W' signatures with odd Higgs particles
FFermilab-PUB-13-549-T W (cid:48) signatures with odd Higgs particles Bogdan A. Dobrescu (cid:63) and Andrea D. Peterson (cid:63) (cid:5) (cid:63)
Theoretical Physics Department, Fermilab, Batavia, IL 60510, USA (cid:5)
Department of Physics, University of Wisconsin, Madison, WI 53706, USA
February 16, 2014
Abstract
We point out that W (cid:48) bosons may decay predominantly into Higgs particles associatedwith their broken gauge symmetry. We demonstrate this in a renormalizable model wherethe W (cid:48) and W couplings to fermions differ only by an overall normalization. This “meta-sequential” W (cid:48) boson decays into a scalar pair, with the charged one subsequently decayinginto a W boson and a neutral scalar. These scalars are odd under a parity of the Higgssector, which consists of a complex bidoublet and a doublet. The W (cid:48) and Z (cid:48) bosons havethe same mass and branching fractions into scalars, and may show up at the LHC in finalstates involving one or two electroweak bosons and missing transverse energy. I. INTRODUCTION
New heavy particles of charge ± W (cid:48) bosons, are predictedin many interesting theories for physics beyond the Standard Model (SM) [1, 2]. Extensivesearches for W (cid:48) bosons at colliders have set limits on the production cross section timesbranching fraction in several final states [1]. The most stringent limit on a W (cid:48) boson thathas the same couplings to quarks and leptons as the SM W boson (“sequential” W (cid:48) ) hasbeen set using the (cid:96)ν channels, where (cid:96) = e or µ ; the current mass limit is 3.8 TeV, set bythe CMS Collaboration [3] using the full data set from the 8 TeV LHC.1 a r X i v : . [ h e p - ph ] F e b n this paper we show that the W (cid:48) boson is likely to decay not only into SM fermions,as often assumed, but also into pairs of scalar particles from the extended Higgs sectorresponsible for the W (cid:48) mass. As a result the existing limits may be relaxed, and differenttypes of searches at the LHC may prove to be more sensitive.Theories that include a W (cid:48) boson embed the electroweak gauge group within an SU (2) × SU (2) × U (1), SU (3) W × U (1), or larger gauge symmetry that is spontaneously broken downto the electromagnetic gauge group, U (1) em . This symmetry breaking pattern is inducedusually by some scalar fields with vacuum expectation values (VEVs). The coupling of the W (cid:48) to these scalars is related to the gauge couplings, and cannot be too small. In perturbativerenormalizable models, the scalars have masses near or below the symmetry breaking scale,because the quartic couplings grow with the energy. The W (cid:48) boson, by contrast, may besignificantly heavier, because large gauge couplings are allowed by the asymptotic freedomof non-Abelian gauge theories. Consequently, it is natural to expect W (cid:48) decays into pairs ofparticles from the extended Higgs sector.We demonstrate the importance of W (cid:48) decays into scalars by analyzing in detail a simplerenormalizable W (cid:48) model: SU (2) × SU (2) × U (1) Y gauge symmetry broken by the VEVsof two complex scalars: a bidoublet ( i.e. , a doublet under each non-Abelian group) of hy-percharge Y = 0 and a doublet under one of the SU (2)’s. This model has been studied indifferent contexts [4, 5], assuming that the Higgs particles are heavy enough to avoid W (cid:48) decays into them. An interesting feature of it is that, up to an overall normalization, the W (cid:48) boson has identical couplings to quarks and leptons as the SM W boson. We refer to itas the “meta-sequential” W (cid:48) .The most general scalar potential has many terms, but it is significantly simplified byimposing a Z symmetry (the bidoublet transforms into its charge conjugate). The lightestHiggs particle that is odd under this parity is stable, and could be a viable dark mattercandidate. Whether or not the Z is exact, it leads to cascade decays of the W (cid:48) that givesignatures 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 heavygauge bosons. Then, in Section III, we compute the branching fractions of the W (cid:48) and Z (cid:48) bosons, and comment on various signatures arising from their cascade decays. In Section2V we discuss the LHC phenomenology assuming that the LOPs escape the detector. Wesummarize 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) × SU (2) × U (1) Y gaugegroup down to U (1) em : a bidoublet complex scalar, ∆, which has 0 hypercharge, and an SU (2) doublet, Φ. We take the SM quarks and leptons to be SU (2) singlets. The scalarand fermion gauge charges are shown in Table I. A. Scalar spectrum
