Relationship between lattice and continuum definitions of the gauge-theory coupling

We generalize the background-field method to lattice gauge theories. By evaluating the lattice partition function in a weak background lattice field for weak coupling we determine the relationship between the lattice definition of the coupling and that of dimensional regularization. This corresponds to a determination of the ratio of the renormalization scale parameters for these definitions. We find, for an SU(N) lattice gauge theory with the Wilson action, that ${({{{Λ}}M̄S̄}{{{Λ}}L})}SU(N)=38.852704exp({-}{3{{π}}²}{11{N}²})$.