{"id":"e3333f19-2bbb-441e-9bbe-c80334fc41e5","arxiv_id":"2412.10497","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"The short-range nuclear contact term in neutrinoless double beta decay depends on sterile neutrino mass more strongly than previously assumed, lengthening predicted half-lives by up to about a factor of six.","lead":"This paper calculates how the strength of a short-range nuclear interaction in neutrinoless double beta decay changes with the mass of a hypothetical sterile neutrino. The result matters for experiments searching for this rare decay, because it can lengthen the predicted half-life by up to a factor of six.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (2.29)'s C~(2)_1 = 12.06 GeV^-2 is a small residual of opposite-sign low- and intermediate-momentum terms of order tens of GeV^-2, yet its NN-potential and inelastic-channel dependence is not checked for ms≠0; the claimed order-of-magnitude enhancement over NDA is the least secure part of the…","rationale":"The paper carefully extends the Cottingham-style matching of Refs. [45,46] to sterile neutrinos with 0 < ms < about 1 GeV, and the machinery is largely sound: the analyticity argument restricting C1(ms) to even powers of ms is correct; the linear-in-ms term from the low-momentum region (Eq. 2.20) is shown in Appendix A to be reproduced by an NLO chiral amplitude; the scale-stability checks (Figs. 2-3) are genuine and non-trivial; and the ms -> 0 limit returns the previously validated result. The strongest-claim number, however, is the ms^2 coefficient itself, advertised as an order-of-magnitude departure from NDA (Ref. [34] used mc = 1 GeV). Equation (2.27) shows that 12.06 GeV^-2 is a residual of a near cancellation between a low-momentum analytic piece (negative, about -1/λ^2 enhanced) and an intermediate-momentum integral with 1/|k|^3 weighting that is dominated by |k| ~ mπ. This is exactly the region where the hadronic model (dipole form factors, Kaplan-Steel half-off-shell amplitude) is least controlled, and - unlike the ms = 0 case that was checked against Reid and AV18 in Ref. [46] - no alternative-potential validation is reported for the ms-dependent coefficients. The λ- and Λ-stability and |p|_ext checks do not probe this model dependence. A 30% spread in C~(2)_1 would translate into a several-GeV^-2 shift, which matters because the NDA baseline is only about 1.4 GeV^-2; the resulting uncertainty would reduce but not erase the qualitative enhancement. The left-right result carries a further, explicitly admitted assumption (Section 3.2.2) that uncalculated N2LO loops cancel the non-analytic ms terms, which is less central to the paper's phenomenological message but should be listed as a caveat on Eq. (3.34). These issues do not invalidate the method or the qualitative conclusion; they do mean the central coefficient should be treated as having a model uncertainty larger than the quoted 30-50% inherited from the ms = 0 case, unless a multi-potential or lattice cross-check is supplied. The reader's CONDITIONAL verdict remains appropriate; I see no reason to move it.","tokens_in":29526,"tokens_out":23991,"duration_ms":208292,"concrete_test":"Recompute C~(2)_1 from Eq. (2.27) and the full matching expression Eq. (2.25) with the Reid-soft-core and Argonne v18 1S0 potentials, the two alternative models already used for the ms = 0 result in Ref. [46], keeping Λ = 2 GeV, λ = 100 MeV, and the same dipole form factors. If the spread in C~(2)_1 across the three potentials exceeds roughly 4 GeV^-2 (about 30% of the central value), the claimed order-of-magnitude enhancement over the NDA estimate (about 1.4 GeV^-2) is not established at the stated precision, and the half-life impact in Fig. 6 needs a correspondingly larger error band. As a complementary check, vary the dipole masses M_V and M_A within their quoted uncertainties and, if feasible, extend the inelastic NNπ analysis of Ref. [71] to nonzero ms to test whether the ms^2 coefficient shifts by a comparable amount.