20.09.2007 S.P.Mikheyev INR RAS1 ``Mesonium and antimesonium’’ Zh. Eksp.Teor. Fiz. 33, 549...

20.09.2007S.P.Mikheyev INR RAS 1

``Mesonium and antimesonium’’Zh. Eksp.Teor. Fiz. 33, 549 (1957)[Sov. Phys. JETP 6, 429 (1957)] translation

B. Pontecorvo

Right time

50 years!

First paper where a possibility

of neutrino mixing and oscillations was mentioned

Right place

III International Pontecorvo Neutrino Physics School

S.P. MikheyevINR RAS



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

~



A. Yu. Smirnov hep-ph/0702061

There are only three types of light neutrinos

Their interactions are described by the Standard electroweak theory

Masses and mixing are generated in vacuum



How neutrino looks (neutrino “image”)

How neutrino oscillations look (graphic representation)



certain neutri

no flavors

e

e

correspond tocertain

charged

leptons 1

2

3

(interact in pairs)

Mass eigenstatesEigenstates of the

CC weak interactions

m1

m2

m3

|fUfi|ii

mixing



2 U = cossin-sincos

( )e = cos1sin

= - sin1cos 1 = cosesin

2 = sinecos

e 1

2

1

2

wavepackets 1

2

coherent mixturesof mass eigenstates

flavor composition of the mass eigenstates

1

2e 1

2

Neutrino “images”:


0 2

A2 + A1 0 2sincos

0

cossinA1

cossinA2


e 1

2

Due to difference of masses 1

and 2 have different phase velocities

Oscillation depth:

Oscillation length:

E2m

v2

ph

tvph

2sin)AA(A 22

21P

2mE4

L



Oscillation probability:

I. Oscillations effect of the phase difference increase between mass

eigenstatesII. 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

Lx

sin2sinL

x2cos1

2A

P 22Pe



x

y

z

2B

(P-1/2)

(Re e+)

(Im e+)

Evolution equation:

P(e e) = e+e = ½(1 + cosZ)

Analogy to equation for the electron spin

precession in magnetic field

Bdtd

21

,Im,Re eeee

2cos,0,2sinL2

B

Lt2



x

y

z



Matter potential

Evolution equation in matter

Resonance

Adiabatic conversion

Adiabaticity violation

Survival probability

Parametric enhancement of oscillations



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) dominate.

Effect of elastic forward scattering is describer by potential

Only difference of e and is important



|H|V int

- the wave function of the system neutrino - mediumHint – Hamiltonian of the weak interaction at low

energye)gg(e)1(

2

GH 5AVe5e

Fint

Unpolarized and isotropic medium: eFnG2V

2e

eeee

2eeee2

e

eF

v11

vvv1v1v1

v1

nG2V

- neutrino velocity e

- vector of polarization

(CC interaction with electrons)(gV = -gA = 1)



Refraction index:

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

Refraction length:

~ 10-20 inside the Earth

< 10-18 inside in the Sun

~ 10-6 inside neutron starpV

1n

eF0 nG

2V2

L



ftotf H

dtd

i

e

f

VHH vactot total Hamiltonian

22

21

vacm0

0m

E21

H

vacuum part

00

0nG2V eF

matter part

e

2

2

eF

2

e

02sinE4m

2sinE4m

nG22cosE2

m

dtd

i



vacuum vs. matter

e

1

2

1m 2m

m

Effective Hamiltonian Hvac Hvac + V

Eigenstates 1, 2 1m, 2m

Eigenvalue H1m, H2mm12/2E, m2

2/2E

Depend on ne, E

Mixing angle determines flavors

of eigenstatea(f)

(f)

(i)

(im)

m



Diagonalization of the Hamiltonian:

2sinm

EnG222cos

2sin2sin

2

2

2eF

2

m2

2sinm

EnG222cos

E2m

HH 2

2

2eF

2

12

2cosE2

mnG2

2

eF

Mixing

Difference of the eigenvalues

At resonance: Resonance condition

12sin m2

2sin

E2m

HH2

12

HHe

mixing is maximal difference of the eigenvalues is minimal

level crossing



sin2 2m = 1 Atsin2 2m

sin2 2 = 0.08

sin2 2 = 0.825

En~LL

e0

2cosLL

0

Resonance half width:

2tan

LL

2sinLL

R00

Resonance energy:

Resonance density:

eF

2

R nG22

2cosmE

EG22

2cosmn

F

2

R

2tgEE RR

2tgnn RR

Resonance layer:

RRe nnn



V. Rubakov, private comm. N. Cabibbo, Savonlinna 1985H. Bethe, PRL 57 (1986) 1271

Dependence of the neutrino eigenvalues on the matter potential (density)

H

2m

1m

e

sin2 2 = 0.08(small mixing)

2m

1m

e

sin2 2 = 0.825(large mixing)

Crossing point - resonance the level split is minimal the oscillation length is maximal

For maximal mixing: nR = 0

En~LL

e0

En~LL

e0

Level crossing:

H

20 m

EV2LL

2cosLL

0



2

1

2

2

2eF

212

m 2sinm

EnG222cos

mE4

HH2

L

Oscillation length in matter:

vacu

umdo

min

ated

matterdominated

E

Lm

2sinL

Lm

eF0 nG2

2L

eF

2

R nG22

2cosmE



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

are different

In 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



m= m=H2 - H1) L

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

and by effective energy split in matter

sin22, L

sin22m, Lm

Flavors of the eigenstates do not change

Constant density

Admixtures of matter eigenstates do not change: no 1m 2m transitions

Monotonous increase of the phase difference between eigenstates Δm

Oscillations as in vacuum

e 1

2

1

2

instead of


sin2 2 = 0.08

e

F(E)Detector


Layer of matter with constant density, length L

e

F0(E)

Source

~E/ER

F (E)F0(E)

thin layer L = L0/

~E/ER

thick layer L = 10L0/

Constant density: Resonance enhancement of oscillations

sin2 2 = 0.824


e

F(E)Detector


e

F0(E)

Source

Instantaneous density change

m = 1 m = 2

n1 n2

x

y

z


x

z

y

e

F(E)Detector


e

F0(E)

Source

Instantaneous density change

m = 1 m = 2

n1 n2



Instantaneous density change: parametric resonancem = 1

m = 2

n1 n2

1 1 1

2 2 21 2 3 4 5 6 7 8

.

..

.

.

..

.12

3

4

5

6

7

8

B1B2

Enhancement associated to certain conditions for the phase of oscillations.

Another way to get strong transition.

No large vacuum mixing and no matter enhancement of mixing or resonance conversion

1 = 2 =



Instantaneous density change: parametric resonance

m = 1m n1 n2 m = 2m 1 2

Resonance condition:

02cos2

cos2

sin2cos2

cos2

sin m212

m121

Simplest realization:

In general, certain correlation between phases

and mixing angles

1 = 2 =



ftotf H

dtd

i

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

Non-uniform density

θm= θm(ne(t))

The Hamiltonian is non-diagonal no split of equations

Transitions 1m 2m



Non-uniform density: AdiabaticityOne can neglect of 1m 2m

transitions if the density changes slowly

enough

Adiabaticity condition:1

HHdt

d

12

m

drdn

n1

12cos2sin

E2m

e

e

22

Adiabaticity parameter:

1



Non-uniform density: Adiabaticity

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

External conditions (density)

change slowly so the system has time to adjust itself

Transitions between the neutrino eigenstates can be neglected

The eigenstatespropagate

independently

Adiabaticity condition:

1HH

dtd

12

m

m2m1

RR Lr

R

RR

dxdn

2tgnr



Non-uniform density: Adiabatic conversionInitial state: )0(sin)0(cos)0( m2

0mm1

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



Resonance

Non-uniform density: Adiabatic conversion

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

Flavors of eigenstates change according to the density change

fixed by mixing in the production point

determined by m

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

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



Non-uniform density: Adiabatic conversion



Non-uniform density: Adiabatic conversionDependence on initial condition

The picture of adiabatic conversion is universal in

terms of variable:R

R

nnn

y

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


13th Lomonosov Conference on Elementary Particle Physics

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

Non-uniform density: Adiabatic conversionSurvive probability (averged over oscillations)

