Matter E ff ect in Long Baseline Neutrino Oscillation Masafumi - - PowerPoint PPT Presentation

Matter Effect in Long Baseline Neutrino Oscillation

Masafumi Koike (Utsunomiya U.)

working with Toshihiko Ota (Saitama U.) Masako Saito (Utsunomiya U.) Joe Sato (Saitama U.)

Neutrino Frontier Workshop 2014 (Fuji Calm, 2014.12.21)

Introduction

1

Neutrino Mixings: Achievements

From discovery to precision measurements

Δ I-N ¼ Parameter Best fit 1σ range 2σ range δm2=10−5eV2 (NH or IH) 7.54 7.32–7.80 7.15–8.00 sin2 θ12=10−1 (NH or IH) 3.08 2.91–3.25 2.75–3.42 Δm2=10−3eV2 (NH) 2.43 2.37–2.49 2.30–2.55 Δm2=10−3eV2 (IH) 2.38 2.32–2.44 2.25–2.50 sin2 θ13=10−2 (NH) 2.34 2.15–2.54 1.95–2.74 sin2 θ13=10−2 (IH) 2.40 2.18–2.59 1.98–2.79 sin2 θ23=10−1 (NH) 4.37 4.14–4.70 3.93–5.52 sin2 θ23=10−1 (IH) 4.55 4.24–5.94 4.00–6.20 δ=π (NH) 1.39 1.12–1.77 0.00 − 0.16 ⊕ 0.86 − 2.00 δ=π (IH) 1.31 0.98–1.60 0.00 − 0.02 ⊕ 0.70 − 2.00

Status of three-neutrino oscillation parameters, circa 2013

• F. Capozzi,1,2 G. L. Fogli,1,2 E. Lisi,2 A. Marrone,1,2 D. Montanino,3,4 and A. Palazzo5

1Dipartimento Interateneo di Fisica “Michelangelo Merlin,” Via Amendola 173, 70126 Bari, Italy

PHYSICAL REVIEW D 89, 093018 (2014)

Neutrino Mixings:
 Challenges

Mass Hierarchy Octant Degeneracy Leptonic CP Violation δm231≷ 0 ? θ23≷ π/4 ? sin δCP = 0 ?

Oscillation experiments with very long baseline (1000~10000 km) Exploiting the matter effect

Evaluating the Matter Effect

Physics potential of neutrino oscillation experiment with a far detector in Oki Island along the T2K baseline

10 20 30 40 50 0.0 1.0 2.0 3.0 Eν (GeV) (a1) OAB1.4(νµ) normal BG+νµ→νe νe→νe+ν

• µ→ν
• e+ν
• e→ν
• e

ν

• µ→ν
• e+ν
• e→ν
• e

ν

• e→ν
• e

10 20 0.0 1.0 2.0 3.0 Eν (GeV) (b1) OAB1.4(ν

• µ) normal

BG+ν

• µ→ν
• e

ν

• e→ν
• e+νµ→νe+νe→νe

νµ→νe+νe→νe νe →νe 10 20 30 40 50 0.0 1.0 2.0 3.0 Eν (GeV) (a2) OAB1.4(νµ) inverted BG+νµ→νe νe→νe+ν

• µ→ν
• e+ν
• e→ν
• e

ν

• µ→ν
• e+ν
• e→ν
• e

ν

• e→ν
• e

10 20 0.0 1.0 2.0 3.0 Eν (GeV) (b2) OAB1.4(ν

• µ) inverted

BG+ν

• µ→ν
• e

ν

• e→ν
• e+νµ→νe+νe→νe

νµ→νe+νe→νe νe →νe

• K. Hagiwara, T. Kiwanami, N. Okamura, K.-i. Senda (2013)

Event number/[2.5x1021 POT] vs Eν/[GeV]

L = 693km

Earth Model

mantle

• uter core

inner core crust

14 12 10 8 6 4 2

Density / [g/cm3]

6000 5000 4000 3000 2000 1000

Length / [km]

Preliminary Reference Earth Model

Density Profile on a Baseline

12 10 8 6 4 2

Density / [g/cm

3]

12000 11000 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000

Length / [km]

L = 12,000 km L = 7,500 km

Core Crust & Mantle Crust & Mantle

Matter Density Profile

12 10 8 6 4 2

Density / [g/cm3]

12000 10000 8000 6000 4000 2000

Length / [km] L = 12000km N = 0

12 10 8 6 4 2

Density / [g/cm3]

