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An update on the determination of the sphaleron rate in finite temperature QCD

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read New lattice data in quenched SU(3) gauge theory show the topological charge density correlator is suppressed at nonzero spatial momentum, evidence that the momentum-dependent sphaleron rate decays at large momenta.

desk verdict First look at the momentum dependence of the topological correlator in quenched SU(3): plausible trend, but the high-momentum extrapolation is not yet under control. read the letter →

arxiv 2412.13685 v1 pith:7UZCITI3 submitted 2024-12-18 hep-lat hep-phhep-th

Nicolò Bellini , Claudio Bonanno , Francesco D'Angelo , Massimo D'Elia , Andrea Giorgieri , Lorenzo Maio This is my paper · ORCID
classification hep-lathep-phhep-th
keywords sphaleronratetopologicalchargedensitylatticeQCDfinitetemperaturequenchedSU(3)gaugetheoryaxionthermalproductionEuclideancorrelatorspectralfunctioninversion
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper reports an extension of the lattice determination of the strong sphaleron rate to nonzero spatial momentum. The sphaleron rate controls topological transitions in the quark–gluon plasma and enters both the Chiral Magnetic Effect and, through the momentum-dependent topological rate, the thermal production of QCD axions in the early universe. Working in quenched SU(3) gauge theory at $T \simeq 1.24\,T_c$, the authors compute the spatial Fourier transform of the Euclidean topological charge density correlator and find that, after a continuum extrapolation followed by a linear zero-smoothing extrapolation, the correlator is negative and suppressed as $|\vec p|/T$ increases, with strong suppression near $|\vec p|/T \sim 10$. They interpret this as evidence that $\Gamma_{\rm top}^>(|\vec p|)$ decays for large momenta, as expected from the sphaleron-size argument of Ref. [5]; the planned next step is to invert these correlators with the HLT Backus–Gilbert method to obtain the momentum-dependent rate directly.

What carries the argument

The machinery is the spatial Fourier transform of the clover-discretized topological charge density, Eq. (13), whose Euclidean time profile is measured on the lattice and then converted to the correlator $\mathcal{G}_{\vec p}(tT)/T^5$ of Eq. (14). To remove ultraviolet fluctuations, the correlator is computed with several cooling radii and passed through a double extrapolation: first an $O(1/N_t^2)$ continuum limit at fixed smoothing radius, Eq. (15), then a linear zero-smoothing limit in $n_{\rm cool}/N_t^2$, Eq. (16). The final object is the input for the HLT Backus–Gilbert inversion formula, Eq. (11), which would give $\Gamma_{\rm top}^>(|\vec p|)$; the present paper stops at the correlator level and uses the scaling of the correlator to infer the rate's behaviour.

What would settle it

A future calculation on the same ensembles with a wider set of momenta (for example $k=8$ and $k=10$) that found the double-extrapolated correlator at large $|\vec p|/T$ comparable to the zero-momentum value, or a zero-smoothing fit that is strongly nonlinear within the accessible range, would contradict the claimed suppression of the momentum-dependent sphaleron rate.

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Extended reading notes

Core claim

The central claim is that, in the pure SU(3) gauge theory at $T \simeq 1.24\,T_c$, the double-extrapolated Euclidean correlator $\mathcal{G}_{\vec p}(tT)/T^5$ of the topological charge density decreases monotonically as the spatial momentum $|\vec p|/T$ grows from 0 to about 9.4, remaining negative at all nonzero time separations in agreement with reflection positivity. Since the nonzero-momentum topological rate $\Gamma_{\rm top}^>(|\vec p|)$ is obtained from these correlators through the Kubo-type inversion formula of Eq. (11), the observed suppression in the correlator implies, on the paper's logic, a decreasing behaviour of $\Gamma_{\rm top}^>(|\vec p|)$ at large momenta rather than a momentum-independent rate. The paper presents this as preliminary but as a direct lattice check of the expectation from Ref. [5] that the rate is flat up to the sphaleron momentum scale and decays above it.

