Oscillations and propagation of Neutrinos through Magnetized GRB Fireball Yong-Yeon Keum IEU, Ewha Womans University GRB2010, Kyoto in Japan April 18, 2010 Collaboration with S. Sahu and N. Fraija: PRD 80, 033009 (2009), arXiv: 0909.3003(JCAP,2009) April 18, 2010 (page 1) Yong-Yeon Keum Oscillations and propagation of Neutrinos through Magnetized GRB Fireball seminar IEU, Ewha Womans University
Magentic Filed in the GRBs Questions of Magnetic Field in the GRBs: • There is no way to get the magnetic field information directly from the fireball. • It is strongly believed in the GRB community that the non-thermal γ -rays which we detect are mostly due to the sychrotron radiation of charged particles in the magnetic field although the strength of it is still unkonwn. • Large magnetic field is expected if the progenitor is highly magnetized. Also amplification of small field due to turbulent dynamo mechanism, compression or shearing. Decreasing of magnetic field due to the expansion at large radii ( B ( r ) = B 0 /r 2 ) • Here we take the weak field approximation and study the oscillation of neutrinos in the fireball environment. • The fireball model explains the temporal strucutre of the bursts and nonthermal spctral behavior. • Expanding fireball runs into the surrounding ISM to give afterglow. April 18, 2010 (page 2) Yong-Yeon Keum Oscillations and propagation of Neutrinos through Magnetized GRB Fireball seminar IEU, Ewha Womans University
Neutrino Oscillations in the Fireball: Sources for the Neutrinos: • Neutrinos of about 5-20 MeV are generated due to the stellar collapse or merger of compact binaries that trigger the burst. • From nucleonic bremsstrahlung: N + N − → N + N + ν + ¯ ν From e + e − annihilation : e + + e − − → ν + ¯ ν • In the fireball: p + e − − → n + ν e • All these neutrinos have the energy in the MeV range and will propagate though the fireball. Effect of Heat Bath on Particle Properties: • Massive Photons (Plasmon) • Plasmon decay: γ L → ν ¯ ν • Massless neutrino acquires an effective mass • MSW effect of neutrinos–flavor conversion • Modification of dispersion relation in the medium with and without magnetic field: p 2 − m 2 � = 0 April 18, 2010 (page 3) Yong-Yeon Keum Oscillations and propagation of Neutrinos through Magnetized GRB Fireball seminar IEU, Ewha Womans University
Neutrino Oscillations in the Fireball: Effective Potential for the Neutrinos: W ( k − p ) Z ( k − p ) ν e ( k ) e ( p ) ν e ( k ) ν e ( k ) ν e ( p ) ν e ( k ) (a) (b) f ( q ) Z ν e ( k ) ν e ( k ) (c) Figure 1: One-loop diagrams for the neutrino self-energy in a medium. April 18, 2010 (page 4) Yong-Yeon Keum Oscillations and propagation of Neutrinos through Magnetized GRB Fireball seminar IEU, Ewha Womans University
