Importance of pseudoscalar–axial-vector mixing in calculation of the properties of the π, η, andη′mesons

In this work we resolve a problem that arises in the calculation of the two-photon decay of the ${\pi }^0,$ η, and ${\eta }^{\prime }$ mesons, when use is made of the Nambu–Jona-Lasinio (NJL) model. It has been found that satisfactory results for the widths are obtained if a momentum cutoff is not introduced in the evaluation of the (convergent) integrals obtained from an analysis of the triangle diagram, while only about one-half of the experimental value is obtained, if the cutoff used for the (divergent) loop integrals of the NJL model is introduced. The resolution of this problem lies in the introduction of pseudoscalar–axial-vector mixing for both the ${\pi }^0$ and the $\eta -{\eta }^{\prime }$ system. (In addition, one has singlet-octet mixing for the η and ${\eta }^{\prime },$ requiring the study of T matrices of dimension four in that case.) Since we have used a relativistic quark model that includes a (covariant) confinement model, we are able to treat the ${\eta }^{\prime }$ and η mesons on the same basis. We found that the ’t Hooft interaction does not work well in our study of $\eta -{\eta }^{\prime }$ mixing, while the parametrization of nonperturbative effects due to coupling to gluonic modes, which affects the strength of the interaction for singlet states only, yields satisfactory results. In the case of a pion, our results may be understood by the observation that the vertex $\mathrm{iP}/{\gamma }_5/\sqrt{P^2}$ yields a much larger amplitude for two-photon decay of a meson of momentum P than the vertex $i{\gamma }_5.$ Therefore, quite small mixing angles can yield a large modification of the calculated width. We have found that our results are sensitive to the form of regulator used. For example, with a sharp cutoff, ${\Gamma }_{{\eta }^{\prime }\to \gamma \gamma }$ is overestimated by about a factor of 3, while the use of a Gaussian regulator leads to a satisfactory fit to the experimental value.