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We examine the renormalisation of flavour-diagonal vector currents in latticeQCD with the aim of understanding and quantifying the systematic errors fromnonperturbative artefacts associated with the use of intermediatemomentum-subtraction schemes. Our study uses the Highly Improved StaggeredQuark (HISQ) action on gluon field configurations that include $n_f=2+1+1$flavours of sea quarks, but our results have applicability to other quarkactions. Renormalisation schemes that make use of the exact lattice vectorWard-Takahashi identity for the conserved current also have renormalisationfactors, $Z_V$, for nonconserved vector currents that are free of contaminationby nonperturbative condensates. We show this by explicit comparison of two suchschemes: that of the vector form factor at zero momentum transfer and theRI-SMOM momentum-subtraction scheme. The two determinations of $Z_V$ differonly by discretisation effects (for any value of momentum-transfer in theRI-SMOM case). The RI$^{\prime}$-MOM scheme, although widely used, does notshare this property. We show that $Z_V$ determined in the standard way in thisscheme has $\mathcal{O}(1\%)$ nonperturbative contamination that limits itsaccuracy. Instead we define an RI$^{\prime}$-MOM $Z_V$ from a ratio of local toconserved vector current vertex functions and show that this $Z_V$ is a safeone to use in lattice QCD calculations. We also perform a first study of vectorcurrent renormalisation with the inclusion of quenched QED effects on thelattice using the RI-SMOM scheme.

Renormalising vector currents in lattice QCD using momentum-subtraction schemes