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

Siddharth P. Maharathy<EMAIL_ADDRESS>Institute of Physics,
Bhubaneswar, Sachivalaya Marg, Sainik School, Bhubaneswar 751005, India  Homi
Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai
400094, India  Manimala Mitra<EMAIL_ADDRESS>Institute of Physics,
Bhubaneswar, Sachivalaya Marg, Sainik School, Bhubaneswar 751005, India  Homi
Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai
400094, India

###### Abstract

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 [6, 5, 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.

Since the triplet scalar has gauge coupling, hence it can be produced
abundantly and then decays to SM particles. Depending on the triplet vev the
doubly-charged scalars can decay to same-sign di-lepton, same-sign gauge
bosons and can also decay through cascade decay. The phenomenology of the
doubly-charged scalar has extensively been studied for LHC [9, 10, 11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32,
33, 34, 35, 36, 37, 38, 39, 40, 41, 42], $e^{+}$$e^{-}$ [43, 44, 45, 46, 47,
48] collider and also for $e$$p$ machine [49, 50, 51, 52]. Several searches
have been performed at the LHC by both ATLAS and CMS collaboration, and the
lack of any excess from the SM signal imposes stringent bounds on the
parameter space [53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63]. The ATLAS multi
lepton search [58] sets a limit of 770-870 GeV and 450 GeV on the mass of
$H^{\pm\pm}$ considering $B(H^{\pm\pm}\rightarrow l^{\pm}l^{\pm})=100\%$ and
$B(H^{\pm\pm}\rightarrow l^{\pm}l^{\pm})=10\%$ respectively. Taking into
account the $W^{\pm}W^{\pm}$ or $W^{\pm}Z$ decay mode of the charged scalars,
the ATLAS search [62] excludes $H^{\pm\pm}$ in the range 200- 350 GeV and 230
GeV, for pair and associated production modes. For 3000 $\textrm{fb}^{-1}$
luminosity, with the aforementioned decay mode of the charged scalars,
expected reach of the doubly-charged scalar increases up to 640 GeV [41]. The
most stringent bound from CMS is from the CMS multilepton search [60],
limiting $H^{\pm\pm}$ mass to 535-820 GeV. With the assumption of equal
branching ratios to each possible leptonic final states,
$Br(H^{\pm\pm}\rightarrow e^{\pm}e^{\pm})=Br(H^{\pm\pm}\rightarrow
e^{\pm}\mu^{\pm})=Br(H^{\pm\pm}\rightarrow
e^{\pm}\tau^{\pm})=Br(H^{\pm\pm}\rightarrow\mu^{\pm}\mu^{\pm})=Br(H^{\pm\pm}\rightarrow\mu^{\pm}\tau^{\pm})=Br(H^{\pm\pm}\rightarrow\tau^{\pm}\tau^{\pm})$,
the ATLAS multi-lepton search [63] imposes a lower limit of 1080 GeV on doubly
charged scalars mass.

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
$\sim\mathcal{O}$(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 doubly 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 ($\sqrt{s}$) = 3 TeV and integrated luminosity ($\mathcal{L}$) = 1000
$\textrm{fb}^{-1}$ 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, $\Phi=(\Phi^{+}\,\,\Phi^{0})^{T}$, a $SU(2)$ triplet
scalar with hyper-charge Y=1 is also present.

$\displaystyle\Delta=\begin{pmatrix}\frac{\Delta^{+}}{\sqrt{2}}&\Delta^{++}\\\
\Delta^{0}&-\frac{\Delta^{+}}{\sqrt{2}}\end{pmatrix}$ (1)

The neutral components of the scalars are parameterised as
$\Phi^{0}=(v_{\Phi}+\phi^{0}+iZ_{1})/\sqrt{2}$ and
$\Delta^{0}=(v_{\Delta}+\delta^{0}+iZ_{2})/\sqrt{2}$. The vacuum expectation
values (vevs) of the SM doublet and the BSM triplet are $v_{\Phi}$ and
$v_{\Delta}$ respectively, and they satisfy the relation
$v=\sqrt{v^{2}_{\Phi}+2v^{2}_{\Delta}}=246\,\,\rm{GeV}$.

