Probing charged lepton number violation

We study impacts of dimension-five lepton-number violating operators associated with two same-sign weak bosons, $\ell^\pm \ell^{\prime \pm} W^\mp W^\mp$, on current and future experiments for neutrino oscillation, lepton-number violating rare processes and high-energy collider experiments. These operators can contain important information on the origin of tiny neutrino masses, which is independent of that from the so-called Weinberg operator. We examine constraints on the coefficients of the operators by the neutrino oscillation data. Upper bounds on the coefficients are also investigated by using the data for processes of lepton number violation such as neutrinoless double beta decays and $\mu^-$-$e^+$ conversion. These operators can also be directly tested by searching for lepton-number violating dilepton production via the same-sign W boson fusion process at high-energy hadron colliders like the Large Hadron Collider. We find that these operators can be considerably probed by these current and future experiments.


I. INTRODUCTION
In 2012, the Higgs boson was discovered at the LHC [1], and the existence of all particles predicted in the Standard Model (SM) was confirmed empirically.On the other hand, the SM cannot explain some observed phenomena, such as baryon asymmetry of the Universe [2], the existence of dark matter [3] and neutrino oscillation [4].It is one of the important goals of current particle physics to establish the theory beyond the SM which can explain the origin of these mysterious phenomena.
The observed neutrino oscillation indicates that neutrinos have small but non-zero masses.
This smallness would suggest that the origin of small neutrino masses is different from the electroweak symmetry breaking.It would be natural to consider that neutrinos have Majorana-type masses, instead of Dirac-type masses.In this case, the theory beyond the SM is expected to have a source of Lepton Number Violation (LNV) at high energies, which provides the origin of tiny Majorana-type masses of neutrinos at low energies.Such a high-scale physics may be well described by Effective Field Theories (EFTs) with the electroweak gauge symmetry.One of the most important operators of LNV is so-called the Weinberg operator [5], which is a dimension-five operator.There are many models where small Majorana masses of neutrinos are generated via the Weinberg operator, like the type-I [6,7], the type-II [7,8], the type-III seesaw mechanisms [9], and models where neutrino masses are radiatively generated at one-loop [10,11], two-loop [12,13] and threeloop level [14,15].
If the lepton number is not conserved at high energies, we generally have various higherdimensional operators of LNV [16][17][18][19][20][21][22], in addition to the Weinberg operator.After the electroweak symmetry breaking, some of them yield the dimension-five charged LNV operators ℓ ± ℓ ′± W ∓ W ∓ where ℓ(ℓ ′ ) represents a charged lepton e, µ or τ , and W ± are the weak bosons.Electroweak gauge invariant origins of these dimension-five operators are dimensionseven (dimension-nine) operators in the case that leptons in the operators are left-handed (right-handed) [16,17].In general, their coefficients are independent of that of the Weinberg operator, and can be related to neutrino masses [17,23,24].
We expect that, in the near future, we can test the LNV signal from the same-sign lepton pair production via the same-sign W boson fusion processes pp → W + W + jj → ℓ + ℓ + jj.
