]>PLB34574S0370-2693(19)30254-010.1016/j.physletb.2019.04.022The AuthorsPhenomenologyFig. 1Predicted correlations involving the three mixing angles in pattern I. “Solar” and “accelerator” angles sin2θ12 (lower, black line) and sin2θ23 (upper, blue line) correlate with the “reactor” mixing parameter sin2θ13. The yellow and the magenta boxes represent the current 3σ ranges of the mixing angles [21]. The cross symbols correspond to ϕ = 0,π,±π/4,±π/2,±3π/4, respectively.Fig. 1Fig. 2Predicted correlation between the Dirac phase δCP and the “reactor” mixing parameter sin2θ13 in pattern I. The yellow box is the current 3σ range from the global fit [21]. The crosses correspond to ϕ = 0,π,±π/4,±π/2,±3π/4, respectively.Fig. 2Fig. 3Predicted correlations involving the three mixing angles in pattern II. “Solar” and “accelerator” angles sin2θ12 (lower, black line) and sin2θ23 (upper, blue line) correlate with the “reactor” mixing parameter sin2θ13. The yellow and the magenta boxes represent the current 3σ ranges of the mixing angles [21]. The cross symbols correspond to ϕ = 0,π,±π/4,±π/2,±3π/4, respectively.Fig. 3Fig. 4Predicted correlation between the Dirac phase δCP and the “reactor” mixing parameter sin2θ13 in pattern II. The yellow box is the current 3σ range from the global fit [21]. The crosses correspond to ϕ = 0,π,±π/4,±π/2,±3π/4, respectively.Fig. 4Fig. 5The νμ → νe transition probability versus energy for pattern I for the T2K, NOνA and DUNE experiments. The mixing angles and Dirac CP phase are taken within the currently allowed 3σ range.Fig. 5Fig. 6νμ → νe transition probability versus distance taking the mixing angles and Dirac CP phase within the currently allowed 3σ range. The broad band (red) refer to a generic scenario, whereas the thin band (blue) corresponds to the bi-large pattern I prediction.Fig. 6Fig. 7The CP asymmetry Aμe=P(νμ→νe)−P(ν¯μ→ν¯e)P(νμ→νe)+P(ν¯μ→ν¯e) versus distance taking the mixing angles and Dirac CP phase within the currently allowed 3σ range. The broad band refer to a generic scenario, whereas the thin band refers to the bi-large pattern I prediction.Fig. 7Predicting neutrino oscillations with “bi-large” lepton mixing matricesPengChena⁎pche@mail.ustc.edu.cnGui-JunDingbdinggj@ustc.edu.cnRahulSrivastavacrahulsri@ific.uv.esJosé W.F.Vallecvalle@ific.uv.esaCollege of Information Science and Engineering, Ocean University of China, Qingdao 266100, ChinaCollege of Information Science and EngineeringOcean University of ChinaQingdao266100ChinabInterdisciplinary Center for Theoretical Study and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, ChinaInterdisciplinary Center for Theoretical StudyDepartment of Modern PhysicsUniversity of Science and Technology of ChinaHefeiAnhui230026ChinacAHEP Group, Institut de Física Corpuscular – CSIC/Universitat de València, Parc Científic de Paterna, C/Catedrático José Beltrán, 2 E-46980 Paterna (Valencia) - SpainAHEP GroupInstitut de Física Corpuscular – CSIC/Universitat de ValènciaParc Científic de PaternaC/Catedrático José Beltrán, 2Paterna (Valencia)E-46980Spain⁎Corresponding author.Editor: A. RingwaldAbstractWe propose two schemes for the lepton mixing matrix U=Ul†Uν, where U=Ul refers to the charged sector, and Uν denotes the neutrino diagonalization matrix. We assume Uν to be CP conserving and its three angles to be connected with the Cabibbo angle in a simple manner. CP violation arises solely from the Ul, assumed to have the CKM form, Ul≃VCKM, suggested by unification. Oscillation parameters depend on a single parameter, leading to narrow ranges for the “solar” and “accelerator” angles θ12 and θ23, as well as for the CP phase, predicted as δCP∼±1.3π.1IntroductionAfter decades of hard work, the origin of flavor mixing and CP violation remains one of the most important challenges in particle physics. Understanding the flavor problem would help us to get a glimpse on physics beyond the standard model. Several approaches have been pursued to find an adequate and predictive description of lepton mixing. Using flavor symmetries from first principles [1] one can obtain “top down” restrictions on neutrino mixing within fundamental theories of neutrino mass [2–6]. Alternatively, one may make educated phenomenological guesses as to what the pattern of lepton mixing should look like. Specially influential were the ideas of mu-tau symmetry and the Tri-Bimaximal (TBM) lepton mixing ansatz proposed by Harrsion, Perkins and Scott [7–9]. The latter predicts the three mixing angles as sin2θ23=1/2, sin2θ12=1/3, sin2θ13=0, while the Dirac CP phase vanishes. However, the precise measurements of the reactor angle θ13∼8.5∘ in Daya