# Soft pions and transport near the chiral critical point

Eduardo Grossi<EMAIL_ADDRESS>Center for Nuclear Theory,
Department of Physics and Astronomy, Stony Brook University, Stony Brook, New
York 11794, USA  Alexander Soloviev<EMAIL_ADDRESS>Center
for Nuclear Theory, Department of Physics and Astronomy, Stony Brook
University, Stony Brook, New York 11794, USA  Derek Teaney
<EMAIL_ADDRESS>Center for Nuclear Theory, Department of Physics
and Astronomy, Stony Brook University, Stony Brook, New York 11794, USA
Fanglida Yan<EMAIL_ADDRESS>Center for Nuclear Theory,
Department of Physics and Astronomy, Stony Brook University, Stony Brook, New
York 11794, USA

###### Abstract

Background: During the expansion of a heavy ion collision, the system passes
close to the $O(4)$ critical point of QCD, and thus the fluctuations of the
order parameter $(\sigma,\vec{\pi})$ are expected to be enhanced. Purpose: Our
goal is to compute how these enhanced fluctuations modify the transport
coefficients of QCD near the pseudo-critical point. We also make a
phenomenological estimate for how chiral fluctuations could effect the
momentum spectrum of soft pions. Method: We first formulate the appropriate
stochastic hydrodynamic equations close to the $O(4)$ critical point. Then,
working in mean field, we determine the correlation functions of the stress
tensor and the currents which result from this stochastic real time theory,
and use these correlation functions to determine the scaling behavior of the
transport coefficients. The hydrodynamic theory also describes the propagation
of pion waves, fixing the scaling behavior of the dispersion curve of soft
pions. Results: We present scaling functions for the shear viscosity and the
charge conductivities near the pseudo-critical point, and estimate the
absolute magnitude of the critical fluctuations to these parameters and the
bulk viscosity. Using the calculated pion dispersion curve, we estimate the
expected critical enhancement of soft pion yields, and this estimate provides
a plausible explanation for the excess seen in experiment relative to ordinary
hydrodynamic computations. Conclusions: Our results motivate further
phenomenological and numerical work on the implications of chiral symmetry on
real time properties of thermal QCD near the pseudo-critical point.

###### pacs:

##  I Introduction

Measurements on heavy ion collisions at the Relativistic Heavy Ion Collider
(RHIC) and Large Hadron Collider (LHC) are remarkably well described by
viscous hydrodynamics, which predicts the measured flow harmonics and their
correlations in exquisite detail Jeon and Heinz (2015); Heinz and Snellings
(2013). These hydrodynamic simulations are based on a theory of ordinary
hydrodynamics, which ignores chiral symmetry breaking at low temperature and
the associated chiral phase transition. This is reasonable at finite quark
mass, where chiral symmetry is always explicitly broken. Nevertheless, if the
quark mass is small enough, one would expect that the pattern of chiral
symmetry breaking would provide a useful organizing principle for
hydrodynamics, increasing its predictive power.

As a starting point for this reorganization, let us describe the appropriate
hydrodynamic theory in the limit of two exactly massless quark flavors. In
this limit the symmetry group of the microscopic theory is $U(1)\times
SU_{L}(2)\times SU_{R}(2)$. At high temperatures where the symmetry of the
Lagrangian is reflected in the symmetry of the thermal state, the hydrodynamic
variables are simply the conserved charges $Q$, i.e. the energy and momentum,
the iso-vector and iso-axial-vector charges, and the baryon number. At low
temperatures the symmetry of the thermal state is spontaneously broken to
$U(1)\times SU_{V}(2)$, and the three massless Goldstone modes associated with
broken symmetry (the pions) must be added to the original list of of
hydrodynamic variables, $\\{Q,\pi\\}$ Son (2000). The theory in this case is
akin to a non-abelian superfluid. The hydrodynamic theories at high and low
temperatures are separated by the chiral critical point, which is somewhat
analogous to the critical point separating the normal and superfluid phases of
helium Rajagopal and Wilczek (1993); Son and Stephanov (2002a). At the
critical point the hydrodynamic variables consist of the conserved charges $Q$
and a four component order parameter field
$\Sigma\sim\langle\bar{q}_{R}q_{L}\rangle$. For $T\gg T_{c}$, $\Sigma$ can be
consistently integrated out, leaving an ordinary fluid state with only the
conserved charges, while for $T\ll T_{c}$ the phase of $\Sigma$ fluctuates,
reducing the hydrodynamics to a superfluid theory consisting of conserved
charges and the Goldstone modes, $\\{Q,\pi\\}$.

In the presence of a small but finite quark mass the theory is only
approximately invariant under $SU_{L}(2)\times SU_{R}(2)$. The iso-axial
vector charge is only approximately conserved, and the $\pi$ fluctuations are
only approximately massless. In addition, the system never passes directly
through the chiral phase transition, and the correlation length remains
finite, but large. Thus at large enough distances, the theory asymptotes to
ordinary hydrodynamics, and the usual approach based on ordinary hydrodynamics
is fine. However, at shorter distances (but still macroscopic) the
fluctuations of the order parameter $\Sigma$ need to be taken into account to
accurately model the system with hydrodynamics. The thermal fluctuations of
the $\Sigma$ field are incorporated into the equation of state and the
transport coefficients of the ordinary fluid theory. By writing down the
hydrodynamics theory including the $\Sigma$, and then integrating out these
modes, one can precisely determine how the critical modes affect the equation
of state, and modify the transport coefficients of the ordinary theory, such
as the shear and bulk viscosities. This computation will determine the
behavior of these parameters in the vicinity of the chiral critical point. Our
goal in this paper to perform this computation, albeit in a mean-field
approximation.

The validity of the approach relies on the smallness of quark mass and the
proximity of the $O(4)$ critical point in real world QCD. We are encouraged by
Euclidean lattice QCD simulations Ding  _et al._ (2019); Kaczmarek  _et al._
(2020) at the physical pion mass and smaller, which show that aspects of QCD
thermodynamics, such as the chiral susceptibility, can be qualitatively, and
even quantitatively, understood using $O(4)$ scaling functions. These scaling
functions dictate the behavior of the singular part of the temperature
dependence (at fixed quark mass) of the equation of state near the pseudo-
critical point. It seems reasonable to expect that the real time $O(4)$
scaling functions can be used to prescribe the temperature dependence of the
transport parameters in the critical region with similar precision.

The singular parts of the equation of state can be determined by simulating an
appropriate $O(4)$ symmetric Landau-Ginzburg field theory on a 3D lattice. In
effect, this means that the singular part is captured by a classical effective
field theory (EFT) describing the equilibrium fluctuations of a classical
order parameter field. In practice, the classical EFT is replaced by a spin
model and lattice techniques are used to determine the scaling functions with
high precision Engels and Karsch (2014, 2012); Engels and Vogt (2010). For
dynamical quantities the appropriate classical real time EFT is stochastic
hydrodynamics Hohenberg and Halperin (1977). The hydrodynamic equations of
motion were written down many years ago in an insightful paper by Wilczek and
Rajagopal Rajagopal and Wilczek (1993). We will present a somewhat different
derivation of their equations of motion in Sect. III. A useful
phenomenological model which tracks the amplitude of the chiral condensate
(but not the phase) within hydrodynamics was presented in Nahrgang  _et al._
(2012).

A numerical simulation of the critical theory could be used to find the two
point functions of the conserved currents, which in turn determine the scaling
functions for the transport coefficients near the critical point. In the
current paper we will work in a mean field approximation, in order to get a
qualitative understanding for the expected scaling functions from such
simulations, and to estimate the absolute magnitude of critical contributions
from the $\Sigma$ field to the transport coefficients. We will reserve a
numerical simulation for future work.

Currently, there is no experimental evidence for the long wavelength
fluctuations of the chiral condensate, which are the hallmark of the chiral
phase transition111Note, however, that there is an observed enhancement of
thermal dileptons in a specific mass range, which can be taken as evidence
that the vector and axial-vector correlation functions are becoming
degenerate, as expected when chiral symmetry is partially restored
[Forareviewsee:][]Rapp:2009yu.. As a first attempt to remedy the situation,
the current paper will point out an enhancement of soft pions seen in the
experimental data, and recall that such an enhancement is an expected
signature of the $O(4)$ critical point. An estimate for the magnitude of the
enhancement expected from critical fluctuations encourages us to explore this
explanation for the observed excess in future work. In addition, the proposed
upgrade to the ALICE detector Colella (2019) will be more sensitive to low
$p_{T}$ pions, and this new experimental thrust provides us with additional
encouragement.

This paper builds upon our earlier work Grossi  _et al._ (2020), which
computed the contributions of soft pions to the transport coefficients of QCD
in the broken phase, and then estimated how these contributions would evolve
as one approaches the critical point from below. We will recover these earlier
results as a low temperature limit of the more general expressions presented
here. However, while the current paper works with mean field theory, the
previous results are more general and are expected match the full numerical
simulations of stochastic hydrodynamics.

An outline of the paper is as follows: to set notation we will first describe
the thermodynamics of the $O(4)$ scaling theory, and compare results from
previous numerical simulations with the mean field expectations used in this
work. Then in Sect. III, we will provide a general formulation of the
hydrodynamic equations of motion, and compute the linearized propagators for
the theory. These propagators will then be used in Sect. IV to compute the
scaling behavior of the transport coefficients in a mean field approximation,
and the results are analyzed. Finally, in Sect. V we estimate the enhanced
yield of soft pions near the chiral critical point and outline future
directions.

##  II Thermodynamic preliminaries

###  II.1 The magnetic equation of state at mean field

The order parameter of the chiral phase transition is a four component field
$\phi_{a}$ transforming in the defining representation of $O(4)$, and reflects
the fluctuations of the chiral condensate,
$\Sigma(x)\equiv-\bar{q}_{R}q_{L}(x)/F^{2}_{0}$ where $F_{0}$ is the vacuum
pion decay constant. $\Sigma$ is expanded in terms of the four component
field222 Roman indices at the beginning of the alphabet $a,b,c\ldots$ are
$O(4)$ indices. Isospin indices are denoted as
$s,s^{\prime},s^{\prime\prime},\ldots$ etc, and are notated with a vector
$\vec{\pi}$. Minkowski indices are $\mu,\nu,\rho,\ldots$ etc, while spatial
indices are $i,j,k,\ldots$. To lighten the notation, contraction of flavor
indices are denoted by a dot, e.g. $H\cdot\phi=H_{a}\phi_{a}$ and $\mu\cdot
n=\mu_{ab}\cdot n_{ab}$. More explicitly, the chiral condensate is
$\left[\Sigma\right]^{\ell_{1}}_{\;\ell_{2}}=-\bar{q}_{R\ell_{2}}\,q_{L}^{\ell_{1}}(x)/F_{0}^{2}$,
where $q^{\ell}=(u,d)$, and $\Sigma$ transforms as $\Sigma\rightarrow
g_{L}\Sigma g_{R}^{\dagger}$ under a chiral rotation.

$\Sigma\equiv\phi_{a}\tau_{a}=\sigma\,\mathbb{I}+i\vec{\pi}\cdot\vec{\lambda}\,,$
(1)

where the matrices of the Clifford algebra
$\tau_{a}=(\mathbb{I},-i\vec{\lambda})$ are an amalgamation of the unit matrix
and the Pauli matrices, $\vec{\lambda}$, transforming together as a vector
under $O(4)$. The components of $\phi_{a}$ are the sigma and pion fields

$(\phi_{a})\equiv(\sigma,-\vec{\pi})\,,$ (2)

where the minus sign appearing in (2) is a slightly inconvenient convention.
Given the approximate $O(4)$ symmetry of the microscopic theory, there are
approximately conserved charge densities, $n_{ab}$, transforming as an
antisymmetric tensor under $O(4)$. $n_{ij}$ is the conserved iso-vector
charge, while $n_{0i}$ is the partially conserved iso-axial-vector charge. The
associated chemical potential is $\mu_{ab}$, and we also adopt the notation
$\mu^{2}=\mu_{ab}\mu_{ab}$.

