{"id":"f222d880-4cfe-4743-aabc-1d4ae808c8cf","arxiv_id":"2608.06610","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"The N4LO contact three-nucleon force couplings extracted from three-nucleon scattering produce excessively large contributions to nuclear and neutron matter energies, indicating they are not directly transferable to infinite matter.","lead":"A short nuclear theory study checks whether recently fitted contact three-nucleon force strengths from nucleon-deuteron scattering are compatible with the energy of nuclear and neutron matter. It finds they are not, with contributions far too large, and revisits the older c_D and c_E couplings in nuclei.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The incompatibility claim depends on an unverified regulator/scheme match between the nd-scattering E_i LECs and Kaiser's density-dependent potentials; without the E_i table and a cutoff check, the large contact energies in Eq. (3) may be artifacts.","rationale":"I read the paper as a cautiously worded phenomenology note, and the central observation is plausible: the E_i extracted from the three-nucleon continuum in Ref. [1] produce unexpectedly large first-order contact contributions in infinite matter, in tension with empirical binding and with naive dimensional analysis. The strongest way this claim could fail is not the absence of the N3LO 3NF alone, which the authors explicitly acknowledge, but the unverified assumption that the fitted E_i can be inserted directly into Kaiser's local density-dependent potentials. If those LECs are strongly regulator- and scheme-dependent, the reported tens-of-MeV contributions could be artifacts. The paper supplies neither the E_i values nor a cutoff-dependence test, so the quantitative incompatibility claim is not reproducible from the manuscript. The finite-nuclei section is exploratory and does not carry the central weight. This concern overlaps with the reader's weakest-assumption statement, which also flags regulator/scheme consistency; I narrow the focus to that single issue and propose a direct check. If the check confirms regulator independence and the E_i table is supplied, the caution would be strengthened; if it fails, the paper should be revised to present the result only as a scheme-dependent observation. The existing CONDITIONAL verdict remains appropriate, so I recommend no change to the reader's verdict.","tokens_in":8345,"tokens_out":5138,"duration_ms":55192,"concrete_test":"Ask the authors to (1) tabulate all 13 E_i values and their fit uncertainties from Ref. [1]; (2) recompute the SNM and NM first-order energies using a regularized version of the N4LO contact interaction with the same regulator and cutoff as used in the nd scattering fit, and compare the result to Eq. (3); (3) repeat the calculation with a second regulator cutoff differing by roughly 100 MeV. If E/A at normal density changes sign or by more than a factor of 2 between these choices, the reported incompatibility is regulator-driven and should not be presented as a quantitative result without a matching/renormalization analysis.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is that inserting the 13 E_i LECs from Ref. [1] into Kaiser's density-dependent contact potentials yields first-order SNM and NM energies per particle that are orders of magnitude too large, so the N4LO contact strengths are incompatible with nuclear and neutron matter. The load-bearing premise is that the E_i of Ref. [1] are the same dimensionless couplings defined in the operator basis of Eqs. (2) and (4), and that they can be used directly in an unregularized local Born calculation. This is never demonstrated. Ref. [1] fits a regularized N4LO three-nucleon contact interaction to nd scattering observables over a finite energy range, while Kaiser's in-medium potentials are derived for local contact operators in a particular scheme. Chiral LECs are scheme- and regulator-dependent; without a matching calculation, a table of the fitted E_i values, or a check of cutoff sensitivity, the large k_F^8-weighted contributions producing the blue curve in Fig. 3 could be artifacts of using regulator-dependent couplings outside the fitted Hilbert space. A second, explicitly acknowledged caveat is the omitted N3LO 3NF: the paper infers incompatibility from the N4LO contact contribution alone, without a full EoS calculation including 2NF plus N2LO and N3LO 3NF, so a compensating contribution of opposite sign is not excluded by the calculation as presented. Both issues point to the same need: the quantitative incompatibility claim must be shown to be independent of regulator and scheme choices.