Abstract
We study the space of all kinematically allowed four photon and four graviton S-matrices, polynomial in scattering momenta. We demonstrate that this space is the permutation invariant sector of a module over the ring of polynomials of the Mandelstam invariants s, t and u. We construct these modules for every value of the spacetime dimension D, and so explicitly count and parameterize the most general four photon and four graviton S-matrix at any given derivative order. We also explicitly list the local Lagrangians that give rise to these S-matrices. We then conjecture that the Regge growth of S-matrices in all physically acceptable classical theories is bounded by s2 at fixed t. A four parameter subset of the polynomial photon S-matrices constructed above satisfies this Regge criterion. For gravitons, on the other hand, no polynomial addition to the Einstein S-matrix obeys this bound for D ≤ 6. For D ≥ 7 there is a single six derivative polynomial Lagrangian consistent with our conjectured Regge growth bound. Our conjecture thus implies that the Einstein four graviton S-matrix does not admit any physically acceptable polynomial modifications for D ≤ 6. A preliminary analysis also suggests that every finite sum of pole exchange contributions to four graviton scattering also violates our conjectured Regge growth bound, at least when D ≤ 6, even when the exchanged particles have low spin.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Minwalla, Two questions about gravity, talk at Strings 2018, https://indico.oist.jp/indico/event/5/picture/141.pdf, (2018).
X.O. Camanho, J.D. Edelstein, J. Maldacena and A. Zhiboedov, Causality constraints on corrections to the graviton three-point coupling, JHEP 02 (2016) 020 [arXiv:1407.5597] [INSPIRE].
M. Kulaxizi, A. Parnachev and A. Zhiboedov, Bulk phase shift, CFT Regge limit and Einstein gravity, JHEP 06 (2018) 121 [arXiv:1705.02934] [INSPIRE].
A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi, Causality, analyticity and an IR obstruction to UV completion, JHEP 10 (2006) 014 [hep-th/0602178] [INSPIRE].
B. Bellazzini, C. Cheung and G.N. Remmen, Quantum gravity constraints from unitarity and analyticity, Phys. Rev. D 93 (2016) 064076 [arXiv:1509.00851] [INSPIRE].
N. Afkhami-Jeddi, T. Hartman, S. Kundu and A. Tajdini, Einstein gravity 3-point functions from conformal field theory, JHEP 12 (2017) 049 [arXiv:1610.09378] [INSPIRE].
N. Afkhami-Jeddi, T. Hartman, S. Kundu and A. Tajdini, Shockwaves from the operator product expansion, JHEP 03 (2019) 201 [arXiv:1709.03597] [INSPIRE].
M. Kologlu, P. Kravchuk, D. Simmons-Duffin and A. Zhiboedov, Shocks, superconvergence and a stringy equivalence principle, arXiv:1904.05905 [INSPIRE].
N. Arkani-Hamed, T.-C. Huang and Y.-T. Huang, Scattering amplitudes for all masses and spins, arXiv:1709.04891 [INSPIRE].
B. Henning, X. Lu, T. Melia and H. Murayama, Hilbert series and operator bases with derivatives in effective field theories, Commun. Math. Phys. 347 (2016) 363 [arXiv:1507.07240] [INSPIRE].
B. Henning, X. Lu, T. Melia and H. Murayama, Operator bases, S-matrices and their partition functions, JHEP 10 (2017) 199 [arXiv:1706.08520] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
S. Caron-Huot, Analyticity in spin in conformal theories, JHEP 09 (2017) 078 [arXiv:1703.00278] [INSPIRE].
D. Simmons-Duffin, D. Stanford and E. Witten, A spacetime derivation of the Lorentzian OPE inversion formula, JHEP 07 (2018) 085 [arXiv:1711.03816] [INSPIRE].
P. Kravchuk and D. Simmons-Duffin, Light-ray operators in conformal field theory, JHEP 11 (2018) 102 [arXiv:1805.00098] [INSPIRE].
I. Halder, Global symmetry and maximal chaos, arXiv:1908.05281 [INSPIRE].
M. Mezei and G. Sárosi, Chaos in the butterfly cone, JHEP 01 (2020) 186 [arXiv:1908.03574] [INSPIRE].
R.R. Poojary, BTZ dynamics and chaos, arXiv:1812.10073 [INSPIRE].
V. Jahnke, K.-Y. Kim and J. Yoon, On the chaos bound in rotating black holes, JHEP 05 (2019) 037 [arXiv:1903.09086] [INSPIRE].
L. Cornalba, Eikonal methods in AdS/CFT: Regge theory and multi-reggeon exchange, arXiv:0710.5480 [INSPIRE].
I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from conformal field theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].
S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132 [arXiv:1412.6087] [INSPIRE].
G.J. Turiaci and A. Zhiboedov, Veneziano amplitude of Vasiliev theory, JHEP 10 (2018) 034 [arXiv:1802.04390] [INSPIRE].
