\documentclass[proceedings,conferenceproceedings,submit,moreauthors,pdftex,10pt,a4paper]{Definitions/mdpi} 

%=================================================================
\firstpage{1} 
\makeatletter 
\setcounter{page}{\@firstpage} 
\makeatother
\pubvolume{xx}
\issuenum{1}
\articlenumber{5}
\pubyear{2018}
\copyrightyear{2018}

%=================================================================

%\externaleditor{Academic Editor: name}
\history{Received: date; Accepted: date; Published: date}
%\updates{yes} % If there is an update available, un-comment this line


%------------------------------------------------------------------
% The following line should be uncommented if the LaTeX file is uploaded to arXiv.org
%\pdfoutput=1

%=================================================================
% Add packages and commands here. The following packages are loaded in our class file: fontenc, calc, indentfirst, fancyhdr, graphicx, lastpage, ifthen, lineno, float, amsmath, setspace, enumitem, mathpazo, booktabs, titlesec, etoolbox, amsthm, hyphenat, natbib, hyperref, footmisc, geometry, caption, url, mdframed, tabto, soul, multirow, microtype, tikz

\def\s{s}
\newcommand{\M}{{\cal M}}

\newcommand{\eq}[1]{(\ref{#1})}
\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\bea{\begin{eqnarray}}
\def\eea{\end{eqnarray}}
\def\bean{\begin{eqnarray*}}
\def\eean{\end{eqnarray*}}

\def\bm#1{\mbox{\boldmath$#1$}}
\def\u#1{\underline{#1}}
\def\D{\displaystyle}
\def\T{\textstyle}
\def\l{\left}
\def\r{\right}
\def\eg{e.\,g.}
\def\ie{i.\,e.}

\def\nf{n_{\!f}}
\def\LO{{_{\mathrm{LO}}}}
\def\NLL{{_{\mathrm{NLL}}}}
\def\gsim{\mathrel{\rlap{\lower0.2em\hbox{$\sim$}}\raise0.2em\hbox{$>$}}}
\def\lsim{\mathrel{\rlap{\lower0.2em\hbox{$\sim$}}\raise0.2em\hbox{$<$}}}
\def\lg{\mathrel{\rlap{\lower0.25em\hbox{$>$}}\raise0.25em\hbox{$<$}}}
\newcommand{\figscale}{0.62}

%=================================================================
%% Please use the following mathematics environments: Theorem, Lemma, Corollary, Proposition, Characterization, Property, Problem, Example, ExamplesandDefinitions, Hypothesis, Remark, Definition
%% For proofs, please use the proof environment (the amsthm package is loaded by the MDPI class).

%=================================================================
\Title{Low-shear QCD plasma from perturbation theory$^{\dagger}$}

\newcommand{\orcidauthorA}{0000-0003-1283-3264}
\newcommand{\orcidauthorB}{0000-0002-7186-2607}

\Author{Greg Jackson $^{1,}$*\orcidA{} and Andr\'{e} Peshier $^{2}$\orcidB{}}

\AuthorNames{Greg Jackson, Andre Peshier}

\address{%
$^{1}$ \quad AEC, Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland\\
$^{2}$ \quad Department of Physics, University of Cape Town, Rondebosch 7700, South Africa}

\corres{Correspondence: jackson@itp.unibe.ch; Tel.: +41-31-631-8532}

\firstnote{Presented at Hot Quarks 2018 - Workshop for young scientists on the physics of ultrarelativistic nucleus-nucleus collisions, Texel, The Netherlands, September 7-14 2018}

\abstract{
We argue that the inferred ratio of shear viscosity to entropy density of the quark-gluon plasma, $\eta/s \lsim 0.5$ near the deconfinement temperature $T_c$, can be understood from perturbative QCD.
To rebut opposite views, we first show that the existing leading order result should not be expanded in logarithms. 
After then settling the question of scale for the running coupling, we establish a temperature dependence of $\eta/s$ which agrees well with constraints from hydrodynamics.}

\keyword{quark-gluon plasma; viscosity; perturbation theory}

\begin{document}

Experiments at the RHIC and LHC have provided substantial evidence that the quark-gluon plasma (QGP) behaves as an almost ideal fluid \cite{Experiment}, with an upper bound on the ratio of shear viscosity to entropy density, $\eta/s \lsim 0.5$. While this remarkably low value clearly indicates a `strongly coupled' system, it remains a theoretical challenge to understand better {\em why} it is so low.

