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W' signatures with odd Higgs particles

We point out that W' bosons may decay predominantly into Higgs particles associated with their broken gauge symmetry. We demonstrate this in a renormalizable model where the W' and W couplings to fermions differ only by an overall normalization. This"meta-sequential"W' boson decays into a scalar pair, with the charged one subsequently decaying into a W boson and a neutral scalar. These scalars are odd under a parity of the Higgs sector, which consists of a complex bidoublet and a doublet. The W' and Z' bosons have the same mass and branching fractions into scalars, and may show up at the LHC in final states involving one or two electroweak bosons and missing transverse energy.

In this paper we show that the W boson is likely to decay not only into SM fermions, as often assumed, but also into pairs of scalar particles from the extended Higgs sector responsible for the W mass. As a result the existing limits may be relaxed, and different types of searches at the LHC may prove to be more sensitive.
Theories that include a W boson embed the electroweak gauge group within an SU (2) 1 × SU (2) 2 ×U (1), SU (3) W ×U (1), or larger gauge symmetry that is spontaneously broken down to the electromagnetic gauge group, U (1) em . This symmetry breaking pattern is induced usually by some scalar fields with vacuum expectation values (VEVs). The coupling of the W to these scalars is related to the gauge couplings, and cannot be too small. In perturbative renormalizable models, the scalars have masses near or below the symmetry breaking scale, because the quartic couplings grow with the energy. The W boson, by contrast, may be significantly heavier, because large gauge couplings are allowed by the asymptotic freedom of non-Abelian gauge theories. Consequently, it is natural to expect W decays into pairs of particles from the extended Higgs sector.
We demonstrate the importance of W decays into scalars by analyzing in detail a simple renormalizable W model: SU (2) 1 × SU (2) 2 × U (1) Y gauge symmetry broken by the VEVs of two complex scalars: a bidoublet (i.e., a doublet under each non-Abelian group) of hypercharge Y = 0 and a doublet under one of the SU (2)'s. This model has been studied in different contexts [4,5], assuming that the Higgs particles are heavy enough to avoid W decays into them. An interesting feature of it is that, up to an overall normalization, the W boson has identical couplings to quarks and leptons as the SM W boson. We refer to it as the "meta-sequential" W .
The most general scalar potential has many terms, but it is significantly simplified by imposing a Z 2 symmetry (the bidoublet transforms into its charge conjugate). The lightest Higgs particle that is odd under this parity is stable, and could be a viable dark matter candidate. Whether or not the Z 2 is exact, it leads to cascade decays of the W that give signatures with one or two electroweak bosons and two of these lightest odd particles (LOPs).
In Section II we study the masses and couplings of the Higgs particles, and of the heavy gauge bosons. Then, in Section III, we compute the branching fractions of the W and Z bosons, and comment on various signatures arising from their cascade decays. In Section IV we discuss the LHC phenomenology assuming that the LOPs escape the detector. We summarize our results in Section V.

II.
AN SU (2) × SU (2) × U (1) Y MODEL WITH ODD HIGGS SECTOR Let us focus on a simple Higgs sector that breaks the SU (2) 1 × SU (2) 2 × U (1) Y gauge group down to U (1) em : a bidoublet complex scalar, ∆, which has 0 hypercharge, and an SU (2) 1 doublet, Φ. We take the SM quarks and leptons to be SU (2) 2 singlets. The scalar and fermion gauge charges are shown in Table I.

A. Scalar spectrum
We require the Lagrangian to be symmetric under the interchange ∆ ↔∆, where∆ is the charge conjugate of ∆. The most general renormalizable scalar potential exhibiting this To avoid runaway directions, we impose λ Φ , λ ∆ > 0. Theλ and λ 0 quartic couplings must be real so that the potential is Hermitian. Theλ quartic coupling may be complex, but its phase can be rotated away by a redefinition of ∆; we then takeλ to be real without loss of generality.
