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Low-energy theorem for $γ\to 3π$: surface terms against $πa_1$-mixing

We reconsider the contribution due to $πa_1$-mixing to the anomalous $γ\toπ^+π^0π^-$ amplitude from the standpoint of the low-energy theorem $F^π=e f_π^2 F^{3π}$, which relates the electromagnetic form factor $F_{π^0\toγγ}=F^π$ with the form factor $F_{γ\toπ^+π^0π^-}=F^{3π}$ both taken at vanishing momenta of mesons. Our approach is based on a recently proposed covariant diagonalization of $πa_1$-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 $F^π$ and $F^{3π}$ are not affected by the $πa_1$-mixing, but the concept of vector meson dominance (VMD) fails for $γ\toπ^+π^0π^-$.