1. What does μ-τsymmetry imply about leptonic CP violation? *) Teppei Baba Tokai University In collaboration with Masaki Yasue * ) based on hep-ph/0612034 to be published in Physical Reviews D (March. 2007)

2. Content 1. Introduction 2. What’s μ-τsymmetry ? 3. μ-τsymmetry-breakings and CP phase 4. Summary

3. 1. Introduction We don’t know which masses give Dirac CP phase (*) Charged lepton masses are diagonalized Dirac CP phase ? ? ? However, there is an ambiguity, where phases of M ij (ij=e, ,  ) are not uniquely determined because of the redefinition of phases of the neutrinos. Observed quantities such as the mixing angles and the Dirac phase are independent of this ambiguity. We can give the Dirac phase in terms of phases M ij (ij=e, ,  ).

4. Experimental data give useful constraints on M ij. Constraints on M ij ⇒ Constraints on δ We study general property of leptonic CP violation without referring to specific relations among M ij. The mixing angles and Dirac CP phase δ are to be given as functions of M ij.

5. Problem μ-τ symmetry gives consistent results with experimental data. But, It can not give Dirac CP Violation. Why? 2. What ’ s μ-τsymmetry ? μ-τsymmetry gives a constraint that Lagrangian is invariant under transformation of ν μ →  σν τ, ν τ →  σν μ (σ=±1) (*)  sign is just our convention.

6. μ-τ Symmetric Part + μ-τSymmery Breaking Part μ-τ symmetry CP Violation (  ) can not be obtained We clarify which flavor neutrino mass determines  as general as possible. extended to experiment: Why doesμ-τsymmetry give no Dirac CP violation? We need μ-τSymmery Breaking Part

7. Definition of mass matrix μ-τ symmetric part μ-τsymmetry breaking part We can fomally divide M into:

8. diagonalized by U sym μ-τsymmetric part U sym gives U PMNS

9. 3. μ-τsymmetry-breaking and CP phase We estimate Dirac CP violation induced by  –  symmetry breakings 1.First, we use perturbation with M b treated as a perturbative part to estimate . 2.Next, we perform exact estimation of  that gives the perturbative results.

10. The phase structure of |3> suggests Δ and γ : and 3-1. Perturbation with

11. These δ 、 Δ and γ consistently describe |1> and |2> Δ and γ can be calculated 3-1. Perturbation with

12. Suggested U PMNS We guess the appropriate form of the PMNS matrix This expression gives perturbative result If

13. 3-2. Exact results We have used redefined masses to control phase-ambiguities of γ: which gives the following formula: Another redundant phase ρcan also be removed by the redefinition of masses. But we keep ρ to see its trace in CP violation.

14. Exact result for a Re part : Maximal atmospheric mixing ⇒ x=0(s 13 cos  ’=0) & D_=0(M  =M  ) ⇒ Maximal CP violation if M  =M 

15. If the textures are approximately  –  symmetric Which masses give which phases δ depends on B － ρ depends on B + Δ depends on D － γ depends on E － ( ＊ ) δ+ρ is Dirac CP Violating phase

16. ・ ・ We can determine the phase of δ and ρ, and θ 23 ・ ・ Maximal atmospheric mixing conditions are given by 4.Summary ・ Redefined flavor masses given by ・ give the weak-base independence of the Jarlskog invariant:

17. ・ ・ The phases of M ν are so constrained to give δ and ρ via B + and B －. ・ ・ We can determine which masses provide which phases. The work to discuss phases of M ν is in progress. ・ δ depends on B － ・ ρ depends on B + ・ Δ depends on D － ・ γ depends on E －

18. END

19. Three versions of M and U PMNS There are other two versions For the redefined masses, we have the PDG version of U PMNS : 2) The intermediate one (γ is excluded from U PMNS ): 1) The original one:

20. ・ Redefined flavor masses given by reassure the weak-base independence of the Jarlskog invariant: Now, we study which masses of Mν give which phases.

21. ・ ・ We can determine θ 23, and the phase of ρand δ ・ ・ Maximal atmospheric mixing conditions are given by ・ ・ We can determine which masses provide which phases. ・ δ depends on B － ・ ρ depends on B + ・ γ depends on E － 5. Summary a ・ Redefined flavor masses given by ・ reassure the weak-base independence of the Jarlskog invariant: