# Same-Sign Tetralepton Signature in Type-II Seesaw at Lepton Colliders

Xu-Hong Bai  Zhi-Long Han<EMAIL_ADDRESS>Yi Jin  Hong-Lei Li  Zhao-Xia
Meng<EMAIL_ADDRESS>School of Physics and Technology, University of
Jinan, Jinan, Shandong 250022, China

###### Abstract

The same-sign tetralepton signature via mixing of neutral Higgs bosons and
their cascade decays to charged Higgs bosons is a unique signal in the type-II
seesaw model. In this paper, we study this signature at future lepton
colliders, such as ILC, CLIC, and MuC. Constrained by direct search,
$H^{\pm\pm}\to W^{\pm}W^{\pm}$ is the only viable decay mode for
$M_{A^{0}}=400$ GeV at $\sqrt{s}=1$ TeV ILC. With an integrated luminosity of
$\mathcal{L}=8~{}\mathrm{ab}^{-1}$, the promising region with about 150 signal
events corresponds to a narrow band in the range of
$10^{-4}~{}\text{GeV}\lesssim v_{\Delta}\lesssim 10^{-2}$ GeV. For heavier
triplet scalars $M_{A^{0}}\gtrsim 900$ GeV, although the
$H^{\pm\pm}\to\ell^{\pm}\ell^{\pm}$ decay mode is allowed, the cascade decays
are suppressed. A maximum event number $\sim 16$ can be obtained around
$v_{\Delta}\sim 4\times 10^{-4}$ GeV and $\lambda_{4}\sim 0.26$ for
$M_{A^{0}}=1000$ GeV with $\mathcal{L}=5~{}\mathrm{ab}^{-1}$ at $\sqrt{s}=3$
TeV CLIC. Meanwhile, we find that this signature is not promising for
$M_{A^{0}}=1500$ GeV at $\sqrt{s}=6$ TeV MuC.

##  I Introduction

The discovery of neutrino oscillations Fukuda:1998mi ; Ahmad:2002jz ;
An:2012eh  confirms that neutrinos have sub-eV masses. Meanwhile, the
underlying mechanism accounting for such tiny neutrino mass is still an open
question. Regarding the standard model (SM) as a low energy effective field
theory, the simplest pathway to generate neutrino mass is via the Weinberg
operator $LL\Phi\Phi/\Lambda$ Weinberg:1979sa . There are three possible ways
at tree level to realize this operator Ma:1998dn , which correspond to the
canonical type-I Minkowski:1977sc ; Mohapatra:1979ia , type-II Magg:1980ut ;
Cheng:1980qt ; Lazarides:1980nt ; Mohapatra:1980yp , and type-III Foot:1988aq
seesaw. To verify whether these scenarios are realized in nature, the
signatures of seesaw models at colliders have been extensively studied
delAguila:2008cj ; Deppisch:2015qwa ; Cai:2017mow . Since the conventional
type-I seesaw requires the right-hand neutrinos $N$ to be quite heavy
($\gtrsim 10^{14}$ GeV), it is far beyond the reach of current and future
planed colliders. Therefore, we consider the type-II seesaw in this work.
Other possible low scale approaches to generate tiny neutrino mass have been
summarized in Ref. Boucenna:2014zba ; Cai:2017jrq .

The type-II seesaw introduces a scalar triplet $\Delta$ with hypercharge
$Y=+2$, where neutrino mass is generated by the Yukawa interaction between the
lepton doublets and scalar triplet. After the spontaneous symmetry breaking of
SM Higgs doublet $\Phi$, the trilinear term
$\mu\Phi^{T}i\tau_{2}\Delta^{\dagger}\Phi$ induces a vacuum expectation value
for the neutral component of scalar triplet with $v_{\Delta}\sim\mu
v^{2}/M_{\Delta}^{2}$. Since the scalar triplet $\Delta$ also carries the
lepton number $+2$, the $\mu$-term breaks the lepton number by two units. In
particular, this trilinear term is the only source of lepton number violation,
thus it should be naturally small. Then, for $\mu\sim v_{\Delta}$, we can
naturally have $M_{\Delta}\sim v$, i.e., the mass of scalar triplet at the
electroweak scale Arhrib:2011uy .

A distinct feature of this model is the presence of doubly charged Higgs
$H^{\pm\pm}$. Assuming degenerate mass spectrum of the scalar triplet, the
typical channels to hunt for $H^{\pm\pm}$ are the same-sign dilepton channel
$H^{\pm\pm}\to\ell^{\pm}\ell^{\pm}$ and the same-sign diboson channel
$H^{\pm\pm}\to W^{\pm}W^{\pm}$ Perez:2008ha . For non-degenerate case, cascade
decay channel $H^{\pm\pm}\to H^{\pm}W^{\pm}$ is also possible Melfo:2011nx ;
Aoki:2011pz ; Han:2015hba ; Han:2015sca . Corresponding signatures have been
extensively studied at LHC Akeroyd:2011zza ; Akeroyd:2012nd ; Chun:2013vma ;
Babu:2016rcr ; Li:2018jns ; Primulando:2019evb , HE-LHC Du:2018eaw ;
deMelo:2019asm ; Padhan:2019jlc ; Fuks:2019clu , $e^{+}e^{-}$ collider
Blunier:2016peh ; Agrawal:2018pci , and $ep$ colliders Dev:2019hev ;
Yang:2021skb . When $v_{\Delta}<10^{-4}$ GeV, the
$H^{\pm\pm}\to\ell^{\pm}\ell^{\pm}$ is the dominant decay mode, and direct
search at LHC has already excluded the region $M_{H^{\pm\pm}}<870$ GeV
Aaboud:2017qph . In this case, the branching ratios of
$H^{\pm\pm}\to\ell^{\pm}\ell^{\pm}$ are only correlated with neutrino
oscillation parameters Akeroyd:2007zv . When $v_{\Delta}>10^{-4}$ GeV, the
$H^{\pm\pm}\to W^{\pm}W^{\pm}$ mode becomes the dominant one, and searches for
pair production of $H^{\pm\pm}$ in this diboson channel have excluded
$M_{H^{\pm\pm}}<350$ GeV Aaboud:2018qcu ; Aad:2021lzu .

