Low-energy theorem for $gammato 3pi$: surface terms against $pi a_1$-mixing

We reconsider the contribution due to $pi a_1$-mixing to the anomalous $gammatopi^+pi^0pi^-$ amplitude from the standpoint of the low-energy theorem $F^{pi}=e f_pi^2 F^{3pi}$, which relates the electromagnetic form factor $F_{pi^0togammagamma}=F^pi$ with the form factor $F_{gammatopi^+pi^0pi^-}=F^{3pi}$ 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 $F^{3pi}$ are not affected by the $pi a_1$-mixing, but the concept of vector meson dominance (VMD) fails for $gammatopi^+pi^0pi^-$.