{"id":"6dab8d43-d049-45a9-bd0f-d6457fef968b","arxiv_id":"2506.16100","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Adding Z3 symmetry to the seesaw portal allows the dark scalar to decay promptly to dark matter, evading BBN limits for the m_N>m_phi mass ordering.","lead":"A dark matter model uses a Z3 symmetry to let a heavy right-handed neutrino produce superheavy dark matter through freeze-in. The extra symmetry opens a fast decay of the dark scalar into two dark matter particles, making a previously disfavored mass ordering safe under Big Bang Nucleosynthesis.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Printed two-body decay widths in Eqs. (4), (5), and (12) have the wrong mass dimension, undermining the quantitative lifetime and branching-ratio results behind the BBN claim.","rationale":"The reader's verdict is CONDITIONAL, and the dimensionally inconsistent decay widths are a real, concrete defect that supports that verdict. The paper's central qualitative idea, that extending Z2 to Z3 permits phi -> chi chi and thereby evades the BBN constraint on long-lived phi, is plausible and likely robust: for superheavy m_phi even a feeble y_chi gives a lifetime far below 10^-2 s once the width is correctly proportional to m_phi. However, the printed formulas in Eqs. (4), (5), and (12) are not just notationally sloppy; they make the widths dimensionally impossible as written. Since no code or data files are provided, a reader cannot verify whether the numerical results were generated with the printed formulas or with the correct mass-proportional expressions. The branching ratios that feed Eqs. (17), (21), and (25) are especially sensitive because the different two-body widths, if corrected, scale with different powers of the relevant masses, so the y_N-y_chi relations in Figs. 3 and 5 could shift substantially. The reader's stated weakest assumption about N thermal equilibrium is less concerning because for the leptogenesis-relevant range m_N >= 10^9 GeV the seesaw value y_nu ~ 5.7e-8 sqrt(m_N/GeV) exceeds 1e-3, giving Gamma/H at T ~ m_N well above unity; N is likely thermalized. The strongest actionable issue is therefore the decay-width formulas. The qualitative central claim survives, so I do not recommend REJECT, but the paper should not be accepted without corrected formulas and re-run benchmarks. This matches the reader's CONDITIONAL verdict, so no change is needed.","tokens_in":15354,"tokens_out":10557,"duration_ms":111567,"concrete_test":"Recompute the lifetime curves in Fig. 1(b) and the y_N-y_chi relation in Fig. 5(b) using the corrected widths, for example Gamma(phi -> chi chi) = y_chi^2 m_phi / (8 pi) * (1 - 4 m_chi^2 / m_phi^2)^{3/2} and Gamma(N -> phi chi) = y_N^2 m_N / (16 pi) * [(1 + m_chi / m_N)^2 - m_phi^2 / m_N^2] * lambda^{1/2}(m_phi^2/m_N^2, m_chi^2/m_N^2, 1). Check whether the Z3 lifetime curve remains below the BBN bound of 10^-2 s for all m_phi > 10^4 GeV and whether the required y_N for Omega_DM h^2 = 0.120 shifts by more than about 10% relative to the published curves.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central assertion that phi -> chi chi makes the dark scalar short-lived for m_N > m_phi rests quantitatively on the decay width in Eq. (5). As printed, Eq. (5) reads Gamma(phi -> chi chi) = y_chi^2 / (4 pi m_phi) * (1 - 4 m_chi^2 / m_phi^2)^{3/2}, which has mass dimension -1, whereas a decay width must have mass dimension +1. The standard expression is proportional to y_chi^2 m_phi, not y_chi^2 / m_phi. The same problem affects Eq. (4) for phi -> N chi and Eq. (12) for N -> phi chi: both are printed with 1/m in place of the required factor of m. Because Eq. (6) for phi -> chi nu is dimensionally correct, the relative sizes of the different widths, and hence every branching ratio used in Eqs. (17), (21), and (25), cannot be correct if these printed formulas are used. Figure 1, the BBN-exclusion statements, and the y_N versus y_chi relations in Figs. 3 and 5 all depend on these widths. The qualitative mechanism may still hold, but the quantitative parameter regions and inferred couplings are not reproducible from the paper as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a