{"id":"23578ffd-f508-4c66-8fb4-e4223f9e51bb","arxiv_id":"2412.14121","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A seesaw model with spontaneously broken lepton number can simultaneously explain dark matter as a freeze-in Majoron and the baryon asymmetry through resonant leptogenesis, with one set of dimension-five operators controlling both.","lead":"Two nearly degenerate heavy neutrinos and a very light Majoron from broken lepton number symmetry are put together in one seesaw model. The paper shows the same higher-order interactions can produce the dark matter abundance by freeze-in and the matter-antimatter asymmetry by resonant leptogenesis, linking three puzzles.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The combined DM and leptogenesis claim is contingent on the assumed scale hierarchy vϕ > TRH > Mi, which is asserted rather than derived; if this ordering fails, both the freeze-in yield and the BAU calculation change, so the common parameter region is not robust to relaxing it.","rationale":"We independently checked the core calculations. The matrix element |M|^2=(s-4M_i^2)/Λ^2 in Eq. (26) gives a constant cross section σ≈1/(16πΛ^2) at high temperature; the resulting yield Yχ∝TRH/Λ^2 is consistent with Eq. (22) and reproduces the benchmark abundance within order one. The annihilation rate Γ(NiNi→χχ)/H is O(0.01) for the BAU-compatible region, so the freeze-in approximation is valid there. The inverse-decay thermalization rate for RHNs is roughly K~O(20) for the entire mass range because yν^2 ∝ M, which supports TRH>Mi when it is imposed. We found no internal inconsistency in the seesaw construction or in the identification of the same dimension-5 operators controlling splitting and DM production. The genuine soft spot is the unquantified dependence on the hierarchy vϕ>TRH>Mi: the paper asserts it, uses it for both DM and leptogenesis, and does not test alternative orderings. This matches the reader's weakest-assumption analysis, so the CONDITIONAL verdict is appropriate.","tokens_in":19556,"tokens_out":57551,"duration_ms":522076,"concrete_test":"Solve the coupled Boltzmann system for RHNs and Majorons with zero initial RHN abundance at T=TRH, for both the benchmark M1=1.4e10 GeV and M1=1e6 GeV with f=0.005, and compare Yχ and the final Y_B. Then repeat the DM and BAU calculation in the alternative hierarchy TRH > vϕ > Mi, with a thermal phase transition after reheating; if the common parameter region survives both tests, the hierarchy is not load-bearing, and if it shifts or disappears, the central claim must be restricted to the assumed ordering.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The weakest point is the scale ordering vϕ > TRH > Mi introduced in Sec. III B. It is used in three places: (i) the U(1)L-breaking dimension-5 operators in Eq. (6) must be active before reheating so the RHN masses and the χχNN coupling are present at T=TRH; (ii) the freeze-in source in Eq. (22) uses the equilibrium yield Y_N^eq, which requires TRH > Mi and thermalized RHNs; (iii) the leptogenesis equations (29) and (38) assume a radiation-dominated era with Mi < TRH. The paper states this hierarchy but does not derive it from the inflaton or scalar sector. In particular, the RHN thermalization check (Sec. III B) is performed only for the benchmark M1=1.4e10 GeV; at the lower end of the claimed range M1~1e6 GeV the effective Yukawa is O(1e-5), and one should verify Y_N reaches Y_N^eq before the DM production peak. If TRH were below Mi, the RHN abundance would be non-thermal and the Yχ ∝ TRH/Λ^2 scaling and the resonant leptogenesis analysis would need to be redone.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs a type-I seesaw model with two right-handed neutrinos and a global U(1)_L symmetry spontaneously broken by a Z2-odd singlet scalar Phi. The resulting Majoron is proposed as a freeze-in dark matter candidate, produced by right-handed neutrino annihilation through a dimension-five lepton-number-violating operator; the same operator generates the diagonal RHN mass entries that split the otherwise degenerate pair and enables near-degenerate leptogenesis. The authors solve Boltzmann equations for the DM yield and the lepton asymmetry, give a benchmark with M1 = 1.4e10 GeV and CP asymmetry about 6e-6, and scan the TRH-Lambda plane to identify a common region satisfying both the DM relic density and the baryon asymmetry, yielding a correlation between m_chi and