Light Doubly Charged Higgs Boson via the $WW^*$ Channel at LHC

The doubly charged Higgs bosons $H^{\pm\pm}$ searches at the Large Hadron Collider (LHC) have been studied extensively and strong bound is available for $H^{\pm\pm}$ dominantly decaying into a pair of same-sign di-leptons. In this paper we point out that there is a large cavity in the light $H^{\pm\pm}$ mass region left unexcluded. In particular, $H^{\pm\pm}$ can dominantly decay into $WW$ or $WW^*$ (For instance, in the type-II seesaw mechanism the triplet acquires a vacuum expectation value around 1 GeV.), and then it is found that $H^{\pm\pm}$ with mass even below $2m_W$ remains untouched by the current collider searches. Searching for such a $H^{\pm\pm}$ at the LHC is the topic of this paper. We perform detailed signal and background simulation, especially including the non-prompt $t\bar{t}$ background which is the dominant one nevertheless ignored before. We show that such $H^{\pm\pm}$ should be observable at the 14 TeV LHC with 10-30 fb$^{-1}$ integrated luminosity.


I. INTRODUCTION
The discovery of the Higgs boson with mass around 125 GeV [1] kicks off a new era toward the Higgs precision test. The observed Higgs properties are well consistent with the prediction of the SM. However, it does not unquestionably mean that the minimal Higgs sector in the SM is the unique choice. In fact, the SM Higgs sector can be extended naturally in a lot of the new physics models beyond the SM, which are motivated to solve some problems in the SM itself. And we can probe these models at the LHC by exploring the additional Higgs bosons or precisely measuring the SM Higgs properties.
As one of the simple but important extension of the SM Higgs sector, the Higgs triplet is inspired by various new physics: the origin of neutrino mass [2], the existence of dark matter [3], and the anomaly in the SM-like Higgs properties [4][5][6][7] (e.g. Higgs naturalness and di-photon excess), and so on. The generic prediction of such extension is the exotic doubly charged scalar particles H ±± , which lead to a same sign di-lepton (SSDL) signature that has almost no SM background. Such a doubly charged Higgs boson, except typically in a SU(2) L triplet, can also be arranged in a singlet [9] and doublet [10] SU(2) L representation. Moreover, H ±± as a component of the multiplets in non-trivial (non-adjoint) SU(2) L representation could be found in many other new physics models [11][12][13].
From the experimental point of view, the searches for new physics beyond the SM at the LHC have a preference for the colored particles, which partially is due to their sizable production rates even at the well motivated TeV scale. Nevertheless, it is also of importance to investigate the status and prospects of new electroweak (EW) charged particles, which are fairly light, says around the Z−boson mass, but missed by the LEP experiment. Such kind of particles are predicted in a lot of models like the sleptons and netralinos in the supersymmetric SMs, the charged Higgs bosons in the models with extended Higgs sectors, and so on. Generically speaking, at the LHC these particles are thoroughly buried in the huge EW and/or QCD backgrounds, except for those with characterized signatures, e.g., large missing transverse energy or SSDL. The latter frequently arises for particles with a larger electric charge, and the doubly charged Higgs bosons, denoted as H ±± , is a good case in point. Therefore, the SSDL, needless to say, provides a perfect channel for the search of H ±± at the LHC.
A lot of works have been done on the search for H ±± in the SU(2) L triplet representation.
Most of the previous works concentrate on the heavier H ±± , whose dominant decay modes can be either the SSDL [14,15] or di-W [16,17], or the cascade decay among scalar fileds [18,19] depending on the model parameters. For a comprehensive discussion on the relative importance of the decay channels of H ±± , see Ref. [20]. The search for H ±± through the SSDL channel has been peformed at the LHC, which excludes the mass of H ±± up to about 300 GeV [21,22]. However, in the current experimental searches the other decay modes like di-W still allow a relatively light H ±± with m H ±± 2m W [23]. Additionally, the light H ±± is well motivated to enhance the Higgs to di-photon rate in the Type II seesaw models [7,8]. There are a lot of motivated new physics models which has a SU(2) L triplet Higgs boson ∆ with hypercharge Y = ±1. Hereafter we take the simplified model approach and make a assumption in the effective model that new particles other than ∆ are absent or decoupled.
Thus, the relevant terms in the Lagrangian can be written as where L kin , L Y and V (Φ, ∆) are the kinetic term, the Yukawa interaction, and the Higgs potential, respectively. Let us define Higgs fields as

