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With Wilson quarks, on-shell O($a$) improvement of the lattice QCD action isachieved by including the Sheikholeslami-Wohlert term and two further operatorsof mass dimension 5, which amount to a mass-dependent rescaling of the bareparameters. We here focus on the rescaled bare coupling, $\tilde{g}_0^2 =g_0^2(1 + b_{\rm g} am_{\rm q})$, and the determination of $b_{\rm g}(g_0^2)$,which is currently only known to 1-loop order of perturbation theory. We derivesuitable improvement conditions in the chiral limit and in a finite space-timevolume and evaluate these for different pure gauge observables, both with andwithout the gradient flow. The choice of $\beta$-values and the line ofconstant physics are motivated by the ALPHA collaboration's decoupling strategyto determine $\alpha_s(m_Z)$. However, the improvement conditions and someinsight into systematic effects may prove useful in other contexts, too.

Heavy Wilson Quarks and O($a$) Improvement: Nonperturbative Results for $b_{\rm g}$