Real-time topological rate at non-zero momentum in quenched QCD

Alessandra Valentino; Andrea Giorgieri; Claudio Bonanno; Francesco Sanfilippo; Giovanni Villadoro; Giuseppe Gagliardi; Massimo D'Elia; Roberto Dionisio

REVIEW 2 major objections 4 minor 115 references

Real-time topological rate at non-zero momentum in quenched QCD

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The real-time topological rate at non-zero momentum is computable from Euclidean lattice correlators; at T≈1.24T_c the on-shell rate grows linearly in momentum up to p/T≈9.4.

desk verdict First lattice values of the momentum-dependent topological rate, a plausible proof of concept — but the claimed linear rise leans on a smearing-width limit they never take and a subset fit. read the letter →

arxiv 2608.12066 v1 pith:6N3JBN63 submitted 2026-08-12 hep-lat hep-phhep-th

Claudio Bonanno , Massimo D'Elia , Roberto Dionisio , Giuseppe Gagliardi , Andrea Giorgieri , Francesco Sanfilippo , Alessandra Valentino , Giovanni Villadoro This is my paper · ORCID

classification hep-lathep-phhep-th PACS 12.38.Gc11.15.Ha

keywords topologicalratesphaleronlatticeQCDHLTmethodinverseproblemaxioncosmologymomentumdependencequenched

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

Quenched QCD at $T\simeq 1.24\,T_c$ — the pure-gauge theory without dynamical quarks — can be made to yield the momentum-dependent real-time topological rate $\Gamma_{\mathrm{top}}(\omega,p)$, the rate per unit volume at which the vacuum changes topological sector. The paper extracts it from the Euclidean time-correlator of the topological charge density by solving the ill-conditioned inverse problem with the Hansen–Lupo–Tantalo (HLT) method, after taking the continuum limit, the zero-smoothing-radius limit, and checking insensitivity to the HLT smearing width. For the on-shell case $\omega=p$, the data show $\Gamma_{\mathrm{top}}$ rising approximately linearly with momentum up to $p/T\simeq9.4$, with slope $c_1\simeq0.18$–$0.23$ in units of $T^4$. This matters because the momentum-dependent rate is a key input to the Boltzmann equation for axion production in the early universe, and it is the first non-perturbative lattice result for that momentum dependence.

What carries the argument

The argument runs on the Hansen–Lupo–Tantalo (HLT) method, a modification of the Backus–Gilbert approach that solves the inverse problem $G_E(\tau,p)=-\int_0^\infty \frac{d\omega}{\pi}\,\rho(\omega,p)K(\omega,\tau)$ by minimizing a functional that balances deviation of the reconstructed smearing kernel from a target of width $\sigma$ against statistical noise of the correlator. The inversion is done on $\rho(\omega,p)/\omega$, which is finite at $\omega=0$, and the rate is recovered through the Kubo relation $\Gamma_{\mathrm{top}}=f(\omega)\,2T\,(\rho/\omega)$ with $f(\omega)=(\omega/T)/(1-e^{-\omega/T})$. Finite smearing width enters through $\Gamma_{\mathrm{top}}|_\sigma = \Gamma_{\mathrm{top}} + C\sigma^2 + O(\sigma^4)$, which the paper uses to argue that the observed mild $\sigma$-dependence justifies quoting results at $\sigma/T=1.75$. The other two limits — continuum $a\to0$ and zero smoothing radius $R_s\to0$ of the cooled charge density — are taken either before or after the inversion, and the two routes cross-check each other.

What would settle it

Using the same HLT data already computed at $\sigma/T=1.25$ through $2.50$, perform the $\sigma\to0$ quadratic extrapolation $\Gamma_{\mathrm{top}}|_\sigma = \Gamma_{\mathrm{top}} + C\sigma^2$ at fixed $(\omega,p)$. If the extrapolated values move outside the quoted errors — especially at $p/T\gtrsim6$ — or if the linear slope $c_1$ of the on-shell rate changes by more than its error, the central claim of controlled error and linear growth is not established.

