Fast modes require different opening angles for neutrinos and anti-neutrinos. The reason seems to be quite understandable. Forward scattering requires “colliding beams”. In the early studies of neutrino self interactions, the models always have same opening angles for neutrinos and antineutrinos, which means that no colliding for neutrinos and antineutrinos on the same side of emission.
Different opening angles for all beams introduce collisions between all the beams, thus possibly leading to more conversions.
Here we use the four beams model. The linear stability analysis is simply four coupled harmonic oscillators.
How could they be harmonic oscillators?
Here is the confusion. How could
We take out the terms related to the difference in opening angles
The matrix becomes block diagonalized. We assume a solution of the form
The solutions of
where the discrimination
Four Beams Model
There are many things to consider for the four beams line model.
Different emission angle for neutrinos and antineutrinos;
Different density for neutrinos and antineutrinos;
Left and right difference.
Is the system unstable even for
Fig. 1.49 The maximum imaginary part in linear stability analysis. These calculations are for fix
The grid of figure are arranged according to the following values of
Fig. 1.50 The maximum imaginary part in linear stability analysis. These calculations are for fix
The grid of figure are arranged according to the following values of
When we have symmetric geometry, the instability region is gone. Such a result is exactly what we expect. However, different
Fig. 1.51 Linear stability analysis for¶
For convinience, we define some quantities for four beam case.
We define the a parameter
The summation of the two angles
Every quantity is in unit of
First we check the result without matter, without vacuum frequency, and
Fig. 1.52 No matter, no vacuum frequency¶
We can also check the matter effect.
Fig. 1.53 With matter, no vacuum frequency.¶
Fig. 1.54 With matter, no vacuum frequency.¶
Fig. 1.55 With matter, no vacuum frequency.¶
Then we check the result without matter, without vacuum frequency, and
Fig. 1.56 No matter, no vacuum frequency.¶
The effect of
Fig. 1.57 Higher order Fourier modes, without matter, no vacuum frequency.¶
Fig. 1.58 Higher order Fourier modes, without matter, no vacuum frequency.¶
Matter + Fourier modes also has suppression
Fig. 1.59 Higher order Fourier modes, with matter, no vacuum frequency.¶
Fig. 1.60 Higher order Fourier modes, with matter, no vacuum frequency.¶
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