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arxiv: 2605.21721 · v1 · pith:K7AA3745new · submitted 2026-05-20 · ✦ hep-ph · hep-ex· hep-th

The Higgs-top-Z mass coincidence relation after NNLO matching

E. Torrente-Lujan This is my paper

Pith reviewed 2026-05-22 08:37 UTC · model grok-4.3

classification ✦ hep-ph hep-exhep-th
keywords Higgs boson masstop quark massZ bosonmass coincidenceNNLO matchingrunning couplingselectroweak symmetrythreshold corrections
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The pith

After NNLO matching the exact running Higgs-top-Z relation predicts a Higgs mass of 123.2 GeV instead of the observed value.

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

The paper reconsiders the proposed relation connecting the Higgs, top quark, and Z boson masses. Using latest electroweak inputs the physical pole masses satisfy the geometric ratio to within 1.4 standard deviations. When the couplings are converted to the MS-bar scheme and matched at the top scale with NNLO corrections, the corresponding ratio falls to 0.967 and the exact running boundary condition would require a Higgs mass two GeV lighter than measured. This forces any symmetry explanation to act either on the pole-level threshold quantities or to supply a finite electroweak-scale matching factor of about 3.4 percent.

Core claim

With current PDG and ATLAS-CMS inputs the pole-level ratio rho_Zt equals 1.00362 plus or minus 0.00261, so an exact geometric relation predicts either M_H of 125.426 GeV or M_t of 171.898 GeV. After complete NNLO weak-scale MS-bar matching at mu equals M_t the running ratio becomes 0.96714 plus or minus 0.00361. Enforcing the exact boundary condition lambda equals g_Z y_t over 4 sqrt 2 at the top scale therefore yields a predicted Higgs mass of 123.19 plus or minus 0.20 GeV. Any symmetry explanation must therefore act on pole-level threshold quantities or provide a finite matching factor kappa_th of 1.0340 plus or minus 0.0039.

What carries the argument

The complete NNLO weak-scale MS-bar matching formulae for the ratio of gauge and top-Yukawa couplings to the Higgs quartic coupling, evaluated at the renormalization scale equal to the top mass.

Load-bearing premise

The proposed coincidence is assumed to be intended as an exact theoretical equality at a definite scale or after a specific matching procedure rather than an unmotivated numerical accident.

What would settle it

A future electroweak fit that extracts MS-bar parameters making the NNLO-matched running ratio equal to one within its uncertainty of 0.0036 would eliminate the reported incompatibility.

Figures

Figures reproduced from arXiv: 2605.21721 by E. Torrente-Lujan.

Figure 2
Figure 2. Figure 2: Observed and implied Higgs masses. The geometric pole relation [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: Relative Gaussian likelihoods for the pole-level geometric ratio, the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: The exact MS boundary at µ = Mt fails by about 3.4%. A symmetry acting at the running-coupling level must therefore supply, or predict, a finite threshold factor close to κth = 1.034. about 2 GeV below the observed Higgs mass. Conversely, hold￾ing MH, MW and αs fixed gives M MS pred t = 177.81 ± 0.50 GeV, (18) well above the direct top-mass average. The result is robust: the discrepancy is driven by the la… view at source ↗
read the original abstract

The relation $M_H^2\simeq M_ZM_t$, previously proposed as a non-trivial Higgs mass coincidence, is reconsidered with present electroweak inputs and with a scheme-consistent matching analysis. With the 2025 PDG values for $M_Z$, $M_W$ and $M_H$, and the ATLAS-CMS direct top-mass combination, the pole-level ratio is $\rho_{Zt}=M_ZM_t/M_H^2=1.00362\pm0.00261$. Thus an exact pole-level geometric relation predicts either $M_H=125.426\pm0.120\,\mathrm{GeV}$ or $M_t=171.898\pm0.302\,\mathrm{GeV}$, which is still a $1.4\sigma$ test rather than an exclusion. By contrast, the companion arithmetic relation gives $\rho_{Wt}=(M_W+M_t)/(2M_H)=1.00994\pm0.00159$ and is not a viable exact mass sum rule. We then evaluate the complete NNLO weak-scale $\overline{\mathrm{MS}}$ matching formulae at $\mu=M_t$. In the standard convention one obtains $\widehat\rho_{Zt}(M_t)=\sqrt{g_2^2+g_Y^2}\,y_t/(4\sqrt2\lambda)=0.96714\pm0.00361$. Consequently, the exact running-coupling boundary condition $\lambda=g_Zy_t/(4\sqrt2)$ at the top scale would predict $M_H=123.19\pm0.20\,\mathrm{GeV}$, or equivalently $M_t=177.81\pm0.50\,\mathrm{GeV}$ when $M_H$ is held fixed. This is incompatible with the measured point. A possible symmetry explanation must therefore act on pole-level threshold quantities, or provide a finite matching factor $\kappa_{\rm th}=1.0340\pm0.0039$ at the electroweak scale. We formulate this requirement as a target for custodial/top-Higgs or triality-like symmetry extensions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript reconsiders the proposed coincidence M_H² ≃ M_Z M_t with 2025 PDG inputs for M_Z, M_W, M_H and the ATLAS-CMS top-mass combination. At the pole level it obtains ρ_Zt = M_Z M_t / M_H² = 1.00362 ± 0.00261 (1.4σ from unity) while the arithmetic sum rule ρ_Wt is disfavored. It then evaluates the complete NNLO weak-scale MSbar matching at μ = M_t, yielding hat ρ_Zt(M_t) = 0.96714 ± 0.00361. This implies that the exact running boundary condition λ = g_Z y_t / (4 √2) predicts M_H = 123.19 ± 0.20 GeV, incompatible with data; any symmetry explanation must therefore act on pole-level threshold quantities or supply a finite matching factor κ_th = 1.0340 ± 0.0039. The paper formulates the latter as a target for custodial/top-Higgs or triality-like extensions.

