Anomalous decay $f_1(1285)\to {\pi }^{+}{\pi }^{-}\gamma$ in the Nambu - Jona-Lasinio model

Using the Nambu–Jona-Lasinio model with the $U(2)\times U(2)$ chiral symmetric effective four-quark interactions, we derive the amplitude of the radiative decay $f_1(1285)\to {\pi }^{+}{\pi }^{-}\gamma$, find the decay width $\Gamma (f_1\to\pi^+\pi^-\gamma)=346,\mbox{keV}$ and obtain the spectral di-pion effective mass distribution. It is shown that in contrast to the majority of theoretical estimates (which consider the $a_1(1260)$ meson exchange as the dominant one), the most relevant contribution to this process comes out from the ${\rho }^0$-resonance exchange related with the triangle $f_1{\rho }^0\gamma$ anomaly. The spectral function is obtained to be confronted with the future empirical data.