The analytic structure of conformal blocks and the generalized Wilson-Fisher fixed points

TL;DR

This work develops a conformal-field-theory–inspired, Lagrangian-free framework to study Wilson-Fisher–type fixed points near generalized free CFTs across dimensions. By exploiting the analytic structure of conformal blocks as functions of the exchanged dimension $\Delta$, the authors implement multiplet recombination via null states to extract leading $\epsilon$-expansions for anomalous dimensions and OPE coefficients for broad families of operators, including single scalars, $O(N)$ vectors and tensors, and theories with multiple marginal deformations. They connect the pole structure of generic conformal blocks to partially conserved higher-spin currents and generalized Killing tensors, and show how these poles govern the appearance of descendants in deformed theories. The results generalize known WF behavior to multicritical and nonunitary regimes, providing numerous new analytic predictions and a unified scheme for handling nonunitary and higher-spin–related sectors, with potential extensions to fermionic, supersymmetric, and matrix/tensor models. Overall, the paper provides a powerful, symmetry-based toolkit for accessing nontrivial CFT data in the $\epsilon$-expansion without relying on Lagrangians or unitarity.

Abstract

We describe in detail the method used in our previous work arXiv:1611.10344 to study the Wilson-Fisher critical points nearby generalized free CFTs, exploiting the analytic structure of conformal blocks as functions of the conformal dimension of the exchanged operator. Our method is equivalent to the mechanism of conformal multiplet recombination set up by null states. We compute, to the first non-trivial order in the $ε$-expansion, the anomalous dimensions and the OPE coefficients of infinite classes of scalar local operators using just CFT data. We study single-scalar and $O(N)$-invariant theories, as well as theories with multiple deformations. When available we agree with older results, but we also produce a wealth of new ones. Unitarity and crossing symmetry are not used in our approach and we are able to apply our method to non-unitary theories as well. Some implications of our results for the study of the non-unitary theories containing partially conserved higher-spin currents are briefly mentioned.