
$\displaystyle+$ $\displaystyle\zeta\epsilon_{T}+4\eta\epsilon_{T}+3T_{0}\chi
P_{T}\Big{]}+\mathscr{O}(k^{3})\,.$

The width of the Brillouin peaks can be identified as

$\displaystyle\Gamma_{B}$ $\displaystyle=$
$\displaystyle-{9\epsilon_{T}(\epsilon_{0}+P_{0})}\Big{[}3T_{0}^{2}\bar{\phi}C_{\phi\pi}\epsilon_{T}\kappa_{{q\pi}}-3T_{0}^{2}\bar{\phi}C_{{T\pi}}\epsilon_{\phi}\kappa_{{q\pi}}+3\chi\bar{\phi}\epsilon_{\phi}-3T_{0}^{2}C_{{n\pi}}\epsilon_{T}n_{0}\kappa_{{q\pi}}$
(51) $\displaystyle+$ $\displaystyle
3T_{0}^{2}C_{{T\pi}}\epsilon_{n}n_{0}\kappa_{{q\pi}}-3\chi\epsilon_{n}n_{0}-3\epsilon_{\phi}P_{0}\kappa_{{q\pi}}-3\epsilon_{0}\epsilon_{\phi}\kappa_{{q\pi}}-3T_{0}\epsilon_{\phi}P_{T}\kappa_{{q\pi}}+3T_{0}\epsilon_{T}P_{\phi}\kappa_{{q\pi}}$
$\displaystyle+$ $\displaystyle\zeta\epsilon_{T}+4\eta\epsilon_{T}+3T_{0}\chi
P_{T}\Big{]}\,.$

The speed of sound obtained from the dispersion relation is expressed as:

$\displaystyle
c^{2}_{s}=\frac{\epsilon_{T}n_{0}P_{n}-\epsilon_{n}n_{0}P_{T}+\epsilon_{0}P_{T}+P_{0}P_{T}+\bar{\phi}\epsilon_{\phi}P_{T}-\bar{\phi}\epsilon_{T}P_{\phi}}{\epsilon_{T}(\epsilon_{0}+P_{0})}\,.$
(52)

where, in Eqs.(49)-(52), we have used the notation as

$\displaystyle X_{Y}=\Big{(}\frac{\partial X}{\partial Y}\Big{)}\,.$ (53)

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