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

TL;DR

This work presents a comprehensive perturbative analysis of the renormalization of composite operators in the most general scalar $\phi^4$ theory, carried out in $d=4-\varepsilon$ to five loops for a broad class of higher-dimensional and spinning operators. By constructing a robust operator basis (Green's basis and the conformal primary subset) and employing the $R^*$ method for diagram evaluation, the authors obtain multiloop anomalous-dimension matrices and extract primary operator dimensions, accounting for mixing with redundant (EOM and total-derivative) operators. They demonstrate direct EFT applications by deriving SMEFT Higgs-sector RG equations at dim=6 (and dim=8 at two loops) and by mapping the results to custodial symmetry representations, and they translate the perturbative data into conformal-field-theory predictions by resumming the $\varepsilon$-expansion to determine the Ising, $O(n)$ and hypercubic CFT spectra up to dim$\le 6$ and rank-2 tensors. The results enable precision bootstrap analyses across a broad set of theories and provide publicly available data and tooling for extracting spectra in many models beyond those treated explicitly. Overall, the paper establishes a scalable multiloop framework linking EFT operator renormalization to CFT data, with immediate implications for SMEFT phenomenology and conformal bootstrap studies of diverse scalar 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.

Figures (7)

• Figure 1: Scalar spectra for $O(n)$ symmetric theories with $n=1,2,3,4$. The energy levels shown on the very right are results in three dimensions: numerical bootstrap for $n=1$Simmons-Duffin:2016wlq, $n=2$Chester:2019wfx, $n=3$Chester:2020iytChester2020UnpRong:2023owx, and Monte Carlo for $n=2$ (charge 5,6) Banerjee:2017fcx and $n=4$Calabrese:2002bmBanerjee:2019jpwHasenbusch:2021rse. Note that we have removed the representations and operators which vanish identically for low $n$; see Cao:2023psi. The results of this paper concern five-loop anomalous dimensions for all operators with $\Delta(4d)\leqslant6$, together with additional results above this value in specific cases. We plot the spin-1 and spin-2 spectra in Figures \ref{['fig:vectors-all']} and \ref{['fig:tensors-all']}, respectively.
• Figure 2: The parameters of an EFT can be determined by matching to a UV theory and by fitting to experimental data. In the regime of validity of the EFT, the anomalous dimensions determine the running of the couplings with renormalization scale $\mu$.
• Figure 3: Long RG flow in three dimensions versus short RG flow in $4-{\varepsilon}$ dimensions and resummation to reach the 3d IR CFT.
• Figure 4: Comparing perturbative and non-perturbative results for the operator $\phi^5$. Prior to our work, three-loop results enabled the Padé$_{[1,2]}$ estimate. Padé$^*_{[3,4]}$ was obtained from our five-loop results, constrained to go through $5.29068$ for $d=3$Henriksson:2022gpa and the exact 2d value $6.125$.
• Figure 5: Padé approximants and conformal bootstrap results for the operator dimension $T'=\phi^4{\partial}^2$ in the Ising CFT. The three-correlator data are from Henriksson:2022gpa which does not provide error-bars, and the single-correlator data are from Cappelli:2018vir. The star denotes Padé approximants fixed to go through the point $(d,\Delta)=(2,6)$.