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Non-renormalizable grand unification utilizing the leptoquark mechanism of neutrino mass

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Pith's one-line read A non-supersymmetric SU(5) grand unified theory with scalar leptoquark mixing generates radiative Majorana neutrino masses tied to the down-type quark mass matrix and keeps the unification scale above $1.4\times10^{16}$ GeV.

desk verdict Serious new 10S+35S radiative GUT, but the TeV color sextet is an input and the omitted dimension-5 scalar mass operators can move it by fifteen orders of magnitude. read the letter →

arxiv 2502.09333 v1 pith:UJMJZE7C submitted 2025-02-13 hep-ph

Çağlar Doğan This is my paper · ORCID
classification hep-ph
keywords grandunifiedtheorySU(5)leptoquarkmechanismradiativeneutrinomassMajoranaprotondecaycolorsextetscalargaugecouplingunification
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

This paper constructs a grand unified theory without supersymmetry in which the only new ingredients beyond the Georgi-Glashow SU(5) model are two scalar multiplets, a 10 and a 35. Its central claim is that the mixing of the two scalar leptoquarks in these multiplets radiatively generates Majorana masses for neutrinos, and that this mechanism ties the neutrino mass matrix to that of down-type quarks. From that link, together with the requirement of perturbative couplings, the paper derives an upper bound of $2.5\times10^{15}$ GeV on the leptoquark masses and, from the proton decay induced by leptoquark mixing, a lower bound of $2.1\times10^9$ GeV on one of them. The same field content achieves gauge coupling unification at a scale above $1.4\times10^{16}$ GeV, with a distinctive low-energy prediction of a color-sextet, weak isodoublet scalar at 1 or 10 TeV that has no tree-level couplings to Standard Model fermions. The paper matters because it shows that one of the simplest non-supersymmetric SU(5) extensions can simultaneously address neutrino masses, unification, and proton stability.

What carries the argument

The central object is the leptoquark mixing between $S_1^*$, the color-triplet in the $5_H$, and $\widetilde{R}_2$, the $(3,2,1/6)$ in the $10_S$, generated after electroweak symmetry breaking by the operator $\lambda_{5-10}\,10_S\,5_H^*\,5_H^*\,24_H$. This mixing makes the scalar loop diagram that gives neutrinos their Majorana mass, and the resulting relation $M_\nu \propto \lambda_{5-10}\, M_D\,(\text{Yukawa combinations})$ is the paper's main identity; the paper uses it to bound the leptoquark masses and to compute the $B-L$ violating proton decay. A second key ingredient is the split-mass spectrum of the $35_S$ scalars, whose $\eta_3$ multiplet is forced light (1--10 TeV) and whose other multiplets, together with the $10_S$ splits, adjust the running couplings so that unification occurs at a sufficiently high scale. The $\eta_3$ is a color sextet, weak isodoublet with no tree-level fermion couplings, the model's distinctive new-physics signature.

What would settle it

Compute the threshold corrections from the omitted dimension-five mass operators of the $10_S$ and $35_S$, with order-one coefficients and a cutoff at the Planck scale; if they shift $B_{12}$ by more than about 1.25, the GUT scale drops below the proton-decay bound of $5.5\times10^{15}$ GeV and the model fails.

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Extended reading notes

Core claim

The paper establishes that the leptoquark mechanism of radiative Majorana neutrino mass, driven by electroweak mixing of the scalar leptoquarks $S_1^*$ (from the $5_H$) and $\widetilde{R}_2$ (from the $10_S$), produces a neutrino mass matrix proportional to the down-type quark mass matrix $M_D$, up to Yukawa matrices and a loop factor. This proportionality, combined with perturbativity of the couplings, gives a common upper bound $m_{S_1^*},\, m_{\widetilde{R}_2} \le 2.5\times10^{15}$ GeV. The same mixing induces a $B-L$ violating s-channel proton decay, which forces $m_{\widetilde{R}_2} \ge 2.1\times10^9$ GeV if the proton is to survive the Super-Kamiokande bounds. With the $35_S$ scalar's split multiplets, especially the light $\eta_3$ and the heavier $\eta_2$, the gauge couplings unify at two-loop order at a scale $M_{\rm GUT} \ge 1.42\times10^{16}$ GeV, above the lower bound of $5.5\times10^{15}$ GeV needed to avoid rapid proton decay, for all scenarios considered.

