Chiral perturbation theory atO(a2)for lattice QCD

We construct chiral effective Lagrangian for two lattice theories: one with Wilson fermions and the other with Wilson sea fermions and Ginsparg-Wilson valence fermions. For each of these theories we construct the Symanzik action through $\mathcal{O}{(a}^2).\mathrm{}$ The chiral Lagrangian is then derived, including terms of $\mathcal{O}{(a}^2),$ which have not been calculated before. We find that there are only few new terms at this order. Corrections to existing coefficients in the continuum chiral Lagrangian are proportional to $a^2$ and appear in the Lagrangian at $\mathcal{O}{(a}^2{p}^2)$ or higher. Similarly, O(4) symmetry-breaking terms enter the Symanzik action at $\mathcal{O}{(a}^2),$ but contribute to the chiral Lagrangian at $\mathcal{O}{(a}^2{p}^4)$ or higher. We calculate the light meson masses in chiral perturbation theory for both lattice theories. At next-to-leading order, we find that there are no $\mathcal{O}{(a}^2)$ corrections to the valence-valence meson mass in the mixed theory due to the enhanced chiral symmetry of the valence sector.