{"id":"a047162d-8bd7-4034-b789-4323a420c368","arxiv_id":"2504.21404","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":9,"one_line_summary":"A modular A4 flavor symmetry applied to a two-leptoquark radiative seesaw model can reproduce observed quark and lepton data while predicting a normal neutrino mass ordering and specific ranges for Σmν and neutrinoless double beta decay.","lead":"This paper builds a beyond-standard-model theory in which a family symmetry derived from a modular group shapes the masses and mixings of quarks and leptons, with neutrino masses generated via new leptoquark particles at one loop. The authors scan the model's many free parameters and show it can fit current data, with predictions for the sum of neutrino masses and for neutrinoless double beta decay.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Neutrino mass formula Eq. (16) has an index mismatch that leaves Mν ill-defined; the NH/IH fits rest on this formula and must be re-derived/recomputed before the central claim can stand.","rationale":"The reader's weakest_assumption targets the external validity of the non-holomorphic modular framework and the reverse-engineered uR assignment. Those are legitimate concerns about predictivity, but they are not the most load-bearing: even granting the framework, the model's central numerical claims are computed from Eq. (16), and that equation as printed cannot define Mν. The index mismatch is internal and demonstrable; it directly undermines the reported fits. The reader noted an index typo in Eq. (16) in the rationale, so there is partial overlap, but the weakest_assumption selected a different issue. A corrected formula and rerun scan are a minimal condition for the central claim.","tokens_in":10978,"tokens_out":9239,"duration_ms":103511,"concrete_test":"Implement Eq. (16) in a small script with random F̃ and G̃: evaluate the bracketed sum as printed for all (i,j) and verify whether it is symmetric and depends on i in both terms. Then rerun the allowed-parameter scan with the corrected second term F̃_{ai} (and, separately, with V_dL replacing V_uL in Eq. (15)) and compare the resulting allowed regions in Figs. 3–8; if the regions change materially or the best-fit χ² degrades, the reported fits do not follow from the model as stated.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Eq. (16) defines the one-loop neutrino mass matrix that drives the entire numerical analysis, yet as printed it is not a valid matrix equation. The bracketed sum is Σ_a [ (F̃^T)_{ja} m_{d_a} G̃_{ai} + (G̃^T)_{ja} m_{d_a} F̃_{aj} ]. In the second term, (G̃^T)_{ja} = G̃_{aj} multiplies F̃_{aj}, so the index i never appears; the term is a vector labeled by j, not a matrix element (i,j). The RHS therefore cannot equal the (i,j) entry of Mν, and it is not symmetric under i↔j as a Majorana mass matrix must be. The natural correction is F̃_{ai}, but the printed formula is what the scan description refers to, so the NH and IH results in Figs. 3–8 are not reproducible from the paper as written. A related rotation ambiguity in Eq. (15), where left-handed down quarks are rotated with V_uL rather than V_dL, also deserves scrutiny; if wrong, the loop amplitude itself carries an incorrect CKM factor. Either way, the central numerical claim is not yet supported by the displayed equations.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs a radiative neutrino mass model with two scalar leptoquarks under a non-holomorphic modular A4 symmetry. SM leptons and quarks are assigned to A4 representations with various modular weights, and the resulting Yukawa structures fix the charged-fermion mass matrices. Neutrino masses are generated at one-loop order through leptoquark exchange, and the authors scan the model parameters to fit quark masses and CKM observables as well as neutrino oscillation data for both normal and inverted mass hierarchies. They then present predictions for the neutrino mass sum and the neutrinoless double beta decay amplitude.","tokens_in":11431,"tokens_out":6311,"duration_ms":69420,"significance":"If