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REVIEW 3 major objections 4 minor 66 references

Asymptotic grand unification in SO(10) with one extra dimension

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read In a five-dimensional SO(10) GUT, all gauge couplings flow to a common nonzero UV fixed point, independent of their initial values, while Yukawa couplings can become asymptotically free.

desk verdict First realistic SO(10) asymptotic GUT with a complete Higgs sector; the gauge fixed point is clean at one loop, but the Yukawa result rests on an imposed matching condition. read the letter →

arxiv 2505.08068 v2 pith:BURJPMAC submitted 2025-05-12 hep-ph hep-th

Gao-Xiang Fang , Zhi-Wei Wang , Ye-Ling Zhou This is my paper · ORCID
classification hep-phhep-th
keywords asymptoticunificationSO(10)grandunifiedtheoryextradimensionsafetyfreedomPati-SalamsymmetryinverseseesawKaluza-Kleinstates
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 tries to establish that asymptotic grand unification, where gauge couplings converge to a common nonzero value in the deep ultraviolet rather than meeting at a single scale, is realized in a five-dimensional SO(10) model with a small, phenomenologically motivated Higgs content. Above the compactification scale, Kaluza-Klein towers make every gauge coupling flow to the same ultraviolet fixed point, $\tilde{\alpha}_{10}^{\rm UV}=6\pi/19$, independent of its starting value, because the one-loop coefficient $b_{10}=-19/3$ is negative. The same model splits the top, bottom, and tau Yukawa couplings using a complex 10 and a real 120, explains neutrino mass through inverse seesaw, and shows that the Yukawa couplings can become asymptotically free rather than hit a Landau pole, provided the Pati-Salam Yukawa couplings are exactly unified at the compactification scale. The paper also argues that the widely used 126-dimensional Higgs must be avoided, since it pushes $b_{10}$ positive and destroys the fixed point.

What carries the argument

The load-bearing object is the effective 't Hooft coupling, $\tilde{\alpha}_i(t)=\alpha_i(t)S(t)$, with $S(t)=\mu R$ for $\mu$ above the compactification scale; its $\beta$ function is linear plus quadratic, $2\pi\,d\tilde{\alpha}_i/dt = 2\pi\tilde{\alpha}_i + b_{10}\tilde{\alpha}_i^2$, whose nonzero root gives the ultraviolet fixed point. The paper's chosen particle content yields $b_{10}=-19/3$, so the root is $6\pi/19$ and is attractive. The Yukawa side is carried by the coupled RGEs for $\tilde{\alpha}_{y_{10}}$ and $\tilde{\alpha}_{y_{120}}$, whose phase diagram contains an asymptotically free basin around the Gaussian fixed point; the separatrix lines and the condition that the gauge coupling be sufficiently large determine whether the flow lands there. The exact Yukawa unification condition at the compactification scale, Eq. (4.11), is what converts the Pati-Salam-level couplings into the $\mathrm{SO}(10)$ couplings before the ultraviolet running takes over.

What would settle it

Compute the two-loop or functional renormalization-group $\beta$ function for the five-dimensional 't Hooft gauge coupling including brane-localized kinetic terms: if the attractive fixed point at $6\pi/19$ is shifted, becomes complex, or disappears, the gauge claim is wrong. Alternatively, scan the full parameter space without imposing Eq. (4.11) and check whether any trajectory lands in the asymptotically free Yukawa region; if none does, the Yukawa claim is wrong.

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Extended reading notes

Core claim

The central claim is that the five-dimensional SO(10) theory with bulk fields $\Psi_{16}$, $\overline{\Psi}_{16}$, $\nu_S$ and Higgs multiplets $H_{10}$ (complex), $H_{120}$ (real), $H_{16}$ flows, above the compactification scale, to a regime where the effective 't Hooft gauge couplings $\tilde{\alpha}_4$, $\tilde{\alpha}_{2L}$, $\tilde{\alpha}_{2R}$ all approach $6\pi/19$ as $\mu\to\infty$, irrespective of their initial values; this is asymptotic unification through asymptotic safety of the gauge sector. The mechanism is the power-law running induced by Kaluza-Klein states, encoded in $S(t)=\mu R$, which turns the one-loop $\beta$ function into $2\pi\,d\tilde{\alpha}/dt = 2\pi\tilde{\alpha} + b_{10}\tilde{\alpha}^2$; with $b_{10}=-19/3$ the nonzero root is attractive. For the Yukawa couplings, the paper claims that asymptotic freedom is possible: if the negative gauge contributions dominate the positive Yukawa self-interactions, $\tilde{\alpha}_{y_{10}}$ and $\tilde{\alpha}_{y_{120}}$ flow to the Gaussian fixed point, and the necessary condition is exact unification of the Pati-Salam couplings $y_1$, $y'_1$, $y_{15}$ (and their Kaluza-Klein partners) at the compactification scale via Eq. (4.11). The model additionally accounts for top-bottom-tau mass splitting and, through inverse seesaw, a $0.07\,{\rm eV}$ neutrino mass at the benchmark point.

