Loops in AdS from conformal field theory
We propose and demonstrate a new use for conformal field theory (CFT) crossing equations in the context of AdS/CFT: the computation of loop amplitudes in AdS, dual to nonplanar correlators in holographic CFTs. Loops in AdS are largely unexplored, mostly due to technical difficulties in direct calculations. We revisit this problem, and the dual 1=N expansion of CFTs, in two independent ways. The first is to show how to explicitly solve the crossing equations to the first subleading order in 1=N ^{2}, given a leading order solution. This is done as a systematic expansion in inverse powers of the spin, to all orders. These expansions can be resummed, leading to the CFT data for nite values of the spin. Our second approach involves Mellin space. We show how the polar part of the fourpoint, looplevel Mellin amplitudes can be fully reconstructed from the leadingorder data. The anomalous dimensions computed with both methods agree. In the case of $$\phi$$ ^{4} theory in AdS, our crossing solution reproduces a previous computation of the oneloop bubble diagram. We can go further, deriving the fourpoint scalar triangle diagram in AdS, which had never been computed. In the process, we show how to analytically derive anomalous dimensions from Mellin amplitudes with an in nite series of poles, and discuss applications to more complicated cases such as the N = 4 superYangMills theory.
 Authors:

^{[1]};
^{[2]};
^{[3]};
^{[4]}
 Weizmann Inst. of Science, Rehovot (Israel). Dept. of Particle Physics and Astrophysics
 Univ. of Oxford (United Kingdom). Mathematical Inst.
 Harvard Univ., Cambridge, MA (United States). Center for the Fundamental Laws of Nature
 Princeton Univ., NJ (United States). Dept. of Physics
 Publication Date:
 Grant/Contract Number:
 FG0291ER40671; 1937/12; 1200/14; 306260
 Type:
 Accepted Manuscript
 Journal Name:
 Journal of High Energy Physics (Online)
 Additional Journal Information:
 Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2017; Journal Issue: 7; Journal ID: ISSN 10298479
 Publisher:
 Springer Berlin
 Research Org:
 Trustees of Princeton Univ., NJ (United States)
 Sponsoring Org:
 USDOE; Israel Science Foundation
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; AdSCFT Correspondence; Conformal Field Theory; Gaugegravity correspondence; Scattering Amplitudes
 OSTI Identifier:
 1425488
Aharony, Ofer, Alday, Luis F., Bissi, Agnese, and Perlmutter, Eric. Loops in AdS from conformal field theory. United States: N. p.,
Web. doi:10.1007/JHEP07(2017)036.
Aharony, Ofer, Alday, Luis F., Bissi, Agnese, & Perlmutter, Eric. Loops in AdS from conformal field theory. United States. doi:10.1007/JHEP07(2017)036.
Aharony, Ofer, Alday, Luis F., Bissi, Agnese, and Perlmutter, Eric. 2017.
"Loops in AdS from conformal field theory". United States.
doi:10.1007/JHEP07(2017)036. https://www.osti.gov/servlets/purl/1425488.
@article{osti_1425488,
title = {Loops in AdS from conformal field theory},
author = {Aharony, Ofer and Alday, Luis F. and Bissi, Agnese and Perlmutter, Eric},
abstractNote = {We propose and demonstrate a new use for conformal field theory (CFT) crossing equations in the context of AdS/CFT: the computation of loop amplitudes in AdS, dual to nonplanar correlators in holographic CFTs. Loops in AdS are largely unexplored, mostly due to technical difficulties in direct calculations. We revisit this problem, and the dual 1=N expansion of CFTs, in two independent ways. The first is to show how to explicitly solve the crossing equations to the first subleading order in 1=N2, given a leading order solution. This is done as a systematic expansion in inverse powers of the spin, to all orders. These expansions can be resummed, leading to the CFT data for nite values of the spin. Our second approach involves Mellin space. We show how the polar part of the fourpoint, looplevel Mellin amplitudes can be fully reconstructed from the leadingorder data. The anomalous dimensions computed with both methods agree. In the case of $\phi$4 theory in AdS, our crossing solution reproduces a previous computation of the oneloop bubble diagram. We can go further, deriving the fourpoint scalar triangle diagram in AdS, which had never been computed. In the process, we show how to analytically derive anomalous dimensions from Mellin amplitudes with an in nite series of poles, and discuss applications to more complicated cases such as the N = 4 superYangMills theory.},
doi = {10.1007/JHEP07(2017)036},
journal = {Journal of High Energy Physics (Online)},
number = 7,
volume = 2017,
place = {United States},
year = {2017},
month = {7}
}