Signals for vector-like leptons in an $S_3$-symmetric 2HDM at ILC

In this work, we explore the signals of an $S_3$-symmetric two Higgs doublet model with two generations of vector-like leptons (VLLs) at the proposed International Linear Collider (ILC). The lightest neutral component of the VLL in this model provides a viable dark matter (DM) candidate satisfying the current relic density data as well as circumventing all direct and indirect DM search constraints. Some representative benchmark points have been selected with low, medium and high DM masses, satisfying all theoretical and experimental constraints of the model and constraints coming from the DM sector. The VLLs (both neutral and charged) will be produced in pair leading to multi-lepton and multi-jet final states. We show that the ILC will prove to be a much more efficient and useful machine to hunt for such signals compared to LHC. Using traditional cut-based analysis as well as sophisticated multivariate analysis, we perform a detailed analysis of some promising channels containing mono-lepton plus di-jet, di-lepton, four-lepton and four jets along with missing transverse energy in the final state at 1 TeV ILC.


I. INTRODUCTION
The particle spectrum of the Standard Model (SM) being complete with the Higgs boson discovery [1, 2] still leaves unanswered questions on our understanding of Nature. In a more complete picture being established, the SM can be better termed as an effective theory sustainable up to a certain scale. To circumvent the theoretical and experimental shortcomings, several extensions of the SM have been proposed in the literature exploiting new symmetries or modifications to the space-time with additional spatial dimensions to name a few. In each such extension addition of either bosonic or fermionic fields somewhat becomes inevitable.
A typical extension which addresses some of the issues in the flavor sector of particle phenomenology involves vector-like fermions with vector-like leptons (VLL) as a natural entity.
In the present analysis, we study an S 3 -symmetric two Higgs doublet model (2HDM) [46,47], along with two generations of VLLs. The addition of two generations of VLLs in this model helps guarantee an S 3 -symmetric Yukawa Lagrangian providing an aesthetic picture to their inclusion. The main motivation of S 3 -symmetric 2HDM is to provide proper mass hierarchy and mixing among the SM fermions. Besides, the S 3 -symmetric 2HDM incorporates a 125 GeV SM-like Higgs boson in a very simple and natural way, unlike in a general 2HDM [46,47]. In our model we impose an additional Z 2 symmetry, under which all the SM fermions are even and the VLLs are odd [48]. Thus the mixing between the SM fermions and VLLs is forbidden throughout the analysis, and the lightest neutral VLL serves as a viable DM candidate which satisfies correct relic density, direct detection cross-sections and thermally averaged annihilation cross-sections in indirect detection obtained from the experiments [48]. A rigorous collider analysis of the multi-leptons + missing transverse energy final state at high-luminosity (HL) LHC can be found in one of our recent studies [48] where we found that the LHC provided us with a limited sensitivity for the VLLs for large masses as well as when the spectrum satisfying DM results demanded a compressed spectrum. Such spectrums will be more likely to be observable in a cleaner environment of an electron-positron collider such as the International Linear Collider (ILC) [49,50]. The ILC will be an invaluable machine with several exciting physics studies and is hence proposed to run at several center of mass energies, each driven by the physics study it aims to achieve. For our analysis, we have chosen the high energy option of √ s = 1 TeV that allows a larger phase space to produce heavier VLLs. In this work, we therefore study leptonic and hadronic states with missing transverse energy at 1 TeV ILC to highlight the sensitivity. We observed that the benchmarks with high DM masses or with a compressed particle spectrum which were challenging to probe owing to small signal cross-sections at HL-LHC are easily discernible with high significance in specific final states involving hadronic final states. To perform our collider analysis we select some benchmark points with low, medium and high DM masses from the multi-dimensional parameter space satisfying the theoretical, experimental and DM constraints. There exist several searches by ATLAS and CMS involving di-lepton [51], four leptons [52,53] and multi jets [54] along with missing transverse energy in the final states. We have validated all the benchmark points with the limits arising out of these existing studies. To optimize the signal over the SM backgrounds, for each channel we have performed a cut-based analysis and also shown the possible improvement in the analysis employing machine learning with a sophisticated multivariate technique.
The paper is organized as follows. In section II, we discuss the relevant scalar and Yukawa sector of the model. In section III, we present a collider analysis of the leptonic and hadronic final states along with missing transverse energy. Finally we summarize and conclude in Section IV.