The kinetic Lagrangian of the SM scalar doublet $\Phi$ and the scalar triplet
$\Delta$ has the following form,

$\displaystyle\mathcal{L}_{\rm{kin}}(\Phi,\Delta)$ $\displaystyle=$
$\displaystyle(D_{\mu}\Phi)^{\dagger}(D^{\mu}\Phi)+\rm{Tr}[(D_{\mu}\Delta)^{\dagger}(D^{\mu}\Delta)]~{}.$
(2)

In the above, the covariant derivatives are given by,

$\displaystyle D_{\mu}\Phi$ $\displaystyle=$
$\displaystyle\left(\partial_{\mu}+i\frac{g}{2}\tau^{a}W_{\mu}^{a}+i\frac{g^{\prime}}{2}B_{\mu}\right)\Phi~{},$
$\displaystyle D_{\mu}\Delta$ $\displaystyle=$
$\displaystyle\partial_{\mu}\Delta+i\frac{g}{2}[\tau^{a}W_{\mu}^{a},\Delta]+ig^{\prime}B_{\mu}\Delta~{}.$
(3)

The scalar potential of the model is,

$\displaystyle V(\Phi,\Delta)$ $\displaystyle=$ $\displaystyle-
m_{\Phi}^{2}\Phi^{\dagger}\Phi+\tilde{M}^{2}_{\Delta}\rm{Tr}(\Delta^{\dagger}\Delta)$
(4) $\displaystyle+$
$\displaystyle\left(\mu\Phi^{T}i\tau_{2}\Delta^{\dagger}\Phi+\rm{h.c.}\right)+\frac{\lambda}{4}(\Phi^{\dagger}\Phi)^{2}$
$\displaystyle+$
$\displaystyle\lambda_{1}(\Phi^{\dagger}\Phi)\rm{Tr}(\Delta^{\dagger}\Delta)+\lambda_{2}\left[\rm{Tr}(\Delta^{\dagger}\Delta)\right]^{2}$
$\displaystyle+$
$\displaystyle\lambda_{3}\rm{Tr}[(\Delta^{\dagger}\Delta)^{2}]+\lambda_{4}\Phi^{\dagger}\Delta\Delta^{\dagger}\Phi~{}.$

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, $R^{\pm}$, $R^{0}$ and $R^{0}_{A}$,
respectively:

$\displaystyle R^{\pm}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}c_{\beta{{}_{\pm}}}&-s_{\beta{{}_{\pm}}}\\\
s_{\beta{{}_{\pm}}}&c_{\beta{{}_{\pm}}}\end{array}\right),R^{0}=\left(\begin{array}[]{cc}c_{\beta_{0}}&-s_{\beta_{0}}\\\
s_{\beta_{0}}&c_{\beta_{0}}\end{array}\right)$ (9) $\displaystyle R^{0}_{A}$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}c_{\alpha}&-s_{\alpha}\\\
s_{\alpha}&c_{\alpha}\end{array}\right)$ (12)

where $c_{\beta_{\pm}}$ = $\cos\beta_{\pm}$, $c_{\beta_{0}}$ =
$\cos\beta_{0}$, $c_{\alpha}$ = $\cos{\alpha}$. The mixing angles
$\beta_{\pm}$, $\beta_{0}$ and $\alpha$ have the following forms:

$\displaystyle\sqrt{2}\tan\beta_{\pm}=\tan\beta_{0}=\frac{2v_{\Delta}}{v_{\Phi}},~{}~{}~{}\tan
2\alpha=\frac{2B}{A-C}$ (13)

where $A=\frac{\lambda v_{\Phi}^{2}}{2}$, $B=-\sqrt{2}\mu
v_{\Phi}+(\lambda_{1}+\lambda_{4})v_{\Phi}v_{\Delta}$ and $C=\frac{\mu
v_{\Phi}^{2}}{\sqrt{2}v_{\Delta}}+2(\lambda_{2}+\lambda_{3})v_{\Delta}^{2}$.
In the above $R^{\pm}$ represents the rotation matrix between charged scalar
eigenstates, $R^{0},R^{0}_{A}$ represent the rotation matrix between the CP-
even and CP-odd neutral scalar states, respectively.