Load-bearing premise

The load-bearing premise is that, after an $O(1/N_t^2)$ continuum extrapolation and a linear zero-smoothing extrapolation, the lattice correlator equals the physical renormalized topological charge density correlator at every momentum; in particular, the smoothing range is required to be linear down to $n_{\rm cool}/N_t^2 = 0.012$, while for larger momenta the allowed linear range shrinks and the resulting systematic error is not quantified.

Editorial extensions

If this is right

  • A momentum-dependent $\Gamma_{\rm top}^>(|\vec p|)$ that decays at large momenta changes the source term in the axion Boltzmann equation, so the axion distribution computed from these rates will be suppressed at high momenta relative to a constant-rate calculation.
  • The observed drop near $|\vec p|/T \sim 10$ gives a first lattice-side indication of the scale where the sphaleron-size suppression sets in, allowing a direct comparison with the estimate $|\vec p_{\rm sp}| \sim \alpha_s T$ of Ref. [5].
  • Applying the same double-extrapolation and inversion pipeline to $N_f=2+1$ QCD at the temperatures of Ref. [24] would yield the full-QCD momentum-dependent rate that enters axion phenomenology.
  • The sign and monotonic suppression of the correlator at fixed $tT$ mean that future measurements can test the qualitative decay without performing the full spectral inversion.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • One testable extension is to invert the correlators at each $k$ and fit the resulting $\Gamma_{\rm top}^>(|\vec p|)$ to a function that is constant below a scale $\Lambda_s$ and falls above it; the fitted $\Lambda_s/T$ would show whether the drop is controlled by $\alpha_s T$ or by $T$.
  • If the suppression is as steep as the correlator suggests, high-momentum axions are produced less efficiently than a momentum-independent rate would predict, which would shift the hot-dark-matter bound on the QCD axion; quantifying the shift requires the full-QCD rate and is not attempted in this paper.
  • The authors currently orient the external momentum along the $x$-axis; repeating the measurement along the $y$ and $z$ axes, which they plan, would verify that the correlator depends only on $|\vec p|$ and is free of volume or anisotropy artifacts.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This proceedings contribution reports a lattice determination of the spatial Fourier transform of the topological charge density correlator, G_p(tT), in quenched SU(3) gauge theory at T ≈ 1.24 Tc, with the aim of extracting the momentum dependence of the topological rate Gamma_top(p) relevant for axion production. Three ensembles with N_t = 14, 16, 20 and aspect ratio N_s/N_t = 4 are used; the clover topological charge density is smoothed by cooling, then a continuum limit at fixed smearing radius is taken according to Eq. (15), followed by a linear zero-smoothing limit according to Eq. (16). The final double-extrapolated correlators are shown as a function of |p|/T for k = 0,...,6 (|p|/T = 0,...,9.42). The authors observe that the correlator becomes suppressed as the spatial momentum increases and interpret this as evidence that Gamma_top(p) decays for large momenta, consistent with Ref. [5]. The paper is explicitly preliminary; the inversion to Gamma_top(p) is left to future work.

Significance. If the reported suppression is genuine, this is the first lattice evidence for the momentum dependence of the topological charge density correlator, a quantity directly relevant to hot-dark-matter bounds on the QCD axion. The paper has clear strengths: it builds transparently on an established double-extrapolation methodology used in the authors' earlier sphaleron-rate computations, it uses a suitably large aspect ratio to resolve small momenta, and it provides a direct visualization of the continuum-limit fits. However, the central qualitative claim rests on a double extrapolation whose systematic control is weakest precisely in the large-momentum region that drives the suppression, and no uncertainties are quoted for the final results. The result is therefore best viewed as an interesting preliminary indication rather than a measurement with a demonstrated significance.

