\section{Introduction\label{intro}}

Key experiments in all areas of subatomic and condensed matter physics,
spanning over more than a hundred years and designed to investigate aspects of the microscopic interaction between
electromagnetic radiation, charged particles, and collective excitations, prove that the quantum of action $h=6.63\times 10^{-34}$\,Js, introduced by Max
Planck in understanding the spectral radiance of a black-body cavity through eventual
appeal to the statistical thermodynamics of Ludwig Boltzmann \cite{Planck1900}, is a {\sl universal}
constant of Nature. Since its inauguration in 1900 the somewhat mysterious sneak-in of $h$,
somehow being associated with `elementary regions' or `free rooms for action'
\cite{PlancksNobelLecture}, into the
conception and development of quantum mechanics \cite{Planck1900,DeBroglie,Einstein,Bohr,Heisenberg,Schrödinger}
has been broaching a quest for deeper interpretation.

To understand the meaning of $h$ as a mediator of indeterminism in fundamental interactions among
propagating gauge fields, between gauge and charged matter fields, or in stationary binding physics could be of relevance
in addressing as of yet poorly understood phenomena, for example those of
2D strongly correlated electron systems \cite{HighTc}. Quantum Mechanics and perturbatively
accessed Quantum Field Theory are highly successful, accurate, and efficient. However, both suffer
from problems of mere interpretation. While the measurement process in Quantum Mechanics, that is, the interaction of
a macroscopic observer with the subatomic system under investigation, mysteriously and in a
nonlocal, instantaneous way collapses
the evolved wave function of the latter to an eigenstate of the operator-valued, measured quantity
the need for renormalization in Quantum Field Theory, in addition
to not easing the classical Euler-Langrangian ambiguity inherent to the value of vacuum energy, is perceived by
many as a necessary evil in putting perturbation theory to work.
Fortunately, the predictive power of 4D gauge theories is not spoiled
by the process of renormalization \cite{Veltman'Hooft,LeeZinnJustin}, and
radiatively introduced gauge anomalies can be canceled granting the
consistency of the highly successful Standard Model of Particle Physics. Yet one may wonder
whether the subtraction of infinities in the perturbative
loop diagrams of gauge theories enjoys a deeper justification possibly supplied
by full nonabelian gauge dynamics.

In the present paper we consider the (Euclidean) action
of a fundamental, selfdual\footnote{Here the term `selfdual' stands for both negative and positive topological charge.}
gauge field configuration of topological charge modulus unity (trivial-holonomy Harrington-Shepard \cite{HS1977}
or nontrivial-holonomy Lee-Lu-Kraan-Van-Baal caloron/anti\-caloron \cite{LeeLuKraanVanBaal})
and scale parameter $\rho$
in the deconfining phase of thermalized SU(2) Yang-Mills theory.
The action of a caloron/anti\-caloron, that is {\sl just not resolved} in the effective theory,
is determined by the value of an effective coupling calculable
in the according effective theory \cite{HerbstHofmann2004,Hofmann2005,Hofmann2007}. Recall
that calorons and anticalorons are not resolved in this effective theory:
Their contribution to effective dynamics fundamentally separates into a free
part causing the emergence of an effective, inert, and spatially homogeneous
adjoint scalar field $\phi$ \cite{HerbstHofmann2004} and the effects of interactions
with the propagating plane-wave sector (vanishing topological charge).
It is a fact that the energy-momentum tensor vanishes on any selfdual gauge-field configuration,
see for example \cite{Hofmann2011}, and hence also on calorons and anticalorons. Since the latter configurations thus
do not propagate one may wonder \cite{HS1977_II}
what their (potentially important) contribution to the thermodynamics of the
high-temperature Yang-Mills gas actually is.

For the generic nonabelian gauge group SU(2)
this question was addressed in \cite{HerbstHofmann2004,Hofmann2005,Hofmann2007}, see also
\cite{Hofmann2011} for a summarizing presentation: While the emergence of
$\phi$ is sharply dominated by (large) calorons/anticalorons of radius $\rho$ smaller than but
comparable\footnote{The integral over $\rho$ in the space-averaged two-point function of
the fundamental field strength defining in singular gauge the kernel ${\cal K}$ of a differential operator $D$, with $\phi\in {\cal K}$,
exhibits a cubic dependence on the cutoff when the latter is comparable
to $|\phi|^{-1}$, see \cite{HerbstHofmann2004} and Sec.\,\ref{phiandrho} below.}
to the scale of maximal resolution $|\phi|$ in the effective theory \cite{HerbstHofmann2004} interactions between calorons/anti\-calorons and
plane waves manifest in a two-fold way effectively. First, there are small
holonomy changes, induced through momentum transfers
by plane-wave fluctuations that are largely off their
free mass shell\footnote{All unresolved plane-wave fluctuations are at least by $|\phi|$
off their free mass shell.},
which generate negative ground-state pressure by virtue of
a pure-gauge configuration of the effective gauge field \cite{Hofmann2005,Hofmann2011,Hofmann2011_I}.
Second, as we shall make explicit in this paper, just-not-resolved
calorons/anti\-calorons are responsible for pointlike
vertices, associated with their dissociation into screened but unresolved
monopole-antimonopole pairs on one hand \cite{SpatialWilson} and a rapidly converging loop expansion
in the effective theory \cite{Schwarz2011,Kaviani2011,Hofmann2006,KavianiHofmann2007}. We will make clear that
this loop expansion, which can naively be organized into an expansion in powers of $\hbar\equiv\frac{h}{2\pi}$, turns out to be a more
subtle expansion. Still, the action of a just-not-resolved caloron or anticaloron,
which introduces a vertex for effective plane waves, turns out to be $\hbar$.

Calculating loops in effective variables yields a quantitative description of
collective effects as induced by fundamental field configurations
\cite{SpatialWilson,Resum2010,FalquezHofmann2011,LudescherHofmann2009}.
Moreover, since the fields of the effective theory and their interactions do only depend on the contributions of
large calorons/anticalorons we would conclude that, on a fundamental level, the mediation of interactions between unresolved
off-shell plane-wave fluctuations by calorons/anticalorons (vertices) dies off rapidly with decreasing
$\rho$. As a consequence, there are {\sl no} interactions with potentially large
momentum transfers deep within the unresolved physics. We tend to interpret this as a justification for the perturbative
renormalization procedure in nonabelian gauge theories. Finally, it is instructive to illustrate the technicalities of the effective loop
expansion to contrast it with renormalized perturbation theory.

The paper is organized as follows. Because our subsequent discussion relies on it
we review in Sec.\,\ref{phiandrho} the emergence of an
inert, adjoint scalar field $\phi$ by a spatial coarse-graining over noninteracting calorons and anticalorons of
charge modulus unity within the deconfining phase of SU(2) Yang-Mills
thermodynamics. For the same reason, we re-derive in Sec.\,\ref{tsc} the thermodynamically consistent
solution for the temperature dependence of the effective gauge coupling $e$. In Sec.\,\ref{CP} we
identify the Euclidean action of just-not-resolved calorons and anticalorons with the quantum of
action $\hbar$ and deduce, by virtue of the dimensionlessness of the fine-structure
constant $\alpha$, an interpretation of the fundamental unit of
electric charge in terms of the dual to the effective SU(2)
gauge coupling in the deconfining phase. Sec.\,\ref{UPW} discusses the implications of the action of a just-not-resolved
caloron or anticaloron being associated with a local vertex. Namely, from the derivation of the effective
theory we learn that such vertices are sharply suppressed with increasing momentum transfer
(or decreasing caloron/anti\-caloron radius $\rho$) yielding, at {\sl any} externally prescribed resolution,
finite answers for all radiative processes. Focusing on the example of QED, Sec.\,\ref{QED}
discusses how just-not-resolved calorons and anticalorons of
SU(2)$_{\tiny\mbox{CMB}}$ may induce local vertices between the plane waves belonging to electrically
charged particles and propagating electromagnetic waves. To point out the contrast to
renormalized perturbation theory, where the emergence of a scalar field associated with the
maximal resolution and induced by the topologically nontrivial sector can not occur due to an a priori
restriction to the plane-wave sector,
Sec.\,\ref{Dariush} discusses the technicalities of the effective loop expansion of the
pressure. Namely, we review the Feynman rules and constraints on loop variables and
apply them to the (numerical) computation of three- and two-loop contributions
to the pressure.

\section{Selfduality, propagating fields, and powers of $\hbar$\label{S1}}

\subsection{Inert adjoint scalar field $\phi$ and effective action\label{phiandrho}}

In \cite{HerbstHofmann2004} the following definition for the kernel ${\cal K}$ of the differential operator $D$ associated with $\phi$'s
equation of motion in the deconfining phase of SU(2) Yang-Mills thermodynamics was shown
to be unique:
\begin{equation}
\label{definition}
{\cal K}\equiv \{\hat{\phi}^a\}\equiv \sum_{C,A}\mbox{tr}\,\int d^3x \int d\rho
  \,t^a \, F_{\mu\nu} (\tau,\vec 0) \, \left\{ (\tau,\vec 0),(\tau,{\vec x})
\right\}\, F_{\mu\nu} (\tau,{\vec x}) \, \left\{ (\tau,{\vec x}),(\tau,{\vec 0})
\right\} \,,
\end{equation}
where
\begin{equation}
\label{abk}
\begin{split}
\left\{(\tau,\vec 0),(\tau,\vec x)\right\}
&\equiv
{\cal P} \exp \left[ i \int_{(\tau,\vec 0)}^{(\tau,\vec x)} dz_{\mu} \, A_{\mu}(z) \right]\,,  \\
\left\{(\tau,\vec x),(\tau,\vec 0)\right\}
&\equiv
\left\{(\tau,\vec 0),(\tau,\vec x)\right\}^\dagger\,,
\end{split}
\end{equation}
and $F_{\mu\nu}$ denotes the fundamental field-strength tensor. The Wilson lines in Eq.\,(\ref{abk}) are calculated along
the straight spatial line connecting the points $(\tau,\vec 0)$ and $(\tau,\vec x)$, and
${\cal P}$ demands path-ordering symbol. (The euclidean time coordinate is $\tau$.)
In (\ref{definition}) the sum is over a
noninteracting caloron ($C$) and a noninteracting anticaloron ($A$) of trivial holonomy and topological charge modulus unity, and
$\rho$ denotes their (instanton) scale parameter. Both caloron
and anticaloron are spatially centered at $\vec{z}_{C,A}=0$, for a discussion of the distribution of action density in
dependence of $\rho/\beta$ ($\beta\equiv 1/T$, $T$ temperature)
see \cite{GrossPisarskiYaffe}. In evaluating the integrals on the right-hand side of Eq.\,(\ref{definition}), one notices
that only the magnetic-magnetic correlation gives rise to a
nonvanishing result. Going over to spherical coordinates, both the unconstrained integration over $r\equiv|\vec{x}|$ and
$\rho$ generate diverging multiplicative constants in front of a rapidly saturating sinusoidal time dependence separately
arising from both the caloron
as well as the anticaloron contribution. On the other hand, the azimuthal angular integration yields a
muliplicative zero \cite{HerbstHofmann2004,Hofmann2011}. Together this generates an undetermined real prefactor
which is one of the parameters spanning the kernel ${\cal K}$ of a uniquely
determined second-order linear operator $D$. More specifically, one arrives at the following expression for the
kernel ${\cal K}=\{\hat{\phi}^a\}$ of differential operator $D$
\begin{eqnarray}
\label{calantical}
\{\hat{\phi}^a\}&=&\{\Xi_C(\delta^{a1}\cos\alpha_C+\delta^{a2}\sin\alpha_C)\,{\cal
  A}\left(2\pi(\hat{\tau}+\hat{\tau}_C)\right)\nonumber\\
&&+\Xi_A(\delta^{a1}\cos\alpha_A+\delta^{a2}\sin\alpha_A)\,{\cal
  A}\left(2\pi(\hat{\tau}+\hat{\tau}_A)\right)\}\ \ \ \ \ \ \ (a=1,2,3)\,,
\end{eqnarray}
where $\Xi_C$, $\Xi_A$, $\hat{\tau}_C$, and $\hat{\tau}_A$ are undetermined real parameters, and
$\alpha_C$, $\alpha_A$ are angles that reflect the freedom of a global gauge choice. The amplitude function ${\cal A}$
is given as
\begin{equation}
\label{calA}
{\cal A}(2\pi\hat{\tau})=\frac{32\pi^7}{3}\int_0^{\xi} d\hat{\rho}\,\hat{\rho}^4\,
\left[\lim_{\hat{r}\to\infty}\sin(2\hat{g}(\hat{\tau},\hat{r},\hat{\rho}))\right]\frac{\pi^2\hat{\rho}^2+
\cos(2\pi\hat{\tau})+2}{\left(2\pi^2\hat{\rho}^2-\cos(2\pi\hat{\tau})+1\right)^2}\,,
\end{equation}
where $\xi$ represents a cutoff for the $\rho$ integration in units of $\beta$. (All quantities of
dimension length are made dimensionless by division with $\beta$ and are denoted by a hat symbol).
Function $\hat{g}$ saturates rapidly to a finite limit as $\hat{\rho},\hat{r}\to\infty$, the
integral over $\hat{\rho}$ in Eq.\,(\ref{calA}) diverges cubically for
$\xi\to\infty$, and function ${\cal A}(2\pi\hat{\tau})$ rapidly
approaches $\mbox{const}_\infty\times\xi^3\sin(2\pi\hat{\tau})$.
Numerically, one obtains $\mbox{const}_\infty=272.018$. To summarize, the kernel ${\cal K}$, which uniquely
determines operator $D$ as $D\equiv\partial^2_\tau+\left(\frac{2\pi}{\beta}\right)^2$, is sharply
dominated by contributions of the $\hat{\rho}$ integration that lie in the vicinity of
the upper limit $\xi$.