In addition to the three Goldstone bosons $G^{\pm}$ and $G^{0}$ which give
masses to the gauge bosons, there are seven physical mass eigen states
$H^{\pm\pm}$, $H^{\pm}$, $A$, $H$ and $h$. The gauge basic and mass basic for
these scalar states are related as,

$\displaystyle\left(\begin{array}[]{c}\Phi^{\pm}\\\
\Delta^{\pm}\end{array}\right)=R^{\pm}\left(\begin{array}[]{c}G^{\pm}\\\
H^{\pm}\end{array}\right)$ ,
$\displaystyle\left(\begin{array}[]{c}Z_{1}^{0}\\\
Z_{2}^{0}\end{array}\right)=R^{0}_{A}\left(\begin{array}[]{c}G^{0}\\\
A\end{array}\right)$ (22) $\displaystyle\left(\begin{array}[]{c}\phi^{0}\\\
\delta^{0}\end{array}\right)$ $\displaystyle=$ $\displaystyle
R^{0}\left(\begin{array}[]{c}h\\\ H\end{array}\right).$ (27)

A detailed discussion on the mass spectrum of the physical scalars of the
model has been presented in [76]. With the assumption: $v_{\Delta}\ll
v_{\Phi}$, the masses of the physical scalars takes the following simplified
form,

$\displaystyle M^{2}_{H^{\pm\pm}}$ $\displaystyle\simeq$
$\displaystyle\tilde{M}^{2}_{\Delta}-\frac{\lambda_{4}}{2}v_{\Phi}^{2},~{}~{}~{}M^{2}_{H^{\pm}}\simeq\tilde{M}^{2}_{\Delta}-\frac{\lambda_{4}}{4}v_{\Phi}^{2},$
$\displaystyle M^{2}_{h}$ $\displaystyle\simeq$ $\displaystyle
2v_{\Phi}^{2}\lambda,~{}~{}~{}~{}M^{2}_{H}\approx
M^{2}_{H}\simeq\tilde{M}^{2}_{\Delta}$

where $\tilde{M}^{2}_{\Delta}\equiv\frac{\mu
v_{\Phi}^{2}}{\sqrt{2}v_{\Delta}}$ and the mass-squared differences are
obtained as,

$\displaystyle M^{2}_{H^{\pm}}-M^{2}_{H^{\pm\pm}}\approx
M^{2}_{H,A}-M^{2}_{H^{\pm}}\approx\frac{\lambda_{4}}{4}v_{\Phi}^{2}$ (28)

Considering the sign of $\lambda_{4}$, three possible mass spectrum of the
physical scalars can be realised,

  1. $(i)$

Degenerate scenario ($\lambda_{4}\\!=\\!0$):
$M^{2}_{H^{\pm\pm}}\\!=\\!M^{2}_{H^{\pm}}\\!=\\!M^{2}_{H,A}$

  2. $(ii)$

Normal scenario ($\lambda_{4}\\!>\\!0$):
$M^{2}_{H^{\pm\pm}}>M^{2}_{H^{\pm}}>M^{2}_{H,A}$

  3. $(iii)$

Inverted scenario ($\lambda_{4}\\!<\\!0$):
$M^{2}_{H^{\pm\pm}}<M^{2}_{H^{\pm}}<M^{2}_{H,A}$

