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Order of the Chiral and Continuum Limits in Staggered Chiral Perturbation Theory

C. Bernard

TL;DR

This paper investigates whether the noncommutativity of chiral and continuum limits seen in a 2D Schwinger model with rooted staggered fermions extends to 4D QCD within staggered chiral perturbation theory. It identifies explicit noncommuting limits in quenched and partially quenched SχPT driven by taste-violations and infrared dynamics, while arguing that in full unquenched SχPT most standard observables remain safe and thus not prone to large systematic errors in MILC results. The analysis clarifies that noncommutativity is not inherently tied to the rooting procedure and discusses which quantities are robust versus potentially problematic, noting that a full all-orders proof of safety is delicate. Overall, the work suggests that, for practical lattice computations at physical quark masses, the observed noncommutativity should not undermine the reliability of staggered fermion results within the SχPT framework.

Abstract

Durr and Hoelbling recently observed that the continuum and chiral limits do not commute in the two dimensional, one flavor, Schwinger model with staggered fermions. I point out that such lack of commutativity can also be seen in four-dimensional staggered chiral perturbation theory (SChPT) in quenched or partially quenched quantities constructed to be particularly sensitive to the chiral limit. Although the physics involved in the SChPT examples is quite different from that in the Schwinger model, neither singularity seems to be connected to the trick of taking the nth root of the fermion determinant to remove unwanted degrees of freedom ("tastes"). Further, I argue that the singularities in SChPT are absent in most commonly-computed quantities in the unquenched (full) QCD case and do not imply any unexpected systematic errors in recent MILC calculations with staggered fermions.

Order of the Chiral and Continuum Limits in Staggered Chiral Perturbation Theory

TL;DR

This paper investigates whether the noncommutativity of chiral and continuum limits seen in a 2D Schwinger model with rooted staggered fermions extends to 4D QCD within staggered chiral perturbation theory. It identifies explicit noncommuting limits in quenched and partially quenched SχPT driven by taste-violations and infrared dynamics, while arguing that in full unquenched SχPT most standard observables remain safe and thus not prone to large systematic errors in MILC results. The analysis clarifies that noncommutativity is not inherently tied to the rooting procedure and discusses which quantities are robust versus potentially problematic, noting that a full all-orders proof of safety is delicate. Overall, the work suggests that, for practical lattice computations at physical quark masses, the observed noncommutativity should not undermine the reliability of staggered fermion results within the SχPT framework.

Abstract

Durr and Hoelbling recently observed that the continuum and chiral limits do not commute in the two dimensional, one flavor, Schwinger model with staggered fermions. I point out that such lack of commutativity can also be seen in four-dimensional staggered chiral perturbation theory (SChPT) in quenched or partially quenched quantities constructed to be particularly sensitive to the chiral limit. Although the physics involved in the SChPT examples is quite different from that in the Schwinger model, neither singularity seems to be connected to the trick of taking the nth root of the fermion determinant to remove unwanted degrees of freedom ("tastes"). Further, I argue that the singularities in SChPT are absent in most commonly-computed quantities in the unquenched (full) QCD case and do not imply any unexpected systematic errors in recent MILC calculations with staggered fermions.

Paper Structure

This paper contains 4 sections, 11 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Examples from S$\chi$PT
  3. Commutativity in full S$\chi$PT
  4. Errors of existing numerical computations

Figures (1)

  • Figure 1: A "t-channel" scattering diagram for the scattering of two taste-singlet mesons, with internal taste-$\xi_5$ mesons: $\pi^+_I(p_1)+\pi^-_I(p_2)\to\pi^+_I(p_3)+\pi^-_I(p_4)$.