{"id":"e8351313-bd63-4373-8dfb-81f4c1dfe7c0","arxiv_id":"2509.02744","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":7,"one_line_summary":"A vector-like lepton dark matter model with two inert scalar leptoquarks produces a radiative pseudo-Dirac mass splitting that evades direct detection and opens up parameter space for the correct relic density.","lead":"This paper adds scalar leptoquarks to a dark matter model made of a vector-like lepton, generating a mass split that helps the dark matter escape direct detection. The result is a large surviving parameter region where the measured relic abundance and current experimental limits are all satisfied.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (32), the sole source of the pseudo-Dirac splitting, has an inconsistent mixing sum; with the missing S-R_d difference structure and sign convention in Eq. (37), the >250 keV DD-evasion threshold is not established.","rationale":"The reader's weakest assumption already targets Eq. (32), and I agree that this is the right place: the whole escape from LZ is that a Yukawa-induced Majorana splitting of at least 250 keV forbids the Z-mediated inelastic channel, while every other ingredient only narrows the parameter space. If |mL| is below threshold, the model is excluded for TeV-scale Dirac-like DM and no other mechanism restores viability. My read sharpens the concern in two ways. First, the printed mixing sum is not an amplitude: contracting U_RS(i,i) with a row of U_dagger_RS and summing over j has no counterpart in Wick contraction, and the 2x2 structure indicates the correct factor is sin(2theta)[B0(m1)-B0(m2)], which is naturally suppressed by the 128 GeV LQ mass splitting and can shift mL by an order of magnitude. Second, the sign handling in Eq. (37) is inconsistent with the stated mL<0, so the paper has not specified how the numerical splitting was extracted. B0 alone also requires a renormalization convention; no scheme is stated. These issues are fixable by a corrected derivation or a benchmark run, and the numerical results may have been obtained from SARAH/SPheno rather than from the printed formula; therefore I do not escalate to rejection. The verification is a condition for the central claim, which matches the reader's CONDITIONAL verdict and leaves the verdict unchanged.","tokens_in":18706,"tokens_out":15730,"duration_ms":153059,"concrete_test":"Take a benchmark from the scan, e.g. m_DM=1.5 TeV, y_a=y_b=1, theta=45 degrees, m_Sigma_1=2 TeV, m_Sigma_2=2.128 TeV, and independently recompute the one-loop neutral VLL self-energy from the two-component Feynman rules of Eq. (6). Evaluate mL both with the printed sum of Eq. (32) and with the U_iS U_iR_d difference form, subtracting the UV divergence in DR-bar and varying the renormalization scale mu between 0.5 and 2 TeV. Then check the eigenvalues from Eq. (37) with the resulting sign: if the corrected |mL| drops below 250 keV, changes by more than a factor of 2, or forces a negative lighter mass, the DD-evasion claim is not supported. Cross-check the same point with SARAH/SPheno if model files are provided.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Eq. (32) is the linchpin of the DD-evasion claim: it is the only source of the pseudo-Dirac splitting that shuts off Z-mediated scattering. As printed, the mixing factor sum_{i,j} U_RS(i,i) U_dagger_RS(i,j) B0(m_b^2,0,m_Sigma_i^2) is not a Wick-contracted amplitude. The loop function depends only on i, but U_dagger_RS carries a free j; summing over j gives a row sum with no physical meaning. For theta=0 the expression reduces to B0(m_Sigma_1)+B0(m_Sigma_2), although with no S-R_d mixing the two Yukawa vertices cannot be connected by a scalar propagator and mL should vanish. The expected structure is proportional to sum_i U_iS U_iR_d B0(m_Sigma_i^2) = sin(2theta)[B0(m_Sigma_1^2)-B0(m_Sigma_2^2)], which is suppressed by the 128 GeV LQ mass splitting and can differ from Eq. (32) by a large factor. In addition, B0 is UV-divergent and no subtraction scheme is specified, so the numerical size of mL depends on an unstated renormalization convention. Finally, the text states mL<0, but inserting mL<0 into Eq. (37) makes the assigned lighter eigenvalue negative; the sign or absolute-value convention is missing. If the corrected mL lies below 250 keV, the claimed evasion of LUX-ZEPLIN fails.