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Collective oscillations of SN neutrinos :: A three-flavor course ::. Amol Dighe Tata Institute of Fundamental Research, Mumbai. Melbourne Neutrino Theory Workshop, 2-4 June 2008. Collective effects in a nutshell. - PowerPoint PPT Presentation
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Collective oscillations of SN neutrinos
:: A three-flavor course ::
Amol DigheTata Institute of Fundamental Research, Mumbai
Melbourne Neutrino Theory Workshop, 2-4 June 2008
Collective effects in a nutshell• Large neutrino density near the neutrinosphere gives
rise to substantial neutrino-neutrino potential
• Nonlinear equations of motion, give rise to qualitatively and quantitatively new neutrino flavor conversion phenomena
• Effects observed numerically in SN numerical simulations since 2006 (Duan, Fuller, Carlson, Qian)
• Analytical understanding in progress (Pastor, Raffelt, Semikoz, Hannestad, Sigl, Wong, Smirnov, Abazajian, Beacom, Bell, Esteban-Pretel, Tomas, Fogli, Lisi, Marrone, Mirizzi, Dasgupta, Dighe et al.)
• Substantial impact on the prediction of SN neutrino flavor convensions
Equations of motion including collective potential
• Density matrix :
• Eqn. of Motion :
• Hamiltonian :
• Useful convention: Antineutrinos : mass-matrix flips sign , as if p is negative (Sigl, Raffelt: NPB 406: 423, 1993; Raffelt, Smirnov: hep-ph/0705.1830)
• Useful approximation: Neglect three-angle effects: single-angle approximation (reasonably valid: Fogli et al.)
(r)(r)r)(p,
1(r) p
p
,
p alln
],[ ppp
H
dt
di
Mass matrix MSW potential Pantaleone’s - interaction
))(.1(8
q2 qqqqqp3
30p
nnvv
dGVH F
p
m
2
|| 231
Collective neutrino oscillation: two flavors
Pee
L0
1
Pee
L0
1
E E
Synchronized oscillation : Neutrinos with all energies oscillate at the same frequency
Bipolar oscillation : Neutrinos and antineutrinos with all energies convert pairwise; flipping periodically to the other flavor stateSpectral split : Energy spectrum of two flavors gets exchanged above a critical energy
In dense neutrino gases…
2- flavors : Formalism• Expand all matrices in terms of Pauli matrices as
• The following vectors result from the matrices
• EOM resembles spin precession
3,2,1
X2
1
2 iii
IX
DP
LL
B
P
)sgn( )( )(2
2
p
0p
p
fdnnGH
NGV
H
F
eF
PHPDLBP ) (hdr
d
The spinning top analogy• Motion of the average P defined by
• Construct the “Pendulum’’ vector
• EOMs are given by
• Mapping to Top :
• EOMs now become
• Note that these are equations of a spinning top!!! (Hannestad, Raffelt, Sigl, Wong: astro-ph/0608695; Fogli, Lisi, Mirizzi,
Marrone: hep-ph/0707.1998)
PS )( fd
BSQ
avg
QBDQDQ
avg ,
/Q. ,
Q , , /Q1- QD
gBjDrQ
m
avg
grjrrrj
mm ,
Synchronized oscillation• Spin is very large : Top precesses about direction of
gravity
• At large » avg : Q precesses about B with frequency avg
• Therefore S precesses about B with frequency avg
• Large : all P are bound together: same EOM
• Survival probability : r
r
avg
ee
22
z
2
sin2sin1
2/)P1()(
P
x
z
B
Precession = Sinusoidal Oscillation
(Pastor, Raffelt, Semikoz: hep-ph/0109035)
PDLBP ) (dr
d
• Spin is not very large : Top precesses and nutates
• At large ≥ avg : Q precesses + nutates about B
• Therefore S does the same
• All P are still bound together, same EOM:
• Survival probability :
Bipolar oscillation
2/)P1()( z
2ree
PDLBP ) (dr
d
P
x
z
B
Nutation = Inverse elliptic functions
(Hannestad, Raffelt, Sigl, Wong: astro-ph/0608695; Duan, Fuller, Carlson, Qian: astro-ph/0703776)
Adiabatic spectral split• Top falls down when it slows down (when mass
increases)
• If decreases slowly P keeps up with H
• As →0 from its large value : P aligns with hB
• For inverted hierarchy P has to flip, BUT…
