Staggered chiral perturbation theory at next-to-leading order

We study taste and Euclidean rotational symmetry violation for staggered fermions at nonzero lattice spacing using staggered chiral perturbation theory. We extend the staggered chiral Lagrangian to $\mathcal{O}(a^2{p}^2)$, $\mathcal{O}(a^4)$, and $\mathcal{O}(a^{2}m)$, the orders necessary for a full next-to-leading order calculation of pseudo-Goldstone boson masses and decay constants including analytic terms. We then calculate a number of $SO(4)$ taste-breaking quantities, which involve only a small subset of these next-to-leading order operators. We predict relationships between $SO(4)$ taste-breaking splittings in masses, pseudoscalar decay constants, and dispersion relations. We also find predictions for a few quantities that are not $SO(4)$ breaking. All these results hold also for theories in which the fourth root of the fermionic determinant is taken to reduce the number of quark tastes; testing them will therefore provide evidence for or against the validity of this trick.