One popular approach to this question is via the AdS/CFT correspondence \cite{Kovtun:2004de}. 
% which allows one to explore the strong coupling behavior of certain conformal field theories. 
%Although 
The conjectured lower limit $\eta/s \ge 1/(4\pi)$ from supersymmetric Yang-Mills theories does compare favorably with the observations, but a rigorous connection to real-world QCD is lacking.
First attempts to compute $\eta$ by lattice QCD corroborate small values \cite{lattice}, but are hampered by the difficulties of applying a static approach for a non-equilibrium observables.
On the other hand, there is a widespread belief that QCD perturbation theory
fails to explain $\eta/s \lsim 0.5\,$.
This is the perception we will scrutinise here.
	
It appears to be largely based on the {\em next-to-leading log} (NLL) formula 
\be
  \eta_\NLL (\alpha)
  =
  b T^3 / [\alpha^2 \ln(c/\alpha) ] \, ,
  \label{eta NLL}
\ee
where $T$ is the temperature and $\alpha$ the coupling strength. 
The coefficients $b$ and $c$ were extracted from the {\em leading order} (LO) result $\eta_\LO$ computed numerically in a QCD effective kinetic framework \cite{AMY}. 
In the quenched limit ($\nf = 0$ quark flavors), the case we will consider mostly for argument's sake, $b \approx 0.34$ and $c \approx 0.61\,$. 
The viscosity should decrease for stronger interactions, 
which is described by \eq{eta NLL} only for $\alpha < \u{\alpha} = c/\sqrt{e}$, at which point $\eta_\NLL(\alpha)$ has a minimum. 
That minimum 
%${\rm Min} \l[ \eta_\NLL \r] = 2b e T^3 / c^2$ 
turns out to be close to the free entropy $\s_0 = (16 + \frac{21}2 \nf) \frac{4\pi^2}{90} T^3$, see Fig.~\ref{fig1}.
Thus \eq{eta NLL} is incompatible with the bound $\eta/\s \lsim 0.5\,$. 

\begin{figure}[t]
  \begin{minipage}[c]{.5\textwidth}
  \includegraphics[scale=\figscale]{fig1}
  \end{minipage}
  \begin{minipage}[c]{.5\textwidth}
	\caption{
    The viscosity, for $\nf = 0$, to LO and NLL accuracy, and from our estimate \eq{eta simple}. 
    To illustrate that $\eta_\NLL$ cannot explain $\eta/s \lsim 0.5$ (but $\eta_\LO$ may), we also show the constraint for the entropy, $4T^3 \le s \le s_0$ for $T > 1.2T_c$ (see main text).
    Near the deconfinement temperature $T_c$ the entropy of the  QGP is notably smaller than $s_0\,$.
  \label{fig1}}
  \end{minipage}
	\vskip -7mm
\end{figure}

%To weigh up this fact, 
We view the minimum of $\eta_\NLL(\alpha)$ as a precursor to its singularity at $\alpha = c$ (marking the break-down of the NLL approximation) -- which an elementary argument reveals to be unphysical: In kinetic theory we may estimate %\cite{Reif:1964}
%\be
$  \eta \approx \T\frac13 n \bar p \lambda\,$,
%  \label{eta estimate}
%\ee
from the density $n$ of particles that can transport a typical momentum $\bar p$ over a distance $\lambda$.
For binary interactions of relativistic particles $\lambda = (n\sigma_{\rm tr})^{-1}$, where $\sigma_{\rm tr}(s) = \int_{-s}^0 dt\, (\frac12|t|/s)\, d\sigma/dt$ is the transport cross section in terms of Mandelstam variables. (Here $s$ is the centre-of-mass energy squared.)
Although the `transport weight' $\frac12|t|/s = 1-\cos\theta$ suppresses the influence of small-angle scatterings that prevail in gauge theories, $\sigma_{\rm tr}$ would still diverge logarithmically at tree-level due to the $t$-channel gluon exchange term in $d\sigma^{\rm tree}/dt \propto \alpha^2 [-us/t^2-ts/u^2-ut/s^2+3]/s^2$.
Since this would imply zero viscosity for {\em any} value of the coupling, it is a {\em necessity} to go beyond tree-level. 
In a hot QGP, the exchanged gluon acquires a self-energy of the order $\mu^2 \sim \alpha T^2$ and is thus screened, schematically $d\sigma^{\rm scr}/dt \sim \alpha^2/(t-\mu^2)^2$ for small $t$.
The typical invariant energy $s \sim T^2$ is much larger than $\mu^2$ for $\alpha \ll 1$, thus screening can be mimicked by a cut-off on $d\sigma^{\rm tree}/dt$,
\be
  \sigma_{\rm tr}^{\rm scr}
  \to
  \sigma_{\rm tr}^{\rm cut}
  \sim
  \int_{-T^2}^{-\mu^2}\!\! dt\, \frac{|t|}{T^2}\, \frac{\alpha^2}{t^2}
  =
  \frac{\alpha^2}{T^2} \ln\alpha^{-1} + O(\alpha^2) \, .
  \label{sigma cut}
\ee
This reproduces (with $\bar p \sim T$) the parametric $\alpha$-dependence of \eq{eta NLL}, but also shows that the singularity of $\eta_\NLL(\alpha)$ is related to coinciding integration bounds in \eq{sigma cut}. 
Thus the reason why $\eta_\NLL$ cannot be extrapolated to larger $\alpha$ has to do with kinematic simplifications that become illegitimate. 
%rather than an `breakdown' of perturbative QCD. %{\em per se} at $\alpha \simeq c\,$.