II. MODEL
We work in the S 3 -symmetric 2HDM which contains two generations of VLLs. Including two generations of VLLs instead of one allows one to write a Yukawa Lagrangian fully S 3 -symmetric. Each generation of VLL comprises of one left-handed lepton doublet L L i , one right-handed charged lepton singlet e R i and one right-handed singlet neutrino ν R i , accompanied by their mirror counter parts with opposite chirality, i.e. L R i , e L i and ν L i with i = 1, 2. The quantum numbers for the SM and beyond Standard Model (BSM) particles are shown in Table I, while Table II shows the S 3 quantum numbers of the particles. Two Higgs doublets φ 1 and φ 2 together form an S 3 -doublet. In Table I, Q iL , L iL are the SM left-handed quark and lepton doublets, while u iR , d iR , e iR are the right-handed up-type, down-type quark and charged lepton singlets respectively for i = 1, 2, 3. Fields

A. Scalar and Yukawa Lagrangian
Two The neutral components of φ i acquire vacuum expectation value (responsible for the spontaneous symmetry breaking (SSB) of SM gauge symmetry). The doublets can be written as shown below, Here v i 's are VEVs of two doublets with v 1 = v cos β, v 2 = v sin β and v = v 2 1 + v 2 2 = 246 GeV. The ratio of two vacuum expectation values can be denoted by tan β , i.e. tan β = v 2 v 1 . The most general renormalisable scalar potential for S 3 -symmetric 2HDM can be written as the sum of V 2 (φ 1 , φ 2 ) and V 4 (φ 1 , φ 2 ) [46] : with and In Eq.(3) and Eq.(4), the subscripts denote the dimensionality of the terms. The hermiticity of the scalar potential in Eq.(4), forces the quartic couplings λ 1 , λ 2 and λ 3 to be real.
In V 2 (φ 1 , φ 2 ), m 2 11 , m 2 22 are real, m 2 12 can be complex in principle. In this analysis, we shall not consider m 2 12 to be complex to circumvent CP -violation. The configuration m 2 11 = m 2 22 along with m 2 12 = 0 makes the quadratic part of the potential S 3 -symmetric. At the same time this condition results in a massless heavy Higgs boson [46]. Thus to avoid any other massless heavy Higgs boson apart from the Goldstone bosons, we adhere to : m 2 11 = m 2 22 and m 2 12 = 0. Thus the value of tan β is fixed to 1, following the minimisation conditions of the scalar potential in Eq.(2) [46].
The particle spectrum of this model comprises of SM-like Higgs (h), heavy CP-even Higgs (H), pseudoscalar Higgs (A) and charged Higgs (H ± ). The alignment limit, in which h resembles SM Higgs boson, is naturally achieved in this model [46].
The most general Yukawa Lagrangian involving two generations of VLLs is given by, Here the charge conjugated fields are denoted with superscript "c" in Eq. (5). In presence of exact S 3 -symmetry, the masses of the VLLs will be proportional to the product of Yukawa coupling and the electroweak vacuum expectation value (VEV), which in turn will lead to non-perturbative Yukawa couplings (for vector lepton masses ∼ 1 TeV). Thus we introduce Dirac and Majorana mass terms in Eq. (5), that break S 3 -symmetry softly, while the rest of the terms in Eq.(5) are S 3 -symmetric.
Two generations of VLLs comprise of eight neutral and four charged flavor eigenstates.
Thus we can construct eight neutral mass eigenstates (N i , i = 1-8) and four charged mass eigenstates (E + i , i = 1-4) out of the aforementioned flavor eigenstates. The unbroken Z 2 symmetry in the model allows the lightest neutral state of the VLLs to act as the DM candidate. It also ensures that the mixing of VLL with SM fermions is also prohibited. We do note that the model can incorporate tiny neutrino masses radiatively through contributions from the Z 2 odd fermions. We however focus on the collider signals of the VLLs at the ILC and leave that study for future considerations. The mass matrices for the neutral and charged fermions and the details of their diagonalisation can be found in [48].

