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Prediction of Leptonic CP Phase in A symmetric model 4 Sin Kyu Kanga and Morimitsu Tanimotob a Institute for Convergence Fundamental Study, School of Liberal Arts, Seoul-Tech, Seoul 139-743, Korea b Department of Physics, Niigata University, Niigata 950-2181, Japan 5 1 (Dated: January 30, 2015) 0 2 Abstract n a We consider minimal modifications to tribimaximal (TBM) mixing matrix which accommodate J 9 non-zero mixing angle θ and CP violation. We derive four possible forms for the minimal modifi- 2 13 cations to TBM mixing in a model with A flavor symmetry by incorporating symmetry breaking ] 4 h p terms appropriately. We show how possible values of the Dirac-type CP phase δ can be predicted D - p with regards to two neutrino mixing angles in the standard parametrization of the neutrino mixing e h [ matrix. Carrying out numerical analysis based on the recent updated experimental results for neu- 1 trino mixing angles, we predict the values of the CP phase for all possible cases. We also confront v 8 our predictions of the CP phase with the updated fit. 2 4 7 0 . 1 0 5 1 : v i X r a 1 I. INTRODUCTION Establishing leptonic CP violation (LCPV) is one of the most challenging tasks in future neutrino experiments [1]. The relatively large value of the reactor mixing angle measured with a high precision in neutrino epxeriments [2] has opened up a wide range of possibilities to explore CP violation in the lepton sector. The LCPV can be induced by the PMNS neutrino mixing matrix [3] which contains, in addition to the three angles, a Dirac type CP violating phase in general as it exists in the quark sector, and two extra phases if neutrinos are Majorana particles. Although we do not yet have compelling evidence for LCPV, the current global fit to available neutrino data indicates nontrivial values of the Dirac-type CP phase [4, 5]. In this situation, it must deserve to predict possible size of LCPV detectable through neutrino oscillations. From the point of view of calculability, much attention has been paid to the prediction of the Dirac type LCPV phase with regards to some observables [6]. Recently, it has been shown [7] that Dirac-type leptonic CP phase can be particularly predictable in terms of neutrino mixing angles in the standard parameterization of PMNS mixing matrix [8]. Before the measurements of the reactor mixing angle, the fit to neutrino data was con- sistent with the so-called tribimaximal (TBM) neutrino mixing matrix, UTBM, which is 0 theoretically well motivated flavor mixing pattern [9]. However, it should be modified to accommodate non-zero reactor mixing angle as well as CP violation. Although the current neutrino data rule out the exact TBM mixing pattern, it can be regarded as leading order approximation. Among various possible modification to U , as discussed in [7], the TBM minimal modificaton is useful to predict Dirac type CP phase. The minimal modification is to multiply UTBM by a rotation matrix in the (i,j) plane with an angle θ and a CP phase 0 ξ, U (θ,ξ), whose form is given either U†(θ,ξ)UTBM or UTBMU (θ,ξ) [10]. Among them, ij ij 0 0 ij U† (θ,ξ)UTBM and UTBMU (θ,ξ) are ruled out because they lead to zero reactor mixing 23 0 0 12 angle. So, all possible forms of minimal modification to TBM mixing matrix are as follows:  UTBMU (θ,ξ) (Case–A),  0 23   UTBMU (θ,ξ) (Case–B), 0 13 V = (1) U† (θ,ξ)UTBM (Case–C),  12 0   U† (θ,ξ)UTBM (Case–D). 