1. Parametric Resonance by the Matter Effect SATO, Joe (Saitama) Koike, Masafumi (Saitama) Ota, Toshihiko (Würzburg) Saito, Masako (Saitama) with Plan Introduction Two-Flavor Oscillation Parametric Resonance in Neutrino Oscillation More on Parametric Resonance Summary

2. Introduction

3. Parametric resonance of neutrino oscillation (Akhmedov, 1999) Matter with periodic density profile causes parametric resonance of the neutrino oscillation. Neutrino oscillation through the Earth may indicate this effect. We show how the effect of parametric resonance emerges in neutrino oscillation in a general matter profile.

4. Interior of the Earth http://www.math.montana.edu/~nmp/materials/ess/geosphere/expert/activities/planet_earth/

5. crust mantle outer core Interior of the Earth http://www.math.montana.edu/~nmp/materials/ess/geosphere/expert/activities/planet_earth/ inner core Preliminary Reference Earth Model Depth

6. Matter Density Profile

8. Constant-Density Approximation

9. Matter Density Profile

11. Inhomogeneous Matter Koike,Sato 1999 Ota,Sato 2001

12. Parametric Resonance in Neutrino Oscillation Ermilova et al. (1986) Akhmedov, Akhmedov et al. (1988 — Present) others “Castle-wall” matter profile (Akhmedov, 1998) Fourier decomposition (Present approach) Mode 1 Mode 2 Mode 3

13. Two-Flavor Oscillation

14. Second-order equation in dimensionless variables Dimensionless variables Initial conditions, Matter effect Evolution equation of the two-flavor neutrino

15. MSW-resonance peak. Peaks and dips of the oscillation spectrum Simple solution when: Appearance probability at the endpoint of the baseline (n+1)-th oscillation peak. n-th oscillation dip. Constant-Density Oscillation

16. id numbers of the oscillation peaks Constant-Density Oscillation Neutrino Energy / [GeV] Appearance Prob

17. Parametric Resonance in Neutrino Oscillation

18. Matter Density Profile

24. Evolution Equation Inhomogeneity Fourier expansion Effect of the n-th Fourier mode on the oscillation Mathieu Equation

25. Modes of Matter Profile Matter profile in Fourier series Mode 1 Mode 5 Mode 2 Mode 3 Mode 4

26. Pow ! Parametric Resonance Periodic Motion Oscillation of Oscillation Parameter in classical mechanics We kick a swing twice in a period of motion. Mathieu Equation

27. Pow ! Parametric Resonance Periodic Motion Oscillation of Oscillation Parameter Parametric Resonance in classical mechanics We kick a swing twice in a period of motion. Parametric Resonance Condition

28. Parametric Resonance Neutrino Oscillation Fourier modes of matter effect in neutrino oscillation

29. Parametric Resonance Neutrino Oscillation Fourier modes of matter effect Parametric Resonance in neutrino oscillation Parametric Resonance Condition n-th oscillation dip

30. Effect of the Mode 1 Neutrino Energy / [GeV] Appearance Prob Sizable effect at 1st peak (n=0) and 2nd peak (n=1) 0 g/cm 3 0 0.1 g/cm 3 0.231 0.2 g/cm 3 0.462 0.3 g/cm 3 0.693 0.4 g/cm 3 0.925 0.5 g/cm 3 1.16

31. Mode 1: Possible Large Effect Earth models suggestfor a through-Earth path 0 g/cm 3 0 1 g/cm 3 2.31 2 g/cm 3 4.62 3 g/cm 3 6.93 4 g/cm 3 9.25 5 g/cm 3 11.6

32. Mode 1: Possible Large Effect Earth models suggestfor a through-Earth path 0 g/cm 3 0 1 g/cm 3 2.31 2 g/cm 3 4.62 3 g/cm 3 6.93 4 g/cm 3 9.25 5 g/cm 3 11.6

33. Effect of the Mode 2 Neutrino Energy / [GeV] Appearance Prob Sizable at 2nd (n=1) and 3rd (n=2) peaks 0 g/cm 3 0 0.1 g/cm 3 0.231 0.2 g/cm 3 0.462 0.3 g/cm 3 0.693 0.4 g/cm 3 0.925 0.5 g/cm 3 1.16

34. Effect of the Mode 3 Neutrino Energy / [GeV] Appearance Prob Sizable at 3rd (n=2) and 4th (n=3) peaks 0 g/cm 3 0 0.1 g/cm 3 0.231 0.2 g/cm 3 0.462 0.3 g/cm 3 0.693 0.4 g/cm 3 0.925 0.5 g/cm 3 1.16

35. More on the Parametric Resonance

36. Resonant Enhancement

41. Resonant enhancement of apparance probability, even for a small Fourier coefficient n = 1 n = 2 n = 3 Fictious repetition of the matter profile Matter profile (Arbitrary vertical scale) Oscillation “dip” at 0 g/cm 3 0 0.3 g/cm 3 0.693 1 g/cm 3 2.31

42. Large-Scale Oscillation

43. Summary Neutrino oscillation across the Earth Deviation from the constant density Fourier analysis Parametric resonance Frequency matching of the matter distribution and the neutrino energy Mathieu-like equation provides an analytic description Matter distributionn-th Fourier mode Appearance probabilityn-th dip and neighbor