{"id":"11b1146d-1224-403d-9c80-d0324f6753aa","arxiv_id":"2412.13685","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"In quenched QCD at T = 1.24 Tc, the momentum-dependent topological charge correlator is suppressed at large |p|, consistent with a decreasing sphaleron rate for large axion momenta.","lead":"This paper finds, from supercomputer simulations of hot quark-gluon plasma, that the signal underlying the sphaleron rate becomes weaker at larger spatial momenta. This matters because the production of axions in the early universe depends on exactly that momentum dependence, which previous lattice studies had not measured.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The observed momentum suppression may be an artifact of the zero-smoothing extrapolation: at the minimum cooling level the smearing radius is comparable to the inverse momentum for the largest k, so the linear extrapolation in Eq. 16 is uncontrolled there.","rationale":"The reader's weakest assumption – that the double-extrapolation procedure (Eqs. 15 and 16) yields the physical correlator for every momentum – is correct as a general risk. My concern sharpens it with a quantitative scale argument: for the largest momenta, the minimum cooling radius used in the extrapolation is not small compared to the inverse momentum, so the linear zero-smoothing assumption is not justified in exactly the regime where the suppression claim is made. This does not invalidate the paper as a preliminary proceedings contribution; the authors explicitly label the results as preliminary and state that the inversion to the topological rate is planned. The qualitative statement is presented as a suggestion, not a demonstrated result. The reader's CONDITIONAL verdict already captures the need for additional control of the extrapolation, and this concern specifies one concrete condition that should be met before the suppression claim is accepted: the stability of the large-k extrapolated correlator under alternative fit forms and ranges. I therefore do not recommend changing the verdict, but I would make the condition explicit in the report. The paper deserves credit for using a well-established double-extrapolation framework, for showing the continuum extrapolations, and for clearly separating the correlator computation from the future inversion step.","tokens_in":12194,"tokens_out":6379,"duration_ms":60699,"concrete_test":"Reanalyze the existing k = 6 data with two alternative zero-smoothing extrapolations: (i) a quadratic fit in n_cool/N_t^2 over the original range, and (ii) a linear fit restricted to the two smallest n_cool values (0.0281 and 0.042). If the extrapolated G_p(tT = 0.5)/T^5 differs by more than one standard deviation from the published linear-fit value, or if it is no longer suppressed relative to k = 2, then the observed momentum dependence is not robust to the extrapolation assumption.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central qualitative claim – that the topological-charge-density correlator is suppressed with increasing 3-momentum – rests entirely on the double-extrapolated values G_p(tT)/T^5 shown in Fig. 3 (right). The zero-smoothing step uses Eq. 16, a linear fit in n_cool/N_t^2 with a fixed lower bound n_cool_min/N_t^2 = 0.012. However, this lower bound was chosen by demanding that the zero-momentum topological susceptibility exhibit a plateau; it is not a condition on the momentum modes being studied. The smearing radius at this bound is r_sm/a = sqrt((8/3) n_cool_min) = sqrt(0.032) N_t, while the lattice momentum for the largest shown k = 6 is p a = 2πk/N_s = πk/(2N_t) (since N_s/N_t = 4). Hence r_sm * p = (π/2) k sqrt(0.032) ≈ 0.281 k, which is 1.69 for k = 6 and 1.13 for k = 4. At these values the smoothing radius is not small compared to the wavelength of the modes that dominate a large-momentum correlator, so the dependence on n_cool need not be linear down to n_cool = 0. The deviations from linearity visible in Fig. 3 (left) for k = 6 – the shrinking of the usable fit range – are precisely the expected symptom. Because the suppression claim is strongest in exactly this regime, an uncontrolled extrapolation could make the suppression an artifact of cooling rather than a property of the topological rate.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":12567,"tokens_out":4626,"duration_ms":43720,"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":[{"comment":"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.","section":"Sec. 3, Fig. 3"},{"comment":"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.","section":"Sec. 3, Fig. 3"},{"comment":"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.","section":"Sec. 3, Eq. (15), Table 1"}],"minor_comments":[{"comment":"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.","section":"Abstract and Sec. 4"},{"comment":"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.","section":"Sec. 3"},{"comment":"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.","section":"Eq. (14) and caption of Fig. 2"},{"comment":"The bibliography contains typesetting artifacts such as '/zero.alt38' and '/u1D449' in several reference entries; these should be fixed in the final version.","section":"References"},{"comment":"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.","section":"Sec. 3, Fig. 3 (left)"}],"recommendation":"major_revision","confidential_remarks":"This is a proceedings contribution, so the bar for completeness is lower than for a full paper, but the central phenomenological claim — momentum suppression of the topological correlator — is stated without error bars and relies on a zero-smoothing extrapolation that is least controlled exactly where the suppression is strongest. These issues are fixable: the authors should add statistical errors, quantify the systematic from the fit-range choice, and temper the conclusion accordingly. I see no problems with attribution or novelty: the methodology is a direct extension of the group's previous work and the comparison with Ref. [5] is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a well-scoped proceedings contribution that computes, for the first time, the spatial Fourier transform of the topological charge density correlator at nonzero momentum in quenched SU(3) at finite T. The observed suppression with |p|/T is plausible, but the zero-smoothing extrapolation at the largest momenta is not under control, so the trend should be treated as suggestive.