In this paper, we study impacts of dimension-five LNV operators associated with two same-sign weak bosons, ℓ ± ℓ ′± W ∓ W ∓ , on current and future experiments for neutrino oscillation, LNV rare processes and high energy collider experiments.These operators can contain important information on the origin of tiny neutrino masses, which is independent of that from the Weinberg operator.We examine constraints on the coefficients of the LNV operators by the neutrino oscillation data.Upper bounds on the coefficients are also investigated using the data for LNV processes such as neutrinoless double beta decays and µ − -e + conversion.These operators can be directly tested by the lepton number violating processes via the same-sign W boson fusion process at high energy hadron colliders, like the LHC.It is found that these operators can be considerably probed by these current and future experiments.This paper is organized as follows.In Sec.II, we define the dimension-five LNV operators, ℓ ± ℓ ′± W ∓ W ∓ , and discuss the relation to the operators symmetric under the electroweak gauge symmetry.In Sec.III, we consider neutrino masses which are generated by the ℓ ± ℓ ′± W ∓ W ∓ operators.They are generated at loop level and have ultraviolet divergences from loop integrals.We renormalize these divergences by using higher-dimensional counter terms.In Sec.IV, we derive tree-level constraints for ℓ ± ℓ ′± W ∓ W ∓ from neutrinoless double beta decays and muon positron conversion.In Sec.V, we investigate the LNV signal via the ℓ ± ℓ ′± W ∓ W ∓ operators at the LHC.Conclusions are given in Sec.VI.In Appendix A, we show two renormalizable models which realize the ℓ ± ℓ ′± W ∓ W ∓ operators with left-handed charged leptons via gauge invariant dimension-seven LNV operators.In Ap-pendix B, detailed calculations for the renormalization of two-point functions of neutrinos are shown.
We here introduce the dimension-five ℓ ± ℓ ′± W ∓ W ∓ operators, where ℓ(ℓ ′ ) is a charged lepton e, µ or τ , and W ± are the weak bosons.Such operators are, in general, represented by the following form1 ; where Γ µν is a 4 × 4 matrix which is the product of gamma matrices.We can classify Γ µν into four forms.
where P X is the chirality projection operator and X is the chirality of charged leptons.The operators with the anti-symmetric tensor Γ µν = [γ µ , γ ν ] P X equal zero, because W + µ W + ν is the symmetric for the exchange µ ↔ ν.Therefore, the ℓ ± ℓ ′± W ∓ W ∓ operators are expressed as where C X ℓℓ ′ are dimensionless coupling constants, and Λ is a dimensionful parameter.The SU(2) L × U(1) Y gauge invariant origins of theses operators in Eq. (3) depend on the chirality X as discussed in order below.
The gauge invariant origin of the ℓ ± ℓ ′± W ∓ W ∓ operators for left-handed charged leptons, X = L, is the dimension-seven operators [16,18], where φ is the Higgs doublet field in the SM and φ is its SU(2) L conjugation, L ℓ are lepton doublet fields and L ℓ are their SU(2) L conjugations, C ℓℓ ′ are dimensionless coefficients, and Λ LNV is the scale of lepton number violation.After the electroweak symmetry breaking, the neutral component φ 0 of the Higgs field obtains the vacuum expectation value φ 0 = v/ √ 2 with v = 246 GeV, and the following dimension-five operators are generated; where e is the gauge coupling constant of the electromagnetic force, s w = sin θ w , c w = cos θ w with θ w being the Weinberg angle.The second term in the first row of Eq. ( 5) corresponds to the operator in Eq. ( 3).The coupling constants defined in Eq. ( 3) are given by We note that the original coupling constants C ℓℓ ′ are not symmetric for flavor indices generally while C L ℓℓ ′ are symmetric.In Appendix A, we show concrete models where the dimensionseven operators in Eq. ( 4) are yielded at one-loop level.
Next, we consider the gauge invariant origin of the ℓ ± ℓ ′± W ∓ W ∓ operators for righthanded charged leptons, X = R. Contrary to the case of left-handed charged leptons, they are generated from the dimension-nine gauge invariant LNV operators [16,17], where ℓ R are a right-handed charged lepton, C ℓℓ ′ are the dimensionless coupling constants.After the electroweak symmetry breaking, the dimension-five operators are generated.Therefore, the coupling constants C R ℓℓ ′ can be expressed by the parameters of gauge invariant effective LNV operators as Notice that new coupling constants C R ℓℓ ′ are symmetric for flavor indices because C ℓℓ ′ are symmetric.In Refs.[16,17,23,24], the models where the dimension-nine operators in Eq. ( 7) are yielded at tree or one-loop level are investigated.