D. Meltzer and E. Perlmutter, Beyond a = c: gravitational couplings to matter and the stress tensor OPE, JHEP 07 (2018) 157 [arXiv:1712.04861] [INSPIRE].
N. Afkhami-Jeddi, S. Kundu and A. Tajdini, A conformal collider for holographic CFTs, JHEP 10 (2018) 156 [arXiv:1805.07393] [INSPIRE].
P. Kravchuk and D. Simmons-Duffin, Counting conformal correlators, JHEP 02 (2018) 096 [arXiv:1612.08987] [INSPIRE].
A. Dymarsky, On the four-point function of the stress-energy tensors in a CFT, JHEP 10 (2015) 075 [arXiv:1311.4546] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal blocks, JHEP 11 (2011) 154 [arXiv:1109.6321] [INSPIRE].
R. de Mello Koch, P. Rabambi, R. Rabe and S. Ramgoolam, Counting and construction of holomorphic primary fields in free CFT4 from rings of functions on Calabi-Yau orbifolds, JHEP 08 (2017) 077 [arXiv:1705.06702] [INSPIRE].
R. de Mello Koch and S. Ramgoolam, Free field primaries in general dimensions: counting and construction with rings and modules, JHEP 08 (2018) 088 [arXiv:1806.01085] [INSPIRE].
J.M. Maldacena and G.L. Pimentel, On graviton non-Gaussianities during inflation, JHEP 09 (2011) 045 [arXiv:1104.2846] [INSPIRE].
S. Giombi, S. Prakash and X. Yin, A note on CFT correlators in three dimensions, JHEP 07 (2013) 105 [arXiv:1104.4317] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
C. Lanczos, A remarkable property of the Riemann-Christoffel tensor in four dimensions, Annals Math. 39 (1938) 842 [INSPIRE].
D. Lovelock, The Einstein tensor and its generalizations, J. Math. Phys. 12 (1971) 498 [INSPIRE].
B. Sundborg, The Hagedorn transition, deconfinement and N = 4 SYM theory, Nucl. Phys. B 573 (2000) 349 [hep-th/9908001] [INSPIRE].
O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas and M. Van Raamsdonk, The Hagedorn/deconfinement phase transition in weakly coupled large N gauge theories, Adv. Theor. Math. Phys. 8 (2004) 603 [hep-th/0310285] [INSPIRE].
J.H. Schwarz, Superstring theory, Phys. Rept. 89 (1982) 223 [INSPIRE].
S. Deser and D. Seminara, Tree amplitudes and two loop counterterms in D = 11 supergravity, Phys. Rev. D 62 (2000) 084010 [hep-th/0002241] [INSPIRE].
S.A. Fulling, R.C. King, B.G. Wybourne and C.J. Cummins, Normal forms for tensor polynomials. 1: the Riemann tensor, Class. Quant. Grav. 9 (1992) 1151 [INSPIRE].
S.D. Chowdhury and J.R. David, Global gravitational anomalies and transport, JHEP 12 (2016) 116 [arXiv:1604.05003] [INSPIRE].
S. Sannan, Gravity as the limit of the type II superstring theory, Phys. Rev. D 34 (1986) 1749 [INSPIRE].
G.J. Turiaci, An inelastic bound on chaos, JHEP 07 (2019) 099 [arXiv:1901.04360] [INSPIRE].
S.B. Giddings, S. Kachru and J. Polchinski, Hierarchies from fluxes in string compactifications, Phys. Rev. D 66 (2002) 106006 [hep-th/0105097] [INSPIRE].
G. Veneziano, S. Yankielowicz and E. Onofri, A model for pion-pion scattering in large-N QCD, JHEP 04 (2017) 151 [arXiv:1701.06315] [INSPIRE].
L.P.S. Singh and C.R. Hagen, Lagrangian formulation for arbitrary spin 1. The boson case, Phys. Rev. D 9 (1974) 898 [INSPIRE].
R.L. Ingraham, Covariant propagators and vertices for higher spin bosons, Prog. Theor. Phys. 51 (1974) 249 [INSPIRE].
P. Nayak, R.R. Poojary and R.M. Soni, A note on S-matrix bootstrap for amplitudes with linear spectrum, arXiv:1707.08135 [INSPIRE].
X. Bekaert, E. Joung and J. Mourad, On higher spin interactions with matter, JHEP 05 (2009) 126 [arXiv:0903.3338] [INSPIRE].
Open Access
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
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1910.14392
Rights and permissions
Open Access . 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.
About this article
Cite this article
Chowdhury, S.D., Gadde, A., Gopalka, T. et al. Classifying and constraining local four photon and four graviton S-matrices. J. High Energ. Phys. 2020, 114 (2020). https://doi.org/10.1007/JHEP02(2020)114
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2020)114