Investigating New Physics Models with
Signature of Same-Sign Diboson$+\not{E}_{T}$

Cheng-Wei Chiang1,2*** E-mail<EMAIL_ADDRESS>, Sudip Jana3†††
E-mail<EMAIL_ADDRESS>Dibyashree Sengupta1‡‡‡ E-mail:
<EMAIL_ADDRESS>
1Department of Physics, National Taiwan University, Taipei, Taiwan 10617,
R.O.C.
2Physics Division, National Center for Theoretical Sciences, Taipei, Taiwan
10617, R.O.C.
3Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg,
Germany

###### Abstract

We investigate the prospect of searching for new physics via the novel
signature of same-sign diboson + $\not{E}_{T}$ at current and future LHC. We
study three new physics models: (i) natural SUSY models, (ii) type-III seesaw
model and (iii) type-II seesaw/Georgi-Machacek model. In the first two class
of models, this signature arises due to the presence of a singly-charged
particle which has lifetime long enough to escape detection, while in the
third model this signature originates resonantly from a doubly-charged
particle produced along with two forward jets that, most likely, would escape
detection. We analyze in great detail the discovery prospects of the signal in
these three classes of models in the current as well as the upcoming runs of
the LHC (such as HL-LHC, HE-LHC and FCC-hh) by showing a distinction among
these scenarios.

##  1 Introduction

In the past few decades, there have been several major discoveries in particle
physics, culminating in the observation of the Higgs boson in 2012 [1, 2].
Despite this tremendous success of the Standard Model (SM), it is incomplete
in its current form. There is strong theoretical as well as experimental
evidence (such as the hierarchical pattern seen in the fermion masses and
mixings, the origin of neutrino masses, an understanding of dark matter, and
the origin of the matter-antimatter asymmetry in the Universe) which calls for
new physics beyond the Standard Model (BSM).

At the LHC, several searches have been performed to look for clues of these
BSM models. However, we have not seen any clear new physics signals so far. In
this work, we investigate the novel signal of same-sign diboson (SSdB) +
$\not{E}_{T}$ which has been less studied and deserves more attention. This
signal is of interest because it has negligibly small background in the SM.
Hence, an observation of this signal will give a clear sign of BSM physics.
After a careful study, we find that it is possible to observe such a unique
signature in three well-motivated BSM scenarios, namely: (i) natural
supersymmetry models [3, 4, 5, 6, 7, 8], (ii) type-III seesaw model [9], and
(iii) type-II seesaw [10, 11, 12, 13]/Georgi-Machacek model [14] , while still
being consistent with the existing theoretical and experimental limits.

Being a well-motivated BSM framework, supersymmetry (SUSY) provides an elegant
solution to the Higgs mass hierarchy problem, accommodates a valid cold dark
matter candidate, explains electroweak symmetry breaking, and features gauge
coupling unification [15]. Although LHC searches for SUSY particles have
pushed the masses of squarks and gluino high enough to expose weak scale SUSY
to the risk of being unnatural/highly fine-tuned, there exist a class of SUSY
models which can be natural as well as accommodate such highly massive
sparticles well beyond the reach of the current LHC [16]. Since experiments do
not put such high mass bounds on the masses of wino, bino, higgsino (and also
the singlino which appears in some extended SUSY models [17, 18, 19, 20, 21,
22, 23]), these natural SUSY models can have a wino-like or a bino-like
lightest supersymmetric particle (LSP), provided gaugino mass unification is
not considered [24]. However, natural SUSY models can have a higgsino-like LSP
irrespective of whether it assumes gaugino mass unification or not [24]. Ref.
[25] and Ref. [26] study SUSY models with bino-like and singlino-like LSP,
respectively, the latter being an extension of the minimal supersymmetric
standard model (MSSM). Here, we consider a specific class of natural SUSY
models that have a higgsino-like LSPs which, under R-parity conservation,
cannot decay to lighter SM particles, and hence can give rise to the novel
SSdB + $\not{E}_{T}$ signature via the generic process shown in Fig. 1.
Earlier analyses have been done in this regard in Refs. [27, 28]. Our current
SUSY analysis differs from these earlier analyses in several aspects to be
discussed in detail in Sec. 2.1.

Figure 1: A generic Feynman diagram for SSdB + $\not{E}_{T}$ production at the
LHC in BSM models, where $B^{0}$, $A^{\pm}$, $X^{0}$ and $Y^{\pm}$ are new
particles.

We consider another interesting theoretical framework, type-III seesaw model,
which has been proposed [9] to explain the tiny neutrino masses and mixings.
In the type-III seesaw model, the SM particle spectrum is extended by three
generations of $SU(2)_{L}$ triplet fermions with hypercharge $Y=0$, the
lightest of which has a lifetime long enough to escape detection [29],
provided they have mass around a few hundred GeV. Hence, this model can also
give rise to the novel signature of SSdB + $\not{E}_{T}$ via the generic
process shown in Fig. 1.

Another framework that can generate Majorana neutrino mass at tree level is
type-II seesaw model [10, 11, 12, 13]. In addition to the SM particles, the
model is extended by at least one $SU(2)_{L}$ triplet scalar $\Delta$ with
hypercharge $Y=1$. Another model, called Georgi-Machacek (GM) model [14],
further has a real $SU(2)_{L}$ triplet scalar. The doubly-charged scalar from
the complex $SU(2)_{L}$ triplet scalar can be produced via vector boson fusion
(VBF) process and decays into two $W$ bosons with same electric charge along
with two forward jets coming from the initial state. The forward jets may not
be caught by the detector and hence the resultant final state will mimic our
signature of interest. However, later in Sec. 2.3, we will show that due to a
stringent $T$-parameter constraint, the type-II seesaw model cannot give a
sizeable cross section for this signature, whereas the GM model can.

There are several other channels that have potential to lead to the discovery
of the three BSM scenarios considered here. Note that this work does not
intend to make a comparative study between the SSdB+$\not{E}_{T}$ channel of
interest here and the other more promising discovery channels. The gist of
this paper is to point out possible BSM models that can be a potential source
of such a novel signature, if seen in experiments, since it is very unlikely
for this signature process to appear within the SM. Since more than one BSM
scenario qualify, thus a need to distinguish among them is called for and such
a distinction can be accomplished by the use of different sets of cuts.

In this article, we analyze the prospects of discriminating the above-
mentioned BSM models through the SSdB + $\not{E}_{T}$ signature at the LHC. We
find that the HL-LHC ($\sqrt{s}=14$ TeV and an integrated luminosity of 3
ab-1) is insufficient to probe any of these three models through the signature
process. The natural SUSY models can be probed at the HE-LHC ($\sqrt{s}=27$
TeV) with an integrated luminosity of 3 ab-1 while the type-II seesaw/GM model
and the type-III seesaw model can be probed at the HE-LHC with an integrated
luminosity of 15 ab-1. For completeness, we also extend our analysis to the
Future Circular Collider of hadrons (FCC-hh with $\sqrt{s}=100$ TeV) with an
integrated luminosity of 3 ab-1 and 15 ab-1. The rest of the paper is
organized as follows. In Sec. 2, we review the three BSM models and how they
give rise to the SSdB + $\not{E}_{T}$ signature. In Sec. 3, the signals from
all the three BSM models are optimized against the SM background. We show how
each BSM model stands out for a particular set of cuts and discuss the
discovery prospect for each scenario. Finally, we conclude in Sec. 4.

##  2 SSdB + $\not{E}_{T}$ Signature from BSM Models

In this section, we briefly review the three classes of new physics models
considered in this work, and how each of them leads to the SSdB +
$\not{E}_{T}$ signature at the LHC.

###  2.1 Supersymmetry

Although weak scale SUSY is a well-motivated BSM framework, experimental
searches for sparticles have pushed the masses for squarks and gluino above 1
TeV. For example, current LHC data indicate that $m_{\tilde{g}}$ > 2.2 TeV
[30, 31] and $m_{\tilde{t}_{1}}$ > 1.2 TeV [31, 32, 33]. Such large lower
bounds on the masses of sparticles question the naturalness of weak scale SUSY
[34]. According to older notions of naturalness, SUSY models with such heavy
sparticles are highly fine-tuned or unnatural [35, 36, 37]. However, these
earlier notions of naturalness can be updated to a more conservative
electroweak naturalness measure, denoted by $\Delta_{EW}$ [38, 39, 40]. A
numerical expression for $\Delta_{EW}$ is obtained from minimizing the MSSM
scalar potential that equates the $Z$ boson mass to weak scale SUSY parameters
as

$m_{Z}^{2}/2=\frac{m_{H_{d}}^{2}+\Sigma_{d}^{d}-(m_{H_{u}}^{2}+\Sigma_{u}^{u})\tan^{2}\beta}{\tan^{2}\beta-1}-\mu^{2}\\\
\simeq-m_{H_{u}}^{2}-\mu^{2}-\Sigma_{u}^{u}(\tilde{t}_{1,2})~{},$ (2.1)

where $\mu$ is the superpotential higgsino mass parameter, $m_{H_{u}}^{2}$ and
$m_{H_{d}}^{2}$ are the soft SUSY breaking up-type and down-type Higgs mass
parameters, respectively, $\tan\beta$ is the ratio of up-type Higgs vacuum
expectation value (VEV) to the down-type Higgs VEV, and $\Sigma_{u}^{u}$ and
$\Sigma_{d}^{d}$ denote radiative corrections as given in the Appendix of Ref.
[41]).