Load-bearing premise

The load-bearing premise is that the one-loop 't Hooft-coupling fixed point, computed in the continuous-Kaluza-Klein approximation, survives in a non-renormalizable five-dimensional theory once higher-loop and brane-localized corrections are included, and that the exact unification of Yukawa couplings at the compactification scale, Eq. (4.11), holds; if either assumption fails, the central claim collapses.

Editorial extensions

If this is right

  • If the fixed point is real, the five-dimensional $\mathrm{SO}(10)$ gauge sector requires no adjustment of initial couplings: any values at the compactification scale flow to $6\pi/19$ in the ultraviolet.
  • The 126-plet Higgs, standard in many $\mathrm{SO}(10)$ fits, is excluded here because its large positive contribution makes $b_{10}\ge 0$ and eliminates the fixed point; viable Higgs sets must use smaller representations.
  • Yukawa couplings can be made asymptotically free without fine-tuning along separatrices, provided the exact unification condition at the compactification scale holds; otherwise the Yukawa sector hits a Landau pole.
  • The benchmark flow predicts power-law ultraviolet scaling for Yukawa couplings, $\tilde{\alpha}_{y_{10}}\sim\mu^{-19/8}$ and $\tilde{\alpha}_{y_{120}}\sim\mu^{-505/152}$, a distinctive signature of the extra-dimensional mechanism.
  • The same framework gives a concrete neutrino-mass mechanism: inverse seesaw with $\mu_M = 1\,{\rm keV}$ and $y_{16}=10^{-2}$ yields $m_\nu = 0.07\,{\rm eV}$.

Reading between the lines

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

  • Editorial inference: the criterion $b_{10}<0$ acts as a general model-building selection rule for five-dimensional asymptotic GUTs; any Higgs representation with a large positive contribution threatens the fixed point, independent of the specific breaking chain.
  • Editorial inference: the exact Yukawa matching at the compactification scale is stronger than the gauge asymptotic condition; a fully ultraviolet-completed construction would need to derive this matching from the orbifold boundary conditions rather than impose it, and the paper does not provide that derivation.
  • Editorial inference: the same $S(t)$ machinery could be applied to other gauge groups or to six-dimensional setups; the fixed-point value and the sign of the beta coefficient change with the compactification dimension, so the quantitative predictions are specific to one extra dimension.
  • Editorial inference: a systematic search over all Higgs representations that keep $b_{10}<0$ while still fitting flavour data would test whether the asymptotically safe window in five-dimensional $\mathrm{SO}(10)$ is viable beyond the third generation.
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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 / 4 minor

Summary. The manuscript constructs a five-dimensional SO(10) grand unified theory on S1/(Z2×Z2') that is broken by boundary conditions to the Pati-Salam group G422 ≡ SU(4)c×SU(2)L×SU(2)R at the compactification scale MKK, and further to the SM by a spinor Higgs H16 at an intermediate scale MPS. The matter content consists of two bulk spinor fermions per family, a complex H10, a real H120, a chiral H16, and a bulk singlet sterile neutrino. Above MKK, using a continuous-KK approximation S(t)=μR, the gauge couplings are recast into 't Hooft couplings whose one-loop beta functions share a common UV fixed point α̃10^UV = −2π/b10 = 6π/19 for b10 = −19/3, so the gauge couplings are claimed to unify asymptotically for any initial values. The Yukawa sector is analyzed at one loop; the authors impose an 'exact unification condition' at MKK so that PS-level Yukawa couplings match their SO(10) counterparts, and they identify a region of the (y10, y120) plane in which the 't Hooft Yukawa couplings flow to the Gaussian fixed point (asymptotic freedom). A benchmark with MPS = 10^6 GeV, MKK = 10^10 GeV, y16 = 10^−2, and μ_M = 1 keV reproduces the top, bottom, and tau Yukawas and yields mν = 0.07 eV via an inverse seesaw. The paper concludes that the usual H126 Higgs is disfavored because it drives b10 positive and destroys asymptotic safety.

