Evaluating ‘elliptic’ master integrals at special kinematic values: using differential equations and their solutions via expansions near singular points. (English) Zbl 1395.81288

Summary: This is a sequel of our previous paper [the authors, J. High Energy Phys. 2018, No. 3, Paper No. 8, 15 p. (2018; Zbl 1388.81927)]] where we described an algorithm to find a solution of differential equations for master integrals in the form of an \(\epsilon\)-expansion series with numerical coefficients. The algorithm is based on using generalized power series expansions near singular points of the differential system, solving difference equations for the corresponding coefficients in these expansions and using matching to connect series expansions at two neighboring points. Here we use our algorithm and the corresponding code for our example of four-loop generalized sunset diagrams with three massive and tw massless propagators, in order to obtain new analytical results. We analytically evaluate the master integrals at threshold, \(p^2 = 9 m^2\), in an expansion in \(\epsilon\) up to \(\epsilon^1\). With the help of our code, we obtain numerical results for the threshold master integrals in an \(\epsilon\)-expansion with the accuracy of 6000 digits and then use the PSLQ algorithm to arrive at analytical values. Our basis of constants is build from bases of multiple polylogarithm values at sixth roots of unity.


81V05 Strong interaction, including quantum chromodynamics
81U05 \(2\)-body potential quantum scattering theory