The Higgs kinetic terms are
where the covariant derivatives are defined by with (W a µ , g) and (B µ , g ′ ) are, respectively, the SU(2) L and U(1) Y gauge fields and couplings, and τ a = σ a /2 with σ a (a = 1, 2, 3) the Pauli matrices. According to Eqs. (2), (3) and (4), the masses of the W and Z gauge boson at tree level are Then the oblique parameter ρ is Concretely, x 0.01 ≪ 1 is required by the experimental value of ρ [40].
The Yukawa interaction of the triplet field is given by where y ij (i, j = 1, 2, 3) is an arbitrary symmetric complex matrix, C = iγ 0 γ 2 is the charge conjugation operator, and L T i = (ν iL , ℓ iL ) is a left-handed lepton doublet in the SM. After the electroweak symmetry breaking, the Majorana neutrino mass terms can be obtained as The most general potential for the scalars, including the SM If µ = 0, this potential accidently respects the global U(1) symmetry, which preserves the lepton number. When y ij = µ = 0 (and moreover λ 5 > 0 to ensure that δ 0 is the lightest component in ∆, see Eq. (11)), corresponding to a imposed Z 2 symmetry in which ∆ is assigned to be odd and the other fields even, the model can provide a dark matter candidate δ 0 [41].
After minimizing the potential Eq. (8) and considering x 0.01 ≪ 1, one finds that the µ term gives rise to a non-vanishing We can see that there are typically two ways to achieve a sufficiently small v ∆ : (A) µ is around the weak scale, and then the triplet is pushed up to the TeV region; (B) By contrast, the triplet is around the weak scale with M ∆ = v φ , and then µ is forced to lie below the GeV scale as µ = v ∆ [49]. The former case is hard to be detected at colliders owing to the highly suppressed production rate of heavy triplet, while the latter is detectable, especially below the LEP energy threshold, the focus of this work.
Here we give a quick recapitulation of the scalar mass spectrum. In addition to the three So we can see that the quartic λ 5 term is responsible for the masses splittings and satisfies It is shown that there exits three patterns of the mass spectrum for the triplet-like Higgs bosons. When λ 5 = 0 , all the triplet-like Higgs bosons are degenerate in mass. However,

