1. 20.09.2007 S.P.Mikheyev INR RAS1 ``Mesonium and antimesonium’’ Zh. Eksp.Teor. Fiz. 33, 549 (1957) [Sov. Phys. JETP 6, 429 (1957)] translation B. Pontecorvo Right time 50 years! First paper where a possibility of neutrino mixing and oscillations was mentioned Right place

2. III International Pontecorvo Neutrino Physics School S.P. Mikheyev INR RAS

3. 20.09.2007 S.P.Mikheyev INR RAS3 III International Pontecorvo Neutrino Physics School 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

4. 20.09.2007 S.P.Mikheyev INR RAS4 III International Pontecorvo Neutrino Physics School A. Yu. Smirnov hep-ph/0702061 There are only three types of light neutrinos Their interactions are described by the Standard electroweak theory Masses and mixing are generated in vacuum

5. 20.09.2007 S.P.Mikheyev INR RAS5 III International Pontecorvo Neutrino Physics School How neutrino looks (neutrino “image”) How neutrino oscillations look (graphic representation)

6. 20.09.2007 S.P.Mikheyev INR RAS6 III International Pontecorvo Neutrino Physics School certain neutrino flavors   e   e correspond to certain charged leptons 1 2 3 (interact in pairs) Mass eigenstates Eigenstates of the CC weak interactions m1m1 m2m2 m3m3 | f    U fi | i  i mixing

7. 20.09.2007 S.P.Mikheyev INR RAS7 III International Pontecorvo Neutrino Physics School 2 U = cos  sin  -  sin  cos  ( ) e = cos  1  sin    = - sin  1  cos   1 = cos  e  sin   2 = sin  e  cos    e 1 2 1 2 wave packets 1 2 coherent mixtures of mass eigenstates flavor composition of the mass eigenstates  1 2 e 1 2 Neutrino “images”:

8. 20.09.2007 S.P.Mikheyev INR RAS8  0  22 A 2  + A 1  0 2sin  cos  0 III International Pontecorvo Neutrino Physics School e 1 2 Due to difference of masses 1 and 2 have different phase velocities Oscillation depth: Oscillation length:

9. 20.09.2007 S.P.Mikheyev INR RAS9 III International Pontecorvo Neutrino Physics School Oscillation probability: 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

10. 20.09.2007 S.P.Mikheyev INR RAS10 III International Pontecorvo Neutrino Physics School x y z  22 B (P-1/2) (Re e +  ) (Im e +  ) Evolution equation: P( e  e ) = e + e = ½(1 + cos  Z ) Analogy to equation for the electron spin precession in magnetic field

11. 20.09.2007 S.P.Mikheyev INR RAS11 III International Pontecorvo Neutrino Physics School x y z

12. 20.09.2007 S.P.Mikheyev INR RAS12 III International Pontecorvo Neutrino Physics School Matter potential Evolution equation in matter Resonance Adiabatic conversion Adiabaticity violation Survival probability Parametric enhancement of oscillations

13. 20.09.2007 S.P.Mikheyev INR RAS13 III International Pontecorvo Neutrino Physics School 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) dominate. Effect of elastic forward scattering is describer by potential Only difference of e and  is important

14. 20.09.2007 S.P.Mikheyev INR RAS14 III International Pontecorvo Neutrino Physics School  - the wave function of the system neutrino - medium H int – Hamiltonian of the weak interaction at low energy Unpolarized and isotropic medium: - neutrino velocity - vector of polarization (CC interaction with electrons) (g V = -g A = 1)

15. 20.09.2007 S.P.Mikheyev INR RAS15 III International Pontecorvo Neutrino Physics School Refraction index: V ~ 10 -13 eV inside the Earth at E = 10 MeV Refraction length: ~ 10 -20 inside the Earth < 10 -18 inside in the Sun ~ 10 -6 inside neutron star

16. 20.09.2007 S.P.Mikheyev INR RAS16 III International Pontecorvo Neutrino Physics School total Hamiltonian vacuum partmatter part

17. 20.09.2007 S.P.Mikheyev INR RAS17 III International Pontecorvo Neutrino Physics School vacuum vs. matter e  1 2 1m 2m  mm Effective Hamiltonian H vac H vac + V Eigenstates 1, 2 1m, 2m Eigenvalue H 1m, H 2m m 1 2 /2E, m 2 2 /2E Depend on n e, E Mixing angle determines flavors of eigenstatea ( f ) ( i ) ( im )  mm

18. 20.09.2007 S.P.Mikheyev INR RAS18 III International Pontecorvo Neutrino Physics School Diagonalization of the Hamiltonian: Mixing Difference of the eigenvalues At resonance: Resonance condition mixing is maximal difference of the eigenvalues is minimal level crossing

