Fast Neutrino Flavor Conversion at Late Time
FFast Neutrino Flavor Conversion at Late Time
Soumya Bhattacharyya 𝑎, ∗ 𝑎 Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
E-mail: [email protected]
We all know that in the dense anisotropic interior of the star, neutrino-neutrino forward-scatteringcan lead to fast collective neutrino oscillations, which has striking consequences on flavor depen-dent neutrino emission and can be crucial for the evolution of a supernova and its neutrino signal.The flavor evolution of such dense neutrino system is governed by a large number of couplednonlinear partial differential equations which are almost always very difficult to solve. Althoughthe triggering, initial linear growth and the condition for fast oscillations to occur are understoodby a well known trick known as “Linear stability analysis” [1], this fails to answer an importantquestion – what is the impact of fast flavor conversion on observable neutrino fluxes or the su-pernova explosion mechanism? This is a significantly harder problem that requires understandingthe nature of the final state solution in the nonlinear regime. Moving towards this direction wepresent one of the first numerical as well as an analytical study of the coupled flavor evolution ofa non-stationary and inhomogeneous dense neutrino system in the nonlinear regime consideringone spatial dimension and a spectrum of velocity modes. This work gives a clear picture of thefinal state flavor dynamics of such systems specifying its dependence on space-time coordinates,phase space variables as well as the lepton asymmetry and thus can have significant implicationsfor the supernova astrophysics as well as its associated neutrino phenomenology even for the mostrealistic scenario. ∗ Speaker © Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ h e p - ph ] J a n ast Neutrino Flavor Conversion at Late Time Soumya Bhattacharyya -L/2 -L/4 0 L/4 L/2x04080120160 C o n s e r v e d q u a n t i t y A = 0 , t = 50 EM σM (1)1 M (2)1 M (3)1 -L/2 -L/4 0 L/4 L/2x04080120160 C o n s e r v e d q u a n t i t y A < 0 , t = 50 EM σM (1)1 M (2)1 M (3)1 Figure 1:
Spatial variation of 𝐸 , 𝜎 , 𝑀 , M in left. Same for 𝑅 ( , , ) . , in right. All plots are done at 𝑡 =
1. Set-up of the system
Neglecting collisions, vacuum and matter effect the equation of motion in our model governingfast flavor evolution of two flavors of neutrinos moving with velocity 𝑣 in space ( 𝑥 ) and time ( 𝑡 ) isgiven by [2]: (cid:0) 𝜕 𝑡 + 𝑣𝜕 𝑥 (cid:1) P 𝑣 ( 𝑥, 𝑡 ) = 𝜇 ∫ − 𝑑𝑣 (cid:48) ( − 𝑣𝑣 (cid:48) ) P 𝑣 (cid:48) ( 𝑥, 𝑡 ) × P 𝑣 ( 𝑥, 𝑡 ) , (1)where the polarization vector P 𝑣 = 𝐺 𝑣 S 𝑣 = 𝐺 𝑣 (cid:16) 𝑠 ( ) 𝑣 , 𝑠 ( ) 𝑣 , 𝑠 ( ) 𝑣 (cid:17) 𝑇 represents the flavor compositionof neutrinos while 𝐺 𝑣 and 𝜇 encode the electron lepton number distribution in momentum spaceand neutrino self-interaction strength respectively. We present our results for systems with totallepton asymmetry, ∫ − 𝐺 𝑣 𝑑𝑣 = 𝐴 = ≠
2. Results
Temporal Behaviour :
In the limit 𝑡 being very large, our numerical analysis suggests that thefully nonlinear solution of Eq.(1) becomes approximately stationary in time for any 𝐴 . [Fig. 3 ] Spatial Behaviour :
