I Introduction

Fermilab-PUB-13-549-T

signatures with odd Higgs particles

[9mm]

Bogdan A. Dobrescu and Andrea D. Peterson

[4mm] Theoretical Physics Department, Fermilab, Batavia, IL 60510, USA

[2mm] Department of Physics, University of Wisconsin, Madison, WI 53706, USA

February 16, 2014

Abstract

We point out that bosons may decay predominantly into Higgs particles associated with their broken gauge symmetry. We demonstrate this in a renormalizable model where the and couplings to fermions differ only by an overall normalization. This “meta-sequential” boson decays into a scalar pair, with the charged one subsequently decaying into a 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 and 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.

## I Introduction

New heavy particles of charge and spin 1, referred to as bosons, are predicted in many interesting theories for physics beyond the Standard Model (SM) Beringer:1900zz (); Mohapatra:1986uf (). Extensive searches for bosons at colliders have set limits on the production cross section times branching fraction in several final states Beringer:1900zz (). The most stringent limit on a boson that has the same couplings to quarks and leptons as the SM boson (“sequential” ) has been set using the channels, where or ; the current mass limit is 3.8 TeV, set by the CMS Collaboration CMS:2013rca () using the full data set from the 8 TeV LHC.

In this paper we show that the 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 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 boson embed the electroweak gauge group within an , , or larger gauge symmetry that is spontaneously broken down to the electromagnetic gauge group, . This symmetry breaking pattern is induced usually by some scalar fields with vacuum expectation values (VEVs). The coupling of the 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 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 decays into pairs of particles from the extended Higgs sector.

We demonstrate the importance of decays into scalars by analyzing in detail a simple renormalizable model: gauge symmetry broken by the VEVs of two complex scalars: a bidoublet (i.e., a doublet under each non-Abelian group) of hypercharge and a doublet under one of the ’s. This model has been studied in different contexts Barger:1980ix (); Cao:2012ng (), assuming that the Higgs particles are heavy enough to avoid decays into them. An interesting feature of it is that, up to an overall normalization, the boson has identical couplings to quarks and leptons as the SM boson. We refer to it as the “meta-sequential” .

The most general scalar potential has many terms, but it is significantly simplified by imposing a 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 is exact, it leads to cascade decays of the 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 and 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 gauge group down to : a bidoublet complex scalar, , which has 0 hypercharge, and an doublet, . We take the SM quarks and leptons to be singlets. The scalar and fermion gauge charges are shown in Table 1.

### ii.1 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 symmetry and CP invariance is Barger:1980ix ()

 V = (2.1) −~λ2∣∣Tr(Δ†~Δ)∣∣2−[~λ′4(Tr(Δ†~Δ))2+H.c.]  .

To avoid runaway directions, we impose . The and quartic couplings must be real so that the potential is Hermitian. The quartic coupling may be complex, but its phase can be rotated away by a redefinition of ; we then take to be real without loss of generality.

Canonical normalization of the and terms would require an extra factor of ; we do not include it in order to simplify some equations below. Other terms in , such as , , or , would be redundant as they are linear combinations of the , and terms. We recover the potential of Ref. Barger:1980ix () using the identity .

We also impose so that acquires a VEV. In addition, we need or such that also acquires a VEV. We are interested in the vacuum that preserves the and symmetries:

 ⟨Δ⟩=vΔ2diag(1,1),⟨Φ⟩=vϕ√2(01)  . (2.2)

This vacuum is indeed a minimum of the potential for a range of parameters (discussed below). The VEVs and are related to , , and the five quartic couplings by the extremization conditions:

 λ⋆v2Δ+λ0v2ϕ =−2m2Δ  , λ0v2Δ+λΦv2ϕ =−2m2Φ  , (2.3)

where we defined

 λ⋆≡λΔ−~λ−~λ′  . (2.4)

In terms of fields of definite electric charge, the scalars can be written as

 Δ=(η0χ+η−χ0)=⟨Δ⟩+⎛⎜ ⎜ ⎜ ⎜⎝cc1√2(η0r+iη0i)χ+η−1√2(χ0r+iχ0i)⎞⎟ ⎟ ⎟ ⎟⎠  . (2.5)

The charge conjugate state of the bidoublet is then

 ~Δ=σ2Δ∗σ2=(χ0∗−η+−χ−η0∗)  . (2.6)

All odd fields under (which cannot mix with even fields, and thus are already in the mass eigenstate basis) are collected in

 Δ−~Δ=(H0+iA0√2H+√2H−−H0+iA0)  , (2.7)

where the physical states consist of a CP-even scalar (), a CP-odd scalar (), and a charged scalar (). These are related to the and fields by