To validate this insight for QCD,
%beyond the scope of \eq{eta estimate}, 
the viscosity has to be calculated from the energy-momentum tensor of the particle distribution $f(\bm p, \bm x, t)$ governed by the Boltzmann equation, $(\partial_t + \bm v \bm\nabla)f = C[f]$, when set up for the case of a collective small-gradient flow $\bm u$ that drives $f$ slightly out of local equilibrium. 
As detailed in Refs.\ \cite{AMY, Baym:1990uj}, $\eta$ can be obtained by extremizing a functional constructed from the collision term $C[f]$.
The essence of this technical calculation is \cite{JP}
\be
	\frac\eta{T^3}
	\simeq
	\Big[\, \int_0^\infty\!\! ds\, sP(s)
		\int_{-s}^0\! dt\, \frac{|t|}{2s}\, \frac{d \sigma}{d t} \ \Big]^{-1}
	+ \, \ldots \, ,
	\label{eta with P(s)}
\ee
if $d\sigma/dt$ depends only on the Mandelstam variables, and omitting terms sub-leading to the dominant small-angle binary scattering contributions.
With $\sigma_{\rm tr}$ factorised from a positive weight $P(s)\,$,
%(that depends on how the system departs from equilibrium)
the convolution \eq{eta with P(s)} specifies more rigorously the `typical' momentum $\bar p\,$.
%in the elementary {\sl Ansatz} \eq{eta estimate}. 
%Calculated with a {\em screened} cross section $d\sigma/dt$, \eq{eta with P(s)} resums powers of both $1/(\ln\alpha^{-1})$ and $\alpha$ -- as does $\eta_\LO$. 
%The inverse-log expansion of $\eta_\LO$ was shown in \cite{AMY} to have zero radius of convergence.
We now discuss why the expansion in $\alpha$ is ill-defined. To that end, 
%we defer QCD particularities and 
we argue on the basis of \eq{eta with P(s)}
%\footnote{%
%	 even though its assumption is somewhat too restrictive for QCD, which limits its agreement with $\eta_\LO$ to leading log (LL) accuracy.
%}
applied to the simple model $d\sigma^{\rm scr}/dt$ which, now with correct kinematic limits, amends \eq{sigma cut} to
\be
	\sigma_{\rm tr}^{\rm scr}(s)
	\propto
	\int_{-s}^0 dt\, \frac{|t|}s \, \frac{\alpha^2}{(t-\mu^2)^2}
	=
	\frac{\alpha^2}s g(a) \, .
	\label{sigma_tr}
\ee
Here $g(a) = \ln\frac{1+a}a - 1/(1+a)$ is a monotonously decreasing, positive function of $a = \mu^2/s\,$. 
%By contrast, 
Its `NLL' approximation, $g = \ln a^{-1} - 1 + O(a)$, becomes obviously unphysical for $a > 1/e$, leading to the same issues as in \eq{eta NLL} and \eq{sigma cut}.
We note first that this problem cannot be cured by higher order terms in the expansion due to the convergence radius, $a = 1$, set by the pole at $t = \mu^2$ (off the physical sheet) in $d\sigma^{\rm scr}/dt$. 
This feature of a finite radius of convergence will carry over to QCD. 
What is more, expanding $\sigma_{\rm tr}^{\rm scr}$ in $\mu^2/s \propto \alpha$ {\em before} convoluting it in \eq{eta with P(s)} with $P(s)$ is forbidden: The coefficients of $\alpha^n$ (the negative moments of $P$) are IR-divergent, with increasing severity, since $P(0) > 0$ \cite{JP}.
%\be
%	P(s) 
%	=
%	\int_{12}\! f_1^{(0)} \bar{f}_1^{(0)} f_2^{(0)} \bar{f}_2^{(0)}
%	\big[ \, \chi^\prime(p_1) - \chi^\prime(p_2) \, \big]^2 + O(s) \, .
%	\label{P(s)}
%\ee
%We denote by $f_i^{(0)} = [\exp(p_i/T)-1]^{-1}$ the equilibrium distribution for incoming particle $i$ in the local rest frame, $\bar{f}_i^{(0)} = 1 + f_i^{(0)}$, and $\int_i = \int d^3 p_i /[(2\pi)^3 2p_i]$ for the phase space integrals.
%(The possibility $P(0) = 0$ is excluded since $\chi(p)$, which parametrises the solution of the Boltzmann equation in the form $f(p) = f^{(0)} \l[ 1 + \chi \bar{f}^{(0)}( p_k p_l/p^2 - \frac13 \delta_{kl} ) \nabla_k u_l \r]$, cannot be strictly linear in $p$ \cite{Heiselberg:1994vy}.)
%Note also that by crossing symmetry in $d\sigma/dt$, we obtain the same contribution \eq{sigma_tr} from the dressed $u$-channel term.

