Abstract
In this contribution we will discuss a new approach to the derivation of linear relations between Feynman integrals. This new approach uses the mathematical object known as the intersection number to define what amounts to an inner product between Feynman integrals, which can be used to project directly unto the basis of master integrals. In particular we will discuss one perspective to this intersection-based method, which we name the top-down approach. This approach can be seen as an integral level version of the algorithms based on integrand reduction and generalized unitarity cuts, that were revolutionizing NLO scattering computations in the early 2000s.
Based on [1].
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Equation (1) is brushing a lot under the rug. Besides the spurious terms discussed below, it does not include the rational term \(\mathcal {R}\) containing the contributions not captured by four-dimensional cuts, nor does it include the extraction of pentagon-terms.
- 2.
We note that this step has nothing to do with intersection theory, and was done for instance in ref. [45].
- 3.
We note that had we replaced the subtraction of the box-coefficient in Eq. (38) with a free subtraction term (c 1 → κ 1), this κ-fitting step would give a solution only if κ 1 = c 1, so treating κ 1 on equal footing with the other κs, is another approach to fixing the box-coefficient.
References
H. Frellesvig, F. Gasparotto, S. Laporta, M.K. Mandal, P. Mastrolia, L. Mattiazzi, S. Mizera, Decomposition of Feynman Integrals by Multivariate Intersection Numbers. arXiv:2008.04823
K.G. Chetyrkin, F.V. Tkachov, Integration by parts: the algorithm to calculate beta functions in 4 loops. Nucl. Phys. B192, 159–204 (1981)
S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations. Int. J. Mod. Phys. A15, 5087–5159 (2000)
C. Anastasiou, A. Lazopoulos, Automatic integral reduction for higher order perturbative calculations. J. High Energy Phys. 07, 046 (2004)
A.V. Smirnov, F.S. Chuharev, FIRE6: Feynman Integral REduction with Modular Arithmetic. arXiv:1901.07808
A. von Manteuffel, C. Studerus, Reduze 2 - Distributed Feynman Integral Reduction. arXiv:1201.4330
R.N. Lee, Presenting LiteRed: A Tool for the Loop InTEgrals REDuction. arXiv:1212.2685
J. Klappert, F. Lange, P. Maierhöfer, J. Usovitsch, Integral Reduction with Kira 2.0 and Finite Field Methods. arXiv:2008.06494
P. Mastrolia, S. Mizera, Feynman integrals and intersection theory. J. High Energy Phys. 02, 139 (2019). arXiv:1810.03818
Z. Bern, L.J. Dixon, D.C. Dunbar, D.A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes. Nucl. Phys. B435, 59–101 (1995). hep-ph/9409265
Z. Bern, L.J. Dixon, D.C. Dunbar, D.A. Kosower, One loop n point gauge theory amplitudes, unitarity and collinear limits. Nucl. Phys. B425, 217–260 (1994). hep-ph/9403226
R. Britto, F. Cachazo, B. Feng, Generalized unitarity and one-loop amplitudes in N=4 super-Yang-Mills. Nucl. Phys. B 725, 275–305 (2005). hep-th/0412103
G. Ossola, C.G. Papadopoulos, R. Pittau, Reducing full one-loop amplitudes to scalar integrals at the integrand level. Nucl. Phys. B 763, 147–169 (2007). hep-ph/0609007
R. Ellis, W. Giele, Z. Kunszt, A numerical unitarity formalism for evaluating one-loop amplitudes. J. High Energy Phys. 03, 003 (2008). arXiv:0708.2398
C. Anastasiou, R. Britto, B. Feng, Z. Kunszt, P. Mastrolia, D-dimensional unitarity cut method. Phys. Lett. B 645, 213–216 (2007). hep-ph/0609191
D. Forde, Direct extraction of one-loop integral coefficients. Phys. Rev. D 75, 125019 (2007). arXiv:0704.1835
E. Nigel Glover, C. Williams, One-loop gluonic amplitudes from single unitarity cuts. J. High Energy Phys. 12, 067 (2008). arXiv:0810.2964
P. Mastrolia, Double-cut of scattering amplitudes and Stokes’ theorem. Phys. Lett. B678, 246–249 (2009). arXiv:0905.2909
S. Badger, Direct extraction of one loop rational terms. J. High Energy Phys. 01, 049 (2009). arXiv:0806.4600
