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High Energy Physics - Phenomenology · arXiv:1210.2538 v8 · submitted 9 Oct 2012 · 25 references

Bare Higgs mass at Planck scale

Yuta Hamada, Hikaru Kawai, Kin-ya Oda

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The paper says

“We compute one- and two-loop quadratic divergent contributions to the bare Higgs mass in terms of the bare couplings in the Standard Model.”

more of the abstract

We approximate the bare couplings, defined at the ultraviolet cutoff scale, by the MS-bar ones at the same scale, which are evaluated by the two-loop renormalization group equations for the Higgs mass around 126GeV in the Standard Model. We obtain the cutoff scale dependence of the bare Higgs mass, and examine where it becomes zero. We find that when we take the current central value for the top quark pole mass, 173GeV, the bare Higgs mass vanishes if the cutoff is about 10^23GeV. With a 1.3 sigma smaller mass, 170GeV, the scale can be of the order of the Planck scale.

from the abstract

Pith says

Machine review · deepseek-v4.1-flash · 12 Nov 2026 · what was read was not recorded · not checked by a person

Reviewed in 2026 with later literature in view.

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Holds up, with conditions. A real calculation with a real number, but the quoted precision in Eqs. (29)-(30) is tighter than the paper's own Appendix A scheme ambiguity allows. This line is wrong

1 major objection · 5 minor

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The paper says, at §4, Eq. (29), and Appendix A

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Pith says: Objection 1 of 1 · major

The quoted theory error ±0.4 in log10 Λ in Eq. (29) is smaller than the scheme ambiguity that the authors themselves derive in Appendix A, where the shift of the crossing scale is estimated as e^{δt} ≲ 10, i.e. roughly ±1 in log10 Λ.

more of objection 1

The dominant systematic identified by the authors is therefore not propagated into the headline precision. The qualitative claim survives (a factor of 10 in Λ still leaves the crossing 'about 10^23 GeV' and still leaves the m_t ≈ 170 GeV case 'of the order of the Planck scale'), but the error bar in Eq. (29) should either be enlarged to include the scheme ambiguity or the two systematics should be stated separately and explicitly. The same question should be answered for the ±0.02 'th' in Eq. (30): the appendix argues that the correction at fixed Λ is as small as the two-loop term, but the reader needs a clear statement of what the quoted 'th' error does and does not include.

5 minor objections

The paper says, at §2.2, Eq. (17)

Pith says: minor objection 1 of 5

The stated logarithm ln(26/33) is negative, which would make I2 negative although the integrand of Eq. (10) is positive; the numerical coefficient 0.005 also does not follow from this logarithm. Please check the sign and the numerical value, and state explicitly how the proper-time regularization of Eq. (16) produces the quoted result.

The paper says, at §4, Eq. (28)

Pith says: minor objection 2 of 5

The second term appears to multiply m_B,2-loop (which already includes I2 according to Eq. (15)) by I2/I1; please check whether the intended expression is m_B,2-loop/I1. Clarifying this would remove a dimensional ambiguity in the definition of the plotted quantity.

show 3 more

The paper says, at Abstract

Pith says: minor objection 3 of 5

The abstract says 'around 126 GeV' while the paper uses m_H = 125.7 ± 0.6 GeV (Eq. (24)); for consistency, use 125.7 GeV or 'around 125–126 GeV'.

The paper says, at §2.2, footnote 8

Pith says: minor objection 4 of 5

The gauge-invariance argument for the cancellation of the J2 coefficients is stated to hold 'under the assumption' of a gauge-invariant regularization behind the calculation. This caveat deserves a sentence in the main text, not only in a footnote, because it qualifies the gauge-invariant interpretation of Eq. (15).

The paper says, at §2.3

Pith says: minor objection 5 of 5

For the Λ ≈ 10^23 GeV case the graviton contributions of order Λ^4/M_Pl^2 are potentially important, and the paper acknowledges this in the discussion. A short quantitative estimate of their size at 10^23 GeV, even if only order-of-magnitude, would help the reader judge how seriously the 10^23 GeV crossing should be taken.

What would settle it

Compute the finite matching coefficients f_ijk in Eq (37) for a specified cutoff scheme (the sharp-Λ cutoff of Eq 18, or Pauli-Villars) in at least the top-Yukawa and quartic sectors, then evaluate δt in Eq (39) and check whether e^{δt} is near 1 or near 10. If e^{δt} is of order 10, Eq (29)'s ±0.4 understates the true uncertainty by roughly a factor of 2.5 in log10 Λ, and the headline should be quoted as 10^22–10^24 GeV rather than 10^23.5 ± 0.4.

from the skeptic's reading

How the review went

4 readings by deepseek-v4.1-flash · 12 Nov 2026

  1. Reader holds up, with conditions confidence moderate
  2. Skeptic agrees 1 min 46 s · 15,054 tokens read, 5,687 written
  3. Editor wrote the verdict above 19 s · 15,983 tokens read, 1,012 written
  4. Referee major revision: 1 major objection, 5 minor 2 min 54 s · 15,315 tokens read, 9,990 written

Not run on this paper: the simulated authors' defence.