(0) = e = 2m 2



Non-uniform density: Adiabaticity violation

2m

1m

ne

2

1

n0 >> nR

Resonance

Fast density changem1m2

Transitions 1m 2m occur, admixtures of the eigenstates change

Flavors of the eigenstates follow the density change

Phase difference of the eigenstates changes, leading to oscillations

= (H1-H2) t



Non-uniform density: Adiabaticity violation



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



distance

su

rviv

al p

rob

ab

ilit

y

Oscillations


Spatial picture

su

rviv

al p

rob

ab

ilit

y

distance



2e

eeee

2eeee2

e

eF

v11

vvv1v1v1

v1

nG2V

anG2V eF

Unpolarized relativistic medium:

2e

eeF

V

v1

v1nG2V

e

e

v1v1

a

e

e

v1v1

a

e

e

polarized isotropic medium:

eeF 1nG2V 1~e

0V if



The Sun

The Earth

Supernovae



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

electron neutrinos are produced

J.N. Bahcall

Oscillations in matterof the Earth

Oscillationsin vacuum

Adiabatic conversionin matter of the Sun

: (150 0) g/cc

e

Adiabaticity parameter ~ 104



Borexino Collaboration arXiv:0708.2251



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



Density Profile (PREM model)

mantle mantle

core



Akhmedov, Maltoni & Smirnov, 2005Liu, Smirnov, 1998; Petcov, 1998; E.Akhmedov 1998



Supernova Neutrino Fluxes

MeVTe 65

1

)()(

2

iiT

Eoi

e

EEF

MeVTx 97

MeVTe 43

5.20.2 e

20x

53e

G.G. Rafelt, “Star as laboratories for fundamental physics” (1996)

H.-T. Janka & W. Hillebrand, Astron. Astrophys. 224 (1989) 49



Matter effect in Supernova Normal Hierarchy Inverted Hierarchy

Dighe & Smirnov, astro-ph/9907423



2cos)5.0

)(10

)(1

(104.1~2

2

36

eres YE

MeV

eV

m

cm

g

2cos22

1 2

e

N

Fres Y

mEm

G 2tgres

343 )1010(~cm

gH

3)3010(~cm

gL

Neutrino transitions occur far

outside of the star core

39

34

e rcm10

cmg

102Y

Supernova Density Profile



Supernova Density Profile

)(1

1

2cos

2sin

2

22

drdn

nE

m

e

e

nr

A n

N

eF

n

n Am

YG

E

m

n

1

11

2112

)22

(

)2(cos

2sin)(

2

1

1

Adiabaticity parameter:


Weak dependence on A Weak dependence on nnA1

n

1


Pf = 0.9

Pf = 0.1

E = 50 MeV

E = 5 MeV


Supernova Neutrino Oscillations

I – Adiabatic conversion

II – Weak violation of adiabaticity

III – Strong violation of adiabaticity



Original fluxes

After leaving thesupernova envelope

for for

for for

for for

0eF0eF0xF

e

e

,,,

Normal

Inverted

sin2(213)

≲ 10-5

≳ 10-3

Any

Hierarchy

sin2(Q12) 0.3

0 cos2(Q12) 0.7

sin2(Q12) 0.3 cos2(Q12) 0.7

0

)for(p e )for(p e )for(p e )for(p e

0x

0e

0e F)p1(FpF 0

x0e

0e F)p1(FpF

0x

0e

0e F)p1(FpF 0

x0e

0e F)p1(FpF

0e

0e

0xx4

1 F4p1

F4p1

F4

pp2F

0

e0e

0xx4

1 F4p1

F4p1

F4

pp2F

20.09.2007 S.P.Mikheyev INR RAS1 ``Mesonium and antimesonium’’ Zh. Eksp.Teor. Fiz. 33, 549...

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Transcript of 20.09.2007 S.P.Mikheyev INR RAS1 ``Mesonium and antimesonium’’ Zh. Eksp.Teor. Fiz. 33, 549...