Abstract
In this paper we study the calculation of multiloop Feynman integrals that cannot be expressed in terms of multiple polylogarithms. We show in detail how certain types of two- and three-point functions at two loops, which appear in the calculation of higher order corrections in QED, QCD and in the electroweak theory (EW), can naturally be expressed in terms of a recently introduced elliptic generalisation of multiple polylogarithms by direct integration over their Feynman parameter representation. Moreover, we show that in all examples that we considered a basis of pure Feynman integrals can be found.
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References
A.V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158 [INSPIRE].
Z. Bern, L.J. Dixon and D.A. Kosower, Dimensionally regulated pentagon integrals, Nucl. Phys. B 412 (1994) 751 [hep-ph/9306240] [INSPIRE].
E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim. A 110 (1997) 1435 [hep-th/9711188] [INSPIRE].
T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].
E.E. Kummer, Über die Transcendenten, welche aus wiederholten Integrationen rationaler Formeln entstehen, J. Reine Angew. Math. 21 (1840) 74.
N. Nielsen, Der Eulersche Dilogarithmus und seine Verallgemeinerungen, Nova Acta Academiae Caesareae Leopoldino-Carolinae Germanicae Naturae Curiosorum, volume 90, Kaiserlich-Leopoldinisch-Carolinische Deutsche Akademie der Naturforscher, Druck Ehrhardt Karras, Halle Germany (1909).
A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998) 497 [arXiv:1105.2076] [INSPIRE].
E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
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].
A.B. Goncharov, Multiple polylogarithms and mixed Tate motives, math.AG/0103059.
C. Duhr, H. Gangl and J.R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP 10 (2012) 075 [arXiv:1110.0458] [INSPIRE].
C. Duhr, Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes, JHEP 08 (2012) 043 [arXiv:1203.0454] [INSPIRE].
J. Vollinga and S. Weinzierl, Numerical evaluation of multiple polylogarithms, Comput. Phys. Commun. 167 (2005) 177 [hep-ph/0410259] [INSPIRE].
F. Brown, The Massless higher-loop two-point function, Commun. Math. Phys. 287 (2009) 925 [arXiv:0804.1660] [INSPIRE].
E. Panzer, Feynman integrals and hyperlogarithms, Ph.D. Thesis, Institut für Mathematik, Humboldt-Universität zu Berlin, Berlin Germany (2015) [arXiv:1506.07243] [INSPIRE].
J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].
A. Sabry, Fourth order spectral functions for the electron propagator, Nucl. Phys. 33 (1962) 401.
D.J. Broadhurst, The Master Two Loop Diagram With Masses, Z. Phys. C 47 (1990) 115 [INSPIRE].
S. Bauberger, F.A. Berends, M. Böhm and M. Buza, Analytical and numerical methods for massive two loop selfenergy diagrams, Nucl. Phys. B 434 (1995) 383 [hep-ph/9409388] [INSPIRE].
S. Bauberger and M. Böhm, Simple one-dimensional integral representations for two loop selfenergies: The Master diagram, Nucl. Phys. B 445 (1995) 25 [hep-ph/9501201] [INSPIRE].
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].
B.A. Kniehl, A.V. Kotikov, A. Onishchenko and O. Veretin, Two-loop sunset diagrams with three massive lines, Nucl. Phys. B 738 (2006) 306 [hep-ph/0510235] [INSPIRE].
U. Aglietti, R. Bonciani, L. Grassi and E. Remiddi, The Two loop crossed ladder vertex diagram with two massive exchanges, Nucl. Phys. B 789 (2008) 45 [arXiv:0705.2616] [INSPIRE].
M. Czakon and A. Mitov, Inclusive Heavy Flavor Hadroproduction in NLO QCD: The Exact Analytic Result, Nucl. Phys. B 824 (2010) 111 [arXiv:0811.4119] [INSPIRE].
F. Brown and A. Levin, Multiple Elliptic Polylogarithms, arXiv:1110.6917.
S. Bloch and P. Vanhove, The elliptic dilogarithm for the sunset graph, J. Number Theor. 148 (2015) 328 [arXiv:1309.5865] [INSPIRE].
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].
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].
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].
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].
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].
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].
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, JHEP 12 (2016) 096 [arXiv:1609.06685] [INSPIRE].
