Ward–Green–Takahashi identities and the axial-vector vertex

Abstract

The colour-singlet axial-vector vertex plays a pivotal role in understanding dynamical chiral symmetry breaking and numerous hadronic weak interactions, yet scant model-independent information is available. We therefore use longitudinal and transverse Ward–Green–Takahashi (WGT) identities, together with kinematic constraints, in order to ameliorate this situation and expose novel features of the axial vertex: amongst them, Ward-like identities for elements in the transverse piece of the vertex, which complement and shed new light on identities determined previously for components in its longitudinal part. Such algebraic results are verified via solutions of the Bethe–Salpeter equation for the axial vertex obtained using two materially different kernels for the relevant Dyson–Schwinger equations. The solutions also provide insights that suggest a practical Ansatz for the axial-vector vertex.