Testing triplet fermions at the electron-positron and electron-proton colliders using fat jet signatures

The addition of $SU(2)_L$ triplet fermions of zero hypercharge with the Standard Model (SM) helps to explain the origin of the neutrino mass by the so-called seesaw mechanism. Such a scenario is commonly know as the type-III seesaw model. After the electroweak symmetry breaking the mixings between the light and heavy mass eigenstates of the neutral leptons are developed which play important roles in the study of the charged and neutral multiplets of the triplet fermions at the colliders. In this article we study such interactions to produce these multiplets of the triplet fermion at the electron-positron and electron-proton colliders at different center of mass energies. We focus on the heavy triplets, for example, having mass in the TeV scale so that their decay products including the SM the gauge bosons or Higgs boson can be sufficiently boosted, leading to a fat jet. Hence we probe the mixing between light-heavy mass eigenstates of the neutrinos and compare the results with the bounds obtained by the electroweak precision study.


INTRODUCTION
The observations of the neutrino oscillation phenomena and the flavor mixing [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] give a very strong indication of the existence of the tiny neutrino masses. The existence of the neutrino mass is one of the missing pieces in the SM. Therefore to correctly fit the experimental results in, the SM requires to be extended. From the point of view of the low energy theory, one can introduce a dimension-5 operator within the SM which involves the SM Higgs and lepton doublets. Such an operator is known as the very famous Weinberg operator [17] which introduces a violation of the lepton number by two units. The breaking of the electroweak symmetry ensures the generation of the tiny neutrino Majorana masses which is suppressed by the scale of the dimension-5 operator. In the context of the renormalizable theory, the dimension-5 operator is naturally generated by the inclusion of the SM singlet right handed Majorana neutrinos and finally integrating them out. This is called the seesaw mechanism [18][19][20][21][22][23] which can successfully explain the origin of the tiny neutrino mass, however, hitherto there is no experimental evidence of this simple but extraordinary theoretical aspect. Such non observation opens up the pathway for a variety of neutrino mass generation mechanisms.
Type-III seesaw scenario is one of the most simple scenarios where the SM is extended by an SU(2) L right handed triplet fermion with zero hypercharge [24] which effectively generates a lepton number violating dimension-5 Weinberg operator. The triplet fermion has neutral and charged multiplets. The neutral component gets involved in the generation of the Majorana mass term for the light neutrinos after the electroweak symmetry breaking which finally generates the mixing between the light-heavy mass eigenstates as it happens in the canonical or type-I seesaw mechanism. These neutral multiplets can be studied at the energy frontier form their production at the different colliders through the mixing. Apart from the neutral multiplets the charged multiplets can also be produced at the collider in the same fashion, however, the charged multiplets can also be produced directly in pair from the SM gauge interactions where a wide variety of phenomenological aspects can be studied involving the prompt and non-prompt decay of the triplet fermions [25][26][27][28][29][30][31][32].
The rich theoretical and phenomenological aspects of the type-III seesaw scenario has been explored in different ways. An SU(5) realization of this scenario from the 24 representation containing both of a triplet fermion and a singlet fermion has been proposed in [33]. A supersymmetric realization of the SU(5) grand unified theories for the singlet and triplet has been studied in [34,35]. In this case the triplet can reside in the intermediate scale to reproduce the neutrino oscillation data. A nonsupersymmetric implementation of the type-III seesaw in the framework of SU(5) grand unified theory with the inclusion of the adjoint fermionic multiplet has been studied in [36] which mainly predicts a theory of the light SU(2) triplet fremion with mass at the electroweak scale. It has been mentioned in [36] that due to the gauge coupling of the multiplets of the triplets, they can be pair-produced directly through the Drell-Yan process. Being Majorana in nature, the neutral component of the triplets can show up with a distinct lepton number violating signature at the collider.
The grand unified theory inspired non-supersymmetric and supersymmetric renormalizable SU(5) frameworks to study the origin of neutrino masses generated by type-III and type-I seesaw mechanisms have been discussed in [37,38]. Collider phenomenology of the heavy triplet fermions from such models have been studied in [39][40][41]. An inverse seesaw realization under the type-III seesaw framework has been studied in [42] involving a singlet hyperchargeless fermion. Another type of type-III seesaw was studied in [43] where an extra U(1) gauge group has been introduced to the SM under the anomaly free scenario [44][45][46].