%Now, since even the model with $d\sigma^{\rm scr}/dt$ does not have a weak-coupling expansion that allows for extrapolation to larger $\alpha$, we cannot expect so for QCD.% features more accurately.
Unless $\alpha\ll c$, estimates of $\eta$ cannot be based on the NLL formula \eq{eta NLL} but require at least the {\em unexpanded} LO result.
As a function of the coupling, $\eta_\LO(\alpha)$ is monotonously approaching zero, which begs the question for `the' value of $\alpha$.
%\footnote{%
%	For large enough $\alpha$, the conjectured bound $\eta/\s \ge 1/(4\pi)$ would be violated.}
To back up that perturbative QCD can indeed explain $\eta/s \lsim 0.5$, let us point out that $\eta_\LO(\alpha)$ is fairly well reproduced by approximation (\ref{eta with P(s)}-\ref{sigma_tr}).
Without needing further details of $P(s)$ we can simply rewrite the convolution in \eq{eta with P(s)} using the mean value theorem,
\be
	\eta/T^3
	\simeq
  b / [\alpha^2 g(\bar a) ] \, .
	\label{eta simple}
\ee
Here we sidestepped solving the Boltzmann equation for $f$ and infer that $1/\bm(2\int ds P(s)\bm) = b$ 
%(the factor 2 accounts for $t \leftrightarrow u$ crossing symmetry) 
since \eq{eta simple} has to reproduce \eq{eta NLL} at LL accuracy.
Furthermore, $\bar a = \mu^2/\bar s = \kappa \cdot \alpha$ could be determined from a `log moment' of $P(s)$, but we will rather adjust it to match $c$ in \eq{eta NLL}, viz.\ $\kappa \to (ce)^{-1}$. 
%This effectively re-incorporates sub-dominant contributions of $t$, $u$, but also $s$-channel and inelastic scatterings that were omitted in our simple scheme. 
To quantify the uncertainty of this artifice, we vary $\kappa$ by factors $2^{\pm 1/2}$ in Fig.\ \ref{fig1}, which confirms a good agreement
of \eq{eta simple} with $\eta_\LO(\alpha)$ even for $\alpha \gsim \u{\alpha}$ (where the NLL result becomes qualitatively incorrect, as discussed).

Figure \ref{fig1} also depicts the rigorous bound $s > 4T^3$ on the entropy for $T > 1.2T_c$ known from lattice calculations \cite{slQCD}, to affirm that $\eta_\NLL$ cannot explain $\eta/s \lsim 0.5$.
For $\alpha$ large enough $\eta_\LO$ could be compatible with $\eta/s \lsim 0.5$ -- which brings us back to the task of specifying $\alpha$ at a given $T$.
A common prescription is to take $\alpha$ as the running coupling
\be
	\alpha (Q^2)
	=
	[ \beta_0 \ln( |Q^2|/\Lambda^2) ]^{-1} 
	\label{alpha}
\ee
(where $\beta_0 = (11 - \frac23\nf)/(4\pi)$ and $\Lambda$ is the QCD parameter) at a `typical thermal scale', like the lowest Matsubara energy
$Q_T = 2\pi T$.
%To then have $\eta_\LO(\alpha)/s_{\rm latt} \lsim 0.5$ at, \eg, $T = 1.2T_c$ would require $\alpha \gsim 0.4$, see Fig.~\ref{fig1}. 
%While the resulting $\xi$ might be $sim 1$, 
However, quantifying the coupling should be based on firmer grounds.

