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NEUTRINO OSCILLATIONS AND THE MSW EFFECT. S. P. Mikheyev INR RAS Moscow. LEP: Number of light, active neutrinos  3. 35. 25.  (nb). 15. 5. 90. 92. 94. 96. 88.  s = E cm (GeV).

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Neutrino oscillations and the msw effect

NEUTRINO OSCILLATIONS

AND

THE MSW EFFECT

S. P. Mikheyev

INR RAS Moscow


Neutrino oscillations and the msw effect

LEP: Number of light, active neutrinos  3

35

25

(nb)

15

5

90

92

94

96

88

s = Ecm (GeV)

Erice 9 July 2004


Neutrino oscillations and the msw effect

Non-zero neutrino masses introduce lepton mixing matrix,U, which, in general, is not expected to be diagonal. The matrix connects the flavor eigenstates (ne,nm,nt,…)with the mass eigenstates (n1,n2,n3,…).

Neutrino Oscillations

B. Pontecorvo (1957),

Z. Maki, M. Nakagawa, S. Sakata (1962)

Erice 9 July 2004


Neutrino oscillations and the msw effect

Flavors, Masses & Mixing

ne

nm

i

n3

n1

n2

nt

2

)

U = (

cosq sinq

- sinq cosq

Flavor Neutrino States

Mass Eigetstates

m1

m2

m3

m

t

e

  • Correspond to certain charged lepton

Mixing

  • Interact in pairs

na= SUaini

  • Eigenstates of the CC Weak Interactions

Erice 9 July 2004


Neutrino oscillations and the msw effect

Neutrino Oscillations

n2

n1

ne

n2

nm

n1

n2 = sinq ne + cosq nm

ne = cosq n1 + sinq n2

inversely

n1 = cosq ne - sinq nm

nm = - sinq n1 + cosq n2

coherent mixtures of mass eigenstates

flavor composition of the mass eigenstates

n2

n2

ne

n1

wave

packets

n1

n2

inserting

nm

n1

Flavors of eigenstates

The relative phases

of the mass states

in ne and nm

are opposite

Interference of the parts of

wave packets with the same

flavor depends on the

phase difference Df

between n1 and n2

Erice 9 July 2004


Neutrino oscillations and the msw effect

Neutrino Oscillations

Dm2

2E

Dvphase =

Dm2 = m22 - m12

Propagation in vacuum:

Flavors of mass eigenstates do not change

Determined by q

Admixtures of mass eigenstates

do not change: no n1 <-> n2 transitions

n2

ne

n1

Df = Dvphase t

Df = 0

Due to difference of masses n1 and n2

have different phase velocities:

Oscillation length:

oscillations:

ln = 2p/Dvphase= 4pE/Dm2

effects of the phase difference

increase which changes

the interference pattern

Amplitude (depth) of oscillations:

A = sin22q

Erice 9 July 2004


Neutrino oscillations and the msw effect

d

i

(ni)= Mdiag(ni)

dt

d

i

(na)= UMdiagU*(na)

dt

nanbt=SUaiUbie-iE t

i

*

i

<nanbt2= SUaiUbiUajUbje-i(E – E )t

j

i

*

*

i,j

(ni)= U*(na)

Erice 9 July 2004


Neutrino oscillations and the msw effect

2- Oscillations in Vacuum

1.0

0.8

Probability

0.6

0.4

0.2

102

103

104

L/E (km/GeV)

P(ee ) =1 - sin22q sin2 (1.27m2(eV2)L(km)/E(GeV))

(L/E) = 0.1

(L/E) = 0.25

(L/E) = 0.5

Erice 9 July 2004


Neutrino oscillations and the msw effect

Oscillation Measurements

  • Disappearence experiment: P( )

  • Appearence experiment: P( )

P( ) = 1 - P( )

<P> = P( )

Erice 9 July 2004


Neutrino oscillations and the msw effect

Matter Effect

L. Wolfenstein, 1978

ne

e

Elastic forward

scattering

Potentials

Ve, Vm

W

V ~ 10-13 eV inside the Earth for E = 10 MeV

ne

e

Difference of potentials is important

for ne nm :

Ve- Vm = 2 GFne

Refraction index:

n - 1 = V / p

Refraction length:

l0 = 2p / (Ve - Vm)

