Static QCD potential atr<ΛQCD−1: Perturbative expansion and operator-product expansion

Abstract

We analyze the static QCD potential ${V}_{\mathrm{QCD}}(r)$ in the distance region $0.1\text{ }\text{ }\mathrm{fm}\ensuremath{\lesssim}r\ensuremath{\lesssim}1\text{ }\text{ }\mathrm{fm}$ using perturbative QCD and operator-product expansion (OPE) as basic theoretical tools. We assemble theoretical developments up to date and perform a solid and accurate analysis. The analysis consists of three major steps: (I) We study large-order behavior of the perturbative series of ${V}_{\mathrm{QCD}}(r)$ analytically. Higher-order terms are estimated by large-${\ensuremath{\beta}}_{0}$ approximation or by renormalization group, and the renormalization scale is varied around the minimal-sensitivity scale. A $``\mathrm{\text{Coulomb}}''+\mathrm{\text{linear}}$ potential can be identified with the scale-independent and renormalon-free part of the prediction and can be separated from the renormalon-dominating part. (II) In the frame of OPE, we define two types of renormalization schemes for the leading Wilson coefficient. One scheme belongs to the class of conventional factorization schemes. The other scheme belongs to a new class, which is independent of the factorization scale, derived from a generalization of the $\mathrm{\text{Coulomb}}+\mathrm{\text{linear}}$ potential of (I). The Wilson coefficient is free from IR renormalons and IR divergences in both schemes. We study properties of the Wilson coefficient and of the corresponding nonperturbative contribution $\ensuremath{\delta}{E}_{\mathrm{US}}(r)$ in each scheme. (III) We compare numerically perturbative predictions of the Wilson coefficient and lattice computations of ${V}_{\mathrm{QCD}}(r)$ when ${n}_{l}=0$. We confirm either correctness or consistency (within uncertainties) of the theoretical predictions made in (II). Then we perform fits to simultaneously determine $\ensuremath{\delta}{E}_{\mathrm{US}}(r)$ and ${r}_{0}{\ensuremath{\Lambda}}_{\overline{\mathrm{MS}}}^{3\mathrm{\text{\ensuremath{-}}}\mathrm{loop}}$ (relation between lattice scale and ${\ensuremath{\Lambda}}_{\overline{\mathrm{MS}}}$). As for the former quantity, we improve bounds as compared to the previous determination; as for the latter quantity, our analysis provides a new method for its determination. We find that (a) $\ensuremath{\delta}{E}_{\mathrm{US}}(r)=0$ is disfavored, and (b) ${r}_{0}{\ensuremath{\Lambda}}_{\overline{\mathrm{MS}}}^{3\mathrm{\text{\ensuremath{-}}}\mathrm{loop}}=0.574\ifmmode\pm\else\textpm\fi{}0.042$. We elucidate the mechanism for the sensitivities and examine sources of errors in detail.