CP violation in lepton-number-conserving processes through heavy Majorana neutrinos at future lepton colliders

Small neutrino masses confirmed in the neutrino oscillation experiments indicate the need for new physics beyond the standard model. Seesaw mechanism is an interesting way to extend the standard model for explaining the neutrino masses. In a low-scale type-I seesaw mechanism, the tiny masses of neutrinos can be explained by heavy Majorana neutrino masses. Heavy Majorana neutrinos can lead to lepton-number-violating processes and the induced CP violation can contribute to the baryon asymmetry in the Universe. Heavy Majorana neutrinos can also lead to lepton-number-conserving processes and in this paper, we study the CP violation in lepton-number-conserving processes through heavy Majorana neutrinos at future lepton colliders. New possible observations of CP violation can also be connected to evidences of new physics beyond the standard model.


I. INTRODUCTION
In the standard model (SM), the neutrinos are strictly massless due to the absence of righthanded chiral states and the requirement of SU (2) L gauge invariance and renormalizability.The tiny masses of neutrinos confirmed in neutrino oscillation experiments show that the SM may not be complete.In fact, the SM has some open problems, such as the flavor puzzle, the existence of the dark matter (DM), the baryon asymmetry in the Universe.However, by introducing heavy Majorana neutrinos [1][2][3], we can solve some of them.A sector of Majorana neutrinos connected to the SM by mixing with the SM neutrinos could exhibit additional CP violation needed to explain the baryon asymmetry in the Universe.Heavy Majorana neutrinos can be used to explain the tiny masses of neutrinos via the interesting type-I seesaw mechanism [4][5][6].
In this work, we introduce three generations of right-handed heavy Majorana neutrinos in SM and the Dirac neutrino mass terms will be generated after spontaneous gauge symmetry breaking.
The heavy Majorana neutrinos will lead to processes with violation of lepton number by two units ∆L = 2, such as the neutrinoless double-beta decay (0νββ) [7,8].It is an interesting process that the Majorana phase may induce additional CP violation effect.In previous works [9][10][11], we have studied the lepton-number-violating (LNV) processes with heavy Majorana neutrinos and the induced CP violation.Not only that, the heavy Majorana neutrinos can also lead to lepton-numberconserving (LNC) processes, in this work, we study several interesting LNC processes caused by heavy Majorana neutrinos at future lepton colliders.The CP violation stems from the interference of contributions from different heavy Majorana neutrinos, and even one heavy Majorana neutrino can lead to CP violation in these processes by the intereference of contributions from the s-channel processes and the t-channel processes.We also investigate the prospects for searching for these heavy Majorana neutrinos at future lepton colliders like the Muon Collider (MuC) [12] and the International Linear Collider (ILC) [13].We analyse the processes e + e − → νe N i → νe e − q q′ at e + e − collision with ILC running at 500 GeV, 1000 GeV, 3000 GeV and the processes µ + µ − → νµ N i → νµ µ − q q′ at µ + µ − collision with MuC running at 3000 GeV and 10 TeV, where N i represent three generations of heavy Majorana neutrinos This paper is organized as follows.In Section II, we reviewed the model we used to describe heavy Majorana neutrinos.In Section III, we analyse the CP violation in processes e + e − → νe N i → νe e − q q′ and µ + µ − → νµ N i → νµ µ − q q′ .The possibility for measuring CP violation at future lepton colliders is studied in Section IV.Finally, a short summary is given in Section V.

