Abstract
We derive new effective field theory (EFT) positivity bounds on the elastic 2 → 2 scattering amplitudes of massive spinning particles from the standard UV properties of unitarity, causality, locality and Lorentz invariance. By bounding the t derivatives of the amplitude (which can be represented as angular momentum matrix elements) in terms of the total ingoing helicity, we derive stronger unitarity bounds on the s- and u-channel branch cuts which determine the dispersion relation. In contrast to previous positivity bounds, which relate the t-derivative to the forward-limit EFT amplitude with no t derivatives, our bounds establish that the t-derivative alone must be strictly positive for sufficiently large helicities. Consequently, they can provide stronger constraints beyond the forward limit which can be used to constrain dimension-6 interactions with a milder assumption about the high-energy growth of the UV amplitude.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
T.N. Pham and T.N. Truong, Evaluation of the Derivative Quartic Terms of the Meson Chiral Lagrangian From Forward Dispersion Relation, Phys. Rev. D 31 (1985) 3027 [INSPIRE].
B. Ananthanarayan, D. Toublan and G. Wanders, Consistency of the chiral pion pion scattering amplitudes with axiomatic constraints, Phys. Rev. D 51 (1995) 1093 [hep-ph/9410302] [INSPIRE].
M.R. Pennington and J. Portoles, The Chiral Lagrangian parameters, ℓ1, ℓ2, are determined by the ρ-resonance, Phys. Lett. B 344 (1995) 399 [hep-ph/9409426] [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, 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, Positivity bounds for scalar field theories, Phys. Rev. D 96 (2017) 081702 [arXiv:1702.06134] [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].
G.N. Remmen and N.L. Rodd, Signs, Spin, SMEFT: Sum Rules at Dimension Six, arXiv:2010.04723 [INSPIRE].
B. Bellazzini, J. Elias Miró, R. Rattazzi, M. Riembau and F. Riva, Positive moments for scattering amplitudes, Phys. Rev. D 104 (2021) 036006 [arXiv:2011.00037] [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].
S. Caron-Huot and V. Van Duong, Extremal Effective Field Theories, JHEP 05 (2021) 280 [arXiv:2011.02957] [INSPIRE].
A. Sinha and A. Zahed, Crossing Symmetric Dispersion Relations in Quantum Field Theories, Phys. Rev. Lett. 126 (2021) 181601 [arXiv:2012.04877] [INSPIRE].
T. Trott, Causality, unitarity and symmetry in effective field theory, JHEP 07 (2021) 143 [arXiv:2011.10058] [INSPIRE].
X. Li, H. Xu, C. Yang, C. Zhang and S.-Y. Zhou, Positivity in Multifield Effective Field Theories, Phys. Rev. Lett. 127 (2021) 121601 [arXiv:2101.01191] [INSPIRE].
N. Arkani-Hamed, T.-C. Huang and Y.-T. Huang, The EFT-Hedron, JHEP 05 (2021) 259 [arXiv:2012.15849] [INSPIRE].
L.-Y. Chiang, Y.-t. Huang, W. Li, L. Rodina and H.-C. Weng, Into the EFThedron and UV constraints from IR consistency, arXiv:2105.02862 [INSPIRE].
G.F. Chew, S-matrix theory of strong interactions, Benjamin, New York, U.S.A. (1961).
R.J. Eden, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne, S-matrix theory of strong interactions, Cambridge University Press (1966).
J. Ellis, M. Madigan, K. Mimasu, V. Sanz and T. You, Top, Higgs, Diboson and Electroweak Fit to the Standard Model Effective Field Theory, JHEP 04 (2021) 279 [arXiv:2012.02779] [INSPIRE].
SMEFiT collaboration, Combined SMEFT interpretation of Higgs, diboson, and top quark data from the LHC, JHEP 11 (2021) 089 [arXiv:2105.00006] [INSPIRE].
B. Bellazzini, F. Riva, J. Serra and F. Sgarlata, The other effective fermion compositeness, JHEP 11 (2017) 020 [arXiv:1706.03070] [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].
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].
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].
G.N. Remmen and N.L. Rodd, Consistency of the Standard Model Effective Field Theory, JHEP 12 (2019) 032 [arXiv:1908.09845] [INSPIRE].
C. Englert, G.F. Giudice, A. Greljo and M. Mccullough, The \( \hat{H} \)-Parameter: An Oblique Higgs View, JHEP 09 (2019) 041 [arXiv:1903.07725] [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].
