One-loop neutrino mass model without any additional symmetries
aa r X i v : . [ h e p - ph ] S e p KIAS-P18084, APCTP Pre2018 - 012
One-loop neutrino mass model without any additional symmetries
Takaaki Nomura ∗ and Hiroshi Okada † School of Physics, KIAS, Seoul 02455, Republic of Korea Asia Pacific Center for Theoretical Physics (APCTP) - Headquarters San 31,Hyoja-dong, Nam-gu, Pohang 790-784, Korea (Dated: September 11, 2019)
Abstract
We propose a radiative seesaw model at one-loop level, introducing fields with large multipletof SU (2) L . Thanks to large representations, any additional symmetries are not needed. In thisframework, we formulate lepton and new fermion sector such as mass matrices, LFVs, and muon g −
2. Furthermore, we show our cut-off scale via RGEs of SU (2) L , and numerical analysis at abenchmark point included in dark matter candidate. Finally, we briefly discuss processes producingexotic particles in our model via proton-proton collision and possibility of detecting them at thelarge hadron collider experiments. ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTIONS Radiative seesaw models are promising candidates to connect active neutrinos and darkmatter (DM) candidate in addition to the natural explanation of smallness for neutrinomasses. In order to realize such kinds of models, typically additional symmetries such as Z N symmetry are simply needed for stabilizing the DM candidate as well as forbiddingneutrino masses at tree level [1–5]. As an alternative approach, unique charge assignmentsunder SU (2) L [6–9] and/or U (1) Y [10–16] in the standard model (SM) are applied to newfields for restricting interactions, without additional symmetries. In particular, introductionof large SU (2) L multiplets is interesting since their interactions are strongly restricted bygauge symmetry and rich phenomenology is provided by such new particles.In this paper, we introduce several exotic fields with large representations of SU (2) L asshown in tab. I. Then we find that these field-contents lead to a reasonable radiative seesawmodel due to assuring stability of the inert quintet scalar field H in a renormalizable theorywithout any additional symmetries, where this issue has been discussed by the comprehensivepaper [17]. It is also found that cutoff scale of a theory would be much lower than the Planckscale by analyzing running of gauge coupling with renormalization groups since contributionsfrom fields with large SU (2) L representations are sizable. This feature would be dangerous toobtain reliable predictions as we would have several unknown resonances around cutoff scaleother than new particles introduced in our model. Then we find that new fields with large SU (2) L representations should be sufficiently heavy; for example, if its scale is 5 TeV thecutoff scale is larger than 10 TeV and our predictions are reliable at the LHC experiments.On the other hand we might be able to test our model by testing deviation of the SMprediction for running of gauge coupling at around TeV or higher scale.This letter is organized as follows. In Sec. II, we review our model and formulate thelepton and new fermion sector. In Sec. III we show numerical analysis at a benchmarkpoint and discuss collider physics. Finally we devote the summary of our results and theconclusion. 2 aL e aR ψ a Σ aR H H SU (2) L U (1) Y − − SU (2) L × U (1) Y , wherethe upper index a is the number of family that runs over 1-3 and all of them are singlet under SU (3) C . II. MODEL SETUP AND CONSTRAINTS
