Mesonium and antimesonium

1
Right time
B. Pontecorvo
Mesonium and antimesonium
Zh. Eksp.Teor. Fiz. 33, 549 (1957) Sov. Phys.
JETP 6, 429 (1957) translation
First paper where a possibility of neutrino
mixing and oscillations was mentioned
50 years!
Right place
2
Neutrinos in Matter
3
Neutrinos in Matter
Propagation of massive neutrino in matter
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

• Potentials

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

4
Neutrinos in Matter
"Standard model of neutrino"
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

5
Neutrinos in Matter
Introduction

• How neutrino looks (neutrino image)

• How neutrino oscillations look (graphic
  representation)

6
Neutrinos in Matter
Neutrino sates
correspond to
certain charged leptons
certain neutrino flavors
(interact in pairs)
n1
m1
ne
e
nm
m
n2
m2
nt
t
n3
m3
Eigenstates of the CC weak interactions
Mass eigenstates
mixing
7
Neutrinos in Matter
III International Pontecorvo Neutrino Physics
School
Neutrino sates
ne cosq n1 sinq n2
n2 sinq ne cosq nm
nm - sinq n1 cosq n2
n1 cosq ne - sinq nm
coherent mixtures of mass eigenstates
flavor composition of the mass eigenstates
ne
n1
wave packets
n2
n1
nm
n2
n2
n1
Neutrino images
8
Neutrinos in Matter
Neutrino propagation in vacuum
Due to difference of masses ?1 and ?2 have
different phase velocities
Oscillation depth
Oscillation length
9
Neutrinos in Matter
Neutrino propagation in vacuum
Oscillation probability
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
10
Neutrinos in Matter
Graphic representation
?z
(P-1/2)
Evolution equation
?
Analogy to equation for the electron spin
precession in magnetic field
2?
?x
(Re ?e??)
?y
(Im ?e??)
P(?e ??e) ?e?e ½(1 cos?Z)
11
Neutrinos in Matter
Graphic representation
12
Neutrinos in Matter
Matter effect

• Matter potential

• Evolution equation in matter

• Resonance

• Adiabatic conversion

• Adiabaticity violation

• Survival probability

• Parametric enhancement of oscillations

13
Neutrinos in Matter
Matter 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

Elastic forward scattering
14
Neutrinos in Matter
Matter potential
? - the wave function of the system neutrino -
medium
Hint Hamiltonian of the weak interaction at low
energy
(CC interaction with electrons)
(gV -gA 1)
Unpolarized and isotropic medium
15
Neutrinos in Matter
Matter potential
V 10-13 eV inside the Earth at E 10 MeV
Refraction index
Refraction length
16
Neutrinos in Matter
Evolution equation in matter
total Hamiltonian
17
Neutrinos in Matter
Evolution equation in matter
vacuum vs. matter
Depend on ne, E
Mixing angle determines flavors of eigenstatea
(?i)
(?f)
?
(?f)
(?im)
?m
18
Neutrinos in Matter
Evolution equation in matter
Diagonalization of the Hamiltonian

• Mixing

• Difference of the eigenvalues

At resonance

• Resonance condition

difference of the eigenvalues is minimal
mixing is maximal
level crossing
19
Neutrinos in Matter
Resonance
At
sin2 2qm 1
sin2 2qm
Resonance half width
sin2 2q 0.08
sin2 2q 0.825
Resonance energy
Resonance density
Resonance layer
20
Neutrinos in Matter
Resonance
H

sin2 2q 0.825 (large mixing)
Level crossing
ne
n2m
Dependence of the neutrino eigenvalues on the
matter potential (density)
nm
n1m
V. Rubakov, private comm. N. Cabibbo, Savonlinna
1985 H. Bethe, PRL 57 (1986) 1271
H

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

For maximal mixing nR 0
21
Neutrinos in Matter
Resonance
Oscillation length in matter
Lm
E
22
Neutrinos in Matter
Oscillations in matter
Pictures of neutrino oscillations in media with
constant density and variable density are
different
In uniform matter (constant density) mixing is
constant
qm(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)
MSW effect
qm(E, n) F(x)
Transformation of one neutrino type to another is
due to change of mixing or flavor of the neutrino
eigenstates
23
Neutrinos in Matter
Oscillations in matter
Constant density

