Peskin & Schroeder Chap. 7.1 ultraviolet divergence

This post follows very closely reference [1]. A Dirac-spinor $\psi(x)$ (for simplicity the dependence of the spinors on $x=(t,\mathbf{r})$ is suppressed in the following)

$$\psi = \left(\begin{array}{c} \psi_L \\ \psi_R \end{array}\right) $$

can be decomposed in its chiral components $\psi_L$ and $\psi_R$ by the means of the $\gamma^5$ matrix:

$$\gamma^5 = \left(\begin{array}{cc} 0 & \mathbb{1} \\ \mathbb{1} & 0\end{array}\right)$$

(where $\mathbb{1}$ is a 2x2 unit matrix) in the following way:

$$\psi = \psi_L + \psi_R = \frac{1}{2}(1-\gamma^5)\psi + \frac{1}{2}(1+\gamma^5)\psi$$

The Lagrangian of the electron field can be written like:

$${\cal L} = \overline{\psi}(i\gamma^\mu \partial_\mu - m_0)\psi= \overline{\psi}_L i\gamma^\mu\partial_\mu \psi_L + \overline{\psi}_R i\gamma^\mu\partial_\mu \psi_R - m_0(\overline{\psi}_L \psi_R + \overline{\psi}_R \psi_L)$$.

Therefore if $m_0=0$, there is no coupling anymore between the chiral components $\psi_L$ and $\psi_R$. And the mass renormalisation does not destroy this decoupling, if it is then the case, since the perturbative correction is proportional to the bare mass which is zero. The perturbation theory remains valid, but the mass correction will be zero too. So the mass renormalisation does not change a zero mass.

This result actually is very good, because a massless Dirac theory enjoys an additional symmetry, the chiral symmetry that keeps the massless Lagrangian invariant:

$$\psi' = e^{i\gamma^5\phi} \psi$$

The found result shows that the chiral symmetry of the massless Lagrangian would not be destroyed by the perturbation theory.

EDIT: The mass renormalisation for fermions is nice, because $\delta m \propto m_0$ and it is even nice if $m_0\neq 0$ because it makes the mass correction term $\delta m \sim log(\Lambda/m)$. The latter is nice because even if one chooses as cut-off 10 times the mass of the universe, $\delta m$ remains tiny compared to $m_0$.

[1]: A. Zee, Quantum Field Theory in a Nutshell, Princeton University Press (2002)