We determine non-perturbatively the normalisation factor \(r_{\mathrm{m}}\equiv Z_{\mathrm{S}}/Z_{\mathrm{S}}^{0}\), where \(Z_{\mathrm{S}}\) and \(Z_{\mathrm{S}}^{0}\) 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 Schrödinger 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 \(Z \equiv Z_{\mathrm{P}}/(Z_{\mathrm{S}}Z_{\mathrm{A}})\) 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.

我们非微扰地确定归一化因子\(r_{\mathrm{m}}\equiv Z_{\mathrm{S}}/Z_{\mathrm{S}}^{0}\)，其中\(Z_{\ mathrm{S}}\)和\(Z_{\mathrm{S}}^{0}\)分别是风味非单线态和单线态标量密度的重整化参数。在用威尔逊费米子计算夸克质量时需要这个量，例如标量密度的核子矩阵元素的重整化。我们的计算涉及有限体积晶格 QCD 的模拟与树级 Symanzik 改进的规范作用，\(N_{\mathrm{f}}= 3\) mass-degenerate \({\mathrm{O}}(a) \)改进的威尔逊费米子和薛定谔函数边界条件。当前夸克质量的斜率，作为减去的威尔逊夸克质量的函数，在单一设置（其中近手性价和海夸克质量退化）和非单一设置（其中所有价味都是手性的）中提取并且海夸克质量很小）。然后将这些斜率与\(Z \equiv Z_{\mathrm{P}}/(Z_{\mathrm{S}}Z_{\mathrm{A}})\) 组合以获得\(r_{\mathrm {m}}\)。新的手性病房恒等式用于计算归一化因子Z。我们的结果涵盖了对应于低于\(0.1\,\) fm 的晶格间距的规范耦合范围，其中\(N_{\mathrm{f}}= 2+1\) 通常执行具有相同晶格作用的大体积 QCD 模拟。