Evaluation of the chiral anomaly in gauge theories withγ5couplings

A straightforward application of various regularization schemes in perturbation theory generally leads to physically unacceptable results in the evaluation of the chiral anomaly if the gauge theory contains ${\gamma }_5$ couplings. In gauge theory the anomaly-free properties should be imposed on all the gauge vertices, regardless of whether they are vector or axial-vector, and thus the (nongauge) vector vertex of a triangle diagram could contain an anomaly if the gauge couplings involve ${\gamma }_5$. This is illustrated by the fermion number (vector) current in the Weinberg-Salam theory. It is then shown that the path-integral formalism naturally resolves those complications and gives rise to a unique gauge-invariant result. The Hermiticity of eigenvalue equations in Euclidean gauge theory plays an important role in the path-integral formalism. A path-integral treatment of gauge theory with $S$ and $P$ Higgs couplings is also described.