$\mathrm{O}(a)$ improvement of the flavour singlet scalar density in a setup with Wilson fermions

We report on our Ward identity determination of the $\mathrm{O}(a)$ improvement coefficient for the flavour singlet scalar density, namely $g_{\mathrm{S}}$, from three-flavour lattice QCD with Wilson-clover fermions and the tree-level Symanzik improved gauge action. We employ five couplings, $g_0^2\in [1.5,1.77]$, that cover the range used in large-volume CLS simulations. While $g_{\mathrm{S}}$ itself is for instance relevant for the $\mathrm{O}(a)$ improvement of meson and baryon sigma terms, a relation to $b_{\mathrm{g}}$, the $\mathrm{O}(a)$ improvement parameter of the gauge coupling, can also be established, allowing for its non-perturbative extraction as well. With Wilson fermions, $b_{\mathrm{g}}$ is in principle required for full $\mathrm{O}(a)$ improvement at non-vanishing sea quark masses. We outline our procedure for extracting $b_{\mathrm{g}}$.