Probing right handed neutrinos at the LHeC and lepton colliders using fat jet signatures

The inclusion of heavy neutral leptons (right-handed neutrinos) to the Standard Model (SM) particle content is one of the best motivated ways to account for the observed neutrino masses and flavor mixing. The modification of the charged and neutral currents from active-sterile mixing of the neutral leptons can provide novel signatures which can be tested at the future collider experiments. In this article, we explore the discovery prospect of a very heavy right handed neutrino to probe such extensions at the future collider experiments like Large Hadron electron Collider (LHeC) and liner collider. We consider the production of the heavy neutrino via the $t$ and $s$-channel processes and its subsequent decays into the semi-leptonic final states. We specifically focus on the scenario where the gauge boson produced from heavy neutrino decay is highly boosted, leading to a fat-jet. We study the bounds on the sterile neutrino properties from several past experiments and compare with our results.


I. INTRODUCTION
One of the most robust evidence that points out to an important inadequacy of the SM is the existence of the tiny but non-zero neutrino masses. It seems unlikely that the very small neutrino masses are generated by the same Higgs mechanism responsible for the masses of the other SM fermions due to the absence of right-handed neutrinos. Even then, extremely small Yukawa couplings, of the order of 10 −12 , must be invoked. There are various BSM extensions which have been proposed to explain small neutrino masses. Among those, one of the most appealing framework of light neutrino mass generation is the addition of new states that, once integrated out, generate the lepton number violating dimension five Weinberg operator . This is embodied by the so-called seesaw mechanisms. There can be a few different variations of seesaw, Type-I [2], Type-II [3], Type-III [4], inverse [5] and radiative [6] seesaw.
Most of the UV completed seesaw models contain Standard Model (SM) gauge singlet heavy right handed neutrino N . Through the seesaw mechanism, the Majorana type Right Handed Neutrinos (RHNs) impart masses to the SM light neutrinos and hence establishes the fact that SM neutrinos have masses which have been experimentally observed in a several neutrino oscillation experiments [7]. These RHNs can have masses from eV scale to 10 14 GeV scale depending upon the models. For instance, the sterile neutrinos [8] with masses in the eV range could lead to effects in short distance neutrino oscillation experiments by introducing an additional mass squared difference, keV mass sterile neutrinos are potential candidates for "warm" dark matter, MeV scale sterile neutrinos can be possible explanation for MiniBoone [9] and there can be very heavy sterile neutrinos with masses M GU T ∼ 10 14 GeV, close to M GU T ∼ 10 16 GeV in model of grand unified theories (GUTs). These RHNs, originally Standard Model (SM) gauge singlet, being mixed with the SM light neutrinos to interact with the SM gauge bosons. Depending on the mass of the gauge singlet RHNs and their mixings with the active neutrino states, seesaw mechanism can be tested at colliders , as well as, in other non-collider experiments, such as, neutrinoless double beta decay [26,[53][54][55][56][57][58][59], neutrino experiments [8,9,60], rare-meson decays [61][62][63], muon g − 2 [64], lepton flavor violating processes l i → l j γ, µ → 3e, µ → e conversion in nuclei [66,67,94], non-unitarity [70-72, 93, 96], etc.
We are specifically interested in the RHNs at the TeV scale so that they can be tested at the high energy colliders. At the LHC, the production cross section of the RHN decreases as the mass of RHN increases as a result of the properties of the constituent quarks of the proton beams. In the linear collider the electron and positron are collided to produce the RHN in association with a light neutrino through the dominant t-channel process. A subdominant schannel process also contributes [73,74]. Otherwise a variety of RHN productions at the linear collider have been discussed in [75] followed by the bounds on the light heavy mixing angles for the electron flavor at the linear collider with 500 GeV and 1 TeV collider energies. The low mass range of the RHN has been studied in [82] which also predicts the limit on the light heavy mixing and the mass of the RHN up to a mass of 250 GeV. The sterile neutrinos at the circular lepton colliders have been studied in [83] which deals with a comprehensive discussion on the detectors from experimental point of view. Higgs searches from RHN has been studied in [84] where the RHN has been produced from the W and Z mediated processes. Such a RHN decays into a Higgs and SM light neutrino and the Higgs can dominantly decay into a pair of b-quarks.
