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Explaining The CDF-II W-Boson Mass Anomaly in the Georgi-Machacek Extension Models

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Xiao Kang Du, Zhuang Li, Fei Wang, Ying Kai Zhang

High Energy Physics - Phenomenology · Submitted 12 Apr 2022 · 97 references

Pith's verdict: The arithmetic is right and the RH-neutrino loop estimate is a real addition, but the paper never shows that the scalar potential actually produces the 0.8 GeV triplet splitting its headline rests on.

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arXiv ID2204.05760 (version 4)
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CitationFei Wang, Xiao Kang Du, Ying Kai Zhang, Zhuang Li. "Explaining The CDF-II W-Boson Mass Anomaly in the Georgi-Machacek Extension Models." arXiv:2204.05760 (2022). https://pith.science/paper/TYAJ5345
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The paper

The paper claims

The Georgi-Machacek model preserves a custodial symmetry that keeps the W boson mass pinned near its Standard Model value; the paper argues that breaking that symmetry in two small ways — misaligning the two triplet vacuum values by under a

The reading

The CDF-II collaboration reported a W boson mass of $80{,}433.5$ MeV, about $7$ $\sigma$ above the Standard Model expectation of roughly $80{,}357$ MeV. The authors ask whether the Georgi-Machacek model — a Higgs sector with one doublet and two triplets, built precisely so that the tree-level $W$ and $Z$ masses obey the custodial relation $m_W = m_Z c_W$ — can absorb that shift. Their answer is that the ordinary, aligned version cannot: they compute a maximal new-physics contribution of $0.0012$ GeV, far short of the roughly $0.07$ GeV needed for the 1-$\sigma$ range. They then propose two ways to break the custodial symmetry softly: let the two triplet vacuum values differ by a splitting $v_xi - v_chi$ of order a GeV, and add a low-scale right-handed neutrino sector that permits neutrino-triplet Yukawa couplings $h_{ij}$ of order $10^{-2}$ instead of $10^{-13}$. Either effect alone is marginal or insufficient; combined, they report that the needed splitting drops to about $0.7$ GeV for $v_chi$ below about $15$ GeV. If this holds, a model already motivated by neutrino masses and leptogenesis also explains the CDF-II anomaly without new particles above the TeV scale.

What carries the argument

The load-bearing object is the tree-level custodial relation for the $rho$ parameter written in terms of the triplet vacuum values, $rho = (v_phi^2 + 4 v_chi^2 + 4 v_xi^2)/(v_phi^2 + 8 v_chi^2)$, which equals unity exactly when $v_chi = v_xi$ and deviates as $Delta rho approx 4(v_xi^2 - v_chi^2)/v_{EW}^2$ when they differ. Because the $W$ mass shift at tree level is $Delta m_W approx m_W Delta rho / 2$, this single ratio converts a GeV-scale splitting of the triplet vacuum values into a GeV-scale shift of $m_W$, which is the mechanism the whole paper rides on. Around it sit two supporting tools: the oblique-parameter machinery $(S, T, U)$ that captures the loop-level contributions and lets the authors compare against the electroweak fit that includes the new CDF-II value, and a numerical pipeline that generates the model with SARAH, computes spectra with SPheno, and filters surviving points through HiggsBounds, HiggsSignals and GMCALC together with the $kappa_V$, $kappa_F$ and $s_H$ constraints.

Extended reading notes

Core claim

The paper's central claim is that the CDF-II $W$ mass excess can be reproduced inside the Georgi-Machacek framework once two custodial-symmetry-breaking effects are present. The first is a small misalignment between the vacuum expectation values of the complex triplet $chi$ and the real triplet $xi$: the tree-level relation $Delta rho = 4(v_xi^2 - v_chi^2)/v_{EW}^2$ feeds directly into $Delta m_W approx m_W Delta rho/2$, so a splitting as small as $0.8$ GeV with $v_chi lesssim 15$ GeV already lifts $Delta m_W$ to about $0.067$ GeV, the 1-$\sigma$ window of the CDF-II measurement. The authors argue that such a misaligned vacuum is reachable from a custodial-preserving tree-level potential because the one-loop Coleman-Weinberg corrections from hypercharge and Yukawa interactions break the symmetry softly anyway. The second effect is a right-handed neutrino sector that allows moderately large $h_{ij} L_i^T C i tau_2 chi L_j$ couplings while keeping neutrino masses tiny through a mixed type-I plus type-II seesaw with fine-tuned cancellation; this alone gives at most $0.03$ GeV, but it pushes the pseudo-Goldstone modes $H_5^pm$ heavy and, when combined with misalignment, reduces the splitting required for the 1-$\sigma$ range to about $0.7$ GeV. The ordinary aligned model's ceiling of $0.0012$ GeV is the baseline the paper sets out to beat.