13 0 While the study in [7] has not accounted for the origin of such modification to UTBM, 0 2 in this paper, we first study how such a minimally modified TBM mixing pattern can be achieved in a neutrino model with A flavor symmetry by incorporating A symmetry 4 4 breaking terms appropriately. Then, following [7], we investigate how the Dirac type CP phase can be predicted based on the updated fit results for neutrino mixing angles [5]. As shown later, comparing with the results obtained in [7], the Dirac type CP phase predicted based on the updated fit results has different implication particularly at 1σ C.L. II. MINIMAL MODIFICATIONS TO TRI-BIMAXIMAL MIXING IN A SYM- 4 METRIC MODEL In [11], an A symmetric model for neutrino masses and mixing has been proposed to 4 accommodate non-zero mixing angle θ on top of TBM mixing. Based on the A symmetric 13 4 model, we study how the forms given in Eq.(1) can be derived by incorporating appropriate A symmetry breaking terms. 4 A. Case-A As proposed in [11], A flavor symmetry allows the charged-lepton mass matrix to be 4 diagonalized by the Cabibbo-Wolfenstein matrix [12],   1 1 1 1   U = √ 1 ω ω2 , (2) CW   3   1 ω2 ω where ω = e2πi/3, with three independent eigenvalues, m ,m ,m . This can be realized by e µ τ the lepton assignments, L = (ν ,l ) ∼ 3, lc ∼ 1, lc ∼ 1(cid:48), lc ∼ 1(cid:48)(cid:48) with 3 Higgs doublets i i i 1 2 3 Φ = (φ0,φ−) ∼ 3. Introducing 6 heavy A Higgs singlets and triplet: i i i 4 η ∼ 1, η ∼ 1(cid:48), η ∼ 1(cid:48)(cid:48), η ∼ 3, (3) 1 2 3 i(=4,5,6) where η = (η++,η+,η0), one can obtain the neutrino mass matrix in the A basis [11] i i i i 4   a+b+c f e   M =  f a+ωb+ω2c d , (4) ν     e d a+ω2b+ωc 3 where a comes from < η0 >, b from < η0 >, c from < η0 >, d from < η0 >, e from < η0 >, 1 2 3 4 5 f from < η0 >. To achieve TBM mixing pattern of the neutrino mixing matrix, A flavor 6 4 symmetry should be broken to Z in such a way that b = c and e = f = 0. Then, the 2 neutrino mass matrix in the flavor basis where the charged lepton mass matrix is diagonal is given by   a+(2d/3) b−(d/3) b−(d/3)   M(e,µ,τ) = U† M U∗ =  b−(d/3) b+(2d/3) a−(d/3) , (5) ν CW ν CW     b−(d/3) a−(d/3) b+(2d/3) which is diagonalized by the TBM mixing matrix UTBM. To achieve non-zero mixing angle 0 θ so as to accommodate neutrino data from reactor experiments, we take b = c and 13 e = −f ≡ (cid:15) (cid:54)= 0 in Eq.(4), and then the neutrino mass matrix in the A basis is given by 4   a+2b (cid:15) −(cid:15)   M =  (cid:15) a−b d . (6) ν     −(cid:15) d a−b In the flavor basis, the neutrino mass matrix can be rewritten as     a+(2d/3) b−(d/3) b−(d/3) 0 −(cid:15) (cid:15)   i   M(e,µ,τ) =  b−(d/3) b+(2d/3) a−(d/3) + √ −(cid:15) −2(cid:15) 0 . (7) ν     3     b−(d/3) a−(d/3) b+(2d/3) (cid:15) 0 2(cid:15) Rotating the mass matrix given in Eq.(7) by TBM mixing matrix, we get   a−b+d 0 0    0 a+2b X , (8)     0 X b−a+d √ where X = 2i(cid:15) and non-zero entries are complex in general. It can be easily shown that the mass matrix given by Eq.(7) can be diagonalized by   1 0 0   V(cid:48) = UTBM0 cosθ −sinθe−iξ ·P , (9) 0   β   0 sinθeiξ cosθ where P = Diag[eiβ1,eiβ2,eiβ3]. β 4 Now, let us check testability of the cases in this neutrino model by taking into account the sum-rules among the light neutrino masses [13]. In the leading order, the mass eigenvalues are given by m0 = a−b+d, m0 = a+2b, m0 = b−a+d, and thus we get the mass sum 1 2 3 rules m0 = m0 +m0, for a = 0, 3 2 1 m0 = 2m0 +m0, for b = 0. (10) 1 2 3 Inclusing the perturbation given by the second matrix in Eq.