The leptonic di-flavor and di-number violation processes at high energy $\mu^\pm\mu^\pm $ colliders

The leptonic di-flavor violation (LFV) processes $\mu^\pm \mu^\pm \rightarrow e^\pm e^\pm $, $\mu^\pm \mu^\pm \rightarrow \tau^\pm \tau^\pm $ and the leptonic di-number violation (LNV) processes $\mu^\pm \mu^\pm \rightarrow W^\pm _iW^\pm _j$ ($i,\;j=1,\;2$) at the same-sign high energy $\mu^\pm \mu^\pm $ colliders are studied. The new physics (NP) factors that may play roles in these processes are highlighted by cataloging them into three types. Taking into account the experimental constraints, the processes at $\mu^\pm\mu^\pm$ colliders are computed and the results are presented properly. The results lead to the conclusion that observing the NP factors through the LFV and LNV processes at TeV-energy $\mu^\pm\mu^\pm$ colliders has significant advantages that cannot be achieved elsewhere. Therefore, in developing the techniques of muon acceleration and collisions, the option of building the same-sign muon high-energy colliders should be considered seriously too.


I. INTRODUCTION
Neutrino oscillations and flavor mixing among the three generations of neutrinos have been observed for several years [1].These phenomena definitely mean the neutrinos have tiny but nonzero masses, and unambiguously are evidences of new physics (NP) beyond the standard model (SM).Regarding NP, one natural way for neutrinos to acquire tiny masses is through the so-called 'seesaw mechanisms' [2].In these mechanisms, neutrinos, along with newly introduced heavy neutral leptons, acquire Majorana components, allowing for leptonic di-flavor violation (LFV) processes µ ± µ ± → e ± e ± , µ ± µ ± → τ ± τ ± and the leptonic di-number violation (LNV) processes µ ± µ ± → W ± W ± as well.Thus studying these kinds of the same-sign di-lepton and di-boson processes quantitatively to explore how the NP factors (such as the leptonic Majorana components, the right handed W-boson and the doubly charged Higgs etc) play roles is interesting and is the motivations for this paper.Furthermore, as it is known that flavor mixing parameters are strongly constrained by charged lepton flavor violation decays [3][4][5][6], whereas the LFV and LNV processes considered in this study may not depend on the flavor mixing very much for the neutral leptons.
Therefore, precisely observing the contributions of these mechanisms and the NP factors to these processes is particularly intriguing.
√ s ≥ 1.0 TeV)1 [7][8][9][10] and so much important physics at the high energy colliders, we aim to investigate the important processes at high-energy same-sign µ ± µ ± colliders [11][12][13][14].If there is important physics present at high-energy same-sign µ ± µ ± colliders comparable to that of high-energy µ + µ − colliders, it is reasonable to consider both of them as potential future options.From the experiences for building a proton-antiproton (pp) collider (Tevatron) and a proton-proton (pp) collider (LHC), there are no serious problems for building high energy same-sign µ ± µ ± colliders in comparison with building high energy µ + µ − colliders, provided that the necessary techniques for muon sources, muon acceleration, and muon colliding are developed.Therefore, constructing high-energy same-sign µ ± µ ± collider(s) is a natural extension of building high-energy µ + µ − colliders, as long as there is sufficient amount of interesting and significant physics to observe.Furthermore, the studies exploring the NP of same-sign µ ± µ ± colliders are relatively fewer in the literatures compared to those focusing on µ + µ − colliders [15][16][17][18][19][20][21][22][23], so we would like quantitatively to investigate the characteristic processes, the LFV processes µ ± µ ± → e ± e ± , µ ± µ ± → τ ± τ ± and the LNV processes 2), at high-energy same-sign muon colliders in this work.The process e ± e ± → µ ± µ ± was studied in Ref. [24], and the authors presented the theoretical predictions with the minimal type-I seesaw mechanism and analyzed the contributions from the supersymmetric particles.In Refs.[25][26][27] the contributions from the doubly charged Higgs to the process e ± e ± → µ ± µ ± were also examined.The studies of the LNV di-boson process e ± e ± → W ± L W ± L were carried out in Refs.[28][29][30][31][32][33][34][35][36][37][38][39].The theoretical predictions on the cross sections of e ± e ± → W ± L W ± L , e ± e ± → W ± L W ± R in the left-right symmetric model (LRSM) were given in Refs.[40][41][42][43].In this work, we will investigate the NP contributions to the LFV di-lepton, LNV di-boson processes, and explore their phenomenological behaviors at the high energy µ ± µ ± colliders.It is worth noting that the contributions to the LFV and LNV processes from e-flavor heavy neutral lepton are significantly constrained by the recent experimental upper bound on the 0ν2β decay half-life, because the 'core' process of the 0ν2β decays is d + d → u + u + e + e. Namely, all the processes with e ± e ± being in the initial state are also constrained by the 0ν2β experiments [44].
In this work, we mainly focus on exploring the sources of NP which generate the LFV dilepton and LNV di-boson processes, namely a). the neutral leptons' Majorana components with the help of the left-handed W L boson only or with the help of the left-handed and righthanded bosons W L , W R both, and b). the doubly charged Higgs and the interference effects of the Higgs and the neutral leptons' Majorana components2 .To explore the phenomena of the NP factors and their combinations, we categorize them into three types: Type I of NP (TI-NP) which, such as the B − L symmetric SUSY model (B-LSSM) [45][46][47][48], involves the combination of neutral leptons' Majorana components and the left-handed W L boson; Type II of NP (TII-NP) which, such as the left-right symmetric model (LRSM) without doubly charged Higgs, involves the combination of neutral leptons' Majorana components, the lefthanded boson W L , and the right-handed boson W R ; Type III of NP (TIII-NP) which, such as the LRSM [49][50][51][52][53][54][55], encompasses the presence of doubly charged Higgs alongside the W L , W R bosons, and neutral leptons' Majorana components.The behaviors of the LFV di-lepton and LNV di-boson processes at high energy µ ± µ ± colliders resulting from the three types of NP will be computed and discussed in this paper.
The paper is organized as follows: in Sec.II for later applications, the seesaw mechanisms, which give rise to the heavy neutral lepton masses, the tiny neutrino masses, the relevant Majorana components and the interaction vertices are outlined.In Sec.III the theoretical computations of the processes for the three types of NP are given.In Sec.IV the numerical results with suitable input parameters which are constrained by the available experiments are calculated and presented by figures.Finally, in Sec.V the results are discussed and a summary is made.

