Chiral perturbation theory for three-flavour lattice QCD with isospin splitting

An important tool for the analysis of results of numerical simulations of lattice QCD is chiral perturbation theory. In Wilson chiral perturbation theory the effects of the finite lattice spacing $a$ are taken into account. In recent years the effects of isospin splitting on the masses of hadrons have been investigated in Monte Carlo simulations. Correspondingly, in this article we derive the expansions of the masses of the pseudoscalar mesons in chiral perturbation theory at next-to-leading order for twisted mass lattice QCD with three light quark flavours, taking the mass difference between the up and down quarks into account. The results include terms up to orders $m_q^2$ in the quark masses, $\Delta m^2$ in the mass splitting between up- and down quarks, and $a^2$ in the lattice spacing, respectively.