The renormalization group evolution of the effective neutrino mass matrix in the SU(2) L triplet fermion extended SM with emphasis on the threshold effects has been studied in [47]. In the type-III seesaw scenario with degenerate heavy triplets the impact of the renormalization group evolution in the context of perturbativity bounds and vacuum stability of the scalar potential has been studied in [48]. Adding two SU(2) L triplet fermions with zero hypercharge has been studied to probe electroweak vacuum stability with the nonzero neutrino mass, naturalness and lepton flavor violation in [49]. A simple realization of the triplet fermion under SU(2) L with a general U(1) extension of the SM has been discussed in [50] where a neutral gauge boson (Z ) plays an important role for the triplet fermion in addition to the SM gauge boson mediated processes. Over the years at different center of mass energies and integrated luminosities, LHC is also searching for the fermions triplet under the SU(2) L group in [51][52][53][54][55][56][57][58][59] from SM gauge mediated processes using different flavor structures of the Yukawa interaction involving the triplet fermion, SM lepton and Higgs doublets.
Apart from the collider searches, Lepton Flavor Violating (LFV) [60] and non-unitarity effects in the type-III model have been studied in [61,62]. In this context it is necessary to mention that such studies have been performed in the context of the type-I seesaw scenario [63][64][65][66][67][68][69][70]. The bounds on the light heavy neutrino mixings from the Eletroweak Precision Data (EWPD) have been studied in [71,72] which can be considered as the upper limits to constrain the limits on the mass-mixing plane for the triplets.
In this paper we study the production of the TeV scale triplet fermions at electronpositron (e − e + ) collider. At the e − e + collider the production of the neutral multiplet of the triplet fermion will take place through the s and t channel process in association with the SM leptons. Similarly the charged multiplets of the triplet can be produced from the s and t channel processes in association with the SM leptons, however, they can be produced in pair from the s channel process. If the triplets are in the TeV scale and heavy they can sufficiently boost the decay products such as SM gauge bosons (W, Z) and the SM Higgs (h) which can further produce fat jets from the leading hadronic decay modes. Such a fat jet signature can have a distinctive nature to separate the signal from the SM backgrounds. Following this we study the allowed parameter space in the mass-mixing plane from the associated production of the triplets with SM leptons. We also study the significance of several final states coming from the pair production of the charged multiplets. At the e + e − collider we consider two center of mass energies √ s = 1 TeV and 3 TeV with integrated luminosities at L = 1 ab −1 , 3 ab −1 and 5 ab −1 , respectively. In this context we mention that in the seesaw scenario a variety of final states have been studied for the linear collider in the literatures [73][74][75][76][77][78][79][80][81][82][83]. We also study the production of the neutral and charged multiplet of the triplet fermion at the electron-proton (e − p) collider at different center of mass energies, √ s = 1.3 TeV, 1.8 TeV and 3.46 TeV. As we are concentrating in the heavy mass range of the triplet fermions we expect that their decay products will be sufficiently boosted to probe the mass mixing plane.
To do this we fix the collider energy at √ s = 3.46 TeV (FCC-he) with the luminosity L = 1 ab −1 , 3 ab −1 and 5 ab −1 , respectively. Studies on the e − p collider considering the seesaw scenario have been performed in [84][85][86]. In both of these colliders to identify the fat jet from the boosted decay products of the heavy triplet fermions hence we study the signals and the SM backgrounds. In this context we mention that in the e − p collider testing the Majorana nature of a fermion could be very interesting because it can show a lepton number violation signature distinctively. Such a process for the seesaw scenario has been studied in [80,83].
The paper is organized in the following way. In Sec. 2, we discuss the model and the interactions of the triplet fermions with the SM particles. In Sec. 3 we study various production processes of the charged and neutral multiplets of the triplet fermion at the e − e + and e − p colliders. In Sec. 4 we discuss the complete collider study of various possible final states and calculate the bounds on the mixing angles in the mass-mixing plane. We also compare the results with the existing constraints. In Sec. 5 we discuss the calculated bounds. Finally, in Sec. 6 we conclude.

