in Decoherence supernova neutrino transformations suppressed by deleptonization
Andreu Esteban-Pretel
AHEP Group. Institut de F isica Corpuscular CSIC/UVEG In collaboration with R. Tom`s, S. Pastor, G. G. Raffelt and G. Sigl a Physical Review D 76 (2007) 125018 (arXiv:0706.2498)
Introduction
A b s t r a c t
In the dense-neutrino region at 50400 km above the neutrino sphere in a supernova, neutrino-neutrino interactions cause large flavor transformations. We study when the multi-angle nature of the neutrino trajectories leads to flavor decoherence between different angular modes. We consider a two-flavor mixing scenario between e and another flavor x and assume the usual hierarchy Fe > Fe > Fx = Fx for the number fluxes. We define = (Fe - Fe )/(Fe - Fx ) as a measure for the deleptonization flux which is the one crucial parameter. The transition between the quasi single-angle behavior and multi-angle decoherence is abrupt as a function of . For typical choices of other parameters, multi-angle decoherence is always suppressed for 0.3.
Setup of the Problem
E q u a t i o n s o f m o t i o n
We work in a two-flavor scenario between e and another flavor x, characterized by the atmospheric m2 and , the unknown 13-mixing angle. Our fundamental quantities are the flux matrices in flavor space Jr represented by polarization vectors Pr as F (e) + F (x) F (e) - F (x) + Pr ; : Pauli matrices. Jr = 2 2 Normalization at the -sphere: P = |P| = 1 + and P = |P| = 1 . We label different angular modes with u = sin2 R, where R is the zenith angle at the -sphere r = R of a given mode relative to the radial direction. We will consider two generic angular distributions for the modes: multi-angle case, neutrino radiation "half isotropic" above the -sphere; single-angle case, all neutrinos are launched at 45 at the -sphere (u = 1/2).
(-) (-)
Coherence vs. Decoherence
Different forms of evolution
It is interesting to look at some large radius where there is no neutrino-neutrino effect and all modes simply perform vacuum oscillations.
1 = 0.25 1 0.12 0.5 0.5 1 0
0.5 z-component
We show the end state of 500 polarization vectors for antinetrinos in IH, representing modes uniformly spaced in the angular coordinate u.
0
0
0
-0.5
-0.5
-0.5
-1 -1 1 -0.5 = 0.25 0 0.5 1
-1 -1 1 0.12 0.5 -0.5 0 0.5 1
-1 -1 1 0 0.5 -0.5 0 0.5 1
0.5 y-component
0
0
0
-0.5
-0.5
-0.5
G e n e r a l
The crucial phenomenon is a collective mode of pair transformations of the form ee xx where x represents some suitable superposition of and . Collective pair transformations require a large neutrino density and a pair excess of a given flavor. In typical SN models one finds Fe > Fe > Fx = Fx .
R e m a r k s
1.25 1 0.75 0.5 0.25 0 -0.25 -0.5 -0.75 -1 -1.25 50 100 150 200 250 300 350 400
-1 -1 -0.5 0 0.5 x-component 1
-1 -1 -0.5 0 0.5 x-component 1
-1 -1 -0.5 0 0.5 x-component 1
- P
Pz
z
Equations of motion in spherical symmetry (monochromatic energy distribution) r P u,r
(-)
y-component
r [km]
The current-current nature of the weak interaction causes the interaction energy to depend on (1 - cos ) for two trajectories with relative angle . As a result, one would expect kinematical decoherence between different angular modes. In the SN context, however, it has been numerically observed that the evolution is more similar to the single-angle case. We consider different values of an asymmetry parameter that we define as F (e) - F (x) F (e) - F (e) = -1 = ; F (e) - F (x) F (e) - F (x)
1.25 1 0.75 0.5 0.25 0 -0.25 -0.5 -0.75 -1 -1.25 10 1.25 1 0.75 0.5 0.25 0 -0.25 -0.5 -0.75 -1 -1.25 50 1.25 1 0.75 0.5 0.25 0 -0.25 -0.5 -0.75 -1 -1.25 50 1.25 1 0.75 0.5 0.25 0 -0.25 -0.5 -0.75 -1 -1.25 50 Normal Hierarchy
where vu,r = 1 - u R2/r2, = |m2/2E|, B = (sin 2, 0, cos 2) (Bz < 0 NH and Bz > 0 IH), L a unit vector in the z-direction, the matter is density is represented by r = 2GF ne- (r) and the strength of the neutrinoneutrino interaction is parameterized by R R = 2GF Fe - Fx . (2)
B P r L P u,r u,r = (-) + v vu,r u,r 1 R2 Pu ,r - Pu ,r du + 2 r vu,r 0
Pz , - z P
= 0.25. Initially, all polarization vectors are aligned in the flavor direction. Right after rsynch a spiral structure is formed, which evolves almost as in the single-angle case and eventually orients itself in the negative B-direction.
1 0.5 0 -0.5 sin(2) = 0.1 -1 -1 -0.5 0 0.5 x-component 1 -1 -1 -0.5 0 0.5 x-component 1 1 0.5 0 -0.5 sin(2) = 10
-3
(-) (-) u,r P - (Pr - Pr ) P , u,r vu,r
1 0.5 0 -0.5 sin(2) = 10 -1 -1 -0.5 0 0.5 x-component 1
-6
The spiral structure is different depending on . For a large sin 2, the polarization vectors stay close to each other.
= 0.12. Initially the same happens, but at the "decoherence radius" the spiral structure dissolves almost instantaneously. = 0. Decoherence is fast and complete. For a small mixing angle, all polarization vectors are confined to the x-z-plane.