The electroweak naturalness measure, denoted by $\Delta_{EW}$, is defined as

$\Delta_{EW}=|(\mbox{max RHS contribution in
Eq.~{}\eqref{eq:mzs}})|/(m_{Z}^{2}/2)$ (2.2)

It is suggested that a conservative choice for natural SUSY models is
$\Delta_{EW}<30$. Therefore, every point in the parameter space of a SUSY
model that yields $\Delta_{EW}$ < 30 is considered to be natural. As can be
derived from Eq. (2.1) and Eq. (2.2), $\Delta_{EW}$ < 30 demands:

  * •

$\mu$ $\sim$ $100-300\rm~{}GeV$. 111The space of $\mu$ ¡ 100 GeV has been
ruled out by LEP2 experiment [42].

  * •

$m_{H_{u}}^{2}$ should acquire a small negative value $\sim$ $-(100-300)^{2}$
$\rm~{}GeV^{2}$ at the weak scale. This occurs when $m_{H_{u}}^{2}$ is driven
radiatively from high-energy scales to the weak scale augmented by a
combination of statistical and anthropic pull such that electroweak symmetry
is only barely broken [43].

  * •

$\Sigma_{u}^{u}$ should also be below $(300)^{2}$ $\rm~{}GeV^{2}$. This is
attainable with $m_{\tilde{t}_{1}}$ > 1.2 TeV and $m_{\tilde{g}}$ > 2.2 TeV.

Ref. [16] shows several natural SUSY models that satisfy all the above
criteria, with huge parameter space still left to be probed experimentally. As
seen from the above conditions, natural SUSY models, assuming gaugino mass
unification, have a unique property that $\mu$ $\ll$ $M_{1,2}$ $<$ $M_{3}$
where $M_{1}$, $M_{2}$ and $M_{3}$ refer to the masses of bino, wino and
gluino, respectively, at the weak scale. Thus, in these natural SUSY models,
the LSP is almost purely higgsino-like. Under assumed R-parity conservation,
the LSP becomes a good dark matter candidate in the model and manifests as
$\not{E}_{T}$ in collider experiments. However, such a higgsino-like LSP of
mass $100-300$ GeV are thermally under-produced. Hence, such an LSP only
partially contributes to the total dark matter content of the Universe. In
order to account for the entire dark matter content of the Universe, either
other particles suitable to form dark matter must be present in the model or
the rest of the dark matter must be non-thermally produced. It has been shown
in Ref. [44] that the latter case is excluded by experiments. Therefore, the
rest of the dark matter must be formed by some other particles. Axion, which
arises in a completely different context of solving the strong CP problem via
the Peccei-Quinn solution [45, 46, 47, 48, 49, 50, 51, 52], turns out to be an
excellent candidate for serving this purpose. Out of various natural SUSY
models listed in Ref. [16], we choose the two extra parameter non-universal
Higgs (NUHM2) model [3, 4] with $\mu\ll M_{1}<M_{2}<M_{3}$ which can give rise
to a clean SSdB + $\not{E}_{T}$ signature via wino-pair production, as pointed
out in Ref. [27, 28]. The corresponding Feynman diagram is shown in Fig. 2.

Figure 2: Feynman diagram for SSdB production at the LHC in SUSY models with
light higgsinos ($\tilde{W}_{1}^{\mp}$ and $\tilde{Z}_{i}$ with $i=1,2$). Here
$\tilde{Z}_{4}$ and $\tilde{W}_{2}^{\pm}$ in the intermediate step are winos.

In this work, we have analyzed this signal in detail using the most up-to-date
constraints on $m_{\tilde{g}}$ and $\sigma^{SI}(\tilde{z}_{1},p)$ obtained
from the LHC data with an integrated luminosity ($\mathcal{L}$) of 139 fb-1
[30, 31] and the XENON1T experiment [53], respectively. The relevant benchmark
point is given in Table I.

parameter | NUHM2
---|---
$m_{0}$ | 5000 GeV
$A_{0}$ |  $-8000$ GeV
$\tan\beta$ | 12
$M_{1}({\rm GUT})$ | 1250 GeV
$M_{2}({\rm GUT})$ | 895 GeV
$M_{3}({\rm GUT})$ | 1250 GeV
$\mu$ | 150 GeV
$m_{A}$ | 2500 GeV
$m_{\tilde{g}}$ | 2938.2 GeV
$m_{\tilde{u}_{L}}$ | 5458.3 GeV
$m_{\tilde{u}_{R}}$ | 5591.1 GeV
$m_{\tilde{e}_{R}}$ | 4840.1 GeV
$m_{\tilde{t}_{1}}$ | 1820.5 GeV
$m_{\tilde{t}_{2}}$ | 3925.7 GeV
$m_{\tilde{b}_{1}}$ | 3959.9 GeV
$m_{\tilde{b}_{2}}$ | 5301.7 GeV

parameter | NUHM2
---|---
$m_{\tilde{\tau}_{1}}$ | 4728.9 GeV
$m_{\tilde{\tau}_{2}}$ | 5061.8 GeV
$m_{\tilde{\nu}_{\tau}}$ | 5067.2 GeV
$m_{\tilde{w}_{1}}$ | 156.6 GeV
$m_{\tilde{w}_{2}}$ | 762.9 GeV
$m_{\tilde{z}_{1}}$ | 146.0 GeV
$m_{\tilde{z}_{2}}$ | 157.9 GeV
$m_{\tilde{z}_{3}}$ | 559.8 GeV
$m_{\tilde{z}_{4}}$ | 775.4 GeV
$m_{h}$ | 125.1 GeV
$\Omega_{\tilde{z}_{1}}^{\rm std}h^{2}$ | 0.007
$BF(b\to s\gamma)\times 10^{4}$ | $3.06$
$BF(B_{s}\to\mu^{+}\mu^{-})\times 10^{9}$ | $3.8$
$\sigma^{SI}(\tilde{z}_{1},p)$ (pb) | $2.08\times 10^{-9}$
$\sigma^{SD}(\tilde{z}_{1},p)$ (pb) | $8.4\times 10^{-5}$
$\langle\sigma v\rangle|_{v\to 0}$ (cm3/sec) | $2.99\times 10^{-25}$
$\Delta_{EW}$ | 29.6

Table I: Input parameters and masses for a SUSY benchmark point from the NUHM2
model with $m_{t}=173.2$ GeV using Isajet 7.88 [54].

Here we have assumed a more general scenario without gaugino mass unification
[24]. This also has the advantage of having the wino mass $\sim 770$ GeV while
satisfying the LHC constraints on gluino mass [30, 31]. Note that the above
choice of wino mass is for a comparison of this signal with a similar one
obtained from the type-III seesaw model, as discussed in the next subsection.
Also note that even though gaugino mass unification is not assumed, the
benchmark point given in Table I satisfies $\mu\ll M_{1}<M_{2}<M_{3}$ which is
an essential criteria for obtaining the signal of our interest via the process
shown in Fig. 2.

###  2.2 The type-III Seesaw Model

In the type-III seesaw model, the SM particle spectrum is extended by multiple
$SU(2)_{L}$ triplet fermions ($\Sigma$s) which have hypercharge $Y=0$. In
order to generate tiny neutrino mass and proper flavor structure in the
neutrino sector, one needs to introduce at least two generations of
$SU(2)_{L}$ triplet fermions. The tiny neutrino masses are generated at the
tree level and can be expressed as $m_{\nu}\simeq
Y^{2}_{\nu}v^{2}/M_{\Sigma}$, where $Y_{\nu}$ is the Yukawa coupling, $v$ is
the SM Higgs VEV, and $M_{\Sigma}$ is the triplet fermion mass [9]. In
general, the type-III seesaw scenario with $M_{\Sigma}\simeq{\cal O}(1)$ TeV
is technically natural and opens up a plethora of implications in collider
experiments [55, 56, 29, 57, 58, 59, 60, 61, 62, 63].

