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arxiv: 2604.07256 · v2 · submitted 2026-04-08 · ✦ hep-lat · astro-ph.CO· hep-ph· hep-th

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Revisiting the sphaleron and axion production rates in QCD at high temperatures

Sayak Guin , Sayantan Sharma
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Pith reviewed 2026-05-11 01:41 UTC · model grok-4.3

classification ✦ hep-lat astro-ph.COhep-phhep-th
keywords sphaleron rateaxion productionlattice QCDthermal effective field theoryhigh temperature QCDreheating after inflationnon-perturbative effectsSU(N) gluons
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The pith

Lattice calculations in an effective theory of soft gluons yield new sphaleron rates and show axion production deviating from perturbation theory at high temperatures.

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

The paper computes the sphaleron rate on the lattice within a thermal effective field theory for soft gluons in SU(2) and SU(3) QCD. These computations cover temperatures from 0.6 GeV to 10^15 GeV and are compared to rates in non-thermal over-occupied gluon plasmas to estimate thermalization times after inflation. The authors also find that the production rate of relativistic axions from these soft gluons deviates significantly from perturbative expectations, with the difference persisting at the electroweak scale. Sympathetic readers would care because accurate modeling of these rates is essential for describing particle production and topological transitions in the hot early universe.

Core claim

Within the thermal effective field theory for soft SU(N) gluons with momenta below the magnetic scale, lattice computations provide the sphaleron transition rate for N equals 2 and 3 across temperatures from 0.6 GeV up to 10 to the 15 GeV on large volumes. These rates, when contrasted with those in a non-thermal plasma of over-occupied infrared gluons, permit an estimate of the thermalization time for ultra-soft gluons in the reheating phase following inflation. The same non-perturbative gluon interactions lead to a thermal axion production rate that deviates substantially from the perturbative prediction, and this deviation remains large even at the electroweak scale.

What carries the argument

thermal effective field theory of soft SU(N) gluons below the magnetic scale, studied via lattice simulations

Load-bearing premise

The thermal effective field theory of soft SU(N) gluons accurately captures the relevant physics for the sphaleron rate and axion production across the stated temperature range, and that the non-thermal plasma comparison reliably estimates thermalization times.

What would settle it

If the axion production rate computed in full QCD lattice simulations at the electroweak scale matches the perturbative estimate rather than the soft-gluon EFT result, the claim of significant deviation would be falsified.

Figures

Figures reproduced from arXiv: 2604.07256 by Sayak Guin, Sayantan Sharma.

Figure 1
Figure 1. Figure 1: Probability distribution of ∆NCS at different optimal cooling depths τc in a thermal SU(3) gauge plasma at g = 0.58. III. RESULTS A. Sphaleron rate in a thermal non-Abelian plasma: SU(2) vs SU(3) Performing the cooling procedure described in the pre￾ceeding section, we first calculate the distribution of the Chern-Simons number for different choices of the opti￾mal cooling time τc for both SU(2) and SU(3) … view at source ↗
Figure 3
Figure 3. Figure 3: Autocorrelation of the Chern-Simons number change [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: Probability distribution of ∆NCS at different optimal cooling depths τc in a thermal SU(3) gauge plasma at g = 1.12. this optimal depth for g < 1. We observe a similar trend at a lower temperature denoted by g = 1.12 shown in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 5
Figure 5. Figure 5: Sphaleron rates for a thermal SU(2) plasma as a func [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 8
Figure 8. Figure 8: Autocorrelation of the Chern-Simons number change [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: Parametric dependence of the time tth required for sphaleron rates to attain their values in a thermal plasma starting from non-thermal initial conditions, conserving the energy density, shown as a function of inverse gauge coupling. the implications of any non-perturbative collective phe￾nomena that might occur the very early stages of reheat￾ing [37]. We discuss one such possible scenario where a sphale… view at source ↗
Figure 12
Figure 12. Figure 12: Present-day relic axion yields normalized by its ther [PITH_FULL_IMAGE:figures/full_fig_p010_12.png] view at source ↗
read the original abstract

We report our new lattice results for the sphaleron rate calculated within a thermal effective field theory of soft SU(N) gluons whose momenta are below the magnetic scale, where $N=2,3$, for a wide range of temperatures spanning from $0.6$-$10^{15}$ GeV at sufficiently large volumes. Comparing these results with sphaleron rates in a non-thermal SU(N) plasma where the infrared gluons are over-occupied, we estimate the typical thermalization time for these ultra-soft soft gluons during the early stages of reheating after inflation. We also calculate the thermal production rate of relativistic axions due to these non-perturbatively interacting soft gluons which shows a significant deviation from its perturbative estimate even at the electroweak scale.

