Decomposition of Feynman integrals on the maximal cut by intersection numbers. (English) Zbl 1416.81198

Summary: We elaborate on the recent idea of a direct decomposition of Feynman integrals onto a basis of master integrals on maximal cuts using intersection numbers. We begin by showing an application of the method to the derivation of contiguity relations for special functions, such as the Euler beta function, the Gauss \( {}_{2} F_1 \) hypergeometric function, and the Appell \(F_1 \) function. Then, we apply the new method to decompose Feynman integrals whose maximal cuts admit 1-form integral representations, including examples that have from two to an arbitrary number of loops, and/or from zero to an arbitrary number of legs. Direct constructions of differential equations and dimensional recurrence relations for Feynman integrals are also discussed. We present two novel approaches to decomposition-by-intersections in cases where the maximal cuts admit a 2-form integral representation, with a view towards the extension of the formalism to \(n\)-form representations. The decomposition formulae computed through the use of intersection numbers are directly verified to agree with the ones obtained using integration-by-parts identities.


81U05 \(2\)-body potential quantum scattering theory
81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
Full Text: DOI arXiv


[1] Mastrolia, P.; Mizera, S., Feynman integrals and intersection theory, JHEP, 02, 139, (2019) · Zbl 1411.81093
[2] K.G. Chetyrkin and F.V. Tkachov, Integration by parts: the algorithm to calculate β-functions in 4 loops, Nucl. Phys.B 192 (1981) 159 [INSPIRE].
[3] Barucchi, G.; Ponzano, G., Differential equations for one-loop generalized Feynman integrals, J. Math. Phys., 14, 396, (1973)
[4] A.V. Kotikov, Differential equations method: new technique for massive Feynman diagrams calculation, Phys. Lett.B 254 (1991) 158 [INSPIRE].
[5] A.V. Kotikov, Differential equation method: the calculation of N point Feynman diagrams, Phys. Lett.B 267 (1991) 123 [Erratum ibid.B 295 (1992) 409] [INSPIRE].
[6] Z. Bern, L.J. Dixon and D.A. Kosower, Dimensionally regulated pentagon integrals, Nucl. Phys.B 412 (1994) 751 [hep-ph/9306240] [INSPIRE].
[7] E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim.A 110 (1997) 1435 [hep-th/9711188] [INSPIRE].
[8] T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys.B 580 (2000) 485 [hep-ph/9912329] [INSPIRE]. · Zbl 1071.81089
[9] Henn, JM, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett., 110, 251601, (2013)
[10] J.M. Henn, Lectures on differential equations for Feynman integrals, J. Phys.A 48 (2015) 153001 [arXiv:1412.2296] [INSPIRE].
[11] 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].
[12] 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].
[13] S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys.A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE]. · Zbl 0973.81082
[14] S. Laporta, Calculation of Feynman integrals by difference equations, Acta Phys. Polon.B 34 (2003) 5323 [hep-ph/0311065] [INSPIRE].
[15] V.A. Smirnov, Evaluating Feynman integrals, Springer Tracts Mod. Phys. 211, Springer, Berlin Heidelberg, Germany (2005) [INSPIRE].
[16] A.G. Grozin, Integration by parts: an introduction, Int. J. Mod. Phys.A 26 (2011) 2807 [arXiv:1104.3993] [INSPIRE].
[17] Y. Zhang, Lecture notes on multi-loop integral reduction and applied algebraic geometry, arXiv:1612.02249 [INSPIRE].
[18] Kotikov, AV; Teber, S., Multi-loop techniques for massless Feynman diagram calculations, Phys. Part. Nucl., 50, 1, (2019)
[19] 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].
[20] Peraro, T., Scattering amplitudes over finite fields and multivariate functional reconstruction, JHEP, 12, 030, (2016) · Zbl 1390.81631
[21] Böhm, J.; Georgoudis, A.; Larsen, KJ; Schönemann, H.; Zhang, Y., Complete integration-by-parts reductions of the non-planar hexagon-box via module intersections, JHEP, 09, 024, (2018) · Zbl 1398.81264
[22] D.A. Kosower, Direct solution of integration-by-parts systems, Phys. Rev.D 98 (2018) 025008 [arXiv:1804.00131] [INSPIRE].
