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REVIEW 3 major objections 5 minor 86 references

At 1-loop order, the entire Zee-model neutrino mass matrix is generated by RG running inside the two-Higgs-doublet EFT, not by matching at the heavy scale, and percent-level UV rescaling moves predicted oscillation parameters beyond next-ge

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2026-08-03 15:20 UTC pith:EC6MGWUP

load-bearing objection A genuinely new 1-loop EFT treatment of the Zee model with reusable matching conditions; the JUNO-comparison plots should be read as sensitivity studies, since the central beta function comes with an uncorroborated unpublished correction. the 3 major comments →

arxiv 2512.17235 v2 pith:EC6MGWUP submitted 2025-12-19 hep-ph

Running of neutrino mass parameters in the Zee model

Michael A. Schmidt , James Vandeleur This is my paper
classification hep-ph
keywords Zee modelradiative neutrino massestwo-Higgs-doublet effective field theoryrenormalization group runningWeinberg operatorneutrino mixing parametersRG correctionsneutrino oscillation precision
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper aims to show that a proper effective-field-theory treatment changes how the Zee model produces neutrino masses. When the heavy charged singlet and the second Higgs doublet are integrated out sequentially, the one-loop matching contributions to the dimension-5 Weinberg operator cancel, and what remains is a logarithm of the two heavy scales that is generated by one-loop renormalization-group running in the intervening two-Higgs-doublet EFT. The authors derive the one-loop matching conditions, solve the RG equations numerically for four benchmark sets, and find that percent-level rescalings of the fitted high-scale couplings—rescalings that leave the full-theory neutrino mass matrix invariant—shift the predicted mass-squared differences and mixing angles by more than current and projected experimental uncertainties. They conclude that precision phenomenological studies of the Zee model must include RG corrections. The quantitative size of these effects rests on a 1-loop beta function for the dimension-5 coefficients that the paper takes from the literature as corrected by an unpublished thesis, and the calculation stops at 1-loop order with no 2-loop check.

Core claim

Working in the hierarchy m_h > m_H2 ≫ EW scale, the paper matches the Zee model to the two-Higgs-doublet EFT at m_h and then to the SMEFT at m_H2, keeping 1-loop terms. The central observation is that the would-be direct contributions to the SMEFT Weinberg coefficient—κ11 at 1-loop and the tree-level κ12 term combined with charged-lepton Yukawas—cancel exactly at the matching scale, leaving only the log(m_H2^2/m_h^2) term, which is the resummed large logarithm of the full theory. Since the log can only be reproduced by running, the paper states that the entirety of the neutrino mass matrix is generated by RGE running in the 2HDM EFT. In four benchmarks, percent-level variations of UV paramet

What carries the argument

The central object is the set of dimension-5 Weinberg-like coefficients κij (i,j = 1,2 for the two Higgs doublets) in the 2HDM EFT; these are the coefficients of the operators (L Hi)(Hj† L) that after electroweak symmetry breaking become Majorana neutrino masses. The paper's argument is carried by the 1-loop β-function for κij (with gauge, Yukawa, and quartic terms), the derived matching conditions from Zee -> 2HDM and 2HDM -> SMEFT, and the cancellation between κ11 and the term κ12(0)Y1e†Y2e + Y2e†Y1eκ12(0) at the intermediate scale. The running of κ11 relative to κ12 is what produces the low-energy neutrino mass matrix.

Load-bearing premise

The load-bearing input is the 1-loop beta function for the dimension-5 coefficients κij in the general two-Higgs-doublet EFT, taken from the published literature as corrected by an unpublished thesis; the cancellation mechanism, the large-log resummation, and every benchmark sensitivity curve inherit this function, and if it is wrong the size of the running effects and the comparison to experimental precision would change.

