Confronting dark fermion with a doubly charged Higgs in the left-right symmetric model

We consider a fermionic dark matter (DM) in the left-right symmetric framework by introducing a pair of vector-like (VL) doublets in the particle spectrum. The stability of the DM is ensured through an unbroken $\mathcal{Z}_2$ symmetry. We explore the parameter space of the model compatible with the observed relic density and direct and indirect detection cross sections. The presence of charged dark fermions opens up an interesting possibility for the doubly charged Higgs signal at LHC and ILC. The signal for the doubly charged scalar decaying into the dark sector is analyzed in multilepton final states for a few representative parameter choices consistent with DM observations.


Introduction
The new era of search for particles or hints of new physics has been facing its challenges since the discovery of the Higgs boson [1,2]. While direct searches at experiments at the Large hadron Collider (LHC) have not revealed anything new yet, the irrevocable hints for the existence of tiny neutrino mass and dark matter (DM) in the Universe has led to new efforts in non-collider experiments to establish signals for new phenomenon which may point to an extension of the Standard Model (SM) and more crucially provide information about DM. Left-right symmetric models (LRSM) [3][4][5][6][7][8] are one of the most well motivated and widely studied extensions of the SM as it is able to address several phenomena which are not very well understood in the framework of SM, be it Parity or the tiny neutrino masses which have their origin very naturally in the model. Parity symmetry (P) prevents one from writing P and Charge-Parity (CP) violating terms in the Quantum Chromodynamics (QCD) Lagrangian thus resolving the strong CP problem naturally without the need to introduce a global Peccei-Quinn symmetry. The gauge structure of these models force us to have a right-handed neutrino in the lepton multiplet. This right-handed neutrino can generate a light neutrino mass through the seesaw mechanism. Moreover left-right symmetry is also favored by many scenarios of gauge-unification.
We have also established that a significant part of the Universe ( ∼ 26% of total energy budget) which is made of non-luminous, non-baryonic matter and interacts via gravity, popularly known as dark matter (DM) is one of the most fundamental concerns in current days particle physics and cosmology [9,10]. There are several astrophysical evidences like the rotational curve of the galaxy, bullet cluster, gravitational lensing, anisotropy in cosmic microwave background (CMB), etc. [11][12][13][14], which indicate the existence of stable DM in the present Universe. However we do not know much about it, the only information so far we know about DM is its relic abundance measured by WMAP and PLANK to be Ω DM h 2 = 0.120 ± 0.001 [15]. Apart from this the nature of the DM e.g. its spin, non-gravitational interaction and mass still remain an open question.
Depending on the production mechanism in the early Universe, DM particles are broadly classified based on its interaction strengths into WIMP [16][17][18], SIMP [19], FIMP [20], etc. Among these the weakly interacting massive particle (WIMP) is one of the popular DM candidates due to its detection possibility in direct (XENON1T [21], PANDAX 4T [22] etc.), indirect (FERMI LAT and MAGIC [23,24]) and collider (LHC [25] and ILC) search experiments. The weak interaction between WIMP like DM and the visible sector can lead to thermal equilibrium in the early universe at temperatures above the mass scale and freezes out from the thermal bath when the temperature falls below its mass [16]. Many theoretical extension of SM have been formulated to accommodate the particle nature of DM over the past several decades. A novel possibility would be its existence in the framework of left-right symmetric theories. While the supersymmetric versions of the LRSM [26,27] naturally incorporate a WIMP (through conserved R-parity) in the form of the lightest supersymmetric particle, it is rather challenging to invoke a DM candidate in the minimal setup of LRSM. Therefore a natural extension could be to include new particles in the set-up in the non-supersymmetric model [28][29][30][31][32][33][34]. To accommodate DM in LRSM, we consider two vector-like fermion doublets, ψ 1 and ψ 2 which belong to the SU (2) L and SU (2) R respectively and both carry a discrete Z 2 charge of −1 to ensure the stability of the lightest state. This induced dark sector in the LRSM would have enhanced interactions including its participation in both left-handed and right-handed charged currents as well as neutral currents mediated by the electroweak (EW) and new heavy gauge bosons and the numerous scalars present in the model. Thus one expects the DM phenomenology to be very interesting and illuminating in several regions of the model parameter space. A crucial hint of LRSM is the presence of two doubly charged scalars which independently couple to the gauge bosons of SU (2) L and SU (2) R and therefore have different production strengths.
With the inclusion of the dark sector which contains new fermions, the search for these exotics can be a lot different at collider experiments. We shall consider this interesting possibility in our work by studying the signal for the doubly charged scalars at LHC as well as the proposed International Linear Collider (ILC) [35][36][37][38] while highlighting the DM phenomenology of the model in more detail.
The rest of our work is organized as follows. We first briefly describe the proposed model in Section 2. In Section 3, we discuss possible theoretical and experimental constraints on the model parameters which would be applicable for our analysis. In Section 4 we discuss the DM phenomenology where we demonstrate the allowed parameter space compatible with current relic density, direct and indirect search constraints. The collider signature of the doubly charged Higgs in the presence of DM at LHC and ILC in this setup are discussed in Section 5 and 6 respectively.
Finally, we summarize our findings in Section 7.