R. Britto, B. Feng, Solving for tadpole coefficients in one-loop amplitudes. Phys. Lett. B 681, 376–381 (2009). arXiv:0904.2766
R. Ellis, W.T. Giele, Z. Kunszt, K. Melnikov, Masses, fermions and generalized D-dimensional unitarity. Nucl. Phys. B 822, 270–282 (2009). arXiv:0806.3467
P. Mastrolia, E. Mirabella, T. Peraro, Integrand reduction of one-loop scattering amplitudes through Laurent series expansion. J. High Energy Phys. 06, 095 (2012). arXiv:1203.0291 [Erratum: J. High Energy Phys. 11, 128 (2012)]
P. Mastrolia, G. Ossola, On the integrand-reduction method for two-loop scattering amplitudes. J. High Energy Phys. 11, 014 (2011). arXiv:1107.6041
P. Mastrolia, E. Mirabella, G. Ossola, T. Peraro, Scattering amplitudes from multivariate polynomial division. Phys. Lett. B 718, 173–177 (2012). arXiv:1205.7087
D.A. Kosower, K.J. Larsen, Maximal unitarity at two loops. Phys. Rev. D85, 045017 (2012). arXiv:1108.1180
S. Badger, H. Frellesvig, Y. Zhang, Hepta-cuts of two-loop scattering amplitudes. J. High Energy Phys. 04, 055 (2012). arXiv:1202.2019
S. Badger, H. Frellesvig, Y. Zhang, An integrand reconstruction method for three-loop amplitudes. J. High Energy Phys. 08, 065 (2012). arXiv:1207.2976
S. Badger, H. Frellesvig, Y. Zhang, A two-loop five-gluon helicity amplitude in QCD. J. High Energy Phys. 12, 045 (2013). arXiv:1310.1051
Y. Zhang, Integrand-level reduction of loop amplitudes by computational algebraic geometry methods. J. High Energy Phys. 09, 042 (2012). arXiv:1205.5707
M. Søgaard, Global residues and two-loop hepta-cuts. J. High Energy Phys. 09, 116 (2013). arXiv:1306.1496
M. Sogaard, Y. Zhang, Massive nonplanar two-loop maximal unitarity. J. High Energy Phys. 12, 006 (2014). arXiv:1406.5044
M. Søgaard, Y. Zhang, Elliptic functions and maximal unitarity. Phys. Rev. D 91(8), 081701 (2015). arXiv:1412.5577
P. Mastrolia, T. Peraro, A. Primo, Adaptive Integrand Decomposition in parallel and orthogonal space. J. High Energy Phys. 08, 164 (2016). arXiv:1605.03157
S. Badger, G. Mogull, A. Ochirov, D. O’Connell, A complete two-loop, five-gluon helicity amplitude in Yang-Mills theory. J. High Energy Phys. 10, 064 (2015). arXiv:1507.08797
S. Badger, G. Mogull, T. Peraro, Local integrands for two-loop all-plus Yang-Mills amplitudes. J. High Energy Phys. 08, 063 (2016). arXiv:1606.02244
S. Badger, C. Brønnum-Hansen, H.B. Hartanto, T. Peraro, First look at two-loop five-gluon scattering in QCD. Phys. Rev. Lett. 120(9), 092001 (2018). arXiv:1712.02229
S. Badger, C. Brønnum-Hansen, H.B. Hartanto, T. Peraro, Analytic helicity amplitudes for two-loop five-gluon scattering: the single-minus case. J. High Energy Phys. 01, 186 (2019). arXiv:1811.11699
S. Badger, D. Chicherin, T. Gehrmann, G. Heinrich, J. Henn, T. Peraro, P. Wasser, Y. Zhang, S. Zoia, Analytic form of the full two-loop five-gluon all-plus helicity amplitude. Phys. Rev. Lett. 123(7), 071601 (2019). arXiv:1905.03733
H. Ita, Two-loop integrand decomposition into master integrals and surface terms. Phys. Rev. D 94(11), 116015 (2016). arXiv:1510.05626
S. Abreu, F. Febres Cordero, H. Ita, M. Jaquier, B. Page, M. Zeng, Two-loop four-gluon amplitudes from numerical unitarity. Phys. Rev. Lett. 119(14), 142001 (2017). arXiv:1703.05273
S. Abreu, F. Febres Cordero, H. Ita, B. Page, M. Zeng, Planar two-loop five-gluon amplitudes from numerical unitarity. Phys. Rev. D 97(11), 116014 (2018). arXiv:1712.03946
S. Abreu, F. Febres Cordero, H. Ita, B. Page, V. Sotnikov, Planar two-loop five-parton amplitudes from numerical unitarity. J. High Energy Phys. 11, 116 (2018). arXiv:1809.09067
S. Abreu, J. Dormans, F. Febres Cordero, H. Ita, B. Page, V. Sotnikov, Analytic form of the planar two-loop five-parton scattering amplitudes in QCD. J. High Energy Phys. 05, 084 (2019). arXiv:1904.00945
H.A. Frellesvig, Generalized unitarity cuts and integrand reduction at higher loop orders. Ph.D. thesis, Copenhagen U., 2014
