{"id":"10113ddc-c2f3-4ced-a7f7-374873f89a05","arxiv_id":"2502.01760","paper_version":5,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A scalar sextet generates eV-scale Dirac neutrino masses in a 3-3-1 model, and the associated right-handed neutrino population constrains the Z' boson mass to exceed 4.4 TeV.","lead":"This paper builds a 331 gauge extension of the Standard Model in which small Dirac neutrino masses come from a new scalar sextet whose vacuum value is suppressed by a softly broken discrete symmetry. The model then predicts that early-universe right-handed neutrinos force the Z' boson mass above 4.4 TeV, a limit slightly stronger than the current LHC bound.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The m_Z' > 4.4 TeV bound relies on an unvalidated instantaneous-decoupling treatment of right-handed neutrinos at the QCD crossover; a full freeze-out calculation could shift the bound across the LHC limit.","rationale":"The paper's main new result is the cosmological lower bound m_Z' > 4.4 TeV from DeltaNeff. The mass-generating seesaw relation (Eq. 16) is a straightforward adaptation of the type-II seesaw and is not the fragile part; the fragile part is the thermal history of the right-handed neutrinos. For m_Z' = 4.4 TeV, the decoupling temperature from Eq. (52) is around 0.2 GeV, in the QCD crossover region. The paper computes this temperature with g_rho = 106.75 and then converts to DeltaNeff using g_s(T_dec), but at these temperatures g_s and g_rho are both about 60, not 106.75, so the procedure is internally inconsistent at the ten-percent level. Moreover, instantaneous decoupling is assumed without testing it against a full freeze-out calculation; the QCD equation of state changes rapidly in exactly this temperature range, so the final entropy transfer to photons is sensitive to the decoupling history. Because the LHC limit is only 4 TeV, a ten-percent shift in the bound is enough to alter the qualitative conclusion. My recommended check is a full Boltzmann solution with NUDEC_BSM and lattice QCD g_s. If the calculation confirms a bound above 4 TeV, the paper's central claim stands (with a revised number); if not, the abstract's emphasis on N_eff as a competitive probe would need to be softened. The LFV section also contains an arithmetic inconsistency: for the quoted benchmark, Eq. (28) with M_Delta++ ≈ 7 TeV gives Br(mu->eee) ≈ 10^{-12}, not 10^{-17}, and M_Delta++ itself is not 7 TeV for the benchmark parameters, so those predictions require correction. The neutrino-mass mechanism and the Neff bound are unaffected by this, but it lowers confidence. Verdict remains CONDITIONAL.","tokens_in":25456,"tokens_out":49914,"duration_ms":445755,"concrete_test":"Implement the three right-handed neutrinos in a modified NUDEC_BSM code with the Z' s-channel rates of Appendix C and the full lattice-QCD g_s(T), solving coupled Boltzmann equations for T_gamma, T_nuL, and T_nuR at m_Z' = 4.4 TeV. Compare the resulting DeltaNeff with Eq. (60), then locate the m_Z' that saturates the Planck 2-sigma bound DeltaNeff = 0.285. If the saturated m_Z' falls below 4.0 TeV, the LHC-competitive claim in the abstract fails; if above, the headline survives but the numerical bound should be updated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Sec. 5.2 derives the headline m_Z' > 4.4 TeV by combining T_dec from Eq. (52) with the entropy-dilution formula DeltaNeff = 3(10.75/g_s(T_dec))^{4/3} (Eqs. 59-60). For m_Z' = 4.4 TeV, Eq. (52) yields T_dec ≈ 0.2 GeV, inside the QCD crossover. The derivation of Eq. (52) assumes g_rho(T_dec) = 106.75, whereas at T ≈ 0.2 GeV the actual g_rho and g_s are both of order 60; this alone biases T_dec by about ten percent. More serious, the decoupling is treated as instantaneous: the right-handed neutrinos share the bath entropy exactly until T_dec and not at all afterward. Near the QCD crossover, g_s varies steeply over the freeze-out range, and the final temperature ratio (T_nuR/T_nuL)^4 depends on the integrated decoupling history, not just on the value of g_s at one temperature. No comparison with a full solution of the Boltzmann equations is provided, and no uncertainty is quoted. The LHC limit is m_Z' ≥ 4 TeV, very close to the quoted 4.4 TeV; a shift of the freeze-out calculation of ±10-20% can move the bound across the LHC value and change the paper's central conclusion that N_eff is a stronger probe than the LHC. The authors acknowledge a low-reheating caveat but not this QCD-crossover systematics.