We require the Lagrangian to be symmetric under the interchange ∆ ↔ ˜∆, where ˜∆ isthe charge conjugate of ∆. The most general renormalizable scalar potential exhibiting this Z symmetry and CP invariance is [4] V = m Φ † Φ + λ Φ (cid:0) Φ † Φ (cid:1) + (cid:0) m + λ Φ † Φ (cid:1) Tr (cid:0) ∆ † ∆ (cid:1) + λ ∆ (cid:2) Tr (cid:0) ∆ † ∆ (cid:1)(cid:3) − ˜ λ (cid:12)(cid:12)(cid:12) Tr (cid:0) ∆ † ˜∆ (cid:1)(cid:12)(cid:12)(cid:12) − (cid:34) ˜ λ (cid:48) (cid:0) Tr (∆ † ˜∆) (cid:1) + H . c . (cid:35) . (2.1) SU (3) c SU (2) SU (2) U (1) Y ∆ 1 2 ¯2 0Φ 1 2 1 + 12 Q L , L L , , − u R , d R , − e R
3o avoid runaway directions, we impose λ Φ , λ ∆ >
0. The ˜ λ and λ quartic couplingsmust be real so that the potential is Hermitian. The ˜ λ (cid:48) quartic coupling may be complex,but its phase can be rotated away by a redefinition of ∆; we then take ˜ λ (cid:48) to be real withoutloss of generality.Canonical normalization of the ˜ λ and ˜ λ (cid:48) terms would require an extra factor of 1 / V , suchas Tr [(∆ † ∆) ], Tr (∆ † ∆ ˜∆ † ˜∆), or Tr (∆ † ˜∆ ˜∆ † ∆), would be redundant as they are linearcombinations of the λ ∆ , ˜ λ and ˜ λ (cid:48) terms. We recover the potential of Ref. [4] using theidentity Φ † (∆ † ∆ + ˜∆ † ˜∆)Φ = Φ † Φ Tr (cid:0) ∆ † ∆ (cid:1) .We also impose m < m < λ < U (1) em and Z symmetries: (cid:104) ∆ (cid:105) = v ∆ , , (cid:104) Φ (cid:105) = v φ √ (cid:18) (cid:19) . (2.2)This vacuum is indeed a minimum of the potential for a range of parameters (discussedbelow). The VEVs v φ > v ∆ > m , m , and the five quartic couplingsby the extremization conditions: λ (cid:63) v + λ v φ = − m ,λ v + λ Φ v φ = − m , (2.3)where we defined λ (cid:63) ≡ λ ∆ − ˜ λ − ˜ λ (cid:48) . (2.4)In terms of fields of definite electric charge, the scalars can be written asΦ = φ + √ (cid:0) v φ + φ r + iφ i (cid:1) , ∆ = η χ + η − χ = (cid:104) ∆ (cid:105) + √ (cid:0) η r + iη i (cid:1) χ + η − √ (cid:0) χ r + iχ i (cid:1) . (2.5)4he charge conjugate state of the bidoublet is then˜∆ = σ ∆ ∗ σ = χ ∗ − η + − χ − η ∗ . (2.6)All odd fields under Z (which cannot mix with even fields, and thus are already in themass eigenstate basis) are collected in∆ − ˜∆ = H + iA √ H + √ H − − H + iA , (2.7)where the physical states consist of a CP-even scalar ( H ), a CP-odd scalar ( A ), and acharged scalar ( H ± ). These are related to the η and χ fields by A = 1 √ (cid:0) η i + χ i (cid:1) ,H = 1 √ (cid:0) η r − χ r (cid:1) ,H ± = 1 √ (cid:0) η ± + χ ± (cid:1) . (2.8)At tree-level, the Z -odd scalars have masses given by M A = (cid:112) λ (cid:48) v ∆ ,M H + = M H = (cid:112) ˜ λ + ˜ λ (cid:48) v ∆ . (2.9)The are two remaining scalars not eaten by the gauge bosons. These are Z -even, CP-even, and neutral; their mass-squared matrix in the ( χ r + η r ) / √ φ r basis is M = λ (cid:63) v λ v φ v ∆ λ v φ v ∆ λ Φ v φ . (2.10)The Z -even physical scalars, h = φ r cos α h − √ (cid:0) χ r + η r (cid:1) sin α h ,H (cid:48) = φ r sin α h + 1 √ (cid:0) χ r + η r (cid:1) cos α h , (2.11)5ave the following squared masses: M h,H (cid:48) = 12 (cid:18) λ (cid:63) v + λ Φ v φ ∓ (cid:113)(cid:0) λ (cid:63) v − λ Φ v φ (cid:1) + 4 λ v φ v (cid:19) . (2.12)The mixing angle α h satisfies tan 2 α h = 2 λ v ∆ v φ λ (cid:63) v − λ Φ v φ . (2.13)The necessary and sufficient conditions for the vacuum (2.2) to be a minimum of thepotential are ˜ λ (cid:48) > Max {− ˜ λ, } ,λ (cid:63) λ Φ > λ ,λ Φ | m | > − λ m ,λ | m | > − λ (cid:63) m ; (2.14)these follow from imposing that all physical scalars have positive squared masses [seeEqs. (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 andthe data suggests that the Higgs sector is near the decoupling limit v φ (cid:28) v ; adopting thislimit, we can analyze the spontaneous symmetry breaking in two stages. The first one is SU (2) × SU (2) × U (1) Y → SU (2) W × U (1) Y at the scale v ∆ . The effective theory below v ∆ consists of the SM (with the Higgs doublet Φ) plus an SU (2) W -triplet of heavy gaugebosons ( W (cid:48)± , Z (cid:48) ), and five of the scalar degrees of freedom from ∆: four Z -odd scalarscombined into an SU (2) W -triplet ( H ± , H ) and a singlet ( A ), and a Z -even singlet ( H (cid:48) ).The second stage of symmetry breaking is the SM one: SU (2) W × U (1) Y → U (1) em at theweak scale v φ ≈
