Bimaximal Neutrino Mixing in a Zee-type Model with Badly Broken Flavor Symmetry

Transcription

1 University of Shizuoka US August 000 hep-ph/000xxx Bimaximal Neutrino Mixing in a Zee-type Model with Badly Broken Flavor Symmetry Yoshio Koide and Ambar Ghosal Department of Physics, University of Shizuoka 5- Yada, Shizuoka 4-856, Japan Abstract A Zee-type neutrino mass matrix model with a badly broken horizontal symmetry SU(3) H is investigated. By putting a simple ansatz on the symmetry breaking effects of SU(3) H for transition matrix elements, it is demonstrated that the model can give a nearly bimaximal neutrino mixing with the ratio m solar / m atm m e /m µ = , which are in excellent agreement with the observed data. In the near future, the lepton-number violating decay Z µ ± τ will be observed. PACS numbers: 4.60.Pq, 4.60.St,.30.Hv koide@u-shizuoka-ken.ac.jp gp95@mail.a.u-shizuoka-ken.ac.jp

2 Introduction The recent Super-Kamiokande collaboration [] has reported, by comparing the day/night spectrum and results of flux global analysis, that the small mixing angle MSW and justso solutions for active neutrinos are disfavored at 95% C.L. and a mixing with sterile neutrinos is also disfavored at 95% C.L. On the other hands, we have already known that the atmospheric neutrino data suggests a ν µ ν τ mixing with a large mixing angle sin θ []. If we take these experimental results seriously, we are forced to accept only a model which gives a nearly bimaximal mixing among the active neutrinos (ν e,ν µ,ν τ ). We must seek for the origin of the nearly bimaximal mixing. As promising one of such the models, the Zee model [3] is known. In this model, a charged scalar field h + is introduced in addition to the Higgs doublets Φ and Φ ; L = f ij l il iτ l c il h + c Φ T iτ Φ h + i,j i y i l il Φ e ir + h.c. (.) l il = ν il, Φ a = Φ+ a e il Φ 0 a, iτ = 0, (.) 0 where i =,, 3 are family indexes, l c il =(l il) c = Cl T il,andc = c are real mass parameter. The neutrino mass matrix M is radiatively generated as M ij = m 0 f ij (m ej m ei )/m τ, (.3) where m 0 = sin φ tan βm τ 3π v/ ln M, (.4) M sin φ = c v/ M M, (.5) m ei = y i Φ 0 = y i(v/ ) cos β, tanβ = Φ 0 / Φ0, v/ = Φ 0 + Φ 0 = 74 GeV, and M and M are the masses of the charged scalars H + and H +, respectively, which are mass eigenstates of (h +, Φ + ), i.e., H+ H + = cos φ sin φ sin φ cos φ h+ Φ +, (.6) Φ + =Φ + cos β Φ + sin β. (.7)

3 Based on the recent solar and atmospheric neutrino data, Smirnov and Tanimoto [4] have investigated the model in detail, and they have concluded that there is no solution of the solar neutrino problem unless introducing a sterile neutrino ν s, although the model can explain the observed large mixing sin θ atm. However, they have investigated only the case with ε = M / M3 + M3 andtanθ = M 3 /M 3 considering m m 3 (m <m <m 3 ), and they have not considered the case M M 3 M 3.Thecase M M 3 M 3 has been investigated by Jarlskog et al. [5],and they have pointed out that the case can lead to the nearly bimaximal mixing. In the present paper, we put a simple ansatz on the coupling constants f ij and y i under a badly broken flavor symmetry, and thereby we will obtain the nearly bimaximal mixing together with a prediction m solar / m atm m e /m µ = Assumption on the symmetry breaking of SU(3) H We consider a badly broken horizontal symmetry [6] SU(3) H. Weintroduceparameters s i (i =,, 3) as a measure of the symmetry breaking of SU(3) H. In the present paper, we do not touch the origin of the symmetry breaking. Our basic assumption on the magnitudes of the symmetry breaking effects is as follows: The magnitude of the matrix element e i (p) y ij (e i e j ) e j (p), which is a component of 3 3 =+8ofSU(3) H, is proportional to δ ij s i in the limit of p, i.e., y i e i (p) (e i e i ) e i (p) = s i const, (.) while the magnitude of the matrix element ν il (p) f ij (ν c il e jl) e j (p) (i j), which is a component of 3 of SU(3) H, is proportional to Σ k ε ijk s k in the limit of p, i.e., f ij ν il (p) (ν c ile jl ) e j (p) (i j) = k ε ij s k const. (.) Here, note that our requirements are applied in the limit of p, because the state e il (alsoevenν il ) is not eigenstate of the helicity h in finite momentum frame unless the particle is massless. These matrix elements, the left-hand sides of Eqs. (.) and (.), are evaluated as follows: m ei y i u (π) 3 ei (p)u ei (p), (.3) E ei f ij (π) 3 mej E ej mνi E νi u c νil(p)u ejl (p) (i j), (.4) 3

