25 09 2008 S P Mikheyev INR RAS

25. 09. 2008 S. P. Mikheyev (INR RAS) 1

Ø Introduction. Ø Vacuum oscillations. Ø Oscillations in matter. Ø Adiabatic conversion. Ø Graphical representation of oscillations Ø Conclusion 25. 09. 2008 S. P. Mikheyev (INR RAS) 2

There are only 3 types of light neutrinos: 3 flavors and 3 mass states. Their interactions are described by the Standard electroweak theory Neutrino are massive. Neutrino masses are in the sub-e. V range - much smaller than masses of charge leptons and quarks. Neutrinos mix. There are two large mixings and one small or zero mixing. Pattern of lepton mixing strongly differs from that of quarks. Masses and mixing are generated in vacuum e 1 m 2 t 3 mixing | f = U i fi| i A. Yu. Smirnov hep-ph/0702061 25. 09. 2008 S. P. Mikheyev (INR RAS) 3

Mixing matrix U can be parameterized with 3 mixing angles ( 12, 23, 13) Phase of CP violation ( ) с13 0 s 13 ei с12 s 12 0 1 0 -s 12 c 12 0 0 с23 s 23 0 0 0 1 0 -s 23 c 23 -s 13 e-i 0 c 13 U = сij = cos ij sij = sin ij 25. 09. 2008 Pontecorvo – Maki – Nakagava -Saka S. P. Mikheyev (INR RAS) 4

2 U = e = cos 1 + sin 2 ( ) cos sin - sin cos 2 = sin e + cos = - sin 1 + cos 2 1 = cos e - sin coherent mixtures of mass eigenstates flavor composition of the mass eigenstates e 2 1 m 2 1 1 wave packets 2 Neutrino “images”: e 25. 09. 2008 m 2 1 S. P. Mikheyev (INR RAS) 2 1 5

e 2 1 Due to difference of masses 2 1 and have different phase velocities A 2 + A 1 0 2 sin cos 0 Oscillation length: Oscillation depth: Oscillation probability: 25. 09. 2008 S. P. Mikheyev (INR RAS) 6

Periodic (in time and distance) process of transformation (partial or complete) of one neutrino species into effectanother of the phase one difference I. Oscillations 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 25. 09. 2008 S. P. Mikheyev (INR RAS) 7

Schroedinger’s equation M is the mass matrix Mixing matrix in vacuum 25. 09. 2008 S. P. Mikheyev (INR RAS) 8

Disappearance experiments: Atmospheric neutrinos; LBL: K 2 K, MINOS; reactor neutrinos: Kam. LAND Probability as a function of distance (atmospheric neutrinos) energy (K 2 K, MINOS) L/E (atmospheric neutrinos, Kam. LAND) Appearence experiment: LBL: MINOS, OPERA, T 2 K 25. 09. 2008 S. P. Mikheyev (INR RAS) 9

Jennifer Raaf Talk at Neutrino’ 2008 25. 09. 2008 S. P. Mikheyev (INR RAS) 10

K 2 K Hugh Gallagher Talk at Neutrino’ 2008 MINOS Erec (Ge. V) 25. 09. 2008 S. P. Mikheyev (INR RAS) 11

Kam. LAND Patrick Decowski Talk at Neutrino’ 2008 25. 09. 2008 S. P. Mikheyev (INR RAS) 12

Neutrino interactions with matter affect neutrino properties as well as medium itself Incoherent interactions CC & NC inelastic scattering CC quasielastic scattering NC elastic scattering with energy loss Coherent interactions CC & NC elastic forward scattering Potentials Neutrino absorption (CC) Neutrino energy loss (NC) Neutrino regeneration (CC) 25. 09. 2008 S. P. Mikheyev (INR RAS) 13

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 e Elastic forward scattering e. W+ e- Potential: e e, + e, Z 0 e- e- V = Ve - V Unpolarized and isotropic medium: 25. 09. 2008 S. P. Mikheyev (INR RAS) 14

V ~ 10 -13 e. V inside the Earth at E = 10 Me. V Refraction index: ~ 10 -20 inside the Earth < 10 -18 inside in the Sun ~ 10 -6 inside neutron star Refraction length: 25. 09. 2008 S. P. Mikheyev (INR RAS) 15

Diagonalization of the Hamiltonian: Mixing Difference of the eigenvalues Resonance condition At resonance: 25. 09. 2008 S. P. Mikheyev (INR RAS) 16

At sin 2 2 m = 1 sin 2 2 m sin 2 2 = 0. 08 Resonance half width: Resonance energy: sin 2 2 = 0. 825 Resonance density: Resonance layer: 25. 09. 2008 S. P. Mikheyev (INR RAS) 17

(Constant density) 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 F (E) F 0(E) vacuum 25. 09. 2008 ~E/ER matter S. P. Mikheyev (INR RAS) ~E/ER 18

(Non-uniform density) In matter with varying density the Hamiltonian depends on time: Htot = Htot(ne(t)) Its eigenstates, m, do not split the equations of motion θm= θm(ne(t)) The Hamiltonian is non-diagonal no split of equations Transitions 25. 09. 2008 1 m 2 m S. P. Mikheyev (INR RAS) 19

Varying density vs. constant density 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) MSW effect Transformation of one neutrino type to another is due to change of mixing or flavor of the neutrino eigenstates 25. 09. 2008 S. P. Mikheyev (INR RAS) 20

