Asymptotic behavior of the gluon propagator from lattice QCD

We study the flavorless gluon propagator in the Landau gauge from high statistics lattice calculations. Hypercubic artifacts are efficiently eliminated by taking the $\sum p_{\mu }^4\to 0$ limit. The propagator is fitted to the three-loops perturbative formula in an energy window ranging from $\sim 2.5$ GeV up to $\sim 5.5$ GeV. ${\alpha }_s$ is extracted from the best fit in a continuous set of renormalization schemes. The fits are very good, with a ${\chi }^2$ per DOF smaller than 1. We propose a more stringent test of asymptotic scaling based on scheme independence of the resulting ${\Lambda }_{\overline{\mathrm{MS}}}.$ This method shows that asymptotic scaling at three loops is not reached by the gluon propagator although we use rather large energies. We are only able to obtain an effective flavorless three-loops estimate ${\Lambda }_{\overline{\mathrm{MS}}}^{\mathrm{}\left(3\right)\mathrm{}}=353\mathrm{}\pm 2_{-10}^{+25\mathrm{}}$ MeV. We argue that the real asymptotic value for ${\Lambda }_{\overline{\mathrm{}\mathrm{MS}}}$ should plausibly be smaller.