We present a method to evaluate on the lattice the leading isospin breaking effects due to both the small mass difference between the up and down quarks and the QED interaction. Our proposal is applicable in principle to any $\mathrm{QCD}+\mathrm{QED}$ gauge invariant hadronic observable that can be computed on the lattice. It is based on the expansion of the path integral in powers of the small parameters $({\widehat{m}}_d-{\widehat{m}}_u)/{\Lambda }_{\mathrm{QCD}}$ and ${\widehat{\alpha }}_{\mathrm{em}}$, where ${\widehat{m}}_f$ is the renormalized quark mass and ${\widehat{\alpha }}_{\mathrm{em}}$ the renormalized fine structure constant. In this paper we discuss in detail the general strategy of the method and the conventional, although arbitrary, separation of QCD from QED isospin breaking corrections. We obtain results for the pion mass splitting, $M_{{\pi }^{+}}^2-M_{{\pi }^0}^2=1.44(13)(16)\times {10}^3{\mathrm{MeV}}^2$, for the Dashen’s theorem breaking parameter ${\epsilon }_{\gamma }=0.79(18)(18)$, for the light quark masses, $[{\widehat{m}}_d-{\widehat{m}}_u](\overline{\mathrm{MS}},2\mathrm{GeV})=2.39(8)(17)\mathrm{MeV}$, $[{\widehat{m}}_u/{\widehat{m}}_d](\overline{\mathrm{MS}},2\mathrm{GeV})=0.50(2)(3)$, and for the flavor symmetry breaking parameters $R$ and $Q$. We also update our previous results for the QCD isospin breaking corrections to the $K_{\ell 2}$ decay rate and for the QCD contribution to the neutron-proton mass splitting. The numerical results of this paper have been obtained by using the gluon field configurations produced by the ETMC Collaboration with $n_f=2$ dynamical quarks. We treated the dynamical quarks as electrically neutral particles (electroquenched approximation) and neglected a disconnected diagram in the charged and neutral pion mass splitting. We provide all the formulas necessary to remove these approximations and discuss in detail the estimate of the associated systematic uncertainties.

• Received 3 April 2013

DOI:https://doi.org/10.1103/PhysRevD.87.114505