# Testing neutrino mass hierarchy under type-II seesaw scenario in $U(1)_{X}$
at colliders

Arindam Das<EMAIL_ADDRESS>Institute for the Advancement of
Higher Education, Hokkaido University, Sapporo 060-0817, Japan  Department of
Physics, Hokkaido University, Sapporo 060-0810, Japan  Puja Das
<EMAIL_ADDRESS>Department of Physics and Astronomy, University of
Alabama, Tuscaloosa, AL35487, USA  Nobuchika Okada<EMAIL_ADDRESS>Department
of Physics and Astronomy, University of Alabama, Tuscaloosa, AL35487, USA

###### Abstract

The origin of tiny neutrino mass is a long standing unsolved puzzle of the
Standard Model (SM) which allows us to consider scenarios beyond the Standard
Model (BSM) in a variety of ways. One of them being the gauge extension of the
SM could be realized as in the form of an anomaly free, general $U(1)_{X}$
extension of the SM where an $SU(2)_{L}$ triplet scalar being charged under
$U(1)_{X}$ gauge group is introduced through a Dirac Yukawa coupling with the
SM lepton doublet. Once the triplet scalar generates VEV, light neutrinos
could acquire tiny Majorana mass and hence affecting the decay modes of the
triplet scalar involving the neutrino oscillation data for different neutrino
mass hierarchies. After the breaking of $U(1)_{X}$ scenarios, a neutral BSM,
neutral gauge boson $(Z^{\prime})$ acquires mass which interact differently
with the left and right handed fermions. Satisfying the recent LHC bounds on
the triplet scalar and $Z^{\prime}$ production, we study the pair production
of the triplet scalar at LHC, $e^{-}e^{+}$ and $\mu^{-}\mu^{+}$ colliders
followed by its decay into dominant mode depending on the neutrino mass
hierarchy. Generating the SM generic backgrounds, we study the possible signal
significance of four lepton final states. We also compare our results with the
purely SM gauge mediated triplet scalar pair production followed by four
lepton final states which could be significant only in $\mu^{-}\mu^{+}$
collider.

_Introduction_.– Observation of neutrino mass and flavor mixing
ParticleDataGroup:2020ssz  is one of the suitable aspects where Standard Model
(SM) falls short of explaining it where beyond the SM (BSM) scenarios step in.
Among a variety of simple but interesting aspects, neutrino mass can be
explained extending the SM with an $SU(2)_{L}$ triplet scalar field with
hyper-charge $Y=+2$, commonly known as type-II seesaw scenario
Schechter:1981cv ; Magg:1980ut ; Cheng:1980qt ; Lazarides:1980nt ;
Mohapatra:1980yp . Apart from the particle extension of the SM, there is also
a promising aspect where SM can be extended by a $U(1)_{B-L}$ gauge group
Pati:1973uk ; Davidson:1978pm ; Davidson:1979wr ; Marshak:1979fm ;
Mohapatra:1980qe . In the light of $U(1)$ extension of SM, we propose a
general $U(1)$ extension of the SM where $SU(1)_{L}$ triplet scalar is
introduced to participate in the neutrino mass generation mechanism. The
anomalies due to the charges of the SM fermions under general $U(1)$ scenarios
are cancelled by the introduction of three generations of SM-singlet right
handed neutrinos (RHNs) which have inhomogeneous charges under the general
$U(1)$ gauge group preventing them to generate Dirac Yukawa coupling with the
SM Higgs and lepton doublets. However, neutrino mass generation could be
achieved though the presence of an $SU(2)_{L}$ triplet scalar field, charged
under general $U(1)$ gauge group, which simply generates Majorana-type
nonvanishing neutrino mass through the Yukawa coupling with the SM lepton
doublets, not requiring any Dirac mass. Hence the smallness of neutrino mass
can be achievable through the Vacuum Expectation Value (VEV) of the triplet
scalar and the corresponding Yukawa coupling. Therefore the RHNs in this case
does not participate in the neutrino mass generation mechanism, rather
participates in Dark Matter (DM) phenomenology. An additional advantage in a
general $U(1)$ extension is the presence of a neutral BSM gauge boson
$Z^{\prime}$ which has different interactions with the left and right handed
SM fermions manifesting chiral nature of the model. Addressing these facts, we
propose such a scenario not only provides a correct degree of neiutrino mass
and flavor mixing but also a testable scenario at the Large Handron Collider
(LHC) and other future colliders. In this model set-up the triplet scalar
acquires VEV through a trilinear coupling with a BSM doublet scalar field
charged under general $U(1)$ gauge group in such a way, so that it does not
interact with SM fermions. The smallness of the VEV governs tiny neutrino mass
satisfying neutrino oscillation data. It should be noted that smallness of the
neutrino mass also ensures a small VEV of the BSM doublet scalar. General
$U(1)$ symmetry can be broken by the introduction of VEV of an SM-singlet
scalar filed with unit charge under the general $U(1)$ gauge group. It further
generates mass of the pseudo-scalar and the Dirac mass term of the RHNs where
one of the RHNs could be massless at the renormalization level triggering the
fact that it could be considered as a Dark Radiation (DR) and the remaining
massive modes could compose a viable DM candidateOkada:2022cby . DRs could fix
the anomaly between the Hubble parameters DiValentino:2021izs  obtained from
PLANK Planck:2018vyg  and SH0ES delaMacorra:2021hoh  collaborations.

This economical extension of the SM explains the aspect of neutrino mass
generation mechanism which could be tested directly at the high energy
colliders where the doubly charged multiplets of the triplet scalar is
produced from the heavy resonance of $Z^{\prime}$ followed by same-sign
dilepton from each multiplet intriguing BSM scalar multiplet searches directly
from the BSM propagators. Recently ATALS experiment rules out the possibility
of a doubly charged scalar below $1$ TeV at 139 fb-1 ATLAS:2022pbd  from
multilepton final states at $\sqrt{s}=13$ TeV. As a result a four lepton final
state is challanging to observe at the colliders if doubly charged scalars are
produced through SM mediators. Therefore we emphasize on the doubly charged
scalar production through a BSM propagator $Z^{\prime}$, followed by its
multilepton decay. However, recent dilepton searches from $Z^{\prime}$ at the
LHC ATLAS:2019erb ; CMS:2021ctt  provides a limit on $Z^{\prime}$ mass,
$M_{Z^{\prime}}>5.2$ TeV around 140 fb-1 luminosity. Therefore $Z^{\prime}$
mediated doubly charged scalar production is also challenging, however, if the
dilepton bound is relaxed on general $U(1)$ coupling, then a heavy doubly
charged scalar can be dominantly produced from heavy $Z^{\prime}$ dominating
over the processes mediated by SM gauge bosons- a noticeable feature of this
model only. Hence following the constrains on the doubly charged scalars
$\rho=1.00038\pm 0.00020$ providing a limit on triplet VEV $v_{\Delta}\leq
0.78(2.6)$ GeV at 2(3)$\sigma$ ParticleDataGroup:2020ssz  and non-observation
of exotic decay from $Z$ boson ruling out doubly charged scalar lighter than
$Z$ boson mass Kanemura:2013vxa , we study this scenario to test neutrino mass
hierarchy and lepton flavor violating (LFV) processes considering BR($\mu\to
3e$)$<1.0\times 10^{-12}$ from SINDRUM SINDRUM:1987nra  and BR($\mu\to
e\gamma$) $<4.2\times 10^{-12}$ from MEG MEG:2016leq  experiments,
respectively. These LFV processes are proportional to roughly fourth power of
the Dirac Yukawa coupling between triplet scalar and lepton doublet. As a
result it could provide a lower limit on $v_{\Delta}$ for doubly charged
scalar mass around $\mathcal{O}(1)$ TeV and its decay modesDas:2023tna .

Considering a type-II seesaw scenario under the general $U(1)_{X}$ extension
of the SM under the recent limits we study the production of the doubly
charged triplet scalar from $Z^{\prime}$ in the context of LHC, $e^{-}e^{+}$
and $\mu^{+}\mu^{-}$ colliders followed by the decay of doubly charged triplet
scalar into same sign dilepton parametrizing the triplet Yukawa coupling
applying neutrino oscillation data for normal and inverted hierarchies. We
also consider the corresponding SM modes to compare the doubly charged scalar
production. Studying generic backgrounds for different colliders we compare
with signal and background events to investigate the role of triplet scalar in
generating tiny neutrino mass.

