Probing doubly charged scalar bosons from the doublet at future high-energy colliders

The isospin doublet scalar field with hypercharge 3/2 is introduced in some new physics models such as tiny neutrino masses. Detecting the doubly charged scalar bosons from the doublet field can be a good probe of such models. However, their collider phenomenology has not been examined sufficiently. We investigate collider signatures of the doubly and singly charged scalar bosons at the LHC for the high-luminosity upgraded option (HL-LHC) by looking at transverse mass distributions etc. With the appropriate kinematical cuts we demonstrate the background reduction in the minimal model in the following two cases depending on the mass of the scalar bosons. (1) The main decay mode of the singly charged scalar bosons is the tau lepton and missing (as well as charm and strange quarks). (2) That is into a top bottom pair. In the both cases, we assume that the doubly charged scalar boson is heavier than the singly charged ones. We conclude that the scalar doublet field with $Y = 3/2$ is expected to be detectable at the HL-LHC unless the mass is too large.


I. INTRODUCTION
In spite of the success of the Standard Model (SM), there are good reasons to regard the model as an effective theory around the electroweak scale, above which the SM should be replaced by a model of new physics beyond the SM. Although a Higgs particle has been discovered at the LHC [1], the structure of the Higgs sector remains unknown. Indeed, the current data from the LHC can be explained in the SM. However, the Higgs sector in the SM causes the hierarchy problem, which must be solved by introducing new physics beyond the SM. In addition, the SM cannot explain gravity and several phenomena such as tiny neutrino masses, dark matter, baryon asymmetry of the universe, and so on. Clearly, extension of the SM is inevitable to explain these phenomena.
In the SM, introduction of a single isospin doublet scalar field is just a hypothesis without any theoretical principle. Therefore, there is still a room to consider non-minimal shapes of the Higgs sector. When the above mentioned problems of the SM are considered together with such uncertainty of the Higgs sector, it might happen that it would be one of the natural directions to think about the possibility of extended Higgs sectors as effective theories of unknown more fundamental theories beyond the SM. Therefore, there have been quite a few studies on models with extended Higgs sectors both theoretically and phenomenologically.
Additional isospin-multiplet scalar fields have often been introduced into the Higgs sector in lots of new physics models such as models of supersymmetric extensions of the SM, those for tiny neutrino masses [2][3][4][5][6][7][8][9][10][11][12], dark matter [13][14][15], CP-violation [16,17], and the first-order phase transition [18,19]. One of the typical properties in such extended Higgs sector is a prediction of existence of charged scalar states. Therefore, theoretical study of these charged particles and their phenomenological exploration at experiments are essentially important to test these models of new physics.
In this paper, we concentrate on the collider phenomenology of the model with an additional isodoublet field Φ with Y = 3/2 at the high-luminosity-LHC (HL-LHC) with the collision energy of √ s = 14 TeV and the integrated luminosity of L = 3000 fb −1 [38].
Clearly, Φ cannot couple to fermions directly. The component fields are doubly charged scalar bosons Φ ±± and singly charged ones Φ ± . In order that the lightest one is able to decay into light fermions, we further introduce an additional doublet scalar field φ 2 with the same hypercharge as of the SM one φ 1 , Y = 1/2. Then, Y = 3/2 component fields can decay via the mixing between two physical singly charged scalar states. Here, we define this model as a minimal model with doubly charged scalar bosons from the doublet. This minimal model has already been discussed in Ref. [20], where signal events via pp → W + * → Φ ++ H − i have been analyzed, where H ± i (i = 1, 2) are mass eigenstates of singly charged scalar states. They have indicated that masses of all the charged states Φ ±± and H ± i may be measurable form this single process by looking at the Jacobian peaks of transverse masses of several combinations of final states etc. However, they have not done any analysis for backgrounds. In this paper, we shall investigate both signal and backgrounds for this process to see whether or not the signal can dominate the backgrounds after performing kinematical cuts at the HL-LHC. This paper is organized as follows. In Sec. II, we introduce the minimal model with doubly charged scalar bosons from the doublet which is mentioned above, and give a brief comment about current constraints on the singly charged scalars from some experiments.
