Distinguishing nonstandard scalar and fermionic charged particles at future e + e − collider

We investigate the possibility to identify the intrinsic spin of exotic charged particles at the future e + e − collider in l ± + 2 j + (cid:0)(cid:0) 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 ± 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 θ ) 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.


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) [1], High Energy LHC (HE-LHC) [2], International Linear Collider (ILC) [3][4][5][6], Large Hadron electron Collider (LHeC) [7], Future Circular Collider (FCC) [8,9], 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 [10][11][12].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 [10,[13][14][15] of a mediator by looking at the angular distribution in a 2 → 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 ± 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) [16].We choose two well motivated models as examples, the Inert Doublet Model (IDM) [17,18] and the Minimal supersymmetric SM (MSSM) [19,20], having potential dark matter candidates of type scalar (spin-0) and fermion (spin-1/2), respectively to study the collider signature of the process In IDM, the C + and C 0 are Z 2 -odd charged Higgs (H ± ) and neutral Higgs (H 0 ), while for MSSM, they are the chargino ( χ ± ) and neutralino ( χ 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 ± jj + 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 A 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 B. 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.

A. The Inert Doublet Model
In the Inert Doublet Model (IDM) [21], the scalar sector of the Standard Model (SM) is modified with one additional scalar doublet Φ, which is odd (Φ → −Φ) under a new discrete Z 2 symmetry (parity).The SM particles together with the SM Higgs doublet (H) are even under this Z 2 symmetry.The two scalar doublets which transform under SU (2) L can be written as, where v = √ 2 ⟨0|H|0⟩ ≈ 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 ± 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 ± .The neutral inert Higgs (H 0 ) and its charged partner (H ± ) play the role of dark matter and the new exotic charged particle, respectively.
The scalar potential of the model is given by, The masses and interactions of the scalar sector are governed by the scalar-potential parameters where which is obtained by minimizing the scalar potential after EWSB.The IDM parameter space can be expressed in terms of physically more intuitive set where the Higgs and inert scalar masses are given by and the couplings λ L , λ S are defined as The interaction Lagrangian in our signal comprising the production and decay vertices of H ± originating from gauge interactions can be written as, Here, e = √ 4πα with α being fine structure constant, g as the SU (2) L gauge couplings, g z = e/(s W c W ), s W = sin θ W , and c W = cos θ W with θ W being the Weinberg angle.

B. 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) [22] 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, where Here Ŷ and T a respectively represents hypercharge and SU (2) generators in the respective representation of the field over these operator act.and 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 ( B, λ 3 , h1 1 , h2 2 ) and the charge sector comprises of ( h1 After electroweak symmetry breaking, the mass term for the higgsino and Electro-Weakino charged sector becomes In the basis  , the above Lagrangian can be written as As the mass matrix is not symmetric, it has to be diagonalized by a bi-unitary transformation, where M D c is a diagonal matrix with real positive eigenvalues.Weyl fermion eigenstates will be χ + k = V km ψ + m and χ − k = U km ψ − m , which can be written as four component chargino field.The mass for the charginos can be written as where χ+  are chargino fields.The Lagrangian in the neutral electroweakino sector can be written as The mass matrix, in the basis of 2 ) is given by This matrix can be diagonalized by a unitary matrix Z.The physical mass eigenstates and the mass diagonalization matrix are given by χ 0 l = Z ln ψ 0 n and Z * M Z −1 = M n D , respectively.In the four component notation the mass term in the Lagrangian for neutralinos can be written as L n M ass = − 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 (µ) and the ratio of the VEV of the Higgs doublets (tan β).
The lightest chargino mass eigenstate coupling to Z boson depends on the chargino mixing matrix elements V 11 and U 11 [23].For simplicity we keep these matrix elements fixed for all our benchmark points which can be obtained by varying the input parameters M 2 , µ and tan β of the model.

C. Benchmark selection
We choose four benchmark points (BPs), in our analysis, for the masses of exotic charged particles (C ± ) and their neutral partner (C 0 ) in both IDM and MSSM.The benchmark points are listed in Table I.The BPs are chosen in such a way that ∆M = M + − M 0 these criteria.We choose one more BP which has a different M + and ∆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, λ L and λ S , which can affect the SM Higgs signal in our study.The value of λ 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 µ 2 (see, Eq. ( 7)).The value of λ 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 V 11 ≡ cos θ R = 0.9725 and U 11 ≡ cos θ 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 [12].Apart from the dark matter constraint, there are other constraints that are considered while choosing the BPs, which we discuss below.
D. Constraints

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 [24] 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 [24].

