Reference to Notes
Izaguirre, I., Raffelt, G., & Tamborra, I. (2016). Fast Pairwise Conversion of Supernova Neutrinos: Dispersion-Relation Approach, 21101(January), 1–6. https://doi.org/10.1103/PhysRevLett.118.021101
In Raffelt’s paper, they defined the polarization tensor as
For numerical calculations, we lower the second index and multiply on both side
where
Raffelt et al parametrize
and
where
with
Note to self
The actual wave vector that determines the instability is
Since
Density matrix is written as
The perturbation
This assumption indicates that even though we find instabilities, a proper initial condition/boundary condition is required to stimulate this instability.
The polarization tensor is in fact
The equation of motion becomes
where
Since
Then we need to find the solution to
which is simplified to
We can also use the polarization tensor
is the determinant of a matrix
Equivalently, we only need to find the eigenvalues of
Raffelt et al proposed that can now solve the dispersion relation by finding the value of
Here is an example that I calculated.
The axial symmetric system can be calculated easily using this method. The paper gave an example of two polar angle beams with axial symmetry.
Fig. 1.62
Fig. 1.63 Dispersion relation.¶
We can check what happens for multibeams. I can plot the dispersion relation for similar configuration but with different number of beams.
Fig. 1.64 Animition of dispersion relation.¶
dataPltNBeamsPlt[Join[Table[1/beams, {n, 1, beams/2}],
Table[-1/beams, {n, 1, beams/2}]],
Table[Pi/3 + n Pi/2/(beams - 1), {n, 0, beams - 1}], {-10, 10}, 0.049, {{-10, 10}, {-10, 10}}]
I plot the
Similar to the previous example, confining the range of
Fig. 1.65 The code for it¶
pltDiffBeamsConfined[beams_] := dataPltNBeamsPlt[
Join[Table[1/beams, {n, 1, beams/2}],
Table[-1/beams, {n, 1, beams/2}]],
Table[Pi/3 + n Pi/2/(beams - 1), {n, 0, beams - 1}], {-1, 1},
0.049, {{-10, 10}, {-10, 10}}]
This should be the continuous limit?
As a comparison, we can plot the dispersion relation in a larger range of n for 10 beams.
Fig. 1.66 10 beams.¶
On the other hand, we can calculate the continuous limit for the same angle range.
Fig. 1.67 Dispersion relation for 10 beams (
MEH
Vectors Using Spherical Harmonics
In principle, solving the dispersion relation is not easy. Neverthless, symmetries would significantly simplify the problem.
Axial symmetry indicates that the integrals of first orders of
We denote the integral
as
For axial symmetric emission, only terms
To simplify the calcualtion, we denote
which shows is an analytical expression of
The four velocity is
We define
where
Since
the matrix
The equation we are solving is
We find the solutions to omega,
Mathematica Code
We plug in the definition
The questions are
What does each of the solutions mean?
By definition, the meaning of polarization tensor,
which clearly shows that the 1, and 2 component of
Is this related to eigenvectors?
The eigenvalue
I don’t think it is related to eigenvalues. However, eigenvalues set limit on the actual solution. When we write down the solution to
That is to say, the part
are the only elements that determines the whether we have a
In turn, it determines the angle dependence of
We have no
In this case we have to calculate
For discrete emission
Thus
For two sets of beams, we have
which is a conic section. We have already used
Hyperbola
For an quadratic equation [HyperbolaWikipedia]
it is hyperbola if
Center of the hyperbola
Principal axis is tilted away from x axis by angle
We can prove that this is a hyperbola. Simplify the equation to standard form of conic sections
The
The condition for it to be hyperbola is
As long as we have different angles,
A special case for it is
We are interested in gaps, so the asymptotic lines are the lines that we are interested in.
First of all, we need to find out the principal axis. The angle between the principal axis and x axis is defined to be
Suppose we have angles
This indicates that the angle
The other solutions
For the solutions
it becomes much more complicated.
Why is
We kind of see why
In some sense it is a weighted average of
Comparing to electrodynamics, where we have the field
Polarization in Electrodynamics
For consistancy check, we parametrize
Fig. 1.68
Fig. 1.69 Same parameters as above. The orange dashed lines are the singularity lines.¶
Fig. 1.68 shows that the limit of
Similar plots are made for 4,6,8 beams.
Fig. 1.70 2, 4, 6, 8 beams with equal division of cosine of emission angle. For example 4 beams are emission at
Fig. 1.71 Same parameters as above.¶
We also calculated the homogeneous emssion.
Fig. 1.72 2, 4, 6, 8 beams for homogeneous
Fig. 1.73 Same parameters as above.¶
Gap
Whenever a gap in
To illustrate this idea, I shaded the region that
Fig. 1.74 The shaded regions are the region that
This provides an method to determine whether we have a gap in
Fig. 1.75 The shaded region are the region where
The behavor of
Fig. 1.76
Suppose neutrinos are emitted within a angle range
where
Assuming
which becomes
Meanwhile we could write down the MZA/bimodal solution in the form of
For MAA and MZA we can plot
Fig. 1.77
Fig. 1.78
On the other hand, we know
Fig. 1.79 Dispersion relation out of Eq. (1.36). For large k, the relation becomes proportional. The discontinuties are at location of
The Limits
There are several limits in the dispersion relation.
From the figure of
Another limit is
We can calculate
In general, we have the limits
I have to break each of the integral into two parts. I calculate
In other words, we have a box-like spetrum.
For MAA solution we define a function,
The dispersion relation is given by
Then we parametrically plot
Limits
Before we do any numerical calculations, we can calculate the limits first.
For
We also have the large k limit which are
Mathematically, we also have
For simplicity, we choose
Fig. 1.80 Dispersion relation for spectral crossing. The discontinuties are at
I can also plot the MAA and MZA soltions for
Fig. 1.81
Fig. 1.82
The reason we have no real values between
Fig. 1.83 The argument of the ln function. The vertical grid lines are
Some Discussions about
It seems that if we plot
In other words, the slopes of
Fig. 1.84 Function
Fig. 1.85
The corresponding dispersion relations are shown in Fig. 1.86
Fig. 1.86 DR for spectra
Fig. 1.87 Function
Fig. 1.88
Fig. 1.89 Dispersion relations for the above plot.¶
Why
It seems that crossing is important to a change in the number of solution to
And a change of the number of solutions indicates a possible gap. I need some verification about the relation between such non-explicit gap and instabilities.
However, I also notice that the combination
Can I derive some expression for the
As long as we have a point on
Or I can simply consider
Imagary Part in k
For MAA solution, I can try to solve
Didn’t find any weird numbers here.
Fig. 1.86 also indicates that crossing probably change the number of solutions to
I can analyze some points on the
Fig. 1.90 MZA solution for spectrum
Fig. 1.91 for MZA+ solution for spectrum
Fig. 1.92 for MZA- solution. The points for
Fig. 1.93 MZA solution for spectrum
Fig. 1.94 MZA for spectrum
Will we have a continuous case if the number of beams is infinite.
For discrete case
The continuous case is
We notice that Eq. (1.37) and Eq. (1.38) are the same when number of beams becomes large.
Fig. 1.95 Reproducing Fig. 3 of Raffelt’s paper.¶
Fig. 1.96 The spectrum I created and used.
The unstable regions can be calculated exactly by setting
Fig. 1.97 MAA¶
Fig. 1.98 MZA+¶
Fig. 1.99 MZA-¶
© 2021, Lei Ma | Created with Sphinx and . | On GitHub | Physics Notebook Statistical Mechanics Notebook Index | Page Source