Pion as a Longitudinal Axial-Vector Meson $q\bar{q}$ Bound State

The success of the Adler-Bell-Jackiw(ABJ) chiral anomaly prediction for $\pi^{0}\to \gamma\gamma$ decay rate shows that non-anomaly terms would make a negligible contribution to the decay rate, in agreement with the Sutherland-Veltman theorem. Thus the conventional $q\bar{q}$ bound-state description of the pion could not be valid since it would produce a $\pi^{0}\to \gamma\gamma$ decay amplitude not suppressed in the soft pion limit, in contradiction with the Sutherland-Veltman theorem. Therefore, if the pion is to be treated as a $q\bar{q}$ bound state, this bound state would be a longitudinal axial-vector meson. In this paper, we consider the pion to be a longitudinal axial-vector meson $q\bar{q}$ bound state with derivative coupling for the pion $q\bar{q}$ Bethe-Salpeter(BS) wave function. We shall show that this BS wave function could produce a suppressed $\pi^{0}\to \gamma\gamma$ decay amplitude in the soft pion limit, in agreement with the Sutherland-Veltman theorem. This explains the almost perfect agreement of the anomaly prediction with experiment and the suppression of the virtual one-photon exchange contribution in $\eta\to 3\pi$ decay. The Goldstone boson equivalence theorem used for longitudinal gauge bosons scattering in the electroweak standard model then identifies the longitudinal axial-vector meson $q\bar{q}$ bound state with the pion.