Dark Matter and Collider Searches in $S_3$-Symmetric 2HDM with Vector Like Leptons

We study the $S_3$-symmetric two Higgs doublet model by adding two generations of vector like leptons (VLL) which are odd under a discrete $Z_2$ symmetry. The lightest neutral component of the VLL acts as a dark matter (DM) whereas the full VLL set belongs to a dark sector with no mixings allowed with the standard model fermions. We analyse the model in light of dark matter and collider searches. We show that the DM is compatible with the current relic density data as well as satisfying all direct and indirect dark matter search constraints. We choose some representative points in the model parameter space allowed by all aforementioned dark matter constraints and present a detailed collider analysis of multi-lepton signal viz. the mono-lepton, di-lepton, tri-lepton and four-lepton along with missing transverse energy in the final state using both the cut-based analysis and multivariate analysis respectively at the high luminosity 14 TeV LHC run.

In this work, we study the S 3 -symmetric two Higgs doublet model (2HDM) [50,51] augmented with two generations of VLLs. The need to add two generations of VLL instead of one, is to maintain an exact S 3 -symmetry in the Yukawa sector. One of the primary motivations of the S 3 -symmetric 2HDM is aimed at understanding the fermion mass hierarchy within the SM, as it provides a proper description of the mass hierarchy and mixing among the quarks. Non-zero quark masses and non-block-diagonal CKM matrix, compatible with the experiment, makes this special kind of 2HDM endowed with non-abelian S 3 group very attractive. One also notes that unlike the general 2HDM, S 3 -symmetric 2HDM naturally provides a 125 GeV SM-like Higgs boson, which we discuss later. As we want to study the DM phenomenology of the VLL in this model, we impose an additional Z 2 symmetry in the model under which all the VLLs are odd while the SM particles along with the 2HDM is even. In Ref. [43], the authors have used a CP -conserving 2HDM along with one generation of VLLs to study the DM phenomenology. The difference in their study with ours lies not only in the particle spectrum due to the presence of an extra generation of VLL which is mandated by the S 3 -symmetry, but also in the way these VLLs interact with the visible sector particles. To be more specific, the quartic part of the Yukawa Lagrangian being S 3symmetric, there exist additional interactions with respect to the reference [43] due to the presence of an extra generation of VLLs. In addition, in our framework, S 3 -symmetry is softly broken by the dimension-3 Dirac as well as Majorana mass terms unlike reference [43], where the authors considered only Dirac mass terms for soft breaking. In our framework the lightest neutral mass eigenstate of the VLL is a viable DM candidate, which produces the correct relic density, and its direct detection cross-sections and thermally averaged annihilation cross-sections in indirect detection are compatible with that of the experiments.
We choose some representative benchmark points from the multi-dimensional parameter space which satisfy the relic density, direct and indirect search constraints and perform collider analysis for some specific multi-lepton channels containing mono-lepton, di-lepton, tri-lepton and four leptons along with missing transverse energy in the final state. Multilepton signals have already been analysed in the context of additional VLLs [52,53]. In addition, there already exists several searches by ATLAS and CMS, in the context of a few beyond SM models, for the final states comprising of mono-lepton [54-56], di-lepton [57], tri-lepton [58,59] and four leptons along with missing transverse energy [60,61]. We have considered the limits arising out of these studies in our work and highlight how the signals differ in each of our individual cases and the necessity to modify the cuts to optimise our signal events over the SM background.
The paper is structured as follows. In section II we briefly discuss the necessary extensions over the S 3 -symmetric 2HDM [50,51] done in our model. Section III deals with the relevant theoretical and experimental constraints to be considered which is followed by the DM phenomenology in section IV. Then we move on to section V where we present the collider analysis of the model in the leptonic channels, namely the signals having mono-lepton, dilepton, tri-lepton and four-lepton along with missing transverse energy in the final state.
Finally we summarise and conclude in section VI.

