# Shear Viscosity of Asymptotically Hot and Dense QCD Matter:
AMY Formalism vs. A Modified Relaxation Time Approximation in 0-flavor SU(3)
Theory

Noah M. MacKay  0000-0001-6625-2321

###### Abstract

The AMY formalism is widely used to describe the transport coefficients of
asymptotically hot and dense QCD matter, such as shear viscosity $\eta$. In
literature prior to AMY, the viscosity of an asymptotically hot QCD plasma was
expressed by a $q^{2}$ momentum transfer-weighted relaxation time
approximation. Recent studies that compared numerical transport calculations
and analytical expressions for $\eta$ demonstrated that asymptotically high
temperatures and densities induce anisotropic scatterings, which are exhibited
in the quark-gluon plasma produced by relativistic heavy ion collisions. In
these studies, the QGP was treated as a Maxwell-Boltzmann-distributed gluon
gas with added (anti-)quark degrees of freedom. One such method used in the
comparison was the “modified” $q^{2}$ transport-weighted RTA. In this study, a
comparison between the AMY formalism (both numerical calculations and next-
leading-log expression) and the modified RTA expression for $\eta$ is made in
0-flavor SU(3) theory. The comparison between numerical AMY calculations and
the modified RTA method shows perfect agreement under the temperatures
associated with collider-produced QGP. Additionally, AMY is compared with the
Chapman-Enskog method, which is well understood to better describe anisotropic
collider-produced QGP.

###### keywords:

Quark-Gluon Plasma , Thermal Field Theory , Relativistic Heavy Ion Physics

††journal: Nuclear Physics B

[first]organization=Universität Potsdam, Institute of Physics and
Astronomy,addressline=Karl-Liebknecht-Straße 24/25, postcode=14476,
city=Potsdam, country=Germany

##  1 Introduction

Relativistic heavy ion collisions produced at the Relativistic Heavy Ion
Collider (RHIC) and at the Large Hadron Collider (LHC) create a form of high-
temperature and high-density QCD matter, i.e., a quark-gluon plasma (QGP)
containing (anti-)quark and gluon degrees of freedom [1, 2]. The partons are
quasi-free in the plasma, where one can treat the QGP as an ideal thermal gas
at temperatures much greater than the Hagedorn temperature: $T_{H}\simeq 150$
MeV [3]. It is suggested by past comparisons [4, 5, 6] between experimental
anisotropic flow measurements and hydrodynamic models (also kinetic transport
models [7, 8, 9, 10]) that the QGP behaves like a near-perfect fluid with a
very small shear-viscosity to entropy-density ratio $\eta/s$ near the KSS
limit of $1/(4\pi)$ [11]. In this report, natural units are used, i.e.,
$\hbar=k_{B}=c=1$.

In numerical transport models, parton interactions are typically represented
by their interaction cross section(s), including the magnitude and angular
distribution, which then determine plasma properties such as the shear
viscosity $\eta$. While hydrodynamic models treat the $\eta/s$ ratio
(including its temperature dependence) as an input parameter, transport model
calculations can extract calculations of QGP shear viscosity and $\eta/s$
after applying the relation between the parton cross section(s) and $\eta$ [8,
10, 12, 13]. The analytical analogue to numerical transport models is kinetic
theory. For a two-body, i.e., $2\leftrightarrow 2$ collision, the collision
kernel $\mathcal{C}[f]$ from the integro-differential Boltzmann equation,
$p^{\mu}\partial_{\mu}f(\vec{p})=\mathcal{C}[f]$, depends on the the specific
distribution function for each of the two bodies before the collision
$f_{a,b}$ and after the collision $f_{c,d}$, where $f_{i}=f(\vec{p}_{i})$; the
factor $(1+\beta f_{i})$ for each distribution function, where $\beta=1$ for
Bose-Einstein (BE), $-1$ for Fermi-Dirac (FD), and $0$ for Maxwell-Boltzmann
(MB); the relative motions between the two bodies, and the differential cross
section of their shared interaction. Distinctive methods for calculating the
collision kernel, either simply or explicitly, bring rise to various
analytical expressions for the same plasma property, such as the shear
viscosity.

For collider-produced QGP, it is repeatedly demonstrated that the Chapman-
Enskog (CE) method [17, 18] for expressing the shear viscosity (and $\eta/s$)
agrees well with the numerical calculations under both isotropic and
anisotropic scattering cases [14, 15]. In both references, this method was
commonly compared with a modified version of the relaxation time approximation
(labeled as RTA∗), where the relaxation time $\tau$ utilizes the transport
cross section $\sigma_{\mathrm{tr}}=\int d\sigma\,\sin^{2}\theta$ [16] rather
than the total cross section $\sigma=\int d\sigma$ per the traditional
Anderson-Witting expression [19, 20]. This is to account for the $q^{2}$
momentum transfer that the viscosity calculations also depend on. In Ref.
[15], in addition to RTA∗, the hydrodynamic Israel-Stewart [21, 22, 23] and
Navier-Stokes [18] methods were also compared. These methods, used for
comparison with numerical calculations, inherently relied on massless MB
statistics when solving the collision kernel, as well as a one-component
simplification of the QGP as a gluon gas with added $N_{f}$-flavor quark
degrees of freedom [14, 15, 24, 25, 26]. Therefore, the QGP was described by a
differential cross section based on the perturbative QCD (pQCD) gluon-gluon
amplitude (to be seen in Ref. [27], Appendix A).

As demonstrated in Ref. [28], the temperature of the QGP influences the
anisotropy of the scatterings. At higher temperatures, the shear viscosity is
enhanced via the frequency of anisotropic scatterings. As a result, isotropic
scatterings are more frequent at lower temperatures, albeit unphysically low
for the QGP. For asymptotically high temperatures, the formalism framed by
Arnold, Moore and Yaffe [27, 29] is widely used to express the shear viscosity
and other transport coefficients of the QGP. What contributes to its reliable
application is the formalism’s ability to linearize the collision kernel for
both $2\leftrightarrow 2$ scatterings and $1\leftrightarrow 2$
splitting/joining events, and the explicit useage of QFT and “quantum”
statistics, i.e., BE and FD statistics for gluons and (anti-)quarks
respectively with relevant degrees of freedom. These linearization
calculations contribute to viscosity calculations under any number of quark
flavors (up to 6, e.g., the up, down, strange, charm, top, and bottom flavors)
and under any gauge symmetry group, such as the conventional SU(3) group for
QCD. In this report, I will use the acronym “AMY” to describe the formalism
from Arnold, Moore and Yaffe – and the authors themselves – provided in Refs.
[27, 29].

However, in reports prior to AMY that analyzed the viscosity of an
asymptotically hot plasma (i.e., a plasma with small $q^{2}$ momentum
transfer) [30, 31, 32, 33], the relaxation time approximation was used. AMY
acknowledges in the introduction section in Ref. [27] that the so-called
“leading-log calculations” are improvements to the previous RTA approach.
However, there is a question whether AMY, despite its novelty and robustness,
has an inherent RTA∗ behavior under a certain temperature region, especially
in 0-flavor SU(3) theory (i.e., a gluon gas simplification of the QGP). This
is the goal of this study, to compare the AMY formalism with RTA∗, using the
exact pQCD gluon-gluon amplitude for both methods. Any discrepancy between
quantum statistical calculations through AMY and MB statistical calculations
through RTA∗ is negligible, as quantum statistics approximate to MB statistics
under high temperatures and densities [28]. A secondary comparison can also be
made, which compares the AMY and RTA∗ methods with the CE method, which is
well understood to better describe anisotropic, i.e., high-temperature,
collider-produced QGP.