12000 10000 8000 6000 4000 2000

Length / [km] L = 12000km N = 1

Average Cosine

Constant vs. Earth Model

L = 12000 km ρ0 = 7.58 g/cm3 δm2

31 = 2.5 10−3eV2

δ m

2 2 1

= 7 . 9 1

− 5

e V

2

sin2 2θ12 = 0.84 sin2 2θ23 = 1.00 sin

2

2θ

1 3

= 0.05 sin δ = 0.00 ρ1 = −2.16 g/cm3

Matter Profile: Fourier Series

12 10 8 6 4 2

Density / [g/cm3]

12000 10000 8000 6000 4000 2000

Length / [km] L = 12000km N = 0

12 10 8 6 4 2

Density / [g/cm3]

12000 10000 8000 6000 4000 2000

Length / [km] L = 12000km N = 1

12 10 8 6 4 2

Density / [g/cm3]

12000 10000 8000 6000 4000 2000

Length / [km] L = 12000km N = 3

12 10 8 6 4 2

Density / [g/cm3]

12000 10000 8000 6000 4000 2000

Length / [km] L = 12000km N = 5

12 10 8 6 4 2

Density / [g/cm3]

12000 10000 8000 6000 4000 2000

Length / [km] L = 12000km N = 10

12 10 8 6 4 2

Density / [g/cm3]

12000 10000 8000 6000 4000 2000

Length / [km] L = 12000km N = 50

12 10 8 6 4 2

Density / [g/cm3]

12000 10000 8000 6000 4000 2000

Length / [km] L = 12000km N = 100

12 10 8 6 4 2

Density / [g/cm3]

12000 10000 8000 6000 4000 2000

Length / [km] L = 12000km N = 500

Formulation

2

Modeling Density Profiles

Step function Fourier series

Koike-Sato (1998), Ota-Sato (2003), Koike-Ota-Saito-Sato (2009)... Akhmedov (1988), Krastev-Smirnov (1989), Krastev-Smirnov (1989), Liu-Smirnov (1998), Petcov (1998), Chizhov-Petcov (1998), ..., Akhmedov-Maltoni-Smirnov (2005), ...

Two-Flavor Oscillation

Matter effect a(x) = 2 √ 2GFne(x)E

Evolution equation of the two-flavor neutrino

i d dx

• νe(x)

νµ(x) ⇥ = 1 2E ⇤δm2 2

• − cos 2θ

sin 2θ sin 2θ cos 2θ ⇥ +

• a(x)

⇥⌅ νe(x) νµ(x) ⇥

Second-order equation in dimensionless variables

z(ξ) + 1 4 ⇤ ∆m(ξ) − ∆ cos 2θ ⇥2 + ∆2 sin2 2θ + 2i∆

m(ξ)

⌅ z(ξ) = 0

MK-Ota-Saito-Sato, PLB 675, 69 (2009)

Dimensionless variables: Initial conditions ,

z(0) = 0 , z(0) = −i∆ 2 sin 2θ νµ(0) = 1 νe(0) = 0

z(ξ) = νe(ξ) exp ⇥ i 2 ξ ds ∆m(s) ⇤ · · ·

• νe(ξ)
• 2 =
• z(ξ)
• 2

ξ ≡ x L ∆ ≡ δm2L 2E ∆m(ξ) ≡ a(ξ)L 2E

Distance Reciprocal E Matter effect

Constant-Density Matter

Prob(νµ → νe) ∝ sin2 ω0ξ

1 2 3 1 ξ = 1 Dips at ω0 = nπ

Constant density: Δm(ξ) ≡ Δ0 = (const.)

≡ ω2 (const.)

z(ξ) + 1 4

• ∆m(ξ) − ∆ cos 2θ

2 + ∆2 sin2 2θ + 2i∆

m(ξ)

• z(ξ) = 0

Presence of the n-th Fourier mode

Inhomogeneous Matter

Fourier series of inhomogeneous matter

∆m(ξ) =

∞

• n=0

∆mn cos 2nπξ ρ(x) =

∞

• n=0

ρn cos 2nπ L x ,

z(ξ) +

• ω2

0 + αn cos 2nπξ − iβn sin 2nπξ + γn cos 4nπξ

⇥ z(ξ) = 0

γn = 1 8∆2

mn

αn = 1 2(∆m0 − ∆ cos 2θ)∆mn , ω2

0 = 1

4(∆m0 − ∆ cos 2θ)2 + 1 4∆2 sin2 2θ + 1 8∆2

mn ,

βn = nπ∆mn ,

z(ξ) + 1 4 ⇤ ∆m(ξ) − ∆ cos 2θ ⇥2 + ∆2 sin2 2θ + 2i∆

m(ξ)