Having to specify the coupling {\em a posteriori} arises because in Ref.\ \cite{AMY} $\alpha$ is treated as if it was constant. 
{\em Imposing} then $Q_T$ as the relevant scale seems counterintuitive given the importance of a range of momenta, parametrically $[\mu,T]$.
As put forward early \cite{Cutler:1977qm} but rarely taken into account in finite-$T$ QCD, the relevant scale of the running coupling in, say, $t$-channel scattering should be $t$.\footnote{%
	Choosing a different scale $Q^2$ gives correction terms  $\alpha(Q^2)\log(Q^2/t)$ which are higher order in $\alpha(\cdot)$ but can be large.}
This rectifies \eq{sigma cut} to
\[
  \sigma_{\rm tr}^{\rm cut}
  \sim
  \int_{-T^2}^{-\mu^2}\!\! dt\, \frac{|t|}{T^2}\, \frac{\alpha^2(t)}{t^2}
  \,=\,
  \frac{\alpha(\mu^2) \alpha(T^2)}{T^2} \ln\frac{T^2}{\mu^2} \, ,
\]
hence the overall factor $\alpha^{-2}$ in \eq{eta NLL} is to be understood as a geometric mean of the running coupling at $T \sim Q_T$ and at the soft screening scale $\mu$.

%To consolidate this as our second key point: 
Running of the coupling emerges from fluctuations, be they in vacuum or in medium.
Thus for observables that require thermal screening, the `scale setting' for $\alpha(Q^2)$ is unambiguous.
For this, several types of radiative corrections are needed but only the gluon self-energy $\Pi = \Pi^{\rm vac} + \Pi^T$ contributes in Coulomb gauge due to its Abelian-like Ward identities \cite{Grozin:2008yd}.
This noteworthy feature simplifies our argument.
%Although rarely used for vacuum QCD, i
In Coulomb gauge it is evident that dressing \eg\ a $t$-channel Born amplitude $\sim \alpha/t$ with $\Pi^{\rm vac}(Q) = \alpha \beta_0\big[ \epsilon^{-1} + \ln( -Q^2/L^2 ) \big] Q^2$ (in dimensional regularization with scale $L$, and $Q^2=t$) gives the renormalised $\M^{\rm vac} \sim \alpha(t)/t$ with
%, indeed, 
the coupling \eq{alpha} at the scale $t$.
At $T>0$,
%(where Coulomb gauge is customary for other reasons), 
the self-energy receives the finite contribution $\Pi^T = \alpha\,\vartheta$, where 
%the function 
$\vartheta \sim T^2$ depends on $q_0$ and $\bm q$. 
Then the renormalised amplitude becomes $\M \sim \alpha(Q^2)/\bm(Q^2-\alpha(Q^2)\,\vartheta\bm)$ \cite{Peshier:2006hi}.
%where we emphasise that $Q^2$ also emerges as the scale for the coupling in the thermal self-energy.
This dependence of the running coupling on the virtuality carries over to the other scattering channels and then to $d\sigma/dt \sim |\sum \M_i|^2$.
%Juxtapose this 
%consistent renormalization 
%with the common 
%(fixed-$\alpha$) 
%procedure: There the vacuum part in the self-energy is dropped, to give $\M^{\rm fix} \sim \alpha / (Q^2-\alpha\,\vartheta)$ with the value of 
%the bare coupling 
%$\alpha$ left unspecified.

