Tripling down on the W boson mass

Abstract

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.

1 Introduction

Recently, the CDF-II experiment [1] measured the W boson mass to be

$$\begin{aligned} m_{W,\text {CDF-II}} = 80.4335 \pm 0.0094~\text {GeV}. \end{aligned}$$

(1)

This suggests a 7$\sigma$ derivation from the Standard Model (SM) prediction [2],

$$\begin{aligned} m_{W,\text {SM}} = 80.357 \pm 0.006~\text {GeV}. \end{aligned}$$

(2)

The CDF-II measurement of $m_W$ is also in tension with the measurements from the previous collider experiments at $\sim 2.6\sigma$ [2, 3]. The discrepancy might be due to some unknown experimental systematical uncertainties, but it could also be a hint for new physics [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88]. 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 [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]. 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 $m_{W,\mathrm{CDF-II}}$ 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 ${\mathbb {Z}}_2$ 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 Sect. 2, we briefly review the Higgs Triplet Model; in Sect. 3, 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 Sect. 4, the compatibility with the Higgs precision measurement in Sect. 5, the bounds and discovery channels from the LHC direct searches in Sect. 6, and their cosmological implications in Sect. 7. We conclude in Sect. 8. In the appendices, we give details of the self-energy corrections and the SM fitting formula and discuss the soft ${\mathbb {Z}}_2$ breaking limit, the unitarity and vacuum stability bounds, the Landau pole, and the decoupling limit of the HTM.

2 Higgs triplet model

In the HTM, the Higgs sector contains an isospin doublet $\Phi$ with hypercharge $Y=\frac{1}{2}$ and an isospin triplet $\Delta$ with $Y=1$.¹ They can be parameterized as

$$\begin{aligned} \begin{aligned} \Phi&= \left( \begin{array}{c} G^+ \\ \frac{v_{\phi }+h+iG^0}{\sqrt{2}} \end{array} \right) ~,~ \Delta&= \left( \begin{array}{c c} \frac{H^+}{\sqrt{2}}&{}H^{++} \\ \Delta ^0&{}-\frac{H^+}{\sqrt{2}} \end{array} \right) ~~\mathrm{with}~\\ \Delta ^0&=\frac{v_\Delta +H+iA}{\sqrt{2}}, \end{aligned} \end{aligned}$$

(3)

where $v_\phi$ and $v_\Delta$ are the vacuum expectation values (vev’s) of the doublet and triplet field obeying

$$\begin{aligned} v^2\equiv v_\phi ^2+2v_\Delta ^2\approx (246\,\mathrm{GeV})^2. \end{aligned}$$

(4)

In addition to the SM-like Higgs boson, the scalar sector contains six new Higgs bosons (degrees of freedom): the $\mathcal{CP}\mathcal{}$-even H boson, the $\mathcal{CP}\mathcal{}$-odd A boson, the singly-charged $H^\pm$ bosons, and the doubly-charged $H^{\pm \pm }$ bosons.

In this model, the tree level W and Z boson masses are given by

$$\begin{aligned} m_W^2=\frac{g^2}{4}v^2,~\quad m_Z^2=\frac{g^2}{4c_W^2}(v^2+2v_\Delta ^2), \end{aligned}$$

(5)

where $c_W^2 \equiv \cos ^2 \theta _W$ and $\theta _W$ 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

$$\begin{aligned} m_W=m_{W,\mathrm{SM}}^\text {tree}\left( 1-\frac{v_\Delta ^2}{v^2}\right) + \Delta m_{W}, \end{aligned}$$

(6)

where $\Delta m_{W}$ 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., $v_\Delta \ll v$. Assuming the difference between the CDF-II measurement and the SM prediction mainly comes from the loop correction $\Delta m_{W}$, i.e.,

$$\begin{aligned} \frac{v_\Delta ^2}{v^2}\ll \frac{m_{W,\mathrm{CDF-II}}-m_{W,\mathrm{SM}}}{m_{W,\mathrm{SM}}}~, \end{aligned}$$

(7)

this restricts $v_\Delta \ll 7.6~ \mathrm{GeV}$. To be concrete, we assume that $v_\Delta < 1~\mathrm{GeV}$ in the rest of the paper. For simplicity, we will work in the limit $v_\Delta =0$ 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 $v_\Delta ^2/v^2 \lesssim 2\times 10^{-5}$ and will be ignored.

2.1 The Inert Triplet

The limit of $v_\Delta =0$ can be realized in a strict sense by imposing a ${\mathbb {Z}}_2$ symmetry, under which $\Phi$ is ${\mathbb {Z}}_2$-even and $\Delta$ is ${\mathbb {Z}}_2$-odd. This ${\mathbb {Z}}_2$ 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

$$\begin{aligned} V(\Phi ,\Delta )= & {} m^2\Phi ^\dagger \Phi +M^2\mathrm{Tr}(\Delta ^\dagger \Delta )\nonumber \\&\quad +\lambda _1\left( \Phi ^\dagger \Phi \right) ^2 +\lambda _2\left[ \mathrm{Tr}(\Delta ^\dagger \Delta )\right] ^2\nonumber \\&\quad +\lambda _3\mathrm{Tr}\left[ (\Delta ^\dagger \Delta )^2\right] \nonumber \\&\quad +\lambda _4\left( \Phi ^\dagger \Phi \right) \mathrm{Tr}(\Delta ^\dagger \Delta ) +\lambda _5 \Phi ^\dagger \Delta \Delta ^\dagger \Phi , \end{aligned}$$

(8)

where all the parameters in the potential can be taken to be real. The minimization of the potential yields

$$\begin{aligned} m^2=-\lambda _1v^2,\quad v_{\Delta }=0. \end{aligned}$$

(9)

In terms of the physical states, the quadratic part of the Higgs potential is

$$\begin{aligned} \begin{aligned} V(\Phi ,\Delta )&\supset \frac{1}{2} (2\lambda _1 v^2) h^2 + \frac{1}{2} \left( M^2+\frac{\lambda _4 v^2}{2}\right) \\&\quad \times \left( A^2+H^2+ 2H^{++}H^{--}+ 2H^+H^-\right) \\&\quad +\frac{1}{4} \lambda _5 v^2 \left( A^2+H^2+H^+H^-\right) ~. \end{aligned} \end{aligned}$$

(10)

Then, the mass spectrum is given by

$$\begin{aligned} \begin{aligned} m_h^2&=2\lambda _1v^2,\\ m_A^2&=m_H^2=M^2+(\lambda _4+\lambda _5)\frac{v^2}{2},\\ m_{H^+}^2&=M^2+\lambda _4\frac{v^2}{2}+\lambda _5\frac{v^2}{4},\\ m_{H^{++}}^2&=M^2+\lambda _4\frac{v^2}{2}. \end{aligned} \end{aligned}$$

(11)

We can substitute the Higgs potential parameters $m^2$, $M^2$, $\lambda _1$, $\lambda _5$ by v, $m_h^2$, $m_{A,H}^2=m_A^2=m_H^2$, $m_{H^+}^2$, and $m_{H^{++}}^2$, where $v= (\sqrt{2} G_F)^{-1/2}\simeq 246$ GeV and $m_h\simeq 125$ GeV. The free parameters in this model are thus given by

$$\begin{aligned} m_{A,H},\quad m_{H^+},\quad m_{H^{++}},\quad \lambda _2,\quad \lambda _3,\quad \lambda _4 \end{aligned}$$

(12)

with the condition that

$$\begin{aligned} m_{H^{+}}^2 - m_{H^{++}}^2 = m_{A,H}^2 - m_{H^+}^2 = \frac{\lambda _5 v^2}{4}. \end{aligned}$$

(13)

I.e., the coupling $\lambda _5$ controls the splitting of the mass spectrum. For the rest of the discussion, the model with $\lambda _5 >0$ ($\lambda _5 <0$) 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.

3 One-loop corrected W boson mass

If $H^{++}, H^+$ and H, A have sizable mass splittings, i.e., if $|\lambda _5|$ 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 $v_\Delta =0$. Finite values for $v_\Delta$ 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 Appendix A (see also Ref. [89]). Table 1 lists all of the input parameters [2] used in the computation.

Table 1 Input parameters used in computation. Note that $\Delta \alpha$ is the sum of the hadronic contribution $\Delta \alpha _\text {had}^{(5)} (m_Z^2) = 0.02766$ and the leptonic contribution $\Delta \alpha _\text {lept} = 0.031497687$ [90]

To determine the W mass from the measurement of the Fermi coupling constant $G_F$, we note that,

$$\begin{aligned} G_F=\frac{\pi \alpha _{\text {em},0}}{\sqrt{2}m_{W,0}^2 s_{W,0}^2}\left( 1+\frac{{\Pi _{WW}}(0)}{m_W^2}+\delta _{VB}\right) , \end{aligned}$$

(14)

where terms with 0 subscripts are the bare parameters, $\Pi _{WW}$ is the self-energy of the W, and $\delta _{VB}$ 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

$$\begin{aligned} \begin{aligned} G_F&=\frac{\pi \alpha _{\text {em}}}{\sqrt{2}m_{W}^2 s_{W}^2}\left( 1+\frac{\delta \alpha _\text {em}}{\alpha _\text {em}}-\frac{\delta m_W^2}{m_W^2}-\frac{\delta s_W^2}{s_W^2}\right. \\&\left. \quad +\frac{{\Pi }_{WW}(0)}{m_W^2}+\delta _{VB}\right) \equiv \frac{\pi \alpha _{\text {em}}}{\sqrt{2}m_{W}^2 s_{W}^2}(1+\Delta r), \end{aligned} \end{aligned}$$

(15)

where $\delta \alpha _\text {em}$, $\delta m_W^2$, and $\delta s_W^2$ are the counterterms for the fine-structure constant $\alpha _\text {em}$, the W mass, and weak mixing angle $s_W\equiv \sin \theta _W$, 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,

$$\begin{aligned} \delta {s^2_W}=-\delta {c^2_W}=-c^2_W\left( \frac{\delta m_W^2}{m_W^2}-\frac{\delta m_Z^2}{m_Z^2}\right) . \end{aligned}$$