• Flavors of the eigenstates do not change

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

Oscillations as in vacuum

• Monotonous increase of the phase
• difference between eigenstates ??m

n2
n1
Dfm 0
Dfm (H2 - H1) L
sin22qm, Lm
Parameters of oscillations (depth and length)
are determined by mixing in matter and by
effective energy split in matter
instead of
sin22q, Ln
24
Neutrinos in Matter
Oscillations in matter
Constant density Resonance enhancement of
oscillations
sin2 2q 0.08
sin2 2q 0.824
25
Neutrinos in Matter
Oscillations in matter
Instantaneous density change
n1
n2
?m ?1
?m ?2
26
Neutrinos in Matter
Oscillations in matter
Instantaneous density change
n1
n2
?m ?1
?m ?2
27
Neutrinos in Matter
Oscillations in matter
Instantaneous density change parametric resonance
?m ?1
?1
?1
?1
n1
n2
?
1
?2
?2
?2
2
3
4
5
6
7
8
?m ?2
B2
B1
?1 ?2 ?
.
Enhancement associated to certain conditions for
the phase of oscillations.
.
1
2
.
3
.
4
Another way to get strong transition. No large
vacuum mixing and no matter enhancement of mixing
or resonance conversion
.
5
.
6
.
.
7
8
28
Neutrinos in Matter
Oscillations in matter
Instantaneous density change parametric resonance
n1
n2
?m ?1m
?m ?2m
?
?1
?2
Resonance condition
Simplest realization
?1 ?2 ?
In general, certain correlation between phases
and mixing angles
29
Neutrinos in Matter
Oscillations in matter
Non-uniform density
In matter with varying density the Hamiltonian
depends on time Htot Htot(ne(t)) Its
eigenstates, ?m, do not split the equations of
motion
?m ?m(ne(t))
The Hamiltonian is non-diagonal ? no split of
equations
Transitions ?1m ? ?2m
30
Neutrinos in Matter
Oscillations in matter
Non-uniform density Adiabaticity
One can neglect of ?1m ? ?2m transitions if the
density changes slowly enough
Adiabaticity condition
Adiabaticity parameter
31
Neutrinos in Matter
Oscillations in matter
Non-uniform density Adiabaticity
External conditions (density) change slowly so
the system has time to adjust itself
Adiabaticity condition
Transitions between the neutrino eigenstates
can be neglected
The eigenstates propagate independently
LR L?/sin2? is the oscillation length in
resonance
Crucial in the resonance layer - the mixing
angle changes fast - level splitting is minimal
is the width of the resonance layer
32
Neutrinos in Matter
Oscillations in matter
Non-uniform density Adiabatic conversion
Initial state
Adiabatic conversion to zero density
?1m(0) ? ?1 ?2m(0) ? ?2
Final state
Probability to find ?e averaged over oscillations
33
Neutrinos in Matter
Oscillations in matter
Non-uniform density Adiabatic conversion
Resonance
?

• Flavors of eigenstates change
• according to the density change

• determined by ?m

• fixed by mixing in
• the production point

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

• Phase difference increases

• according to the level split
• which changes with density

Effect is related to the change of flavors of
the neutrino eigenstates in matter with varying
density
34
Neutrinos in Matter
Oscillations in matter
Non-uniform density Adiabatic conversion
35
Neutrinos in Matter
Oscillations in matter
Non-uniform density Adiabatic conversion
Dependence on initial condition
The picture of adiabatic conversion is universal
in terms of variable
resonance layer
There is no explicit dependence on oscillation
parameters, density distribution, etc. Only
initial value of y0 is important.
production point y0 - 5
oscillation band
y0 lt -1
Non-oscillatory conversion
survival probability
Interplay of conversion and oscillations
y0 -1?1
averaged probability
resonance
Oscillations with small matter effect
y0 gt 1
y
(distance)
36
Oscillations of Natural Neutrinos
Oscillations in matter
Non-uniform density Adiabatic conversion
Survive probability (averged over oscillations)
sin22? 0.8
Vacuum oscillations P 1 0.5sin22?
Non - adiabatic conversion
?(0) ?e ?2m ? ?2
Adiabatic edge
Adiabatic conversion P lt?e?2gt2 sin2?
200
0.2
2
20
E? (MeV)
(?m2 8?10-5 eV2)
37
Neutrinos in Matter
Oscillations in matter
Non-uniform density Adiabaticity violation
Fast density change
n0 gtgt nR
n2m n1m
n2 n1
ne
Resonance

