Quark-antiquark static energy from a restricted Fourier transform

Abstract

We provide a fully analytical determination of the perturbative quark-antiquark static energy in position space as defined by a restricted Fourier transformation from momentum to position space. Such a determination is complicated by the fact that the static energy genuinely decomposes into a strictly perturbative part (made up of contributions $\sim\alpha_s^n$, with $n\in\mathbb{N}$) which is conventionally evaluated in momentum space, and a so-called ultrasoft part (including terms $\sim\alpha_s^{n+m}\ln^m\alpha_s$, with $n\geq3$ and $m\in\mathbb{N}$) which, conversely, is naturally evaluated in position space. Our approach facilitates the explicit determination of the static energy in position space at the accuracy with which the perturbative potential in momentum space is known, i.e., presently up to order $\alpha_s^4$.