Left-Right Model with Dark Doublets
The model is an extension of the popular left-right symmetry model (LRSM) [3,4] where the only addition is the introduction of two vector-like (VL) fermion doublets [34] in the particle spectrum, The Lagrangian for this model can be written in two separate components: The first part of the Lagrangian in Eqn.1 represents the Z 2 even LRSM Lagrangian while the second part of the Lagrangian represents the proposed dark sector in this setup. Since the left-right symmetric Lagrangian, L LRSM has been comprehensively studied in the literature we do not discuss it in great detail here and refer the readers to Refs. [3][4][5][6][7][8][39][40][41][42][43][44]. Our main motivation in this work is to study the phenomenology of DM in the extended LRSM setup. We, therefore, restrict ourselves to only the relevant part of the Lagrangian of LRSM for our analysis.

Fermion Fields
where I 3 is the third component of isospin and B and L represent the baryon and lepton numbers respectively.

The scalar potential of LRSM
The most general C (charge) − P (parity) invariant scalar potential in the LRSM, invariant under the gauge symmetry SU (3) C ⊗ SU (2) R ⊗ SU (2) L ⊗ U (1) B−L reads as [6,39] 4 Scalar Fields The neutral scalar fields in the multiplets acquire non-zero vacuum expectation value (vev) leading to the symmetry breaking pattern: [6,39] as: The vevs of the scalar fields are parametrized as: where v can be identified as SM vev and is given by The parity symmetry implies g L = g R . Without any loss in generality of the BSM phenomenology, the above scalar potential can be simplified by considering β i = 0, α 2 = 0, λ 4 = 0 and v L → 0. Under these assumptions, the masses and corresponding eigenstates of the scalar and the gauge bosons are tabulated in Table3. Here we have categorized the different scalar types according to their CP properties and electric charge without going into the details, as these have been discussed in the literature [8].
We shall work in the limit of parameter choices in the scalar sector which helps us with a favorable DM phenomenology. This will be obvious when we calculate the observables relevant for DM abundance and its correlation with the scalar spectrum.

Physical State
Mass The mixing angles are identified as:

