NEUTRINO OSCILLATIONS AND THE MSW EFFECT

S. P. MikheyevINR RAS Moscow

S. P. Mikheyev INR Moscow

Erice 9 July 2004 2

s = Ecm (GeV)

5

15

25

35

(n

b)

88 90 92 94 96

LEP: Number of light, active neutrinos 3


Erice 9 July 2004 3

Non-zero neutrino masses introduce

lepton mixing matrix, U, which, in general, is not expected to be diagonal. The matrix connects the flavor eigenstates (e, , , …) with the mass eigenstates (1, , , …).

Neutrino Oscillations

B. Pontecorvo (1957), Z. Maki, M. Nakagawa, S. Sakata (1962)


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Flavor Neutrino States

Eigenstates of the CC Weak Interactions

m1 m2 m3

Mass Eigetstates

Correspond to certain charged

leptonInteract in pairs

= Ui ii

2 cossinsincos

U = ( )

e

e

12 3


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e = cos1sin

= - sin1cos 1 = cosesin

2 = sinecos

e 1

2

1

2

1

2

e

1

2

1

2

coherent mixtures of mass eigenstates

wavepackets

Interference of the parts of wave packets with the sameflavor depends on the phase difference between 1 and 2

flavor composition of the mass eigenstates

The relative phases of the mass states in e and are opposite

Flavors of eigenstates

inserting

inversely


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21

Due to difference of masses 1 and 2 have different phase velocities:

effects of the phase difference increase which changes the interference pattern

=vphase t

Flavors of mass eigenstates do not change

Admixtures of mass eigenstates do not change: no 1transitions

Determined by

=

vphase =m2

2Em2 = m2

2 - m12

Propagation in vacuum:

e

Oscillation length:

lvphase= 4E/m2

Amplitude (depth) of oscillations:

A = sin22


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(i) = U*()

ddt

(i) = Mdiag(i)i

ddt

() = UMdiagU*()i

it =UiUie-iE t* i

<t = UiUiUjUje-i(E – E )t

i,j* i j*


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1.0

0.2

0.4

0.6

0.8

102 103 104

L/E (km/GeV)

Pro

bab

ilit

y(L/E) = 0.1(L/E) = 0.25(L/E) = 0.5

P(ee ) = 1 - sin22 sin2 (1.27m2(eV2)L(km)/E(GeV))

2- Oscillations in Vacuum


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P( ) = 1 - P( )

Oscillation Measurements Disappearence experiment: P( )

Appearence experiment: P( )

<P> = P( )


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Difference of potentials is important for e :

e

e

e

e

W

Ve- V = 2 GFne

Elastic forward scattering

PotentialsVe, V

L. Wolfenstein, 1978

Refraction index:

n - 1 = V / p

~ 10-20 inside the Earth< 10-18 inside the Sun~ 10-6 inside the neutron star

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

n - 1

Refraction length:

l0 = 2 / (Ve - V)

= 2 /GFne



V = Ve - V

is determined with respect to eigenstates in matter

is the mixing angle in matter

1m2m

H = H0 + VEffectiveHamiltonean

Eigenstatesdepend on ne, EEigenvalues

H0

12

m12/2E , m2

2/2Em1, m2

H1m, H2m

m1m, m2m

e

2m

1m

2

1

m m



2m1m

Flavors of the eigenstates do not change

Admixtures of matter eigenstates do not change: no 1m2mtransitions

m= m=H2 - H1) L

Monotonous increase of the phase differencebetween the eigenstates m

Parameters of oscillations (depth and length) are determined by mixing in matter and by effective energy split in matter

In uniform matter (constant density)mixing is constant

m(E, n) = constant

as in vacuum

e

sin22, l sin22m, lm



sin2 2m = 1

Mixing in matter is maximalLevel split is minimal

In resonance:

l = l0 cos 2

Vacuumoscillation length

Refractionlength

~~

For large mixing: cos 2the equality is brokenthe case of strongly coupled system shift of frequencies

l / l0

sin2 2m

sin2 2 = 0.08

sin2 2 = 0.825

~ n E

Resonance width: nR = 2nR tan2Resonance layer: n = nR + nR



Resonance enhancement Resonance enhancement of neutrino oscillationsof neutrino oscillations

Adiabatic Adiabatic (partially adiabatic)(partially adiabatic)neutrino conversionneutrino conversion

