Running coupling constant and correlation length from Wilson loops

Abstract

We consider a definition of the QCD running coupling constant $\alpha(\mu)$ related to Wilson loops of size $r{\times}t$ with arbitrary fixed $t/r$. The schemes defined by these couplings are very close to the $\overline{\rm MS}$ scheme (i.e.\ the one-loop perturbative correction to the coupling is small) for all values of $t/r$; in the $t/r\to\infty$ limit, the ``$q\bar q$ force'' scheme is recovered, where the coupling constant is related to the quark-antiquark force. We discuss the possibility of applying finite-size scaling techniques to the Monte Carlo evaluation of $\alpha(\mu)$ up to very large momentum scales. We propose a definition of correlation length, also related to Wilson loops, which should make such a computation feasible.