#  Characteristic momentum of Hydro+ and a bound on the speed of sound near
the QCD critical point

Navid<EMAIL_ADDRESS>School of Nuclear Science and Technology,
Lanzhou University, 222 South Tianshui Road, Lanzhou 730000, China,  Matthias
<EMAIL_ADDRESS>Department of Physics and Astronomy, University of
Alabama, 514 University Boulevard, Tuscaloosa, AL 35487, USA.

Near the critical point in the QCD phase diagram, hydrodynamics breaks down at
a momentum where the frequency of the fastest hydrodynamic mode becomes
comparable with the decay rate of the slowest non-hydrodynamic mode. Hydro+
was developed as a framework which extends the range of validity of
hydrodynamics beyond that momentum value. This was achieved through coupling
the hydrodynamic modes to the slowest non-hydrodynamic mode. In this work,
analyzing the spectrum of linear perturbations in Hydro+, we find that a slow
mode falls out of equilibrium if its momentum is greater than a characteristic
momentum value. That characteristic momentum turns out to be set by the branch
points of the dispersion relations. These branch points occur at the critical
momenta of so-called spectral curves and are related to the radius of
convergence of the derivative expansion. The existence of such a
characteristic momentum scale suggests that a particular class of slow modes
has no remarkable effect on the flow of the plasma. Based on these results and
previously derived relations to the stiffness of the equation of state, we
find a temperature-dependent upper bound for the speed of sound near the
critical point in the QCD phase diagram.

###### Contents

  1. 1 Introduction
  2. 2 Convergence radius of single-mode Hydro$+$
    1. 2.1 A brief overview of Hydro+
    2. 2.2 A brief overview of spectral curves & hydrodynamic convergence
    3. 2.3 Single mode Hydro$+$
  3. 3 Hydro+ near the critical point in the QCD phase diagram
  4. 4 Constraint on the stiffness of EoS from radius of convergence
  5. 5 Range of applicability of our results
  6. 6 Comment on the gravity dual of Hydro+
  7. 7 Conclusion and Outlook
  8. A Mode collision and the radius of convergence
  9. B Thermodynamics scanning through the critical point

##  1 Introduction

In the previous 15 years, most of the heavy-ion-collision experiments have
been set for high collision energies at RHIC and LHC [3, 34]. In these
experiments, the non-vanishing net baryon charge coming from the incident
nuclei mostly ends up at high rapidity. Meanwhile, the Quark-Gluon-Plasma
(QGP) forming at mid-rapidity carries nearly no baryon charge, i.e.
$\mu_{B}=0$. In order to study the QGP at nonzero $\mu_{B}$, one requires
heavy ion collisions at lower collision energies. This is the goal of the
ongoing Beam Energy Scan (BES) program at RHIC [1, 2] (see also [3] for a
comprehensive review).333High baryon densities are also a focus of study at
the Compressed Baryonic Matter (CBM) experiment at the FAIR facility at GSI,
the Multi-Purpose-Detector (MPD) at the NICA accelerator in Dubna, and the
CSR-External target Experiment (CEE) at the HIAF facility in China. The BES
program provides us with the opportunity to detect signatures of the QCD
critical point, if such a point exists within the region of the phase diagram
which is accessible to experiment [4, 5].

In order to maximize the discovery potential of the experimental efforts, it
is desirable to identify signatures originating from the critical fluctuations
appearing at all length scales near the critical point, a behavior that is
familiar from any second order phase transition. In thermal equilibrium, such
fluctuations have been well-understood for a long time [6]. However, during a
heavy ion collision, critical fluctuations can not possibly stay in thermal
equilibrium [7].

If there is a critical point in the equilibrium phase diagram of QCD, the
phase space trajectory of the droplet of QGP formed in a heavy ion collision
may pass near the critical point as it expands and cools down. Near the
critical point, the equilibration time scale diverges. In other words, the
critical fluctuations fall out-of-equilibrium [9]. One naturally expects these
out-of-equilibrium fluctuations to modify the equation of state (EoS), which
eventually affects the hydrodynamic evolution. The recently developed Hydro+
framework [8] is an approach to self-consistently study the effect of critical
fluctuations on the evolution of hydrodynamic variables such as the fluid
velocity and baryon chemical potential near the critical point.

Coming closer to the critical point, there exists an increasing number of
long-lived non-hydrodynamic modes, which we will refer to as slow modes. This
effect is known as critical slowing down. If the decay rate of these modes is
of order of the frequency of the hydrodynamic modes, i.e.
$\Gamma_{\boldsymbol{Q}}\sim\omega$, the standard hydrodynamic approximation
fails to work. Heuristically, Hydro+ [8] couples these slow non-hydrodynamic
modes, $\phi_{\boldsymbol{Q}}$, to the hydrodynamic modes. Hence, Hydro+ has a
larger regime of validity than hydrodynamics. In this way, Hydro+ is capable
of describing fluctuations closer to the critical point than hydrodynamics is.

The spectrum of linear perturbations around thermal equilibrium in (neutral)
single-mode Hydro+ contains three modes with momentum vector $\boldsymbol{q}$
obeying distinct dispersion relations $\omega(\boldsymbol{q})$: two sound
modes together with a single non-hydrodynamic mode,
$\phi_{\boldsymbol{Q}_{0}}$. The effect of this slow mode is more important at
large values of $q=|\boldsymbol{q}|$, i.e.
$q\gtrsim\Gamma_{\boldsymbol{Q}_{0}}/c_{s}$; the equation of state becomes
stiffer444This terminology refers to the increasing rigidity of the QGP near
the critical point. It behaves more like an incompressible fluid closer to the
critical point. and consequently, the sound velocity increases near the
critical point at such large momenta.

In this work, for the first time, we compute the branch point singularities of
the Hydro+ dispersion relations. These occur at critical values of the
momentum, $q_{c}$, and they can be related to the radius of convergence of
dispersion relations [12, 13, 14, 18]. We further consider the effect of these
singularities on the observable $\Delta c_{s}^{2}$, the shift in the speed of
sound in QCD plasma due to the fluctuations near the critical point. For this
purpose, we explicitly compute the spectrum of linear perturbations,
$\omega(q)$, in a single-mode Hydro+555By a single-mode Hydro+ we mean a toy
model consisting of the coupling between hydrodynamics and
$\phi_{\boldsymbol{Q}_{0}}$ at a fixed $\boldsymbol{Q}_{0}$. , and thereby we
determine the critical momentum $q_{c}$.

We find that at $q<q_{c}$, the $\phi_{\boldsymbol{Q}_{0}}$-mode is still a
fast-decaying mode, that is to say that standard hydrodynamics continues to
hold in this range. However, at $q\gtrsim q_{c}$, the critical slowing down
phenomenon is inevitable. Our work is guided by a recent line of research
studying the radius of convergence of hydrodynamics in holographic models [12,
13, 14, 18, 15, 20, 21, 24, 25, 28, 23, 16, 17, 19, 22, 26, 50, 29, 30]. In
the present work, however, we work entirely in field theory while holography
makes its one and only appearance in section 6.

In the second part of this work, we apply our single-mode Hydro+ results to
QCD plasma near the critical point in the QCD phase diagram. We use a set of
standard assumptions from the literature (see [34] and references therein),
and extract the characteristic momentum $q_{c}$ near the critical point within
the kinetic framework [8]. Let us emphasize that our goal is not to provide
phenomenological values in this work. Instead, we study qualitatively how the
existence of the characteristic momentum scale, the critical momentum $q_{c}$,
affects physical predictions, especially the effective speed of sound near the
critical point of the QCD phase diagram.

This paper is structured as follows. First, in section 2, we introduce Hydro+
as well as the analysis of spectral curves leading to the computation of the
radius of convergence of the hydrodynamic expansion when linearized in the
hydrodynamic field variables. We close that section by elaborating on the
spectrum of linear perturbations in single-mode Hydro+. In section 3, we first
review the effects of fluctuations near the critical point. Then by use of the
results obtained in section 2, the radius of convergence as a function of the
temperature $T$ is computed. Next, section 4 is devoted to investigating the
impact of convergence on the stiffness of the equation of state near the
critical point, and consequently its effect on the speed of sound. In section
5 we discuss the limitations of our study. We discuss a possible gravity dual
of Hydro+ in section 6. Finally, in section 7, we end with the review of our
results and mention some possible follow-up directions.

##  2 Convergence radius of single-mode Hydro$+$

In this section, we first present a brief overview of the relevant aspects of
Hydro+ and subsequently a brief overview of spectral curves and how they
determine the radius of convergence of hydrodynamics. Combining these two
concepts, we then compute the radius of convergence of single-mode Hydro+.

###  2.1 A brief overview of Hydro+

In a system with partially equilibrated states, the evolution is locally
described by hydrodynamics in terms of conserved densities which are referred
to as hydrodynamic fields, energy density $\epsilon(t,\boldsymbol{x})$, charge
density $n(t,\boldsymbol{x})$ and momentum density
$w(t,\boldsymbol{x})\,u^{\mu}$ [41]. Usually, all other microscopic (non-
hydrodynamic) modes, corresponding to non-conserved quantities, decouple from
these densities. However, nearing a critical point in the phase diagram, an
increasing number of non-hydrodynamic (gapped) modes is known to become long-
lived, as their decay rate $\Gamma$ decreases and the correlation length
increases. As a result the standard hydrodynamics fails to work near the
critical point. Hydro+ is a framework that systematically combines the
dynamics of these long-lived modes with that of the conserved densities. A
detailed review of Hydro+ can be found in [8].