What would settle it

Reproduce the paper's numerical procedure with an independent implementation of the 1-loop RGEs for the 2HDM EFT and the two matching steps, apply the same γ scalings of f, Y1_e and Y2_e that preserve the full-theory mass matrix, and compare the resulting displacements with the paper's current and projected experimental bands; a disagreement larger than those bands would indicate an error in the input β-function or matching conditions. A direct two-loop calculation of the Zee-model neutrino mass in the same scheme would settle whether the 1-loop running picture is quantitatively correct.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Precision fits of the Zee model should replace the one-loop full-theory formula with a sequential EFT calculation that includes 1-loop matching at m_h and m_H2 and 1-loop RG running between them.
  • Percent-level freedom in the fitted UV couplings—directions that the full theory does not constrain—changes the predicted neutrino observables by more than current and next-generation experimental errors, so RG corrections must be included when exploiting the new data.
  • The corrections grow with large quartic coupling λ6, sizable second-doublet Yukawa couplings, and a large hierarchy between the charged singlet and second doublet masses; the same qualitative effect appears for a 1 TeV second doublet.
  • The large-log approximation reproduces the full-theory formula at leading log, confirming that the running is the source of the logarithm, but in benchmark 1 the β-function varies enough that a constant-β large-log approximation is not quantitatively valid.
  • The authors expect similarly sized RG corrections in other radiative neutrino mass models, since running enters at the same loop order as the radiative mass itself.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The split between matching and running is scheme- and matching-scale-dependent; in another scheme part of the effect could be shifted into the matching condition. The scheme-independent content is that a full one-loop EFT calculation, not the naive one-loop full-theory formula, is required for precision.
  • A direct check of the quantitative claim would be an independent re-derivation of the 1-loop β-function for κij in the general 2HDM EFT; if the coefficients change, the γ-scaling shifts would change and the comparison to experimental precision would need revision.
  • The flat directions that preserve the full-theory mass matrix could be used in future global fits as a diagnostic: any fitted point that ignores running sits on a degeneracy that the EFT running breaks, effectively constraining the UV parameters at the level of next-generation experiments.
  • The same two-step EFT construction could be applied to other radiative models, such as Zee-Babu or the scotogenic model, to check whether their neutrino mass matrices likewise arise from running in the intermediate theory.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper presents a two-step effective field theory (EFT) treatment of the Zee model for the hierarchy m_h ≫ m_H2 ≫ m_hSM. It derives 1-loop matching conditions for the Weinberg-like operators when integrating out the charged singlet to the 2HDM EFT and then the second doublet to SMEFT, and combines these with 1-loop RG running in the 2HDM EFT. The central claim is that, after the matching, the entire neutrino mass matrix is generated by the RG running that spoils a cancellation between κ11 and the κ12(0) Yukawa combination, and that percent-level changes in fitted high-scale parameters that leave the full-theory prediction invariant induce shifts in neutrino mass parameters exceeding current and JUNO-era projected uncertainties. Four benchmarks illustrate the effect. The paper does not perform a full parameter scan; the benchmarks are fitted to reproduce the low-energy oscillation data using the full-theory formula and then used to illustrate sensitivity of the EFT calculation.

Significance. If the central results hold, the paper makes a useful and timely point: for a radiative neutrino mass model, a naive 1-loop full-theory calculation may miss large logarithms that are properly resummed only by 1-loop matching plus 1-loop running, and this can affect phenomenological comparisons at the precision of JUNO and future experiments. The paper has several genuine strengths: the matching conditions are nontrivial, presented with diagrams, and stated to have been verified by hand; the two possible hierarchies are both discussed; the role of the charged-lepton Yukawa couplings and trace terms in the running of κ11 is analysed in detail; and the numerical setup includes consistency checks of perturbativity, stability, and the mass hierarchy. The comparison against current and projected JUNO sensitivities in Figs. 9 and 10 is a concrete falsifiable demonstration, even though it is sensitivity-based rather than a full statistical scan.

