Why Three-Body Physics Does Not Solve the Proton-Radius Puzzle

The possible involvement of weakly bound three-body systems in the muonic hydrogen spectroscopy experiment, which could resolve the current discrepancy between determinations of the proton radius, is investigated. Using variational calculations with complex coordinate rotation, we show that in the pμe ion, which was recently proposed as a possible candidate, the pμ core fails to bind the outer electron tightly enough to explain the discrepancy. It is also shown that the ppμ molecular ion cannot play any role in the observed line.