Searching for the minimal Seesaw models at the LHC and beyond

The existence of the tiny neutrino mass and the flavor mixing can be naturally explained by type-I Seesaw model which is probably the simplest extension of the Standard Model (SM) using Majorana type SM gauge singlet heavy Right Handed Neutrinos (RHNs). If the RHNs are around the Electroweak(EW)-scale having sizable mixings with the SM light neutrinos, they can be produced at the high energy colliders such as Large Hadron Collider (LHC) and future $100$ TeV proton-proton (pp) collider through the characteristic signatures with same sign di-lepton introducing lepton number violations(LNV). On the other hand Seesaw models, namely inverse Seesaw, with small LNV parameter can accommodate EW-scale pseudo-Dirac neutrinos with sizable mixings with SM light neutrinos while satisfying the neutrino oscillation data. Due to the smallness of the LNV parameter of such models, the `smoking-gun' signature of same-sign di-lepton is suppressed where as the RHNs in the model will manifest at the LHC and future $100$ TeV pp collider dominantly through the Lepton number conserving (LNC) trilepton final state with Missing Transverse Energy (MET). Studying various production channels of such RHNs we give an updated upper bound on the mixing parameters of the light-heavy neutrinos at the 13 TeV LHC and future 100 TeV pp collider.

The experimental evidence of the neutrino oscillation and flavor mixings from neutrino oscillation experiments [1][2][3][4][5][6] indicates that the SM is not enough to explain the existence of the tiny neutrino mass and flavor mixing. After the pioneering discovery of the d = 5 operator [7] within the SM using the SM leptons, SM Higgs doublets and ∆L = 2(L = Lepton number) unit of LNV, it turned out that the Seesaw mechanism [8][9][10][11][12][13][14] could be the simplest idea to explain the small neutrino mass and flavor mixing where the SM can be extended by the SM-gauge singlet Majorana type RHNs. After the Electroweak (EW) symmetry breaking, the light Majorana neutrino masses are generated after the so-called type-I Seesaw mechanism.
In type-I Seesaw, we introduce SM gauge-singlet Majorana RHNs N β R where β is the flavor index. N β R couples with the SM lepton doublets α L and the SM Higgs doublet H. The relevant part of the Lagrangian is where Y αβ D is the Yukawa coupling, α L is the SM SU (2) L lepton doublet and m N is the Majorana mass term. After the spontaneous EW symmetry breaking by the vacuum expectation where M D is the Dirac mass matrix and m N is the Majorana mass term. Diagonalizing M ν we obtain the Seesaw formula for the light Majorana neutrinos as For m N ∼ 100 GeV, we may find Y D ∼ 10 −6 with m ν ∼ 0.1 eV. However, in the general parameterization for the Seesaw formula [15], Y D can be large and sizable (O(1)), and this is the case we consider in this paper.
The searches of the Majorana RHNs can be performed by the 'smoking-gun' of the samesign di-lepton plus di-jet signal which is suppressed by the square of the light-heavy neutrino mixing parameter |V N | 2 |m D m −1 N | 2 . A comprehensive general study on the parameters of |V N | 2 has been given in [17] using the data from the neutrino oscillation experiments [1][2][3][4][5][6], bounds from the Lepton Flavor Violation (LFV) experiments [18][19][20], Large Electron-Positron (LEP) and Electroweak Precision test [90][91][92][93][94][95][96][97] experiments using the non-unitarity effects [23,24] applying the Casas-Ibarra conjecture [15,16,[25][26][27]. At this point we mention that [17] has a good agreement with a previous analysis [28] on the sterile neutrinos. The bounds in [17] has been compared with the existing results in [29-31, 93, 96, 97] considering the degenerate Majorana RHNs. In case of Seesaw mechanism the Dirac Yukawa matrix (Y D ) can carry the flavors where the RHN mass matrix is considered to be diagonal. This case is favored by the neutrino oscillation data as studied in [16,17,[35][36][37][38][39]. Such a scenario for the Seesaw scenario is called the Flavor Non-Diagonal (FND). The other possibility of considering both of the diagonal Y D and m N is not supported by the neutrino oscillation data.
