The unique Polyakov blocks
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Springer
Received: July 6, 2020 Accepted: October 6, 2020 Published: November 16, 2020
The unique Polyakov blocks
a
School of Natural Sciences, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, U.S.A. b Department of Physics, Princeton University, Jadwin Hall, Princeton, NJ 08544, U.S.A. c Dipartimento di Fisica “Ettore Pancini”, Università degli Studi di Napoli Federico II, Via Cintia, Monte S. Angelo, Napoli 80126, Italy d INFN, Sezione di Napoli, Via Cintia, Monte S. Angelo, Napoli 80126, Italy
E-mail: [email protected], [email protected] Abstract: In this work we present a closed form expression for Polyakov blocks in Mellin space for arbitrary spin and scaling dimensions. We provide a prescription to fix the contact term ambiguity uniquely by reducing the problem to that of fixing the contact term ambiguity at the level of cyclic exchange amplitudes — defining cyclic Polyakov blocks — in terms of which any fully crossing symmetric correlator can be decomposed. We also give another, equivalent, prescription which does not rely on a decomposition into cyclic amplitudes. We extract the OPE data of double-twist operators in the direct channel expansion of the cyclic Polyakov blocks using and extending the analysis of [1, 2] to include contributions that are non-analytic in spin. The relation between cyclic Polyakov blocks and analytic Bootstrap functionals is underlined. Keywords: Conformal Field Theory, AdS-CFT Correspondence ArXiv ePrint: 1912.07998
1
Also at the Université Libre de Bruxelles and International Solvay Institutes, Belgium.
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP11(2020)075
JHEP11(2020)075
Charlotte Sleighta,1 and Massimo Taronnab,c,d
Contents 1
2 An alternative derivation for identical external legs
7
3 Conformal block decomposition of Polyakov blocks 3.1 Conformal block decomposition 3.2 Relation with dual bootstrap functionals
11 12 22
A Mack polynomials
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B Continuous Hahn polynomials
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C Contact term ambiguity
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D Leading non-analytic piece
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1
The problem and its resolution
The simplest solutions to the crossing equation in Conformal Field Theory (CFT) that are single-valued in the Euclidean region are given by tree-level exchange amplitudes in the dual anti-de Sitter (AdS) space, which have come to be known as Polyakov blocks [3]. It is well known that it is possible to bootstrap the Witten diagram for the exchange of a field with spin-` and mass m2 R2 = ∆ (∆ − d) − ` simply by requiring [4–7]: crossing symmetry, the presence of a conformal block with scaling dimension ∆ and spin ` in the OPE decomposition and Euclidean single-valuedness. This bootstrap problem is however subject to an ambiguity that is parameterised by solutions to crossing with finite support in spin. These are in one-to-one correspondence with bulk contact terms which, for spinning internal legs, have a better Regge behaviour than the full exchange solution itself. Naively one might expect that there does not exist a canon
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