M1.","tokens_in":19854,"tokens_out":17536,"duration_ms":161546,"significance":"If valid, the model offers a compact single-operator link between neutrino mass, Majoron dark matter, and baryogenesis, with a sub-GeV DM candidate and RHN masses above roughly 1e6 GeV. The analysis uses standard Boltzmann equations and the Casas-Ibarra parametrization, provides a concrete benchmark, and explicitly compares self-energy and vertex contributions to the CP asymmetry. At the same time, the headline correlation is partly constructed rather than predicted, because m_chi is set for each scan point to match Omega_chi h^2 and theta_R is chosen to match the baryon asymmetry; several assumptions and observational constraints are not fully quantified.","major_comments":[{"comment":"The scale hierarchy v_phi > TRH > M_i introduced in Sec. III B is used in three places: the dimension-five operators in Eq. (6) must be active before reheating, the freeze-in source in Eq. (22) is evaluated with the equilibrium yield Y_N^eq, and the leptogenesis Boltzmann equations (29) and (38) assume radiation domination with M_i < TRH. This ordering is asserted rather than derived, and the thermalization check following Fig. 2 is performed only for the benchmark M1 = 1.4e10 GeV. The combined parameter space extends to M1 around 1e6 GeV (lower panel of Fig. 5), where the Casas-Ibarra Yukawa entries are of order 1e-5. Please show explicitly that Y_N tracks Y_N^eq before the DM production peak for representative low-M1 points, and quantify the shift in m_chi when the RHNs start with zero abundance. If TRH were below M_i, both the Y_chi scaling quoted in Sec. III B and the leptogenesis calculation would require re-initialization with a non-thermal RHN abundance.","section":"III B"},{"comment":"The text states that the self-energy CP asymmetry is maximised when M_i^2 - M_j^2 ~ M_i Gamma_Nj, but then says the mass difference is large compared to the decay widths. For the benchmark M1 = 1.4e10 GeV and M2 = 1.36e10 GeV, delta = M2^2 - M1^2 is about -1.1e19 GeV^2, while M_i Gamma_Ni is about 1e14 GeV^2 using Gamma_Ni ~ (y_nu^dagger y_nu)_ii M_i/(8 pi) with diagonal entries of order 1e-5. Thus delta exceeds the width term by about five orders of magnitude, and the self-energy factor in Eq. (36) is S12 ~ M_i/(2 Delta M) ~ 17 rather than the resonant value S ~ M_i/Gamma ~ 1e6. The quoted epsilon ~ 6e-6 therefore arises from off-resonance self-energy enhancement, not from the resonant pole. Please clarify whether 'resonant leptogenesis' is the correct label, and if the resonance condition is imposed, show how Lambda and the correlation plots in Fig. 5 are affected.","section":"IV, Eq. (36)"},{"comment":"The monochromatic neutrino line constraint quoted in Sec. III B ('These experiments restrict the Majoron parameter space for m_chi >~ 4 MeV') is not applied in the relic-satisfied parameter space of Fig. 2 or in the combined BAU plots of Figs. 5 and 7. As shown in Fig. 7, the lower-panel successful region reaches m_chi of order 1 GeV, and Fig. 2 includes relic-satisfying points with m_chi up to 10 GeV. Depending on the exact interpretation of the bound, a substantial part of the claimed common parameter region is either excluded or needs to be re-plotted. Please overlay the Borexino/KamLAND/Super-K/IceCube constraint and state clearly which part of the (TRH, Lambda) and (m_chi, M1) planes survives after all observational cuts.","section":"III B and Figs. 5, 7"}],"minor_comments":[{"comment":"The text says the viable Majoron mass range is 'O(100) GeV >= m_chi >= 1 keV', but the color bar in Fig. 2 has an upper limit of log10 m_chi = 1, i.e. 10 GeV; please make the caption and text consistent.","section":"III B and Fig. 2"},{"comment":"In Eq. (37), the numerator contains m_2^2 - m_3^2 while the denominator contains combinations of m_2 and m_3 without squares; please check the dimensions and clarify the notation, since the quantities quoted are mass-squared differences.","section":"IV, Eq. (37)"},{"comment":"Flavor effects are neglected for leptogenesis, but the BAU patches in Fig. 5 span M1 values from about 1e8 to 1e12 GeV, where the two-flavor and three-flavor regimes are relevant; a short quantitative estimate of the expected uncertainty would strengthen the combined-region claim.","section":"IV, Sec. V"},{"comment":"There is a typographical error in the sentence near Eq. (3): 'as no neglect' should read 'as to neglect'.","section":"II B"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of a hep-ph journal and the central construction is coherent. The major comments are addressable: the hierarchy assumption needs a dedicated check, the 'resonant' terminology should be reconciled with the benchmark parameters, and the observational cuts should be applied consistently to the combined plots. I do not see citation or novelty concerns, but the paper would benefit from stating more explicitly that the m_chi-M1 correlation is a consistency relation obtained after fixing both observed abundances, rather than a sharp prediction."