Axial-Vector Current in Spinor Electrodynamics

Adler has shown that in perturbation theory for spinor electrodynamics, suitably regularized, the divergence of the unrenormalized axial-vector current contains, in addition to the expected mass term, a term of the form ${\epsilon }_{\alpha \beta \gamma \delta }{F}_{\alpha \beta }{F}_{\gamma \delta }$. Motivated by this result, we study here the renormalized axial-vector current ${j_{\mu }}^5$. We explicitly construct the essentially unique finite local ${j_{\mu }}^5$ and find it to be invariant under local gauge transformations. Taking the divergence ${\partial }_{\mu }{j_{\mu }}^5$ gives the renormalized analog of Adler's additional term. Similar "noncanonical" operator terms are seen to occur in equal-time commutators involving ${j_{\mu }}^5$. Although all matrix elements of ${\partial }_{\mu }{j_{\mu }}^5$ are finite, we find that the off-shell Green's function $〈T{\partial }_{\mu }{j_{\mu }}^5\mathrm{}\left(x\right)\mathrm{}\psi \mathrm{}\left(y\right)\mathrm{}\overline{\psi }\mathrm{}\left(z\right)\mathrm{}〉$ is divergent and, correspondingly, that equal-time commutators involving ${j_{\mu }}^5$ are in general singular. We show that Ward identities can nevertheless be given a meaning. The algebraic properties of ${j_{\mu }}^5$ are seen to be reflected in the scattering amplitudes of the theory. Renormalized integral representations of axial-vector vertices are constructed, and radiative corrections to weak interaction are discussed. We conclude with a discussion of the axial-vector currents in other spinor models.