Weak decay constants of the neutral 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 perform a lattice analysis of the weak decay constants of the neutral pseudoscalar mesons including such isospin breaking effects, with particular consideration being given to the state mixing of the ${\pi }^0$, $\eta$ and ${\eta }^{\mathrm{\prime }}$. We also detail extensions to the non-perturbative RI${}^{\mathrm{\prime }}$-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 determine the leptonic decay constants for the ${\pi }^0$ and $\eta$ mesons, demonstrating in principle how precision determinations of all neutral pseudoscalar decay constants could be reached in lattice QCD with QED and strong isospin-breaking accounted for.