The four-point correlation function of the energy-momentum tensor in the free conformal field theory of a scalar field
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Regular Article - Theoretical Physics
The four-point correlation function of the energy-momentum tensor in the free conformal field theory of a scalar field Mirko Serinoa Department of Physics, Ben Gurion University of the Negev, Beer Sheva 8410501, Israel
Received: 22 May 2020 / Accepted: 6 July 2020 © The Author(s) 2020
Abstract We present an explicit momentum space computation of the four-point function of the energy-momentum tensor in 4 spacetime dimensions for the free and conformally invariant theory of a scalar field. The result is obtained by explicit evaluation of the Feynman diagrams by tensor reduction. We work by embedding the scalar field theory in a gravitational background consistently with conformal invariance in order to derive all the terms the correlator consists of and all the Ward identities implied by the requirements of general covariance and anomalous Weyl symmetry. We test all these identities numerically in several kinematic configurations. Mathematica notebooks detailing the step-by-step computation are made publicly available through a GitHub repository (https://github.com/mirkos86/ 4-EMT-correlation-function-in-a-4d-CFT.). To the best of our knowledge, this is the first explicit result for the fourpoint correlation function of the energy-momentum tensor in a conformal and non supersymmetric field theory which is readily numerically evaluable in any kinematic configuration.
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . 2 Setting up the computation . . . . . . . . . . . . . . 2.1 The structure of the correlators: topologies and diagrams . . . . . . . . . . . . . . . . . . . . . 2.2 Organization of the Mathematica files . . . . . 3 Derivation of the transverse and trace Ward identities 4 Counterterms, anomalies and a preliminary test of the 4T correlator . . . . . . . . . . . . . . . . . . . 5 Into the full calculation . . . . . . . . . . . . . . . . In loving memory of Giuseppe Serino * April 1st 1956 † December 4th 2018. a e-mail:
[email protected] (corresponding author)
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5.1 The calculation of the four-point correlator . . . . 5.2 Analytical checks of the Ward identities for the 2T and 3T . . . . . . . . . . . . . . . . . . . . . 5.3 Numerical checks of the Ward identities for the 4T correlator . . . . . . . . . . . . . . . . . . . 5.4 Discussion about numerical stability . . . . . . . 6 Conclusions and perspectives . . . . . . . . . . . . . A Conventions . . . . . . . . . . . . . . . . . . . . . . A.1 Conventions for signs and momentum space correlators definition . . . . . . . . . . . . . . . . . A.2 Weyl invariant and Euler density in 4d . . . . . . A.3 Vanishing of the Euler counterterms . . . . . . . B Functional derivatives . . . . . . . . . . . . . . . . . B.1 Basic functional derivatives . . . . . . . . . . . . B.2 Interaction vertices of the scalar field with gravitons References . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction The energy-momentum tensor is the most universal operator for conf
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