{"id":"463eb6f6-8fe5-438c-ad1a-557503c5db03","arxiv_id":"2505.24121","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"First NLO prediction of the nn→ppee amplitude in relativistic chiral EFT with Bayesian uncertainties and no unknown contact term.","lead":"Physicists calculated, for the first time, the next-to-leading-order quantum correction to the basic two-neutron decay process that underlies neutrinoless double-beta decay searches. The result narrows a key theoretical uncertainty in interpreting experiments hunting for a rare decay that would prove neutrinos are their own antiparticles.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central 'no-unknown-contact-term' claim rests on the asserted O(Λ^-2) convergence of the J^{1,2-loop}_fi integrals; the Letter defers this to a self-cited reference and unpublished Supplemental Material, so the parameter-free NLO amplitude is not yet independently checkable.","rationale":"The reader's weakest_assumption correctly isolates the pivotal enabling claim. My stress-test did not find a stronger objection: the discrepancy with the previous nonrelativistic central value is a substantive physics difference but is not fatal, since the validation and the error bars make it a testable prediction; the 'most accurate' phrasing is softened by the comparable quoted uncertainty, but this does not affect the core result. The finiteness assumption is, however, exactly the place where an undetected error would destroy the parameter-free claim. The paper provides no explicit check of the O(Λ^-2) convergence in the Letter, and the referenced Supplemental Material is not present in this arXiv version. The CIB/CSB validation is genuinely supportive but not a direct test of the two-loop neutrino-exchange integrals. Therefore I agree with the reader's conditional verdict and recommend the same concrete check: an independent regulator/dimension-regularization evaluation of the loop integrals, plus release of the posterior samples. No verdict adjustment is needed beyond the existing CONDITIONAL.","tokens_in":9735,"tokens_out":16227,"duration_ms":172130,"concrete_test":"Recompute the NLO amplitude from Eqs. (2)-(7) with the explicit J^{1,2-loop}_fi integrands (from Ref. [34] or the Supplemental Material) using a hard momentum cutoff at Λ = 2, 4, 8, 16 GeV; verify that the amplitude converges to a finite value with residuals decreasing as O(Λ^-2). Then repeat with a second regulator (e.g., a smooth exponential cutoff or dimensional regularization) and confirm the same finite part. If the limit is absent or regulator-dependent, a counterterm is required and the parameter-free NLO claim fails. The authors should post the LEC posterior samples and the loop-integral derivation so the check can be done independently.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The NLO prediction |A0ν| = 0.0209(7) MeV^-2 is parameter-free only if the neutrino-exchange loop integrals J^{1,2-loop}_fi in Eq. (5) are ultraviolet finite and their finite parts are regulator-independent. The Letter states, after Eq. (7), that 'Following the renormalization analysis in Ref. [34], one finds J^{1,2-loop}_fi are convergent as O(Λ^-2)', but it gives no explicit integrands, no cutoff-dependence plot, and no derivation in the main text. The details are deferred to the Supplemental Material, whose URL is not populated in this preprint. The CIB/CSB validation does not close this gap: it tests a single photon exchange iterated in the scattering equation, not the second-order two-neutrino-exchange loops with the axial-current structure of Vν in Eq. (4). If the claimed O(Λ^-2) convergence fails, or if a different regulator yields a different finite part, an unknown weak-sector contact term must be added at NLO, and the quoted value would be incomplete rather than a prediction. This is the load-bearing point because the novelty claim—'no unknown contact term is required up to NLO'—is exactly what distinguishes this calculation from the nonrelativistic framework of Ref. [18].","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The authors report a next-to-leading-order (NLO) prediction for the elementary neutrinoless double-beta decay subprocess nn -> ppee, computed in a relativistic chiral EFT. The two strong-sector low-energy constants C_R and D_R are fitted to low-energy np scattering data, with uncertainties quantified by a Bayesian analysis, and the same constants are then used to predict the nn -> ppee amplitude. The central result is |A0nu| = 0.0209(7) MeV^-2 at |p_i| = 25 MeV and |p_f| = 30 MeV. The paper argues that no unknown