Abstract
We derive an off-shell recursion relation for correlators that holds at all loop orders. This allows us to prove how generalized amplitudes transform under generic field redefinitions, starting from an assumed behavior of the one-particle-irreducible effective action. The form of the recursion relation resembles the operation of raising the rank of a tensor by acting with a covariant derivative. This inspires a geometric interpretation, whose features and flaws we investigate.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.S.R. Chisholm, Change of variables in quantum field theories, Nucl. Phys. 26 (1961) 469 [INSPIRE].
S. Kamefuchi, L. O’Raifeartaigh and A. Salam, Change of variables and equivalence theorems in quantum field theories, Nucl. Phys. 28 (1961) 529 [INSPIRE].
C. Arzt, Reduced effective Lagrangians, Phys. Lett. B 342 (1995) 189 [hep-ph/9304230] [INSPIRE].
A.V. Manohar, Introduction to Effective Field Theories, arXiv:1804.05863 [https://doi.org/10.1093/oso/9780198855743.003.0002] [INSPIRE].
T. Cohen, N. Craig, X. Lu and D. Sutherland, On-Shell Covariance of Quantum Field Theory Amplitudes, Phys. Rev. Lett. 130 (2023) 041603 [arXiv:2202.06965] [INSPIRE].
L. Lehman and A. Martin, Hilbert Series for Constructing Lagrangians: expanding the phenomenologist’s toolbox, Phys. Rev. D 91 (2015) 105014 [arXiv:1503.07537] [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].
L. Lehman and A. Martin, Low-derivative operators of the Standard Model effective field theory via Hilbert series methods, JHEP 02 (2016) 081 [arXiv:1510.00372] [INSPIRE].
B. Henning, X. Lu, T. Melia and H. Murayama, 2, 84, 30, 993, 560, 15456, 11962, 261485, ...: Higher dimension operators in the SM EFT, JHEP 08 (2017) 016 [Erratum ibid. 09 (2019) 019] [arXiv:1512.03433] [INSPIRE].
A. Kobach and S. Pal, Hilbert Series and Operator Basis for NRQED and NRQCD/HQET, Phys. Lett. B 772 (2017) 225 [arXiv:1704.00008] [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].
A. Kobach and S. Pal, Reparameterization Invariant Operator Basis for NRQED and HQET, JHEP 11 (2019) 012 [arXiv:1810.02356] [INSPIRE].
M. Ruhdorfer, J. Serra and A. Weiler, Effective Field Theory of Gravity to All Orders, JHEP 05 (2020) 083 [arXiv:1908.08050] [INSPIRE].
C.B. Marinissen, R. Rahn and W.J. Waalewijn, ..., 83106786, 114382724, 1509048322, 2343463290, 27410087742, ... efficient Hilbert series for effective theories, Phys. Lett. B 808 (2020) 135632 [arXiv:2004.09521] [INSPIRE].
L. Graf et al., 2, 12, 117, 1959, 45171, 1170086, . . . : a Hilbert series for the QCD chiral Lagrangian, JHEP 01 (2021) 142 [arXiv:2009.01239] [INSPIRE].
L. Gráf et al., Hilbert series, the Higgs mechanism, and HEFT, JHEP 02 (2023) 064 [arXiv:2211.06275] [INSPIRE].
H. Sun, Y.-N. Wang and J.-H. Yu, Hilbert Series and Operator Counting on the Higgs Effective Field Theory, arXiv:2211.11598 [INSPIRE].
D. Kondo, H. Murayama and R. Okabe, 23, 381, 6242, 103268, 1743183, . . . : Hilbert series for CP-violating operators in SMEFT, JHEP 03 (2023) 107 [arXiv:2212.02413] [INSPIRE].
A. Delgado, A. Martin and R. Wang, Constructing operator basis in supersymmetry: a Hilbert series approach, JHEP 04 (2023) 097 [arXiv:2212.02551] [INSPIRE].
H. Sun, M.-L. Xiao and J.-H. Yu, Complete NNLO operator bases in Higgs effective field theory, JHEP 04 (2023) 086 [arXiv:2210.14939] [INSPIRE].
J. Bijnens, S.B. Gudnason, J. Yu and T. Zhang, Hilbert series and higher-order Lagrangians for the O(N) model, JHEP 05 (2023) 061 [arXiv:2212.07901] [INSPIRE].