246 GeV. The lightest CP-even scalar, h , represents the recently discoveredHiggs boson, because its couplings are the same as the SM ones up to small corrections oforder v φ /v . Its mass is given by M h = v φ (cid:18) λ Φ − λ λ (cid:63) (cid:19) / (cid:20) − λ v φ λ (cid:63) v + O (cid:0) v φ /v (cid:1)(cid:21) , (2.15)6nd should be identified with the measured Higgs mass, near 126 GeV. The H (cid:48) even scalarhas the same couplings as the SM Higgs except for an overall suppression bysin α h = λ v φ λ (cid:63) v ∆ + O (cid:0) v φ /v (cid:1) , (2.16)and is significantly heavier: M H (cid:48) = (cid:112) λ (cid:63) v ∆ + O (cid:0) v φ /v ∆ (cid:1) . (2.17)Consequently, its dominant decay modes are W + W − and ZZ .The odd scalars, H ± , H , A , couple exclusively to gauge bosons and scalars, and onlyin pairs. The lightest of them is stable, and a component of dark matter. A is naturallythe lightest odd particle (LOP). because in the ˜ λ (cid:48) → A becomes the Nambu-Goldstone boson of a global U (1) symmetry acting on ∆. We note,however, that H could also be the LOP (for ˜ λ (cid:48) > ˜ λ ) and a viable dark matter candidate.Even though it is part of an SU (2) W triplet that is degenerate at tree-level, electroweakloops split the H ± and H masses [6, 7].In what follows we will assume that A is the LOP. The heavier odd scalars then decayas follows: H ± → W ± A , H → ZA . Even when these two-body decays are kinematicallyforbidden, the three-body decays through an off-shell W ± or Z are the dominant ones. Otherchannels are highly suppressed, either kinematically ( H + → π + π H and H + → (cid:96) + νH ) orby loops ( H → γA and the CP-violating H → h A ). B. Meta-sequential W (cid:48) boson The kinetic terms for the Φ and ∆ scalars,( D µ Φ) † D µ Φ + Tr (cid:2) ( D µ ∆) † D µ ∆ (cid:3) , (2.18)involve the covariant derivative D µ = ∂ µ − ig Y Y B µ − ig (cid:126)T · (cid:126)W µ − ig (cid:126)T · (cid:126)W µ , (2.19)with T , = σ , /
2; notice that (cid:126)T acts from the right on the bidoublet: (cid:126)T · ∆ = − ∆ · (cid:126)σ/ v φ g W +1 µ W − µ + v (cid:0) g W +1 µ − g W +2 µ (cid:1) (cid:0) g W − µ − g W − µ (cid:1) . (2.20)7iagonalizing them gives the physical charged spin-1 states, W µ = W µ cos θ + W µ sin θ ,W (cid:48) µ = − W µ sin θ + W µ cos θ , (2.21)with the following mixing angle, 0 ≤ θ ≤ π/ θ = 2 g g v ( g − g ) v − g v φ . (2.22)The masses of the W and W (cid:48) bosons are M W,W (cid:48) = 12 √ (cid:20)(cid:18) g + g ∓ g g sin 2 θ (cid:19) v + g v φ (cid:21) / . (2.23)Given that the left-handed quarks and leptons transform as doublets only under SU (2) ,their couplings to the W and W (cid:48) bosons are proportional to the respective coefficients of W µ in Eqs. (2.21). The measured W coupling to fermions gives a value for the SU (2) W gauge coupling of g = √ πα/s W ≈ . M Z scale: α ≡ α ( M Z ) ≈ / . s W ≡ sin θ W ≈ √ . SU (2) W gaugecoupling can be expressed as g cos θ = g . (2.24)The W (cid:48) coupling to quarks and leptons, derived from Eq. (2.21) and Table I, is then − g sin θ = − g tan θ . (2.25)Thus, tan θ determines completely the tree-level couplings of W (cid:48) to SM fermions. Imposinga perturbativity condition on the SU (2) × SU (2) gauge couplings, g , / (4 π ) (cid:46)
1, and usingEq. (2.24) we find that 0 . (cid:46) tan θ (cid:46) . (2.26)In the particular case of tan θ = 1, the couplings of W (cid:48) to fermions are identical (at treelevel) to those of the W . This is usually referred to as the sequential W (cid:48) boson, and is acommon benchmark model for W (cid:48) searches at colliders. The most recent limit on the massof a sequential W (cid:48) at CMS, using 20 fb − of 8 TeV data, is 3.8 TeV [3], assuming that W (cid:48) can8ecay only into SM fermions. Note that the relative sign in Eqs. (2.24) and (2.25) impliesconstructive interference between the W and W (cid:48) amplitudes that contribute to processesconstrained by W (cid:48) searches at the LHC. In the next sections we will focus on the region0 . < tan θ <
