Asymptotic scaling of the gluon propagator on the lattice

We pursue the study of the high energy behavior of the gluon propagator on the lattice in the Landau gauge in the flavorless case ${(n}_{f}=0).$ It was shown in a preceding paper that the gluon propagator did not reach three-loop asymptotic scaling at an energy scale as high as 5 GeV. Our present high statistics analysis includes also a simulation at $\ensuremath{\beta}=6.8$ $(a\ensuremath{\simeq}0.03 \mathrm{fm}),$ which allows us to reach $\ensuremath{\mu}\ensuremath{\simeq}10 \mathrm{GeV}.$ Special care has been devoted to the finite lattice-spacing artifacts as well as to the finite-volume effects, the latter being acute at $\ensuremath{\beta}=6.8$ where the volume is bounded by technical limits. Our main conclusion is strong evidence that the gluon propagator has reached three-loop asymptotic scaling, at $\ensuremath{\mu}$ ranging from 5.6--9.5 GeV. We buttress up this conclusion on several demanding criteria of asymptoticity, including scheme independence. Our fit in the 5.6 GeV to 9.5 GeV window yields ${\ensuremath{\Lambda}}^{\overline{\mathrm{MS}}}=319\ifmmode\pm\else\textpm\fi{}{14}_{\ensuremath{-}20}^{+10}$ MeV, in good agreement with our previous result ${\ensuremath{\Lambda}}^{\overline{\mathrm{MS}}}=295\ifmmode\pm\else\textpm\fi{}20 \mathrm{MeV},$ obtained from the three-gluon vertex, but it is significantly above the Schr\"odinger functional method estimate: $238\ifmmode\pm\else\textpm\fi{}19 \mathrm{MeV}.$ The latter difference is not understood. Confirming our previous paper, we show that a fourth loop is necessary to fit the whole $(2.8--9.5) \mathrm{GeV}$ energy window.