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The review in full

The review in brief, machine-written

The thing worth knowing: this paper computes the one- and two-loop quadratic divergence of the SM bare Higgs mass in terms of bare couplings, runs the couplings up with two-loop RGEs, and asks where the bare mass vanishes. The answer is ~1023 GeV for the central top pole mass, or of order the Planck scale for mt about 1.3σ lower (~170 GeV). That numerical statement is the new content, and it is a clean, honest piece of work. The one-loop coefficient is Veltman's; the two-loop coefficient had errors in the earlier version, was corrected after Jones's comment (Note added), and now agrees with Al-sarhi-Jack-Jones. So the two-loop formula is not new — the evaluation is. The paper is also candid in the body: graviton loops matter above M_Pl (Sec. 2.3), and no parameter set within 2σ makes both λ and mB2 vanish at M_Pl. Good, that is the right way to write this up. Where I would push back is on the error bars. Appendix A derives a scheme ambiguity in the crossing scale of eδt ≲ 10, i.e. roughly ±1 in log10 Λ. Eq. (29) then quotes ±0.4 'th' and Eq. (30) quotes ±0.02 'th'. The dominant systematic the authors themselves identify is not carried into the headline numbers. Related: the claim that the two-loop term is negligible (mB,2−loop ≈ -0.005 M_Pl2/16π2) rests on I2/I1 = 0.005 from the specific regularization in Eq. (17), so it is not an independent check. Neither point kills the qualitative claim — a factor of 10 in Λ still leaves 'about 1023' and still leaves the 170 GeV case 'of order the Planck scale' — but the quoted precision is overstated. The abstract also reads cleaner than the body: the classifier found no stated limitations in it, while the body has two substantive ones. Minor, but worth fixing. Who should read it: anyone working on SM vacuum stability, Higgs inflation, or the question of how far the SM extends. The crossing-scale number is a useful reference even if you do not buy the interpretation. I would send this to referees — the calculation is checkable, the inputs are standard measurements, and the authors engage honestly with the literature. My recommendation: accept for review, with a request that the scheme ambiguity be propagated into Eqs. (29)-(30) or the 'th' errors be explicitly flagged as excluding it.

The strongest objection

The central numbers rest on approximating the bare couplings at the cutoff by MS-bar couplings at the same scale (Eq 37, Appendix A). Appendix A estimates the induced shift in the crossing scale as e^{δt} ≲ 10, i.e., roughly ±1 in log10 Λ. Yet Eq (29) quotes ±0.4 'th' and Eq (30) quotes ±0.02 'th'. The dominant systematic identified by the authors themselves is therefore not propagated into the headline numbers. A second, related soft spot: the claim that the two-loop term is negligible (m_B,2-loop ≈ -0.005 M_Pl^2/16π^2) is established using the ratio I2/I1 = 0.005 from the specific regularization of Eq (17); because that ratio is itself scheme-dependent, the smallness check is not an independent verification. Neither point overturns the qualitative claim — a factor of 10 in Λ still leaves the crossing 'about 10^23' and still leaves the m_t ≈ 170 GeV case 'of the order of the Planck scale' — but together they mean the quoted precision is overstated. The authors acknowledge the scheme ambiguity in Appendix A, so this is not a new finding; the unresolved part is that the headline numbers in Sec 4 do not carry it. The Note added (corrected Eq 15, cross-checked against Ref [21]) is a resolved correctness issue, not a live concern.

The paper's core computation is sound: the one-loop coefficient is the standard Veltman result, the two-loop coefficient (Eq 15) was corrected after Ref [20] and now agrees with Ref [21], and the qualitative conclusion — that the quadratic-divergence coefficient crosses zero near 10^23 GeV for the central top mass, or near the Planck scale for m_t ≈ 170 GeV — is robust to the dominant systematics. The reader correctly identifies the MS-bar approximation (Appendix A, Eq 37) as the weakest link, and correctly notes that the factor-of-10 ambiguity in Λ exceeds the ±0.4 quoted in Eq 29. I agree with this identification and with the CONDITIONAL verdict. My only sharpening is to state the concern as an internal inconsistency between the appendix's own error estimate and the headline error bars, rather than as a new discovery: the authors acknowledge the scheme ambiguity, but the numbers in Sec 4 do not reflect it. The two-loop smallness check shares the same scheme dependence, which weakens it as an independent verification but does not change the conclusion because the two-loop term is small in any reasonable scheme. The paper is otherwise candid about its limitations (graviton loops above M_Pl, no parameter set within 2σ making both λ and m_B^2 vanish at M_Pl, the Note added correction). No concern here rises to the level of undermining the central claim; the appropriate verdict remains CONDITIONAL, with the recommendation that the quoted precision be widened to match the paper's own error analysis.

