1.2.1. Fast Modes

1.2.1.1. Why is it fast?

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 ϵ become very large if they are coupled harmonic oscillators? Is this related the fact that ϵ is complex instead of real?

We take out the terms related to the difference in opening angles χ=1cos(θ1θ2). The linearized equation of motion becomes

iz(ϵLϵ¯LϵRϵ¯R)=(μαχsinθ1μαχsinθ100μχsinθ2μχsinθ20000μαχsinθ1μαχsinθ100μχsinθ2μχsinθ2)(ϵLϵ¯LϵRϵ¯R).

The matrix becomes block diagonalized. We assume a solution of the form ϵ=ϵ0eiΩ which is substitued into the equation. The first two components obey the equation.

(μM11+ΩμM12μM21μM22+Ω)(ϵLϵ¯L)=0.

The solutions of Ω has the form

Ω=(M11+M22)±Δ2μμ,

where the discrimination Δ is

Δ=(M11M22)2+4M12M21.

1.2.1.2. Several Numerical Calculations

Four Beams Model

There are many things to consider for the four beams line model.

  1. Different emission angle for neutrinos and antineutrinos;

  2. Different density for neutrinos and antineutrinos;

  3. Left and right difference.

Is the system unstable even for ωv=0 and λ=0?

../../_images/fast-mode-alpha-0.8.png

Fig. 1.49 The maximum imaginary part in linear stability analysis. These calculations are for fix km/E^=1, where E^ is the energy scale used to scale all the quantities.

ωv=0λ=0α=0.8.

The grid of figure are arranged according to the following values of {θ1,θ2}.

{π6,π6}{π6,2π9}{π6,5π18}{π6,π3}{2π9,π6}{2π9,2π9}{2π9,5π18}{2π9,π3}{5π18,π6}{5π18,2π9}{5π18,5π18}{5π18,π3}{π3,π6}{π3,2π9}{π3,5π18}{π3,π3}
../../_images/fast-mode-alpha-1.2.png

Fig. 1.50 The maximum imaginary part in linear stability analysis. These calculations are for fix km/E^=1, where E^ is the energy scale used to scale all the quantities.

ωv=0λ=0α=1.2.

The grid of figure are arranged according to the following values of {θ1,θ2}.

{π6,π6}{π6,2π9}{π6,5π18}{π6,π3}{2π9,π6}{2π9,2π9}{2π9,5π18}{2π9,π3}{5π18,π6}{5π18,2π9}{5π18,5π18}{5π18,π3}{π3,π6}{π3,2π9}{π3,5π18}{π3,π3}

When we have symmetric geometry, the instability region is gone. Such a result is exactly what we expect. However, different

research-notes/collective/assets/fast-modes-fast-mode-no-matter-asymmetric-alpha-0.8.png

Fig. 1.51 Linear stability analysis for

ωv=0λ=0α=0.8θL=2π/9θR=π/6.

1.2.1.3. Regions of Instability

For convinience, we define some quantities for four beam case.

  1. We define the a parameter α=(1a)/(1+a) so that α[0,] is mapped onto a[1,1].

  2. The summation of the two angles Σθ=θ1+θ2 and the difference between two angles Δθ=θ1θ2, where θ1 is for neutrino beams.

  3. Every quantity is in unit of μ.

First we check the result without matter, without vacuum frequency, and Σθ=2π/3.

../../_images/plt2-sigmatheta-2Pi-divided-by-3-mk-divided-by-mu-0-lambda-divided-by-mu-0.png

Fig. 1.52 No matter, no vacuum frequency

We can also check the matter effect.

../../_images/plt2-sigmatheta-2Pi-divided-by-3-mk-divided-by-mu-0-lambda-divided-by-mu-1.png

Fig. 1.53 With matter, no vacuum frequency.

../../_images/plt2-sigmatheta-2Pi-divided-by-3-mk-divided-by-mu-0-lambda-divided-by-mu-10.png

Fig. 1.54 With matter, no vacuum frequency.

../../_images/plt2-sigmatheta-2Pi-divided-by-3-mk-divided-by-mu-0-lambda-divided-by-mu-100.png

Fig. 1.55 With matter, no vacuum frequency.

Then we check the result without matter, without vacuum frequency, and Σθ=2π/3, and mkμ=0.1.

../../_images/plt2-sigmatheta-2Pi-divided-by-3-mk-divided-by-mu-0.1-lambda-divided-by-mu-0.png

Fig. 1.56 No matter, no vacuum frequency.

The effect of mk/μ is also similar to matter effect.

../../_images/plt2-sigmatheta-2Pi-divided-by-3-mk-divided-by-mu-1-lambda-divided-by-mu-0.png

Fig. 1.57 Higher order Fourier modes, without matter, no vacuum frequency.

../../_images/plt2-sigmatheta-2Pi-divided-by-3-mk-divided-by-mu-10-lambda-divided-by-mu-0.png

Fig. 1.58 Higher order Fourier modes, without matter, no vacuum frequency.

Matter + Fourier modes also has suppression

../../_images/plt2-sigmatheta-2Pi-divided-by-3-mk-divided-by-mu-10-lambda-divided-by-mu-100.png

Fig. 1.59 Higher order Fourier modes, with matter, no vacuum frequency.

../../_images/plt2-sigmatheta-2Pi-divided-by-3-mk-divided-by-mu-1-lambda-divided-by-mu-10.png

Fig. 1.60 Higher order Fourier modes, with matter, no vacuum frequency.


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