In this section we formulate our model. As for the fermion sector, we introduce three fam-ilies of vector fermions ψ with (4 , − /
2) charge under the SU (2) L × U (1) Y gauge symmetry,and right-handed fermions Σ R with (5 ,
0) charge under the SU (2) L × U (1) Y gauge symmetry.As for the scalar sector, we add a quintet complex scalar field H with 0 charge under the U (1) Y gauge symmetry, where SM-like Higgs field is denoted as H . Here we write vacuumexpectation value(VEV) of H by h H i ≡ (0 , v/ √ T which induces the spontaneously elec-troweak symmetry breaking. All the field contents and their assignments are summarized inTable I, where the quark sector is exactly the same as the SM. The renormalizable YukawaLagrangian under these symmetries is given by −L ℓ = y ℓ aa ¯ L aL H e aR + f ab [ ¯ L aL H ( ψ R ) b ] + g ab [ ¯ ψ aL H ∗ Σ bR ] + g ′ ab [( ¯ ψ cR ) a H Σ bR ]+ M D aa ¯ ψ aR ψ aL + 12 M Σ aa ( ¯Σ cR ) a Σ aR + h . c ., (1)where SU (2) L index is omitted assuming it is contracted to be gauge invariant inside bracket[ · · · ], upper indices ( a, b ) = 1-3 are the number of families, and ( y ℓ , M D , M Σ ) are assumedto be diagonal matrix with real parameters without loss of generality. The mass matrix ofcharged-lepton is defined by m ℓ = y ℓ v/ √
2. Here we assign lepton number 1 to ψ L ( R ) and 0to Σ R so that the source of lepton number violation is terms with coupling g ab and g ′ ab inthe Lagrangian. Scalar potential and VEVs : The scalar potential in our model is given by V = − µ h | H | + λ H | H | + M | H | + M ′ [ H ] + µ [ H + h . c . ]+ X i λ i [ H H H H + h.c. ] + λ H | H | + λ H | H | + λ H H | H | | H | , (2)3here sum in quartic term of H indicate independent terms corresponding to differentcontraction of SU (2) L index. Applying condition ∂ V /∂v = 0, we obtain the VEV of Higgsfield H as v ∼ s µ h λ H . (3)On the other hand, we require a non-zero VEV of H and stability of the potential byimposing M > µ > { λ i , λ H } > Exotic particles : The scalars and fermions with large SU (2) L multiplet provide exoticcharged particles. Here we write components of multiplets as H = ( φ ++5 , φ +5 , φ , φ ′− , φ ′−− ) T , (4) ψ L ( R ) = ( ψ + , ψ , ψ ′− , ψ −− ) TL ( R ) , (5)Σ R = (Σ ++ , Σ + , Σ , Σ ′− , Σ ′−− ) TR . (6)The mass of component in H is given by ∼ M where charged particles in the same multiplethave degenerate mass at tree level which will be shifted at loop level [17]. For chargedfermions, components from ψ L ( R ) and Σ R can be mixed after electroweak symmetry breakingvia Yukawa coupling. If the Yukawa couplings are negligibly small the charged componentsin ψ L ( R ) have Dirac mass M D while the charged components in Σ R have Dirac mass M Σ where mass terms are constructed by pairs of positive-negative charged components in themultiplet. Note that mass term of neutral component is discussed with neutrino sectorbelow. A. Neutral fermion masses
Heavier neutral sector : After the spontaneously symmetry breaking, neutral fermion massmatrix in basis of (Ψ R ≡ ( ψ R , ψ cL , Σ R ) T is given by M N = M TD m ′ M D mm ′ T m T ( M Σ + M † Σ ) , (7) Due to the trilinear term of µ in the scalar potential, the DM candidates for ψ, Σ , H do not haveany remnant symmetries after the electroweak symmetry breaking. Nevertheless, they could be a DMcandidate that is called ”minimal dark matter” [32]. IG. 1: Feynman diagram to generate active neutrino mass. where m ( ′ ) ≡ g ( ′ ) v/ √
