{"id":"45ebc415-7159-4c9b-bde2-c615546116f9","arxiv_id":"2412.13138","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A variational basis with nucleon-sigma interpolators reduces excited state contamination in direct lattice QCD determinations of nucleon sigma terms, demonstrated on one Nf=3 ensemble at M_pi=429 MeV.","lead":"This lattice QCD study shows that adding a nucleon-sigma-meson operator to the variational basis removes most of the excited state contamination in the extraction of nucleon scalar charges and sigma terms. The method stabilizes at much shorter time separations than the standard approach, but only on one ensemble at an unphysically large pion mass.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The GEVP-improved extraction neglects the Nσ→Nσ diagonal contribution on the untested assumption that is not enhanced; if this matrix element is large, the improved ratios inherit an unmodelled excited-state contamination.","rationale":"The central claim has two components: (i) the dominant excited-state contamination in the standard scalar three-point function is the N→Nσ transition, and (ii) the GEVP-improved correlators, which omit Nσ→Nσ, reliably extract the scalar charges. Component (ii) is the load-bearing one for the method's validity. The paper's justification for omitting Nσ→Nσ is an expectation ('we do not expect to be enhanced'), not a measured bound. The suppression factor (v1_pσ)^2 is not quantified in absolute terms, and the diagonal matrix element of a scalar density on a mesonic state could in principle be sizeable; a Feynman-Hellmann relation <σ|S|σ> = ∂m_σ/∂m_q suggests it has no obvious suppression. If this term is non-negligible, the GEVP-improved ratio (15) is contaminated at early times, and the linear fits (18) from t=4a are biased. The paper's internal cross-checks (agreement with the standard method at large t, stability of the charges under fit range) reduce this concern but do not eliminate it, because those checks use the same neglected term. The Nπ-operator omission is a separate issue that affects interpretation of the 'dominant' claim, but it is less load-bearing: at this pion mass the lowest P-wave Nπ level lies above the Nσ level (Table I gives ΔE_Nπ ≈ 678 MeV versus ΔE_Nσ ≈ 554 MeV), so Nσ is the first excited state in energy, and the observed flattening of the improved ratio already indicates that the remaining higher-state contamination is relatively small. Thus the most serious gap is the unmodelled Nσ→Nσ diagonal contribution. The proposed test—computing directly, or adding it to the fit—would settle whether the neglect is numerically safe. In the absence of that test, a conditional verdict is appropriate.","tokens_in":14846,"tokens_out":14976,"duration_ms":145613,"concrete_test":"Compute the diagonal three-point function and its ratio to to obtain an estimate of . Combine this with the measured eigenvector component v1_pσ (and the overlap factors entering O_imp) to evaluate the size of the neglected term in C_imp,3pt relative to the ground-state term. If this ratio is >1% of the extracted charge, the Nσ→Nσ neglect introduces a bias comparable to or larger than the quoted statistical errors (~6%), and the fit model should include it. Alternatively, add this term to the fit of the improved summed ratios with the Nσ energy fixed from the GEVP and check whether the fitted amplitude is consistent with zero and whether the charges shift.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section III defines the improved three-point function (14) from the GEVP ground-state eigenvector. The spectral decomposition of C_imp,3pt contains a diagonal Nσ→Nσ term proportional to |v1_pσ|^2 (and the corresponding time-ordered contributions), and the paper states: 'We neglect the N σ to N σ contributions ... suppressed by the second power of the small eigenvector component ... we do not expect to be enhanced relative to .' The suppression is only quantitative if the eigenvector component v1_pσ is genuinely small and the matrix element ratio is O(1); neither is demonstrated. Figure 1 shows the multi-particle component of v1 is about two orders of magnitude below the single-nucleon component, but the physical amplitude of O_imp to create the Nσ state also involves the overlap of Opσ with Nσ, which is not reported. Moreover, the 'dominant contamination' claim is inferred from the flatness of the GEVP-improved ratio, not from a direct extraction of ; the flattening could be driven by the large overlap of the standard interpolator with Nσ (v2) rather than by an intrinsically large transition matrix element. If were enhanced (e.g., by the scalar density of the σ meson), the neglected term would bias R_GEVP at early times and hence the summed-ratio fits, which the paper assumes are linear from t=4a. The paper provides no estimate of the size of this term or of the residual higher-state contamination, so the systematic error on the method is not bounded.