Since any number of singlets can be added in a gauge theory without contributing to anomalies, one can utilize such freedom to find a natural alternative of the low-scale realization of the Seesaw mechanism. Simplest among such scenarios is commonly known as the inverse Seesaw mechanism [40,41] where a small Majorana neutrino mass originates from tiny LNV parameters rather than being suppressed by the RHN mass as done in the case of conventional Seesaw mechanism. In the inverse Seesaw model two sets of SM-singlet RHNs are introduced which are pseudo-Dirac by nature and their Dirac Yukawa couplings can be even order one, while reproducing the neutrino oscillation data. Therefore at the high energy colliders such pseudo-Dirac neutrinos can be produced through a sizable mixing with the SM light neutrinos [32,34,[42][43][44][45][46]49].
In the inverse Seesaw mechanism the relevant part of the Lagrangian is given by where N α R and S β L are two SM-singlet heavy neutrinos with the same lepton numbers, m N is the Dirac mass matrix, and µ is a small Majorana mass matrix violating the lepton numbers.
After the EW symmetry breaking we obtain the neutrino mass matrix as Diagonalizing this mass matrix we obtain the light neutrino mass matrix Note that the smallness of the light neutrino mass originates from the small LNV term µ.
The smallness of µ allows the m D m −1 N parameter could be on the order one even for an EW scale RHNs. In the inverse Seesaw case we can consider Y D as non-diagonal when µ and m N are diagonal which is called the Flavor Non-Diagonal (FND) case. On the other hand we can also consider the diagonal Y D , m N when µ will be non-diagonal. This situation is called the Flavor Democratic scenario. For the inverse Seesaw both of the FND and FD cases are supported by the neutrino oscillation data. In this article we will consider the FND case from the Seesaw and the FD case from the inverse Seesaw mechanisms.
Through the Seesaw mechanism, a flavor eigenstate (ν) of the SM neutrino is expressed in terms of the mass eigenstates of the light (ν m ) and heavy (N m ) Majorana neutrinos such Using the mass eigenstates, the charged current (CC) interaction for the heavy neutrino is given by where e denotes the three generations of the charged leptons in the vector form, and P L = 1 2 (1 − γ 5 ) is the projection operator. Similarly, the neutral current (NC) interaction is given by where c w = cos θ w with θ w being the weak mixing angle and W µ , Z µ are the SM gauge bosons.
The main decay modes of the heavy neutrino are N → W , ν Z, ν h. The corresponding partial decay widths [42] are respectively given by The decay width of heavy neutrino into charged gauge bosons being twice as large as neutral one owing to the two degrees of freedom (W ± ). We plot the branching ratios BR i (= Γ i /Γ total ) of the respective decay modes (Γ i ) with respect to the total decay width (Γ total ) of the heavy neutrino into W , Z and Higgs bosons in Fig. 1 as a function of the heavy neutrino mass (m N ). Note that for larger values of m N such as m N ≥ 1500 GeV, the branching ratios can be obtained as In this paper we study the RHN production from various initial states such as the quarkquark (qq), quark-gluon(qg) and gluon-gluon(gg) at the 13 TeV LHC and future 100 TeV pp collider. We consider the photon mediated processes as well as the gluon-gluon fusion This paper is is organized as follows. In Sec. II we calculate the production cross sections at the 13 TeV LHC and future 100 TeV pp collider [47] with a variety of initial states. In Sec. III we study the multilepton decay modes of the RHNs at the 13 TeV LHC and future 100 TeV pp collider. For both of the cases we take the luminosity as 3000 fb −1 . In Sec. IV we put the bounds on the mixing angle as function of the RHN mass. Sec. V is dedicated for conclusions.

