{"id":"e728d7b2-4f1d-4778-8b0c-a0a62004f4d8","arxiv_id":"2412.10254","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":8,"one_line_summary":"For four flavour-symmetry cases, this paper maps heavy-neutrino lifetimes, decay flavour ratios, and the parameter space where leptogenesis works and colliders can test it.","lead":"An extension of the Standard Model with three right-handed neutrinos and discrete flavour and CP symmetries is studied to see if heavy neutrinos could be seen at accelerators and explain the matter-antimatter asymmetry. The paper maps out where the model can produce the observed baryon asymmetry and be tested by experiments like SHiP, MATHUSLA, and FCC-ee.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Testability maps and lifetime ratios depend on the ad hoc form of ΔMR; a different symmetry-breaking structure could alter Table 1 and the leptogenesis regions.","rationale":"The reader's weakest assumption identifies the same load-bearing concern: the form of the Majorana-mass corrections, particularly the ad hoc λ term. I agree this is the most critical point because the central claim explicitly invokes non-zero splittings, and the quantitative content — which experiments can probe which regions, and how lifetime ratios distinguish cases — is derived from the interplay of δMR and ΔMR in the heavy-neutrino mass matrix and the leptogenesis CP-violating combinations. The paper is honest that λ is ad hoc, but this does not reduce the sensitivity of the conclusions to that choice. The other points raised by the reader (mass matrices quoted without derivation, generic-case flavour ratios reducing to PMNS matrix elements, and reduced convergence in one supplementary scan) are less load-bearing: the mass matrices can in principle be verified by independent derivation, the PMNS-dependent flavour ratios are a known limiting behaviour of the framework, and the convergence issue affects only a supplementary panel. The ad hoc λ, by contrast, feeds directly into Table 1, the resonance condition, and the boundaries of the testable parameter space that the summary highlights. A concrete check — recomputing the key observables for a different form of ΔMR — would settle whether the conclusions are accidental features of the chosen perturbation or robust properties of the low-scale seesaw with flavour and CP symmetries. Given that the reader already assigned a CONDITIONAL verdict, my read does not change that recommendation; the concern reinforces the need for the authors to address the sensitivity to the structure of λ before the quantitative maps are taken at face value.","tokens_in":61761,"tokens_out":15709,"duration_ms":644522,"concrete_test":"Replace Eq. (23) with an alternative splitting, e.g. ΔMR = λ M diag(1,0,-1), and re-derive Eq. (62) and Table 1 for Case 1, then recompute the leptogenesis-viable region in the |κ|–U²·M plane for M = 10 GeV (analogous to the upper-left panel of Fig. 14). If the mixing ratios U²₁:U²₂:U²₃ in the |λ|≫U² and U²≫|κ|,|λ| limits change, or if the boundary of the viable parameter space shifts by more than a factor of two, the ad hoc structure of ΔMR is load-bearing and the testability claim requires qualification.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that a sizeable portion of the leptogenesis-viable parameter space is testable, and that heavy-neutrino lifetime and flavour ratios distinguish the symmetry cases, rests on the assumed form of the Majorana-mass corrections, especially ΔMR = λ M diag(0,1,1) in Eq. (23). The paper admits λ is introduced ad hoc, unlike δMR which is motivated by the residual charged-lepton symmetry. The quantitative results — the mixing ratios in Table 1, the resonance line Γ_naive = ΔM in Figs. 14 and 24, and the grey-shaded excluded regions from conditions such as Eq. (47) — all follow from the diagonalization in Eqs. (62)–(71) and the CP-violating combinations in Eqs. (141)–(143). If the symmetry-breaking term that splits the three RH neutrino masses had a different flavour structure, e.g. a component mixing the first and second states, these matrices and the derived lifetimes, resonances, and testable-region boundaries would change. Because the paper does not explore sensitivity to the structure of λ, the quantitative maps