MODEL
The type-III seesaw model is a simple extension of the SM with an SU (2) L triplet fermion (Σ) with zero hypercharge. A three generation triplet fermion of this kind can be introduced in this model to generate the neutrino mass. For simplicity we suppress the generation indices. The relevant part of the Lagrangian can be written as where the first term is the kinetic interaction of the triplet and D µ represents the covariant derivative. L SM is relevant part of the SM Lagrangian. In the second term of Eq. 1, M Σ represents the triplet mass parameter. For simplicity we consider M Σ as a real parameter and the triplets are degenerate in nature. Therefore M Σ is a real diagonal matrix. Y in the third term of Eq. 1 is the Yukawa coupling between the SM lepton doublet ( L ), SM Higgs doublet (H) and the triplet fermion (Σ). In this analysis we represent the the relevant SM candidates in the following way After the symmetry breaking φ 0 acquires the vacuum expectation value and we can express it as φ 0 = v+h √ 2 with v = 246 GeV. The explicit form ofΣ and its charge conjugateΣ c ≡ CΣ T (C is the charge conjugation operator) are given bỹ whereas D µ is given by To study the mixing between the SM charged leptons and the charged multiplets of the triplet fermions, it is convenient to express the four degrees of freedom of each of the charged multiplets of the triplet fermions in terms of a single Dirac spinor as Σ = Σ − R + Σ +c R . The neutral multiplets of the triplet fermions are two component fermions. Therefore they have the two degrees of freedom. Finally mixing with the SM light neutrinos, these neutral fermions generate tiny neutrino mass through the seesaw mechanism after the electroweak symmetry breaking. With this convention we re-write the Eq. 1 as After the electroweak symmetry breaking we can derive the mass matrix for charged and neutral sectors. The mass term of the charged leptons shows the usual nature of the Dirac particles and it can be written as where m represents the Dirac mass of the SM charged lepton coming from the well known SM Lagrangian. Similarly the mass term for the neutral fermions can be given as In case of a Dirac mass, the charged lepton mass matrix can be diagonalized by a bi-unitary transformation which can be written as  The transformation matrices in Eq. 8 can be written as The symmetric neutral lepton mass matrix can be diagonalized by a single unitary which can be written as  The transformation matrix in Eq. 10 can be written as In the above expressions and U PMNS is the lowest order unitary neutrino mixing matrix from [28,60] in accordance with [61]. According to [28,60] and are the small quantities where we can neglect the effects of the higher powers (above 1) of them in the calculations. For a three generation case U L , U R and V are the 6 × 6 unitary matrices. The neutrino mass can be generated by the seesaw mechanism after the diagonalization of the neutral lepton mass matrix in Eq. 7 can be written as Due to the Eqs. 9 and 11, a mixing parameter between the SM lepton and the triplet is generated which affects the interaction of the triplet fermion with the SM leptons through the W , Z and H bosons. The modified charged current (CC) interaction can be written as 13) and the modified neutral current (NC) interaction for the charged sector can be written as The modified NC interaction for the neutral sector of the leptons can be written as where θ W is the Weinberg angle or weak mixing angle. The complete NC interaction can be written as L NC = L 1 NC + L 2 NC . Finally we write the interaction Lagrangian of the SM leptons, charged and neutral multiplets of the the triplet fermions with the SM Higgs (h) boson. The interaction between the charged sector and the h can be written as and that same between the neutral sector and the SM Higgs can be written as The quantities m and m ν are the SM charged lepton and tiny light neutrino mass matrices which are real and diagonal, too. The effect of these masses in the collider study will be negligible. Therefore in our further analyses we do not consider the effects coming from them. The complete Higgs interaction can be written as L H = L 1 H + L 2 H . We want to make a comment that the general expressions calculated independently from Eq. 6 to Eq. 17 in this section match exactly with the expressions given in [28,60]. The charged multiplets of the triplet fermions can interact with photons (A µ ). The corresponding Lagrangian derived from Eq. 5 can be written as At this point we want to mention that in the rest of the text we express the light- which can easily be obtained from the expressions given in the interactions between Eq. 13 to Eq. 17. The limit on V for the electron flavor in the type-III seesaw scenario is V e < 0.019 from the electroweak precision data as stated in [26,72]. For the time being we are considering the e − e + and e − p colliders therefore we probe V e from a variety of the final states including electron.