_Model_.– We work on SM$\otimes U(1)_{X}$ framework where the SM colored
fields transform as
$q_{L}^{i}=\\{3,2,\frac{1}{6},x_{q}=\frac{1}{6}x_{H}+\frac{1}{3}\\}$,
$u_{R}=\\{3,1,\frac{2}{3},x_{u}=\frac{2}{3}x_{H}+\frac{1}{3}\\}$,
$d_{R}^{i}=\\{3,1,-\frac{1}{3},x_{d}=-\frac{1}{3}x_{H}+\frac{1}{3}\\}$,
respectively whereas SM lepton fields transforms as
$\ell_{L}^{i}=\\{1,2,-\frac{1}{2},x_{\ell}=-\frac{1}{2}x_{H}-1\\}$,
$e_{R}^{i}=\\{1,1,-1,x_{e}=-x_{H}-1\\}$, respectively where $i$ represents
three generations. In this framework SM Higgs filed transforms as
$H=\\{1,2,\frac{1}{2},x_{h}=\frac{1}{2}x_{H}\\}$. We introduce three SM-
singlet RHNs to cancel gauge and mixed gauge-gravity anomalies which transform
as $N_{R}^{\\{j=1,2\\}}=\\{1,1,0,x_{\nu}^{\\{j=1,2\\}}=-4\\}$,
$N_{R}^{3}=\\{1,1,0,\tilde{x_{\nu}}^{3}=5\\}$ with inhomogeneity in $U(1)_{X}$
charges and an SM-singlet $U(1)_{X}$ scalar which transforms as
$\Phi=\\{1,1,0,1\\}$ to break the $U(1)_{X}$ symmetry. Finally we write the
transformations of the additional scalar fields
$\tilde{H}=\\{1,2,\frac{1}{2},\frac{1}{2}x_{H}+1\\}$ and
$\Delta=\\{1,3,1,x_{\Delta}=x_{H}+2\\}$, respectively. The $U(1)_{X}$ charge
of the fields can be defined as the linear combination of $U(1)_{B-L}$ and
$U(1)_{Y}$. Due to this charge assignment and $U(1)_{X}$ symmetry, RHNs cannot
have Dirac Yukawa coupling with $\ell_{L}$ and $H$. Therefore in this model
set-up light neutrino mass can not be generated through RHNs. As a result we
economically introduce $\Delta$ to originate a Yukawa coupling with $\ell_{L}$
which generates Majorana type light neutrino mass through the VEV of $\Delta$
which is commonly known as type-II seesaw mechanism maintaining a minimal
scalar extension. The other scalar doublet $\tilde{H}$ does not have any
Yukawa coupling with the SM lepton and quark sectors like the type-I 2HDM
Branco:2011iw , however, it is unavoidable to generate a small VEV for
$\Delta$. Hence we write down the Yukawa interactions for the triplet and
singlet scalar fields from the relevant part of the Lagrangian as

$\displaystyle\mathcal{L}^{Y}=-\frac{1}{\sqrt{2}}Y^{ij}\overline{{\ell_{L}^{i}}^{C}}.\Delta\ell_{L}^{j}-\sum_{i=1,2}\tilde{Y}^{i}\Phi\overline{{N_{R}^{i}}^{C}}N_{R}^{3}+h.c.$
(1)

where in the first term $C$ is the charge conjugate, dot represents
antisymmetric product $SU(2)$ gauge group. Here $Y$ and $\tilde{Y}$ are the
Yukawa couplings of the left-handed lepton doublet with $\Delta$ and RHNs with
$\Phi$, respectively. After the VEV of $\Delta$ is generated Majorana-type
left handed neutrino mass is also generated as
$\mathcal{M}_{\nu}=Y^{ij}v_{\Delta}$ where $v_{\Delta}$ is the triplet VEV.
After the breaking of $U(1)_{X}$ symmetry through the VEV of $\Phi$, Dirac
masses of the RHNs can be generated where one of the three components is
massless and can be considered as DR and the remaining two components can form
a Dirac spinor which could be considered as a viable DM candidate. We note
that due to $U(1)_{X}$ gauge symmetry there is no Dirac Yukawa coupling
between $N_{R}^{i}$ and $\Phi$. Therefore Majorana mass term of the RHNs can
not be generated in this case. As a result in a minimal set-up neutrino mass
can only be generated only through type-II seesaw mechanism. Due to general
$U(1)_{X}$ gauge symmetry we find the left and right handed fermions interact
differently with the $Z^{\prime}$ manifesting the chiral nature of the model.
Hence corresponding Lagrangian can be written as

$\displaystyle\mathcal{L}=g_{X}\sum_{i}Q^{i}_{L,R}\overline{f^{i}_{L,R}}\gamma_{\mu}{Z^{\prime}}^{\mu}f^{i}_{L,R}$
(2)

where $i$ corresponds to the sum over quark and lepton states and their three
generations, $Q_{L,R}$ are the charges for the left and right handed fermions
under $U(1)_{X}$ and $g_{X}$ is the $U(1)_{X}$ coupling. Now we discuss about
the scalar sector whose potential is given by Eq. (16). The $U(1)_{X}$
symmetry spontaneously breaks at a high scale due to
$<\Phi>=\frac{v_{\Phi}}{\sqrt{2}}=\frac{m_{\Phi}^{2}}{\lambda_{\Phi}}$ where
$v_{\Phi}\gg 246$ GeV. Hence $U(1)_{X}$ gauge boson $Z^{\prime}$ and physical
state from the SM-singlet scalar acquire mass as
$M_{Z^{\prime}}=g_{X}^{2}v_{\Phi}^{2}$ and
$\tilde{m}_{\Phi}^{2}=\lambda_{\Phi}v_{\Phi}^{2}$, respectively. Now setting
$<\Phi>=\frac{v_{\Phi}}{\sqrt{2}}$ in Eq. (16) and rearranging terms we obtain
the low energy effective potential below $v_{\Phi}$ which is given by Eq.
(17). From the final term in the low energy effective potential we find
$-\tilde{m}^{2}_{H\tilde{H}}=\lambda\frac{\Phi}{\sqrt{2}}$. In contrast to the
2HDM, $(\tilde{H}^{\dagger}H)^{2}$ term is absent due to the $U(1)_{X}$
symmetry, however, the term containing $m_{H\tilde{H}}^{2}$ is essential for
the generation of the CP odd scalar boson mass removing dangerous Nabu-
Goldstone bosons from the mass spectrum. Using the stationary conditions on
Eq. (17) we can express $\tilde{m}_{i}^{2}$ with $i=H,\tilde{H},\Delta$ in
terms of the model parameters given by Eq. (18). Now from Eq. (19), we infer
that if trilinear mass term between $\tilde{H}$ and $\Delta$ and VEV of the
BSM scalar doublet are small, then type-II seesaw mechanism naturally occurs
through small $v_{\Delta}$ further naturally fitting constraints from $\rho$
parameter. This enhances the same sign dilepton branching mode from
$\Delta^{\pm\pm}$. The mass eigenvalues of neutral, charged scalars and
pseudoscalars are given in Eq. (20).

Results and Discussions.– We show the branching ratios of $Z^{\prime}$ into
different modes in the upper panel of Fig. 1 and the ratio of branching ratios
of $Z^{\prime}\to\Delta^{++}\Delta^{--}$ to $e^{-}e^{+}$ and
$(e^{+}e^{-}+\mu^{-}\mu^{+})$ in the lower panel, respectively. We find that
BR$(Z^{\prime}\to\Delta^{++}\Delta^{--})$ is 1.3 times greater than
BR$(Z^{\prime}\to e^{-}e^{+})$ mode and that is 0.65 times the
BR$(Z^{\prime}\to(e^{-}e^{+}+\mu^{-}\mu^{+})$ mode at $x_{H}=-1$.

Figure 1: Branching ratios of $Z^{\prime}$ into different modes(upper panel)
and ratio of branching ratios of the triplet to the electron positron (red)
and electron positron plus muons (lower panel).