the numerical results are correct, the paper demonstrates a single modular-symmetry framework that jointly accommodates quark and lepton flavor data while generating radiative neutrino masses, with testable predictions for Σmν and m_ee. The construction is concrete, the group-theoretic setup is clearly stated, and the predictions are falsifiable: for example, the inverted-hierarchy points populate a narrow band Σmν ≈ 115–180 meV and m_ee ≈ 28–41 meV, which is in tension with the combined CMB+DESI bound but testable by next-generation experiments. However, the central one-loop formula contains an index error that makes the printed neutrino mass matrix ill-defined, and the quark-sector fit is not quantitatively documented. These issues must be resolved before the claimed simultaneous fit can be accepted.","major_comments":[{"comment":"As printed, Eq. (16) is not a valid matrix equation. The second term in the bracketed sum, (G̃^T)_{ja} m_{d_a} F̃_{aj}, has no free index i and is not symmetric under i ↔ j, so the right-hand side cannot define the (i,j) element of a Majorana mass matrix. Since Sec. III.B states that the numerical scan uses this formula, the NH and IH results in Figs. 3–8 are not reproducible from the manuscript as written. The authors must correct the index (presumably to (G̃^T)_{ia} m_{d_a} F̃_{aj}, i.e. G̃_{ai} F̃_{aj}) and re-run the analysis.","section":"Eq. (16), Sec. II.D"},{"comment":"The rotation in Eq. (15), \tilde F_{ai} ≡ (V^T_{uL})_{aj} F_{ji}, uses the left-handed up-quark rotation matrix V_uL, but the interaction in Eq. (11) involves the left-handed quark doublet Q_L and down-type quarks propagate in the loop. If this is not a typo, the loop amplitude carries an incorrect CKM-type factor; if it is a typo and V_dL (or the appropriate component of the doublet rotation) is intended, the numerical results must be recomputed with the corrected rotation. The paper should clarify which quark mass-basis rotation enters the loop formula.","section":"Eq. (15), Sec. II.D"},{"comment":"The quark-sector fit is not quantitatively documented. The only result shown is Fig. 2, a region in the τ plane, with no Δχ² values, no best-fit point, no number of scanned points, and no per-observable comparison. The trace conditions (6)–(8) fix only Tr, Det, and the sum of principal minors of M†M, so the six quark masses are imposed by construction rather than independently predicted; the nontrivial content of the fit is the CKM matrix and the CP phase, and the paper should report how well those are reproduced. Without this information, the claim that the model fits all quark data is not quantitatively supported.","section":"Sec. III.A, Eqs. (24)–(25)"}],"minor_comments":[{"comment":"In the second line of Eq. (1), the last term is written with f_A in the coupling proportional to d_R η L_L; from the definitions in Eq. (13) this should be g_A, not f_A.","section":"Eq. (1), Sec. II"},{"comment":"The phases α2 and α3 in the expression for ⟨m_ee⟩ are not defined; they presumably correspond to α21 and α31 introduced earlier, but this should be stated explicitly.","section":"Eq. (23), Sec. II.D"},{"comment":"The modular forms Y_3^(6) are essential for the up-quark mass matrix, but their explicit components are not given; since the entire quark fit depends on them, the paper should either reproduce them from Ref. [2] or give the precise definition used in the numerical scan.","section":"Sec. II.B, Eq. (5)"},{"comment":"Fig. 2 shows only a binary allowed region; a color scale indicating the Δχ² value inside the region would make the quality of the quark fit much more transparent.","section":"Fig. 2, Sec. III.A"}],"recommendation":"major_revision","confidential_remarks":"The manuscript depends heavily on the non-holomorphic modular framework of Ref. [2], and the correctness of the numerical scan cannot be checked without resolving the index error in Eq. (16) and the rotation ambiguity in Eq. (15). The authors should also be asked to provide a reproducible account of the quark fit, including χ² values and the handling of the trace conditions. If the index and rotation issues are simple typos, the paper may become acceptable after a careful revision; if they reflect a deeper misunderstanding, the numerical results would need to be rederived."