Heavy Majorana neutrino pair production from $Z^\prime$ at hadron and lepton colliders

A gauged U$(1)$ extension of the Standard Model (SM) is a simple and anomaly free framework where three generations of Majorana type right-handed neutrinos (RHNs) are introduced to generate light neutrino mass and flavor mixings through the seesaw mechanism. We investigate such models at different hadron and lepton colliders via $Z^\prime$ induced Majorana type RHNs pair production. We derive bounds on U$(1)$ gauge coupling $(g^\prime)$ comparing the model cross sections with experimentally observed data for different $Z^\prime$ mass $(M_{Z^\prime})$ and RHNs mass $(M_N)$. Using these limits we estimate the allowed RHN pair production cross section which can be manifested by lepton number violating signatures in association with fat-jets at the hadron colliders depending on the mass of the RHNs. Hence we study dilepton and trilepton modes with fat-jet/s of the signal for different benchmark values of $M_{Z^\prime}$ and $M_N$. Using fat-jet signatures and studying the signal and corresponding SM backgrounds, we estimate bounds on $M_N-M_{Z^\prime}$ plane at different center of mass energies which could be probed at different hadron colliders. In the context of the lepton colliders we consider electron positron initial states where Majorana type RHNs can be produced from $Z^\prime$ manifesting same sign dilepton plus jets signature and trilepton plus jets in association with missing energy. Studying the signal and corresponding SM backgrounds we estimate the bounds on the $M_N-M_{Z^\prime}$ plane for different center of mass energies. In the context of U$(1)$ extension of the SM there is an SM singlet BSM scalar which couples with the RHNs. We can probe the Majorana nature of RHNs via this BSM scalar production at electron positron colliders.


I. INTRODUCTION
The experimental observations of the SM have set it on a stable foundation, however, the observations of the light neutrino mass and the flavor mixing [1] give a strong indication of physics beyond the Standard Model (BSM). This leads us to extend the SM. From a perspective of the low energy effective theory, one can introduce a dimension-5 operator [2] involving the Higgs and lepton doublets within the SM framework which violates lepton number by two units. After the breaking of electroweak (EW) symmetry, the neutrinos acquire tiny Majorana masses which are suppressed by the scale of the dimension-5 operator. In the context of a renormalizable theory, the dimension-5 operator can be naturally generated by introducing SM singlet heavy Majorana RHNs. This is the well known type-I seesaw mechanism [3][4][5][6][7][8].
RHNs in the TeV scale or lighter can be produced at the Large Hadron Collider (LHC) or other hadron colliders with a distinctive same-sign di-lepton plus jets final state which manifest the lepton number violation. The RHNs are SM-singlet therefore they can only be produced at the colliders through the mixings with the light neutrinos. The estimated light-heavy neutrino mixing becomes naturally small (∼ 10 −6 ) when the TeV scale RHNs reproduce the observed light neutrino mass around 0.1 eV. This mixing parameter can be comparatively large when the Dirac mass matrix is generally parametrized [9] in order to satisfy the neutrino oscillation data, electroweak precision measurements and the lepton flavor violating processes [10]. The study of the heavy neutrinos have been a point of interest for a long period of time at different high energy colliders from a variety of production modes to study different final states to estimate bounds on light-heavy neutrino mixing as a function of heavy neutrino mass .
In addition to the RHN productions through the light-heavy neutrino mixing [62][63][64][65][66][67], this model provides a new mechanism for the RHN production in pair through Z at the colliders. Once produced these RHNs can decay into the SM particles through the usual light-heavy neutrino mixings.
In this paper we consider a general U(1) X scenario which includes three generations of Majorana type RHNs to cancel the gauge and mixed gauge gravity anomalies. After the U(1) X symmetry breaking, the light neutrino masses are generated by the type-I seesaw mechanism. The U (1) X symmetry can be identified as the linear combination of the U (1) Y in SM and the U (1) B−L gauge groups, hence the U (1) X scenario is the generalization of the U (1) B−L extension of the SM. A suitable choice of the U(1) X charges can even enhance the RHN pair production cross section from the Z compare to U (1) B−L [68][69][70], which can further increase the discovery potential of the RHNs. In addition to that, there is an alternative U (1) X scenario where two of the RHNs have U(1) X charge as −4 and the third one has the U(1) X charge as +5 [71]. Note that, although the RHNs production is light-heavy neutrino mixing independent, RHN decays to SM final sates such as W, νZ and νh through the mixing. For small enough values of the mixing, the heavy neutrinos can be long-lived, leading to displaced decays. The displaced decays of the RHNs in this model have been studied in [72,73].
If the Z and Majorana type RHN masses reside in TeV scale, they can be produced at high energy colliders. The U(1) X coupling can be constrained from the existing and prospective experimental results. Using the allowed parameters, pair production of RHNs can be possible from Z at hadron colliders. Subsequently, each of the RHNs will dominantly decay into a charged lepton and SM W boson. The W can decay dominantly into quarks (qq ) or subdominantly into leptons ( ν ). Depending on the mass of the RHNs, the W boson produced can be sufficiently boosted so that the hadrons produced from the W can form a fat-jet. Hence from each RHN we obtain ( ± ) and a fat-jet (J). As a result each event contains two same sign charged leptons and two fat-jets ( ± ± + 2J) due to the Majorana nature of the RHNs. On the other hand there is another possibility where one of the W bosons can decay leptonically which shows a trilepton plus one fat-jet accompanied by missing momentum ( ± ± ∓ + J + p miss T ). In this paper, analyzing these signals we investigate the discovery potential of the RHNs. Majorana type RHN can be produced at the electron positron colliders from Z which can also produce W in pair. Hence same sign dilepton signature can be produced in association with four jets ( ± ± + 4j). We can also study the trilepton mode in association with two jets and missing energy when one of the W from RHN decays hadronically ( ± ± ∓ + 2j + MET). General U(1) extension of the SM contains SM singlet scalars which has Yukawa interaction withe Majorana type RHNs. In this article we consider production of the BSM scalar in e − e + collision by Z association and vector boson fusion process. The BSM scalar can decay into a pair of RHNs which can further decay into same sign dilepton mode in association with jets and missing energy.
We arrange this paper in the following way. In Sec. II we describe the different gauged U (1) extensions of the the SM and study the constraints on the U(1) coupling as a function of M Z . We give the relation on neutrino mass, mixing and partial widths in Sec. III. The bounds on g − M Z plane has been discussed for two cases (general U (1) X and alternative U (1) X ) in Sec. IV where we compare with the existing and prospective limits. We estimate the production cross sections of the heavy neutrino pair using the experimental bounds and propose theoretically estimated density plots on M N − M Z plane in Sec. V at hadron colliders. In this section we choosing different benchmarks values of M Z and M N to study the same sign dilepton and trilepton modes in association with fat-jets. Simulating the signal, backgrounds and applying kinematic cuts we postulate a 2−σ exclusion contour on M N − M Z plane which could be probed at different hadron colliders. In Sec. VI we study the RHN pair production at the electron positron collider and study multilpeton modes in association with jets and missing energy. We study the pair production of the RHNs from the BSM scalar at the electron positron colliders in Sec VII. Finally we conclude the article in Sec. VIII. U (1)X sector is given by where C stands for the charge-conjugation. The Higgs potential of this model is given by In the limit where λ is small we can analyze separately the Higgs potential for H and Φ as a good approximation.