II. MODEL
We consider the S 3 -symmetric 2HDM augmented with two generations of VLL. The reason for adding two generations of VLL is to ensure an S 3 -symmetric Yukawa Lagrangian.
Each generation of VLL consists of one left-handed lepton doublet L L i , one right-handed charged lepton singlet e R i , one right-handed singlet neutrino ν R i and their mirror counter parts with opposite chirality but same gauge charges, i.e. L R i , e L i and ν L i with i = 2. These two generations of VLLs are doublets under S 3 -symmetry. Different quantum numbers associated with the particles are shown in Table I and Table II. In Table I, Q iL , L iL are left-handed quark and lepton doublets respectively in SM with i = 1, 2, 3. u iR , d iR are right-handed up-type and down-type quark singlets respectively with i = 1, 2, 3. Fields quantum numbers assigned to the particles in the model.
For this specific doublet representation, the elements of S 3 is given by [62], After symmetry breaking, φ i can be expressed as, Here v i 's are vacuum expectation values (VEV) and v GeV. tan β can be defined as the ratio of two vacuum expectation values : tan β = v 2 v 1 . The quartic part of the most general renormalisable S 3 -symmetric scalar potential is given by [50], 1 The hyper-charge Y has been computed by using the relation : Here T 3 and Q are the weak isospin and electric charge respectively.
The most general quadratic part of the scalar potential is [50] : In Eq.(3), the quartic couplings λ 1 , λ 2 and λ 3 are real owing to the hermiticity of the scalar potential. In the quadratic part of the potential in Eq.(4), m 2 11 , m 2 22 are real, m 2 12 can be complex in principle. Throughout the analysis, we have taken m 2 12 to be real to avoid CP -violation. Though m 2 11 = m 2 22 and m 2 12 = 0 ensure that the quadratic part of the potential is S 3 -symmetric, this configuration ends up with a massless scalar [50]. Thus for our analysis, we stick to the configuration m 2 11 = m 2 22 and m 2 12 = 0 (which breaks the S 3symmetry softly), which fixes the value of tan β = 1, following the minimisation condition of the scalar potential: 2 The scalar sector of this model consists of SM-like Higgs (h), heavy Higgs (H), pseudoscalar Higgs (A) and charged Higgs (H ± ). The limit at which h behaves as SM-like Higgs boson is defined as the alignment limit. This limit is naturally achieved in this model [50].
Transformations from flavour basis to mass basis of the scalars occur through the following 2 × 2 orthogonal matrix : w ± and z being the charged and neutral Golstone boson respectively.
The light Higgs and the heavy Higgs of the model are connected to there flavour eigenstates via, 2 In Ref.
[63], it has been shown that with the configuration m 2 11 = m 2 22 and m 2 12 = 0, one can still achieve the correct mass hierarchy in the quark sector by computing the correction to the eigenvalues using first order (non-degenerate) perturbation theory.

B. Yukawa Lagrangian
The dimension-4 terms in S 3 -symmetric Yukawa Lagrangian involving two generations of VLLs can be written as, Next we write down the dimension-3 Dirac and Majorana mass terms present in the Yukawa Lagrangian, which break S 3 -symmetry softly: 3 Here the fields with superscript "c" are the charge conjugated fields. The subscripts of "L" in Eq.(9) and Eq.(10) denote the mass dimensions of the operators. Thus whole Yukawa lagrangian can be written as the sum of L 3 and L 4 as : 3 Note that with exact S 3 -symmetry, the mass of the vector like leptons 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). With S 3 softly broken we can write down gauge-invariant bilinear dimension-3 interaction terms in the Lagrangian.
Here we can construct eight neutral mass eigenstates (N i , i = 1..8) out of two generations of vector leptons. To ensure that the lightest VLL N 1 is the DM candidate, we impose a Z 2 -symmetry (mentioned in Table I) under which all the VLLs are odd and all the SM leptons are even. This forbids the mixing between the SM leptons and VLLs 4 . The two Higgs doublets are assumed to be even under this Z 2 -symmetry.
