The four-loop string effective action from the bosonic σ-model

Abstract

We discuss the formulation of the string effective action, in terms of quantities defined within the bosonic σ-model, at four loops or O( σ′ 4). In particular we show that a recent construction given by Osborn correctly yields the effective action at this order for the minimal subtraction renormalisation scheme. We demonstrate a renormalisation scheme for which Osborn's action reduces to a simpler form proposed by Tseytlin at this order. As part of our analysis, we calculate, using indirect methods, various O( σ′ 4) quantities for the σ-model, in particular the five-loop dilaton β-function.