Parametric Resonance by the Matter Effect SATO, Joe (Saitama) Koike, Masafumi (Saitama) Ota, Toshihiko (Würzburg) Saito, Masako (Saitama) with Plan Introduction.

Parametric Resonance

by the Matter EffectSATO, Joe (Saitama)

Koike, Masafumi (Saitama)Ota, Toshihiko (Würzburg)

Saito, Masako (Saitama)

with

PlanIntroduction

Two-Flavor Oscillation

Parametric Resonance in Neutrino Oscillation

More on Parametric Resonance

Summary

Introduction

Interior of the Earth

http://www.math.montana.edu/~nmp/materials/ess/geosphere/expert/activities/planet_earth/

cru

st

man

tle

ou

ter

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

Matter Density Profile

Matter Density Profile

Constant-Density Approximation

Matter Density Profile

Matter Density Profile

Inhomogeneous Matter Koike,Sato

1999Ota,Sato 2001

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

Two-Flavor Oscillation

Two-Flavor Oscillation

• Second-order equation in dimensionless variables

• Dimensionless variables

• Initial conditions ,

•

• Matter effect

• Evolution equation of the two-flavor neutrino

• 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

id numbers of the oscillation peaks

Constant-Density Oscillation

Neutrino Energy / [GeV]

Ap

peara

nce P

rob

Parametric Resonancein Neutrino Oscillation

Matter Density Profile

Matter Density Profile

Matter Density Profile

Matter Density Profile

Matter Density Profile

Matter Density Profile

Evolution Equation

• Inhomogeneity

• Fourier expansion

• Effect of the n-th Fourier mode on the oscillation

Mathieu Equation

Pow!

Pow!

Parametric Resonance

Periodic Motion

Oscillation of Oscillation Parameter

in classical

mechanicsin classical

mechanics

We kick a swing twice in a period of motion.

Mathieu Equation

Pow!

Pow!

Parametric Resonance

Periodic Motion

Oscillation of Oscillation Parameter

Para

metr

ic

Reso

nance

in classical

mechanicsin classical

mechanics

We kick a swing twice in a period of motion.

Parametric Resonance Condition

Parametric Resonance

Neutrino Oscillation

Fourier modes of

matter effect

in neutrino

oscillationin neutrino

oscillation

Parametric Resonance

Neutrino Oscillation

Fourier modes of

matter effect

Para

metr

ic

Reso

nance

in neutrino

oscillationin neutrino

oscillation

Parametric Resonance Condition n-th oscillation

dip

Effect of the Mode 1

Neutrino Energy / [GeV]

Ap

peara

nce P

rob

Sizable effect at1st peak (n=0) and 2nd

peak (n=1) 0 g/cm3 0

0.1 g/cm3

0.231

0.2 g/cm3

0.462

0.3 g/cm3

0.693

0.4 g/cm3

0.925

0.5 g/cm3

1.16

Mode 1: Possible Large EffectEarth models suggest for a through-Earth

pathEarth models suggest for a through-Earth path

0 g/cm3 0

1 g/cm3 2.31

2 g/cm3 4.62

3 g/cm3 6.93

4 g/cm3 9.25

5 g/cm3 11.6

Mode 1: Possible Large EffectEarth models suggest for a through-Earth

pathEarth models suggest for a through-Earth path

0 g/cm3 0

1 g/cm3 2.31

2 g/cm3 4.62

3 g/cm3 6.93

4 g/cm3 9.25

5 g/cm3 11.6

Effect of the Mode 2

Neutrino Energy / [GeV]

Ap

peara

nce P

rob

Sizable at2nd (n=1) and 3rd (n=2)

peaks 0 g/cm3 0

0.1 g/cm3

0.231

0.2 g/cm3

0.462

0.3 g/cm3

0.693

0.4 g/cm3

0.925

0.5 g/cm3

1.16

Effect of the Mode 3

Neutrino Energy / [GeV]

Ap

peara

nce P

rob

Sizable at3rd (n=2) and 4th (n=3)

peaks 0 g/cm3 0

0.1 g/cm3

0.231

0.2 g/cm3

0.462

0.3 g/cm3

0.693

0.4 g/cm3

0.925

0.5 g/cm3

1.16

More on the Parametric Resonance

Resonant Enhancement

Resonant Enhancement

Resonant Enhancement

Resonant Enhancement

Resonant Enhancement

Resonant EnhancementResonant enhancement of apparance probability, even for a small Fourier coefficientResonant enhancement of apparance probability, even for a small Fourier coefficient

n = 1

n = 2

n = 3Fictious repetition of the matter profile

Matter profile (Arbitrary vertical scale)

Oscillation “dip” at

0 g/cm3 0

0.3 g/cm3

0.693

1 g/cm3 2.31

Large-Scale Oscillation

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 distribution n-th Fourier mode

Appearance probability

n-th dip and neighbor