Canonical normalization of theλ andλ terms would require an extra factor of 1/2; we do not include it in order to simplify some equations below. Other terms in V , such as Tr [(∆ † ∆) 2 ], Tr (∆ † ∆∆ †∆ ), or Tr (∆ †∆∆ † ∆), would be redundant as they are linear combinations of the λ ∆ ,λ andλ terms. We recover the potential of Ref. [4] using the We also impose m 2 ∆ < 0 so that ∆ acquires a VEV. In addition, we need m 2 Φ < 0 or λ 0 < 0 such that Φ also acquires a VEV. We are interested in the vacuum that preserves the U (1) em and Z 2 symmetries: This vacuum is indeed a minimum of the potential for a range of parameters (discussed below). The VEVs v φ > 0 and v ∆ > 0 are related to m 2 Φ , m 2 ∆ , and the five quartic couplings by the extremization conditions: In terms of fields of definite electric charge, the scalars can be written as The charge conjugate state of the bidoublet is theñ All odd fields under Z 2 (which cannot mix with even fields, and thus are already in the mass eigenstate basis) are collected in where the physical states consist of a CP-even scalar (H 0 ), a CP-odd scalar (A 0 ), and a charged scalar (H ± ). These are related to the η and χ fields by At tree-level, the Z 2 -odd scalars have masses given by The are two remaining scalars not eaten by the gauge bosons. These are Z 2 -even, CPeven, and neutral; their mass-squared matrix in the (χ 0 r + η 0 r )/ √ 2 , φ 0 r basis is (2.10) The Z 2 -even physical scalars, have the following squared masses: (2.12) The mixing angle α h satisfies (2.13) The necessary and sufficient conditions for the vacuum (2.2) to be a minimum of the potential areλ (2.14) these follow from imposing that all physical scalars have positive squared masses [see Eqs. (2.9) and (2.12)], and that the extremization conditions (2.3) have solutions.
All above results are valid for any v φ /v ∆ . The agreement between SM predictions and the data suggests that the Higgs sector is near the decoupling limit v 2 φ v 2 ∆ ; adopting this limit, we can analyze the spontaneous symmetry breaking in two stages. The first one is The effective theory below v ∆ consists of the SM (with the Higgs doublet Φ) plus an SU (2) W -triplet of heavy gauge bosons (W ± , Z ), and five of the scalar degrees of freedom from ∆: four Z 2 -odd scalars combined into an SU (2) W -triplet (H ± , H 0 ) and a singlet (A 0 ), and a Z 2 -even singlet (H 0 ).
The second stage of symmetry breaking is the SM one: SU (2) W ×U (1) Y → U (1) em at the weak scale v φ ≈ 246 GeV. The lightest CP-even scalar, h 0 , represents the recently discovered Higgs boson, because its couplings are the same as the SM ones up to small corrections of order v 2 φ /v 2 ∆ . Its mass is given by and should be identified with the measured Higgs mass, near 126 GeV. The H 0 even scalar has the same couplings as the SM Higgs except for an overall suppression by and is significantly heavier: Consequently, its dominant decay modes are W + W − and ZZ.
The odd scalars, H ± , H 0 , A 0 , couple exclusively to gauge bosons and scalars, and only in pairs. The lightest of them is stable, and a component of dark matter. A 0 is naturally the lightest odd particle (LOP). because in theλ → 0 limit the symmetry is enhanced: A 0 becomes the Nambu-Goldstone boson of a global U (1) symmetry acting on ∆. We note, however, that H 0 could also be the LOP (forλ >λ) and a viable dark matter candidate.
Even though it is part of an SU (2) W triplet that is degenerate at tree-level, electroweak loops split the H ± and H 0 masses [6,7].
In what follows we will assume that A 0 is the LOP. The heavier odd scalars then decay as follows: Even when these two-body decays are kinematically forbidden, the three-body decays through an off-shell W ± or Z are the dominant ones. Other channels are highly suppressed, either kinematically (H + → π + π 0 H 0 and H + → + νH 0 ) or by loops (H 0 → γA 0 and the CP-violating H 0 → h 0 A 0 ).

B. Meta-sequential W boson
The kinetic terms for the Φ and ∆ scalars, involve the covariant derivative with T 1,2 = σ 1,2 /2; notice that T 2 acts from the right on the bidoublet: After symmetry breaking, the electrically-charged gauge bosons acquire mass terms: (2.20) Diagonalizing them gives the physical charged spin-1 states, with the following mixing angle, 0 ≤ θ ≤ π/2: The masses of the W and W bosons are Given that the left-handed quarks and leptons transform as doublets only under SU (2) The W coupling to quarks and leptons, derived from Eq. (2.21) and Table I, is then − g 1 sin θ = −g tan θ . (2.25) Thus, tan θ determines completely the tree-level couplings of W to SM fermions. Imposing a perturbativity condition on the SU (2) 1 ×SU (2) 2 gauge couplings, g 2 1,2 /(4π) 1, and using Eq. (2.24) we find that 0.2 tan θ 5 . (2.26) In the particular case of tan θ = 1, the couplings of W to fermions are identical (at tree level) to those of the W . This is usually referred to as the sequential W boson, and is a common benchmark model for W searches at colliders. The most recent limit on the mass of a sequential W at CMS, using 20 fb −1 of 8 TeV data, is 3.8 TeV [3], assuming that W can decay only into SM fermions. Note that the relative sign in Eqs. (2.24) and (2.25) implies constructive interference between the W and W amplitudes that contribute to processes constrained by W searches at the LHC. In the next sections we will focus on the region 0.2 < tan θ < 1, where the LHC limits are relaxed. Given that the W boson in this model has couplings to fermions proportional to the SM W ones (by an overall factor of − tan θ), we refer to it as a "meta-sequential W boson".