Now leaving behind the `toy' models discussed, this analysis allows us to re-instate running in the fixed-coupling calculation \cite{AMY},
where IR sensitive terms in $d\sigma^{\rm tree}/dt$ were screened with {\em hard thermal loop} (HTL) insertions, replacing \eg
\be
  - \alpha^2\, (us/t^2)
	\to
	| \alpha D_{\mu\nu}^\star(Q) Y^{\mu\nu} |^2 + \T\frac14\alpha^2 \, .
	\label{AMY screening}
\ee
Here $Y^{\mu\nu} = (P_1-\frac12 Q)^\mu(P_2+\frac12 Q)^\nu$, and $D^\star_{\mu\nu} = (D_0^{-1} - \Pi_\star^T)_{\mu\nu}^{-1}$ is the Coulomb HTL propagator.
The matrix element $\alpha D^\star$ %, which corresponds to $\M^{\rm fix}$, 
separates into transverse and longitudinal contributions ($i = \{t,\ell\}$), with $D_i^\star = 1/(Q^2 - \alpha \vartheta_i^\star)$.
Promoting  $\alpha$ to be $Q^2$-dependent gives the {\em renormalised} amplitude
\be
  \alpha D_i^\star(Q)
  \to
  {\alpha(Q^2)}/ [Q^2 - \alpha(Q^2)\, \vartheta_i^\star ] \, .
  \label{JP screening}
\ee
%The same goes for the $\alpha^2 (-ts)/u^2$ contribution in $d\sigma^{\rm tree}/dt$, with $Q^2 \to u$ in \eq{JP screening}.
%It remains to discuss the terms $\alpha^2 (3-ut/s^2)$ in $d\sigma^{\rm tree}/dt$ and $\alpha^2/4$ in \eq{AMY screening}, which only give sub-leading (finite) contributions to $\sigma_{\rm tr}\,$. 
%even without thermal screening.
%Accordingly, the scale for the running coupling in these terms is  irrelevant; we set it to $(stu)^{1/3}$.
%Inelastic scatterings are on par -- we neglect them altogether as they affect $\eta_\LO$ by merely a few percent \cite{AMY}.
%We note that although \eq{alpha} becomes unphysical in the far-IR, $|Q^2| \lsim \Lambda^2$, this effect is rendered unimportant by screening $\sim \vartheta^\star_i$ \cite{JP}.
The HTL screening in (\ref{AMY screening}, \ref{JP screening}) is justified for soft momenta $|Q^2| \lsim T^2$ (which is sufficient for LO accuracy). 
%Adapting the Braaten-Yuan method \cite{Braaten:1991dd} (as done in \cite {AMY}), 
To probe the sensitivity of higher order contributions,
we omit screening for $|Q^2|>|t^\star|$ and then vary $|t^\star| \in [\frac12, 2]T^2\,$.
Figure \ref{fig2} shows a factor of two uncertainty of $\eta$ for relevant $T$, which makes our estimate based on the scale setting and omission of inelastic scatterings sufficiently robust \cite{JP}.

%Such improved estimates of the viscosity only depend on the QCD scale $\Lambda$ which is of the order of $T_c$. 
In light of the overbearing sensitivity of $\eta$ on $t^\star$ we set $\Lambda \to T_c$ for the viscosity shown in Fig.~\ref{fig2}, normalised by the {\em interacting} entropy from lattice QCD calculations \cite{slQCD}.
For $\nf=\{0,3\}$ our results are compatible with existing lattice calculations \cite{lattice} and
%which  may give some guidance despite their limitations.
%Interestingly, $\eta/s$ hardly changes when including quarks; apparently the increased interaction rate is compensated by the density.
%Our results compare favorably to 
also recent constraints from hydrodynamics \cite{hydro}. 
%testing the average value and the $T$-dependence of the viscosity.
A mild increase in the ratio
%$\eta(T)/s(T)$ 
reflects the QCD feature of an effective coupling which weakens logarithmically.
Figure \ref{fig2} also illustrates that the LO result, with $\alpha(Q_T^2)$ 
%$Q_T = 2\pi T \cdot \xi$ with $\xi \in [\frac12,2]$, 
still overestimates $\eta/s\,$. % by half an order of magnitude.
\begin{figure*}[h]
	\includegraphics[scale=\figscale]{fig2}
	\vskip -2mm
	\caption{
    The viscosity in units of the entropy \cite{slQCD}; full lines show our result with 
    running coupling, the bands give the uncertainty from $t^\star \in [\frac12,2]T^2$, see text.
    The left panel, for the quenched limit, shows lattice results \cite{lattice}, 
    and by the dashed line the LO result with $Q_T = 2\pi T$ in running coupling.
	Overlayed on the right, for $\nf = 3\,$, are 
	estimates from hydrodynamics \cite{hydro}. Hatched region: $\eta/s \leq 1/(4\pi)$.
  }
  \label{fig2}
\end{figure*}

%We have demonstrated that this estimate is misleading for two reasons, namely due to compromising the fixed-$\alpha$ LO (resummed) result by another (log) expansion and an {\em ad hoc} choice for the value of $\alpha$.
%In fact, both issues are closely related: Resummation accounts for thermal screening which results from loop corrections to tree level amplitudes -- as does running coupling.

