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Neutrino mass generation in asymptotically safe gravity

T0 review · 4 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read In asymptotically safe gravity, the Standard Model plus gravity cannot give neutrinos a mass: the Weinberg operator's coupling is forced to zero at all scales, so new degrees of freedom are required.

desk verdict A clear, significant no-go for the Weinberg operator in asymptotically safe gravity, but the central beta-function calculation is not shown and the printed critical exponent has a sign inconsistency; the seesaw bound is a useful corollary. read the letter →

arxiv 2505.01422 v1 pith:RKN5PKYJ submitted 2025-05-02 hep-ph gr-qchep-th

Gustavo P. de Brito , Astrid Eichhorn , Antonio D. Pereira , Masatoshi Yamada This is my paper · ORCID
classification hep-phgr-qchep-th PACS 04.60.-m14.60.Pq11.10.Hi
keywords asymptoticallysafegravityneutrinomassWeinbergoperatortype-Iseesawpseudo-Diracrenormalizationgroupfunctionalswampland
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

Neutrinos are known to have mass, but the mechanism remains unknown. This paper asks which mechanisms can survive inside asymptotically safe quantum gravity, the proposal that gravity and matter are defined by an interacting fixed point at trans-Planckian energies. Its central answer is a negative one for the most economical option: the Weinberg operator, a dimension-five interaction built from Standard Model fields only, cannot generate neutrino masses there, because its coupling must sit exactly at zero at the Planck scale and stay zero. New degrees of freedom beyond the Standard Model are therefore required. The paper also derives an upper bound on the type-I seesaw scale, about $6\times10^{13}$ GeV for a $10^{-10}$ GeV visible neutrino, and shows that pseudo-Dirac neutrinos remain a viable option.

What carries the argument

The machinery is the functional renormalization group, an exact RG flow equation for the effective action with an infrared cutoff scale $k$, used to compute $\beta$ functions and fixed points for the coupled gravity-matter system. The load-bearing object is the $\beta$ function for the Weinberg coupling, whose gravitational term $\frac{17}{18\pi}G\zeta$ screens the coupling and keeps the critical exponent $\theta_\zeta=-1+\frac{17}{18\pi}G$ negative at the fixed point. The seesaw bound is carried by the gravitational contribution $f_y$ inside the neutrino Yukawa $\beta$ function, which generates the upper bound on $y_\nu$.

What would settle it

Compute the critical exponent $\theta_\zeta$ at the interacting gravitational fixed point in an extended truncation, for example including higher-derivative gravity operators or a momentum-dependent gravity-matter vertex. If $\theta_\zeta>0$ at the fixed point, then $\zeta$ is relevant and a non-zero low-energy Weinberg operator can result, directly refuting the paper's central claim; the same check applies to $f_y$, where a vanishing or sign-flipped gravitational coefficient would remove the seesaw upper bound.

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Extended reading notes

Core claim

The paper's central claim is that the dimensionless coupling $\zeta$ of the Weinberg operator obeys a $\beta$ function that is linear in $\zeta$ and contains a gravitational contribution $\frac{17}{18\pi}G\zeta$, so the only fixed point is $\zeta_*=0$, with critical exponent $\theta_\zeta=-1+\frac{17}{18\pi}G$. At the near-perturbative gravitational fixed point this exponent is negative, so $\zeta$ is irrelevant and cannot move away from zero; since $\zeta=0$ preserves lepton number, the coupling stays zero at every lower scale. As a consequence, the Weinberg operator cannot give neutrinos mass in asymptotic safety. For the type-I seesaw, the same machinery gives $m_R \lesssim y_{\nu,\mathrm{upper}}^2 v_H^2/(2m_2)$, numerically about $6\times10^{13}$ GeV for $m_2=10^{-10}$ GeV, and if the seesaw scale is taken at the Planck scale the visible neutrino mass is bounded by about $10^{-15}$ GeV. Pseudo-Dirac neutrinos, with $m_R \sim 10^{-2} m_D$, are realized by explicit RG trajectories.

Load-bearing premise

The load-bearing premise is that the gravitational contribution to the Weinberg-operator $\beta$ function, $\frac{17}{18\pi}G$, has the size and sign found in the paper's truncation; if an extended truncation turned the critical exponent $\theta_\zeta=-1+\frac{17}{18\pi}G$ positive, the operator could be nonvanishing and the no-go would collapse.

