The Effects of an Extended Neutrino Sphere on Supernova Neutrino Oscillations

Transkript

1 The Effects of an Extended Neutrino Sphere on Supernova Neutrino Oscillations max-planck-institut für Kernphysik Heidelberg NDM 2018, IBS, Korea July 2, / 24

2 Janka et al / 24

3 Impact of neutrino oscillations Supernova explosion mechanism: Modify the neutrino signal: E(ν µ/τ ) > E(ν e ) ν e ν µ/τ ν µ/τ ν e Nucleosynthesis depends on the neutron fraction: p + ν e n + e + n + ν e p + e (See also talk by Baha Balantekin on Friday) 3 / 24

4 Matter effect Electron background shifts the energy eigenvalues. ( ) H = m2 c2θ s 2θ + ( ) 1 0 2G F n e 2E 0 0 s 2θ c 2θ ν 2,eff ν e m 2 eff. m 2 2 vacuum ν 1,eff ν x solar centre m 2 1 n e 4 / 24

5 Density matrix formalism In the mean field approximation: ( ) ρee ρ ρ = ex = 1 ( ) P0 + P z P x ip y = 1 ρ xe ρ xx 2 P x + ip y P 0 P z 2 (P 0 + P σ). Similarly for the Hamiltonian: H = 1 ( ) Vz V x iv y = 1 V 2 V x + iv y V z 2 σ. Equation of motion (in absence of collisions, see also poster by Hirokazu Sasaki): i ρ = [H, ρ] P = V P 5 / 24

6 Collective oscillations Normal oscillations: prob(ν e ν e ) cos 2 ( m 2 L/4E). prob(νe νe) E0 5E0 3E0 Neutrino background: H νν = 2G F dp(ρ ρ). 0.0 E0. L Conversion independent of E. Non-linear problem hard to solve in a realistic setting. Hannestad et al / 24

7 Linear equations, RSLH and Smirnov, 2018 Observation Considering the individual neutrino, its oscillations in a SN is a linear problem. Idea How much of the neutrino-neutrino refraction can we describe using linear equations. Ultimate goal General conclusions about the behaviour of neutrino oscillations in presence of neutrino-neutrino refraction. Methods - Solve equations from first principles, analytic and numeric. - Describe complicated systems using effective potentials. 7 / 24

8 General equations Probe neutrino in arbitrary neutrino and matter background. H (p) = 1 ( c2θ ω p + V e + V ν s 2θ ω p + 2 V ν e iφ ) B 2 s 2θ ω p + 2 V ν e iφ, (1) B c 2θ ω p V e V ν V ν = dkvν 0 (k) [ρ ee (k) ρ ττ (k))], V ν e iφ B = dkv 0 ν (k)ρ eτ (k) V 0 ν (k) = 2G F n(k) (1 v bg v p ) 8 / 24

9 General equations - rotated Rewrite the off-diagonal as V e iφ = s 2θ ω p + 2 V ν e iφ B ( ). Apply the transformation U = diag e iφ /2, e iφ /2. H (p) = 1 2 ( V r V V V r ), where V = 4 V 2 ν + 4s 2θ ω p cos φ B Vν + s 2 2θ ω2 p, V r = V e + V ν + φ c 2θ ω p. 9 / 24

10 Conditions for a large conversion Our Ansatz is that every case where a large conversion happens can be described in terms of at least one of these frameworks: 1 Resonance Vanishing diagonal: 2 Adiabatic conversion V e + V ν + φ c 2θ ω p = 0. Fast oscillations in V r and V can be removed by going to a rotating frame. This can result in a Hamiltonian describing adiabatic evolution. 3 Parametric enhancement Present if the period of oscillation equals the period of change of mixing angle. 10 / 24

11 Energy spectrum ω k = 0.5ω p ω k = ω p ω k = 1.5ω p Box spectrum Analytic Peτ tω p RSLH and Smirnov, / 24

12 Neutrino emission ν µ/τ have: Number sphere: ν µ/τ decouple first, then ν e and last ν e. Energy sphere: e + e ν µ/τ ν µ/τ e ν µ/τ e ν µ/τ Transport sphere: N ν µ/τ N ν µ/τ 12 / 24

13 Neutrino emission ν µ/τ decouple first, then ν e and last ν e. ν µ/τ have: Number sphere: Energy sphere: e + e ν µ/τ ν µ/τ e ν µ/τ e ν µ/τ Transport sphere: N ν µ/τ N ν µ/τ ν e and ν e are dominated by absorption and emission from nucleons. (n n > n p ) Mean free path: 1 λ G 2 F π (g 2 V + 3g 2 A )E 2 n N. Emissivity: j = 1 λ exp( E/T ) 12 / 24

14 Extended source Rough estimates: Neutrino sphere: 10km. Width of neutrino sphere: 1km. Oscillation length: 1 G F n e km. Average over emission region suppresses oscillatory terms by Parametric resonance is not removed as such. 13 / 24