S. Bloch, M. Kerr and P. Vanhove, Local mirror symmetry and the sunset Feynman integral, Adv. Theor. Math. Phys. 21 (2017) 1373 [arXiv:1601.08181] [INSPIRE].
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].
G. Passarino, Elliptic Polylogarithms and Basic Hypergeometric Functions, Eur. Phys. J. C 77 (2017) 77 [arXiv:1610.06207] [INSPIRE].
A. von Manteuffel and L. Tancredi, A non-planar two-loop three-point function beyond multiple polylogarithms, JHEP 06 (2017) 127 [arXiv:1701.05905] [INSPIRE].
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].
E. Remiddi and L. Tancredi, An Elliptic Generalization of Multiple Polylogarithms, Nucl. Phys. B 925 (2017) 212 [arXiv:1709.03622] [INSPIRE].
J. Broedel, C. Duhr, F. Dulat and L. Tancredi, Elliptic polylogarithms and iterated integrals on elliptic curves. Part I. General formalism, JHEP 05 (2018) 093 [arXiv:1712.07089] [INSPIRE].
J. Ablinger et al., Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams, J. Math. Phys. 59 (2018) 062305 [arXiv:1706.01299] [INSPIRE].
L.-B. Chen, Y. Liang and C.-F. Qiao, NNLO QCD corrections to γ + η c(η b) exclusive production in electron-positron collision, JHEP 01 (2018) 091 [arXiv:1710.07865] [INSPIRE].
M. Hidding and F. Moriello, All orders structure and efficient computation of linearly reducible elliptic Feynman integrals, JHEP 01 (2019) 169 [arXiv:1712.04441] [INSPIRE].
J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series, JHEP 08 (2018) 014 [arXiv:1803.10256] [INSPIRE].
L. Adams, E. Chaubey and S. Weinzierl, Planar Double Box Integral for Top Pair Production with a Closed Top Loop to all orders in the Dimensional Regularization Parameter, Phys. Rev. Lett. 121 (2018) 142001 [arXiv:1804.11144] [INSPIRE].
L. Adams, E. Chaubey and S. Weinzierl, Analytic results for the planar double box integral relevant to top-pair production with a closed top loop, JHEP 10 (2018) 206 [arXiv:1806.04981] [INSPIRE].
J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, From modular forms to differential equations for Feynman integrals, in proceedings of the KMPB Conference: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, Zeuthen, Germany, 23–26 October 2017, arXiv:1807.00842 [INSPIRE].
L. Adams and S. Weinzierl, On a class of Feynman integrals evaluating to iterated integrals of modular forms, in proceedings of the KMPB Conference: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, Zeuthen, Germany, 23–26 October 2017, arXiv:1807.01007 [INSPIRE].
J. Blümlein, A. De Freitas, M. Van Hoeij, E. Imamoglu, P. Marquard and C. Schneider, The ρ parameter at three loops and elliptic integrals, PoS(LL2018)017 (2018) [arXiv:1807.05287] [INSPIRE].
J. Blümlein, Iterative Non-iterative Integrals in Quantum Field Theory, in proceedings of the KMPB Conference: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, Zeuthen, Germany, 23–26 October 2017, arXiv:1808.08128 [INSPIRE].
P. Vanhove, Feynman integrals, toric geometry and mirror symmetry, in proceedings of the KMPB Conference: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, Zeuthen, Germany, 23–26 October 2017, arXiv:1807.11466 [INSPIRE].
J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, Elliptic Feynman integrals and pure functions, JHEP 01 (2019) 023 [arXiv:1809.10698] [INSPIRE].
J. Broedel, C. Duhr, F. Dulat and L. Tancredi, Elliptic polylogarithms and iterated integrals on elliptic curves II: an application to the sunrise integral, Phys. Rev. D 97 (2018) 116009 [arXiv:1712.07095] [INSPIRE].
L.J. Dixon, J.M. Drummond and J.M. Henn, Analytic result for the two-loop six-point NMHV amplitude in N = 4 super Yang-Mills theory, JHEP 01 (2012) 024 [arXiv:1111.1704] [INSPIRE].
L.J. Dixon, J.M. Drummond and J.M. Henn, Bootstrapping the three-loop hexagon, JHEP 11 (2011) 023 [arXiv:1108.4461] [INSPIRE].
