Baikov-Lee representations of cut Feynman integrals. (English) Zbl 1380.81132
Summary: We develop a general framework for the evaluation of \( d\)-dimensional cut Feynman integrals based on the Baikov-Lee representation of purely-virtual Feynman integrals. We implement the generalized Cutkosky cutting rule using Cauchy’s residue theorem and identify a set of constraints which determine the integration domain. The method applies equally well to Feynman integrals with a unitarity cut in a single kinematic channel and to maximally-cut Feynman integrals. Our cut Baikov-Lee representation reproduces the expected relation between cuts and discontinuities in a given kinematic channel and furthermore makes the dependence on the kinematic variables manifest from the beginning. By combining the Baikov-Lee representation of maximally-cut Feynman integrals and the properties of periods of algebraic curves, we are able to obtain complete solution sets for the homogeneous differential equations satisfied by Feynman integrals which go beyond multiple polylogarithms. We apply our formalism to the direct evaluation of a number of interesting cut Feynman integrals.
MSC:
| 81Q30 | Feynman integrals and graphs; applications of algebraic topology and algebraic geometry |
| 81U05 | \(2\)-body potential quantum scattering theory |
| 81V05 | Strong interaction, including quantum chromodynamics |
Software:
FIRE; Reduze; JaxoDraw; Mincer; Wolfram Functions Site; pySecDec; FIRE5; HyperInt; Forcer; LiteRed; AxodrawReferences:
| [1] | R.P. Feynman, Space-time approach to quantum electrodynamics, Phys. Rev.76 (1949) 769 [INSPIRE]. · Zbl 0038.13302 · doi:10.1103/PhysRev.76.769 |
| [2] | A.V. Kotikov, Differential equations method: new technique for massive Feynman diagrams calculation, Phys. Lett.B 254 (1991) 158 [INSPIRE]. · doi:10.1016/0370-2693(91)90413-K |
| [3] | A.V. Kotikov, Differential equations method: the calculation of vertex type Feynman diagrams, Phys. Lett.B 259 (1991) 314 [INSPIRE]. · doi:10.1016/0370-2693(91)90834-D |
| [4] | 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]. · Zbl 1020.81734 |
| [5] | Z. Bern, L.J. Dixon and D.A. Kosower, Dimensionally regulated one loop integrals, Phys. Lett. B 302 (1993) 299 [Erratum ibid.B 318 (1993) 649] [hep-ph/9212308] [INSPIRE]. · Zbl 1007.81512 |
| [6] | Z. Bern, L.J. Dixon and D.A. Kosower, Dimensionally regulated pentagon integrals, Nucl. Phys.B 412 (1994) 751 [hep-ph/9306240] [INSPIRE]. · Zbl 1007.81512 |
| [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] | F.V. Tkachov, A theorem on analytical calculability of four loop renormalization group functions, Phys. Lett.B 100 (1981) 65 [INSPIRE]. · doi:10.1016/0370-2693(81)90288-4 |
| [10] | 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]. · doi:10.1016/0550-3213(81)90199-1 |
| [11] | 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 |
| [12] | A. von Manteuffel and C. Studerus, Reduze 2 — distributed Feynman integral reduction, arXiv:1201.4330 [INSPIRE]. · Zbl 1219.81133 |
| [13] | 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]. · Zbl 1330.81151 · doi:10.1016/j.physletb.2015.03.029 |
| [14] | A.V. Smirnov, FIRE5: a C++ implementation of Feynman Integral REduction, Comput. Phys. Commun.189 (2015) 182 [arXiv:1408.2372] [INSPIRE]. · Zbl 1344.81030 · doi:10.1016/j.cpc.2014.11.024 |