In order to introduce the key concepts behind Hydro+ which are of relevance to
our analysis, let us consider a general out-of-equilibrium two-point function,
$G({\boldsymbol{x}_{1},\boldsymbol{x}_{2}})$, of a conserved operator, for
example the energy-momentum tensor. If the scale at which the midpoint
$(\boldsymbol{x}_{1}+\boldsymbol{x}_{2})/2$ varies is much larger than the
scale of the $|\boldsymbol{x}_{1}-\boldsymbol{x}_{2}|$-dependence, then $G$
can be replaced with a continuous set of local modes,
$G_{\boldsymbol{Q}}(\boldsymbol{x})$. Here $\boldsymbol{Q}$ is a continuous
index which denotes the momentum associated with the critical fluctuation. The
separation of scales mentioned above can also be written as $q\ll Q$ where $q$
is the momentum of the hydrodynamic perturbation.

Among the critical fluctuations, the two-point function of $m=s/n$, with the
entropy density $s$, is referred to as
$G_{m}(\boldsymbol{x}_{1},\boldsymbol{x}_{2})$, and it is special. The
corresponding local modes $\phi_{\boldsymbol{Q}}$ are the slowest-varying non-
conserved modes in the system. The presence of such long-lived non-
hydrodynamic modes near the critical point is what is referred to as the
critical slowing down. If the decay rate of these modes is of order of the
frequency of the conserved modes, i.e. $\Gamma_{\boldsymbol{Q}}\sim\omega$,
the standard hydrodynamic description fails to work. Then Hydro+ comes into
play. It actually extends the regime of validity of hydrodynamics near the
critical point [8] through coupling the dynamics of $\phi_{\boldsymbol{Q}}$
with that of the hydrodynamic modes.

###  2.2 A brief overview of spectral curves & hydrodynamic convergence

Recently, methods from complex analysis have been developed in order to
compute the radius of convergence of the hydrodynamic derivative expansion
from the so-called spectral curve of a given theory [13, 14, 12]. In the
present paper we will extend these methods to Hydro+, but before that we start
with a short review.

In hydrodynamics, the spectral curve arises from the determinant of a system
of hydrodynamic perturbation equations that encodes the hydrodynamic
dispersion relations. It is an implicit function of frequency $\omega$ and
momentum $\boldsymbol{q}$ taking the form $F(\omega,\boldsymbol{q})=0$. Here
we consider rotation-invariant theories and states, hence the dependence on
$\boldsymbol{q}$ is through $|\boldsymbol{q}|^{2}=q^{2}$. As an example,
consider the spectral curve
$F(\omega,q^{2})=c_{s}^{2}\omega^{2}-q^{2}=0$666This is actually the general
structure of the spectral curve in gapless theories, expanded to leading order
about $\omega(0)=0$. at small momentum and frequency, corresponding to the
hydrodynamic description in the absence of any derivative (viscous)
correction; it encodes the sound dispersion relation $\omega=\pm c_{s}q$, with
the speed of sound, $c_{s}$. At arbitrary values of the momentum and
frequency, the analytic structure of spectral curves can be very complicated.
Let us limit our discussion to a description whose spectral curve is an
analytic function, which is the case for Hydro+ truncated at leading
order777Considering Hydro+ to subleading order can be viewed similar to
Mueller-Israel-Stewart theory (MIS). Similar to the dissipative shear tensor
in MIS, we here may interpret the slow mode as a resummed version of
contributions of all non-hydrodynamic modes to all orders in
derivatives.888See [18] for theories with non-analytic spectral curve..

Given an analytic spectral curve, there is a relation between the regime of
validity of linear hydrodynamics and critical points of the spectral
curve.999These critical points of spectral curves have nothing to do with the
critical point which occurs in the QCD phase diagram. The identical naming is
a coincidence. To elaborate on this relation, let us recall that the critical
points of spectral curves can be computed as those points
$(\omega^{*},q^{*})$, which satisfy

$F(\omega,q)|_{(\omega^{*},q^{*})}=0\,,\quad\partial_{\omega}(\omega,q)|_{(\omega^{*},q^{*})}=0\,.$
(2.1)

Note that, in general, $q^{*}$ and $\omega^{*}$ are complex-valued. In
general, solutions $\omega(q)$ of eq. (2.1) are referred to as dispersion
relations; one example, the sound dispersion relation was mentioned above. A
subset of all critical points are also branch points in the sense of complex
analysis, singularities of the dispersion relations. Therefore, it is no
surprise that those critical points limit the radius of convergence of the
hydrodynamic derivative expansion in momentum space and also in position space
[19].

In a gapless theory, the spectral curve encodes the dispersion relations
$\omega(q)$ that pass through $(\omega=0,q=0)$. These are the so-called
hydrodynamic dispersion relations. If $F$ is analytic, such $\omega(q)$ can be
found as a Puiseux series about $(\omega=0,q=0)$ [13, 14]. This series then
may have a finite radius of convergence, $q_{c}$, in the complex $q-$plane.
This radius of convergence is set by one of the critical points of $F$, namely
the one closest to the origin; i.e. it is that $q^{*}$ which has the minimal
magnitude $q_{c}=\min\\{|q^{*}|\\}$. This $q_{c}$ is what is referred to as
the radius of convergence of the derivative expansion of $\omega(q)$. In other
words, $\omega(q)$ is an analytic function within the disc $q<q_{c}$, and
develops a non-analyticity at $q=q_{c}$.

In summary, in order to determine the radius of convergence of linear
hydrodynamics from the spectral curve, one has to first find all branches of
Puiseux series passing through $(\omega=0,q=0)$ [15]. Each of these branches
corresponds to one of the hydrodynamic dispersion relations $\omega(q)$. Then
for a particular $\omega(q)$ of interest, e.g. the sound dispersion
$\omega(q)$, the convergence radius of the derivative expansion is the
distance from the nearest singularity of $\omega(q)$ to the origin. That
singularity may be located at a complex-valued momentum.

###  2.3 Single mode Hydro$+$

As mentioned above, there are situations where one (or a set of) non-
hydrodynamic modes decays so slowly that it couples to the hydrodynamic
densities, for example, near the critical point in the QCD phase diagram. Let
us focus on one single mode and call it $\phi$. Then the partial-equilibrium
states are those not only satisfying the constraint related to the conserved
densities, but also satisfying the constraint that the expectation value of
the slow mode takes on a particular value $\phi$. Accordingly, the partial
equilibrium entropy density $s(\epsilon,n)$ should be modified to include the
$\phi$ mode as $s_{(+)}(\epsilon,n,\phi)$ [8]. Then one finds

$ds_{+}=\,\beta_{(+)}d\epsilon-\alpha_{(+)}dn-\pi d\phi\,,$ (2.2)

with $\alpha_{(+)}$ and $\beta_{(+)}$ generalizing $\alpha=\mu/T$ and
$\beta=1/T$, respectively, to the partial-equilibrium states now including the
slow mode $\phi$. Here, $\pi(\epsilon,n,\phi)$ is the thermodynamic force
returning $\phi$ to its equilibrium value. In a complete equilibrium, where
$\phi$ reaches its equilibrium value $\bar{\phi}(\epsilon,n)$,

$\pi(\epsilon,n,\bar{\phi}(\epsilon,n))=0\,.$ (2.3)

The hydrodynamic equations coupled to the relaxation equation of $\phi$ are
then given by

$\displaystyle D\epsilon$ $\displaystyle=$ $\displaystyle-
w_{(+)}\theta-\partial_{(\mu}u_{\nu)}\,\Pi^{\mu\nu},$ (2.4) $\displaystyle Dn$
$\displaystyle=$ $\displaystyle-n\theta-\partial\cdot\Delta J,$ (2.5)
$\displaystyle w_{(+)}Du^{\nu}$ $\displaystyle=$
$\displaystyle-\partial^{\nu}_{\perp}p-\delta_{\perp\lambda}^{\nu}\,\partial_{\mu}\Pi^{\mu\lambda},$
(2.6) $\displaystyle D\phi$ $\displaystyle=$ $\displaystyle-\gamma_{\pi}\pi-
A_{\phi}\theta+\cdots\,,$ (2.7)

with $D=u\cdot\partial,\,\theta=\partial\cdot u$, the fluid velocity
$u^{\mu}$, the viscous stress tensor $\Pi^{\mu\nu}$, the partial equilibrium
enthalpy $w_{(+)}=\epsilon+p_{(+)}$, $A_{\phi}$ is the compression/expansion
susceptibility of $\phi$, $\gamma_{\pi}$ parametrizes the strength of the
returning force driving $\phi$ towards its equilibrium value, and dots stand
for the higher order derivative corrections. Imposing the constraints induced
by the second law of thermodynamics and substituting $\phi(\epsilon,n,\pi)$
into eq.(2.7), this equation takes the form

$D\pi=-\Gamma_{\pi}\pi-\frac{\beta p_{\pi}}{\phi_{\pi}}\theta+\cdots\,,$ (2.8)

where the relaxation rate is defined as
$\Gamma_{\pi}\equiv\gamma_{\pi}/\phi_{\pi}$, with
$\phi_{\pi}=\left(\frac{\partial\phi}{\partial\pi}\right)_{\epsilon,n}$, and
$p_{\pi}=\left(\frac{\partial p}{\partial\pi}\right)_{\epsilon,n}$.

