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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