The above results are valid for any v φ /v ∆ . It is instructive to expand these results in The W coupling to fermions, relative to the W one is where we defined , the values of tan θ 0 span essentially the same range as tan θ. The W and W masses, given in Eq. (2.23), have simple expressions to leading order in v φ /v ∆ : The low-energy charged current interactions are mediated in this model by both W and W exchange. Consequently, the Fermi constant is related to our parameters by where we used Eq. (2.24), which defines g as the tree-level W coupling to leptons and quarks. This shows that the measurements of the weak coupling in low-energy processes and in collider processes involving W bosons should agree up to tiny corrections of order Defining the weak scale v ≈ 246 GeV through G F = 2 1/2 v −2 , and using Eq. (2.29), we obtain the relation between the Φ VEV and the weak scale (2.32)

C. Z mass and couplings
Electrically-neutral gauge bosons also acquire mass terms in the vacuum (2.2): It is convenient to diagonalize these in two steps. First, we define some intermediate fields where the angleθ W is defined in terms of coupling ratios: The gauge boson orthogonal toẐ µ andẐ µ is the photon (A µ = W 3 1µ cos θ 0 sinθ W +B µ cosθ W ), already in the physical eigenstate. The measured electromagnetic coupling, e = √ 4πα ≈ 0.313, is related to the original gauge couplings through The mass-squared matrix forẐ µ andẐ µ takes the form (2.37) In the second step, we rotateẐ µ andẐ µ by an angle Z , given by in order to obtain the mass eigenstate Z and Z bosons: The masses of the heavy neutral spin-1 particles are The tree-level results (2.33)-(2.40) have been obtained without approximations. Expand- (2.42) The original five parameters from the gauge sector (g 1 , g 2 , g Y , v φ , v ∆ ) can be traded for three observables (e.g., e, s W , M W ) and two parameters that can be measured once the W or Z boson is discovered (M W , tan θ), using Eqs. (2.27), (2.29), (2.30), (2.36) and Furthermore, the Z couplings to fermions are modified at order v 2 φ /v 2 ∆ compared to the SM. Thus, the current agreement between electroweak measurements and the SM imposes an upper limit on v 2 φ /v 2 ∆ , or equivalently, a lower limit on the W mass for a fixed tan θ. The lower limit at the 95% CL given by the global fit performed in Ref. [5] increases from M W 600 GeV for tan θ = 0.2, to M W 2 TeV for tan θ = 1 (i.e., sequential W ).
The relative mass splitting between W and Z is very small: Neglecting corrections of O(v 2 φ /v 2 ∆ ), the W and Z coupling to fermion doublets is given by g tan θ. The partial widths for decays to leptons (without summing over flavors) are suppressed for 0.2 < tan θ < 1. By contrast, the W and Z couplings to pairs of odd Higgs particles are enhanced by 1/ tan θ: where we ignored corrections of order v 2 φ /v 2 ∆ . These couplings lead to the following partial widths: The W can also decay into W Z and W h 0 final states, but these partial widths are suppressed by v 4 φ /v 4 ∆ . Figure 1 shows the branching fractions of the W and Z as a function of tan θ for the The heavier odd scalars decay into the LOP (taken to be A 0 ) and an electroweak boson, so that W and Z can each undergo two cascade decays: If the Z 2 symmetry discussed in Section II is exact, then A 0 is a component of dark matter. We will not explore here the constraints on the parameter space from the upper limit on relic density, nor from direct detection experiments (nuclear scattering would occur through Higgs exchange and gauge boson loops); these constraints can be in any case relaxed by allowing a tiny Z 2 violation in the scalar potential. While an in-depth exploration of this model as an explanation for dark matter is left for future work, we note that it shares many features with inert doublet [8] and minimal dark matter scenarios [7].