In this set up, 8 × 8 neutral VLL mass matrix in the basis Since M ν is hermitian, it can be brought to diagonal form by the following transformation via unitary matrix V : Among all states N 1 is the lightest and M N j+1 > M N j for j = 1, 2, ...7.
In the charged VLL sector, the mass matrix is, where, The non-hermitian M c can be diagonalised by using the following bi-unitary transformation, with the unitary matrices U L and U R , Here we follow the same convention as the neutral sector, i.e. M E + i+1 > M E + i for i = 1, 2, 3.

III. CONSTRAINTS TO BE CONSIDERED
The S 3 -symmetric 2HDM model has an extended scalar sector and we have included VLLs in our model with some being SU (2) L doublets . The addition of particles under a new symmetry which are not singlets under the SM gauge symmetry would lead to interactions and mixings that could affect several existing experimental observations. In addition, the new parameters in the model would also have to adhere to theoretical constraints to make the model mathematically consistent. We look at the most relevant ones and extract the constraints they could put on the model parameters.
A. Theoretical constraints • Perturbativity: The quartic couplings λ 1 .λ 2 , λ 3 are taken to be perturbative: For Yukawa couplings the corresponding bound reads: • Stability conditions of the potential: The quartic couplings λ 1 , λ 2 and λ 3 are also constrained from the stability conditions, so that the potential remains bounded from below in any field direction: • Higgs mass : We keep the SM-like Higgs mass within the range: 125.1 ± 0.14 GeV GeV, the allowed ranges [67] are ∆S = 0.04 ± 0.11, ∆T = 0.09 ± 0.14, ∆U = −0.02 ± 0.11 (17) Notably the deviations in the T -parameter from its SM value enforces the mass splitting between the neutral and the charged scalar to be less than ∼ 50 − 100 GeV. Regarding the contributions coming from the VLL counterpart, the differences between the Yukawa couplings |y 2 −y 2 | and |y 4 −y 4 | should be small to evade the bound coming from T -parameter [35].

C. Higgs signal strength
Since we demand that the lightest CP -even scalar h is the SM like Higgs, it is imperative that we should check whether the production and decay of h in our model is consistent with the current experimental data. The compatibility can be checked by computing the signal strength in the ith decay mode of h as, Assuming gluon-gluon fusion to be the most dominant Higgs production process at the LHC, one can rewrite the signal strength µ i as, where Γ tot stands for the total decay width of SM like Higgs.
Since the Alignment limit is maintained naturally in this model, the signal strengths in the decay channels of h into W W [68,69], ZZ [70,71], bb [72,73], τ + τ − [74,75] are satisfied without any loss of generality. On the other hand, the loop-induced decay mode of the h → γγ will get additional contribution from the charged vector leptons and the non-standard charged scalars. For the chosen benchmark points, the h → γγ signal strength remains within 2σ-range of the current experimental value [76,77]. The detailed formula for the decay width of h → γγ channel is relegated to Appendix A.

IV. DARK MATTER PHENOMENOLOGY
As mentioned earlier, the lightest neutral VLL N 1 is a viable DM candidate due to its stable nature ensured by the imposed Z 2 -symmetry. In this section we investigate the parameter space spanned by relevant and independent model parameters which are compatible with relic density [78], direct and indirect DM searches. Setting the mass of the SM-like Higgs to 125 GeV, we scan the independent parameters of the model in the following range For the analysis, we derive the interactions, mass and mixings in the model which is then implemented in FeynRules [79].