Close to the critical point, the Euclidean action that determines the
fluctuations in the order parameter $\phi_{a}$ at fixed temperature $T$ and
chemical potential $\mu_{ab}$ is

$\displaystyle{S}_{E}=$ $\displaystyle\beta\int
d^{3}x\,\left(p_{0}(T)+\frac{1}{2}\chi_{0}\mu^{2}-\frac{1}{2}\partial_{i}\phi_{a}\,\partial^{i}\phi_{a}-V({\Phi})+H_{a}\phi_{a}\right)\,,$
(3)

where the scalar potential is of Landau-Ginzburg form

$V({\Phi})=\frac{1}{2}m_{0}^{2}(T){\Phi}^{2}+\frac{\lambda}{4}{\Phi}^{4}\,.$
(4)

Here we have defined

${\Phi}\equiv\sqrt{\phi_{a}\phi_{a}}\,,$ (5)

and

$m_{0}^{2}(T)\equiv{\mathfrak{m}}^{2}\,\frac{(T-T_{c})}{T_{c}}\equiv{\mathfrak{m}}^{2}t,$
(6)

where $t$ is the reduced temperature, and ${\mathfrak{m}}$ is of order the
vacuum sigma mass or higher and is a constant. $H_{a}\equiv(H,0,0,0)$ is the
applied magnetic field or quark mass. At this point $T$ and $\mu$ are simply
constants but have been brought inside the integral in eq. (3) to motivate the
hydrodynamic analysis of Sect. III, where $T,\mu$ depend slowly space and
time.

The full partition function takes the form

$Z=\int D\phi\,e^{{S}_{E}[\phi,H]}\,,$ (7)

and reproduces the critical behavior of the equation of state. In spite of its
well known shortcomings, we will work in a mean field approximation. The mean
field takes the form

$\left\langle\phi_{a}\right\rangle=(\bar{\sigma},0)\,,$ (8)

where $\bar{\sigma}$ is the real solution to

$m_{0}^{2}(T)\,\bar{\sigma}+\lambda\,\bar{\sigma}^{3}-H=0\,.$ (9)

It is straightforward to show that the solution to (9) takes the scaling form

$\bar{\sigma}=\frac{{\mathfrak{m}}}{\sqrt{\lambda}}\,h^{1/3}f_{G}(z)\,,\qquad\text{with}\quad
z=th^{-2/3}\,,$ (10)

where we have defined reduced field

$\displaystyle h\equiv$
$\displaystyle\frac{H\sqrt{\lambda}}{{\mathfrak{m}}^{3}}.$ (11)

$z$ is the mean field scaling variable, and $f_{G}(z)$ is the (mean field)
scaling function for the magnetic equation of state. As we will see in the
next section, the pion screening mass on the critical line, $z=0$, is given
by333Here and below the subscript $c$, such as $m_{c}$ and $m_{\sigma c}$,
indicates that the quantity is being evaluated on the critical line $z=0$.
Later we will introduce $v_{c}^{2}$ and $u_{c}^{2}$ (in eqs. (24) and (73)).

$m_{c}^{2}={\mathfrak{m}}^{2}\,h^{2/3}\,,$ (12)

and is a temperature independent constant which parametrizes $h$. It is
convenient to express all lengths in terms of $m_{c}$. The scaling variable
and equation of state take the form

$\bar{\sigma}=\frac{m_{c}}{\sqrt{\lambda}}f_{G}(z)\,,\qquad
z=\frac{m_{0}^{2}(T)}{m_{c}^{2}}=\frac{{\mathfrak{m}}^{2}}{m_{c}^{2}}\frac{(T-T_{c})}{T_{c}}\,.$
(13)

Parametrically, ${\mathfrak{m}}^{2}/m_{c}^{2}=h^{-2/3}$ is a large parameter,
and thus $T$ must be close to $T_{c}$ in order to have an order one scaling
variable, $z\sim 1$.

Outside of the mean field approximation, the expectation value of the order
parameter also takes the scaling form

$\bar{\sigma}=B\,h^{1/\delta}f_{G}(z)\,,\qquad z=th^{-1/\Delta}\,,$ (14)

with $B$ a non-universal constant. $\delta$ and $\Delta$ are known critical
exponents, and $f_{G}(z)$ is a known universal function Engels and Karsch
(2014, 2012); Engels and Vogt (2010). Table 1 compares the mean field
expectations for the critical exponents to the $O(4)$ scaling theory, and Fig.
1(a) compares the mean field $f_{G}(z)$ to the scaling theory.

Exponent or ratio | Mean field |  $O(4)$ scaling theory
---|---|---
$\beta$ | 1/2 | 0.38
$\delta$ | 3 | 4.8
$\Delta=\beta\delta$ | 3/2 | 1.83
$\nu_{c}=\nu/\beta\delta$ | 1/3 | 0.40
$m_{\sigma c}^{2}/m_{c}^{2}$ | 3 | 4.0
Table 1:  A comparison of mean field theory and the $O(4)$ scaling theory (see
for example Parisen Toldin  _et al._ (2003); Engels  _et al._ (2003) for
current estimates of the $O(4)$ exponents). $m_{c}$ and $m_{\sigma c}$ are the
pion and sigma screening masses on the critical line, $z=0$, and this ratio
was taken from Engels  _et al._ (2003).

Figure 1: (a) A comparison of the mean field magnetic EOS to numerical results
from lattice methods taken from Engels and Karsch (2014, 2012). (b) The sigma
and pion screening masses (inverse correlation lengths), $m_{\sigma}$ and $m$,
compared to results from lattice methods. The lattice curves were obtained by
digitizing the numerical data from Fig. 8 and Fig. 9 of Engels  _et al._
(2003), which was subsequently fit with a parametrized form.

###  II.2 Static correlation functions in mean field

Given the mean value $\bar{\sigma}$, we can evaluate the action (3) to
quadratic order

${S}_{E}=\beta\int
d^{3}x\,\left[p_{\sigma}(T)+\frac{1}{2}\chi_{0}\mu^{2}-\frac{1}{2}\left(\partial_{i}\delta\sigma\,\partial^{i}\delta\sigma+m_{\sigma}^{2}\delta\sigma^{2}\right)-\frac{1}{2}\left(\partial_{i}\vec{\pi}\cdot\partial^{i}\vec{\pi}+m^{2}\vec{\pi}^{2}\right)\right],$
(15)

where

$p_{\sigma}(T)=p_{0}(T)-\left(\frac{1}{2}m_{0}^{2}(T)\bar{\sigma}^{2}(T)+\frac{\lambda}{4}\bar{\sigma}(T)^{4}-H\bar{\sigma}(T)\right)\,.$
(16)

The sigma and pion screening masses are

$\displaystyle m_{\sigma}^{2}$ $\displaystyle\equiv
m_{c}^{2}\left(z+3f_{G}^{2}(z)\right)\,,$ (17a) $\displaystyle m^{2}$
$\displaystyle\equiv\frac{H}{\bar{\sigma}(T)}=\frac{m_{c}^{2}}{f_{G}(z)}\,.$
(17b)

As in the previous section, the screening masses (or inverse correlation
lengths) are also defined outside of mean field theory. These are expected to
scale as

$\displaystyle m_{\sigma}=$
$\displaystyle{\mathfrak{m}}_{L}\,h^{\nu_{c}}\,g_{L}(z)\,,$ (18)
$\displaystyle m=$ $\displaystyle{\mathfrak{m}}_{T}\,h^{\nu_{c}}\,g_{T}(z)\,,$
(19)

where ${\mathfrak{m}}_{L}$ and ${\mathfrak{m}}_{T}$ are non-universal
constants, and $g_{L}(z)$ and $g_{T}(z)$ are universal scaling functions. As
before, $g_{L}$ and $g_{T}$ are normalized to unity for $z=0$. The ratio
between $m_{\sigma}$ and $m$ is also universal, and can be parameterized by
$m_{\sigma}/m$ on the critical line, i.e. $m_{\sigma c}^{2}/m_{c}^{2}$. This
universal ratio is compared to the mean field prediction of three in Table 1.
$m(z)$ and $m_{\sigma}(z)$ are extracted from the numerical work of Ref.
Engels  _et al._ (2003) and compared to mean field theory in Fig. 1(b).

The quadratic action predicts the equal time correlation functions

$\displaystyle\frac{1}{V}\left\langle\delta\sigma({\bm{k}})\delta\sigma(-{\bm{k}})\right\rangle$
$\displaystyle=\frac{T}{k^{2}+m_{\sigma}^{2}}\,,$ (20a)
$\displaystyle\frac{1}{V}\left\langle\varphi_{s}({\bm{k}})\varphi_{s^{\prime}}(-{\bm{k}})\right\rangle$
$\displaystyle=\frac{T}{\bar{\sigma}^{2}(k^{2}+m^{2})}\delta_{ss^{\prime}}\,.$
(20b)

Finally, we can use the general theory of thermodynamics fluctuations to
recognize that444 The easiest way to see this in the current framework is to
recognize that the thermodynamic fluctuations in $n_{ab}$ are Gaussian and
summed over in the grand canonical ensemble. The factor
$\tfrac{1}{4}\chi_{0}\mu^{2}$ reflects the integration over $n$ with the
Lagrange multiplier $\mu$ $e^{\beta\int{\rm
d}^{3}x\,\tfrac{1}{4}\chi_{0}\mu^{2}}=\int[Dn]\,{\rm exp}\left(-\beta\int{\rm
d}^{3}x\left(\tfrac{1}{4\chi_{0}}n^{2}-\frac{1}{2}n\cdot\mu\right)\right)\,.$
(21) This integral implies eq. (22).

$\displaystyle\frac{1}{V}\left\langle
n_{ab}({\bm{k}})n_{cd}(-{\bm{k}})\right\rangle$
$\displaystyle=T\chi_{0}\,(\delta_{ac}\delta_{bd}-\delta_{ad}\delta_{bc})\,.$
(22)

Well below $T_{c}$, the pion mass is small and soft pions are long lived
quasi-particles Son (2000); Son and Stephanov (2002a, b). From this context we
introduce a number of definitions following Son and Stephanov Son and
Stephanov (2002b). The phase of the condensate is

$\varphi_{s}\equiv\frac{\pi_{s}}{\bar{\sigma}}\,,$ (23)

while the associated the pion velocity squared is

$v^{2}(T)\equiv\frac{\bar{\sigma}^{2}(T)}{\chi_{0}}.$ (24)

The pole mass is defined as $m_{p}^{2}\equiv v^{2}m^{2}$, and the soft pion
dispersion curve takes the form

$\omega^{2}_{q}=v^{2}q^{2}+m_{p}^{2},$ (25)

which is parameterized by two Euclidean quantities $v^{2}(T)$ and $m^{2}(T)$.

In the next section we will develop the hydrodynamic theory for the $O(4)$
model. The real time correlation functions constructed from this theory will
reproduce (20) and (22) after integrating over frequency.

##  III Hydrodynamics

Having discussed the thermodynamics, we are ready to derive the corresponding
hydrodynamic theory. The resulting equations of motion are equivalent to those
derived previously by Rajagopal and Wilczek using Poisson bracket methods
Rajagopal and Wilczek (1993). Well below $T_{c}$, the equations of motion
resemble a non-abelian superfluid theory and also have been analyzed Son
(2000); Son and Stephanov (2002b); Jain (2017); Grossi  _et al._ (2020). The
methodology here follows closely our previous work Grossi  _et al._ (2020).

###  III.1 Ideal hydrodynamics and the Josephson constraint

To derive the ideal hydrodynamic expressions we follow an expedient procedure
procedure outlined in Jensen  _et al._ (2012) and take as hydrodynamic action

${S}[g_{\mu\nu},A_{\mu},H]=\int
d^{4}x\sqrt{-g}\,p_{\Sigma}(T,\mu,(\partial_{\perp}\phi)^{2},\phi^{2},H\cdot\phi)\,,$
(26)

where the redefined pressure takes the same form as its Euclidean counterpart

$p_{\Sigma}(T,\mu,(\partial_{\perp}\phi)^{2},\phi^{2},H\cdot\phi)\equiv
p(T)+\frac{1}{4}\chi_{0}\,\mu\cdot\mu-\frac{1}{2}\Delta^{\mu\nu}D_{\mu}\phi\cdot
D_{\nu}\phi-V({\Phi})+H\cdot\phi\,,$ (27)

but we have replaced the integration over thermal circle with an integration
over time

$\beta=\int_{0}^{\beta}d\tau\rightarrow\int dt\,.$ (28)

We also have added external gauge and gravitational fields, $(A_{\mu})_{ab}$
and $g_{\mu\nu}$, for the purpose of deriving the stress tensor and currents,
and ultimately these sources will be set to zero. In these expressions
$T\equiv(-\beta^{\mu}g_{\mu\nu}\beta^{\mu})^{-1/2}$, and then we define
$u^{\mu}\equiv T\beta^{\mu}$, and $\Delta^{\mu\nu}\equiv
g^{\mu\nu}+u^{\mu}u^{\nu}$. The chemical potential can be written

$\mu_{ab}=\left(u^{\rho}\tilde{\mu}_{\rho}+u^{\rho}A_{\rho}\right)_{ab}\,,$
(29)

where $(\tilde{\mu}_{\rho})_{ab}$ is the contact chemical potential and is
independent of $A_{\rho}$. The covariant derivative is