Zbl 1388.81927
Full Text: DOI arXiv


[1] E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys.A 15 (2000) 725 [hep-ph/9905237] [INSPIRE]. · Zbl 0951.33003
[2] Goncharov, AB, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett., 5, 497, (1998) · Zbl 0961.11040
[3] D. Maître, HPL, a mathematica implementation of the harmonic polylogarithms, Comput. Phys. Commun.174 (2006) 222 [hep-ph/0507152] [INSPIRE].
[4] J. Vollinga and S. Weinzierl, Numerical evaluation of multiple polylogarithms, Comput. Phys. Commun.167 (2005) 177 [hep-ph/0410259] [INSPIRE]. · Zbl 1196.65045
[5] 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
[6] Kotikov, AV, Differential equations method: new technique for massive Feynman diagrams calculation, Phys. Lett., B 254, 158, (1991)
[7] 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
[8] E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim.A 110 (1997) 1435 [hep-th/9711188] [INSPIRE].
[9] 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
[10] 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].
[11] 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].
[12] Henn, JM, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett., 110, 251601, (2013)
[13] Lee, RN, Reducing differential equations for multiloop master integrals, JHEP, 04, 108, (2015) · Zbl 1388.81109
[14] 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 1025.81002
[15] Gituliar, O.; Magerya, V., Fuchsia: a tool for reducing differential equations for Feynman master integrals to epsilon form, Comput. Phys. Commun., 219, 329, (2017)
[16] M. Prausa, epsilon: A tool to find a canonical basis of master integrals, Comput. Phys. Commun.219 (2017) 361 [arXiv:1701.00725] [INSPIRE].
[17] Meyer, C., Transforming differential equations of multi-loop Feynman integrals into canonical form, JHEP, 04, 006, (2017) · Zbl 1378.81064
[18] Meyer, C., Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA, Comput. Phys. Commun., 222, 295, (2018)
[19] R.N. Lee and A.A. Pomeransky, Normalized Fuchsian form on Riemann sphere and differential equations for multiloop integrals, arXiv:1707.07856 [INSPIRE].
[20] 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]. · Zbl 1151.81364
[21] 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]. · Zbl 1342.81139
[22] 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
[23] 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
[24] 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
[25] Adams, L.; Weinzierl, S., Feynman integrals and iterated integrals of modular forms, Commun. Num. Theor. Phys., 12, 193, (2018) · Zbl 1393.81015
[26] 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 1245.81033
[27] Remiddi, E.; Tancredi, L., An elliptic generalization of multiple polylogarithms, Nucl. Phys., B 925, 212, (2017) · Zbl 1375.81109
[28] J. Ablinger et al., Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams, J. Math. Phys.59 (2018) 062305 [arXiv:1706.01299] [INSPIRE]. · Zbl 1394.81164
[29] M. Hidding and F. Moriello, All orders structure and efficient computation of linearly reducible elliptic Feynman integrals, arXiv:1712.04441 [INSPIRE].
[30] J. Broedel, C. Duhr, F. Dulat and L. Tancredi, Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism, JHEP05 (2018) 093 [arXiv:1712.07089] [INSPIRE].
[31] 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].
[32] J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series, arXiv:1803.10256 [INSPIRE].
[33] Lee, RN; Smirnov, AV; Smirnov, VA, Solving differential equations for Feynman integrals by expansions near singular points, JHEP, 03, 008, (2018) · Zbl 1388.81927
[34] Ferguson, HRP; Bailey, DH; Arno, S., Analysis of PSLQ, an integer relation finding algorithm, Math. Comput., 68, 351, (1999) · Zbl 0927.11055
[35] J.M. Henn, A.V. Smirnov and V.A. Smirnov, Evaluating Multiple Polylogarithm Values at Sixth Roots of Unity up to Weight Six, Nucl. Phys.B 919 (2017) 315 [arXiv:1512.08389] [INSPIRE]. · Zbl 1361.81105
[36] M. Czakon, Tops from Light Quarks: Full Mass Dependence at Two-Loops in QCD, Phys. Lett.B 664 (2008) 307 [arXiv:0803.1400] [INSPIRE].
[37] Bärnreuther, P.; Czakon, M.; Fiedler, P., Virtual amplitudes and threshold behaviour of hadronic top-quark pair-production cross sections, JHEP, 02, 078, (2014)
[38] 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
[39] Kniehl, BA; Pikelner, AF; Veretin, OL, Three-loop massive tadpoles and polylogarithms through weight six, JHEP, 08, 024, (2017)
[40] Mueller, R.; Öztürk, DG, On the computation of finite bottom-quark mass effects in Higgs boson production, JHEP, 08, 055, (2016)
[41] K. Melnikov, L. Tancredi and C. Wever, Two-loop ggHg amplitude mediated by a nearly massless quark, JHEP11 (2016) 104 [arXiv:1610.03747] [INSPIRE].
[42] Smirnov, AV, Algorithm FIRE — Feynman integral reduction, JHEP, 10, 107, (2008) · Zbl 1245.81033
[43] Smirnov, AV; Smirnov, VA, FIRE4, litered and accompanying tools to solve integration by parts relations, Comput. Phys. Commun., 184, 2820, (2013) · Zbl 1344.81031
[44] Smirnov, AV, FIRE5: a C++ implementation of Feynman integral reduction, Comput. Phys. Commun., 189, 182, (2015) · Zbl 1344.81030
[45] R.N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE]. · Zbl 0961.11040
[46] 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].
[47] Lee, RN; Pomeransky, AA, Critical points and number of master integrals, JHEP, 11, 165, (2013) · Zbl 1342.81139
[48] Smirnov, AV, FIESTA4: optimized Feynman integral calculations with GPU support, Comput. Phys. Commun., 204, 189, (2016) · Zbl 1378.65075
[49] Marquard, P.; Smirnov, AV; Smirnov, VA; Steinhauser, M., Quark mass relations to four-loop order in perturbative QCD, Phys. Rev. Lett., 114, 142002, (2015)
[50] P. Marquard, A.V. Smirnov, V.A. Smirnov, M. Steinhauser and D. Wellmann, \( \overline{\text{MS}} \)-on-shell quark mass relation up to four loops in QCD and a general SU(\(N\)) gauge group, Phys. Rev.D 94 (2016) 074025 [arXiv:1606.06754] [INSPIRE].
[51] K.G. Chetyrkin, Operator Expansions in the Minimal Subtraction Scheme. 1: The Gluing Method, Theor. Math. Phys.75 (1988) 346 [INSPIRE]. · Zbl 1378.81064
[52] K.G. Chetyrkin, Operator Expansions in the Minimal Subtraction Scheme. 2: Explicit Formulas for Coefficient Functions, Theor. Math. Phys.76 (1988) 809 [INSPIRE].
[53] Smirnov, VA, Asymptotic expansions in limits of large momenta and masses, Commun. Math. Phys., 134, 109, (1990) · Zbl 0729.47069
[54] Smirnov, VA, Applied asymptotic expansions in momenta and masses, Springer Tracts Mod. Phys., 177, 1, (2002) · Zbl 1025.81002
[55] F.A. Berends, A.I. Davydychev and N.I. Ussyukina, Threshold and pseudothreshold values of the sunset diagram, Phys. Lett.B 426 (1998) 95 [hep-ph/9712209] [INSPIRE]. · Zbl 1388.81109
[56] A.I. Davydychev and V.A. Smirnov, Threshold expansion of the sunset diagram, Nucl. Phys.B 554 (1999) 391 [hep-ph/9903328] [INSPIRE].
[57] J. Fleischer and M.Yu. Kalmykov, Single mass scale diagrams: Construction of a basis for the\( ϵ \)-expansion, Phys. Lett.B 470 (1999) 168 [hep-ph/9910223] [INSPIRE].
[58] A.I. Davydychev and M.Yu. Kalmykov, New results for the\( ϵ \)-expansion of certain one, two and three loop Feynman diagrams, Nucl. Phys.B 605 (2001) 266 [hep-th/0012189] [INSPIRE]. · Zbl 0969.81598
[59] M.Yu. Kalmykov and B.A. Kniehl, ’Sixth root of unity’ and Feynman diagrams: Hypergeometric function approach point of view, Nucl. Phys. Proc. Suppl.205-206 (2010) 129 [arXiv:1007.2373] [INSPIRE].
[60] D.J. Broadhurst, Massive three-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity, Eur. Phys. J.C 8 (1999) 311 [hep-th/9803091] [INSPIRE].
[61] F. Moriello, Linearization and symmetrization of generalized harmonic polylogarithms, Ph.D. Thesis (2013).
[62] R.N. Lee, A.V. Smirnov and V.A. Smirnov, Master Integrals for Four-Loop Massless Propagators up to Transcendentality Weight Twelve, Nucl. Phys.B 856 (2012) 95 [arXiv:1108.0732] [INSPIRE]. · Zbl 1246.81057
[63] M. Czakon, The Four-loop QCD β-function and anomalous dimensions, Nucl. Phys.B 710 (2005) 485 [hep-ph/0411261] [INSPIRE]. · Zbl 1115.81400
[64] Y. Schröder and A. Vuorinen, High-precision\( ϵ \)-expansions of single-mass-scale four-loop vacuum bubbles, JHEP06 (2005) 051 [hep-ph/0503209] [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.