In Table III, we present five representative points BP1, BP2, BP3, BP4 and BP5 with increasing DM mass, which satisfy the relic density constraints as well as the direct and indirect detection bounds. Corresponding values of σ SI , σ SD and σv are also tabulated in  the same table. Since the DM is Majorana-like, due to the Z-mediated process, for all the benchmarks σ SI < σ SD . As mentioned earlier, from the minimisation conditions of the scalar potential of the S 3 -symmetric 2HDM, with m 2 11 = m 2 22 and m 2 12 = 0, tan β is fixed to 1. Now Hf f and Af f ("f " is SM fermion) couplings being proportional to (cos β − sin β), vanish at tan β = 1 limit. Thus s-channel annihilations into SM fermions mediated by H or A are absent in this framework. The only surviving s-channel annihilation to SM fermions is mediated by h. For the first two points, since M N 1 < M Z , the DM pair dominantly annihilate into W + W − . After crossing the ZZ-threshold, the major annihilation occurs to the final state ZZ along with W + W − . Since the H, A-mediated s-channel annihilation to W + W − and ZZ are forbidden at alignment limit, t-channel annihilation to W + W − and ZZ via E ± i becomes the major contributor. Moderate ZN i N j couplings (with i = j) participating in the annihilation come out to be the main reason behind this dominance. To put this in perspective, we list the dominant annihilation modes for the aforementioned five benchmark points in Table III too. We note that since the alignment limit is maintained naturally in this model, the s-channel H, A-mediated processes leading to W + W − and ZZ final state will not contribute to σv . The scattering of DM with the nuclei within the detector material mediated by Z or h, gives rise to spin-dependent and spin-independent cross-sections (σ SD and σ SI ) respectively, which in turn are constrained from direct detection experiments. This forces hN 1 N 1 and ZN 1 N 1 couplings to be small enough to circumvent the direct detection bound. This is merely a choice and the smallness of the aforementioned couplings is achieved by tuning relevant parameters of the model. Due to the Majorana nature of the DM, the WIMP-nucleon cross-section is dominated by spin-dependent interactions mediated by Z boson. Hence we have to consider the direct detection bound on the σ SD coming from the PICO experiment [85].
In Fig.1, we depict the variation of spin-independent cross-section (σ SI ) with DM mass predicted by our model (magenta curve). The black line and the green band correspond to 90% confidence level (C.L.) and 2σ sensitivity predicted by Xenon-1T experiment. We can conclude that for the dark matter mass range allowed by relic density constraint, σ SI are smaller and allowed by the experimental limit shown by the black line in Fig.1. Therefore the spin-independent cross-sections for all the chosen benchmarks evade the constraint coming from the direct detection experiments. As mentioned earlier, the strongest bound on spin-dependent cross-section comes from PICO experiment [85]. For the chosen benchmark points, the spin-dependent cross-section remain below the experimental limit as can be seen from Fig. 2. Indirect detection experiments look for annihilation of the DM pair to SM particles through various channels that could produce distinctive signatures in cosmic rays. In Fig. 3, we show the variation of the thermally averaged annihilation cross-section as a function of dark matter mass. The magenta curve signifies the variation of annihilation cross-section for our model. Combined results from the FERMI-LAT and MAGIC experiments [87] are represented by the dashed lines. Here the blue, black, red and green dashed curves show the variation of σv with DM masses for the annihilation to µµ, τ τ, bb and W + W − respectively. We find that the parameter space characterised by our benchmarks survive the bounds coming from the indirect detection experiments. We however note that for the DM mass range of 100-200 GeV lies quite close to the experimental bounds from the indirect detection and may become sensitive to future data from indirect detection experiments. We have also incorporated the experimental results obtained from PLANCK data [78] in our analysis, though we have not shown it in Fig.3. We have checked that the curve representing our model in Fig.3 lies well below the experimental limits from PLANCK.  70 100 200 300 500 Colour coding is expressed in legends.

