Tripling down on the $W$ boson mass

A new precision measurement of the $W$ boson mass has been announced by the CDF collaboration, which strongly deviates from the Standard Model prediction. In this article, we study the implications of this measurement on the parameter space of the $SU(2)_L$ triplet extension (with hypercharge $Y=1$) of the Standard Model Higgs sector, focusing on a limit where the new triplet is approximate $\mathbb{Z}_2$-odd while the SM is $\mathbb{Z}_2$-even. We study the compatibility of the triplet spectrum preferred by the $W$ boson mass measured by the CDF-II experiment with other electroweak precision observables and Higgs precision data. We comprehensively consider the signals of new Higgs states at the LHC and highlighted the promising search channels. In addition, we also investigate the cosmological implications of the case in which the lightest new Higgs particle is either late decaying or cosmologically stable.

The CDF-II measurement of m W is also in tension with the measurements from the previous collider experiments at ∼ 2.6σ [2,3]. The discrepancy might be due to some unknown experimental systematical uncertainties, but it could also be a hint for new physics . A class of new physics solutions contain extensions to the Standard Model (SM) Higgs sector, whereby the new Higgs states provide additional sources of custodial symmetry breaking . In a particular class of models, the correction to the W mass from new physics enters at the one-loop level. The new physics scale is then predicted to be around a few hundreds of GeV. This is particularly interesting since it could give rise to signals in LHC new physics searches. The new physics in this class are generically in some SU (2) L multiplet. The correction to the W mass requires that the masses of the different members of the multiplet receive different custodial symmetry breaking contributions from the electroweak symmetry breaking. Hence, some of the couplings between the new physics and the Higgs need to be sizable, and there need to be significant mass splittings within the multiplet. Both of these features have interesting phenomenological consequences.
In this paper, we explore the phenomenology of the Higgs Triplet Model (HTM), with hypercharge Y = 1, in the context of electroweak precision measurements, direct collider searches, and Higgs precision measurement in light of the CDF-II W mass measurement. The prediction of this model for the W mass has been investigated in Ref. [17]. We go beyond the existing works by investigating the compatibility of the m W,CDF−II preferred triplet spectra with the measurements of the effective weak mixing angle and Higgs precision data as well as by providing a comprehensive analysis of possible signatures at the Large Hadron Collider (LHC). Furthermore, we explore the situation that the new Higgs triplet is approximately inert. This can be achieved naturally by imposing an approximate Z 2 symmetry, which can be broken softly. In this case, its lightest neutral states can be candidates for a fraction of stable dark matter or decaying dark matter. We explore the CDF-II measurement's impact on those dark matter candidates.
The paper is organized as follows: in Section II, we briefly review the Higgs Triplet Model; in Section III, we calculate the HTM's correction to the W mass at the one loop and give the preferred mass spectra for the new Higgses from the CDF-II measurement. We explore the phenomenology of this spectra in various aspects, including their contributions to the effective weak mixing angle in Section IV, the compatibility with the Higgs precision measurement in Section V, the bounds and discovery channels from the LHC direct searches in Section VI, and their cosmological implications in Section VII. We conclude in Section VIII. In the appendices, we give details of the self-energy corrections and the SM fitting formula and discuss the soft Z 2 breaking limit, the unitarity and vacuum stability bounds, the Landau pole, and the decoupling limit of the HTM.

II. Higgs Triplet Model
In the HTM, the Higgs sector contains an isospin doublet Φ with hypercharge Y = 1 2 and an isospin triplet ∆ with Y = 1. 1 They can be parameterized as where v φ and v ∆ are the vacuum expectation values (vev's) of the doublet and triplet field obeying In addition to the SM-like Higgs boson, the scalar sector contains six new Higgs bosons (degrees of freedom): the CP-even H boson, the CP-odd A boson, the singly-charged H ± bosons, and the doubly-charged H ±± bosons.
In this model, the tree level W and Z boson masses are given by where c 2 W ≡ cos 2 θ W and θ W is the weak mixing angle. If we take the Z boson mass as an input, the expected W boson mass is naively smaller than the SM prediction at the tree level where ∆m W denotes loop corrections.
However, as we shall see below, a mass splitting between the new Higgs states can correct the W mass at the loop level with an opposite sign compared to the tree level correction. To explain the CDF-II result, it is preferred that the 1-loop correction dominates over the tree level correction, i.e., v ∆ v. Assuming the difference between the CDF-II measurement and the SM prediction mainly comes from the loop correction ∆m W , i.e., this restricts v ∆ 7.6 GeV. To be concrete, we assume that v ∆ < 1 GeV in the rest of the paper.
For simplicity, we will work in the limit v ∆ = 0 for the calculation of the W mass correction, effective weak mixing angle, the Higgs di-photon rate, and the trilinear Higgs coupling (see below). Note that deviations from this limit will be suppressed by powers of v 2 ∆ /v 2 2 × 10 −5 and will be ignored.

