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For an accurate theoretical description of muonic bound systems, it is crucial to consistently treat relativistic and recoil corrections to vacuum polarization. The one-loop vacuum-polarization effect is by far the dominant quantum electrodynamic (QED) energy correction for bound muons, being of order (Z) ² mᵣ, where is the fine-structure constant, Z is the nuclear charge number, and mᵣ is the reduced mass. Gauge invariance of the relativistic and recoil corrections to vacuum polarization of order (Z) ⁴ mᵣ is investigated with respect to nonretarded and standard, renormalized variants of Coulomb gauge. The invariance is shown after including three-photon exchange diagrams. Our derivation is based on an adapted form of Nonrelativistic Quantum Electrodynamics for bound muon systems (NRQED_), which is a version of NRQED where the hard scale is set at the muon mass instead of the electron mass. Updated values for the gauge-independent corrections for one-muon ions with nuclear charge numbers Z = 1, 2, 6 are presented.
Adkins et al. (Thu,) studied this question.