Point-particle effective field theory III: relativistic fermions and the Dirac equation

We formulate pointparticle effective field theory (PPEFT) for relativistic spin half fermions interacting with a massive, charged finitesized source using a firstquantized effective field theory for the heavy compact object and a secondquantized language for the lighter fermion with which it interacts. This description shows how to determine the nearsource boundary condition for the Dirac field in terms of the relevant physical properties of the source, and reduces to the standard choices in the limit of a point source. Using a firstquantized effective description is appropriate when the compact object is sufficiently heavy, and is simpler than (though equivalent to) the effective theory that treats the compact source in a secondquantized way. As an application we use the PPEFT to parameterize the leading energy shift for the bound energy levels due to finitesized source effects in a modelindependent way, allowing these effects to be fit in precision measurements. Besides capturing finitesourcesize effects, the PPEFT treatment also efficiently captures how other shortdistance source interactions can shift boundstate energy levels, such as due to vacuum polarization (through the Uehling potential) or strong interactions for Coulomb bound states of hadrons, or any hypothetical new short-range forces sourced by nuclei.