Dark Sector Lagrangian:
In this set-up the lightest neutral state which is an admixture of the neutral component of SU (2) L fermion doublet (ψ 1 ) and SU (2) R fermion doublet (ψ 2 ) after symmetry breaking, gives rise to a viable candidate of DM due to the unbroken discrete symmetry Z 2 . The Lagrangian of the dark sector in the extended LRSM can be written as, A very similar extension of the LRSM was studied in Ref. [34]. Although our model is the same with similar particle content, our study differs in how the Yukawa structure of the model has been give additional freedom to treat the dark sector fermions independently. We consider a more natural and aesthetic approach by choosing the Yukawa couplings to be identical and therefore allows us to correlate the nature of the DM based on its composition. We note that this also allows a more interesting signature for the doubly charged scalar in the model. Our choice also leads to a very different DM analysis and allowed parameter space for the model.
In the above Lagrangian (in Eqn. 5) M L and M R are the bare masses of ψ 1 and ψ 2 respectively and Y 1 and Y 2 are the Dirac Yukawa couplings. The other two Majorana type Yukawa couplings, y L and y R are responsible for generating the mass splitting between the physical eigenstates after mixing. In addition the non-zero values of both y L and y R lead to interesting collider signatures of the doubly charged scalars (H ±± L,R ) in the model that can affect the search strategies at LHC and ILC which forms a major motivation to study this model.
After symmetry breaking, the dark sector Lagrangian in Eqn.5 leads to mixing between the neutral components ψ 0 1 and ψ 0 2 and also leads to the mixing among the charged components ψ ± 1 and ψ ± 2 thanks to the Yukawa interactions: The mass matrices of the neutral and charged fermion states can be expressed in the interactions basis of respectively as, 2 in the limit of tan β → 0. The phenomenology of DM depends on the following parameters in dark sector: along with the free parameters of LRSM. The nature of DM i.e. whether the DM is SU (2) L type or SU (2) R type or an admixture of them is mainly decided by the choice of the above parameters.
Depending on these parameter choices, the model offers three different type of DM scenarios which we shall discuss now.
For the given mass hierarchy, M L M R , the light neutral states, χ 1,2 and the light charged state, χ ± 1 dominantly behave like the fermion doublet, ψ 1 . The presence of Majorana type Yukawa interaction with ψ 1 (y L ψ 1 ∆ † L iσ 2 ψ c 1 ), leads to the mass splittings (generated after symmetry breaking) between the light neutral and charged fermion states, χ 1,2 and χ ± 1 as: whereas the remaining physical states χ 3,4 and χ ± 2 are very heavy (O(M R )) and do not play any role in contributing to number density of DM. Note here that the light neutral and charged states are nearly degenerate (O(M L )) as v L is small ( 8 GeV [45], constrained from ρ parameter). In this setup the light dark states interact mostly with the SU (2) L fields of LRSM.
Unlike the previous case, here the right triplet (∆ R ) vev, v R plays a significant role. In addition, for the reversed case where M L M R , the light neutral and charged physical states, χ 1 , 2 and χ ± 1 mostly behave like the second fermion doublet, ψ 2 belonging to SU (2) R . Similar to the previous scenario, ψ 2 interacts with the SU (2) R triplet (∆ R ) with the Majorana type Yukawa interactions: which leads to the mass splitting between light neutral and charged physical states, χ 1 , χ 2 and χ ± 1 as: 8 whereas the remaining physical states χ 3,4 and χ ± 2 are much heavier (O(M L )), which mostly decay into light states before the time of DM freeze-out from thermal bath. Hence they do not affect todays DM density. It is important to mention here that the right triplet vev, v R , can easily generate a large mass splitting between the light physical states, obvious from the mass expressions in Eqn.9. The light physical states in this case are SU (2) R like in nature and dominantly interact with SU (2) R fields.
When both M L and M R are of similar order of magnitude and the Yukawa couplings, y R , Y 1 and Y 2 have non-zero values, the physical dark states are admixture of both ψ 1 and ψ 2 . We henceforth refer this as mixed DM scenario. In order to obtain the physical neutral states, χ i (i = 1, 2, 3, 4) one needs to diagonalize the mass matrix M N in Eqn.6 by a unitarity matrix, U N 4×4 . The mass diagonalization leads to a relation between the physical and interactions states given by where χ 1 , χ 2 , χ 3 and χ 4 are the physical eigenstates with mass M 1 , M 2 , M 3 and M 4 respectively following the mass hierarchy |M 1 | < |M 2 | < |M 3 | < |M 4 |. Thus the lightest neutral state, χ 1 of mass M 1 (≡ m DM ) is the stable DM candidate in this setup. The mass eigenvalues of the mass matrix, M N are given in the limit of v L → 0, tan β → 0 as The order of magnitude of the above eigenvalues depend on the model parameters. Here all the neutral physical states, χ i (i = 1, 2, 3, 4) behave like Majorana states and are defined as Similarly the mass matrix, M C for the charged fermion states mentioned in Eqn.6 can be diagonalized by a unitarity matrix, U C 2×2 . The corresponding mass diagonalization relation and the relation between physical and interaction states are expressed as Here χ ± 1 and χ ± 2 are the physical charged eigenstates with mass M ± 1 and M ± 2 respectively with mass hierarchy |M ± 1 | < |M ± 2 |. The eigenvalues of the mass matrix, M C in the limit of v L → 0, tan β → 0 9 are given as In our analysis we have used the numerical tool SPheno [46] to diagonalize the mass matrices and generate the mass spectrum.

