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

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 $\epsilon$-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.

38 pages, no figures; v2: 40 pages, typos corrected, references added; v3: matches published JHEP version, one reference added

References in corpus (13)

• Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents
• Bootstrapping Mixed Correlators in the 3D Ising Model
• Bounds on 4D Conformal and Superconformal Field Theories
• Conformal Bootstrap in Mellin Space
• The $\mathcal{N}=8$ Superconformal Bootstrap in Three Dimensions
• Critical exponents of the 3d Ising and related models from Conformal Bootstrap
• Five dimensional $O(N)$-symmetric CFTs from conformal bootstrap
• Closure of the Operator Product Expansion in the Non-Unitary Bootstrap
• Truncatable bootstrap equations in algebraic form and critical surface exponents
• Higher-Spin Theory and Space-Time Metamorphoses
• Generalized Wilson-Fisher critical points from the conformal OPE
• Flato-Fronsdal theorem for higher-order singletons
• $\epsilon$-Expansion in Critical ${\varphi }^3$-Theory on Real Projective Space from Conformal Field Theory

Cited by in corpus (39)

• The Conformal Bootstrap: Theory, Numerical Techniques, and Applications
• Weight Shifting Operators and Conformal Blocks
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• Casimir recursion relations for general conformal blocks
• Simplifying large spin bootstrap in Mellin space
• Scalar CFTs and Their Large N Limits
• Lightcone Bootstrap at higher points
• Leading CFT constraints on multi-critical models in d>2
• Anomalous Dimensions in the WF O($N$) Model with a Monodromy Line Defect
• Higher-Point Conformal Blocks in the Comb Channel
• O(N) and O(N) and O(N)
• Functional perturbative RG and CFT data in the $\epsilon$-expansion
• Critical O(N) model to order ${\epsilon }^4$ from analytic bootstrap
• Aspects of CFTs on Real Projective Space
• Scaling dimensions in QED${}_3$ from the $\epsilon$-expansion
• New Methods for Conformal Correlation Functions
• Inverse Bootstrapping Conformal Field Theories
• Critical Ising Model in Varying Dimension by Conformal Bootstrap
• Leading order CFT analysis of multi-scalar theories in d>2
• Towards a Bootstrap approach to higher orders of epsilon expansion
• Spectrum of anomalous dimensions in hypercubic theories
• Bands and gaps in Nekrasov partition function
• Higher dimensional higher derivative ${\varphi }^4$ theory
• Three ways to solve critical ${\varphi }^4$ theory on $4-\epsilon$ dimensional real projective space: perturbation, bootstrap, and Schwinger-Dyson equation
• Ising model close to $d=2$
• Efficient Rules for All Conformal Blocks
• Anomalous dimensions of spinning operators from conformal symmetry
• Uncovering novel phase structures in ${\square }^k$ scalar theories with the renormalization group
• Multi-critical ${\square }^k$ scalar theories: A perturbative RG approach with $\epsilon$-expansion
• Navigating through the O(N) archipelago
• Conformal Higher-Spin Gravity: Linearized Spectrum = Symmetry Algebra
• Anomalous dimensions at large charge for $U(N)\times U(N)$ theory in three and four dimensions
• Anomalous dimensions of partially-conserved higher-spin currents from conformal field theory: bosonic ${\varphi }^{2n}$ theories
• Mellin Bootstrap for Scalars in Generic Dimension
• Properties of the $\epsilon$-Expansion, Lagrange Inversion and Associahedra and the $O(1)$ Model
• Anomalous dimensions from conformal field theory: Generalized ${\varphi }^{2n+1}$ theories
• Boundary anomalous dimensions from BCFT: O($N$)-symmetric ${\varphi }^{2n}$ theories with a boundary and higher-derivative generalizations