~ 10-20 inside the Earth

< 10-18 inside the Sun

n - 1

= 2 p/GFne

~ 10-6 inside the neutron star

Erice 9 July 2004


Neutrino oscillations and the msw effect

ne

n1

n2m

n1m

q

n2

nm

qm

Neutrino eigenstates in matter

in vacuum:

in matter:

Effective

Hamiltonean

H = H0 + V

H0

V = Ve - Vm

n1m, n2m

n1, n2

Eigenstates

depend

on ne, E

m1m, m2m

m1, m2

Eigenvalues

m12/2E, m22/2E

H1m, H2m

Mixing in matter

is determined with respect

to eigenstates in matter

qm

is the mixing angle in matter

Erice 9 July 2004


Neutrino oscillations and the msw effect

Oscillations in matter

In uniform matter (constant density)

qm(E, n) = constant

mixing is constant

Flavors of the eigenstates do not change

Admixtures of matter eigenstates

do not change: no n1m <-> n2mtransitions

Oscillations

Monotonous increase of the phase difference

between the eigenstates Dfm

as in vacuum

n2m

ne

n1m

Dfm= (H2 - H1) L

Dfm= 0

Parameters of oscillations (depth and length)

are determined by mixing in matter

and by effective energy split in matter

sin22qm, lm

sin22q, ln

Erice 9 July 2004


Neutrino oscillations and the msw effect

Resonance

In resonance:

sin2 2qm

sin2 2qm = 1

n

n

Mixing in matter is maximal

Level split is minimal

sin2 2q= 0.08

sin2 2q= 0.825

ln = l0 cos 2q

~

Refraction

length

Vacuum

oscillation

length

~

For large mixing: cos 2q = 0.4 - 0.5

the equality is broken

the case of strongly coupled system

shift of frequencies

ln / l0

~ n E

Resonance width:DnR = 2nR tan2q

Resonance layer: n = nR + DnR

Erice 9 July 2004


Neutrino oscillations and the msw effect

Two effects

Adiabatic

(partially adiabatic)

neutrino conversion

Resonance enhancement

of neutrino oscillations

Density

profiles:

Variable density

Constant density

Change of mixing, or

flavor of the neutrino

eigenstates

Change of the phase

difference between

neutrino eigenstates

Degrees of

freedom:

In general:

Interplay of oscillations

and adiabatic conversion

MSW

Erice 9 July 2004


Neutrino oscillations and the msw effect

Resonance enhancement

n

ne

ne

F0(E)

F(E)

Source

Layer of matter with constant density, length L

Detector

k = p L/ l0

thin layer

thick layer

k = 1

k = 10

F (E)

F0(E)

sin2 2q = 0.824

sin2 2q = 0.824

E/ER

E/ER

Erice 9 July 2004


Neutrino oscillations and the msw effect

Resonance enhancement

n

ne

ne

F0(E)

F(E)

Source

Layer of matter with constant density, length L

Detector

thick layer

thin layer

k = 1

k = 10

F (E)

F0(E)

sin2 2q = 0.08

sin2 2q = 0.08

E/ER

E/ER

Erice 9 July 2004


Neutrino oscillations and the msw effect

Resonance enhancement

resonance

layer

Continuity:

neutrino and antineutrino semiplanes

normal and inverted hierarchy

P

Oscillations (amplitude of oscillations)

are enhanced in the resonance layer

ln / l0

E = (ER - DER) -- (ER + DER)

DER = ERtan 2q = ER0sin 2q

ER0= Dm2 / 2V

P

With increase of mixing:

q -> p/4

ER-> 0

ln / l0

DER-> ER0

Erice 9 July 2004


Neutrino oscillations and the msw effect

MSW: adiabatic conversion

H = H(t) depends on time

Non-uniform matter

density changes on

the way of neutrinos:

n1m n2m are no more the

eigenstates of propagation

-> n1m <-> n2m transitions

qm = qm(ne(t))

mixing changes in the

course of propagation

ne = ne(t)

However

if the density changes slowly enough (adiabaticity condition)

n1m <->n2m transitions can be neglected

Flavors of eigenstates change

according to the density change

determined by qm

Admixtures of the eigenstates,

n1m n2m, do not change

fixed by mixing in

the production point

Phase difference increases according to the level split

which changes with density

MSW

Effect is related to the change of flavors

of the neutrino eigenstates in matter with varying density

Erice 9 July 2004


Neutrino oscillations and the msw effect

Adiabaticity

External conditions (density)

change slowly

so the system has time to

adjust itself

dqm

dt

Adiabaticity condition

<< 1

H2 - H1

The eigenstates

propagate

independently

transitions between

the neutrino eigenstates

can be neglected

n1m <--> n2m

Crucial in the resonance layer:

