s.p.mikheyev (inr ras). 25.09.2008s.p.mikheyev (inr ras)2 introduction. vacuum oscillations. ...
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S.P.Mikheyev (INR RAS)
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25.09.2008 S.P.Mikheyev (INR RAS)
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Introduction.
Vacuum oscillations.
Oscillations in matter. Adiabatic conversion.
Graphical representation of oscillations
Conclusion
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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.
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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
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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
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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
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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
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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
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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)
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Jennifer Raaf Talk at Neutrino’2008
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K2K
Erec (GeV)
MINOS
Hugh GallagherTalk at Neutrino’2008
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Patrick DecowskiTalk at Neutrino’2008
KamLAND
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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
~
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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) dominates.
Effect of elastic forward scattering is describer by potential
Only difference of e and is important
Unpolarized and isotropic medium: eFnG2V
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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
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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
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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
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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
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(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
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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
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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
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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
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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
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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
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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
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distance
su
rviv
al p
rob
ab
ilit
y
Oscillations
Adiabatic conversion
Spatial picture
su
rviv
al p
rob
ab
ilit
y
distance
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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
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SNO
Hamish RobertsonTalk at Neutrino’2008
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Cristano GalbiatiTalk at Neutrino’2008
Cl-Ar data
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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
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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
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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
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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
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x
y
z
2B
(P-1/2)
(Re e+x)
(Im e+x)
P
Lt2
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Non-uniform density: Adiabatic conversion
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Non-uniform density: Adiabaticity violation
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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
![Page 38: S.P.Mikheyev (INR RAS). 25.09.2008S.P.Mikheyev (INR RAS)2 Introduction. Vacuum oscillations. Oscillations in matter. Adiabatic conversion. Graphical](https://reader036.vdocuments.us/reader036/viewer/2022062315/5697bff01a28abf838cba9b7/html5/thumbnails/38.jpg)
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“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.
![Page 39: S.P.Mikheyev (INR RAS). 25.09.2008S.P.Mikheyev (INR RAS)2 Introduction. Vacuum oscillations. Oscillations in matter. Adiabatic conversion. Graphical](https://reader036.vdocuments.us/reader036/viewer/2022062315/5697bff01a28abf838cba9b7/html5/thumbnails/39.jpg)
25.09.2008 S.P.Mikheyev (INR RAS)
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“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!!!