Шестые Марковские чтения (К 100-летию со дня рождения М.А.Маркова) 14 – 16 мая 2008 года Москва - Дубна - Москва

С.П.Михеев (ИЯИ РАН), А.Ю.Смирнов (ICTP и ИЯИ РАН) Шестые Марковские чтения (К 100-летию со дня рождения М.А.Маркова) 14 – 16 мая 2008 года

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 3 Шестые Марковские чтения Preliminary remarks. Matter effect. Oscillations in matter. Adiabatic conversion. Propagation neutrinos in real media: solar neutrinos; supernova neutrinos; high energy neutrinos from hidden sources; atmospheric neutrinos in the Earth.

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 4 A. Yu. Smirnov hep-ph/ There are only three types of light neutrinos Their interactions are described by the Standard electroweak theory Masses and mixing are generated in vacuum Шестые Марковские чтения ( "Standard model of neutrino ) e | f U fi | i i mixing Neutrino are massive and mixed

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 5 2 U = cos sin - sin cos ( ) e 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 Шестые Марковские чтения

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 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: Шестые Марковские чтения Propagation in vacuum

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & 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 Oscillation probability: Шестые Марковские чтения Propagation in vacuum

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 8 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 Шестые Марковские чтения Propagation in matter

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 9 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 Шестые Марковские чтения Unpolarized and isotropic medium: Potential

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 10 V ~ eV inside the Earth at E = 10 MeV Refraction index: ~ inside the Earth < inside in the Sun ~ inside neutron star Refraction length: Шестые Марковские чтения Potential

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 11 total Hamiltonian vacuum partmatter part Шестые Марковские чтения Evolution equation

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 12 vacuum vs. matter e 1 2 1m 2m m Effective Hamiltonian H vac H vac + V Eigenstates 1, 2 1m, 2m Eigenvalue H 1m, H 2m Depend on n e, E Mixing angle determines flavor of eigenstates ( f ) ( i ) ( im ) m Шестые Марковские чтения Evolution equation

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 13 Diagonalization of the Hamiltonian: Mixing Difference of the eigenvalues At resonance: Resonance condition mixing is maximal difference of the eigenvalues is minimal level crossing Шестые Марковские чтения Evolution equation

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 14 V. Rubakov, private comm. N. Cabibbo, Savonlinna 1985 H. Bethe, PRL 57 (1986) 1271 Dependence of the neutrino eigenvalues on the matter potential (density) Crossing point - resonance the level split is minimal the oscillation length is maximal For maximal mixing: n R = 0 Level crossing: 2m 1m e sin 2 2 = 0.08 (small mixing) H Шестые Марковские чтения Resonance 2m 1m e sin 2 2 = (large mixing) H

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 15 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 = Шестые Марковские чтения Resonance

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 16 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 Шестые Марковские чтения Constant density ~E/E R F (E) F 0 (E) vacuum ~E/E R F (E) F 0 (E) matter

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 17 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 Шестые Марковские чтения Non-uniform density

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 18 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 Шестые Марковские чтения Varying density vs. constant density

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 19 Adiabaticity condition One can neglect of 1m 2m transitions if the density changes slowly enough Adiabaticity condition: Adiabaticity parameter: Шестые Марковские чтения

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 20 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: Шестые Марковские чтения Adiabatic conversion

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 21 Initial state: Adiabatic conversion to zero density: 1m (0) 1 2m (0) 2 Final state: Probability to find e averaged over oscillations: Шестые Марковские чтения Adiabatic conversion

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 22 Resonance 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 Шестые Марковские чтения Adiabatic conversion

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & 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 Шестые Марковские чтения Adiabatic conversion

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 24 sin 2 2 = E (MeV) ( m 2 = 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 Шестые Марковские чтения Adiabatic conversion

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 25 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 Шестые Марковские чтения

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 26 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 Шестые Марковские чтения

