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

Abstract

AbstractWe derive chiral Ward identities for lattice QCD with Wilson quarks and $$N_{\mathrm{f}}\ge 3$$Nf≥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 $$Z_{\mathrm {S}}/Z_{\mathrm {P}}$$ZS/ZP of the renormalisation parameters of these operators. We obtain results for $$N_{\mathrm{f}}=3$$Nf=3 QCD with tree-level Symanzik-improved gluons and Wilson-Clover quarks, for bare gauge couplings which cover the typical range of large-volume $$N_{\mathrm{f}}= 2+1$$Nf=2+1 simulations with Wilson fermions at lattice spacings below $$0.1\,$$0.1fm. The precision of our results varies from 0.3 to 1%, except for the coarsest lattice, where it is 2%. We discuss how the $$Z_{\mathrm {S}}/Z_{\mathrm {P}}$$ZS/ZP ratio can be used in the non-perturbative calculations of $${\mathrm {O}}(a)$$O(a) improved renormalised quark masses.