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague —\n\nQuick take: standard modular-flavor model building, but with one genuinely new hook: the quark fit pins the modulus to Im τ ≈ 2.3–2.4, and the same τ then gives acceptable neutrino fits for both hierarchies. The model itself — non-holomorphic modular A4 plus two scalar leptoquarks — is a reasonable combination of known ingredients. The paper is honest that the natural uR assignment fails and that weight -6 singlets are the minimal workaround (footnote 1). The mass-matrix construction is coherent and the trace fixes are legitimate.\n\nThe soft spots are real. Eq. (16) as printed is not a matrix equation: the second term in the bracket has no free i, so it can't be the (i,j) element of Mν, and the result isn't symmetric. The natural fix is to replace F̃_{aj} with F̃_{ai}, which gives the symmetric F̃^T m G̃ + G̃^T m F̃ structure you'd expect from Ref [13]. Similarly, Eq. (15) defines \\tilde{F} with V_uL instead of V_dL; since the loop quark is down-type, that can't be right. If these are just typos in the write-up, the physics could survive, but as displayed the central formula is wrong, and the fits are not reproducible from the paper.\n\nThe numerical reporting is also thinner than I'd like. The quark fit is shown only as a patch in the τ plane, with no χ² value or best-fit point. The lepton scan is described by broad ranges, and the Σmν and m_ee predictions are largely dictated by fixing κ to the atmospheric splitting — so they are generic consequences of the oscillation data, not sharp model predictions. And \"all points in tension with CMB+DESI\" is a noteworthy conclusion that should be in the abstract, not buried in the figure.\n\nAll that said, the authors know what they're doing; this is serious work in the modular-flavor program. I'd send it to a referee, but the referee will need the corrected equations, the actual scan statistics, and a clearer statement of what the model predicts beyond the known constraints. Fixable, but not citable in its current form.","headline":"A solid modular-flavor leptoquark model with a genuinely new quark-sector constraint, but the central neutrino-mass formula is misprinted and the numerics need transparency.","tokens_in":11943,"tokens_out":12434,"would_cite":true,"duration_ms":122793,"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 non-holomorphic modular A4 symmetry acting on a two-leptoquark radiative seesaw fits all measured quark and lepton flavor data and produces concrete predictions for the neutrino mass sum and neutrinoless double beta decay.","keywords":["neutrino mass","radiative seesaw","leptoquark","modular A4 symmetry","non-holomorphic modular forms","fermion mass matrices","neutrinoless double beta decay","sum of neutrino masses"],"falsifier":"A cosmological measurement that establishes $\\sum m_\\nu < 72\\,\\mathrm{meV}$ would exclude all fitted parameter sets, since the paper reports that every allowed point for both normal and inverted ordering violates the combined CMB and baryon-acoustic-oscillation bound. A second check: if future oscillation data pinned $\\sin^2\\theta_{12}$ below about 0.30 in the normal hierarchy, the NH allowed region would be excluded, because the paper finds a preference for larger solar mixing.","tokens_in":10780,"feed_emoji":"⚛️","tokens_out":12201,"duration_ms":110913,"temperature":0.7,"pith_summary":"This paper constructs a model in which a non-holomorphic modular $A_4$ symmetry dictates the Yukawa couplings of the Standard Model fermions, while two scalar leptoquarks generate neutrino masses at one loop. The authors scan the free parameters and report that the model can fit the measured quark masses and CKM mixings, the charged-lepton masses, and the neutrino oscillation