1, where the LHC limits are relaxed. Given that the W (cid:48) boson in this modelhas 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 (cid:48) boson”.The above results are valid for any v φ /v ∆ . It is instructive to expand these results inpowers of ( v φ /v ∆ ) (cid:28)
1. The W (cid:48) coupling to fermions, relative to the W one istan θ = tan θ (cid:18) − v φ v cos θ (cid:19) + O (cid:0) v φ /v (cid:1) , (2.27)where we defined tan θ ≡ g g . (2.28)For v φ (cid:28) v , the values of tan θ span essentially the same range as tan θ . The W and W (cid:48) masses, given in Eq. (2.23), have simple expressions to leading order in v φ /v ∆ : M W = g v φ sin θ (cid:20) − v φ v sin θ + O (cid:0) v φ /v (cid:1)(cid:21) , (2.29) M W (cid:48) = g v ∆ θ (cid:20) v φ v sin θ + v φ v (cid:0) θ − (cid:1) sin θ + O (cid:0) v φ /v (cid:1)(cid:21) . (2.30)The low-energy charged current interactions are mediated in this model by both W and W (cid:48) exchange. Consequently, the Fermi constant is related to our parameters by4 √ G F = ( g cos θ ) M W + ( g sin θ ) M W (cid:48) = g M W (cid:20) v φ v sin θ + O (cid:0) v φ /v (cid:1)(cid:21) , (2.31)where we used Eq. (2.24), which defines g as the tree-level W coupling to leptons andquarks. This shows that the measurements of the weak coupling in low-energy processesand in collider processes involving W bosons should agree up to tiny corrections of order( v φ /v ∆ ) sin θ . Defining the weak scale v ≈
246 GeV through G F = 2 / v − , and using9q. (2.29), we obtain the relation between the Φ VEV and the weak scale v = v φ (cid:20) − v φ v sin θ + O (cid:0) v φ /v (cid:1)(cid:21) . (2.32) C. Z (cid:48) mass and couplings Electrically-neutral gauge bosons also acquire mass terms in the vacuum (2.2): v φ (cid:0) g W µ − g Y B µ (cid:1) + v (cid:0) g W µ − g W µ (cid:1) . (2.33)It is convenient to diagonalize these in two steps. First, we define some intermediate fieldsdenoted with hats: ˆ Z (cid:48) µ = W µ cos θ − W µ sin θ , ˆ Z µ = (cid:0) W µ sin θ + W µ cos θ (cid:1) cos ˆ θ W − B µ sin ˆ θ W , (2.34)where the angle ˆ θ W is defined in terms of coupling ratios:tan ˆ θ W = g Y g sin θ . (2.35)The gauge boson orthogonal to ˆ Z µ and ˆ Z (cid:48) µ is the photon ( A µ = W µ cos θ sin ˆ θ W + B µ cos ˆ θ W ),already in the physical eigenstate. The measured electromagnetic coupling, e = √ πα ≈ . g Y cos ˆ θ W = e . (2.36)The mass-squared matrix for ˆ Z µ and ˆ Z (cid:48) µ takes the form M Z = g θ v φ cos ˆ θ W − v φ tan θ cos ˆ θ W − v φ tan θ cos ˆ θ W v sin θ + v φ tan θ . (2.37)In the second step, we rotate ˆ Z µ and ˆ Z (cid:48) µ by an angle (cid:15) Z , given by10an 2 (cid:15) Z = v φ sin 2 θ sin θ cos ˆ θ W v cos ˆ θ W + v φ sin θ (cid:16) cos ˆ θ W − cot θ (cid:17) , (2.38)in order to obtain the mass eigenstate Z and Z (cid:48) bosons: Z µ = ˆ Z µ cos (cid:15) Z + ˆ Z (cid:48) µ sin (cid:15) Z ,Z (cid:48) µ = − ˆ Z µ sin (cid:15) Z + ˆ Z (cid:48) µ cos (cid:15) Z . (2.39)The masses of the heavy neutral spin-1 particles are M Z,Z (cid:48) = g √ (cid:20) v cos θ + v φ sin θ (cid:18) ˆ θ W + tan θ ∓ θ sin 2 (cid:15) Z cos ˆ θ W (cid:19)(cid:21) / . (2.40)The tree-level results (2.33)-(2.40) have been obtained without approximations. Expand-ing now in v φ /v , we find M Z = g v φ sin θ θ W (cid:20) − v φ v sin θ + O (cid:0) v φ /v (cid:1)(cid:21) , (2.41) M Z (cid:48) = g v ∆ θ (cid:20) v φ v sin θ + v φ v (cid:18) θ cos ˆ θ W − (cid:19) sin θ + O (cid:0) v φ /v (cid:1)(cid:21) . (2.42)The original five parameters from the gauge sector ( g , g , g Y , v φ , v ∆ ) can be traded for threeobservables ( e.g. , e, s W , M W ) and two parameters that can be measured once the W (cid:48) or Z (cid:48) boson is discovered ( M W (cid:48) , tan θ ), using Eqs. (2.27), (2.29), (2.30), (2.36) and s W = sin ˆ θ W (cid:20) − v φ v sin θ cos θ + O (cid:0) v φ /v (cid:1)(cid:21) . (2.43)Eqs. (2.41) and (2.42), combined with the above equation, show that the tree-level relation M Z c W = M W , where c W ≡ cos θ W , is satisfied only up to corrections of order v φ /v .Furthermore, the Z couplings to fermions are modified at order v φ /v compared to theSM. Thus, the current agreement between electroweak measurements and the SM imposesan upper limit on v φ /v , or equivalently, a lower limit on the W (cid:48) 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 (cid:48) (cid:38)
