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Communications in Physics, Vol. 26, No. 1 (2016), pp. 1-9 DOI:10.15625/0868-3166/26/1/7860 ON THE CP VIOLATION PHASE IN A NEUTRINO MIXING MODEL WITH AN A4 FLAVOR SYMMETRY PHI QUANG VAN AND NGUYEN THI HONG VAN† Institute of Physics, Vietnam Academy of Science and Technology † E-mail: nhvan@iop.vast.vn Received 09 March 2016 Accepted for publication 14 April 2016 Abstract. Neutrino masses and mixing in an extended standard model acquiring an A4 flavor symmetry are considered. The corresponding three-neutrino mixing matrix obtained via a perturbative method allows us to determine the Dirac CP violation phase (δCP ) as a function of the mixing angles (θ12 , θ23 , θ13 ). Then, numerical values and distributions of δCP are given. The latter values are quite close to the global fits of the experimental data for both the normal ordering and inverse ordering of the neutrino masses. Keywords: neutrino mass and mixing, flavor symmetries, models beyond the standard model. Classification numbers: 14.60.Pq, 11.30.Fs, 12.60.-i. I. INTRODUCTION Standard model (SM) [1–4] has been confirmed as an excellent model of elementary particles and their interactions, especially after the discovery of the Brout-Englert-Higgs boson (Higg boson) by ATLAS and CMS [5, 6], which are the two biggest LHC collaborations (see [7] for a review on the discovering of the Higgs boson). The SM, however, cannot explain a number of phenomena including that of the neutrino masses and mixing [8, 9]. They, therefore, call for an extension of the SM. One of the extensions of the SM which has attracted great interest for last ten years is that with a flavor symmetry. A widely investigated flavor symmetry is based on the discrete group A4 (see, for example, [10,11]). This symmetry can make a neutrino mixing scheme tribi-maximal (TBM). The recent experimental data [12,13], however, shows that the mixing angle θ13 and the Dirac CP-violation phase (CPV) δCP are non-zero. That means that the TBM scheme is no longer valid. Consequently, many attempts to explain these new phenomena have been made. Here, in this paper, as reported in [14], we use a perturbation method to diagonalize a general neutrino mass matrix and obtain a Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino mixing matrix from which a relation between δCP and the mixing angles θi j can be derived. This relation c 2016 Vietnam Academy of Science and Technology 2 PHI QUANG VAN AND NGUYEN THI HONG VAN allows us, using experimental data of the mixing angles, to calculate δCP numerically, for both the normal ordering (NO) and inverse ordering (IO) of neutrino masses. This means that the present paper is an extended work of [14] in which only δCP for an NO was considered. The plan of this paper is the following. In the next section we will make a quick introduction to the representations of A4 group and an A4-symmetric neutrino mass model. Section 3 deals with a perturbation approach to the problem of neutrino masses and mixing for this model, from where a mass spectrum and a relation between δCP and θi j can be obtained. Then, numerical values and distributions of δCP in both NO and IO are presented. II. NEUTRINO MASS MODEL WITH AN A4 FLAVOR SYMMETRY Quick introduction to representations of A4 group Let us give a concise introduction to representations of A4 which are