{"id":"d85266ce-fee5-4e9a-a121-fc573a51b0a1","arxiv_id":"2412.17442","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A distillation-based GEVP analysis resolves nucleon, nucleon-pion, and, for the first time, nucleon-pion-pion states in lattice QCD, with a physical-point nucleon mass of 0.927(21)(05) GeV.","lead":"Lattice QCD calculations that include nucleon-pion and nucleon-pion-pion states are needed to predict neutrino-nucleon cross sections for DUNE and Hyper-K. This exploratory study shows distillation can resolve these multi-hadron states and that the nucleon mass extrapolates close to the experimental value.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The distillation truncation systematic is real and unquantified; whether it biases the positive-parity GEVP energies and the Npi/Npipi assignment is the decisive question.","rationale":"The reader's weakest_assumption is exactly the distillation truncation systematic, and the supporting evidence in the full text is strong: the negative-parity section itself flags insufficient modes as the explanation for the degraded nucleon signal (Sec. V and Sec. VI: 'the number of distillation modes is too small for most ensembles to perform sophisticated analyses'), and the positive-parity multi-hadron energies are compared only to non-interacting energies. The concern is that the same truncation could affect the positive-parity GEVP energies and hence the central claim that Npi/Npipi states are resolved and that their contamination is negligible. The test is concrete and would settle the issue. I agree with the reader's verdict of CONDITIONAL rather than REJECT, because there is independent support: the physical-point nucleon mass (0.927(21)(05) GeV) lies close to experiment, the FV-correction check for volume pairs improves from 2.8 to 0.5 sigma, and the negative-parity failure is acknowledged and localized. The strongest claim remains conditional on the truncation effect, but the paper is honest about the limitation and the evidence otherwise supports the workflow. The headline and attack focus on the single decisive missing check: a direct quantification of the mode-truncation effect on the positive-parity GEVP.","tokens_in":28043,"tokens_out":3255,"duration_ms":25270,"concrete_test":"Recompute the positive-parity GEVP for at least one ensemble with a substantially larger truncation, e.g., Nd=200-400 or with a stochastic all-to-all correction for omitted modes; if the extracted N, Npi, and Npipi energies shift by more than the quoted statistical errors, or if the N2pt-minus-GEVP-mode-0 difference (Eq. 50) changes sizeably, the central identification and the negligible-contamination conclusion are not yet established. If the energies are stable, the concern is resolved.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper's central claim is that the positive-parity GEVP cleanly separates N, Npi, and Npipi states using Nd=60 (or 120) lowest Laplacian eigenmodes, enabling a physical-point nucleon mass and the conclusion that Npi/Npipi contaminations are negligible. The load-bearing premise is that this truncation preserves the low-lying baryon spectrum. The paper's own negative-parity section shows the same truncation substantially degrades the nucleon signal (Sec. V: 'One possible explanation is that we do not include sufficiently many high modes in our distillation smearing'; Sec. VI: 'the number of distillation modes is too small for most ensembles to perform sophisticated analyses'). The negative-parity analysis uses nearly the same operators (N and Npi), so the failure is not obviously restricted to one parity. In the positive-parity sector, the extracted Npi and Npipi energies are compared only to non-interacting benchmarks (Figs. 3, 6, 12); no Luscher-type or other interaction check is performed. If mode truncation shifts the multi-hadron energy levels by an amount comparable to the level spacing, the identification of GEVP modes 1 and 2 as Npi and Npipi could be incorrect, and the smallness of the per-mill difference between GEVP mode 0 and the nucleon 2pt function (Eq. 50, Fig. 4) would not establish negligible contamination. The claim that the positive-parity sector is unaffected needs a direct quantitative test, not an argument from the observed plateau.