To break the electroweak and the U (1) X gauge symmetries we consider the parameters of the potential for the scalar fields H and Φ to develop their VEVs at the potential minimum where v h 246 GeV is the electroweak scale and v Φ is a free parameter. After the symmetry breaking, the mass term of the U (1) X gauge boson (Z ) is generated The process of U (1) X symmetry breaking also induces the Majorana mass term for the RHNs from the second term of the Eq. 1 and followed by the electroweak symmetry breaking the neutrino Dirac mass term is generated In this model x H and x Φ are real parameters and U(1) X coupling g is a free parameter. Without the loss of generality we consider the basis where Y 2 is a diagonal matrix. With the Majorana and Dirac neutrino mass terms in Eqs. 5 and 6 respectively the seesaw mechanism becomes accountable for the generation of the tiny Majorana masses of the light neutrino mass eigenstates.

B. Case-II
There is another interesting U (1) X extension of the SM whose minimal particle content is shown in Tab. II. We call it an alternative U(1) X scenario. The U(1) X charge x H is a real parameter and the U(1) X coupling g is a free parameter. The RHNs in this model are differently charged under the U(1) X . In this model first two generations of RHNs have charge −4 whereas the third one has a charge +5. This non-universal charge assignment is a unique choice in order to cancel all the anomalies [74]. In this model we introduce two Higgs doublets (H 1 , H 2 ) and three additional SM-singlet scalars (Φ 1,2,3 ). The Higgs doublet H 2 is responsible for the generation of the Dirac mass term for N R1,2 . The SM-singlet scalar Φ 1 is responsible for the generation of the Majorana mass term of N R1,2 after the U(1) X breaking. The Majorana mass term of N R3 is generated from the VEV of Φ 2 , however, there is no Dirac mass term for N R3 due to the preservation of U(1) X symmetry. Hence N R3 does not participate in the neutrino mass generation mechanism. The relevant part of the interaction Lagrangian of the RHNs is given by The charged-current interactions can be expressed in terms of the neutrino mass eigenstates where e represents the three generations of the charged leptons, and P L = 1 2 (1 − γ 5 ) is the projection operator. Similarly, in terms of the mass eigenstates, the neutral-current interactions are written as where c w ≡ cos θ w with θ w being the weak mixing angle. From the Eqs. 18 and 19 we see that the heavy neutrinos (N ) decay into W , νZ and νh respectively. Here h is the SM Higgs boson. For sterile neutrinos heavier than W , Z and h, the decays are on-shell, i.e., tow body followed by the further decays of the SM bosons otherwise there will be three-body decays of the heavy neutrinos, with the off-shell SM bosons. For simplicity we assume that the U (1) X Higgs bosons in Cases-I and II are heavier than the RHNs. In these models we consider heavy Z in the TeV scale. As a result, RHNs decay through the off-shell Z will be negligibly small due to the Z mass suppression. The coupling between the Z , light neutrino and RHN arises after the U (1) X breaking, however, this is proportional to the light-heavy mixing and suppressed. Therefore we neglect all the Z mediated off-shell decays of the RHNs into the SM fermions. The allowed decay modes of the RHNs are written in the following. When we consider the RHNs are heavier than the SM bosons so that they can decay into W , ν Z, and ν h on-shell modes. The corresponding partial decay widths are When RHNs are lighter than the SM bosons they decay into three body modes where partial decay widths of N i are approximately given by where respectively which come from the Z boson interaction with the quarks and N c = 3 is the color factor. Eq. 23 is the interference between the Z and W mediated channels where all flavors are same. We estimate the partial decay widths in a limit neglecting the SM fermion masses.

IV. Z PRODUCTION AND BOUNDS ON THE U(1)X GAUGE COUPLING
In the U(1) X models, the RHNs have non-zero U(1) X charges like all SM fermions. It ensures the production of the RHNs through the Z at the hadron collider. First Z boson is resonantly produced and subsequently decay into a pair of RHNs, if kinematically allowed. The production cross section depends on the choice of the parameter which is consistent with the dilepton [84][85][86], dijet [87,88] searches from the LHC and constraints obtained from LEP [89][90][91][92]. To estimate the cross section we calculate the partial decay widths into a pair of SM fermions neglecting their masses and a pair of Majorna RHNs as respectively where N c = 3(1) for SM quarks (leptons), Q f L (R) are the U(1) X charges of the left (right) handed SM fermions and Q N is the U(1) X charge of the heavy neutrino. In the bottom panel of Fig. 1 we plot the ratio of the branching ratios of Z → N N to Z → − + which is defined as where Q N is the U(1) X charge of the RHNs under consideration from the Cases-I and II respectively. This property is valid for any RHN mass below M Z 2 for Cases-I and II. To estimate the bounds on the U(1) X gauge coupling we calculate the dilepton production cross section from the Z at the 13 TeV LHC. We compare our cross section with the observed dilepton cross sections from the LHC where sequential standard model (SSM) [93] was studied assumming Γ m = 3%. Using the following master equation we find the bounds on the U(1) X gauge coupling. The Z production cross sections can be calculated using narrow width approximation as where q(x, Q) and q(x, Q) are the parton distribution functions of the quark and antiquark respectively andŝ = xys is the invariant mass squared of the colliding quark at the center of mass energy √ s. Due to the narrow width approximation the cross section of the colliding quarks to produce Z boson iŝ With a factorization scale set at Q = M Z and employing CTEQ6L [94], one can estimate σ Model = σ(pp → Z ) BR(Z → 2 ) or σ(pp → Z ) BR(Z → 2j) for dilepton or dijet final states. For the dijet production cross section we use 58% and 70% acceptance respectively to compare with the LHC results at the ATLAS and CMS respectively. which are shown by the dashed lines (Red/ Green/ Blue) to represented the LHC dilepton (ATLAS2l/ CMS2l/ ATLAS-TDR) bounds. Due to this fact the Z can not decay into the RHNs. Hence the other decay modes of Z will be enhanced providing the strongest bound on g ; (ii) when two generations of the RHN masses are below M Z 2 leading to the decay of the Z into the kinematically allowed RHN pairs. We have chosen two benchmark scenarios in this case for at least two generations of the degenerate RHNs with M N1,2 = 500 GeV which are shown by the dotted lines (Red/ Green/ Blue) to represent the LHC dilepton (ATLAS2l-2/ CMS2l-2/ ATLAS-TDR-2) bounds; (iii) similar bounds for M N1,2 = 1 TeV are represented by dot-dashed lines (Red/ Green/ Blue) to represent the LHC dilepton (ATLAS2l-3/ CMS2l-3/ ATLAS-TDR-3) bounds. The di-jet bounds are shown by the magenta lines for the case (i). It is important to notice that the dilepton bound is the strongest one with respect to the di-jet. Therefore we did not extract the dijet bounds considering the RHN masses at 500 GeV and 1 TeV. The lines with different RHN masses match with the case where RHNs are heavier than M Z 2 for the heavier M Z mass because of factor (1 − 4 in the Z → N N partial mode. It becomes almost 1 when M Z >> M N . The bounds on the g − M Z plane for Case-II are shown in the right panel of Fig. 2, the branching ratio of Z → N N is more than one order of magnitude higher than the Case-I due the U(1) X charges. Such a behavior has been reflected in the nature of the constraints on g for different M Z . The bounds from the LHC are estimated using narrow width approximation at 139(140) fb −1 luminosity at the LHC where the Z production cross section is proportional to g 2 . Hence we naively estimate the future or prospective upper bound on g scaling the luminosity following The bounds are calculated from the dilepton channels at the ATLAS data [84], CMS [85,95], ATLAS-TDR [86]. The bounds from the LEP-II data have been calculated from [91]. The bounds from the dijet have been estimated from the ATLAS [87] and CMS [88] respectively. The shaded region is ruled out by the current experimental data.
