Trace and dilatation anomalies in gauge theories

Abstract

The form of the anomaly in the trace of the energy-momentum tensor in a general theory of interacting fermions and non-Abelian gauge bosons is dervied. The result is shown to involve precisely those gaugevariant operatos which are known to mix with the naive trace under renormalization. The trace is shown to be soft on the mass shell if and only if the theory is at an eigenvalue of the Callan-Symanzik $\ensuremath{\beta}$ function. The dilatation anomaly in the matrix element of $\ensuremath{\theta}_{}^{\ensuremath{\lambda}}{}_{\ensuremath{\lambda}}{}^{}$ with two electromagnetic currents (to lowest order in electromagnetism, but including all orders of the strong gauge interaction) is derived and shown to be infinitely renormalized in finite orders of strong perturbation theory. This anomaly is then shown to be canonical and given precisely by the lowest-order result provided the strong interactions are summed to all orders before going to the limit of physical space-time dimensions.