{"id":"e1e61f92-1067-4235-be18-2f627d8b1e3d","arxiv_id":"2411.11640","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"Tree-level improvement of flowed Wilson loops reduces lattice and flow-time errors, giving preliminary 1/m_Q and 1/m_Q^2 corrections to the static quark-antiquark potential.","lead":"This paper uses lattice QCD with gradient flow to compute the first relativistic corrections to the force between a heavy quark and antiquark. The authors show that a tree-level correction removes most flow and lattice errors, and their preliminary checks satisfy known consistency relations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Tree-level improvement of field-strength correlators rests on an untested assumption that the coefficient \\tilde c extracted from the static potential equals the correlator's artifact coefficient.","rationale":"The reader's weakest_assumption already identified the transferability of \\tilde c from the static potential to the field-strength correlators as the critical point, and my concern is precisely that. The paper gives qualitative evidence that the improved correlators look better, but no quantitative check that the coefficient used in Eq. (12) is the correct one. Since the reader's verdict is CONDITIONAL, and the missing comparison is a reproducibility/verification item rather than a demonstrated logical contradiction, the verdict should remain CONDITIONAL. I do not see an internal inconsistency in the static-potential improvement itself; the Gromes/BBMP relation check is a positive cross-check for the field-strength correlator approach. The weakness is the unverified universality of the tree-level subtraction coefficient, which is exactly the kind of assumption that must be tested before the proposed method can support controlled continuum and zero-flow-time extrapolations of the 1/m_Q corrections. The concrete test proposed here would settle the matter with the data already in hand.","tokens_in":7748,"tokens_out":9742,"duration_ms":100508,"concrete_test":"From the same unimproved correlator data used in Fig. 5 (all three ensembles and the four flow times), perform a single global fit of Eq. (12) with \\tilde c as a free parameter, using the same fit ansaetze as in Fig. 3 for the improved correlator. Report the best-fit \\tilde c with its uncertainty, and compare quantitatively with \\tilde c^{(0)} from the static-potential fit of Eq. (7). Check stability of the fitted \\tilde c under variations of the fit range, in particular excluding r < 2 r_f and t < r_f where flow effects are strongest. If the two coefficients agree within combined errors and the improved correlator is independent of the fit range, the transferability assumption is supported; if they disagree, the tree-level improved correlators carry an uncontrolled systematic error of order \\alpha_s.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the identification of \\tilde c in Eq. (12) with \\tilde c^{(0)} from the static-potential fit of Eq. (7). The proportionality implied by Eq. (12) is exact only at tree level. Beyond tree level, the coefficient multiplying the tree-level lattice-flow artifact shape can differ between V^{(0)} and the field-strength correlator: the flowed operator insertions carry their own flow-time and renormalization dependence, and the fitted \\tilde c^{(0)} can absorb contamination from the \\sigma r term or from higher-order corrections specific to the static potential. The paper states that a global fit of \\tilde c from the correlator data is possible, and that using \\tilde c = \\tilde c^{(0)} gives 'similar quality', but it does not report the fitted \\tilde c value, its uncertainty, or a quantitative comparison with \\tilde c^{(0)}. Without such a comparison, the improved small-r/t correlator data in Fig. 5 may be biased by an O(\\alpha_s) coefficient-mismatch that is not visible in the plots. This matters because the actual 1/m_Q corrections in Sec. 4.3 are not yet tree-level improved; the central claim for controlled extrapolations of those corrections therefore depends on the transferability being valid, which is presently an assumption rather than a demonstrated result.