applicable. Similar scenario could be obtained for the other choices of the heavy neutrino masses, however, those bounds will be weaker than the case M N > M Z 2 . These prospective limits could be verified in the near future. LEP-II bounds have been estimated from [89][90][91][92] for x H = −1.2 and has been shown by the solid cyan line in Fig. 2. We estimated bounds on the g − M Z plane from ILC at √ s =250 GeV, 500 GeV and 1 TeV respectively from [92] considering M Z > √ s. Hence from [92] for x H = −1.2 we find, the limit is M Z g > 2.68 TeV from LEP-II and the prospective limits on this quantity are 13.85 TeV, 23.74 TeV and 41.96 TeV respectively from the ILC at √ s = 250 GeV, 500 GeV and 1 TeV respectively. Hence we estimate the upper bounds on g -M Z plane from e − e + colliders which are represented by LEP-II (cyan), ILC250 (brown, dashed), ILC500 (brown, dotted) and ILC1000 (brown, dot-dashed) respectively in Fig. 2 for Case-I (II) in the left (right) panel. The LEP-II limit is weaker compared to the current LHC bounds, however, the prospective bounds obtained from the ILC are stronger than the existing LHC limits for heavier Z . In Tab. III we show some allowed benchmark values of M Z , M N and g which will be used in the RHN pair production from Z .  BR(Z → N N ) plays an important role to study RHN pair production from Z at the hadron colliders. From Fig. 1 we find that it is maximum for x H = −1.2 over any other decay modes including the otherwise stronger dilepton modes whereas for other values of x H , BR(Z → N N ) is subdominant. Therefore we choose x H = −1.2 for RHN pair production at the hadron colliders. Also note that different values of x H for x Φ = 1 have implications in the theory which has been studied in [92] in detail manifesting the chiral nature of the model. We have studied the low energy aspect of the model for light Z which could be tested at the neutrino experiments like DUNE to probe the chiral nature of such a scenario in [96]. On the other hand variation of x Φ can manifest the periodic nature of the model which is beyond the scope of this article, however, will be tested in future to study different aspects of this model. On the other hand we will use the aspect M Z >> √ s for the e − e + colliders to produce the RHN pair from Z where we will consider different values of x H other than −1.2 which will also manifest the chiral nature of the Z for RHN pair production.

V. HEAVY NEUTRINO PAIR PRODUCTION AT pp COLLIDERS
The RHNs can be produced in pair at √ s = 14 TeV, 27 TeV and 100 TeV proton proton colliders respectively through the Z production for After the RHNs are pair produced, they dominantly decay into W mode followed by the hadronic decay of the W boson. Hence due to the pair production of the Majorana RHNs, same sign dilepton (SSDL) plus four jet signal is produced. We show the density plots on the . The bar charts in the density plots represent the cross sections from low to high (bottom to top) in fb. The colored area shows the allowed cross sections of the trilepton pus two jet in association with missing energy in M N − M Z plane. In this case we consider the Z mass from 2 TeV to 10 TeV. For the Case-I at 14 TeV we did not obtain any points for M Z > 6 TeV, at 27 TeV did not obtain any data point for M Z > 8 TeV. However, at the 100 TeV we obtain larger cross sections for the trilepton case from the RHN pair production even for large M Z . The density plots for the Case-II are shown in the lower panel of Fig. 4. Similar behavior like the Case-I can be observed, however, the cross sections increase due to the U(1) X charge of the RHNs and the cross sections increase with center of mass energy.
Another important motivation of this paper is to study the boosted objects from the RHNs pair. After the RHNs are produced, they can decay through the dominant mode W , followed by the hadronic decay of the W . The Majorana nature of the RHNs will manifest a distinct ± ± signature along with the hadronic decay of the W boson. We consider the RHNs which are sufficiently heavy, e. g., M N ≥ 500 GeV which allow the W boson to be boosted so that the hadronic decay mode of the W can be collimated to produce fat-jet. Hence SSDL plus two fat-jet and trilepton plus one fat-jet shown in Fig. 5 could be interesting to study.
FIG. 5. Heavy neutrino pair production processes at the hadron collider from the Z . The heavy neutrinos (N ) decay into the SSDL plus two fat-jets (left) and trilepton plus one fat-jet in association with missing energy (right).

A. Same sign dilepton (SSDL) with fat-jets
The resonant production of the Majorana RHN pair from the Z can show a distinct signature of the lepton number violation at the collider. In this case RHNs will produce same sign leptons and W bosons. Due to the heavy mass of the RHNs the W boson will be boosted to make a fat-jet. The production process is where J 1 and J 2 are the fat-jets, shown in the left panel of Fig. 5. In our scenarios there are three RHNs out of them we consider the first two generations are degenerate. We consider two benchmark scenarios M N1,2 = 500 GeV and M N1,2 = 1 TeV respectively for the U(1) scenarios stated in Cases-I and II.
The dominant SM background in the case of SSDL scenario comes from the same sign W boson production in association with jets. The same sign W will decay leptonically to produce SSDL environment and the two jets will resemble the W ± like fat-jets. Another significant background will be contributed from W ± Z+ jets where W ± and Z decay leptonically. In addition to that, another important contribution will come from W ± W ∓ Z+ jets background where one W and Z will decay leptonically where as the remaining W will decay hadronically. In these cases there will be an SSDL pair in association with an opposite sign third lepton. A third lepton veto will reduce these backgrounds. Other significant contributions will come from ttW ± and ttZ channels. The SSDL events will come from either t or t and the leptonic decay of W ± . A similar scenario can be observed for ttZ for the SSDL signature, however, there will be an additional lepton. In this case, the choice of the two same sign lepton and third lepton veto are very important to reduce the background. The W like fat-jets can be produced from the additional jets in the event or from the remaining t or t. We use the veto on b-jets to reject events with b-jets.