weak-sector contact term is needed up to NLO because the neutrino-exchange loop integrals J^{1,2-loop}_{fi} are ultraviolet finite, in contrast to the nonrelativistic heavy-baryon framework. The approach is validated by comparing the predicted CIB and CSB contributions in NN scattering with experiment.","tokens_in":10024,"tokens_out":10296,"duration_ms":104338,"significance":"If the central finiteness claim is correct, this is an important step: it would provide the first NLO EFT prediction of the nn -> ppee amplitude without a weak-sector counterterm, with a quantified Bayesian uncertainty, and it would give a direct constraint for the short-range operator in 0nubb nuclear matrix element calculations. The calculation is not circular, since the target amplitude is not used in the fit; the LECs come from np scattering data, and the CIB/CSB comparison is a genuine external check. The paper makes a specific, falsifiable prediction. The main risk is not internal inconsistency but the reliance of the 'no contact term' claim on a convergence statement that is deferred to a self-cited reference and to an unavailable supplemental file.","major_comments":[{"comment":"The no-unknown-contact-term claim is the load-bearing result of the Letter, and it rests on the finiteness of J^{1,2-loop}_{fi} in Eq. (5). The manuscript asserts, after Eq. (7), that 'Following the renormalization analysis in Ref. [34], one finds J^{1,2-loop}_{fi} are convergent as O(Λ^-2)', and it defers details to the Supplemental Material [43]. Neither the integrands nor the regularization procedure are shown in the main text, the placeholder '[URL]' in Ref. [43] means the supplement is not available in the current submission, and the cited Ref. [34] is the authors' own previous work. This is not merely a presentation issue: if one of the two J loops has a logarithmic or power divergence, or if its finite part is regulator-dependent, an unknown weak-sector contact term enters at NLO, and Eq. (11) would no longer be a parameter-free prediction. I ask the authors to supply the explicit derivation of the loop ultraviolet behavior and a cutoff-convergence check, or at least a complete and accessible supplemental derivation, before the claim can be independently checked.","section":"Relativistic chiral EFT for 0νββ at NLO (after Eq. (7))"},{"comment":"The validation in the 'Validation' section is useful but does not close the gap identified above. The CIB/CSB comparison tests one-photon exchange iterated in the NN scattering equation, which has a different ultraviolet structure from the neutrino-exchange loops in Fig. 1(a); it does not test the second-order insertions of Vν with the axial-current structure of Eq. (4). The statement that the nn and pp phase shifts are cutoff-independent is asserted without a cutoff-dependence plot. I recommend either adding such a plot for the J loops or explicitly limiting the validation claim to the strong sector.","section":"Validation"}],"minor_comments":[{"comment":"The units of the quoted amplitude are inconsistent in the comparison with the previous model estimate: Eq. (11) gives MeV^-2 while the following paragraph quotes the previous value in MeV. Please correct this and also support the claim that the present result is 'by far the most accurate' given that the quoted Bayesian uncertainty is numerically comparable to the earlier model estimate.","section":"Summary (Eq. (11))"},{"comment":"Please report the explicit prior covariance for (C_R, D_R), the treatment of correlations among the np scattering data, and the Metropolis-Hastings acceptance rate and effective sample size for the 10^4 samples. The phrase 'mildly encodes the expected scaling' is not sufficient for reproducibility.","section":"Bayesian uncertainty quantification"},{"comment":"For the nonrelativistic generalized-Cottingham result shown as empty diamonds, please state explicitly whether the error bars are 68% intervals and which model inputs are varied in producing them.","section":"NLO predictions for nn → ppee (Fig. 5)"},{"comment":"Ref. [43] currently appears as 'Supplemental Material at [URL]'. This placeholder must be replaced with the actual link before publication; in the present submission the supplemental material is not accessible.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The central structural claim of the Letter is inherited from the authors' prior Ref. [34] and from a supplement that is not currently available. I recommend that the editor verify that the prior paper explicitly contains the claimed renormalization analysis for the same J loops before accepting the 'no contact term' result. The paper is within the scope of the journal and the reported application is significant, but the missing derivation is load-bearing rather than cosmetic."