OPE Convergence in Conformal Field Theory

Abstract

We clarify questions related to the convergence of the OPE and conformal block decomposition in unitary Conformal Field Theories (for any number of spacetime dimensions). In particular, we explain why these expansions are convergent in a finite region. We also show that the convergence is exponentially fast, in the sense that the operators of dimension above Delta contribute to correlation functions at most exp(-a Delta). Here the constant a>0 depends on the positions of operator insertions and we compute it explicitly.

26 pages, 6 figures; v2: a clarifying note and two refs added; v3: note added concerning an extra constant factor in the main error estimate, misprint corrected

References in corpus (18)

• Bounding scalar operator dimensions in 4D CFT
• Holography from Conformal Field Theory
• Solving the 3D Ising Model with the Conformal Bootstrap
• Conformal Four Point Functions and the Operator Product Expansion
• Conformal Partial Waves and the Operator Product Expansion
• The Analytic Bootstrap and AdS Superhorizon Locality
• Spinning Conformal Blocks
• Bounds on 4D Conformal and Superconformal Field Theories
• Carving Out the Space of 4D CFTs
• Emergent Spacetime and Holographic CFTs
• Effective Conformal Theory and the Flat-Space Limit of AdS
• Universal Constraints on Conformal Operator Dimensions
• Analyticity and the Holographic S-Matrix
• Bounds in 4D Conformal Field Theories with Global Symmetry
• Central Charge Bounds in 4D Conformal Field Theory
• Rigorous Limits on the Interaction Strength in Quantum Field Theory
• Improved bounds for CFT's with global symmetries
• Entanglement Entropy of 3-d Conformal Gauge Theories with Many Flavors

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