Point-Particle Effective Field Theory II: Relativistic Effects and Coulomb/Inverse-Square Competition

We apply point-particle effective field theory (PPEFT) to compute the leading shifts due to finite-sized source effects in the Coulomb bound energy levels of a relativistic spinless charged particle; the analogue for spinless electrons of the charge-radius of the proton, and disagrees with standard calculations. We find there are {\em two} effective interactions with the same dimension that contribute to leading order in the nuclear size, one of which captures the standard charge-radius contribution. The other effective operator is a contact interaction whose leading contribution to $\delta E$ arises linearly (rather than quadratically) in the small length scale, $\epsilon$, characterizing the finite-size effects, and is suppressed by $(Z \alpha)^5$. We argue that standard calculations err in their choice of boundary conditions at the source for the wave-function of the orbiting particle. PPEFT predicts how this boundary condition depends on the source's charge radius, as well as on the orbiting particle's mass. Its contribution turns out to be crucial if the charge radius satisfies $\epsilon < (Z\alpha)^2 a_B$, where $a_B$ is the Bohr radius, because then relativistic effects become important for the boundary condition. A similar enhancement is not predicted for the hyperfine structure, due to its spin-dependence. We show how the charge-radius effectively runs due to classical renormalization effects, and why the resulting RG flow is central to predicting the size of the energy shifts (and is responsible for its being linear in the source size). We comment on the relevance of these results to the proton-radius problem. We argue in passing how this flow can also be relevant to describing `catalysis' within an EFT (ie why scattering from very small objects sometimes produces much larger-than-geometric cross sections), such as in monopole-catalyzed scattering.