P.A. Baikov, Explicit solutions of the multiloop integral recurrence relations and its application. Nucl. Instrum. Meth. A389, 347–349 (1997). hep-ph/9611449
R.N. Lee, Calculating multiloop integrals using dimensional recurrence relation and D-analyticity. Nucl. Phys. Proc. Suppl. 205–206, 135–140 (2010). arXiv:1007.2256
A.G. Grozin, Integration by parts: an introduction. Int. J. Mod. Phys. A26, 2807–2854 (2011). arXiv:1104.3993
K.J. Larsen, Y. Zhang, Integration-by-parts reductions from unitarity cuts and algebraic geometry. Phys. Rev. D93(4), 041701 (2016). arXiv:1511.01071
J. Bosma, K.J. Larsen, Y. Zhang, Differential Equations for Loop Integrals in Baikov Representation. arXiv:1712.03760
H. Frellesvig, C.G. Papadopoulos, Cuts of Feynman integrals in Baikov representation. J. High Energy Phys. 04, 083 (2017). arXiv:1701.07356
M. Harley, F. Moriello, R.M. Schabinger, Baikov-Lee representations of cut Feynman integrals. J. High Energy Phys. 06, 049 (2017). arXiv:1705.03478
H. Frellesvig, F. Gasparotto, S. Laporta, M.K. Mandal, P. Mastrolia, L. Mattiazzi, S. Mizera, Decomposition of Feynman integrals on the maximal cut by intersection numbers. J. High Energy Phys. 05, 153 (2019). arXiv:1901.11510
K. Cho, K. Matsumoto, Intersection theory for twisted cohomologies and twisted Riemann’s period relations I. Nagoya Math. J. 139, 67–86 (1995)
K. Matsumoto, Intersection numbers for logarithmic k-forms. Osaka J. Math. 35(4), 873–893 (1998)
S. Mizera, Scattering amplitudes from intersection theory. Phys. Rev. Lett. 120(14), 141602 (2018). arXiv:1711.00469
S. Mizera, Aspects of scattering amplitudes and moduli space localization. Ph.D. thesis, Princeton, Inst. Advanced Study, 2020. arXiv:1906.02099
H. Frellesvig, F. Gasparotto, M.K. Mandal, P. Mastrolia, L. Mattiazzi, S. Mizera, Vector space of Feynman integrals and multivariate intersection numbers. Phys. Rev. Lett. 123(20), 201602 (2019). arXiv:1907.02000
A.V. Smirnov, FIRE5: a C++ implementation of Feynman integral reduction. Comput. Phys. Commun. 189, 182–191 (2015). arXiv:1408.2372
S. Weinzierl, On the Computation of Intersection Numbers for Twisted Cocycles. arXiv:2002.01930
K. Matsumoto, Relative Twisted Homology and Cohomology Groups Associated with Lauricella’s F D. arXiv:1804.00366
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, J. Trnka, Local integrals for planar scattering amplitudes. J. High Energy Phys. 06, 125 (2012). arXiv:1012.6032
J.M. Henn, Multiloop integrals in dimensional regularization made simple. Phys. Rev. Lett. 110, 251601 (2013). arXiv:1304.1806
S. Mizera, A. Pokraka, From infinity to four dimensions: higher residue pairings and Feynman integrals. J. High Energy Phys. 02, 159 (2020). arXiv:1910.11852
J. Chen, X. Xu, L.L. Yang, Constructing Canonical Feynman Integrals with Intersection Theory. arXiv:2008.03045
S. Abreu, R. Britto, C. Duhr, E. Gardi, J. Matthew, From positive geometries to a coaction on hypergeometric functions. J. High Energy Phys. 02, 122 (2020). arXiv:1910.08358
Acknowledgements
I would like to thank my collaborators on the work presented here, Federico Gasparotto, Stefano Laporta, Manoj K. Mandal, Pierpaolo Mastrolia, Luca Mattiazzi, and Sebastian Mizera. This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No. 847523 ‘INTERACTIONS’. The work of HF has been partially supported by a Carlsberg Foundation Reintegration Fellowship.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Frellesvig, H. (2021). Top-Down Decomposition: A Cut-Based Approach to Integral Reductions. In: Blümlein, J., Schneider, C. (eds) Anti-Differentiation and the Calculation of Feynman Amplitudes. Texts & Monographs in Symbolic Computation. Springer, Cham. https://doi.org/10.1007/978-3-030-80219-6_8
Download citation
DOI: https://doi.org/10.1007/978-3-030-80219-6_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-80218-9
Online ISBN: 978-3-030-80219-6
eBook Packages: Computer ScienceComputer Science (R0)