Hence a 2b plus missing momentum will be a signal from this process. In this paper the RHN up to a 500 GeV mass have been tested where the maximum center of mass energy is also taken up to 500 GeV. The distinct and interesting signature of the RHN can be displaced vertex search if the mixing between the light and heavy neutrinos become extremely small. Such a scenario has been tested in [85] for the colliders 240 GeV, 350 GeV and 500 GeV. Another interesting work on the RHNs has been found in the form of [86] where a variety of the colliders have been considered to test the observability of the RHN production. They have discussed several production modes of the RHNs at the LHC, lepton-Hadron collider (LHeC) 1 [87] and linear collider. They have studied all possible modes of the RHN production in these colliders and compared the bounds on the light-heavy neutrino mixing angles. In the linear collider, the references [82][83][84][85][86] did not go further than 500 GeV as they constrained themselves within the center of mass energy of 500 GeV. However, none of these papers studied the boosted object at the LHeC and linear collider respectively.
In our analysis we consider the following things: 1. We study the prospect of discovery of RHNs at LHeC considering the boosted objects for the first time. In the LHeC we concentrate on the lepton number violating (LNV) and lepton number conserving (LNC) channels to produce the RHN in association with a jet (j 1 ). Hence the RHN will decay into the dominant W and the W will decay into a pair of jets. The daughter W coming from the heavy RHN will be boosted and its hadronic decay products, jets, of the W will be collimated such that they can form a fat jet (J).Hence a signal sample of + j 1 + J can be studied thoroughly at this collider.
In this process people have mostly studied the lepton number conserving channel where as the lepton number violating will be potentially background free. However, for clarity we study the combined channel and the corresponding SM backgrounds. We consider two scenarios at the LHeC where the electron and proton beams will have 60 GeV and 7 TeV energies where the center of mass energy becomes √ s = 1.3 TeV. We have also considered another center of mass energy at the √ s = 1.8 TeV where the proton beam energy is raised up to the 13.5 TeV. For both of the colliders we consider the luminosity at 1 ab −1 . Here the RHN is a first generation RHN (N 1 ) and is electron (e). Finally we study up to 3 ab −1 luminosity.
2. At the linear collider the production of the RHNs is occurring from the s-and t-channel processes in association with a SM light neutrino (ν). We consider the linear collider at two different center of mass energies, such as √ s = 1 TeV and √ s = 3 TeV which can probe up to a high mass of the RHNs such as 900 GeV (at the 1 TeV linear collider) and 2.9 TeV (at the 3 TeV linear collider) due to the almost constant cross section for the N ν production. For both of the center of mass energies we consider 1 ab −1 luminosity.
At this mass scale, the RHNs will be produced at rest, however, the daughter particles can be sufficiently boosted. We consider N → W, W → jj and N → hν, h → bb modes at the linear collider where h is the SM Higgs boson. If the RHN is sufficiently heavy, such the, M N ≥ 400 GeV, the W and h can be boosted because M W and M h << M N 2 . As a result W and h will produce a fat jet (J) and a fat b jet (J b ) respectively. Therefore the signal will be + J plus missing momentum and J b plus missing momentum in the W and h modes respectively at the linear collider. Therefore studying the signals and the backgrounds for each process we put the bounds in the mass-mixing plane of the RHNs.
3. We want to comment that studying e − e + → N 2 ν µ /N 3 ν τ mode in the Z mediated s-channel will be interesting where N 2 (N 3 ) will be the second (third) generation RHN. Studying the signal events and the corresponding SM backgrounds one can also calculate the limits on the mixing angles involved in these processes. Such a process will be proportional to . In these processes the signal will be µ(τ ) + jj plus missing momentum followed by the decay of N 2 (N 3 ) → µjj(τ jj). One can also calculate the bounds on the mass-mixing plane for different significances. A boosted analysis could be interesting, however, a non-boosted study might be more useful as the cross-section goes down with the rise in collider energy in these processes. Such signals can also be studied if the RHNs can decay through the LFV modes, such as e − e + → N ν e , N → µW, W → jj, however, µ → eγ process will make this process highly constrained due to the strong limit Br(µ + → e + γ) < 4.2 × 10 −13 at the 90% C. L. [88]. The corresponding limits on τ are weaker [89,90]. Such final states have been studied in [73] for M N = 150 GeV, a high mass test with using boosted object will be interesting in future. A comprehensive LHC study has been performed in [91].
