Lattice Perturbation Theory for $O(N)$-Symmetric $sigma$-Models with General Nearest-Neighbour Action I. Conventional Perturbation Theory

摘要

We compute the beta-function and the anomalous dimension of all thenon-derivative operators of the theory up to three-loops for the most generalnearest-neighbour O(N)-invariant action together with some contributions to thefour-loop beta-function. These results are used to compute the first analyticcorrections to various long-distance quantities as the correlation length andthe general spin-$n$ susceptibility. It is found that these corrections areextremely large for $RP^{N-1}$ models (especially for small values of N), sothat asymptotic scaling can be observed in these models only at very largevalues of beta. We also give the first three terms in the asymptotic expansionof the vector and tensor energies.