Editorial extensions

If this is right

  • If the central claim is correct, any asymptotically safe theory of the Standard Model plus gravity must include new degrees of freedom beyond the Standard Model to reproduce observed neutrino oscillations.
  • The Weinberg operator is predicted to be exactly zero at all scales, so lepton-number-violating processes generated purely by it, such as neutrinoless double-beta decay mediated by the Weinberg operator, are absent.
  • Type-I seesaw models remain viable only with $m_R$ below the quantum-gravity upper bound, for example $\lesssim6\times10^{13}$ GeV for $m_2=10^{-10}$ GeV; heavier right-handed neutrinos lie in the asymptotic-safety swampland.
  • If one insists on a natural seesaw scale near the Planck mass, the model predicts an upper bound on the visible neutrino mass of about $10^{-15}$ GeV, which future cosmological and laboratory bounds could confront.
  • Pseudo-Dirac neutrinos, with a tiny Majorana mass splitting, sit in the asymptotically safe landscape, so searches for active-sterile oscillations can probe this scenario.

Reading between the lines

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

  • A sharp way to test the paper's central no-go is to compute the same critical exponent in an extended truncation: if higher-derivative gravity or a momentum-dependent coupling changes the coefficient of $G$ so that $\theta_\zeta$ becomes positive at the fixed point, the Weinberg operator becomes relevant and the main conclusion reverses; the paper's own Fig. 6 indicates this would require unusuall
  • The same functional-RG machinery could be applied to other higher-dimensional operators, for example proton-decay operators or dimension-six four-fermion operators, to map out which SMEFT directions asymptotic safety leaves open and which it forces to vanish.
  • If the no-go survives, asymptotic safety becomes empirically distinguishable from other quantum-gravity approaches: a purely Weinberg-operator origin of neutrino mass would count against it, whereas a seesaw origin at the predicted scale would support it.
  • Combining the new upper bound with the standard leptogenesis lower bound, $m_R\gtrsim10^8$-$10^9$ GeV, leaves a finite but narrow window for thermal-leptogenesis seesaw models; the paper notes the leptogenesis range can be accommodated but does not perform this combined constraint analysis.
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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 / 4 minor

Summary. The paper addresses three questions about neutrino mass generation in asymptotically safe quantum gravity. Using the functional RG with an Einstein-Hilbert truncation for gravity plus Standard Model (SM) fields, it claims: (i) the Weinberg operator coupling zeta has only the Gaussian fixed point zeta*=0 and is irrelevant there, so zeta(M_Pl)=0 and remains zero at all lower scales, implying that new degrees of freedom beyond the SM are necessary for neutrino masses; (ii) in the type-I seesaw, an upper bound on the right-handed neutrino mass m_R <= y_nu,upper^2 v_H^2/(2 m_2) exists, numerically about 6e13 GeV for m_2=1e-10 GeV; and (iii) pseudo-Dirac neutrinos can be accommodated. The central tool is the beta function (2) for zeta, which contains a new gravitational term, while the seesaw analysis uses beta functions for SM fermion Yukawas plus gravity with fixed-point inputs from earlier work.

Significance. If correct, the no-go result would be a substantive step: it would show that, within the asymptotic-safety paradigm, the SM plus gravity alone cannot produce neutrino masses and that the seesaw scale is bounded from above. The paper is clearly written, the logical chain from the stated beta functions to the conclusions is internally consistent, and the provision of an ancillary notebook for the seesaw beta functions and a gauge-parameter robustness check (Fig. 6) is commendable. However, the load-bearing gravitational term in Eq. (2) is not derived in the manuscript, and the printed critical exponent and the content of Fig. 6 are in tension with the paper's own conventions and with each other. The secondary bound also inherits truncation-dependent fixed-point inputs whose extended-truncation behavior is deferred to a 'To appear' reference. These issues leave the central claims insufficiently supported as the paper stands.