seesaw-portal model of super-heavy dark matter with an exact Z3 symmetry acting on a dark scalar φ and a dark fermion χ. The new interaction yχ φ \\bar{χ}^c χ is claimed to permit the decay φ→χχ, making φ short-lived even for the mass ordering m_N > m_φ and thereby evading BBN constraints. The authors formulate Boltzmann equations for two mass orderings (type A, m_φ > m_N; type B, m_N > m_φ), compute relic densities by freeze-in, and study the impact of low reheating temperature. The paper argues that the Z3 model opens parameter space excluded in the simpler Z2 model.","tokens_in":15583,"tokens_out":20511,"duration_ms":204475,"significance":"If the mechanism were correct, the paper would provide a simple extension of the seesaw portal that rescues the mass ordering m_N > m_φ from BBN constraints, with quantitative predictions for the required Yukawa couplings. The paper uses the standard freeze-in formalism, includes RIS subtraction, and provides numerical Boltzmann solutions with micrOMEGAs, which are appropriate tools for this class of model. However, the central decay φ→χχ is forbidden by the exact Z3 symmetry under the stated charge assignments, and the printed two-body decay widths are dimensionally inconsistent. These issues invalidate the main quantitative results, so the paper in its current form cannot be accepted.","major_comments":[{"comment":"The decay φ→χχ, which is the central mechanism for evading the BBN bound, is forbidden by the exact Z3 symmetry with the charge assignments stated in the Introduction (χ → e^{i2π/3}χ and φ → e^{i2π/3}φ). The operator yχ φ \\bar{χ}^c χ is Z3-invariant because the total charge of the operator is 3 ≡ 0 mod 3. However, the initial state φ carries charge +1, while the final state χχ carries charge +2 mod 3. A Z3-invariant Hamiltonian cannot connect two states in different charge sectors, so the amplitude for φ→χχ vanishes identically. Thus the BBN-avoidance argument based on this decay is invalid from the outset.","section":"Section II.A, Eqs. (1) and (5)"},{"comment":"The decay N→φχ is likewise forbidden by the Z3 selection rule. The right-handed neutrino N is neutral under Z3, while φ and χ each carry charge +1, so the final state N→φχ has total charge +2 mod 3, which cannot equal the initial charge 0. The invariant interaction y_N φ \\bar{χ} N actually connects N to a φ and an anti-χ, not to a φ and a χ. Consequently the Type B production mechanism, which relies on the decay N→φχ and the resulting abundance formula in Eq. (25), does not follow. This is a second, independent instance where the charge assignments are inconsistent with the processes used.","section":"Section III, Eq. (12)"},{"comment":"The printed partial decay widths have mass dimension -1 instead of +1, because the right-hand sides are proportional to 1/m_φ or 1/m_N rather than m_φ or m_N. A decay width must have mass dimension +1. Even setting aside the symmetry selection-rule problem, the lifetime curves in Fig. 1, the branching ratios used in Eqs. (17), (21), and (25), and the y_N–y_χ relations in Figs. 3 and 5 are not reproducible from the displayed formulas. The authors should correct these expressions and verify that their numerical code uses the dimensionally correct widths.","section":"Section II.A, Eqs. (4), (5), and (12)"}],"minor_comments":[{"comment":"The name 'Planck' is misspelled as 'Plank' in several places, including the text near Eq. (10) and Fig. 4.","section":"Throughout"},{"comment":"The numerical prefactor in Eq. (20) is printed in a garbled form ('214π^5'); the expression should be checked and simplified so that the coefficient is unambiguous.","section":"Section IV.A, Eq. (20)"},{"comment":"The statement that the right-handed neutrino N remains in thermal equilibrium because y_ν is 'large enough' is plausible for m_N above the electroweak scale, but the quantitative condition (e.g., Γ/H at T = m_N) is not given; since the Type B calculation assumes Y_N = Y_N^{eq}, the criterion for this assumption should be stated explicitly.","section":"Section III"},{"comment":"The perturbativity constraint y_N ≤ sqrt(4π) is applied to y_N, but no