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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:
 [1];  [2];  [3];  [4]
  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:
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

Citation Formats

Aharony, Ofer, Alday, Luis F., Bissi, Agnese, and Perlmutter, Eric. Loops in AdS from conformal field theory. United States: N. p., 2017. 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. Mon . "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 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=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 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.},
doi = {10.1007/JHEP07(2017)036},
journal = {Journal of High Energy Physics (Online)},
number = 7,
volume = 2017,
place = {United States},
year = {2017},
month = {7}
}

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Works referenced in this record:

Comments on 4-point functions in the CFT/AdS correspondence
journal, April 1999


Comments on operators with large spin
journal, November 2007


Conformal Bootstrap in Mellin Space
journal, February 2017


Causality constraints in conformal field theory
journal, May 2016

  • Hartman, Thomas; Jain, Sachin; Kundu, Sandipan
  • Journal of High Energy Physics, Vol. 2016, Issue 5
  • DOI: 10.1007/JHEP05(2016)099

Order 1/N2 test of the Maldacena conjecture: cancellation of the one-loop Weyl anomaly
journal, December 2000


The analytic bootstrap and AdS superhorizon locality
journal, December 2013

  • Fitzpatrick, A. Liam; Kaplan, Jared; Poland, David
  • Journal of High Energy Physics, Vol. 2013, Issue 12
  • DOI: 10.1007/JHEP12(2013)004

General scalar exchange in AdSd+1
journal, June 1999


Conformal Regge theory
journal, December 2012

  • Costa, Miguel S.; Goncalves, Vasco; Penedones, João
  • Journal of High Energy Physics, Vol. 2012, Issue 12
  • DOI: 10.1007/JHEP12(2012)091

Gauge theory correlators from non-critical string theory
journal, May 1998


Higher spins in AdS5 at one loop: vacuum energy, boundary conformal anomalies and AdS/CFT
journal, November 2014

  • Beccaria, Matteo; Tseytlin, Arkady A.
  • Journal of High Energy Physics, Vol. 2014, Issue 11
  • DOI: 10.1007/JHEP11(2014)114

An algebraic approach to the analytic bootstrap
journal, April 2017

  • Alday, Luis F.; Zhiboedov, Alexander
  • Journal of High Energy Physics, Vol. 2017, Issue 4
  • DOI: 10.1007/JHEP04(2017)157

Mellin transforming the minimal model CFTs: AdS/CFT at strong curvature
journal, September 2016


Bounding the space of holographic CFTs with chaos
journal, October 2016


Anti de Sitter space and holography
journal, January 1998


Factorization of Mellin amplitudes
journal, October 2015

  • Gonçalves, Vasco; Penedones, João; Trevisani, Emilio
  • Journal of High Energy Physics, Vol. 2015, Issue 10
  • DOI: 10.1007/JHEP10(2015)040

A new spin on causality constraints
journal, October 2016

  • Hartman, Thomas; Jain, Sachin; Kundu, Sandipan
  • Journal of High Energy Physics, Vol. 2016, Issue 10
  • DOI: 10.1007/JHEP10(2016)141

Causality constraints on corrections to the graviton three-point coupling
journal, February 2016

  • Camanho, Xián O.; Edelstein, José D.; Maldacena, Juan
  • Journal of High Energy Physics, Vol. 2016, Issue 2
  • DOI: 10.1007/JHEP02(2016)020

Analyticity and the holographic S-matrix
journal, October 2012

  • Fitzpatrick, A. Liam; Kaplan, Jared
  • Journal of High Energy Physics, Vol. 2012, Issue 10
  • DOI: 10.1007/JHEP10(2012)127

5 loops in 24/5 dimensions
journal, August 2010

  • Björnsson, Jonas; Green, Michael B.
  • Journal of High Energy Physics, Vol. 2010, Issue 8
  • DOI: 10.1007/JHEP08(2010)132

Conformal bootstrap with slightly broken higher spin symmetry
journal, June 2016

  • Alday, Luis F.; Zhiboedov, Alexander
  • Journal of High Energy Physics, Vol. 2016, Issue 6
  • DOI: 10.1007/JHEP06(2016)091