(16)

The $\alpha _\text {em}$ counterterm, which we also define in the on-shell scheme, is given by

$$\begin{aligned} \frac{\delta \alpha _\text {em}}{\alpha _\text {em}}=\Pi '_{\gamma \gamma }(0)+ 2\frac{{c}_W}{{s}_W}\frac{\Pi _{Z\gamma }^{\mathrm{1PI}}(0)}{m_Z^2}, \end{aligned}$$

(17)

where $\Pi '_{\gamma \gamma }(0)\equiv d \Pi _{\gamma \gamma }(p^2)/dp^2|_{p^2=0}$. Combining everything, $\Delta r$ is at the one-loop level given by

$$\begin{aligned} \Delta r&=\Pi '_{\gamma \gamma }(0) +\frac{\Pi ^{\mathrm{1PI}}_{WW}(0)-\mathrm{Re}\Pi ^{\mathrm{1PI}}_{WW}(m_W^2)}{m_W^2}\nonumber \\&\quad +\frac{{c}_W^2}{{s}_W^2}\left( \frac{\mathrm{Re}\Pi ^{\mathrm{1PI}}_{WW}(m_W^2)}{m_W^2}\right. \nonumber \\&\left. \quad -\frac{\mathrm{Re}\Pi ^{\mathrm{1PI}}_{ZZ}(m_Z^2)}{m_Z^2} \right) \nonumber \\&\quad +2\frac{{c}_W}{{s}_W}\frac{\Pi _{Z\gamma }^{\mathrm{1PI}}(0)}{m_Z^2}+\delta _{VB}. \end{aligned}$$

(18)

The vertex and box diagram corrections to the muon decay, $\delta _{VB}$, are given by (see e.g. Ref. [91])

$$\begin{aligned} \delta _{VB} = \frac{\alpha _\text {em}}{4\pi {s}_W^2}\left( 6+\frac{7c_W^2+3 {s}_W^2}{2s_W^2}\ln c_W^2\right) , \end{aligned}$$

(19)

where we neglected the contributions proportional to the electron and muon Yukawa couplings.

Based on Eq. (15), we can then write

$$\begin{aligned} {m_W^2}=m_Z^2 \times \left( \frac{1}{2}+\sqrt{\frac{1}{4}-\frac{\pi \alpha _{\mathrm{em}}}{\sqrt{2}G_Fm_Z^2}\left( 1+\Delta r(m_{W})\right) } \right) , \end{aligned}$$

(20)

which we can iterate to solve for $m_W$.

For the numerical implementation, we follow the procedure outlined e.g. in [92]. We split $\Delta r$ into three parts: the one-loop SM contributions that depend on $m_W$ as an input ($\Delta r_{\text {SM}, W}$), the remaining one-loop and higher-order SM contributions ($\Delta r_{\text {SM, rest}}$), and the beyond-the-Standard-Model (BSM) contributions ($\Delta r_\text {BSM}$),

$$\begin{aligned} \Delta r = \Delta r_{\text {SM}, W}(m_{W, \text {BSM}}) + \Delta r_{\text {SM, rest}} + \Delta r_\text {BSM}. \end{aligned}$$

(21)

The quantity $\Delta r_\text {SM, rest}$ is given by

$$\begin{aligned} \Delta r_{\text {SM, rest}} =\Delta r_\text {SM}- \Delta r_{\text {SM}, W}(m_{W, \text {SM}}), \end{aligned}$$

(22)

where $m_{W,\text {SM}}$ is computed from the fitting formula given in Ref. [93] (see Appendix B). The fitting formula can also be used to obtain a number for $\Delta r_\text {SM}$ (i.e., $\Delta r_\text {SM} \simeq 0.03807$). Combining the Eqs. (21) and (22) together yields

$$\begin{aligned} \Delta r (m_{W, \text {BSM}})&= \Delta r_\text {SM} - \Delta r_{\text {SM}, W}(m_{W, \text {SM}}) \nonumber \\&\quad + \Delta r_{\text {SM}, W}(m_{W, \text {BSM}}) + \Delta r_\text {BSM}, \end{aligned}$$

(23)

This equation consistently combines the full HTM one-loop corrections with the SM higher-order corrections, which are crucial for a precise result.

Fig. 1

One-loop corrected W boson mass $m_W$ as a function of the coupling $|\lambda _5| = 4|m_{H^+}^2 - m_{H^{++}}^2|/v^2$ for various masses of the lightest state in the HTM. We assume the mass hierarchy of the new states following $m_{H^{++}}< m_{H^+}< m_A$ ($m_{H^{++}}> m_{H^+}> m_A$) in the left (right) panel. Different curves in each panel represent different masses for the lightest state. The brown (dark purple) line represent the CDF II measured (PDG) value and the yellow (purple)/gray band shows the $1\sigma$/$2\sigma$ intervals

In the left (right) panel of Fig. 1, we show the resulting numerical value for $m_W$ as a function of $|\lambda _5|=\lambda _5$ ($|\lambda _5|=-\lambda _5$) and $m_\text {lightest}=m_{H^{++}}$ ($m_\text {lightest}=m_{A,H}$) for the type-I (II) HTM.² In both panels, we depict the CDF measured (PDG) value as a brown (dark purple) line, the 1$\sigma$ region as a yellow (purple) band, and the 2$\sigma$ region as gray bands. For a fixed value of the lightest BSM state $m_\text {lightest}$, the one-loop corrected W boson mass increases with $|\lambda _5|$. For a fixed shift in the W mass, a heavier $m_\text {lightest}$ requires a larger value of $|\lambda _5|$. With the same $m_\text {lightest}$, the type-I model needs a larger value of $|\lambda _5|$ to obtain the same W mass shift compared to the type II.

The largish value of $|\lambda _5|$ required for large choices of $m_\text {lightest}$ could potentially cause the appearance of a Landau pole close to the electroweak scale. As we show in Appendix E, no Landau pole appears below $\sim 10 \text { TeV}$. This makes the additional contribution from the UV completion above the Landau pole subleading in comparison to those considered here.

Fig. 2

Mass spectrum of the new Higgs states for a given $|\lambda _5|$ that yields the CDF-II measured central values of $m_W$, $m_W-2\sigma _{m_W}$, or $m_W+2\sigma _{m_W}$ for the type-I and type-II HTM. We exclude the mass spectrum that corresponds $m_{H^{++}} < 42.9\,\text {GeV}$ (excluded by Z decays [94]) for the type-I model and that corresponds to $m_{A,H} < m_h/2 = 62.5\,\text {GeV}$ (excluded by Higgs precision measurement, see Sect. 5.2) for the type-II model. In each panel, we explicitly show three sets of benchmark values

Furthermore, we scan $m_\text {lightest}$ and $\lambda _5$ to pinpoint the parameter regions predicting a $m_W$ value close to the CDF-II measurement. The resulting mass spectra for the new Higgs states are shown in Fig. 2. The first, third, and fifth (second, fourth, and sixth) panels respectively represent the spectra for the type-I (II) HTM that yield the $m_W$ value measured by CDF-II, the CDF-II value plus two times the experimental one-sigma uncertainty, and the CDF-II value minus two times the experimental one-sigma uncertainty. The blue, purple, and red lines in each panel represent the corresponding values of $m_{A,H}$, $m_{H^{+}}$, and $m_{H^{++}}$, respectively. We also explicitly show three sets of benchmark values for each scenario. For the type-I HTM, we do not show the mass spectrum corresponding to $m_{H^{++}} < 42.9\,\text {GeV}$ since it is excluded by the measurement of Z boson decays [94]. For the type-II HTM, we do not show the mass spectrum corresponding to $m_{H,A} < m_h/2 =62.5\,\text {GeV}$ given it is excluded by the precision measurement of exotic Higgs decays as we discuss in Sect. 5.2. 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 Sect. 6.

4 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, $\sin ^2\theta _\text {eff}$. In this computation, $\alpha _\text {em},~M_Z,$ and $G_F$ are chosen as inputs. Experimentally, $\sin ^2\theta _\text {eff}$ 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, $s_W^2$, can be parameterized by $\Delta \kappa$, where

$$\begin{aligned} \sin ^2\theta _\text {eff}=s_W^2(1+\Delta \kappa ). \end{aligned}$$

(24)

At one-loop, $\Delta \kappa$ obtains contributions from $A-Z$ mixing, corrections to the weak mixing angle, and corrections to the axial/vector vertices,

$$\begin{aligned} \Delta \kappa&= -\frac{c_W}{s_W}\left( \frac{\text {Re}\Pi ^{1\text {PI}}_{Z\gamma }(m_Z^2)}{m_Z^2}\right) \nonumber \\&\quad -\frac{{c}_W^2}{{s}_W^2}\left( \frac{\mathrm{Re}\Pi ^{\mathrm{1PI}}_{WW}(m_W^2)}{m_W^2}-\frac{\mathrm{Re}\Pi ^{\mathrm{1PI}}_{ZZ}(m_Z^2)}{m_Z^2} \right) \nonumber \\&\quad +\frac{v_l}{v_l-a_l}\left( \frac{F^l_V(m_z^2)}{v_l}-\frac{F_A^l(m_z^2)}{a_l}\right) , \end{aligned}$$

(25)

where $v_l$ and $a_l$ are the tree-level vector and axial couplings respectively, and $F_{V,A}^l$ are the form-factors for the leptonic vector/axial currents. Since the extra Higgs bosons do not couple to the SM fermions, they do not contribute to $F_{V,A}^l$.

