The four-dimensional Yang–Mills partition function in the vicinity of the vacuum

Abstract

The partition function of four-dimensional Euclidean, non-supersymmetric textrm{SU}(2) Yang–Mills theory is calculated in the perturbative and weak coupling regime, i.e., in a small open ball about the flat connection and when the gauge coupling constant acquires a small but finite value. The computation is based on various known inequalities, valid only in four dimensions, providing two-sided estimates for the exponentiated Yang–Mills action in terms of the L^2-norm of the derivative of the gauge potential only; these estimates then give rise to Gaußian-like infinite- dimensional formal integrals involving the Laplacian and hence can be computed via zeta-function and heat kernel techniques. It then turns out that these formal integrals give a sharp value for the partition function in the aforementioned perturbative and weak coupling regime of the theory. In the resulting expression for the partition function, the original classical value of the coupling constant is shifted to a smaller one which can be interpreted as the manifestation, in this approach, of a non-trivial beta -function and asymptotic freedom in pure non-Abelian gauge theories.