  We compute the renormalisation factors (Z-matrices) of the $\Delta F=2$
four-quark operators needed for Beyond the Standard Model (BSM) kaon mixing. We
work with nf=2+1 flavours of Domain-Wall fermions whose chiral-flavour
properties are essential to maintain a continuum-like mixing pattern. We
introduce new RI-SMOM renormalisation schemes, which we argue are better
behaved compared to the commonly-used corresponding RI-MOM one. We find that,
once converted to MS, the Z-factors computed through these RI-SMOM schemes are
in good agreement but differ significantly from the ones computed through the
RI-MOM scheme. The RI-SMOM Z-factors presented here have been used to compute
the BSM neutral kaon mixing matrix elements in the companion paper [1]. We
argue that the renormalisation procedure is responsible for the discrepancies
observed by different collaborations, we will investigate and elucidate the
origin of these differences throughout this work.