are not robust to this unconstrained choice. The qualitative conclusion may survive, but the specific predictions that constitute the paper's novelty are conditional on a particular ad hoc perturbation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the low-scale type-I seesaw with three right-handed neutrinos, a flavour symmetry Δ(3n²) or Δ(6n²), and CP, extending the earlier work of ref. [39] to a comprehensive phenomenological analysis. For the four symmetry-defined mixing cases (Case 1 through Case 3 b.1), it derives the heavy-neutrino mass matrices, the ratios of active-sterile mixings U_i²/U², the lifetimes, and the branching ratios U_α²/U², and then scans the parameter space for resonant leptogenesis with both vanishing and thermal initial conditions. The central claim is that a sizeable portion of the leptogenesis-viable parameter space is testable at SHiP, MATHUSLA, (HL-)LHC and FCC-ee/CEPC, especially for vanishing initial conditions and non-zero mass splittings κ or λ, and that lifetime and flavour ratios can distinguish the symmetry cases.","tokens_in":62165,"tokens_out":12756,"duration_ms":111831,"significance":"If correct, the paper provides concrete, falsifiable targets for a well-motivated flavour-symmetric low-scale seesaw: specific lifetime ratios, restricted flavour branching ratios, and experimentally accessible leptogenesis regions. The strengths are the comprehensive treatment of four symmetry cases, the inclusion of both vanishing and thermal initial conditions, analytic formulae for the CP-violating combinations (Appendix B.2), and the frank admission of limitations, including the ad hoc nature of the λ splitting and a non-converged supplementary scan. The flavour-ratio predictions, while partly inherited from fits to NuFIT data, are still useful discriminants between the cases. However, the quantitative lifetime and leptogenesis predictions in the λ-sensitive regimes rest on an assumed, unconstrained perturbation structure, which limits the robustness of the claimed discriminating power without further analysis.","major_comments":[{"comment":"The form ΔMR = λ M diag(0,1,1) is introduced in Eq. (23) and the paper later states explicitly that the splitting λ is 'ad hoc'. Despite this, the quantitative predictions in the λ-sensitive regimes—the exact mixing-ratio formulae in Eqs. (64) and (68), the rows of Table 1 in the |λ| ≫ U² limit, the resonance lines in Figs. 14 and 24, and the λ-scan regions in Figs. 23, 24, and 26—all depend on this specific diagonal structure. The paper varies the magnitude of λ but never the flavour structure; a different symmetry-breaking perturbation, e.g. one mixing the first and second heavy-neutrino states, would change the diagonalisation of M_R and hence the lifetime ratios and leptogenesis parameter space. Since the summary claims that lifetime ratios distinguish the cases and that the testable regions are sizeable, the authors should either demonstrate robustness under variations of the λ structure or explicitly qualify these claims as benchmark-dependent.","section":"Section 2.1, Eq. (23); Section 5.2, 'Impact of the splitting λ'"},{"comment":"The central heavy-neutrino mass matrices (Eqs. (62), (66), (71)) and the resulting mixing ratios in Table 1 are stated without derivation; the text simply says 'Its form reads' and 'we get'. These matrices underlie the lifetime-ratio predictions that are a principal novelty of the paper, and the limiting ratios (e.g. 2:1:3 vs 1:0:1 vs 4:11:9) are used in Fig. 1 to claim experimental distinguishability. A derivation, or at least a sketch of the diagonalisation procedure, should be provided in an appendix so that the reader can verify the structure and the limits; if this is already available in the companion paper [39], a precise pointer is needed.","section":"Section 3.1, Eqs. (62)-(71) and Table 1"},{"comment":"The caption of Fig. 25 (right plot) states that 'the angular line shapes ... are due to a reduced convergence, since the scan has not been optimised, unlike for the other results shown in this work.' In the main text this plot is used to support the conclusion that for Case 3 b.1) with IO there is 'only a mild enlargement of the allowed parameter space' for m0 = 0.015 eV. A non-converged scan cannot