Operator $D$ is explicitly
dependent on temperature and thus, by itself, cannot be taken to be associated with a
temperature-independent effective action\footnote{This action is not explicitly temperature dependent because the
weights of selfdual configurations in the fundamental partition function are solely associated with
their topological charges.} for the field $\phi$. The latter, however, arises by
allowing a potential $V$ to absorb the explicit temperature dependence introduced by $D$. Appealing to the BPS
condition inherited from the selfduality of the fundamental gauge-field configurations entering into the
definitition (\ref{definition}), a first-order equation for $\phi$ is derived whose solution is parametrized
by one constant of integration -- the Yang-Mills scale $\Lambda$. Perturbative renormalizability \cite{Veltman'Hooft,LeeZinnJustin}, constraining
the interactions between effective plane-wave fields $a_\mu=a_\mu^a\,t_a$ ($t^a$ with $a=1,2,3$ the generators of SU(2) in fundamental representation,
normalized to tr\,$t^a t^b=\frac12\delta^{ab}$), gauge invariance, and the fact that the field $\phi$ cannot, by definition,
absorb or emit any energy-momentum yield the following unique result for the effective Euclidean action
density ${\cal L}_{\mbox{\tiny eff}}$ \cite{HerbstHofmann2004,Hofmann2011}
\begin{equation}
\label{fullactden}
{\cal L}_{\mbox{\tiny eff}}[a_\mu]=\mbox{tr}\,\left(\frac12\,
  G_{\mu\nu}G_{\mu\nu}+(D_\mu\phi)^2+\frac{\Lambda^6}{\phi^2}\right)\,,
\end{equation}
where $G_{\mu\nu}=\partial_\mu a_\nu-\partial_\nu
a_\mu-ie[a_\mu,a_\nu]\equiv G^a_{\mu\nu}\,t_a$ denotes the field
strength for gauge field $a_\mu$, $D_\mu\phi=\partial_\mu\phi-ie[a_\mu,\phi]$, and $e$ is the effective
gauge coupling to be determined, see Sec.\,\ref{tsc}. Action
density ${\cal L}_{\mbox{\tiny eff}}$ yields a highly accurate tree-level
ground-state estimate and, as easily deduced in unitary gauge $\phi=2|\phi|\,t_3$, tree-level
mass $m=2\,e|\phi|=2\,\sqrt{\frac{\Lambda^3}{2\pi T}}$ for propagating
gauge modes $a_\mu^{1,2}$ rendering them thermal quasiparticles.

\subsection{Thermodynamical consistency\label{tsc}}

For action density ${\cal L}_{\mbox{\tiny eff}}$ to be thermodynamically
consistent implied quantities like pressure and energy density must be related by
Legendre transformations. On the level of free thermal
fluctuations\footnote{In the physical unitary-Coulomb gauge $\phi=2|\phi|\,t_3\,,\ \partial_i a_i^3=0$ the
off-shellness of quantum fluctuations in the field $a_\mu$ is constrained by the maximal resolution
$|\phi|$. Their contribution turns out to be negligible compared to
that of thermal fluctuations \cite{Hofmann2005,Hofmann2011}.} we obtain the following evolution equation \cite{Hofmann2005}
 \begin{equation}
\label{evoleqsu2}
\partial_a\lambda=-\frac{24\lambda^4
  a}{(2\pi)^6}\frac{D(2a)}{1+\frac{24\lambda^3a^2}{(2\pi)^6}D(2a)}\,.
\end{equation}
Eq.\,(\ref{evoleqsu2}) is equivalent to \cite{GiacosaHofmann2007}
\begin{equation}
\label{evalambdasu2}
1=-\frac{24\lambda^3}{(2\pi)^6}\left(\lambda\frac{da}{d\lambda}+a\right)a\,D(2a)\,,
\end{equation}
where $\lambda\equiv\frac{2\pi T}{\Lambda}$, $a\equiv\frac{m}{2T}$ and thus $a=2\pi e\lambda^{-3/2}$,
and $D(y)\equiv \int_0^\infty dx\,\frac{x^2}{\sqrt{x^2+y^2}}\,\frac{1}{\mbox{e}^{\sqrt{x^2+y^2}}-1}$. The right-hand side of
Eq.\,(\ref{evoleqsu2}) is negative definite. As a consequence, its solution $\lambda(a)$ is strictly
monotonic decreasing and so is its inverse $a(\lambda)$. Hence there is a regime in temperature $\lambda>\lambda_1$
where $a(\lambda)<1$, and Eq.\,(\ref{evalambdasu2}) can be approximated as
\begin{equation}
\label{evalambdasu2sim}
1=-\frac{\lambda^3}{(2\pi)^4}\left(\lambda\frac{da}{d\lambda}+a\right)a\,.
\end{equation}
The solution to Eq.\,(\ref{evalambdasu2sim}), subject to the initial condition $a(\lambda_i)=a_i\ll 1$, is given as
\begin{equation}
\label{solasu2}
a(\lambda)=4\sqrt{2}\pi^2\lambda^{-3/2}\left(1-\frac{\lambda}{\lambda_i}\left[1-\frac{a^2_i\lambda_i^3}{32\pi^4}\right]\right)^{1/2}\,.
\end{equation}
Thus for $\lambda_1<\lambda\ll\lambda_i$ function $a(\lambda)$ runs into the
attractor $a(\lambda)=4\sqrt{2}\pi^2\lambda^{-3/2}$. Using $a\equiv\frac{m}{2T}=2\pi
e\lambda^{-3/2}$, this attractor corresponds to the plateau $e\equiv\sqrt{8}\pi$. Because the
attractor increases with
decreasing $\lambda$ the condition $a\ll 1$ will be violated at small
temperatures. The estimate $14.61>\lambda_1$ is obtained by setting the
attractor equal to unity. Lowering $\lambda$ further, the solution to the
full Eq.\,(\ref{evalambdasu2}) runs into a thin pole at $\lambda_c$ of the form \cite{Hofmann2011}
\begin{equation}
\label{lambdapole}
e(\lambda)=-4.59\,\log(\lambda-\lambda_c)+18.42\,,
\end{equation}
where $\lambda_c=13.87$. Now, setting the upper integration limit
$\xi$ in Eq.\,(\ref{calA}) equal to $1/(\beta |\phi|^{-1})$ and invoking
$\lambda_c=13.87$, yields $\xi\ge (13.87)^{3/2}/(2\pi)\sim 8.22$.
But the approach of function ${\cal A}(2\pi\hat{\tau})$ to a sine is
perfect for $\xi\sim 8.22$ \cite{HerbstHofmann2004,Hofmann2011}, and
for $\xi\sim 8.22$ the integral in Eq.\,(\ref{calA}) is
strongly dominated by its upper limit. We conclude
that calorons and anticalorons that are just not
resolved dominate the emergence of the field $\phi$ and
its consequences for the effective gauge dynamics
(adjoint Higgs mechanism and thermal ground state estimate).

The divergence of $e$ at $\lambda_c$ initiates a transition to a phase
where thermal quasiparticles decouple, the ground state is a condensate of magnetic monopoles, and
a formerly massless gauge mode acquires mass by the dual Meissner effect
\cite{Hofmann2005,Hofmann2011}.

We conclude that {\sl almost everywhere}
in the deconfining phase the effective coupling assumes the constant value
$e\equiv\sqrt{8}\pi$.

\subsection{Counting of powers of $\hbar$ and action of just-not-resolved caloron\label{CP}}

Concerning the counting of powers in $\hbar$,
loop expansions in the effective theory are carried out as in conventional
perturbation theory. Additional subtleties arise because of the existence
of a maximal resolution $|\phi|$ which enforces constraints on momentum
transfers in four-vertices and on the off-shellness of the massless
mode. These constraints have an obvious form in physical unitary-Coulomb gauge,
see Sec.\,\ref{Dariush}. To make the power counting in $\hbar$ most explicit, we work in
units where $k_B=c=\epsilon_0=\mu_0=1$ but $\hbar$ is re-instated as an action. (So far we have
worked in supernatural units $\hbar=c=k_B=1$).
The (dimensionless) exponential
\begin{equation}
\label{expweight}
-\frac{\int_0^{\beta} d\tau d^3x\,{\cal L}^\prime_{\mbox{\tiny eff}}[a_\mu]}{\hbar}\,,
\end{equation}
in the weight belonging to fluctuating fields in the partition function\footnote{Recall that $\phi$ is inert.
As a consequence, the factor, whose exponent is the potential-part of the effective action, can be pulled out of the partition
function and needs not be considered in a discussion of the effective loop expansion.} thus can, in unitary gauge, be re-cast as
\begin{equation}
\label{expweightdiml}
-\int_0^{\beta} d\tau d^3x\,\mbox{tr}\,
\Big(\frac12(\partial_\mu \tilde{a}_\nu-\partial_\nu
\tilde{a}_\mu-ie\sqrt{\hbar}[\tilde{a}_\mu,\tilde{a}_\nu])^2-e^2\hbar[\tilde{a}_\mu,\tilde{\phi}]^2\Big)\,,
\end{equation}
where $\tilde{a}_\mu\equiv a_\mu/\sqrt{\hbar}$ and $\tilde{\phi}\equiv \phi/\sqrt{\hbar}$ are assumed to not depend on $\hbar$ \cite{Brodsky2011}  ,
see also \cite{DonoghuePapers}. Notice that because of the terms
$\propto \hbar^0$ in (\ref{expweightdiml}) the unit of $\tilde{a}_\mu$ is
length$^{-1}$. Thus the coupling $e$ must have the unit of $1/\sqrt{\hbar}$
(and the unit of $\tilde{\phi}$ is also length$^{-1}$). Together with the results reviewed
in Sec.\,\ref{tsc} we thus have
\begin{equation}
\label{plateaucoupl}
e=\frac{\sqrt{8}\pi}{\sqrt{\hbar}}
\end{equation}
almost everywhere in the deconfining phase.
Because fundamental field configurations are smoothly connected
to the effective dynamics\footnote{The spatial coarse-graining rapidly saturates:
$\frac{|\phi|^{-1}}{\beta}\ge
8.221\times\left(\frac{\lambda}{\lambda_c}\right)^{3/2}$ and practically no alteration of $\phi$'s
time dependence occurs when varying $r,\rho\sim |\phi|^{-1}$ \cite{Hofmann2011}.} the action $S_{C,A;\rho\sim |\phi|^{-1}}$ of a
just-not-resolved caloron/anti\-caloron reads
\begin{equation}
\label{actioncalantcal}
S_{C,A;\rho\sim |\phi|^{-1}}=\frac{8\pi^2}{e^2}=\hbar\,.
\end{equation}
Equation (\ref{actioncalantcal}) has implications. First, it suggests that the quantum of action
$\hbar$, whose introduction in (\ref{expweight}) is motivated by the laws of Quantum Mechanics and which should enable a systematic accounting of quantum
corrections (number of loops) in the effective theory, coincides with
the Euclidean action of a just-not-resolved selfdual (classical) field configuration whose
topological charge is fixed to be of modulus
unity \cite{HerbstHofmann2004,Hofmann2011}. That is, the reason why
$\hbar$ really is a constant can be traced to the universal constancy of $e$ (almost no dependence on $T$ and
no dependence on the specific Yang-Mills scale $\Lambda$ of a given
SU(2) gauge theory) and the
fact that only charge-modulus-unity calorons/anticalorons contribute to the
effective thermodynamical consistency of interacting, fundamental
topological and plane-wave configurations \cite{HerbstHofmann2004}. Moreover, rather than multiplying first and second
powers of $\hbar^{1/2}$ onto the powers $e$ and $e^2$
of an a priori unknown coupling constant $e$ in three- and four-vertices of a Yang-Mills
theory we would infer that these vertices exist only because of the presence of just-not-resolved
calorons and anticalorons. Second, for the fine-structure constant $\alpha$ of
Quantum Electrodynamics (QED) to be dimensionless,
\begin{equation}
\label{FSC}
\alpha=N^{-1}\frac{g^2}{4\pi\hbar}\,,
\end{equation}
the coupling $g$ in Eq.\,(\ref{FSC}) must have the unit of $\sqrt{\hbar}$. ($N^{-1}$ denotes a numerical
factor related to the mixing of the massless modes belonging to several SU(2)
groups, see \cite{Hofmann2011,GiacosaHofmann2005}.) This is guaranteed if $g$ is taken to be the electric-magnetically dual coupling
to $e$:
\begin{equation}
\label{elch}
g=\frac{4\pi}{e}\propto \sqrt{\hbar}\,.
\end{equation}
That is, a magnetic monopole liberated by the dissociation of a large-holonomy SU(2) caloron and other incarnations of this
magnetic charge in the preconfining and confining phase\footnote{For example, the isolated charge situated in the center of the flux
eddy associated with the selfintersection of a center-vortex loop in the confining phase \cite{Hofmann2011,MoosmannHofmann2008},
see also discussion in Sec.\,\ref{QED}.} are interpreted as {\sl electric} charges in the real world. Third,
by virtue of Eq.\,(\ref{plateaucoupl}) the factors $e\sqrt{\hbar}$ and $e^2\hbar$ in (\ref{expweightdiml})
do not depend on $\hbar$. Thus, by assumption the weight of the partition function associated with the effective theory is independent of
$\hbar$, and thus the effective loop expansion is {\sl not} an expansion in powers of $\hbar$. This is in contrast to renormalized perturbation
theory where the value of the coupling is a priori unknown: The perturbative coupling constant needs to
be determined by an empirical boundary condition.