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript extends the 3-3-1 model with right-handed neutrinos by adding a scalar sextet and a softly broken Z2 symmetry, implementing a type-II seesaw mechanism for Dirac neutrino masses. The resulting small neutrino masses arise from the induced VEV v_Phi ~ M v_eta v_chi' / (2 sqrt(2) mu_S^2). The paper then studies lepton-flavor-violating decays mu -> e gamma and mu -> eee mediated by the sextet scalars, and computes the contribution of right-handed neutrinos to N_eff through Z' interactions. The headline result is a lower bound m_Z' > 4.4 TeV, slightly above the LHC limit of 4 TeV, which the authors present as evidence that N_eff can be a stronger probe of the 3-3-1 breaking scale than current colliders.","tokens_in":25711,"tokens_out":16614,"duration_ms":145135,"significance":"If the central claims hold, the model offers a simple low-scale Dirac seesaw in a 3-3-1 context with distinctive LFV and cosmological signatures. The seesaw formula is clean and the use of N_eff to bound m_Z' is a valuable complement to collider searches. The paper provides explicit analytic expressions for the relevant cross sections and uses numerical integration (Cuba) and a modified NUDEC_BSM code for the left-handed neutrino contribution, which are strengths. However, the headline m_Z' bound rests on an approximate instantaneous-decoupling treatment at the QCD crossover, and the quoted mu -> eee branching ratio contains a large numerical error; both issues must be addressed before the quantitative conclusions can be taken at face value.","major_comments":[{"comment":"The bound m_Z' > 4.4 TeV is derived from an instantaneous-decoupling formula that assumes g_rho(Tdec) = 106.75. For m_Z' = 4.4 TeV, Eq. (52) gives Tdec of order 0.2 GeV, which is inside the QCD crossover, where g_rho and g_s are both significantly smaller and vary steeply (g_s ~ 60, not 106.75). The paper does not solve the Boltzmann equation for the right-handed neutrinos and provides no estimate of the systematic uncertainty from the sudden-decoupling approximation. Because the final (T_nuR/T_nuL)^4 depends on the integrated decoupling history across the crossover, a shift of order 10-20% in Delta N_eff is plausible; since the quoted bound is only about 10% above the LHC limit, this systematics is load-bearing for the paper's central conclusion. I recommend repeating the calculation with temperature-dependent g_* in the Hubble rate and with a smooth freeze-out treatment, or at least providing a quantified two-sided estimate of the approximation error. The low-reheating caveat is acknowledged, but this QCD-crossover systematics is not.","section":"Sec. 5.2, Eqs. (52), (59)-(60)"},{"comment":"The quoted Br(mu -> eee) = 9.89e-17 for the inverted ordering is inconsistent with the authors' own inputs. Using G_nu_IO from Eq. (24), G_F = 1.166e-5 GeV^-2, and M_Delta++ from Eq. (61) with v_eta = 178 GeV, v_chi' = 10 TeV, M = 0.1 keV, and v_Phi = 1 eV gives M_Delta++ ~ 9.4 TeV and Br(mu -> eee) ~ 4e-13; if the 7 TeV mass quoted in Appendix A is used instead, the value is ~ 1.4e-12. The quoted number is about four orders of magnitude smaller. The second form of Eq. (28) also appears to omit the square on |G_nu_12|. Although the conclusion that the prediction respects the current experimental upper bound may survive, the numerical values and the comparison with prospective sensitivity need to be corrected.","section":"Sec. 4, Eqs. (28)-(29)"}],"minor_comments":[{"comment":"The second equality in Eq. (28) should read |G_nu_11|^2 |G_nu_12|^2, not |G_nu_11|^2 |G_nu_12|.","section":"Eq. (28)"},{"comment":"The benchmark values are inconsistent: the text quotes v_eta = 178 GeV in the LFV calculation but v_eta ~ 10^2 GeV elsewhere, and Appendix A says the sextet scalars have mass around 7 TeV while Eq. (61) with the stated inputs gives about 9.4 TeV. These numbers should be harmonized.","section":"Sec. 4 and Appendix A"},{"comment":"The value of the Fermi constant is quoted as G_F = 1.116e-5 GeV^-2; the standard value is 1.166e-5 GeV^-2. Please check and use a consistent