• B.D is conserved so all P
can’t flip• Low energy modes anti-align• All P with < c flip over• Spectral Split
x
P
z
B
0)(
QBB.DB.B.D avgdr
d
(Raffelt, Smirnov:hep-ph/0705.1830)
3- flavors : Formalism• Expand all matrices in terms of Gell-Mann matrices
as
• The following vectors result from the matrices
• EOM formally resembles spin precession
81
X2
1
3 iii
IX
DP
LL
B
P
)sgn( )( )(2
2
p
0p
p
fdnnGH
NGV
H
F
eF
PHPDLBP ) (dr
d
Motion of the polarization vector P• P moves in eight-dimensional space, inside the
“Bloch sphere” (All the volume inside a 8-dim sphere is not accessible)
• Flavor content is given by diagonal elements: e3 and e8 components (allowed projection: interior of a triangle)
Some observations about 3- case• When ε = ∆m21
2 /∆m312 is taken to zero, the problem
must reduce to a 2- flavor problem• That problem is solved easily by choosing a useful
basis• When we have 3- flavors
• Each term by itself reduces to a 2- flavor problem• Hierarchical ``precession frequencies’’, so
factorization possible
• Enough to look at the e3 and e8 components of P
)3(13
)2()1( BBBB hhh
The e3 - e8 triangle
xyexey hhh BBBB 13/21213
-13/2 Rsin2 R
eyh B-13/2R xyh B1
3/21213 Rsin2
e
y
x
P
exh B
e3
e8
The 2-flavorslimit
eyh BB -13/2R
)0(P
)0(P
10
0)(
)(P
)(P
8
31
8
3 Rr
Rr
r ey
)2/(sin2sin21)( 213
2 rhrey
e
y
x
P Bip
olar
Vac
uum
/Mat
ter/
Sync
hron
ized
Osc
illat
ions
Spec
tral
Spl
it
e 3ey
e 8ey
Mass matrix gives only
Evolution function looks like
So that,
3-flavors and factorization
Neutrinos trace something like Lissajous figures in the e3-e8 triangle
e
y
x
P
• Each sub-system has widely different frequency• Interpret motion as a product of successive precessions in different subspaces of SU(3)• To first order,
)0(P
)0(P
10
0)(
10
0)(
)(P
)(P
8
31
8
3 rR
rR
r
r exey
Solar
Atmospheric
(Opposite order for bipolar)
Synchronized oscillations
e
y
x
P
All energies have same trajectory, but different speeds
Bipolar oscillations
e
y
x
P
Petal-shaped trajectories due to bipolar oscillations
Spectral splits
e
y
x
P
Two lepton number conservation laws : B.D conserved (Duan, Fuller, Qian: hep-ph/0801.1363; Dasgupta, Dighe, Mirizzi, Raffelt hep-ph/0801.1660)
A typical SN scenario
Order of events :
(1) Synchronization (2) Bipolar (3) Split Collective effects
(4) MSW resonances (5) Shock wave Traditional effects
(6) Earth matter effects
Spectral splits in SN spectraB
efo
reA
fter
Split Swap
Neutrinos Antineutrinos
Survival probabilities after collective+MSW
Hierarchy 13p pbar
A Normal Large 0 Sin2 sol
B Inverted Large Cos2 sol | 0 Cos2 sol
C Normal small Sin2 sol Cos2 sol
D Inverted small Cos2 sol | 0 0
• Spectral split in neutrinos for inverted hierarchy• All four scenarios are in principle distinguishable
Presence / absence of shock effectsHierarchy 13 e Anti- e
A Normal Large √ √
B Inverted Large X √
C Normal smallX X
D Inverted small X X
Condition for shock effects:Neutrinos: p should be different for A and CAntineutrinos: pbar should be different for B and D
Presence / absence of Earth matter effects
Hierarchy 13 e Anti- e
A Normal Large X √
B Inverted Large X √
C Normal small √ √
D Inverted small X X
Conditions for Earth matter effects:Neutrinos: p should be nonzero Antineutrinos: pbar should be nonzero
State of the CollectiveFor “standard” SN,
flavor conversion can be predicted more-or-less robustly
(Talks of Basudeb Dasgupta, Andreu Esteban-Pretel, Sergio Pastor)
Some open issues still to be clarified are:
• How multi-angle decoherence is prevented• Behaviour at extremely small 13 values• Possible nonadiabaticity in spectral splits• Possible interference between MSW resonances
and bipolar oscillations
Collective efforts are in progress !