We find that at $x_{H}=0$ BR$(Z^{\prime}\to\Delta^{++}\Delta^{--})\simeq
7.5\%$ which is higher than all other modes except the dilepton one. Therefore
we choose $x_{H}=0$ which is B$-$L scenario. We find that the ratios of the
branching ratios of $Z^{\prime}\to\Delta^{++}\Delta^{--}$ to $e^{-}e^{+}$ and
$(e^{+}e^{-}+\mu^{-}\mu^{+})$ for this charge are 0.63 and 0.31, respectively
making B$-$L scenario quiet reasonable to study the doubly charged triplet
production from $Z^{\prime}$ induced processes and followed by its decay into
same sign dilepton mode. We also find that for $x_{H}=0$ only the branching
ratio of the dilepton mode from $Z^{\prime}$ is higher than that of the doubly
charged triplet scalar mode from $Z^{\prime}$ keeping other SM modes from
$Z^{\prime}$ subdominant.

Hence for B$-$L scenario we evaluate constraints on $g_{X}-M_{Z^{\prime}}$
plane comparing the dilepton $(\ell=e,\leavevmode\nobreak\ \mu)$ production
cross section $(\sigma^{\prime})$ in our model with those
$(\sigma^{\rm{ATLAS,CMS}})$ from LHC ATLAS:2019erb ; CMS:2021ctt  following
$g_{X}=g^{\prime}\sqrt{\frac{\sigma^{\rm{ATLAS/CMS}}}{\sigma^{\ell\ell}_{\rm
LHC}}}$ where $g^{\prime}$ is the trial value of $U(1)_{X}$ gauge coupling to
calculate $\sigma^{\ell\ell}_{\rm LHC}$ using narrow width approximation as

$\displaystyle\sigma_{\rm LHC}^{\ell\ell}$ $\displaystyle=$ $\displaystyle
3.89\times
10^{8}\times\frac{2\pi^{2}}{3}\int_{\frac{M_{Z^{\prime}}^{2}}{xE_{\rm
LHC}^{2}}}^{1}\frac{dx}{xE_{\rm
LHC}^{2}}\sum_{q,\overline{q}}\Big{[}f_{q}(x,M_{Z^{\prime}})$ (3)
$\displaystyle f_{\overline{q}}(\frac{M_{Z^{\prime}}^{2}}{xE_{\rm
LHC}^{2}},M_{Z^{\prime}})\Big{]}\frac{\Gamma_{Z^{\prime}(Z^{\prime}\to
q\overline{q})}}{M_{Z^{\prime}}}\delta(\hat{s}-M_{Z^{\prime}}^{2})f(x_{H})\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $

where
$f(x_{H})=\Big{(}\frac{8+12x_{H}+5x_{H}^{2}}{13+16x_{H}+10x_{H}^{2}}\Big{)}$
and $f_{q(\bar{q})}$ being parton distribution function from CTEQ6L
Pumplin:2002vw  for (anti)quark with a suitable $k-$factor of $0.947$ to match
ATLAS ATLAS:2019erb  result which is slightly stronger than CMS CMS:2021ctt
prediction. The $x_{H}$ dependence is evolved from the interactions of SM
quarks and leptons with $Z^{\prime}$ manifesting the chiral nature of the
model and further influencing the partial decay widths of $Z^{\prime}$ given
in Eq. (21). We consider $x_{H}=0$ for the B$-$L scenario and estimated
constrains are shown in Fig. 2.

Figure 2: Limits on $g_{X}-M_{Z^{\prime}}$ plane from LHC and $e^{-}e^{+}$
colliders for $x_{H}=0$.

Limits on $g_{X}$ from LEP-II data using $M_{Z^{\prime}}\gg\sqrt{s}$ can be
calculated for the B$-$L case considering the contact interaction
$e^{-}e^{+}\to f\bar{f}$ as

$\displaystyle{\cal L}_{\rm
eff}=\frac{g_{X}^{2}}{(1+\delta_{ef})(\Lambda_{lm}^{f\pm})^{2}}\sum_{l,m=L,R}C_{lm}(\overline{e}\gamma^{\mu}P_{l}e)(\overline{f}\gamma_{\mu}P_{m}f)\;\;$
(4)

where $g_{X}^{2}/4\pi$ is taken to be unity using the convention
$\delta_{ef}=1\ (0)$ for $f=e$ ($f\neq e$), $C_{lm}=\pm 1$ or 0, and
$\Lambda_{lm}^{f\pm}$ is the scale of contact interaction where constructive
and destructive interference with SM processes $e^{+}e^{-}\to f\bar{f}$
Kroha:1991mn ; Carena:2004xs  are represented by plus and minus signs in this
account respectively. Now we evaluate the $Z^{\prime}$ mediated matrix element
under B$-$L scenario as

$\displaystyle\frac{g_{X}^{2}}{{M_{Z^{\prime}}}^{2}-s}[\overline{e}\gamma^{\mu}(\tilde{x}_{\ell}P_{L}+\tilde{x}_{e}P_{R})e][\overline{f}\gamma_{\mu}(\tilde{x}_{f_{L}}P_{L}+\tilde{x}_{f_{R}}P_{R})f]$
(5)

where $\tilde{x}_{f_{L}}$ and $\tilde{x}_{f_{R}}$ are B$-$L charges of $f_{L}$
and $f_{R}$ respectively which can be found from our model with $x_{H}=0$. Now
we match Eqs. (4) and (5) and obtain

$\displaystyle M_{Z^{\prime}}^{2}\ \gtrsim\
\frac{{g_{X}}^{2}}{4\pi}|{x_{e_{l}}}x_{f_{m}}|(\Lambda_{lm}^{f\pm})^{2}$ (6)

considering $M_{Z^{\prime}}^{2}\gg s$ where $\sqrt{s}=209$ GeV for LEP-II.
Hence we estimate bounds on $M_{Z^{\prime}}/g_{X}$ from LEP using different
values of $C_{lm}^{f\pm}$ for the B$-$L case. To do that we use $95\%$ bounds
on $\Lambda_{lm}^{f\pm}$ from ALEPH:2013dgf  for leptons and quarks
considering $lm=LL,\ RR,\ LR,\ RL,\ VV$ and $AA$ assuming universality in the
contact interactions. Similarly we estimate prospective limits on
$M_{Z^{\prime}}/g_{X}$ for the B$-$L case at ILC with $\sqrt{s}=250$ GeV,
$500$ GeV and $1$ TeV using the bounds on $\Lambda_{lm}^{f\pm}$ from
LCCPhysicsWorkingGroup:2019fvj . Hence we estimate the bounds on
$M_{Z^{\prime}}/g_{X}$ for B$-$L scenario as 7.0 TeV for LEP-II and
prospective bounds from ILC are 48.2 TeV, 81.6 TeV and 137.2 TeV for
$\sqrt{s}=250$ GeV, 500 GeV and 1 TeV respectively. Corresponding limits on
$g_{X}-M_{Z^{\prime}}$ plane are shown in Fig. 2 at $95\%$ C. L.

Now we consider the production and decay of the doubly charged scalar at LHC
considering $pp\to\Delta^{++}\Delta^{--}$ process taking $Z^{\prime}$ mediated
scenario using Narrow Width Approximation (NWA). The production cross section
being normalized by $g_{X}^{2}$ can be written as

$\displaystyle\sigma_{\rm BSM}^{\rm LHC}$ $\displaystyle=$ $\displaystyle
3.89\times
10^{8}\times\frac{8\pi^{2}}{3}\int_{\frac{M_{Z^{\prime}}^{2}}{xE_{\rm
LHC}^{2}}}^{1}dx\sum_{q,\bar{q}}\Big{[}f_{q}(x,M_{Z^{\prime}})f_{\overline{q}}$
(7) $\displaystyle(\frac{M_{Z^{\prime}}^{2}}{xE_{\rm
LHC}^{2}},M_{Z^{\prime}})\Big{]}\frac{\Gamma(Z^{\prime}\to
q\overline{q})}{g_{X}^{2}M_{Z^{\prime}}}BR(Z^{\prime}\to\Delta^{--}\Delta^{++})\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $

where $f_{q(\bar{q})}$ being parton distribution function from CTEQ6L
Pumplin:2002vw  for (anti)quark. We show the variation of the cross section as
a function of $M_{Z^{\prime}}$ for $x_{H}=0$ at $\sqrt{s}=14$ TeV considering
$m_{\Delta^{\pm\pm}}=1.03$ TeV in Fig. 3.

