Action principle for OPE

Abstract

We formulate an "action principle" for the operator product expansion (OPE) describing how a given OPE coefficient changes under a deformation induced by a marginal or relevant operator. Our action principle involves no ad-hoc regulator or renormalization and applies to general (Euclidean) quantum field theories. It implies a natural definition of the renormalization group flow for the OPE coefficients and of coupling constants. When applied to the case of conformal theories, the action principle gives a system of coupled dynamical equations for the conformal data. The last result has also recently been derived (without considering tensor structures) independently by Behan (arXiv:1709.03967) using a different argument. Our results were previously announced and outlined at the meetings "In memoriam Rudolf Haag" in September 2016 and the "Wolfhart Zimmermann memorial symposium" in May 2017.

29 pages, 5 figures, based on conference talks at the meetings "In memoriam Rudolf Haag" in September 2016 and the "Wolfhart Zimmermann memorial symposium" in May 2017; v2: details added concerning geometry of field redefinitions, discussion of degeneracies and normalization issues, references edited, other minor editorial changes, v3: edited para on invariant 2-point tensor structures

References in corpus (25)

• Exact evolution equation for the effective potential
• Solving the 3D Ising Model with the Conformal Bootstrap
• EPFL Lectures on Conformal Field Theory in D>= 3 Dimensions
• Spinning Conformal Blocks
• OPE Convergence in Conformal Field Theory
• Axiomatic quantum field theory in curved spacetime
• Conformal Blocks for Arbitrary Spins in Two Dimensions
• Radial expansion for spinning conformal blocks
• Harmony of Spinning Conformal Blocks
• Casimir recursion relations for general conformal blocks
• Full field algebras
• In search of conformal theories
• The Renormalization Group According to Balaban - I. Small fields
• The operator product expansion converges in perturbative field theory
• The Renormalization Group According to Balaban - III. Convergence
• The Renormalization Group According to Balaban - II. Large fields
• Renormalization of SU(2) Yang-Mills theory with Flow Equations
• Recursive construction of operator product expansion coefficients
• Operator product expansion algebra
• All-order bounds for correlation functions of gauge-invariant operators in Yang-Mills theory
• Quantum field theory in terms of consistency conditions I: General framework, and perturbation theory via Hochschild cohomology
• The operator product expansion converges in massless ${\phi }_4^4$-theory
• All-order existence of and recursion relations for the operator product expansion in Yang-Mills theory
• Associativity of the operator product expansion
• All-order uniform momentum bounds for the massless phi^4 theory in four dimensional Euclidean space

Cited by in corpus (16)

• A Study of Quantum Field Theories in AdS at Finite Coupling
• Decoding a Three-Dimensional Conformal Manifold
• On Higher-Spin Points and Infinite Distances in Conformal Manifolds
• Holographic Evidence for Non-Supersymmetric Conformal Manifolds
• New Methods for Conformal Correlation Functions
• Renormalization of Galilean Electrodynamics
• The unitary subsector of generalized minimal models
• Conformal Manifolds with Boundaries or Defects
• First-order conformal perturbation theory by marginal operators
• Conformal Two-Point Correlation Functions from the Operator Product Expansion
• Marginal deformations from type IIA supergravity
• Holographic 3d $\mathcal{N}=1$ Conformal Manifolds
• Recursive construction of the operator product expansion in curved space
• Wilson Loops with Lagrangians: Large Spin OPE and Cusp Anomalous Dimension Dictionary
• The Operator Product Expansion in Quantum Field Theory
• Constructing CFTs from AdS flows