Constraints
In this section we briefly address the existing theoretical and experimental constraints on the model parameters which become relevant for our analysis.
Perturbativity: The quartic couplings in the scalar potential, gauge couplings and the Yukawa couplings are bounded from above as: Relic and Direct Search: The current observation from PLANCK [15] puts a stringent bound on DM number density in the Universe : We will implement this constraint on our model parameter space. Along with this astrophysical observation, the DM-nucleon scattering cross-section also faces severe constraints from nonobservation of DM at direct search experiments like XENON 1T [21] and PANDAX 4T [22]. We also include this limit on the model parameter space when we scan over the free parameters.
Higgs invisible decay: When the DM mass is below M h /2, the SM Higgs can decay to DM.
The invisible decay width of the SM Higgs is measured at LHC [47] which gives a constraint on the parameter space that leads to m DM < M h 2 as well as the coupling strength of DM with the SM like Higgs boson through an upper bound: LEP constraint: LEP has excluded exotic charged fermion masses below ∼ 102.7 GeV [48].
We implement this constraint on the dark charged fermions (χ ± 1,2 ) mass M ± 1,2 > 102.7 GeV. FCNC constraint: The bi-doublet structure of the scalar gives rise to tree-level flavor changing neutral current (FCNC) interactions with SM quark in LRSM which is mediated by the heavy neutral scalars H, A and can contribute to flavor observables such as [8,49]. The flavor observable data puts very stringent lower bound on heavy neutral scalars in the model given by M H,A 15 TeV. This upper bound further translates on to the quartic coupling α 3 in the scalar potential and on the triplet scalar vev ∆ 0 R = v R . Using the approximate form of the heavy neutral scalar mass M H , the FCNC constraint can be expressed as: Collider constraint on M W R : The dominant bound on heavy charged gauge boson W come from its decay to dijet. However, in the LRSM model the presence of a heavy right-handed neutrino leads to the possibility of W R decay to a charged lepton and a heavy N R when kinematically allowed.
This leads to a final state with same-sign leptons and jets, which has suppressed SM background and leads to stronger bounds. The decay of W R into a boosted right handed neutrino N R yields same-sign lepton pair plus jets final states at collider as W R → N R → jets + 2 . The current search by ATLAS [50] with integrated luminosity of 80 fb −1 for √ s = 13 TeV, excludes M W R smaller than 5 TeV. This lower bound can be also expressed in terms of v R as: In addition the doubly charged scalars, H ±± L,R are also constrained from existing searches at LHC which we discuss in detail in the collider analysis section. For our analysis we pick up a set of benchmark points (BPs) which are consistent with the above mentioned constraints. The BPs are tabulated in Table-4 and 5 BPs Input parameters Mass spectrum generated from SPheno (in GeV) BP1 v R = 12 TeV

