Multi-loop spectra in general scalar EFTs and CFTs

TL;DR

The paper addresses the problem of obtaining high-loop anomalous dimensions for the most general scalar EFT, including higher-dimensional and spinning operators, to map the full operator spectrum of IR CFTs in $d=4-\varepsilon$. It develops a primary operator basis to remove redundancies and employs an improved $R^*_{\text{ME}}$ renormalisation method with a small-momentum expansion, enabling multi-loop computations up to dimension-six operators. The authors deliver extensive spectra for Ising ($Z_2$), $O(n)$, and cubic ($S_n\ltimes(\mathbb{Z}_2)^n$) CFTs, including a five-loop result for a rank-three operator in the $O(3)$ CFT, with strong agreement to bootstrap data where available and new predictions elsewhere. This work provides high-precision CFT data derived from EFT, supporting conformal bootstrap applications and offering datasets to advance non-perturbative studies across dimensions and symmetries.

Abstract

We consider the most general effective field theory (EFT) Lagrangian with scalar fields and derivatives, and renormalise it to substantially higher loop order than existing results in the literature. EFT Lagrangians have phenomenological applications, for example by encoding corrections to the Standard Model from unknown new physics. At the same time, scalar EFTs capture the spectrum of Wilson--Fisher conformal field theories (CFTs) in $4-\varepsilon$ dimensions. Our results are enabled by a more efficient version of the $R^*$ method for renormalisation, in which the IR divergences are subtracted via a small-momentum asymptotic expansion. In particular, we renormalise the most general set of composite operators up to engineering dimension six and Lorentz rank two. We exhibit direct applications of our results to Ising ($Z_2$), $O(n)$, and hypercubic ($S_n \ltimes (Z_2)^n$) CFTs, relevant for a plethora of real-world critical phenomena. The computed scaling dimensions agree well with known non-perturbative results, and they lead to new predictions where such results do not yet exist. We thereby expand the understanding of generic EFTs and open new possibilities in diverse fields, such as the numerical conformal bootstrap.

Figures (2)

• Figure 1: Overview of our results for all operators in the general theory at spin $\ell$ and mass dimension $[\mathcal{O}]$. Beyond dimension six, we provide results for the single scalar EFT ($Z_2$) and the $O(n)$ model (singlets only).
• Figure 2: Scalar spectrum in the Ising CFT with $\Delta<8$ for spacetime dimension $2.6\!\leq \! d\!<\!4$, using results up to $O({\varepsilon}^n)$. Dashed lines show previous leading determinations at $O(\textcolor{gray}{{\varepsilon}^m})$. Numerical bootstrap results (the dots) are from Simmons-Duffin:2016wlq (${d=3}$) and Henriksson:2022gpa. These works did not detect the operator $\phi^7$.