Heavy Higgs bosons and the 2 TeV $W'$ boson

The hints from the LHC for the existence of a $W'$ boson of mass around 1.9 TeV point towards a certain $SU(2)_L\times SU(2)_R\times U(1)_{B-L}$ gauge theory with an extended Higgs sector. We show that the decays of the $W'$ boson into heavy Higgs bosons have sizable branching fractions. Interpreting the ATLAS excess events in the search for same-sign lepton pairs plus $b$ jets as arising from $W'$ cascade decays, we estimate that the masses of the heavy Higgs bosons are in the 400--700 GeV range.


Introduction
Using LHC data at √ s = 8 TeV, the ATLAS and CMS Collaborations have reported deviations from the Standard Model (SM) of statistical significance between 2σ and 3σ in several final states, indicating mass peaks in the 1.8-2 TeV range [1]- [5]. The cross sections required for producing these mass peaks are consistent with the properties of a W boson in an SU (2) L × SU (2) R × U (1) B−L gauge theory with right-handed neutrinos that have Dirac masses at the TeV scale [6].
The spontaneous breaking of SU (2)×SU (2)×U (1) gauge groups requires an extended Higgs sector. For large regions of parameter space, the W boson has large branching fractions into heavy scalars from the Higgs sector [7] [8]. We show here that the W boson hinted by the LHC data may also decay into H + A 0 and H + H 0 , where H + , A 0 and H 0 are heavy spin-0 particles present in Two-Higgs-Doublet models. We compute the branching fractions for these decays and present some evidence that signals for the W → H + A 0 /H 0 processes may already be visible in the 8 TeV LHC data.
There are numerous and diverse studies of SU (2) L × SU (2) R × U (1) B−L models, spanning four decades [9]. An interesting aspect of the left-right symmetric models is that they can be embedded in the minimal SO(10) grand unified theory. This scenario must be significantly modified due to the presence of Dirac masses for right-handed neutrinos required by the CMS e + e − jj events. 1 The theory introduced in [6] involves at least one vectorlike fermion transforming as a doublet under SU (2) R . This may be part of an additional SO(10) multiplet, but it may also be associated with entirely different UV completions.
In Section 2 we write down the Higgs potential and analyze its implications for the scalar spectrum. In Section 3 we derive the interactions of the W boson with scalars, and compute all W branching fractions. The couplings of heavy Higgs bosons to quarks are discussed in Section 4. LHC signals of heavy scalars produced in W decays are the subject of Section 5. Our conclusions are summarized in Section 6.

Extended Higgs sector
The Higgs sector of the SU (2) L ×SU (2) R ×U (1) B−L gauge theory discussed in [6] consists of two complex scalar fields: an SU (2) R triplet T of B − L charge +2, and an SU (2) L × SU (2) R bidoublet Σ of B − L charge 0.

Scalar potential
The renormalizable Higgs potential is given by where the triplet-only potential is (2. 2) The bidoublet-only potential V (Σ) is chosen such that by itself it does not generate a VEV for Σ. This is discussed later, together with V (T, Σ), which collects all the terms that involve both scalars.
For M 2 T > 0 and λ T + λ T > 0, the T scalar acquires a VEV, which upon an SU (2) R × U (1) B−L transformation can be written as where the minimization of V (T ) gives to large masses for the W and Z bosons. The value of the T VEV is related to the parameters of the W boson. In the next section we will show that the parameters indicated by the LHC mass peaks near 2 TeV imply u T ≈ 3−4 TeV.
The triplet field includes 6 degrees of freedom, and can be written as T = (T 1 , T 2 , T 3 ), with T i (i = 1, 2, 3) complex scalars, or more explicitly as where σ i are the Pauli matrices, and the factors of √ 2 are required for canonical normalization of the kinetic terms. The fields of definite electric charge, which are combinations of the T i components, include three Nambu-Goldstone bosons (G ± R , G 0 R ). These become the longitudinal degrees of freedom of the W ± and Z bosons. The three remaining fields are a real scalar T 0 , a doubly-charged scalar T ++ and its charge conjugate state T −− . These have masses given by For quartic couplings in the 0.1-1 range and in the absence of fine tuning, the T 0 and T ++ particles have masses comparable to, or heavier than W .
The bidoublet-only potential includes the following terms: where Σ is the charge conjugate state of the bidoublet. Other terms, such as Tr (Σ † Σ) 2 can be written as linear combinations of the terms in Eq. (2.7). We take M 2 Σ ,M 2 Σ > 0, which is a sufficient condition for the minimum of V (Σ) to be at Σ = 0. The mixed terms, which involve both the T and Σ scalars, will induce a nonzero VEV. In terms of fields of definite electric charge, the bidoublet scalar can be written as and its charge conjugate state is The mixed terms in the potential are: For λ T Σ > M 2 Σ /u 2 T the VEV of T induces a negative squared mass for (Σ 0 2 , Σ − 2 ), given by −λ T Σ u 2 T + M 2 Σ < 0. Likewise, the squared mass for (Σ 0 1 , Σ − 1 ) turns negative, due to the second term in Eq. (2.10), when −λ T Σ u 2 T + M 2 Σ < 0. In addition, a Σ 0 1 Σ 0 2 term is induced forλ T Σ = 0. Thus, for a range of parameters, the VEV of Σ takes the form where v H 174 GeV is the electroweak scale. We are interested in the case where The effect of the Σ VEV on the T 0 and T ++ masses and couplings is thus negligible. At energy scales below the T 0 and T ++ masses, the scalar sector consists only of Σ, which is the same as two Higgs doublets.