Implementing the model 1 in FeynRules [97,98] we generate the events using MadGraph [99,100] and parton distribution function CTEQ6L [94] fixing the factorization scale µ F at the default MadGraph option. The showering, fragmentation and hadronization of the signal and SM backgrounds were performed by the PYTHIA8 [101]. The detector simulation of the showered events was performed using the detector simulation package Delphes [102] equipped with the Cambridge-Achen (C/A) jet clustering algorithm [103,104]. In our analysis, we have taken jet cone radius, R = 1.0. The jets and associated sub-jet variables are constructed using the softdrop procedure. The softdrop procedure depends on two parameters, an asymmetry cut Z cut , and an angular exponents β describe here [105][106][107]. For β = 0, the Softdrop algorithm to find sub-jets is the same as the modified mass-drop tagger [107]. The algorithm goes as, (i) By undoing the last step of the C/A clustering algorithm of fatjet we will get two subjets, for example, j 1 and j 2 , (ii) Now, if these two subjets pass the asymmetry cut > Z cut (here we use Z cut = 0.1) we call the fat-jet as a softdrop jet. (iii) If the last condition is not satisfied, we consider the largest p T jet as a fat-jet and do the same procedure from step 1. In this algorithm, the asymmetry cut helps us to discriminate the jets created from the decays of the SM heavy resonance particles (top quark, W , Z and Higgs bosons) with respect to the quark and gluon jets which have pure QCD origin. This asymmetry cut also removes contamination from ISR, FSR, etc. Because of that, the softdrop jet mass is close to the heavy resonance particle mass if the source of the fat-jet is a decay of heavy resonance particle.
Backgrounds processes SSDL+jets 14 TeV 27 TeV 100 TeV  Generating the µ ± µ ± + 2J and e ± e ± + 2J signals at 14 TeV we study Cases-I and II. The corresponding SM backgrounds are W W jj, W Zjj, W W Z+jets, ttW and ttZ respectively. The backgrounds are generated using the selection cuts H T > 300 GeV where H T is the scalar sum of the transverse momentum of the jets (p j T ). The jets are selected with the transverse momentum p j T > 10 GeV, pseudorapidity of jets |η j | < 2.5. The transverse momentum of the leptons p T > 10 GeV, pseudorapidity of the leptons |η | < 2.5, separation between the leptons in the η − φ plane ∆R > 0.4 and lepton and jets ∆R j > 0.4 are among them. The partonic cross sections after the basic cuts are given in Tab. IV for √ s = 14 TeV. To generate the W W Z + jets, ttW , and ttZ backgrounds, we slightly change our generation level cuts. Here we applied the same H T cut as above for all jets, including jets coming from standard model heavy resonance decay. However we did not use any p T , η, and ∆ R cuts to leptons and jets coming from standard model resonance decay. After the primary cuts the p T distributions of the fat-jets and the leptons are shown in Fig. 6. The distributions of the jet masses and missing energy are shown in Fig. 7. After selecting the signal events we use advanced cuts for the signals and SM backgrounds. To study the SM background from W W Z+ jets process we consider the leptonic decay of one W and the hadronic decay of the other whereas the Z boson decays into charged leptons. Such leptonic events allow us to use a third lepton veto in the advanced cuts. A similar procedure has been adopted for ttZ. To study SM backgrounds from ttZ process, we consider the leptonic decay of one W from the top quark and the hadronic decay of the other from the remaining top quark, whereas the Z boson decays leptonically. From Fig. 6 we find the peak of the p J1 T distribution situated in the high p T region whereas the backgrounds are distributed mainly in the low p T region. We consider this as the leading fat-jet and the remaining fat-jet (J 2 ) as the sub-leading one. The sub-leading fat-jet has a peak coinciding with the SM backgrounds. For the heavier RHNs, these jets are more energetic, and their distribution shifted to the higher p T region. The leptons from the RHNs are more energetic than those from SM backgrounds. Hence leptons' p T will be another important discriminator along with the p T of the fat-jets. From the distribution of the jet masses in the upper panel of Fig. 7, we find peaks from the signal and some SM backgrounds like W W Z, ttW and ttZ around the W boson mass; however, the backgrounds peaks are not that prominent as the signal peak. These allow us to choose a window of 15 GeV around the W boson mass to suppress the SM backgrounds further and get rid of other low energy hadronic effects below 60 GeV. From the missing energy distributions shown in the lower panel of Fig. 7 we find a conservative cut below 150 GeV. We have noticed that many trilepton events coming from the SM processes W W Z, ttW and ttZ where we use a criteria of at least two leptons having the same sign. In addition to that we use a third lepton veto which reduces such SM background events further. The backgrounds coming from the top quark induced events contain b-jets. In the final selection we use a b-veto to reject such events.
Selecting the signal and backgrounds with basic cuts we employ the post selection cuts or advanced cuts. The transverse momentum of each fat jet has been considered to be p J T > 180 GeV (C-I). We select the events with SSDL with transverse momenta p 1 T , p 2 T > 100 GeV (C-II). Both of the jets have sub-jet number greater than 1 (C-III). We have considered that the invariant mass of the two leading sub-jet is within a window of ±15 GeV around the W boson mass for the leading fat-jets (C-IV). Finally we consider that both jets are not b-jets (C-V). We apply a conservative cut on the missing energy as E miss T < 150 GeV (C-VI). The cut flow has been shown in Tab. V with the corresponding efficiencies for the signals and corresponding backgrounds for µ ± µ ± and e ± e ± samples in the upper and lower halves of the table respectively. The upper part of each row represents the + + signal and the lower part represents the − − signal respectively. The combined significance of the SSDL mode has been calculated using S (S) and background (B) events. Solving for a particular significance we estimate the required luminosity for Cases-I and II. We find the significance is below 1−σ up to the total luminosity of the LHC for both of the benchmark RHN masses whereas an achievable 5−σ significance can be reached for Case-II around 1.88 ab −1 for M N = 1 TeV and 1.52 ab −1 luminosities for M N = 500 GeV respectively.
We analyze the SSDL signal with two fat jets consisting µ ± µ ± + 2J and e ± e ± + 2J signals at √ s = 27 TeV. After that we generate the corresponding SM backgrund processes W W jj, W Zjj, W W Z+jets, ttW and ttZ to compare the signals for M N1,2 = 500 GeV and M N1,2 = 1 TeV both for Case-I and II. To generate the backgrounds we impose the selection cuts as H T > 300 GeV, p j T > 10 GeV, |η j | < 2.5, p T > 10 GeV, |η | < 2.5, ∆R > 0.4 and ∆R j > 0.4 respectively. The parton level cross sections of the SM backgrounds after the selection cuts are given in Tab. IV. By studying the signal and the backgrounds we finalize the advanced cuts for the signals and backgrounds. We do not show the histograms, apart from the higher energy reaches of the kinematic variables, their nature will remain almost the same as the previous case. The advanced cuts are used selecting at least two fat-jets (J) with transverse momentum p J T > 180 GeV (C-I). Only two same sign leptons with p T > 100 GeV are selected (C-II) vetoing the third lepton. Both of the jets have sub-jet number > 1 (C-III). The Invariant mass of the leading two sub-jets is within ±15 GeV window around the W boson mass (C-IV). We ensure that both jets are not b-jets (C-V). The missing energy is conservatively considered as / E T < 150 GeV (C-VI). The cut flow and the corresponding efficiencies for µ ± µ ± + 2J and e ± e ± + 2J signals and SM backgrounds are given in the top, right panel of Tab. VI.
