#Today I Read#
The authors consider collision less neutrino propagation. The convention for Hamiltonian is
The idea is that axial symmetry can be spontaneously broken in multiangle approach or continuous angle distribution. That is to say, even we start from a axial symmetric configuration, the axial symmetric for flavor instabilities can be broken locally.
Comment: This result seems to be weird. We have axial symmetry of initial conditions. Meanwhile the equation of motion preserves the axial symmetry.
Notations
Here is a list of all the definitions and notations.

The vacuum Hamiltonian

Matter contribution to Hamiltonian
 $\theta_R$ position of emission point on neutrino sphere.
 $u=\sin^2\theta_R$ is a parameter that is used to work out the radial velocity $v_{r,u}=\sqrt{1u R^2/r^2}$ and transverse velocity $\beta_{r,u}=\sqrt{u}R/r$.

Flux matrix is related to density matrix
Flux matrix can be used instead of density matrix to describe the dynamics,

Coherent scattering becomes

Linearization is done using flux matrix

Matter potential
and

Normalized selfinteraction potential
Linearized Equation of Motion
Assuming the solution to perturbation $S$ has the form $S = Q_\Omega e^{i\Omega r}$, where $\Omega=\gamma + i \kappa$.
Question: What happens if
Then the equation of motion becomes the rhs integral being 0. Either the integrand is zero, or the integrand has positive and negative regions and they sum up to zero.
I don’t think we can solve this equation.
The integral on the right hand side should have the form
The solution to $Q_\Omega$ should be of the form
Then we plug this result back into the equation of motion and find out a matrix equation for the unknown coefficients $a,b,c,d$. We require that the equation has nontrivial solutions. So the problem becomes an eigenvalue problem.
Axial Symmetric Emission
The spectrum $g$ doesn’t depend on $\phi$. But the solution to frequency can depend on $\phi$.
About Equation 13
In the beginning, it wasn’t so clear how could it be true.
Mathematically, with $I$ defined,
In equation 12,
becomes
Move them to the same side, we have
from which we solve $1/I$.
From the meaning of $a,b,c,d$ we know that $c,d$ are associated with angle $\phi$ so they are related to the MAA solution. We also know that the solution $I_1=1$ is the one related to $c,d$ which indicates that this is the MAA solution.
The paper also says that
is bimodal solution while
is the MZA solution.
Not sure why.
Single Energy Model
Energy spectrum is
The author solved system
 Inverted hierarchy ($\omega_0>0$), bimodal instability.
 NH ($\omega_0<0$), MZA & MAA instability.
Matter Effect
At large matter density, instability occurs at larger $\sim\lambda/\lvert q_j\rvert\sim \lambda$. Matter suppression of instability.
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