In Sec. III, we investigate decays of doubly and singly charged scalars and a production of doubly charged scalars at hadron colliders. In Sec. IV, results of numerical evaluations for the process pp → W + * → Φ ++ H − i are shown. Final states of the process depend on mass spectrums of the charged scalars, and we investigate two scenarios with a benchmark value.
Conclusions are given In Sec. V. In Appendix A, we show analytic formulae for decay rates of two-body and three-body decays of the charged scalars.

II. MODEL OF THE SCALAR FIELD WITH Y = 3/2
We investigate the model whose scalar potential includes three isodoublet scalar fields φ 1 , φ 2 , and Φ [20]. Gauge groups and fermions in the model are same with those in the SM.
Quantum numbers of scalar fields are shown in Table I. The hypercharge of two scalars φ 1 and φ 2 is 1/2, and that of the other scalar Φ is 3/2. In order to forbid the flavor changing neutral current (FCNC) at tree level, we impose the softly broken Z 2 symmetry, where φ 2 and Φ have odd parity and φ 1 has even parity [39]. The scalar potential of the model is given by where V THDM is the scalar potential in the two Higgs doublet model (THDM), and it is given by The Z 2 symmetry is softly broken by the terms of µ 2 3 φ † 1 φ 2 and its hermitian conjugate. Three coupling constants µ 3 , λ 5 and κ can be complex number generally. After redefinition of phases of scalar fields, either µ 3 or λ 5 remains as the physical CP-violating parameter.
In this paper, we assume that this CP-violating phase is zero and all coupling constants are real for simplicity.
Component fields of the doublet fields are defined as follows.
Mass terms for the neutral scalars h i and z i are generated by V THDM . Thus, mass eigenstates of the neutral scalars are defined in the same way with those in the THDM (See, for example, Ref. [40]). Mass eigenstates h, H, A, and z are defined as where α and β (= Tan −1 (v 2 /v 1 )) are mixing angles, and R(θ) is the two-by-two rotation matrix for the angle θ, which is given by The scalar z is the Nambu-Goldstone (NG) boson, and it is absorbed into the longitudinal component of Z boson. Thus, the physical neutral scalars are h, H, and A. For simplicity, we assume that sin(β − α) = 1 so that h is the SM-like Higgs boson.
On the other hand, the mass eigenstates of singly charged scalars are different from those in the THDM, because the field Φ ± is mixed with ω ± 1 and ω ± 2 . The singly charged mass eigenstates ω ± , H ± 1 , and H ± 2 are defined as  The scalar ω ± is the NG boson, and it is absorbed into the longitudinal component of W ± boson. Thus, there are two physical singly charged scalars H ± 1 and H ± 2 . The doubly charged scalar Φ ±± is mass eigenstate without mixing.
The doublet Φ does not have the Yukawa interaction with the SM fermions because of its hypercharge. 1 Therefore, Yukawa interactions in the model is same with those in the THDM.
They are divided into four types according to the Z 2 parities of each fermion (Type-I, II, 1 If we consider higher dimensional operators, interactions between Φ and leptons are allowed [32]. X, and Y [41]). In the following, we consider the Type-I Yukawa interaction where all lefthanded fermions have even parity, and all right-handed ones have odd-parity. The type-I Yukawa interaction is given by where Q iL (L iL ) is the left-handed quark (lepton) doublet, and u jR , d jR , and ℓ jR are the right-handed up-type quark, down-type quark and charged lepton fields, respectively. The Yukawa interaction of the singly charged scalars are given by [16,42], δ ij is the Kroneker delta, and P L (P R ) is the chirality projection operator for lefthanded (right-handed) chirality. In addition, (u 1 , u 2 , u 3 ) = (u, c, t) are the up-type quarks, are the down-type quarks, (ℓ 1 , ℓ 2 , ℓ 3 ) = (e, µ, τ ) are the charged leptons, and (ν 1 , ν 2 , ν 3 ) = (ν e , ν µ , ν τ ) are the neutrinos. The symbols m u i , m d i , and m ℓ i are the masses for u i , d i , and ℓ i , respectively. In the following discussions, we neglect non-diagonal terms of the CKM matrix.
Finally, we discuss constraints on some parameters in the model from various experiments.