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper computes the first-order (Born) contribution of the N4LO three-nucleon contact interaction to the energy per particle of symmetric nuclear matter and neutron matter, using the density-dependent in-medium potentials of Kaiser (Eqs. (2) and (4)) together with the 13 contact couplings E_i extracted from elastic nucleon-deuteron scattering in Ref. [1]. Using Eq. (3) and Eq. (5), the authors find contributions that are orders of magnitude larger than the empirical binding energy (blue curve in Fig. 3 and Table III) and conclude that the contact strengths obtained from the three-nucleon continuum are incompatible with nuclear and neutron matter. The paper also revisits the c_D,c_E couplings of the leading three-nucleon force and performs an explicitly exploratory finite-nuclei study in which a modified equation of state is used to improve binding energies of 16O, 40Ca, and 208Pb.","tokens_in":8737,"tokens_out":2974,"duration_ms":29745,"significance":"If the incompatibility claim is robust, it is a significant result: it would show that N4LO three-nucleon contact couplings fitted to the continuum cannot be transferred directly to infinite matter, with consequences for nuclear-structure calculations and for the interpretation of the N4LO contact force. The paper's first-order derivation is transparent and easy to follow, and the authors are appropriately cautious in describing the finite-nuclei part as exploratory. However, the central numerical claim is not reproducible from the manuscript as written, and its validity depends on regulator and scheme consistency between Ref. [1] and Kaiser's density-dependent potentials, which is not established. The concern raised in the stress-test note is genuine and lands: without the E_i values and a cutoff/scheme check, the huge k_F^8-weighted energies in Eq. (3) could be artifacts of mixing regularized scattering-fit couplings with unregularized local Born expressions.","major_comments":[{"comment":"The central quantitative claim is that Eq. (3) gives contributions of tens of MeV per nucleon, but the 13 input values E_i from Ref. [1] are not listed anywhere in the manuscript. Without these numbers, the blue curve in Fig. 3 and the entire incompatibility statement cannot be reproduced or checked. A table of the E_i values (including the two that relate to c_D and c_E) is essential and is the first thing a reader needs.","section":"Section II.B, Eq. (3) and Fig. 3"},{"comment":"The manuscript does not establish that the E_i couplings from the regularized nd-scattering fit of Ref. [1] can be inserted directly into Kaiser's density-dependent potentials, which are derived for local contact operators without the same regulator. Chiral low-energy constants are scheme- and regulator-dependent; the k_F^8 scaling of Eq. (3) amplifies any mismatch. The authors should either provide a matching calculation, a cutoff-sensitivity study with the regulator of Ref. [1], or at minimum a clear statement of the scheme used in each input and a quantitative estimate of the resulting uncertainty. As written, the huge values in Fig. 3 could be artifacts of using the couplings outside their fitted Hilbert space.","section":"Sections II.B and III, Eqs. (2)-(5)"},{"comment":"The paper's abstract and main conclusion state that the N4LO contact strengths are 'incompatible' with nuclear and neutron matter, yet the calculation omits the N3LO 3NF (acknowledged in the introduction as 'presently not on stable grounds') and the complete N4LO 3NF. The first-order N4LO contact contribution alone does not exclude a compensating contribution of opposite sign from the omitted terms. The authors should soften the claim to a statement about the N4LO contact contribution in isolation, or perform a complete equation-of-state calculation including the known 2NF and 3NF pieces to demonstrate that compensation cannot occur.","section":"Introduction and Sections II.B-V"},{"comment":"The finite-nuclei test is transparently exploratory, but as presented it is partly post hoc: the blue EoS is constructed by selecting c_D=6.0 and c_E=0.50 and then artificially rejoining the red EoS at rho~0.12 fm^-3 to preserve saturation. The resulting improvement in binding energies in Table IV is therefore built into the construction rather than being a prediction. This does not invalidate the section, but the authors should clearly state that the blue-curve agreement is not evidence for the N4LO contact strengths or for a particular c_D,c_E pair; it is only a sensitivity study of the mass formula to low-density attraction.","section":"Section IV, Fig. 5 and Table IV"},{"comment":"No uncertainties are given for any of the computed energies. Since the main claim is that the N4LO contact contribution is orders of magnitude too large, the authors should at least indicate the sensitivity of Eqs. (3) and (5) to the E_i values, for example by providing the individual E_i contributions or a range from the fit covariance of Ref. [1] if available. Without any uncertainty estimate, the reader cannot distinguish a robust incompatibility from a large but poorly constrained number.","section":"Tables I-III"}],"minor_comments":[{"comment":"There are several typos, including 'fi?ts' in the abstract, 'strucure' in the introduction, 'a a wide range' in the introduction, and 'conributions' in Section