 A0=1√2(η0i+χ0i)  , H0=1√2(η0r−χ0r)  , H±=1√2(η±+χ±)  . (2.8)

At tree-level, the -odd scalars have masses given by

 MA=√2~λ′vΔ  , MH+=MH0=√~λ+~λ′vΔ  . (2.9)

The are two remaining scalars not eaten by the gauge bosons. These are -even, CP-even, and neutral; their mass-squared matrix in the , basis is

 M2even=(λ⋆v2Δλ0vϕvΔλ0vϕvΔλΦv2ϕ)  . (2.10)

The -even physical scalars,

 h0 =ϕ0rcosαh−1√2(χ0r+η0r)sinαh  , H′0 =ϕ0rsinαh+1√2(χ0r+η0r)cosαh  , (2.11)

have the following squared masses:

 M2h,H′=12(λ⋆v2Δ+λΦv2ϕ∓√(λ⋆v2Δ−λΦv2ϕ)2+4λ20v2ϕv2Δ)  . (2.12)

The mixing angle satisfies

 tan2αh=2λ0vΔvϕλ⋆v2Δ−λΦv2ϕ  . (2.13)

The necessary and sufficient conditions for the vacuum (2.2) to be a minimum of the potential are

 ~λ′ >Max{−~λ,0}  , λ⋆λΦ >λ20  , λΦ|m2Δ| >−λ0m2Φ  , λ0|m2Δ| >−λ⋆m2Φ  ; (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 . The agreement between SM predictions and the data suggests that the Higgs sector is near the decoupling limit ; adopting this limit, we can analyze the spontaneous symmetry breaking in two stages. The first one is at the scale . The effective theory below consists of the SM (with the Higgs doublet ) plus an -triplet of heavy gauge bosons (), and five of the scalar degrees of freedom from : four -odd scalars combined into an -triplet () and a singlet (), and a -even singlet ().

The second stage of symmetry breaking is the SM one: at the weak scale GeV. The lightest CP-even scalar, , represents the recently discovered Higgs boson, because its couplings are the same as the SM ones up to small corrections of order . Its mass is given by

 Mh=vϕ(λΦ−λ20λ⋆)1/2[1−λ20v2ϕ2λ2⋆v2Δ+O(v4ϕ/v4Δ)]  , (2.15)

and should be identified with the measured Higgs mass, near 126 GeV. The even scalar has the same couplings as the SM Higgs except for an overall suppression by

 sinαh=λ0vϕλ⋆vΔ+O(v3ϕ/v3Δ)  , (2.16)

and is significantly heavier:

 MH′=√λ⋆vΔ+O(v2ϕ/vΔ)  . (2.17)

Consequently, its dominant decay modes are and .

The odd scalars, , , , couple exclusively to gauge bosons and scalars, and only in pairs. The lightest of them is stable, and a component of dark matter. is naturally the lightest odd particle (LOP). because in the limit the symmetry is enhanced: becomes the Nambu-Goldstone boson of a global symmetry acting on . We note, however, that could also be the LOP (for ) and a viable dark matter candidate. Even though it is part of an triplet that is degenerate at tree-level, electroweak loops split the and masses Dodelson (); Cirelli:2005uq ().

In what follows we will assume that 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 or are the dominant ones. Other channels are highly suppressed, either kinematically ( and ) or by loops ( and the CP-violating ).

### ii.2 Meta-sequential W′ boson

The kinetic terms for the and scalars,

 (DμΦ)†DμΦ+Tr[(DμΔ)†DμΔ]  , (2.18)

involve the covariant derivative

 Dμ=∂μ−igYYBμ−ig1→T1⋅→W1μ−ig2→T2⋅→W2μ  , (2.19)

with ; notice that acts from the right on the bidoublet: . After symmetry breaking, the electrically-charged gauge bosons acquire mass terms:

 v2ϕ4g21W+1μW−μ1+v2Δ4(g1W+1μ−g2W+2μ)(g1W−μ1−g2W−μ2)  . (2.20)

Diagonalizing them gives the physical charged spin-1 states,

 Wμ =W1μcosθ+W2μsinθ  , W′μ =−W1μsinθ+W2μcosθ  , (2.21)

with the following mixing angle, :

 tan2θ=2g1g2v2Δ(g22−g21)v2Δ−g21v2ϕ  . (2.22)

The masses of the and bosons are

 MW,W′=12√2[(g22+g21∓2g1g2sin2θ)v2Δ+g21v2ϕ]1/2  . (2.23)