Solving the signal and background relation for a particular significance we can estimate the luminosity to achieve that significance. Hence we calculate the significance for 1 TeV RHN from Case-I around 5−σ at 14.23 ab −1 luminosity and that for 1 TeV reaches up to 3.5 − σ at 15 ab −1 luminosity. The results for Case-II are also improved compared to the previous case. A significance of 5 − σ can be attained at 144 fb −1 luminosity for 500 GeV RHN whereas that can be attained at 130.5 fb −1 luminosity for 1 TeV RHN respectively.
Finally, we study the SSDL signal from Cases-I and II at √ s = 100 TeV proton proton collider considering M Z = 3 TeV. All primary selection cuts are the same as the previous two cases except H T . Here we use H T > 500 GeV for all the background processes. The partonic cross sections of the SM backgrounds are given in Tab. IV. After selecting the signal events we use the advanced cuts for the signals and SM backgrounds. The advanced cuts are used by selecting at least two fat-jets (J) with transverse momentum p J T > 250 GeV (C-I). The other advanced cuts are the same as µ ± µ ± + 2J W ± W ± jj W ± Zjj W ± W ∓ Zj ttW ± ttZ sig(500 GeV) sig(1 TeV) sig(500 GeV) sig(1 TeV)  the previous two cases. In this case we have only considered two degenerate RHNs with mass M N = 1 TeV. The cut flow table for SSDL scenario is given in Tab. VII with the efficiencies after each level of cuts for the µ ± µ ± + 2J and e ± e ± + 2J channels. We find that the 5 − σ significance could be achievable for the Case-I around 534 fb −1 luminosity and for the Case-II, the same benchmark point can be achievable with 22.2 fb −1 luminosity.

B. Trilepton plus fat-jet in association with missing energy
After the pair production of the RHNs from Z , each of the RHNs will dominantly decay into W mode. One of the W can decay hadronically producing a fat-jet and the other W can decay leptonically. Hence a trilepton signal will be generated with a fat-jet in association with missing energy. In the leptonic decay of the W boson we do not consider tau lepton in this analysis. In this case the selection and advanced cuts will remain the same as the SSDL case. Hence we do not show any histograms. The fat-jet will be created from the boosted W produced from the decay of one of the RHNs. We consider M Z =3 TeV and RHNs are degenerate with two benchmark masses at M N1,2 = 500 GeV and M N1,2 = 1 TeV. The trilepton signal consists of two same sign leptons and the other one must have the opposite sign. Therefore the combination of three lepton system has either ' + 1' or ' − 1' charge taking all possible combinations of the leptons.
We generate SM backgrounds with pre-selection cuts including the transverse momentum of the jets p j T > 10 GeV and pseudo-rapidity of the jets |η j | < 2.5. We consider the separation cuts in the η − φ plane between jets and leptons as ∆R j > 0.4 and lepton-lepton as ∆R > 0.4. At the 27 TeV and 100 TeV we consider the transverse momentum of the jets as p j T > 20 GeV with other pre-selection cuts. We generate the backgrounds W ± h+ jets, W ± Z+jets, ZZ+jets, W ± W ± W ∓ +jets, ZZW ± , W ± W ∓ Z, ttW and ttZ, to study trilepton final state plus one fat-jet in association with missing energy. For these backgrounds we consider leptonic decay modes of the gauge bosons. From these backgrounds we consider µ ± µ ± µ ± (e ± ) and e ± e ± e ± (µ ± ) modes respectively in association with a fat-jet and missing energy which will mimic the trilepton plus missing energy and fat-jet signal from the RHN pair. The SM background cross sections are given in Tab. VIII at proton colliders for different √ s after the application of the basic cuts. In this trilepton + jet + MET final state, the leading two leptons in p T order are expected to come directly from µ ± µ ± + 2J W ± W ± jj W ± Zjj W ± W ∓ Zj ttW ± ttZ sig(500 GeV) sig(1 TeV) sig(500 GeV) sig(1 TeV)  the decay of the RHN (N → W ) and the third lepton is mostly coming from further decays of one of the W boson (we call it a trailing lepton). As the RHN is very heavy, the leading two leptons have larger p T than the third lepton.
Here we use a large p T cut for leading and sub-leading leptons and a low p T cut for the third lepton. Another W boson comes from one of the RHN if decays hadronically form collimated jet or fat-jet as this W boson is coming from heavy RHN decay, so it is boosted. This fat-jet can have a large p T . Using softdrop algorithm, we estimate the jet mass of the fat-jet for the signal and background. The nature of the distributions regarding the leading and subleading leptons and fat-jet are almost the same as those of the SSDL case. Therefore we do not add the histograms for trilepton mode. At the 14 TeV and 27 TeV hadron colliders we select the events with at least one fat-jet with p J T > 180 GeV (C-I). Three leptons are selected as leading, sub-leading and trailing respectively with p 1 T > 200 GeV, p 2 T > 100 GeV and p 3 T > 20 GeV (C-II). Finally the softdrop jet mass is selected within a window of ±15 GeV around the W boson mass (C-III). At the 100 TeV collider along with C-I and C-III we add different lepton p T cuts. At the 100 TeV three leptons are selected with p 1 T > 200 GeV, p 2 T > 200 GeV and p 3 T > 20 GeV respectively (C-II). The cut flow with the efficiencies for µ ± µ ± ∓ and e ± e ± ∓ signals and backgrounds are shown in the upper and lower panels of Tab. IX for √ s = 14 TeV. We study 500 GeV and 1 TeV RHNs as benchmark points for Cases-I and II. Like the SSDL case here also we consider the degenerate RHNs. The leading background is originated from W Z+jets process where as the sub-leading contribution from the background is coming from W W Z process. The other backgrounds are 2-3 orders of magnitude smaller than the leading background. Solving the signal and background relation for a particular significance applied in Sec. V A, we estimate the luminosity to achieve that significance. We find that the significance of the two benchmarks in Case-I are very close and roughly below 1 − σ. In Case-II, 5 − σ significance can be attained at 3 ab −1 luminosity for M N = 500 GeV and for M N = 1 TeV 5 − σ significance could be attained at 1.5 ab −1 . The cut flow and the efficiencies for µ ± µ ± ∓ signal and backgrounds at √ s = 27 TeV are shown in the upper panel and those for the e ± e ± ∓ events are shown in the lower panel of Tab. X. We study 1 TeV RHN as a benchmark for Cases-I and II. The leading background is originated from W Z+jets processes, whereas the sub-leading contribution to the background comes from W W Z process, respectively. The other backgrounds are 2-3 orders of magnitude smaller than the leading background. We find that significance is below 2 − σ up to a high luminosity for Case-I, however, that for Case-II can be 5 − σ at 200 fb −1 luminosity. The cut flow and the efficiencies for µ ± µ ± ∓ and e ± e ± ∓ ∓ events at √ s = 100 TeV are shown in the upper and lower panels of Tab. XI. We notice that a 5 − σ significance can be µ ± µ ± + 2J W ± W ± jj W ± Zjj W ± W ∓ Zj ttW ± ttZ sig(    Applying the constraints on the g − M Z plane we produce the RHN pair from Z using Cases-I and II. We study the boosted objects from the RHNs. Considering the SSDL final state manifesting the Majorana nature of the RHNs we perform a scan over 3 TeV ≤ M Z < 7 TeV considering 500 GeV ≤ M N < M Z 2 . We consider the leading decay mode W from each RHN and the heavy mass will boost each W boson from the RHN to produce fat-jet. Hence we µ ± µ ± µ ∓ /e ∓ ttW ± ttZ W ± hj W ± Zj W ± W ± W ∓ j ZZj ZZW ± W ± W ∓ Z Sig(500 GeV) Sig(1 TeV) Sig(500 GeV) Sig(1 TeV)  study the SSDL+ 2 J signature at hadron colliders with √ s = 14 TeV, 27 TeV and 100 TeV respectively. Following the selection and advanced cuts described in Sec. V A, we produce the signal and study the backgrounds to produce a 2 − σ exclusion limit in the M N − M Z plane. The contours are produced using 3 ab −1 luminosity for 14 TeV and 27 TeV colliders and 30 fb −1 luminosity for 100 TeV collider respectively. These contours for Case-I (II) are shown in the left (right) panel of Fig. 8. Hence we infer SSDL plus two fat-jets provides an interesting handle to study the RHN pair production from Z . In this context, we mention that trilepton plus single fat-jet signature is another interesting aspect that could be tested from the RHN pair production, however, to test this channel we require a high luminosity at 27 TeV and 100 TeV colliders. In that case, we could probe a relatively smaller parameter space compared to the SSDL scenario in the proton colliders.
