Lepton family symmetry and possible application to the Koide mass formula

Physics Letters B 649 (2007) 287–291 www.elsevier.com/locate/physletb

Lepton family symmetry and possible application to the Koide mass formula Ernest Ma Physics and Astronomy Department, University of California, Riverside, CA 92521, USA Received 6 December 2006; received in revised form 7 March 2007; accepted 13 April 2007 Available online 18 April 2007 Editor: M. Cvetiˇc

Abstract A finite group generated by four Z3 transformations is applied to lepton families in a supersymmetric model, resulting in the charged-lepton masses mi being proportional to vi2 , where vi are three vacuum expectation values. This may be relevant in obtaining the Koide formula me + √ √ √ mμ + mτ = (2/3)( me + mμ + mτ )2 . © 2007 Published by Elsevier B.V.

The Koide mass formula [1–4] √ 2 2 √ √ (1) me + mμ + mτ 3 is remarkably accurate, and cries out for a possible theoretical explanation. Koide himself has proposed [5] to understand it as me + mμ + mτ =

2 v12 + v22 + v32 = (v1 + v2 + v3 )2 . 3

(2)

In that case, the condition mi ∝ vi2 is required, and must be valid to great precision. To facilitate such a relationship, the following non-Abelian discrete family symmetry is proposed. Consider the basis (a1 , a2 , a3 ) and four separate Z3 transformations: a1 → a2 → a3 → a1 ,

(3) ai → ωai → ω2 ai → ai (i = 1, 2, 3), √ where ω = exp(2πi/3) = −1/2 + i 3/2. The resulting group, call it Σ(81), has 81 elements, divided into 17 equivalence classes. It has 9 one-dimensional irreducible representations 1i (i = 1, . . . , 9) and 8 three-dimensional ones 3A , 3¯ A , 3B , 3¯ B , 3C , 3¯ C , 3D , 3¯ D . Its character table is given in Appendix A, together with the 81 matrices of the defining representation 3A . The key property of Σ(81) which allows this to work is that given just the representations 3A and 3¯ A , there are only three invariants: a1 a1 a1 + a2 a2 a2 + a3 a3 a3 ,

a1 a¯ 1 + a2 a¯ 2 + a3 a¯ 3 ,

a¯ 1 a¯ 1 a¯ 1 + a¯ 2 a¯ 2 a¯ 2 + a¯ 3 a¯ 3 a¯ 3 .

(4)

Consider now the supersymmetric standard model, with lepton superfields Li = (νi , li ),

lic ∼ 3A .

Add the following Higgs superfields:     ξi = ξi+ , ξi0 ∼ 3A , ηi = ηi0 , ηi− , E-mail address: [email protected]. 0370-2693/$ – see front matter © 2007 Published by Elsevier B.V. doi:10.1016/j.physletb.2007.04.020

(5)

(6)

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  ζi = ζi0 , ζi− , σi = σ 0 ∼ 3¯ A , i

  ψi = ψi+ , ψi0 ∼ 3¯ A ,   φ = φ 0 , φ − ∼ 1.

(7) (8)

The relevant superpotential is then given by W = f Li lic ηi + M1 ηi ψi + h1 ζi ψi σi + M2 ζi ξi + h2 ξi σi φ, resulting in the scalar potential  2 V = M1 ψi + f Li lic  + |M2 ξi + h1 ψi σi |2 + |M1 ηi + h1 ζi σi |2 + |M2 ζi + h2 σi φ|2 .

