Crossing symmetry, transcendentality and the Regge behaviour of 1d CFTs
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Springer
Received: January 2, Revised: June 1, Accepted: June 21, Published: July 23,
2020 2020 2020 2020
Pietro Ferrero,a Kausik Ghosh,b Aninda Sinhab and Ahmadullah Zahedb a
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, U.K. b Centre for High Energy Physics, Indian Institute of Science, C.V. Raman Avenue, Bangalore 560012, India
E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: We develop the technology for Polyakov-Mellin (PM) bootstrap in onedimensional conformal field theories (CFT1 ). By adding appropriate contact terms, we bootstrap various effective field theories in AdS2 and analytically compute the CFT data to one loop. The computation can be extended to higher orders in perturbation theory, if we ignore mixing, for any external dimension. We develop PM bootstrap for O(N ) theories and derive the necessary contact terms for such theories (which also involves a new higher gradient contact term absent for N = 1). We perform cross-checks which include considering the diagonal limit of the 2d Ising model in terms of the 1d PM blocks. As an independent check of the validity of the results obtained with PM bootstrap, we propose a suitable basis of transcendental functions, which allows to fix the four-point correlators of identical scalar primaries completely, up to a finite number of ambiguities related to the number of contact terms in the PM basis. We perform this analysis both at tree level (with and without exchanges) and at one loop. We also derive expressions for the corresponding CFT data in terms of harmonic sums. Finally, we consider the Regge limit of one-dimensional correlators and derive a precise connection between the latter and the large-twist limit of CFT data. Exploiting this result, we study the crossing equation in the three OPE limits and derive some universal constraints for the large-twist limit of CFT data in Regge-bounded theories with a finite number of exchanges. Keywords: 1/N Expansion, AdS-CFT Correspondence, Conformal Field Theory, Effective Field Theories ArXiv ePrint: 1911.12388
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP07(2020)170
JHEP07(2020)170
Crossing symmetry, transcendentality and the Regge behaviour of 1d CFTs
Contents 1 Introduction
1 5 7 12
3 Implementing bootstrap 3.1 Deforming away from GFF 3.1.1 Contact term without derivatives 3.1.2 Generic contact interactions 3.2 Theories with O(N ) global symmetry 3.2.1 With contact term degree 1 in s and t 3.2.2 Contact term of degree 2 in s and t 3.3 Effective field theory — exchange interaction 3.4 Tower of exchange operators
18 18 19 22 25 25 27 27 29
4 Transcendentality ansatz — tree level 4.1 Contact terms, single field 4.2 Contact terms, O(N ) global symmetry 4.3 Exchanges 4.3.1 Large ∆E : EFT expansion
30 33 36 41 45
5 Intermezzo: Regge limit in 1d CFTs 5.1 The Regge limit of conformal blocks 5.2 OPE limits and crossin
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