Fig. 3

Effective weak mixing angle vs. W boson mass for the type I (upper row) and II (lower row) mass hierarchies. For each panel, the different curves represent models with different mass values for the lightest state. For each curve, we vary $|\lambda _5|$ from 0 (corresponds to the SM values) to 10. This finite range of $|\lambda _5|$ 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 $1\sigma$/$2\sigma$ range. The dark purple line represent the PDG value for the W boson mass with the purple/gray band showing the $1\sigma$/$2\sigma$ range. The dark green line in the left (right) column represent the world averaged value $0.23153\pm 0.00016$ [2, 95] (SLD measured value $0.23098\pm 0.00026$ [95]) of $\sin ^2 \theta _\text {eff}$ with the green/gray band shows $1\sigma$/$2\sigma$ range

We compute the SM contribution to $\sin ^2 \theta _\text {eff}$ with the help of the SM fitting formula. Similar to the treatment of $\Delta r$, we split $\Delta \kappa$ into three pieces,

$$\begin{aligned} \Delta \kappa = \Delta \kappa _{\text {SM},W} (m_{W, \text {BSM}})+ \Delta \kappa _\text {SM, rest} + \Delta \kappa _\text {BSM}~. \end{aligned}$$

(26)

$\Delta \kappa _\text {SM, rest}$ is determined via

$$\begin{aligned} \Delta \kappa _\text {SM} = \Delta \kappa _{\text {SM},W} (m_{W, \text {SM}})+ \Delta \kappa _\text {SM, rest}, \end{aligned}$$

(27)

where $\Delta \kappa _\text {SM} \simeq 0.03640$ is computed from the fitting formula given in Ref. [96] (see Appendix B) and $\Delta \kappa _{\text {SM},W} (m_{W, \text {SM}}) \simeq 0.03628$ is Eq. (25) restricted to the SM contribution only, which explicitly depends on $m_W$.

In Fig. 3, we check whether the parameter space of the HTM that predicts $m_W$ close to the CDF-II value is compatible with the measured effective mixing angle. The upper (lower) row of Fig. 3 shows the resulting $\sin ^2 \theta _\text {eff}$ vs. $m_W$ plot for a given $m_\text {lightest}$ in the interval $[100~\text {GeV}, 1000~\text {GeV}]$ and $|\lambda _5|\in [0, 10]$ for the type-I (II) HTM. For each panel, we highlight the CDF-II (PDG) $m_W$ value as the brown (dark purple) vertical lines while the yellow (purple) and gray vertical bands show the $1\sigma$ and $2\sigma$ ranges, respectively. The dark green horizontal lines in the left column represent the world-average value for the effective weak mixing angle [2, 95] while the green and gray horizontal band shows 1$\sigma$ and 2$\sigma$ 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 $|\lambda _5|=0$, 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 $|\lambda _5|$ increases, the resulting $m_W$ increases while the resulting $\sin ^2 \theta _\text {eff}$ decreases.³ On the other hand, a change in $m_\text {lightest}$ has a less significant impact (at least for $m_\text {lightest} > rsim 400\,\text {GeV}$). Note that a heavier $m_\text {lightest}$ yields a larger deviation from the world average for $\sin ^2\theta _\text {eff}$ for the type-I model while it yields a smaller departure for type II. For the type-I model, the parameter space that explains $m_{W, \text {CDF-II}}$ is consistent with the world averaged value of $\sin ^2\theta _\text {eff}$ within $2\sigma$ level. For the type-II model, the two measurements are inconsistent at the $2\sigma$ level for the $m_\text {lightest}$–$|\lambda _5|$ parameter space that we scanned. If we instead compare $\sin ^2 \theta _\text {eff}$ to the value measured by the SLD collaboration [95], we find that the parameter space explaining $m_{W, \text {CDF-II}}$ is consistent with the measured $\sin ^2 \theta _\text {eff}$ within the $2\sigma$ 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 $\sin ^2\theta _\text {eff}$ 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.

5 Precision measurement of the standard model Higgs

For $v_\Delta \simeq 0$, 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.

5.1 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

$$\begin{aligned} \kappa _\gamma ^2 \equiv \frac{\Gamma _{H\rightarrow \gamma \gamma }}{\Gamma ^\text {SM}_{H\rightarrow \gamma \gamma }}. \end{aligned}$$

For the triplet model, it is given by

$$\begin{aligned} \kappa _\gamma ^2&= \frac{\left| \frac{4}{3}F_{1/2}(\tau _t)+\cdots +Q_{H^{\pm \pm }}^2\frac{v^2\lambda _{h H^{\pm \pm }H^{{\mp }{\mp }}}}{m_h^2} F_{\pm }(\tau _{H^{\pm \pm }})+Q_{H^{\pm }}^2\frac{v^2\lambda _{h H^{\pm }H^{{\mp }}}}{m_h^2} F_{\pm }(\tau _{H^{\pm }})\right| ^2}{\left| \frac{4}{3}F_{1/2}(\tau _t)+\cdots \right| ^2}, \end{aligned}$$

(28)

where Q denotes the electric charge; $\tau _f \equiv m_h^2/(4 m_f^2)$; and the ellipsis denotes subleading SM contributions. The scalar couplings are given by

$$\begin{aligned} \lambda _{h H^{\pm \pm }H^{{\mp }{\mp }}}&= - v\lambda _4, \lambda _{h H^{\pm }H^{{\mp }}} = -v(\lambda _4 + \lambda _5/2). \end{aligned}$$

(29)

The loop functions $F_{1/2}$ and $F_\pm$ have the form

$$\begin{aligned} F_{1/2}(\tau )&= \frac{(\tau - 1) f(\tau ) + \tau }{\tau ^2},\quad F_{\pm }(\tau ) = \frac{\tau - f(\tau )}{\tau } \end{aligned}$$

(30)

with

$$\begin{aligned} f(\tau ) = {\left\{ \begin{array}{ll} \arcsin ^2(\sqrt{\tau }) &{} \text {if } \tau \le 1, \\ -\frac{1}{4}\left( \ln \frac{1+\sqrt{1-1/\tau }}{1-\sqrt{1-1/\tau }} - i\pi \right) ^2 &{} \text {if } \tau > 1. \end{array}\right. } \end{aligned}$$

(31)

We evaluate the LHC constraints set on the triplet couplings through modifications of the $H\rightarrow \gamma \gamma$ rate by employing HiggsSignals [97, 98].

In addition to the di-photon rate, we also evaluate loop corrections to the trilinear Higgs self-coupling, 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, $\kappa _\lambda$, to the strongest current bound of $-1.0\le \kappa _\lambda \le 6.6$ [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. While quantum corrections to double-Higgs production are not only induced by corrections to the trilinear Higgs self-coupling, evaluating the one-loop corrections to the trilinear Higgs self-coupling takes into account all one-loop corrections to double Higgs production leading in powers of scalar couplings. Since the scalar couplings are responsible for the dominant deviation from the SM, this justifies applying the bound of Ref. [104].

Fig. 4

Constraints on $(m_{H^{++}},m_{H^{+}})$ parameter space for the HTM from the measurement of Higgs-photon coupling ($\kappa _\gamma$, blue shaded region), Higgs self-coupling ($\kappa _\lambda$, orange shaded region), perturbative unitarity (green hatched region), and meta-stability condition of the vacuum (red hatched region). We set $\lambda _4=0$ and $\lambda _\Delta =1$ for all panels. The left (right) column shows the $2\sigma$ favored parameter space that explained the measured $m_W$ by CDF-II for the type-I (-II) model as red (blue) narrow bands. Note that we do not show the parameters for $|\lambda _5| > 10$ in drawing the narrow bands. The parameter space with $m_{H^{++}} > rsim 250\,\text {GeV}$ ($m_{H^{++}} \lesssim 350\,\text {GeV}$) for the type-I (-II) HTM remains unconstrained. We also show the LHC constraints on $m_{H^{++}}$ for type-I if $H^{++}$ decays promptly (gray band) or if it is detector stable (left of the gray dash line). See Sect. 6.2 for more details

The constraints in the $(m_{H^{++}},m_{H^{+}})$ parameter plane due to modifications of $\kappa _\gamma$ and $\kappa _\lambda$ are shown in Fig. 4. The blue shaded region shows the excluded parameter space by measurements of the Higgs di-photon rate (demanding compatibility at 95% CL); the orange shaded region is excluded by the constraint on the trilinear Higgs couplings (at 95% CL). Moreover, we show the constraints set by perturbative unitarity (green hashed region) and by the (meta-)stability of the electroweak vacuum (red hashed region), which we evaluate as detailed in Appendix D. Note that we set $\lambda _4 = 0$ and $\lambda _2 = \lambda _3 = 1$ in drawing the plots.⁴

For the left panel of Fig. 4, we concentrate on the type-I hierarchy. Almost the complete lower right half of the parameter plane is excluded by requiring metastability of the electroweak vacuum. Perturbative unitarity excludes large differences between $m_{H^{++}}$ and $m_{H^+}$. Measurements of the Higgs to di-photon rate additionally exclude a portion of the parameter space around $m_{H^{++}}\sim 300\, \text {GeV}$ and $m_{H^{++}}\sim 450\, \text {GeV}$ unconstrained by perturbative unitary and vacuum stability. The experimental measurements of the Higgs trilinear coupling are, so far, not precise enough to probe parameter space unconstrained by perturbative unitarity and vacuum stability in the considered scenario. We find, the parameter space favored by the CDF-II measurement of $m_W$ (red narrow band) with $m_{H^{++}} > rsim 250\, \text {GeV}$, which lies close to the diagonal, to not lie in the parameter space excluded by the above mentioned constraints. In addition to the constraints discussed above, we also show the LHC constraints on $m_{H^{++}}$ if $H^{++}$ decays promptly (gray band) or if it is detector stable (gray dash line). These constraints are discussed in detail in Sect. 6.2 below.

For type II (see the right panel of Fig. 4), the constraints set by the Higgs couplings, perturbative unitarity and vacuum stability are unchanged. The parameter space favored by the CDF-II $m_W$ measurements (blue narrow bands) is, however, shifted downwards with respect to the type-I hierarchy. As a result, the parameter space favored by the CDF-II $m_W$ measurements lies at the boundary of the region excluded by demanding vacuum stability. The parameter space favored by $m_{W,\text {CDF-II}}$ is only accessible for $m_{H^{++}}\lesssim 350\,\text {GeV}$. Note, however, that the evaluating of the vacuum stability constraint relies on various assumptions (see Appendix D).

5.2 Exotic decays of the Higgs boson

For the type-I HTM, the branching ratio for $h\rightarrow H^{++} H^{--}$ depends on $\lambda _4$ (see Eq. 29). This coupling needs to be small ($\lesssim 1$ for $m_\text {lightest} \sim {\mathcal {O}}(100)\;\text {GeV}$) in order to evade constraints from the di-photon decay rate of the SM-like Higgs boson. This leads to negligibly small exotic decay modes for the SM-like Higgs boson (if at all kinematically accessible). The situation is quite different for the type-II HTM. In this case, the exotic decay modes of the SM-like Higgs boson are mainly given by $h\rightarrow HH$, $h\rightarrow AA$, and $h\rightarrow H^+ H^-$ once they are kinetically accessible. Their branching ratios mostly depend on $\lambda _5$ (29), which needs to be large to explain the CDF-II measurement of $m_W$.

Fig. 5

(Upper) Main production channels of the exotic Higgs states at the LHC. (Lower) Decay chains for the heaviest new Higgs state in type I (left) and type II (right). In type II, the lightest state H/A can decay to both SM fermions (as shown on the right) and the SM-like Higgs and gauge bosons

Fig. 6

NLO pair production cross sections for Type-I (left) and Type-II (right) as a function of $m_\text {lightest}$ at the 14 TeV run of the LHC

We compute the branching ratio for the SM-like Higgs boson decays to the BSM Higgs states for the type-II model as a function of $m_{A,H}$ for the parameter space that explains $m_{W, \text {CDF-II}}$. We find the resulting branching ratio to lie between $80\%-97\%$ if the decay modes are kinematically accessible ($m_{A,H} < m_h/2$). Such a large branching ratio for the exotic decays is in tension with the Higgs precision measurements from the LHC. For example, the ATLAS experiment places a 95% CL constraint of $\text {Br}(h\rightarrow \text {BSM}) = \text {Br}(h\rightarrow \text {inv}) + \text {Br}(h\rightarrow \text {undetected}) < 49\%$ [105]. A similar constraint has also been placed by CMS experiment [106]. These constraints exclude the type-II model if neutral states are lighter than $m_h/2$.

6 Direct searches at the LHC

In this Section, we study potential LHC signals of the new scalars with a spectrum preferred by $m_{W,\mathrm{CDF-II}}$. As we have demonstrated in the previous sections, an explanation of the W mass deviation in the context of the HTM points to new Higgs bosons below a TeV which makes them targets for direct searches at the LHC. As a guide for the dedicated experimental searches in the future, the main goal of this section is to highlight the promising search channels with distinct signatures. Of course, some of the LHC searches designed to look for different signal processes would also have sensitivity 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.

6.1 Production

In the absence of additional Yukawa-type interaction terms and for $v_\Delta \ll v$, 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 v2.9.10 [108]. The dependence of the production cross sections on the lightest BSM state mass in the respective model type is shown in Fig. 6. Here, we have chosen the mass spectrum such that we can reproduce the CDF-II central value for $m_W$ as shown in Fig. 2.

Fig. 7

Lifetime (times the speed of light) of the lightest exotic Higgs state for type I (left) and type II (right) as a function of mass and $v_\Delta$ (setting $\lambda _2=\lambda _3=\lambda _4=0$). The remaining parameters are chosen such that CDF-II $m_{W}$ value is explained. The sharp drop at around 160 GeV correspond to the threshold at which the WW decay becomes on shell. For type II (right), additional sharp drops occur at around 250 (215) GeV where $H\rightarrow hh$ ($A\rightarrow Zh$) becomes on shell; furthermore, $m_H$ is restricted to be $\notin (120,130)$ GeV since H maximally mixes with h there (see Appendix C for more details). For reference, we have drawn dashed lines representing $c\tau = 10^{-4}$ meter (corresponding to $\sim 10^{-12}$ 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

For type I (see left panel of Fig. 6), the $pp\rightarrow H^{\pm \pm }H^{{\mp }}$ channel mediated by a W boson has the largest cross section of up to $\sim 1$ pb for $m_{H^{++}}\sim 100\;\text {GeV}$. The $pp\rightarrow H^{\pm \pm }H^{{\mp }{\mp }}$ production cross section is of similar size (especially for lower mass values). Less important are the $pp\rightarrow H^\pm H + H^\pm A$, $pp\rightarrow HA$, and the $pp\rightarrow H^\pm H^{\mp }$ production channels.

The overall behavior is similar for type II (see right panel of Fig. 6). As a consequence of H and A being the lightest BSM Higgs bosons, the $pp\rightarrow H^\pm H + H^\pm A$ and $pp\rightarrow HA$ channels have, however, now the largest cross sections given their larger phase spaces. Their cross sections reach $\sim 1$ pb for $m_{H,A}\sim 100\;\text {GeV}$.

In our discussion of potential search strategies at the LHC below, we will only focus on the production channels with the largest cross sections.

6.2 Detection signatures

In order to correctly reproduce the W mass measured by CDF-II, the triplet vev $v_\Delta$ generically needs to be small. Given the size of $v_\Delta$ 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 $v_\Delta$ directly affects the lifetime of the lightest state, thus affecting the detection signature at the LHC. We discuss the cosmological implications in Sect. 7.

In Fig. 7, we show this lifetime of the lightest state for different choices of $v_\Delta$ ranging from $10^{-8}\,\mathrm{GeV}$ to 1 GeV for the type-I/II HTM. For $v_\Delta \sim 10^{-4}$ 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 $v_\Delta \sim 10^{-8}$ 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.

Fig. 8

Branching ratios for H (left) and A (right) in the type-II HTM for $\lambda _2=\lambda _3=\lambda _4=0$, $v_\Delta =1$ GeV, with a mass spectrum explaining the CDF-II $m_W$ measurement. Branching ratios for two-body (three-body) decays are shown as solid (dashed) lines. Since all decay widths are proportional to $v_\Delta ^2$, the branching ratios do not depend on the choice of $v_\Delta$ provided that it is nonzero

We further show the decay branching ratio of H and A for type-II model in Fig. 8 (assuming a small but finite $v_\Delta$).⁵ Generically, the branching ratio of the dominant decay mode is always very close to one. This dominant decay mode depends on whether or not the preferred final state is kinematically accessible. For the $\mathcal{CP}\mathcal{}$-even BSM Higgs boson, H, the important thresholds are the hh and WW mass thresholds. For the $\mathcal{CP}\mathcal{}$-odd Higgs boson, A, the important threshold is the Zh mass threshold. Below the lowest mass threshold, they both predominantly decay to $b{\bar{b}}$ due to the bottom Yukawa inherited from its mixing with the SM Higgs doublet.

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.

6.2.1 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 Table 2.

Table 2 Summary of main channels and example search signatures for additional Higgs bosons of the HTM that promptly decay at the LHC. The upper and lower tables consider the type-I and II HTM, respectively. See text for more details

For type-I HTM, the production process with the largest cross section is $pp\rightarrow H^{\pm \pm }H^{{\mp }}$. The singly-charged Higgs boson then decays to a doubly-charged Higgs boson via emission of a W boson, $H^{{\mp }}\rightarrow H^{{\mp }{\mp }} W^{\pm }$. All doubly-charged Higgs bosons will then promptly decay into a pair of W bosons, $H^{\pm \pm } \rightarrow W^{\pm } W^{\pm }$, with branching ratio $\approx 1$. (c.f., the lower left diagram of Fig. 5.) As such, the corresponding search channel will be a final state of five W bosons. These W bosons could be off-shell depending on the masses.

No dedicated searches for this channel exist so far. To nevertheless gain an estimate for the LHC sensitivity for this signature, we use CheckMATE 2.2 [109,110,111,112,113,114,115] to recast a large set of existing searches on a set of benchmark points. CheckMATE will generically summarize the result with $r=S/S_{95}$, where 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 $\sqrt{\int {\mathcal {L}}dt}$. 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 $84\, \text {GeV}\le m_{H^{++}}\le 200 \,\text {GeV}$ 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, $pp\rightarrow H^{\pm \pm }H^{{\mp }{\mp }}$. 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] . Their search excludes doubly charged Higgs promptly decaying into W bosons with masses $200\, \text {GeV}\le m_{H^{++}}\le 350 \,\text {GeV}$. Studies recasting 8 TeV ATLAS data excludes the mass range $m_{H^{++}} < 84\,\text {GeV}$ [94, 119].

Another significant production process for the type-I HTM is $pp\rightarrow H^\pm H/A$. The neutral Higgs boson in the type-I HTM decays to a singly-charged Higgs boson via $W^{(*)}$ emission with a branching ratio close to one. ($H/A \rightarrow W^{{\mp }} H^{\pm }$, c.f., the lower left diagram of Fig. 5.) Fully decaying all of the extra Higgs bosons will generate a final state of seven $W^{(*)}$ bosons. The corresponding experimental final state will contain various jets, leptons and missing transverse energy. We have checked a benchmark point with $m_{H^{++}}=350$ GeV using the search in Ref. [120] in signal region SR12, and found that it is not sensitive to this point.

For the type-II HTM, the production process with the largest cross section is $pp\rightarrow H^\pm H/A$. The singly-charged Higgs boson will decay to H/A via W boson emission, $H^\pm \rightarrow W^\pm H/A$; both H and A have roughly the same probability of being produced. From Fig. 8, the neutral Higgs boson will likely decay to either to a heavy fermion pair or a pair of SM bosons. As before, all of these SM bosons could be off-shell. For this scenario, we ran CheckMATE for both $m_H=100$ GeV and $m_H=300$ GeV. We find both benchmark values to be allowed using the built-in 13 TeV run analyses. The $m_H=100$ GeV benchmark point scenario yielded $r\approx 0.6$ using the search of Ref. [121] in the 3b1j signal region. As this study only used 3.2 fb$^{-1}$ of 13 TeV data, one could potentially exclude the benchmark (i.e., cases where di-boson decays are kinematically forbidden) at $2\sigma$ using a dedicated search with existing data. The $m_H=300$ GeV benchmark point yielded $r\approx 0.1$ using the search of Ref. [120]. Even with the full HL-LHC dataset, it seems unlikely that a re-analysis could exclude this parameter point based on naive rescaling alone. This analysis is not dedicated to this particular search. It does not make use of the h or Z in the final state.

For the type-II HTM with a light H/A, $pp\rightarrow HA$ production can be sizable. For $m_H=100$ GeV, recasting Ref. [121] in the same signal regions as the previous production mode yielded $r\approx 0.4$. Once again, naive luminosity based scaling indicates that existing data is potentially sufficient to exclude this. For $m_H=300$ GeV, we obtained $r\approx 0.02$ 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.

6.2.2 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 $v_\Delta \lesssim 10^{-8}\text { GeV}$.

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 $\lambda _5$ to give the desired shift in the W mass as shown in Fig. 2.

In type II, starting with the $pp\rightarrow H^\pm H/A$ production, a generic final state consists of $W^*$ and missing transverse energy (MET). As such, the search channels are either monolepton + MET or dijet + MET. Recasting existing searches using CheckMATE did not yield any exclusions for the $m_H=100$ GeV benchmark point. $pp\rightarrow H^{\pm \pm }H^{{\mp }}$ production leads to a different final state with more visible particles. The final state consists now out of three $W^{(*)}$ boson. The final state signature could be three charged leptons + MET, two charged leptons + jets + MET, monolepton + jets + 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 $pp\rightarrow HA$ production, the main search channel is a mono-jet or mono-photon + 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.

6.2.3 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 multi-lepton/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 $H^{\pm \pm }$ 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.

7 Cosmological implications

For sufficiently small $v_\Delta$, 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 $|\lambda _5|$ 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.

Fig. 9

(Upper left) Sum of the relative relic abundances for H and A in the type-II HTM with respect to that of cold dark matter $f_\chi = \Omega _{H+A}/\Omega _c$ as a function of $m_{H,A}=m_H=m_A$ for parameters that explain the measured $m_{W,\text {CDF-II}}$ within $2\sigma$. (Lower left) 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 $2\sigma$ (blue band) (assuming $v_\Delta < 10^{-16}\,\text {eV}$). 95% CL constraints from the LZ experiment with 5.5 ton$\cdot$ 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 $\gamma$-ray backgrounds [132] are shown as yellow, green, and brown shaded regions. We highlighted the range of $f_\chi$ that explains $m_{W,\text {CDF-II}}$ 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. In the upper axes, we show the corresponding values of $v_\Delta$ for $m_A = 65\,\text {GeV}$ and $m_A=1.5\,\text {TeV}$ as red and blue ticks, respectively. See text for more details

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 of cold dark matter, $f_\chi \equiv \Omega _{H+A}/\Omega _c$ as a function of $m_{A,H}\equiv m_H\simeq m_A$⁶ for model parameters that explains the CDF-II measured $m_W$ within $2\sigma$. The relative relic abundance of A and H ranges from $10^{-7}$ to $10^{-1}$ of the total dark matter abundance. It reaches a maximum of $7\%$ around $m_{A,H}\approx m_W$ and converges to $\sim 1\%$ for $m_{A,H} > 700\,\text {GeV}$. The dips around $m_{A,H} =m_Z/2$ and $m_{A,H} =m_h/2$ correspond to the resonant enhancement of the annihilation cross sections. Note that the parameter space below $m_{A,H} < m_h/2$ (shaded in gray) is excluded by Higgs precision measurements at the LHC (see Sect. 5.2). If we restrict $m_{H,A}$ to be away from the resonant region of $m_h/2$, i.e. $m_{H,A} > 63\, \text {GeV}$, the relative abundance $f_\chi$ varies from $0.08\%$ to $7\%$.

To discuss the observational signatures of the massive relic, we consider two scenarios according to the lifetime of H and A: (i) H and A are cosmologically stable and (ii) H and A are cosmologically unstable.⁷ For the parameters that explain $m_{W,\text {CDF-II}}$, the lifetime of H and A mostly depends on the size of $v_\Delta$ and weakly depends on $m_{H,A}$. Besides the two parameters, the observational signature of the massive relic additionally depends on $f_\chi$, which is determined by $m_{A,H}$ as shown in the upper left panel of Fig. 9.

7.1 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 $\tau _{H,A} > rsim \tau _U=10^{18}\,\text {sec}$. A stronger constraints on $\tau _{A,H}$ comes from the observations of the diffused $\gamma$-ray backgrounds [132, 134,135,136]. Observations from Fermi-LAT telescope restrict the decaying time of dark matter $\tau _\chi > rsim 10^{28}\,\text {sec}$ if it consists all the dark matter [132]. We translate this bound into $\tau _{H,A} > rsim 10^{28} f_\chi \,\text {sec}$ if only an $f_\chi$ 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, $v_\Delta$ needs to be small. In the right panel of Fig. 9, we explicitly show the value of $v_\Delta$ for a given lifetime for A with mass $m_A=1.5\,\text {TeV}$ as the blue upper ticks. For the $m_{A,H}$ parameter space we consider, we find that setting

$$\begin{aligned} v_\Delta \lesssim 10^{-16}\,\text {eV} \end{aligned}$$

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 $f_\chi$ between the massive relic and cold dark matter stays the same for the local dark matter environment (with cold dark matter density $\rho _{\text {local}} = 0.3\,\text {GeV}/\text {cm}^3$). Besides, the two share the same velocity distribution. The resulting weighted cross section (blue band), which is favored to explain $m_{W,\text {CDF-II}}$, is ranging from $10^{-49}$ to $10^{-44}\,\text {cm}^2$ 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$\cdot$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 $m_{W,\text {CDF-II}}$ 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.

7.2 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 $\gamma$-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 $\gamma$-ray background [132] as yellow, green, and brown shaded regions, respectively. To get the BBN constraints, we take the constraints on massive relic $\chi$ with $m_\chi = 1\,\text {TeV}$ that decaying to $b{\bar{b}}$ from Ref. [129] ⁸. This constraints are representative for massive relic that mostly decaying to hadronic energy. As shown in Ref. [129], lighter relic ($m_\chi = 30 \,\text {GeV}$ and $m_\chi =100\,\text {GeV}$) or other hadronic energy-dominant decay channels ($\chi \rightarrow {{\bar{u}}} u, {{\bar{t}}} t, gg, WW$) share similar constraints. Constraints for massive relics decaying to electromagnetic energy, e.g. $\chi \rightarrow e^+e^-$, 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 $b{\bar{b}}$, WW, 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 $m_{H,A} > 63\,\text {GeV}$ (away from the fine-tuned mass region) whose corresponding parameters explain $m_{W,\text {CDF-II}}$. For such an abundance range (0.08%–7%), the strongest constraints for the decaying relic come from BBN, which restrict $\tau _{A,H} \lesssim 50\,\text {sec}$. To satisfy this constraint, the value of $v_\Delta$ needs to be large. In the right panel of Fig. 9, we explicitly show the value of $v_\Delta$ for a given lifetime of A with mass $m_A=65\,\text {GeV}$ as the red upper ticks. For the $m_{A,H}$ parameter space we consider, we find that

$$\begin{aligned} v_\Delta > rsim 1\,\text {eV} \end{aligned}$$

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 $v_\Delta \ge 1\,\text {eV}$ corresponds to $c\tau _{H,A} \lesssim 1\,\text {km}$. Such decay signal could be searched at the long-lived particle search facilities at the LHC.

8 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 $m_{H^{++}} > rsim 250\,\text {GeV}$ explain the observed $m_{W,\text {CDF-II}}$ 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 $62.5\,\text {GeV}\lesssim m_{H,A}\lesssim 350\,\text {GeV}$ explain the observed $m_{W,\text {CDF-II}}$ 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 $\sin ^2_{\theta _{\mathrm{eff}}}$ at the $2\sigma$ 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 $v_\Delta$ and $m_\text {lightest}$ (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 Table 2). Current LHC searches are most sensitive to the type-I HTM with a prompt decay of the lightest state (excluding $200\,\text {GeV}< m_{H^{++}} < 350\,\text {GeV}$) or detector-stable lightest state (excluding $m_{H^{++}} < 1050\,\text {GeV}$). A dedicated analysis using current data can also exclude a promptly decaying doubly-charged Higgs with $m_{H^{++}}\sim 150$ GeV by studying the $H^{\pm \pm }H^{{\mp }}$ production channel. The case of a long-lived lightest state is so far largely unconstrained for type I. Dedicated searches with the existing data could effectively cover the parameter space for the type-II HTM, especially given the constrained mass range for which the CDF-II measurement can be explained while evading other constraints (see above).

Furthermore, we explored the scenario that the new Higgs triplet is approximately inert. In this case, its lightest neutral state can be a candidate for a sub-dominate fraction of stable dark matter if $v_\Delta \lesssim 10^{-16}\, \text {eV}$ or decaying dark matter if $v_\Delta > rsim 1\,\text {eV}$. The former scenario is almost fully constrained by current dark matter direct detection experiments such as the LZ experiment. The later scenario remains possible.

Appendices

Appendix A: Self-energy corrections

We take the one-loop contributions to self energies from Ref. [89], setting $v_\Delta =0$, in the computation of $m_W$ and $\sin ^2\theta _\text {eff}$. We listed all the relevant formula here for readers’ convenience. The corrections are parameterized in terms of $g^2 = e^2/s_W^2$ and $g_Z^2 = e^2/(s_W^2 c_W^2)$.

