6.2. Conservations¶

6.2.1. Conservation of Lepton Numbers¶

In the paper [Duan2009], Duan mentioned the conservation of neutrino lepton numbers.

In forward scattering, total neutrino number is conserved for each momentum. So the first conservation law we have is

(6.1)¶

(\partial_{t} + \hat{v} \cdot \nabla) n_{p} (t, x) = 0,

where $n_{p} (t, x)$ is the total neutrino number density for momentum $p$ at time $t$ and space point $x$ .

[Duan2009] defined the traceless flavor matrix $ρ_{p} (t, x)$ which describes the flavor states at each time and space point.

We will work in mass basis. In mass basis, we expect the vacuum Hamiltonian doesn’t change the two populations of density matrix thus preserving the number density of different mass states.

We also notice that for vacuum oscillations, the unitary transformation

ρ_{p} (t, x) = \exp (i ϕ σ_{3} / 2) ρ_{p} (t, x) \exp (- i ϕ σ_{3} / 2)

will not change the equation of motion. This is a global phase transformation which leads to a global conservation law by Noether’s theorem. The conserved quantity is the lepton numbers. This is also true with neutrino coherent scattering.

(6.2)¶

\partial_{t} n_{L} (t, x) + \nabla \cdot J_{L} (t, x) = 0,

where

\begin{aligned} n_{L} (t, x) & = \int_{p} n_{p} (t, x) \Tr σ_{3} ρ_{p} (t, x) \\ J_{L} (t, x) & = \int_{p} \hat{v} n_{p} (t, x) \Tr σ_{3} ρ_{p} (t, x) . \end{aligned}

Derivation of Conservation of Lepton Numbers

\Tr (\partial_{t} + \hat{v} \cdot \nabla) ρ_{p} (t, x) = 0.

To summarize, we found

conservation of total number of neutrino for each momentum (6.1), which is true only for forward scattering;
conservation of population in different mass states (6.2), which is valid for vacuum and coherent scattering;

6.2.2. References and Notes¶

Duan2009(1,2): Duan, H., Fuller, G. M., & Qian, Y.-Z. (2009). Neutrino Flavor Spin Waves. Journal of Physics G: Nuclear and Particle Physics, 36(10), 105003. https://doi.org/10.1088/0954-3899/36/10/105003