Neutrino quantum kinetic equations: The collision term

We derive the collision term relevant for neutrino quantum kinetic equations in the early universe and compact astrophysical objects, displaying its full matrix structure in both flavor and spin degrees of freedom. We include in our analysis neutrino-neutrino processes, scattering and annihilation with electrons and positrons, and neutrino scattering off nucleons (the latter in the low-density limit). After presenting the general structure of the collision terms, we take two instructive limiting cases. The one-flavor limit highlights the structure in helicity space and allows for a straightforward interpretation of the off-diagonal entries in terms of the product of scattering amplitudes of the two helicity states. The isotropic limit is relevant for studies of the early universe: in this case the terms involving spin coherence vanish and the collision term can be expressed in terms of two-dimensional integrals, suitable for computational implementation.

Figure 1

Feynman graphs contributing to $\mathrm{\Sigma }(x)$. External lines represent neutrinos. Internal lines represent $\nu$, $e$, $n$, $p$ propagators. We represent each 4-fermion interaction vertex from Eqs. (2) in terms of two displaced fermionic current vertices.

Figure 2

Feynman graphs contributing to $\mathrm{\Pi }(k,x)$. We represent each 4-fermion interaction vertex from Eqs. (2) in terms of two displaced fermionic current vertices. External lines represent neutrinos. Internal lines represent $\nu$, $e$, $n$, $p$ propagators (left diagram) and $\nu$ propagators (right diagram).