# Scattering amplitudes versus potentials in nuclear effective field theory: search for a potential compromise

## TL;DR

This paper investigates whether expanding nuclear potentials in effective field theory ensures that the resulting amplitudes follow the same power counting, revealing potential discrepancies and exploring solutions for consistent expansions.

## Contribution

It explicitly examines the relationship between potential expansion and amplitude power counting in nuclear effective field theory, proposing potential strategies for consistency.

## Key findings

- Amplitudes sometimes violate the expected power counting.
- Expanding potentials does not always guarantee consistent amplitude expansion.
- Constructing specific potentials can ensure amplitudes follow the desired power counting.

## Abstract

In effective field theory physical quantities, in particular observables, are expressed as a power series in terms of a small expansion parameter. For non-perturbative systems, for instance nuclear physics, this requires the non-perturbative treatment of at least part of the interaction (or the potential, if one is dealing with a non-relativistic system), while the rest of the interaction is included as perturbations. This is not entirely trivial and as a consequence different interpretations on how to treat these systems have appeared. A practical approach is to expand the effective potential, where this potential is later fully iterated in the Schroedinger equation for obtaining amplitudes and observables. The expectation is that this will lead to observables that will have an implicit power counting expansion. Here I explicitly check whether the amplitudes (when expanded according to the counting) are actually following the same power counting as the potential. It happens that reality does not necessarily conform to expectations and the amplitudes will sometimes violate the power counting with which the potential has been expanded. A more formal approach is to formulate the expansion directly in terms of amplitudes and observables, which is the original aim of the effective field theory idea. Yet this second approach is technically complicated. I explore here the possibility of constructing potentials that when fully iterated will make sure that amplitudes are indeed expansible in terms of a small expansion parameter.