major comments (3)
  1. [§3.2, Eq. (31) / App. B.1, Eq. (72)] The engine of the quantitative claim is the 1-loop β-function for the dimension-five coefficients κij. It is taken from Ref. [40] as corrected by the unpublished Honours thesis [85]; no independent derivation, ancillary file, or machine-checkable calculation is provided. The hand-verification mentioned in Sec. 3.1 covers the matching conditions, not this RGE. An O(1) error in any trace or gauge term, e.g. the signs of Tkiκkj or the g2 terms, would change the relative running of κ11 and κ12 and hence the size and sign of the shifts in Figs. 9–10. Because the abstract and Sec. 5 make quantitative statements about exceeding JUNO-era uncertainties, the paper should make the derivation of Eq. (31) fully available, either by reproducing it in the paper or by including a public notebook/appendix that verifies it.
  2. [§3, paragraph after Eq. (40)] The authors note that 2-loop RGEs for the dimension-five operators in the general 2HDM EFT are not available and assert that they 'do not expect any issues' with scheme dependence. For a claim that running effects exceed percent-level experimental uncertainties, this expectation needs to be quantified. Concretely, the matching-scale dependence of the final prediction should be tested by varying ¯µ around m_h and m_H2 (e.g. by factors of e^{±1/2}) and showing that the shifts in Fig. 9 are stable, or by estimating the size of the missing 2-loop contributions. Without such a test, the percent-level statement is not established at the claimed precision.
  3. [§4.2–4.4, Figs. 9–10] The benchmark f and Y2 are fitted so that the full-theory tree-level formula reproduces the same low-energy neutrino parameters that the EFT calculation then shifts. The curves in Figs. 9–10 are therefore sensitivity illustrations along a particular high-scale parameter subspace, not evidence that realistic Zee-model parameter points actually produce these shifts. The text partly acknowledges this ('qualitatively demonstrate', 'will not reproduce quantitatively correct values'), but the abstract and Sec. 5 phrase the conclusion as 'quantum corrections have to be included'. The paper should separate these two statements more sharply and state explicitly that a full prior- or parameter-scan analysis is needed to establish whether the effect is generic in the allowed Zee-model parameter space.
minor comments (5)
  1. [Abstract / Sec. 5] The wording 'quantum corrections have to be included in studies of neutrino mass parameters' overstates what is demonstrated. The analysis shows that within the chosen benchmarks and fitted subspaces, running effects can exceed experimental uncertainties; it does not show that this happens for all viable Zee-model parameters. Consider phrasing the conclusion as 'can be important' rather than 'have to be included' unless a scan is performed.
  2. [Figure 9 caption] The caption says 'shaded regions represent current and future estimated experimental uncertainties' and that γ increases 'in the direction of the arrows marked', but the arrows are not visible in the arXiv text version and the distinction between current and JUNO-projected bands is not labelled. Please add explicit labels to the figure and mention the different shadings in the caption.
  3. [Table 1] The table lists m_t = 168.62 GeV while the caption says the inputs are at the scale of the top quark pole mass m_t = 172.4 GeV. The relationship between the two conventions should be stated explicitly to avoid confusion (probably m_t(MS-bar) at the pole-mass scale).
  4. [Reference [85] / footnote 3] The β-function correction is attributed to an unpublished Honours thesis. If this source remains the only reference for a load-bearing expression, please provide a web link, an institutional repository URL, or include the derivation as a supplementary appendix so that readers can check it.
  5. [General copy-editing] There are several typographical and grammatical slips, e.g. 'We will first in detail the matching procedure' (Sec. 3.1) and 'the hierarchy ... that we eluded to schematically' (Sec. 2.2, should be 'alluded to'). These do not affect the physics but should be corrected.

Circularity Check

0 steps flagged

No circularity in the central EFT derivation; benchmark sensitivity plots are demonstrations, not fitted predictions; self-citations are non-load-bearing.