The Yukawa interaction of the model responsible for neutrino mass is,

$\displaystyle-\mathcal{L}=Y_{ij}L_{i}^{T}Ci\sigma_{2}\Delta L_{j}+h.c.$ (29)

where $Y$ is a $3\times 3$ complex symmetric matrix and
$L=(\nu_{L},l_{L})^{T}$ is the left-handed SM lepton doublet. After the
triplet scalar, $\Delta$, acquires the vev ($v_{\Delta}$), a tiny neutrino
mass can be obtained from Yukawa term in Eq. 29 as,

$\displaystyle m_{\nu}=\sqrt{2}\,Yv_{\Delta}$ (30)

The above $3\times 3$ matrix, $m_{\nu}$, can further be diagonalised by
unitary transformation by Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix
$U_{PMNS}$ as
$U^{T}_{PMNS}m_{\nu}U_{PMNS}=m^{d}_{\nu}=diag(m_{1},m_{2},m_{3})$, where
$m_{1},m_{2},m_{3}$ are the three mass eigen values. The matrix $U_{PMNS}$ is
parameterized by three mixing angles ($\theta_{12},\theta_{23},\theta_{13}$),
one Dirac phase ($\delta$) and two Majorana phases ($\phi,\phi^{\prime}$).

Various constraints on the model parameter space are the following,

  * •

Constraints from electroweak precision data (EWPD): The $\rho$-parameter
($m_{W}^{2}/m_{Z}^{2}\cos^{2}_{\theta_{W}}$) of the model in terms of the
doublet and triplet vev is defined as [40, 41] ,

$\rho=(v_{\Phi}^{2}+2v_{\Delta}^{2})/(v_{\Phi}^{2}+4v_{\Delta}^{2})$

the present EWPD [77] sets the value of $\rho$ parameter as,
$\rho=1.00038(20)$ which is 1.9$\sigma$ away from SM tree-level value
($\rho=1$) and this leads to an upper bound of $\mathcal{O}(1)$ GeV on
$v_{\Delta}$.

  * •

Constraints from oblique parameters: The mass splitting, $\Delta m$, 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 $\Delta m<40$ GeV [9, 24].

  * •

Constraints from lepton flavour violation(LFV): From the Yukawa interaction
shown in Eq. 29, LFV decays like $\mu\rightarrow e\gamma$ at loop-level and
$\mu\rightarrow 3e$ at tree-level can be possible. The branching fractions can
be calculated as [78, 79],

$\hskip 14.22636ptBR(\mu\rightarrow
e\gamma)=\frac{\alpha|(Y^{\dagger}Y)_{e\mu}|^{2}}{192\pi
G_{F}^{2}}\left(\frac{1}{m_{H^{\pm}}^{2}}+\frac{8}{m_{H^{\pm\pm}}^{2}}\right)$
$BR(\mu\rightarrow 3e)=\frac{|(Y^{\dagger}_{ee}Y_{\mu
e})|^{2}}{4G_{F}^{2}m_{H^{\pm\pm}}^{4}}$

where $\alpha$ is the electromagnetic fine-structure constantand $G_{F}$ is
the Fermi constant. The upper bounds on the above processes, $4.2\times
10^{-13}$ for $\mu\rightarrow e\gamma$ [80] and $1.0\times 10^{-12}$ for
$\mu\rightarrow 3e$ [81] limit the lower value of the triplet
vev($v_{\Delta}$), which can be expressed as [41],

$\hskip 22.76228ptv_{\Delta}\gtrsim 0.78-1.5(0.69-2.3)\,\text{GeV}\times
10^{-9}\times\frac{1~{}\text{TeV}}{M_{H^{\pm\pm}}}$

for NH(IH).

  * •

Constraints from Colliders: For $\Delta m=0$ with large(small) $v_{\Delta}$,
doubly-charged scalar mass below 420(955) GeV has already been excluded [41].
The limit extends upto 1115 GeV for $\Delta m<0$ and moderate $v_{\Delta}$.
However for moderate $v_{\Delta}$ with $\Delta m>0$, doubly-charged scalar as
light as 200 GeV are still allowed by the LHC results.