(9)

(10)

Assuming φ 0  = 0 and σi  = 0 (to be discussed later), the supersymmetric minimum V = 0 is obtained for 

   ψi0 = ξi0 = 0,

 0  −h1 ζi0 σi   0  −h2 σi φ 0  ηi = , ζi = , M1 M2 resulting thus in the charged-lepton mass relationship   f h1 h2 σi 2 φ 0  mi = f ηi0 = , M1 M2 exactly as advertised. To obtain Eq. (2), let [5]      1 1 1 χ0 σ1 1 1 ω2 ω χ1 = √ σ2 , 3 1 ω ω2 χ2 σ3

(11)

(12)

(13)

so that σ12 + σ22 + σ32 = χ02 + 2χ1 χ2 ,  1  σ13 + σ23 + σ33 = √ χ03 + χ13 + χ23 + 6χ0 χ1 χ2 . 3 Consider the superpotential

(14) (15)

 1 1 1 1  W = m0 χ02 + m1 χ12 + m2 χ22 + m3 χ1 χ2 + λ χ03 + χ13 + χ23 + 6χ0 χ1 χ2 , (16) 2 2 2 3 which breaks Σ(81) only softly with its bilinear terms. [It is implicitly assumed that all other possible bilinear terms are zero.] The resulting scalar potential is given by     2  2  2 V = m0 χ0 + λ χ02 + 2χ1 χ2  + m1 χ1 + m3 χ2 + λ χ12 + 2χ0 χ2  + m2 χ2 + m3 χ1 + λ χ22 + 2χ0 χ1  . (17) The aim now is to find a supersymmetric minimum, i.e. V = 0, for which χ0 2 = 2χ1 χ2  =

 1 χ0 2 + 2χ1 χ2  , 2

(18)

which is equivalent to [5]  2 1  1 σ1  + σ2  + σ3  = σ1 2 + σ2 2 + σ3 2 , 3 2 i.e. Eq. (2), and then use Eq. (12) to obtain the Koide formula of Eq. (1). This turns out to be extremely simple, i.e. m3 = m0 = 8m1 m2 ,

(19)

(20)

for which −m0 −m1 −m2 (21) , χ1  = , χ2  = . 2λ λ λ The validity of Eq. (20) is of course not understood, just as the absence of other possible bilinear terms in Eq. (16) is not understood. It may be due to some underlying symmetry or dynamics, but whatever the origin, once it is established, Eq. (2) is protected by the unbroken supersymmetry of the theory. Assuming that χi acquire vacuum expectation values not far above the supersymmetry breaking scale, the effective Yukawa couplings of the leptons at the electroweak scale will not deviate very much from their values which predict Eq. (1). As for the neutrino sector [6], it is clear that having only superfields which transform as 3A and 3¯ A will not work, because that necessarily yields a diagonal mass matrix. To obtain a realistic neutrino mass matrix, consider for example scalar superfields χ0  =

E. Ma / Physics Letters B 649 (2007) 287–291

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+ 0 (Δ++ i , Δi , Δi ) which transform as 3A and 3B under Σ(81). Using the multiplication rule

3A × 3A = 3¯ A + 3¯ B + 3¯ B ,

(22)

it is possible to have Mν of the most general form, i.e.   a f e Mν = f b d , e d c

(23)

where (a, b, c) come from 3A and (d, e, f ) from 3B . This makes no prediction, but if the vacuum structure can be restricted, say b = c and e = f , as for example in a model [7] based on S4 , then the approximate tribimaximal mixing of neutrinos may be obtained. Acknowledgements I thank Yoshio Koide for correspondence. This work was supported in part by the US Department of Energy under Grant No. DE-FG03-94ER40837. Appendix A The proposed finite group Σ(81) contains Δ(27) [8,9]. The latter’s character table and defining representation have been given elsewhere [10]. Here the 17 × 17 character table of Σ(81) and the 81 matrices of its defining representation 3A are also given (Tables 1 and 2). The matrices of the 3A representation of Σ(81) are given by      2  1 0 0 ω 0 0 ω 0 0 2 C2 : C3 : C1 : (A.1) 0 1 0 , 0 ω 0 , 0 , 0 ω 0 0 1 0 0 ω 0 0 ω2       2 ω 0 0 1 0 0 ω 0 0 C4 : (A.2) 0 ω 0 , 0 1 0 , 0 ω2 0 , 0 0 ω2 0 0 1 0 0 ω       2 ω 0 0 1 0 0 ω 0 0 2 C5 : (A.3) 0 ω 0 , 0 1 0 , 0 ω 0 , 0 0 ω2 0 0 ω 0 0 1           0 ω2 0 0 ω 0 0 1 0 0 ω2 0 0 1 0 C6 : 0 0 1 , 0 0 ω , 0 0 ω2 , 0 0 ω , 0 0 1 , 2 ω 0 0 ω2 0 0 1 0 0 ω 0 0 ω 0 0 Table 1 Character table of Σ(81) for the 9 one-dimensional representations; n is the number of elements and h is the order of each element Class