It is easy to show that from the linearized hydrodynamic equations the
dispersion relations of the two sound modes together with the slow mode are
given as the three roots of the following equation [8]

$F(\omega,q^{2})=\,\omega^{2}-q^{2}\left(c_{s}^{2}+\frac{\omega}{\omega+i\Gamma_{\pi}}\,\frac{\beta
p_{\pi}}{\phi_{\pi}w}\right)=\,0\,.$ (2.9)

$F$ is actually the spectral curve of Hydro+ leading order in derivative
corrections. The real part of the expression inside the parentheses is the
square of the effective velocity of sound. Thus the presence of the slow mode
leads to an enhancement of the value of $c_{s}^{2}$ given by [8]

$\Delta c_{s}^{2}=\frac{\omega^{2}}{\omega^{2}+\Gamma_{\pi}^{2}}\,\Delta
c_{s}^{2}(\infty),\,\,\,\,\,\,\,\,\,\,\Delta c_{s}^{2}(\infty)=\,\frac{\beta
p_{\pi}}{\phi_{\pi}w}\,.$ (2.10)

We now define the dimensionless quantities

${\mathfrak{w}}=\frac{\omega}{\Gamma_{\pi}},\,\,\,\,\,\,\,\,{\mathfrak{q}}=\frac{c_{s}\,q}{\Gamma_{\pi}},\,\,\,\,\,\,\,\,\alpha=\frac{\Delta
c_{s}^{2}(\infty)}{c_{s}^{2}}\,,$ (2.11)

through which the spectral curve (2.9) simplifies to

$F({\mathfrak{w}},{\mathfrak{q}}^{2})=\,{\mathfrak{w}}^{2}-{\mathfrak{q}}^{2}\,\frac{i+{\mathfrak{w}}+\,\alpha\,{\mathfrak{w}}}{i+{\mathfrak{w}}}=\,0\,.$
(2.12)

To our knowledge, the spectral curve has not been written in the form (2.12)
before. We will see that this form has several advantages. This spectral curve
is a polynomial of order three. Thus we find here, for the first time, the
dispersion relations which are given by the three algebraic roots
${\mathfrak{w}}({\mathfrak{q}})$ of (2.12), namely

$\begin{split}{\mathfrak{w}}_{1}({\mathfrak{q}})=&\,-\frac{i}{12}\left(4+\frac{2^{7/3}(-1+3(1+\alpha){\mathfrak{q}}^{2})}{3\,\mathcal{D}({\mathfrak{q}})}-\frac{2^{2/3}}{3}\mathcal{D({\mathfrak{q}})}\right)\,,\\\
{\mathfrak{w}}_{2}({\mathfrak{q}})=&\,-\frac{i}{12}\left(4+\frac{2^{4/3}(-i+\sqrt{3})(-1+3(1+\alpha){\mathfrak{q}}^{2})}{\mathcal{D}({\mathfrak{q}})}-2^{2/3}(i+\sqrt{3})\mathcal{D({\mathfrak{q}})}\right)\,,\\\
{\mathfrak{w}}_{3}({\mathfrak{q}})=&\,-\frac{i}{12}\left(4+\frac{2^{4/3}(-i-\sqrt{3})(-1+3(1+\alpha){\mathfrak{q}}^{2})}{\mathcal{D}({\mathfrak{q}})}-2^{2/3}(1+i\sqrt{3})\mathcal{D({\mathfrak{q}})}\right)\,,\end{split}$
(2.13)

where

$\mathcal{D}({\mathfrak{q}})=\,\left(2i+9i(2-\alpha){\mathfrak{q}}^{2}+\,3\sqrt{3}\sqrt{-4-4{\mathfrak{q}}^{4}(1+\alpha^{3})+{\mathfrak{q}}^{2}(-8+20\alpha+\alpha^{2})}\right)^{1/3}.$
(2.14)

Continuing these dispersion relations to complex momenta, there are four
square-root singularities, branch points, in the dispersion relation of each
mode. These stem from the polynomial of order four under the square-root in
$\mathcal{D}({\mathfrak{q}})$. These branch points are located at the roots of

$\begin{split}({\mathfrak{q}}_{1}^{*})^{2}=&\frac{\alpha^{2}+20\alpha-8+\sqrt{\alpha-8}\,(\alpha^{3/2}-8\alpha^{1/2})}{8(1+\alpha)^{3}}\,,\\\
\,\,\,\,\,\,\,\,({\mathfrak{q}}_{2}^{*})^{2}=&\frac{\alpha^{2}+20\alpha-8-\sqrt{\alpha-8}\,(\alpha^{3/2}-8\alpha^{1/2})}{8(1+\alpha)^{3}}\,.\end{split}$
(2.15)

Depending on whether $\alpha<8$ or $\alpha>8$, the branch points occur at
complex or real momenta, respectively, distinguishing the following cases:

  * (i)

When $\alpha<8$, $({\mathfrak{q}}^{*}_{1})^{2}$ and
$({\mathfrak{q}}^{*}_{2})^{2}$ are complex, however
$|{\mathfrak{q}}^{*}_{1}|^{2}=|{\mathfrak{q}}^{*}_{2}|^{2}$.

  * (ii)

At $\alpha=8$, $({\mathfrak{q}}^{*}_{1})^{2}=\,({\mathfrak{q}}^{*}_{2})^{2}$
and both become real.

  * (iii)

When $\alpha>8$, $({\mathfrak{q}}^{*}_{1})^{2}$ and
$({\mathfrak{q}}^{*}_{2})^{2}$ are both real and
$({\mathfrak{q}}^{*}_{1})^{2}<({\mathfrak{q}}^{*}_{2})^{2}$.

In the next subsections, we discuss the spectrum for the above three cases in
detail.

Obviously, the existence of branch points (2.15) limits the range of
analyticity of the dispersion relations (2.13). About the point
$({\mathfrak{w}},{\mathfrak{q}})=(0,0)$, each of the dispersion relations
(2.13) is analytic within the disc
$|{\mathfrak{q}}|\leq\min\\{|{\mathfrak{q}}^{*}_{1}|,|{\mathfrak{q}}^{*}_{2}|\\}$
in the complex ${\mathfrak{q}}$-plane. Then, according to the Puiseux theorem
[13], ${\mathfrak{w}}({\mathfrak{q}})$ can be computed as a convergent series
about ${\mathfrak{q}}=0$; the radius of convergence of this series is set by

${\mathfrak{q}}_{c}=\min\\{|{\mathfrak{q}}^{*}_{1}|,|{\mathfrak{q}}^{*}_{2}|\\}\,.$
(2.16)

We see that ${\mathfrak{q}}_{c}$ is a characteristic momentum scale in the
theory which depends only on $\alpha$.

### Asymptotics of the spectrum

Consider the asymptotic behavior of the dispersion relations (2.13) for the
two cases ${\mathfrak{q}}\ll 1$ and ${\mathfrak{q}}\gg 1$:

  * •

For $\,{\mathfrak{q}}\ll 1$, Hydro+ dispersion relations reduce to the
standard hydrodynamic ones, given by

$\begin{split}{\mathfrak{w}}_{1,2}=&\,\pm{\mathfrak{q}}-\frac{1}{2}\,i\,\alpha\,{\mathfrak{q}}^{2}+O({\mathfrak{q}}^{3})\,,\\\
{\mathfrak{w}}_{3}=&-i+\,i\,\alpha\,{\mathfrak{q}}^{2}+O({\mathfrak{q}}^{4})\,.\end{split}$
(2.17)

The first two modes are nothing but the ordinary sound modes; $\omega=\pm
c_{s}q$ (see (2.11)). The third mode is a fast decaying non-hydrodynamic mode
whose dynamics decouples from the conserved quantities.

  * •

For $\,{\mathfrak{q}}\gg 1$, hydrodynamics seizes to work. The slow mode comes
to couple with the sound modes, yielding

$\begin{split}{\mathfrak{w}}_{1,2}=&\,\pm\sqrt{1+\alpha}\,{\mathfrak{q}}-\frac{i\,\alpha}{2(1+\alpha)}+O(\frac{1}{{\mathfrak{q}}})\,,\\\
{\mathfrak{w}}_{3}=&-\frac{i}{1+\alpha}+O(\frac{1}{{\mathfrak{q}}^{2}})\,.\end{split}$
(2.18)

The first two modes are the sound modes which do now propagate faster than the
ordinary sound modes. This is equivalent to saying that the equation of state
(EoS) has become stiffer. The third mode is a non-hydrodynamic (gapped) mode.
It is important to note that:

    1. 1.

For $\alpha>2$, the gapped slow mode is the longest-lived mode. This implies
that Hydro+ is necessary in this range.

    2. 2.

For $0<\alpha<2$, the dissipative sound modes are the longest-lived modes.
Nevertheless, the decay rate of the slow mode is comparable with that of sound
modes. Therefore, although the slow mode decays faster than the sound modes in
this range, Hydro+ should still be applied.

    3. 3.

For $\alpha=2$, the sound modes and the slow modes decay at the same rate for
${\mathfrak{q}}\gg 1$, see figure 1.

### Spectrum at $\alpha<8$

As it was sown earlier, for $\alpha<8$, each of the three dispersion relations
has two complex-valued singularities. For concreteness, let us consider the
case $\alpha=2$.101010This special case has been studied in [8]. It is special
in the sense that in the large-${\mathfrak{q}}$ limit all modes have the same
imaginary part for $\alpha=2$, as can be seen in figure 1. The location of
branch points is given by

$\mathcal{Q}_{1,2}\equiv{\mathfrak{q}}^{*2}=\,0.166667\pm\,0.096225\,i\,.$
(2.19)

Note that $|\mathcal{Q}_{1}|=|\mathcal{Q}_{2}|$. The corresponding spectrum of
modes has been illustrated in figure 1. As it can be seen in the right panel,
in this special case, $\phi$ mode decays at the same rate as the sound modes
decay at large momenta.

Figure 1: Dispersion relations for $\alpha<8$ (displayed is the example
$\alpha=2$).  The slow mode (green curves) is purely imaginary. Sound modes
(blue and red curves) have complex frequencies, whose velocities are $c_{s}$
at ${\mathfrak{q}}\ll 1$ while they tend to $\sqrt{1+\alpha}\,c_{s}$ at
${\mathfrak{q}}\gg 1$. This is due to the backreaction of the slow mode on the
hydrodynamic evolution. The sound modes are the longest-lived modes for all
values of momentum ${\mathfrak{q}}$. Note that away from this very special
case, for any $2<\alpha<8$ the slow mode will be longer lived than the two
sound modes, i.e. the green curve will be closer to the ${\mathfrak{q}}$-axis
than the blue and red curves. Compare this with figure 2 in [8].

