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

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.