Watch

Extended reading notes

Core claim

The central claim is that the spectral density $\rho(\omega,p)$ of the topological charge density correlator can be recovered at non-zero momentum with controlled systematics by applying the HLT method to $G_E(\tau,p)$, and that the resulting rate $\Gamma_{\mathrm{top}}(\omega,p)=2\rho(\omega,p)/(1-e^{-\omega/T})$ is trustworthy for $\omega/T\lesssim10$ and $p/T\lesssim9.4$. The paper demonstrates this in quenched QCD at $T\simeq1.24\,T_c$ using three lattice spacings and an aspect ratio $LT=4$, in two independent orders of limits: first double-extrapolate the correlator then invert, or first invert then extrapolate the rate; both agree. The on-shell rate $\Gamma_{\mathrm{top}}(\omega=p,p)$ is found to grow approximately linearly with $p$, a behavior the paper shows arises because the growth of $\Gamma_{\mathrm{top}}$ with $\omega$ at fixed $p$ wins over its suppression with $p$ at fixed $\omega$. This matches the semiclassical expectation for asymptotically large momenta and temperatures, and it is the first lattice benchmark for the momentum dependence of the topological rate.

Load-bearing premise

The paper quotes all rates at a single HLT smearing width, $\sigma/T=1.75$, having seen only mild dependence on $\sigma$ and having not performed the $\sigma\to0$ extrapolation that its own quadratic scaling formula would allow.

Editorial extensions

If this is right

At $T\simeq1.24\,T_c$ the on-shell topological rate increases approximately linearly with $p$ up to $p/T\simeq9.4$, with a best-fit slope between $0.158$ and $0.226$ in units of $T^4$ depending on the fitting range.
Off-shell data show $\Gamma_{\mathrm{top}}$ growing with energy at fixed momentum and decreasing with momentum at fixed energy; the on-shell rise is the net balance of these two effects.
The two independent orders for taking the continuum and zero-smoothing limits give compatible determinations, indicating that the main systematic effects are under control.
The continuum extrapolation of the topological susceptibility is independent of the smoothing radius and agrees with a previous gradient-flow determination at the same temperature, validating the chosen smoothing radii.

Reading between the lines

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

If the linear growth persists in full QCD, axion production in the early universe has to be treated with a momentum-dependent rate: high-momentum axions would be produced relatively faster than the zero-momentum sphaleron rate alone would suggest.
The HLT pipeline demonstrated here is portable: any Euclidean correlator with resolvable spectral density, such as energy-momentum or conserved-current correlators at non-zero momentum, could be subjected to the same three-limit control.
A direct test of the observed linearity would be a second temperature, e.g., $T\simeq2\,T_c$: if the slope does not follow the semiclassical scaling $p\,(\alpha_s T)^3$ with $\alpha_s(T)$, the linear rise at $T\simeq1.24\,T_c$ is an infrared phenomenon rather than the onset of the asymptotic regime.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Real talk: this is the first time anyone has gotten non-zero-momentum values of Γ_top from the lattice, and that alone makes it worth a careful look. The pipeline is inherited from Refs. [71,72] — HLT inversion plus cooling — but the extension to p≠0 is real new content, and they do it honestly: two independent routes (invert-then-extrapolate vs extrapolate-then-invert) agree, the continuum limit over Nτ=12,14,16 is smooth, and the topological-susceptibility check against Ref. [106] anchors the smoothing-radius window. The off-shell data are a nice bonus; they show Γ_top grows with ω at fixed p and falls with p at fixed ω, which explains why the on-shell rate can rise even though the Euclidean correlator is suppressed in p.