If the coupling constant κ in the scalar potential is zero, the model have a new discrete Z 2 symmetry where the doublet Φ is odd and all other fields are even. This Z 2 symmetry stabilizes Φ ±± or Φ ± , and their masses and interactions are strongly constrained. Thus, κ = 0 is preferred, and it means that sin χ = 0. In this paper, we assume that χ = π/4 just for simplicity. Since the charged scalars H ± 1 and H ± 2 have Type-I Yukawa interaction, it is expected that the constraints on H ± 1 and H ± 2 are almost same with those on the charged Higgs boson in the Type-I THDM and the difference is caused by the factor sin χ or cos χ in Eq. (8). In the case where sin χ = cos χ = 1/ √ 2, the constraints are as follows. For tan β 1.4, the lower bound on the masses of H ± 1 and H ± 2 are given by flavor experiments. This lower bound depends on the value of tan β, and it is about 400 GeV for tan β = 1 [43][44][45]. In the region that 1.4 tan β 5.7, the lower bound on the mass is given by the search for the decay of the top quark into the bottom quark and the singly charged scalar at the LHC Run-I. This lower bound is about 170 GeV [45,46]. For tan β 5.7, the direct search at LEP gives the lower bound on the mass. It is about 80 GeV [47]. From Eq. (8), it is obvious that if we think the case where | sin χ| > | cos χ|, (| sin χ| < | cos χ|) the constraints on H ± 1 (H ± 2 ) are relaxed, and those on H 2 (H ± 1 ) become more stringent.

III. PRODUCTION AND DECAYS OF CHARGED SCALAR STATES
In this section, we investigate the decay of the new charged scalars and the production of the doubly charged scalar at hadron colliders. In the following discussion, we assume that Φ ±± , H, and A are heavier than H 1 ± and H 2 ± . Then, H ± 1,2 cannot decay into Φ ±± , H, and A. In addition, the masses of H ± 1 , H ± 2 , and Φ ±± are denoted by m H 1 m H 2 , and m Φ , respectively.

A. Decays of charged scalar sates
First, we discuss the decays of the singly charged scalars H ± 1 and H ± 2 . They decay into the SM fermions via Yukawa interaction in Eq. (8). Since they are lighter than Φ ±± , H, and A, their decays into Φ ±± W ∓( * ) , HW ±( * ) , and AW ±( * ) are prohibited. On the other hand, the decay of the heavier singly charged scalars into the lighter one and Z ( * ) is allowed, and it is generated via the gauge interaction. In the following, we assume that H ± 2 is heavier than H ± 1 (m H 2 > m H 1 ). In Fig. 1, the branching ratio for each decay channel of H ± 1 is shown. Since we assume that H ± 1 is lighter than H ± 2 , it decays via the Yukawa interaction [41] 2 . In the region where m H 1 140 GeV, the decay into cs and that into τ ν are dominant. When we consider a little heavier H ± 1 , which are in the mass region between 140 GeV and m t + m b ≃ 180 GeV, the branching ratio for H ± 1,2 → t * b → W ± bb is dominant [48]. 3 In the mass region m t +m b < m H 1 , the branching ratio for H ± 1 → tb is almost 100 %. The decays into cs, τ ν, and t ( * ) b are all induced by the Yukawa interaction. Since we consider the Type-I Yukawa interaction, the dependence on tan β of each decay channel is same. Thus, the branching ratio in Fig. 1 hardly depends on the value of tan β. Analytic formulae of decay rates for each decay channel are shown in Appendix A 1.
The singly charged scalar H ± 2 also decays into the SM fermions via the Yukawa interaction. H ± 2 → H ± 1 Z * become negligible small, and the branching ratio for H ± 2 → tb is almost 100 %. If we consider larger tan β, the decays via the Yukawa interaction is suppressed, and the branching ratio for H ± 2 → H ± 1 Z * increases. Thus, the crossing point of the branching ratio for H ± 2 → tb(t * b) and that for H ± 2 → H ± 1 Z * move to the point at heavier m H 2 . Analytic formulae of decay rates for each decay channel are shown in Appendix A 1.