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper reports preliminary lattice QCD results for the spin-dependent and spin-independent O(1/m_Q^2) corrections to the static potential, obtained from generalized Wilson loops with two field-strength insertions renormalized via gradient flow. The main methodological content is a tree-level improvement prescription: for the static potential, the lattice data are fitted with the Cornell ansatz plus a correction term proportional to the difference between lattice and continuum tree-level expressions, Eq. (7); for field-strength correlators, an analogous subtraction is proposed in Eq. (12), with the coefficient \\tilde c either fitted globally or identified with the static-potential parameter \\tilde c^{(0)}. The static-potential improvement is demonstrated on three ensembles and four flow times. For the 1/m_Q^2 potentials, the paper shows fits for four selected contributions on one ensemble (and, for one of them, all ensembles and flow times) and checks the Gromes and BBMP relations at one flow time. The authors explicitly state that the results are preliminary and that continuum and zero-flow-time extrapolations of the potential corrections are still ongoing.","tokens_in":8065,"tokens_out":7679,"duration_ms":80448,"significance":"If the tree-level improvement program works as claimed, it would provide a practical route to controlled small-r and small-t data for the 1/m_Q and 1/m_Q^2 static-potential corrections, addressing a long-standing difficulty with renormalization and signal-to-noise in field-strength correlators. The static-potential demonstration in Sec. 4.2, where flow-time dependence is visibly reduced after the subtraction, is encouraging and is supported by data on three ensembles. The use of gradient flow also avoids the approximate Huntley-Michael renormalization that was a concern in earlier work. However, the central new step for the correlators, Eq. (12), rests on an assumption about the transferability of the coefficient \\tilde c from the static potential to the field-strength correlators, and the manuscript does not yet provide the quantitative evidence needed to validate that assumption. As it stands, this is a promising proceedings contribution rather than a completed validation of the method.","major_comments":[{"comment":"The identification of the correlator-improvement coefficient \\tilde c with the static-potential fit parameter \\tilde c^{(0)} from Eq. (7) is not demonstrated. The text states that using \\tilde c = \\tilde c^{(0)} leads to improved correlators of 'similar quality' to a global fit of \\tilde c, but it reports neither the fitted value of \\tilde c, its uncertainty, nor a quantitative comparison with \\tilde c^{(0)}. Since Eq. (12) is an exact relation only at tree level, and the flowed field-strength insertions carry their own flow-time and renormalization dependence, the equality \\tilde c = \\tilde c^{(0)} is a load-bearing assumption for the claim that the improved small-t correlator data are trustworthy. Please report the global correlator fit for \\tilde c and compare it with \\tilde c^{(0)} explicitly, or recast the improved-correlator results as conditional on this transferability assumption.","section":"Sec. 4.4, Eq. (12)"},{"comment":"The statement that NRQCD matching coefficients 'differ from 1 at O(α2)' appears inconsistent with the cited NLO matching calculation of Ref. [19], which computes one-loop, i.e. O(α_s), corrections to the relevant matching coefficients. If the sentence is intended to justify ignoring matching coefficients in a tree-level improvement, it should be phrased as a tree-level statement; as written, it is a statement about the matching coefficients themselves and is inaccurate.","section":"Sec. 4.4, first paragraph"}],"minor_comments":[{"comment":"The paper claims that the long-range parameter g_Λ' is 'determined quite accurately' and that the long-range term is 'crucial' for V_LS^(1,1), but it does not report the fitted parameter values, uncertainties, or χ²/DOF for any of the ansaetze in Fig. 3. Please provide these numbers in a table or in the text so that the claims can be checked.","section":"Sec. 4.3, Fig. 3"},{"comment":"The conclusion that violations of the Gromes relation in Ref. [14] 'were mostly caused by approximations in the Huntley-Michael renormalization prescription' goes beyond what can be concluded from a single ensemble and a single flow time without continuum extrapolation. The authors correctly note the lack of extrapolation, but the interpretive statement should be softened or accompanied by a controlled comparison.","section":"Sec. 4.3, Fig. 4 and surrounding text"},{"comment":"The tree-level improved correlator data are shown for only one spatial separation, r ≈ 0.24 fm, and the claim about reliable data at 'significantly smaller r and t' is explicitly deferred to future work. It would be helpful to state clearly in Sec. 4.4 that the small-r part of the claim is not yet demonstrated in this paper.","section":"Sec. 4.4, Fig. 5"},{"comment":"The two-step fitting procedure involving first a fit with fixed c~(0) = 0 (Fig. 2, left) and then the full ansatz with c~(0) free (Fig. 2, right) is described only briefly. A more explicit description of how c~(0) is determined and whether the small-r points are included in that determination would help the reader assess the significance of the collapse shown in the right panel.","section":"Sec. 4.2 and Fig. 2"},{"comment":"There are minor typographical errors, e.g. 