DM Phenomenology
We now discuss the phenomenology of our proposed fermionic dark matter in the extended LRSM.
The lightest neutral Majorana state χ 1 which can be an admixture of SU (2) L like fermion (ψ 1 ) and SU (2) R like fermion (ψ 2 ) or purely SU (2) L like fermion ψ 1 or purely SU (2) R like fermion ψ 2 , is the stable DM candidate under the extended symmetry group G. In this section we review the region of parameter space which is allowed by observed DM density from WMAP-PLANCK data [14,15], latest upper bound on DM-nucleon scattering cross-section from direct search experiments [21,22] and also from indirect search constraints [23,24].
The lightest neutral Majorana state, χ 1 assumed to be the viable candidate of DM can be produced at early time of the Universe via thermal freeze-out mechanism [16]. The dark sector particles were connected with visible sector via gauge and scalar mediated interactions of the LRSM and freezes out when the interaction rate (Γ = σv n DM ) falls below the expansion rate of the universe (H). Apart from χ 1 the dark sector also has heavy neutral and charged fermion states, 3,4) and χ ± j (j = 1, 2) respectively. When the mass of these heavy states lie close to the DM mass, the number density of DM also gets affected by the number changing processes due to these heavy states. So the relic density of DM, χ 1 is guided by the different type of gauge and Higgs mediated number changing processes as: where X and Y are the light states in LRSM. The evolution of DM number density with time can be described by solving the Boltzmann equation given by [16]: where n DM n χ 1 denotes the number density of DM and n eq = g DM ( is the equilibrium density. The mass of DM is defined as m DM (i.e m DM = M 1 ). σv eff denotes the effective thermal averaged cross-section where all annihilation and co-annihilation type number changing processes are taken into account [51,52] and which can be expressed as follows: . The g 1 , g i and g k are the internal degrees of freedom of χ 1 , χ i and χ ± k state respectively and i, j = 2, 3, 4; k, l = 1, 2. The parameter x is defined as T where T is the thermal bath temperature. Using the above expression of σv eff , one can express the number density of DM approximately as [51,52]: where the SM degrees of freedom g * = 106.7 and [16]. T f denotes here the freeze-out temperature of DM. The first term of Eqn.20 represents the standard DM annihilation while the rest of the terms are part of DM co-annihilation. Note that the co-annihilation contribution reduces with large mass splitting, ∆M = M i − M 1 , M ± i − M 1 , due to the Boltzmann suppression of exp(−∆M/T ). It is worth mentioning here the different tools/packages that have been used for our study. We first implement the model in the public code SARAH [53] for generating numerical modules for SPheno [46] and model files for MicrOmegas [54]. We then use SPheno to compute the mass spectrum and branching ratios which is passed on to MicrOmegas to calculate DM relic density as well as obtain the direct and indirect search cross-section.
We shall now illustrate the behavior of DM number density with the following independent free parameters in the dark sector which are relevant for DM phenomenology: The phenomenology of DM also significantly depends on the free parameters like mass of the light states which behave as mediator between dark and visible sector, vevs of the scalar fields v R , v L and tan β in LRSM. It is difficult to study the role of all the free parameters at a time. In addition varying all the parameters at the same time makes it difficult to determine the physics implications of any particular set. We therefore choose to fix the well studied LRSM model parameters which simplifies things but allows us to study the DM phenomenology based on the dark sector parameters.
We first fix the LRSM sector as mentioned in BP1 listed in Table4 As stated earlier in section 2, the DM (χ 1 ) along with heavier components χ 2 and χ ± 1 behave like the SU (2) L doublet dark fermion (ψ 1 ) for the mass parameter M L M R . Since the physical 13 states belong to the SU (2) L doublet, the relic density of DM is mainly governed by SM gauge boson mediated interactions along with the new scalar triplet ∆ L involved through the Yukawa interaction: which is also responsible for generating the small mass splittings between χ 1 , χ 2 and χ ± 1 . Note here that all other states, χ 3 , 4 , χ ± 2 have mass O(M R ) and do not play any relevant role in the DM density. GeV [45], hence the maximum splitting available for this type of scenario is ∼ 8 GeV for y L = 1.
As a result the relic becomes under abundant due to the large gauge mediated co-annihilation cross-sections (χ i χ ± 1 → X Y) as depicted in the left panel of Fig.1. The co-annihilation contribution will get suppressed with increase in DM mass and we find that the correct density is obtained when the DM mass is around 1.2 TeV. As we further increase y L , the light scalars, The spin-independent (SI) DM-nucleon scattering cross-section with DM mass is shown in the right panel of Fig.1 for different values of y L . Due to Majorana nature of DM, the χ 1 χ 1 Z interaction leads to a vanishing contribution to DM-nucleon scattering cross-section. However the presence of the Yukawa interaction y L ψ 1 ∆ † L iσ 2 ψ c 1 can still lead to SI DM-nucleon scattering through Φ − ∆ L mixed t-channel diagrams which depend on the coupling y L . The DD cross-section increases with increase of y L which can be easily seen in the right panel of Fig.1.
With the mass parameter hierarchy M L M R , the DM (χ 1 ) along with χ 2 and χ ± 1 now behave like the SU (2) R dark doublet fermion, ψ 2 . The relic density in this case is mainly governed by right handed gauged mediated interactions along with scalar triplet ∆ R via the Yukawa interaction: The Yukawa interaction splits the state, ψ 0 2 into two physical Majorana states χ 1,2 and generates a much larger mass splitting between χ 1 , χ 2 and χ ± 1 , thanks to the large v R . The other heavy states, χ 3 , 4 and χ ± 2 which are SU (2) L like with mass O(M L ) decay into the less heavier dark states much before DM freeze-out and do not alter the observed DM density. This can be understood from the DM mass formula which is defined as Let us now discuss the mixed scenario where the dark states, χ i and χ ± j are an admixture of both SU (2) L type doublet (ψ 1 ) and SU (2) R type doublet (ψ 2 ). The mixing between the neutral states,  fixed value of v R . As a consequence of that, the DM becomes more ψ 0 2 dominant. Similarly with the increase of y R , the splitting between DM and heavy state increases which corresponds to less co-annihilation contribution to DM density. Therefore the relic density spans the region between under abundance to over abundance with increase in y R as shown in the left top panel Fig.3. This

Parameter space scan
Here we investigate the allowed region of DM parameter space for the mixed scenario. We perform a numerical scan over the following region : while the remaining parameters in LRSM are kept fixed as specified in BP1. Note here that the choices of Y 1 = Y 2 and y L = y R does not affect much in DM phenomenology but it does matter for the collider study which we will discuss in detail in a later section.

Direct search constraint
Non-observation of DM signal at direct search experiments like XENON-1T [21], PANDAX 4T [22] has set a stringent constraint on DM-nucleon scattering cross-section for WIMP like DM. Here we will apply those constraints on our model parameters space which satisfy the observed relic density constraint.     Fig.8 which corresponds to M ± 1 > 102.7 GeV is excluded by the LEP data [48]. We separate the parameter space in two regions along the ∆M direction with ∆M = M W (black dashed line). The light purple region with ∆M > M W , the light charged dark fermion, χ ± 1 (of mass M ± 1 ) can decay to DM, χ 1 via on shell W . Whereas the region with ∆M < M W is shown by light green shaded region, the light charged dark fermion, χ ± where we focus on the doubly charged scalar, we shall choose a different set of BP in the LRSM sector which we discuss in the next section.

