# zbMATH — the first resource for mathematics

Multiloop integrand reduction for dimensionally regulated amplitudes. (English) Zbl 1331.81218
Summary: We present the integrand reduction via multivariate polynomial division as a natural technique to encode the unitarity conditions of Feynman amplitudes. We derive a recursive formula for the integrand reduction, valid for arbitrary dimensionally regulated loop integrals with any number of loops and external legs, which can be used to obtain the decomposition of any integrand analytically with a finite number of algebraic operations. The general results are illustrated by applications to two-loop Feynman diagrams in QED and QCD, showing that the proposed reduction algorithm can also be seamlessly applied to integrands with denominators appearing with arbitrary powers.

##### MSC:
 81T18 Feynman diagrams 81V10 Electromagnetic interaction; quantum electrodynamics 81V05 Strong interaction, including quantum chromodynamics
##### Software:
Macaulay2; FeynArts; FormCalc; FORM ; Axodraw; CutTools; FeynCalc; SAMURAI
Full Text:
##### References:
 [1] Ossola, G.; Papadopoulos, C. G.; Pittau, R., Reducing full one-loop amplitudes to scalar integrals at the integrand level, Nucl. Phys. B, 763, 147-169, (2007) · Zbl 1116.81067 [2] Mastrolia, P.; Ossola, G., On the integrand-reduction method for two-loop scattering amplitudes, J. High Energy Phys., 1111, 014, (2011) · Zbl 1306.81357 [3] Ellis, R. K.; Giele, W. T.; Kunszt, Z., A numerical unitarity formalism for evaluating one-loop amplitudes, J. High Energy Phys., 0803, 003, (2008) [4] Badger, S.; Frellesvig, H.; Zhang, Y., Hepta-cuts of two-loop scattering amplitudes, J. High Energy Phys., 1204, 055, (2012) · Zbl 1348.81340 [5] Mastrolia, P.; Mirabella, E.; Peraro, T., Integrand reduction of one-loop scattering amplitudes through Laurent series expansion [6] Zhang, Y., Integrand-level reduction of loop amplitudes by computational algebraic geometry methods, J. High Energy Phys., 1209, 042, (2012) [7] Mastrolia, P.; Mirabella, E.; Ossola, G.; Peraro, T., Scattering amplitudes from multivariate polynomial division, Phys. Lett. B, 718, 173-177, (2012) [8] Badger, S.; Frellesvig, H.; Zhang, Y., An integrand reconstruction method for three-loop amplitudes, J. High Energy Phys., 1208, 065, (2012) [9] Mastrolia, P.; Mirabella, E.; Ossola, G.; Peraro, T., Integrand-reduction for two-loop scattering amplitudes through multivariate polynomial division, Phys. Rev. D, 87, 085026, (2012) [10] Feng, B.; Huang, R., The classification of two-loop integrand basis in pure four-dimension, J. High Energy Phys., 1302, 117, (2013) · Zbl 1342.81228 [11] Caron-Huot, S.; Larsen, K. J., Uniqueness of two-loop master contours, J. High Energy Phys., 1210, 026, (2012) [12] Huang, R.; Zhang, Y., On genera of curves from high-loop generalized unitarity cuts, J. High Energy Phys., 1304, 080, (2013) · Zbl 1342.81353 [13] Ossola, G.; Papadopoulos, C. G.; Pittau, R., Cuttools: a program implementing the OPP reduction method to compute one-loop amplitudes, J. High Energy Phys., 0803, 042, (2008) [14] Mastrolia, P.; Ossola, G.; Reiter, T.; Tramontano, F., Scattering amplitudes from unitarity-based reduction algorithm at the integrand-level, J. High Energy Phys., 1008, 080, (2010) · Zbl 1290.81151 [15] Broadhurst, D. J.; Fleischer, J.; Tarasov, O., Two loop two point functions with masses: asymptotic expansions and Taylor series, in any dimension, Z. Phys. C, 60, 287-302, (1993) [16] Anastasiou, C.; Melnikov, K., Higgs boson production at hadron colliders in NNLO QCD, Nucl. Phys. B, 646, 220-256, (2002) [17] Grayson, D. R.; Stillman, M. E., Macaulay2, a software system for research in algebraic geometry, Available at [18] Kuipers, J.; Ueda, T.; Vermaseren, J.; Vollinga, J., FORM version 4.0, Comput. Phys. Commun., 184, 1453-1467, (2013) · Zbl 1317.68286 [19] Nogueira, P., Automatic Feynman graph generation, J. Comput. Phys., 105, 279-289, (1993) · Zbl 0782.68091 [20] Hahn, T., Generating Feynman diagrams and amplitudes with feynarts 3, Comput. Phys. Commun., 140, 418-431, (2001) · Zbl 0994.81082 [21] Mertig, R.; Bohm, M.; Denner, A., FEYN CALC: computer algebraic calculation of Feynman amplitudes, Comput. Phys. Commun., 64, 345-359, (1991) [22] Agrawal, S.; Hahn, T.; Mirabella, E., Formcalc 7, J. Phys. Conf. Ser., 368, 012054, (2012)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.