Analytical properties of the quark polarization operator in an external self-dual field

Abstract

It is shown that in the lowest orders in α s, light quark polarization operators in an arbitrary multi-instanton external field have the asymptotic form Π( q 2) ∼ Π (0) + C/ q 2 + exponential terms and the terms ∼1/ q 4, ∼1/ q 6, are absent. We use the operator expansion language that corresponds to the vanishing of the contributions of higher-dimension operators to the polarization operator in the lowest order in α s in a self-dual external field. It is shown that in a self-dual field quark and gluon vacuum condensate densities are strictly connected: m〈 ψψ〉 0=− 1 8π 〈α sG a μνG a μν〉 0. This relation is strongly violated in experiment, displaying the inapplicability of the saddle approximation in calculating continual integrals. The abelian case is also discussed. The physical pictures in the abelian and non-abelian cases are very similar. It is shown, in particular, that in the constant electromagnetic self-dual field, zero-mode solutions are present.