The Leptonic Dirac CP-Violating Phase from Sum Rules

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1 The Leptonic Dirac CP-Violating Phase from Sum Rules Arsenii Titov in collaboration with I. Girardi and S.T. Petcov SISSA and INFN, Trieste, Italy XIV International Conference on Topics in Astroparticle and Underground Physics 9 September 2015, Turin, Italy

2 Outline 3-neutrino mixing General setup Sum rules Predictions Conclusions Based on I. Girardi, S.T. Petcov, A.T., NPB 894 (2015) 733 [arxiv: ] I. Girardi, S.T. Petcov, A.T., EPJC 75 (2015) 7, 345 [arxiv: ] 2

3 3-neutrino mixing U is the Pontecorvo-Maki-Nakagawa- Sakata neutrino mixing matrix Best fit 3σ range sin 2 θ sin 2 θ 23 (NO) sin 2 θ 23 (IO) sin 2 θ 13 (NO) sin 2 θ 13 (IO) δ/π (NO) θ 12 π/ θ 23 π/ θ Symmetry? δ/π (IO) Capozzi et. al., PRD 89 (2014)

4 General setup M e is the charged lepton mass matrix M ν is the neutrino Majorana mass matrix Ũ e and Ũ ν are CKM-like 3 3 unitary matrices Frampton, Petcov, Rodejohann, NPB 687 (2004) 31 Ũ ν is assumed to have a symmetry form which is dictated by, or associated with, a flavour (discrete) symmetry, e.g., A 4, S 4, A 5, T 4

5 General setup Symmetry forms of Ũ ν : bimaximal, tri-bimaximal, golden ratio, hexagonal where Tri-bimaximal (TBM) A 4 /T θ ν 12 = arcsin (1/ 3) 35 Bimaximal (BM) S 4 θ ν 12 = π/4 = 45 Golden ratio A (GRA) A 5 θ ν 12 = arcsin (1/ (2+r)) 31 Golden ratio B (GRB) D 10 θ ν 12 = arcsin ( (3 r)/2) = 36 Hexagonal (HG) D 12 θ ν 12 = π/6 = 30 θ ν 23 = π/4 for all these symmetry forms r is the golden ratio: r = (1 + 5)/2 5

6 General setup Charged lepton (CL) corrections: 1 rotation from the CL sector and 2 rotations from the neutrino sector 2 rotations from the CL sector and 2 rotations from the neutrino sector 1 rotation from the CL sector and 3 rotations from the neutrino sector => sum rules for cos δ, i.e., cos δ as a function of the observable mixing angles θ 12, θ 23, θ 13 and the angles θ ν ij, whose values are fixed 6

7 Sum rules 1 rotation from the CL sector and 2 rotations from the neutrino sector R 12 (θ e 12) R 13 (θ e 13) 7

8 Sum rules 2 rotations from the CL sector and 2 rotations from the neutrino sector R 12 (θ e 12) R 23 (θ e 23) Petcov, NPB 892 (2015) 400 R 13 (θ e 13) R 23 (θ e 23) 8

9 Sum rules 1 rotation from the CL sector and 3 rotations from the neutrino sector R 12 (θ e 12) R 13 (θ e 13) % 9

10 Predictions for cos δ Using best fit values of mixing angles for NO neutrino mass spectrum θ ν 23 = π/4 [θ ν 13, θ ν 12] a = arcsin (1/3) b = arcsin (1/ (2+r)) c = arcsin (1/ 3) d = arcsin ( (3 r)/2) Non-zero values of θ ν 13: Bazzocchi, arxiv: , Toorop et. al., PLB 703 (2011) 447, Rodejohann and Zhang, PLB 732 (2014)

11 Predictions for J CP : statistical analysis R 12 (θ e 12) R 23 (θ e 23) J CP determines the magnitude of CP-violating effects in neutrino oscillations Krastev and Petcov, PLB 205 (1988) 84 NO scheme IO scheme NO global fit IO global fit Relatively large CP-violating effects in neutrino oscillations in the cases of TBM, GRA, GRB, HG: J CP 0.03, J CP 3σ and suppressed ones in the case of BM: J CP 0 Using latest results on sin 2 θ ij and δ, obtained in global analysis of neutrino oscillation data in Capozzi et. al., PRD 89 (2014)

12 Predictions for cos δ: statistical analysis Likelihood function R 12 (θ e 12) R 23 (θ e 23) Using current best fit values and prospective 1σ uncertainties for sin 2 θ ij : 0.7% for sin 2 θ 12 (JUNO), 3% for sin 2 θ 13 (Daya Bay), 5% for sin 2 θ 23 (NOvA and T2K) + Gaussian approximation 12

13 Predictions for cos δ: statistical analysis Likelihood function R 13 (θ e 13) R 23 (θ e 23) Using current best fit values and prospective 1σ uncertainties for sin 2 θ ij : 0.7% for sin 2 θ 12 (JUNO), 3% for sin 2 θ 13 (Daya Bay), 5% for sin 2 θ 23 (NOvA and T2K) + Gaussian approximation 13

14 Extension of the study Today on arxiv: I. Girardi, S.T. Petcov, A.J. Stuart, A.T., arxiv: G f Flavour symmetry group (non-abelian discrete) G e G ν Residual symmetries of the CL and neutrino mass matrices => Sum rules for cos δ 14

15 Conclusions Exact (within the schemes considered) sum rules for cos δ Relatively large CP-violating effects in neutrino oscillations in the cases of TBM, GRA, GRB, HG and suppressed ones in the case of BM Sufficiently precise measurement of δ combined with prospective precision on the neutrino mixing angles can provide information about the existence of a new type of fundamental symmetry in the lepton sector 15