Higgs bosons
Using the notation of Two-Higgs-Doublet models [11], the components of Σ defined in Eq. (2.8) are related to the Higgs doublets H 1 and H 2 as follows: where h 0 is the SM-like Higgs boson, and G ± , G 0 are the Nambu-Goldstone bosons, which become the longitudinal W and Z. We have not included here the effects of the CPviolating phase α Σ , which would lead to H 0 − A 0 mass mixing. The measurements of h 0 couplings are in good agreement with the SM predictions, implying an alignment limit [12], α = β − π/2. In addition, the observed ATLAS events [1] consistent with the W → W Z process indicate sin 2β 0.8 (see Ref. [6]), which gives 0.5 tanβ 2.
We have obtained a Two-Higgs-Doublet model with a potential formed of the 4 quartic terms of V (Σ), and 3 independent mass terms from V (Σ) + V (T, Σ) with T replaced by its VEV. The most general renormalizable potential for two Higgs doublets includes three more quartic terms. The allowed potential in our model is a special case of the general Two-Higgs-Doublet model, where λ 1 = λ 2 = λ 3 ≡ λ Σ and λ 6 = λ 7 ≡λ Σ , using the standard notation [11]. The usual −m 2 12 H 1 H 2 mass mixing term arises here from the Tr( Σ † Σ) term in V (Σ) and the last term in V (T, Σ), so that (2.13) The squared masses of the charged Higgs boson and of the CP-odd scalar take a simple form: 14) The value of m 12 given by Eq. (v H /m 12 ) 2 1. The heavy CP-even scalar, H 0 , has a squared mass Note that the relative mass splittings between H ± , A 0 and H 0 are small. These four states approximately form a heavy SU (2) W doublet of zero VEV.
The mass of the SM-like Higgs boson is related to the quartic couplings by This provides an estimate of the typical values of the quartic couplings: for tan β → 1, 2λ Σ +λ Σ +λ Σ ≈ 0.5. The departure from the alignment limit is given by (2.17)

W decays into heavy Higgs bosons
Given the Higgs sector discussed in the previous section, which breaks the SU (2) L × SU (2) R × U (1) B−L gauge symmetry down to the U (1) em group of electromagnetism, we can now derive the W couplings to bosons and the ensuing decay widths.

Couplings of the W to bosons
The SU (2) L and SU (2) R gauge bosons include electrically-charged states, W ±µ L and W ±µ R , respectively. The kinetic terms for the T and Σ scalars give rise to the following mass terms for the charged gauge boson TeV. The SU (2) L gauge coupling is given by the SM SU (2) W gauge coupling g, while the SU (2) R gauge coupling is given by the W coupling g R to the ud quarks in the gauge eigenstate basis: where the range for g R is required in order to explain [6] the LHC mass peaks near 2 TeV.
The physical bosons, W and W , are admixtures of W ±µ L and W ±µ R : and the W and W masses are given by For M W ≈ 1.9 TeV and g R ≈ 0.45-0.6 (as determined in [6], by comparing the W production cross section to the CMS dijet excess [4]), we find the SU (2) R breaking scale u T ≈ 3-4 TeV. This set of parameters is compatible with constraints from electroweak precision observable [13].
The W W Z interactions are given by Here c W ≡ cos θ W is the usual SM parameter, and [µ, ν] represents commutation (µν − νµ) of the two Lorentz indices. Both W + L W − L Z and W + R W − R Z terms contribute to the above W W Z interactions through W L -W R mixing, with coupling strengths gc W and gs 2 W /c W , respectively. Note that the coefficient of Eq. (3.6) can also be written as Comparing this form with Eq. (4) of [6] gives ξ Z = sin 2β.
The W interactions with a W and a neutral Higgs boson arise from the Σ kinetic term: also a W ± interaction with ZH ∓ , which gets contributions from the kinetic terms of Σ The W ± couplings to H ∓ and one of the neutral Higgs bosons are the following: (3.9) In the alignment limit with tan β → 1 the above W ± H ∓ h 0 coupling vanishes, while the W ± H ∓ H 0 coupling reaches its maximum. Finally, the W ± H ∓ A 0 coupling is