II. THE SEESAW MECHANISMS AND THE RELEVANT INTERACTIONS
In this section, we briefly review the mechanisms which make the neutrinos to acquire tiny masses and mixtures.Furthermore, we outline the necessary interactions which relate to the three types of NP and are required for the computation of the LFV di-lepton and LNV di-boson processes.
where M D is the Dirac mass matrix of 3 by 3, and M R is the Majorana mass matrix of 3 by 3. We can define ξ ij = (M T D M −1 R ) ij , then the mass matrix in Eq. ( 1) can be diagonalized approximately as ) denoting the i−th generation of heavy neutral lepton masses, and Then we have In the following analysis, the 3 by 3 matrix U is taken as the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix which is measured from the neutrino oscillation experiments.
The relevant Lagrangian for the l − W − ν and l − W − N (l = e, µ, τ ) interactions in the model becomes where P L,R = (1 ∓ γ 5 )/2, and ν, N are the four-component forms of mass eigenstates corresponding to light, heavy neutral leptons respectively.The LFV di-lepton processes will be calculated in the Feynman gauge later on, hence the l − G − ν and l − G − N interactions where G denotes the Goldstone boson will occur, and the relevant Lagrangian is written as where U, S, T, V are matrices of 3 by 3 defined in Eq. ( 3 are involved and The Yukawa Lagrangian for the lepton sector is given by where the family indices i, j are summed over, Φ = σ 2 Φ * σ 2 .The tiny neutrino masses are obtained by both of the type-I and type-II seesaw mechanisms when the Higgs φ 0 1 , φ 0 2 , ∆ 0 L , ∆ 0 R achieve VEVs [49][50][51][52][53]. Then the mass matrix of neutral leptons can be written as where The Lagrangian for the l − − ∆ −− L,R − l − interactions is The mass matrix in Eq. ( 9) can be diagonalized in terms of a unitary matrix U ν , whereas the matrix U ν can be expressed similarly as that in the above case of the B-LSSM Eq. ( 3).
Then we can obtain [44] The second formula in Eq. ( 12) are obtained by using the approximation Due to the SU(2) R gauge group, there are charged right-handed gauge bosons W ± R additionally and the two kinds of bosons W L and W R may be mixed.As the result of W L − W R mixing, the masses of physical W 1 (dominated by the left-handed W L ) and W 2 (dominated by the right-handed W R ) can be written as [44]: Then the Lagrangian for the l − W − ν and l − W − N interactions for the LRSM is where tan 2ζ denotes the mixing between W L and W R , the matrices U, S, T, V are defined in Eq. (3), ν, N are the four-component fermion fields of the mass eigenstates corresponding to light and heavy neutral leptons respectively.The Lagrangian for the l − G − ν and l − G − N interactions may be written as [49] L where G L , G R are the unphysical Goldstone bosons when the Feynman gauge is applied3 , and Finally, the relevant Lagrangian for the W − ∆ −− L,R − W interactions is Hence now, the theoretical predictions of the considered processes for TII-NP can be calculated based on the Lagrangian L LR W , L LR G , and the ones for TIII-NP can be calculated based on the Lagrangian L LR ∆ll , L LR W , L LR G , L LR ∆W W .Note that there is a possible model which contains doubly charged Higgs in its scalar triplet with Type-II seesaw [56][57][58][59], and the doubly charged component of the triplet Higgs can also cause the LFV and LNV processes.But the contributions to the processes are highly suppressed (which are much smaller than those considered here) as the relevant interactions are proportional to the light neutrino masses.Thus the contributions to the LFV and LNV processes from the three types of NP considered here are much great that we will not discuss the model here.