16 Backup 16

17 Statistical details 1) χ i2 being extracted from Capozzi et. al., PRD 89 (2014) ) Gaussian approximation: Future: and are b.f.v. and 1σ uncertainties are parameters of the scheme 17

18 Statistical details: comparison 1) 2) 18

Predictions for cos δ: statistical analysis Likelihood function R 12 (θ e 12) R 23 (θ e 23) Using latest results on sin 2 θ ij and δ, obtained in global analysis of neutrino oscillation data in

19 Predictions for cos δ: statistical analysis Likelihood function R 12 (θ e 12) R 23 (θ e 23) Using latest results on sin 2 θ ij and δ, obtained in global analysis of neutrino oscillation data in Capozzi et. al., PRD 89 (2014) Using prospective 1σ uncertainties on sin 2 θ ij : 0.7% for sin 2 θ 12 (JUNO) 3% for sin 2 θ 13 (Daya Bay) 5% for sin 2 θ 23 (NOvA and T2K) + Gaussian approximation 19

20 Dependence on best fit values Likelihood function R 12 (θ e 12) R 23 (θ e 23) (sin 2 θ 12 ) bf = (sin 2 θ 23 ) bf = (sin 2 θ 13 ) bf = (sin 2 θ 12 ) bf = (sin 2 θ 23 ) bf = (sin 2 θ 13 ) bf = IO neutrino mass spectrum Gonzalez-Garcia et. al., JHEP 1411 (2014)

21 Predictions for δ: statistical analysis R 12 (θ e 12) R 23 (θ e 23) NO scheme IO scheme NO global fit IO global fit Using latest results on sin 2 θ ij and δ, obtained in global analysis of neutrino oscillation data in Capozzi et. al., PRD 89 (2014)

22 Results for sin 2 θ 23 : statistical analysis R 12 (θ e 12) R 23 (θ e 23) NO scheme IO scheme NO global fit IO global fit Using latest results on sin 2 θ ij and δ, obtained in global analysis of neutrino oscillation data in Capozzi et. al., PRD 89 (2014)

Predictions for cos δ: statistical analysis Likelihood function R 13 (θ e 13) R 23 (θ e 23) Using latest results on sin 2 θ ij and δ, obtained in global analysis of neutrino oscillation data in

23 Predictions for cos δ: statistical analysis Likelihood function R 13 (θ e 13) R 23 (θ e 23) Using latest results on sin 2 θ ij and δ, obtained in global analysis of neutrino oscillation data in Capozzi et. al., PRD 89 (2014) Using prospective 1σ uncertainties on sin 2 θ ij : 0.7% for sin 2 θ 12 (JUNO) 3% for sin 2 θ 13 (Daya Bay) 5% for sin 2 θ 23 (NOvA and T2K) + Gaussian approximation 23

24 Results for sin 2 θ 23 : statistical analysis R 13 (θ e 13) R 23 (θ e 23) NO scheme IO scheme NO global fit IO global fit Using latest results on sin 2 θ ij and δ, obtained in global analysis of neutrino oscillation data in Capozzi et. al., PRD 89 (2014)

Predictions for cos δ: statistical analysis Likelihood function R 12 (θ e 12) Using latest results on sin 2 θ ij and δ, obtained in global analysis of neutrino oscillation data in

25 Predictions for cos δ: statistical analysis Likelihood function R 12 (θ e 12) Using latest results on sin 2 θ ij and δ, obtained in global analysis of neutrino oscillation data in Capozzi et. al., PRD 89 (2014) [θ ν 13, θ ν 12]: Case I = [π/10, π/4] Case II = [π/20, arcsin (1/ (2+r))] Case III = [π/20, π/4] Case IV = [arcsin (1/3), π/4] Case V = [π/20, π/6] 25

26 Predictions for cos δ: statistical analysis R 12 (θ e 12) [θ ν 13, θ ν 12]: Case I = [π/10, π/4] Case II = [π/20, arcsin (1/ (2+r))] Case III = [π/20, π/4] Case IV = [arcsin (1/3), π/4] Case V = [π/20, π/6] 26

Predictions for cos δ: statistical analysis Likelihood function R 13 (θ e 13): Using latest results on sin 2 θ ij and δ, obtained in global analysis of neutrino oscillation data in F.

27 Predictions for cos δ: statistical analysis Likelihood function R 13 (θ e 13): Using latest results on sin 2 θ ij and δ, obtained in global analysis of neutrino oscillation data in F. Capozzi et. al., PRD 89 (2014) [θ ν 13, θ ν 12]: Case I = [π/20, π/4] Case II = [arcsin (1/3), π/4] Case III = [π/20, arcsin (1/ 3)] Case IV = [π/10, π/4] Case V = [π/20, arcsin ( (3 r)/2)] 27

28 Predictions for cos δ: statistical analysis R 13 (θ e 13) [θ ν 13, θ ν 12]: Case I = [π/20, π/4] Case II = [arcsin (1/3), π/4] Case III = [π/20, arcsin (1/ 3)] Case IV = [π/10, π/4] Case V = [π/20, arcsin ( (3 r)/2)] 28

CP Violation Predictions from Flavour Symmetries

CP Violation Predictions from Flavour Symmetries Arsenii V. Titov in collaboration with Ivan Girardi and Serguey T. Petcov SISSA and INFN, Trieste, Italy Neutrino Oscillation Workshop 016 September 6,