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Causal representation of multi-loop Feynman integrands within the loop-tree duality. (English) Zbl 1459.81044
Summary: The numerical evaluation of multi-loop scattering amplitudes in the Feynman representation usually requires to deal with both physical (causal) and unphysical (non-causal) singularities. The loop-tree duality (LTD) offers a powerful framework to easily characterise and distinguish these two types of singularities, and then simplify analytically the underling expressions. In this paper, we work explicitly on the dual representation of multi-loop Feynman integrals generated from three parent topologies, which we refer to as Maximal, Next-to-Maximal and Next-to-Next-to-Maximal loop topologies. In particular, we aim at expressing these dual contributions, independently of the number of loops and internal configurations, in terms of causal propagators only. Thus, providing very compact and causal integrand representations to all orders. In order to do so, we reconstruct their analytic expressions from numerical evaluation over finite fields. This procedure implicitly cancels out all unphysical singularities. We also interpret the result in terms of entangled causal thresholds. In view of the simple structure of the dual expressions, we integrate them numerically up to four loops in integer space-time dimensions, taking advantage of their smooth behaviour at integrand level.

81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
81T18 Feynman diagrams
81V05 Strong interaction, including quantum chromodynamics
81U05 \(2\)-body potential quantum scattering theory
81P42 Entanglement measures, concurrencies, separability criteria
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[1] Mangano, M., LHC at 10: the physics legacy, CERN Cour., 60, 40 (2020)
[2] FCC collaboration, FCC physics opportunities: Future Circular Collider conceptual design report volume 1, Eur. Phys. J. C79 (2019) 474 [INSPIRE].
[3] FCC collaboration, FCC-ee: the lepton collider: Future Circular Collider conceptual design report volume 2, Eur. Phys. J. ST228 (2019) 261 [INSPIRE].
[4] FCC collaboration, FCC-hh: the hadron collider: Future Circular Collider conceptual design report volume 3, Eur. Phys. J. ST228 (2019) 755 [INSPIRE].
[5] FCC collaboration, HE-LHC: the High-Energy Large Hadron Collider: Future Circular Collider conceptual design report volume 4, Eur. Phys. J. ST228 (2019) 1109 [INSPIRE].
[6] A. Blondel et al., Theory for the FCC-ee: Report on the 11^thFCC-ee Workshop Theory and Experiments, CYRM-2020-003 (2019).
[7] P. Bambade et al., The International Linear Collider: a global project, arXiv:1903.01629 [INSPIRE].
[8] CLIC and CLICdp collaborations, The Compact Linear e^+e^−Collider (CLIC): physics potential, arXiv:1812.07986 [INSPIRE].
[9] CEPC Study Group collaboration, CEPC conceptual design report: volume 2 — Physics & detector, arXiv:1811.10545 [INSPIRE].
[10] Bollini, CG; Giambiagi, JJ, Dimensional renormalization: the number of dimensions as a regularizing parameter, Nuovo Cim. B, 12, 20 (1972)
[11] G. ’t Hooft and M.J.G. Veltman, Regularization and renormalization of gauge fields, Nucl. Phys. B44 (1972) 189 [INSPIRE].
