Abstract
Effective field theories (EFT) parameterize the long-distance effects of short-distance dynamics whose details may or may not be known. Previous work showed that EFT coefficients must obey certain positivity constraints if causality and unitarity are satisfied at all scales. We explore those constraints from the perspective of 2 → 2 scattering amplitudes of a light real scalar field, using semi-definite programming to carve out the space of allowed EFT coefficients for a given mass threshold M. We point out that all EFT parameters are bounded both below and above, effectively showing that dimensional analysis scaling is a consequence of causality. This includes the coefficients of s2 + t2 + u2 and stu type interactions. We present simple 2 → 2 extremal amplitudes which realize, or “rule in”, kinks in coefficient space and whose convex hull span a large fraction of the allowed space.
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References
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].
A. Martin, Scattering theory: unitarity, analyticity and crossing, Lecture Notes in Physics volume 3, Springer (1969) [INSPIRE].
S. M. Roy, Exact integral equation for pion pion scattering involving only physical region partial waves, Phys. Lett. B 36 (1971) 353 [INSPIRE].
G. Colangelo, J. Gasser and H. Leutwyler, ππ scattering, Nucl. Phys. B 603 (2001) 125 [hep-ph/0103088] [INSPIRE].
I. Caprini, G. Colangelo, J. Gasser and H. Leutwyler, On the precision of the theoretical predictions for ππ scattering, Phys. Rev. D 68 (2003) 074006 [hep-ph/0306122] [INSPIRE].
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].
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].
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].
C. Cheung and G. N. Remmen, Positive signs in massive gravity, JHEP 04 (2016) 002 [arXiv:1601.04068] [INSPIRE].
N. Arkani-Hamed and Y.-T. Huang, Positive geometry of effective field theory, lectures given at the Cern Winter School on supergravity, strings and gauge theory, February 3–7, CERN, Switzerland (2020).
N. Arkani-Hamed and Y.-T. Huang, New positivity bounds from the EFT hedron, talk at the 24th Rencontres Itzykson of the IPHT of CEA-Saclay, June 5–7, IPhT CEA-Saclay, France (2019).
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].
B. Bellazzini, J. Elias Miró, R. Rattazzi, M. Riembau and F. Riva, Positive moments for scattering amplitudes, arXiv:2011.00037 [INSPIRE].
A. J. Tolley, Z.-Y. Wang and S.-Y. Zhou, New positivity bounds from full crossing symmetry, 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].
M. Correia, A. Sever and A. Zhiboedov, An analytical toolkit for the S-matrix bootstrap, arXiv:2006.08221 [INSPIRE].
M. Gell-Mann, M. L. Goldberger and W. E. Thirring, Use of causality conditions in quantum theory, Phys. Rev. 95 (1954) 1612 [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. Caron-Huot, Lorentzian and analytic bootstrap lecture 3, tlak given at 2020 Bootstrap School, June 8–12, Harvard University, U.K. (2020).
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].
R. J. Eden, P. V. Landshoff, D. I. Olive and J. C. Polkinghorne, The analytic S-matrix, Cambridge University Press, Cambridge U.K. (1966).
S. Caron-Huot, D. Mazac, L. Rastelli and D. Simmons-Duffin, Dispersive CFT sum rules, arXiv:2008.04931 [INSPIRE].
J. Penedones, J. A. Silva and A. Zhiboedov, Nonperturbative Mellin amplitudes: existence, properties, applications, JHEP 08 (2020) 031 [arXiv:1912.11100] [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].
D. Simmons-Duffin, A semidefinite program solver for the conformal bootstrap, JHEP 06 (2015) 174 [arXiv:1502.02033] [INSPIRE].
S. M. Chester et al., Carving out OPE space and precise O(2) model critical exponents, JHEP 06 (2020) 142 [arXiv:1912.03324] [INSPIRE].
L. Córdova, Y. He, M. Kruczenski and P. Vieira, The O(N) S-matrix monolith, JHEP 04 (2020) 142 [arXiv:1909.06495] [INSPIRE].
M. F. Paulos, J. Penedones, J. Toledo, B. C. van Rees and P. Vieira, The S-matrix bootstrap. Part III. Higher dimensional amplitudes, JHEP 12 (2019) 040 [arXiv:1708.06765] [INSPIRE].
A. L. Guerrieri, J. Penedones and P. Vieira, Bootstrapping QCD using pion scattering amplitudes, Phys. Rev. Lett. 122 (2019) 241604 [arXiv:1810.12849] [INSPIRE].
A. L. Guerrieri, A. Homrich and P. Vieira, Dual S-matrix bootstrap. Part I. 2D theory, JHEP 11 (2020) 084 [arXiv:2008.02770] [INSPIRE].
Z. Komargodski and A. Schwimmer, On renormalization group flows in four dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [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. 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].
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. 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].
S. D. Chowdhury, A. Gadde, T. Gopalka, I. Halder, L. Janagal and S. Minwalla, Classifying and constraining local four photon and four graviton S-matrices, JHEP 02 (2020) 114 [arXiv:1910.14392] [INSPIRE].
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Caron-Huot, S., Van Duong, V. Extremal effective field theories. J. High Energ. Phys. 2021, 280 (2021). https://doi.org/10.1007/JHEP05(2021)280
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DOI: https://doi.org/10.1007/JHEP05(2021)280