A. Yu. Smirnov

1. Supernova Neutrinos and Non-linear neutrino effects A. Yu. Smirnov International Centre for Theoretical Physics, Trieste, Italy Institute for Nuclear Research, RAS, Moscow, Russia Heidelberg, March 10, 2008

2. Flavor of effects 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

3. Supernova neutrinos

4. Physics picture Diffusion Collective effects Flavor conversion inside the star Propagation in vacuum Oscillations Inside the Earth

5. Supernova neutrinos r ~ (1011 - 10 12 ) g/cc 0 E (ne) < E (ne) < E ( nx )

6. Density profiles T. Janka, 2006 neutrinosphere Collective effects 0.5 s 1 s 5 s 3 s 7 s 9 s 107 108 109 1010 > 3 – 5 s G. Fuller et al

7. Neutrino density neutrinosphere usual matter potential: l = V = 2 GF ne R = 20 – 50 km neutrino potential: m = 2 GF (1 – cos x) nn n r nn ~ 1/r2 x x ~ 1/r for large r n m ~ 1/r4 in neutrinosphere in all neutrino species: nn ~ 1033 cm-3 electron density: ne ~ 1035 cm-3 l >> m

8. History Pre-history? 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 ZhETF 91, 7-13, 1986 (Sov. Phys. JETP 64, 4-7, 1986) ArXiv: 0706.0454 (hep-ph) ``Neutrino oscillations in a variable density medium and neutrino burst due to the gravitational collapse of star’’ Conversion in SN can probe: Dm2 = (10-6 - 107 ) eV2 sin2 2q = (10-8 - 1)

9. Polarization vectors Grapical representation

10. Neutrino polarization vectors ne nt, Re ne+ nt, P = Im ne+ nt, ne+ ne - 1/2 Polarization vector: y = P = y+s/2 y Evolution equation: d Y d t d Y d t i = H Y i = w (Bs ) Y B = (sin 2q, 0, cos2q) w = D m2 /2E Differentiating P and using equation of motion dP dt = w ( B x P) Coincides with equation for the electron spin precession in the magnetic field

11. Graphical representation • = P = (Re ne+ nt, Im ne+ nt, ne+ ne - 1/2) ne 2p lm B = (sin 2qm, 0, cos2qm) nt, Evolution equation dn dt = ( B x n) f= 2pt/ lm - phase of oscillations probability to find ne P = ne+ne = nZ+ 1/2 = cos2qZ/2

12. Oscillations

18. Conversion effects Partialy adiabatic conversion Pure adiabatic conversion ne in matter nm If initial mixing is small: P ~ Bm

19. Non-linear effects

20. nn -scattering ne ne Refraction in neutrino gases ne nb Z0 Z0 nb nb nb ne A = 2 GF (1 – ve vb ) velocities ne ne (p) t-channel elastic forward scattering nb nb (q) ne J. Pantaleone ne (p) u-channel can lead to the coherent effect Momentum exchange  flavor exchange ne nb (q)  flavor mixing nb Collective flavor transformations

21. Flavor exchange J. Pantaleone S. Samuel V.A. Kostelecky ne projection If the background is in the mixed state: ne nb |nib =Fie |ne + Fit |nt coherent nb Bet ~ SiFie*Fit background ne 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. sum over particles of bg. w.f. give projections projection nt Contribution to the Hamiltonian in the flavor basis Flavor exchange between the beam (probe) and background neutrinos A. Friedland C. Lunardini |Fie|2 Fie*Fit Hnn = 2 GF Si (1 – ve vib ) FieFit * |Fit|2

22. Evolution Total Hamiltonian for individual neutrino state: • w cos 2q + V + B w sin 2q + Bet • w sin 2q + Bet* w cos 2q - V - B H = w = D m2 /2E V – describes scattering on electrons Bet ~ nnFie*Fit - non-linear problem Collective flavor conversion effects Two classes of collective effects: Kostelecky & Samuel Pastor, Raffelt, Semikoz Synchronized oscillations Bipolar oscillations S. Samuel , H. Duan, G. Fuller, Y-Z Qian

