Point-particle effective field theory II: relativistic effects and Coulomb/inverse-square competition
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We apply point-particle effective field theory (PPEFT) to compute the leading
shifts due to finite-size source effects in the Coulomb bound energy levels of
a relativistic spinless charged particle. This is the analogue for spinless
electrons of the contribution of the charge-radius of the source to these
levels, and we disagree with standard calculations in several ways. Most
notably we find there are two effective interactions with the same dimension
that contribute to leading order in the nuclear size. One is the standard
charge-radius contribution, while the other is a contact interaction whose
leading contribution to $\delta E$ arises linearly in the small length scale,
$\epsilon$, characterizing the finite-size effects, and is suppressed by
$(Z\alpha)^5$. We argue that standard calculations miss the contributions of
this second operator because they 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 $\epsilon \lesssim (Z\alpha)^2 a_B$, with $a_B$ the
Bohr radius, since then relativistic effects become important. We show how the
problem is equivalent to solving the Schr\"odinger equation with competing
Coulomb, inverse-square and delta-function potentials, which we solve
explicitly. 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. We discuss how this flow is
relevant to systems having much larger-than-geometric cross sections, and the
possible relevance to catalysis of reactions through scattering with monopoles.