Weak decays and finite-volume QED

The precision goal on the Cabibbo-Kobayashi-Maskawa matrix elements $|V_{us}|$ and $|V_{ud}|$ requires the inclusion of isospin-breaking effects. These effects, associated to QED and the light-quark mass difference, can be included in lattice QCD+QED simulations. The long-range nature of QED, however, poses a problem for finite-volume lattice simulations that necessitates the adoption of a prescription for the definition of QED in a spacetime of finite size. In this talk we discuss the novel prescription QED$_{\textrm{r}}$, a specific choice within the class of infrared-improved prescriptions from Davoudi et al. 2019. We in particular emphasise how QED$_{\textrm{r}}$ circumvents previous bottlenecks in analytical predictions of the finite-volume dependence in hadron masses and leptonic decays. These observables are in particular highly relevant for precision determinations of Cabibbo-Kobayashi-Maskawa elements and associated unitarity tests.