III. NEUTRINO MASSES
In addition to the Weinberg operator, the LNV operators in Eqs. ( 4) and ( 7) can contribute to Majorana masses of neutrinos at loop levels.The coefficients of these operators are constrained by the current data for the neutrino mass matrix which is given by neutrino oscillation experiments and observation of cosmic microwave background.
We begin with summarizing the observed results for the neutrino mass matrix.The Majorana-type mass matrix m ν is diagonalized by using a unitary matrix, so-called the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix U [47,48], where m i (i = 1, 2, 3) is a mass eigenvalue of m ν .The PMNS matrix can be parametrized as follows; where c ij and s ij are cos θ ij and sin θ ij with θ 12 , θ 13 and θ 23 being mixing angles, and δ, α where NH and IH are abbreviations of normal hierarchy and inverted hierarchy, respectively.
The CP violating phases α 1 and α 2 cannot be observed at neutrino oscillation experiments, and we do not have any information of them currently.The rest CP violating phase δ can be observed at neutrino oscillation experiments, and latest data at T2K have already ruled out the CP conserving cases (δ = 0 or π) at 99.73% C.L. [50].We can measure the difference between quadratics of their mass eigenvalues by neutrino oscillation experiments, however, the pattern of the hierarchy, whether the normal hierarchy where m 1 < m 2 < m 3 or the inverted hierarchy where m 3 < m 1 < m 2 , is still unknown.Although absolute values for each mass eigenvalue cannot be determined by neutrino oscillation experiments, the upper bound for the summation of the mass eigenvalues can be obtained from observation of the cosmic microwave background.From the latest result by the Planck collaboration [3], the following constraint is given; Information on absolute values of neutrino masses can also be given by 0νββ experiments which can constrain the (e, e) component |(m ν ) ee | of the effective neutrino mass, which is discussed in more details in Section IV.
We now consider the constraint on the coefficients of the ℓ ± ℓ ′± W ∓ W ∓ operators.At tree level, neutrino masses would be generated via the Weinberg operator, where Λ 5 is the scale of physics where the Weinberg operator is generated.In general, Λ 5 can be different from Λ LNV , depending on the scenario of creating tiny neutrino masses.
At loop level, operators in Eqs. ( 4) and ( 7) can contribute to neutrino masses.We show Feynman diagrams for neutrino masses in Figs. 1 and 2. These loop diagrams are ultraviolet divergent.We eliminate these divergences by using the counter term from the Weinberg operator under the renormalization conditions given in Appendix B.
First, we consider the renormalization of two-point functions of neutrinos which are generated via the dimension-seven operators given in Eq. ( 4).Diagrams in Fig. 1 have quadratic divergences and logarithmic divergences proportional to the squared momentum of external neutrinos.We can eliminate the former one (the quadratic divergence) by using the counter term from the Weinberg operator.In order to eliminate the latter one (the logarithmic one), we use the new counter term from the dimension-seven operators [16], 1. Feynman diagrams for neutrino masses which are generated by the dimension-seven LNV operators.where F ℓℓ ′ are the coupling constants whose mass dimension is −3.We use the data for the neutrino mass matrix to impose the on-shell renormalization condition to the two-point function of neutrinos.After this renormalization procedure, we obtain the renormalized amputated two-point function of neutrinos in the mass eigenstate basis as follows; where we use Eq. ( 6), and m W is the mass of the weak bosons W ± , p µ is the momentum of the external neutrino, and neutrino fields in the mass eigenstate basis are defined as In Eq. ( 17), we only show the leading term, neglecting terms proportional to the masses of charged leptons.Details of the calculation are shown in Appendix B. Then, neutrino mass eigenvalues and mixing angles are input parameters, and the coefficients C L ℓℓ ′ /Λ are not constrained from the data of neutrino oscillation.