$(D_{\mu}\phi)_{a}=\partial_{\mu}\phi_{a}-\tfrac{i}{2}(A_{\mu}\cdot\mathcal{J})_{ab}\phi_{b}\,,$
(30)

where $\mathcal{J}_{cd}$ are the generators $O(4)$ rotation group

$(i\mathcal{J}_{cd})_{ab}=\delta_{ca}\delta_{db}-\delta_{cb}\delta_{da}\,.$
(31)

Setting the external fields to zero for simplicity, the differential of
pressure at fixed $H$ follows from the form of $p_{\Sigma}$

$dp_{\Sigma}=s_{\Sigma}\,dT+\frac{1}{2}n_{ab}\,d\mu_{ab}-\frac{1}{2}d(\partial_{\perp}^{\mu}\phi)^{2}+\left(-\frac{\partial
V}{\partial\phi_{a}}+H_{a}\right)d\phi_{a}\,,$ (32)

which defines the entropy density, $s_{\Sigma}\equiv\partial
p_{\Sigma}/\partial T$, and the number densities, $n_{ab}\equiv 2\,\partial
p_{\Sigma}/\partial\mu_{ab}$, respectively555 More explicitly,
$s_{\Sigma}(T)=s(T)-\frac{1}{2T_{c}}{\mathfrak{m}}^{2}\,\Phi^{2}\,$. The
factor of two in the definition of $n_{ab}$, leading to
$n_{ab}=\chi_{0}\mu_{ab}$, is a symmetry factor, i.e. $n_{ab}\equiv\partial
p_{\Sigma}/\partial\mu_{ab}-\partial p_{\Sigma}/\partial\mu_{ba}$ with
$\mu_{12}$ and $\mu_{21}$ treated as independent variables. Similar symmetry
factors for symmetric and antisymmetric tensors are present in the definitions
of $T^{\mu\nu}$ and $J^{\mu}_{ab}$. . Here and below $d\equiv
u^{\mu}\partial_{\mu}$ and
$\partial_{\perp}^{\mu}=\Delta^{\mu\nu}\partial_{\nu}$.

Varying the action with respect to the metric yields the conserved stress
tensor $\partial_{\mu}T^{\mu\nu}=0$. Recognizing that both the temperature and
the chemical potential depend implicitly on the metric after they are written
in terms of $\beta^{\mu}$, the variation of the action gives

$\displaystyle T^{\mu\nu}=\left.\frac{2}{\sqrt{-g}}\frac{\delta{S}}{\delta
g_{\mu\nu}}\right|_{g=A=0}=(\varepsilon_{\Sigma}+p_{\Sigma})\,u^{\mu}u^{\nu}+p_{\Sigma}g^{\mu\nu}+\partial^{\mu}\phi\cdot\partial^{\nu}\phi-u^{\mu}u^{\nu}(u^{\sigma}\partial_{\sigma}\phi)\cdot(u^{\rho}\partial_{\rho}\phi),$
(33)

where in this expression the energy density has been defined through the
Gibbs-Duhem relation

$\varepsilon_{\Sigma}=-p_{\Sigma}+Ts_{\Sigma}+\tfrac{1}{2}\mu_{ab}\cdot
n_{ab}\,.$ (34)

We can find the partial current conservation equation by requiring that the
action in (26) be invariant under gauge transformations. We will limit the
discussion to weak fields and switch off the gravitational field for this
purpose. Under an infinitesimal $O(4)$ rotation with parameters
$\omega_{cd}(x)$, the gauge fields and magnetic field transform as

$\displaystyle A_{\mu,cd}\rightarrow$ $\displaystyle
A_{\mu,cd}+\partial_{\mu}\omega_{cd}\,,$ (35a) $\displaystyle\delta
H_{a}\rightarrow$ $\displaystyle
H_{a}+\tfrac{i}{2}(\omega\cdot{\mathcal{J}})_{ab}H_{b}\,.$ (35b)

Then, requiring invariance of the action under the rotation,

$\displaystyle\delta{S}=$ $\displaystyle\int d^{4}x\,\frac{\delta{S}}{\delta
A_{\mu,ab}}\,\delta A_{\mu,ab}+\frac{\delta{S}}{\delta H_{a}}\delta H_{a}=0,$
(36)

and inserting the transformation rules (35), we find partial current
conservation

$\partial_{\mu}J^{\mu}_{cd}=\phi_{c}H_{d}-\phi_{d}H_{c}\,.$ (37)

Here the currents are given by

$\displaystyle J^{\mu}_{ab}\equiv 2\frac{\delta{S}}{\delta
A_{\mu,ab}}=\chi_{0}\mu_{ab}u^{\mu}+(J^{\mu}_{\perp})_{ab}\,,$ (38)

where the first term is the normal component and the second term is the
superfluid component, given by

$(J_{\perp})_{ab}^{\mu}=\Delta^{\mu\nu}(\partial_{\nu}\phi_{a}\phi_{b}-\partial_{\nu}\phi_{b}\phi_{a}).$
(39)

To complete the equations of motion of ideal hydrodynamics, we need to specify
a relationship between the phase of the condensate and the chemical potential
known as the Josephson constraint. The Josephson constraint is the requirement
that the field $\phi_{a}$ is stationary under the evolution generated by the
grand potential, $\Omega=H-\frac{1}{2}\mu_{bc}N_{bc}$, i.e. the stability of
the thermal state. This reasoning leads to a requirement on the classical
Poisson bracket between $\phi$ and $\Omega$

$\displaystyle\\{\phi_{a},\Omega\\}=\\{\phi_{a},-u^{\mu}P_{\mu}-{\textstyle\frac{1}{2}}\mu_{bc}N_{bc}\\}=0\,.$
(40)

Recalling that $P_{\mu}$ and $N_{ab}$ generate translations and rotations
respectively (which determines their Poisson brackets with $\phi$), we find

$u^{\mu}\partial_{\mu}\phi_{a}+{\textstyle\frac{1}{2}}(\mu_{ab}\phi_{b}-\phi_{b}\mu_{ba})=0\,.$
(41)

Alternatively, but ultimately equivalently, the Josephson constraint can be
derived by requiring entropy conservation at ideal order Jain (2017)

$\partial_{\mu}(s_{\Sigma}u^{\mu})=0.$ (42)

Appendix A uses the conservation laws together with the Gibbs-Duhem relation
(34) and the pressure differential (32) to show that entropy is only conserved
if the Josephson constraint is satisfied. When viscous corrections are
included at subsequent orders in the gradient expansion, the Josephson
constraint will need to be modified.

Finally, it is useful to express the Josephson constraint in terms of the
amplitude, $\Phi$, and $SU(2)$ phase, $U$. Writing the chiral condensate as

$\Sigma=\phi_{a}\tau_{a}=\Phi U\,,$ (43)

the Josephson constraint can be written

$\displaystyle u^{\mu}\partial_{\mu}\Phi$ $\displaystyle=0\,,$ (44a)
$\displaystyle iu^{\mu}\partial_{\mu}UU^{-1}$
$\displaystyle=\mu_{L}-U\mu_{R}U^{\dagger}\,,$ (44b)

where $\mu_{L}=\tfrac{1}{2}\mu_{ab}\tau_{ab}$ and
$\mu_{R}=\tfrac{1}{2}\mu_{ab}\bar{\tau}_{ab}$ are the left and right chemical
potentials666The Clifford algebra of $O(4)$ is generated by
$\tau_{a}=(\mathbb{I},-i\vec{\lambda})$ and
$\bar{\tau}_{a}=(\mathbb{I},i\vec{\lambda})$. The generators of the (1/2, 0)
and $(0,1/2)$ representations of $O(4)$ are
$\tau_{ab}=-i[\tau_{a},\bar{\tau}_{b}]/4$ and
$\bar{\tau}_{ab}=-i[\bar{\tau}_{a},\tau_{b}]/4$, respectively. . The last
relation between the phase and the chemical potentials is familiar from non-
abelian superfluids Son (2000); Grossi  _et al._ (2020)

###  III.2 Viscous corrections

So far we have considered only the ideal equations of motion. In the
dissipative case the energy-momentum tensor, the charge current, and the
Josephson constraint will acquire new terms that correspond to dissipation
into the system. The energy-momentum tensor and the conserved currents are
modified due to dissipative effects in the usual way:

$\displaystyle T^{\mu\nu}$
$\displaystyle=T^{\mu\nu}_{\text{ideal}}+\Pi^{\mu\nu},$ (45) $\displaystyle
J_{ab}^{\mu}$ $\displaystyle=J_{ab,\text{ideal}}^{\mu}+q^{\mu}_{ab}.$ (46)

We will work in the Landau frame, where the dissipative contributions to the
stress tensor and the diffusion current are taken to be orthogonal to the four
velocity $u^{\mu}$, i.e.

$\Pi^{\mu\nu}u_{\mu}=0,\quad q^{\mu}_{ab}u_{\mu}=0.$ (47)

The stress tensor can be further decomposed into a symmetric-traceless and
transverse part, $\pi^{\mu\nu}$, and a trace part, $\Pi$,

$\Pi^{\mu\nu}=\pi^{\mu\nu}+\Pi\Delta^{\mu\nu}.$ (48)

In addition to the dissipative corrections to the energy-momentum tensor and
the current, the evolution equation of the chiral condensate, $\phi_{a}$, gets
modified by dissipative effects. Therefore it is useful to define

$u^{\mu}\partial_{\mu}\phi_{a}+\mu_{ab}\phi_{b}=\Xi_{a},$ (49)

where $\Xi_{a}$ is a Lorentz scalar that encodes the dissipative contribution
to the scalar field equation of motion.

Using the conservation of the energy-momentum tensor and the partial
conservation of the charge, the Gibbs-Duhem relation in (34), and the pressure
differential in (32) we can derive the entropy production as

$\displaystyle\partial_{\mu}(s_{\Sigma}u^{\mu}-\frac{\mu_{ab}}{2T}q^{\mu}_{ab})=\frac{\Xi_{a}}{T}\Theta_{a}-\partial_{\mu}\left(\frac{u_{\nu}}{T}\right)\Pi^{\mu\nu}-\partial_{\mu}\left(\frac{\mu_{ab}}{2T}\right)q^{\mu}_{ab}.$
(50)

where we have defined the scalar quantity

$\Theta_{a}=\partial^{2}_{\perp}\phi_{a}-\frac{\partial
V}{\partial\phi_{a}}+H_{a},$ (51)

with
$\partial_{\perp}^{2}\phi_{a}\equiv\partial_{\mu}\partial^{\mu}_{\perp}\phi_{a}$.

Up to now we have not specified an expansion scheme; the equations are just a
consequence of the definition of entropy, the conservation of energy and
momentum, and the partial conservation charge. The positivity of entropy
production in the tensor sector can be enforced with

$\pi^{\mu\nu}=-\eta_{\Sigma}\sigma^{\mu\nu},\quad\text{with}\quad\eta_{\Sigma}\geq
0,$ (52)

where $\eta_{\Sigma}$ is the shear viscosity of the $O(4)$ theory. In the
vector sector we have

$q_{ab}^{\mu}=-T\sigma_{\Sigma}\partial^{\mu}\left(\frac{\mu_{ab}}{T}\right),\quad\text{with}\quad\sigma_{\Sigma}\geq
0\,,$ (53)

where $\sigma_{\Sigma}$ is the $O(4)$ conductivity.

