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

Using a recently proposed gauge covariant diagonalization of $pi a_1$-mixing we show that the low energy theorem $F^{pi}=e f_pi^2 F^{3pi}$ of current algebra, relating the anomalous form factor $F_{gamma to pi^+pi^0pi^-} = F^{3pi}$ and the anomalous neutral pion form factor $F_{pi^0 to gammagamma}=F^pi$, 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.