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript reports a distillation-based GEVP spectroscopy study of baryon states on seven RBC/UKQCD domain-wall fermion ensembles with pion masses from 139 to 279 MeV, two lattice spacings, and several volumes. In the positive-parity channel a 3x3 GEVP is built from nucleon, p-wave nucleon-pion, and s-wave nucleon-pion-pion interpolating operators; in the negative-parity channel a 3x3 GEVP uses a nucleon and two nucleon-pion operators. The authors introduce an automated Wick-contraction algorithm for nucleon-plus-multipion correlation functions, apply All Mode Averaging, and use Wigner-Eckart relations to reduce the number of contractions. The central results are stable positive-parity GEVP plateaus with energies close to non-interacting N, Npi, and Npipi benchmarks, a per-mil difference between GEVP mode 0 and the nucleon two-point function, a finite-volume-corrected nucleon mass extrapolated to the physical point as M_N = 0.927(21)(05) GeV, and a negative-parity analysis in which the multi-hadron states are resolved but the nucleon two-point function suffers from signal degradation attributed to insufficient distillation modes.","tokens_in":28409,"tokens_out":6914,"duration_ms":69387,"significance":"If the identification of the three positive-parity GEVP modes as N, Npi, and Npipi is accepted, the paper provides the first Npipi baryon operator in a distillation GEVP analysis and an explicit demonstration that multi-hadron contaminations in the nucleon two-point function are at the per-mil level, validating a workflow for excited-state-free nucleon operators in future form-factor calculations. The automatic contraction tool and the Wigner-Eckart reduction are useful methodological contributions, and the physical-point nucleon mass obtained from AIC model averaging is a nontrivial cross-check. However, the central demonstration is not yet quantitative: the distillation-mode truncation is shown by the paper's own negative-parity results to be a limiting systematic, and the positive-parity multi-hadron levels are compared only with non-interacting energies without a direct truncation control. Several preferred fits also violate the paper's stated extrapolation-check criterion.","major_comments":[{"comment":"The central claim that the positive-parity GEVP cleanly separates N, Npi, and Npipi states depends on the unquantified effect of truncating the distillation Laplacian to Nd=60 (or 120) modes. The paper's own negative-parity analysis (Sec. V; Fig. 15) shows that the same truncation is insufficient for the nucleon two-point function, with the authors explicitly attributing the degradation to missing high modes (Sec. VI: \"the number of distillation modes is too small for most ensembles to perform sophisticated analyses\"). Because the positive-parity Npi and Npipi levels are tested only against non-interacting energies, a mode-truncation shift comparable to the level spacing cannot currently be excluded. I request a concrete control, for example repeating the positive-parity GEVP with a larger Nd on at least one ensemble, or comparing the GEVP ground-state mass with a conventional smeared or point-source nucleon mass, together with an estimate of the resulting shift in the Npi and Npipi energies.","section":"Sec. V, Sec. VI, Figs. 3, 6, 12"},{"comment":"The paper states that fits are selected with an extrapolation-check tension sigma_{tau0-1} < 2, but several preferred fits in Table V violate this criterion. Positive-parity examples include Ensemble-4 GEVP0 with sigma_{tau0-1}=2.23, Ensemble-9 GEVP2 with -3.28, and Ensemble-3 GEVP0 with 2.15 and N2N with 2.17; negative-parity entries show many more violations. Since Table V reports the fits used in the analysis, either these fits should be replaced by ranges satisfying the stated criterion, or the criterion and its application should be revised and documented. As written, the selection rule is not consistently enforced, which weakens the fit-quality argument for the preferred mass estimates.","section":"Sec. III E, Table V, Eq. (49)"},{"comment":"The finite-volume correction is applied only to the nucleon masses; the Npi and Npipi energies are compared with non-interacting finite-volume energies without an estimate of the finite-volume interaction shift. For ensemble 9 both multi-hadron energies lie about 2 sigma below the non-interacting benchmarks, and the discussion in Sec. VI treats this as a marginal fluctuation. Since the assignment of GEVP modes 1 and 2 to Npi and Npipi is part of the central claim, the size of the missing finite-volume corrections for the multi-hadron states should be estimated or explicitly argued to be negligible.","section":"Sec. IV B, Sec. IV A, Figs. 6 and 7"}],"minor_comments":[{"comment":"The phase factor in the definition of the momentum insertion P^{nm}(t,p) is written as e^{ix·y}, but y is not defined and the momentum variable should appear in the exponent; presumably e^{ix·p} is intended.","section":"Eq. (28)"},{"comment":"The caption of Fig. 5 states that eigenvectors are shown at t/a = 12, while the text discusses the eigenvectors at t/a = 8; the caption should be corrected to match the actual time slice.","section":"Fig. 5 and Sec. IV A"},{"comment":"The tension is defined as sigma[X]/E[X], which is the inverse of the usual standardized residual; the values reported in Table V suggest that E[X]/sigma[X] is meant, so the definition should