The BSM contributions to the vector-boson self energies are given by

$$\begin{aligned} \Pi _{WW}^\text {1PI, BSM}(p^2)&=\frac{g^2}{16\pi ^2}\Big ( B_5(p^2, m_{H^{++}}^2, m_{H^+}^2) \nonumber \\&\quad +\frac{1}{2} B_5(p^2, m_{H^+}^2, m_H^2)\nonumber \\&\quad +\frac{1}{2} B_5(p^2, m_{H^+}^2, m_A^2)\Big ), \end{aligned}$$

(A1)

$$\begin{aligned} \Pi _{ZZ}^\text {1PI, BSM}(p^2)&=\frac{g_Z^2}{16\pi ^2} \Big ((c_W^2-s_W^2)^2 B_5(p^2, m_{H^{++}}^2, m_{H^{++}}^2) \nonumber \\&\quad + s_W^4 B_5(p^2, m_{H^{+}}^2, m_{H^{+}}^2)\nonumber \\&\quad + B_5(p^2, m_H^2, m_A^2)\Big ), \end{aligned}$$

(A2)

$$\begin{aligned} \Pi _{\gamma \gamma }^\text {1PI, BSM}(p^2)&=\frac{e^2}{16\pi ^2}\big (4 B_5(p^2, m_{H^{++}}^2, m_{H^{++}}^2)\nonumber \\&\quad +B_5(p^2, m_{H^{+}}^2, m_{H^{+}}^2)\big ), \end{aligned}$$

(A3)

$$\begin{aligned} \Pi _{\gamma \gamma }^{\prime \text {1PI, BSM}}(p^2)&=\frac{e^2}{16\pi ^2}\big (4 B'_5(p^2, m_{H^{++}}^2, m_{H^{++}}^2)\nonumber \\&\quad +B'_5(p^2, m_{H^{+}}^2, m_{H^{+}}^2)\big ), \end{aligned}$$

(A4)

$$\begin{aligned} \Pi _{Z \gamma }^\text {1PI, BSM}(p^2)&=-\frac{e g_Z}{16 \pi ^2}\Big (2(c_W^2-s_W^2)B_5(p^2, m_{H^{++}}^2, m_{H^{++}}^2) \nonumber \\&\quad +\frac{1}{2}(c_W^2-s_W^2-1)B_5(p^2, m_{H^{+}}^2, m_{H^{+}}^2)\Big ) \end{aligned}$$

(A5)

where $B_{0,1,00,11}(p^2, m_1^2,m_2^2)$ are the Passarino-Veltman two-point functions, which we evaluate using LoopTools 2.16 [101]. The remaining loop-functions are given by

$$\begin{aligned} B_3(p^2, m_1^2, m_2^2)&= -B_1(p^2, m_1^2, m_2^2) \nonumber \\&\quad - B_{11}(p^2, m_1^2, m_2^2), \end{aligned}$$

(A6)

$$\begin{aligned} B_4(p^2, m_1^2, m_2^2)&= -m_1^2 B_1(p^2, m_2^2, m_1^2) \nonumber \\&\quad - m_2^2 B_{1}(p^2, m_1^2, m_2^2) \end{aligned}$$

(A7)

$$\begin{aligned} B_5(p^2, m_1^2, m_2^2)&= A_0(m_1^2)+ A_0(m_2^2)\nonumber \\&\quad -4B_{00}(p^2, m_1^2, m_2^2), \end{aligned}$$

(A8)

where $A_0(m^2)$ is the Passarino–Veltman one-point function.

The SM contributions to $\Delta r$ (and $\sin ^2 \theta _\text {eff}$) can be separated into three classes: those from scalar bosons, fermions, and gauge bosons – i.e., $\Pi _{i}^\text {1PI, SM}(p^2) = \Pi _{i,S}^\text {1PI, SM}(p^2) +\Pi _{i,F}^\text {1PI, SM}(p^2) + \Pi _{i,V}^\text {1PI, SM}(p^2)$ where $i= WW, ZZ, \gamma \gamma , Z\gamma$. The scalar contributions are given by

$$\begin{aligned} \Pi _{WW}^\text {1PI, SM}(p^2)_S&=\frac{g^2}{64\pi ^2}\left( B_5(p^2,m_W^2, m_h^2)\right. \nonumber \\&\left. \quad + B_5(p^2,m_W^2, m_Z^2)\right) , \end{aligned}$$

(A9)

$$\begin{aligned} \Pi _{ZZ}^\text {1PI, SM}(p^2)_S&=\frac{g_Z^2}{64\pi ^2}\left( (c_W^2-s_W^2)^2 B_5 (p^2, m_W^2, m_W^2)\right. \nonumber \\&\quad \left. + B_5(p^2, m_h^2, m_Z^2) \right) , \end{aligned}$$

(A10)

$$\begin{aligned} \Pi _{Z\gamma }^\text {1PI, SM}(p^2)_S&= -\frac{e g_Z}{32\pi ^2}(c_W^2-s_W^2) B_5(p^2, m_W^2, m_W^2), \end{aligned}$$

(A11)

$$\begin{aligned} \Pi _{\gamma \gamma }^\text {1PI, SM}(p^2)_S&=\frac{e^2}{16\pi ^2} B_5(p^2, m_W^2, m_W^2), \end{aligned}$$

(A12)

$$\begin{aligned} {\Pi '}_{\gamma \gamma }^\text {1PI, SM}(p^2)_S&=\frac{e^2}{16\pi ^2} B'_5(p^2, m_W^2, m_W^2). \end{aligned}$$

(A13)

The fermionic contributions are given by

$$\begin{aligned} \Pi _{WW}^\text {1PI, SM}(p^2)_F&= \frac{g^2}{16\pi ^2} N_c^f (2 p^2 B_3(p^2, m_f^2, m_{f'}^2)\nonumber \\&\quad -B_4(p^2, m_f^2, m_{f'}^2)), \end{aligned}$$

(A14)

$$\begin{aligned} \Pi _{ZZ}^\text {1PI, SM}(p^2)_F&= \frac{g_Z^2}{8\pi ^2}N_c^f \left( 2 p^2 (2 s_W^4 Q_f^2- 2 s_W^2 Q_f I_f+ I_f^2)\right. \nonumber \\&\quad B_3(p^2, m_f^2, m_f^2)\nonumber \\&\quad \left. - I_f^2 m_f^2 B_0(p^2, m_f^2, m_f^2)\right) , \end{aligned}$$

(A15)

$$\begin{aligned} \Pi _{Z\gamma }^\text {1PI, SM}(p^2)_F&= \frac{e g_Z}{4\pi ^2} N_c^f p^2 (2 s_W^2 Q_f^2 -I_f Q_f) B_3 (p^2, m_f, m_f), \end{aligned}$$

(A16)

$$\begin{aligned} \Pi _{\gamma \gamma }^\text {1PI, SM}(p^2)_F&= \frac{e^2}{2\pi ^2} N_c^f Q_f^2 p^2 B_3(p^2, m_f^2, m_f^2), \end{aligned}$$

(A17)

$$\begin{aligned} \Pi _{\gamma \gamma }^{\prime \text {1PI, SM}}(p^2)_F&= \frac{e^2}{2\pi ^2} N_c^f Q_f^2 \left( B_3(p^2, m_f^2, m_f^2)\right. \nonumber \\&\left. \quad + p^2 B'_3(p^2, m_f^2, m_f^2)\right) \end{aligned}$$

(A18)

where $m_f$, $Q_f$, $I_f$, and $N_c^f$ 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

$$\begin{aligned}&\Pi _{WW}^\text {1PI, SM}(p^2)_V\nonumber \\&\quad = \frac{g^2}{16\pi ^2}\bigg (m_W^2 \left( B_0(p^2, m_h^2, m_W^2)+s_W^2 B_0(p^2, m_W^2, 0)\right. \nonumber \\&\qquad \left. +\frac{s_W^4}{c_W^2}B_0(p^2, m_W^2, m_Z^2)\right) \nonumber \\&\qquad -c_W^2(6 D-8) B_{00}(p^2, m_Z^2, m_W^2) \nonumber \\&\qquad - 2 p^2 c_W^2 B_{11}(p^2, m_Z^2, m_W^2) \nonumber \\&\qquad - 2 p^2 c_W^2 B_{1}(p^2, m_Z^2, m_W^2) \nonumber \\&\qquad - 5 p^2 c_W^2 B_{0}(p^2, m_Z^2, m_W^2) \nonumber \\&\qquad +c_W^2 (D-1) A_0(m_Z^2)\nonumber \\&\qquad +(D-1) A_0(m_W^2)-s_W^2 (6 D-8) B_{00}(p^2, 0, m_W^2)\nonumber \\&\qquad -2 p^2 s_W^2 B_{11}(p^2, 0, m_W^2) \nonumber \\&\qquad - 2 p^2 s_W^2 B_1(p^2, 0, m_W^2)\nonumber \\&\qquad - 5 p^2 s_W^2 B_0(p^2, 0, m_W^2) \bigg ) \nonumber \\&\qquad -\frac{g^2}{4 \pi ^2}(p^2 -m_W^2)\left( c_W^2 B_0 (p^2, m_Z^2, m_W^2)\right. \nonumber \\&\left. \qquad + s_W^2 B_0(p^2, 0, m_W^2)\right) , \end{aligned}$$

(A19)

$$\begin{aligned}&\Pi _{ZZ}^\text {1PI, SM}(p^2)_V\nonumber \\&\quad =\frac{g_Z^2}{16 \pi ^2}\bigg (m_Z^2 B_0(p^2, m_h^2, m_Z^2) \nonumber \\&\qquad + 2m_W^2 s_W^4 B_0(p^2, m_W^2, m_W^2)\nonumber \\&\qquad -c_W^4(6D-8)B_{00}(p^2, m_W^2, m_W^2)\nonumber \\&\qquad - 2 p^2 c_W^4 B_{11}(p^2, m_W^2, m_W^2) \nonumber \\&\qquad - 2 p^2 c_W^4 B_1(p^2, m_W^2, m_W^2) \nonumber \\&\qquad - 5 p^2 c_W^4 B_0(p^2, m_W^2, m_W^2)\nonumber \\&\qquad +2(D-1)c_W^4 A_0(m_W^2)\bigg ) \nonumber \\&\qquad -\frac{g_Z^2}{4\pi ^2}(p^2-m_Z^2)c_W^4 B_0(p^2, m_W^2, m_W^2), \end{aligned}$$