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Honest read: the paper does what it claims within its assumptions. The genuinely new piece is the observation that one dimension-five U(1)_L-breaking operator does double duty: it controls the UV freeze-in yield of the Majoron and generates the RHN mass splitting that makes resonant leptogenesis work. That link is not in Ref. 26 or Ref. 33, and the combined scans in Fig. 5 are the real contribution. The treatment of the CP asymmetry, separating self-energy and vertex pieces, is clean, and the benchmark is internally consistent: the mass splitting is much larger than the widths, and the quoted epsilon_N ~ 6e-6 does produce the observed BAU in their Boltzmann setup. The regime switch near M1 ~ 1e11 GeV, where the vertex contribution starts to dominate, is a nice physical feature worth keeping.\n\nThe soft spots are real but not fatal. The headline m_chi-M1 correlation is partly constructed: m_chi is fit point-by-point to the relic density and theta_R is chosen at the value that maximizes CP violation. That makes the paper a viability study rather than a prediction, and it should say so more prominently. The scale ordering v_phi > T_RH > M_i is assumed, not derived; the paper is upfront about this in the summary, which I appreciate. The stress-test worry about RHN thermalization at low M1 is legitimate but only partly lands: the BAU-satisfying patches in Fig. 5 sit at M1 ~ 1e8-1e12 GeV where the effective Yukawa is sizeable, so the central combined claim likely survives. Still, the equilibrium check in Sec. III B is done only for the benchmark; a plot of Y_N / Y_N^eq across the parameter plane would close the gap. Flavor effects are neglected at M1 ~ 1e9-1e12 GeV, where they can change washout by O(1); citing the literature is not the same as estimating the error. No code or data files accompany the scans, which makes reproduction harder than it should be.\n\nWho is this for: anyone working on Majoron dark matter or seesaw leptogenesis. It is a competent, well-referenced model paper that gives a testable sub-GeV Majoron mass range. It deserves a serious referee. I would send it to review and ask for the low-M1 thermalization check, a flavor-effect estimate, and a reproducibility appendix. This is not desk-reject material.","headline":"Same dimension-five operator drives Majoron freeze-in and resonant leptogenesis; the correlation is partly fitted, but the paper is a solid, referee-able viability study.","tokens_in":20389,"tokens_out":3052,"would_cite":true,"duration_ms":28410,"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":"One seesaw model with a spontaneously broken lepton number explains both dark matter and the baryon asymmetry through the same higher-dimensional operator.","keywords":["Majoron","dark matter","freeze-in","resonant leptogenesis","type I seesaw","lepton number violation","right-handed neutrinos","baryon asymmetry"],"falsifier":"A concrete check: measure or exclude the monochromatic neutrino line from $\\chi\\to\\nu\\nu$. A detected line fixes $m_\\chi$; feeding that mass into the relic-satisfying Boltzmann solutions fixes $\\Lambda$, $T_{\\rm RH}$, and the right-handed neutrino mass $M_1$ and splitting $\\Delta M$, so the observed line energy either lands on the predicted $m_\\chi$–$M_1$ track or rules the common region out. Alternatively, a global fit of neutrino oscillation data that leaves no complex $\\theta_R$ able to produce $\\epsilon_N \\gtrsim 7\\times10^{-6}$ at the relic-fixed splitting would falsify the leptogenesis side.","tokens_in":19314,"feed_emoji":"🌌","tokens_out":18910,"duration_ms":149044,"temperature":0.7,"pith_summary":"Two of the biggest unsolved numbers in cosmology—the dark matter abundance and the baryon asymmetry—could come from the same small symmetry-violating sector. This paper aims to show that in a type I seesaw model (light