Point-particle effective field theory II: relativistic effects and Coulomb/inverse-square competition

Abstract

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. This is the analogue for spinless electrons of calculating the contribution of the charge-radius of the source to these levels, and our calculation disagrees 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 of which captures the standard charge-radius contribution. The other effective operator is a contact interaction whose leading contribution to δE arises linearly (rather than quadratically) in the small length scale, , characterizing the finite-size effects, and is suppressed by (Zα) 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 (Zα) 2 a B , where a B is the Bohr radius, because then relativistic effects become important for the boundary condition. We show how the problem is equivalent to solving the Schrödinger 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 (and is responsible for its being linear in the source size). We discuss how this flow is relevant to systems having much larger-than-geometric cross sections, such as those with large scattering lengths and perhaps also catalysis of reactions through scattering with monopoles. Experimental observation of these effects would require more precise measurement of energy levels for mesonic atoms than are now possible.

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

“…Effects like this, scaling linearly with R, are unusual and so lead to the question of whether similar shifts occur for the spin-half electrons and muons that arise in conventional and muonic atoms. We here address this question by extending the discussion of [5,6] to spin-half systems, finding that although many of the features of the Klein-Gordon problem of [6] also carry over to the Dirac field studied here the scaling of (1.1) does not: the corresponding leading Dirac expression instead gives the standard result:…”

“…We identify how the couplings for two-body contact interactions run, even at the classical level, and how this running goes over to the running found in [5,6] in the non-relativistic limit. This running properly captures how effective theories can sometimes generate scattering lengths that are much larger than the size R of the underlying object, and corresponds to the first-quantized version of a similar discussion found in [15][16][17][18][19].…”

“…The study in [5,6] considered both nonrelativistic and spinless relativistic particles orbiting the massive compact object, focussing in particular on unusual effects that arise if the compact source size, R, is small enough that relativistic kinematics is relevant for the matching problem to the interior physics of the source even for bound states whose total energy, ω, is nonrelativistic: m − ω m. This mixed relativistic/nonrelativistic regime occurs when mR v 1, where v ∼ Zα is the speed of the orbiting particle (whose mass is m). (Here we take the source charge to be Ze and α = e 2 /4π is the usual fine-structure constant.…”

“…Another result from [5,6] carries over to fermions: the fixed point of the running is not at c s = c v = 0 for charged sources (for which Zα = 0). It turns out this nontrivial…”

“…Our companion papers [5,6] explore how to formulate such effective theories using instead a first-quantized language for the heavy compact object, reserving the secondquantized language for the lighter particles with which it interacts. 1 In this mixed firstquantized/second-quantized (one-two) language, if the heavy (χ) particle is in a position eigenstate situated at x = 0 then the two-body contact interaction mentioned above instead has the form g(ψ * ψ) δ 3 (x).…”