reliably support this quantitative statement; either rerun the scan to converged results or explicitly soften the conclusion.","section":"Appendix B.1, Fig. 25 (right plot)"}],"minor_comments":[{"comment":"The caption of Fig. 2 says 'both κ and λ do not impact the value of the ratios U_α²/U² as long as the condition in Eq. (48) is fulfilled', but Eq. (48) is specific to Case 1) with cos 2θ_R ≈ 0; the analogous conditions for other cases are only 'alike' (as stated in the text). Please rephrase to refer to 'conditions such as Eq. (48)'.","section":"Fig. 2 caption and Section 2.2.2"},{"comment":"The flavour-ratio predictions in this section are mod-squares of PMNS matrix elements whose inputs are taken from the NuFIT global fit. The paper does note that the mixing angle is fitted, but it would be useful to state explicitly that these are not parameter-free predictions; their discriminating power is conditional on the fitted values of θL (or eθL) and the group parameters.","section":"Section 4.1, Eqs. (84)-(91)"},{"comment":"The row 'Case 3 b.1), κ = 0 ... 6.1 / 2.5 /' is ambiguous: the caption says '/' marks situations where the BAU always vanishes, but the row contains only three entries for a four-column table. Please clarify the formatting so that each of the NO/IO and VIC/TIC entries is unambiguous.","section":"Table 3"},{"comment":"The statement that 'the mean lifetime is 1/3 of the quantity Γ_N expected from Eq. (52)' appears to assume equal production of three heavy-neutrino species; as written it could be read as a general result. Please add the qualifying assumption.","section":"Section 3.3, discussion of mean lifetime"}],"recommendation":"major_revision","confidential_remarks":"This is a solid phenomenological paper that extends a known symmetry framework to collider and leptogenesis observables. The main concern is the robustness of the λ-dependent predictions to the explicitly ad hoc form of ΔMR; this is a fixable issue if the authors add a sensitivity study or sharpen their claims. The supplementary non-converged plot (Fig. 25 right) is used in support of a quantitative conclusion and should be addressed. I recommend major revision rather than rejection, since the core framework and the κ-based results are sound and the paper is transparent about its limitations."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does what it says: it takes the flavour/CP-symmetric low-scale seesaw framework from the authors' earlier JHEP paper and runs it through lifetime, branching-ratio, flavour-ratio, and full leptogenesis scans for all four symmetry cases, including Case 3 a) which had not been studied numerically. The new content is real: the heavy-neutrino lifetime ratios in Table 1, the flavour-ratio ternary plots for all cases, the mu-e conversion constraint study, and the marginalised scans over the symmetry parameters. It is also unusually honest about its own limitations, flagging the ad hoc nature of the λ splitting and noting where a supplementary scan has not fully converged.\n\nThe central results look plausible. The mass matrices in Eqs. (62)–(71) are stated without derivation, which is a readability cost rather than a correctness problem—a patient reader can reproduce them from Eq. (61) and the given Ω(3') forms. The flavour-ratio predictions in Section 4 do reduce, in the generic case, to mod-squares of PMNS elements, so they are not independent of the global fit; the sharper value is in the case-specific regions, where the symmetry constrains the mixing enough to produce distinct ternary locations. That distinction is a genuine result.\n\nThe one soft spot that deserves more attention—and it is partly self-inflicted—is the sensitivity to the assumed form of ΔMR. λ enters as an ad hoc diag(0,1,1) perturbation, and Table 1, the resonance lines, and the grey-shaded excluded regions all follow from diagonalising the combined mass matrix with that specific structure. If the symmetry-breaking term that splits the three RH neutrinos had off-diagonal components, the lifetime ratios and the testability boundaries would shift. The paper acknowledges this in passing, but it does not explore how robust the quantitative maps are to that choice. I do not think this is fatal: κ alone already breaks the degeneracy and is motivated by the residual charged-lepton symmetry, and the qualitative conclusion that sizeable testable regions exist for all cases will likely survive. But readers should treat the specific numbers as benchmark predictions of a particular ansatz, not as robust model-independent statements.