What does μ-τsymmetry imply about leptonic CP violation? *)

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)

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

1.Introduction ? ? Dirac CP phase ? (*) Charged lepton masses are diagonalized We don’t know which masses give Dirac CP phase However, there is an ambiguity, where phases of Mij (ij=e,m,t) 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 Mij (ij=e,m,t).

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

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

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

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

μ-τsymmetric part diagonalized by Usym Usym gives UPMNS

3.μ-τsymmetry-breaking and CP phase We estimate Dirac CP violation induced by m–t symmetry breakings • First, we use perturbation with Mb treated as a perturbative part to estimate d. • Next, we perform exact estimation of dthat gives the perturbative results.

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

3-1. Perturbation with Δandγcan be calculated These δ、 Δand γconsistently describe |1> and |2>

Suggested UPMNS If We guess the appropriate form of the PMNS matrix This expression gives perturbative result

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

Exact result for a Exact result for a Re part : Maximal atmospheric mixing ⇒ x=0(s13cosd’=0) & D_=0(Mmm=Mtt) ⇒ Maximal CP violation if Mmm=Mtt

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

4.Summary ・We can determine the phase of δ and ρ, and θ23 ・Maximal atmospheric mixing conditions are given by ・Redefined flavor masses given by ・give the weak-base independence of the Jarlskog invariant:

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

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

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

5.Summary a ・We can determine θ23, 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－・Redefined flavor masses given by ・reassure the weak-base independence of the Jarlskog invariant:

What does μ-τsymmetry imply about leptonic CP violation? *)