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 27 distance survival probability Oscillations Adiabatic conversion Spatial picture survival probability distance Шестые Марковские чтения

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 28 J.N. Bahcall Oscillations in matter of the Earth Oscillations in vacuum 4p + 2e - 4 He + 2 e MeV electron neutrinos are produced Adiabatic conversion in matter of the Sun : (150 0) g/cc e Adiabaticity parameter ~ 10 4 Шестые Марковские чтения Solar neutrinos

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 29 Borexino Collaboration arXiv: Шестые Марковские чтения Solar neutrinos

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & 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 Шестые Марковские чтения Solar neutrinos

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 31 с ij = cos ij s ij = sin ij с 23 s s 23 c с 23 s s 23 c 23 с 13 0 s 13 e i s 13 e -i 0 c 13 с 13 0 s 13 e i s 13 e -i 0 c 13 с 12 s s 12 c с 12 s s 12 c U = Atmospheric neutrinos m 2 ( ) eV 2 Sin 2 2 > m 32 m Solar neutrinos m 2 ( ) eV 2 Sin 2 2 ( ) Шестые Марковские чтения 2 mass differences ( ) 3 mixing angles ( 12, 23, 13 ) Phase of CP violation ( )

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 32 sin CP phase Mass hierarchy Unknown parameters O. Mena and S. Parke, hep-ph/ Шестые Марковские чтения

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 33 Fluxes: G.G. Rafelt, Star as laboratories for fundamental physics (1996) H.-T. Janka & W. Hillebrand, Astron. Astrophys. 224 (1989) 49 Шестые Марковские чтения Supernova neutrinos

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 34 Matter effect in Supernova Normal Hierarchy Inverted Hierarchy Dighe & Smirnov, astro-ph/ Шестые Марковские чтения Supernova neutrinos

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 35 Neutrino transitions occur far outside of the star core Шестые Марковские чтения Supernova neutrinos Density Profile:

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 36 Density Profile: Adiabaticity parameter: Adiabatic conversion Weak dependence on AWeak dependence on n Шестые Марковские чтения Supernova neutrinos

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 37 P f = 0.9 P f = 0.1 E = 50 MeV E = 5 MeV Oscillations I – Adiabatic conversion II – Weak violation of adiabaticity III – Strong violation of adiabaticity Шестые Марковские чтения Supernova neutrinos

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 38 After leaving the supernova envelope Original fluxes for for for Normal Inverted sin 2 (2 13 ) Any Hierarchy sin 2 ( 12 ) cos 2 ( 12 ) 0.7 sin 2 ( 12 ) 0.3 cos 2 ( 12 ) Шестые Марковские чтения Supernova neutrinos

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 39 Шестые Марковские чтения Neutrinos from hidden sources O. Mena, I. Mocioiu & S. Razzaque, Phys.Rev. D75 (2007) Neutrino fluxes: Pion, kaon and muon decay neutrinos from hadronic interactions e : : = 1:2:0 (initial) e : : = 1:1:1 (at the Earth) vacuum oscillations

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 40 Шестые Марковские чтения Neutrinos from hidden sources R 0 – stars radius n = 3 Density Profile: Adiabaticity parameter >> 1 up to E ~ 1 TeV O. Mena, I. Mocioiu & S. Razzaque, Phys.Rev. D75 (2007)

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 41 Шестые Марковские чтения Neutrinos from hidden sources O. Mena, I. Mocioiu & S. Razzaque, Phys.Rev. D75 (2007) Observable: normal Hierarchy: invert

S.P.Mikheyev (INR RAS) and A.Yu.Smirnov (ICTP & INR RAS) 42 Шестые Марковские чтения Atmospheric neutrinos; the Earth E. Akhmedov, M. Maltoni & A. Smirnov, arXiv: v3 [hep-ph] Contours of equal osc. probabilities in (,E ) plane Neutrino oscillograms of the Earth Density Profile (PREM model)