#  Distinguishing nonstandard scalar and fermionic charged particles at future
$e^{+}e^{-}$ collider

Anjan Kumar Barik<EMAIL_ADDRESS>Rafiqul Rahaman
<EMAIL_ADDRESS>Santosh Kumar Rai<EMAIL_ADDRESS>Regional Centre
for Accelerator-based Particle Physics, Harish-Chandra Research Institute,
A CI of Homi Bhabha National Institute, Chhatnag Road, Jhunsi, Prayagraj
211019, India

###### Abstract

We investigate the possibility to identify the intrinsic spin of exotic
charged particles at the future $e^{+}e^{-}$ collider in
$l^{\pm}+2j+\cancel{E}_{T}$ final state. We choose the Inert Doublet Model
(IDM) and Minimal supersymmetric Standard Model (MSSM), as examples for the
new physics models with scalar and fermionic exotic charged particles,
respectively. The signal arises when these exotics charged particles are pair
produced and then decay to a $W^{\pm}$ boson and the lightest neutral stable
particle in the new physics model. We choose four benchmarks for the mass
parameters which give significant deviation from the dominant Standard Model
(SM) $W^{+}W^{-}$ background. We find that an asymmetry in the cosine of the
scattering angle ($\cos\theta$) of one of the $W$ bosons reconstructed from
$jj$ pair as well as the charged lepton have the potential to identify the
MSSM and IDM signal over SM with longitudinally polarized initial beams. A
more robust distinction is seen in the shape of the azimuthal angle
distribution of the $W$ boson and charged lepton, which can identify and
distinguish the IDM signal from MSSM further if the initial beams are
transversely polarized.

††preprint: HRI-RECAPP-2022-009

##  I Introduction

The Standard Model (SM) of particle physics has been a great success in
explaining most of the phenomena in Nature. Despite its huge success,
phenomena such as the existence of non-zero neutrino mass and their
oscillation, the presence of dark matter (DM) in the universe, the obvious
matter-antimatter asymmetry vis-a-vis Baryogenesis, stability of the
electroweak scale or the gauge hierarchy problem, etc., requires one to think
of physics beyond the SM (BSM). A plethora of candidate BSM models exist in
the literature to address such non-standard phenomena. These BSM models are
being probed at the current Large Hadron Collider (LHC), and strategies are
being set up to probe them with more precision at future colliders such as
High Luminosity LHC (HL-LHC) Apollinari:2015wtw , High Energy LHC (HE-LHC)
Azzi:2019yne , International Linear Collider (ILC) Djouadi:2007ik ;
Baer:2013cma ; Behnke:2013xla ; Bambade:2019fyw , Large Hadron electron
Collider (LHeC) LHeC:2020van , Future Circular Collider (FCC) FCC:2018evy ;
FCC:2018vvp , etc. A very obvious and well known aspect of having so many
different BSM theories is the so called inverse problem, where different
models lead to overlapping outputs in the signal space, and it becomes
challenging to map it to any given BSM scenario. Therefore the chances of
observing that rare event of new physics, which has proven to be so elusive at
LHC may have too many BSM candidates to claim as their own, since similar
collider signatures can arise from different BSM models. Not much attention
has been paid to identify the type of BSM models by looking at collider
signatures Asano:2011aj ; Ginzburg:2014ora ; Belyaev:2021ngh . We try to fill
up a gap in this direction by showing through this work how to distinguish two
types of BSM models by looking at similar signatures at colliders based on the
spins of exotic particles. Typically one expects to be able to identify spin
Boudjema:2009fz ; Asano:2011aj ; Christensen:2013sea ; Belyaev:2016pxe  of a
mediator by looking at the angular distribution in a $2\to 2$ scattering
process where the final state particles are SM fields. We consider a more
common configuration which appears in models that have a DM candidate that
would escape detection. The lightest neutral stable particles (LNSP) in these
models can play a good dark matter candidate. These models also predict
charged particles that decay to LNSP. These BSM charged particles together
with the LNSP can have different spins ($0$, $1/2$, $1$) in different models.
Understanding the spin nature of dark matter (LNSP) experimentally is of
particular importance not only for particle physics but also for astrophysics
and cosmology.

In this article, we investigate the possibility of identifying the spin of BSM
charged particles or the candidate dark matter through the collider signature
of pair production of BSM charged particles followed by their decay to LNSP
and $W^{\pm}$ boson. It isn’t easy to do such precision measurements at the
LHC as it is overwhelmed by the huge QCD background. A future $e^{+}e^{-}$
collider or ILC, on the other hand, offers a great possibility in this
direction for having a clean signature with a very low background and extra
handles such as beam polarization (longitudinal and transverse)
MoortgatPick:2005cw . We choose two well motivated models as examples, the
Inert Doublet Model (IDM) Deshpande:1977rw ; Majumdar:2006nt  and the Minimal
supersymmetric SM (MSSM) Haber:1993wf ; Wess:1992cp , having potential dark
matter candidates of type scalar (spin-$0$) and fermion (spin-$1/2$),
respectively to study the collider signature of the process

$e^{+}e^{-}\to C^{+}\,C^{-}\to C^{0}\,C^{0}\,\,W^{+}\,W^{-}.$ (1)

In IDM, the $C^{+}$ and $C^{0}$ are $Z_{2}$ \- odd charged Higgs ($H^{\pm}$)
and neutral Higgs ($H^{0}$), while for MSSM, they are the chargino
($\widetilde{\chi}^{\pm}$) and neutralino ($\widetilde{\chi}^{0}$)
respectively. Note that the charged and neutral scalars (fermions) can appear
in several BSM set-ups involving $Z_{2}$ odd parity, and therefore this
analysis will be applicable in all such scenarios. We use the potential of
beam polarization (both longitudinal and transverse) of an $e^{+}e^{-}$
collider to discriminate between the two models by looking at various angular
distributions in the $l^{\pm}jj+\cancel{E}_{T}$ final states.

The rest of the article is organized as follows. In the next section (section
II), we briefly review the models IDM and MSSM, along with benchmark points
for the new physics parameters such as masses and couplings. In section III we
discuss the signal and corresponding SM background, followed by the analysis
setup. In section IV.1 we study various kinematic and angular distributions
with longitudinal beam polarization and perform an analysis with a simple cut
and count on the variables. We then try to determine the spin of the exotic
charged particles and their partner with the help of transverse beam
polarization in section IV.2. Finally, we conclude in section V.

##  II Representative model

We choose two well-motivated models, IDM and MSSM, as examples of having
exotic charged particle and their neutral partner (potential dark matter) of
scalar type and fermionic type, respectively. These models are briefly
described below to self contain this article.

###  II.1 The Inert Doublet Model

In the Inert Doublet Model (IDM) Barbieri:2006dq , the scalar sector of the
Standard Model (SM) is modified with one additional scalar doublet $\Phi$,
which is odd ($\Phi\to-\Phi$) under a new discrete $\mathbb{Z}_{2}$ symmetry
(parity). The SM particles together with the SM Higgs doublet ($H$) are even
under this $\mathbb{Z}_{2}$ symmetry. The two scalar doublets which transform
under $SU(2)_{L}$ can be written as,

$H=\left(\begin{array}[]{c}G^{+}\\\
\frac{1}{\sqrt{2}}\left(v+h+iG^{0}\right)\end{array}\right),\
\Phi=\left(\begin{array}[]{c}H^{+}\\\
\frac{1}{\sqrt{2}}\left(H^{0}+iA^{0}\right)\end{array}\right),$ (2)

where $v=\sqrt{2}\leavevmode\nobreak\ \langle 0|H|0\rangle\approx 246$ GeV is
the vacuum expectation value of the neutral component of $H$. The $h$ state
corresponds to the physical SM-like Higgs boson, whereas $G^{0}$ and $G^{\pm}$
are the Goldstone bosons. The “inert” sector consists of a neutral CP-even
scalar $H^{0}$, a pseudo-scalar $A^{0}$, and a pair of charged scalars
$H^{\pm}$. The neutral inert Higgs ($H^{0}$) and its charged partner
($H^{\pm}$) play the role of dark matter and the new exotic charged particle,
respectively.

The scalar potential of the model is given by,

$\displaystyle V$
$\displaystyle=\mu_{1}^{2}|H|^{2}+\mu_{2}^{2}|\Phi|^{2}+\lambda_{1}|H|^{4}+\lambda_{2}|\Phi|^{4}+\lambda_{3}|H|^{2}|\Phi|^{2}+\lambda_{4}|H^{\dagger}\Phi|^{2}+\frac{\lambda_{5}}{2}\Bigl{[}(H^{\dagger}\Phi)^{2}+\mathrm{h.c.}\Bigr{]}.$
(3)

The masses and interactions of the scalar sector are governed by the scalar-
potential parameters

$\left\\{\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4},\lambda_{5},\mu_{2}\right\\},$
(4)

where $\mu_{1}^{2}$ is eliminated by
$M_{h}^{2}=-2\mu_{1}^{2}=2\lambda_{1}v^{2}$ which is obtained by minimizing
the scalar potential after EWSB. The IDM parameter space can be expressed in
terms of physically more intuitive set

$\left\\{M_{h},\ M_{H^{0}},\ M_{A^{0}},\ M_{H^{\pm}},\ \lambda_{L},\
\lambda_{2}\right\\},$ (5)

where the Higgs and inert scalar masses are given by

$\displaystyle M_{h}^{2}$ $\displaystyle=\mu_{1}^{2}+3\lambda_{1}v^{2},$ (6)
$\displaystyle M_{H^{0}}^{2}$ $\displaystyle=\mu_{2}^{2}+\lambda_{L}v^{2},$
(7) $\displaystyle M_{A^{0}}^{2}$
$\displaystyle=\mu_{2}^{2}+\lambda_{S}v^{2},$ (8) $\displaystyle
M_{H^{\pm}}^{2}$ $\displaystyle=\mu_{2}^{2}+\frac{1}{2}\lambda_{3}v^{2},$ (9)

and the couplings $\lambda_{L},\lambda_{S}$ are defined as

$\displaystyle\lambda_{L}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\lambda_{3}+\lambda_{4}+\lambda_{5}\right)$
$\displaystyle\lambda_{S}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\lambda_{3}+\lambda_{4}-\lambda_{5}\right).$
(10)

The interaction Lagrangian in our signal comprising the production and decay
vertices of $H^{\pm}$ originating from gauge interactions can be written as,

$\displaystyle{\mathscr{L}}$ $\displaystyle=$ $\displaystyle
i\left[g_{Z}(1/2-s_{W}^{2})Z^{\mu}+eA^{\mu}\right]\left[\left(\partial_{\mu}H^{+}_{S}\right)H^{-}_{S}-\left(\partial_{\mu}H^{-}_{S}\right)H^{+}_{S}\right]$
(11)
$\displaystyle+(g/2)\left[-\left(\partial^{\mu}H^{+}_{S}\right)H^{0}_{S}W^{-}_{\mu}+\left(\partial^{\mu}H^{0}_{S}\right)H^{+}_{S}W^{-}_{\mu}+h.c.\right].$

Here, $e=\sqrt{4\pi\alpha}$ with $\alpha$ being fine structure constant, $g$
as the $SU(2)_{L}$ gauge couplings, $g_{z}=e/(s_{W}c_{W})$,
$s_{W}=\sin\theta_{W}$, and $c_{W}=\cos\theta_{W}$ with $\theta_{W}$ being the
Weinberg angle.