Load-bearing premise

The load-bearing premise is that the dimension-five operators that are not written down in the paper, the ones that give mass to the $10_S$ and $35_S$ scalars, are suppressed enough that they do not change the predicted neutrino masses, proton decay rate, or unification scale by more than the paper's claimed margins.

Editorial extensions

If this is right

  • The model predicts a color-sextet, weak isodoublet scalar $\eta_3$ with a mass of 1 or 10 TeV and no tree-level couplings to Standard Model fermions, making it a concrete search target at the LHC and future colliders.
  • The neutrino mass matrix is, to good approximation, proportional to the down-type quark mass matrix, so the model generically predicts a strong hierarchy in neutrino masses tied to $m_b \gg m_s \gg m_d$ and can accommodate both normal and inverted ordering for particular choices of Yukawa couplings.
  • The $B-L$ violating proton decay mode induced by leptoquark mixing sets $m_{\widetilde{R}_2} > 2.1\times10^9$ GeV, so a lighter scalar would make the proton decay too fast; the bound tightens as the mixing grows.
  • Gauge coupling unification occurs at $M_{\rm GUT}$ between $1.42\times10^{16}$ and $4.53\times10^{16}$ GeV in the scenarios tabulated, meaning gauge-boson-mediated proton decay is just beyond the current Super-Kamiokande limit and within reach of Hyper-Kamiokande.
  • The two-loop analysis fixes $\eta_2$ at masses of order $10^9$--$10^{11}$ GeV for the chosen $\eta_3$ masses, giving a well-defined intermediate mass scale.

Reading between the lines

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

  • The proportionality of the neutrino mass matrix to the down-type quark mass matrix suggests that, within this framework, the atmospheric neutrino mass scale is essentially inherited from the bottom quark mass, so a precise measurement of the neutrino mixing pattern could probe the structure of the down-type Yukawa matrices beyond what the paper assumes.
  • The light color-sextet $\eta_3$, despite having no tree-level fermion couplings, could be produced at colliders through its gauge and Higgs couplings, and its decay patterns would provide an indirect check of the leptoquark mixing parameters that govern proton decay.
  • Because the paper omits the dimension-five operators that give mass to the $10_S$ and $35_S$ scalars, a natural extension is to include them and re-run the unification analysis; if those operators are not suppressed at the Planck-scale cutoff, the claimed GUT scale window could shift or close.
  • The same leptoquark mechanism could be transplanted to other grand unified groups, such as SO(10), where the down-quark-neutrino connection might be constrained by additional gauge interactions and could sharpen or alter the upper bound on leptoquark masses.
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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

4 major / 5 minor

Summary. The paper proposes a non-supersymmetric SU(5) grand unified theory whose field content is the Georgi-Glashow model plus scalar multiplets in the 10 and 35 representations, with non-renormalizable operators used to modify fermion masses. Neutrino masses are generated radiatively through the leptoquark mechanism, yielding a claimed relation between the Majorana neutrino mass matrix and the down-type quark mass matrix. From this relation the author derives an upper bound m_{S_1^*}, m_{\widetilde R_2} ≤ 2.5×10^15 GeV, a lower bound m_{\widetilde R_2} ≥ 2.1×10^9 GeV from proton decay, and a unification scale M_GUT ≥ 1.4×10^16 GeV, together with a predicted light color-sextet scalar η3 at 1 or 10 TeV.

Significance. If the claims were correct, the model would be a minimal non-supersymmetric GUT with radiative neutrino masses, a TeV-scale color sextet, and an explicit connection between neutrino and down-quark mass matrices—an attractive and nontrivial package. The paper does contain a two-loop unification analysis with threshold corrections and a transparent analytical derivation of the neutrino mass formula. However, the central quantitative claims are currently not established: there is a numerical inconsistency in the main neutrino-mass formula, an admitted omission of dimension-5 scalar-mass operators that can destroy the mass spectrum used in the unification analysis, and a 'prediction' for the η3 mass that is in fact an input. These issues affect the paper's headline results and require substantive revision.