VI. HEAVY NEUTRINO PAIR PRODUCTION AT THE e + e − COLLIDER Pair production of heavy neutrinos at the electron positron collider is another important aspect of this model which can be studied at √ s = 250 GeV, 500 GeV and 1 TeV for Cases-I and II. We calculate the heavy neutrino pair production at e − e + collider in terms of M Z , N N and x H as  where Q N is the U (1) X charge of the heavy neutrino under the U (1) X gauge group and Γ Z is the total decay width of Z in Cases-I and II respectively. In the limit of M Z > √ s the pair production of the heavy neutrinos from Eq. 34 reduces to We note that the cross section is directly proportional to g M Z . Before estimating the cross sections at different √ s we calculate bounds on g depending on M Z using the dilepton cross sections from ATLAS [84], CMS [85] and ATLAS technical design report [86] and dijet cross sections from ATLAS [87] and CMS [88] using Eq. 30 considering M N > M Z 2 which exerts strongest limit on the U (1) X coupling. The constraints from LEP [89][90][91][92] are obtained using M Z > √ s. Hence we obtain the limits on the quantity M Z g in terms of M Z for different x H which measures the VEV of the U (1) X theory. The Z phenomenology at the e − e + collider been studied in [92] where it has been shown that for x H = −2 there is no interaction between the left handed fermion doublet and Z which is U (1) R scenario. It can also be noted that for x H = −1 the interaction between the right handed electron and Z vanishes. Therefore we consider these two charges as they directly affect the interaction between electron and Z for the heavy neutrino pair production. In addition to that we consider x H = 1 where left handed lepton doublet and right handed where L future = 3000 fb −1 . These prospective bounds estimated by scaling the ATALS (CMS) bounds are shown by the orange dashed (dot-dashed) line in Fig. 9. We calculate the cross section for the heavy neutrino pair production being normalized by the U (1) X gauge coupling for M Z = 7.5 TeV (left panel) and 10 TeV (right panel) depending on x H for different √ s in Fig. 10 fixing M N = 100 GeV. We find that the cross sections attain certain values at x H = −2 and decreases with the increase in x H attaining a minimum at x H = −1.2. Finally the cross section increases with the increase in x H and attains a maximum at x H = 1 and maintains a constant value with x H ≥ 1. We show the Case-I (II) in the upper (lower) panel of Fig. 10 for √ s = 250 GeV, 500 GeV and 1 TeV in each panel from bottom to top respectively. In this calculation we consider only one generation of the heavy neutrino for simplicity. Production of more than one generation of heavy neutrino pair can be possible, however, due to the universal U (1) X gauge coupling of our model, the corresponding cross sections can be multiplied by the number of generations.

A. SSDL+2j signal
In electron positron colliders we can produce the heavy neutrinos in pair and each heavy neutrino decay into the leading mode following N → W . In this analysis we consider SSDL plus four jet signal. The jets are coming from the hadronic decay of one of the W bosons. To study this signal we consider x H = −2 and 1 for √ s = 250 GeV, 500 GeV and 1 TeV respectively for 5 TeV ≤ M Z ≤ 20 TeV. From Fig. 10 we find that heavy neutrino pair production cross In electron positron colliders we can produce the heavy neutrinos in pair and each heavy neutrino decay into the leading mode following N → W . In this analysis we consider a trilepton signal in association with two jets and missing energy. The jets are coming from the hadronic decay of one of the W bosons whereas the third lepton and missing momentum is coming the leptonic decay of the other W boson. In this case we consider two generations of the heavy neutrinos comprising the trilepton signal where the leptons include all possible combinations with electron and muon with trilepton charge combination as +1 and −1. To study this signal we consider x H = −2 and 1 for √ s = 250 GeV, 500 GeV and 1 TeV respectively for 5 TeV ≤ M Z ≤ 20 TeV. We considered 100 GeV ≤ M N ≤ 125 GeV for √ s = 250 GeV, 100 GeV ≤ M N ≤ 250 GeV for √ s = 500 GeV, and 100 GeV ≤ M N ≤ 500 GeV for √ s = 1 TeV respectively to produce the density plots for Cases-I and II in Figs. 13 and 14 respectively. The results in Case-II is roughly more than one order of magnitude higher than the results of Case-I due to different U (1) X charges of the heavy neutrinos. Due to the heavy neutrino pair production process, the cross section becomes negligibly small at the threshold M N ∼ sharply to zero. According to the choice of x H we obtain the maximum cross section at x H = 1. The cross section increases with the increase in √ s.

