Axial-Vector Current Consisting of Pseudoscalar Octet

Abstract

The boson part of the axial-vector current which satisfies the $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(3\right)\mathrm{}$ commutator algebra is constructed in terms of the pseudoscalar fields alone. This axial-vector current consists of linear and triple terms in the pseudoscalar fields, all of which are linear in their space-time derivatives. It is shown that there is no octet of such a current which one can construct in terms of the pseudoscalar octet alone. In fact, no such octet satisfies even the $\mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$ portion of the algebra. However, one can construct an octet of the axial-vector current in terms of the pseudoscalar octet and a pseudoscalar singlet. This axial-vector current satisfies the $U\left(3\right)\mathrm{}\times U\left(3\right)\mathrm{}$ commutator algebra and is the only current of this kind. Even when the pseudoscalar-singlet part of the current is dropped, this axial-vector current can maintain the $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(3\right)\mathrm{}$ algebra or the $\mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$ algebra in two limits of possible interest. The most general axial-vector current of this type which satisfies the $\mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$ algebra is also constructed in terms of the members of the pseudoscalar octet.