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper extends the minimal SU(2)_L doublet vector-like lepton (VLL) dark matter model by adding two inert scalar leptoquarks, an SU(2) singlet S and an SU(2) doublet R, both odd under the stabilizing Z2 symmetry. The authors argue that one-loop diagrams involving bottom quarks and the down-type leptoquark mass eigenstates induce a Majorana mass for the neutral VLL component, splitting it into two pseudo-Dirac states; if the splitting exceeds roughly 250 keV, the Z-mediated elastic scattering that excludes the pure VLL model becomes kinematically forbidden. They further show that new (co)annihilation channels involving leptoquarks allow the correct relic abundance for DM masses roughly in the 1-2.5 TeV range, while the predicted spin-independent direct detection cross sections remain below the LUX-ZEPLIN bound. The numerical analysis uses SARAH, SPheno, CalcHEP and micrOMEGAs and applies constraints from vacuum stability, perturbativity, the T parameter, h->γγ, flavour physics and LHC searches.","tokens_in":19062,"tokens_out":10895,"duration_ms":103946,"significance":"If correct, the paper is significant: it offers a radiative, triplet-vev-free mechanism for converting a Z-coupled Dirac VLL into a pseudo-Dirac dark matter candidate, reviving a well-motivated minimal dark matter setup in the TeV range and connecting it to leptoquark and LHC missing-transverse-energy searches. The paper is also useful for its broad treatment of constraints (EWPO, h->γγ, flavour, collider bounds) and for the decomposition of the relic density into Type-I/II/III channels. Credit is due for using standard public codes and for presenting a falsifiable phenomenological profile, including future DARWIN/LZ/XENONnT projections. However, the printed one-loop Majorana-mass formula that anchors the direct-detection evasion claim is not a valid Wick contraction, and no benchmark points or public model files are supplied, so the numerical results cannot currently be verified from the manuscript.","major_comments":[{"comment":"The mixing factor in Eqs. (31) and (32), sum_{i,j} U_RS(i,i) U^dagger_RS(i,j) B(... , m_Sigma_i^2), is not a Wick-contracted amplitude: the loop function depends only on i, so summing over j is just a row sum of U^dagger_RS and has no physical meaning. The correct projection of the two Yukawa vertices onto the physical LQ mass eigenstates should produce a factor proportional to U_{iS} U_{iR_d}, i.e. a sin(2theta) [B(m_Sigma_2^2) - B(m_Sigma_1^2)] structure. In particular, for theta -> 0 the printed expression does not vanish, whereas the physical Majorana mass must vanish because y_a and y_b couple to two different unmixed fields and cannot be connected by a single scalar propagator. Since the splitting in Fig. 7(B) and the entire LUX-ZEPLIN evasion argument in Sec. 6.3 are driven by Eq. (32), this is a load-bearing issue; please provide a proper derivation, correct the printed formulas, and confirm that the numerical scans were based on the corrected expression.","section":"Sec. 5, Eqs. (31) and (32)"},{"comment":"Even after the mixing structure is corrected, the expression mL ~ 6 m_b y_a y_b^dagger ... B0(m_b^2, 0, m_Sigma_i^2) is not numerically meaningful as written: B0 is UV divergent and no subtraction scheme is specified, and the choice of external momentum p^2 = m_b^2 rather than a scale tied to the DM mass is not justified. The statement in the text that mL is negative is also incomplete, because inserting mL < 0 into Eq. (37) gives a negative value for m_{f^0_1}; physical masses require an absolute value or an explicit phase convention, and the mass splitting shown in Fig. 7(B) must be defined