major comments (4)
  1. [Section 2, Eqs. (14) and (15) with Eq. (4)] The passage from Eq. (14) to Eq. (15) contains a numerical factor error. Using Eq. (4) with M_V = M_GUT gives v24 = M_GUT sqrt(3/(5π α_GUT)). Substituting into the coefficient of Eq. (14), the prefactor 3/(32π²) · (1/2) sqrt(5/6) v24 becomes 3 M_GUT/(64π²) · 1/sqrt(2π α_GUT). Equation (15), however, has 3 M_GUT/(64π²) · sqrt(2π)/sqrt(α_GUT), which is larger by a factor 4π. Consequently the upper bound m_{S_1^*}, m_{\widetilde R_2} ≤ 2.5×10^15 GeV derived from Eq. (15) and any other quantitative use of this formula need to be recomputed. This is a load-bearing discrepancy, not a typographical nuance.
  2. [Section 3, after Eq. (30)] The text explicitly states: 'we have not listed the dimension 5 operators giving mass to the 10S and 35S that must be included for consistency.' This omission is not harmless. With <24H> ~ M_GUT ~ 10^16 GeV and cutoff Λ ≤ M_Pl, a generic operator such as 10S†10S 24H^3/Λ contributes δm² ~ v24^3/Λ ~ 10^29 GeV², which is sixteen orders of magnitude larger than m_η3 = 1–10 TeV and substantial compared to m_η2 ~ 10^9–10^11 GeV. Unless a symmetry forbids these operators or their coefficients are tuned to extreme smallness—neither of which is argued in the paper—the split masses in Eqs. (27), (29), and Tables 2–3 are not stable under the omitted operators. The unification threshold corrections, and therefore the claimed M_GUT ≥ 1.4×10^16 GeV, are not established. The same operators can also generate new baryon-number-violating couplings and additional leptoquark mixings, affecting the proton decay and neutrino mass calculations.
  3. [Abstract and Section 3, Tables 2–3] The abstract states as a prediction that the color sextet η3 has mass 1 TeV (10 TeV), but the body says: 'η3 prefers to be light, that is why we placed the mass of η3, which is the scale of new physics as well, at 1 (10) TeV.' The unification analysis in Tables 2 and 3 treats m_η3 as a free input and solves for m_η2 and M_GUT. For η3 to be a genuine prediction, the author must show that successful unification is possible only if m_η3 is in this range (e.g., that larger m_η3 pushes M_GUT below the proton-decay bound). As written, the headline 'prediction' is an assumption, which is a circularity in the argument.
  4. [Section 2, after Eq. (16)] The paper claims that the neutrino mass matrix from Eq. (14) can reproduce the observed neutrino oscillation parameters for both normal and inverted hierarchies, with couplings in the range 0.1 ≤ Y^ν, Y^D, λ_{5-10} ≤ sqrt(4π), but no explicit numerical fit is presented. The texture of M_ν in Eq. (15) is highly restricted (it is proportional to combinations of M_D and the Yukawa matrices), and it is far from obvious that the required mixing angles and mass-squared differences are attainable. A concrete point in parameter space, listing the Yukawa entries and the resulting PMNS parameters, is necessary to support this central claim.
minor comments (5)
  1. [Abstract] The word 'contrains' should be 'constrains'.
  2. [Section 2] The phrase 'This hiearchy holds' contains a typo: 'hiearchy' should be 'hierarchy'.
  3. [Section 3] The text contains 'parantheses' (should be 'parentheses') and the broken string 'MGU T' in the paragraph after Eq. (32).
  4. [Section 2, Eq. (11)] The use of the same symbol R_{-1/3} for the rotation matrix and for the 'b' decorated mass eigenstates is confusing; a clearer basis notation would help.
  5. [Section 3, Tables 2–3] The tables would be more informative if they included the values of α_GUT and the corresponding proton lifetime for each scenario, rather than only M_GUT and m_η2.

Circularity Check

1 steps flagged · score 6.0 of 10

The advertised color-sextet mass 'prediction' is an input: η3 is placed at 1 TeV (10 TeV) by hand, not derived.

  1. fitted input called prediction [Abstract (first sentence); Section 3, paragraph after Table 1; Conclusion, first paragraph]
    "Abstract: 'A prediction of our model is a color sextet, weak isodoublet whose mass lies at 1 TeV (10 TeV)...' Section 3: 'Among these split multiplets η3 prefers to be light, that is why we placed the mass of η3, which is the scale of new physics as well, at 1 (10) TeV.' Conclusion: 'η3 particle which came out to be light as a result of unification constraints was placed at 1 (10) TeV.'"