II. PRODUCTION CROSS SECTIONS
The RHNs can be produced at the 13 TeV LHC and future 100 TeV pp collider from various initial states. We consider the combined production of the heavy neutrinos from various initial states including quark-(anti)quark (qq ), quark-gluon (qg) and gluon-gluon (gg) interactions. We also study contributions coming from the gluon-gluon fusion (ggF), photon-proton interactions and Vector Boson Fusion (VBF) processes to produce the RHNs at the LHC and beyond. In this section we consider the RHN productions in association with SM leptons and jets. To obtain the cross sections and generate the events we implement our model in MadGraph [62] bundled with PYTHIA [67] and DELPHES [76]. The production modes of the heavy neutrinos in association with leptons and jets are proportional to the square of the mixing angle, |V N | 2 . In this section we use |V N | 2 = 10 −4 to estimate the cross sections 1 .
(1) Charged current interaction mediated by W : along with the charge conjugates where j stands for the jets.
In this process the the s-channel quark anti-quark pair of different flavors (qq ) will interact through the W exchange and finally follow Eq. 8 to produce the RHN (N ) in 1 The EWPD bounds |V N | 2 have been discussed in [90,96,97]. The universal bound has been considered to be 9 × 10 −4 which rules out the possibility of the mixing angles above this value. Therefore we used a value of |V N | 2 < 9 × 10 −4 to calculate the cross sections. association with a lepton ( ). The corresponding Feynman diagram for qq → N is given in Fig. 2.
The contributions from the quark-quark interaction (qq ) to one-jet (N j) are given in Fig. 3 for the t -channel process and additional contributions to the one-jet process from the quark-gluon (qg) interaction as shown in Fig. 4 form the s, t-channel processes.
In both of the cases the W is radiated form the quarks/ anti-quarks and hence follow the CC to to produce N in association with .
two-jet contributions (N jj) coming from the qq process are shown in Fig. 5. In this case, the s, t channel contributions between the quarks are involved to produce N with in association with two jets following the CC interactions at the N production vertex. The quark-gluon (qg) processes contributing in the two-jet case are shown in Fig. 6 where the s, t channel contributions between the quarks and gluons are involved to produce N with in association with two jets following the CC interactions at the N production vertex. In addition to that gluon-gluon (gg) processes contributed in N jj are shown in Fig. 7 in the s, t-channel followed by the N production from the CC interactions with in association with two jets.
We combine all these processes to compute the production cross section of the RHNs through the charged current interaction for N X final state where X stands for n-jet and n = 0, 1, 2, 3. To produce such processes we use the following trigger cuts (a) transverse momentum of the jets, p j T > 30 GeV, (b) transverse momentum of the leptons, p T > 10 GeV, (c) pseudo-rapidity of the jets, |η j | < 2.5,    After the rewight, each final state parton can find the node where it was generated.
This was n-parton shower can be generated (µ 2 g ) in the directions of the primary partons so that the initial condition for each parton shower is kept at µ 2 g . Hence running the k T clustering algorithm one can find all the jets at the resolution scale (τ r ). If all the jets match with the primary partons then the event is accepted otherwise rejected.
In MadGraph the switch ickkw = 1 stands for the MLM scheme in MadGraph [62] using are used, while if xqcut > 0, k T jet matching is assumed. In this case, transverse momentum of jet (p j T ) and separation between the jets (∆R jj ) should be set to zero. For most processes, the generation speed can be improved by setting p j T = xqcut which is done automatically if the switch auto − ptj − mjj is set to 'T' at the time of event generation using MadGraph to control the transverse momentum of the jets (p j T ) and the invariant mass of the jets (m jj ). If some jets should not be restricted this way (as in single top or Vector Boson Fusion (VBF) production, where some jets are not radiated from QCD) in that case the switch auto − pTj − mjj should be set to 'F' in the MagGraph at the time of event generation. QCUT is used for the matching with the k T scheme, this is case the jet measure cutoff is used by Pythia. If the value is not given, it will be set to max(xqcut + 5, xqcut * 1.2), where xqcut is taken from the MadGraph [50,51]. We use such cuts to keep the analysis collinear safe. The (2) Neutral current interaction mediated by Z: > add process pp → N νjj followed by the same production procedure as we discussed in (1). Using quark antiquark pair of same flavor (qq) at the s-channel by Eq. 9 N can be produced in association with a light neutrino (ν) as shown in Fig. 2. The N νj (in Figs. 3,4) and N νjj (in Figs. 5, 6, 7) processes are same as those have been discussed in (1). The only difference is the production vertex of N ν comes from the NC from Eq. 9. Combining these final states we can denote as N νX where X stands for n-jet with n = 0, 1, 2, 3.