\subsection{Unresolved, interacting plane waves: Local vertices by caloron/anticaloron mediation\label{UPW}}

From the results of Sec.\,\ref{CP} and the facts reviewed in Sec.\,\ref{intro} we may conclude that in the effective
theory the pure-gauge configuration, associated with the a priori estimate of the thermal ground state,
describes the collective effect of interacting (typically small-holonomy) calorons/anticalorons of
action $\hbar$ that are just not resolved. Recall that the holonomy of calorons/anticalorons is elevated from small to large by effective plane-wave
interactions which collectively account for dissociation into screened but unresolved
magnetic monopole-antimonopole pairs \cite{SpatialWilson,Diakonov2004}. Extrapolating the finding,
that it is unresolved calorons/anticalorons which induce local vertices
between plane waves, deeply into the unresolved domain, we would by the (ultraviolet as well as infrared)
finiteness of all quantities calculable in the effective theory and the fact that the emergence of the field
$\phi$ is dominated by just-not-resolved calorons/anticalorons reason (cubic dependence of the prefactor of $\phi$'s
time dependence on the cutoff for the $\rho$ integration, see Eq.\,(\ref{calA}) and \cite{HerbstHofmann2004,Hofmann2011})
that the strength of interactions between
fundamental plane waves, which are largely off their
free mass shell, rapidly ceases with increasing momentum transfer. Recall that the effective pure-gauge configuration
of the thermal ground-state estimate,
which collectively describes these interactions,
is {\sl sourced} by the field $\phi$ \cite{Hofmann2005,Hofmann2007,Hofmann2011}.
But again, the content of fundamental charge-modulus unity selfdual gauge-field configuration in
$\phi$ is {\sl strongly dominated} by those calorons/anti\-calorons that are {\sl just not resolved}.

The point of view, see Fig.\,\ref{Fig-1}, that the emergence of local vertices between plane waves in a Quantum Yang-Mills theory
is a consequence of the existence of nonpropagating, topologically stabilized and selfdual Euclidean
field configurations of the same theory, compares in an interesting way to pure perturbation theory
where such local vertices are taken for granted \footnote{For the effective and local interaction between fundamentally
charged chiral fermions via their zero modes, which are localized on instantons, the 't Hooft vertex represents
an example of plane waves interacting by the mediation of selfdual gauge-field configurations \cite{'tHooft1976I,'tHooft1976II}.}. Namely, while perturbation theory requires a
renormalization programme to sweep ultraviolet divergences under the carpet the latter
do not occur to begin with when vertices are understood to be induced by unresolved, nonpropagating, topological,
and selfdual field configurations: Because potential plane-wave fluctuations, that are extremely off-shell compared
to an externally imposed resolution\footnote{In the case of thermodynamics this resolution is
prescribed by temperature to be $|\phi|$.}, do not interact they are not generated in
the first place. To the authors' minds this situation does by no means contradict the renormalization programme
of perturbation theory but rather supplies it with a deeper justification: The subtraction of
infinities in perturbative loop diagrams, which ignore the presence of
the nontrivially selfdual sector of gauge-field configurations, simulates the nonoccurrence of
such divergences in the full Yang-Mills theory.
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.124in,
width=5.6628in]{Fig-1.ps}
\caption{(a) Plane waves (lines) of a Yang-Mills theory interacting via local vertices (solid circles), (b) generation of a vertex
(local crumble within the ensemble of noninteracting plane waves)
by a nonpropagating and selfdual gauge-field configuration of topological charge modulus unity in the same Yang-Mills theory.}
\label{Fig-1}
\end{center}
\end{figure}

\subsection{Gauge-field-matter interactions\label{QED}}

So far we have discussed the situation of a pure SU(2) Yang-Mills theory. The undisputed
successes of perturbatively accessed QED in particular (and the Standard Model of Particle Physics as a whole) demand to put the results of
Secs.\,\ref{CP} and \ref{UPW} into the perspective of electrically charged particle interacting with photons.
Let us undertake a perhaps speculative discussion of the situation in QED.

To attribute caloron/anti\-caloron mediation to a QED vertex, the U(1) gauge group must be considered a dynamically broken
SU(2). As explained in Sec.\,\ref{intro} and investigated in \cite{HerbstHofmann2004,Hofmann2005,Hofmann2007,Hofmann2011}
this gauge-symmetry breaking is collectively mediated by calorons
and anticalorons in terms of the field $\phi$. In \cite{Hofmann2009} the critical temperature
$T_c$ of this SU(2) theory (and thus its Yang-Mills scale) was, by interpretation of the observed low-frequency
excess in line temperature (see references in \cite{Hofmann2009}),
extracted to be the present baseline temperature 2.725\,Kelvin
of the Cosmic Microwave Background (CMB). Thus the name SU(2)$_{\tiny\mbox{CMB}}$.

Radiative corrections in QED are organized
into a power series in the fine-structure constant
$\alpha$. As already mentioned in Sec.\,\ref{CP} $\alpha$ measures the strength of interaction
of the massless mode of SU(2)$_{\tiny\mbox{CMB}}$ (the propagating photon) with a screened yet unresolved
monopole \cite{SpatialWilson}. Such a monopole becomes isolated and
thus resolvable when trapped in the core of the selfintersection
of a center-vortex loop \cite{MoosmannHofmann2008II}. Notice that in the presence of propagating photons
a center-vortex loop of selfintersection number unity
is the only absolutely stable degree of freedom in the confining phase of an
SU(2) Yang-Mills theory. (The nonselfintersecting species shrinks to a round point \cite{MoosmannHofmann2008}
and vortex loops of intersection number larger than unity decay by charge annihilation or
repulsion \cite{MoosmannHofmann2008II,Hofmann2007CVL}.) For the SU(2)$_{\tiny\mbox{CMB}}$ photon to see
the charge of a monopole belonging to another SU(2) theory, which presently is in its confining phase,
the two theories must mix. So the crumble in the photonic plane wave, induced by the
presence of the electric charge of, say, an electron, occurs by having the latter couple
to its very massive photon (confining phase) which in turn mixes into the SU(2)$_{\tiny\mbox{CMB}}$ plane wave
by caloron/anti\-caloron mediation of the latter theory. Recall that,
according to the results of Sec.\,\ref{CP}, this mediation is accompanied
by a unit $\hbar$ of `free rooms for action'.

Interestingly, since the QED action, involving the electron field $\psi$ and $a_\mu^3$, is given as \cite{Brodsky2011}
\begin{equation}
\label{QEDact}
S_{\tiny\mbox{QED}}=\int d^4x\,\left(\bar{\psi}(\gamma^\mu(i\hbar\partial_\mu-g\,a_\mu^3 )-m\right)\psi\,,
\end{equation}
where now $\psi$, $\tilde{m}\equiv\frac{m}{\hbar}$, and $\tilde{a}_\mu^3$ are considered independent
of $\hbar$, one arrives at

\begin{equation}
\label{QEDactscal}
-\frac{S_{\tiny\mbox{QED}}}{\hbar}=-\int d^4x\,\left(\bar{\psi}(\gamma^\mu(i\partial_\mu-\frac{1}{\sqrt{\hbar}} g\,a_\mu^3 )-\tilde{m}\right)\psi\,.
\end{equation}
By virtue of Eq.\,(\ref{elch}) we would thus conclude from Eq.\,(\ref{QEDactscal}) that the interaction $\frac{1}{\sqrt{\hbar}} g
\bar{\psi}\gamma^\mu a_\mu^3\psi$ does not exhibit an $\hbar$ dependence. So even in
QED the loop expansion can no longer be considered an expansion in powers of $\hbar$ if the
vertex is interpreted to be induced by a caloron/anticaloron of SU(2)$_{\tiny\mbox{CMB}}$.

\section{Effective and irreducible, planar 3-loop corrections\label{Dariush}}

After `scratching into unresolved physics' in Sec.\,\ref{S1} to associate interactions
between plane waves with the presence of calorons/anti\-calorons we would here
like to re-iterate \cite{KavianiHofmann2007,HerbstHofmannRohrer2004,SchwarzHofmannGiacosa2007} on technicalities in performing the effective loop expansion
of the pressure in deconfining SU(2) Yang-Mills thermodynamics \cite{Hofmann2006}.

\subsection{Feynman rules and constraints}

Following the extraction of the effective gauge coupling $e(\lambda)$
from the thermodynamical consistency of effective, noninteracting quasiparticles,
one may investigate the magnitude of radiative effects implied by the so-obtained function
$e(\lambda)$. This strategy turns out to be selfconsistent. To perform the
calculation we go back to supernatural units and work within the
real-time formulation \cite{Hofmann2005,Hofmann2007,Hofmann2011}.
Thus $\tilde{a}_\mu\to a_\mu$ etc.