neutrino masses generated by heavy right-handed neutrinos) with two right-handed neutrinos and a spontaneously broken global lepton number, the same dimension-five operators that split the degenerate right-handed neutrino masses (enabling resonant leptogenesis) also mediate the annihilation $N_i N_i \\to \\chi\\chi$ that produces the Majoron dark matter by freeze-in. Solving the coupled Boltzmann equations, the authors identify a parameter region where the observed relic density $\\Omega_\\chi h^2 \\simeq 0.12$ and baryon asymmetry $Y_B \\simeq 8.75\\times10^{-11}$ are both obtained, with a sub-GeV Majoron and right-handed neutrinos above about $10^6$ GeV. If correct, neutrino mass, dark matter, and baryogenesis are not three independent problems but three outputs of one seesaw sector.","feed_headline":"One mechanism sets dark matter and the matter-antimatter gap","feed_subtitle":"Dark matter and baryon asymmetry emerge from one symmetry-breaking term.","key_machinery":"The load-bearing objects are the dimension-five $U(1)_L$-breaking operators of Eq. (6): $(c_1/2\\Lambda)(\\Phi^2 + \\Phi^{*2})N_1^c N_1$, $(c_2/2\\Lambda)(\\Phi^2 + \\Phi^{*2})N_2^c N_2$, and $y_{\\alpha 2}\\Lambda^{-1} L_\\alpha \\tilde H N_2 (\\Phi + \\Phi^*)$. After $\\Phi$ develops a vev, the first two generate the diagonal mass splitting $\\kappa = v_\\phi^2/\\Lambda$ that breaks the degeneracy of the two right-handed neutrinos, while the resulting $\\chi^2 N_i N_i$ coupling gives the matrix element $|\\mathcal{M}|^2 = (s-4M_i^2)/\\Lambda^2$ for $N_iN_i\\to\\chi\\chi$, a UV freeze-in process whose yield scales as $T_{\\rm RH}/\\Lambda^2$. On the leptogenesis side, the same splitting controls the self-energy CP asymmetry $S_{ij}$, which resonantly enhances $\\epsilon_{N_i}$ when $M_i^2 - M_j^2 \\sim M_i \\Gamma_{N_j}$; the orthogonal-matrix parametrization of the neutrino Yukawa coupling supplies the CP-violating phases through the complex angle $\\theta_R = z_R + i z_I$.","core_discovery":"The central claim is that the same lepton-number-violating terms control both sides of the cosmological puzzle. At the renormalizable level the two right-handed neutrinos couple off-diagonally through $f\\Phi N_1^c N_2$, producing exactly degenerate masses $M = f v_\\phi/\\sqrt{2}$; the dimension-five operators $(\\Phi^2 + \\Phi^{*2}) N_i^c N_i / 2\\Lambda$ add diagonal entries that split them, $M_{1,2} = f v_\\phi/\\sqrt{2} \\pm v_\\phi^2/\\Lambda$, with $\\Delta M = 2v_\\phi^2/\\Lambda$. This same splitting sets the resonance condition for leptogenesis, while the interaction $\\chi^2(N_1^c N_1 - N_2^c N_2)/2\\Lambda$ opens the annihilation channel $N_iN_i \\to \\chi\\chi$ whose yield $Y_\\chi \\propto T_{\\rm RH}/\\Lambda^2$ produces the Majoron. A combined scan over $\\Lambda$, $T_{\\rm RH}$, $m_\\chi$, and the complex angle $\\theta_R$ of the neutrino Yukawa parametrization yields a common region in which both the dark matter relic density and the baryon asymmetry are reproduced; the required CP asymmetry is $\\epsilon_N \\gtrsim 7\\times10^{-6}$, and the right-handed neutrino masses fall in the $10^6$–$10^{13}$ GeV range. The authors therefore conclude that the model offers a unified origin for neutrino mass, dark matter, and the baryon asymmetry.","pith_inferences":["If a monochromatic neutrino line from $\\chi\\to\\nu\\nu$ is ever observed, it overdetermines the model: the line fixes $m_\\chi$, the relic condition fixes $\\Lambda$ and $T_{\\rm RH}$, and leptogenesis then predicts $M_1$, so the observed line energy can be checked against the predicted $m_\\chi$–$M_1$ track.","The high-scale common region evades TeV-scale collider tests but may be reachable through gravitational waves radiated by the heavy right-handed neutrinos during leptogenesis, a signature the paper notes but does not calculate.","A flavour-resolved calculation, which the paper drops for simplicity, could modify the washout efficiency and the required $\\epsilon_N$; whether the common region survives flavour effects is a natural next check.","Relaxing the CP symmetry that forces equal $\\Phi$ and $\\Phi^*$ couplings introduces additional Majoron decay and production channels, turning the Majoron decay rate into a direct probe of CP violation in the lepton-number-breaking sector."],"forward_implications":["Matching the observed relic density fixes the right-handed neutrino mass splitting, so in the common region the baryon asymmetry is not an independent input: the