# 7.2. Equation of Motion¶

The most important reference about the equation of motion for neutrinos is [sigl1993].

 [sigl1993] Sigl, G., & Raffelt, G. (1993). General kinetic description of relativistic mixed neutrinos. Nuclear Physics B, 406(1–2), 423–451. http://doi.org/10.1016/0550-3213(93)90175-O

Without much effort, we know that the equation of motion is the liouville’s equation

$i \frac{d}{dt}\rho = [H,\rho].$

For the purpose of neutrino oscillations, we always assume they travel with speed of light thus

$\frac{d}{dt} = \frac{d}{dr},$

where $$r$$ is the distance travelled by the neutrino. However, in general, what we should have is

$\frac{d}{dt} = \partial_t + \mathbf v\cdot \boldsymbol{\nabla}.$

The Hamiltonian is composed of three different terms,

$H = H_v + H_m + H_{\nu\nu},$

where

$\begin{split}H_v =& -\frac{1}{2}\beta\eta \omega_0 \sigma_3\\ H_m =& \frac{1}{2} \sqrt{2}G_F n_e \sigma_3 \\ H_{\nu\nu} =& \sqrt{2}G_F \int d\omega d\Omega_{\hat v'} n(\omega,\hat v')\beta(\hat v')\rho(\omega,\hat v') (1-\hat v \cdot \hat v'),\end{split}$

where $$\eta=\pm 1$$ for Normal Hierarchy and Inverted Hierarchy respectively, $$\beta=1$$ for neutrinos and $$\beta=-1$$ for antineutrinos. In other words, the vacuum frequency is $$\omega_v = \eta \omega_0$$. $$\beta(\hat v')$$ indicates wether the density matrix $$\rho(\omega,\hat v')$$ is for neutrinos or antineutrinos. If $$\rho(\omega,\hat v')$$ is for antineutrinos, $$\beta(\hat v')=-1$$, otherwise $$\beta(\hat v')=1$$. More explicitly, the neutrino-neutrino interaction Hamiltonian is

$\begin{split}H_v =& \begin{cases} -\frac{1}{2}\eta \omega_0 \sigma_3 & \text{for neutrinos}\\ \frac{1}{2}\eta \omega_0 \sigma_3 & \text{for antineutrinos} \end{cases}\\ H_{\nu\nu} =& \begin{cases} \sqrt{2}G_F \int d\omega d\Omega_{\hat v'} n(\omega,\hat v')\rho(\omega,\hat v') (1-\hat v \cdot \hat v') & \text{interacting with neutrinos} \\ - \sqrt{2}G_F \int d\omega d\Omega_{\hat v'} n(\omega,\hat v')\bar\rho(\omega,\hat v') (1-\hat v \cdot \hat v') & \text{interacting with antineutrinos} \end{cases}\end{split}$

Please note that in this notion,

1. $$\omega_0$$ is meant to be the absolute value of the frequency, since $$\eta$$ takes care of the signs.
2. the integral in $$H_{\nu\nu}$$ must take care of both interactions with neutrinos and anti-neutrinos, thus the density matrix is not only for neutrinos.

For the simplicity of notions, we define some new quantities.

1. We define $$\lambda$$ to measure the matter interactions

$\lambda = \sqrt{2} G_F n_e.$
2. Angle distribution of number density is defined as

$f(\hat v) = \frac{n(\omega,\hat v)}{n_{total}},$

where $$n_{total}$$ is the total number density of neutrinos for all energies. It can also be defined for anti-neutrinos

$\bar f(\hat v) = \frac{n(\omega,\hat v)}{\bar n_{total}},$

where $$\bar n_{total}$$ is the total number density of anti-neutrinos.

In fact, the direction of momentum $$\hat v$$ depends only on an angle for line models, hence $$f(\theta)$$. With this definition, we know that the number density of neutrinos within an angle $$[\theta, \theta + d\theta]$$ can be calculated

$n_{total} f(\theta) d\theta.$

Similarly, the the number density of antineutrinos within angle $$[\theta, \theta+d\theta]$$ is

$\bar n_{total} \bar f(\theta) d\theta.$
3. Total number density of neutrinos and anti-neutrinos are related through a asymmetry parameter

$\alpha = \frac{\bar n_{total} }{n_{total}}.$

With the two definitions we simplify the matter effect and neutrino self-interaction

$\begin{split}H_m =& \frac{1}{2} \lambda \sigma_3 \\ H_{\nu\nu} =& \sqrt{2}G_F n_{total} \int d\omega d\Omega_{\hat v'} f(\omega,\hat v)\rho(\omega,\hat v') (1-\hat v \cdot \hat v') \\ & - \sqrt{2}G_F \bar n_{total} \int d\omega d\Omega_{\hat v'} \bar f(\omega,\hat v)\bar\rho(\omega,\hat v') (1-\hat v \cdot \hat v') \\ =& \frac{1}{2}\mu \int d\omega d\Omega_{\hat v'} f(\omega, \hat v)\rho(\omega,\hat v') (1-\hat v \cdot \hat v') \\ & - \frac{1}{2}\alpha \mu \int d\omega d\Omega_{\hat v'} \bar f(\omega, \hat v)\bar\rho(\omega,\hat v') (1-\hat v \cdot \hat v') ,\end{split}$

where

$\mu = 2\sqrt{2} G_F n_{total}.$

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