Heavy Majorana Neutrino Production at Future $ep$ Colliders

The heavy singlet Majorana neutrinos are introduced to generate the neutrino mass in the framework of canonical type-I seesaw mechanism. The phenomena induced by the heavy Majorana neutrinos are important to search for new physics. In this paper, we explore the heavy Majorana neutrino production and decay at future $e^{-}p$ colliders. The corresponding cross sections via $W$ and photon fusion are predicted for different collider energies. Combined with the results of the heavy Majorana neutrino production via single $W$ exchange, this work can provide helpful information to search for heavy Majorana neutrinos at future $e^{-}p$ colliders.


Introduction 2 The model
Within the standard model, neutrinos are massless in the absence of right-handed neutrinos. However, recent neutrino oscillation experiments have clearly shown that neutrinos are massive. In order to explain the smallness of neutrino masses, many new physics models have been proposed. A simple extension of the standard model is to introduce three heavy right-handed neutrino singlets N R and the gauge-invariant Lagrangian relevant for the neutrino masses can be written as where ℓ L andH ≡ iσ 2 H * respectively denote the left-handed lepton doublet and Higgs doublet, and N R the right-handed neutrino singlet. Y ν is the 3 × 3 neutrino Yukawa coupling matrix and M R the symmetric right-handed Majorana neutrino mass matrix. After the spontaneous gauge symmetry breaking, the neutrino mass terms appear as Here ( Moreover, the relation between the neutrino flavor eigenstates ν α (for α = e, µ, τ ) and the mass eigenstates ν i and N i (for i = 1, 2, 3) can be given by  Therefore the standard weak charged-current interaction Lagrangian of leptons in terms of the mass eigenstates can be written as It is worth mentioning that we have already chosen the basis where the flavor eigenstates of three charged-leptons are identified with their mass eigenstates. The matrix V in Eq. (6) is the so-called Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [29,30], denotes the mixing between charged-leptons and light Majorana neutrinos and can be measured from the oscillation experiments. The standard parametrization of the matrix V can be expressed as [31] where c ij ≡ cos θ ij , s ij ≡ sin θ ij , δ is the Dirac CP violation phase and ρ, σ are two additional Majorana CP violation phases. The matrix R indicates the mixing between charged-leptons and heavy Majorana neutrinos, which can be determined from the 0νββ-decay experiments or possible collider experiments. In our previous work [32], we proposed a simple scenario to explicitly break the S 3L × S 3R flavor symmetry in the framework of the canonical seesaw model, where we can naturally explain the realistic lepton mass spectra and flavor mixing pattern as well as the cosmological matterantimatter asymmetry via resonant leptogenesis. In that case, however, the mass scale of M R is too high to be detected experimentally. In order to probe the heavy Majorana neutrino production signal, the scale of M R must be at the order of O(1) TeV or smaller and the strength of matrix R, which scales as O(M D /M R ), must be large enough. In Ref. At present, the constraint on the mixing between heavy Majorana neutrinos and electrons can be derived from the 0νββ-decay experiments [24,25] For the mixing between heavy Majorana neutrinos and muons, the most stringent bound comes from the LHC experiments [34,35] From the lepton universality tests [36], the mixing between heavy Majorana neutrinos and chargedleptons can be restricted to As shown in Ref. [9,37], the total decay width of the heavy Majorana neutrino can be expressed approximately as