A. The Inert Triplet
The limit of v ∆ = 0 can be realized in a strict sense by imposing a Z 2 symmetry, under which Φ is Z 2 -even and ∆ is Z 2 -odd. This Z 2 can also be used to forbid the neutrino yukawa term typically seen in the Type-II seesaw model. The general gauge invariant potential is then given by where all the parameters in the potential can be taken to be real. The minimization of the potential yields In terms of the physical states, the quadratic part of the Higgs potential is Then, the mass spectrum is given by We can substitute the Higgs potential parameters m 2 , with the condition that  (5) had (m 2 Z ) = 0.02766 and the leptonic contribution ∆α lept = 0.031497687 [90].
I.e., the coupling λ 5 controls the splitting of the mass spectrum. For the rest of the discussion, the model with λ 5 > 0 (λ 5 < 0) will be referred to as the type-I (II) Higgs triplet model, which has a mass ordering of m H ++ < m H + < m A,H (m A,H < m H + < m H ++ ), respectively.

III. One-loop corrected W boson mass
If H ++ , H + and H, A have sizable mass splittings, i.e., if |λ 5 | is large, the HTM provides additional sources of custodial symmetry breaking, therefore correcting the W mass differently than the Z mass.
We summarize our results below. Note that we perform this calculation in the limit v ∆ = 0. Finite values for v ∆ compatible with the upper bound of Eq. (7) will only induce negligible small shifts of m W . All necessary self-energy corrections are listed in App. A (see also Ref. [89]). Tab. I lists all of the input parameters [2] used in the computation.
To determine the W mass from the measurement of the Fermi coupling constant G F , we note that, where terms with 0 subscripts are the bare parameters, Π W W is the self-energy of the W , and δ V B are the vertex and the box diagram corrections to the muon decay process. Rewriting this expression in terms of the physical parameters, one gets at the one-loop level that where δα em , δm 2 W , and δs 2 W are the counterterms for the fine-structure constant α em , the W mass, and weak mixing angle s W ≡ sin θ W , respectively. The counterterm of s 2 W can be expressed in terms of the W and Z mass counterterms, which we define in the on-shell scheme, The α em counterterm, which we also define in the on-shell scheme, is given by where Π γγ (0) ≡ dΠ γγ (p 2 )/dp 2 | p 2 =0 . Combining everything, ∆r is at the one-loop level given by The vertex and box diagram corrections to the muon decay, δ V B , are given by (see e.g. Ref. [91]) where we neglected the contributions proportional to the electron and muon Yukawa couplings.
Based on Eq. (15), we can then write which we can iterate to solve for m W .
For the numerical implementation, we follow the procedure outlined e.g. in [92]. We split ∆r into The quantity ∆r SM, rest is given by where m W,SM is computed from the fitting formula given in Ref. [93] (see App. B). The fitting formula can also be used to obtain a number for ∆r SM (i.e., ∆r SM 0.03807). Combining the Eqs. (21) and (22) This equation consistently combines the full HTM one-loop corrections with the SM higher-order corrections, which are crucial for a precise result.
In the left (right) panel of Fig. 1, we show the resulting numerical value for m W as a function of  [94].
For the type-II HTM, we do not show the mass spectrum corresponding to m H,A < m h /2 = 62.5 GeV given it is excluded by the precision measurement of exotic Higgs decays as we discuss in Sec. V B.
Note that there are stronger yet model-dependent constraints on the HTM mass spectrum from direct collider searches. We will summarize them in detail in Sec. VI.