Among various possible collider signatures of the type-II seesaw, a unique one
is the same-sign tetralepton signature Chun:2012zu ; Chun:2019hce , which
arises from the mixing of neutral Higgs bosons and their cascade decays to
singly and doubly charged Higgs bosons. Previous studies Chun:2012zu ;
Chun:2019hce  focus on the hadron colliders as LHC and FCC-hh with
$\sqrt{s}=100$ TeV. In this paper, we will analyze this signature at future
lepton colliders. Considering current lower bound on doubly charged Higgs
$M_{H^{\pm\pm}}>350$ GeV, this signature is beyond the reach of CEPC
CEPCStudyGroup:2018ghi . In order to pair produce $H^{\pm\pm}$, the collision
energy should be at least higher than 700 GeV. Therefore, we take the
following three benchmark scenarios to illustrate, i.e., $M_{H^{\pm\pm}}\sim
400$ GeV at $\sqrt{s}=1$ TeV ILC Barklow:2015tja ; Fujii:2017vwa ,
$M_{H^{\pm\pm}}\sim 1000$ GeV at $\sqrt{s}=3$ TeV CLIC Linssen:2012hp ;
Robson:2018zje , and $M_{H^{\pm\pm}}\sim 1500$ GeV at $\sqrt{s}=6$ TeV Muon
Collider (MuC) Delahaye:2019omf ; Long:2020wfp .

In our paper, the type-II seesaw model will be briefly introduced in Sec. II.
The branching ratios of the scalar triplet components are also discussed in
Sec. II. The same-sign tetralepton signals at ILC, CLIC, and MuC are analyzed
in Sec. III. Finally, the conclusion is presented in Sec. IV.

##  II The Model

We concisely review the type-II seesaw in this section. Besides the SM Higgs
doublet $\Phi$, a scalar triplet $\Delta$ is also employed, which can be
denoted as

$\Phi=\left(\begin{array}[]{c}\phi^{+}\\\
\Phi^{0}\end{array}\right),\quad\Delta=\left(\begin{array}[]{cc}\frac{\Delta^{+}}{\sqrt{2}}&\Delta^{++}\\\
\Delta^{0}&-\frac{\Delta^{+}}{\sqrt{2}}\end{array}\right),$ (1)

where after spontaneous symmetry breaking, the neutral components can be
further written as
$\Phi^{0}=\frac{1}{\sqrt{2}}\left(v+\phi^{0}+i\chi^{0}\right)$ and
$\Delta^{0}=\frac{1}{\sqrt{2}}\left(v_{\Delta}+\delta^{0}+i\eta^{0}\right)$,
respectively. The Yukawa interaction that generates tiny neutrino mass is
given by

$\mathcal{L}_{Y}=Y_{\Delta}\overline{L_{L}^{c}}i\tau_{2}\Delta L_{L}+\text{
h.c. }$ (2)

The scalar potential involving $\Phi$ and $\Delta$ is

$\displaystyle V(\Phi,\Delta)$
$\displaystyle=m_{\Phi}^{2}\Phi^{\dagger}\Phi+M^{2}\operatorname{Tr}\left(\Delta^{\dagger}\Delta\right)+\left(\mu\Phi^{\mathrm{T}}i\tau_{2}\Delta^{\dagger}\Phi+\mathrm{h.c.}\right)+\frac{\lambda_{0}}{4}\left(\Phi^{\dagger}\Phi\right)^{2}$
(3)
$\displaystyle+\lambda_{1}\left(\Phi^{\dagger}\Phi\right)\operatorname{Tr}\left(\Delta^{\dagger}\Delta\right)+\lambda_{2}\left[\operatorname{Tr}\left(\Delta^{\dagger}\Delta\right)\right]^{2}+\lambda_{3}\operatorname{Tr}\left[\left(\Delta^{\dagger}\Delta\right)^{2}\right]+\lambda_{4}\Phi^{\dagger}\Delta\Delta^{\dagger}\Phi.$

Mixing between the doublet and triplet scalars leads to seven physical
scalars,i.e., doubly charged Higgs $H^{\pm\pm}$, singly charged Higgs
$H^{\pm}$, CP-even Higgs bosons $h$ and $H^{0}$, and CP-odd Higgs $A^{0}$,
with the mixing angles specified by

$\tan\beta_{\pm}=\frac{\sqrt{2}v_{\Delta}}{v},\quad\tan\beta_{0}=\frac{2v_{\Delta}}{v},\quad\tan
2\alpha=\frac{4v_{\Delta}}{v}\frac{v^{2}\left(\lambda_{1}+\lambda_{4}\right)-2M_{\Delta}^{2}}{v^{2}\lambda-2M_{\Delta}^{2}-4v_{\Delta}^{2}\left(\lambda_{2}+\lambda_{3}\right)},$
(4)

where $M^{2}_{\Delta}=\mu v^{2}/(\sqrt{2}v_{\Delta})$. The masses of the
doubly and singly charged Higgs bosons $H^{++}$ and $H^{+}$ is given by