The scalar sector requires a bit more care as there are two Lorentz scalars,
$\Xi_{a}\Theta_{a}$ and $\Pi\,\partial_{\mu}u^{\mu}$. Generally we have as the
constitutive relations for $\Pi$ and $\Xi_{a}$

$\displaystyle\Pi$
$\displaystyle=-\zeta_{\Sigma}\,\partial_{\mu}u^{\mu}-\zeta^{(1)}_{\Sigma}\,\phi_{a}\Theta_{a},$
(54) $\displaystyle\Xi_{a}$
$\displaystyle=\zeta^{(1)}_{\Sigma}\,\phi_{a}\partial_{\mu}u^{\mu}+\Gamma\,\Theta_{a},$
(55)

where $\zeta_{\Sigma}$ is the bulk viscosity, $\zeta^{(1)}_{\Sigma}$ and
$\Gamma$ are the transport coefficients regulating the dissipative effects of
the scalar field dynamics. The positivity of the associated quadratic form is
enforced if

$\zeta_{\Sigma}\geq 0,\quad\Gamma\geq 0,\quad\text{ and
}\quad\zeta_{\Sigma}\,\Gamma-(\zeta^{(1)}_{\Sigma})^{2}\,\phi^{2}\geq 0.$ (56)

Having specified the dissipative fluxes, it is possible to write down the
energy-momentum tensor, the current, and the scalar field equation, including
the first gradient corrections. The scalar field obeys a relaxation-type
equation where the ideal part is the Josephson constraint

$\displaystyle
u^{\mu}\partial_{\mu}\phi_{a}+\mu_{ab}\phi_{a}=\Gamma\left[\partial^{2}_{\perp}\phi_{a}-\frac{\partial
V}{\partial\phi_{a}}+H_{a}\right]+\zeta^{(1)}_{\Sigma}\phi_{a}\,\partial_{\mu}u^{\mu}.$
(57)

The energy momentum tensor now includes dissipative contributions due to
chiral condensate

$\displaystyle\Pi^{\mu\nu}=-\eta_{\Sigma}\sigma^{\mu\nu}-\Delta^{\mu\nu}\left[\zeta_{\Sigma}\partial_{\mu}u^{\mu}-\zeta^{(1)}_{\Sigma}\phi_{\alpha}\left(\partial^{2}_{\perp}\phi_{a}-\frac{\partial
V}{\partial\phi_{a}}+H_{a}\right)\right].$ (58)

Finally the current has the form

$\displaystyle(J^{\mu})_{ab}=n_{ab}u^{\mu}+(J^{\mu}_{\perp})_{ab}-T\sigma_{\Sigma}\,\Delta^{\mu\nu}\partial_{\nu}\left(\frac{\mu_{ab}}{T}\right)\,,$
(59)

and is partially conserved as in (37). The coefficient $\zeta^{(1)}_{\Sigma}$
is an independent transport coefficient that couple the expansion rate
$\partial_{\mu}u^{\mu}$ to the Josephson constraint and vice versa. Near the
phase transition $\phi_{\alpha}$ is approximately zero, which means this term
is subdominant and can be neglected.

Let us compare these equations to a number of results in the literature. The
equations are equivalent to those of Rajagopal and Wilczek written down almost
thirty years ago Rajagopal and Wilczek (1993); our notation for
$\sigma_{\Sigma}$ and $\Gamma$ follows theirs. The current reformulation is
covariant and includes the coupling to the background flow. In the low
temperature limit the equations match those of our previous paper Grossi  _et
al._ (2020) (which includes a discussion of earlier work Son (2000); Son and
Stephanov (2002b)), provided one identifies some of the
coefficients777Specifically, we have $\Gamma\rightarrow D_{m}$ and
$\Gamma+D\rightarrow D_{A}$ where $D=\sigma_{\Sigma}/\chi$. .

###  III.3 Linear response

Knowing the equations of motion we can determine the hydrodynamic predictions
for the retarded Green functions of the system. In the axial channel we can
consider the coupled equations of motion for $\varphi_{s}$ and the axial
chemical potential $\mu_{0s}$, when $\phi_{a}$ is linearized around
equilibrium

$\displaystyle\phi_{a}=$
$\displaystyle(\bar{\sigma},-\bar{\sigma}\varphi_{s})\,.$ (60)

To find the response function for $(\omega_{k}\varphi_{s},\mu_{0s})$, we
introduce a pseudoscalar source $H_{a}=(H,\delta H_{s}(x))$ and a gauge field
$(A_{0}(x))_{0s}$ which are conjugate to $-\sigma\varphi_{s}$ and
$\chi_{0}\mu_{0s}$ respectively. Due to the $O(4)$ symmetry, the external
gauge field can appear in the time derivative of $\varphi_{s}$ and spatial
gradient of the chemical potential

$\displaystyle\partial_{t}\phi_{s}$ $\displaystyle\to
D_{t}\phi_{s}=\partial_{t}\phi_{s}-(A_{0})_{s0}\bar{\sigma},$ (61a)
$\displaystyle\partial_{i}\mu_{0s}$
$\displaystyle\rightarrow\partial_{i}\mu_{0s}-(E_{i})_{0s},$ (61b)

where $(E_{i})_{0s}=(\partial_{i}A_{0})_{0s}$. Applying these transformations
and Fourier transforming leads us to the linearized equations in matrix form

$\displaystyle\begin{pmatrix}-i\omega+\Gamma(k^{2}+m^{2})&\omega_{k}\\\
-\omega_{k}&-i\omega+Dk^{2}\\\
\end{pmatrix}\begin{pmatrix}\omega_{k}\varphi_{s}\\\
\mu_{0s}\end{pmatrix}=\frac{1}{\chi_{0}}\begin{pmatrix}\Gamma(k^{2}+m^{2})&\omega_{k}\\\
-\omega_{k}&Dk^{2}\\\ \end{pmatrix}\begin{pmatrix}-\bar{\sigma}\delta
H_{s}/\omega_{k}\\\ \chi_{0}(A_{0})_{0s}\end{pmatrix},$ (62)

where we have defined the diffusion coefficient $D=\sigma_{\Sigma}/\chi_{0}$
of the $O(4)$ symmetric theory. The linearized equations can be solved to find
the retarded propagator

$\begin{pmatrix}\omega_{k}\varphi_{s}\\\
\mu_{0s}\end{pmatrix}=\frac{1}{\chi_{0}}\frac{1}{(-\omega^{2}+\omega_{k}^{2}+g_{1}g_{2})-i\omega\Gamma_{k}}\\\
\times\begin{pmatrix}g_{1}(g_{2}-i\omega)+\omega_{k}^{2}&-i\omega\omega_{k}\\\
i\omega\omega_{k}&g_{2}(g_{1}-i\omega)+\omega_{k}^{2}\\\
\end{pmatrix}\begin{pmatrix}-\bar{\sigma}\delta H_{s}/\omega_{k}\\\
\chi_{0}\,(A_{0})_{0s}\end{pmatrix}\,,$ (63)

where, for compactness, we define the following shorthand for the dissipative
rates:

$\displaystyle g_{1}$ $\displaystyle\equiv\Gamma(k^{2}+m^{2})\,,$ (64a)
$\displaystyle g_{2}$ $\displaystyle\equiv Dk^{2}\,,$ (64b)
$\displaystyle\Gamma_{k}$
$\displaystyle\equiv\Gamma(k^{2}+m^{2})+Dk^{2}=g_{1}+g_{2}\,.$ (64c)

$\Gamma_{k}$ determines the damping rate of soft pions in the broken phase Son
and Stephanov (2002b).

To compute the hydrodynamic loops in the next section, it will be necessary to
use the symmetrized propagator

$[G_{\rm
sym}(\omega)]=\frac{T}{\omega}\frac{[G_{R}(\omega)]-[G_{A}(\omega)]}{i}\equiv\frac{T}{\omega}[\rho(\omega,k)],$
(65)

where the advanced propagator is

$\displaystyle[G_{A}(\omega)]=[G_{R}(\omega)]^{\dagger},$ (66)

and $\rho$ notates the spectral density. Thus, the symmetrized propagator is

$[G_{\rm
sym}(\omega)]=\frac{2T}{\chi_{0}}\frac{1}{(-\omega^{2}+\omega_{k}^{2}+g_{1}g_{2})^{2}+(\omega\Gamma_{k})^{2}}\begin{pmatrix}g_{1}(\omega^{2}+g_{2}^{2})+g_{2}\omega_{k}^{2}&-i\omega_{k}\omega\Gamma_{k}\\\
i\omega_{k}\omega\Gamma_{k}&g_{2}(\omega^{2}+g_{1}^{2})+g_{1}\omega_{k}^{2}\end{pmatrix}.$
(67)

In Fig. 2 we exhibit the spectral density for several values of the scaling
variable $z$ and a specific choice of parameters discussed below. It is
instructive to analyze the spectral density in the pion-axial charge channel
in two different limits, the broken phase, $z\to-\infty$, and the symmetric
phase, $z\to\infty$.

In the broken phase $z\to-\infty$, the field expectation value $\bar{\sigma}$
is large and $\omega_{k}\gg g_{1,2}$. In this limit the spectral density
approaches a Breit-Wigner form with the peaks given by the quasi-particle
dispersion relation $\omega=\pm\omega_{k}$ and a width given by $\Gamma_{k}$
Son and Stephanov (2002b). Then the denominator in the spectral density can be
approximated as

$\frac{\Gamma_{k}}{(-\omega^{2}+\omega_{k}^{2}+g_{1}g_{2})^{2}+(\omega\Gamma_{k})^{2}}\sim\frac{1}{4\omega_{k}^{2}}\left[\rho(\omega,\omega_{k})+\rho(\omega,-\omega_{k})\right],$
(68)

where $\rho(\omega,\omega_{k})$ notates the Breit-Wigner form

$\rho(\omega,\omega_{k})=\frac{\Gamma_{k}}{(-\omega+\omega_{k})^{2}+(\Gamma_{k}/2)^{2}}.$
(69)

In this limit, we can simplify the expression of the spectral density, leading
to

$[\rho(\omega)]=\frac{\omega}{2\chi_{0}}\left[\rho(\omega,\omega_{k})+\rho(\omega,-\omega_{k})\right]\begin{pmatrix}1&-i\\\
i&1\end{pmatrix}+\mathcal{O}\left(\frac{\Gamma_{k}}{\omega_{k}}\right).$ (70)

Thus, there is a relation between the spectral density of pions and the axial
charge888 The response function in (63) gives the retarded function and
spectral density of the chemical potential $\rho_{\mu_{A}\mu_{A}}$. Since
$n_{A}=\chi_{0}\mu_{A}$, the density-density spectral function can be obtained
including the appropriate power of $\chi_{0}$, e.g.
$\rho_{AA}=\chi_{0}^{2}\rho_{\mu_{A}\mu_{A}}$.

$\displaystyle\rho_{AA}(\omega,k)=i\chi_{0}\omega_{k}\,\rho_{\varphi
A}(\omega,k)=(\chi_{0}\omega_{k})^{2}\,\rho_{\varphi\varphi}\,,$ (71)

which is a manifestation of the PCAC relations. These relations are the direct
consequences of the Josephson equation, $-\partial_{t}\varphi=n_{A}/\chi_{0}$.
Indeed, due to the ideal equation of motion for the field $\varphi_{s}$, we
have

$\displaystyle\rho_{AA}(\omega,k)=-\chi_{0}\,\rho_{\partial_{t}\varphi
A}(\omega,k)=\chi_{0}^{2}\,\rho_{\partial_{t}\varphi\partial_{t}\varphi}(\omega,k)\,,$
(72)

which highlights that the axial charge and the time derivative of the pion
field are two interchangeable concepts.

As the temperature increases, the real and imaginary parts of the propagator
become of the same order of magnitude $\omega_{k}\sim g_{1}g_{2}$, and the
parameter that governs their relative importance can be taken as

$u^{2}=\frac{\omega^{2}_{k}}{g_{1}g_{2}}\Big{|}_{k=m}=\frac{v^{2}}{\Gamma
Dm^{2}}.$ (73)

At the phase transition, $z=0$, where $m=m_{c}$ and $v=v_{c}$ it is natural to
assume that the real and imaginary part are the same order of magnitude Son
and Stephanov (2002b), and therefore here we will consider the case where

$u^{2}_{c}=\frac{v_{c}^{2}}{\Gamma Dm_{c}^{2}}=1.$ (74)

The propagator also depends on a another dimensionless parameter

$r^{2}=\frac{\Gamma}{\Gamma+D},$ (75)

which expresses the relative strengths of the axial diffusion and the order
parameter relaxation. Calculations from chiral perturbation theory found the
value $r^{2}=3/4$ Teaney  _et al._ , and we will adopt this number as an
estimate for this order one constant.

In the symmetric phase $z\to\infty$, the field expectation value is very small
$\bar{\sigma}\sim 0$ and $v^{2}\sim 0$. Thus, the spectral density matrix
becomes diagonal

$[\rho(\omega)]_{z\to\infty}=\frac{2\omega}{\chi_{0}}\frac{1}{(\omega^{2}+g_{2}^{2})(\omega^{2}+g_{1}^{2})}\begin{pmatrix}g_{1}(\omega^{2}+g_{2}^{2})&0\\\
0&g_{2}(\omega^{2}+g_{1}^{2})\end{pmatrix}\,,$ (76)

and the pion and axial charge are completely decoupled. The pion field simply
relaxes to zero and the axial charge is purely diffusive. The spectral density
of axial charge is therefore

$\rho_{AA}(\omega,k)=\frac{2\omega\chi_{0}Dk^{2}}{(\omega^{2}+(Dk^{2})^{2})},$
(77)

while in the pion channel we have

$\bar{\sigma}^{2}\rho_{\varphi\varphi}(\omega,k)=\frac{2\omega\Gamma}{\omega^{2}+\Gamma^{2}(k^{2}+m^{2})^{2}},$
(78)

which exhibits a simple relaxation pole with relaxation rate
$\Gamma(k^{2}+m^{2})$. In the symmetric phase the axial charge and the
pions999In the symmetric phase there are no Goldstone bosons, the “pions” are
the pseudo-scalar fluctuations of the chiral condensate and have a very large
mass.  are completely disentangled, and their dissipative dynamics is
controlled by two distinct transport coefficients.