Significance. If the central fixed-point claim survives beyond the one-loop continuous-KK approximation, the paper is a genuine step for asymptotic GUTs: it provides the first realistic SO(10) realization with an economical Higgs sector capable of splitting quark and lepton masses, and it shows a coexistence of asymptotically safe gauge couplings and asymptotically free Yukawa couplings in a single 5D construction. Several strengths deserve credit: the one-loop RGEs are derived in unusual detail in Appendices B–D with explicit Γ-matrix algebra and Feynman rules; the gauge fixed point α̃10^UV = 6π/19 is an output of the ODE with no fitted constants, so the gauge unification claim is not circular; the 5D loop-factor estimate in Eq. (3.15) at least demonstrates that the fixed point is numerically small when measured with the d=5 loop factor; and Eq. (4.17) gives explicit power-law scaling exponents for the UV Yukawa behaviour that are, in principle, falsifiable. The main limitations are that the gauge fixed point is computed within a one-loop continuous-KK approximation, and the Yukawa phenomenology relies on an exact matching condition at MKK that is assumed rather than derived.

major comments (3)
  1. [Section 3, Eq. (3.11) and Eqs. (3.7)–(3.15)] The UV fixed point α̃10^UV = 6π/19 is derived from the one-loop 't Hooft equation built on the continuous-KK approximation S(t)=μR of Eq. (3.8). The paper's response to the possible destabilizing effect of higher-loop and brane-localized corrections is the 5D loop-factor estimate in Eq. (3.15), which checks that α̃* is small in the d=5 loop factor but does not compute the two-loop 't Hooft terms or the radiatively generated brane-localized gauge kinetic operators ci δ(y) F^2 that are generic in orbifold GUTs. Such operators can change the effective 4D couplings and the KK sum S(t), and can therefore shift or destroy the one-loop zero of the beta function. Because the central gauge-unification claim rests entirely on this fixed point, the paper should either provide an estimate or bound for these omitted corrections or explicitly state that the claim holds at one-loop order in the continuous-KK approximation.
  2. [Section 4.1, Eq. (4.11)] The 'exact unification condition' for the Yukawa couplings at MKK is imposed, not derived. The paper correctly observes that without exact matching, ratios of PS Yukawa couplings that share gauge contributions stay constant in the UV, but the condition itself is an additional assumption about the threshold structure at the compactification scale. KK threshold corrections and brane-localized interactions would turn exact matching into a model-dependent relation. This assumption is load-bearing for the asymptotic freedom of the Yukawa sector and for the benchmark values in Table 3. The manuscript should present Eq. (4.11) explicitly as an assumption and discuss how the asymptotically free region changes if the matching is relaxed by a few percent.
  3. [Section 4.3, Table 3 and Fig. 3] The 'predicted' values of yb(MKK), yτ(MKK), yt(MKK), and yν(MKK) are outputs of a scan in which y10(MKK) and y120(MKK) are adjusted so that the fixed EW inputs in Eq. (4.14) are matched through Eqs. (2.11) and (2.12). In other words, the fermion masses are accommodated by the scan, not predicted in an independent sense. The caption of Fig. 3, which states that 'the predicted charged fermion masses ... are calculated at the compactification scale', overstates the status of these quantities. Please clarify that these are fitted outputs of the parameter scan and identify which quantities, if any, are genuinely predicted by the model rather than chosen to reproduce the low-energy data.
minor comments (4)
  1. [Throughout] There are numerous typos, including 'motiviation' (p. 3), 'diagoinalisation' (p. 7), 'paragdim' (p. 19), 'acount' (p. 8), 'dstinct' (p. 21), 'phenpomenologically' (p. 20), 'sclae' (p. 28), 'ajoint' (p. 24), and 'separtirx' (Fig. 2 caption).
  2. [Fig. 2 caption] The sentence 'In the left panel, apart from a non-physical region where α̃y120 becomes negative and the flow appears asymptotically safe, no asymptotically free region is observed' is confusing; please clarify whether the negative-α̃ region is a plot artifact, a mathematical solution that is physically excluded, or a real phase.
  3. [Section 2, Eq. (2.10)] The VEV parametrization lists the same symbol cd'120 twice and cd120 twice; please introduce distinct labels for the doublets originating from h'1 and h15 (e.g., cd'120 and cd120 with an explanatory notation), and move the normalization constraint (4.15) to the first introduction of these coefficients.
  4. [Section 4.3, Eq. (4.17)] The power-law exponents in Eq. (4.17) are stated without derivation; please add one sentence explaining that they follow from linearizing Eq. (4.8) around the gauge fixed point, and check the numerical values against the stated formula.

