Production of Heavy neutrino in next-to-leading order QCD at the LHC and beyond

Majorana and pseudo-Dirac heavy neutrinos are introduced into the type-I and inverse seesaw models, respectively, in explaining the naturally small neutrino mass. TeV scale heavy neutrinos can also be accommodated to have a sizable mixing with the Standard Model light neutrinos, through which they can be produced and detected at the high energy colliders. In this paper we consider the Next-to-Leading Order QCD corrections to the heavy neutrino production, and study the scale variation in cross-sections as well as the kinematic distributions with different final states at $14$ TeV LHC and also in the context of $100$ TeV hadron collider. The repertoire of the Majorana neutrino is realized through the characteristic signature of the same-sign dilepton pair, whereas, due to a small lepton number violation, the pseudo-Dirac heavy neutrino can manifest the trileptons associated with missing energy in the final state. Using the $\sqrt{s}=8$ TeV, $20.3$ fb$^{-1}$ and $19.7$ fb$^{-1}$ data at the ATLAS and CMS respectively, we obtain prospective scale dependent upper bounds of the light-heavy neutrino mixing angles for the Majorana heavy neutrinos at the $14$ TeV LHC and $100$ TeV collider. Further exploiting a recent study on the anomalous multilepton search by CMS at $\sqrt{s}=8$ TeV with $19.5$ fb$^{-1}$ data, we also obtain the prospective scale dependent upper bounds on the mixing angles for the pseudo-Dirac neutrinos. We thus project a scale dependent prospective reach using the NLO processes at the $14$ TeV LHC.


Introduction
The journey of Large Hadron Collider (LHC) in its 7 and 8 TeV run has been extremely successful in discovering, and further constraining the properties of longawaited Higgs boson [1,2] of Standard Model (SM). However, LHC is still lacking any clinching signature yet from the beyond the Standard Model (BSM) physics. With general wisdom, there exists a broad agreement in admitting the SM as, at most a very accurate description for low energy effective theory of particle physics. This notion is embolden from the fact, that the SM falls short to explain divers outstanding issues both in theory and in explaining some crucial experimental observations.
The most recent observations on the neutrino oscillation phenomena [3][4][5][6][7][8] have established that tiny neutrino mass and the flavor mixing of the SM neutrinos which is one of the divers mysteries in SM. The existence of such a tiny neutrino mass requires us to extend the SM. The seesaw mechanism [9][10][11][12][13][14][15] is probably the simplest idea to extend the SM, which can explain the small neutrino mass naturally. The SMsinglet heavy right-handed Majorana neutrinos induce the dimension five operators [16] leading to very small light Majorana neutrino masses. If such heavy neutrino mass lies in the electroweak scale, then the heavy neutrinos can be produced in the high energy colliders  from various initial states. The heavy neutrinos are singlet under the SM gauge group therefore they can be coupled to the SM gauge bosons through the mixing with the light SM neutrinos vis Dirac Yukawa coupling. The Dirac Yukawa coupling can be sizable for the electroweak scale heavy neutrinos in general Casas-Ibarra parameterization [58], while reproducing the neutrino oscillation data.
There is another kind of seesaw mechanism, commonly known as inverse seesaw [59,60], where the tiny Majorana mass is generated from the small lepton number violating parameters, rather than being suppressed by the heavy neutrino mass in conventional seesaw mechanism. In case of inverse seesaw the heavy neutrinos are pseudo-Dirac and the Dirac Yukawa coupling could be of order one, satisfying the neutrino oscillation data. Thus at the high energy colliders the pseudo-Dirac heavy neutrinos can be produced through the sizable mixing with the SM neutrinos. In our analysis we choose the LHC at the center of mass energy √ s = 14 TeV and a proposed proton-proton collider at the center of mass energy √ s = 100 TeV [61] which can enlighten the new physics era including the heavy neutrino physics more precisely with its higher fact finding ability.
