Lepton Flavor Violating Z l I lj in SO(3) Gauge Extension of SM

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1 Commun. Theor. Phys. (Beijing, China) 54 (2010) pp c Chinese Physical Society and IOP Publishing Ltd Vol. 54, No. 4, October 15, 2010 Lepton Flavor Violating Z l I lj in SO(3) Gauge Extension of SM WANG Wen-Yu ( ), WANG Yu-Xia ( ), and XIONG Zhao-Hua ( Ù) Institute of Theoretical Physics, Beijing University of Technology, Beijing , China (Received January 27, 2010; revised manuscript received March 29, 2010) Abstract The SO(3) gauge extension of SM, which is proposed to present a successful explanation for the observed small masses of neutrino and the nearly tri-bimaximal neutrino mixing, predicted the vector-lie SO(3) triplet Majorana neutrinos and SU L(2) double Higgs bosons. In this wor we calculate branching ratios of the charged lepton flavor violating decays l I ljv (V = γ, Z) induced by these Majorana neutrinos and Higgs bosons. We find that under the model parameters constrained by experimental bounds on the decays Z l I lj, the branching ratio of decays l I l Jγ can be up to 10 10, which may be accessible at the future experiments. PACS numbers: St, w, Bt Key words: new physics beyond SM, neutrino, gauge theory 1 Introduction Neutrino Oscillations [1 4] prove that neutrinos have mass and present the first evidence for new physics beyond the Standard Model (SM). [5 6] The current experimental data [7] indicate that the neutrino mixing is nearly the tribimaximal mixing, [8] which implies mystic symmetry in lepton flavor sector. Great theoretical efforts have been made to obtain such a mixing matrix via imposing various symmetries. [9 26] In order to solve the puzzle that why neutrino masses are so tiny and their mixing are so large, a simple extension of the standard model with three families and Majorana neutrinos was proposed recently. [26] The author taes a non-abelian gauge family symmetry SO(3) instead of discrete symmetries discussed widely in earlier literature, and constructs a model with gauge symmetry SO(3) SU(3) c SU L (2) UY (1), As a successful explanation for both the tiny masses and the large mixing of the neutrinos, the model can tae the place of the seesaw mechanism. [27 29] It is well nown that the SM predicts an unobservably small branching ratio for any lepton flavor violating (LFV) process, such as l I l J γ or Z l I lj, thus the LFV processes give very stringent constraint on the new physics beyond SM. [30 32] However, in seesaw mechanism, considering that the Majorana neutrinos are so large (> GeV) that it nearly has no effect to the low energy LFV process, it is impossible to extract information of the Majorana. The situation is different in the SO(3) gauge extension of SM in which predicts a minimal set of new particles such as the majorana neutrinos N i and the SU(2) singlet vector-lie charged leptons E i with masses at the order of ( ) GeV. It is thus expected that such light particles in the model may have a reasonable implications in the low energy LFV process. Aim to constrain parameters of the model by analyzing the LFV processes in the model, we will give a brief description of the model in Sec. 2, and explain how the model predicted the lepton flavor violating and calculate the branching ratio of Z l I lj and l I l J γ. In Sec. 3, we will constrain parameters of the model in different initial conditions by the process Z l I lj and then analyze the branching ratio of l I l J γ with the constrained parameters. The conclusion will be given at Sec SO(3) Gauge Extension of SM and New LFV Loop 2.1 SO(3) Gauge Extension of SM It is noted that only SO(3) rather than SU(3) is allowed due to the Majorana feature of neutrinos. SO(3) family symmetry can easily explain the maximal mixing between three generations neutrino. As the current experimental data favor a nearly tri-bimaximal mixing, the paper [26] constructs a model with gauge symmetry SO(3) SU(3) c SU L (2) UY (1). The model introduces some new particles, which include vector-lie SO(3) triplet and SU L (2) singlet charged leptons E i (i = 1, 2, 3 are the flavor number), vector-lie triplet and SU L (2) singlet Majorana neutrinos N i with Ni c = N i, two SO(3) tri-triplet Higgs bosons Φ ν and Φ e, two SU L (2) Higgs doublets (H, H N ) with H = τ 2 H, a singlet Higgs boson φ s, and the right-handed neutrinos ν Ri with ν c Ri = c ν T Ri the charge conjugated ones. The invariant Lagrangian for Yuawa interactions of leptons with Majorana neutrinos is as follows: L Y = y ν l HνR + y N lhn N ξ N NΦ ν N M R ν R ν c R + y e lhe + ξe φ s Ēe R ξ EĒΦ ee + H.c. (1) wywang@bjut.edu.cn wangyuxia1985@ s.bjut.edu.cn xiongzh@ihep.ac.cn

