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

Abstract

Using a recently proposed gauge covariant diagonalization of πa 1 -mixing we show that the low energy theorem F π = ef 2 π F 3π of current algebra, relating the anomalous form factor F γ→π + π 0 π − = F 3π and the anomalous neutral pion form factor F π 0 →γγ = F π , 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 γπ qq 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.