Instanton-based vacuum from the Feynman variational principle

Abstract

We investigate the possibility that the most essential non-perturbative gauge field configurations in QCD are of an instanton-anti-instanton type. Since these are not exact solutions of classical equations, we use a modification of the Feynman variational principle in order to calculate in a mathematically consistent way the contribution of given trial gauge field configurations (with small oscillations around them) to the QCD partition function. We observe an effective repulsion of instantons which removes the usual infrared problems of the instanton calculus. The repulsion leads to the stabilization of the pseudoparticle “liquid” at distances being a factor of ∼ 3 larger than their average sizes. We calculate the upper bound for the vacuum energy, the non-perturbative gluon condensate 〈 F μν 2〉, the topological susceptibility, etc. in terms of renormalization-invariant combinations. We find our numerical results reasonable from the phenomenological point of view. The variational principle permits one to improve the upper bound for the vacuum energy by varying the profile function of instantons. We find that the power fall-off of instantons is replaced at large distances by a Yukawa-type one. Therefore, a mass gap for glueballs is naturally obtained at the classical level.