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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
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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
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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
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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…
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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
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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 ,
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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
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• 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)
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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)
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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
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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)
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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
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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
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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,
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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)
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Synchronized oscillations
e
y
x
P
All energies have same trajectory, but different speeds
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Bipolar oscillations
e
y
x
P
Petal-shaped trajectories due to bipolar oscillations
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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)
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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
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Spectral splits in SN spectraB
efo
reA
fter
Split Swap
Neutrinos Antineutrinos
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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
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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
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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
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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 !