S.P.Mikheyev (INR RAS). 25.09.2008S.P.Mikheyev (INR RAS)2 Introduction. Vacuum oscillations. ...

S.P.Mikheyev (INR RAS)

25.09.2008 S.P.Mikheyev (INR RAS)

2

Introduction.

Vacuum oscillations.

Oscillations in matter. Adiabatic conversion.

Graphical representation of oscillations

Conclusion


3

Neutrino are massive. Neutrino masses are in the sub-eV range - much smaller than masses of charge leptons and quarks.

A. Yu. Smirnov hep-ph/0702061

There are only 3 types of light neutrinos: 3 flavors and 3 mass states.

Their interactions are described by the Standard electroweak theory

Masses and mixing are generated in vacuum

e 1

2

3

|fUfi|ii

mixing Neutrinos mix. There are two

large mixingsand one small or zero mixing. Pattern of leptonmixing strongly differs from that of quarks.


4

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 =

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

Mixing matrix U can be parameterized with

Pontecorvo – Maki – Nakagava -Sakata


5

2 U = cossin-sincos

( )

e 1

2

1

2

wavepackets

e = cos1sin = - sin1cos

coherent mixturesof mass eigenstates

1 = cosesin

2 = sinecos

flavor composition of the mass eigenstates

1

2e 1

2

Neutrino “images”:

1

2


6

0 2

A2 + A1 0 2sincos

0

cossinA1

cossinA2e 1

2

Due to difference of masses 1

and 2 have different phase velocities

E2m

v2

ph

tvph

Oscillation depth: 2sin)AA(A 22

21P Oscillation length: 2m

E4L

Oscillation probability:

Lx

sin2sinL

x2cos1

2A

P 22Pe

ee

P1P e


7

I. Oscillations effect of the phase difference increase between mass

eigenstates

II. Admixtures of the mass eigenstates i in a given neutrino state do not change during

propagation III. Flavors (flavor composition) of the

eigenstates are fixed by the vacuum mixing angle

Periodic (in time and distance) process of transformation (partial or complete) of one neutrino species into another one


8

M is the mass matrix

Schroedinger’s equation

Mixing matrix in vacuum

HΨdtdΨ

i

Ψ

Ψ

Ψ

Ψ μ

e

2EMM

H

),m,mdiag(mM

UUMMM23

22

21

2diag

2diag

)tjEii(E

βjUαjUβiUji,

αiU)βναP(ν e


9

E(GeV)L(km))(eVΔm1.27

sin2θsin1)νP(ν22

22αα

E(GeV)L(km))(eVΔm1.27

sin2θsin1)νP(ν22

22αα

Disappearance experiments:

Appearence experiment:

E(GeV)

L(km))(eVΔm1.27sin2θsin)νP(ν

2222

α

E(GeV)

L(km))(eVΔm1.27sin2θsin)νP(ν

2222

α

Atmospheric neutrinos; LBL: K2K, MINOS; reactor neutrinos: KamLAND

LBL: MINOS, OPERA, T2K

Probability as a function of distance (atmospheric neutrinos) energy (K2K, MINOS) L/E (atmospheric neutrinos, KamLAND)


10

Jennifer Raaf Talk at Neutrino’2008


11

K2K

Erec (GeV)

MINOS

Hugh GallagherTalk at Neutrino’2008


12

Patrick DecowskiTalk at Neutrino’2008

KamLAND


13

Neutrino interactions with matter affect neutrino properties as well as

medium itself

Incoherent interactions Coherent interactions CC & NC inelastic scattering CC quasielastic scattering NC elastic scattering with energy loss

CC & NC elastic forward scattering

Neutrino absorption (CC) Neutrino energy loss (NC) Neutrino regeneration (CC)

Potentials

2243

2F

MeVE

cm10~sG

~


14

Elastic forward scattering +e e,

e-

W+ Z0

e-

e- e-e

e,

V = Ve - V Potential:

At low energy elastic forward scattering (real part of amplitude) dominates.

