Type-II see-saw at $\mu^+$$\mu^-$ collider

Doubly-charged Higgs bosons have extensively been searched at the LHC. In this work, we study the sensitivity reach of the doubly-charged scalar ($H^{\pm\pm}$) in muon collider for the well-known Type-II seesaw scenario. First, we perform a cut-based analysis to predict the discovery prospect in the muon collider operating with 3 TeV center of mass energy. In addition to this, we have also performed a multivariate analysis and compare the cut-based result with the result obtained from the multivariate analysis. We find that the cut-based analysis is more significant as compared to the multivariate analysis in the large doubly-charged scalar mass region. We predict that a doubly-charged scalar mass, $M_{H^{\pm\pm}}$, upto 1450 GeV can be probed with $5\sigma$ significance for center of mass $\sqrt{s}= 3$ TeV and integrated luminosity $\mathcal{L} = 1000\,\textrm{fb}^{-1}$.


I. INTRODUCTION
The discovery of the Higgs boson at the Large Hadron Collider (LHC) assured the Brout-Englert-Higgs (BEH) mechanism to be the most accurate formalism responsible for the generation of the Standard Model (SM) fermions and gauge-bosons masses. The BEH mechanism can generate Dirac mass for neutrino by extending the SM with right-handed neutrinos, however, to explain eV scale small neutrino masses, a very tiny Yukawa 10 −12 is required, enhancing the fine-tuning problem of the SM to a multi-fold level. One of the most appealing mechanisms to describe tiny neutrino mass is via seesaw, where light neutrino masses are generated from higher dimensional d = 5 Weinberg operator [1].
The tree level realization of the Weinberg operator are Type-I [2][3][4][5], Type-II [5][6][7] and Type-III [8] seesaw mechanisms, where the SM has been extended with a SU (2) L singlet fermion, SU (2) L triplet scalar with hyper-charge Y = 1 and a SU (2) L triplet fermion with hyper-charge Y = 0 respectively. Among these, in the second variant of the seesaw mechanism, referred as the Type-II seesaw, generated light neutrino mass is proportional to the vev of the triplet scalar folded with the respective Yukawa coupling.
Other than a pp machine, the doubly-charged Higgs can also be searched at future lepton colliders: Linear or Circular electron positron collider [64][65][66][67] and muon colliders [68][69][70][71][72]. For higher TeV scale masses of the doubly charged Higgs for which cross-section at the LHC is ∼ O(fb) or even lower, and to probe hadronic final states, lepton collider can be most useful, as depending upon the centre of mass energy, the cross-section for dou-bly charged Higgs production is typically large until the kinematic threashold. Additionally, a lepton collider can be more effective because of its cleaner signal. In case of muon collider the collision is free from any pile-up events. An additional benefit is, contrary to the circular e + e − colliders, a muon collider suppresses the loss of energy due to synchrotron radiation because of the heavier mass of muon, thereby making both high energy as well as high luminosity achievable. In this work we choose a particular configuration of muon collider: center-of-mass energy ( √ s) = 3 TeV and integrated luminosity (L) = 1000 fb −1 and explore the sensitivity reach for the doubly charged scalar. Some recent works on BSM particle searches at muon collider can be found in [73][74][75].
The paper is arranged in the following way. In Sec.
[II] we discuss the model description of Type-II seesaw. We discuss the collider analysis that includes both the cut based as well as multivariate analysis in Sec. [III]. Finally in Sec.
[IV] we summarize our outcomes.

