Lattice calculation of $\alpha_s$ in momentum scheme

Abstract

We compute the flavorless running coupling constant of QCD from the three gluon vertex in the (regularisation independent) momentum subtraction renormalisation scheme. This is performed on the lattice with high statistics. The expected color dependence of the Green functions is verified. There are significant $O(a^2\mu^2)$ effects which can be consistently controlled. Scaling is demonstrated when the renormalisation scale is varied between 2.1 GeV and 3.85 GeV. Scaling when the lattice spacing is varied is very well satisfied. The resulting flavorless conventional two loop $\Lambda^{(c)}_{\bar{MS}}$ is estimated to be, respectively for the MOM and $\widetilde{MOM}$ scheme, 361(6) MeV and 345(6) MeV, while the three loop results are, depending on $\beta_2: \Lambda^{(3)}_{\bar{MS}} = (412 - 59 \frac{\beta_2} {\beta_{2, \bar{MS}}} \pm 6 MeV) $ and $\Lambda^{(3)}_{\bar{MS}} = (382- 46\frac{\beta_2} {\beta_{2, \bar{MS}}}\pm 5 MeV) $. A preliminary computation of $\beta_2$ in the $\widetilde{MOM}$ scheme leads to $\Lambda^{(3)}_{\bar{MS}} = 303(5)\frac {a^{-1}(\beta=6.0)} {1.97 GeV}$ MeV.