$\eta^{(\prime)}\to\pi^+\pi^-l^+l^-$ decays in the NJL model

Abstract

The branching ratios of dilepton anomalous decays $\eta^{(\prime)}\to\pi^+\pi^-l^+l^-$, where $l=e,\mu$, are calculated in the framework of the Nambu -- Jona-Lasinio model. Particular attention is paid to studying the hadronic perturbative part of the full decay amplitudes, parameterized by two low-energy constants $\delta^{(\prime)}$ and $\alpha^{(\prime)}$. It is shown that due to the mixing of $0^{-+}$--$1^{++}$ $(J^{PC})$ states, the NJL model does not allow one to calculate the low-energy parameter $\delta^{(\prime)}$, which, however, can be fixed by the experimental data on the $\eta^{(\prime)}\to \pi^+\pi^-\gamma$ decay width. We then estimate another low-energy parameter $\alpha^{(\prime)}$. The impact of different schemes for describing $\eta$-$\eta^\prime$ mixing on the value of $\delta^{(\prime)}$ and $\alpha^{(\prime)}$ is discussed. It is shown that the model predictions are in full accordance with the available experimental data.