Observations on staggered fermions at nonzero lattice spacing

Abstract

We show that the use of the fourth-root trick in lattice QCD with staggered fermions corresponds to a non-local theory at non-zero lattice spacing, but argue that the non-local behavior is likely to go away in the continuum limit. We give examples of this non-local behavior in the free theory, and for the case of a fixed topologically non-trivial background gauge field. In both special cases, the non-local behavior indeed disappears in the continuum limit. Our results invalidate a recent claim that at nonzero lattice spacing an additive mass renormalization is needed because of taste-symmetry breaking.1 One reason is that breaking the taste degeneracy requires additional hopping terms in the lattice action, which, for a generic choice, make the fermion determinant complex. Also, the existence of a partially-conserved continuous chiral symmetry depends on the choice of mass term.2 In the isospin limit, the up-down sector is represented by a square root of a staggered determinant with the common light quark mass.

NeutralB-meson mixing from three-flavor lattice quantum chromodynamics: Determination of theSU(3)-breaking ratioξ<

“…The configurations we use in this analysis were generated using the fourth-root procedure for eliminating extra degrees of freedom originating from fermion doubling. Despite the nonlocal violations of unitarity of the rooted theory at non-zero lattice spacing [45,46], there are strong theoretical arguments [47][48][49][50], as well as other analytical and numerical evidence [51][52][53][54], that the local, unitary theory of QCD is recovered in the continuum limit. This gives us confidence that the rooting procedure yields valid results.…”

Rooting issue for a lattice fermion formulation similar to staggered fermions but without taste mixing

“…Since this formulation is not manifestly a local lattice field theory there is a danger that it might not be in the right universality class for QCD. (In fact it has been argued [4] that this lattice theory is necessarily non-local but with locality being restored in the continuum limit [5]. 1 ) Because of the high stakes, this has become a prominent, hotly debated issue in the lattice community.…”

“…For an ensemble of lattice gauge fields generated at sufficiently small bare coupling (or with a sufficiently improved lattice gauge action) the low-lying real eigenvalues of DW cluster around a critical (positive) value m c -see, e.g., [27]. We henceforth 4 The robustness of low-lying real eigenvalue modes and index of the Wilson-Dirac operator is ensured by the property that in sufficiently smooth backgrounds the eigenvalues cannot vary arbitrarily under deformations of the background but are constrained to be close to zero. An upper bound can be analytically derived when the plaquette variables satisfy a bound ||1 − U µν − 1|| < ǫ [32,33].…”

Staggered chiral perturbation theory in the two-flavor case and SU(2) analysis of the MILC data

“…Throughout this work, we always assume that the fourth-root procedure is legitimate [11,8], and in practice it is done by setting n r = 1/4 at the end of the calculations. The results are…”