2 710 WANG Wen-Yu, WANG Yu-Xia, and XIONG Zhao-Hua Vol. 54 Note that in the subsection the generation index is suppressed. For each generation, the Yuawa coupling constants y ν, y e, y N, ξ e, ξ N, and ξ E is supposed to be the real, and l i = ( ν Li, ē Li) are SU L (2) doublet leptons, which the SM has introduced. Both SO(3) and SU L (2) gauge symmetries are the discrete symmetries so they can broe down spontaneously. We tae the vacuum expectation values of H, H N, φ s as: H(x) = v, H N (x) = v N, φ s (x) = v s. (2) With such a vacuum structure after spontaneous symmetry breaing, the mass matrix of neutrinos and charged leptons are given by the following generalized seesaw mechanism [27 29] with M ν = m D ν M 1 R md ν + md N M 1 N md N, M e = V e m D E M 1 E md E V e, (3) m D ν = y ν v, m D N = y N v N, m D E = y e ξ e vv s, (4) and V e is unitary, which can be parameterized as c e 12c e 13 s e 12c e 13 s e 13 V e Pδ e s e 12 ce 23 ce 12 se 23 se 13 c e 12 ce 23 se 12 se 23 se 13 s e 23 ce 13, (5) s e 12s e 23 c e 12c e 23s e 13 c e 12s e 23 s e 12c e 23s e 13 c e 23c e 13 with Pδ e = diag (eiδe 1, e iδ e 2, e iδ e 3 ), where δ e i are CP phases arising from spontaneous symmetry breaing. c e ij cosθij e and se ij sin θe ij. θe ij are charged lepton mixing angles, which given as functions of three rotational angles of SO(3). If the Majorana neutrino masses M R and M N to be large, the interactions with Majorana neutrinos will decouple from the theory. Namely, the Yuawa interactions possess approximate global U(1) family symmetries. This can be seen from the following effective interactions mediated via the Majorana neutrinos y 2 ν M R l H HT l c, y 2 N M N lhn H T Nl c 0, for M R, M N. (6) It is a possible explanation why the observed left-handed neutrinos are so light. Based on seesaw mechanism and such a vacuum structure after spontaneous symmetry breaing, the charged lepton flavor violating can be obtained directly by the vetexes y N lhn N, y e lhe, and ξe φ s Ēe R, which come from L = y ei l I P L E φ + (y e y N, φ φ N, E N) + (y e ξ e, φ φ s, P L P R ) + H.c., (7) with φ, φ N being Higgs scalar fields, which engendered by H, H N spontaneous symmetry breaing. These interactions can contribute the (E i and N i ) loop effect for LFV. Note the Yuawa coefficients have the relation: ye j y ei = 0, ξe j ξ ei = 0, yn j y Ni = 0. (8) In the following subsection we will calculate the LFV processes such as l I l J γ, Z l I lj and elevate virtual E i, N i effects in the LFV processes. 2.2 New LFV Calculations The Feynman diagrams of l I Fig. 1. Since l J γ are given by p 2 1 = m2 l I, p 2 2 = m2 l J, p 1 p 2 M 2 φ, M2 E, where p 1, p 2, q are the momentum of the external lepton lines and gauge boson, respectively, the results of l I l J γ can be obtained by the approximate loop function safely. Using the loop integral functions and considering unitarity given by Eq. (8), the total matrix element for the decay l I l J γ is obtained a sum of M a, M b, M c, which are the contributions of (a), (b), (c) in Fig. 1 respectively. M = em li ε µ ū lj [( y e j y ei F + yn j y Ni F )/qγ µ P R + with P R,L = (1 ± γ 5 )/2, ε µ is the polarization vector of photon, and F = 3 C 23 C 11, F = 3 C 23 C 11, F ξ e j ξ ei F /qγµ P L ]u li, (9) = 3 C 23 C 11, C ij = C ij(p 1, p 2 ; m 2 φ s, m 2 E, m 2 E ) are the where C ij = C ij (p 1, p 2 ; m 2 φ, m2 E, m 2 E ), C ij = C ij(p 1, p 2 ; m 2 φ N, m 2 N, m 2 N ), and loop integral functions. The symbols and conventions are in accordance with Refs. [33 34] and the terms proportional to the lepton masses m lj are neglected. Note that the amplitude Eq. (9) satisfies Ward Identities naturally. The branching ratio of l I l J γ is then expressed as Br (l I l J γ) = α m 5 [ l I ye 4 Γ j y ei F + yn 2 2] j y Ni F + li ξe j ξ ei F. (10)

