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 $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ lattice gauge theory with the Wilson action, that ${\left(\frac{{\Lambda }_{\overline{\mathrm{M}}\overline{\mathrm{S}}}}{{\Lambda }_L}\right)}_{\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}}=38.852\mathrm{}704\exp (-\frac{3{\pi }^2}{11N^2})$.