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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 0S , where Z S and Z 0S 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 S Z A ) in order to obtain r m . A novel chiral Ward