Reducing the O(3) model as an effective field theory

TL;DR

This paper analyzes the O(3) nonlinear sigma model as an effective field theory, demonstrating how to eliminate instabilities at the four-derivative level and revealing new dynamical instabilities at higher orders.

Contribution

It provides a systematic classification of operators in the O(3) model's effective Lagrangian and shows how to avoid Ostrogradsky instabilities through field redefinitions, uncovering new dynamical instabilities.

Findings

Four-derivative theory avoids Ostrogradsky instability

Six-derivative order introduces dynamical instabilities

Field redefinitions can eliminate certain higher-derivative dependencies

Abstract

We consider the O(3) or CP(1) nonlinear sigma model as an effective field theory in a derivative expansion, with the most general Lagrangian that obeys O(3), parity and Lorentz symmetry. We work out the complete list of possible operators (terms) in the Lagrangian and eliminate as many as possible using integrations by parts. We further show at the four-derivative level, that the theory can be shown to avoid the Ostrogradsky instability, because the dependence on the d'Alembertian operator or so-called box, can be eliminated by a field redefinition. Going to the six-derivative order in the derivative expansion, we show that this can no longer be done, unless we are willing to sacrifice Lorentz invariance. By doing so, we can eliminate all dependence on double time derivatives and hence the Ostrogradsky instability or ghost, however, we unveil a remaining dynamical instability that takes the form either as a spiral instability or a runaway instability and estimate the critical field norm, at which the instability sets off.