B. Possible constraints
There are various possible theoretical and experimental constraints on the triplet Higgs model or Type II seesaw model [42][43][44][45]. Here, we only highlight some constraints which are closely relevant to our study.
1. The magnitude of v ∆ As discussed above, the VEV v ∆ = 0 modifies the tree-level relation for the electroweak However, this mass splittings between the component of ∆ will induce an additional positive contribution, with proportional to mass splitting, to ρ to cancel the effect lead by v ∆ , for example, an upper limit from perturbativity (λ 5 3) to be v ∆ 7 GeV, for m H = 120 GeV [20]. Conservatively, we take the upper bound v ∆ 2 GeV, which is corresponding to The lepton flavor violations involving µ and τ provide the strongest constraint on the y ij and thus v ∆ ∽ (M ν ) ij /y ij . To accommodate the currently favored experimental constraints, a combined limit on v ∆ times the mass of the doubly charged Higgs M H ±± is derived as follows [46] v ∆ M H ±± 100 eVGeV.
A relevant constraint comes from the neutrino masses. If the Yukawa coupling of triplet scalar is the unique origin for neutrino mass, the current observations from the neutrino oscillation experiments and cosmological bounds give [40]: It is easily seen that y ⋍ 10 −10 can accommodate neutrino masses, if v ∆ = 1GeV. Although the µ terms in V (Φ, ∆) is usually seemed as a lepton number violating term, we can also do an assignment for lepton number in which the Yukawa coupling of triplet scalar is lepton number violated. Then a small y can be naturally attributed to the lepton number violation.
Furthermore, we consider an effective model here, and the other mechanisms or contributions to neutrino masses are possible in the extended models. Thus, the constraint in Eq. (13) can be relaxed.
Since we concentrate on the search for a light (∽ 100GeV) H ±± through di-W channel at the LHC, v ∆ can be safely selected with a relatively lager magnitude as v ∆ = 1GeV in this paper without spoiling the above constraints.
From [40] we know and this puts stringent constraint on the mass of doubly charged scalar. The lower mass bound can be obtained M H ±± > 42.9 GeV at 95% confidential level.
The mass bound on M ±± H can also be taken through its direct searches at the LHC. The ATLAS Collaboration has searched for doubly-charged Higgs bosons via pair production in the SSDL final states. Based on the data sample corresponding to an integrated luminosity of 4.7 fb −1 at √ s = 7 TeV, the masses below 409 GeV, 375 GeV and 398 GeV have been excluded respectively for e ± e ± , e ± µ ± and µ ± µ ± by assuming a branching ratio of 100% for each final state [22]. Besides pair production, the CMS Collaboration also considered the associated production pp → H ±± H ∓ , in which the masses of H ±± and H ∓ are assumed to be degenerate. Using three or more isolated charged lepton final states, the upper limit on M H ±± is driven under specific assumptions on branching ratios [21]. However, other decay modes for H ±± such as di-W will become dominant under some conditions. The search for doubly-charged Higgs boson based on this channel is also studied in Ref. [23].
By fully utilizing the result of the SSDL search by the ATLAS Collaboration (with 4.7fb −1 integrated luminosity at √ s = 7 TeV), the lower limit is obtained to be 60 GeV at the 95% C.L.. Moreover, considering the integrated luminosity of 20fb −1 , the lower bound is evaluated to 85 GeV. Since the treatment for backgrounds and signals in di-W mode will be in principle different from the SSDL case, a detailed analysis on this topic is necessary. In this article, we concentrate on this scenario and elaborate the search for such a H ±± .

A. Production
The prospect for the production of doubly charged scalar H ±± has been widely studied at the hadron colliders such as Tevatron and LHC. For an elaborate discussion on this topic, please see [15]. Although the VEV of triplet v ∆ can be chosen as large as 1 GeV without spoiling the experiment constraints, the main production process for H ±± at the LHC is yet the pair production via Drell-Yan process pp → γ * /Z → H ±± H ∓∓ and the associated production pp → W ± * → H ±± H ∓ . Note that these production processes are independent on v ∆ and only depend on the mass of the doubly charged Higgs m H ±± . The next-to-leading (NLO) QCD corrections to the pair production can increase the cross section by about 20 − 30% [47]. Moreover, the authors have calculated the two-photon fusion process and found its contribution to pair production can be comparable with NLO QCD corrections to Drell-Yan process [16]. For conservatively, we only consider the leading-order (LO) cross section in this work.  In Fig 1, we show the LO production cross sections for the corresponding charged Higgs pair productions at the 14 TeV LHC. The production rate ranges from a few fb to a few pb in mass range of [50,500] GeV. We have also shown in this figure the production rate of H ±± H ∓ associated production, assuming mass degeneracy between H ±± and H ± , whose rate is a few times larger than H ±± H ∓∓ pair production. Hereafter, we only consider the H ±± H ∓∓ pair production as a more conservative study.

B. Decays
In this effective model, the possible decay channels for a relatively light H ±± include: (1) The lepton-number violating (LNV) decays H ±± → ℓ ± i ℓ ± j ; (2) The di-W mode H ±± → W ± W ± * → W ± ff ; (3) The cascade decays H ±± → H ± W ± * → W ± ff . The corresponding decay rates can be found in Appendix A. The LNV decays are apparently proportional to Yukawa coupling y ij , and then inversely proportional to v ∆ due to v ∆ = M v /y. However, the di-W mode is proportional to v ∆ , which means that the higher the value of v ∆ is, the more important we expect the di-W mode to be, with a corresponding decrease in the LNV.
In the last, the cascade decays of H ±± are driven by the gauge coupling and only depend on the mass splitting ∆M = M H ±± − M H ± . To be quantitative, the above mentioned fact has been demonstrated in Fig. 2 and Fig. 3.
In Fig. 2, it is shown that, with the degenerate mass spectrum of triplet-like Higgs bosons, a relatively large v ∆ with v ∆ = 1GeV will lead the di-W mode to be the dominant decay  Fig. 3. It is found that, for a relatively light H ±± , a mass splitting ∆M = 5GeV makes the cascade decays rapidly overcome the di-W mode and become the dominant channel. Additionally, due to a relatively large v ∆ chosen here, the branching ratios for the LNV decays of H ±± is always vanishingly small.