Given that the left-handed quarks and leptons transform as doublets only under , their couplings to the and bosons are proportional to the respective coefficients of in Eqs. (2.21). The measured coupling to fermions gives a value for the gauge coupling of , where the electromagnetic coupling constant and the weak mixing angle are evaluated at the scale: and . In terms of the parameters of this model, the gauge coupling can be expressed as

 g1cosθ=g  . (2.24)

The coupling to quarks and leptons, derived from Eq. (2.21) and Table I, is then

 −g1sinθ=−gtanθ  . (2.25)

Thus, determines completely the tree-level couplings of to SM fermions. Imposing a perturbativity condition on the gauge couplings, , and using Eq. (2.24) we find that

 0.2≲tanθ≲5  . (2.26)

In the particular case of , the couplings of to fermions are identical (at tree level) to those of the . This is usually referred to as the sequential boson, and is a common benchmark model for searches at colliders. The most recent limit on the mass of a sequential at CMS, using 20 fb of 8 TeV data, is 3.8 TeV CMS:2013rca (), assuming that can decay only into SM fermions. Note that the relative sign in Eqs. (2.24) and (2.25) implies constructive interference between the and amplitudes that contribute to processes constrained by searches at the LHC. In the next sections we will focus on the region , where the LHC limits are relaxed. Given that the boson in this model has couplings to fermions proportional to the SM ones (by an overall factor of ), we refer to it as a “meta-sequential boson”.

The above results are valid for any . It is instructive to expand these results in powers of . The coupling to fermions, relative to the one is

 tanθ=tanθ0(1−v2ϕv2Δcos2θ0)+O(v4ϕ/v4Δ)  , (2.27)

where we defined

 tanθ0≡g1g2  . (2.28)

For , the values of span essentially the same range as . The and masses, given in Eq. (2.23), have simple expressions to leading order in :

 MW=g22vϕsinθ0[1−v2ϕ2v2Δsin4θ0+O(v4ϕ/v4Δ)]  , (2.29) MW′=g2vΔ2cosθ0⎡⎣1+v2ϕ2v2Δsin4θ0+v4ϕ8v4Δ(4cot2θ0−1)sin8θ0+O(v6ϕ/v6Δ)⎤⎦  . (2.30)

The low-energy charged current interactions are mediated in this model by both and exchange. Consequently, the Fermi constant is related to our parameters by

 4√2GF =(g1cosθ)2M2W+(g1sinθ)2M2W′ =g2M2W[1+v2ϕv2Δsin4θ0+O(v4ϕ/v4Δ)]  , (2.31)

where we used Eq. (2.24), which defines as the tree-level coupling to leptons and quarks. This shows that the measurements of the weak coupling in low-energy processes and in collider processes involving bosons should agree up to tiny corrections of order . Defining the weak scale GeV through , and using Eq. (2.29), we obtain the relation between the VEV and the weak scale

 v=vϕ[1−v2ϕv2Δsin2θ0+O(v4ϕ/v4Δ)]  . (2.32)

### ii.3 Z′ mass and couplings

Electrically-neutral gauge bosons also acquire mass terms in the vacuum (2.2):

 v2ϕ8(g1W31μ−gYBμ)2+v2Δ8(g2W32μ−g1W31μ)2  . (2.33)

It is convenient to diagonalize these in two steps. First, we define some intermediate fields denoted with hats:

 ^Z′μ =W32μcosθ0−W31μsinθ0  , ^Zμ =(W32μsinθ0+W31μcosθ0)cos^θW−Bμsin^θW  , (2.34)

where the angle is defined in terms of coupling ratios:

 tan^θW=gYg2sinθ0  . (2.35)

The gauge boson orthogonal to and is the photon (), already in the physical eigenstate. The measured electromagnetic coupling, , is related to the original gauge couplings through

 gYcos^θW=e  . (2.36)

The mass-squared matrix for and takes the form

 M2Z=g224sin2θ0⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝[]ccv2ϕcos2^θW−v2ϕtanθ0cos^θW−v2ϕtanθ0cos^θW4v2Δsin22θ0+v2ϕtan2θ0⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠  . (2.37)

In the second step, we rotate and by an angle , given by

 tan2ϵZ=v2ϕsin2θ0sin2θ0cos^θWv2Δcos2^θW+v2ϕsin4θ0(cos2^θW−cot2θ0)  , (2.38)

in order to obtain the mass eigenstate and bosons:

 Zμ =^ZμcosϵZ+^Z′μsinϵZ  , Z′μ =−^ZμsinϵZ+^Z′μcosϵZ  . (2.39)