Now the soft spots, in proportion. The abstract says they control the 'limit of vanishing smearing width', but they never take it. They scan σ/T in [1.25,2.5], see mild dependence, and quote σ/T=1.75. Eq. (34) gives the functional form for a σ→0 extrapolation; they just don't do it. The stress-test note is right: C(p) need not be small or p-independent, and since the HLT kernel smears in ω, the on-shell point gets positive admixture from ω>p. If C(p) grows with p, the unremoved dC/dp·σ² could masquerade as the observed linear slope. Their flatness observation over a factor-4 range in σ², with 15-40% errors, cannot bound a common offset of order 0.1-0.3 T⁴. This is the load-bearing assumption, and it deserves a real response.

Second: the 'linear increase' claim is shakier than the abstract suggests. The all-data linear fit has reduced χ²≈2 (10.2/5); you need to drop p/T≤1 to get something acceptable, and the power-law fit gives α=1.20(11), which is compatible with linear but also with a mild bend. So the data support 'approximately linear, at least for p/T≳2' — not a clean linear law across the board.

Minor stuff: no data/code release, and Method 2 misses the highest momentum. Both are annoying but not fatal.

Bottom line: this is a solid proof of concept that deserves a serious referee. The σ→0 issue is addressable — either do the extrapolation or bound C(p) — and the wording on linearity should be softened. If that gets fixed, it is a genuinely useful stepping stone toward the full-QCD calculation that axion cosmology actually needs. Yes, send it to review.

Referee Report

2 major / 4 minor

Summary. The manuscript presents a proof-of-concept lattice calculation of the real-time topological rate Γ_top(ω,p) at T≈1.24T_c in quenched QCD, using the HLT inverse-problem method applied to Euclidean correlators of the smoothed topological charge density at nonzero spatial momentum, with LT=4 and Nτ=12,14,16. Two extraction routes are used: Method 1 double-extrapolates the correlator to the continuum and to zero smoothing radius before the HLT inversion; Method 2 inverts finite-lattice correlators and takes the limits directly on Γ_top. The main physics result is the on-shell rate Γ_top(ω=p,p), which the authors find to grow approximately linearly with p/T up to p/T≈9.4, with off-shell results showing growth in ω at fixed p and suppression in p at fixed ω. The paper also reports a continuum-extrapolated topological susceptibility χ/T_c^4=0.0363(87) consistent with Ref. [106], and an appendix cross-checks the time-reversal-breaking sequential cooling implementation against gradient flow and randomized cooling.

Significance. If the extracted values are reliable, this is the first non-perturbative lattice estimate of the momentum-dependent topological rate, and it provides useful input for momentum-dependent axion-production calculations. The study has genuine strengths: two independent analysis routes agree within errors; the continuum limits are smooth and nearly flat; the smoothing-radius window is anchored by a susceptibility cross-check against an independent determination; the error budget includes statistical and HLT λ-stability fluctuations; and the cooling-systematics check in Appendix A is careful. The main caveat is that the absolute values and the 'linear increase' conclusion rest on a fixed HLT smearing width rather than on the σ→0 extrapolation advertised in the method, and the linear-fit evidence is less stable than the abstract suggests; for these reasons the central claim is plausible but not yet fully established.

major comments (2)

[Sec. III B, Eq. (34), Fig. 8] The paper's stated protocol is to control the vanishing-smearing-width limit, and Eq. (34) explicitly provides the quadratic-in-σ extrapolation Γ_top|_σ = Γ_top + Cσ² + O(σ⁴), yet the final values are quoted at σ/T=1.75 solely because the dependence over σ/T∈[1.25,2.5] is described as 'rather mild'. This is not a control of the σ→0 limit. Because the off-shell rate grows steeply with ω at fixed p (Fig. 9), the smeared on-shell value receives positive contamination from ω>p, and the coefficient C in Eq. (34) should be treated as a function of p; a p-dependent O(σ²) shift could therefore directly mimic or distort the claimed linear rise in Sec. IV. The observed flatness over a factor-four range in σ² with relative errors of 15–40% cannot exclude a common offset of order 0.1–0.3 T⁴. The authors should either perform the σ→0 extrapolation that Eq. (34) allows, or explicitly quote the results as σ/T=1.75-smoothed rates with an estimated σ-systematic propagated through the momentum-dependence analysis.
[Sec. III D, Eqs. (43)-(47), Table II] The evidence for the headline linear increase is not as robust as the abstract's wording suggests. The fit to all seven points has reduced χ²=10.2/5, while the slopes for p/T≥0, ≥1, and ≥2 are c1=0.180(13), 0.226(24), and 0.158(43), respectively, central values that vary by more than the quoted errors on individual fits; the power-law fit gives α=1.20(11) with χ²/dof=6.76/4. This fit-range dependence, combined with the fact that the selected fit in Fig. 12 excludes the p=0 point, means the 'linear increase' conclusion is dominated by a subset of the data. The paper should report a scan of c1 as a function of the minimum p/T and add a model-averaged or fit-range systematic to the slope before claiming a linear increase.