same $\\Lambda$ and $v_\\phi$ set both the freeze-in yield and the leptogenesis resonance.","Resonant leptogenesis in this construction happens at high scale, with right-handed neutrino masses from about $10^6$ GeV to $10^{13}$ GeV, rather than at the TeV scale usually invoked for resonant leptogenesis.","The successful Majoron is sub-GeV and stable on cosmological timescales, but decays to neutrino pairs; for $m_\\chi \\gtrsim 4$ MeV it is within reach of monochromatic neutrino searches, and its two-loop decay to photons can be probed by gamma-ray telescopes.","The relative weight of self-energy versus vertex contributions to the CP asymmetry changes near $M_1 \\sim 10^{11}$ GeV, giving the predicted $m_\\chi$–$M_1$ correlation a distinctive shape that could be tested observationally.","If the right-handed neutrinos start with zero abundance, inverse decays bring them into equilibrium slightly below $T_{\\rm RH}$; the Majoron yield is mildly suppressed and the required $m_\\chi$ shifts upward, leaving the overall conclusion intact."],"supporting_citations":[{"why":"Type I seesaw mechanism that generates light neutrino masses from heavy right-handed neutrinos; the foundation of the model.","marker":"[1–3]"},{"why":"Resonant leptogenesis formalism and the orthogonal parametrization of the neutrino Yukawa matrix used for the CP asymmetry.","marker":"[6, 7]"},{"why":"Origin of the Majoron as the massless Nambu-Goldstone boson of spontaneously broken global lepton number.","marker":"[8–10]"},{"why":"Freeze-in mechanism, the framework used to compute the Majoron dark matter relic abundance.","marker":"[21]"},{"why":"Earlier demonstration that dimension-five U(1)_L-breaking operators allow a keV-to-GeV Majoron FIMP; the production process this paper couples to leptogenesis.","marker":"[26]"},{"why":"Provides the parametrization of the neutrino Yukawa matrix from oscillation data and the complex angle theta_R.","marker":"[48]"},{"why":"The cosmological measurement of the dark matter relic density used as the target Omega_chi h^2 = 0.12.","marker":"[57]"},{"why":"Hierarchical thermal leptogenesis bound on the right-handed neutrino mass that motivates the degenerate resonant regime.","marker":"[76]"},{"why":"Sphaleron conversion of the final lepton asymmetry into the baryon asymmetry, Y_B = (28/51) Y_L.","marker":"[83]"}],"fun_headline_variants":["One operator drives dark matter and baryon asymmetry","Same symmetry term seeds DM and matter-antimatter gap","Unified origin for dark matter and the baryon excess","A single mechanism sets DM and cosmic imbalance"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole result assumes the lepton-number-breaking scale sits above the reheating temperature, which in turn sits above both right-handed neutrino masses; if reheating were colder than those neutrinos, both the dark matter yield and the leptogenesis calculation would have to be redone.","fun_headline_variants_meta":{"raw":{"variants":["One operator drives dark matter and baryon asymmetry","Same symmetry term seeds DM and matter-antimatter gap","Unified origin for dark matter and the baryon excess","A single mechanism sets DM and cosmic imbalance"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000255,"raw_usage":{"total_tokens":1603,"prompt_tokens":1011,"completion_tokens":592,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":627,"completion_tokens_details":{"reasoning_tokens":530}},"tokens_in":627,"tokens_out":592,"duration_ms":6325,"temperature":1.0,"reasoning_tokens":530,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:28:46.165285+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check: measure or exclude the monochromatic neutrino line from $\\chi\\to\\nu\\nu$. A detected line fixes $m_\\chi$; feeding that mass into the relic-satisfying Boltzmann solutions fixes $\\Lambda$, $T_{\\rm RH}$, and the right-handed neutrino mass $M_1$ and splitting $\\Delta M$, so the observed line energy either lands on the predicted $m_\\chi$–$M_1$ track or rules the common region out. Alternatively, a global fit of neutrino oscillation data that leaves no complex $\\theta_R$ able to produce $\\epsilon_N \\gtrsim 7\\times10^{-6}$ at the relic-fixed splitting would falsify the leptogenesis side.","supporting_citations":[{"cited_title":"Pilaftsis, Phys","cited_arxiv_id":null,"evidence_quote":"Sphaleron conversion of the final lepton asymmetry into the baryon asymmetry, Y_B = (28/51) Y_L."}],"review_version":1}