In our analysis, we consider three generations of $SU(2)_{L}$ triplet
fermions, $\Sigma_{i}$ ($i=1,2,3$), with a non-degenerate mass spectrum. The
relevant Lagrangian is given by

$\mathcal{L}_{\Sigma}={\rm
Tr}\left[{\overline{\Sigma}}_{i}{\not{D}}\Sigma_{i}\right]-\left(\frac{1}{2}M_{\Sigma}^{ij}{\rm
Tr}\left[\overline{\Sigma^{c}}_{i}\Sigma_{j}\right]+{\rm
h.c}\right)-\left({\sqrt{2}}Y_{\Sigma}^{ij}{\overline{L}}_{i}\Sigma_{j}{H}+{\rm
h.c}\right)~{},$ (2.3)

where $D_{\mu}$ is the covariant derivative for $\Sigma_{i}$, $M_{\Sigma}$
denotes the triplet fermions mass matrix, and $Y_{\Sigma}$ is the Yukawa
coupling matrix. For the rest of our paper, we refer to the lightest
generation of heavy fermions as $\tilde{\Sigma}$ and their masses as
$m_{\tilde{\Sigma}}$. Depending on a normal or inverted hierarchy, the
lightest fermion triplet will be $\Sigma_{1}$ or $\Sigma_{3}$, respectively.
For simplicity, we set the other two generations of heavy fermions to be
almost degenerate.

Figure 3: Feynman diagram for the SSdB + $\not{E}_{T}$ signature at the LHC in the type-III seesaw model, where $\tilde{\Sigma}^{0}$ and $\tilde{\Sigma}^{\pm}$ are members of the lightest fermionic triplets.  parameter | Type III Seesaw
---|---
$m(\Sigma_{i}^{0})$ | 770.39 GeV
$m(\Sigma_{i}^{\pm})$ | 770.57 GeV
$m(\tilde{\Sigma}^{0})$ | 670.39 GeV
$m(\tilde{\Sigma}^{\pm})$ | 670.57 GeV
$c\tau(\Sigma_{i}^{0})$ |  $5.34\times 10^{-13}$ cm
$c\tau(\Sigma_{i}^{\pm})$ |  $1.034\times 10^{-12}$ cm
$c\tau(\tilde{\Sigma}^{0})$ |  $1.2\times 10^{5}$ cm
$c\tau(\tilde{\Sigma}^{\pm})$ | 44.37 cm
Table II: Masses and decay widths for a Type III Seesaw benchmark point.

As can be inferred from a detailed calculation of partial decay widths of
these $SU(2)_{L}$ triplet fermions in Ref. [29], depending on the neutrino
parameters, the electically neutral member of the lightest generation of
fermionic triplets $\tilde{\Sigma}^{0}$ of mass around a few hundred GeV can
have lifetime long enough to escape detection and hence shows up as large
$\not{E}_{T}$ in collider experiments. $\tilde{\Sigma}^{\pm}$, being only a
few MeV heavier than its neutral partner $\tilde{\Sigma}^{0}$, travels a short
distance before primarily decaying into $\tilde{\Sigma}^{0}$ and a charged
pion of momentum low enough to be reconstructed as a track. This results in a
disappearing track signature from $\tilde{\Sigma}^{\pm}$ as can also be seen
in Ref. [29]. There are several dedicated searches for the disappearing track
signature at the LHC [64]. We recast a recent LHC limit in Ref. [64] to derive
a bound on the charged heavy fermion in the type-III seesaw model. We find the
lower bound on mass of $\tilde{\Sigma}^{\pm}$ to be around 670 GeV in order to
be consistent with the collider data. In our analysis, we set the other two
pairs of heavy fermions to have mass at 770 GeV, so that they primarily decay
to a $W^{\pm}$ boson and a $\tilde{\Sigma}^{\pm,0}$ particle through a tiny
mixing. This leads to a clean SSdB + $\not{E}_{T}$ signature from pair
production of $\Sigma_{i}^{\pm,0}$ at the LHC via the Feynman diagram shown in
Fig. 3. This particular benchmark point is shown in Table. II.

###  2.3 Type-II seesaw/Georgi-Machacek model

In this subsection, we focus on the scenario where the SSdB signature
originates from the decay of a doubly-charged scalar. Generally, these doubly-
charged scalars appear in several BSM frameworks [10, 11, 13, 65, 66, 67, 68,
69, 70, 71, 72, 73, 74, 75, 76, 14, 77, 78, 79, 80, 81]. One such framework is
the simplest type-II seesaw model [10, 11, 12, 13] which introduces an
$SU(2)_{L}$ triplet scalar $\Delta=(\Delta^{++},\Delta^{+},\Delta^{0})$ with
hypercharge $Y=1$. Tiny neutrino masses are generated while the neutral
component of the $SU(2)_{L}$ triplet, $\Delta^{0}$, acquires a small VEV,
$v_{\Delta}$. This type of $SU(2)_{L}$ triplet scalar which contains a doubly-
charged scalar $\Delta^{++}$ also appears in Minimal Left-Right Symmetric
Model [66, 67, 68] as well as GM model [14].

At the LHC, these doubly-charged scalars ($\Delta^{\pm\pm}$) can be pair-
produced via the Drell-Yan process (s-channel $Z/\gamma$ exchanges). There are
several extensive phenomenological studies as well as experimental studies on
pair-production of doubly-charged scalars which also look at the
$W^{+}W^{+}W^{-}W^{-}/\ell^{+}\ell^{+}\ell^{-}\ell^{-}$ final state signatures
(for a review, see Ref. [82]). However, we are focusing on the resonant
production of the doubly-charged scalar through the VBF process here, as shown
in Fig. 4. This production rate is proportional to $v_{\Delta}^{2}$ and
becomes more dominant than the Drell-Yan process for $v_{\Delta}\sim{\cal
O}(10)$ GeV and $m_{\Delta^{\pm\pm}}\sim{\cal O}(100)$ GeV [83, 84]. Note that
$v_{\Delta}$ in the simplest type-II seesaw model [10, 11, 12, 13] is tightly
bounded by the electroweak $T$ parameter, giving $v_{\Delta}\lesssim 3$ GeV
[85]. In the GM model that also contains $SU(2)_{L}$ scalar triplet fields,
$\xi$ with hypercharge $Y=0$ and $\chi$ with hypercharge $Y=1$, to preserve
the custodial symmetry at tree level, $v_{\Delta}$ can be as high as $\sim 50$
GeV [86, 87]. As a consequence, the resonant production rate could be much
larger than in the simplest type-II seesaw scenario.

Figure 4: Feynman diagram for SSdB + forward jets production at LHC in the type-II seesaw models. parameter | Type II Seesaw
---|---
$m(\Delta^{\pm\pm})$ | 300.0 GeV
$m(\Delta^{\pm})$ | 302.0 GeV
$\Gamma(\Delta^{\pm\pm})$ |  $3.83\times 10^{-4}$ GeV
$v_{\Delta}$ | 1 GeV
Table III: Masses and decay widths for a Type II Seesaw benchmark point.

In general the doubly-charged scalar can be either lightest or heaviest
depending on the sign of the quartic coupling in the potential. In the rest of
our analysis, we consider the scenario where $\Delta^{++}$ is the lightest
among all members in the triplet fields. In this scenario, $\Delta^{++}$
dominantly decays into same-sign dilepton (SSdL)
($\Delta^{\pm\pm}\to\ell^{\pm}\ell^{\pm}$) or SSdB ($\Delta^{\pm\pm}\to
W^{\pm}W^{\pm}$), depending on the value of $v_{\Delta}$ [88, 89, 90]. In Fig.
5, we show a complete decay phase diagram of the doubly-charged scalar of mass
300 GeV. As shown in the plot, $\Delta^{\pm\pm}$ dominantly decays to two
same-sign $W$ bosons for $v_{\Delta}$ $\gtrsim$ 1 MeV, provided the mass
splitting $\Delta m\equiv m_{\Delta^{\pm\pm}}-m_{\Delta^{\pm}}$ $\lesssim$ 5
GeV. For our analysis, we set the mass splitting $\Delta m$ = 2 GeV and
$v_{\Delta}\sim$ 1 GeV so that this benchmark point (shown in Table III) lies
in the blue shaded region of Fig. 5 that is of our interest. Thus, after being
produced at the LHC along with two forward jets, $\Delta^{\pm\pm}$ decays
primarily to two same-sign $W$ bosons. These jets may escape detection,
especially in the forward region with lower detector efficiency. Assuming
leptonic decay of the $W$ bosons, we obtain SSdL + $\not{E}_{T}$ in the final
state. In this case, the final state mimics the signature of our interest.