###  II.2 Minimal supersymmetric SM

We briefly discuss the basic setup of the model and the relevant spectrum used
in our analysis. The Minimal Supersymmetric SM (MSSM) Drees:2004jm  is the
supersymmetric extension of SM where for every SM fermion (boson), there is a
boson (fermion) superpartner. The MSSM has two Higgs doublet superfields with
opposite hypercharge (needed to cancel the resulting gauge anomaly).
Supersymmetry (SUSY) cannot be an exact symmetry as it would lead to similar
mass for the superpartners of the SM particles. SUSY is softly broken, and in
MSSM through explicit mass terms for the superpartners of the SM particles.
The Lagrangian in MSSM has a discrete global symmetry called R-Parity
($R_{P}$), defined as $(-1)^{3B-L+2S}$ whereas all SM particles are even under
$R_{P}$ while their superpartners have $R_{P}=-1$. This makes the lightest
SUSY particle (LSP) stable and is considered as a dark matter (DM) candidate.
For our analysis, we consider the LSP as a composition of the higgsino and
gaugino states of MSSM. The Lagrangian containing only higgsinos and gauginos
is given by,

$\displaystyle\mathscr{L}$ $\displaystyle\supset$ $\displaystyle
i\tilde{H}_{1i}\sigma^{\mu}{\Delta_{\mu}}_{ij}\bar{\tilde{H}}_{1j}+i\tilde{H}_{2i}\sigma^{\mu}{\Delta_{\mu}}_{ij}\bar{\tilde{H}}_{2j}+\mu\epsilon_{ab}\tilde{H}_{1a}\tilde{H}_{2b}+i\tilde{B}\sigma^{\mu}\partial_{\mu}\bar{\tilde{B}}+i\lambda_{i}\sigma^{\mu}{\Delta_{\mu}}_{ij}\bar{\lambda}_{j}$
(12) $\displaystyle-$
$\displaystyle\frac{1}{2}M_{1}\tilde{B}\tilde{B}-\frac{1}{2}M_{2}\lambda_{i}\lambda_{i}-\sqrt{2}g_{2}\tilde{H}_{1i}\lambda_{a}\frac{\sigma^{a}_{ij}}{2}{H}_{1j}-\sqrt{2}g_{2}\tilde{H}_{2i}\lambda_{a}\frac{\sigma^{a}_{ij}}{2}{H}_{2j}$
$\displaystyle-$
$\displaystyle\sqrt{2}g_{1}\frac{\hat{Y}_{i}}{2}\tilde{B}\tilde{H}_{1i}{H}_{1i}-\sqrt{2}g_{1}\frac{\hat{Y}_{i}}{2}\tilde{B}\tilde{H}_{2i}{H}_{2i}+h.c.,$

where

$\displaystyle\Delta^{\mu}_{ij}=\delta_{ij}\partial^{\mu}+ig_{1}\frac{\hat{Y}_{i}}{2}\delta_{ij}\,B_{\mu}+ig_{2}W^{\mu
a}\,T^{a}_{ij}$

Here $\hat{Y}$ and $T^{a}$ respectively represents hypercharge and $SU(2)$
generators in the respective representation of the field over these operator
act.

and

$\displaystyle\tilde{H}_{1}=\left(\begin{array}[]{c}\tilde{h^{1}_{1}}\\\
\tilde{h^{2}_{1}}\end{array}\right),$
$\displaystyle\tilde{H}_{2}=\left(\begin{array}[]{c}\tilde{h^{1}_{2}}\\\
\tilde{h^{2}_{2}}\end{array}\right),$ $\displaystyle\tilde{B},$
$\displaystyle\lambda_{s}=(\lambda_{1},\lambda_{2},\lambda_{3})$

are the fermionic super partner of two Higgs fields $H_{1}$, $H_{2}$ and the
super partners of $U(1)$ , $SU(2)$ gauge fields, respectively. Here the
neutral sector of SUSY fermionic partners of the SM bosons consist of neutral
Weyl fermions ($\tilde{B}$, $\lambda_{3}$, $\tilde{h}^{1}_{1}$,
$\tilde{h}^{2}_{2}$) and the charge sector comprises of ($\tilde{h^{1}_{2}}$,
$\tilde{h^{2c}_{1}}$,
$\lambda^{+}\equiv\frac{\lambda_{1}-i\lambda_{2}}{\sqrt{2}}$,
$\lambda^{-c}\equiv\frac{\lambda^{c}_{1}-i\lambda^{c}_{2}}{\sqrt{2}}$,
superscript ‘c’ being the charge conjugation operator) two component Weyl
fermions.

Spontaneous symmetry breaking is realised through vacuum expectation value
(VEV) for the two Higgs fields $H_{1}$ and $H_{2}$ with VEV
$\langle\frac{v_{1}}{\sqrt{2}}\rangle$ and
$\langle\frac{v_{2}}{\sqrt{2}}\rangle$ respectively. Here, we define Electro-
Weak VEV $v=\sqrt{v^{2}_{1}+v^{2}_{2}}$ with $\tan\beta=\frac{v_{2}}{v_{1}}$.

After electroweak symmetry breaking, the mass term for the higgsino and
Electro-Weakino charged sector becomes

$\mathscr{L}^{c}_{Mass}=-\frac{g_{2}}{\sqrt{2}}(v_{1}\lambda^{+}\tilde{h^{2}_{1}}+v_{2}\lambda^{-}\\\
\tilde{h^{1}_{2}}+h.c.)-(M_{2}\lambda^{+}\lambda^{-}+\mu\tilde{h^{2}_{1}}\tilde{h^{1}_{2}}+h.c.).$
(13)

In the basis $\psi^{+}=\left(\begin{array}[]{c}\lambda^{+}\\\
\tilde{h^{1}_{2}}\end{array}\right)$ and
$\psi^{-}=\left(\begin{array}[]{c}\lambda^{-}\\\
\tilde{h^{2}_{1}}\end{array}\right)$, the above Lagrangian can be written as

$-\mathscr{L}^{c}_{Mass}=(\psi^{-})^{T}\begin{pmatrix}M_{2}&\sqrt{2}M_{W}sin\beta\\\
\sqrt{2}M_{W}cos\beta&\mu\end{pmatrix}\psi^{+}.$ (14)

As the mass matrix is not symmetric, it has to be diagonalized by a bi-unitary
transformation,

$M^{D}_{c}=\mathscr{U}^{*}\begin{pmatrix}M_{2}&\sqrt{2}M_{W}sin\beta\\\
\sqrt{2}M_{W}cos\beta&\mu\end{pmatrix}\mathscr{V}^{-1},$ (15)

where $M^{D}_{c}$ is a diagonal matrix with real positive eigenvalues. Weyl
fermion eigenstates will be $\chi^{+}_{k}=\mathscr{V}_{km}\psi^{+}_{m}$ and
$\chi^{-}_{k}=\mathscr{U}_{km}\psi^{-}_{m}$, which can be written as four
component chargino field. The mass for the charginos can be written as

$-\mathscr{L}^{c}_{Mass}=\tilde{M_{1}}\overline{\tilde{\chi}^{+}_{1}}\tilde{\chi}^{+}_{1}+\tilde{M_{2}}\overline{\tilde{\chi}^{+}_{2}}\tilde{\chi}^{+}_{2},$
(16)

where $\tilde{\chi}^{+}_{1}\equiv\begin{pmatrix}\tilde{\chi}^{+}_{1}\\\
\overline{\tilde{\chi}^{-}_{1}}^{T}\end{pmatrix}$ and
$\tilde{\chi}^{+}_{2}\equiv\begin{pmatrix}\tilde{\chi}^{+}_{2}\\\
\overline{\tilde{\chi}^{-}_{2}}^{T}\end{pmatrix}$ are chargino fields. The
Lagrangian in the neutral electroweakino sector can be written as

$\mathscr{L}^{n}_{Mass}=-\frac{g_{2}}{2}\lambda_{3}(v_{1}\tilde{h}^{1}_{1}-v_{2}\tilde{h}^{2}_{2})+\frac{g_{1}}{2}\tilde{B}(v_{1}\tilde{h}^{1}_{1}-v_{2}\tilde{h}^{2}_{2})+\mu\tilde{h}^{1}_{1}\tilde{h}^{2}_{2}-\frac{1}{2}M_{2}\lambda_{3}\lambda_{3}-\frac{1}{2}M_{1}\tilde{B}\tilde{B}+h.c.$
(17)

The mass matrix, in the basis of
$\psi^{0}=(\tilde{B},\lambda_{3},\tilde{h}^{1}_{1},\tilde{h}^{2}_{2})$ is
given by

$\mathscr{M}^{n}=\begin{pmatrix}M_{1}&0&-M_{z}c_{\beta}s_{W}&M_{z}s_{\beta}s_{W}\\\
0&M_{2}&M_{z}c_{\beta}c_{W}&-M_{z}s_{\beta}c_{W}\\\
-M_{z}c_{\beta}s_{W}&M_{z}c_{\beta}c_{W}&0&-\mu\\\
M_{z}s_{\beta}s_{W}&-M_{z}s_{\beta}c_{W}&-\mu&0\end{pmatrix}.$ (18)

This matrix can be diagonalized by a unitary matrix $Z$. The physical mass
eigenstates and the mass diagonalization matrix are given by
$\chi^{0}_{l}=Z_{ln}\psi^{0}_{n}$ and $Z^{*}\mathscr{M}Z^{-1}=M^{n}_{D}$,
respectively. In the four component notation the mass term in the Lagrangian
for neutralinos can be written as
$\mathscr{L}^{n}_{Mass}=-\frac{1}{2}\sum_{l}\tilde{M}^{n}_{l}\overline{\tilde{\chi}^{0}_{l}}\tilde{\chi}^{0}_{l}$,
where $\tilde{\chi}^{0}_{l}=\begin{pmatrix}\chi^{0}_{1}\\\
{\overline{\chi}^{0}_{l}}^{T}\end{pmatrix}$ are the four Majorana neutralino
fields. The lowest mass eigenstate of four neutralinos will be the LSP and
represents the DM candidate.

The masses and vertices relevant for our analysis except the gauge couplings
depend only on four parameters which are the soft breaking gaugino mass
parameters ($M_{1}$ and $M_{2}$), the higgsino mass parameter ($\mu$) and the
ratio of the VEV of the Higgs doublets ($\tan\beta$). The lightest chargino
mass eigenstate coupling to $Z$ boson depends on the chargino mixing matrix
elements $\mathscr{V}_{11}$ and $\mathscr{U}_{11}$ Choi:1998ut . For
simplicity we keep these matrix elements fixed for all our benchmark points
which can be obtained by varying the input parameters $M_{2}$, $\mu$ and
$\tan\beta$ of the model.

###  II.3 Benchmark selection

Benchmark | Masses
---|---
BP1 |  $M_{\pm}=160$ GeV, $M_{0}=60$ GeV
BP2 |  $M_{\pm}=220$ GeV, $M_{0}=100$ GeV
BP3 |  $M_{\pm}=220$ GeV, $M_{0}=120$ GeV
BP4 |  $M_{\pm}=300$ GeV, $M_{0}=10$ GeV
Table 1:  Benchmark points for the MSSM and IDM.

We choose four benchmark points (BPs), in our analysis, for the masses of
exotic charged particles ($C^{\pm}$) and their neutral partner ($C^{0}$) in
both IDM and MSSM. The benchmark points are listed in Table 1. The BPs are
chosen in such a way that $\Delta M=M^{+}-M^{0}$ ($M^{+}=M_{C^{+}}$,
$M^{0}=M_{C^{0}}$) remain fixed for two BPs with different $M^{+}$ and $M^{0}$
(BP1 and BP3); Two BPs have the same $M^{+}$ but with a different $\Delta M$
(BP2 and BP3); Three BPs satisfy these criteria. We choose one more BP which
has a different $M^{+}$ and $\Delta M$ compared to the other three (BP4).

The mass parameters are kept the same in both models in order to have similar
kinematic behavior. Through all four BPs, the mass difference between charged
odd particle with respective DM particle is kept higher than the mass of
$W$-boson, so that both $W$ boson and DM particle can be produced on-shell
from their parent charged dark sector particle. In IDM, there are two coupling
parameters, $\lambda_{L}$ and $\lambda_{S}$, which can affect the SM Higgs
signal in our study. The value of $\lambda_{L}$ is kept very small so that the
second neutral CP-even Higgs mass ($m_{H_{0}}$) which is the DM, is nearly
equal to $\mu_{2}$ (see, Eq. (7)). The value of $\lambda_{S}$ is chosen such
that $m_{A_{0}}-m_{H^{+}}=1$ GeV. These two choices help us in evading the
bound on Higgs invisible decay as well as electroweak precision observables.
For all four BPs of MSSM we have kept the chargino mixing matrices to be the
same with $\mathscr{V}_{11}\equiv\cos\theta_{R}=0.9725$ and
$\mathscr{U}_{11}\equiv\cos\theta_{L}=0.9168$. We do not focus on whether the
benchmarks satisfy the requirements of dark matter relic density. However, we
keep BP1 as a reference point which does satisfy dark matter constraints
Belyaev:2021ngh . Apart from the dark matter constraint, there are other
constraints that are considered while choosing the BPs, which we discuss
below.