Asympotically high temperatures enable ultra-relativistic mechanics in the
QGP, where the rest mass of partons is negelected due to $T\gg m$. In
addition, the running gauge coupling $g$ is small due to asymptotic freedom
[34], allowing the scattering amplitude $|\mathcal{M}|^{2}$ of parton
interactions to be inserted into the differential cross section, where
$d\sigma/d\Omega\propto|\mathcal{M}|^{2}$. With such differential cross
sections, we can extract the shear viscosity and make comparisons.

##  2 Methods

###  2.1 The Gluon-Gluon Interaction

In previous studies [9, 10, 14, 15, 16, 25, 26, 35], numerical transport
models, such as the ZPC parton cascade and a multi-phase transport (AMPT)
model, utilized a cross section that is based on pQCD gluon-gluon scatterings.
In this study, the pQCD gluon-gluon scattering amplitude in SU(3) [27, 36] is
explicitly used:

$|\mathcal{M}|^{2}=1152g^{4}\left(3-\frac{su}{(t-m_{D}^{2})^{2}}-\frac{st}{(u-m_{D}^{2})^{2}}-\frac{tu}{s^{2}}\right).$
(1)

In the above, $s$, $t$ and $u$ are the so-called Mandelstam variables from
Feynman diagram calculations (${s}$ relates to the square of the center-of-
momentum, or CM, energy; $t$ and $u$ both relate to the square of the momentum
transfer). In some literature, the Mandelstam variables are given the hat
notation $\hat{s},~{}\hat{t},~{}\hat{u}$; in this report I will not use the
hat notation. As the external gluon lines are massless, $u=-t-s$; this defines
Eq. (1) as purely dependent on the $s$\- and $t$-Mandelstam variables.

In addition, $g$ and $m_{D}$ are the QCD gauge coupling and the Debye
screening masses respectively. As a running parameter, the QCD gauge coupling
can be expressed as a function of $T$ [37]:

$\frac{1}{g^{2}(T)}=\frac{9}{8\pi^{2}}\ln\left(\frac{T}{\Lambda_{T}}\right)+\frac{4}{9\pi^{2}}\ln\left[2\ln\left(\frac{T}{\Lambda_{T}}\right)\right],$
(2)

where $\Lambda_{T}=30$ MeV, and one can define
$g=\left[1/g^{2}(T)\right]^{-1/2}$. The Debye screening mass is defined to be
proportional to the product $gT$ in $N_{f}$-flavor SU(3) theory [27]:

$m_{D}^{2}=\frac{1}{3}\left(3+\frac{1}{2}N_{f}\right)g^{2}T^{2}.$ (3)

For $N_{f}=0$, $m_{D}=gT$ straight-forwardly. As $g$ is a running parameter
via Eq. (2), the Debye mass is also a running parameter.

Eq. (1) is readily summed over all spin-color states; however, it is not
averaged by the incoming gluons’ degrees of freedom. As gluons have 2 spin
states and 8 gluon numbers, the averaging factor is $(2\times 8)^{2}=256$.
Thus, we have an averaged $gg\leftrightarrow gg$ amplitude, with $u=-t-s$:

$\langle|\mathcal{M}|^{2}\rangle=\frac{9g^{4}}{2}\left(3+\frac{s(t+s)}{(t-m_{D}^{2})^{2}}-\frac{st}{(t+s+m_{D}^{2})^{2}}+\frac{t(t+s)}{s^{2}}\right).$
(4)

Provided in Ref. [38], a mass-independent differential cross section
$d\sigma/d\Omega$ in the CM frame is defined in terms of the spin-color
averaged amplitude $\langle|\mathcal{M}|^{2}\rangle$. Based on the Mandelstam
variables $s$ and $t$, the differential cross section is proportional to
$1/s^{2}$ as it is differentiated by the momentum transfer variable $t$:

$\frac{d\sigma}{dt}=\frac{S}{16\pi s^{2}}\langle|\mathcal{M}|^{2}\rangle.$ (5)

Here, $S$ is a statistical factor that corrects the over-counting of identical
final-state particles. In the most general case for $n$ identical final-state
particles of $m$ different species,

$S=\prod_{i=1}^{m}\frac{1}{n_{i}!}.$ (6)

In the case of no identical particles in the final state, $S=1$. In our case
for $gg\leftrightarrow gg$ ($n=2$ and $m=1$), $S=1/2$. Therefore,

$\begin{split}\frac{d\sigma}{dt}&=\frac{1}{32\pi
s^{2}}\langle|\mathcal{M}|^{2}\rangle\\\ &=\frac{9g^{4}}{64\pi
s^{2}}\left(3+\frac{s(t+s)}{(t-m_{D}^{2})^{2}}-\frac{st}{(t+s+m_{D}^{2})^{2}}+\frac{t(t+s)}{s^{2}}\right).\end{split}$
(7)

###  2.2 Shear Viscosity

Allowing a MB-distributed system with no chemical potential to be slightly
perturbed out of equilibrium, the distribution function in the Boltzmann
equation $f(\vec{p})$ can be expressed as an expansion containing the local
equilibrium distribution and a small deviation:

$f(\vec{p})=f_{0}(\vec{p})[1+\phi].$ (8)

Here, $f_{0}(\vec{p})=\exp(-U_{\nu}p^{\nu}/T)$ is the local equilibrium
distribution with $U_{\nu}p^{\nu}=p^{0}\equiv\sqrt{|\vec{p}|^{2}+m^{2}}$ under
the metric $g_{\mu\nu}=\mathrm{diag}(+1,-1,-1,-1)$ ($U^{\mu}$ is the 4-flow
velocity whereby $U_{\mu}U^{\mu}=1$, and $p^{\mu}$ is the 4-momentum of a
particle). Throughout the derivations, ultra-relativistic (i.e., massless)
particles, such as the QGP partons, will be considered. In addition,

$\phi=\varepsilon_{0}+\varepsilon_{\mu}p^{\mu}+\varepsilon_{\mu\nu}p^{\mu}p^{\nu}+\dots$
(9)

is a polynomial deviation function that satisfies $|\phi|\ll 1$, where the
choice of the highest order in the polynomial (i.e., the highest moment) fine-
tunes the evaluation of the collision kernel $\mathcal{C}[f]$. Through this
linearization procedure in the distribution function and in the deviation
polynomial, the kernel is linearized as an operator acting on $\phi$ and using
the local equilibrium distribution:

$p^{\mu}\partial_{\mu}f_{0}(\vec{p})=-f_{0}(\vec{p})\mathcal{L}[\phi].$ (10)

This is where solving the Boltzmann equation can go on different paths, based
on the preferred moment chosen to linearize the collision kernel. This is how
we have the various approaches – namely the relaxation time approximation and
the Israel-Stewart, Navier-Stokes and Chapman-Enskog methods (obtaining the
latter three is discussed in A) – that each assigns a specific Ansatz for
defining $\phi$ and obtaining $\mathcal{L}[\phi]$. However, all methods have a
common step in solving the left-hand side of Eq. (10). Using
$f_{0}(\vec{p})=\exp(-U_{\nu}p^{\nu}/T)$, the left-hand side of Eq. (10) comes
to be

$p^{\mu}\partial_{\mu}f_{0}(\vec{p})=-f_{0}(\vec{p})\frac{p^{\mu}p^{\nu}}{T}\left(\partial_{\mu}U_{\nu}-\frac{1}{T}U_{\nu}\partial_{\mu}T\right),$
(11)

where the term $T^{-1}\partial_{\mu}T$ is defined by the conservation laws
$\partial_{\nu}T^{\mu\nu}=0$ and $\partial_{\mu}S^{\mu}=0$ for the energy-
momentum and entropy tensors, respectively. For the case of ultra-relativistic
partons with a square of the sound velocity of $v_{s}^{2}\equiv
dP/d\epsilon=1/3$ (c.f. Eqs. (42) and (43) in Ref. [39]), one can obtain the
following for the left-hand side of Eq. (10):

$\begin{split}&p^{\mu}\partial_{\mu}f_{0}(\vec{p})=-f_{0}(\vec{p})\frac{p^{\mu}p^{\nu}}{2T}\left[\nabla_{\langle\mu}U_{\nu\rangle}+\frac{2}{3}g_{\mu\nu}\partial_{\rho}U^{\rho}\right],\\\
&\mathrm{where}~{}~{}~{}\nabla_{\langle\mu}U_{\nu\rangle}=\left(D_{\mu}U_{\nu}+D_{\nu}U_{\mu}+\frac{2}{3}\Delta_{\mu\nu}\partial_{\rho}U^{\rho}\right),\end{split}$
(12)

$D_{\mu}=\partial_{\mu}-U_{\mu}U^{\nu}\partial_{\nu}$, and
$\Delta_{\mu\nu}=U_{\mu}U_{\nu}-g_{\mu\nu}$.