[23] X. Liu and Y.-Q. Ma, Determining arbitrary Feynman integrals by vacuum integrals, Phys. Rev.D 99 (2019) 071501 [arXiv:1801.10523] [INSPIRE].
[24] A. Kardos, A new reduction strategy for special negative sectors of planar two-loop integrals without Laporta algorithm, arXiv:1812.05622 [INSPIRE].
[25] H.A. Chawdhry, M.A. Lim and A. Mitov, Two-loop five-point massless QCD amplitudes within the integration-by-parts approach, Phys. Rev.D 99 (2019) 076011 [arXiv:1805.09182] [INSPIRE].
[26] A. von Manteuffel and C. Studerus, Reduze 2 — distributed Feynman integral reduction, arXiv:1201.4330 [INSPIRE].
[27] R.N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE].
[28] A.V. Smirnov, Algorithm FIREFeynman Integral REduction, JHEP10 (2008) 107 [arXiv:0807.3243] [INSPIRE].
[29] P. Maierhöfer, J. Usovitsch and P. Uwer, Kiraa Feynman integral reduction program, Comput. Phys. Commun.230 (2018) 99 [arXiv:1705.05610] [INSPIRE].
[30] Georgoudis, A.; Larsen, KJ; Zhang, Y., Azurite: an algebraic geometry based package for finding bases of loop integrals, Comput. Phys. Commun., 221, 203, (2017)
[31] P. Maierhöfer and J. Usovitsch, Kira 1\(.\)2 release notes, arXiv:1812.01491 [INSPIRE].
[32] A.V. Smirnov and F.S. Chuharev, FIRE6: Feynman Integral REduction with modular arithmetic, arXiv:1901.07808 [INSPIRE].
[33] Argeri, M.; etal., Magnus and Dyson series for master integrals, JHEP, 03, 082, (2014) · Zbl 1333.81379
[34] Lee, RN, Reducing differential equations for multiloop master integrals, JHEP, 04, 108, (2015) · Zbl 1388.81109
[35] T. Gehrmann, A. von Manteuffel, L. Tancredi and E. Weihs, The two-loop master integrals for\( q\overline{q}\to VV \), JHEP06 (2014) 032 [arXiv:1404.4853] [INSPIRE].
[36] Gituliar, O.; Magerya, V., Fuchsia: a tool for reducing differential equations for Feynman master integrals to epsilon form, Comput. Phys. Commun., 219, 329, (2017) · Zbl 1411.81015
[37] R.N. Lee and A.A. Pomeransky, Normalized Fuchsian form on Riemann sphere and differential equations for multiloop integrals, arXiv:1707.07856 [INSPIRE].
[38] L. Adams and S. Weinzierl, The ε-form of the differential equations for Feynman integrals in the elliptic case, Phys. Lett.B 781 (2018) 270 [arXiv:1802.05020] [INSPIRE]. · Zbl 1398.81097
[39] S. Laporta, High-precision calculation of the 4-loop contribution to the electron g − 2 in QED, Phys. Lett.B 772 (2017) 232 [arXiv:1704.06996] [INSPIRE].
[40] Anastasiou, C.; Duhr, C.; Dulat, F.; Herzog, F.; Mistlberger, B., Higgs boson gluon-fusion production in QCD at three loops, Phys. Rev. Lett., 114, 212001, (2015)
[41] R. Bonciani, V. Del Duca, H. Frellesvig, J.M. Henn, F. Moriello and V.A. Smirnov, Next-to-leading order QCD corrections to the decay width H → \(Z\)\(γ\), JHEP08 (2015) 108 [arXiv:1505.00567] [INSPIRE].
[42] T. Gehrmann, S. Guns and D. Kara, The rare decay H → \(Z\)\(γ\)in perturbative QCD, JHEP09 (2015) 038 [arXiv:1505.00561] [INSPIRE].