Collider signatures of H ±± in presence of dark fermion doublets
The LRSM gives us some unique collider signatures in the form of new gauge bosons, a righthanded charged current interaction, heavy Majorana neutrino production, lepton number violations, etc. Thus each of them can be a test of the model. In addition to the above, the presence of doubly charged scalars in the theory which when produced give a smoking gun signal in terms of resonances in the same sign dilepton final state. In fact, this signal is one of the well studied cases at LHC [55][56][57] which leads to very strong bounds on the mass of the doubly charged scalar. This signal is however shared with other models which also predict doubly charged scalars, for example the Higgs Triplet model which leads to Type-II seesaw for neutrino masses. A significant part of the parameter space for sub-TeV doubly charged scalar is ruled out, when it decays dominantly in the leptonic mode. In LRSM, we have two copies of the doubly charged Higgs where one predominantly couples to the Z while the other couples to the Z R . This in turn affects the production rates for the two incarnates at LHC. However, in the process, both could have a significantly off-shell Z or Z R and conspire to give a cross section of nearly similar strengths for a given mass. The case where the Yukawa couplings that dictate the branching ratios in the leptonic mode being very small for the H ++ L and H ++ R makes the diboson mode (W W ) as the other possibility. Since the W R is relatively heavy, the H ++ R decays mostly via the Yukawa coupling to charged leptons. For the H ++ L the decay is to on-shell W boson. Thus, we get the possibility of 4W or 4 + / E T signal from the doubly charged scalar pair production at LHC. It is noteworthy that this decay mode also relaxes the bound on the doubly charged scalars significantly [55,57].

25
We look at the following signal subprocesses: Figure 9: Feynman diagram for doubly charged scalar production at LHC.
The W ± decays to leptons and/or jets giving rise to the following different final states: We note that the most promising signal would involve the larger multiplicity of charged leptons in the final state which will also be suggestive of the doubly charged scalar as the parent particle. The final states with increasing jet multiplicities would provide complementary signals, hitherto with reduced sensitivity as the SM background would be large compared to an all lepton final state. We therefore restrict ourselves to the first two channels involving n ≥ 3 charged leptons in the final state for our analysis. We also note that this signal overlaps with the 4W final state coming from pair produced doubly charged Higgs when the H ±± → W ± W ± is the dominant decay channel, which could give us an idea on the improvement of the signal over that of the traditional 4W signal.
We use the publicly available package SARAH [58] to write the model files and create the Universal Feynman Object (UFO) [59] files. The mass spectrum and mixings are generated using SPheno [60,61]. We have used the package MadGraph5@aMCNLO (v2.6.7) [62,63] to calculate the scattering process and generate parton-level events at LHC with √ s = 14 TeV which were then showered with the help of Pythia 8 [64]. We simulate detector effects using the fast detector simulation in Delphes-3 [65] and have used the default ATLAS card. The reconstructed events were finally analyzed using the analysis package MadAnalysis5 [66].