%Let us conclude with two general comments, discussing first the applicability of weak-coupling methods at `larger coupling', as often relevant for heavy-ion phenomenology. 
%Perturbation theory may at best give asymptotic expansions, hence higher loop corrections are not guaranteed to improve accuracy (due to lack of convergence).
%Since the optimal order is expected to {\em decrease} with the characteristic $\alpha$ \cite{ItzyksonZuber}, low-order approximations can make for useful and in fact more reliable estimates. 
%In the case of $\eta$, the first (and only available) candidate is the LO result for which we have emphasised the importance of running coupling: After all, $\alpha(Q^2)$ varies most where it is large.
%With our second remark we justify {\em a posteriori} the use of kinetic theory  which relies on the mean interparticle distance $\bar r sim n^{-1/3}$ being sufficiently smaller than the transport mean free path $\lambda$ \cite{Danielewicz:1984ww}.
%The latter can be calculated systematically from the gain (or loss) term of our renormalised collision operator $C$, with the result that $\lambda/\bar r$ remains larger than one (although only by a small margin) even near $T_c$ \cite{JP}.  Apparently, the interactions of a few partons is sufficient to maintain local equilibrium.

We have demonstrated that many estimates for $\eta$ are misleading 
for two reasons, namely due to compromising the fixed-$\alpha$ LO (resummed) result by another (log) expansion and an {\em ad hoc} choice for the value of $\alpha$.
In fact, both issues are closely related: Resummation accounts for thermal screening which results from loop corrections to tree level amplitudes -- as does running coupling.
%To conclude: the vacuum and thermal parts of loop corrections ought to be treated on the same footing. 
Treating them on an equal footing, 
we arrive at a consistent position regarding a long-standing question: The reckoned constraint $\eta \lsim 0.5\,s$ for the QGP produced in heavy-ion collisions can be {\em understood} on the basis of the LO viscosity -- rather than being a genuinely non-perturbative effect. 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{6pt} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\funding{Please add: ``This research received no external funding'' or ``This research was funded by NAME OF FUNDER grant number XXX.'' and  and ``The APC was funded by XXX''. Check carefully that the details given are accurate and use the standard spelling of funding agency names at \url{https://search.crossref.org/funding}, any errors may affect your future funding.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\acknowledgments{In this section you can acknowledge any support given which is not covered by the author contribution or funding sections. This may include administrative and technical support, or donations in kind (e.g. materials used for experiments).}

%=====================================
% References, variant A: internal bibliography
%=====================================
\reftitle{References}
\begin{thebibliography}{999}
\bibitem{Experiment} 
  K.~Aamodt {\it et al.} [ALICE Collaboration],
  %``Rapidity and transverse momentum dependence of inclusive J$/\psi$
  %production in $pp$ collisions at $\sqrt{s} = 7$ TeV,''
  Phys.\ Lett.\ B {\bf 704}, 442 (2011);
  A.~Adare {\it et al.} [PHENIX Collaboration],
  %``Heavy Quark Production in $p+p$ and Energy Loss and Flow of Heavy Quarks
  %in Au+Au Collisions at $\sqrt{s_{NN}}=200$ GeV,''
  Phys.\ Rev.\ C {\bf 84}, 044905 (2011).
  
\bibitem{Kovtun:2004de} 
  P.~Kovtun, D.\,T.~Son and A.\,O.~Starinets,
  %``Viscosity in strongly interacting quantum field theories from black hole
  %physics,''
  Phys.\ Rev.\ Lett.\  {\bf 94}, 111601 (2005).
  %[hep-th/0405231].

\bibitem{lattice} 
  H.\,B.~Meyer,
  %``A Calculation of the bulk viscosity in SU(3) gluodynamics,''
  Phys.\ Rev.\ Lett.\  {\bf 100}, 162001 (2008);
  % [arXiv:0710.3717 [hep-lat]];
  %``Transport properties of the quark-gluon plasma from lattice QCD,''
  Nucl.\ Phys.\ A {\bf 830}, 641C (2009);
  % [arXiv:0907.4095 [hep-lat]];
  S.\,W.~Mages {\em et al.},
  %``Shear Viscosity from Lattice QCD,''
  PoS LATTICE {\bf 2014}, 232 (2015);
  N.~Astrakhantsev, V.~Braguta and A.~Kotov,
  %``Temperature dependence of shear viscosity of $SU(3)$--gluodynamics within lattice simulation,''
  JHEP {\bf 1704}, 101 (2017).

\bibitem{AMY} 
  P.\,B.~Arnold, G.\,D.~Moore and L.\,G.~Yaffe,
  %``Transport coefficients in high temperature gauge theories. 1. Leading log
  %results,''
  JHEP {\bf 0011}, 001 (2000);
  %[hep-ph/0010177]; 
  {\em ibid.} {\bf 0305}, 051 (2003).
  %[hep-ph/0302165].

