Perturbative Power Counting, Lowest-Index Operators and Their Renormalization in Standard Model Effective Field Theory

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.