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The possible involvement of weakly bound three-body systems in the muonichydrogen spectroscopy experiment [1], which could resolve the currentdiscrepancy between determinations of the proton radius, is investigated. Usingvariational calculations with complex coordinate rotation, it is shown that the$p\mu e$ ion, which was recently proposed as a possible candidate [2], has noresonant states in the energy region of interest. QED level shifts are includedphenomenologically by including a Yukawa potential in the three-body CoulombHamiltonian before diagonalization. It is also shown that the $pp\mu$ molecularion cannot play any role in the observed line.

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