2. Then M N are Ψ R are respectively rotated by the unitary matrix asΨ R = V T ( ψ − ) R , D N ≡ diag( M , ..., M ) = V M N V T , (8)where ( ψ − ) R and D N are respectively mass eigenvectors and mass eigenvalues. We thushave 9 exotic Majorana fermions with mass eigenvalue M − . Active neutrino sector : In our scenario, we assume lepton number conservation is violatedonly through Yukawa interactions with coupling g and g ′ in Eq. (1), and physics at scalehigher than cut-off does not contribute lepton number violation. In such a case, activeneutrino mass can only be induced via interactions in Eq. (1) and (2) even if cut-off scaleis not very high as we discuss below. Then active neutrino mass is dominantly generatedat one-loop level where ψ α and H propagate inside a loop diagram as shown Fig. 1. As aresult the active neutrino mass matrix is obtained such that m ν = X α =1 − F iα F Tαj M α π (cid:20) m R m R − M α ln (cid:20) m R M α (cid:21) − m I m I − M α ln (cid:20) m I M α (cid:21)(cid:21) , (9)where F iα ≡ P k =1 f ik V Tkα and m R ( I ) is mass of φ R ( I ) which comes from real/imaginarycomponents of φ . Diagonalizing the matrix, neutrino mass eigenvalues ( D ν ) are found as D ν = U MNS m ν U T MNS , where U MNS is the MNS matrix. Once we define m ν ≡ F RF T , one can5ewrite f in terms of the other parameters [18, 19] as follows: f ik = X α =1 U † ij q D ν jj O jα p R αα V ∗ αk , (10)where O is a three by nine arbitrary matrix, satisfying OO T = 1. B. Charged fermion masses
Singly-charged fermion sector : The singly-charged fermion mass matrix, in basis of Ψ − R ≡ ( ψ − R ( ≡ ( ψ + L ) c ) , ψ ′− R , Σ ′− R ) T and Ψ − L ≡ ( ψ − L , ψ ′− L , Σ − L ( ≡ (Σ + R ) c )) T , is given by L M ± = ¯Ψ − L M ± Ψ − R , M ± = M TD − m ′ M D √ m − m ′ T √ m T ( M Σ + M T Σ ) . (11)When M ± is symmetric, M ± are Ψ ± L ( R ) and respectively rotated by the unitary matrix asΨ ± L ( R ) = V TC ψ ± L ( R ) − , D ± ≡ diag( M C , ..., M C ) = V C M ± V TC , (12)where ψ ± R − and D ± are respectively mass eigenvectors and mass eigenvalues of Dirac type. Doubly-charged fermion sector : The doubly-charged fermion mass matrix, in basis of Ψ −− R ≡ ( ψ −− R , Σ ′−− R ) T and Ψ −− L ≡ ( ψ −− L , Σ −− L ( ≡ (Σ ++ R ) c )) T , is given by L M ±± = ¯Ψ −− L M ±± Ψ −− R , M ±± = M D mm ′ T ( M Σ + M T Σ ) . (13)When M ±± is symmetric, then M ±± and Ψ ±± L ( R ) are respectively rotated by the unitary matrixas Ψ ±± L ( R ) = V TCC ψ ±± L ( R ) − , D ±± ≡ diag( M CC , ..., M CC ) = V CC M ±± V TCC , (14)where ψ ±± L ( R ) − and D ±± are respectively mass eigenvectors and mass eigenvalues of Diractype. C. Constraints from running of gauge coupling and LFV
Beta function of SU (2) L and U (1) Y gauge coupling g and g Y Here it is worth discussingthe running of gauge couplings of g and g Y in the presence of several new multiplet fields6f SU (2) L and new U (1) Y charged fields. The new contribution to g for an SU (2) L quintetfermion(boson) ψ ( H ), septet fermion Σ R , and quartet boson H are respectively given by∆ b ψg = 103 , ∆ b Σ R g = 203 , ∆ b H g = 103 . (15)In addition, new contribution to g Y from ψ a is given by∆ b ψg Y = 2 (16)Then one finds the energy evolution of the gauge coupling g and g Y as [8, 20]1 g ( µ ) = 1 g ( m in ) − b SMg (4 π ) ln (cid:20) µ m in (cid:21) − θ ( µ − m th ) ( N f ψ ∆ b ψg + N f Σ ∆ b Σ R g ) + ∆ b H g (4 π ) ln (cid:20) µ m th (cid:21) , g Y ( µ ) = 1 g Y ( m in ) − b SMg Y (4 π ) ln (cid:20) µ m in (cid:21) − θ ( µ − m th ) N f ψ ∆ b ψg Y (4 π ) ln (cid:20) µ m th (cid:21) , (17)where N f ψ/ Σ = 3 is the number of ψ and Σ R , µ is a reference energy, b SMg Y = 41 / b SMg = − /
