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.
Highlights
The scope of this paper is to provide a method for the determination of ZS/ZP based on Ward identities on physically small lattices with Schrödinger functional boundary conditions and realising a line of constant physics (LCP) in parameter space
We prefer to plot the results as functions of a3 in Fig. 7, where we show a one-parameter fit of the form 1 + c3a3; for this ansatz we obtain χ 2/d.o.f = 0.300, c3 = 206(14) for LCP-0 and χ 2/d.o.f = 0.170, c3 = 169(12) for LCP-1.11 We interpret this as confirmation that the two methods are compatible w.r.t. the expected lattice spacing ambiguities and that the effects of O(a2) are subdominant compared to the higher order
In order to ensure a smooth dependence of the renormalisation constant ratio on the bare gauge coupling, we have enforced a constant physics condition by working with an approximately fixed physical volume of spatial extent L ≈ 1.2 fm and T /L ≈ 3/2
Summary
Under the small axial variations (A.8) of the fermion fields the formal, continuum QCD action in Euclidean space-time transforms as follows: δA S = d4x (∂μ a(x)) Aaμ(x) + i a(x)ψ (x){T a, M}γ5ψ(x). We will derive chiral Ward identities which relate correlation functions of non-singlet scalar and pseudoscalar composite operators (densities). These enable us to compute non-perturbatively the ratio ZS/ZP, which determines the relative normalisation of these scalar and pseudoscalar densities when the regularisation (Wilson fermion action) breaks chiral symmetry. In the lattice regularisation this implies that the contact term contributes an O(am) discretisation effect to the Ward identity, even in a Symanzik-improved setup
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