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 [24] where where The experimental values of S and T with U = 0 is given by [25] S = 0.04 ± 0.08, and T = 0.08 ± 0.07.
In our case, BP4 with the largest ∆M = M H ± − M H 0 in IDM gives the maximum deviation in EWPO with ∆S = − 0.0218 and ∆T = − 0.0017.These values are well within the 1σ limit of experimental uncertainty.

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. [26].For MSSM, our branching fraction (BR(h → 2 χ0 )) is around 2 % and 4 % for BP1 and BP4 respectively.In IDM, the value of λ 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 → H 0 H 0 below 1 % for both BP1 and BP4.

Higgs Signal
The charged Higgs and chargino being at the electroweak scale, can contribute to the loop-induced h → γγ 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 [27] as For IDM, the maximum contribution to BR(h → γγ) comes from BP1 which has the lightest The signal for our analysis comes from the pair production of the exotic charged particles (C ± ) followed by their decay to their neutral partner (C 0 ) and charged gauge boson W ± .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., forming a final state of l ± 2j + / E T .The Feynman diagram up to W ± production is shown in Fig. 1.The signal process in IDM and MSSM, along with the SM background, are described below.
a. IDM Signal: In IDM, H + H − is produced via two s-channel processes with Z boson and γ as mediators from initial e − e + .Then H ± decay to W ± 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 ± .Since there are no other lighter particles which are Z 2 odd, branching ratio (BR) of H ± to W ± H 0 channel will be 100%.The two DM particles H 0 and neutrino act as the source of missing energy.

b. MSSM Signal:
In MSSM, similar s-channel diagrams can produce the lightest chargino χ ± 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 ± χ 0 and further decay of W ± 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 χ ± 1 which again gives its decay BR to W ± χ 0 as 100%.Here too the missing energy gets contribution from the two DM states χ 0 and a neutrino.
c. 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 → ν ν 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 → ν ν 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 [28] at ILC with √ 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 [29] for energy smearing effects.After the hadronization of the final state jets we perform the fast detector simulation in Delphes-3 [30] with the International Linear Detector (ILD) card.
Events are selected at the detector level with the following selection cuts, 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 [31].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 annihila-tion 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.
A. Analysis with longitudinal beam polarization We perform our collider analysis in the chosen final state with longitudinal beam polarization (LPol) of initial e ± 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 W W 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. ( 24).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 II, 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 II, it follows that IDM has O(1) less cross section than MSSM.This can be understood in the following way.In the massless limit (boost β → 1), for the photon mediated diagram, the total pair production cross section of H ± is four times smaller than the pair production cross section of χ ± 1 at an electron-positron collider.The reason for the enhanced cross section of χ ± 1 pair production is that there are four ways of combining helicity states of χ ± 1 and e ± , while for scalar there is only one degree of freedom for H ± 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 for unpolarized initial beams (see Eqs. (B25), (B35), and (B16)).Similarly for the diagram with Z-boson propagator, the factor associated with Z-boson vertex (T 3 −sin 2 θ 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 24 for unpolarized beam in β → 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 β → 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.
a. Kinematic variables The role of kinematic variables to analyse the signal against the SM background, which naturally arises in the final state l ± 2j + E T are missing transverse energy ( 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 for the SM background compared to the signal.This is because the source of 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 / E T shifts toward higher values as the splitting between the mass of the charged exotic and the stable neutral particle (∆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 ∆M = 100 GeV are at the same energy, while the peak for BP2 (∆M = 120 GeV) and BP4 (∆M = 290 GeV) are at the higher energy side in 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 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 √ s ef f ≥ 2M W of the process.On the other hand, in the case of signal, the W boson is produced from the decay of C ± , 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 / 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.Table III: The cut-flow information on the e − e + → l ± + 2j + / 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.
b. Angular variables 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 − → f 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 θ of W (jj pair) and lepton and the isolation variable ∆R = ∆η 2 + ∆ϕ 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 θ W sharply peaks at ±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 θ 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 θ W distribution has a small peak near ±1 with an overall flat shape throughout.The angular behavior of W transfers to its decay product thus, lepton's cos θ distribution is of similar nature to that of W , see right-top panel.So an inclusion cut of | cos θ| < 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 ± together with C 0 s.Hence the ∆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 ∆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 L = 100 fb −1 with the formulae 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 III 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 σ 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.
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 θ W and cos θ 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 χ 2 using the asymmetries.We have already discussed the difference in the shapes of cos θ W/l distribution, given in Fig. 4 for the MSSM and IDM signal events.Based on the symmetric shape in cos θ = 0, we define an asymmetry for both the variables as The difference is then estimated by a χ 2 , calculated as i runs on the variables; σ SM ×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, 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 IV (top two rows).
Although the A S for cos θ W/l seems to be (−)ve for IDM signal and (+)ve for MSSM signal by looking Fig. 4, the 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 III.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 θ W/l | < 0.9.Moreover, the cross sections are larger in MSSM compared to SM by roughly a factor of 2, see Table III.This makes the A S+B to be (−)ve for most of the BP.
The changes in the asymmetries w.r.t.background are above 1σ statistical errors shown in the second column in Table IV.Differences between the signals are calculated in terms of 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 1σ 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.(1/σ)dσ/dφ We observe in the previous section that though the angular variables cos θ W and cos θ 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 [32][33][34][35][36][37][38].Here, we use the facility of transverse polarization of e ± beams to showcase the distribution of azimuthal angle ϕ, 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 ϕ 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 ϕ distribution for the C + analytically in e − e + → 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 ϕ C are also shown with solid lines computed from events generated in WHIZARD 1 .The differential cross section in ϕ C has the form

