Abstract
Sum rules in effective field theories, predicated upon causality, place restrictions on scattering amplitudes mediated by effective contact interactions. Through unitarity of the S-matrix, these imply that the size of higher dimensional corrections to transition amplitudes between different states is bounded by the strength of their contributions to elastic forward scattering processes. This places fundamental limits on the extent to which hypothetical symmetries can be broken by effective interactions. All analysis is for dimension 8 operators in the forward limit. Included is a thorough derivation of all positivity bounds for a chiral fermion in SU(2) and SU(3) global symmetry representations resembling those of the Standard Model, general bounds on flavour violation, new bounds for interactions between particles of different spin, inclusion of loops of dimension 6 operators and illustration of the resulting strengthening of positivity bounds over tree-level expectations, a catalogue of supersymmetric effective interactions up to mass dimension 8 and 4 legs and the demonstration that supersymmetry unifies the positivity theorems as well as the new bounds.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C.P. Burgess, Introduction to Effective Field Theory, Ann. Rev. Nucl. Part. Sci. 57 (2007) 329 [hep-th/0701053] [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].
G. Dvali, A. Franca and C. Gomez, Road Signs for UV-Completion, arXiv:1204.6388 [INSPIRE].
P. Cooper, S. Dubovsky and A. Mohsen, Ultraviolet complete Lorentz-invariant theory with superluminal signal propagation, Phys. Rev. D 89 (2014) 084044 [arXiv:1312.2021] [INSPIRE].
S.B. Giddings and R.A. Porto, The Gravitational S-matrix, Phys. Rev. D 81 (2010) 025002 [arXiv:0908.0004] [INSPIRE].
L. Keltner and A.J. Tolley, UV properties of Galileons: Spectral Densities, arXiv:1502.05706 [INSPIRE].
J. Tokuda, Extension of positivity bounds to non-local theories: IR obstructions to Lorentz invariant UV completions, JHEP 05 (2019) 216 [arXiv:1902.10039] [INSPIRE].
T. Hartman, S. Jain and S. Kundu, Causality Constraints in Conformal Field Theory, JHEP 05 (2016) 099 [arXiv:1509.00014] [INSPIRE].
Z. Komargodski, M. Kulaxizi, A. Parnachev and A. Zhiboedov, Conformal Field Theories and Deep Inelastic Scattering, Phys. Rev. D 95 (2017) 065011 [arXiv:1601.05453] [INSPIRE].
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].
S. Caron-Huot, Analyticity in Spin in Conformal Theories, JHEP 09 (2017) 078 [arXiv:1703.00278] [INSPIRE].
N. Afkhami-Jeddi, S. Kundu and A. Tajdini, A Bound on Massive Higher Spin Particles, JHEP 04 (2019) 056 [arXiv:1811.01952] [INSPIRE].
M. Kologlu, P. Kravchuk, D. Simmons-Duffin and A. Zhiboedov, Shocks, Superconvergence, and a Stringy Equivalence Principle, JHEP 11 (2020) 096 [arXiv:1904.05905] [INSPIRE].
S. Caron-Huot, D. Mazac, L. Rastelli and D. Simmons-Duffin, Dispersive CFT Sum Rules, JHEP 05 (2021) 243 [arXiv:2008.04931] [INSPIRE].
B. Bellazzini, L. Martucci and R. Torre, Symmetries, Sum Rules and Constraints on Effective Field Theories, JHEP 09 (2014) 100 [arXiv:1405.2960] [INSPIRE].
B. Bellazzini, Softness and amplitudes’ positivity for spinning particles, JHEP 02 (2017) 034 [arXiv:1605.06111] [INSPIRE].
C. de Rham, S. Melville, A.J. Tolley and S.-Y. Zhou, UV complete me: Positivity Bounds for Particles with Spin, JHEP 03 (2018) 011 [arXiv:1706.02712] [INSPIRE].
C. de Rham, S. Melville, A.J. Tolley and S.-Y. Zhou, Positivity Bounds for Massive Spin-1 and Spin-2 Fields, JHEP 03 (2019) 182 [arXiv:1804.10624] [INSPIRE].
Z. Komargodski and A. Schwimmer, On Renormalization Group Flows in Four Dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [INSPIRE].
B. Bellazzini, J. Elias Miró, R. Rattazzi, M. Riembau and F. Riva, Positive Moments for Scattering Amplitudes, arXiv:2011.00037 [INSPIRE].
S. Caron-Huot and V. Van Duong, Extremal Effective Field Theories, JHEP 05 (2021) 280 [arXiv:2011.02957] [INSPIRE].
A.J. Tolley, Z.-Y. Wang and S.-Y. Zhou, New positivity bounds from full crossing symmetry, JHEP 05 (2021) 255 [arXiv:2011.02400] [INSPIRE].