600 GeV for tan θ = 0 .
2, to M W (cid:48) (cid:38) θ = 1 ( i.e. , sequential W (cid:48) ).11he relative mass splitting between W (cid:48) and Z (cid:48) is very small: M Z (cid:48) M W (cid:48) − s W c W tan θ (cid:18) M W M W (cid:48) (cid:19) + O (cid:0) M W /M W (cid:48) (cid:1) , (2.44)which is less than 6 × − for M W (cid:48) > θ <
1. This implies that the W (cid:48) mass and tan θ will be constrained by both Z (cid:48) and W (cid:48) searches. The Z (cid:48) interacts with theleft-handed fermion doublets, with a coupling given by g tan θ T plus corrections of order v φ /v that are different for quarks and leptons.. The Z (cid:48) couplings to SU (2) W singlets aresuppressed by v φ /v . III. W (cid:48) AND Z (cid:48) DECAYS
The new gauge bosons interact with SM fermions and gauge bosons, as well as with theHiggs particles. Usually, resonance searches for new gauge bosons rely on sizable branchingfractions of the W (cid:48) and Z (cid:48) decays into SM fermions. However, if the scalars are lighter thanthe vector bosons than the decays into SM fermions may be suppressed. In our model, theleft-handed fermion doublets transform under SU (2) , while all fermions are singlets under SU (2) . Thus, the W (cid:48) and Z (cid:48) couplings to fermions are induced through mixing with the W and Z , so that for small tan θ decays to heavy scalars become important.Neglecting corrections of O ( v φ /v ), the W (cid:48) and Z (cid:48) coupling to fermion doublets is givenby g tan θ . The partial widths for decays to leptons (without summing over flavors)Γ( W (cid:48) → (cid:96)ν ) ≈ Z (cid:48) → (cid:96) + (cid:96) − ) ≈ α s W tan θ M W (cid:48) , (3.1)are suppressed for 0 . < tan θ <
1. By contrast, the W (cid:48) and Z (cid:48) couplings to pairs of oddHiggs particles are enhanced by 1 / tan θ : g W (cid:48) H ± A = g Z (cid:48) H A = g sin 2 θ ,g W (cid:48) H ± H = g Z (cid:48) H + H − = g tan 2 θ , (3.2)where we ignored corrections of order v φ /v . These couplings lead to the following partial12 .2 0.4 0.6 0.8 1.00102030405060 tan Θ B (cid:72) W ' (cid:174) X Y (cid:76)(cid:72) (cid:37) (cid:76) q q ' t be (cid:43) Ν H (cid:43) H H (cid:43) A tan Θ B (cid:72) Z ' (cid:174) X Y (cid:76)(cid:72) (cid:37) (cid:76) q qt t e (cid:43) e (cid:45) H (cid:43) H (cid:45) H A FIG. 1. W (cid:48) and Z (cid:48) branching fractions as a function of mixing angle, for M W (cid:48) = 3 TeV, M H + = 300GeV, M A = 200 GeV. widths:Γ( W (cid:48) → H ± A ) ≈ Γ( Z (cid:48) → H A ) ≈ αM W (cid:48) s W sin θ (cid:18) − M H + + M A M W (cid:48) + ( M H + − M A ) M W (cid:48) (cid:19) / , Γ( W (cid:48) → H ± H ) ≈ Γ( Z (cid:48) → H + H − ) ≈ αM W (cid:48) s W tan θ (cid:18) − M H + M W (cid:48) (cid:19) / . (3.3)The W (cid:48) can also decay into W Z and
W h final states, but these partial widths are suppressedby v φ /v .Figure 1 shows the branching fractions of the W (cid:48) and Z (cid:48) as a function of tan θ for thedominant channels. As a benchmark point, we have used M W (cid:48) = 3 TeV, M H + = 300 GeVand M A = 200 GeV (as shown in Section II, M W (cid:48) = M Z (cid:48) and M H + = M H to a goodaccuracy). For tan θ (cid:46) .
