Ratio of flavour non-singlet and singlet scalar density renormalisation parameters in N_f=3 QCD with Wilson quarks

We determine non-perturbatively the normalisation factor r_m≡ Z_S/Z_S^0 , where Z_S and Z_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_f= 3 mass-degenerate O(a) improved Wilson fermions and Schrödinger 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 ≡ Z_P/(Z_SZ_A) in order to obtain r_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_f= 2+1 QCD simulations in large volumes with the same lattice action are typically performed.