1. Collective oscillations of SN neutrinos :: A three-flavor course :: Amol Dighe Tata Institute of Fundamental Research, Mumbai Melbourne Neutrino Theory Workshop, 2-4 June 2008

2. 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

3. 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.) Mass matrix MSW potential Pantaleone’s - interaction

4. Collective neutrino oscillation: two flavors P ee L 0 1 L 0 1  EE 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 state Spectral split : Energy spectrum of two flavors gets exchanged above a critical energy In dense neutrino gases…

5. 2- flavors : Formalism Expand all matrices in terms of Pauli matrices as The following vectors result from the matrices EOM resembles spin precession

6. 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 )

7. 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 :  PP x z B Precession = Sinusoidal Oscillation ( Pastor, Raffelt, Semikoz: hep-ph/0109035 )

8. 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  PP x z B Nutation = Inverse elliptic functions (Hannestad, Raffelt, Sigl, Wong: astro-ph/0608695; Duan, Fuller, Carlson, Qian: astro-ph/0703776)

9. 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 h  B 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P z BB (Raffelt, Smirnov:hep-ph/0705.1830)

10. 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

11. 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: e 3 and e 8 components (allowed projection: interior of a triangle)

12. Some observations about 3- case When ε = ∆m 21 2 /∆m 31 2 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 e 3 and e 8 components of P 

13. The e3 - e8 triangle e y x P e3e3 e8e8

14. The 2-  flavors  limit e y x P Bipolar Vacuum/Matter/Synchronized Oscillations Spectral Split e 3 ey e 8 ey Mass matrix gives only Evolution function looks like So that,

15. 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, SolarAtmospheric (Opposite order for bipolar)

16. Synchronized oscillations e y x P All energies have same trajectory, but different speeds

17. Bipolar oscillations e y x P Petal-shaped trajectories due to bipolar oscillations

18. 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)

19. 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

20. Spectral splits in SN spectra Before After SplitSwap Neutrinos Antineutrinos

21. Survival probabilities after collective+MSW Hierarchy  13 ppbar ANormalLarge0 Sin 2  sol BInvertedLarge Cos 2  sol | 0 Cos 2  sol CNormal small Sin 2  sol Cos 2  sol DInverted small Cos 2  sol | 0 0 Spectral split in neutrinos for inverted hierarchy All four scenarios are in principle distinguishable

22. Presence / absence of shock effects Hierarchy  13 e Anti- e ANormalLarge√√ BInvertedLargeX√ CNormal small XX DInverted small XX Condition for shock effects: Neutrinos: p should be different for A and C Antineutrinos: pbar should be different for B and D

23. Presence / absence of Earth matter effects Hierarchy  13 e Anti- e ANormalLargeX√ BInvertedLargeX√ CNormal small √√ DInverted small XX Conditions for Earth matter effects: Neutrinos: p should be nonzero Antineutrinos: pbar should be nonzero

24. State of the Collective For “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 !