[43] M. Bonetti, K. Melnikov and L. Tancredi, Three-loop mixed QCD-electroweak corrections to Higgs boson gluon fusion, Phys. Rev.D 97 (2018) 034004 [arXiv:1711.11113] [INSPIRE].
[44] Borowka, S.; etal., Full top quark mass dependence in Higgs boson pair production at NLO, JHEP, 10, 107, (2016)
[45] J. Baglio, F. Campanario, S. Glaus, M. Mühlleitner, M. Spira and J. Streicher, Gluon fusion into Higgs pairs at NLO QCD and the top mass scheme, arXiv:1811.05692 [INSPIRE].
[46] Lindert, JM; Melnikov, K.; Tancredi, L.; Wever, C., Top-bottom interference effects in Higgs plus jet production at the LHC, Phys. Rev. Lett., 118, 252002, (2017)
[47] Jones, SP; Kerner, M.; Luisoni, G., Next-to-leading-order QCD corrections to Higgs boson plus jet production with full top-quark mass dependence, Phys. Rev. Lett., 120, 162001, (2018)
[48] F. Maltoni, M.K. Mandal and X. Zhao, Top-quark effects in diphoton production through gluon fusion at NLO in QCD, arXiv:1812.08703 [INSPIRE].
[49] T. Gehrmann, J.M. Henn and N.A. Lo Presti, Analytic form of the two-loop planar five-gluon all-plus-helicity amplitude in QCD, Phys. Rev. Lett.116 (2016) 062001 [Erratum ibid.116 (2016) 189903] [arXiv:1511.05409] [INSPIRE].
[50] S. Badger, C. Brønnum-Hansen, H.B. Hartanto and T. Peraro, First look at two-loop five-gluon scattering in QCD, Phys. Rev. Lett.120 (2018) 092001 [arXiv:1712.02229] [INSPIRE].
[51] Abreu, S.; Febres Cordero, F.; Ita, H.; Page, B.; Sotnikov, V., Planar two-loop five-parton amplitudes from numerical unitarity, JHEP, 11, 116, (2018)
[52] S. Abreu, J. Dormans, F. Febres Cordero, H. Ita and B. Page, Analytic form of planar two-loop five-gluon scattering amplitudes in QCD, Phys. Rev. Lett.122 (2019) 082002 [arXiv:1812.04586] [INSPIRE].
[53] Chicherin, D.; Gehrmann, T.; Henn, JM; Wasser, P.; Zhang, Y.; Zoia, S., Analytic result for a two-loop five-particle amplitude, Phys. Rev. Lett., 122, 121602, (2019)
[54] S. Abreu, L.J. Dixon, E. Herrmann, B. Page and M. Zeng, The two-loop five-point amplitude in N = 4 super-Yang-Mills theory, Phys. Rev. Lett.122 (2019) 121603 [arXiv:1812.08941] [INSPIRE].
[55] Badger, S.; Brønnum-Hansen, C.; Hartanto, HB; Peraro, T., Analytic helicity amplitudes for two-loop five-gluon scattering: the single-minus case, JHEP, 01, 186, (2019) · Zbl 1409.81155
[56] D. Chicherin, T. Gehrmann, J.M. Henn, P. Wasser, Y. Zhang and S. Zoia, The two-loop five-particle amplitude in N = 8 supergravity, JHEP03 (2019) 115 [arXiv:1901.05932] [INSPIRE].
[57] S. Abreu, L.J. Dixon, E. Herrmann, B. Page and M. Zeng, The two-loop five-point amplitude in N = 8 supergravity, JHEP03 (2019) 123 [arXiv:1901.08563] [INSPIRE].
[58] 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].
[59] Lee, RN; Pomeransky, AA, Critical points and number of master integrals, JHEP, 11, 165, (2013) · Zbl 1342.81139
[60] Marcolli, M., Motivic renormalization and singularities, Clay Math. Proc., 11, 409, (2010) · Zbl 1218.81081
[61] M. Marcolli, Feynman motives, World Scientific, Singapore (2010).
[62] Bitoun, T.; Bogner, C.; Klausen, RP; Panzer, E., Feynman integral relations from parametric annihilators, Lett. Math. Phys., 109, 497, (2019) · Zbl 1412.81141
[63] 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].
