Title: Loops in AdS from conformal field theory

Abstract

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 non-planar 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 four-point, loop-level Mellin amplitudes can be fully reconstructed from the leading-order 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 one-loop bubble diagram. We can go further, deriving the four-point 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 super-Yang-Mills theory.

Authors:

Aharony, Ofer ^[1]; Alday, Luis F. ^[2]; Bissi, Agnese ^[3]; Perlmutter, Eric ^[4]

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+ Show Author Affiliations

1. Weizmann Inst. of Science, Rehovot (Israel). Dept. of Particle Physics and Astrophysics
2. Univ. of Oxford (United Kingdom). Mathematical Inst.
3. Harvard Univ., Cambridge, MA (United States). Center for the Fundamental Laws of Nature
4. Princeton Univ., NJ (United States). Dept. of Physics

Publication Date:

2017-07-10

Research Org.:

Trustees of Princeton Univ., NJ (United States)

Sponsoring Org.:

USDOE; Israel Science Foundation

OSTI Identifier:

1425488

Grant/Contract Number:  

FG02-91ER40671; 1937/12; 1200/14; 306260

Resource 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 1029-8479

Publisher:

Springer Berlin

Country of Publication:

United States

Language:

English

Subject:

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; AdS-CFT Correspondence; Conformal Field Theory; Gauge-gravity correspondence; Scattering Amplitudes