II. HEAVY MAJORANA NEUTRINOS BEYOND THE SM
The heavy neutrinos can only interact with the SM through mixing effects, which come from a mass matrix between the electroweak doublet neutrinos and Majorana neutrinos.In this work we extend the standard model with three right-handed heavy Majorana neutrinos.The Lagrangian of the model [14] we used in our process is given by: where L N is a sum of kinetic and mass terms for heavy Majorana neutrinos: where i = 1, 2, 3, stand for three heavy Majorana neutrinos.The L W N ℓ corresponds to heavy neutrino interactions with a W boson: The L ZN ν to interactions with a Z boson: then the L HN ν to interactions with a Higgs boson: Finally we write the weak charged-current interaction Lagrangian as: Here V ℓm is the neutrino mixing matrix that can be measured from the neutrino oscillation experiments.The R ℓi indicates the mixing between heavy Majorana neutrinos and charged-leptons, which can be parameterized as [15].
The mixing between heavy Majorana neutrinos and Z boson is different from those between heavy neutrinos and W bosons, so the phases ϕ ℓi should be set to different values.In this work we have at least 6 different phases as free parameters.For convenience, we set the phases between heavy neutrinos and Z boson to ϕ ℓi = 0, then we will set three free parameters of phases ϕ ℓ1 = ϕ a , ϕ ℓ2 = ϕ b , ϕ ℓ3 = ϕ c which are from mixing between heavy neutrinos and W bosons to study each process and induced CP violation at ILC and MuC.Now we give the mixing relations between the neutrino flavor eigenstates and mass eigenstates as follows [17] : In this work, we consider the case that the heavy Majorana neutrinos are nearly degenerate, so we and we set m N 1 in the range of 300 GeV-3000 GeV for ILC, 300 GeV-10 TeV for MuC according to Ref. [14] and Ref. [16], for these ranges, at ILC, the parameters of the mixing between heavy neutrinos and leptons are set as 0003, and at MuC, the mixing parameters |R µi | 2 are obviously dependent on m N 1 , we set the values of |R µi | 2 according to the constraint in Fig. 10 in Ref. [16].
We simplify the widths of heavy Majorana neutrinos Γ N 1 , Γ N 2 , Γ N 3 according to Refs.[17,18] in expression: In the range of the mass m N 1 we consider, we take A = 4, so that the widths can be simplified as: We set the Cabibbo-Kobayashi-Maskawa (CKM) matrix as diagonal with unit entries for simplicity in our calculation.We put all these parameters in the model with F EYN R ULES [19], the M ATHEMATICA package to calculate Feynman rules associated with the Lagrangian of a given model, and use the model to generate the cross sections of the process with MadGraph5 aMC@NLO [20].