III. COMPUTATIONS OF THE PROCESSES µ
In this section we will provide the calculations of the processes µ ± µ ± → l ± l ± (j = e, τ ), ) within TI-NP, TII-NP and TIII-NP.The results for TI-NP and TII-NP can be obtained by switching off certain interactions from the relevant results for TIII-NP.In all computations, the flavor mixing parameters of the heavy neutral leptons are ignored, because the flavor mixing parameters are constrained strongly by the charged lepton flavor violating decays [3][4][5][6].
The leading-order Feynman diagrams for the LFV di-lepton processes µ ± µ ± → l ± l ± (l = e, τ ), incorporating the contributions from Majorana neutral leptons in TI-NP, TII-NP processes via neutrino and neutral heavy lepton exchanges.
and TIII-NP are collected in Fig. 1, but in the case of TI-NP, the Feynman diagrams with ) and charged lepton masses.Then the amplitude of the processes µ ± µ ± → l ± l ± for TIII-NP can be formulated as where where u is the Dirac spinor of the leptons, u c ≡ C ūT is its charge conjugation, the charge conjugation operator C ≡ iγ 2 γ 0 , and γ 0 , γ 2 are the Dirac matrices.The momenta p 1 , p 2 , k 1 , k 2 are defined as shown in Fig. 1, the coefficients C XY i in Eq. ( 18) can be read out from the amplitudes relating to the Feynman diagrams Fig. 1.
To show how the coefficients are read out, now let us take the Feynman diagram Fig. 1 (1) as an example, where ν k , ν l , W 1 , W 2 appear in the loop.Firstly, according to the interactions in Sec.II, the amplitude can be written as where Γ W 1 , Γ W 2 are the total decay widthes of W 1 , W 2 bosons respectively.The integrals appearing in Eq. ( 20) can be calculated by using Loop-Tools [61,62], hence for the further calculations, we define the functions following the conventions in Loop-Tools as where l 1 , l 2 , l 3 are combinations of out-leg particles' momentum (for example, for the loop integral in Eq. ( 20), we may have denoting the mass and total decay width of loop particle i respectively, and According to the definitions in Eq. ( 21), the amplitude M(ννW 1 W 2 ) can be simplified by neglecting the tiny neutrino mass terms in the numerators of the neutrino propagators and the terms proportional to charged lepton masses which appear after applying the on-shell condition for the leptons.Then the amplitude M(ννW 1 W 2 ) becomes where D 00 , D 0 , D 1 , D 2 , D 12 can be computed numerically by using Loop-Tools.Then from M(ννW 1 W 2 ), the coefficients 7), defined in Eq. ( 19), can be read out as The Feynman diagrams for µ ± µ ± → l ± l ± (l = e, τ ) due to doubly charged Higgs.
are applied to O XY i (i = 1, ..., 7) defined in Eq. (19).Because each amplitude of the above Feynman diagrams may be treated in the same way as Fig. 1 (1), so all the coefficients C XY i in Eq. ( 18) for the relevant processes may be obtained.
The contributions from doubly charged Higgs (∆ ±± ) to the LFV di-lepton processes ) can be estimated from the Feynman diagram in Fig. 2. The amplitude corresponding to Fig. 2 can be formulated into that as Eq. ( 18), and the nonzero coefficients are are left, where j = e, τ , Γ is the total decay width of the Higgs ∆ −− L,R .Based on the total amplitude formulated as Eq. ( 18), the amplitude can be squared absolutely by summing up the lepton spins in the initial and final states.Neglecting the charged lepton masses which is turned out from the square of O XY i (i = 1, ..., 7), those in Eq. ( 19), the result can be simplified as Now the cross section of LFV di-lepton processes can be written as where the factor 1 4 comes from averaging the lepton spins in the initial state, θ is the angle between the direction of the outgoing lepton l with the collision axis, √ s is the total collision energy of µ ± µ ± colliders.
The Feynman diagrams contributing to the LNV processes µ ± µ ± → W ± i W ± j , i, j = 1, 2 at the tree-level are plotted in Fig. 3, where the final sates Summing up the fermions' spin and gauge bosons' polarizations, the squared amplitude for the processes µ ± µ ± → W ± i W ± j can be simplified by neglecting the terms proportional to O(sin 2 ζ) and the charged lepton mass m µ as where Note that the mixing parameter sin ζ is not present in Eqs.(31,32,33), because here in the calculations the further approximation, keeping the contributions up-to O(sin ζ) and setting the mass of the initial charged lepton to be zero is made.This approximation also disregards the contributions of doubly charged Higgs ∆ ±± L,R to the process For TIII-NP, Eq. (31) and Eq.(33) show that ∆ ±± L primarily contributes to the process µ ± µ ± → Consequently, the distinct characteristics of ∆ ±± L and ∆ ±± R can be identified by observing the LNV di-boson processes at µ ± µ ± colliders.
The results of µ ± µ ± → W ± L W ± L for TI-NP can be acquired by setting Y L,22 = 0 of Eq. ( 31), and the results of for TII-NP can be acquired by setting Y L,22 = Y R,22 = 0 in Eqs.(31,33).Based on the computations of the LNV di-boson processes, one may realize that the results of µ ± µ ± → W ± 1 W ± 1 cross section for TII-NP is similar to the ones of µ ± µ ± → W ± L W ± L cross section for TI-NP, the results of µ ± µ ± → W ± 1 W ± 2 cross section for TIII-NP is similar to the ones of µ ± µ ± → W ± 1 W ± 2 cross section for TII-NP.The cross sections of LNV di-boson processes can be written as where the factor 1 4 comes from averaging the lepton spins in the initial state, θ is the angle between the momentum of outgoing W i and the collision axis.Considering the phase space integration factor of identified particles, A = 2 for

