Electromagnetic decays of the neutral pion

Abstract

We complement studies of the neutral pion transition form factor ${\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}{\ensuremath{\gamma}}^{(*)}{\ensuremath{\gamma}}^{(*)}$ with calculations for the electromagnetic decay widths of the processes ${\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}{e}^{+}{e}^{\ensuremath{-}}$, ${\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}{e}^{+}{e}^{\ensuremath{-}}\ensuremath{\gamma}$ and ${\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}{e}^{+}{e}^{\ensuremath{-}}{e}^{+}{e}^{\ensuremath{-}}$. Their common feature is that the singly or doubly virtual transition form factor serves as a vital input that is tested in the nonperturbative low-momentum region of QCD. We determine this form factor from a well-established and symmetry-preserving truncation of the Dyson-Schwinger equations. Our results for the three- and four-body decays match results of previous theoretical calculations and experimental measurements. For the rare decay we employ a numerical method to calculate the process directly by deforming integration contours, which in principle can be generalized to arbitrary integrals as long as the analytic structure of the integrands are known. Our result for the rare decay is in agreement with dispersive calculations but still leaves a $2\ensuremath{\sigma}$ discrepancy between theory and experiment.