In this case we use the following trigger cuts (a) transverse momentum of the jets, p j T > 30 GeV, (b) pseudo-rapidity of the jets, |η j | < 2.5 and the same matching scheme used in (1). Such p j T cuts will keep the analysis collinear safe specially in (1) and (2) to calculate the cross sections using MLM matching [48,[50][51][52][53][58][59][60][61] schemes. The production cross sections at the 13 TeV LHC and future 100 TeV pp collider are shown in Figs. 11 and 12 respectively as a function of m N .
(3) Gluon-fusion channel (ggF): In the ggF top loop produces the SM Higgs and Higgs can decay into N in association with ν [32,34,68]. The corresponding Feynman diagram is given in the left panel of Fig. 8. There is another complementary production channel of N in association with ν which comes from the Z [32,33]. The corresponding Feynman diagram is given in the right panel of Fig. 8 The production process depends upon m N , one can produce them promptly from the Higgs decay when m N < m h where m h is the Higgs mass and also from the off-shell Z (Z * ). When m N < m h , Higgs can decay into RHNs and the partial decay width can be written as This has additive contribution to the total decay width of the SM Higgs boson keeping Y D as a free parameter. Constraints on Y D and |V N | 2 have recently been studied in [29,32,34]. The cross section goes down at the 13 TeV LHC for m h < m N due to the off-shell decay. However, at the future 100 TeV pp collider the cross section again rises around m N = 250 GeV due to large contributions coming from the gluon PDFs.
In this channel we are testing m N = 100GeV − 1 TeV. The production cross sections are shown in Figs. 11 and 12 respectively with respect to m N . At the future 100 TeV, the large gluon PDF starts contributing to the RHN production resulting a rise in the ggF curve compared to the 13 TeV result.
(4) Photon initiated processes: The photon (denoted by a in MadGraph) initiated processes also have important contributions in the RHNs production which have been studied in [44,[69][70][71][72][73][74]. In case of the the photon initiated process, the photon can be radiated from the proton and also from a parton. Those production channels will give additive contributions at the colliders once the RHNs decay into multi-lepton modes. The corresponding Feynman diagrams for these processes are given in Figs. 9. The N j production process is possible as γ → W W vertex is present which will make the production cross section for N j higher in comparison to that of N νj where γ → ZZ vertex is not present under the SM gauge group. N νj will be produced from the Z → N ν vertex under the NC interaction according to Eq. 9. For the photon initiated production processes we use     GeV in this section for different production processes. We use the data files after the hadronization using PYTHIA [67] and detector simulation using DELPHES [76] bundled with MadGrpah [62] and use the MLM matched results as prescribed in the previous section. In this analysis the leading particle is the particle having longest transverse momentum (p T ) distribution where as the trailing particle is that which has shorter transverse momentum   We study the VBF processes to produce the heavy neutrinos and hence study the trilepton plus MET final state. We have used the VBF prescriptions written in the previous section where the jets j 1 and j 2 are widely separated. The other two jets are initial state radiations (ISR) which also are not populating the central region. The rapidity distributions of the jets are shown in Fig. 19. This is a striking feature of the VBF process. The leading lepton Another important contribution is coming from the N νjj + X process from the NC interaction between the RHNs and Z bosons. Here X are the radiated jets. The final state consists of a single lepton, two jets and MET. The produced jets can be used to reconstruct the W boson. The distribution shows a peak around the W mass. Similarly from the RHN decay we get N → jj and the invariant mass distribution of the jj system can show a peak around the RHN mass (m N ). The invariant mass distribution of m jj is shown in Fig. 21 and that of m jj are shown in Fig. 22.
