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Analysis of General Power Counting Rules in Effective Field Theory

We derive the general counting rules for a quantum effective field theory (EFT) in $\mathsf{d}$ dimensions. The rules are valid for strongly and weakly coupled theories, and predict that all kinetic energy terms are canonically normalized. They determine the energy dependence of scattering cross sections in the range of validity of the EFT expansion. We show that the size of cross sections is controlled by the $Λ$ power counting of EFT, not by chiral counting, even for chiral perturbation theory ($χ$PT). The relation between $Λ$ and $f$ is generalized to $\mathsf{d}$ dimensions. We show that the naive dimensional analysis $4π$ counting is related to $\hbar$ counting. The EFT counting rules are applied to $χ$PT, low-energy weak interactions, Standard Model EFT and the non-trivial case of Higgs EFT.