[64] J. Bosma, K.J. Larsen and Y. Zhang, Differential equations for loop integrals in Baikov representation, Phys. Rev.D 97 (2018) 105014 [arXiv:1712.03760] [INSPIRE].
[65] Frellesvig, H.; Papadopoulos, CG, Cuts of Feynman integrals in Baikov representation, JHEP, 04, 083, (2017) · Zbl 1378.81039
[66] Zeng, M., Differential equations on unitarity cut surfaces, JHEP, 06, 121, (2017) · Zbl 1380.81135
[67] Bosma, J.; Sogaard, M.; Zhang, Y., Maximal cuts in arbitrary dimension, JHEP, 08, 051, (2017) · Zbl 1381.81146
[68] Harley, M.; Moriello, F.; Schabinger, RM, Baikov-Lee representations of cut Feynman integrals, JHEP, 06, 049, (2017) · Zbl 1380.81132
[69] Lee, RN; Smirnov, VA, The dimensional recurrence and analyticity method for multicomponent master integrals: using unitarity cuts to construct homogeneous solutions, JHEP, 12, 104, (2012) · Zbl 1397.81073
[70] E. Remiddi and L. Tancredi, Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral, Nucl. Phys.B 907 (2016) 400 [arXiv:1602.01481] [INSPIRE]. · Zbl 1336.81038
[71] A. Primo and L. Tancredi, On the maximal cut of Feynman integrals and the solution of their differential equations, Nucl. Phys.B 916 (2017) 94 [arXiv:1610.08397] [INSPIRE]. · Zbl 1356.81136
[72] A. Primo and L. Tancredi, Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph, Nucl. Phys.B 921 (2017) 316 [arXiv:1704.05465] [INSPIRE]. · Zbl 1370.81073
[73] C. Anastasiou and K. Melnikov, Higgs boson production at hadron colliders in NNLO QCD, Nucl. Phys.B 646 (2002) 220 [hep-ph/0207004] [INSPIRE].
[74] S. Laporta and E. Remiddi, Analytic treatment of the two loop equal mass sunrise graph, Nucl. Phys.B 704 (2005) 349 [hep-ph/0406160] [INSPIRE]. · Zbl 1119.81356
[75] Manteuffel, A.; Tancredi, L., A non-planar two-loop three-point function beyond multiple polylogarithms, JHEP, 06, 127, (2017)
[76] Adams, L.; Chaubey, E.; Weinzierl, S., Analytic results for the planar double box integral relevant to top-pair production with a closed top loop, JHEP, 10, 206, (2018)
[77] K. Cho and K. Matsumoto, Intersection theory for twisted cohomologies and twisted Riemanns period relations I, Nagoya Math. J.139 (1995) 67.
[78] Matsumoto, K., Intersection numbers for logarithmic k-forms, Osaka J. Math., 35, 873, (1998) · Zbl 0937.32013
[79] Mizera, S., Scattering amplitudes from intersection theory, Phys. Rev. Lett., 120, 141602, (2018)
[80] K. Aomoto and M. Kita, Theory of hypergeometric functions, Springer Monogr. Math., Springer, Japan (2011).
[81] K. Aomoto, On the structure of integrals of power product of linear functions, Sci. Papers College Gen. Ed. Univ. Tokyo27 (1977) 49.
[82] Gelfand, IM, General theory of hypergeometric functions, Dokl. Akad. Nauk SSSR, 288, 14, (1986)
[83] M. Yoshida, Hypergeometric functions, my love: modular interpretations of configuration spaces, Aspects of Mathematics, Vieweg+Teubner Verlag, Germany (2013).
[84] Mizera, S., Inverse of the string theory KLT kernel, JHEP, 06, 084, (2017) · Zbl 1380.81424
[85] Mizera, S., Combinatorics and topology of Kawai-Lewellen-Tye relations, JHEP, 08, 097, (2017) · Zbl 1381.83126
[86] Cruz, L.; Kniss, A.; Weinzierl, S., Properties of scattering forms and their relation to associahedra, JHEP, 03, 064, (2018) · Zbl 1388.81929
[87] P. Tourkine, On integrands and loop momentum in string and field theory, arXiv:1901.02432 [INSPIRE].
