Families of sum rules from current algebra

Abstract

By making use of a simple method we obtain different families of sum rules from current algebra. The method allows to understand the relation of the reference frame in which equal-time commutators are considered with the form of the sum rules obtained and dispersion relations. The different contributions to the sum rules are then analysed and it is recognized that besides the contributions which are generally considered, there are others which come from disconnected intermediate states, which play an important role. By discussing in detail theP→∞ sum rule we put in evidence the physical content of these extra terms and how they are responsible for the automatic cancellation of the singularities in the variables which, due to the locality of equal-time commutators, do not appear in the right-hand side of the sum rules. The origin and value of these terms is purely dynamical, they have nothing to do with Schwinger terms and their presence is not related to convergence of the sum rule integrals. It is shown that theassumption that they vanish (so to recover the Fubini sum rule for theP→∞ family) implies superconvergence so that it appears clearly that superconvergence is a purely dynamical assumption and is not related to current algebra or locality of equal-time commutators. The structure of other families of sum rules is also discussed and, in particular, that corresponding to the “Δ→∞” family which appears to be of particular interest.