Conformal 3-Point Functions and the Lorentzian OPE in Momentum Space
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Communications in
Mathematical Physics
Conformal 3-Point Functions and the Lorentzian OPE in Momentum Space Marc Gillioz1,2 1 SISSA, Via Bonomea 265, 34136 Trieste, Italy 2 Theoretical Particle Physics Laboratory, Institute of Physics, EPFL, Lausanne, Switzerland.
E-mail: [email protected] Received: 1 October 2019 / Accepted: 29 May 2020 © The Author(s) 2020
Abstract: In conformal field theory in Minkowski momentum space, the 3-point correlation functions of local operators are completely fixed by symmetry. Using Ward identities together with the existence of a Lorentzian operator product expansion (OPE), we show that the Wightman function of three scalar operators is a double hypergeometric series of the Appell F4 type. We extend this simple closed-form expression to the case of two scalar operators and one traceless symmetric tensor with arbitrary spin. Time-ordered and partially-time-ordered products are constructed in a similar fashion and their relation with the Wightman function is discussed. Contents 1. 2.
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Introduction . . . . . . . . . . . . . . . . . . . . . 1.1 Strategy and main result . . . . . . . . . . . . . The Wightman 3-Point Function of Scalar Operators 2.1 Momentum eigenstates and support . . . . . . . 2.2 OPE limits in momentum space . . . . . . . . . 2.3 Conformal Ward identities . . . . . . . . . . . 2.4 Analytic continuation and normalization . . . . 2.5 Generalized free field theory . . . . . . . . . . 2.6 Holomorphic factorization in two dimensions . Adding Spin: Traceless Symmetric Tensor . . . . . 3.1 Poincaré and scale symmetry . . . . . . . . . . 3.2 Conformal Ward identities . . . . . . . . . . . 3.3 Analytic continuation and normalization . . . . Time-Ordered Products . . . . . . . . . . . . . . . 4.1 Partial time-ordering . . . . . . . . . . . . . . 4.2 Relationship with the Wightman function . . . 4.3 The fully time-ordered 3-point function . . . .
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M. Gillioz
5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Conformal Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Direct Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction Conformal field theory can be formulated algebraically in terms of a set of primary operators and of rules that define the operator product expansion (OPE). Equivalently, all correlation functions of a conformal field theory can be obtained from 2- and 3-point functions, which are themselves fixed by conformal symmetry up to a small number of numerical coefficients. This statement applies both to correlation functions in position space and in mo
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