    The abstract advertises mη3 = 1 TeV (10 TeV) as a prediction of the model. The model's own analysis, however, takes this value as an input: Section 3 fixes mη3 at 1 (10) TeV and then solves for mη2 and MGUT from the two-loop unification equations (Tables 2 and 3); the Conclusion says the particle 'was placed' at that mass. Nothing in the renormalization-group or neutrino-mass derivation produces the value 1 TeV or 10 TeV; the headline 'prediction' is therefore the assumed parameter by construction. This is not a statistical fit to data, but it is the same logical reduction: the claimed output is the input.

full rationale

Most of the derivation chain is self-contained and non-circular. The Mν ∝ MD formula (Eq. (15)) follows from a one-loop leptoquark-mixing calculation with the Yukawa structures of Eq. (1); it is a genuine model relation, not a restatement of an assumption. The upper bound m(S1*), m(R2) ≤ 2.5×10^15 GeV is a constraint obtained by inverting that formula with measured neutrino masses and perturbativity; the lower bound m(R2) ≥ 2.1×10^9 GeV follows from the proton-lifetime bound with the mixing Lagrangian (Eqs. (5)–(7), (17)). The unification scale is an output of two-loop RGE running subject to chosen scalar masses. There are no load-bearing self-citations by the author; the cited leptoquark-mechanism and mass-constraint results are independent prior work. The one clear circular element is the advertised color-sextet mass 'prediction': Section 3 says the η3 mass was 'placed' at 1 (10) TeV, and the Conclusion repeats 'was placed', while the Abstract calls it a prediction. That is an input presented as an output, hence partial circularity. Separately, the paper admits after Eq. (30) that dimension-5 operators giving mass to 10S and 35S 'must be included for consistency' but are not listed; this could shift the split masses used as unification inputs and thereby MGUT. That is a completeness/correctness risk, not itself a circular reduction, and per the hard rules it is not scored as circularity.

Assumptions & free parameters 7 free parameters · 7 assumptions · 2 invented entities

This ledger shows that the model's quantitative output depends on many unmeasured inputs: the TeV-scale eta3 mass, leptoquark masses, Yukawa matrices, scalar quartic couplings, and splitting parameters are all chosen, scanned, or constrained by data rather than derived. The key relation Mnu proportional to MD is a genuine model consequence, but the headline 'predictions' are mostly inputs or bounds set by already measured neutrino and proton-decay data.

free parameters (7)
  • m_eta3 (color sextet mass) = 1 TeV or 10 TeV
    Set by hand in Section 3 ('we placed the mass of eta3 ... at 1 (10) TeV') and used as an input to solve unification; the abstract labels it a prediction.
  • m_S1* (scalar leptoquark mass) = 2.8e11 to 2.5e15 GeV (scanned)
    Free input controlling the neutrino mass loop and B12; lower bound from proton decay, upper bound from neutrino mass perturbativity.
  • m_R2 tilde (second leptoquark mass) = 2.4e9 to 2.5e15 GeV (scanned)
    Free input; lower bound from B-L violating proton decay, upper bound from the neutrino mass relation.
  • Yukawa matrices Ynu, YD, YDtilde, YDbar, Ynutilde = not specified; entries assumed in [0.1, sqrt(4pi)]
    Arbitrary complex 3x3 matrices; no explicit values are given, so neutrino masses and mixings are not actually fit but assumed reproducible.
  • Leptoquark-Higgs quartic couplings lambda5-10, lambda10-35 = assumed real, order 0.1 to sqrt(4pi)
    Control leptoquark mixing angles and neutrino mass; values are not derived.
  • Scalar mass-splitting parameters M10^2, g10, lambda10, lambdag,10 and M35^2, g35, lambda35, lambdag,35 = not specified
    Adjusted to place split multiplets at desired masses; enough freedom to satisfy unification, so constraints are partly empirical.
  • Cutoff Lambda for non-renormalizable operators = M_GUT << Lambda <= M_Pl (not specified)
    Scale of dimension-5 terms; affects Yukawa terms and omitted mass operators; no value is chosen.
assumptions (7)
  • standard math Two-loop beta functions and matching conditions from refs. [43]-[46] are sufficient for the unification analysis.
    Used in Section 3 and Appendices A and B; treated as established background.
  • domain assumption SM fermions are embedded in SU(5) as 5F and 10F, with no new fermions below the GUT scale.
    Georgi-Glashow embedding; the model deliberately avoids adding fermions.
  • ad hoc to paper The only relevant non-renormalizable corrections are the displayed dimension-5 operators; higher orders are negligible.
    The model is non-renormalizable but no power-counting or operator classification is given; this is assumed throughout.
  • ad hoc to paper Unlisted dimension-5 operators giving mass to 10S and 35S do not affect proton decay, mixing, or unification.
    Stated after Eq. (30): 'we have not listed the dimension 5 operators giving mass to the 10S and 35S that must be included for consistency.'
  • ad hoc to paper Gauge-boson-mediated proton decay dominates over scalar-leptoquark-mediated decays.
    Section 3: 'Here we assume the gauge boson mediated decay processes to be dominant over those mediated by scalar leptoquarks.'
  • ad hoc to paper Yukawa coupling matrices have no strong hierarchy between generations, so the bottom quark dominates the neutrino mass sum.
    Invoked before Eq. (15) to justify the MD contribution dominated by mb; this enables the leptoquark mass upper bound.
  • domain assumption All dimensionless couplings remain perturbative, with entries in [0.1, sqrt(4pi)].
    Used to turn the neutrino mass formula into an upper bound on leptoquark masses.
invented entities (2)
  • 10S scalar multiplet of SU(5) (contains R2 tilde, phi1, phi3)
    purpose: Supplies the second leptoquark R2 tilde that mixes with S1* to generate radiative Majorana neutrino masses.
    New scalar not present in the GG model; no observed signals; constraints are indirect, via proton decay and unification.
  • 35S scalar multiplet of SU(5) (contains eta1-eta4, including color sextet eta3) independent evidence
    purpose: Raises the unification scale through split-multiplet contributions and supplies the light TeV-scale color sextet eta3.
    eta3 is predicted at 1-10 TeV and carries color, so it could be pair-produced at colliders through gauge interactions even without tree-level SM fermion couplings; no direct search is presented.