Neutrino Self-Energy-I: The total self-energy of neutrino in a magnetized medium: Σ( k ) = Σ W ( k ) + Σ Z ( k ) + Σ t ( k ) . (1) with d 4 p � − ig � � − ig � � γ ν L iW µν ( q ) , √ √ − i Σ W ( k ) = γ µ L iS ℓ ( p ) (2) (2 π ) 4 2 2 d 4 p � � � � − ig − ig � γ ν L iZ µν ( q ) , − i Σ Z ( k ) = √ γ µ L iS ν ℓ ( p ) √ (3) (2 π ) 4 2 cos θ W 2 cos θ W and d 4 p � 2 g � � R γ µ iZ µν (0) − i Σ t ( k ) = − (2 π ) 4 Tr [ γ ν ( C V + C A γ 5 ) iS ℓ ( p )] . (4) 2 cos θ W W-boson diagram contributions: � � Re Σ W ( k ) = R a W ⊥ / k ⊥ + b W / u + c W / b L, (5) √ 2 G F �� � E ν e ( N e − ¯ e − ¯ N e ) + k 3 ( N 0 N 0 a W ⊥ = − e ) M 2 W � ∞ ∞ � m 2 + eB − H � � ( f e,n + ¯ e � dp 3 (2 − δ n, 0 ) f e,n ) , (6) 2 π 2 E n E n 0 n =0 April 18, 2010 (page 5) Yong-Yeon Keum Oscillations and propagation of Neutrinos through Magnetized GRB Fireball seminar IEU, Ewha Womans University
Neutrino Self-Energy-II: b W 0 + ˜ b W = b W � eB √ m 2 + E 2 1 + 3 + E ν e k 3 �� � � ( N e − ¯ e − ¯ e ν e ( N 0 N 0 = 2 G F N e ) + e ) M 2 M 2 M 2 M 2 2 W W W W � ∞ ∞ E n + m 2 eB � � �� � ( f e,n + ¯ e � − dp 3 (2 − δ n, 0 ) 2 k 3 E n δ n, 0 + 2 E ν e f e,n ) 2 π 2 M 2 2 E n 0 W n =0 c W = c W 0 + ˜ c W � eB √ m 2 − k 2 1 + 1 − E ν e k 3 �� � � e − ¯ ( N e − ¯ e 3 ( N 0 N 0 = 2 G F e ) + N e ) M 2 M 2 M 2 M 2 2 W W W W � ∞ ∞ E n − m 2 eB � � � e � − dp 3 (2 − δ n, 0 ) 2 E ν e δ n, 0 2 π 2 M 2 2 E n 0 W n =0 m 2 E n − 3 − H � �� � ( f e,n + ¯ e +2 k 3 f e,n ) . 2 E n E n April 18, 2010 (page 6) Yong-Yeon Keum Oscillations and propagation of Neutrinos through Magnetized GRB Fireball seminar IEU, Ewha Womans University
Neutrino Self-Energy-III: Z-boson diagram contributions: Re Σ Z ( k ) = R ( a Z / k + b Z / u ) L, (8) where √ � E ν e N ν e ) + 2 1 � �� ( N ν e − ¯ � E ν e � N ν e + � ¯ E ν e � ¯ a Z = 2 G F N ν e , (9) M 2 M 2 3 Z Z and √ N ν e ) − 8 E ν � � �� ( N ν e − ¯ � E ν e � N ν e + � ¯ E ν e � ¯ b Z = 2 G F N ν e . (10) 3 M 2 Z Tadpole diagram contributions: √ �� C V e ( N e − ¯ N e ) + C V p ( N p − ¯ N p ) + C V n ( N n − ¯ N n ) + ( N ν e − ¯ Re Σ t ( k ) = 2 G F R N ν e ) � � +( N ν µ − ¯ N ν µ ) + ( N ν τ − ¯ e − ¯ u − C A e ( N 0 N 0 N ν τ ) / e ) / b L. (11) April 18, 2010 (page 7) Yong-Yeon Keum Oscillations and propagation of Neutrinos through Magnetized GRB Fireball seminar IEU, Ewha Womans University
Neutrino Oscillations in the Fireball: Effective Potential for the Neutrinos: In a relativistic and non-degenerate e + − e − plasma, • the effective potential for the (anti-)electron neutrino is: (if the fireball is charge neutral: L e = L p ): � 2 T 2 √ � 7 ξ (4) ± L e ∓ 1 . V ν e , ¯ ν e = 2 G F N γ 2 L n − (12) M 2 ξ (3) W • For the muon (tau)-neutrinos: √ V ν µ,τ ≃ 2 G F N γ L µ,τ . (13) where the particle asymmetry is defined as L i ≡ N i − ¯ N i N γ April 18, 2010 (page 8) Yong-Yeon Keum Oscillations and propagation of Neutrinos through Magnetized GRB Fireball seminar IEU, Ewha Womans University