The referee's report

If it holds, the result connects low-energy Standard Model parameters to the Planck scale and suggests either supersymmetry restoration or an almost flat Higgs potential suitable for inflation. The explicit two-loop calculation is a genuine technical contribution, and the correction and cross-check recorded in the Note added are transparent and strengthen confidence in Eq. (15). The prediction is falsifiable: the crossing scale depends on the top pole mass, α_s, and m_H in a definite way.

Pith's reading of the paper (a machine's summary, not the authors')

The Higgs boson's mass is famously sensitive to quantum corrections. In the Standard Model, loop diagrams involving the Higgs itself, the top quark, and the gauge bosons add a piece to the Higgs mass that grows with the square of the energy cutoff, the scale at which the theory is assumed to break down. This is the hierarchy problem: to keep the physical Higgs mass at 125 GeV, this quadratically divergent piece must be cancelled by a bare mass parameter tuned to many decimal places. This paper does not try to solve that tuning problem. Instead it asks a narrower question: if we compute the bare Higgs mass as a function of the cutoff, at what cutoff does it pass through zero? The authors compute the one-loop and two-loop coefficients of the quadratic divergence in terms of the Standard Model couplings, then run those couplings up to high scales using two-loop renormalization group equations. They find that the bare mass decreases monotonically with the cutoff and crosses zero at about 10^23 GeV if the top quark mass is at its measured central value of 173.3 GeV. If the top mass is about 1.3 sigma lower, around 170 GeV, the crossing happens near the Planck scale, 10^19 GeV. The result is a numerical coincidence, not a solution to the hierarchy problem, and the authors are careful to say so. They also note that the precise crossing scale is uncertain by up to a factor of ten because the bare couplings are approximated by MS-bar couplings, and that graviton loops matter once the cutoff exceeds the Planck scale.

Core claim. The Standard Model bare Higgs mass, computed to one and two loops in terms of bare couplings and evaluated with two-loop RGE running, vanishes at a UV cutoff of about 10^23 GeV for the central top pole mass 173.3 GeV, and at the Planck scale for a top mass about 1.3 sigma lower, near 170 GeV (Eqs 29, 30).

The premise it rests on. The bare couplings at the cutoff are approximated by the MS-bar couplings at the same scale (Appendix A, Eq 37). The authors estimate this introduces an ambiguity of up to a factor of ten (e^delta t less than about 10) in the scale where the bare mass vanishes, which is larger than the plus or minus 0.4 quoted in Eq 29 and directly affects the headline number.

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The paper

Abstract

We compute one- and two-loop quadratic divergent contributions to the bare Higgs mass in terms of the bare couplings in the Standard Model. We approximate the bare couplings, defined at the ultraviolet cutoff scale, by the MS-bar ones at the same scale, which are evaluated by the two-loop renormalization group equations for the Higgs mass around 126GeV in the Standard Model. We obtain the cutoff scale dependence of the bare Higgs mass, and examine where it becomes zero. We find that when we take the current central value for the top quark pole mass, 173GeV, the bare Higgs mass vanishes if the cutoff is about 10^23GeV. With a 1.3 sigma smaller mass, 170GeV, the scale can be of the order of the Planck scale.

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. No sentence in this record was written by a model.

abstract compiled · 2026-09-23

The authors' main claim, as the classifier picked it out 0.82

“With a 1.3 sigma smaller mass, 170GeV, the scale can be of the order of the Planck scale.”

How the main claim is stated: hedged possibility 0.87

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.35 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.08 · mentions code or data availability 0.02

The abstract, sentence by sentence

This table labels the abstract only (arXiv listing abstract v8). The number is the probability the classifier gave the label.