Load-bearing premise

The argument assumes that the Georgi-Machacek scalar potential, once one-loop corrections are included, actually has a stable or long-lived vacuum in which the two triplet vacuum values differ by about $0.7$ to $0.8$ GeV; the paper asserts this is reasonable but presents no effective-potential minimum, no tunneling rate and no lifetime estimate.

Editorial extensions

If this is right

  • A triplet splitting of roughly $0.7$ to $0.8$ GeV with $v_chi lesssim 15$ GeV is enough to place $Delta m_W$ in the 1-sigma CDF-II window, so the required custodial breaking is small and does not require a new mass threshold.
  • The doubly charged Higgs $H^{++}$ is pushed above about $600$ GeV in the misaligned case, and can drop below $400$ GeV once the right-handed neutrino sector is added, which reopens Barr-Zee contributions relevant to the muon anomalous magnetic moment.
  • Neutrino-triplet Yukawa couplings of order $10^{-2}$ become admissible only with the right-handed neutrino extension, since the ordinary Georgi-Machacek model would need $h_{ij} sim 10^{-13}$ at $v_Delta sim 10$ GeV.
  • The right-handed neutrino mass scale of $10^2$ to $10^4$ TeV that the fit favours is compatible with resonant leptogenesis, so baryogenesis and the $W$ mass shift can share one sector.
  • Because the two breaking sources add constructively, a future tightening of the CDF-II central value would be met by increasing either the splitting or the $h_{ij}$ coupling rather than by abandoning the framework.

Reading between the lines

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

  • The paper never computes the one-loop effective potential minimum or a tunneling rate, so an obvious follow-up is a numerical scan of $V_{CW}$ in the $v_chi$-$v_xi$ plane to check whether the misaligned vacuum is the global minimum or a sufficiently long-lived metastable one at the quoted splittings.
  • The pseudo-Goldstone modes $H_5^pm$ from breaking custodial $SU(2)_c$ down to $U(1)$ are the natural place to look for a collider signature that distinguishes this proposal from a plain alignment scenario, since their masses are pushed up by the same $h_{ij}$ couplings that raise $Delta m_W$.
  • Because the $W$ mass shift here is dominated by a tree-level $rho$ parameter effect rather than by oblique loops, the same trick should transfer to any extended Higgs sector with two or more multiplets whose vacuum values can drift apart by a GeV, which makes the mechanism broader than the Georgi-Machacek setting.
  • The fine-tuned cancellation between type-I and type-II seesaw contributions, needed to keep neutrino masses at $10^{-3}$ eV while $h_{ij} sim 10^{-2}$, is a testable liability: a future measurement pinning the absolute neutrino mass scale would constrain how much of the $W$ mass shift can come from this channel.
Original abstract

Original Georgi-Machacek model can preserve the custodial symmetry at tree level after the electroweak symmetry breaking. Unless additional $SU(2)_c$ custodial symmetry breaking effects are significant, the new physics contributions to $\Delta m_W$ are always very small. Our numerical results show that ordinary GM model can contribute to $\Delta m_W$ a maximal amount $0.0012$ GeV, which can not explain the new CDF-II anomaly on W boson mass. We propose firstly to introduce small misalignment among the triplet VEVs to increase $\Delta m_W$, which can give large new physics contributions to $\Delta m_W$. Such slightly misaligned triplet VEVs from custodial symmetry preserving scalar potential can still be allowed. Our numerical results indicate that the resulting $\Delta m_W$ can easily reach the $1\sigma$ range of CDF-II $m_W$ data and the splitting among the triplet VEVs $\Delta v$($\equiv v_\xi-v_\chi$) can be as small as $0.8$ GeV for $v_\chi\lesssim 15$ GeV. We also propose to introduce an additional custodial symmetry breaking source by extending the GM model with a low scale RH neutrino sector, which can adopt the leptogenesis mechanism and allow moderately large coupling strength $h_{ij}$ even for triplet VEVs of order GeV. With low scale RH neutrino mass scale of order $10^2\sim 10^4$ TeV, the new physics contribution to $\Delta m_W$ can reach $0.03$ GeV and is much larger than that of ordinary GM model. Combining both custodial $SU(2)_c$ symmetry breaking effects, the small misalignment among the triplet VEVs and the moderately large $h_{ij}$ couplings allowed with RH neutrino sector, the value of $\Delta m_W$ can still easily reach the $1\sigma$ range of CDF-II $m_W$ data, with a minimum splitting (among the triplet VEVs) approximately $0.7$ GeV for $v_\chi\lesssim 15$ GeV.

The compiled record

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. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 7,488 input tokens.

