Leading infrared logarithms for sigma-model with fields on arbitrary Riemann manifold

TL;DR

This paper derives a recursive equation to compute leading infrared logarithms in four-dimensional sigma-models on arbitrary Riemann manifolds, enabling high-order calculations based on geometric properties.

Contribution

It introduces a non-linear recursion relation for leading infrared logs in sigma-models on arbitrary Riemann manifolds, simplifying complex calculations at high loop orders.

Findings

Derived recursion equation for infrared logs

Reduced SU(oo) principal chiral field solution to simple recursion

Enabled computation of logs to unlimited loop order

Abstract

We derive non-linear recursion equation for the leading infrared logarithms (LL) in four dimensional sigma-model with fields on an arbitrary Riemann manifold. The derived equation allows one to compute leading infrared logarithms to essentially unlimited loop order in terms of geometric characteristics of the Riemann manifold. We reduce the solution of the SU(oo) principal chiral field in arbitrary number of dimensions in the LL approximation to the solution of very simple recursive equation. This result paves a way to the solution of the model in arbitrary number of dimensions at N-->oo