### Spectrum at $\alpha>8$

It turns out that for $\alpha>8$, the dispersion relations (2.13) have two
unequal real-valued singularities. In other words, ${\mathfrak{q}}^{*2}_{1}$
and ${\mathfrak{q}}^{*2}_{2}$ in (2.15) will be two real values in this range
of $\alpha$. In figure 2 we have illustrated the spectrum of modes at
$\alpha=12>8$ case. We find

$\mathcal{Q}_{1}={\mathfrak{q}}^{*2}_{1}=\,0.03125,\,\,\,\,\mathcal{Q}_{1}={\mathfrak{q}}^{*2}_{2}=\,0.03200\,.$
(2.20)

Figure 2:  Dispersion relations for $\alpha>8$ (displayed is the example
$\alpha=12$). There is a universal feature in this sector (for all
$\alpha>8$): for ${\mathfrak{q}}>{\mathfrak{q}}_{2}^{*}$ it is the slow mode
of Hydro$+$ (green curves) which is the longest-lived one. Within the range
${\mathfrak{q}}^{*}_{1}<{\mathfrak{q}}<{\mathfrak{q}}^{*}_{2}$, all three
modes become purely imaginary. For this reason, one cannot distinguish between
the type of modes in this range. Hence, within this range, in the right panel
we display all three modes as gray curves.

As it can be seen in the right panel of Figure.2, the $\phi$ mode is the
slowest mode at large momenta, say at
${\mathfrak{q}}\gtrsim{\mathfrak{q}}_{c}$.

### Spectrum at $\alpha=8$

At this special value of $\alpha$, the magnitudes of the square of the
critical momenta coincide

$\mathcal{Q}_{1}=\mathcal{Q}_{2}=\frac{1}{27}\,.$ (2.21)

All four of the branch points coincide at this value of $\alpha$, see figure
3.

Figure 3:  Dispersion relations for the degenerate case $\alpha=8$. At
$\alpha=8$, the four branch point singularities become degenerate, which leads
to $|{\mathfrak{q}}^{*}_{1}|=|{\mathfrak{q}}^{*}_{2}|$.

##  3 Hydro+ near the critical point in the QCD phase diagram

The important point about QCD plasma near the critical point in its phase
diagram is that its partial-equilibrium states cannot be described only by the
conserved densities. Since the correlation length $\xi$ is much larger than
the thermal equilibrium scale $\sim 1/T$ there, the two point function of the
density or even the higher point functions may be different from the
corresponding values in complete equilibrium. Then the dynamics is given by
the set of equations governing the evolution of conserved quantities together
with those of their two and higher point functions.

In the simplest setup, one only considers one and two point functions of the
conserved densities, $\bar{\Psi}(\boldsymbol{x})$ and
$\bar{G}(\boldsymbol{x}_{1},\boldsymbol{x}_{2})$. It is also important to note
that in such states, entropy is a functional of $\bar{\Psi}(\boldsymbol{x})$
and $\bar{G}(\boldsymbol{x}_{1},\boldsymbol{x}_{2})$:

$S\equiv\,S\left[\bar{\Psi},\bar{G}\right]$ (3.1)

In partial-equilibrium states, $\bar{\Psi}(\boldsymbol{x})$ is a slowly-
varying function of $\boldsymbol{x}$ and similarly
$\bar{G}(\boldsymbol{x}_{1},\boldsymbol{x}_{2})$ is slow on
$(\boldsymbol{x}_{1}+\boldsymbol{x}_{2})/2$, associated with scales larger
than $\ell$, which is much larger than the scale $\xi$ of
$|\boldsymbol{x}_{1}-\boldsymbol{x}_{2}|$ dependence.

The above separation of scales is used to perform a Wigner transformation of
$G$:

$G_{\boldsymbol{Q}}(\boldsymbol{x})=\,\int_{\Delta\boldsymbol{x}}G\left(\boldsymbol{x}+\frac{\Delta\boldsymbol{x}}{2},\boldsymbol{x}-\frac{\Delta\boldsymbol{x}}{2}\right)e^{i\boldsymbol{Q}\Delta{\boldsymbol{x}}}.$
(3.2)

In fact, $G_{\boldsymbol{Q}}(\boldsymbol{x})$ characterizes states which vary
slowly with respect to $\boldsymbol{x}$ compared to $\boldsymbol{Q}$. In other
words

$1/\ell\,\ll\,Q\,.$ (3.3)

For such partial-equilibrium states, the problem simplifies in the sense that
the entropy functional (3.1), or equivalently the 2PI effective action, can be
written as a local functional of $G_{\boldsymbol{Q}}(\boldsymbol{x})$ [8]:

$S_{2}[\bar{\Psi},G]\approx
S_{1}[\bar{\Psi}]+\frac{1}{2}\int_{\boldsymbol{x}}\int_{\boldsymbol{Q}}\text{Tr}(1-C_{\boldsymbol{Q}}G_{\boldsymbol{Q}}+\log
C_{\boldsymbol{Q}}G_{\boldsymbol{Q}})$ (3.4)

where $S_{1}$ is the 1PI effective action and
$C_{\boldsymbol{Q}}=\bar{G}_{\boldsymbol{Q}}^{-1}$.

Now we can consider the slowest eigenmode of
$G_{\boldsymbol{Q}}(\boldsymbol{x})$ locally and call it

$\phi_{\boldsymbol{Q}}(\boldsymbol{x})=\,\text{the slowest eigenmode
of}\,\,G_{\boldsymbol{Q}}(\boldsymbol{x})$ (3.5)

Substituting it back into (3.4), one finds

$S_{2}[\bar{\Psi},G]\approx
S_{1}[\bar{\Psi}]+\frac{1}{2}\int_{\boldsymbol{x}}\int_{\boldsymbol{Q}}(1-\phi_{\boldsymbol{Q}}/\bar{\phi}_{\boldsymbol{Q}}+\log(\phi_{\boldsymbol{Q}}/\bar{\phi}_{\boldsymbol{Q}}))$
(3.6)

Now, in addition to the conserved densities $\bar{\Psi}$, there is another
degree of freedom $\phi_{\boldsymbol{Q}}$. The local equilibrium value of this
mode is denoted by $\bar{\phi}_{\boldsymbol{Q}}$ which is related to local
value of the hydrodynamic degrees of freedom $\bar{\Psi}$.

The equation of motion for $\phi_{\boldsymbol{Q}}$ is given by [8]

$D\phi_{\boldsymbol{Q}}=\,-\Gamma_{\boldsymbol{Q}}\left(\phi_{\boldsymbol{Q}}-\bar{\phi}_{\boldsymbol{Q}}\right)$
(3.7)

where111111Note that equation (3.7) can be also written in the form (2.7).
$\bar{\phi}_{\boldsymbol{Q}}$ can be approximated by [34]

$\bar{\phi}_{\boldsymbol{Q}}\approx\frac{c_{M}\,\xi^{2}}{1+(Q\xi)^{2}}\,.$
(3.8)

In this equation, $c_{M}$ is a constant and $\xi$ denotes the equilibrium
correlation length. The leading behavior of $Q$-dependent of
$\Gamma_{\boldsymbol{Q}}$ near the critical point is given by [35]

$\Gamma(\boldsymbol{Q})=\frac{2D_{0}\xi_{0}}{\xi^{3}}K(Q\xi)\,,$ (3.9)

where $\xi_{0}$ is the value equilibrium correlation length far from the
critical point and $K(x)=\frac{3}{4}[1+x^{2}+(x^{3}-x^{-1})\arctan(x)]$ [8].
In our later computations, we will consider two distinct values for $D_{0}$:

$D_{0}=\,0.1,\,0.5\,\text{fm}\,.$ (3.10)

In order to parameterize $\xi$, we follow [34] and write

$\left(\frac{\xi}{\xi_{0}}\right)^{-2}=\sqrt{\tanh^{2}\left(\frac{T-T_{c}}{\Delta
T}\right)\left(1-\left(\frac{\xi_{\text{max}}}{\xi_{0}}\right)^{-4}\right)+\left(\frac{\xi_{\text{max}}}{\xi_{0}}\right)^{-4}}\,,$
(3.11)

with $\xi_{0}=0.5\,\text{fm}$. In this ansatz, the equilibrium correlation
length, $\xi$, increases to a maximum value $\xi_{\text{max}}$, then decreases
to its value at freeze-out. The finite value of $\xi_{\text{max}}$ indicates
that the trajectory of the QGP droplet is close to the critical point, instead
of running directly through it. We will consider two distinct values for
$\xi_{\text{max}}$:

$\frac{\xi_{\text{max}}}{\xi_{0}}=\,2,6\,.$ (3.12)

It is clear that by approaching the critical point, $\xi_{\text{max}}$
increases. Very close to the critical point, namely in the scaling limit,
$\xi_{\text{max}}$ diverges (see section B for comparison with the case of
passing through the critical point).

In summary, we see that near the QCD critical point, a slowly varying variable
$\phi_{\boldsymbol{Q}}$ indexed by a continuous index $\boldsymbol{Q}$ does
not take its equilibrium value $\bar{\phi}_{\boldsymbol{Q}}(\epsilon,n)$ for
given $\epsilon$ and $n$. In the following subsections we briefly review what
exactly this mode is and how it back-reacts on the QGP droplet near the
critical point.

### What is the lowest mode?

The eigenmodes of the two point function of conserved densities, namely $G$,
can be found by studying the linear perturbations on top of thermal
equilibrium. Considering the linearized hydrodynamic equations as

$D\Psi=\,-L\delta\Psi+\mathcal{O}(\delta\Psi^{2}),$ (3.13)

the linearized evolution equation for 2-point function is given by the
following matrix form equation:

$\partial_{t}G=\,-L(G-\bar{G})-(G-\bar{G})L^{\dagger}+\mathcal{O}(G-\bar{G})^{2}$
(3.14)

As discussed earlier, in the limit $1/\ell\ll Q$, G can be replaced with local
$G_{\boldsymbol{Q}}$ modes. Then equation (3.14) indicates that the slowest
$G_{\boldsymbol{Q}}$ mode, namely $\phi_{\boldsymbol{Q}}$, corresponds to the
smallest eigenvalue of the matrix $L$. It is easy to show that the smallest
eigenvalue of $L$ is $0$. Then the projection of an arbitrary perturbation
$\delta\Psi$ on the corresponding eigenvector turns out to be proportional to
the function $m=s/n$, i.e. the ratio of the entropy density to the baryon
density [8]. Thus one finds that the $\phi_{\boldsymbol{Q}}$ mode is
proportional to the two point function of fluctuations of $\delta m$:

$\phi_{\boldsymbol{Q}}(\boldsymbol{x})\sim\int_{\Delta\boldsymbol{x}}\bigg{\langle}\delta
m\left(\boldsymbol{x}+\frac{\Delta\boldsymbol{x}}{2}\right)\delta
m\left(\boldsymbol{x}-\frac{\Delta\boldsymbol{x}}{2}\right)\bigg{\rangle}\,.$
(3.15)

See Appendix C of reference [8] for an interesting discussion about the
arbitrariness in normalization of $\phi_{\boldsymbol{Q}}$.