be corrected.","section":"Eq. (49), Sec. III E"},{"comment":"The heading \"POSITVE PARITY RESULTS\" contains a typo and should read \"POSITIVE PARITY RESULTS\".","section":"Sec. IV heading"},{"comment":"The ensemble naming is inconsistent: Table I uses the label \"4\" while the text and Fig. 6 refer to \"Ens 4\"; a uniform naming convention would improve readability.","section":"Table I and Fig. 6"},{"comment":"The sentence in Sec. III F describes a normalization of GEVP eigenvectors, but the figure captions and table headers use different notations for the same objects; a single definition and consistent notation across Secs. IV, V, and the appendix would help the reader.","section":"Sec. III F and Table V"}],"recommendation":"major_revision","confidential_remarks":"The paper is a methods-focused exploratory study with a sound overall workflow and a useful set of algorithmic contributions. The main risk is that the central positive-parity claim rests on an unquantified distillation-truncation systematic that the paper's own negative-parity section shows to be real. I would not reject the manuscript, but I would require a direct truncation control or a substantially weakened central claim before publication. The fit-selection inconsistencies in Table V should also be resolved, as they currently undermine confidence in the preferred mass estimates."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick read: this is a competent exploratory lattice study that gives the distillation method a fair test on baryon multi-hadron spectroscopy. What's genuinely new is the inclusion of an s-wave N-pi-pi operator in the positive-parity GEVP on near-physical domain-wall ensembles, plus an automatic contraction tool that handles arbitrary pion multiplicities. The authors also show, as advertised, that the N-pi/N-pi-pi contamination in the nucleon two-point function is small at the per-mil level when a GEVP basis is used.\n\nCredit where due: the positive-parity spectra are clean for all seven ensembles, the eigenvector decompositions match the expected N, N-pi, N-pi-pi assignments, and the physical-point nucleon mass from the model average, 0.927(21)(05) GeV, agrees with experiment. The negative-parity section is honestly reported as limited by the truncation, which is exactly the right thing to flag.\n\nThe soft spot is the one the authors themselves expose: the distillation truncation at Nd = 60 or 120 Laplacian modes is known to degrade the negative-parity nucleon signal, and there is no quantitative test of its effect on the positive-parity multi-hadron energies. The N-pi and N-pi-pi levels are compared only to non-interacting dispersion estimates; no Luscher-type or other interaction check is attempted. That leaves open the possibility, even if not the likelihood, that the mode truncation shifts the excited levels and distorts the assignment. The supporting evidence - eigenvector overlaps and the coincidence of the exponential gap with the N-pi-pi energy - is suggestive but not a systematic error budget. A rerun of one ensemble with a larger Nd, or a comparison of the extracted N-pi energy with a known phase-shift prediction, would settle this.\n\nMinor additional issues: several preferred fits violate the paper's own sigma_{tau0-1} < 2 criterion (Table V), and the AutoWick code is referenced but not linked, which makes the automatic contraction hard to reuse in practice.\n\nBottom line: this paper deserves a serious referee. It's a solid validation of a workflow that others will want to build on, and the main caveat is addressable rather than fatal. For someone working on nucleon structure or baryon spectroscopy with distillation, it's worth reading and citing.","headline":"Competent exploratory distillation study; new Npi-pi GEVP analysis is plausible, but the mode-truncation systematic is real and unquantified.","tokens_in":28968,"tokens_out":2750,"would_cite":true,"duration_ms":26846,"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":"A three-operator GEVP with a new Nππ operator resolves the positive-parity nucleon spectrum and shows multi-pion contamination of the nucleon mass is negligible.","keywords":["lattice QCD","distillation","nucleon spectrum","nucleon-pion states","nucleon-pion-pion states","generalized eigenvalue problem","domain-wall fermions","automatic contractions"],"falsifier":"Repeat the positive-parity GEVP on ensemble C with a substantially larger mode count (for example, $N_d > 200$) or with a distillation-plus-all-to-all completion, and check whether the $N\\pi$ and $N\\pi\\pi$ energies and the inferred nucleon mass move outside the quoted uncertainties; a systematic shift would indicate that the clean reported spectrum is a truncation artifact.","tokens_in":27790,"feed_emoji":"⚛️","tokens_out":5600,"duration_ms":48543,"temperature":0.7,"pith_summary":"This paper uses lattice