Lepton flavor violation in inverse seesaw model
Ke-Sheng Sun, Tai-Fu Feng, Guo-Hui Luo, Xiu-Yi Yang, Jian-Bin Chen
aa r X i v : . [ h e p - ph ] D ec August 30, 2018 20:52 WSPC/INSTRUCTION FILE mpla˙sun
Modern Physics Letters Ac (cid:13)
World Scientific Publishing Company
LEPTON FLAVOR VIOLATION IN INVERSE SEESAW MODEL
KE-SHENG SUN a,b, ∗ , TAI-FU FENG b , GUO-HUI LUO a , XIU-YI YANG c,d ,JIAN-BIN CHEN ea Department of Physics, Dalian University of Technology, Dalian 116024, China b Department of Physics, Hebei University, Baoding 071002,China c School of Physics, Shandong University, Jinan 250100,China d School of Science, University of Science and Technology Liaoning, Anshan 114051,China e College of Physics and Optoelectronics, Taiyuan University of Technology, Taiyuan 030024,China ∗ [email protected] Received (Day Month Year)Revised (Day Month Year)We analyze the lepton flavor violation processes µ − e conversion, l i → l j γ and l i → l j in framework of the Standard Model (SM) extended with inverse seesaw mechanism, as afunction of ˜ η = 1 − | Det ( ˜ U PMNS ) | that parameterizes the departure from unitary of thelight neutrino mixing sub-matrix ˜ U PMNS . In a wide range of ˜ η , the predictions on the µ − e conversion rates and the branching ratio of µ → eγ are sizeable to be compatiblewith the experimental upper limits or future experimental sensitivities. For large scaleof ˜ η , the predictions on branching ratios of other lepton flavor processes can also bereach the experimental upper limits or future experimental sensitivities. The value of˜ η depends on the determinant of the Majorana mass term M µ . Finally, searching forlepton flavor violation processes in experiment provides us more opportunities for thesearches of seesaw nature of the neutrino masses. Keywords : Lepton flavor violating; Inverse seesaw; Non-unitary.
1. Introduction
Neutrino oscillation experiments 1 , , , l i → l j γ ) and leptonic threebody decays ( l i → l j ), remain highly suppressed, see Table.15 for current exper-imental upper bounds, making them difficult to observe. The limit on branchingratio of µ → eγ is the most recent result given by the MEG experiments at the 90%confidence level 6. Nevertheless, various extensions of the SM, such as the seesawmodel with or without GUT, supersymmetry, Z ′ models, etc., have predicted en-hanced branching ratios of LFV processes to be accessible in current experiments.Thus, searching for LFV processes are a powerful way to prove physics beyond theSM. ugust 30, 2018 20:52 WSPC/INSTRUCTION FILE mpla˙sun Ke-Sheng Sun,etc.