[88] R.N. Lee, Calculating multiloop integrals using dimensional recurrence relation and D-analyticity, Nucl. Phys. Proc. Suppl.205-206 (2010) 135 [arXiv:1007.2256] [INSPIRE].
[89] D.A. Kosower and K.J. Larsen, Maximal unitarity at two loops, Phys. Rev.D 85 (2012) 045017 [arXiv:1108.1180] [INSPIRE].
[90] Caron-Huot, S.; Larsen, KJ, Uniqueness of two-loop master contours, JHEP, 10, 026, (2012)
[91] S. Abreu, R. Britto, C. Duhr and E. Gardi, Algebraic structure of cut Feynman integrals and the diagrammatic coaction, Phys. Rev. Lett.119 (2017) 051601 [arXiv:1703.05064] [INSPIRE]. · Zbl 1383.81321
[92] S. Abreu, R. Britto, C. Duhr, E. Gardi and J. Matthew, Coaction for Feynman integrals and diagrams, PoS(LL2018)047 (2018) [arXiv:1808.00069] [INSPIRE].
[93] E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys.A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
[94] Y. Goto and K. Matsumoto, The monodromy representation and twisted period relations for Appells hypergeometric function F_{4}, Nagoya Math. J.217 (2015) 61.
[95] K. Matsumoto, Monodromy and Pfaffian of Lauricellas F_{\(D\)}in terms of the intersection forms of twisted (co)homology groups, Kyushu J. Math.67 (2013) 367.
[96] Y. Goto, Twisted period relations for Lauricellas hypergeometric functions F_{\(A\)}, Osaka J. Math.52 (2015) 861.
[97] Y. Goto, Contiguity relations of Lauricellas F_{\(D\)}revisited, Tohoku Math. J.69 (2017) 287.
[98] K. Matsumoto, Relative twisted homology and cohomology groups associated with Lauricellas F_{\(D\)}, arXiv:1804.00366.
[99] A.V. Smirnov, FIRE5: a C++ implementation of Feynman Integral REduction, Comput. Phys. Commun.189 (2015) 182 [arXiv:1408.2372] [INSPIRE].
[100] S. Laporta, High precision ϵ-expansions of massive four loop vacuum bubbles, Phys. Lett.B 549 (2002) 115 [hep-ph/0210336] [INSPIRE]. · Zbl 1001.81063
[101] S. Laporta and E. Remiddi, The analytical value of the electron (\(g\) − 2) at order\(α\)3in QED, Phys. Lett.B 379 (1996) 283 [hep-ph/9602417] [INSPIRE].
[102] S. Laporta, High precision ϵ-expansions of three loop master integrals contributing to the electron g − 2 in QED, Phys. Lett.B 523 (2001) 95 [hep-ph/0111123] [INSPIRE].
[103] D.J. Broadhurst, J. Fleischer and O.V. Tarasov, Two loop two point functions with masses: asymptotic expansions and Taylor series, in any dimension, Z. Phys.C 60 (1993) 287 [hep-ph/9304303] [INSPIRE].
[104] U. Aglietti, R. Bonciani, G. Degrassi and A. Vicini, Analytic results for virtual QCD corrections to Higgs production and decay, JHEP01 (2007) 021 [hep-ph/0611266] [INSPIRE].
[105] C. Anastasiou, S. Beerli, S. Bucherer, A. Daleo and Z. Kunszt, Two-loop amplitudes and master integrals for the production of a Higgs boson via a massive quark and a scalar-quark loop, JHEP01 (2007) 082 [hep-ph/0611236] [INSPIRE].
[106] V.A. Smirnov, Analytical result for dimensionally regularized massless on shell double box, Phys. Lett.B 460 (1999) 397 [hep-ph/9905323] [INSPIRE].
[107] V.A. Smirnov and O.L. Veretin, Analytical results for dimensionally regularized massless on-shell double boxes with arbitrary indices and numerators, Nucl. Phys.B 566 (2000) 469 [hep-ph/9907385] [INSPIRE].
