Abstract
We explore maximal unitarity for nonplanar two-loop integrals with up to four massive external legs. In this framework, the amplitude is reduced to a basis of master integrals whose coefficients are extracted from maximal cuts. The hepta-cut of the nonplanar double box defines a nodal algebraic curve associated with a multiply pinched genus-3 Riemann surface. All possible configurations of external masses are covered by two distinct topological pictures in which the curve decomposes into either six or eight Riemann spheres. The procedure relies on consistency equations based on vanishing of integrals of total derivatives and Levi-Civita contractions. Our analysis indicates that these constraints are governed by the global structure of the maximal cut. Lastly, we present an algorithm for computing generalized cuts of massive integrals with higher powers of propagators based on the Bezoutian matrix method.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys. B 435 (1995) 59 [hep-ph/9409265] [INSPIRE].
Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One loop n point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys. B 425 (1994) 217 [hep-ph/9403226] [INSPIRE].
R. Britto, F. Cachazo and B. Feng, New recursion relations for tree amplitudes of gluons, Nucl. Phys. B 715 (2005) 499 [hep-th/0412308] [INSPIRE].
R. Britto, F. Cachazo, B. Feng and E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett. 94 (2005) 181602 [hep-th/0501052] [INSPIRE].
Z. Bern and A.G. Morgan, Massive loop amplitudes from unitarity, Nucl. Phys. B 467 (1996) 479 [hep-ph/9511336] [INSPIRE].
Z. Bern, L.J. Dixon and D.A. Kosower, One loop amplitudes for e + e − to four partons, Nucl. Phys. B 513 (1998) 3 [hep-ph/9708239] [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].
R. Britto, F. Cachazo and B. Feng, Computing one-loop amplitudes from the holomorphic anomaly of unitarity cuts, Phys. Rev. D 71 (2005) 025012 [hep-th/0410179] [INSPIRE].
Z. Bern, N.E.J. Bjerrum-Bohr, D.C. Dunbar and H. Ita, Recursive calculation of one-loop QCD integral coefficients, JHEP 11 (2005) 027 [hep-ph/0507019] [INSPIRE].
S.J. Bidder, N.E.J. Bjerrum-Bohr, D.C. Dunbar and W.B. Perkins, One-loop gluon scattering amplitudes in theories with N < 4 supersymmetries, Phys. Lett. B 612 (2005) 75 [hep-th/0502028] [INSPIRE].
R. Britto, E. Buchbinder, F. Cachazo and B. Feng, One-loop amplitudes of gluons in SQCD, Phys. Rev. D 72 (2005) 065012 [hep-ph/0503132] [INSPIRE].
R. Britto, B. Feng and P. Mastrolia, The Cut-constructible part of QCD amplitudes, Phys. Rev. D 73 (2006) 105004 [hep-ph/0602178] [INSPIRE].
P. Mastrolia, On Triple-cut of scattering amplitudes, Phys. Lett. B 644 (2007) 272 [hep-th/0611091] [INSPIRE].
A. Brandhuber, S. McNamara, B.J. Spence and G. Travaglini, Loop amplitudes in pure Yang-Mills from generalised unitarity, JHEP 10 (2005) 011 [hep-th/0506068] [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].
C. Anastasiou, R. Britto, B. Feng, Z. Kunszt and P. Mastrolia, Unitarity cuts and Reduction to master integrals in d dimensions for one-loop amplitudes, JHEP 03 (2007) 111 [hep-ph/0612277] [INSPIRE].
Z. Bern, L.J. Dixon and D.A. Kosower, On-Shell Methods in Perturbative QCD, Annals Phys. 322 (2007) 1587 [arXiv:0704.2798] [INSPIRE].
D. Forde, Direct extraction of one-loop integral coefficients, Phys. Rev. D 75 (2007) 125019 [arXiv:0704.1835] [INSPIRE].
S.D. Badger, Direct Extraction Of One Loop Rational Terms, JHEP 01 (2009) 049 [arXiv:0806.4600] [INSPIRE].
W.T. Giele, Z. Kunszt and K. Melnikov, Full one-loop amplitudes from tree amplitudes, JHEP 04 (2008) 049 [arXiv:0801.2237] [INSPIRE].
R. Britto and B. Feng, Unitarity cuts with massive propagators and algebraic expressions for coefficients, Phys. Rev. D 75 (2007) 105006 [hep-ph/0612089] [INSPIRE].
R. Britto and B. Feng, Integral coefficients for one-loop amplitudes, JHEP 02 (2008) 095 [arXiv:0711.4284] [INSPIRE].
Z. Bern, J.J. Carrasco, T. Dennen, Y.-t. Huang and H. Ita, Generalized Unitarity and Six-Dimensional Helicity, Phys. Rev. D 83 (2011) 085022 [arXiv:1010.0494] [INSPIRE].