III. COLLIDER SEARCHES
By the virtue of Z 2 -symmetry, the lightest neutral VLL N 1 cannot decay and becomes a possible DM candidate. To explore the model parameter space compatible with relic density (Ω DM h 2 ) 1 , direct and indirect DM searches, we implement the model Lagrangian in FeynRules [55] to generate the interaction vertices and mass matrices. The CALCHEP [56] compatible model files obtained from FeynRules is then included in micrOMEGAs [57], which helps us to calculate the DM observables like relic density (Ω DM h 2 ), spin-dependent (σ SD ) and spin-independent (σ SI ) cross-sections, thermally averaged annihilation cross-sections ( σv ), etc. We choose four representative benchmark points BP1 BP2, BP3 and BP4 according to low, medium and high DM masses, that we found consistent with observed relic abundance obtained from the PLANCK experiment [58], and also allowed by the stringent bounds coming from the direct detection experiments like LUX [59] for spin-independent cross-sections and PICO [60] for spin-dependent cross-sections, and from the indirect detection bounds coming from FERMI-LAT [61], MAGIC [62] and PLANCK [58] experiments. The masses of the neutral (N i s) and charged (E + i s) VLLs for our chosen benchmark points can be found in Table III. All these four benchmark points represent model parameters which satisfy theoretical constraints like stability of the scalar potential, perturbativity and constraints coming from electroweak precision data and Higgs signal strengths 2 .
Ω DM h 2 , σ SD , σ SI , σv and dominant annihilation modes for indirect detection 3 along with 1 Ω DM is defined as the ratio of non-baryonic DM density to the critical density of the universe and h is the reduced Hubble parameter (not to be confused with SM Higgs h). 2 More details can be found in our earlier work [48]. 3 The 6th and 7th columns of Table IV actually refer to the < σv > relevant for indirect detection, and, the corresponding annihilation channels respectively. The aforementioned indirect detection annihilation cross sections alone cannot lead to an estimate for the relic density since there might be other annihilation/coannihilation channels entering the relic density calculation.
corresponding dark matter masses for the aforementioned benchmark points (BPs) are tabulated in Table IV. Our choice of BP's is envisaged to cover varied and complementary features of the model. For example, BP1 corresponds to low DM mass, BP2 corresponds to a slightly heavier DM mass with substantial mass splitting with the charged VLLs, BP3 corresponds to a compressed spectrum where the mass difference between the components of the VLL's are ∼ 20 GeV while BP4 corresponds to an overall heavy spectrum with higher DM mass which renders them close to threshold value of the ILC center of mass energy.  We focus on the √ s = 1 TeV of ILC and present our analysis for some specific processes in the S 3 -symmetric model which gives us the semi-leptonic 1 + 2j + E T / , fully leptonic 2 + E T / and 4 + E T / 4 and the fully hadronic 4j + E T / final states. The 1 + 2j + E T / and 4j + E T / channel containing multi-jets prove to be promising signals at the ILC, compared to LHC where huge SM backgrounds would supersede the signal. The spectrum with higher DM mass also proved hard to search at LHC [48] even with high integrated luminosity, since corresponding signal cross-section was too small to yield significant signal significance.
To generate the signal and SM background at leading order (LO), we use the public package MG5aMC@NLO [63]. We first use the following acceptance cuts : Here p j( ) T , |η j( ) | are the transverse momentum and pseudo-rapidity of jets (leptons). ∆R ij is defined as : This is due to the fact that mass difference between E ± 1 and N 1 is much less than the mass of W ± . Here we consider the following processes that can lead to the 2 + E T / final state: where i, j = 1... GeV. At the same time, we reject any third lepton in the final state. Since our signal does not contain any jet, we veto the light jets as well as b jets.
The dominant background comes from the e + e − → + − + E T / final state comprising of the following possible subprocesses that lead to a similar final state :  Table V. From now on, we shall only present the distributions and calculate the signal significances for the unpolarised incoming beams using cut-based as well as BDT analysis.
Cross section for Cross section for Cross section for  We carry out the cut-based analysis by looking at some relevant kinematic variables which can help design proper cuts (A 1 , A 2 , A 3 ) on them to improve the signal over the backgrounds.
• A 1 : We depict the normalized pseudorapidity distributions of the leading and subleading leptons in Fig. 1(a) and Fig. 1(b). For the background, the leptons can be produced via s-channel exchange of γ, Z as well as t-channel exchange of ν's, which results in the peaks at higher η values. However, for the signal, the leptons are produced from the decay of the W ± , which are generated from the decay of heavier VLLs, produced via s-channel exchange of γ, Z. As a result, the η distribution for the signal is more centrally peaked. We note that, choosing |η l 1,2 | < 1.0 helps to reduce the background significantly.