IV. NUMERICAL RESULTS
In this section, we will calculate numerical results for the cross-sections of the LFV and LNV processes associated with TI-NP, TII-NP, and TIII-NP, using the formulas derived in Sec.III.To carry out the numerical evaluations, a lot of parameters in the NP which are constrained by available experiments need to be fixed, so let us explain how the parameters are fixed.The PDG [1] have collected a lot of the parameters such as 08 GeV and α em (m Z ) = 1/128.9;the charged lepton masses m e = 0.511 MeV, m µ = 0.105 GeV, m τ = 1.77GeV etc, and we adopt them all.The sum of neutrino masses is limited in the range i m νi < 0.12 eV by Plank [63], it indicates that the contributions of the neutrino mass terms are negligible, regardless whether the neutrino masses are normal hierarchy (m ν 1 < m ν 2 < m ν 3 ) or inverse hierarchy (m ν 3 < m ν 1 < m ν 2 ).Hence we will ignore the contributions proportional to neutrino masses in the numerical evaluations.The matrix U (the upper-left sub-matrix of the whole matrix U ν in Eq. ( 3)) is taken as the PMNS mixing matrix [1] to describe the mixing of the light neutrinos.
For the light-heavy neutral lepton mixing (LHM) matrix S ij , we set S 2 i ≡ j |S ij | 2 (i, j = e, µ, τ ) to describe the strength of LHM.The direct constraint on S 2 e comes from the 0ν2β decay searches as that S 2 e < ∼ 10 −5 [44].The direct constraint on S 2 µ is given by CMS as S 2 µ < ∼ 0.4 for TeV-scale heavy neutral lepton masses [64][65][66].Recently, the constraint on S 2 µ from the future high-luminosity Large Hadron Collider (HL-LHC) is analyzed in Ref. [67], and the authors claim S 2 µ < ∼ 0.06 for TeV-scale heavy neutral lepton masses.There is no direct constraint presently on S 2 τ for TeV-scale heavy neutral lepton masses.In the following analysis, we will show the proposed µ ± µ ± colliders are more sensitive to S 2 µ , S 2 τ than LHC and future HL-LHC for TeV-scale heavy neutral lepton masses, hence we set lepton and LNV di-boson processes at µ ± µ ± colliders can be used to identify them, and the signatures may well show their mass and width accordingly.According to Ref. [74][75][76][77], the total decay widthes of ∆ ±± L and ∆ ±± R can be written as where and The relevant Yukawa coupling Y R,ll is not a free parameter 4 , therefore we only have the Yukawa coupling Y L of the left-handed doubly charged Higgs to the leptons need to be set.
Generally it take the formulation below: Y ee is constrained strongly by the 0ν2β decay experiments in the range Y ee < ∼ 0.04.In addition, a small VEV v L of ∆ 0 L , v L < ∼ 5.0 GeV, is constrained by the ρ−parameter [1], so later on we will set it as v L = 0.1 GeV to simplify the numerical evaluations.

A. Numerical results for the LFV di-lepton processes
In this subsection, the numerical results about the LFV di-lepton processes µ ± µ ± → e ± e ± and µ ± µ ± → τ ± τ ± for TI-NP, TII-NP, TIII-NP are presented.The characters of LFV di-lepton processes are clear and practically free from SM background at high energy samesign muon colliders [26] 5 .Firstly, let us focus lights on the effects of LHM parameter S 2 µ and the heavy neutral lepton mass M N 2 .TeV scale µ ± µ ± colliders in Fig. 4, and the numerical results for various √ s are investigated in Fig. 5.