W branching fractions
The dominant decay modes of W are into fermion pairs, mainly quark-antiquark pairs.
W can also decay into bosons, leading to challenging and interesting phenomenology. Amongst many possible bosonic decay modes, W → W Z and W → W h 0 are inevitable, originating from the kinetic term of the bidoublet scalar field Σ. Unlike other bosonic decays discussed below, the widths of these two modes do not depend on the unknown masses of heavy scalars. Eqs. (3.6) and (3.7) imply where terms of order (M W,Z,h /M W ) 2 have been neglected, and the approximate relation c W M Z M W has been used. We use full expressions for numerical study later in this section, which lead to a slight difference between the partial widths of these two W decay modes, arising from differences in underlying dynamics and kinematics.
The equivalence theorem requires that W decays into fields that are part of the same Higgs doublet (the longitudinal Z and h 0 for α → β − π/2 in this case) have equal decay widths up to electroweak symmetry breaking effects and phase-space factors. As mentioned in Ref. [6], in the alignment limit Γ Besides h 0 and the longitudinal W and Z, the bidoublet field Σ includes the heavy scalars H ± , H 0 , A 0 . The range of allowed masses for these particles spans more than an order of magnitude, from the weak scale to the SU (2) R breaking scale. If they are lighter than the W boson, then W decays may provide the main mechanisms for production of these scalar particles at hadron colliders. The W decays into a heavy scalar and an electroweak boson have widths where φ labels the heavy neutral scalars, and ξ φ follows from the couplings in Eq. (3.7): and a similar replacement for W ± → ZH ± , where the function F is defined by and f is the phase-space suppression factor: Similarly, the W decay width into a charged Higgs boson and the SM-like Higgs particle is The terms of order M h /M W , neglected here, are taken into account by replacing the last factor in the above equation by For M H ± < M W /2, the W can also decay into a pair of heavy scalars, with widths where ξ 2 φ is defined in Eq. (3.13). Here we used the M A = M H 0 = M H ± limit; the exact expression is obtained by replacing the last factor with f 3 Note that the leading order in M 2 H ± /M 2 W is independent of tan β. The dominant decay modes of W are into quarks, and have the following widths: for light flavors, and for heavy flavors. QCD corrections increase these two widths by about 3%. The decay widths into a τ lepton or an electron and the N τ right-handed neutrino are where s θe is the coefficient of the g R W νē R γ ν N τ R interaction term in the Lagrangian, and the analogous coefficient for the muon satisfies s θµ s θe . The baseline W model used in [6] to explain the excess events near 2 TeV reported in several channels by ATLAS and CMS, including e + e − jj [2], has s θe ≈ 0.5, and is consistent with all current constraints on flavor-changing processes. 2 We emphasize that taking into account the e + e − jj excess was crucial in identifying our baseline W model (without it, leptophobic W models [15] are interesting alternatives). If the W boson will be discovered in Run 2, then the eejj process will allow various tests of the underlying couplings [16].
The W → jj and W → tb channels have a combined branching fraction of approximately 86%, as shown in Figure 1. The branching fractions for W → τ N τ R and W → eN τ R add up to 6% for the benchmark value of M N τ R = 1 TeV. The remaining decay modes are into bosons, which together have a branching fraction of approximately 8% for M H ± = 500 GeV, as indicated by the dashed line in Figure 1. Note that the difference between tan β = 1 and 2 cannot be resolved, as expected from Eq. as favored by the CMS and ATLAS mass peaks near M W = 1.9 TeV discussed in [6].
The dependence on the CP-even Higgs mixing angle α is very weak, due to the alignment limit Eq.

