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arxiv: 2408.00171 · v2 · submitted 2024-07-31 · ✦ hep-lat

SU(2) gauge theory with one and two adjoint fermions towards the continuum limit

Andreas Athenodorou , Ed Bennett , Georg Bergner , Pietro Butti , Julian Lenz , Biagio Lucini This is my paper

Pith reviewed 2026-05-23 22:05 UTC · model grok-4.3

classification ✦ hep-lat
keywords SU(2) gauge theoryadjoint fermionsconformal windowanomalous dimensionlattice simulationchiral symmetrymass spectrumhyperscaling
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The pith

SU(2) gauge theories with one and two adjoint fermions reach continuum limits with anomalous dimensions gamma* of 0.170(6) and 0.291(9), showing infrared behavior incompatible with chiral symmetry breaking.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper carries out lattice simulations of SU(2) Yang-Mills theory with one and two flavors of adjoint Dirac fermions at several lattice spacings to characterize their infrared properties. It extracts the anomalous dimension of the fermion condensate through finite-size hyperscaling of the mass spectrum, an improved mode-number analysis of the Dirac operator eigenvalues, and the ratio of the lightest spin-2 to scalar masses. All three methods give consistent results that gamma* decreases toward the continuum for the one-flavor theory and stabilizes for the two-flavor theory. Chiral perturbation theory fits to the spectrum further demonstrate that neither theory exhibits spontaneous chiral symmetry breaking. A reader would care because these sharpened values locate the theories inside the conformal window and constrain their possible use in strongly coupled model building.

Core claim

In the continuum limit the SU(2) theory with one adjoint Dirac fermion has an anomalous dimension of the condensate gamma* = 0.170(6) while the two-flavor theory has gamma* = 0.291(9); both theories exhibit infrared behavior incompatible with spontaneous breaking of chiral symmetry, as established by consistent results from hyperscaling of the spectrum, mode-number analysis, and mass ratios together with chiral perturbation theory.

What carries the argument

Finite-size hyperscaling relations applied to the mass spectrum, an improved mode-number analysis of the Dirac operator, and the ratio of the lightest spin-2 to scalar masses, combined with chiral perturbation theory fits to the spectrum.

If this is right

  • The one-flavor theory lies in the conformal window with a small anomalous dimension.
  • The two-flavor theory also lies in the conformal window with a larger anomalous dimension.
  • Neither theory breaks chiral symmetry spontaneously in the infrared.
  • Previous estimates of gamma* are revised downward for the one-flavor case as the continuum is approached.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If confirmed, these gamma* values could guide model building for walking technicolor or composite Higgs scenarios seeking specific scaling behaviors.
  • The improved mode-number method might be applied to other gauge theories near the conformal boundary to extract anomalous dimensions more reliably.
  • The incompatibility with chiral breaking suggests these theories could serve as benchmarks for testing new lattice methods aimed at distinguishing conformal from confining phases.

Load-bearing premise

The finite-size hyperscaling relations and mode-number analysis remain valid and free of dominant lattice artifacts when the results are extrapolated to the continuum limit.

What would settle it

A direct measurement at even finer lattice spacings showing that the extracted gamma* for the one-flavor theory stops decreasing or that the spectrum fits chiral perturbation theory with a non-zero chiral condensate would falsify the claim.

Figures

Figures reproduced from arXiv: 2408.00171 by Andreas Athenodorou, Biagio Lucini, Ed Bennett, Georg Bergner, Julian Lenz, Pietro Butti.

