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REVIEW 3 major objections 5 minor 2 cited by

Ultraviolet Completion of a Two-loop Neutrino Mass Model

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Embedding the Zee-Babu model in SU(5) forces TeV-scale scalars and a proton decay signal within Hyper-Kamiokande's reach.

desk verdict Real SU(5) embedding of the Zee-Babu model with sharp collider and proton-decay predictions, but the paper never demonstrates the model fits neutrino oscillation data, and the asymptotic-freedom claim outruns what is shown. read the letter →

arxiv 2412.14562 v2 pith:HLESMAAS submitted 2024-12-19 hep-ph

K.S. Babu , Shaikh Saad This is my paper · ORCID
classification hep-ph
keywords Zee-Babumodeltwo-loopneutrinomassSU(5)grandunificationgaugecouplingprotondecayvector-likequarkradiativecolor-sextetscalar
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Radiative neutrino mass models usually live at the TeV scale without a reason to be connected to higher energies. This paper claims that the Zee-Babu two-loop model can be embedded into an SU(5) grand unified theory in a way that is realistic, asymptotically free, and consistent with gauge coupling unification. In that embedding the colored GUT partners of the original Zee-Babu scalars contribute to neutrino masses just as much as the color-neutral ones, and the combined constraints of unification, proton decay, and perturbativity force all of them below roughly $10^{3}$ TeV. If true, the model turns proton decay into a near-term experimental target: the Markov chain Monte Carlo analysis finds a high likelihood that Hyper-Kamiokande sees p→e+π0 in its first decade.

What carries the argument

The load-bearing machinery is the two-loop Majorana neutrino mass formula combining the original color-neutral Zee-Babu diagram with the colored-partner diagram: $M_{\nu}^{\rm loop} = 16\mu \hat{Y}_A M_E^{\rm diag} \hat{Y}_S M_E^{\rm diag} \hat{Y}_A \hat{I} + 48\mu \tilde{Y}_A M_D^{\rm diag} \tilde{Y}_S M_D^{\rm diag} \tilde{Y}_A \tilde{I}$. The submultiplet mass relations of the 50H representation, for example $m_{\chi_4}^2 = 3m_{\chi_2}^2 - 2m_{\chi_3}^2$, are what allow the colored partners to remain light while the color-triplet states that mediate proton decay are pushed above $10^{12}$ GeV.

What would settle it

A numerical scan of the Yukawa parameters entering Eq. (3.41) that either reproduces or fails to reproduce $\Delta m^2_{21}$, $\Delta m^2_{31}$, and the three PMNS angles would settle the model's realism; the paper's own Sec. 3.4 leaves this check undone.

Watch

Extended reading notes

Core claim

The central discovery claimed is a specific SU(5) embedding of the Zee-Babu model, with scalar content 10H + 50H and one vector-like 5F + 5F fermion pair, in which the original singly charged η1 and doubly charged χ1 scalars are accompanied by colored partners η3 (color triplet) and χ5 (color sextet). The paper shows from two-loop RGE running with the 50H mass relations that gauge coupling unification selects a spectrum where both the color-neutral and colored Zee-Babu states sit near the TeV scale, so the two-loop neutrino mass formula contains two comparable contributions. It further shows that the vector-like down-type quark must sit at the TeV scale, the vector-like lepton near the GUT scale, and that the unified coupling αGUT is enhanced to about 1/14, which shortens the proton lifetime by roughly an order of magnitude relative to typical non-supersymmetric unified theories. The paper does not provide a numerical fit to neutrino oscillation data; it argues only that the lightest neutrino mass can be nonzero and small, while the heavier two are compatible with the observed hierarchy.

Load-bearing premise

The load-bearing premise is that the neutrino mass matrix of Eq. (3.41) can reproduce the measured neutrino mass-squared differences and mixing angles; the paper does not demonstrate this numerically, only that the lightest neutrino mass can be nonzero and small.

