Abstract
The causal representation of multi-loop scattering amplitudes, obtained from the application of the loop-tree duality formalism, comprehensively elucidates, at integrand level, the behaviour of only physical singularities. This representation is found to manifest compact expressions for multi-loop topologies that have the same number of vertices. Interestingly, integrands considered in former studies, with up-to six vertices and L internal lines, display the same structure of up-to four-loop ones. The former is an insight that there should be a correspondence between vertices and the collection of internal lines, edges, that characterise a multi-loop topology. By virtue of this relation, in this paper, we embrace an approach to properly classify multi-loop topologies according to vertices and edges. Differently from former studies, we consider the most general topologies, by connecting vertices and edges in all possible ways. Likewise, we provide a procedure to generate causal representation of multi-loop topologies by considering the structure of causal propagators. Explicit causal representations of loop topologies with up-to nine vertices are provided.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Blondel, J. Gluza, S. Jadach, P. Janot and T. Riemann, eds., Theory for the FCC-ee: Report on the 11th FCC-ee Workshop Theory and Experiments, CYRM-2020-003 [arXiv:1905.05078] [INSPIRE].
P. Banerjee et al., Theory for muon-electron scattering @ 10 ppm: A report of the MUonE theory initiative, Eur. Phys. J. C 80 (2020) 591 [arXiv:2004.13663] [INSPIRE].
C. F. Berger et al., An Automated Implementation of On-Shell Methods for One-Loop Amplitudes, Phys. Rev. D 78 (2008) 036003 [arXiv:0803.4180] [INSPIRE].
F. Cascioli, P. Maierhofer and S. Pozzorini, Scattering Amplitudes with Open Loops, Phys. Rev. Lett. 108 (2012) 111601 [arXiv:1111.5206] [INSPIRE].
S. Badger, B. Biedermann, P. Uwer and V. Yundin, Numerical evaluation of virtual corrections to multi-jet production in massless QCD, Comput. Phys. Commun. 184 (2013) 1981 [arXiv:1209.0100] [INSPIRE].
G. Cullen et al., GoSam-2.0: a tool for automated one-loop calculations within the Standard Model and beyond, Eur. Phys. J. C 74 (2014) 3001 [arXiv:1404.7096] [INSPIRE].
S. Actis, A. Denner, L. Hofer, J.-N. Lang, A. Scharf and S. Uccirati, RECOLA: REcursive Computation of One-Loop Amplitudes, Comput. Phys. Commun. 214 (2017) 140 [arXiv:1605.01090] [INSPIRE].
J. Alwall et al., The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations, JHEP 07 (2014) 079 [arXiv:1405.0301] [INSPIRE].
G. Heinrich, Collider Physics at the Precision Frontier, arXiv:2009.00516 [INSPIRE].
D. E. Soper, QCD calculations by numerical integration, Nucl. Phys. B Proc. Suppl. 79 (1999) 444.
D. E. Soper, Techniques for QCD calculations by numerical integration, Phys. Rev. D 62 (2000) 014009 [hep-ph/9910292] [INSPIRE].
T. Binoth and G. Heinrich, An automatized algorithm to compute infrared divergent multiloop integrals, Nucl. Phys. B 585 (2000) 741 [hep-ph/0004013] [INSPIRE].
D. E. Soper, Choosing integration points for QCD calculations by numerical integration, Phys. Rev. D 64 (2001) 034018 [hep-ph/0103262] [INSPIRE].
M. Krämer and D. E. Soper, Next-to-leading order numerical calculations in Coulomb gauge, Phys. Rev. D 66 (2002) 054017 [hep-ph/0204113] [INSPIRE].
S. Becker, C. Reuschle and S. Weinzierl, Numerical NLO QCD calculations, JHEP 12 (2010) 013 [arXiv:1010.4187] [INSPIRE].
S. Becker, C. Reuschle and S. Weinzierl, Efficiency Improvements for the Numerical Computation of NLO Corrections, JHEP 07 (2012) 090 [arXiv:1205.2096] [INSPIRE].