C. Bounds on MN − M Z plane
We simulate the SSDL plus four jet signal the generic backgrounds using MadGraph [99,100], hadronizing the events by PYTHIA8 [101] followed by the detector simulation using the ILD card in Delphes [102] for different x H and √ s varying M N and M Z according to Figs. 11 and 12 for Cases-I and II respectively to prepare a 2−σ limit plot in the M N − M Z plane. We apply p j T > 20 GeV, p T > 10 GeV, |η j,l | < 2.5 to estimate the SSDL background from W ± W ± and 4Z channels. The SSDL plus four jet process can be generated from W ± W ± process in association with missing energy where two same sign W will decay leptonically into same flavor and the remaining ones will decay hadronically. The e ± e ± + 4j background has extremely small cross section at 250 GeV, however, the cross section becomes 0.0015 fb and 0.0082 fb at 500 GeV and 1 TeV respectively. The µ ± µ ± + 4j background also has extremely small cross section at 250 GeV, however, the cross section becomes 0.0015 fb and 0.0082 fb at 500 GeV and 1 TeV respectively. From 4Z we consider two of the Z bosons decay leptonically and rest of the two decay hadronically. The cross section for this process at 500 GeV is 3 × 10 −6 fb and 1.2 × 10 −5 fb respectively with the electrons. Similar cross sections can be obtained for muons. We also produce the ZZ+jets background where each Z decays into e − e + (µ − µ + ) modes giving rise to SSDL pairs. The cross sections for the e − e + modes at 250 GeV, 500 GeV and 1 TeV are 1.5 × 10 −5 fb, 0.00126 fb and 0.00132 fb respectively. The cross sections for the µ − µ + modes at 250 GeV, 500 GeV and 1 TeV are 1.5 × 10 −5 fb, 0.00127 fb and 0.00133 fb respectively. We estimate the tt backgrounds at √ s = 500 GeV and 1 TeV applying p j T > 20 GeV, p T > 10 GeV, |η j,l | < 2.5. We can not generate this background at √ s = 250 GeV which is energetically disallowed. We consider the final state 4j2b which has the cross section 222.4 fb (68.0 fb) at √ s = 500 GeV (1 TeV). The cross section of 2e2ν2b final state at √ s = 500 GeV (1 TeV) is 6.23 fb (1.88 fb). We find that a final state of 2µ2ν2b has a cross section of 6.23 fb (1.9 fb) at √ s = 500 GeV (1 TeV). Finally we consider another combination of final state eµ2ν2b form tt process having cross section of 12.4 fb (3.76 fb) at √ s = 500 GeV (1 TeV). We ensure that the final signal and SM backgrounds have SSDL pair only and impose the azimuthal angular cut on the leptons as | cos θ | < 0.95 defining θ = tan −1 p T p Z where p T as the transverse momentum and p z is the z− component of the three momentum of the lepton respectively. We impose a missing energy cut for the events such that E miss T < 80 GeV. The backgrounds coming from the top quark pair production do not survive after the application of these cuts. Considering the signals and backgrounds we estimate the 2 − σ significance limit using S √ S+B where S stands for signal events and B stands for backgrounds using luminosities as 2 ab −1 , 4 ab −1 and 8 ab −1 at 250 GeV, 500 GeV and 1 TeV e − e + colliders following Ref. [109]. Corresponding limit plots are shown in Fig. 15  We simulate the trilepton plus two jet signal in association with missing energy and the generic backgrounds using MadGraph [99,100], hadronizing the events by PYTHIA8 [101] followed by the detector simulation using the ILD card in Delphes [102] for different x H and √ s varying M N and M Z according to Figs. 13 and 14 for Cases-I and II respectively to prepare a 2−σ limit plot in the M N − M Z plane. We apply p j T > 20 GeV, p T > 10 GeV, |η j,l | < 2.5 to estimate the three electron plus two jets in association with missing energy generic SM background and the cross sections are 0.111 fb at √ s = 250 GeV, 1.05 fb at √ s = 500 GeV and 3.53 fb at √ s = 1 TeV respectively. Using the same cuts we simulate the three muon plus two jet background in association with missing energy and obtain the corresponding generic SM backgrounds cross sections as 0.1 fb at √ s = 250 GeV, 0.2 fb at √ s = 500 GeV and 0.2 fb at √ s = 1 TeV respectively. Using the mentioned cuts we produce the two electron and one muon generic SM background in association with two jets and missing momentum. The cross sections at √ s = 250 GeV, 500 GeV and 1 TeV are obtained as 0.061 fb, 0.6 fb and 1.63 fb respectively. Similarly we generate the two muon and one electron generic SM background events in association with two jets and missing energy at √ s = 250 GeV, 500 GeV and 1 TeV. We checked that ttZ channel gives extremely low cross sections O(10 −4 fb) after the application of the kinematic cuts. Hence we do not consider this process in further analysis. After applying the cuts we obtain the cross sections as 0.05 fb, 0.18 fb and 0.352 fb respectively. To estimate the SM backgrounds we considered charge combinations of the three charged leptons as +1 and −1. Using the azimuthal angular cut on the leptons as | cos θ | < 0.95. To estimate the signal and generic SM background events we consider the luminosities as 2 ab −1 , 4 ab −1 and 8 ab −1 at 250 GeV, 500 GeV and 1 TeV e − e + colliders following [109]. We estimate the 2−σ contours on the M N − M Z plane in the upper (lower) panel of Fig. 16  There is another interesting aspect of the Majorana heavy neutrino pair production from scalar involved in a general U (1) X scenario. In Case-I for simplicity, applying the stationary conditions on the scalar potential in Eq. 2 we find that the mass matrix of the scalars as Hence diagonalizing the mass matrix in Eq. 36 we obtain the mass eigenvalues of the physical scalars as (37) and the scalar quartic couplings in Eq. 2 of Case-I can be written as λ = sin 2α where α is the mixing angle between the two scalars required to diagonalize the mass matrix given in Eq. 36. The BSM scalar h 2 can decay into SM particles or gauge bosons through the scalar mixing and the partial decay width of h 2 can be written as where Γ h→X SM X SM is the partial decay widths of the SM Higgs. The BSM scalar can decay into SM Higgs (m h1 = 125 GeV) through the interaction and s α (c α ) = sin α(cos α). The decay width for h 2 → h 1 h 1 is given by The BSM scalar interacts with the pair of heavy neutrinos through the Yukawa interaction in Eq. 1. The partial decay process of h 2,1 → N N can be written as where Hence we find a relation between the Z and heavy neutrino mass as g = . The complete form of the partial decay widths of h 1 and h 2 are given in the Appendix. We show the branching ratios of h 2 into different modes including a pair of RHNs in Fig. 17 as a function of m h2 considering M Z = 5 TeV, g = 0.1 and M Ni = 50 GeV. The allowed benchmark value of g is taken from [92] satisfying LHC dilepton, dijet and LEP-II constraints. We consider three benchmarks of sin α = 0.1, 0.05 and 0.025 respectively which is taken from [110] which are allowed benchmark points after the application of LEP and LHC limits. In this case we consider x H = 0 which is the B−L case. We consider two different modes of the h 2 production one at the electron positron collider. One is e − e + → Zh 2 and the other is e + e − → h 2 νν. In Fig. 18 we show the cross sections of Zh 2 (upper panel) and h 2 νν (lower panel) respectively considering sin α = 0.1 and m h2 = 150 GeV and 300 GeV respectively as a function of √ s. In this study we consider three choices of the polarizations P e − = P e + = 0, P e − = −0.3, P e + = 0.8 and P e − = 0.3, P e + = −0.8 respectively. producing h 2 in the above modes we consider the heavy neutrino pair production followed by the decay h 2 → N i N i . Note that h 2 can be produced via e + e − → e + e − h 2 via Z boson fusion but its production cross section is smaller compared to the other modes. As a result we did not consider it in this analysis. In our following analysis masses of extra scalar and heavy neutrinos are fixed to be m h2 = 150 GeV, M N1 = M N2 = 50 GeV for illustration. We also choose M Z = 6.5 TeV and g = 1 to enhance h 2 → N i N i decay mode. We consider two generations of the heavy neutrinos where N 1 dominantly decays into electron and N 2 dominantly decays into muon. Here we consider the dominant decay modes of the heavy neutrinos where N i → jj where two jets come from the hadronic decay mode of W ± * .