| [15] | S.G. Gorishnii, S.A. Larin, L.R. Surguladze and F.V. Tkachov, Mincer: program for multiloop calculations in quantum field theory for the Schoonschip system, Comput. Phys. Commun.55 (1989) 381 [INSPIRE]. · doi:10.1016/0010-4655(89)90134-3 |
| [16] | 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]. |
| [17] | 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]. |
| [18] | R.N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE]. · Zbl 1196.81130 |
| [19] | B. Ruijl, T. Ueda and J. Vermaseren, The diamond rule for multi-loop Feynman diagrams, Phys. Lett.B 746 (2015) 347 [arXiv:1504.08258] [INSPIRE]. · Zbl 1343.81104 · doi:10.1016/j.physletb.2015.05.015 |
| [20] | H. Ita, Two-loop integrand decomposition into master integrals and surface terms, Phys. Rev.D 94 (2016) 116015 [arXiv:1510.05626] [INSPIRE]. |
| [21] | 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]. |
| [22] | T. Ueda, B. Ruijl and J.A.M. Vermaseren, Calculating four-loop massless propagators with Forcer, J. Phys. Conf. Ser.762 (2016) 012060 [arXiv:1604.08767] [INSPIRE]. · doi:10.1088/1742-6596/762/1/012060 |
| [23] | S. Abreu, F. Febres Cordero, H. Ita, M. Jaquier and B. Page, Subleading poles in the numerical unitarity method at two loops, Phys. Rev.D 95 (2017) 096011 [arXiv:1703.05255] [INSPIRE]. |
| [24] | S. Abreu, F. Febres Cordero, H. Ita, M. Jaquier, B. Page and M. Zeng, Two-loop four-gluon amplitudes with the numerical unitarity method, arXiv:1703.05273 [INSPIRE]. · Zbl 1114.68598 |
| [25] | R.N. Lee, Calculating multiloop integrals using dimensional recurrence relation and D-analyticity, Nucl. Phys. Proc. Suppl.B 205-206 (2010) 135 [arXiv:1007.2256] [INSPIRE]. · doi:10.1016/j.nuclphysbps.2010.08.032 |
| [26] | C. Anastasiou, C. Duhr, F. Dulat and B. Mistlberger, Soft triple-real radiation for Higgs production at N3LO, JHEP07 (2013) 003 [arXiv:1302.4379] [INSPIRE]. · doi:10.1007/JHEP07(2013)003 |
| [27] | R. Kumar, Covariant phase-space calculations of n-body decay and production processes, Phys. Rev.185 (1969) 1865 [INSPIRE]. · doi:10.1103/PhysRev.185.1865 |
| [28] | A.G. Grozin, Integration by parts: an introduction, Int. J. Mod. Phys.A 26 (2011) 2807 [arXiv:1104.3993] [INSPIRE]. · Zbl 1247.81138 · doi:10.1142/S0217751X11053687 |
| [29] | R.E. Cutkosky, Singularities and discontinuities of Feynman amplitudes, J. Math. Phys.1 (1960) 429 [INSPIRE]. · Zbl 0122.22605 · doi:10.1063/1.1703676 |
| [30] | R.N. Lee and V.A. Smirnov, The dimensional recurrence and analyticity method for multicomponent master integrals: using unitarity cuts to construct homogeneous solutions, JHEP12 (2012) 104 [arXiv:1209.0339] [INSPIRE]. · Zbl 1397.81073 · doi:10.1007/JHEP12(2012)104 |
| [31] | 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 · doi:10.1016/j.nuclphysb.2016.12.021 |
| [32] | H. Frellesvig and C.G. Papadopoulos, Cuts of Feynman integrals in Baikov representation, JHEP04 (2017) 083 [arXiv:1701.07356] [INSPIRE]. · Zbl 1378.81039 · doi:10.1007/JHEP04(2017)083 |
| [33] | J.A. Lappo-Danilevsky, Théorie algorithmique des corps de Riemann (in French), Rec. Math. Moscou6 (1927) 113. · JFM 53.0405.01 |
| [34] | A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett.5 (1998) 497 [arXiv:1105.2076] [INSPIRE]. · Zbl 0961.11040 · doi:10.4310/MRL.1998.v5.n4.a7 |