R. Pittau, A four-dimensional approach to quantum field theories, JHEP 11 (2012) 151 [arXiv:1208.5457] [INSPIRE].
A. M. Donati and R. Pittau, Gauge invariance at work in FDR: H → γγ, JHEP 04 (2013) 167 [arXiv:1302.5668] [INSPIRE].
R. A. Fazio, P. Mastrolia, E. Mirabella and W. J. Torres Bobadilla, On the Four-Dimensional Formulation of Dimensionally Regulated Amplitudes, Eur. Phys. J. C 74 (2014) 3197 [arXiv:1404.4783] [INSPIRE].
A. Primo and W. J. Torres Bobadilla, BCJ Identities and d-Dimensional Generalized Unitarity, JHEP 04 (2016) 125 [arXiv:1602.03161] [INSPIRE].
P. Mastrolia, A. Primo, U. Schubert and W. J. Torres Bobadilla, Off-shell currents and color-kinematics duality, Phys. Lett. B 753 (2016) 242 [arXiv:1507.07532] [INSPIRE].
R. J. Hernandez-Pinto, G. F. R. Sborlini and G. Rodrigo, Towards gauge theories in four dimensions, JHEP 02 (2016) 044 [arXiv:1506.04617] [INSPIRE].
G. F. R. Sborlini, F. Driencourt-Mangin, R. Hernandez-Pinto and G. Rodrigo, Four-dimensional unsubtraction from the loop-tree duality, JHEP 08 (2016) 160 [arXiv:1604.06699] [INSPIRE].
G. F. R. Sborlini, F. Driencourt-Mangin and G. Rodrigo, Four-dimensional unsubtraction with massive particles, JHEP 10 (2016) 162 [arXiv:1608.01584] [INSPIRE].
C. Gnendiger et al., To d, or not to d: recent developments and comparisons of regularization schemes, Eur. Phys. J. C 77 (2017) 471 [arXiv:1705.01827] [INSPIRE].
Z. Capatti, V. Hirschi, A. Pelloni and B. Ruijl, Local Unitarity: a representation of differential cross-sections that is locally free of infrared singularities at any order, JHEP 04 (2021) 104 [arXiv:2010.01068] [INSPIRE].
W. J. Torres Bobadilla et al., May the four be with you: Novel IR-subtraction methods to tackle NNLO calculations, Eur. Phys. J. C 81 (2021) 250 [arXiv:2012.02567] [INSPIRE].
R. M. Prisco and F. Tramontano, Dual Subtractions, arXiv:2012.05012 [INSPIRE].
H. A. Chawdhry, M. L. Czakon, A. Mitov and R. Poncelet, NNLO QCD corrections to three-photon production at the LHC, JHEP 02 (2020) 057 [arXiv:1911.00479] [INSPIRE].
F. Caola, A. Von Manteuffel and L. Tancredi, Diphoton Amplitudes in Three-Loop Quantum Chromodynamics, Phys. Rev. Lett. 126 (2021) 112004 [arXiv:2011.13946] [INSPIRE].
S. Kallweit, V. Sotnikov and M. Wiesemann, Triphoton production at hadron colliders in NNLO QCD, Phys. Lett. B 812 (2021) 136013 [arXiv:2010.04681] [INSPIRE].
S. Badger, H. B. Hartanto and S. Zoia, Two-loop QCD corrections to \( Wb\overline{b} \) production at hadron colliders, arXiv:2102.02516 [INSPIRE].
B. Agarwal, F. Buccioni, A. von Manteuffel and L. Tancredi, Two-loop leading colour QCD corrections to \( q\overline{q} \) → γγg and qg → γγq, arXiv:2102.01820 [INSPIRE].
K. G. Chetyrkin and F. V. Tkachov, Integration by Parts: The Algorithm to Calculate β-functions in 4 Loops, Nucl. Phys. B 192 (1981) 159 [INSPIRE].