I. Brivio and M. Trott, The Standard Model as an Effective Field Theory, Phys. Rept. 793 (2019) 1 [arXiv:1706.08945] [INSPIRE].
G.N. Remmen and N.L. Rodd, Consistency of the Standard Model Effective Field Theory, JHEP 12 (2019) 032 [arXiv:1908.09845] [INSPIRE].
G.N. Remmen and N.L. Rodd, Flavor Constraints from Unitarity and Analyticity, Phys. Rev. Lett. 125 (2020) 081601 [arXiv:2004.02885] [INSPIRE].
Q. Bi, C. Zhang and S.-Y. Zhou, Positivity constraints on aQGC: carving out the physical parameter space, JHEP 06 (2019) 137 [arXiv:1902.08977] [INSPIRE].
C. Zhang and S.-Y. Zhou, Positivity bounds on vector boson scattering at the LHC, Phys. Rev. D 100 (2019) 095003 [arXiv:1808.00010] [INSPIRE].
K. Yamashita, C. Zhang and S.-Y. Zhou, Elastic positivity vs extremal positivity bounds in SMEFT: a case study in transversal electroweak gauge-boson scatterings, JHEP 01 (2021) 095 [arXiv:2009.04490] [INSPIRE].
C. Zhang and S.-Y. Zhou, Convex Geometry Perspective on the (Standard Model) Effective Field Theory Space, Phys. Rev. Lett. 125 (2020) 201601 [arXiv:2005.03047] [INSPIRE].
B. Bellazzini and F. Riva, New phenomenological and theoretical perspective on anomalous ZZ and Zγ processes, Phys. Rev. D 98 (2018) 095021 [arXiv:1806.09640] [INSPIRE].
Y.-t. Huang, J.-Y. Liu, L. Rodina and Y. Wang, Carving out the Space of Open-String S-matrix, JHEP 04 (2021) 195 [arXiv:2008.02293] [INSPIRE].
J.-Y. Liu and Z.-M. You, The supersymmetric spinning polynomial, arXiv:2011.11299 [INSPIRE].
Q. Bonnefoy, E. Gendy and C. Grojean, Positivity bounds on Minimal Flavor Violation, JHEP 04 (2021) 115 [arXiv:2011.12855] [INSPIRE].
X. Li, C. Yang, H. Xu, C. Zhang and S.-Y. Zhou, Positivity in Multi-Field EFTs, arXiv:2101.01191 [INSPIRE].
J. Bros, H. Epstein and V.J. Glaser, Some rigorous analyticity properties of the four-point function in momentum space, Nuovo Cim. 31 (1964) 1265 [INSPIRE].
M. Gell-Mann, M.L. Goldberger and W.E. Thirring, Use of causality conditions in quantum theory, Phys. Rev. 95 (1954) 1612 [INSPIRE].
A. Martin, Extension of the axiomatic analyticity domain of scattering amplitudes by unitarity. 1., Nuovo Cim. A 42 (1965) 930 [INSPIRE].
D. Simmons-Duffin, CFT in Lorentzian Signature, https://physicslearning.colorado.edu/tasi/tasi_2019/tasi_2019.html (2019).
M. Froissart, Asymptotic behavior and subtractions in the Mandelstam representation, Phys. Rev. 123 (1961) 1053 [INSPIRE].
D. Olive, Unitarity and evaluation of discontinuities, Nuovo Cim. A 26 (1962) 3905.
C. Cheung and G.N. Remmen, Infrared Consistency and the Weak Gravity Conjecture, JHEP 12 (2014) 087 [arXiv:1407.7865] [INSPIRE].
J. Tokuda, K. Aoki and S. Hirano, Gravitational positivity bounds, JHEP 11 (2020) 054 [arXiv:2007.15009] [INSPIRE].
L. Alberte, C. de Rham, S. Jaitly and A.J. Tolley, Positivity Bounds and the Massless Spin-2 Pole, Phys. Rev. D 102 (2020) 125023 [arXiv:2007.12667] [INSPIRE].
B.M. Gavela, E.E. Jenkins, A.V. Manohar and L. Merlo, Analysis of General Power Counting Rules in Effective Field Theory, Eur. Phys. J. C 76 (2016) 485 [arXiv:1601.07551] [INSPIRE].
A.V. Manohar and V. Mateu, Dispersion Relation Bounds for ππ Scattering, Phys. Rev. D 77 (2008) 094019 [arXiv:0801.3222] [INSPIRE].
V. Mateu, Universal Bounds for SU(3) Low Energy Constants, Phys. Rev. D 77 (2008) 094020 [arXiv:0801.3627] [INSPIRE].
Y.-J. Wang, F.-K. Guo, C. Zhang and S.-Y. Zhou, Generalized positivity bounds on chiral perturbation theory, JHEP 07 (2020) 214 [arXiv:2004.03992] [INSPIRE].