III. CP VIOLATION IN LEPTON-NUMBER-CONSERVING PROCESSES CONTRIBUTED BY HEAVY MAJORANA NEUTRINOS
The Feynman diagrams for our process are given in Fig. 1, where ℓ ± = e ± , µ ± .We can see there are two main diagrams, one is an s-channel diagram with Z boson rare decay and the other one is a t-channel diagram with W boson exchange.The N i respect three heavy Majorana neutrinos N 1 , N 2 , N 3 .We take q, q ′ = c, s or u, d.The total cross section of this process can be expressed as where the M ℓ + ℓ − → νℓ ℓ − q q′ 2 represents the squared matrix elements averaged (summed) over the MuC, and the full line represents √ s = 10 TeV.We can see that the cross section decrease quickly as the m N 1 increase.The difference between the rates of ℓ + ℓ − → νℓ ℓ − q q′ and ℓ − ℓ + → ν ℓ ℓ + qq′, where ℓ = e, µ may induce the CP asymmetry, which can be defined as As mentioned before, we have three CP phases ϕ a , ϕ b , ϕ c as free parameters, they will cause CP violation in the processes ℓ + ℓ − → νℓ ℓ − q q′, for the case that if there is only one generation of heavy neutrinos, its s-channel diagram will give a CP phase but the t-channel diagram will not, so that this heavy Majorana neutrino will cause CP violation in this proces.We study the CP violation for cases that there is only one heavy Majorana neutrino, two generations of heavy Majorana neutrinos, and three generations of heavy Majorana neutrinos respectively.The results are in following pictures.We show the results of CP violation as functions of mass m N 1 and CP phase ϕ a for case with only one generation of heavy Majorana neutrino here.find that the CP violation is independent of the heavy Mjaorana neutrino mass m N 1 , in Fig. 4(a) and Fig. 4(c) we can see that the CP violation is related to CP phase, and the maximum value can reach near 1.25 × 10 −4 at ϕ a = ±π/2.This shows that only one generation of Majorana neutrino can lead to CP violation in LNC process even though it is small, which is different from that in LNV process, and the results for µ + µ − case is not the same as that in e + e − case, the function have the same shape but the values in µ + µ − case are much smaller than those in e + e − , the differences come from the different value of mixing |R µi | 2 we take at ILC and MuC.
The results with two generations of Majorana neutrinos N 1 and N 2 are shown in Fig. 5. Fig. 5(a) violation is nearly the same with that when there are two heavy neutrinos but a little bit higher.
The results show that the feature of CP violation caused by heavy Majorana neutrinos in LNV processes which we studied in previous works [9][10][11] are different from those in the LNC processes, CP violation in LNC processes are obviously smaller but nonzero.In LNV processes, the CP violation is obviously independent on mass of heavy Majorana neutrinos m N 1 , but in LNC process, when there are more than one generation of heavy Majorana neutrino, the CP violation will be influenced by m N 1 , it will decrease as the m N 1 goes up.In LNV processes, the CP violation exists when there are at least two generations of heavy Majorana neutrinos, but in LNC process, only one generation of heavy Majorana neutrino will cause nozero CP violation though it is small.For the case there are three heavy Majorana neutrinos, we search for the maximum value of CP violation when we running all CP phase ϕ a , ϕ b , ϕ c from −π to +π where we take the interval of each point as π/10, and the maximum value of total CP violation will reach GeV, 1000 GeV, 3000 GeV ILC and 3000 GeV, 10 TeV MuC.In order to identify the isolated lepton or jet, we indentify isolated jets and leptons by angular separation, which can be defined as where ∆ϕ ij (∆η ij ) is the azimuthal angle (rapidity) difference of the corresponding particles.We apply some basic acceptance cuts (referred as cut-I) In order to purify the signal, the missing transverse energy is required to satisfy (referred as cut-II) There are too much diagrams of backgrounds in the SM for this process, so we simulate all of them with the SM model by MadGraph5 aMC@NLO, we also simulate the signals via Mad-Graph5 aMC@NLO with the New Physics model generated by F EYN R ULES .The parton shower and hadronization are performed with Pythia-8.2[21].We also give another cut for our signal process, a W boson will decay hadronically, and we can reconstructed it from the two jets (j 1 , j 2 ).Their invariant mass should be closest to m W .This leads to a new cut (referred as cut-III): where j 1 , j 2 are the two jets decayed by W boson.
In this process, for a better cut at background the transverse momentum of jets H T is required to satisfy (referred as cut-IV) It is clear that at 500 GeV ILC, a 3σ discovery can be made for near 440 GeV ≤ m N 1 ≤ 450 GeV with L = 1600 fb −1 , at 1000 GeV ILC, a 5σ discovery can be made for near 650 GeV ≤ m N 1 ≤ 950 GeV with L = 3200 fb −1 , a 5σ discovery can be made for near 2650 ≤ m N 1 ≤ 2950 GeV with L = 4000 fb −1 .It's hard to have a 3σ discovery at 3 TeV, 10 TeV MuC, the results are too small that we don't put the results in this paper.Finally, we give the total cross sections for signal processes in Table .I.

σ 16 L
total [fb] at ILC L = 1.6ab −1 , √ s = 500 GeV 3.48e + e − → νe e − q q′ L = 3.2ab −1 , √ s = 1000 GeV 5.47L = 4ab −1 , √ s = 3000 GeV 6.= 1ab −1 , √ s = 3000 GeV 1.01 × 10 −2 µ + µ − → νµ µ − q q′ L = 10ab −1 , √ s = 10 TeV 0.241 V. SUMMARYThe small neutrino masses show that we need to expand the SM for explaining the tiny neutrino masses.An interesting model is the type-I seesaw mechanism which introduced heavy Majorana neutrinos that can lead to the CP violation in LNV process, the CP violation can give a new source to explain the baryon asymmetry in the Universe via leptogenesis.We have studied the heavy Majorana neutrinos in LNV process and there shold be at least two generations of heavy Majorana neutrinos to generate CP violation by the interference of contributions from different heavy Majorana neutrinos.In this work, we investigate the heavy Majorana neutrinos in an interesting LNC process, and find that only one generation of heavy Majorana neutrino can lead to CP violation, and several features of CP violation in the LNC process are different from that in LNV process.The CP violation in the LNC process is caused by the interference of contributions from different generations of heavy Majorana neutrinos and from the interference of contribution of the s-channel diagram and t-channel diagram.We consider three generations of heavy Majorana neutrinos N 1 , N 2 , N 3 , and the possibility for searching it at future lepton colliders is studied at future lepton colliders 500,1000,3000 GeV ILC, and 3000 GeV,10 TeV MuC.The results show that there are great chances to explore the CP violation effects at future lepton colliders, and the exploring of CP violation can be a probe for studying the underlying new physics.