Table 1. Current limits and future expectations for µ − e con-version, l i → l j γ and l i → l j .Channel Limit Future CR ( µ − e, Al ) - 2 . × − ,7,10 − ,8 CR ( µ − e, T i ) < . × − − ,9 CR ( µ − e, SiC ) - 10 − ,10 CR ( µ − e, Au ) < × − - CR ( µ − e, P b ) < . × − - BR ( µ → eγ ) < . × − × − ,11 BR ( τ → eγ ) < . × − . × − ,12 BR ( τ → µγ ) < . × − × − ,13,1 . × − ,12 BR ( µ → e ) < . × − − ,14,10 − ,15 BR ( τ → e ) < . × − × − ,12 BR ( τ → µ ) < . × − − ,13,2 × − ,12 The seesaw mechanisms have been recognized as the most natural scenario forunderstanding the smallness of neutrino mass up to now. In canonical Type(I)seesaw, three right-handed neutrinos are introduced, and to achieve sub-eV rangeof light neutrino masses, Grand Unified (GUT) scale (i.e., 10 GeV) of the right-handed neutrinos is required and that makes the LHC study of the new physicsscale difficult. In order to make the right-handed neutrino masses down to theTeV scale, the small neutrino masses have to be effectively suppressed via othermechanisms rather than the GUT scale, such as radiative generation, small leptonnumber breaking, or neutrino masses from a higher than dimension-five effectiveoperator 16. Another option to relate small neutrino mass to TeV scale physics isthe inverse seesaw mechanism 17 ,
18. The smallness of the light neutrino masses canbe ascribed to the smallness of M µ , which breaks the lepton number by two unity.The smallness of M µ is a key element of the inverse seesaw models. So far, a veryappealing picture is the radiative origin of the two unity lepton number-breakingparameter as it has been proposed in Ref.19: it is induced at two-loop level, thusexplaining its smallness with respect to the electroweak scale (EW). Introducing newscalar fields, the two unity lepton number-breaking term can also be induced at two-loop level and is naturally around the keV scale, while righthanded neutrinos are atthe TeV scale 20. In the supersymmetric inverse seesaw mechanism, the smallnessof M µ was related to vanishing trilinear susy soft terms at the grand unified theory(GUT) scale 21. In warped extra dimension, one can have the M µ smallness dictatedby parameters of order one that govern the location of the 5D profile of the S fieldsin the bulk 22.The effective mass matrix for the light neutrinos is given by m ν ≈ M TD ( M TR ) − M µ ( M R ) − M D . (1)So that scale of M R can be made small and many phenomena due to the non-unitaryfeature of the neutrino mixing matrix can be manifested, such as LFV, CP violationand non standard effects in neutrino propagation 23. Non-unitary mixing betweenugust 30, 2018 20:52 WSPC/INSTRUCTION FILE mpla˙sun LEPTON FLAVOR VIOLATION IN INVERSE SEESAW MODEL light and heavy particles can be large and can be probed at colliders 24 , , , , , , , , ,
34, SM with B - L extension 35 and supersymmetricmodels 36 , , ,
