# Relativistic corrections to the static energy in terms of Wilson loops at weak coupling

## TL;DR

This paper calculates relativistic corrections to the static energy of heavy quarkonium using Wilson loops, providing perturbative results at short distances and comparisons with lattice data and string models.

## Contribution

It offers the first perturbative determination of ${
m O}(1/m)$ and ${
m O}(1/m^2)$ quasi-static energies, including ultrasoft contributions, for heavy quarkonium with unequal masses.

## Key findings

- Calculated short-distance behavior of quasi-static energies to ${
m O}(\alpha^3)$ and ${
m O}(\alpha^2)$.
- Resummed ultrasoft contributions to quasi-static energies.
- Compared lattice data with effective string model predictions.

## Abstract

We consider the ${\mathcal O}(1/m)$ and the spin-independent momentum-dependent ${\mathcal O}(1/m^2)$ quasi-static energies of heavy quarkonium (with unequal masses). They are defined nonperturbatively in terms of Wilson loops. We determine their short-distance behavior through ${\mathcal O}(\alpha^3)$ and ${\mathcal O}(\alpha^2)$, respectively. In particular, we calculate the ultrasoft contributions to the quasi-static energies, which requires the resummation of potential interactions. Our results can be directly compared to lattice simulations. In addition, we also compare the available lattice data with the expectations from effective string models for the long-distance behavior of the quasi-static energies.