Figure 5: Decay phase diagram of doubly-charged scalar ($\Delta^{\pm\pm}$)
with mass = 300 GeV. The solid, dashed, dot-dashed and dotted contours
indicate 99$\%$, 90$\%$, 50$\%$ and 10$\%$ branching ratios respectively, for
the bosonic, leptonic or cascade decays. The mass splitting $\Delta m$ is
defined in the main text.

Being proportional to $v_{\Delta}^{2}$, the cross section obtained for this
signature in the type-II seesaw models, even before any cuts, is negligibly
small even for $v_{\Delta}$ = 1 GeV. As stated above, the GM model can
accommodate $v_{\Delta}$ as high as 50 GeV. Hence, from now on, we will be
considering the GM model, instead of the simplest type-II seesaw model,
assuming $v_{\Delta}$ = 10 GeV, which is an arbitrary choice. The cross
section for the stated signal is now sufficiently large to be detectable at
the LHC and can scale easily with $v_{\Delta}^{2}$.

##  3 Signal and Background Evaluations

Here in this section we systematically investigate the signal and the
background for the aforementioned models. Considering leptonic decays of the
$W$ bosons, the signal of interest here has a final state of SSdL +
$\not{E}_{T}$, where the leptons include the electron and muon. As stated in
Sec. 2, we can obtain such a signal from wino pair production in NUHM2 model
in SUSY, pair production of heavy $SU(2)_{L}$ triplet in type-III seesaw
model, and the resonant production of the doubly-charged scalar in the GM
model when the forward jets go undetected. Note that, a final state of SSdL +
$\not{E}_{T}$ can also be obtained from gluino/squark pair production in SUSY
models [91, 92, 93, 94]. However, this signature can be distinguished from the
signal studied here because the SSdL + $\not{E}_{T}$ from gluino/squark pair
production appears along with large number of hard central jets. We evaluate
the signal from all the three models and the background from the SM, and
optimize cuts to efficiently reduce the background. As compared to an earlier
analysis in Ref. [28], three new backgrounds, namely, $ZZ$, $W^{\pm}W^{\mp}Z$
and $W^{\pm}ZZ$ have been included here. Therefore, the relevant SM background
processes are: $t\bar{t}$, $t\bar{t}t\bar{t}$, $t\bar{t}W^{\pm}$, $t\bar{t}Z$,
$W^{\pm}W^{\pm}jj$, $W^{\pm}W^{\pm}W^{\mp}$, $W^{\pm}Z$, $ZZ$,
$W^{\pm}W^{\mp}Z$ and $W^{\pm}ZZ$.

As discussed in Sec. 2.2, for our analysis the mass of
$\tilde{\Sigma}^{\pm,0}$ is taken to be around 670 GeV so as to satisfy the
mass constraint in Ref. [64] and we take ${\Sigma}_{i}^{\pm,0}$ of mass 770
GeV to ensure that ${\Sigma}_{i}^{\pm,0}$ primarily decay to $W^{\pm}$ and
$\tilde{\Sigma}^{\pm,0}$. Hence, we take a suitable benchmark point in the
NUHM2 model as well, with the wino-like particles ($\tilde{Z}_{4}$ and
$\tilde{W}_{2}^{\pm}$) also attaining a mass of around 770 GeV, so that the
signals from the type-III seesaw model and the NUHM2 model are at par with
each other, as stated earlier in Sec. 2.1. However, for the GM model, we have
considered $m(\Delta^{\pm\pm})$ = $300\rm~{}GeV$ since the limit is less
stringent on the mass of $\Delta^{\pm\pm}$ ($>200\rm~{}GeV$ [83]) while
looking for the bosonic final state signatures at the LHC. Using the same
argument, we could have taken lower mass for the wino-like particles
($\tilde{Z}_{4}$ and $\tilde{W}_{2}^{\pm}$) in the NUHM2 model but then the
NUHM2 benchmark point would not satisfy various constraints such as the mass
limit on gluino from the LHC, dark matter constraints from direct detection
experiments, etc.

For simulations, we have used MadGraph5$\\_$aMC@NLO [95, 96] for event
generation, interfaced with Pythia 8.2 [97] for parton showering and
hadronization, followed by Delphes 3.4.2 [98] for detector simulation where
the default Delphes card is employed. Therefore, in the detector level
simulation:

  * •

The anti-$k_{T}$ jet algorithm [99] has been used with $R=0.5$ and $p_{T}({\rm
jet})>20$ GeV.

  * •

The default jet flavor association module has been used that identifies a jet
containing a $b$-hadron as a potential $b$-jet if the $b$-hadron lies within
$\Delta R=0.5$ of the jet axis. Such $b$-jets are tagged with $85\%$
efficiency with a mistagging probability of $10\%$ for other lighter jets
following CMS $b$-tagging algorithm [100].222The $b$-tagging and
misidentification rates considered here are from those used for $\sqrt{s}=7$
TeV collisions. We have also used the $b$-tagging and misidentification rates
from those used for $\sqrt{s}=13$ TeV collisions, where $b$-jets are tagged
with an efficiency of $68\%$ and a mistagging probability of $1\%$ for other
lighter jets following a more recent CMS $b$-tagging algorithm [101] for the
$t\bar{t}$ process and found similar results after the same cuts.

  * •

The $\tau$ lepton is identified with an efficiency of $60\%$ (fixed in Delphes
3.4.2 [98]) if it lies within $\Delta R=0.5$ of the jet axis [102, 103].

  * •

The $e$ and $\mu$ leptons are isolated following the criterion that the ratio
of the sum of transverse momenta higher than 0.5 GeV of all particles that lie
within a cone of $\Delta R<0.5$ around the lepton to the transverse momentum
of that lepton is less than 0.25.

We have used Isajet 7.88 [54] to generate the Les Houches Accord (LHA) file
for the NUHM2 signal and pass it through the above-mentioned simulation chain.

We have used Prospino [104] to calculate the leading-order (LO) and next-to-
leading-order (NLO) cross sections for the NUHM2 signal process and type-III
seesaw signal process for 14 TeV LHC. Since Prospino is designed specifically
for calculating NLO cross sections of SUSY processes, using it to calculate
the same for the type-III seesaw model is made possible by utilizing the
analogy between the type-III seesaw model and the minimal anomaly-mediated
SUSY breaking (mAMSB) model [6, 105, 106]. The mAMSB model has a wino-like LSP
($\tilde{Z_{1}}$) and a wino-like next-to-LSP (NLSP) ($\tilde{W_{1}^{\pm}}$),
analogous to the type-III seesaw model with its lightest and next-to-lightest
particles being $\tilde{\Sigma}^{0}$ and $\tilde{\Sigma}^{\pm}$, respectively.
Thus, a suitable mAMSB parameter space point has been used to calculate the LO
and NLO cross sections for the type-III seesaw model.

We have used the K-factor for the type-II seesaw models (and hence the GM
model), as done in Ref. [107]. The K-factors for the SM background processes
are used as in Ref. [28]. We obtain the K-factor for the $ZZ$ process from
Ref. [108]. The K-factor for the $W^{\pm}W^{\mp}Z$ process has been used as in
Ref. [109, 110, 111]. For the $W^{\pm}ZZ$ process, we use the same K-factor as
for $W^{\pm}W^{\mp}Z$, as suggested in Ref. [112]. We generate $10^{7}$ events
for the $t\bar{t}$ process and $10^{6}$ events for all the other background
and signal processes for $\sqrt{s}=27$ TeV. While for $\sqrt{s}=100$ TeV, we
generate $10^{7}$ events for the $t\bar{t}$ and $W^{\pm}Z$ processes and
$10^{6}$ events for all the other background and signal processes.

Motivated by the earlier analyses [27, 28], we put a set of basic cuts, dubbed
the S1-cuts, to reduce the SM backgrounds. Explicitly, the S1-cuts include:

  * •

Require exactly two same-sign isolated leptons, where the isolated leptons are
defined as those with $p_{T}(\ell)>10$ GeV and $\eta(\ell)<2.5$.

  * •

Veto events with any identified $b$-jet.

  * •

Require $p_{T}(\ell_{1})>20$ GeV, where $\ell_{1}$ denotes the leading lepton.

In the following subsections, we show how each BSM model stands out by further
imposing a particular set of additional cuts.