4l + / E T signal
The 4l + / E T is one of the cleanest signal because of the low SM background. In our study this signal will appear when all four W bosons produced in the cascade decay of the pair of double charged Higgs, decays leptonically as shown in Fig. 9. The dominant background for the above final state would come from the SM subprocesses producing ttZ, ZZ and V V V [67]. Additional sources of background events could also emerge from tt and W Z production, where additional charged leptons can come from misidentification of jets. Although such events would be small, the sheer size of the cross section of the aforementioned processes could lead to significant events mimicking the signal.
However, these backgrounds can be eliminated by choosing specific selection cuts. The signal and the background process are generated using the same Monte Carlo event generator and then the cross section of the backgrounds are scaled with their respective k-factors. The k-factor for ZZ , ttZ, V V V and W Z considered here are 1.72, 1.38, 2.27 and 2.01 respectively [68][69][70][71]. Here the k-factor for ZZ scales it to next-to-next-to-leading order (NNLO), while the rest of the backgrounds are at next-to-leading order (NLO) cross section.
To consider the four charged lepton final state coming from the 4W we choose events which have exactly N l = 4 isolated charged leptons (l = e, µ) in the final state. As the final state will still be littered with jets coming from initial state radiations, we therefore choose a more inclusive final state where all jets are vetoed with a relatively large transverse momenta of 40 GeV. As basic acceptance cuts, we therefore demand that all reconstructed objects are isolated (∆R ab > 0.4). In addition, • all charged leptons must have p T l > 10 GeV and lie within the rapidity gap satisfying |η l | < 2.5.
• We impose additional conditions to demand a hadronically quite environment by putting veto on events with light jets and b jets with p T b/j > 40 GeV and |η b/j | < 2.5. This helps in suppressing a significant part of the background coming from tt(Z) production.
• The largest contribution to the SM background comes from ZZ. To suppress it and bring it down, we choose a missing / E T > 30 GeV selection cut. Since the ZZ decaying to give four leptons will have very little missing energy in the final state the cut will throw away a significant part of the background events. This cut does not affects our signal much since   it has decay products consisting of dark matter and neutrinos leading to a larger / E T in the signal events. Hence this cut becomes very efficient in improving the signal sensitivity.  • At this point the background is almost at a comparable level with the signal and most of the remaining SM background contribution is from the W Z channel where additional jets/photons can be misidentified as an additional charged lepton. But the events from W Z will give softer decay products and we find the use a strong p T cut helpful in suppressing them significantly.
In our case the sub-leading lepton with p T [l 2 ] separates the signal from background when compared to the same observable for other charged leptons. Thus we choose a p T [l 2 ] > 30 GeV selection cut to help reduce the W Z background.
• There is one more kinematic variable of interest which can be used to distinguish the signal from the background. It is the invariant mass of same-sign (SS) charged lepton pairs which can be used to reduce the background even further. Even though the doubly charged scalar does not decay directly to SS leptons, we expect that in the all lepton channel the SS leptons would come from the same primary scalar. As the SS leptons in our signal come from the decay of same parent particle (H ±± L ), we expect a maximum invariant mass for such lepton the background will be much broader compared to the signal. We can remove this tail in the observable for the SM background without rejecting any significant signal events.
All the cuts used above have been shown through a cut-flow chart in Table6. The final surviving events (after the selection cuts) are shown in Table7 for an integrated luminosity of 3000 fb −1 and the signal significance is estimated using where b stands for the SM background and s represents the new physics signal events respectively.  • all charged leptons must have p T > 5 GeV and lie within the rapidity gap satisfying |η | < 2.5.
• We impose additional conditions and demand no b-tagged jets by putting veto on events with b jets with p T b > 40 GeV and |η b | < 2.5. This helps in suppressing a significant part of the background coming from tt and tt(Z) production.    At this stage the background coming from W Z and tt is massive compared to other background sources and signal. Hence to suppress these two, we divide our event data set for both signal and background in two mutually exclusive sets.
SetA : This corresponds to events which have two positively charged leptons and one negatively charged lepton.
SetB : This set consists of events with one positively charged lepton and two negatively charged leptons.
• To reduce background sources which do (may) not have any / E T , such as ZZ and V V V in contrast to our signal which has both neutrinos and dark matter as source for missing  transverse energy, we use a selection cut of / E T > 30 GeV. We also use as before, a rejection cut on the invariant mass of opposite sign same flavor lepton pairs near the Z mass pole to further suppress the background events.
• In our signal two same sign leptons come from same parent doubly charged Higgs via the W bosons while the other doubly charged Higgs gives the opposite sign lepton and two jets in its decay chain. Hence a strong correlation in invariant mass shows up for decay products of these two doubly charged Higgs. So we can put a rejection cut on SetA, with M (l + l + ) , This cut is very effective in reducing the background coming from tt and W Z.   We therefore conclude that with high enough integrated luminosity we can discover a doubly charged Higgs of mass around 300 GeV in the multi-lepton final state with at least N l = 3 leptons.
As the lepton multiplicity decreases we find that the large SM background is more difficult to suppress and give enough sensitivity to observe a doubly charged Higgs decaying to the dark fermions.
In addition, a similar analysis of 4l and 3l final states for the third benchmark point (BPC3) shown in Table5, which represents a compressed mass spectrum for the lighter states in the dark fermion sector, yields a very low (< 1σ) signal significance for a 300 GeV doubly charged scalar. The 3-body decay of the ψ ± leads to softer final states which make it more difficult to distinguish from the SM background leading to lower signal sensitivity. give a mass bound as low as between 230-350 GeV [57], the modified decay modes in our model lead to a much weaker sensitivity at current integrated luminosities. Even with the full high luminosity LHC (HL-LHC), we find that a discovery of such a doubly charged scalar would still be limited to sub-400 GeV masses. It would therefore seem that while the LHC energies would probe a much higher energy scale of models such as LRSM and restrict very heavy W R and Z R , it would lack in efficiency for these doubly charged exotics. It would be interesting to find out the sensitivity for such particles at the proposed ILC which may be restricted by its energy reach but would prove beneficial for such particles in general which become more elusive at LHC as they develop newer decay channels in their fold. We choose to highlight just a simple comparison with one of the signals studied at the LHC here and leave a more dedicated ILC study for later work. We however present a slight variation in the spectrum to include a compressed scenario, which has very clear challenges in LHC searches.