Neutrino Oscillations in the Fireball: Effective Potential for the Neutrinos: Including the (weak) magnetic field contributions ( B << m 2 /e = B c ≃ 10 13 G ), The effective potentials is given: � m � 2 E ν e √ m 3 � � Φ 1 − Φ 2 − 4 V = V ν e − V ν µ ≃ 2 G F m (Φ 3 − Φ 4 ) . (14) π 2 π 2 M W e = m 3 ∞ ( − 1) l sinh α K 1 ( σ ) = m 3 B e − ¯ N 0 N 0 � π 2 Φ 1 , (15) π 2 B c l =0 � 2 N e = m 3 ∞ = m 3 σK 2 ( σ ) − B � ( − 1) l sinh α N e − ¯ � K 1 ( σ ) π 2 Φ 2 , (16) π 2 B c l =0 �� 3 � K 1 ( σ ) ∞ � σ 2 − 1 B 1 + 6 � � ( − 1) l cosh α � Φ 3 = K 0 ( σ ) + , (17) σ 2 4 B c σ l =0 ∞ ( − 1) l cosh α 1 K 0 ( σ ) + 2 � � � Φ 4 = σK 1 ( σ ) , (18) σ 2 l =0 April 18, 2010 (page 9) Yong-Yeon Keum Oscillations and propagation of Neutrinos through Magnetized GRB Fireball seminar IEU, Ewha Womans University
Neutrino Oscillations ν e ↔ ν µ,τ : Neutrino Oscillation ν e ↔ ν µ,τ (I): The evolution equation for the propagation of neutrinos is: � ˙ �� ν e � ∆ V − ∆ cos 2 θ 2 sin 2 θ ν e � � i = , ∆ ν µ ˙ 2 sin 2 θ 0 ν µ ∆ = δm 2 / 2 E ν , V = V ν e − V ν µ , E ν is the neutrino energy (5-20 MeV)and θ is the neutrino mixing angle. The conversion probability at a given time t is given by P ν e → ν µ ( ν τ ) ( t ) = ∆ 2 sin 2 2 θ � ωt � sin 2 , (19) ω 2 2 with � ( V − ∆ cos 2 θ ) 2 + ∆ 2 sin 2 2 θ. ω = (20) the effective potential for the (anti-)electron neutrino is: The oscillation length for the neutrino is given by L v L osc = , (21) ∆ cos 2 θ ) 2 + sin 2 2 θ � cos 2 2 θ (1 − V where L v = 2 π/ ∆ is the vacuum oscillation length. April 18, 2010 (page 10) Yong-Yeon Keum Oscillations and propagation of Neutrinos through Magnetized GRB Fireball seminar IEU, Ewha Womans University
Neutrino Oscillations ν e ↔ ν µ,τ : Neutrino Oscillation ν e ↔ ν µ,τ (II): At Resonance, V = ∆ cos 2 θ. (22) The resonance condition is ˜ δm 2 Φ 1 − Φ 2 − 3 . 196 × 10 − 11 E MeV (Φ 3 − Φ 4 ) = 2 . 26 cos 2 θ, (23) E MeV The left had side depends on the chemical potential µ of the background electrons and positrons, temperature T of the plasma and the neutrino energy. On the other hand the right hand side depends only on the neutrino energy (for a given set of neutrino mass square difference and the mixing angle). The resonance condition can be written as δm 2 cos 2 θ ˜ L e T 3 MeV = 0 . 124 (24) E MeV Baryon Load: M b ∼ 2 . 23 × 10 − 4 R 3 7 T 3 MeV L e M ⊙ . (25) April 18, 2010 (page 11) Yong-Yeon Keum Oscillations and propagation of Neutrinos through Magnetized GRB Fireball seminar IEU, Ewha Womans University
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