Effect of elastic forward scattering is describer by potential

Only difference of e and is important

Unpolarized and isotropic medium: eFnG2V


15

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

Refraction index:

~ 10-20 inside the Earth

< 10-18 inside in the Sun

~ 10-6 inside neutron starpV

1n

Refraction length:eF

0 nG2

V2

L


16

HΨdtdΨ

i V2E

MMH

Diagonalization of the Hamiltonian:

2sinm

EnG222cos

2sin2sin

2

2

2eF

2

m2

Mixing

02

eF

L

L

m

EnG222cos

Resonance condition

2sinm

EnG222cos

E2m

HH 2

2

2eF

2

12 Difference of the eigenvalues

At resonance: 12sin m2

2sin

E2m

HH2

12


17

sin2 2m = 1 At 2cosLL

0

Resonance half width:

2tan

LL

2sinLL

R00

Resonance energy: eF

2

R nG22

2cosmE

2tgEE RR

Resonance density:

EG22

2cosmn

F

2

R

2tgnn RR

Resonance layer:

RRe nnn

sin2 2m

sin2 2 = 0.08

sin2 2 = 0.825

En~LL

e0


18

Pictures of neutrino oscillations in media with constant density and vacuum are

identicalIn uniform matter (constant density) mixing is constant

m(E, n) = constant

As in vacuum oscillations are due to change of the phase difference between neutrino eigenstates

(Constant density)

~E/ER

F (E)F0(E)

vacuum

~E/ER

F (E)F0(E)

matter


19

(Non-uniform

density)ftot

f Hdt

di

e

f

m2

m1

12m

m

m2

m1

HHdt

di

dtd

i0

dtd

immf )(U

In matter with varying density the Hamiltonian depends on time: Htot = Htot(ne(t))Its eigenstates, m, do not split the equations of motion

m2

m1m

θm= θm(ne(t))

The Hamiltonian is non-diagonal no split of equations

Transitions 1m 2m


20

Pictures of neutrino oscillations in media with constant density and variable density

are differentIn uniform matter (constant density) mixing is constant

m(E, n) = constant

As in vacuum oscillations are due to change of the phase difference between neutrino eigenstates

In varying density matter mixing is function of distance

(time)

m(E, n) = F(x)

Transformation of one neutrino type to another is due to change of mixing or flavor of the neutrino eigenstates

MSWeffect

Varying density vs. constant density


21

One can neglect of 1m 2m

transitions if the density changes slowly

enough

Adiabaticity condition: 1HH

dtd

12

m

External conditions (density)change slowly so the system has time to adjust itself Transitions between

the neutrino eigenstates can be neglected

The eigenstatespropagate

independentlym2m1

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

LR = L/sin2 is the oscillation length

in resonance

is the width of the resonance

layer

RR Lr

R

RR

dxdn

2tgnr


22

Initial state: )0(sin)0(cos)0( m20mm1

0me

Adiabatic conversion to zero density:

1m(0) 1

2m(0) 2

Final state: 20m1

0m sincos)f(

Probability to find e averaged over oscillations:

0m

2220m

20m

2

e cos2cossinsinsincoscos)f(|P


23

R

R

nnn

y

Dependence on initial condition

The picture of adiabatic conversion is universal in

terms of variable:

There is no explicit dependence on oscillation parameters, density distribution, etc.