Figure 1:  Branching ratios of doubly-charged scalar: $H^{\pm\pm}$ (left
panel), singly-charged scalar: $H^{\pm}$ (2nd panel), CP-even scalar: $H$ (3rd
panel) and CP-odd scalar: $A$ (right panel) versus triplet vev ($v_{\Delta}$)
for an illustrative benchmark point $M_{H^{\pm\pm}}$ = $M_{H^{\pm\pm}}$ =
$M_{H}$ = $M_{A}$ = 1000 GeV.

###  II.1 Decay Widths and Branching Ratios

In this section, we discuss the different decay modes of the scalars:
$H^{\pm\pm}$, $H^{\pm}$, $A$ and $H$. We are mainly interested 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 be found in [17,
20, 82, 83, 84]. In Fig. 1, we have shown the branching fractions of all the
aforementioned scalars having a degenerate mass of 1000 GeV. From the figure
it is quite evident that for $v_{\Delta}<10^{-4}$ GeV, all the scalars
dominantly decay into leptonic final states ($l^{\pm}l^{\pm}$, $l^{\pm}\nu$
and $\nu\nu$). However, for $v_{\Delta}>10^{-4}$ GeV the hadronic decay modes
($t\bar{b},t\bar{t}~{}\text{and}~{}b\bar{b}$) along with the di-boson decay
modes ($WW$, $WZ$, $hZ$ and $hh$ ) starts to dominate over the leptonic decay
mode.

When the non-degenerate scenario is realized, viz $\Delta m\neq 0$, 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 $\mu^{+}$$\mu^{-}$
collider in probing doubly-charged scalars and exploring its sensitivity reach
mainly in the high mass regime. A center of mass energy of $\sqrt{s}$ = 3 TeV
is being considered for our collider analysis. The overall analysis is focused
mainly in the large triplet vev region, $v_{\Delta}$ $>$ $10^{-4}$, where the
gauge boson mode of the doubly-charged scalars is dominant. At muon collider
the doubly-charged scalars, $H^{\pm\pm}$, 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, $\sqrt{s}$ = 3, and 6 TeV along
with the production cross section at 13 TeV LHC. Being an s-channel 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
$M_{H^{\pm\pm}}\sim\frac{\sqrt{s}}{2}$. As can be seen from the figure that
after $\mathcal{O}(400)$ 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 $H^{\pm\pm}$, as
all hadronic final state in a leptonic collider can provide a better
sensitivity reach.

Figure 2:  Production cross-section of doubly-charged scalars ($H^{\pm\pm}$)
as a function of $M_{H^{\pm\pm}}$ for 3, and 6 TeV $\mu^{+}\mu^{-}$ collider
along with 13 TeV LHC.  Figure 3: Normalized $H_{T}$ distribution for signal
(benchmark point $M_{H^{\pm\pm}}$ = 1 TeV) and background at 3 TeV
$\mu^{+}$$\mu^{-}$ collider.

###  III.1 Multijet signature

The analysis in the paper aim to probe the high $v_{\Delta}$ regime of the
parameter space, the region dominated by the gauge production of $H^{\pm\pm}$
and its subsequent decay into multijet final states. The parameter space of
our interest is the higher mass region of $H^{\pm\pm}$ where the production of
$W^{\pm}$ is on-shell. In this region the jets resulted from $W^{\pm}$ are
highly collimated, and fat-jets reconstruction is more favorable. Thus our
signal comprises of up to 4 fat-jets as shown in the Fig. 4. The production
process for this signal is,

$\displaystyle\mu^{+}\mu^{-}\rightarrow H^{\pm\pm}H^{\mp\mp}\rightarrow
4W^{\pm}(W^{\pm}\rightarrow jj)$ (31)

Figure 4: The Feynman diagram for pair production of doubly-charged scalars
($H^{\pm\pm}$) via Drell-Yan process and their consecutive decays leading to
multijet final state.

