Phenomenology of the new light Higgs bosons in Gildener-Weinberg model

Eric Pilon; Kenneth Lane

REVIEW 3 major objections 5 minor 2 cited by

Phenomenology of the new light Higgs bosons in Gildener-Weinberg model

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Gildener-Weinberg models predict new Higgs bosons below 500 GeV, with the 125 GeV Higgs self-couplings at twice and four times the Standard Model values, so direct scalar searches are the decisive LHC test.

desk verdict A clean and honest one-loop calculation of GW Higgs self-couplings; the main caveat is the unpropagated O(100 GeV) uncertainty in the 540 GeV sum rule. read the letter →

arxiv 1909.02111 v2 pith:K7DZDDUK submitted 2019-09-04 hep-ph

Kenneth Lane , Eric Pilon This is my paper · ORCID

classification hep-ph

keywords Gildener-Weinbergmodeltwo-Higgs-doubletscaleinvarianceHiggstrilinearcouplingquarticdi-HiggsproductionscalarmasssumruleLHCsearches

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

Gildener-Weinberg models explain the 125 GeV Higgs as the would-be massless boson of spontaneously broken scale symmetry, and that symmetry keeps its mass and couplings close to Standard Model values. This paper establishes two consequences of that structure. A mass sum rule, $(M_{H'}^4+M_A^4+2M_{H^\pm}^4)^{1/4}=540$ GeV, forces the new charged and neutral Higgs bosons of any GW model to lie below about 500 GeV, within reach of LHC data in hand or soon to come. The same scale-symmetric logic makes the Higgs trilinear and quartic self-couplings vanish at tree level; the one-loop values are $\lambda_{HHH}\simeq64$ GeV and $\lambda_{HHHH}\simeq0.129$, about two and four times the Standard Model. These couplings put $\sigma(pp\to HH)$ at its minimum of 15–20 fb at 13–14 TeV, too small for the HL-LHC, so the paper argues that direct LHC searches for the new light scalars are the surest test this decade.

What carries the argument

The central mechanism is the Gildener-Weinberg two-Higgs-doublet model, in which a scale-invariant quartic potential has a flat direction and a massless tree-level dilaton (the scale-symmetry Goldstone boson) that is exactly aligned with Standard Model couplings. The one-loop Coleman-Weinberg potential gives this dilaton its 125 GeV mass. Two identities carry the argument: the sum rule $(M_{H'}^4+M_A^4+2M_{H^\pm}^4)^{1/4}=540$ GeV, obtained by inserting tree-level masses into the one-loop mass formula, which extends the low-mass prediction to all GW models; and the vanishing of the dilaton's tree-level cubic and quartic self-couplings, which follows from the homogeneity of the scale-invariant potential and its vanishing along the flat direction. These vanishings force $\lambda_{HHH}$ and $\lambda_{HHHH}$ to start at one-loop order, producing the numerical ratios near $2$ and $4$ times the Standard Model.

What would settle it

Measure $\sigma(pp\to HH)$ at the 27 TeV HE-LHC and run dedicated LHC searches for $H^\pm\to t\bar b$ and $A/H_2\to b\bar b$ over 200–500 GeV at $\tan\beta\simeq0.3$; a di-Higgs rate well above 20 fb, or exclusion of the new scalars at their predicted masses and rates, would contradict the sum-rule and coupling forecasts.

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Extended reading notes

Core claim

On its own terms, the paper claims that every Gildener-Weinberg model of electroweak symmetry breaking shares a sum rule $(M_{H'}^4+M_A^4+2M_{H^\pm}^4)^{1/4}=540$ GeV, so at least some of the new Higgs bosons must be light enough for the LHC. It further claims that, because the classical potential is scale invariant and homogeneous of degree four, the trilinear and quartic self-couplings of the 125 GeV Higgs vanish at tree level and first appear in the Coleman-Weinberg loop expansion. In the GW-2HDM the one-loop calculation gives $\lambda_{HHH}\simeq2(\lambda_{HHH})_{\rm SM}=64$ GeV and $\lambda_{HHHH}\simeq4(\lambda_{HHHH})_{\rm SM}=0.129$. Because the sum rule applies to any GW model, the paper concludes these coupling values and the resulting cross sections apply to all GW models, not only the two-doublet example. The collider consequence is that di-Higgs production stays near its theoretical minimum of 15–20 fb at 13–14 TeV, tri-Higgs production needs a 100 TeV machine, and direct production of $H^\pm$, $A$, and $H_2$ is the realistic discovery channel.