4, the W (cid:48) decays dominantly to pairs of odd Higgs particles. It isimportant to investigate collider signatures of these decays.The heavier odd scalars decay into the LOP (taken to be A ) and an electroweak bo-son, so that W (cid:48) and Z (cid:48) can each undergo two cascade decays: W (cid:48) → H + A → W A A , W (cid:48) → H + H → W + A ZA (see Figure 2), and Z (cid:48) → H A → Z A A , Z (cid:48) → H + H − → W + A W − A .If the Z symmetry discussed in Section II is exact, then A is a component of darkmatter. We will not explore here the constraints on the parameter space from the upperlimit on relic density, nor from direct detection experiments (nuclear scattering would occur13 ! W ! q ¯ q ! q ¯ q ! A A A A WWZH + H + H W ! W ! q ¯ q ! q ¯ q ! A A A A WWZH + H + H FIG. 2. W (cid:48) production and cascade decays through odd Higgs particles. through Higgs exchange and gauge boson loops); these constraints can be in any case relaxedby allowing a tiny Z violation in the scalar potential. While an in-depth exploration of thismodel as an explanation for dark matter is left for future work, we note that it shares manyfeatures with inert doublet [8] and minimal dark matter scenarios [7].The possibility that the Z symmetry is violated by terms in the scalar potential of thetype Tr (cid:0) ∆ † ˜∆ (cid:1) , Φ † ˜∆ † ˜∆Φ , Tr (cid:0) ∆ † ∆∆ † ˜∆ (cid:1) , Tr (cid:0) ∆ † ∆ (cid:1) Tr (cid:0) ∆ † ˜∆ (cid:1) , (3.4)is also worth considering. The weak-triplet scalar ( H ± , H ) as well as the singlet A wouldmix with the Φ doublet, allowing direct two-body decays of A , H and H ± to SM particles.Furthermore, the three CP-even neutral scalars ( H , h, H (cid:48) ) would then mix, so that W (cid:48) and Z (cid:48) decays involving the SM-like Higgs boson are possible. These include W (cid:48) → H + h with H + → t ¯ b (this channel is analyzed in [9]), as well as W (cid:48) → H + h → W + A h and Z (cid:48) → h A with A → b ¯ b (or t ¯ t if kinematically allowed). There are, however, various constraints ondeviations from the SM Higgs couplings, implying that the Z violating mixing is small, sothat we expect that the above final states have relatively small branching fractions.It is also interesting to consider the intermediate case, where the violation of Z is verysmall, i.e. , the coefficients of the operators (3.4) are much less than one. In that case all W (cid:48) and Z (cid:48) cascade decays through the odd Higgs particles proceed as before, but the A would decay to a pair of heaviest fermions of mass below M A /
2. This leads to a variety ofnoteworthy final states: W (cid:48) → W Z + 4 b , Z (cid:48) → Z + 4 b , or W (cid:48) → W Zt ¯ tt ¯ t , Z (cid:48) → Zt ¯ tt ¯ t , etc.For a range of parameters, the decays of A may be displaced but still within the detector,leading to potentially confusing events. In what follows we will consider only the case where A is stable enough to escape the detector. 14 V. LHC SIGNATURES WITH STABLE A At the LHC, the W (cid:48) boson would be mainly produced in the s channel from quark-antiquark initial state, even for small tan θ . In the narrow width approximation, the leading-order cross section for W (cid:48) production followed by decay into H + A or H + H is σ ( pp → W (cid:48) → H + A , H + H ) ≈ α tan θ s W s w ( M W (cid:48) /s, M W (cid:48) ) B ( W (cid:48) → H + A , H + H ) (4.1)where w ( z, µ ) = (cid:90) x dxx (cid:104) u ( x, µ ) ¯ d ( zx , µ ) + ¯ u ( x, µ ) d ( zx , µ ) (cid:105) . (4.2)The functions u ( x, µ ) and d ( x, µ ) are the proton parton distribution functions for up- anddown- quarks of the at factorization scale µ . Although QCD corrections to W (cid:48) productionare usually significant [10], in our case they are somewhat reduced due to the smaller α s atthe large values of M W (cid:48) that are relevant here.Figure 3 shows the total cross section for the pp → W (cid:48) → H + A and pp → W (cid:48) → H + H processes at √ s = 8 and 14 TeV, with tan θ = 0 .
25. To compute these cross sections, we usedFeynRules [11] for generating vertices from our Lagrangian, and input these into Madgraph 5 M W ' (cid:72) TeV (cid:76) Σ (cid:180) B (cid:72) f b (cid:76) T e V T e V W' (cid:174) H (cid:43) A W' (cid:174) H (cid:43) H Z' (cid:174) H A Z' (cid:174) H (cid:43) H (cid:45) FIG. 3. Leading-order cross sections times branching fractions for the processes pp → W (cid:48) → H + A , H + H and pp → Z (cid:48) → H A , H + H − at √ s = 8 TeV and 14 TeV. We have chosentan θ = 1 / M H + = 300 GeV and M A = 200 GeV.