Figure 3: $\Delta^{\pm\pm}$ production under B$-$L scenario at the LHC
normalized by $g_{X}^{2}$.

We calculate the total BSM cross section within the invariant mass windows
$M_{Z^{\prime}}\pm 200$ GeV and $M_{Z^{\prime}}\pm 250$ GeV for
$M_{Z^{\prime}}=4$ TeV and $5$ TeV are 0.69 fb and 0.14 fb . In addition to
that we estimate $\Delta^{++}\Delta^{--}$ production cross section at the LHC
from the purely SM process mediated by $Z$ and photon following

$\displaystyle\sigma_{\rm SM}^{\rm LHC}$ $\displaystyle=$
$\displaystyle\int_{\frac{M_{\rm inv}^{2}}{E_{\rm LHC}^{2}}}^{1}\frac{4M_{\rm
inv}}{xE_{\rm LHC}^{2}}\sum_{q,\overline{q}}\Big{[}f_{q}(x,M_{\rm
inv})f_{\bar{q}}(\frac{M_{\rm inv}^{2}}{xE_{\rm LHC}^{2}},M_{\rm inv})\Big{]}$
(8) $\displaystyle\hat{\sigma}dx\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $

where

$\displaystyle\hat{\sigma}$ $\displaystyle=$ $\displaystyle\frac{3.89\times
10^{8}}{144\pi
s}\Big{(}1-4\frac{m_{\Delta^{\pm\pm}}^{2}}{s}\Big{)}^{\frac{3}{2}}s^{2}\Big{(}A^{2}Q_{u(d)}^{2}e^{2}$
(9)
$\displaystyle+B^{2}(C_{V_{u(d)}}^{2}+C_{A_{u(d)}}^{2})+2ABeQ_{u(d)}C_{V_{u(d)}}\Big{)}$

where $A=\frac{2e}{s}$, $B=\frac{g_{Z}(1-2\sin^{2}\theta_{W})}{s-m_{Z}^{2}}$,
$e=\sqrt{\frac{4\pi}{128}}$, $Q_{u(d)}=\frac{2}{3}(-\frac{1}{3})$,
$C_{V_{u(d)}}=g_{Z}((-)\frac{1}{4}-Q_{u(d)}\sin^{2}\theta_{W})$,
$C_{A_{u}(d)}=g_{Z}(-\frac{1}{4})$ and $g_{Z}=2\frac{m_{Z}}{v}$, respectively.
Integrating Eq. (8) over the ranges 3800 GeV $\leq M_{\rm inv}\leq 4200$ GeV
and 4750 GeV $\leq M_{\rm inv}\leq 5250$ GeV we find the cross sections as
0.0015 fb and 0.00026 fb, respectively. Here $m_{Z}=91.2$ GeV is the $Z$ boson
mass and $v=246$ GeV is the electroweak VEV.

We calculate the production cross section for
$e^{-}e^{+}\to\Delta^{++}\Delta^{--}$ process as

$\displaystyle\sigma_{\rm BSM}^{ee}$ $\displaystyle=$
$\displaystyle\frac{3.89\times 10^{8}}{48\pi
s}(1-4\frac{m_{\Delta}^{2}}{s})^{\frac{3}{2}}s^{2}[\\{A^{2}{Q_{e}}^{2}{e}^{2}+$
(10) $\displaystyle
B^{2}({C_{V_{e}}}^{2}+{C_{A_{e}}}^{2})+2ABeQ_{e}C_{V_{e}}\\}+2\Re(B_{1})\times$
$\displaystyle\\{AQ_{e}{e}^{2}Q_{V_{e}}+B(C_{V_{e}}Q_{V_{e}}+C_{A_{e}}Q_{A_{e}})\\}$
$\displaystyle+\\{\frac{g_{X}^{2}(x_{H}+2)^{2}({Q_{V_{e}}}^{2}+{Q_{A_{e}}}^{2})}{(s-M_{Z^{\prime}}^{2})^{2}+M_{Z^{\prime}}^{2}\Gamma_{Z^{\prime}}^{2}}\\}]$

which is shown in the upper panel of Fig. 4 as a function of $\sqrt{s}$ where
purely SM contribution (mediated by $Z$ and photon) is shown in red and the
BSM contribution from $Z^{\prime}$ and its interference between
$\\{Z^{\prime},Z\\}$ and $\\{Z^{\prime},\gamma\\}$ is shown in blue which
provide a sharp resonance at $M_{Z^{\prime}}=3$ TeV .

Figure 4: $\Delta^{\pm\pm}$ production at $e^{-}e^{+}$ (upper panel) and muon
(lower panel) colliders.

We find $Z^{\prime}$ induced production cross section is 0.69 pb for $x_{H}=0$
and $g_{X}=0.03$ while that for purely SM process is 0.005 pb considering
$B_{1}=\frac{g_{X}^{2}(x_{H}+2)^{2}}{(s-M_{Z^{\prime}}^{2})+\sqrt{-1}m_{Z^{\prime}}\Gamma_{Z^{\prime}}}$,
$Q_{e}=-1$, $C_{V_{e}}=g_{Z}(-\frac{1}{4}-Q_{e}\sin^{2}\theta_{W})$,
$C_{A_{e}}=g_{Z}\frac{1}{4}$, $Q_{V_{e}}=(-\frac{3}{4}x_{H}-1)g_{X}$ and
$Q_{A_{e}}=-\frac{1}{2}x_{H}g_{X}$, respectively.

We calculate the production cross section of $\Delta^{++}\Delta^{--}$ process
at muon collider for $x_{H}=0$ using NWA as

$\displaystyle\sigma_{\rm BSM}^{\mu\mu}$ $\displaystyle=$ $\displaystyle
3.89\times
10^{8}\times\frac{4\pi^{2}}{3}\frac{\Gamma(Z^{\prime}\to\mu^{+}\mu^{-})}{M_{Z^{\prime}}}\times$
(11) $\displaystyle BR(Z^{\prime}\to\Delta^{++}\Delta^{--}).$

which is shown in the lower panel of Fig. 4 as a function of where total
production cross sections for $M_{Z^{\prime}}=4$ TeV within the invariant mass
$M_{Z^{\prime}}\pm 200$ GeV is 407.32 pb with $g_{X}=0.01$ whereas the
production cross section from the purely SM gauge boson mediated processes is
1.86 pb.

We study decay of the doubly charged scalars using partial modes into same
sign dilepton $(\ell^{\pm}\ell^{\pm})$ and same sign gauge boson
$(W^{\pm}W^{\pm})$, respectively from Eq. (22). We consider
$\ell^{\pm}\ell^{\pm}$ mode due to its dominance over $W^{\pm}W^{\pm}$ mode
for $m_{\Delta^{\pm\pm}}=1.03$ TeV for $v_{\Delta}<10^{-4}$ GeV. From Eq. (1)
we find that neutrino mass can be generated from $v_{\Delta}$ in terms of
Dirac Yukawa coupling $Y^{ij}$. It further helps in generating Majorana mass
term for the light neutrinos after triplet scalar acquires VEV $v_{\Delta}$ as
$\mathcal{L}^{Y}\supset m_{\nu}\bar{\nu_{L}^{c}}\nu_{L}$ where
$m_{\nu}=\frac{Y^{ij}}{\sqrt{2}}v_{\Delta}$ neutrino mass and $Y^{ij}$ is a
$3\times 3$ matrix. Now we diagonalize the neutrino mass matrix to obtain the
mass eigenvalues following $U_{\rm MNS}^{T}m_{\nu}U_{\rm
MNS}=diag(m_{1},m_{2},m_{3})$ where $U_{\rm MNS}$ is a $3\times 3$ unitary
matrix being a function of neutrino mixing angles
$\theta_{12}=\frac{\sin^{-1}{\sqrt{0.87}}}{2}$,
$\theta_{23}=\frac{\sin^{-1}{\sqrt{1}}}{2}$,
$\theta_{13}=\frac{\sin^{-1}{\sqrt{0.092}}}{2}$ and Dirac CP phase
$\delta_{CP}=\frac{3\pi}{2}$ ParticleDataGroup:2020ssz .

