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Discretisation effects of gradient flows in QCD-like theories on the lattice

Pietro Butti, Michele Della Morte, Benjamin Jäger, Sofie Martins, J. Tobias Tsang

Abstract

Recent software advances now allow large-scale lattice studies of the Corrigan--Ramond large-$N_C$ limit of Yang-Mills theory coupled with a two-index antisymmetric fermion, providing a path to SUSY Yang-Mills. We are currently generating ensembles for $N_C=4,5,6$ for lattice spacings in the range $0.11 - 0.08$ fm. We report on two aspects of our work: the study of topological properties as well as estimates of discretisation effects. The first aspect is relevant since naively, fractional topological charges might be expected in our simulations. Using a gluonic definition of the topological charge combined with gradient flow, we perform an analysis of the effect of different discretisations of the kernel action, from which we identify and interpret quantitative differences between Wilson and over-improved flows such as DBW2. The second aspect is addressed by considering ratios of different reference flow times. We conclude that our current simulations might be affected by discretisation effects of order 10\%.

Discretisation effects of gradient flows in QCD-like theories on the lattice

Abstract

Recent software advances now allow large-scale lattice studies of the Corrigan--Ramond large- limit of Yang-Mills theory coupled with a two-index antisymmetric fermion, providing a path to SUSY Yang-Mills. We are currently generating ensembles for for lattice spacings in the range fm. We report on two aspects of our work: the study of topological properties as well as estimates of discretisation effects. The first aspect is relevant since naively, fractional topological charges might be expected in our simulations. Using a gluonic definition of the topological charge combined with gradient flow, we perform an analysis of the effect of different discretisations of the kernel action, from which we identify and interpret quantitative differences between Wilson and over-improved flows such as DBW2. The second aspect is addressed by considering ratios of different reference flow times. We conclude that our current simulations might be affected by discretisation effects of order 10\%.
Paper Structure (6 sections, 5 equations, 4 figures, 2 tables)

This paper contains 6 sections, 5 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Exploring different flows
  3. Numerical setup
  4. Results and discussion
  5. A first look at lattice artefacts
  6. Conclusions and Outlook

Figures (4)

  • Figure 1: Topological charge determined using the Wilson flow (top) and the DBW2 flow (bottom) on the coarsest (left) and finest (right) $N_C=4$ ensembles. Figure taken from Ref. Butti:2025rnu.
  • Figure 2: Topological charge (top) as well as $h_\text{avg}$ and $h_\text{max}$ (bottom) for the Wilson flow (red, dashed and dotted) and the DBW2 flow (cyan, solid and dash-dotted) as a function of very long flow times and on a single configuration of the N4L12 ensemble. The vertical dotted lines correspond to the location of local maxima in $h^\text{Wilson}_\text{max}$ which correlate with transitions of $Q^\mathrm{Wilson}$. Figure taken from Ref. Butti:2025rnu.
  • Figure 3: Left-hand side panel: ratio of scales $R=t_0/t_1$ vs lattice spacing in units of $\sqrt{t_0^{\text{Wilson}}}$ for the ensembles in Tab. \ref{['tab:ens']} for different kernel action in the gradient flow discretisation. Right-hand side panel: difference from unity of the ratio $R^{\text{Wilson}}/R^{\text{DBW2}}$ for the same ensembles vs $a/\sqrt{t_0}$. Qualitative extrapolation performed as constrained linear + quadratic term in $a$.
  • Figure 4: Ratio of scales $t_0/t_1$ for representative ensembles marked in bold in Tab. \ref{['tab:current_ensembles']}. The leftmost $N_C=3$ point (yellow circle corresponding to $a/\sqrt{t_0}\simeq 0.39$) has been taken from Ref. DellaMorte:2023ylq.