• A 2 : The normalized distribution of the invariant mass of the opposite sign (OS) lepton pair (same or different flavor) M + − is shown in Fig. 1(c). Since the 2 + E T / background consists of the contributions from ZZ and γ * Z, to exclude the Z-peak, we reject events which lie within the window : |M + − − M Z | < 15 GeV. At the same time, we also demand M + − > 12 GeV to reduce the γ * Z background contribution.
• A 3 : The variable M eff is constructed as the scalar sum of the lepton p T and E T / .
The distribution is shown in Fig. 1(d). Instead of giving cuts on the lepton p T and E T / separately, it is useful to put a cut on M eff which helps to reject the background more efficiently. Since for BP3 and BP4, the mass difference between the charged and neutral component of the VLL's are smaller compared to BP1 and BP2, the lepton p T is less. As a result, M eff peaks at a smaller value for BP3 and BP4. We impose an upper cut : M eff < 500 (350) GeV for BP1 and BP2 (BP3 and BP4) to reduce the background.
We tabulate the number of signal and background events surviving after the application of each cut for each benchmark at an integrated luminosity 100 fb −1 in  To showcase further improvement of the signal sensitivity from the cut-based analysis, we carry out the multivariate analysis (MVA) using Decorrelated Boosted Decision Tree (BDTD) algorithm within the Toolkit for Multivariate Data Analysis (TMVA) framework.
A detailed description of the method has already been described in one of our earlier work [48]. According to the discerning ability between the signal and the backgrounds of this channel, the most important kinematic variables turn out to be 5 : The BDTD parameters like NTrees, MinNodeSize, MaxDepth, nCuts and KS-scores [48] for both signal and backgrounds are tabulated in Table VII. The first four input parameters are regulated in such a way, that the KS-scores for both signal and backgrounds become stable [48]. The next task is to tune the BDT cut value or BDT score to maximise the     Fig.2(a)), we show the degree of background rejection against signal efficiency. It can clearly be inferred that the degree of background rejection is maximum for BP4 (magenta curve in Fig.2(a)).
The signal and background yields for 2 + E T / channel at L = 100 fb −1 along with the integrated luminosity required to achieve a 5σ significance for each benchmark points using MVA, have been tabulated in where σ B = σ sys_un × N B .
Cross section for Cross section for Cross section for (P e − , P e + = 0, 0) (P e − , P e + = 80%L, 30%R) (P e − , P e + = 80%R, 0)   Since the signal consists of one lepton and two jets along with transverse missing energy, we reject any second lepton or any third jet in the final state for the backgrounds. This helps us to suppress ZZ background. Finally we are left with W + W − and W + W − Z background.
Apart from the basic acceptance cuts mentioned in Eq.(6), we implement the following cuts to enhance the signal over backgrounds.
• B 1 : The pseudorapidity distributions for lepton and jets are different for the signal and background as can be seen in Fig 3(a),(b),(c) due to the t-channel dominant background. Choosing the pseudo-rapidity of the lepton and jets within the range : |η 1 |, |η j 1,2 | < 1.0, helps to reduce the background drastically.
• B 2 : The normalized E T / distribution is shown in Fig 3(d). For the background, the missing energy comes from the neutrinos and the distribution peaks at a lower E T / value. On the other hand, for the signal, apart from the neutrinos, missing energy can arise from the DM candidates and as a result it will peak relatively at a higher value.
A lower cut of E T / > 50 GeV helps to diminish the background.
• B 3 : We use the kinematic variable transverse mass M T 6 to distinguish the signal and background. As expected for the background, M T will peak at the W ± mass while for , where ∆φ ,E T / is the azimuthal angle between the lepton and transverse missing energy. the signal the corresponding distribution is smeared as seen in Fig 3(e). As there is additional source of E T / in the signal, we get a tail in M T distribution for the signal.
We observe that putting a cut of M T > 90 GeV helps to suppress the background.
• B 4 : We depict the ∆R 1 j 1,2 distributions in Fig 4(a),(b). It can be seen that an upper cut of ∆R 1 j 1,2 < 3.2 helps to enhance the signal significance.
We show the effect of the each cut in Table X. It is noticed that after putting all the cuts, we merely need 4, 3 and 14 fb −1 integrated luminosity to probe BP1, BP2 and BP3 for achieving 5σ significance. However, due to small production cross-section, to probe BP4 we need comparatively higher (220 fb −1 ) luminosity.