The possibility that the Z 2 symmetry is violated by terms in the scalar potential of the type is also worth considering. The weak-triplet scalar (H ± , H 0 ) as well as the singlet A 0 would mix with the Φ doublet, allowing direct two-body decays of A 0 , H 0 and H ± to SM particles. For a range of parameters, the decays of A 0 may be displaced but still within the detector, leading to potentially confusing events. In what follows we will consider only the case where A 0 is stable enough to escape the detector.

IV. LHC SIGNATURES WITH STABLE A 0
At the LHC, the W boson would be mainly produced in the s channel from quarkantiquark initial state, even for small tan θ. In the narrow width approximation, the leadingorder cross section for W production followed by decay into H  is included for pp → W χχ through aqγ µ qχγ µ χ contact interaction (for m χ = 100 GeV).
[12] (with parton distribution functions CTEQ6L1 [13]), which includes interference between the W and W contributions. We have set M H + = 300 GeV, M A = 200 GeV; the cross sections are only weakly sensitive to the scalar masses as long as W is much heavier. Figure   3 also shows the cross sections for pp → Z → H 0 A 0 and pp → Z → H + H − , for the same parameters.
We assume that the Z 2 symmetry discussed in Section II is sufficiently preserved so that the LOP escapes the detector. As noted there, A 0 is most likely the LOP, so that each of the above processes includes two A 0 in the final state, which appear as missing transverse carrying energy roughly equal of M W /4. This implies that hadronic decays of the W or Z boson lead to an interesting signature with the two jets collimated into a single wide jet with substructure, plus / E T .
The ATLAS collaboration [14] has searched for this type of signature in the case of DM particles pair produced through a contact interaction to quarks [15,16]. Compared to our model, the processes pp → Wχχ and pp → Zχχ give rise to a smaller transverse momentum for the electroweak boson, which is radiated from an initial state quark. In Figure 4 we show the p T distributions for the W arising from W → H + A 0 → W + A 0 A 0 , as well as from initial state radiation in the case of aqγ µ qχγ µ χ contact interaction (for a Dirac fermion χ of mass m χ = 100 GeV). It is clear that the efficiency for a stringent p T (W ) cut is much higher for our W decays than in the case of contact interactions.
The cascade decays to two highly boosted electroweak bosons plus / E T . Hadronic decays of these W and Z bosons allow the use of substructure techniques to reduce the QCD background.
The boosted W and Z "jets" plus / E T channels have the largest branching fractions.
Nevertheless, leptonic decays of the boosted W and Z are also promising due to small backgrounds. These lead to final states with one, two or three leptons, plus / E T .
The mono-lepton signature has been studied theoretically [17] and searched for at the LHC [18] in the case of contact interactions. Again, in our case the W producing the lepton is generically more boosted. Unlike W decays directly to a lepton-neutrino pair, there will be no Jacobian peak in the missing transverse energy distribution, as the A 0 's carry away a substantial fraction of the energy of the W . In fact, the distribution will be peaked at low-p T . Furthermore, if the masses of the A 0 and H + are similar, the transverse momenta of the two final-state A 0 particles will have similar magnitudes but opposite directions, so their contribution to the / E T of the event is reduced. In this case, the missing energy distribution could look like a SM W decay. This problem is mitigated if the A 0 is substantially lighter than the Higgs triplet states, in which case the / E T distribution will have a longer tail.
We simulate W signals using Madgraph 5 [12], including showering and hadronization with Pythia 6.4 [19], and PGS detector simulation [20]; then we analyze the events with the MadAnalysis package [21]. A better observable for the single-lepton process W → H + A 0 → W + + / E T is the separation in azimuthal angle between the missing transverse energy and the lepton transverse momentum, ∆φ p T , / E T . When a W or W decays directly to a lepton-neutrino pair, the decay products are nearly back-to-back; for both the W and dark matter mono-lepton analyses, CMS requires that ∆φ p T , / E T > 0.8π [3]. However, the kinematics for the decay W → A 0 A 0 lν are substantially different, with the ∆φ p T , / E T distribution peaked at moderate-to-small values of ∆φ p T , / E T ; see Figure 6. In the rest frame of the W , p l T = − p miss T , but in the lab frame, the W transverse momentum is distributed among the four decay products and the There are also two processes leading to + − + / E T . One of them is the Z → H 0 A 0 → ZA 0 A 0 cascade decay, with Z → + − ; the related process in the case of contact interactions has been discussed in [23]. The other one is Z → H + H − → W + A 0 W − A 0 with leptonic W decays; a similar final state, but without s-channel resonance, arises from chargino pair production Ref. [24].