G.N. Remmen and N.L. Rodd, Flavor Constraints from Unitarity and Analyticity, Phys. Rev. Lett. 125 (2020) 081601 [Erratum ibid. 127 (2021) 149901] [arXiv:2004.02885] [INSPIRE].
Q. Bonnefoy, E. Gendy and C. Grojean, Positivity bounds on Minimal Flavor Violation, JHEP 04 (2021) 115 [arXiv:2011.12855] [INSPIRE].
J. Distler, B. Grinstein, R.A. Porto and I.Z. Rothstein, Falsifying Models of New Physics via WW Scattering, Phys. Rev. Lett. 98 (2007) 041601 [hep-ph/0604255] [INSPIRE].
L. Vecchi, Causal versus analytic constraints on anomalous quartic gauge couplings, JHEP 11 (2007) 054 [arXiv:0704.1900] [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].
C. Cheung and G.N. Remmen, Positivity of Curvature-Squared Corrections in Gravity, Phys. Rev. Lett. 118 (2017) 051601 [arXiv:1608.02942] [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].
A. Gruzinov and M. Kleban, Causality Constrains Higher Curvature Corrections to Gravity, Class. Quant. Grav. 24 (2007) 3521 [hep-th/0612015] [INSPIRE].
C. Cheung and G.N. Remmen, Positive Signs in Massive Gravity, JHEP 04 (2016) 002 [arXiv:1601.04068] [INSPIRE].
J. Bonifacio, K. Hinterbichler and R.A. Rosen, Positivity constraints for pseudolinear massive spin-2 and vector Galileons, Phys. Rev. D 94 (2016) 104001 [arXiv:1607.06084] [INSPIRE].
B. Bellazzini, F. Riva, J. Serra and F. Sgarlata, Beyond Positivity Bounds and the Fate of Massive Gravity, Phys. Rev. Lett. 120 (2018) 161101 [arXiv:1710.02539] [INSPIRE].
C. de Rham, S. Melville and A.J. Tolley, Improved Positivity Bounds and Massive Gravity, JHEP 04 (2018) 083 [arXiv:1710.09611] [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].
L. Alberte, C. de Rham, A. Momeni, J. Rumbutis and A.J. Tolley, Positivity Constraints on Interacting Spin-2 Fields, JHEP 03 (2020) 097 [arXiv:1910.11799] [INSPIRE].
L. Alberte, C. de Rham, A. Momeni, J. Rumbutis and A.J. Tolley, Positivity Constraints on Interacting Pseudo-Linear Spin-2 Fields, JHEP 07 (2020) 121 [arXiv:1912.10018] [INSPIRE].
Z.-Y. Wang, C. Zhang and S.-Y. Zhou, Generalized elastic positivity bounds on interacting massive spin-2 theories, JHEP 04 (2021) 217 [arXiv:2011.05190] [INSPIRE].
K. Hinterbichler, A. Joyce and R.A. Rosen, Massive Spin-2 Scattering and Asymptotic Superluminality, JHEP 03 (2018) 051 [arXiv:1708.05716] [INSPIRE].
J. Bonifacio and K. Hinterbichler, Bounds on Amplitudes in Effective Theories with Massive Spinning Particles, Phys. Rev. D 98 (2018) 045003 [arXiv:1804.08686] [INSPIRE].
B. Bellazzini, F. Riva, J. Serra and F. Sgarlata, Massive Higher Spins: Effective Theory and Consistency, JHEP 10 (2019) 189 [arXiv:1903.08664] [INSPIRE].
A. Nicolis, R. Rattazzi and E. Trincherini, Energy’s and amplitudes’ positivity, JHEP 05 (2010) 095 [Erratum ibid. 11 (2011) 128] [arXiv:0912.4258] [INSPIRE].
H. Elvang, D.Z. Freedman, L.-Y. Hung, M. Kiermaier, R.C. Myers and S. Theisen, On renormalization group flows and the a-theorem in 6d, JHEP 10 (2012) 011 [arXiv:1205.3994] [INSPIRE].
C. de Rham, S. Melville, A.J. Tolley and S.-Y. Zhou, Massive Galileon Positivity Bounds, JHEP 09 (2017) 072 [arXiv:1702.08577] [INSPIRE].
V. Chandrasekaran, G.N. Remmen and A. Shahbazi-Moghaddam, Higher-Point Positivity, JHEP 11 (2018) 015 [arXiv:1804.03153] [INSPIRE].