Bay [10], RENO [11] and Double Chooz [12] now exclude TBM as a realistic lepton mixing pattern. The discrepancy between experiment and the prediction of TBM led people to pursue new lepton mixing structures. One method is to modify the TBM pattern based on flavor or CP symmetries such as in ref. [13,14].A more phenomenological approach is to explore new neutrino mixing patterns [15–17]. Recently a “bi-large” mixing scheme has been proposed in Ref. [18] assuming that sinθ13≃λ where λ is the Cabibbo angle. A generalization of this pattern was proposed in Ref. [19], taking the Cabibbo angle as a universal seed for quark and lepton mixing. Such schemes may emerge from Grand Unified Theories (GUTs) and flavor symmetry [20]. The good features of such bi-large mixing patterns deserve further investigation.In this paper we will propose two “bi-large” lepton mixing schemes and investigate their phenomenological implications. For definiteness, we assume normal ordered neutrino masses throughout this paper, since inverted ordering is disfavored at more than 3σ [21].11A slightly stronger limit can be obtained from cosmology [22]. As in Ref. [19], we assume that the charged lepton diagonalization matrix is CKM-like, given in terms of the Wolfenstein parameters λ and A, whose values we take from the PDG as λ=0.22453 and A=0.836 [23]. In our ansatz the three mixing angles characterizing the neutrino diagonalization matrix are related with λ in a very simple manner. We obtain tight predictions for the physical lepton mixing angles and the CP phase. These are contrasted with current experiments and used to make projections for upcoming long baseline oscillation experiments.2Bi-large: pattern IIn this section we propose our first “bi-large” lepton mixing pattern. Within the standard parameterization the three angles of the neutrino diagonalization matrix are assumed to be given as(1)sinθ23ν=1−λ,sinθ12ν=2λ,sinθ13ν=λ, and the Dirac CP phase is taken as δCPν=π.22The Majorana phases are taken to be zero, hence our ansatz is just for oscillation physics, with hardly any predictivity for neutrinoless double beta decay experiments. In this case the neutrino diagonalization matrix can be approximated as(2)Uν≃(1−5λ222λ−λλ−22λ322λ−λ32221−λ−λ222λ+2λ32−1+λ2λ−λ3222). If the charged leptons are taken diagonal then this will imply that the leptonic mixing parameters are the same as eq. (1): sin2θ23=sin2θ23ν≃0.601, sin2θ12=sin2θ12ν≃0.202 and sin2θ13=sin2θ13ν≃0.0504, and lie outside the 3σ experimental range [21]. However, corrections are expected from the charged lepton diagonalization matrix [24]. Following Ref. [19] we assume that the bi-large pattern arises from the simplest SO(10) model where the charged and the down-type quarks have roughly the same mass. Then the lepton diagonalization matrix is naturally of the CKM type [25](3)Ul=R23(θ23CKM)ΦR12(θ12CKM)Φ†≃(1−λ22λe−iϕ0−λeiϕ1−λ22Aλ2Aλ3eiϕ−Aλ21), where sinθ23CKM=Aλ2 and sinθCKM12=λ, where λ and A are the Wolfenstein parameters, Rij is the i-j real rotation matrix, and Φ=diag(e−iϕ/2,eiϕ/2,1) where ϕ is a free phase. For convenience we set ϕ∈(−π,π] throughout this paper. The elements of the lepton mixing matrix U=Ul†Uν are all given in terms of just one free parameter ϕ, leading to a very high degree of predictivity.To leading order the three mixing angles and the Jarlskog invariant JCP are given by(4)sin2θ13≃4λ2(1−λ)cos2ϕ2,sin2θ12≃2λ2(2−22λcosϕ+λ),sin2θ23≃(1−λ)2−22Aλ52−2λ3(1+2cosϕ),JCP≃−22λ52sinϕ. The fact that the above parameters depend on just one free parameter ϕ, leads to strong correlations.The predictions for the oscillation parameters are shown in Fig. 1 and Fig. 2. Requiring sin2θ13 to lie inside the allowed 3σ range implies that the value of ϕ/π should be inside the range of ±[0.749,0.779]. This severely restricts the allowed ranges for the θ12, θ23 and δCP. The resulting ranges for the oscillation parameters become(5)0.0196≤sin2θ13≤0.0241,0.302≤sin2θ12≤0.309,0.572≤sin2θ23≤0.574,1.303≤δCP/π≤1.307, where we have chosen the positive solution of ϕ in order to be compatible with the present data of δCP. One sees that, given θ13, we find that the resulting allowed ranges for the other mixing angles and CP violation phase are very narrow. We perform a conventional χ2 analysis including the information on δCP. The χ2 takes the minimum value χmin2=2.366 when ϕ=0.766π, leading to the following values for the physical mixing parameters,(6)sin2θ23=0.573,sin2θ13=0.0216,sin2θ12=0.306,δCP=1.305π, where sin2θ13, sin2θ12 and δCP are inside the 1σ range while sin2θ23 is inside the 2σ range of [21], hence fitting very well the experimental results. One sees