A. V. Kotikov, Differential equation method: The Calculation of N point Feynman diagrams, Phys. Lett. B 267 (1991) 123 [Erratum ibid. 295 (1992) 409] [INSPIRE].
S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].
A. von Manteuffel and C. Studerus, Reduze 2 — Distributed Feynman Integral Reduction, arXiv:1201.4330 [INSPIRE].
J. M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].
M. Argeri et al., Magnus and Dyson Series for Master Integrals, JHEP 03 (2014) 082 [arXiv:1401.2979] [INSPIRE].
P. Maierhöfer, J. Usovitsch and P. Uwer, Kira — A Feynman integral reduction program, Comput. Phys. Commun. 230 (2018) 99 [arXiv:1705.05610] [INSPIRE].
S. Borowka, G. Heinrich, S. P. Jones, M. Kerner, J. Schlenk and T. Zirke, SecDec-3.0: numerical evaluation of multi-scale integrals beyond one loop, Comput. Phys. Commun. 196 (2015) 470 [arXiv:1502.06595] [INSPIRE].
A. V. Smirnov, FIESTA4: Optimized Feynman integral calculations with GPU support, Comput. Phys. Commun. 204 (2016) 189 [arXiv:1511.03614] [INSPIRE].
K. J. Larsen and Y. Zhang, Integration-by-parts reductions from unitarity cuts and algebraic geometry, Phys. Rev. D 93 (2016) 041701 [arXiv:1511.01071] [INSPIRE].
P. Mastrolia and S. Mizera, Feynman Integrals and Intersection Theory, JHEP 02 (2019) 139 [arXiv:1810.03818] [INSPIRE].
S. Abreu et al., Caravel: A C++ Framework for the Computation of Multi-Loop Amplitudes with Numerical Unitarity, arXiv:2009.11957 [INSPIRE].
H. Ita, Two-loop Integrand Decomposition into Master Integrals and Surface Terms, Phys. Rev. D 94 (2016) 116015 [arXiv:1510.05626] [INSPIRE].
S. Catani, T. Gleisberg, F. Krauss, G. Rodrigo and J.-C. Winter, From loops to trees by-passing Feynman’s theorem, JHEP 09 (2008) 065 [arXiv:0804.3170] [INSPIRE].
I. Bierenbaum, S. Catani, P. Draggiotis and G. Rodrigo, A Tree-Loop Duality Relation at Two Loops and Beyond, JHEP 10 (2010) 073 [arXiv:1007.0194] [INSPIRE].
I. Bierenbaum, S. Buchta, P. Draggiotis, I. Malamos and G. Rodrigo, Tree-Loop Duality Relation beyond simple poles, JHEP 03 (2013) 025 [arXiv:1211.5048] [INSPIRE].
S. Buchta, G. Chachamis, P. Draggiotis, I. Malamos and G. Rodrigo, On the singular behaviour of scattering amplitudes in quantum field theory, JHEP 11 (2014) 014 [arXiv:1405.7850] [INSPIRE].
S. Buchta, G. Chachamis, P. Draggiotis and G. Rodrigo, Numerical implementation of the loop-tree duality method, Eur. Phys. J. C 77 (2017) 274 [arXiv:1510.00187] [INSPIRE].
C. Bogner and S. Weinzierl, Feynman graph polynomials, Int. J. Mod. Phys. A 25 (2010) 2585 [arXiv:1002.3458] [INSPIRE].
J. L. Jurado, G. Rodrigo and W. J. Torres Bobadilla, From Jacobi off-shell currents to integral relations, JHEP 12 (2017) 122 [arXiv:1710.11010] [INSPIRE].
F. Driencourt-Mangin, G. Rodrigo and G. F. R. Sborlini, Universal dual amplitudes and asymptotic expansions for gg → H and H → γγ in four dimensions, Eur. Phys. J. C 78 (2018) 231 [arXiv:1702.07581] [INSPIRE].