We produced the RHNs from the ggF process where the top loop becomes dominant to produce the Higgs which can promptly decay into the RHN when m N < m H . In our case we consider the m N = 100 GeV and the Higgs will decay promptly into N ν and the RHN will decay into W followed by its hadronic decay. The invariant mass distribution of the two jet system (m jj ) is shown in Fig. 23 where as that of the lepton and two jet system are shown in Fig. 24. The m jj distribution peaks around the W mass where as the m jj distribution peaks around the RHN mass. For detailed results regarding the Higgs decaying into RHNs can be found in [29,32,34] with pervious and updated limits.

IV. BOUNDS ON THE MIXING ANGLE
For m N < M Z , the RHN can be produced from the on-shell Z-decay through the NC interaction in association with missing energy, however, if m N > M Z off-shell production will take place with the same final state. The heavy neutrino can decay according to the CC and NC interactions. Different production processes, different decay modes of the heavy neutrinos and various phenomenologies have been discussed in [46,49,73,[77][78][79][80][81][82][83][84][85][86][87]. RHN production from various initial states and as well as scale dependent production cross sections at the Leading Order (LO) and Next-to-Leading-Oder QCD (NLO QCD) of N ν have been studied at the 14 TeV LHC and future 100 TeV pp collider [46,49]. The L3 collaboration [91] has performed a search on such heavy neutrinos directly from the LEP data and found a limit on B(Z → νN ) < 3 × 10 −5 at the 95% CL for the mass range up to 93 GeV. The exclusion limits from L3 are given in Fig. 25 where the red dot-dashed line stands for the limits obtained from electron (L3-e) and the red dashed line stands for the exclusion limits coming from µ (L3-µ). The corresponding exclusion limits on |V ( =e)N | at the 95% CL [92,93] have been drawn from the LEP2 data which have been denoted by the dark magenta line.
In this analysis they searched for 80 GeV ≤ M N ≤ 205 GeV with a center of mass energy between 130 GeV to 208 GeV [93].
The DELPHI collaboration [94] had also performed the same search from the LEP-I data high luminosity e + e − collision (FCC-ee) with a center-of-mass energy around 90 GeV to 350 GeV [95]. According to this report, a sensitivity down to |V N | 2 ∼ 10 −11 could be achieved from a range of the heavy neutrino mass, 10 GeV ≤ m N ≤ 80 GeV. The darker cyan-solid line in Fig. 25 shows the prospective search reaches by the FCC-ee. A sensitivity down to a mixing of |V N | 2 ∼ 10 −12 can be obtained in FCC-ee [95], covering a large phase space for m N from 10 GeV to 80 GeV.
The heavy neutrinos can participate in many electroweak (EW) precision tests due to the active-sterile couplings. For comparison, we also show the 95% CL indirect upper limit on the mixing angle, |V N | < 0.030, 0.041 and 0.065 for = µ, e, τ respectively derived from a global fit to the electroweak precision data (EWPD), which is independent of m N for m N > M Z , as shown by the horizontal pink dash, solid and dolled lines respectively in and for muon it has probed down to 10 −5 for |V eN | 2 and |V µN | 2 respectively. The ATLAS bounds from the 8 TeV LHC are weaker than the CMS bounds. The ATLAS [30] bounds are represented by the brown dashed lines for µ (|V µN | 2 ) and brown dotted line represents the bounds from e (|V eN | 2 ) in Fig. 25.
The relevant existing upper limits at the 95% CL are also shown to compare with the experimental bounds using the LHC Higgs boson data in [29,43] using [98][99][100][101][102]. The darker green dot-dashed line named Higgs boson shows the relevant bounds on the mixing angle.
In this analysis we will compare our results taking this line as one of the references. We  A. Same-sign di-lepton plus di-jet signal For simplicity we consider the case that only one generation of the heavy neutrino is light and accessible to the LHC which couples to only the second generation of the lepton flavor.
