{"id":"99fa81b6-81e9-4753-98c3-b07e88935331","arxiv_id":"2412.10215","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A new lattice QCD determination of the hadronic scales r0 = 0.4729(75) fm and r1 = 0.3127(40) fm in 2+1 flavor QCD, together with r0 Λ_MSbar^(3) = 0.820(28).","lead":"This paper measures two standard length scales of the strong nuclear force, r0 and r1, from supercomputer simulations of quantum chromodynamics with 2+1 quark flavors, and reports their values in physical units. These scales are used across lattice QCD to convert simulation results into physical units, so a new precise determination serves as a cross-check and input for many other calculations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The per-lattice-spacing gap model in eq. (4.10) is underdetermined at β=3.85, where only one stable ensemble contributes; J501's r1 may inherit an unconstrained mass-dependence term.","rationale":"The central result is r1 and r0/r1 from the static force at the physical point, and the extraction on fine lattices near r1 is the least secure step. The reader's weakest assumption correctly identifies eq. (4.10) as a fitted input, but focuses on E300. I sharpen this: at β=3.85, with J501 problematic in the r1 range, the model's mass-dependence term cannot be determined from the single stable ensemble J500, so the Δ used for J501 (if the constrained linear fit was used) depends on an unconstrained parameter. The paper's robustness checks—the pencil-of-functions method, the E300-excluded analysis, and the log-correction study—are genuine, but none of them directly demonstrates that the β=3.85 mass slope is constrained. The abstract/body inconsistency for r0 (0.4757 vs 0.4729) is a real presentation error that should be corrected, but it does not undermine the argument itself. The underdetermination of b̂0 at β=3.85 is the most load-bearing concern because it directly affects the finest-lattice input to the continuum extrapolation of r1. The recommended verdict remains CONDITIONAL: the concern is concrete and testable, but not shown to be fatal, and the existing cross-checks bound the likely impact to the tens-of-percent-of-error level.","tokens_in":24119,"tokens_out":7944,"duration_ms":73791,"concrete_test":"Refit J501's effective masses at β=3.85 using Δ from eq. (4.10) with b̂0 free (jointly constrained across β values, or fixed to the β=3.70 value) and also using pencil-of-functions; compare the resulting r1/a to 8.226(32). If the spread exceeds the quoted error, the underdetermination of b̂0 is not negligible. Additionally, report which method was used for B450 and J501 in the final analysis.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4.3 introduces eq. (4.10), a four-parameter model for the gap Δ, fit once per lattice spacing using ensembles where the three-parameter fit (4.9) is stable. At β=3.85 the only fully stable ensemble appears to be J500; J501 is described in §4.4 as problematically lacking a plateau in the r1 range. With a single input mπ value, the mass-dependent coefficient b̂0 in eq. (4.10) is degenerate with the constant Δ̂0 and cannot be constrained. If J501's r1 was obtained from the constrained linear fit with Δ read off from this model, its Δ is extrapolated in mπ using an unconstrained slope; if instead the pencil-of-functions method was used for B450 and J501, the manuscript does not say so. The E300 cut shows a ~0.002 fm shift in r1, but no analogous test is shown for J501, whose fine lattice spacing has strong weight in the continuum limit.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper determines the hadronic scales r0, r1 and their ratio in Nf=2+1 QCD from the static force measured with Wilson loops on 18 CLS ensembles covering five lattice spacings and pion masses from 134 to 420 MeV. The analysis uses a tree-level improved distance, a GEVP with HYP smearing, and a two-step procedure in which an empirical model for the excited-state gap (eq. 4.10) is fitted per lattice spacing and used to stabilize linear fits for problematic ensembles. The scales are extrapolated to the continuum and physical pion mass with four fit ansatze, several data cuts, logarithmic correction tests, and AIC averaging. The final results are r0=0.4729(57)(48) fm, r1=0.3127(24)(32) fm, r0/r1=1.532(12), plus a byproduct r0 Lambda_MSbar^(3)=0.820(28). The paper also studies the shape parameter c(r) and compares it to potential models.","tokens_in":24390,"tokens_out":9680,"duration_ms":89214,"significance":"If the identified gap-model issue for the finest lattice spacing is resolved, the paper provides a competitive, independent determination of r0 and r1 with a transparent error budget. Its strengths are the multiple cross-checks: three interpolation forms for the force, a pencil-of-functions analysis for the difficult E300 ensemble, four continuum/chiral fit ansatze, data cuts, logarithmic correction tests, and AIC weighting. The direct continuum extrapolation of r0/Lhad to obtain r0 Lambda is elegant and avoids a double continuum extrapolation. The results agree with the FLAG averages within errors, and the r0/r1 ratio is a valuable constraint for scale