Determination of the charm quark mass in lattice QCD with $2+1$ flavours on fine lattices

We present a determination of the charm quark mass in lattice QCD with three active quark flavours. The calculation is based on PCAC masses extracted from $N_\mathrm{f}=2+1$ flavour gauge field ensembles at five different lattice spacings in a range from 0.087 fm down to 0.039 fm. The lattice action consists of the $\mathrm{O}(a)$ improved Wilson-clover action and a tree-level improved Symanzik gauge action. Quark masses are non-perturbatively $\mathrm{O}(a)$ improved employing the Symanzik-counterterms available for this discretisation of QCD. To relate the bare mass at a specified low-energy scale with the renormalisation group invariant mass in the continuum limit, we use the non-pertubatively known factors that account for the running of the quark masses as well as for their renormalisation at hadronic scales. We obtain the renormalisation group invariant charm quark mass at the physical point of the three-flavour theory to be $M_\mathrm{c} = 1486(21)\,\mathrm{MeV}$. Combining this result with five-loop perturbation theory and the corresponding decoupling relations in the $\overline{\mathrm{MS}}$ scheme, one arrives at a result for the renormalisation group invariant charm quark mass in the four-flavour theory of $M_\mathrm{c}(N_\mathrm{f}=4) = 1548(23)\,\mathrm{MeV}$. In the $\overline{\mathrm{MS}}$ scheme, and at finite energy scales conventional in phenomenology, we quote $m^{\overline{\mathrm{MS}}}_{\mathrm{c}}(m^{\overline{\mathrm{MS}}}_{\mathrm{c}}; N_\mathrm{f}=4)=1296(19)\,\mathrm{MeV}$ and $m^{\overline{\mathrm{MS}}}_{\mathrm{c}}(3\,\mathrm{GeV}; N_\mathrm{f}=4)=1007(16)\,\mathrm{MeV}$ for the renormalised charm quark mass