Taking possible M N
The numerical results in Fig. 4 indicate that when an integrated luminosity of 500 fb −1 is accumulated at a TeV scale µ ± µ ± collider, the process µ ± µ ± → τ ± τ ± predicted by TI-NP, TII-NP and the processes µ ± µ ± → τ ± τ ± , µ ± µ ± → e ± e ± predicted by TIII-NP have great opportunities to be observed.Because their cross section can be larger than 0.1 fb in a reasonable chosen parameter space, that means more than 50 signal events/year can be collected, so it indicates that the µ ± µ ± collider is more sensitive to S 2 µ and S 2 τ than the future HL-LHC.The contributions from Majorana neutral leptons to σ(µ ± µ ± → τ ± τ ± ), σ(µ ± µ ± → e ± e ± ) are proportional to S 2 µ S 2 τ , S 2 µ S 2 e respectively.And the contributions from doubly charged Higgs to σ(µ ± µ ± → τ ± τ ± ), σ(µ ± µ ± → e ± e ± ) are proportional to Y µµ Y τ τ , Y µµ Y ee respectively.Moreover, S 2 e and Y ee are limited to be small by the 0ν2β experiments while S 2 τ and Y τ τ are not, which leads to the contributions to σ(µ ± µ ± → e ± e ± ) are suppressed.The fact can be see clearly by comparing Fig. 4 (a) with Fig. 4 (b).From the figures one may see that the three blue curves merge together in the logarithmic coordinate, and the predicted cross-sections σ(µ ± µ ± → τ ± τ ± ), σ(µ ± µ ± → e ± e ± ) for TIII-NP are much larger than the ones predicted by TI-NP and TII-NP.It is because that the leading contributions to the LFV di-lepton processes for TIII-NP come from the s−channel mediation of the doubly charged Higgs at the tree level, while the ones for TI-NP and TII-NP start with the oneloop level which involves the Majorana neutral leptons.The black and red curves in Fig. 4 (a), (b) show that σ(µ ± µ ± → τ ± τ ± ) and σ(µ ± µ ± → e ± e ± ) are dominated by right-handed gauge boson for TII-NP if S 2 l (l = e, µ, τ ) are small.The results of σ(µ ± µ ± → τ ± τ ± ) for TI-NP, TII-NP are similar when S 2 µ > ∼ 10 −4 , and σ(µ ± µ ± → e ± e ± ) for TII-NP is always larger than that for TI-NP owing to the fact that there are only the contributions from W L mediation for TI-NP.The contributions to the LFV di-lepton processes for TIII-NP are dominated by the doubly charged Higgs so the dependence on S 2 l may be ignorable.A large M N 2 plays an enhancing role on σ(µ ± µ ± → τ ± τ ± ) and σ(µ ± µ ± → e ± e ± ) for TI-NP and TII-NP, because the goldstone component (in Feynman gauge) makes dominant contributions to the LFV di-lepton processes for TI-NP and TII-NP, the corresponding couplings increase with the increasing of heavy neutral lepton masses for a given S 2 l .The results of σ(µ ± µ ± → τ ± τ ± ), σ(µ ± µ ± → e ± e ± ) versus the √ s are presented in Fig. 5 (a) and in Fig.
The small "hill" in red curves in Fig. 5 is the result due to the W 2 boson being on-shell.doubly charged Higgs depends on their total widthes, and the height of the resonance peak depends on the relevant Yukawa coupling Y ll (see Eq. ( 39)) of the doubly charged Higgs to the leptons.And how the coupling Y ll affects the cross sections σ(µ ± µ ± → τ ± τ ± ) and σ(µ ± µ ± → e ± e ± ) are computed and presented in Fig. 6.
In the plotting of Fig. 6 increases with increasing of Y µµ and Y τ τ ; the cross-section σ(µ ± µ ± → e ± e ± ) increases with the increasing of Y µµ and Y ee .And the behavior shown in Fig. 6 is that as expects indeed.
Comparing the bound on relevant couplings from the muonium to anti-muonium transition  The picture shows that the angular distributions of the processes µ ± µ ± → τ ± τ ± , µ ± µ ± → e ± e ± are flat in these three types of NP models.

