The KTY formalism and the neutrino oscillation probability including nonadiabatic contributions Tokyo Metropolitan niversity Osamu Yasuda Based on Phys.Rev. D89 (2014) 093023 21 December 2014@Miami2014 1/18
Contents 1. Introduction 2. Extension of Kimura-Takamura- Yokomakura s formulation 3. Examples of non-adiabatic cases 4. Summary 2/17
1. Introduction 1.1 Scheme of 3 flavor ν oscillation Both hierarchy patterns are allowed Mixing matrix Functions of mixing angles θ 12, θ 23, θ 13, and CP phase δ ν ν ν e = μ τ e1 μ1 τ1 e2 μ2 τ2 e3 μ3 τ3 All 3 mixing angles have been measured (2012): ν ν ν 1 2 3 Normal Hierarchy Inverted Hierarchy ν solar +KamLAND (reactor) θ π 6 2 5 2 12, m21 8 10 ev ν atm +K2K,MINOS(accelerators) π 2 3 2 DCHOOZ+Daya Bay+Reno (reactors), T2K+MINOS, others θ 23, m32 2.5 10 ev 4 θ 13 π / 20 3/17
1.2 sefulness of analytical expressions of ν oscillation Analytical expressions for ν oscillation probability are useful to discuss qualitative behaviors in various cases. ν oscillation probability in matter is complicated beyond the 2-flavor case. The results which are obtained so far are mainly for the case of matter with constant density The treatment of more generalized cases has been developed (KTY formalism described later). 4/17
1.3 Nonadiabatic contributions in ν oscillation Nonadiabatic contributions can be important for ν oscillation probability for a small mixing angle. In the case (e.g., solar ν) where the baseline length is large so that rapid oscillations are averaged, probability is In the ν oscillation in the presence of New Physics, the effective mixing angle could be small. The results which are obtained so far are mainly for the standard case. In this talk treatment of more general cases is discussed. 5/17
1.4 Exact oscillation probability in matter with constant density matter effect 2 E p + j effective mixing matrix elements in matter m 2 j e1 = μ1 τ1 e2 μ2 τ2 e3 μ3 τ3 Probability of ν oscillation can be expressed in terms of the energy eigenvalues and bilinear forms of effective mixing matrix elements in matter 6/17
1.5 Formulation by Kimura-Takamura- Yokomakura (KTY PLB537:86,2002) Simultaneous equation for can be solved Thus the problem of obtaining the exact analytical oscillation probability is reduced to obtaining only the eigenvalues! 7/17
2.Extension of KTY s formulation To what extent can we generalize the result of the case in matter with constant density? The case in which density varies adiabatically The case in which density varies nonadiabatically In these cases the mixing matrix element themselves is required. In KTY s formulation the bilinear form can be obtained, but itself cannot be. It turns out that can be obtained from the bilinear form : the main result of this talk 8/17
In the following, the notation used; we assume is known from KTY is From trivial identities, we have: p to phases, we obtain the following : 9/17
To get the standard parameterization, we multiply a diagonal phase matrix both from left and right as follows: At a locally given density, we can obtain the analytical expression for the effective mixing angles. 10/17
3. Examples of non-adiabatic cases 3.1 Supernova ν with Non-Standard Interactions in matter For simplicity self interactions of ν are not considered λ 3 λ 2 λ 3 A ρ λ 2 λ 2 λ 1 λ 1 ν from inside in general undergo non-adiabatic transition at High and Low resonance x 11/17
λ 9 =diag(1,0,-1) γ=arg(ε eτ ) Constraints from high energy ν atm Due to NSI, the effective mixing angles are modified: 12/17
From these, we get the probability which takes into account non-adiabatic transitions 13/17
3.2 The case with magnetic transitions & magnetic field In this example we have magnetic interaction terms instead of matter effects. μ αβ : Majorana type magnetic transitions 14/17
We consider the situation in which B(t=0) 0 B(t=L)=0 occurs adiabatically or non-adiabatically. For simplicity we assume the following: no CP phase from the charged lepton sector: β, γ =0 θ 13, Δm 2 21 0 15/17
The probability w/ non-adiabatic transitions 1 level crossing 2 3 x For simplicity we assume χ <<1 : χ plays a role of the vacuum angle Nonadiabaticity becomes significant when χ <<1 16/17
4. Summary From trivial identities, we can (in principle) obtain the analytical expression for the mixing angles and the CP phase. sing such expressions, probability for ν flavor transitions including non-adiabatic processes can be analytically obtained. As a demonstration, two examples were discussed: (i) Supernova ν with Non- Standard Interactions in matter; (ii) ν with magnetic transitions in a large magnetic field. 17/17