19. 20.09.2007 S.P.Mikheyev INR RAS19 III International Pontecorvo Neutrino Physics School sin 2 2  m = 1 At sin 2 2  m sin 2 2  = 0.08 sin 2 2  = 0.825 Resonance half width: Resonance energy: Resonance density: Resonance layer:

20. 20.09.2007 S.P.Mikheyev INR RAS20 III International Pontecorvo Neutrino Physics School V. Rubakov, private comm. N. Cabibbo, Savonlinna 1985 H. Bethe, PRL 57 (1986) 1271 Dependence of the neutrino eigenvalues on the matter potential (density) H 2m 1m  e sin 2 2  = 0.08 (small mixing) 2m 1m e  sin 2 2  = 0.825 (large mixing) Crossing point - resonance the level split is minimal the oscillation length is maximal For maximal mixing: n R = 0 Level crossing: H

21. 20.09.2007 S.P.Mikheyev INR RAS21 III International Pontecorvo Neutrino Physics School Oscillation length in matter: vacuum dominated matter dominated E LmLm

22. 20.09.2007 S.P.Mikheyev INR RAS22 III International Pontecorvo Neutrino Physics School 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

23. 20.09.2007 S.P.Mikheyev INR RAS23 III International Pontecorvo Neutrino Physics School  m  =   m  =  H 2 - H 1 ) L Parameters of oscillations (depth and length) are determined by mixing in matter and by effective energy split in matter sin 2 2 , L sin 2 2  m, L m Flavors of the eigenstates do not change Constant density Admixtures of matter eigenstates do not change: no 1m  2m transitions Monotonous increase of the phase difference between eigenstates Δ  m Oscillations as in vacuum e 1 2 1 2 instead of

24. 20.09.2007 S.P.Mikheyev INR RAS24 sin 2 2  = 0.08 e F(E) Detector III International Pontecorvo Neutrino Physics School Layer of matter with constant density, length L e F 0 (E) Source ~E/E R F (E) F 0 (E) thin layer L = L 0 /  ~E/E R thick layer L = 10L 0 /  Constant density: Resonance enhancement of oscillations sin 2 2  = 0.824

25. 20.09.2007 S.P.Mikheyev INR RAS25 e F(E) Detector III International Pontecorvo Neutrino Physics School e F 0 (E) Source Instantaneous density change  m =  1  m =  2 n1n1 n2n2 x y z

26. 20.09.2007 S.P.Mikheyev INR RAS26 x z y e F(E) Detector III International Pontecorvo Neutrino Physics School e F 0 (E) Source Instantaneous density change  m =  1  m =  2 n1n1 n2n2

27. 20.09.2007 S.P.Mikheyev INR RAS27 III International Pontecorvo Neutrino Physics School Instantaneous density change: parametric resonance  m =  1  m =  2 n1n1 n2n2 11 11 11 22 22 22 1 2345678........ 1 2 3 4 5 6 7 8 B1B1 B2B2 Enhancement associated to certain conditions for the phase of oscillations. Another way to get strong transition. No large vacuum mixing and no matter enhancement of mixing or resonance conversion  1 =  2 = 

28. 20.09.2007 S.P.Mikheyev INR RAS28 III International Pontecorvo Neutrino Physics School Instantaneous density change: parametric resonance  m =  1m n1n1 n2n2  m =  2m 11 22 Resonance condition: Simplest realization: In general, certain correlation between phases and mixing angles  1 =  2 = 

29. 20.09.2007 S.P.Mikheyev INR RAS29 III International Pontecorvo Neutrino Physics School 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 Non-uniform density θ m = θ m (n e (t)) The Hamiltonian is non-diagonal  no split of equations Transitions 1m  2m

30. 20.09.2007 S.P.Mikheyev INR RAS30 III International Pontecorvo Neutrino Physics School Non-uniform density: Adiabaticity One can neglect of 1m  2m transitions if the density changes slowly enough Adiabaticity condition: Adiabaticity parameter:

31. 20.09.2007 S.P.Mikheyev INR RAS31 III International Pontecorvo Neutrino Physics School Non-uniform density: Adiabaticity 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 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 Adiabaticity condition:

32. 20.09.2007 S.P.Mikheyev INR RAS32 III International Pontecorvo Neutrino Physics School Non-uniform density: Adiabatic conversion Initial state: Adiabatic conversion to zero density: 1m (0)  1 2m (0)  2 Final state: Probability to find e averaged over oscillations:

33. 20.09.2007 S.P.Mikheyev INR RAS33 III International Pontecorvo Neutrino Physics School Resonance Non-uniform density: Adiabatic conversion Admixtures of the eigenstates, 1m 2m, do not change Flavors of eigenstates change according to the density change  fixed by mixing in the production point  determined by  m Effect is related to the change of flavors of the neutrino eigenstates in matter with varying density Phase difference increases  according to the level split which changes with density