15 Extended source - non-linear ρee = 1 2 (1 + P z) z = 0 3π sin β z = λ 7π sin β z = λ 21π sin β z = λ emission region ρex Hex z 14 / 24

16 Extended source - non-linear ρee = 1 2 (1 + P z) z = 0 3π sin β z = λ 7π sin β z = λ 21π sin β z = λ emission region ρex Hex z 14 / 24

17 Changing background - only matter Coordinate system with V along z-axis: ( r ) ρ 12 (r) = ρ 12,initial exp i ω m (r )dr. r e Average over emission point: ρ 12 (r) = r 0 p(r e ) 1 2 sin 2θ m(r e )exp ( r ) i ω m (r )dr dr e, r e where p(r e ) = 1 ( ) 1 exp( E/T ) exp λ(r e ) r e λ(r ) dr. 15 / 24

18 Changing background - only matter p r[km] p sin(2θ) r[km] Include effect of damping, D: P = V P D P T. ρ averaging damping p(r) = δ(r rνsphere) r[km] For V z and D large and P z 1: P x V xv z V 2 z + D 2, P y V xd V 2 z + D 2. Bell et al. 1998, Hannestad et al / 24

19 Effect of non-adiabaticity Solve P = V P D PT numerically: ρ numerical adiabatic non-adiabatic r[km] Non-adiabatic effects for D=0: V x Vx r Vz V z + sin ( V Vz 3 z (r )dr ) ( V P = x r V z ( V 1 cos z 3 Vz (r )dr )) 1 + V x 2 r V z sin (. V Vz 4 z (r )dr ) ρ 12 V x r V z 2Vz 3 V x 2Vz 2 r 0 17 / 24

20 Linear stability analysis Banerjee, Dighe, Raffelt 2011 Linear analysis demonstrate stability or instability. ρ = f ν e + f νx + f ( ) ν e f νx s S 2 2 S s S = P x + ip y 1, s 2 + S 2 = 1 s = P z 1. Linearised equation: iṡ = (ω + λ + µ)s µ dγ (1 v v )S, 18 / 24

21 Linear stability analysis In Fourier space: S = e iωt Q ΩQ = (ω + λ + µ)q µ dγ (1 v v )Q Unstable if Im(Ω) 0. (See also Capozzi et al. 2017) Discrete modes: solve matrix equation. (Continious modes: Decompose in independent functions.) Can also be formulated as a dipersion relation. (Izaguirre, Raffelt and Tamborra, 2016) 19 / 24

22 Simple model Linearised equation: (ω = 1, λ = 30, µ = 3) i ( ) ω + λ µ µ S = µ ω + λ + µ S Eigenvalues: Ω = λ ± ω(2µ + ω) Growth rate = Im(Ω). (1 + Pz) ρee = z = π sin β z = 0.5 λ 7π sin β 0.4 z = λ 21π sin β 0.3 z = λ 0.2 emission region Emission point = 2 3 z. Start value = 1 zλ sin 2θ m. Does not work for large z. ρex Hex z 20 / 24

23 Multiple angles, in-homogeneous Linear stability analysis of a more realistic model: (Ω + v k)q = (ω + λ + µ(ɛ v φ))q µ dγ (1 v v )g Q µ and λ functions of r. Multi angle matter effect. Homogeneous mode: k = 0 Chakraborty, RSLH, Izaguirre and Raffelt, / 24

24 Very fast flavour conversion R. F. Sawyer Chakraborty, RSLH, Izaguirre and Raffelt, 2016 Conversion on meter-scale. Can also occur in a supernova. Dasgupta, Mirizzi and Sen, / 24

25 Summary The non-negligible width of the neutrino sphere affects the neutrino state due to averaging over different emission points. The angle between the neutrino state and the Hamiltonian in polarization space is reduced by a factor 10 8 at the neutrino sphere by the averaging. A small adiabaticity violation increases the angle significantly as the neutrino propagates out through the supernova. The onset of neutrino conversion can be analysed using linear stability analysis, and for a given model, it can be calculated if conversion has the potentially to occur. 23 / 24

26 Thanks for your attention! 24 / 24

27 Backup Slides Neutrino emission. Deleptonisation Accretion Cooling Lang et al / 24

28 Backup Slides Neutrino mixing Normal ν 3 ν 2 Inverted L int = g 2 W + µ ν l L γµ l L + h.c. m 2 atm ν 1 m 2 sun Interaction states and mass states are different: ν 2 m 2 atm ν l = Uν i. ν e m 2 sun ν 1 ν µ ν τ ν 3 Mixing matrix: c 13 0 s 13 e iδ CP c 12 s 12 0 U = 0 c 23 s s 12 c s 23 c 23 s 13 e iδ CP 0 c / 24

29 Backup Slides Collective oscillations Can collective oscillations still occur? YES! ρ r[km] 27 / 24

30 Backup Slides Very fast flavour conversion Dasgupta, Mirizzi and Sen, / 24