major comments (4)
  1. [§IV, Eq. (2) and following paragraph] The critical exponent is misstated. With the convention of Sec. III, beta_zeta = (-1 + 17G/(18π) + ...) ζ implies θ_ζ = 1 - 17G/(18π), which at G*=4.6 is about -0.38 (irrelevant), not θ_ζ = -1 + 17G/(18π) ≈ +0.38 (relevant). As printed, the text's conclusion of irrelevance contradicts its own formula. This is more than a typographical slip, because the sign of the gravitational term determines the no-go: a positive coefficient in Eq. (2) counteracts the canonical suppression, whereas the prose states that 'gravity fluctuations also screen the coupling.' The reader cannot determine which sign the actual calculation produced.
  2. [§IV, Eq. (2)] The gravitational contribution 17G/(18π) ζ is asserted as a new result, but no derivation is shown in the text or the appendix, and no ancillary notebook is provided for this beta function (in contrast to the seesaw beta functions). Since the entire no-go result rests on this coefficient, the calculation must be presented in a reproducible way or a detailed reference must be supplied.
  3. [Appendix A, Fig. 6 and footnote 5] The robustness check does not support the claim as presented. The caption states that the gravitational contribution is positive and 'the correct one to make the coupling relevant,' and that relevance is achieved only for very large G, with the lowest values around G≈30. The fixed-point value used in Sec. V is G*=4.6, far below that range. Thus Fig. 6 highlights the sensitivity of the central premise to truncation rather than demonstrating robustness; the deferral in footnote 5 to reference [105] ('To appear') confirms that extended-truncation behavior is not settled.
  4. [§V, Eqs. (7)-(8) and footnote 5] The seesaw upper bound inherits the truncation-dependent fixed-point values G*=4.6 and Λ*=-6.8 from prior work, and the paper itself states in footnote 5 that in extended truncations the physics generating the relevant direction 'may be encoded in other ways [105].' Without that reference, the existence of the bound cannot be assessed beyond the present truncation. Since the quantitative claim in Eq. (10) is a headline result, the derivation of f_y and of the upper bound on y_nu should be shown in the text or the ancillary notebook, and the truncation dependence should be stated as a caveat in the main conclusions.
minor comments (4)
  1. [Abstract and §V, Eq. (10)] The abstract quotes a numerical bound of 10^14 GeV, while Eq. (10) gives approximately 6×10^13 GeV; the order-of-magnitude rounding should be stated consistently.
  2. [§V, Fig. 5 caption] The gauge-parameter robustness estimate for m_R is described as using the fixed-point value of y_nu as the initial condition at the Planck scale and neglecting transplanckian running, whereas the main bound in Eq. (10) uses y_nu(k=mt)<0.45. The relation between these two estimates should be clarified.
  3. [§V, text near Eq. (6)] The normalization of the gauge coupling g_2 is not defined. In the action (A8) the gauge kinetic term is written as 1/(4 g_2^2) F^2, while the beta function (2) uses a conventional 3/(16π^2) g_2^2 term; specifying the convention would help the reader reproduce the non-gravitational part.
  4. [§VII, Conclusions] The phrase 'first unequivocal evidence' is stronger than warranted given the truncation dependencies identified above; a more cautious formulation would better match the evidence presented.

Circularity Check

0 steps flagged · score 2.0 of 10

No constructional circularity: the no-go and seesaw bound follow from stated beta functions, with self-cited truncation results acting as independent inputs rather than fitted outputs.

full rationale

The central claim that the Weinberg operator vanishes in asymptotically safe SM+gravity is a direct consequence of Eq. (2): because beta_zeta is linear in zeta, the only fixed point is zeta*=0, and the stated sign/magnitude of the coefficient determines its (ir)relevance. This is a derivation from an assumed RG equation, not a reduction of the conclusion to fitted neutrino data. The gravitational term in Eq. (2) is asserted as a new computation; its derivation is omitted from the manuscript, and there is an apparent sign inconsistency between the critical exponent quoted after Eq. (2) and the definition of critical exponents in Sec. III, as well as tension with Fig. 6 and footnote 5. These are correctness and robustness concerns, not circularity. The seesaw bound Eq. (9) is an algebraic rearrangement of the seesaw relation m2 = m_D^2/m_R with m_D = y_nu v_H/sqrt(2); no fitted parameter is renamed as a prediction. The numerical inputs G*=4.6, Lambda*=-6.8 and f_y are taken from prior work by the same group (Refs. [51], [67], [101]), but those are parameter-free truncation results with stated assumptions, not quantities fitted to the neutrino masses predicted here; under the criterion that independent, assumption-stated prior calculations constitute real evidence, these self-citations are not load-bearing circularity. Footnote 5 explicitly defers extended-truncation behavior to Ref. [105] ('To appear'), which is a limitation statement rather than an imported uniqueness theorem. Overall, the paper's derivation chain is self-contained conditional on its truncation inputs; the main risk is truncation and systematic uncertainty, not circularity.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

No data are fitted in this paper. The quantitative outputs inherit scheme-dependent fixed-point inputs (G*=4.6, Lambda*=-6.8) and truncation choices. The seesaw bound is a rearrangement of the seesaw formula with a previously computed Yukawa bound, so it adds no new free parameters beyond those inputs.