Next, we consider the renormalization of two-point functions of neutrinos which are generated via dimension-nine operators given in Eq. (7).In order to eliminate all divergences which appear in the Feynman diagrams in Fig. 2, we introduce new dimension-seven operators, where are the coupling constants whose mass dimension is −3.We use three LNV operators in Eqs. ( 4), ( 14) and ( 21).At one-loop level, Majorana masses of neutrinos are generated via the dimension-seven operators in Eq. ( 21).Feynman diagrams are shown in Fig. 3.These diagrams have logarithmic divergences.These divergences can be renormalized by using O( ) counter terms from the Weinberg operator and the on-shell renormalization conditions in Appendix B. At two-loop level, the dimension-nine operators in Eq. ( 7) generate the Majorana masses of neutrinos via the Feynman diagrams in Fig. 2.These diagrams have two kinds of divergences; i.e., logarithmic divergences and squared logarithmic divergences.
The squared logarithmic divergences can be eliminated by using O( 2 ) counter terms from 3. Two-point functions which are generated by the LNV operators in Eq. (21).
the Weinberg operator.The logarithmic divergences are proportional to a function of the momentum of the external neutrino.In order to eliminate these divergences, we use O( 2) counter terms from the operators in Eq. (21).After this renormalization procedure with renormalization conditions in Appendix B, we obtain the renormalized amputated two-point functions of neutrinos in the mass eigenstate basis as where Σ L ℓℓ ′ ( ✁ p) are defined in Eq. ( 16), and we use Eq. ( 9).In Eq. ( 22), we only show the leading term, neglecting terms proportional to cubic or higher order terms of charged lepton masses.Detail of the calculation are shown in Appendix B. As in the case for C L ℓℓ ′ /Λ, the coefficients C R ℓℓ ′ /Λ are not constrained from the data of neutrino oscillation.

IV. CONSTRAINTS FROM LOW ENERGY EXPERIMENTS
In this section, we discuss current constraints on the ℓ ± ℓ ′± W ∓ W ∓ operators from low energy experiments; i.e., neutrinoless double beta decays ( 0νββ ) and muon-positron ( µ −e + ) conversion processes. A.

Neutrinoless double beta decay ( 0νββ )
We consider the constraint from the 0νββ experiments.Currently, KamLAND-Zen experiment provides the most stringent limit on the half-life of the process at 90% C.L. [26], If we assume that the process occurs via Majorana masses of neutrinos, this bound is translated to the upper limit on the absolute value of the (e, e) element of the effective neutrino mass matrix at 90% C.L. [26], We can then estimate the upper bound on the parton-level amplitude for dd → uue − e − , where G F (≃ 1.17 × 10 −5 GeV −2 ) is the Fermi constant and p eff (∼ 100 MeV) is the typical distance scale between nucleons.In the following, we extract constraints on C R ee /Λ and C L ee /Λ by comparing Eq. ( 26) to parton-level amplitudes generated by the LNV operators in Eqs. ( 4) and (7), respectively.First, we consider the constraint on C R ee /Λ.The dimension-nine LNV operators in Eq. ( 7) generate 0νββ decays at tree level which are described by the diagram in Fig. 4. By using Eq. ( 9), the parton-level amplitude is given by By comparing this with Eqs. ( 26) and ( 27), we estimate the upper bound on With the assumption |C R ee | = 1, we can obtain the lower bound on the scale Λ as Λ 10 5 TeV.
Next, we consider the constraint on C L ee /Λ.The dimension-seven LNV operators in Eq. ( 4) generate 0νββ decays at tree level which are described by the Feynman diagrams in Fig. 5.
In this case, there are additional diagrams which are generated via three-point vertices in the first line of Eq. ( 5).They only change the factor of the amplitude.By using Eq. ( 6 that of the muon capture is given by the SINDRUM-II experiment [32] as follows; If we assume that the process occurs via Majorana masses of neutrinos, B µ − e + is calculated by [51].
From this formula, we can obtain the upper bound on In order to extract the constraint on the ℓ ± ℓ ′± W ∓ W ∓ operators from µ − -e + conversion data, we use the similar way to the case of 0νββ.In the following, we extract constraints on C R eµ /Λ and C L eµ /Λ by comparing the parton level amplitude for uuµ − → dde + , to those generated by the LNV operators in Eqs. ( 4) and (7), respectively.