IV. Effective weak mixing angle
After obtaining the preferred spectra of the HTM, we assess whether these are compatible with the electroweak precision data by computing the effective weak mixing angle, sin 2 θ eff . In this computation, α em , M Z , and G F are chosen as inputs. Experimentally, sin 2 θ eff is defined as the ratio of the leptonic vector current to the leptonic axial current at the Z pole. The deviation from the tree-level value of the mixing angle, s 2 W , can be parameterized by ∆κ, where At one-loop, ∆κ obtains contributions from A − Z mixing, corrections to the weak mixing angle, and corrections to the axial/vector vertices, where v l and a l are the tree-level vector and axial couplings respectively, and F l V,A are the formfactors for the leptonic vector/axial currents. Since the extra Higgs bosons do not couple to the SM fermions, they do not contribute to F l V,A . We compute the SM contribution to sin 2 θ eff with the help of the SM fitting formula. Similar to the treatment of ∆r, we split ∆κ into three pieces, ∆κ = ∆κ SM,W (m W,BSM ) + ∆κ SM, rest + ∆κ BSM .  10. This finite range of |λ 5 | scanned results in the endpoints of the contours. The brown line represent the CDF II measured W boson mass and the yellow/gray band shows 1σ/2σ range. The dark purple line represent the PDG value for the W boson mass with the purple/gray band showing the 1σ/2σ range. The dark green line in the left (right) column represent the world averaged value 0.23153 ± 0.00016 [2,95] (SLD measured value 0.23098 ± 0.00026 [95]) of sin 2 θ eff with the green/gray band shows 1σ/2σ range. ∆κ SM, rest is determined via where ∆κ SM 0.03640 is computed from the fitting formula given in Ref. [ and 2σ ranges, respectively. The dark green horizontal lines in the left column represent the worldaverage value for the effective weak mixing angle [2,95] while the green and gray horizontal band shows 1σ and 2σ range respectively. For comparison, we show in the right column the value of the single most precise effective weak mixing angle measurement obtained by the SLD collaboration [95].
In the limit of |λ 5 | = 0, the type-I/II HTM predicts a W boson mass that agrees well with the world-averaged value. The effective weak mixing angle also agrees well with its world-average. As |λ 5 | increases, the resulting m W increases while the resulting sin 2 θ eff decreases. 3 On the other hand, a change in m lightest has a less significant impact (at least for m lightest 400 GeV). Note that a heavier m lightest yields a larger deviation from the world average for sin 2 θ eff for the type-I model while it yields a smaller departure for type II. For the type-I model, the parameter space that explains m W,CDF-II is consistent with the world averaged value of sin 2 θ eff within 2σ level. For the type-II model, the two measurements are inconsistent at the 2σ level for the m lightest -|λ 5 | parameter space that we scanned.
If we instead compare sin 2 θ eff to the value measured by the SLD collaboration [95], we find that the parameter space explaining m W,CDF-II is consistent with the measured sin 2 θ eff within the 2σ level for both type-I and -II mass hierarchies.
In the Two-Higgs-doublet model (2HDM), for which also large upwards shift of m W with respect to the SM prediction can be realized, a quite similar correlation between the predictions for m W and sin 2 θ eff is known to exist (see e.g. Ref. [9]). In comparison to the 2HDM, the type-I triplet model provides a slightly better fit of the effective weak mixing angle measurements if the the lightest BSM state is close to the electroweak scale; in contrast, the type-II triplet model provides a slightly worse fit if the lightest BSM state is close to the electroweak scale.

V. Precision measurement of the Standard Model Higgs
For v ∆ 0, the tree-level couplings of the SM-like Higgs boson are only modified negligibly with respect to the SM. Significant effects can, however, occur a the loop level or through the presence of new exotic decay modes.

