The absence of cut--off effects for the fixed point action in 1--loop perturbation theory

In order to support the formal renormalization group arguments that the fixed point action of an asymptotically free model gives cut–off independent physical predictions in 1–loop perturbation theory, we calculate the finite volume mass–gap $m(L)$ in the non–linear $\sigma$–model. No cut–off effect of the type $g^4{\left(a\mathrm{/}L\right)}^n$ is seen for any $n$. The results are compared with those of the standard and tree level improved Symanzik actions.