IV. THE LHC PROSPECT OF LIGHT H ±±
In this Section, we employ a detailed study of the discovery prospect for light H ±± at the future LHC. It is found that the 14 TeV LHC is able to cover all the mass region of light H ±± , using the SSDL signature, aided by multi-jets and missing energy.  peaked around the mass of H ±± , which is a very efficient cut for H ±± → l ± i l ± j , no longer works well here, as shown in the middle left of Fig 4. Furthermore, the rate of SSDL in our scenario is suppressed by the W boson decay breaching ratio. So, there are no bound from those searches as well.
Until recently, the CMS Collaboration has searched for H ±± using the SSDL signals with jets in low E miss T and low H T region both with and without b-tagging [29]. Following the similar procedure as in [30], we recast the analysis that proposed in [29]. We find the search can reach our benchmark processes only if the production rates are about 4 times larger. The ratio needed for corresponding benchmark points are shown in the first row of Table III. It will be interesting to consider the associated production H ±± H ∓ , with H ± subsequently decaying into H ±± . The distribution of H ±± can be similar with the direct H ±± H ∓∓ pair production as long as the mass splitting ∆M keeps small. What's more, as we can see from Fig 1, the cross section of associate production is about 2 times larger than pair production. Thus, when ∆M is small, the extra contribution from associated production will possibly help the discovery of H ±± while keep our benchmark points safe from the current LHC searches. Even though we only consider the direct H ±± H ∓∓ pair production in the following discussion, in technical view, our result can be generalized to include the associated production by rescaling.

A. Backgrounds
The background for SSDL signature can be divided into three categories: real SSDL from rare SM processes, non-prompt lepton backgrounds, and opposite-sign dilepton events with charge misidentifications. The non-prompt lepton backgrounds, which are the dominate background for SSDL, arise from events either with jets misidentifying as leptons or with leptons resulting from heavy flavor decay (HF fake). To suppress the non-prompt lepton backgrounds caused by jet misidentification, in our simulation we require the leptons in the final state to be both "tight" [31] and isolated, where the isolated lepton final state means the scalar sum the transverse momentum of calorimeter energy within a cone of R = 0.3 around the lepton excluding the lepton itself must be less than 16% of lepton's p T . In the following analysis, we only need to consider non-prompt background from heavy flavor decay, concretely, semi-leptonic tt events with a non-prompt lepton from b-quark decays.
The dominate processes that give the SSDL in SM and their production cross section at the 14 TeV LHC are listed in Table I. The NLO production cross sections are calculated by MCFM-6.6 [32,33]. However, for ttZ and W + W + jj, the NLO cross section are not calculated by MCFM, so we simply assume their K-factor is 1.5. The background from charge mis-identification are dominated by Drell-Yan processes, di-leptonic tt and W + W − by electrons which have undergone hard bremsstrahlung with subsequent photon conversion.
As pointed out in [34], this kind of backgrounds usually contribute less than 5% of the total backgrounds. We will ignore the charge mis-identification backgrounds in the following discussion, although this backgrounds can also be suppressed by the requirement of isolated lepton final state.