$M_{H^{++}}^{2}=M_{\Delta}^{2}-v_{\Delta}^{2}\lambda_{3}-\frac{\lambda_{4}}{2}v^{2},\quad
M_{H^{+}}^{2}=\left(M_{\Delta}^{2}-\frac{\lambda_{4}}{4}v^{2}\right)\left(1+\frac{2v_{\Delta}^{2}}{v^{2}}\right).$
(5)

The masses of CP-even Higgs bosons $h$, and $H^{0}$ can be written as

$\displaystyle
M_{h}^{2}=\mathcal{T}_{11}^{2}\cos^{2}\alpha+\mathcal{T}_{22}^{2}\sin^{2}\alpha-\mathcal{T}_{12}^{2}\sin
2\alpha,$ (6) $\displaystyle
M_{H^{0}}^{2}=\mathcal{T}_{11}^{2}\sin^{2}\alpha+\mathcal{T}_{22}^{2}\cos^{2}\alpha+\mathcal{T}_{12}^{2}\sin
2\alpha,$ (7)

where $\mathcal{T}_{11}$, $\mathcal{T}_{22}$, and $\mathcal{T}_{12}$ are of
the form

$\mathcal{T}_{11}^{2}=\frac{\lambda_{0}}{2}v^{2},\quad\mathcal{T}_{22}^{2}=M_{\Delta}^{2}+2v_{\Delta}^{2}\left(\lambda_{2}+\lambda_{3}\right),\quad\mathcal{T}_{12}^{2}=-\frac{2v_{\Delta}}{v}M_{\Delta}^{2}+v_{\Delta}v\left(\lambda_{1}+\lambda_{4}\right).$
(8)

Finally, the CP-odd Higgs $A^{0}$ has the following mass

$M_{A^{0}}^{2}=M_{\Delta}^{2}\left(1+\frac{4v_{\Delta}^{2}}{v^{2}}\right).$
(9)

Constrained by the $\rho$ parameter, $v_{\Delta}\lesssim 1$ GeV should be
satisfied. Neglecting the contributions from $v_{\Delta}$, masses of triplet
scalars have the relation

$M_{H^{++}}^{2}-M_{H^{+}}^{2}\approx
M_{H^{+}}^{2}-M_{H^{0},A^{0}}^{2}\approx-\frac{1}{4}\lambda_{4}v^{2}.$ (10)

In this paper, we consider the scenario with $\lambda_{4}>0$, which leads to
the mass spectrum $M_{H^{++}}<M_{H^{+}}<M_{H^{0}}\simeq M_{A^{0}}$. The mass
difference between $H^{0}$ and $A^{0}$ plays a vital important role in the
production of the same-sign trilepton signature, which is controlled by
$v_{\Delta}$ as

$M_{H^{0}}^{2}-M_{A^{0}}^{2}\sim
2(\lambda_{2}+\lambda_{3})v_{\Delta}^{2}-4\frac{M_{\Delta}^{2}}{v^{2}}v_{\Delta}^{2}.$
(11)

Figure 1: The branching ration of $H^{++}$ (upper panels) and $H^{+}$ (lower
panels) for masses $M_{A^{0}}=400$ GeV (left), 1000 GeV (middle), and 1500 GeV
(right). The other relevant parameters are fixed as $\lambda_{0}=0.52$,
$\lambda_{1,2,3}=0.1$, and $\lambda_{4}=0.3$.

Here, we briefly discuss the decay properties of triplet scalars with the mass
spectrum $M_{H^{++}}<M_{H^{+}}<M_{H^{0}}\simeq M_{A^{0}}$. Expressions of
partial decay widths of triplet scalars can be found in Ref. Aoki:2011pz . In
this scenario, the doubly charged Higgs $H^{\pm\pm}$ is the lightest. The
possible decay channels are same-sign dilepton
$H^{\pm\pm}\to\ell^{\pm}\ell^{\pm}$ and same-sign diboson $H^{\pm\pm}\to
W^{\pm}W^{\pm}$. The branching ratios are plotted in Fig. 1 for three
benchmark cases with $M_{A^{0}}=$400, 1000, and 1500 GeV. The decay widths of
dilepton $H^{\pm\pm}\to\ell^{\pm}\ell^{\pm}$ channel is proportional to
$1/v_{\Delta}^{2}$, while that of diboson $H^{\pm\pm}\to W^{\pm}W^{\pm}$ is
proportional to $v_{\Delta}^{2}$. Therefore, we have
BR($H^{\pm\pm}\to\ell^{\pm}\ell^{\pm})\simeq 1$ for $v_{\Delta}\lesssim
10^{-5}$ GeV, BR($H^{\pm}\to H^{\pm\pm}W^{*})\simeq 1$ for $v_{\Delta}\gtrsim
10^{-3}$ GeV. Increasing the mass of $H^{\pm\pm}$ do not have a large impact
on the results of BR($H^{\pm\pm}$). As for the singly charged Higgs $H^{\pm}$,
possible decay channels are leptonic $H^{\pm}\to\ell^{\pm}\nu$, bosonic
$H^{\pm}\to W^{\pm}Z/W^{\pm}h$, quarks $H^{\pm}\to tb/cs$, and cascade
$H^{\pm}\to H^{\pm\pm}W^{*}$. Here, we focus on the same-sign tetralepton
signature related channel, i.e., the cascade decay $H^{\pm}\to
H^{\pm\pm}W^{*}$. This channel is the dominant one in the range of
$10^{-6}~{}\text{GeV}\lesssim v_{\Delta}\lesssim 10^{-3}$ GeV when
$M_{A^{0}}=400$ GeV. As the mass of triplet scalars increase to about 1000
GeV, the dominant range of this channel shrinks to $v_{\Delta}\sim 5\times
10^{-5}$ GeV, and the corresponding branching ratio never reaches one.
Meanwhile, this channel can not become the dominant one when $M_{A^{0}}=1500$
GeV. This is because of as the increase of triplet scalar masses, the phase
space of cascade decay is suppressed. It has been shown that the dominant
range of cascade decays $H^{0}\to H^{\pm}W^{*}$ and $A^{0}\to H^{\pm}W^{*}$
are similar to the channel $H^{\pm}\to H^{\pm\pm}W^{*}$ Han:2015hba .

