Partial flavor symmetry restoration for chiral staggered fermions

We study the leading discretization errors for staggered fermions by first constructing the continuum effective Lagrangian, including terms of ${O(a}^2),$ and then constructing the corresponding effective chiral Lagrangian. The terms of ${O(a}^2)$ in the continuum effective Lagrangian completely break the $\mathrm{SU}\left(4\right)\mathrm{}$ flavor symmetry down to the discrete subgroup respected by the lattice theory. We find, however, that the ${O(a}^2)$ terms in the potential of the chiral Lagrangian maintain an $\mathrm{SO}\left(4\right)\mathrm{}$ subgroup of $\mathrm{SU}\left(4\right).$ It follows that the leading discretization errors in the pion masses are $\mathrm{SO}\left(4\right)\mathrm{}$ symmetric, implying three degeneracies within the seven lattice irreducible representations. These predictions hold also for perturbatively improved versions of the action. These degeneracies are observed, to a surprising degree of accuracy, in existing data. We argue that the $\mathrm{SO}\left(4\right)\mathrm{}$ symmetry does not extend to the masses and interactions of other hadrons (vector mesons, baryons, etc.) or to higher order in $a^2.$ We show how it is possible that, for physical quark masses of ${O(a}^2),$ the new $\mathrm{SO}\left(4\right)\mathrm{}$ symmetry can be spontaneously broken, leading to a staggered analogue of the Aoki phase of Wilson fermions. This does not, however, appear to happen for presently studied versions of the staggered action.