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

TL;DR

This paper analyzes the impact of $_1$-mixing on the anomalous $ o^-$ amplitude, showing that surface terms are fixed by the low-energy theorem and that VMD fails for this process.

Contribution

It demonstrates that $_1$-mixing does not affect certain form factors and clarifies the role of surface terms in the low-energy theorem within a covariant meson Lagrangian framework.

Findings

Surface terms are fixed by the low-energy theorem.

Form factors $F^$ and $F^{3}$ are unaffected by $_1$-mixing.

Vector meson dominance fails for $ o^-$.

Abstract

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, but the concept of vector meson dominance (VMD) fails for $\gamma\to\pi^+\pi^0\pi^-$.