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this is the first NLO calculation of the nn→ppee amplitude in relativistic chiral EFT, and the structural result—no unknown contact term enters at NLO—is genuinely new and potentially important for 0νββ nuclear matrix element calculations. The authors fit the two strong-sector LECs to np data and then predict the weak amplitude with no weak-sector parameters, so there is no circularity in the central number. The CIB/CSB validation is a real plus: one-photon exchange is structurally similar to neutrino exchange, and reproducing those scattering data parameter-free at NLO is a nontrivial check.\n\nThe paper does what it says. The derivation follows a documented EFT program, the Bayesian uncertainty quantification is standard and clearly described, and the quoted amplitude 0.0209(7) MeV^-2 at |pi|=25, |pf|=30 MeV is accompanied by posterior distributions rather than a single number. That is good practice.\n\nThe soft spots are real. The central claim that the J^{1,2-loop} integrals are convergent as O(Λ^-2) is asserted by reference to the authors' own Ref. [34] and to Supplemental Material whose URL is not populated in this preprint. That is the load-bearing point: if this convergence fails or the finite part is regulator-dependent, an unknown counterterm enters at NLO and the parameter-free prediction collapses. The CIB/CSB check does not close this gap, because it tests iterated single-photon exchange, not the two-loop two-neutrino-exchange diagrams with the axial-current structure of Eq. (4). The referee needs to see explicit integrands or a cutoff-dependence plot.\n\nSecond, the \"most accurate\" phrasing is oversold. The quoted NLO error (0.0007) is comparable to the generalized Cottingham model error (0.0005) at the same kinematics, and the two central values differ by about 10%, with the nonrelativistic result sitting outside the NLO 68% interval. That tension deserves discussion rather than a gloss. The 50% difference at |pf|=100 MeV also suggests the truncation error estimate should be scrutinized.\n\nBottom line: this is a serious EFT Letter with a credible derivation and an honest validation attempt. It deserves a proper referee, but the referee should be sent specifically to the finiteness claim and the comparison with the Cottingham model. If the finiteness derivation checks out, this is a useful step forward for the 0νββ program.","headline":"First NLO nn→ppee amplitude in relativistic chiral EFT with no new contact term, but the key finiteness claim is deferred and the accuracy claim is slightly oversold.","tokens_in":10589,"tokens_out":2331,"would_cite":true,"duration_ms":25015,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper reports the first next-to-leading-order prediction of the $nn\\to ppee$ amplitude in relativistic chiral effective field theory, with no unknown contact term required at this order, quoting $|A^{0\\nu}| = 0.0209(7)\\…","keywords":["neutrinoless double-beta decay","nn to ppee amplitude","relativistic chiral effective field theory","next-to-leading order","Bayesian uncertainty quantification","charge independence breaking","charge symmetry breaking","nuclear matrix elements"],"falsifier":"One decisive check is to evaluate the two-loop neutrino-exchange integral $J^{2\\text{-loop}}_{fi}$ numerically at increasing momentum cutoffs $\\Lambda$: the central claim stands only if it converges as $O(\\Lambda^{-2})$ with no residual cutoff dependence. A future lattice QCD determination of the $nn\\to ppee$ amplitude at physical pion mass that disagrees with $|A^{0\\nu}|=0.0209(7)\\ \\mathrm{MeV}^{-2}$ by more than the quoted 68% uncertainty would also falsify the prediction.","tokens_in":9491,"feed_emoji":"☢️","tokens_out":11523,"duration_ms":102309,"temperature":0.7,"pith_summary":"The paper reports the first next-to-leading-order (NLO) prediction of the elementary neutrinoless double-$\\beta$ decay subprocess $nn\\to ppee$ in relativistic chiral effective field theory, quoting $|A^{0\\nu}|=0.0209(7)\\ \\mathrm{MeV}^{-2}$ at one kinematic point. The central advance is that no unknown short-range contact term enters at NLO in this framework, so the amplitude is fixed entirely by two nucleon-nucleon low-energy constants that are already constrained by scattering data. The same framework reproduces, in a parameter-free way, the charge-independence and