The limits on our model set by current LHC results are already stronger than those from electroweak fits mentioned in Section II. The searches in the W → ν channel, although affected by suppressed branching fraction for small tan θ, set relevant bounds. In Figure   7, we reinterpret the 95% CL limit set by the CMS Collaboration [3] on σ excl. /σ SSMW as a limit on tan θ.
Existing LHC searches for other processes set less stringent limits. Figure 4 of [18] to estimate the number of background events with 1 < ∆φ p T , / E T < 1.5 in the muon channel, then set an upper limit on W → µ + / E T events in the same region assuming no excess is observed. This limit is at most M W > 1.05 TeV for any tan θ ≤ 1.
For the process W → H + H 0 → W + A 0 ZA 0 → + + − + / E T , we use results from leptonic searches for charginos and neutralinos in [22]. We consider the search for a same-flavor, opposite sign electron or muon pair on the Z peak (75 GeV < M < 105 GeV), plus an additional electron or muon. We sum over M T bins, then we use a Poisson likelihood function multiplied over / E T bins to set an upper bound on the number of events. The upper limits on the number of W events in each channel are then translated to a limit on tan θ as a function of M W . This limit is rather weak: for M W = 1 TeV, only values of tan θ > 3.5 are excluded.
We see that the search for direct decays to lepton plus / E T final states still provides the most stringent constraint on our model. The reason is that the 3-lepton rate is suppressed by both the W → ν and Z → + − branching fractions. Furthermore, for the monolepton search, the selection cuts that optimize signal over background for W → ν cut out a substantial portion of the H ± A 0 events. A new analysis focusing on the small ∆φ p T , / E T region, using both the electron and muon channels, would provide a stronger limit.
Given the mass degeneracy between W and Z , limits set by searches for Z → + − can also be plotted in the M W − tan θ plane. However, they are weaker than those from W → ν because both the production cross section and the leptonic branching fraction are smaller for Z than for W .

V. CONCLUSIONS
The SU (2) 1 × SU (2) 2 × U (1) Y model with a bidoublet and a doublet complex scalars is a simple renormalizable model that can serve as a benchmark for various LHC searches. It includes a meta-sequential W boson whose s-channel production interferes constructively with the W contribution, and depends on only two parameters: M W and the overall coupling normalization, tan θ. It also includes a Z boson (degenerate in mass with W ) which couples, to a good approximation, only to left-handed fermions.
The potential for the bidoublet (∆) and doublet (Φ) scalars is chosen to be invariant under a Z 2 transformation that interchanges the bidoublet and its charge conjugate. The physical scalar spectrum then consists of four odd Higgs particles (a mass-degenerate weaktriplet H + , H 0 , H − , and a CP-odd singlet A 0 ), the recently discovered Higgs boson (h 0 ), and a heavier scalar (H ) whose couplings to SM fields are the same as those of h 0 except for an universal suppression. The A 0 is naturally the LOP because a global U (1) symmetry becomes exact in the M A → 0 limit.
The phenomenology of the scalars is worth exploring whether or not the W and Z bosons are light enough to be produced at the LHC. Electroweak production of the triplet scalars, for example, would lead to final states involving one or two weak bosons and two LOPs.
The range of parameters where A 0 is a viable dark matter particle remains to be studied.
For the present work we focused on the case where A 0 is sufficiently long-lived to escape the detector, but we also mentioned possible signatures in the case where A 0 decays (promptly or with a displaced vertex) into fermion pairs. This model illustrates nicely the possibility that the W and Z bosons may decay predominantly (with branching fraction as large as 96%) into the scalars responsible for breaking the extended gauge symmetry. Generically, the high-energy behavior of any W boson requires it to be associated with a non-Abelian gauge symmetry (or else it must be a bound state with the compositeness scale not much higher than its mass), which in turn implies a larger Higgs sector. The non-Abelian gauge coupling can be significantly larger than the Higgs quartic couplings, implying vector bosons much heavier than the scalars.
In our model, the W and Z couplings to the odd Higgs particles are enhanced for tan θ 1 by 1/ tan θ. Consequently, the usual ud → W → ν or tb channels currently used in searches at the LHC are suppressed both in production and in braching fractions, the combined effect being of order tan 6 θ. The mass limits on a sequential W , currently around 3.8 TeV, are relaxed for tan θ ≈ 0.2 (the lower perturbativity bound) to M W > 1 TeV. At the same time, the cascade decays through odd Higgs particles, W → H + A 0 → W + A 0 A 0 , interesting searches at the LHC, with boosted W and Z bosons decaying either hadronically or leptonically.