From recent groundbreaking experiments, it is now known that the Pontecorvo-Maki-Nakagawa-Sakata mixing differs significantly from the tribimaximal model in which theta_13=0 and theta_23=pi/4. Flavor symmetry can require that the departures from these two equations are linearly related. T' and A_4, which successfully accommodated the pre-T2K Pontecorvo-Maki-Nakagawa-Sakata matrix, predict that 38.07 deg.< theta_23 <39.52 deg. at 95% C.L.. The best fit values, combining the model predictions with T2K, MINOS, Double Chooz, Daya Bay, and RENO data, are theta_23=38.7 deg. and theta_13=8.9 deg.

Eby, David A.

Frampton, Paul H.

Carolina Digital Repository

arXiv:1112.2675v5  [hep-ph]  16 Jan 2013Nonzero θ13 signals nonmaximal atmospheric neutrino mixingDavid A. Eby and Paul H. FramptonDepartment of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599-3255, USA(Dated: November 10, 2018)From recent groundbreaking experiments, it is now known that the Pontecorvo-Maki-Nakagawa-Sakata mixing differs significantly from the tribimaximal model in which θ13 = 0 and θ23 = π/4.Flavor symmetry can require that the departures from these two equations are linearly related. T′and A4, which successfully accommodated the pre-T2K Pontecorvo-Maki-Nakagawa-Sakata matrix,predict that 38.07◦ ≤ θ23 ≤ 39.52◦ at 95% C.L.. The best fit values, combining the model predictionswith T2K, MINOS, Double Chooz, Daya Bay, and RENO data, are θ23 = 38.7◦ and θ13 = 8.9◦.Of the parameters in the standard model of particletheory, we will focus on the mixing matrices for down-type quarks and for neutrinos, named respectively forCabibbo, Kobayashi, and Maskawa (CKM) [1, 2] and forPontecorvo, Maki, Nakagawa, and Sakata (PMNS) [3, 4].Without losing generality, we choose a basis in which theflavor and mass eigenstates coincide for the three up-typequarks and all three charged leptons.This investigation will consider one of three mixingangles of CKM quark mixing (Θ12) and two of the threemixing angles of PMNS neutrino mixing (θ13 and θ23),ignoring for the moment the CP-violating phases in bothcases.We recall the values of the angles θ13 and θ23 listed inthe 2010 Review of Particle Physics1 [5] since these twoare, we suggest, both changed by the T2K measurement[6–11]. The values then were36.8◦ . θ23 ≤ 45.0◦, 0.0◦ ≤ θ13 . 11.4◦ (1)consistent with vanishing θ13 and maximal θ23.The other angles are not considered to be variablesin this analysis, although the superior experimental ac-curacy of the CKM Gell-Mann-Lévy quark mixing an-gle [12],Θ12 = (13.03± 0.06)◦, (2)played an important role in our investigation of flavorsymmetry.To accommodate the new data, we invoke broken bi-nary tetrahedral (T′) flavor symmetry as a promising ap-proach to explaining the mixing angles [13–22].This flavor symmetry was first used in Ref. [13] solelyas a symmetry for quarks, because neutrinos were still be-lieved to be massless. After neutrino masses and mixingswere discovered [23], the mixing matrix for neutrinos wasmeasured and found to be very different from the CKMmixing matrix for quarks. A number of theories arose[24–28] to explain this. Eventually, a useful approxima-tion to the empirical PMNS mixing was determined tobe the tribimaximal (TBM) matrix [29],1 The reader is directed to the references summarized in RPP.UTBM =√2/3√1/3 0−√1/6√1/3 −1/√2−√1/6√1/3 1/√2 . (3)Flavor symmetry based on the tetrahedral group, A4 =T , was introduced by Ref. [30] to underpin TBM neutrinomixing. Further investigation revealed that this modelcould not be extended to quarks because a viable CKMmatrix could not be obtained [31]. A4 is not a subgroup ofits double cover [20], T′, nevertheless from the viewpointof kronecker products used in model building [14], A4behaves as if it were a subgroup. This explains why thelarger group can act as a successful flavor symmetry forboth quarks and leptons.We shall consider only the projection on the two-dimensional θ23 - θ13 plane of the three-dimensional θ12 -θ23 - θ13 space. At leading order, requiring sinα ∼ α2for θ13 and (π4− θ23), the calculation of the perturba-tion of this projection from the TBM matrix in Eq.(3)is independent of the solar neutrino mixing angle θ12.The relevant perturbation away from Eq.