Abstract
We present the integrand decomposition of multiloop scattering amplitudes in parallel and orthogonal space-time dimensions, d = d ∥ + d ⊥, being d ∥ the dimension of the parallel space spanned by the legs of the diagrams. When the number n of external legs is n ≤ 4,thecorrespondingrepresentationofmultiloopintegralsexposesasubsetofintegration variables which can be easily integrated away by means of Gegenbauer polynomials orthogonality condition. By decomposing the integration momenta along parallel and orthogonal directions, the polynomial division algorithm is drastically simplified. Moreover, the orthogonality conditions of Gegenbauer polynomials can be suitably applied to integrate the decomposed integrand, yielding the systematic annihilation of spurious terms. Consequently, multiloop amplitudes are expressed in terms of integrals corresponding to irreducible scalar products of loop momenta and external ones. We revisit the one-loop decomposition, which turns out to be controlled by the maximum-cut theorem in different dimensions, and we discuss the integrand reduction of two-loop planar and non-planar integrals up to n = 8 legs, for arbitrary external and internal kinematics. The proposed algorithm extends to all orders in perturbation theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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 U.K. (1984).
D. Kreimer, One loop integrals revisited. 1. The two point functions, Z. Phys. C 54 (1992) 667 [INSPIRE].
D. Kreimer, The two loop three point functions: general massive cases, Phys. Lett. B 292 (1992) 341 [INSPIRE].
A. Czarnecki, U. Kilian and D. Kreimer, New representation of two loop propagator and vertex functions, Nucl. Phys. B 433 (1995) 259 [hep-ph/9405423] [INSPIRE].
A. Frink, U. Kilian and D. Kreimer, New representation of the two loop crossed vertex function, Nucl. Phys. B 488 (1997) 426 [hep-ph/9610285] [INSPIRE].
D. Kreimer, XLOOPS: an introduction to parallel space techniques, Nucl. Instrum. Meth. A 389 (1997) 323 [INSPIRE].
P. Mastrolia and G. Ossola, On the integrand-reduction method for two-loop scattering amplitudes, JHEP 11 (2011) 014 [arXiv:1107.6041] [INSPIRE].
S. Badger, H. Frellesvig and Y. Zhang, Hepta-cuts of two-loop scattering amplitudes, JHEP 04 (2012) 055 [arXiv:1202.2019] [INSPIRE].
Y. Zhang, Integrand-level reduction of loop amplitudes by computational algebraic geometry methods, JHEP 09 (2012) 042 [arXiv:1205.5707] [INSPIRE].
P. Mastrolia, E. Mirabella, G. Ossola and T. Peraro, Scattering amplitudes from multivariate polynomial division, Phys. Lett. B 718 (2012) 173 [arXiv:1205.7087] [INSPIRE].
P. Mastrolia, E. Mirabella, G. Ossola and T. Peraro, Integrand-reduction for two-loop scattering amplitudes through multivariate polynomial division, Phys. Rev. D 87 (2013) 085026 [arXiv:1209.4319] [INSPIRE].
B. Feng and R. Huang, The classification of two-loop integrand basis in pure four-dimension, JHEP 02 (2013) 117 [arXiv:1209.3747] [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, S. Caron-Huot and J. Trnka, The all-loop integrand for scattering amplitudes in planar N = 4 SYM, JHEP 01 (2011) 041 [arXiv:1008.2958] [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka, Local integrals for planar scattering amplitudes, JHEP 06 (2012) 125 [arXiv:1012.6032] [INSPIRE].
P. Mastrolia, E. Mirabella, G. Ossola and T. Peraro, Multiloop integrand reduction for dimensionally regulated amplitudes, Phys. Lett. B 727 (2013) 532 [arXiv:1307.5832] [INSPIRE].
G. Ossola, C.G. Papadopoulos and R. Pittau, Reducing full one-loop amplitudes to scalar integrals at the integrand level, Nucl. Phys. B 763 (2007) 147 [hep-ph/0609007] [INSPIRE].
G. Ossola, C.G. Papadopoulos and R. Pittau, Numerical evaluation of six-photon amplitudes, JHEP 07 (2007) 085 [arXiv:0704.1271] [INSPIRE].
R.K. Ellis, W.T. Giele and Z. Kunszt, A numerical unitarity formalism for evaluating one-loop amplitudes, JHEP 03 (2008) 003 [arXiv:0708.2398] [INSPIRE].
R.K. Ellis, W.T. Giele, Z. Kunszt and K. Melnikov, Masses, fermions and generalized D-dimensional unitarity, Nucl. Phys. B 822 (2009) 270 [arXiv:0806.3467] [INSPIRE].
G. Ossola, C.G. Papadopoulos and R. Pittau, On the rational terms of the one-loop amplitudes, JHEP 05 (2008) 004 [arXiv:0802.1876] [INSPIRE].
P. Mastrolia, G. Ossola, C.G. Papadopoulos and R. Pittau, Optimizing the reduction of one-loop amplitudes, JHEP 06 (2008) 030 [arXiv:0803.3964] [INSPIRE].
R.K. Ellis, Z. Kunszt, K. Melnikov and G. Zanderighi, One-loop calculations in quantum field theory: from Feynman diagrams to unitarity cuts, Phys. Rept. 518 (2012) 141 [arXiv:1105.4319] [INSPIRE].
