Axial-vector current and dimensional regularization

Abstract

The properties of the axial-vector current are investigated using the dimensional-regularization scheme. The problem of defining an appropriate generalization of ${\ensuremath{\gamma}}_{5}$ in $n$ dimensions is discussed, and previous work is briefly reviewed. For the $\mathrm{VVA}$ triangle, in QED, we find that the dimensional scheme provides for vector current conservation, with the divergence of the axial-vector current anomalous. This is shown unambiguously without specifying the anticommuting nature of ${\ensuremath{\gamma}}_{5}$ in $n$ dimensions. If one arranges to have two species of fermions with different masses and equal but opposite couplings to the axial-vector current, the $\mathrm{VVA}$ anomaly is proportional to $n\ensuremath{-}4$, being fully canceled only at $n=4$. However, the behavior of the triangle amplitude for large external momenta is reduced by two powers, and the resulting softened triangle does not give rise to any finite (as $n\ensuremath{\rightarrow}4$) anomalies when inserted in higher-order diagrams. Finally, the appropriate generalization of ${\ensuremath{\gamma}}_{5}$ for even-parity fermion loops is shown to be totally anticommuting, and the validity of Ward identities for two-point functions is demonstrated.