###  II.4 Constraints

####  II.4.1 Vacuum stability and unitarity

In IDM, the scalar potential must be bounded from below. The condition on
various quartic couplings to satisfy this constraint are given by
Belyaev:2016lok

$\displaystyle\lambda_{1}>0,\,\,\,\,\lambda_{2}>0,$ $\displaystyle
2\sqrt{\lambda_{1}\lambda_{2}}+\lambda_{3}>0,$ $\displaystyle
2\sqrt{\lambda_{1}\lambda_{2}}+\lambda_{3}+\lambda_{4}-|\lambda_{5}|>0.$ (19)

We have checked that all our four BPs for IDM satisfy the above conditions. In
addition, we have also checked that the BPs satisfy unitarity bounds
Belyaev:2016lok .

####  II.4.2 Electroweak precision observables

The BSM particles affect the electroweak (EW) observables via oblique
correction and these corrections are parameterized by three observables
$S,\,T$ and $U$ as electroweak precision observables (EWPO). The contribution
to EWPO ($S$ and $T$) from IDM are given by Belyaev:2016lok

$S=\dfrac{1}{{72\pi\left(x_{2}^{2}-x_{1}^{2}\right)^{3}}}\left[x_{2}^{6}f_{a}(x_{2})-x_{1}^{6}f_{a}(x_{1})+9x_{2}^{2}x_{1}^{2}\left(x_{2}^{2}f_{b}(x_{2})-x_{1}^{2}f_{b}(x_{1})\right)\right],$
(20)

where $x_{1}=\frac{M_{H^{0}}}{M_{H^{\pm}}}$,
$x_{2}=\frac{M_{A}}{M_{H^{\pm}}}$, $f_{a}(x)=-5+12\log(x)$,
$f_{b}(x)=3-4\log(x)$ and

$T=\dfrac{1}{{32\pi^{2}\alpha
v^{2}}}\left[f_{c}\left(M_{H^{+}}^{2},M_{A}^{2}\right)-f_{c}\left(M_{A}^{2},M_{H^{0}}^{2}\right)+f_{c}\left(M_{H^{+}}^{2},M_{H^{0}}^{2}\right)\right].$
(21)

where

$\displaystyle
f_{c}(x,y)=\begin{cases}\frac{x+y}{2}-\frac{xy\log\left(\frac{x}{y}\right)}{x-y},&{x\neq
y}\\\ 0,&{x=y}.\end{cases}$

The experimental values of $S$ and $T$ with $U=0$ is given by Haller:2018nnx

$S=0.04\pm 0.08,\,\,{\rm and}\,\,T=0.08\pm 0.07.$ (22)

In our case, BP4 with the largest $\Delta M=M_{H^{\pm}}-M_{H^{0}}$ in IDM
gives the maximum deviation in EWPO with $\Delta S=-\,0.0218$ and $\Delta
T=-\,0.0017$. These values are well within the $1\sigma$ limit of experimental
uncertainty.

####  II.4.3 Higgs invisible decay

We note that only BP1 and BP4 have $m_{h}>2\,m_{DM}$. Hence the SM Higgs can
decay to two DM particles for these benchmark points. The observed value of
Higgs invisible branching fraction is less than 0.18 at $95\%$ C.L.
CMS:2022qva . For MSSM, our branching fraction ($BR(h\to 2\bar{\chi}^{0})$) is
around $2\,\%$ and $4\,\%$ for BP1 and BP4 respectively. In IDM, the value of
$\lambda_{L}$ which enters into the interaction strength is kept fixed at
$10^{-4}$ for all BPs, which helps us keep the invisible branching ratio (BR)
of $h\to H_{0}\,H_{0}$ below $1\,\%$ for both BP1 and BP4.

####  II.4.4 Higgs Signal

The charged Higgs and chargino being at the electroweak scale, can contribute
to the loop-induced $h\to\gamma\gamma$ decay. This can affect the Higgs signal
observations at the LHC in this channel. The observed value of the signal
strength is constrained by ATLAS ATLAS:2022tnm  as

$\mu_{\gamma\gamma}=\frac{BR_{BSM}(h\to\gamma\gamma)}{BR_{SM}(h\to\gamma\gamma)}=1.04_{-0.09}^{+0.10}\,\,.$

For IDM, the maximum contribution to $BR(h\to\gamma\gamma)$ comes from BP1
which has the lightest $H^{+}$, and the total Higgs signal strength in this
particular channel with respect to the SM predictions i.e.,
$\mu_{\gamma\gamma}^{\rm IDM}=0.975$. The deviation from the SM prediction
goes down as the charged Higgs mass increases. Similarly for MSSM, BP1 gives
the maximum contribution to $BR(h\to\gamma\gamma)$ with
$\mu_{\gamma\gamma}^{\rm MSSM}=1.21$. We find that the Higgs signal strength
for both IDM and MSSM are within $1\sigma$ and $2\sigma$ of the allowed signal
strength, respectively.

##  III Signal and background

Figure 1:  Representative Feynman diagram for the signal process
$e^{+}\,e^{-}\rightarrow C^{+}\,C^{-}\rightarrow W^{+}\,W^{-}\,2C^{0}$.

The signal for our analysis comes from the pair production of the exotic
charged particles ($C^{\pm}$) followed by their decay to their neutral partner
($C^{0}$) and charged gauge boson $W^{\pm}$. It is worth pointing out here
that most models, which are proposed to give a DM candidate in the spectrum
exhibit a similar chain of production where the final state involves a pair of
DM particles. The visible particle multiplicity associated with the two DM
final states depends on the particle being produced in the hard scattering
process. The challenging aspect of obtaining good signal sensitivity then
relies on the strength with which the hard scattering process takes place.
Producing the electroweak strength particles at LHC leads to very weak
sensitivities and therefore makes the spin determination very difficult. We,
therefore, consider their production at ILC in the simplest and most obvious
mode of production process. We choose the channel where one of the $W$ decays
leptonically while the other one decays hadronically, i.e.,

$e^{+}e^{-}\to C^{+}C^{-},\leavevmode\nobreak\ C^{\pm}\to
C^{0}W^{\pm},\leavevmode\nobreak\ W^{\pm}\to
l^{\pm}\nu_{l}/\bar{\nu_{l}},\leavevmode\nobreak\ W^{\mp}\to jj,$ (23)

forming a final state of $l^{\pm}2j+\not{E}_{T}$. The Feynman diagram up to
$W^{\pm}$ production is shown in Fig. 1. The signal process in IDM and MSSM,
along with the SM background, are described below.

##### IDM Signal:

In IDM, $H^{+}H^{-}$ is produced via two $s$-channel processes with $Z$ boson
and $\gamma$ as mediators from initial $e^{-}e^{+}$. Then $H^{\pm}$ decay to
$W^{\pm}H_{0}$ where $H_{0}$ is stable while the $W$ boson decays to lepton
and hadrons producing one lepton, two jets and missing energy final states. We
have kept the pseudoscalar $A_{0}$ mass higher than $H^{\pm}$. Since there are
no other lighter particles which are $Z_{2}$ odd, branching ratio (BR) of
$H^{\pm}$ to $W^{\pm}H_{0}$ channel will be $100\%$. The two DM particles
$H_{0}$ and neutrino act as the source of missing energy.

##### MSSM Signal:

In MSSM, similar $s$-channel diagrams can produce the lightest chargino
$\chi^{\pm}_{1}$ at ILC. Although there is a possibility of a $t$-channel
diagram due to a sneutrino, we assume that all scalar superpartner masses are
very heavy including the sneutrino, which is kept at a mass of 12 TeV. Hence
the contribution from the sneutrino exchange will be much suppressed due to
the large mass of sneutrino. The charginos then decay to $W^{\pm}\chi_{0}$ and
further decay of $W^{\pm}$ as before, giving the desired final states with the
neutralinos now playing the role of DM. For our analysis, we have taken all
$R_{P}$ odd particles except the lightest neutralino heavier than the mass of
$\chi^{\pm}_{1}$ which again gives its decay BR to $W^{\pm}\,\chi_{0}$ as
$100\%$. Here too the missing energy gets contribution from the two DM states
$\chi_{0}$ and a neutrino.

##### SM Background:

The dominant background for our signal is the $W^{+}W^{-}$ production in SM
where one $W$ boson decays leptonically while the other one decays
hadronically. There are three major sub-processes in SM contributing to
$W^{-}W^{+}$ production, with two being $s$-channel processes having photon
and $Z$ boson propagator and the other and most dominant $t$-channel process
with the light neutrino running in the propagator. The $W^{-}W^{+}Z$
production with $Z\to\nu\bar{\nu}$ also produces the same final state, but
this background is negligibly small (roughly a hundred times smaller in cross
section than $W^{+}W^{-}$ production) due to an additional electroweak
coupling and phase-space suppression. Additionally the small branching for
$Z\to\nu\bar{\nu}$ decay also makes the contributions from this process
weaker. We can therefore safely ignore the $W^{-}W^{+}Z$ contributions for the
background in our analysis.

We wish to study the polarization and spin of the intermediate exchanged as
well as produced particles. We have generated the signal and background events
in the package WHIZARD Kilian:2007gr  at ILC with $\sqrt{s}=1$ TeV center of
mass energy. We generate the events with the initial state radiation (ISR)
effect switched on. During the event generation, we have put a set of
inclusive cuts with $P_{{T}_{j_{1},j_{2},l}}>10$ GeV and the invariant mass of
jets $M_{jj}$ to lie between $60$ GeV to $100$ GeV. Then the simulated events
are showered in Pythia8 Sjostrand:2014zea  for energy smearing effects. After
the hadronization of the final state jets we perform the fast detector
simulation in Delphes-3 deFavereau:2013fsa  with the International Linear
Detector (ILD) card. Events are selected at the detector level with the
following selection cuts,

$\displaystyle p_{T_{j}}>30\leavevmode\nobreak\
\text{GeV},\leavevmode\nobreak\ p_{T_{l}}>30\leavevmode\nobreak\
\text{GeV},\leavevmode\nobreak\ \cancel{E}_{T}>30\leavevmode\nobreak\
\text{GeV},\leavevmode\nobreak\ 60\leavevmode\nobreak\
\text{GeV}<M_{jj}<100\leavevmode\nobreak\ \text{GeV},$
$\displaystyle\leavevmode\nobreak\
|\eta_{j}|<4.5,|\eta_{l}|<2.5,\leavevmode\nobreak\ \Delta
R_{j,j}>0.5\leavevmode\nobreak\ ,\Delta R_{l,l}>0.5,\leavevmode\nobreak\
\Delta R_{l,j}>0.5.$ (24)

We name these cuts as Sel_cut.

##  IV Signal analysis at ILC with beam polarization

We now present our signal analysis using polarized beams in the initial
states. Initial-state polarization is a useful diagnostic at the ILC and by
adjusting initial-state polarizations, one can select specific states
preferentially. In this work, we present our analysis by using the option of
using both $e^{-}$ and $e^{+}$ beams longitudinally polarized as well as
having them transversely polarized. Since we want to show how the spin of the
newly produced particles could show up in some kinematic variables, it is
useful to find out which polarization option will be best suited to help
identify such states with better confidence.

Unlike the LHC, where the parton collisions are over a range in center of mass
energies, we know that ILC will have a more or less fixed center of mass
energy for the $e^{-}e^{+}$ collisions. A small spread in the collision energy
however arises from Initial State Radiation (ISR) at the ILC, which is the
most important QED correction to the Born cross section Yennie:1961ad .
Bremsstrahlung effects are an important source of ISR, and it is the radiation
caused by the interaction between the electron and positron participating in
the annihilation event at $e^{+}e^{-}$colliders. Thus its effects must be
considered for realistic simulation to study physics signals at future linear
colliders such as the ILC. The radiative corrections to processes with
arbitrary final states need evaluation to achieve precision measurements. The
ISR photons are generally soft with small transverse momenta, so they
eventually escape detection. However, their effect is imprinted in the physics
analysis through modification of the colliding beam energies and an effective
boost along the beam axis for the final states. For a realistic analysis we,
therefore, include the ISR effects in our study. A somewhat subdominant
correction for the center of mass energy also comes from another phenomenon
called beamstrahlung which we have neglected.