Shear viscosity $\eta$, in part with the bulk viscosity $\zeta$, comes from
the dissipative part of the energy-momentum tensor $\pi_{\mu\nu}$, which is
generally defined as a linear combination of a shearing and bulking factor
[39]:

$\pi_{\mu\nu}=\eta\nabla_{\langle\mu}U_{\nu\rangle}-\zeta\Delta_{\mu\nu}\partial_{\rho}U^{\rho}.$
(13)

The left-hand side of Eq. (10) can therefore be expressed in terms of the
dissipative tensor and the viscosities. However, setting the condition that
$\partial_{\rho}U^{\rho}=0$ in the local rest frame, only the shearing
contribution takes precedence:

$\frac{-p^{\mu}p^{\nu}}{2T}\frac{\pi_{\mu\nu}}{\eta}f_{0}(p)=-f_{0}(p)\mathcal{L}[\phi].$
(14)

From this revision of the Boltzmann equation, one can define the shear
viscosity $\eta$ as inversely proportional to the linearized collision kernel,
i.e., $\eta\propto\mathcal{L}[\phi]^{-1}$. Furthermore, as shear viscosity
describes the thermodynamic flow of the off-diagonal shear stress tensor
$\pi_{ij}$, the spacetime indices $\mu\nu$ only need to be the orthogonal
3-space indices $ij$:

$\eta=\frac{1}{2T}\frac{\pi_{ij}p^{i}p^{j}}{\mathcal{L}[\phi]}.$ (15)

Now we define the shear stress tensor. The energy-momentum tensor in the local
rest frame is ideally defined as
$T_{\mu\nu}=(\epsilon+P)U_{\mu}U_{\nu}-Pg_{\mu\nu}$. As the energy density
$\epsilon$ and ultra-relativistic pressure $P=\epsilon/3$ are defined as
momentum-space integrals in kinetic theory, the tensor itself can be written
as a momentum-space integral, under any distribution with quantum degrees of
freedom $d_{j}$:

$T^{\mu\nu}=d_{j}\int_{-\infty}^{\infty}\frac{d^{3}\vec{p}}{(2\pi)^{3}}\frac{p^{\mu}p^{\nu}}{p^{0}}f(\vec{p}).$
(16)

For a slightly perturbed system, we use the expansion of the distribution
function provided as Eq. (8) in the integral definition of $T^{\mu\nu}$. This
is to define the tensor at local equilibrium and its small deviation, i.e.,
$T^{\mu\nu}=T^{\mu\nu}_{(0)}+\pi^{\mu\nu}$:

$T^{\mu\nu}=d_{j}\int_{-\infty}^{\infty}\frac{d^{3}\vec{p}}{(2\pi)^{3}}\frac{p^{\mu}p^{\nu}}{p^{0}}f_{0}(\vec{p})+d_{j}\int_{-\infty}^{\infty}\frac{d^{3}\vec{p}}{(2\pi)^{3}}\frac{p^{\mu}p^{\nu}}{p^{0}}f_{0}(\vec{p})|\phi|.$
(17)

Here, $\pi^{\mu\nu}$ is the dissipative part of the energy-momentum tensor,
which is readily defined as an integral:

$\begin{split}\pi^{\mu\nu}=d_{j}\int_{-\infty}^{\infty}\frac{d^{3}\vec{p}}{(2\pi)^{3}}\frac{p^{\mu}p^{\nu}}{p^{0}}f_{0}(\vec{p})|\phi|.\end{split}$
(18)

This integral expression matches Eq. (36) in Ref. [39], however having set
$d_{j}=1$ for semi-classical particles, and the integral is summed over all
relevant particle types in the gas. For a one-component mixture, such as the
gluon gas simplification for the QGP, the summation is dropped, as readily
defined.

For Eq. (15), the spacetime indices of $\pi^{\mu\nu}$ only need to be the off-
diagonal 3-space indicies $ij$. This defines, therefore, an integral
definition for the shear viscosity using $|\phi|=2/15$:

$\eta=\frac{d_{j}}{15T}\int_{-\infty}^{\infty}\frac{d^{3}\vec{p}}{(2\pi)^{3}}\frac{|\vec{p}|^{4}}{p^{0}}\frac{f_{0}(\vec{p})}{\mathcal{L}[\phi]},$
(19)

which is analogous to Eq. (39) in Ref. [39]. Specifically for massless partons
where $p^{0}=|\vec{p}|$, one has

$\eta=\frac{d_{j}}{15T}\int_{-\infty}^{\infty}\frac{d^{3}\vec{p}}{(2\pi)^{3}}|\vec{p}|^{3}\frac{f_{0}(\vec{p})}{\mathcal{L}[\phi]}.$
(20)

####  2.2.1 The Relaxation Time Approximation

RTA was first used by Bhatnagar, Gross and Krook [40], which is designed to
solve the linearized collision kernel by assigning a characteristic frequency
at which collisions in local equilibrium occur. Its inverse, therefore, is
characterized as the time between consecutive collisions in local equilibrium.
In terms of the linearized kernel, RTA addresses the first moment, i.e., the
first-order linear term of the $\phi$ polynomial as the particle energy in
local equilibrium, $\varepsilon_{\mu}p^{\mu}=U_{\mu}p^{\mu}\equiv p^{0}$, and
the characteristic frequency (defined as an integral via the linearized
kernel) is factored in:

$\mathcal{L}[\phi]=\frac{p^{0}}{\tau}.$ (21)

Here, $\tau=\omega^{-1}$ is the relaxation time, which is for now left
undefined. Specifically with $p^{0}=|\vec{p}|$, Eq. (20) is defined in the
massless RTA case as

$\eta=d_{j}\frac{\tau}{15T}\int_{-\infty}^{\infty}\frac{d^{3}\vec{p}}{(2\pi)^{3}}|\vec{p}|^{2}f_{0}(\vec{p}).$
(22)

Eq. (22) can be reduced even further for a simpler definition. Directly from
statistical mechanics, the number density and the average of a given quantity
$a$ under any distribution (with quantum degeneracy and in local equilibrium)
are defined as follows:

$\begin{split}&n=d_{j}\int_{-\infty}^{\infty}\frac{d^{3}\vec{p}}{(2\pi)^{3}}f_{0}(\vec{p}),~{}~{}~{}\langle
a\rangle=\frac{1}{Z_{0}}\int_{-\infty}^{\infty}d^{3}\vec{p}\,a(\vec{p})f_{0}(\vec{p}),\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}Z_{0}\equiv\int_{-\infty}^{\infty}d^{3}\vec{p}\,f_{0}(\vec{p})=\frac{(2\pi)^{3}}{d_{j}}n.\end{split}$
(23)

From these definitions, the massless RTA shear viscosity can be straight-
forwardly redefined as

$\eta=\frac{\langle|\vec{p}|^{2}\rangle}{15T}n\tau.$ (24)

Under massless MB statistics, $\langle|\vec{p}|^{2}\rangle=12T^{2}$. Also
under massless MB statistics, the energy density is defined as $\epsilon=3nT$,
which is introduced through $\langle|\vec{p}|^{2}\rangle$:

$\eta=\frac{4}{15}\epsilon\tau.$ (25)

Eq. (25) was previously obtained in the numerical Green-Kubo context in Ref.
[26], whereby the relaxation time $\tau$ was numerically calculated under a
ZPC parton cascade model. In this provided context, $\tau$ is an integral
expression; traditionally, the Anderson-Witting relaxation time [19, 20] is
used, which expresses $\tau$ in terms of the number density, an energy-
independent total cross section, and the relative velocity between two
particles in a two-distribution, i.e., thermal average:

$\tau=\frac{1}{n\sigma\langle v_{\mathrm{rel}}\rangle}.$ (26)

For massless particles, $\langle v_{\mathrm{rel}}\rangle=1$.