major comments (3)
  1. [Sec. 3, Fig. 3] The zero-smoothing extrapolation is uncontrolled for the largest momenta that carry the suppression claim. The lower fit bound n_cool^min/N_t^2 = 0.012 is fixed by the zero-momentum topological susceptibility plateau, not by a convergence test of the individual momentum modes. At this bound the product of smearing radius and lattice momentum is r_sm * p = sqrt((8/3)*0.012) * (pi/2) * k ≈ 0.28 k, giving approximately 1.7 for k = 6 and 1.1 for k = 4. The smoothing radius is therefore not small compared to the wavelength of the modes that dominate the correlator at these momenta, so the linear dependence assumed in Eq. (16) is not justified down to the lower bound for the largest k. The manuscript itself notes that the usable fit range shrinks at large k because deviations from linearity appear; the resulting systematic error is never quantified. I recommend a stability analysis that varies the lower bound, shows the effect of excluding the largest k values, or uses a non-linear ansatz to assess how much of the observed suppression is an artifact of the extrapolation.
  2. [Sec. 3, Fig. 3] No statistical or systematic errors are reported for any of the double-extrapolated correlators shown in the right panel of Fig. 3, nor for the continuum-limit fits in Fig. 2. The conclusion in Sec. 4 that the suppression is 'significant' for |p|/T ≈ O(10) cannot be checked without uncertainties. The revision should quote errors on the fits of Eqs. (15) and (16), propagate them through both extrapolations, and add the systematic uncertainty from the zero-smoothing fit-range choices. Without this, the central claim is not quantitatively supported.
  3. [Sec. 3, Eq. (15), Table 1] The continuum limit uses only three lattice spacings (N_t = 14, 16, 20) at fixed n_cool/N_t^2, and the scale uncertainties quoted in Table 1 (approximately 1.5% on the lattice spacing) are not propagated to the dimensionless ratio G_p/T^5. With three points and no quoted chi-squared or residuals, the statement that Eq. (15) 'well describes our data' is a weak test of the assumed O(1/N_t^2) behavior. The authors should provide the fit parameters with uncertainties, the chi^2/dof, and the sensitivity to including or excluding the coarsest lattice spacing.
minor comments (5)
  1. [Abstract and Sec. 4] The abstract and Sec. 4 present the computation as 'an extension of our recent determination' without flagging that the reported results are explicitly preliminary; I recommend adding a sentence stating that the numerical results are preliminary and that a full analysis including errors is in progress.
  2. [Sec. 3] There is a typo in Sec. 3 ('le left hand side plot' should be 'the left-hand side plot'); the text also uses 'the le left' earlier in the same passage.
  3. [Eq. (14) and caption of Fig. 2] The normalization of G_p(tT)/T^5 could be stated more explicitly: the relation between the integer vector k in Eq. (13), the lattice momentum p = 2π k/N_s, and the continuum momentum |p|/T = (π/2) k for N_s/N_t = 4 is only implicit; writing it out would improve reproducibility.
  4. [References] The bibliography contains typesetting artifacts such as '/zero.alt38' and '/u1D449' in several reference entries; these should be fixed in the final version.
  5. [Sec. 3, Fig. 3 (left)] For k = 6, the figure shows only a few filled points and a very short linear fit range; reporting the number of points and the chi^2/dof of each fit in the caption or text would help judge the quality of the extrapolation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the lattice measurement of the momentum-dependent topological correlator is independent of the external expectation it is compared with.