| [35] | E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys.A 15 (2000) 725 [hep-ph/9905237] [INSPIRE]. · Zbl 0951.33003 |
| [36] | T. Gehrmann and E. Remiddi, Two loop master integrals for γ∗ → 3 jets: the planar topologies, Nucl. Phys.B 601 (2001) 248 [hep-ph/0008287] [INSPIRE]. · Zbl 1353.81097 |
| [37] | A.V. Kotikov, The property of maximal transcendentality in the N = 4 supersymmetric Yang-Mills, arXiv:1005.5029 [INSPIRE]. · Zbl 1286.81136 |
| [38] | J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett.110 (2013) 251601 [arXiv:1304.1806] [INSPIRE]. · doi:10.1103/PhysRevLett.110.251601 |
| [39] | J.M. Henn, A.V. Smirnov and V.A. Smirnov, Analytic results for planar three-loop four-point integrals from a Knizhnik-Zamolodchikov equation, JHEP07 (2013) 128 [arXiv:1306.2799] [INSPIRE]. · Zbl 1342.81352 · doi:10.1007/JHEP07(2013)128 |
| [40] | K.-T. Chen, Iterated path integrals, Bull. Am. Math. Soc.83 (1977) 831 [INSPIRE]. · Zbl 0389.58001 · doi:10.1090/S0002-9904-1977-14320-6 |
| [41] | A.B. Goncharov, Galois symmetries of fundamental groupoids and noncommutative geometry, Duke Math. J.128 (2005) 209 [math/0208144] [INSPIRE]. · Zbl 1095.11036 |
| [42] | F. Brown, On the decomposition of motivic multiple zeta values, arXiv:1102.1310 [INSPIRE]. · Zbl 1321.11087 |
| [43] | C. Duhr, H. Gangl and J.R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP10 (2012) 075 [arXiv:1110.0458] [INSPIRE]. · Zbl 1397.81355 · doi:10.1007/JHEP10(2012)075 |
| [44] | C. Duhr, Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes, JHEP08 (2012) 043 [arXiv:1203.0454] [INSPIRE]. · Zbl 1397.16028 · doi:10.1007/JHEP08(2012)043 |
| [45] | T. Gehrmann, A. von Manteuffel and L. Tancredi, The two-loop helicity amplitudes forqq¯′→V1V2→\[4 q\overline{q}^{\prime}\to{V}_1{V}_2\to\ 4\] leptons, JHEP09 (2015) 128 [arXiv:1503.04812] [INSPIRE]. · doi:10.1007/JHEP09(2015)128 |
| [46] | A. von Manteuffel and L. Tancredi, The two-loop helicity amplitudes for gg → V1V2 → 4 leptons, JHEP06 (2015) 197 [arXiv:1503.08835] [INSPIRE]. · doi:10.1007/JHEP06(2015)197 |
| [47] | 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 → Zγ, JHEP08 (2015) 108 [arXiv:1505.00567] [INSPIRE]. · doi:10.1007/JHEP08(2015)108 |
| [48] | A. von Manteuffel and R.M. Schabinger, Numerical multi-loop calculations via finite integrals and one-mass EW-QCD Drell-Yan master integrals, JHEP04 (2017) 129 [arXiv:1701.06583] [INSPIRE]. · doi:10.1007/JHEP04(2017)129 |
| [49] | H. Frellesvig, D. Tommasini and C. Wever, On the reduction of generalized polylogarithms to Linand Li2,2and on the evaluation thereof, JHEP03 (2016) 189 [arXiv:1601.02649] [INSPIRE]. · Zbl 1388.33001 · doi:10.1007/JHEP03(2016)189 |
| [50] | F. Brown and A. Levin, Multiple elliptic polylogarithms, arXiv:1110.6917. · Zbl 1344.81024 |
| [51] | S. Bloch and P. Vanhove, The elliptic dilogarithm for the sunset graph, J. Number Theor.148 (2015) 328 [arXiv:1309.5865] [INSPIRE]. · Zbl 1319.81044 · doi:10.1016/j.jnt.2014.09.032 |
| [52] | S. Bloch, M. Kerr and P. Vanhove, A Feynman integral via higher normal functions, Compos. Math.151 (2015) 2329 [arXiv:1406.2664] [INSPIRE]. · Zbl 1365.81090 · doi:10.1112/S0010437X15007472 |
| [53] | L. Adams, C. Bogner and S. Weinzierl, The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms, J. Math. Phys.55 (2014) 102301 [arXiv:1405.5640] [INSPIRE]. · Zbl 1298.81204 · doi:10.1063/1.4896563 |