####  2.2.2 Modified RTA (RTA∗)

On general physical grounds, the viscosity also depends on the $q^{2}$
momentum transfer that the collisions produce on average [14, 15]. This
momentum transfer would therefore infleunce the effective measurement on the
scattering cross section. In literature sometimes, the transport cross section
$\sigma_{\mathrm{tr}}$ is considered in lieu of the total cross section in the
Anderson-Witting formula, where [16]

$\sigma_{\mathrm{tr}}\equiv\int
d\sigma\,\sin^{2}\theta=\int_{-s}^{0}\frac{d\sigma}{dt}\left(-\frac{4t^{2}}{s^{2}}-\frac{4t}{s}\right)dt.$
(27)

The latter definition is in terms of the $s$\- and $t$-Mandelstam variables,
and $d\sigma/dt$ is the momentum-transfer differential cross section, such as
Eq. (7).

To account for all values of $q^{2}$ momentum transfer in the transport cross
section, it is placed in the thermal average to be evaluated with the relative
velocity. In Ref. [14], the following expression was dubbed the modified
relaxation time:

$\tau^{*}=\frac{1}{n\langle\sigma_{\mathrm{tr}}v_{\mathrm{rel}}\rangle},$ (28)

where, for the case of massless partons [41],

$\langle\sigma_{\mathrm{tr}}v_{\mathrm{rel}}\rangle=\frac{1}{16}\int_{0}^{\infty}\sigma_{\mathrm{tr}}\left[\tilde{u}^{4}K_{1}(\tilde{u})\right]d\tilde{u},$
(29)

$K_{n}$ is the modified Bessel function of the second kind, and
$\tilde{u}=\sqrt{{s}}/T$ is the integration variable. This leads to the
definition of shear viscosity under a modified version of RTA, or RTA∗ [14,
15]:

$\eta^{\mathrm{RTA^{*}}}=\frac{4}{5}\frac{T}{\langle\sigma_{\mathrm{tr}}v_{\mathrm{rel}}\rangle}.$
(30)

####  2.2.3 $\eta$ in pQCD

The AMY formalism is widely used to express the shear viscosity of a QGP in
pQCD [27]. In this regime, the plasma temperature is asymptotically high and
QFT is applied to the Boltzmann equation. In reports prior to AMY that
analyzed the viscosity of an asymptotically hot plasma [29, 30, 31, 32, 33],
RTA with $q^{2}$ momentum transfer-weighting was used. Specifically in Refs.
[30, 31, 32, 33], the obtained relaxation rate of a weak interacting QCD
plasma, $\tau^{-1}$, is proportional to the inverse log of the QCD structure
constant $\alpha_{s}$:

$\frac{1}{\tau}\propto\alpha_{s}^{2}\ln\left(1/\alpha_{s}\right)T+\mathcal{O}(\alpha_{s}^{2})\propto
g^{4}T\ln\left(1/g\right),$ (31)

where $g=\sqrt{4\pi\alpha_{s}}$. If one applies Eq. (31) in place of $1/\tau$
in Eq. (25) with $\epsilon\propto T^{4}$, the obtained expression for
viscosity is the following:

$\eta=\kappa\frac{T^{3}}{g^{4}\ln(1/g)}.$ (32)

Here, $\kappa$ is a proportionality constant. This expression for shear
viscosity is provided as Eq. (1.14) in Ref. [29]. Table 2 in Ref. [29] offers
a list of specific values for $\kappa$ depending on number of quark flavors in
SU(3) theory.

Should one follow this formula directly as provided with Eq. (2) as $g$, a
negative curve of $\eta/\kappa$ is produced for all temperatures. This led AMY
in Ref. [27] to use expansion of inverse logs to provide a correction.
Firstly, AMY considers a collision kernel $\mathcal{C}[f]$ that includes both
$2\leftrightarrow 2$ and $1\leftrightarrow 2$ interactions. The
$2\leftrightarrow 2$ kernel is generally defined for the collision
$a+b\rightarrow c+d$, whether elastic or inelastic, and under quantum
statistics:

$\begin{split}\mathcal{C}[f]=\frac{1}{4|\vec{p}_{a}|\nu_{a}}\sum_{bcd}\int&|\mathcal{M}|^{2}(2\pi)^{4}\frac{\delta^{4}(p_{a}+p_{b}-p_{c}-p_{d})}{2|\vec{p}_{b}|2|\vec{p}_{c}|2|\vec{p}_{d}|}\frac{d^{3}\vec{p}_{b}}{(2\pi)^{3}}\frac{d^{3}\vec{p}_{c}}{(2\pi)^{3}}\frac{d^{3}\vec{p}_{d}}{(2\pi)^{3}}\\\
&\times\Big{(}f_{a}f_{b}[1\pm f_{c}][1\pm f_{d}]-f_{c}f_{d}[1\pm f_{a}][1\pm
f_{b}]\Big{)}.\end{split}$ (33)

Here, $f_{i}=f(\vec{p}_{i})$ for the particle types $a,~{}b,~{}c$ and $d$; the
$a+b\rightarrow c+d$ interaction is depicted in the summed (but not averaged)
scattering amplitude $|\mathcal{M}|^{2}$. The kernel itself is summed only
over the types $b,~{}c,~{}d$. In addition, $\nu_{a}$ is the spin-color states
of particle type $a$, i.e., 6 for (anti-)quarks and 16 for gluons. In 0-flavor
SU(3) theory, i.e., for the pure gluon gas, all particles are of the same
species and the summation is dropped. In addition, BE statistics are utilized,
where only the Bose enhancement $[1+f_{i}]$ is applied in $\mathcal{C}[f]$;
and $|\mathcal{M}|^{2}$ is the unaveraged gluon-gluon amplitude (Eq. [1]).

AMY minimizes the linearized collision kernel $\mathcal{C}(m_{D}/T)$ by
driving $m_{D}/T\rightarrow 0$ in Eq. (33) via a variational method of basis
functions. This effectively considers asymptotically high temperatures, and
with it forward-peaking scatterings. After minimization, the kernel is then
subjugated in a log expansion; as the viscosity is inversely proportional to
the linearized kernel, the log expansion acting on the linearized kernel acts
as an inverse log expansion on the viscosity. This defines a geometric sum for
the viscosity; only the first term is considered for the “next-leading-log”
expression of the viscosity:

$\eta^{\mathrm{NLL}}=\frac{T^{3}}{g^{4}}\left[\frac{\eta_{1}}{\ln(\mu_{*}/m_{D})}\right].$
(34)

Here, $\mu_{*}/T$ is an arbitrary scale (so that $\mu_{*}/m_{D}\propto 1/g$
rather than $=1/g$), whose value is specific for number of quark flavors and
gauge symmetry (see Table 1 in Ref. [27]). Obtaining $\eta_{1}$ depends on the
calculation of the linearized collision kernel $\mathcal{C}(m_{D}/T)$, which
in turn also depends on the number of quark flavors and the gauge symmetry
implied in $|\mathcal{M}|^{2}$. For 0-flavor SU(3), AMY calculates
$\eta_{1}=27.126$ with the scaling $\mu_{*}/T=2.765$ (obtaining $\eta_{1}$
under MB statistics, and how it closely relates to AMY’s $\eta_{1}$ under BE
statistics, is provided in B).

