Abstract
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 ϵ.
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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].
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].
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].
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].
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].
J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].
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].
R.N. Lee, Reducing differential equations for multiloop master integrals, JHEP 04 (2015) 108 [arXiv:1411.0911] [INSPIRE].
O. Gituliar and V. Magerya, Fuchsia and master integrals for splitting functions from differential equations in QCD, PoS(LL2016)030 [arXiv:1607.00759] [INSPIRE].
O. Gituliar and V. Magerya, Fuchsia: a tool for reducing differential equations for Feynman master integrals to epsilon form, Comput. Phys. Commun. 219 (2017) 329 [arXiv:1701.04269] [INSPIRE].
M. Prausa, epsilon: a tool to find a canonical basis of master integrals, Comput. Phys. Commun. 219 (2017) 361 [arXiv:1701.00725] [INSPIRE].
C. Meyer, Transforming differential equations of multi-loop Feynman integrals into canonical form, JHEP 04 (2017) 006 [arXiv:1611.01087] [INSPIRE].
C. Meyer, Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA, Comput. Phys. Commun. 222 (2018) 295 [arXiv:1705.06252] [INSPIRE].
E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998) 497 [arXiv:1105.2076] [INSPIRE].
D. Maître, HPL, a Mathematica implementation of the harmonic polylogarithms, Comput. Phys. Commun. 174 (2006) 222 [hep-ph/0507152] [INSPIRE].
J. Vollinga and S. Weinzierl, Numerical evaluation of multiple polylogarithms, Comput. Phys. Commun. 167 (2005) 177 [hep-ph/0410259] [INSPIRE].
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].
R.N. Lee and A.A. Pomeransky, Normalized Fuchsian form on Riemann sphere and differential equations for multiloop integrals, arXiv:1707.07856 [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].
R. Bonciani et al., Two-loop planar master integrals for Higgs → 3 partons with full heavy-quark mass dependence, JHEP 12 (2016) 096 [arXiv:1609.06685] [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].
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].
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].
L. Adams and S. Weinzierl, Feynman integrals and iterated integrals of modular forms, arXiv:1704.08895 [INSPIRE].
E. Remiddi and L. Tancredi, An elliptic generalization of multiple polylogarithms, Nucl. Phys. B 925 (2017) 212 [arXiv:1709.03622] [INSPIRE].
M. Czakon, Tops from light quarks: full mass dependence at two-loops in QCD, Phys. Lett. B 664 (2008) 307 [arXiv:0803.1400] [INSPIRE].
P. Bärnreuther, M. Czakon and P. Fiedler, Virtual amplitudes and threshold behaviour of hadronic top-quark pair-production cross sections, JHEP 02 (2014) 078 [arXiv:1312.6279] [INSPIRE].
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].
B.A. Kniehl, A.F. Pikelner and O.L. Veretin, Three-loop massive tadpoles and polylogarithms through weight six, JHEP 08 (2017) 024 [arXiv:1705.05136] [INSPIRE].
R. Mueller and D.G. Öztürk, On the computation of finite bottom-quark mass effects in Higgs boson production, JHEP 08 (2016) 055 [arXiv:1512.08570] [INSPIRE].
J.M. Henn, A.V. Smirnov and V.A. Smirnov, Analytic results for planar three-loop integrals for massive form factors, JHEP 12 (2016) 144 [arXiv:1611.06523] [INSPIRE].
W. Wasow, Asymptotic expansions for ordinary differential equations, John Wiley & Sons, Inc., New York U.S.A. (1965).
K. Melnikov, L. Tancredi and C. Wever, Two-loop gg → Hg amplitude mediated by a nearly massless quark, JHEP 11 (2016) 104 [arXiv:1610.03747] [INSPIRE].
A.V. Smirnov, Algorithm FIRE — Feynman Integral REduction, JHEP 10 (2008) 107 [arXiv:0807.3243] [INSPIRE].
A.V. Smirnov and V.A. Smirnov, FIRE4, LiteRed and accompanying tools to solve integration by parts relations, Comput. Phys. Commun. 184 (2013) 2820 [arXiv:1302.5885] [INSPIRE].
A.V. Smirnov, FIRE5: a C++ implementation of Feynman Integral REduction, Comput. Phys. Commun. 189 (2015) 182 [arXiv:1408.2372] [INSPIRE].
R.N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE].
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].
A.V. Smirnov, FIESTA4: optimized Feynman integral calculations with GPU support, Comput. Phys. Commun. 204 (2016) 189 [arXiv:1511.03614] [INSPIRE].
J.M. Henn, A.V. Smirnov and V.A. Smirnov, Evaluating single-scale and/or non-planar diagrams by differential equations, JHEP 03 (2014) 088 [arXiv:1312.2588] [INSPIRE].
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Lee, R.N., Smirnov, A.V. & Smirnov, V.A. Solving differential equations for Feynman integrals by expansions near singular points. J. High Energ. Phys. 2018, 8 (2018). https://doi.org/10.1007/JHEP03(2018)008
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DOI: https://doi.org/10.1007/JHEP03(2018)008