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

Starting from the 2-dimensional nonlinear σ-model living on a lattice Λ of lattice spacing a with action S[φ]=−12β∫zφΔφ, φ(z)∈SN we compute a manifestly covariant closed form expression for the Wilson effective action Seff[Φ] on a lattice of lattice spacing ã in a 1-loop approximation for a Gaussian choice of blockspin, where Cφ(x)≡Cφ(x)/|Cφ(x)| fluctuates around Φ(x). C is averaging of φ(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 lnã/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 Seffʹs domain of validity in field space. It turns out that Hasenfratz and Niedermayerʹs choice of a low value of the parameter κ which governs the width of the Gaussian is optimal also in this respect.