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central result, C~(2)_1 = 12.06 GeV^-2 in Eq. (2.29), is quoted as roughly an order of magnitude above the NDA-based value (mc = 1 GeV) used in Ref. [34]. The structure of Eq. (2.27) shows that this coefficient is the sum of an analytic low-momentum piece, (1+2g_A^2)/4 (2 d_NLO/λ - 1/λ^2) with λ = 100 MeV (about -20 to -40 GeV^-2 depending on d_NLO), and an intermediate-momentum integral -α_<2/2 = (1/2) ∫_λ^Λ dk g_full(k^2) r(|k|)/|k|^3 (of order +30 to +50 GeV^-2), which nearly cancel. Because the integrand scales as 1/|k|^3, the integral is dominated by |k| just above λ ~ mπ, precisely the regime in which the full amplitude is modeled by dipole form factors (g_full) times the half-off-shell factor r(|k|) obtained from the Kaplan-Steel three-Yukawa 1S0 potential. In contrast to the ms = 0 analysis of Ref. [46], which was validated across the Reid, AV18, and Kaplan-Steel potentials, no alternative-potential cross-check and no inelastic NNπ estimate (cf. Ref. [71]) is reported for the ms-dependent coefficients. The λ-stability check in Fig. 3 tests the internal consistency of the momentum split and the NLO truncation of the low-region integrand; it does not constrain the physics content of r(|k|) or g_full, so it cannot bound the model error in C~(2)_1. If the potential dependence of C~(2)_1 is comparable to the 12 GeV^-2 value itself, the order-of-magnitude claim and the factor-of-six half-life impact (Fig. 6) would both need large upward revisions of the uncertainty. A secondary admitted gap is in Section 3.2.2: for the left-right result Eq. (3.34), non-analytic ms terms are subtracted from Eq. (3.24) under the explicit assumption that uncalculated N2LO EFT loops cancel them, leaving C~(2)_{1+2} = 7.99 GeV^-2 with an uncontrolled N2LO shift and a ΛS-dependent residual.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives the sterile-neutrino-mass dependence of the leading short-range nn→pp couplings in chiral EFT for 0νββ, for the minimal νSM (LL) scenario and for left-right (LR) scenarios. Using a Cottingham-style matching between a modeled full hadronic amplitude and chiral EFT, it obtains Eq. (2.29) for C1(ms) and Eq. (3.34) for C1+C2(ms), and uses these results to update the prediction of 0νββ half-lives in a simplified 3+1 model, finding a lengthening by up to a factor of roughly six for ms between 200 and 800 MeV.","tokens_in":30109,"tokens_out":3707,"duration_ms":35586,"significance":"If correct, the extracted ms^2 coefficient C~(2)_1 ≈ 12 GeV^-2 is an order of magnitude larger than the NDA estimate used in Ref. [34], and it materially changes the sterile-neutrino interpretation of 0νββ searches in a mass range that is otherwise difficult to probe. The paper's strengths include explicit analytic matching expressions, a transparent polynomial expansion in ms (Eqs. (2.29) and (3.34)), stability checks against the arbitrary scales λ and Λ and against the external momenta, and a falsifiable prediction that can be confronted with future lattice-QCD or alternative NN-potential calculations. The paper also identifies a small (≈ -0.6) shift in the ms=0 LR contact term relative to Ref. [46], which is a useful technical advance. The main limitation is that the full hadronic amplitude is modeled rather than computed from QCD, and the LR extraction relies on an assumed cancellation of non-analytic terms, so the accuracy of the headline coefficients rests on model assumptions that are not yet fully quantified.","major_comments":[{"comment":"","section":"§2.2, Eqs. (2.27)–(2.29)"},{"comment":"","section":"§3.2.2, Eq. (3.28)"},{"comment":"","section":"§4, Eq. (4.5) and Fig. 6"}],"minor_comments":[{"comment":"","section":"§2.2, Eq. (2.29)"},{"comment":"","section":"§3.4, Eq. (3.34)"},{"comment":"","section":"§4, Eq. (4.4)"},{"comment":"","section":"§5, Conclusions"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid extension of the authors' earlier matching framework and the technical derivation is explicit and careful in its treatment of the arbitrary scales. The main question for the journal is whether the model dependence of the 'full' amplitude and the unproven LR cancellation are sufficiently quantified for the headline numerical claims. I would like the authors to add an error estimate based on alternative NN potentials and an explicit discussion of the NNπ inelastic uncertainty, and to either compute or clearly justify the cancellation assumption in the LR case. These are fixable within the manuscript's scope, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper supplies the first real calculation of the ms^2 (and ms^4) dependence of the leading short-range 0νββ couplings for light sterile neutrinos. That is genuinely new—previous work used NDA or simple interpolations. If you work on 0νββ interpretations, this is the number you need.