Classical Nonrelativistic Effective Field Theories for a Real Scalar Field

TL;DR

This paper corrects errors in previous derivations of nonrelativistic effective Lagrangians for a real scalar field, demonstrating their equivalence through T-matrix calculations and field redefinitions.

Contribution

It identifies and corrects errors in existing effective Lagrangians and proves their equivalence using T-matrix elements and field redefinitions.

Findings

Corrected effective Lagrangians for a real scalar field.

Established the equivalence of the two corrected Lagrangians.

Validated the equivalence through T-matrix element comparisons.

Abstract

A classical nonrelativistic effective field theory for a real Lorentz-scalar field $\phi$ is most conveniently formulated in terms of a complex scalar field $\psi$. There have been two derivations of effective Lagrangians for the complex field $\psi$ in which terms in the effective potential were determined to order $(\psi^* \psi)^4$. We point out an error in each of the effective Lagrangians. After correcting the errors, we demonstrate the equivalence of the two effective Lagrangians by verifying that they both reproduce $T$-matrix elements of the relativistic real scalar field theory and by also constructing a redefinition of the complex field $\psi$ that transforms terms in one effective Lagrangian into the corresponding terms of the other.