widely given in the literature (see, for example, [15, 16]). This group is composed of 12 elements and generated by two basic permutations S and T, S2 = T 3 = (ST )3 = 1. (1) It has a three-dimensional unitary representation generated by     1 0 0 0 1 0 S =  0 −1 0  , T =  0 0 1  , (2) 0 0 −1 1 0 0 0 00 and three one-dimensional unitary representations 1, 1 and 1 generated by 1 : S = 1 T = 1, (3a) 0 1 : S = 1 T = ei2π/3 ≡ ω, 00 (3b) 2 i4π/3 ≡ω . (3c) 1 : S=1 T =e As in the case of an arbitrary group, applications of A4 often require to know the multiplication rule of its (irreducible) representations 1 × 1 = 1, 0 00 1 × 1 = 1, 00 0 1 × 1 = 1, 0 0 0 00 1 ×1 = 1 , 00 00 0 1 ×1 = 1 , (4a) 00 3 × 3 = 1 + 1 + 1 + 31 + 32 . (4b) Let us explain the rule (4b) in more details, while the first ones are trivial. For two triplets, say 3a ∼ (a1 , a2 , a3 ) and 3b ∼ (b1 , b2 , b3 ), their direct product can be decomposed into irreducible representations in the following way: 0 1 = a1 b1 + a2 b2 + a3 b3 , (5a) 2 (5b) 1 = a1 b1 + ω a2 b2 + ωa3 b3 , 00 2 1 = a1 b1 + ωa2 b2 + ω a3 b3 , (5c) 31 ∼ (a2 b3 , a3 b1 , a1 b2 ), (5d) 32 ∼ (a3 b2 , a1 b3 , a2 b1 ). (5e) The information given here is used in constructing an A4-invariant action (Lagrangian) of the considered model, including the Yukawa terms (6). ON THE CP VIOLATION PHASE IN A NEUTRINO MIXING MODEL WITH AN A4 FLAVOR SYMMETRY 3 The model generality In principle, an A4 flavor symmetry can be applied to a neutrino mixing model with an arbitrary number of Majorana neutrinos but here we will consider, as an example, a model with three active Majorana neutrinos, for which the neutrino mass matrix is symmetric and parametrized by 6 independent parameters. Similar models have been studied widely before (see, for example, in [17–19]) but the main goal of this paper (and [14]) is to concentrate on considering the Dirac CPV phase (see also [20]). Besides the SM leptons transforming now also under A4 as a triplet (left-handed leptons lL , l = e, µ, τ) or singlets (right-handed leptons eR , µR and τR ), the lepton sector of this model includes an A4 triplet N which is an SU(2)L singlet (right-handed neutrino). The scalar sector consists of the SM Higgs φh which is an A4 singlet, and five new SU(2)L -singlet 0 00 scalars including two A4-triplets (denoted as φE , φN ) and three A4-singlets (denoted as ξ , ξ , ξ ). The A4- and SU(2)L transformation rules of the leptons and the scalars in the present model are listed in the table given below. It is enough for us to construct an A4-invariant Lagrangian. Table 1. A4 and SU(2)L symmetry of leptons and scalars. A4 SU(2)L lL eR 3 1 2 1 µR τR N 0 00 1 1 3 1 1 1 φE 3 1 φN 3 1 ξ 1 1 0 00 ξ 0 1 1 ξ 00 1 1 φh 1 2 The Lagrangian of the model includes the new Yukawa term
00 φE
0 φE φE −LYnew = λe (l L φh )eR + λµ l L φh µR + λτ l L φh τR + λN l L φ̃h N Λ Λ Λ


00 0
0 00 + cN N c N φN + cξ N c N ξ + cξ 0 N c N ξ + cξ 00 N c N ξ + H.c. (6) From here, by denoting the VEVs of the scalars as 0 00 hξ i = ua , hξ i = ub , hξ i = uc , hφE i = (u1 , u2 , u3 ) , hφN i = (v1 , v2 , v3 ) , hφh i = v, the mass matrix of the charged leptons has the form   κe u1 κe u2 κe u3 Mlept = v  κµ u1 ωκµ u2 ω 2 κµ u3  , κτ u1 ω 2 κτ u2 ωκτ u3 (7) (8) where λµ λe λτ , κµ = , κτ = . (9) Λ Λ Λ To keep maximally the SM Higgs VEV structure, we can assume hφE i = (u, u, u), thus, the charged lepton