Load-bearing premise

The load-bearing premise is that first-order Coleman-Weinberg perturbation theory, evaluated with tree-level masses, fixes the 125 GeV Higgs mass and the 540 GeV sum rule closely enough that higher-order corrections shift the scalar masses by only $O(100)$ GeV or less.

Editorial extensions

If this is right

If the sum rule holds, every GW model with only $W^\pm$, $Z$, and top quark in the loop has at least one new scalar boson below about 500 GeV, and models with more scalars must put some of them even lower.
The predicted trilinear coupling $\lambda_{HHH}\simeq64$ GeV gives $\sigma(pp\to HH)\simeq15$--$20$ fb at 13–14 TeV, so an HL-LHC observation of di-Higgs production would be a surprise; the 27 TeV HE-LHC is needed to see it.
The quartic coupling $\lambda_{HHHH}\simeq0.129$ is four times the Standard Model value, yet $pp\to HHH$ remains unobservable until a 100 TeV hadron collider.
Searches for $H^\pm\to t\bar b$, $A/H_2\to b\bar b,t\bar t$, and $A/H_2\to ZH_2,ZA\to \ell^+\ell^- b\bar b$ in the 200–500 GeV range are the decisive near-term tests; null results at the expected rates would strongly constrain the class.
Because near alignment suppresses $H_2,A\to WW,ZZ$ and $H^\pm\to WZ$, observing unsuppressed decays of this type from a new scalar would be a significant, possibly fatal, blow to GW models.

Reading between the lines

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

The paper leaves implicit that the sum rule's universality cuts both ways: if LHC searches cover the 200–500 GeV window and find nothing at the expected $\tan\beta\sim0.5$ rates, the entire GW class is disfavored, not just the two-doublet example.
Because the one-loop sum rule could shift by $O(100)$ GeV, the robust forecast is better read as 'some new scalar below roughly a TeV' than as a sharp 540 GeV cut, so low-mass searches discriminate GW models from decoupled Higgs sectors.
The sharp rise of $\kappa_\lambda$ and $\mu_\lambda$ near the sum-rule endpoint suggests a testable corner: if $M_{H^\pm}=M_A$ is close to 400 GeV with a light $H_2$, $H_2$-associated multi-Higgs final states could have enhanced rates that the present paper does not quantify.
A natural extension is to compute $\sigma(pp\to H_2H_2)$ using the sizable $\lambda_{H_1H_2H_2}$ coupling found here, a rate the paper identifies as interesting but leaves for future work.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

The thing to know: this paper identifies a clean, apparently new reason why triple and quartic Higgs self-couplings in Gildener-Weinberg models are near the Standard Model values. A scale-invariant quartic potential is homogeneous of degree four, so at the flat-direction minimum the cubic and quartic couplings involving Goldstone bosons vanish at tree level. The authors demonstrate this in the aligned basis for a two-Higgs-doublet GW model, then compute the one-loop corrections. The calculation is transparent and the numbers — κλ ≈ 1.6, μλ ≈ 3.6 for MH± below 370 GeV — agree with the earlier Coleman-Weinberg-like estimate of Agrawal et al. (Ref. [6]), which they cite fairly.

The paper also does a real service by pushing the sum rule (M_H'^4 + M_A^4 + 2M_H±^4)^(1/4) = 540 GeV as the main LHC handle. That gives concrete search channels and mass ranges for new scalars below ~500 GeV, which is useful for Run 3.

The soft spot is exactly what the authors concede in passing. The 540 GeV value comes from first-order Coleman-Weinberg perturbation theory; they say higher-order corrections could shift it by O(100 GeV). But every downstream number — the 'below 500 GeV' bound, the mass assignments in Fig. 3 and Table 2, and the κλ/μλ values used for the di-Higgs forecast — is computed from that one-loop relation without propagating the shift. If the two-loop correction is near the upper end of their estimate, the qualitative conclusions survive but the specific mass ranges and cross sections move. I don't think this sinks the paper; it is an honest first-order calculation. But the authors should state which results are robust under a ±100 GeV change in the sum-rule scale, and which are not.

The 'applies to all GW models' claim is asserted more than derived. The likely argument is that the dominant V1 contribution depends only on the 125 GeV Higgs field, not on the rest of the scalar sector, so the sum rule is sufficient. That is plausible, and it is consistent with the stress-test note, but it deserves a sentence of justification.