full rationale

The paper's central, explicitly preliminary observation is that the double-extrapolated lattice correlator G_p(tT)/T^5 is suppressed as |p|/T increases, and it compares this with the external theoretical expectation of Ref. [5]. The input data are raw lattice measurements of the clover topological charge density correlator at finite n_cool and N_t; the output is a smooth extrapolated curve. Nothing in Eqs. (15)-(16) defines the momentum suppression into the input, and no fitted parameter is renamed as a prediction: the authors do not even perform the Backus-Gilbert inversion to extract Gamma_top(p), only report the correlator. The zero-smoothing lower bound n_cool_min/N_t^2 = 0.012 is taken from Ref. [25], a self-citation, but that prior work was benchmarked against independent quenched determinations (Kotov, Mancha-Moore) and the current paper's Fig. 1 shows consistency with external results. Even if the extrapolation at large k is uncontrolled, as the skeptic notes, that is a systematic/correctness risk, not a circularity: the conclusion is not enforced by construction, by an unverified self-citation chain, or by renaming a known result. The comparison to Ref. [5] is an external, falsifiable benchmark, not an input. Hence no circular step is present.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The analysis uses standard spectral representation and cooling-based renormalization assumptions, plus an isotropy assumption to restrict to one momentum direction. The only fitted quantities are per-momentum extrapolation coefficients and a smoothing lower bound chosen by a plateau criterion; no physical free parameters or new entities are introduced.

free parameters (3)
  • Continuum extrapolation coefficient C_p(tT, rho_s) = not quoted
    Fitted coefficient in the O(1/Nt^2) continuum extrapolation, Eq. 15, for each momentum, time separation and smoothing radius; it determines the extrapolated correlator.
  • Zero-smoothing extrapolation slope s_p(tT) = not quoted
    Fitted slope in the linear zero-smoothing extrapolation, Eq. 16, for each momentum and time separation; the extrapolated value depends directly on it.
  • Minimal smoothing n_cool_min/Nt^2 = 0.012
    Chosen as the common value where the topological susceptibility starts exhibiting a plateau, setting the lower bound of the zero-smoothing fit range; the same value as Ref. [25].
assumptions (4)
  • domain assumption The Euclidean correlator can be related to the real-time rate through the Kubo formula and the spectral representation in Eqs. 4 and 10.
    Standard analytic continuation assumption in lattice thermal computations; the paper uses it to interpret the correlator as a proxy for the sphaleron rate.
  • domain assumption Cooling, followed by O(1/Nt^2) continuum extrapolation and linear zero-smoothing extrapolation in n_cool/Nt^2, yields the physical renormalized topological charge density correlator.
    Inherited from Refs. [25] and [43]; no independent renormalization check is performed in this paper.
  • domain assumption The lattice system is isotropic, so a momentum along the x-axis is representative of all directions.
    Section 3 states this to justify setting k_y = k_z = 0 without loss of generality.
  • domain assumption Reflection positivity implies the topological charge density correlator is negative for tT > 0.
    Used in Section 3 to validate the sign of the double-extrapolated correlator, citing Ref. [44].

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Cite this review

Pith. "Pith review of An update on the determination of the sphaleron rate in finite temperature QCD." pith.science (2026). https://pith.science/paper/7UZCITI3

@misc{pith2026241213685,
  author       = {Pith},
  title        = {Pith review of: An update on the determination of the sphaleron rate in finite temperature QCD},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7UZCITI3}},
  note         = {Machine review of arXiv:2412.13685}
}
read the original abstract

The sphaleron rate is a key phenomenological quantity both for the axion thermal production in the Early Universe and the Chiral Magnetic Effect occurring in the Quark-Gluon Plasma in presence of a background magnetic field. In this talk we present an extension of our recent determination of the sphaleron rate, in the SU(3) gauge theory, based on the determination of the two-point function of the topological charge density at finite temperature.

Figures

Figures reproduced from arXiv: 2412.13685 by the authors.

Figure 1
Figure 1. = 2 + 1 QCD sphaleron rate behaviour, represented with diamond points taken from Ref. [24], as a function of the temperature. A comparison with some previous quenched results is also done: square points are taken from Refs. [21, 22], round markers from Ref. [23] and, finally, the starred one from Ref. [25]. Left: x-axis expressed in terms of the absolute temperature in MeV. Right: x-axis expressed in terms of / , wi… view at source ↗
Figure 2
Figure 2. Continuum limit of the correlator ® [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Left: zero smoothing limit of ® (, cool/ 2 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

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Reference graph

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Reviewed August 11, 2026 · model on record in the stance chip above.