setting.","major_comments":[{"comment":"The gap model in Eq. (4.10) contains two terms, -delta_hat0/(r sqrt(t0)) and -gamma_hat0 sqrt(t0)/r, which are linearly dependent for any single ensemble because sqrt(t0) is a fixed number in lattice units. With only one stable ensemble at beta=3.85 (J500), the parameters b_hat0, delta_hat0, and gamma_hat0 cannot be identified from the data. The manuscript does not state how Delta was assigned to J501, whose r1 plateau is described as problematic in Section 4.4, and no J501-excluded analysis is provided analogous to the E300 cut. If J501's r1/a in Table 5 was obtained with the constrained linear fit using Delta from this model, the finest-spacing continuum point inherits an unconstrained extrapolation in m_pi; please clarify the fitting procedure at beta=3.85 and add a J501-removed cross-check.","section":"4.3, Eq. (4.10)"},{"comment":"The abstract quotes r0 = 0.4757(64) fm, while Eq. (6.1) quotes r0 = 0.4729(57)(48) fm, and Table 6 indicates that the AIC average gives the latter value. These differ by about 0.003 fm, which is within the combined errors but is still an internal inconsistency in the primary result; the abstract should be updated to the final value or the discrepancy should be explained.","section":"Abstract vs. Section 6.1"},{"comment":"For r1, all the global fits have chi^2/dof around 2.0 (e.g., Fit 1 with no cuts gives 30.4/15), which is poor for a fit with this number of degrees of freedom and indicates that the mass and lattice-spacing ansatz does not fully describe the r1 data. Since r1 is one of the two main results, the authors should either identify the source of the poor chi^2 and show that the fit is stable without the offending ensemble, or add a fit-systematic uncertainty that reflects the poor goodness of fit.","section":"Table 6 and Section 5"}],"minor_comments":[{"comment":"The caption says 'On the left the values r0/sqrt(t0) and the right r0/sqrt(t0)' but the right panel appears to show r1/sqrt(t0); please correct the typo.","section":"Figure 8 caption"},{"comment":"The title 'Evalution of exceptionally problematic ensembles' contains a misspelling and should read 'Evaluation'.","section":"Section 4.4 title"},{"comment":"The phrase 'inter1 -extrapolation' appears to be a typo for 'inter-/extrapolation'.","section":"Section 5, first paragraph"},{"comment":"Please define the parameters delta_hat0, gamma_hat0, and b_hat0 in Eq. (4.10), and state the r-range used in the global fit, since the text notes that the first datapoints with large lattice artifacts were excluded from the fit.","section":"Section 4.3"},{"comment":"The entries for H102-1 and N300-1 do not list the hopping parameters (kappa_u,d, kappa_s) even though those values are needed to reproduce the ensembles; please add them or state that they are the same as the preceding row.","section":"Table 1"},{"comment":"The citation sequence in the introduction, '[18, 19, 19, 20] [21, 22]', contains a duplicated reference number and an unusual gap; please check the bibliography and correct the citation order.","section":"Introduction, references"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and the central extraction is sound overall, but the gap-model identifiability at beta=3.85 and the missing J501 cross-check are load-bearing issues that should be addressed before publication. The abstract/main-text value inconsistency suggests the manuscript was revised without updating all parts. Note also that the authors share authorship with the scale-setting reference [48] and the ALPHA Lambda-parameter reference [100]; the r0 Lambda result is presented as a re-expression of [100] rather than an independent determination, which is appropriate but should be kept explicit in the final version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThis is a careful, standard lattice QCD determination of r0 and r1 from the static force on the CLS 2+1 ensembles. The new numbers—r0 = 0.4729(57)(48) fm, r1 = 0.3127(24)(32) fm, r0/r1 = 1.532(12)—are new measurements, not a re-analysis of existing results, and the r0Λ result (0.820(28)) is a useful byproduct obtained by directly continuum-extrapolating r0/Lhad.\n\nThe extraction is well cross-checked: tree-level improved force, multiple interpolation forms, several continuum/chiral fit ansätze (Fits 1–4), AIC averaging, log-correction tests, and an explicit E300-deletion study. The fit tables are transparent and the error budget is sensible.\n\nSoft spots, in rough order:\n\n1. The abstract quotes r0 = 0.4757(64) fm, while the main text gives r0 = 0.4729(57)(48) fm (eq. 6.1). The abstract value looks like the Fit 1 no-cuts result from Table 6; the body quotes the AIC average with the coarsest spacing excluded. This inconsistency will confuse readers and should be fixed.\n\n2. The gap model in eq. (4.10) is the weakest link. It is fitted once per lattice spacing, and at β = 3.85 there appears to be only one stable ensemble (J500) for the three-parameter fit. That leaves the mass-dependent coefficient b̂0 effectively unconstrained at that β. J501 is described as problematic in the r1 range, and the text does not explicitly state whether its r1 came from the constrained linear fit (with Δ read off the model) or from the pencil-of-functions method. The E300 cut shows a ~0.002 fm shift, but no analogous J501 cut is shown. This deserves a referee question, but the evidence that it matters much is thin.