34. 20.09.2007 S.P.Mikheyev INR RAS34 III International Pontecorvo Neutrino Physics School Non-uniform density: Adiabatic conversion

35. 20.09.2007 S.P.Mikheyev INR RAS35 III International Pontecorvo Neutrino Physics School Non-uniform density: Adiabatic conversion 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

36. 20.09.2007 S.P.Mikheyev INR RAS36 13th Lomonosov Conference on Elementary Particle Physics 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 Non-uniform density: Adiabatic conversion Survive probability (averged over oscillations) (0) = e = 2m  2

37. 20.09.2007 S.P.Mikheyev INR RAS37 III International Pontecorvo Neutrino Physics School Non-uniform density: Adiabaticity violation 2m 1m nene 2 1 n 0 >> n R Resonance Fast density change Transitions 1m  2m occur, admixtures of the eigenstates change Flavors of the eigenstates follow the density change Phase difference of the eigenstates changes, leading to oscillations  = (H 1 -H 2 ) t

38. 20.09.2007 S.P.Mikheyev INR RAS38 III International Pontecorvo Neutrino Physics School Non-uniform density: Adiabaticity violation

39. 20.09.2007 S.P.Mikheyev INR RAS39 III International Pontecorvo Neutrino Physics School 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

40. 20.09.2007 S.P.Mikheyev INR RAS40 III International Pontecorvo Neutrino Physics School distance survival probability Oscillations Adiabatic conversion Spatial picture survival probability distance

41. 20.09.2007 S.P.Mikheyev INR RAS41 III International Pontecorvo Neutrino Physics School Unpolarized relativistic medium:  e   e  polarized isotropic medium: if

42. 20.09.2007 S.P.Mikheyev INR RAS42 III International Pontecorvo Neutrino Physics School The Sun The Earth Supernovae

43. 20.09.2007 S.P.Mikheyev INR RAS43 III International Pontecorvo Neutrino Physics School 4p + 2e - 4 He + 2 e + 26.73 MeV electron neutrinos are produced J.N. Bahcall Oscillations in matter of the Earth Oscillations in vacuum Adiabatic conversion in matter of the Sun  : (150 0) g/cc e Adiabaticity parameter  ~ 10 4

44. 20.09.2007 S.P.Mikheyev INR RAS44 III International Pontecorvo Neutrino Physics School Borexino Collaboration arXiv:0708.2251

45. 20.09.2007 S.P.Mikheyev INR RAS45 III International Pontecorvo Neutrino Physics School 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

46. 20.09.2007 S.P.Mikheyev INR RAS46 III International Pontecorvo Neutrino Physics School Density Profile (PREM model) mantle core

47. 20.09.2007 S.P.Mikheyev INR RAS47 III International Pontecorvo Neutrino Physics School

48. 20.09.2007 S.P.Mikheyev INR RAS48 III International Pontecorvo Neutrino Physics School Akhmedov, Maltoni & Smirnov, 2005 Liu, Smirnov, 1998; Petcov, 1998; E.Akhmedov 1998

49. 20.09.2007 S.P.Mikheyev INR RAS49 III International Pontecorvo Neutrino Physics School Supernova Neutrino Fluxes G.G. Rafelt, “Star as laboratories for fundamental physics” (1996) H.-T. Janka & W. Hillebrand, Astron. Astrophys. 224 (1989) 49

50. 20.09.2007 S.P.Mikheyev INR RAS50 III International Pontecorvo Neutrino Physics School Matter effect in Supernova Normal Hierarchy Inverted Hierarchy Dighe & Smirnov, astro-ph/9907423

51. 20.09.2007 S.P.Mikheyev INR RAS51 III International Pontecorvo Neutrino Physics School Neutrino transitions occur far outside of the star core Supernova Density Profile

52. 20.09.2007 S.P.Mikheyev INR RAS52 III International Pontecorvo Neutrino Physics School Supernova Density Profile Adiabaticity parameter: Adiabatic conversion Weak dependence on AWeak dependence on n

53. 20.09.2007 S.P.Mikheyev INR RAS53 P f = 0.9 P f = 0.1 E = 50 MeV E = 5 MeV III International Pontecorvo Neutrino Physics School Supernova Neutrino Oscillations I – Adiabatic conversion II – Weak violation of adiabaticity III – Strong violation of adiabaticity

54. 20.09.2007 S.P.Mikheyev INR RAS54 III International Pontecorvo Neutrino Physics School Original fluxes After leaving the supernova envelope for for for Normal Inverted sin 2 (2  13 ) ≲ 10 -5 ≳ 10 -3 Any Hierarchy sin 2 (Q 12 )  0.3 0 cos 2 (Q 12 )  0.7 sin 2 (Q 12 )  0.3 cos 2 (Q 12 )  0.7 0