Number of Events after cuts (L = 100 fb −1 )     We now perform the multivariate analysis for 1 + 2j + E T / channel. According to the degree of differentiating potential between the signal and backgrounds, the most important variables turn out to be : ∆R 1 j 1 , ∆R 1 j 2 , M T , E T / , η 1 , η j 1 , η j 2 , M j 1 j 2 , ∆φ 1 j 1 , ∆φ 1 j 2 Here ∆φ 1 j 1 , (∆φ 1 j 2 ) are the azimuthal angle between 1 and j 1 (j 2 ), while the other variables have been defined earlier. The tuned BDT parameters for each benchmark points are listed in Table XI. The signal and background yields for an integrated luminosity 100 fb −1 are shown in Table XII. The same table contains the necessary integrated luminosities to attain 5σ significance for all benchmarks. Fig.5(a) and 5(b) depict the ROC curves and variation of significances with BDT-scores for all benchmarks respectively. The BDT scores for the four benchmarks are 0.13,0.165, 0.193, 0.141 respectively.

C. 4 + E T final state
In this section, we analyse the final state comprising 4 + E T / . The 4 + E T / final state for the signal can be obtained from the following processes: The SM backgrounds [52] that give rise to the similar final state is V V V, (V = W ± , Z) production along with additional contribution coming from ZZ production. Demanding  Table XIII.
Cross section for Cross section for Cross section for (P e − , P e + = 0, 0) (P e − , P e + = 80%L, 30%R) (P e − , P e + = 80%R, 0) Along with the basic cuts (Eq.6), we implement the following cuts to maximise the signal significance : • C 1 : Out of the four leptons, we choose two pairs of OS same flavor leptons ((M + − ) 1,2 ) which have invariant mass close to the Z-mass. We reject all events where |(M + − ) 1,2 − M Z | < 15 GeV to exclude the Z-peak of ZZ-background.
• C 2 : The pseudo-rapidity distributions of the leading and sub-leading leptons look similar to Fig. 1(a),(b) as the t-channel contribution dominates. The |η 1,2 | < 1.0 cut helps to suppress the background.
• C 3 : The E T / distribution for the background peaks at a lower value as it gets contribution only from neutrinos unlike the signal that also gets contribution from the heavy dark matter. A lower cut on E T / > 30 GeV helps to enhance the signal significance.
• C 4 : The normalized M eff distribution is similar to Fig. 1(d), except a larger tail. This is due to the fact that instead of two leptons, here M eff includes the scalar sum of four lepton p T 's and the missing transverse energy. We have optimized M eff < 600(500) GeV for BP1(rest of the BPs) to suppress the background significantly.
We tabulate the surviving events for the signal and backgrounds after each cut in Ta-ble XIV at an integrated luminosity 4 ab −1 . It can be seen that to probe benchmark BP1 and BP2 at 5σ significance, we need 550 fb −1 ) and 150 fb −1 ) luminosity and BP3 and BP4 are beyond the ILC projected luminosity [49,50].
This final state originates from e + e − → E + 1 E − 1 → W + W − E T / process, where both the W ± decay hadronically. The background for this process comes from e + e − → 4j + E T / which is dominated by di-boson (in that part of the phase space where E T / measurement is not important) and tri-boson production. However, due to small cross-section, ZZZ contributes insignificantly, while W + W − , ZZ and W + W − Z act as the irreducible backgrounds for this signal. We have demanded a b-veto to reduce the tt background (tt production cross-section is one order of magnitude less than ZZ production cross-section and two orders of magnitude  less than W + W − production cross-section) as the efficiency of mistagging a b jet as light jet is 1%. The effective cross-sections for the signal and backgrounds are shown in Table XV.
Cross section for Cross section for Cross section for (P e − , P e + = 0, 0) (P e − , P e + = 80%L, 30%R) (P e − , P e + = 80%R, 0)  To ensure that our signal contains exactly four jets, we veto any fifth jet with p j T > 20 GeV along with the basic cuts described in Eq. (6). In addition to these cuts, we put the following set of cuts to suppress the background.
• D 1 : We draw normalized pseudo-rapidity distribution in Fig. 6(a) for the leading jet. For the signal, the jets are much more centralised. Therefore putting a cut of |η j | < 1.2, j = 1...4, helps to suppress the background very efficiently.
• D 2 : For the background, the source of E T / is only the neutrinos coming from the decay of W ± or Z. For the signal, the additional source is the massive dark matter. By examining the distribution as depicted in Fig. 6(b), we put a cut of E T / > 30 GeV to enhance the signal.