Editorial extensions

If this is right

  • New scalars η1, χ1, η3, and χ5 have masses below roughly 10^3 TeV, within reach of future collider experiments.
  • Proton decay p→e+π0 should be observable by Hyper-Kamiokande within about ten years if the model is correct.
  • A vector-like down-type quark B with mass near 3 TeV decays as B→W−t, B→Zb, and B→hb with relative rates 2:1:1.
  • The color-neutral and colored two-loop diagrams must contribute comparably to neutrino masses, and the lightest neutrino mass is nonzero but much smaller than the two heavier ones.
  • The theory is asymptotically free and can be extrapolated to the Planck scale.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the model is right, the original Zee-Babu picture was incomplete: the colored partners are not optional but required by unification, so collider searches for the leptoquark-like η3 and diquark-like χ5 become as important as searches for the charged scalars.
  • The correlation between proton decay and the TeV-scale scalar masses means a null result at Hyper-Kamiokande would not kill the model but would push the spectrum toward its upper end, while a positive signal would pin down the unification scale and strengthen the case for TeV-scale scalars.
  • A decisive next step the paper leaves implicit is a full parameter scan of the neutrino sector: the claim that det(Mν)≠0 with m1≪m2,3 must be checked against the measured mass-squared differences and mixing angles.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This paper embeds the two-loop Zee-Babu neutrino mass model into a non-supersymmetric SU(5) GUT by adding 10H and 50H scalars and a vector-like 5F + 5bar_F fermion pair. The authors compute the two-loop neutrino mass matrix from both color-neutral and colored scalars, present a benchmark fit to charged-fermion masses and CKM parameters, derive perturbativity bounds on the new Yukawa couplings, and perform a two-loop gauge-coupling-unification analysis with an MCMC scan over multiplet masses. From this they conclude that the Zee-Babu scalars and their colored partners must lie near or below the TeV scale, that a vector-like down-type quark should be at the TeV scale, and that p -> e+ pi0 should be observable in the first decade of Hyper-Kamiokande. The abstract and conclusions further claim that the model is realistic, asymptotically free, and can be extrapolated to the Planck scale.

Significance. If the central claims are correct, the paper would provide an interesting and testable UV completion of a radiative neutrino mass model, with concrete collider and proton-decay predictions. The technical apparatus is substantial: the two-loop RGE coefficients for all new multiplets are given explicitly, the 50H mass relations are used systematically, and the MCMC exploration of the unification parameter space is a useful step beyond single benchmark studies. The explicit benchmark charged-fermion fit and the flavor-violation checks are also commendable. However, the significance is conditional on three load-bearing gaps: the neutrino sector is never fitted to oscillation data, the charged-fermion fit is overparameterized and is presented with incorrect degrees of freedom, and the asymptotic-freedom/Planck-scale-extrapolation claim is not demonstrated. These issues must be addressed before the 'realistic and UV-complete' conclusion can be accepted.

major comments (3)
  1. [Sec. 3.4 and Eq. (3.41)] The central claim that the model is realistic and that the TeV-scale states are 'required from ... neutrino oscillation data' is not supported, because the two-loop mass matrix in Eq. (3.41) is never confronted with the measured neutrino mass splittings and mixing angles. The text only argues that det(A) != 0 and m1 << m2,3, and Eq. (4.14) uses a single one-entry estimate m_nu ~ 0.05 eV. No numerical scan or fit of YA, YS, and the scalar/fermion masses to Delta m^2_21, Delta m^2_31, and the PMNS angles is presented. If this flavor structure cannot accommodate oscillation data, the model is not realistic and the subsequent proton-decay and collider predictions lose their motivation. This is the load-bearing gap in the paper's central claim.
  2. [Sec. 3.2] The charged-fermion fit is overparameterized in a way that weakens the claim of realistic fermion masses. The paper states that there are 13 magnitudes and 6 phases, i.e. 19 parameters, for 11 observables, and then says the number of degrees of freedom is 8. With 11 observables and 19 fitted parameters the number of degrees of freedom is actually 11 - 19 = -8, so the reported total chi^2 = 0.1 is not a meaningful goodness-of-fit statistic. This fit should be presented as an existence proof or benchmark, not as evidence that the model successfully predicts the charged-fermion sector.
  3. [Sec. 4 and Abstract] The abstract and introduction claim that the model is asymptotically free and can be extrapolated to the Planck scale, but this is not demonstrated anywhere in the paper. Section 4 only evolves the Yukawa couplings from the TeV scale to the GUT scale of 2 x 10^16 GeV and imposes |y| <= sqrt(4 pi); no gauge or Yukawa running from the GUT scale to M_Planck is shown, and no two-loop or higher-order analysis establishes asymptotic freedom. The Planck-scale extrapolation claim should either be supported by an explicit calculation or removed from the abstract and conclusions.
minor comments (5)
  1. [Sec. 3.2] The text says the number of degrees of freedom is 8 for the charged-fermion fit; as noted in the major comments, this is arithmetically incorrect and should be corrected.
  2. [Throughout] There are several typographical errors: 'Oklahom State University' on the title page, 'F ermion' in the Section 3 heading, 'vectorike' in Table 1, 'T able 1' in the caption, and an extra '+' in Eq. (2.2) ('++Phi4').
  3. [Sec. 2 and Sec. 5] The notation for the Zee-Babu scalars is inconsistent: the introduction uses eta^+ and k^{++}, while Section 2 switches to eta1 and chi1; later text also uses 'm_h+ = m_k++' where eta1 and chi1 are meant. A single notation should be used throughout.
  4. [Sec. 3.1 and Table 1] The GUT-scale inputs from Ref. [51] are computed assuming only SM particles run between M_Z and M_GUT, whereas the model contains new TeV-scale states. The paper says it 'expects' the modifications to be insignificant, but this is not quantified; the systematic uncertainty in the fermion-mass inputs should at least be discussed quantitatively.
  5. [Eq. (5.17)] The proton decay rate formula uses alpha_U, which is not explicitly defined; the text later refers to alpha_GUT = g_GUT^2/(4 pi). Please define alpha_U consistently in Eq. (5.17).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's predictions are outputs of a parameter scan constrained by external data, not re-imported fitted inputs.

