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

Starting from the 2-dimensional nonlinear σ-model living on a lattice Λ of lattice spacing a with action S[phi]=-1/2β∫zphiΔphi, phi(z) is a subset of SN we compute a manifestly covariant closed form expression for the Wilson effective action Seff[PHI] on a lattice of lattice spacing a-tilde in a 1-loop approximation for a Gaussian choice of blockspin, where Cphi(x)≡Cphi(x)/ vertical barCphi(x) vertical bar fluctuates around PHI(x). C is averaging of phi(z) over a block x. The limiting case of a δ-function is also considered. The result extends Polyakov which had furnished those contributions to the effective action which are of order lna-tilde/a. The additional terms 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 Seff's domain of validity in field space. It turns out that Hasenfratz and Niedermayer's choice of a low value of the parameter κ which governs the width of the Gaussian is optimal also in this respect