###  3.1 Supersymmetry Analysis

In the NUHM2 signal, the LSP $\tilde{Z}_{1}$, due to R-parity conservation, is
stable and shows up as $\not{E}_{T}$ at the LHC. The particles $\tilde{Z}_{2}$
and $\tilde{W}_{1}^{\pm}$, being not much heavier than the LSP, promptly decay
into very soft leptons and the LSP. These leptons are so soft that they pass
the detector undetected. Hence, $\tilde{Z}_{2}$ and $\tilde{W}_{1}^{\pm}$ also
show up as $\not{E}_{T}$ at the LHC. Hence the NUHM2 signal has large missing
transverse energy ($\not{E}_{T}$). Similarly, the NUHM2 signal also has large
minimum transverse mass ($m_{T_{\rm min}}$) which is defined as:

$m_{T_{\rm min}}={\rm
min}(m_{T}(\ell_{1},\not{E}_{T}),m_{T}(\ell_{2},\not{E}_{T}))$ (3.4)

It turns out that we cannot gain a sufficient cross section for $\sqrt{s}=14$
TeV. Therefore, we extend the analysis to $\sqrt{s}=27$ TeV. Although
$\sqrt{s}=27$ TeV is sufficient to render a significance above 5$\sigma$, we
also show an analysis for $\sqrt{s}=100$ TeV. Inspired by the earlier analyses
done in this context [27, 28], we further impose the following set of cuts:

  * •

Require $\not{E}_{T}>200$ GeV.

  * •

Require $m_{T_{\rm min}}>175$ GeV.

Together with the S1-cuts, we call the entire set of cuts as the A1-cuts.
After the A1-cuts, we plot the $m_{T_{\rm min}}$ distribution in Fig. 6(a) for
$\sqrt{s}=27$ TeV and in Fig. 6(b) for $\sqrt{s}=100$ TeV.

(a)

(b)

Figure 6: $m_{T_{\rm min}}$ distribution after A1-cuts for (a) $\sqrt{s}=27$
TeV and (b) $\sqrt{s}=100$ TeV.

As suggested by Fig. 6(a), a cut of $m_{T_{\rm min}}>200$ GeV, if employed
after the A1-cuts, would reduce the SM background to some extent. Therefore,
we have the A2-cuts:

  * •

A1-cuts + $m_{T_{\rm min}}>200$ GeV.

Similarly, Fig. 6(b) suggests that a cut of $m_{T_{\rm min}}>325$ GeV would be
highly beneficial in significantly reducing the SM background at
$\sqrt{s}=100$ TeV. Therefore, we have the A2′-cuts:

  * •

A1-cuts + $m_{T_{\rm min}}>325$ GeV.

After the A2-cuts and A2′-cuts, we plot the $\not{E}_{T}$ distribution for
$\sqrt{s}=27$ TeV and $\sqrt{s}=100$ TeV in Fig. 7(a) and Fig. 7(b),
respectively.

As shown in Fig. 7(a) (7(b)), a cut of $\not{E}_{T}>250$ GeV
($\not{E}_{T}>350$ GeV), applied after the A2 (A2′)-cuts, would result in a
cleaner NUHM2 signal with efficiently reduced SM background as well as heavily
reduced signal cross sections for the other two BSM models at $\sqrt{s}=27$
TeV ($100$ TeV). Therefore, we finally have the A3-cuts:

  * •

A2-cuts + $\not{E}_{T}>250$ GeV.

and the A3′-cuts:

  * •

A2′-cuts + $\not{E}_{T}>350$ GeV.

(a)

(b)

Figure 7: $\not{E}_{T}$ distribution after (a) A2-cuts at $\sqrt{s}=27$ TeV
and (b) A2′-cuts at $\sqrt{s}=100$ TeV.

The cut flow is summarized in Table IV.

|  |  $\sqrt{s}=27$ TeV
---|---|---
Process | K-factor | $\sigma$ (NLO) [ab] | A1 [ab] | A2 [ab] | A3 [ab] | Significance
$\mathcal{L}=$ 3 ab-1 |  $\mathcal{L}=$ 15 ab-1
NUHM2 | 1.17 | $4.2\cdot 10^{4}$ | 60.9 | 53.3 | 46.1 |  $8.06$ ($7.8$) |  $18.01$ ($15.4$)
type-III | 1.16 | $4.36\cdot 10^{4}$ | 9.33 | 5.7 | 5.3 |  $1.21$ ($1.14$) |  $2.71$ ($2.12$)
GM | 1.26 | $5.6\cdot 10^{4}$ | 0.28 | 0.28 | 0.0562 |  $0.0135$ ($0.013$) |  $0.03$ ($0.023$)
$t\bar{t}$ | 1.72 | $4.1\cdot 10^{9}$ | 0 | 0 | 0 | - | -
$t\bar{t}t\bar{t}$ | 1.27 | $1.1\cdot 10^{5}$ | 0.8 | 0.4 | 0.3 | - | -
$t\bar{t}W^{\pm}$ | 1.24 | $1.5\cdot 10^{6}$ | 12.1 | 6.03 | 6.03 | - | -
$t\bar{t}Z$ | 1.39 | $4.4\cdot 10^{6}$ | 17.6 | 4.4 | 0 | - | -
$W^{\pm}W^{\pm}jj$ | 1.04 | $2.5\cdot 10^{6}$ | 78.4 | 48.04 | 30.3 | - | -
$W^{\pm}W^{\pm}W^{\mp}$ | 2.45 | $8.0\cdot 10^{5}$ | 20.8 | 15.2 | 8.0 | - | -
$W^{\pm}Z$ | 1.88 | $1.2\cdot 10^{8}$ | 0 | 0 | 0 | - | -
$ZZ$ | 1.7 | $4.1\cdot 10^{7}$ | 0 | 0 | 0 | - | -
$W^{\pm}W^{\mp}Z$ | 2.0 | $5.2\cdot 10^{5}$ | 8 | 7 | 4.2 | - | -
$W^{\pm}ZZ$ | 2.0 | $1.6\cdot 10^{5}$ | 4.7 | 3.7 | 3.2 | - | -
Total BG | $-$ | $4.3\cdot 10^{9}$ | 142.4 | 84.77 | 52.1 | - | -
|  |  $\sqrt{s}=100$ TeV
Process | K-factor | $\sigma$ (NLO) [ab] | A1 [ab] | A2′ [ab] | A3′ [ab] | Significance
$\mathcal{L}=$ 3 ab-1 |  $\mathcal{L}=$ 15 ab-1
NUHM2 | 1.17 | $3.71\cdot 10^{5}$ | 374.4 | 170.5 | 121.9 |  $13.6$ ($12.7$) |  $30.5$ ($22.9$)
type-III | 1.16 | $4.2\cdot 10^{5}$ | 112.55 | 10.1 | 9.7 |  $1.5$ ($1.3$) |  $3.3$ ($2.1$)
GM | 1.26 | $3.7\cdot 10^{5}$ | 5.2 | 1.5 | 0.4 |  $0.064$ ($0.056$) |  $0.14$ ($0.1$)
$t\bar{t}$ | 1.72 | $4.6\cdot 10^{10}$ | 4570.0 | 0 | 0 | - | -
$t\bar{t}t\bar{t}$ | 1.27 | $3.75\cdot 10^{6}$ | 48.8 | 4.0 | 4.0 | - | -
$t\bar{t}W^{\pm}$ | 1.24 | $9.3\cdot 10^{6}$ | 83.5 | 18.5 | 18.5 | - | -
$t\bar{t}Z$ | 1.39 | $6.51\cdot 10^{7}$ | 325.5 | 65.1 | 0 | - | -
$W^{\pm}W^{\pm}jj$ | 1.04 | $1.6\cdot 10^{7}$ | 1315.2 | 82.0 | 49.3 | - | -
$W^{\pm}W^{\pm}W^{\mp}$ | 2.45 | $4.1\cdot 10^{6}$ | 151.5 | 49.0 | 24.6 | - | -
$W^{\pm}Z$ | 1.88 | $5.2\cdot 10^{8}$ | 104.5 | 0 | 0 | - | -
$ZZ$ | 1.7 | $1.8\cdot 10^{8}$ | 0 | 0 | 0 | - | -
$W^{\pm}W^{\mp}Z$ | 2.0 | $2.8\cdot 10^{6}$ | 94 | 20 | 11.4 | - | -
$W^{\pm}ZZ$ | 2.0 | $8.7\cdot 10^{5}$ | 38.3 | 14.8 | 9.6 | - | -
Total BG | $-$ | $4.72\cdot 10^{10}$ | 6731.3 | 253.4 | 117.4 | - | -

Table IV: Cut flow table for cleaner NUHM2 signal.