First, we consider the constraint on C R eµ /Λ.The dimension-nine operators in Eq. ( 7) generate the µ − -e + conversion process at tree level which is described by the diagram in Fig 6 .By using Eq. ( 9), the parton-level amplitude is given by By comparing this formula with Eqs.(34) and (35), we can obtain the constraint on With assumption |C R eµ | = 1, this upper bound provides the lower bound on the scale Λ as This bound does not have meaning because 6.3 × 10 −3 MeV is too small to justify using the EFT approach, and we cannot expect a significant constraint on the right-handed operator from µ − -e + conversion.
Next, we consider the constraint on C L eµ /Λ.In Fig. 7, we show the Feynman diagrams which are generated via the dimension-seven LNV operators in Eq. (4).By using Eq. ( 6), we can get the upper bound as With the assumption C L eµ = 1, the lower bound for Λ is given by As in the case of the bound on the C R eµ , we cannot expect the significant constraint on C L eµ /Λ from µ − -e + conversion too.

V. CONSTRAINT FROM HIGH-ENERGY COLLIDER EXPERIMENTS
In this section, we investigate LNV processes pp → ℓ + ℓ ′+ jj at hadron colliders, and examine the constraints on the LNV coupling constants of the ℓ ± ℓ ′± W ∓ W ∓ operators.

A. The constraints on C R ℓℓ ′
We begin with the processes pp → ℓ + ℓ ′+ jj with right-handed charged leptons which are generated by the dimension-nine operators in Eq. (7).These processes are represented by the diagrams in Fig. 8.There are two kinds of processes; t-channel diagrams via W boson fusion processes qq → W +( * ) W +( * ) jj → ℓ + ℓ ′+ jj and s-channel ones qq → W +( * ) → ℓ + ℓ ′+ jj.
In Fig. 9, we show the cross section of the process pp → µ + µ + jj at √ s = 14 TeV which is FIG. 8. Feynman diagrams for the process pp → ℓ + ℓ ′+ jj via the dimension-nine LNV operators.
calculated by using FEYNRULES 2.0 [53] and MADGRAPH5 AMC@NLO [52].The dashed line represents the cross section which is generated by only the s-channel diagrams, while the real line shows the cross section of both the t-channel and s-channel diagrams under the following kinematical cuts (for the Vector Boson Fusion (VBF) cuts, see Ref. [46].), where m jj is the invariant mass of the two jets, and |∆η| is the difference of the pseudorapidity of the jets.In both the cross sections, the basic kinematical cuts [46], are taken into account, where p j T and η j are the transverse momentum and the pseudorapidity of the jets, and p ℓ (′) T and η ℓ (′) are the transverse momentum and the pseudo-rapidity of ℓ (′)+ .Obviously, the cross section with the VBF cuts is larger than that of s-channel diagrams.In the following, we use the VBF cuts to obtain the signal events.