V. COLLIDER SEARCHES
In this section we focus on the collider phenomenology of our model. We study the most likely signals of the model that may manifest itself at the current and future runs of the large hadron collider (LHC). As the model consists of an extension of the spectrum in the electroweak and leptonic sector, it becomes quite clear that the production of these new exotic particles would be limited by their cross section if they are too heavy. In fact, the limits on weakly interacting BSM particles are still quite weak from LHC. In this model, VLLs with unbroken Z 2 symmetry have no mixing with the SM fermions. Thus, the production of these VLL's will have to be in pairs and they would decay to a SM particle and a lighter component of the VLL. We therefore focus on the relatively lighter spectrum of the exotics whose lightest neutral component is the DM candidate, represented by the states for the five benchmark points (BP) shown in Table III and consistent with the DM phenomenology presented in the previous section. The mass of the remaining VLL components which correspond to the same five BP's viz. M N i , with i = 1, 2, ..., 8 and M E ± j with j = 1, 2, ..., 4 are tabulated in Table IV.  The pair production of the VLL would give rise to lepton rich final states, that may include mono-lepton, di-leptons, tri-leptons and four-leptons along with E T / in the final states. Note that in the absence of any mixing between the SM leptons and VLL's, the all hadronic multi-jet +E T / is the dominant signal. However this signal would be swamped by huge SM backgrounds, which leads us to consider multi-lepton final states starting with at least one charged lepton (e/µ) as a more useful signal for this model. We shall perform the analysis for the collider signals based on five benchmark points (BP1, BP2, BP3, BP4, BP5) given in Table IV. We tabulate the two-body and three-body decay branching ratios of the charged and neutral VLLs in Appendix B (Table XVIII and Table XIX and XX). We must however note that for all benchmark points, the relative mass splittings among the mother and daughter particles of the VLL in the cascade decays are not very large, leading to a somewhat compressed spectrum. This would imply relatively softer decay products in the final state for some of the benchmark points leading to challenges in signal-background discrimination, as we will see in our analysis. We therefore try to use machine learning methods in a few channels to check what kind of improvement one may achieve over the traditional cut-based analysis. For the chosen benchmark points, we implement the model using FeynRules [79], which gives the required UFO that is fed in MG5aMC@NLO [97] to generate the signal and background events with the cross-section at the leading order (LO). The LO production cross-sections at the LHC for signal and SM backgrounds are calculated using the NNPDF3.0 parton distribution functions (PDF). To simulate the showering and hadronisation, the parton level events are passed through Pythia8 [98]. Finally, we implement the detector effects in our analysis using the default CMS detector simulation card for LHC available in Delphes-3.4.1 [99].
For jet reconstruction, the anti-k t clustering algorithm has been used throughout. Besides where σ S(B) , L, S ( B ) 6 denote the signal (background) cross-section, integrated luminosity and signal (background) cut-efficiency respectively. Following this strategy, let us proceed to perform the collider analysis of the aforementioned channels at 14 TeV high luminosity (HL)-LHC.

A. Mono-lepton final state
To include all possible processes leading to a signal containing mono-lepton and missing transverse energy (E T / ) in the final state, we take into account the pair production and associated production of the VLL's: background to negligible values. Thus in the study, we can afford to ignore this background completely.
To generate the signal and backgrounds, we apply the following criterion to identify the isolated objects (∆ R ij > 0.4): In  To perform the cut-based analysis, we apply the following selection cuts on chosen kinematic variables to disentangle the signal from SM backgrounds: 7 • A 1 : From Fig.4(a) it can be seen that the p T -distribution of the lepton for the SM background coming from the decay of W boson has the sharp Jacobian peak at ∼ M W /2, whereas the corresponding distribution is smeared for the signal where the charged lepton comes from the cascade decays of the heavy VLLs. However, a large part of the signal overlaps where the background peaks. Thus we demand that the charged lepton has a minimum transverse momentum p T > 20 GeV to exclude a significant part of the sharply peaked background (magenta line) without losing too many signal events. 7 We plot the relevant kinematic distributions for only BP1 and BP3 as representative points while only the dominant SM background via pp → W ± → ± E T / which is ∼ O(10 4 ) bigger than the rest is shown. which is the end-product of all cascades giving rises to a much larger E T / in the signal distribution.