###  IV.1 Analysis with longitudinal beam polarization

| IDM Cross section (fb) | MSSM Cross section (fb)
---|---|---
Benchmark | Mass($M^{\pm}$, $M_{0}$) (GeV) | Un-pol | Lpol | Un-pol | Lpol
BP1 | $(160,60)$ | $5.911$ | $14.331$ | $55.163$ | $158.900$
BP2 | $(220,100)$ | $4.996$ | $12.103$ | $54.959$ | $158.136$
BP3 | $(220,120)$ | $5.006$ | $12.146$ | $54.799$ | $157.702$
BP4 | $(300,10)$ | $3.699$ | $8.966$ | $54.445$ | $156.363$

Table 2: The cross sections of signal for the final state
$e^{-}\,e^{+}\rightarrow l^{\pm}+2j+\not{E}_{T}$ after the basic generation
level cuts.

We perform our collider analysis in the chosen final state with longitudinal
beam polarization (LPol) of initial $e^{\pm}$ beams primarily because of the
larger cross sections than with the unpolarized (Un-Pol) beams; The degree of
polarization for ($e^{-}$, $e^{+}$) beams are chosen to be ($-\,80\,\%$ ,
$+\,60\,\%$ ). The SM $WW$ background has a cross section of about $680.954$
fb in Un-Pol, and $1957.84$ fb in LPol with generation level cuts given in Eq.
(III). The larger cross sections are effectively due to the choice of
polarization that enhances left-handed current contribution.

The cross sections of the signal are larger in LPol too, compared to Un-Pol in
all four BPs, shown in Table 2, by roughly the same factor by which the
background cross section is larger in LPol compared to Un-Pol. Thus, signal
significance improves in LPol than in Un-Pol, even without any kinematic cuts
to reduce SM background. From Table 2, it follows that IDM has
$\mathscr{O}(1)$ less cross section than MSSM. This can be understood in the
following way. In the massless limit (boost $\beta\to 1$), for the photon
mediated diagram, the total pair production cross section of $H^{\pm}$ is four
times smaller than the pair production cross section of $\chi_{1}^{\pm}$ at an
electron-positron collider. The reason for the enhanced cross section of
$\chi_{1}^{\pm}$ pair production is that there are four ways of combining
helicity states of $\chi_{1}^{\pm}$ and $e^{\pm}$, while for scalar there is
only one degree of freedom for $H^{\pm}$ each. The production cross section
for a pair of fermions again goes up when we consider them to be massive. For
the photon mediated diagram, the cross sections follow as

$\sigma(e^{-}e^{+}\to
H^{-}H^{+})\propto\frac{2}{3}e^{4}\beta^{3}\,\,\text{and}\,\,\sigma(e^{-}e^{+}\to\chi_{1}^{-}\chi_{1}^{+})\propto-\frac{4}{3}e^{4}\beta(-3+\beta^{2})$
(25)

for unpolarized initial beams (see Eqs. (56), (B.0.3), and (47)). Similarly
for the diagram with $Z$-boson propagator, the factor associated with
$Z$-boson vertex $(T_{3}-\sin^{2}\theta_{W}\,Q)$ is larger for chargino than
the charged Higgs. In our case, the lightest chargino is wino dominated.
Following the notation of Appendix B, the ratio between the production cross
section of the chargino pair and the charged Higgs pair is
$4\left(a_{\chi}^{2}+v_{\chi}^{2}\right)/\left(g_{2}\cos{\theta_{W}}-g_{1}\sin{\theta_{W}}\right)^{2}=27.24$
for unpolarized beam in $\beta\rightarrow 1$ limit. The individual cross
section from the photon and the $Z$ diagrams are of the same order and they
interfere destructively. Considering both the photon and the $Z$ mediated
diagram, we get chargino pair production cross section nearly $8$ times larger
than the charged Higgs pair production cross section for unpolarized beam in
$\beta\rightarrow 1$ limit. This ratio increases further when we consider the
final state particles to be massive. However, it should be noted that in
generic scenarios not restricted to SUSY where interactions with such exotics
emerge from suppressed mixing angles, smaller branching fractions in the decay
modes or additional subprocesses contributing destructively can lead to
somewhat comparable cross sections with the scalar production. Nevertheless,
the large cross section for the fermion production is always a good
discriminator when compared to scalar production in simple setups.

##### Kinematic variables

Figure 2:  Normalized distributions of Kinematic variables in
$e^{-}\,e^{+}\rightarrow l^{\pm}+2j+\not{E}_{T}$ with $\sqrt{s}=1$ TeV and
longitudinal polarized beams for BP1.

Figure 3:  Normalized distribution of $\cancel{E}_{T}$ for all four BPs of the
IDM and MSSM along with SM background in $e^{-}\,e^{+}\rightarrow
l^{\pm}+2j+\not{E}_{T}$ with $\sqrt{s}=1$ TeV and longitudinal polarized
beams.

The role of kinematic variables to analyse the signal against the SM
background, which naturally arises in the final state
$l^{\pm}2j+\cancel{E}_{T}$ are missing transverse energy ($\cancel{E}_{T}$),
transverse momentum of lepton and jets ($p_{T_{l}}$, $p_{T_{j_{1}}}$,
$p_{T_{j_{2}}}$), total visible mass ($M_{lj_{1}j_{2}}$), and energy of $jj$
pair which reconstructs the $W$ boson energy ($E_{W}$) and lepton ($E_{l}$).
Normalized distribution of these kinematic variables are shown in Fig. 2 for
BP1 as representative with longitudinally polarized beams at $1$ TeV ILC set
up for both IDM (in dashed/blue line) and MSSM (in dotted/red line) signal and
SM background (in solid/green line).

The $\cancel{E}_{T}$ distribution (left-top panel in Fig. 2) shows a peak in
the lower energy values for the SM background compared to the signal. This is
because the source of $\cancel{E}_{T}$ in the SM is the neutrino coming from
the $W$ boson decay, while for the signal, we have additional contributions
coming from the undetected stable exotics $C^{0}$ along with a neutrino.
Further, in the case of signal, the peak of $\not{E}_{T}$ shifts toward higher
values as the splitting between the mass of the charged exotic and the stable
neutral particle ($\Delta M$) increases, which can be seen in Fig. 3 , where
the SM background along with all four BPs of IDM and MSSM are shown. The peak
for BP1 and BP3 with $\Delta M=100$ GeV are at the same energy, while the peak
for BP2 ($\Delta M=120$ GeV) and BP4 ($\Delta M=290$ GeV) are at the higher
energy side in $\cancel{E}_{T}$ distribution due to more availability of
energy for missing $C^{0}$ in the rest frame of it’s charged partner.

We note that the two jets in the signal as well as the SM background come from
the $W$ boson decay. Therefore it is possible to reconstruct the energy of the
$W$ using the two jets. Ideally, for the SM background the $W$ bosons should
be produced back to back with $E_{W}=\sqrt{s}/2=500$ GeV. However, the ISR
effect gives an effective boost as well as changes the hard scattering
collision energy, which along with hadronization and detector effects, result
in the smearing of energies for the $W$ boson ($jj$ pair) and lepton as well
as a longish tail in the energy distribution of $E_{W}$ extending to the
kinematic threshold of $\sqrt{s_{eff}}\geq 2M_{W}$ of the process. On the
other hand, in the case of signal, the $W$ boson is produced from the decay of
$C^{\pm}$, along with missing $C^{0}$. The available energy for $W$ is much
lower than that of SM and is highlighted in the energy distribution for
$E_{W}$. The difference between the endpoint of $E_{W}$ in SM and BSM models
will be at least the $C^{0}$ mass. Similarly, the presence of missing $C^{0}$
can be seen from its imprint in other kinematic variables as well, given by
$E_{l}$, $M_{lj_{1}j_{2}}$ and $P_{T}$’s of jets and lepton, where the
endpoint in these distributions for the signal shifts towards the low energy
values as compared to SM background. Hence, rejection cuts near the endpoint
of these variables help reduce the SM background significantly while keeping
enough statistics for the signal.

We thus, implement a set of cuts on these kinematic variables as
$\not{E}_{T}>50$ GeV, $P_{T_{l,j_{1}}}<400$ GeV, $M_{l,j_{1},j_{2}}<700$ GeV,
$E_{l}<400$ GeV and $E_{W}<500$ GeV to reduce the SM background while keeping
enough statistics for the signal. We name these sets of cuts as Kin_cut.

##### Angular variables

Figure 4:  Normalized distribution of Angular variables in
$e^{-}\,e^{+}\rightarrow l^{\pm}+2j+\not{E}_{T}$ with $\sqrt{s}=1$ TeV and
longitudinal polarized beams for BP1.

| SM background | IDM signal | MSSM signal
---|---|---|---
Cuts | $W^{+}W^{-}$ |  BP1 |  BP2 |  BP3 |  BP4 |  BP1 |  BP2 |  BP3 |  BP4
Sel_cut | 22709.6 | 558.29 | 462.28 | 456.62 | 321.47 | 5517.66 | 5337.85 | 4948.52 | 5743.17
Sel_cut+Kin_cut | 3256.22 | 491.49 | 420.57 | 400.05 | 295.14 | 4738.46 | 4789.66 | 4224.54 | 5153.8
Sel_cut+Kin_cut +Ang_cut | 1942.75 | 479.75 | 410.05 | 392.64 | 274.70 | 4318.11 | 4352.4 | 3924.35 | 4497.23
| Significance ($\mathscr{S}$) | 10.47 | 9.00 | 8.63 | 6.09 | 77.56 | 78.08 | 71.55 | 80.25

Table 3:  The cut-flow information on the $e^{-}\,e^{+}\rightarrow
l^{\pm}+2j+\not{E}_{T}$ process for both the signals and background along with
the significances for all four BPs at the $1$ TeV ILC for $100$ fb-1 of
integrated luminosity.

It is quite well known that particle spin dictates the Lorentz structure of
interaction vertices, and an efficient way of identifying these properties is
through observing the kinematics in the angular variables of final state
particles. For a scattering process such as $e^{+}e^{-}\to f\bar{f}$, where
$f$ is a stable SM fermion; it is clearly highlighted in the polar
distributions of the final state fermion. However, it becomes more challenging
if the final state particles come from cascade decay of unstable particles
produced as primaries in the $e^{+}e^{-}$ collisions, as in our case. We,
therefore, explore the properties by looking at the normalized distributions
of some angular variables such as $\cos\theta$ of $W$ ($jj$ pair) and lepton
and the isolation variable $\Delta R=\sqrt{\Delta\eta^{2}+\Delta\phi^{2}}$
between the lepton and $W$, which are shown in Fig. 4 for the SM background
and signal (IDM and MSSM) for BP1 as representative. Owing to a $t$-channel
sub-process in the SM process, $\cos\theta_{W}$ sharply peaks at $\pm 1$,
i.e., along with the beam directions: The $W^{-}$ tends to remain toward
$e^{-}$ direction, while the $W^{+}$ tends to remain toward $e^{+}$ direction,
see left-top panel in Fig. 4. On the other hand, there are only two
$s$-channel sub-process in the IDM signal, which makes the $\cos\theta_{W}$
peak at $0$, i.e., transverse to the beam directions. In the case of the MSSM
signal, there could be a $t$-channel sub-process due to sneutrino, which is
kept heavy not to contribute to our analysis. Thus the $\cos\theta_{W}$
distribution has a small peak near $\pm 1$ with an overall flat shape
throughout. The angular behavior of $W$ transfers to its decay product thus,
lepton’s $\cos\theta$ distribution is of similar nature to that of $W$, see
right-top panel. So an inclusion cut of $|\cos\theta|<0.9$ for lepton and the
$W$ will be effective to suppress the background with less effect on signals.
In SM, the two intermediate $W$s are boosted in opposite directions, while in
the case of signals, the $W$s are less boosted and also not in opposite
directions as they are decayed from $C^{\pm}$ together with $C^{0}$s. Hence
the $\Delta R_{l,W}$ attains higher values as compared to that of signals, see
bottom panel in Fig. 4. We, thus, put a rejection cut of $\Delta R_{lw}>5.0$
to reduce SM background with less effect on the signals. We name the set of
cuts on the angular variables as Ang_cut.

We estimate signal significance with an integrated luminosity of ${\cal
L}=100$ fb-1 with the formulae

$\mathscr{S}=\sqrt{2\left[\left(S+B\right)\log\left(1+\frac{S}{B}\right)-S\right]}$
(26)

with $S$ being the number of signal events and $B$ being the number of
background events at a given luminosity. The signal significance of the IDM
signal and MSSM signal are shown in Table 3 for the successive cuts on
kinematic variables and angular variables in all four BPs. The significance
values confirm that all four BPs are above $5\leavevmode\nobreak\ \sigma$
discovery limits in our analysis.