Using Eq. 13 to Eq. 17 and the expression for the mixing we calculate the partial decay widths of the neutral multiplet (Σ 0 ) of the triplet fermion as respectively. M W , M Z and M h are the W , Z and Higgs boson masses respectively in SM.
If the mass splitting (∆M ) between the charged (Σ ± ) and neutral (Σ 0 ) multiplets induced by the SM gauge bosons is of the order of the pion mass [87], Σ ± can show the following additional decay modes which are independent of the free parameters. The value of the Fermi Constant, G F is 1.1663787 × 10 −5 GeV −2 , the CKM matrix element V ud is 0.97420 ± 0.00021 and the π meson decay constant, f π , can be taken as 0.13 GeV from [88].     for V e = V µ = 0.016 and V τ = 0..

TRIPLETS
The production of the triplet fermion at the e − e + and e − p colliders will be followed by its interactions with the SM gauge bosons as described in the Sec. 2. At the e − e + collider the triplet fermion can be produced from the s and t channel processes being mediated by the photon (γ), W boson and Z boson respectively. The corresponding production modes are given in Fig. 3. The associate production of Σ 0 and Σ + (Σ − ) with the SM leptons (electron and neutrino) are suppressed by the light-heavy mixing square (|V e | 2 ). These production In the lower panel we show the production cross section as a function of M Σ fixing the center of mass energy at 1 TeV (lower, left) and 3 TeV (lower, right) respectively.
The triplet fermion can be produced at the e − p collider in association with a jet through the t channel process exchanging the W and Z bosons. The production process is shown in Fig. 5. We consider this process at √ s = 1.3 TeV, 1.8 TeV and 3.46 TeV, respectively.
At the e − p collider the electron beam energy has been fixed at 60 GeV but proton beam energies are 7 TeV, 13.5 TeV and 50 TeV, respectively. The production processes of the Σ 0 j and Σ − j are suppressed by the corresponding light-heavy mixing. The production cross sections are shown in Fig. 6 as a function of the M Σ for fixed center of mass energy ( √ s).     To calculate the associate production of the Σ ± , Σ 0 with the SM leptons we consider V e = 0.019 [26,72], however, the pair production of Σ + Σ − is direct.
After the production of Σ 0 and Σ ± at the e − e + and e − p colliders, the particles will decay into the respective modes according to Eqs. 19 and 20. In this article we consider the heavier triplet fermion so that it can sufficiently boost the decay products to form a fat jet hence we study the signal and the SM backgrounds.    V e = 0.019 [26,72] to calculate the production cross sections.

COLLIDER ANALYSIS
To find the discovery prospect we first implement the model in FeynRules [89] framework, generated signal and the SM backgrounds using the Monte Carlo event generator MadGraph5-aMCNLO [90]. For the subsequent decay, initial state radiation, final state radiation and hadronisation, we have used Pythia6 [91] for e − p and Pythia8 [92] for e − e + colliders respectively. We have considered the high mass regime of the triplet so that the daughter particles from triplet can be sufficiently boosted. Due to large mass gap between triplet fermion and the SM gauge bosons (W, Z) and SM Higgs (h), the hadronic decay modes of W , Z and h can be collimated so that we will have a single jet called fat jet (J).
The fat jet topology is a very powerful tool to significantly reduce the SM backgrounds. We perform the detector simulation using Delphes-3.4.1 [93]. The detector card for the e − p collider has been obtained from [94]. We use the ILD card for the e − e + collider in Delphes. In our analysis the jets are reconstructed by Cambridge-Achen algorithm [95,96] implemented in Fastjet [97,98] package with the radius parameter as R = 0.8. We study the production of the triplet fermion and its subsequent decay modes at the e − e + and e − p colliders respectively. We consider two scenarios at e − e + collider where the center of mass energies are √ s = 1 TeV and 3 TeV. For e − p collider, we consider the case of where √ s = 3.46 TeV.
In this case the electron and proton beam energies are 60 GeV and 50 TeV respectively.