Significance. If the numerical results hold, the work supplies a sharpened phenomenological target for symmetry-based explanations of the Higgs-top-Z mass relation by cleanly separating pole-level and running-coupling versions and quantifying the required threshold correction. Credit is due for the direct use of published NNLO matching expressions together with standard electroweak inputs, which makes the central values reproducible from the stated sources.

major comments (1)
  1. [Abstract and matching section] Abstract and the paragraph following Eq. (the NNLO matching result): the incompatibility statement (M_H = 123.19 ± 0.20 GeV) is presented without an explicit breakdown of the uncertainty budget for terms beyond NNLO or for residual scheme/scale dependence in the matching coefficients. Because this budget directly affects the significance of the 1.4σ pole-level test versus the running-level discrepancy, an explicit estimate (even if small) is load-bearing for the claim that a symmetry must supply κ_th or act at the pole level.
minor comments (2)
  1. [Abstract] Notation: the symbol hat ρ_Zt is introduced without an explicit definition in the abstract; a parenthetical reminder of its meaning (√(g₂² + g_Y²) y_t / (4 √2 λ)) would aid readability.
  2. [Pole-level discussion] The text states that the arithmetic relation ρ_Wt is 'not a viable exact mass sum rule' but does not quantify how far it deviates from unity relative to the geometric one; a one-sentence comparison would clarify the contrast.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive overall assessment and the recommendation for minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract and matching section] Abstract and the paragraph following Eq. (the NNLO matching result): the incompatibility statement (M_H = 123.19 ± 0.20 GeV) is presented without an explicit breakdown of the uncertainty budget for terms beyond NNLO or for residual scheme/scale dependence in the matching coefficients. Because this budget directly affects the significance of the 1.4σ pole-level test versus the running-level discrepancy, an explicit estimate (even if small) is load-bearing for the claim that a symmetry must supply κ_th or act at the pole level.

    Authors: We agree that an explicit uncertainty budget strengthens the presentation. In the revised manuscript we insert a new paragraph immediately after the NNLO matching result. This paragraph estimates the size of O(α^3) corrections by comparing the relative magnitude of the NNLO shift to the NLO shift in the published matching coefficients and by quoting the residual scale dependence obtained when the matching scale is varied by a factor of two around M_t. The combined theoretical uncertainty is found to be ≲ 0.2 % on hat ρ_Zt(M_t), which shifts the predicted M_H by at most 0.25 GeV and leaves the incompatibility with the measured value intact. The abstract is also updated with a brief reference to this estimate. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper performs a numerical update of an externally proposed coincidence relation M_H² ≈ M_Z M_t using 2025 PDG inputs and evaluates standard NNLO MSbar matching formulas at μ = M_t. The derived quantities ρ_Zt, hat rho_Zt, and the implied M_H predictions follow directly from the quoted experimental values and perturbative expressions without any reduction to self-defined parameters, fitted inputs renamed as predictions, or load-bearing self-citations. The requirement for a finite matching factor κ_th or pole-level action is a straightforward logical consequence of the observed mismatch, not a circular re-derivation of the input relation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the previously proposed mass coincidence is a meaningful target for symmetry rather than a numerical accident. No new free parameters are introduced; the masses themselves are taken from external measurements. No new entities are postulated.

axioms (1)
  • domain assumption The coincidence relation M_H² ≈ M_Z M_t is a non-trivial relation worth testing for an exact underlying symmetry rather than an accidental numerical proximity.
    This premise is stated in the opening sentence of the abstract and motivates the entire matching analysis.

pith-pipeline@v0.9.0 · 5908 in / 1530 out tokens · 27502 ms · 2026-05-22T08:37:30.268355+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · 6 internal anchors

  1. [1]

    The Higgs mass coincidence problem: why is the higgs mass $m_H^2=m_Z m_t$?

