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 !