# 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:

- 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:

- 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}

}

*Citation information provided by*

Web of Science

Web of Science

Works referenced in this record:

##
Comments on 4-point functions in the CFT/AdS correspondence

journal, April 1999

- Freedman, Daniel Z.; Mathur, Samir D.; Matusis, Alec
- Physics Letters B, Vol. 452, Issue 1-2

##
Comments on operators with large spin

journal, November 2007

- Alday, Luis F.; Maldacena, Juan
- Journal of High Energy Physics, Vol. 2007, Issue 11

##
Conformal Bootstrap in Mellin Space

journal, February 2017

- Gopakumar, Rajesh; Kaviraj, Apratim; Sen, Kallol
- Physical Review Letters, Vol. 118, Issue 8

##
Causality constraints in conformal field theory

journal, May 2016

- Hartman, Thomas; Jain, Sachin; Kundu, Sandipan
- Journal of High Energy Physics, Vol. 2016, Issue 5

##
Order 1/N2 test of the Maldacena conjecture: cancellation of the one-loop Weyl anomaly

journal, December 2000

- Mansfield, Paul; Nolland, David
- Physics Letters B, Vol. 495, Issue 3-4

##
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

##
General scalar exchange in AdSd+1

journal, June 1999

- D'Hoker, Eric; Freedman, Daniel Z.
- Nuclear Physics B, Vol. 550, Issue 1-2

##
Conformal Regge theory

journal, December 2012

- Costa, Miguel S.; Goncalves, Vasco; Penedones, João
- Journal of High Energy Physics, Vol. 2012, Issue 12

##
Gauge theory correlators from non-critical string theory

journal, May 1998

- Gubser, S. S.; Klebanov, I. R.; Polyakov, A. M.
- Physics Letters B, Vol. 428, Issue 1-2

##
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

##
An algebraic approach to the analytic bootstrap

journal, April 2017

- Alday, Luis F.; Zhiboedov, Alexander
- Journal of High Energy Physics, Vol. 2017, Issue 4

##
Mellin transforming the minimal model CFTs: AdS/CFT at strong curvature

journal, September 2016

- Lowe, David A.
- Physics Letters B, Vol. 760

##
Bounding the space of holographic CFTs with chaos

journal, October 2016

- Perlmutter, Eric
- Journal of High Energy Physics, Vol. 2016, Issue 10

##
Anti de Sitter space and holography

journal, January 1998

- Witten, Edward
- Advances in Theoretical and Mathematical Physics, Vol. 2, Issue 2

##
Factorization of Mellin amplitudes

journal, October 2015

- Gonçalves, Vasco; Penedones, João; Trevisani, Emilio
- Journal of High Energy Physics, Vol. 2015, Issue 10

##
A new spin on causality constraints

journal, October 2016

- Hartman, Thomas; Jain, Sachin; Kundu, Sandipan
- Journal of High Energy Physics, Vol. 2016, Issue 10

##
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

##
Analyticity and the holographic S-matrix

journal, October 2012

- Fitzpatrick, A. Liam; Kaplan, Jared
- Journal of High Energy Physics, Vol. 2012, Issue 10

##
5 loops in 24/5 dimensions

journal, August 2010

- Björnsson, Jonas; Green, Michael B.
- Journal of High Energy Physics, Vol. 2010, Issue 8

##
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

##
One-loop partition functions of 3D gravity

journal, August 2008

- Giombi, Simone; Maloney, Alexander; Yin, Xi
- Journal of High Energy Physics, Vol. 2008, Issue 08

##
More holography from conformal field theory

journal, September 2010

- Heemskerk, Idse; Sully, James
- Journal of High Energy Physics, Vol. 2010, Issue 9

##
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

##
Holography from conformal field theory

journal, October 2009

- Heemskerk, Idse; Penedones, Joao; Polchinski, Joseph
- Journal of High Energy Physics, Vol. 2009, Issue 10

##
AdS/CFT four-point functions: How to succeed at z-integrals without really trying

journal, November 1999

- D'Hoker, Eric; Freedman, Daniel Z.; Rastelli, Leonardo
- Nuclear Physics B, Vol. 562, Issue 1-2

##
Fusing gauge theory tree amplitudes into loop amplitudes

journal, February 1995

- Bern, Zvi; Dixon, Lance; Dunbar, David C.
- Nuclear Physics B, Vol. 435, Issue 1-2

