Quark-antiquark interaction at all momentum transfers

Abstract

The running coupling constant of quantum chromodynamics ${\alpha }_s\left(q^2\right)$ is obtained for all $q^2$ by integrating the renormalization-group equation. The $\beta$ function in the asymptotic- freedom region is given by perturbation theory through order $g^5$ and at large coupling by the string model with string tension related to the Regge slope ${\alpha }^{\prime }$. We incorporate these features into a Padé approximant to the $\beta$ function thereby obtaining it for all $g$. The constant of integration $\Lambda$ of the renormalization-group equation is chosen to be about 500 MeV, as required by deep-inelastic phenomenology. ${\alpha }_s$ is thus completely determined by ${\alpha }^{\prime }$, $\Lambda$, and the first two terms of the $\beta$ function. The resulting charmonium and $\Upsilon$ spectroscopy is in excellent agreement with experiment. In particular the ${\Upsilon }^{\prime }-\Upsilon$ mass difference is forced to be nearly equal to the ${\psi }^{\prime }-\psi$ mass difference. In addition, it is shown that the structure of ${\alpha }_s\mathrm{}\left(q\right)\mathrm{}$ signals the presence of instantons.