Abstract
We compute the spectrum of anomalous dimensions of non-derivative composite operators with an arbitrary number of fields $n$ in the $O(N)$ vector model with cubic anisotropy at the one-loop order in the $\epsilon$-expansion. The complete closed-form expression for the anomalous dimensions of the operators which do not undergo mixing effects is derived and the structure of the general solution to the mixing problem is outlined. As examples, the full explicit solution for operators with up to $n=6$ fields is presented and a sample of the OPE coefficients is calculated. The main features of the spectrum are described, including an interesting pattern pointing to the deeper structure.
Highlights
Conformal field theories (CFTs) play a fundamental role in our understanding of the Universe with applications ranging from the discovery of the asymptotic freedom in QCD [1,2] to the potential applications in gravity in the scenario of asymptotic safety [3]
For a given symmetry group, CFT is specified via CFT data, i.e., the scaling dimensions of all the primary operators and the set of operator product expansion coefficients, defined as the constants appearing in the three-point functions of the theory
We study the CFT data of the theory invariant under the hypercubic symmetry group HN realized as a group of symmetries of an N-dimensional hypercube
Summary
Conformal field theories (CFTs) play a fundamental role in our understanding of the Universe with applications ranging from the discovery of the asymptotic freedom in QCD [1,2] to the potential applications in gravity in the scenario of asymptotic safety [3]. Hypercubic models are usually investigated in the perturbative ε expansion [6] using standard diagrammatic techniques, and present-day results involve the computation of anomalous dimensions (γφ, γm2) and beta functions to six-loop order [7,8]. This theory was explored nonperturbatively via an exact. The aim of this paper is to compute the spectrum of anomalous dimensions for composite operators with an arbitrary number of fields n but no derivatives in the HN critical theory to OðεÞ. Having introduced the model under consideration, the step is to unveil the operator spectrum at the cubic fixed point, i.e., to find the irreducible representations of the hypercubic group
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