Next, we discuss the decay of the doubly charged scalar Φ ±± . The doubly charged scalar Φ ±± does not couple to fermions via Yukawa interaction 4 . Therefore, it decays via the weak 2 In this paper, we neglect the effects of one-loop induced decays H ± i → W ± γ and H i ± → W ± Z [49]. 3 In Ref [48], Type-II Yukawa interaction is investigated, and the condition tan β 1 is needed to make the decay H ± 1,2 → t * b dominant. In our case (Type-I), this condition is not necessary because all fermions couple to φ 2 universally. 4 This is different from doubly charged Higgs boson in the triplet model in which dilepton decays of doubly charged Higgs bosons are important signature to test the model [36]. In the left figure, we assume that ∆m(≡ m H 2 −m H 1 ) = 20 GeV and tan β = 10. In the right figure, we assume that ∆m = 50 GeV and tan β = 3 gauge interaction 5 . We consider the following three cases.
considered. In this case, Φ ±± cannot decay into the on-shell H ± 1,2 , and three-body decays are dominant. In the upper left figure of Fig. 3, the branching ratio of Φ ±± in this case is shown.
We assume that tan β = 3, ∆m 1 < 20 GeV, ∆m 2 < 10 GeV. In the small mass region, With increasing of m Φ , the masses of H ± 1,2 also increase because the mass differences between them are fixed. Thus, the branching ratio for Φ ±± → W ± f f is dominant in the large mass region. At the point m Φ ≃ 260 GeV, the branching ratio for Φ ±± → W ± f f changes rapidly. It is because that at this point, the decay channel Φ ±± → W ± tb is open. If we consider the large tan β, the decay rates of Φ ±± → W ∓ f f becomes small because this process includes H ± * 1,2 → f f via Yukawa interaction which is proportional to cot β. However, the decays Φ ±± → H ± 1,2 f f are generated via only the gauge interaction. Thus, for tan β 3, the branching ratio for Φ ±± → W ± f f becomes small.
Second, the case where ∆m 1 > 80 GeV and ∆m 2 < 80 GeV is considered. In this case, In the upper right figure of Fig. 3, the branching ratio of Φ ±± in this case is shown. We assume that tan β = 3, ∆m 1 < 100 GeV, ∆m 2 < 50 GeV. In all mass region displayed in the figure, the branching ratio for Φ ±± → H ± 1 W ± are almost 100 %, and those for other channels are at most about 0.1 %. At the point m Φ ≃ 260 GeV, the branching ratio for Φ ±± → W ± f f changes rapidly.
It is because that at this point, the decay channel Φ ±± → W ± tb is open.
Third, the case where ∆m 1 > 80 GeV and ∆m 2 > 80 GeV is considered. and both of Φ ±± → H ± 1,2 W ± are allowed. In the lower figure of Fig. 3, the branching ratio in this case is shown. We assume that tan β = 3, ∆m 1 = 100 GeV, ∆m 2 = 90 GeV. In all mass region displayed in the figure, the branching ratio does not change because the mass differences between Φ ±± and H ± 1,2 are fixed. The branching ratio for Φ ±± → H ± 1 W ± is about 75 %, and that for Φ ±± → H ± 2 W ± is about 25 %. These decays are generated via only the gauge interaction. Thus, the branching ratios of them do not depend on tan β, and they are determined by only the mass differences between Φ ±± and m H 1,2 . The bottom one corresponds to the case that ∆m 1 = 100 GeV and ∆m 2 = 90 GeV.

B. Production of Φ ±± at hadron colliders
We here discuss the production of the doubly charged scalar Φ ±± . In our model, production processes of charged scalar states are pp → W In the THDM, the first and second processes (the singly charged scalar production) can also occur [50,51] However, doubly charged scalar bosons are not included in the THDM 6 . In the model with the isospin triplet scalar with Y = 1 [3,4,8,26,27], all of these production processes can appear. However, the main decay mode of doubly charged scalar is different from our model. In the triplet model, the doubly charged scalar from the triplet mainly decays into dilepton [36] or diboson [31].
In our model, on the other hand, Φ ±± mainly decays into the singly charged scalar and W boson.
