Polygonal Derivation of the Neutrino Mass Matrix

Transcription

1 UCRHEP-T38 September 4 arxiv:hep-ph/4988 v 9 Sep 4 Polygonal Derivation of the Neutrino Mass Matrix Ernest Ma Physics Department, University of California, Riverside, California 951, USA Abstract Representations of the symmetry group D n of the n sided regular polygon have generic multiplication rules if n is prime. Using D n with n = 5 or greater, a particular well-known form of the Majorana neutrino mass matrix is derived. **To appear in Fizika B (Zagreb) memorial issue dedicated to Dubravko Tadic.

2 The form of the 3 3 Majorana neutrino mass matrix M ν has been the topic of theoretical study for some time. If M ν has less than the full 6 parameters, then there exists at least one relationship among masses and mixing angles, which may be tested against the increasingly more precise experimental data from neutrino oscillations. However, even if such a comparison is successful, the question still remains as to why it has such a form. A possible answer is that it comes from an underlying symmetry. In this paper, it is shown how a c d M (e,µ,τ) ν = c b (1) d b may be derived from D n, the symmetry group of the n sided regular polygon, where n is a prime number, equal to or greater than 5. Consider D 5, the symmetry group of the regular pentagon. It has 1 elements, 4 equivalence classes, and 4 irreducible representations. Its character table is given by Table 1: Character Table of D 5. class n h χ 1 χ χ 3 χ 4 C C C φ 1 φ C φ φ 1 Here n is the number of elements and h is the order of each element. The number φ is the Golden Ratio (or Divine Proportion) known to the ancient Greeks: φ = , () and satisfies the equation φ = φ + 1, (3)

3 which implies that φ k+1 = φf k+1 + F k, (4) where F k are the Fibonacci numbers. [Zadar on the Dalmatian coast in Croatia is an ancient city with a rich history and a university whose origin dates back to One person who taught there was Luca Pacioli, whose famous work Divina Proportione (159) was illustrated by Leonardo da Vinci.] The character of each representation is its trace and must satisfy the following two orthogonality conditions: n i χ ai χ bi = nδ ab, n i χ ai χ aj = nδ ij, (5) C i χ a where n is the total number of elements. The number of irreducible representations must be equal to the number of equivalence classes. The two irreducible two-dimensional representations of D 5 may be chosen as follows. For, let ( ) ( ) 1 ω k C 1 :, C :, (k =, 1,, 3, 4); 1 ω 5 k ( ) ( ) ( ) ( ) ω ω 4 ω ω 3 C 3 :,, C ω 4 4 :,, (6) ω ω 3 ω where ω = exp(πi/5), then is simply obtained by interchanging C 3 and C 4. Note that cos(π/5) = φ 1, cos(4π/5) = φ, (7) as expected. For D n with n prime, there are n elements divided into (n+3)/ equivalence classes: C 1 contains just the identity, C has the n reflections, C k from k = 3 to (n+3)/ has elements each of order n. There are one-dimensional representations and (n 1)/ two-dimensional ones. For D 3 = S 3, the above reduces to the complex representation with ω = exp(πi/3) discussed in a recent review [1]. 3

4 The group multiplication rules of D 5 are: 1 1 = 1, 1 =, 1 =, (8) = , = , = +. (9) In particular, let (a 1, a ), (b 1, b ), then a 1 b + a b 1 1, a 1 b a b 1 1, (a 1 b 1, a b ). (1) Similarly, in the decomposition of, (a b, a 1 b 1 ), and in the decomposition of, (a a 1, a 1 a ), and (a a, a 1 a 1). The most natural assignment of the 3 lepton families under D 5 is (ν i, l i ), l c i 1 +. (11) Assuming two Higgs doublets Φ 1 1, Φ 1, the charged-lepton mass matrix is then of the form a M l = b c, (1) b + c where a, b come from φ 1, and c from φ 1. Redefining l c,3 as l c 3,, M l becomes diagonal with m e = a, m µ = b c, m τ = b + c. Assuming that neutrino masses are Majorana and that they come from the naturally small vacuum expectation values [] of heavy Higgs triplets ξ 1 1, ξ,3, then a c d M ν = c b (13) d b as advertised, where a, b come from ξ1, and c = f ξ 3, d = f ξ. The two texture zeros are the result of the absence of a Higgs triplet transforming as. In the case of D 3 = S 3, there is only one two-dimensional representation, hence these zeros cannot be maintained without also making c = d =. 4

5 The decomposition = holds not only in D 5, but also in D n with n prime and n > 5. For example in D 7, there are 3 two-dimensional irreducible representations, corresponding to the 3 cyclic permutations of C 3 :, ω 6 C 4 :, ω 5 3 C 5 :, ω 4, (14) ω 6 5, (15) ω 4, (16) ω 3 where ω = exp(πi/7). It is clear that 1 1 = , = , (17) etc. Hence Eq. (13) is valid in all these symmetries. Phenomenologically, Eq. (13) has been studied [3] as an example of the class of neutrino mass matrices with two texture zeros. It was first derived from a symmetry (Q 8 or D 4 ) only recently [4]. Whereas Q 8 or D 4 allows other forms, D n with n prime and n 5 allows only Eq. (13). Models based on D 4 Z have also been proposed [5]. The 4 parameters of Eq. (13) imply that m 1,,3 are related to the mixing angles. Given the present global experimental constraints [6]: m atm = ( ) 1 3 ev, sin θ atm >.9, (18) m sol = ( ) 1 5 ev, tan θ sol =.33.49, (19) and sinθ 13 <., the allowed region in the m 3 m plane has been obtained in Ref. [4]. That figure is reproduced here for the convenience of the reader. It shows that there are lower bounds on m and m 3 and that m 3 < m up to about.1 ev. The parameter a in Eq. (13) measures neutrinoless double beta decay and has a lower bound of about. ev in this case. 5

6 This work was supported in part by the U. S. Department of Energy under Grant No. DE-FG3-94ER4837. It is dedicated to the memory of Professor Dubravko Tadic. References [1] E. Ma, talk at SI4, Fuji-Yoshida, Japan, hep-ph/4975. [] E. Ma and U. Sarkar, Phys. Rev. Lett. 8, 5716 (1998); E. Ma, Phys. Rev. Lett. 81, 1171 (1998). [3] P. H. Frampton, S. L. Glashow, and D. Marfatia, Phys. Lett. B536, 79 (); Z.-Z. Xing, Phys. Lett. B53, 159 (); 539, 85 (); M. Frigerio and A. Yu. Smirnov, Phys. Rev. D67, 137 (3). [4] M. Frigerio, S. Kaneko, E. Ma, and M. Tanimoto, hep-ph/ [5] W. Grimus and L. Lavoura, Phys. Lett. B57, 189 (3); W. Grimus, A. S. Joshipura, S. Kaneko, L. Lavoura, and M. Tanimoto, hep-ph/4711. [6] See for example the recent update of M. Maltoni, T. Schwetz, M. A. Tortola, and J. W. F. Valle, New J. Phys. 6, 1 (4). 6

7 m 3 ev m ev.15. Figure 1: Allowed region in m m 3 plane for Eq. (13) 7