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 new lattice results for the sphaleron rate in a thermal effective field theory of soft SU(N) gluons (N=2,3) with momenta below the magnetic scale, covering temperatures from 0.6 GeV to 10^15 GeV at large volumes. These are compared to sphaleron rates in a non-thermal over-occupied SU(N) plasma to estimate the thermalization timescale for ultra-soft gluons during post-inflation reheating. The work also computes the thermal production rate of relativistic axions from the non-perturbatively interacting soft gluons, finding a significant deviation from the perturbative estimate even at the electroweak scale.

Significance. If the lattice results are robust and the mapping to reheating dynamics holds, the findings would supply valuable non-perturbative data on sphaleron processes and axion production over an unusually broad temperature range, with direct relevance to early-universe cosmology, baryogenesis, and axion dark matter models. The lattice approach in the thermal EFT and the direct comparison to non-thermal simulations are positive features that go beyond purely perturbative estimates.

major comments (2)
  1. The central estimate of the typical thermalization time for ultra-soft gluons during reheating is obtained by taking the ratio of the sphaleron rate in the thermal EFT lattice calculation to the rate in the non-thermal over-occupied plasma. This mapping assumes a static configuration whose evolution is governed solely by the rate difference; it does not incorporate Hubble expansion or continuous gluon injection from inflaton decay, both of which can shift the effective timescale by an order of magnitude or more in a realistic expanding background.
  2. The manuscript provides no explicit details on the lattice discretization (action, spacing, volume sizes in physical units), the precise implementation of the thermal EFT, the method used to extract the sphaleron rate, or the error controls and finite-volume extrapolations. Without these, it is impossible to assess whether the quoted results remain reliable across the 15-order-of-magnitude temperature interval.
minor comments (2)
  1. The axion production rate section should state the precise axion-gluon coupling and the momentum integration limits used when comparing to the perturbative estimate.
  2. Clarify whether the same lattice ensembles are used for both the sphaleron rate and the axion production rate, or whether separate runs are required.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for highlighting these important points. We address each major comment below and will revise the manuscript to improve clarity and completeness.

read point-by-point responses

  1. Referee: The central estimate of the typical thermalization time for ultra-soft gluons during reheating is obtained by taking the ratio of the sphaleron rate in the thermal EFT lattice calculation to the rate in the non-thermal over-occupied plasma. This mapping assumes a static configuration whose evolution is governed solely by the rate difference; it does not incorporate Hubble expansion or continuous gluon injection from inflaton decay, both of which can shift the effective timescale by an order of magnitude or more in a realistic expanding background.

    Authors: We agree that the mapping used to estimate the thermalization timescale is simplified and assumes a static setup without Hubble expansion or continuous gluon injection. This provides an order-of-magnitude indication of the timescale in the early reheating phase rather than a full dynamical prediction. In the revised manuscript we will add a dedicated paragraph in the discussion section explicitly stating these limitations, noting that the reported value should be interpreted as a baseline estimate, and suggesting that incorporating expansion and inflaton decay would require separate cosmological simulations. revision: partial

  2. Referee: The manuscript provides no explicit details on the lattice discretization (action, spacing, volume sizes in physical units), the precise implementation of the thermal EFT, the method used to extract the sphaleron rate, or the error controls and finite-volume extrapolations. Without these, it is impossible to assess whether the quoted results remain reliable across the 15-order-of-magnitude temperature interval.

    Authors: We acknowledge that the current version does not present these technical details with sufficient explicitness. In the revised manuscript we will expand the methods and appendix sections to include the lattice action, the range of lattice spacings and corresponding physical volumes for each temperature, the precise formulation of the thermal effective field theory, the procedure for extracting the sphaleron rate, and the full error budget together with finite-volume extrapolation details. This will allow readers to evaluate the robustness of the results over the reported temperature range. revision: yes

Circularity Check

0 steps flagged

No significant circularity; direct lattice computations

full rationale

The paper presents new lattice results for sphaleron rates in a thermal EFT of soft SU(N) gluons and axion production rates across a wide temperature range. These are obtained via direct numerical simulation on sufficiently large volumes rather than any analytical derivation. The subsequent comparison to non-thermal over-occupied plasma is used only to estimate a thermalization timescale; this interpretive mapping does not reduce any claimed result to a fitted parameter or self-referential definition by construction. No load-bearing self-citations, ansatzes smuggled via prior work, or uniqueness theorems imported from the same authors are invoked in the provided text. The central claims remain independent numerical outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on abstract only; no specific free parameters, axioms or invented entities identifiable without full text. The EFT itself may involve standard assumptions from prior literature.

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Cited by 1 Pith paper

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