M. Herrero-Valea, I. Timiryasov and A. Tokareva, To Positivity and Beyond, where Higgs-Dilaton Inflation has never gone before, JCAP 11 (2019) 042 [arXiv:1905.08816] [INSPIRE].
C. Cheung and G.N. Remmen, Infrared Consistency and the Weak Gravity Conjecture, JHEP 12 (2014) 087 [arXiv:1407.7865] [INSPIRE].
C. Cheung, J. Liu and G.N. Remmen, Proof of the Weak Gravity Conjecture from Black Hole Entropy, JHEP 10 (2018) 004 [arXiv:1801.08546] [INSPIRE].
Y. Hamada, T. Noumi and G. Shiu, Weak Gravity Conjecture from Unitarity and Causality, Phys. Rev. Lett. 123 (2019) 051601 [arXiv:1810.03637] [INSPIRE].
C. Cheung, J. Liu and G.N. Remmen, Entropy Bounds on Effective Field Theory from Rotating Dyonic Black Holes, Phys. Rev. D 100 (2019) 046003 [arXiv:1903.09156] [INSPIRE].
B. Bellazzini, M. Lewandowski and J. Serra, Positivity of Amplitudes, Weak Gravity Conjecture, and Modified Gravity, Phys. Rev. Lett. 123 (2019) 251103 [arXiv:1902.03250] [INSPIRE].
A.M. Charles, The Weak Gravity Conjecture, RG Flows, and Supersymmetry, arXiv:1906.07734 [INSPIRE].
S. Melville and J. Noller, Positivity in the Sky: Constraining dark energy and modified gravity from the UV, Phys. Rev. D 101 (2020) 021502 [Erratum ibid. 102 (2020) 049902] [arXiv:1904.05874] [INSPIRE].
C. de Rham, S. Melville and J. Noller, Positivity bounds on dark energy: when matter matters, JCAP 08 (2021) 018 [arXiv:2103.06855] [INSPIRE].
D. Baumann, D. Green, H. Lee and R.A. Porto, Signs of Analyticity in Single-Field Inflation, Phys. Rev. D 93 (2016) 023523 [arXiv:1502.07304] [INSPIRE].
T. Grall and S. Melville, Inflation in motion: unitarity constraints in effective field theories with (spontaneously) broken Lorentz symmetry, JCAP 09 (2020) 017 [arXiv:2005.02366] [INSPIRE].
T. Grall and S. Melville, Positivity Bounds without Boosts, arXiv:2102.05683 [INSPIRE].
K. Aoki, S. Mukohyama and R. Namba, Positivity vs. Lorentz-violation: an explicit example, JCAP 10 (2021) 079 [arXiv:2107.01755] [INSPIRE].
L. Alberte, C. de Rham, S. Jaitly and A.J. Tolley, QED positivity bounds, Phys. Rev. D 103 (2021) 125020 [arXiv:2012.05798] [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].
J. Tokuda, K. Aoki and S. Hirano, Gravitational positivity bounds, JHEP 11 (2020) 054 [arXiv:2007.15009] [INSPIRE].
M. Herrero-Valea, R. Santos-Garcia and A. Tokareva, Massless positivity in graviton exchange, Phys. Rev. D 104 (2021) 085022 [arXiv:2011.11652] [INSPIRE].
S. Caron-Huot, D. Mazac, L. Rastelli and D. Simmons-Duffin, Sharp Boundaries for the Swampland, JHEP 07 (2021) 110 [arXiv:2102.08951] [INSPIRE].
Z. Bern, D. Kosmopoulos and A. Zhiboedov, Gravitational effective field theory islands, low-spin dominance, and the four-graviton amplitude, J. Phys. A 54 (2021) 344002 [arXiv:2103.12728] [INSPIRE].
S. Melville, A New Spin on Effective Field Theory Sum Rules (and the Massive Gravity Island), to appear.
M. Porrati and R. Rahman, A Model Independent Ultraviolet Cutoff for Theories with Charged Massive Higher Spin Fields, Nucl. Phys. B 814 (2009) 370 [arXiv:0812.4254] [INSPIRE].
J. Bonifacio and K. Hinterbichler, Universal bound on the strong coupling scale of a gravitationally coupled massive spin-2 particle, Phys. Rev. D 98 (2018) 085006 [arXiv:1806.10607] [INSPIRE].
A.V. Manohar and V. Mateu, Dispersion Relation Bounds for pi pi Scattering, Phys. Rev. D 77 (2008) 094019 [arXiv:0801.3222] [INSPIRE].