In this paper, we investigate the associated production pp → W + * → Φ ++ H − i (i = 1, 2). In this process, informations on masses of all the charged states Φ ±± and H ± i appear in the Jacobian peaks of transverse masses of several combinations of final states [20]. Pair productions are also important in searching for Φ ±± and H ± i , however we focus on the associated production in this paper. The parton-level cross section of the process qq ′ → where s is the square of the center-of-mass energy, G F is the Fermi coupling constant, and V qq ′ is the (q, q ′ ) element of CKM matrix. In addition, χ i in Eq. (9) is defined as In Fig. 4, we show the cross section for pp → W + * → Φ ++ H − 1 in the case that √ s = 14 TeV and χ = π/4. The cross section is calculated by using MADGRAPH5 AMC@NLO [58] and FeynRules [59]. The black, red, blue lines are those in the case that ∆m 1 = 0, 50, and 100 GeV, respectively. The results in Fig. 4  to be generated at the HL-LHC in the case that m Φ = 800 GeV. The cross section increases with increasing of the mass difference ∆m 1 . Since we assume that χ = π/4, the cross section of the process pp → W + * → Φ ++ H − 2 is same with that in Fig. 4 if m H 2 = m H 1 . If we consider the case that | sin χ| > | cos χ| (| cos χ| > | sin χ|), the cross section of pp

IV. SIGNAL AND BACKGROUNDS AT HL-LHC
In this section, we investigate the detectability of the process pp → W + * → Φ ++ H − i (i = 1, 2) in two benchmark scenarios. In the first scenario (Scenario-I), the masses of H ± 1 and H ± 2 are set to be 100 GeV and 120 GeV, so that they cannot decay into tb. In this case, their masses are so small that the branching ratio for three body decay H ± 1,2 → W ± bb is less than 5 % approximately. Thus, their main decay modes are H ± 1,2 → cs and H ± 1,2 → τ ν. In the second scenario (Scenario-II), masses of H ± 1 and H ± 2 are set to be 200 GeV and 250 GeV, and they predominantly decay into tb with the branching ratio to be almost 100 %.
In our analysis below, we assume the collider performance at HL-LHC as follows [38].
where √ s is the center-of-mass energy and L is the integrated luminosity. Furthermore, we use the following kinematical cuts (basic cuts) for the signal event [58]; where p j T (p ℓ T ) and η j (η ℓ ) are the transverse momentum and the pseudo rapidity of jets (charged leptons), respectively, and ∆R jj , ∆R ℓj , and ∆R ℓℓ in Eq. (12) are the angular distances between two jets, charged leptons and jets, and two charged leptons, respectively. In this scenario, the singly charged scalars decay into cs or τ ν dominantly. (See Figs. 1 and 2.) We investigate the process pp → W + * → Φ ++ H − 1,2 → τ + ℓ + ννjj (ℓ = e, µ). The Feynman diagram for the process is shown in Fig. 5. In this process, the doubly charged scalar Φ ++ and one of the singly charged scalars H − 1,2 are generated via s-channel W + * . The produced singly charged scalar decays into a pair of jets, and Φ ++ decays into τ + ℓ + νν through the on-shell pair of the singly charged scalar and W + . Thus, in the distribution of the transverse mass of τ + ℓ + E T , where E T is the missing transverse energy, we can see the Jacobian peak whose endpoint corresponds to m Φ [20] 7 . In the present process, furthermore, in the distribution of the transverse mass of two jets, we can basically see twin Jacobian peaks at m H 1 and m H 2 [20]. Therefore, by using the distributions of M T (τ + ℓ + E T ) and M T (jj), we can obtain the information on masses of all the charged scalars H ± 1 , H ± 2 , and Φ ±± . This is the characteristic feature of the process in this model. When we consider the decay of the tau lepton, the transverse mass of the decay products of the tau lepton and ℓ + νν can be used instead of M T (τ + ℓ + νν).
In the following, we discuss the kinematics of the process at HL-LHC with the numerical evaluation. For input parameters, we take the following benchmark values for Scenario-I; From the LEP data [47], the singly charged scalars are heavier than the lower bound of the mass (80 GeV). In addition, we take the large tan β(=10), so that they satisfy the constraints from flavor experiments [43,44] and LHC Run-I [45,46].
