Geometry and causal flux in multi-loop Feynman diagrams
DOI:
https://doi.org/10.31349/SuplRevMexFis.3.020703Keywords:
Perturbative QFT, Scattering amplitudes, Higher-orders, Loop calculationsAbstract
In this review, we discuss recent developments concerning efficient calculations of multi-loop multi-leg scattering amplitudes. Inspired by the remarkable properties of the Loop-Tree Duality (LTD), we explain how to reconstruct an integrand level representation of scattering amplitudes which only contains physical singularities. These so-called causal representations can be derived from connected binary partitions of Feynman diagrams, properly entangled according to specific rules. We will focus on the detection of flux orientations which are compatible with causality, describing the implementation of a quantum algorithm to identify such configurations.
References
A. Abada et al., (The FCC Collab.), FCC Physics Opportunities: Future Circular Collider Conceptual Design Report Volume 1, Eur. Phys. J. C 79 (2019) 474, http://dx.doi.org/10.1140/epjc/s10052-019-6904-3
M. Dong et al., CEPC Conceptual Design Report: Volume 2 - Physics & Detector, http://arxiv.org/abs/1811.10545
J. F. Ashmore, A Method of Gauge Invariant Regularization, Lett. Nuovo Cim. 4 (1972) 289, http://dx.doi.org/10.1007/BF02824407
G. M. Cicuta and E. Montaldi, Analytic renormalization via continuous space dimension, Lett. Nuovo Cim. 4 (1972) 329, http://dx.doi.org/10.1007/BF02756527.
G. ’t Hooft and M. J. G. Veltman, Regularization and Renormalization of Gauge Fields, Nucl. Phys. B 44 (1972) 189, http://dx.doi.org/10.1016/0550-3213(72)90279-9.
C. Gnendiger et al., To d, or not to d: recent developments and comparisons of regularization schemes, Eur. Phys. J. C 77 (2017) 471, http://dx.doi.org/10.1140/epjc/s10052-017-5023-2.
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, http://dx.doi.org/10.1140/epjc/s10052-021-08996-y.
T. Kinoshita, Mass singularities of Feynman amplitudes, J. Math. Phys. 3 (1962) 650, http://dx.doi.org/10.1063/1.1724268.
T. D. Lee and M. Nauenberg, Degenerate Systems and Mass Singularities, Phys. Rev. 133 (1964) B1549, http://dx.doi.org/10.1103/PhysRev.133.B1549.
G. Altarelli and G. Parisi, Asymptotic Freedom in Parton Language, Nucl. Phys. B 126 (1977) 298, http://dx.doi.org/10.1016/0550-3213(77)90384-4.
G. F. R. Sborlini, D. de Florian and G. Rodrigo, Double collinear splitting amplitudes at next-to-leading order, JHEP 01 (2014) 018, http://dx.doi.org/10.1007/JHEP01(2014)018.
G. F. R. Sborlini, D. de Florian and G. Rodrigo, Triple collinear splitting functions at NLO for scattering processes with photons, JHEP 10 (2014) 161, http://dx.doi.org/10.1007/JHEP10(2014)161.
G. F. R. Sborlini, D. de Florian and G. Rodrigo, Polarized triple-collinear splitting functions at NLO for processes with photons, JHEP 03 (2015) 021, http://dx.doi.org/10.1007/JHEP03(2015)021.
J. C. Collins, Renormalization: an introduction to renormalization, the renormalization group, and the operator-product expansion. Cambridge monographs on mathematical physics. Cambridge Univ. Press, Cambridge, 1984.
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, http://dx.doi.org/10.1088/1126-6708/2008/09/065.