Circularity Check

1 steps flagged · score 2.0 of 10

Gauge fixed-point derivation is self-contained; only a minor neutrino-mass benchmark output is an accommodation of chosen inputs.

  1. fitted input called prediction [Sec. 4.2 (neutrino-mass discussion), Sec. 4.3/Table 3, with Eq. (2.15)]
    "Thus, to produce the sub-eV neutrino mass,y16 must remain sufficiently large to compete with the suppression fromµM. Specifically, the doubly-suppressed term mD/y16vS ∼ mt/y16MPS, together with a small Majorana mass µM, can yield viable light neutrino mass. For instance, choosing µM = 1 keV gives y16 = 10−2. ... With y16 = 10−2, a light neutrino mass is found to be mν = 0.07 eV."

    The benchmark output mν = 0.07 eV is not an independent prediction: y16 and μ_M are explicitly chosen to 'produce the sub-eV neutrino mass', and the output is then recovered from the same input through Eq. (2.15), mν = μ_M m_D^2/m_S^2 with m_S = y16 v_S. The target observable is used to fix the free parameters, so the quoted neutrino mass is an accommodation by construction rather than a test of the model. This step is peripheral; the central gauge asymptotic-safety claim (Eqs. 3.11–3.13) involves no fitted constants and is not affected.

full rationale

The central gauge claim is self-contained: Eq. (3.11) is a one-loop ODE for the 't Hooft couplings, and its solution (3.12) approaches -2π/b10 = 6π/19 with b10 = -19/3 computed from the stated field content, independently of the constants c_i. No fitted parameter is renamed as a gauge prediction. The Yukawa asymptotic-freedom analysis is similarly an honest phase-space calculation, with the exact unification condition Eq. (4.11) explicitly imposed rather than derived; an imposed assumption is not circular. Self-citations to [10,16–20,28,29] supply context and previous examples but are not load-bearing here. The only notable circularity is the neutrino-mass benchmark, where y16 and μ_M are tuned to produce a sub-eV mass and the resulting mν = 0.07 eV is presented as an output. That is a minor accommodation peripheral to the paper's main result, so the overall circularity score is 2.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central claim adds one model realization; it is built on standard one-loop RG plus three load-bearing assumptions (continuous KK approximation, one-loop UV fixed point, exact Yukawa matching at MKK) and six fitted or selected numerical inputs. No new particles beyond known SO(10) multiplets and a sterile neutrino are introduced.

free parameters (6)
  • y10(MKK) = 0.348 (benchmark); valid range (0.3460, 0.4223)
    Fundamental SO(10) Yukawa coupling input; scanned to satisfy low-energy fermion matching and asymptotic freedom.
  • y120(MKK) = 0.035 (benchmark); valid range (0.0353, 0.6187)
    Fundamental SO(10) Yukawa coupling input; scanned alongside y10 to satisfy matching and stay in the asymptotically free region.
  • MPS = 10^6 GeV
    Pati-Salam breaking scale; chosen at the lower bound from rare meson decay processes.
  • MKK = 10^10 GeV
    Compactification scale; chosen 'without loss of generality' and controls the KK-tower running.
  • muM = 1 keV
    Brane-localized Majorana mass; chosen together with y16 to obtain mnu = 0.07 eV.
  • y16(MKK) = 10^-2
    Neutrino-sector Yukawa coupling; chosen to make inverse seesaw yield a sub-eV neutrino mass.
assumptions (5)
  • standard math One-loop RGE framework for gauge and Yukawa couplings.
    Used throughout for gauge running (Eq. 3.1) and Yukawa running (Eqs. 4.2, 4.3, 4.12).
  • domain assumption Continuous KK approximation S(t)=muR above MKK.
    Eq. (3.8) replaces the discrete KK sum by a continuum; centrally used to obtain the gauge fixed point in Eq. (3.13).
  • domain assumption The one-loop interacting UV fixed point in a 5D non-renormalizable theory is physical.
    Assumes higher-loop and brane-localized corrections do not remove the fixed point; used to claim asymptotic safety in Sec. 3.
  • ad hoc to paper Exact Yukawa unification at MKK.
    Eq. (4.11) imposes matching of all PS Yukawa couplings to their SO(10) values at MKK; necessary for Yukawa asymptotic freedom and not derived from a symmetry.
  • domain assumption Neglect of y16 running and MPS threshold effects.
    Section 4 sets y16 to be small and Section 3 ignores threshold effects; justified as approximations but not quantified.