The final state include the tau lepton, and we consider the case that the tau lepton decays into π + ν. In this case, π + flies in the almost same direction of τ + in the Center-of-Mass (CM) frame because of the conservation of the angular momentum [51]. The branching ratio for τ + → π + ν is about 11 % [60], and we assume that the efficiency of tagging the hadronic decay of tau lepton is 60 % [61]. Under the above setup, we carry out the numerical evaluation of the signal events by using MADGRAPH5 AMC@NLO [58], FeynRules [59], and TauDecay [62]. As a result, about 600 signal events are expected to be produced at HL-LHC.
The distributions of the signal events for M T (π + ℓ + E T ) and M T (jj) are shown in red line in the left figure of Fig. 6 and in the right one, respectively.
Next, we discuss the background events and their reduction. The main background process is pp → W + W + jj → τ + ℓ + ννjj. The leading order of this background process is O(α 6 ) and O(α 4 α 2 s ). For O(α 6 ), the vector boson fusion (VBF) and tri-boson production pp → W + W + W − → W + W + jj are important. On the other hand, for O(α 4 α 2 s ), the main process is t-channel gluon mediated pp → q * q ′ * → W + W + jj, where q and q ′ are quarks in internal lines. The number of the total background events under the basic cuts in Eq. (12) 7 In general, the transverse mass M T of n particles is defined as follows.
where p T i and m i are the transverse momentum and the mass of i-th particle, respectively. is shown in Table II. Transverse mass distributions of background events for M T (π + ℓ + E T ) and M T (jj) are shown in the blue line in the left figure of Fig. 6 and in the right one, respectively. The number of the background events is larger than that of the signal. Clearly, background reduction has to be performed by additional kinematical cuts.
First, we impose the pseudo-rapidity cut for a pair of two jets (∆η jj ). The ∆η jj distributions of the signal and background processes are shown in the upper left figure in Fig. 7.
For the signal events, the distribution has a maximal value at ∆η jj = 0 as they are generated via the decay of H − 1 or H − 2 . On the other hand, for the VBF background, two jets fly in the almost opposite directions, and each jet flies almost along the beam axis. Large |∆η jj | is then expected to appear [63], so that we can use |∆η jj | < 2.5 to reduce the VBF background. We note that this kinematical cut is not so effective to reduce other O(α 6 ) and O(α 4 α 2 s ) processes because in these background, the distribution are maximal at ∆η jj = 0. Second, we impose the angular distance cut for a pair of two jets (∆R jj ). The ∆R jj distributions of the signal and background processes are shown in the upper right figure in Fig. 7. For the signal events, the distribution has a maximal value at ∆R jj ≃ 1.0. On the other hand, for the O(α 4 α 2 s ) background events, ∆R jj has a peak at ∆R jj ∼ π. In addition, in the O(α 6 ) ones, ∆R jj has large values between 3 and 6. Therefore, for ∆R jj < 2, the background events are largely reduced while the almost all signal events remains.
Third, we impose invariant mass cut for a pair of two jets (M jj ). The M jj distributions of the signal and background processes are shown in the bottom figure in Fig. 7. For the signal events, as they are generated via the decay of the singly charged scalars, the distribution has twin peaks at the masses of H ± 1 and H ± 2 (100 GeV and 120 GeV). On the other hand, for the background events, the jets are generated via on-shell W or t-channel diagrams. Then, the distribution of the background has a peak at the W boson mass (∼ 80 GeV). Thus, the kinematical cut 90 GeV < M jj < 180 GeV is so effective to reduce the background events.
We note that this reduction can only be possible when we already know some information on the masses of the singly charged scalars.
We summarize three kinematical cuts for the background reduction. signal process can be detected at HL-LHC in Scenario-I of Eq. (15). However, the endpoint of the signal is unclear due to the background events, so that it would be difficult to precisely decide the mass of Φ ++ . On the other hand, we can see the twin Jacobian peaks of M T (jj) in the right figure of Fig. 8. Therefore, we can also obtain information on masses of both the singly charged scalars. In this way, all the charged scalar states Φ ±± , H ± 1 , and H ± 2 can be detected and their masses may be obtained to some extent.  The significance is also improved as S/ √ S + B = 20. Distributions for M T (π + ℓ + E T ) and M T (jj) are shown in Fig 9. In the left figure of Fig 9, we can see that there are only few background events around the end point of Jacobian peak M T (π + ℓ + E T ). Thus, it would be expected we obtain the more clear information on m Φ than that from the case where only (i) and (ii) are imposed as additional kinematical cuts. We can also clearly see the twin Jacobian peaks in the right figure of Fig 9, and a large improvement can be achieved for the determination of the masses of both the singly charged scalar states.