\n\nThe citation pattern is appropriate—this builds on the authors' own earlier work and cites the relevant experimental and global-fit literature. Who is this for? Anyone working on heavy neutral lepton searches, low-scale seesaw models, or leptogenesis will get value from the scans and the analytic limits. It deserves a serious referee: the conditions raised—derivation of the mass matrices, clarifying the PMNS-element dependence, and a sensitivity study for λ—are addressable in revision. I would send it to review and, after those points are handled, take the maps as useful reference points.","headline":"The paper is a solid, honest extension of the authors' earlier low-scale seesaw framework, with real new results in lifetime ratios and full scans; the main caveat is the ad hoc ΔMR structure, which the authors flag but whose impact on the quantitative maps they do not quantify.","tokens_in":62624,"tokens_out":3391,"would_cite":true,"duration_ms":33872,"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":"This paper shows that discrete flavour and CP symmetries make a large slice of leptogenesis-viable parameter space testable at planned colliders, with heavy-neutrino decay flavours and lifetimes identifying the symmetry case.","keywords":["low-scale seesaw","resonant leptogenesis","flavour and CP symmetries","heavy neutral leptons","collider searches","lepton flavour ratios","lepton mixing patterns","baryon asymmetry of the Universe"],"falsifier":"At a Z-pole machine such as FCC-ee/CEPC with roughly $10^5$ reconstructed heavy-neutrino decays, measure the flavour ratios $U_e^2/U^2$, $U_\\mu^2/U^2$, $U_\\tau^2/U^2$ and the decay-length distribution: detecting $U_e^2/U^2 > 0.35$ would falsify Case 3 b.1) as analysed here, and a single-exponential distribution instead of the predicted mixture with ratios such as $2:1:3$ or $1:0:1$ would falsify the lifetime predictions. In the $\\kappa$-$\\lambda$ plane, observing successful leptogenesis with splittings that violate the consistency conditions of Eqs. (47)–(49) would break the framework's central assumption.","tokens_in":61553,"feed_emoji":"⚫️","tokens_out":13156,"duration_ms":122900,"temperature":0.7,"pith_summary":"This paper tries to establish that a low-scale type-I seesaw with three right-handed neutrinos, whose flavour structure is dictated by a discrete symmetry $\\Delta(3 n^2)$ or $\\Delta(6 n^2)$ combined with CP, is not just a neutrino-mass mechanism but a testable origin of the baryon asymmetry of the Universe. For four symmetry cases that produce distinct lepton mixing patterns, it maps the region of the heavy-neutrino mass–mixing plane where resonant leptogenesis can generate the observed baryon asymmetry, and overlays the projected reach of SHiP, MATHUSLA, (HL-)LHC, and FCC-ee/CEPC. The paper also shows that the lifetimes and flavour ratios of heavy-neutrino decays are predicted differently in each case, so a future collider that observes these decays could identify which symmetry case is realised and, in several cases, the neutrino mass ordering and the lightest neutrino mass. A sympathetic reader should care because the conclusion is that a large part of the leptogenesis-viable parameter space is experimentally accessible whenever at least one of the small mass splittings $\\kappa$ or $\\lambda$ is non-zero.","feed_headline":"Flavour and CP symmetry puts leptogenesis in collider reach","feed_subtitle":"A sizeable share of baryogenesis-viable parameter space is testable at SHiP, MATHUSLA, HL-LHC, and FCC-ee.","key_machinery":"The load-bearing object is the factorised Yukawa parametrisation $Y_D = \\Omega^{(3)} R_{ij}(\\theta_L) \\operatorname{diag}(y_1,y_2,y_3) P^{ij}_{kl} R_{kl}(-\\theta_R) \\Omega^{(3')\\dagger}$, which encodes the residual flavour and CP symmetries in fixed matrices $\\Omega^{(3)}$, $\\Omega^{(3')}$ and in the fixed rotation planes, leaving three couplings, two angles and the mass scale $M$ as free parameters. On top of the degenerate