{"id":"c7d7249b-5fe2-4ceb-be44-97395cd62138","arxiv_id":"2412.08015","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Radiative corrections to neutrino masses can increase the e−e− → W−W− cross section by about 15% for TeV-scale heavy neutral leptons.","lead":"This paper calculates how quantum corrections to neutrino masses change the rate of a lepton-number-violating process, e−e− → W−W−, in a minimal seesaw model. It finds the cross section can grow by about 15% for TeV-scale heavy neutral leptons, a shift that could matter for future e−e− colliders.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 15% enhancement rests on the unquantified neglect of one-loop corrections to M_D and M_M; in the large-Xω benchmark of Fig. 4 these can be comparable to the retained δMLL effect.","rationale":"The paper is a coherent one-loop study; I checked the linear algebra in Eqs. (7)–(13) and it is internally consistent provided δM_D and δM_M are negligible. The reader's weakest-assumption identification matches mine. I do not see an independent algebraic error in the cross-section formula; the numerical 15% enhancement is plausible. However, the paper does not quantify the size of the dropped corrections, and in the large-Xω regime needed for observable HNL mixing, the loop suppression factor is only Y^2/16π^2 ~ 10^-3, not several orders of magnitude below the retained δMM effect. Since the i0νββ enhancement is a relative statement, it is sensitive to exactly this type of O(10^-3) shift in the mixing elements, especially if the maximum sits near the meff-cancellation region discussed in Sec. 3. A full one-loop matching would settle the question. Therefore the verdict should stay CONDITIONAL: the claim is plausible but not yet established to the accuracy implied by the 15% statement.","tokens_in":8266,"tokens_out":24804,"duration_ms":264592,"concrete_test":"Recompute the benchmark of Fig. 4 (M1=3 TeV, M2=10M1, Reω=π/4, Xω chosen to saturate |meff|=122 meV and |Θ_e|^2=2.1×10^-3) with the full one-loop-corrected 5×5 neutral-fermion mass matrix, adding the one-loop self-energy contributions to the 12/21 and 22 blocks that are dropped after Eq. (9), and then diagonalize numerically to obtain the physical HNL masses and mixings. Use these in Eq. (26) to compute σ_with/σ_without at √s=3 TeV. If the ratio remains ≈1.15, the neglected terms are genuinely subdominant; if it moves by more than ~5 percentage points, the central claim is not robust.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central result is the ~15% increase in σ(e−e−→W−W−) at √s=3 TeV. It is computed from the modified HNL mixing elements Θ^(1) in Eq. (13), which follow from Eq. (9) after dropping the one-loop corrections to M_D and M_M with the statement that they are 'subdominant due to the loop suppression.' This is the load-bearing assumption. In the region used for the maximal cross section (|Θ_eI|^2 near 2.1×10^-3, Xω≫1), the Dirac Yukawa entries are not small: with M1=3 TeV and Θ_e~0.04, M_D=Θ M1 is O(10^2) GeV and Y is O(0.5). The corresponding one-loop corrections are δM_D/M_D ∼ g^2/(16π^2) ~ few×10^-3 and δM_M/M_M ∼ Y^2/(16π^2) ~ few×10^-3, whereas the retained correction changes Θ by only δMM/2 ~ 1–2%. The neglected terms are therefore not parametrically negligible, and if the enhancement arises near a cancellation between ν and N amplitudes (the paper's own meff=0 mechanism), even a 0.5% shift in Θ can move the 15% number substantially. The paper provides no estimate of this error.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the impact of one-loop radiative corrections to the left-handed neutrino Majorana masses (δMLL) on lepton-number-violating processes in the minimal seesaw model with two right-handed neutrinos. The authors derive the corrected HNL mixing elements Θ^(1) via a modified Casas-Ibarra parametrization, express the 0νββ effective mass including the correction, extend their previous meff=0 cancellation condition, and compute the cross section for e−e−→W−W−. They find that the maximal cross section at √s=3 TeV increases by about 15% when the one-loop correction is included for TeV-scale