1. S.P.Mikheyev (INR RAS)

2. 25.09.2008S.P.Mikheyev (INR RAS)2  Introduction.  Vacuum oscillations.  Oscillations in matter.  Adiabatic conversion.  Graphical representation of oscillations  Conclusion

3. 25.09.2008S.P.Mikheyev (INR RAS)3 Neutrino are massive. Neutrino masses are in the sub-eV range - much smaller than masses of charge leptons and quarks. A. Yu. Smirnov hep-ph/0702061 There are only 3 types of light neutrinos: 3 flavors and 3 mass states. Their interactions are described by the Standard electroweak theory Masses and mixing are generated in vacuum   e 1 2 3 | f    U fi | i  i mixing Neutrinos mix. There are two large mixings and one small or zero mixing. Pattern of lepton mixing strongly differs from that of quarks.

4. 25.09.2008S.P.Mikheyev (INR RAS)4 с ij = cos ij s ij = sin ij 1 0 0 0 с 23 s 23 0 -s 23 c 23 1 0 0 0 с 23 s 23 0 -s 23 c 23 с 13 0 s 13 e i  0 1 0 -s 13 e -i  0 c 13 с 13 0 s 13 e i  0 1 0 -s 13 e -i  0 c 13                     с 12 s 12 0 -s 12 c 12 0 0 0 1 с 12 s 12 0 -s 12 c 12 0 0 0 1 U =               3 mixing angles ( 12, 23, 13 ) Phase of CP violation (  ) Mixing matrix U can be parameterized with Pontecorvo – Maki – Nakagava -Sakata

5. 25.09.2008S.P.Mikheyev (INR RAS)5 2 U = cos  sin  -  sin  cos  ( )  e 1 2 1 2 wave packets e = cos  1  sin    = - sin  1  cos   coherent mixtures of mass eigenstates 1 = cos  e  sin   2 = sin  e  cos   flavor composition of the mass eigenstates  1 2 e 1 2 Neutrino “images”: 1 2

6. 25.09.2008S.P.Mikheyev (INR RAS)6  0  22 A 2  + A 1  0 2sin  cos  0 e 1 2 Due to difference of masses 1 and 2 have different phase velocities Oscillation depth: Oscillation length: Oscillation probability:

7. 25.09.2008S.P.Mikheyev (INR RAS)7 I. Oscillations  effect of the phase difference increase between mass eigenstates II. Admixtures of the mass eigenstates i in a given neutrino state do not change during propagation III. Flavors (flavor composition) of the eigenstates are fixed by the vacuum mixing angle Periodic (in time and distance) process of transformation (partial or complete) of one neutrino species into another one

8. 25.09.2008S.P.Mikheyev (INR RAS)8 M is the mass matrix Schroedinger’s equation Mixing matrix in vacuum

9. 25.09.2008S.P.Mikheyev (INR RAS)9 Disappearance experiments: Appearence experiment: Atmospheric neutrinos; LBL: K2K, MINOS; reactor neutrinos: KamLAND LBL: MINOS, OPERA, T2K Probability as a function of distance (atmospheric neutrinos) energy (K2K, MINOS) L/E (atmospheric neutrinos, KamLAND)

10. 25.09.2008S.P.Mikheyev (INR RAS)10 Jennifer Raaf Talk at Neutrino’2008

11. 25.09.2008S.P.Mikheyev (INR RAS)11 K2K E rec (GeV) MINOS Hugh Gallagher Talk at Neutrino’2008

12. 25.09.2008S.P.Mikheyev (INR RAS)12 Patrick Decowski Talk at Neutrino’2008 KamLAND

13. 25.09.2008S.P.Mikheyev (INR RAS)13 Neutrino interactions with matter affect neutrino properties as well as medium itself Incoherent interactionsCoherent interactions CC & NC inelastic scattering CC quasielastic scattering NC elastic scattering with energy loss CC & NC elastic forward scattering Neutrino absorption (CC) Neutrino energy loss (NC) Neutrino regeneration (CC) Potentials

14. 25.09.2008S.P.Mikheyev (INR RAS)14 Elastic forward scattering + e e,  e-e- W+W+ Z0Z0 e-e- e-e- e-e- e V = V e - V  Potential: At low energy elastic forward scattering (real part of amplitude) dominates. Effect of elastic forward scattering is describer by potential Only difference of e and  is important Unpolarized and isotropic medium:

15. 25.09.2008S.P.Mikheyev (INR RAS)15 V ~ 10 -13 eV inside the Earth at E = 10 MeV Refraction index: ~ 10 -20 inside the Earth < 10 -18 inside in the Sun ~ 10 -6 inside neutron star Refraction length:

16. 25.09.2008S.P.Mikheyev (INR RAS)16 Diagonalization of the Hamiltonian: Mixing Resonance condition Difference of the eigenvalues At resonance:

17. 25.09.2008S.P.Mikheyev (INR RAS)17 sin 2 2  m = 1 At Resonance half width: Resonance energy:Resonance density: Resonance layer: sin 2 2  m sin 2 2  = 0.08 sin 2 2  = 0.825

18. 25.09.2008S.P.Mikheyev (INR RAS)18 Pictures of neutrino oscillations in media with constant density and vacuum are identical In uniform matter (constant density) mixing is constant  m (E, n) = constant As in vacuum oscillations are due to change of the phase difference between neutrino eigenstates (Constant density) ~E/E R F (E) F 0 (E) vacuum ~E/E R F (E) F 0 (E) matter

19. 25.09.2008S.P.Mikheyev (INR RAS)19 (Non-uniform density) In matter with varying density the Hamiltonian depends on time: H tot = H tot (n e (t)) Its eigenstates, m, do not split the equations of motion θ m = θ m (n e (t)) The Hamiltonian is non-diagonal  no split of equations Transitions 1m  2m

20. 25.09.2008S.P.Mikheyev (INR RAS)20 Pictures of neutrino oscillations in media with constant density and variable density are different In uniform matter (constant density) mixing is constant  m (E, n) = constant As in vacuum oscillations are due to change of the phase difference between neutrino eigenstates In varying density matter mixing is function of distance (time)  m (E, n) = F(x) Transformation of one neutrino type to another is due to change of mixing or flavor of the neutrino eigenstates MSW effect Varying density vs. constant density

21. 25.09.2008S.P.Mikheyev (INR RAS)21 One can neglect of 1m  2m transitions if the density changes slowly enough Adiabaticity condition: External conditions (density) change slowly so the system has time to adjust itself Transitions between the neutrino eigenstates can be neglected The eigenstates propagate independently Crucial in the resonance layer: - the mixing angle changes fast - level splitting is minimal L R = L /sin2  is the oscillation length in resonance is the width of the resonance layer

22. 25.09.2008S.P.Mikheyev (INR RAS)22 Initial state: Adiabatic conversion to zero density: 1m (0)  1 2m (0)  2 Final state: Probability to find e averaged over oscillations:

23. 25.09.2008S.P.Mikheyev (INR RAS)23 Dependence on initial condition The picture of adiabatic conversion is universal in terms of variable: There is no explicit dependence on oscillation parameters, density distribution, etc. Only initial value of y 0 is important. survival probability y (distance) resonance layer production point y 0 = - 5 resonance averaged probability oscillation band y 0 < -1 Non-oscillatory conversion y 0 = -1  1 y 0 > 1 Interplay of conversion and oscillations Oscillations with small matter effect

24. 25.09.2008S.P.Mikheyev (INR RAS)24 sin 2 2  = 0.8 0.2220 200 E (MeV) (  m 2 = 8  10 -5 eV 2 ) Vacuum oscillations P = 1 – 0.5sin 2 2  Adiabatic conversion P =| | 2 = sin 2  Adiabatic edge Non - adiabatic conversion Survive probability (averged over oscillations) (0) = e = 2m  2

25. 25.09.2008S.P.Mikheyev (INR RAS)25 Both require mixing, conversion is usually accompanying by oscillations Oscillation Adiabatic conversion Vacuum or uniform medium with constant parameters Phase difference increase between the eigenstates Non-uniform medium or/and medium with varying in time parameters Change of mixing in medium = change of flavor of the eigenstates In non-uniform medium: interplay of both processes  θmθm

26. 25.09.2008S.P.Mikheyev (INR RAS)26 distance survival probability Oscillations Adiabatic conversion Spatial picture survival probability distance