J.D. Richman, An Experimenter’s Guide to the Helicity Formalism, CALT-68-1148 [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].
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].
S. Mizera, Bounds on Crossing Symmetry, Phys. Rev. D 103 (2021) 081701 [arXiv:2101.08266] [INSPIRE].
S. Mizera, Crossing symmetry in the planar limit, Phys. Rev. D 104 (2021) 045003 [arXiv:2104.12776] [INSPIRE].
T.L. Trueman and G.C. Wick, Crossing relations for helicity amplitudes, Annals Phys. 26 (1964) 322 [INSPIRE].
G. Cohen-Tannoudji, A. Morel and H. Navelet, Kinematical singularities, crossing matrix and kinematical constraints for two-body helicity amplitudes, Annals Phys. 46 (1968) 239 [INSPIRE].
Y. Hara, On crossing relations for helicity amplitudes, J. Math. Phys. 11 (1970) 253 [INSPIRE].
Y. Hara, Crossing relations for helicity amplitudes, Prog. Theor. Phys. 45 (1971) 584 [INSPIRE].
H.J. Bremermann, R. Oehme and J.G. Taylor, Proof of Dispersion Relations in Quantized Field Theories, Phys. Rev. 109 (1958) 2178 [INSPIRE].
N.N. Bogoliubov, D.V. Shirkov and S. Chomet, Introduction to the theory of quantized fields, vol. 59, Interscience New York, U.S.A. (1959).
K. Hepp, On the analyticity properties of the scattering amplitude in relativistic quantum field theory, Helv. Phys. Acta (Switzerland) 37 (1964).
Y.S. Jin and A. Martin, Number of Subtractions in Fixed-Transfer Dispersion Relations, Phys. Rev. 135 (1964) B1375 [INSPIRE].
A. Martin, Extension of the axiomatic analyticity domain of scattering amplitudes by unitarity. 1, Nuovo Cim. A 42 (1965) 930 [INSPIRE].
G. Mahoux and A. Martin, Extension of axiomatic analyticity properties for particles with spin, and proof of superconvergence relations, Phys. Rev. 174 (1968) 2140 [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].
M. Froissart, Asymptotic behavior and subtractions in the Mandelstam representation, Phys. Rev. 123 (1961) 1053 [INSPIRE].
A. Martin, Unitarity and high-energy behavior of scattering amplitudes, Phys. Rev. 129 (1963) 1432 [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].
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. Nicolis, R. Rattazzi and E. Trincherini, The Galileon as a local modification of gravity, Phys. Rev. D 79 (2009) 064036 [arXiv:0811.2197] [INSPIRE].
L. Keltner and A.J. Tolley, UV properties of Galileons: Spectral Densities, arXiv:1502.05706 [INSPIRE].
S. Bhattacharya and J. Wudka, Dimension-seven operators in the standard model with right handed neutrinos, Phys. Rev. D 94 (2016) 055022 [Erratum ibid. 95 (2017) 039904] [arXiv:1505.05264] [INSPIRE].
F. del Aguila, A. Aparici, S. Bhattacharya, A. Santamaria and J. Wudka, Effective Lagrangian approach to neutrinoless double beta decay and neutrino masses, JHEP 06 (2012) 146 [arXiv:1204.5986] [INSPIRE].
S. Melville, D. Roest and D. Stefanyszyn, UV Constraints on Massive Spinning Particles: Lessons from the Gravitino, JHEP 02 (2020) 185 [arXiv:1911.03126] [INSPIRE].
M.T. Grisaru and H.N. Pendleton, Soft Spin 3/2 Fermions Require Gravity and Supersymmetry, Phys. Lett. B 67 (1977) 323 [INSPIRE].
J. Henriksson, B. McPeak, F. Russo and A. Vichi, Rigorous Bounds on Light-by-Light Scattering, arXiv:2107.13009 [INSPIRE].
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].
H. Elvang and Y.-t. Huang, Scattering Amplitudes, arXiv:1308.1697 [INSPIRE].
G. Cohen-Tannoudji, A. Kotański and P. Salin, Kinematical singularities in cross-sections and density matrices, Phys. Lett. B 27 (1968) 42 [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: 2108.06334
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
Davighi, J., Melville, S. & You, T. Natural selection rules: new positivity bounds for massive spinning particles. J. High Energ. Phys. 2022, 167 (2022). https://doi.org/10.1007/JHEP02(2022)167
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2022)167