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,

$\displaystyle\mu^{+}\mu^{-}$ $\displaystyle\rightarrow$ $\displaystyle jjjj$
$\displaystyle\mu^{+}\mu^{-}$ $\displaystyle\rightarrow$ $\displaystyle
V+2j\quad(V\rightarrow jj)$ $\displaystyle\mu^{+}\mu^{-}$
$\displaystyle\rightarrow$ $\displaystyle V+3j\quad(V\rightarrow jj)$
$\displaystyle\mu^{+}\mu^{-}$ $\displaystyle\rightarrow$ $\displaystyle
VV+2j\quad(V\rightarrow jj)$ $\displaystyle\mu^{+}\mu^{-}$
$\displaystyle\rightarrow$ $\displaystyle VVV+2j\quad(V\rightarrow jj)$ (32)

where V = ($W^{\pm},Z,h$).

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,

  1. $(i)$

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 $\Delta R(j,j)>0.6$ and $p_{T}^{j}>60$
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. $p_{T}\geq 20$
GeV. For all other background processes the jet-jet separation is $\Delta
R(j,j)>0.4$ and the leading 4-jets have a minimum $p_{T}$ value,
$p_{T}^{j}>60$ 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 Fig. 5.

  2. $(ii)$

Fig. 3 corresponds to scalar sum of transverse momenta of all visible objects,
$H_{T}$, of both signal and backgrounds. Most of the signals are populated at
high $H_{T}$ region as compared to the background. Hence we have also imposed
a minimum $H_{T}$ criteria of $H_{T}>1000$ GeV on all the background processes
at the generation level i.e. during the production of backgrounds at Madgraph.

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 ($E_{T}^{miss}$). 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 $|\eta|<2.5$ and to have $p_{T}>20$ GeV. All the
leptons, both electrons and muons, have $p_{T}>10$ GeV and $|\eta|<2.5$. The
missing transverse momenta ($p_{T}^{miss}$) 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.

Figure 5: Normalized $p_{T}$ distribution of leading 4-fat-jets for signal
(benchmark point $M_{H^{\pm\pm}}$ = 1 TeV) at 3 TeV $\mu^{+}$$\mu^{-}$
collider.

####  III.1.1 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, $M_{H}^{\pm\pm}=1000\,\text{GeV}$ and
$M_{H}^{\pm\pm}=1400\,\text{GeV}$ are given in Table. 1. 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.

Figure 6: Normalized invariant mass distribution of $j_{1}$ and $j_{4}$ for
signal ($M_{H^{\pm\pm}}$ = 1 TeV) and background at 3 TeV center of mass
energy of $\mu^{+}$$\mu^{-}$ collider.

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 $M_{W^{\pm}}$. 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
($H^{\pm\pm}$), out of the final state signal 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,
$M_{H^{\pm\pm}}=1000\,\text{GeV}$, 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 ($|M_{W^{\pm}}-M_{ji}|=\Delta M_{ji}<20\,\text{GeV}$) around
$M_{W^{\pm}}$ for the leading three fat-jets. We also select events lying
within a 100 GeV mass gap around the mass of $H^{\pm\pm}$
($|M_{H^{\pm\pm}}-M^{i}_{jj}|=\Delta M^{i}_{jj}<100\,\text{GeV}$) to reduce
the background further. We have summarized the results in cut flow Table. 1.