Before closing Subsection A, we give a comment about the detector resolution. In the process, the transverse momenta of jets (p j T ) are mainly distributed between 0 and 200 GeV, and the typical value of them is about 100 GeV. According to Ref. [64], at the current ATLAS detector, the energy resolution for p j T ≃ 100 GeV is about 10 %. In Figs. 6-9, we take the width of bins as 10 GeV. Therefore, it would be possible that the twin Jacobian peaks in the distribution for M T (jj) overlap each other and they looks like one Jacobian peak with the unclear endpoint at the ATLAS detector if the mass differences is not large enough. Then, it would be difficult to obtain the information on both m H 1 and m H 2 from the transverse momentum distribution. Even in this case, it would be able to obtain the hint for the masses by investigating the process. In our analysis, we did not consider the background where the Z boson decays into dijet such as qq → Z * → Zh → jjτ τ → jjπ + ν τ ℓ − ν τ ν ℓ , which can be expected to be reduced by veto the events of M jj at the Z boson mass and the cut of the transverse mass M T (π + ℓ + E T ) below 125 GeV. It does not affect the Jacobian peak and the endpoint at the mass of doubly charged scalar boson Φ ±± .

B. Scenario-II
In this scenario, the singly charged scalars predominantly decay into tb with the branching ratio almost 100 %. We investigate the signal pp → W + * → Φ ++ H − 1,2 → ttbbℓ + ν → bbbbℓ + ℓ ′+ ννjj (ℓ, ℓ ′ = e, µ). The Feynman diagram for the process is shown in Fig. 10. The decay products of Φ ++ and H ± 1,2 are bbℓ + ℓ ′+ νν and bbjj, respectively. Therefore, in the same way as Scenario-I, we can obtain information on masses of all the charged scalars by investigating the transverse distributions of signal and background events for M T (bbℓ + ℓ ′+ νν) and M T (bbjj). However, in the Scenario-II, decay products of both Φ ++ and H − 1,2 include a bb pair, and it is necessary to distinguish the origin of the two bb pairs. We suggest the following two methods of the distinction.
In the first method, we use the directions of b and b. In the process, Φ ++ and H − the typical value of the transverse momentum of b from H − 1,2 is larger than that of b from the top quark. In the same way, the typical value of transverse momentum of b from H + 1,2 is larger than that of b from the anti-top quark. Therefore, in this case, we can construct the bb pair which mainly comes from the decay of Φ ++ by selecting b with the smaller transverse momentum and b with the larger transverse momentum. The other bb pair comes from the decay of H − 1,2 . On the contrary, when the singly charged scalars are light enough to satisfy the inequality, Then, the typical values of the transverse momenta of two b are similar, and those of two b are also similar. Therefore, we can construct the correct bb pair only partly by using the above method, and it is not so effective. In this case, the first method explained in the previous paragraph is needed.
In the following, we discuss the signal and the background events at HL-LHC with the numerical calculation. In the numerical evaluation, we take the following benchmark values as Scenario-II.
For tan β = 3, the lower bound on the masses of singly charged scalars is about 170 GeV as mentioned in the end of Sec. II. Then, this benchmark values satisfy the experimental constraints on singly charged scalars. In addition, we adopt the assumption about the collider performance at HL-LHC in Eq. (11), and we use the basic kinematical cuts in Eq. (12). The final state of the signal includes two bottom quarks and two anti-bottom quarks, and we assume that the efficiency of the b-tagging is 70 % per one bottom or antibottom quark [65]. Thus, the total efficiency of the b-tagging in the signal event is about 24 %. In the numerical calculation, we use MADGRAPH5 AMC@NLO [58], FeynRules [59].