\n\nWhat the paper does well: the matching framework is explicit, the scale checks in λ, Λ, and external momentum are reasonable, and the polynomial fits in Eqs. (2.29) and (3.34) are easy to use. The paper also identifies a ~ -0.6 shift in the vector combination relative to Ref. [46], which is the kind of detail that matters. The half-life illustration (factor ~6 for ms in 200-800 MeV) makes clear why the result matters.\n\nSoft spots: the stress-test is right that C~(2)_1 = 12.06 GeV^-2 is a small residual of large opposite-sign terms—the low-momentum analytic piece is around -20 to -40 GeV^-2 and the intermediate-momentum integral around +30 to +50 GeV^-2. The integrand is dominated by |k| just above λ ~ mπ, exactly where the full amplitude is modeled with dipole form factors and the Kaplan-Steel three-Yukawa potential. The ms=0 analysis was cross-checked against Reid and AV18; this paper does not do that for the ms-dependent pieces. The inelastic NNπ contribution is not included, and the cited Ref. [71] addresses only the ms=0 case. So the central coefficient could carry a model error comparable to its value. The left-right part has a second soft spot: the subtraction of non-analytic ms terms in Eq. (3.28) assumes that uncalculated N2LO loops cancel them. The authors say this explicitly, but it is not demonstrated, and the ΛS dependence is a residual.\n\nThese are real caveats, not fatal flaws. The paper is honest about the ~30-50% uncertainty, and the central result is likely correct within the adopted model. What is not yet established is whether the ms^2 coefficient is robust against NN-potential and inelastic-channel choices. That should be probed before the factor-of-six half-life impact is used quantitatively.\n\nRecommendation: send to peer review. The referees should push for an alternative-potential check and a bound on the N2LO cancellation, but this is a serious calculation that belongs in the literature. I would cite it if I were writing on sterile-neutrino 0νββ.","headline":"First real calculation of the ms-dependence of the 0νββ short-range couplings—likely right within the model, but the central ms^2 coefficient is a delicate cancellation and deserves a robustness check before it drives phenomenology.","tokens_in":30718,"tokens_out":2474,"would_cite":true,"duration_ms":23504,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Sterile neutrino masses below 1 GeV shift the 0νββ short-range coupling by an order of magnitude more than assumed, lengthening predicted half-lives in the 200–800 MeV range.","keywords":["neutrinoless double beta decay","sterile neutrinos","chiral effective field theory","short-range contact terms","Cottingham matching","left-right symmetric models","low-energy constants"],"falsifier":"A concrete check is to re-run the matching with a different representation of the full amplitude—for instance, including inelastic $NN\\pi$ intermediate states or using a different $NN$ potential such as Reid or CD-Bonn—and compare the extracted $m_s^2$ coefficient of $C_1(m_s)$ with the 12.062 GeV$^{-2}$ of Eq. (2.29); a shift larger than the paper's stated 30–50% uncertainty would falsify the prediction. A direct lattice QCD extraction of the $nn\\to pp$ short-range amplitude at a few sterile masses between 100 and 500 MeV would settle the same question without the model assumption.","tokens_in":29254,"feed_emoji":"☢️","tokens_out":10138,"duration_ms":68937,"temperature":0.7,"pith_summary":"This paper establishes how the short-range nuclear contact term that controls neutrinoless double $\\beta$ decay (0νββ) depends on the mass $m_s$ of light sterile neutrinos in two motivated extensions of the Standard Model. Generalizing the Cottingham-style matching used for the massless case, the authors derive the polynomial $m_s^2$ and $m_s^4$ coefficients of the $nn\\to pp$ coupling $C_1(m_s)$ in the minimal neutrino-extended Standard Model, and the $m_s^2$ coefficient of $C_1+C_2$ in left-right symmetric models. The central quantitative claim is that the $m_s^2$ coefficient is about an order of magnitude larger than the naive dimensional analysis estimate assumed in earlier rate calculations. If this is right, the predicted 0νββ half-life in the 3+1 sterile-neutrino scenario grows by up to a factor of six when $m_s$ lies between 200 and 800 MeV, which would change how experimental limits on sterile-neutrino contributions to 0νββ are