HNLs, a result they argue is relevant for future collider searches.","tokens_in":8582,"tokens_out":26004,"duration_ms":231591,"significance":"The central claim is specific and falsifiable: a 15% enhancement of the i0νββ cross section at CLIC-like energies due to radiative corrections to active-neutrino Majorana masses. If the result is robust, it sharpens the projected sensitivity of future e−e− colliders and demonstrates that radiative corrections cannot be ignored in LNV searches. The analytic relations are internally consistent: Eq. (20) reduces to the expected δMM fβ term, and the loop function δMM in Fig. 1 grows with the HNL mass as expected. The numerical estimate is based on explicitly defined constraints and external inputs rather than a fit to data, and the meff=0 cancellation is extended to the radiative case in closed form. The main weakness is the unquantified neglect of one-loop corrections to M_D and M_M in the parameter region used for the maximal cross section, which needs to be addressed before the central number can be considered reliable.","major_comments":[{"comment":"The statement just after Eq. (9) that corrections to the Dirac masses M_D and the Majorana masses M_M are 'subdominant due to the loop suppression' is load-bearing for the central result, because the 15% enhancement in Sec. 3 is obtained by changing only the mixing elements via Eq. (13). In the parameter region that maximizes the cross section in Fig. 4 (|ΘeI|^2 near 2.1e-3 and M1 = 3–5 TeV), the electron-flavor Yukawa entry is Y_eI = Θ_eI M_I/v ≈ 0.6–1. The one-loop fractional corrections to M_M and M_D are then of order (Y†Y)_II/(16π^2) log(M_I^2/m_H^2), which is several percent, i.e., comparable to or larger than the retained δMM of about 1–2% from Eq. (8). The loop-suppression argument does not discriminate between retained and neglected terms because both are one-loop; the M^2/(16π^2 v^2) enhancement that makes δMLL sizable is absent in δM_D/M_D and δM_M/M_M. Since the enhancement arises near the active–HNL cancellation described by Eqs. (21)–(23), a neglected few-percent shift in Θ can change the 15% result substantially. The authors should include the δM_D and δM_M contributions in the one-loop matching or specify an on-shell scheme that removes them, and demonstrate the numerical stability of the central result.","section":"Sec. 2, Eq. (9)"},{"comment":"The claimed 15% increase is not reproducible from the information given. The text says that the 'maximal cross section' is estimated by imposing |meff|<122 meV and |Θe|^2<2.1e-3, but it does not state the values of Xω and Reω (or the scan procedure) used for the solid and dashed curves, nor whether the maximum is taken over all free parameters. Because the cross section depends sensitively on these parameters through the cancellation condition, please specify the parameter choices and the maximization procedure so that the central number can be verified.","section":"Sec. 3, Fig. 4"}],"minor_comments":[{"comment":"The constraint in the caption reads '|Θe|2 < 2.1×103'; it should read '|Θe|^2 < 2.1×10^-3'.","section":"Fig. 4 caption"},{"comment":"The notation 'Xω = 104' and '10 3' should be '10^4' and '10^3' with proper superscripts; in several places the superscripts are lost in the text.","section":"Fig. 2 caption"},{"comment":"The abbreviation 'i0νββ' is used in the abstract and in Sec. 3 without a formal definition at first use in the main text; please define it explicitly.","section":"Sec. 3, Eq. (24)"},{"comment":"The loop function fδLL depends on mZ and mH; please state explicitly that these are the physical masses taken from Ref. [30] and specify the renormalization scheme or scale used for the one-loop correction.","section":"Sec. 2, Eq. (8)"},{"comment":"The cross section formula is a tree-level expression for the scattering process; please state clearly that radiative corrections to the process itself are not included, to avoid ambiguity with the title of the paper.","section":"Sec. 3, Eq. (26)"}],"recommendation":"major_revision","confidential_remarks":"The main technical concern is the missing quantitative estimate of the neglected one-loop corrections to M_D and M_M; this should be addressed before publication. The self-citations are not problematic; they are limited to the meff=0 discussion and do not affect the i0νββ estimate. The paper is within the scope of hep-ph and of potential interest to the LNV and future-collider community."