$\mu^{+}\mu^{-}\to H^{++}H^{--}\to W^{+}W^{+}W^{-}W^{-}\to Nj_{\rm{fat}}$
$M_{H^{\pm\pm}}$ (GeV) $\geq 4j$ $p_{T}^{j1}$ $>450$ $p_{T}^{j2}$ $>350$
$p_{T}^{j3}$ $>200$ $p_{T}^{j4}$ $>100$ $\Delta M_{j_{1}}<20$ $\Delta
M_{j_{2}}<20$ $\Delta M_{j_{3}}<20$ $\Delta M^{1}_{jj}$ $<100$ $\Delta
M^{2}_{jj}$ $<100$ 1000 [0.825] 0.739 0.736 0.730 0.679 0.540 0.504 0.442
0.388 0.380 0.220 1400 [0.0922] 0.064 0.0623 0.062 0.060 0.053 0.050 0.046
0.041 0.039 0.019 Backgrounds BG $\geq 4j$ $p_{T}^{j1}$ $>450$ $p_{T}^{j2}$
$>350$ $p_{T}^{j3}$ $>200$ $p_{T}^{j4}$ $>100$ $\Delta M_{j_{1}}<20$ $\Delta
M_{j_{2}}<20$ $\Delta M_{j_{3}}<20$ $\Delta M^{1}_{jj}$ $<100$ $\Delta
M^{2}_{jj}$ $<100$ $VV+2j\,[4.680]$ 1.527 1.390 1.256 0.351 0.156 0.066 0.028
0.0092 0.0053 0.0005 $V+3j\,[5.450]$ 3.613 3.319 3.077 1.815 0.907 0.284 0.088
0.0265 0.0151 0.002 $4j\,[6.677]$ 5.123 4.830 4.560 3.038 1.840 0.448 0.113
0.0288 0.0163 0.0019 $V+2j\,[5.360]$ 1.787 1.669 1.554 1.055 0.486 0.185 0.063
0.0195 0.0109 0.0013 $VVV+2j\,[0.049]$ 0.048 0.034 0.027 0.021 0.015 0.0036
0.002 0.0017 0.001 0.00009

Table 1: The cut-flow for the signal and backgrounds. For signal two benchmark
$M_{H^{\pm\pm}}$ = 1000 GeV and $M_{H^{\pm\pm}}$ = 1400 GeV are considered.
The values mentioned in square brackets are the production cross section of
each processes. The cross sections are in fb.

Figure 7:  Normalized invariant mass distribution $M_{jj}^{1}$ (first pair)
and $M_{jj}^{2}$ (second pair) for signal ($M_{H^{\pm\pm}}$ = 1 TeV) and
background at 3 TeV $\mu^{+}$$\mu^{-}$ collider.

####  III.1.2 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]

$\displaystyle\sigma_{dis}$ $\displaystyle=$
$\displaystyle\Big{[}2\Big{(}(s+b)ln\Big{[}1+\frac{s}{b}\Big{]}-s\Big{)}\Big{]}^{1/2}$
(33)

For a $5\sigma$ discovery we assume $\sigma_{dis}$ $\geq$ 5\. The luminosity
required for $5\sigma$ discovery for different benchmark mass of doubly
charged scalar have been shown in Fig. 8.

Figure 8: Required luminosity for $5\sigma$ discovery of the doubly-charged
scalars at 3 TeV $\mu^{+}$$\mu^{-}$ collider.

####  III.1.3 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 discussion 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,

$\displaystyle\left<S^{2}\right>=\frac{1}{2}\int
dy\frac{(\hat{y}_{S}(y)-\hat{y}_{B}(y))^{2}}{\hat{y}_{S}(y)+\hat{y}_{B}(y)}$
(34)

where $\hat{y}_{S}$ and $\hat{y}_{B}$ 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. 2, 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,

$\displaystyle\rho(x,y)=\frac{\left<xy\right>-\left<x\right>\left<y\right>}{\sigma_{x}\sigma_{y}}$
(35)

where $\left<x\right>$ and $\sigma_{x}$ are the expectation value and standard
deviation respectively for the dataset $x$.

In Table. 3, 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. 2, we also present
the method-specific ranking of the input variables, which shows the importance
of the used variables in separating signal from background.