Fe > Fe > F (x) = F (x) . (1) The flavor content decoheres quickly for both NH and IH.
Schematic supernova model
Assuming E = 20 MeV and m2 = 2.410-3 eV2 the oscillation frequency is fixed to m2 = 0.3 km-1 (3) = 2E
Parameter sin 2 Standard value 0.25 7 105 km-1 0.3 km-1 10-3 Definition Eq. (1) Eq. (2) Eq. (3) --
- Pz - |P |
=0 10.05 10.1 10.15 10.2
1.25 1 0.75 0.5 0.25 0 -0.25 -0.5 -0.75 -1 -1.25 10 1.25 1 0.75 0.5 0.25 0 -0.25 -0.5 -0.75 -1 -1.25 250 300 1.25 1 0.75 0.5 0.25 0 -0.25 -0.5 -0.75 -1 -1.25 250 300 1.25 1 0.75 0.5 0.25 0 -0.25 -0.5 -0.75 -1 -1.25 350 400 50 50 50
Inverted Hierarchy
- Pz - |P |
Measures
of
decoherence
As shown in the figure of panel 2, the length of P can be used to discriminate between different modes of evolution. Another global measure of decoherence is the length of the line joining neighboring polarization vectors on the unit sphere. It is simply the sum of the angles between them. At rsynch the length on the unit sphere quickly increases from 0 to a value that is almost independent of , but depends on the mixing angle, for smaller sin 2 it is larger. Later, this length stays practically constant, once the spiral structure is established it does not change much except tilting toward the negative B-direction and precessing around it.
60 =0.35 =0.30 =0.25 =0.20 =0.15 =0.10
- - Pz , |P|
=0 10.1 10.05 10.15 10.2
Default values for our model parameters.
- Pz - |P |
- Pz - |P |
= 0.06 100 150 200
= 0.06 100 150 200 250 300
NH: large and abrupt decoherence far beyond rsynch. IH: at first analogous to = 0.25, at some larger radius P shrinks significantly. Partial decoherence. NH: the length P also shrinks, but closely tracks Pz . IH: qualitatively equivalent to = 0.06.
250 300
Assuming an average luminosity per flavor of 0.5 1052 erg/s we use for the strength of the neutrino-neutrino interaction = 7 10 km
5 -1
- - Pz , |P|
Lx Le - Ee Ex
15 MeV 1052 erg/s
10 km R
2
.
Length on unit sphere
= 0.12 100 150 200
- Pz - |P |
= 0.12 100 150 200
- Pz - |P |
Collective pair conversions only begin at rsynch where is so small that synchronization ends. The synchronization condition is 2 > . 2 ( 1 + - 1) Since the effective neutrino-neutrino interaction strength varies at large distances as eff (r) = R4/2r4, the synchronization radius is 1/2 1/4 rsynch 1+-1 1/4 = . R 2 2 The second part assumes 1.
We fix the -sphere at the radius R = 10 km.
50
40
30
- - Pz , |P|
20
10
- Pz - |P |
- Pz - |P |
= 0.25 100 150 200 250 300
= 0.25 100 150 200 250 300 350 400
NH: nothing visible happens, in analogy to the single-angle case. IH: similar to the single-angle case. The nutations wash out quickly. P shrinks a bit after rsynch.
0 50 100 150 200 r [km] 250 300 350
- - Pz , |P|
When is smaller than a critical value, at the decoherence radius there is a sudden second growth phase. For smaller , the final length is longer, representing a more "phase-space filling" line on the unit sphere.
r [km]
r [km]
Role of Model Parameters
O r d i n a r y M a t t e r
108 106 104 102 100 10-2 10 s 10-4 101 10
2
Summary
C o n c l u s i o n s
We have explored numerically the range of parameters where different forms of behavior dominate in a realistic SN scenario. "Multi-angle effects" mean one of two clearly separated forms of behavior. The flavor content of the system can evolve in a quasi single-angle form. On the level of the polarization vectors this means that they fill only a restricted volume of the available phase space and maintain a coherent structure. On the other hand, nearly complete flavor equilibrium can arise where the available phase space is more or less uniformly filled. The switch between these modes of evolution is set by the degree of asymmetry between the neutrino and antineutrino fluxes as defined in Eq. (1). The critical value of that is enough to suppress decoherence depends on the type of neutrino mass hierarchy, the average energies, luminosities, and on the mixing angle. We have found that for 0.3, decoherence is suppressed for the entire range of plausible parameters, but a value smaller than 0.1 may be enough, depending on the combination of other parameters.
103 104 105 [km-1] 106 107
E n e r g y
d i s t r i b u t i o n
Typical matter density profiles from numerical simulations (Garching group). For comparison we also show our benchmark value = 0.3 km-1 and (r). We assume that - refraction and ordinary matter effects do not interfere.
1.25 1 0.75 0.5 0.25 - Pz 0 -0.25 -0.5 -0.75 -1 -1.25 50 1.25 1 0.75 0.5 0.25 - Pz 0 -0.25 -0.5 -0.75 -1 -1.25 50 100 150 200 250 300 350 400 r [km] 0.8 50 75 100 1 0.9 100 150 200 250 300
-4
= 0.3 km-1 (r)
0.5 s
2s
We have studied several numerical examples of quasi single-angle behavior and of multi-angle decoherence, taking different neutrino and antineutrino energy spectra, such as flat or thermal and with equal or different temperatures. We always found that the evolution of the global polarization vectors is almost identical to the equivalent monochromatic cases.
105
[km-1]
10 r [km]
3
104
0 =0 1 0.9 ...