L.J. Dixon, J.M. Drummond, M. von Hippel and J. Pennington, Hexagon functions and the three-loop remainder function, JHEP 12 (2013) 049 [arXiv:1308.2276] [INSPIRE].
L.J. Dixon and M. von Hippel, Bootstrapping an NMHV amplitude through three loops, JHEP 10 (2014) 065 [arXiv:1408.1505] [INSPIRE].
L.J. Dixon, J.M. Drummond, C. Duhr and J. Pennington, The four-loop remainder function and multi-Regge behavior at NNLLA in planar N = 4 super-Yang-Mills theory, JHEP 06 (2014) 116 [arXiv:1402.3300] [INSPIRE].
J.M. Drummond, G. Papathanasiou and M. Spradlin, A Symbol of Uniqueness: The Cluster Bootstrap for the 3-Loop MHV Heptagon, JHEP 03 (2015) 072 [arXiv:1412.3763] [INSPIRE].
L.J. Dixon, M. von Hippel and A.J. McLeod, The four-loop six-gluon NMHV ratio function, JHEP 01 (2016) 053 [arXiv:1509.08127] [INSPIRE].
S. Caron-Huot, L.J. Dixon, A. McLeod and M. von Hippel, Bootstrapping a Five-Loop Amplitude Using Steinmann Relations, Phys. Rev. Lett. 117 (2016) 241601 [arXiv:1609.00669] [INSPIRE].
L.J. Dixon, J. Drummond, T. Harrington, A.J. McLeod, G. Papathanasiou and M. Spradlin, Heptagons from the Steinmann Cluster Bootstrap, JHEP 02 (2017) 137 [arXiv:1612.08976] [INSPIRE].
S. Caron-Huot and K.J. Larsen, Uniqueness of two-loop master contours, JHEP 10 (2012) 026 [arXiv:1205.0801] [INSPIRE].
J.L. Bourjaily, A.J. McLeod, M. von Hippel and M. Wilhelm, Bounded Collection of Feynman Integral Calabi-Yau Geometries, Phys. Rev. Lett. 122 (2019) 031601 [arXiv:1810.07689] [INSPIRE].
S. Bloch, M. Kerr and P. Vanhove, A Feynman integral via higher normal functions, Compos. Math. 151 (2015) 2329 [arXiv:1406.2664] [INSPIRE].
J.L. Bourjaily, A.J. McLeod, M. Spradlin, M. von Hippel and M. Wilhelm, Elliptic Double-Box Integrals: Massless Scattering Amplitudes beyond Polylogarithms, Phys. Rev. Lett. 120 (2018) 121603 [arXiv:1712.02785] [INSPIRE].
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].
N. Matthes, Elliptic Multiple Zeta Values, Ph.D. Thesis, Universität Hamburg, Hamburg Germany (2016).
J. Broedel, C.R. Mafra, N. Matthes and O. Schlotterer, Elliptic multiple zeta values and one-loop superstring amplitudes, JHEP 07 (2015) 112 [arXiv:1412.5535] [INSPIRE].
R. Bonciani, G. Degrassi, P.P. Giardino and R. Gröber, A Numerical Routine for the Crossed Vertex Diagram with a Massive-Particle Loop, arXiv:1812.02698 [INSPIRE].
H. Cheng and T.T. Wu, Expanding protons: scattering at high-energies, MIT Press, Cambridge U.S.A. (1987) [INSPIRE].
J. Bosma, M. Sogaard and Y. Zhang, Maximal Cuts in Arbitrary Dimension, JHEP 08 (2017) 051 [arXiv:1704.04255] [INSPIRE].
M. Harley, F. Moriello and R.M. Schabinger, Baikov-Lee Representations Of Cut Feynman Integrals, JHEP 06 (2017) 049 [arXiv:1705.03478] [INSPIRE].
U. Aglietti and R. Bonciani, Master integrals with 2 and 3 massive propagators for the 2-loop electroweak form-factor — planar case, Nucl. Phys. B 698 (2004) 277 [hep-ph/0401193] [INSPIRE].
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Broedel, J., Duhr, C., Dulat, F. et al. Elliptic polylogarithms and Feynman parameter integrals. J. High Energ. Phys. 2019, 120 (2019). https://doi.org/10.1007/JHEP05(2019)120
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DOI: https://doi.org/10.1007/JHEP05(2019)120