Analytic calculation of the 1-loop effective action for the O(N+1)-symmetric 2-dimensional nonlinear sigma-model

Starting from the 2-dimensional nonlinear sigma -model living on a lattice Lambda of lattice spacing a with action S[phi] = -1/2 beta integral (z) phi Delta phi, phi (z) is an element of S-N we compute a manifestly covariant closed form expression for the Wilson effective action S-eff[Phi] on a latt ice of lattice spacing (a) over tilde in a 1-loop approximation for a Gauss ian choice of blockspin, where C phi (x) equivalent to C phi (x)/\C phi (x) \ fluctuates around phi (x). C is averaging of phi (z) over a block x. The limiting case of a delta -function is also considered. The result extends Polyakov which had furnished those contributions to the effective action which are of order In (a) over tilde /a. The additional te rms which remain finite as a --> 0 include corrections other than coupling constant renormalization: a current-current interaction and a contribution from an augmented Jacobian which has a field dependence of a different kind than S has. Particular attention is paid to S-eff's domain of validity in field space. It turns out that Hasenfratz and Niedermayer's choice of a low value of the parameter k which governs the width of the Gaussian is optimal also in thi s respect. (C) 2001 Elsevier Science B.V. All rights reserved.