Majorana mass matrix $M_R^0$ the symmetry-breaking splittings $\\delta M_R = \\kappa M \\operatorname{diag}(2,0,-1;\\,0,-1,0)$ and $\\Delta M_R = \\lambda M \\operatorname{diag}(0,1,1)$ control the resonance condition for leptogenesis and, through the ratios $U_1^2:U_2^2:U_3^2$, the pattern of heavy-neutrino lifetimes. The argument is carried by the CP-violating combinations $C_{\\mathrm{LFV},\\alpha}$, $C_{\\mathrm{LNV},\\alpha}$, $C_{\\mathrm{DEG},\\alpha}$ together with the flavoured washout parameter $f_\\alpha$, which decide whether the asymmetry is generated by lepton-number violation or by flavoured washout. The comparison with experiments uses analytic Z-pole event numbers and projected displaced-vertex and beam-dump sensitivities, with branching ratios fixed by the flavour ratios $U_\\alpha^2/U^2$ shown in ternary plots.","core_discovery":"On the paper's own terms, the central discovery is that the symmetry-fixed structure of the heavy neutrino sector remains predictive once symmetry breaking is switched on. In the limit of large active-sterile mixing, realised when the angle $\\theta_R$ sits near a special value so that one Yukawa coupling dominates, the physical heavy-neutrino masses are set by the degenerate scale $M$, the Higgs-induced contribution $\\Delta \\hat{M}_{\\theta\\theta}$, and two small splittings: $\\delta M_R$ parametrised by $\\kappa$ (motivated by the residual charged-lepton symmetry) and $\\Delta M_R$ parametrised by $\\lambda$ (admitted as a generic perturbation). Depending on which splitting dominates, the mixing ratios $U_1^2:U_2^2:U_3^2$ collapse to discrete patterns such as $2:1:3$, $4:1:3$, $1:0:1$ or $4:11:9$, producing decay-length distributions that deviate from a single exponential and can be fitted to recover the splittings. In the same regime the flavour ratios $U_\\alpha^2/U^2$ equal the moduli squared of columns of the symmetry-determined PMNS matrix, up to case-specific permutations, giving sharp bounds such as $U_e^2/U^2 \\leq 0.35$ for Case 3 b.1). Solving the quantum kinetic equations for flavoured resonant leptogenesis, the paper finds viable parameter space in all four cases, and shows that a sizeable portion of it lies within the projected reach of current and future accelerator experiments, especially for vanishing initial heavy-neutrino abundances and non-zero $\\kappa$ or $\\lambda$; for Case 2 and Case 3 b.1) the baryon asymmetry can even be generated at exact degeneracy through flavoured washout.","pith_inferences":["(Beyond the paper) A precise measurement of the decay-length distribution would effectively measure a combination of the two splitting parameters, turning the ad hoc $\\lambda$ correction into an observable.","(Beyond the paper) The same ternary-plot logic could be applied to seesaw variants with two heavy neutrinos or non-degenerate masses: because the flavour ratios fix columns of the PMNS matrix, any measured ratio lying outside every predicted region would point to a different symmetry structure.","(Beyond the paper) A determination of $U_e^2/U^2$ at a Z-pole collider would act as a complementary probe of the neutrino mass ordering and of $m_0$, cross-checking cosmological and oscillation bounds, since Case 1 correlates these quantities tightly.","(Beyond the paper) The sensitivity forecasts assume idealised reconstruction efficiencies; realistic efficiencies will shrink the absolute reach, but the relative ordering of the cases and the qualitative distinction between single-exponential and multi-exponential decay distributions should survive."],"forward_implications":["If the central claim is right, a large fraction of the leptogenesis-viable region in the mass–mixing plane is within the projected reach of SHiP, MATHUSLA, (HL-)LHC, and FCC-ee/CEPC, especially for vanishing initial heavy-neutrino abundances and non-zero $\\kappa$ or $\\lambda$.","Heavy-neutrino decay-length distributions will not be simple exponentials: predicted ratios like $2:1:3$ or $1:0:1$ mean that the distribution of displacements carries information about the combination $3\\kappa-\\lambda$ even when the three masses cannot be resolved.","Measuring the flavour ratios $U_e^2/U^2$, $U_\\mu^2/U^2$, $U_\\tau^2/U^2$ at the one-percent level, plausible with roughly $10^5$ Z-pole events, can distinguish the four symmetry cases and, for Case 1 and parts of Case 2, the neutrino mass ordering and the value of the lightest neutrino mass $m_0$.","Even with exactly degenerate heavy neutrinos ($\\kappa = \\lambda = 0$), Case 2 and Case 3 b.1) can still generate the baryon asymmetry through flavoured washout, leaving a smaller but partly FCC-ee-testable region.","In the strong-inverted-ordering version of Case 3 a) and the strong-normal-ordering version of Case 3 b.1), the model effectively reduces to a two-heavy-neutrino framework for both collider searches and leptogenesis."],"supporting_citations":[{"why":"It defines the underlying scenario, the special values of $\\theta_R$, and the CP-violating combinations that the present scan extends and marginalises over.","marker":"[39]"},{"why":"It derives the constraints on flavour-group parameters that select the viable ranges for each of the four mixing cases.","marker":"[45]"},{"why":"It establishes the semi-direct flavour-plus-CP framework in which the PMNS matrix depends on a single free angle.","marker":"[23]"},{"why":"It shows that with three heavy neutrinos a large active-sterile mixing is compatible with light neutrino masses, underpinning the testability premise.","marker":"[14]"},{"why":"It supplies the quantum kinetic equations and interaction rates used to compute the generated baryon asymmetry.","marker":"[122]"},{"why":"It provides the non-relativistic rate extrapolation and resonant-leptogenesis treatment adopted in the numerical scans.","marker":"[123]"},{"why":"It gives the Z-pole event-number estimates and FCC-ee/CEPC sensitivity contours used for the testable regions.","marker":"[89]"},{"why":"It provides the semi-leptonic event formula and the statistical accuracy with which flavour ratios $U_\\alpha^2/U^2$ could be measured.","marker":"[96]"},{"why":"It supplies the displaced-vertex search sensitivities at LHC and HL-LHC that define part of the experimental reach.","marker":"[69]"}],"fun_headline_variants":["Symmetry ties leptogenesis to collider tests","Low-scale seesaw puts leptogenesis in collider sights","Flavour and CP symmetries map leptogenesis to colliders","Collider reach for leptogenesis from neutrino symmetries","Predictive neutrino model links baryogenesis to colliders"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The assumed flavour structure of the tiny right-handed-neutrino mass splittings, namely the $\\kappa$ term tied to the residual charged-lepton symmetry and the more ad hoc $\\lambda$ term that separates the second and third masses, carries the quantitative predictions for lifetime ratios, the leptogenesis resonance, and the borders of the testable region; if the actual symmetry-breaking terms had a different flavour structure, those predictions would change.","fun_headline_variants_meta":{"raw":{"variants":["Symmetry ties leptogenesis to collider tests","Low-scale seesaw puts leptogenesis in collider sights","Flavour and CP symmetries map leptogenesis to colliders","Collider reach for leptogenesis from neutrino symmetries","Predictive neutrino model links baryogenesis to colliders"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000782,"raw_usage":{"total_tokens":3529,"prompt_tokens":1095,"completion_tokens":2434,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":711,"completion_tokens_details":{"reasoning_tokens":2356}},"tokens_in":711,"tokens_out":2434,"duration_ms":18198,"temperature":1.0,"reasoning_tokens":2356,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T16:02:00.334985+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"At a Z-pole machine such as FCC-ee/CEPC with roughly $10^5$ reconstructed heavy-neutrino decays, measure the flavour ratios $U_e^2/U^2$, $U_\\mu^2/U^2$, $U_\\tau^2/U^2$ and the decay-length distribution: detecting $U_e^2/U^2 > 0.35$ would falsify Case 3 b.1) as analysed here, and a single-exponential distribution instead of the predicted mixture with ratios such as $2:1:3$ or $1:0:1$ would falsify the lifetime predictions. In the $\\kappa$-$\\lambda$ plane, observing successful leptogenesis with splittings that violate the consistency conditions of Eqs. (47)–(49) would break the framework's central assumption.","supporting_citations":[],"review_version":1}