data for both normal and inverted mass orderings. The quark-sector fit pins the modular parameter to $\\mathrm{Im}[\\tau]\\sim 2.3$–$2.4$ with $|\\mathrm{Re}[\\tau]|\\lesssim 0.4$, and the neutrino fit then gives definite allowed regions for the sum of neutrino masses and the neutrinoless double $\\beta$ decay amplitude. If the construction is right, flavor in both sectors would trace back to one modulus $\\tau$ and a handful of order-one couplings, with testable consequences for cosmology and double $\\beta$ decay.","feed_headline":"One modular symmetry plus two leptoquarks fit all fermion data","feed_subtitle":"Radiative neutrino masses from the model land in ranges that cosmology and double-beta searches can test.","key_machinery":"The load-bearing machinery is the non-holomorphic modular $A_4$ framework and its Maaß-form triplets: fixed triplets of modular functions $Y_3^{(0)}$ and two independent $Y_3^{(6)}$ of the modulus $\\tau$ that transform under $A_4$ and supply every Yukawa coupling in the Lagrangian. The two leptoquarks $\\eta$ and $S$ mix through the term $H^\\dagger\\eta S$, giving mass eigenstates $\\rho_{1/3}$ and $\\chi_{1/3}$ with mixing angle $\\alpha$; the one-loop neutrino mass formula $$M_\\nu = \\frac{f_1 g_1 N_c s_\\$\\alpha$ c_\\$\\alpha$}{2(4\\pi)^2}\\left(1-\\frac{m_\\$rho^{2}$}{m_\\$chi^{2}$}\\right)\\sum_a \\left[(\\tilde F^T)_{ja} m_{d_a} \\tilde G_{ai} + (\\tilde G^T)_{ja} m_{d_a} \\tilde F_{aj}\\right] F_I(r_\\rho, r_{d_a})$$ then converts the modular-symmetric Yukawa textures into the observed neutrino mass matrix. The machinery does two jobs at once: it reduces the Standard Model's arbitrary Yukawa matrices to a handful of parameters times fixed modular forms, and it explains the smallness of neutrino masses by loop suppression rather than by very heavy right-handed neutrinos.","core_discovery":"The paper's central claim is that a non-holomorphic modular $A_4$ symmetry can serve as the flavor principle behind all quark and lepton masses. In the setup, left-handed leptons, right-handed charged leptons, left-handed quarks and right-handed down quarks are $A_4$ triplets with modular weight zero, while the three right-handed up quarks are $A_4$ singlets of weight $-6$; the two scalar leptoquarks $\\eta$ (an $SU(2)_L$ doublet) and $S$ (a singlet) carry no nontrivial $A_4$ charge. The neutrino mass matrix is produced radiatively by a one-loop diagram in which the mixed leptoquarks and down-type quarks run, with the loop integral controlled by the leptoquark mixing angle $\\alpha$ and the masses $m_\\rho$, $m_\\chi$. A numerical scan over the remaining couplings finds parameter sets consistent with all measured fermion data. In the normal hierarchy the model slightly restricts the solar angle ($\\sin^2\\theta_{12}\\gtrsim 0.30$–$0.32$) and disfavors $\\delta_{CP}\\in[-130^\\circ,-70^\\circ]$, while in the inverted hierarchy it predicts a narrow band $\\sum m_\\nu \\sim [115,180]\\,\\mathrm{meV}$ and $\\langle m_{ee}\\rangle \\sim [28,41]\\,\\mathrm{meV}$. In both cases the authors find that every allowed point violates the stricter combined cosmological bound on $\\sum m_\\nu$ of about $72\\,\\mathrm{meV}$, while some (normal) or all (inverted) points remain within reach of current double $\\beta$ decay limits.","pith_inferences":["A sharp test the authors do not state: since the quark sector fixes $\\tau$, the predicted correlations among $\\delta_{CP}$, $\\alpha_{21}$, and $\\sin^2\\theta_{12}$ could be combined into a single frequentist test of the modular hypothesis, rather than separate scans.","The same modular-symmetric couplings that produce neutrino masses also generate charged-lepton flavor violation through leptoquark exchange; computing $\\mu\\to e\\gamma$ rates for the allowed parameter points would give an independent, low-energy probe of the same flavor structure.","Because all fitted points hit the 72 meV cosmological bound, a natural extension would add a small additional contribution to neutrino masses from a different loop or tree-level