that the three mixing angles and the Dirac phase are in very good agreement with the current experimental values [21]. It is also remarkable that, starting from a CP conserving Uν in eq. (2), we obtain a CP violating phase that lies very close to the best fit value.3Bi-large: pattern IIWe now turn to our second example. Again we take the Dirac CP phase as δCPν=π but now assume the neutrino mixing angles in the standard parameterization to be given by(7)sinθ13ν=1λ,sinθ12ν=2λ,sinθ23ν=3λ. To order λ2 the neutrino mixing matrix of such “1-2-3” bi-large mixing pattern is written as(8)Uν≃(1−52λ22λ−λ−2λ+3λ21−132λ23λλ+6λ2−3λ+2λ21−5λ2). As theoretical motivation this time we consider the framework of SU(5) Grand Unified models. In the simplest SU(5) GUTs the lepton and down quark mass matrices obey the relation Me∼MdT. As in the previous section, this suggests us to adopt a CKM-type lepton diagonalization matrix(9)Ul=Φ†R12T(θ12CKM)ΦR23T(θ23CKM)≃(1−λ22−λeiϕAλ3eiϕλe−iϕ1−λ22−Aλ20Aλ21), with ϕ∈(−π,π]. Then to leading order, the mixing angles and JCP obtained from the lepton mixing matrix U=Ul†Uν are given by(10)sin2θ13≃λ2−6λ3cosϕ,sin2θ12≃λ2(5+4cosϕ),sin2θ23≃9λ2+6λ3(A+cosϕ),JCP≃−3λ3sinϕ. As before, requiring sin2θ13 to lie in the allowed 3σ range severely restricts the consistency ranges for the other oscillation parameters θ12, θ23 and δCP, as follows(11)0.0196≤sin2θ13≤0.0241,0.315≤sin2θ12≤0.323,0.511≤sin2θ23≤0.513,1.266≤δCP/π≤1.274. Taking into account the current experimental data, the χ2 takes the minimum value χmin2=3.162 when ϕ=0.233π, and the mixing parameters are(12)sin2θ23=0.512,sin2θ13=0.0216,sin2θ12=0.319,δCP=1.270π. One sees that sin2θ13, sin2θ12 and δCP are inside the 1σ range, while sin2θ23 is inside the 2σ range given by current global oscillation fits. The results are displayed in Figs. 3 and 4. As before, one sees that the predictions fit very well with the observed oscillation parameter values.4long baseline oscillationsThe lepton mixing matrix in both cases discussed above only depends on one free parameter ϕ. As we saw, the one-parameter nature of both anzatze leads to tight correlations amongst the oscillation parameters and predict very narrow ranges for the “solar” and “accelerator” angles θ12 and θ23. This translates into phenomenological implications for the expected neutrino and anti-neutrino appearance probabilities in neutrino oscillation experiments [26,27]. To illustrate the implications of our mixing patterns for future long baseline oscillation experiments we present the resulting oscillation probabilities in Figs. 5 and 6.One sees that indeed the expected oscillation probabilities are tightly restricted, indicating that our bi-large mixing patterns should be testable at the upcoming long baseline oscillation experiments. In particular, the CP asymmetry, displayed in Fig. 7, is very tightly predicted as compared to the generic three-neutrino oscillation scheme. This is seen by comparing the thin band (blue) with the broad band (red) in the figure.5ConclusionIn this letter we have proposed two bi-large-type lepton mixing schemes. They make definite assumptions on the two factors that comprise the lepton mixing matrix U=Ul†Uν, where U=Ul comes from the charged sector while Uν describes the neutrino diagonalization matrix. We assume Uν to be CP conserving and its three angles to be related with the Cabibbo angle in a simple way, given as sinθ13ν=λ, sinθ12ν=2λ and sinθ23ν=1−λ (pattern I) or 3λ (pattern II), with the Dirac CP phase taken at the CP conserving value δCPν=π. CP violation arises only from the Ul factor, assumed to have the CKM form, Ul≃VCKM, as expected in the simplest Grand Unified models. The Dirac CP phase is predicted as δCP∼±1.3π, the positive value being very close to its current best fit value. The mixing angles also depend on a single parameter ϕ. The good measurement of the “reactor” angle leads to tight correlations that predict narrow ranges for the “solar” and “atmospheric” angles θ12 and θ23 in good agreement with current oscillation data. The predictions should be testable at the upcoming long baseline oscillation experiments. Moreover, the structure of the two patterns is very simple, consistent with unification scenarios, and suggestive of novel model building approaches involving Abelian family symmetries [20].AcknowledgementsPC is supported by National Natural Science Foundation of China under Grant No 11847240 and China Postdoctoral Science Foundation Grant No 2018M642700. GJD acknowledges the support of the National Natural Science Foundation of China under Grant No 11835013. 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