\n\nWhat is new: G_p(tT) for nonzero spatial momentum has not appeared before in SU(3) at finite temperature. The authors motivate it via the axion hot dark matter bound in Ref. [5] and give the Kubo formula they plan to invert to get the momentum-dependent rate. The data are genuinely new. The framework, cooling, and double extrapolation are inherited from their earlier zero-momentum work, so this is an incremental extension rather than a conceptual advance. Credit where due: the paper is honest about its preliminary status, says the result \"suggests\" rather than proves, and defers the inversion and full QCD to future work.\n\nThe soft spots are real and mostly concentrated in the high-momentum endpoints. No errors are quoted on the double-extrapolated correlator, the continuum limit uses only three lattice spacings, and the usable zero-smoothing range shrinks as k grows. The stress-test concern is quantitatively right: at n_cool/N_t^2 = 0.012, r_sm * p ≈ 0.28 k, which is 1.13 for k=4 and 1.69 for k=6. At those values the smoothing radius is not small compared to the mode wavelength, so a linear extrapolation in n_cool is uncontrolled there. The deviations from linearity in Fig. 3 (left) are the expected symptom, and the authors do not quantify the resulting systematic. On the other hand, the suppression is already visible at k=2 and k=3, where r_sm * p is 0.56 and 0.84 and the extrapolation is more believable, so I would not call the central claim an artifact. It needs a systematic study of the lower bound, a check with gradient flow, or a larger aspect ratio before I'd trust the magnitude.\n\nWho is this for: people working on sphaleron rates and axion thermal production will want to know this measurement exists. It is a useful progress report, not a final result. I'd tell the authors to put errors and a stability analysis into the next iteration. If this were submitted as a full journal paper, a serious referee should engage with it, but the current version is appropriately presented as a proceedings talk.","headline":"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.","tokens_in":13105,"tokens_out":3910,"would_cite":false,"duration_ms":33764,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["sphaleron rate","topological charge density","lattice QCD","finite temperature","quenched SU(3) gauge theory","axion thermal production","Euclidean correlator","spectral function inversion"],"falsifier":"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.","tokens_in":12026,"feed_emoji":"⚛️","tokens_out":8479,"duration_ms":68429,"temperature":0.7,"pith_summary":"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.","feed_headline":"Sphaleron rate drops as axion momentum grows","feed_subtitle":"New lattice data on topological charge correlations support a decaying axion production rate at high momenta.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the sphaleron-size expectation that the topological rate is flat below the sphaleron momentum and decays above it, which motivates the whole nonzero-momentum study.","marker":"[5]"},{"why":"Earlier quenched lattice determination of the sphaleron rate whose temperature and setup the present work extends.","marker":"[21]"},{"why":"Companion quenched determination used as a comparison point for the rate.","marker":"[22]"},{"why":"Independent Euclidean-lattice computation of the quenched sphaleron rate used as a baseline in Fig. 1.","marker":"[23]"},{"why":"The $N_f=2+1$ QCD sphaleron-rate determination that this extension builds on; its temperatures are the target of the planned full-QCD extension.","marker":"[24]"},{"why":"Introduces the modified Backus–Gilbert inversion and the double-extrapolation criteria (continuum at fixed smoothing radius, linear zero-smoothing) adopted here.","marker":"[25]"},{"why":"Provides the nonzero-momentum Kubo-type formula used to relate the Euclidean correlator to the momentum-dependent topological rate.","marker":"[26]"},{"why":"Establishes the linear zero-smoothing extrapolation in $n_{\\rm cool}/N_t^2$ used to remove the residual smoothing-radius dependence.","marker":"[43]"}],"fun_headline_variants":["Lattice QCD: sphaleron rate drops with momentum","Sphaleron rate in QCD not flat, falls at high p","Axion production check: sphaleron rate declines at large momenta","Momentum-dependent sphaleron rate from lattice QCD"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Lattice QCD: sphaleron rate drops with momentum","Sphaleron rate in QCD not flat, falls at high p","Axion production check: sphaleron rate declines at large momenta","Momentum-dependent sphaleron rate from lattice QCD"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000346,"raw_usage":{"total_tokens":1831,"prompt_tokens":814,"completion_tokens":1017,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":430,"completion_tokens_details":{"reasoning_tokens":941}},"tokens_in":430,"tokens_out":1017,"duration_ms":8948,"temperature":1.0,"reasoning_tokens":941,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:53:20.331306+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Notari, F","cited_arxiv_id":null,"evidence_quote":"Supplies the sphaleron-size expectation that the topological rate is flat below the sphaleron momentum and decays above it, which motivates the whole nonzero-momentum study."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier quenched lattice determination of the sphaleron rate whose temperature and setup the present work extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Companion quenched determination used as a comparison point for the rate."},{"cited_title":"Barroso Mancha and G","cited_arxiv_id":null,"evidence_quote":"Independent Euclidean-lattice computation of the quenched sphaleron rate used as a baseline in Fig. 1."},{"cited_title":"Bonanno, F","cited_arxiv_id":null,"evidence_quote":"The $N_f=2+1$ QCD sphaleron-rate determination that this extension builds on; its temperatures are the target of the planned full-QCD extension."},{"cited_title":"Bonanno, F","cited_arxiv_id":null,"evidence_quote":"Introduces the modified Backus–Gilbert inversion and the double-extrapolation criteria (continuum at fixed smoothing radius, linear zero-smoothing) adopted here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the nonzero-momentum Kubo-type formula used to relate the Euclidean correlator to the momentum-dependent topological rate."},{"cited_title":"Altenkort, A","cited_arxiv_id":null,"evidence_quote":"Establishes the linear zero-smoothing extrapolation in $n_{\\rm cool}/N_t^2$ used to remove the residual smoothing-radius dependence."}],"review_version":1}