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Pith. "Pith review of Asymptotic grand unification in SO(10) with one extra dimension." pith.science (2026). https://pith.science/paper/BURJPMAC

@misc{pith2026250508068,
  author       = {Pith},
  title        = {Pith review of: Asymptotic grand unification in SO(10) with one extra dimension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BURJPMAC}},
  note         = {Machine review of arXiv:2505.08068}
}
read the original abstract

Asymptotic grand unification provides an alternative approach to gradually unify gauge couplings in the UV limit, where they reach a non-trivial UV fixed point. Using an economical and realistic particle content setup, we demonstrate that asymptotic grand unification can be achieved in a 5D SO(10) model with one extra dimension. The top, bottom and tau masses are split, and the smallness of the neutrino mass is explained via inverse seesaw. One intermediate scale, the Pati-Salam symmetry breaking scale, is included below the compactification scale. Due to the absence of large-dimensional Higgs representations, gauge couplings exhibit asymptotic safety and are thus asymptotically unified, regardless of their initial values. In contrast, Yukawa couplings can achieve asymptotic freedom if the negative gauge contributions dominate over the positive Yukawa terms, requiring exact unification at the compactification scale. The widely-used 126-dimensional Higgs is not recommended in this 5D asymptotic SO(10) GUT, as it tends to drive the gauge beta function positive, compromising asymptotic safety.

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Works this paper leans on

66 extracted references · 17 canonical work pages

  1. [30]

    M. O. Khojali, A. S. Cornell, A. Deandrea, G. Cacciapaglia, A. Abdalgabar and C. Cot, [arXiv:2210.03596 [hep-ph]]

    arXiv

  2. [1]

    J. C. Pati and A. Salam, Phys. Rev. D 8 (1973), 1240-1251 doi:10.1103/PhysRevD.8.1240

    doi:10.1103/physrevd.8.1240 1973

  3. [2]

    J. C. Pati and A. Salam, Phys. Rev. D 10 (1974), 275-289 [erratum: Phys. Rev. D 11 (1975), 703-703] doi:10.1103/PhysRevD.10.275

    doi:10.1103/physrevd.10.275 1974

  4. [3]

    Georgi and S

    H. Georgi and S. L. Glashow, Phys. Rev. Lett. 32 (1974), 438-441 doi:10.1103/PhysRevLett.32.438 – 29 –

    doi:10.1103/physrevlett.32.438 1974

  5. [4]

    Fritzsch and P

    H. Fritzsch and P. Minkowski, Annals Phys. 93 (1975), 193-266 doi:10.1016/0003-4916(75)90211-0

    doi:10.1016/0003-4916(75)90211-0 1975

  6. [5]

    S. M. Barr, Phys. Lett. B 112 (1982), 219-222 doi:10.1016/0370-2693(82)90966-2

    work page doi:10.1016/0370-2693(82)90966-2 1982

  7. [6]

    J. P. Derendinger, J. E. Kim and D. V. Nanopoulos, Phys. Lett. B 139 (1984), 170-176 doi:10.1016/0370-2693(84)91238-3

    work page doi:10.1016/0370-2693(84)91238-3 1984

  8. [7]

    ”Ultraviolet divergences in quantum theories of gravitation”

    Weinberg, Steven (1979). ”Ultraviolet divergences in quantum theories of gravitation”. In S. W. Hawking; W. Israel (eds.). General Relativity: An Einstein centenary survey. Cambridge University Press. pp. 790–831

    work page 1979

Show all 66 references
  1. [8]

    Bonanno, A

    A. Bonanno, A. Eichhorn, H. Gies, J. M. Pawlowski, R. Percacci, M. Reuter, F. Saueressig and G. P. Vacca, Front. in Phys. 8 (2020), 269 doi:10.3389/fphy.2020.00269 [arXiv:2004.06810 [gr-qc]]

    2020

  2. [9]

    D. F. Litim and F. Sannino, JHEP 12 (2014), 178 doi:10.1007/JHEP12(2014)178 [arXiv:1406.2337 [hep-th]]

    2014 arXiv

  3. [10]

    R. Mann, J. Meffe, F. Sannino, T. Steele, Z. W. Wang and C. Zhang, Phys. Rev. Lett. 119 (2017) no.26, 261802 doi:10.1103/PhysRevLett.119.261802 [arXiv:1707.02942 [hep-ph]]

    2017 arXiv

  4. [11]

    Antipin and F

    O. Antipin and F. Sannino, Phys. Rev. D 97 (2018) no.11, 116007 doi:10.1103/PhysRevD.97.116007 [arXiv:1709.02354 [hep-ph]]

    2018 arXiv

  5. [12]

