Effective action and vacuum expectations in nonlinear $\sigma$ model

TL;DR

This paper derives effective action equations for the nonlinear sigma model using DeWitt's method, providing loop-expansion solutions and a way to calculate vacuum expectations of various quantities, highlighting a singular limit as coupling tends to zero.

Contribution

It introduces a detailed derivation of effective action equations for the nonlinear sigma model and demonstrates how to compute vacuum expectations using these equations.

Findings

Vacuum expectation values can be calculated via the derived equations.

Loop-expansion solutions to the effective action equations are obtained.

The zero coupling limit is shown to be singular.

Abstract

The equations for effective action for nonlinear $\sigma$ model are derived using DeWitt method in two forms - for generator of vertex parts $\Gamma$ and for generator of weakly connected parts $W$. Loop-expansion solutions to these equations are found. It is shown that vacuum expectation values for various quantities including divergence of a N\"{o}ther current, trace of the energy-momentum tensor and so on, can be calculated by this method. Also it is shown that vacuum expectation to the sigma-field is determined by an explicit combination of tree Green function and classical solution. It is shown that the limit when coupling constant tends to zero is singular one.