interpreted.","feed_headline":"Sterile neutrino mass lengthens 0νββ half-life up to sixfold","feed_subtitle":"The short-range nn→pp coupling grows with sterile mass far more than assumed, so 0νββ limits must be reinterpreted.","key_machinery":"The central object is the matching condition for the dimensionless low-energy constant $\\widetilde{C}_1(\\mu_\\chi, m_s)$, defined from the $nn\\to pp$ contact term by Eq. (2.6). The authors build a 'full' amplitude $a_<(|k|, m_s)$ for neutrino virtualities up to an arbitrary scale $\\Lambda$ from chiral EFT at low momentum, dipole form factors and a Kaplan–Steele three-Yukawa potential for the two-nucleon half-off-shell amplitude (the momentum dependence of the short-range $^1S_0$ scattering amplitude away from the energy shell) at intermediate momentum, and an operator-product-expansion tail above $\\Lambda$; they then equate this to the chiral EFT amplitude, whose singular piece contains the same topology plus the counterterm $\\widetilde{C}_1$. The key structural step is subtracting the infrared behavior: the full amplitude contains powers of $m_s \\tan^{-1}(\\lambda/m_s)$ and logarithms of $m_s^2+|p|_{\\rm ext}^2$, while the EFT amplitude reproduces them only after including the NLO term $d_{\\rm NLO}\\,\\pi\\, m_s$, so that the difference is the polynomial in $m_s^2$ that defines $C_1(m_s)$. The same machinery, with the $\\pi^-\\to\\pi^+$ amplitude and the vector-like current $J_\\mu^V=J_\\mu^L+J_\\mu^R$, extracts $C_1+C_2$ for left-right models via Eq. (3.28).","core_discovery":"On the paper's own terms, the discovery is an explicit, numerically evaluated mass dependence for the leading-order short-range $nn\\to pp$ operators of 0νββ. For the minimal $\\nu$SM, the dimensionless coupling evaluated at $\\mu_\\chi=m_\\pi$ is $C_1(m_s) \\simeq 1.377 + (12.062/\\mathrm{GeV}^2)\\, m_s^2 - (16.735/\\mathrm{GeV}^4)\\, m_s^4$, Eq. (2.29); for the left-right scenario, $C_1+C_2 \\simeq 2.253 + (7.993/\\mathrm{GeV}^2)\\, m_s^2$, Eq. (3.34). These polynomials are extracted by matching a modeled full amplitude, split into low-, intermediate-, and high-momentum regions, against the chiral EFT amplitude, after including the NLO linear-in-$m_s$ term that carries the infrared behavior. The paper demonstrates that the extracted coupling is independent of the arbitrary matching scales and external momenta to within about 1%, and that the non-analytic $m_s$ logarithms cancel between the full and EFT amplitudes, so the coupling is the polynomial in $m_s^2$ that EFT principles require. The numerical consequence is that the $m_s^2$ coefficient is roughly an order of magnitude larger than the NDA-based estimate used in Ref. [34], which substantially alters half-life predictions when $200\\ \\mathrm{MeV} \\lesssim m_s \\lesssim 800\\ \\mathrm{MeV}$.","pith_inferences":["If the assumed cancellation of non-analytic $m_s$ terms in Section 3.2.2 were found to fail in a complete N2LO calculation, the left-right coefficient $C_1+C_2$ quoted in Eq. (3.34) could shift at the same order as its $m_s^2$ term; a full two-loop EFT calculation would be the direct test.","The same matching approach could be applied to sterile masses above O(1 GeV) with the $1/m_s^2$ dimension-nine operator description restored, in principle connecting the low-mass polynomial and the heavy-mass tail by a single hadronic model.","Because the contact term is isospin-symmetric and nuclear-structure independent, comparing 0νββ rates across different isotopes in the $m_s=200$–$800\\ \\mathrm{MeV}$ window could separate the $C_1(m_s)$ contribution from the long-range potential, providing an observable cross-check on the nuclear matrix elements used in the interpolation.","If a future lattice QCD calculation confirms the large $m_s^2$ coefficient found here, then previous sterile-neutrino interpretations of 0νββ limits—which used the NDA-sized coefficient—will need to be re-evaluated, and the allowed parameter space will shift toward larger mixings."],"forward_implications":["For sterile neutrino masses between 200 and 800 MeV, the half-life of 0νββ in the 3+1 νSM scenario is up to six times longer than earlier NDA-based estimates, so interpreting a null experimental signal as a bound on sterile-neutrino parameters requires the new $C_1(m_s)$.","The $m_s^2$ coefficient of $C_1+C_2$ in left-right symmetric models is also several times larger than NDA would suggest, shifting the predicted 0νββ amplitude