A critical subject in fully differential QED calculations originates from numerical instabilities due to small fermion masses that act as regulators of collinear singularities. At next-to-next-to-leading order (NNLO) a major challenge is therefore to find a stable implementation of numerically delicate real-virtual matrix elements. In the case of Bhabha scattering this has so far prevented the development of a fixed-order Monte Carlo at NNLO accuracy. In this paper we present a new method for stabilising the real-virtual matrix element. It is based on the expansion for soft photon energies including the non-universal subleading term calculated with the method of regions. We have applied this method to Bhabha scattering to obtain a stable and efficient implementation within the McMule framework. We therefore present for the first time fully differential results for the photonic NNLO corrections to Bhabha scattering.

全微分 QED 计算中的一个关键问题源于作为共线奇点调节器的小费米子质量引起的数值不稳定性。因此，在紧邻前导顺序 (NNLO) 中，一个主要挑战是找到数值精细的实虚矩阵元素的稳定实现。在 Bhabha 散射的情况下，到目前为止，这阻止了 NNLO 精度的固定阶蒙特卡罗的发展。在本文中，我们提出了一种稳定实虚矩阵元素的新方法。它基于软光子能量的扩展，包括使用区域方法计算的非普遍次领先项。我们已将此方法应用于 Bhabha 散射，以在McMule 中获得稳定有效的实现框架。因此，我们首次提出了对 Bhabha 散射的光子 NNLO 校正的全微分结果。