3 No. 4 Lepton Flavor Violating Z l I lj in SO(3) Gauge Extension of SM 711 Neglecting the high order terms in loop-integral functions, we get a simple form as: 16π 2 [ 1 5δ 2δ 2 m F 2 E = 12(1 δ ) 3 δ2 lnδ ] 2(1 δ ) 4 with δ = 16π 2 [ 2δ + 5δ 2 δ3 m 2 E 12(1 δ ) 3 + δ2 lnδ ] 2(1 δ ) 4 F ( mφ m E ) 2, if mφ < m E, with δ = ( me m φ ) 2, if mφ > m E. and F have the similar form with F. The Feynman diagrams of Z l I lj are similar to l I l J γ and the total matrix element amplitude for the decay l I l J γ is obtained a sum of M a, M b, M c too: with M = eε µ y e j y ei ū lj[( y e j y ei F + y N j y Ni F ) γ µ P L + ( F = tanθ W m 2 ZC 23 C ) C 0m 2 E cot2θ W (B 0 + B 1 ), B i = B i (q; m 2 φ N, m 2 N ). Similarly, we can obtain F, F. (11) ξ e j ξ ei F γ µ P R ]u li, (12) Fig. 1 Feynman diagrams of the model contributions to the LFV processes l I l Jγ. Note that in the decay mode Z l I lj, the dot product p 1 p 2 m 2/2, which is at the same order with Z M2 φ and ME 2, thus we cannot use the approximate loop integral functions. Considering the unitarity condition Eq. (8), the ultraviolet divergent parts of the loop function cancel each other. The branching ratio of Z l I lj is then given by: Br (Z l I lj ) = α { m 2 Z 2 2} ye 4 Γ j y ei F + yn j y Ni F Z + ξe j ξ ei F. (13) In the following section, we will numerically analysis for the LFV processes with the formulas list above to constrain the parameter space and to see the implication on the colliders. 3 Numerical Results and Discussions Before going into detailed numerical analysis, we consider the constraints from experimental measurements of neutrino masses and mixings: [26] (i) The Dirac type neutrino mass m D in Eq. (3) is chosen to be at the same order of electron mass: m D (0.1 N N 1.0) MeV; (ii) The vector-lie Majorana neutrino masses are taen as m N2 m N O(500) GeV O(50) TeV,

4 712 WANG Wen-Yu, WANG Yu-Xia, and XIONG Zhao-Hua Vol. 54 m N1 = m N3 = ( )m N2 O(250) GeV O(25) TeV. (14) (iii) The masses of m EI (I = 1, 2, 3) m E1 ( ) GeV, m E2 ( ) GeV, m E3 ( ) GeV. (15) In deriving the masses we have used the relation m EI = (md E )2 m li, (16) and too m D (15 25) GeV. E (iv) The masses of higgs φ, φ s, φ N are chosen as ( ) GeV. Now we consider the current experiment data of LFV lepton decays given by [7] Br (µ eγ) < , Br (τ (e, µ)γ) < (1.1, 0.68) 10 7, CL = 90%, Br (Z τ + µ ) < , Br (Z (µ, τ) + e ) < (1.7, 9.8) 10 6, CL = 95%. (17) Since there exist many parameters in the expressions for the modes Z l I lj and l I l J γ, the better way is focusing on attention on the part, which will give dominate contribution to the LFV progresses. Considering the fact that m E3 ranges at a few hundred GeV, the part seems to be loops with internal line E 3 ((a) and (b) of Fig. 1). To chec this we abide following rule in getting the numerical results and show the contribution of (a), (b), (c) of Fig. 1 separately in Fig. 2: (i) We just calculate each type diagram of Fig. 1 separately and neglect the cross-terms for the purpose of comparison; (ii) Parameters are scanned randomly in ranges given by Eqs. (14) (15) etc.; (iii) Because m E1, m E2 m E3, we ignore the contribution from diagrams with E 1, E 2 in loops. Fig. 2 Scatter plot of branching ratio Br(µ eγ) (a) and Br(Z µ + e ) (b) changes with the lightest SO(3) triplet E. The circle, star, cross points are the contributions of Feynman diagram (a), (b), (c) in Fig. 1 respectively. Circles are total contribution. Comparing the contributions of (a), (b), (c) in Fig. 2, we can see that the contribution of (c) is much smaller than (a) and (b), thus the contribution of Feynman diagram (c) can be neglected. Furthermore, from the Eqs. (10) and (13), one can see clearly that the contributions of (a) and (b) are so similar that for demonstrate how the branching ratios changing with model parameters, considering one of them is sufficient. Results of Br(Z l I lj ) via parameter λ = ye e3 y eµ3 are displayed in Fig. 3. The dot dash line in each plots stands for the experiment upper bound of Br (Z l I lj ) From Fig. 3 we can get the following information: (i) The branching ratios of l I l J γ decrease obviously as m E and m φ getting larger. The conclusion is consistent with the simple form by using the expression in Eq. (11). (ii) The branching ratios of Z l I lj decrease slowly as m φ getting larger but do not decrease when m E getting larger. This can be understood that m Z is still comparable to the mass parameter we scanned, thus the decoupling effects is not so obvious. Note that since the approximate loop functions cannot be used, we used LoopTools [35] to calculation of the branching ratios of Z l I lj. (iii) Under the limits from Z mode, the branching ratio of τ (e, µ)γ can still occur at order 10 10, which are smaller than the current experimental data but may be accessible in future measurements. Of course, this situation is much better than that in the seesaw models.