operator, lowering $\\sum m_\\nu$ without changing the quark-sector fit."],"forward_implications":["The quark sector fixes $\\mathrm{Im}[\\tau]\\sim 2.3$–$2.4$, so any future measurement of lepton mixing or CP violation is a test of the same modulus, not an independent parameter.","If the fit is correct, neutrino masses are radiatively generated by leptoquark exchange; no right-handed neutrinos or high seesaw scale are required.","The inverted-hierarchy region predicts $\\sum m_\\nu\\in[115,180]\\,\\mathrm{meV}$ and $\\langle m_{ee}\\rangle\\in[28,41]\\,\\mathrm{meV}$, placing the model within reach of upcoming neutrinoless double beta decay experiments.","Every currently allowed parameter point violates the stricter $\\sum m_\\nu\\lesssim 72\\,\\mathrm{meV}$ cosmological bound, so the model would be excluded if that bound is confirmed.","Both mass orderings remain viable, but the normal-hierarchy region has lower $\\sum m_\\nu$ and $\\langle m_{ee}\\rangle$ below current double beta limits, leaving some points for near-future tests."],"supporting_citations":[{"why":"Defines the non-holomorphic modular A4 framework and the Maaß-form triplets that supply every Yukawa coupling used in the mass matrices.","marker":"[2]"},{"why":"Provides the one-loop neutrino mass formula for the two-leptoquark radiative seesaw that the model adopts.","marker":"[13]"},{"why":"Supplies the measured quark and lepton masses and CKM parameters used to fix the Yukawa couplings.","marker":"[30]"},{"why":"Supplies the neutrino oscillation data against which the lepton-sector parameter scan is fit.","marker":"[37]"},{"why":"Provides the current upper bound on the neutrinoless double beta decay effective mass used to test the predictions.","marker":"[36]"},{"why":"Provides the combined CMB and baryon-acoustic-oscillation upper bound on the sum of neutrino masses that all fitted points violate.","marker":"[33]"},{"why":"Gives the updated combined cosmological bound near 72 meV quoted in the analysis.","marker":"[34]"}],"fun_headline_variants":["Modular A4 with leptoquarks fits all fermion data","Leptoquarks and modular symmetry fix neutrino masses","Loop neutrinos from leptoquarks under modular A4","Radiative neutrino mass via leptoquarks and modular symmetry"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the non-holomorphic modular $A_4$ framework used here is valid and that its two weight-6 Maaß-form triplets for the up-quark sector are the complete set of such forms; the alternative natural assignment of the up-type singlets failed to fit data, so the successful scheme is partly reverse-engineered from observations.","fun_headline_variants_meta":{"raw":{"variants":["Modular A4 with leptoquarks fits all fermion data","Leptoquarks and modular symmetry fix neutrino masses","Loop neutrinos from leptoquarks under modular A4","Radiative neutrino mass via leptoquarks and modular symmetry"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000199,"raw_usage":{"total_tokens":1413,"prompt_tokens":1026,"completion_tokens":387,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":642,"completion_tokens_details":{"reasoning_tokens":318}},"tokens_in":642,"tokens_out":387,"duration_ms":4485,"temperature":1.0,"reasoning_tokens":318,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T05:05:39.745271+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A cosmological measurement that establishes $\\sum m_\\nu < 72\\,\\mathrm{meV}$ would exclude all fitted parameter sets, since the paper reports that every allowed point for both normal and inverted ordering violates the combined CMB and baryon-acoustic-oscillation bound. A second check: if future oscillation data pinned $\\sin^2\\theta_{12}$ below about 0.30 in the normal hierarchy, the NH allowed region would be excluded, because the paper finds a preference for larger solar mixing.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the non-holomorphic modular A4 framework and the Maaß-form triplets that supply every Yukawa coupling used in the mass matrices."}],"review_version":1}