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A. Yu. Smirnov International Centre for Theoretical Physics, Trieste, Italy Institute for Nuclear Research, RAS, Moscow, Russia

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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:0709.4641 Pei Hong Gu, A.Yu. S. in preparation

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Diffusion Flavor conversion inside the star inside the star Propagation in vacuum Oscillations Inside the Earth Collectiveeffects

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E ( e ) < E ( e ) < E ( x ) 10 11 - 10 12 g/cc 0

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0.5 s 1 s 3 s 5 s 7 s 9 s G. Fuller et al > 3 – 5 s 10 7 10 9 10 810 T. Janka, 2006 neutrinosphere Collective effects

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r = 2 G F (1 – cos ) n neutrinosphere n ~ 1/r 2 ~ 1/r ~ 1/r 4 for large r n ~ 10 33 cm -3 in neutrinosphere in all neutrino species: electron density: n e ~ 10 35 cm -3 = V = 2 G F n e usual matter potential: neutrino potential: R = 20 – 50 km

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``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) m 2 = (10 -6 - 10 7 ) eV 2 sin 2 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 neutrinos in vacuum L. Wolfenstein 1978 Matter effects suppress oscillations inside the star P r e - h i s t o r y ?

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Re e + , P = Im e + , e + e - 1/2 B = (sin 2 , 0, cos2 ) = ( B x P ) d P 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 m 2 /2E

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= P = (Re e + , Im e + , e + e - 1/2) B = (sin 2 m, 0, cos2 m ) 2 l m = ( B x ) d dt Evolution equation = 2 t/ l m - phase of oscillations P = e + e = Z + 1/2 = cos 2 Z /2 probability to find e e ,

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Pure adiabatic conversion Partialy adiabatic conversion e P ~ B m If initial mixing is small: P ~ B m in matter

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Collective flavor transformations e e e e b b b b Z0Z0 Z0Z0 J. Pantaleone Refraction in neutrino gases e b b e e A = 2 G F (1 – v e v b ) 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

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e b b Flavor exchange between the beam (probe) and background neutrinos J. Pantaleone S. Samuel V.A. Kostelecky e e background coherent A. Friedland C. Lunardini projection B e ~ i ie * i ib = ie e + i If the background is in the mixed state: w.f. give projections sum over particles of bg. Contribution to the Hamiltonian in the flavor basis H = e 2 G F i (1 – v e v ib ) ie ie i * 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.

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Total Hamiltonian for individual neutrino state: H = - cos 2 + V + B sin 2 + B e sin 2 + B e * cos 2 - V - B m 2 /2E V – describes scattering on electrons B e ~ n ie * i - non-linear problem Two classes of collective effects: S. Samuel, H. Duan, G. Fuller, Y-Z Qian Kostelecky & Samuel Pastor, Raffelt, Semikoz Synchronized oscillations Bipolar oscillations

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H H = (H ) H d t P = H 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:

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d t P =(- B + L + D) x P d t P =(+ B + L + D) x P = V = 2 G F n e = 2 G F n D = P - 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 m 2 /2E - collective vector

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In rotating frame B ``trapping cone’’ P L B rotates with high frequency Without interactions, P B P would precess around 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 -

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In rotating frame ``trapping cone’’ P In presence of both neutrinos and antineutrinos interactions, produce a force which pushes P outside the trapping cone P F F = D x P = 0 PB If P is outside the trapping cone quick rotation of B can be averaged In the original frame one can understand this ``escaping evolution’’ as a kind of parametric resonance.

