# Finite size effects in the free energy density for Abelian (anti-)self-dual gluon field in $SU(3)$ gluodynamics

## TL;DR

This paper investigates finite-size effects on the free energy density of Abelian (anti-)self-dual gluon fields in SU(3) gluodynamics, emphasizing the role of gluon quasi-zero modes and their impact on the effective potential.

## Contribution

It introduces a method to account for gluon quasi-zero modes beyond one-loop approximation to accurately describe the strong-field behavior in finite regions.

## Key findings

- Conditions for the global minimum of free energy at finite field strength and size are identified.
- Effective potential behavior aligns with asymptotic freedom when quasi-zero modes are properly treated.
- Finite-size effects significantly influence the stability of gluon field configurations.

## Abstract

Finite-size effects in the free energy density for Abelian (anti-)self-dual gluon field are investigated within $SU(3)$ gluodynamics. In particular, the role of gluon quasi-zero modes is studied. The effective potential is calculated within the framework of zeta function regularization for finite spherical four-dimensional region of radius $R$ in Euclidean space-time. In order to obtain the correct strong-field behavior of the effective potential which is determined by the asymptotic freedom, the quasi-zero gluon modes have to be treated beyond one-loop approximation in line with the argumentaion of Leutwyler. Conditions for appearance of the global minimum of the free energy density at finite nonzero values of both field strength and region size are discussed.