What the abstract itself states, as probabilities: claims first 0.46 · states limitation 0.48 · human participants 0.01 · controlled comparison 0.29 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.93 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; .

Original Georgi-Machacek model can preserve the custodial symmetry at tree level after the electroweak symmetry breaking. Unless additional $SU(2)_c$ custodial symmetry breaking effects are significant, the new physics contributions to $\Delta m_W$ are always very small. Our numerical results show that ordinary GM model can contribute to $\Delta m_W$ a maximal amount $0.0012$ GeV, which can not explain the new CDF-II anomaly on W boson mass. We propose firstly to introduce small misalignment among the triplet VEVs to increase $\Delta m_W$, which can give large new physics contributions to $\Delta m_W$. Such slightly misaligned triplet VEVs from custodial symmetry preserving scalar potential can still be allowed. Our numerical results indicate that the resulting $\Delta m_W$ can easily reach the $1\sigma$ range of CDF-II $m_W$ data and the splitting among the triplet VEVs $\Delta v$($\equiv v_\xi-v_\chi$) can be as small as $0.8$ GeV for $v_\chi\lesssim 15$ GeV. We also propose to introduce an additional custodial symmetry breaking source by extending the GM model with a low scale RH neutrino sector, which can adopt the leptogenesis mechanism and allow moderately large coupling strength $h_{ij}$ even for triplet VEVs of order GeV. With low scale RH neutrino mass scale of order $10^2\sim 10^4$ TeV, the new physics contribution to $\Delta m_W$ can reach $0.03$ GeV and is much larger than that of ordinary GM model. Combining both custodial $SU(2)_c$ symmetry breaking effects, the small misalignment among the triplet VEVs and the moderately large $h_{ij}$ couplings allowed with RH neutrino sector, the value of $\Delta m_W$ can still easily reach the $1\sigma$ range of CDF-II $m_W$ data, with a minimum splitting (among the triplet VEVs) approximately $0.7$ GeV for $v_\chi\lesssim 15$ GeV.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Original Georgi-Machacek model can preserve the custodial symmetry at tree level after the electroweak symmetry breaking. background 1.00 0.06 0.02 asserted 0.86
s1 Unless additional $SU(2)_c$ custodial symmetry breaking effects are significant, the new physics contributions to $\Delta m_W$ are always very small. background 0.85 0.12 0.06 asserted 0.93
s2 Our numerical results show that ordinary GM model can contribute to $\Delta m_W$ a maximal amount $0.0012$ GeV, which can not explain the new CDF-II anomaly on W boson mass. result 0.87 0.92 0.13 asserted 1.00
s3 We propose firstly to introduce small misalignment among the triplet VEVs to increase $\Delta m_W$, which can give large new physics contributions to $\Delta m_W$. contribution 0.92 0.77 0.20 asserted 0.49
s4 Such slightly misaligned triplet VEVs from custodial symmetry preserving scalar potential can still be allowed. interpretation 0.60 0.42 0.04 hedged possibility 0.60
s5 Our numerical results indicate that the resulting $\Delta m_W$ can easily reach the $1\sigma$ range of CDF-II $m_W$ data and the splitting among the triplet VEVs $\Delta v$($\equiv v_\xi-v_\chi$) can be as small as $0.8$ GeV for $v_\chi\lesssim 15$ GeV. result 1.00 0.95 0.26 hedged possibility 0.90
s6 We also propose to introduce an additional custodial symmetry breaking source by extending the GM model with a low scale RH neutrino sector, which can adopt the leptogenesis mechanism and allow moderately large coupling strength $h_{ij}$ even for triplet VEVs of order GeV. contribution 0.90 0.75 0.08 asserted 0.51
s7 With low scale RH neutrino mass scale of order $10^2\sim 10^4$ TeV, the new physics contribution to $\Delta m_W$ can reach $0.03$ GeV and is much larger than that of ordinary GM model. result 0.99 0.82 0.13 asserted 0.88
s8 Combining both custodial $SU(2)_c$ symmetry breaking effects, the small misalignment among the triplet VEVs and the moderately large $h_{ij}$ couplings allowed with RH neutrino sector, the value of $\Delta m_W$ can still easily reach the $1\sigma$ range of CDF-II $m_W$ data, with a minimum splitting (among the triplet VEVs) approximately $0.7$ GeV for $v_\chi\lesssim 15$ GeV. result 1.00 0.88 0.72 hedged possibility 0.52

What kind of paper the classifier reports this is: theoretical result 0.95 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v4
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

