Solving differential equations for Feynman integrals by expansions near singular points. (English) Zbl 1388.81927

Summary: We describe a strategy to solve differential equations for Feynman integrals by powers series expansions near singular points and to obtain high precision results for the corresponding master integrals. We consider Feynman integrals with two scales, i.e. non-trivially depending on one variable. The corresponding algorithm is oriented at situations where canonical form of the differential equations is impossible. We provide a computer code constructed with the help of our algorithm for a simple example of four-loop generalized sunset integrals with three equal non-zero masses and two zero masses. Our code gives values of the master integrals at any given point on the real axis with a required accuracy and a given order of expansion in the regularization parameter \(\epsilon\).


81V05 Strong interaction, including quantum chromodynamics
81U05 \(2\)-body potential quantum scattering theory
81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
Full Text: DOI arXiv


[1] Kotikov, AV, Differential equations method: new technique for massive Feynman diagrams calculation, Phys. Lett., B 254, 158, (1991)
[2] 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
[3] Remiddi, E., Differential equations for Feynman graph amplitudes, Nuovo Cim., A 110, 1435, (1997)
[4] 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
[5] 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].
[6] T. Gehrmann and E. Remiddi, Two loop master integrals for γ\^{}{*} → 3 jets: the nonplanar topologies, Nucl. Phys.B 601 (2001) 287 [hep-ph/0101124] [INSPIRE].
[7] Henn, JM, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett., 110, 251601, (2013)
[8] Chetyrkin, KG; Tkachov, FV, Integration by parts: the algorithm to calculate β-functions in 4 loops, Nucl. Phys., B 192, 159, (1981)
[9] Lee, RN, Reducing differential equations for multiloop master integrals, JHEP, 04, 108, (2015) · Zbl 1388.81109
[10] O. Gituliar and V. Magerya, Fuchsia and master integrals for splitting functions from differential equations in QCD, PoS(LL2016)030 [arXiv:1607.00759] [INSPIRE]. · Zbl 1378.65075
[11] Gituliar, O.; Magerya, V., Fuchsia: a tool for reducing differential equations for Feynman master integrals to epsilon form, Comput. Phys. Commun., 219, 329, (2017)
[12] M. Prausa, epsilon: a tool to find a canonical basis of master integrals, Comput. Phys. Commun.219 (2017) 361 [arXiv:1701.00725] [INSPIRE]. · Zbl 1344.81030
[13] Meyer, C., Transforming differential equations of multi-loop Feynman integrals into canonical form, JHEP, 04, 006, (2017) · Zbl 1378.81064
[14] Meyer, C., Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA, Comput. Phys. Commun., 222, 295, (2018)
[15] E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys.A 15 (2000) 725 [hep-ph/9905237] [INSPIRE]. · Zbl 0951.33003
[16] Goncharov, AB, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett., 5, 497, (1998) · Zbl 0961.11040
[17] D. Maître, HPL, a Mathematica implementation of the harmonic polylogarithms, Comput. Phys. Commun.174 (2006) 222 [hep-ph/0507152] [INSPIRE].
[18] J. Vollinga and S. Weinzierl, Numerical evaluation of multiple polylogarithms, Comput. Phys. Commun.167 (2005) 177 [hep-ph/0410259] [INSPIRE]. · Zbl 1196.65045
[19] C.W. Bauer, A. Frink and R. Kreckel, Introduction to the GiNaC framework for symbolic computation within the C++ programming language, J. Symb. Comput.33 (2000) 1 [cs/0004015] [INSPIRE]. · Zbl 1017.68163
[20] R.N. Lee and A.A. Pomeransky, Normalized Fuchsian form on Riemann sphere and differential equations for multiloop integrals, arXiv:1707.07856 [INSPIRE]. · Zbl 1353.81097
[21] Aglietti, U.; Bonciani, R.; Grassi, L.; Remiddi, E., The two loop crossed ladder vertex diagram with two massive exchanges, Nucl. Phys., B 789, 45, (2008) · Zbl 1151.81364
[22] Bonciani, R.; etal., Two-loop planar master integrals for Higgs → 3 partons with full heavy-quark mass dependence, JHEP, 12, 096, (2016)
[23] Primo, A.; Tancredi, L., On the maximal cut of Feynman integrals and the solution of their differential equations, Nucl. Phys., B 916, 94, (2017) · Zbl 1356.81136
[24] 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
[25] Adams, L.; Bogner, C.; Schweitzer, A.; Weinzierl, S., The kite integral to all orders in terms of elliptic polylogarithms, J. Math. Phys., 57, 122302, (2016) · Zbl 1353.81097
[26] L. Adams and S. Weinzierl, Feynman integrals and iterated integrals of modular forms, arXiv:1704.08895 [INSPIRE]. · Zbl 1393.81015
[27] Remiddi, E.; Tancredi, L., An elliptic generalization of multiple polylogarithms, Nucl. Phys., B 925, 212, (2017) · Zbl 1375.81109
[28] Czakon, M., Tops from light quarks: full mass dependence at two-loops in QCD, Phys. Lett., B 664, 307, (2008)
[29] Bärnreuther, P.; Czakon, M.; Fiedler, P., Virtual amplitudes and threshold behaviour of hadronic top-quark pair-production cross sections, JHEP, 02, 078, (2014)
[30] S. Pozzorini and E. Remiddi, Precise numerical evaluation of the two loop sunrise graph master integrals in the equal mass case, Comput. Phys. Commun.175 (2006) 381 [hep-ph/0505041] [INSPIRE]. · Zbl 1196.81075
[31] Kniehl, BA; Pikelner, AF; Veretin, OL, Three-loop massive tadpoles and polylogarithms through weight six, JHEP, 08, 024, (2017)
[32] Mueller, R.; Öztürk, DG, On the computation of finite bottom-quark mass effects in Higgs boson production, JHEP, 08, 055, (2016)
[33] Henn, JM; Smirnov, AV; Smirnov, VA, Analytic results for planar three-loop integrals for massive form factors, JHEP, 12, 144, (2016) · Zbl 1390.81179
[34] W. Wasow, Asymptotic expansions for ordinary differential equations, John Wiley & Sons, Inc., New York U.S.A. (1965). · Zbl 0133.35301
[35] Melnikov, K.; Tancredi, L.; Wever, C., Two-loop gg → hg amplitude mediated by a nearly massless quark, JHEP, 11, 104, (2016)
[36] Smirnov, AV, Algorithm FIRE — Feynman integral reduction, JHEP, 10, 107, (2008) · Zbl 1245.81033
[37] Smirnov, AV; Smirnov, VA, FIRE4, litered and accompanying tools to solve integration by parts relations, Comput. Phys. Commun., 184, 2820, (2013) · Zbl 1344.81031
[38] Smirnov, AV, FIRE5: a C++ implementation of Feynman integral reduction, Comput. Phys. Commun., 189, 182, (2015) · Zbl 1344.81030
[39] R.N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE].
[40] R.N. Lee, LiteRed 1.4: a powerful tool for reduction of multiloop integrals, J. Phys. Conf. Ser.523 (2014) 012059 [arXiv:1310.1145] [INSPIRE].
[41] Smirnov, AV, FIESTA4: optimized Feynman integral calculations with GPU support, Comput. Phys. Commun., 204, 189, (2016) · Zbl 1378.65075
[42] Henn, JM; Smirnov, AV; Smirnov, VA, Evaluating single-scale and/or non-planar diagrams by differential equations, JHEP, 03, 088, (2014)
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.