Who this is for: BSM phenomenologists and LHC searchers. It deserves a serious referee. The math is internally consistent, the citations are honest, and the central observation is worth publishing.

Referee Report

3 major / 5 minor

Summary. This paper studies the Gildener-Weinberg (GW) mechanism in a two-Higgs-doublet model (GW-2HDM). The authors review the one-loop effective-potential calculation of Ref. [2], which yields a sum rule for the new scalar masses, (M_H'^4 + M_A^4 + 2M_H±^4)^(1/4) = 540 GeV, once the 125 GeV Higgs mass is used as input. They then compute the one-loop triple and quartic Higgs self-couplings λ_HHH and λ_HHHH, finding κλ ≈ 1.6–3.6 and μλ ≈ 3.6–5.6 relative to the SM, with representative values near κλ ≈ 2 and μλ ≈ 4. These are translated into a forecast that σ(pp→HH) ≈ 15–20 fb, near the minimum of the di-Higgs cross section at 13–14 TeV, and that λ_HHHH would require a 100 TeV collider. Because the sum rule is independent of the number and type of Higgs multiplets, the authors claim these conclusions apply to all GW models. The paper concludes by advocating direct LHC searches for the new charged and neutral scalars below about 500 GeV.

Significance. If the central claims hold, the paper provides a concrete and falsifiable phenomenological target for GW models: new scalars below roughly 500 GeV, while di-Higgs and tri-Higgs rates remain at or below the SM values, so that direct scalar searches are the most promising probe. The one-loop effective-potential machinery is standard, the formulas for the self-couplings are given explicitly, and the numerical values in Table 1 are internally consistent with the stated expressions. The paper also usefully connects the sum rule to existing LHC searches and identifies the most sensitive channels. However, the strength of the conclusions depends on two aspects that are not fully established: the numerical impact of the acknowledged O(100 GeV) higher-order uncertainty in the sum rule, and the validity of the 'all GW models' generalization given the logarithmic mass dependence of the one-loop couplings.

major comments (3)

[Sec. II (after Eq. (15)) and Secs. III–IV] The paper explicitly states that higher-order corrections may change the right-hand side of Eq. (15) by O(100 GeV), but this uncertainty is never propagated into any of the downstream results. The mass assignments in Fig. 3 and Table 2, the coupling ratios in Fig. 4 and Table 1, and the di-Higgs and tri-Higgs forecasts in Sec. III all use the one-loop sum-rule constant 540 GeV. Because the scalar contribution to λ^(1)_HHH in Eq. (37) is proportional, up to logarithms, to M_H'^4 + M_A^4 + 2M_H±^4, a shift of the constant from 540 GeV to, say, 440 GeV reduces that scalar fourth-power sum by a factor (440/540)^4 ≈ 0.44 and changes the one-loop scalar contribution by tens of GeV, which can move κλ by order one. The authors should either quantify the resulting spread in κλ, μλ, and σ(pp→HH), or explicitly state that the quoted forecasts are conditional on the one-loop value of the sum rule being exact.
[Sec. III, Eqs. (37)–(41), and the universal claim in the abstract] The claim that the results apply to all GW models because of the sum rule is stronger than what Eqs. (37)–(41) establish. The non-logarithmic scalar term in λ^(1)_HHH is fixed by Σ M_H^4 = (540 GeV)^4, but the logarithmic terms Σ M_H^4 ln(M_H^2/Λ^2) depend on the distribution of the new scalar masses. In the GW-2HDM scan of this paper these logarithms are numerically significant and are comparable to the non-log term. In a general GW model with a different number of scalars or a different mass hierarchy, the logarithmic contribution can shift λ_HHH by tens of GeV. The universal statement should therefore be either restricted to the GW-2HDM or backed by an explicit demonstration that the logarithmic dependence is negligible over the full GW parameter space.
[Sec. III, text after Fig. 4 and Table 1] The abstract and Sec. III quote λ_HHH ≈ 2(λ_HHH)_SM = 64 GeV and σ(pp→HH) = 15–20 fb as the minimum, but Fig. 4 and Table 1 show that κλ ranges from about 1.6 to 3.6 over the allowed MH± = MA range. Since the di-Higgs cross section is a function of κλ (and μλ), the paper should present σ(pp→HH) as a function of MH±, using the parametrizations of Refs. [16–18], and specify for which value of κλ the minimum applies. Quoting a single 15–20 fb range together with κλ ∈ [1.6, 3.6] requires justification, as the cross section is not generally flat over that interval.