After the A3 and A3′-cuts for $\sqrt{s}=27$ TeV and $100$ TeV, respectively, a
sufficient NUHM2 signal cross section is retained, while the SM background and
signals from the other two BSM models are greatly reduced.

The statistical significance of the signal has been computed using the
relation $S/\sqrt{S+B}$. We have shown the significance for center of mass
energy ($\sqrt{s}$) at $27$ TeV and $100$ TeV with an integrated luminosity
$\mathcal{L}$ = 3 ab-1 and 15 ab-1 in Table IV. Note that, on calculating the
significance at $\sqrt{s}=14$ TeV and $\mathcal{L}$ = 3 ab-1, the NUHM2 BM
point chosen here yields $S/\sqrt{S+B}=4.7$. This result is different from
that obtained in an earlier analysis in Ref. [28] mainly due to differences in
the detector-level simulations in these two cases. Moreover, in order to show
the impact of systematic uncertainties, we compute the significance
considering 3$\%$ systematic uncertainties using the relation
$S/\sqrt{S+B+\Delta^{2}B^{2}}$ with $\Delta=0.03$. We find that the
significance drops down while we consider systematic uncertainties, as shown
in parenthesis in Table IV. From Table IV, we can see that after the A3 and
A3′-cuts for $\sqrt{s}=27$ TeV and $100$ TeV, respectively, the NUHM2 signal
yields a significance above 5$\sigma$ while the type-III and GM models don’t,
with or without considering the 3$\%$ systematic uncertainties.

We have plotted the cluster transverse mass (MCT) distribution and the
$\not{E}_{T}$ distribution after A3 and A3′-cuts at the respective energies
for the total SM background and various signals on top of it in Fig. 8.

(a)

(b)

(c)

(d)

Figure 8: MCT distribution after (a) A3-cuts at $\sqrt{s}=27$ TeV and (b)
A3′-cuts at $\sqrt{s}=100$ TeV; and $\not{E}_{T}$ distribution after (c)
A3-cuts at $\sqrt{s}=27$ TeV and (d) A3′-cuts at $\sqrt{s}=100$ TeV.

###  3.2 Type-III Seesaw Analysis

We cannot gain a sufficient cross section for $\sqrt{s}=14$ TeV for the type-
III seesaw signal either, and therefore consider $\sqrt{s}=27$ TeV and
$\sqrt{s}=100$ TeV. We start with S1-cuts as defined earlier.

It is expected that after the S1-cuts all the SM backgrounds, considered here,
should have numerous jets while the type-III seesaw signal can have jets only
from initial state QCD radiation. Therefore, requiring only those events that
have less than two jets would significantly reduce the SM background while
retaining enough type-III seesaw signal.

Similar to the NUHM2 signal, the type-III signal will also show high
$\not{E}_{T}$. However, since in type-III signal, the mass difference between
the intermediate state and the final state is not as high as that in the NUHM2
signal, hence the $\not{E}_{T}$ distribution does not tail out as high as the
NUHM2 signal. Therefore, for $\sqrt{s}=27$ TeV ($100$ TeV), we apply a cut of
$\not{E}_{T}>100$ GeV ($>120$ GeV), a less stringent cut on $\not{E}_{T}$ as
compared to that applied in case of NUHM2 signal.

For the same reason as mentioned above, we employ a less stringent cut on
$m_{T_{\rm min}}$ as well: a cut of 105 GeV $<m_{T_{\rm min}}<195$ GeV.

A cut on the upper limit of $m_{T_{\rm min}}$ has been applied in order to
reduce the NUHM2 signal and yet retain most of the type-III signal as for
NUHM2 signal the $m_{T_{\rm min}}$ distribution tails out to a much higher
value than that in the type-III signal. The cut on the upper limit of
$m_{T_{\rm min}}$ is thus necessary to differentiate between the NUHM2 and the
type-III signal. Therefore after S1-cuts we apply three additional cuts,
namely, njet $\leq 1$, $\not{E}_{T}>100$ GeV and 105 GeV $<m_{T_{\rm
min}}<195$ GeV and name this entire set of cut as B1-cuts:

  * •

S1-cuts + njet $\leq 1$ \+ $\not{E}_{T}>100$ GeV + 105 GeV $<m_{T_{\rm
min}}<195$ GeV.

At $\sqrt{s}=100$ TeV, we use a slightly tougher cut on $\not{E}_{T}$ and
apply B1′-cuts, which include:

  * •

S1-cuts + njet $\leq 1$ \+ $\not{E}_{T}>120$ GeV + 105 GeV $<m_{T_{\rm
min}}<195$ GeV.

After the B1-cuts and B1′-cuts for $\sqrt{s}=27$ TeV and $100$ TeV,
respectively, the MCT distribution is plotted in Fig. 9(a) and in Fig. 9(b).
From Fig. 9(a) we see that a cut of 200 GeV < MCT < 325 GeV can further reduce
the SM background and NUHM2 signal as well while retaining enough type-III
signals to be visible. Therefore, next we apply the B2-cuts defined as:

  * •

B1-cuts + 200 GeV < MCT < 325 GeV.

Similarly, for $\sqrt{s}=100$ TeV, Fig. 9(b) shows that a suitable cut would
be 200 GeV < MCT < 350 GeV. Therefore, next we apply the B2′-cuts defined as:

  * •

B1′-cuts + 200 GeV < MCT < 350 GeV.

(a)

(b)

Figure 9: MCT distribution after (a) B1-cuts for $\sqrt{s}=27$ TeV (b)
B1′-cuts for $\sqrt{s}=100$ TeV .

The cut flow for this scenario is summarized in Table V.

|  |  $\sqrt{s}=27$ TeV
---|---|---
Process | K-factor | $\sigma$ (NLO) [ab] | B1 [ab] | B2 [ab] | Significance
$\mathcal{L}=$ 3 ab-1 |  $\mathcal{L}=$ 15 ab-1
NUHM2 | 1.17 | $4.2\cdot 10^{4}$ | 13.2 | 2.2 |  $0.52$ ($0.5$) |  $1.2$ ($0.9$)
type-III | 1.16 | $4.36\cdot 10^{4}$ | 22.8 | 16.6 |  $3.5$ ($3.3$) |  $7.8$ ($6.3$)
GM | 1.26 | $5.6\cdot 10^{4}$ | 2.6 | 1.9 |  $0.45$ ($0.42$) |  $1.0$ ($0.8$)
$t\bar{t}$ | 1.72 | $4.1\cdot 10^{9}$ | 0 | 0 | - | -
$t\bar{t}t\bar{t}$ | 1.27 | $1.1\cdot 10^{5}$ | 0 | 0 | - | -
$t\bar{t}W^{\pm}$ | 1.24 | $1.5\cdot 10^{6}$ | 12.1 | 1.5 | - | -
$t\bar{t}Z$ | 1.39 | $4.4\cdot 10^{6}$ | 4.4 | 4.4 | - | -
$W^{\pm}W^{\pm}jj$ | 1.04 | $2.5\cdot 10^{6}$ | 25.3 | 10.1 | - | -
$W^{\pm}W^{\pm}W^{\mp}$ | 2.45 | $8.0\cdot 10^{5}$ | 76.0 | 26.4 | - | -
$W^{\pm}Z$ | 1.88 | $1.2\cdot 10^{8}$ | 0 | 0 | - | -
$ZZ$ | 1.7 | $4.1\cdot 10^{7}$ | 41 | 0 | - | -
$W^{\pm}W^{\mp}Z$ | 2.0 | $5.2\cdot 10^{5}$ | 10.9 | 5.2 | - | -
$W^{\pm}ZZ$ | 2.0 | $1.6\cdot 10^{5}$ | 9.8 | 4 | - | -
Total BG | $-$ | $4.3\cdot 10^{9}$ | 179.5 | 51.6 | - | -
|  |  $\sqrt{s}=100$ TeV
Process | K-factor | $\sigma$ (NLO) [ab] | B1′ [ab] | B2′ [ab] | Significance
$\mathcal{L}=$ 3 ab-1 |  $\mathcal{L}=$ 15 ab-1
NUHM2 | 1.17 | $3.71\cdot 10^{5}$ | 36.3 | 5.6 |  $0.8$ ($0.7$) |  $1.8$ ($1.1$)
type-III | 1.16 | $4.2\cdot 10^{5}$ | 50.2 | 33 |  $4.3$ ($3.8$) |  $9.6$ ($6.0$)
GM | 1.26 | $3.7\cdot 10^{5}$ | 18.9 | 10 |  $1.4$ ($1.2$) |  $3.1$ ($1.9$)
$t\bar{t}$ | 1.72 | $4.6\cdot 10^{10}$ | 0 | 0 | - | -
$t\bar{t}t\bar{t}$ | 1.27 | $3.75\cdot 10^{6}$ | 0 | 0 | - | -
$t\bar{t}W^{\pm}$ | 1.24 | $9.3\cdot 10^{6}$ | 9.3 | 0 | - | -
$t\bar{t}Z$ | 1.39 | $6.51\cdot 10^{7}$ | 0 | 0 | - | -
$W^{\pm}W^{\pm}jj$ | 1.04 | $1.6\cdot 10^{7}$ | 213.7 | 65.8 | - | -
$W^{\pm}W^{\pm}W^{\mp}$ | 2.45 | $4.1\cdot 10^{6}$ | 135.1 | 65.5 | - | -
$W^{\pm}Z$ | 1.88 | $5.2\cdot 10^{8}$ | 156.7 | 0 | - | -
$ZZ$ | 1.7 | $1.8\cdot 10^{8}$ | 0 | 0 | - | -
$W^{\pm}W^{\mp}Z$ | 2.0 | $2.8\cdot 10^{6}$ | 17.1 | 5.7 | - | -
$W^{\pm}ZZ$ | 2.0 | $8.7\cdot 10^{5}$ | 14.0 | 6.1 | - | -
Total BG | $-$ | $4.72\cdot 10^{10}$ | 545.9 | 143.1 | - | -