Figure 5: Branching ratio of $\Delta^{--}$ into a pair of same sign, same
(black) and different (red) flavor dileptons for NH (IH) case in upper(lower)
panel as a function of lightest light neutrino mass $m_{1(3)}$. In NH and IH
cases of same sign same flavor modes
$(\Delta^{--}\to\ell_{i}^{-}\ell_{i}^{-})$ branching ratios for
$\Delta^{--}\to\mu^{-}\mu^{-}$ (dashed) and $\Delta^{--}\to\tau\tau$ (dotted)
coincide with each other whereas $\Delta^{--}\to e^{-}e^{-}$ (solid) does not.
In case of same sign different flavor modes
$(\Delta^{--}\to\ell_{i}^{-}\ell_{j}^{-})$ branching ratios for $\Delta\to
e\mu$ (solid) and $\Delta\to e\tau$ (dashed) coincide with each other whereas
$\Delta\to\mu^{-}\tau^{-}$ (dotted) does not. Shaded regions are excluded by
PLANCK data for NH (blue) and IH (purple) cases.

In this analysis we consider the neutrino mass eigenvalues form their mass
squared differences $\Delta m_{12}^{2}=7.6\times 10^{-5}$ eV-2 and $\Delta
m_{23}^{2}=2.4\times 10^{-3}$ eV-2 where the mass eigenvalues could be ordered
following either of $m_{1}<m_{2}<m_{3}$ the normal hierarchy (NH) or
$m_{3}<m_{1}<m_{2}$ the inverted hierarchy (IH) taking the lightest mass
eigenvalue $m_{1}(m_{3})$ in NH(IH) to be a free parameter. Hence the total
decay width of the triplet scalar can be written in terms of neutrino
oscillation data through the Yukawa coupling $Y^{ij}$ for the NH(IH) case as

$\displaystyle\Gamma_{\rm Total}^{\rm NH(IH)}=\sum_{i=1}^{3}{|[m_{\rm
NH(IH)_{\nu}}]_{ii}|^{2}}+2\sum_{i<j=1}^{3}{|[m_{\rm
NH(IH)_{\nu}}]_{ij}|^{2}}\leavevmode\nobreak\ \leavevmode\nobreak\ $ (12)

where the first term represents same flavor and the second term represents
different flavors of dilepton scenarios coming from the diagonal and off-
diagonal entries of the Yukawa coupling. Hence we estimate the branching ratio
of the doubly charged scalar into a pair of same sign different flavor leptons
as

$\displaystyle\rm{Br}_{\rm NH(IH)}^{off-diag}=\frac{2\sum_{i<j=1}^{3}{|[m_{\rm
NH(IH)_{\nu}}]_{ij}|^{2}}}{\Gamma_{\rm tot}^{\rm NH(IH)}}=\rm
Br(\Delta^{--}\to\ell_{\alpha}\ell_{\beta})\leavevmode\nobreak\
\leavevmode\nobreak\ $ (13)

and the branching ratio of the doubly charged scalar into a pair of same sign
same flavor lepton as

$\displaystyle\rm{Br}_{\rm NH(IH)}^{\rm diag}=\frac{\sum_{i=1}^{3}{|[m_{\rm
NH(IH)_{\nu}}]_{ii}|^{2}}}{\Gamma_{\rm tot}^{\rm NH(IH)}}=\rm
Br(\Delta^{--}\to\ell_{\alpha}\ell_{\alpha}),\leavevmode\nobreak\
\leavevmode\nobreak\ $ (14)

respectively. In this analysis considering mass eigenvalues for the NH case
can be written as $diag(m_{1},\sqrt{m_{12}^{2}+{m_{1}}^{2}_{\rm
NH}}),\sqrt{m_{23}^{2}+{m_{2}}^{2}_{\rm NH}})$ whereas those for the IH case
can be written as $diag(\sqrt{{m_{2}}^{2}_{\rm
NH}-m_{12}^{2}}),\sqrt{m_{23}^{2}+{m_{3}}^{2}_{\rm NH}},m_{2})$, respectively.

We show the branching ratios of $\Delta^{--}$ into a pair of same sign lepton
in the upper (lower) panel of Fig. 5 for the NH(IH) case for the same and
different flavors as a function of lightest neutrino mass eigenvalue
$m_{1(3)}$.

Figure 6: Branching ratio of $\Delta^{--}$ into a pair of same sign same
(different) flavor dileptons for NH (red) and IH (blue) case in upper(lower)
panel as a function of lightest light neutrino mass. In case of same sign same
flavor modes $(\Delta^{--}\to\ell_{i}^{-}\ell_{i}^{-})$ branching ratios for
$\Delta^{--}\to\mu^{-}\mu^{-}$ (dashed) and $\Delta^{--}\to\tau^{-}\tau^{-}$
(dotted) coincide with each other whereas $\Delta^{--}\to e^{-}e^{-}$ (solid)
does not. Decay of $\Delta^{--}$ into same sign different flavor modes are
represented by the branching ratio of $\Delta^{--}\to e^{-}\mu^{-}$ in the
lower panel for NH (red) and IH (blue) cases. Shaded regions are excluded by
PLANCK data for NH (blue) and IH (purple) cases.

In NH case branching ratio of $\Delta^{--}$ into same sign same flavor
dilepton are shown by black solid $(ee)$, dashed ($\mu\mu$) and dotted
$(\tau\tau)$ lines where $\mu\mu$ and $\tau\tau$ cases coincide with each
other. We find that the branching ratio into $ee$ mode increases with $m_{1}$
whereas those of $\mu\mu$ and $\tau\tau$ modes remain almost same over the
range of $m_{1}$ under consideration. In this case we consider the PLANCK
limit on the lightest neutrino mass eigenvalue as $m_{1}>0.03026$ eV is ruled
out in NH case Planck:2018vyg  and it is shown by blue shaded region
satisfying the upper limit on the sum of the neutrino mass eigenvalues
$\sum_{i}m_{i}<0.12$ eV. On the other hand we find branching ratios of
$\Delta^{--}$ in different flavors involving $e\mu$ and $e\tau$ modes being
denoted by red solid and dashed lines coincide with each other and remain
almost same through out the range of $m_{1}$ under consideration except near
the vicinity of the PLANCK exclusion limit $m_{1}>0.003$ eV whereas throughout
the mass range of $m_{1}$, however, $\mu\tau$ mode remains almost constant
which is denoted by red dotted line.

In the lower panel of Fig. 5 we show the IH case where same sign same flavor
cases of by solid black $(ee)$, dashed $(\mu\mu)$ and dotted $(\tau\tau)$
lines, respectively where $\mu\mu$ and $\tau\tau$ modes coincide with each
other. Same sign different flavor modes are shown by the red solid $(e\mu)$,
dashed $(e\tau)$ and dotted $(\mu\tau)$ lines, respectively where $e\mu$ and
$e\tau$ modes coincide. These different flavor scenarios evolve from the off-
diagonal entries of $Y^{ij}$. In this case branching ratios of respective
modes remain almost constant at their respective values up to the vicinity of
the PLANCK exclusion limit $m_{3}>0.01701$ eV which is represented by the
shaded region in purple.

In the upper panel of Fig. 6 we compare branching ratio of same sign same
flavor dilepton modes $ee$, $\mu\mu$ and $\tau\tau$ from NH (IH) represented
by red (blue) solid, dashed and dotted lines respectively with the PLANCK
exclusion limits for NH(IH) case shown by blue(purple) shaded regions. We find
that for NH and IH cases $\mu\mu$ and $\tau\tau$ modes coincides. It is found
that branching ratios of $ee$, $\mu\mu$ and $\tau\tau$ modes vary between
$12\%-48.3\%$ when lightest neutrino mass eigenvalue is lower than the PLANCK
exclusion limit for NH and IH case. Branching ratio of $ee$ mode from NH case
is below $1\%$ for $m_{\rm lightest}<0.0047$ eV, however, it reaches up to
$16\%$ for $m_{\rm lightest}=0.03026$ eV, that is, at the boundary of the
PLANCK exclusion limit. The branching ratios of $\mu\mu$ and $\tau\tau$ remain
almost fixed at $30\%$ for for $m_{1}$ below PLANCK exclusion limit and
$34.4\%$ for $m_{1}=0.01$ eV. We show the leading $\Delta^{--}\to
e^{-}\mu^{-}$ mode for the NH(IH) case in the lower panel by red(blue) where
the corresponding branching ratio decreases at the vicinity of the PLANCK
exclusion limit, however, before that it remains constant at $36(27)\%$ for
$m_{\rm lightest}<\mathcal{O}(0.001)$ eV. Considering three generations of
neutrinos with non-zero light neutrino mass eigenvalues satisfying PLANCK
exclusion limits we could calculate the number of events of multilepton
signals from $\Delta^{\pm\pm}$ pair produced at pp, $e^{-}e^{+}$ and
$\mu^{-}\mu^{+}$ colliders, respectively. Further we could compare these
events with those obtained from the SM backgrounds to estimate prospective
signal significance for the NH and IH cases.