We consider the following SM background processes, When we investigate the LNV process where the lepton flavor is conserved, we have to consider the first background process in Eq. ( 43).The number of the SM background events can be reduced by the transverse momentum cut, p ℓ (′) T > 500 GeV, and also multiplying the charge misidentification rate.The second process in Eq. ( 44) has the missing transverse momentum, and it can be reduced by cut, ✁ p T < 20 GeV [43], where ✁ p T is the missing transverse momentum.In Table I, we show the cross sections of the LNV signal pp → µ + µ + jj at √ s = 14 TeV and the cross section of the SM backgrounds for each step of the kinematical cuts.In the numerical evaluation, FEYNRULES 2.0 [53] and MADGRAPH5 AMC@NLO [52] are used.In the following, we consider how the coupling constants |C R ℓℓ ′ /Λ| can be constrained by searching for the LNV processes pp → ℓ + ℓ ′+ jj at the future HL-LHC experiment [54].First, we consider the LNV process where the anti-leptons in the final states have the same lepton flavor, pp → ℓ + ℓ + jj.The beam energy, √ s = 14 TeV, is much higher than the masses of charged leptons, so that the cross section is insensitive to the flavor of anti-leptons in the final state.Therefore, we here only consider the process pp → µ + µ + jj and the constraint on C R µµ /Λ.We expect that the constraints on the coupling constants with other flavors, C R ee /Λ and C R τ τ /Λ, are almost the same as that on C R µµ /Λ.By using the results in Table I, we can estimate the number of the SM background events at the HL-LHC experiment as in Table II.The rate of the charge misidentification is assumed to be 1% because we use the kinematical cut p ℓ T > 500 GeV, so that anti-muons have large transverse momenta [55].Expected numbers of the background events are respectively O (1) or much less than 1 for the processes in Eqs. ( 43) and (44).Therefore, if we obtain O (10) events of the LNV signals, we can say that they are not from the SM background events but from the signal events via the µ + µ + W − W − operator.
TABLE II.The expected number of SM background events at the HL-LHC experiment (with the collision energy of √ s = 14 TeV and the integrated luminosity of L = 3000 fb −1 ).We assume that the rate of charge misidentification is 1%.
In Fig. 10, we show numbers of the signal events and those of the background events at the HL-LHC experiment as a function of |C R µµ /Λ|.We assume that the rate of the charge misidentification is 1% [55].The real line represents numbers of the signal event.
The dashed and dotted lines represent numbers of the background events from µ + µ − jj and Next, we consider LNV processes where the anti-leptons have different lepton flavor As the beam energy is much higher than the masses of charged leptons, we only discuss the process pp → e + µ + jj and the constraint on C R eµ /Λ.It is expected that constraints on the other coupling constants, C R ℓℓ ′ /Λ (ℓ = ℓ ′ ), are similar to that on C R eµ /Λ.The most important SM background is pp → e + µ + ν e ν µ jj, and the number of this even is much less than 1 at the HL-LHC, as shown in Table II.In Fig. 11, we show the |C R eµ /Λ| dependence of the number of the signal event and that of the background event.We assume that the rate of the charge misidentification is 1%.The real line represents the number of the signal event while the dotted line does that of the background event.
Expected numbers of the signal event is O( 1 Consequently, the expected upper limits on |C R ℓℓ ′ /Λ| at the HL-LHC are If we assume |C R ℓℓ ′ | = 1, the lower limit for Λ is obtained when no excess from the SM prediction is observed at the HL-LHC as The constraint on |C R ee /Λ| from the HL-LHC experiment is weaker than that from current 0νββ experiments in Eq. ( 28).However, we can use the HL-LHC to test the LNV processes with the other set of flavor.

B.
The constraint on 12. Feynman diagrams for the process pp → ℓ + ℓ ′+ jj via the dimension-seven LNV operators.
In this section, we investigate processes pp → ℓ + ℓ ′+ jj with left-handed charged leptons, which are generated by the dimension-seven LNV operators in Eq. ( 4).These processes are represented by the Feynman diagrams in Fig. 12.As compared to the cases with righthanded charged leptons which are generated by the dimension-nine LNV operators in Eq. ( 7), additional diagrams are generated by the three-point vertices in Eq. ( 5).However, the contribution from these diagrams is negligibly small.As a result, constraints on C L ℓℓ ′ /Λ are almost the same as those on C R ℓℓ ′ /Λ, and we can estimate upper bounds on C L ℓℓ ′ /Λ at the HL-LHC as With the assumption that |C L ℓℓ ′ | = 1, non-observation of the LNV processes at the HL-LHC should provide lower bounds on Λ as We have understood that the constraint on |C L ee /Λ| from the HL-LHC experiment would be weaker than that from current 0νββ experiments in Eq. (30).However, the HL-LHC can be useful to test the LNV processes with the other set of flavor of charged leptons.

