Complete High Temperature Expansions for One-Loop Finite Temperature Effects

TL;DR

The paper addresses the high-temperature behavior of one-loop finite-temperature contributions to the effective potential in odd spatial dimensions in the presence of a non-trivial Polyakov loop. It introduces exact closed-form high-$T$ expansions by proving Bessel-function identities that resum the low-temperature series, yielding explicit expressions for the bosonic and fermionic functions $V_B(\theta)$ and $V_F(\theta)$ and preserving $\theta$-periodicity. A key result is the explicit $d=3$ formulation, along with representations for various gauge-group representations, which connect to free energies and pressures in gauge theories. The work enhances practical finite-temperature calculations in QCD-like theories and informs deconfinement studies and related vacuum models by providing efficient, accurate, and periodic high-$T$ expressions.

Abstract

We develop exact, simple closed form expressions for partition functions associated with relativistic bosons and fermions in odd spatial dimensions. These expressions, valid at high temperature, include the effects of a non-trivial Polyakov loop and generalize well-known high temperature expansions. The key technical point is the proof of a set of Bessel function identities which resum low temperature expansions into high temperature expansions. The complete expressions for these partition functions can be used to obtain one-loop finite temperature contributions to effective potentials, and thus free energies and pressures.