QED at NNLO and beyond for precision experiments

Abstract: Low-energy experiments allow for some of the most precise measurements in particle physics, such as $g-2$. To make the most of these experiments, theory needs to match the experimental precision. Over the last decade, this meant that even in QED next-to-next-to-leading order calculations (or even more in some cases) became necessary. McMule (Monte Carlo for MUons and other LEptons) is a framework that we have developed to obtain NNLO predictions for a number of processes, such as $e\mu \to e\mu$, $ee\to ee$, and $\mu\to e\nu\bar\nu$. I will discuss some of the challenges faced when dealing with QED corrections and some possible solutions we have implemented in McMule, namely the subtraction scheme FKS$^\ell$, massification, and next-to-soft stabilisation. I will also demonstrate how to calculate the three-loop massification constant that will be required at N$^3$LO.