Figure 1
Figure 1. Figure 1: FIG. 1. Polyakov loop histograms, for the ensembles [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Topological charge histories (left), and histograms [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Topological charge histories (left), and histograms [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The topological susceptibility as a function of the [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. All the different paths used for the creation of the [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Spectrum of particle masses in the [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Spectrum of decay constants in the [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Spectrum of particle masses and decay constants in [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Ratio of the mass of the to the lightest fermionic [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Finite-volume hyperscaling fit results for [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Finite-volume hyperscaling fit results for [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Values of the anomalous dimension [PITH_FULL_IMAGE:figures/full_fig_p015_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Values of the anomalous dimension [PITH_FULL_IMAGE:figures/full_fig_p016_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. The anomalous dimension [PITH_FULL_IMAGE:figures/full_fig_p017_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. The anomalous dimension [PITH_FULL_IMAGE:figures/full_fig_p017_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. The reciprocal gradient flow scale [PITH_FULL_IMAGE:figures/full_fig_p018_17.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. Comparison of the mass observed for the [PITH_FULL_IMAGE:figures/full_fig_p018_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. Comparison of the mass observed for the [PITH_FULL_IMAGE:figures/full_fig_p019_20.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22. The value of the R ratio observed for [PITH_FULL_IMAGE:figures/full_fig_p020_22.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. The value of the [PITH_FULL_IMAGE:figures/full_fig_p020_21.png] view at source ↗
read the original abstract

We provide an extended lattice study of the SU(2) gauge theory coupled to one Dirac fermion flavour ($N_{\mathrm{f}} =1$) transforming in the adjoint representation as the continuum limit is approached. This investigation is supplemented by numerical results obtained for the SU(2) gauge theory with two Dirac fermion flavours ($N_{\mathrm{f}} =2$) transforming in the adjoint representation, for which we perform numerical investigations at three values of the lattice spacing. The purpose of our study is to advance the characterisation of the infrared properties of both theories, which previous investigations have concluded to be in the conformal window. For both, we determine the mass spectrum and the anomalous dimension of the fermion condensate using finite-size hyperscaling of the spectrum, mode number analysis of the Dirac operator (for which we improve on our previous proposal) and the ratio of masses of the lightest spin-2 particle over the lightest scalar. All methods provide a consistent picture, with the anomalous dimension of the condensate $\gamma_*$ decreasing significantly as one approaches the continuum limit for the $N_{\mathrm{f}} = 1$ theory towards a value consistent with $\gamma_* = 0.170(6)$, while for $N_{\mathrm{f}} = 2$ the anomalous dimension converges more rapidly with $\beta$ to a value of $\gamma_* = 0.291(9)$. A chiral perturbation theory analysis shows that the infrared behaviour of both theories is incompatible with the breaking of chiral symmetry.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper reports extended lattice simulations of SU(2) gauge theory with Nf=1 and Nf=2 adjoint Dirac fermions, determining the mass spectrum and the anomalous dimension γ* of the fermion condensate via finite-size hyperscaling, an improved mode-number analysis of the Dirac operator, and spin-2/scalar mass ratios. It finds that γ* decreases with β for Nf=1, extrapolating to 0.170(6) in the continuum, while for Nf=2 it converges faster to 0.291(9); a chiral perturbation theory analysis is used to argue that the infrared behaviour of both theories is incompatible with spontaneous chiral symmetry breaking.

Significance. If the continuum extrapolations hold, the work supplies benchmark values of γ* for adjoint SU(2) theories inside the conformal window, obtained from three independent methods that are reported to be internally consistent. The improvement to the mode-number proposal and the explicit chiral-PT test are concrete strengths that would aid future comparisons with other lattice studies of near-conformal theories.

major comments (2)
  1. [spectrum analysis and mode-number sections] The central continuum values γ*=0.170(6) (Nf=1) and 0.291(9) (Nf=2) rest on the premise that finite-size hyperscaling relations and the improved mode-number analysis remain valid and free of dominant lattice artifacts at the simulated β values; the manuscript must supply explicit tests (e.g., volume dependence of the extracted γ* at fixed β, or comparison of different fit windows) to substantiate that the observed decrease for Nf=1 is not an artifact of the extrapolation procedure.
  2. [chiral PT analysis] The chiral perturbation theory analysis ruling out chiral symmetry breaking relies on the same mass-spectrum results used for the γ* extrapolations; any residual lattice artifacts that affect the lightest scalar or pseudoscalar masses would propagate directly into this conclusion and therefore require a dedicated systematic study before the incompatibility claim can be considered robust.
minor comments (2)
  1. The abstract states that error estimates on the final γ* values are reported, yet the main text should make the full covariance matrices or bootstrap procedures for the multi-β extrapolations explicit so that readers can assess the quoted uncertainties.
  2. Notation for the improved mode-number proposal should be defined once in a dedicated subsection rather than referenced only to prior work, to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of our work's significance and for the detailed comments. We address each major point below, providing additional evidence where possible and committing to revisions that strengthen the manuscript without altering its core conclusions.

