Coherence of Supernova Neutrinos Jörn Kersten University of Hamburg Based on work done in collaboration with A.Yu. Smirnov Jörn Kersten (Uni Hamburg) Coherence of Supernova Neutrinos 1 / 13
Neutrino Sources Jörn Kersten (Uni Hamburg) Coherence of Supernova Neutrinos 2 / 13
Neutrino Oscillations and Decoherence Normal (coherent) 2-flavor oscillation probability P ( ν e → ν e ) = | cos 2 θ + sin 2 θ e i φ | 2 = 1 − sin 2 2 θ sin 2 ∆ m 2 L 4 E Oscillation phase φ = − ∆ m 2 L 2 E Mass eigenstates have different velocities � Wave packets cease to overlap ν 1 ν 1 ν 2 ν 2 � Probability is incoherent sum P ( ν e → ν e ) = | cos 2 θ | 2 + | sin 2 θ e i φ | 2 = cos 4 θ + sin 4 θ � No oscillations between supernova and Earth Jörn Kersten (Uni Hamburg) Coherence of Supernova Neutrinos 3 / 13
Observable Effects with Supernova Neutrinos Oscillations inside the Earth MSW and collective effects inside the supernova Initial neutrino and antineutrino fluxes Final fluxes in inverted hierarchy (single-angle) 1.0 1.0 ν e 0.8 0.8 ν Flux (a.u.) 0.6 0.6 x − → 0.4 0.4 ν ν x 0.2 0.2 e 0.0 0.0 0 10 20 30 40 50 0 10 20 30 40 50 E (MeV) E (MeV) Is coherence preserved in these cases? Jörn Kersten (Uni Hamburg) Coherence of Supernova Neutrinos 4 / 13
Coherence Length of Supernova Neutrinos Wave packets overlap up to coherence length L coh ∼ σ ∆ v Depends on Size of wave packets σ Velocity difference ∆ v of mass eigenstates Jörn Kersten (Uni Hamburg) Coherence of Supernova Neutrinos 5 / 13
Size of Wave Packets Determined by length of production process Electron capture pe − → n ν e Time scale ∼ time electron needs to cross proton, τ ∼ σ e / v e Temperature ∼ 5 MeV ⇒ electron relativistic, v e ∼ 1 Size of electron wave packet σ e ∼ mean free path Jörn Kersten (Uni Hamburg) Coherence of Supernova Neutrinos 6 / 13
Size of Wave Packets Determined by length of production process Electron capture pe − → n ν e Time scale ∼ time electron needs to cross proton, τ ∼ σ e / v e Temperature ∼ 5 MeV ⇒ electron relativistic, v e ∼ 1 Size of electron wave packet σ e ∼ mean free path � − 1 / 3 ∼ 10 − 11 cm 4 πα 2 n � Result: σ ∼ σ e ∼ For comparison: Atmospheric neutrinos: σ ∼ 1 cm Reactor neutrinos: σ ∼ 10 − 8 cm σ E ∼ 1 /σ ∼ 1 MeV not much smaller than E ∼ 10 MeV Jörn Kersten (Uni Hamburg) Coherence of Supernova Neutrinos 6 / 13
Coherence Length of Supernova Neutrinos Wave packets overlap up to coherence length L coh ∼ σ ∆ v Depends on Size of wave packets σ ∼ 10 − 11 cm Velocity difference ∆ v of mass eigenstates Jörn Kersten (Uni Hamburg) Coherence of Supernova Neutrinos 7 / 13
Coherence Length of Supernova Neutrinos Wave packets overlap up to coherence length L coh ∼ σ ∆ v Depends on Size of wave packets σ ∼ 10 − 11 cm atm ∼ 10 − 3 eV 2 or ∆ m 2 Mass difference ∆ m 2 sol ∼ 10 − 5 eV 2 Neutrino energy E ∼ 10 MeV Jörn Kersten (Uni Hamburg) Coherence of Supernova Neutrinos 7 / 13
