A determination of the mass gap in the O(n) sigma model

We calculate the finite volume mass gap $M(L)$ at 3-loop level in the non-linear O($n$) $\sigma$-model in two dimensions in small volumes. By applying the Monte Carlo measurements of the running coupling $\bar g^2(L)=2nM(L)L/(n-1)$ by L"uscher, Weisz and Wolff measured in units of the physical mass gap $m$, the result is used to determine $m$ in units of the $\mathrm{\Lambda }$-parameter in the O(3) and O(4) models. Our determinations show good agreement with those by Hasenfratz, Maggiore and Niedermayer in both models. We note that this manuscript has been revised in our paper hep-lat/9810025 by using the corrected four-loop $\beta$-function on the lattice.