A. Delgado, A. Martin and R. Wang, Counting operators in N = 1 supersymmetric gauge theories, JHEP 07 (2023) 081 [arXiv:2305.01736] [INSPIRE].
C. Grojean, J. Kley and C.-Y. Yao, Hilbert series for ALP EFTs, JHEP 11 (2023) 196 [arXiv:2307.08563] [INSPIRE].
T. Cohen, H. Elvang and M. Kiermaier, On-shell constructibility of tree amplitudes in general field theories, JHEP 04 (2011) 053 [arXiv:1010.0257] [INSPIRE].
C. Cheung and C.-H. Shen, Nonrenormalization Theorems without Supersymmetry, Phys. Rev. Lett. 115 (2015) 071601 [arXiv:1505.01844] [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].
N. Arkani-Hamed, T.-C. Huang and Y.-T. Huang, Scattering amplitudes for all masses and spins, JHEP 11 (2021) 070 [arXiv:1709.04891] [INSPIRE].
Y. Shadmi and Y. Weiss, Effective Field Theory Amplitudes the On-Shell Way: Scalar and Vector Couplings to Gluons, JHEP 02 (2019) 165 [arXiv:1809.09644] [INSPIRE].
N. Christensen and B. Field, Constructive standard model, Phys. Rev. D 98 (2018) 016014 [arXiv:1802.00448] [INSPIRE].
G. Durieux, T. Kitahara, Y. Shadmi and Y. Weiss, The electroweak effective field theory from on-shell amplitudes, JHEP 01 (2020) 119 [arXiv:1909.10551] [INSPIRE].
G. Durieux and C.S. Machado, Enumerating higher-dimensional operators with on-shell amplitudes, Phys. Rev. D 101 (2020) 095021 [arXiv:1912.08827] [INSPIRE].
Z. Bern, J. Parra-Martinez and E. Sawyer, Nonrenormalization and Operator Mixing via On-Shell Methods, Phys. Rev. Lett. 124 (2020) 051601 [arXiv:1910.05831] [INSPIRE].
N. Christensen, B. Field, A. Moore and S. Pinto, Two-, three-, and four-body decays in the constructive standard model, Phys. Rev. D 101 (2020) 065019 [arXiv:1909.09164] [INSPIRE].
T. Ma, J. Shu and M.-L. Xiao, Standard model effective field theory from on-shell amplitudes, Chin. Phys. C 47 (2023) 023105 [arXiv:1902.06752] [INSPIRE].
R. Aoude and C.S. Machado, The Rise of SMEFT On-shell Amplitudes, JHEP 12 (2019) 058 [arXiv:1905.11433] [INSPIRE].
B. Bachu and A. Yelleshpur, On-Shell Electroweak Sector and the Higgs Mechanism, JHEP 08 (2020) 039 [arXiv:1912.04334] [INSPIRE].
B. Henning and T. Melia, Constructing effective field theories via their harmonics, Phys. Rev. D 100 (2019) 016015 [arXiv:1902.06754] [INSPIRE].
Z. Bern, J. Parra-Martinez and E. Sawyer, Structure of two-loop SMEFT anomalous dimensions via on-shell methods, JHEP 10 (2020) 211 [arXiv:2005.12917] [INSPIRE].
G. Durieux et al., Constructing massive on-shell contact terms, JHEP 12 (2020) 175 [arXiv:2008.09652] [INSPIRE].
J. Elias Miró, J. Ingoldby and M. Riembau, EFT anomalous dimensions from the S-matrix, JHEP 09 (2020) 163 [arXiv:2005.06983] [INSPIRE].
P. Baratella, C. Fernandez and A. Pomarol, Renormalization of Higher-Dimensional Operators from On-shell Amplitudes, Nucl. Phys. B 959 (2020) 115155 [arXiv:2005.07129] [INSPIRE].
A. Falkowski, G. Isabella and C.S. Machado, On-shell effective theory for higher-spin dark matter, SciPost Phys. 10 (2021) 101 [arXiv:2011.05339] [INSPIRE].
M. Jiang, T. Ma and J. Shu, Renormalization Group Evolution from On-shell SMEFT, JHEP 01 (2021) 101 [arXiv:2005.10261] [INSPIRE].
Q. Jin, K. Ren and G. Yang, Two-Loop anomalous dimensions of QCD operators up to dimension-sixteen and Higgs EFT amplitudes, JHEP 04 (2021) 180 [arXiv:2011.02494] [INSPIRE].