Constant density Variable density

Change of mixing, or flavor of the neutrinoeigenstates

Change of the phase difference between neutrino eigenstates

Degrees of freedom:

Interplay of oscillations Interplay of oscillations and adiabatic conversionand adiabatic conversion

Density profiles:

In general:



Layer of matter with constant density, length LSource Detector

F0(E) F(E)

F (E)F0(E)

E/ER E/ER

thin layer thick layerk = L/ l0

sin2 2 = 0.824

k = 1 k = 10

e e

sin2 2 = 0.824



Layer of matter with constant density, length LSource Detector

F0(E) F(E)

F (E)F0(E)

E/ER E/ER

thin layer thick layer

sin2 2 = 0.08

k = 1 k = 10

e e

sin2 2 = 0.08



Continuity: neutrino and antineutrino semiplanes normal and inverted hierarchy

Oscillations (amplitude of oscillations) are enhanced in the resonance layer

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

resonancelayer

ER0

= m2 / 2V

ER = ERtan 2 = ER0sin 2

With increase of mixing:

ER 0

ER ER0

l / l0

l / l0

P

P



Admixtures of the eigenstates,1m 2m, do not change

Flavors of eigenstates change according to the density change

fixed by mixing inthe production point

determined by m

1m 2m are no more the eigenstates of propagation 1m2m transitions

Effect is related to the change of flavors of the neutrino eigenstates in matter with varying density

if the density changes slowly enough (adiabaticity condition) 1m <-> 2m transitions can be neglected

Non-uniform matter density changes on the way of neutrinos:ne = ne(t)

mm(ne(t)) mixing changes in the course of propagation

H = H(t) depends on time

However

Phase difference increases according to the level split which changes with density



dm

dt Adiabaticity conditionH2 - H1

Crucial in the resonance layer: - the mixing angle changes fast - level splitting is minimal

rR > lR

lR = l/sin2is the oscillation width in resonance rR = nR / (dn/dx)R tan2is the width of the resonance layer

External conditions (density)change slowlyso the system has time to adjust itself

<< 1

transitions between the neutrino eigenstates can be neglected

1m2mThe eigenstatespropagate independently

if vacuum mixing is small

If vacuum mixing is large the point of maximal adiabaticity violation is shifted to larger densities

n(a.v.) nR0 > nR

nR0 = m 2/ 2 2 GF E



The picture of conversion depends on how far from the resonance layer in the density scale the neutrino is produced

n0 > nR

n0 - nR >> nR

n0 ~ nR nR - n0 >> nR

n0 < nR

Non-oscillatory conversion

Oscillations with small matter effect

Interplay of conversion and oscillations

All three possibilities are realized for the solar neutrinos in different energy ranges

nR ~ 1/ E



2m

1m

2m

1m

ne

2

1

2m

1m

Non-oscillatory transition

Adiabatic conversion + oscillationsn0 > nR

n0 >> nR

n0 < nR

2

1

2

1

Small matter corrections

Resonance

P = sin2

interference suppressedMixing suppressed

1m 2m



2m

1m

2m 1m

ne

2

1

Admixture of 1m increases

n0 >> nR

Resonance

Fast density change



y = (nR - n) / nR

surv

ival

pro

babi

lity

resonance

productionpointy0 = - 5

averagedprobability

oscillationband

(distance)

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

(no explicit dependence on oscillation parameters density distribution, etc.)Only initial value y0 matters.

resonance layer

For zero final density:y = 1/tan 2

LMA



Oscillations in matterof the Earth

Oscillationsin vacuum

Adiabatic conversionin matter of the Sun

: (150 0) g/cc



tan2m2 = 7.3 10-5 eV2

y

surv

ival

pro

babi

lity

surv

ival

pro

babi

lity

surv

ival

pro

babi

lity

surv

ival

pro

babi

lity

coreE = 14 MeV

E = 6 MeV E = 2 MeV

distancey

y y

resonance E = 10 MeV

surface

sin2



tan2m2 = 7.3 10-5 eV2

y

surv

ival

pro

babi

lity

surv

ival

pro

babi

lity

E = 0.4 MeVE = 0.86 MeV

distance distance

y

Low energy part of the spectrum: vacuum oscillations with small matter corrections

Dashed lines: for pure vacuum oscillations



p, He, …

N

N

A

(K

e

e

R = = 2

e

e

Atmospheric Neutrinos: Flux FeaturesAtmospheric Neutrinos: Flux Features

E3 d

F/d

E

E(GeV)

E(GeV)



1

102

103

104

Cos

Path

len

gth

(k

m)

-1 -.6 .6-.2 .2

up horizontal down

Atmospheric Neutrinos: Flux FeaturesAtmospheric Neutrinos: Flux Features

L =

20km

L =

10 0

00km

L = 500km

cos = 0

cos

= -0

.8

cos

= 0

.8

Flux is up/down symmetricbut isn’t isotropic Fn~ 1/CosQ

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

to study neutrino oscillations.