One can neglect of 1 m 2 m transitions if the density changes slowly enough External conditions (density) change slowly so the system has time to adjust itself Adiabaticity condition: 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 25. 09. 2008 S. P. Mikheyev (INR RAS) LR = L /sin 2 is the oscillation length in resonance is the width of the resonance layer 21

Initial state: Adiabatic conversion to zero density: 1 m(0) 1 2 m(0) 2 Final state: Probability to find e averaged over oscillations: 25. 09. 2008 S. P. Mikheyev (INR RAS) 22

Dependence on initial condition survival probability The picture of adiabatic conversion is universal in terms of variable: resonance layer There is no explicit dependence on oscillation parameters, density distribution, etc. Only initial value of y 0 is important. production point y 0 = - 5 y 0 < -1 oscillation band resonance y 0 = -1 1 averaged probability y 0 > 1 25. 09. 2008 y (distance) S. P. Mikheyev (INR RAS) Non-oscillatory conversion Interplay of conversion and oscillations Oscillations with small matter effect 23

Survive probability (averged over oscillations) sin 22 = 0. 8 Vacuum oscillations P = 1 – 0. 5 sin 22 (0) = e = 2 m 2 Adiabatic edge Non - adiabatic conversion Adiabatic conversion P =|< e| 2>|2 = sin 2 0. 2 2 20 200 E (Me. V) ( m 2 = 8 10 -5 e. V 2) 25. 09. 2008 S. P. Mikheyev (INR RAS) 24

Both require mixing, conversion is usually accompanying by oscillations Adiabatic conversion Oscillation Vacuum or uniform medium with constant parameters Non-uniform medium or/and medium with varying in time parameters Phase difference increase between the eigenstates Change of mixing in medium = change of flavor of the eigenstates θm In non-uniform medium: interplay of both processes 25. 09. 2008 S. P. Mikheyev (INR RAS) 25

Spatial picture survival probability Oscillations 25. 09. 2008 survival probability Adiabatic conversion distance S. P. Mikheyev (INR RAS) 26

4 p + 2 e- + 2 e + 26. 73 Me. V electron neutrinos are produced Adiabatic conversion in matter of the Sun r : (150 4 He J. N. Bahcall 0) g/cc e Adiabaticity parameter ~ 104 25. 09. 2008 S. P. Mikheyev (INR RAS) 27

SNO 25. 09. 2008 S. P. Mikheyev (INR RAS) Hamish Robertson Talk at Neutrino’ 2008 28

Cl-Ar data 25. 09. 2008 Cristano Galbiati Talk at Neutrino’ 2008 S. P. Mikheyev (INR RAS) 29

Solar neutrinos vs. Kam. LAND Adiabatic conversion (MSW) Vacuum oscillations Matter effect dominates (at least in the HE part) Matter effect is very small Non-oscillatory transition, or averaging of oscillationsthe oscillation phase is irrelevant Oscillation phase is crucialfor observed effect Adiabatic conversion formula Vacuum oscillations formula Coincidence of these parameters determined from the solar neutrino data and from Kam. LAND results testifies for the correctness of theory (phase of oscillations, matter potential, etc. . ) 25. 09. 2008 S. P. Mikheyev (INR RAS) 30

Known parameters Solar neutrinos m 221 (5. 4 10 -5 9. 5 10 -5) e. V 2 Sin 22 12 (0. 71 0. 95) с13 0 s 13 ei с12 s 12 0 1 0 -s 12 c 12 0 0 с23 s 23 0 0 0 1 0 -s 23 c 23 -s 13 e-i 0 c 13 U = 25. 09. 2008 Atmospheric neutrinos 2 m 32 (1. 3 10 -3 3. 0 10 -3) e. V 2 Sin 22 23 > 0. 9 S. P. Mikheyev (INR RAS) 31

sin 22 13 0. 2 Unknown parameters - CP phase Mass hierarchy O. Mena and S. Parke, hep-ph/0312131 Sin 2 13 = 0. 016 0. 010 G. L. Fogli, E. Lisi, A. Marrone, A. Palazzo, A. M. Rotunno ar. Xiv: 0806. 2649 25. 09. 2008 S. P. Mikheyev (INR RAS) 32

Polarization vector: ( - Pauli matrices) Evolution equation: d. Y dt Differentiating P and using equation of motion Coincides with equation for the electron spin precession in the magnetic field 25. 09. 2008 S. P. Mikheyev (INR RAS) 33

z (P-1/2) B 2 x (Re e+ x) y (Im e+ x) 25. 09. 2008 S. P. Mikheyev (INR RAS) 34

Non-uniform density: 25. 09. 2008 Adiabatic conversion S. P. Mikheyev (INR RAS) 35

Non-uniform density: Adiabaticity violation 25. 09. 2008 S. P. Mikheyev (INR RAS) 36

Collective effects related to neutrino self-interactions ( - scattering) e e e Z 0 b t-channel b b u-channel e (q) b (q) elastic forward scattering e (p) b b Momentum exchange flavor mixing Collective flavor transformations J. Pantaleone 25. 09. 2008 b e e (p) e b Z 0 S. P. Mikheyev (INR RAS) can lead to the coherent effect 37

“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 CPphases, as well as hopefully to establish type of neutrino hierarchy, nature of neutrino and neutrino mass. 25. 09. 2008 S. 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 CPphases, as well as hopefully to establish type of neutrino hierarchy, nature of neutrino and neutrino mass. 25. 09. 2008 However neutrinos gave us many puzzles in past and one can expect more in future!!! S. P. Mikheyev (INR RAS) 39