Although there is a good distinction between the background and the signals,
it is ambiguous about the spin nature of particles contained in signals apart
from the fact that the MSSM signal has more significance than the IDM signal
with the same mass parameters. We try to estimate how the IDM signal is
different from the MSSM signal based on the asymmetry of some angular
variables in the following sub-section.

####  IV.1.1 Identifying the nature of signal

We now try and identify the spin nature of the exotic charged particle
($C^{+}$) and its partner ($C^{0}$) by taking account of the observations made
in the previous subsection. We try to identify the type of signal based on
asymmetries constructed from the angular variables $\cos\theta_{W}$ and
$\cos\theta_{l}$ for the signal inclusive of the background after effectively
reducing the background contribution maximally as discussed earlier. Then we
estimate the difference between the signal from the two models based on
$\chi^{2}$ using the asymmetries. We have already discussed the difference in
the shapes of $\cos\theta_{W/l}$ distribution, given in Fig. 4 for the MSSM
and IDM signal events. Based on the symmetric shape in $\cos\theta=0$, we
define an asymmetry for both the variables as

$\mathscr{A}\left(\cos\theta_{W/l}\right)=\frac{\sigma\left(|\cos\theta_{W/l}|>0.5\right)-\sigma\left(|\cos\theta_{W/l}|<0.5\right)}{\sigma\left(|\cos\theta_{W/l}|>0.5\right)+\sigma\left(|\cos\theta_{W/l}|<0.5\right)}.$
(27)

The difference is then estimated by a $\chi^{2}$, calculated as

$\chi^{2}=\sum_{i}\left|\dfrac{\Delta{\cal A}^{i}={\cal
A}_{\text{IDM+SM}}^{i}-{\cal
A}_{\text{MSSM+SM}}^{i}}{\delta\mathscr{A}_{\text{\text{SM}}}^{i}}\right|^{2},$
(28)

$i$ runs on the variables;
$\delta\mathscr{A}_{\text{\text{SM}}}=\sqrt{\frac{{1-\mathscr{A}^{2}_{\text{\text{SM}}}}}{\sigma_{\text{\text{SM}}}\times\mathscr{L}}}$
is the statistical error on asymmetries due SM background. The asymmetries in
each signal are estimated including the background after putting all the cuts
(Sel_cut+Kin_cut +Ang_cut), i.e., reducing maximum background so as to
highlight only the signal. The asymmetry of a signal ($S$) including
background ($B$) will be given as,

$\mathscr{A}_{S+B}=\frac{\mathscr{A}_{S}\sigma_{S}+\mathscr{A}_{B}\sigma_{B}}{\sigma_{S}+\sigma_{B}},$
(29)

i.e., weighted by their respective cross sections. The estimated values of
asymmetries of the variables in each signal mixed with the background are
shown in Table 4 (top two rows).

| $\mathscr{A}_{\text{SM}}$ | $\mathscr{A}_{\text{SM+IDM}}$ | $\mathscr{A}_{\text{SM+MSSM}}$
---|---|---|---
Variables |  $W^{+}W^{-}$ |  BP1 |  BP2 |  BP3 |  BP4 |  BP1 |  BP2 |  BP3 |  BP4
$\cos\theta_{W}$ | 0.1682 | 0.0632619 | 0.0663791 | 0.0573263 | 0.118111 | 0.0260318 | -0.030412 | -0.0775697 | -0.00320661
$\cos\theta_{l}$ | 0.291 | 0.169782 | 0.182949 | 0.180954 | 0.225528 | 0.0212554 | -0.0130694 | -0.0379408 | 0.0206557
| $\delta\mathscr{A}_{\text{SM}}$ | $\left|\frac{\Delta\mathscr{A}}{\delta\mathscr{A}_{\text{SM}}}\right|$
|  $W^{+}W^{-}$ |  BP1 |  BP2 |  BP3 |  BP4
$\cos\theta_{W}$ | 0.0223 | 1.66469 | 4.32789 | 6.0317 | 5.42458
$\cos\theta_{l}$ | 0.0217 | 6.84267 | 9.03066 | 10.0846 | 9.43857
$\chi^{2}$ | 49.5933 | 100.2823 | 138.0805 | 118.5126

Table 4:  Asymmetries for both the signals and background and difference
between the signals in the $e^{-}\,e^{+}\rightarrow\ell+2j+\not{E}_{T}$
process for longitudinally polarized beams for ${\cal L}=100$ fb-1 of
integrated luminosity.

Although the ${\cal A}_{S}$ for $\cos\theta_{W/l}$ seems to be $(-)$ve for IDM
signal and $(+)$ve for MSSM signal by looking Fig. 4, the ${\cal A}_{S+B}$ are
$(+)$ve for IDM, while they are $(-)$ve for MSSM in most of the BP. The
reasons are the following. The asymmetries in SM are large $(+)$ve, and the SM
cross section is larger by roughly a factor of $5$ compared to the cross
sections in IDM after all cuts, see Table 3. Hence the asymmetries in IDM
signal reduces but remain positive when combined with the SM background
weighted by cross sections in accordance with Eq. (29). For the case of MSSM,
individual asymmetries in signal become $(-)$ve after the inclusion cut
$|\cos\theta_{W/l}|<0.9$. Moreover, the cross sections are larger in MSSM
compared to SM by roughly a factor of $2$, see Table 3. This makes the ${\cal
A}_{S+B}$ to be $(-)$ve for most of the BP. The changes in the asymmetries
w.r.t. background are above $1\sigma$ statistical errors shown in the second
column in Table 4. Differences between the signals are calculated in terms of
$\chi^{2}$ and shown in Table 4 in the last three rows for all BPs for an
integrated luminosity of ${\cal L}=100$ fb-1. The two signals have a
difference of $\sim 10\sigma$ in all the four BPs based on the total
$\chi^{2}$ shown in the last row.

These asymmetries can help to identify the nature of the signal if observed
above the background. The asymmetries remain $(+)$ve for IDM, while they
become $(-)$ve and above statistical errors for BP2 and BP3 due to having
large $M_{0}$, and $(+)$ve and within $1\sigma$ error for other BPs.
Nevertheless, the characteristics of the asymmetries which can be useful to
distinguish between the signals are benchmark dependent. We require a variable
that can distinguish between the signals irrespective of the benchmark masses;
we will explore such a possibility in the following section by using
transverse beam polarizations.

###  IV.2 Analysis with transverse beam polarization

Figure 5:  Normalized distribution of the azimuthal angle $\phi$ of the
charged particle ($C^{+}$) at production level for the background and the
signals in BP1 in $e^{-}e^{+}$ collider with transversely polarized beams at
$\sqrt{s}=1$ TeV.

We observe in the previous section that though the angular variables
$\cos\theta_{W}$ and $\cos\theta_{l}$ are useful in identifying the nature of
the signal, they are limited to being dependent on benchmark selections. The
transversely polarized beams have the potential of exploring new physics from
the azimuthal distributions of final state particles Ananthanarayan:2004xf ;
Burgess:1990ba ; Rizzo:2002ww ; Ananthanarayan:2004eb ; Rindani:2004ue ;
Hikasa:1985qi ; Godbole:2009qv . Here, we use the facility of transverse
polarization of $e^{\pm}$ beams to showcase the distribution of azimuthal
angle $\phi$, which has a very distinctive nature for two signals irrespective
of the benchmark selections as well as the background. Before looking at the
distribution of $\phi$ for the final states, we first observe the distribution
at the production level and make an ansatz at the final state particle. We
calculate the normalized $\phi$ distribution for the $C^{+}$ analytically in
$e^{-}e^{+}\to C^{+}C^{-}$ process for SM as well as for the signals with
transverse beam polarization of ($80\%$, $60\%$) for ($e^{-}$, $e^{+}$) beams
and show them in Fig. 5 for BP1 with dotted lines. The same distributions for
$\phi_{C}$ are also shown with solid lines computed from events generated in
WHIZARD111ISR effect has been neglected here for simplicity.. The differential
cross section in $\phi_{C}$ has the form

$\frac{d\sigma}{d\phi_{C}}=\frac{\sigma}{2\pi}+\eta_{T}\xi_{T}f(\beta_{C})\cos\left(2\phi_{C}-\phi_{e^{-}}+\phi_{e^{+}}\right),$
(30)

where $f(\beta_{C})$ is a function of boost ($\beta$) for the particle $C$
with different form for different physics model, see appendix B for details.
We choose the spin direction for both the initial beams along the $(+)$ve
$x$-axis as an example; making the spin directions opposite for the two beams
would result in an overall phase-shift for all models not affecting our
findings. The explicit form of $f(\beta_{C})$ (Shown in Eqs. (49), (60) and
(70) for analytical expressions) makes the amplitude large for the IDM signal,
while negligibly small for the MSSM signal and the $W^{+}W^{-}$ background,
see Fig. 5. If the $\phi_{C}$ distribution in MSSM attains an amplitude
comparative to IDM in some other benchmark scenario, the nature will be
different due to having a relative $\pi$ phase shift compared to IDM, as can
be seen in Fig. 5.

Figure 6:  Normalized distribution of $\phi_{W}$ (left-panel ) and $\phi_{l}$
(right-panel) for the background and the signals in BP1 in $e^{+}e^{-}$
collider for $\sqrt{s}=1$ TeV with transversely polarized beams.

| $\mathscr{A}_{\text{SM}}$ | $\delta\mathscr{A}_{\text{SM}}$ | $\mathscr{A}_{\text{SM+IDM}}$ | $\mathscr{A}_{\text{SM+MSSM}}$
---|---|---|---|---
Variables |  |  |  BP1 |  BP2 |  BP3 |  BP4 |  BP1 |  BP2 |  BP3 |  BP4
$\phi_{W}$ | -0.0029 | 0.0379 | 0.0471 | 0.0382 | 0.0387 | 0.0116 | -0.0053 | -0.0032 | -0.0057 | -0.0038
$\phi_{l}$ | 0.0113 | 0.0379 | 0.0475 | 0.0378 | 0.0375 | 0.0222 | -0.0018 | 0.0009 | 0.0001 | 0.0027

Table 5:  Azimuthal asymmetries for both the signals and background in the
$e^{-}\,e^{+}\rightarrow\ell+2j+\not{E}_{T}$ process for transversely
polarized initial beams. The error in the SM is shown for ${\cal L}=100$ fb-1
of integrated luminosity.

This behavior of $\phi$ distribution for the mother particle $C^{+}$ is
transferred to its decay products which in this case are the lepton and
$W(jj)$ even after including the detector level effects as well as with ISR
effect, as seen in Fig. 6. Here also, the amplitude in IDM is large enough
compared to MSSM and SM, making the $\phi_{W/l}$ a good discriminator for
identifying the nature of signal. To depict a quantitative measure of
distinction between the two signals, we again construct asymmetries for the
$\phi_{W/l}$ given by

$\mathscr{A}\left(\phi_{W/l}\right)=\frac{\sigma\left(\cos\left(2\phi_{W/l}\right)>0\right)-\sigma\left(\cos\left(2\phi_{W/l}\right)<0\right)}{\sigma\left(\cos\left(2\phi_{W/l}\right)>0\right)+\sigma\left(\cos\left(2\phi_{W/l}\right)<0\right)}\leavevmode\nobreak\
.$ (31)

We have listed the asymmetry values in Table 5 for the SM background as well
as all four BPs combined with the background after the cuts Sel_cut+Kin_cut
+Ang_cut, as used in the previous section for our analysis with longitudinally
polarized beams. The asymmetries for both $\phi_{W}$ and $\phi_{l}$ are nearly
zero (much smaller than the $1\sigma$ statistical error given in the first and
second rows of the second column) for the background (shown in the second
column) and the MSSM signal. The asymmetries for the IDM combined with the
background, however, remain large enough even after mixing with the background
with a larger cross section compared to the signal. Although the asymmetry in
IDM reduces as the $m_{H}$ increases (see Eq. (60) for phi analytic) mainly
due to smaller cross sections, asymmetry can still be significant enough with
higher integrated luminosity. Thus these two distributions and their
asymmetries are able to identify the spin nature of $C^{+}$/$C^{0}$ contained
in the signal if a significant deviation is observed at future ILC. The signal
identification can be quantified with higher significance for higher
luminosity.