4 respectively, where the spinor u i (p) is normalized as u i (p)u i (p) =. Since in the limit of p,weobtainu ei (p)u ei (p) =and lim p uc νil (p)u ejl(p) = m ej m νi m ej, (.5) so that the assumptions (.) and (.) require y i m ei = s i const, (.6) and f ij (m ej + m ei )= k ε ijk s k const, (.7) respectively, where const includes p. Recalling y i = m ei / Φ 0, we obtain the symmetry breaking effect on the coupling constants f ij f ij ε ijks k m ej + m ei ε ijkm ek m ej + m ei, (.8) i.e., m τ m µ m e f eµ = ε 3 f, f eτ = ε 3 f, f µτ = ε 3 f. (.9) m µ + m e m τ + m e m τ + m µ 3 Mass spectrum and bimaximal mixing By using the results (.9), we obtain the following mass matrix elements M ij : Since m τ m µ m e,weobtain M eµ = ε 3 m τ (m µ m e )fm 0 /m τ, M eτ = ε 3 m µ (m τ m e )fm 0 /m τ, M µτ = ε 3 m e (m τ m µ )fm 0 /m τ. (3.) 0 a a M a 0 b, (3.) a b 0 where a = m µ m τ fm 0, b = m e m τ fm 0. (3.3) 4

5 The matrix form (3.), except for the sign of M 3, is identical with the neutrino mass matrix which has recently been proposed by one of the authors (A.G.) [7] on the basis of discrete Z 3 Z 4 symmetries, and it is know that the matrix form (3.) can lead to the nearly bimaximal mixing. The mixing matrix U and mass eigenvalues m νi are as follows: U = cos θ sin θ 0 sin θ cos θ sin θ cos θ, (3.4) tan θ = ( / 8a + b + b mν =, (3.5) 8a + b b) m ν m ν = ( 8a + b + b ), m ν = ( 8a + b b ), m ν3 = b. (3.6) Since a b in the present model, we obtain m = m ν m ν = b 8a + b ab, (3.7) m 3 = m ν m ν3 a, (3.8) so that we can predict m m 3 together with the (nearly) bimaximal mixing b a = m e m µ = , (3.9) U 0. (3.0) We regard m and m 3 as m solar and m atm, respectively. (3.9) is in excellent agreement with the observed value [8, ] The predicted value ( ) m solar m atm exp. 0 5 ev ev (3.) Note that the neutrino mass hierarchy in the present model is m ν m ν m ν3. Since m 3 m ν, we can obtain the value of m ν (and m ν ) as follows m ν m ν m 3 m atm ev (3.) 5

6 so that we also obtain m ν3 b a m ν = m e mµ m ν = ev. (3.3) It will be hard to detect such small masses of m νi directly. Furthermore, since m ν i m i Uei = 0 due to M = 0 in the present model, it is also impossible to detect the effective mass m ν in the neutrinoless double beta decay. From ( ) m atm = m 3 a mµ fm 0, (3.4) m τ we obtain the numerical result fm ev, (3.5) f sin φ tan β ln M M (3.6) 4 Constraints from the electroweak data The sensitive upper bound on f ij is, at present, given from the µ eν e ν µ decay as derived by Smirnov and Tanimoto [4] f eµ < G F M, (4.) where M is defined by (/M )=(/M )cos φ+(/m )sin φ and G F M = [M/(v/ )], (v/ = 74 GeV). (Our definition of the coupling constants f ij are different from that in Ref. [4] by a factor.) From the relation sin φ = M Φ M M M, (4.) where M Φ is the squared mass of the charged scalar Φ + defined by Eq. (.7) [the M component of the mass matrix M for (Φ +,h + )], we estimate M = M M M cos φ + M sin φ = M M M Φ M, (4.3) for M Φ M 04 GeV. Therefore, we can read the constraint (4.) as f eµ < M v/ 3 0. (4.4) 6

7 Since we expect visible effects of the Zee boson, we consider a value of f eµ as large as possible. For example, we suppose f eµ 0 0,whichmeansf 0 and M 0 4 GeV. We consider that it is likely that the Zee boson has such a intermediate mass scale. The contribution of the Zee boson to the radiative decays µ eγ, τ eγ and τ µγ are proportional to (f µτ f eτ ),(f τµ f µe ) and (f τe f eµ ), respectively, where (f µτ f eτ ) ( me m µ ) f f 4, ( ) (f τµ f µe ) me m µ f f 4, m τ (f τe f eµ ) f 4. (4.5) Although the dominant mode of the radiative decay in the present model is τ µγ, the constraint on the Zee boson contribution in the µ eγ decay is still severe compared with that in the τ µγ, because the present experimental upper limits [9] of the partial decay widths Γ(τ µγ) andγ(µ eγ) areb(τ µγ)/τ(τ) < / s= s and B(µ eγ)/τ(µ) < /. 0 6 s=. 0 5 s, respectively. However, even in the µ eγ decay, as discussed in Refs. [4] and [5], the present experimental upper limit of the decay rate B(µ eγ) cannot give a severe constraint on the magnitudes of f ij. We think that the most promising test of the present model is the observation of a lepton-number violating decay Z τ ± µ, which is caused through Z ν e + ν e and a triangle loop with exchange of the Zee boson [and also through Z H a + + H a (a =, ) ]. Similar lepton-number violating decays Z e i e j (i j) can be caused by the exchange of scalar fermions in a minimal SUSY standard model with explicitly broken R-parity via L-violation [0]. In the present model, it is characteristic that only the dominant mode is Z τ ± µ, which is proportional to (f eµ f eτ ) f 4. We roughly estimate the ratio R = B(Z τ ± µ )/B(Z e + e )as R = Γ(Z µ± τ ( ) Γ(Z e + e ) feµ f eτ 6π ln M ) 0 6, (4.6) m Z wherewehavesupposedf eµ, i.e., f 0. (We must calculate all related diagrams in order to remove the logarithmic divergence. More details of the Z e i e j decays will be given elsewhere.) The present experimental upper limit [9] is B(Z τµ)/b(z ee) <. 0 5 / = We think that the value R 0 6 is within the reach of our near future experiment. 7

