M. V. Polyakov, A. A. Vladimirov, “Leading infrared logarithms for the <nobr>$\sigma$</nobr>-model with fields on an arbitrary Riemann manifold”, TMF, 2011, Volume 169, Number 1,Pages <nobr>158

This article is cited in 11 papers

Leading infrared logarithms for the $\sigma$-model with fields on an arbitrary Riemann manifold

M. V. Polyakov^ab, A. A. Vladimirov^b

^a Konstantinov Petersburg Nuclear Physics Institute, Gatchina, Leningrad Oblast, Russia
^b Institut für Theoretische Physik II, Ruhr--Universität, Bochum, Germany

Abstract: We derive a nonlinear recurrence equation for the infrared leading logarithms (LLs) in the four-dimensional $\sigma$-model with fields on an arbitrary Riemann manifold. The derived equation allows computing the LLs to an essentially unlimited loop order in terms of the geometric characteristics of the Riemann manifold. We reduce solving the $SU(\infty)$ principal chiral field in an arbitrary number of dimensions in the LL approximation to solving a very simple recurrence equation. This result prepares a way to solve the model in an arbitrary number of dimensions as $N\to\infty$.

Keywords: renormalization group, sigma model, large $N$.

Received: 20.10.2011

DOI: 10.4213/tmf6717