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Renormalization-group analysis of the validity of staggered-fermion QCD with the fourth-root recipe

Yigal Shamir

TL;DR

The paper develops a renormalization-group blocking framework for lattice QCD with staggered fermions to address the continuum validity of the fourth-root trick. By performing successive RG blockings and constructing local reweighted theories that approximate the blocked rooted theory, the author argues that taste-violating effects are irrelevant and vanish in the continuum, yielding locality and unitarity in the physical subspace. A multi-gauge-field representation enables a perturbative scaling analysis of the taste-violating operator $\Delta_n$, connecting the rooted theory to the local staggered theory via convergent expansions. The results provide a plausible, testable nonperturbative justification for the fourth-root recipe used in simulations, while outlining concrete avenues for numerical and analytical verification. Overall, the work lays out a rigorous, albeit assumption-dependent, pathway to reconcile nonlocal rooted formulations with the established continuum behavior of QCD.

Abstract

I develop a renormalization-group blocking framework for lattice QCD with staggered fermions. Under plausible, and testable, assumptions, I then argue that the fourth-root recipe used in numerical simulations is valid in the continuum limit. The taste-symmetry violating terms, which give rise to non-local effects in the fourth-root theory when the lattice spacing is non-zero, vanish in the continuum limit. A key role is played by reweighted theories that are local and renormalizable on the one hand, and that approximate the fourth-root theory better and better as the continuum limit is approached on the other hand.

Renormalization-group analysis of the validity of staggered-fermion QCD with the fourth-root recipe

TL;DR

The paper develops a renormalization-group blocking framework for lattice QCD with staggered fermions to address the continuum validity of the fourth-root trick. By performing successive RG blockings and constructing local reweighted theories that approximate the blocked rooted theory, the author argues that taste-violating effects are irrelevant and vanish in the continuum, yielding locality and unitarity in the physical subspace. A multi-gauge-field representation enables a perturbative scaling analysis of the taste-violating operator , connecting the rooted theory to the local staggered theory via convergent expansions. The results provide a plausible, testable nonperturbative justification for the fourth-root recipe used in simulations, while outlining concrete avenues for numerical and analytical verification. Overall, the work lays out a rigorous, albeit assumption-dependent, pathway to reconcile nonlocal rooted formulations with the established continuum behavior of QCD.

Abstract

I develop a renormalization-group blocking framework for lattice QCD with staggered fermions. Under plausible, and testable, assumptions, I then argue that the fourth-root recipe used in numerical simulations is valid in the continuum limit. The taste-symmetry violating terms, which give rise to non-local effects in the fourth-root theory when the lattice spacing is non-zero, vanish in the continuum limit. A key role is played by reweighted theories that are local and renormalizable on the one hand, and that approximate the fourth-root theory better and better as the continuum limit is approached on the other hand.

Paper Structure

This paper contains 35 sections, 112 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The RG blocking framework
  3. The blocking transformations
  4. A convenient representation
  5. Pull-back mapping
  6. Lessons from the free theory
  7. Overview of the argument
  8. Continuum limit of the ordinary staggered theory
  9. Continuum-limit observables
  10. Scaling of irrelevant operators
  11. Recovery of taste symmetry in the continuum limit
  12. Reweighted theories at weak coupling
  13. Renormalizability
  14. Staggered theory
  15. Reweighted theory
  16. ...and 20 more sections

Figures (1)

  • Figure 1: Rotations. The two-dimensional example shows how fine- and coarse-lattice rotations are related. The small circle marks the origin. The point $X=({1\over 2},{1\over 2})$ is marked by a cross. Thick squares show the blocking pattern. Left panel: Counter-clockwise $90^0$ rotation about the point $X$. The blocked squares are mapped onto themselves: centers are mapped to centers; corners undergo a rotation with respect to the square's center. Right panel: The same effect is achieved by a rotation about the origin, followed by a translation that brings the point $X$ back into its original position.