Majorana Phase in Minimal S_3 Invariant Extension of the Standard Model

The leptonic sector in a recently proposed minimal extension of the standard model, in which the permutation symmetry S_3 is assumed to be an exact flavor symmetry at the weak scale, is revisited. We find that S_3 with an additional Z_{N} symmetry allows CP violating phases in the neutrino mixing. The leptonic sector contains six real parameters with two independent phases to describe charged lepton and neutrino masses and the neutrino mixing. The model predicts: an inverted spectrum of neutrino mass, tan theta_{23}=1+O(m_e^2/m_mu^2) and sin theta_{13}=m_e/sqrt{2}m_mu+O(m_em_mu/m_{tau}^2) simeq 0.0034. Neutrino mass as well as the effective Majorana massin the neutrinoless double-beta decay can be expressed in a closed form as a function of phi_nu, Delta m^2_{21},Delta m^2_{23} and tantheta_{12}, where phi_nu is one of the independent phases. The model also predictsgeq (0.036 - 0.066) eV.

The Yukawa sector of the standard model (SM), which is responsible for the generation of the mass of leptons and quarks, and their mixing, has too many redundant parameters. This not only weakens the predictivity of the SM, but also makes ambiguous how to go beyond the SM. An exact flavor symmetry could reduce this redundancy, thereby giving useful hints about how to unify the flavor structure of the SM.
Recently, a minimal S 3 invariant extension of the SM was suggested in [1], while assuming that the Higgs, quark and lepton including the right-handed neutrino fields belong to the threedimensional reducible representation of the permutation group S 3 1 . This smallest nonabelian symmetry based on S 3 is only spontaneously broken, because the electroweak gauge symmetry SU(2) L × U(1) Y is spontaneously broken. It was found in [1] that this flavor symmetry is consistent with experiments, and that in the leptonic sector an additional discrete symmetry Z 2 can be introduced. It was argued there that due the additional discrete Z 2 symmetry the neutrino mixing matrix V MNS can not contain any CP violating phase 2 . We now believe this is incorrect, and we would like to re-investigate the leptonic sector of the model in this letter.
We will find that it is possible to introduce two independent CP violating phases [19] in the neutrino mixing even with an additional Z N symmetry in the leptonic sector. The permutation symmetry S 3 with Z N allows three real mass parameters for the charged lepton mass matrix, and three real parameters and two phases for the neutrino mass matrix. The model predicts 3 : an inverted spectrum of neutrino mass, tan θ 23 = 1 + O(m 2 e /m 2 µ ) and sin θ 13 = m e /m µ √ 2+O(m e m µ /m 2 τ ). Neutrino mass as well as the effective Majorana mass < m ee > in the neutrinoless double-β decay can be expressed in a closed form as a function of φ ν , ∆m 2 21 , ∆m 2 23 and tan θ 12 , where φ ν is one of the independent phases. We find that the minimum of m ν 2 as well as < m ee > occurs at φ ν = 0, which is approximately ∆m 2 23 / sin 2θ 12 . Before we will come to our main purpose of the letter, let us briefly summarize the basic ingredient of the S 3 invariant SM of [1]. The quark, lepton and Higgs fields are denoted by Q T = (u L , d L ) , u R , d R , L T = (ν L , e L ) , e R , ν R , H. Each of them forms a reducible representation 1 + 2 of S 3 . The doublets carry capital indices I, J which run from 1 to 2, and the singlets are denoted by Q 3 , u 3R , u 3R , L 3 , e 3R , ν 3R , H 3 . The most general renormalizable Yukawa interactions are given by where 1 A partial list for permutation symmetries is [2]- [17]. See for instance [8] for a review. The basic idea of [1] is similar to that of [2,5,6].
2 See for instance [18] for recent reviews on CP violation in the leptonic sector. 3 Similar but different predictions are obtained from different types of discrete symmetry [10]- [15]. See also [16] and [17].
and the Yukawa coupling matrices are given by [1] Y Further, the Majorana mass terms for the right-handed neutrinos is given by where C is the charge conjugation matrix 4 . Pakvasa and Sugawara [2] analyzed the Higgs potential. The potential they analyzed has not only an abelian discrete symmetry (which we will use for selection rules of the Yukawa couplings), but also a permutation symmetry S 2 : H 1 ↔ H 2 , which is not a subgroup of the flavor group S 3 of the model. We assume throughout this letter that the vacuum can respect this accidental symmetry of the Higgs potential, and is satisfied. [< H 1 >= − < H 2 > would yield the same physics.] Then the Yukawa interactions (1) yield the mass matrices of the general form The Majorana mass for ν L can be obtained from the see-saw mechanism [20], and the corresponding mass matrix is given by . The mass matrices are diagonalized by the unitary matrices U ′ s as The diagonal masses can be complex, and so the physical masses are their absolute values, which we denote by m ν 1 , m ν 2 , m ν 3 , m e , m µ , m τ , etc. It would be certainly desirable to classify, in a similar way as in [21,22], all possible mass matrices that are consistent with an additional discrete abelian symmetry and with experimental data. We, however, leave this program to feature work. Here we simply adopt the result of [1] that 4 Supersymmetrization of the present model has been proposed in [9]. and consequently follows from a Z 2 symmetry. We emphasize that there are a number of different charge assignments of Z N that can yield (7) modulo N should be satisfied to forbid Y e 1 , Y e 3 , Y ν 1 and Y ν 5 . Unfortunately, none of the abelian discrete symmetries above is a symmetry in the quark sector. Note that if Z N is chiral, it is broken by QCD anyway (S 3 is not broken by QCD, because it is not a chiral symmetry.) The symmetry violating effect of the quark sector appears first at the two-loop level in the leptonic sector, so that the violation of Z N in the leptonic sector may be assumed to be negligibly small. Therefore, we throughout neglect that violating effect 6 .