B. Numerical results for the LNV di-boson processes
In this subsection, we compute the LNV di-boson processes µ ± µ ± → W ± i W ± j , (i, j = 1, 2) for TI-NP, TII-NP, TIII-NP and present the numerical results properly.For the collider search of µ − µ − → W − W − (the case of µ + µ + → W + W + collisions is similar), the different decay channels of the final W bosons corresponding to different SM background processes.
For the pure leptonic channel µ − µ − + E / T and µ − e − + E / T where E / T denotes the missing energy carried by neutrinos, we compute the SM background processes µ − µ − → W − µ − ν µ and µ − µ − → Zµ − µ − in terms of MadGraph5 [79], and the results at √ s = 15 TeV are so large as It indicates they may make significant backgrounds when the pure leptonic channel µ − µ − + E / T , µ − e − + E / T are considered in observing the LNV processes, and the signals for the LNV processes are very hard to be picked up from the backgrounds [38].However there is a technique way to avoid the problem at least, namely ignoring all the events in which the decays W − → µ − νµ and/or W − → τ − ντ with τ − → µ − ν τ νµ are involved, and taking into account only the events in which the W − bosons decay either to eν e or τ ντ (except the τ lepton decay τ − → µ − ν τ νµ ) or quarks (q q′ ).In this way, although the efficiency of identifying W -boson(s) in the final states of the LNV processes will be lost a bit, the accuracy for identifying the LNV processes will be ensured.Therefore later on we will not concern this kind of possible SM backgrounds for the LNV any more.As analyzed in Ref. [38] (which focus on the same-sign electron colliders and the case of same-sign muon colliders considered in this work is similar), the dominant SM background process is µ − µ − → W − W − ν µ ν µ for the pure leptonic channel e − e − + E / T .The W bosons in the final state have effective missing energies and momenta due to the involving neutrinos in the final state, it indicates that the background process can be highly suppressed by cutting the invariant-mass of the two outgoing electrons.In addition, it is concluded in Ref. [38] that the semi-leptonic channel , where the solid, dashed, dotted curves denote the results for the processes when  e − + j W + E / T and the pure hadronic channel 2j W also have great potential to observe the signal processes.
We stated in Sec.III that the cross section of µ ± µ ± → W ± 1 W ± 1 for TII-NP is similar to the one of µ ± µ ± → W ± L W ± L for TI-NP, and the one of µ ± µ ± → W ± 1 W ± 2 for TIII-NP is similar to the one of µ ± µ ± → W ± 1 W ± 2 for TII-NP, therefore for simplicity and avoiding repeats, below we will not present the results about µ ± µ ± → W ± 1 W ± 1 for TII-NP and the results about µ ± µ ± → W ± 1 W ± 2 for TIII-NP.The results on the cross-sections σ(µ ± µ ± → W ± i W ± j ), (i, j = 1, 2) versus √ s are presented in Fig. 8 for M N 2 = 2.0 TeV, S 2 µ = 10 −4 , where the solid, dashed, dotted curves denote the results for the processes when  The enhancement as a hill on blue curve for σ(µ and the observed resonance enhancement appears in µ ± µ ± → W ± 2 W ± 2 when the s−channel ∆ ±± R plays roles.In addition, in Fig. 8 the 'valley' which appears on the blue solid or blue dotted curve due to the interference effect between the contributions from Majorana neutral leptons and those from the doubly charged Higgs. To see the interference effects indicated by the 'hill-valley' structure for TIII-NP in the blue curves in Fig. 8(a) and in Fig. 8 √ s in Fig. 9, where (a) for i = 1, (b) for i = 2, and the solid, dashed, dotted curves in the figures denote the results with M N 2 = 1.0, 2.0, 3.0 TeV respectively.In The heights of the peaks on the three blue curves in Fig. 9 (a) are similar, because the contributions to σ(µ ± µ ± → W ± 1 W ± 1 ) are dominated by ∆ ±± L contributions at the resonance, and varying M N 2 does not affect the heights of the resonance peaks at all.However Fig. 9 (b) shows that the heights of the peaks of the three blue curves vary with M N 2 , because M N 2 is related to the Yukawa coupling ∆ ±± R µ ∓ µ ∓ and the resonance heights are dominated by the Yukawa coupling.
To explore and to see the effects of S 2 µ to the cross-sections σ(µ ± µ ± → W ± i W ± j ), we set M N 2 = 2.0 TeV, √ s = 15.0TeV and the rest relevant parameters as the same as those in Fig. 9, and plot σ(µ ± µ ± → W ± i W ± j ) versus S 2 µ in Fig. 10.The black solid curve denotes the is not sensitive to S 2 µ , and this fact can be read out from Eq. (33).The results described by the black solid curve and the red dashed curve show that σ(µ ± µ ± → W ± 1 W ± 1 ) for TI-NP, TII-NP and σ(µ ± µ ± → W ± 1 W ± 2 ) for TII-NP, TIII-NP increase with increasing S 2 µ .As shown by the blue solid curves in the figures, σ(µ ± µ ± → W ± 1 W ± 1 ) for TIII-NP decreases with S 2 µ increasing first and then increases with S 2 µ increasing.It is due to the fact that the contributions to the process µ ± µ ± → W ± 1 W ± 1 for TIII-NP are dominated by the doubly charged Higgs when S 2 µ is small (S 2 µ < ∼ 10 −5 ), while the interference effects of the contributions from Majorana neutral leptons and from the doubly charged Higgs become important when S 2 µ ≈ 10 −4 (the interference effects can be seen more clearly in Fig. 9), and the Majorana neutral lepton contributions play dominant roles when S 2 µ is large enough (S 2 µ > ∼ 10 −3 ).It is pointed in Ref. [80] that a high energy µ + µ − collider running at the collision energy ) if a integrated luminosity 10 4 fb −1 is accumulated.At the same sign high energy muon colliders with a collision energy such as √ s ≃ 15 TeV, the solid black curve in Fig. 10 shows that the total cross section of the LNV process µ − µ − → W − 1 W − 1 may reach up-to about 0.01 fb by the prediction of TI-NP with S 2 µ ≈ 10 −5 .Note from the blue and red curves in Fig. 10 that the cross sections of the LNV processes µ ± µ ± → W ± 1 W ± 1 predicted by TII-NP and TIII-NP are larger than the one of predicted by TI-NP when S 2 µ < 10 −4 , hence the LNV processes predicted by TII-NP and TIII-NP at a high energy µ ± µ ± collider have much greater opportunities to be observed.The observation of these processes must offers unambiguous evidences of NP, which is a totally different way to fix the parameter S 2 µ from that at the high energy µ + µ − colliders [80].Moreover the same-sign muon colliders have very special advantages in observing the doubly charged Higgs, and setting constraints on the couplings of doubly charged Higgs with charged leptons etc. From the facts mentioned here particularly, the complementarity of the high energy same-sign muon colliders with the high energy µ + µ − are flat when the contributions are dominated by doubly charged Higgs, and σ(µµ → W 2 W 2 ) takes the minimum value at cos θ = 0 when the contributions are dominated by Majorana neutral leptons.
In summary, observing the leptonic di-flavor violation (LFV) and di-number violation (LNV) processes represents one of the most important physics aspects at high-energy µ ± µ ± colliders.The quantitative investigations in this paper lead us to conclude that the LFV process µ ± µ ± → τ ± τ ± predicted by TI-NP, TII-NP and TIII-NP is highly expected to be observed at very high-energy µ ± µ ± colliders, while the process µ ± µ ± → e ± e ± is expected to be observed only for TIII-NP.And all of the LNV di-boson processes predicted by TI-NP, TII-NP and TIII-NP have great opportunities to be observed at such colliders.Based on the numerical results analyzed in Sec.IV, one can have more insights into the characteristics of the processes due to NP: TI, TII, TIII respectively, and will achieve relevant constraints on parameters such as It should be emphasized that at such high energy µ ± µ ± collider, there are significant opportunities to observe the NP, such as the leptonic Majorana components, right-handed W -bosons and the two doubly charged Higgs and their properties etc.Therefore, owing to the fact that there are so many important physics and no serious problems in building high energy same-sign muon colliders as explored in this paper we believe that in the future high-energy µ + µ − colliders and same-sign µ ± µ ± colliders both will be built when the techniques on muon acceleration, muon beam storage in a circular ring and collisions of two counter-propagating muon beams etc are matured.