(7), we get the following sum rule, mˆ +mˆ = mˆ0 +mˆ0, (11) 2 3 2 3 where mˆ ≡ m e−iξ, mˆ ≡ m eiξ, mˆ0 ≡ m0e−iξ, mˆ0 ≡ m0eiξ with m representing the 2 2 3 3 2 2 3 3 i(=1,2,3) mass eigenvalues obtained by diagonalizing the mass matrix Eq.(8). Plugging Eq.(10) into Eq.(11), we can get the following sum rules for ξ = 0, π, 2π, m +m −m = 2 δm , for a = 0, 1 2 3 2 2m +m −m = δm , for b = 0, (12) 2 3 1 2 where δm ≡ m −m0 and we have used m = m0. The sum rules for ξ = π/2,(3π/2) are 2 2 2 1 1 m +m = m , for a = 0, 1 2 3 2m +m −m = 3 δm , for b = 0. (13) 2 3 1 2 B. Case-B To realize the case B, we add the breaking terms δM to M in the A basis, which is ν ν 4 given by     g +h 0 0 0 0 0     δM =  0 ωg +ω2h 0  = 0 A 0 , (14) ν         0 0 ω2g +ωh 0 0 −A √ where g = −h and A = 3ig. 5 Then, the matrix given in Eq.(14) becomes in the flavor basis as follows:   0 g −g    g −g 0 . (15)     −g 0 g Then, the mass matrix M +δM can be diagonalized by ν ν   cosθ 0 −sinθe−iξ   V = UTBM 0 1 0 ·P . (16) 0   β   sinθeiξ 0 cosθ For the case B, the sum rules at the leading order are the same as Eq.(10). Including the perturbation given by the second matrix in Eq.(15), we get the following sum rule mˆ +mˆ = mˆ0 +mˆ0, (17) 1 3 1 3 wheremˆ ≡ m e−iξ, mˆ ≡ m eiξ, mˆ0 ≡ m0e−iξ, mˆ0 ≡ m0eiξ,withm representingthe 1 1 3 3 1 1 3 3 i(=1,2,3) mass eigenvalues obtained by diagonalizing the mass matrix M +δM . Plugging Eq.(10) ν ν into Eq.(17), we can get the following sum rules for ξ = 0, π, 2π, m −m −m = 2 δm , for a = 0, 3 2 1 3 m +2m −m = 2 δm , for b = 0, (18) 3 2 1 3 where δm ≡ m −m0 and we have used m = m0. The sum rules for ξ = π/2,(3π/2) are 3 3 3 2 2 m −m +m = 0, for a = 0, 1 3 2 m −2m −m = 0, for b = 0. (19) 1 2 3 C. Case-C The case C can be realized by adding the A breaking term δM to the charged lepton 4 l mass matrix M : l   g v g v 0 1 1 2 1   δM =  g ωv g v 0. (20) l  1 2 2 2    g ω2v g v 0 1 3 3 3 6 Taking v = v = v and g = g = g, the matrix given by (20) becomes 1 2 3 1 2     gv gv 0 0 g 0    √ δM =  gωv gv 0 = U g 0 0 3v. (21) l   CW       gω2v gv 0 0 0 0 DuetotheadditionofδM , thePMNSmixingmatrixshouldbechangedtoU† (θ,ξ)UTBMP . l 12 0 β For Case C, the sum rules are given by Eq.(10). D. Case-D Similarily, the case D can be achieved by adding the following matrix δM to the charged l lepton mass matrix M : l   g v 0 g v 1 1 2 1   δM = g ω2v 0 g v . (22) l  1 2 2 2    g ωv 0 g v 1 3 3 3 Taking v = v = v and g = g = g, the matrix given in (22) becomes 1 2 3 1 2     gv 0 gv 0 0 g    √ δM = gω2v 0 gv  = U  0 0 0  3v. (23) l   CW       gωv 0 gv g 0 0 The addition of δM causes the PMNS mixing matrix changed to U† (θ,ξ)UTBMP . l 13 0 β For Case D, the sum rules are given by Eq.(10). III. PREDICTIONS OF DIRAC-TYPE CP PHASE Now, let us review how to predict Dirac-type CP phase in PMNS mixing matrix with regards to neutrino mixing angles presented in [7]. Multiplying V given in Eq.(1) by phase matrices P and P that can be arisen from the charged lepton sector and neutrino sector, α β respectively, we can equate it with the standard parameterization of the PMN mixing matrix 7 as follows: P ·V ·P = UST = UPMNS ·P α β 0 φ   c c s c s e−iδD 12 13 12 13 13   = −s12c23 −c12s23s13eiδD c12c23 −s12s23s13eiδD s23c13 Pφ. (24)   s s −c c s eiδD −c s −s c s eiδD c c 12 23 12 23 13 12 23 12 23 13 23 13 The equivalence between both parameterizations dictates the following relations, V ei(αi+βj) = UST = (UPMNS) eiφj . (25) ij ij 0 ij A. Case A and B Applying |V | = |UST| and |V /V | = |UST/UST|, we obtain the relations 13 13 11 12 11 12 sin2θ= 3s2 , 13  2tan2θ (CaseA), cos2θ= 12 (26)  1 cot2θ (CaseB), 2 12 which lead to the relation between the solar and reactor mixing angles,  1− 2 (CaseA), s2 = 3(1−s213) (27) 12  1 (CaseB). 