- the mixing angle changes fast

- level splitting is minimal

if vacuum mixing

is small

DrR > lR

lR = ln/sin2q is the oscillation width in resonance

DrR = nR / (dn/dx)Rtan2q is the width of the resonance layer

If vacuum mixing is large

the point of maximal adiabaticity violation

is shifted to larger densities

n(a.v.) -> nR0 > nR

nR0 = Dm 2/ 2 2 GF E

Erice 9 July 2004


Neutrino oscillations and the msw effect

Adiabatic conversion

and initial condition

The picture of conversion depends on how far from the resonance layer in the density

scale the neutrino is produced

n0 < nR

n0 > nR

n0 ~ nR

nR - n0 >> DnR

n0 - nR >> DnR

Interplay of

conversion and

oscillations

Oscillations with

small matter effect

Non-oscillatory

conversion

nR ~ 1/ E

All three possibilities are realized for the solar neutrinos

in different energy ranges

Erice 9 July 2004


Neutrino oscillations and the msw effect

Adiabatic conversion

n1m <-->n2m

P = sin2q

n0 >> nR

Non-oscillatory transition

n2m

n1m

n2

n1

interference suppressed

Resonance

Mixing suppressed

n0 > nR

Adiabatic conversion + oscillations

n2m

n1m

n2

n1

n0 < nR

Small matter corrections

n2m

n1m

n2

n1

ne

Erice 9 July 2004


Neutrino oscillations and the msw effect

Adiabaticity violation

n2m n1m

Fast density change

n0 >> nR

n2m

n1m

n2

n1

ne

Resonance

Admixture of n1m increases

Erice 9 July 2004


Neutrino oscillations and the msw effect

The MSW effect

The picture of adiabatic conversion is universal in terms of variable y = (nR - n ) / DnR

(no explicit dependence on oscillation parameters density distribution, etc.)

Only initial value y0 matters.

resonance layer

production

point

y0 = - 5

resonance

survival probability

For zero

final density:

y = 1/tan 2q

oscillation

band

averaged

probability

LMA

y = (nR - n) / DnR

(distance)

Erice 9 July 2004


Neutrino oscillations and the msw effect

Solar neutrinos

Adiabatic conversion

in matter of the Sun

r : (150 0) g/cc

Oscillations

in vacuum

n

Oscillations

in matter

of the Earth

Erice 9 July 2004


Neutrino oscillations and the msw effect

Conversion inside the Sun

tan2q = 0.41,

Dm2 = 7.3 10-5 eV2

survival probability

survival probability

sin2q

E = 14 MeV

resonance

E = 10 MeV

core

y

y

distance

surface

survival probability

survival probability

E = 6 MeV

E = 2 MeV

y

y

Erice 9 July 2004


Neutrino oscillations and the msw effect

Conversion inside the Sun

tan2q = 0.41,

Dm2 = 7.3 10-5 eV2

Low energy part of the spectrum:

vacuum oscillations with small matter corrections

E = 0.4 MeV

E = 0.86 MeV

survival probability

survival probability

y

y

distance

distance

Dashed lines: for pure vacuum oscillations

Erice 9 July 2004


Neutrino oscillations and the msw effect

Atmospheric Neutrinos: Flux Features

p, He, …

A

N

E3dF/dE

N

p± (K ±)

E(GeV)

nm

ne

nm

(nm + )

nm

R = = 2

(ne+ )

ne

E(GeV)

Erice 9 July 2004


Neutrino oscillations and the msw effect

Atmospheric Neutrinos: Flux Features

cos = 0.8

L = 20km

104

Path length (km)

L = 500km

cos = 0

103

L = 10 000km

102

-1

-.6

-.2

.2

.6

1

Cos

cos = -0.8

up

horizontal

down

L/E values from 0.1 (10 km/1000 GeV) to 104 (104 km/1 GeV) can be explored to study neutrino oscillations.

