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Interpretation of 95 GeV Excess within the Georgi-Machacek Model in Light of Positive Definiteness Constraints

T0 review · 2 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read The Georgi-Machacek model can still explain the 95 GeV excesses once vacuum stability is checked at large field values, where the positive-definiteness conditions actually widen the allowed parameter space.

desk verdict The GM-model interpretation of the 95 GeV excess is extended to the authors' positive-definiteness criterion, but the benchmark points violate the small-ρ5, σ4 approximation, so the claimed expansion of parameter space is not yet demonstrated. read the letter →

arxiv 2502.06444 v2 pith:IVQN25T4 submitted 2025-02-10 hep-ph

Xiaokang Du , Huiling Liu , Qin Chang This is my paper · ORCID
classification hep-ph
keywords 95GeVexcessGeorgi-MachacekmodelpositivedefinitenessconstraintsvacuumstabilitycustodialsingletHiggsdi-photonbottomquarkpairRG-improvedeffectivepotential
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

The paper asks whether the Georgi-Machacek (GM) model, an extension of the Standard Model with extra triplet scalars and a protected custodial symmetry, can still account for the observed ~95 GeV excesses in the di-photon and b\bar{b} channels if vacuum stability is judged by the new "positive definiteness" conditions at large field values instead of the usual tree-level bounded-from-below conditions at the electroweak scale. The authors answer yes. They find a light custodial-singlet Higgs at about 95 GeV that fits the CMS-ATLAS di-photon and LEP b\bar{b} signals within 2\$\sigma$, while a heavier singlet plays the 125 GeV Higgs. They further argue that the positive-definiteness criterion is the one that should be applied first, and that it opens up parameter regions the older tree-level bounds had excluded. If they are right, the GM model remains a live explanation of the excesses and the standard stability test for triplet-extended Higgs sectors needs revision.

What carries the argument

The machinery is the one-loop RG-improved tree-level potential of the most general gauge-invariant GM scalar potential, evolved from the electroweak scale up to the Planck scale or a Landau pole. The custodial-symmetric quartic couplings at the electroweak scale are translated into the general couplings by the boundary relations (4.16), and stability is certified by checking that the potential remains positive definite along all large-field directions. In practice the authors reduce the potential to the quadratic form (4.19) under the assumption that the couplings $\rho_5$ and $\sigma_4$ are small and then require all leading principal minors of the associated $5\times 5$ matrix to be positive via Sylvester's criterion. This criterion replaces the traditional bounded-from-below analysis of the quartic part of the potential and is what lets formerly excluded parameter points survive.

What would settle it

Compute the full one-loop effective potential, including intermediate field scales and vacuum tunneling, for benchmark point 1 in Table 1 (where $\rho_5=2.7191$ and $\sigma_4=0.7862$); if a deeper minimum appears or the decay time is shorter than the age of the universe, the positive-definiteness check as applied is not sufficient to guarantee the stability claim.

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Extended reading notes

Core claim

The central claim is that the stability of the custodial electroweak vacuum in the GM model should be tested by the positive definiteness of the one-loop RG-improved tree-level scalar potential at asymptotically large field values, not by the tree-level bounded-from-below bounds evaluated with electroweak-scale couplings. The paper's numerical scan exhibits parameter points that fail the old bounds yet pass the new ones and reproduce the observed excesses; the opposite also happens, so the two criteria are genuinely inequivalent. Two benchmark points are given in which the lighter CP-even custodial singlet has mass 95.59 GeV or 96.11 GeV, the heavier singlet is the SM-like Higgs at 125.55 GeV or 125.35 GeV, and the predictions for $\mu_{\gamma\gamma}$ and $\mu_{b\bar{b}}$ sit inside the 2\$\sigma$ experimental ranges. The conclusion stated by the authors is that positive-definiteness constraints should be taken into account first and that they expand the allowed parameter space relative to the traditional bounded-from-below conditions.

Load-bearing premise

Everything hinges on the assumption that checking the quantum-corrected scalar potential only at very large field values, using a particular translation between the two ways of writing the couplings and assuming two of those couplings are small, correctly tells whether the electroweak vacuum is stable.

