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Using the Landau gauge gluon propagator to set the lattice physical scales and understanding the finite size effects

Orlando Oliveira, Paulo J Silva

TL;DR

The paper tackles scale setting in lattice QCD by exploiting the Landau-gauge gluon propagator to establish a relative lattice spacing between β values. It introduces a perturbative-inspired, effective description of ultraviolet lattice artefacts to connect high-momentum lattice data with continuum perturbation theory, validating the approach with large ensembles and renormalization at μ = 4 GeV. The main contributions are a robust procedure for aligning D(p^2) across β values, an explicit UV artefact parameterization with small, lattice-spacing-dependent coefficients, and a framework extendable to other Green functions. This work enhances the reliability of lattice predictions at high momenta and provides a practical bridge to continuum QCD analyses, with implications for precise scale setting and cross-checks against perturbation theory.

Abstract

A crucial step in extracting physical predictions from lattice QCD simulations is the scale setting, i.e. the determination of the lattice spacing ($a$) in physical units. Herein, the relative scale setting for different $β$'s is discussed, using the Landau gauge gluon propagator computed with large statistical ensembles. After setting the relative scales, finite size effects are observed in the ultraviolet regime and handled in an effective description, inspired in perturbation theory. The new devised procedure is efficient in handling the finite size effects, linking the lattice simulations with continuum perturbation theory for the high momenta regime. Furthermore, the procedure can be extended to handle other Green functions computed within lattice QCD simulations.

Using the Landau gauge gluon propagator to set the lattice physical scales and understanding the finite size effects

TL;DR

The paper tackles scale setting in lattice QCD by exploiting the Landau-gauge gluon propagator to establish a relative lattice spacing between β values. It introduces a perturbative-inspired, effective description of ultraviolet lattice artefacts to connect high-momentum lattice data with continuum perturbation theory, validating the approach with large ensembles and renormalization at μ = 4 GeV. The main contributions are a robust procedure for aligning D(p^2) across β values, an explicit UV artefact parameterization with small, lattice-spacing-dependent coefficients, and a framework extendable to other Green functions. This work enhances the reliability of lattice predictions at high momenta and provides a practical bridge to continuum QCD analyses, with implications for precise scale setting and cross-checks against perturbation theory.

Abstract

A crucial step in extracting physical predictions from lattice QCD simulations is the scale setting, i.e. the determination of the lattice spacing () in physical units. Herein, the relative scale setting for different 's is discussed, using the Landau gauge gluon propagator computed with large statistical ensembles. After setting the relative scales, finite size effects are observed in the ultraviolet regime and handled in an effective description, inspired in perturbation theory. The new devised procedure is efficient in handling the finite size effects, linking the lattice simulations with continuum perturbation theory for the high momenta regime. Furthermore, the procedure can be extended to handle other Green functions computed within lattice QCD simulations.
Paper Structure (7 sections, 17 equations, 8 figures, 4 tables)

This paper contains 7 sections, 17 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Definitions
  3. Lattice setup
  4. Setting the relative scale
  5. Lattice Artefacts
  6. An effective description of the lattice artefacts
  7. Summary and conclusions

Figures (8)

  • Figure 1: The dimensionless lattice bare dressing function $p^2 D(p^2)$. The data reported here complies the cylindrical plus conical momentum cuts as defined in Leinweber:1998uu. The full lines are the curves that interpolate, with cubic splines, the lattice data points. See text for details.
  • Figure 2: The lattice bare gluon propagator in terms of the improved momenta in GeV. In the conversion into physical units the scale setting as in Oliveira:2012eh was used. The figure also shows the interpolation of the lattice data. The dashed vertical lines corresponds to the momenta used to renormalize $D(p^2)$.
  • Figure 3: The lattice dressing function renormalized at $\mu = 4$ GeV for all the data sets. For the conversion to physical units we used for the lattice spacing the numbers quoted in Oliveira:2012eh; see Tab. \ref{['tab:setup']}.
  • Figure 4: The lattice dressing function, renormalized at $\mu = 4$ GeV, for the $\beta = 6.2$ data sets, using the relative scale that sets $1/a(\beta = 6.0) = 1.943$ GeV and $1/a(\beta = 6.2) = 2.7052$ GeV.
  • Figure 5: The lattice dressing function, renormalized at $\mu = 4$ GeV, for all data sets, using the relative normalization that sets $1/a(\beta = 6.0) = 1.943$ GeV and $1/a(\beta = 6.2) = 2.7052$ GeV.
  • ...and 3 more figures