• D 3 : ∆R between the jets become an important variable. We put a cut of ∆R j i j k < 3.5, i = k = 1...4, to suppress the background.
• D 4 : The invariant mass for two jet pair becomes an efficient variable. For BP1 and BP2, since the mass difference between the charged and neutral VLL's is higher compared to BP3 and BP4, the invariant mass distribution has a larger tail. As seen from Fig. 6(d), M j i j k < 300(150) GeV, i = k = 1...4, helps to enhance the signal for benchmark BP1 and BP2 (BP3 and BP4) over the background.
We present the cut-flow for the signal and backgrounds at integrated luminosity 100 fb −1 in Table XVI. We observe that compared to all the remaining aforementioned channels, this channel performs the best.
Number of Events after cuts (L = 100 fb −1 )  GeV cut for the background. As we can achieve 5σ significance with low integrated luminosity with cut-based analysis already, we refrain ourselves in performing the multivariate analysis. Comparing with the previous channels, this channel fares the best among all at 1 TeV ILC .
We conclude this section by a thorough comparison of the four aforementioned channels.
We have quoted the required luminosity to probe the benchmark points with 5σ significance by cut-based analysis only. From  respectively in this mode. The third best performing channel is 2 + E T / . To probe the first three benchmark points with 5σ significance one requires luminosity < 25 fb −1 . However, due to small effective cross-section, probing the BP4 at 5σ, one requires 570 fb −1 luminosity.
The 4 + E T / channel performs the worst at ILC although the background cross-sections are small. This is due to the fact that the effective production-cross section of signals in 4 + E T / channel is very small. Only for BP1 and BP2, 5σ significance can be achieved with 500 fb −1 and 150 fb −1 integrated luminosity respectively. Thus one can conclude that, the final states containing one or more jets, which are challenging to probe at LHC due to large backgrounds, turn out to be promising at ILC due to clean environment. We would like to mention here that these results can be improved by using the multivariate analysis.

IV. CONCLUSION
In this work we study the signals for an S 3 -symmetric 2HDM extended with two generations of VLLs. We impose an additional Z 2 -symmetry through which the mixing between the SM fermions and VLLs is disallowed, since the SM fermions are even and the VLLs are odd under the aforementioned symmetry. Thus the lightest neutral VLL turns out to be a viable DM candidate owing to the Z 2 -symmetry.
We choose four representative benchmark points BP1, BP2, BP3 and BP4 corresponding to low, medium and high DM masses, which satisfy the constraints coming from vacuum stability, perturbative unitarity, electroweak precision variables and Higgs signal strength along with the DM constraints coming from relic density, direct and indirect detections. We have presented detailed collider analysis for four distinct channels to probe these BPs at 1 TeV ILC, namely 2 + E T / , 1 + 2j + E T / , 4 + E T / and 4j + E T / .
For 2 + E T / and 1 + 2j + E T / channels, we perform both cut-based and BDT analysis.
All these channels originate from the pair production of the charged and neutral VLLs. In the 2 + E T / channel, both the W ± decay leptonically and in the 1 + 2j + E T / channel, one W ± decays leptonically and the other hadronically and for 4j + E T / channel, both the W ± decay hadronically. However, it is seen that one can probe 4j + E T / channel with 5σ significance with integrated luminosity < 2 fb −1 for the first 3 benchmark points and with 27 fb −1 luminosity for BP4 even with the cut-based analysis. This is the best performing channel at 1 TeV ILC. 1 + 2j + E T / channel at 1 TeV ILC is the second best performing channel as it requires only O(1) fb −1 luminosity to probe the first three benchmark points and 8.6 fb −1 luminosity to probe BP4 using MVA. The third well-performing channel is 2 + E T / , where 5σ significance is achieved for the first three BPs with luminosity < 25 fb −1 .
However, due to small effective cross-section, one requires 216.3 fb −1 luminosity to attain 5σ significance for BP4. The 4 + E T / channel performs the worst among all channels at ILC. Only for BP1 and BP2 with luminosity 550 fb −1 and 150 fb −1 respectively, one can attain 5σ significance. We find that a better sensitivity to heavier VLLs with high DM masses can be obtained at ILC in both the leptonic and hadronic channels, which proved more challenging and nearly impossible at HL-LHC due to smaller signal cross sections as well as large hadronic backgrounds. Thus ILC will prove to be a better hunting ground for such particles which have electroweak strength interactions.