SentenceRoleThe authors' own workStates the main conclusionStrength
We compute one- and two-loop quadratic divergent contributions to the bare Higgs mass in terms of the bare couplings in the Standard Model. contribution 0.76 0.95 0.05 asserted 0.99
We approximate the bare couplings, defined at the ultraviolet cutoff scale, by the MS-bar ones at the same scale, which are evaluated by the two-loop renormalization group equations for the Higgs mass around 126GeV in the Standard Model. method 1.00 0.90 0.04 asserted 0.98
We obtain the cutoff scale dependence of the bare Higgs mass, and examine where it becomes zero. contribution 0.68 0.94 0.06 asserted 0.83
We find that when we take the current central value for the top quark pole mass, 173GeV, the bare Higgs mass vanishes if the cutoff is about 10^{23}GeV. result 1.00 0.95 0.41 asserted 1.00
With a 1.3 sigma smaller mass, 170GeV, the scale can be of the order of the Planck scale. result 0.99 0.89 0.43 hedged possibility 0.87

What kind of paper the classifier reports this is: theoretical result 0.97 · evidence from simulation or computation 0.63

What was examined, and what was not

Sources read

  • arXiv listing abstract v8

Not examined

  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers

Levels run

  • P abstract

Not run

  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

How the labels were made

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,352 input tokens, $0.0002 reported.

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. This record as JSON

Figures (4)
Figure 1 of the paper
Figure 1. Nonvanishing two-loop Feynman diagrams. Arrows are omitted. The dashed,
Figure 2 of the paper
Figure 2. Left: MS running of the quartic coupling λ. The band corresponds to the 1σ deviation m pole t = 173.3±2.8 GeV. Right: The scale µmin at which λ(µ) takes its minimum value, as a function of m pole t . In both panels, low energy inputs are given by the central values αs(mZ) = 0.1184 and mH = 125.7 GeV. two-loop corrections [5] λ(m pole t ) = 0.12577+0.00205 mH GeV−125 −0.00004 m pole t GeV−173.15 ±0.00140th, (23) where mH is the observed Higgs mass which we read off from Eq. (1) as mH = 125.7 ± 0.6 GeV. (24) The MS top Yukawa coupling at the scale m pole t is given by taking into…
Figure 3 of the paper
Figure 3. Left: The bare Higgs mass m2 B in units of Λ2/16π 2 vs the UV cutoff scale Λ. The blue (narrower) and pink (wider) bands represent the one and two sigma deviations of m pole t , respectively. Right: The UV cutoff scale at which the bare mass m2 B becomes zero as a function of m pole t . The solid (dashed) line corresponds to the scale where m2 B (m2 B, 1-loop) becomes zero. In both panels, we have taken the central values αs(mZ) = 0.1184 and mH = 125.7 GeV. asymptotically free. At the intermediate scale, λ can become negative but it is shown that a metastability condition can be met…
Figure 4 of the paper
Figure 4. The blue solid (dashed) line corresponds to the one-plus-two-loop (one-loop)

Figures are the authors', reproduced from the paper.

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  2. Weak Gravity Conjecture, Multiple Point Principle and the Standard Model Landscape hep-th · Jul 2017
  3. Non-minimal coupling in Higgs-Yukawa model with asymptotically safe gravity hep-th · Oct 2015
  4. Eternal Higgs inflation and cosmological constant problem hep-ph · Jan 2015
  5. Minimal Higgs inflation hep-ph · Aug 2013
  6. Simultaneous interpretation of $K$ and $B$ anomalies in terms of chiral-flavorful vectors hep-ph · Jun 2018
  7. Quantum gravity fluctuations flatten the Planck-scale Higgs potential hep-th · Dec 2017
  8. Asymptotic safety of higher derivative quantum gravity non-minimally coupled with a matter system hep-th · Mar 2017
  9. Meaning of the field dependence of the renormalization scale in Higgs inflation hep-th · Oct 2016
  10. Constraints on inert dark matter from metastability of the electroweak vacuum hep-ph · Mar 2015
  11. Higgs inflation from Standard Model criticality hep-ph · Aug 2014
  12. Predictions on mass of Higgs portal scalar dark matter from Higgs inflation and flat potential hep-ph · Apr 2014
  13. Higgs inflation still alive hep-ph · Mar 2014
  14. An O(750) GeV Resonance and Inflation hep-ph · Dec 2015
  15. On the difference between the pole and the MSbar masses of the top quark at the electroweak scale hep-ph · Dec 2012
  16. Cosmological constant and vacuum energy: old and new ideas gr-qc · Jun 2013
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  • Let us explain our treatment of the bare mass by taking a simple example of the $ϕ^4$ theory with the bare Lagrangian: [equation] In the mass independent renormalization scheme,[ See, e.g., the introduction and the subsequent section of Ref. [ref] for a recent review of the discussion explained in this paragraph.

    Remark on the subtractive renormalization of quadratically divergent scalar mass · Bare mass in $ϕ^4$ theory · 0.93

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