39. It is shown that LFV decay µ → eγ can be sizeable to be ob-served in experiment, where the scale of M R is fixed to 10 GeV and the scale of M µ varies in the range of [10 − , − ] GeV, and to be compatible with the experimentlimit on µ → eγ , large value setup of M µ is favored 23. Assuming ∆ L = 2 interac-tions are absent from the model, i.e., M µ = 0, Ref.31 estimate the BR ( τ → e ) or BR ( τ → eµµ ) can be large as 10 − and the limits are out of date. In the inverseseesaw model, the limits on degenerate values of M R and M µ from the photoniccontribution are much more stringent than from the non-photonic contribution for µ − e conversion in nucleus, and the rates arising from virtual photon exchangeare generically correlated to the µ → eγ decay 34. It is also shown that predictionon the branching ratio of µ → eγ can be within the reach of MEG experiment inB - L extension of the SM with inverse seesaw mechanism 35. In supersymmetricinverse seesaw model, the LFV decays can be enhanced by flavour violating sleptoncontributions, the non-unitary of the charged current mixing matrix or the Higgsmediated processes 36 , ,
38. In the framework of a supersymmetric SO (10) modelwith inverse seesaw 39, the expected branching ratio for ( l i → l j γ ) are several or-ders of magnitude below the future sensitivity in experiment with TeV scale sleptonmass, and for ( l i → l j ) and µ − e conversion, the predictions are much smaller thanwhat can be probed in planned experiments.In SM, the LFV decays mainly originate from the charged current with themixing among three lepton generations. The fields of the flavor neutrinos in chargedcurrent weak interaction Lagrangian are combinations of three massive neutrinos: L = − g √ X l = e,µ,τ l L ( x ) γ µ ν lL ( x ) W µ ( x ) + h.c.,ν lL ( x ) = X i =1 (cid:16) U P MNS (cid:17) li ν iL ( x ) , (2)where g denotes the coupling constant of gauge group SU(2), ν lL are fields of theflavor neutrinos, ν iL are fields of massive neutrinos, and U P MNS corresponds to theunitary neutrino mixing matrix 40 , , l i → l j γ , l i → l j and µ − e conversionas a function of non-unitary parameter ˜ η , which is firstly introduced in Ref.43 inthe SM extended with inverse seesaw mechanism. Moreover, we also investigate thethe dependence of ˜ η on M µ . From this point of view, the paper proposed is differentfrom others. We perform a scan over non-degenerate parameters M R and M µ , whichvary in region of [1 , ] GeV and [10 − , − ] GeV, respectively, by taking accountof the constraints from neutrino oscillation data and several rare decays. We haveugust 30, 2018 20:52 WSPC/INSTRUCTION FILE mpla˙sun Ke-Sheng Sun,etc. give a discussion about the parameter spaces, which is more narrow than Ref.34. For CR ( µ − e, N ucleus ), both photonic and non-photonic contribution are consideredin this paper.The paper is organized as follows. In Section.2, we review the inverse seesawmechanism and give the expressions for the unitary violating parameter ˜ η . Thenumerical results and discussions are presented in Section.3. The conclusion is drawnin Section.4.