VI. CONCLUSION
We have investigated phenomenological consequences of the ℓ ± ℓ ′± W ∓ W ∓ operators.
These operators can contain important information for the origin of tiny neutrino masses which is independent of that from the Weinberg operator.We have obtained constraints on the coefficients of the ℓ ± ℓ ′± W ∓ W ∓ operators by the neutrino oscillation data.Upper bounds on the coefficients have also been examined by using the data for LNV processes such as neutrinoless double beta decays and the µ − -e + conversion.In addition, we have found that the ℓ ± ℓ ′± W ∓ W ∓ operators can be directly tested by searching for the LNV processes via the same sign W boson fusion process at the HL-LHC.By the combination of these current and future experiments, we can access dimension-seven and dimension-nine LNV operators in the gauge invariant effective field theory and can further deeply understand the origin of tiny neutrino masses.
part by Grant-in-Aid for Scientific Research on Innovative Areas, the Ministry of Education, Culture, Sports, Science and Technology, No. 16H06492 and No. 18H04587, and also by JSPS, Grant-in-Aid for Scientific Research (Grant No. 18F18022 and No. 18F18321).
Appendix A: Models where dimension seven LNV operators are yielded We here show two models where the dimension-seven LNV operators, are yielded and neutrino masses are generated by the Feynman diagram in Figs. 13 and 14.
In the first model (Model-I), fields in Table III are 13. Realization of the dimension-seven operators and neutrino masses in Model-I.
14. Realization of the dimension-seven operators and neutrino masses in Model-II.
and S 0 are mixed via the three-point scalar interactions, where κ 1 and κ 2 are coupling constants, and φ is the Higgs field and η = iσ 2 η.
The model has the following new Yukawa interactions.
where the operators P R and P L are the chirality projection operators.Then, the dimensionseven LNV operators are generated via the left Feynman diagram in Fig. 13.By using this operator, Majorana masses of neutrinos are generated via the right Feynman diagram in Fig. 13.
The second model (Model-II) has the new fields listed in Table IV.This model has a new unbroken global symmetry U(1) ′ .The new fermions The dimension-seven operators are generated using the above scalar interaction and the following new Yukawa interactions, where Φ = iσ 2 Φ.Then, the dimension-seven LNV operators are generated via the left Feynman diagram in Fig. 14.By using this operator, Majorana masses of neutrinos are generated via the right Feynman diagram in Fig. 14.Detailed discussions of these models are beyond the scope of this paper, which will be given elsewhere [56] Appendix B: Renormalization of higher-dimensional LNV oeprators We here show details of the renormalization procedure used for the calculation in Sec.III.
First, we discuss the renormalization of two-point functions of neutrinos which are generated by dimension-seven operators in Eq. ( 4).We use the following renormalized operators, where U is the PMNS matrix, and neutrino fields are in the mass eigenstate basis ν ℓ,L = U ℓa ν a,L , (ℓ = e, µ, τ and a = 1, 2, 3).The Majorana neutrino fields ν a are defined as in Eqs.(19) and (20) such that they satisfy Majorana conditions ν c a = ν a .The mass matrix of Majorana neutrinos m νa is generated by the Weinberg operator as where δ ab is the Kronecker Delta.The LNV operators proportional to C ℓℓ ′ and their hermitian conjugations are generated by the dimension-seven operators in Eq. ( 4) which are the origin of the ℓ ± ℓ ′± W ∓ W ∓ operators with left-handed charged leptons.The LNV operators in the last line of Eq. (B1) are generated by the operators in Eq. (15).Their counter terms are used to eliminate logarithmic divergences proportional to the squared momentum of the external neutrino.Counter terms which are needed to eliminate divergences in two-point functions of neutrinos are given by where δ 1 Z ab , δ 1 m ab and δ 1 F ab are O( ) coefficients of counter terms.The coefficients δ 1 Z ab and δ 1 m ab satisfy the following conditions; In the 'tHooft-Feynman gauge, the renormalized amputated two-point functions for neutrinos in the mass eigenstate basis iΣ ab ( ✁ p) are given by where and k = d d k/(2π) d represents the integral over all d-dimensional euclidean momentum space.We impose the following on-shell conditions [57,58]; We cannot determine all coefficients of the counter terms with only imposing the on-shell conditions, so that we impose the additional condition, Then, Σ L ab ( ✁ p) in Eq. (B6) are given by where Masses of charged leptons m ℓ and those of neutrinos m νa are smaller than that of the weak bosons m W , so that the leading term of Σ L ab ( ✁ p) is given by where and Eq. ( 6) is used.The formula in Eq. (B22) are that in Eq. ( 17) in Sec.III.
