{"id":"f991d61b-abb3-4967-ad31-e6ac20a9381b","arxiv_id":"2608.12066","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A first-principles lattice calculation in quenched QCD extracts the momentum-dependent topological rate and finds approximate linear growth of the on-shell rate with momentum.","lead":"This paper reports the first lattice-QCD computation of the real-time topological transition rate at non-zero momentum, using the Hansen-Lupo-Tantalo inversion method at a temperature of about 360 MeV. It finds the on-shell rate grows roughly linearly with momentum up to p/T about 10, a step toward momentum-dependent axion production inputs.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Fixed-σ smearing is the load-bearing assumption: without a σ→0 extrapolation, a p-dependent O(σ²) shift could mimic the claimed linear rise.","rationale":"The reader's CONDITIONAL verdict is appropriate, and the σ→0 issue is indeed the weakest load-bearing assumption. I agree with the reader that the absence of a σ→0 extrapolation is the primary unresolved systematic; my pass sharpens it: the O(σ²) coefficient may depend on p, and through that p-dependence it can directly shape the central physics claim of a linear rise. The paper deserves credit for the cross-checks it does include: Method 1 and Method 2 agree where both are available; the topological susceptibility extrapolates to a continuum value consistent with Ref. [106]; the imaginary-part and cooling-implementation checks in Appendix A are careful; and the off-shell results make the on-shell interpretation more transparent than a single-curve extraction would be. Those checks strengthen confidence in the proof of concept, but they do not remove the fixed-smearing systematic, since both methods quote results at the same σ/T=1.75. A secondary concern is the fragility of the linearity claim: the all-data linear fit has reduced χ² ≈ 2.0, the slope moves from 0.180(13) to 0.226(24) to 0.158(43) as low-momentum points are excluded, and the power-law exponent α=1.20(11) is compatible with but not a sharp test of linearity. I do not treat this as a separate fatal objection, because the paper is explicitly a proof of concept and the quoted rate values remain the primary deliverable; however, it reinforces the need for the σ→0 check before the linear-rise statement is used. The proposed quadratic-in-σ² test can be run entirely on data the authors already possess, and it directly determines whether the fixed-σ results and the p-dependent slope survive an extrapolation of Eq. (34).","tokens_in":24703,"tokens_out":9550,"duration_ms":95093,"concrete_test":"For each (ω,p), including all on-shell points and representative off-shell pairs, recompute Γ_top at σ/T = 1.25, 1.50, 1.75, 2.00, 2.25, 2.50 with the same bootstrap resampling, then fit Γ_top(σ;p) = A(p) + B(p)(σ/T)², propagating the full covariance across σ. Compare A(p) with the quoted σ/T=1.75 values; if A(p) differs by more than the quadrature sum of statistical and λ-systematic errors for any p, the fixed-σ numbers are biased. Then recompute the on-shell slope c1 from A(p), e.g. the p/T≥2 fit reported as c1 = 0.158(43) in Sec. III D; if the slope shifts by more than about one standard deviation, the claimed linear increase is not established by the current extraction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Sec. III B quotes final Γ_top(ω,p) at σ/T=1.75 after observing a mild dependence on σ/T∈[1.25,2.5], but does not perform the σ→0 extrapolation allowed by Eq. (34), Γ_top|_σ = Γ_top + Cσ² + O(σ⁴). This is the load-bearing point for both the absolute values and the 'linear increase' conclusion. The coefficient C is not necessarily small, nor is it necessarily common to all (ω,p). Because the HLT target kernel smears in ω while the on-shell extraction sets ω*=p, C is a function of p, C=C(p). At fixed p, the paper's own off-shell results show Γ_top growing steeply with ω (Fig. 9), so the smeared on-shell value receives positive admixture from ω>p. If C(p) grows with p, the observed slope dΓ_top/dp contains an unremoved contribution dC/dp·σ². The quoted flatness across σ/T∈[1.25,2.5] tests only a quadratic over a factor-4 range in σ² and, with relative errors of 15–40%, cannot bound a common offset of order 0.1–0.3 T⁴. Method 2 also uses σ/T=1.75, so the Method 1/Method 2 agreement does not mitigate this concern.