II. MODEL
The Type-II seesaw model has an extended scalar sector, where in addition to the SM scalar doublet, Φ = (Φ + Φ 0 ) T , a SU (2) triplet scalar with hyper-charge Y=1 is also present.
The neutral components of the scalars are parameterised as Φ 0 = (v Φ +φ 0 +iZ 1 )/ √ 2 and ∆ 0 = (v ∆ +δ 0 +iZ 2 )/ √ 2. The vacuum expectation values (vevs) of the SM doublet and the BSM triplet are v Φ and v ∆ respectively, and they satisfy the relation v = v 2 Φ + 2v 2 ∆ = 246 GeV. The kinetic Lagrangian of the SM scalar doublet Φ and the scalar triplet ∆ has the following form, In the above, the covariant derivatives are given by, The scalar potential of the model is, After the symmetry breaking the charged scalars and neutral scalars mix resulting several physical mass eigen states. The masses can be obtained by diagonalising with rotation matrices, R ± , R 0 and R 0 A , respectively: where c β± = cos β ± , c β0 = cos β 0 , c α = cos α. The mixing angles β ± , β 0 and α have the following forms: In the above R ± represents the rotation matrix between charged scalar eigenstates, R 0 , R 0 A represent the rotation matrix between the CPeven and CP-odd neutral scalar states, respectively.
In addition to the three Goldstone bosons G ± and G 0 which give masses to the gauge bosons, there are seven physical mass eigen states H ±± , H ± , A, H and h. The gauge basic and mass basic for these scalar states are related as, A detailed discussion on the mass spectrum of the physical scalars of the model has been presented in [76]. With the assumption: v ∆ v Φ , the masses of the physical scalars takes the following simplified form, 2v∆ and the mass-squared differences are obtained as, Considering the sign of λ 4 , three possible mass spectrum of the physical scalars can be realised, The Yukawa interaction of the model responsible for neutrino mass is, where Y is a 3 × 3 complex symmetric matrix and L = (ν L , l L ) T is the left-handed SM lepton doublet. After the triplet scalar, ∆, acquires the vev (v ∆ ), a tiny neutrino mass can be obtained from Yukawa term in Eq. 10 as, The above 3 × 3 matrix, m ν , can further be di- Various constraints on the model parameter space are the following, • Constraints from electroweak precision data (EWPD): The ρ-parameter (m 2 W /m 2 Z cos 2 θ W ) of the model in terms of the doublet and triplet vev is defined as [40,41] , the present EWPD [77] sets the value of ρ parameter as, ρ = 1.00038 (20) which is 1.9σ away from SM tree-level value (ρ = 1) and this leads to an upper bound of O(1) GeV on v ∆ .
• Constraints from oblique parameters: The mass splitting, ∆m, between the different physical scalars affects the EWPD observable, named as oblique parameters (S, T and U parameters). These parameters constraints the mass splitting as ∆m < 40 GeV [9,24].

• Constraints from lepton flavour violation(LFV):
From the Yukawa interaction shown in Eq. 10, LFV decays like µ → eγ at loop-level and µ → 3e at treelevel can be possible. The branching fractions can be calculated as [78,79], where α is the electromagnetic fine-structure constantand G F is the Fermi constant. The upper bounds on the above processes, 4.2 × 10 −13 for µ → eγ [80] and 1.0 × 10 −12 for µ → 3e [81] limit the lower value of the triplet vev (v ∆ ), which can be expressed as [41], • Constraints from Colliders: For ∆m = 0 with large(small) v ∆ , doubly-charged scalar mass below 420(955) GeV has already been excluded [41]. The limit extends upto 1115 GeV for ∆m < 0 and moderate v ∆ . However for moderate v ∆ with ∆m > 0, doubly-charged scalar as light as 200 GeV are still allowed by the LHC results.

A. Decay Widths and Branching Ratios
In this section, we discuss the different decay modes of the scalars: H ±± , H ± , A and H. We are mainly inter-ested on the degenerate mass spectrum of these scalars and hence we have not given much attention to the other mass spectrums (Normal and Inverted scenario). The decay width of the scalars are extensively studied and can GeV, all the scalars dominantly decay into leptonic final states (l ± l ± , l ± ν and νν). However, for v ∆ > 10 −4 GeV the hadronic decay modes (tb, tt and bb) along with the di-boson decay modes (W W , W Z, hZ and hh ) starts to dominate over the leptonic decay mode.
When the non-degenerate scenario is realized, viz ∆m = 0, cascade decay modes turn out to be important. This scenario is beyond the scope of our work, however the interested readers can look for this scenario in [41].

