A perturbative determination of O(a) boundary improvement coefficients for the Schr\"odinger Functional coupling at 1-loop with improved gauge actions

We perturbatively determine O($a$) boundary improvement coefficients at 1-loop for the Schr\"odinger Functional coupling with improved gauge actions. These coefficients are required to implement the 1-loop O($a$) improvement in full QCD simulations for the coupling with the improved gauge actions. To this order, lattice artifacts of the step scaling function (SSF) for each improved gauge action are also investigated. Furthermore, we investigate the effect of the 1-loop O($a$) improvement to the SSF in numerical simulations of the pure SU(3) gauge theory.