From Fig. 5 considering the NH case and $m_{1}=0.01(0.001)$ eV we find
branching ratios of $\Delta^{--}$ into same sign dilepton modes as: (i)
BR$(ee)$: $3.0(0.21)\%$, (ii) BR$(\mu\mu/\tau\tau)$ $34.0(30.0\%)$, (iii)
BR$(e\mu/e\tau)$ $3.0(3.0)\%$ and (iv) BR$(\mu\tau)$: $23.0(33.79)\%$,
respectively. Now for the IH case taking $m_{3}=0.01(0.001)$ eV we find
dilepton branching ratios of $\Delta^{--}$ into same sign dilepton modes as:
(i) BR$(ee)$: $46.0(48.0)\%$, (ii) BR$(\mu\mu/\tau\tau)$: $17.0(12.3\%)$,
(iii) BR$(e\mu/e\tau)$ $1.5(1.2)\%$ and (iv) BR$(\mu\tau)$: $17.0(25.5)\%$,
respectively. Now we use these branching ratios to estimate the number of
signal events from $\Delta^{--}$ pair production at different colliders using
$Z^{\prime}$ induced and purely SM processes for comparison using

$\displaystyle\rm{Events}_{\rm BSM/SM}^{NH/IH}$ $\displaystyle=$
$\displaystyle\sigma_{\rm BSM/SM}^{\rm LHC/e^{-}e^{+}/\mu^{-}\mu^{-}}\times$
(15) $\displaystyle\rm(BR(\Delta^{--}\to\ell\ell)^{\rm
NH/IH})^{2}\times\mathcal{L}$

where $\mathcal{L}$ is the luminosity of colliders such as 3 ab-1 (LHC) and 1
ab-1 $(e^{-}e^{+}/\mu^{-}\mu^{+})$, respectively and $\sigma_{\rm BSM/SM}^{\rm
LHC/e^{-}e^{+}/\mu^{-}\mu^{-}}$ will be obtained from Eqs. (7)-(11),
respectively.

At the LHC after $\Delta^{\pm\pm}$ production using $g_{X}=0.1$ and $0.3$ for
$M_{Z^{\prime}}=4$ TeV and 5 TeV from Fig. 2 and Eq. (7), we consider
considering the leading decay modes of $\Delta^{--}$ from the NH case using
$m_{1}=0.01(0.001)$ eV and obtain $4\mu$ final state with 2.4(6.0) events for
$M_{Z^{\prime}}=4$ TeV and 0.5(1.3) events for $M_{Z^{\prime}}=5$ TeV from Eq.
(15). In NH case $4e$ events from $Z^{\prime}$ induced process will be
$\mathcal{O}(2)$ of magnitude small compared to the $4\mu$ process for
$m_{1}=0.01$ eV, however, that for $m_{1}=0.001$ will be negligibly small
following the smallness of BR$(\Delta^{--}\to e^{-}e^{-})$ mode at
$m_{1}=0.001$ eV. On the other hand for IH case and using $m_{3}=0.01(0.001)$
eV we obtain the number of events for $4e$ final states as 4.4(14.4) and
1.0(3.0) for $M_{Z^{\prime}}=4$ TeV and 5 TeV, respectively. Due to the
dependence of $\Delta^{--}\to\mu^{-}\mu^{-}$ on $m_{3}$ in IH case, $4\mu$
final state in this case could be $<1$ for $M_{Z^{\prime}}=4$ TeV and 5 TeV,
respectively. From Eq. (8) we have calculated that $\Delta^{--}$ pair
production cross section will be $\mathcal{O}(3)$ of magnitude small compared
to the $Z^{\prime}$ induced scenarios. Hence at 3 ab-1 luminosity number of
events could be $<1$. Simulating the generic SM backgrounds for four lepton
final states from $pp\to\ell^{+}\ell^{-}\ell^{+}\ell^{-}$ using MadGraph
Alwall:2011uj  we find the cross sections for LHC at $\sqrt{s}=14$ TeV as
0.0048 pb for $\ell=e,\leavevmode\nobreak\ \mu$. Imposing invariant mass
$(M_{\rm inv}^{\ell\ell})$ cuts for opposite sign dilepton of same flavor 980
GeV $\leq M_{\rm inv}^{\ell\ell}\leq$ 1.08 TeV we find that the background
cross section reduces to $\mathcal{O}(10^{-15})$ pb which is very small. Hence
we find that the significance of these signals at LHC using $\rm
Significance=Signal/\sqrt{Signal+SM\leavevmode\nobreak\ background}$ could be
2.4$\sigma$ for $m_{1}=0.001$ eV in NH case for $4\mu$ final state, 2.1
(3.8)$\sigma$ for $m_{3}=0.01(0.001)$ eV in IH case for $4e$ final state,
respectively. At the LHC we do not consider the cases with different flavors
because of BR$(\Delta^{--}\to e^{-}\mu^{-}/e^{-}\tau^{-})\simeq 3.0(1.5)\%)$
for NH(IH) causing low signal efficiency followed by the decay of $\tau$
lepton in both $Z^{\prime}$ induced and purely SM cases. Similar scenario
could be observed for
BR$(\Delta^{--}\to\tau^{-}\tau^{+}/\mu^{-}\tau^{-})\simeq)$ case where
application of $\tau$ lepton decay might reduce the signal strength.

Using Eq. (15) we estimate signal events at $e^{-}e^{+}$ colliders considering
1 ab-1 luminosity for $4\mu$ final state in NH(IH) as 79760 (20271) for
$m_{1}=0.01$ eV which becomes 62100(10609) for $m_{1}=0.001$ eV. The
corresponding number of events from the purely SM scenario are roughly
$\mathcal{O}(3)$ of magnitude smaller than the $Z^{\prime}$ induced scenario.
Due to the dependence of BR$(\Delta^{--}\to\ell^{-}\ell^{-})$ on
$m_{1(3)}=0.01$ eV we find number of events for $4e$ final state from NH(IH)
cases is 621(148554), however, that for $m_{1(3)}=0.001$ eV is 3(158976),
respectively. Considering both the $\Delta^{--}$ decaying into $e^{-}\mu^{-}$
produce 621(155) events for NH(IH) case taking $m_{1(3)}=0.01$ eV whereas that
for $m_{1(3)}=0.001$ eV is 621(99), respectively. In case of $e^{-}e^{+}$
collider we generate the generic SM backgrounds from
$e^{-}e^{+}\to\ell^{+}\ell^{-}\ell^{+}\ell^{-}$ process with
$\ell=e,\leavevmode\nobreak\ \mu$ at $\sqrt{s}=3$ TeV where the cross sections
are $8\times 10^{-4}$ pb, $1.44\times 10^{-5}$ pb and $8.05\times 10^{-4}$ pb
for $4e$, $4\mu$ and $2e2\mu$ final states, respectively. Due to comparatively
small number of SM background we find that $4e$, $4\mu$ and $2ee\mu$ final
states could be observed with a significance of more than $5\sigma$ which is
significantly dominant over the $4e$, $4\mu$ and $2e2\mu$ final states coming
from purely SM processes. In this analysis we did not consider $4\tau$,
$\mu\tau$ and $e\tau$ modes for simplicity, however, considering the decay of
$\tau$ lepton such modes could be studied in $e^{-}e^{+}$ collider.