##  III Same-Sign Tetalepton Signature

Figure 2: Production cross section of $H^{0}A^{0}$ at various colliders. The
solid red, green, and blue lines are the results at 1 TeV ILC, 3 TeV CLIC, and
6 TeV MuC, respectively. The dashed cyan and pink lines are the results at 14
TeV LHC and 100 TeV FCC-hh.

In this section, we explore the same-sign tetralepton signature resulting from
the neutral Higgs decay. First, let’s consider the production cross section of
$H^{0}A^{0}$. The results are shown in Fig. 2, where the cross section
$\sigma(H^{0}A^{0})$ at 14 TeV LHC and 100 TeV FCC-hh are also illustrated for
comparison. All the results are computed by using Madgraph5_aMC@NLO
Alwall:2014hca . For lepton colliders, the neutral Higgs pair $H^{0}A^{0}$ can
be produced when $M_{A^{0}}<\sqrt{s}/2$. At the 1 TeV ILC, the cross section
$\sigma(H^{0}A^{0})$ is larger than at 14 TeV LHC in the range of
$300~{}\text{GeV}\lesssim M_{A^{0}}\lesssim 500$ GeV. For
$500~{}\text{GeV}\lesssim M_{A^{0}}\lesssim 1300$ GeV, the 3 TeV CLIC
generates the largest cross section among lepton colliders. Notably,
$\sigma(H^{0}A^{0})$ at 3 TeV CLIC can be two orders of magnitudes larger than
at LHC for $M_{A^{0}}\sim 1000$ GeV. When $M_{A^{0}}\gtrsim 1300$ GeV, the 6
TeV MuC becomes one of the best options. Especially, in the range of
$1700~{}\text{GeV}\lesssim M_{A^{0}}\lesssim 2700$ GeV, $\sigma(H^{0}A^{0})$
at 6 TeV MuC is even larger than at 100 TeV FCC-hh.

At the 1 TeV ILC with $M_{A^{0}}=400$ GeV, this signal is generated via the
tetraboson process

$\displaystyle e^{+}e^{-}\to H^{0}A^{0}$ $\displaystyle\to$ $\displaystyle
H^{\pm}W^{*}H^{\pm}W^{*}\to H^{\pm\pm}W^{*}H^{\pm\pm}W^{*}+W^{*}W^{*}\to
4W^{\pm}+X,$ (12)

with the leptonic decay $W^{\pm}\to\ell^{\pm}\nu~{}(\ell=e,\mu)$. Note that
the dilepton decay $H^{\pm\pm}\to\ell^{\pm}\ell^{\pm}$ has already been
excluded by direct search at LHC. Since the typical mass splitting between
triplet scalars for the same-sign tetralepton signature is at the order of
$\mathcal{O}(\text{GeV})$, the final states from off-shell $W$ decay are hard
to be detected. Such signature occurs due to the interference effect between
$H^{0}$ and $A^{0}$, which is sizable when $\delta
M=M_{H^{0}}-M_{A^{0}}\sim\Gamma_{H^{0}/A^{0}}$. The cross section for this
signal is calculated as Chun:2012zu

$\displaystyle\sigma_{W}(4\ell^{\pm}+X)$ $\displaystyle=$
$\displaystyle\sigma\left(e^{+}e^{-}\rightarrow
H^{0}A^{0}\right)\times\left(\frac{2+x^{2}}{1+x^{2}}\frac{x^{2}}{1+x^{2}}\right)\times\text{BR}\left(H^{0}/A^{0}\rightarrow
H^{\pm}W^{*}\right)^{2}$ $\displaystyle\times\text{BR}\left(H^{\pm}\rightarrow
H^{\pm\pm}W^{*}\right)^{2}\times\text{BR}\left(H^{\pm\pm}\rightarrow
W^{\pm}W^{\pm}\right)^{2}\times\text{BR}(W^{\pm}\to\ell^{\pm}\nu)^{4},$

where $x=\delta M/\Gamma_{H^{0}/A^{0}}$. The initial cross section
$\sigma(e^{+}e^{-}\to H^{0}A^{0})$ is about 10 fb at the 1 TeV ILC with
$M_{A^{0}}=400$ GeV. In the left panel of Fig. 3, we show the product of BRs
in the above process. As shown in Fig. 1, BR$(H^{\pm\pm}\to W^{\pm}W^{\pm})$
is quickly suppressed for $v_{\Delta}<10^{-4}$ GeV, which corresponds to the
left boundary. While the right one is determined by the cascade decay
branching ratios as BR$(H^{\pm}\to H^{\pm\pm}W^{*})$. In this way, a larger
$\lambda_{4}$ leads to a larger mass splitting, hence a wider range of
$v_{\Delta}$ Han:2015sca .