4. The RHN produced at the linear collider may decay in to another interesting mode, namely, N → Zν, Z → bb. Which can be another interesting channel where boosted objects can be stated. However, precision measurements at the Z-boson resonance using electron-positron colliding beams at LEP experiment strongly constrains Z boson current, and hence, Zbb coupling. This channel also suffers from larger QCD background compared to the leptonic decay of Z boson, and hence, leptonic decay of Z boson has better discovery prospect for this particular mode of RHN decay. On the other hand, SM Higgs , h, mostly decays (∼ 60%) to bb due to large hbb coupling. Due to this, we focus on the Higgs decay mode of RHN, N → hν, h → bb to study the fat jet signature. For the time being, we mainly focus on the first two items. The investigation of the mode, N → Zν, Z → bb is beyond the scope of this article and shall be presented in future work in detail.
The paper is organised as follows. in Sec. II, we discuss the model and the interactions of the heavy neutrino with SM particles and also calculate the production cross sections at different colliders. In Sec.III we discuss the complete collider study. In Sec. IV we calculate the bounds on the mixing angles and compare them with the existing results. Finally, we conclude in Sec. V.

II. MODEL AND THE PRODUCTION MODE
In type-I seesaw [2], SM gauge-singlet right handed Majorana neutrinos N β R are introduced, where β is the flavor index. N β R have direct coupling with SM lepton doublets α L and the SM Higgs doublet H. The relevant part of the Lagrangian can be written as : After diagonalizing this matrix, 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 [92], Dirac Yukawa term Y D can be as large as 1, and this scenario is considered in this paper.
There is another seesaw mechanism, so-called inverse seesaw [5], where the light Majorana neutrino mass is generated through tiny lepton number violation. The relevant part of the Lagrangian is given by where M N is the Dirac mass matrix, N α R and S β L are two SM-singlet heavy neutrinos with the same lepton numbers, and µ is a small lepton number violating Majorana mass matrix. After the electroweak symmetry breaking the neutrino mass matrix is obtained as After diagonalizing this mass matrix, we obtain the light neutrino mass matrix Note that the small lepton number violating term µ is responsible for the tiny neutrino mass generation. The smallness of µ allows the M D M −1 N parameter to be order one even for an EW scale heavy neutrino. Since the scale of µ is much smaller than the scale of M N , the heavy neutrinos become the pseudo-Dirac particles. This is the main difference between the type-I and the inverse seesaw. where with = R * R T , and U M N S is the usual neutrino mixing matrix by which the mass matrix m ν is diagonalized as In the presence of , the mixing matrix N is not unitary [93][94][95][96]. Considering the mass eigenstates, the charged current interaction in the Standard Model 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 interaction is given by where c w = cos θ w is the weak mixing angle. Because of non-unitarity of the matrix N , N † N = 1 and the flavor-changing neutral current occurs.
The dominant decay modes of the heavy neutrino are N → W , ν Z, ν h and the corresponding partial decay widths 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 , the branching ratios can be obtained as

A. Production cross section at LHeC
The LHeC can produce the RHN in the process e p → N 1 j 1 through the t-channel exchanging the W boson. In this case the first generation RHN (N 1 ) will be produced. The corresponding Feynman diagram is given in Fig. 2. The total differential production cross section for this process is calculated as where C = C q = g section asσ LHeC and finally convoluting the PDF (CTEQ5M) [97] we get the total cross section where E CM is the center of mass energy of the LHeC and i runs over the quark flavors. For different center of mass energies E will be different. In Fig. 3 we plot the total production cross sections of N 1 at the three different collider energies such as process and the right panel is s channel process to produce the e + e − → N 1 ν 1 . To produce N 2 ν 2 and N 3 ν 3 , the Z mediated s channel process will act.

B. Production cross section at linear collider
The linear collider can produce the heavy neutrino in the process e + e − → ν 1 N 1 through t and s-channels exchanging the W and Z bosons, respectively. The corresponding Feynman diagrams are given in Fig. 4. The total differential production cross section for this process is calculated as The total production cross section for the process e + e − → ν 1 N 1 from the t and s channel processes at the linear collider at different center of mass energies are shown in Fig. 5.