G. Passarino and M.J.G. Veltman, One loop corrections for e + e − annihilation into μ + μ − in the Weinberg model, Nucl. Phys. B 160 (1979) 151 [INSPIRE].
G. Ossola, C.G. Papadopoulos and R. Pittau, CutTools: a program implementing the OPP reduction method to compute one-loop amplitudes, JHEP 03 (2008) 042 [arXiv:0711.3596] [INSPIRE].
P. Mastrolia, G. Ossola, T. Reiter and F. Tramontano, Scattering amplitudes from unitarity-based reduction algorithm at the integrand-level, JHEP 08 (2010) 080 [arXiv:1006.0710] [INSPIRE].
P. Mastrolia, E. Mirabella and T. Peraro, Integrand reduction of one-loop scattering amplitudes through Laurent series expansion, JHEP 06 (2012) 095 [Erratum ibid. 11 (2012) 128] [arXiv:1203.0291] [INSPIRE].
T. Peraro, Ninja: automated integrand reduction via Laurent expansion for one-loop amplitudes, Comput. Phys. Commun. 185 (2014) 2771 [arXiv:1403.1229] [INSPIRE].
H. van Deurzen, G. Luisoni, P. Mastrolia, G. Ossola and Z. Zhang, Automated computation of scattering amplitudes from integrand reduction to Monte Carlo tools, Nucl. Part. Phys. Proc. 267-269 (2015) 140 [INSPIRE].
S. Badger, H. Frellesvig and Y. Zhang, A two-loop five-gluon helicity amplitude in QCD, JHEP 12 (2013) 045 [arXiv:1310.1051] [INSPIRE].
S. Badger, G. Mogull, A. Ochirov and D. O’Connell, A complete two-loop, five-gluon helicity amplitude in Yang-Mills theory, JHEP 10 (2015) 064 [arXiv:1507.08797] [INSPIRE].
R. Britto, F. Cachazo and B. Feng, Generalized unitarity and one-loop amplitudes in N = 4 super-Yang-Mills, Nucl. Phys. B 725 (2005) 275 [hep-th/0412103] [INSPIRE].
H. Ita, Two-loop integrand decomposition into master integrals and surface terms, arXiv:1510.05626 [INSPIRE].
O.V. Tarasov, Connection between Feynman integrals having different values of the space-time dimension, Phys. Rev. D 54 (1996) 6479 [hep-th/9606018] [INSPIRE].
R.N. Lee, Space-time dimensionality D as complex variable: calculating loop integrals using dimensional recurrence relation and analytical properties with respect to D, Nucl. Phys. B 830 (2010) 474 [arXiv:0911.0252] [INSPIRE].
J. Gluza, K. Kajda and D.A. Kosower, Towards a basis for planar two-loop integrals, Phys. Rev. D 83 (2011) 045012 [arXiv:1009.0472] [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.A. Baikov, Explicit solutions of the multiloop integral recurrence relations and its application, Nucl. Instrum. Meth. A 389 (1997) 347 [hep-ph/9611449] [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].
P. Kant, Finding linear dependencies in integration-by-parts equations: a Monte Carlo approach, Comput. Phys. Commun. 185 (2014) 1473 [arXiv:1309.7287] [INSPIRE].
S. Borowka et al., Higgs boson pair production in gluon fusion at next-to-leading order with full top-quark mass dependence, Phys. Rev. Lett. 117 (2016) 012001 [Erratum ibid. 117 (2016) 079901] [arXiv:1604.06447] [INSPIRE].
W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann, Singular 4-0-2 — a computer algebra system for polynomial computations, http://www.singular.uni-kl.de, (2015).
D. Maître and P. Mastrolia, S@M, a mathematica implementation of the spinor-helicity formalism, Comput. Phys. Commun. 179 (2008) 501 [arXiv:0710.5559] [INSPIRE].
D. Cox, J.B. Little and D. O’ Shea, Ideals, varieties, and algorithms — an introduction to computational algebraic geometry and commutative algebra, second ed., Springer, Germany (1997).
D. Cox, J.B. Little and D. O’ Shea, Using algebraic geometry, second ed., Springer, Germany (2005).
B. Buchberger, Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems (in German), Aequat. Math. 4 (1970) 374.
B. Sturmfels, Solving systems of polynomial equations, Amer. Math. Soc., U.S.A. (2002).
G. Heinrich, G. Ossola, T. Reiter and F. Tramontano, Tensorial reconstruction at the integrand level, JHEP 10 (2010) 105 [arXiv:1008.2441] [INSPIRE].
V. Hirschi and T. Peraro, Tensor integrand reduction via Laurent expansion, JHEP 06 (2016) 060 [arXiv:1604.01363] [INSPIRE].
F. del Aguila and R. Pittau, Recursive numerical calculus of one-loop tensor integrals, JHEP 07 (2004) 017 [hep-ph/0404120] [INSPIRE].
Z. Bern and A.G. Morgan, Massive loop amplitudes from unitarity, Nucl. Phys. B 467 (1996) 479 [hep-ph/9511336] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1605.03157
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Mastrolia, P., Peraro, T. & Primo, A. Adaptive integrand decomposition in parallel and orthogonal space. J. High Energ. Phys. 2016, 164 (2016). https://doi.org/10.1007/JHEP08(2016)164
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2016)164