n

h

11

12

13

14

15

16

17

18

19

C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17

1 1 1 3 3 9 9 3 3 3 3 3 3 9 9 9 9

1 3 3 3 3 3 3 3 3 3 3 3 3 9 9 9 9

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 ω ω2 1 1 1 1 1 1 ω ω2 ω2 ω

1 1 1 1 1 ω2 ω 1 1 1 1 1 1 ω2 ω ω ω2

1 1 1 1 1 1 1 ω2 ω2 ω2 ω ω ω ω2 ω2 ω ω

1 1 1 1 1 ω2 ω ω2 ω2 ω2 ω ω ω ω 1 ω2 1

1 1 1 1 1 ω ω2 ω2 ω2 ω2 ω ω ω 1 ω 1 ω2

1 1 1 1 1 1 1 ω ω ω ω2 ω2 ω2 ω ω ω2 ω2

1 1 1 1 1 ω ω2 ω ω ω ω2 ω2 ω2 ω2 1 ω 1

1 1 1 1 1 ω2 ω ω ω ω ω2 ω2 ω2 1 ω2 1 ω

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Table 2 Character table of Σ(81) for the 8 three-dimensional representations Class

n

h

3A

3¯ A

3B

3¯ B

3C

3¯ C

3D

3¯ D

C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17

1 1 1 3 3 9 9 3 3 3 3 3 3 9 9 9 9

1 3 3 3 3 3 3 3 3 3 3 3 3 9 9 9 9

3 3ω 3ω2 0 0 0 0 √ −i √3 −i √3ω 2 −i √ 3ω i √3 i √3ω i 3ω2 0 0 0 0

3 3ω2 3ω 0 0 0 0√ i √3 i √3ω2 i √ 3ω −i √3 −i √3ω2 −i 3ω 0 0 0 0

3 3ω 3ω2 0 0 0 0 √ −i √3ω2 −i √3 −i √ 3ω i √3ω i √3ω2 i 3 0 0 0 0

3 3ω2 3ω 0 0 0 0√ i √3ω i √3 i √ 3ω2 −i √3ω2 −i √3ω −i 3 0 0 0 0

3 3ω 3ω2 0 0 0 0 √ −i √3ω −i √3ω2 −i √ 3 i √3ω2 i √3 i 3ω 0 0 0 0

3 3ω2 3ω 0 0 0 0√ i √3ω2 i √3ω i √ 3 −i √3ω −i √3 −i 3ω2 0 0 0 0

3 3 3 3ω 3ω2 0 0 0 0 0 0 0 0 0 0 0 0

3 3 3 3ω2 3ω 0 0 0 0 0 0 0 0 0 0 0 0



C7 :

C8 : C9 : C10 : C11 : C12 : C13 : C14 :

C15 :

0 0 1  0 1 0  0 ω2 0  2 ω 0 0  ω 0 0  ω 0 0  ω 0 0  2 ω 0 0  2 ω 0 0  0 0 1  0 0 ω2  0 0 1