Table V: Cut flow table for cleaner type-III signal.

After the B2 and B2′-cuts at $\sqrt{s}=27$ TeV and $100$ TeV, respectively, we
list the significance $S/\sqrt{S+B}$ for the type-III seesaw signal, the NUHM2
signal and the GM model signal for $\mathcal{L}$ = 3 ab-1 and 15 ab-1.
Besides, we also consider the impact of 3$\%$ systematic uncertainties and
calculate the significance which we show in parenthesis in Table V. We see
that on considering 3$\%$ systematic uncertainties the significance for each
signal reduces, for example, at $\sqrt{s}=27$ TeV and for $\mathcal{L}$ = 15
ab-1, the type-III seesaw signal BM point yields a significance of 7.8 which
reduces to 6.3 on considering 3$\%$ systematic uncertainties. The table shows
that the type-III seesaw signal yields a significance higher than 5$\sigma$ at
both energies for $\mathcal{L}$ = 15 ab-1 while the other two BSM scenarios
don’t. We have plotted the MCT distribution and the $\not{E}_{T}$ distribution
for the total SM background and various signals on top of it, after imposing
the B2 and B2′-cuts at the respective energies in Fig. 10.

(a)

(b)

(c)

(d)

Figure 10: MCT distribution after (a) B2-cuts at $\sqrt{s}=27$ TeV and (b)
B2′-cuts at $\sqrt{s}=100$ TeV; and $\not{E}_{T}$ distribution after (c)
B2-cuts at $\sqrt{s}=27$ TeV and (d) B2′-cuts at $\sqrt{s}=100$ TeV.

###  3.3 Type-II Seesaw/Georgi-Machacek Model Analysis

In the GM model, since the SSdL and $\not{E}_{T}$ originate from
$\Delta^{\pm\pm}$ of mass 300 GeV, the MCT distribution should peak and then
sharply fall around 300 GeV. Therefore, an efficient cut after S1-cuts to
scoop out the GM model signal would be to require the MCT to be $\leq$ 300
GeV.

Since we explicitly have two forward jets in the GM model signal, hence
requiring the number of jets $\geq$ 2 would be a suitable cut to retain most
of the GM model signal. Hence after S1- cuts, we apply two additional cuts,
namely, MCT $\leq 300$ GeV and njet $\geq$ 2 and this entire set of cut is
called the C1-cuts:

  * •

S1-cuts + MCT $\leq 300$ GeV + njet $\geq$ 2.

After applying the C1-cuts, we plot the distribution of the pseudorapidity
($\eta$) difference between the two leading jets, $\Delta\eta(j_{1},j_{2})$,
in Fig. 11(a) and 11(b) at $\sqrt{s}=27$ TeV and $100$ TeV, respectively. Due
to the presence of two explicit forward jets in the GM model signal, we expect
$\Delta\eta(j_{1},j_{2})$ should peak towards higher values and it indeed
peaks around 5 in Fig. 11. One naively expects that the $W^{\pm}W^{\pm}jj$
process in the SM should also peak towards higher value of
$\Delta\eta(j_{1},j_{2})$. The reason that Fig. 11 does not feature this is
the following. The $W^{\pm}W^{\pm}jj$ process includes two types of processes:
(a) those of ${\cal O}(\alpha^{2}\alpha_{S}^{0})$, and (b) those of ${\cal
O}(\alpha_{S}\alpha)$, both at the amplitude level. The processes of type (a)
do peak towards higher value of $\Delta\eta(j_{1},j_{2})$ after certain cuts
as depicted in Ref. [113]. But here we have included both types of diagrams
and our choice of cuts are very different from those used in Ref. [113].
Hence, the peak due to type (a) processes is overshadowed once type (b)
processes are included and becomes less prominent.

(a)

(b)

Figure 11: $\Delta\eta(j_{1},j_{2})$ distribution after C1-cuts at (a)
$\sqrt{s}=27$ TeV (b) and $\sqrt{s}=100$ TeV .

Therefore, requiring $\Delta\eta(j_{1},j_{2})>5$ is an extremely efficient cut
to not only reduce the SM background but also to almost eliminate the signals
from the other two BSM models. We now call this full set of cuts to be the
C2-cuts:

  * •

C1-cuts + $\Delta\eta(j_{1},j_{2})>5$.

After the C2-cuts, we plot the $m_{T_{\rm min}}$ distribution at $\sqrt{s}=27$
TeV and $100$ TeV in Fig. 12(a) and 12(b), respectively.

(a)

(b)

Figure 12: $m_{T_{\rm min}}$ distribution after C2-cuts for (a) $\sqrt{s}=27$
TeV (b) and $\sqrt{s}=100$ TeV .

As can be inferred from Fig. 12(a), a further cut of $m_{T_{\rm min}}>105$ GeV
will make the largest SM backgrounds $t\bar{t}$ and $W^{\pm}Z$ vanish. We also
impose a small $\not{E}_{T}$ cut of $\not{E}_{T}>50$ GeV just to ensure we
pick up those events with some non-zero $\not{E}_{T}$ in the final state since
we are looking for SSdB + $\not{E}_{T}$ signature. We therefore propose the
C3-cuts:

  * •

C2-cuts + $m_{T_{\rm min}}$ > 105 GeV + $\not{E}_{T}$ > 50 GeV.

Similarly, Fig. 12(b) shows that a further cut of $m_{T_{\rm min}}>120$ GeV
would be needed to make $t\bar{t}$ and $W^{\pm}Z$ backgrounds vanish. We
therefore propose the C3′-cuts:

  * •

C2-cuts + $m_{T_{\rm min}}$ > 120 GeV + $\not{E}_{T}$ > 50 GeV.

The cut flow for this scenario is summarized in Table VI.

After the final sets of cuts, namely the C3 and C3′-cuts for $\sqrt{s}=$ 27
and 100 TeV, respectively, we calculate and list the significance
$S/\sqrt{S+B}$ for the GM model signal, the type-III seesaw signal, and the
NUHM2 signal for $\mathcal{L}$ = 3 ab-1 and 15 ab-1. Additionally, we also
show the impact of 3$\%$ systematic uncertainties on the significance of the
signal in parenthesis in Table VI.