Note that Eq. (34) is a “high-temperature,” i.e., $m_{D}/T\rightarrow 0$
approximation to the exact numerical calculations from the basis functions. As
shown in Figure 1 for 0-flavor SU(3) theory, the mutual agreement between Eq.
(34) and the exact calculations is present under $m_{D}/T\leq 1$, as per
design. However, under arguments of $m_{D}/T>1$, the inverse-log curve begins
to diverge away from the basis function curve. The 0-flavor SU(3) basis
function curve is readily offered in Figure 2 in Ref. [27]. Alternatively, one
can avoid the basis function calculations by deducing a sufficient amount of
data points along the 0-flavor curve, and from the dataset interpolate a
function to smoothen the list-line plot into a flowing curve.

Figure 1:  0-flavor SU(3) AMY curves for $\eta\times g^{4}/T^{3}$ vs.
$m_{D}/T$; the NLL log expression is drawn as the red solid curve, and the
basis function calculations are interpolated as the black dashed curve.

Divergence in Eq. (34) occurs as $\mu_{*}/m_{D}\rightarrow 1$, i.e., as
$m_{D}/T\rightarrow\mu_{*}/T$. For 0-flavor SU(3), $m_{D}/T\equiv g$, meaning
divergence occurs as $g\rightarrow 2.765$. This corresponds to a threshold
temperature of $T=74.5$ MeV via Eq. (2), which is much below the Hagedorn
temperature. For collider-produced QGP, where the temperature window is on
average between $T=550$ MeV and $T=150$ MeV [15, 24], this relates to a
relevant x-axis region of $m_{D}/T\in[1.56,2.06]$. In Figure 1, it is the
region along the x-axis where the NLL formula begins to diverge from the basis
function curve.

Therefore, for a proper comparison between the AMY and RTA∗ viscosities for a
collider-produced QGP, the numerical calculations via the basis functions will
be compared with RTA∗. Eq. (34) will also be provided for comparison in the
x-region of $m_{D}/T\leq 1$.

##  3 $\eta\times g^{4}/T^{3}$ vs. $m_{D}/T$: AMY vs. RTA∗

Using Eq. (7), the transport cross section for the pQCD gluon-gluon
interaction is defined as follows, given the variable combinations $w=m_{D}/T$
and $\tilde{u}=\sqrt{s}/T$:

$\begin{split}\sigma_{\mathrm{tr}}&=\frac{9g^{4}}{8\pi
T^{2}\tilde{u}^{2}}\left[-3\left(\frac{w^{2}}{\tilde{u}^{2}}+\frac{34}{45}\right)+\left(\frac{w^{2}}{\tilde{u}^{2}}+1\right)\left(\frac{3w^{2}}{\tilde{u}^{2}}+1\right)\ln\left(1+\frac{\tilde{u}^{2}}{w^{2}}\right)\right]\\\
&\equiv\frac{9g^{4}}{8\pi
T^{2}\tilde{u}^{2}}H\left(\frac{w^{2}}{\tilde{u}^{2}}\right).\end{split}$ (35)

The last line is to redefine the gluon-gluon transport cross section as a
short-hand function of the x-axis variable $w$ from Figure 1 and the
integration variable $\tilde{u}$ from Eq. (29). Furthermore, one can
reformulate Eq. (30) to make it comparable to Eq. (34):

$\begin{split}\eta^{\mathrm{RTA^{*}}}=\frac{T^{3}}{g^{4}}\left[\frac{35.744}{\int_{0}^{\infty}H\left({w^{2}}/{\tilde{u}^{2}}\right)\cdot[\tilde{u}^{2}K_{1}(\tilde{u})]d\tilde{u}}\right].\end{split}$
(36)

Figure 2 depicts the log-log plot of $\eta\times g^{4}/T^{3}$ vs. $m_{D}/T$
under RTA∗ and 0-flavor SU(3) AMY formalism (both inverse-log expression and
basis function calculations). As seen in the figure, small arguments of
$m_{D}/T$ show that RTA∗ severely underestimates the highly anisotropic QGP
viscosity. The same conclusion was made in Refs. [14, 15], which instead used
a “large $s$” gluon-gluon amplitude based on Eq. (1). However, for
$m_{D}/T>1$, the AMY basis function calculations for 0-flavor SU(3) agree well
with the RTA∗ curve, especially within the relevant region of $m_{D}/T\equiv
g$ that corresponds to gauge coupling values for a collider-produced QGP.

Figure 2:  $\eta\times g^{4}/T^{3}$ vs. $m_{D}/T$ under RTA∗ (blue solid) and
0-flavor SU(3) AMY formalism (red solid via NLL, black dashed from the basis
functions). The pink region along the x-axis highlights the range of values of
running coupling $g$ relevant to a collider-produced QGP.

##  4 Discussion

To reiterate, the question proposed in Section 1 was whether the 0-flavor
SU(3) AMY formalism has an inherent $q^{2}$ momentum transfer-weighted RTA
behavior under a certain temperature region. After the discussions and
derivations provided in Section 2.2 – for the shear viscosity under RTA∗ via
Eq. (30) and under AMY’s inverse-log expression via Eq. (34) –, and after a
brief discussion on AMY’s exact calculations of $\eta$ using basis functions
and how it compares with Eq. (34), Figure 2 displays the comparison between
the AMY formalism and the RTA∗ expression.

The results of the comparison show that the AMY calculations from the basis
functions mutually agree with RTA∗ for large $m_{D}/T$, while for small
$m_{D}/T$ the numerical results agree well with the inverse-log expression.
More importantly, RTA∗ and AMY’s numerical results mutually agree in the
relevant range of $m_{D}/T$ that is associated with gauge coupling values for
a collider-produced QGP. This is only under the pretense that the QGP is
simplified as a gluon gas, where $m_{D}/T=g$.

###  4.1 AMY and RTA∗ vs. CE

Using a differential cross section based on Eq. (7) for transport models, the
Chapman-Enskog method is well understood to agree well with the numerical
Green-Kubo calculations under both isotropic and anisotropic scatterings [14,
15]. It was shown in both references that the CE method demonstrates an
anisotropic enhancement of the shear viscosity, which correlates to high
temperatures [28]. With this in consideration, a secondary comparison can be
made between the AMY formalism (both numerical and analytical) and the RTA∗
and CE methods, which is shown in Figure 3.

Figure 3:  $\eta\times g^{4}/T^{3}$ vs. $m_{D}/T$ under RTA∗ (blue solid), AMY
0-flavor formalism (black dashed from basis functions, red solid via NLL) and
CE (green solid). Like in Figure 2, the pink region highlights the range of
values of running coupling $g$ relevant to a collider-produced QGP.

With the inclusion of the CE curve for $\eta\times g^{4}/T^{3}$ in the
comparison, one can see that the CE curve is larger than the other comparable
methods over all $m_{D}/T$, except for the inverse-log expression as it
diverges towards $m_{D}/T\rightarrow 2.765$. One can even deduce that the CE
curve is greater than the numerical AMY calculations roughly by a factor of 2
over all $m_{D}/T$. As anisotropy is understood to enhance the shear
viscosity, one can claim that the CE method is the preferrable method for an
asymptotically hot QGP, at least in 0-flavor SU(3) theory.

###  4.2 $N_{f}$-flavor AMY and $N$-Component RTA∗

The reliability of the AMY formulism is contributed to the treatment of
$N_{f}$-flavor quarks in the linearized collision kernel, i.e., the explicit
utilization of FD statistics, Pauli blocking factors, and fermion kinematics
via $|\mathcal{M}|^{2}$ in Eq. (33). In SU(3), the calculations of $\eta_{1}$
and the scale $\mu_{*}/T$ for $N_{f}>0$ are provided in Table 1.