    E. Torrente-Lujan, The Higgs mass coincidence problem: why is the Higgs massM 2 H =M Z Mt?, Eur. Phys. J. C 74 (2014) 2744, arXiv:1209.0474 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  2. [2]

    Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC

    G. Aad et al. [ATLAS Collaboration], Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B 716 (2012) 1, arXiv:1207.7214 [hep-ex]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  3. [3]

    Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC

    S. Chatrchyan et al. [CMS Collaboration], Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC, Phys. Lett. B 716 (2012) 30, arXiv:1207.7235 [hep-ex]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  4. [4]

    Navas et al

    S. Navas et al. [Particle Data Group], Review of Particle Physics, Phys. Rev. D 110 (2024) 030001, 2025 update, Gauge and Higgs Bosons summary tables

    work page 2024

  5. [5]

    Aad et al

    G. Aad et al. [ATLAS and CMS Collaborations], Com- bination of measurements of the top quark mass from data collected by the ATLAS and CMS experiments at√s=7 and 8 TeV , Phys. Rev. Lett. 132 (2024) 261902, arXiv:2402.08713 [hep-ex]

    work page arXiv 2024

  6. [6]

    Navas et al

    S. Navas et al. [Particle Data Group], Review of Particle Physics, Phys. Rev. D 110 (2024) 030001, 2025 update, Top Quark review

    work page 2024

  7. [7]

    Navas et al

    S. Navas et al. [Particle Data Group], Review of Particle Physics, Phys. Rev. D 110 (2024) 030001, 2025 update, Quantum Chromodynamics review

    work page 2024

  8. [8]

    Aaltonen et al

    T. Aaltonen et al. [CDF Collaboration], High-precision measurement of theWboson mass with the CDF II de- tector, Science 376 (2022) 170

    work page 2022

  9. [9]

    CMS Collaboration, High-precision measurement of the Wboson mass with the CMS experiment, Nature (2026), DOI: 10.1038/s41586-026-10168-5

    work page doi:10.1038/s41586-026-10168-5 2026

  10. [10]

    Higgs mass and vacuum stability in the Standard Model at NNLO

    G. Degrassi, S. Di Vita, J. Elias-Miro, J.R. Espinosa, G.F. Giudice, G. Isidori and A. Strumia, Higgs mass and vac- uum stability in the Standard Model at NNLO, JHEP 08 (2012) 098, arXiv:1205.6497 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  11. [11]

    Investigating the near-criticality of the Higgs boson

    D. Buttazzo, G. Degrassi, P.P. Giardino, G.F. Giudice, F. Sala, A. Salvio and A. Strumia, Investigating the near- criticality of the Higgs boson, JHEP 12 (2013) 089, arXiv:1307.3536 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  12. [12]

    Sikivie, L

    P. Sikivie, L. Susskind, M.B. V oloshin and V .I. Zakharov, Isospin breaking in technicolor models, Nucl. Phys. B 173 (1980) 189

    work page 1980

  13. [13]

    Arkani-Hamed, A.G

    N. Arkani-Hamed, A.G. Cohen, E. Katz and A.E. Nel- son, The Littlest Higgs, JHEP 07 (2002) 034, arXiv:hep- ph/0206021

    work page arXiv 2002

  14. [14]

    Schmaltz and D

    M. Schmaltz and D. Tucker-Smith, Little Higgs review, Ann. Rev. Nucl. Part. Sci. 55 (2005) 229, arXiv:hep- ph/0502182

    work page arXiv 2005

  15. [15]

    Dimension-Six Terms in the Standard Model Lagrangian

    B. Grzadkowski, M. Iskrzynski, M. Misiak and J. Rosiek, Dimension-six terms in the Standard Model Lagrangian, JHEP 10 (2010) 085, arXiv:1008.4884 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  16. [16]

    Jenkins, A.V

    E.E. Jenkins, A.V . Manohar and M. Trott, Renormaliza- tion group evolution of the Standard Model dimension six operators I: formalism and lambda dependence, JHEP 10 (2013) 087, arXiv:1308.2627 [hep-ph]. 4

    work page arXiv 2013