value.","section":"Sec. 4"},{"comment":"References [51] and [52] appear to be identical; please distinguish the MEG II current result from its future sensitivity projection or remove the duplicate.","section":"References [51] and [52]"},{"comment":"The text says 'we assume g_rho(T_nuR) = 106.75' but the derived Tdec lies in the QCD crossover; please clarify whether g_rho or g_s is meant and justify the constant value used.","section":"Sec. 5.2, Eq. (52)"},{"comment":"The labeling of the NO and IO Yukawa matrices should be checked for consistency with Eq. (22); in particular, the null entries and the signs should be cross-validated against the stated PMNS parametrization.","section":"Eq. (24)"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the journal's scope and the Dirac type-II seesaw construction in the 3-3-1 model is a reasonable contribution. The central m_Z' > 4.4 TeV claim is interesting but currently rests on an unquantified instantaneous-decoupling approximation at the QCD crossover, and the LFV section contains a clear numerical error. I do not see a circularity problem: the neutrino oscillation parameters are external inputs, and the Delta N_eff and LFV predictions follow from the model's gauge and Yukawa structure. The issues appear fixable within the manuscript's scope, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a competent application of an existing Dirac type-II seesaw idea to the 331RHN, with one new phenomenological result, a DeltaNeff-derived lower bound mZ' > 4.4 TeV. The model-building is clean and the paper is honest about prior work [41,42]. It will not reshape the field, but it is a useful data point: Neff can compete with LHC bounds on Z' in this framework.\n\nWhat is actually new and good: the Z2 soft-breaking realization and the formula vPhi ~ M v_eta v_chi'/(2 sqrt(2) muS^2) are clearly derived, and the benchmark M = 0.1 keV, muS = v_chi' = 10 TeV gives vPhi ~ eV, so small Dirac masses emerge naturally. The cosmology section modifies NUDEC_BSM and provides appendices with cross sections, which is reproducible enough. The seesaw argument itself holds up, and there is no circularity in the DeltaNeff bound.\n\nSoft spots, in proportion. First, the LFV section has internal arithmetic problems. Using their Eq. (28) with their benchmark and G matrices gives Br(mu -> eee) for inverted ordering near 1e-12 if M_Delta++ = 7 TeV, four orders above the quoted 9.89e-17. Also, the Appendix A claim that the sextet scalars sit around 7 TeV is inconsistent with their own formula m^2 ~ M v_ew v_chi'/(4 vPhi) for M = 0.1 keV, v_chi' = 10 TeV, vPhi = 1 eV, which gives roughly 250 TeV. The mu -> e gamma numbers do not match a single consistent mass either. The qualitative conclusion, that the model satisfies current LFV bounds, may survive, but the numbers as reported are not reliable.\n\nSecond, the headline Neff bound. The derivation uses sudden decoupling with g_rho = 106.75 to compute T_dec and then g_s(T_dec) for the entropy release. For mZ' = 4.4 TeV, T_dec ~ 0.2 GeV, right in the QCD crossover, where g_s varies steeply. The error from not solving the full Boltzmann equations is not quantified. A 10-20% shift in T_dec could change g_s enough to move the bound below the LHC 4 TeV limit, which would undo the paper's central selling point. The authors mention the low-reheating caveat but not this systematics, and that needs to be addressed.\n\nWho this is for: people working on 3-3-1 models, Dirac neutrino mass mechanisms, or Neff constraints on extended gauge bosons. They will want to know this bound exists, even if they should treat the exact number with caution.\n\nRecommendation: send to peer review. It deserves referee time, not a desk rejection, but the report should ask for corrected LFV numbers and a robustness check of the decoupling approximation, or at least an error estimate from the QCD crossover.","headline":"A competent, incremental extension of the 331RHN with a Dirac type-II seesaw; the new DeltaNeff bound on mZ' is plausible but rests on an unquantified sudden-decoupling approximation, and the LFV numbers contain internal arithmetic errors that need correcting.","tokens_in":26405,"tokens_out":7961,"would_cite":true,"duration_ms":72842,"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":"A scalar sextet turns the