Flux is up/down symmetric

but isn’t isotropic Fn~ 1/CosQ

Erice 9 July 2004


Neutrino oscillations and the msw effect

Atmospheric Neutrino Detection

K.Nishikawa’s talk

Erice 9 July 2004


Neutrino oscillations and the msw effect

Atmospheric Neutrino Detection

Super-Kamiokande

Erice 9 July 2004


Neutrino oscillations and the msw effect

Atmospheric Neutrino Events

m

Partially contained

1/10t/year

Acceptance 4p

nm

m

e

Fully

contained

Ed Kearns for the SK Collaboration Neutrino2004, June 15, Paris

0.8

0.6

Events/day

0.4

ne

nm

0.2

100

103

101

102

Neutrino energy (GeV)

Erice 9 July 2004


Neutrino oscillations and the msw effect

Atmospheric Neutrino Events

Upward through-going

m

1/10m2/year

Acceptance 2p

Ed Kearns for the SK Collaboration Neutrino2004, June 15, Paris

0.18

Upwardstopping

0.14

m

0.10

Events/day

0.06

nm

nm

0.02

100

103

101

102

Neutrino energy (GeV)

Erice 9 July 2004


Neutrino oscillations and the msw effect

Atmospheric Neutrinos

Zenith Angle Distributions

Sub-GeV Evis < 1.3 GeV

Ed Kearns for the SK Collaboration Neutrino2004, June 15, Paris

Multi-GeV Evis > 1.3 GeV

Erice 9 July 2004


Neutrino oscillations and the msw effect

Atmospheric Neutrinos

 oscillations

Ed Kearns for the SK Collaboration Neutrino2004, June 15, Paris

Erice 9 July 2004


Neutrino oscillations and the msw effect

Atmospheric Neutrinos

 oscillations L/E analysis

Ed Kearns for the SK Collaboration Neutrino2004, June 15, Paris

Oscillation deep @ L/E  500 km/GeV

Erice 9 July 2004


K2k experiment

K2K is the first accelerator-based long-baseline neutrino oscillation experiment to investigate the neutrino oscillation observed in atmospheric neutrinos.

K2K experiment

Super-Kamiokande

KEK-

12GeV PS

L=250 km

Erice 9 July 2004


Neutrino oscillations and the msw effect

K2K experiment

Neutrino Beam Production

p+Al  p+  m+ +nm

e+ + ne (1.3%)+ nm (0.5%)

Near Detector

Far Detector

 - disappearance

Erice 9 July 2004


Neutrino oscillations and the msw effect

Erec (GeV)

K2K experiment

Shape of energy spectrum

CC – quasielastic reactions

Erice 9 July 2004


Neutrino oscillations and the msw effect

K2K experiment

Both  disappearance and energy spectrum distortion have the consistent result

Erice 9 July 2004


Neutrino oscillations and the msw effect

Solar Neutrinos

G.Fogli et al. hep-ph/0106247

Erice 9 July 2004


Neutrino oscillations and the msw effect

Solar Neutrinos

All solar neutrino data

Erice 9 July 2004


Neutrino oscillations and the msw effect

Experiment KamLAND

Erice 9 July 2004


Neutrino oscillations and the msw effect

Experiment KamLAND

Erice 9 July 2004


Neutrino oscillations and the msw effect

Experiment KamLAND

Erice 9 July 2004


Neutrino oscillations and the msw effect

Experiment KamLAND

Erice 9 July 2004


Neutrino oscillations and the msw effect

Experiment KamLAND

Erice 9 July 2004


Neutrino oscillations and the msw effect

Solar neutrinos

m2(5.410-59.510-5) eV2

Sin22 (0.710.95)

m21

12

с12 s12 0

-s12 c12 0

0 0 1

с13 0 s13 ei

0 1 0

-s13 e-i 0 c13

1 0 0

0 с23s23

0 -s23 c23

U =

Atmospheric neutrinos

m2 (1.310-3 3.010-3) eV2

Sin22> 0.9

m32

23

3 – neutrino mixing

3 mixing angles(12,23,13)

&phase of CP violation ()

сij = cosij

sij = sinij

Erice 9 July 2004


Neutrino oscillations and the msw effect

m21

m31

2

2

Known parameters

( ,, sin2212, sin2223)

… and limit forsin2213

Erice 9 July 2004


Neutrino oscillations and the msw effect

normal

inverted

3

2

1

2

Sign ofm31

2

1

3

Unknownparameters

sin2213  0.2

 - phase of CP violation

 Mass hiererchy

Erice 9 July 2004