8 Another interesting observable quantity is the mixing matrix element U e3. The direct numerical calculation from the expression (3.) [not the approximate expression (3.)] gives the value of the mixing matrix element U e3 U e3 = (4.7) However, the value (4.7) is too small to detect even in the near future, since the present experimental upper bound [] is U e3 < (0. 0.4). 5 Conclusion and discussion In conclusion, we have investigated a neutrino mass matrix based on the Zee model with a badly broken horizontal symmetry. A simple ansatz for the symmetry breaking effects leads to f eµ (m τ /m µ )f, f eτ (m µ /m τ )f and f µτ (m e /m τ )f for the Zee bosonlepton coupling constants f ij. The Zee mass matrix with such coupling constants f ij gives the nearly bimaximal mixing (3.0) and the ratio of the squared mass differences m solar / m atm = m e /m µ = Since the coupling constant c of the Φ T iτ Φ h term in the Lagrangian (.) has a dimension of mass, we think that the Lagrangian (.) is not a fundamental one, but an effective one. In the present model, the horizontal symmetry SU(3) H is badly broken. We do not consider that the broken symmetry in the effective Lagrangian is brought by a spontaneous symmetry breaking. We consider that there is no horizontal symmetry in the fundamental Lagrangian from the beginning. Nevertheless, we have used the prescription of the broken symmetry only for the convenience of the phenomenological treatments. Usually, the assumptions for the symmetry breaking are put on the coupling constants directly, while in the present paper, the requirements are put on the transition matrix elements including the coupling constants. The present prescription is somewhat unfamiliar and strange if quarks and leptons are fundamental entities. The present assumption may be understood by a composite model picture of quarks and leptons in future. The present phenomenological success seems to suggest that the Zee model should be taken seriously. Then, our future tasks will be as follows: What is the meaning of the present prescription for the symmetry breaking? How can the Zee model we embedded into a unification scenario? Acknowledgments We would like to thank Professor M. Matsuda for his helpful discussions, especially, for offering us his detailed calculation on the µ eγ decay. One of the authors (Y.K.) thanks M. Tanimoto for his valuable comments and for informing useful references. One 8

9 of the authors (A.G.) is supported by the Japan Society for Promotion of Science (JSPS), Postdoctoral Fellowship for Foreign Researches in Japan (Grant No. 99). References [] Y. Suzuki, Talk presented at Neutrino 000, Sudbury, Canada, June 000 ( [] Y. Fukuda et al., Phys. Lett. B335, 37 (994); Super-Kamiokande collaboration, Y. Fukuda, et. al., Phys. Rev. Lett. 8, 56 (998); H. Sobel, Talk presented at Neutrino 000, Sudbury, Canada, June 000 ( [3] A. Zee, Phys. Lett. 93B, 389 (980). [4] A. Yu. Smirnov and M. Tanimoto, Phys. Rev. D55, 665 (997). [5] C. Jarlskog, M. Matsuda, S. Skadhauge, M. Tanimoto, Phys. Lett. B449, 40 (999). [6] K. Akama and H. Terazawa, Univ. of Tokyo, Report No.57, 976 (unpublished); T. Maehara and T. Yanagida, Prog. Theor. Phys. 60, 8 (978); F. Wilczek and A. Zee, Phys. Rev. Lett. 4, 4 (979) ; A. Davidson, M. Koca and K. C. Wali, Phys. Rev. D0, 95 (979); J. Chakrabarti, ibid. D0, 4 (979). [7] A. Ghosal, US-00-0 (000) (hep-ph/00047), to be published in Phys. Rev. D. [8] M. Gonzalez-Garcia, Talk presented at Neutrino 000, Sudbury, Canada, June 000 ( [9] Particle data group, Eur. Phys. Jour. C5, (000) ( [0] M. A. Mughal, M. Sadiq and K. Ahmed, Phys. Lett. B47, 87 (998). [] CHOOZ Collaboration, M. Apollonio et al., Report No. hep-ex/