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper presents a lattice QCD determination of the nucleon sigma terms on a single N_f=3 ensemble at M_pi=429 MeV, using a variational (GEVP) analysis with a basis of nucleon and nucleon-sigma interpolating operators. The authors find that the second GEVP level is close to the non-interacting S-wave N-sigma energy and that the GEVP-improved ratios for the scalar charges plateau at significantly smaller source-sink separations than the standard ratios. They extract the scalar charges from summed-ratio fits and compare with standard-method fits, obtaining consistent results. They conclude that the dominant excited-state contamination in the standard three-point function is due to the N to N-sigma transition and that this contamination is substantially removed by the variational approach.","tokens_in":15248,"tokens_out":6376,"duration_ms":60512,"significance":"The variational approach demonstrated here could be valuable for reducing excited-state contamination in nucleon matrix elements, and the consistency between the GEVP-improved and standard results on this ensemble is encouraging. The paper is clearly written and provides detailed supplementary material on the contractions and diagram topologies. However, as a proof-of-concept on a single ensemble at an unphysically large pion mass, the significance is limited; the physical-point applicability is speculative and rests on assumptions about the sigma meson width and the stability of the approach.","major_comments":[{"comment":"The neglect of the Nσ→Nσ diagonal contribution to the GEVP-improved three-point function is load-bearing for the central claim that the scalar charges can be reliably extracted from t=4a. The only justification given is the expectation that is not enhanced relative to , but no quantitative estimate or test is provided. Since the summed-ratio fits are assumed to be linear from t=4a, an enhanced excited-to-excited matrix element would introduce an unmodelled curvature and bias the extracted charges. Please either (i) include an excited-to-excited term in the fit ansatz for the GEVP-improved summed ratios and report the resulting constraint (even if consistent with zero), or (ii) provide a model-based estimate of (e.g., via the Feynman-Hellmann relation applied to the sigma meson mass) to justify the suppression.","section":"Section III, Eq. (14) and following paragraph"},{"comment":"The headline claim that the dominant excited-state contamination is due to the N→Nσ transition is inferred rather than directly demonstrated. The basis B2/B3 contains no Nπ interpolators, and the transition matrix element is not extracted. The flattening of R_GEVP shows that the chosen basis removes a large excited-state contribution, but it does not by itself identify the state as Nσ or establish that the transition matrix element is intrinsically large. To support the claim, the authors could either include an Nπ operator in the basis on this ensemble or directly extract from the standard three-point functions using the GEVP eigenvectors and compare its magnitude with residual contamination. As written, the wording overstates the directness of the evidence.","section":"Abstract and Section IV"}],"minor_comments":[{"comment":"The value of the sigma mass is quoted as 554(49) MeV in the Introduction but 554(39) MeV in the Supplemental Material; please harmonize the quoted uncertainty.","section":"Introduction and Supplemental Material"},{"comment":"The identification of the second GEVP level with the non-interacting S-wave Nσ energy ignores the finite-volume interaction shift, which is not estimated. This does not affect the extracted charges but weakens the interpretation; a brief comment on the expected size of the shift would be helpful.","section":"Section III, Fig. 1"},{"comment":"The choice of eigenvector evaluation time t'=6a and reference time t0=3a is not varied to demonstrate stability of the improved operator; a short discussion of the sensitivity to these choices would strengthen the analysis.","section":"Section III, Eq. (12)"},{"comment":"For the summed-ratio fits using Eq. (17), only a range of χ2/Ndf is quoted, not the fit ranges themselves; please specify the fit ranges used for each model so the results are reproducible.","section":"Table I"},{"comment":"The claim that the new method is 'cost effective' is not quantified; please compare the total propagator counts (or computational cost) of the GEVP analysis with the standard analysis at comparable precision.","section":"Section V"},{"comment":"The caption does not define the solid versus open symbols; please clarify which data points are used in the fits.","section":"Figure 2"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid proof-of-concept with a promising technique, and the numerical analysis appears careful. My main concern is that the headline claim ('dominant contamination is due to N→Nσ') is stated more strongly than the evidence supports; the authors should be asked to either add a direct test or soften the claim. The paper fits the journal scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe short version: this is a genuine proof of concept, not a high-precision result. The load-bearing claim is that Nσ is the dominant excited-state contamination in the scalar three-point function. That claim is reasonable but indirect. Still, the paper deserves a serious referee.\n\nThe new thing is adding S-wave nucleon-sigma interpolators to the GEVP basis for scalar matrix elements. The improved ratios plateau by t≈0.4 fm, while standard ratios remain contaminated at 0.8 fm, and the GEVP and standard extractions agree once the standard fits are given enough time separation. That is a nice internal cross-check. The eigenvector analysis places the second level near M_N + M_σ, which supports the interpretation. The paper is honest about being a single-ensemble, unphysical-pion-mass study, and the Wick contraction analysis is thorough.