Figure 2:  The spectral density $\rho_{AA}(\omega,q)$ for the axial charge
density-density correlator with the scaling variable $z\equiv th^{-2/3}$
taking values $z=-16,-4,-1,0,1,4,16$. For large positive $z$ the distribution
asymptotes to the simple diffusive pole, $\rho_{AA}/\omega\propto
Dk^{2}/(\omega^{2}+(Dk^{2})^{2})$, reflecting the diffusion of quarks. For
large negative $z$ the pair of peaks reflects the propagating pions. We have
rescaled the axes, defining $\bar{\omega}\equiv\omega/\Gamma m_{c}^{2}$ and
$\bar{\rho}_{AA}(\omega)=\rho_{AA}(\omega,q)/2\chi_{0}$, and chosen
$q/m_{c}=1$ for illustration. For definiteness, we have set $D/\Gamma=1/3$,
and $v^{2}_{c}/\Gamma Dm^{2}_{c}=1$, and the motivation for these constants is
given in the text surrounding eq. (73).

Moving to the $\sigma$ contribution, we see that the linearized equation of
motion is

$\partial_{t}\delta\sigma=\Gamma(\nabla^{2}-m_{\sigma}^{2})\,\delta\sigma+\Gamma\delta
H\,,$ (79)

where we have added an external source to the scalar field, $H\rightarrow
H+\delta H$. Solving in Fourier space, we see that the retarded Green’s
function is

$G_{R}^{\sigma\sigma}(\omega,k)=\frac{\Gamma}{-i\omega+\Gamma(k^{2}+m_{\sigma}^{2})}\,,$
(80)

and the symmetrized propagator is

$\displaystyle G_{\rm
sym}^{\sigma\sigma}=\frac{2T\Gamma}{\omega^{2}+\Gamma^{2}(k^{2}+m_{\sigma}^{2})^{2}}.$
(81)

In the symmetric case ($z\to\infty$) the propagator of $\delta\sigma$ and
$\varphi$ become degenerate and $O(4)$ symmetric

$\rho_{\sigma\sigma}(\omega,k)=\bar{\sigma}^{2}\rho_{\varphi\varphi}(\omega,k),$
(82)

with $m^{2}=m_{\sigma}^{2}$.

##  IV Transport coefficients

In this section we will use the response functions calculated in the previous
section to determine the current-current and stress-stress correlation
functions. This will determine the critical behavior of the transport
coefficients, which is analyzed and estimated in Sect. IV.2.

###  IV.1 Hydrodynamic loops

In the critical hydrodynamic theory we outlined in Sect. III, we have
integrated out modes of order $k\sim T$. These modes are incorporated into the
transport coefficients such as $\eta_{\Sigma}$ and its associated noise,
$\xi_{\eta_{\Sigma}}^{\mu\nu}$. Modes of order $k\sim m_{\sigma}$ are
explicitly propagated in the theory, and the critical hydrodynamic theory is
defined with a cutoff $\Lambda_{T}$

$k\sim m_{\sigma}\ll\Lambda_{T}\ll T.$ (83)

In normal hydrodynamics, modes with $k\sim m_{\sigma}$ are integrated out and
incorporated into the transport coefficients of the normal theory, such as
$\eta$ and its noise. The only modes which are explicitly propagated are the
conserved charges, and the theory is defined with a cutoff $\Lambda_{\sigma}$

$k\ll\Lambda_{\sigma}\ll m_{\sigma}\,.$ (84)

The two transport coefficients $\eta_{\Sigma}$ and $\eta$ may be related by
integrating out modes between $k\in[\Lambda_{\sigma},\Lambda_{T}]$.

The $xy$ components of the stress tensor in the critical hydrodynamic theory
is

$\displaystyle T^{xy}=$
$\displaystyle\partial^{x}\delta\sigma\,\partial^{y}\delta\sigma+\bar{\sigma}^{2}\partial^{x}\varphi_{s}\partial^{y}\varphi_{s}+\xi^{xy}_{\eta_{\Sigma}},$
(85)

where the noise satisfies

$\left\langle\xi_{\eta_{\Sigma}}^{xy}(x_{1})\xi_{\eta_{\Sigma}}^{xy}(x_{2})\right\rangle=2T\eta_{\Sigma}\delta^{4}(x_{1}-x_{2})\,.$
(86)

It is understood that the noise in the critical hydrodynamic theory is only
local on scales with $k\ll\Lambda_{T}$, i.e. the $\delta$-function in (86)
should be cutoff at the scale $\Lambda_{T}$ and associated with a scale-
dependent parameter, $\eta_{\Sigma}(\Lambda_{T})$. The stress tensor in the
normal hydrodynamic theory is simply the noise (in the absence of external
flow)

$T^{xy}_{\rm hydro}=\xi^{xy}_{\eta}(x)\,,$ (87)

and satisfies

$\left\langle\xi_{\eta}^{xy}(x_{1})\xi_{\eta}^{xy}(x_{2})\right\rangle=2T\eta\delta^{4}(x_{1}-x_{2})\,.$
(88)

Matching the two effective theories yields Kubo formulas, which require that
the integrated variances of the fluctuations are equal in the two theories
Forster (1995):

$2T\eta=\int d^{4}x\left\langle T^{xy}_{\rm hydro}(t,\bm{x})T^{xy}_{\rm
hydro}(0,{\bm{0}})\right\rangle=\int d^{4}x\left\langle
T^{xy}(t,\bm{x})T^{xy}(0,{\bm{0}})\right\rangle\,.$ (89)

Incorporating the fluctuations of $\sigma$ and $\varphi$ at one loop, this
evaluates to101010 Here and below $d_{A}=3$ and $T_{A}=2$ denote the dimension
and trace of the adjoint representation of the unbroken $SU(2)$ iso-vector
subgroup. The “extra” factor of $1/\omega_{k}^{4}$ multiplying $G_{\rm
sym}^{\varphi\varphi}$ is because of the way $G_{\rm sym}^{\varphi\varphi}$
was defined in (67) as the symmetric correlator of $\omega_{k}\varphi$.

$\displaystyle 2T\eta$
$\displaystyle=2T\eta_{\Sigma}(\Lambda_{T})+2\int^{\Lambda_{T}}\frac{d^{3}k}{(2\pi)^{3}}\frac{d\omega}{2\pi}\left[(k^{x}k^{y}G_{\rm
sym}^{\sigma\sigma})^{2}+d_{A}\frac{\bar{\sigma}^{4}}{\omega_{k}^{4}}(k^{x}k^{y}G_{\rm
sym}^{\varphi\varphi})^{2}\right],$ (90)

where we have anticipated a divergence which is regulated at the scale
$\Lambda_{T}$, as is appropriate for the critical hydrodynamic theory.

The other transport coefficients of interest here are expressed similarly as

$\displaystyle 2T\sigma_{I}=$ $\displaystyle\int
d^{4}x\frac{1}{d_{A}}\left\langle{\textstyle\frac{1}{2}}\\{J_{V,s}^{x}(t,{\bm{x}}),J_{V,s}^{x}(0,{\bm{0}})\\}\right\rangle,$
(91) $\displaystyle 2T\zeta=$ $\displaystyle\int
d^{4}x\left\langle{\textstyle\frac{1}{2}}\left\\{{\mathcal{O}}_{\rm
bulk}(t,{\bm{x}}),{\mathcal{O}}_{\rm bulk}(0,{\bm{0}})\right\\}\right\rangle.$
(92)

where ${\mathcal{O}}_{\rm bulk}=\tfrac{1}{3}T^{i}_{i}+c_{s}^{2}T^{0}_{0}$. The
bulk viscosity is significantly more complicated, and quite susceptible to
physics which goes beyond the mean field approach adopted here. Therefore we
will evaluate the bulk viscosity only in the high temperature symmetric
regime. The relevant operators appearing in the conductivity computation and
the bulk viscosity are

$\displaystyle J^{x}_{V,s}$
$\displaystyle=\bar{\sigma}^{2}\epsilon_{ss^{\prime}s^{\prime\prime}}\varphi_{s^{\prime}}\partial^{x}\varphi_{s^{\prime\prime}},$
(93) $\displaystyle O_{\rm bulk,\infty}$
$\displaystyle=\frac{1}{2}c_{s}^{2}{\mathfrak{m}}^{2}(\delta\sigma^{2}+\pi^{2}).$
(94)

Here and below we use the $\infty$ subscript to indicate that we have made
approximations of $O_{\rm bulk}$ appropriate only in the symmetric phase where
$z$ is large. We have also recognized that near $T_{c}$ the terms stemming
from $c_{s}^{2}T^{0}_{0}$ are parametrically large compared to $T^{i}_{i}$.
Evaluating the relevant Feynman diagrams leads to

$\displaystyle 2T\sigma_{I}$
$\displaystyle=2T\sigma_{\Sigma}+2T_{A}\bar{\sigma}^{4}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{d\omega}{2\pi}\frac{1}{\omega_{k}^{4}}(k^{x}G_{\rm
sym}^{\varphi\varphi})^{2}\,,$ (95) $\displaystyle 2T\zeta_{\infty}$
$\displaystyle\approx
2T\zeta_{\Sigma}+2\left(\tfrac{1}{2}c_{s}^{2}{\mathfrak{m}}^{2}\right)^{2}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{d\omega}{2\pi}\left[(G_{\rm
sym}^{\sigma\sigma})^{2}+d_{A}(G_{\rm sym}^{\pi\pi})^{2}\right].$ (96)

The propagators in these expressions can be read from (67) and (81). The
$\sigma$ and $\pi$ propagators at large $z$ which are used in (96) are the
same and are given in (81). To make the results of the above integrations more
explicit we recall the dimensionless variables of Sect. III.3

$r^{2}=\frac{\Gamma}{\Gamma+D},\quad\text{and}\quad u^{2}=\frac{v^{2}}{\Gamma
Dm^{2}},$

and introduce the symmetric, dimensionless function $f_{n}(r,u)=f_{n}(u,r)$,
defined by

$\displaystyle f_{n}(r,u)$
$\displaystyle=\frac{16}{15\pi}\int_{0}^{\infty}\frac{dk}{m}\frac{k^{2n}}{(k^{2}+m^{2})^{3}}\frac{m^{8-2n}k^{2}}{(k^{2}+r^{2}m^{2})(k^{2}+u^{2}m^{2})}.$
(97)

For the transport coefficients in question, we will need only the following
explicit expressions

$\displaystyle f_{3}(r,u)$
$\displaystyle=\frac{1}{15\left(r^{2}-u^{2}\right)}\left[\frac{r^{2}\left(8r^{2}+9r+3\right)}{(r+1)^{3}}-\frac{u^{2}\left(8u^{2}+9u+3\right)}{(u+1)^{3}}\right],$
(98) $\displaystyle f_{2}(u,r)$
$\displaystyle=\frac{1}{15\left(r^{2}-u^{2}\right)}\left[\frac{r^{2}(3r+1)}{(r+1)^{3}}-\frac{u^{2}(3u+1)}{(u+1)^{3}}\right].$
(99)

More details can be found in Appendix B. Then the conductivity, shear
viscosity, and asymptotic bulk viscosity are given by

$\displaystyle\sigma_{I}(z)$
$\displaystyle=\sigma_{\Sigma}+\frac{TT_{A}}{32\pi
m\Gamma}\left(1-5u^{2}(1-r^{2})f_{2}(r,u)\right)\,,$ (100a)
$\displaystyle\eta(z)$
$\displaystyle=\eta_{\Sigma}-\frac{T}{32\pi\Gamma}(m_{\sigma}+md_{A}+md_{A}u^{2}(1-r^{2})f_{3}(r,u)),$
(100b) $\displaystyle\zeta_{\infty}(z)$
$\displaystyle=\zeta_{\Sigma}+\frac{T}{8\pi\Gamma
m_{\sigma}^{3}}\left(\tfrac{1}{2}c_{s}^{2}{\mathfrak{m}}^{2}\right)^{2}.$
(100c)

In these expressions the shear viscosity has been renormalized

$\displaystyle\eta_{\Sigma}$
$\displaystyle=\eta_{\Sigma}(\Lambda)+\delta_{aa}\,\frac{T\Lambda}{30\pi^{2}\Gamma},$
(101a)

and the parameters $m(z)$, $m_{\sigma}(z)$, and $u^{2}(z)$, depend on the
scaling variable $z$.