A. Higgs-photon coupling and Higgs self-coupling
We define the ratio of the coupling between the SM-like Higgs boson and photon to the SM predicted coupling by For the triplet model, it is given by where Q denotes the electric charge; τ f ≡ m 2 h /(4m 2 f ); and the ellipsis denotes subleading SM contributions. The scalar couplings are given by The loop functions F 1/2 and F ± have the form We evaluate the LHC constraints set on the triplet couplings through modifications of the H → γγ rate by employing HiggsSignals [97,98].
In addition to the di-photon rate, we also evaluate loop corrections to the trilinear Higgs selfcoupling, which can receive large quantum corrections in the presence of large scalar couplings potentially excluding otherwise unconstrained parameter space (see e.g. Ref. [99]).
We compute the one-loop correction using FeynArts [100] and FormCalc [101] with the necessary model file derived using FeynRules [102,103]. For this calculation, we renormalize the SM-like Higgs boson mass in the on-shell scheme. The SM-like vev is also renormalized in the on-shell scheme by renormalizing the W and Z boson masses as well as the electric charge in the on-shell scheme.
We compare the predicted value for the trilinear Higgs self-coupling normalized to the SM tree-level value, κ λ , to the strongest current bound of −1.0 ≤ κ λ ≤ 6.6 [104] (at 95% CL). This bound is based on searches for the production of two Higgs bosons and assumes that this production mechanism is only affected by a deviation of the trilinear Higgs self-coupling from its SM value.  For type II (see the right panel of Fig. 4 to the signals considered here. To this end, we recast some of the most relevant searches. Instead of providing detailed limits on the model, our focus is to obtain an indication whether the parameter space has been thoroughly covered. As we will show later in this section, most of the parameter space remains open. We expect dedicated searches designed specifically for the signature described in this section will be much more sensitive. For the rest of this section, we start by discussing the various production channels for the BSM states. We then differentiate between three situations for the decays of the BSM Higgs states resulting in distinct collider signatures: a promptly-decaying lightest BSM state, a detector-stable lightest BSM state, a long-lived lightest BSM state.

A. Production
In the absence of additional Yukawa-type interaction terms and for v ∆ v, the exotic Higgs states are dominantly produced via electroweak pair production as shown in the upper row of Fig. 5.
In order to obtain an overview of the rate of the various production channels, we computed the next-leading-order (NLO) pair production cross sections for both mass hierarchies using a modified version of the Type-II Seesaw model file [107] (derived using FeynRules 2.3 [103]) and MG5aMC@NLO    In our discussion of potential search strategies at the LHC below, we will only focus on the production channels with the largest cross sections. . For reference, we have drawn dashed lines representing cτ = 10 −4 meter (corresponding to ∼ 10 −12 sec, which is the typical B meson lifetime) and 10 meter. This is the range in which long lived particle searches at the LHC could be sensitive.

B. Detection signatures
In order to correctly reproduce the W mass measured by CDF-II, the triplet vev v ∆ generically needs to be small. Given the size of v ∆ is controlled by the amount of soft breaking, a small value can be naturally achieved. If the triplet vev is exactly zero, the lightest triplet state is stable. This implies that the choice of v ∆ directly affects the lifetime of the lightest state, thus affecting the detection signature at the LHC. We discuss the cosmological implications in Sec. VII.
In Fig. 7, we show this lifetime of the lightest state for different choices of v ∆ ranging from 10 −8 GeV to 1 GeV for the type-I/II HTM. For v ∆ ∼ 10 −4 GeV, the lifetime of the lightest state is generically of the order of the B-meson lifetime. As such, any decay products of the lightest state will be tagged as displaced. For v ∆ ∼ 10 −8 GeV, the lifetime is generically orders of magnitude greater than the radius of the detector. In this case, the lightest state is unlikely to decay within the detector volume.
We further show the decay branching ratio of H and A for type-II model in Fig. 8   In the remainder of the section, we discuss the qualitatively different LHC signatures for the three different lifetime domains: prompt decay of the lightest state, detector-stable lightest state, long-lived lightest state.

The lightest state promptly decays
An overview of the main LHC search channels for a promptly decaying lightest BSM state for the type-I and type-II HTM can be found in Tab. II.  S is the number of signal events and S 95 is the 95% C.L. limit on the number of signal events for the given analysis. For statistically limited searches, one would expect r to scale as Ldt. We will use this naive scaling to make statements about potential reach with searches involving more data.
We find that m H ++ = 150 GeV can be excluded by recasting the multi-lepton final state search of Ref. [116] (i.e., by the B02 signal region). Based on this channel, one could potentially expect to fully close the gap of 84 GeV ≤ m H ++ ≤ 200 GeV between the searches for doubly-charged Higgs boson pair production based, as described below. We also checked a benchmark point of m H ++ = 350 GeV. Here, we expect four on-shell W bosons and one off-shell W boson in the final state. This benchmark is not constrained, for example, by using the search of Ref. [116] in the G05 signal region.
Applying the naive integrated luminosity based rescaling indicates that the full high-luminosity (HL)-LHC dataset (3 ab −1 ) can exclude this mass point; albeit with an analysis that is not dedicated to searching for a doubly-charged Higgs.
In the type-I HTM, the process with the second largest cross section is doubly-charged Higgs boson pair production, pp → H ±± H ∓∓ . The corresponding search channel involves a final state of four W bosons. A dedicated search for this signature has been performed by ATLAS using 13 TeV data [117,118] . For the type-II HTM with a light H/A, pp → HA production can be sizable. For m H = 100 GeV, recasting Ref. [121] in the same signal regions as the previous production mode yielded r ≈ 0.4. Once again, naive luminosity based scaling indicates that existing data is potentially sufficient to exclude this. For m H = 300 GeV, we obtained r ≈ 0.02 using [121] in the 4b1j signal region. Accounting for the differences in integrated luminosity used in this and Ref. [120], the exclusion reach comparable to the previous production mode. It should be noted that this production mode ensures a Z boson in the final state. Reconstructing it can potentially reduce the background.