\n\nThe soft spots are real but not fatal. The neglect of Nσ→Nσ in the improved three-point function is justified by the small eigenvector component and by the expectation that is not enhanced. The suppression is plausible because the GEVP eigenvector makes the improved operator nearly orthogonal to the Nσ state, but the residual term is not bounded or estimated. If is larger than expected, the early-time plateaus carry an unmodelled bias. The 'dominant contamination' claim is also inferential: there is no Nπ operator in the basis, so the flattening shows only that the added operator captures something, not that it is the physically dominant contamination. A larger basis (Nπ, Nππ) and a direct extraction of would settle this. There is also a minor inconsistency: the text quotes M_σ0 = 554(49) MeV while the Supplemental fit gives 554(39) MeV; the larger error should be used.\n\nFor a proof of concept, this is well executed. The target audience is lattice practitioners working on nucleon matrix elements and dark matter direct-detection inputs. I would send it to review, but the authors should be asked to test or at least bound the neglected Nσ→Nσ term, and to either soften or better support the 'dominant' claim. With that revision, the paper would be a solid contribution.\n\nRecommendation: accept with revision.","headline":"A careful proof-of-concept that S-wave nucleon-sigma operators in the GEVP basis remove most excited-state contamination in scalar charges; the neglected Nσ→Nσ term is the main unresolved systematics.","tokens_in":15756,"tokens_out":3458,"would_cite":false,"duration_ms":34295,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["12.38.Gc","14.20.Dh"],"model":"deepseek-v4-flash","headline":"The dominant contamination of nucleon sigma-term determinations comes from transitions to a nucleon-sigma scattering state, and a variational basis containing nucleon-sigma interpolators removes it.","keywords":["lattice QCD","nucleon sigma terms","scalar current","excited-state contamination","generalised eigenvalue problem","nucleon-sigma scattering state","variational analysis","Wilson fermions"],"falsifier":"A direct calculation of $\\langle N\\sigma|S^q|N\\sigma\\rangle$ on this ensemble, or a GEVP basis that also includes $N\\pi$ and $N\\pi\\pi$ operators, would settle it: if the plateau values shift by more than the quoted uncertainties, the omission is not harmless.","tokens_in":14627,"feed_emoji":"⚛️","tokens_out":18188,"duration_ms":137811,"temperature":0.7,"pith_summary":"The paper asks what contaminates lattice-QCD determinations of the nucleon $\\sigma$ terms, the quark-mass contributions to the nucleon mass that set the Higgs-boson coupling to nucleons and matter for dark-matter searches. It claims that the dominant excited-state contamination in the three-point function with a scalar current comes not from pion-nucleon states but from the transition between the nucleon and an S-wave scattering state (zero relative orbital angular momentum) made of a nucleon and a scalar ($\\sigma$) meson. Using a generalised eigenvalue (GEVP) analysis whose interpolator basis includes nucleon-$\\sigma$ operators, the contamination is reduced enough that the scalar charges can be extracted from source-sink separations up to 1.1 fm, with plateaus setting in near 0.4 fm. On the single $N_f=3$ ensemble studied, at pion mass 429 MeV, the improved values agree with the standard method and convert to $\\sigma_{\\pi N} = (235 \\pm 24)$ MeV and $\\sigma_{sN} = (42 \\pm 16)$ MeV.","feed_headline":"Nucleon-sigma state is the dominant contaminant of sigma-term fits","feed_subtitle":"A variational basis with nucleon-sigma operators removes the contamination, so scalar charges plateau early.","key_machinery":"The load-bearing object is the generalised eigenvalue problem (GEVP) for the matrix of two-point functions built from a basis of interpolating operators: the standard proton operator $O_p$ and either one SU(2) proton-$\\sigma$ operator $O_{p\\sigma}$ (basis B2) or the SU(3) singlet and octet versions $O_{p\\sigma_0}$ and $O_{p\\sigma_8}$ (basis B3). The lowest eigenvector defines an improved interpolator as a linear combination of basis operators, and the same combination builds improved two- and three-point functions whose ratios are then fitted with the summation method. The crucial omission is the transition between two nucleon-$\\sigma$ states, which the paper argues is suppressed by the square of the small eigenvector component and is not expected to be enhanced relative to the nucleon-to-nucleon matrix element.","core_discovery":"On its own terms, the paper's finding is that the S-wave nucleon-$\\sigma$ scattering state, rather than the P-wave pion-nucleon state that standard multi-state fits typically include, is the dominant source of excited-state contamination in the nucleon three-point function with a scalar current. The evidence is that the second generalised eigenvalue sits at the non-interacting sum of the nucleon and $\\sigma$ masses, and that adding nucleon-$\\sigma$ interpolators to the basis makes the improved ratios plateau at short times while the standard ratios still bend. From the plateaus the paper extracts lattice-scheme scalar charges $g_S^{\\mathrm{lat},u+d} = 30.1 \\pm 1.8$ and $g_S^{\\mathrm{lat},s} = 11.7 \\pm 0.7$ with the two-operator basis, consistent results with the