S. Andriolo, T.-C. Huang, T. Noumi, H. Ooguri and G. Shiu, Duality and axionic weak gravity, Phys. Rev. D 102 (2020) 046008 [arXiv:2004.13721] [INSPIRE].
K. Fukuda, Lecture: Polyhedral computation, http://www-oldurls.inf.ethz.ch/personal/fukudak/lect/pclect/notes2016/PolyComp2016.pdf (2016).
D. Avis, lrs: A Revised Implementation of the Reverse Search Vertex Enumeration Algorithm, in Polytopes — Combinatorics and Computation, G. Kalai and G.M. Ziegler eds., DMV Seminar 29 (2000) 177.
H.K. Dreiner, H.E. Haber and S.P. Martin, Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry, Phys. Rept. 494 (2010) 1 [arXiv:0812.1594] [INSPIRE].
S. Weinberg, Feynman Rules for Any Spin. 2. Massless Particles, Phys. Rev. 134 (1964) B882 [INSPIRE].
M.K. Gaillard and B. Zumino, Duality Rotations for Interacting Fields, Nucl. Phys. B 193 (1981) 221 [INSPIRE].
H. Elvang and Y.-t. Huang, Scattering Amplitudes, arXiv:1308.1697 [INSPIRE].
H. Elvang, D.Z. Freedman and M. Kiermaier, A simple approach to counterterms in N = 8 supergravity, JHEP 11 (2010) 016 [arXiv:1003.5018] [INSPIRE].
S. Lal and S. Raju, The Next-to-Simplest Quantum Field Theories, Phys. Rev. D 81 (2010) 105002 [arXiv:0910.0930] [INSPIRE].
H. Elvang, Y.-t. Huang and C. Peng, On-shell superamplitudes in N < 4 SYM, JHEP 09 (2011) 031 [arXiv:1102.4843] [INSPIRE].
M. Srednicki, Quantum field theory, Cambridge University Press (2007) [DOI].
M. Dine, G. Festuccia and Z. Komargodski, A Bound on the Superpotential, JHEP 03 (2010) 011 [arXiv:0910.2527] [INSPIRE].
M. Ramana and A.J. Goldman, Some geometric results in semidefinite programming, J. Glob. Optim. 7 (1995) 33.
D. Simmons-Duffin, The Conformal Bootstrap, in Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings, (2016) [DOI] [arXiv:1602.07982] [INSPIRE].
A. Hebbar, D. Karateev and J. Penedones, Spinning S-matrix Bootstrap in 4d, arXiv:2011.11708 [INSPIRE].
M.B. Green and C. Wen, Superstring amplitudes, unitarily, and Hankel determinants of multiple zeta values, JHEP 11 (2019) 079 [arXiv:1908.08426] [INSPIRE].
G.N. Remmen and N.L. Rodd, Signs, Spin, SMEFT: Positivity at Dimension Six, arXiv:2010.04723 [INSPIRE].
J. Gu and L.-T. Wang, Sum Rules in the Standard Model Effective Field Theory from Helicity Amplitudes, JHEP 03 (2021) 149 [arXiv:2008.07551] [INSPIRE].
A. Falkowski, S. Rychkov and A. Urbano, What if the Higgs couplings to W and Z bosons are larger than in the Standard Model?, JHEP 04 (2012) 073 [arXiv:1202.1532] [INSPIRE].
J. Gu, L.-T. Wang and C. Zhang, An unambiguous test of positivity at lepton colliders, arXiv:2011.03055 [INSPIRE].
B. Fuks, Y. Liu, C. Zhang and S.-Y. Zhou, Positivity in electron-positron scattering: testing the axiomatic quantum field theory principles and probing the existence of UV states, Chin. Phys. C 45 (2021) 023108 [arXiv:2009.02212] [INSPIRE].
S. Alioli, R. Boughezal, E. Mereghetti and F. Petriello, Novel angular dependence in Drell-Yan lepton production via dimension-8 operators, Phys. Lett. B 809 (2020) 135703 [arXiv:2003.11615] [INSPIRE].
A. Azatov, R. Contino, C.S. Machado and F. Riva, Helicity selection rules and noninterference for BSM amplitudes, Phys. Rev. D 95 (2017) 065014 [arXiv:1607.05236] [INSPIRE].
A. Falkowski and G. Isabella, Matter coupling in massive gravity, JHEP 04 (2020) 014 [arXiv:2001.06800] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2011.10058
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
Trott, T. Causality, unitarity and symmetry in effective field theory. J. High Energ. Phys. 2021, 143 (2021). https://doi.org/10.1007/JHEP07(2021)143
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2021)143