Ratio of flavour non-singlet and singlet scalar density renormalisation parameters in $N_\mathrm{f}=3$ QCD with Wilson quarks

TL;DR

This paper non-perturbatively determines the ratio of scalar density renormalisation parameters in $N_f=3$ QCD with Wilson quarks, aiding quark mass calculations and nucleon scalar matrix element renormalisations.

Contribution

It introduces a novel method using a chiral Ward identity to compute the ratio of scalar density renormalisation parameters in lattice QCD with Wilson fermions.

Findings

Provides non-perturbative values of $r_m$ for lattice spacings below 0.1 fm.

Employs both unitary and non-unitary setups for robust results.

Results are relevant for $N_f=2+1$ QCD simulations in large volumes.

Abstract

We determine non-perturbatively the normalisation factor $r_\mathrm{m}\equiv Z_{\rm S}/Z_{\rm S}^{0}$, where $Z_{\rm S}$ and $Z_{\rm S}^{0}$ are the renormalisation parameters of the flavour non-singlet and singlet scalar densities, respectively. This quantity is required in the computation of quark masses with Wilson fermions and for instance the renormalisation of nucleon matrix elements of scalar densities. Our calculation involves simulations of finite-volume lattice QCD with the tree-level Symanzik-improved gauge action, $N_\mathrm{f} = 3$ mass-degenerate $\mathrm{O}(a)$ improved Wilson fermions and Schr\"odinger functional boundary conditions. The slope of the current quark mass, as a function of the subtracted Wilson quark mass is extracted both in a unitary setup (where nearly chiral valence and sea quark masses are degenerate) and in a non-unitary setup (where all valence flavours are chiral and the sea quark masses are small). These slopes are then combined with $Z \equiv Z_{\rm P}/(Z_{\rm S}Z_{\rm A})$ in order to obtain $r_\mathrm{m}$. A novel chiral Ward identity is employed for the calculation of the normalisation factor $Z$. Our results cover the range of gauge couplings corresponding to lattice spacings below $0.1\,$fm, for which $N_\mathrm{f} = 2+1$ QCD simulations in large volumes with the same lattice action are typically performed.