We reformulate the scattering amplitudes of 4D flat space gauge theory and gravity in the language of a 2D CFT on the celestial sphere. The resulting CFT structure exhibits an OPE constructed from 4D collinear singularities, as well as infinite-dimensional Kac-Moody and Virasoro algebras encoding the asymptotic symmetries of 4D flat space. We derive these results by recasting 4D dynamics in terms of a convenient foliation of flat space into 3D Euclidean AdS and Lorentzian dS geometries. Tree-level scattering amplitudes take the form of Witten diagrams for a continuum of (A)dS modes, which are in turn equivalent to CFT correlators via the (A)dS/CFT dictionary. The Ward identities for the 2D conserved currents are dual to 4D soft theorems, while the bulk-boundary propagators of massless (A)dS modes are superpositions of the leading and subleading Weinberg soft factors of gauge theory and gravity. In general, the massless (A)dS modes are 3D Chern-Simons gauge fields describing the soft, single helicity sectors of 4D gauge theory and gravity. Consistent with the topological nature of Chern-Simons theory, Aharonov-Bohm effects record the “tracks” of hard particles in the soft radiation, leading to a simple characterization of gauge and gravitational memories. Soft particle exchanges between hard processes define the Kac-Moody level and Virasoro central charge, which are thereby related to the 4D gauge coupling and gravitational strength in units of an infrared cutoff. Finally, we discuss a toy model for black hole horizons via a restriction to the Rindler region.
P.D. Francesco, P. Mathieu and D. Sénéchal, Conformal field theory, Grad. Texts Contemp. Phys., Springer, New York U.S.A. (1997) [INSPIRE].
H. Bondi, M.G.J. van der Burg and A.W.K. Metzner, Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems, Proc. Roy. Soc. Lond. A 269 (1962) 21 [INSPIRE].
R.K. Sachs, Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times, Proc. Roy. Soc. Lond. A 270 (1962) 103 [INSPIRE].
S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516 [INSPIRE].
F.E. Low, Bremsstrahlung of very low-energy quanta in elementary particle collisions, Phys. Rev. 110 (1958) 974 [INSPIRE].
T.H. Burnett and N.M. Kroll, Extension of the low soft photon theorem, Phys. Rev. Lett. 20 (1968) 86 [INSPIRE].
Y. Zeldovich and A. Polnarev, Radiation of gravitational waves by a cluster of superdense stars, Sov. Astron. 18 (1974) 17.
V.B. Braginsky and K.S. Thorne, Gravitational-wave bursts with memory and experimental prospects, Nature 327 (1987) 123.
D. Christodoulou, Nonlinear nature of gravitation and gravitational wave experiments, Phys. Rev. Lett. 67 (1991) 1486 [INSPIRE].
Y. Aharonov and D. Bohm, Significance of electromagnetic potentials in the quantum theory, Phys. Rev. 115 (1959) 485 [INSPIRE].
G.W. Moore and N. Read, Nonabelions in the fractional quantum Hall effect, Nucl. Phys. B 360 (1991) 362 [INSPIRE].
X.G. Wen, Non-Abelian statistics in the fractional quantum Hall states, Phys. Rev. Lett. 66 (1991) 802 [INSPIRE].
P.A.M. Dirac, Wave equations in conformal space, Annals Math. 37 (1936) 429 [INSPIRE].
V.P. Nair, A current algebra for some gauge theory amplitudes, Phys. Lett. B 214 (1988) 215 [INSPIRE].
E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
S. Elitzur, G.W. Moore, A. Schwimmer and N. Seiberg, Remarks on the canonical quantization of the Chern-Simons-Witten theory, Nucl. Phys. B 326 (1989) 108 [INSPIRE].
E. Witten, On holomorphic factorization of WZW and coset models, Commun. Math. Phys. 144 (1992) 189 [INSPIRE].
E. Witten, (2+ 1)-dimensional gravity as an exactly soluble system, Nucl. Phys. B 311 (1988) 46 [INSPIRE].
N.D. Birrell and P.C.W. Davies, Quantum fields in curved space, Camb. Monogr. Math. Phys., Cambridge Univ. Press, Cambridge U.K. (1984) [INSPIRE].
P. Breitenlohner and D.Z. Freedman, Positive energy in anti-de Sitter backgrounds and gauged extended supergravity, Phys. Lett. B 115 (1982) 197 [INSPIRE].
P. Breitenlohner and D.Z. Freedman, Stability in gauged extended supergravity, Annals Phys. 144 (1982) 249 [INSPIRE].
P.K. Townsend, K. Pilch and P. van Nieuwenhuizen, Selfduality in odd dimensions, Phys. Lett. B 136 (1984) 38 [Addendum ibid. B 137 (1984) 443] [INSPIRE].
S. Axelrod and I.M. Singer, Chern-Simons perturbation theory, in Differential geometric methods in theoretical physics. Proceedings, 20th International Conference, New York U.S.A. June 3–7 1991, pg. 3 [hep-th/9110056] [INSPIRE].
H. Leutwyler, A (2 + 1)-dimensional model for the quantum theory of gravity, Nuovo Cim. A 42 (1966) 159.
Staruszkiewicz, Gravitation theory in three-dimensional space, Acta Phys. Polon. 24 (1963) 735 [INSPIRE].
H. Kawai, D.C. Lewellen and S.-H. Henry Tye, A relation between tree amplitudes of closed and open strings, Nucl. Phys. B 269 (1986) 1 [INSPIRE].
Stewart and C. Bauer, The soft-collinear effective theory, https://courses.edx.org/c4x/MITx/8.EFTx/asset/notes_scetnotes.pdf.
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
ArXiv ePrint: 1609.00732
About this article
Cite this article
Cheung, C., de la Fuente, A. & Sundrum, R. 4D scattering amplitudes and asymptotic symmetries from 2D CFT. J. High Energ. Phys. 2017, 112 (2017). https://doi.org/10.1007/JHEP01(2017)112
- AdS-CFT Correspondence
- Conformal Field Theory
- Scattering Amplitudes