charge-symmetry breaking contributions to low-energy $NN$ scattering, which the authors use to validate the approach. If correct, this gives the nuclear-matrix-element community a statistically meaningful, EFT-based value for the amplitude that anchors the short-range operator in $0\\nu\\beta\\beta$ searches.","feed_headline":"Double-beta decay amplitude predicted at next-to-leading order","feed_subtitle":"First uncertainty-quantified EFT value for the elementary decay process could shrink nuclear matrix element errors.","key_machinery":"The central object is the relativistic two-nucleon propagator $G_0(k;E)= \\frac{M}{k^2+M^2}\\frac{1}{E+2M-2\\sqrt{k^2+M^2}+i0^+}$ inside the scattering equation (2). Because this propagator is milder in the ultraviolet than its nonrelativistic reduction, the neutrino-exchange loop integrals $J^{1,2\\text{-loop}}_{fi}$ converge as $O(\\Lambda^{-2})$, which removes the need for an unknown contact term at NLO. The amplitude is then expressed through the renormalized low-energy constants $C_R$ and $D_R$, which map onto the scattering length and effective range, the NLO neutrino potential, and the resummed pion-exchange Yukawa propagator.","core_discovery":"Within the standard picture of long-range Majorana-neutrino exchange, the authors derive the NLO amplitude from the renormalized couplings $C_R$ and $D_R$ together with the loop integrals $J^{1,2\\text{-loop}}_{fi}$. Their central finding is that the relativistic nucleon propagator makes these integrals ultraviolet finite, converging as $O(\\Lambda^{-2})$, in contrast to the logarithmic divergence that appears in nonrelativistic heavy-baryon treatments; hence no weak-sector counterterm is needed up to NLO. The two low-energy constants are determined from $np$ scattering data through a Bayesian posterior, and the maximum-likelihood prediction is $|A^{0\\nu}|=0.0209(7)\\ \\mathrm{MeV}^{-2}$ at $|p_i|=25$ MeV and $|p_f|=30$ MeV. This is roughly 7% larger than the previous nonrelativistic LO estimate from the generalized Cottingham model at the same kinematics, and about 50% larger at 100 MeV, where the NLO correction is naturally more significant.","pith_inferences":["If the ultraviolet-finiteness claim survives closer scrutiny, the same relativistic machinery should extend to other two-body electroweak amplitudes where nonrelativistic EFT needs counterterms, making the NLO amplitude a template for lepton-number-violating processes more generally.","The growing gap between the relativistic NLO and nonrelativistic LO amplitudes at higher momenta suggests that nuclear-structure calculations using the older amplitude may systematically underestimate the short-range contribution; recomputing matrix elements with the new amplitude would test this.","A direct extension would be to apply the same Bayesian analysis to $pp$ and $nn$ scattering data at higher energies, which would sharpen the truncation-error estimate and check whether the 68% band stays realistic."],"forward_implications":["The NLO amplitude gives nuclear-matrix-element calculations a parameter-free anchor for the short-range contact operator, replacing model-based inputs with an EFT value whose uncertainty carries a statistical meaning.","Because the only free inputs are two $NN$ low-energy constants, improved $np$ scattering data directly tighten the posterior and reduce the quoted amplitude uncertainty.","The NLO corrections grow with momentum, differing from the previous nonrelativistic LO result by about 10% at 30 MeV and about 50% at 100 MeV, so the short-range piece of nuclear matrix elements should be evaluated at NLO rather than LO.","The parameter-free reproduction of charge-independence and charge-symmetry breaking in $NN$ scattering supports the framework's use for the weak amplitude and its uncertainty estimate."],"supporting_citations":[{"why":"Builds the relativistic chiral EFT for $nn\\to ppee$ and supplies the renormalization analysis used to conclude the loop integrals are UV finite.","marker":"[34]"},{"why":"The nonrelativistic EFT treatment where the two-loop integral is logarithmically divergent; it defines the contact-term problem this paper avoids.","marker":"[18]"},{"why":"Identifies the leading-order contact decay operator in $0\\nu\\beta\\beta$ that the $nn\\to ppee$ amplitude is meant to constrain.","marker":"[17]"},{"why":"Supplies the nonrelativistic LO $nn\\to ppee$ amplitude from the generalized Cottingham model used for comparison.","marker":"[32]"},{"why":"Companion derivation of the generalized Cottingham model that provides the comparison value and the CIB input