# Chirality Transition Rate

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The mass term of a fermion mixes the left- and right-handed chiral states, and in the limit that the mass goes to zero, the chiralities no longer mix at all and can be treated independently.

\mathcal{L} = \begin{pmatrix}\overline \psi_R & \overline \psi_L \end{pmatrix}
\begin{pmatrix} -m & i \sigma^\mu \partial_\mu \\
i \overline \sigma^\mu \partial_\mu & -m \end{pmatrix}
\begin{pmatrix} \psi_L \\ \psi_R \end{pmatrix}

What I wish to consider is density of the chiral asymmetry, $n_5 := n_R - n_L$.  This quantity should be well defined in the massless limit as the left- and right-handed components are independent; however, I'm wondering whether one can define this if a mass term does exist.

If it can be defined, then how does this asymmetry evolve in time?  Specifically, I wish to compute

$$\frac{dn_5}{dt}$$

in the context of cosmology.  I would naively expect that when $T \gg m$, then $\dot n_5 \to 0$ as the fermions are effectively massless compared to their energies, but I do not know how to compute this rate, nor was I able to find much literature on this.

So in all:

1. Is it valid to consider this chiral asymmetry density for a massive fermion?  And if so,
2. How does this chiral asymmetry density evolve in the context of cosmology?
asked Oct 28, 2019

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