One-loop partition functions of 3D gravity
journal, August 2008


More holography from conformal field theory
journal, September 2010


Aspects of the conformal operator product expansion in $\mathrm{AdS}$/$\mathrm{CFT}$ correspondence
journal, January 2000

  • Hoffmann, Laurent; Petkou, Anastasios C.; Rühl, Werner
  • Advances in Theoretical and Mathematical Physics, Vol. 4, Issue 3
  • DOI: 10.4310/ATMP.2000.v4.n3.a3

Holography from conformal field theory
journal, October 2009


AdS/CFT four-point functions: How to succeed at z-integrals without really trying
journal, November 1999


Fusing gauge theory tree amplitudes into loop amplitudes
journal, February 1995


terms in 11 dimensions and conformal anomaly of () theory
journal, September 2000


Writing CFT correlation functions as AdS scattering amplitudes
journal, March 2011


Universality of long-distance AdS physics from the CFT bootstrap
journal, August 2014

  • Fitzpatrick, A. Liam; Kaplan, Jared; Walters, Matthew T.
  • Journal of High Energy Physics, Vol. 2014, Issue 8
  • DOI: 10.1007/JHEP08(2014)145

A bound on chaos
journal, August 2016

  • Maldacena, Juan; Shenker, Stephen H.; Stanford, Douglas
  • Journal of High Energy Physics, Vol. 2016, Issue 8
  • DOI: 10.1007/JHEP08(2016)106

Rigid holography and six-dimensional N = 2 0 $$ \mathcal{N}=\left(2,0\right) $$ theories on AdS5 × S $$ \mathbb{S} $$ 1
journal, March 2015

  • Aharony, Ofer; Berkooz, Micha; Rey, Soo-Jong
  • Journal of High Energy Physics, Vol. 2015, Issue 3
  • DOI: 10.1007/JHEP03(2015)121

1/N 2 corrections to the holographic Weyl anomaly
journal, January 2014

  • Ardehali, Arash Arabi; Liu, James T.; Szepietowski, Phillip
  • Journal of High Energy Physics, Vol. 2014, Issue 1
  • DOI: 10.1007/JHEP01(2014)002

One loop tests of higher spin AdS/CFT
journal, December 2013

  • Giombi, Simone; Klebanov, Igor R.
  • Journal of High Energy Physics, Vol. 2013, Issue 12
  • DOI: 10.1007/JHEP12(2013)068

A natural language for AdS/CFT correlators
journal, November 2011

  • Fitzpatrick, A. Liam; Kaplan, Jared; Penedones, Joao
  • Journal of High Energy Physics, Vol. 2011, Issue 11
  • DOI: 10.1007/JHEP11(2011)095

Non-abelian binding energies from the lightcone bootstrap
journal, February 2016

  • Li, Daliang; Meltzer, David; Poland, David
  • Journal of High Energy Physics, Vol. 2016, Issue 2
  • DOI: 10.1007/JHEP02(2016)149

Harmonic Polylogarithms
journal, February 2000

  • Remiddi, E.; Vermaseren, J. A. M.
  • International Journal of Modern Physics A, Vol. 15, Issue 05
  • DOI: 10.1142/S0217751X00000367

The operator product expansion of SYM and the 4-point functions of supergravity
journal, November 2000


Convexity and liberation at large spin
journal, November 2013

  • Komargodski, Zohar; Zhiboedov, Alexander
  • Journal of High Energy Physics, Vol. 2013, Issue 11
  • DOI: 10.1007/JHEP11(2013)140

On Feynman rules for Mellin amplitudes in AdS/CFT
journal, May 2012

  • Nandan, Dhritiman; Volovich, Anastasia; Wen, Congkao
  • Journal of High Energy Physics, Vol. 2012, Issue 5
  • DOI: 10.1007/JHEP05(2012)129

Graviton exchange and complete four-point functions in the AdS/CFT correspondence
journal, November 1999


One-loop n-point gauge theory amplitudes, unitarity and collinear limits
journal, August 1994


A Mellin space approach to the conformal bootstrap
journal, May 2017

  • Gopakumar, Rajesh; Kaviraj, Apratim; Sen, Kallol
  • Journal of High Energy Physics, Vol. 2017, Issue 5
  • DOI: 10.1007/JHEP05(2017)027

