An O(a) modified lattice set-up of the Schr\"odinger functional in SU(3) gauge theory

The set-up of the QCD Schr\"odinger functional (SF) on the lattice with staggered quarks requires an even number of points $L/a$ in the spatial directions, while the Euclidean time extent of the lattice, $T/a$, must be odd. Identifying a unique renormalisation scale, $L=T$, is then only possible up to O($a$) lattice artefacts. In this article we study such lattices in the pure SU(3) gauge theory, where we can also compare to the standard set-up. We consider the SF coupling as obtained from the variation of an SU(3) Abelian and spatially constant background field. The O($a$) lattice artefacts can be cancelled by the existing O($a$) boundary counterterm. However, its coefficient, $\ct$, differs at the tree-level from its standard value, so that one first needs to re-determine the induced background gauge field. The perturbative one-loop correction to the coupling allows to determine $\ct$ to one-loop order. A few numerical simulations serve to demonstrate that residual cutoff effects in the step scaling function are small in both cases, $T=L\pm a$ and comparable to the standard case with $T=L$.