##  V Conclusion

In this article, we make an effort to identify the spin nature of exotic
charged particles and their neutral partner, possibly a dark matter candidate,
in $l^{\pm}2j+\cancel{E}_{T}$ final state at the future $e^{+}e^{-}$ collider.
We chose two well known BSM models such as IDM and MSSM, containing exotic
charged particles and neutral partners of spin zero (scalar) and half
(fermion), respectively, for the demonstration. First, we devised some
rectangular cuts on some kinematic as well as angular variables to reduce the
SM background in order to find a significant deviation from the SM background.
We then use the shape of angular variables and their asymmetries to identify
the nature of the new physics signals. The $\cos\theta_{W/l}$ distribution and
their asymmetries help distinguish and identify the MSSM and the IDM signals
over the SM backgrounds with longitudinally polarized beams depending on the
benchmark points. The transverse beam polarization helps to identify the IDM
signal robustly through the azimuthal ($\phi_{W/l}$) distribution of the final
state particles. Thus both longitudinal polarization and transverse
polarization prove useful in their respective ways in identifying the spin
nature of exotic charged particles and their partner contained in a new
physics signal if observed at the future $e^{-}e^{+}$ collider in
$l^{\pm}2j+\cancel{E}_{T}$ final state. We note that while the identification
gets help from the enhanced cross section in the longitudinal polarization due
to the large cross section, a more clear and more distinct identification is
established with the use of transversely polarized beam.

## Acknowledgments

The authors would like to acknowledge the support from DAE, India for the
Regional Centre for Accelerator based Particle Physics (RECAPP), Harish
Chandra Research Institute.

##  Appendix A Feynman rules

The Feynman rules for the vertices involved in the SM, IDM and MSSM processes
are

  * •

$H^{+}(k_{1})H^{-}(k_{2})A_{\mu}$ : $-i\,e\,(k_{2}-{k_{1}})_{\mu}$,

  * •

$H^{+}(k_{1})H^{-}(k_{2})Z_{\mu}$ :
$-i\,(-g_{1}\sin\theta_{W}+g_{2}\cos\theta_{W})\,(k_{2}-{k_{1}})_{\mu}$,

  * •

$\chi^{+}_{1}\chi^{-}_{1}A_{\mu}$ : $-i\,e\,\gamma_{\mu}$,

  * •

$\chi^{+}_{1}\chi^{-}_{1}Z_{\mu}$ :
$i\,\gamma_{\mu}(v_{\chi}-a_{\chi}\gamma_{5})=i\,g_{Z}\,\gamma_{\mu}\left((g_{L}+g_{R})+(g_{R}-g_{L})\,\gamma_{5}\right)$,

where
$g_{L}=\left(\sin^{2}{\theta_{W}}-\frac{3}{4}-\frac{1}{4}(2\cos^{2}{\theta_{L}-1})\right),\leavevmode\nobreak\
g_{R}=\left(\sin^{2}{\theta_{W}}-\frac{3}{4}-\frac{1}{4}(2\cos^{2}{\theta_{R}-1})\right)$,
and $\cos{\theta_{L}}=\mathscr{U}_{11}$, $\cos{\theta_{R}}=\mathscr{V}_{11}$
are the cosine of left and right chargino mixing angles.

##  Appendix B Azimuthal angular distribution with transversely polarized
beams

####  B.0.1 SM: $e^{-}(p_{1},s_{1})e^{+}(p_{2},s_{2})\to
W^{-}(k_{2},\lambda_{2})W^{+}(k_{1},\lambda_{1})$

The amplitude for the diagram with the photon in the $s$-channel is given by

$\displaystyle\mathscr{M}_{\gamma}(s_{1},s_{2},\lambda_{1},\lambda_{2})=-\frac{\,ie^{2}}{s}\,\bar{v}(s_{2},p_{2})\gamma^{\mu}u(s_{1},p_{1})\epsilon^{\nu}(\lambda_{1},k_{1}){\epsilon^{\rho}}^{*}(\lambda_{2},k_{2})G_{\mu\nu\rho}(k_{1},k_{2}),$
(32)

where

$G_{\mu\nu\rho}(k_{1},k_{2})\equiv((k_{2}-{k_{1}})_{\mu}g_{\nu\rho}-(2k_{2}+k_{1})_{\nu}g_{\rho\nu}+(2k_{1}+k_{2})_{\rho}g_{\mu\nu}).$
(33)

Following are the expressions for the amplitudes of the diagrams containing
$Z$ boson and neutrino propagator, respectively,

$\displaystyle\mathscr{M}_{Z}(s_{1},s_{2},\lambda_{1},\lambda_{2})=$
$\displaystyle\frac{ie^{2}}{2\,(s-M^{2}_{Z})\,\sin^{2}{\theta_{W}}}\,\bar{v}(s_{2},p_{2})\gamma^{\mu}(v_{f}-a_{f}\gamma_{5})u(s_{1},p_{1})$
(34)
$\displaystyle\times\epsilon^{\nu}(\lambda_{1},k_{1}){\epsilon^{\rho}}^{*}(\lambda_{2},k_{2})G_{\mu\nu\rho}(k_{1},k_{2}),$
$\mathscr{M}_{\nu}(s_{1},s_{2},\lambda_{1},\lambda_{2})=-\frac{i{g_{2}}^{2}}{(p_{1}-k_{1})^{2}}\,\bar{v}(s_{2},p_{2})\gamma^{\nu}P_{L}(\not{p}_{1}-\not{k}_{1})\gamma^{\mu}P_{L}u(s_{1},p_{1})\epsilon^{\mu}(\lambda_{1},k_{1}){\epsilon^{\nu}}^{*}(\lambda_{2},k_{2}).$
(35)

The square amplitude with polarized initial beams is given by

$|\mathscr{M}|^{2}=\sum_{s_{i},\lambda_{j}}\mathscr{M}(s_{1},s_{2},\lambda_{1},\lambda_{2})\mathscr{P}_{e^{-}}(s_{1},s_{3})\mathscr{P}_{e^{+}}(s_{2},s_{4})\mathscr{M}^{\dagger}(s_{3},s_{4},\lambda_{1},\lambda_{2}),$
(36)

where

$\mathscr{M}=\mathscr{M}_{\gamma}+\mathscr{M}_{Z}+\mathscr{M}_{\nu}.$ (37)

The spin density matrix for electron and positron
($\mathscr{P}_{e^{-}}$,$\mathscr{P}_{e^{+}}$) are written as

$\mathscr{P}_{e^{-}}=\left(\begin{array}[]{cc}\frac{1}{2}&\frac{1}{2}e^{-i\phi_{e^{-}}}\eta_{T}\\\
\frac{1}{2}e^{i\phi_{e^{-}}}\eta_{T}&\frac{1}{2}\\\ \end{array}\right),$ (38)
$\mathscr{P}_{e^{+}}=\left(\begin{array}[]{cc}\frac{1}{2}&\frac{1}{2}e^{-i\phi_{e^{+}}}\xi_{T}\\\
\frac{1}{2}e^{i\phi_{e^{+}}}\xi_{T}&\frac{1}{2}\\\ \end{array}\right)$ (39)

with their degree of polarization along transverse direction
($\theta=\frac{\pi}{2}$, $\phi_{e^{-}/e^{+}}$) as ($\eta_{T}$, $\xi_{T}$),
respectively,

Throughout the calculation we use $c\theta$ and $s\theta$ as shorthand
notations for $\cos{\theta}$ and $\sin{\theta}$ respectively. The total square
amplitude can be written as

$|\mathscr{M}|^{2}=|\mathscr{M}_{\gamma}|^{2}+|\mathscr{M}_{Z}|^{2}+|\mathscr{M}_{\nu}|^{2}+2Re\left(\mathscr{M}_{\gamma}\mathscr{M}_{Z}^{\dagger}\right)+2Re\left(\mathscr{M}_{\gamma}\mathscr{M}_{\nu}^{\dagger}\right)+2Re\left(\mathscr{M}_{\nu}\mathscr{M}_{Z}^{\dagger}\right).$
(40)

The expressions for the square amplitude of various diagrams and their
interference are given by

$\displaystyle|\mathscr{M}_{\gamma}|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2\left(\beta^{2}-1\right)^{2}}\Bigg{[}\beta^{2}e^{4}\left(3\beta^{4}-18\beta^{2}-\left(3\beta^{4}-2\beta^{2}+3\right)c\theta^{2}+19\right)$
(41) $\displaystyle-$
$\displaystyle\beta^{2}\left(3\beta^{4}-2\beta^{2}+3\right)\left(c\theta^{2}-1\right)e^{4}\eta_{T}\xi_{T}\cos(2\phi)\cos\left(\phi_{e^{-}}-\phi_{e^{+}}\right)$
$\displaystyle-$
$\displaystyle\beta^{2}\left(3\beta^{4}-2\beta^{2}+3\right)\left(c\theta^{2}-1\right)e^{4}\eta_{T}\xi_{T}\sin(2\phi)\sin\left(\phi_{e^{-}}-\phi_{e^{+}}\right)\Bigg{]},$
$\displaystyle|\mathscr{M}_{Z}|^{2}$ $\displaystyle=$
$\displaystyle\frac{2\beta^{2}E_{e}^{4}e^{4}}{\sin^{4}{\theta_{W}}\left(\beta^{2}-1\right)^{2}\left(s-M^{2}_{Z}\right)^{2}}\Bigg{[}\left(a_{f}^{2}+v_{f}^{2}\right)\left(3\beta^{4}-18\beta^{2}-\left(3\beta^{4}-2\beta^{2}+3\right)c\theta^{2}+19\right)$
$\displaystyle+\left(3\beta^{4}-2\beta^{2}+3\right)\left(c\theta^{2}-1\right)\eta_{T}\xi_{T}\left(a_{f}^{2}-v_{f}^{2}\right)\cos(2\phi)\cos\left(\phi_{e^{-}}-\phi_{e^{+}}\right)$
$\displaystyle-\left(3\beta^{4}-2\beta^{2}+3\right)\left(c\theta^{2}-1\right)\eta_{T}\xi_{T}\left(a_{f}^{2}-v_{f}^{2}\right)\sin(2\phi)\sin\left(\phi_{e^{-}}-\phi_{e^{+}}\right)\Bigg{]},$

$\displaystyle|\mathscr{M}_{\nu}|^{2}$ $\displaystyle=$
$\displaystyle\frac{e^{4}}{4\sin^{4}\theta_{W}\left(\beta^{4}-2\beta^{3}c\theta+2\beta
c\theta-1\right)^{2}}\Bigg{[}\left(\beta^{6}-10\beta^{4}+9\beta^{2}-4\beta^{4}c\theta^{4}+4\left(\beta^{5}+\beta^{3}\right)c\theta^{3}\right)$
(43) $\displaystyle-$
$\displaystyle\left(\left(5\beta^{6}+6\beta^{4}-11\beta^{2}\right)c\theta^{2}+4\left(3\beta^{4}-\beta^{2}-4\right)\beta
c\theta+4\right)\Bigg{]},$