###  IV.2 Discussion

Figure 3: The critical contribution to the hydrodynamic transport
coefficients, $\Delta\eta$and $\Delta\sigma_{I}$, as a function of the scaling
variable $z=th^{-2/3}$. The asymptotic forms at large $z$ and small $z$, e.g.
$\Delta\eta_{\infty}(z)$ and “pion kinetics” respectively, are discussed in
text surrounding eq. (103). For the viscosity we have normalized the curve by
a positive constant, $\eta^{\rm pc}_{\infty}\equiv|\Delta\eta_{\infty}(z_{\rm
pc})|$, so that at the pseudo-critical point, $z_{\rm pc}=1.19$, the orange
dashed asymptotic curve passes through minus one. We have defined
$\sigma_{I\infty}^{\rm pc}$ with an analogous notation. The absolute
magnitudes of these normalization constants are discussed in the text
surrounding eq. (109). The curves depend weakly on two order one parameters,
which we take to be $r^{2}$ and $u_{c}^{2}$ (see Fig. 2).

To gain an appreciation for the results of the previous section, in Fig. 3 we
plot the critical contribution to the transport coefficients $\Delta\eta$ and
$\Delta\sigma$ as a function of the scaling variable, $z$. The normalization
of the curves and the asymptotics at large and small $z$ will be discussed
shortly. We emphasize that Fig. 3 contains just the contribution from critical
modes, e.g. the full shear viscosity takes the form

$\eta(z)=\eta_{\Sigma}+\Delta\eta(z)\,,$ (102)

where $\eta_{\Sigma}$ is a $z$ independent constant (the regular contribution
to the shear viscosity).

At large positive $z$ the propagators for the $\sigma$ and $\pi$ fields become
degenerate and take a simple form (see Sect. III.3). This greatly simplifies
the computation of the hydrodynamic loop, leading to some simple forms for the
critical transport corrections. Expanding our results in (100) for large $z$,
or $u\rightarrow 0$, we find

$\displaystyle\Delta\sigma_{\infty}(z)$ $\displaystyle\equiv\frac{T}{16\pi
m\Gamma},$ (103a) $\displaystyle\Delta\eta_{\infty}(z)$
$\displaystyle\equiv-\frac{Tm_{\sigma}}{8\pi\Gamma},$ (103b)
$\displaystyle\Delta\zeta_{\infty}(z)$ $\displaystyle\equiv\frac{T}{8\pi\Gamma
m_{\sigma}^{3}}\left(\tfrac{1}{2}c_{s}^{2}{\mathfrak{m}}^{2}\right)^{2}.$
(103c)

These large $z$ asymptotics are presented as the (orange) dashed curves in
Fig. 3. In these expressions $T$, ${\mathfrak{m}}^{2}$, $c_{s}$, and $\Gamma$
are constants near $T_{c}$, while the remaining functions, $m(z)$,
$m_{\sigma}(z)$, are scaling functions which are determined by the equilibrium
magnetic equation of state. Outside of the mean field approximation used here,
$\Gamma$ is not a constant, but is expected to grow (fairly weakly) near the
critical point as $\Gamma\sim m_{\sigma}^{d/2-2}\sim m_{\sigma}^{-1/2}$
Rajagopal and Wilczek (1993); Son and Stephanov (2002b). Treating $\Gamma$ and
$D$ as constants is known in the literature as the van Hove approximation
Hohenberg and Halperin (1977).

The asymptotic form of the transport coefficients sets the overall scale for
our results. Thus in Fig. 3 we have divided each transport coefficient by a
$z$-independent constant, the magnitude of the asymptotic result at the
pseudo-critical point

$\displaystyle\sigma_{\infty}^{\rm pc}\equiv$
$\displaystyle\Delta\sigma_{\infty}(z_{\rm pc})\,,$ (104a)
$\displaystyle\eta_{\infty}^{\rm pc}\equiv$
$\displaystyle|\Delta\eta_{\infty}(z_{\rm pc})|\,,$ (104b)
$\displaystyle\zeta_{\infty}^{\rm pc}\equiv$
$\displaystyle\Delta\zeta_{\infty}(z_{\rm pc})|\,.$ (104c)

Estimates for these scale coefficients in absolute units are given below. We
also find that the simple asymptotic forms in (103) provide a useful order of
magnitude estimate over the whole range in $z$, and in Fig. 4 we present the
ratio between the full result and these forms. We expect that our asymptotic
expression for the critical bulk viscosity in (103c) can provide a similarly
good estimate over the whole range in $z$.

Figure 4: Ratio of the singular part of the transport coefficients to the
asymptotic formulas (103) over the full range in $z$.

For large negative $z$, the $\sigma$ is significantly heavier than the pions,
$m_{\sigma}\gg m$. We can integrate out the heavy sigma modes and pions with
$p\sim m_{\sigma}$, leaving a local effective theory for soft pions with
$p\sim m$. A hydrodynamic theory can be worked out for these soft pion modes
coupled to the background stress Grossi  _et al._ (2020). The (stochastic)
hydrodynamic equations for the soft pions are equivalent to a Boltzmann
equation in a “fluid metric”, which describes how the soft pions propagate in
the background fluid Grossi  _et al._ (2020). The collision terms of the
kinetic equation are determined by the transport coefficients of the
hydrodynamic theory. The computation of $\sigma$, $\eta$ and $\zeta$ at large
negative $z$ could thus be done in two steps: first one would match the
hydrodynamic equations at the critical point given in Sect. III to the soft-
pion hydro-kinetic theory, and then one would use the soft-pion kinetic theory
to determine the transport coefficients as a function of the temperature. For
large negative $z$, the results predicted by the (matched) pion hydro-kinetic
theory are shown by the black dotted curves in Fig. 3. The pion kinetic theory
gives a reasonable description of the results of the full theory up to its
boundary of applicability, $z\sim 0$. We will use the pion kinetic theory to
estimate soft pion yields in Sect. V. Further details about the pion kinetic
theory are given in Appendix C.

Now we will make several estimates for the absolute scales of the critical
contribution to the transport coefficients, i.e. we wish to estimate
$\eta_{\infty}^{\rm pc}$, $\zeta_{\infty}^{\rm pc}$, and
$\sigma_{I\infty}^{\rm pc}$ defined in (104). These formulas have a number of
physical quantities that need to be estimated, which we will do in the next
paragraphs.

First, we consider the thermodynamic quantities, which are precisely
determined by lattice measurements. To present each transport coefficient, we
will first divide by the corresponding susceptibilities: $sT$ in the shear and
bulk cases (the momentum susceptibility), and $T\chi_{Q}$ for the conductivity
(the charge susceptibility). The pseudo-critical point is at $T_{\rm pc}\simeq
155\,{\rm MeV}$ Borsanyi  _et al._ (2020). From lattice measurements of QCD
thermodynamics at $T=155$ we have Borsanyi  _et al._ (2010, 2014); Bazavov
_et al._ (2012, 2014):

$sT^{-3}=5.4\,,\qquad\chi_{Q}T^{-2}=0.4\,.$ (105)

We will also need to estimate the screening masses, $m_{\sigma}$ and $m$, at
these temperatures. At a temperature of $T_{\rm pc}=155$, we take from Table X
of Ref. Bazavov  _et al._ (2019)

$m_{\sigma}(T_{\rm pc})=0.271\,{\rm GeV},\,\quad\text{and}\quad m(T_{\rm
pc})=0.198\,{\rm GeV}\,.$ (106)

The mean field predictions for $m_{\sigma}$ and $m$ are described in Sect. II;
the one free mass parameter is adjusted so that the pion screening mass at the
mean field pseudo-critical point at $z_{\rm pc}=1.19$ matches the lattice. The
corresponding mean field $\sigma$ mass at $z_{\rm pc}=1.19$ is
$m_{\sigma}=0.24\,{\rm GeV}$, which is slightly lower than the lattice
results. To summarize, in our estimates below we take

$\displaystyle m_{\sigma}/T=1.56,\quad\text{and}\quad m/T=1.28\,.$ (107)

Finally, in order to evaluate the bulk viscosity we need to estimate
${\mathfrak{m}}^{2}$. In mean field theory we have111111All of these relations
follow with minor algebra from (9) and (17).

$\frac{{\mathfrak{m}}^{2}}{T^{2}}=\frac{m_{\sigma}^{2}}{T^{2}}\left(-\frac{d\log\bar{\sigma}}{d\log
T}\right)=\frac{m_{\sigma}^{2}}{T^{2}}\left(\frac{T_{c}}{T-T_{c}}\right)\left(-\frac{d\log
f_{G}}{d\log z}\right)\simeq 7.0.$ (108)

In making this estimate we have taken $T\simeq 155\,{\rm MeV}$ and
$T_{c}\simeq 132\,{\rm MeV}$ Ding  _et al._ (2019); Kaczmarek  _et al._
(2020), and used the mean field equation of state. In absolute units
${\mathfrak{m}}\simeq 0.410\,{\rm GeV}$, which seems somewhat too low for a
cutoff scale. Indeed $O(4)$ fits to lattice data suggest a somewhat higher
value Kaczmarek  _et al._ (2020).

The real time quantities in the transport coefficients are comparatively
poorly determined. The two real time parameters are the order parameter
relaxation coefficient $\Gamma$ and the diffusion coefficient $D$, which set
the critical relaxation rates, $\Gamma m_{\sigma}^{2}\sim Dm_{\sigma}^{2}$.
$D$ is regular near $T_{c}$ and determines the charge diffusion coefficient
well above $T_{c}$. We will therefore adopt the strong coupling estimate,
$D=1/2\pi T$ Son and Starinets (2007); Schäfer and Teaney (2009); Heinz and
Snellings (2013), and we take $r^{2}=\Gamma/(\Gamma+D)=3/4$ and
$v_{c}^{2}/\Gamma Dm_{c}^{2}=1$ as in Fig. 2 (see Sect. III.3 for further
discussion).

Figure 5: The yields for soft pions due to a critical modification of the
dispersion curve, relative to an expectation based on the vacuum dispersion
curve (see eq. (121)). The results are shown for two different values of the
cutoff $\Lambda$.

With these preliminaries we can estimate the scale factors for each transport
coefficient.

$\displaystyle\frac{\sigma_{I\infty}^{\rm pc}}{\chi_{Q}}$
$\displaystyle=\frac{0.50}{2\pi T}\,\left[\left(\frac{1.5}{\pi
T\Gamma}\right)\left(\frac{1.27}{m/T}\right)\left(\frac{0.4}{\chi
T^{-2}}\right)\right],$ (109a) $\displaystyle\frac{\eta_{\infty}^{\rm
pc}}{sT}$ $\displaystyle=\frac{0.3}{4\pi T}\,\left[\left(\frac{1.5}{\pi
T\Gamma}\right)\left(\frac{5.4}{s/T^{3}}\right)\left(\frac{m_{\sigma}/T}{1.56}\right)\right],$
(109b) $\displaystyle\frac{\zeta_{\infty}^{\rm pc}}{sT}$
$\displaystyle=\frac{0.025}{4\pi T}\,\left[\left(\frac{1.5}{\pi
T\Gamma}\right)\left(\frac{c_{s}^{2}}{0.2}\right)^{2}\left(\frac{{\mathfrak{m}}^{2}}{7.0\,T^{2}}\right)^{2}\left(\frac{5.4}{s/T^{3}}\right)\left(\frac{1.56}{m_{\sigma}/T}\right)^{3}\right].$
(109c)

We have rescaled each transport coefficient by a value which is typical of
strongly coupled plasmas Son and Starinets (2007); Schäfer and Teaney (2009).
Thus, we see that the correction to the shear viscosity is small even in units
of $1/4\pi$. The correction to the charge diffusion coefficient
$D_{Q}=\sigma_{I}/\chi_{Q}$ is also modest, which is surprising given that
this parameter diverges in the chiral limit. Evidently this parametrically
large enhancement does not compensate for the overall kinematics of the loop
integral. Similarly, the bulk viscosity is also parametrically enhanced by
$m_{\sigma}^{-3}$, but in practice this also does not compensate for other
kinematic factors. A similar observation about the bulk viscosity has been
made previously in a somewhat different context Martinez  _et al._ (2019).

##  V Outlook: chiral critical dynamics in heavy ion data?

The previous section estimated the influence of critical chiral modes on the
transport coefficients of QCD. Given the rather large pion mass, these
corrections are modest, and probably can not be observed in practice. However,
it may be possible to observe the critical chiral fluctuations by directly
measuring soft pions, rather than indirectly through their influence on the
kinetics of the system. The approach in this section bears some similarities
with Bluhm  _et al._ (2020), which investigated how a reduced chiral
condensate could influence the thermal fits over a wide range of collision
energies.