The Feynman rules are formulated in the completely fixed (and physical) unitary-Coulomb gauge which avoids the consideration of
Faddeev-Popov ghosts.
$\\$
$\bullet$ For the three-vertex $\Gamma^{\mu\nu\rho}_{[3]abc} (p,k,q)$ and for the
four-vertex $\Gamma^{\mu\nu\rho\delta}_{[4]abcd}(p,q,r,s)$, we read off from Eq.\,(\ref{expweightdiml}) that
\begin{eqnarray}
\label{D1}
\Gamma^{\mu\nu\rho}_{[3]abc} (p,k,q) & = & e(2\pi)^4 \delta (p+q+k) \varepsilon_{abc}[g^{\mu\nu}(q-p)^{\rho}+g^{\nu\rho}(k-q)^{\mu}+ \notag\\[1ex]
&& g^{\rho\mu}(p-k)^{\nu}], \\ \Gamma^{\mu\nu\rho\delta}_{[4]abcd}(p,q,r,s) & = & -ie^2(2\pi)^4\delta(p+q+s+r)
[\varepsilon_{fab}\varepsilon_{fcd}(g^{\mu\rho}g^{\nu\sigma}-g^{\mu\sigma}g^{\nu\rho})+ \nonumber  \\
&& \varepsilon_{fac}\varepsilon_{fdb}(g^{\mu\sigma}g^{\rho\nu}-g^{\mu\nu}g^{\rho\sigma})+
\varepsilon_{fad}\varepsilon_{fbc}(g^{\mu\nu}g^{\sigma\rho}-g^{\mu\rho}g^{\sigma\nu})]\,,\nonumber\\
\end {eqnarray}
where $\varepsilon_{abc}$ are the structure constants of SU(2).
$\\$
$\bullet$  The propagator of the tree-level heavy (TLH) modes propagates thermalized on-shell
particles (no energy-momentum tranfer to $\phi$ in the summation of the
Dyson series for the interaction of $a_\mu^{1,2}$ with $\phi$) of mass $m=2\,e|\phi|$ only \cite{Hofmann2011}
\begin{eqnarray}
\label{D2}
D^{\text{TLH}}_{\mu\nu,ab}(p) & = & -2\pi\delta_{ab}\tilde D_{\mu\nu}\delta(p^2-m^2)n_B\left(\frac{|p^0|}{T}\right)\,, \\
\tilde D_{\mu\nu} & = & \bigg(g_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{m^2}\bigg)\,.
\end{eqnarray}
The propagation of tree-level massless (TLM) modes splits into a vacuum and a thermal part:
\begin{eqnarray}
\label{D3}
D^{\text{TLM}}_{\mu\nu,ab}(p) &=& -\delta_{ab}\left\{P^{T}_{\mu\nu}\left[\frac{i}{p^2} +
2\pi\delta (p^2)n_B\left(\frac{|p^0|}{T}\right)\right] - \frac{iu_{\mu}u_{\nu}}{\vec{p}^2}\right\}\,,
\end{eqnarray}
where
\begin{equation}
\label{D4}
P^{00}_T = P^{0i}_T = P^{i0}_T \equiv 0,\;\;\;\ P^{ij}_T = \delta^{ij} - \frac{p^{i}p^{j}}{\vec{p}^2}\,.
\end{equation}
Here $n_B(x)=\frac{1}{e^x-1}$ denotes the Bose-Einstein distribution function. Notice that TLM modes have a color index $a=3$
and TLH modes $a=1$ or $a=2$. Moreover, the term $\propto u_{\mu}u_{\nu}$ in Eq.\,(\ref{D3}) expresses the instantaneous
``propagation'' of the $a^3_0$ field (Coulomb term\footnote{However, one-loop radiatively induced, the emergence of
longitudinally propagating $E$-field (charge-density) waves is observed \cite{FalquezHofmann2011}.}), and $u_{\mu} = (1,0,0,0)$ represents the four-velocity of the heat bath.
$\\$
$\bullet$ The maximal off-shellness of a TLM mode with four-momentum $p$ is constrained as \cite{Hofmann2005,Hofmann2006,Hofmann2011}:
\begin{equation}
\label{D5}
|p^2|  \leq  |\phi|^2 \quad  \mbox{(TLH modes)}\quad\quad |p^2| \leq  |\phi|^2  \quad \mbox{(TLM modes)}\,.
\end{equation}
For a four-vertex with ingoing four-momenta $p_1,p_2$ and outgoing four-momenta $p_3,p_4$
we have \cite{Hofmann2007,Hofmann2006,Hofmann2011}
\begin {eqnarray}
\label{D8}
|(p_1 + p_2)^2| &\leq& |\phi|^2 \quad \mbox{(s-channel)}, \quad |(p_3 - p_1)^2| \leq |\phi|^2 \quad \mbox{(t-channel)}, \notag \\
|(p_3 - p_2)^2| &\leq& |\phi|^2 \quad \mbox{(u-channel)}.
\end{eqnarray}
As we shall review \cite{Hofmann2006,KavianiHofmann2007} in Sec.\,\ref{GELC},
conditions (\ref{D5}), (\ref{D8}), and the requirement of on-shellness of TLH modes imply a rapid growth of the
number of constraints over the number of a priori noncompact (radial) integration variables with an increasing number of
loops in an irreducible diagram.

\subsection{Expectations for convergence properties of loop expansion from
two- and three-loop corrections to the pressure \label{GELC}}

As discussed in the previous section, the momentum transfer in all Mandelstam variables s,t, and u
in a four-vertex with ingoing four-momenta $p_1, p_2$ and outgoing
four-momenta $p_3, p_4$ should not exceed the scale $|\phi|$.
For three-loop diagrams, and in particular the ones of Fig.\,\ref{Fig-3}, where the cutting of any
line yields a 1Pi irreducible contibution to a polarization tensor,
the number $\tilde{K}$ of independent, radial loop-momentum variables
$(p_{0},|\vec{p}|)_{i}$ for $i=1,2,3$ is $\tilde{K}=6$. To judge the
number $K$ of independent constraints imposed on them we,
in addition to (\ref{D5}) and (\ref{D8}), to consider on-shellness constraints for the TLH modes:
\begin{equation}
\label{D10}
p_1^2=m^2,\;\;\;\ p_{2}^2=m^2,\;\;\;\ p_{3}^2=m^2,\;\;\;\,p_{4}^2=(p_{1}+p_{2}-p_{3})^2=m^2\,.
\end{equation}
Conditions (\ref{D8}) and (\ref{D10}) together yield for irreducible three-loop diagrams A and B a total of $K=3+4=7$ constraints.

According to (\ref{D5}), the TLM modes in the irreducible three-loop diagram C are subject to the maximal off-shellness conditions
\begin {equation}
\label{D11}
|p_1^2| \leq |\phi|^2,\;\;\;\ |p_2^2| \leq |\phi|^2\,,
\end{equation}
and the on-shellness relations for the TLH modes in diagram C are
\begin{equation}
\label{D12}
p_3^2=m^2,\;\;\;\,p_{4}^2=(p_{1}+p_{2}-p_{3})^2=m^2.
\end{equation}
Thus, again, we have $K=3+4=7$ constraints for diagram C.
Hence for all irreducible three-loop diagrams we have $K=7$.
Therefore, the number of constraints exceeds the number of radial variables:
\begin{equation}
\label{D13}
\tilde{K}<K\,.
\end{equation}
This relation implies that irreducible three-loop integrations have either
compact or empty supports.

For the two-loop diagrams of Fig.\,\ref{Fig-2}, we have $\tilde{K}>K$.
Namely, the number of $\tilde{K}$ of independent radial
loop momenta $(p_{0},|\vec{p}|)_{i}$ for $i=1,2$ is $\tilde{K}=4$ whereas the constraints plus the on-shellness conditions are
\begin{figure}
[ptb]
\begin{center}
\includegraphics[height=1.63in, width=3.5in]{Fig-2loopmod.ps}
\caption{Two out of three nonvanishing two-loop diagrams with TLH  (solid) and TLM (dashed) modes in the loops.}
\label{Fig-2}
\end{center}
\end{figure}
\begin{figure}
[ptb]
\begin{center}
\includegraphics[height=1.63in, width=4.5in]{Fig-3loopmod.ps}
\caption{The three irreducible, planar three-loop diagrams with TLH  (solid) and TLM (dashed) modes in the loops.}
\label{Fig-3}
\end{center}
\end{figure}
\begin{equation}
\label{D14}
|(p_{1}+p_{2})^2| \leq |\phi|^2,\;\;\;\ p_1^2=m^2,\;\;\;\ p_2^2=m^2.
\end{equation}
This implies $K=1+2=3$. Thus, unlike the case of irreducible three-loop integrations, we expect the support of two-loop
integrations to be noncompact. In Ref.\,\cite{Hofmann2006} a counting of $\tilde{K}$ versus
$K$ was performed for an arbitrary loop number in the case of planar and nonplanar
irreducible loop diagrams which gives rise to the
conjecture that, modulo resummations of irreducible contributions to the
polarization tensors in the lines of a given bubble diagram,
the expansion of the pressure {\sl terminates} at a finite loop order.
The fact that $\tilde{K}<K$ \cite{KavianiHofmann2007} is already in agreement with this conjecture.

In the remaining subsections of Sec.\,\ref{Dariush}
we discuss how numerical results are obtained and that, compared to the free quasiparticle pressure, two-loop and
irreducible three-loop diagrams exhibit a hierarchical suppression of their contributions to the thermodynamical
pressure \cite{KavianiHofmann2007}. In fact, diagram C of Fig.\,\ref{Fig-3} is shown to vanish.

\subsection{Irreducible 3-loop integrations}

For the irreducible three-loop bubble diagrams in Fig.\,\ref{Fig-3}, we have in case A
two four-vertices and four propagators and the correction to the pressure is
\begin{eqnarray}
\label{D15}
\Delta P_A &=& \frac{1}{48}\int \frac{d^4p_1d^4p_2d^4p_3}{(2\pi)^4(2\pi)^4)(2\pi)^4} \Gamma_{[4]abcd}^{\mu\nu \rho \sigma}
\Gamma_{[4]\bar{a}\bar{b}\bar{c}\bar{d}}^{\bar{\mu}\bar{\nu} \bar{\rho}\bar{\sigma}}D_{\rho \bar{\rho}, c \bar{c}}(p_1)
D_{\sigma \bar{\sigma}, d \bar{d}}(p_2)\\ && \times D_{\mu \bar{\mu}, a \bar{a}}(p_3) D_{\nu \bar{\nu}, b \bar{b}}(p_4)\nonumber,
\end {eqnarray}
where the momenta are subject to the constraints (\ref{D8}) and (\ref{D10}) and overall momentum
conservation has been used ($p_4$ is a function of $p_1$, $p_2$, $p_3$.).

For diagram B we have one four-vertex,
two three-vertices and five propagators. Therefore, the correction to the pressure is
\begin{eqnarray}
\label{D16}
\Delta P_B &=& \frac{1}{2}\int \frac{d^4p_1d^4p_2d^4p_3}{(2\pi)^4(2\pi)^4)(2\pi)^4} \Gamma_{[4]hijk}^{\alpha \beta \gamma \lambda}\Gamma_{[3]abc}^{\mu \nu \rho}
\Gamma_{[3] \bar{a} \bar{b} \bar{c}}^{\bar{\mu} \bar{\nu} \bar{\rho}} D_{\mu \alpha, ah}(p_1)D_{\bar{\mu} \beta, \bar{a} i}(p_2) \\ && \times D_{\gamma \rho, jc}(p_3) D_{\lambda \bar{\rho}, k \bar{c}}(p_4)D_{\mu \bar{\nu}, b \bar{b}}(p_5)\nonumber,
\end{eqnarray}
where the momenta are subject to same constrains as in diagram A. It should be noticed that for the massless mode,
$p_{5}=p_{1}-p_{3}$, we do not need to consider an additional constraint $|p_{5}|^2 \leq |\phi|^2$ because this constraint is
automatically satisfied by the t--channel condition in (\ref{D8}).

For diagram C, the Feynman rules are similar to diagram A and the correction to the pressure is
\begin{eqnarray}
\label{D17}
\Delta P_C &=& \frac{1}{48}\int \frac{d^4p_1d^4p_2d^4p_3}{(2\pi)^4(2\pi)^4)(2\pi)^4} \Gamma_{[4]abcd}^{\mu \nu \rho \sigma}
\Gamma_{[4]\bar{a}\bar{b}\bar{c}\bar{d}}^{\bar{\mu} \bar{\nu} \bar{\rho} \bar{\sigma}}
D_{\rho \bar{\rho}, c \bar{c}}(p_1)D_{\sigma \bar{\sigma}, d \bar{d}}(p_2) \nonumber\\
&& \times D_{\mu \bar{\mu}, a \bar{a}}(p_3) D_{\nu \bar{\nu}, b \bar{b}}(p_4)\,,
\end{eqnarray}
where the momenta are subject to the constraints (\ref{D8}), (\ref{D11}), and
(\ref{D12}).