full rationale

The derivation chain is self-contained conditional on external RGE literature. The matching conditions (Eqs. 23, 26, 53, 56) are derived from the Zee-model Lagrangian with Matchete and hand verification (footnote 1), and the κij RGE (Eq. 31) is attributed to Ref. [40] as corrected by Ref. [85] and cross-checked with RGBeta (Appendix B); no step is defined in terms of the target neutrino observables. The claim that the neutrino mass matrix is 'generated due to the running in the 2HDM EFT' (Sec. 3.1) is an EFT reorganization: Eq. (28) reproduces the full-theory logarithm of Eq. (20), and Eq. (39) resums the same logarithm from the RGEs, which is a consistency check rather than a reduction of the prediction to its input. The benchmark couplings are fitted to measured neutrino parameters using the full-theory formula (Eqs. 20, 46), but the paper explicitly labels this as a qualitative demonstration, stating 'we argue that this will not reproduce quantitatively correct values for the UV parameters' (Sec. 4.2); the γ-scalings (Eqs. 50-52) are constructed to leave the full-theory mass matrix invariant, so the plotted displacements are sensitivity diagnostics isolating running effects, not independent predictions that are then fitted. Self-citations [17,44,45,59,83,99] are contextual or motivational and are not load-bearing: e.g., [83] motivates the hierarchy but both hierarchies are derived and the qualitative conclusion is stated to apply to both. The main support gap is the reliance on the unpublished Honours thesis [85] for corrections to the κij β-function (Eq. 31, footnote 3); if that β-function is wrong, the numerical size of the running shifts changes. This is a correctness/verification risk, not circularity. Similarly, the assertion 'We verified the matching conditions in full by hand' (footnote 1) is an omitted proof but not a circular step. Overall, the central comparison (full theory vs. EFT) does not reduce to a fitted value or to a self-citation chain.

Axiom & Free-Parameter Ledger

6 free parameters · 6 axioms · 0 invented entities

All benchmark inputs are hand-chosen or fitted; the matching and RGE equations themselves are derived from the model or taken from external literature. The paper introduces no invented entities. The main intended contribution—the running-induced mass generation and its phenomenological size—is not built into the input assumptions.

free parameters (6)
  • f (antisymmetric singlet Yukawa-like coupling) = Table 3 (e.g., f_eμ ~ 7.2e-6 for BM1, 9.3e-3 for BM2)
    Chosen by fitting the full-theory Zee formula (20) to reproduce observed neutrino mixing parameters; central to benchmark inputs.
  • Y2_e (second Higgs doublet charged-lepton Yukawa) = Table 3; complex 3x3 matrices with entries ~O(0.01-0.7)
    Fitted simultaneously with f to reproduce low-energy neutrino oscillation data while respecting cLFV bounds (Sec. 4.2).
  • Quartic couplings λ1,...,λ7 = BM1/2: λ1=λ2=0.25, λ3-7=0; BM3: λ6=-λ7=0.25; low-scale: λ1=λ2=λ6=λ7=0.2, λ4=λ5=0.4
    Chosen by hand to satisfy stability bounds and illustrate λ6 effects; not fitted to neutrino data.
  • α (quark Yukawa alignment parameter) = 1 (BM1,2, low-scale) or 0 (BM3)
    Introduced to reduce quark-sector freedom; affects running through trace terms T12.
  • Mass scales m_h, m_H2, μ_Zee = m_h=μ_Zee=10^4 TeV, m_H2=10^2 TeV for BM1-3; 100 TeV/1 TeV for low-scale
    Chosen to satisfy EFT hierarchy and perturbativity; results are claimed to be qualitatively independent of scale.
  • λ_h and λ8,9,10 = λ_h=1e-4, λ8=λ9=λ10=0
    Set to simplify matching; small λ_h ensures potential stability, but contributions to κij matching are neglected.
axioms (6)
  • domain assumption The 1-loop RGEs for the dimension-5 Weinberg-like coefficients κij in the general 2HDM EFT (Eq. 31/72) are correct as given by Ref. [40] as corrected by the unpublished Honours thesis [85].
    The central cancellation-and-running mechanism and all benchmark numerics rely on these β-functions; the published source was found to need correction, and no public peer-reviewed re-derivation is supplied.
  • ad hoc to paper A 1-loop matching + 1-loop running calculation is scheme-independent even though 2-loop RGEs for the 2HDM Weinberg-like operators are not computed.
    Authors state 'we do not expect any issues with scheme dependence' (Sec. 3) but cite Refs. [80-82] showing that one-loop-higher running is normally needed; this is asserted, not quantified.
  • domain assumption The hierarchy m_h >> m_H2 >> m_hSM is preserved under RG running and physical masses can be identified with Lagrangian mass parameters (alignment limit).
    Section 2.2 uses Ref. [72] to justify expansion in μ2; Section 4.1 checks the ratio μ1^2/μ2^2 runs slowly for the chosen quartic couplings, but the identification is assumed for the general EFT statement.
  • domain assumption Perturbativity λ_i ≲ sqrt(4π) and stability conditions (Eqs. 6-7) hold at all scales.
    Section 4.1 verifies this numerically for each benchmark set; it is not a proof, and the claimed generality of the conclusion relies on it.
  • domain assumption The SMEFT RGEs and the SM input parameters from Ref. [100] are correct at the stated precision.
    The low-scale SMEFT running and matching uses these published results; standard, but assumed.
  • ad hoc to paper Neglecting λ8=λ9=λ10=0 and λh=1e-4 does not change the qualitative conclusion about running.
    These couplings contribute to κij matching (Eq. 23c) and to scalar-sector running; the paper sets them to zero to reduce parameters and argues their effect is uninteresting/loop-suppressed.