(3) was explic-itly calculated in Ref. [18, 19].Before T2K, the neutrino mixing angles were all empir-ically consistent with the TBM values. However, as theexperimental accuracy has now improved in recent datafrom T2K [6–11], MINOS [32–38], Double Chooz [39–43],Daya Bay [44, 45], and RENO [46, 47], this situation haschanged dramatically, as discussed in the global fits ofRefs. [48–50]; of these we shall use Fogli et al. [49]. Thesefive remarkable experiments have provided us with a richnew perspective on mixing angles. From flavor symmetry,it is then possible to predict quantitatively how depar-tures from the TBM values,θ12 = tan−1(1√2), θ23 = (π/4) , θ13 = 0, (4)are related. The model allows one to address this ques-tion by relating the perturbations around TBM,θij = (θij)TBM + ǫk, (5)2 This is a < 1% approximation for θ13 and (π4−θ23) since both an-gles are less than α = 12◦ = 0.2094 radians with sinα = 0.2079.FIG. 1. The global analysis of Ref. [49], incorporating SBL, LBL, solar, and atmospheric neutrino observations, excludes thered-shaded region at 2σ. The same assessment excludes the orange-shaded region at 1σ. The best fit value for θ13 is indicatedby the vertical green line at θ13 = 8.9◦. Extreme values of the linear correlation coefficient, η, are indicated by dashed lines atη = 0.902 and η = 3.29, while our predicted correlation of η =√2 is indicated by the solid dark blue line. The intersection ofour correlation prediction and the θ13 best fit occurs at θ13 = 8.9◦ and θ23 = 38.7◦, a close match to the current experimentalbest fit of θ23 = 38.4◦.(where ǫ3 corresponds to θ12, and so on) to the analogousperturbations around the minimal model’s prediction forthe CKM Gell-Mann-Lévy quark mixing angle,tan 2(Θ12) =(√23). (6)The data from KamLAND, LBL accelerators (like T2Kand MINOS), solar experiments, SBL accelerators (suchas Double Chooz, Daya Bay, and RENO), and Super-Kamiokande, as combined in Ref. [49] indicate (account-ing for CP violation)sin2 θ13 = 0.0241+0.0049−0.0048 with 95% C.L. (7)for a normal neutrino mass hierarchy, as favored by T′.Because Eq. (6) yields a value of Θ12 = 12.62◦, whichwhile close is significantly below the experimental value,Eq. (2), it is possible to perturb to the empirical Θ12and to track the deviations in the PMNS mixing matrixto the linear relationship,3θ13 = η(π4− θ23), (8)with the sharp prediction4 that η =√2. Thus,θ13 =√2(π4− θ23), (9)This prediction is derived in further detail in Ref. [19].Considering the result, Eq. (8), it requires that, if ηis finite as expected, any departure from θ13 = 0 sig-nals that θ23 < π/4. As shown in Fig. (1), the re-cent experimental data, combined with theory, suggest3 A4 is also capable of producing Eq.(8) with η =√2, though wegive preference in this paper to T′for its capacity to explainCKM mixing.4 It is notable that Eq.(8) with η ≃√2 appears en passant inRef. [51]; see also Ref. [52] which implies that η ∼ 2. Another,model-independent correlation was developed in Ref. [53], includ-ing the three PMNS mixing angles and the CP-violating phase.2that (θ13, θ23) are respectively closer to (8.9◦, 38.7◦)than to (0.0◦, 45.0◦). Before T2K, η was unconstrained,0 ≤ η < ∞. With the current global fit data, we find0.902 ≤ η ≤ 3.29.This is in sharp contrast to the previously widespreadacceptance of a maximal θ23 = π/4, which fitted so wellwith vanishing θ13 = 0 in the TBM context.As the measurement of θ13 sharpens experimentally,so will the prediction for θ23 from Eq. (9), and mea-surement of the atmospheric neutrino mixing’s departurefrom maximality will provide an interesting test of the bi-nary tetrahedral flavor symmetry.Several years ago Super-Kamiokande showed θ23 >36.8◦ [54], and current analysis places it at θ23 ≃40.7◦ [55]. Once combined in a global fit of 3ν oscillation,Ref. [49] states the best fit of θ23 = 38.4◦, tantalizinglyclose to our central value of θ23 = 38.7◦.This suggests to us that the T′flavor symmetry, in-troduced in Ref. [13], should now be taken much moreseriously. As errors in θ13 and θ23 diminish even further,it will be interesting to see how the prediction of Eq. (9)by T′perseveres, as it would inspire further investigationinto other mixing angles for quarks and leptons. This, inturn, may show that T′, first mentioned in physics asan example of an SU(2) subgroup [56], is actually a use-ful approximate symmetry in the physical application ofquark and lepton flavors.This work was supported by DOE Grant No.DE-FG02-05ER41418 and DOE-GAANN Award No.P200A090135.[1] N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963).[2] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49,652 (1973).[3] B. Pontecorvo, Sov. Phys. JETP 7, 172 (1958).[4] Z. Maki, M. Nakagawa, and S. Sakata, Prog. Theor. Phys.28, 870 (1962).[5] K. Nakamura et al. (Particle Data Group), J. Phys. G37, 075021 (2010).[6] K. Abe et al. [T2K Collaboration], Phys. Rev. Lett. 107,041801 (2011) [arXiv:1106.2822 [hep-ex]].[7] F. Dufour [T2K Collaboration], J. Phys. Conf. Ser. 335,012053 (2011).[8] M. Hartz [T2K Collaboration], arXiv:1201.1846 [hep-ex].[9] E. 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Nonzero theta_13 signals nonmaximal atmospheric neutrino mixing

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Nonzero theta_13 signals nonmaximal atmospheric neutrino mixing

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