Charmless chiral perturbation theory for $N_f=2+1+1$ twisted mass lattice QCD

The chiral Lagrangian describing the low-energy behavior of $N_f=2+1+1$ twisted mass lattice QCD is constructed through $\mathrm{O}(a^2)$. In contrast to existing results the effects of a heavy charm quark are consistently removed. This Lagrangian is used to compute the pion and kaon masses to one loop in a regime where the pion mass splitting is large and taken as a leading order effect. In comparison with continuum chiral perturbation theory additional chiral logarithms are present in the results. In particular, chiral logarithms involving the neutral pion mass appear. These predict rather large finite volume corrections in the kaon mass which roughly account for the finite volume effects observed in lattice data.