# Coulomb expectation values in $D=3$ and $D=3-2\epsilon$ dimensions

## TL;DR

This paper provides detailed methods and formulas for calculating expectation values in Coulomb bound states across three and near-three dimensions, aiding dimensional regularization in quantum electrodynamics corrections.

## Contribution

It introduces new techniques and formulas for evaluating expectation values in D=3 and D=3-2ε dimensions, including divergent and finite cases, enhancing dimensional regularization methods.

## Key findings

- Derived formulas for expectation values involving Laguerre polynomials.
- Presented perturbative expansions in ε for bound state energies.
- Provided regularized expressions for divergent expectation values.

## Abstract

We explore the quantum Coulomb problem for two-body bound states, in $D=3$ and $D=3-2\epsilon$ dimensions, in detail, and give an extensive list of expectation values that arise in the evaluation of QED corrections to bound state energies. We describe the techniques used to obtain these expectation values and give general formulas for the evaluation of integrals involving associated Laguerre polynomials. In addition, we give formulas for the evaluation of integrals involving subtracted associated Laguerre polynomials--those with low powers of the variable subtracted off--that arise when evaluating divergent expectation values. We present perturbative results (in the parameter $\epsilon$) that show how bound state energies and wave functions in $D=3-2\epsilon$ dimensions differ from their $D=3$ dimensional counterparts and use these formulas to find regularized expressions for divergent expectation values such as $\big \langle \bar V^3 \big \rangle$ and $\big \langle (\bar V')^2 \big \rangle$ where $\bar V$ is the $D$-dimensional Coulomb potential. We evaluate a number of finite $D$-dimensional expectation values such as $\big \langle r^{-2+4\epsilon} \partial_r^2 \big \rangle$ and $\big \langle r^{4\epsilon} p^4 \big \rangle$ that have $\epsilon \rightarrow 0$ limits that differ from their three-dimensional counterparts $\big \langle r^{-2} \partial_r^2 \big \rangle$ and $\big \langle p^4 \big \rangle$. We explore the use of recursion relations, the Feynman-Hellmann theorem, and momentum space brackets combined with $D$-dimensional Fourier transformation for the evaluation of $D$-dimensional expectation values. The results of this paper are useful when using dimensional regularization in the calculation of properties of Coulomb bound systems.