At the e − e + collider the following set of signals after the production of the triplet fermion can be found: where J is the fat jet coming from boosted W boson. The corresponding Feynman diagram is shown in Fig. 7.
from the boosted SM Higgs decay. The corresponding Feynman diagram is shown in Fig. 13.
4. e − e + → Σ + Σ − , Σ ± → W ± ν, W ± → J. The corresponding Feynman diagram is shown in Fig. 17. At e − p collider we study the signal e − p → Σ 0 j followed by Σ 0 → e ± W ∓ , W ∓ → J. The corresponding Feynman diagram is shown in Fig. 19. As the production cross section for Σ 0 j at ep collider quickly decreases with increasing triplet mass, we decide to study the signal coming from the dominant decay mode Σ 0 → eW .
For the analyses of the signal and background events we use the following set of basic cuts: 1. electrons in the final state should have the following transverse momentum (p e T ) and pseudo-rapidity (|η e |): p e T > 10 GeV, |η e | < 2.5 (for ep collider, |η e | < 5).
2. jets are ordered in p T and they should have p j T > 10 GeV and |η j | < 2.5 (for e − p collider, |η j | < 5).
3. leptons should be separated by ∆R > 0.2. 4. the jets and leptons should be separated by ∆R j > 0.3. 5. fat Jet is constructed with radius parameter R = 0.8. The final state e ± +J +p miss T arises from the production of e − e + → Σ 0 ν and the subsequent decay of Σ 0 to its dominant channel e ± W ∓ at the e − e + collider. The corresponding Feynman diagram is given in Fig. 7. The W boson can further decay to pair of jets. As we are considering heavy mass region of the triplet fermion, the W boson will be boosted and its hadronic decay products, jets, will be collimated such that they can form a fat jet (J).
There are a number of SM process like ν e eW , W W , ZZ and tt which can mimic this final state which are the significant SM backgrounds. Among these channels ν e eW and W W Due to the heavy mass of the triplet fermion, the leading lepton and the fat jet p T distributions for the signal will be in the high values than the SM backgrounds. Hence the high p T cut for leading lepton and fat jet will be effective to reduce SM background. At this point we would like to mention that the same final state can also be obtained from e − e + → Σ ± e ∓ . We have found that the p T distribution for the electron is mostly in the low momentum range for this channel. The application of the high p T cut for the electron applied in the Σ 0 ν channel will significantly cut this channel out so that its contribution becomes negligible. The high p T cut for the electron from the Σ 0 ν process is required because the electron in this process is coming from the heavy triplet in contrary to the Σ ± e ∓ process.
Hence in the further analyses we neglect events from the Σ ± e ∓ process.
The polar angle variable for the electron cos θ e in Fig. 9 is defined as θ e = tan −1 p e T p e z where p e z is the z component of the three momentum of the electron. At the e − e + collider the polar angle cut is very effective to reduce the SM backgrounds.
To study this process we have chosen the the triplet mass M Σ = 800 GeV-950 GeV for √ s = 1 TeV and M Σ = 800 GeV-2.9 TeV for √ s = 3 TeV. In view of the distributions in Figs. 8-12, we have used the following set of advanced selection cuts to reduce the SM backgrounds further: • Advanced cuts for M Σ = 800 GeV-900 GeV at the √ s = 1 TeV e − e + collider after the detector simulation: 1. polar angle of the lepton and the fat jet |cos θ e | < 0.9. 2. transverse momentum for the fat jet p J T > 300 GeV.
3. transverse momentum for the leading lepton p e ± T > 300 GeV.
• Advanced cuts for M Σ = 800 GeV-2.9 TeV at the √ s = 3 TeV e − e + collider after the detector simulation: 1. polar angle of the lepton and the fat jet |cos θ e | < 0.9.      2. transverse momentum for J b , p J b T > 300 GeV for 800 GeV ≤ M Σ ≤ 1.5 TeV and p J b T > 500 GeV for 1.6 TeV ≤ M Σ ≤ 2.9 TeV.
