# Graviton scattering and a sum rule for the c anomaly in 4D CFT

## Abstract

4D CFTs have a scale anomaly characterized by the coefficient c, which appears as the coefficient of logarithmic terms in momentum space correlation functions of the energy-momentum tensor. By studying the CFT contribution to 4-point graviton scattering amplitudes in Minkowski space we derive a sum rule for c in terms of TT O OPE coefficients. The sum rule can be thought of as a version of the optical theorem, and its validity depends on the existence of the massless and forward limits of the $$\langle$$TTTT$$\rangle$$ correlation functions that contribute. The finiteness of these limits is checked explicitly for free scalar, fermion, and vector CFTs. The sum rule gives c as a sum of positive terms, and therefore implies a lower bound on c given any lower bound on TT O OPE coefficients. We compute the coefficients to the sum rule for arbitrary operators of spin 0 and 2, including the energy-momentum tensor.

- Authors:

- Ecole Polytechnique Federale Lausanne (Switzlerland). Theoretical Particle Physics Lab.
- Univ. of California, Davis, CA (United States). Center for Quantum Mathematics and Physics (QMAP)

- Publication Date:

- Research Org.:
- Univ. of California, Davis, CA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1505197

- Grant/Contract Number:
- [FG02-91ER40674]

- 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: 2018; Journal Issue: 9]; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Anomalies in Field and String Theories; Conformal Field Theory

### Citation Formats

```
Gillioz, Marc, Lu, Xiaochuan, and Luty, Markus A. Graviton scattering and a sum rule for the c anomaly in 4D CFT. United States: N. p., 2018.
Web. doi:10.1007/jhep09(2018)025.
```

```
Gillioz, Marc, Lu, Xiaochuan, & Luty, Markus A. Graviton scattering and a sum rule for the c anomaly in 4D CFT. United States. doi:10.1007/jhep09(2018)025.
```

```
Gillioz, Marc, Lu, Xiaochuan, and Luty, Markus A. Wed .
"Graviton scattering and a sum rule for the c anomaly in 4D CFT". United States. doi:10.1007/jhep09(2018)025. https://www.osti.gov/servlets/purl/1505197.
```

```
@article{osti_1505197,
```

title = {Graviton scattering and a sum rule for the c anomaly in 4D CFT},

author = {Gillioz, Marc and Lu, Xiaochuan and Luty, Markus A.},

abstractNote = {4D CFTs have a scale anomaly characterized by the coefficient c, which appears as the coefficient of logarithmic terms in momentum space correlation functions of the energy-momentum tensor. By studying the CFT contribution to 4-point graviton scattering amplitudes in Minkowski space we derive a sum rule for c in terms of TT O OPE coefficients. The sum rule can be thought of as a version of the optical theorem, and its validity depends on the existence of the massless and forward limits of the $\langle$TTTT$\rangle$ correlation functions that contribute. The finiteness of these limits is checked explicitly for free scalar, fermion, and vector CFTs. The sum rule gives c as a sum of positive terms, and therefore implies a lower bound on c given any lower bound on TT O OPE coefficients. We compute the coefficients to the sum rule for arbitrary operators of spin 0 and 2, including the energy-momentum tensor.},

doi = {10.1007/jhep09(2018)025},

journal = {Journal of High Energy Physics (Online)},

number = [9],

volume = [2018],

place = {United States},

year = {2018},

month = {9}

}

*Citation information provided by*

Web of Science

Web of Science

Works referenced in this record:

##
Anomalies, conformal manifolds, and spheres

journal, March 2016

- Gomis, Jaume; Hsin, Po-Shen; Komargodski, Zohar
- Journal of High Energy Physics, Vol. 2016, Issue 3

##
On scale and conformal invariance in four dimensions

journal, October 2015

- Dymarsky, Anatoly; Komargodski, Zohar; Schwimmer, Adam
- Journal of High Energy Physics, Vol. 2015, Issue 10

