A. Yu. Smirnov
description
Transcript of A. Yu. Smirnov
A. Yu. Smirnov
International Centre for Theoretical Physics, Trieste, Italy Institute for Nuclear Research, RAS, Moscow, Russia
Supernova neutrinos
Graphical representation
Non-linear collective effects. Evolution
Spectral splits
Observational consequences
G.Raffelt, A.Yu. S. Phys. Rev. D76:081301, 2007, arXiv:0705.3221 Phys. Rev. D76:125008, 2007 arXiv:0709.4641Pei Hong Gu, A.Yu. S. in preparation
Diffusion Diffusion
Flavor conversion Flavor conversion inside the starinside the star
Propagation Propagation in vacuumin vacuum
Oscillations Oscillations Inside the EarthInside the Earth
Collective Collective effects effects
E (e) < E (e) < E ( x )
1011 - 10 12g/cc 0
0.5 s
1 s
3 s 5 s7 s
9 s
G. Fuller et al > 3 – 5 s
107 109108 1010
T. Janka, 2006neutrinosphere
Collective effects
r
= 2 GF (1 – cos ) n
neutrinosphere
n ~ 1/r2
~ 1/r
~ 1/r4
for large r
n ~ 1033 cm-3 in neutrinosphere in all neutrino species:
electron density: ne ~ 1035 cm-3
= V = 2 GF ne usual matter potential:
neutrino potential:
R = 20 – 50 km
``Neutrino oscillations in a variable density medium and neutrino burst due to the gravitational collapse of star’’
ZhETF 91, 7-13, 1986 (Sov. Phys. JETP 64, 4-7, 1986)ArXiv: 0706.0454 (hep-ph)
m2 = (10-6 - 107 ) eV2
sin2 2 = (10-8 - 1)
Conversion in SN can probe:
Ya. B. Zeldovich : Neutrino fluxes from gravitational collapses
G. T. Zatsepin: Detection of supernova neutrinos
L. Stodolsky G Zatsepin, O. Ryazhskaya A. Chudakov
Oscillations of SN neutrinosin vacuum
L. Wolfenstein 1978 Matter effects suppress oscillations inside the star
Pre-history?
Re e
+ , P = Im e
+ , e
+ e - 1/2
B = (sin 2, 0, cos2)
= ( B x P ) dP dt
Coincides with equation for the electron spin precession in the magnetic field
= e ,
Polarization vector:
P =+
Evolution equation:
i = H d d t
d d t i = (B )
Differentiating P and using equation of motion
m2 /2E
= P = (Re e
+ , Im e+ , e
+ e - 1/2)
B = (sin 2m, 0, cos2m) 2 lm
= ( B x ) d dt
Evolution equation
= 2t/ lm - phase of oscillations
P = e+e = Z + 1/2 = cos2Z/2 probability to find e
e
,
Pure adiabatic conversion Partialy adiabatic conversion
e
If initial mixing is small: P ~ BP ~ Bmm
in matter
Collective flavor transformations
ee
e
e
b b b
b
Z0Z0
J. Pantaleone
Refraction in neutrino gases
e
bb
e
A = 2 GF (1 – vee vb )
e
e
e
b
b
u-channel
t-channel (p)
(q)
(p)
(q)
can lead to the coherent effect
Momentum exchange flavor exchange
flavor mixing
elastic forward scattering
velocities
e
b
b
Flavor exchange between the beam (probe) and background neutrinos
J. PantaleoneS. SamuelV.A. Kostelecky
e
e
backgroundcohere
nt
A. FriedlandC. Lunardini
projection
projection
Be ~ i ie*i
ib = ie e + i
If the background is in the mixed state:
w.f. give projectionssum over particles of bg.
Contribution to the Hamiltonianin the flavor basis
H = 2 GF i (1 – vee vib ) ie
iei*
ie*i
i
The key point is that the background
should be in mixed flavor state.
For pure flavor state the off-diagonal
terms are zero. Flavor evolution
should be triggered by some other effect.
Total Hamiltonian for individual neutrino state:
H = - cos 2 + V + B sin 2 + Be sin 2 + Be* cos 2 - V - B
m2 /2EV – describes scattering on electrons
Be ~ n ie*i - non-linear problem
Two classes of collective effects:
S. Samuel , H. Duan, G. Fuller, Y-Z Qian
Kostelecky & SamuelPastor, Raffelt, SemikozSynchronized oscillations
Bipolar oscillations
H = (HH)
dt P= HH x P
Suppose we know the Hamiltonian H for neutrino state with frequency
Represent it in the form
- is Pauli matrices
Then equation for the polarization vector:
dt P=(- B + L + D) x P
dt P=(+ B + L + D) x P = V = 2 GF ne
= 2 GF n
D= P - P
P= d P P= d P
where
(in single angle approximation)
Ensemble of neutrino polarization vectors P
L = (0, 0, 1)
inf 0
inf 0
Total polarization vectors for neutrinos and antineutrinos
m2 /2E
- collective vector
In rotating frame
BB
``trapping cone’’
PPLL
B rotates with high frequency
Without interactions, P P would precess around B B with frequency
P has no time to follow B
B precesses with small angle ~ near the initial position
With interactions, D D provides with the force which pushes P outside the trapping cone
transition to the rotating system around L with frequency -
In rotating frame
``trapping cone’’
PP
In presence of both neutrinosand antineutrinos interactions, produce a force which pushes P outside the trapping cone
PP F
F = D x P = 0
If PP is outside the trapping cone quick rotation of B B can be averaged
In the original frame one can understand this ``escaping evolution’’ as a kind of parametric resonance.
