Comparison of the heavy-fermion and Foldy-Wouthuysen formalisms at third order

TL;DR

This paper compares the heavy-fermion and Foldy-Wouthuysen non-relativistic reduction schemes, providing explicit transformations up to second order in 1/m and analyzing scheme-dependent corrections to low-energy constants.

Contribution

It explicitly derives the transformation between the two NR reduction schemes to second order, clarifying scheme dependence and the role of field renormalization.

Findings

Transformations between schemes are explicitly obtained to O(1/m^2).

Scheme-dependent corrections to low-energy constants are identified.

Field renormalization aligns fixed coefficient corrections across schemes.

Abstract

We compare two non-relativistic (NR) reduction schemes (heavy-fermion and Foldy-Wouthuysen) that are used to derive low-energy effective-field-theory Lagrangians. We give the explicit transformation between the two types of fields to O(1/m^2), derived from a quite general, relativistic Lagrangian. Beyond leading order the NR reductions always involve the smaller components of the Dirac spinors that are to be integrated out to formulate the NR theory. Even so, the transformation between the NR Lagrangians can be carried out explicitly to O(1/m^2) using a field renormalization, as long as the lower components of the Lagrangian are known. The fixed coefficient corrections to some low-energy constants at O(1/m^2) will depend on the particular scheme chosen, but will match after the field renormalization.