$N_{f}>0$ | $\eta_{1}$ | $\mu_{*}/T$
---|---|---
2 | 86.470 | 2.954
3 | 106.66 | 2.957
4 | 122.96 | 2.954
5 | 136.38 | 2.947
6 | 147.63 | 2.940
Table 1:  AMY calculations of $\eta_{1}$ and $\mu_{*}/T$ in $N_{f}$-flavor
SU(3) theory. Flavor numbers provided range from 2 to 6.

While Figure 2 of Ref. [27] maps the $\eta\times g^{4}/T^{3}$ vs. $m_{D}/T$
for flavor numbers $N_{f}=0,~{}2$–$6$ via the basis function calculations,
Figure 4 maps the quantity via the inverse-log expression (Eq. [34]) with the
parameters provided in Table 1, and how it compares to the 0-flavor RTA∗
curve.

Figure 4:  RTA∗ versus $N_{f}$-flavor SU(3) AMY for $\eta\times g^{4}/T^{3}$
vs. $m_{D}/T$.

$N$-component, i.e., $N$-species expansions on the shear viscosity have been
previously addressed using various methods [18, 42, 43, 44, 45]. For
asymptotically hot, i.e., ultra-relativistic anisotropic scatterings, the CE
method was recently addressed [45]. Specific useage of the traditional
Anderson-Witting RTA viscosity for $N$-components was addressed in Ref. [46],
where a binary ($N=2$) hadronic gas (i.e., using FD statistics) was analyzed.
However, the binary viscosity was treated as a linear combination of the “pure
gas” viscosities of the components, and thus neglected the viscosity that
would be produced by cross-species interference.

Naively, one can address $N$-component RTA∗ in the similar treatment as
$N$-component RTA in the Anderson-Witting form (c.f. Eq. (59) in Ref. [39]),
however with the so-called molar fraction $x_{i}=n_{i}/\sum_{j=1}^{N}n_{j}$
[18, 44, 45] that defines $\sum_{i=1}^{N}x_{i}=1$. For the specific case of
massless MB-distributed partons with $\epsilon_{a}=3n_{a}T$,

$\begin{split}&\eta^{\mathrm{RTA^{*}}}_{~{}~{}~{}~{}N}=\frac{4T}{5}\sum_{a=1}^{N}x_{a}\tau^{*}_{a}n_{a},\\\
&\mathrm{where}~{}~{}~{}\tau^{*}_{a}=\sum_{b=1}^{N}\frac{x_{b}}{n_{b}\langle\sigma^{ab}_{\mathrm{tr}}v_{\mathrm{rel}}\rangle}\end{split}$
(37)

and $\langle\sigma^{ab}_{\mathrm{tr}}v_{\mathrm{rel}}\rangle$ is the thermal
average in the form of Eq. (29) that includes the transport cross section for
a purely elastic $a+b\rightarrow a+b$ interaction. The procedure would be to
expand $\tau_{a}^{*}$ first, and then expand
$\eta^{\mathrm{RTA^{*}}}_{~{}~{}~{}~{}N}$.

One can easily recover the pure gas ($N=1$) formula as Eq. (30), and $N>1$
expansion is able to consider cross-species interactions. Just as the 0-flavor
SU(3) comparison utilized the pure gas viscosity for the gluon-gluon
interaction, the $N$-component RTA∗ formula should be compared with the AMY
$N_{f}$-flavor basis function calculations for $m_{D}/T>1$ and under relevant
ranges of $m_{D}/T=g\sqrt{(3+N_{f}/2)/3}$ for a collider-produced QGP.

##  5 Conclusion

The AMY formalism for expressing the shear viscosity is widely used in pQCD
for its specific treatment of gluons and (anti-)quarks kinematically and
statistically. As kinetic transport analysis on a collider-produced quark-
gluon plasma utilizes 0-flavor SU(3) theory (i.e., a gluon gas approximation
with added quark degrees of freedom), various analytical methods were used to
express its shear viscosity within a confined temperature window. Such methods
include a modified version of the relaxation time approximation and the
Chapman-Enskog method. In the historical context, asymptotically hot QCD
plasma was approached by RTA with an accounted $q^{2}$ momentum transfer. As
briefly shown in Section 3, the comparison between the 0-flavor SU(3) AMY
formalism and RTA∗ expression for shear viscosity demonstrated that the
numerical AMY calculations using basis functions mutually agree with the
analytical RTA∗ expression within the relevant temperature window for a
collider-produced QGP.

Discussions of the results are offered in Section 4, which addressed a second
comparison that included the CE method under the exact gluon-gluon amplitude,
and extension of AMY vs. RTA∗ analysis for larger flavor numbers (i.e.,
introduction of different particle types). In respective order, the CE curve
has the similar but enhanced profile as the numerical AMY results, enhanced
roughly by a factor of 2. Consideration of additional partons in a multi-
component ($N_{f}>0$) QGP supposes the $N$-component expansion of the
viscosity expression. For distinctive comparison between numerical
$N_{f}$-flavor SU(3) AMY and analytical RTA∗, $N$-component expansion of the
RTA∗ expression involves series expansion on the modified relaxation time to
consider like- and cross-species interactions, as provided as Eq. (37) in
Section 4.2.

A continued comparison between the AMY formalism and RTA∗ is encouraged in
order to verify the RTA∗-like behavior of AMY viscosity calculations within
the temperatures for a collider-produced QGP. To extend the comparison into
larger quark flavors in SU(3) theory, this requires an $N$-component expansion
on RTA∗. Furthermore, the $N$-component CE expression can also be used in the
comparison, in order to determine if CE still exhibits an enhanced profile of
the numerical AMY calculations under larger quark flavors.

##  Appendix A Other Methods of Obtaining Shear Viscosity

Given Eq. (19), one can also obtain the other analytical methods of expressing
shear viscosity.

###  A.1 Israel-Stewart

The Israel-Stewart method was independently derived by the two namesakes,
respectively in Refs. [21, 22]. I will refer to Ref. [22] from Stewart,
especially for the linearization of the collision kernel
$\mathcal{C}[f]\propto\mathcal{L}[\phi]$.

All of Chapter 5 in Ref. [22] is dedicated to solving the collision kernel
using the second moment term $\varepsilon_{\mu\nu}p^{\mu}p^{\nu}$ as the
square of the energy:
$\varepsilon_{\mu\nu}p^{\mu}p^{\nu}=U_{\mu}p^{\mu}U_{\nu}p^{\nu}$. Ref. [22]
would use the “non-quantum” kernel, i.e., MB kernel with no quantum
degeneracy, and the parton masses were kept to be non-zero. In summary, the
linearized kernel is vastly shortened to incorporate the $\sin^{2}\theta$
transport of the relative momentum $q$. Thus, the Israel-Stewart method
utilizes the transport cross section $\sigma_{\mathrm{tr}}=\int
d\sigma\sin^{2}\theta$:

$\mathcal{C}[f]=\iint\frac{d^{3}\vec{p}_{b}}{(2\pi)^{3}}(f_{c}f_{d}-f_{a}f_{b})(U_{\mu}p^{\mu}U_{\nu}p^{\nu})v_{\mathrm{rel}}d\omega_{q}\int\frac{d\sigma}{d\Omega}\sin^{2}\theta
d\Omega.$ (38)

Here, $f_{i}=f(\vec{p}_{i})$ and $d\omega_{q}$ is an area element on the unit
mass-shell $q^{\mu}q_{\mu}=-1$.

