Physics ================== Reactions Related to Neutrinos ----------------------------------------- Before we actually reach a list of reactions, here are some most common examples which neutrinos participate in. 1. Beta decays, :math:`n\to p + e^- +\bar \nu` and :math:`p\to n + e^+ +\nu` 2. Electron capture and positron capture, :math:`e^- + p\to n+\nu` and :math:`e^+ + n \to p + \bar \nu`. 3. Inverse beta decays, :math:`\nu+ n \to p+e^-` and :math:`\bar\nu + p \to n + e^+`. 4. Inverses of beta decays, :math:`\bar\nu + e^- + p \to n` and :math:`n+e^++\nu \to n`. As for a list of possible reactions, here we have this table. .. figure:: assets/physics/neutrino-related-reactions.png :align: center Neutrino related nuclear or leptonic reactions. Another view of this is through drawing Feynman diagrams. There are six different kinds of Feynman diagrams that is related to neutrinos, * Elastic scattering * Quasi-elastic scattering * Resonance * Deep inelastic scattering * Coherent neutral current neutrino-neucleus elastic scattering * Neutrino-neutrino self-interaction. Examples of elastic scattering: .. image:: assets/physics/elastic-scattering-cc.png :width: 45% .. image:: assets/physics/elastic-scattering-nc.png :width: 45% .. image:: assets/physics/tau-neutrino-and-electron-nc.png :width: 45% .. image:: assets/physics/elastic-scattering-neutrino-different-family.png :width: 45% Example of quasi-elastic scattering .. image:: assets/physics/quasi-elastic-scattering-neutrinos.png :width: 45% Example of resonance .. image:: assets/physics/resonance-neutrinos.png :width: 45% Example of deep inelastic scattering .. image:: assets/physics/deep-inelastic-scattering-neutrinos.png :width: 45% Example of coherent neutral current scattering .. image:: assets/physics/coherent-nc-neutrinos.png :width: 45% What is a Neutrino Particle? ------------------------------------- As Wigner said, a physical particle is an irreducible representation of the Poincaré group. A characteristic of Poincaré group is that mass comes in. A neutrino particle is better recognized as its mass eigenstate. In QFT, there are 3 different forms of neutrino mass term, left-handed Majorana, right-handed Majorana and Dirac mass terms. Chirality and Helicity ----------------------------- .. index:: Helicity Helicity ~~~~~~~~~~~~~~ **Helicity** is the projection of spin onto direction of momentum, .. math:: h = \vec J\cdot\hat p = \vec L\cdot\hat p + \vec S\cdot \hat p = \vec S\cdot \hat p, where .. math:: \hat p = \frac{\vec p}{\left|\vec p\right|} A state is called **right-handed** if helicity is positive, i.e., spin has the same direction as momentum. .. index:: Chirality Chirality ~~~~~~~~~~~~ **Chirality** is the eigenstate of the Dirac :math:`\gamma_5` matrix, which is explicitly, [1]_ .. math:: \gamma^5 &= \begin{pmatrix} \mathbf 0 & \mathbf I \\ \mathbf I & \mathbf 0 \end{pmatrix} \\ & = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix}. Majorana or Dirac -------------------------------- Double Beta Decay ~~~~~~~~~~~~~~~~~ States -------------------------------------- .. index:: Wigner Function Wigner Function ~~~~~~~~~~~~~~~~~~~~~~~~ .. figure:: assets/physics/classicalProbDist.jpg :align: center A ensemble of classical harmonic oscillators can be described using such phase-space probability distribution. Wigner function is an analogue of the classical phase-space probability distribution function though it is not really probability. [3]_ The mean of Wigner function lies in the two quadratures, i.e., space distribution and momentum distribution. There is a collection of Wigner functions on this site. [3]_ .. [3] `http://www.iqst.ca/quantech/wigner.php `_ .. admonition:: Question :class: note How do one describe a system of neutrinos using Wigner function? The effect of statistics? Statistics ------------------------------- Fermi-Dirac distribution .. math:: f(p,\xi) = \frac{1}{1+\exp (p/T-\xi)}, where :math:`\xi=\mu/T` is the degeneracy parameter. The neutrino-neutrino forward scattering is [2]_ .. math:: \nu_\alpha (p) + \nu_\beta (k) \to \nu_\alpha (k)+\nu_\beta (p). .. admonition:: Question :class: warning Meaning of each term in Liouville equation. .. [2] Pantaleone (1992), Friedland & Lunardini (2003). Refs & Notes ------------------- .. [1] `*Chirality and Helicity In Depth* by Robert D. Klauber `_