full rationale

The paper's central derivation chain is self-contained with respect to its inputs. The neutrino mass formula in Eq. (3.41) is written out explicitly from the Lagrangian of Eq. (2.13), with the loop integrals given in Eq. (3.42) and the cubic coupling in Eq. (3.43); the loop functions are standard and are cited to independent literature, but the paper itself displays the needed expressions. The charged fermion mass fit in Sec. 3.2 is an explicit fit to external low-energy observables, not a claim of prediction, and the vector-like fermion mass matrices in Eqs. (3.7)-(3.10) are derived in the text. The proton lifetime calculation in Eq. (5.17) uses well-established hadronic matrix elements and the MCMC-determined MGUT and alpha_GUT, which are outputs of the gauge coupling unification scan, not inputs fitted to proton decay data. The TeV-scale range for eta1 and chi1 is stated as an assumption ('we restrict the Zee-Babu states, namely eta1 and chi1 in the range m_eta1,chi1 in (1,10) TeV'), but this is an input scenario, not a fitted parameter disguised as a prediction; the colored partner masses and the proton decay reach are derived consequences shown in Figs. 4 and 5. Self-citations appear (Refs. [24], [35], [49], [62]), but in each case the cited result is either independently authored, explicitly re-derived in this paper, or a standard group-theoretic identity; no load-bearing step invokes a self-citation as its sole justification, and no uniqueness theorem is imported from the authors' prior work. The absence of a numerical neutrino oscillation fit is a completeness gap in demonstrating full realism, not a circularity: the model's neutrino mass matrix is not fitted to the oscillation observables, so the unrealized claim is under-support, not self-reference. Overall, the central claims do not reduce to their own inputs by construction.

Assumptions & free parameters 6 free parameters · 6 assumptions · 3 invented entities

The model's predictions rest on a large number of freely chosen masses and couplings plus imported GUT results. The most consequential inputs are the TeV-scale prior on the original Zee-Babu scalars and the 50H mass relations, which together force the colored partners to be light. The absence of a neutrino-data fit and the undemonstrated asymptotic-freedom claim are the main gaps.