R. Nagai, M. Tanabashi, K. Tsumura and Y. Uchida, Scalar and fermion on-shell amplitudes in generalized Higgs effective field theory, Phys. Rev. D 104 (2021) 015001 [arXiv:2102.08519] [INSPIRE].
Z.-Y. Dong, T. Ma and J. Shu, Constructing on-shell operator basis for all masses and spins, Phys. Rev. D 107 (2023) L111901 [arXiv:2103.15837] [INSPIRE].
M. Accettulli Huber and S. De Angelis, Standard Model EFTs via on-shell methods, JHEP 11 (2021) 221 [arXiv:2108.03669] [INSPIRE].
S. De Angelis, Amplitude bases in generic EFTs, JHEP 08 (2022) 299 [arXiv:2202.02681] [INSPIRE].
S. Chang, M. Chen, D. Liu and M.A. Luty, Primary observables for indirect searches at colliders, JHEP 07 (2023) 030 [arXiv:2212.06215] [INSPIRE].
Z.-Y. Dong, T. Ma, J. Shu and Y.-H. Zheng, Constructing generic effective field theory for all masses and spins, Phys. Rev. D 106 (2022) 116010 [arXiv:2202.08350] [INSPIRE].
R. Balkin et al., On-shell Higgsing for EFTs, JHEP 03 (2022) 129 [arXiv:2112.09688] [INSPIRE].
I. Low, J. Shu, M.-L. Xiao and Y.-H. Zheng, Amplitude/operator basis in chiral perturbation theory, JHEP 01 (2023) 031 [arXiv:2209.00198] [INSPIRE].
H. Liu, T. Ma, Y. Shadmi and M. Waterbury, An EFT hunter’s guide to two-to-two scattering: HEFT and SMEFT on-shell amplitudes, JHEP 05 (2023) 241 [arXiv:2301.11349] [INSPIRE].
L. Bradshaw and S. Chang, Primary observables for top quark collider signals, Phys. Rev. D 108 (2023) 015019 [arXiv:2304.06063] [INSPIRE].
C. Arzate, S. Chang and G. Jacobo, Primary observables for gauge boson collider signals, Phys. Rev. D 109 (2024) 075046 [arXiv:2312.03821] [INSPIRE].
J. Honerkamp, Chiral multiloops, Nucl. Phys. B 36 (1972) 130 [INSPIRE].
L. Tataru, One Loop Divergences of the Nonlinear Chiral Theory, Phys. Rev. D 12 (1975) 3351 [INSPIRE].
L. Alvarez-Gaume, D.Z. Freedman and S. Mukhi, The Background Field Method and the Ultraviolet Structure of the Supersymmetric Nonlinear Sigma Model, Annals Phys. 134 (1981) 85 [INSPIRE].
L. Alvarez-Gaume and D.Z. Freedman, Geometrical Structure and Ultraviolet Finiteness in the Supersymmetric Sigma Model, Commun. Math. Phys. 80 (1981) 443 [INSPIRE].
G.A. Vilkovisky, The Unique Effective Action in Quantum Field Theory, Nucl. Phys. B 234 (1984) 125 [INSPIRE].
B.S. DeWitt, The spacetime approach to quantum field theory, in the proceedings of the Les Houches Summer School on Theoretical Physics: Relativity, Groups and Topology, Les Houches, France, June 27 – August 04 (1983) [INSPIRE].
M.K. Gaillard, The Effective One Loop Lagrangian With Derivative Couplings, Nucl. Phys. B 268 (1986) 669 [INSPIRE].
B.S. DeWitt, The Effective Action, in Les Houches School of Theoretical Physics: Architecture of Fundamental Interactions at Short Distances (1987) pp. 1023–1058.
R. Alonso, E.E. Jenkins and A.V. Manohar, A Geometric Formulation of Higgs Effective Field Theory: Measuring the Curvature of Scalar Field Space, Phys. Lett. B 754 (2016) 335 [arXiv:1511.00724] [INSPIRE].
R. Alonso, E.E. Jenkins and A.V. Manohar, Sigma Models with Negative Curvature, Phys. Lett. B 756 (2016) 358 [arXiv:1602.00706] [INSPIRE].
R. Alonso, E.E. Jenkins and A.V. Manohar, Geometry of the Scalar Sector, JHEP 08 (2016) 101 [arXiv:1605.03602] [INSPIRE].