100 200 300 400 500 600 7000.010.020.050.100.200.50 p T , W (cid:72) GeV (cid:76) (cid:72) G e V (cid:144) Σ (cid:76) d Σ (cid:144) dp T M W' (cid:61) W' (cid:61) Χ Χ
FIG. 4. Transverse momentum distribution of the W produced in pp → W (cid:48) → H + A → W A A when M W (cid:48) = 2 TeV (solid blue line) and M W (cid:48) = 3 TeV (dashed red line), for √ s = 8 TeV, M H + = 300 GeV and M A = 200 GeV. For comparison, the W p T distribution (dotted black line)is included for pp → W χ ¯ χ through a ¯ qγ µ q ¯ χγ µ χ contact interaction (for m χ = 100 GeV). [12] (with parton distribution functions CTEQ6L1 [13]), which includes interference betweenthe W (cid:48) and W contributions. We have set M H + = 300 GeV, M A = 200 GeV; the crosssections are only weakly sensitive to the scalar masses as long as W (cid:48) is much heavier. Figure3 also shows the cross sections for pp → Z (cid:48) → H A and pp → Z (cid:48) → H + H − , for the sameparameters.We assume that the Z symmetry discussed in Section II is sufficiently preserved so thatthe LOP escapes the detector. As noted there, A is most likely the LOP, so that each ofthe above processes includes two A in the final state, which appear as missing transverseenergy ( /E T ) in the detector. If M W (cid:48) (cid:29) M H + , then the W or Z boson emmited in thecascade decays W (cid:48) → H + A → W + A A and Z (cid:48) → H A → ZA A is highly boosted,carrying energy roughly equal of M W (cid:48) /
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 jetwith substructure, plus /E T .The ATLAS collaboration [14] has searched for this type of signature in the case of DMparticles pair produced through a contact interaction to quarks [15, 16]. Compared to our16odel, the processes pp → W ¯ χχ and pp → Z ¯ χχ give rise to a smaller transverse momentumfor the electroweak boson, which is radiated from an initial state quark. In Figure 4 we showthe p T distributions for the W arising from W (cid:48) → H + A → W + A A , as well as from initialstate radiation in the case of a ¯ qγ µ q ¯ χγ 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 forour W (cid:48) decays than in the case of contact interactions.The cascade decays W (cid:48) → H + H → W + A ZA and Z (cid:48) → H + H − → W + A W − A leadto 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 smallbackgrounds. 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 theLHC [18] in the case of contact interactions. Again, in our case the W producing the leptonis generically more boosted. Unlike W (cid:48) decays directly to a lepton-neutrino pair, there willbe no Jacobian peak in the missing transverse energy distribution, as the A ’s carry awaya substantial fraction of the energy of the W (cid:48) . In fact, the distribution will be peaked atlow- p T . Furthermore, if the masses of the A and H + are similar, the transverse momenta ofthe two final-state A particles will have similar magnitudes but opposite directions, so theircontribution to the /E T of the event is reduced. In this case, the missing energy distributioncould look like a SM W decay. This problem is mitigated if the A is substantially lighterthan the Higgs triplet states, in which case the /E T distribution will have a longer tail.We simulate W (cid:48) signals using Madgraph 5 [12], including showering and hadronizationwith Pythia 6.4 [19], and PGS detector simulation [20]; then we analyze the events with theMadAnalysis package [21]. Figure 5 (left panel) shows missing transverse energy distribu-tions for M H = 300 GeV and M H = 1 TeV, all other parameters constant. The transversemass distribution, which is used in LHC W (cid:48) searches, is also peaked at small M T . Moreover,the distribution does not change substantially for different values of the Higgs masses, asshown in Figure 5 (right panel). Therefore, the transverse mass is not the best observablefor a W (cid:48) decaying through odd Higgs particles.17 .0 0.2 0.4 0.6 0.8 1.0 1.2 1.410 (cid:45) (cid:45) (cid:45) (cid:45) E T miss (cid:72) TeV (cid:76) (cid:72) G e V (cid:144) Σ (cid:76) d Σ (cid:144) d E T m i ss M H (cid:61) M H (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) M T (cid:72) GeV (cid:76) (cid:72) G e V (cid:144) Σ (cid:76) d Σ (cid:144) d M T M H (cid:61) M H (cid:61) FIG. 5. /E T distribution (left) and transverse mass (right) distributions for M H = 300 GeV and M H = 1 TeV, all other parameters constant ( M W (cid:48) = 3 TeV, M A = 200 GeV, tan θ = 1 / Φ rad . r a d Σ d Σ d Φ FIG. 6. ∆ φ p (cid:96)T , /E T distribution, for M H + = 300 GeV, M A = 200 GeV, and tan θ = 0 . A better observable for the single-lepton process W (cid:48) → H + A → W + + /E T is the sepa-ration in azimuthal angle between the missing transverse energy and the lepton transversemomentum, ∆ φ p (cid:96)T , /E T . When a W or W (cid:48) decays directly to a lepton-neutrino pair, the decayproducts are nearly back-to-back; for both the W (cid:48) and dark matter mono-lepton analyses,CMS requires that ∆ φ p (cid:96)T , /E T > . π [3]. However, the kinematics for the decay W (cid:48) → A A lν are substantially different, with the ∆ φ p (cid:96)T , /E T distribution peaked at moderate-to-small val-ues of ∆ φ p (cid:96)T , /E T ; see Figure 6. In the rest frame of the W (cid:48) , (cid:126)p lT = − (cid:80) (cid:126)p missT , but in the labframe, the W (cid:48) transverse momentum is distributed among the four decay products and the18 .0 1.5 2.0 2.5 3.0 3.5 4.00.20.40.60.81.0 M W ' (cid:72) TeV (cid:76) t a n Θ FIG. 7. Exclusion limit in the M W (cid:48) − tan θ plane, derived from the CMS W (cid:48) → (cid:96)ν search [3], for M H = 300 GeV and M