### Feedback from the slow mode on $c_{s}^{2}$

Following the discussion in section 2, we simply find that in the present
case, equation (2.10) takes the following form [34]

$\Delta
c_{s}^{2}(\omega)\approx\,\frac{c_{s}^{4}}{2s}\int\frac{d^{3}\boldsymbol{Q}}{(2\pi)^{3}}\,\,[f_{2}(Q\xi)]^{2}\,\left(\frac{\xi}{\xi_{0}}\right)^{4}\,\left(T\frac{\partial}{\partial
T}\left(\frac{\xi}{\xi_{0}}\right)^{-2}\right)^{2}\frac{\omega^{2}}{\omega^{2}+\Gamma^{2}_{\boldsymbol{Q}}}\,,$
(3.16)

with the decay rate $\Gamma_{\boldsymbol{Q}}$ given by (3.9).

Ultimately, what we want to do is to show how the characteristic momentum of
Hydro+ discussed in section 2 limits the range of momenta of the critical
fluctuations contributing to $\Delta c_{s}^{2}$, discussed in the previous
paragraph.

Our strategy is to think of any of the points near the critical point (in the
phase diagram) as one distinct equilibrium state. In other words, we neglect
the time-evolution of the QGP droplet on a specific trajectory through the
phase space. Then for each of these equilibrium states, the calculation of
section 2.3 is applicable. One may consider this an adiabatic approximation.
Needless to say that in this case, the temperature dependence of single-mode
Hydro+ quantities will be through the special parameterization of $\xi$ in
(3.11) as well as the equation of state we will construct below. It should
also be emphasized that (3.11) ignores the dependence of the state on the
baryon density. We just consider states near $n_{B}=0$ in the phase diagram.

### Radius of convergence of derivative expansion near the QCD critical point

Near the QCD critical point, equation (2.7) is replaced with (3.7).
Correspondingly, $\alpha$, given by equation (2.11) becomes a function of
$\boldsymbol{Q}$ in this case; we will refer to it as
$\alpha_{\boldsymbol{Q}}$. This function can be simply extracted from (3.16)
by reading off the individual contribution of each mode to the integral

$\boxed{\alpha_{\boldsymbol{Q}}(\omega\gg\Gamma_{\boldsymbol{Q}})=\frac{\Delta
c_{s,\boldsymbol{Q}}^{2}(\infty)}{c_{s}^{2}}\approx\,\frac{c_{s}^{2}}{2s}\,\frac{Q^{2}\,\Delta
Q}{2\pi^{2}}\,[f_{2}(Q\xi)]^{2}\,\left(\frac{\xi}{\xi_{0}}\right)^{4}\left(T\frac{\partial}{\partial
T}\left(\frac{\xi}{\xi_{0}}\right)^{-2}\right)^{2}}\,,$ (3.17)

where $\Delta Q$ is the range over which we take $Q$ to be approximately
constant.

In order to use our theoretical results obtained from single-mode Hydro$+$,
namely (2.15), we need to know whether $\phi_{\boldsymbol{Q}}$ falls into the
$\alpha<8$ or into the $\alpha>8$ sub-sector. So our main task is to evaluate
$\alpha_{\boldsymbol{Q}}$ near the QCD critical point. To this end, three
things need to be determined: 1) $\Delta Q$, 2) the entropy density $s$ and 3)
the sound velocity $c_{s}$. In our numerical computations, we take $\Delta
Q=0.002\,\text{fm}^{-1}$.

### The entropy density $s$ and the sound velocity $c_{s}$

In order to compute the entropy density, one needs to construct the
thermodynamic equation of state. Without taking into account the effect of the
critical point, $c_{V}$ may be expressed by the following ansatz [34]

$\frac{c_{V}^{\text{no\,C.P.}}}{T^{3}}=\,\left[\left(\frac{a_{H}+a_{L}}{2}\right)+\left(\frac{a_{H}-a_{L}}{2}\right)\tanh\left(\frac{T-T_{\text{C.O.}}}{\Delta
T_{\text{C.O.}}}\right)\right]\,,$ (3.18)

which interpolates between two temperature independent asymptotic values
$a_{L}$ and $a_{H}$, corresponding to the value of $c_{V}/T^{3}$ at
$T_{L}=T_{c}-\Delta T$ and $T_{H}=T_{c}+\Delta T$, respectively. We assume
$\Delta T=0.2T_{c}$. In (3.18), the data corresponding to the crossover is
given by

$T_{\text{C.O.}}=\,T_{c}\,\,\,\,\text{and}\,\,\,\,\Delta
T_{\text{C.O.}}=\,0.6\,T_{c}\,.$ (3.19)

Similar to reference [34], we also choose

$a_{L}=\,0.1\,a_{\text{QGP}},\,\,\,\,\,a_{H}=\,0.8\,a_{\text{QGP}}\,,$ (3.20)

where $a_{\text{QGP}}$ is the value of $c_{V}/T^{3}$ for the non-interacting
ideal gas QGP, given by

$a_{\text{QGP}}=\,\frac{4\pi^{2}(N_{c}^{2}-1)+21\pi^{2}N_{f}}{15}\,,$ (3.21)

with $N_{c}=3$ and $N_{f}=3$ corresponding to the number of colors and
flavors, respectively. Having specified $c_{V}$, one can then directly compute
$s$ and then $c_{s}^{2}$ as the following

$\begin{split}s(T)=&\,\int_{0}^{T}dT^{\prime}\frac{c_{V}(T^{\prime})}{T^{\prime}}\,,\\\
c_{s}^{2}=&\,\frac{s}{c_{V}}\,.\end{split}$ (3.22)

These functions are displayed as blue-dashed curves in figure 4. However, we
have not considered the effect of the critical point so far. For this reason
we refer to these curves as the no-critical-point (no C.P.) result.

Figure 4:  The left (right) vertical dashed line shows the location of
$T_{L}=T_{c}-\Delta T\,(T_{H}=T_{c}+\Delta T)$. In all panels, solid red
curves represent thermodynamic quantities taking the critical contributions
into account, dashed blue curves represent results without critical
contributions. In the bottom left panel, the red dashed line indicates the
critical temperature, $T_{c}=0.16\,\text{GeV}$.

In order to construct $c_{V}$ near the critical point, we need to include the
critical contribution. We follow [34] and take the textbook result
$C_{V}\propto\xi$ [38] to write

$c_{V}^{\text{crit}}(T)=\frac{1}{2}\frac{1}{\xi_{0}^{3}}\frac{\xi(T)}{\xi_{0}}\,.$
(3.23)

with $\xi(T)/\xi_{0}$ given by (3.11) 121212See section B for another
construction of the EoS near the critical point.. It is expected that the
effect of critical fluctuations on $c_{V}$ is only important near the critical
point. In addition, the value of $c_{V}$ should approach that without a
critical point far from $T_{c}$. So, $c_{V}$ can be constructed as follows
[34]

$c_{V}(T)=\begin{cases}c_{V}^{\text{no C.P.}}(T)\,,\quad&\,T\leq T_{L}\\\
c_{V}^{\text{cirt}}(T)+T^{3}\sum_{n=0}a_{n}\left(\frac{T-T_{c}}{\Delta
T}\right)^{n}\,,\quad&\,T_{L}\leq T\leq T_{H}\\\ c_{V}^{\text{no
C.P.}}(T)\,,\quad&\,T\geq T_{H}\end{cases}\,.$ (3.24)

Demanding $c_{V}/T^{3}$ and its derivatives to be continuous at $T=T_{L,H}$,
one can specify coefficients $a_{n}$ to any arbitrary order. We choose to go
up to the second order in derivatives. We then find six equations leading to
six non-zero coefficients131313For $\xi_{\text{max}}=6\xi_{0}=3\,\text{fm}$,
we find
$a_{0}=17.16,\,\,a_{1}=13.80,\,\,a_{2}=1.29\,\,a_{3}=-1.54,\,\,a_{4}=-1.16,\,\,a_{5}=0.64$..
Having specified $c_{V}$ (see the red curve in the top right panel in figure
4), one can use (3.22) to find $s(T)$ and $c_{s}^{2}$. The result is given by
red curves in the two bottom panels of figure 4.

Note that in our present case the sound velocity does not approach zero as
$T\rightarrow T_{c}$. The reason is that the states we are considering here
are near (never at) the critical point in the phase diagram. In appendix B, we
will show how the behavior of above quantities change when passing through the
critical point.

### Computing $\alpha_{\boldsymbol{Q}}$

Considering the specifications mentioned above, in figure 5 we display
$\alpha$ as a function of $T$ for various $\phi_{\boldsymbol{Q}}$ modes in two
cases. The left panel corresponds to $\xi_{\text{max}}=1\,\text{fm}^{-1}$
while the right panel corresponds to $\xi_{\text{max}}=3\,\text{fm}^{-1}$.