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"For your desk: arXiv:2412.08015 is a focused, competent phenomenological paper on how one-loop radiative corrections to the light neutrino Majorana masses affect lepton-number-violating processes in the minimal two-right-handed-neutrino seesaw. The new element is the numerical estimate for the inverse 0νββ process e−e− → W−W−: with HNL masses at a few TeV, the maximal cross-section at √s = 3 TeV rises by about 15% once the δMLL correction is encoded in the HNL mixing elements. That number is genuinely new; earlier i0νββ studies used tree-level mixing.\n\nWhat the paper does well: the algebra connecting Eq. (7) to Eq. (20) is coherent, the δMM function is correctly derived, and the authors are clear about the input constraints (KamLAND-Zen bound, EW precision bound on |Θ_e|²). The meff = 0 cancellation discussion is a useful extension of their previous work, and it is not circular — the cross-section estimate uses Θ^(1) from the modified seesaw relation, not the cancellation condition.\n\nThe soft spot is exactly the one the stress-test flags. In Eq. (9) the authors drop the one-loop corrections to M_D and M_M as 'subdominant due to the loop suppression.' In the large-Xω region that maximizes the cross-section, the Dirac Yukawa entries are O(0.5) or larger, and a simple estimate gives δM_D/M_D and δM_M/M_M of order (0.5²/16π²)×log(M/m_H) ≈ 0.5–1%, while the retained δMM/2 effect is only about 1.5% at M = 3 TeV. So the omitted corrections are not an order of magnitude below the retained one; they could shift the 15% enhancement by a meaningful amount. The paper gives no error estimate for this. A referee should ask the authors to either compute the δM_D, δM_M corrections in their benchmark or to justify the hierarchy with explicit numbers. This is a revision request, not a rejection.\n\nThere is a minor point: the 0νββ constraint uses the approximate fβ with ⟨p²⟩^{1/2} = 200 MeV; that choice propagates into the allowed parameter region, but since the i0νββ cross-section itself has no nuclear matrix element, the impact is indirect.\n\nBottom line: this is a serious paper, worth sending to a competent referee. The qualitative result — an O(10%) enhancement from radiative corrections — is likely to survive; the exact 15% needs an error estimate. I'd bring it to the group with a caution about the unquantified corrections.","headline":"A competent numerical study pointing to a ~15% radiative enhancement in e−e− → W−W−, but the central number sits on an unquantified neglect of δM_D and δM_M that could be as large as the effect itself.","tokens_in":9153,"tokens_out":10206,"would_cite":true,"duration_ms":96253,"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":"The minimal seesaw model's one-loop correction to active-neutrino Majorana masses enlarges HNL mixing enough to raise the maximal $e^-e^- \\to W^-W^-$ cross section by about 15% at $\\sqrt{s} = 3$ TeV.","keywords":["seesaw mechanism","heavy neutral leptons","radiative corrections","lepton number violation","neutrinoless double beta decay","inverse neutrinoless double beta decay","electron-positron collisions","Majorana neutrino masses"],"falsifier":"Compute the full one-loop radiative corrections to $M_D$ and $M_M$ in the same two-right-handed-neutrino model and compare the resulting mixing elements with $\\Theta^{(1)}$ from Eq. (13). If the full result changes the mixing by more than a few percent for TeV-scale heavy neutral leptons, the 15% cross-section enhancement is not robust; a future $e^-e^-$ collider measurement of $\\sigma(e^-e^- \\to W^-W^-)$ at $\\sqrt{s}=3$ TeV could also settle the question empirically.","tokens_in":8048,"feed_emoji":"⚛️","tokens_out":8190,"duration_ms":87386,"temperature":0.7,"pith_summary":"This paper argues that one-loop radiative corrections