Heavy Majorana neutrino phenomena at ep colliders
We start by considering the process The relevant process at the parton level ( Fig. 1) is where p i (for i = 1, · · · , 7) is the four-momentum of the corresponding particle. The photon is emitted from the electron and can be described by the photon density function [38] Here x = E γ /E e with E γ and E e the energies of the photon and electron, respectively. α is the fine structure constant and m e the mass of electron. Q 2 min = m 2 e x 2 /(1 − x) and Q 2 max = (θ c E e ) 2 (1 − x) + Q 2 min with θ c the cut of the electron scattering angle. Figure 1: Feynman diagrams at the parton level for the process γq → ℓ ± α ℓ ± β + X.
The cross section for the process in Eq. (12) can be written as where f q/p (x 2 , µ 2 ) is the parton distribution function with x 2 the energy fraction of q, and µ the factorization scale. Here, we employ the CT14QED [39] for the photon distribution function and parton distribution functions in proton.σ is the partonic cross section Hereŝ = x 1 x 2 s is the flux factor with √ s the electron-proton center-of-mass energy. dLips 5 represents the five-body Lorentz invariant phase space of the final particles, and |M| 2 is the squared scattering amplitude averaged (summed) over the initial (final) particles. For the convenience of the discussions on the numerical results, all the input parameters used in our numerical analysis are listed as follows, In this work, we only consider the contribution of a single heavy Majorana neutrino and concentrate on the di-muon production channel. We obtain the cross section for the process in Eq. (12) at LHeC with √ s = 1.3 TeV, FCC-ep with √ s = 3.5 TeV and ILC ⊗ FCC with √ s = 10 TeV. The cross sections in final states with like-sign dileptons are shown in Fig. 2 as a function of the heavy Majorana neutrino mass. The difference between σ(e − p → e − µ + µ + + X) and σ(e − p → e − µ − µ − +X) can be attributed to the role of parton distribution function f q/p (x, µ 2 ) and induce the charge asymmetry. To investigate this charge asymmetry, we define The numerical results of δ as a function of m N are listed in Table 1.           We also investigate the process e − p → e − µ − µ + + X of the unlike-sign dileptons, the corresponding cross sections as a function of m N are displayed in Fig. 3. For this process, the standard model process for the µ − µ + production via a Z 0 or a virtual photon is the dominant background and can be greatly reduced by the constraint for the invariant mass of the µ − µ + pair.
Similar as Eq. (12), we study the process e − + p → ν e + ℓ − α + N + X → ν e + ℓ − α + ℓ ± β + X, its cross section can be written as where the photon is emitted from the proton and can be described by the photon distribution function f γ/p (x, µ 2 ). The results of the corresponding cross sections related to µ − µ ± channels are shown in Fig. 4. In the following, taking the process Eq. (12) as an example, we investigate the reconstructed invariant mass distributions and transverse momentum distributions of the final state like-sign dileptons and jets for the process e − p → e − µ ± µ ± + X. The invariant mass of the heavy Majorana neutrino can be reconstructed from the four-momenta of the final state charge-leptons and jets. Since the final state charged-leptons (ℓ ± α and ℓ ± β ) are indistinguishable, we define the normalized differential distribution 1/σdσ/dM ℓjj = 1/σ(dσ/dM ℓαjj + dσ/dM ℓ β jj )/2 for the reconstructed invariant mass. In Fig. 5, the results of the normalized invariant mass distributions 1/σdσ/dM ℓjj for various heavy Majorana neutrino mass are displayed. The peak positions imply the mass of heavy Majorana neutrino and can be reconstructed effectively. We also calculate the normalized distributions for the invariant mass of the lepton pair in Fig. 6. When m N is much lower than m W , e.g. m N = 20 GeV, a peak of m ℓℓ appears due to the resonant production of W boson. Analogously, we define the normalized differential distribution 1/σdσ/dp ℓ,j T = 1/σ(dσ/dp ℓα,j 1 T + dσ/dp ℓ β ,j 2 T )/2 for the transverse momentum of the final state charged-leptons and jets, and display the normalized transverse momentum distributions 1/σdσ/dp ℓ,j T for m N = 60 GeV (Fig. 7).

Summary
The heavy Majorana neutrinos N R are introduced to explain the minor neutrino mass in the framework of type-I seesaw mechanism. Due to the existence of the Majorana neutrino mass term, we can search for the Majorana neutrinos via the lepton-number violating processes, which have been studied in various experiments. In this work, we explore the heavy Majorana neutrino production and decay in the context of W * γ interaction and investigate the related dilepton production process at future e − p colliders. The cross sections for the processes e − p → e − µ ± µ ± +X and e − p → ν e µ − µ ± + X at future LHeC, FCC-ep and ILC⊗FCC are predicted. We further investigate the process e − p → e − µ ± µ ± + X in detail, and obtain several differential distributions. Combined with the results of the heavy Majorana neutrino production via single W exchange, this work can provide helpful information to search for the heavy Majorana neutrinos at future e − p colliders.