• A 3 : The next kinematic variables used for separating signal from background is transverse mass (M T ) which is defined as [56], where ∆φ 1 ,E T / is the azimuthal angle between the lepton and E T / . In Fig.4(c), the M T distribution sharply peaks around M W for the background as expected, while the signal has a comparatively smeared distribution. Thus we demand M T > 100 GeV to eliminate the sharp background peak which in turn enhances the signal significance.
• A 4 : Distribution of M eff is depicted in Fig.4(d). M eff is defined by the scalar sum of the lepton p T and E T / . We find that putting a lower cut M eff > 110 GeV for all the benchmark points helps enhance the signal over background.  Having completed the cut-based analysis, we now proceed to perform the multivariate analysis (MVA) using Decorrelated Boosted Decision Tree (BDTD) algorithm within the Toolkit for Multivariate Data Analysis (TMVA) framework, with the hope of improvement in signal significance compared to the cut-based one. Before doing the BDTD analysis of the channels, let us present a brief overview of the method.

Number of Events after cuts (L
To classify the signal-like or background-like events, decision trees are used as classifier.
One discriminating variable with an optimised cut value applied on it is associated with each node of the decision tree, to provide best possible separation between the signal-like  According to the degree of discriminatory power between the signal and backgrounds, 8 The purity p can be defined as : p = S S+B . 9 KS-score > 0.01 will also serve the purpose if it remains stable even after changing the internal parameters of the algorithm.  following are the kinematic variables of importance : These relevant kinematic variables are constructed for each and every channel to discriminate between the signal and the backgrounds.    Fig.6(b). It can be clearly seen that the signal significance attains a maximum value for each benchmark at a particular value of BDT score.
Signal and background yields with 3 ab −1 integrated luminosity after performing BDTD analysis have been tabulated in

B. Di-lepton final state
We now consider the final states containing same or different flavour and opposite sign (OS) di-leptons along with E T / that can arise from the following subprocesses in our model: where i, j = 1, 2, ...4, k = 2, 3, ...8. The dominant signal contribution comes from the pair production of the charged VLLs followed by their decay to DM and a lepton. Production of the vector like neutrino along with the DM can also give rise to the similar final state albeit small cross-section. However for the sake of completeness we take into account all such processes that may give rise to a di-lepton final state. The major SM background for the signal comes from the inclusive 2 + E T / process which includes contributions from W + W − and ZZ pair production. Due to large cross-section, tt followed by the leptonic decay of topquark (leading to 2b + 2 + E T / final state) also contributes as one of the major background.
Even after a b-jet veto along with a jet-veto, the small fraction of events surviving from the tt process can still lead to a significant number of events in the 2 + E T / final state. In addition, processes with smaller cross-sections such as W ± Z, and W + W − Z followed by the leptonic decay of W ± and Z can also be a possible source of background for the 2 + E T / final state, if one or more leptons escape detection. For the analysis, we consider the above three SM subprocesses as major contributions to the SM background. In Table IX  To generate the signal and backgrounds we apply the same set of generation cuts as mentioned in subsection V A. We select events with exactly two charged leptons with p T > 10 GeV and |η | < 2.5 and reject any additional lepton with p T > 10 GeV. To ensure a hadronically quiet final state, we veto all light-jets, b-jets and τ -jets with p T > 20 GeV.
We then analyse the signal containing OS di-leptons and compute the signal significance using traditional cut-based method. To differentiate our signal from the SM background, we focus on the following kinematic variables: p 1 T , p 2 T , E T / and invariant mass of two OS same or different flavoured leptons M + − . We define the cuts applied on them as B 1 , B 2 , B 3 , B 4 respectively and we describe them below : • B 1 : In Fig. 7(a) and 7(b), we depict normalised p T distribution for the leading and sub-leading leptons 1 and 2 for both signal and SM background. In can be seen that the distributions have a significant overlap owing to their origin being from W decay.