2. Inverse seesaw model
The inverse seesaw mechanism can be accommodated in SM by adding two kindof singlet fermions, N iR and S iR , and one gauge singlet scalar Φ to the SM fieldcontent, where N iR ( i = 1,2,3) stand for the usual right-handed neutrinos, S iR ( i =1,2,3) stand for the additional gauge singlet neutrinos, and these two kind fermionsshare opposite lepton number (-1 and 1, respectively). The relevant gauge invariantLagrangian for neutrino masses is given by 17 , , , L = N cR Y ν e Hl L + N cR Y ′ ν S R Φ + 12 S cR M µ S R + h.c., (3)where l L stands for the SU (2) L lepton doublet, e H ≡ iσ H ∗ stands for the Higgsdoublets, Y ν and Y ′ ν are the 3 × M µ is a symmetricMajorana mass matrix. In this mechanism, it introduces an extra U (1) gauge sym-metry into the electroweak model, under which the right-handed neutrino must bea non-singlet. After spontaneous gauge symmetry breaking, the extra U (1) gaugegroup breaks into U (1) Y , the weak hypercharge of the standard model. The invari-ant Lagrangian in Eq.(3) would be: L = ν L M D N cR + N cR M R S R + 12 S cR M µ S R + h.c., (4)where M D = Y ν h H i = υ √ Y ν and M R = Y ′ ν h Φ i are 3 × υ thevacuum expectation value of the SM Higgs boson. It shows that the right-handedneutrino mass term M R conserves lepton number and the Majorana mass term M µ violates the lepton number by two units.The neutrino mass matrix in the flavor basis defined by ( ν L , N cR , S cR ) is given by M = M TD M D M R M TR M µ , (5)where M is a 9 × M D , M R and M µ in Eq.(5) maynaturally have a hierarchy M R ≫ M D ≫ M µ
45. In reality, a small non-vanishing M µ can be viewed as a slight breaking of a global U (1) symmetry. The 9 × M can be diagonalized by the unitary mixing matrix U :ˆ M = U T M U, (6)ugust 30, 2018 20:52 WSPC/INSTRUCTION FILE mpla˙sun LEPTON FLAVOR VIOLATION IN INVERSE SEESAW MODEL and it yields nine mass eigenstates N i . The light neutrino flavour states ν lL couldbe given in terms of the mass eigenstates via the unitary matrix U as ν lL = X i =1 ( U ) li N i . (7)It is obvious that the mixing matrix would be simply the rectangular matrix formedby the first three rows of U in Eq.(6) and the matrix ˜ U P MNS describing the mixingbetween the charged leptons and light neutrinos in inverse seesaw mechanism couldbe written by: ˜ U P MNS = U U U U U U U U U . (8)In inverse seesaw mechanism, U in Eq.(6) is unitary. However, ˜ U P MNS is not unitary.To parametrize this departure from unitary, we could define ˜ η as in Ref.43 by:˜ η = 1 − | Det ( ˜ U P MNS ) | . (9)It has been shown in Ref.43 that large value of ˜ η is responsible for the lepton flavouruniversality violation in K + and π + leptonic decays in SM extended with inverseseesaw mechanism.The diagonalization of M leads to an effective mass matrix for the light neutrinosin the leading order approximation 46, m ν = M TD ( M TR ) − M µ ( M R ) − M D , (10)which indicates that the light neutrino masses vanish in the limit M µ → m ν is diagonalized bythe physical neutrino mixing matrix U P MNS ,ˆ m ν = U TP MNS m ν U P MNS , (11)and, in the standard parametrization 5, U P MNS is given by U P MNS = c c c s s e − iδ − c s s e iδ − c s c c − s s s e iδ c s s s − c s c e iδ c s − s c s e iδ c c × diag (cid:16) e i Φ / , , e i Φ / (cid:17) , (12)where s ( c ) = sin(cos) θ , s ( c ) = sin(cos) θ , s ( c ) = sin(cos) θ , and the exper-imental limits on the mixing angles are given in Table.2. The phase δ is the DiracCP phase, and Φ i are the Majorana phases. The remaining six heavy states havemasses approximately given by M ν ≃ M R .Without loss of generality, we work in a basis where M R is assumed as diagonalmatrix. Using a modified Casas-Ibarra parametrisation 47, which is automaticallyreproducing the light neutrino data, Y ν can be written by Y ν = √ υ V † p ˆ M R p ˆ m ν U † P MNS , (13)ugust 30, 2018 20:52 WSPC/INSTRUCTION FILE mpla˙sun Ke-Sheng Sun,etc. with υ the vacuum expectation value of the SM Higgs boson. ˆ M is the relevantdiagonal matrix of M = M R M − µ M TR , which is diagonalized by matrix V :ˆ M = V M V T = V M R M − µ M TR V T , (14)and R is a 3 × α , α , α : R = c c − c s − s s c s s − c s c c s c c − s s s − s c − c s s s s c c c , (15)with the notation c i = cos α i and s i = sin α i , with i = 1,2,3. For simplify, we willassume R is real in our calculation.The interactions of the nine neutrino mass eigenstates, N i,j , and charged leptons, l i , with the gauge bosons, W ± and Z, are correspondingly given by the Lagrangians: L W ± = g √ U ij ¯ l i γ µ P L N j W − µ + h.c., ( i = 1 , ..., , j = 1 , ..., , (16) L Z = g c w C ij ¯ N i γ µ P L N j Z µ , ( i, j = 1 , ..., , (17)where g is the coupling constant of gauge group SU(2), and c w is the cosine of theweak mixing angle. P L/R = (1 ∓ γ ). C ij is defined as C ij = X α =1 U † iα U αj . (18)Here, C ij is also not unitary.
3. Numerical Analysis
To quantitatively study the non-unitary effect on various LFV processes, we performa scan over the parameter space described as following.Before the calculation, it is clear that present data on neutrino masses andmixing should be accounted for, which are listed in Table.2 5.
Table 2. Neutrino oscillation data from PDG.Parameter Value Parameter Valuesin θ . ± .