$\displaystyle Re\left(\mathscr{M}_{\gamma}\mathscr{M}_{Z}^{\dagger}\right)$
$\displaystyle=$
$\displaystyle\dfrac{2\beta^{2}E_{e}^{2}e^{4}}{2\sin^{2}{\theta_{W}}\left(\beta^{2}-1\right)^{2}\left(s-M^{2}_{Z}\right)}\Bigg{[}v_{f}\left(18\beta^{2}-3\beta^{4}+\left(3\beta^{4}-2\beta^{2}+3\right)c\theta^{2}-19\right)$
$\displaystyle+$
$\displaystyle\left(3\beta^{4}-2\beta^{2}+3\right)\left(c\theta^{2}-1\right)\eta_{T}\xi_{T}v_{f}\cos(2\phi)\cos\left(\phi_{e^{-}}-\phi_{e^{+}}\right)$
$\displaystyle+$
$\displaystyle\left(3\beta^{4}-2\beta^{2}+3\right)\left(c\theta^{2}-1\right)\eta_{T}\xi_{T}v_{f}\sin(2\phi)\sin\left(\phi_{e^{-}}-\phi_{e^{+}}\right)\Bigg{]},$
$\displaystyle Re\left(\mathscr{M}_{\gamma}\mathscr{M}_{\nu}^{\dagger}\right)$
$\displaystyle=$
$\displaystyle\dfrac{\beta^{2}e^{4}}{4\sin^{2}{\theta_{W}}\left(\beta^{2}-1\right)^{2}\left(2\beta
c\theta-\beta^{2}-1\right)}\Bigg{[}\left(2\beta^{2}-3\beta^{4}-3\right)c\theta^{2}$
(45) $\displaystyle+$
$\displaystyle\left(3\beta^{4}-18\beta^{2}+19\right)+2\left(\beta^{3}+\beta^{1}\right)c\theta^{3}$
$\displaystyle+$
$\displaystyle\beta^{2}e^{4}\eta_{T}\xi_{T}\left(2\left(\beta^{3}+\beta\right)c\theta^{3}+\left(3\beta^{4}-2\beta^{2}+3\right)s\theta^{2}\right)\cos(2\phi)\cos\left(\phi_{e^{-}}-\phi_{e^{+}}\right)$
$\displaystyle+$
$\displaystyle\eta_{T}\xi_{T}\left(2\left(\beta^{3}+\beta\right)c\theta^{3}+\left(3\beta^{4}-2\beta^{2}+3\right)s\theta^{2}\right)\sin(2\phi)\sin\left(\phi_{e^{-}}-\phi_{e^{+}}\right)$
$\displaystyle-$
$\displaystyle\eta_{T}\xi_{T}\left(2\left(\beta^{3}+\beta\right)c\theta\right)\cos(2\phi)\cos\left(\phi_{e^{-}}-\phi_{e^{+}}\right)+\left(6\beta^{4}-2\beta^{2}-8\right)c\theta$
$\displaystyle-$
$\displaystyle\eta_{T}\xi_{T}\left(2\left(\beta^{3}+\beta\right)c\theta\right)\sin(2\phi)\sin\left(\phi_{e^{-}}-\phi_{e^{+}}\right)\Bigg{]},$

$\displaystyle Re\left(\mathscr{M}_{\nu}\mathscr{M}_{Z}^{\dagger}\right)$
$\displaystyle=$ $\displaystyle\dfrac{\beta
E_{e}^{2}e^{4}}{2\sin^{4}{\theta_{W}}\left(\beta^{2}-1\right)^{2}\left(2\beta
c\theta-\beta^{2}-1\right)\left(s-M^{2}_{Z}\right)}\Bigg{[}\beta(a_{f}+v_{f})\left(2\left(\beta^{3}+\beta\right)c\theta^{3}\right)$
(46) $\displaystyle+$
$\displaystyle\beta(a_{f}+v_{f})3\beta^{4}s\theta^{2}-\left(18\beta^{2}-19\right)$
$\displaystyle+$
$\displaystyle\eta_{T}\xi_{T}(a_{f}-v_{f})\left(\left(3\beta^{4}-2\beta^{2}+3\right)s\theta^{2}\right)\cos(2\phi)\cos\left(\phi_{e^{-}}-\phi_{e^{+}}\right)$
$\displaystyle+$
$\displaystyle\beta\eta_{T}\xi_{T}(a_{f}-v_{f})\left(\left(3\beta^{4}-2\beta^{2}+3\right)s\theta^{2}\right)\sin(2\phi)\sin\left(\phi_{e^{-}}-\phi_{e^{+}}\right)$
$\displaystyle+$
$\displaystyle(a_{f}+v_{f})\left(\left(2\beta^{3}-3\beta\right)c\theta^{2}+\left(6\beta^{4}-2\beta^{2}-8\right)c\theta\right)$
$\displaystyle-$
$\displaystyle\beta\eta_{T}\xi_{T}\left(2\left(\beta^{3}+\beta\right)c\theta
s\theta^{2}\right)\sin(2\phi)\sin\left(\phi_{e^{-}}-\phi_{e^{+}}\right)$
$\displaystyle-$
$\displaystyle\beta\eta_{T}\xi_{T}\left(2\left(\beta^{3}+\beta\right)c\theta
s\theta^{2}\right)\cos(2\phi)\cos\left(\phi_{e^{-}}-\phi_{e^{+}}\right)\Bigg{]}.$

Here, $\beta$ is the boost of the $W$ bosons and $\sqrt{s}=2E_{e}$ is the
center of mass energy. Integrating over the polar angle $\theta$, we get the
differential cross section with respect to the azimuthal angle ($\phi$) as

$\frac{d\sigma}{d\phi}=\dfrac{\beta}{64\pi^{2}s}\int
d\cos\theta\leavevmode\nobreak\ |\mathscr{M}|^{2}\times\left(3.894\times
10^{11}\right)\leavevmode\nobreak\ \leavevmode\nobreak\ \text{fb}.$ (47)

In this case,

$\displaystyle\frac{d\sigma}{d\phi}$ $\displaystyle=$
$\displaystyle\frac{\beta\times 3.894\times
10^{11}}{64\pi^{2}s}\Bigg{(}\frac{E_{e}^{2}e^{4}(a_{f}+v_{f})\left(4\beta\left(3\beta^{6}-23\beta^{4}+\beta^{2}+27\right)\right)}{12\beta
M^{4}_{W}\left(s-M^{2}_{Z}\right)}$ (48) $\displaystyle+$
$\displaystyle\frac{E_{e}^{2}\eta_{T}e^{4}\xi_{T}(a_{f}-v_{f})\left(3\left(\beta^{2}-1\right)^{4}\log\left(\frac{1+\beta}{1-\beta}\right)^{2}\right)\cos(2\phi-\phi_{e^{-}}+\phi_{e^{+}})}{12\beta
M^{4}_{W}\left(s-M^{2}_{Z}\right)\sin^{4}{\theta_{W}}}$ $\displaystyle+$
$\displaystyle\frac{16\beta^{3}e^{4}\left(3\beta^{4}-26\beta^{2}+\left(3\beta^{4}-2\beta^{2}+3\right)\eta_{T}\xi_{T}\cos(2\phi-\phi_{e}+\phi_{p})+27\right)}{24\beta
M^{4}_{W}}$ $\displaystyle+$
$\displaystyle\frac{8\beta^{2}E_{e}^{4}e^{4}\left(\left(3\beta^{4}-26\beta^{2}+27\right)\left(a_{f}^{2}+v_{f}^{2}\right)\right)}{3M^{4}_{W}\left(s-M^{2}_{Z}\right)^{2}\sin^{4}{\theta_{W}}}$
$\displaystyle-$
$\displaystyle\frac{8\beta^{2}E_{e}^{2}e^{4}v_{f}\left(3\beta^{4}-26\beta^{2}+\left(3\beta^{4}-2\beta^{2}+3\right)\eta_{T}\xi_{T}\cos(2\phi-\phi_{e}+\phi_{p})+27\right)}{3M^{4}_{W}\left(s-M^{2}_{Z}\right)\sin^{2}{\theta_{W}}}$
$\displaystyle+$
$\displaystyle\frac{e^{4}\eta_{T}\xi_{T}\left(3\left(\beta^{2}-1\right)^{4}\log\left(\frac{1+\beta}{1-\beta}\right)^{2}-4\beta\left(3\beta^{6}+\beta^{4}+\beta^{2}+3\right)\right)\cos(2\phi-\phi_{e}+\phi_{p})}{24\beta
M^{4}_{W}\sin^{2}{\theta_{W}}}$ $\displaystyle-$
$\displaystyle\frac{e^{4}\left(3\beta^{5}-\beta^{3}+3\left(\beta^{2}-1\right)^{2}\left(\beta^{2}+1\right)\tanh^{-1}(\beta)-3\beta\right)}{3\beta
M^{4}_{W}\sin^{4}{\theta_{W}}}-\frac{e^{4}\left(4\beta\left(3\beta^{6}-23\beta^{4}+\beta^{2}+27\right)\right)}{24\beta
M^{4}_{W}}$ $\displaystyle-$
$\displaystyle\frac{e^{4}\left(3\left(\beta^{2}-9\right)\left(\beta^{2}-1\right)^{3}\log\left((\beta-1)^{2}\right)-6\left(\beta^{2}-9\right)\left(\beta^{2}-1\right)^{3}\log(\beta+1)\right)}{24\beta
M^{4}_{W}}$ $\displaystyle-$
$\displaystyle\frac{E_{e}^{2}e^{4}(a_{f}+v_{f})\left(3\left(\beta^{2}-9\right)\left(\beta^{2}-1\right)^{3}\log\left(\frac{1+\beta}{1-\beta}\right)^{2}\right)}{12\beta
M^{4}_{W}\left(s-M^{2}_{Z}\right)}$ $\displaystyle-$
$\displaystyle\frac{E_{e}^{2}\eta_{T}e^{4}\xi_{T}(a_{f}-v_{f})\left(4\left(3\beta^{6}+\beta^{4}+\beta^{2}+3\right)\right)\cos(2\phi-\phi_{e^{-}}+\phi_{e^{+}})}{12M^{4}_{W}\left(s-M^{2}_{Z}\right)\sin^{4}{\theta_{W}}}$
$\displaystyle-$
$\displaystyle\frac{8\beta^{2}E_{e}^{4}e^{4}\left(\left(3\beta^{4}-2\beta^{2}+3\right)\eta_{T}\xi_{T}(a_{f}^{2}-v_{f}^{2})\cos(2\phi-\phi_{e^{-}}+\phi_{e^{+}})\right)}{3M^{4}_{W}\left(s-M^{2}_{Z}\right)^{2}\sin^{4}{\theta_{W}}}\Bigg{)}.$

After putting the values of masses, couplings and energy of the beam in the
non-oscillatory part of the above equation, while keeping the coefficient of
the oscillatory part as a function of $\eta_{T}$ , $\xi_{T}$ , and
$f(\beta_{W})$, we find the following expression for the differential cross
section

$\displaystyle\frac{d\sigma}{d\phi}=427.968+\eta_{T}\xi_{T}f(\beta)\cos(2\phi-\phi_{e^{-}}+\phi_{e^{+}}),$
(49)

where

$\displaystyle f(\beta)$ $\displaystyle=$
$\displaystyle\frac{\left(0.0261478\beta^{8}-0.104591\beta^{6}+0.156887\beta^{4}-0.104591\beta^{2}+0.0261478\right)\log\left((1-\beta)^{2}\right)}{M^{4}_{W}}$
(50) $\displaystyle-$
$\displaystyle\frac{\left(0.0522956\beta^{8}-0.209182\beta^{6}+0.313774\beta^{4}-0.209182\beta^{2}+0.0522956\right)\log(1+\beta)}{M^{4}_{W}}$
$\displaystyle-$
$\displaystyle\frac{0.105566\beta^{7}-0.174968\beta^{5}+0.175293\beta^{3}-0.104591\beta}{M^{4}_{W}}.$

####  B.0.2 IDM: $e^{-}(p_{1},s_{1})e^{+}(p_{2},s_{2})\to
H^{-}(k_{2})H^{+}(k_{1})$

The amplitudes, in this case, for the $s$-channel diagrams containing photon
and $Z$ boson are given below as

$\mathscr{M}_{\gamma}(s_{1},s_{2})=-\frac{ie^{2}}{s}\,\bar{u}(s_{2},p_{2})\gamma^{\mu}u(s_{1},p_{1})(k_{1}-k_{2})_{\mu},$
(51)
$\mathscr{M}_{Z}(s_{1},s_{2})=\frac{ig_{Z}(g_{2}\cos{\theta_{W}}-g_{1}\sin{\theta_{W}})}{4\,(s-M^{2}_{Z})}\,\bar{u}(s_{2},p_{2})\gamma^{\mu}(v_{f}-a_{f}\gamma_{5})u(s_{1},p_{1})(k_{1}-k_{2})_{\mu}.$
(52)

The square amplitude for the process is given by

$|\mathscr{M}|^{2}=\sum_{s_{i}}\mathscr{M}(s_{1},s_{2})\mathscr{P}_{e^{-}}(s_{1},s_{3})\mathscr{P}_{e^{+}}(s_{2},s_{4})\mathscr{M}^{\dagger}(s_{3},s_{4}),$
(53)

where

$\mathscr{M}=\mathscr{M}_{\gamma}+\mathscr{M}_{Z}.$ (54)

The expressions for square amplitudes of the two diagrams along with their
interference are written as

$|\mathscr{M}|^{2}=|\mathscr{M}_{\gamma}|^{2}+|\mathscr{M}_{Z}|^{2}+2Re\left(\mathscr{M}_{\gamma}\mathscr{M}_{Z}^{\dagger}\right).$
(55)