\n\n3. The AIC average excludes the coarsest lattice spacing, and that exclusion is not folded into the quoted systematic error. The spread across cuts in Table 6 is larger than the statistical error (r0 ranges from 0.4716 to 0.4778). The authors mention the exclusion but do not quantify the resulting systematic. This is minor but should be stated.\n\nNone of these undermine the central result. The values are consistent with FLAG and previous determinations within errors. The paper is solid and should go to peer review; it needs a revised abstract and a clearer description of the J501 treatment.","headline":"New r0 and r1 values for CLS 2+1 QCD are trustworthy, but the abstract disagrees with the main text on r0 and the finest-spacing gap model needs explicit clarification.","tokens_in":24929,"tokens_out":3837,"would_cite":true,"duration_ms":31482,"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":"Lattice QCD determines the hadronic scales r0 and r1 from the static force at the physical point, with r0 = 0.4729(57)(48) fm, r1 = 0.3127(24)(32) fm, and r0/r1 = 1.532(12).","keywords":["lattice QCD","static potential","hadronic scale r0","hadronic scale r1","Wilson loops","static force","tree-level improvement","continuum extrapolation"],"falsifier":"Repeat the analysis on the same ensembles with a purely data-driven determination of the first excited state, for example a full multi-exponential fit with a larger variational basis or a multilevel algorithm for dynamical fermions, and compare the resulting r1 at the physical point with 0.3127(24)(32) fm. A deviation larger than the combined error would show that the assumed gap model biases the ground-state potential.","tokens_in":23894,"feed_emoji":"⚛️","tokens_out":6686,"duration_ms":65456,"temperature":0.7,"pith_summary":"The paper aims to pin down the hadronic length scales r0 and r1, defined by the strength of the static quark-antiquark force at two reference distances, directly from lattice QCD with dynamical up, down, and strange quarks. These scales are widely used to convert lattice measurements into physical units, so precise values improve the comparability of many lattice results. The analysis extracts the static force from Wilson loops, uses a tree-level improved distance to suppress lattice artifacts, and extrapolates data at five lattice spacings and pion masses from 134 to 420 MeV to the continuum and physical mass limits. It reports r0 = 0.4729(57)(48) fm, r1 = 0.3127(24)(32) fm, and r0/r1 = 1.532(12), and as a byproduct converts the three-flavor QCD Lambda parameter to r0 units, obtaining r0 Lambda_MSbar^(3) = 0.820(28).","feed_headline":"Lattice QCD sets hadronic scales r0=0.4729 fm, r1=0.3127 fm","feed_subtitle":"A tree-level improved static force yields r0/r1=1.532(12) and r0 Lambda=0.820(28) at the physical point.","key_machinery":"The central object is the static quark-antiquark force F(r) = dV(r)/dr, computed from Wilson loops through a generalized eigenvalue problem, with lattice artifacts removed at tree level by replacing the separation r with an improved distance r_I chosen so that the tree-level continuum relation $r_I^{2}$ F_tree(r_I) = C_F $g0^{2}$/(4 pi) holds exactly. The scales r0 and r1 are defined through $r_i^{2}$ F(r_i) = c_i. The argument is carried by this improved force together with a two-step excited-state analysis: a global model for the energy gap $\\Delta$ between the ground and first excited state is fitted on stable ensembles and then used to stabilize effective-mass fits on all ensembles, including the problematic near-physical-mass fine ensemble where the direct fit was unstable. Continuum and chiral limits are obtained by Akaike-weighted averages over four fit forms and several data cuts, with explicit checks of logarithmic scaling violations.","core_discovery":"The central claim is that the hadronic scales r0 and r1 can be computed at the physical point of 2+1 flavor QCD with controlled statistical and systematic errors, using the tree-level improved static force F(r) extracted from Wilson loops. The scale r_i is defined by $r_i^{2}$ F(r_i) = c_i with c0 = 1.65 and c1 = 1, so these distances mark where the force takes a reference strength. Combining measurements on ensembles with five lattice spacings down to 0.039 fm and pion masses from 134 to 420 MeV, continuum and chiral extrapolations yield the values quoted above, with agreement across different fit ansatze and data cuts. The paper also shows that the shape parameter of the potential, c(r) = (1/2) $r^{3}$ V''(r), is distance-dependent in the probed range and follows a screened-logarithmic potential curve rather than the constant-shape model, indicating sensitivity to sea-quark effects.","pith_inferences":["If the quoted values hold, the r0/r1 ratio becomes a sharper cross-check of scale-setting than either scale alone, because the ratio is independent of the overall fm conversion and of sqrt(t0). A direct re-determination