3l + 2j + / E signal analysis for BPC3 at ILC
The benchmark point BPC3 is chosen as it represent a scenario where the mass gap between χ ± 1 and dark matter (χ 1 ) is less than the mass of W such that the decay χ ± 1 → W ± χ 1 is energetically forbidden. Here χ + 1 has a three-body decay to l + i ν j χ 1 or 2j χ 1 . In this case the leptons and jets will be soft and one can no longer put a strong requirement on the p T of jets as required to avoid large hadronic debris that can affect any analysis at LHC. This makes an analysis for such a compressed spectrum leading to soft final states at LHC very challenging. Since ILC has a much cleaner environment the jets can be triggered upon with much lower energies and we can put weaker jet tagging conditions (P T (j) > 10 GeV), better suited for benchmarks such as BPC3.
For analyzing this signal at ILC we consider the dominant background coming from W W Z which gives us 3l + 2j + / E final state where one W decays hadronically while the other W and Z decay leptonically producing three leptons and missing energy. As before, our signal comes from H ++ L H −− L pair production via photon or Z mediator (the heavy Z R contribution is negligible). Each H ±± L then decays to two χ ± 1 followed by one χ ± 1 decaying to χ 1 and 2 j and rest of the dark charged fermion decaying leptonically via the W boson. For the analysis we choose only those events as signals which have exactly three charged leptons (i.e., e and µ) and exactly two jets. The basic acceptance cuts for all isolated objects (i.e., ∆R ab > 0.2) are chosen similar to that for LHC.
We list the selection cuts for the signal and background events in Table10 along with the events surviving each cut. Here the first cut is on invariant mass of opposite sign same flavor lepton at Z mass pole which suppresses the background significantly because W W Z has one Z always decaying leptonically. The next cut is on the invariant mass of SS leptons which is a prominent characteristic 35 Cuts (GeV) 85 < M e + e − < 95 85 < M µ + µ − < 95 M l ± i l ± j > 80 M l 3 j 1 j 2 > 110 significance  of the signal (as discussed earlier) since these lepton pairs come from the decay chain of the same doubly charged Higgs. The final cut is on the invariant mass of visible decay product of one doubly charged Higgs where one χ ± 1 decays hadronically while the other decays to give a charged lepton. We find that for a mass of H ++ similar to BPC1 and a compressed spectrum in the dark sector which gives a 3-body decay for the charged dark fermion, HL-LHC gives a signal sensitivity of less than 1σ for BPC3 while the same final state is able to achieve a much higher sensitivity at ILC with √ s = 1 TeV for a much smaller luminosity, and gives a 4.5σ signal with an integrated luminosity of 1000 fb −1 . In fact, the cleaner environment at ILC would open up the same benchmark for a hadronic rich final state with much larger effective cross section. As a multi-particle final state with hadrons improves the size of signal events and one can effectively control the background easily since all processes will be through electroweak interactions, ILC will give a much higher significance in the final states with smaller lepton multiplicity. The same-sign lepton pairs however provide a symbolism of the produced doubly charged scalar and therefore to establish its presence, the channel with 2 ± 2j / E would be a more appropriate channel for the study at ILC.

Conclusions
In this work we have used the well motivated left-right symmetry model and invoke a dark matter show the region of parameter space of the mixed DM scenario which is consistent with relic density observations and also satisfy direct detection constraints as well as indirect detection constraints.
The dark sector also contains charged states which on one hand can play a major role in DM phenomenology by contributing to the number density through co-annihilations with the DM when the mass splitting is very low between them, while they could be directly produced at experiments through gauge interactions. Their phenomenology would be very similar to a pair produced VLL that decays to a W boson and a DM. We are however more interested in the signal for the more unique doubly charged Higgs present in the model in the presence of a dark fermion sector which couples to it directly. As in the case of the lepton doublets having a Majorana interaction with the triplet scalars of the model leading to a seesaw mechanism for neutrinos, the dark fermion sector would also have a similar seesaw mechanism. Thus in a significant region of parameter space consistent with DM observations, the doubly charged scalar can decay to the pair of charged dark fermions. We consider this interesting possibility in our work and perform a detailed collider analysis of its signal at the LHC in multi-lepton final state. We find that the bounds on the doubly charged scalar become weaker compared to the more standard leptonic and bosonic decay modes and LHC sensitivity for a doubly charged Higgs in such case would be sub-400 GeV even with an integrated luminosity of 3000 fb −1 . We then show how this reach can be improved at the proposed ILC with a center of mass energy of 1 TeV by comparing a similar final state which gives less than 1σ sensitivity at LHC but an improved 4.8σ sensitivity at ILC.