# Perturbative violation of the unitarity: what is it?

Consider the Fermi theory:

$$\mathcal{L} = \frac{G_{F}}{2\sqrt{2}}\bar{n}\gamma_{\mu}(1-\gamma_{5})p \bar{\nu}\gamma^{\mu}(1-\gamma_{5})e$$

The cross section of $$2 \to 2$$ scattering calculated within the leading order of the perturbation theory grows with energy: $$\sigma_{\bar{p}n \to e\bar{\nu}} \sim G_{F}^{2}s$$ Thus, as is typically said, it violates the unitarity since it breaks down the "tree level unitarity" at high energies.

My question is the following: why the first statement follows from the second statement? Maybe the perturbation theory cannot be considered for energies $$s\gtrsim G_{F}^{-1}$$, since the effective coupling is $$G_{F}s$$?

• One assumes you are cool with this? – Cosmas Zachos Jun 25 '19 at 19:30