⌅ z(ξ) = 0

ρ(x) = ρ0 + ρn cos 2nπ L x ,

∆m(ξ) = ∆m0 + ∆mn cos 2nπξ

Mathieu Equation z(t) + (ω2 − 2ε cos t)z(t) = 0

Parametric Resonance

• Periodic perturbation
• Twice in a period
• Grows amplitude of
• scillation
• Matter effect as a bunch of

periodic perturbations

Pow! Pow!

Ermilova et al. (1986), Akhmedov (1988), Krastev- Smirnov (1989), Liu-Smirnov (1998), Petcov (1998), Chizhov-Petcov (1998), ..., Akhmedov-Maltoni- Smirnov (2005), ...

2ω0

ω0

z(ξ) +

• ω2

0 + αn cos 2nπξ − iβn sin 2nπξ + γn cos 4nπξ

⇥ z(ξ) = 0

ω0 = nπ

E = E(±)

n

≡ δm2L 2 1 ∆m0 cos2θ ±

• 4n2π2 − ∆2

m0 sin2 2θ

The values of other parameters used in this plot found in arXiv:0902.1597

Mode 1 Mode 2 Mode 3

Resonance Condition

Parametric Resonance Condition

n-th mode n-th dip

1st dip 2nd dip 3rd dip

Mode 1 Mode 2 Mode 3

Matter Profile

Effect of the Mode 1

ρ1 = (0→5)g/cm3

1st Dip

Effect of the Mode 2

2nd Dip

ρ2 = (0→5)g/cm3

Matter-Profile Effects

3

Oscillogram: Full Profile

12500 12000 11500 11000 10500

Baseline Length / [km]

7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0

Neutrino Energy / [GeV]

0.4 0.4 0.4 0.4 0.3 0.3 0.3 0.3 . 3 . 2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 Full (PREM)

Density / [g/cm3]

Length / [km] L = 12000km N = 0

Fourier Coefficients

8 6 4 2

• 2

Coefficient / [g cm

–3]

12500 12000 11500 11000 10500

Baseline Length / [km]

0th (Average) 1st 2nd 3rd

Oscillogram: First Few Modes

12500 12000 11500 11000 10500

Baseline Length / [km]

7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.1 0.1 0.1

• Const. (0th)

12500 12000 11500 11000 10500 7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.4 0.4 . 4 . 4 0.4 0.3 0.3 0.3 0.3 0.3 0.3 . 2 . 2 0.2 0.2 0.1 . 1 0.1 0.1 Up to 1st 12500 12000 11500 11000 10500

Baseline Length / [km]

7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0

Neutrino Energy / [GeV]

0.4 0.4 0.4 0.3 0.3 0.3 0.3 0.3 . 2 . 2 0.2 0.2 . 1 . 1 0.1 0.1 0.1 Up to 2nd 12500 12000 11500 11000 10500 7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0

Neutrino Energy / [GeV]

0.4 0.4 0.3 0.3 . 3 0.3 . 3 . 2 0.2 0.2 0.2 0.1 0.1 . 1 . 1 0.1 0.1 0.1 0.1 Up to 3rd

νµ → νe

Baseline Length / [km]

Neutrino Energy / [GeV]

Oscillogram: Residues

12500 12000 11500 11000 10500

Baseline Length / [km]

7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.3 0.3 0.3 0.3 0.3 . 2 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 0.05 0.05

• Const. (0th)

12500 12000 11500 11000 10500 7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.1 0.05 0.05 0.05 0.05 0.05 Up to 1st 12500 12000 11500 11000 10500

Baseline Length / [km]

7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0

Neutrino Energy / [GeV]

0.05 0.05 0.05 Up to 2nd 12500 12000 11500 11000 10500 7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0

Neutrino Energy / [GeV]

Up to 3rd

νµ → νe

Neutrino Frontier Workshop 2014 (Fuji Calm, 2014.12.21)

Summary & Outlook

Fourier analysis is powerful to account for the matter-profile effects in neutrino oscillation.

• n-th Fourier mode ↔ n-th dip of the appearance probability
• Inhomogeneity

→ Parametric resonance

• Systematic improvement

Low Eν ↔ Small-size structure of matter

Backup slides

First-mode effect

Second-mode effect