Soft photon theorems from CFT Ward identites in the flat limit of AdS/CFT
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Springer
Received: May 27, Revised: July 31, Accepted: October 3, Published: November 4,
2020 2020 2020 2020
Eliot Hijanoa and Dominik Neuenfeldb a
Department of Physics, Princeton University, Princeton, NJ 08544, U.S.A. b Perimeter Institute for Theoretical Physics, 31 Caroline Street N., Waterloo, Ontario N2L 2Y5, Canada
E-mail: [email protected], [email protected] Abstract: S-matrix elements in flat space can be obtained from a large AdS-radius limit of certain CFT correlators. We present a method for constructing CFT operators which create incoming and outgoing scattering states in flat space. This is done by taking the flat limit of bulk operator reconstruction techniques. Using this method, we obtain explicit expressions for incoming and outgoing U(1) gauge fields. Weinberg soft photon theorems then follow from Ward identites of conserved CFT currents. In four bulk dimensions, gauge fields on AdS can be quantized with standard and alternative boundary conditions. Changing the quantization scheme corresponds to the S-transformation of SL(2, Z) electricmagnetic duality in the bulk. This allows us to derive both, the electric and magnetic soft photon theorems in flat space from CFT physics. Keywords: AdS-CFT Correspondence, Conformal and W Symmetry, Duality in Gauge Field Theories, Scattering Amplitudes ArXiv ePrint: 2005.03667
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP11(2020)009
JHEP11(2020)009
Soft photon theorems from CFT Ward identites in the flat limit of AdS/CFT
Contents 1 Introduction
2 3 4 5 7 8 10 12 13
3 Photon scattering states 3.1 Inequivalent quantization schemes of scalar fields in AdS 3.2 Electric-magnetic duality and SL(2, Z) action on CFT3 ’s 3.3 Flat limit: magnetic boundary conditions 3.4 Flat limit: electric boundary conditions 3.5 Coulombic source terms
15 15 16 17 20 21
4 From conformal Ward identities to Weinberg soft theorems 4.1 Ward identities in CFT 4.2 Weinberg soft theorems as symmetries of the S-matrix 4.3 From CFT physics to soft theorems 4.4 From CFT physics to magnetic soft theorems
22 22 23 25 27
5 Conclusions
29
A Complex coordinates on S 2
31
B Scalar HKLL operators
31
C Reconstruction of U(1) gauge fields in global AdS4 C.1 Reconstruction with “magnetic” boundary conditions C.2 Reconstruction with “electric” boundary conditions
33 33 36
D Parameter α(x)
37
E AdS Liénard-Wiechert potentials
39
F Sample calculations F.1 One-to-one scattering F.2 One particle-state normalization
41 42 42
–1–
JHEP11(2020)009
2 Scattering states revisited: flat limit of AdS/CFT 2.1 Scattering amplitudes in flat space-time 2.2 Lorentzian AdS/CFT 2.3 Flat limit of AdS/CFT 2.4 Global reconstruction of scalar operators in AdS/CFT 2.4.1 Flat limit of HKLL operator: massless 2.4.2 Flat limit of HKLL operator: massive 2.5 Mapping of asymptotic regions
1
Introduction
1
This is not true in certain cases, most prominently if the particles live in four dimensions and couple to long range forces. In that case the
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