The masses of the heavy neutral spin-1 particles are

 MZ,Z′=g22√2[v2Δcos2θ0+v2ϕsin2θ0(1cos2^θW+tan2θ0∓2tanθ0sin2ϵZcos^θW)]1/2  . (2.40)

The tree-level results (2.33)-(2.40) have been obtained without approximations. Expanding now in , we find

 MZ=g2vϕsinθ02cos^θW[1−v2ϕ2v2Δsin4θ0+O(v4ϕ/v4Δ)]  , (2.41) MZ′=g2vΔ2cosθ0⎡⎣1+v2ϕ2v2Δsin4θ0+v4ϕ8v4Δ(4cot2θ0cos2^θW−1)sin8θ0+O(v6ϕ/v6Δ)⎤⎦  . (2.42)

The original five parameters from the gauge sector () can be traded for three observables (e.g., ) and two parameters that can be measured once the or boson is discovered (), using Eqs. (2.27), (2.29), (2.30), (2.36) and

 sW=sin^θW[1−v2ϕv2Δsin2θ0cos2θ0+O(v4ϕ/v4Δ)]  . (2.43)

Eqs. (2.41) and (2.42), combined with the above equation, show that the tree-level relation , where , is satisfied only up to corrections of order . Furthermore, the couplings to fermions are modified at order compared to the SM. Thus, the current agreement between electroweak measurements and the SM imposes an upper limit on , or equivalently, a lower limit on the mass for a fixed . The lower limit at the 95% CL given by the global fit performed in Ref. Cao:2012ng () increases from GeV for , to TeV for (i.e., sequential ).

The relative mass splitting between and is very small:

 MZ′MW′−1=s2W2c2Wtan2θ(MWMW′)4+O(M6W/M6W′)  , (2.44)

which is less than for TeV and . This implies that the mass and will be constrained by both and searches. The interacts with the left-handed fermion doublets, with a coupling given by plus corrections of order that are different for quarks and leptons.. The couplings to singlets are suppressed by .

## Iii W′ and Z′ decays

The new gauge bosons interact with SM fermions and gauge bosons, as well as with the Higgs particles. Usually, resonance searches for new gauge bosons rely on sizable branching fractions of the and decays into SM fermions. However, if the scalars are lighter than the vector bosons than the decays into SM fermions may be suppressed. In our model, the left-handed fermion doublets transform under , while all fermions are singlets under . Thus, the and couplings to fermions are induced through mixing with the and , so that for small decays to heavy scalars become important.

Neglecting corrections of , the and coupling to fermion doublets is given by . The partial widths for decays to leptons (without summing over flavors)

 Γ(W′→ℓν)≈2Γ(Z′→ℓ+ℓ−)≈α6s2Wtan2θMW′  , (3.1)

are suppressed for . By contrast, the and couplings to pairs of odd Higgs particles are enhanced by :

 gW′H±A0=gZ′H0A0 =gsin2θ  , gW′H±H0=gZ′H+H− =gtan2θ  , (3.2)

where we ignored corrections of order . These couplings lead to the following partial widths:

 Γ(W′→H±A0) ≈Γ(Z′→H0A0)≈αMW′12s2Wsin22θ(1−2M2H++M2A0M2W′+(M2H+−M2A0)2M4W′)3/2, Γ(W′→H±H0) ≈Γ(Z′→H+H−)≈αMW′12s2Wtan22θ(1−4M2H+M2W′)3/2  . (3.3)

The can also decay into and final states, but these partial widths are suppressed by .

Figure 1 shows the branching fractions of the and as a function of for the dominant channels. As a benchmark point, we have used TeV, GeV and GeV (as shown in Section II, and to a good accuracy). For , the decays dominantly to pairs of odd Higgs particles. It is important to investigate collider signatures of these decays.

The heavier odd scalars decay into the LOP (taken to be ) and an electroweak boson, so that and can each undergo two cascade decays: , (see Figure 2), and , .

If the symmetry discussed in Section II is exact, then 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 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 Barbieri:2006dq () and minimal dark matter scenarios Cirelli:2005uq ().

The possibility that the symmetry is violated by terms in the scalar potential of the type

 (3.4)

is also worth considering. The weak-triplet scalar () as well as the singlet would mix with the doublet, allowing direct two-body decays of , and to SM particles. Furthermore, the three CP-even neutral scalars () would then mix, so that and decays involving the SM-like Higgs boson are possible. These include with (this channel is analyzed in poster ()), as well as and with