minor comments (5)

[Abstract and Fig. 4] The abstract presents κλ ≈ 2 and μλ ≈ 4 as clean values, whereas Fig. 4 and Table 1 show a range κλ ≈ 1.6–3.6 and μλ ≈ 3.6–5.6; the abstract should state that these are representative values rather than predictions valid across the whole allowed parameter space.
[Sec. III, after Eq. (26)] There is a duplicated word in 'the nonzero cubic terms terms in the tree-level potential'; this should be corrected.
[Sec. II, Eq. (15)] The phrase 'below about 500 GeV' is looser than what the sum rule strictly implies: the sum rule bounds the fourth-power combination, and in principle one scalar could be as heavy as 540 GeV if the others are very light. The wording should be adjusted to describe a bound on mass combinations rather than a strict upper bound on every individual scalar mass.
[Sec. IV, Table 2] The ATLAS and GW-2HDM entries in Table 2 include B(Z→ℓ+ℓ−) implicitly through the cross-section definition, while the CMS entry includes B(Z→e+e−, μ+μ−); the text notes this, but a footnote or a column header making the normalization identical would improve comparability.
[Sec. IV, after item (3)] The statement that there appear to be no dedicated searches for H±→W±H2 and H2→W±H∓ would be more useful with a brief estimate of the expected yields in the GW-2HDM parameter region, or with a dedicated search reference if one now exists.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the self-coupling predictions follow from the one-loop effective potential with measured masses as inputs, and self-citations are not load-bearing.

full rationale

The central quantitative claims do not reduce to their inputs. The mass formula Eq. (14) is a one-loop expression relating the 125 GeV Higgs mass to the known gauge-boson, top-quark, and scalar masses; Eq. (15) is obtained by inserting M_H = 125 GeV, so the 540 GeV sum-rule combination is a derived constraint, not a fitted input. The triple and quartic Higgs couplings are then computed as third and fourth derivatives of the one-loop improved potential, Eqs. (33)-(41), with scalar masses satisfying that sum rule; no observed Higgs self-coupling data are used to fix parameters. The choice tan beta = 0.5 is taken from a prior paper by one of the authors, but the paper states there is no discernible effect on kappa_lambda and mu_lambda for any plausible tan beta, so that self-citation is not load-bearing. The agreement with Agrawal et al. (Ref. [6]) is an external cross-check, not the source of the prediction. The authors' explicit caveat that higher-order corrections may shift the right side of Eq. (15) by O(100 GeV) is an honest limitation of first-order perturbation theory, not a circular reduction; it affects the reliability of the numerical forecasts but does not make the derivation self-referential.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central prediction rests on standard effective-potential methods plus the GW model assumptions above. No constant is fitted to Higgs-pair data; the only hand-set inputs are the benchmark choices tanβ=0.5 and MH±=MA.

free parameters (2)

tan β = 0.50 (current experimental upper limit)
Chosen as current upper limit from charged-Higgs searches; the authors state there is no discernible effect on cubic and quartic couplings for plausible tanβ>0, but it enters production cross-section estimates.
MH± = MA bench-points = 200, 400, 410 GeV (scanned)
The equality is imposed to make scalar contributions to the T parameter vanish; the scan values are chosen within the sum-rule allowed region and determine all predicted couplings.

assumptions (5)

domain assumption The tree-level scalar potential V0 is exactly scale invariant and contains only quartic couplings, with no quadratic or cubic terms.
Definition of the Gildener-Weinberg framework in Sec. II; the vanishing-coupling argument depends on V0 being homogeneous of degree four.
domain assumption The only heavy fermion is the top quark and the only weak bosons are W and Z; no other states contribute appreciably to V1.
Stated in Sec. II before Eq. (15); the 540 GeV sum rule follows only under this restriction.
domain assumption First-order (one-loop) Coleman-Weinberg perturbation theory is accurate enough that tree-level masses can be inserted on the right side of Eq. (14).
The authors say first-order perturbation theory is reliable and note higher-order corrections could shift Eq. (15) by O(100 GeV), which is the main uncertainty in the mass bound.
domain assumption The 125 GeV Higgs H1 is the nearly aligned dilaton with decay constant f=v=246 GeV.
Central GW setup in Sec. II; alignment suppresses many decay modes and is used for cross-section estimates.
ad hoc to paper MH±=MA is assumed so scalar contributions to the T parameter vanish.
Adopted in Sec. II and used for all numerical benchmarks; not forced by a symmetry.