|  |  $\sqrt{s}=27$ TeV
---|---|---
Process | K-factor | $\sigma$ (NLO) [ab] | C1 [ab] | C2 [ab] | C3 [ab] | Significance
$\mathcal{L}=$ 3 ab-1 |  $\mathcal{L}=$ 15 ab-1
NUHM2 | 1.17 | $4.2\cdot 10^{4}$ | 77.0 | 0.97 | 0 | 0 | 0
type-III | 1.16 | $4.36\cdot 10^{4}$ | 117.86 | 8.6 | 1.4 |  $0.224$ ($0.2$) |  $0.5$ ($0.3$)
GM | 1.26 | $5.6\cdot 10^{4}$ | 976.52 | 428.4 | 16.35 |  $2.5$ ($2.2$) |  $5.5$ ($3.6$)
$t\bar{t}$ | 1.72 | $4.1\cdot 10^{9}$ | 131044.0 | 2850.0 | 0 | - | -
$t\bar{t}t\bar{t}$ | 1.27 | $1.1\cdot 10^{5}$ | 40.0 | 0.32 | 0 | - | -
$t\bar{t}W^{\pm}$ | 1.24 | $1.5\cdot 10^{6}$ | 2938.5 | 15.1 | 0 | - | -
$t\bar{t}Z$ | 1.39 | $4.4\cdot 10^{6}$ | 2260.2 | 22.0 | 4.4 | - | -
$W^{\pm}W^{\pm}jj$ | 1.04 | $2.5\cdot 10^{6}$ | 21638.0 | 2693.0 | 108.7 | - | -
$W^{\pm}W^{\pm}W^{\mp}$ | 2.45 | $8.0\cdot 10^{5}$ | 4622.6 | 41.61 | 1.6 | - | -
$W^{\pm}Z$ | 1.88 | $1.2\cdot 10^{8}$ | 69464.0 | 2126.4 | 0 | - | -
$ZZ$ | 1.7 | $4.1\cdot 10^{7}$ | $1.1\cdot 10^{4}$ | 408.8 | 0 | - | -
$W^{\pm}W^{\mp}Z$ | 2.0 | $5.2\cdot 10^{5}$ | $1.1\cdot 10^{3}$ | 13.04 | 1.0 | - | -
$W^{\pm}ZZ$ | 2.0 | $1.6\cdot 10^{5}$ | 351.5 | 6.7 | 0.3 | - | -
Total BG | $-$ | $4.3\cdot 10^{9}$ | $2.4\cdot 10^{5}$ | $8.2\cdot 10^{3}$ | 116.0 | - | -
|  |  $\sqrt{s}=100$ TeV
Process | K-factor | $\sigma$ (NLO) [ab] | C1 [ab] | C2 [ab] | C3′ [ab] | Significance
$\mathcal{L}=$ 3 ab-1 |  $\mathcal{L}=$ 15 ab-1
NUHM2 | 1.17 | $3.71\cdot 10^{5}$ | 415.13 | 11.1 | 1.9 |  $0.22$ ($0.17$) |  $0.48$ ($0.24$)
type-III | 1.16 | $4.2\cdot 10^{5}$ | 900.8 | 85.6 | 11.0 |  $1.23$ ($1.0$) |  $2.7$ ($1.4$)
GM | 1.26 | $3.7\cdot 10^{5}$ | 4135.6 | 1663.6 | 28.0 |  $3.02$ ($2.4$) |  $6.75$ ($3.5$)
$t\bar{t}$ | 1.72 | $4.6\cdot 10^{10}$ | $9.5\cdot 10^{5}$ | $1.4\cdot 10^{4}$ | 0 | - | -
$t\bar{t}t\bar{t}$ | 1.27 | $3.75\cdot 10^{6}$ | 860 | 26.3 | 0 | - | -
$t\bar{t}W^{\pm}$ | 1.24 | $9.3\cdot 10^{6}$ | $1.1\cdot 10^{4}$ | 111.3 | 9.3 | - | -
$t\bar{t}Z$ | 1.39 | $6.51\cdot 10^{7}$ | $2.5\cdot 10^{4}$ | $1.3\cdot 10^{3}$ | 0 | - | -
$W^{\pm}W^{\pm}jj$ | 1.04 | $1.6\cdot 10^{7}$ | $8.0\cdot 10^{4}$ | $1.5\cdot 10^{4}$ | 214.0 | - | -
$W^{\pm}W^{\pm}W^{\mp}$ | 2.45 | $4.1\cdot 10^{6}$ | $1.4\cdot 10^{4}$ | 327.5 | 4.1 | - | -
$W^{\pm}Z$ | 1.88 | $5.2\cdot 10^{8}$ | $3.1\cdot 10^{5}$ | $1.7\cdot 10^{4}$ | 0 | - | -
$ZZ$ | 1.7 | $1.8\cdot 10^{8}$ | $4.6\cdot 10^{4}$ | $1.1\cdot 10^{3}$ | 0 | - | -
$W^{\pm}W^{\mp}Z$ | 2.0 | $2.8\cdot 10^{6}$ | $4.2\cdot 10^{3}$ | 119.5 | 2.8 | - | -
$W^{\pm}ZZ$ | 2.0 | $8.7\cdot 10^{5}$ | $1.2\cdot 10^{3}$ | 34.8 | 0 | - | -
Total BG | $-$ | $4.72\cdot 10^{10}$ | $1.4\cdot 10^{6}$ | $4.9\cdot 10^{4}$ | 230.2 | - | -

Table VI: Cut flow table for cleaner GM model signal.

As can be inferred from Table VI, with the C3 (C3′)-cuts at $\sqrt{s}=27$
($100$) TeV, the GM model signal can be observed above the 5$\sigma$ level
with IL = 15 ab-1 while the other two BSM scenarios yield a significance below
5$\sigma$. In Fig. 13, we show the MCT distribution and the $\not{E}_{T}$
distribution for the total SM background and the various signals on top of it,
after the C3 and C3′-cuts at the respective energies.

(a)

(b)

(c)

(d)

Figure 13: MCT distribution after (a) C3-cuts at $\sqrt{s}=27$ TeV and (b)
C3′-cuts at $\sqrt{s}=100$ TeV; and $\not{E}_{T}$ distribution after (c)
C3-cuts at $\sqrt{s}=27$ TeV and (d) C3′-cuts at $\sqrt{s}=100$ TeV.

As seen in the above background and signal evaluations, the dominant SM
backgrounds, $t\bar{t}$ and $W^{\pm}Z$ vanish completely after all the final
cuts. This happens because after the S1-cuts, the surviving events from the
$t\bar{t}$ background will have one lepton originating from real $W$, another
lepton coming from semi-leptonic decay of one of the $b$-jets and the
$\not{E}_{T}$ would also dominantly come from leptonic decays of $W$. Some of
these surviving events will survive the C2-cuts also as these surviving events
are accompanied by two or more jets originating from either semi-leptonic
decay of $b$-jets or misidentification of the $b$-jet as some lighter jet. In
that case, $m_{T_{\rm min}}$ would be mostly bounded by $m_{W}$. Therefore, a
cut of $m_{T_{\rm min}}>m_{W}$ would reduce the $t\bar{t}$ background as well
as the $W^{\pm}Z$ background to a great extent which indeed is the case as
seen in the above analyses.

Note that, we have taken $m(\Delta^{\pm\pm})=300$ GeV (not at par with the
mass of intermediate states in the other two new physics model signature) in
order to highlight the fact that experimental limits on $m(\Delta^{\pm\pm})$
do allow us to take such low mass of $\Delta^{\pm\pm}$. However, even with
such small $m(\Delta^{\pm\pm})$, we still do not obtain $5\sigma$ significance
unless for the scheme of IL = 15 ab-1 for both 27 and 100 TeV because the SM
background of $W^{\pm}W^{\pm}jj$ is very much similar to the GM model signal.
Thus, the cuts employed to extract the GM model do not reduce the
$W^{\pm}W^{\pm}jj$ background as efficiently as they reduce the other SM
backgrounds. Since the GM model signal is produced via the VBF process, taking
$m(\Delta^{\pm\pm})$ $\sim$ $m(\Sigma_{i}^{\pm})$ $\sim$ $770\rm~{}GeV$, would
yield a much lower cross section and one might need to resort to a higher-
energy collider to obtain $5\sigma$ significance for the GM model signal.

##  4 Conclusions

In this paper, our goal is to catalogue various BSM scenarios that can give
rise to the SSdB+$\not{E}_{T}$ (precisely SSdL+$\not{E}_{T}$, as considering
leptonic decays of the diboson gets rid of the large SM QCD backgrounds)
signature in experiments and extract these signals from SM background by
imposing suitable cuts. Since more than one BSM scenario qualify, we also
focus on devising suitable cuts to distinguish these BSM models from one
another. We have analyzed three new physics models: the NUHM2 scenario of
natural SUSY models, the type-III seesaw model, and the GM model. We carefully
select the imposed cuts for each model to obtain a sufficiently large
significance for its signal. Assuming $\mathcal{L}$ = 15 ab-1 and
$\sqrt{s}=27$ ($100$) TeV, the C3 (C3′)-cuts and the B2 (B2′)-cuts are needed
to observe clean GM model and type-III seesaw model signals, respectively, at
a level above $5\sigma$ significance. For the NUHM2 model, a clean signal at a
level above $5\sigma$ significance can be seen with the A3 (A3′)-cuts for data
collected from $\mathcal{L}$ = 3 ab-1 and $\sqrt{s}=27$ ($100$) TeV.

## Acknowledgments

We thank Howard Baer for useful discussions. The work of CWC and DS was
supported in part by the Ministry of Science and Technology (MOST) of Taiwan
under Grant Nos. 108-2112-M-002-005-MY3 and 109-2811-M-002-570.

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