Here we consider associated production process e + e − → Zh 2 for √ s = 250 GeV since the process provides the largest h 2 production cross section at the center of mass energy. In this case our signal process is e + e − → h 2 Z → N N jj → ± ± + 6j where two of these jets are coming from the hadronic decay of the Z boson. We assume that the Yukawa coupling involved in this process is diagonal. The signal indicates lepton number violation and number of SM background (BG) events can be reduced by same sign charged lepton tagging. For the BG from SM processes at the LHC 250 GeV, we consider following processes + − W + W − (BG1), + − ZZ (BG2), ± νW ± Z (BG3, BG4) and the cross sections of these processes are 0.33 fb, 0.094 fb and 2.5 fb respectively at √ s = 250 GeV. These backgrounds can mimic our signals providing same sign charged lepton events. We consider M N = 50 GeV so that heavy neutrinos can decay off-shell.
We generate signal and SM background events using MadGraph [100]. Then PYTHIA8 [111] is applied to deal with hadronization effects, the initial-state radiation (ISR) and final-state radiation (FSR) effects and the decays of SM particles, and Delphes [102] is used for detector level simulation using {P e − , P e + } = {0, 0}. At the detector level, event selection is imposed with kinematical cuts demanding an SSDL pair for electrons and muons as: (i) transverse momenta of leptons: p ± T > 10 GeV and those of the jets are: p j T > 20 GeV, (ii) pseudo-rapidity: η ,j < 2.5 and (iii) missing energy: / E T < 20 GeV. We evaluate number of signals and SM baground events imposing selection cuts where N event = L int σ N Selected N Generated , N Select is number of events after selection, N Generated is number of generated events, σ is a cross section for each process and L int =2 ab −1 is integrated luminosity. We combine the electron and muon events claiming at least 3 jets n j ≥ 3 in the events. After the application of the selection cuts we evaluate the significance if the process using In Table XII, we summarize number of signal and SM background events after selection choosing sin α = 0.1 and 0.2 as benchmark values. Hence we notice that a sizable significance can be observed at the 250 GeV electron positron collider.
B. Signal from e + e − → h2νν Here we consider W boson fusion for h 2 production, e + e − → h 2ν ν, with √ s = 1 TeV since the cross section for h 2 production increases with √ s. We use polarization of the e + and e − beam as {P e − , P e + } = {−0.8, 0.3}, since this gives the largest cross section as shown in the lower panel of Fig. 18. Our signal process is e + e − → h 2 νν → N N νν → ± ± + 4j + νν. We simulate the signal and the SM backgrounds events using MadGraph [100] followed by hadronization using PYTHIA8 [111] and detector simulation using Delphes [102]. The SM backgrounds + − W + W − (BG1), + − ZZ (BG2), ± νW ± Z (BG3, BG4) have the cross sections 44.0 fb, 0.48 fb and 39.0 fb at √ s = 1 TeV respectively. We use the selection cuts for the signal and SM background events as: p T ( ± ) > 10 GeV, η ,j < 2.5, p j T > 20 GeV and / E T > 30 GeV respectively. As in the previous case we summarize number of events after the selection cuts and significance in Table XIII where we write the events combining the signals with electron and muon. We find that the significance can be sizable at 1 TeV electron positron collider using 2 ab −1 luminosity.

VIII. CONCLUSIONS
We have considered two general U(1) extensions of the SM where neutrino mass can be generated by the seesaw mechanism. To cancel the gauge and mixed gauge-gravity anomalies, these models include three generations of the Majorana type heavy neutrinos which could be produced at the high energy colliders. Studying the existing constraints on the U(1) X gauge coupling as a function of the M Z for different M N , we produce the RHNs in pair from Z in hadron colliders at different center of mass energies. We study the SSDL and trilepton modes manifesting the Majorana nature of the RHNs considering the leading decay mode of the RHNs. Considering sufficiently heavy RHNs we find that the W boson from the RHN decay can be sufficiently boosted. Selecting the signal and backgrounds and passing through advanced cuts we find that these signals can be obtained with reasonable significance for different benchmark scenarios of M N and M Z at hadron colliders. Hence scaning over a range of M Z and M N we estimate 2 − σ exclusion contours at hadron colliders with different luminosities for the SSDL signal. Finally we conclude that the SSDL signal with two fat-jets can be probed at hadron colliders with different luminosities in the near future. Majorana heavy neutrinos can be produced in pair at the electron positron colliders where heavy Z can be probed from the SSDL and trilepton signature. To do this we estimate the limits on the scale of the vacuum expectation value of the U(1) breaking from LHC and LEP. Using those limits we simulate the SSDL and trilepton events. Applying the kinematic cuts on the signal and SM backgrounds on the combined electron and muon events with jets, we estimate a 2−σ contours on the M N − M Z plane at 250 GeV, 500 GeV and 1 TeV respectively depending on the choice of the U(1) charges of the particles. In addition to the Z induced Majorana type heavy neutrino pair production we consider the production of a BSM scalar under the general U(1) X scenario considering the current bounds on the scalar mixing angle form LHC and LEP at different center of mass energies and polarizations. This BSM scalar can decay into a pair of Majorana type heavy neutrinos through the Yukawa interaction which can further decay into SSDL modes in association with jets and missing energy. Studying the signals and corresponding SM backgrounds we find that such process can also be probed at the electron positron colliders with sizable significance in the near future. Furthermore studying the neutrino mass generation mechanism in the context of an inverse seesaw mechanism we can probe the pseudo-Dirac heavy neutrinos. The lepton number violating and conserving modes are different in the Majorana and pseudo-Dirac cases in colliders which will be studied in detail in our upcoming work (in progress) introducing heavy neutrino pair production mechanism from Z to distinguish between Majorana and Dirac nature. Currently it is beyond the scope of this article.

APPENDIX
The partial decay widths of the SM like Higgs boson of mass m h1 into various modes are given below from [112]: (i) SM fermions (f): N c = 1 and 3 for SM leptons and quarks respectively.
(ii) on-shell gauge bosons (V = W ± or Z ): where C V = 1 and 2 for V = Z or W ± gauge boson, respectively.
(iii) gluon (g) via top-quark loop: where (iv) one off-shell gauge boson : where sin 2 θ W = 0.231 and the loop functions can be represented as where 1/4 < x < 1, for energetically allowed decays. For the U (1) X scalar h 2 , cos α will be replaced by sin α.

ACKNOWLEDGMENTS
The work of S.M. is supported by KIAS Individual Grants (PG086001) at Korea Institute for Advanced Study. S.S acknowledge the support of the SAMKHYA: High Performance Computing Facility provided by IOPB.