Abstract
We consider constraints on the S-matrix of any gapped, Lorentz invariant quantum field theory in 3+1 dimensions due to crossing symmetry, analyticity and unitarity. We extremize cubic couplings, quartic couplings and scattering lengths relevant for the elastic scattering amplitude of two identical scalar particles. In the cases where our results can be compared with the older S-matrix literature they are in excellent agreement. We also extremize a cubic coupling in 2+1 dimensions which we can directly compare to a universal bound for a QFT in AdS. This paper generalizes our previous 1+1 dimensional results of [1] and [2].
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M.F. Paulos, J. Penedones, J. Toledo, B.C. van Rees and P. Vieira, The S-matrix bootstrap. Part I: QFT in AdS, JHEP11 (2017) 133 [arXiv:1607.06109] [INSPIRE].
M.F. Paulos, J. Penedones, J. Toledo, B.C. van Rees and P. Vieira, The S-matrix bootstrap II: two dimensional amplitudes, JHEP11 (2017) 143 [arXiv:1607.06110] [INSPIRE].
M. Creutz, Rigorous bounds on coupling constants in two-dimensional field theories, Phys. Rev.D 6 (1972) 2763 [INSPIRE].
M. Hogervorst and S. Rychkov, Radial coordinates for conformal blocks, Phys. Rev.D 87 (2013) 106004 [arXiv:1303.1111] [INSPIRE].
X.O. Camanho, J.D. Edelstein, J. Maldacena and A. Zhiboedov, Causality constraints on corrections to the graviton three-point coupling, JHEP02 (2016) 020 [arXiv:1407.5597] [INSPIRE].
S. Weinberg, The quantum theory of fields. Volume 1: foundations, Cambridge University Press, Cambridge, U.K. (2005) [INSPIRE].
J. Bros, H. Epstein and V. Glaser, A proof of the crossing property for two-particle amplitudes in general quantum field theory, Commun. Math. Phys.1 (1965) 240 [INSPIRE].
A. Martin, Rigorous results from theory and unitarity, CERN-TH-1181, CERN, Geneva, Switzerland (1970).
J.D. Bjorken and S.D. Drell, Relativistic quantum fields, (1965) [INSPIRE].
A. Martin, Scattering theory: unitarity, analyticity and crossing, Lect. Notes Phys.3 (1969) 1 [INSPIRE].
F.J. Yndurain, Rigorous constraints, bounds and relations for scattering amplitudes, Rev. Mod. Phys.44 (1972) 645 [INSPIRE].
S. Caron-Huot, Z. Komargodski, A. Sever and A. Zhiboedov, Strings from massive higher spins: the asymptotic uniqueness of the Veneziano amplitude, JHEP10 (2017) 026 [arXiv:1607.04253] [INSPIRE].
A. Sever and A. Zhiboedov, On fine structure of strings: the universal correction to the Veneziano amplitude, JHEP06 (2018) 054 [arXiv:1707.05270] [INSPIRE].
F. Forstnerivc, Stein manifolds and holomorphic mappings: the homotopy principle in complex analysis, Springer, (2011).
V.F. Müller, An upper bound on the coupling constant of scalar mesons, Nuovo Cim.A 42 (1966) 185.
G. Auberson, L. Epele and F.R.A. Simao, Almost optimality of an axiomatic bound for π 0π 0scattering, Nucl. Phys.B 133 (1978) 266 [INSPIRE].
C. Lopez and G. Mennessier, Bounds on the π 0π 0amplitude, Nucl. Phys.B 118 (1977) 426 [INSPIRE].
G. Auberson and G. Mennessier, On the optimality of axiomatic constraints for π 0π 0scattering: the case of the lower bounds, Nucl. Phys.B 162 (1980) 440 [INSPIRE].
S.R. Coleman, More about the massive Schwinger model, Annals Phys.101 (1976) 239 [INSPIRE].
I. Caprini and P. Dita, A new method for deriving rigorous results on ππ scattering, J. Phys.A 13 (1980) 1265 [INSPIRE].
C. Lopez and G. Mennessier, A new absolute bound on the π 0π 0S-wave scattering length, Phys. Lett.B 58 (1975) 437 [INSPIRE].
L. Lukaszuk and A. Martin, Absolute upper bounds for ππ scattering, Nuovo Cim.A 52 (1967) 122 [INSPIRE].
B. Bonnier and R.V. Mau, Connection between the Wigner inequalities and analyticity and unitarity, Phys. Rev.165 (1968) 1923 [INSPIRE].
B. Bonnier, Derivation and implications of rigorous absolute bounds for ππ partial waves up to 1 GeV, Nucl. Phys.B 95 (1975) 98 [INSPIRE].
P. Dorey, Exact S matrices, in Conformal field theories and integrable models. Proceedings, Eotvos Graduate Course, Budapest, Hungary, 13–18 August 1996, pg. 85 [hep-th/9810026] [INSPIRE].
S.O. Aks, Proof that scattering implies production in quantum field theory, J. Math. Phys.6 (1965) 516.
A.J. Dragt, Amount of four-particle production required in S-matrix theory, Phys. Rev.156 (1967) 1588.
S. Mandelstam, Analytic properties of transition amplitudes in perturbation theory, Phys. Rev.115 (1959) 1741 [INSPIRE].
M.A. Ruderman and S. Gasiorowicz, Limits on coupling constants in field theories with finite sources, Nuovo Cim.8 (1958) 861.
V. Gribov, Y.B. Zel’dovich and A. Perelomov, On the maximal charge for a given mass of the bound state, Sov. Phys. JETP13 (1961) 836 [Zh. Eksp. Teor. Fiz.40 (1961) 1190] [INSPIRE].
D. Simmons-Duffin, A semidefinite program solver for the conformal bootstrap, JHEP06 (2015) 174 [arXiv:1502.02033] [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: 1708.06765
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Paulos, M.F., Penedones, J., Toledo, J. et al. The S-matrix bootstrap. Part III: higher dimensional amplitudes. J. High Energ. Phys. 2019, 40 (2019). https://doi.org/10.1007/JHEP12(2019)040
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2019)040