Conundrum for the free energy of a holonomous gluonic plasma at cubic order

TL;DR

This paper investigates the cubic order free energy in a gluonic plasma with nonzero holonomy, revealing a universal gauge invariance constraint that nullifies charged gluon contributions when holonomy is present.

Contribution

It demonstrates that the charged gluon contribution at order g^3 vanishes in the presence of nonzero holonomy due to gauge invariance, a novel insight.

Findings

Charged gluon contributions at order g^3 are zero with nonzero holonomy.

The vanishing of these contributions is mandated by gauge invariance.

The result is universal, independent of how holonomy is generated.

Abstract

We compute the term $\sim g^3$ in the free energy for a $SU(N)$ gauge theory with nonzero holonomy at nonzero temperature. If the holonomy is generated kinematically by the introduction of gauge invariant sources coupled to Polyakov loops, the contribution of charged (off-diagonal) gluons to the free energy at order $g^3$, ${\cal F}^{\left( 3\rm{:c.g.} \right)}$, is singular: ${\cal F}^{\left( 3\rm{:c.g.} \right)} \neq 0 $ without holonomy, but ${\cal F}^{\left( 3 \rm{:c.g.} \right)}= 0$ when the holonomy is nonzero, even infinitesimally. We show that the absence of the charged gluon contribution is required by gauge invariance alone and is therefore a universal feature.