In the right panel of Fig. 3, we show the expected event number for the same-
sign tetralepton signature at the 1 TeV ILC with an integrated luminosity of
$\mathcal{L}=8~{}\text{ab}^{-1}$. A detector level simulation with Delphes
deFavereau:2013fsa  is also performed, where we only require
$p_{T}(\ell^{\pm})>10$ GeV and $|\eta(\ell^{\pm})|<2.5$. The total cut
efficiency we applied is $c_{eff}=0.6$ for $M_{A^{0}}=400$ GeV. The promising
region in the $\lambda_{4}-v_{\Delta}$ plane fills a narrow band, where the
maximum event number can reach about 160. Such a narrow band is formed mainly
due to the interference effect between $H^{0}$ and $A^{0}$. For fixed value of
$v_{\Delta}$, the mass splitting $\delta M$ is then determined. A certain
value $\lambda_{4}^{M}$ resulting suitable cascade decay width, i.e.,
$x=\delta M/\Gamma_{H^{0}/A^{0}}\sim 1$, leads to the maximum event number. If
$\lambda_{4}>\lambda_{4}^{M}$, then $\Gamma_{H^{0}/A^{0}}$ will increase, thus
$x$ will decrease, and the final event number also will decrease. Considering
the fact that for a small mass splitting of triplet scalars $\Delta
M\sim\lambda_{4}v^{2}/(8M_{A^{0}})$, the cascade decay dominant width
$\Gamma_{H^{0}/A^{0}}\propto\Delta M^{5}$, and $\delta M\propto
v_{\Delta}^{2}$, it is easy to derive the relation $\lambda_{4}\propto
v_{\Delta}^{2/5}$ by taking $\delta M\sim\Gamma_{H^{0}/A^{0}}$.

Figure 3: Left panel: Product of branching ratios
$\operatorname{BR}\left(H^{0}/A^{0}\rightarrow
H^{\pm}W^{-*}\right)^{2}\times\operatorname{BR}\left(H^{\pm}\rightarrow
H^{\pm\pm}W^{-*}\right)^{2}\times\operatorname{BR}\left(H^{\pm\pm}\rightarrow
W^{\pm}W^{\pm}\right)^{2}\times\operatorname{BR}\left(W^{\pm}\rightarrow\ell\nu\right)^{4}$
for the process $e^{+}e^{-}\rightarrow H^{0}A^{0}$ with mass of $A^{0}$ being
fixed as $M_{A^{0}}=400~{}\mathrm{GeV}$. Right panel: Event number of the
same-sign tetralepton signature $4\ell^{\pm}+X$ for the mass
$M_{A^{0}}=400~{}\mathrm{GeV}$ from $e^{+}e^{-}\to H^{0}A^{0}$ and subsequent
decays at the $\sqrt{s}=1~{}\mathrm{TeV}$ ILC with
$\mathcal{L}=8~{}\mathrm{ab}^{-1}$.

Now, let’s consider the same-sign tetralepton signature at the 3 TeV CLIC. In
this scenario, we set $M_{A^{0}}=1000$ GeV, and the same-sign dilepton decay
$H^{\pm}\to\ell^{\pm}\ell^{\pm}$ is still allowed. Therefore, in addition to
the tetraboson process in Eqn. (12), we also have the direct tetralepton
channel

$\displaystyle e^{+}e^{-}\to H^{0}A^{0}$ $\displaystyle\to$ $\displaystyle
H^{\pm}W^{*}H^{\pm}W^{*}\to H^{\pm\pm}W^{*}H^{\pm\pm}W^{*}+W^{*}W^{*}\to
4\ell^{\pm}+X.$ (14)

The corresponding cross section is then calculated as

$\displaystyle\sigma_{\ell}(4\ell^{\pm}+X)$ $\displaystyle=$
$\displaystyle\sigma\left(e^{+}e^{-}\rightarrow
H^{0}A^{0}\right)\times\left(\frac{2+x^{2}}{1+x^{2}}\frac{x^{2}}{1+x^{2}}\right)\times\text{BR}\left(H^{0}/A^{0}\rightarrow
H^{\pm}W^{*}\right)^{2}$ $\displaystyle\times\text{BR}\left(H^{\pm}\rightarrow
H^{\pm\pm}W^{*}\right)^{2}\times\text{BR}\left(H^{\pm\pm}\rightarrow\ell^{\pm}\ell^{\pm}\right)^{2}.$

In the left panel of Fig. 4, we show the product of BRs in the direct
tetralepton decay process. As shown in Fig. 1, the cascade decays are
suppressed for $v_{\Delta}\lesssim 10^{-5}$ GeV with $M_{A^{0}}=1000$ GeV, so
we do not show the region $v_{\Delta}<10^{-5}$ GeV. The right boundary
corresponds to the area where BR($H^{\pm\pm}\to\ell^{\pm}\ell^{\pm}$) is
suppressed. For $\lambda_{4}>0.5$, there are large parameter space where the
product of BRs reaches the maximum, i.e., 0.25. In the middle panel of Fig. 4,
the product of BRs in the diboson process is also shown. Comparing with the
region of $M_{A^{0}}=400$ GeV in Fig. 3, the region of $M_{A^{0}}=1000$ GeV is
much smaller. For instance, when product of BRs is larger than 0.002, one
needs $\lambda_{4}\gtrsim 0.5$ and $10^{-4}~{}\text{GeV}\lesssim
v_{\Delta}\lesssim 10^{-3}$ GeV. This is because for heavier scalar triplet,
the branching ratios of cascade decays are suppressed.

