NEUTRINO OSCILLATIONS IN EXTERNAL FIELDS IN CURVED SPACE-TIME. Maxim Dvornikov University of São Paulo, Brazil IZMIRAN, Russia

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1 NEUTRINO OSCILLATIONS IN EXTERNAL FIELDS IN CURVED SPACE-TIME Maxim Dvornikov University of São Paulo, Brazil IZMIRAN, Russia

2 Outline Brief review of the neutrino properties Neutrino interaction with matter Neutrino spin precession in Minkowski space Generalized equation for the neutrino spin evolution in matter and electromagnetic field in curved space-time Spin oscillations in Schwarzschild and Kerr metrics Oscillations of UHE neutrinos in realistic accretion disk Summary

3 References M. Dvornikov, Neutrino spin oscillations in matter under the influence of gravitational and electromagnetic fields, JCAP 06 (013) 015, arxiv: [hep-ph]. M. Dvornikov, Neutrino spin oscillations in gravitational fields, Int. J. Mod. Phys. D 15, 1017 (006), hep-ph/ M. Dvornikov and A. Studenikin, Neutrino spin evolution in presence of general external fields, JHEP 09 (00) 016, hep-ph/00113.

4 Some facts about neutrinos Neutrinos are massive and mixed particles (Daya Bay 01, Double Chooz 01, RENO 01) The absolute values of the neutrino masses are unknown (Troitsk 011: m ν <. ev) It is unclear whether neutrinos are Dirac of Majorana particles (GERDA 013, EXO 01) Although neutrinos are electrically neutral particles, they can interact with an electromagnetic field owing to their magnetic moments (GEMMA 01: µ ν < µ B )

5 Why do we need to study the neutrino spin evolution (spin oscillations)? Standard model neutrinos are massless and maintain their polarization: neutrinos are left-polarized and anti-neutrinos are rightpolarized. If neutrinos are massive they can change their polarization in external fields (e.g., in a magnetic field). The change of the neutrino polarization will result in the effective reduction of the neutrino flux since right-polarized neutrinos interact much weaker with background particles in a detector. The observation of neutrino spin oscillations will be the indication of the nonzero neutrino mass and magnetic moment. We shall study the evolution of the neutrino polarization within one neutrino eigenstate (ν αl è ν αr ) assuming that other channels of neutrino oscillations (e.g., neutrino flavor oscillation ν αl è ν βl, where polarization is conserved) are suppressed.

6 Interaction of neutrinos with background matter Charged currents interaction ( CC) GF 5 5 Leff = νγ µ e (1 γ ) ν e eγµ (1 γ ) e Neutral currents interaction ( NC G ) F 5 ( ( f (1 ) ) 5 ( f L ) eff = νγ µ α γ ν α fγµ I3L (1 γ) Q sin θw ) f After the averaging over the background fermions, we get: G L G G q J q F µ 5 (1) ( f ) () ( f ) eff = νγ α (1 γ) να µ, µ = f µ f µ, + Λ f = e, p, n U( ζu) J nu n U U 1+ U µ µ µ µ 0 = 0, Λ = 0 ( ζu), ζ +, = (, U) 0

7 Relativistic spin operator We shall study a Dirac neutrino with mass m and energy E propagating in background matter and interacting with an external electromagnetic field F µν = (E,B) Relativistic spin operator Dirac equation in presence of external fields Heisenberg equation for spin operator Elimination of zitterbewegung (Schrödinger 1930) Averaging over the neutrino state ˆ 0 5 p 0 p( Σp) O = γ Σ + γ γ E E( E+ m) ν i = Hˆν Hˆ = Hˆ + Hˆ + Hˆ Hˆ = ˆ + m t ˆ GF 0 µ 5 ˆ µ 0 µν Hmat = γγ (1 γ) Gµ, Hem = γσ Fµν 0, 0 mat em, 0 ( αp) γ, doˆ dt = i Hˆ, ˆ O doˆ 1 doˆ, Hˆ 0 dt E dt ( ζ p ) p ( ζ p ) ˆ,, µ O = ζ S = ζ+ m m( m+ E), +

8 Spin dynamics in flat space-time Quasi-classical evolution equation for the 3-vector of the neutrino spin in in matter and electromagnetic field The effective Schrödinger equation can be associated with the dynamics of 3- vector of spin This equation can be rewritten in the covariant form (MD & Studenikin 00) dζ = ( ζ Ω), dt µ G Ω= B + M γ γ i ν t = Ĥ eff ν, ν = Ĥ eff = ( σ Ω) ds dτ + µ F 0 0 ν R ν L, ( µν µ νλ F S U U F S ) ν ν λ = µ G F ε µνλρ GU S ν λ ρ