| [54] | L. Adams, C. Bogner and S. Weinzierl, The two-loop sunrise integral around four space-time dimensions and generalisations of the Clausen and Glaisher functions towards the elliptic case, J. Math. Phys.56 (2015) 072303 [arXiv:1504.03255] [INSPIRE]. · Zbl 1320.81059 · doi:10.1063/1.4926985 |
| [55] | L. Adams, C. Bogner, A. Schweitzer and S. Weinzierl, The kite integral to all orders in terms of elliptic polylogarithms, J. Math. Phys.57 (2016) 122302 [arXiv:1607.01571] [INSPIRE]. · Zbl 1353.81097 · doi:10.1063/1.4969060 |
| [56] | C. Anastasiou and K. Melnikov, Higgs boson production at hadron colliders in NNLO QCD, Nucl. Phys.B 646 (2002) 220 [hep-ph/0207004] [INSPIRE]. · Zbl 0122.22605 |
| [57] | C. Anastasiou, L.J. Dixon, K. Melnikov and F. Petriello, Dilepton rapidity distribution in the Drell-Yan process at NNLO in QCD, Phys. Rev. Lett.91 (2003) 182002 [hep-ph/0306192] [INSPIRE]. · Zbl 1247.81138 |
| [58] | C. Anastasiou, L.J. Dixon, K. Melnikov and F. Petriello, High precision QCD at hadron colliders: electroweak gauge boson rapidity distributions at NNLO, Phys. Rev.D 69 (2004) 094008 [hep-ph/0312266] [INSPIRE]. |
| [59] | M. Caffo, H. Czyz, S. Laporta and E. Remiddi, The master differential equations for the two loop sunrise selfmass amplitudes, Nuovo Cim.A 111 (1998) 365 [hep-th/9805118] [INSPIRE]. |
| [60] | 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 |
| [61] | L. Adams, C. Bogner and S. Weinzierl, The iterated structure of the all-order result for the two-loop sunrise integral, J. Math. Phys.57 (2016) 032304 [arXiv:1512.05630] [INSPIRE]. · Zbl 1333.81283 · doi:10.1063/1.4944722 |
| [62] | S. Bloch, M. Kerr and P. Vanhove, Local mirror symmetry and the sunset Feynman integral, arXiv:1601.08181 [INSPIRE]. · Zbl 1390.14123 |
| [63] | 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 |
| [64] | 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]. · doi:10.1007/JHEP12(2016)096 |
| [65] | A. von Manteuffel and L. Tancredi, A non-planar two-loop three-point function beyond multiple polylogarithms, arXiv:1701.05905 [INSPIRE]. · Zbl 1356.81136 |
| [66] | A. Primo and L. Tancredi, Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph, arXiv:1704.05465 [INSPIRE]. · Zbl 1370.81073 |
| [67] | J. Bosma, M. Sogaard and Y. Zhang, Maximal cuts in arbitrary dimension, arXiv:1704.04255 [INSPIRE]. · Zbl 1381.81146 |
| [68] | G. ’t Hooft and M.J.G. Veltman, Diagrammar, NATO Sci. Ser.B 4 (1974) 177 [INSPIRE]. |
| [69] | A. Gehrmann-De Ridder, T. Gehrmann and G. Heinrich, Four particle phase space integrals in massless QCD, Nucl. Phys.B 682 (2004) 265 [hep-ph/0311276] [INSPIRE]. · Zbl 1045.81558 |
| [70] | S. Abreu, R. Britto, C. Duhr and E. Gardi, From multiple unitarity cuts to the coproduct of Feynman integrals, JHEP10 (2014) 125 [arXiv:1401.3546] [INSPIRE]. · Zbl 1333.81148 · doi:10.1007/JHEP10(2014)125 |
| [71] | V.A. Smirnov, Evaluating Feynman integrals, Springer Tracts Mod. Phys.211 (2004) 1 [INSPIRE]. · Zbl 1098.81003 |
| [72] | A.V. Smirnov and A.V. Petukhov, The number of master integrals is finite, Lett. Math. Phys.97 (2011) 37 [arXiv:1004.4199] [INSPIRE]. · Zbl 1216.81076 · doi:10.1007/s11005-010-0450-0 |