Here,

$\displaystyle|\mathscr{M}_{\gamma}|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\beta^{2}s\theta^{2}e^{4}e^{-i(\phi_{e^{-}}+\phi_{e^{+}})}\Bigg{(}2e^{i(\phi_{e^{-}}+\phi_{e^{+}})}-i\eta_{T}\xi_{T}\sin(2\phi)\left(e^{2i\phi_{e^{-}}}-e^{2i\phi_{e^{+}}}\right)$
(56) $\displaystyle+$
$\displaystyle\eta_{T}\xi_{T}\cos(2\phi)\left(e^{2i\phi_{e^{-}}}+e^{2i\phi_{e^{+}}}\right)\Bigg{)},$
$\displaystyle|\mathscr{M}_{Z}|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\left(s-M^{2}_{Z}\right)^{2}\tan^{2}{2\theta_{W}}}\Bigg{[}\beta^{2}s\theta^{2}E_{e}^{4}g_{Z}^{2}e^{-i(\phi_{e^{-}}+\phi_{e^{+}})}e^{2}\left(2\left(a_{f}^{2}+v_{f}^{2}\right)e^{i(\phi_{e^{-}}+\phi_{e^{+}})}\right)$
(57) $\displaystyle-$
$\displaystyle\beta^{2}\left(s\theta^{2}\right)E_{e}^{4}g_{Z}^{2}e^{-i(\phi_{e^{-}}+\phi_{e^{+}})}e^{2}\left(\eta_{T}\xi_{T}\left(a_{f}^{2}-v_{f}^{2}\right)\cos(2\phi)\left(e^{2i\phi_{e^{-}}}+e^{2i\phi_{e^{+}}}\right)\right)$
$\displaystyle+$ $\displaystyle
i\beta^{2}s\theta^{2}E_{e}^{4}g_{Z}^{2}e^{-i(\phi_{e^{-}}+\phi_{e^{+}})}e^{2}\left(\eta_{T}\xi_{T}\left(a_{f}^{2}-v_{f}^{2}\right)\sin(2\phi)\left(e^{2i\phi_{e^{-}}}-e^{2i\phi_{e^{+}}}\right)\right)\Bigg{]},$
$\displaystyle Re\left(\mathscr{M}_{\gamma}\mathscr{M}_{Z}^{\dagger}\right)$
$\displaystyle=$
$\displaystyle\dfrac{1}{2\left(s-M^{2}_{Z}\right)\tan{2\theta_{W}}}\Bigg{[}\beta^{2}\left(c\theta^{2}-1\right)E_{e}^{2}e^{3}g_{Z}v_{f}e^{-i(\phi_{e^{-}}+\phi_{e^{+}})}\left(2e^{i(\phi_{e^{-}}+\phi_{e^{+}})}\right)$
(58) $\displaystyle+$
$\displaystyle\beta^{2}\left(s\theta^{2}\right)E_{e}^{2}e^{3}g_{Z}v_{f}e^{-i(\phi_{e^{-}}+\phi_{e^{+}})}\left(i\eta_{T}\xi_{T}\sin(2\phi)\left(e^{2i\phi_{e^{-}}}-e^{2i\phi_{e^{+}}}\right)\right)$
$\displaystyle+$
$\displaystyle\beta^{2}\left(s\theta^{2}\right)E_{e}^{2}e^{3}g_{Z}v_{f}e^{-i(\phi_{e^{-}}+\phi_{e^{+}})}\left(\eta_{T}\xi_{T}\cos(2\phi)\left(e^{2i\phi_{e^{-}}}+e^{2i\phi_{e^{+}}}\right)\right)\Bigg{]}.$

In this case, the $\beta$ is the boost for the $H^{\pm}$. From the above
square amplitude, we obtain the differential cross section, after integrating
over the variable $\theta$, as

$\displaystyle\frac{d\sigma}{d\phi}$ $\displaystyle=$
$\displaystyle\frac{3.894\times
10^{11}}{64\pi^{2}s}\Bigg{(}\frac{8\beta^{3}E_{e}^{4}g_{Z}^{2}e^{2}\left(a_{f}^{2}-\eta_{T}\xi_{T}(a_{f}-v_{f})(a_{f}+v_{f})\cos(2\phi-\phi_{e^{-}}+\phi_{e^{+}})+v_{f}^{2}\right)}{3\left(s-M^{2}_{Z}\right)^{2}\tan^{2}{2\theta_{W}}}$
$\displaystyle+$
$\displaystyle\frac{2}{3}\beta^{3}e^{4}(\eta_{T}\xi_{T}\cos(2\phi-\phi_{e^{-}}+\phi_{e^{+}})+1)-\frac{8\beta^{3}E_{e}^{2}e^{3}g_{Z}v_{f}(\eta_{T}\xi_{T}\cos(2\phi-\phi_{e^{-}}+\phi_{e^{+}})+1)}{3\left(s-M^{2}_{Z}\right)\tan{2\theta_{W}}}\Bigg{)}.$

With the values of masses, couplings and energy of the beam, we find the
differential cross section to be a function of $\eta_{T}$ , $\xi_{T}$ , and
$f(\beta)$ as follows

$\displaystyle\frac{d\sigma}{d\phi}=4.64277\beta^{3}+3.42967\,\eta_{T}\xi_{T}\beta^{3}\cos(2\phi-\phi_{e^{-}}+\phi_{e^{+}}).$
(60)

####  B.0.3 MSSM:
$e^{-}(p_{1},s_{1})e^{+}(p_{2},s_{2})\to\chi^{-}(k_{1},s_{3})\chi^{+}(k_{2},s_{4})$

The amplitudes of two $s$-channel processes have the following expressions

$\mathscr{M}_{\gamma}(s_{1},s_{2},\lambda_{3},\lambda_{4})=\frac{ie^{2}}{s}\,\bar{v}(s_{2},p_{2})\gamma_{\mu}u(s_{1},p_{1})\bar{u}_{\chi}(\lambda_{3},k_{1})\gamma^{\mu}v_{\chi}(\lambda_{4},k_{2}),$
(61)
$\mathscr{M}_{Z}(s_{1},s_{2},\lambda_{3},\lambda_{4})=\frac{ig^{2}_{Z}}{4\,(s-M^{2}_{Z})}\,\overline{v}(s_{2},p_{2})\gamma^{\mu}(v_{f}-a_{f}\gamma_{5})u(s_{1},p_{1})\bar{u}_{\chi}(\lambda_{3},k_{1})\gamma_{\mu}(v_{\chi}-a_{\chi}\gamma_{5})v_{\chi}(\lambda_{4},k_{2}).$
(62)

The square of the amplitude is written as

$|\mathscr{M}|^{2}=\sum_{s_{i},\lambda_{i}}\mathscr{M}(s_{1},s_{2},\lambda_{3},\lambda_{4})\mathscr{P}_{e^{-}}(s_{1},s_{3})\mathscr{P}_{e^{+}}(s_{2},s_{4})\mathscr{M}^{\dagger}(s_{3},s_{4},\lambda_{3},\lambda_{4}),$
(63)

where

$\mathscr{M}=\mathscr{M}_{\gamma}+\mathscr{M}_{Z}.$ (64)

The expressions for the square amplitude of two $s$-channel diagrams and their
cross term are given as

$|\mathscr{M}|^{2}=|\mathscr{M}_{\gamma}|^{2}+|\mathscr{M}_{Z}|^{2}+2Re\left(\mathscr{M}_{\gamma}\mathscr{M}_{Z}^{\dagger}\right).$
(65)

Here,

$\displaystyle|\mathscr{M}_{\gamma}|^{2}$ $\displaystyle=$
$\displaystyle\frac{e^{4}}{2}\left(e^{-i(2\phi+\phi_{e^{-}}+\phi_{e^{+}})}\beta^{2}\left(c\theta^{2}-1\right)\eta_{T}\xi_{T}\left(e^{2i(2\phi+\phi_{e^{+}})}+e^{2i\phi_{e^{-}}}\right)-\left(2\left(\beta^{2}s\theta^{2}-2\right)\right)\right),$

$\displaystyle|\mathscr{M}_{Z}|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{{\left(s-M^{2}_{Z}\right)^{2}}}\Bigg{[}E_{e}^{4}g_{z}^{2}v_{f}^{2}\left(a_{\chi}^{2}\beta^{2}\left(c\theta^{2}+1\right)+v_{\chi}^{2}\left(\beta^{2}\left(c\theta^{2}-1\right)+2\right)\right)$
$\displaystyle+$ $\displaystyle
E_{e}^{4}g_{z}^{2}\left(2a_{f}^{2}\left(a_{\chi}^{2}\beta^{2}\left(c\theta^{2}+1\right)+v_{\chi}^{2}\left(\beta^{2}\left(c\theta^{2}-1\right)+2\right)\right)+16a_{f}a_{\chi}\beta
c\theta v_{f}v_{\chi}\right)$ $\displaystyle+$
$\displaystyle\left(a_{f}^{2}-v_{f}^{2}\right)E_{e}^{4}g_{z}^{2}e^{-i(2\phi+\phi_{e^{-}}+\phi_{e^{+}})}\beta^{2}\left(s\theta^{2}\right)\eta_{T}\xi_{T}\left(a_{\chi}^{2}+v_{\chi}^{2}\right)\left(e^{2i(2\phi+\phi_{e^{+}})}+e^{2i\phi_{e^{-}}}\right)\Bigg{]},$
$\displaystyle Re\left(\mathscr{M}_{\gamma}\mathscr{M}_{Z}^{\dagger}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2\left(s-M^{2}_{Z}\right)}\Bigg{[}2E_{e}^{2}e^{2}g_{z}\beta^{2}\left(c\theta^{2}-1\right)\eta_{T}\xi_{T}v_{f}v_{\chi}e^{2i\phi_{e^{-}}}$
(68) $\displaystyle+$ $\displaystyle
2E_{e}^{2}e^{2}g_{z}\left(2a_{f}a_{\chi}\beta
c\theta+v_{f}v_{\chi}\left(\beta^{2}\left(c\theta^{2}-1\right)+2\right)\right)$
$\displaystyle+$ $\displaystyle
E_{e}^{2}e^{2}g_{z}e^{-i(2\phi+\phi_{e^{-}}+\phi_{e^{+}})}\left(\beta^{2}\left(c\theta^{2}-1\right)\eta_{T}\xi_{T}v_{f}v_{\chi}e^{2i(2\phi+\phi_{e^{+}})}\right)\Bigg{]}.$

Here, the $\beta$ stands for the boost of $\chi^{\pm}$. Using the above matrix
amplitude square the differential cross section is found to be

$\displaystyle\frac{d\sigma}{d\phi}$ $\displaystyle=$
$\displaystyle\frac{3.894\times
10^{11}\,\beta}{64\,\pi^{2}s}\Bigg{(}\frac{8E_{e}^{2}e^{2}g_{z}v_{f}v_{\chi}\left(\beta^{2}\eta_{T}\xi_{T}\cos(2\phi-\phi_{e^{-}}+\phi_{e^{+}})+\beta^{2}-3\right)}{3\left(s-M^{2}_{Z}\right)}$
(69) $\displaystyle+$
$\displaystyle\frac{4E_{e}^{4}g^{2}_{z}\left(\beta^{2}\eta_{T}\xi_{T}(a^{2}_{f}-v^{2}_{f})(a_{f}+v_{f})\left(a_{\chi}^{2}+v_{\chi}^{2}\right)\cos(2\phi-\phi_{e^{-}}+\phi_{e^{+}})\right)}{3\left(s-M^{2}_{Z}\right)^{2}}$
$\displaystyle-$
$\displaystyle\frac{4}{3}e^{4}\left(\beta^{2}\eta_{T}\xi_{T}\cos(2\phi-\phi_{e^{-}}+\phi_{e^{+}})+\beta^{2}-3\right)$
$\displaystyle+$
$\displaystyle\frac{4E_{e}^{4}g^{2}_{z}\left(\left(a_{f}^{2}+v_{f}^{2}\right)\left(\left(3-\beta^{2}\right)v_{\chi}^{2}+2a_{\chi}^{2}\beta^{2}\right)\right)}{3\left(s-M^{2}_{Z}\right)^{2}}\Bigg{)}.$

Substituting the values of all the couplings, masses and beam energy except
the $m_{\chi}$, we obtain the expression for the differential cross section as

$\displaystyle\frac{d\sigma}{d\phi}=52.4371\beta-17.457\beta^{3}-0.966784\beta^{3}\eta_{T}\xi_{T}\cos\left(2\phi+\phi_{e^{+}}-\phi_{e^{-}}\right).$
(70)

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