consistently with that convention. Please supply the renormalized loop expression and demonstrate that the >250 keV splitting, and hence the direct-detection evasion, survives.","section":"Sec. 5, Eqs. (32) and (37)"},{"comment":"The numerical pipeline is described only in general terms ('SARAH ... SPheno ... CalcHEP ... micrOMEGAs') and no benchmark points are provided. Because the printed one-loop formula is inconsistent as written, the reader cannot tell whether Figs. 7, 11, 13 and 15 were generated from Eq. (32) or from a different, correct implementation. Please provide at least one benchmark table with the Lagrangian parameters (m_f, y_a, y_b, y_c, sin(theta), m_Sigma_{d1}, m_Sigma_{d2} - m_Sigma_{d1}) and the corresponding outputs (m_{f^0_1}, Delta m0, Omega h^2, sigma_SI, <sigma v>), and ideally make the model files public. This is necessary to validate the central claims.","section":"Sec. 6, numerical pipeline"},{"comment":"The scan is restricted to 0.5 <= y_a,b <= 1.5 and 0.5 <= y_c <= 1, and the text motivates these lower bounds by the direct-detection requirement that Delta m0 ~ y_a y_b sin(2theta) exceed the inelastic-scattering threshold. This means the DD-evading region is imposed as a prior rather than emerging as a prediction. Please show explicitly, with the corrected mL formula, how large the allowed parameter region is for smaller Yukawas (e.g., y ~ 0.1-0.3) and how much of the parent parameter space is excluded by LUX-ZEPLIN; otherwise the claim of a 'large parameter space' should be appropriately qualified.","section":"Sec. 6, scan ranges and Eq. (32)"}],"minor_comments":[{"comment":"The vacuum stability conditions as printed, e.g. lambda5 = lambda5 + sqrt(lambda1 + lambda2), appear to be a typographical corruption of the standard conditions lambda5 + 2 sqrt(lambda1 lambda2) > 0 (and similarly for lambda7 and lambda4); please correct these expressions and ensure that the conditions used in the numerical analysis are the correct ones.","section":"Sec. 4, Eq. (12)"},{"comment":"The formula for <sigma v>_eff contains a parenthesis error in the last term: the factors (1 + Delta chi_i)^{3/2} and (1 + Delta chi_j)^{3/2} should be written with explicit brackets, as the current expression is ambiguous.","section":"Sec. 6.1, Eq. (39)"},{"comment":"The decomposition of the effective annihilation rate into Type-I, Type-II and Type-III processes is shown in Fig. 12, but the text does not explain how this decomposition was performed within micrOMEGAs; a short description of the procedure would improve reproducibility.","section":"Sec. 6.1, Fig. 12"},{"comment":"References [55] and [59] are incomplete (no arXiv number or journal/volume information); please update them.","section":"References"},{"comment":"In Eq. (30), the Higgs-leptoquark coupling g_Sigma_{d1,2} is expressed in terms of the dimensionful parameter mu_RS with no explicit mass-dimensional factor; please clarify the normalization used in Eq. (24) so that the loop amplitude is dimensionless.","section":"Sec. 5, Eq. (30)"},{"comment":"The horizontal threshold at 250 keV that is central to the direct-detection argument is mentioned in the text but is not marked in Fig. 7(B); adding it would make the viability condition immediately visible.","section":"Sec. 5, Fig. 7(B)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's central mechanism is plausible and the scope is appropriate for a phenomenological hep-ph journal, but the printed one-loop Majorana-mass formula is demonstrably incorrect as a Wick contraction, and the DD-evasion claim rests entirely on that formula. The revision should be judged on whether the corrected formula still yields Delta m0 > 250 keV and on whether benchmark points are supplied; if the authors cannot provide a corrected derivation and reproducible numerical results, I would not recommend publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nYou should know two things before deciding what to do with this paper. First, the physics idea is legitimate: inert scalar