6, and we assume to be m in (= m Z ) < m th , being m th threshold masses of exoticfermions and bosons. The resulting flow of g ( µ ) and g Y ( µ ) are then given by the Fig. 2.For g Y , we find it is relevant up to Planck scale. For g , the figure shows that the red lineis relevant up to the mass scale µ = O (100) TeV in case of m th =0.5 TeV, while the blue isrelevant up to the mass scale µ = O (1) PeV in case of m th =5 TeV. Thus our theory doesnot spoil, as far as we work on at around the scale of TeV; see also ref. [21]. Lepton flavor violations(LFVs)
LFVs arise from the term f at one-loop level, and its formcan be given by [22, 23]BR( ℓ i → ℓ j γ ) = 48 π α em C ij G m ℓ i (cid:0) | a R ij | + | a L ij | (cid:1) , (18)where a R ij = m ℓ i (4 π ) (cid:20) F jα F † αi G ( ψ α , H − ) + F ( ± ) jβ F ( ± ) † βi [2 G ( ψ ± β , H ±± ) + G ( H ±± , ψ ± β )] − F ′ ( ± ) jγ F ′ ( ± ) † γi G ( H , ψ ± γ ) − F ( ±± ) jρ F ( ±± ) † ρi [ G ( ψ ±± ρ , H ± ) + 2 G ( H ± , ψ ±± ρ )] (cid:21) , (19) a L = a R ( m ℓ i → m ℓ j ), F ( ± ) iβ ≡ P j =1 f ij ( V TC ) jβ , F ′ ( ± ) iγ ≡ P j =4 f ij ( V TC ) jγ , F ( ±± ) iρ ≡ P j =1 f ij ( V TCC ) jρ , and G ( a, b ) ≡ Z dx Z − x dy xy ( x − x ) m ℓ i + xm a + (1 − x ) m b . (20)7 th = m th =
100 10 Μ @ GeV D g Y m th = m th =
100 1000 10 Μ GeV G a ug ec oup li ngo f g FIG. 2: The running of g Y and g in terms of a reference energy of µ , where the red line correspondsto m th =0.5 TeV, while the blue one does m th =5 TeV. LFV with higher SU (2) L representation in radiative neutrino mass model are also discussedin ref. [24, 25]. New contributions to the muon anomalous magnetic moment (muon g − : ∆ a µ ) In ourmodel ∆ a µ arises from the same interactions inducing LFVs and can be formulated bythe following expression: ∆ a µ ≈ − m µ [ a L µµ + a R µµ ] = − m µ a L µµ , (21)where we use the fact that a L µµ = a R µµ . Thus first two terms in Eq. (19) provide negativecontribution while the last two terms give positive contribution. We will compare our pre-8iction to the 1 σ range of current estimation, ∆ a µ = (26 . ± . × − , in our numericalanalysis below. D. perturbation of heavier fermions
Before numerical analysis, let us introduce mass eigenvectors and eigenvalues with per-turbation theory up to the first order. In this case, one finds analytical formulations forcomplicated mass matrix. First of all, we assume related mass matrices m, m ′ , M D , M Σ to be real and diagonal for simplicity. Moreover, we impose hierarchy among them; m, m ′ << M D , M Σ , which is a reasonable assumption to satisfy the oblique parame-ters [26, 27]. Then the neutral, singly-, doubly mass eigenvalues and eigenstates are re-spectively given by [28] D N ≈ diag( M D − (2 mm ′ ) / , M D + (2 mm ′ ) / , M Σ ) , V ≈ − i √ i √ i ( − δ ′ + δ ) √ √ δ ′ + δ − δ − δ ′ , (22) D ± ≈ diag( M D − √ m + m ′ , M D + 12 √ m + m ′ , M Σ ) , V C ≈ δ ′ − √ δ δ ′ − √ δ , (23) D ±± ≈ diag( M D , M Σ ) , V CC ≈ − δ ′ δ , (24)where we consider one generation, δ ( ′ ) ≡ m ( ′ ) / ( M Σ − M D ); ( m ( ′ ) << M Σ − M D ), and thesingly-charged sector does not have nonzero perturbations at the first order, Although this simplification is ad hoc, any phenomenologies that we focus on would not be spoiled! II. NUMERICAL ANALYSES
Her we have numerical analysis in a benchmark point. First of all, we fix the followingparameters: ( m , m , m ) = (1 , ,
10) GeV , ( m ′ , m ′ , m ′ ) = (2 , ,
50) GeV , ( M D , M D , M D ) = ( M Σ + 500 , M Σ + 500 , M Σ + 500) GeV . (25)Note that here we adopt m − and m ′ − to be around 1 GeV to 50 GeV that corresponds to g ( g ′ ) ∼ O (0 . −O (0 .