II.A. The manuscript should be carefully proofread.","section":"Abstract and Introduction"},{"comment":"In the paragraph after Eq. (3), the green curve is said to use c_E=0.13 with reference [1], but the correct source for c_E=0.13 appears to be Ref. [6] (Drischler et al.), as stated earlier in Section II.A. Please check and correct the citation.","section":"Fig. 3 caption and text"},{"comment":"Equation (6) contains '=≈' and should be either '=' or '≈'. Also, the expansion in alpha is standard, but the definition of e_sym(rho) is not given explicitly; a one-line definition would improve clarity.","section":"Eq. (6)"},{"comment":"The statement 'the saturation properties of SNM ... should be naturally related to the energy and density distributions of nucleons in nuclei' is repeated twice (once in the Introduction and once at the start of Section IV). One occurrence should be removed.","section":"Section IV"},{"comment":"The notation for the Fermi momentum k_f and k_f,n is introduced inconsistently: in Eq. (2) and Eq. (4) the symbols are used without a clear definition in the text. Since the paper is short, a brief definitions sentence in Section II or III would help.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"This is a borderline case. The first-order computation is simple and likely correct in isolation, and the authors hedge many of their statements. However, the main claim cannot be assessed without the E_i table, and the regulator/scheme mismatch is a real correctness risk that could turn the headline numbers into artifacts. I recommend major revision rather than rejection because the missing pieces are concrete and fixable: report E_i, test cutoff sensitivity, and rephrase the incompatibility claim to account for the omitted N3LO 3NF. The finite-nuclei section should be framed more explicitly as an illustrative sensitivity study, not as a nuclear-structure prediction."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a small, honestly hedged phenomenology note that does one genuinely new thing. It takes the 13 N4LO contact couplings that Witala et al. fitted to nd scattering and plugs them into Kaiser's density-dependent in-medium potentials. In first-order Born approximation the resulting energy per nucleon in symmetric nuclear matter and neutron matter is tens to hundreds of MeV—orders of magnitude above the empirical binding. That qualitative incompatibility is almost certainly correct, and it is worth saying out loud. The paper says it clearly and does not oversell.\n\nWhat it does well: the authors are upfront about what is missing. They state that the N3LO 3NF is not included, that the complete N4LO 3NF is unavailable, and that the finite-nuclei exercise is exploratory. The leading-order comparison with cE sets a useful scale, and the figures are easy to read. The arithmetic in Eqs. (3) and (5) is simple and transparent.\n\nThe soft spots are real but not fatal to the qualitative claim. The E_i values from Ref. [1] are not listed in the paper, so a reader cannot reproduce Eq. (3) or see which coupling drives the huge repulsion. More importantly, the regulator/scheme compatibility between the nd-scattering fit and Kaiser's local density-dependent potentials is not discussed. Chiral LECs are scheme-dependent; without a cutoff check or a matching argument, the large k_F^8 contribution could be partly an artifact. The paper also gives no uncertainty estimate. And the omitted N3LO 3NF could, in principle, bring compensating attraction, though that is speculation. All of these are acknowledged or at least consistent with the paper's cautious tone.\n\nThe finite-nuclei section is the weakest part. Choosing cD=6.0, cE=0.50 to mimic extra attraction and then rejoining the other EoS is a post hoc exercise. It is labeled exploratory, so I do not count it heavily, but it does not add much beyond what is already known about cD,cE systematics.\n\nBottom line: this is a cautionary note for anyone who wants to use the N4LO contact LECs from nd scattering in nuclear matter or nuclear structure. It deserves peer review. I would ask the authors to include a table of the E_i values, run a quick cutoff-dependence check, and be explicit about the scheme-matching limitation. With those additions it would be a solid, useful short paper.","headline":"A careful, honestly hedged plug-in calculation shows the N4LO contact LECs from nd scattering give absurdly large nuclear-matter energies; the qualitative caution is worth taking seriously, but the missing E_i table and regulator discussion keep it from being a quantitative result.","tokens_in":9236,"tokens_out":2969,"would_cite":false,"duration_ms":28041,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["21.30.Fe","21.65.Mn"],"model":"deepseek-v4-flash","headline":"Contact three-nucleon-force couplings fitted to nd scattering give nuclear-matter energy shifts far too large to be compatible with empirical binding.","keywords":["three-nucleon force","N4LO","contact interactions","nuclear matter","neutron