minor comments (4)

[Sec. III C] The text says the results of Sec. III A (Method 1) are being cross-checked, but Method 1 is described and applied in Sec. III B; the cross-reference should be corrected.
[Table II, note] The note contains a typo: 'For method 2, we could not provide a result with Method 2 due to...' should read 'For method 2, we could not provide a result for p/T≈9.4 because this momentum is close to the cutoff scale for Nτ=12'.
[Fig. 12 caption] The right panel divides the data by f(ω); the caption should state explicitly that this division removes the kinematic prefactor of Eq. (20) so that Method 1 and Method 2 are compared on the same quantity.
[Sec. II C] The sentence introducing Eq. (34) states that one can use the quadratic prediction to perform a σ→0 extrapolation, but Sec. III B does not perform it; the wording should be aligned with what is actually done, or the extrapolation should be added.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the non-zero-momentum topological rate is obtained by inverting lattice Euclidean correlators, with no target result fed back into the extraction.

full rationale

The central claim, the momentum dependence of the real-time topological rate, is produced by a genuine inverse-problem pipeline: lattice Monte Carlo data for the Euclidean correlator G_E(τ,p) are the independent input; the HLT inversion reconstructs ρ(ω,p)/ω; and Γ_top(ω,p) follows from the Kubo-type formula in Eq. (5). No fitted parameter is renamed as a prediction, and no target observable is inserted into the inversion. The quoted values at σ/T=1.75 without a σ→0 extrapolation are a systematic-uncertainty concern, not a circularity, because the σ-dependence is checked empirically and Eq. (34) is only used to motivate a mild quadratic dependence. The self-citations to Refs. [71,72] supply the method, but that method's zero-momentum control was previously validated against the independent determination in Ref. [36], so the citation is real evidence rather than a load-bearing self-citation. The perturbative formula Eq. (42) is used only for qualitative comparison after the fact; the reported linear slope c1=0.158(43) for p/T≥2 comes from a fit to the lattice-extracted on-shell points, not from Eq. (42). The off-shell results (Fig. 9) are also independent outputs used to interpret the on-shell behavior, not inputs that enforce it. Accordingly, the derivation chain is self-contained with respect to the claimed predictions.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard finite-temperature field theory (Kubo formula), the HLT inversion method with user-chosen smearing width, and two extrapolation ansätze. No invented physical entities are introduced. The main free choices are analysis parameters (σ, λ) and the linear/quadratic extrapolation forms.

free parameters (4)

HLT target smearing width σ/T = 1.75 (center of [1.25, 2.5])
Input to the inverse problem; the paper checks that results are flat in σ and quotes at 1.75 instead of extrapolating to σ=0.
HLT regularization parameters λ1, λ2 = chosen per reconstructed-kernel figure of merit dλ
λ1 sets the central value and statistical error; λ2 sets the systematic error via Eq. (31). Values vary by (ω,p); they are analysis choices, not physics inputs.
Continuum extrapolation coefficient C1(τ,p;R_s) = fitted, not quoted
Linear-in-(aT)^2 fit coefficient in Eq. (37); a nuisance parameter of the double-limit procedure.
Zero-smoothing extrapolation coefficient C2(τ,p) = fitted, not quoted
Quadratic-in-(R_s T)^2 fit coefficient in Eq. (38); nuisance parameter of the double-limit procedure.