In the right panel of Fig. 4, we show the expected event number for the same-
sign tetralepton signature at the 3 TeV CLIC with an integrated luminosity of
$\mathcal{L}=5~{}\text{ab}^{-1}$. Here, the expected event number is the sum
of both diboson decay process in Eqn. (III) and the dilepton decay process in
Eqn. III. In a small area around $v_{\Delta}\sim 4\times 10^{-4}$ GeV and
$\lambda_{4}\sim 0.26$, we have the maximum number $\sim 16$, where the
dominant contribution is from $H^{\pm\pm}\to\ell^{\pm}\ell^{\pm}$. Meanwhile,
the $H^{\pm\pm}\to W^{\pm}W^{\pm}$ dominant tail region with
$10^{-4}~{}\text{GeV}\lesssim v_{\Delta}\lesssim 10^{-3}$ GeV only predicts a
total event number less than three, thus this long tail region is not
promising.

Figure 4: Left panel: Product of branching ratios
$\operatorname{BR}\left(H^{0}/A^{0}\rightarrow
H^{\pm}W^{-*}\right)^{2}\times\operatorname{BR}\left(H^{\pm}\rightarrow
H^{\pm\pm}W^{-*}\right)^{2}\times\operatorname{BR}\left(H^{\pm\pm}\rightarrow\ell^{\pm}\ell^{\pm}\right)^{2}$
for the process $e^{+}e^{-}\rightarrow H^{0}A^{0}$ with
$M_{A^{0}}=1000\mathrm{GeV}$. Middle panel: Product of branching ratios
$\operatorname{BR}\left(H^{0}/A^{0}\rightarrow
H^{\pm}W^{-*}\right)^{2}\times\operatorname{BR}\left(H^{\pm}\rightarrow
H^{\pm\pm}W^{-*}\right)^{2}\times\operatorname{BR}\left(H^{\pm\pm}\rightarrow
W^{\pm}W^{\pm}\right)^{2}\times\operatorname{BR}\left(W^{\pm}\rightarrow\ell\nu\right)^{4}$.
Right panel: Event number of the same-sign tetralepton signature
$4\ell^{\pm}+X$ for the mass $M_{H^{0}}\sim M_{A^{0}}=1000\mathrm{GeV}$ from
$e^{+}e^{-}\to H^{0}A^{0}$ and subsequent decays at the
$\sqrt{s}=3~{}\mathrm{TeV}$ CLIC with luminosity
$\mathcal{L}=5~{}\mathrm{ab}^{-1}$.

At last, we consider the same-sign tetralepton signature at the 6 TeV MuC. The
corresponding production processes at muon collider are

$\displaystyle\mu^{+}\mu^{-}\to H^{0}A^{0}$ $\displaystyle\to$ $\displaystyle
H^{\pm}W^{*}H^{\pm}W^{*}\to H^{\pm\pm}W^{*}H^{\pm\pm}W^{*}+W^{*}W^{*}\to
4W^{\pm}(\to\ell^{\pm}\nu)+X,$ (16) $\displaystyle\mu^{+}\mu^{-}\to
H^{0}A^{0}$ $\displaystyle\to$ $\displaystyle H^{\pm}W^{*}H^{\pm}W^{*}\to
H^{\pm\pm}W^{*}H^{\pm\pm}W^{*}+W^{*}W^{*}\to 4\ell^{\pm}+X.$ (17)

The production cross section is then obtained by simply replace
$\sigma(e^{+}e^{-}\to H^{0}A^{0})$ in Eqn. III and Eqn. III with
$\sigma(\mu^{+}\mu^{-}\to H^{0}A^{0})$. In the left and middle panel of Fig.
5, we show the product of BRs in the direct tetralepton and tetraboson decay
process with $M_{A^{0}}=1500$ GeV. To realize a relatively large value of BRs,
$\lambda_{4}$ has to be larger than $0.8$. However, such large $\lambda_{4}$
leads to too large mass splitting of triplet scalars that the interference
factor $x$ is suppressed. In the right panel of Fig. 5, we show the total
event number for the same-sign tetralepton signature at the 6 TeV MuC with an
integrated luminosity of $\mathcal{L}=10~{}\text{ab}^{-1}$. It is obvious that
the event number is always smaller than three. Therefore, the same-sign
tetralepton signature is not promising at the MuC for $M_{A^{0}}=1500$ GeV.

Figure 5: Same as Fig. 4, but for $M_{H^{0}}\sim M_{A^{0}}=1500\mathrm{GeV}$
from the process $\mu^{+}\mu^{-}\to H^{0}A^{0}$ at the
$\sqrt{s}=6~{}\mathrm{TeV}$ MuC with luminosity
$\mathcal{L}=10~{}\mathrm{ab}^{-1}$.

##  IV Conclusion

In this paper, we study the novel same-sign tetra-lepton signature in type-II
seesaw at the future lepton colliders (including 1 TeV ILC, 3 TeV CLIC, and 6
TeV MuC). The signature arises from the mixing of associated production of
Higgs fields $H^{0}A^{0}$ followed by the cascade decays
$H^{0}/A^{0}\rightarrow H^{\pm}W^{*}$, $H^{\pm}\rightarrow H^{\pm\pm}W^{*}$,
and $H^{\pm\pm}\to\ell^{\pm}\ell^{\pm}/W^{\pm}W^{\pm}$ with
$W^{\pm}\to\ell^{\pm}\nu$. There are two important parameters $\lambda_{4}$
and $v_{\Delta}$ closely related to this signature, where $\lambda_{4}$
controls the mass splitting of triplet scalars and $v_{\Delta}$ determines the
decay mode of $H^{\pm\pm}$.