9 Spinning particles in curved space-time µν DS µ ν ν µ = pv pv, Dτ µ Dp 1 µ ν ρσ = R νρσ v S, Dτ 1 Sρ = gε µνλρp S m µ νλ Papapetrou (1951) eqns for a spinning body in a gravitational field. Here v µ is the unit tangent vector to the center-of-mass world line and R µ νρσ is Riemann tensor. The motion of a spinning body deviates from geodesics. Rietdijk and van Holten (1993) showed that for point like particles this deviation is negligible. We get the new spin evolution equation for a neutrino interacting with background matter under the influence of electromagnetic and gravitational fields (MD 013): DS Dτ µν µ νλ ε DU = µ ( F Sν U UνF Sλ) + GF GU ν λsρ, = 0 g Dτ µ µνλρ µ

10 Spin evolution in vierbein frame Metric is Minkowskian in the vierbein frame All the objects should be transformed to the vierbein frame ds dt a a b gµν = eµ eνηab, ηab = diag( + 1, 1, 1, 1) s a = e µ a S µ, u a = e µ a U µ, Neutrino spin evolution equation in the vierbein frame: a 1 ab ab a bc abcd du 1 ab = G s µ ( f s u u f s ) G ε g u s, G u γ + + = dt γ b b b c F b c d b G u e e e U f c d 0 ab = γabc = ( eg, bg ), γabc = η µ ν ad µν ; b c, γ =, ab = ( e, b) Evolution of the three vector of the neutrino spin: ( gu ) uub ( ) dζ G F 0 0 = ( ζ Ω), Ω= g 0 ( g ) g µ u 0 0 ( ) dt γ b + e u u u g b e u 1+ u 1+ u

11 Schwarzschild metric r g r g d τ = 1 dt 1 dr r d sin d r r θ + θ φ 1 ( ) We shall study a circular orbit with the radius R. In this case a neutrino moves along the third axis line in the vierbein frame. The spin rotates around the second axis. ( ) PL R( t) = sin Ωt vφ Ω =, Ω 1 =Ω 3 = 0, γ v φ 1/ 1/ rg 3rg =, γ 1 3 = R R

12 Kerr metric dτ = 1 rr g Σ dt + rr g sin θ dtdφ Σ Σ Δ dr Σdθ Ξ Σ sin θdφ, Δ = r rr g + a, Σ = r + a cos θ, Ξ = ( r + a )Σ + rr g a sin θ We shall take that BH is surrounded by an accretion disk. The asymptotic value of the magnetic field, B 0, is constant and directed along the rotation axis of BH. Then (Wald 1974): rrg B rr 0 ga Fµν = µ Aν µ Aν, At = B0a 1 ( 1+ cos θ), Aφ = r + a ( 1+ cos θ) sin θ Σ Σ Circular neutrino motion Ω ( ), P(t) = Ω + Ω sin Ω + Ω 3 t 3 Ω = 1 γ r g Ω 3 = G F n eff γ 1 x µb r x (x 1) ± α x(x 1) + α 3/ 0 g, x 3/ x 3 3x ± α x 3/ ±U 0 φ f r g U f ( x 3/ ± α ), x = R, α = a x 3 3x ± α x 3/ r g r g

13 Radial propagation of UHE neutrinos Because of the symmetry reasons a purely radial motion in Kerr metric is possible only along the rotation axis of BH. If U 0 >> 1 (UHE neutrinos), the approximate treatment of spin oscillations is possible. We shall study the motion in the equatorial plane. A neutrino moves along the first axis in the vierbein frame. dν x 1 1 i = rg( σω ) ν, νr, L =, dx x 1 ± 1 Gn F eff 0 1 φ α Ω 1 = U f 1 + ru g f, x x 1 α x 3 Ω = µ B 1 +, x 4r x x 1 0 7/ g Gn Ω = α x F eff 3 1 x 0 1 U f, Gravity does not produce a sizable effect on neutrino spin oscillations

14 Interaction of UHE neutrinos with a Motivation relativistic accretion disk Deficit of UHE ν- s was reported by IceCube (01). In 013 some UHE ν e -s were observed by IceCube. Still there is a lack of signal for UHE ν µ,τ -s. Barranco et al. (01) : neutrino spin oscillations in strong magnetic field è the neutrino flux is reduced Input data UHE ν-s emitted in GRB Dipole magnetic field: B 0 = 10 1 G / x 3 Magnetic moment: µ = 10-1 µ B (Kuznetsov et al. 009) Accretion disk density: ρ = 10 g/cm 3 (MacFadyen & Woosley 1999) Result: Spin oscillations cannot explain the deficit of UHE ν-s. The neutrino interaction the a realistic relativistic accretion disk will suppress the transition probability even if magnetic field is strong.

15 Summary The new covariant equation for neutrino spin evolution in matter and electromagnetic field in Minkowski space was derived. For the first time this equation was generalized to include the effects of gravity. Neutrino spin oscillations in Schwarzschild and Kerr metrics were described. Spin oscillations of UHE neutrinos in dense magnetized relativistic accretion disk were studied. It was shown that spin oscillations cannot solve the problem of the UHE neutrinos deficit.

16 Acknowledgments To the organizers of IWARA 013 for the invitation. To FAPESP (Brazil) for the financial support.