##
On graviton non-gaussianities during inflation

journal, September 2011

- Maldacena, Juan M.; Pimentel, Guilherme L.
- Journal of High Energy Physics, Vol. 2011, Issue 9

##
Observations on conformal anomalies

journal, July 1977

- Duff, M. J.
- Nuclear Physics B, Vol. 125, Issue 2

##
Averaged null energy condition from causality

journal, July 2017

- Hartman, Thomas; Kundu, Sandipan; Tajdini, Amirhossein
- Journal of High Energy Physics, Vol. 2017, Issue 7

##
Another odd thing about unparticle physics

journal, July 2007

- Georgi, Howard
- Physics Letters B, Vol. 650, Issue 4

##
A conformal truncation framework for infinite-volume dynamics

journal, July 2016

- Katz, Emanuel; Khandker, Zuhair U.; Walters, Matthew T.
- Journal of High Energy Physics, Vol. 2016, Issue 7

##
The a-theorem and the asymptotics of 4D quantum field theory

journal, January 2013

- Luty, Markus A.; Polchinski, Joseph; Rattazzi, Riccardo
- Journal of High Energy Physics, Vol. 2013, Issue 1

##
Spinning conformal correlators

journal, November 2011

- Costa, Miguel S.; Penedones, João; Poland, David
- Journal of High Energy Physics, Vol. 2011, Issue 11

##
Beyond a = c : gravitational couplings to matter and the stress tensor OPE

journal, July 2018

- Meltzer, David; Perlmutter, Eric
- Journal of High Energy Physics, Vol. 2018, Issue 7

##
Remarks on the convergence properties of the conformal block expansion

journal, February 2016

- Rychkov, Slava; Yvernay, Pierre
- Physics Letters B, Vol. 753

##
Modular Hamiltonians for deformed half-spaces and the averaged null energy condition

journal, September 2016

- Faulkner, Thomas; Leigh, Robert G.; Parrikar, Onkar
- Journal of High Energy Physics, Vol. 2016, Issue 9

##
Implications of Conformal Invariance in Field Theories for General Dimensions

journal, May 1994

- Osborn, H.; Petkou, A.
- Annals of Physics, Vol. 231, Issue 2

##
Scale anomalies, states, and rates in conformal field theory

journal, April 2017

- Gillioz, Marc; Lu, Xiaochuan; Luty, Markus A.
- Journal of High Energy Physics, Vol. 2017, Issue 4

##
On renormalization group flows in four dimensions

journal, December 2011

- Komargodski, Zohar; Schwimmer, Adam
- Journal of High Energy Physics, Vol. 2011, Issue 12

##
The 3d stress-tensor bootstrap

journal, February 2018

- Dymarsky, Anatoly; Kos, Filip; Kravchuk, Petr
- Journal of High Energy Physics, Vol. 2018, Issue 2

##
Can we change c in four-dimensional CFTs by exactly marginal deformations?

journal, July 2017

- Nakayama, Yu
- Journal of High Energy Physics, Vol. 2017, Issue 7

##
Markov Property of the Conformal Field Theory Vacuum and the $a$ Theorem

journal, June 2017

- Casini, Horacio; Testé, Eduardo; Torroba, Gonzalo
- Physical Review Letters, Vol. 118, Issue 26

##
Projectors, shadows, and conformal blocks

journal, April 2014

- Simmons-Duffin, David
- Journal of High Energy Physics, Vol. 2014, Issue 4

##
On the four-point function of the stress-energy tensors in a CFT

journal, October 2015

- Dymarsky, Anatoly
- Journal of High Energy Physics, Vol. 2015, Issue 10

##
Bounds on OPE coefficients from interference effects in the conformal collider

journal, November 2017

- Córdova, Clay; Maldacena, Juan; Turiaci, Gustavo J.
- Journal of High Energy Physics, Vol. 2017, Issue 11

##
Implications of conformal invariance in momentum space

journal, March 2014

- Bzowski, Adam; McFadden, Paul; Skenderis, Kostas
- Journal of High Energy Physics, Vol. 2014, Issue 3