of r0/r1 on the same ensembles with a completely independent excited-state treatment would confirm whether the assumed gap model biases the force near r1.","The observation that the shape parameter c(r) is distance-dependent rather than constant suggests that phenomenological potential models used for heavy-quarkonium spectra may need a distance-dependent curvature term; this could be tested against charmonium spectra or against future 2+1+1 flavor determinations with a dynamical charm quark.","The E300-excluded analysis shifts r1 by about 0.002 fm, within the quoted error. A targeted reanalysis of that ensemble with larger statistics or multilevel algorithms for dynamical fermions would settle whether the outlier is a statistical fluctuation or a sign of a small systematic effect."],"forward_implications":["The reported r0 and r1 values give an absolute scale in fm for 2+1 flavor ensembles, so other lattice quantities measured on them can be converted from lattice units to physical units with these numbers.","Since r0/sqrt(t0) = 3.277(39) and r1/sqrt(t0) = 2.167(16), potential scales and gradient flow scales are now cross-calibrated, allowing direct comparison of scale-setting between the two conventions.","The value r0 Lambda_MSbar^(3) = 0.820(28) ties the nonperturbatively computed three-flavor Lambda parameter to a hadronic scale, offering a scale-setting-free comparison point for determinations of the strong coupling.","The ratio r0/r1 = 1.532(12) is a purely gluonic observable nearly independent of quark masses, so it can serve as a stringent test for other lattice actions and for effective field theory descriptions of the static potential."],"supporting_citations":[{"why":"Defines the r0 scale through r^2 F(r) = 1.65 and introduces the tree-level improved distance used to remove lattice artifacts from the static force.","marker":"[1]"},{"why":"Supplies the Wilson-loop measurement method, the improved distance for the force and shape parameter, and the comparison of sea-quark effects on the potential.","marker":"[24]"},{"why":"Provides the physical values of sqrt(t0) and pion masses used to set the absolute scale and to locate the physical point.","marker":"[48]"},{"why":"Gives the nonperturbative three-flavor Lambda parameter results that the paper converts into r0 units.","marker":"[100]"},{"why":"Provides the FLAG averages against which the new r0, r1 and r0/r1 values and r0 Lambda are compared.","marker":"[34]"},{"why":"Previous determination of r0 and r1 by the same authors that this paper extends with new analysis methods.","marker":"[68]"},{"why":"Introduces the pencil-of-functions method used to handle the problematic effective-mass plateaus on the fine near-physical-mass ensemble.","marker":"[62]"}],"fun_headline_variants":["Lattice QCD fixes hadronic scales r0=0.4757 fm and r1","Potential scales r0 and r1 from 2+1 flavor QCD at physical point","Tree-level improved static force yields r0=0.4757 fm","Hadronic scales r0, r1, and r0/r1 from CLS ensembles"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The analysis assumes that the energy gap between the ground state and first excited state of the Wilson-loop correlation functions follows a single four-parameter curve as the quark separation, pion mass, and lattice spacing vary; if that curve is wrong on the ensembles where the direct fit is unstable, the extracted potential, and therefore r0 and r1, would be biased.","fun_headline_variants_meta":{"raw":{"variants":["Lattice QCD fixes hadronic scales r0=0.4757 fm and r1","Potential scales r0 and r1 from 2+1 flavor QCD at physical point","Tree-level improved static force yields r0=0.4757 fm","Hadronic scales r0, r1, and r0/r1 from CLS ensembles"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000342,"raw_usage":{"total_tokens":1899,"prompt_tokens":978,"completion_tokens":921,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":594,"completion_tokens_details":{"reasoning_tokens":829}},"tokens_in":594,"tokens_out":921,"duration_ms":8567,"temperature":1.0,"reasoning_tokens":829,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T16:12:45.202060+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Repeat the analysis on the same ensembles with a purely data-driven determination of the first excited state, for example a full multi-exponential fit with a larger variational basis or a multilevel algorithm for dynamical fermions, and compare the resulting r1 at the physical point with 0.3127(24)(32) fm. A deviation larger than the combined error would show that the assumed gap model biases the ground-state potential.","supporting_citations":[{"cited_title":"The determination of $r_0$ and $r_1$ in $N_f=2+1$ QCD","cited_arxiv_id":"2312.14726","evidence_quote":"Previous determination of r0 and r1 by the same authors that this paper extends with new analysis methods."},{"cited_title":"On the generalised eigenvalue method and its relation to prony and generalised pencil of function methods,","cited_arxiv_id":null,"evidence_quote":"Introduces the pencil-of-functions method used to handle the problematic effective-mass plateaus on the fine near-physical-mass ensemble."}],"review_version":1}