R. Nagai, M. Tanabashi, K. Tsumura and Y. Uchida, Symmetry and geometry in a generalized Higgs effective field theory: Finiteness of oblique corrections versus perturbative unitarity, Phys. Rev. D 100 (2019) 075020 [arXiv:1904.07618] [INSPIRE].
A. Helset, A. Martin and M. Trott, The Geometric Standard Model Effective Field Theory, JHEP 03 (2020) 163 [arXiv:2001.01453] [INSPIRE].
T. Cohen, N. Craig, X. Lu and D. Sutherland, Is SMEFT Enough?, JHEP 03 (2021) 237 [arXiv:2008.08597] [INSPIRE].
T. Cohen, N. Craig, X. Lu and D. Sutherland, Unitarity violation and the geometry of Higgs EFTs, JHEP 12 (2021) 003 [arXiv:2108.03240] [INSPIRE].
R. Alonso and M. West, Roads to the Standard Model, Phys. Rev. D 105 (2022) 096028 [arXiv:2109.13290] [INSPIRE].
I. Banta et al., Non-decoupling new particles, JHEP 02 (2022) 029 [arXiv:2110.02967] [INSPIRE].
J. Talbert, The geometric νSMEFT: operators and connections, JHEP 01 (2023) 069 [arXiv:2208.11139] [INSPIRE].
R. Alonso, J.C. Criado, R. Houtz and M. West, Walls, bubbles and doom — the cosmology of HEFT, JHEP 05 (2024) 049 [arXiv:2312.00881] [INSPIRE].
K. Finn, S. Karamitsos and A. Pilaftsis, Frame Covariance in Quantum Gravity, Phys. Rev. D 102 (2020) 045014 [arXiv:1910.06661] [INSPIRE].
K. Finn, S. Karamitsos and A. Pilaftsis, Frame covariant formalism for fermionic theories, Eur. Phys. J. C 81 (2021) 572 [arXiv:2006.05831] [INSPIRE].
C. Cheung, A. Helset and J. Parra-Martinez, Geometric soft theorems, JHEP 04 (2022) 011 [arXiv:2111.03045] [INSPIRE].
R. Alonso and M. West, On the effective action for scalars in a general manifold to any loop order, Phys. Lett. B 841 (2023) 137937 [arXiv:2207.02050] [INSPIRE].
A. Helset, E.E. Jenkins and A.V. Manohar, Geometry in scattering amplitudes, Phys. Rev. D 106 (2022) 116018 [arXiv:2210.08000] [INSPIRE].
A. Helset, E.E. Jenkins and A.V. Manohar, Renormalization of the Standard Model Effective Field Theory from geometry, JHEP 02 (2023) 063 [arXiv:2212.03253] [INSPIRE].
A. Pilaftsis, K. Finn, V. Gattus and S. Karamitsos, Geometrising the Micro-Cosmos on a Supermanifold, PoS CORFU2021 (2022) 080 [arXiv:2204.00123] [INSPIRE].
B. Assi et al., Fermion geometry and the renormalization of the Standard Model Effective Field Theory, JHEP 11 (2023) 201 [arXiv:2307.03187] [INSPIRE].
E.E. Jenkins, A.V. Manohar, L. Naterop and J. Pagès, An algebraic formula for two loop renormalization of scalar quantum field theory, JHEP 12 (2023) 165 [arXiv:2308.06315] [INSPIRE].
E.E. Jenkins, A.V. Manohar, L. Naterop and J. Pagès, Two loop renormalization of scalar theories using a geometric approach, JHEP 02 (2024) 131 [arXiv:2310.19883] [INSPIRE].
V. Gattus and A. Pilaftsis, Minimal supergeometric quantum field theories, Phys. Lett. B 846 (2023) 138234 [arXiv:2307.01126] [INSPIRE].
R. Alonso, A primer on Higgs Effective Field Theory with Geometry, arXiv:2307.14301 [INSPIRE].
C. Cheung, A. Helset and J. Parra-Martinez, Geometry-kinematics duality, Phys. Rev. D 106 (2022) 045016 [arXiv:2202.06972] [INSPIRE].
N. Craig, Y.-T. Lee, X. Lu and D. Sutherland, Effective field theories as Lagrange spaces, JHEP 11 (2023) 069 [arXiv:2305.09722] [INSPIRE].
N. Craig and Y.-T. Lee, Effective Field Theories on the Jet Bundle, Phys. Rev. Lett. 132 (2024) 061602 [arXiv:2307.15742] [INSPIRE].
