Weyl consistency conditions and a local renormalisation group equation for general renormalisable field theories

Abstract

A local renormalisation group equation which realises infinitesimal Weyl rescalings of the metric and which is an extension of the usual Callan-Symanzik equation is described. In order to ensure that any local composite operators, with dimensions so that on addition to the basic lagrangian they preserve renormalisability, are well defined for arbitrarily many insertions into correlation functions the couplings are assumed to depend on x. Local operators are then defined by functional differentiation with respect to the couplings just as the energy-momentum tensor is given by functional differentiation with respect to the metric. The local renormalisation group equation contains terms depending on derivatives of the couplings as well as the curvature tensor formed from the metric, constrained by power counting. Various consistency relations arising from the commutativity of Weyl transformations are derived, extending previous one-loop results for the trace anomaly to all orders. In two dimensions the relations give an alternative derivation of the c-theorem and similar extensions are obtained in four dimensions. The equations are applied in detail to general renormalisable σ-models in two dimensions. The Curci-Paffuti relation is derived without any commitment to a particular regularisation scheme and further equations used to construct an action for the vanishing of the β-functions are also obtained. The discussion is also extended to σ-models with a boundary, as appropriate for open strings, and relations for the additional β-functions present in such models are obtained.