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Possible values of S, T with U = 0 that can give ∆mW up to 1σ − 3σ range of new CDF-II data. The box with red line denotes the range S = 0.00 ± 0.07 and T = 0.05 ± 0.06 [84] while the box with blue line denotes the range S = 0.14 ± 0.08 and T = 0.26 ± 0.06 [85]. Higgs mH++ mass are also shown in the right panel with mH++ lying near 900 GeV for ∆mW ∼ 0.0012 GeV. Such a heavy H++ can give very small contributions to the muon anomalous magnetic moment ∆aµ by the Barr-Zee diagrams, which can not [94] account for the recently reported muon g − 2 anomaly [95]
Figure 2. We show the new physics contributions to ∆mW in ordinary custodial symmetry pre￾serving GM model. The dependence of ∆mW on the mass of doubly charged scalar is also shown. The T parameter can be increased in GM type models if additional SU(2)c custodial symmetry breaking effects are included. The most economical recipe without putting in by hand new SU(2)c breaking terms is to split slightly the triplet VEVs, that is, requir￾ing misalignment between vξ and vχ by a small amount. Such a small misalignment with real non-custodial vacuum from SU(2)L × SU(2)R invariant scalar potential…
Figure 3. We show the new physics contribution ∆mW in the GM model with a small misalignment among the triplet VEVs. The dependences of ∆mW on the mass of doubly charged scalar, the values of κW , κZ, κt and the S parameter are also shown in the panels. We scan the parameter space of GM model with slightly misaligned triplet VEVs and show our numerical results in fig.3. From our numerical results, we can see in the upper left panel that the new physics contribution ∆mW is indeed correlated to the SU(2)c breaking parameter ∆v ≡ vξ − vχ. New contribution to ∆mW can easily reach 0.067 GeV for ∆v…
Figure 4. We show the new physics contribution ∆mW in the RH neutrino extended GM model without (the upper panels) and with (the middle and the lower panels) small misalignment among the triplet VEVs. The corresponding ranges of κW , κZ, MR and the doubly charged scalar masses mH++ etc are also shown. hij coupling can be large after the introduction of RH neutrinos. It can be seen in the middle and lower panels of fig.4 that large new physics con￾tributions to ∆mW can still be easily given in this RH neutrino extended GM model (accompanied with small misalignment among the triplet VEVs).…

Figures reproduced from arXiv:2204.05760 by the authors.

References
How the literature uses it

50 arXiv papers cite this in 57 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 24 sentences without a confident label

Cites it as support 1 paper
Cites it as background 32 papers

What it cites · 24 sentences without a confident label

Uses the method or tool of 4 papers
Uses data or values from 1 paper
  • Recently, electroweak precision fit of the oblique parameters from the analysis of precision electroweak data, including the new CDF-II result of $m_W$, gives [ref] S = 0.06 ±0.10, T = 0.11±0.12 , U = 0.13±0.09 , S = 0.14 ±0.08, T = 0.26±0.06 , (fix U=0)with (and without) fix $U=0$.

    Electroweak Precision Fit and New Physics in light of $W$ Boson Mass · Contributions to $Δm_W$ with slightly misaligned triplet VEVs · 0.92

Cites as support 1 paper
  • Because of large cancelations between the type-I and type-II seesaw contributions to the neutrino masses, both (type-I and type-II) contributions to lepton asymmetry by leptogenesis are important [ref] .

    Consequences of Triplet Seesaw for Leptogenesis · RH-neutrino Extended GM Model · 0.96

Cites as background 47 papers
  • In the GM model [ref] , the Higgs sector contains the ordinary SM $SU(2)_L$ doublet Higgs field $ϕ$ with hypercharge $Y=1/2$ and two $SU(2)_L$ triplet Higgs fields: complex triplet Higgs $χ$ with $Y=1$ and real triplet $ξ$ with $Y=0$.

    Testing the custodial symmetry in the Higgs sector of the Georgi-Machacek model · GM model and its extension with slightly misaligned triplet VEVs · 0.94

  • Georgi-Machacek (GM) model [ref] , which augments the SM Higgs sector with a complex $SU(2)_L$ triplet of hypercharge $Y=1$ and a real $SU(2)_L$ triplet of $Y=0$, can protect the relation $ρ_tree=1$ with custodial symmetry preserving Higgs potential and vacuum alignment between the complex and real triplets.

    Probing exotic Higgs sectors from the precise measurement of Higgs boson couplings · Introduction · 0.86

  • Georgi-Machacek (GM) model [ref] , which augments the SM Higgs sector with a complex $SU(2)_L$ triplet of hypercharge $Y=1$ and a real $SU(2)_L$ triplet of $Y=0$, can protect the relation $ρ_tree=1$ with custodial symmetry preserving Higgs potential and vacuum alignment between the complex and real triplets.

    Triplet Higgs boson collider phenomenology after the LHC · Introduction · 0.86

Discussion

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