Electromagnetic interactions of mesons and the $\pi a_1$ diagonalization

It has been demonstrated that, contrary to the commonly accepted viewpoint, electromagnetic interactions cannot be consistently included in the Nambu-Jona-Lasinio model within vector meson dominance. This refers to the procedure for removing the $\pi a_1$ transition in the effective meson Lagrangian. By the example of the $a_1(1260)\to\gamma\pi^+\pi^-$ radiative decay, it has been demonstrated that the gauge invariance condition for the amplitude gives rise to an electromagnetic vertex describing the direct emission of a photon and a pion by a quark-antiquark pair. Surface terms appearing in the calculation of an anomalous triangular quark diagram, where one of the vertices is the interaction violating the vector-dominance hypothesis, play a key role in satisfying the Ward identities.