1) since m ( m ′ ) = g ( g ′ ) v/ √ m ( m ′ ) so that the corresponding Yukawa couplings are not very small. Alsowe assume all the components of H to be degenerate. Here we consider the case of fermioncDM which is Σ R in the first generation. . Then we have just refer to the detailed resultsin ref. [17, 32] for relic density of Σ R . Note here that since the quintet Majorana fermionsΣ R does not have Yukawa interactions with SM particles at tree level, the result is basicallythe same as the case of minimal quintet fermion [17, 32]. Also as we assumed above, leptonnumber is conserved at energy scale higher than cut-off so that non-renormalizable operatorinducing decay of Σ R such as ([ ¯ ψ L H ∗ Σ R ] H † H ) does not appear, guaranteeing stability ofDM [33, 34]. Here we neglect the mixing between the neutral components of ψ and Σ that isa natural assumption to satisfy the constraint of oblique parameters. Even though there isthe mixing, the decaying process is induced via 8 mass dimension which can be sufficientlytiny.The typical DM mass is M Σ ∼ O , , = π + 5 . i (6 . i ) , m I = M H + 10 GeV , ( M Σ , M Σ ) = ( M Σ + 1000 , M Σ + 2000) GeV , (26)where O , , are arbitral mixing matrix with complex values that are introduced in theneutrino sector , and M H is considered as a free parameter. Here the values of O , , Since a bosonic DM candidate of φ R dominant can decay into 4 SM Higgs via 5 dimensional operator λH † H H † H H , it could be shorter than the age of Universe even though cut-off scale is Planck scale if λ = O (1) In general, there are 21 free parameters. But here we simply reduce them to be three. article Ψ Q Ψ Q Ψ Q Ψ Q Ψ Q Ψ Q mass [TeV] 4.4(10.0) 5.4(11.0) 6.4(12.0) 4.9(10.5) 5.9(11.5) 6.9(12.5)TABLE II: The benchmark mass spectrum for new fermions in the model where Q = {±± , ± , } . M H GeV M a x @ f D
10 000 20 000 30 000 40 000 50 0000.40.50.60.70.80.91.0 M H GeV M a x @ f D FIG. 3: A line of the maximum absolute value of f (=Max[f]) in terms of M H , where Max[ f ] is0.41(1.0) in the left(right) side that satisfies the perturbation limit ( ∼ √ π ). are chosen so that new physics contributions to muon g − f (=Max[f]) in terms of M H , which suggests Max[ f ] is0.41(1.0) for M Σ ∼ ∼ √ π ) very well. On the other hand, Fig. 4 represents various LFV processesand ∆ a µ in terms of M Σ for M Σ ∼ µ → eγ ),BR( τ → µγ ), BR( τ → eγ ), and − ∆ a µ . The figure suggests 6.6(16) TeV ≤ M H that comesfrom the bound on BR( µ → eγ ). Notice that our muon g − − ∆ a µ ∼ O (10 − ) when we assume degenerate masses of H components; the terms givingnegative contribution in Eq. (19) dominate the terms inducing positive contribution andthis situation could be modified considering hierarchy among masses of H components.The predicted value of ∆ a µ is negative and its absolute value is very small compared withthe discrepancy between experimental data and the SM prediction, ∆ a µ ∼ − . Thus thecontributions in our new particles are not suitable to explain the discrepancy.Here let us briefly comments possible collider physics of our model. Rich phenomenology11 ® e Γ BR H Μ ® e Γ L exp Τ ® ΜΓΤ® e Γ - D a Μ ´ - ´ - ´ - ´ - ´ - M H GeV L F V s a nd D a Μ Μ® e ΓΤ ® ΜΓ Τ® e Γ - D a Μ BR H Μ ® e Γ L exp
12 000 14 000 16 000 18 000 20 00002. ´ - ´ - ´ - ´ - ´ - ´ - ´ - M H GeV L F V s a nd D a Μ FIG. 4: Lines of LFV processes and ∆ a µ in terms of M Σ , where blue, magenta, brown, and redlines respectively denote the theoretical bounds on BR( µ → eγ ), BR( τ → µγ ), BR( τ → eγ ), and-∆ a µ , while the black horizontal line is the experimental bound on BR( µ → eγ ). The left(right)side corresponds to M Σ ∼ at collider experiments can be induced since there are many exotic charged particles fromlarge SU (2) multiplet scalars and fermions as we show in Eq. (4)-(6). The charged particlesin the multiplets can be produced through electroweak interactions at hadron collider exper-iments. For O (1) TeV mass scale, production cross sections can be O (1) fb scale for doublycharged fermion at the LHC 13 TeV [8] and sizable number of events will be obtained withintegrated luminosity of O (100) fb − . We also show doubly charged fermion and scalar