G. Rodrigo, S. Catani, T. Gleisberg, F. Krauss and J.-C. Winter, From multileg loops to trees (by-passing Feynman’s Tree Theorem), Nucl. Phys. B Proc. Suppl. 183 (2008) 262, http://dx.doi.org/10.1016/j.nuclphysbps.2008.09.114
I. Bierenbaum, S. Catani, P. Draggiotis and G. Rodrigo, A Tree-Loop Duality Relation at Two Loops and Beyond, JHEP 10 (2010) 073, http://dx.doi.org/10.1007/JHEP10(2010)073
I. Bierenbaum, S. Buchta, P. Draggiotis, I. Malamos and G. Rodrigo, Tree-Loop Duality Relation beyond simple poles, JHEP 03 (2013) 025, http://dx.doi.org/10.1007/JHEP03(2013)025
J. Aguilera-Verdugo et al., A Stroll through the Loop-Tree Duality, Symmetry 13 (2021) 1029, http://dx.doi.org/10.3390/sym13061029
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, http://dx.doi.org/10.1007/JHEP11(2014)014
R. J. Hernandez-Pinto, G. F. R. Sborlini and G. Rodrigo, Towards gauge theories in four dimensions, JHEP 02 (2016) 044, http://dx.doi.org/10.1007/JHEP02(2016)044
G. F. R. Sborlini, F. Driencourt-Mangin, R. Hernandez-Pinto and G. Rodrigo, Four-dimensional unsubtraction from the looptree duality, JHEP 08 (2016) 160, http://dx.doi.org/10.1007/JHEP08(2016)160
G. F. R. Sborlini, F. Driencourt-Mangin and G. Rodrigo, Four-dimensional unsubtraction with massive particles, JHEP 10 (2016) 162, http://dx.doi.org/10.1007/JHEP10(2016)162
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, http://dx.doi.org/10.1007/JHEP02(2019)143
F. Driencourt-Mangin, G. Rodrigo and G. F. Sborlini, Universal dual amplitudes and asymptotic expansions for gg → H and H → γγ in four dimensions, Eur. Phys. J. C 78 (2018) 231, http://dx.doi.org/10.1140/epjc/s10052-018-5692-5
J. Plenter, Asymptotic Expansions Through the Loop-Tree Duality, Acta Phys. Polon. B 50 (2019) 1983. http://dx.doi.org/10.5506/APhysPolB.50.1983
J. Plenter and G. Rodrigo, Asymptotic expansions through the loop-tree duality, Eur. Phys. J. C 81 (2021) 320, http://dx.doi.org/10.1140/epjc/s10052-021-09094-9
J. L. Jurado, G. Rodrigo and W. J. Torres Bobadilla, From Jacobi off-shell currents to integral relations, JHEP 12 (2017) 122, http://dx.doi.org/10.1007/JHEP12(2017)122
S. Buchta, G. Chachamis, P. Draggiotis and G. Rodrigo, Numerical implementation of the loop–tree duality method, Eur. Phys. J. C77 (2017) 274, http://dx.doi.org/10.1140/epjc/s10052-017-4833-6.
S. Buchta, Theoretical foundations and applications of the Loop-Tree Duality in Quantum Field Theories. PhD thesis, Valencia U., 2015. http://arxiv.org/abs/1509.07167.
J. J. Aguilera-Verdugo et al., Causality, unitarity thresholds, anomalous thresholds and infrared singularities from the looptree duality at higher orders, JHEP 12 (2019) 163, http://dx.doi.org/10.1007/JHEP12(2019)163.
F. Driencourt-Mangin, G. Rodrigo, G. F. Sborlini and W. J. Torres Bobadilla, On the interplay between the loop-tree duality and helicity amplitudes, http://arxiv.org/abs/1911.11125.
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, http://dx.doi.org/10.1103/PhysRevLett.124.211602
R. Runkel, Z. Szor, J. P. Vesga and S. Weinzierl, Causality and loop-tree duality at higher loops, Phys. Rev. Lett. 122 (2019) 111603, http://dx.doi.org/10.1103/PhysRevLett.122.111603.