024 ∆ m (7 . ± . × − eV sin θ > . | ∆ m | (0 . +0 . − . ) eV sin θ . ± . In calculation, we have randomly varied the values of sin θ , sin θ , ∆ m and ∆ m within 3 σ experimental errors and set the value of sin θ equal to 1.The light neutrino mass spectrum is assumed to be normal ordering, i.e., ∆ m > δ , Φ and Φ are set to zero. The lightest neutrino massugust 30, 2018 20:52 WSPC/INSTRUCTION FILE mpla˙sun LEPTON FLAVOR VIOLATION IN INVERSE SEESAW MODEL would vary in region of [10 − ,
1] eV. We also assume the R matrix angles in Eq.(15)are taken to be real (thus no contributions to lepton electric dipole moments areexpected), and randomly vary in the range [0 , π ]. The use of Y ν in Eq.(13) ensuresus the above neutrino oscillation data satisfied.In SM with inverse seesaw mechanism, the relevant input parameters are theright-handed neutrino mass matrix M R and Majorana mass matrix M µ . Here, asmentioned before Eq.(13), M R is diagonal matrix. We will make the minimal flavorviolation hypothesis which consists in assuming that flavor is violated only in thestandard Dirac Yukawa coupling. Under this simplification the 3 × M µ must be also diagonal. We have randomly varied the entries of ( M R ) ii in the rangeof [1 , ] GeV and ( M µ ) ii in the range of [10 − , − ] GeV. Table 3. Constraints used in the scan over free parameters.Channel Fraction or Limit Channel Fraction or Limit W → eν . ± . W → µν . ± . W → τν . ± . B → eν < . × − B → τν (1 . ± . × − B → µν < . × − D s → µν (5 . ± . × − D s → eν < . × − D s → τν . ± . π → eν (1 . ± . × − K → µν . ± . π → µν . ± . K → eν (1 . ± . × − Z → µτ < . × − Z → eµ < . × − Z → eτ < . × − The experimental measurements of several rare decays should be also consideredcause the parameter spaces are strongly constrained by such measurements. Theserare decays have been investigated in literatures 31 , , ,
48. The non-unitary natureof the neutrino mixing matrix can manifest itself in tree level processes like leptonicdecays of W boson and mesons ( B + , D + s , K + and π + ), and invisible decay of Z boson.It can also manifest in LFV decays of Z boson, LFV rare charged lepton decays like l i → l j γ , l i → l j , and LFV process µ − e conversion in an atom, which proceed viaone loop processes, and hence can be constrained. The current experimental limitsare listed in Table.1 and Table.3 at 1 σ level. Current experimental limits are listedat the 90% confidence level 5(except for Z → e ± µ ± , Z → e ± τ ± and Z → µ ± τ ± for which the 95% C.L bounds are given). We will use these limits to bound theparameter spaces. For the channels listed in Table.3, we require that our numericalresults are compatible with the experimental values within 3 σ experimental errors.We have studied the possible constraints on the mass matrix inputs ( M R ) ,( M R ) , ( M R ) , ( M µ ) , ( M µ ) and ( M µ ) . It has been studied in Ref.34 thatthe region of M µ M R . − has been excluded by considering the constraints from µ − e conversion in various nucleus, where the eigenvalues of both M R and M µ aredegenerate. We display the area plot of ( M µ ) ( M R ) versus ( M R ) in Fig.1. It shows that,without the assumption the eigenvalues of both M R and M µ are degenerate andby considering more constraints, the excluded region is narrowed to M µ M R . − . Inugust 30, 2018 20:52 WSPC/INSTRUCTION FILE mpla˙sun Ke-Sheng Sun,etc. -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 ( M ) / ( M R ) (M R ) Fig. 1. Area plot of ( M µ ) ( M R ) versus ( M R ) . The shadow region is compatible with constraints inTable.1 and Table.3. addition, the blank area in the upper right corner in Fig.1 is also excluded, which isnot displayed in Ref.34. With fixed value of the scale of M R to 10 GeV, the lowervalue of the scale of M µ is approximately given by M µ ∼ − or M µ M R ∼ − in Ref.23 by considering the constraint from µ → eγ , which is also less restrictivethan our result. There are similar correlations for ( M µ ) ( M R ) versus ( M R ) and ( M µ ) ( M R ) versus ( M R ) , which are not displayed to shorten the length of text.We also investigate the dependence of ˜ η on sin θ , sin θ , ∆ m , ∆ m , m ν e , ( M R ) ii and ( M µ ) ii . It displays ˜ η strongly depends on ( M µ ) ii . In Fig.2, wedisplay the determinant Det ( M µ ) versus Log [˜ η ] from a scan over few 10 points inparameter space in the inverse seesaw mechanism. Here, Det ( M µ ) = Q i =1 ( M µ ) ii .It shows that large values of unitary violation ˜ η (e.g., 10 − ) correspond to smallscales of Det ( M µ ) (e.g., 10 − Ge V ) or ( M µ ) ii (e.g., 10 − GeV). In models wherelepton number is spontaneously broken by a vacuum expectation value h σ i
46 onehas ( M µ ) ii = ( λ ) ii h σ i , where M µ is diagonal as assumed. For typical Yukawas( λ ) ii ∼ − one sees that ( M µ ) ii ∼ − GeV corresponds to a scale of leptonnumber violation value h σ i ∼ − GeV 34. Thus, if the LFV processes are observedin experiment, the vacuum expectation value h σ i should be the scale of (1 − − )GeV, under the assumption of typical Yukawas.In Fig.3, we show the area plot of CR ( µ − e, Au ) versus Log [˜ η ] in the inverse see-saw mechanism from the scan over few 10 points in parameter space. The expectedconversion rates CR ( µ → e, Au ) are sizeable to compatible with the experimentalupper limit and future experimental sensitivities in range of 10 − < ˜ η < − . For˜ η < − , the upper limit of the CR ( µ − e, Au ) decreases. The expected conversionrates CR ( µ → e, Au ) could be very small in the whole region of 10 − < ˜ η < − .The area plots for CR ( µ − e, Al ), CR ( µ − e, T i ) and CR ( µ − e, P b ) versus Log [˜ η ]have the same behavior.In Fig.4, we display area plot of BR ( µ → eγ ) and BR ( τ → eγ ) versus Log [˜ η ]ugust 30, 2018 20:52 WSPC/INSTRUCTION FILE mpla˙sun LEPTON FLAVOR VIOLATION IN INVERSE SEESAW MODEL -18 -16 -14 -12 -10 -8 -6 -410 -23 -22 -21 -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 D e t ( M ) Log[ ]
Fig. 2. Area plot of
Det ( M µ ) versus Log [˜ η ]. The shadow region is compatible with constraintsin Table.1 and Table.3. -18 -16 -14 -12 -10 -8 -6 -410 -21 -19 -17 -15 -13 -11 CR ( e , A u ) Log[ ]
Fig. 3. Area plot of CR ( µ − e, Au ) versus Log [˜ η ]. The horizontal solid line denotes the experi-mental bound. The shadow region is compatible with constraints in Table.1 and Table.3. in the inverse seesaw mechanism from the scan over few 10 points in parameterspace. It shows most predictions of BR ( µ → eγ ) are just below the experimentalupper limit in range of 10 − < ˜ η < − . In a narrow range of 10 − < ˜ η < − ,the upper limit of the predictions decreases. The prediction of BR ( τ → eγ ) canreach to the current limits only when ˜ η is large (˜ η > − ). The upper limit ofthe predictions decreases when ˜ η < − . It is noteworthy that µ → eγ is