Now, insert the Feynman rules (\ref{D1}) - (\ref{D3}) in the above integrals (\ref{D15}) - (\ref{D17}),
consider the dimensionless variables $x_i \equiv \frac{|\vec{p_i}|}{|\phi|}$ $(i = 1, 2, 3)$ for rescaling the
radial loop momenta, and consider the following steps:
$\\$
$\bullet$ Carry out Lorenz and color contractions.
$\\$
$\bullet$ Go over to 3D spherical coordinates.
$\\$
$\bullet$ Trivially integrate the temporal components of four-momenta by appealing to the delta functions of the propagators.
$\\$
$\bullet$ Apply the triangle inequality to obtain upper bounds for the modulus of each correction.
$\\$
For the corrections (\ref{D15}) and (\ref{D16}) we obtain the following expressions
\begin {eqnarray}
\label{D18}
|\Delta P_{A(B)}| &\leq& (24)\,\frac{e^4 \Lambda^4\lambda^{-2}}{3\times 2^7 \times (2\pi)^6} \sum_{l,m,n=1}^2 \!\!\int dx_1 \!\!\int dx_2 \!\!\int dx_3
\!\!\int dz_{12} \!\!\int dz_{13} \!\!\int_{z_{23,l}}^{z_{23,u}} \!\!\!\!\!\!dz_{23} \nonumber \\
&&\times \frac{1}{\sqrt{(1-z_{12}^2)(1-z_{13}^2)-(z_{23}-z_{12}z_{13})^2}}\nonumber \\
&&\times \frac{x_1^2x_2^2x_3^2}{\sqrt{x_1^2+4e^2}\sqrt{x_2^2+4e^2}\sqrt{x_3^2+4e^2}}\nonumber \\
&&\times \delta(4e^2+((-1)^{l+m}\sqrt{x_1^2+4e^2}\sqrt{x_2^2+4e^2} - x_1x_2z_{12}-\nonumber \\
&& ((-1)^{l+n}\sqrt{x_1^2+4e^2}\sqrt{x_3^2+4e^2} - x_1x_3z_{13})-\nonumber \\
&& ((-1)^{m+n}\sqrt{x_2^2+4e^2}\sqrt{x_3^2+4e^2} - x_2x_3z_{23})\nonumber \\
&&\times |P_{A(B)}(x,z,l,m,n)| n_B(2\pi\lambda^{-3/2} \sqrt{x_1^2 + 4e^2}\nonumber\times \\
&& n_B(2\pi\lambda^{-3/2} \sqrt{x_2^2 + 4e^2} n_B (2\pi\lambda^{-3/2} \sqrt{x_3^2 + 4e^2}\nonumber\times \\
&& n_B(2\pi\lambda^{-3/2} |(-1)^l \sqrt{x_1^2 + 4e^2} + (-1)^m \sqrt{x_2^2 + 4e^2}\nonumber + \\
&&(-1)^n \sqrt{x_3^2 + 4e^2}|\,,
\end {eqnarray}
where $z_{12} \equiv \cos \angle (\vec{x_1}, \vec{x_2}), z_{13} \equiv \cos \angle (\vec{x_1}, \vec{x_3}), z_{23} \equiv
\cos \angle (\vec{x_2}, \vec{x_3})$. The extremely lengthy polynomials $P_A$ and $P_B$ arise from Lorenz
and color contractions and are regular at $x_1 = x_2 = x_3 = 0$. We also define \cite{Hofmann2006}:
\begin{equation}
\label{D19}
z_{23,u}\equiv\cos\left|\arccos z_{12}-\arccos z_{13}\right|\,,\ \ \ z_{23,l}\equiv\cos\left|\arccos z_{12}+\arccos z_{13}\right|\,.\\[2ex]
\end{equation}

Recalling (\ref{D10}) and the fact that only the minus sign is relevant $(e > \frac{1}{2\sqrt{2}})$ \cite{SchwarzHofmannGiacosa2007}, which makes the
expression within the absolute-value signs strictly negative, we have
\begin{eqnarray}
\label{D20}
z_{12} & \leq & \frac{1}{x_1x_2}(4e^2 - \sqrt{x_1^2 + 4e^2} \sqrt{x_2^2 + 4e^2} + \frac{1}{2}) \equiv g(x_1, x_2) \\
\label{DD20}
z_{13} & \geq & \frac{1}{x_1x_3}(-4e^2 + \sqrt{x_1^2 + 4e^2} \sqrt{x_3^2 + 4e^2} - \frac{1}{2}) \equiv g(x_1, x_3)  \\
\label{DDD20}
z_{23} & \geq & \frac{1}{x_2x_3}(-4e^2 + \sqrt{x_2^2 + 4e^2} \sqrt{x_3^2 + 4e^2} - \frac{1}{2}) \equiv g(x_2, x_3)\,.
\end {eqnarray}
Repeating the same steps as before, for the vacuum-vacuum contribution (\ref{D17}) of diagram C we obtain
\begin {eqnarray}
\label{D21}
|\Delta P_C| & \leq & \frac{e^4\Lambda^4\lambda^{-2}}{3\times 2^5 \times (2\pi)^8} \sum_{l,m}^2 \int dy_1 \!\!\int dx_1 \!\!\int dx_2
\!\!\int dx_3 \!\!\int dz_{12} \!\!\int dz_{13} \!\!\int_{z_{23,l}}^{z_{23,u}} \!\!\!\!\!\!dz_{23}\nonumber \\
& & \times \frac{x_1^2x_2^2x_3^2}{\sqrt{(1-z_{12}^2)(1-z_{13}^2)-(z_{23}-z_{12}z_{13})}^2}\nonumber \\
& & \times|P_{C}({\bf x},{\bf z}, y_1,l,m)| n_B(2\pi\lambda^{-3/2} \sqrt{x_3^2 + 4e^2})\nonumber \\
& & \times\frac{n_B(2\pi\lambda^{-3/2}|(-1)^l \sqrt{x_3^2 + 4e^2} + (-1)^m f_2({\bf x},{\bf z})|)}{f_2({\bf x},{\bf z}, y_1)\sqrt{x_3^2 + 4e^2}}\,,
\end {eqnarray}
where
\begin{equation}
\label{D22}
f_2({\bf x},{\bf z}) \equiv \sqrt{x_1^2 + x_2^2 +x_3^2+2x_1x_2z_{12}-2x_1x_3z_{13}-2x_2x_3z_{23}},\;\;\;\;\;\;\ y_1 \equiv \frac{p_1^{o}}{|\phi|}\,,
\end {equation}
with $z_{12}, z_{13}, z_{23}, z_{23,u}$, and $z_{23,l}$ defined in (\ref{D20}),(\ref{DD20}),(\ref{DDD20}) and (\ref{D19}), respectively.
The polynomial $P_C$ emerges from Lorenz and color
contractions and is regular at $x_1 = x_2 = x_3 = 0$.
According to relation (\ref{D8}) - (\ref{D11}) the rescaled constraints read:
\begin{eqnarray}
\label{D23}
|y_1^2 + y_2^2-x_1^2-x_2^2 + y_1 y_2-2x_1x_2z_{12}| &\leq & 1, \\
\label{DD23}
|y_2^2-x_2^2 + 4e^2-(-1)^l \, 2y_2 \sqrt{x_3^2+4e^2}+2x_2x_3z_{23}| &\leq & 1, \\
\label{DDD23}
|y_1^2-x_1^2 + 4e^2-(-1)^l \ 2y_1 \sqrt{x_3^2+4e^2}+2x_1x_3z_{13}| & \leq & 1, \\
\label{DDDD23}
|y_1^2 - x_1^2| \leq 1,\qquad |y_2^2 - x_2^2| &\leq& 1.
\end {eqnarray}
Moreover, $y_2$ is replaced by
\begin{equation}
\label{D24}
y_2 = -y_1 + 2(-1)^l \sqrt{x_3^2 + 4e^2} + (-1)^m f_2({\bf x}, {\bf z}) ,
\end{equation}
through delta function integration, and $(l, m = 1, 3)$.

\subsection{Monte-Carlo integration of irreducible 3-loop diagrams}

\subsubsection{Monte-Carlo integration in general}

In the following we briefly discuss how the integral of a function $f$  over a region $G \subset \mathbb{R}^n $ can
be calculated using statistical methods. Du to the complexity of the region (algebraic varieties) $G$ that support\footnote{The region $G$ is determined by a
set of inequalities.}
a three-loop integration, statistical methods are required in a numerical computations of the
values of bubble diagrams A, B, and C in Fig.\,\ref{Fig-3}. Its characteristic
function is denoted by $\chi_{G}$. The integral then reads
\begin{equation}
\label{D25}
\int_{G} f(x) dx =\int_{\mathbb{R}^n} \chi_{G}(x)  f(x) dx.
\end {equation}
If $G$ is compact then it can be included in a box $B=[x_1,X_1] \times [x_2,X_2] \times ... \times [x_n,X_n]$.
Therefore the integral in Eq. (\ref{D25}) can be written as
\begin {equation}
\label{D26}
\int_{\mathbb{R}^n} \chi_{G}(x)  f(x) dx =\int_B \chi_{G}(x)  f(x) dx.
\end {equation}
If $B$ has volume $V$, $\frac{1}{V} \chi_B$ can be considered as the probability density function of a
random variable $X$, which is equally distributed thoughout the box $B$. The integral then is exactly the expectation value
$E(V \chi_{G}(X)  f(X) )$
\begin {equation}
\label{D27}
\int_B \chi_{G}(x)  f(x) dx= V \int_B \chi_{G}(x)  f(x) \frac{1}{V} \chi_B dx = V E(\chi_{G}(X) f(X)).
\end {equation}
The Monte-Carlo method to calculate this integral consists of a statistical estimate of the expectation value. It
is well known that the mean value $\bar{X}$  of a sample is an unbiased estimator for the expected value. If one draws
a random sample $(x_1,x_2...,x_n)$ of points from the box, the estimate becomes:
\begin {equation}
\label{D28}
\int_{G} f(x) dx \approx V \frac{1}{n}\sum_{i=1}^n \chi_{G}(x_i)  f(x_i).
\end {equation}
The Monte-Carlo-Method is particularly useful to determine integrals over high-dimensional integration regions,
where deterministic methods would be too time consuming. Unfortunately, the precision of this statistical estimate increases
only with the root of the sample size. For example, this implies that the Monte-Carlo method is a relatively
fast way to achieve a result with
a relative precision of about 1 percent, but to achieve one more decimal place in the result the sample size must
be increased by a factor of 100.

