On conservation of the axial current in massless electrodynamics

Abstract

Conservation of the axial current in massless electrodynamics is considered in connection with the existence of the so-called anomalous terms in the equation for the divergence of the axial current. It is shown that in the limit of vanishing electron mass m transitions forbidden by γ 5 invariance give a non-vanishing contribution to the imaginary part of the triangle graph. This phenomenon is analogous in some respect to the breaking of the formal consequences from γ 5 invariance for the probability of the transition between electrons with right and left helicity through the emission of a photon. As was noticed by Lee and Nauenberg, this probability does not vanish in the limit of m=0. In this respect the appearance of an ultraviolet divergence in the triangle graph is irrelevant to the breaking of γ 5 invariance. We give some examples of well defined loop graphs and tree graphs which nevertheless violate conservation of the axial current at m=0 for some particular configuration of external momenta. We consider also another limiting procedure when m is chosen to be equal to zero from the very beginning but the photon mass is chosen to be finite. It then becomes possible to construct a conserved axial current but the matrix element associated with the triangle graph proves to be singular in the photon mass. In the case of m=0 a recipe to calculate the triangle graph is given which implies the validity of the divergence equation without any anomalous term. However, the matrix element of the pseudoscalar density j 5 is singular in m in this case.