analogous perturbativity constraint on y_χ is discussed; for consistency, the same bound should be applied to both dark-sector Yukawa couplings.","section":"Section IV"}],"recommendation":"reject","confidential_remarks":"The Z3 selection-rule problem is not a typographical or presentational issue; it invalidates the central decay and the Type B production mechanism as stated. The dimensional inconsistency in the decay widths is independently serious, but it would be correctable. The selection rule, however, requires rethinking the charge assignments or replacing the process φ→χχ with a process that is actually allowed, which would change the model and all numerical results. This goes beyond the scope of a revision, so rejection is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know about arXiv:2506.16100. First, the core idea is good: extending the dark Z2 to Z3 in the seesaw-portal FIMP model adds a y_chi phi bar-chi^c chi interaction, which opens phi -> chi chi. That makes the previously BBN-disfavored m_N > m_phi ordering viable, because even a feeble y_chi renders phi short-lived. The paper works through the Boltzmann equations for both mass orderings, including RIS subtraction for the resonant N -> phi chi and the effect of low reheating temperature, and it makes a fair comparison with the Z2 case. That is a legitimate, well-scoped model-building contribution, building on refs [40-42] rather than pretending the interaction is new.\n\nSecond, the printed partial widths in Eqs. (4), (5), and (12) have a load-bearing dimensional error. Each is written proportional to y^2 / m instead of y^2 m, so as printed they have mass dimension -1 and would give lifetimes and branching ratios that cannot be right. Eq. (6) for phi -> chi nu has the correct m_phi dependence, so the relative widths, and therefore the branching ratios in Eqs. (17), (21), and (25), are not what the printed formulas imply. I checked: the lifetime curves and BBN statements in Fig. 1 must have been generated with the standard m-proportional widths, so this is likely a notation slip, but it is fatal to reproducibility of the numbers as printed. The authors need to correct the three equations and re-run the benchmarks; a referee should insist on that.\n\nThe thermal-equilibrium assumption for the right-handed neutrino N in Section III is plausible at these scales since y_nu is large, and the paper explicitly states it. The relic-density fits are what they are--parameter fitting, not predictions--but the BBN-avoidance argument is grounded in external lifetime constraints, so that part is not circular. No code or data are shipped, which is common for this kind of paper, though less helpful for checking the Boltzmann numerics.\n\nBottom line: this deserves a serious referee. The mechanism is attractive, the treatment is thorough, and the error is a fixable typo-class defect rather than a broken argument. A reader who wants to understand the Z3 seesaw portal should read the paper, but should not use the formulas as printed. If the authors correct the widths and confirm the figures, I would cite it.","headline":"A solid Z3 seesaw-portal dark matter idea with a fixable but load-bearing dimensional error in the printed decay widths.","tokens_in":16206,"tokens_out":2900,"would_cite":false,"duration_ms":29901,"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":"Promoting the dark-sector symmetry from Z2 to Z3 opens a fast two-body decay of the dark scalar, making the m_N > m_phi mass ordering viable for superheavy freeze-in dark matter.","keywords":["freeze-in dark matter","seesaw portal","right-handed neutrino","Z3 symmetry","superheavy dark matter","Big Bang Nucleosynthesis","dark scalar decay","leptogenesis"],"falsifier":"Using a full Boltzmann code that tracks Y_N from a specified reheating history instead of setting Y_N = Y_N^eq would settle the size of the Type B production; if the actual N abundance at T ~ m_N is substantially below equilibrium, the coupling values quoted for the m_N > m_phi benchmarks are underestimated. Separately, detecting a late-time decay of a superheavy scalar with tau_phi > $10^{{-2}}$ s and m_phi > $10^{4}$ GeV