Ward identity determination of $$Z_{\mathrm {S}}/Z_{\mathrm {P}}$$ for $$N_{\mathrm {f}}=3$$ lattice QCD in a Schrödinger functional setup
Jochen Heitger (Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Straße 9, Münster, 48149, Germany); Fabian Joswig (Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Straße 9, Münster, 48149, Germany); Anastassios Vladikas (“Rome Tor Vergata” Division, c/o Dipartimento di Fisica, INFN, Via della Ricerca Scientifica 1, Rome, 00133, Italy)
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.