K.Nishikawa’s talkAtmospheric Neutrino DetectionAtmospheric Neutrino Detection



Atmospheric Neutrino DetectionAtmospheric Neutrino DetectionSuper-Kamiokande



Atmospheric Neutrino EventsAtmospheric Neutrino Events

e

e

Partially contained

Fullycontained

1/10t/yearAcceptance 4

100 101 102 103

0.2

0.4

0.6

0.8

Neutrino energy (GeV)

Even

ts/

day

Ed Kearns for the SK Collaboration Neutrino2004,

June 15, Paris



Atmospheric Neutrino EventsAtmospheric Neutrino Events

Upward through-going

Upward stopping

1/10m2/yearAcceptance 2

100 101 102 103

0.02

0.06

0.10

0.14

Neutrino energy (GeV)

Even

ts/

day

0.18


June 15, Paris




June 15, Paris

Zenith Angle Distributions

Sub-GeV Evis < 1.3 GeV

Multi-GeV Evis > 1.3 GeV

Atmospheric NeutrinosAtmospheric Neutrinos



Atmospheric NeutrinosAtmospheric Neutrinos oscillations


June 15, Paris



Atmospheric NeutrinosAtmospheric Neutrinos

oscillations L/E analysis Ed Kearns for the SK

Collaboration Neutrino2004, June 15, Paris

Oscillation deep @ L/E 500 km/GeV



K2K experimentK2K experimentK2K experimentK2K experiment• K2K is the first accelerator-based long-baseline neutrino

oscillation experiment to investigate the neutrino oscillation observed in atmospheric neutrinos.

KKEK-EK-12GeV 12GeV

PSPS

Super-Super-KKamiokanamiokan

dedeAtm.- K2K

L 10~104km 250km(fix.)

E 0.1~100GeV ~ 1.3GeV

m2 10-1 ~10-4eV2 > 2 ・ 10-3 eV2

e 0.50 ~0.01

L=250 kmL=250 km



p+Al + e+ + e (1.3%) + (0.5%)

Neutrino Beam Production

K2K experimentK2K experimentK2K experimentK2K experiment

Near Detector Far Detector

- disappearance



K2K experimentK2K experimentK2K experimentK2K experimentShape of energy spectrum

CC – quasielastic reactions

Erec (GeV)



K2K experimentK2K experimentK2K experimentK2K experimentBoth disappearance and energy spectrum distortion have the consistent result



Solar NeutrinosSolar NeutrinosG.Fogli et al. hep-ph/0106247



Solar NeutrinosSolar Neutrinos

All solar neutrino data



Experiment KamLANDExperiment KamLANDExperiment KamLANDExperiment KamLAND



3 mixing angles (12,23,13)& phase of CP violation ()

сij = cosij

sij = sinij

1 0 0 0 с23 s23

0 -s23 c23

1 0 0 0 с23 s23

0 -s23 c23

с13 0 s13 ei

0 1 0 -s13 e-i 0 c13

с13 0 s13 ei

0 1 0 -s13 e-i 0 c13

с12 s12 0 -s12 c12 0 0 0 1

с12 s12 0 -s12 c12 0 0 0 1

U = U =

Atmospheric neutrinosm2 (1.310-3 3.010-3) eV2

Sin22 > 0.923

m32

m21

12

Solar neutrinosm2 (5.410-5 9.510-5) eV2

Sin22 (0.71 0.95)

3 – neutrino mixing



( , , sin2212, sin2223)( , , sin2212, sin2223)m2

1 m2

1

22 m3

1 m3

1

22

… and limit for sin2213… and limit for sin2213

Known parameters



sin2213 0.2sin2213 0.2

Sign of m31 Sign of m31 22

- phase of CP violation - phase of CP violation

Mass hiererchy Mass hiererchy

1

2

3

normal

3

1

2

inverted

Unknown parameters

NEUTRINO OSCILLATIONS AND THE MSW EFFECT

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