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.

\begin{equation}

\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}

\end{equation}

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

\begin{equation}\frac{dn_5}{dt}\end{equation}

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:

- Is it valid to consider this chiral asymmetry density for a massive fermion? And if so,
- How does this chiral asymmetry density evolve in the context of cosmology?