Firstly, as a
representative relating to TI-NP let us consider the B-LSSM.Its gauge group is extended by adding a local group U(1) B−L to SM, where B, L represent the baryon number and lepton number respectively.In this model, three right-handed neutral leptons and two singlet scalars (Higgs), possessing a non-zero B − L charge, are introduced.The Majorana masses of the right-handed neutral leptons arise when the two singlet scalars (Higgs) acquire vacuum expectation values (VEVs).Combining the Majorana mass terms with the Dirac mass terms, tiny neutrino masses can be obtained by the type-I seesaw mechanism.Namely the mass matrix of neutral leptons reads ), ml = diag(m e , m µ , m τ ) with m e , m µ , m τ denoting the charged lepton masses, G L being the goldstone boson which is "eaten" by W L in unitary gauge, M W L is the W L boson mass.Having the relevant lagrangian L BL W and L BL G , the theoretical predictions of the considered processes for TI-NP can be calculated.As a representative relating to the TII-NP and TIII-NP, let us consider the left-right symmetric model (LRSM).Its gauge group is SU(3) C SU(2) L SU(2) R U(1) B−L .The additional three generations of the right-handed neutral leptons which with the three generations of the right-handed charged leptons form doublets, the di-doublet scalar (Higgs) and the two triplet scalars (Higgs)

W 2
and/or G R line(s) should be moved away.Note that, here for simplification, the Feynman diagrams with the gauge boson W 1,2 lines and/or the Goldstone boson G L,R lines being crossed are not presented in the figure, in fact the contributions from these crossing diagrams to the processes should be well taken into account.Hence when doing the calculations, we do consider the contributions from these crossing diagrams.The mixing parameter ζ between the left-handed boson W L and the right-handed boson W R is constrained in the range ζ < ∼ 7.7 × 10 −4 [60], and the charged lepton masses are much smaller than the center-of-mass energy of the collisions.So in the calculations we ignore the contributions proportional to O(ζ 2

)
Namely corresponding to the amplitude for the Feynman diagram Fig. 1 (1), only the coefficients C LL 1 and C LL 2 receive nonzero contributions.When we calculate the other Feynman diagrams of Fig. 1 where the two outgoing charged leptons l (l = e, τ ) of the Feynman diagrams are alternated, and the identity formulas

FIG. 3 :
FIG. 3: The Feynman diagrams for the LNV processes µ ± µ ± → W ± i W ± j as those in the LRSM: figure (1, 2) are relating to the neutral Majorana lepton contributions and figure (3) is relating to the doubly charged Higgs contributions.Here the final sates W± i W ± j denote W ± 1 W ± 1 or W ± 1 W ±