A = 200 GeV. correlation in azimuthal angle is lost.There are also two processes leading to (cid:96) + (cid:96) − + /E T . One of them is the Z (cid:48) → H A → ZA A cascade decay, with Z → (cid:96) + (cid:96) − ; the related process in the case of contact interactionshas been discussed in [23]. The other one is Z (cid:48) → H + H − → W + A W − A with leptonic W decays; a similar final state, but without s -channel resonance, arises from chargino pairproduction Ref. [24].The limits on our model set by current LHC results are already stronger than those fromelectroweak fits mentioned in Section II. The searches in the W (cid:48) → (cid:96)ν channel, althoughaffected by suppressed branching fraction for small tan θ , set relevant bounds. In Figure7, we reinterpret the 95% CL limit set by the CMS Collaboration [3] on σ excl. /σ SSM W (cid:48) as alimit on tan θ .Existing LHC searches for other processes set less stringent limits. For W (cid:48) → H + A → W + A A → (cid:96) + /E T , we use Figure 4 of [18] to estimate the number of background eventswith 1 < ∆ φ p (cid:96)T , /E T < . W (cid:48) → µ + /E T events in the same region assuming no excess is observed. This limit is at most M W (cid:48) > . θ ≤ W (cid:48) → H + H → W + A ZA → (cid:96) + (cid:96) + (cid:96) − + /E T , we use results from leptonicsearches for charginos and neutralinos in [22]. We consider the search for a same-flavor,19pposite sign electron or muon pair on the Z peak (75 GeV < M (cid:96)(cid:96) <
105 GeV), plus anadditional electron or muon. We sum over M T bins, then we use a Poisson likelihood functionmultiplied over /E T bins to set an upper bound on the number of events. The upper limitson the number of W (cid:48) events in each channel are then translated to a limit on tan θ as afunction of M W (cid:48) . This limit is rather weak: for M W (cid:48) = 1 TeV, only values of tan θ > . /E T final states still provides themost stringent constraint on our model. The reason is that the 3-lepton rate is suppressedby both the W → (cid:96)ν and Z → (cid:96) + (cid:96) − branching fractions. Furthermore, for the mono-lepton search, the selection cuts that optimize signal over background for W → (cid:96)ν cut outa substantial portion of the H ± A events. A new analysis focusing on the small ∆ φ p (cid:96)T , /E T region, using both the electron and muon channels, would provide a stronger limit.Given the mass degeneracy between W (cid:48) and Z (cid:48) , limits set by searches for Z (cid:48) → (cid:96) + (cid:96) − can also be plotted in the M W (cid:48) − tan θ plane. However, they are weaker than those from W (cid:48) → (cid:96)ν because both the production cross section and the leptonic branching fraction aresmaller for Z (cid:48) than for W (cid:48) . V. CONCLUSIONS
The SU (2) × SU (2) × U (1) Y model with a bidoublet and a doublet complex scalars isa simple renormalizable model that can serve as a benchmark for various LHC searches. Itincludes a meta-sequential W (cid:48) boson whose s -channel production interferes constructivelywith the W contribution, and depends on only two parameters: M W (cid:48) and the overall couplingnormalization, tan θ . It also includes a Z (cid:48) boson (degenerate in mass with W (cid:48) ) which couples,to a good approximation, only to left-handed fermions.The potential for the bidoublet (∆) and doublet (Φ) scalars is chosen to be invariantunder a Z transformation that interchanges the bidoublet and its charge conjugate. Thephysical scalar spectrum then consists of four odd Higgs particles (a mass-degenerate weak-triplet H + , H , H − , and a CP-odd singlet A ), the recently discovered Higgs boson ( h ),and a heavier scalar ( H (cid:48) ) whose couplings to SM fields are the same as those of h except20or an universal suppression. The A is naturally the LOP because a global U (1) symmetrybecomes exact in the M A → W (cid:48) and Z (cid:48) bosonsare 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 is a viable dark matter particle remains to be studied.For the present work we focused on the case where A is sufficiently long-lived to escape thedetector, but we also mentioned possible signatures in the case where A decays (promptlyor with a displaced vertex) into fermion pairs.This model illustrates nicely the possibility that the W (cid:48) and Z (cid:48) bosons may decay predom-inantly (with branching fraction as large as 96%) into the scalars responsible for breakingthe extended gauge symmetry. Generically, the high-energy behavior of any W (cid:48) boson re-quires it to be associated with a non-Abelian gauge symmetry (or else it must be a boundstate with the compositeness scale not much higher than its mass), which in turn impliesa larger Higgs sector. The non-Abelian gauge coupling can be significantly larger than theHiggs quartic couplings, implying vector bosons much heavier than the scalars.In our model, the W (cid:48) and Z (cid:48) couplings to the odd Higgs particles are enhanced fortan θ (cid:28) / tan θ . Consequently, the usual u ¯ d → W (cid:48) → (cid:96)ν or t ¯ b channels currently usedin searches at the LHC are suppressed both in production and in braching fractions, thecombined effect being of order tan θ . The mass limits on a sequential W (cid:48) , currently around3.8 TeV, are relaxed for tan θ ≈ . M W (cid:48) > W (cid:48) → H + A → W + A A , W (cid:48) → H + H → W + A ZA , Z (cid:48) → H A → ZA A and Z (cid:48) → H + H − → W + A W − A allowinteresting searches at the LHC, with boosted W and Z bosons decaying either hadronicallyor leptonically. Acknowledgments:
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