III. COLLIDER ANALYSIS
We implement our model in FeynRules [98], generate the UFO file of the model for MadGraph5-aMC@NLO [99] to calculate the signals and the backgrounds. Further we use PYTHIA6 [100] for LHeC as used in [87] and PYTHIA8 [101] for the linear colliders, where subsequent decay, initial state radiation, final state radiation and hadronisation have been carried out. We have indicated in [14,15] that if the RHNs are sufficiently heavy, the daughter particles can be boosted. We prefer the hadronic decay mode of the W where the jets can be collimated so that we can call it a fat-jet (J). Such a topology is very powerful to discriminate the signal from the SM backgrounds. We perform the detector simulation using DELPHES version 3.4.1 [102]. The detector card for the LHeC has been used from [103]. We use the ILD card for the linear collider. In our analysis the jets are reconstructed by Cambridge-Achen algorithm [104,105] implemented in Fastjet package [106,107] with the radius parameter as We study the production of the first generation RHN (N 1 ) and its subsequent leading decay mode (e p → N 1 j 1 , N 1 → W e, W → J) at the LHeC with √ s = 1.3 TeV and 1.8 TeV center of mass energies. The corresponding Feynman diagram is given in Fig. 7. We also study the RHN production at the linear collider (International Linear Collider, ILC) at √ s = 1 TeV and CLIC at √ s = 3 TeV collider energies. However, for simplicity we will use the term linear collider unanimously. At the linear collider we consider two sets of signals after the production of the RHN, such that, e + e − → N 1 ν, N 1 → W e, W → J and e + e − → N 1 ν, 1 σ dσ dp 1 σ dσ dp A. LHeC analysis for the signal e − p → jN 1 → e ± + J + j 1 Producing N 1 at the LHeC and followed by its decay into leading mode to study the boosted objects, we consider the final state e ± + J + j 1 . In this case we have two different processes, one is them is the e + + J + j 1 and the other one is e − + J + j 1 . The first one is the Lepton Number Violating (LNV) channel and the second one is the Lepton Number Conserving (LNC). At the time of showing the results we combine LNV and LNC channels to obtain the final state as The LNV signal is almost background free until some e + +jets events appear from some radiations, however, that effect will be negligible. Therefore for completeness we include the LNC channel where the leading SM backgrounds will come from e − jjj, e − jj and e − j including initial state and final state radiations. For completeness we include both of the LNV and LNC channels. Further we use the fat-jet algorithm to reduce the SM backgrounds. We have shown the distributions of the transverse momentum of the leading jet (p j 1 T ), lepton (p e T ) and fat-jet (p J T ) in Figs.9-11. The fat-jet mass distribution (M J ) has been shown in Figs.12. The invariant mass distribution of the lepton and fat-jet system (M eJ ) has been shown in Fig. 13. We have also compared the signals with the corresponding SM backgrounds. As a sample we consider • Transverse momentum for fat-jet p J T > 175 GeV.
• Invariant mass window of e ± and fat-jet J, |M eJ − M N | ≤ 20 GeV. We have noticed that M J > 70 GeV cuts out the low energy peaks (M J ≤ 25 GeV) which come from the hadronic activity of the low energy jets. Similarly, the p J T and p e T cuts are also very effective. Due to the presence of the RHN, these distributions from the signal will be in the high values than the SM backgrounds. Therefore selecting such cuts at high values, as we have done here, will be extremely useful to reduce the SM backgrounds.
We have noticed that ej background can completely be reduced with the application of the kinematic cuts on p e T , p J T and M J . It is difficult to obtain a fat jet from this process because the t channel exchange of the Z boson and photon will contribute to this process, however, the other low-energy jets may come from the radiations at the initial and final states. These jets do not help to make the fat jets sufficiently energetic. Therefore p J T > 175 GeV (p J T > 400 GeV) at the LHeC (HE-LHeC) are very useful. Similarly the ejjj is the irreducible background in this case which will contribute most among the backgrounds. Whereas ejj is the second leading background in this case. However, both of these backgrounds can be reduced using the invariant mass cut of the RHN. As the RHN will decay according to N → eJ, therefore the invariant mass of the eJ system with an window of 20 GeV (|M eJ − M N | ≤ 20 GeV) will be extremely useful to reduce the backgrounds further in these colliders. In Tab the square of the mixing.        fat-jet p J T in Figs. 14-16 for the linear colliders. The fat-jet mass M J distribution has been shown in Fig. 17. We construct the polar angle variable in Fig. 18  • Transverse momentum for leading lepton p e ± T > 100 GeV for M N mass range 400 GeV-600 GeV and p e ± T > 200 GeV for M N mass range 700 GeV-900 GeV.