C. Anastasiou, R. Britto, B. Feng, Z. Kunszt and P. Mastrolia, D-dimensional unitarity cut method, Phys. Lett. B 645 (2007) 213 [hep-ph/0609191] [INSPIRE].
P. Mastrolia, Double-Cut of Scattering Amplitudes and Stokes’ Theorem, Phys. Lett. B 678 (2009) 246 [arXiv:0905.2909] [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].
C.F. Berger, Z. Bern, L.J. Dixon, F. Febres Cordero, D. Forde et al., An Automated Implementation of On-Shell Methods for One-Loop Amplitudes, Phys. Rev. D 78 (2008) 036003 [arXiv:0803.4180] [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, C.G. Papadopoulos and R. Pittau, Optimizing the Reduction of One-Loop Amplitudes, JHEP 06 (2008) 030 [arXiv:0803.3964] [INSPIRE].
W.T. Giele and G. Zanderighi, On the Numerical Evaluation of One-Loop Amplitudes: The Gluonic Case, JHEP 06 (2008) 038 [arXiv:0805.2152] [INSPIRE].
C.F. Berger, Z. Bern, L.J. Dixon, F. Febres Cordero, D. Forde et al., Precise Predictions for W + 3 Jet Production at Hadron Colliders, Phys. Rev. Lett. 102 (2009) 222001 [arXiv:0902.2760] [INSPIRE].
S. Badger, B. Biedermann and P. Uwer, NGluon: A Package to Calculate One-loop Multi-gluon Amplitudes, Comput. Phys. Commun. 182 (2011) 1674 [arXiv:1011.2900] [INSPIRE].
C.F. Berger, Z. Bern, L.J. Dixon, F. Febres Cordero, D. Forde et al., Precise Predictions for W + 4 Jet Production at the Large Hadron Collider, Phys. Rev. Lett. 106 (2011) 092001 [arXiv:1009.2338] [INSPIRE].
V. Hirschi, R. Frederix, S. Frixione, M.V. Garzelli, F. Maltoni et al., Automation of one-loop QCD corrections, JHEP 05 (2011) 044 [arXiv:1103.0621] [INSPIRE].
Z. Bern, J.S. Rozowsky and B. Yan, Two loop four gluon amplitudes in N = 4 super Yang-Mills, Phys. Lett. B 401 (1997) 273 [hep-ph/9702424] [INSPIRE].
Z. Bern, L.J. Dixon and D.A. Kosower, A Two loop four gluon helicity amplitude in QCD, JHEP 01 (2000) 027 [hep-ph/0001001] [INSPIRE].
E.W.N. Glover, C. Oleari and M.E. Tejeda-Yeomans, Two loop QCD corrections to gluon-gluon scattering, Nucl. Phys. B 605 (2001) 467 [hep-ph/0102201] [INSPIRE].
Z. Bern, A. De Freitas and L.J. Dixon, Two loop helicity amplitudes for gluon-gluon scattering in QCD and supersymmetric Yang-Mills theory, JHEP 03 (2002) 018 [hep-ph/0201161] [INSPIRE].
C. Anastasiou, E.W.N. Glover, C. Oleari and M.E. Tejeda-Yeomans, Two-loop QCD corrections to the scattering of massless distinct quarks, Nucl. Phys. B 601 (2001) 318 [hep-ph/0010212] [INSPIRE].
C. Anastasiou, E.W.N. Glover, C. Oleari and M.E. Tejeda-Yeomans, Two loop QCD corrections to massless identical quark scattering, Nucl. Phys. B 601 (2001) 341 [hep-ph/0011094] [INSPIRE].
C. Anastasiou, E.W.N. Glover, C. Oleari and M.E. Tejeda-Yeomans, Two loop QCD corrections to massless quark gluon scattering, Nucl. Phys. B 605 (2001) 486 [hep-ph/0101304] [INSPIRE].
E.I. Buchbinder and F. Cachazo, Two-loop amplitudes of gluons and octa-cuts in N = 4 super Yang-Mills, JHEP 11 (2005) 036 [hep-th/0506126] [INSPIRE].
F. Cachazo, Sharpening The Leading Singularity, arXiv:0803.1988 [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].
R.M. Schabinger, A New Algorithm For The Generation Of Unitarity-Compatible Integration By Parts Relations, JHEP 01 (2012) 077 [arXiv:1111.4220] [INSPIRE].
D.A. Kosower and K.J. Larsen, Maximal Unitarity at Two Loops, Phys. Rev. D 85 (2012) 045017 [arXiv:1108.1180] [INSPIRE].