free parameters (6)
  • mη1 and mχ1 (original Zee-Babu singlet scalars) = 1-10 TeV (benchmark 1.0 TeV)
    Assumed light for collider-visible Zee-Babu phenomenology; this prior drives the conclusion that colored partners must also be light via the 50H mass relations.
  • Threshold masses of the other BSM multiplets = Table 2: mη2=1.09e5 GeV, mη3=1.34e3 GeV, mχ3=2.98e3 GeV, mχ5=4.10e3 GeV, mΦ3=1.12e3 GeV, md4=1.41e3 GeV, mL4=6.18e15…
    Fitted/scan parameters used to achieve gauge coupling unification in the MCMC; the specific mass spectrum is not derived from a principle.
  • Charged fermion fit parameters (yi, ξEi/ξE4, ξDa, CKM-like angles/phases) = Eqs. (3.12)-(3.14), (3.23)
    19 real parameters fitted to 11 observables; the overparameterization makes the chi2=0.1 benchmark uninformative.
  • Yukawa couplings Yh, Yk, Yη, Yχ = Up to ~2.2 within perturbativity bounds of Sec. 4
    Used in the neutrino mass estimate; the max-mass bound assumes they saturate the perturbativity limits.
  • Scalar cubic coupling µ = Assumed ~ mmax (100-400 TeV)
    Appears in the neutrino mass formula; constrained only by perturbative unitarity to a few times the largest loop mass.
  • Scalar potential coefficients for doublet-triplet splitting (fine-tuning) = Fine-tuned; e.g., μ+κv24 ~ TeV^2/MGUT
    Acknowledged fine-tuning in Sec. 5 to keep Φ3 light while Φ2 is superheavy.
assumptions (6)
  • domain assumption The three mass relations among the 50H submultiplets, m^2_χ4 = 3m^2_χ2 - 2m^2_χ3, m^2_χ5 = 2m^2_χ3 - m^2_χ1, m^2_χ6 = (3/2)m^2_χ2 - (1/2)m^2_χ1, hold.
    Eqs. (2.9)-(2.11), taken from Refs [35,49]; these relations are essential for linking light color-neutral and colored scalar masses.
  • domain assumption The vector-like 5F+5bar_F mechanism of Ref [24] corrects the wrong SU(5) fermion mass relations without spoiling other predictions.
    Used in Sec. 3.1-3.2; the analytic mass matrix formulas (3.7)-(3.9) rely on this earlier result.
  • domain assumption Yukawa couplings are neglected in the two-loop gauge beta functions, and threshold corrections are approximated by step functions.
    Sec. 5, Eqs. (5.1)-(5.3); this is a standard simplifying assumption, but it limits the precision of the unification predictions.
  • ad hoc to paper The one-loop Zee-type neutrino mass diagram is negligible because the vector-like lepton L4 and the triplet Φ2 are near the GUT scale.
    Sec. 3.4 after Eq. (3.49); this assumption is needed to attribute neutrino mass entirely to the two-loop diagrams.
  • ad hoc to paper The GUT-scale input masses from Ref [51] (computed with SM running) remain valid in the presence of the new TeV-scale states.
    Sec. 3.2: 'we expect that the modifications needed to correct for this running will not be significant.' No calculation is provided.
  • ad hoc to paper The theory remains perturbative and asymptotically free up to the Planck scale.
    Stated in the abstract and Sec. 1 but not demonstrated anywhere in the text.
invented entities (3)
  • 10H scalar representation of SU(5) independent evidence
    purpose: Contains η1 (the singly-charged Zee-Babu scalar), η2, and η3; η1 and η3 both participate in two-loop neutrino mass generation.
    Mass of η3 is predicted near 1.3 TeV (Table 2), which is within reach of LHC searches for color-triplet scalars; η1 has known collider signatures.
  • 50H scalar representation of SU(5) independent evidence
    purpose: Contains χ1 (the doubly-charged Zee-Babu scalar), the color-triplet χ2, color-sextet χ5, and other states; needed for neutrino mass, fermion mass corrections, and gauge coupling unification.
    Mass of χ5 is predicted near 4.1 TeV (Table 2), testable in collider searches for color-sextet scalars; χ1 has known Zee-Babu signatures such as same-sign dileptons.
  • Vector-like fermion pair 5F + 5bar_F independent evidence
    purpose: Adds the vector-like down-type quark d4 and lepton doublet L4; corrects SU(5) fermion mass relations and helps achieve gauge coupling unification.
    The vector-like down quark is predicted at 1.4-3 TeV (Table 2 and Sec. 3.2), within the reach of LHC searches for vector-like B quarks.

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Cite this review

Pith. "Pith review of Ultraviolet Completion of a Two-loop Neutrino Mass Model." pith.science (2026). https://pith.science/paper/HLESMAAS

@misc{pith2026241214562,
  author       = {Pith},
  title        = {Pith review of: Ultraviolet Completion of a Two-loop Neutrino Mass Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HLESMAAS}},
  note         = {Machine review of arXiv:2412.14562}
}
abstract

The Zee-Babu model is an economical framework for neutrino mass generation as two-loop quantum corrections. In this work, we present a UV completion of this model by embedding it into an $SU(5)$ unified framework. Interestingly, we find that loop-induced contributions to neutrino masses arising from colored scalars are just as important as those from color-neutral ones. These new states, which are required from gauge coupling unification and neutrino oscillation data to have masses below $\mathcal{O}(10^3)$ TeV, may be accessible to future collider experiments. Additionally, the model can be probed in proton decay searches. Our Markov chain Monte Carlo analysis of model parameters shows a high likelihood of observable $p \rightarrow e^+ \pi^0$ decay signal in the first decade of Hyper-Kamiokande operation. The model predicts a vector-like down-type quark at the TeV scale, utilized for realistic fermion mass generation and gauge coupling unification. The model is UV-complete in the sense that it is a unified theory which is realistic and asymptotically free that can be extrapolated to the Planck scale.

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.