Editorial extensions

If this is right

  • If the positive-definiteness criterion is the right stability test, the GM model remains a viable simultaneous explanation of the 95 GeV di-photon and $b\bar{b}$ excesses with a light custodial-singlet Higgs.
  • The tree-level bounded-from-below conditions at the electroweak scale should no longer be used as the primary vacuum-stability filter for the GM model; parameter regions they exclude can be legitimately reopened.
  • The surviving parameter space makes concrete predictions: doubly charged Higgs masses between about 200 and 340 GeV, growing with triplet VEVs in the 3-30 GeV range, giving clear search targets for colliders.
  • Future precision measurements of the 125 GeV Higgs couplings would cut the allowed region sharply, and the paper notes that only a small portion would survive the sensitivities of a future lepton collider, so the interpretation is decisively testable.

Reading between the lines

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

  • The stability test is applied only along asymptotically large field directions and tunneling is not calculated; because benchmark point 1 has $\rho_5=2.7191$ and $\sigma_4=0.7862$, values that strain the small-coupling reduction behind the quadratic form, a full intermediate-scale and tunneling analysis could shrink the claimed parameter space. The paper explicitly defers those checks.
  • The claim that the allowed region expands may depend on the boundary conditions used at the electroweak scale to map the custodial potential into the general one; a different ultraviolet completion that keeps custodial symmetry at high energies could change the running and therefore the positive-definiteness verdict.
  • The same substitution of large-field positive definiteness for tree-level bounded-from-below conditions could be applied to other triplet-extended Higgs sectors, where it might either reopen or close parameter space; the direction of the effect is model-dependent.
  • The 95 GeV excess itself may still be a statistical fluctuation, so the concrete falsifiable content is the correlated prediction of the 125 GeV Higgs couplings and the charged-scalar masses; accurate future measurements of those would test the GM interpretation independently of the excess's fate.
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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

2 major / 5 minor

Summary. The paper investigates whether the Georgi-Machacek (GM) model can simultaneously explain the ~95 GeV di-photon excess reported by CMS and ATLAS and the ~98 GeV b-bbar excess reported by LEP, subject to theoretical and experimental constraints. After reviewing the GM scalar sector and the relevant excess signal strengths, the authors impose perturbative unitarity, HiggsBounds/HiggsSignals, electroweak precision, and B-physics constraints, and compare two stability criteria: the traditional tree-level bounded-from-below conditions at the electroweak scale and the recently proposed 'positive definiteness' constraints on the one-loop RG-improved tree-level potential at large field values. The numerical scan yields parameter points that satisfy the positive-definiteness constraints, including two benchmark points with a ~95 GeV custodial singlet Higgs, and the paper concludes that the positive-definiteness constraints not only permit a 95 GeV explanation but actually enlarge the allowed parameter region relative to the tree-level bounded-from-below conditions.

Significance. If the central claim were established, it would be a useful contribution to the 95 GeV excess literature: it would show that the GM model remains viable under a scale-improved stability criterion and that the customary tree-level bounded-from-below constraints are not the most restrictive ones. The paper is also constructive in providing benchmark spectra and in using public tools (GMCALC, SPheno/SARAH, HiggsBounds, HiggsSignals) for a realistic phenomenological analysis. However, the central numerical comparison is not yet supported, because the positive-definiteness criterion is applied through a quadratic-form reduction whose stated validity condition is violated by the benchmark points used to demonstrate the claim.