Figure 9: The linear correlation coefficients for signal (left) and background (right) between the different kinematic variables used in multivariate analysis. Feature | Method-unspecific | Method-specific
---|---|---
| separation | ranking
$M_{j3}$ | 0.393 | 0.144
$M_{j1}$ | 0.391 | 0.142
$M_{jj}^{2}$ | 0.384 | 0.102
$M_{jj}^{1}$ | 0.336 | 0.095
$M_{j2}$ | 0.289 | 0.116
$M_{j4}$ | 0.257 | 0.096
$p_{T}^{j3}$ | 0.241 | 0.063
$p_{T}^{j4}$ | 0.229 | 0.045
$H_{T}$ | 0.218 | 0.066
$p_{T}^{j1}$ | 0.141 | 0.057
$p_{T}^{j2}$ | 0.137 | 0.069
Table 2:  Method unspesific and Method spesific relative importance of the different discriminating variables used in MVA. BDT hyperparameter | Optimised choice
---|---
NTrees | $500$
MinNodeSize |  $5.0$%
MaxDepth | $5$
BoostType | AdaBoost
AdaBoostBeta | $0.1$
UseBaggedBoost | True
BaggedSampleFraction | $0.5$
SeparationType | GiniIndex
nCuts | 20
Table 3: Summary of optimised BDT hyperparameters.

We present the BDT response and cut efficiency at a benchmark mass value,
$M_{H^{\pm\pm}}$ = 1000 GeV, in Fig. 10. We have used the optimised
hyperparameters at different benchmark mass to obtain the statistical
significance,

$\displaystyle\mathcal{Z}=\frac{N_{S}}{\sqrt{N_{S}+N_{B}}}$ (36)

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. 4, we present the
obtained statistical significance $\mathcal{Z}$ for different benchmark
doubly-charged scalar mass $M_{H^{\pm\pm}}$.

Mass[GeV] | Significance
---|---
1000 | 22.834
1100 | 19.303
1200 | 15.256
1300 | 10.959
1400 | 6.259
1450 | 3.640
Table 4: Statistical significance, $\mathcal{Z}$, for different doubly-charged
scalar mass obtained from the multivariate analysis for muon collider with
center of mass $\sqrt{s}$ = 3 TeV and integrated luminosity $\mathcal{L}$ =
1000 $\textrm{fb}^{-1}$.

Figure 10: (Left): Normalised distribution of the BDT response for both
signal(blue) and background(red) classes (both training and testing samples).
(Right): Signal(blue) and background(red) efficiencies along with the
statistical significance(green) of the signal as function of applied BDT cut
value.

##  IV Summary

We have discussed the discovery prospects of the doubly-charged scalar,
$H^{\pm\pm}$, present in Type-II seesaw model in $\mu^{+}\mu^{-}$ collider. We
mainly focus our attention to that part of the parameter space where the
produced doubly-charged scalar mainly decays into $W^{\pm}W^{\pm}$ 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 $\mu^{+}\mu^{-}$
collider. Among these two, the $\mu^{+}\mu^{-}$ 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
$\sqrt{s}=3$ 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 $\textrm{fb}^{-1}$ luminosity up to 1450 GeV massive doubly-charged
scalar can be discovered with $5\sigma$ confidence level, see Fig. 8. The
result from the multivariate analysis for different $M_{H}^{\pm\pm}$ value has
been given in Table. 4, from which it is evident that, apart from a higher
mass of 1.450 TeV, BDT analysis offers a larger statistical significance,
greater than $5\sigma$.

##  V Acknowledgments

The authors acknowledge SAMKHYA: high Performance Computing Facility provided
by Institute of physics, for the performed simulations. The authors
acknowledge the support from the Indo-French Centre for the Promotion of
Advanced Research (Grant no: 6304-2). SPM thank Saiyad Ashanujjaman, Rojalin
Padhan and Agnivo Sarkar for the useful discussions.

Figure 11: Distributions of different kinematic variables used for MVA for
signal as well as for all back grounds. The normalised distributions for
signal is given by the solid blue line and the back ground distributions are
given by solid red line.

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