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