Analytic and numerical study of the free energy in gauge theory

TL;DR

This paper derives bounds and asymptotic behaviors of the free energy in SU(N) gauge theory, revealing that the gluon propagator vanishes at zero momentum under certain conditions, with implications for lattice data and gauge response.

Contribution

It provides exact bounds, asymptotic expressions, and numerical evaluations of the free energy and gluon propagator in SU(N) gauge theory, highlighting a jump in propagator behavior at zero source and momentum.

Findings

Proves the inequality int_0^inf dh D(k,h)<=√2 k, implying D(k,h)→0 as k→0 for h>0.

Lattice data suggest a finite D(k,0) at zero momentum, indicating a discontinuity at h=0.

Numerical evaluations support the theoretical bounds and behaviors derived.

Abstract

We derive some exact bounds on the free energy W(J) in an SU(N) gauge theory, where J_mu^b is a source for the gluon field A_mu^b in the minimal Landau gauge, and W(J) is the generating functional of connected correlators, exp W(J) = <exp(J, A)>. We also provide asymptotic expressions for the free energy W(J) at large J and for the quantum effective action Gamma(A) at large A. We specialize to a source J(x)=h cos(kx) of definite momentum k and source strength h, and study the gluon propagator D(k,h) in the presence of this source. Among other relations, we prove int_0^inf dh D(k,h)<=2^1/2 k, which implies lim_(k->0) D(k,h) = 0, for all positive h>0. Thus the system does not respond to a static color probe, no matter how strong. Recent lattice data in minimal Landau gauge in d =3 and 4 dimensions at h=0 indicate that the gluon propagator in the minimum Landau gauge is finite, lim_(k->0) D(k,0)>0. Thus these lattice data imply a jump in the value of D(k,h) at h=0 and k=0, and the value of D(k,h) at this point depends on the order of limits. We also present numerical evaluations of the free energy W(k,h) and the gluon propagator D(k,h) for the case of SU(2) Yang-Mills theory in various dimensions which support all of these findings.