We first consider a low mass benchmark scenario with $M_{A^{0}}=400$ GeV at 1
TeV ILC. In this scenario, $H^{\pm\pm}\to W^{\pm}W^{\pm}$ is the only viable
decay mode. The production cross section of the process $e^{+}e^{-}\to
H^{0}A^{0}$ varies around 10 fb. The promising region corresponds to a narrow
band in the range of $10^{-4}~{}\text{GeV}\lesssim v_{\Delta}\lesssim 10^{-2}$
GeV. With an integrated luminosity of $\mathcal{L}=8~{}\mathrm{ab}^{-1}$, we
find that a neutral Higgs of mass around $400~{}\mathrm{GeV}$ can lead to
around $150$ events at ILC. For heavier triplet scalars, we then consider
$M_{A^{0}}=1000$ GeV at 3 TeV CLIC, where the cross section
$\sigma(e^{+}e^{-}\to H^{0}A^{0})$ is about 2 fb. Although this value is about
two orders of magnitudes larger than at 14 TeV LHC, the cascade decay
branching ratios are suppressed for small $\lambda_{4}$. This leads to a miss
match between cascade decays and the interference effect. A maximum event
number $\sim 16$ can be obtained around $v_{\Delta}\sim 4\times 10^{-4}$ GeV
and $\lambda_{4}\sim 0.26$ with an integrated luminosity of
$\mathcal{L}=5~{}\mathrm{ab}^{-1}$ at CLIC. In this high mass scenario, the
$H^{\pm\pm}\to\ell^{\pm}\ell^{\pm}$ decay mode is the dominant contribution to
the same-sign tetralepton signature. If the triplet scalars are even heavier
than 1 TeV, e.g., $M_{A^{0}}=1500$ GeV, the cascade decays will be heavily
suppressed. With an integrated luminosity of
$\mathcal{L}=10~{}\mathrm{ab}^{-1}$ at 6 TeV MuC, there are at best have 3
signal events. Therefore, this signature is not promising at MuC.

## Acknowledgments

This work is supported by the National Natural Science Foundation of China
under Grant No. 11605074 and 11805081, Natural Science Foundation of Shandong
Province under Grant No. ZR2019QA021 and ZR2018MA047, Joint Large-Scale
Scientific Facility Funds of the NSFC and CAS under Contracts Nos. U1732263
and U2032115.