    Antipin, N

    O. Antipin, N. A. Dondi, F. Sannino, A. E. Thomsen and Z. W. Wang, Phys. Rev. D 98 (2018) no.1, 016003 doi:10.1103/PhysRevD.98.016003 [arXiv:1803.09770 [hep-ph]]

    2018 arXiv

  6. [13]

    G. M. Pelaggi, A. D. Plascencia, A. Salvio, F. Sannino, J. Smirnov and A. Strumia, Phys. Rev. D 97 (2018) no.9, 095013 doi:10.1103/PhysRevD.97.095013 [arXiv:1708.00437 [hep-ph]]

    2018 arXiv

  7. [14]

    Kowalska and E

    K. Kowalska and E. M. Sessolo, JHEP 04 (2018), 027 doi:10.1007/JHEP04(2018)027 [arXiv:1712.06859 [hep-ph]]

    2018 arXiv

  8. [15]

    Bajc and F

    B. Bajc and F. Sannino, JHEP 12 (2016), 141 doi:10.1007/JHEP12(2016)141 [arXiv:1610.09681 [hep-th]]

    2016 arXiv

  9. [16]

    Molinaro, F

    E. Molinaro, F. Sannino and Z. W. Wang, Phys. Rev. D 98 (2018) no.11, 115007 doi:10.1103/PhysRevD.98.115007 [arXiv:1807.03669 [hep-ph]]

    2018 arXiv

  10. [17]

    Sannino, J

    F. Sannino, J. Smirnov and Z. W. Wang, Phys. Rev. D 100 (2019) no.7, 075009 doi:10.1103/PhysRevD.100.075009 [arXiv:1902.05958 [hep-ph]]

    2019 arXiv

  11. [18]

    Z. W. Wang, A. Al Balushi, R. Mann and H. M. Jiang, Phys. Rev. D 99 (2019) no.11, 115017 doi:10.1103/PhysRevD.99.115017 [arXiv:1812.11085 [hep-ph]]

    2019 arXiv

  12. [19]

    Cacciapaglia, A

    G. Cacciapaglia, A. S. Cornell, C. Cot and A. Deandrea, Phys. Rev. D 104 (2021) no.7, 075012 doi:10.1103/PhysRevD.104.075012 [arXiv:2012.14732 [hep-th]]

    2021 arXiv

  13. [20]

    Cacciapaglia, A

    G. Cacciapaglia, A. Deandrea, R. Pasechnik and Z. W. Wang, Phys. Lett. B 852 (2024), 138629 doi:10.1016/j.physletb.2024.138629 [arXiv:2302.11671 [hep-th]]

    2024

  14. [21]

    Cacciapaglia, JHEP 12 (2023) 162

    G. Cacciapaglia, JHEP 12 (2023) 162

    2023

  15. [22]

    Gies, Phys

    H. Gies, Phys. Rev. D 68 (2003), 085015 doi:10.1103/PhysRevD.68.085015 [arXiv:hep-th/0305208 [hep-th]]

    2003 arXiv

  16. [23]

    T. R. Morris, JHEP 01 (2005), 002 doi:10.1088/1126-6708/2005/01/002 [arXiv:hep-ph/0410142 [hep-ph]]. – 30 –

    2005 arXiv

  17. [24]

    Pastor-Guti´ errez and M

    ´A. Pastor-Guti´ errez and M. Yamada, SciPost Phys.15 (2023) no.3, 101 doi:10.21468/SciPostPhys.15.3.101 [arXiv:2205.13250 [hep-th]]

    2023 arXiv

  18. [25]

    Pastor-Guti´ errez and M

    ´A. Pastor-Guti´ errez and M. Yamada, Phys. Rev. D109 (2024) no.7, 7 doi:10.1103/PhysRevD.109.076018 [arXiv:2401.07908 [hep-th]]

    2024 arXiv

  19. [26]

    K. R. Dienes, E. Dudas and T. Gherghetta, Phys. Lett. B 436 (1998), 55-65 doi:10.1016/S0370-2693(98)00977-0 [arXiv:hep-ph/9803466 [hep-ph]]

    1998 arXiv

  20. [27]

    K. R. Dienes, E. Dudas and T. Gherghetta, Phys. Rev. Lett. 91 (2003), 061601 doi:10.1103/PhysRevLett.91.061601 [arXiv:hep-th/0210294 [hep-th]]

    2003 arXiv

  21. [28]

    Cacciapaglia, A

    G. Cacciapaglia, A. S. Cornell, A. Deandrea, W. Isnard, R. Pasechnik, A. Preda and Z. W. Wang, [arXiv:2409.16137 [hep-ph]]

    arXiv

  22. [29]

