Effective Field Theory for the Bound States and Scattering of a Heavy Charged Particle and a Neutral Atom

TL;DR

This paper develops an effective field theory for a system of a heavy charged particle and a neutral atom, accurately describing scattering and bound states by focusing on the dominant $1/r^4$ potential and including higher-order corrections.

Contribution

It introduces the Induced-dipole EFT (ID-EFT), a novel low-energy framework that captures the scattering phase shifts and bound states of the system with systematic corrections.

Findings

ID-EFT reproduces phase shifts of the microscopic potential over a wide energy range.

At leading order, it describes the highest-lying bound states accurately.

Next-to-leading order corrections improve the description of lower-lying bound states.

Abstract

We show the system of a heavy charged particle and a neutral atom can be described by a low-energy effective field theory where the attractive $1/r^4$ induced dipole potential determines the long-distance/low-energy wave functions. The $1/r^4$ interaction is renormalized by a contact interaction at leading order. Derivative corrections to that contact interaction give rise to higher-order terms. We show that this ``Induced-dipole EFT'' (ID-EFT) reproduces the $\pi^+$-hydrogen phase shifts of a more microscopic potential, the Temkin-Lamkin potential, over a wide range of energies. Already at leading order it also describes the highest-lying excited bound states of the pionic-hydrogen ion. Lower-lying bound states receive substantial corrections at next-to-leading order, with the size of the correction proportional to their distance from the scattering threshold. Our next-to-leading order calculation shows that the three highest-lying bound states of the Temkin-Lamkin potential are well-described in ID-EFT.