leptoquarks radiatively split the neutral vector-like lepton into pseudo-Dirac states, suppressing Z-mediated direct detection, while the new Yukawa channels help fix the relic density. That specific combination is not in the literature and is worth taking seriously. Second, the printed formula for the splitting, Eq. (32), is not something you can trust as written. The index sum over U_RS(i,i) U^dagger_RS(i,j) with the loop function depending only on i is not a Wick-contracted amplitude. At theta=0 it would give B0(m1)+B0(m2), but with no S–R_d mixing the two Yukawa vertices cannot be connected by a propagator; the term should vanish. The expected structure is sin2theta [B0(m1^2)-B0(m2^2)], which can differ from Eq. (32) by a large factor. This matters because Eq. (32) is the only source of the >250 keV pseudo-Dirac splitting that lets the model evade LUX-ZEPLIN.\n\nWhat the paper does well: the setup is simple, the phenomenological survey is broad (relic density, direct and indirect detection, T parameter, h->gamma gamma, flavour assumptions), and the numerical pipeline uses standard public codes (SARAH, SPheno, MicrOmegas). The qualitative mechanism is sound, and the parameter scan is honestly described rather than hidden.\n\nThe soft spots are concentrated in the loop section. Besides the index issue, there is no subtraction scheme for the UV-divergent B0, so the numerical size of m_L is undefined absent a renormalization convention. Eq. (37) with m_L<0 gives a negative 'lighter' eigenvalue unless an absolute value is intended. And the scan choices (ya,b > 0.5, LQ mass splitting fixed at 128 GeV) are made with the direct-detection constraint in view, so the claim of a large viable parameter space is partly built into the priors. No code or benchmark points are shipped, so the printed formula cannot be checked against their implementation.\n\nIf Eq. (32) is a typo and the numerical implementation is correct, the central claim survives. If it is not, the DD evasion threshold is not established. This is exactly what a referee should pin down. I would send it to review and ask for a corrected derivation of the loop amplitude, a stated renormalization scheme, and benchmark points. The idea deserves referee time; the paper as printed needs revision.","headline":"A plausible and new way to rescue VLL dark matter via scalar-leptoquark-induced pseudo-Dirac splitting, but the printed loop formula for that splitting is inconsistent and must be pinned down before the direct-detection claim can be trusted.","tokens_in":19630,"tokens_out":2331,"would_cite":false,"duration_ms":22439,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["95.35.+d","12.60.-i","14.80.Sv"],"model":"deepseek-v4-flash","headline":"Adding two inert scalar leptoquarks splits the neutral vector-like lepton into two pseudo-Dirac states, suppressing its Z-mediated scattering with nuclei and opening a viable dark-matter window near 1–2.5 TeV.","keywords":["vector-like lepton","dark matter","scalar leptoquark","pseudo-Dirac fermion","inelastic dark matter","direct detection","relic density","radiative mass splitting"],"falsifier":"Re-evaluate the one-loop Majorana mass with all flavor and mixing indices carried explicitly at a benchmark point such as $m_{f^0_1}\\simeq 1.5$ TeV, $y_a=y_b=1$, $\\sin2\\theta=1$; if the resulting splitting falls below 250 keV, the direct-detection evasion fails. Experimentally, a search for inelastic up-scattering of $f_1^0$ into $f_2^0$ -- the recoil spectrum and possible de-excitation signal of the heavier state -- would test the claimed splitting independently of the loop calculation.","tokens_in":18458,"feed_emoji":"🌌","tokens_out":16107,"duration_ms":137555,"temperature":0.7,"pith_summary":"The paper claims that adding two inert scalar leptoquarks to the Standard Model can rescue the otherwise-excluded vector-like lepton doublet as a dark matter candidate. In the minimal model, the neutral lepton is a Dirac fermion that scatters elastically off nuclei through Z exchange, overshooting