Current hydrodynamic codes underestimate the yield of pions at small
transverse momenta, see for example Fig. 3 of Devetak  _et al._ (2020) where
the data to model ratios are approximately $1.5$ for $p_{T}\lesssim\pi T$.
Comparable discrepancies are also found in Mazeliauskas and Vislavicius
(2020); Acharya  _et al._ (2020); Guillen and Ollitrault (2020). In the broken
phase, the critical dynamics is characterized by the formation of light
Goldstone bosons, which is reflected in the spectral density of axial charge
by the formation of two quasiparticle peaks (see Sect. III.3). The dynamics of
the heavy scalar field can be neglected well below $T_{c}$. With this in mind,
it is reasonable to search for effects of the chiral crossover in soft pions.
Well below $T_{c}$, we have previously shown that the phase-space density of
pions with momentum $q\ll\pi T$ is approximately governed by a simple kinetic
equation Grossi  _et al._ (2020). We will use this kinetic equation right at
its boundary of applicability (the pseudo-critical point) to make an estimate
for the critical soft pion yield.

The kinetic equation in the rest frame of the fluid is121212For simplicity we
will limit the discussion to the rest frame of the fluid, leaving the more
general case to the references Grossi  _et al._ (2020).

$\displaystyle\frac{\partial f_{\pi}}{\partial
t}+\frac{\partial\omega_{0}(q)}{\partial q_{i}}\frac{\partial
f_{\pi}}{\partial x^{i}}-\frac{\partial\omega_{0}(q)}{\partial
x^{i}}\frac{\partial f_{\pi}}{\partial
q_{i}}=-\Gamma_{q}\left(f_{\pi}-\frac{T}{\omega_{0}(q)}\right)\,,$ (110)

where $f_{\pi}(t,{\bm{x}},{\bm{q}})$ is the phase-space density of pions, and
the soft pion dispersion curve is Son and Stephanov (2002b)

$\omega^{2}_{0}(q)=v^{2}_{0}\,q^{2}+m^{2}_{0p}\,.$ (111)

Here and in the remainder of this section we have attached the “zero”
subscript to $v^{2}_{0}(T)$ and $m^{2}_{0p}(T)$ as a reminder that (111) holds
only for nearly zero momenta, $q\ll\pi T$. The equilibrium phase-space
distribution in this limit is the classical part of the Bose-Einstein
distribution

$f_{\pi}\big{|}_{\rm eq}=\frac{T}{\omega_{0}(q)}\,.$ (112)

Since $v^{2}_{0}(T)$ and $m_{0p}^{2}(T)$ both drop near $T_{c}$, it is natural
to expect an enhancement of soft pions Son and Stephanov (2002a). Here we will
given estimate of this enhancement by estimating the critical modifications of
(111).

The dispersion curve in (111) is valid only for soft pions $q\ll\pi T$, and at
higher momenta one expects higher derivative corrections, i.e.

$\omega^{2}_{0}(q)=v^{2}_{0}q^{2}+m_{0p}^{2}+\mathcal{O}\left(\frac{q^{4}}{\Lambda^{2}},\frac{m_{0p}^{2}q^{2}}{\Lambda^{2}}\right)\,,$
(113)

with $\Lambda\sim\pi T$. At large momentum the dispersion curve should
approach its vacuum form131313In this formula and in (116), $c=1$ is the speed
of light and $m^{2}_{\rm vac}\simeq 140\,{\rm MeV}$ is the vacuum pion mass.

$\omega^{2}_{\rm vac}(q)=c^{2}q^{2}+m_{\rm vac}^{2}\,.$ (114)

In the future, it might be possible to constrain the dispersion curve at
fourth order in momenta using second order chiral hydrodynamics and lattice
QCD measurements Grossi  _et al._ (2020); Son and Stephanov (2002b). For now,
we will adopt an ansatz for the pion dispersion curve at all momenta which
interpolates between the low and high momentum limits, by writing

$\omega^{2}(q)=v^{2}(q)q^{2}+m_{p}^{2}(q)\,,$ (115)

with $v^{2}(q)$ and $m_{p}^{2}(q)$ taking the rough form

$\displaystyle v^{2}(p)=$ $\displaystyle
c^{2}\,(1-F(p/\Lambda))+v^{2}_{0}\,F(p/\Lambda)\,,$ (116) $\displaystyle
m^{2}_{p}(p)=$ $\displaystyle m^{2}_{\rm
vac}(1-F(p/\Lambda))+m^{2}_{0p}\,F(p/\Lambda)\,.$ (117)

Here $F(p/\Lambda)$ is any cutoff function which has a Taylor series,
$F(y)\simeq 1-y^{2}/2$, at small $y$ and approaches zero for $y\sim 1$. In
Fig. 5, we take

$F(y)=\frac{1}{1+y^{2}/2+y^{4}}\,,$ (118)

although qualitatively similar results were found with a simple cutoff,
$F(y)={\rm max}(1-y^{2}/2,0)$.

In order to have a prediction for the dispersion curve, we still need to
specify $v_{0}^{2}$ and $m^{2}_{0p}=v_{0}^{2}m_{0}^{2}$. These choices should
be approximately consistent with lattice data on screening masses. The lattice
finds that the pion screening mass is approximately its vacuum value for a
temperature of $135\,{\rm MeV}$, and approximately $198\,{\rm MeV}$ at the
pseudo-critical point Bazavov  _et al._ (2019). The temperature of $135\,{\rm
MeV}$ is when the chiral susceptibility has reached approximately 60% of its
maximum and defines $z_{60}$. In mean-field theory $z_{60}=-0.79$ and the
pseudo critical point is at $z_{\rm pc}=1.19$, which is determined from the
maximum of the susceptibility. At $z_{60}=-0.79$, we will choose the pion’s
pole and screening masses to be equal to the vacuum pion mass, and the
velocity to be $c$. The mean-field the scaling curves then dictate the
screening mass at $z_{\rm pc}=1.19$, yielding:

$m_{0}(z_{\rm pc})\simeq 0.197\,{\rm GeV}\,,$ (119)

which is nicely consistent with lattice measurements on the pion screening
mass at $T=155\,{\rm MeV}$. The same mean field scaling curves then give the
values of the pole mass and the pion velocity at the pseudo-critical point:

$\displaystyle m_{0p}(z_{\rm pc})$ $\displaystyle\simeq 0.1\,{\rm GeV}\,,$
(120a) $\displaystyle v_{0}^{2}(z_{\rm pc})$ $\displaystyle\simeq 0.25\,.$
(120b)

In the future it would be nice to measure $v_{0}$ and $m$ very precisely on
the lattice (they are Euclidean quantities) and to verify their critical
scaling behavior in the chiral limit.

We have now fully specified the dispersion curve $\omega^{2}(q)$ with eqs.
(115), (116), (117) and (120). Given the dispersion curve we can estimate the
expected enhancement of yields

$\frac{\frac{dN^{\rm crit}}{d^{3}p}}{\frac{dN_{\rm
vac}}{d^{3}p}}=\frac{\omega_{\rm vac}(p)}{\omega(p)}\,.$ (121)

This prediction is shown in Fig. 5 for two different choices of $\Lambda$. We
note that using the full Bose-Einstein distribution instead of its classical
limit $T/\omega$ produces only minor differences, which a slightly increases
ratio shown in Fig. 5.

The ratio estimated in Fig. 5 is roughly inline with the observed enhancement,
although strong conclusions about the chiral critical point can not be made at
this time. Nevertheless, we find the result encouraging and it strongly
motivates further research. The most obvious deficiency in our estimate is the
lack of resonance decays at a naive level. Resonances are a way of encoding
interactions, and these interactions are already incorporated into the
dispersion curve. It is therefore difficult “include” resonances without
double counting. From a phenomenological perspective, it would be good to know
if the fluctuations in the soft pion yield are correlated with rest of the
pion $p_{T}$ spectrum, or if the variance of the soft yield has an independent
component. This correlation measurement certainly can be done, and is ideally
suited to the proposed ITS3 detector by the ALICE collaboration ALI (2018).
Additional clarifying measurements could include a direct measurement of the
correlations between two soft pions. It should be possible to provide good
theoretical predictions for these correlations using $O(4)$ scaling ideas.
These predictions can be contrasted with the (presumably) rather different
predictions of the hadron resonance gas. Finally, it would be interesting to
see if the velocity of the soft pions could be measured directly with non-
identical particle correlations. We hope to address these and other topics in
the future.

###### Acknowledgements.

We thank Anirban Lahiri and Rob Pisarski for discussions. This work is
supported by the U.S. Department of Energy, Office of Science, Office of
Nuclear Physics, grants Nos. DE-FG-02-08ER41450. AS is supported by the
Austrian Science Fund (FWF), project no. J4406.

##  Appendix A Entropy production

In this appendix we compute entropy production with guidance from Bhattacharya
_et al._ (2011) and the insightful eightfold way classification scheme Haehl
_et al._ (2015). Repeating eq. (34) and eq. (32) for convenience, the entropy
is given by the Gibbs-Duhem relation

$s_{\Sigma}=\frac{1}{T}(e_{\Sigma}+p_{\Sigma}-{\textstyle\frac{1}{2}}\mu_{ab}n_{ab}),$
(122)

and the pressure differential follows from the action

$dp_{\Sigma}=s_{\Sigma}dT+\frac{1}{2}n_{ab}d\mu_{ab}-\frac{1}{2}d(\partial_{\perp}\phi)^{2}+\left(-\frac{\partial
V}{\partial\phi_{a}}+H_{a}\right)\,d\phi_{a}\,.$ (123)

Here $d\equiv u^{\mu}\partial_{\mu}$, and below we define $\partial
u\equiv\partial_{\mu}u^{\mu}$.

Differentiating (122) and using (123), the differential of the entropy density
$ds_{\Sigma}$ can be written as

$\displaystyle Tds_{\Sigma}$
$\displaystyle=de_{\Sigma}-\frac{1}{2}\mu_{ab}dn_{ab}-\frac{1}{2}d(\partial_{\perp}\phi)^{2}+\left(-\frac{\partial
V}{\partial\phi_{a}}+H_{a}\right)d\phi_{a}\,.$ (124)

The divergence of the entropy current is then:

$\displaystyle\partial_{\mu}(s_{\Sigma}u^{\mu})$
$\displaystyle=ds_{\Sigma}+s_{\Sigma}\,\partial u$ (125)
$\displaystyle=\frac{1}{T}[de_{\Sigma}+(e_{\Sigma}+p_{\Sigma})\partial
u]-\frac{\mu_{ab}}{2T}[dn_{ab}+n_{ab}\partial
u]-\frac{1}{2T}d(\partial_{\perp}\phi)^{2}+\left(-\frac{\partial
V}{\partial\phi_{a}}+H_{a}\right)\,\frac{d\phi_{a}}{T}.$ (126)

We will now evaluate the first two terms in square brackets using energy-
momentum and charge conservation respectively.

###  A.1 Energy conservation

Energy conservation follows from the timelike projection of the conservation
law, $u_{\nu}\partial_{\mu}T^{\mu\nu}=0$, and yields

$\displaystyle-
de_{\Sigma}-(e_{\Sigma}+p_{\Sigma})\partial_{\mu}u^{\mu}=-u_{\nu}\partial_{\mu}[\partial^{\mu}\phi\cdot\partial^{\nu}\phi]+u_{\nu}\partial_{\mu}[u^{\mu}u^{\sigma}u^{\nu}u^{\rho}\partial_{\rho}\phi\cdot\partial_{\sigma}\phi]\,.$
(127)

To simplify the notation, we introduce the shorthand

$\xi^{\mu}_{a}=\partial^{\mu}\phi_{a},\quad\xi^{\mu}_{a}=-d\phi_{a}\,u^{\mu}+\partial_{\perp}^{\mu}\phi,$
(128)

and then rhs of 127 can be rewritten as

$\displaystyle
u_{\nu}\partial_{\mu}(\xi^{\mu}\cdot\xi^{\nu}-u^{\mu}u^{\nu}(d\phi)^{2})$
$\displaystyle=d\phi\cdot\partial_{\mu}\xi^{\mu}+\frac{1}{2}d\xi^{2}+u_{\nu}\xi^{\mu}\cdot(\partial_{\mu}\xi^{\nu}-\partial^{\nu}\xi_{\mu})-u_{\nu}\partial_{\mu}(u^{\mu}u^{\nu}(d\phi)^{2}).$
(129)

The curl vanishes due to the definition of $\xi$, and then using
$d\xi^{2}=d(d\phi)^{2}+d(\partial_{\perp}\phi^{2})$ this evaluates to

$\displaystyle
u_{\nu}\partial_{\mu}(\xi^{\mu}\cdot\xi^{\nu}-u^{\mu}u^{\nu}(d\phi)^{2})=d\phi\cdot\partial_{\mu}\partial_{\perp}^{\mu}\phi+\frac{1}{2}d\,(\partial_{\perp}\phi)^{2}\,.$
(130)