Thus we apply p 1 T > 20 GeV suppress the SM backgrounds.
• B 2 : The normalised distribution of missing transverse energy is shown in Fig. 7(c).
The E T / distributions for the BP1 and BP3 (green and blue lines) are much harder as in the mono-lepton case. Thus we demand E T / > 40 GeV, which helps to reduce the 2 + E T / background. As the mass splitting between the VLLs become smaller for heavier DM, the E T / distribution is shifted towards the softer side.
• B 3 : The normalised distribution of M + − is shown in Fig.7(d). The distribution for the W Z background (red line) shows a peak at M Z , since two same flavour opposite sign leptons out of the three in the final state, originate from the Z-boson decay. As the signal does not have a Z peak in its distribution, we reject events 75 < (M + − ) 1,2 < 105 GeV to exclude the Z-peak. This cut helps in suppressing the W Z background.
We sum up the number of surviving signal and background events after applying the aforementioned cuts in Table X   After discussing the cut-based analysis, let us move on to the multivariate (BDTD) analysis, which improves the signal significance by enhancing the discriminatory power between the signal and the backgrounds. For this analysis, we consider the following kinematic   variables with maximal discerning ability : Using these variables we train the signal and backgrounds so that the signal significance is maximized.
We present the set of tuned BDT parameters for all the benchmarks in Table XI to make the KS-score stable following the criteria mentioned in Sec. V A. The KS-scores for BP1 and BP3 (both for signal and background) are given in Fig.8. In the sixth column of Table XI KS-scores for all benchmarks have been quoted. Having fixed the KS-score, we next proceed to tune the BDT score to yield maximum significance. Background rejection efficiency vs.
signal efficiency have been plotted in the ROC curves in Fig.9(a) using the aforementioned kinematic variables. From the ROC curves of the 2 + E T / channel, it is evident that the background rejection efficiency is somewhat poor compared to the 1 + E T / channel. The significances have been plotted against BDT score for all benchmarks in Fig.9(b).
Signal and background yields with 3 ab −1 integrated luminosity for our chosen benchmark points along with the significances are listed in Table XII. From Table XII it can be inferred that the signal significance has improved a bit compared to the cut-based counter part. For BP1, BP2, BP3, BP4 and BP5 the improvements in signal significance are 16.4%, 60.0%, 30.8%, 12.5% and 6.7% respectively.
We generate the events with tri-lepton final state using the same generation-level cuts and following the method discussed in subsection V A. Among all possible decay products of the pair produced neutral and charged VLLs, we select only those events which have three charged leptons and missing transverse energy in the final state. We consider pp → 3 + E T / with zero jets as the dominant irreducible SM background for our signal, which includes both on-shell and off-shell contributions from diboson and triboson production. In addition, the pair production of Z boson where ZZ → 4 can also give rise to a similar final state if one of the leptons is missed. All LO cross-sections for this signal and backgrounds at 14 TeV LHC are given in  For this channel with more leptons, which is cleaner with smaller SM background, we restrict ourselves to the cut-based analysis only. To discriminate the signal from background, we demand our final state to have exactly three charged leptons with p T > 10 GeV out of which two leptons are of the same sign and the third lepton is of opposite sign. Among these three leptons, at least two are expected to be of same flavour. We also impose b-jet veto (reject p T (b) > 20 GeV) to eliminate the b-jets in the final state coming from the tt background. Next we identify a few kinematic variables which would help to discriminate the signal from background as follows: • C 1 : Out of two same sign leptons and one opposite sign lepton in the final state, one can construct two invariant mass system (M + − ) 1,2 , considering one same and one 11 Before proceeding towards the analysis at 14 TeV, we first validate the chosen benchmarks using the existing search for chargino-neutralino pair production in final states with three leptons and missing transverse momentum at √ s = 13 TeV performed by the ATLAS detector [59]. TeV HL-LHC.