5 No. 4 Lepton Flavor Violating Z l I lj in SO(3) Gauge Extension of SM 713 Fig. 3 A comparison constraints on parameter λ = y e e3 y eµ3 (Z µ + e and µ eγ) in unit of 10 3 with different higgs mass. In every plot dot, solid, dash line corresponds to m E3 = 100, 250, 400 GeV, respectively, the dot dash line is the current experimental data. 4 Conclusion The SO(3) gauge extension of SM explained the observed small masses of neutrino and the nearly tri-bimaximal neutrino mixing successfully, as well as predicted the vector-lie SO(3) triplet Majorana neutrinos and SU L (2) double Higgs bosons inducing the charged lepton flavor violating decays l I lj V (V = γ, Z). In this paper we constraint the model parameters by using the experimental bounds on the decays Z l I lj. After a series of calculations we find that, the branching ratio of τ (e, µ)γ can occur an order as large as These orders are smaller than the current experimental bound but may be accessible in future detectors. Although different forms of lepton couplings may lead to different size of LFV, we emphasize that it is important to study the order of the rate for the LFV in some typical cases and analyze the possibility to observe l I l J γ and Z l I lj in future experiments. By our calculation results we conclude that, subject to the constraints from the experimental bounds on Z l I lj the decays of l I l J γ can still be sizable in the SO(3) gauge extension of SM, among, which τ (e, µ)γ can occur with branching ratio of and thus maybe accessible at the future experiments such ILC etc. References [1] J.M. Conrad, arxiv:hep-ex/ ; L.Di Lella, in Proc. of the 19th Intl. Symp. on Photon and Lepton Interactions at High Energy LP99, ed. J.A. Jaros and M.E. Pesin, In the Proceedings of 19th International Symposium on Lepton and Photon Interactions at High-Energies (LP 99), Stanford, California, 9-14 Aug. (1999) pp [arxiv:hep-ex/ ]; W.A. Mann, in Proc. of the 19th Intl. Symp. on Photon and Lepton Interactions at High Energy LP99, eds. J.A. Jaros and M.E. Pesin, In the Proceedings of 19th International Symposium on Lepton and Photon Interactions at High-Energies (LP 99), Stanford, California, 9-14 Aug. (1999) pp ; [arxiv:hepex/ ]; E.K. Ahmedov, [arxiv:hep-ph/ ]. [2] Bing-Lin Young, Neutrino Oscillations Lectures Given at the Beijing Summer School in Particle Physics (2000). [3] C. Jarlsog, Phys. Rev. Lett. 55 (1985) [4] Z.Z. Xing, Int. J. Mod. Phys. A 19 (2004) 1. [5] S.L. Glashow, Nucl. Phys. 22 (1961) 579; S. Weinberg, Phys. Rev. Lett. 19 (1967) [6] A. Djouadi, Phys. Rept. 457 (2008) 1. [7] Particle Data Group, the Review of Particle Physics, C. Amsler, et al., Phys. Lett. B 667 (2008) 1. [8] P.F. Harrison, D.H. Perins, and W.G. Scott, Phys. Lett. B 530 (2002) 167. [9] C.I. Low and R.R. Volas, Phys. Rev. D 68 (2003) [10] E. Ma, Phys. Rev. D 70 (2004) R.

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