Axial-Vector Current Consisting of Pseudoscalar Octet. II

Abstract

Some critical discussions are presented on the set of vector and axial-vector currents which were constructed previously in terms of the (spinor) baryons and pseudoscalar bosons alone, and whose time components satisfy the $\mathrm{SU}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(3)$ algebra upon equal-time commutation. The boson parts of the above axial-vector currents contain triple products of pseudoscalar fields and, therefore, contain obvious divergences of the self-energy type. It is shown that these divergences can be made to cancel among themselves so that the octet currents are free from such divergences, if all the pseudoscalar bosons have the same bare mass. It is furthermore shown that, if this condition for the bare mass is satisfied, the leptonic decays of the pion and kaon can be understood simultaneously when the Cabibbo angle satisfies $\frac{tan\ensuremath{\theta}\ensuremath{\lesssim}{m}_{\ensuremath{\pi}}}{{m}_{k}}$. Another novel feature of the above currents is that the equal-time commutators between the time components of the axial-vector currents and the space components of all the currents contain, in addition to the Schwinger terms, noncovariant terms which are proportional to the space $\ensuremath{\delta}$ function. These noncovariant terms affect virtually all the applications of current commutators to the electromagnetic and weak matrix elements. In particular, the Kroll-Ruderman limit of pion photoproduction on nucleons is generally violated by such noncovariant terms, implying that the pion fields cannot, in general, be proportional (as operators) to the divergences of the axial-vector currents in a field theory underlying the currents under discussion. However, it is pointed out that there seems to be a specific dynamical mechanism which suppresses the contribution of the above noncovariant terms to the Kroll-Ruderman limit. Thus, the usefulness of the current-commutator approach in pion photoproduction is not expected to be vitiated by such noncovariant terms.