Low-energy theorem for $\gamma \to 3\pi$: surface terms against $\pi a_1$-mixing

We reconsider the contribution due to $\pi a_1$-mixing to the anomalous $\gamma \to {\pi }^{+}{\pi }^0{\pi }^{-}$ amplitude from the standpoint of the low-energy theorem $F^{\pi }=e{f}_{\pi }^2{F}^{3\pi }$, which relates the electromagnetic form factor $F_{{\pi }^0\to \gamma \gamma }=F^{\pi }$ with the form factor $F_{\gamma \to {\pi }^{+}{\pi }^0{\pi }^{-}}=F^{3\pi }$ both taken at vanishing momenta of mesons. Our approach is based on a recently proposed covariant diagonalization of $\pi a_1$-mixing within a standard effective QCD-inspired meson Lagrangian obtained in the framework of the Nambu-Jona-Lasinio model. We show that the two surface terms appearing in the calculation of the anomalous triangle quark diagrams or AVV- and AAA-type amplitudes are uniquely fixed by this theorem. As a result, both form factors $F^{\pi }$ and $F^{3\pi }$ are not affected by the $\pi a_1$-mixing,buttheconceptofvectormesondominance(VMD)failsfor$\gamma\to\pi^+\pi^0\pi^-$.