R. Runkel, Z. Szor, J. P. Vesga and S. Weinzierl, Integrands of loop amplitudes within loop-tree duality, Phys. Rev. D 101 (2020) 116014, http://dx.doi.org/10.1103/PhysRevD.101.116014
Z. Capatti, V. Hirschi, D. Kermanschah, A. Pelloni and B. Ruijl, Numerical Loop-Tree Duality: contour deformation and subtraction, JHEP 04 (2020) 096, http://dx.doi.org/10.1007/JHEP04(2020)096
Z. Capatti, V. Hirschi, D. Kermanschah and B. Ruijl, LoopTree Duality for Multiloop Numerical Integration, Phys. Rev. Lett. 123 (2019) 151602, http://dx.doi.org/10.1103/PhysRevLett.123.151602.
Z. Capatti, V. Hirschi, D. Kermanschah, A. Pelloni and B. Ruijl, Manifestly Causal Loop-Tree Duality, http://arxiv.org/abs/2009.05509
J. Aguilera-Verdugo, R. J. Hernandez-Pinto, G. Rodrigo, G. F. R. Sborlini and W. J. Torres Bobadilla, Mathematical properties of nested residues and their application to multiloop scattering amplitudes, JHEP 02 (2021) 112, http://dx.doi.org/10.1007/JHEP02(2021)112
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, http://dx.doi.org/10.1007/JHEP01(2021)069
G. F. R. Sborlini, Geometry and causality for efficient multiloop representations, in 15th International Symposium on Radiative Corrections: Applications of Quantum Field Theory to Phenomenology and LoopFest XIX: Workshop on Radiative Corrections for the LHC and Future Colliders, September 2021. http://arxiv.org/abs/2109.07808.
E. Tomboulis, Causality and Unitarity via the Tree-Loop Duality Relation, JHEP 05 (2017) 148, http://dx.doi.org/10.1007/JHEP05(2017)148
J. Aguilera-Verdugo et al., Manifestly Causal Scattering Amplitudes, in Snowmass 2021 - Letter of Intention, August 2020.
W. J. Torres Bobadilla, Loop-tree duality from vertices and edges, JHEP 04 (2021) 183, http://dx.doi.org/10.1007/JHEP04(2021)183
G. F. R. Sborlini, Geometrical approach to causality in multiloop amplitudes, Phys. Rev. D 104 (2021) 036014, http://dx.doi.org/10.1103/PhysRevD.104.036014
W. J. Torres Bobadilla, Lotty The loop-tree duality automation, Eur. Phys. J. C 81 (2021) 514, http://dx.doi.org/10.1140/epjc/s10052-021-09235-0
S. Ram´ırez-Uribe, R. J. Hernandez-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, http://dx.doi.org/10.1007/JHEP04(2021)129
S. Ram´ırez-Uribe, A. E. Renter´ıa-Olivo, G. Rodrigo, G. F. R. Sborlini and L. Vale Silva, Quantum algorithm for Feynman loop integrals, http://arxiv.org/abs/2105.087032105.08703.
R. E. Cutkosky, Singularities and discontinuities of Feynman amplitudes, J. Math. Phys. 1 (1960) 429, http://dx.doi.org/10.1063/1.1703676
S. Even and G. Even, Graph Algorithms, second edition, (2011) 9780521517188. 10.1017/CBO9781139015165.
L. K. Grover, A fast quantum mechanical algorithm for database search, in Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, STOC ’96, (New York, NY, USA), p. 212, Association for Computing Machinery, 1996. http://dx.doi.org/10.1145/237814.237866DOI.
C. H. Bennett, E. Bernstein, G. Brassard and U. Vazirani, Strengths and weaknesses of quantum computing, SIAM J. Comput. 26 (1997) 1510, http://dx.doi.org/10.1137/S0097539796300933
L. K. Grover, Quantum computers can search rapidly by using almost any transformation, Phys. Rev. Lett. 80 (1998) 4329, http://dx.doi.org/10.1103/PhysRevLett.80.4329
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