23. Equations for polarization vectors Suppose we know the Hamiltonian Hw for neutrino state with frequency w Represent it in the form Hw = (Hw s ) s - is Pauli matrices Then equation for the polarization vector: dt Pw = H x Pw

24. Evolution equations Ensemble of neutrino polarization vectorsPw w = D m2 /2E (in single angle approximation) Total polarization vectors for neutrinos and antineutrinos inf 0 inf 0 P= dwPw P= dwPw L = (0, 0, 1) dt Pw =(+ wB + lL + mD) x Pw l = V = 2 GF ne dt Pw =(- wB + lL + mD) x Pw m = 2 GF nn where - collective vector D= P - P

25. Eliminating matter effect  transition to the rotating system around L with frequency -l In rotating frame B rotates with high frequency l P B L Without nn-interactions, P would precess around B with frequency w ``trapping cone’’ w << l P has no time to follow B B precesses with small angle ~ w/l near the initial position With nn-interactions, D provides with the force which pushes P outside the trapping cone

26. Eliminating matter effect In rotating frame P P F In presence of both neutrinos and antineutrinos nn-interactions, produce a force which pushes P outside the trapping cone ``trapping cone’’ F = D x Pw = 0 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. Combining neutrinos and antineutrinos Pw = P- w w > 0 Introducing negative frequencies for antineutrinos dt Pw =(wB + mD) x Pw + inf - inf D= dw swPw wheresw= sign(w) Equation of motion for D: integrating equation of motion with sw inf - inf dt D= B x M M= dw swwPw where In another form: dt Pw = Hw(m) x Pw Hw =(wB + mD) where

28. Synchronization B If m |D| >> w - the individual vectors form large the self-interaction term dominates dt Pw ~ m D x Pw D does not depend on w - evolution is the same for all modes – Pware pinned to each other M = wsynD synchronization frequency dw sww Pw wsyn = dwsw P w dt D= wsynB x D D - precesses around B with synchronization frequency

29. Lepton number conservation If B = const, from equation dt D= B x M dt (DB ) = 0 DB = (DB ) = const Strictly: B is the mass axis – so the total n1 - number is conserved For small effective angle DB ~ Dz - total electron lepton number is conserved Play crucial role in evolution and split phenomenon

30. General picture dt Pw = Hw(m) x Pw Hw = wB + mD dt D= wcB x D Hw D precesses around B with frequency wc Hw precesses around B with the same frequency as D mD Pw precesses around Hw in general, wB weff (Pw ) ~ weff(Hw) adiabaticity is not satisfied weff (Pw ) >> weff(Hw)

31. Spectral splits

32. Spectral split thin lines – initial spectrum thick lines – after split ne n n nm w = Dm2/2E neutrinos antineutrinos

33. Split and density change Spectral split: result of the adiabatic evolution of ensemble of neutrinos propagating from large neutrino densities to small neutrino densities m = 2 GF (1 – cos x) nn n r x nn ~ 1/r2 for large r x ~ 1/r n neutrinosphere m ~ 1/r4

34. Explaining spectral splits Split is a consequence of existence of special frame in the flavor space, the adiabatic frame, which rotates around B with frequency wC change (decrease) of the neutrino density: m 0 adiabatic evolution of the neutrino ensemble in the adiabatic frame wsplit= wC( m  0 ) Split frequency: It is determined by conservation of lepton number Spectral split exists also in usual MSW case without self-interaction with zero split frequency

35. Adiabatic frame & solution Adiabatic frame: co-rotating frame formed by Dand B wC – is its frequency Hw =(w - wC)B + mD In the adiabatic frame: Since D is at rest, motion of Hw in this plane is due to change m(t) only. If m changes slowly enough  adiabatic evolution Relative motion of Pw and Hw can be adiabatic: Pw follow Hw(m(t))