D= ds P
dt P=(B + D) x P
Introducing negative frequencies for antineutrinos P= P > 0
wheres = sign()
Equation of motion for D: integrating equation of motion with sw
dt D= B x M M= ds P where
+ inf
- inf
dt P= H() x P
H=(B + D)
inf
- inf
where
In another form: In another form:
If |D| >> - the individual vectors form largethe self-interaction term dominates
M = syn D
ds P
syn = dsP
synchronization frequency
dt D= synB x D
D - precesses around B with synchronization frequency
dt P~ D x P
- evolution is the same for all modes – P are pinned to each other
does not depend on D
B
dt D= B x M If B = const, from equation
dt ( D B ) = 0
DB = ( D B ) = const
For small effective angle DB ~ Dz
- total electron lepton number is conserved
Strictly: B is the mass axis – so the total 1 - number is conserved
Play crucial role in evolution and split phenomenon
H= B + D
B
H
D
dt P= H() x P
dt D= cB x D
D precesses around B with frequency c
H precesses around B with the same frequency as D P precesses around H
eff ( P) ~ eff(H)
adiabaticity is not satisfied
in general,
eff ( P) >> eff(H)
= m2/2E
e
thin lines – initial spectrumthick lines – after split
neutrinosantineutrinos
Spectral split: result of the adiabatic evolution of ensemble of neutrinos propagating from large neutrino densities to small neutrino densities
r
= 2 GF (1 – cos ) n
neutrinosphere
n ~ 1/r2
~ 1/r
~ 1/r4
for large r
Split is a consequence of
existence of special frame in the flavor space, the adiabatic frame, which rotates around B with frequency C
change (decrease) of the neutrino density: 0
adiabatic evolution of the neutrino ensemble in the adiabatic frame
Split frequency: split= C()
Spectral split exists also in usual MSW case without self-interactionwith zero split frequency
It is determined by conservation of lepton number
Relative motion of Pand H can be adiabatic:
Adiabatic frame: co-rotating frame formed by Dand B
Since DD is at rest, motion of Hin this plane is due to change (t) only.If changes slowly enough adiabatic evolution
C – is its frequency
Pfollow H((t))
H=(-C)B + DIn the adiabatic frame:In the adiabatic frame:
C - frequency of the co-rotating frame
Individual Hamiltonians in the co-rotating frame
H=(-C)B + D
Pfollow H((t))
Initial mixing angle is very small: P~ H((t)) Pare co-linear with H((t))
P= H() P
H= H/|H| - unit vector in the direction of HamiltonianP=|P| - frequency spectrum of neutrinos given by initial condition
one needs to find C and Dperp P= H() P
DB is conserved and given by the initial condition
Pperp= (Hperp/ H) P
PB= (HB/ H) P
HB
Hperp
H
Hperp = Dperp
HB = -C+ DB
Projecting:
Inserting this into the previous equations and integrating over s d
From the expression for H()
B
DB = ds ( -C+ DB) P
(-C+ DB)2 + ( Dperp)2
1 = ds P
[(-C)/ + DB]2 + Dperp2
Equations forC and Dperp
``sum rules’’
H=(-C)B + D
In the limit 0 H (–C0)B
C0 = split
>C0
<C0
0
(-C)B
H
D
H
H
initial e
initial
C0 = C ( = 0)
In adiabatic (rotating) frame
Is determined by the lepton number conservation (and initial energy spectrum)
Flux of neutrinos is larger than flux of antineutrinos – split in the neutrino channel
DB> 0
DB(initial) = DB(final) + continuity
In final state the non-zero lepton number is due to high frequency tail of the neutrino spectrum >split
DB= d P inf split
or lepton number in antineutrinos is compensated by the low frequency part of the neutrino spectrum
d P= d P split
0
-inf
original spectrum(mixed state)
final spectrum (exact)
final spectrum (adiabatic)
P
B
initial state
Adiabatic solution: sharp splitspread – due to adiabaticity violation
Adiabaticity is violated for modes with frequencies near the split
1~e
2~
0.5
PB
final state
Adiabatic solution
Exact solution
PB
density decreases for 51 modes
adiabaticity violation
split
PB
PB
Pp
erp
PB
D
pP
initial
final
PB
initial spectrum
final spectrum
Adiabatic solutionExact numerical calculations
Wiggles:“nutations’’
PB
PB
Sharpness is determined by degree of adiabaticity violation
the variance of root mean square width ~ width on the half height
universal function
PB
Pp
erp
Wiggles - nutations
Solid lines – adiabatic solution
Pp
erp
PB
evolution of 25 modes
Spinning top
e
l/ l0
e
anti-neutrinos neutrinos
1
-1
anti-neutrinos neutrinos
0
Electron neutrinos are convertedantineutrinos - not
split= 0
Adiabaticityviolation
Further evolution
Conversion in the mantle of the star
Earth matter effect
Determination of the neutrino mass hierarchy
B Dasgupta, A. Dighe, A Mirizzi,arXiv: 0802.1481
B Dasgupta, A. Dighe, A Mirizzi, G. RaffeltarXiv: 0801.1660
Neutronization burst:
G. Fuller et al.
SN bursts have enormous potential to study the low energy (< 100 MeV) physics phenomena
Standard scenario: sensitivity to sin2 13< 10-5 , mass hierarchy
Non-linear effects related to neutrino self-interactions; Can lead to new phenomena: syncronized oscillations, bi-polar flipsspectral splits
Spectral splits: concept of adiabatic (co-rotating) frame splits are result of the adiabatic evolution in the adiabatic frame
Observable effects