Staggered Chiral Perturbation Theory and the Fourth-Root Trick

TL;DR

This paper demonstrates that under certain assumptions, the fourth-root trick in staggered chiral perturbation theory is valid for representing the reduction of taste degrees of freedom in lattice QCD, ensuring no fundamental problems in the continuum limit.

Contribution

It provides a theoretical justification for the validity of the fourth-root trick within staggered chiral perturbation theory under specific assumptions.

Findings

The fourth-root trick reduces four flavors to one in the degenerate case.

Assumptions on analyticity and decoupling extend validity to fewer flavors.

The trick does not cause unitarity violations in the continuum limit.

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.