the LUX-ZEPLIN bound; the leptoquarks generate a one-loop Majorana mass that splits it into two pseudo-Dirac states, and the splitting kinematically forbids that scattering. The same extension opens new co-annihilation channels that bring the relic density into the Planck-observed range for dark matter masses roughly between 1 and 2.5 TeV. The paper further finds that the viable parameter space satisfies indirect-detection limits from H.E.S.S. and lies partly within reach of next-generation direct-detection experiments.","feed_headline":"Leptoquark loops make vector-like lepton dark matter viable","feed_subtitle":"A radiative mass splitting suppresses the Z-mediated signal and opens a viable 1–2.5 TeV window.","key_machinery":"The load-bearing object is the one-loop Majorana mass $m_L$ of the neutral vector-like lepton, obtained from a bottom-quark/leptoquark loop and proportional to $m_b y_a y_b^\\dagger \\sum_{i,j} U_{RS}(i,i)U_{RS}^\\dagger(i,j) B_0(m_b^2,0,m_{\\Sigma^d_i}^2)$, together with the physical masses $m_{f^0_1}$ and $m_{f^0_2}$ obtained from Eq. (37). Here $B_0$ is the Passarino-Veltman scalar loop integral, and $U_{RS}$ rotates the down-type leptoquark gauge states into mass eigenstates, so the mixing angle $\\theta$ controls the size of the splitting. A pseudo-Dirac fermion is a Dirac fermion that has received a small Majorana mass, splitting it into two nearly degenerate Majorana eigenstates; the paper's key observation is that the splitting is typically larger than about 250 keV, the inelastic-scattering threshold that closes the $Z$-mediated direct-detection channel. The companion Dirac-mass correction $\\delta m_D$ from the same leptoquark loops also enlarges the charged-neutral splitting in the doublet.","core_discovery":"The central claim is that two inert scalar leptoquarks -- an $SU(2)_L$ doublet $R$ and an $SU(2)_L$ singlet $S$, both odd under the stabilizing $Z_2$ -- change the nature of the vector-like lepton dark matter candidate. Through one-loop diagrams involving bottom quarks and down-type leptoquarks, the neutral component $f^0$ of the doublet receives a Majorana mass, so the single Dirac state splits into two pseudo-Dirac mass eigenstates $f_1^0$ and $f_2^0$. The lighter state is the dark matter; the splitting is typically above 250 keV across the parameter scan, which is enough to make the $Z$-mediated inelastic transition $f_1^0\\to f_2^0$ kinematically forbidden at LUX-ZEPLIN. The same leptoquark couplings open new annihilation and co-annihilation channels that lower the relic density to the Planck-observed value for dark matter masses from about 1 to 2.5 TeV. The parameter space that survives relic-density and indirect-detection constraints also lies below the current LZ and H.E.S.S. limits, with part of it accessible to the next generation of direct-detection experiments.","pith_inferences":["The mechanism should generalize: any $Z_2$-odd scalar coupling a vector-like doublet fermion to quarks can generate the same pseudo-Dirac splitting, so the core idea is not tied to the particular leptoquark charges chosen here.","Because the splitting scales as $y_a y_b \\sin2\\theta$ and is suppressed by the bottom-quark mass, the model effectively predicts strong third-generation quark couplings for the leptoquarks, which could be probed through associated top/bottom plus missing-energy events at the LHC.","If the splitting sits only slightly above 250 keV, the heavier pseudo-Dirac state could be excited by up-scattering in direct-detection experiments, producing a distinctive inelastic signature rather than the usual elastic recoil; current and future detectors could search for that signature directly."],"forward_implications":["The minimal pure doublet VLL model is excluded through $Z$-mediated scattering; with the two inert scalar leptoquarks, the model becomes viable for $m_{f^0_1}$ between roughly 1 and 2.5 TeV.","The loop-induced splitting is typically above 250 keV, putting the model in the inelastic-dark-matter regime and