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

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.