For Review Only. General Structure of Democratic Mass Matrix of Lepton Sector in E 6 Model. Canadian Journal of Physics

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1 General Structure of Democratic Mass Matrix of Lepton Sector in E 6 Model Journal: Canadian Journal of Physics Manuscript ID cjp r1 Manuscript Type: Article Date Submitted by the Author: 08-Jan-2018 Complete List of Authors: Ciftci, A.K.; Izmir Ekonomi Universitesi Ciftci, R.; Ege Universitesi, Physics Keyword: Grand Unified Theory, Standard Model, Democratic Mass Matrix, E6, Leptons Is the invited manuscript for consideration in a Special Issue? : 33rd International Physics Conference of Turkish Physical Society

2 Page 1 of 8 Canadian Journal of Physics General Structure of Democratic Mass Matrix of Lepton Sector in E 6 Model A. K. Çiftçi 1 Dept. of Physics, Faculty of Arts and Sciences, Izmir University of Economics, Balçova, Izmir, Turkey R. Çiftçi 2 Dept. of Physics, Faculty of Science, Ege University, Bornova, Izmir, Turkey Abstract An extension of the Standard Model (SM) fermion sector, which is inspired by the E 6 Grand Unified Theory (GUT) model, might be a good candidate to explain a number of unanswered questions in SM. Existence of the E 6 leptons might explain great mass difference of charged and neutral leptons. Also, democracy on mass matrix elements is a natural approach in SM. In this study, we have given general structure of Democratic Mass Matrix (DMM) of lepton sector in E 6 model. 1 kenan.ciftci@ieu.edu.tr 2 rena.ciftci@ege.edu.tr 1

3 Page 2 of 8 I. INTRODUCTION A. Question of mass spectra Mass spectra and mixings of quarks and leptons are one of the most important unsolved problems of particle physics. These masses and mixings originate from interactions with Higgs doublet via spontaneous symmetry breaking. In the Standard Model (SM), a large number of parameters are employed by hand (such as Yukawa couplings) in order to explain the reality. In the framework of the SM, fermions with the same quantum number (electric charge, weak isospin, etc.) are identical before the symmetry breaking applied. Therefore, in the fermion- Higgs interaction, the Lagrangian terms corresponding to fermions with the same quantum numbers should come with equal strength. As a result, one deals with singular mass matrices before the spontaneous symmetry breaking. As a consequence, elements of the mass matrices ought to be equal. B. Democracy in 3-Family SM Since before the spontaneous symmetry breaking, all quarks are massless, there are no differences between d (0), s (0), and b (0) (this is the first assumption). Consequently, mass matrices of quarks are = and = After spontaneous symmetry breaking, mass matrices are = and = 0 0 0, where η η 250GeV is electroweak breaking scale. These equations mean that while all masses of the first and the second family quarks are zero, masses of the b an t quarks are 3 and 3, respectively. Here we assume either only one Higgs doublet gives mass to both up and down sector or different Higgs doublets have almost same electroweak breaking scale (this is the second assumption). Since electroweak breaking scales responsible from Dirac masses to two types of quarks are almost same, it seems natural to make the third assumption; namely, the Yukawa constants for different types of quarks should naturally be equal. However, =10 in 3 family SM. It is even smaller for lepton sector: l =10. To satisfy the third assumption, a fourth generation SM fermions were introduced in number of studies [1-4]. However, usual fourth generation fermions are excluded by experiments on LHC. Therefore, we have to find an 2

4 Page 3 of 8 Canadian Journal of Physics underlying theory to satisfy the third assumption. Actually, it is possible to find a number of models for this approach. In the following, we explain one of those: (3, 3, 1) model of E 6 group. C. E 6 Model as an Extension of SM All the experimental data up to today imposes local gauge group G SM SU(3) c SU(2) L U(1) Y called the Standard Model (SM). However, the SM cannot explain some issues like hierarchical masses and mixing angles, charge quantization, CP violation, etc., and many physicists believe that it does not represent the final theory, but serves as an effective theory, originating from a more fundamental one. Let us consider as an example the extension of the SM fermion sector, which is inspired by the E 6 grand unified theory (GUT) model initially suggested by F. Gursey and collaborators [5,6]. Also, it is known that this model is strongly favored in the framework of SUGRA. The local gauge group SU(3) c SU(3) L U(1) X is a subgroup of the electroweak-strong unification group E 6. The best approach maybe to depart from SM as little as possible. In this regard, SU(3) L U(1) X SU(2) L U(1) Y as a flavor group has been introduced at several times in the literature. Various fermion structures given in a review article [7] are possible in this symmetry. One of those models given on it has been studied number of times in the literature [8-11] (and studied in connection with neutrino oscillations [12]). The definition of quantum number of X arising in the electric charge generators of the gauge group is = + +, where are the Gell-Mann matrices for SU(3) L and I 3 is 3x3 identity matrix. The quark content and quantum numbers of the model are = 3,3,0 3,1, 2 3 3,1, 1 3 3,1, 1 3 for α = 1, 2 corresponding to two of the three families and = 3,3, 1 3 3,1, 2 3 3,1, 1 3 3,1,

