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

2. 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

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

5. E ( e ) < E ( e ) < E ( x )  10 11 - 10 12  g/cc 0

6. 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

7. 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

8. ``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 ?

10. 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

11. = 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 ,

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

20. 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

21. 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.

22. 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

23. 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:

24. 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

25. 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 -

26. 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.

27. 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:

28. 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

29. 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

30. H  =  B +  D BB HH DD 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  )

32.  =  m 2 /2E e  thin lines – initial spectrum thick lines – after split neutrinos antineutrinos

33. 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

34. 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

35. 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:

36.  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

37. 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’’

38. H  =(  -  C )  B +  D In the limit   0 H   (  –  C 0 )B  C 0 =  split  >  C 0  <  C 0   0 (  -  C )B HH DD HH HH initial  e initial    C 0 =  C (  = 0) In adiabatic (rotating) frame

39. 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

40. 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

41. Adiabatic solution Exact solution P  B density decreases  for 51 modes adiabaticity violation split P  B P  perp

42. P  B

43. D p P initial final P  B initial spectrum final spectrum

44. Adiabatic solution Exact numerical calculations Wiggles: “nutations’’ P  B

45. 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

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

47. e  l  / l 0  e anti-neutrinos neutrinos 1 anti-neutrinos neutrinos 0  Electron neutrinos are converted antineutrinos - not  split  = 0 Adiabaticity violation

48. 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.

50. 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