[108] Becchetti, M.; Bonciani, R., Two-loop master integrals for the planar QCD massive corrections to di-photon and di-jet hadro-production, JHEP, 01, 048, (2018)
[109] D.H. Bailey and J.M. Borwein, PSLQ: an algorithm to discover integer relations, tech. rep., (2009).
[110] Henn, JM; Smirnov, VA, Analytic results for two-loop master integrals for Bhabha scattering I, JHEP, 11, 041, (2013)
[111] K. Melnikov, L. Tancredi and C. Wever, Two-loop ggHg amplitude mediated by a nearly massless quark, JHEP11 (2016) 104 [arXiv:1610.03747] [INSPIRE].
[112] R. Bonciani, V. Del Duca, H. Frellesvig, J.M. Henn, F. Moriello and V.A. Smirnov, Two-loop planar master integrals for Higgs → 3 partons with full heavy-quark mass dependence, JHEP12 (2016) 096 [arXiv:1609.06685] [INSPIRE].
[113] Papadopoulos, CG; Tommasini, D.; Wever, C., The pentabox master integrals with the simplified differential equations approach, JHEP, 04, 078, (2016)
[114] L. Mattiazzi, Multiparticle scattering amplitudes at two-loop, master thesis, Univ. of Padua, Padua, Italy (2018).
[115] F. Gasparotto, A modern approach to Feynman integrals and differential equations, master thesis, Univ. of Padua, Padua, Italy (2018).
[116] D. Chicherin, T. Gehrmann, J.M. Henn, N.A. Lo Presti, V. Mitev and P. Wasser, Analytic result for the nonplanar hexa-box integrals, JHEP03 (2019) 042 [arXiv:1809.06240] [INSPIRE]. · Zbl 1414.81255
[117] P. Orlik and H. Terao, Arrangements of hyperplanes, Grund. math. Wiss., Springer, Berlin Heidelberg, Germany (2013).
[118] P. Mastrolia, Double-cut of scattering amplitudes and Stokestheorem, Phys. Lett.B 678 (2009) 246 [arXiv:0905.2909] [INSPIRE].
[119] P. Mastrolia, Unitarity-cuts and Berrys phase, Lett. Math. Phys.91 (2010) 199 [arXiv:0906.3789] [INSPIRE].
[120] Ellis, RK; Kunszt, Z.; Melnikov, K.; Zanderighi, G., One-loop calculations in quantum field theory: from Feynman diagrams to unitarity cuts, Phys. Rept., 518, 141, (2012)
[121] Duhr, C., Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes, JHEP, 08, 043, (2012) · Zbl 1397.16028
[122] Abreu, S.; Britto, R.; Duhr, C.; Gardi, E., Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case, JHEP, 12, 090, (2017) · Zbl 1383.81321
[123] Broedel, J.; Duhr, C.; Dulat, F.; Penante, B.; Tancredi, L., Elliptic Feynman integrals and pure functions, JHEP, 01, 023, (2019) · Zbl 1409.81162
[124] J.L. Bourjaily, Y.-H. He, A.J. Mcleod, M. Von Hippel and M. Wilhelm, Traintracks through Calabi-Yau manifolds: scattering amplitudes beyond elliptic polylogarithms, Phys. Rev. Lett.121 (2018) 071603 [arXiv:1805.09326] [INSPIRE].
[125] J. Milnor, Morse theory, Ann. Math. Stud.51, Princeton University Press, Princeton, NJ, U.S.A. (2016).
[126] Y. Zhang, private communication.
[127] R. Hwa and V. Teplitz, Homology and Feynman integrals, Mathematical Physics Monograph Series, W.A. Benjamin, U.S.A. (1966).
[128] D. Binosi, J. Collins, C. Kaufhold and L. Theussl, JaxoDraw: a graphical user interface for drawing Feynman diagrams. Version 2\(.\)0 release notes, Comput. Phys. Commun.180 (2009) 1709 [arXiv:0811.4113] [INSPIRE].
[129] J.A.M. Vermaseren, Axodraw, Comput. Phys. Commun.83 (1994) 45 [INSPIRE].
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.