Precision renormalisation and improvement of $N_{\rm f}=3$ lattice QCD with Wilson fermions

Abstract

We renormalise (and improve) the flavour non-singlet axial current, pseudo-scalar density, vector current and tensor current, as well as quark masses, in O(a) improved lattice QCD with three massless flavours and lattice spacings down to 0.01 fm. To this end, we tune a number of lattices with Schrödinger functional boundary conditions and resolutions $8\leq L/a\leq 64$ to lines of constant physics with massless quarks and fixed gradient flow coupling $\bar{g}_\mathrm{GF}^2(L_i),\; i=0,1,2$, corresponding to $L_0 \approx 0.25$ fm, $L_1=2L_0$ and $L_2=4L_0$. We further renormalise and improve the quark mass of additional heavy quarks for use in the B-physics programme of the collaboration (arXiv:2312.09811). Our somewhat technical results enable first-principles strategies for solving multi-scale problems involving, e.g., the b-quark mass (arXiv:2312.10017) or a large temperature (arXiv:2501.11603). Comparing also to other determinations of the axial current renormalisation constant $Z_{\rm A}$, we have a precise confirmation of how renormalisation and the restoration of chiral symmetry work out with Wilson fermions at small $a$. In particular, the accurate restoration of chiral symmetry and the exact flavour symmetry lead to practically negligible uncertainties in observables determined from Ward identities: four to five significant digits are achieved for $Z_{\rm A},Z_{\rm V}$. We provide an explanation for the strong suppression of their statistical variances.