It was initially assumed in Ref. [22] that the transport cross section is a
constant, although claiming it is not always the case for all collisions.
Israel [21] would make the same claim. Therefore, the Israel-Stewart method
was derived under the specific case of isotropic scatterings, where
$\sigma_{\mathrm{tr}}=2\sigma/3$. Therefore, the collision kernel was solved
to be $\mathcal{C}[f]\propto n\sigma_{\mathrm{tr}}$. Alternatively, the
linearized kernel operator is defined as

$\mathcal{L}[\phi]=p^{0}n\sigma_{\mathrm{tr}}\langle v_{\mathrm{rel}}\rangle.$
(39)

Implying massless partons on Eq. (39), e.g., $p^{0}=|\vec{p}|$ and $\langle
v_{\mathrm{rel}}\rangle=1$ and placing it in Eq. (20), we have the massless
Israel-Stewart viscosity:

$\begin{split}\eta^{\mathrm{IS}}&=\frac{d_{j}}{15nT}\int_{-\infty}^{\infty}\frac{d^{3}\vec{p}}{(2\pi)^{3}}\frac{|\vec{p}|^{2}}{\sigma_{\mathrm{tr}}}f_{0}(p)\\\
&=\frac{\langle|\vec{p}|^{2}\rangle}{15T\sigma_{\mathrm{tr}}}=\frac{4}{5}\frac{T}{\sigma_{\mathrm{tr}}}.\end{split}$
(40)

Had the transport cross section been left non-constant, $\sigma_{\mathrm{tr}}$
would be influenced by the momentum transfer. Therefore, the non-constant
transport cross section would need to be evaluated with the relative velocity
in the thermal average. Therefore, a “generalized” yet massless Israel-Stewart
expression would hold the exact same form as Eq. (30).

####  A.1.1 Navier-Stokes

A similar method is the hydrodynamic Navier-Stokes method [18], which defines
the shear viscosity for isotropic scatterings as

$\eta^{\mathrm{NS}}\simeq
0.8433\frac{T}{\sigma_{\mathrm{tr}}}=1.265\frac{T}{\sigma}.$ (41)

###  A.2 First-Order Chapman-Enskog

Ref. [20] derives the $N$th-order Chapman-Enskog expression for $\eta$ of a
massive one-component gas. In this method, all moments above the first in the
polynomial deviation function are considered, i.e.,
$\phi=\sum_{j=2}\varepsilon_{j}p^{j}$. The order of the CE expression $N$,
however, is ranked as $j-1$, meaning “first-order” CE uses the second moment
of $\phi$. In Ref. [20], the highest order to be (numerically) calculated was
16th order, albeit for ultra-relativistic (massless) partons. It is shown in
previous studies that first-order CE is sufficent enough to show good
agreement with numerical transport calculations and theoretical expectation.
Therefore, we will only consider deriving the first-order expression.

Defining the linearized kernel in Eq. (19), the first order CE viscosity is
generally defined with parton mass and the so-called “relativistic omega
integrals” $c_{00}$, which take on the role as an effective thermal average of
the transport cross section [14, 20]:

$\begin{split}&\mathcal{L}[\phi]=n\left[\frac{50}{3}\left(\frac{\langle|\vec{p}|^{2}\rangle}{Tz}\right)^{2}\Big{\langle}\frac{(p^{0})^{2}}{|\vec{p}|^{4}}\Big{\rangle}\right]^{-1}c_{00},~{}~{}\mathrm{where}\\\
&c_{00}=\frac{16z^{3}}{K_{2}^{2}(z)}\int_{1}^{\infty}dy(y^{2}-1)^{3}\left[\left(y^{2}+\frac{1}{3z^{2}}\right)K_{3}(2zy)-\frac{y}{z}K_{2}(2zy)\right]\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\int
d\sigma(1-\cos^{2}\theta),\end{split}$ (42)

$y=\sqrt{s}/(2m)$ is the integration variable, and $z=m/T$ is a mass term for
a particle. Thus, $Tz=m$ considers the rest energy of a specific particle. One
should note that $2zy$ couples into a mass-independent variable:
$\sqrt{s}/T=\tilde{u}$. Even at first-order, CE is already a method that is
more compatable for non-constant $\sigma_{\mathrm{tr}}$.

Under MB statistics with non-zero mass, Eq. (19) becomes

$\eta=\frac{10}{9T^{3}z^{2}}\left(\frac{T^{2}}{z^{2}K_{2}(z)}\int_{z}^{\infty}r(r^{2}-z^{2})^{3/2}\exp(-r)dr\right)^{2}\frac{1}{c_{00}}\Rightarrow\frac{10T}{c_{00}}\left(\frac{K_{3}(z)}{K_{2}(z)}\right)^{2}.$
(43)

In the above, $r=\sqrt{X^{2}+z^{2}}$ is the integration variable that varies
with $X=|\vec{p}|/T$. Let us define the Bessel function ratio
$K_{3}(z)/K_{2}(z)=\gamma_{0}/10$, which relates to the enthalpy. This defines
the first-order Chapman-Enskog viscosity elegantly as

$\eta^{\mathrm{CE}}_{\,\mathrm{I}}=\frac{T}{10}\frac{\gamma_{0}^{2}}{c_{00}}.$
(44)

If a constant $\sigma_{\mathrm{tr}}$ were used and the partons were massless,
we quickly recover Eq. (40).

##  Appendix B 0-flavor SU(3) $\eta_{1}$ under MB Statistics

While AMY defined the 0-flavor $\eta_{1}$ factor under BE statistics and the
unaveraged gluon-gluon amplitude, one can alternatively calculate 0-flavor
$\eta_{1}$ under MB statistics and the averaged111In this procedure, the
amplitude is averaged as the exit channel $I_{E}$ in Eq. (45) relates to a
differential cross section evaluation, where
$d\sigma/d\Omega\propto\langle|\mathcal{M}|^{2}\rangle$. amplitude. This is to
serve as a demonstration that quantum statistics approximate to MB statistics
under asymptotically high temperatures.

Eq. (33) is simplified for only one species type with different momenta and
the Bose/Pauli factors removed, i.e., $[1\pm f_{i}]=1$:

$\begin{split}&\mathcal{C}[f]=\frac{1}{4|\vec{p}_{1}|\nu_{a}}\int\Big{(}f(\vec{p}_{1})f(\vec{p}_{2})-f(\vec{p}_{3})f(\vec{p}_{4})\Big{)}I_{E}\frac{d^{3}\vec{p}_{2}}{(2\pi)^{3}2|\vec{p}_{2}|},\\\
&\mathrm{where}\,\,\,\,\,I_{E}=\int|\mathcal{M}|^{2}(2\pi)^{4}\frac{\delta^{4}(P_{1}+P_{2}-P_{3}-P_{4})}{2|\vec{p}_{3}|2|\vec{p}_{4}|}\frac{d^{3}\vec{p}_{3}}{(2\pi)^{3}}\frac{d^{3}\vec{p}_{4}}{(2\pi)^{3}}\end{split}$
(45)

(only in this section, $P$ is the 4-momentum). The last line defines an exit
channel $I_{E}$, which can be defined in the two-parton CM frame to feature
the differential cross section. This was done in Ref. [20] as Eqs. (93 – 95),
where $I_{E}$ is easily defined in terms of $d\sigma/d\Omega$, the solid angle
element $d\Omega$, and the $s$-Mandelstam variable. However, to keep the
averaged matrix element, we use the definition
$\langle|\mathcal{M}|^{2}\rangle=(64\pi^{2}s)^{-1}d\sigma/d\Omega$ to have

$\begin{split}I_{E}=\int\frac{1}{16\pi}\langle|\mathcal{M}|^{2}\rangle\sin\theta
d\theta.\end{split}$ (46)

To write the exit channel in terms of the $s$\- and $t$-Mandelstam variables,
$\sin\theta d\theta=-2dt/s$; the kernel is now defined as

$\begin{split}\mathcal{C}[f]=\frac{1}{32\pi|\vec{p}_{1}|\nu_{a}}\int\Big{(}f(\vec{p}_{1})f(\vec{p}_{2})-f(\vec{p}_{3})f(\vec{p}_{4})\Big{)}\frac{d^{3}\vec{p}_{2}}{(2\pi)^{3}2|\vec{p}_{2}|}\int_{-s}^{0}\frac{dt}{s}\langle|\mathcal{M}|^{2}\rangle.\end{split}$
(47)