B. Analysis with transverse beam polarization
where f (β C ) is a function of boost (β) 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 (β C ) (Shown in Eqs.(B18), (B29) and (B39) 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 ϕ 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 π phase shift compared to IDM, as can be seen in Fig. 5.This behavior of ϕ W -0.0029 0.0379 0.0471 0.0382 0.0387 0.0116 -0.0053 -0.0032 -0.0057 -0.0038 ϕ l 0.0113 0.0379 0.0475 0.0378 0.0375 0.0222 -0.0018 0.0009 0.0001 0.0027 Table V: Azimuthal asymmetries for both the signals and background in the e − e + → ℓ + 2j + / E T process for transversely polarized initial beams.The error in the SM is shown for L = 100 fb −1 of integrated luminosity.
1 ISR effect has been neglected here for simplicity.ϕ 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 ϕ 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 ϕ W/l given by We have listed the asymmetry values in Table V 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 ϕ W and ϕ l are nearly zero (much smaller than the 1σ 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. (B29) 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 ± 2j + 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 θ 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 (ϕ 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 ± 2j + 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. where The amplitude for the diagram with the photon in the s-channel is given by where Following are the expressions for the amplitudes of the diagrams containing Z boson and neutrino propagator, respectively, The square amplitude with polarized initial beams is given by where The spin density matrix for electron and positron (P e − ,P e + ) are written as with their degree of polarization along transverse direction (θ = π 2 , ϕ e − /e + ) as (η T , ξ T ), respectively, Throughout the calculation we use cθ and sθ as shorthand notations for cos θ and sin θ respectively.The total square amplitude can be written as The expressions for the square amplitude of various diagrams and their interference are given by In this case, 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 η T , ξ T , and f (β W ), we find the following expression for the differential cross section The amplitudes, in this case, for the s-channel diagrams containing photon and Z boson are given below as The square amplitude for the process is given by M (s 1 , s 2 )P e − (s 1 , s 3 )P e + (s 2 , s 4 )M † (s 3 , s 4 ), where The expressions for square amplitudes of the two diagrams along with their interference are written as The amplitudes of two s-channel processes have the following expressions M γ (s 1 , s 2 , λ 3 , λ 4 ) = ie 2 s v(s 2 , p 2 )γ µ u(s 1 , p 1 )ū χ (λ 3 , k 1 )γ µ v χ (λ 4 , k 2 ), (B30) v(s 2 , p 2 )γ µ (v f −a f γ 5 )u(s 1 , p 1 )ū χ (λ 3 , k 1 )γ µ (v χ −a χ γ 5 )v χ (λ 4 , k 2 ).


are the four Majorana neutralino fields.The lowest mass eigenstate of four neutralinos will be the LSP and represents the DM candidate.

1 Figure 1 :
Figure 1: Representative Feynman diagram for the signal process

χ 2
and shown in Table IV in the last three rows for all BPs for an integrated luminosity of L = 100 fb −1 .The two signals have a difference of ∼ 10σ in all the four BPs based on the total χ 2 shown in the last row.

Figure 5 :
Figure 5: Normalized distribution of the azimuthal angle ϕ 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 √ s = 1 TeV.

Figure 6 :
Figure 6: Normalized distribution of ϕ W (left-panel ) and ϕ l (right-panel) for the background and the signals in BP1 in e + e − collider for √ = 1 TeV with transversely polarized beams.
) , and cos θ L = U 11 , cos θ R = V 11 are the cosine of left and right chargino mixing angles.Appendix B: Azimuthal angular distribution with transversely polarized beams a. SM: e − (p 1 , s 1

Table II :
The cross sections of signal for the final state e − e Cross section (fb) MSSM Cross section (fb)Benchmark Mass(M ± , M 0 ) (GeV) Un-+ → l ± + 2j + / E T after the basic generation level cuts.

Table IV :
Asymmetries for both the signals and background and difference between the signals in the e − e + → ℓ + 2j + / E T process for longitudinally polarized beams for L = 100 fb −1 of integrated luminosity.