Separate the modes of flavor isospins.
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 $m=\pm 1$.