27. 25.09.2008S.P.Mikheyev (INR RAS)27 J.N. Bahcall 4p + 2e - 4 He + 2 e + 26.73 MeV electron neutrinos are produced Adiabatic conversion in matter of the Sun  : (150 0) g/cc e Adiabaticity parameter  ~ 10 4

28. 25.09.2008S.P.Mikheyev (INR RAS)28 SNO Hamish Robertson Talk at Neutrino’2008

29. 25.09.2008S.P.Mikheyev (INR RAS)29 Cristano Galbiati Talk at Neutrino’2008 Cl-Ar data

30. 25.09.2008S.P.Mikheyev (INR RAS)30 Solar neutrinos vs. KamLAND Adiabatic conversion (MSW)Vacuum oscillations Matter effect dominates (at least in the HE part) Non-oscillatory transition, or averaging of oscillationsthe oscillation phase is irrelevant Matter effect is very small Oscillation phase is crucialfor observed effect Coincidence of these parameters determined from the solar neutrino data and from KamLAND results testifies for the correctness of the theory (phase of oscillations, matter potential, etc..) Adiabatic conversion formula Vacuum oscillations formula

31. 25.09.2008S.P.Mikheyev (INR RAS)31 1 0 0 0 с 23 s 23 0 -s 23 c 23 1 0 0 0 с 23 s 23 0 -s 23 c 23 с 13 0 s 13 e i  0 1 0 -s 13 e -i  0 c 13 с 13 0 s 13 e i  0 1 0 -s 13 e -i  0 c 13                     с 12 s 12 0 -s 12 c 12 0 0 0 1 с 12 s 12 0 -s 12 c 12 0 0 0 1 U =               Atmospheric neutrinos  m 2 (1.3  10 -3  3.0  10 -3 ) eV 2 Sin 2 2  > 0.9 2323  m 32 m21m21  12 Solar neutrinos  m 2 (5.4  10 -5  9.5  10 -5 ) eV 2 Sin 2 2  (0.71  0.95) Known parameters

32. 25.09.2008S.P.Mikheyev (INR RAS)32 sin 2 2 13  0.2  - CP phase Mass hierarchy Unknown parameters O. Mena and S. Parke, hep-ph/0312131 G.L. Fogli, E. Lisi, A. Marrone, A. Palazzo, A.M. Rotunno arXiv:0806.2649 Sin 2  13 = 0.016  0.010

33. 25.09.2008S.P.Mikheyev (INR RAS)33 Coincides with equation for the electron spin precession in the magnetic field Polarization vector: Evolution equation: d  d t Differentiating P and using equation of motion (  - Pauli matrices)

34. 25.09.2008S.P.Mikheyev (INR RAS)34 x y z  22 B (P-1/2) (Re e + x ) (Im e + x )

35. 25.09.2008S.P.Mikheyev (INR RAS)35 Non-uniform density: Adiabatic conversion

36. 25.09.2008S.P.Mikheyev (INR RAS)36 Non-uniform density: Adiabaticity violation

37. 25.09.2008S.P.Mikheyev (INR RAS)37 Collective effects related to neutrino self-interactions ( - scattering) e e e e b b b b Z0Z0 Z0Z0 b b e e t-channel (p) (q) elastic forward scattering e e b b u-channel (p) (q) Collective flavor transformations J. Pantaleone can lead to the coherent effect Momentum exchange  flavor exchange  flavor mixing

38. 25.09.2008S.P.Mikheyev (INR RAS)38 “Standard neutrino scenario” gives complete description of neutrino oscillation phenomena. But it tells us nothing what physics is behind of neutrino masses and mixing. New experiments will allow us to measure the 1-3 mixing, deviation of 2-3 mixing from maximal, and CP- phases, as well as hopefully to establish type of neutrino hierarchy, nature of neutrino and neutrino mass.

39. 25.09.2008S.P.Mikheyev (INR RAS)39 “Standard neutrino scenario” gives complete description of neutrino oscillation phenomena. But it tells us nothing what physics is behind of neutrino masses and mixing. New experiments will allow us to measure the 1-3 mixing, deviation of 2-3 mixing from maximal, and CP- phases, as well as hopefully to establish type of neutrino hierarchy, nature of neutrino and neutrino mass. However neutrinos gave us many puzzles in past and one can expect more in future!!!