Staggered chiral perturbation theory and the fourth-root trick

Staggered chiral perturbation theory ($\mathrm{S}\chi \mathrm{PT}$) takes into account the “fourth-root trick” for reducing unwanted (taste) degrees of freedom with staggered quarks by multiplying the contribution of each sea quark loop by a factor of $1/4$. In the special case of four staggered fields (four flavors, $n_F=4$), I show here that certain assumptions about analyticity and phase structure imply the validity of this procedure for representing the rooting trick in the chiral sector. I start from the observation that, when the four flavors are degenerate, the fourth root simply reduces $n_F=4$ to $n_F=1$. One can then treat nondegenerate quark masses by expanding around the degenerate limit. With additional assumptions on decoupling, the result can be extended to the more interesting cases of $n_F=3$, 2, or 1. An apparent paradox associated with the one-flavor case is resolved. Coupled with some expected features of unrooted staggered quarks in the continuum limit, in particular, the restoration of taste symmetry, $\mathrm{S}\chi \mathrm{PT}$ then implies that the fourth-root trick induces no problems (for example, a violation of unitarity that persists in the continuum limit) in the lowest energy sector of staggered lattice QCD. It also says that the theory with staggered valence quarks and rooted staggered sea quarks behaves like a simple, partially-quenched theory, not like a mixed theory in which sea and valence quarks have different lattice actions. In most cases, the assumptions made in this paper are not only sufficient but also necessary for the validity of $\mathrm{S}\chi \mathrm{PT}$, so that a variety of possible new routes for testing this validity are opened.