3. fat-b mass, m J b > 100 GeV. We have shown the cut flow for two benchmark points M Σ = 1 TeV and 2 TeV at √ s = 3 TeV in Table. IV and V, respectively. In this section we discuss the potential to test Σ + in e − e + → e − Σ + mode (conjugate process implied) followed by Σ + → e + Z → e − e + J at the √ s = 3 TeV e − e + collider. The corresponding Feynman diagram is given in Fig 15. There exist several SM backgrounds for this process. The dominant background arises from Z/γjj whereas, tt, W W Z, W W jj TeV and 2.2 TeV respectively. Note that as in before these distributions are generated with basic cuts.
Our focus of interest is 800 GeV ≤ M Σ ≤ 2.9 TeV. Based on the normalized distributions in Fig. 16, we have applied following set of advanced selection cuts: 1. the fat jet transverse momentum is required to be p J T > 150 GeV.
3. the p T of the leading electron is chosen to be p e 1 T > 350 GeV, while p T of the subleading electron, p e 2 T assumed to be > 200 GeV for 900 GeV ≤ M Σ ≤ 1.4 TeV, > 100 GeV for 1.6 TeV ≤ M Σ ≤ 2.4 TeV and, > 40 GeV for 2.6 TeV ≤ M Σ ≤ 3.0 TeV.
For the chosen M Σ range the subleading recoils against the Σ + . Therefore we found that for the heavier Σ + the significances improves significantly with lower p e 2 T . The impact of the event selection cuts are provided in Table VI for M Σ = 1 TeV for illustration.     ab −1 luminosities, however, the impressive significance well above 3-σ can reach at 1 ab −1 luminosity.
At this point we mention that we have also studied the 2J b + e − e + signal from Fig. 17 where Σ ± decays into ± h. Each of h boson can further decay into a pair of collimated b-jets. These collimated b jets form a fat b-jet. Using a 70% b-tagging efficiency for the signal and SM backgrounds we give only the significance of this process just as a reference in right panel of Fig. 18. The significance of this process can prospectively reach above the 5-σ level at the 1 ab −1 luminosity at 3 TeV e − e + collider. Higher luminosities at 3 ab −1 and 5 ab −1 can improve the significance of the heavier triplets leading to a prospect of above 5-σ significance. In this context we mention that there is J b + e − e + final state which can be obtained from the single Higgs production as given in Fig. 13. Due to the single fat b-jet this channel is less efficient compared to the 2J b + e − e + signal. e − + J + j 1 . The first one is Lepton number violating (LNV) and second one is Lepton number conserving (LNC) process at the e − p collider. We have combined both these LNV and LNC processes to obtain the final state e ± + J + j 1 .
We expect the LNV signal to be almost background free, unless some e + + jets events appear from the radiation effects, which, one expects to be negligible. For the LNC channel,  It is difficult to obtain fat jet for the background process ej because of the t channel exchange of the Z boson and photon. Initial and final state radiations can give low energy jets which can produce soft fat jet. Therefore ej background can completely be reduced with high p e T , p J T cuts and fat jet mass 70 GeV ≤ m J ≤ 90 GeV. These cuts will not be enough to reduce the irreducible backgrounds coming from the process ejj and ejjj. However, both of these backgrounds can be reduced using the invariant mass cut of the Σ 0 . As Σ 0 decays according to Σ 0 → eW, W → J, the invariant mass of eJ system should be close to mass of the Σ 0 . Therefore an invariant mass window cut |M eJ − M Σ | ≤ 20 GeV will be extremely useful to reduce these two SM backgrounds further. We have shown the cut flow for two benchmark points M Σ = 1 TeV and 1.5 TeV in Table. VIII and IX.

DISCUSSIONS
After studying the signals and the SM backgrounds for the triplet fermion production at the e − e + collider and e − p at different energies and luminosities we calculate the bounds on the mass mixing plane at 3-σ and 5-σ significance. To calculate the bounds on the mixings from the e − e + collider we use two different center of mass energies like 1 TeV and 3 TeV. To do the same at the e − p collider we use 3.46 TeV center of mass energy. At these colliders we have used 1 ab −1 , 3 ab −1 and 5 ab −1 luminosities respectively. We compare our  results with the bounds obtained from [72]. We use the bounds on the mixing V e = 0.019 and the universal bounds 0.016 as studied from the Electroweak Precision Data (EWPD).
We represent the bounds as EWPD-e and EWPD-U in Figs. 21 and 22 by the horizontal dot-dashed and dotted lines respectively.