The lightest state is detector stable
In this section, we consider the case in which the lightest member of the Higgs triplet is stable on detector timescales. This can be achieve with a small v ∆ 10 −8 GeV.
In type I, if the lightest state is detector stable, charged tracks in multiple subsystems of the detector are a generic signature. ATLAS presented a search for such tracks excluding doubly-charged particles masses below 1050 GeV [122]. The unexcluded mass regions will typically require very large values of λ 5 to give the desired shift in the W mass as shown in Fig. 2 MET, or jets + MET. We will focus on the three charged lepton signature. Our recasting with this benchmark show that current searches, such as the one in Ref. [116], is not yet sensitive. Naively rescaling based on the full HL-LHC integrated luminosity shows that this analysis barely misses the exclusion. Lastly, for pp → HA production, the main search channel is a mono-jet or monophoton + MET signature (with the jet or photon originating from initial-state radiation). Current available searches, such as Ref. [121] in the MET1j signal region, are not sensitive. This scenario can potentially be excluded using the full HL-LHC dataset.

The lightest state is long-lived
If the charged particle decays before reaching the muon spectrometer, the previously mentioned ATLAS charged track search [122] is not sensitive to it. If the particle decays in the inner tracker, the signal caused by doubly-charged Higgs bosons will be disappearing tracks plus delayed multilepton/multi-jet final states. Depending on the initial state, one may also expect prompt off-shell W bosons. These prompt jets/leptons could be used to tag the events provided that the intrinsic jet time spread is sufficiently low [123]. It should also be noted that recently ATLAS found am anomalously large ionization energy loss [124]. A highly boosted, long-lived, doubly-charged particle is a potential explanation to explain this excess [125] suggesting that H ±± could be a good candidate.
A large partonic center-of-mass energy could provide the desired boost. A detailed study should be performed to determine the viability of the HTM as an explanation for the dE/dx anomaly.
For the neutral Higgs states, Ref. [126] could be recasted for pair production of the neutral Higgs.
However, the only hard objects in this production mode are delayed objects. Generically, we expect a search strategy involving prompt jets/lepton tagging + delayed jets/leptons to be better. Furthermore, for m A > 215 GeV, the dominant decay mode involves an on-shell Z boson. Reconstructing a delayed Z boson will be a good signal to search for.

VII. Cosmological implications
For sufficiently small v ∆ , the lifetime of the lightest states in type-II, H and A, could be longer than the age of the Universe. Given H and A are electrically neutral, they could provide a good candidate for dark matter or a massive relic. To explain the m W value measured by CDF-II, a large |λ 5 | is needed. This requires H and A to strongly couple to h. Such strong couplings yield a small relic density for H/A if they are produced through the standard thermal freeze-out. The large couplings also lead to large scattering cross sections between H/A and nucleons as well as the production of significant amounts of electromagnetic or hadronic energy if they are not cosmologically stable.
We first compute the thermal relic density for A and H using MadDM 3.2 [133]. The upper panel of Fig. 9 shows the resulting sum of the relative relic abundances for H and A with respect to that   Sum of the direct detection cross sections times the relative abundance for cosmologically stable H and A for parameters that explain the measured m W by CDF-II within 2σ (blue band) (assuming v ∆ < 10 −16 eV). 95% CL constraints from the LZ experiment with 5.5 ton· 60 day exposure [127] are shown as the orange shaded region. The neutrino background for a xenon target [128] is shown as the yellow shaded region. For both panels, we added constraints from the exotic decays of the SM-like Higgs as the gray shaded region. (Right panel) Constraints on the relative abundance of visibly decaying relic with respect to cold dark matter as a function of their lifetime. 95% CL constraints from BBN [129], CMB [130,131], and isotropic γ-ray backgrounds [132] are shown as yellow, green, and brown shaded regions. We highlighted the range of f χ that explains m W,CDF-II as the blue band with arrows indicating the allowed lifetimes for stable massive relic and decaying massive relic. In the lifetime axes, we indicate the age of Universe at recombination and today with black arrows.  Fig. 9.