    Cacciapaglia, A

    G. Cacciapaglia, A. S. Cornell, A. Deandrea, W. Isnard, R. Pasechnik, A. Preda and Z. W. Wang, Eur. Phys. J. C 85 (2025) no.5, 496 doi:10.1140/epjc/s10052-025-14215-9 [arXiv:2501.13118 [hep-ph]]

    2025 arXiv

  23. [31]

    S. L. Glashow, NATO Sci. Ser. B 61 (1980), 687 doi:10.1007/978-1-4684-7197-7 15

    1980 doi

  24. [32]

    Magg and C

    M. Magg and C. Wetterich, Phys. Lett. B 94 (1980), 61-64 doi:10.1016/0370-2693(80)90825-4

    1980 doi

  25. [33]

    R. N. Mohapatra and G. Senjanovic, Phys. Rev. D 23 (1981), 165 doi:10.1103/PhysRevD.23.165

    1981 doi

  26. [34]

    Lazarides, Q

    G. Lazarides, Q. Shafi and C. Wetterich, Nucl. Phys. B 181 (1981), 287-300 doi:10.1016/0550-3213(81)90354-0

    1981 doi

  27. [35]

    NEUTRINO MASS AND UNIFICATION,

    G. Senjanovic, “NEUTRINO MASS AND UNIFICATION,” BNL-29092

  28. [36]

    J. R. Ellis and M. K. Gaillard, Phys. Lett. B 88 (1979), 315-319 doi:10.1016/0370-2693(79)90476-3

    1979 doi

  29. [37]

    Weinberg, Phys

    S. Weinberg, Phys. Rev. Lett. 43 (1979), 1566-1570 doi:10.1103/PhysRevLett.43.1566

    1979 doi

  30. [38]

    Shafi and C

    Q. Shafi and C. Wetterich, Phys. Rev. Lett. 52 (1984), 875 doi:10.1103/PhysRevLett.52.875

    1984 doi

  31. [39]

    Dutta, Y

    B. Dutta, Y. Mimura and R. N. Mohapatra, Phys. Lett. B 603 (2004), 35-45 doi:10.1016/j.physletb.2004.09.076 [arXiv:hep-ph/0406262 [hep-ph]]

    2004 arXiv

  32. [40]

    Dutta, Y

    B. Dutta, Y. Mimura and R. N. Mohapatra, Phys. Rev. D 72 (2005), 075009 doi:10.1103/PhysRevD.72.075009 [arXiv:hep-ph/0507319 [hep-ph]]

    2005 arXiv

  33. [41]

    Dutta, Y

    B. Dutta, Y. Mimura and R. N. Mohapatra, Phys. Rev. Lett. 94 (2005), 091804 doi:10.1103/PhysRevLett.94.091804 [arXiv:hep-ph/0412105 [hep-ph]]

    2005 arXiv

  34. [42]

    Grimus and H

    W. Grimus and H. Kuhbock, Phys. Lett. B 643 (2006), 182-189 doi:10.1016/j.physletb.2006.10.038 [arXiv:hep-ph/0607197 [hep-ph]]

    2006 arXiv

  35. [43]

    Grimus and H

    W. Grimus and H. Kuhbock, Eur. Phys. J. C 51 (2007), 721-729 doi:10.1140/epjc/s10052-007-0324-5 [arXiv:hep-ph/0612132 [hep-ph]]

    2007 arXiv

  36. [44]

    A. S. Joshipura and K. M. Patel, Phys. Rev. D 83 (2011), 095002 doi:10.1103/PhysRevD.83.095002 [arXiv:1102.5148 [hep-ph]]

    2011 arXiv

  37. [45]

    Altarelli and D

    G. Altarelli and D. Meloni, JHEP 08 (2013), 021 doi:10.1007/JHEP08(2013)021 [arXiv:1305.1001 [hep-ph]]. – 31 –

    2013 arXiv

  38. [46]

    Dueck and W

    A. Dueck and W. Rodejohann, JHEP 09 (2013), 024 doi:10.1007/JHEP09(2013)024 [arXiv:1306.4468 [hep-ph]]

    2013 arXiv

  39. [47]

    K. S. Babu, B. Bajc and S. Saad, JHEP 02 (2017), 136 doi:10.1007/JHEP02(2017)136 [arXiv:1612.04329 [hep-ph]]

    2017 arXiv

  40. [48]