'apprroximately' in Sec. 4.3, and the caption of Fig. 2 refers to a vertical grey line without defining its meaning in the caption; these should be corrected.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"This is a proceedings-style preliminary report, and the main methodological gap is easy to fix: report the global correlator fit for \\tilde c and compare it with \\tilde c^{(0)}. If the fitted values agree within uncertainties, the central claim about tree-level improved correlators would be substantially strengthened. If they do not agree, the improved-correlator results should be presented as relying on an untested assumption. The comparison with Koma and Koma should also be moderated. The paper is within scope for a lattice proceedings volume."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a worthwhile proceedings paper. The genuinely new pieces are the tree-level improvement of two-field-strength correlators (Eq. 12) and the first extraction of g_Lambda' with an error estimate. The static-potential side is the most convincing: the ansatz (7) removes flow-time dependence at small r across three ensembles and four flow times, and the Gromes/BBMP consistency checks for one ensemble are a nice sanity check. That part deserves credit.\n\nThe soft spots are the usual ones for a LATTICE proceedings, plus one more specific gap. The 1/m_Q corrections themselves are not yet tree-level improved, there is no continuum or zero-flow-time extrapolation, and fit ansaetze and parameters are only shown in figures without quoted uncertainties. That is fine if you treat this as a methods report, and the authors say it is.\n\nThe specific worry I'd flag is the identification of tilde c in Eq. (12) with tilde c^(0) from the static-potential fit. The paper says the improved correlators are of 'similar quality' using that identification, but it never reports the fitted value of tilde c from the correlator data or a quantitative comparison with tilde c^(0). Since the whole point is to subtract the tree-level artifact shape, and beyond tree level the coefficient can in principle differ between the static potential and the correlators, this is an untested assumption. It is not a fatal flaw, and it is fixable — report the fitted coefficient and show the comparison — but until then the improved small-r correlator data carry an unquantified systematic.\n\nThe citation pattern looks fine: prior work by the same group and by Brambilla et al. is acknowledged appropriately. No data release accompanies the paper, which limits reproducibility but is normal for a proceedings note.\n\nBottom line: if you work on quarkonium potentials or gradient-flow methods, this is worth a quick read and worth engaging at the conference level. As a referee, I'd send it out; the method is plausible, the presentation is honest, and the open question is a clear request for more information rather than a sign of a broken argument.","headline":"A solid, clearly preliminary proceedings paper showing tree-level improvement works for the static potential and plausibly extends to field-strength correlators; the main open question is whether the improvement coefficient transfers.","tokens_in":8584,"tokens_out":2393,"would_cite":true,"duration_ms":22392,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["11.15.Ha","12.38.Gc"],"model":"deepseek-v4-flash","headline":"Tree-level improvement removes the dominant lattice and flow-time errors from the Wilson loops behind the 1/m_Q^2 corrections to the static potential.","keywords":["static potential","heavy quark mass corrections","gradient flow","tree level improvement","generalized Wilson loops","lattice gauge theory","NRQCD","Gromes relation"],"falsifier":"Compute the tree-level-improved correlator (12) on a fourth lattice ensemble with roughly half the lattice spacing of the coarsest ensemble used here at the same physical $r$ and $t_f$; if the small-$t$ data still shift by more than the statistical errors when $a$ and $t_f$ are reduced, the tree-level subtraction is not capturing