A. Stable massive relic
For scenario (i), H/A could enter dark matter direct detection experiments on Earth and leave imprints even if they are subdominant components of dark matter. Given H/A are thermally produced in the early universe, their lifetime coincides with the age of the Universe. To realize the stable relic scenario, the lifetime for H/A needs to be longer than the age of universe today τ H,A τ U = 10 18 sec. A stronger constraints on τ A,H comes from the observations of the diffused γ-ray backgrounds [132,[134][135][136]. Observations from Fermi-LAT telescope restrict the decaying time of dark matter τ χ 10 28 sec if it consists all the dark matter [132]. We translate this bound into τ H,A 10 28 f χ sec if only an f χ fraction of dark matter decays visibly. This is shown as the brown shaded region in the right panel of Fig. 9. To satisfy the constraint, v ∆ needs to be small. In the right panel of Fig. 9, we explicitly show the value of v ∆ for a given lifetime for A with mass m A = 1.5 TeV as the blue upper ticks. For the m A,H parameter space we consider, we find that setting v ∆ 10 −16 eV guarantees the cosmological stability.
We use MadDM 3.2 [133] to compute the spin-independent direct detection cross section for A and H. The lower panel of Fig. 9 shows the corresponding sum of the spin-independent direct detection cross section between the nucleon and H/A, weighted by the relative abundance. In the computation, we assume the relative abundance f χ between the massive relic and cold dark matter stays the same for the local dark matter environment (with cold dark matter density ρ local = 0.3 GeV/cm 3 ). Besides, 7 We do not consider the scenario where A is stable and H is unstable given the small difference in their lifetimes for a fixed v ∆ compared to the cosmological timescales.
the two share the same velocity distribution. The resulting weighted cross section (blue band), which is favored to explain m W,CDF-II , is ranging from 10 −49 cm 2 to 10 −44 cm 2 for m A,H ranging from 30 GeV to 1.5 TeV. In the same panel, we also show 95% CL constraint from the LUX-ZEPLIN (LZ) experiment with 5.5 ton·60 day exposure, where we scale up the cross section by 1.96/1.64 to estimate 95% CL limit based on the 90% CL limit reported in [127]. Note that most of the parameter space to explain m W,CDF-II is excluded by the LZ experiment together with the Higgs precision measurement.
One exception is a fine-tuned parameter space with m A,H slightly above m h /2, which could be excluded by future direct detection experiments or Higgs precision measurements. Otherwise, an additional mechanism is needed to further deplete its relic abundance to make this case viable.

B. Decaying massive relic
If H and A are not cosmologically stable, they could decay into the Standard Model particles through their couplings to the SM-like Higgs boson. The decays could inject significant amount of electromagnetic or hadronic energy into the Standard Model plasma in the early universe or intergalactic medium in the late universe, depending on the their lifetimes. This could lead to various observational signature in astrophysics and cosmology, such as those from Big Bang Nucleosynthesis (BBN) [129,130,137], Cosmic Microwave Background (CMB) [130,131,[138][139][140], and galactic and extragalactic diffuse γ-ray background observations [132,[134][135][136], even if they are subdominant components of dark matter.
In the right panel of Fig. 9, we summarize current cosmological constraints on visibly-decaying massive relic from BBN [129], CMB (combining constraints from anisotropy [130] from Planck 2018 and spectra distortion [131] from COBE/FIRAS), and isotropic γ-ray background [132] as yellow, green, and brown shaded regions, respectively. To get the BBN constraints, we take the constraints on massive relic χ with m χ = 1 TeV that decaying to bb from Ref. [129] 8 . This constraints are representative for massive relic that mostly decaying to hadronic energy. As shown in Ref. [129], lighter relic (m χ = 30 GeV and m χ = 100 GeV) or other hadronic energy-dominant decay channels (χ →ūu,tt, gg, W W ) share similar constraints. Constraints for massive relics decaying to electromagnetic energy, e.g. χ → e + e − , are generically weaker than those for relics decaying to hadronic energy. In our scenario explaining the CDF-II m W measurement, the dominant decay channels of H (A) are bb, W W , and hh (bb and Zh), depending on the kinematic accessibility (c.f. Fig. 8). All these decay channels generate significant amount of hadronic energy. Hence the BBN constraint we quoted are applicable.
In the same panel, we highlight a light blue band to show the range of the relative abundance for m H,A > 63 GeV (away from the fine-tuned mass region) whose corresponding parameters explain m W,CDF-II . For such an abundance range (0.08%-7%), the strongest constraints for the decaying relic come from BBN, which restrict τ A,H 50 sec. To satisfy this constraint, the value of v ∆ needs to be large. In the right panel of Fig. 9, we explicitly show the value of v ∆ for a given lifetime of A with mass m A = 65 GeV as the red upper ticks. For the m A,H parameter space we consider, we find that v ∆ 1 eV guarantees that A and H evade all the cosmological constraints for a visibly-decaying massive relic in the scenario which explains the CDF-II m W measurement. Note that v ∆ ≥ 1 eV corresponds to cτ H,A 1 km. Such decay signal could be searched at the long-lived particle search facilities at the LHC.

