Leading isospin breaking effects on the lattice

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 QCD+QED gauge invariant hadronic observable which can be computed on the lattice. It is based on the expansion of the path-integral in powers of the small parameters (m_d - m_u)/Lambda_{QCD} and alpha_{em}, where m_f is the renormalized quark mass and alpha_{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_{pi0}^2= 1.44(13)(16) x 10^3 MeV^2, for the Dashen's theorem breaking parameter epsilon_{gamma}= 0.79(18)(18), for the light quark masses, [m_d - m_u](MSbar,2 GeV)= 2.39(8)(17) MeV, [m_u / m_d](MSbar,2 GeV)= 0.50(2)(3) and for the flavour symmetry breaking parameters R and Q. We also update our previous results for the QCD isospin breaking corrections to the Kl2 decay rate and for the QCD contribution to the neutron-proton mass splitting.