The heavy neutrino can be produced at the high energy colliders from various initial states among them the leading contributions come from the processes generated from the quark-quark (qq ), quark-gluon (qg) and gluon-gluon (gg) initial states. Among these processes the qq initial state is the commonly studied leading order (LO) production channel for the heavy neutrinos, where as, the other channels can contribute in its Next-to-Leading-Order (NLO) and Next-to-Next-to-Leading-Order (NNLO) QCD corrections together with the corresponding LO processes. In this paper we concentrate on the QCD NLO production processes including the virtual correction contributions and the real emission processes. For the LO processes we demonstrate the production of the heavy neutrino for different factorization (µ F ) scales associated to the parton density functions (PDFs) considering the 14 TeV LHC and in the context of proposed 100 TeV hadron collider. On the other hand, NLO processes are studied with different choices of factorization (µ F ) as well as renormalization (µ R ) scales juxtaposing together with LO contributions.
The paper is organized in the following way. In Sec. 2 we introduce the type-I seesaw and inverse seesaw models. These are the primary models we concentrate in our present analysis. In Sec. 3 we calculate the production cross-section of the heavy neutrino at the high energy colliders. We discuss the methodology followed with different choice of parameters in estimating in leading order and next-to-leading order production of heavy neutrinos. We also opened up discussion on the scale variation related to these production cross-sections. In Sec. 4 we study the scale dependent kinematic distributions of different kinematic measurable quantities in the heavy neutrino production from the trilepton plus missing energy final state. In Sec. 5 we utilize the current Large Hadron Collider (LHC) data from ATLAS and CMS to put scale dependent upper bounds on the mixing angles between the lightheavy neutrinos. Sec. 6 is dedicated to the conclusion.

Neutrino Mass Mechanism
In type-I seesaw [9][10][11][12][13][14][15], we introduce SM gauge-singlet right handed Majorana neutrinos N β R , where β is the flavor index. N β R couple with SM lepton doublets α L and the SM Higgs doublet H. The relevant part of the Lagrangian is After the spontaneous electroweak symmetry breaking by the vacuum expectation Using the Dirac and Majorana mass matrices we can write the neutrino mass matrix as Diagonalizing this matrix we obtain the seesaw formula for the light Majorana neutrinos as For m N ∼ 100 GeV, one may find that the extremely minuscule Y D ∼ 10 −6 is needed to construct some light neutrino mass of order m ν ∼ 0.1 eV. However, using the general parameterization based on Casas-Ibarra [58], one gets the Yukawa coupling expressed in terms of a orthogonal matrix which remains completely arbitrary and hence can be large. Following this mechanism Y D can be phenomenologically viable, as large as order one, and this is the case we consider in our present work. There is another seesaw mechanism, so-called inverse seesaw [59,60], where the light Majorana neutrino mass is generated through tiny lepton number violation. The relevant part of the Lagrangian is given by 4) 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 electroweak symmetry breaking we obtain the neutrino mass matrix as (2.5) Diagonalizing this mass matrix we obtain the light neutrino mass matrix Note that the smallness of the light neutrino mass originates from the small lepton number violating term µ. The smallness of µ allows the m D m −1 N parameter to be order one even for an electroweak 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 seesaws.
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 as where V N is the mixing between the SM neutrino and the SM-singlet heavy neutrino, and we have assumed a small mixing, |V N | 1. Using the mass eigenstates, the charged current interaction for the heavy neutrino is given by where 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 with θ w being the weak mixing angle. The main decay modes of the heavy neutrino are N → W , ν Z, ν h. The corresponding partial decay widths are respectively given by (2.10) 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 decay width (Γ total ) of the heavy neutrino into W , Z and Higgs bosons in Fig. 1 where ξ is the scale factor varying between 0.1 to 10. Whereas the NLO cross-section 1 In our final drafting phase Ref. [56] appeared with the NLO prediction of the heavy neutrino production using MadGraphMC@NLO which overlaps with a part our result consistently depending upon the choices of scale and selection cuts. It is important to mention that we have used our independently developed code for the type-I and Inverse seesaw mechanisms in MadGraphMC@NLO, fixing bugs with the active support from the MadGraph team [97] and finally with private communications. The implementation of scale variations were suggested in [98], although we finally used the straightforward method bypassing the SyaCalc.  