free parameters (4)
  • Gravitational fixed-point value G* = 4.6
    Input from reference [101] used in Eq. (8), Fig. 1 and Eq. (10). It sets the size of the Yukawa bound and hence the seesaw bound; it is scheme-dependent and is varied by about 33% in Fig. 1.
  • Cosmological fixed-point value Lambda* = -6.8
    Input from reference [101]; enters f_y in Eq. (7) and thus the neutrino Yukawa bound.
  • Light neutrino mass m_2 (example value) = 10^-10 GeV
    Chosen as an example in Eq. (10) and Fig. 2 because only an upper bound on neutrino masses is known; the seesaw bound scales as 1/m_2.
  • IR neutrino Yukawa y_nu and heavy mass m_R (example trajectory) = 0.1 and 2.9x10^12 GeV at k=173 GeV
    Concrete example trajectory for Fig. 3; not part of the general bound.
assumptions (7)
  • domain assumption Asymptotic safety: gravity and matter possess an interacting UV fixed point, and RG trajectories emanating from it describe nature.
    Invoked in Sec. II and used for all three results; unproven and outside current consensus.
  • domain assumption The truncation of the effective action to Einstein-Hilbert gravity, SM gauge/Yukawa/Higgs operators, and the specified neutrino sector is adequate for the conclusions.
    The paper itself labels this a truncation and tests only gauge-parameter sensitivity in Appendix 3.
  • domain assumption The functional RG flow with Litim regulator and Landau-DeWitt gauge approximates the exact flow for the operators considered.
    Stated in Sec. III and Appendix 1; regulator and gauge dependence is used as an error estimate.
  • domain assumption For the Weinberg-operator no-go, no degrees of freedom beyond SM plus gravity are present.
    This is the scenario being tested in Sec. IV, not a result.
  • standard math Standard type-I seesaw mass matrix and the light-mass formula m_2 approx m_D^2 / m_R for m_R >> m_D.
    Used in Sec. V, Eqs. (3)-(4) and Eq. (9).
  • domain assumption One-generation simplification and neglect of neutrino mixing are adequate for the bound.
    Stated in Sec. V; the quoted numerical value is for this simplified case.
  • domain assumption The previously computed Yukawa upper bound y_nu,upper and the gravitational coefficient f_y from references [51,67] are reliable inputs.
    The seesaw bound is built directly on these inputs; they are not re-derived in this paper.

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Cite this review

Pith. "Pith review of Neutrino mass generation in asymptotically safe gravity." pith.science (2026). https://pith.science/paper/RKN5PKYJ

@misc{pith2026250501422,
  author       = {Pith},
  title        = {Pith review of: Neutrino mass generation in asymptotically safe gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RKN5PKYJ}},
  note         = {Machine review of arXiv:2505.01422}
}
abstract

There exist several distinct phenomenological models to generate neutrino masses. We explore, which of these models can consistently be embedded in a quantum theory of gravity and matter. We proceed by invoking a minimal number of degrees of freedom beyond the Standard Model. Thus, we first investigate whether the Weinberg operator, a dimension-five-operator that generates neutrino masses without requiring degrees of freedom beyond the Standard Model, can arise in asymptotically safe quantum gravity. We find a negative answer with far-reaching consequences: new degrees of freedom beyond gravity and the Standard Model are necessary to give neutrinos a mass in the asymptotic-safety paradigm. Second, we explore whether the type-I Seesaw mechanism is viable and discover an upper bound on the Seesaw scale. The bound depends on the mass of the visible neutrino. We find a numerical value of $10^{14}\, \rm GeV$ for this bound when neglecting neutrino mixing for a visible mass of $10^{-10}\, \rm GeV$. Conversely, for the most ``natural" value of the Seesaw scale in a quantum-gravity setting, which is the Planck scale, we predict an upper bound for the neutrino mass of the visible neutrino of approximately $10^{-15}\, \rm GeV$. Third, we explore whether neutrinos could also be Pseudo-Dirac-neutrinos in asymptotic safety and find that this possibility can be accommodated.

Figures

Figures reproduced from arXiv: 2505.01422 by the authors.

Figure 1
Figure 1. FIG. 1. We show the upper bound on [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. We show the dependence of the upper bound for [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. We show the RG flow of the SM couplings and the [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Gauge-parameter ( [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. We show the gravitational contribution to the critical exponent at the free fixed point in [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

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Forward citations

Cited by 3 Pith papers

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

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