Only initial value of y0 is important.

surv

ival

pro

babi

lity

y (distance)

resonance layer

productionpoint y0 = - 5

resonance averagedprobability

oscillationband

y0 < -1 Non-oscillatory conversion

y0 = -11

y0 > 1

Interplay of conversion and oscillationsOscillations with small matter effect


24

sin22 = 0.8

0.2 2 20 200 E (MeV)

(m2 = 810-5 eV2)

Vacuum oscillationsP = 1 – 0.5sin22

Adiabatic conversionP =|<e|2>|2 = sin2

Adiabatic edgeNon -

adiabatic conversion

Survive probability (averged over oscillations)

(0) = e = 2m 2


25

Both require mixing, conversion is usually accompanying by oscillations

Oscillation Adiabatic conversion Vacuum or uniform

medium with constant parameters

Phase difference increase between the eigenstates

Non-uniform medium or/and medium with varying in time parameters

Change of mixing in medium = change of flavor of the eigenstates

In non-uniform medium: interplay of

both processes

θm


26

distance

su

rviv

al p

rob

ab

ilit

y

Oscillations

Adiabatic conversion

Spatial picture

su

rviv

al p

rob

ab

ilit

y

distance


27

J.N. Bahcall

4p + 2e- 4He + 2e + 26.73 MeV

electron neutrinos are producedAdiabatic conversionin matter of the Sun

: (150 0) g/cc

e

Adiabaticity parameter ~ 104


28

SNO

Hamish RobertsonTalk at Neutrino’2008


29

Cristano GalbiatiTalk at Neutrino’2008

Cl-Ar data


30

Solar neutrinos vs. KamLANDAdiabatic conversion (MSW)

Vacuum oscillations

Matter effect dominates (at least in the HE part)

Non-oscillatory transition, or averaging of oscillationsthe oscillation phase is irrelevant

Matter effect is very small

Oscillation phase is crucialfor observed effect

Coincidence of these parameters determined from the solar neutrino data and from KamLAND results testifies for the correctness of the theory (phase of oscillations, matter

potential, etc..)

;m2Adiabatic conversion formula Vacuum oscillations formula


31

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)

Known parameters


32

sin2213 0.2sin2213 0.2 - CP phase - CP phase

Mass hierarchy Mass hierarchy

Unknown parameters

O. Mena and S. Parke, hep-ph/0312131

G.L. Fogli, E. Lisi, A. Marrone, A. Palazzo, A.M. Rotunno arXiv:0806.2649

Sin213 = 0.0160.010


33

Coincides with equation for the electron spin precession in the magnetic field

Polarization vector:

Evolution equation:

d d t

Differentiating P and using equation of motion

x

e

2P

21

xe

xeIm

xeRe

P

( - Pauli matrices)

H

dtd

i

)(dtd

i B )2cos,0,2(sinl2

mmm

B

)(dt

d PBP


34

x

y

z

2B

(P-1/2)

(Re e+x)

(Im e+x)

P

Lt2


35

Non-uniform density: Adiabatic conversion


36

Non-uniform density: Adiabaticity violation


37

Collective effects related to neutrino self-interactions ( -

scattering)ee

e

e

b b b

b

Z0Z0

bb

e

e

t-channel (p)

(q)

elastic forward scattering

e

e

b

b

u-channel (p)

(q)

Collective flavor transformations

J. Pantaleonecan lead to the coherent effect

Momentum exchange flavor exchange flavor mixing


38

“Standard neutrino scenario” gives complete description of neutrino oscillation phenomena.

But it tells us nothing what physics is behind of neutrino masses and mixing.

New experiments will allow us to measure the 1-3 mixing, deviation of 2-3 mixing from maximal, and CP-phases, as well as hopefully to establish type of neutrino hierarchy, nature of neutrino and neutrino mass.


39

“Standard neutrino scenario” gives complete description of neutrino oscillation phenomena.

But it tells us nothing what physics is behind of neutrino masses and mixing.

New experiments will allow us to measure the 1-3 mixing, deviation of 2-3 mixing from maximal, and CP-phases, as well as hopefully to establish type of neutrino hierarchy, nature of neutrino and neutrino mass.

However neutrinos gave us many puzzles in past and one can expect more in future!!!

S.P.Mikheyev (INR RAS). 25.09.2008S.P.Mikheyev (INR RAS)2 Introduction. Vacuum oscillations. ...

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