F. Driencourt-Mangin, G. Rodrigo, G. F. R. Sborlini and W. J. Torres Bobadilla, Universal four-dimensional representation of H → γγ at two loops through the Loop-Tree Duality, JHEP 02 (2019) 143 [arXiv:1901.09853] [INSPIRE].
F. Driencourt-Mangin, G. Rodrigo, G. F. R. Sborlini and W. J. Torres Bobadilla, On the interplay between the loop-tree duality and helicity amplitudes, arXiv:1911.11125 [INSPIRE].
J. Plenter and G. Rodrigo, Asymptotic expansions through the loop-tree duality, arXiv:2005.02119 [INSPIRE].
J. J. Aguilera-Verdugo et al., Causality, unitarity thresholds, anomalous thresholds and infrared singularities from the loop-tree duality at higher orders, JHEP 12 (2019) 163 [arXiv:1904.08389] [INSPIRE].
J. J. Aguilera-Verdugo et al., Open Loop Amplitudes and Causality to All Orders and Powers from the Loop-Tree Duality, Phys. Rev. Lett. 124 (2020) 211602 [arXiv:2001.03564] [INSPIRE].
J. J. Aguilera-Verdugo, R. J. Hernandez-Pinto, G. Rodrigo, G. F. R. Sborlini and W. J. Torres Bobadilla, Causal representation of multi-loop Feynman integrands within the loop-tree duality, JHEP 01 (2021) 069 [arXiv:2006.11217] [INSPIRE].
S. Ramírez-Uribe, R. J. Hernández-Pinto, G. Rodrigo, G. F. R. Sborlini and W. J. Torres Bobadilla, Universal opening of four-loop scattering amplitudes to trees, JHEP 04 (2021) 129 [arXiv:2006.13818] [INSPIRE].
J. Jesús Aguilera-Verdugo, R. J. Hernández-Pinto, G. Rodrigo, G. F. R. Sborlini and W. J. Torres Bobadilla, Mathematical properties of nested residues and their application to multi-loop scattering amplitudes, JHEP 02 (2021) 112 [arXiv:2010.12971] [INSPIRE].
E. T. Tomboulis, Causality and Unitarity via the Tree-Loop Duality Relation, JHEP 05 (2017) 148 [arXiv:1701.07052] [INSPIRE].
R. Runkel, Z. Szőr, J. P. Vesga and S. Weinzierl, Integrands of loop amplitudes within loop-tree duality, Phys. Rev. D 101 (2020) 116014 [arXiv:1906.02218] [INSPIRE].
R. Runkel, Z. Szőr, J. P. Vesga and S. Weinzierl, Causality and loop-tree duality at higher loops, Phys. Rev. Lett. 122 (2019) 111603 [Erratum ibid. 123 (2019) 059902] [arXiv:1902.02135] [INSPIRE].
Z. Capatti, V. Hirschi, D. Kermanschah, A. Pelloni and B. Ruijl, Manifestly Causal Loop-Tree Duality, arXiv:2009.05509 [INSPIRE].
Z. Capatti, V. Hirschi, D. Kermanschah, A. Pelloni and B. Ruijl, Numerical Loop-Tree Duality: contour deformation and subtraction, JHEP 04 (2020) 096 [arXiv:1912.09291] [INSPIRE].
Z. Capatti, V. Hirschi, D. Kermanschah and B. Ruijl, Loop-Tree Duality for Multiloop Numerical Integration, Phys. Rev. Lett. 123 (2019) 151602 [arXiv:1906.06138] [INSPIRE].
L. de la Cruz, A scattering amplitudes approach to hard thermal loops, arXiv:2012.07714 [INSPIRE].
O. Steinmann, Über den Zusammenhang Zwischen den Wightmanfunktionen und den Retardierten Kommutatoren, Helv. Phys. Acta 33 (1960) 257.