SU(3) singlet/octet basis, and converts these to $\\sigma_{\\pi N} = (235 \\pm 24)$ MeV and $\\sigma_{sN} = (42 \\pm 16)$ MeV at this unphysically large pion mass.","pith_inferences":["If the S-wave nucleon-sigma state is indeed the dominant contaminant, earlier variational studies that added only P-wave nucleon-pion operators to scalar matrix elements may have targeted the wrong state; nucleon-sigma operators are the natural next ingredient for other flavour-diagonal channels.","The one explicitly neglected term, the transition between two nucleon-sigma states, can be quantified by computing $\\langle N\\sigma|S^q|N\\sigma\\rangle$ on this ensemble, or by including it in the improved three-point function, so the assumption is directly testable.","If the method succeeds at the physical point, it could reduce the excited-state systematics in direct sigma-term determinations enough to sharpen the comparison between lattice and phenomenological values, with consequences for predicted dark-matter-nucleon scattering rates.","The same operator-selection logic, matching the basis to the flavour content of the current, could be applied to strangeness-content determinations of other baryons and to other flavour-diagonal currents."],"forward_implications":["The scalar charges can be extracted reliably from source-sink separations up to about 1.1 fm, where the standard method still shows sizeable excited-state contamination.","Because the plateau sets in near 0.4 fm, the method avoids the exponential noise-over-signal growth that makes long source-sink separations expensive, with roughly 132 propagators per configuration for the whole GEVP analysis.","The approach may carry over to physical quark masses: although the sigma meson is unstable there, its width is comparable to its energy, so a scalar bilinear interpolator should still couple to the relevant scattering states.","The GEVP-improved values agree with the standard method and with other determinations at similar pion masses, so the contribution is an improved control of systematics rather than a shift in the central value."],"supporting_citations":[{"why":"Demonstrates the variational removal of nucleon-pion excited-state contamination in nucleon matrix elements, the method this paper extends to the scalar current.","marker":"[28]"},{"why":"Extends the variational treatment to scalar matrix elements at the physical pion mass but finds no improvement from a P-wave nucleon-pion operator, motivating the nucleon-sigma basis.","marker":"[29]"},{"why":"Provides precedent for nucleon-sigma interpolators in variational scattering studies, supporting the basis construction used here.","marker":"[31]"},{"why":"A direct sigma-term determination whose chiral-perturbation-theory-inspired excited-state priors set the comparison point for the contamination mechanism proposed here.","marker":"[11]"},{"why":"Supplies the quark mass, nucleon mass, and scale of the ensemble used for the extraction and for converting charges to sigma terms.","marker":"[17]"},{"why":"Provides the ratio of singlet to non-singlet scalar-density renormalisation parameters needed to convert the lattice charges into sigma terms.","marker":"[33]"},{"why":"Previous baryon-octet sigma-term determination at a similar pion mass used for consistency comparison.","marker":"[41]"}],"fun_headline_variants":["Variational basis kills nucleon-sigma contamination","Nucleon-sigma scattering state dominates excited-state bias","New method zeros in on sigma terms by adding sigma operators","Why sigma-term fits fail: it's the N-sigma scattering state","Scalar charges plateau early once N-sigma states are included"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The analysis assumes that the second level the GEVP resolves is the non-interacting S-wave nucleon-$\\sigma$ state, that the transition between two such states is no stronger than the nucleon's own scalar matrix element, and that states left out of the basis (notably $N\\pi$) are not contaminating; if any of these fails, the extracted charges carry unmodelled contamination.","fun_headline_variants_meta":{"raw":{"variants":["Variational basis kills nucleon-sigma contamination","Nucleon-sigma scattering state dominates excited-state bias","New method zeros in on sigma terms by adding sigma operators","Why sigma-term fits fail: it's the N-sigma scattering state","Scalar charges plateau early once N-sigma states are included"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0003,"raw_usage":{"total_tokens":1697,"prompt_tokens":872,"completion_tokens":825,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":488,"completion_tokens_details":{"reasoning_tokens":742}},"tokens_in":488,"tokens_out":825,"duration_ms":7650,"temperature":1.0,"reasoning_tokens":742,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T13:23:13.609858+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct calculation of $\\langle N\\sigma|S^q|N\\sigma\\rangle$ on this ensemble, or a GEVP basis that also includes $N\\pi$ and $N\\pi\\pi$ operators, would settle it: if the plateau values shift by more than the quoted uncertainties, the omission is not harmless.","supporting_citations":[{"cited_title":"Sigma terms of the baryon octet in $N_\\mathrm{f} = 2+1$ QCD with Wilson quarks","cited_arxiv_id":"2301.03871","evidence_quote":"Previous baryon-octet sigma-term determination at a similar pion mass used for consistency comparison."}],"review_version":1}