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":24988,"tokens_out":6112,"duration_ms":58443,"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":[{"comment":"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.","section":"Sec. III B, Eq. (34), Fig. 8"},{"comment":"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.","section":"Sec. III D, Eqs. (43)-(47), Table II"}],"minor_comments":[{"comment":"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.","section":"Sec. III C"},{"comment":"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'.","section":"Table II, note"},{"comment":"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.","section":"Fig. 12 caption"},{"comment":"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.","section":"Sec. II C"}],"recommendation":"major_revision","confidential_remarks":"To the editor: The paper is a solid proof-of-concept and the technical apparatus is appropriate, but the headline 'linear increase' is at risk of being overclaimed relative to the missing σ→0 control and the fit-range sensitivity. The revision should be judged on whether the authors either perform the σ→0 extrapolation suggested by Eq. (34) or explicitly downgrade the claim to 'compatible with linear growth at fixed σ/T=1.75'. I see no circularity or novelty problem: the self-references [71,72] provide the method, and the application to p≠0 is a genuine extension."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nNow 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.\n\nSecond: 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.\n\nMinor stuff: no data/code release, and Method 2 misses the highest momentum. Both are annoying but not fatal.\n\nBottom 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.","headline":"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.","tokens_in":25587,"tokens_out":2812,"would_cite":true,"duration_ms":25167,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["12.38.Gc","11.15.Ha"],"model":"deepseek-v4-flash","headline":"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.","keywords":["topological rate","sphaleron rate","lattice QCD","HLT method","inverse problem","axion cosmology","momentum dependence","quenched QCD"],"falsifier":"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.","tokens_in":24488,"feed_emoji":"🌀","tokens_out":16699,"duration_ms":127633,"temperature":0.7,"pith_summary":"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.","feed_headline":"Topological rate grows linearly with momentum, lattice QCD shows","feed_subtitle":"It is the first non-perturbative lattice determination of these rates up to p/T ~ 10, a key input for axion cosmology.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduces the Hansen–Lupo–Tantalo inversion method, the algorithm used to recover the spectral density from the Euclidean correlator.","marker":"[55]"},{"why":"Establishes the zero-momentum HLT strategy for the sphaleron rate and the line of constant physics that this paper extends to non-zero momentum.","marker":"[71]"},{"why":"Shows the same HLT strategy works in full QCD for the sphaleron rate, providing the precedent for the momentum-dependent extension.","marker":"[72]"},{"why":"Supplies the physical target: the momentum-dependent topological rate as the key input to the Boltzmann equation for axion thermal production.","marker":"[1]"},{"why":"Provides the previous pure-gauge sphaleron rate determination against which the HLT zero-momentum strategy was cross-checked.","marker":"[36]"},{"why":"Supplies the continuum topological susceptibility at the same temperature, used to validate the range of smoothing radii.","marker":"[106]"},{"why":"Fixes the lattice spacing through the Sommer scale, setting the physical temperature and momentum scales.","marker":"[73]"},{"why":"Provide the asymptotic perturbative formula $\\Gamma_{\\mathrm{top}}(p)\\propto p\\,(\\alpha_s T)^3[A\\ln(pT/m_D^2)+B]$ that the lattice data are compared with for the linear-in-$p$ growth.","marker":"[113, 114]"}],"fun_headline_variants":["Lattice QCD: topological rate rises linearly with momentum","First lattice measure of topological rate at high momentum","Quenched QCD: topological rate scales with momentum","Topological rate's momentum growth seen in lattice QCD","Non-perturbative benchmark: topological rate vs momentum"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Lattice QCD: topological rate rises linearly with momentum","First lattice measure of topological rate at high momentum","Quenched QCD: topological rate scales with momentum","Topological rate's momentum growth seen in lattice QCD","Non-perturbative benchmark: topological rate vs momentum"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000502,"raw_usage":{"total_tokens":2490,"prompt_tokens":1021,"completion_tokens":1469,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":637,"completion_tokens_details":{"reasoning_tokens":1391}},"tokens_in":637,"tokens_out":1469,"duration_ms":9458,"temperature":1.0,"reasoning_tokens":1391,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:17:58.182947+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}