depends on both the scale, namely, the factorization scale (µ F ) through PDFs and the renormalisation scale (µ R ) through NLO partonic cross-section (mainly due to the couplings renormalisation). For simplicity, throughout the present analysis we have considered to vary both these scales as, We have produced the scale dependent cross-sections normalized by the square of the mixing angle |V N | 2 for a fixed choice of heavy neutrino mass m N at 100 GeV, 400 GeV, 800 GeV and 1 TeV at the 14 TeV LHC for the LO and NLO processes with varying ξ between 0.1 to 10. This scale dependence are shown in Fig. 3. In the same plot, we also display the theoretical scale uncertainty in NLO calculation due to µ F alone by fixing the renormalisation scale at the corresponding heavy neutrino mass (µ R = m N ). Since later dependence only enters at the the NLO level in the form of α s (µ R ), one expects the µ F scale dependence which actually soften in NLO calculation. The other scenario by fixing the factorization scale at the corresponding heavy neutrino mass (µ F = m N ) is also shown by changing only the µ R scale in the same plot. For m N = 100 GeV, the leading order cross-section σ LO varies sharply and in-  creasing almost monotonically by a factor of two approximately with increase in scale factor (ξ), which indicates a substantial amount of theoretical uncertainty present in the LO result. This is because of only the LO quark-antiquark flux with varying scale factor ξ 2 . Whereas in NLO, it is three fold -the scale dependent logarithmic terms present in partonic cross-sections, the NLO PDF fluxes (namely, quark-antiquark, quark-gluon and antiquark-gluon) as well as strong coupling constant and hence the strong scale dependent part cancels among themselves. Therefore this strong scale dependence has been soften by including the NLO calculation which varies slowly with the scale. At the larger choices of heavy neutrino mass, both the LO and the NLO cross sections decrease with ξ variation, although the basic feature of softening of NLO variations are evident in all such examples.
The scale dependent results for 100 TeV collider considering m N at 100 GeV, 400 GeV, 1 TeV and 1.5 TeV are also shown in Fig. 4. Here the scale variation of the leading order cross-section varies rather sharply especially for lower value of m N , which provides the LO cross-section dominating over the NLO prediction for ξ > 2. This steep rising of LO cross-section is mainly due to the LO PDF (CTEQ6L1) sets. On the other hand the NLO cross-sections reduce this PDF scale uncertainty significantly so that the NLO cross-section remains almost flat with respect to ξ for all m N at the 100 TeV hadron collider.
The scale variations of the heavy neutrino production cross-sections, normalized by the square of the mixing matrix, is further demonstrated as a function of m N at 14 TeV LHC and 100 TeV hadron collider in Fig. 5 and Fig. 6. In these figures, the blue (red) bands shows the scale dependence of the σ LO (σ N LO ). It is clear from these figures that the strong scale dependence reduces significantly at NLO cross-section as compared to LO one.
We also study the next-to-leading order predictions for N ν final state mediated by the Z boson. The scale dependent cross-sections at 14 TeV LHC and 100 TeV hadron collider are given in Fig. 7. We consider the heavy neutrino mass at 100 GeV and 800 GeV. We used the same scale dependence for the N ν final state as we did in N final state for LO and NLO respectively. At 14 TeV the LO cross-section at m N = 100 GeV increases with ξ at a faster rate than the NLO process whereas for m N = 800 GeV the LO cross-section decreases with the increase in ξ with a faster rate than that in NLO. On the other hand at 100 TeV collider the LO process at m N = 100 GeV takes over the NLO for ξ > 1.9. For m N = 800 GeV the LO crosssection increases with the increase in ξ but the NLO cross-section remains more or less same for ξ > 0.5. Following our earlier demonstration on N , we have also shown

Scale dependent kinematic distributions in trilepton channel
In this section we consider the pseudo-Dirac heavy neutrino production at the hadron collider and study its decay process for the 14 TeV LHC and 100 TeV collider. For further demonstration we choose the heavy neutrino with mass m N = 400 GeV. Different kinematic distributions are constructed both for the LO and NLO calculations by choosing both the factorization scale as well as renormalisation scale varying simultaneously (as in Eq.3.2) with a scale factor ξ between 0.1 and 10. The pseudo-Dirac heavy neutrino is involved in the inverse seesaw mechanism to generate the neutrino mass. The collider phenomenology of inverse seesaw mechanism has been studied in [22][23][24]26] for the LO process with trilepton final state for a fixed scale.
