Why three-body physics do not solve the proton radius puzzle

TL;DR

This paper investigates whether weakly bound three-body systems like pμe or ppμ could explain the proton radius puzzle, concluding they do not have resonant states or influence in the relevant energy region.

Contribution

It provides a detailed variational analysis showing that proposed three-body systems lack resonant states and cannot resolve the proton radius discrepancy.

Findings

pμe ion has no resonant states in the energy region of interest

ppμ molecular ion cannot influence the observed spectral line

Three-body systems do not resolve the proton radius puzzle

Abstract

The possible involvement of weakly bound three-body systems in the muonic hydrogen spectroscopy experiment [1], which could resolve the current discrepancy between determinations of the proton radius, is investigated. Using variational calculations with complex coordinate rotation, it is shown that the $p\mu e$ ion, which was recently proposed as a possible candidate [2], has no resonant states in the energy region of interest. QED level shifts are included phenomenologically by including a Yukawa potential in the three-body Coulomb Hamiltonian before diagonalization. It is also shown that the $pp\mu$ molecular ion cannot play any role in the observed line.