[12] Gnendiger, C., To d, or not to d: recent developments and comparisons of regularization schemes, Eur. Phys. J. C, 77, 471 (2017)
[13] Landau, LD, On analytic properties of vertex parts in quantum field theory, Nucl. Phys., 13, 181 (1960) · Zbl 0088.22004
[14] Catani, S.; Gleisberg, T.; Krauss, F.; Rodrigo, G.; Winter, J-C, From loops to trees by-passing Feynman’s theorem, JHEP, 09, 065 (2008) · Zbl 1245.81117
[15] Bierenbaum, I.; Catani, S.; Draggiotis, P.; Rodrigo, G., A tree-loop duality relation at two loops and beyond, JHEP, 10, 073 (2010) · Zbl 1291.81381
[16] Bierenbaum, I.; Buchta, S.; Draggiotis, P.; Malamos, I.; Rodrigo, G., Tree-loop duality relation beyond simple poles, JHEP, 03, 025 (2013)
[17] Buchta, S.; Chachamis, G.; Draggiotis, P.; Malamos, I.; Rodrigo, G., On the singular behaviour of scattering amplitudes in quantum field theory, JHEP, 11, 014 (2014) · Zbl 1333.81149
[18] Buchta, S.; Chachamis, G.; Draggiotis, P.; Rodrigo, G., Numerical implementation of the loop-tree duality method, Eur. Phys. J. C, 77, 274 (2017)
[19] Aguilera-Verdugo, JJ, Causality, unitarity thresholds, anomalous thresholds and infrared singularities from the loop-tree duality at higher orders, JHEP, 12, 163 (2019) · Zbl 1431.81156
[20] Hernandez-Pinto, RJ; Sborlini, GFR; Rodrigo, G., Towards gauge theories in four dimensions, JHEP, 02, 044 (2016) · Zbl 1388.81329
[21] Sborlini, GFR; Driencourt-Mangin, F.; Hernandez-Pinto, R.; Rodrigo, G., Four-dimensional unsubtraction from the loop-tree duality, JHEP, 08, 160 (2016)
[22] Sborlini, GFR; Driencourt-Mangin, F.; Rodrigo, G., Four-dimensional unsubtraction with massive particles, JHEP, 10, 162 (2016)
[23] Fazio, RA; Mastrolia, P.; Mirabella, E.; Torres Bobadilla, WJ, On the four-dimensional formulation of dimensionally regulated amplitudes, Eur. Phys. J. C, 74, 3197 (2014)
[24] W.J. Torres Bobadilla, Generalised unitarity for dimensionally regulated amplitudes within FDF, PoS(RADCOR2015)063 [arXiv:1601.05742] [INSPIRE].
[25] Baeta Scarpelli, AP; Sampaio, M.; Hiller, B.; Nemes, MC, Chiral anomaly and CPT invariance in an implicit momentum space regularization framework, Phys. Rev. D, 64 (2001)
[26] Baeta Scarpelli, AP; Sampaio, M.; Nemes, MC, Consistency relations for an implicit n-dimensional regularization scheme, Phys. Rev. D, 63 (2001)
[27] Pittau, R., A four-dimensional approach to quantum field theories, JHEP, 11, 151 (2012) · Zbl 1397.81177
[28] Page, B.; Pittau, R., NNLO final-state quark-pair corrections in four dimensions, Eur. Phys. J. C, 79, 361 (2019)
[29] Gnendiger, C.; Signer, A., γ_5in the four-dimensional helicity scheme, Phys. Rev. D, 97 (2018)
[30] A.M. Bruque, A.L. Cherchiglia and M. Pérez-Victoria, Dimensional regularization vs. methods in fixed dimension with and without γ_5, JHEP08 (2018) 109 [arXiv:1803.09764] [INSPIRE]. · Zbl 1396.83046
[31] Pozzorini, S.; Zhang, H.; Zoller, MF, Rational terms of UV origin at two loops, JHEP, 05, 077 (2020)
[32] Tomboulis, ET, Causality and unitarity via the tree-loop duality relation, JHEP, 05, 148 (2017) · Zbl 1396.81184
[33] 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].
[34] Runkel, R.; Szőr, Z.; Vesga, JP; Weinzierl, S., Integrands of loop amplitudes within loop-tree duality, Phys. Rev. D, 101, 116014 (2020)
[35] Capatti, Z.; Hirschi, V.; Kermanschah, D.; Ruijl, B., Loop-tree duality for multiloop numerical integration, Phys. Rev. Lett., 123, 151602 (2019)
[36] S. Buchta, Theoretical foundations and applications of the loop-tree duality in quantum field theories, Ph.D. thesis, Valencia University, Valencia, Spain (2015), arXiv:1509.07167 [INSPIRE].