Coherence Length of Supernova Neutrinos Wave packets overlap up to coherence length L coh ∼ σ ∆ v Depends on Size of wave packets σ ∼ 10 − 11 cm atm ∼ 10 − 3 eV 2 or ∆ m 2 Mass difference ∆ m 2 sol ∼ 10 − 5 eV 2 Neutrino energy E ∼ 10 MeV Matter density Mixing angle θ 13 ∼ 9 ◦ or θ 12 ∼ 30 ◦ Jörn Kersten (Uni Hamburg) Coherence of Supernova Neutrinos 7 / 13
Coherence Length of Supernova Neutrinos Wave packets overlap up to coherence length L coh ∼ σ ∆ v Depends on Size of wave packets σ ∼ 10 − 11 cm atm ∼ 10 − 3 eV 2 or ∆ m 2 Mass difference ∆ m 2 sol ∼ 10 − 5 eV 2 Neutrino energy E ∼ 10 MeV Matter density Mixing angle θ 13 ∼ 9 ◦ or θ 12 ∼ 30 ◦ Supernova, inner region: L coh ∼ 100 km � length scale of collective effects Earth: L coh ∼ 1000 km Jörn Kersten (Uni Hamburg) Coherence of Supernova Neutrinos 7 / 13
Coherence Length of Supernova Neutrinos Wave packets overlap up to coherence length L coh ∼ σ ∆ v Depends on Size of wave packets σ ∼ 10 − 11 cm atm ∼ 10 − 3 eV 2 or ∆ m 2 Mass difference ∆ m 2 sol ∼ 10 − 5 eV 2 Neutrino energy E ∼ 10 MeV Matter density Mixing angle θ 13 ∼ 9 ◦ or θ 12 ∼ 30 ◦ Supernova, inner region: L coh ∼ 100 km � length scale of collective effects Earth: L coh ∼ 1000 km Supernova neutrinos are special Very short wave packets � short coherence length Jörn Kersten (Uni Hamburg) Coherence of Supernova Neutrinos 7 / 13
Sometimes Decoherence Is Irrelevant P ( ν e → ν e ) = | cos 2 θ + sin 2 θ e i φ | 2 = cos 4 θ + sin 4 θ + 2 cos 2 θ sin 2 θ cos φ Measured probability: average over energy resolution ∆ E of detector Jörn Kersten (Uni Hamburg) Coherence of Supernova Neutrinos 8 / 13
Sometimes Decoherence Is Irrelevant P ( ν e → ν e ) = | cos 2 θ + sin 2 θ e i φ | 2 = cos 4 θ + sin 4 θ + 2 cos 2 θ sin 2 θ cos φ Measured probability: average over energy resolution ∆ E of detector ∆ E too large � interference term vanishes Jörn Kersten (Uni Hamburg) Coherence of Supernova Neutrinos 8 / 13
Sometimes Decoherence Is Irrelevant P ( ν e → ν e ) = | cos 2 θ + sin 2 θ e i φ | 2 = cos 4 θ + sin 4 θ + 2 cos 2 θ sin 2 θ cos φ Measured probability: average over energy resolution ∆ E of detector ∆ E too large � interference term vanishes � Need sufficient energy resolution to observe oscillations 1.0 0.8 P � Ν e �Ν e � 0.6 0.4 0.2 0.0 0 5 10 15 20 25 30 E � MeV ∆ E max Jörn Kersten (Uni Hamburg) Coherence of Supernova Neutrinos 8 / 13
Sometimes Decoherence Is Irrelevant P ( ν e → ν e ) = | cos 2 θ + sin 2 θ e i φ | 2 = cos 4 θ + sin 4 θ + 2 cos 2 θ sin 2 θ cos φ Measured probability: average over energy resolution ∆ E of detector ∆ E too large � interference term vanishes � Need sufficient energy resolution to observe oscillations Corresponding uncertainty of arrival time ∆ t ∼ 1 / ∆ E Coherence preserved for wave packets arriving within ∆ t , even if they are spacially separated � Detector restores coherence Jörn Kersten (Uni Hamburg) Coherence of Supernova Neutrinos 8 / 13