the dominant artifact.","tokens_in":7504,"feed_emoji":"🌀","tokens_out":12748,"duration_ms":114181,"temperature":0.7,"pith_summary":"These proceedings describe a way to compute the $O(1/m_Q)$ and $O(1/m_Q^2)$ corrections to the static quark-antiquark potential that appear in heavy-quark effective theories. The corrections are encoded in generalized Wilson loops with two field-strength insertions, which are difficult because the insertions require renormalization and the correlators have large fluctuations. The authors use gradient flow to make the field-strength operators finite without extra renormalization, and then apply tree-level improvement: for each quantity they subtract the difference between its lattice and continuum tree-level expressions at finite flow time. After this subtraction the static potential from three lattice spacings and four flow times agrees down to $r \\approx 0.05$ fm, and a chromoelectric-field correlator shows almost no flow-time dependence at small times. The paper concludes that tree-level improvement is a successful strategy for combined continuum and zero-flow-time extrapolations of the potential and of its mass corrections.","feed_headline":"Tree-level subtraction cancels artifacts in quark-potential loops","feed_subtitle":"Flow-time and lattice-spacing errors drop out, so O(1/m_Q^2) corrections reach small r and t.","key_machinery":"The central object is the generalized Wilson loop with two field-strength insertions, Eq. (3), whose ground-state limit yields the correlator of clover-defined chromoelectric or chromomagnetic fields in the flux tube of a static quark-antiquark pair. The argument is carried by tree-level improvement. For the static potential the ansatz (7) adds the correction term $\\tilde c^{(0)}(4\\pi G(r,t_f)-1/r)$, with $G(r,t_f)$ the lattice tree-level gluon propagator at flow time $t_f$; subtracting this term and the fitted $r$-independent shift $V_c(t_f,a)$ removes most of the discretization and flow dependence. For two-field-strength correlators the same idea enters through Eq. (12), where $\\tilde c (4\\pi/C_F g^2)(C^{\\mathrm{lattice}}_{\\mathrm{tree\\ level}} - C^{\\mathrm{continuum}}_{\\mathrm{tree\\ level}})$ is subtracted from the numerical correlator, with the coefficient $\\tilde c$ taken from the static-potential fit. Gradient flow is the enabling regulator: flowed correlators need no extra renormalization, and the flow time $t_f$ regulates the logarithmic divergence of chromomagnetic insertions.","core_discovery":"The central claim is that the dominant finite-lattice-spacing and finite-flow-time artifacts in generalized Wilson loops with two field-strength insertions are tree-level effects, and that subtracting the difference between the lattice and continuum tree-level results removes them. For the static potential the paper fits the ansatz (7), $V^{(0)}(r,t_f) = -c^{(0)}/r + \\sigma r + V_c(t_f,a) + \\tilde c^{(0)}(4\\pi G(r,t_f) - 1/r)$, where $G(r,t_f)$ is the flowed lattice tree-level propagator; removing the shift $V_c$ and the correction term collapses data from three ensembles and four flow times onto one curve down to $r \\approx 0.05$ fm. The same strategy is applied to the two-chromoelectric-field correlator $\\langle \\Sigma^+_{g,r}|E_z(t,0)E_z(0,0)|\\Sigma^+_{g,r}\\rangle_c$ via Eq. (12), producing improved correlators whose small-$t$ behavior is essentially independent of flow time. The paper also reports that the Gromes and first BBMP relations are satisfied within statistical errors at finite flow time, in contrast to an earlier multilevel computation that used an approximate multiplicative renormalization; this supports the interpretation that the earlier violations came mostly from the renormalization prescription rather than from lattice artifacts.","pith_inferences":["If the fitted coefficient $\\tilde c^{(0)}$ is genuinely transferable from the static potential to every two-field-strength correlator, the same subtraction can be applied to the chromomagnetic correlators that enter the spin-dependent potentials, extending the method beyond the $\\langle E_z E_z \\rangle_c$ example shown.","The agreement of improved data at $r \\approx 0.05$ fm suggests the residual nonperturbative lattice artifacts are subleading at surprisingly short distances; a test would be to check whether $\\tilde c^{(0)}$ stays constant when computed on a finer lattice.","Because gradient-flow regularization avoids the leftover $O(g^4)$ and $O(g^6)$ terms of the earlier multiplicative renormalization, the Gromes and BBMP relations could be promoted from