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Pith. "Pith review of Phenomenology of the new light Higgs bosons in Gildener-Weinberg model." pith.science (2026). https://pith.science/paper/K7DZDDUK

@misc{pith2026190902111,
  author       = {Pith},
  title        = {Pith review of: Phenomenology of the new light Higgs bosons in Gildener-Weinberg model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K7DZDDUK}},
  note         = {Machine review of arXiv:1909.02111}
}

abstract

Gildener-Weinberg (GW) models of electroweak symmetry breaking are especially interesting because the low mass and nearly Standard Model couplings of the $125\,{\rm GeV}$ Higgs boson, $H$, are protected by approximate scale symmetry. Another important but so far under-appreciated feature of these models is that a sum rule bounds the masses of the new charged and neutral Higgs bosons appearing in {\em all} these models to be below about $500\,{\rm GeV}$. Therefore, they are within reach of LHC data currently or soon to be in hand. Also so far unnoticed of these models, certain cubic and quartic Higgs scalar couplings vanish at the classical level. This is due to spontaneous breaking of the scale symmetry. These couplings become nonzero from explicit scale breaking in the Coleman-Weinberg loop expansion of the effective potential. In a two-Higgs doublet GW model, we calculate $\lambda_{HHH} \simeq 2(\lambda_{HHH})_{\rm SM} = 64\,{\rm GeV}$. This corresponds to $\sigma(pp \to HH) \cong 15$--$20\,{\rm fb}$, its {\em minimum} value for $\sqrt{s} = 13$--$14\,{\rm TeV}$ at the LHC. It will require at least the $27\,{\rm TeV}$ HE-LHC to observe this cross section. We also find $\lambda_{HHHH} \simeq 4(\lambda_{HHHH})_{\rm SM} = 0.129$, whose observation in $pp \to HHH$ requires a $100\,{\rm TeV}$ collider. Because of the above-mentioned sum rule, these results apply to {\em all} GW models. In view of this unpromising forecast, we stress that LHC searches for the new relatively light Higgs bosons of GW models are by far the surest way to test them in this decade.

Figures

Figures reproduced from arXiv: 1909.02111 by the authors.

**Figure 1.** Left: The CP-even Higgs one-loop mass eigenvalues MH1 and MH2 , the tree-level mass MH0 = √ −λ345 v and the one-loop mass MH from Eq. (14) as functions of λ3 = (2M2 H± − M2 H0)/v2 . Here, tan β = 0.50 and MH± = MA = 390 GeV corresponding to λ4 = λ5 = −2.513. The input H ∼= H1 mass is MH = 125.0 GeV, the corresponding initial MH0 = 353 GeV and λ3 = 2.966. MH0 vanishes at λ3 = 2M2 H± /v2 = 5.027. Right: The angle δ = … view at source ↗

**Figure 2.** The mass of the neutral Higgs [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** The tree-approximation mass MH0 of the CP-even Higgs calculated from the sum rule (15) and the larger eigenvalue MH2 of the one-loop corrected CP-even mass matrix MH0+ . Both are calculated as a function of MH± = MA. MH0 starts to dive to zero at MH± ∼= 370 GeV and becomes zero at MH± ∼= 410.22 GeV. and decay branching ratios in Sec. IV. We recommend this approach for searches by ATLAS and CMS. For example, in a se… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: The ratios κλ = λ (0)+(1) H1H1H1 /(λHHH)SM (with (λHHH)SM ∼= 32 GeV) and µλ = λ (0)+(1) H1H1H1H1 /(λHHHH)SM (with (λHHHH)SM ∼= 0.0323) as a function of MH± = MA. The sharp rise starting near MH± = 370 GeV is an artifact of MH0 starting its dive to zero. potential of th…

**Figure 5.** Figure 5: The cross sections for √ s = 13 TeV at the LHC for single Higgs production processes in the alignment limit (δ → 0) of the GW-2HDM with the dependence on tan β scaled out. Both charged Higgs states are included in pp → tH−. From Ref. [5]. suming as we have that MH± = M…

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Reviewed August 14, 2026 · model on record in the stance chip above.

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