5 Page 4 of 8 for the other family (numbers in the parenthesis correspond to (SU(3) c, SU(3) L, U(1) X )). Anomaly free lepton content and quantum numbers in this model are ψ = 1,3, 1 1,1,1 1,1,0 1,1,0 3 for =,,. Where D αl (U 3L ) is a singlet exotic quark of 1/3 (2/3) electric charge and is a singlet exotic lepton of zero electric charge. New gauge bosons for charged currents are K +, K, K 0 and its anti-particle. While K + and K gauge bosons mediate between s and α s, K 0 and it antiparticle will mediate between and [13]. II. LEPTONIC MASS MATRICES IN (3,3,1) E 6 MODEL Similar basis was used for quarks and quark mass matrices were derived and presented for E 6 model in BPU conference [14]. In this section, the leptonic mass matrix for one-family will be written firstly for the E 6 model. Later, it will be generalized for the three-family case. A. Mass Matrix for One-Family E 6 Model To discuss democracy in E 6 (3,3,1) Model, it is beneficial to consider three bases: SU(3) L U(1) X basis {f (0) }, Mass basis {f (m) } and Weak basis {f (w) } Before the spontaneous symmetry breaking, leptons are grouped into following multiplets:,,, ; In one family case, all bases are equal.,,, ; 4,,,. For the lepton sector, we can write the following Lagrangian with the Yukawa terms for only one-family case: L l =Ψ h..

6 Page 5 of 8 Canadian Journal of Physics where,,, and are Yukawa couplings in the SU(3) L U(1) X basis and C is the charge conjugate operator. Vacuum Expectation Values (VEV) of the Higgs fields are: =0,0,, =0,,0 and =,0,0. As it is seen, three massive Higgs bosons are uncoupled from each other completely. In this case, we obtain = for the charged lepton sector (η = η is taken) and a mass matrix form for the neutrino sector in the, basis: = / / For the special case of = = =, mass eigenvalues of the above mass matrix are =0 and =+. To have with a small mass, we have to depart a small amount from the special case given above, such as = = and = =, where ε is chosen very close to one. Now, the neutrino sector has following mass matrix in the, basis: = / /. In this case, mass eigenvalues are obtained by diagonalizing of the above mass matrix = 0, 0 where = and = If one expands terms in square-root to series, masses become = 1 + and =+. For example, since = for one family case, ε = should be taken to obtain correct e and ν e masses. 5

7 Page 6 of 8 B. Mass Matrix for Three-Family E 6 Model Now, we can write 3-family leptonic Yukawa Lagrangian in the SU(3) L U(1) X basis L l =Ψ l l l + + +, + +h.. The diagonalization of mass matrix of each type of leptons and neutrinos, or in other words transition from SU(3) L U(1) X basis to mass basis, is performed by well-known bi-unitary transformation. Then, the transition from mass basis to weak basis result in PMNS matrix = l. This mixing matrix is mediated by bosons. Other possible mixing matrix is = l mediated by bosons. And another possible mixing matrix is = mediated by boson. Before the spontaneous symmetry breaking, all leptons and neutrinos are massless and there is no difference between, and, Consequently, leptons (and neutrinos) with same quantum number are indistinguishable. Therefore, Yukawa couplings should be equal: l l = and =. Another assumption of Yukawa couplings of different type of fermions to be equal is only natural. Therefore, we can generalize neutrino sector mass matrix to three family case: = Moreover, we can write charged lepton sector mass matrix: l =. Mass spectrum of the charged lepton sector is,, =0,0,3. Mass spectrum of the neutrino sector is 6

8 Page 7 of 8 Canadian Journal of Physics,,,,, =0,0, ,0,0, These two mass matrices have nonzero eigenvalues for τ, ν τ and singlet N τ leptons. For example, for ε = , η = η = 250GeV, a = and M = 2000GeV, one obtains: =1.7771GeV, =3.477keV and =4200GeV. To obtain correct mass spectrum and PMNS matrix, one has to depart from democracy slightly. III. CONCLUSION It is shown a correct democratic mass matrix form for an E 6 inspired model. It is demonstrated that big mass difference of τ neutrino and τ lepton comes naturally in this model. For having charged and neutral neutrinos of the first and the second generations to correct mass values, one has to break democracy on {a} Yukawa coupling values slightly. However, this will be considered on future study. [1] A. Celikel, A. K. Ciftci, and S. Sultansoy, Phys. Lett. B 342, 257 (1995). [2] S. Atağ et al., Phys. Rev. D 54, 5745 (1996). [3] A. K. Ciftci, R. Ciftci, and S. Sultansoy, Phys. Rev. D 65, (2002). [4] A. K. Ciftci, R. Ciftci, and S. Sultansoy, Phys. Rev. D 72, (2005). [5] F. Gursey, P. Ramond, and P. Sikivie, Phys. Lett. 60B, 177 (1976). [6] F. Gursey and M. Serdaroglu, Lett. Nuovo Cimento Soc. Ital. Fis. 21, 28 (1978). [7] W. A. Ponce, J. B. Flórez and L. A. Sánchez, Int. J. of Mod. Phys. A, 17, 5, (2002). [8] M. Singer, J. W. F. Valle and J. Schechter, Phys. Rev. D, 22, 738 (1980). [9] R. Foot, H. N. Long and T. A. Tran, Phys. Rev. D, 50, R34 (1994). [10] H. N. Long, Phys. Rev. D, 53, 437 (1996); ibid 54, 4691 (1996). [11] V. Pleitez, Phys. Rev D, 53, 514 (1996). 7

9 Page 8 of 8 [12] T. Kitabayashi and M. Yasué, Phys. Rev. D, 63, (2001). [13] L. A. Sánchez, W. A. Ponce and R. Martínez, Phys. Rev. D 64, (2001). [14] R. Ciftci, A. K. Ciftci, AIP Conf. Proc. 1722, 1, (2016). 8