In case of inverse seesaw mechanism the Yukawa coupling could be high enough to enhance the heavy neutrino production. However, due to the small lepton number violating parameter 2 the heavy neutrino becomes pseudo-Dirac. In this analysis we           generation of the lepton flavor 3 . As a result the golden channel for the final state signal is the trilepton plus missing energy which is given by Within the same set of the model parameters as described in the last section, we show the differential distributions of scattering cross-section as a function of the  (fb/0.1) (fb/0.1)  (fb/0.1) (fb/0.1) (fb/5 Degrees) (fb/5 Degrees)  (fb/0.1) (fb/0.1) s =100 TeV Figure 10. Scale variation of the differential scattering cross-section distribution with respect to cos θ i j of the leptons with i = 1, 2, 3, j = 1, 2, 3 and i = j and ∆φ i j of the leptons with i = 1, 2, 3, j = 1, 2, 3 and i = j for the trilepton production channel for m N = 400 GeV. The first row corresponds to cos θ i j at √ s = 14 TeV LHC whereas the second row represents the √ s = 100 TeV collider. The third row corresponds to ∆φ i j at √ s = 14 TeV LHC whereas the fourth row represents the same at √ s = 100 TeV collider. The differential cross-section distributions are normalized by |V N | 2 .
transverse momenta for these three leptons separately, p i T for i = 1, 2, 3 in Fig. 9 for the 14 TeV LHC and 100 TeV collider. In case of 14 TeV, the NLO distributions dominate over the LO distributions in the high transverse momentum region for 1 and 2 . Whereas for 3 , the NLO distributions dominate over the LO distributions quite impressively in the low transverse momentum region. This behavior is due to inclusion of extra radiation at the NLO process as well as showering effect. In the 100 TeV case, similar situations are demonstrated, however leading order scale uncertainties are exceptionally large, again due to LO PDF sets as mentioned earlier.
We exhibit the pseudo-rapidity distributions in Fig. 9. The η 2,3 distributions at the 14 TeV are sharper than that of η 1 , resulting the production of these two To study the different dilepton correlating observables in terms of LO and NLO calculation with their corresponding scale uncertainties, in Fig. 10 we display the differential distributions for scattering cross-section with respect to the angles between the leptons, cos θ i j for i = 1, 2, 3, j = 1, 2, 3 and i = j. The production of the leptons with large polar angular separation is abundant, although tend to choose smaller one since main production channel involves the contributions from both the valence and sea quarks. This effect is quite more evident at 100 TeV machine, where the leptons are mostly produced at small polar angle. The difference between the azimuthal angle between the two leptons, namely, ∆φ i j of the leptons with i = 1, 2, 3, j = 1, 2, 3 and i = j are also shown in Fig. 10. One notice that both the ∆φ 1 2 and ∆φ 1 3 distributions show some enhancement when leptons are produced back to back, supported by the leptons produced in boosted heavy neutrino decay. This peak is further enhanced in the NLO calculation. However, ∆φ 1 2 remains flat in both calculations. Scale uncertainty is also shown to be substantially controlled in NLO estimates. The scale dependent differential scattering cross-section as a function of missing transverse energy, E miss T , are given Fig. 11 where the LO and NLO variations have good agreements at 14 TeV LHC. NLO distribution enhances at larger E miss T . We can make the same observation at the NLO and LO cases for the 100 TeV collider.

Scale dependent prospective upper bound on the mixing angles
In this section, we study the prospective upper bounds on the mixing angles between the heavy neutrino and the SM light neutrinos. We consider two different scenario such as type-I seesaw which involves a heavy Majorana neutrino and inverse seesaw which introduces a pseudo-Dirac heavy neutrino. Due to the large lepton number violation, the type-I seesaw scenario is observed through the characteristic same sign dilepton plus dijet final state. On the other hand due to a very small lepton number violating parameter, optimal observable being the trilepton plus missing energy signal for inverse seesaw case. Using these mechanisms and existing searches done by LHC at √ s = 8 TeV(Run−1) we obtain prospective search reaches at the LHC at √ s = 14 TeV(Run−2) and proposed proton-proton collider at √ s = 100 TeV.

