  We examine the renormalisation of flavour-diagonal vector currents in lattice
QCD with the aim of understanding and quantifying the systematic errors from
nonperturbative artefacts associated with the use of intermediate
momentum-subtraction schemes. Our study uses the Highly Improved Staggered
Quark (HISQ) action on gluon field configurations that include $n_f=2+1+1$
flavours of sea quarks, but our results have applicability to other quark
actions. Renormalisation schemes that make use of the exact lattice vector
Ward-Takahashi identity for the conserved current also have renormalisation
factors, $Z_V$, for nonconserved vector currents that are free of contamination
by nonperturbative condensates. We show this by explicit comparison of two such
schemes: that of the vector form factor at zero momentum transfer and the
RI-SMOM momentum-subtraction scheme. The two determinations of $Z_V$ differ
only by discretisation effects (for any value of momentum-transfer in the
RI-SMOM case). The RI$^{\prime}$-MOM scheme, although widely used, does not
share this property. We show that $Z_V$ determined in the standard way in this
scheme has $\mathcal{O}(1\%)$ nonperturbative contamination that limits its
accuracy. Instead we define an RI$^{\prime}$-MOM $Z_V$ from a ratio of local to
conserved vector current vertex functions and show that this $Z_V$ is a safe
one to use in lattice QCD calculations. We also perform a first study of vector
current renormalisation with the inclusion of quenched QED effects on the
lattice using the RI-SMOM scheme.

High Energy Physics - Lattice