Renormalization Group Equations for the SMEFT Operators up to Dimension Seven

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 $\mathcal{O} \left( \Lambda^{-3} \right)$.