Heavy-particle formalism with Foldy-Wouthuysen representation

TL;DR

This paper develops a method to construct heavy-baryon Lagrangians using the Foldy-Wouthuysen representation, providing a clear $1/m$ expansion that maintains Lorentz invariance and simplifies effective field theory applications.

Contribution

It introduces a bottom-up approach to derive heavy-baryon Lagrangians without starting from a relativistic Lagrangian, avoiding spurious degrees of freedom and clarifying coupling redundancies.

Findings

Reproduces known pion-nucleon couplings

Identifies redundant $ ext{π}N ext{Δ}$ couplings in literature

Suggests dropping redundant couplings for accurate low-energy constants

Abstract

Utilizing the Foldy-Wouthuysen representation, we use a bottom-up approach to construct heavy-baryon Lagrangian terms, without employing a relativistic Lagrangian as the starting point. The couplings obtained this way feature a straightforward $1/m$ expansion, which ensures Lorentz invariance order by order in effective field theories. We illustrate possible applications with two examples in the context of chiral effective field theory: the pion-nucleon coupling, which reproduces the results in the literature, and the pion-nucleon-delta coupling, which does not employ the Rarita-Schwinger field for describing the delta isobar, and hence does not invoke any spurious degrees of freedom. In particular, we point out that one of the subleading $\pi N \Delta$ couplings used in the literature is, in fact, redundant, and discuss the implications of this. We also show that this redundant term should be dropped if one wants to use low-energy constants fitted from $\pi N$ scattering in calculations of $NN\to NN\pi$ reactions.