       ω 0 0 1 0 0 ω 0 0 ω2 0 2 2 0 ω , 0 0 ω , 0 0 1 , 0 0 ω , 0 0 ω 0 0 ω2 0 0 1 0 0          0 1 0 0 ω 0 0 ω2 0 0 1 0 0 ω2 2 0 0 , ω 0 0 , ω 0 0 , ω 0 0 , 1 0 0 , 1 0 0 ω 0 0 ω2 0 0 ω2 0 0 ω 0        2 0 ω 0 0 ω 0 0 1 0 0 ω 0 0 , ω2 0 0 , 1 0 0 , ω 0 0 , 1 0 0 ω 0 0 ω2 0 0 1 0   2    ω 0 0 0 0 1 0 0 ω2 0 , 0 1 0 , 0 ω2 0 , 0 1 0 0 ω2 0 0 ω2      0 0 1 0 0 1 0 0 1 0 , 0 ω 0 , 0 1 0 , 0 1 0 0 1 0 0 ω     2  0 0 ω 0 0 ω 0 0 2 ω 0 , 0 ω 0 , 0 ω 0 , 0 ω2 0 0 ω 0 0 ω      0 0 ω 0 0 1 0 0 ω 0 , 0 1 0 , 0 ω 0 , 0 1 0 0 ω 0 0 ω      2 ω 0 0 ω 0 0 0 0 2 2 0 , 0 ω 0 , 0 ω 0 , ω 0 ω 0 0 ω2 0 0 ω2      1 0 0 0 0 1 0 0 2 1 0 , 0 ω 0 , 0 1 0 , 0 0 1 0 0 ω2 0 1          ω 0 0 1 0 0 1 0 0 ω2 0 0 ω2 0 0 1 , 0 0 ω , 0 0 1 , 0 0 ω2 , 0 0 1 , 0 0 1 0 0 ω 0 0 1 0 0 ω2 0 0        1 0 0 ω 0 0 ω 0 0 ω2 0 2 2 0 ω , 0 0 ω , 0 0 ω , 0 0 ω , 0 0 ω2 0 0 ω 0 0 ω 0 0          ω2 0 0 1 0 0 ω 0 0 ω 0 0 1 0 0 1 , 0 0 ω2 , 0 0 1 , 0 0 ω , 0 0 1 , 1 0 0 ω2 0 0 1 0 0 ω 0 0 0 0

(A.4)

(A.5)

(A.6)

(A.7)

(A.8)

(A.9)

(A.10)

(A.11)

(A.12)

E. Ma / Physics Letters B 649 (2007) 287–291



C16 :

C17 :

0 0 ω  0 1 0  0 ω2 0  0 1 0  0 ω 0

       0 0 ω2 0 0 ω2 0 0 ω 0 ω , 0 0 ω2 , 0 0 ω , 0 0 ω2 , 2 0 ω 0 0 0 0 ω2 0 0 ω         ω 0 0 1 0 0 1 0 0 ω2 0 0 2 0 , ω 0 0 , 1 0 0 , ω 0 0 , 1 0 0 0 1 0 0 ω 0 0 1 0 0 ω2        2 0 0 ω 0 0 ω 0 0 ω 0 1 2 0 0 , ω 0 0 , ω 0 0 , ω 0 0 , 0 ω2 0 ω2 0 0 ω 0 0 ω 0         0 ω2 0 0 ω 0 0 0 0 1 0 0 1 2 0 0 , ω 0 0 , 1 0 0 , ω 0 0 , 1 0 0 1 0 0 ω 1 0 0 1 0 0 ω2 0        2 0 0 ω 0 1 0 0 ω 0 0 ω2 2 2 0 0 , ω 0 0 , ω 0 0 , ω 0 0 . ω 0 0 ω 0 0 ω2 0 0 ω2 0 1 0 0 0 0 1

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

Y. Koide, Lett. Nuovo Cimento 34 (1982) 201. Y. Koide, Phys. Rev. D 28 (1983) 252. Y. Koide, Mod. Phys. Lett. A 5 (1990) 2319. For a recent review and other references, see Y. Koide, hep-ph/0506247. Y. Koide, Phys. Rev. D 73 (2006) 057901. Y. Koide, hep-ph/0605074. E. Ma, Phys. Lett. B 632 (2006) 352. G.C. Branco, J.-M. Gerard, W. Grimus, Phys. Lett. B 136 (1984) 383. I. de Medeiros Varzielas, S.F. King, G.G. Ross, hep-ph/0607045. E. Ma, Mod. Phys. Lett. A 21 (2006) 1917.

291

(A.13) ω2 0 0

 ,

(A.14)  ω 0 , 0 (A.15)