III. COLLIDER ANALYSIS
In this section, we discuss the potential strength of µ + µ − collider in probing doubly-charged scalars and exploring its sensitivity reach mainly in the high mass regime. A center of mass energy of √ s = 3 TeV is being considered for our collider analysis. The overall analysis is focused mainly in the large triplet vev region, v ∆ > 10 −4 , where the gauge boson mode of the doubly-charged scalars is dominant. At muon collider the doubly-charged scalars, H ±± , is produced by photon and Z mediated Drell-Yan (DY) processes, shown in Fig. 4. The DY pair production cross section of the doubly-charged scalars has been shown in Fig. 2 for two different configuration of muon colliders, √ s = 3, and 6 TeV along with the production cross section at 13 TeV LHC. Being an schannel process the production cross section decreases with the increase in center-of-mass energy. A sharp fall in the production cross section can be seen around the kinematic threshold of each configurations of muon collider i.e. around M H ±± ∼ √ s 2 . As can be seen from the figure that after O(400) GeV mass range, clearly the cross-section in the muon collider is substantially large.
In the following analysis we mainly focus into the multi jet final state resulting from the decay of H ±± , as all hadronic final state in a leptonic collider can provide a better sensitivity reach.

A. Multijet signature
The analysis in the paper aim to probe the high v ∆ regime of the parameter space, the region dominated by the gauge production of H ±± and its subsequent decay into multijet final states. The parameter space of our interest is the higher mass region of H ±± where the production of W ± is on-shell. In this region the jets resulted from W ± are highly collimated, and fat-jets reconstruction is more favorable. Thus our signal comprises of up to  Fig. 4. The production process for this signal is,

fat-jets as shown in the
A significant number of SM background can mimic these final states with multiple fat-jets. We consider the following sets of backgrounds in our analysis, where V = (W ± , Z, h).

MadGraph5aMC@NLO
[85] is used to generate parton level signal and background processes. During the generation process some pre-selection cuts have been implemented to reduce the size of the background. The following enumerated points discuss about these pre-selection cuts, (i) Since the signal contains at-least 4-jets (fat-jets) we imposed an increased jet-jet separation criteria at the production level, mainly for 4j background. The 4j background is generated with ∆R(j, j) > 0.6 and p j T > 60 GeV. The p T requirement is only on the leading 4-jets and the rest of the jets are required to have relaxed minimum p T value i.e. p T ≥ 20 GeV. For all other background processes the jet-jet separation is ∆R(j, j) > 0.4 and the leading 4-jets have a minimum p T value, p j T > 60 GeV. The minimum p T requirement on the background processes is justified, as most of the signal is mainly populated in the high p T region, as can be seen from

After the parton level event generation in
MadGraph5aMC@NLO [85], we then pass the generated events into Pythia8 [86] for showering and hadronization.
For simulating detector effects we use Delphes3 [87], and reconstructed jets, electrons, muons and missing energy (E miss T ). The purpose is accomplished by using the Delphes ILD card. We use FastJet [88] for the clustering of jets and consider Cambridge-Aachen (CA) jet clustering algorithm [89] with radius parameter R = 0.8. All the jets are required to be in the pseudorapidity interval |η| < 2.5 and to have p T > 20 GeV. All the leptons, both electrons and muons, have p T > 10 GeV and |η| < 2.5. The missing transverse momenta (p miss T ) is calculated from the momentum imbalance along the transverse direction for all the reconstructed objects. As we have mainly focused on the multi-jet final state we did not bother about the isolation requirement of the leptons.

