  In this paper, we propose a Green's basis and also a new physical basis for
dimension-seven (dim-7) operators, which are suitable for the matching of
ultraviolet models onto the Standard Model effective field theory (SMEFT) and
the deviation of renormalization group equations (RGEs) for dim-7 operators in
the SMEFT. The reduction relations to convert operators in the Green's basis to
those in the physical basis are achieved as well, where some redundant dim-6
operators in the Green's basis are involved if the dim-5 operator exists.
Working in these two bases for dim-7 operators and with the help of the
reduction relations, we work out the one-loop RGEs resulting from the mixing
among different dimensional operators for the dim-5 and dim-7 operators up to
$\mathcal{O} \left( \Lambda^{-3} \right)$ in the SMEFT. These new results
complete the previous results for RGEs of the dim-5 and dim-7 operators and
hence can be used for a consistent one-loop analysis of the SMEFT at
$\mathcal{O} \left( \Lambda^{-3} \right)$.
