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We perform the non-perturbative renormalization of matrix elements of thestatic-light axial current by a computation of its scale dependence in latticeQCD with two flavours of massless O(a) improved Wilson quarks. Theregularization independent factor that relates any running renormalized matrixelement of the axial current in the static effective theory to therenormalization group invariant one is evaluated in the Schroedinger functionalscheme, where in this case we find a significant deviation of thenon-perturbative running from the perturbative prediction. An importanttechnical ingredient to improve the precision of the results consists in theuse of modified discretizations of the static quark action introduced earlierby our collaboration. As an illustration how to apply the renormalization ofthe static axial current presented here, we connect the bare matrix element ofthe current to the B_s-meson decay constant in the static approximation for onevalue of the lattice spacing, a ~ 0.08 fm, employing large-volume N_f=2 data atbeta=5.3.Comment: 33 pages including figures and tables, latex2e, uses JHEP3.cls;  version published in JHEP, small additions, results unchange

Non-perturbative renormalization of the static axial current in two-flavour QCD