pith-pipeline@v1.3.0-alltime-deepseek · 39219 in / 18170 out tokens · 168635 ms · 2026-08-03T15:20:14.257839+00:00 · methodology

0 comments
read the original abstract

We analyse the size of quantum corrections in the Zee model using effective field theory techniques. We derive the relevant 1-loop matching conditions and use them together with the existing renormalisation group equations in the two Higgs doublet model to calculate quantum corrections to the neutrino mass squared differences, mixing angles, and phases. Using four benchmark scenarios, we demonstrate when quantum corrections have to be included in studies of neutrino mass parameters in the Zee model.

Figures

Figures reproduced from arXiv: 2512.17235 by James Vandeleur, Michael A. Schmidt.

Figure 1
Figure 1. Figure 1: 1-loop Feynman diagrams that generate neutrino masses in the Zee model. In the Higgs [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Tree and 1-loop level contributions from the Zee model to the Weinberg like [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Tree level contribution in the Zee model to the quartic couplings [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Tree and 1-loop level contributions from the 2HDM EFT to the SMEFT Weinberg operator [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Running of κ 11 , κ 12 , Y 1 e and Y 2 e in the 2HDM EFT for the first set of high-scale benchmark parameters. The real component of the diagonal elements of each parameter matrix is plotted. The values of the Weinberg-like operator κ are multiplied by v 2 to represent the scale of the resulting neutrino masses. 19 [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Running of κ 11 and Y 1 e in the 2HDM EFT for the second set of high-scale benchmark parameters (Y 1 e ∼ Y 2 e ). The real component of the diagonal elements of each parameter matrix is plotted. An additional feature of this benchmark set is that the suppression of Y 2 e to approximately the same order of magnitude as Y 1 e allows the cLFV constraints in appendix C to be satisfied with a lower scale of new… view at source ↗
Figure 7
Figure 7. Figure 7: Running of κ 11 and Y 1 e in the 2HDM EFT for the set of low-scale benchmark parameters with mH2 = 1TeV, mh = µZee = 100TeV, and (Y 1 e ∼ Y 2 e ). The real component of the diagonal elements of each parameter matrix is plotted. As predicted at the end of section 4.2, we observe exactly the same qualitative behaviour with the lower scale benchmark parameters in figure 7 as with the higher scale benchmark in… view at source ↗
Figure 8
Figure 8. Figure 8: Running of κ 11 and Y 1 e in the 2HDM EFT for the third set of high-scale benchmark parameters (α = 0). The real component of the diagonal elements of each parameter matrix is plotted. Indeed, we see in figure 8b that there is extremely little running of Y 1 e . Nevertheless, the running of κ 11 shows a shift to the second gradient as per the first benchmark case. This similar behaviour is not entirely sur… view at source ↗
Figure 9
Figure 9. Figure 9: Sensitivity of the solar neutrino mixing parameters to the high-scale model parameters in the [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Sensitivity of the neutrino mixing parameters to the UV model parameters of the low scale [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Representative contributions to cLFV processes from the high scale Higgs fields. [PITH_FULL_IMAGE:figures/full_fig_p034_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Sensitivity of the normal-ordered atmospheric neutrino mixing parameters to the UV model [PITH_FULL_IMAGE:figures/full_fig_p038_12.png] view at source ↗

discussion (0)

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