• 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
38
Neutrinos in Matter
Oscillations in matter
Non-uniform density Adiabaticity violation
39
Neutrinos in Matter
Oscillations versus conversion
Both require mixing, conversion is usually
accompanying by oscillations
Oscillation
Adiabatic conversion

• Vacuum or uniform medium with constant
  parameters

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

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

• Phase difference increase between the
  eigenstates

?m
?
In non-uniform medium interplay of both processes
40
Neutrinos in Matter
Oscillations versus conversion
Adiabatic conversion
Spatial picture
survival probability
Oscillations
distance
survival probability
distance
41
Neutrinos in Matter
Matter potential
Unpolarized relativistic medium
? ? e ?
? ? e ?
polarized isotropic medium
if
42
Neutrinos in Matter
Neutrino propagation in real media

• The Sun

• The Earth

• Supernovae

43
Neutrinos in Matter
The Sun
4p 2e- 4He 2ne 26.73 MeV
electron neutrinos are produced
Adiabatic conversion in matter of the Sun
r (150 0) g/cc
J.N. Bahcall
Oscillations in vacuum
n
Oscillations in matter of the Earth
?e
Adiabaticity parameter ? 104
44
Neutrinos in Matter
The Sun
Borexino Collaboration arXiv0708.2251
45
Neutrinos in Matter
The Sun
Solar neutrinos vs. KamLAND
Adiabatic conversion (MSW)
Vacuum oscillations
Matter effect dominates (at least in the HE part)
Matter effect is very small
Non-oscillatory transition, or averaging of
oscillationsthe oscillation phase is irrelevant
Oscillation phase is crucialfor observed effect
Adiabatic conversion formula
Vacuum oscillations formula
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..)
46
Neutrinos in Matter
The Earth
Density Profile (PREM model)
core
mantle
mantle
47
Neutrinos in Matter
The Earth
48
Neutrinos in Matter
The Earth
Liu, Smirnov, 1998 Petcov, 1998 E.Akhmedov 1998
Akhmedov, Maltoni Smirnov, 2005
49
Neutrinos in Matter
Supernovae
Supernova Neutrino Fluxes
H.-T. Janka W. Hillebrand, Astron. Astrophys.
224 (1989) 49
G.G. Rafelt, Star as laboratories for
fundamental physics (1996)
50
Neutrinos in Matter
Supernovae
Matter effect in Supernova
Normal Hierarchy
Inverted Hierarchy
Dighe Smirnov, astro-ph/9907423
51
Neutrinos in Matter
Supernovae
Supernova Density Profile
Neutrino transitions occur far outside of the
star core
52
Neutrinos in Matter
Supernovae
Supernova Density Profile
Adiabaticity parameter
Adiabatic conversion
Weak dependence on A
Weak dependence on n
53
Neutrinos in Matter
Supernovae
Supernova Neutrino Oscillations
I Adiabatic conversion
Pf 0.9
E 50 MeV
E 5 MeV
II Weak violation of adiabaticity
Pf 0.1
III Strong violation of adiabaticity
54
Neutrinos in Matter
Supernovae
for
Original fluxes
for
for
After leaving the supernova envelope
sin2(2?13)
Hierarchy
Normal
? 10-3
0
cos2(Q12) ? 0.7
Inverted
sin2(Q12) ? 0.3
0
? 10-5
Any
sin2(Q12) ? 0.3
cos2(Q12) ? 0.7