• Fat-jet mass M J > 70 GeV. • Transverse momentum for leading lepton p e ± T > 200 GeV for M N mass range 700 − 900 GeV and p e ± T > 250 GeV for M N mass range 1 − 2.9 TeV.

IV. CURRENT BOUNDS
The bounds on the light-heavy neutrino mixing for the electron flavor comes from a variety of searches. As we are interested on the RHN of mass M N ≥ 100 GeV, therefore we will compare our results with such bounds which are important for that mass range. The Electroweak Precision Data (EWPD) bounds have been calculated in [108][109][110] which obtains the bound on |V eN | 2 as 1.681 × 10 −3 at the 95% C. L., the LEP2 [111], calculated at the 95% C.L., bounds are rather weaker except M N = 108 GeV where it touches the EWPD line. The strongest bounds are coming from the GERDA [112] 0ν2β study where the limits as calculated in [13] up to M N = 959 GeV. The lepton universality limits from [113] set bounds on |V eN | 2 at 6.232 × 10 −4 up to M N = 1 TeV at the 95% C. L. These bounds are plotted in Figs. 22 -27. Apart from the above mentioned indirect searches, the recent collider searches for the LHC also set bounds |V eN | 2 at the √ s = 8 TeV at 95% C. L. from same sign dilepton plus dijet search. The bounds on |V eN | 2 from ATLAS (ATLAS8-ee) [114] and CMS (CMS8 − ee) [115] are obtained at 23.3 fb −1 and 19.7 fb −1 luminosities respectively for the e ± e ± + 2j sample. The ATLAS limit is weaker than the CMS limits for 100 GeV ≤ M N ≤ 500 GeV. The LHC has also published the recent results at √ s = 13 TeV with 35.9 fb −1 luminosity which set stronger bounds on |V eN | 2 than the previous direct searches for 100 GeV ≤ M N ≤ 500 GeV. The bounds on |V eN | 2 from the e ± e ± + 2j signal in CMS (CMS13-ee) [116] and from trilepton search at CMS (CMS13-3 ) [117] are also competitive, however, weaker than the EWPD for 100 GeV ≤ M N ≤ 1.2 TeV. These limits are also plotted in Figs. 22 -27. We have explored that at the LHeC with √ s = 1.3 TeV collider energy and 1 ab −1 luminosity, the bound on |V eN | 2 for M N N = 600 GeV with 1-σ C.L. is better than the 0ν2β limit from GERDA-low where as M N ≥ 959 GeV at 1-σ limit can be probed better than the GERDA-low and high limit [13,112]. The results are shown in Fig. 26. We have also studied the linear colliders at 1(3) TeV center of mass energy with 3(5) ab −1 luminosity. We can find the improved results in Fig. 27. Finally we comment that our results at the linear collider are stronger than the limits obtained from HE-LHeC (red band) at the 1 ab −1 luminosity compared to EWPD [108][109][110], LEP2 [111], GERDA [112] 0ν2β study from [13], ATLAS (ATLAS8-ee) [114], CMS (CMS8 − ee) [115] at the 8 TeV LHC, 13 TeV CMS search for e ± e ± + 2j (CMS13-ee) [116] and 13 TeV CMS search for 3 (CMS13-ee) [116] respectively.

V. CONCLUSION
We have studied the RHNs which can be responsible for the generation of the tiny light neutrino masses. We have calculated the production cross sections for the RHNs at the LHeC and linear collider at various center of mass energies and followed by that we have tested the discovery prospects of this RHNs. We have chosen √ s = 1.  Note added: While in final drafting phase, we noticed Ref. [118] appeared in arXiv which also studied fat jet signatures from RHNs at the linear colliders. We have studied LHeC and linear collider at different center of mass energies using detailed cut based analyses. We have compared our results with all the existing bounds using the decay modes of the RHNs to W  [108][109][110], LEP2 [111], GERDA [112] 0ν2β study from [13], ATLAS (ATLAS8-ee) [114], CMS (CMS8 − ee) [115] at the 8 TeV LHC, 13 TeV CMS search for e ± e ± + 2j (CMS13-ee) [116] and 13 TeV CMS search for 3 (CMS13-ee) [116] respectively. and SM h bosons. The 0ν2β bound became very strong up to M N = 959 GeV. At the linear collider the polar angle variable for the lepton became very useful for us. In our analysis we have showed that high mass RHNs can be observed at 5-σ significance or more in these colliders. regarding the linear collider card in DELPHES.