Tensor decomposition of $e^+e^-\to\pi^+\pi^-\gamma$ to higher orders in the dimensional regulator

TL;DR

This paper develops a tensor decomposition and analytical evaluation of one-loop amplitudes for the process e+e- to pi+pi-gamma, advancing towards NNLO precision for radiative return studies.

Contribution

It introduces a complete four-dimensional tensor decomposition and analytical higher-order dimensional regulator evaluation for the process, enabling NNLO accuracy.

Findings

Analytical evaluation of one-loop polarized amplitudes to higher orders in dimensional regulator.

Efficient numerical framework for stable evaluation of five-point Feynman integrals.

Results suitable for implementation in Monte Carlo event generators.

Abstract

We present a first study of the scattering process $e^+ e^-\to\pi^+\pi^-\gamma$ beyond next-to-leading order, aimed at providing preliminary insights required for future NNLO predictions for radiative return processes. A complete four-dimensional tensor decomposition of the amplitude is developed, and the associated one-loop polarised amplitudes are evaluated analytically to higher orders in the dimensional regulator, as required for NNLO accuracy. The calculation is complemented by an efficient numerical framework for the evaluation of the resulting five-point Feynman integrals, enabling stable and fast evaluations across the physical production region with evaluation times of a few hundreds of milliseconds. These results are suitable for implementation in Monte Carlo event generators.