Looking for a bulk point
journal, January 2017

  • Maldacena, Juan; Simmons-Duffin, David; Zhiboedov, Alexander
  • Journal of High Energy Physics, Vol. 2017, Issue 1
  • DOI: 10.1007/JHEP01(2017)013

Holographic Derivation of Entanglement Entropy from the anti–de Sitter Space/Conformal Field Theory Correspondence
journal, May 2006


Eikonal approximation in AdS/CFT: resumming the gravitational loop expansion
journal, September 2007


A proof of the conformal collider bounds
journal, June 2016

  • Hofman, Diego M.; Li, Daliang; Meltzer, David
  • Journal of High Energy Physics, Vol. 2016, Issue 6
  • DOI: 10.1007/JHEP06(2016)111

Emergent spacetime and holographic CFTs
journal, October 2012

  • El-Showk, Sheer; Papadodimas, Kyriakos
  • Journal of High Energy Physics, Vol. 2012, Issue 10
  • DOI: 10.1007/JHEP10(2012)106

Witten diagrams revisited: the AdS geometry of conformal blocks
journal, January 2016

  • Hijano, Eliot; Kraus, Per; Perlmutter, Eric
  • Journal of High Energy Physics, Vol. 2016, Issue 1
  • DOI: 10.1007/JHEP01(2016)146

The heat kernel on AdS 3 and its applications
journal, April 2010

  • David, Justin R.; Gaberdiel, Matthias R.; Gopakumar, Rajesh
  • Journal of High Energy Physics, Vol. 2010, Issue 4
  • DOI: 10.1007/JHEP04(2010)125

Spinning AdS propagators
journal, September 2014

  • Costa, Miguel S.; Gonçalves, Vasco; Penedones, João
  • Journal of High Energy Physics, Vol. 2014, Issue 9
  • DOI: 10.1007/JHEP09(2014)064

Unitarity and the holographic S-Matrix
journal, October 2012

  • Fitzpatrick, A. Liam; Kaplan, Jared
  • Journal of High Energy Physics, Vol. 2012, Issue 10
  • DOI: 10.1007/JHEP10(2012)032

AdS field theory from conformal field theory
journal, February 2013

  • Fitzpatrick, A. Liam; Kaplan, Jared
  • Journal of High Energy Physics, Vol. 2013, Issue 2
  • DOI: 10.1007/JHEP02(2013)054

Quantum corrections to holographic entanglement entropy
journal, November 2013

  • Faulkner, Thomas; Lewkowycz, Aitor; Maldacena, Juan
  • Journal of High Energy Physics, Vol. 2013, Issue 11
  • DOI: 10.1007/JHEP11(2013)074

Correlation functions and massive Kaluza–Klein modes in the AdS/CFT correspondence
journal, August 2003


Towards Feynman rules for Mellin amplitudes in AdS/CFT
journal, October 2011


Mellin Amplitudes for Supergravity on AdS 5 × S 5
journal, March 2017


Exploring perturbative conformal field theory in Mellin space
journal, January 2017

  • Nizami, Amin A.; Rudra, Arnab; Sarkar, Sourav
  • Journal of High Energy Physics, Vol. 2017, Issue 1
  • DOI: 10.1007/JHEP01(2017)102

The Amplituhedron
journal, October 2014

  • Arkani-Hamed, Nima; Trnka, Jaroslav
  • Journal of High Energy Physics, Vol. 2014, Issue 10
  • DOI: 10.1007/JHEP10(2014)030

Lessons from crossing symmetry at large N
journal, June 2015

  • Alday, Luis F.; Bissi, Agnese; Lukowski, Tomasz
  • Journal of High Energy Physics, Vol. 2015, Issue 6
  • DOI: 10.1007/JHEP06(2015)074

    Works referencing / citing this record:

    Higher spin ANEC and the space of CFTs
    journal, July 2019


    Bootstrapping the (A1, A2) Argyres-Douglas theory
    journal, March 2018

    • Cornagliotto, Martina; Lemos, Madalena; Liendo, Pedro
    • Journal of High Energy Physics, Vol. 2018, Issue 3
    • DOI: 10.1007/jhep03(2018)033