M. Alminawi, I. Brivio and J. Davighi, Jet Bundle Geometry of Scalar Field Theories, arXiv:2308.00017 [INSPIRE].
H. Neufeld, J. Gasser and G. Ecker, The one loop functional as a Berezinian, Phys. Lett. B 438 (1998) 106 [hep-ph/9806436] [INSPIRE].
S. Weinberg, The quantum theory of fields. Vol. 2: Modern applications, Cambridge University Press (2013) [https://doi.org/10.1017/CBO9781139644174] [INSPIRE].
M.D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press (2014).
A.V. Manohar and E. Nardoni, Renormalization Group Improvement of the Effective Potential: an EFT Approach, JHEP 04 (2021) 093 [arXiv:2010.15806] [INSPIRE].
H. Lehmann, K. Symanzik and W. Zimmermann, On the formulation of quantized field theories, Nuovo Cim. 1 (1955) 205 [INSPIRE].
H. Lehmann, K. Symanzik and W. Zimmermann, On the formulation of quantized field theories. II, Nuovo Cim. 6 (1957) 319 [INSPIRE].
F.A. Berends and W.T. Giele, Recursive Calculations for Processes with n Gluons, Nucl. Phys. B 306 (1988) 759 [INSPIRE].
L.S. Brown, Summing tree graphs at threshold, Phys. Rev. D 46 (1992) R4125 [hep-ph/9209203] [INSPIRE].
R. Monteiro and D. O’Connell, The Kinematic Algebra From the Self-Dual Sector, JHEP 07 (2011) 007 [arXiv:1105.2565] [INSPIRE].
A.A. Rosly and K.G. Selivanov, On amplitudes in selfdual sector of Yang-Mills theory, Phys. Lett. B 399 (1997) 135 [hep-th/9611101] [INSPIRE].
K.G. Selivanov, SD perturbiner in Yang-Mills + gravity, Phys. Lett. B 420 (1998) 274 [hep-th/9710197] [INSPIRE].
K. Lee, Quantum off-shell recursion relation, JHEP 05 (2022) 051 [arXiv:2202.08133] [INSPIRE].
K. Cho, K. Kim and K. Lee, Binary black holes and quantum off-shell recursion, JHEP 05 (2024) 050 [arXiv:2311.01284] [INSPIRE].
T. Cohen, X. Lu and Z. Zhang, Functional Prescription for EFT Matching, JHEP 02 (2021) 228 [arXiv:2011.02484] [INSPIRE].
T. Cohen, X. Lu and Z. Zhang, Snowmass White Paper: Effective Field Theory Matching and Applications, in the proceedings of the Snowmass 2021, Seattle, U.S.A., July 17–26 (2022) [arXiv:2203.07336] [INSPIRE].
B.S. DeWitt, The global approach to quantum field theory. Vol. 1, 2, Int. Ser. Monogr. Phys. 114 (2003) [INSPIRE].
B.S. DeWitt and G. Esposito, An introduction to quantum gravity, Int. J. Geom. Meth. Mod. Phys. 5 (2008) 101 [arXiv:0711.2445] [INSPIRE].
Y. Kluth, P. Millington and P. Saffin, Renormalization group flows from the Hessian geometry of quantum effective actions, arXiv:2311.17199 [INSPIRE].
A.S. Arvanitakis, The L∞-algebra of the S-matrix, JHEP 07 (2019) 115 [arXiv:1903.05643] [INSPIRE].
T. Macrelli, C. Sämann and M. Wolf, Scattering amplitude recursion relations in Batalin-Vilkovisky–quantizable theories, Phys. Rev. D 100 (2019) 045017 [arXiv:1903.05713] [INSPIRE].
Acknowledgments
We thank Nathaniel Craig for collaboration during the early stages of this work. We would like to thank Andreas Helset and Aneesh Manohar for useful conversations. T. Cohen is supported by the U.S. Department of Energy under grant number DE-SC0011640. X. Lu is supported by the U.S. Department of Energy under grant number DE-SC0009919. D. Sutherland acknowledges support from the Institute for Particle Physics Phenomenology Associateship Scheme.
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: 2312.06748
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
Cohen, T., Lu, X. & Sutherland, D. On amplitudes and field redefinitions. J. High Energ. Phys. 2024, 149 (2024). https://doi.org/10.1007/JHEP06(2024)149
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2024)149