moreconstraining than τ → eγ in most cases from a compare between figures in Fig.4.However, there is still probability that both predictions of these processes are veryclose to the experimental upper limit (˜ η > − ). The area plot for BR ( τ → µγ )versus Log [˜ η ] has the same behavior with BR ( τ → eγ ) versus Log [˜ η ].In Fig.5, we display area plot of BR ( µ → e ) and BR ( τ → e ) versus Log [˜ η ] inthe inverse seesaw mechanism. It displays that the predictions of BR ( µ → e ) andugust 30, 2018 20:52 WSPC/INSTRUCTION FILE mpla˙sun Ke-Sheng Sun,etc. -18 -16 -14 -12 -10 -8 -6 -410 -17 -16 -15 -14 -13 -12 -11 B R ( e ) Log[ ] -18 -16 -14 -12 -10 -8 -6 -410 -19 -17 -15 -13 -11 -9 -7 B R ( e ) Log[ ]
Fig. 4. Area plot of BR ( τ → eγ ) versus Log [˜ η ] and BR ( µ → eγ ) versus Log [˜ η ]. The horizontalsolid line denotes the experimental bound. The shadow region is compatible with constraints inTable.1 and Table.3. -18 -16 -14 -12 -10 -8 -6 -410 -21 -19 -17 -15 -13 -11 B R ( e ) Log[ ] -18 -16 -14 -12 -10 -8 -6 -410 -26 -24 -22 -20 -18 -16 -14 -12 -10 -8 -6 B R ( e ) Log[ ]
Fig. 5. Area plot of BR ( τ → e ) versus Log [˜ η ] and BR ( µ → e ) versus Log [˜ η ]. The horizontalsolid line denotes the experimental bound. The shadow region is compatible with constraints inTable.1 and Table.3. BR ( τ → e ) can reach the experimental limit at large values of ˜ η (about ˜ η > − and ˜ η > − ). Also, the upper limits of predictions decrease when ˜ η < − and ˜ η < − , respectively. The observation of LFV decays µ → e and τ → e indicate the large violation of unitary in light neutrino mixing matrix and a lowervacuum expected value h σ i . The area plot for BR ( τ → µ ) versus Log [˜ η ] has thesame behavior with BR ( τ → e ) versus Log [˜ η ].
4. Conclusions
The non-unitary mixing matrix in seesaw mechanism is a generic feature for theorieswith mixing between neutrinos and heavy states and provides a window to probenew physics at TeV scale.In this paper we have studied lepton flavor violation decays l i → l j γ , l i → l j and µ − e conversion as a function of non-unitary parameter ˜ η in the SM extendedugust 30, 2018 20:52 WSPC/INSTRUCTION FILE mpla˙sun LEPTON FLAVOR VIOLATION IN INVERSE SEESAW MODEL with inverse seesaw mechanism through a scan over the parameter spaces definedfrom the right-handed neutrino mass matrix M R and Majorana mass matrix M µ .Taking account of the constraints from neutrino oscillation and various rare decays,the relevant parameter spaces are more narrow than that in Ref.34. The result showsthat large values of unitary violation ˜ η are related to small scales of Det ( M µ ) or asmall vacuum expectation value h σ i in spontaneously lepton number broken models.In range of 10 − < ˜ η < − , the upper limits of predictions of CR ( µ − e, N ucleus )and BR ( µ → eγ ) can reach the sensitivity of experiment, and is promising to detectdirectly in experiment in near future. In range of 10 − < ˜ η < − , the upper limitsof BR ( τ → e ( µ ) γ ), BR ( µ → e ) and BR ( τ → e ( µ )) can also reach the sensitivityof experiment. Finally, searching for LFV processes can serve as a window to thenew physics of seesaw nature of neutrino masses. Acknowledgements
The work has been supported by the National Natural Science Foundation of China(NNSFC) with Grants No.11275036 and 11047002 and Natural Science Fund ofHebei University with Grant No. 2011JQ05, No. 2012-242.
