Effect of the Schr\"odinger functional boundary conditions on the convergence of step scaling

Recently several lattice collaborations have studied the scale dependence of the coupling in theories with different gauge groups and fermion representations using the Schr\"odinger functional method. This has motivated us to look at the convergence of the perturbative step scaling to its continuum limit with gauge groups SU(2) and SU(3) with Wilson fermions in the fundamental, adjoint or sextet representations. We have found that while the improved Wilson action does remove the linear terms from the step scaling, the convergence is extremely slow with the standard choices of the boundary conditions for the background field. We show that the situation can be improved by careful choice of the boundary fields.