consistency checks to constraints used in the fits that determine the spin-dependent potentials."],"forward_implications":["The static potential can be extrapolated to $a \\to 0$ and $t_f \\to 0$ while including data at $r \\approx 0.05$ fm, where unimproved data are dominated by flow effects.","The tree-level improved $\\langle E_z E_z \\rangle_c$ correlators agree across flow times at small $t$, so the integrals defining the potential corrections can be evaluated reliably at short separations.","The Gromes and first BBMP relations, which need no matching coefficients, are satisfied within statistical errors at finite flow time, indicating that earlier violations came mostly from the approximate renormalization prescription.","Spin-independent $1/m_Q^2$ corrections can be obtained directly from a combined continuum and zero-flow-time extrapolation, while spin-dependent corrections still require NLO matching coefficients to convert from the gradient-flow scheme to the $\\overline{\\mathrm{MS}}$ scheme."],"supporting_citations":[{"why":"Defines the 1/m_Q and 1/m_Q^2 decomposition of the heavy-quark potential and the integrals over field-strength correlators that the paper computes.","marker":"[5]"},{"why":"Introduces the gradient flow equation used to evolve the gauge field and render flowed observables finite.","marker":"[16]"},{"why":"Shows that correlation functions of flowed fields need no additional renormalization, the basis for using gradient flow on field-strength insertions.","marker":"[17]"},{"why":"Supplies the tree-level improvement idea for the static force and the flowed lattice tree-level propagator G(r,t_f) used in Eq. (8).","marker":"[18]"},{"why":"One of the two sources of the correction-term parametrization adopted in the static-potential ansatz (7).","marker":"[12]"},{"why":"The companion source of the correction-term parametrization in the static-potential ansatz (7).","marker":"[21]"},{"why":"Establishes the flow-radius criterion beyond which flow effects on operator pairs are suppressed.","marker":"[20]"},{"why":"Provides the lattice perturbation theory expressions used to compute the tree-level correlator subtracted in Eq. (12).","marker":"[32]"},{"why":"Earlier multilevel computation whose approximate renormalization caused a few-percent violation of the Gromes relation, the comparison point for the gradient-flow check.","marker":"[14]"}],"fun_headline_variants":["Tree-level subtraction cancels lattice artifacts in quark loops","Matching tree level removes flow-time errors in potential loops","Heavy-quark potential gains from tree-level artifact subtraction","Flow-time corrected loops sharpen static potential measurement","Tree-level matching tames lattice errors in quark potential"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method assumes the dominant finite-lattice-spacing and finite-flow-time artifacts are exactly the tree-level lattice-versus-continuum difference, with the remaining deviation absorbed by one r-independent shift and a single fitted coefficient that is the same for the static potential and for the two-field-strength correlators.","fun_headline_variants_meta":{"raw":{"variants":["Tree-level subtraction cancels lattice artifacts in quark loops","Matching tree level removes flow-time errors in potential loops","Heavy-quark potential gains from tree-level artifact subtraction","Flow-time corrected loops sharpen static potential measurement","Tree-level matching tames lattice errors in quark potential"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000782,"raw_usage":{"total_tokens":3437,"prompt_tokens":910,"completion_tokens":2527,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":526,"completion_tokens_details":{"reasoning_tokens":2451}},"tokens_in":526,"tokens_out":2527,"duration_ms":15930,"temperature":1.0,"reasoning_tokens":2451,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T18:17:46.192842+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the tree-level-improved correlator (12) on a fourth lattice ensemble with roughly half the lattice spacing of the coarsest ensemble used here at the same physical $r$ and $t_f$; if the small-$t$ data still shift by more than the statistical errors when $a$ and $t_f$ are reduced, the tree-level subtraction is not capturing the dominant artifact.","supporting_citations":[{"cited_title":"Gradient-flowed thermal correlators: how much flow is too much?","cited_arxiv_id":"1802.04562","evidence_quote":"Establishes the flow-radius criterion beyond which flow effects on operator pairs are suppressed."}],"review_version":1}