# #Today I Read#

**Author:**Lei Ma

**Summary:**Just some key ideas of Flavor Oscillation Modes in Dense Neutrino Media, not a complete story.

**Categories:**{ collective oscillations }

**Tags:**#collective oscillations #symmetries in neutrino oscillations

**References:**- Flavor Oscillation Modes in Dense Neutrino Media

## Method

Separate the modes of flavor isospins.

## Results

### Two Colliding Beams

- Plus mode (summation of d modes for the two beams) is unstable in IH; Minus mode is unstable in NH.

### With Axial Distribution

- For some axial distribution of emission, axial part of spherical harmonic expansion of the modes shows that the equation of motions for the modes are decoupled. And the $\lvert m\rvert > 1$ modes do not depend on the number density of neutrinos and are the same.
- The $m=0$ mode, aka axial symmetric mode, is just like the plus mode in two colliding beams model, which means it’s unstable in IH.
- The $m=\pm 1$ modes, behaves like the minus mode in two colliding beams model, which means it is unstable in NH.

### A More General Angular Distribution

- Expand the modes using spherical harmonics shows that the $l$, $m$ modes are decoupled and $l=0$ works like $m=0$ in previous case and l=1 works like $m=\pm 1$.

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