major comments (2)
  1. [Sec. 4.2, Eqs. (4.18)-(4.20), and Table 1] The positive-definiteness criterion is implemented by reducing the scalar potential to the quadratic form (4.19), which the text states is valid only 'when the parameters ρ5, σ4 are sufficiently small.' The benchmark points in Table 1 do not satisfy this condition: BP1 has ρ5 = 2.7191 and σ4 = 0.7862, and BP2 has ρ5 = -1.3651. For these O(1) values, the dropped terms ρ5(v_{ξ0}v_{χ+} - v_{ξ+}v_{χ0})^2 and (σ4/2√2)v_φ^2(v_{χ+}v_{ξ+} + v_{χ0}v_{ξ0}) are of the same order as the retained terms; the σ4 term is not sign-definite and can be negative along directions with v_{χ+}v_{ξ+} + v_{χ0}v_{ξ0} < 0. Consequently, the Sylvester conditions (4.20) are neither necessary nor sufficient for positive definiteness of the full potential (4.18). Since the scan points and benchmarks are selected with this approximate criterion, the paper's central claim that the positive-definiteness constraints 'expand the allowed parameter regions' is not demonstrated. The authors should either impose and enforce a quantitative smallness bound on ρ5 and σ4, or redo the analysis using the full quartic form (4.18).
  2. [Sec. 6 and Sec. 4.2, final paragraph] The conclusions state that the GM model 'remains a viable explanation' of the 95 GeV excesses while 'satisfying theoretical and experimental constraints.' The final paragraph of Sec. 4.2 explicitly defers tunneling-rate estimates, stability at intermediate field values, and gravitational effects. As presented, the positive-definiteness constraint is a necessary asymptotic condition, not a full vacuum-stability statement. The 'viable' wording in the conclusions therefore overstates what the analysis establishes; please qualify the claim or supplement the benchmark points with metastability estimates.
minor comments (5)
  1. [Eq. (2.8)] The second tadpole equation contains a typesetting error: the term involving µ1 appears malformed, with an operator apparently missing before 'µ1 - 6µ2v∆'; please correct it.
  2. [Sec. 3.2, Eq. (3.6) and Eq. (3.7)] The text says the κ_i parameters are defined as the square of the coupling ratio, but Eq. (3.7) defines κ_i as the ratio itself, with the square appearing explicitly in Eq. (3.6); please align the wording with the equations.
  3. [Table 1 and Eq. (5.1)] Benchmark point 1 has sin(θH) = 0.5069, which exceeds the stated scan range 0 < sin θ < 0.45 in Eq. (5.1); please clarify whether this benchmark is meant to lie outside the scan range or whether the stated range is incorrect.
  4. [Sec. 5 and Fig. 1] The text refers to 'the lower three panels of Fig. 1', but each row of Fig. 1 contains two panels; the intended reference is presumably the lower two panels.
  5. [Sec. 3.2, Eq. (2.16), and Table 1] The notation for m_h and m_H is inconsistent: Eq. (2.16) gives m_h as the lighter eigenvalue, while Sec. 3.2 refers to the lighter state as H and the heavier as h; Table 1 lists input masses m_h = 126.82 GeV and m_H = 121.97 GeV but then reports mh1 (m_H) = 95.59 GeV and mh2 (m_h) = 125.55 GeV. Please standardize the naming conventions so the benchmarks can be reproduced.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the positive definiteness constraints are external inputs and the 95 GeV parameter space is the output, not the input.

full rationale

The paper applies the positive definiteness (PD) constraints developed in the authors' prior work [145] to the question of whether the Georgi-Machacek model can explain the 95 GeV excesses. This is a self-citation, but it is not circular: the cited work derives constraints on the scalar potential from the RG-improved tree-level potential, and it does not use the 95 GeV excess data as an input. The present paper's numerical scan searches for parameter points that satisfy the PD constraints and then checks whether those points can explain the excesses; the observed signal strengths are not used to set the PD constraints. The claimed expansion of parameter space is therefore an output of the scan rather than an input. The main caveat identified in the text is that the quadratic-form reduction in Eq. (4.19) is stated to be valid only when rho5 and sigma4 are sufficiently small, while benchmark point 1 has rho5 = 2.7191 and sigma4 = 0.7862. That is a correctness and validity concern about the approximation, not a circularity, because the approximation is not defined in terms of the 95 GeV result and the benchmark points are not used to define the constraints. The self-citation to [145] is load-bearing for the method, but the method is independently derived in Section 4.2 and the cited work does not presuppose the present conclusion, so no reduction of the central claim to its inputs is present.