References
1. Y. Fukuda et al., Phys. Lett. B (1994)237.2. Y. Fukuda et al., Phys. Rev. Lett. (1998)1562.3. Q. R. Ahmad et al., Phys. Rev. Lett. (2002)011301.4. K. Eguchi et al., Phys. Rev. Lett. (2003)021802.5. J. Beringer et al. (Particle Data Group), Phys. Rev. D (2012)010001.6. J. Adam et al. [MEG Collaboration], hep-ex/1303.0754.7. R. J. Abrams et al. [Mu2e Collaboration], arXiv:1211.7019 [physics.ins-det].8. Y. Kuno, Nucl. Phys. Proc. Suppl. (2012) 228.9. R. J. Barlow, ”The PRISM/PRIME project,” Nucl. Phys. Proc. Suppl. (2011)44.10. M. Aoki [DeeMe Collaboration], AIP Conf. Proc. (2012) 599.11. J. Adam et al. [MEG Collaboration], arXiv:1301.7225[physics.ins-det]..12. M. Bona et al.[SuperB Collaboration].hep-ph/0709.0451.13. T. Abe. et al. (Belle II).hep-ph/1011.0352.14. M. Yoshida, ”The MUSIC Project,” AIP Conf. Proc. (2010)400.15. A. Blondel et al., ”Letter of Intent for an Experiment to Search for the Decay µ → e (2009)076.17. R. N. Mohapatra and J. W. F. Valle, Phys. Rev. D (1986)1642.18. R. N. Mohapatra, Phys. Rev. Lett. (1986)561.19. E. Ma, Phys. Rev. D 80, 013013 (2009).20. Federica Bazzocchi. Phys. Rev. D (2011)093009.21. F. Bazzocchi, D. G. Cerdeno, C. Munoz, and J.W. F. Valle, Phys. Rev. D 81, 051701(2010).22. Chee Sheng Fong, Rabindra N. Mohapatra and Ilmo Sung. Phys. Lett. B (2011)171.23. D.V. Forero, S. Morisi, M.Tortola and J.W.F. Valle.JHEP (2011)142. ugust 30, 2018 20:52 WSPC/INSTRUCTION FILE mpla˙sun Ke-Sheng Sun,etc.
24. ISS Physics Working Group collaboration, A. Bandyopadhyay et al.,Rep. Prog.Phys. (2009)106201.25. A. Das and N. Okada, arXiv:1207.3734.26. P. S. B. Dev, R. Franceschini and R. N. Mohapatra.Phys.Rev. D86 (2012) 093010.27. P. Bandyopadhyay, E. J. Chun, H. Okada and J. -C. Park.JHEP 1301 (2013) 079.28. E. Fernandez-Martinez, M. B. Gavela, J. Lopez-Pavon and O. Yasuda, Phys.Lett.B649 (2007) 427.29. S. Goswami and T. Ota, Phys. Rev. D 78, 033012 (2008).30. S. Antusch, M. Blennow, E. Fernandez-Martinez and J. Lopez-Pavon,Phys. Rev. D80, 033002 (2009).31. A. Ilakovac and A. Pilaftsis, Nucl. Phys. B (1995)491.32. S. Antusch, C. Biggio, E. Fernandez-Martinez, M.B. Gavela, J. Lopez-Pavon.JHEP (2006)084.33. R. Alonso, M. Dhen, M. B. Gavela and T. Hambye,hep-ph/1209.2679.34. F. Deppisch, T. S. Kosmas, J. W. F. Valle.Nucl.Phys.B (2006)80.35. W. Abdallah, A. Awad, S. Khalil, H. Okada.Eur. Phys. J. C (2012)2108.36. F. Deppisch and J. Valle.Phys. Rev. D (2005)036001.37. A. Abada, D. Das, A. Vicentea and C. Weiland.JHEP (2011)015.38. A. Abada, D. Das and C. Weiland.JHEP (2012)100.39. P. S. Bhupal Dev and R. N. Mohapatra.Phys. Rev. D (2010)01300140. B.Pontecorvo, Zh. Eksp. Teor. Fiz.JETP (1957)549.41. B.Pontecorvo, Zh. Eksp. Teor. Fiz.JETP (1958)247.42. Z.Maki, M.Nakagawa and S.Sakata, Prog. Theor. Phys. (1962)870.43. A. Abada, D. Das, A.M. Teixeira, A. Vicentea and C. Weiland.JHEP (2013)048.44. Aik Hui Chan, Hwee Boon Low, Zhi-zhong Xing. Phys. Rev. D (2009)073006.45. G. t Hooft, in Proceedings of 1979 Cargese Institute on Recent Developments in GaugeTheories, edited by G. t Hooft et al. (Plenum Press, New York, 1980), p. 135.46. M. C. Gonzalez-Garcia, J. W. F. Valle, Phys. Lett. B (1989)360.47. J. A. Casas, A. Ibarra, Nucl. Phys. B (2001)171.48. A.G. Akeroyd and F. Mahmoudi.JHEP10