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

Abstract

We derive chiral Ward identities for lattice QCD with Wilson quarks and $N_{\mathrm{f}}\ge 3$ flavours, on small lattices with Schrödinger 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 $Z_{\mathrm{S}}/Z_{\mathrm{P}}$ 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 $Z_{\mathrm{S}}/Z_{\mathrm{P}}$ ratio can be used in the non-perturbative calculations of $\mathrm{O}(a)$ improved renormalised quark masses.

24 pages, 8 figures, corrected a few misprints of the published version