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
20.09.2007S.P.Mikheyev INR RAS 3
III International Pontecorvo Neutrino Physics School
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
~
20.09.2007S.P.Mikheyev INR RAS 4
III International Pontecorvo Neutrino Physics School
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
20.09.2007S.P.Mikheyev INR RAS 5
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How neutrino looks (neutrino “image”)
How neutrino oscillations look (graphic representation)
20.09.2007S.P.Mikheyev INR RAS 6
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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
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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”:
20.09.2007S.P.Mikheyev INR RAS 8
0 2
A2 + A1 0 2sincos
0
cossinA1
cossinA2
III International Pontecorvo Neutrino Physics School
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
20.09.2007S.P.Mikheyev INR RAS 9
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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
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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
20.09.2007S.P.Mikheyev INR RAS 11
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x
y
z
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Matter potential
Evolution equation in matter
Resonance
Adiabatic conversion
Adiabaticity violation
Survival probability
Parametric enhancement of oscillations
20.09.2007S.P.Mikheyev INR RAS 13
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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
20.09.2007S.P.Mikheyev INR RAS 14
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|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)
20.09.2007S.P.Mikheyev INR RAS 15
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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
20.09.2007S.P.Mikheyev INR RAS 16
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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
20.09.2007S.P.Mikheyev INR RAS 17
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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
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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
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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
20.09.2007S.P.Mikheyev INR RAS 20
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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
20.09.2007S.P.Mikheyev INR RAS 21
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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
20.09.2007S.P.Mikheyev INR RAS 22
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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
20.09.2007S.P.Mikheyev INR RAS 23
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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
20.09.2007S.P.Mikheyev INR RAS 24
sin2 2 = 0.08
e
F(E)Detector
III International Pontecorvo Neutrino Physics School
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
20.09.2007S.P.Mikheyev INR RAS 25
e
F(E)Detector
III International Pontecorvo Neutrino Physics School
e
F0(E)
Source
Instantaneous density change
m = 1 m = 2
n1 n2
x
y
z
20.09.2007S.P.Mikheyev INR RAS 26
x
z
y
e
F(E)Detector
III International Pontecorvo Neutrino Physics School
e
F0(E)
Source
Instantaneous density change
m = 1 m = 2
n1 n2
20.09.2007S.P.Mikheyev INR RAS 27
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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 =
20.09.2007S.P.Mikheyev INR RAS 28
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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 =
20.09.2007S.P.Mikheyev INR RAS 29
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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
20.09.2007S.P.Mikheyev INR RAS 30
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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
20.09.2007S.P.Mikheyev INR RAS 31
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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
20.09.2007S.P.Mikheyev INR RAS 32
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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
20.09.2007S.P.Mikheyev INR RAS 33
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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
20.09.2007S.P.Mikheyev INR RAS 34
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Non-uniform density: Adiabatic conversion
20.09.2007S.P.Mikheyev INR RAS 35
III International Pontecorvo Neutrino Physics School
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
20.09.2007S.P.Mikheyev INR RAS 36
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
20.09.2007S.P.Mikheyev INR RAS 37
III International Pontecorvo Neutrino Physics School
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
20.09.2007S.P.Mikheyev INR RAS 38
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Non-uniform density: Adiabaticity violation
20.09.2007S.P.Mikheyev INR RAS 39
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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
20.09.2007S.P.Mikheyev INR RAS 40
III International Pontecorvo Neutrino Physics School
distance
su
rviv
al p
rob
ab
ilit
y
Oscillations
Adiabatic conversion
Spatial picture
su
rviv
al p
rob
ab
ilit
y
distance
20.09.2007S.P.Mikheyev INR RAS 41
III International Pontecorvo Neutrino Physics School
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
20.09.2007S.P.Mikheyev INR RAS 42
III International Pontecorvo Neutrino Physics School
The Sun
The Earth
Supernovae
20.09.2007S.P.Mikheyev INR RAS 43
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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
20.09.2007S.P.Mikheyev INR RAS 44
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Borexino Collaboration arXiv:0708.2251
20.09.2007S.P.Mikheyev INR RAS 45
III International Pontecorvo Neutrino Physics School
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
20.09.2007S.P.Mikheyev INR RAS 46
III International Pontecorvo Neutrino Physics School
Density Profile (PREM model)
mantle mantle
core
20.09.2007S.P.Mikheyev INR RAS 47
III International Pontecorvo Neutrino Physics School
20.09.2007S.P.Mikheyev INR RAS 48
III International Pontecorvo Neutrino Physics School
Akhmedov, Maltoni & Smirnov, 2005Liu, Smirnov, 1998; Petcov, 1998; E.Akhmedov 1998
20.09.2007S.P.Mikheyev INR RAS 49
III International Pontecorvo Neutrino Physics School
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
20.09.2007S.P.Mikheyev INR RAS 50
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Matter effect in Supernova Normal Hierarchy Inverted Hierarchy
Dighe & Smirnov, astro-ph/9907423
20.09.2007S.P.Mikheyev INR RAS 51
III International Pontecorvo Neutrino Physics School
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
20.09.2007S.P.Mikheyev INR RAS 52
III International Pontecorvo Neutrino Physics School
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:
Adiabatic conversion
Weak dependence on A Weak dependence on nnA1
n
1
20.09.2007S.P.Mikheyev INR RAS 53
Pf = 0.9
Pf = 0.1
E = 50 MeV
E = 5 MeV
III International Pontecorvo Neutrino Physics School
Supernova Neutrino Oscillations
I – Adiabatic conversion
II – Weak violation of adiabaticity
III – Strong violation of adiabaticity
20.09.2007S.P.Mikheyev INR RAS 54
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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
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