%\bibitem{Reif:1964} 
  %F.~Reif,
  %{\em Fundamentals of Statistical and Thermal Physics},
  %(McGraw-Hill, 1965).

\bibitem{Baym:1990uj}
  %G.~Baym, H.~Monien, C.~J.~Pethick and D.~G.~Ravenhall,
  G.~Baym {\em et al.}, 
  %``Transverse Interactions and Transport in Relativistic Quark - Gluon and
  %Electromagnetic Plasmas,''
  Phys.\ Rev.\ Lett.\ {\bf 64}, 1867 (1990).

\bibitem{JP}
  G.~Jackson and A.~Peshier,
  %``Re-running the QCD shear viscosity,''
  J.\ Phys.\ G {\bf 45} (2018) no.9,  095001 (2018) .
  %%CITATION = doi:10.1088/1361-6471/aad374;%%


%\bibitem{Heiselberg:1994vy}
  %H.~Heiselberg,
  %``Viscosities of quark - gluon plasmas,''
  %Phys.\ Rev.\ D {\bf 49}, 4739 (1994).
  %[hep-ph/9401309].

\bibitem{slQCD} 
  S.~Borsanyi {\em et al.},
  %``Precision SU(3) lattice thermodynamics for a large temperature range,''
  JHEP {\bf 1207}, 056 (2012);
  %[arXiv:1204.6184 [hep-lat]];
  %``Full result for the QCD equation of state with 2+1 flavors,''
  Phys.\ Lett.\ B {\bf 730}, 99 (2014).
  %[arXiv:1309.5258 [hep-lat]].

\bibitem{Cutler:1977qm} 
  R.~Cutler and D.\,W.~Sivers,
  %``Quantum Chromodynamic Gluon Contributions to Large p(T) Reactions,''
  Phys.\ Rev.\ D {\bf 17}, 196 (1978).
  %doi:10.1103/PhysRevD.17.196.

\bibitem{Grozin:2008yd}
  A.~Grozin,
  %``Asymptotic freedom: History and interpretation,''
  [arXiv:0803.2589 [hist-ph]].

\bibitem{Peshier:2006hi} 
  A.~Peshier,
  %``The QCD collisional energy loss revised,''
  Phys.\ Rev.\ Lett.\  {\bf 97}, 212301 (2006).
  %doi:10.1103/PhysRevLett.97.212301 [hep-ph/0605294].

%\bibitem{Braaten:1991dd} 
  %E.~Braaten and T.\,C.~Yuan,
  %``Calculation of screening in a hot plasma,''
  %Phys.\ Rev.\ Lett.\  {\bf 66}, 2183 (1991).

\bibitem{hydro}
  J.~E.~Bernhard, J.~S.~Moreland, S.~A.~Bass, J.~Liu and U.~Heinz,
  %``Applying Bayesian parameter estimation to relativistic heavy-ion collisions: simultaneous characterization of the initial state and quark-gluon plasma medium,''
  Phys.\ Rev.\ C {\bf 94}, no. 2, 024907 (2016).
  %doi:10.1103/PhysRevC.94.024907
  %[arXiv:1605.03954 [nucl-th]].
  %%CITATION = doi:10.1103/PhysRevC.94.024907;%%
  %127 citations counted in INSPIRE as of 22 Oct 2018
  %H.~Niemi, K.\,J.~Eskola and R.~Paatelainen,
  %``Event-by-event fluctuations in a perturbative QCD + saturation +
  %hydrodynamics model: Determining QCD matter shear viscosity in
  %ultrarelativistic heavy-ion collisions,''
  %Phys.\ Rev.\ C {\bf 93}, 024907 (2016).
  %[arXiv:1505.02677 [hep-ph]].

%\bibitem{ItzyksonZuber} 
%  C.~Itzykson and J.\,B.~Zuber,
%  {\em Quantum Field Theory},
%  (Dover, 2006).

%\bibitem{Danielewicz:1984ww}
%  P.~Danielewicz and M.~Gyulassy,
  %``Dissipative Phenomena in Quark Gluon Plasmas,''
%  Phys.\ Rev.\ D {\bf 31}, 53 (1985).
  %doi:10.1103/PhysRevD.31.53
\end{thebibliography}


%=====================================
% References, variant B: external bibliography
%=====================================
%\externalbibliography{yes}
%\bibliography{your_external_BibTeX_file}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% optional
%\sampleavailability{Samples of the compounds ...... are available from the authors.}

%% for journal Sci
%\reviewreports{\\
%Reviewer 1 comments and authors’ response\\
%Reviewer 2 comments and authors’ response\\
%Reviewer 3 comments and authors’ response
%}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}