Including the dissipative part of the energy-momentum tensor, energy
conservation yields finally

$\displaystyle de_{\Sigma}+(e_{\Sigma}+p_{\Sigma})\,\partial
u=d\phi\,\cdot\partial_{\mu}\partial^{\mu}_{\perp}\phi+\frac{1}{2}d\,(\partial_{\perp}\phi)^{2}+u_{\nu}\,\partial_{\mu}\Pi^{\mu\nu}\,.$
(131)

###  A.2 Charge Conservation

The equation of (partial) current conservation reads

$\displaystyle\partial_{\mu}J^{\mu}_{ab}=\phi_{a}H_{b}-\phi_{b}H_{a},$ (132)

where the current is defined as

$J^{\mu}_{ab}=n_{ab}u^{\mu}+J^{\mu}_{\perp ab}+q^{\mu}_{ab}.$ (133)

Here $n_{ab}$ is the charge, $J^{\mu}_{\perp ab}$ is the superfluid current in
(39), and $q^{\mu}_{ab}$ is the dissipative part of the current,
$q^{\mu}_{ab}u_{\mu}=0$. We then contract the eom with the antisymmetric
tensor $\mu_{ab}$ and find

$\displaystyle-\frac{1}{2}\mu_{ab}\,(dn_{ab}+n_{ab}\,\partial
u)=\frac{1}{2}\mu_{ab}\,\partial_{\mu}q^{\mu}_{ab}+\frac{1}{2}\mu_{ab}\,\partial_{\mu}J^{\mu}_{\perp
ab}+\mu_{ab}\,\phi_{b}H_{a}\,.$ (134)

Using the superfluid current in (39), we find finally

$\displaystyle-\frac{1}{2}\mu_{ab}\,(dn_{ab}+n_{ab}\,\partial
u)=\frac{1}{2}\mu_{ab}\,\partial_{\mu}q^{\mu}_{ab}+\mu_{ab}\phi_{b}\left(\partial_{\mu}\partial_{\perp}^{\mu}\phi_{a}-\frac{\partial
V}{\partial\phi_{a}}+H_{a}\right)\,,$ (135)

where we have inserted, $\phi_{b}\,\partial
V/\partial\phi_{a}-\phi_{a}\,\partial V/\partial\phi_{b}$, which vanishes due
to the $O(4)$ symmetry of the potential.

###  A.3 Synthesis

After substitutions using (131) and (135), we find the final expression for
the entropy production quoted in the text

$\displaystyle\partial_{\mu}(s_{\Sigma}u^{\mu}-\frac{\mu}{2T}\cdot q^{\mu})=$
$\displaystyle\frac{1}{T}\left(d\phi_{a}+\mu_{ab}\phi_{b}\right)\,[\partial_{\mu}\partial^{\mu}_{\perp}\phi_{a}-\frac{\partial
V}{\partial\phi_{a}}+H_{a}]-\Pi^{\mu\nu}\partial_{\mu}\beta_{\nu}-q^{\mu}\cdot\partial_{\mu}\left(\frac{\mu}{2T}\right)\,.$
(136)

##  Appendix B Computing the transport coefficients near the critical point

In this appendix, we gather the details of the computation of the transport
coefficients. First, we note that the dimensionless function introduced in
(97) can be integrated exactly

$\displaystyle f_{n}(r,u)$
$\displaystyle=\frac{16}{15\pi}\frac{m^{7-2n}}{r^{2}-u^{2}}\int_{0}^{\infty}dk\frac{k^{2n}}{(k^{2}+m^{2})^{3}}\left[\frac{r^{2}}{k^{2}+r^{2}m^{2}}-\frac{u^{2}}{k^{2}+u^{2}m^{2}}\right],$
(137) $\displaystyle=\frac{\sec(\pi
n)}{15\left(r^{2}-u^{2}\right)}\Big{[}\frac{4n^{2}\left(r^{2}-1\right)^{2}-8r^{2n+1}+8n\left(r^{2}-1\right)-r^{4}+6r^{2}+3}{\left(r^{2}-1\right)^{3}}$
$\displaystyle\qquad\qquad\qquad-\frac{4n^{2}\left(u^{2}-1\right)^{2}-8u^{2n+1}+8n\left(u^{2}-1\right)-u^{4}+6u^{2}+3}{\left(u^{2}-1\right)^{3}}\Big{]}.$
(138)

Next, we take a closer look at the shear viscosity computation. We see from
(90) that the shear viscosity will have a contribution from the $\sigma$ and
$\varphi$ propagators:

$\displaystyle\langle T^{xy}(x)T^{xy}(z)\rangle$
$\displaystyle=\langle\partial^{x}\delta\sigma(x)\partial^{y}\delta\sigma(x)\partial^{x}\delta\sigma(z)\partial^{y}\delta\sigma(z)\rangle+\sigma_{0}^{4}\langle\partial^{x}\varphi_{a}(x)\partial^{y}\varphi_{a}(x)\partial^{x}\varphi_{b}(z)\partial^{y}\varphi_{b}(z)\rangle,$
$\displaystyle\equiv I_{\sigma\sigma}^{xy}+I_{\varphi\varphi}^{xy}\,.$ (139)

The contribution from the $\sigma\sigma$ propagator reads

$\displaystyle I_{\sigma\sigma}^{xy}$
$\displaystyle=\frac{2T^{2}}{(30\pi^{2}\Gamma)}\int_{0}^{\Lambda}\frac{k^{6}dk}{(k^{2}+m^{2}_{\sigma})^{3}}=\frac{T^{2}\Lambda}{15\pi^{2}\Gamma}-\frac{T^{2}m_{\rm\sigma}}{16\pi\Gamma}.$
(140)

Similarly, we need to evaluate the contribution to the shear viscosity due to
the $\varphi\varphi$ propagator:

$\displaystyle
I^{xy}_{\varphi\varphi}=2d_{A}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{d\omega}{(2\pi)}\frac{\bar{\sigma}^{4}}{\omega_{k}^{4}}(k^{x}k^{y}G^{\varphi\varphi}_{\rm
sym})^{2}=2T^{2}d_{A}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{(k^{x}k^{y})^{2}}{(k^{2}+m^{2})^{2}}\frac{g_{2}^{2}+(g_{1}g_{2}+\omega_{k}^{2})}{(g_{1}+g_{2})(g_{1}g_{2}+\omega_{k}^{2})}\,.$
(141)

We can evaluate the expression neatly by adding and subtracting the leading
divergent piece

$\displaystyle
I^{xy}_{\varphi\varphi}=\frac{2T^{2}d_{A}}{30\pi^{2}\Gamma}\int_{0}^{\Lambda}\frac{k^{6}}{(k^{2}+m^{2})^{3}}+\frac{2T^{2}}{30\pi^{2}}\int
dk\frac{k^{6}}{(k^{2}+m^{2})^{2}}\left(\frac{g_{2}^{2}+(g_{1}g_{2}+\omega_{k}^{2})}{(g_{1}+g_{2})(g_{1}g_{2}+\omega_{k}^{2})}-\frac{1}{g_{1}}\right),$
(142)

and by using (73) and (75), we can evaluate the above expression to find

$\displaystyle
I^{xy}_{\varphi\varphi}=\frac{2T^{2}d_{A}\Lambda}{30\pi^{2}\Gamma}-\frac{2T^{2}md_{A}}{32\pi\Gamma}\left(1+u^{2}(1-r^{2})f_{3}(r,u)\right).$
(143)

Combining the ingredients, we find that the shear viscosity is given by
(100b).

##  Appendix C Comparison with pion kinetics

Our purpose in this appendix is to explain the (black dashed) “$\pi$-kinetics”
curves in Fig. 3. As discussed in Sect. IV, when writing down the hydrodynamic
theory with the $\Sigma$ field we have integrated out modes with $k\sim T$,
which are then incorporated into the dissipative transport coefficients of the
hydrodynamic theory such as $\eta_{\Sigma}$. Modes with $k\sim m_{\sigma}$ are
explicitly propagated in the theory.

At large negative $z$ (well in the broken phase), the $\sigma$ is heavy is
compared to the pions, and can be consistently integrated out by exploiting
the mass hierarchy

$m\ll m_{\sigma}\ll T\,.$ (144)

The resulting hydrodynamic effective theory consists of energy, momentum, and
light pions, which are parameterized by the unitary matrix, $U=e^{i2\varphi}$
Grossi  _et al._ (2020). Modes with $k\sim m_{\sigma}$ are now incorporated
into the new transport coefficients of this theory such as $\eta_{U}$,
$\eta_{U}$ differs from $\eta_{\Sigma}$ due to the contribution of these
modes.

At the longest distances with $k\ll m$, the pion hydrodynamic theory reduces
to ordinary hydrodynamics with the familiar transport coefficients $\eta$,
$\zeta$ and $\sigma_{I}$. Matching the pion effective theory to normal
hydrodynamics determines the contribution of soft pions to these normal
coefficients. This computation gives Grossi  _et al._ (2020)

$\displaystyle\eta=$
$\displaystyle\eta_{U}-\frac{d_{A}Tm}{120\pi(\Gamma+D_{0})}\left[\frac{2r^{3}+4r^{2}+6r+3}{(1+r)^{2}}\right],$
(145a) $\displaystyle\sigma_{I}=$
$\displaystyle(\sigma_{I})_{U}+\frac{T_{A}T}{24\pi
m(\Gamma+D_{0})}\left[\frac{1+2r}{(1+r)^{2}}\right],$ (145b)
$\displaystyle\zeta=$
$\displaystyle\zeta_{U}-\frac{d_{A}Tm}{8\pi(\Gamma+D_{0})}\left(\frac{\beta
c_{s0}^{2}}{t}\right)^{2}\left[\frac{8r^{3}+16r^{2}+16r+7}{4(1+r)^{2}}\right].$
(145c)

Here $r=\Gamma/(\Gamma+D_{0})$, and $\eta_{U}$, $\zeta_{U}$, and
$(\sigma_{I})_{U}$ are the dissipative parameters of the soft-pion effective
theory141414In Grossi  _et al._ (2020), the (renormalized) dissipative
parameters $\eta_{U},\zeta_{U},(\sigma_{I})_{U}$ where called $\eta^{(0)}_{\rm
phys}$, $\zeta^{(0)}_{\rm phys}$, and $(\sigma_{I})^{(0)}_{\rm phys}$. The
transport coefficients $\Gamma$ and $D_{0}$ in this work were called $D_{m}$
and $D_{A}-D_{m}$ in Grossi  _et al._ (2020). .

Expanding our results in eq. (100) for $\eta$ and $\sigma_{I}$ at large
negative $z$ (where the parameter $u$ tends to infinity), we find that our
expressions match with the pion EFT results in (145), provided we identify

$\displaystyle(\sigma_{I})_{U}$
$\displaystyle=\sigma_{\Sigma}+\frac{T_{A}T}{12\pi\Gamma
m_{\sigma}}\left(\frac{\Gamma+D}{D}\right)\left(\frac{\sqrt{\Gamma
Dm_{\sigma}^{2}}}{v}\right)_{-\infty},$ (146a) $\displaystyle\eta_{U}$
$\displaystyle=\eta_{\Sigma}-\frac{Tm_{\sigma}}{32\pi\Gamma}-\frac{Td_{A}m_{\sigma}}{60\pi\Gamma}\left(\frac{D}{\Gamma+D}\right)\left(\frac{v}{\sqrt{\Gamma
Dm_{\sigma}^{2}}}\right)_{-\infty}.$ (146b)

Here we have defined the constant

$\left(\frac{v}{\sqrt{\Gamma
Dm_{\sigma}^{2}}}\right)_{-\infty}\equiv\lim_{z\to-\infty}\frac{v}{\sqrt{\Gamma
Dm_{\sigma}^{2}}}=\frac{u_{c}}{\sqrt{2}}\,,$ (147)

where $u_{c}^{2}\equiv v^{2}_{c}/\Gamma Dm_{c}^{2}$ is a dimensionless
combination of parameters evaluated on the critical line (see Sect. III.3 for
a physical explanation). Throughout the paper we have taken $u_{c}=1$ as in
Fig. 2. As discussed above, the difference between $\eta_{U}$ and
$\eta_{\Sigma}$ (and similarly for the conductivity) comes from integrating
out modes with $k\sim m_{\sigma}$. Thus, for instance, the second term in
(146b) stems from integrating out the $\sigma$ field, while the third term
stems from integrating out hard pions with $k\sim m_{\sigma}$. The
“$\pi$-kinetics” curves in Fig. 3 are the asymptotics given in (145) with
parameters identified in (146).

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