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 $\overline{\mathrm{MS}}$ 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.

Comments: 8 pages, latex (revtex)