Running coupling and the Lambda parameter from SU(3) lattice simulations.

Abstract

We present new results on the static $q\overline{q}$ potential from high-statistics simulations on ${32}^{4}$ and smaller lattices, using the standard Wilson action at $\ensuremath{\beta}=6.0, 6.4, \mathrm{and} 6.8$ on the Connection Machine CM-2. Within our statistical errors (\ensuremath{\approx} 1%) we do not observe any finite-size effects affecting the potential values, on varying the spatial lattice extent from 0.9 fm up to 3.3 fm. We are able to see and quantify the running of the coupling from the Coulomb behavior of the interquark force. From this we extract the ratio $\frac{\sqrt{\ensuremath{\sigma}}}{{\ensuremath{\Lambda}}_{L}}$. We demonstrate that scaling violations on the string tension can be considerably reduced by introducing effective coupling schemes, which allow for a safe extrapolation of ${\ensuremath{\Lambda}}_{L}$ to its continuum value. Both methods yield consistent values for $\ensuremath{\Lambda}:{\ensuremath{\Lambda}}_{\stackrel{-}{\mathrm{MS}}}={0.555}_{\ensuremath{-}0.017}^{+0.019}\ifmmode\times\else\texttimes\fi{}\sqrt{\ensuremath{\sigma}}={244}_{\ensuremath{-}7}^{+8}$ MeV, where $\stackrel{-}{\mathrm{MS}}$ denotes the modified minimal subtraction scheme. At the highest-energy scale attainable to us we find $\ensuremath{\alpha}(5 \mathrm{GeV})=0.150(3)$.