S. Caron-Huot and K.J. Larsen, Uniqueness of two-loop master contours, JHEP 10 (2012) 026 [arXiv:1205.0801] [INSPIRE].
H. Johansson, D.A. Kosower and K.J. Larsen, Two-Loop Maximal Unitarity with External Masses, Phys. Rev. D 87 (2013) 025030 [arXiv:1208.1754] [INSPIRE].
H. Johansson, D.A. Kosower and K.J. Larsen, Maximal Unitarity for the Four-Mass Double Box, Phys. Rev. D 89 (2014) 125010 [arXiv:1308.4632] [INSPIRE].
M. Søgaard, Global Residues and Two-Loop Hepta-Cuts, JHEP 09 (2013) 116 [arXiv:1306.1496] [INSPIRE].
M. Søgaard and Y. Zhang, Multivariate Residues and Maximal Unitarity, JHEP 12 (2013) 008 [arXiv:1310.6006] [INSPIRE].
M. Sogaard and Y. Zhang, Unitarity Cuts of Integrals with Doubled Propagators, JHEP 07 (2014) 112 [arXiv:1403.2463] [INSPIRE].
K.J. Larsen, Global Poles of the Two-Loop Six-Point N = 4 SYM integrand, Phys. Rev. D 86 (2012) 085032 [arXiv:1205.0297] [INSPIRE].
S. Badger, H. Frellesvig and Y. Zhang, Hepta-Cuts of Two-Loop Scattering Amplitudes, JHEP 04 (2012) 055 [arXiv:1202.2019] [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, An Integrand Reconstruction Method for Three-Loop Amplitudes, JHEP 08 (2012) 065 [arXiv:1207.2976] [INSPIRE].
Y. Zhang, Integrand-Level Reduction of Loop Amplitudes by Computational Algebraic Geometry Methods, JHEP 09 (2012) 042 [arXiv:1205.5707] [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].
B. Feng and R. Huang, The classification of two-loop integrand basis in pure four-dimension, JHEP 02 (2013) 117 [arXiv:1209.3747] [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].
P. Mastrolia, E. Mirabella, G. Ossola, T. Peraro and H. van Deurzen, The Integrand Reduction of One- and Two-Loop Scattering Amplitudes, PoS(LL2012)028 [arXiv:1209.5678] [INSPIRE].
R.H.P. Kleiss, I. Malamos, C.G. Papadopoulos and R. Verheyen, Counting to One: Reducibility of One- and Two-Loop Amplitudes at the Integrand Level, JHEP 12 (2012) 038 [arXiv:1206.4180] [INSPIRE].
R. Huang and Y. Zhang, On Genera of Curves from High-loop Generalized Unitarity Cuts, JHEP 04 (2013) 080 [arXiv:1302.1023] [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].
B. Feng, J. Zhen, R. Huang and K. Zhou, Integral Reduction by Unitarity Method for Two-loop Amplitudes: A Case Study, JHEP 06 (2014) 166 [arXiv:1401.6766] [INSPIRE].
Z. Bern, J.J.M. Carrasco, H. Ita, H. Johansson and R. Roiban, On the Structure of Supersymmetric Sums in Multi-Loop Unitarity Cuts, Phys. Rev. D 80 (2009) 065029 [arXiv:0903.5348] [INSPIRE].
M. Sogaard, Supersums for all supersymmetric amplitudes, Phys. Rev. D 84 (2011) 065011 [arXiv:1106.3785] [INSPIRE].
F. Caola, J.M. Henn, K. Melnikov and V.A. Smirnov, Non-planar master integrals for the production of two off-shell vector bosons in collisions of massless partons, JHEP 09 (2014) 043 [arXiv:1404.5590] [INSPIRE].
A.V. Smirnov and V.A. Smirnov, FIRE4, LiteRed and accompanying tools to solve integration by parts relations, Comput. Phys. Commun. 184 (2013) 2820 [arXiv:1302.5885] [INSPIRE].
P. Griffiths, J. Harris, Principles of Algebraic Geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978.
R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, 1977, graduate texts in mathematics, no. 52.
D.R. Grayson and M.E. Stillman, Macaulay2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2/.
E. Cattani and A. Dickenstein, Introduction to residues and resultants: Solving polynomial equations, Springer Berlin Heidelberg, 2005.
D.A. Cox, J. Little and Donal. O’Shea, Using algebraic geometry, Springer, New York, 2005.
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: 1406.5044
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
Søgaard, M., Zhang, Y. Massive nonplanar two-loop maximal unitarity. J. High Energ. Phys. 2014, 6 (2014). https://doi.org/10.1007/JHEP12(2014)006
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2014)006