Landau gauge ghost propagator and running coupling in SU(2) lattice gauge theory

We study finite (physical) volume and scaling violation effects of the Landau gauge ghost propagator as well as of the running coupling $\alpha_s(p)$ in the $SU(2)$ lattice gauge theory. We consider lattices with physical linear sizes between $aL \simeq 3$ fm and $7$ fm and values of lattice spacing between $a=0.2$ fm and $0.07$ fm. To fix the gauge we apply an efficient gauge fixing method aimed at finding extrema as close as possible to the global maximum of the gauge functional. We find finite volume effects to be small for the lattice size $aL \simeq 3$ fm at momenta $|p|\, \ge \, 0.6$ GeV. For the same lattice size we study extrapolations to the continuum limit of the ghost dressing function as well as for the running coupling with momenta chosen between $|p| = 0.41$ GeV and $3.2$ GeV. We present fit formulae for the continuum limit of both observables in this momentum range. Our results testify in favor of the decoupling behavior in the infrared limit.