Staggered Chiral Perturbation Theory and the Fourth-Root Trick

TL;DR

This paper defends the fourth-root trick for rooted staggered quarks by constructing a chiral effective theory, staggered chiral perturbation theory (SχPT), that accounts for rooting via the replica trick. It proves that for four degenerate flavors the rooted theory is equivalent, order by order, to the unrooted chiral theory and argues that, under analyticity and decoupling assumptions, this equivalence extends to fewer flavors, including the physically important three- and two-flavor cases. A key resolution of the one-flavor paradox shows that light, unphysical pions decouple in the continuum limit, leaving only heavy flavor-singlet state contributions. Consequently, if taste symmetry is restored in the continuum, the low-energy sector of rooted staggered QCD coincides with ordinary χPT, and the rooted theory behaves like a partially quenched system rather than a mixed one. The work also identifies testable assumptions and outlines routes to numerical verification, with implications for heavy-light, baryon sectors, and universality.

Abstract

Staggered chiral perturbation theory (schpt) 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, nF=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 nF=4 to nF=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 nF=3, 2, or 1. A 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, schpt 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 schpt, so that a variety of possible new routes for testing this validity are opened.

Figures (3)

• Figure 1: Valence quark contractions in the scalar propagator $G(x-y)$, corresponding to Eq. (\ref{['eq:Gxy-valence']}). The solid dots represent the source $\sigma$. Only the valence quark lines are shown; completely disconnected contributions to (b) should be omitted.
• Figure 2: Lowest order S$\chi$PT meson diagrams coming from Eq. (\ref{['eq:Gxy-chiral']}), and corresponding to Fig. \ref{['fig:contractions']}. As in Fig. \ref{['fig:contractions']}, a solid dot is a source, $\sigma$. The cross represents one or more insertions of a "hairpin" vertex, and hence indicates a meson propagator that is disconnected as a quark-flow diagram.
• Figure 3: Quark flow diagrams corresponding to the S$\chi$PT contributions of Fig. \ref{['fig:mesons']}. Not shown are two additional diagrams that are very similar to (b) and (c) but have the roles of valence quarks $\alpha$ and $\beta$ interchanged. Diagrams (a) and (d) have no hairpin vertices and correspond to Fig. \ref{['fig:mesons']}(a); diagrams (b) and (c) have one hairpin vertex and correspond to Fig. \ref{['fig:mesons']}(b); while diagram (e), with two hairpin vertices, corresponds to Fig. \ref{['fig:mesons']}(c). In meson lines with hairpin vertices, a summation of sea-quark loop insertions is implied.