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-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 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. 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.

LaTeX, 22 pages plus appendices, v3: revised appendices, made more precise and concise discussion about proton radius for mesonic systems

References in corpus (17)

• Introduction to Effective Field Theory
• The Proton Radius Puzzle
• K^- p scattering length from scattering experiments
• Model independent extraction of the proton charge radius from electron scattering
• Muonic hydrogen and MeV forces
• New Parity-Violating Muonic Forces
• Proton size anomaly
• Model independent extraction of the proton magnetic radius from electron scattering
• Proton radius from electron-proton scattering and chiral perturbation theory
• SO(2,1) conformal anomaly: Beyond contact interactions
• Efimov physics from a renormalization group perspective
• Point-Particle Effective Field Theory I: Classical Renormalization and the Inverse-Square Potential
• The Hierarchy Problem and the Self-Localized Higgs
• Model independent determination of the muonic hydrogen Lamb shift and proton radius
• Point-Particle Effective Field Theory III: Relativistic Fermions and the Dirac Equation
• On the Inequivalence of Renormalization and Self-Adjoint Extensions for Quantum Singular Interactions
• Kaonic hydrogen atom and kaon-proton scattering length

Cited by in corpus (16)

• Review of experimental and theoretical status of the proton radius puzzle
• Effective Field Theory of Black Hole Echoes
• Point-Particle Effective Field Theory I: Classical Renormalization and the Inverse-Square Potential
• Constraining Fundamental Physics with the Event Horizon Telescope
• Point-Particle Effective Field Theory III: Relativistic Fermions and the Dirac Equation
• Gravity, Horizons and Open EFTs
• Fall to the Centre in Atom Traps and Point-Particle EFT for Absorptive Systems
• Reduced Theoretical Error for QED Tests with 4He+ Spectroscopy
• Nuclear Predictions for $H$ Spectroscopy without Nuclear Errors
• Duality between the quantum inverted harmonic oscillator and inverse square potentials
• A perturbative method for resolving contact interactions in quantum mechanics
• Precision Nuclear-Spin Effects in Atoms: EFT Methods for Reducing Theory Errors
• On the EFT of Dyon-Monopole Catalysis
• Non-relativistic Effective Quantum Mechanics of the Coulomb Interaction
• Influence Through Mixing: Hotspots as Benchmarks for Basic Black-Hole Behaviour
• Point-Particle Catalysis