in the nonrelativistic approach.","marker":"[33]"},{"why":"Gives the perturbative treatment of the effective-range correction at NLO that underlies the amplitude formula.","marker":"[36]"},{"why":"Supplies the NLO neutrino-exchange potential used in constructing the amplitude.","marker":"[42]"},{"why":"Provides the low-energy $np$ scattering data and scattering length used to constrain $C_R$ and $D_R$ through the Bayesian likelihood.","marker":"[47]"},{"why":"Provides the lattice QCD calculation of $nn\\to ppee$ at unphysical pion mass used for external validation.","marker":"[31]"},{"why":"Shows agreement between the relativistic EFT and the lattice QCD result, supporting the framework's reliability.","marker":"[60]"}],"fun_headline_variants":["First NLO amplitude for double-beta decay, uncertainty quantified","Relativistic EFT predicts double-beta decay amplitude at NLO","NLO double-beta decay amplitude: no unknown contact terms needed","Bayesian NLO prediction for double-beta decay subprocess","Double-beta decay amplitude: NLO with quantified uncertainty"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the neutrino-exchange loop integrals in the relativistic framework are ultraviolet finite, converging as $O(\\Lambda^{-2})$, so that no unknown contact term appears at NLO; the paper defers the details of that convergence argument to the Supplemental Material, and if a counterterm were actually required, an undetermined low-energy constant would enter and the parameter-free prediction would fail.","fun_headline_variants_meta":{"raw":{"variants":["First NLO amplitude for double-beta decay, uncertainty quantified","Relativistic EFT predicts double-beta decay amplitude at NLO","NLO double-beta decay amplitude: no unknown contact terms needed","Bayesian NLO prediction for double-beta decay subprocess","Double-beta decay amplitude: NLO with quantified uncertainty"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000297,"raw_usage":{"total_tokens":1717,"prompt_tokens":935,"completion_tokens":782,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":551,"completion_tokens_details":{"reasoning_tokens":699}},"tokens_in":551,"tokens_out":782,"duration_ms":7366,"temperature":1.0,"reasoning_tokens":699,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:34:47.338694+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One decisive check is to evaluate the two-loop neutrino-exchange integral $J^{2\\text{-loop}}_{fi}$ numerically at increasing momentum cutoffs $\\Lambda$: the central claim stands only if it converges as $O(\\Lambda^{-2})$ with no residual cutoff dependence. A future lattice QCD determination of the $nn\\to ppee$ amplitude at physical pion mass that disagrees with $|A^{0\\nu}|=0.0209(7)\\ \\mathrm{MeV}^{-2}$ by more than the quoted 68% uncertainty would also falsify the prediction.","supporting_citations":[{"cited_title":"Yang and P","cited_arxiv_id":null,"evidence_quote":"Builds the relativistic chiral EFT for $nn\\to ppee$ and supplies the renormalization analysis used to conclude the loop integrals are UV finite."},{"cited_title":"Cirigliano, W","cited_arxiv_id":null,"evidence_quote":"The nonrelativistic EFT treatment where the two-loop integral is logarithmically divergent; it defines the contact-term problem this paper avoids."},{"cited_title":"Cirigliano, W","cited_arxiv_id":null,"evidence_quote":"Identifies the leading-order contact decay operator in $0\\nu\\beta\\beta$ that the $nn\\to ppee$ amplitude is meant to constrain."},{"cited_title":"Cirigliano, W","cited_arxiv_id":null,"evidence_quote":"Supplies the nonrelativistic LO $nn\\to ppee$ amplitude from the generalized Cottingham model used for comparison."},{"cited_title":"Cirigliano, W","cited_arxiv_id":null,"evidence_quote":"Companion derivation of the generalized Cottingham model that provides the comparison value and the CIB input in the nonrelativistic approach."},{"cited_title":"Long and C.-J","cited_arxiv_id":null,"evidence_quote":"Gives the perturbative treatment of the effective-range correction at NLO that underlies the amplitude formula."},{"cited_title":"Cirigliano, W","cited_arxiv_id":null,"evidence_quote":"Supplies the NLO neutrino-exchange potential used in constructing the amplitude."},{"cited_title":"Davoudi, W","cited_arxiv_id":null,"evidence_quote":"Provides the lattice QCD calculation of $nn\\to ppee$ at unphysical pion mass used for external validation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows agreement between the relativistic EFT and the lattice QCD result, supporting the framework's reliability."}],"review_version":1}