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Pith. "Pith review of Non-renormalizable grand unification utilizing the leptoquark mechanism of neutrino mass." pith.science (2026). https://pith.science/paper/UJMJZE7C

@misc{pith2026250209333,
  author       = {Pith},
  title        = {Pith review of: Non-renormalizable grand unification utilizing the leptoquark mechanism of neutrino mass},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UJMJZE7C}},
  note         = {Machine review of arXiv:2502.09333}
}
abstract

We analyze a non-supersymmetric, non-renormalizable grand unified theory whose particle content is that of the Georgi-Glashow model augmented only by scalars from the \textbf{10} and \textbf{35} representations. A prediction of our model is a color sextet, weak isodoublet whose mass lies at 1 TeV (10 TeV) and that does not couple to Standard Model fermions at tree level. The leptoquark mechanism through which Majorana neutrinos radiatively acquire their masses relates the neutrino mass matrix to that of the down-type quarks. A consequence of this relation and perturbativity of coupling constants is the upper bound of $2.5 \times 10^{15}$ GeV on the masses of the scalar leptoquarks $S_1^*$ and ${\widetilde R}_{2}$. Electroweak mixing of these leptoquarks induces a B-L violating decay of the proton which indirectly contrains the mass of ${\widetilde R}_{2}$ to be greater than $2.1 \times 10^9$ GeV. We find the grand unification scale to exceed $1.4 \times 10^{16}$ GeV in all scenarios considered.

Figures

Figures reproduced from arXiv: 2502.09333 by the authors.

Figure 1
Figure 1. The Yukawa terms of SM matter fields, LY, is given in Eq.(1). −LY = Y ν ab10ij S 5F ai5 C F bj + Y D ab 10ij F a5F bi5 ∗ Hj + ϵijklm10ij F aY U ab10kl F b5 m H + 1 Λ  Ye D ab 10ik F a5F bi5 ∗ Hj24j Hk + Y D ab10kj F a5F bi5 ∗ Hj24i Hk + Yeν ab10ij S 5F ai5 C F bk24k Hj + H.c.(1) where Y ν ab = −Y ν ba, Y U ab = Y U ba and subscripts a, b = 1, 2, 3 of Yukawa coupling matrices denote generation indices. Y ν can be t… view at source ↗
Figure 1
Figure 1. Feynman diagram for Majorana neutrino mass [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. B-L violating s channel contribution to proton decay induced by the EW mixing [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figures from the paper (2 more)
Figure 3
Figure 3. Figure 3: Unification scenario with the highest GUT scale of [PITH_FULL_IMAGE:figures/full_fig_p017_3.png]
Figure 4
Figure 4. Figure 4: Unification scenario with the lowest GUT scale of [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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    The paper shows that non-renormalizable SU(5) models can host light scalar leptoquarks with suppressed proton decay while explaining neutrino masses and unifying gauge couplings.

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