On-mass-shell current algebra and theπ0→2γdecay and partial conservation of axial-vector current anomaly

Abstract

Using the algebra of the axial-vector current which is defined as a source for the pion field, we study the ${\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}2\ensuremath{\gamma}$ decay rate and the Adler anomaly. Our treatment is identical to the one previously applied to other interactions which are mentioned in the text. It consists of expressing the reduction formula in terms of the above current and evaluating it directly on the mass shell. In the process, we have a set of intermediate states from which we choose the one- and two-meson states. The $3\ensuremath{\pi}$ continuum is taken into account effectively by a pion isobar, ${\ensuremath{\pi}}^{\ensuremath{'}}$. Hence the ${\ensuremath{\pi}}^{0}$ amplitude is expressed in terms of the parameters of the $\ensuremath{\rho}$, $\ensuremath{\omega}$, $\ensuremath{\varphi}$, and ${\ensuremath{\pi}}^{\ensuremath{'}}$ and a coefficient $\ensuremath{\beta}$ which represents the momentum dependence of the coupling constants of these mesons at the $\ensuremath{\pi}\ensuremath{\gamma}$ vertex. A feature of our method is that it does not explicitly depend on the anomaly operator. However, by writing our current in terms of the modified PCAC (partial conservation of axial-vector current) operator, the ${\ensuremath{\pi}}^{0}$ lifetime is linked with the anomaly via these meson data and two unknown parameters, in addition to the $\ensuremath{\beta}$, which represent the ${\ensuremath{\pi}}^{\ensuremath{'}}$ contributions in the ${\ensuremath{\pi}}^{0}$ lifetime and in the chiral-symmetry-breaking terms underlying the PCAC relation. We give the most appropriate combinations of these parameters for obtaining the observed ${\ensuremath{\pi}}^{0}$ lifetime in accord with the anomaly given by either of the existing quark-theoretical models. We find the correction to the Adler anomaly, in terms of the ${\ensuremath{\pi}}^{\ensuremath{'}}$ parameters and according to the quark model we choose. Assuming that the ${\ensuremath{\pi}}^{\ensuremath{'}}$ contribution to the decay amplitude is small, our results favor the integral charge quark schemes.