    B. Fu, S. F. King, L. Marsili, S. Pascoli, J. Turner and Y. L. Zhou, JHEP 11 (2022), 072 doi:10.1007/JHEP11(2022)072 [arXiv:2209.00021 [hep-ph]]

    2022 arXiv

  41. [49]

    B. Fu, S. F. King, L. Marsili, S. Pascoli, J. Turner and Y. L. Zhou, Phys. Rev. D 109 (2024) no.5, 5 doi:10.1103/PhysRevD.109.055025 [arXiv:2308.05799 [hep-ph]]

    2024 arXiv

  42. [50]

    Meloni, T

    D. Meloni, T. Ohlsson and S. Riad, JHEP 03 (2017), 045 doi:10.1007/JHEP03(2017)045 [arXiv:1612.07973 [hep-ph]]

    2017 arXiv

  43. [51]

    Ohlsson and M

    T. Ohlsson and M. Pernow, JHEP 11 (2018), 028 doi:10.1007/JHEP11(2018)028 [arXiv:1804.04560 [hep-ph]]

    2018 arXiv

  44. [52]

    Hebecker and J

    A. Hebecker and J. March-Russell, Nucl. Phys. B 625 (2002), 128-150 doi:10.1016/S0550-3213(02)00016-0 [arXiv:hep-ph/0107039 [hep-ph]]

    2002 arXiv

  45. [53]

    Alezraa, G

    T. Alezraa, G. Cacciapaglia and A. Deandrea, [arXiv:2503.14397 [hep-ph]]

    arXiv

  46. [54]

    H. c. Han and Z. z. Xing, Nucl. Phys. B 973 (2021), 115609 doi:10.1016/j.nuclphysb.2021.115609 [arXiv:2110.12705 [hep-ph]]

    2021

  47. [55]

    Wyler and L

    D. Wyler and L. Wolfenstein, Nucl. Phys. B 218 (1983), 205-214 doi:10.1016/0550-3213(83)90482-0

    1983 doi

  48. [56]

    R. N. Mohapatra and J. W. F. Valle, Phys. Rev. D 34 (1986), 1642 doi:10.1103/PhysRevD.34.1642

    1986 doi

  49. [57]

    E. K. Akhmedov, M. Lindner, E. Schnapka and J. W. F. Valle, Phys. Lett. B 368 (1996), 270-280 doi:10.1016/0370-2693(95)01504-3 [arXiv:hep-ph/9507275 [hep-ph]]

    1996 arXiv

  50. [58]

    E. K. Akhmedov, M. Lindner, E. Schnapka and J. W. F. Valle, Phys. Rev. D 53 (1996), 2752-2780 doi:10.1103/PhysRevD.53.2752 [arXiv:hep-ph/9509255 [hep-ph]]

    1996 arXiv

  51. [59]

    S. M. Barr, Phys. Rev. Lett. 92 (2004), 101601 doi:10.1103/PhysRevLett.92.101601 [arXiv:hep-ph/0309152 [hep-ph]]

    2004 arXiv

  52. [60]

    Valencia and S

    G. Valencia and S. Willenbrock, Phys. Rev. D 50 (1994), 6843-6848 doi:10.1103/PhysRevD.50.6843 [arXiv:hep-ph/9409201 [hep-ph]]

    1994 arXiv

  53. [61]

    R. R. Volkas, Phys. Rev. D 53 (1996), 2681-2698 doi:10.1103/PhysRevD.53.2681 [arXiv:hep-ph/9507215 [hep-ph]]

    1996 arXiv

  54. [62]

    TASI lectures on extra dimensions and branes,

    C. Csaki, “TASI lectures on extra dimensions and branes,” [arXiv:hep-ph/0404096 [hep-ph]]

    arXiv

  55. [63]

    G. y. Huang and S. Zhou, Phys. Rev. D 103 (2021) no.1, 016010 doi:10.1103/PhysRevD.103.016010 [arXiv:2009.04851 [hep-ph]]

    2021 arXiv

  56. [64]

    Rajpoot, Phys

    S. Rajpoot, Phys. Rev. D 22 (1980), 2244-2254 doi:10.1103/PhysRevD.22.2244

    1980 doi

  57. [65]

    Aspects of symmetry breaking in Grand Unified Theories,

    L. Di Luzio, “Aspects of symmetry breaking in Grand Unified Theories,” [arXiv:1110.3210 [hep-ph]]

    arXiv

  58. [66]

    C. S. Aulakh and A. Girdhar, Int. J. Mod. Phys. A 20 (2005), 865-894 doi:10.1142/S0217751X0502001X [arXiv:hep-ph/0204097 [hep-ph]]. – 32 –

    2005 arXiv

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