[37] Capatti, Z.; Hirschi, V.; Kermanschah, D.; Pelloni, A.; Ruijl, B., Numerical loop-tree duality: contour deformation and subtraction, JHEP, 04, 096 (2020) · Zbl 1436.81144
[38] Driencourt-Mangin, F.; Rodrigo, G.; Sborlini, GFR, Universal dual amplitudes and asymptotic expansions for gg → H and H → γγ in four dimensions, Eur. Phys. J. C, 78, 231 (2018)
[39] Jurado, JL; Rodrigo, G.; Torres Bobadilla, WJ, From Jacobi off-shell currents to integral relations, JHEP, 12, 122 (2017)
[40] Driencourt-Mangin, F.; Rodrigo, G.; Sborlini, GFR; Torres Bobadilla, WJ, Universal four-dimensional representation of H → γγ at two loops through the loop-tree duality, JHEP, 02, 143 (2019)
[41] F. Driencourt-Mangin, Four-dimensional representation of scattering amplitudes and physical observables through the application of the Loop-Tree Duality theorem, Ph.D. thesis, Valencia University, Valencia, Spain (2019), arXiv:1907.12450 [INSPIRE].
[42] 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]. · Zbl 1431.81156
[43] Plenter, J., Asymptotic expansions through the loop-tree duality, Acta Phys. Polon. B, 50, 1983 (2019)
[44] J. Plenter and G. Rodrigo, Asymptotic expansions through the loop-tree duality, arXiv:2005.02119 [INSPIRE].
[45] M. Beneke and V.A. Smirnov, Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys. B522 (1998) 321 [hep-ph/9711391] [INSPIRE].
[46] Aguilera-Verdugo, JJ, Open loop amplitudes and causality to all orders and powers from the loop-tree duality, Phys. Rev. Lett., 124, 211602 (2020)
[47] 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, arXiv:2006.13818 [INSPIRE].
[48] von Manteuffel, A.; Schabinger, RM, A novel approach to integration by parts reduction, Phys. Lett. B, 744, 101 (2015) · Zbl 1330.81151
[49] Peraro, T., Scattering amplitudes over finite fields and multivariate functional reconstruction, JHEP, 12, 030 (2016) · Zbl 1390.81631
[50] Chetyrkin, KG; Tkachov, FV, Integration by parts: the algorithm to calculate β-functions in 4 loops, Nucl. Phys. B, 192, 159 (1981)
[51] S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A15 (2000) 5087 [hep-ph/0102033] [INSPIRE]. · Zbl 0973.81082
[52] Peraro, T., FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs, JHEP, 07, 031 (2019)
[53] Hepp, K., Proof of the Bogolyubov-Parasiuk theorem on renormalization, Commun. Math. Phys., 2, 301 (1966) · Zbl 1222.81219
[54] M. Roth and A. Denner, High-energy approximation of one loop Feynman integrals, Nucl. Phys. B479 (1996) 495 [hep-ph/9605420] [INSPIRE].
[55] T. Binoth and G. Heinrich, An automatized algorithm to compute infrared divergent multiloop integrals, Nucl. Phys. B585 (2000) 741 [hep-ph/0004013] [INSPIRE]. · Zbl 1042.81565
[56] Heinrich, G., Sector decomposition, Int. J. Mod. Phys. A, 23, 1457 (2008) · Zbl 1153.81522
[57] S. Borowka et al., SecDec-3.0: numerical evaluation of multi-scale integrals beyond one loop, Comput. Phys. Commun.196 (2015) 470 [arXiv:1502.06595] [INSPIRE]. · Zbl 1360.81013
[58] Smirnov, AV, FIESTA4: optimized Feynman integral calculations with GPU support, Comput. Phys. Commun., 204, 189 (2016) · Zbl 1378.65075
[59] J.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 multi-loop scattering amplitudes, arXiv:2010.12971 [INSPIRE].
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