For any mode, each plot features two peaks around the critical temperature.
The presence of the two peaks can be understood as follows. Approaching the
critical temperature, the correlation length $\xi$ increases. At the same
time, the rate of change of $\xi$, namely $T\partial_{T}(\xi/\xi_{0})$
increases, too. However, at some point, the rate of the latter begins to
decrease, resulting in $T\partial_{T}(\xi/\xi_{0})=0$ at the critical
temperature. This can be obviously seen in the top left panel of figure 4.
Therefore, this is the competition between the last two factors in eq. (3.17)
that leads to the appearance of the peaks in figure 5. On the other hand, at
larger values of $\xi_{\text{max}}$, the maximum value of $\xi/\xi_{0}$ in
figure 4 will tend to become a singularity. Then the interval between the two
peaks becomes smaller and the peaks become sharper.

Figure 5:  The contribution, $\alpha_{\boldsymbol{Q}}$, of each mode
$\phi_{\boldsymbol{Q}}$ (labelled by the magnitude $Q$) to the stiffness as a
function of temperature. The left panel corresponds to
$\xi_{max}=1\,\text{fm}$, and the right panel to
$\xi_{\text{max}}=3\,\text{fm}$. Values of $\alpha_{\boldsymbol{Q}}$ range
from $\alpha_{\boldsymbol{Q}}<8$ to $\alpha_{\boldsymbol{Q}}>8$ with a
transition between these two regimes occurring at $\alpha_{\boldsymbol{Q}}=8$,
as described in section 2.3.

##  4 Constraint on the stiffness of EoS from radius of convergence

Having found the value of $\alpha_{\boldsymbol{Q}}$ as a function of
temperature, we are now ready to evaluate (2.15). We want to apply these
equations to each of the $\phi_{\boldsymbol{Q}}$-modes and find the
characteristic momentum $q_{c}$, associated with that mode, as a function of
temperature.

Because of the transition between the $\alpha<8$ and $\alpha>8$ cases, shown
in figure 5, we should be careful how to use (2.15). As discussed earlier, for
$\alpha<8$, the convergence radius is set as
${\mathfrak{q}}_{c}=|{\mathfrak{q}}^{*}_{1}|=|{\mathfrak{q}}^{*}_{2}|$. For
$\alpha>8$, however, the correct value of ${\mathfrak{q}}_{c}$ at each
temperature is determined by the minimum value of $|{\mathfrak{q}}^{*}_{1}|$
and $|{\mathfrak{q}}^{*}_{2}|$ at that temperature:

${\mathfrak{q}}_{c}=\min\\{|{\mathfrak{q}}^{*}_{1}|,|{\mathfrak{q}}^{*}_{2}|\\}\,.$
(4.1)

Note that in the present case, $|{\mathfrak{q}}^{*}_{1}|$ and
$|{\mathfrak{q}}^{*}_{2}|$ are functions of $T$ and $Q$ through the dependence
on $\alpha_{\boldsymbol{Q}}(T)$.

Let us recall that ${\mathfrak{q}}_{c}$ is a dimensionless momentum (see
(2.11)); thus having $\alpha_{\boldsymbol{Q}}$ is not enough to find a
sensible result for it in QCD. The corresponding dimensionful quantity is
given by (see (2.11))

$q_{c}=\frac{\Gamma_{{\pi}}\,{\mathfrak{q}}_{c}}{c_{s}^{2}}\,.$ (4.2)

In this equation,

  1. 1.

$\Gamma_{\pi}$ is exactly $\Gamma_{\boldsymbol{Q}}$ defined by (3.9) (see
equation (94) in reference [8]).

  2. 2.

$c_{s}^{2}$ is found in the bottom left panel of figure 4.

  3. 3.

${\mathfrak{q}}_{c}$ is found via applying (2.15) together with (4.1) to the
$\alpha_{\boldsymbol{Q}}(T)$ found in figure 5.

In Figure 6, we have shown $q_{c}$ for various $\phi_{\boldsymbol{Q}}$ modes
at four temperatures near $T_{c}$. In each row, we keep the value of
$\xi_{\text{max}}$ fixed and consider two different values of $D_{0}$. From
the top panels to the bottom ones, we change the value of $\xi_{\text{max}}$.
One observes that

  * •

The larger the value of $\xi_{\text{max}}$ becomes, the smaller the
characteristic momentum gets. It can be intuitively attributed to the fact
that a larger $\xi_{\text{max}}$ corresponds to an equilibrium state closer to
the critical point and consequently to an earlier breakdown of standard
hydrodynamics.

  * •

From these plots it seems that by approaching the critical temperature, either
from above or from blow, $q_{c}$ decreases. As we will show in figure.7, this
behavior continues to persist up to temperatures very close to $T_{c}$.

Figure 6: The characteristic momentum $q_{c}$ at four temperatures near
$T_{c}$ as a function of the momentum $Q$ labeling the critical fluctuation.
From the left to the right panel, $\xi_{\text{max}}$ has tripled. As expected,
this caused the value of $q_{c}$ to decrease. This is because the relaxation
rate of modes in the right panels is four time larger than that of the
corresponding modes in the left panels.

As mentioned earlier, specifically in eq. (3.16), all $\phi_{\boldsymbol{Q}}$
modes seem to contribute to the stiffness of the EoS, and consequently to the
enhancement in magnitude of $c_{s}^{2}$. However, our analysis of the Hydro+
spectrum and its characteristic momentum proves that it is necessary to narrow
this range, resulting in a significant decrease in the value of $\Delta
c_{s}^{2}$, compared to the case in which all modes contribute.

To investigate this issue in detail, let us first list some of our assumptions
and results:

  1. 1.

In all three cases shown in figures 1, 2 and 3, the enhancement in $c_{s}^{2}$
becomes remarkable when ${\mathfrak{q}}$ exceeds ${\mathfrak{q}}_{c}$, namely
${\mathfrak{q}}\gtrsim{\mathfrak{q}}_{c}$. Here ${\mathfrak{q}}$ is the
dimensionless momentum of flow in the single-mode Hydro+. It corresponds to
the length scale over which sound modes and the slow $\phi$ mode vary. The
critical momentum ${\mathfrak{q}}_{c}$ is the dimensionless characteristic
momentum of the theory. In a single-mode Hydro+, ${\mathfrak{q}}_{c}$ depends
on $T$: ${\mathfrak{q}}_{c}\equiv{\mathfrak{q}}_{c}(T)$.

  2. 2.

In the case of QCD near the critical point, we deal with a spectrum of slow
modes, i.e. $\phi_{\boldsymbol{Q}}$. In order to apply Hydro+ to this case, it
is necessary to explicitly distinguish between $\boldsymbol{Q}$ and
$\boldsymbol{q}$. To this end, let us denote that each mode
$\phi_{\boldsymbol{Q}}$ can be decomposed into a set of Fourier modes
$\hat{\phi}_{\boldsymbol{Q}}$ with momenta $\boldsymbol{q}$:

$\phi_{\boldsymbol{Q}}(\boldsymbol{x})=\int_{\boldsymbol{q}}\hat{\phi}_{\boldsymbol{Q}}(\boldsymbol{q})e^{i\boldsymbol{\boldsymbol{q}}\cdot\boldsymbol{x}}$
(4.3)

Here $\boldsymbol{q}$ is the momentum of the flow while $\boldsymbol{Q}$
denotes the momentum of the critical fluctuation. Let us recall that the main
assumption based on which we include $\phi_{\boldsymbol{Q}}$ as a local slow
mode is the separation of scales given by (3.3), or equivalently

$q\ll Q\,.$ (4.4)

  3. 3.

Based on (4.4), we can think of each $\phi_{\boldsymbol{Q}}$ as the slow mode
of a single-mode Hydro+. Then one naturally expects $q_{c}$ to become a
function of both $Q$ and $T$: $q_{c}\equiv q_{c}(Q,T)$.

  4. $\Rightarrow$

Considering the three items above, we conclude that the modes contributing to
the enhancement of $c_{s}^{2}$ near the critical point are only those
satisfying

$q_{c}(Q,T)\ll Q\,.$ (4.5)

In order to understand why this is the case, we define $Q_{j}^{*}(T)$ as the
two roots of the following equation:

$q_{c}(Q^{*}_{j},T)=Q^{*}_{j},\,\,\,\,\,j=1,2\,,$ (4.6)

It turns out that at any temperature around $T_{c}$, only the modes within the
interval $[Q^{*}_{1},Q^{*}_{2}]$ satisfy (4.5). If (4.5) is not satisfied,
namely $Q<q_{c}(Q,T)$, then (4.4) requires the flow to vary slowly, i.e. $q\ll
q_{c}$. This is indeed the standard hydrodynamic range, and there is no need
to consider the impact of critical slowing down. Therefore (4.5) is a
necessary condition for a mode with magnitude $Q$ to be regarded as a slow
mode.

Figure 7, displays $q_{c}(Q,T)$ for exactly the four situations discussed in
figure 6. The blue plane shows $q_{c}=Q$. Then the modes satisfying (4.5)
exist in the regions where the blue plane is located above the orange surface.
The modes corresponding to the intersection of the blue plane with the orange
surface are the $Q^{*}$ modes defined by eq. (4.6).

Figure 7: For any mode $\boldsymbol{Q}$, $q_{c}$ indicates when a single mode
Hydro+ with the slow mode $\phi_{\boldsymbol{\phi}}$ must be applied. We have
illustarated $q_{c}$ as a function of $T$ and $Q$ for four cases:
Top left panel: $\xi_{\text{max}}=1\,\text{fm}$ and $D_{0}=0.1\,\text{fm}$.
Top right panel: $\xi_{\text{max}}=3\,\text{fm}$ and $D_{0}=0.1\,\text{fm}$.
Bottom left : $\xi_{\text{max}}=1\,\text{fm}$ and $D_{0}=0.5\,\text{fm}$.
Bottom right : $\xi_{\text{max}}=3\,\text{fm}$ and $D_{0}=0.5\,\text{fm}$. The
blue plane shows $q_{c}=Q$. According to (4.5), regions where this blue plane
is higher than the orange surface correspond to modes that may contribute to
the stiffness. The large value of $q_{c}$ near $T_{c}$ in the two right panel
plots is due to very small rate of change of the slow mode with respect to
temperature there. On the other hand, the large values of $q_{c}$ at the two
end of the temperate interval the fact that the corresponding $\xi$ is much
smaller than $\xi_{\text{max}}$ there.