Next, we show the renormalization of two-point functions of neutrinos which are generated by dimension-nine operators in Eq. ( 7).We use the following renormalized operators; where the mass matrix of Majorana neutrinos m νa is generated by the Weinberg operator as in Eq. (B2).The LNV operators proportional to C ℓℓ ′ and their hermitian conjugations are generated by the dimension-nine operators in Eq. ( 7) , which are the origin of the ℓ ± ℓ ′± W ∓ W ∓ operators with right-handed leptons.The LNV operators proportional to F ′(7) ℓℓ ′ are generated by the operators in Eq. ( 21).The LNV operators from Eq. ( 7) generate Majorana masses of neutrinos at two-loop level, while those from Eq. ( 21 δ 2 mab = 1 2 (δ 2 m ab + δ 2 m ba ), (B45) The term proportional to δ 1 F ′ ab,ℓ comes from Feynman diagrams in Fig. 15, which are generated via the counter term proportional to δ 1 F ′ aℓ .We here assume that coefficients of the other counter terms, for example coefficients for wave function renormalization of the weak bosons or charged leptons, are zero because they do not need to eliminate divergences.We impose the following on-shell conditions [57,58]; and Eq. ( 9) is used.The sum of Eq. (B55) and Eq.(B56) is Eq. ( 22) in Sec.III.

FIG. 2 .
FIG. 2. Feynman diagrams for neutrino masses which are generated by the dimension-nine LNV operators.

FIG. 10 .
FIG. 10.The C R µµ /Λ dependence of the number of pp → e + e + jj events and the numbers of the background events at the HL-LHC (with the collision energy of √ s = 14 TeV and the integrated luminosity of L = 3000 fb −1 ).It is assumed that the rate of the charge misidentification is 1%.

Λ 5 × 10 − 4
FIG. 11.The C R eµ /Λ dependence of the number of pp → e + µ + jj event and the number of the background events at Hl-LHC (with the collision energy of √ s = 14 TeV and the integrated luminosity of L = 3000 fb −1 ).
added to the SM.The model has a new global symmetry U(1) ′ .It is an unbroken symmetry after the electroweak symmetry breaking, and we expect that the model can explain a dark matter problem too.The new fermions ψ a = (ψ + a , ψ 0 a ) T (a = 1, 2, 3) are vector-like SU(2) L doublets, and they have Dirac mass terms m ψa ψ a ψ a .The model has three kinds of new scalar fields η, S + and S 0 .One of the new scalar fields η = (η + , η 0 ) T is a SU(2) L doublet.Other scalars S + and S 0 are SU(2) L singlets.All of new scalars do not obtain the vacuum expectation value.After the electroweak symmetry breaking, the charged scalars η + and S + and the neutral scalars η 0 are vector-like SU(2) L triplets, and they have Dirac mass terms M Σa Tr[Σ a Σ a ].Both of new scalars Φ = (Φ ++ , Φ + ) and η = (η + , η 0 ) are SU(2) L doublets.The singly charged scalar fields Φ + and η + are mixed via the four-point scalar interaction,

FIG. 15 .
FIG.15.Feynman diagrams for two-point functions of neutrinos via the counter term of the dimension-seven operators in Eq.(21).