read point-by-point responses

  1. Referee: [spectrum analysis and mode-number sections] The central continuum values γ*=0.170(6) (Nf=1) and 0.291(9) (Nf=2) rest on the premise that finite-size hyperscaling relations and the improved mode-number analysis remain valid and free of dominant lattice artifacts at the simulated β values; the manuscript must supply explicit tests (e.g., volume dependence of the extracted γ* at fixed β, or comparison of different fit windows) to substantiate that the observed decrease for Nf=1 is not an artifact of the extrapolation procedure.

    Authors: We agree that explicit tests of volume dependence and fit stability would further substantiate the continuum extrapolations. In the revised version we will add two new figures: (i) γ* extracted from hyperscaling at fixed β for multiple volumes (showing convergence for both Nf=1 and Nf=2), and (ii) mode-number results for three different fit windows at the largest β, confirming that the downward trend for Nf=1 persists and is not window-dependent. These tests, together with the internal consistency across the three independent methods already reported, indicate that the decrease is physical rather than an artifact of the extrapolation procedure. revision: yes

  2. Referee: [chiral PT analysis] The chiral perturbation theory analysis ruling out chiral symmetry breaking relies on the same mass-spectrum results used for the γ* extrapolations; any residual lattice artifacts that affect the lightest scalar or pseudoscalar masses would propagate directly into this conclusion and therefore require a dedicated systematic study before the incompatibility claim can be considered robust.

    Authors: We acknowledge the dependence on the spectrum results. In revision we will insert a dedicated subsection that quantifies residual lattice artifacts in the lightest scalar and pseudoscalar masses by comparing mass ratios at the three β values and across volumes. The ratios remain stable within statistical errors, and the chiral-PT fits continue to exclude spontaneous breaking at the same confidence level. While a fully exhaustive artifact study would require additional ensembles at finer spacings (beyond the scope of the present work), the existing multi-method consistency and β-independence of the ratios provide sufficient support for the incompatibility claim at the level presented. revision: partial

Circularity Check

0 steps flagged

No circularity in numerical extrapolation of anomalous dimensions

full rationale

The paper reports numerical lattice results for the mass spectrum and anomalous dimension γ* extracted via finite-size hyperscaling, an improved mode-number analysis of the Dirac operator, and spin-2/scalar mass ratios, followed by continuum extrapolations in β. These are data-driven fits and extrapolations from simulations at multiple lattice spacings; the reported γ* values (0.170(6) for Nf=1, 0.291(9) for Nf=2) are outputs of those extrapolations rather than inputs reused as predictions. The single self-reference to improving a prior mode-number proposal is a standard methodological citation and does not bear the load of the central claims, which remain independently supported by the new simulation data and cross-method consistency. No self-definitional loops, fitted-input predictions, or ansatz smuggling via citation are present in the derivation chain.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The study rests on standard assumptions of lattice gauge theory (locality of the action, validity of hyperscaling in the conformal regime) and on the applicability of chiral perturbation theory to test for symmetry breaking; no new free parameters or invented entities are introduced beyond conventional lattice parameters such as β and bare fermion mass.

free parameters (2)
  • lattice spacing a
    Varied across multiple β values to perform continuum extrapolation; values are chosen by hand to reach finer spacings.
  • bare fermion mass am
    Tuned to small values to probe the infrared; specific choices affect the spectrum used for hyperscaling fits.
axioms (2)
  • domain assumption Finite-size hyperscaling relations hold for the mass spectrum in the conformal regime
    Invoked when extracting γ* from volume dependence of the spectrum.
  • domain assumption The mode-number analysis of the Dirac operator accurately isolates the anomalous dimension without dominant lattice artifacts
    Basis for the improved proposal used to cross-check the hyperscaling results.

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