| [73] | E. Panzer, Feynman integrals and hyperlogarithms, Ph.D. thesis, Humboldt U., Berlin Germany, (2015) [arXiv:1506.07243] [INSPIRE]. · Zbl 1344.81024 |
| [74] | F. Brown, The massless higher-loop two-point function, Commun. Math. Phys.287 (2009) 925 [arXiv:0804.1660] [INSPIRE]. · Zbl 1196.81130 · doi:10.1007/s00220-009-0740-5 |
| [75] | F.C.S. Brown, On the periods of some Feynman integrals, arXiv:0910.0114 [INSPIRE]. · Zbl 1282.81193 |
| [76] | E. Panzer, On the analytic computation of massless propagators in dimensional regularization, Nucl. Phys.B 874 (2013) 567 [arXiv:1305.2161] [INSPIRE]. · Zbl 1282.81091 · doi:10.1016/j.nuclphysb.2013.05.025 |
| [77] | E. Panzer, On hyperlogarithms and Feynman integrals with divergences and many scales, JHEP03 (2014) 071 [arXiv:1401.4361] [INSPIRE]. · doi:10.1007/JHEP03(2014)071 |
| [78] | E. Panzer, Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals, Comput. Phys. Commun.188 (2015) 148 [arXiv:1403.3385] [INSPIRE]. · Zbl 1344.81024 · doi:10.1016/j.cpc.2014.10.019 |
| [79] | A. von Manteuffel, E. Panzer and R.M. Schabinger, A quasi-finite basis for multi-loop Feynman integrals, JHEP02 (2015) 120 [arXiv:1411.7392] [INSPIRE]. · Zbl 1388.81378 · doi:10.1007/JHEP02(2015)120 |
| [80] | A. von Manteuffel, E. Panzer and R.M. Schabinger, On the computation of form factors in massless QCD with finite master integrals, Phys. Rev.D 93 (2016) 125014 [arXiv:1510.06758] [INSPIRE]. |
| [81] | M. Kompaniets and E. Panzer, Renormalization group functions of ϕ4theory in the MS-scheme to six loops, PoS(LL2016)038 [arXiv:1606.09210] [INSPIRE]. |
| [82] | C. Bogner and S. Weinzierl, Feynman graph polynomials, Int. J. Mod. Phys.A 25 (2010) 2585 [arXiv:1002.3458] [INSPIRE]. · Zbl 1193.81072 · doi:10.1142/S0217751X10049438 |
| [83] | I.M. Gel’fand and G.E. Shilov, Generalized functions, volume I: properties and operations, AMS Chelsea Publishing 377, (1964), pg. 1. · Zbl 0115.33101 |
| [84] | G.B. Folland, Fourier analysis and its applications, Wadsworth & Brooks/Cole Mathematics Series, (1992), pg. 1. · Zbl 0786.42001 |
| [85] | S. Abreu, R. Britto, C. Duhr and E. Gardi, Cuts from residues: the one-loop case, arXiv:1702.03163 [INSPIRE]. · Zbl 1380.81421 |
| [86] | Y. Li, A. von Manteuffel, R.M. Schabinger and H.X. Zhu, N3LO Higgs boson and Drell-Yan production at threshold: the one-loop two-emission contribution, Phys. Rev.D 90 (2014) 053006 [arXiv:1404.5839] [INSPIRE]. |
| [87] | Z. Bern, V. Del Duca, L.J. Dixon and D.A. Kosower, All non-maximally-helicity-violating one-loop seven-gluon amplitudes in N = 4 super-Yang-Mills theory, Phys. Rev.D 71 (2005) 045006 [hep-th/0410224] [INSPIRE]. |
| [88] | 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]. · Zbl 1178.81202 · doi:10.1016/j.nuclphysb.2005.07.014 |
| [89] | J. Carlson, S. Müller-Stach and C. Peters, Period mappings and period domains, Cambridge Studies in Advanced Mathematics 85, Cambridge University Press, Cambridge U.K., (2003), pg. 1. · Zbl 1030.14004 |
| [90] | M. Kontsevich and D. Zagier, Periods, Springer, Germany, (2001), pg. 771. · Zbl 1039.11002 |
| [91] | S. Müller-Stach, S. Weinzierl and R. Zayadeh, A second-order differential equation for the two-loop sunrise graph with arbitrary masses, Commun. Num. Theor. Phys.6 (2012) 203 [arXiv:1112.4360] [INSPIRE]. · Zbl 1275.81069 · doi:10.4310/CNTP.2012.v6.n1.a5 |