To proceed with our discussion, we calculate the unitary matrix U eL from and all the mass parameters appearing in (11) are real. We find that U eL can be approximately written as [9] U eL ≃ where x = m e 5 /m e 2 ≃ m τ /m µ and y = m e 4 /m e 2 ≃ √ 2m e /m µ . The Majorana masses of the right-handed neutrinos, M 1 and M 3 in (4) which may be complex, can be absorbed by a redefinition of m ν 2 , m ν 4 and m ν 3 , and we may therefore assume that M 1 and M 3 are real. After rescaling of m ν 2 , m ν 4 and m ν 3 as we obtain All the phases in (14), except for one, can be absorbed. Without loss of generality, we may assume that ρ ν 3 is complex. We find that M ν can be diagonalized as where and c 12 = cos θ 12 and s 12 = sin θ 12 . The mixing angle is given by from which we find where t 12 = tan θ 12 , r = ∆m 2 21 /∆m 2 23 . As in [1], we find that only an inverted mass spectrum is consistent with the experimental constraint |∆m 2 21 | < |∆m 2 23 | in the present model. To see this, we first derive where A 1 = sin 2θ 12 + cos 2 θ 12 / tan 2θ 12 , A 2 = sin 2θ 12 − sin 2 θ 12 / tan 2θ 12 .
In fig. 1 we plot m ν 2 versus sin θ 12 for ∆m 2 21 = 6.9 × 10 −5 eV 2 , ∆m 2 23 = 2.5 × 10 −3 eV 2 (best-fit values reported in [25,26,27]) and sin φ ν = 0 (solid), 0.6 (dotted) and 0.96 (dashed). The sin φ ν dependence of m ν 2 is shown in fig. 2 for tan θ 12 = 0.68 and the same values of ∆m 2 21 and ∆m 2 23 as in fig. 1. As we see from (20) and also from fig. 2, m ν 2 assumes at sin φ ν = 0 its minimal value where we have used ∆m 2 23 = (1.3 − 3.0) × 10 −3 eV 2 and sin 2θ 12 = 0.83 − 1.0 [25]- [28]. Now the product U † eL P U ν with P = diag.(1, 1, exp iarg(Y ν 4 )) defines a neutrino mixing matrix, which we bring by an appropriate phase transformation to the popular form We find: for φ 1 + φ 2 ∼ ±π, where φ 1 , φ 2 and φ ν are defined in (15). Since sin 2 2θ 13 ≃ 4.6 × 10 −5 , future oscillation experiments such as J-Park experiment [28] can easily exclude the model. In fig. 3 we plot sin 2α (solid) and sin 2β (dotted) as a function of sin φ ν . As we can see, sin 2α reaches its maximal value 1 at sin φ ν ≃ 0.94. Similarly, the maximal value of sin 2β, which is about 1, occurs at sin φ ν ≃ 0.85. We then consider the effective Majorana mass which can be measured in neutrinoless double β decay experiments. (α is given in (29).) In fig. 4 we plot < m ee > as a function of sin φ ν . As we can see from fig. 4, the effective Majorana mass stays at about its minimal value < m ee > min for a wide range of sin φ ν . Since < m ee > min is approximately equal to m ν 2 ,min (which is given in (25)), it is consistent with recent experiments [29,30] and is within an accessible range of future experiments [31]. An experimental verification of (20), (21) and (27)-(33) would strongly indicate the existence of the smallest nonabelian symmetry based on the permutation group S 3 along with an abelian discrete symmetry Z N at the electroweak scale, where Z N is only an approximate symmetry of the whole theory, but the effect of its violation is of two-loop order in the leptonic sector.   2 21 dependence is very small. and quarks. S 3 , of course, can not explain the hierarchy of the fermion mass spectrum, but S 3 with Z N in the leptonic sector can relate the mass spectrum and mixing in this sector, making testable predictions, which have been re-investigated in the present letter. Therefore, S 3 solves partially the flavor problem of the SM. Since there are three SU(2) L doublet Higgs fields in the model, there exit FCNC processes at the tree level. In [1] the magnitude of various tree level FCNC amplitudes have been estimated, and it has been found that they are sufficiently suppressed. The suppression follows from the smallness of the corresponding Yukawa couplings, where S 3 plays an important role for that smallness. However, we find that ∆m K , the difference of the mass of K L and K S , exceeds the experimental value, unless the mixing of the Higgs fields is fine tuned. This problem is currently under investigation, and we will report the result elsewhere.
It is straightforward to keep the discrete flavor symmetries, S 3 in the hadronic sector and S 3 × Z N in the leptonic sector, in a supersymmetric extension of the standard model [9]. The supersymmetric flavor problem has been investigated there, and it has been explicitly found that thanks to the flavor symmetries the dangerous FCNC and CP violating processes, that originate from soft supersymmetry breaking terms, are sufficiently suppressed, in a similar manner as it was found in [32].