(A20)

$$\begin{aligned}&\Pi _{Z\gamma }^\text {1PI, SM}(p^2)_V = \frac{e g_Z}{16\pi ^2}\bigg (c_W^2(6D-8)B_{00}(p^2, m_W^2, m_W^2)\nonumber \\&\qquad +2 p^2 c_W^2 B_{11}(p^2, m_W^2, m_W^2)\nonumber \\&\qquad +2 p^2 c_W^2 B_1(p^2, m_W^2, m_W^2)\nonumber \\&\qquad + 5 p^2 c_W^2 B_0(p^2, m_W^2, m_W^2)\nonumber \\&\qquad -2 c_W^2(D-1)A_0(m_W^2) \nonumber \\&\qquad +2 m_W^2 s_W^2 B_0(p^2, m_W^2, m_W^2)\bigg )\nonumber \\&\qquad +\frac{e g_Z}{8\pi ^2} \left( 2p^2-m_Z^2\right) c_W^2 B_0(p^2, m_W^2, m_W^2), \end{aligned}$$

(A21)

$$\begin{aligned}&\Pi _{\gamma \gamma }^\text {1PI, SM}(p^2)_V \nonumber \\&\quad =-\frac{e^2}{16 \pi ^2}\bigg ((6D-8) B_{00}(p^2, m_W^2, m_W^2)\nonumber \\&\qquad + 2 p^2 B_{11}(p^2, m_W^2, m_W^2) + 2 p^2 B_1(p^2, m_W^2, m_W^2)\nonumber \\&\qquad + 5 p^2 B_0(p^2, m_W^2, m_W^2)- 2(D-1)A_0(m_W^2)\nonumber \\&\qquad - 2 m_W^2 B_0(p^2, m_W^2, m_W^2)\bigg ) \nonumber \\&\qquad -\frac{ e^2}{4\pi ^2} p^2 B_0(p^2, m_W^2, m_W^2), \end{aligned}$$

(A22)

$$\begin{aligned}&\Pi _{\gamma \gamma }^{\prime \text {1PI, SM}}(p^2)_V \nonumber \\&\quad =-\frac{e^2}{16 \pi ^2}\bigg ((6D-8) B'_{00}(p^2, m_W^2, m_W^2)\nonumber \\&\qquad + 2 B_{11}(p^2, m_W^2, m_W^2) + 2 p^2 B'_{11}(p^2, m_W^2, m_W^2) \nonumber \\&\qquad + 2 B_1(p^2, m_W^2, m_W^2) \nonumber \\&\qquad + 2 p^2 B'_1(p^2, m_W^2, m_W^2)\nonumber \\&\qquad + 5 B_0(p^2, m_W^2, m_W^2) \nonumber \\&\qquad + 5 p^2 B'_0(p^2, m_W^2, m_W^2) \nonumber \\&\qquad - 2 m_W^2 B'_0(p^2, m_W^2, m_W^2)\bigg ) \nonumber \\&\qquad -\frac{ e^2}{4\pi ^2} \left( B_0(p^2, m_W^2, m_W^2)+ p^2 B'_0(p^2, m_W^2, m_W^2)\right) , \end{aligned}$$

(A23)

where $D=4-2\epsilon$ with $\epsilon$ being the dimensional regulator. The divergences of the loop functions are given by

$$\begin{aligned} A_0(m^2)_\text {div}&= m^2 \Delta , \quad B_0(p^2, m_1^2, m_2^2)_\text {div} =\Delta \nonumber \\ B_1(p^2, m_1^2, m_2^2)_\text {div}&= -\frac{\Delta }{2}, \quad B_{11}(p^2, m_1^2, m_2^2)_\text {div} =\frac{\Delta }{3} \nonumber \\ B_{00}(p^2, m_1^2, m_2^2)_\text {div}&= \left( \frac{m_1^2+m_2^2}{4}-\frac{p^2}{12}\right) \Delta , \nonumber \\ B'_{00}(p^2, m_1^2, m_2^2)_\text {div}&= -\frac{\Delta }{12}, \nonumber \\ B_3(p^2, m_1^2, m_2^2)_\text {div}&=\frac{\Delta }{6}, B_4(p^2, m_1^2, m_2^2)_\text {div} =\frac{m_1^2 +m_2^2}{2} \Delta \nonumber \\ B_5(p^2, m_1^2, m_2^2)_\text {div}&= \frac{p^2}{3} \Delta , \quad B'_5 (p^2, m_1^2, m_2^2)_\text {div} =\frac{\Delta }{3}, \end{aligned}$$

(A24)

where $\Delta = \frac{1}{\epsilon } + \ln \mu ^2$ with $\mu$ being the renormalization scale.

Appendix B: The SM fitting formula

The SM prediction for the W boson mass, $m_W$, and the leptonic effective mixing angle, $\sin ^2 \theta _{W}$, are parameterized by the fitting formula in Ref. [93] and Ref. [96], respectively. We list them here for readers’ convenience. The fitting formula for $m_{W, \text {SM}}$ is given by

$$\begin{aligned} m_{W,\text {SM}}&= m_{W}^0 - c_1 dH - c_2 dH^2 +c_3 dH^4\nonumber \\&\quad +c_4 (dh-1) -c_5 d\alpha + c_6 dt \nonumber \\&\quad -c_7 dt^2 - c_8 dH dt +c_9 dh dt - c_{10} d\alpha _s + c_{11} dZ, \end{aligned}$$

(B1)

where

$$\begin{aligned} dH&= \ln \left( \frac{m_h}{100~\text {GeV}}\right) ,\quad&dh =\left( \frac{m_h}{100~\text {GeV}}\right) ^2, \nonumber \\ dt&=\left( \frac{m_t}{174.3~\text {GeV}}\right) ^2-1, \quad&dZ = \frac{m_Z}{91.1875~\text {GeV}} -1, \nonumber \\ d\alpha&= \frac{\Delta \alpha }{0.05907}-1, \quad&d\alpha _s = \frac{\alpha _s(m_Z^2)}{0.119}-1 \end{aligned}$$

(B2)

and the coefficients are given by

$$\begin{aligned} m_{W}^0&= 80.3799 \;\text {GeV}, \nonumber \\ c_1&= 0.05263 \;\text {GeV}, \nonumber \\ c_2&= 0.010239 \;\text {GeV}, \nonumber \\ c_3&= 0.000954 \;\text {GeV}, \nonumber \\ c_4&= -0.000054 \;\text {GeV}, \nonumber \\ c_5&= 1.077 \;\text {GeV}, \nonumber \\ c_6&= 0.5252 \;\text {GeV}, \nonumber \\ c_7&= 0.0700 \;\text {GeV}, \nonumber \\ c_8&= 0.004102 \;\text {GeV}, \nonumber \\ c_9&= 0.000111 \;\text {GeV}, \nonumber \\ c_{10}&= 0.0774 \;\text {GeV}, \nonumber \\ c_{11}&= 115.0 \;\text {GeV}. \end{aligned}$$

(B3)

This fitting formula includes the complete one-loop and two-loop results [96, 141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157]. Moreover, partial higher-order corrections up to four-loop order are included [158,159,160,161,162,163,164,165,166,167].

The fitting formula for $\sin ^2 \theta _{\text {eff, SM}}^{\ell }$ is given by

$$\begin{aligned} \sin ^2 \theta _{\text {eff, SM}}^{\ell }&= s_0 + d_1 L_H + d_2 L_H^2 + d_3 L_H^4 \nonumber \\&\quad +d_4 (\Delta _H^2-1)+ d_5 \Delta _\alpha +d_6 \Delta _t \nonumber \\&\quad + d_7 \Delta _t^2 + d_8 \Delta _t (\Delta _H-1)+ d_9 \Delta _{\alpha _s} + d_{10} \Delta _Z, \end{aligned}$$

(B4)

where

$$\begin{aligned} L_H&= \ln \left( \frac{m_h}{100~\text {GeV}}\right) ,\quad \Delta _H = \frac{m_h}{100~\text {GeV}}, \nonumber \\ \Delta _\alpha&=\frac{\Delta \alpha }{0.05907}-1,\quad \Delta _t = \left( \frac{m_t}{178.0~\text {GeV}}\right) ^2-1, \nonumber \\ \Delta _{\alpha _s}&=\frac{\alpha _s (m_Z^2)}{0.117}-1, \quad \Delta _Z = \frac{m_Z}{91.1876~\text {GeV}}-1 \end{aligned}$$

(B5)

and the coefficients

$$\begin{aligned} s_0&= 0.2312527,\nonumber \\ d_1&= 4.729\times 10^{-4},\nonumber \\ d_2&= 2.07\times 10^{-5},\nonumber \\ d_3&= 3.85 \times 10^{-6},\nonumber \\ d_4&= -1.85\times 10^{-6},\nonumber \\ d_5&= 2.07\times 10^{-2}, \nonumber \\ d_6&= -2.851\times 10^{-3},\nonumber \\ d_7&= 1.82\times 10^{-4}, \nonumber \\ d_8&= -9.74 \times 10^{-6}, \nonumber \\ d_9&= 3.98\times 10^{-4},\nonumber \\ d_{10}&= -0.655. \end{aligned}$$

(B6)

This fitting formula is based on the full one-, and two-loop corrections as well as the leading three- and four-loop corrections computed in Refs. [143,144,145,146,147,148, 158,159,160, 162, 163, 168]. The corresponding $\Delta \kappa _\text {SM}$ can then be derived by

$$\begin{aligned} \Delta \kappa _\text {SM} = \frac{\sin ^2 \theta _{\text {eff, SM}}^{\ell }}{1-m_{W,\text {SM}}^2/m_Z^2}-1. \end{aligned}$$

(B7)

Appendix C: Soft ${\mathbb {Z}}_2$ breaking

Here we introduce a soft ${\mathbb {Z}}_2$ breaking term in the Higgs potential

$$\begin{aligned} \begin{aligned} \Delta V&=\mu \Phi ^{\mathrm{T}}i\sigma _2\Delta ^\dagger \Phi +h.c. =-\sqrt{2}\mu Hh^2\\&\quad +\mu (-\sqrt{2}H^-G^+G^0+H^{--}G^+G^++h.c.) \end{aligned} \end{aligned}$$