O. Steinmann, Wightman-Funktionen und Retardierte Kommutatoren. II, Helv. Phys. Acta 33 (1960) 347.
H. Araki, Generalized Retarded Functions and Analytic Function in Momentum Space in Quantum Field Theory, J. Math. Phys. 2 (1961) 163.
D. Ruelle, Connection between Wightman Functions and Green Functions in p-Space, Nuovo Cim. 19 (1961) 356.
H. P. Stapp, Inclusive cross-sections are discontinuities, Phys. Rev. D 3 (1971) 3177 [INSPIRE].
M. Lassalle, Analyticity Properties Implied by the Many-Particle Structure of the N Point Function in General Quantum Field Theory. 1. Convolution of n Point Functions Associated with a Graph, Commun. Math. Phys. 36 (1974) 185 [INSPIRE].
K. E. Cahill and H. P. Stapp, Optical theorems and steinmann relations, Annals Phys. 90 (1975) 438 [INSPIRE].
S. Caron-Huot, L. J. Dixon, A. McLeod and M. von Hippel, Bootstrapping a Five-Loop Amplitude Using Steinmann Relations, Phys. Rev. Lett. 117 (2016) 241601 [arXiv:1609.00669] [INSPIRE].
S. Caron-Huot, L. J. Dixon, F. Dulat, M. Von Hippel, A. J. McLeod and G. Papathanasiou, The Cosmic Galois Group and Extended Steinmann Relations for Planar \( \mathcal{N} \) = 4 SYM Amplitudes, JHEP 09 (2019) 061 [arXiv:1906.07116] [INSPIRE].
P. Benincasa, A. J. McLeod and C. Vergu, Steinmann Relations and the Wavefunction of the Universe, Phys. Rev. D 102 (2020) 125004 [arXiv:2009.03047] [INSPIRE].
S. Caron-Huot et al., The Steinmann Cluster Bootstrap for N = 4 Super Yang-Mills Amplitudes, PoS CORFU2019 (2020) 003 [arXiv:2005.06735] [INSPIRE].
J. L. Bourjaily, H. Hannesdottir, A. J. McLeod, M. D. Schwartz and C. Vergu, Sequential Discontinuities of Feynman Integrals and the Monodromy Group, JHEP 01 (2021) 205 [arXiv:2007.13747] [INSPIRE].
A. von Manteuffel and R. M. Schabinger, A novel approach to integration by parts reduction, Phys. Lett. B 744 (2015) 101 [arXiv:1406.4513] [INSPIRE].
T. Peraro, Scattering amplitudes over finite fields and multivariate functional reconstruction, JHEP 12 (2016) 030 [arXiv:1608.01902] [INSPIRE].
T. Peraro, FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs, JHEP 07 (2019) 031 [arXiv:1905.08019] [INSPIRE].
J. Klappert and F. Lange, Reconstructing rational functions with FireFly, Comput. Phys. Commun. 247 (2020) 106951 [arXiv:1904.00009] [INSPIRE].
M. Heller and A. von Manteuffel, MultivariateApart: Generalized Partial Fractions, arXiv:2101.08283 [INSPIRE].
W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann, Singular 4-2-0 — A computer algebra system for polynomial computations, http://www.singular.uni-kl.de (2020).
J. Boehm, M. Wittmann, Z. Wu, Y. Xu and Y. Zhang, IBP reduction coefficients made simple, JHEP 12 (2020) 054 [arXiv:2008.13194] [INSPIRE].
W. J. Torres Bobadilla, Lotty — The loop-tree duality automation, arXiv:2103.09237 [INSPIRE].
G. F. R. Sborlini, A geometrical approach to causality in multi-loop amplitudes, arXiv:2102.05062 [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: 2102.05048
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
Torres Bobadilla, W.J. Loop-tree duality from vertices and edges. J. High Energ. Phys. 2021, 183 (2021). https://doi.org/10.1007/JHEP04(2021)183
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2021)183