We present the results of a non-perturbative determination of the improvement coefficient \(c_\mathrm{V}\) and the renormalisation factor \(Z_\mathrm{V}\), which define the renormalised vector current in three-flavour \(\mathrm{O}(a)\) improved lattice QCD with Wilson quarks and tree-level Symanzik-improved gauge action. In case of the improvement coefficient, we consider both lattice descriptions of the vector current, the local as well as the conserved (i.e., point-split) one. Our improvement and normalisation conditions are based on massive chiral Ward identities and numerically evaluated in the Schrödinger functional setup, which allows to eliminate finite quark mass effects in a controlled way. In order to ensure a smooth dependence of the renormalisation constant and improvement coefficients on the bare gauge coupling, our computation proceeds along a line of constant physics, covering the typical range of lattice spacings \(0.04\,\mathrm{fm}\lesssim a\lesssim 0.1\,\mathrm{fm}\) that is useful for phenomenological applications. Especially for the improvement coefficient of the local vector current, we report significant differences between the one-loop perturbative estimates and our non-perturbative results.

我们给出了改进系数\（c_ \ mathrm {V} \）和重归一化因子\（Z_ \ mathrm {V} \）的非摄动确定的结果，它们定义了三味\ （\ mathrm {O}（a）\）威尔逊夸克和树级Symanzik改进的轨距动作改进了晶格QCD。在改善系数的情况下，我们考虑矢量电流的晶格描述，局部和守恒（即点分裂）。我们的改进和归一化条件基于大量的手性Ward身份，并在Schrödinger功能设置中进行了数值评估，从而可以以受控方式消除有限的夸克质量效应。为了确保重归一化常数和改进系数对裸规耦合的平稳依赖，我们的计算沿恒定物理线进行，覆盖了典型的晶格间距范围\（0.04 \，\ mathrm {fm} \ lesssim a \ lesssim 0.1 \，\ mathrm {fm} \）对于现象学应用很有用。特别是对于局部矢量电流的改善系数，我们报告了单环摄动估计与非摄动结果之间的显着差异。