The bounds obtained from the e − e + collider studying a variety of final states are shown in Fig. 21. We have studied the e ± + J + p miss T final state from Fig. 7 at the 1 TeV (top, left) and 3 TeV (top, right) e − e + colliders. One can probe the mixing up to 8 × 10 −5 at the 1 TeV collider at 3-σ significance with the luminosity of 5 ab −1 . With the same luminosity, the bounds remain below the EWPD-e and EWPD-U up to 900 GeV (EWPD-U) and 945 GeV (EWPD-e) at 5-σ significance. The same has been studied at the 3 TeV e − e + collider where the results can impressively be improved even with 1 ab −1 luminosity with 5-σ significance up to 2.5 TeV triplets which can be further improved at the higher luminosity probing the heavy triplets up to 2.9 TeV. We have studied the J b + p miss T final state from Fig. 13 and the corresponding bounds at the 3 TeV e − e + collider are given in Fig. 21 (bottom, left). The lowest mixing 3.5 × 10 −5 could be reached 1.725 TeV heavy triplet mass. At the collider energy threshold the cross section decreases at the e − e + collider which in turn does not help the further heavier triplets in getting lower bounds, however, analyzing the signal and the corresponding SM backgrounds we find that sightly heavier triplets than 2.5 TeV can be probed with 1 ab −1 luminosity with 5-σ significance. Higher luminosity will make the results better. We have studied e + e − + J final state from Fig. 15 and the bound obtained from this final state have been shown in Fig. 21 (bottom, right) where the best results can be obtained at the 3 TeV e − e + collider with 5 ab −1 luminosity just below the limits obtained from the EWPD at 3-σ significance.
Bounds obtained from the e − p collider have been shown in Fig. 22 studying the e ± +J +j 1 final state from Fig. 19 which has a visible particle in the final state from the leading decay mode compared to the invisible one from the other possibilities. In this case we find that the heavy triplets are favored to study the fat jets due to their better efficiency for boosting the W boson coming from the leading decay mode of the Σ 0 . Comparing our results with the bounds obtained from the EWPD we find that higher luminosity with 3 ab −1 and 5 ab −1 will be useful to probe heavier triplets. Triplets heavier than 1.4 TeV can be probed up to the mixings 3 × 10 −5 at 3-σ significance and 10 −4 at the 5-σ significance at 5 ab −1 luminosity respectively. The bounds obtained for M Σ > 1. 4 TeV is better than those obtained for M Σ < 1.4 TeV. This is because of the heavier triplets which produced fat jets better than the comparatively lighter ones. ab −1 at 3σ and 5σ significance respectively from the final state e ± + J + j 1 at √ s = 3.46 TeV of the e − p collider.

CONCLUSION
We have studied the triplet fermion initiated seesaw model which is commonly known as the type-III seesaw scenario which is responsible for the light neutrino mass generation through the seesaw mechanism. As a result a light-heavy mixing appears in the model. We consider the production of the charged and neutral multiplets of the triplet fermion at the e − e + and e − p colliders at different center of mass energies. Being produced in association with the SM particles and as well as in pair the charged and neutral multiplets can decay into a variety of final states.
In this article we consider mostly the leading decay modes of the triplets and some nextto-leading modes, too. As the mass of the triplet is a free parameter, we consider the heavier mass of the triplet which can sufficiently boost its decay products. As a results the decay products including SM gauge bosons and Higgs boson can manifest fat jet through their leading hadronic decay modes. Finally we study a variety of final states and the SM backgrounds to probe the light-heavy mixing as a function of the triplet mass for different center of mass energies at the above mentioned colliders using different integrated luminosities.
Comparing our results with the bounds on the light-heavy mixing obtained from the electroweak precision test results we find that the heavier triplets can be successfully probed and the bounds on the light-heavy mixing as a function of the triplet mass can be better than the results from the electroweak test.
Apart from the associate production of the triplet we study the pair production of the heavy triplets at the e − e + collider followed by the decays into the SM bosons. Due to the heavier mass, the triplets can easily boost the daughter particles so that the hadronic decay products of the SM gauge bosons among them can manifest fat jet signatures which can efficiently segregate the signals from the SM backgrounds leading to the test above 5-σ significance in the near future.