type-II seesaw into a Dirac-neutrino mechanism, and the thermal history of the resulting right-handed neutrinos forces the new Z' boson above 4.4 TeV.","keywords":["type-II seesaw","Dirac neutrino masses","3-3-1 model","scalar sextet","right-handed neutrinos","effective number of neutrino species","lepton flavor violation","Z' boson"],"falsifier":"A decisive test: measure $m_{Z'}$ in dilepton searches and find it below 4.4 TeV while a future high-precision CMB experiment reports $\\Delta N_{\\mathrm{eff}}$ consistent with the Standard Model value; that combination would rule out the model's thermalized right-handed-neutrino history, since the predicted extra radiation would have to be present.","tokens_in":25163,"feed_emoji":"🌌","tokens_out":10371,"duration_ms":85914,"temperature":0.7,"pith_summary":"The paper argues that the type-II seesaw mechanism, normally used for Majorana neutrinos, can be adapted to generate naturally small Dirac neutrino masses inside a 3-3-1 model. It adds a scalar sextet and a softly broken $Z_2$ symmetry; the unique soft-breaking term drives the sextet's neutral component to an eV-scale vacuum expectation value, so neutrino masses of the right size emerge without tiny Yukawa couplings. The same construction produces calculable rates for the rare decays $\\mu \\to e\\gamma$ and $\\mu \\to \\bar e e e$, both below current limits. The paper then follows the right-handed neutrinos through the early universe: $Z'$ interactions thermalize them, and the radiation they contribute before decoupling is controlled by the $Z'$ mass. Requiring this extra radiation to respect current CMB and BAO bounds yields $m_{Z'} > 4.4$ TeV, a lower bound slightly above today's collider exclusion.","feed_headline":"Cosmology pushes new Z' boson past 4.4 TeV","feed_subtitle":"A Dirac-neutrino seesaw makes right-handed neutrinos add radiation, so N_eff data beat today's collider limit.","key_machinery":"The load-bearing object is the scalar sextet $S$ with the soft $Z_2$-breaking term $-\\frac{M}{\\sqrt{2}}\\eta^T S^\\dagger \\chi$; this produces the seesaw suppression $v_\\Phi \\approx \\frac{M v_\\eta v_{\\chi'}}{2\\sqrt{2}\\mu_S^2}$. The second load-bearing piece is the $Z'$-mediated annihilation of right-handed neutrinos: the thermally averaged cross sections into left-handed neutrinos, charged leptons, and quarks set the decoupling temperature $T_{\\mathrm{dec}}^{\\nu_R}$, and the ratio of entropic degrees of freedom at that temperature converts it into $\\Delta N_{\\mathrm{eff}}$. The mass relation $m_{W'} \\approx m_{U^0} \\approx 0.72\\, m_{Z'}$ carries the bound from $Z'$ to all exotic gauge bosons.","core_discovery":"The paper's central claim is that the soft $Z_2$-breaking term $-\\frac{M}{\\sqrt{2}}\\eta^T S^\\dagger \\chi + \\mathrm{H.c.}$, which preserves lepton number, induces the small vacuum expectation value $v_\\Phi \\approx \\frac{M v_\\eta v_{\\chi'}}{2\\sqrt{2}\\mu_S^2}$; for TeV-scale 3-3-1 breaking and $M \\approx 0.1$ keV this yields $v_\\Phi \\approx 1$ eV and hence Dirac masses $m_\\nu = G_\\nu v_\\Phi$ at the observed sub-eV scale. With $v_\\Phi = 1$ eV and neutrino mixing data, the model predicts $\\mathrm{Br}(\\mu \\to e\\gamma) = 3.52\\times 10^{-19}$ (normal ordering) and $3.52\\times 10^{-15}$ (inverted ordering), and $\\mathrm{Br}(\\mu \\to \\bar e e e) = 9.89\\times 10^{-17}$ (inverted), all below current limits. In cosmology, the right-handed neutrinos thermalize for essentially any $m_{Z'}$ below $10^{16}$ GeV; their decoupling temperature is set by $Z'$-mediated annihilations, and entropy dilution after decoupling gives $\\Delta N_{\\mathrm{eff}} = 3(10.75/g_s(T_{\\mathrm{dec}}))^{4/3}$. The observed $\\Delta N_{\\mathrm{eff}} < 0.285$ ($2\\sigma$) then implies $m_{Z'} > 4.4$ TeV, corresponding to $v_{\\chi'} > 9.9$ TeV.","pith_inferences":["The same $\\Delta N_{\\mathrm{eff}}$ calculation transfers to any 3-3-1 variant in which a light $Z'$ couples to right-handed leptons, so the bound may constrain other embeddings, not just this specific sextet construction.","A future CMB experiment at the few-times-$10^{-2}$ sensitivity level on $\\Delta N_{\\mathrm{eff}}$ would sharpen the lower bound on $m_{Z'}$ considerably; the paper's formulas can be inverted once the temperature dependence of the entropy degrees of freedom