VIII. Summary
In this work, we studied the HTM with hypercharge Y = 1 in light of the recent CDF-II W mass measurement. The HTM can be realized with two distinct types of spectra: type I for which m H ++ < m H + < m H,A , and type II for which m H ++ > m H + > m H,A . First, we derived the mass spectrum of the additional Higgs bosons (for both type I and type II) preferred by the CDF-II m W measurement.
For this mass spectra, we then checked the compatibility with experimental measurements of the effective weak mixing angle and Higgs precision data (i.e., measurements of the Higgs di-photon rate, constraints on the Higgs trilinear coupling and constraints on exotic decay channels of the SM-like Higgs boson). For the type-I HTM, we find that mass spectra (as shown in the first, third, and fifth panel of Fig. 2) with the lightest state mass m H ++ 250 GeV explain the observed m W,CDF-II while being consistent with the measurements of the effective weak mixing angle and Higgs precision measurements, while also satisfying the theoretical constraints of perturbative unitarity and vacuum stability. For the type-II HTM, we find that mass spectra (as shown in the second, fourth, and sixth panel of Fig. 2) with the lightest state mass 62.5 GeV m H,A 350 GeV explain the observed m W,CDF-II while being consistent with the Higgs precision measurements, perturbative unitarity, and vacuum stability. For type II, we, however, find a mild tension with the world average measurement of sin 2 θ eff at the 2σ level, while still being well consistent with the single-most precise measurement of the effective weak mixing angle by the SLD collaboration.
Direct searches at the LHC provide stronger yet model-dependent constraints on the HTM. The model dependence mainly originates from the decay length of the lightest state, which is mostly controlled by the value of v ∆ and m lightest (c.f. Fig. 7). We classified the LHC signatures according to if the lightest state promptly decays, if it is detector-stable, or it is long-lived. We investigated the collider phenomenology for each of these cases and pointed out a number of promising discovery channels that the LHC could be sensitive to (summaries of those channels can be found in Tabs

A. Self-energy corrections
We take the one-loop contributions to self energies from Ref. [89], setting v ∆ = 0, in the computation of m W and sin 2 θ eff . We listed all the relevant formula here for readers' convenience. The corrections are parameterized in terms of g 2 = e 2 /s 2 W and g 2 Z = e 2 /(s 2 W c 2 W ). The BSM contributions to the vector-boson self energies are given by where B 0,1,00,11 (p 2 , m 2 1 , m 2 2 ) are the Passarino-Veltman two-point functions, which we evaluate using LoopTools 2.16 [101]. The remaining loop-functions are given by where A 0 (m 2 ) is the Passarino-Veltman one-point function.
The SM contributions to ∆r (and sin 2 θ eff ) can be separated into three classes: those from scalar bosons, fermions, and gauge bosons -i.e., Π 1PI, SM where i = W W, ZZ, γγ, Zγ. The scalar contributions are given by The fermionic contributions are given by where m f , Q f , I f , and N f c are the mass, electric charge, isospin, and color numbers of the SM fermion f , respectively. Here, we sum over all the SM quarks and leptons.
Finally, the gauge boson contributions are given by where D = 4 − 2 with being the dimensional regulator. The divergences of the loop functions are given by where ∆ = 1 + ln µ 2 with µ being the renormalization scale.
The fitting formula for sin 2 θ eff, SM is given by where