36. Adiabatic solution Individual Hamiltonians in the co-rotating frame Hw =(w - wC)B + mD wC - frequency of the co-rotating frame Adiabaticity: Pw follow Hw(m(t)) Pw ~ Hw(m(t)) Initial mixing angle is very small: Initial condition: Pw are co-linear with Hw(m(t)) Adiabatic solution: Pw = Hw(m) Pw Hw = Hw/|Hw| - unit vector in the direction of Hamiltonian Pw =|Pw| - frequency spectrum of neutrinos given by initial condition

37. Explicit solution B Hw Pw = Hw(m) Pw Hw B one needs to find wC and Dperp DB is conserved and given by the initial condition Projecting: Pw perp= (Hw perp/ Hw) Pw Pw B= (Hw B/ Hw) Pw Hw perp Hw perp = m Dperp From the expression for Hw(m) Hw B = w - wC+ m DB Inserting this into the previous equations and integrating over sw dw ( w - wC+ m DB) Pw DB = dw sw (w - wC+ m DB)2 + (m Dperp)2 Equations for wC(m) and Dperp(m) Pw 1 = dw sw ``sum rules’’ [(w - wC)/m + DB]2 + Dperp2

38. Explaining splits Hw =(w - wC)B + mD In adiabatic (rotating) frame Hw  (w – wC0)B In the limit m 0 wC0 = wC (m = 0) Hw ninitial  ne Hw w > wC0 mD wC0 = wsplit m 0 ninitial  nm (w - wC)B Hw w < wC0

39. Split frequency Is determined by the lepton number conservation (and initial energy spectrum) + continuity DB(initial) = DB(final) Flux of neutrinos is larger than flux of antineutrinos – split in the neutrino channel DB> 0 In final state the non-zero lepton number is due to high frequency tail of the neutrino spectrum w > wsplit inf wsplit DB= dw Pw wsplit 0 or lepton number in antineutrinos is compensated by the low frequency part of the neutrino spectrum 0 -inf dwP w = dw Pw

40. Neutrinos only B final spectrum (exact) final spectrum (adiabatic) 0.5 P n1~ ne original spectrum (mixed state) initial state final state n2~ nm Pw B Adiabaticity is violated for modes with frequencies near the split Adiabatic solution: sharp split spread – due to adiabaticity violation

41. Evolution of modes Pw B adiabaticity violation Pw B Exact solution split Pw perp Pw B Adiabatic solution for 51 modes density decreases 

42. Cross-over Pw B

43. Neutrinos and antineutrinos D Pw B p P initial spectrum initial final spectrum final

44. Nu + anti-nu: split Exact numerical calculations Adiabatic solution Wiggles: “nutations’’ Pw B Pw B

45. Sharpness of split Sharpness is determined by degree of adiabaticity violation Pw B Pw perp Dw the variance of root mean square width ~ width on the half height universal function

46. Transient phenomena evolution of 25 modes Wiggles - nutations Pw B Solid lines – adiabatic solution Pw perp Spinning top

47. Spectral split in the MSW conversion ne ne ln / l0 nm anti-neutrinos neutrinos 1 Adiabaticity violation anti-neutrinos neutrinos Electron neutrinos are converted antineutrinos - not -1 0 w wsplit= 0

48. Observational consequences Further evolution Conversion in the mantle of the star B Dasgupta, A. Dighe, A Mirizzi, arXiv: 0802.1481 Earth matter effect Determination of the neutrino mass hierarchy G. Fuller et al. B Dasgupta, A. Dighe, A Mirizzi, G. Raffelt arXiv: 0801.1660 Neutronization burst:

49. Conclusions

50. Collective effects and supernova neutrinos SN bursts have enormous potential to study the low energy (< 100 MeV) physics phenomena Standard scenario: sensitivity to sin2 q13< 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