QCD with distillation and domain-wall fermions to compute the spectrum of the nucleon together with nucleon-pion and, for the first time, nucleon-pion-pion states in the positive-parity channel, plus nucleon-pion states in the negative-parity channel. The authors build a generalized eigenvalue problem (GEVP) from these operators and automate the large number of quark contractions with a new algorithm. They find stable, well-separated energy levels on seven ensembles, the multi-hadron energies matching their non-interacting values, and they extrapolate the nucleon mass to the physical point, obtaining $M_N = 0.927(21)(05)$ GeV. The paper argues that direct $N\\pi$ and $N\\pi\\pi$ contributions to the nucleon two-point function are negligible, consistent with chiral perturbation theory, which would justify using such operators to control excited-state contamination in future baryon matrix-element calculations. The negative-parity analysis is more limited: the same distillation truncation that works for positive parity degrades the nucleon signal, indicating a mode-truncation systematic.","feed_headline":"First nucleon-pion-pion spectrum from lattice QCD distillation","feed_subtitle":"Multi-pion contamination of the nucleon mass is at the per-mil level, enabling cleaner form-factor calculations.","key_machinery":"The machinery is distillation: smearing quark fields with the lowest $N_d$ eigenvectors of the three-dimensional lattice Laplacian (Eqs. 19–20), which converts correlation functions into contractions of perambulators, momentum insertions, and modified elementals. To handle the factorial growth of Wick contractions for $N\\pi\\pi$ correlation functions, the paper introduces an automatic contraction algorithm that classifies each diagram as trace-full or trace-less and generates the corresponding tensor contractions. Energies are extracted through a GEVP with a fixed time separation $t-t_0=a$, and the paper defines an “extrapolation check” $\\sigma_{\\tau_0-1}$ to select fit ranges where the excited-state model describes the first excluded time slice.","core_discovery":"In the positive-parity sector, the central discovery is that a $3\\times 3$ GEVP built from a nucleon operator, a p-wave nucleon-pion operator, and a newly introduced s-wave nucleon-pion-pion operator yields clear, stable energy levels on all ensembles: the ground state tracks the nucleon mass, and the two excited levels sit at the non-interacting $N\\pi$ and $N\\pi\\pi$ thresholds. The difference between the nucleon two-point effective mass and the GEVP ground-state effective mass is at the per-mil level, and an exponential fit to this difference gives an energy gap consistent with the $\\Delta E_{N\\pi\\pi}$ gap, showing that within the considered operator set $N\\pi\\pi$ is the dominant multi-hadronic contamination of the nucleon two-point function. The paper concludes that $N\\pi$ and $N\\pi\\pi$ contributions to the nucleon two-point function are negligible, consistent with chiral perturbation theory, and that the distillation setup, though originally designed for the muon $g-2$ program, is well suited to baryon multi-hadron spectroscopy.","pith_inferences":["If the positive-parity mode-truncation systematic is comparable to what the paper observes in the negative-parity sector, the reported per-mil-level contamination might be an artifact of the specific distillation mode count, and the true $N\\pi$/$N\\pi\\pi$ contamination could be larger with more modes.","The same GEVP framework could be extended to resonances such as the Roper and $N(1535)$, but the negative-parity results caution that the distillation basis must be enlarged for those channels.","A testable extension is to compute the nucleon axial-vector form factor using the GEVP-constructed excited-state-free nucleon operator from this paper and compare the excited-state contamination with that of the plain nucleon two-point function.","The mode-truncation systematic could be probed directly by varying $N_d$ on a single ensemble and tracking the trend of the nucleon mass and the level splittings."],"forward_implications":["The new $N\\pi\\pi$ operator can be included in future GEVP analyses for form factors, such as the nucleon axial-vector charge, to remove excited-state contamination from the nucleon interpolating operator.","The automatic Wick-contraction algorithm generalizes to correlation functions with an arbitrary number of pions at source and sink, making multi-hadron baryon spectroscopy with many contractions feasible in distillation.","The physical-point nucleon mass $M_N = 0.927(21)(05)$ GeV, within $1\\sigma$ of the experimental value, supports the continuum and physical-mass extrapolation strategy based on Akaike-model averaging.","The observation that $N\\pi\\pi$ contamination dominates over $N\\pi$ in the nucleon two-point function, based on the