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

With Wilson quarks, on-shell O($a$) improvement of the lattice QCD action is achieved by including the Sheikholeslami-Wohlert term and two further operators of mass dimension 5, which amount to a mass-dependent rescaling of the bare parameters. 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 derive suitable improvement conditions in the chiral limit and in a finite space-time volume and evaluate these for different gluonic observables, both with and without the gradient flow. The choice of $\beta$-values and the line of constant physics are motivated by the ALPHA collaboration's decoupling strategy to determine $\alpha_s(m_Z)$. However, the improvement conditions and some insight into systematic effects may prove useful in other contexts, too.