A New Way to Set the Energy Scale in Lattice Gauge Theories and its Application to the Static Force and $α_s$ in SU(2) Yang--Mills Theory

We introduce a hadronic scale $R_0$ through the force $F(r)$ between static quarks at intermediate distances $r$. The definition $F(R_0)R_0^2=1.65$ amounts to $R_0 \simeq 0.5$~fm in phenomenological potential models. Since $R_0$ is well defined and can be calculated accurately in a Monte Carlo simulation, it is an ideal quantity to set the scale. In SU(2) pure gauge theory, we use new data (and $R_0$ to set the scale) to extrapolate $F(r)$ to the continuum limit for distances $r=0.18$~fm to $r=1.1$~fm. Through $R_0$ we determine the energy scale in the recently calculated running coupling, which used the recursive finite size technique to reach large energy scales. Also in this case, the lattice data can be extrapolated to the continuum limit. The use of one loop Symanzik improvement is seen to reduce the lattice spacing dependence significantly. Warning: The preprint is not completely fresh, but maybe you haven't seen it...