\subsubsection{Monte-Carlo integration for diagrams A and B}

The calculation of $\Delta P_A$ and $\Delta P_B$ includes an integration over a
six-dimensional region $G$ with variables of integration
\footnote{Notice that $x_1$ is integrated analytically through the delta function in $\Delta P_A$ and $\Delta P_B$.}
$x_1,x_2,x_3,z_{12},z_{13},z_{23}$. As $z_{ij}\equiv\cos\angle(\vec{x}_i,\vec{x}_j)$, their range is restricted to the interval $[-1,1]$.
The exact shape of $G$ is then determined by the constraints (\ref{D20}) - (\ref{DDD20}) and an additional
 angular condition (\ref{D19}) on $z_{23}$. This angular condition allows for a further reduction of the range of $z_{23}$ to the interval $[z_{23,l},z_{23,u}]$.
Also, the constraints (\ref{D20}) - (\ref{DDD20}) and condition (\ref{D20}) restrict
the $x_i$ to the interval $[0,3]$ \cite{Hofmann2006}. Therefore a box $B$ which contains $G$ is
identified. The constraints define the characteristic function $\chi_{G}$ whereas $z_{23,l}$ and $z_{23,u}$ are used directly as integration limits.
According to this, the ratios of the moduli $\Delta P_A$ and $\Delta P_B$ to the one-loop (ground-state subtracted) approximation
are computed as a function of the dimensionless temperature $\lambda_c = 13.87\le\lambda\equiv\frac{2\pi T}{\Lambda}\le 140$, and
the results are represented in Fig.\,\ref{Fig-4}--\ref{Fig-7}. For the effective gauge coupling the solution of the one-loop evolution
equation (\ref{evalambdasu2}) is used. Figs.\,\ref{Fig-4} - \ref{Fig-7} show that at the critical
temperature $\lambda_c = 13.87$ there is no loop correction (decoupling of TLH modes), and $\Delta P_A$ and $\Delta P_B$ approach zero for large temperatures.
For the three-loop diagrams A and B,
the maxima of the ratio of the estimates of their moduli to the one-loop correction are peaked between $\lambda = 17.5$ and $\lambda = 20$. They are
$6 \cdot 10^{-14}$ and $2 \cdot 10^{-7}$, respectively. This shows the dominance of diagram B. (The next section suggests that diagram C is vanishing).
Comparing Fig.\,\ref{Fig-5} with Fig.\,\ref{Fig-6} and Fig.\,\ref{Fig-7},
shows the dominance of Coulomb fluctuations $(10^{-7})$ over
quantum fluctuation $(10^{-12})$ for in the TLM propagation contributing to diagram B.
\vspace{.5cm}
\begin{figure}
\begin{center}
\includegraphics[scale=0.9]{OnshellGraphik4_gr1.eps}
\caption{Upper estimate for the modulus $\frac{1}{24}|\Delta P_A|/P_{\mbox{1-loop}}$ as a function of $\lambda$ for diagram A in Fig.\,\ref{Fig-3}.}
\label{Fig-4}
\end{center}
\end{figure}
\vspace{.5cm}
\begin{figure}
\begin{center}
\includegraphics[scale=0.9]{Graphik1Photon4_gr1.eps}
\caption{Total upper estimate for the modulus $\frac{1}{24}|\Delta P_B|/P_{\mbox{1-loop}}$ as a function of $\lambda$ for diagram B in Fig.\,\ref{Fig-3}.}
\label{Fig-5}
\end{center}
\end{figure}
\vspace{.5cm}
\vspace{.5cm}
\begin{figure}
\begin{center}
\includegraphics[scale=0.9]{Graphik1PhotonC4_gr1.eps}
\caption{Upper estimate for the modulus of $\frac{1}{24}|\Delta P_B|/P_{\mbox{1-loop}}$ due to the Coulomb part of the TLM propagator as a function
of $\lambda$ for diagram B.}
\label{Fig-6}
\end{center}
\end{figure}
\vspace{.5cm}
\vspace{.5cm}
\begin{figure}
\begin{center}
\includegraphics[scale=0.9]{Graphik1PhotonQ4_gr1.eps}
\caption{Upper estimate for the modulus of $\frac{1}{24}|\Delta P_B|/P_{\mbox{1-loop}}$ due to TLM quantum fluctuations as a
function of $\lambda$ for diagram B.}
\label{Fig-7}
\end{center}
\end{figure}

\subsubsection{Monte-Carlo integration of diagram C}

The region of support for the radial loop integration the irreducible three-loop diagram C
cannot be determined in the same way as for diagrams A and B.
This has mainly to do with the occurrence of additional off-shell variables.
In order to estimate the region of loop integration $G$
for diagram C, we consider a box with volume
$10 \times 10 \times 10 \times 20 \times 2 \times 2 \times 2$
for the loop and angular variables $x_1,x_2,x_3,y_1,z_{12},z_{13},z_{23}$, respectively.
Through the condition $|y_1^2-x_1^2| \leq 1$, and the definition
of $z_{23,(u,l)}$ in Eq.\,(\ref{D19}) a subregion of this volume is determined,
which roughly represents 2 \% of the volume of the box.
In this subset $150,\!000,\!000$ points are randomly picked. Subsequently,
it is checked whether the conditions (\ref{D23}) - (\ref{D24}). With $600,\!000,\!000$ tests we have not found a single
point to satisfy all these conditions. This yields an estimate of a fraction of about
$2\times 10^{-5}$ of the box volume for the region of integration. The typical distance between the sampled point
is $0.2\sim\sqrt[7]{2 \times 10^{-5}}$. Thus it is highly probable that the region of integration is empty.
A similar analysis applies to the vacuum-thermal combination of TLM propagators,
and the constraints imply that the integration region is empty provided
the region of integration for the vacuum-vacuum combination is empty. Again, we have gathered
compelling evidence for the latter to be true.

\subsection{Comparison to 2-loops results, hierarchy}

In order to check the reliability of our numerical method in case of three-loop
integrations, we compute the pressure of the two-loop diagram B in Fig.\,\ref{Fig-2}
with the Monte-Carlo method and compare the result with results obtained by deterministic means
\cite{HerbstHofmannRohrer2004,SchwarzHofmannGiacosa2007}.

For the two-loop diagram B in Fig.\,\ref{Fig-2}, we have two-propagators
and one four-vertex at which the momenta
are subject to the s--channel constraint constraint in (\ref{D8}). Thus the two-loop correction
to the free quasiparticle pressure arising from diagram B takes the form
\begin{eqnarray}
\Delta P_{B} & = & \frac{1}{i}\int \frac{d^4p \, d^4k}{(2\pi)^8}\Gamma_{[4]abcd}^{\mu\nu\rho\delta} D_{\mu\nu,ab}(p)D_{\rho\delta,cd}(k)\,,
\end{eqnarray}
where
\begin{equation}
\label{3_g10}
|(p+k)^2| \leq {|\phi|^2}\,.
\end{equation}
As before, we can rescale the momenta as $x\equiv\frac{|\vec{p}|}{|\phi|}$ and $y\equiv\frac{|\vec{k}|}{|\phi|}$ and obtain
\begin{eqnarray}
\label{3_g11}
\Delta P_{B} & = & -\frac{e^2\Lambda^4\lambda^{-2}}{2 (2\pi)^4} \int dx \int dy \int dz_{xy} \frac{x^2y^2}{\sqrt{x^2+4e^2}\sqrt{y^2+4e^2}}\nonumber \\
&& \times P_{B}(x,y,z_{xy})n_B (2\pi\lambda^{-3/2} \sqrt{x^2+4e^2})n_B(2\pi\lambda^{-3/2}\sqrt{y^2+4e^2})\,,
\end{eqnarray}
where
\begin{equation}
\label{3_g12}
z_{xy} \leq \frac{1}{xy}(4e^2-\sqrt{x^2+4e^2}\sqrt{y^2+4e^2} + \frac{1}{2}) \equiv g(x,y)\,.
\end{equation}
Notice that
\begin{equation}
\label{3_g14}
\lim_{x,y \to \infty} g(x,y) = -1\,.
\end{equation}
In (\ref{3_g10}) only the minus sign is relevant since $e>\frac{1}{2\sqrt{2}}$, and
therefore the expression within the absolute-value signs is strictly negative. This leads to (\ref{3_g12}).
Apart from a small compact region, where $g(x,y) \geq 1$ and which includes the origin $x = y = 0$ in the $(x \geq 0, y \geq 0)$-quadrant,
the admissible region of $x,y$-integration $(-1 \leq g(x,y) \leq +1)$ is an infinite strip bounded by two functions:
\begin{equation}
y^{u} (x)\equiv\frac{x+8e^2+\sqrt{1+16e^2}\sqrt{x^2+4e^2}}{8e^2}\,,
\end{equation}
\begin{equation}
y^{l} (x)\equiv\frac{x+8e^2-\sqrt{1+16e^2}\sqrt{x^2+4e^2}}{8e^2}\,.
\end{equation}
We now use the Monte-Carlo Method to calculate diagram B.
In order to bound the region we restrict the arguments of the Bose-factors to the following interval:
\vspace{.3cm}
\begin {equation}
\label{5_g5}
0\leq 2\pi \lambda^{-3/2} \sqrt{x^2+4e^2}\leq 10
\end {equation}
and
\begin {equation}
\label{5_g6}
0\leq 2\pi \lambda^{-3/2} \sqrt{y^2+4e^2}\leq 10\,.
\end {equation}
These restrictions ensure that no Boltzmann tails are included in the region of radial loop integration.
The integrand of (\ref{3_g11}) is plotted in Fig.\,\ref{5_f5} below.
It is clear from Fig.\,\ref{5_f5} that the support for the $x$ and $y$ integrations is an infinite strip in the $x$-$y$ plane.
From $x = y = 150$ on suppressions are severe justifying the limits taken in (\ref{5_g5}) and (\ref{5_g6}).
For instance, by taking $x = y = 150, e=\sqrt{8}\pi$ and $\lambda = 30$, we obtain $2\pi\lambda^{-3/2}\sqrt{x^2 + 4e^2} \sim 5.77$.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.7]{3DGraphikx_gr1.eps}
\caption{The integrand in Eq.\,(\ref{3_g11}) plotted as a function of $x$ and $y$ for fixed $z_{xy} \equiv \cos\angle(\vec{x},\vec{y})$. The horizontal plane
represents the $x$-$y$ plane where the integrand (mountain-shaped object) stands on an infinite strip.}
\label{5_f5}
\end{center}
\end{figure}
The ratio of two-loop to one-loop, as computed by the Monte-Carlo method, is
represented in Fig.\,\ref{5_f6}. There are no significant deviations from the deterministic
results computed in \cite{HerbstHofmannRohrer2004,SchwarzHofmannGiacosa2007}. Both estimates
are of order $10^{-6}$ at temperatures $\lambda = 20 \pm 5$.
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=0.7]{2LoopGraphik4_gr1.eps}
\caption{The ratio of $\Delta P_B$ (2-loop) and $P_{1-loop}$ plotted for $13.87 \leq \lambda \leq 140$.}
\label{5_f6}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=0.7]{3DGraphiky_gr1.eps}
\caption{The result of $x_1$-integration and summation over a grid of cosine
values \mbox{$z_{12} \equiv \cos \angle (\vec{x}_1, \vec{x}_2)$}, \mbox{$z_{13} \equiv \cos \angle
(\vec{x}_1, \vec{x}_3)$}, and \mbox{$z_{23} \equiv \cos \angle (\vec{x}_2, \vec{x}_3)$} of the integrand in
Eq.\,(\ref{D18}) plotted as a function of $x_2$ and $x_3$, see text.}
\label{5_f7}
\end{center}
\end{figure}
Let us now compare the integrand of (\ref{3_g11}), illustrated in Fig.\,\ref{5_f5} for the
two-loop diagram B, with the integrand of (\ref{D18}), illustrated in Fig.\,\ref{5_f7} for the three-loop diagram B.

To obtain an idea about the shape of the region of radial loop integration for the
three-loop diagram B we adopt the following procedure. We use a grid of cosine values for $z_{12}$, $z_{13}$, and $z_{23}$ of 0.15 separation. This
amounts to approximatively 2300 points. For each point
we compute the modulus of the integrand as a function of $x_2$ and $x_3$; the $x_1$-integration can be performed trivially by appealing to the
$\delta$-function in the integrand.
Subsequently, a sum over all grid points
is performed, and Fig.\,\ref{5_f7} depicts the result of this summation and the $x_1$ integration. The $x_2$-$x_3$ region
of integration is the area where this result does not vanish. Fig.\,\ref{5_f7} supports the
claim in \cite{Hofmann2006} that the region of
radial loop integration in the three-loop case is compact. The fact that the upper limits of integration
for $x_2$ and $x_3$ have been set to 3 shows that this region is fully covered in the
Monte-Carlo process.

Recall that Fig.\,\ref{5_f5} shows the integrand for the two-loop diagram for a particular
choice of the cosine value $z_{xy}$ close to $-1$. The region of radial loop integration is
obviously not compact. However, the integrand is Boltzmann suppressed for large $x$ and $y$.