Number of Events after cuts (L = 3 ab −1 )   GeV one can get rid of the Z-peak, which in turn reduces the W ± Z, ZZ background drastically. We also impose a lower cut (M + − ) 1,2 > 12 GeV to suppress the Drell-Yan background [102].
• C 2 : We define a variable M eff as the scalar sum of all the lepton p T 's and the E T / .
In Fig.10(a) the distribution of 3 + E T / background (magenta line) is flatter and smeared with respect to the distributions of the signal (green and blue lines) and other background ZZ (brown line). Setting M eff < 500 GeV helps in reducing the background.
• C 3 : Since the background ZZ does not have E T / in the final state explicitly, corresponding E T / distribution peaks at lower value than the signal as can be seen from Fig.10(b). Thus a minimum cut of E T / > 30 GeV helps to reduce the ZZ background drastically as can be found in Table XIV. • C 4 : We choose the vector sum of three leptons (p vector T,3 ) and the scalar sum of the same (p scalar T,3 ) and show their distributions in Fig.10(c) and 10(d) respectively. We find that kinematic selections of p vector T,3 < 200 GeV and p scalar T,3 < 250 GeV helps to reduce the 3 + E T / background efficiently.
• C 5 : We also construct the azimuthal angle between the unpaired third lepton out of total three leptons in the final state and E T / as ∆φ ,E T / . Corresponding distributions are shown in Fig.10(e). We find that the choice ∆φ ,E T / > 1.5 on the events help in eliminating the SM background further.
The number of events for signal and background, surviving after imposing the selection cuts on the aforementioned variables with 3 ab −1 integrated luminosity are quoted along with the significances in Table XIV. For the five benchmarks BP1, BP2, BP3, BP4, BP5, using the cut-based analysis, the signal significances are 11.1, 4.5, 3.5, 1.2, 0.5 respectively. This is a substantial improvement over the previous two final state topologies considered earlier. In fact for BP1, L ∼ 609 fb −1 of integrated luminosity is enough to achieve a 5σ significance in the tri-lepton channel.

D. Four-lepton final state
In this section, we analyse the final state comprising of 4 + E T / . The 4 + E T / final state for the signal can be obtained from the following processes: The events are generated using the same generation-level cuts and following the same    Table XV we have tabulated the LO cross-sections for signal and background at 14 TeV LHC.
To disentangle the signal and background, we select four leptons with p T > 10 GeV and |η | < 2.5 and reject any additional charged lepton satisfying the same. We also apply a veto on light-jets and b−jet in the final state. We consider the following set of kinematic variables to improve the the signal sensitivity over the background: The cuts applied on the aforementioned kinematic variables along with the significances are listed in Table XVI. For the five benchmarks, the significances at the integrated luminosity 3 ab −1 are 10.1, 7.1, 5.2, 3.2, 1.0 respectively. Note that the first four benchmarks seem to achieve a significance > 3σ (the first three having S > 5σ). Thus we find that the higher lepton multiplicity of the final states tend to achieve a more significant signal sensitivity in our model which is expected due to the addition of vector like fermions which decay to charged leptons.
Number of Events after cuts (L = 3 ab −1 )   With 3 ab −1 luminosity, first four benchmark points can be probed with significance > 3σ.
Next better performing channel after 4 + E T / is 3 + E T / for last four benchmarks. In fact for BP1, 3 + E T / channel turns out to be best performing with signal significance 11.1 at 3 ab −1 integrated luminosity. For 1 + E T / and 2 + E T / channel the significance for BP1 are 7.0 and 6.7 respectively with cut-based analysis, which is improved to 7.8 using the BDTD analysis.

VI. CONCLUSION
In this work, we extend the S 3 -symmetric 2HDM with two generations of VLLs. To start with, we first analyse the final state containing 1 + E T / , which can originate from the pair production of the charged VLLs and neutral VLLs as well as from the associated production of the charged and neutral VLLs. The major background for this channel is