EFT meets CFT: Multiloop renormalization of higher-dimensional operators in general $\boldsymbol{φ^4}$ theories

Abstract: The renormalization of composite operators is a fundamental aspect of quantum field theory, relevant for the description of phase transitions and high energy phenomenology. We calculate the anomalous dimensions of a large set of operators in any scalar $φ^4$ theory in $d=4-\varepsilon$ dimensions, up to five loops in most cases. The results have applications in both effective field theory (EFT) and conformal field theory (CFT). As an EFT application, we extract the five-loop renormalization group (RG) equations of the Higgs sector of the Standard Model EFT at dimension six, and up to two loops at dimension eight, aligning our operator basis with custodial symmetry violation. Additionally, for CFT, by resumming the $\varepsilon$-expanded results at the fixed-point, we determine the entire low-lying spectrum (i.e. up to dimension six and Lorentz rank two) of the Ising, $O(n)$ and hypercubic scalar CFTs. Our work enables future conformal bootstrap studies for numerous theories of interest. We include introductions to EFT and CFT, and we illustrate our method and the structure in RG mixing matrices in several illuminating examples, which may also be of general interest. All results in the general theory are publicly available and we describe a systematic path towards applying them to more complicated CFTs.