Sometimes Decoherence Is Irrelevant P ( ν e → ν e ) = | cos 2 θ + sin 2 θ e i φ | 2 = cos 4 θ + sin 4 θ + 2 cos 2 θ sin 2 θ cos φ Measured probability: average over energy resolution ∆ E of detector ∆ E too large � interference term vanishes � Need sufficient energy resolution to observe oscillations Corresponding uncertainty of arrival time ∆ t ∼ 1 / ∆ E Coherence preserved for wave packets arriving within ∆ t , even if they are spacially separated � Detector restores coherence Always the case in vacuum and matter with slowly changing density (adiabatic case) Does this change in non-adiabatic case? Jörn Kersten (Uni Hamburg) Coherence of Supernova Neutrinos 8 / 13
Adiabaticity Violation Simplest case: density step Each wave packet splits up into two ρ ρ ′ ρ ′′ ν 1 Jörn Kersten (Uni Hamburg) Coherence of Supernova Neutrinos 9 / 13
Adiabaticity Violation Simplest case: density step Each wave packet splits up into two ρ ρ ′ ρ ′′ ν ′ 1 ν 1 ν ′ 2 Jörn Kersten (Uni Hamburg) Coherence of Supernova Neutrinos 9 / 13
Adiabaticity Violation Simplest case: density step Each wave packet splits up into two ρ ρ ′ ρ ′′ ν ′ ν ′ 1 1 ν 1 ν ′ ν ′ 2 2 Jörn Kersten (Uni Hamburg) Coherence of Supernova Neutrinos 9 / 13
Adiabaticity Violation Simplest case: density step Each wave packet splits up into two ρ ρ ′ ρ ′′ ν ′′ 1 ν ′ ν ′ 1 1 ν ′′ 2 ν 1 ν ′′ 1 ν ′ ν ′ 2 2 ν ′′ 2 Jörn Kersten (Uni Hamburg) Coherence of Supernova Neutrinos 9 / 13
Adiabaticity Violation Simplest case: density step Each wave packet splits up into two ρ ρ ′ ρ ′′ ν ′′ ν ′′ 1 1 ν ′ ν ′ 1 1 ν ′′ ν ′′ 2 2 ν 1 ν ′′ ν ′′ 1 1 ν ′ ν ′ 2 2 ν ′′ ν ′′ 2 2 Jörn Kersten (Uni Hamburg) Coherence of Supernova Neutrinos 9 / 13
Adiabaticity Violation Simplest case: density step Each wave packet splits up into two Different oscillation phase in each layer Complete coherence: 2 � � a + b e i φ 1 + c e i φ 2 + . . . � P = � � � 0.70 0.65 P � Ν 1 �Ν e � 0.60 0.55 0.50 6 8 10 12 14 16 18 20 E � MeV Jörn Kersten (Uni Hamburg) Coherence of Supernova Neutrinos 9 / 13
Adiabaticity Violation Simplest case: density step Each wave packet splits up into two Different oscillation phase in each layer Complete coherence: 2 � � a + b e i φ 1 + c e i φ 2 + . . . � P = � � � Complete decoherence: P = a 2 + b 2 + c 2 + . . . Energy resolution good enough to resolve all oscillation features ⇒ detector restores coherence as before Jörn Kersten (Uni Hamburg) Coherence of Supernova Neutrinos 9 / 13
Incomplete Averaging 2 � � a + b e i φ 1 + c e i φ 2 + . . . � P = � � � = a 2 + b 2 + c 2 + · · · + 2 ab cos φ 1 + 2 bc cos ( φ 2 − φ 1 ) 0.70 0.65 P � Ν 1 �Ν e � 0.60 0.55 0.50 6 8 10 12 14 16 18 20 E � MeV Energy resolution may be too bad to observe cos φ 1 term good enough to observe cos ( φ 2 − φ 1 ) term if φ 1 ≈ φ 2 Jörn Kersten (Uni Hamburg) Coherence of Supernova Neutrinos 10 / 13
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