The remarkable contrast between the two-loop and irreducible, planar
three-loop diagrams can be seen from Fig.\,\ref{5_f5} and Fig.\,\ref{5_f7}: in agreement
with our counting of constraints in Sec.\,3.2, the region of radial loop integration is
compact in the latter while it is non-compact in the former case.
The numerical results \cite{SchwarzHofmannGiacosa2007,KavianiHofmann2007}, reviewed in Fig.\,\ref{Fig-4} - \ref{Fig-7} and Fig.\,\ref{5_f6},
show that one-loop, two-loop, and irreducible three-loop corrections are hierarchically ordered with a pronounced
suppression at increasing loop order. In particular,
comparing the modulus of the dominant irreducible three-loop correction with
the smallest two-loop correction, we infer that they are comparable at their maxima in
$\lambda$. On the other hand, comparing it with the dominant two-loop
correction, shows a minimal suppression by a factor of $\sim 10^{-3}$. The dominant irreducible
three-loop contribution is significantly dominated by Coulomb fluctuations, and the other nonvanishing
irreducible three-loop contribution is minimally suppressed by a factor $10^{-11}$ compared to
the dominant two-loop contribution. Also, recall that one three-loop diagram appears to vanish exactly.
These results provide numerical evidence for the conjecture \cite{Hofmann2006} on the termination of
expansion of thermodynamical quantities or polarization tensors into irreducible loops
(modulo resummation of reducible diagrams) to be true \footnote{As discussed in \cite{Hofmann2006}, constraints on the loop-integration variables in
nonplanar, irreducible bubble diagrams are more severe than in the here-discussed planar case.}.

\section{Summary}

In this work we have addressed the question how the universality of the quantum of action $h$ may
find an explanation in terms of the Euclidean action of nonpropagating, selfdual gauge-field
configurations of an SU(2) Yang-Mills theory and, after interpreting these configurations as the inducers of
local vertices between plane waves, how the radiative correction
of the full theory become finite. Recall that in perturbation theory, where the topologically nontrivial
sector of field configurations is ignored, ultraviolet divergences are
removed by a conditioned subtraction. We have pointed out that this pragmatism of renormalization theory,
which was proved a long time ago to be physically consistent \cite{Veltman'Hooft,LeeZinnJustin}, is justified by solely a small window
in scale parameter, associated with the just-not-resolved, relevant topological sector of the theory
(calorons/anticalorons of topological charge modulus unity),
inducing local vertices between plane waves. We also have discussed
how the coupling of photons to electric charges is related to the dual of the effective gauge coupling in the
deconfining phase. Finally, to illustrate the workings of effective loops
we have made explicit certain technicalities of the effective pressure expansion in deconfining SU(2) Yang-Mills thermodynamics. Numerically, the
hierarchical suppression of
increasing loop orders supports the conjecture of a termination at a finite order of the
expansion into irreducible diagrams \cite{Hofmann2006}.


\section{The four vertex theorem}

The classic 4-vertex theorem states that {\it the curvature of a smooth closed convex planar curve has at least four critical points}, see Figure \ref{ellipse} for an illustration.

\begin{figure}[hbtp]
\centering
\includegraphics[height=2in]{ellipse}
\caption{An ellipse and its evolute, the envelope of its normals or, equivalently, the locus of the centers of curvature. The cusps of the evolute correspond to the vertices of the curve.}
\label{ellipse}
\end{figure}

Since its discovery by  S. Mukhopadhyaya in 1909, this theorem has generated a large literature, comprising various generalizations and variants of this result; see  \cite{DGPV,GTT,OT3,Pak} for a sampler.

One approach to the proof of the 4-vertex theorem is based on the following observation: if a $2\pi$-periodic function $f(x)$ is $L^2$-orthogonal to the first harmonics, that is, to the functions $1, \sin x, \cos x$, then $f(x)$ must have at least four sign changes over the period.

The proof is simple: since $\int_0^{2\pi} f(x) dx =0$, the function $f(x)$ must change sign. If there are only two sign changes, one can find a linear combination $g(x)=c+a\cos x + b \sin x$ that changes sign at the same points as $f(x)$. Since the first harmonic $g(x)$ cannot have more than two sign changes, $f(x) g(x)$ has a constant sign, and $\int_0^{2\pi} f(x) g(x) dx \neq 0$, a contradiction. Discrete versions of this argument are in the hearts of our proofs presented below.

(In the 4-vertex theorem, one takes $f(x)=p'(x)+p'''(x)$, where $p(x)$ is the support function of the curve; then $p(x)+p''(x)$ is the curvature radius.)

The above observation is a particular case of the Sturm-Hurwitz theorem: {\it the number of zeros of a  periodic function is not less than the number of zeros of its first non-trivial harmonic}, see \cite{OT3} for five proofs and applications of this remarkable result.

\begin{figure}[hbtp]
\centering
\includegraphics[height=2in]{Mukhopadhyaya}\quad
\includegraphics[height=2in]{Sturm}\quad
\includegraphics[height=2in]{Hurwitz}
\caption{Syamadas Mukhopadhyaya, Jacques Charles Fran\c{c}ois Sturm, and Adolf Hurwitz.}
\label{MSH}
\end{figure}

\section{Frieze patterns}

A frieze pattern is an array of numbers consisting  of finitely many bi-infinite rows; each next row is offset half-step from the previous one. The top two rows consist of 0s and of 1s, respectively, the bottom two rows are the row of 1s and 0s as well, and every elementary diamond
$$
 \begin{matrix}
 &N&
 \\
W&&E
 \\
&S&
\end{matrix}
$$
satisfies the unimodular relation $EW-NS=1$. The number of non-trivial rows is called the width of a frieze pattern. Denote the width by $w$ and set $n=w+3$.

For example,  a general frieze pattern with $w=2, n=5$ looks like this:
$$
 \begin{array}{ccccccccccc}
\cdots&&1&& 1&&1&&\cdots
 \\[4pt]
&x_1&&\frac{x_2+1}{x_1}&&\frac{x_1+1}{x_2}&&x_2&&
 \\[4pt]
\cdots&&x_2&&\frac{x_1+x_2+1}{x_1x_2}&&x_1&&\cdots
 \\[4pt]
&1&&1&&1&&1&&
\end{array}
$$
where the rows of 0s are omitted. These formulas appeared in the paper by Gauss ``Pentagramma Mirificum", published posthumously; Gauss calculated  geometric quantities characterizing spherical self-polar pentagons, see Figure \ref{miri}. See also A. Cayley's paper  \cite{Cay}. (According to Coxeter \cite{Cox} -- the very paper where frieze patterns were introduced -- the story goes further back, to N. Torporley, who in 1602 investigated the five ``parts" of a right-angled spherical triangle, anticipating by a dozen years the rule of J. Napier in spherical trigonometry.)

\begin{figure}[hbtp]
\centering
\includegraphics[height=2.5in]{mirificum1}
\caption{Pentagramma mirificum of Carl Friedrich Gauss.}
\label{miri}
\end{figure}

And here is a frieze pattern of width four whose entries are natural numbers:
$$
 \begin{array}{ccccccccccccccccccccccc}
&&1&&1&&1&&1&&1&&1&&1&&1
 \\[4pt]
&1&&3&&2&&2&&1&&4&&2&&1&
 \\[4pt]
&&2&&5&&3&&1&&3&&7&&1&&2&
 \\[4pt]
&1&&3&&7&&1&&2&&5&&3&&1&
\\[4pt]
&&1&&4&&2&&1&&3&&2&&2&&1&
\\[4pt]
&1&&1&&1&&1&&1&&1&&1&&1
\end{array}
$$
The very existence of such frieze patterns is surprising: the unimodular rule $EW-NS=1$ does not agree easily with the property of being a positive integer!

The  frieze patters consisting of positive integers were classified by Conway and Coxeter \cite{CoCo}: they are in 1-1 correspondence with the triangulations of a convex $n$-gons by diagonals, and there are
$\frac{(2(w+1))!}{(w+1)!(w+2)!}$ (Catalan number) of them;  see \cite{Hen} for an exposition of this beautiful theorem. For example, the above frieze pattern corresponds to the triangulation in Figure \ref{heptagon}.

\begin{figure}[hbtp]
\centering
\includegraphics[height=2in]{heptagon}
\caption{A triangulation of a heptagon: the labels are the number of the triangles adjacent to each vertex. These numbers comprise the first row of the frieze pattern.}
\label{heptagon}
\end{figure}

For a while, frieze patterns remained a relatively esoteric subject, but recently they have attracted much attention due of their significance in algebraic combinatorics and the theory of cluster algebras. I recommend a  comprehensive contemporary survey of this subject \cite{Mor}.

\begin{figure}[hbtp]
\centering
\includegraphics[height=1.5in]{Conway}\quad
\includegraphics[height=1.5in]{Coxeter}
\caption{John Horton Conway and Harold Scott MacDonald Coxeter.}
\label{ConCox}
\end{figure}

Let us summarize the basic properties of frieze patterns relevant to this article. Denote by $a_i$  the entries of the first non-trivial row.
\begin{enumerate}
\item The NE diagonals of a frieze pattern satisfy the 2nd order linear recurrence (discrete Hill's equation)
$$V_{i+1} = a_i V_i - V_{i-1}$$
 with $n$-periodic coefficients whose all solutions are antiperiodic, i.e., $V_{i+n}=-V_i$ for all $i$:
 $$
\begin{array}{ccccccccccc}
0\ \ &&0&&0&&0\\
&1&&1&&1\\
&&a_1&&a_{2}&&a_{3}\\
&&&a_1a_2-1&&a_2a_3-1\\
&&\cdots&&a_1a_2a_3-a_1-a_3&&\cdots\\
\end{array}
$$
\item The solutions of the discrete Hill's equation can be thought of as  polygonal lines $\ldots,V_1, V_2, \ldots$ in ${\mathbb R}^2$, with $\det (V_i,V_{i+1})=1$ and  $V_{i+n}=-V_i$. Such polygonal line is well defined up to $\operatorname{SL}(2,{\mathbb R})$-action. The projections of the vectors $V_i$ to ${\mathbb {RP}}^1$ form an $n$-gon therein, well-defined up to a M\"obius transformation. For odd $n$, this correspondence between frieze patterns of width $n-3$ and projective equivalence classes of $n$-gons in the projective line is 1-1.
\item Label the entries as follows:
$$
\begin{array}{ccc}
&v_{i,j}&\\
v_{i,j-1}&&v_{i+1,j}\\
&v_{i+1,j-1}&
\end{array}
$$
with $a_i=v_{i-1,i+1}$.
Then one has $v_{i,j}=\det(V_i,V_j)$, explaining the glide reflection symmetry of the entries: $v_{i,j} = v_{j,i+n}$

The Conway-Coxeter article \cite{CoCo} starts with a description of the seven ornamental frieze patterns  where the glide reflection symmetry is represented by
$
{\bf \large
\ldots b\quad p\quad b\quad p\quad b\quad p\ldots
}
$
and described as ``the relation between successive footprints when one walks along a straight path covered with snow". In Conway's nomenclature, this ornamental frieze pattern is called ``step", see Figure \ref{step}.

\begin{figure}[hbtp]
\centering
\includegraphics[height=.7in]{step}
\caption{An ornamental frieze pattern with the glide reflection symmetry.}
\label{step}
\end{figure}

\item The entries of a frieze pattern are given by the 3-diagonal determinants
$$
\det\left|\begin{array}{cccccc}
a_{j}&1&&&\\\
1&a_{j+1}&1&&\\
&\ddots&\ddots&\ddots&\\
&&1&a_{i-1}&1\\
&&&1&a_{i}
\end{array}\right|,
$$
the continuants (called so because of their relation with continued fractions; see \cite{CO} for an intriguing history of this name).
\end{enumerate}

\section{A problem, a theorem, and a counter-example}

I shall be concerned with frieze patterns whose entries are positive real numbers. Given two such frieze patterns of the same width $w$,  choose a row and consider the $n$-periodic sequence of the differences of  the respective entries of the two friezes. I am interested in the number of sign changes in this sequence over the period.

More precisely, let $1\le k \le [w/2]$ be the number of a row (we don't need to go beyond $[w/2]$ due to the glide symmetry), and let
$v_{i,i+k+1}$ and $u_{i,i+k+1}$ be the entries of $k$th rows of the two frieze patterns. I am interested in the sign changes of $v_{i,i+k+1}-u_{i,i+k+1}$ as $i$ increases by 1 (not excluding the case when either of these differences vanishes).
\medskip

\noindent {\bf Problem 1.}
{\it
For which $k$ must the cyclic sequence $v_{i,i+k+1}-u_{i,i+k+1}$ have at least four sign changes?
}
\medskip

As a partial answer, one has
\medskip

\noindent {\bf Theorem.}
{\it
Four sign changes must occur for $k=1$ and for $k=2$.  In addition, for every $k$, four sign changes must occur in the
infinitesimal version of the problem.
}
\medskip

Let me explain the last statement.

