``Mesonium and antimesonium’’

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. PontecorvoRight 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 itselfIncoherent 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

MeVEcm10~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 2A2 + 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:

E2mv

2

ph

tvph

2sin)AA(A 22

21P

2mE4L



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

Lxsin2sin

Lx2cos1

2AP 22P

e



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,0,2sinL2B

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 energy

e)gg(e)1(2

GH 5AVe5eF

int

Unpolarized and isotropic medium: eFnG2V

2e

eeee

2eeee2

e

eF

v11vvv1v1v1

v1nG2V

- 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 starpV1n

eF0 nG

2V2L



ftotf H

dtdi

ef

VHH vactot total Hamiltonian

2

2

21

vac m00m

E21H

vacuum part

000nG2V eF

matter part

e2

2

eF

2

e

02sinE4m

2sinE4mnG22cos

E2m

dtdi



vacuum vs. matter

e

1

2

1m 2m

m

Effective Hamiltonian Hvac Hvac + V

Eigenstates 1, 2 1m, 2m

Eigenvalue H1m, H2mm12/2E, m2

2/2EDepend on ne, E

Mixing angle determines flavors

of eigenstatea(f)

(f)

(i)

(im)

m



Diagonalization of the Hamiltonian:

2sinm

EnG222cos

2sin2sin2

2

2eF

2

m2

2sinm

EnG222cosE2

mHH 2

2

2eF

2

12

2cosE2

mnG22

eF

Mixing

Difference of the eigenvalues

At resonance: Resonance condition

12sin m2

2sin

E2mHH

2

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

LL2sin

LL

R00

Resonance energy:

Resonance density:

eF

2

R nG222cosmE

EG222cosmn

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



21

2

2

2eF

212

m 2sinm

EnG222cosm

E4HH

2L

Oscillation length in matter:

vacu

umdo

min

ated

matterdominated

E

Lm

2sinLLm

eF0 nG2

2L

eF

2

R nG222cosmE



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

21

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

78

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

dtdi

ef

m2

m1

12m

m

m2

m1

HHdt

didt

di0

dtdimmf )(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: 1

HHdt

d

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

2e 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

nnny

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 edge Non - 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 change m1m2

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



distancesu

rviv

al p

roba

bilit

y

Oscillations

Adiabatic conversion

Spatial picture

surv

ival

pro

babi

lity

distance



2e

eeee

2eeee2

e

eF

v11vvv1v1v1

v1nG2V

anG2V eF

Unpolarized relativistic medium:

2e

eeF

V

v1v1nG2V

e

e

v1v1a

e

e

v1v1a

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 smallOscillation 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

iiTE

oi

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

MeVeVm

cmg

2cos221 2

e

N

Fres Y

mEm

G 2tgres

343 )1010(~cmg

H 3)3010(~cmg

L

Neutrino transitions occur far

outside of the star core

39

34

e rcm10

cmg102Y

Supernova Density Profile



Supernova Density Profile

)(11

2cos2sin

2

22

drdn

nEm

e

e

nrA

n

N

eF

n

n Am

YGEm

n

1

11

2112

)22()2(cos

2sin)(21

1

Adiabaticity parameter:

Adiabatic conversion

Weak dependence on A Weak dependence on nnA1

n1


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

0x0e0e F)p1(FpF

0x0e0e F)p1(FpF

0e0e0xx41 F4

p1F4p1F4

pp2F

Documents

``Mesonium and antimesonium’’