As a result, 145 events are expected to appear at HL-LHC as shown in Table III is the other. In the left figure of Fig. 11, the endpoint of the Jacobian peak is not so sharp because the selection of the bb pairs do not work well in the associated production of Φ ++ and H − 2 . In the right figure of Fig. 11, we can see the twin Jacobian peaks at the masses of the singly charged scalars. However, the number of events around the Jacobian peaks, especially the one due to H ± 2 , are small, and it would be difficult to obtain information on masses form the distribution for M T (b 2 b 1 jj). In order to obtain the clearer information on  Next, we discuss the background events at HL-LHC. We consider the process pp → ttbbW + → bbbbW + W + W − → bbbbℓ + ℓ ′+ ννjj as the background. As a result of the numerical calculation, 40 events are expected to appear at HL-LHC as shown in Table. III. This is the same order with the signal events. In Fig. 11, the distributions of M T (b 1 b 2 ℓ + ℓ ′+ E T ) and much larger than those of the background events.
In summary, it would be possible that we obtain information on masses of all the charged scalars H ± 1 , H ± 2 , and Φ ±± by investigating the transverse mass distribution for M T (b 2 b 1 ℓ + ℓ ′+ E T ) and M T (b 1 b 2 jj) and the invariant mass distribution for M(b 1 b 2 jj) at HL-LHC.
Before closing Subsection B, we give a comment about the detector resolution. In the process of Scenario-II, the typical value of the transverse momenta of jets and bottom quarks is about 100 GeV. As mentioned in the end of the section for Scenario-I, at the ATLAS detector, the energy resolution for p j T ≃ 100 GeV is about 10 % [64]. In Figs. 11 and 12, we take the width of bins as 10 GeV. Therefore, it would be possible that the twin Jacobian peaks in the distribution for M T (jj) or M(jj) overlap each other and they looks like one Jacobian peak with the unclear endpoint at the ATLAS detector if the mass differences is not large enough. Then, it would be difficult to obtain the information on both m H 1 and m H 2 from the transverse momentum distribution. Even in this case, it would be able to obtain the hint for masses by investigating the process.

V. SUMMARY AND CONCLUSION
We have investigated collider signatures of the doubly and singly charged scalar bosons at the HL-LHC by looking at the transverse mass distribution as well as the invariant mass distribution in the minimal model with the isospin doublet with the hypercharge Y = 3/2.
We have discussed the background reduction for the signal process pp → W + * → Φ ++ H − 1,2 in the following two cases depending on the mass of the scalar bosons with the appropriate kinematical cuts . (1) The main decay mode of the singly charged scalar bosons is the tau lepton and missing (as well as charm and strange quarks). (2) That is into a top bottom pair.
In the both cases, we have assumed that the doubly charged scalar boson is heavier than the singly charged ones. It has been concluded that the scalar doublet field with Y = 3/2 is expected to be detectable for these cases at the HL-LHC unless the masses of Φ ±± and H ± 1,2 are too large.
where N f c is the color degree of freedom of a fermion f , r Z and r j are defined same with that in Eq. (A7), and r Γ Z is the ratio of the squared decay rate of Z boson to squared mass of H ± i : In addition, the coeffitient C f V (C f A ) in Eq. (A10) is the coupling constant of the vector (axial vector) current: where g L is the gauge coupling constant of the gauge group SU(2) L , and θ W is the Weinberg angle. In Eq. (A10), mass of fermions are neglected.

a. 2-body decay
If m Φ ±∓ > m H i + m W , the decay Φ ±± → H ± i W ± (i = 1, 2) is allowed. The decay rate is given by where χ i is defined in Eq. (10), the function F (x, y) is defined in Eq. (A4), and R i and R W is defined as follows.

3-body decay
In the case that where the mass differences between Φ ±± and H ± i is so small that decays Φ ±± → H ± i W ± are prohibited, three-body decays Φ ±± → H ± i f f ′ , where f and f ′ are SM fermions, are dominant in small m Φ region. (See Fig. 3.) The branching ratio for where R Γ W is the squared ratio of the decay width of W boson (Γ W ) to m Φ ; In Eq. (A15), we neglect the masses of f and f ′ .
In the large m Φ region, Φ ±± → W ± f f ′ is also important. The decay rate is given by where the function G(x) is defined as follows. (A18) The symbols R f , R f ′ , R i , and R Γ i (i = 1, 2) are given by