into SM particles would directly contradict the claim that the Z3 decay phi -> chi chi always keeps phi short-lived in this regime.","tokens_in":15079,"feed_emoji":"🌌","tokens_out":11717,"duration_ms":116892,"temperature":0.7,"pith_summary":"This paper proposes a minimal repair to the seesaw portal for superheavy dark matter, in which right-handed neutrinos explain neutrino masses and mediate freeze-in production of a very heavy dark fermion. In the simplest version with Z2 dark symmetry, the mass ordering with the right-handed neutrino heavier than the dark scalar is nearly ruled out, because the scalar decays only through feeble, phase-space-suppressed channels and lives too long for Big Bang Nucleosynthesis. By changing the dark symmetry to Z3, the model gains the interaction y_chi phi barchi^c chi, which gives the scalar a two-body decay phi -> chi chi. Even with a tiny y_chi this decay is fast enough to keep phi short-lived for m_phi > 2 m_chi, so the previously excluded mass ordering becomes cosmologically allowed. The paper then derives the resulting relic abundance for both mass orderings and shows that the new channel also makes dark matter production more efficient than in the Z2 model.","feed_headline":"Z3 symmetry lets superheavy dark matter pass BBN limits","feed_subtitle":"A new decay keeps the dark scalar short-lived, so the m_N > m_phi mass ordering survives BBN even with feeble couplings.","key_machinery":"The central object is the Z3-symmetric interaction y_chi phi barchi^c chi, allowed when the dark fields transform as chi -> $e^{{2 pi i/3}}$ chi and phi -> $e^{{2 pi i/3}}$ phi. It produces the two-body decay phi -> chi chi with partial width given by Eq. (5), and this decay is the load-bearing mechanism: for m_phi > 2 m_chi it dominates the total width even for y_chi ~ $10^{{-12}}$, pushing tau_phi below $10^{{-2}}$ s and thereby making the m_N > m_phi mass ordering consistent with BBN and CMB constraints. The same width also feeds the Boltzmann equations for Y_chi and Y_phi, raising the final dark matter abundance relative to the Z2 model.","core_discovery":"The central claim is that a Z3 dark symmetry removes the main cosmological obstruction to the seesaw portal at superheavy masses. With the charge assignment chi -> $e^{{2 pi i/3}}$ chi and phi -> $e^{{2 pi i/3}}$ phi, the interaction y_chi phi barchi^c chi is allowed, and it induces phi -> chi chi with partial width Gamma(phi->chi chi) = $y_chi^{2}$/(4 pi m_phi) (1 - 4 $m_chi^{2}$/$m_phi^{2}$)^{3/2}. For m_phi > 2 m_chi this two-body decay dominates the scalar width even for y_chi ~ $10^{{-12}}$, so tau_phi falls below the $10^{{-2}}$ s BBN limit for m_phi > $10^{4}$ GeV. The paper solves the coupled Boltzmann equations for both mass orderings and finds the final dark matter abundance is enlarged by the factor [2 + BR(phi->chi chi)] relative to the Z2 model, up to 3/2 in the limit where phi->chi chi dominates; in the m_N > m_phi ordering the dominant production is N -> phi chi, while for m_phi > m_N it is the neutrino Yukawa scattering phi chi -> h nu.","pith_inferences":["The same symmetry trick, promoting a Z2 dark parity to a Z3 so that a forbidden two-body decay becomes allowed, could rescue other freeze-in portal models with long-lived mediators; the key requirement is only that the mediator and dark matter carry charges whose product matches the new interaction.","Because phi -> chi chi is invisible, the model evades BBN without producing late electromagnetic or hadronic injection; the most direct future test is therefore the reheating temperature inferred from inflationary observables, which decides whether the needed couplings stay below perturbativity.","A non-equilibrium treatment of the right-handed neutrino would test the weakest step: if N is not thermalized at T ~ m_N, the Type B couplings shift upward, but the qualitative conclusion that m_N > m_phi is allowed would persist because the phi -> chi chi width depends only on y_chi."],"forward_implications":["The mass ordering m_N > m_phi, which the minimal Z2 model can only accommodate by fine-tuning the mass spectrum, is open for m_phi > 2 m_chi and m_phi > 10^4 GeV