| [92] | S. Müller-Stach, S. Weinzierl and R. Zayadeh, Picard-Fuchs equations for Feynman integrals, Commun. Math. Phys.326 (2014) 237 [arXiv:1212.4389] [INSPIRE]. · Zbl 1285.81029 · doi:10.1007/s00220-013-1838-3 |
| [93] | A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge U.K., (2002), pg. 1. · Zbl 1044.55001 |
| [94] | L. Adams, C. Bogner and S. Weinzierl, The two-loop sunrise graph with arbitrary masses, J. Math. Phys.54 (2013) 052303 [arXiv:1302.7004] [INSPIRE]. · Zbl 1282.81193 · doi:10.1063/1.4804996 |
| [95] | T. Binoth and G. Heinrich, An automatized algorithm to compute infrared divergent multiloop integrals, Nucl. Phys.B 585 (2000) 741 [hep-ph/0004013] [INSPIRE]. · Zbl 1042.81565 |
| [96] | C. Bogner and S. Weinzierl, Resolution of singularities for multi-loop integrals, Comput. Phys. Commun.178 (2008) 596 [arXiv:0709.4092] [INSPIRE]. · Zbl 1196.81010 · doi:10.1016/j.cpc.2007.11.012 |
| [97] | S. Borowka et al., pySecDec: a toolbox for the numerical evaluation of multi-scale integrals, arXiv:1703.09692 [INSPIRE]. · Zbl 07693053 |
| [98] | S. Abreu, R. Britto, C. Duhr and E. Gardi, The algebraic structure of cut Feynman integrals and the diagrammatic coaction, arXiv:1703.05064 [INSPIRE]. · Zbl 1333.81148 |
| [99] | Z. Bern, L.J. Dixon and D.A. Kosower, A two loop four gluon helicity amplitude in QCD, JHEP01 (2000) 027 [hep-ph/0001001] [INSPIRE]. |
| [100] | D. Binosi and L. Theussl, JaxoDraw: a graphical user interface for drawing Feynman diagrams, Comput. Phys. Commun.161 (2004) 76 [hep-ph/0309015] [INSPIRE]. |
| [101] | J.A.M. Vermaseren, Axodraw, Comput. Phys. Commun.83 (1994) 45 [INSPIRE]. · Zbl 1114.68598 · doi:10.1016/0010-4655(94)90034-5 |
| [102] | N.N. Lebedev, Special functions and their applications, in Selected russian publications in the mathematical sciences, R.A. Silverman ed., (1965), pg. 1. · Zbl 1298.81204 |
| [103] | M.J. Schlosser, Multiple hypergeometric series: Appell series and beyond, in Computer algebra in quantum field theory — integration, summation, and special functions, C. Schneider and J. Blümlein eds., Springer-Verlag, Vienna Austria, (2013), pg. 305 [arXiv:1305.1966]. · Zbl 1343.81104 |
| [104] | L.J. Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge U.K., (1966), pg. 1. · Zbl 0135.28101 |
| [105] | Wolfram Research Inc., The Wolfram Functions website, http://functions.wolfram.com/. |
| [106] | T.H. Koornwinder, Identities of non-terminating series by Zeilberger’s algorithm, J. Comput. Appl. Math.99 (1998) 449 [math.CA/9805010]. · Zbl 0945.65015 |
| [107] | J. Letessier and G. Valent, Some integral relations involving hypergeometric functions, SIAM J. Appl. Math.48 (1988) 214. · Zbl 0634.33002 · doi:10.1137/0148010 |
| [108] | I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series, and products, seventh edition, A. Jeffrey and D. Zwillinger eds., Academic Press, U.S.A., (2007), pg. 1. · Zbl 1208.65001 |
| [109] | W.L. van Neerven, Dimensional regularization of mass and infrared singularities in two loop on-shell vertex functions, Nucl. Phys.B 268 (1986) 453 [INSPIRE]. · doi:10.1016/0550-3213(86)90165-3 |
| [110] | R.J. Gonsalves, Dimensionally regularized two loop on-shell quark form-factor, Phys. Rev.D 28 (1983) 1542 [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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