Using Eq. (15) we estimate signal events at $\mu^{-}\mu^{+}$ collider
considering 1 ab-1 luminosity for $4\mu$ final state in NH(IH) as
$\mathcal{O}(10^{7})$ of magnitude for $m_{1}=0.01$ eV and $0.001$ eV for
$M_{Z^{\prime}}=4$ TeV. In the same line we will find that if both the
$\Delta^{\pm\pm}$ decay into $e^{-}\mu^{-}$ mode then number of signal events
will be 366588 (58654) for NH(IH) case considering $m_{1(3)}=0.01$ eV,
however, BR$(\Delta\to e^{-}\mu^{-})$ for $m_{1(3)}=0.001$ eV is almost same
for $m_{1(3)}=0.01$ eV in NH(IH) case, then number of signal events will be
also close to the previous one. In the same line we could find that signal
events for the decay of $\Delta^{--}$ into remaining same sign same flavor
modes and same sign different flavor modes will be high. In case of
$\mu^{-}\mu^{+}$ collider we generate the generic SM backgrounds from
$\mu^{-}\mu^{+}\to\ell^{+}\ell^{-}\ell^{+}\ell^{-}$ process with
$\ell=e,\leavevmode\nobreak\ \mu$ at $\sqrt{s}=10$ TeV and the cross sections
are $10^{-4}$ pb, $10^{-6}$ pb and $10^{-4}$ pb for $4\mu$, $4e$ and $2e2\mu$
final states, respectively which are very weak compared to the signal.
Therefore in muon collider there is a possibility that all same flavor and
different flavor modes with equal sign could be observed with reasonably high
significance. Besides $Z^{\prime}$ induced scenario we study the purely SM
gauge boson mediated $\Delta^{\pm\pm}$ production which could produce
$\mathcal{O}(2)$ of magnitude less events compared to the $Z^{\prime}$ induced
scenarios. However, in this case same sign same flavor and different flavor
scenarios could also be significant to study.

Figure 7: Effective neutrino mass for the neutrinoless double beta decay in NH
(red) and IH (blue) cases as a function of lightest light neutrino mass.
Shaded regions are excluded by PLANCK data for NH (blue) and IH (purple)
cases.

In Fig. 7 we show the effective neutrino mass from the $(1,1)$ element of the
neutrino mass matrix, responsible for the neutrinoless double beta decay, with
respect to the lightest light neutrino mass eigen value depending for NH(IH)
case by red (blue) solid line. We further compare our results with PLANCK
exclusion limits. We find that $(1,1)$ element of the neutrino mass matrix is
stronger than that of the IH case which could be probed by neutrino
oscillation experiments in future.

Conclusions.– We study a type-II seesaw scenario under an anomaly free,
general $U(1)_{X}$ extension of the SM where Majorana type light neutrino mass
is generated by the VEV of an $SU(2)_{L}$ triplet scalar being charged under
the general $U(1)_{X}$ gauge group. In this scenario, after the breaking of
the general $U(1)_{X}$ scenario, a BSM neutral gauge boson is evolved which
interacts with the left and right handed fermions separately. Due to the
general $U(1)_{X}$ charge of the triplet scalar, it interacts with
$Z^{\prime}$ boson in addition to the SM gauge bosons. The charge multiplets
of the triplet scalar, that is, doubly charged scalar can be produced at the
high energy colliders like LHC, $e^{-}e^{+}$ and $\mu^{-}\mu^{+}$ at 14 TeV, 3
TeV and 10 TeV center of mass energies, respectively. Due to the interaction
between the triplet scalar and SM lepton doublet and doubly charged scalar can
dominantly decay into same sign lepton considering the triplet VEV to be less
than $10^{-4}$ GeV. These same sign dilpetons could be of same flavor and
different flavors depending on the diagonal and off-diagonal entries of the
Dirac Yukawa coupling between triplet scalar and SM lepton doublet. This
Yukawa coupling is related to the Majorana type neutrino mass and hence can be
formulated using the neutrino oscillation data depending on neutrino mass
hierarchies(NH and IH). Hence branching ratio of the doubly charged triplet
scalar into same sign same flavor and different flavor dilepton will also
depedn on neutrino mass hierarchies. Therefore after the production of doubly
charged scalar from $Z^{\prime}$ induced processes followed by its decay into
same sign dilepton we study $4e$, $4\mu$ and $2e2\mu$ signals at 3 ab-1
luminosity at LHC and 1 ab-1 luminosity at $e^{-}e^{+}$ and $\mu^{-}\mu^{+}$
colliders respectively. After producing the generic four lepton SM backgrounds
we find that $4\mu$ and $4e$ final states could observed at the LHC with more
than $2\sigma$ depending on the NH and IH scenarios whereas same signals from
the purely SM processes will be suppressed. In both the cases same sign
different flavor dilepton signals will be less significant. In $e^{-}e^{+}$
colliders after we study the $4e$, $4\mu$ and $2e2\mu$ signals and the
corresponding generic SM backgrounds which could be observed with a
significance of more than $5\sigma$ for NH and IH cases, however, the same
final states from the doubly charged scalar production from the purely SM
processes will be suppressed due to small cross section. Finally we find that
in case of $\mu^{-}\mu+$ collider the signals with same sign same flavor and
same sign different flavor could be observed with a high significance form
$Z^{\prime}$ induced and purely SM processes depending on the neutrino mass
hierarchy which could be probed in future.

###### Acknowledgements.

Acknowledgments.– The works of PD and NO are supported by the United States
Department of Energy Grant No. DE-SC0012447.

##  Appendix A Scalar potential, partial decay widths of $Z^{\prime}$ and
$\Delta^{--/++}$

$\displaystyle V$ $\displaystyle=$ $\displaystyle-
m_{H}^{2}|H|^{2}+m_{\tilde{H}}^{2}|\tilde{H}|^{2}-m_{\Phi}^{2}|\Phi|^{2}+m_{\Delta}^{2}Tr(\Delta^{\dagger}\Delta)+\frac{1}{2}\lambda_{H}|H|^{4}+\frac{1}{2}\lambda_{\tilde{H}}|\tilde{H}|^{4}+\frac{1}{2}\lambda_{\Phi}|\Phi|^{4}+\frac{1}{2}\lambda_{\Delta}(Tr(\Delta^{\dagger}\Delta))^{2}$
(16) $\displaystyle+$
$\displaystyle\frac{1}{2}\tilde{\lambda}_{\Delta}((Tr(\Delta^{\dagger}\Delta))^{2}-Tr((\Delta^{\dagger}\Delta)^{2}))+\lambda_{H\tilde{H}}|H|^{2}|\tilde{H}|^{2}+\tilde{\lambda}_{H\tilde{H}}|\tilde{H}^{\dagger}H^{\prime\prime}|^{2}+\lambda_{\Phi
H}|\Phi|^{2}|H|^{2}+\lambda_{\Phi\tilde{H}}|\Phi|^{2}|\tilde{H}|^{2}$
$\displaystyle+$
$\displaystyle\lambda_{\Phi\Delta}|\Phi|^{2}Tr(\Delta^{\dagger}\Delta)+\\{\lambda_{HH}|H|^{2}+\lambda_{\tilde{H}\tilde{H}}|\tilde{H}|^{2}\\}Tr(\Delta^{\dagger}\Delta)+\lambda_{\tilde{H}\Delta}\tilde{H}^{\dagger}[\Delta^{\dagger},\Delta]\tilde{H}+\lambda_{H\Delta}H^{\dagger}[\Delta^{\dagger},\Delta]H$
$\displaystyle-$
$\displaystyle\frac{\mu_{1}}{\sqrt{2}}(\tilde{H}^{T}.\Delta\tilde{H}+H.c.)+(\lambda\Phi(\tilde{H}^{\dagger}H)+H.c.)$

$\displaystyle V$ $\displaystyle=$
$\displaystyle-\tilde{m}_{H}^{2}|H|^{2}+\tilde{m}_{\tilde{H}}^{2}|\tilde{H}|^{2}+\tilde{m}_{\Delta}^{2}Tr(\Delta^{\dagger}\Delta)+\frac{1}{2}\lambda_{H}|H|^{4}+\frac{1}{2}\lambda_{\tilde{H}}|\tilde{H}|^{4}+\frac{1}{2}\lambda_{\Delta}(Tr(\Delta^{\dagger}\Delta))^{2}+\frac{1}{2}\tilde{\lambda}_{\Delta}((Tr(\Delta^{\dagger}\Delta))^{2}$
(17) $\displaystyle-$ $\displaystyle
Tr((\Delta^{\dagger}\Delta)^{2}))+\lambda_{H\tilde{H}}|H|^{2}|\tilde{H}|^{2}+\tilde{\lambda}_{H\tilde{H}}|\tilde{H}^{\dagger}H|^{2}+\\{\lambda_{HH}|H|^{2}+\lambda_{\tilde{H}\tilde{H}}|\tilde{H}|^{2}\\}Tr(\Delta^{\dagger}\Delta)+\lambda_{\tilde{H}\Delta}\tilde{H}^{\dagger}[\Delta^{\dagger},\Delta]\tilde{H}$
$\displaystyle+$
$\displaystyle\lambda_{H\Delta}H^{\dagger}[\Delta^{\dagger},\Delta]H-\frac{\mu_{1}}{\sqrt{2}}(\tilde{H}^{T}.\Delta\tilde{H}+H.c.)+(\tilde{m}_{H\tilde{H}}^{2}(\tilde{H}^{\dagger}H)+H.c.)$