##
terms in 11 dimensions and conformal anomaly of () theory

journal, September 2000

- Tseytlin, A. A.
- Nuclear Physics B, Vol. 584, Issue 1-2

##
Writing CFT correlation functions as AdS scattering amplitudes

journal, March 2011

- Penedones, Joao
- Journal of High Energy Physics, Vol. 2011, Issue 3

##
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

##
A bound on chaos

journal, August 2016

- Maldacena, Juan; Shenker, Stephen H.; Stanford, Douglas
- Journal of High Energy Physics, Vol. 2016, Issue 8

##
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

##
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

##
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

##
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

##
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

##
Harmonic Polylogarithms

journal, February 2000

- Remiddi, E.; Vermaseren, J. A. M.
- International Journal of Modern Physics A, Vol. 15, Issue 05

##
The operator product expansion of SYM and the 4-point functions of supergravity

journal, November 2000

- D'Hoker, Eric; Mathur, Samir D.; Matusis, Alec
- Nuclear Physics B, Vol. 589, Issue 1-2

##
Convexity and liberation at large spin

journal, November 2013

- Komargodski, Zohar; Zhiboedov, Alexander
- Journal of High Energy Physics, Vol. 2013, Issue 11

##
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

##
Graviton exchange and complete four-point functions in the AdS/CFT correspondence

journal, November 1999

- D'Hoker, Eric; Freedman, Daniel Z.; Mathur, Samir D.
- Nuclear Physics B, Vol. 562, Issue 1-2

##
One-loop n-point gauge theory amplitudes, unitarity and collinear limits

journal, August 1994

- Bern, Zvi; Dixon, Lance; Dunbar, David C.
- Nuclear Physics B, Vol. 425, Issue 1-2

##
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

##
Looking for a bulk point

journal, January 2017

- Maldacena, Juan; Simmons-Duffin, David; Zhiboedov, Alexander
- Journal of High Energy Physics, Vol. 2017, Issue 1

##
Holographic Derivation of Entanglement Entropy from the anti–de Sitter Space/Conformal Field Theory Correspondence

journal, May 2006

- Ryu, Shinsei; Takayanagi, Tadashi
- Physical Review Letters, Vol. 96, Issue 18

##
Eikonal approximation in AdS/CFT: resumming the gravitational loop expansion

journal, September 2007

- Cornalba, Lorenzo; Costa, Miguel S.; Penedones, João
- Journal of High Energy Physics, Vol. 2007, Issue 09

##
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

##
Emergent spacetime and holographic CFTs

journal, October 2012

- El-Showk, Sheer; Papadodimas, Kyriakos
- Journal of High Energy Physics, Vol. 2012, Issue 10

##
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

##
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

##
Spinning AdS propagators

journal, September 2014

- Costa, Miguel S.; Gonçalves, Vasco; Penedones, João
- Journal of High Energy Physics, Vol. 2014, Issue 9

##
Unitarity and the holographic S-Matrix

journal, October 2012

- Fitzpatrick, A. Liam; Kaplan, Jared
- Journal of High Energy Physics, Vol. 2012, Issue 10

##
AdS field theory from conformal field theory

journal, February 2013

- Fitzpatrick, A. Liam; Kaplan, Jared
- Journal of High Energy Physics, Vol. 2013, Issue 2

##
Quantum corrections to holographic entanglement entropy

journal, November 2013

- Faulkner, Thomas; Lewkowycz, Aitor; Maldacena, Juan
- Journal of High Energy Physics, Vol. 2013, Issue 11

##
Correlation functions and massive Kaluza–Klein modes in the AdS/CFT correspondence

journal, August 2003

- Arutyunov, G.; Dolan, F. A.; Osborn, H.
- Nuclear Physics B, Vol. 665

##
Towards Feynman rules for Mellin amplitudes in AdS/CFT

journal, October 2011

- Paulos, Miguel F.
- Journal of High Energy Physics, Vol. 2011, Issue 10

##
Mellin Amplitudes for Supergravity on ${\mathrm{AdS}}_{5}\times {\mathrm{S}}^{5}$

journal, March 2017

- Rastelli, Leonardo; Zhou, Xinan
- Physical Review Letters, Vol. 118, Issue 9

##
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

##
The Amplituhedron

journal, October 2014

- Arkani-Hamed, Nima; Trnka, Jaroslav
- Journal of High Energy Physics, Vol. 2014, Issue 10

##
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

Works referencing / citing this record:

##
Higher spin ANEC and the space of CFTs

journal, July 2019

- Meltzer, David
- Journal of High Energy Physics, Vol. 2019, Issue 7

##
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