Using the following definitions and normalized identities for massless partons
with Lorentz frame invariance [28, 41]:

$\begin{split}&\frac{d^{3}\vec{p}_{i}}{2|\vec{p}_{i}|}=d^{4}P_{i}\,\delta_{0}(P_{i}^{2}),\\\
&1=\frac{1}{8\pi T^{3}}\int
f(\vec{p}_{i})d^{3}\vec{p}_{i}~{}~{}~{}\mathrm{and}~{}~{}~{}1=\int
ds\,\delta(P_{12}^{2}-s),\\\ &\mathrm{where}~{}~{}~{}P_{12}=P_{1}+P_{2},\\\
&f_{1}f_{2}=\int d^{4}P_{12}e^{-P_{12}/T}\delta^{4}(P_{12}-P_{1}-P_{2}),\\\
&\int
d^{4}P_{1}d^{4}P_{2}\delta^{4}(P_{12}-P_{1}-P_{2})\delta_{0}(P_{1}^{2})\delta_{0}(P_{2}^{2})=\frac{\pi}{2},\\\
&\int d^{4}P_{12}e^{-P_{12}/T}\delta(P_{12}^{2}-s)=2\pi
T\sqrt{s}K_{1}\left(\frac{\sqrt{s}}{T}\right);\end{split}$ (48)

and implying purely elastic collisions (i.e., $P_{1}+P_{2}=P_{3}+P_{4}$) and
the $q^{2}$ momentum transfer weighting, the collision kernel is linearized
and reduced to

$\begin{split}\Rightarrow\mathcal{C}(m_{D}/T)&=\frac{f_{a}(\vec{p}_{a})}{16\nu_{a}(4\pi)^{3}T}\int_{0}^{\infty}\frac{\sqrt{s}}{T}K_{1}\left(\frac{\sqrt{s}}{T}\right)ds\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\int_{-s}^{0}\langle|\mathcal{M}|^{2}\rangle\left(-\frac{4t^{2}}{s^{2}}-\frac{4t}{s}\right)\frac{dt}{s}\\\
&=f_{a}(\vec{p}_{a})\mathcal{L}_{a}\left[\frac{m_{D}}{T}\right].\end{split}$
(49)

Because of the $\sin^{2}\theta$ transport weighting and the singular Bessel
function in the collision kernel, the resulting $\mathcal{L}[m_{D}/T]$ takes
on a form similar to Eq. (29), showing an analogy to RTA∗. This readily
suggests the mutual comparison between the AMY formalism and RTA∗.

Recall in Section 2.2.3 that the viscosity calculations were extracted by a
maximized inverse kernel via a variational method. Because
$\eta\propto\mathcal{L}^{-1}$ ($\mathcal{L}$ is the linearized kernel), the
maximization of viscosity comes from the minimization of the linearized
kernel. As this relates to asymptotically high temperatures and implied
anisotropic scatterings, one can use a “large $s$” asymptotic form of
$\langle|\mathcal{M}|^{2}\rangle$ in Eq. (49):

$\lim_{w\rightarrow
0}\mathcal{C}(w)\propto\lim_{s\rightarrow\infty}\langle|\mathcal{M}|^{2}\rangle,$
(50)

where $w=m_{D}/T$.

AMY defines the next-leading-log viscosity as follows (c.f. Eq. (2.25c) in
Ref. [27]):

$\eta^{\mathrm{NLL}}=\frac{2}{15}Q_{\mathrm{max}},$ (51)

where $Q_{\mathrm{max}}$ maximizes the expectation of the inverse linearized
collision kernel, provided as a sum in Eq. (4.11) in Ref. [27]. For the first
order ($n=1$) term, one has

$Q_{\mathrm{max}}=\frac{1}{2g^{4}T}\frac{(\mathcal{S}_{i\dots
j},\mathcal{A}^{-1}\mathcal{S}_{i\dots j})}{\ln(\mu_{*}/m_{D})},$ (52)

where

$\mathcal{A}=\lim_{w\rightarrow 0}\left\\{-w\frac{\partial}{\partial
w}\left[\frac{\mathcal{C}(w)}{g^{4}T}\right]\right\\}$ (53)

and this parentheses notation defines (for all relevant species $a$)

$\Big{(}g(\vec{p}),h(\vec{p})\Big{)}:=\frac{1}{T^{3}}\sum_{a}\nu_{a}\int\frac{d^{3}\vec{p}_{a}}{(2\pi)^{3}}g^{a}(\vec{p}_{a})*h^{a}(\vec{p}_{a}).$
(54)

For a pure gas, i.e., one-component gas, the species summation is dropped.

The factor $\mathcal{A}$ has the implied minimizing condition of
$m_{D}/T\rightarrow 0$. Considering the discussion leading to Eq. (50), one
can alternatively define

$\mathcal{A}\simeq-w\frac{\partial}{\partial
w}\left[\frac{1}{g^{4}T}\lim_{w\rightarrow 0}\mathcal{C}(w)\right].$ (55)

Eqs. (2.11 – 2.12) in Ref. [27] define $\mathcal{S}_{i\dots j}$ for an
arbitrary transport coefficent and under quantum statistics. For shear
viscosity under MB statistics, we explicitly define

$\mathcal{S}_{i\dots
j}=-T|\vec{p}_{a}|f_{a}(\vec{p}_{a})\sqrt{\frac{3}{2}}\left(\hat{p}_{i}\cdot\hat{p}_{j}-\frac{1}{3}\delta_{ij}\right),$
(56)

where $\hat{p}_{i,j}$ is the directional vector of the $i$-th momentum (for
particle $a$) and the $j$-th momentum (for particle $b$), and $\delta_{ij}$ is
the Kronecker delta dependent on the momentum directions, i.e., it is equal to
0 for all directions other than mutually parallel. In the CM frame, the two
partons have anti-parallel directions: $\hat{p}_{i}\cdot\hat{p}_{j}=-1$ with
$\delta_{ij}=0$.

In defining the shear viscosity as a series, we equate Eq. (51) with the
definitions above with Eq. (34):

$\frac{1}{15g^{4}T}\frac{(\mathcal{S}_{i\dots
j},\mathcal{A}^{-1}\mathcal{S}_{i\dots
j})}{\ln(\mu_{*}/m_{D})}=\frac{T^{3}}{g^{4}}\left[\frac{\eta_{1}}{\ln(\mu_{*}/m_{D})}\right],$
(57)

defining therefore the coefficient $\eta_{1}$ explicitly as

$\eta_{1}=\frac{\nu_{a}}{10T^{5}}\int\frac{d^{3}\vec{p}_{a}}{(2\pi)^{3}}\frac{|\vec{p}_{a}|^{2}}{\mathcal{A}}f_{a}(\vec{p}_{a})^{2}.$
(58)

Under MB statistics, we have the following calculations with the averaged
gluon-gluon amplitude given as Eq. (4) and $\nu_{a}=16$:

$\begin{split}&\frac{1}{g^{4}T}\lim_{w\rightarrow
0}\mathcal{C}(w)=-f_{a}(\vec{p}_{a})\frac{3}{40\pi^{3}}\left[19+30\gamma_{E}-15\ln\left(\frac{4}{w^{2}}\right)\right],\\\
&\mathcal{A}=\frac{36}{16\pi^{3}}f_{a}(\vec{p}_{a}),\\\
&\eta_{1}=26.808.\end{split}$ (59)

In the above, $\gamma_{E}\simeq 0.57722$ is the Euler gamma constant, and the
MB-statistical calculation of $\eta_{1}$ is close to the AMY calculation of
$\eta_{1}=27.126$ via a 1:1 ratio of 0.9883. This relates to a percentage
error of 1.19%.

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