Exact spectral-function sum rules

Abstract

A general procedure for extracting exact spectral-function sum rules is presented. The short-distance behavior of products of vector and axial-vector currents is related to the convergence (or superconvergence) of the original first and second spectral sum rules together with a third sum rule involving only the spin-0 spectral function. The operator-product expansion is then applied to determine all (and only) those linear combinations of current propagators for which the short-distance behavior is sufficiently soft to yield superconvergent sum rules for the corresponding combinations of spectral functions. Our method is applied to determine the complete set of sum rules for a theory defined by a global chiral SU(4)×SU(4) symmetry, broken (a) explicitly by hadron (quark) masses and (b) by dynamical symmetry breaking to any subgroup containing the symmetry group of the mass matrix. Our derivation is strictly true only for asymptotically free theories, but the results are expected to apply for a range of other theories. The method is easily extended to deal with current propagators involving scalar and pseudoscalar densities (not necessarily divergences of vector or axial-vector currents)—the relevant sum rules in the context of the SU(4)×SU(4) model are derived. Finally, we compare our approach and results to those of several recent studies of the spectral-function sum rules. An appendix presents a proof that Wilson functions exhibit the full symmetries of any theory, whether or not these are spontaneously broken.