3(1−s2 ) 13 Those relations indicate that non-zero values of s2 lead to s2 < 1/3 for Case A and 13 12 s2 > 1/3 for Case B . From |V /V | = |UST/UST|, we also get the relations 12 23 33 23 33   c213(√s223−c223) (CaseA),  |cosη| = 2s13 2−6s213 (28)  c213√(c223−s223) (CaseB).  s13 2−3s213 Now, we demonstrate how to derive δ in terms of neutrino mixing angles in the standard D parametrization. From the components of the neutrino mixing matrix for Case–A, we see that V +V V 23 33 13 = . (29) V +V V 22 32 12 From the relation (25), we get the relations UST UST +USTe−i(α3−α2) 13 = 23 33 , (30) UST UST +USTe−i(α3−α2) 12 22 32 UST V 3i = 3iei(α3−α2). (31) UST V 2i 2i 8 Since V = V , 21 31 UST ei(α3−α2) = 31 . (32) UST 21 Plugging Eq.(32) into Eq.(30), we finally obtain the relation UST USTUST +USTUST 13 = 23 31 33 21 . (33) UST USTUST +USTUST 12 22 31 32 21 NoticethattheMajoranaphasesinEq.(33)arecancelled. PresentingUST explicitlyinterms ij of the neutrino mixing angles as well as δ , we get the equation for δ as D D −1 1−5s2 cosδ = · 13 . (34) D (cid:112) 2tan2θ s 2−6s2 23 13 13 Notice that the imaginary part in Eq. (33) is automatically cancelled. Similarily, we get the relation for Case B, 1 2−4s2 cosδ = · 13 . (35) D (cid:112) 2tan2θ s 2−3s2 23 13 13 B. Case C and D Applying |V | = |UST| and |V /V | = |UST/UST|, we obtain the relations 13 13 23 33 23 33 sin2θ= 2s2 , 13  tan2θ (CaseC), cos2θ= 23 (36)  cot2θ (CaseD), 23 which lead to the relation between the atmospheric and reactor mixing angles,  1− 1 (CaseC), s2 = 2(1−s213) (37) 23  1 (CaseD). 2(1−s2 ) 13 Those relations indicate that non-zero values of s2 lead to s2 < 1/2 for Case C and 13 23 s2 > 1/2 for Case D . From |V /V | = |UST/UST|, we also get the relation 23 11 12 11 12 3c2 s2 −1 |cosη| = 13 12 . (38) (cid:112) 2s 2−4s2 13 13 We note that both cases lead to the same relation for |cosη|. Following the same procedures for obtaining Eqs.(34, 35), we get the relations   s213−(1−3s21√2)(1−3s213) (CaseC),  cosδ = 6s12c12s13 1−2s213 (39) D  (1−3s212)(1−√3s213)−s213 (CaseD).  6s12c12s13 1−2s213 Sustituting experimental values for neutrino mixing angles into Eqs.(34,35,39), we can estimate the values of δ for each cases. D 9 IV. NUMERICAL RESULTS For our numerical analysis, we take the current experimental data for three neutrino mixing angles as inputs, which are given at 1σ − 3σ C.L., as presented in Ref. [5]. This analysis is, in fact, to update the numerical results for the prediction of δ given in [7] by D taking the new fit to the data [5]. However, as shown later, the results based on 1σ data is completely different from those in [7]. Here, we perform numerical analysis and present results only for normal hierarchical neutrino mass spectrum. It is straight-forward to get numerical results for the inverted hierarchical case, and we anticipate that the conclusion is not severly changed in the inverted hierarchical case. Using experimental results for three neutrino mixing angles, we first check if the relations Eqs.(27,37) hold and then estimate the values of δ in terms of neutrino mixing angles for those four cases. D 2Π 2Π 3Π 3Π 2 2 D Π Π ∆ Π Π 2 2 0 0 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34 s2 12 FIG. 1. Prediction of δ in terms of s2 for Case C based on 1σ experimental data. D 12 10