Non-perturbative renormalization of the flavour singlet local vector current with O($a$)-improved Wilson fermions

We determine non-perturbatively the renormalization constant of the flavour singlet local vector current with O($a$)-improved Wilson fermions. The renormalization constant is fixed by comparing the expectation values (one-point function) of the local vector current and of the conserved one in thermal QCD in a moving reference frame with a non-zero imaginary chemical potential and in the chiral limit. We implement the method in QCD with 3 flavours discretized by the standard Wilson action for gluons and the non-perturbatively O($a$)-improved Wilson fermions. By carrying out extensive numerical simulations, the renormalization constant is determined with a permille precision for values of the bare coupling constant in the range $0.52 \leq g_0^2 \leq 1.13$.