To generate the events in the MadGraph we use the CTEQ6L1 PDF [75] with xqcut= p j T = 20 GeV and QCUT= 25 GeV. We calculate the cross sections for the combined processes N X, N → jj as functions of m N from various initial states as described in Fig. 11 and 12. Comparing our generated events with the recent ATLAS results [30] at the 8 TeV LHC with the luminosity 20.3 fb −1 , we obtain an upper limit on the mixing angles between the Majorana type heavy neutrino and the SM leptons as a function of m N . In the ATLAS analysis the upper bound of the production cross section (σ ATLAS/CMS ) is obtained for the final state with the same-sign di-muon plus di-jet as a function of m N . For σ ATLAS results we use the cross sections given in [30] and for σ CMS we use [88]. Implementing our model in MadGraph we generate the signal event and compare the experimental cross sections from fb −1 luminosity. Recently the CMS has performed the same-sign di-lepton plus di-jet search [88]. Using this result and adopting the same procedure for the ATLAS result we calculate the upper bound on the mixing angles at the 13 TeV LHC with 3000 fb −1 luminosity using Eq. 13. The results are shown in Fig. 25.
A significant improvement for the upper bound on the mixing angle is obtained by combining all the processes with jets and various initial states as described in this paper.
We can see that at 91.2 GeV≤ m N ≤ 200 GeV, the bounds obtained using the ATLAS (ATLAS13 − µ@3000fb −1 ) analysis is better than the recent exclusions limits shaded in  where only one heavy neutrino is light and accessible to the LHC which couples to only the first or second generation of the lepton flavor using the CTEQ6L PDF [75]. In this analysis we consider N X final state followed by N → ν . We generate the combined parton level events using MadGraph and then gradually hadronize and perform detector simulations with xqcut= p j T = 30 GeV and QCUT= 36 GeV for the hadronization. In our analysis we use the matched cross section after the detector level analysis. After the signal events are generated we adopt the following basic criteria,  (iii) the jet transverse momentum: p j T > 30 GeV; (iv) the pseudo-rapidity of leptons: |η | < 2.4 and of jets: |η j | < 2.5; (v) the lepton-lepton separation: ∆R > 0.1 and the lepton-jet separation: ∆R j > 0.3; (vi) the invariant mass of each OSSF (opposite-sign same flavor) lepton pair: m + − < 75 GeV or > 105 GeV to avoid the on-Z region which was excluded from the CMS search.
Events with m + − < 12 GeV are rejected to eliminate background from low-mass Drell-Yan processes and hadronic decays; (vii) the scalar sum of the jet transverse momenta: H T < 200 GeV; (viii) the missing transverse energy: / E T < 50 GeV.
The additional trilepton contributions come from N → Zν, hν, followed by the Z, h decays into a pair of OSSF leptons. However, the Z contribution is rejected after the implementation of the invariant mass cut for the OSSF leptons to suppress the SM background and the h    EWPD and Higgs data. Prospective bounds from the Higgs data had been calculated in [32] for the LHC at the high luminosity (3000 fb −1 ) and future 100 TeV pp collider.

V. CONCLUSIONS
In this paper we have studied both of the type-I and inverse Seesaw models where SM singlet RHNs are involved. The RHNs involved in the type-I Seesaw mechanism are Majorana type where as those are present in the inverse Seesaw model are pseudo-Dirac in nature.
We have studied the production mechanisms of the RHNs from various initial states at the 13 TeV LHC and future 100 TeV pp collider considering the square of the mixing angle as Higgs data. Prospective bounds from the Higgs data had been calculated in [32] for the LHC at the high luminosity (3000 fb −1 ) and future 100 TeV pp collider. trilepton plus MET respectively.
Using the recent searches by the CMS and ATLAS for the type-I Seesaw with same-sign di-muon and di-jet final state we put a prospective upper bound on the |V µN | 2 for the 13 TeV LHC and future 100 TeV pp collider at 3000 fb −1 luminosity. Applying the cuts used in the anomalous multi-lepton search done by the LHC, we can put upper bounds on the square of the mixing angle between the SM light and of the degenerate RHNs. We consider the SF and FD cases where the limits in FD case is twice stronger than the SF case.
We have noticed that our currently given projected limits on the square of the mixing angles are better than those obtained from different experiments and even at the current stage of the LHC. We expect improvements in the status at the HL-LHC and future 100 TeV pp collider when multi-lepton final states will be studied from the decays of the RHNs which will lead us to a more optimistic conclusion.

VI. CONFLICTS OF INTEREST
The author declares that there is no conflict of interest regarding the publication of this paper.