pro-duction cross section as functions of their mass in Fig. 5 where we included both Drell-Yanprocess and photon fusion process γγ → Σ ++ Σ −− ( φ ++5 φ −− ), and the center of mass energy √ s = 14 TeV is applied. Then produced charged particles decay into lighter component inthese large multiplets with SM gauge bosons W ± /Z which will be off-shell state as the massdifference is smaller than W ± /Z boson mass. The signatures of our new particles couldbe obtained as multi-particle states including charged leptons, jets and missing transversemomentum. Note, however, that in our benchmark point the production cross section willbe too small. In such a case, collider experiment with higher energy is required such as 100TeV hadron collider. Since the signal is complicated, detailed analysis is beyond the scopeof this paper . The mass scale can be O (1) TeV if we do not require neutral component ofthe multiplet to satisfy observed relic density of DM, and we would get detectable number Collider phenomenology of charged particles in large multiplet is discussed in, for example, refs. [8, 29–31]. p ® Ψ ++ Ψ --
14 TeV
800 1000 1200 1400 1600 1800 20000.0010.010.1110 M S ±± @ GeV D Σ @ f b D pp ® Φ ++ Φ --
14 TeV
800 1000 1200 1400 1600 1800 20000.0010.010.1110 M Φ ±± @ GeV D Σ @ f b D FIG. 5: Doubly charged fermion and scalar production cross sections as functions of their massesat the LHC 14 TeV including both Drell-Yan and photon fusion processes. of signal event at the LHC 14 TeV in near future. Note that in lower mass scale the relicdensity of the neutral component becomes much smaller than observed relic density and it isnot excluded by cosmological constraints. In addition we expect displaced vertex signaturesince lifetime of charged component could be long as τ & cm; for M Σ ∼ . τ ∼ . IV. SUMMARY AND CONCLUSIONS
We have constructed a model in which neutrino mass is induced at one-loop level intro-ducing fields with large multiplet of SU (2) L . Thanks to large representations, our modelhas been realized without any additional symmetries, and we have formulated lepton andnew fermion sector such as mass matrices, LFVs, and muon g −
2. Furthermore, our modelis valid up to 100 TeV at most via RGEs, which could be tested by current experimentssuch as collider. Due to small mixings among neutral fermions that are required by theoblique parameters and heavier extra masses that come from lower mass bound on DMmass 4.4(10) TeV estimated by perturbative(including non-perturbative effect) calculation,13ur LFV processes are enough suppressed but muon g − Acknowledgments
This research was supported by an appointment to the JRG Program at the APCTPthrough the Science and Technology Promotion Fund and Lottery Fund of the KoreanGovernment. This was also supported by the Korean Local Governments - Gyeongsangbuk-do Province and Pohang City (H.O.) H. O. is sincerely grateful for KIAS and all the members. [1] E. Ma, Phys. Rev. D , 077301 (2006) [hep-ph/0601225].[2] L. M. Krauss, S. Nasri and M. Trodden, Phys. Rev. D , 085002 (2003)[arXiv:hep-ph/0210389].[3] M. Aoki, S. Kanemura and O. Seto, Phys. Rev. Lett. , 051805 (2009) [arXiv:0807.0361].[4] M. Gustafsson, J. M. No and M. A. Rivera, Phys. Rev. Lett. , no. 21, 211802 (2013)Erratum: [Phys. Rev. Lett. , no. 25, 259902 (2014)] [arXiv:1212.4806 [hep-ph]].[5] Y. Kajiyama, H. Okada and K. Yagyu, Nucl. Phys. B , 198 (2013) [arXiv:1303.3463 [hep-ph]].[6] T. Nomura, H. Okada and Y. Orikasa, Phys. Rev. D , no. 5, 055012 (2016) [arXiv:1605.02601[hep-ph]].[7] T. Nomura, H. Okada and Y. Orikasa, Phys. Rev. D , no. 11, 115018 (2016)[arXiv:1610.04729 [hep-ph]].[8] T. Nomura and H. Okada, Phys. Rev. D , no. 9, 095017 (2017) [arXiv:1708.03204 [hep-ph]].[9] G. Anamiati, O. Castillo-Felisola, R. M. Fonseca, J. C. Helo and M. Hirsch, arXiv:1806.07264[hep-ph].[10] A. Zee, Phys. Lett. B , 389 (1980) [Erratum-ibid. B , 461 (1980) ].[11] T. P. Cheng and L. F. Li, Phys. Rev. D , 2860 (1980).[12] A. Zee, Nucl. Phys. B , 99 (1986); K. S. Babu, Phys. Lett. B , 132 (1988).[13] K. Cheung, T. Nomura and H. Okada, Phys. Rev. D , no. 11, 115024 (2016)[arXiv:1610.02322 [hep-ph]].
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