assumptions (5)

standard math Kubo formula connecting the Euclidean correlator to the spectral density and the topological rate (Eqs. 5-7)
Standard finite-temperature field theory relation; the inversion is applied to it.
domain assumption Reflection positivity of the continuum topological-charge correlator, G_E(τ,p)<0 for τ>0
Used in Sec. II B to justify the ordering of limits: continuum first, then R_s→0. Fails for smeared lattice sources at τ<R_s.
domain assumption Gradient-flow theorem: the topological-charge density correlator at small flow time receives linear corrections in t_GF, hence quadratic in R_s (Refs. [109,110])
Basis for the R_s→0 extrapolation form in Eq. (38), assumed to hold for cooling after the matching Eq. (19).
domain assumption Hansen-Lupo-Tantalo inversion reconstructs the smeared spectral density with controlled bias when dλ is small
The whole extraction rests on HLT delivering a kernel ∆(ω,ω*) close to the target δσ; validated only through internal stability analysis, not by an external benchmark in this paper.
domain assumption On-shell identification ω=p for axion-like particles (Eq. 35)
Phenomenological input from axion physics (mass neglected), used to select ω*=p; not needed for the raw lattice extraction but for the final physical interpretation.

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Cite this review

Pith. "Pith review of Real-time topological rate at non-zero momentum in quenched QCD." pith.science (2026). https://pith.science/paper/6N3JBN63

@misc{pith2026260812066,
  author       = {Pith},
  title        = {Pith review of: Real-time topological rate at non-zero momentum in quenched QCD},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6N3JBN63}},
  note         = {Machine review of arXiv:2608.12066}
}

abstract

We present a proof-of-concept numerical study of the real-time topological rate at non-zero momentum in quenched lattice QCD at a temperature $T\simeq 1.24 \, T_c \simeq 360$ MeV, as an important step toward the determination of this quantity in full QCD. Our strategy, already applied to compute the sphaleron rate in pure Yang--Mills and in full QCD, extracts the rate from the resolution of an appropriate inverse problem, solved applying the Hansen--Lupo--Tantalo (HLT) method to the thermal Euclidean time-correlator of the topological charge density. This method requires to control three different limits: continuum limit, limit of vanishing smearing width used in the HLT inverse problem resolution, and limit of vanishing smoothing radius used in the topological charge density correlator computation. Our lattice calculation is based on the standard Wilson discretization for the gauge action, and on three gauge ensembles with up to $N_\tau=16$ temporal points to achieve a controlled continuum limit. In all cases we employed an aspect ratio $LT=4$, which allowed us to compute the topological rate up to momenta as large as $p/T \sim 10$.

Figures

Figures reproduced from arXiv: 2608.12066 by the authors.

**Figure 2.** As it can be seen, in all cases we obtain a smooth continuum limit which turns out, as expected, to be independent of Rs . Our final result reads χ/T 4 c = 0.0363(87). 0.2 0.3 0.4 0.5 RsT 0.00 0.01 0.02 0.03 0.04 0.05 0.06 χ/T 4 c Nτ = 12 Nτ = 14 Nτ = 16 FIG. 1: Dependence of the finite-lattice-spacing determinations of the topological susceptibility χ/T 4 c on the smoothing radius RsT. 0.000 0.002 0.004 0.006 0.0… view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 9.** Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]

**Figure 7.** Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

**Figure 8.** Figure 8: This is in agreement with the general theoreti [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]

**Figure 10.** Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]

**Figure 11.** Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]

**Figure 12.** Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p013_12.png]

**Figure 13.** Figure 13: (central and right panels). Thus, we conclude that the breaking of time-reversal symmetry of sequential cooling does not introduce any artifact in the calculation of the real part of the correlator at finite smoothing radius [PITH_FULL_IMAGE:figures/full_fig_p014_13.png]

Discussion (0). Continue with ORCID to comment.

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