Cut based analysis
After reconstruction of all the physical objects (mostly jets), we than follow the cut and count method to discriminate our signal from the background. The cut flow for two benchmark mass points, M ±± H = 1000 GeV and M ±± H = 1400 GeV are given in Table. I. From Fig. 5 it can be seen that the 4th jet is quite soft. Hence imposing a hard cut on all the reconstructed 4 fat-jets, we found it more effective to put large p T cuts on the leading jets and a relaxed p T cut on the sub-leading ones. In Fig. 6, we plot the invariant mass of j 1 (M j1 ) and j 4 (M j4 ) out of the leading 4 fat-jets. The distribution of M j1 and M j4 shows a clear peak around M W ± . From this it is clear that the two prong W-jet submerged to form a single fat-jets. The small peak of M j4 distribution signifies that the low p T jets are not pure W-jet. We also tried to reconstruct the invariant mass of the doubly-charged scalars (H ±± ), out of the final state sig-nal fat-jets. In Fig. 7, we plot the two invariant mass reconstructed out of the final state fat-jets. M 1 jj and M 2 jj are the first and second pair of reconstructed invariant mass out of the four final fat-jets respectively. During the selection process of the second pair, we remove the those fat-jets which are already considered in construction the first pair, which takes care of the double counting. As we have shown the distribution for our benchmark point, M H ±± = 1000 GeV, we can see the invariant mass distributions clearly peaks around 1000 GeV for signal process.
To discriminate the signal from the background, we select events within 20 GeV mass window (|M W ± − M ji | = ∆M ji < 20 GeV) around M W ± for the leading three fatjets. We also select events lying within a 100 GeV mass gap around the mass of H ±± (|M H ±± − M i jj | = ∆M i jj < 100 GeV) to reduce the background further. We have summarized the results in cut flow Table. I Normalized invariant mass distribution M 1 jj (first pair) and M 2 jj (second pair) for signal (M H ±± = 1 TeV) and background at 3 TeV µ + µ − collider.

Cut-based Results
With the optimized signal and background events, that survive the selection cuts, we analyse the signal sensitivity. With s and b being the signal cross section and background cross section after all the selection cuts, the statistical significance is defined as [90,91] For a 5σ discovery we assume σ dis ≥ 5. The luminosity required for 5σ discovery for different benchmark mass of doubly charged scalar have been shown in Fig. 8.

Multivariate Analysis (MVA)
In this section we discuss the results of the multivariate analysis (MVA) [92] that we have performed. Fig. 11 shows the kinematic variables used for the MVA analysis assuming the variables are having a good discriminating power between the signal and background, and have low correlations among themselves. A detailed discus-sion about the variables used has already been presented in the text. The method-unspecific separation is a good measure of the discriminating power of any variable, for a given feature y [92]. This is defined as, whereŷ S andŷ B are the probability functions for the signal and background for the particular feature y. The quantity is equal to zero for similar signal and background distributions, and 1 for distributions with no overlaps. We have considered those variables having method-unspecific separation value greater than 1%. In Table. II, we present the method-unspecific separation for the input variables. In Fig. 9, we have shown the Pearson's linear correlation coefficients between the input variables, defined as, where x and σ x are the expectation value and standard deviation respectively for the dataset x.   In Table. III, we have summarised the relevant BDT hyperparameters. We have optimised the BDT hyperparameters with adaptive boost algorithm with a learning rate of 0.1. The so-called Gini index has been used for the separation between the nodes in decision trees. In Table. II, we also present the method-specific ranking of the input variables, which shows the importance of the used variables in separating signal from background.
We present the BDT response and cut efficiency at a benchmark mass value, M H ±± = 1000 GeV, in Fig. 10. We have used the optimised hyperparameters at different benchmark mass to obtain the statistical significance, where N S and N B are the number of signal and background events after the optimal cut is applied on the BDT response. In Table. IV, we present the obtained statistical significance Z for different benchmark doublycharged scalar mass M H ±± .

IV. SUMMARY
We have discussed the discovery prospects of the doubly-charged scalar, H ±± , present in Type-II seesaw model in µ + µ − collider. We mainly focus our attention to that part of the parameter space where the produced doubly-charged scalar mainly decays into W ± W ± final state and subsequently we consider the hadronic final states from the W decays. All hadronic final states are not favourable channel to probe at the LHC due to the presence of the towering QCD backgrounds. Hence this region is more favorable for linear colliders, viz e + e − and µ + µ − collider. Among these two, the µ + µ − collider has a low energy loss due to synchrotron radiation. Firstly, we have performed the trivial cut-based analysis and predict the statistical significance of the muon collider at √ s = 3 TeV. Secondly, we have also performed a multivariate analysis and compared both the results. From the cut-based analysis we have concluded that at 1000 fb −1 luminosity up to 1450 GeV massive doubly-charged scalar can be discovered with 5σ confidence level, see Fig. 8. The result from the multivariate analysis for different M ±± H value has been given in Table. IV, from which it is evident that, apart from a higher mass of 1.450 TeV, BDT analysis offers a larger statistical significance, greater than 5σ.