to the Majorana masses of left-handed neutrinos can measurably alter lepton-number-violating signals in the minimal seesaw model, even when the heavy neutral leptons are at the TeV scale. The corrections enlarge the effective heavy-neutral-lepton mixing with electrons, and the authors find that the cross section of the inverse neutrinoless double $\\beta$ decay $e^-e^- \\to W^-W^-$ increases by about 15% at $\\sqrt{s}=3$ TeV under the current bounds. If this is right, precision predictions for future electron-positron colliders must include the one-loop shift, and the search for lepton number violation gains a several-percent boost in reach.","feed_headline":"One-loop neutrino corrections raise e−e−→W−W− rate by 15%","feed_subtitle":"Seesaw-scale heavy neutrinos mix more strongly after loop corrections, boosting the signal future colliders can hunt.","key_machinery":"The load-bearing object is the one-loop corrected seesaw block, whose upper-left entry is $\\delta M_{LL} = M_D M_M^{-1} \\delta_{LL} M_D^T$ with $\\delta_{LL}$ diagonal in the heavy-neutrino basis. Inserting this correction into the seesaw relation replaces the heavy Majorana mass matrix $M_M$ by $\\tilde M_M = M_M(1+\\delta_{MM})$, and in the standard parametrization of the neutrino Yukawa matrix the mixing elements become $\\Theta^{(1)}_{\\alpha I} = i[U D_\\nu^{1/2}\\Omega \\tilde M_M^{1/2} M_M^{-1}]_{\\alpha I}$. This shifted mixing enters the $t$- and $u$-channel heavy-neutrino amplitudes that dominate the $e^-e^- \\to W^-W^-$ cross section, which is what produces the $\\mathcal{O}(10)$% enhancement.","core_discovery":"Starting from the two-right-handed-neutrino seesaw model with electroweak-scale heavy neutral leptons, the paper claims that the one-loop correction $\\delta M_{LL}$ to the active-neutrino Majorana mass matrix is not negligible: it modifies the seesaw relation and hence the heavy-neutral-lepton mixing elements, even though it enters only through the diagonal matrix $\\delta_{MM}$. With this correction included, the maximal cross section of $e^-e^- \\to W^-W^-$, computed while imposing the neutrinoless double $\\beta$ decay bound on the effective mass and the electroweak precision bound on mixing, increases by about 15% at $\\sqrt{s}=3$ TeV for heavy lepton masses near a few TeV. The same correction shifts the effective mass in ordinary $0\\nu\\beta\\beta$ decay by less than a few percent, and the special parameter choice that makes the effective mass vanish remains available once the loop parameter in the cancellation condition is updated.","pith_inferences":["This suggests the same one-loop shift will appear in other lepton-number-violating observables that grow quadratically with heavy-neutrino mixing, such as same-sign dilepton searches, where the correction may be larger than the few-percent effect seen in $0\\nu\\beta\\beta$.","The functional form of $\\delta_{MM}$ in the paper indicates the enhancement likely grows as the heavy lepton mass rises toward $\\gtrsim 10$ TeV, so the 15% figure may be a lower-end estimate for heavier spectra probed by future colliders.","A natural next step, not taken in the paper, is a full one-loop computation of the Dirac and heavy Majorana mass corrections; depending on their size, the 15% prediction could move by a comparable amount.","Because the relative phases between active and heavy contributions are known to control interference in the cross section, the corrected mixing may shift the angular distribution beyond a simple overall normalization change."],"forward_implications":["The maximal $e^-e^- \\to W^-W^-$ cross section at $\\sqrt{s}=3$ TeV is about 15% larger than the tree-level prediction after including the one-loop correction to left-handed neutrino masses.","The loop correction tightens the upper bound on the heavy-neutrino mixing parameter $X_\\omega$ set by $0\\nu\\beta\\beta$ by roughly 3% for the benchmark $M_1=5\\times10^3$ GeV, $M_2=10^2 M_1$, and $\\mathrm{Re}\\,\\omega=\\pi/4$.","The cancellation point $m_{\\mathrm{eff}}=0$ survives the radiative corrections, with the parameter $\\delta$ in the cancellation condition replaced by $\\tilde\\delta$, so the model still admits a vanishing effective mass despite the loop shift.","For heavy neutral lepton masses above about 10 GeV the correction $\\delta_{MM}$ becomes sizable, so any precision prediction of lepton-number-violating observables in this model should include it.","A 15% larger signal improves the discovery reach of future $e^-e^-$ colliders for heavy neutral leptons compared with tree-level estimates."],"supporting_citations":[{"why":"Gives the one-loop corrected parametrization of heavy-neutral-lepton mixing elements used to define $\\Theta^{(1)}$.","marker":"[28]"},{"why":"Supplies the tree-level parametrization of the neutrino Yukawa matrix that the corrected version extends.","marker":"[29]"},{"why":"Provides the cross-section formula for $e^-e^- \\to W^-W^-$ on which the numerical estimate is based.","marker":"[24]"},{"why":"Sets the neutrinoless double beta decay upper bound on the effective mass imposed in the parameter scan.","marker":"[11]"},{"why":"Sets the electroweak-precision bound on the heavy-lepton mixing elements used in the maximal-cross-section estimate.","marker":"[36]"},{"why":"Derives the one-loop radiative correction to the left-handed Majorana mass matrix that drives the effect.","marker":"[25]"},{"why":"Established the condition for a vanishing effective mass in the same model, which the paper re-derives with radiative corrections.","marker":"[33]"},{"why":"Gave the cancellation parameter that the paper replaces by $\\tilde\\delta$ when loop effects are included.","marker":"[34]"}],"fun_headline_variants":["Loop corrections boost inverse neutrinoless double beta decay by 15%","Radiative seesaw corrections raise e−e−→W−W− cross section 15%","One-loop Majorana mass shift lifts inverse 0νββ rate at 3 TeV","Heavy neutrino loop effect boosts e−e−→W−W− by 15%","Radiative correction to seesaw boosts e−e−→W−W− cross section 15%"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes that radiative corrections to the Dirac mass matrix $M_D$ and to the heavy Majorana masses $M_M$ are negligible compared with the correction $\\delta M_{LL}$, so that $\\delta M_{LL}$ alone controls the loop-level change in the mixing elements; if those other corrections were comparable, the predicted 15% enhancement could shrink or disappear.","fun_headline_variants_meta":{"raw":{"variants":["Loop corrections boost inverse neutrinoless double beta decay by 15%","Radiative seesaw corrections raise e−e−→W−W− cross section 15%","One-loop Majorana mass shift lifts inverse 0νββ rate at 3 TeV","Heavy neutrino loop effect boosts e−e−→W−W− by 15%","Radiative correction to seesaw boosts e−e−→W−W− cross section 15%"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000672,"raw_usage":{"total_tokens":3050,"prompt_tokens":924,"completion_tokens":2126,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":540,"completion_tokens_details":{"reasoning_tokens":2004}},"tokens_in":540,"tokens_out":2126,"duration_ms":15499,"temperature":1.0,"reasoning_tokens":2004,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T18:19:41.749483+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the full one-loop radiative corrections to $M_D$ and $M_M$ in the same two-right-handed-neutrino model and compare the resulting mixing elements with $\\Theta^{(1)}$ from Eq. (13). If the full result changes the mixing by more than a few percent for TeV-scale heavy neutral leptons, the 15% cross-section enhancement is not robust; a future $e^-e^-$ collider measurement of $\\sigma(e^-e^- \\to W^-W^-)$ at $\\sqrt{s}=3$ TeV could also settle the question empirically.","supporting_citations":[{"cited_title":"Hiding neutrinoless double beta decay in the minimal seesaw mechanism","cited_arxiv_id":"2012.12564","evidence_quote":"Established the condition for a vanishing effective mass in the same model, which the paper re-derives with radiative corrections."},{"cited_title":"What if a specific neutrinoless double beta decay is absent","cited_arxiv_id":"2012.13186","evidence_quote":"Gave the cancellation parameter that the paper replaces by $\\tilde\\delta$ when loop effects are included."}],"review_version":1}