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D = d s P d t P =( B + D) x P Introducing negative frequencies for antineutrinos P = P > 0 where s = sign( ) Equation of motion for D: integrating equation of motion with s w d t D = B x M M = d s P where + inf - inf d t P = H ( ) x P H =( B + D) inf - inf where In another form:

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If |D| >> - the individual vectors form large the self-interaction term dominates M = syn D d s P syn = d s P synchronization frequency d t D = syn B x D D - precesses around B with synchronization frequency d t P ~ D x P - evolution is the same for all modes – P are pinned to each other does not depend on D B

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d t D = B x M If B = const, from equation d t ( D B ) = 0 D B = ( D B ) = const For small effective angleD B ~ D z - 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

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H = B + D BB HH DD d t P = H ( ) x P d t D = c B 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 )

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= m 2 /2E e thin lines – initial spectrum thick lines – after split neutrinos antineutrinos

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Spectral split: result of the adiabatic evolution of ensemble of neutrinos propagating from large neutrino densities to small neutrino densities r = 2 G F (1 – cos ) n neutrinosphere n ~ 1/r 2 ~ 1/r ~ 1/r 4 for large r

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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-interaction with zero split frequency It is determined by conservation of lepton number

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Relative motion of P and H can be adiabatic: Adiabatic frame: co-rotating frame formed by D and B D Since D is at rest, motion of H in this plane is due to change (t) only. If changes slowly enough adiabatic evolution C – is its frequency P follow H ( (t)) H =( - C ) B + D In the adiabatic frame:

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C - frequency of the co-rotating frame Individual Hamiltonians in the co-rotating frame H =( - C ) B + D P follow H ( (t)) Initial mixing angle is very small: P ~ H ( (t)) P are co-linear with H ( (t)) P = H ( ) P H = H /|H | - unit vector in the direction of Hamiltonian P =|P | - frequency spectrum of neutrinos given by initial condition

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one needs to find C and D perp P = H ( ) P D B is conserved and given by the initial condition P perp = (H perp / H ) P P B = (H B / H ) P H B H perp H H H perp = D perp H B = - C + D B Projecting: Inserting this into the previous equations and integrating over s d From the expression for H ( ) B D B = d s ( - C + D B ) P ( - C + D B ) 2 + ( D perp ) 2 1 = d s P P [( - C )/ + D B ] 2 + D perp 2 Equations for C and D perp ``sum rules’’

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H =( - C ) B + D In the limit 0 H ( – C 0 )B C 0 = split > C 0 < C 0 0 ( - C )B HH DD HH HH initial e initial C 0 = C ( = 0) In adiabatic (rotating) frame

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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 D B > 0 D B (initial) = D B (final) + continuity In final state the non-zero lepton number is due to high frequency tail of the neutrino spectrum > split D B = 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

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original spectrum (mixed state) final spectrum (exact) final spectrum (adiabatic) P B initial state Adiabatic solution: sharp split spread – due to adiabaticity violation Adiabaticity is violated for modes with frequencies near the split 1 ~ e 2 ~ 0.5 P B final state

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Adiabatic solution Exact solution P B density decreases for 51 modes adiabaticity violation split P B P perp

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P B

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D p P initial final P B initial spectrum final spectrum

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Adiabatic solution Exact numerical calculations Wiggles: “nutations’’ P B

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Sharpness is determined by degree of adiabaticity violation the variance of root mean square width ~ width on the half height universal function P B P perp

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Wiggles - nutations Solid lines – adiabatic solution P perp P B evolution of 25 modes Spinning top

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e l / l 0 e anti-neutrinos neutrinos 1 anti-neutrinos neutrinos 0 Electron neutrinos are converted antineutrinos - not split = 0 Adiabaticity violation

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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. Raffelt arXiv: 0801.1660 Neutronization burst: G. Fuller et al.

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SN bursts have enormous potential to study the low energy (< 100 MeV) physics phenomena Standard scenario: sensitivity to sin 2 13 < 10 -5, mass hierarchy Non-linear effects related to neutrino self-interactions; Can lead to new phenomena: syncronized oscillations, bi-polar flips spectral splits Spectral splits: concept of adiabatic (co-rotating) frame splits are result of the adiabatic evolution in the adiabatic frame Observable effects

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