Same sign dilepton plus two jet production signal
For simplicity we consider the case that only one generation of the Majorana heavy neutrino is lighter enough 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 for the LO and CTEQ6M for the NLO(µ F = µ R ) cases respectively. We study the scale dependent same sign dilepton plus dijet signal cross-section as a function of the heavy neutrino mass (m N ). The signal cross-section at the level of LO and NLO are calculated as σ(ξ) LO and σ(ξ) N LO respectively for the same sign dimuon production, pp → N µ ± → µ ± µ ± jj.
We notice that the scale dependence at LO is very clear for m N 300 GeV in comparison to the NLO case at the 14 TeV LHC. The LO and NLO results could be comparable to the EWPD at the 14 TeV. Which we can easily verify using the LHC results in Run−2 at the 14 TeV. We have also studied the prospective upper bounds on the mixing angles at the 100 TeV collider for the LO and NLO cases at 20.3 fb −1 luminosity. We have noticed that for m N 250 GeV, the mixing angle could be a factor better than those given by EWPD. An improved prospective search reach by an order of magnitude (more) for 300 fb −1 (1000 fb − 1) luminosity is also given in Fig. 12 for the 14 TeV LHC. We have also calculated a prospective search reach for  [104][105][106] for electron (cyan, EWPD-e(old)) and muon (cyan, EWPD-µ(old)) flavors(new values can be found from [107] , for tau (dotted, EWPDτ ) electron (solid, EWPDe) and muon (dashed, EWPDµ) flavors). The shaded region is excluded by the 8 TeV data. the 100 TeV Collider at 20.3 fb −1 , 300 fb −1 and 1000 fb −1 luminosities in Fig. 13. The improvement in search reach of the mixing angle from a factor to an order of magnitude with respect to the EWPD can be obtained at the 100 TeV.
Recently the CMS has performed the same-sign dilepton plus dijet search [103]. Using this result and adopting the same procedure for the ATLAS result we calculate the prospective upper bound on the mixing angles at the 14 TeV LHC for the LO and NLO cases at 19.7 fb −1 . The results are shown in Fig. 13. A clear scale dependence is observed for m N 300 GeV for the LO case whereas the mixing angle around 120 GeV is comparable to the EWPD. The scale dependence is not very high in the NLO case in comparison to the LO case. We can easily verify using the LHC  [104][105][106] for electron (cyan, EWPD-e(old)) and muon (cyan, EWPD-µ(old)) flavors(new values can be found from [107] , for tau (dotted, EWPDτ ) electron (solid, EWPDe) and muon (dashed, EWPDµ) flavors). The shaded region is excluded by the 8 TeV data. results in Run−2 at the 14 TeV. An improved prospective search reach by an order of magnitude (more) for 300 fb −1 (1000 fb −1 ) luminosity is also given in Fig. 14 for the 14 TeV LHC. We have also depicted the prospective search reach in Fig. 15 for the 100 TeV Collider at 19.7 fb −1 , 300 fb −1 and 1000fb −1 luminosities where we can improve the upper bound on the mixing angle from a factor up to an order of magnitude with respect to the EWPD from low to high luminosities.

Trilepton associated with missing transverse energy signal
We consider two cases in this analysis. One is the Flavor Diagonal case (FD), where we employ three generations of the degenerate heavy neutrinos and each generation couples with the single, corresponding lepton flavor. The other one is the Single Flavor case (SF) where only one of the heavy neutrinos is accessible to the LHC and having mass in the Electroweak scale being coupled to only the first or second generation of the lepton flavor. In this analysis we use the CTEQ6M PDF [112] for generating the NLO(µ F = µ R ) processes to compute the scale dependent trilepton plus missing energy events with ξ = 0.1, 1.0 and 10.0 at √ s = 14 TeV LHC using MadGreph-aMC@NLO bundled with PYTHIA6Q using anti-k T algorithm for jet clustering in FastJet. We use the hadronized events in Delphes [113] to produce events after the detector simulation. The trilepton plus missing energy mode is given in Eq. 4.1. After the detector simulation we have considered the events with 3 + E miss T + n−jets where n = 0,1 and 2.
Recently the CMS has studied the anomalous multilepton plus missing energy final state at the 8 TeV [114]. We adopt their search result for out trilepton analysis and compare our trilepton plus missing energy final state after the detector simulation to put a prospective upper limit on the mixing angle at the 14 TeV. The cuts we used for this analysis according to [114] are itemized below: (i) The transverse momentum of each lepton: p T > 10 GeV.
