# Electromagnetic decays of the neutral pion

## TL;DR

This paper calculates electromagnetic decay widths of the neutral pion using a non-perturbative QCD approach, providing results consistent with previous theories and experiments, and explores a numerical method for complex integrals.

## Contribution

It introduces a symmetry-preserving Dyson-Schwinger equation approach to compute pion decay widths and employs a novel numerical contour deformation method for rare decay calculations.

## Key findings

- Results agree with previous theoretical and experimental data.
- The rare decay calculation aligns with dispersive methods but shows a 2 sigma discrepancy with experiment.
- The numerical method can be generalized to other integrals with known analytic structure.

## Abstract

We complement studies of the neutral pion transition form factor pi^0 --> gamma^(*) gamma^(*) with calculations for the electromagnetic decay widths of the processes pi^0 --> e^+ e^-, pi^0 --> e^+ e^- gamma and pi^0 --> e^+ e^- e^+ e^-. Their common feature is that the singly- or doubly-virtual transition form factor serves as a vital input that is tested in the non-perturbative 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 sigma discrepancy between theory and experiment.