Ward identity determination of $Z_{\mathrm{S}}\mathrm{/}{Z}_{\mathrm{P}}$ for $N_{\mathrm{f}}=3$ lattice QCD in a Schr\"odinger functional setup

We derive chiral Ward identities for lattice QCD with Wilson quarks and $N_{\mathrm{f}}\ge 3$ flavours, on small lattices with Schr"odinger functional boundary conditions and vanishingly small quark masses. These identities relate the axial variation of the non-singlet pseudoscalar density to the scalar one, thus enabling the non-perturbative determination of the scale-independent ratio $Z_{\mathrm{S}}\mathrm{/}{Z}_{\mathrm{P}}$ of the renormalisation parameters of these operators. We obtain results for $N_{\mathrm{f}}=3$ QCD with tree-level Symanzik-improved gluons and Wilson-Clover quarks, for bare gauge couplings which cover the typical range of large-volume $N_{\mathrm{f}}=2+1$ simulations with Wilson fermions at lattice spacings below $0.1$fm. The precision of our results varies from 0.3% to 1%, except for the coarsest lattice, where it is 2%. We discuss how the $Z_{\mathrm{S}}\mathrm{/}{Z}_{\mathrm{P}}$ ratio can be used in the non-perturbative calculations of $\mathrm{O}(a)$ improved renormalised quark masses.