mass matrix becomes     ye v 0 0 me 0 0 Mlept = UL  0 yµ v 0  ≡ UL  0 mµ 0 , (10) 0 0 yτ v 0 0 mτ κe = 4 PHI QUANG VAN AND NGUYEN THI HONG VAN where   1 1 1 UL = √13  1 ω ω 2  , 1 ω2 ω ye = uκe , yµ = uκµ , yτ = uκτ , and me , mµ , mτ are the charged lepton masses. (11) (12) In the neutrino subsector, we obtain the following Majorana mass matrix     ca + cb + cd ε3 ε2 a b c  ≡  b e d , ε3 ca + ωcb + ω 2 cd ε1 MN =  2 c d f ε2 ε1 ca + ω cb + ωcd (13) where ca = cξ ua , cb = cξ 0 ub , cd = cξ 00 uc , ε1 = cN v1 , ε2 = cN v2 , ε3 = cN v3 , and Dirac mass matrix     λN1 v 0 0 x 0 0 λN2 v 0  ≡  0 y 0 . MD =  0 (14) 0 0 λN3 v 0 0 z The seesaw mechanism gives us the neutrino mass matrix, Mν = −MDT MN−1 M D , (15) which, with (13) and (14) taken into account, can be written as
  d 2 − e f x2 (−cd + b
f )xy (−bd + ce)xz −1  , (−cd + b f )xy c2 − a f y2 (−bc + ad)yz Mν =
2 det(M) 2 (−bd + ce)xz (−bc + ad)yz b − ae z
with det(M) = 2bcd − c2 e − b2 f − a d 2 − e f being the determinant of MN . (16) Diagonalizing a neutrino mass matrix, such as the one in (16), is often a difficult task. To solve this problem different methods and tricks have been proposed. Here we will use a perturbation approach to solving this problem. III. NEUTRINO MIXING AND THE CP VIOLATION PHASE Let us work in the basis where the lepton mass matrix is diagonal. In this basis Mν becomes   A B C Mν = ULT Mν UL ≡  B E D  . (17) C D F The current experimental data have shown that the PMNS neutrino mixing matrix UPMNS is a small deviation from the TBM form and we have (see [14] and also [21]) Mν = M0 + λV, with   A B −B E −(A − E + B) , M0 =  B −B −(A − E + B) E (18)   0 0 e1 λV =  0 0 e3  , e1 e3 e2 (19) ON THE CP VIOLATION PHASE IN A NEUTRINO MIXING MODEL WITH AN A4 FLAVOR SYMMETRY 5 where λ is a perturbation, i.e., small, parameter and M0 is a non-perturbative (TBM level) mass matrix which can be diagonalized by the TBM neutrino mixing matrix q  q  2 1 0  q3 q3 q 
 1 1 1  × P ∼ |10 i, |20 i, |30 i × P, UT BM =  − 6 (20)   q q3 q 2  1 1 − 13 6 2 as diag(m01 , m02 , m03 ) = UT†BM M0UT BM . (21) Here P is a matrix of Majorana phases which are not given explicitly because they play no role in the CP violation process. Using the perturbation expansion of the basis |ni around the nonperturbative one, |n0 i, |ni = |n0 i + λ Vkn ∑ |k0 i m0 − m0 + O(λ 2 ); n k6=n n, k = 1, 2, 3, (22) k where Vkn = hk|V |ni, we can diagonalize the matrix Mν by the PMNS matrix  q   UPMNS =   q 2 1 3+ q 3X q q 1 − 6 + 13 X + 12 Y q q q 1 1 1 − X + 6 3 2Y q q 1 2 3− 3X q q q 1 1 1 + X + Z q2 q3 q6 − 13 − 16 X + 12 Z (23) q q − 23 Y − 13 Z q q q 1 1 1 + Y − Z q2 q6 q3 1 1 1 2− 6Y + 3Z     × P + ∆U, (24)  where ∆U is a higher order perturbative correction to UPMNS and √  √      2 2e3 − e1 − e2 3 2e1 + e2 1 e1 − e2 √ X= , Y= , Z= . 6 6 m01 − m03 m01 − m02 6 m02 − m03 (25) Here we discuss only the CP-violation Dirac phase δCP , while the mass spectrum derived by the above-described diagonalization of the neutrino mass matrix Mν is a subject of a future work because more analysis on the VEV’s of the scalars and the Yukawa coupling coefficients as well as different phenomena and experimental results is needed. Since the CP violation is expected to be small the symmetric matrix Mν is near to a Hermitian matrix, therefore, the matrix (24) is near to an unitary one. Denoting the matrix elements of (24) by Ui j , i, j = 1, 2, 3, we get the relation √ √ √ U21 + 2U22 −U31 − 2U32 = 2U11 − 2U12 + ∆Ui j , (26) where ∆Ui j is a higher order perturbative correction. Fitting the matrix (24) with the elements of the matrix UPMNS in the canonical form   c12 c13 s12 c13 s13 e−iδ UPMNS =  −c23 s12 − s13 s23 c12 eiδ c23 c12 − s13 s23 s12 eiδ (27) s23 c13  × P, iδ iδ s23 s12 − s13 c23 c12 e −s23 c12 − s13 c23 s12 e c23 c13 , 6 PHI QUANG VAN AND NGUYEN THI HONG VAN where si j = sin θi j , ci j = cos θi j ; i, j = 1, 2, 3, we obtain from (26) the following relation between δCP ≡ δ and the mixing angles θi j : ! √ √ 2 1 + tan θ23 ( 2 − tan θ12 ) √ cos δ tan θ13 = − . (28) c13 (1 + 2 tan θ12 )(1 − tan θ23 ) c23 Solving (28) for δ ∈ [0, 2π] we can get two solutions: if a value δ0 is a solution of (28), the value 2π − δ0 is the other solution. Therefore, we can choose to discuss one of these solutions, for example, the bigger one. In general, both theoretically and experimentally, it is not very easy to find δCP but the relation (28) allows us to determine δCP numerically via the experimental data on the mixing angles θi j . The distributions of δCP are plotted in Fig. 1 and Fig. 2 for a normal- and an inverseordering. Here, a value of δCP is calculated event by event using equation (28) with the value of sin θi j taken randomly (within 3σ range) based on a Gaussian distribution having the mean and the standard deviation (σ ) to be the best fit value and the sigmas, respectively, of sin θi j determined experimentally and given in [12, 13]. These figures show that δCP distributes in the region 3.1 < δCP < 5.8 (for an NO) and 3.1 < δCP < 6.3 (for an IO). This distributions have a mean value at δCP = 4.28 (for an NO) and at δCP = 4.56 (for an IO) which is close, between 1σ , to the global fit values (GFV’s) to the experimental data and have a maximum density around δCP = 4.45 (for an NO) and δCP = 4.55 (for an IO) between 1σ region from the GFV [12,13]. For more information, δCP versus sin2 θ13 are depicted in Fig. 3 for an NO and in Fig. 4 for an IO. Entries Distribution of δ CP (for an NO) Mean RMS 1.999 Mean 0.2568 RMS 4.284 0.2549 2000 1800 1600 1400 1200 1000 800 600 400 200 0 0 1 2 3 4 5 6 7 δCP Fig. 1. Distribution of δCP for a normal ordering. ON THE CP VIOLATION PHASE IN A NEUTRINO MIXING MODEL WITH AN A4 FLAVOR SYMMETRY Entries Distribution of δ CP (for an IO) Mean RMS Mean RMS 1.722 0.5166 4.561 0.5157 1200 1000 800 600 400 200 0 0 1 2 3 4 5 6 7 δCP Fig. 2. Distribution of δCP for an inverse ordering. δCP δ CP versus sin2θ13 (for an NO) Mean x 0.02339 Mean y 3.142 RMS x 0.00198 RMS y 1.171 7 6 5 4 3 2 1 0 0.01 0.015 0.02 0.025 0.03 0.035 0.04 sin θ13 2 2 Fig. 3. δCP versus sin θ13 for a normal ordering. 7 8 PHI QUANG VAN AND NGUYEN THI HONG VAN δCP δ CP versus sin2θ13 (for an IO) Mean x 0.02395 Mean y 3.141 RMS x 0.002081 RMS y 1.511 7 6 5 4 3 2 1 0 0.01 0.015 0.02 0.025 0.03 0.035 0.04 sin θ13 2 2 Fig. 4. δCP versus sin θ13 for an inverse ordering. IV. CONCLUSION A model of neutrino mass and mixing with an A4 flavor symmetry is considered. The mass matrix is diagonalized and the mixing matrix is found by using a perturbation method. Then, a relation between the CP-violation Dirac phase δCP and the mixing angles θi j can be established. It allows us to determine δCP via the experimental data on θi j . 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