matter","chiral effective field theory","density-dependent potential","low-energy constants"],"falsifier":"Recompute the energy per nucleon with the same contact strengths but including the omitted N3LO three-nucleon force and regulator-consistent density-dependent potentials; if the resulting $E/A$ at $\\rho\\approx0.16$ fm$^{-3}$ comes within a few MeV of the empirical $-16$ MeV, the claimed incompatibility would not survive. Alternatively, refit the 13 $E_i$ to simultaneously reproduce nd scattering, the triton binding energy, and nuclear-matter saturation and check whether the nd-scattering observables remain equally well described.","tokens_in":8100,"feed_emoji":"⚛️","tokens_out":14874,"duration_ms":115250,"temperature":0.7,"pith_summary":"The paper argues that the 13 contact couplings of the N4LO (fifth-order chiral) three-nucleon force, as extracted from fits to elastic nucleon-deuteron scattering, cannot simply be transferred to infinite nuclear matter. Evaluated through density-dependent in-medium potentials in first-order Born approximation, these couplings produce positive energy shifts in symmetric nuclear matter and neutron matter that are far larger than the empirical binding energy per nucleon. The same incompatibility appears for the leading-order $c_D,c_E$ contact couplings when $c_E$ is taken from the three-nucleon continuum rather than from triton-plus-saturation fits. If the claim holds, any practical use of these contact terms in nuclei requires additional constraints beyond the three-nucleon continuum, and the system-dependence of the low-energy constants must be understood rather than ignored.","feed_headline":"Scattering-fitted 3N forces fail the nuclear-matter test","feed_subtitle":"Contact couplings that fit nd scattering give energy shifts far beyond the empirical ~16 MeV binding.","key_machinery":"The load-bearing object is the density-dependent in-medium nucleon-nucleon potential obtained by averaging the three-nucleon force over the filled Fermi sea. For the N4LO contact force this potential (Eq. (2)) is a quadratic polynomial in the relative momentum $p$, the momentum transfer $q$, and the Fermi momentum $k_F$, with 13 couplings $E_1,\\ldots,E_{13}$. Its first-order expectation value in symmetric nuclear matter collapses to the single combination $2E_1+2E_2+2E_3+6E_4-E_9-3E_{10}-E_{11}-3E_{12}+E_{13}$ multiplying $k_F^8/(10\\pi^4)$ (Eq. (3)); neutron matter gives the analogous combination in Eq. (5). These closed-form expressions are what turn the scattering-fitted contact strengths into a direct prediction for the equation of state.","core_discovery":"Using the density-dependent potentials derived from subleading chiral three-nucleon forces, the authors compute the first-order contribution to the energy per particle from the N4LO contact interaction. With the 13 strengths $E_i$ taken from the nucleon-deuteron scattering fit of Ref. [1], the resulting energy shift in symmetric nuclear matter and neutron matter is incompatible with the empirical equation of state: the magnitude is far too large, and the sign is wrong for binding. The authors also show that the leading-order contact contribution changes from $-0.72$ MeV to $+7.05$ MeV at normal density when $c_E=0.13$ (triton/saturation constrained) is replaced by $c_E=-1.27$ (continuum fitted), illustrating how strongly the short-range couplings depend on the system used to fix them. They then use a mass-formula estimate to show that moving the equation of state toward more attraction at densities around $0.07$--$0.12$ fm$^{-3}$ improves the binding energies of $^{16}$O, $^{40}$Ca, and $^{208}$Pb, suggesting where the missing attraction would need to come from.","pith_inferences":["Because Eq. (3) is a single linear combination of the 13 couplings, nuclear and neutron matter data constrain only one direction in the $E_i$ parameter space; the remaining 12 combinations could be adjusted to restore agreement with nd scattering without changing this energy shift.","The paper's suggestion that off-shell components matter is testable: refitting the $E_i$ with matter saturation included should move the couplings along the combination that leaves on-shell nd-scattering observables unchanged.","A direct extension is to compute the N4LO contact contribution to the symmetry energy and neutron-skin thickness; those observables are sensitive to the same linear combinations and could discriminate between alternative $E_i$ sets.","If the omitted N3LO three-nucleon force turns out to be repulsive at saturation, it would cancel part of the large contact contribution reported here; checking that cancellation is a clear target for the next generation of chiral potentials."],"forward_implications":["Contact strengths fixed by 3N scattering data alone cannot be used reliably in nuclear-matter or finite-nucleus calculations; a refit that includes matter observables is needed.","The $c_D,c_E$ couplings of the leading three-nucleon force depend strongly on the system used to determine