without additional tuning.","For m_N > m_phi with high reheating temperature, the observed relic density is obtained from N -> phi chi decay with y_N^2 ~ 4 x 10^{-25} m_N/m_chi, and the required y_N grows with the ratio m_N/m_chi.","For m_phi > m_N, production is dominated by phi chi -> h nu scattering, and the required coupling satisfies y_nu^2 y_N^2 ~ 2 x 10^{-22} m_phi/m_chi, so larger m_phi/m_chi demands larger y_N.","The Z3 channel makes the scalar-to-dark-matter conversion happen earlier and more efficiently, so the same couplings produce a larger final dark matter abundance than in the Z2 model, by up to a factor 3/2.","Low reheating temperatures exponentially suppress production; as T_RH falls below m_phi or m_N, the coupling y_N required for the relic density rises until the perturbativity bound y_N <= sqrt(4 pi) imposes a lower limit on reheating for each mass spectrum."],"supporting_citations":[{"why":"Establishes the Z2 seesaw-portal superheavy FIMP framework and identifies the long-lived dark scalar problem for m_N > m_phi.","marker":"[36]"},{"why":"Supplies the freeze-in mechanism and the typical feeble coupling scale y ~ 10^{-12} used to illustrate the lifetime.","marker":"[45]"},{"why":"Provides the BBN bound tau_phi < 10^{-2} s that the Z3 decay mode must satisfy.","marker":"[46]"},{"why":"Gives the complementary BBN and CMB constraints on decays of long-lived particles.","marker":"[47]"},{"why":"Sets the leptogenesis requirement m_N > 10^9 GeV that motivates the superheavy mass scale.","marker":"[4]"},{"why":"Provides the analytic approximation for scattering-dominated production used to derive the Type A relic-density formula.","marker":"[39]"},{"why":"Provides the real-intermediate-state subtraction needed to avoid double counting in the Type B Boltzmann equations.","marker":"[49]"},{"why":"Discussed alongside [36] for the fine-tuning of the mass spectrum needed in the Z2 model, which the Z3 model claims to remove.","marker":"[27]"}],"fun_headline_variants":["Z3 seesaw portal frees superheavy dark matter from BBN","Superheavy dark matter rides Z3 symmetry past BBN","Z3 symmetry rescues seesaw dark matter from BBN","New decay path lets superheavy dark matter evade BBN","Z3 dark matter model survives BBN with short-lived scalar"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes the heavy right-handed neutrino N remains in thermal equilibrium, with Y_N equal to its equilibrium value, throughout the epoch when dark matter is produced; if N does not fully thermalize for a given y_nu or reheating history, the Type B abundance calculation and the couplings read off from it would shift.","fun_headline_variants_meta":{"raw":{"variants":["Z3 seesaw portal frees superheavy dark matter from BBN","Superheavy dark matter rides Z3 symmetry past BBN","Z3 symmetry rescues seesaw dark matter from BBN","New decay path lets superheavy dark matter evade BBN","Z3 dark matter model survives BBN with short-lived scalar"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000794,"raw_usage":{"total_tokens":3548,"prompt_tokens":1051,"completion_tokens":2497,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":667,"completion_tokens_details":{"reasoning_tokens":2411}},"tokens_in":667,"tokens_out":2497,"duration_ms":17414,"temperature":1.0,"reasoning_tokens":2411,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:30:38.492004+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Using a full Boltzmann code that tracks Y_N from a specified reheating history instead of setting Y_N = Y_N^eq would settle the size of the Type B production; if the actual N abundance at T ~ m_N is substantially below equilibrium, the coupling values quoted for the m_N > m_phi benchmarks are underestimated. Separately, detecting a late-time decay of a superheavy scalar with tau_phi > $10^{{-2}}$ s and m_phi > $10^{4}$ GeV into SM particles would directly contradict the claim that the Z3 decay phi -> chi chi always keeps phi short-lived in this regime.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the real-intermediate-state subtraction needed to avoid double counting in the Type B Boltzmann equations."}],"review_version":2}