  We study two aspects of higher dimensional operators in standard model
effective field theory. We first introduce a perturbative power counting rule
for the entries in the anomalous dimension matrix of operators with equal mass
dimension. The power counting is determined by the number of loops and the
difference of the indices of the two operators involved, which in turn is
defined by assuming that all terms in the standard model Lagrangian have an
equal perturbative power. Then we show that the operators with the lowest index
are unique at each mass dimension $d$, i.e., $(H^\dagger H)^{d/2}$ for even
$d\geq 4$, and $(L^T\epsilon H)C(L^T\epsilon H)^T(H^\dagger H)^{(d-5)/2}$ for
odd $d\geq 5$. Here $H,~L$ are the Higgs and lepton doublet, and $\epsilon,~C$
the antisymmetric matrix of rank two and the charge conjugation matrix,
respectively. The renormalization group running of these operators can be
studied separately from other operators of equal mass dimension at the leading
order in power counting. We compute their anomalous dimensions at one loop for
general $d$ and find that they are enhanced quadratically in $d$ due to
combinatorics. We also make connections with classification of operators in
terms of their holomorphic and anti-holomorphic weights.
