Two-loop computation of a finite volume running coupling on the lattice

In pure SU(2) gauge theory we compute the two-loop coefficient in the relation between the lattice bare coupling and the running coupling defined through the Schrödinger functional. This result is required to relate the latter to the MS-coupling in our programme to compute αs. In addition it allows us to implement O(a) improvement of the Schrödinger functional to two-loop order. The two-loop β-function is verified in a perturbative computation on the lattice, and the behavior of expansions in the standard and in the Parisi-improved bare couplings are investigated beyond one loop.