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

The branching ratios of dilepton anomalous decays ${\eta }^{(\mathrm{\prime })}\to {\pi }^{+}{\pi }^{-}{l}^{+}{l}^{-}$, 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 ${\delta }^{(\mathrm{\prime })}$ and ${\alpha }^{(\mathrm{\prime })}$. 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 ${\delta }^{(\mathrm{\prime })}$, which, however, can be fixed by the experimental data on the ${\eta }^{(\mathrm{\prime })}\to {\pi }^{+}{\pi }^{-}\gamma$ decay width. We then estimate another low-energy parameter ${\alpha }^{(\mathrm{\prime })}$. The impact of different schemes for describing $\eta$-${\eta }^{\mathrm{\prime }}$ mixing on the value of ${\delta }^{(\mathrm{\prime })}$ and ${\alpha }^{(\mathrm{\prime })}$ is discussed. It is shown that the model predictions are in full accordance with the available experimental data.