energy-gap fit, provides guidance on which multi-hadron operators to prioritize in future, larger operator bases.","In the negative-parity channel, the significant overlap between the nucleon two-point function and the nucleon-pion states implies that multi-hadron operators are essential there, but a larger distillation-mode count or an all-to-all completion is needed first."],"supporting_citations":[{"why":"Supplies the distillation method used to construct all quark correlation functions via smeared fields and perambulators.","marker":"[20]"},{"why":"Provides the generalized eigenvalue problem formalism used to extract energies from the correlation matrix.","marker":"[33]"},{"why":"Predicts in chiral perturbation theory that $N\\pi$ contamination of the nucleon two-point function is negligible, a result the paper reproduces.","marker":"[7]"},{"why":"Extends the chiral perturbation theory prediction to $N\\pi\\pi$ contamination, matching the paper's new operator and its findings.","marker":"[8]"},{"why":"Supplies the multi-hadron operator construction and the Wigner–Eckart reduction used for the $N\\pi$ and $N\\pi\\pi$ operators.","marker":"[15]"},{"why":"Provides the modified elemental tensor for baryon contractions in distillation, which the paper adapts for its own objects.","marker":"[27]"},{"why":"Supplies the finite-volume correction formula from baryon chiral perturbation theory used to correct the nucleon masses.","marker":"[43]"}],"fun_headline_variants":["Nucleon mass contamination at per-mil level in lattice QCD","First nucleon-pion-pion levels from lattice QCD distillation","Per-mil precision nucleon mass from distillation lattice QCD","Distillation resolves nucleon-pion-pion states in lattice QCD","Clean nucleon spectrum: multi-pion contamination under 0.1%"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The claim that the positive-parity spectrum is clean rests on the assumption that keeping only the lowest 60 (or 120) Laplacian eigenmodes preserves the low-lying baryon states; the paper's own negative-parity results, where the same truncation degrades the nucleon signal, show this assumption is fragile.","fun_headline_variants_meta":{"raw":{"variants":["Nucleon mass contamination at per-mil level in lattice QCD","First nucleon-pion-pion levels from lattice QCD distillation","Per-mil precision nucleon mass from distillation lattice QCD","Distillation resolves nucleon-pion-pion states in lattice QCD","Clean nucleon spectrum: multi-pion contamination under 0.1%"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000234,"raw_usage":{"total_tokens":1516,"prompt_tokens":981,"completion_tokens":535,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":597,"completion_tokens_details":{"reasoning_tokens":441}},"tokens_in":597,"tokens_out":535,"duration_ms":5541,"temperature":1.0,"reasoning_tokens":441,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T05:28:06.971178+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Repeat the positive-parity GEVP on ensemble C with a substantially larger mode count (for example, $N_d > 200$) or with a distillation-plus-all-to-all completion, and check whether the $N\\pi$ and $N\\pi\\pi$ energies and the inferred nucleon mass move outside the quoted uncertainties; a systematic shift would indicate that the clean reported spectrum is a truncation artifact.","supporting_citations":[{"cited_title":"Peardon, J","cited_arxiv_id":null,"evidence_quote":"Supplies the distillation method used to construct all quark correlation functions via smeared fields and perambulators."},{"cited_title":"collaboration, B","cited_arxiv_id":null,"evidence_quote":"Provides the generalized eigenvalue problem formalism used to extract energies from the correlation matrix."},{"cited_title":"B¨ ar, Physical Review D92 (2015), 10.1103/physrevd.92.074504","cited_arxiv_id":null,"evidence_quote":"Predicts in chiral perturbation theory that $N\\pi$ contamination of the nucleon two-point function is negligible, a result the paper reproduces."},{"cited_title":"B¨ ar, Physical Review D97 (2018), 10.1103/physrevd.97.094507","cited_arxiv_id":null,"evidence_quote":"Extends the chiral perturbation theory prediction to $N\\pi\\pi$ contamination, matching the paper's new operator and its findings."},{"cited_title":"Barca, G","cited_arxiv_id":null,"evidence_quote":"Supplies the multi-hadron operator construction and the Wigner–Eckart reduction used for the $N\\pi$ and $N\\pi\\pi$ operators."},{"cited_title":"Egerer, D","cited_arxiv_id":null,"evidence_quote":"Provides the modified elemental tensor for baryon contractions in distillation, which the paper adapts for its own objects."},{"cited_title":"Ali Khan, T","cited_arxiv_id":null,"evidence_quote":"Supplies the finite-volume correction formula from baryon chiral perturbation theory used to correct the nucleon masses."}],"review_version":1}