## References

  * (1) Y. Fukuda et al. [Super-Kamiokande], Phys. Rev. Lett. 81, 1562-1567 (1998) [arXiv:hep-ex/9807003 [hep-ex]].
  * (2) Q. R. Ahmad et al. [SNO], Phys. Rev. Lett. 89, 011301 (2002) [arXiv:nucl-ex/0204008 [nucl-ex]].
  * (3) F. P. An et al. [Daya Bay], Phys. Rev. Lett. 108, 171803 (2012) [arXiv:1203.1669 [hep-ex]].
  * (4) S. Weinberg, Phys. Rev. Lett. 43, 1566-1570 (1979)
  * (5) E. Ma, Phys. Rev. Lett. 81, 1171-1174 (1998) [arXiv:hep-ph/9805219 [hep-ph]].
  * (6) P. Minkowski, Phys. Lett. B 67, 421-428 (1977)
  * (7) R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44, 912 (1980)
  * (8) M. Magg and C. Wetterich, Phys. Lett. B 94, 61-64 (1980)
  * (9) T. P. Cheng and L. F. Li, Phys. Rev. D 22, 2860 (1980)
  * (10) G. Lazarides, Q. Shafi and C. Wetterich, Nucl. Phys. B 181, 287-300 (1981)
  * (11) R. N. Mohapatra and G. Senjanovic, Phys. Rev. D 23, 165 (1981)
  * (12) R. Foot, H. Lew, X. G. He and G. C. Joshi, Z. Phys. C 44, 441 (1989)
  * (13) F. del Aguila and J. A. Aguilar-Saavedra, Nucl. Phys. B 813, 22-90 (2009) [arXiv:0808.2468 [hep-ph]].
  * (14) F. F. Deppisch, P. S. Bhupal Dev and A. Pilaftsis, New J. Phys. 17, no.7, 075019 (2015) [arXiv:1502.06541 [hep-ph]].
  * (15) Y. Cai, T. Han, T. Li and R. Ruiz, Front. in Phys. 6, 40 (2018) [arXiv:1711.02180 [hep-ph]].
  * (16) S. M. Boucenna, S. Morisi and J. W. F. Valle, Adv. High Energy Phys. 2014, 831598 (2014) [arXiv:1404.3751 [hep-ph]].
  * (17) Y. Cai, J. Herrero-García, M. A. Schmidt, A. Vicente and R. R. Volkas, Front. in Phys. 5, 63 (2017) [arXiv:1706.08524 [hep-ph]].
  * (18) A. Arhrib, R. Benbrik, M. Chabab, G. Moultaka, M. C. Peyranere, L. Rahili and J. Ramadan, Phys. Rev. D 84, 095005 (2011) [arXiv:1105.1925 [hep-ph]].
  * (19) P. Fileviez Perez, T. Han, G. y. Huang, T. Li and K. Wang, Phys. Rev. D 78, 015018 (2008) [arXiv:0805.3536 [hep-ph]].
  * (20) A. Melfo, M. Nemevsek, F. Nesti, G. Senjanovic and Y. Zhang, Phys. Rev. D 85, 055018 (2012) [arXiv:1108.4416 [hep-ph]].
  * (21) M. Aoki, S. Kanemura and K. Yagyu, Phys. Rev. D 85, 055007 (2012) [arXiv:1110.4625 [hep-ph]].
  * (22) Z. L. Han, R. Ding and Y. Liao, Phys. Rev. D 91, 093006 (2015) [arXiv:1502.05242 [hep-ph]].
  * (23) Z. L. Han, R. Ding and Y. Liao, Phys. Rev. D 92, no.3, 033014 (2015) [arXiv:1506.08996 [hep-ph]].
  * (24) A. G. Akeroyd and H. Sugiyama, Phys. Rev. D 84, 035010 (2011) [arXiv:1105.2209 [hep-ph]].
  * (25) A. G. Akeroyd, S. Moretti and H. Sugiyama, Phys. Rev. D 85, 055026 (2012) [arXiv:1201.5047 [hep-ph]].
  * (26) E. J. Chun and P. Sharma, Phys. Lett. B 728, 256-261 (2014) [arXiv:1309.6888 [hep-ph]].
  * (27) K. S. Babu and S. Jana, Phys. Rev. D 95, no.5, 055020 (2017) [arXiv:1612.09224 [hep-ph]].
  * (28) T. Li, JHEP 09, 079 (2018) [arXiv:1802.00945 [hep-ph]].
  * (29) R. Primulando, J. Julio and P. Uttayarat, JHEP 08, 024 (2019) [arXiv:1903.02493 [hep-ph]].
  * (30) Y. Du, A. Dunbrack, M. J. Ramsey-Musolf and J. H. Yu, JHEP 01, 101 (2019) [arXiv:1810.09450 [hep-ph]].
  * (31) T. B. de Melo, F. S. Queiroz and Y. Villamizar, Int. J. Mod. Phys. A 34, no.27, 1950157 (2019) [arXiv:1909.07429 [hep-ph]].
  * (32) R. Padhan, D. Das, M. Mitra and A. Kumar Nayak, Phys. Rev. D 101, no.7, 075050 (2020) [arXiv:1909.10495 [hep-ph]].
  * (33) B. Fuks, M. Nemevšek and R. Ruiz, Phys. Rev. D 101, no.7, 075022 (2020) [arXiv:1912.08975 [hep-ph]].
  * (34) S. Blunier, G. Cottin, M. A. Díaz and B. Koch, Phys. Rev. D 95, no.7, 075038 (2017) [arXiv:1611.07896 [hep-ph]].
  * (35) P. Agrawal, M. Mitra, S. Niyogi, S. Shil and M. Spannowsky, Phys. Rev. D 98, no.1, 015024 (2018) [arXiv:1803.00677 [hep-ph]].
  * (36) P. S. B. Dev, S. Khan, M. Mitra and S. K. Rai, Phys. Rev. D 99, no.11, 115015 (2019) [arXiv:1903.01431 [hep-ph]].
  * (37) X. H. Yang and Z. J. Yang, [arXiv:2103.11412 [hep-ph]].
  * (38) M. Aaboud et al. [ATLAS], Eur. Phys. J. C 78, no.3, 199 (2018) [arXiv:1710.09748 [hep-ex]].
  * (39) A. G. Akeroyd, M. Aoki and H. Sugiyama, Phys. Rev. D 77, 075010 (2008) [arXiv:0712.4019 [hep-ph]].
  * (40) M. Aaboud et al. [ATLAS], Eur. Phys. J. C 79, no.1, 58 (2019) [arXiv:1808.01899 [hep-ex]].
  * (41) G. Aad et al. [ATLAS], [arXiv:2101.11961 [hep-ex]].
  * (42) E. J. Chun and P. Sharma, JHEP 08, 162 (2012) [arXiv:1206.6278 [hep-ph]].
  * (43) E. J. Chun, S. Khan, S. Mandal, M. Mitra and S. Shil, Phys. Rev. D 101, no.7, 075008 (2020) [arXiv:1911.00971 [hep-ph]].
  * (44) J. B. Guimarães da Costa et al. [CEPC Study Group], [arXiv:1811.10545 [hep-ex]].
  * (45) T. Barklow, J. Brau, K. Fujii, J. Gao, J. List, N. Walker and K. Yokoya, [arXiv:1506.07830 [hep-ex]].
  * (46) K. Fujii, C. Grojean, M. E. Peskin, T. Barklow, Y. Gao, S. Kanemura, H. Kim, J. List, M. Nojiri and M. Perelstein, et al. [arXiv:1710.07621 [hep-ex]].
  * (47) L. Linssen, A. Miyamoto, M. Stanitzki and H. Weerts, [arXiv:1202.5940 [physics.ins-det]].
  * (48) A. Robson and P. Roloff, [arXiv:1812.01644 [hep-ex]].
  * (49) J. P. Delahaye, M. Diemoz, K. Long, B. Mansoulié, N. Pastrone, L. Rivkin, D. Schulte, A. Skrinsky and A. Wulzer, [arXiv:1901.06150 [physics.acc-ph]].
  * (50) K. Long, D. Lucchesi, M. Palmer, N. Pastrone, D. Schulte and V. Shiltsev, [arXiv:2007.15684 [physics.acc-ph]].
  * (51) J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao, T. Stelzer, P. Torrielli and M. Zaro, JHEP 07, 079 (2014) [arXiv:1405.0301 [hep-ph]].
  * (52) J. de Favereau et al. [DELPHES 3], JHEP 02, 057 (2014) [arXiv:1307.6346 [hep-ex]].