(C1)

where $\mu$ is assumed to be real. Such a term breaks the degeneracy between the two neutral states H, A, therefore it is expected that $m_H^2-m_A^2\propto \mu ^2$. $\Delta V$ generates a non-zero $v_\Delta$:

$$\begin{aligned} \begin{aligned} v_\Delta \approx&\frac{\sqrt{2}}{2M^2/v^2+\lambda _{4}+\lambda _5}\mu +{\mathcal {O}}(\mu ^3),\\ v^2_\phi \approx&-\frac{m^2}{\lambda _1}+{\mathcal {O}}(\mu ^2)=v^2-2v_\Delta ^2. \end{aligned} \end{aligned}$$

(C2)

Defining $\epsilon \equiv \sqrt{2}v_\Delta /v\ll 1$, the explicit ${\mathbb {Z}}_2$ breaking mixes the states of $\Phi$ and $\Delta$ with the same quantum numbers at ${\mathcal {O}}(\epsilon )$. To avoid confusion, we write the weak eigenstates as

$$\begin{aligned} \begin{aligned} \Phi '= \left( \begin{array}{c} H_1^+ \\ \frac{v_{\phi }+H_1+iA_1}{\sqrt{2}} \end{array} \right) ~,~ \Delta '&= \left( \begin{array}{c c} \frac{H_2^+}{\sqrt{2}}&{}H^{++} \\ \frac{v_\Delta +H_2+iA_2}{\sqrt{2}}&{}-\frac{H_2^+}{\sqrt{2}} \end{array} \right) \end{aligned} \end{aligned}$$

(C3)

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:

$$\begin{aligned} h\approx & {} H_1 +\frac{M^2}{M_\Delta ^2-2\lambda _1v^2}\sqrt{2}\epsilon H_2~,~ \nonumber \\ G^+\approx & {} H_1^++\epsilon H_2^+~,~ \nonumber \\ G^0\approx & {} A_1 +\sqrt{2}\epsilon A_2~, \end{aligned}$$

(C4)

where $M_\Delta ^2=M^2+\frac{1}{2}(\lambda _{4}+\lambda _5)v^2$. 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 $\epsilon$, the physical states have masses given by

$$\begin{aligned} m_h^2= & {} 2\lambda _1v^2-\left( 2\lambda _1v^2+2M^2\frac{M^2}{M_\Delta ^2-2\lambda _1v^2}\right) \epsilon ^2,\nonumber \\ m_H^2= & {} M_\Delta ^2+\left( v^2(\lambda _{2}+\lambda _3)+M^2\frac{M^2}{M_\Delta ^2-2\lambda _1v^2}\right) \epsilon ^2,\nonumber \\ m_A^2= & {} M_\Delta ^2+2M_\Delta ^2\epsilon ^2,\nonumber \\ m_{H^+}^2= & {} M_\Delta ^2-\frac{\lambda _5}{4}v^2+M_\Delta ^2 \epsilon ^2,\nonumber \\ m_{H^{++}}^2= & {} M_\Delta ^2-\frac{\lambda _5}{2}v^2+\frac{\lambda _5-\lambda _3}{2}v^2\epsilon ^2. \end{aligned}$$

(C5)

From (C4), when $m_H^2\rightarrow m_h^2$, the mixing parameter between $H_1$ and $H_2$ diverges. This means that h and H can be maximally mixed even when $\epsilon \ll 1$. In this limit, the $H-h$ mixing depends on the details of the Higgs potential parameters.

Appendix 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 $2\rightarrow 2$ scattering matrix

$$\begin{aligned} |x_i|< 8\pi ,\quad i=1,2,3 \end{aligned}$$

(D1)

where

$$\begin{aligned} \begin{aligned} x_1&=3\lambda _1+7\lambda _{\Delta }+\sqrt{(3\lambda _1-7\lambda _\Delta )^2+\frac{3}{2}(2\lambda _4+\lambda _5)^2}~,~\\ x_2&=\frac{1}{2}(2\lambda _4+3\lambda _5)~,~ x_3=\frac{1}{2}(2\lambda _4-\lambda _5) \end{aligned}. \end{aligned}$$

(D2)

Taking $\lambda _\Delta \equiv \lambda _2=\lambda _3>0$, the necessary and sufficient condition for the Higgs potential to be bounded from below is

$$\begin{aligned} \begin{aligned} \lambda _1>0,~\lambda _\Delta>0,~ 2\sqrt{2\lambda _1\lambda _\Delta }+\lambda _4+\mathrm{min}(0,\lambda _5)>0 \end{aligned} \end{aligned}$$

(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

$$\begin{aligned}&\frac{m_h}{v}\sqrt{\lambda _\Delta }+\frac{m_{H^{++}}^2-M^2}{v^2}\nonumber \\&\quad +\mathrm{min}\left( 0,2\frac{m_{H^+}^2-m_{H^{++}}^2}{v^2}\right) >0 \end{aligned}$$

(D4)

This can be easily satisfied if $\lambda _4\ge 0$ 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 $H^{++}$:

$$\begin{aligned} m_h v\sqrt{\lambda _\Delta }+2m_{H^+}^2-m_{H^{++}}^2>M^2>0 \end{aligned}$$

(D5)

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 $v_\Delta \ll v$ recovering approximately the SM vacuum structure.

The conditions in Eq. (D5) imposes absolute stability. One can slightly relax this assumption by demanding metastability with a lifetime longer than the age of the Universe. For large field values, the quartic term dominates allowing to solve analytical for the bounce action B [170]. Following Refs. [171, 172], we then demand that $B < 440$ resulting in the condition

$$\begin{aligned}&\frac{1}{4}\lambda _1 \cos ^4\varphi + (\lambda _2 + \lambda _3)\sin ^4\varphi \nonumber \\&\quad + \frac{1}{8}(\lambda _4 + \lambda _5)\cos ^2\varphi \sin ^2\varphi < - \frac{\pi ^2}{165}\simeq -0.06 \end{aligned}$$

(D6)

for any $\varphi \in [0,2\pi ]$.

Appendix E: Landau pole

From Sect. 3, we see that a larger choice of $m_\text {lightest}$ generically requires a larger value of $\lambda _5$. This generally tells us that the Landau pole could potentially be very close to $m_\text {lightest}$. For our purposes, we will denote the Landau pole as the scale at which the running coupling $\lambda _5(\mu )$ grows to $4\pi$.

Here, we will compute the one-loop beta function for $\lambda _5$. For simplicity, we will only compute the leading $\lambda _5^2$ term. The one-loop counterterm for $\lambda _5$ in $d=4-2\epsilon$ is given by

$$\begin{aligned} \delta ^{(1)}\lambda _5=\frac{\lambda _5^2}{8\pi ^2}\left( \frac{1}{\epsilon }+\text {finite}\right) . \end{aligned}$$

(E1)

At leading order in $\lambda _5$, the wave functions of $\Phi$ and $\Delta$ are not renormalized at the one-loop level. As such, the one-loop beta function for $\lambda _5$ is simply

$$\begin{aligned} \beta _{\lambda _5}= \frac{d\lambda _5}{d\ln \mu }=\frac{\lambda _5^2}{4\pi ^2} \end{aligned}$$

(E2)

Solving the RGE yields

$$\begin{aligned} \Lambda _\text {Landau-pole}=m_\text {lightest}\exp \left( \frac{4\pi ^2}{\lambda _5(m_\text {lightest})}-\pi \right) . \end{aligned}$$

(E3)

The curve for $m_\text {lightest} = 1$ TeV in Fig. 1 crosses the CDF-II band at $|\lambda _5| \sim 7$. Inserting this value into Eq. (E3) implies a Landau pole at $\sim 12 m_\text {lightest} = 12$ 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.

Fig. 10

1-loop corrected W boson mass $m_W$ as a function of $|\delta m| = |m_{H^+}- m_{H^{++}}|$ for various masses of the lightest state in the HTM. We assume the mass hierarchy of the new states following $m_{H^{++}}< m_{H^+}< m_A$ ($m_{H^{++}}> m_{H^+}> m_A$) in the left (right) panel. We do not show the parameters for $|\lambda _5|>10$ in drawing the curves. The color bands are the same as those in Fig. 1

Decoupling

In the literature, the one-loop corrected $m_W$ is often presented as a function of $|\delta m| \equiv |m_{H^+} - m_{H^{++}}|$, as shown in Fig. 10. At first sight, it is confusing that to reach a given amount of $m_W$ increment, $|\delta m|$ stays almost the same as $m_\text {lightest}$ increases (for the type-I case, it even decreases). Naturally, one would expect that the BSM corrections go to zero in the limit $M\rightarrow \infty$.

To understand such behavior, we should notice that decoupling behavior is only manifest if $\Delta m^2 \equiv m_{H^{\pm }}^2 - m_{H^{\pm \pm }}^2$ or $\lambda _5=4 \Delta m^2 /v^2$ is fixed. If, however, $|\delta m|$ is fixed and M is increased, no decoupling occurs. This happens because

$$\begin{aligned} |\delta m|&= |m_{H^{\pm }} - m_{H^{\pm \pm }}| \nonumber \\&= \left| \sqrt{M^2 - \frac{1}{4}\lambda _5 v^2} - \sqrt{M^2 - \frac{1}{2}\lambda _5 v^2}\right| \nonumber \\&= \frac{1}{8}|\lambda _5| \frac{v^2}{M} \left( 1 + {\mathcal {O}}\left( \frac{v^2}{M^2}\right) \right) \end{aligned}$$

(F1)

and therefore $|\lambda _5| \sim M \delta m/v^2$ leads to $|\lambda _5|\rightarrow \infty$ in the limit $M\rightarrow \infty$. Consequently, this limit will unavoidably violate perturbative unitarity. (We truncated all the curves once $|\lambda _5|$ grows to 10 in Fig. 10.)