Quark masses
The SU (2) L × SU (2) R × U (1) B−L gauge symmetry allows quark masses to be generated by Yukawa couplings to the bidoublet Σ: Here i, j = 1, 2, 3 label the SM fermion generations, Q i L = (u i L , d i L ) is the SU (2) L quark doublet of the ith generation, and Q i R = (u i R , d i R ) is the corresponding SU (2) R doublet. The Yukawa couplings y ij andỹ ij are complex numbers. The mass terms for the up-and down-type quarks are then These terms highlight a problem. As we are interested in tan β = O(1), the above terms generically induce masses of the same order of magnitude for up-and down-type quarks.
Even though the fermion mass hierarchies are not understood in the SM, they can be fitted by appropriately small Yukawa couplings. In the case of Eq. (4.2), the known ratio of the bottom and top quark masses can be fitted only by tuning various parameters.
For example, m b v H can be achieved by imposingỹ 33 /y 33 − tanβ. In the absence of an explanation for the tuning of these independent parameters, it is useful to explore alternative mechanisms for fermion mass generations.
Consider the case where y ij andỹ ij are negligibly small. A different type of gaugeinvariant operator that can generate the quark masses is where the SU (2) R indices of the scalar triplet T are contracted with T † such that T † T belongs to the triplet representation of SU (2) R . The flavor-dependent coefficients C ij are complex dimensionless parameters, and form a 3 × 3 mass matrix C. We assume that the analogous operator with Σ replaced by Σ has negligibly small coefficients. Replacing T by its VEV given in Eq. (2.3), the dimension-6 operator of Eq. (4.3) generates a 3 × 3 mass matrix for the up-type quarks, while the down-type quarks remain massless at this stage.
There is, however, an analogous effective operator with T replaced by T ≡ σ 2 T * σ 2 , This generates a mass matrix only for down-type quarks: Given that the coefficientsC ij are different than C ij , we obtain mass matrices for the up-and down-type quarks that are independent of each other, as in the SM. Various possibilities for the origin of the effective operators (4.3) and (4.5) will be discussed elsewhere.
We have obtained the Yukawa couplings of the Two-Higgs-doublet model of Type II. This automatically avoids tree-level flavor-changing processes (see [17] and references therein).
The couplings of the heavy Higgs bosons to the physical eigenstates of the quarks are proportional to the quark masses, with overall coefficients of sinβ for up-type quarks and cosβ for down-type quarks. For 0.5 tanβ 2 and close to the alignment limit, as indicated [6] by the W signals near 2 TeV, the branching fractions for H + → tb and H 0 , A 0 → tt are almost 100%.