Figure 8: Each panel is related to the corresponding panel in figure 7,
showing the line of intersection between the blue and orange surfaces. The
bottom left figure illustrates that there is no intersection between the blue
and orange surfaces in figure 7, which implies that there is not any slow mode
in this case. The dashed (solid) curves indicate the value of $Q_{1}^{*}(T)$
(of $Q_{2}^{*}(T)$) defined in eq. (4.6), representing the lower (upper)
integral boundary on which eq. (3.16) has to be evaluated.

Considering the discussion above, we understand that at any temperature $T$,
only fluctuations with $Q^{*}_{1}(T)\ll Q\ll Q^{*}_{2}(T)$ contribute to
$\Delta c_{s}^{2}$. Thus the actual value of $\Delta c_{s}^{2}$ cannot exceed
the following upper bound

$\boxed{\Delta
c_{s}^{2}\,\,\,<\,\,\,\text{eq.}~{}\eqref{Deltacs2}\,\text{integrated
over}\,\big{[}Q^{*}_{1}(T),Q^{*}_{2}(T)\big{]}}\,.$ (4.7)

In order to determine the range of integration on the right side of (4.7), we
have found the roots of the equation $q_{c}(Q,T)=Q$, shown in figure 8. The
blue curves correspond to the intersection of the magenta plane with the
orange surfaces in figure 7. As a result, the range of integration in the
right side of (4.7) is determined by the interval between the $Q^{*}_{1}$\-
and $Q^{*}_{2}$-curves.

As one would expect, all panels of figure 7 illustrate that we find only modes
within an interval near the critical point, $T=T_{c}$, may contribute to
$\Delta c_{s}^{2}$. Why this is the case can be understood as follows. At a
temperature sufficiently far from $T_{c}$, there are indeed three length
scales in the system, separated as

$\xi(T)\ll\frac{1}{T}\ll\ell\,.$ (4.8)

Here $1/T$ represents the size of the smallest hydrodynamic cell. The right
inequality indicates that hydrodynamics is valid over the scale $\ell$ while
the left one tells us that no effect of critical slowing down is seen; all
critical fluctuations fall in equilibrium faster than the local equilibration
time. However, when approaching the critical temperature from above, both
$\xi(T)$ and $1/T$ will increase, while the former will change faster. Then at
some temperature near $T_{c}$, $\xi(T)$ will exceed $1/T$. It is clear that by
approaching $T_{c}$ from below, $\xi(T)$ will exceed $1/T$ much earlier.
Because in this case $\xi(T)$ is increasing while $1/T$ is
decreasing141414This is actually the reason why in all panels of figure.8 the
bounded region with $T<T_{c}$ is larger than that with $T>T_{c}$.. The
competition between $\xi(T)$ and $1/T$ on the two sides of $T_{c}$ will
finally end up in a new ordering of scales

$\frac{1}{T}\ll\xi(T)\lesssim\ell\,.$ (4.9)

The appearance of a bounded region in the panels of figure 8 is related to the
switch between the ordering of $\xi(T)$ and $1/T$ from (4.8) to (4.9). Inside
these regions, standard hydrodynamics breaks down, the critical slowing down
is inevitable and Hydro+ must be applied.

An important observation regarding figure 8 is that the critical slowing down
is less important for larger values of $D_{0}$. For this reason, from now on,
we only focus on the two top panels of this figure, namely on the case
$D_{0}=0.1\,\text{fm}$. In figure 9, we compare the upper bound found in (4.7)
with the estimate based on eq. (3.16). We have also plotted the ratio of the
latter two, in the right panel.

Figure 9:  Top panel: The blue curve shows the upper bound of the stiffness of
EoS near the QCD critical point. The red curve corresponds to the known
estimate in which all modes $0<Q$ contribute to the stiffness. Bottom panel:
The ratio of the upper bound to the known estimate of $\Delta c_{s}^{2}$.

Top panel of figure 9: Our results show that naively treating all modes as
valid modes in the stiffness calculation would result in a small enhancement
in the speed of sound. When considering the effect of characteristic momentum,
discussed in this paper, we will find that the enhancement is even smaller
than simply found by including all patterns. We also see that the stiffness
only appears in the interval $[T_{L},T_{H}]$, as defined in section 3.

Bottom panel of figure 9: Our analysis reveals that the upper bound of
enhancement could be significantly smaller than the known estimates resulting
from [34]. According to (4.5), only a limited amount of modes has momenta
greater than $q_{c}$. As a result, the bound given in (4.7) becomes smaller
than the value that would be obtained from integration over all modes in eq.
(3.16).

In order to show the magnitude of the stiffness, in figure 10, we illustrate
the enhanced speed of sound as a function of temperature. Comparing the left
and the right figure, one observes that increasing the value of
$\xi_{\text{max}}$, the enhancement of the speed of sound also increases. For
larger values of $D_{0}$, the enhancement will become smaller. The enhancement
of the speed of sound in any case is small, which is similar to the case of
the bulk viscosity enhancement being small [52], however, there authors
integrated over the whole range of $Q$ considering the Ising equation of
state.

Figure 10:  Dashed blue curves: The speed of sound squared when neglecting the
influence of the critical point. Red curves: The speed of sound squared when
taking into account the influence of the critical point on the thermodynamic
quantities. Dashed black curves: The speed of sound squared when taking into
account the influence of the critical point on both, the thermodynamic and
hydrodynamic quantities by extending hydrodynamics to Hydro+.

##  5 Range of applicability of our results

There are three important physical scales in our problem that compete with
each other [8]:

  * •

The background evolution rate $\omega\sim c_{s}\,q$.

  * •

The slow mode decay rate $\Gamma_{\boldsymbol{Q}}\sim\xi^{-3}$. In fact
$\Gamma_{\boldsymbol{Q}}$ corresponds to $\Gamma_{\pi}$ in single-mode Hydro+
(see eq.(2.8)).

  * •

The shear and diffusion relaxation rate $\Gamma_{\eta}\sim\xi^{-2}$.

Far from any critical point, $q\ll\xi^{-1}$ and therefore

$\omega\ll\Gamma_{\eta}\lesssim\Gamma_{\boldsymbol{Q}}\,.$ (5.1)

Thus standard hydrodynamics is the valid picture at long wave length, and the
next major correction comes from statistical fluctuations, i.e. the long time
tails [40, 41, 42, 43, 44, 45]. In this case, there is no any slowing down
mode.
However, near a critical point, $\xi^{-1}$ can be so small that
$\Gamma_{\boldsymbol{Q}}\lesssim\Gamma_{\eta}$. This is the well-known
critical slowing down limit, discussed in previous sections. Now depending on
the value of $\omega$, different scenarios can take place. When

$\Gamma_{\boldsymbol{Q}}\lesssim\omega\ll\Gamma_{\eta}\,.$ (5.2)

hydrodynamics breaks down and Hydro+ applies. When passing by the QCD critical
point, depending on the trajectory that QGP droplet follows in the $T-\mu$
phase diagram, it may satisfy (5.2). This is exactly the regime of
applicability of our result. It is worth noting that equations (2.4)-(2.7) are
consistent with (5.2). Since $\Gamma_{\boldsymbol{Q}}\ll\Gamma_{\eta}$, the
shear tensor is completely equilibrated and tracks the hydrodynamic variables
and slow mode.

In a more rapidly evolving background, i.e.

$\Gamma_{\boldsymbol{Q}}\ll\Gamma_{\eta}\lesssim\omega,$ (5.3)

Hydro+ breaks down as well (see [39] for a review). In this regime, it is
necessary to take into account the effect of hydrodynamic fluctuations, the
long time tails [10]. Then the problems translates to studying the radius of
convergent of hydrodynamics near the critical point and in the presence of
long-time tails [46]. It is actually beyond the scope of our analysis in the
current study.

##  6 Comment on the gravity dual of Hydro+

In reference [31], the hydrodynamic theory incorporating slowly damping gapped
modes has been developed. The so-called quasi-hydrodynamics theory of [31]
describes those systems in which one (or a finite number of gapped) mode(s) is
(are) parametrically slow, in the sense that their life-time at
${\mathfrak{q}}\ll 1$ is comparable with that of hydrodynamic modes. The
equations of quasi-hydrodynamics then describe the dynamics of hydrodynamic
modes together with the slow mode(s) at small momenta.

Authors of reference [31] introduce several examples for the quasi-
hydrodynamics. One of their examples is the Muller-Israel-Stewart
theory151515We assume that the decay rate of the shear mode in the MIS theory
is parametrically larger then that of other infinitely many non-hydrodynamic
modes. This assumption is necessary to extend the range of applicability of
MIS theory beyond the standard hydrodynamic approximation [8].. They find the
following equation for the sound modes in this theory 161616The example
illustrated in [31] is similar to both our reference1 and reference1 in [8].
Indeed, in all these three cases, $\alpha=2$. In reference [31] the authors
choose $\eta/(\epsilon+p)\tau=1/2$ together with $c_{s}^{2}=1/3$. Then by
using (6.3) one finds $\alpha=2$.:

$\text{MIS}:\,\,\,\,\,\,\omega^{3}+\frac{i}{\tau}\omega^{2}-\left(c_{s}^{2}+\frac{4}{3}\frac{\eta}{(\epsilon+p)\tau}\right)\omega\,q^{2}-\frac{ic_{s}^{2}}{\tau}q^{2}=\,0\,.$
(6.1)

Let us compare it with (2.9) in its simplified form:

$\text{Hydro}+:\,\,\,\,\,\omega^{3}+i\Gamma\,\omega^{2}-\big{(}c_{s}^{2}+\,\Delta
c_{s}^{2}(\infty)\big{)}\omega\,q^{2}-\,i\,c_{s}^{2}\,\Gamma\,q^{2}=\,0\,.$
(6.2)

One immediately notices that at the linearized level, MIS theory is exactly
the same as Hydro$+$ by the following identifications

$\Gamma\equiv\frac{1}{\tau},\,\,\,\,\,\,\Delta
c_{s}^{2}(\infty)\equiv\frac{4}{3}\frac{\eta}{(\epsilon+p)\tau}\,.$ (6.3)

As it was expected, we see that Hydro$+$ is one another example of quasi-
hydrodynamics. Interestingly, reference [31] uncovers the equations of MIS
theory from a holographic higher-derivative Einstein-Gauss-Bonnet gravity
theory, which was previously shown to be able to exhibit long-lived massive
excitations within the low-energy (hydrodynamic) regime [32]. Then we find out
that linearized Hydro$+$ can be derived from Einstein-Gauss-Bonnet gravity
theory, too.