B. Event generation and analysis
The signals and backgrounds are generated by MadGraph5 v1 5 11 [35], where Pythia6 [36] and Delphes 3.0.9 [37] have been packed to implement parton shower and detector simulation. We implement the effective model for doubly charged Higgs in Feyn-Rules [38] to generate the UFO format of this model for MadGraph. Some important details in our simulation are summarized here. While generating the backgrounds from rare SM processes involving weak gauge bosons, the results are listed at matrix element level to maintain helicity information. From Table I, the resulting cross sections can be obtained by rescaling according to the corresponding branching ratios. Additionally, in our simulation only gauge boson decaying into the first two generations of leptons are considered. We use the MLM matching adopted in MadGraph5 to avoid double counting between matrix element and parton shower generation of additional jets. In the last, the matrix element of signals and all backgrounds, except for W ± W ± jj, are generated up to 2 jets.
With the backgrounds and signals from simulation, we consider the event selection procedure in the following: • Events should contain exactly a pair of SSDL and those with additional lepton are vetoed. The leptons are required to satisfy: • We require at least one jet and non of jets is b-tagged in our signal events. The jets are required to have p T > 20 GeV, |η| < 4.5 .
• The LNV decays of H ±± will give small missing energy and the hadronic decay of H ±± will give H T with magnitude proportional to H ±± mass. Thus we require • The invariant mass of SSDL pair should be smaller than H ±± mass m ll < 75 GeV (19) • Since H ±± is light, it can be fairly boosted when it's produced at the 14 TeV LHC.
As a result, the SSDL pair and the missing transverse momentum will tend to align with each other. Therefore, we impose cuts ∆R(l 1 , l 2 ) < 1.5, |∆φ(l 1 , l 2 )| < 1.5, |∆φ(ll, p miss where R(l, l), ∆φ(l, l) and ∆φ(ll, p miss T ) corresponding to angle difference, azimuthal difference between the SSDL, and azimuthal difference between the SSDL system and missing transverse momentum, respectively.
• In H ±± decay, two hadronically decaying W bosons produce many jets, especially at large M H ±± region. We require that there be at least three jets in the signal events, whose invariant mass should be smaller than 150 GeV.
The cuts efficiencies for backgrounds and signals are listed in Table II and Table III, respectively. In these tables, we can see that the SSDL cut is the most efficient one to suppress the huge tt background, which produces SSDL owing to the heavy flavour quark decay. Even though the requirement of SSDL suppress the tt by more than 3 order of magnitude, it still stay as the most dominate background for SSDL signal because of its larger production rate. After SSDL, non-b-tagged jet is imposed to further reduce the backgrounds. We also apply the most well studied E miss As for different signal benchmark points, within the mass range that we have considered, larger M H ±± can have higher rate of SSDL due to the higher energy of lepton it can give.
Furthermore, heavier H ±± can have more energetic jets, which have better sensitivity after N j > 2 cut. On the other hand, larger M H ±± will also give relatively larger m ll , which makes the cut less efficient. Furthermore, heavier H ±± is difficult to be boosted, the angular difference cuts are also slightly weaken with increasing m H ++ .
To have an impression on the discover potential, we calculate the signal significance with the systematic error β = 5%. The signal significance for all benchmark points are given in the last row of Table III. We find that H ±± should be discovered at the 14 TeV LHC with 10-30 fb −1 integrated luminosity for the different masses of H ±± among 100 − 150 GeV.
We choose the cuts such that our search is most conservative in the whole mass range that we are interested in. As for a specific benchmark point, we can fine tune the corresponding cuts to get a better search sensitivity. For example, m ll cut may be needed to become looser for heavier H ±± to have a better signal significance. What's more, the m jjj can even be dropped for m H ±± = 100 GeV. In this case, the signal significance can be as high as 5.3.
In the last of this Section, some further discussions are necessary on the sensitivity for the searches of H ±± at the LHC. Here we only carry out a more conservative study on this topic. In fact, the inclusion of H ±± H ∓ associated production and the non-degenerate mass spectrum will influence our result here. Concretely, the additional associated production will increase the sensitivity of H ±± due to more H ±± production. Additionally, the nondegenerate mass spectrum with H ±± the lightest particle in triplet-like Higgs bosons will also push up the discovery capacity of H ±± [18]. On the other hand, if H 0 is the lightest particle in triplet-like Higgs bosons, the cascade decay of H ±± will open, we can naively expect that the search sensitivity of H ±± will deteriorate due to the decrease of H ±± to di-W decay [19].   first row shows the ratios needed for the production rate so that the benchmark points can be discovered by the CMS search [29]. In the last row, we show the corresponding signal significances for those benchmark points in our search.