Asymptotic Limits and Sum Rules for Propagators in Quantum Chromodynamics

Abstract

In gauge field theories with asymptotic freedom, the short distance properties of Green's functions can be obtained on the basis of weak coupling perturbation expansions. Within this framework, the large momentum behavior of the structure functions for gluon, quark and ghost propagators is derived. The limits are found for general, covariant, linear gauges, and in all directions of the complex $k^2-$plane. Except for the coefficients, the functional forms of the leading asymptotic terms for the various structure functions are independent of the gauge parameter. They are determined exactly in terms of one-loop expressions (two-loop expressions in cases where one-loop terms vanish). With the exception of the Landau gauge, the asymptotic expressions for the gauge field propagator play an important rôle for the corresponding limits of quark and ghost propagators. For {\it all} gauges considered, it is the sign of the one-loop anomalous dimension coefficient of the gluon field in Landau gauge (as a fixed point of the gauge parameter) which is of considerable relevance for the asymptotics of the various propagators. The bounds obtained from the asymptotic expressions, together with the analytic properties of the structure functions, generally lead to un-subtracted dispersion representations. In special cases, for a limited number of flavors, sum rules are obtained for the discontinuities along the real axis. The sum rule for the gluon propagator is a generalization of the superconvergence relation derived previously in the Landau gauge.