Stationary conditions from Eq. (17) at the point of symmetry breaking can be
given by

$\displaystyle\tilde{m}_{H}^{2}$ $\displaystyle=$
$\displaystyle\frac{\lambda_{H}v_{H}^{3}+v_{\tilde{H}}^{2}v_{H}(\lambda_{H\tilde{H}}+\tilde{\lambda}_{H\tilde{H}})+v_{H}v_{\Delta}^{2}(\lambda_{HH}-\lambda_{H\Delta})-2\tilde{m}_{H\tilde{H}}^{2}v_{\tilde{H}}}{2v_{H}},$
$\displaystyle\tilde{m}_{\tilde{H}}^{2}$ $\displaystyle=$
$\displaystyle\frac{2\tilde{m}_{H\tilde{H}}^{2}v_{H}^{2}-v_{\tilde{H}}(\lambda_{\tilde{H}}v_{\tilde{H}}^{2}+v_{H}^{2}(\lambda_{H\tilde{H}}+\tilde{\lambda}_{H\tilde{H}})+v_{\Delta}^{2}(\lambda_{\tilde{H}\tilde{H}}-\lambda_{\tilde{H}\Delta})-2\mu_{1}v_{\Delta})}{2v_{\tilde{H}}},$
$\displaystyle\tilde{m}_{\Delta}^{2}$ $\displaystyle=$
$\displaystyle\frac{\mu_{1}v_{\tilde{H}}^{2}-v_{\Delta}(\lambda_{H\tilde{H}}v_{\Delta}^{2}+v_{\tilde{H}}^{2}(\lambda_{\tilde{H}\tilde{H}}-\lambda_{\tilde{H}\Delta})+v_{H}^{2}(\lambda_{HH}-\lambda_{H\Delta}))}{2v_{\Delta}}.$
(18)

where $v_{H,\tilde{H}}$ are the VEVs of $H$ and $\tilde{H}$ respectively,
satisfying $v=\sqrt{v_{H}^{2}+v_{\tilde{H}}^{2}}=246$ GeV and we can also
write

$\displaystyle\mu_{1}=\frac{v_{\Delta}(\lambda_{\tilde{H}}v_{\Delta}^{2}+v_{\tilde{H}}^{2}(\lambda_{\tilde{H}\tilde{H}}-\lambda_{\tilde{H}\Delta})+v_{H}^{2}(\lambda_{HH}-\lambda_{H\Delta})+2\tilde{m}_{\Delta}^{2})}{v_{\tilde{H}}^{2}}$
(19)

where type-II seesaw scenario can be achieved from the approximation
$\tilde{m}_{\Delta}^{2}\gg v$ reducing to
$v_{\Delta}\simeq\frac{\mu_{1}v_{\tilde{H}}^{2}}{2\tilde{m}_{\Delta}^{2}}$
where small $\mu_{1}$ and $v_{\tilde{H}}^{2}$ produces a small $v_{\Delta}$.
Hence we can write the masses of the scalars as

$\displaystyle m_{h/\tilde{h}}^{2}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\lambda_{\tilde{H}}v_{\tilde{H}}^{2}+m_{\tilde{H}H}^{2}\frac{v_{H}}{v_{\tilde{H}}}&v_{\tilde{H}v_{H}}(\lambda_{H\tilde{H}}+\tilde{\lambda}_{H\tilde{H}})-m_{\tilde{H}H}^{2}\\\
v_{\tilde{H}v_{H}}(\lambda_{H\tilde{H}}+\tilde{\lambda}_{H\tilde{H}})-m_{\tilde{H}H}^{2}&\lambda_{H}v_{H}^{2}+m_{\tilde{H}H}^{2}\frac{v_{\tilde{H}}}{v_{H}}\end{pmatrix},$
$\displaystyle m_{H}^{2}$ $\displaystyle=$ $\displaystyle
m_{H^{\pm}}^{2}-\frac{1}{2}(\lambda_{\tilde{H}\Delta}v_{\tilde{H}}^{2}+\lambda_{H\Delta}v_{H}^{2})=m_{A}^{2},\;\;\;m_{\tilde{A}}^{2}=m_{\tilde{H}H}^{2}(\frac{v_{\tilde{H}}}{v_{H}}+\frac{v_{H}}{v_{\tilde{H}}})$
$\displaystyle m_{\tilde{H}^{\pm}}^{2}$ $\displaystyle=$ $\displaystyle
m_{\tilde{A}}^{2}-\frac{\tilde{\lambda}_{H\tilde{H}}v^{2}}{2},\;\;\;m_{H^{\pm}}^{2}=\frac{1}{2}(\lambda_{\tilde{H}\tilde{H}}v_{\tilde{H}}^{2}+\lambda_{HH}v_{H}^{2}+2m_{\Delta}^{2}),\;\;\;m_{\Delta^{\pm\pm}}=m_{H^{\pm}}^{2}+\frac{1}{2}(\lambda_{\tilde{H}\Delta}v_{\tilde{H}}^{2}+\lambda_{H\Delta}v_{H}^{2})$
(20)

Now we show the partial decay widths of $Z^{\prime}$ and doubly charged
multiplet of the triplet scalar $(\Delta^{\pm\pm})$ as

$\displaystyle\sum_{f=\rm quarks,leptons}\Gamma(Z^{\prime}\to f\overline{f})$
$\displaystyle=$
$\displaystyle\frac{g_{X}^{2}}{24\pi}M_{Z^{\prime}}(13+16x_{H}+10x_{H}^{2})$
$\displaystyle\Gamma(Z^{\prime}\to\Delta^{\pm\pm}\Delta^{\mp\mp}/\Delta^{+}\Delta^{-})$
$\displaystyle=$
$\displaystyle\frac{g_{X}^{2}(x_{H}+2)^{2}M_{Z^{\prime}}}{48\pi}\Big{[}1-4\frac{m_{\Delta^{++}}^{2}}{M_{Z^{\prime}}^{2}}\Big{]}^{\frac{3}{2}}$
$\displaystyle\Gamma(Z^{\prime}\to N_{R}\overline{N_{R}})$ $\displaystyle=$
$\displaystyle\frac{g_{X}^{2}}{24\pi}M_{Z^{\prime}}x_{N_{R}}^{2}\Big{[}1-4\frac{m_{N}^{2}}{s}\Big{]}^{\frac{3}{2}},(x_{N_{R}}=-4,5),$
$\displaystyle\Gamma(Z^{\prime}\to\chi\chi)$ $\displaystyle=$
$\displaystyle\frac{41g_{X}^{2}}{24\pi}M_{Z^{\prime}}\sqrt{1-4\frac{m_{\chi}^{2}}{M_{Z^{\prime}}^{2}}}(41-120m_{\chi}^{2})\
$ (21) $\displaystyle\Gamma(\Delta^{\pm\pm}\to\ell^{+}\ell^{-})$
$\displaystyle\simeq$ $\displaystyle
C_{ij}\frac{|Y^{ij}|^{2}}{2}\frac{m_{\Delta^{\pm\pm}}}{4\pi},\;\;\;\Gamma(\Delta^{\pm\pm}\to
W^{\pm}W^{\pm})\simeq\frac{g_{X}^{4}v_{\Delta}^{2}m_{\Delta^{\pm\pm}}^{2}}{64\pi
m_{W}^{4}}\Big{(}3\frac{m_{W}^{4}}{m_{\Delta^{\pm\pm}}^{4}}-\frac{m_{W}^{2}}{m_{\Delta^{\pm\pm}}^{2}}+\frac{1}{4}\Big{)}$
(22)

where $C_{ij}=1(\frac{1}{2})$ for $i\neq j$ $(i=j)$.

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