is refined.","If a collider discovers a $Z'$ below 4.4 TeV with no accompanying radiation excess, the thermalization assumption would be the first thing to give way, and the low-reheating-temperature scenario is a concrete way the model could survive.","The lepton-flavor-violation predictions depend strongly on the neutrino mass ordering, so a positive $\\mu \\to e\\gamma$ signal would point toward the inverted-ordering parameter region of this model."],"forward_implications":["Dirac neutrino masses below the eV scale are generated naturally at TeV-scale 3-3-1 breaking, without introducing tiny Yukawa couplings or Majorana mass terms.","The predicted branching ratios for $\\mu \\to e\\gamma$ and $\\mu \\to \\bar e e e$ lie below current limits; the inverted-ordering $\\mu \\to e\\gamma$ prediction sits within roughly an order of magnitude of the planned experimental sensitivity.","$N_{\\mathrm{eff}}$ provides a lower bound on the 3-3-1 breaking scale, $m_{Z'} > 4.4$ TeV (equivalently $v_{\\chi'} > 9.9$ TeV), which already exceeds the current collider exclusion and is comparable to future high-luminosity collider projections.","The bound extends to the exotic charged gauge bosons $W'$ and $U^0$ because their masses are locked to $m_{Z'}$ by the gauge structure.","If the reheating temperature is below the right-handed-neutrino decoupling temperature, the thermal population never forms and the cosmological bound disappears, leaving collider searches as the main test."],"supporting_citations":[{"why":"supplies the canonical type-II seesaw construction whose VEV suppression mechanism is adapted to Dirac neutrinos.","marker":"[8]"},{"why":"defines the 331RHN gauge model and its fermion and scalar content that the extension builds on.","marker":"[16, 17, 18]"},{"why":"earlier constructions of Dirac seesaw mechanisms in 3-3-1 models that this work extends.","marker":"[41, 42]"},{"why":"provides the gauge boson spectrum and the mass relations among $Z'$, $W'$, and $U^0$ used to translate bounds.","marker":"[44]"},{"why":"supplies the Boltzmann solver used to obtain the left-handed neutrino contribution to $N_{\\mathrm{eff}}$.","marker":"[61]"},{"why":"provides the energy-transfer-rate formalism and the Standard Model neutrino decoupling equations modified here.","marker":"[69]"},{"why":"documents the low-reheating-temperature caveat under which the right-handed neutrinos fail to thermalize and the bound is relaxed.","marker":"[70]"},{"why":"gives the current and projected collider lower bounds on $m_{Z'}$ used for comparison with the cosmological limit.","marker":"[78, 79]"}],"fun_headline_variants":["Dirac seesaw tightens Z' mass to 4.4 TeV","Neutrino cosmology bounds Z' beyond LHC","Type-II seesaw sets Z' floor above 4.4 TeV","N_eff data lifts Z' mass past 4.4 TeV","3-3-1 model: Z' must exceed 4.4 TeV"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The $m_{Z'} > 4.4$ TeV bound assumes the right-handed neutrinos actually thermalize and then decouple suddenly with the Standard Model having 106.75 energy degrees of freedom; if the reheating temperature is below their decoupling temperature, they never reach equilibrium and the bound evaporates.","fun_headline_variants_meta":{"raw":{"variants":["Dirac seesaw tightens Z' mass to 4.4 TeV","Neutrino cosmology bounds Z' beyond LHC","Type-II seesaw sets Z' floor above 4.4 TeV","N_eff data lifts Z' mass past 4.4 TeV","3-3-1 model: Z' must exceed 4.4 TeV"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000639,"raw_usage":{"total_tokens":3037,"prompt_tokens":1134,"completion_tokens":1903,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":750,"completion_tokens_details":{"reasoning_tokens":1804}},"tokens_in":750,"tokens_out":1903,"duration_ms":13210,"temperature":1.0,"reasoning_tokens":1804,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T14:37:41.960920+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A decisive test: measure $m_{Z'}$ in dilepton searches and find it below 4.4 TeV while a future high-precision CMB experiment reports $\\Delta N_{\\mathrm{eff}}$ consistent with the Standard Model value; that combination would rule out the model's thermalized right-handed-neutrino history, since the predicted extra radiation would have to be present.","supporting_citations":[],"review_version":1}