Signals of heavy scalars produced in W decays
Even though the W decays into heavy scalars have only percent-level branching fractions (see Figure 2), the large W production cross section at the LHC [6] (200-350 fb at √ s = 8 TeV, and 1-2 pb at √ s = 13 TeV) makes these decays promising discovery channels. The pp → W → H ± A 0 and H ± H 0 processes lead to cascade decays that end up with 3W + 4b final states, including where = e or µ. The charge conjugate processes lead to a number of events smaller by a factor of almost 2, due to the smaller W − production in pp collisions. The cross section times branching fraction relevant for these processes is shown in Figure 3 for the LHC at 8 and 13 TeV. We used there tan β = 1 and a W production cross section of 300 fb at √ s = 8 TeV for M W = 1.9 TeV (implying a cross section of 1.7 pb at √ s = 13 TeV). An additional W decay mode that contributes to final states with same-sign leptons or three leptons and b jets is The branching fraction for the N τ decay used here is B(N τ → τ − tb) ≈ 23%, while similar final states arise from B(N τ → e − tb) ≈ 8%. The cascade decays W + → e + N τ with N τ → τ tb or etb (the latter process is mentioned in [18]) also contribute.
The heavy scalars may also be produced directly, without W decays. A promising channel is the production of A 0 and H 0 in association with a tt pair [19]: Compared to the SM tth 0 process at large M h , each of the above two processes has a cross section scaled by (tan β) −2 . In Figure 3 we show the cross section for this process, computed at leading order using MadGraph5 aMC@NLO [20] with the MSTWnlo2008 [21] parton distribution functions. We see that the processes listed in Eqs. deviations above the background. We note that the W -produced events include more leptons with positive charge than with negative charge; the same feature is seen in the ATLAS events (Tables 11 and 12 of [22]).
If the heavy scalar masses are decreased to M H ± ≈ 400 GeV, the number of signal events in Table 1 increases by a factor of about 3, so that the number of predicted events becomes too large (at tan β = 2 it may still be acceptable, given the uncertainties in the event selection efficiencies). For M H ± 700 GeV the number of predicted events becomes too small to account for the ATLAS excess. Thus, the preferred mass range for the heavy Higgs bosons is 400-700 GeV.
A CMS search [23] in a similar final state with same-sign leptons and b jets has yielded a smaller excess. The sum over the number of events with two or more b jets, large H T and high lepton p T from Table 3 of [23] gives 11 observed events for a background of roughly 6 ± 2 events. The compatibility of the ATLAS and CMS results in these channels needs further scrutiny. We note though that the CMS event selection includes a veto of a third lepton, while the events reported by ATLAS include events with three leptons. 3 3 In addition, the CMS jets are required to have p T > 40 GeV, while for the ATLAS search a jet p T > 25 GeV cut is imposed. Note that the energy released in the W decay is shared between 10 or more objects, so that the stronger jet p T cut imposed by CMS may reduce the number of observed events.
3-3.5 6 (1.1±0.9 ± 0.4) ttA 0 , ttH 0 → 4t 4.1 × 10 −2 0.7 Table 1: Contributions from W cascade decays and ttA 0 /H 0 production to the same-sign leptons plus b-jets signals at the 8 TeV LHC. The last column gives the observed and expected number of events in the ATLAS search [22]. For the upper and lower blocks of the table tan β = 1 and 2, respectively. The range of predicted signal events corresponds to the 200-350 fb range for the W production rate. The parameters are fixed as follows: heavy scalar masses are 500 GeV, the N τ mass is 1 TeV, s θe = 0.5, and M W = 1.9 TeV.
Searches for final states with 3 leptons may also test the presence of the heavy Higgs particles, as follows from the processes in Eqs. (5.1)-(5.3). The CMS search [24] for events with 3 leptons and one or more b jets has yielded a deficit of events compared to the SM prediction. In particular, the large H T category with no hadronic τ decays, E T / > 100 GeV and no e + e − or µ + µ − pairs includes a single observed event for an expected background of 5.5 ± 2.2 events ( would pass the criterion of 3 leptons and no e + e − or µ + µ − pairs. Similar suppressions apply to the number of 3 events contributed by the processes listed in the other rows of Table 1. Thus, searches in 3-lepton final states do not appear to be able for now to differentiate between the SM and our theory with a W and heavy Higgs bosons.
The event selection employed in the four-top ATLAS search [22] has a few additional categories, including those with lower H T , one b-jet, lower E T / , which have not lead to significant excesses as the categories "SR4t3" and "SR4t4" discussed above. The heavy Higgs contributions from W decays to these other categories are much smaller. For instance, the heavy W renders the H T large (> 700 GeV), with harder E T / (> 100 GeV).
Due to the use of "loose" b-tagging in the search and the presence of four b quarks in the heavy Higgs signal, the number of events in the one b-jet categories is a factor of 3-5 smaller than in the two b-jet ("SR4t3") and three-or-more b-jet ("SR4t4') categories.

Conclusions and outlook
A W boson of mass near 1.9 TeV with properties detailed in Ref. [6] provides a compelling explanation for the excess events reported by the ATLAS and CMS Collaborations [1]- [5] in the W Z, W h 0 , e + e − jj, and jj channels. In this article we have shown that the gauge structure associated with that W boson predicts the existence of heavy Higgs bosons of masses between the weak scale and a few TeV. We have derived the branching fractions for all W decays, including six channels with heavy Higgs bosons (see Figures 1 and 2).
The main decay modes for the heavy Higgs bosons are A 0 , H 0 → tt and H + → tb.
If their masses are below M W /2, then the cascade decay W → H ± A 0 /H 0 → 3t + b → 3W +4b has a branching fraction of up to 3% and provides a promising way for discovering all these particles. An excess of events, with a statistical significance of about 3σ, has been reported by the ATLAS Collaboration [22] in the final state with two leptons of same charge and two or more b jets. We have shown that this can be explained by the cascade decays of W if the heavy Higgs bosons have masses in the 400-700 GeV range.
The SU (2) L × SU (2) R × U (1) B−L gauge theory presented here depends on only a few parameters, whose ranges are already determined by accounting for the deviations from the SM mentioned above. The various phenomena predicted by this gauge theory can thus be confirmed or ruled out in the near future. In Run 2 of the LHC, the W production cross section is large, in the 1-2 pb range at √ s = 13 TeV. Besides resonant production of W Z, W h 0 , jj and tb, there are several W discovery modes: W → τ N τ → τ τ jj, τ τ tb, eτ jj, eτ tb and W → eN τ → eejj, eetb would test the existence of the heavy right-handed neutrino, while W → H ± A 0 /H 0 → 3t + b, W → W A 0 /H 0 → W tt, W → H ± h 0 → tbh 0 and others would test the existence of the heavy Higgs bosons.
Another promising search channel for the heavy neutral Higgs bosons, independent of the W , follows from production in association with a tt pair, which has a cross section of the order of 10 fb at √ s = 13 TeV. With more data, the Z boson analyzed in [6] will also be accessible in a variety of channels.