them, so constraints from the triton and from saturation must be combined with continuum data in any consistent chiral EFT application.","The N4LO contact contribution is not naturally small at normal density; the first-order Born estimate is large enough that regulator and convergence questions cannot be ignored.","A softer two-nucleon force plus extra attraction in the density region $0.07$--$0.12$ fm$^{-3}$ would bring mass-formula binding energies for $^{16}$O, $^{40}$Ca, and $^{208}$Pb closer to experiment.","Neutron matter shows the same incompatibility, which matters for neutron-star equations of state because the $c_D,c_E$ terms vanish there and the N4LO contacts carry the entire short-range 3N contribution."],"supporting_citations":[{"why":"Supplies the 13 N4LO contact strengths $E_i$ fitted to elastic nucleon-deuteron scattering; these are the values whose incompatibility with matter energies is under test.","marker":"[1]"},{"why":"Provides the SMS N4LO+ two-nucleon potential used as the 2NF baseline throughout the study.","marker":"[3]"},{"why":"Derives the density-dependent in-medium potential for the leading three-nucleon contact force used for the $c_E$-dependent leading-order estimate.","marker":"[5]"},{"why":"Supplies the triton- and saturation-constrained $c_E=0.13$ value used as the comparison benchmark against the continuum-fitted $c_E=-1.27$.","marker":"[6]"},{"why":"Derives the closed-form first-order energy-per-particle expressions from subleading chiral three-nucleon interactions, including Eq. (3) and Eq. (5) that carry the calculation.","marker":"[8]"},{"why":"Establishes the density-dependent nucleon-nucleon potential from subleading chiral three-neutron forces used for the neutron-matter contact potential.","marker":"[9]"},{"why":"Provides the N2LO450 two-nucleon potentials used to build the equations of state in the finite-nuclei exploratory study.","marker":"[7]"}],"fun_headline_variants":["N4LO contact force: scattering fit fails matter test","3N force couplings: nd fit clashes with nuclear matter","Scatter-fitted 3N strengths wrong for matter binding","c_E fit changes sign: matter vs scattering mismatch","Contact 3N force: one fit can't satisfy both systems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The calculation assumes the 13 contact strengths fitted to nucleon-deuteron scattering can be inserted directly into the density-dependent in-medium potentials and evaluated at first order, with the regularization used in the scattering fit matching the one used for matter and with no compensating contribution from the omitted N3LO three-nucleon force.","fun_headline_variants_meta":{"raw":{"variants":["N4LO contact force: scattering fit fails matter test","3N force couplings: nd fit clashes with nuclear matter","Scatter-fitted 3N strengths wrong for matter binding","c_E fit changes sign: matter vs scattering mismatch","Contact 3N force: one fit can't satisfy both systems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000208,"raw_usage":{"total_tokens":1369,"prompt_tokens":875,"completion_tokens":494,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":491,"completion_tokens_details":{"reasoning_tokens":411}},"tokens_in":491,"tokens_out":494,"duration_ms":5222,"temperature":1.0,"reasoning_tokens":411,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T04:16:39.011895+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the energy per nucleon with the same contact strengths but including the omitted N3LO three-nucleon force and regulator-consistent density-dependent potentials; if the resulting $E/A$ at $\\rho\\approx0.16$ fm$^{-3}$ comes within a few MeV of the empirical $-16$ MeV, the claimed incompatibility would not survive. Alternatively, refit the 13 $E_i$ to simultaneously reproduce nd scattering, the triton binding energy, and nuclear-matter saturation and check whether the nd-scattering observables remain equally well described.","supporting_citations":[{"cited_title":"It underbinds the nuclei in Table IV due to insufficient attraction at the appropriate density","cited_arxiv_id":null,"evidence_quote":"Supplies the 13 N4LO contact strengths $E_i$ fitted to elastic nucleon-deuteron scattering; these are the values whose incompatibility with matter energies is under test."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the SMS N4LO+ two-nucleon potential used as the 2NF baseline throughout the study."},{"cited_title":"Chiral NN model and Ay puzzle","cited_arxiv_id":"nucl-th/0111033","evidence_quote":"Derives the density-dependent in-medium potential for the leading three-nucleon contact force used for the $c_E$-dependent leading-order estimate."},{"cited_title":"Density-dependent effective nucleon-nucleon interaction from chiral three-nucleon forces","cited_arxiv_id":"0910.1249","evidence_quote":"Derives the closed-form first-order energy-per-particle expressions from subleading chiral three-nucleon interactions, including Eq. (3) and Eq. (5) that carry the calculation."}],"review_version":1}