Weak decay constants of the pseudoscalar mesons from lattice QCD+QED

With increasing requirements for greater precision, it becomes essential to describe the effects of isospin breaking induced by both quark masses and electro-magnetic effects. In this work we have performed a lattice analysis of the weak decay constants of the pseudoscalar mesons including such isospin breaking effects, with particular consideration being given to the state mixing of the $\pi^0$, $\eta$ and $\eta^\prime$. We also detail extensions to the non-perturbative RI$^\prime$-MOM renormalization scheme for application to non-degenerate flavour-neutral operators which are permitted to mix, and present initial results. Using flavour-breaking expansions in terms of quark masses and charges we reach decay constant determinations at physical quark masses for all but the $\eta^\prime$, demonstrating in principle how precision determinations of all pseudoscalar decay constants could be reached in lattice QCD with QED and strong isospin-breaking.