C. Soft Z 2 breaking
Here we introduce a soft Z 2 breaking term in the Higgs potential where µ is assumed to be real. Such a term breaks the degeneracy between the two neutral states Defining ≡ √ 2v ∆ /v 1, the explicit Z 2 breaking mixes the states of Φ and ∆ with the same quantum numbers at O( ). To avoid confusion, we write the weak eigenstates as The physical 125 GeV Higgs boson, and the Goldstone bosons that would become the longitudinal W and Z all have small mixtures of the corresponding component of the triplet: where M 2 ∆ = M 2 + 1 2 (λ 4 + λ 5 )v 2 . This mixture allows the mass eigenstates H and A to decay to fermions even though no Yukawa interactions are explicitly introduced in the triplet model. Up to quadratic order in , the physical states have masses given by From (C4), when m 2 H → m 2 h , the mixing parameter between H 1 and H 2 diverges. This means that h and H can be maximally mixed even when 1. In this limit, the H − h mixing depends on the details of the Higgs potential parameters.

D. Unitarity and Vacuum Stability Bounds
We follow the analysis of vacuum stability and unitarity constraints as given by [89,169].
Demanding perturbative unitarity impose an upper bound on the eigenvalues of the 2 → 2 scattering matrix |x i | < 8π, i = 1, 2, 3 where Taking λ ∆ ≡ λ 2 = λ 3 > 0, the necessary and sufficient condition for the Higgs potential to be bounded from below is λ 1 > 0, λ ∆ > 0, 2 2λ 1 λ ∆ + λ 4 + min(0, λ 5 ) > 0 (D3) The first two conditions are trivially satisfied. In terms of masses of physical states and M 2 , the last condition can be written as This can be easily satisfied if λ 4 ≥ 0 and the mass spectrum is that m A(H) > m H + > m H ++ . If the spectrum is m A(H) < m H + < m H ++ , the vacuum stability condition places an upper bound on the mass of H ++ : While boundedness-from-below is only a necessary condition for vacuum stability, we do not expect a second minimum deeper than the electroweak vacuum to exists since we always assume that v ∆ v recovering approximately the SM vacuum structure.

E. Landau Pole
From Sec. III, we see that a larger choice of m lightest generically requires a larger value of λ 5 . This generally tells us that the Landau pole could potentially be very close to m lightest . For our purposes, we will denote the Landau pole as the scale at which the running coupling λ 5 (µ) grows to 4π.
Here, we will compute the one-loop beta function for λ 5 . For simplicity, we will only compute the leading λ 2 5 term. The one-loop counterterm for λ 5 in d = 4 − 2 is given by At leading order in λ 5 , the wave functions of Φ and ∆ are not renormalized at the one-loop level. As such, the one-loop beta function for λ 5 is simply Solving the RGE yields The curve for m lightest = 1 TeV in Fig. 1 crosses the CDF-II band at |λ 5 | ∼ 7. Inserting this value into Eq. (E3) implies a Landau pole at ∼ 12m lightest = 12 TeV. In this situation, additional BSM physics preventing the appearance of the Landau pole should appear in the multi-TeV range. For a more precise estimate also subleading RGE effects would need to be taken into account.

F. Decoupling
In the literature, the one-loop corrected m W is often presented as a function of |δm| ≡ |m H + − m H ++ |, as shown in Fig. 10. At first sight, it is confusing that to reach a given amount of m W increment, |δm| stays almost the same as m lightest increases (for the type-I case, it even decreases).
Naturally, one would expect that the BSM corrections go to zero in the limit M → ∞.
To understand such behavior, we should notice that decoupling behavior is only manifest if ∆m 2 ≡ m 2 H ± − m 2 H ±± or λ 5 = 4∆m 2 /v 2 is fixed. If, however, |δm| is fixed and M is increased, no decoupling occurs. This happens because and therefore |λ 5 | ∼ M δm/v 2 leads to |λ 5 | → ∞ in the limit M → ∞. Consequently, this limit will unavoidably violate perturbative unitarity. (We truncated all the curves once |λ 5 | grows to 10 in Fig. 10.)