The SF running coupling with four flavours of staggered quarks

In order to study the running coupling in four-flavour QCD, we review the set-up of the Schr\"odinger functional (SF) with staggered quarks. Staggered quarks require lattices which, in the usual counting, have even spatial lattice extent $L/a$ while the time extent $T/a$ must be odd. Setting $T=L$ is therefore only possible up to ${\rm O}(a)$, which introduces different cutoff effects already in the pure gauge theory. We re-define the SF such as to cope with this situation and determine the corresponding classical background field. A perturbative calculation yields the coefficient of the pure gauge ${\rm O}(a)$ boundary counterterm to one-loop order.