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

No new particles or forces are introduced; the GM model's additional scalars are pre-existing. The central claim rests on the free parameters of the scan and on the validity of the positive definiteness criterion as a vacuum stability check, which is the main load-bearing theoretical input drawn from the authors' prior work.

free parameters (7)
  • lambda2 = scanned in (-sqrt(4pi), sqrt(4pi)); BP1: -0.9263, BP2: -0.2818
    Quartic coupling scanned in the numerical analysis; the central claim depends on the existence of values passing all constraints.
  • lambda3 = scanned in (-sqrt(4pi), sqrt(4pi)); BP1: 0.6798, BP2: -0.3413
    Quartic coupling scanned; affects scalar masses and vacuum stability.
  • lambda4 = scanned in (-sqrt(4pi), sqrt(4pi)); BP1: 0.2608, BP2: 0.2597
    Quartic coupling scanned; relevant for bounded-from-below and positive definiteness.
  • lambda5 = scanned in (-sqrt(4pi), sqrt(4pi)); BP1: -0.5559, BP2: -0.0295
    Quartic coupling scanned; enters the singlet mass matrix and the positive definiteness matrix.
  • sin(theta) = scanned in (0, 0.45); BP1 listed as 0.5069 (inconsistent), BP2: 0.2067
    Triplet-doublet mixing parameter; controls coupling modifications and the allowed triplet VEV.
  • mh, mH = scanned in (50, 200) GeV; BP1: 126.8, 122.0 GeV; BP2: 128.2, 120.6 GeV
    Input singlet masses (before physical mixing) used to fix the dimensionful potential parameters.
  • alpha = scanned in (0, pi); BP1: 0.1736, BP2: 1.9891
    Mixing angle between the two CP-even singlets; determines the 95 GeV scalar couplings to fermions and vectors.
assumptions (4)
  • domain assumption One-loop RG-improved tree-level scalar potential is a valid approximation of the effective potential at large field values
    Used in Sec. 4.2 to derive the positive definiteness constraints; standard in SM vacuum stability analyses but an approximation for the full GM potential.
  • domain assumption Custodial symmetry is exact at the EW scale and the boundary conditions (4.16) map the GM potential to the general potential
    Eq. (4.16) in Sec. 4.2; the entire RGE analysis starts from this mapping, which is only valid at the matching scale.
  • domain assumption Positive definiteness at asymptotically large field values is sufficient to exclude dangerous deeper vacua
    Sec. 4.2 final paragraph explicitly defers tunneling-rate and intermediate-scale stability analyses; without them the vacuum stability conclusion is incomplete.
  • ad hoc to paper The parameters rho5 and sigma4 are sufficiently small for the quadratic-form reduction (4.19) to apply
    The matrix reduction in Sec. 4.2 assumes small rho5 and sigma4, but Table 1 benchmark 1 has rho5 = 2.7191 and sigma4 = 0.7862, so the assumption is not satisfied for the benchmark.

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Pith. "Pith review of Interpretation of 95 GeV Excess within the Georgi-Machacek Model in Light of Positive Definiteness Constraints." pith.science (2026). https://pith.science/paper/IVQN25T4

@misc{pith2026250206444,
  author       = {Pith},
  title        = {Pith review of: Interpretation of 95 GeV Excess within the Georgi-Machacek Model in Light of Positive Definiteness Constraints},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IVQN25T4}},
  note         = {Machine review of arXiv:2502.06444}
}
abstract

The recent observation of a di-photon excess around 95 GeV by the CMS and ATLAS Collaborations, along with the $b\bar{b}$ excess reported by the LEP Collaboration in the same mass region, has drawn significant interest in the possibility of new physics beyond the Standard Model (SM). The Georgi-Machacek (GM) model, which extends the Higgs sector of the SM by introducing additional triplet scalars while preserving custodial symmetry at tree level, provides a compelling framework to explain both excesses simultaneously via a light custodial singlet Higgs. In this work, we investigate whether the GM model can still accommodate these excesses when taking into account newly proposed vacuum stability constraints, particularly the positive definiteness conditions. Our numerical analysis not only confirms the existence of a viable parameter space capable of explaining the 95 GeV excesses, but also demonstrates that, compared to traditional tree-level constraints at the electroweak scale, the positive definiteness conditions further expand the allowed parameter space, thereby enhancing the viability of the GM model. Furthermore, we emphasize that future collider experiments will play a crucial role in testing this interpretation by refining Higgs coupling measurements and searching for additional Higgs bosons.

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Reviewed August 8, 2026 · model on record in the stance chip above.