(ii) The transverse momentum of at least one lepton: p ,leading T > 20 GeV.
(iii) The jet transverse momentum: p j T > 30 GeV.
(v) The lepton-lepton separation: ∆R > 0.1 and the lepton-jet separation: ∆R j > 0.3. To derive the limits on |V N | 2 , we calculate the signal cross-section normalized by the square of the mixing angle as a function of the heavy neutrino mass m N for both SF and FD cases, by imposing the CMS selection criteria listed above for different scale values of ξ at the NLO process. 4 Passing the generated detector events through all the  [101] (Higgs), (ii) from a direct search at LEP [102](L3), valid only for the electron flavor, (iii) ATLAS limits from √ s = 7 TeV LHC data [123] (ATLAS7) and √ s =8 TeV LHC data [100] (ATLAS8), valid for a heavy Majorana neutrino of the muon flavor, (iv) CMS limits from √ s =8 TeV LHC data [103] (CMS8), for a heavy Majorana neutrino of the muon flavor and (v) indirect limits from the global fit to the electroweak precision data (EWPD) from [104][105][106] for electron (cyan, EWPD-e(old)) and muon (cyan, EWPD-µ(old)) flavors(new values can be found from [107] , for tau (dotted, EWPDτ ) electron (solid, EWPDe) and muon (dashed, EWPDµ) flavors). Here SF 75 and F D 75 are the single flavor and flavor diagonal cases below the Z-pole whereas SF 105 and F D 105 are the same above the Z-pole.
cuts we compare them with the observed number of events at the 19.5 fb −1 luminosity [114]. For the selection criteria listed above, the CMS experiment observed: (a) 10 events with the SM background expectation of 11±3.8 events for m + − < 75 GeV.
In case (a) we have an upper limit of 2.8 signal events, while in case (b) leads to an upper limit of 0.6 signal events. Using these limits, we can set an upper bound on |V N | 2 for a given value of m N for the scale dependent NLO case. In Fig. 14 we plot our results of the prospective upper bounds for the SF and FD cases for the scale dependent NLO case at the 14 TeV. In [23] different H T and E miss T regions are considered to calculate the upper bounds on the mixing angle which has been improved in [24] for the LO processes.
In this work we have considered a different region for H T and E miss T to evaluate the upper bounds on the mixing angles which has not been tested before. In this new region we can put the prospective upper bounds on the mixing angle for the 14 TeV LHC for m N = 91.2 − 400 GeV for the scale dependent NLO processes. We notice that for the trilepton case for the NLO processes at 14 TeV the scale dependent prospective bounds on mixing angle well coincide with each other. An estimation at the 100 TeV collider for the same study can make a legitimate improvement by approximately one order of magnitude or more and this will be tested in future. A prospective search reach for the 300 fb −1 luminosity at 14 TeV LHC is also given in Fig. 16 for which we can get up to order one improvement in the upper bounds of the mixing angles.

Conclusion
In this paper we have discussed the generation of the SM neutrino mass through the type-I and inverse seesaw mechanisms which involve the Majorana and the pseudo-Dirac heavy neutrinos respectively. Such heavy neutrinos, residing in the eleactroweak scale, can be produced at LHC and proposed 100 TeV hadron collider through a large mixing angle with the SM light neutrinos. To produce such heavy neutrinos at such high energy hadronic colliders it is important to discuss the scale dependent production cross-sections and distributions at the LO and also at the NLO QCD.
We have studied two different channels for the heavy neutrino production; one is the W mediated for the associated production of lepton and the other one is the Z mediated process with associated light neutrino. We have demonstrated that the heavy neutrino production cross-sections at the next-to-leading order QCD accuracy are quite stable against the scale variations, where as leading order estimated can change substantially. We also exhibit the scale dependance in different differential distributions related with the leptons and correlations between them.
We have obtained the prospective scale dependent search reach at the 14 TeV LHC and as well as at the 100 TeV collider for the Majorana heavy neutrino through the same sign dilepton plus dijet final state. Using the pseudo-Dirac heavy neutrinos we have studied the trilepton plus missing energy final state with jets and obtained the prospective search reach at the 14 TeV. A collider with a higher energy can probe the mixing angle more precisely improving the 14 TeV result up to an order of magnitude or more.