S 2 e
≤ 10 −5 , S 2 µ ≤ 0.01, S 2 τ ≤ 0.01.(35) where S 2 µ and S 2 τ are set a smaller value than that HL-LHC can reach at.Considering the direct observations on the right-handed weak gauge boson W R (W 2 ), the lower bound for the mass of W 2 boson reads M W 2 > ∼ 4.8 TeV [68-71], and its total decay width can be estimated as Γ W 2 ≈ 0.028M W 2 [52] roughly.On the doubly charged Higgs masses, the most fresh limits from the LHC [72, 73] are M ∆ ±± L > ∼ 800 GeV, M ∆ ±± R > ∼ 650 GeV.As pointed out above, the signals of ∆ ±± L and ∆ ±± R produced in the LFV di-

Fig. 5 (FIG. 6 :
Fig. 5 (b) shows the fact that the cross-section σ(µ ± µ ± → e ± e ± ) for TII-NP is always larger than the one for TI-NP as analyzed above.The explicit resonance enhancements (the peaks) appearing on the blue curves for TIII-NP in Fig. 5 are owing to √ s crossing the doubly charged Higgs ∆ ±± L or ∆ ±± R mass value as √ s increasing, it indicates the LFV processes µ ± µ ± → τ ± τ ± and µ ± µ ± → e ± e ± are very good channels to observe the two doubly charged Higgs at a µ ± µ ± collider by scanning the collision energy.Namely if the resonance enhancements in the LFV processes appear, it means that the signals may be used to observe the doubly charged Higgs ∆ ±± L and ∆ ±± R .The enhancement signal of the , we set √ s = 5.0 TeV, M W 2 = 5.0 TeV (in fact for TIII-NP M W 2 affects the numerical results of LFV processes slightly, hence fixing the value of M W 2 does not lose the general features which we are interested in), M N 1 = 1.0 TeV, M N 2 = 2.0 TeV, M N 3 = 3.0 TeV, M ∆ ±± L = 10.0 TeV, M ∆ ±± R = 11.0TeV, S 2 e = 10 −5 , S 2 µ = 10 −4 , S 2 τ = 10 −2 .Fig. 6 (a) is for l = τ and the blue solid, blue dashed, blue dotted curves denote the results with various Y τ τ = 0.1, 0.5, 1.0 respectively.Fig. 6 (b) is for l = e and the blue solid, blue dashed, blue dotted curves denote the results with various Y ee = 0.01, 0.025, 0.04 respectively.The contributions from doubly charged Higgs are proportional to (

Figure
Figure (b) is about those additionally to have M ∆ ±± L = 5.0 TeV, M ∆ ±± R = 15.0TeV for TIII-NP.The black curves denote the results for TI-NP, the red curves denote the results for TII-NP with M W 2 = 5.0 TeV, the blue curves denote the results for TIII-NP with M W 2 = 5.0 TeV, Y µµ = 1.0.

Fig. 8 (
Fig. 8 (b) is about those, additionally to have M ∆ ±± L = 5.0 TeV, M ∆ ±± R = 15.0TeV for TIII-NP.The black curve denotes the results for TI-NP, the red curves denote the results for TII-NP with M W 2 = 5.0 TeV, the blue curves denote the results for TIII-NP with M W 2 = 5.0 TeV, Y µµ = 1.0.In Fig. 8 the blue dotted curve and the red dotted curve merge together at √ s ≃ 10.0 TeV, is the resonance signal due to the s−channel ∆ ±± L or ∆ ±± R , and it corresponds explicitly to the blue solid and blue dotted curves respectively.From the figure one may see the resonance signal which is caused by ∆ ±± L or ∆ ±± R i.e. the resonance peak appears in the process µ

Fig. 9 (Fig. 9 ,
Fig. 9 (a) the black curves denote the results for TI-NP, the red curves in Fig. 9 (b) denote the results for TII-NP with M W 2 = 5.0 TeV, and the blue curves in both the figures denote the results for TIII-NP with M W 2 = 5.0 TeV, Y µµ = 1.0, M ∆ ±± L = 10.0 TeV, figure (b) is for W i W j = W 2 W 2 .The solid, dashed, dotted curves in the figures denote the results for M N 2 = 1.0, 2.0, 3.0 TeV respectively.The black curves denote the results for TI-NP, the red curves denote the results for TII-NP with M W 2 = 5.0 TeV.The blue curves denote the results for TIII-NP with M W 2 = 5.0 TeV, Y µµ = 1.0, M ∆ ±± L = 10.0 TeV, M ∆ ±± R = 11.0TeV.

FIG. 10 :
FIG.10: σ(µ ± µ ± → W ± i W ± j ) versus S 2 µ with M N 2 = 2.0 TeV, √ s = 15.0TeV (the rest relevant parameters are set the same as those in Fig. 9.The black solid curve denotes the result of W i W j = W 1 W 1 for TI-NP.The red dashed, red dotted curves denote the results ofW i W j = W 1 W 2 , W i W j = W 2 W2 respectively for TII-NP with M W 2 = 5.0 TeV.The blue solid, blue dotted curves denote the results ofW i W j = W 1 W 1 , W i W j = W 2 W2 respectively for TIII-NP with M W 2 = 5.0 TeV, Y µµ = 1.0, M ∆ ±± L = 10.0 TeV, M ∆ ±± R = 11.0TeV.