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We formulate point-particle effective field theory (PPEFT) for relativisticspin-half fermions interacting with a massive, charged finite-sized sourceusing a first-quantized effective field theory for the heavy compact object anda second-quantized language for the lighter fermion with which it interacts.This description shows how to determine the near-source boundary condition forthe Dirac field in terms of the relevant physical properties of the source, andreduces to the standard choices in the limit of a point source. Using afirst-quantized effective description is appropriate when the compact object issufficiently heavy, and is simpler than (though equivalent to) the effectivetheory that treats the compact source in a second-quantized way. As anapplication we use the PPEFT to parameterize the leading energy shift for thebound energy levels due to finite-sized source effects in a model-independentway, allowing these effects to be fit in precision measurements. Besidescapturing finite-source-size effects, the PPEFT treatment also efficientlycaptures how other short-distance source interactions can shift bound-stateenergy levels, such as due to vacuum polarization (through the Uehlingpotential) or strong interactions for Coulomb bound states of hadrons, or anyhypothetical new short-range forces sourced by nuclei.

Point-Particle Effective Field Theory III: Relativistic Fermions and the Dirac Equation