Ratio of flavour non-singlet and singlet scalar density renormalisation parameters in $N_\mathrm{f}=3$ QCD with Wilson quarks

We determine non-perturbatively the normalisation factor $r_\mathrm{m}\equiv Z_{\rm S}/Z_{\rm S}^{0}$, where $Z_{\rm S}$ and $Z_{\rm S}^{0}$ are the renormalisation parameters of the flavour non-singlet and singlet scalar densities, respectively. This quantity is required in the computation of quark masses with Wilson fermions and for instance the renormalisation of nucleon matrix elements of scalar densities. Our calculation involves simulations of finite-volume lattice QCD with the tree-level Symanzik-improved gauge action, $N_\mathrm{f} = 3$ mass-degenerate $\mathrm{O}(a)$ improved Wilson fermions and Schr\"odinger functional boundary conditions. The slope of the current quark mass, as a function of the subtracted Wilson quark mass is extracted both in a unitary setup (where nearly chiral valence and sea quark masses are degenerate) and in a non-unitary setup (where all valence flavours are chiral and the sea quark masses are small). These slopes are then combined with $Z \equiv Z_{\rm P}/(Z_{\rm S}Z_{\rm A})$ in order to obtain $r_\mathrm{m}$. A novel chiral Ward identity is employed for the calculation of the normalisation factor $Z$. Our results cover the range of gauge couplings corresponding to lattice spacings below $0.1\,$fm, for which $N_\mathrm{f} = 2+1$ QCD simulations in large volumes with the same lattice action are typically performed.