From Eq.(1) using the steady state approximation in time and also goingto the multipole space [3] we obtain the following set of equations governing the system’s late timeflavor dynamics in a frame rotating with constant velocity √ M · M w.r.t the fixed lab frame: 𝑑 𝑥 P 𝑣 ( 𝑥 ) = M ( 𝑥 ) 𝑣 × P 𝑣 ( 𝑥 ) , M ( 𝑥 ) × 𝑑 𝑥 M ( 𝑥 ) + 𝜎 𝑑 𝑥 M ( 𝑥 ) = 𝑀 B ( 𝑥 ) × M ( 𝑥 ) , 𝑑 𝑥 M ( 𝑥 ) = M 𝑛 = ∫ − 𝐿 𝑛 ( 𝑣 ) P 𝑣 𝑑𝑣 denotes the 𝑛 𝑡ℎ multipole moment of P 𝑣 . This indicates at latetimes each P 𝑣 has a spatial precession of frequency 𝑣 around a common axis, M , [See Fig. 2]which shows gyroscopic pendulum motion in space under the action of a spatially varying magneticfield , B ( 𝑥 ) = (cid:205) ∞ 𝑟,𝑛 = 𝑐 𝑟𝑛 M 𝑛 ( 𝑥 ) maintaining fixed length 𝑀 = √ M · M , fixed angular momentum 𝜎 = M · D and conserved energy 𝐸 = B · M + D · D [See Fig. 1]. 𝑐 𝑟𝑛 ’s are ( 𝑥, 𝑡, 𝑣 ) independentconstants but depend only on the value of 𝑟 and 𝑛 . Eq.(2) also indicates a non-separable steady-state solution (non-collective) for P 𝑣 ( 𝑥 ) in position and velocity coordinates [4]. We numericallychecked this via plotting the spatial variation of 𝑅 ( 𝑖 ) 𝑣 ,𝑣 ( 𝑥 ) = 𝑠 ( 𝑖 ) 𝑣 ( 𝑥 ) 𝑠 ( 𝑖 ) 𝑣 ( 𝑥 ) at 𝑡 =
50 for 𝑖 = ( , , ) andfixed ( 𝑣 , 𝑣 ) which will be 𝑥 independent if the solution is collective otherwise not [See Fig. 1].We find the phase relationship ( 𝜙 𝑣 ) between the transverse components of the polarization vector2 ast Neutrino Flavor Conversion at Late Time Soumya Bhattacharyya - π4 - π2 -π - π4 π2 π ϕ v P D F ( ϕ v ) t = 50, A = 0 −1.0−0.50.00.51.0v - π4 - π2 -π - π4 π2 π ϕ v P D F ( ϕ v ) t = 50, A < 0 −1.0−0.50.00.51.0v -L/2 -L/4 0 L/4 L/2x−2−1012 s ( ) , ( ) , ( ) v ( x , t ) t = 50, A = 0, v = 0.5 s (1)v (x,t)s (2)v (x,t)s (3)v (x,t) -L/2 -L/4 0 L/4 L/2x−2−1012 s ( ) , ( ) , ( ) v ( x , t ) t = 50, A < 0, v = 0.5 s (1)v (x,t)s (2)v (x,t)s (3)v (x,t) Figure 2:
Spatial distribution of 𝜙 𝑣 (cid:12)(cid:12) 𝑡 = in left. 𝑠 𝑖𝑣 (cid:12)(cid:12) 𝑡 = vs 𝑥 in right shows oscillations indicating precession ⟨ s ( ) v ( t ) ⟩ A = 0 −1.0−0.50.00.51.0v 0 10 20 30 40 50t−1.0−0.50.00.51.0 ⟨ s ( ) v ( t ) ⟩ A < 0 −1.0−0.50.00.51.0v −1.0 −0.5 0.0 0.5 1.0v−1.0−0.50.00.51.0 ⟨ s ( ) , ( ) , ( ) v ⟩ t = 50, A = 0 ⟨ s (1)v ⟩⟨ s (2)v ⟩⟨ s (3)v ⟩ −1.0 −0.5 0.0 0.5 1.0v−1.0−0.50.00.51.0 ⟨ s ( ) , ( ) , ( ) v ⟩ t = 50, A < 0 ⟨ s (1)v ⟩⟨ s (2)v ⟩⟨ s (3)v ⟩ Figure 3: (cid:104) 𝑠 ( ) 𝑣 (cid:105) vs 𝑡 in left for various 𝑣 shows steady solution in time. (cid:104) 𝑠 ( , , ) 𝑣 (cid:105) vs 𝑣 at 𝑡 =
50 in right. at different spatial locations becomes randomly distributed over [− 𝜋, 𝜋 ] at late times for any 𝐴 and 𝑣 [Fig. 2]. Dependence on lepton asymmetry :
Integrating both sides of Eq.(1) w.r.t all velocity modesand then taking spatial average give rise to three conservation equations: ∫ − 𝐺 𝑣 (cid:104) S 𝑣 (cid:105) = A with A = ( , , 𝐴 ) . This implies, for 𝐴 = synchronized behaviour inmomentum space . But for 𝐴 ≠ velocity dependence at late times to conserve the total lepton asymmetry. [See Fig. 3] Momentum dependence :
Naively, the spatial averaged version of Eq.(1) in some corotatingframe looks like, 𝑑 𝑡 (cid:104) S 𝑣 (cid:105) = (cid:104) H 𝑣 (cid:105) × (cid:104) S 𝑣 (cid:105) with (cid:104) H 𝑣 (cid:105) = − A − 𝑣 G and G = (cid:104) M (cid:105) . This indicates that (cid:104) 𝑠 ( ) 𝑣 (cid:105) for specific 𝑣 satisfying, (cid:104) 𝐻 ( ) 𝑣 ( 𝑡 𝑖𝑛𝑖 )(cid:105)(cid:104) 𝐻 ( ) 𝑣 ( 𝑡 𝑓 𝑖𝑛 )(cid:105) <
0, or (cid:16) 𝐴 + 𝑣 𝐺 ( ) 𝑖𝑛𝑖 (cid:17) (cid:16) 𝐴 + 𝑣 𝐺 ( ) 𝑓 𝑖𝑛 (cid:17) < may flip its sign in the same spirit as the spectral swaps seen in collective oscillations [5]. Clearly,this condition can not be satisfied for 𝐴 = 𝐴 ≠
3. Acknowledgments
S.B would like to thank the organizers of “40th International Conference on High Energyphysics- ICHEP2020” for giving an opportunity to present this work as a poster.
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