in that scenario for $m_s$ up to about 1 GeV.","The extracted low-energy constant remains polynomial in $m_s^2$ up to the breakdown scale $m_s\\sim\\Lambda_\\chi$, confirming that the EFT description of Refs. [32–34] is consistent; only the numerical interpolation of $C_1(m_s)$ needs to be updated.","The new interpolation of $C_1(m_s)$ follows the polynomial expansion up to about $3m_\\pi$ before turning over to the $1/m_s^2$ behavior, which changes where the contact term's contribution is maximal relative to the long-range and ultrasoft contributions.","Varying the arbitrary matching scales $\\lambda$, $\\Lambda$, and external momenta changes the extracted coupling by only about 1%, so the numerical result is stable within the chosen model."],"supporting_citations":[{"why":"Supplies the original Cottingham-style matching method for the massless $nn\\to pp$ contact term that this paper generalizes to nonzero $m_s$.","marker":"[45]"},{"why":"Provides the detailed full-amplitude representation and matching framework, including the Kaplan–Steele potential, that is extended here.","marker":"[46]"},{"why":"Gives the previous NDA-based estimate of $C_1(m_s)$ and the interpolation and half-life framework whose predictions this paper corrects.","marker":"[34]"},{"why":"Defines the three-Yukawa nucleon-nucleon potential used as the baseline model for the intermediate-momentum region of the full amplitude.","marker":"[51]"},{"why":"Defines the renormalized chiral EFT formulation in which $C_1$ appears as the counterterm that removes the UV divergence of the $nn\\to pp$ amplitude.","marker":"[44]"},{"why":"Supplies the shell-model short-range nuclear matrix element $M_{F,sd}$ used to translate $C_1(m_s)$ into 0νββ half-life predictions for $^{136}$Xe.","marker":"[70]"}],"fun_headline_variants":["Sterile neutrino mass up to 800 MeV shifts 0νββ half-life","Contact terms for 0νββ grow with sterile mass, altering limits","0νββ matching: sterile mass dependence larger than assumed","New short-range couplings change 0νββ predictions for 200-800 MeV","Sterile neutrinos modify 0νββ half-life via contact terms"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the model chosen for the full hadronic amplitude that is matched onto the EFT — chiral EFT at low momentum, dipole form factors with a three-Yukawa nucleon-nucleon potential in the intermediate region, and an OPE tail at high momentum — so if that interpolation misses an important intermediate state such as inelastic $NN\\pi$, or if the assumed cancellation of non-analytic $m_s$ terms in the left-right extraction fails, the quoted polynomial coefficients shift.","fun_headline_variants_meta":{"raw":{"variants":["Sterile neutrino mass up to 800 MeV shifts 0νββ half-life","Contact terms for 0νββ grow with sterile mass, altering limits","0νββ matching: sterile mass dependence larger than assumed","New short-range couplings change 0νββ predictions for 200-800 MeV","Sterile neutrinos modify 0νββ half-life via contact terms"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000231,"raw_usage":{"total_tokens":1519,"prompt_tokens":1015,"completion_tokens":504,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":631,"completion_tokens_details":{"reasoning_tokens":402}},"tokens_in":631,"tokens_out":504,"duration_ms":559154,"temperature":1.0,"reasoning_tokens":402,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T15:54:24.512981+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check is to re-run the matching with a different representation of the full amplitude—for instance, including inelastic $NN\\pi$ intermediate states or using a different $NN$ potential such as Reid or CD-Bonn—and compare the extracted $m_s^2$ coefficient of $C_1(m_s)$ with the 12.062 GeV$^{-2}$ of Eq. (2.29); a shift larger than the paper's stated 30–50% uncertainty would falsify the prediction. A direct lattice QCD extraction of the $nn\\to pp$ short-range amplitude at a few sterile masses between 100 and 500 MeV would settle the same question without the model assumption.","supporting_citations":[{"cited_title":"Impact of the leading-order short-range nuclear matrix element on the neutrinoless double-beta decay of medium-mass and heavy nuclei","cited_arxiv_id":"2107.13354","evidence_quote":"Supplies the shell-model short-range nuclear matrix element $M_{F,sd}$ used to translate $C_1(m_s)$ into 0νββ half-life predictions for $^{136}$Xe."}],"review_version":1}