1-Loop improved lattice action for the nonlinear sigma-model

In this paper we show the Wilson effective action for the 2-dimensional O(N+1)-symmetric lattice nonlinear sigma-model computed in the 1-loop approximation for the nonlinear choice of blockspin $\mathrm{\Phi }(x)$, $\Phi(x)= \Cav\phi(x)/{|\Cav\phi(x)|}$,where$\Cav$isaveragingofthefundamentalfield$\phi(z)$overasquare$x$ofside$\tilde a$. The result for $S_{eff}$ is composed of the classical perfect action with a renormalized coupling constant ${\beta }_{eff}$, an augmented contribution from a Jacobian, and further genuine 1-loop correction terms. Our result extends Polyakov’s calculation which had furnished those contributions to the effective action which are of order $\ln \tilde{a}\mathrm{/}a$, where $a$ is the lattice spacing of the fundamental lattice. An analytic approximation for the background field which enters the classical perfect action will be presented elsewhere.