Consider a frieze pattern whose first row is constant: $a_i=2x$ for all $i$. Then each next row is constant as well, and their entries, denoted by $U_k(x)$, satisfy $U_{k+1} = 2x U_k (x)- U_{k-1}(x)$ with $U_0(x)=1, U_1(x)=2x$. That is, $U_k(x)$ is the Chebyshev polynomial of the second kind:
$$
U_k(\cos \alpha) = \frac{\sin(k+1)\alpha}{\sin \alpha}.
$$
For this constant frieze pattern to have width $n-3$, set $\alpha=\pi/n$.

For the infinitesimal version of Problem 1,  take this constant frieze pattern and its infinitesimal deformation in the class of frieze patterns.

Originally, I hoped that Problem 1 had an affirmative answer for all values of $k$. However, this conjecture was over-optimistic. The following counter-example is provided by Michael Cuntz; in this example, $w=5$ (the smallest possible not to contradict Theorem), all entries are positive rational numbers, and the differences of the entries of  the third row are all positive (this row is 4-periodic due to the glide symmetry). I present only the first lines of the two frieze patterns; these are 8-periodic sequences:
$$
\left(2,\ 2,\ 4,\ 2,\ 3,\ \frac{18}{41},\ 41, \frac{30}{41}\right) \ \ {\rm and}\ \
\left( 5,\ \frac{21}{97},\ 194,\ \frac{36}{97},\ 3,\ 5,\ 1,\ 5 \right).
$$
It still may be  possible that the bold conjecture holds for Conway-Coxeter frieze patterns that consist of positive integers.

\section{Proofs}

\paragraph{Case $k=1$.}
 Let $a_i$ and $b_i$ be the entries of the first rows of the two frieze patterns. Consider the respective discrete Hill's equations
$$
V_{i+1} = a_i V_i - V_{i-1},\  U_{i+1} = b_i U_i - U_{i-1}.
$$
Let $U_i$ and $V_i$ be some solutions.I claim that the sequence $a_i-b_i$ is $\ell_2$-orthogonal to $U_i V_i$:
\begin{equation} \label{orth}
\sum _1^n (a_i - b_i) U_i V_i = 0.
\end{equation}
Indeed,
$$
\sum _1^n (a_i - b_i) U_i V_i = \sum _1^n [U_{i+1} + U_{i-1}] V_i - U_i [V_{i+1} + V_{i-1}] = 0,
$$
due to antiperiodicity.

Note that the space of solutions of a discrete Hill equation is 2-dimensional, and that its solutions are non-oscillating  in the sense that they change sign only once over the period (since the entries of the frieze pattern are positive).

Assume that $a_i - b_i$ does not change sign at all. Choose the initial conditions for solutions $U_{i}$ and $V_i$ as follows:
$$
U_1=-1,U_2=1,V_1=-1,V_2=1.
$$
That is, both solution change sign from $i=1$ to $i=2$, and then, due to the non-oscillating property, there are no other sign changes. Hence $U_i V_i >0$ for all $i$, contradicting (\ref{orth}).

Likewise, if $a_i - b_i$ changes sign only twice, from $i_1$ to $i_1+1$, and from $i_2$ to $i_2+1$, choose the initial conditions for solutions $U_i$ and $V_i$ as follows:
$$
U_{i_1}=-1,U_{i_1+1}=1,V_{i_2}=-1,V_{i_2+1}=1.
$$
Then $(a_i - b_i) U_i V_i$ has a constant sign, again contradicting  (\ref{orth}).
$\Box$\bigskip

This result, along with its proof, is a discrete version of the following theorem from \cite{OT1} concerning Hill's equations $\varphi''(x) = k(x) \varphi(x)$ whose solutions are $\pi$-antiperiodic (and hence the potential $k(x)$ is $\pi$-periodic) and disconjugate, meaning that every solution changes sign only once on the period $[0,\pi)$. The claim is that, {\it given two such equations, the function $k_1(x) - k_2(x)$ has at least four zeroes on $[0,\pi)$}.

This theorem is equivalent to the beautiful theorem of E. Ghys: {\it the Schwartzian derivative of a diffeomorphism of ${\mathbb {RP}}^1$ has at least four distinct zeroes}, see \cite{OT3} for the relation of the Schwartzian derivative with the Hill equation, and an explanation why zeroes of the Schwartzian derivative are the vertices of a curve in  Lorentzian geometry.

\paragraph{Case $k=2$.}
As I mentioned, to a frieze pattern there corresponds an $n$-gon in ${\mathbb {RP}}^1$. The entries of the second row of the frieze pattern are the cross-ratios of the consecutive quadruples of the vertices of this $n$-gon, where cross-ratio is defined as
$$
[a,b,c,d]_1 = \frac{(d-a)(c-b)}{(d-c)(b-a)},
$$
see \cite{Mor}.

On the other hand, one of the results in \cite{OT2}, another discretization of Ghys's theorem, states that, given two cyclically ordered  $n$-tuples of points $x_i$ and $y_i$ in ${\mathbb {RP}}^1$, the difference of the cross-ratios $[x_i,x_{i+1},x_{i+2},x_{i+3}]_2  - [y_i,y_{i+1},y_{i+2},y_{i+3}]_2$ changes sign at least four times; here the cross-ratio is defined by
$$
[a,b,c,d]_2 = \frac{(d-b)(c-a)}{(d-c)(b-a)}.
$$
To complete the proof, observe that $[a,b,c,d]_2 - [a,b,c,d]_1 =1$.
$\Box$\bigskip

\paragraph{Infinitesimal version, $k$ arbitrary.}
Consider the polygonal line
$$
V_i = \frac{1}{\sqrt{\sin \frac{\pi}{n}}} \left(\cos \frac{i\pi}{n}, \sin \frac{i\pi}{n}  \right),
$$
so that $[V_i,V_{i+1}]=1$ and $V_{i+n}=-V_i$ hold.

Let
$$
W_i = V_i + {\varepsilon} E_i,\ [W_i,W_{i+1}]=1
$$
be an infinitesimal deformation of this polygon $V_i$. I assume in our calculations that ${\varepsilon}^2=0$.

Let
\begin{equation} \label{EtoV}
E_i=p_iV_i+\bar p_i V_{i+1} = q_i V_i + \bar q_i V_{i-1}.
\end{equation}
We shall express the $n$-periodic coefficients $p_i,\bar p_i, \bar q_i$ via the coefficients $q_i$, that solely determine the deformation.

To do so, use the fact that $V_{i+1}=cV_i-V_{i-1}$ with $c= 2 \cos (\pi/{n})$.
This linear relation must be equivalent to the second equality in (\ref{EtoV}), hence
$$
q_i-p_i=c \bar p_i,\ \bar p_i = - \bar q_i.
$$
We also have $[W_i,W_{i+1}]=1$,  implying $[V_i,E_{i+1}]+[E_i,V_{i+1}]=0$ and, using (\ref{EtoV}),
$p_i=-q_{i+1}$. Thus
\begin{equation} \label{viaq}
p_i = -q_{i+1},\ \bar p_i = \frac{1}{c} (q_i+q_{i+1}),\ \bar q_i = -\frac{1}{c} (q_i+q_{i+1}).
\end{equation}

Now fix $k$ and consider the deformation of  $(k-1)$st row of the frieze pattern:
$$
[W_i,W_{i+k}]=[V_i,V_{i+k}] + {\varepsilon} ([V_i,E_{i+k}]+[E_i,V_{i+k}]).
$$
Using (\ref{EtoV}) and (\ref{viaq}), one finds
\begin{equation*}
\begin{split}
[V_i,E_{i+k}]+[E_i,V_{i+k}] = (q_{i+k}-q_{i+1}) \left([V_i,V_{i+k}] - \frac{1}{c} [V_i,V_{i+k-1}]\right)&\\
 - \frac{1}{c} (q_{i+k+1}-q_i) [V_i,V_{i+k-1}]&\\
= \frac{1}{\sin \frac{2\pi}{n}} \left[ (q_{i+k}-q_{i+1}) \sin \frac{\pi (k+1)}{n}  -  (q_{i+k+1}-q_i) \sin \frac{\pi (k-1)}{n}  \right]&.
\end{split}
\end{equation*}
We want to show that the sequence
\begin{equation} \label{test}
c_i := (q_{i+k}-q_{i+1}) \sin \frac{\pi (k+1)}{n}  -  (q_{i+k+1}-q_i) \sin \frac{\pi (k-1)}{n}
\end{equation}
must change sign at least four times.

First, observe that $c_i$ is $\ell_2$-orthogonal to the constant sequence $(1,\ldots,1)$, that is, $\sum_{i=1}^n  c_i =0$; hence $c_i$ must have sign changes.

Next, I claim that $c_i$ is $\ell_2$-orthogonal to the  sequence $\sin (2\pi i/n)$.
Indeed
\begin{equation*}
\begin{split}
\sum_{i=1}^{n} c_i \sin \frac{2\pi i}{n} = \sum_{i=1}^{n} &q_i \sin \frac{\pi (k+1)}{n}
\left( \sin \frac{2\pi (i-k)}{n} - \sin \frac{2\pi (i-1)}{n} \right)\\
- &q_i  \sin \frac{\pi (k-1)}{n} \left( \sin \frac{2\pi (i-k-1)}{n}  + \sin \frac{2\pi i}{n} \right).
\end{split}
\end{equation*}
Hence twice the coefficient of $q_i$ on the right hand side equals
\begin{equation*}
\begin{split}
&\sin \frac{\pi (k+1)}{n} \sin \frac{\pi (1-k)}{n} \cos \frac{\pi (2i-k-1)}{n}\\
&+ \sin \frac{\pi (k-1)}{n} \sin \frac{\pi (1+k)}{n} \cos \frac{\pi (2i-k-1)}{n} =0,
\end{split}
\end{equation*}
as needed.

Similarly, $c_i$ is $\ell_2$-orthogonal to the  sequence $\cos (2\pi i/n)$.

Finally, if $c_i$ changes sign only twice, one can find a linear combination
$$
c+ a \sin \frac{2\pi i}{n} + b \cos \frac{2\pi i}{n},
$$
a discrete first harmonic, that changes sign at the same positions as $c_i$. This ``first harmonic" has no other sign changes, so its signs coincide with those of $c_i$. But it is also orthogonal to $c_i$, a contradiction.
$\Box$\bigskip

\section{Back to four vertices, and another problem}
Perhaps the oldest result in the spirit of  the four vertex-like theorem is the Legendre-Cauchy Lemma (which is about 100 years older than the theorem of Mukhopadhyaya):
{\it if two convex polygons in the plane have equal respective side length, then the cyclic sequence of the differences of their respective angles has at least four sign changes}.

A  version of this lemma in spherical geometry is the main ingredient of the proof of the Cauchy rigidity theorem ({\it convex polytopes with congruent corresponding faces are congruent to each other}); interestingly,  its original proof contained an error that remained unnoticed for nearly a century, see, e.g., chapters 22 and 26 of \cite{Pak}.

The values of the angles in the formulation of the Legendre-Cauchy Lemma can be replaces by the lengths of the short, skip-a-vertex, diagonals of the respective polygons: with fixed side lengths, the angles depend monotonically on these diagonals.

In particular, one may assume that the polygons are equilateral, e.g., each side has unit length. In this formulation, the Legendre-Cauchy Lemma becomes an analog of the $k=1$ case of  Theorem above, with the determinants $a_i=\det (V_{i-1},V_{i+1})$  replaced by the lengths $|V_{i+1}-V_{i-1}|$. This prompts to ask another question.
\medskip

\noindent {\bf Problem 2.}
{\it
Given two equilateral convex $n$-gons, for which $k$ must the cyclic sequence $|V_{i+k}-V_{i-1}|$ have at least four sign changes?
}

\bigskip
{\bf Acknowledgements}. It is a pleasure to acknowledge stimulating discussions with S. Morier-Genoud, V. Ovsienko, I. Pak, and R. Schwartz.
Many thanks to M. Cuntz for providing his (counter)-examples. This work was supported by NSF grant DMS-1510055.