Anomalous decay $f_1(1285)topi^+pi^-gamma$ 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 $f_1(1285) topi^+pi^-gamma$, find the decay width $Gamma (f_1topi^+pi^-gamma)=346,mbox{keV}$ 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 $f_1rho^0gamma$ anomaly. The spectral function is obtained to be confronted with the future empirical data.