closing the LUX-ZEPLIN scattering channel.","The new leptoquark-mediated (co)annihilation channels, dominated by co-annihilations among the nearly degenerate dark-sector states, bring the relic density into agreement with the Planck measurement over a wide parameter range.","The viable parameter space also satisfies the H.E.S.S. bound on the $W^+W^-$ annihilation channel.","A substantial part of the predicted spin-independent cross-section is below the current LZ limit but within the projected reach of DARWIN, the upgraded LZ, and XENONnT."],"supporting_citations":[{"why":"It supplies the Planck relic abundance Ωh² = 0.12 that the model must reproduce.","marker":"[1]"},{"why":"It gives the LUX-ZEPLIN first-result limit that the minimal vector-like lepton model fails to satisfy.","marker":"[2]"},{"why":"It provides the H.E.S.S. indirect-detection bound on the W+W- annihilation channel used to constrain the viable parameter space.","marker":"[5]"},{"why":"It supplies the electroweak radiative mass-splitting formula that the leptoquark corrections build on.","marker":"[30]"},{"why":"It provides the 4.2 tonne-year LUX-ZEPLIN spin-independent bound used as the direct-detection benchmark.","marker":"[31]"},{"why":"It establishes the inelastic dark matter mechanism that makes a large enough mass splitting suppress direct detection.","marker":"[32]"},{"why":"It defines the Passarino-Veltman B0 and B1 integrals appearing in the leptoquark-induced mass formulas.","marker":"[43]"},{"why":"It gives the effective co-annihilation cross-section formalism used in the relic-density calculation.","marker":"[48]"}],"fun_headline_variants":["Leptoquark loops create dark matter mass gap","Scalar leptoquarks resurrect vector-like DM model","Mass splitting from leptoquarks bypasses direct tests","Fermionic DM viable with leptoquark-induced splitting","Leptoquark assistance unlocks 1-2.5 TeV DM window"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The direct-detection evasion rests on the loop-generated pseudo-Dirac splitting being genuinely larger than about 250 keV in the claimed parameter region; if the one-loop Majorana-mass calculation has the wrong sign, wrong mixing factors, or wrong loop kinematics, the splitting could be far smaller and the $Z$-mediated scattering would reappear.","fun_headline_variants_meta":{"raw":{"variants":["Leptoquark loops create dark matter mass gap","Scalar leptoquarks resurrect vector-like DM model","Mass splitting from leptoquarks bypasses direct tests","Fermionic DM viable with leptoquark-induced splitting","Leptoquark assistance unlocks 1-2.5 TeV DM window"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000889,"raw_usage":{"total_tokens":3831,"prompt_tokens":938,"completion_tokens":2893,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":554,"completion_tokens_details":{"reasoning_tokens":2810}},"tokens_in":554,"tokens_out":2893,"duration_ms":19899,"temperature":1.0,"reasoning_tokens":2810,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:34:56.644838+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Re-evaluate the one-loop Majorana mass with all flavor and mixing indices carried explicitly at a benchmark point such as $m_{f^0_1}\\simeq 1.5$ TeV, $y_a=y_b=1$, $\\sin2\\theta=1$; if the resulting splitting falls below 250 keV, the direct-detection evasion fails. Experimentally, a search for inelastic up-scattering of $f_1^0$ into $f_2^0$ -- the recoil spectrum and possible de-excitation signal of the heavier state -- would test the claimed splitting independently of the loop calculation.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It gives the LUX-ZEPLIN first-result limit that the minimal vector-like lepton model fails to satisfy."},{"cited_title":"Collaboration collaboration, Search for dark matter annihilation signals in the h.e.s.s","cited_arxiv_id":null,"evidence_quote":"It provides the H.E.S.S. indirect-detection bound on the W+W- annihilation channel used to constrain the viable parameter space."}],"review_version":2}