Gauge-covariant diagonalization of $\pi$-$a_1$ mixing and the resolution of a low energy theorem

TL;DR

This paper demonstrates that a gauge covariant diagonalization approach within the NJL model satisfies a low energy theorem relating pion form factors, resolving a longstanding issue with vector and axial-vector meson extensions.

Contribution

It introduces a gauge covariant diagonalization method that ensures the low energy theorem holds in the NJL model with vector and axial-vector mesons.

Findings

The low energy theorem is fulfilled in the NJL model with the new diagonalization.

A novel gamma-pi-qq vertex is identified, absent in conventional treatments.

The approach resolves previous inconsistencies in extending the model with vector and axial-vector mesons.

Abstract

Using a recently proposed gauge covariant diagonalization of $\pi a_1$-mixing we show that the low energy theorem $F^{\pi}=e f_\pi^2 F^{3\pi}$ of current algebra, relating the anomalous form factor $F_{\gamma \to \pi^+\pi^0\pi^-} = F^{3\pi}$ and the anomalous neutral pion form factor $F_{\pi^0 \to \gamma\gamma}=F^\pi$, is fulfilled in the framework of the Nambu-Jona-Lasinio (NJL) model, solving a long standing problem encountered in the extension including vector and axial-vector mesons. At the heart of the solution is the presence of a $\gamma \pi {\bar q} q $ vertex which is absent in the conventional treatment of diagonalization and leads to a deviation from the vector meson dominance (VMD) picture. It contributes to a gauge invariant anomalous tri-axial (AAA) vertex as a pure surface term.