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

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.

Comments: prepared for eQCD 2020 proceedings