What is the consequence of the above discussion? In last sections we discussed
the spectral curve of Hydro$+$ in details. We analytically showed that the
singularities of modes were all square-root type. When combined with the
discussion of the previous paragraph, one concludes that the spectral curve of
holographic theories at finite coupling and at small momentum is analytic
except for some square-root singularities. It is actually in agreement with
well-known examples in the literature [33]. It may then help us to better
understand the full nature of the spectral curve in holographic theories which
is still an open question.

##  7 Conclusion and Outlook

Motivated by the recent studies on the convergence of the derivative expansion
in holographic theories, in this work we applied the same idea to
hydrodynamics near the QCD critical point. For the first time, we computed the
radius of convergence, $q_{c}$, of Hydro+, as well as its effect on the
observable $\Delta c_{s}^{2}$, the shift in the speed of sound in QCD plasma
due to the fluctuations near the critical point. We obtained three distinct
results:

  1. 1.

We computed and analyzed the full spectrum of linear perturbations in single-
mode Hydro+. We computed the critical point of the spectral curve and
identified the radius of convergence of the derivative expansion, given by the
critical momentum $q_{c}$, see equations (2.15) and (2.16). The key message is
that $q_{c}$ distinguishes between the regime of validity of standard
hydrodynamics and the regime where Hydro+ needs to be employed.

  2. 2.

For the hydrodynamic description of QCD near the critical point, $q_{c}$ is a
function of the momentum of critical fluctuations, namely $\boldsymbol{Q}$, as
well as the temperature, $T$, as visualized in figure 7.

  3. 3.

At any temperature, the competition between $Q$ and $q_{c}(Q)$ determines
whether the mode $\phi_{\boldsymbol{Q}}$ actually should be regarded as a slow
mode in single-mode Hydro+ or not. If $Q<q_{c}(Q)$, standard hydrodynamics
suffices. By finding $Q_{1,2}^{*}(T)$ as the roots of the equation
$Q^{*}=q_{c}(Q^{*})$, we discussed that at any temperature only critical modes
within the interval $(Q_{1}^{*},Q_{2}^{*})$ may contribute to $\Delta
c_{s}^{2}$. This leads to an upper-bound for $\Delta c_{s}^{2}/c_{s}^{2}$ near
the critical point, which is calculated numerically and given in Fig 9.

We think that the upper-bound given in reference9 is not saturated. The reason
is that our desired separation of scale (4.4) demands $Q$ to be much larger
than $q$. So as we found in (4.5), $Q$ is much larger than $q_{c}$. Thus not
all the modes within the interval $(q_{c}(Q^{*}),\xi^{-1})$ are able to
contribute significantly. In this sense, the bound which is the result of
integration over the whole interval, is not saturated.

In this work we did not consider any non-linear effects in the time-evolution
of fluctuations. As discussed in section 5, in more rapidly evolving
backgrounds than what we study here, the presence of non-linear effects is
inevitable. Including such effects in the present study requires extending our
knowledge about hydrodynamics in two directions. First, the convergence of the
derivative expansion in the presence of long-time tails should be explored
[46]. Second, the non-linear structure of Hydro+ should be understood [11].
Then, combing the latter two points will provide us with the background on
which we can take into account the effect of non-linearities, which we leave
to future work. It would be also very interesting to pursue this discussion in
the framework of Schwinger-Keldysh effective field theory .

In a more realistic situation, one should also include the effect of the
evolution of the QGP droplet, itself. Following the study of long-time tails
[42] and the evolution of critical fluctuations [37] in a Bjorken flow, it
would be very interesting to explore what the effect of the evolution of the
background fluid on the results of our paper would be. More realistically, one
should also include the fact that the QGP is a highly vortical plasma [47,
48]. Finally, hydrodynamics is the description of deviations from a known
state, for example Bjorken flow [49], and more generally it is necessary to
consider nonlinear flows beyond Bjorken flow as well as higher orders in the
hydrodynamic derivative expansion [50, 51].

## Acknowledgment

We would like to thank Misha Stephanov and Yi Yin for valuable discussions,
and Sašo Grozdanov for helpful comments on a draft of this work. This work was
supported, in part, by the U.S. Department of Energy grant DE-SC-0012447, and
by grant number 561119208 “Double First Class” start-up funding of Lanzhou
University, China.

##  Appendix A Mode collision and the radius of convergence

According to refs.[12, 13, 14, 18], in order to find the radius of
convergence, one has to find the singularity of dispersion relations, the
nearest to the origin. Since in our present case, the spectral curve (2.12) is
analytic, the singular points are those at which
$\partial_{{\mathfrak{w}}}F=\,0$. One can simply check that indeed, singular
points given by (2.15) satisfy this equation.

On the other hand, the points at which $\partial_{{\mathfrak{w}}}F=\,0$, are
the critical points of $F$. In the following, by studying the modes at complex
momenta in the single-mode Hydro+, we explicitly demonstrate that complexified
modes collide at the critical points.

Figure 11 is devoted to the case $\alpha<8$. As it is seen, at small values of
$|{\mathfrak{q}}|$, the two sound modes are more slowly decaying than the slow
mode. The sound modes can be still described with the standard hydrodynamics.
They actually form a low energy decoupled sector in the system. However, by
increasing $|{\mathfrak{q}}|$, their dynamics becomes gradually coupled with
the dynamics of the slow mode. Then Hydro+ must be considered as the
alternate. As it is seen in the figure, at ${\mathfrak{q}}_{c}\approx 0.43$,
the two trajectories collide. In fact the slow mode collide with the two sound
modes at two different values of $\theta$. We see that after the collision,
the sound modes cannot be considered as a decoupled low energy sector anymore.
We conclude that at $\alpha<8$, the critical momentum ${\mathfrak{q}}_{c}$
corresponds to collision of the slow mode with the sound modes.

Figure 11: modes for $\alpha=2$ in the complex ${\mathfrak{w}}-$plane, at
various values of the complexified momentum
${\mathfrak{q}}^{2}=|{\mathfrak{q}}^{2}|e^{i\theta}$. Large dots and
correspond to the poles with purely real momentum (i.e. at $\theta=0$). As
$\theta$ increases from $0$ to $2\pi$, each pole moves counter-clockwise
following the trajectory whose color changes continuously from blue to red. In
all panels, the two blue dots located in the 3rd and 4th quadrants indicate
the two sound modes and the single blue dot on the vertical axis illustrates
the slow mode, at $\theta=0$.

Figure 12: modes for $\alpha=12$ in the complex ${\mathfrak{w}}-$plane, at
various values of the complexified momentum
${\mathfrak{q}}^{2}=|{\mathfrak{q}}^{2}|e^{i\theta}$. Large dots correspond to
the poles with purely real momentum (i.e. at $\theta=0$). As $\theta$
increases from $0$ to $2\pi$, each pole moves counter-clockwise following the
trajectory whose color changes continuously from blue to red. In all panels,
the two blue dots located in the 3rd and 4th quadrants indicate the two sound
modes and the single blue dot on the vertical axis illustrates the slow mode,
at $\theta=0$.

Figure 12 is devoted to the case $\alpha>8$. As it is seen, at small values of
$|{\mathfrak{q}}|$, the two sound modes are more slowly decaying than the slow
mode. The sound modes can be still described with the standard hydrodynamics.
However, by increasing $|{\mathfrak{q}}|$, their dynamics becomes gradually
coupled with the dynamics of the slow mode. Then Hydro+ must be considered as
the alternate. At ${\mathfrak{q}}_{c}\approx 0.145$, the two sound modes
collide, although the trajectory of slow modes does not collide with that of
sound modes yet. Thus at $\alpha>8$, the critical momentum
${\mathfrak{q}}_{c}$ corresponds to collision of the two sound modes.

As we showed earlier, the QCD near the critical point falls in $\alpha<8$
sector. Thus it is the collision between two sound modes with the slow mode
that determine the regime of validity of hydrodynamics, or equivalently the
characteristic momentum $q_{c}$, there.

##  Appendix B Thermodynamics scanning through the critical point

We follow [36, 37] and map the critical point of QCD to that of the Ising
model. Doing so, it turns out that

$\frac{\xi}{\xi_{0}}=\,\left(\frac{|T-T_{c}|}{T_{c}}\right)^{-a\nu},\,\,\,\,a=1.12,\,\,\nu=0.63$
(B.1)

with $T_{c}=0.16\,\text{GeV}$. Applying this to (3.24), we can simply fix the
first six non-vanishing coefficients $a_{n}$ as:

$a_{0}=23.12,\,\,a_{1}=10.25,\,\,a_{2}=1.85,\,\,a_{3}=-1.56,\,\,a_{4}=-0.69,\,\,a_{5}=0.43\,.$
(B.2)

Having specified $c_{V}$, one can easily find $c_{s}^{2}$ and $s$, as shown in
(13). It can be clearly seen that $c_{s}^{2}\rightarrow 0$ at the critical
point.

Figure 13:  The left (right) vertical dashed line shows the location of
$T=T_{c}-\Delta T\,(T=T_{c}+\Delta T)$. In all panels, red curves represent
thermodynamic quantities by considering the critical contributions.

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