Abstract
We elaborate on the method of differential equations for evaluating Feynman integrals. We focus on systems of equations for master integrals having a linear dependence on the dimensional parameter. For these systems we identify the criteria to bring them in a canonical form, recently identified by Henn, where the dependence of the dimensional parameter is disentangled from the kinematics. The determination of the transformation and the computation of the solution are obtained by using Magnus and Dyson series expansion. We apply the method to planar and non-planar two-loop QED vertex diagrams for massive fermions, and to non-planar two-loop integrals contributing to 2 → 2 scattering of massless particles. The extension to systems which are polynomial in the dimensional parameter is discussed as well.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A.V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158 [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].
M. Argeri and P. Mastrolia, Feynman Diagrams and Differential Equations, Int. J. Mod. Phys. A 22 (2007) 4375 [arXiv:0707.4037] [INSPIRE].
V.A. Smirnov, Analytic tools for Feynman integrals, Springer Tracts. Mod. Phys. 250 (2012) 1.
K.G. Chetyrkin, A.L. Kataev and F.V. Tkachov, New Approach to Evaluation of Multiloop Feynman Integrals: The Gegenbauer Polynomial x Space Technique, Nucl. Phys. B 174 (1980) 345 [INSPIRE].
F.V. Tkachov, A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions, Phys. Lett. B 100 (1981) 65 [INSPIRE].
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].
E. Remiddi and L. Tancredi, Schouten identities for Feynman graph amplitudes; The Master Integrals for the two-loop massive sunrise graph, Nucl. Phys. B 880 (2014) 343 [arXiv:1311.3342] [INSPIRE].
S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].
J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].
T. Gehrmann, J.M. Henn and T. Huber, The three-loop form factor in N = 4 super Yang-Mills, JHEP 03 (2012) 101 [arXiv:1112.4524] [INSPIRE].
J.M. Henn, A.V. Smirnov and V.A. Smirnov, Analytic results for planar three-loop four-point integrals from a Knizhnik-Zamolodchikov equation, JHEP 07 (2013) 128 [arXiv:1306.2799] [INSPIRE].
J.M. Henn and V.A. Smirnov, Analytic results for two-loop master integrals for Bhabha scattering I, JHEP 11 (2013) 041 [arXiv:1307.4083] [INSPIRE].
J.M. Henn, A.V. Smirnov and V.A. Smirnov, Evaluating single-scale and/or non-planar diagrams by differential equations, arXiv:1312.2588 [INSPIRE].
W. Magnus, On the exponential solution of differential equations for a linear operator, Comm. Pure Appl. Math. VII (1954) 649.
F.J. Dyson, The Radiation theories of Tomonaga, Schwinger and Feynman, Phys. Rev. 75 (1949) 486 [INSPIRE].
S. Blanes, F. Casas, J.A. Oteo and J. Ros, The Magnus expansion and some of its applications, Phys. Rep. 470 (2009) 151 [arXiv:0810.5488].
R. Bonciani, P. Mastrolia and E. Remiddi, Vertex diagrams for the QED form-factors at the two loop level, Nucl. Phys. B 661 (2003) 289 [Erratum ibid. B 702 (2004) 359] [hep-ph/0301170] [INSPIRE].
R. Bonciani, P. Mastrolia and E. Remiddi, QED vertex form-factors at two loops, Nucl. Phys. B 676 (2004) 399 [hep-ph/0307295] [INSPIRE].
J.B. Tausk, Nonplanar massless two loop Feynman diagrams with four on-shell legs, Phys. Lett. B 469 (1999) 225 [hep-ph/9909506] [INSPIRE].
C. Anastasiou, T. Gehrmann, C. Oleari, E. Remiddi and J.B. Tausk, The Tensor reduction and master integrals of the two loop massless crossed box with lightlike legs, Nucl. Phys. B 580 (2000) 577 [hep-ph/0003261] [INSPIRE].
M. Czakon, J. Gluza and T. Riemann, Master integrals for massive two-loop bhabha scattering in QED, Phys. Rev. D 71 (2005) 073009 [hep-ph/0412164] [INSPIRE].
R. Bonciani and A. Ferroglia, Two-loop Bhabha scattering in QED, Phys. Rev. D 72 (2005) 056004 [hep-ph/0507047] [INSPIRE].
M. Czakon, J. Gluza and T. Riemann, The Planar four-point master integrals for massive two-loop Bhabha scattering, Nucl. Phys. B 751 (2006) 1 [hep-ph/0604101] [INSPIRE].
C. Studerus, Reduze-Feynman Integral Reduction in C++, Comput. Phys. Commun. 181 (2010) 1293 [arXiv:0912.2546] [INSPIRE].
A. von Manteuffel and C. Studerus, Reduze 2 - Distributed Feynman Integral Reduction, arXiv:1201.4330 [INSPIRE].
R. Bonciani, 2-loop radiative corrections to Bhabha scattering in QED, Ph.D. thesis, University of Bologna, Italy (2001).
R. Bonciani, A. Ferroglia, P. Mastrolia, E. Remiddi and J.J. van der Bij, Planar box diagram for the (N(F) = 1) two loop QED virtual corrections to Bhabha scattering, Nucl. Phys. B 681 (2004) 261 [Erratum ibid. B 702 (2004) 364] [hep-ph/0310333] [INSPIRE].
E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
M. Argeri, P. Mastrolia and E. Remiddi, The Analytic value of the sunrise selfmass with two equal masses and the external invariant equal to the third squared mass, Nucl. Phys. B 631 (2002) 388 [hep-ph/0202123] [INSPIRE].
D. Maître, HPL, a mathematica implementation of the harmonic polylogarithms, Comput. Phys. Commun. 174 (2006) 222 [hep-ph/0507152] [INSPIRE].
D. Maître, Extension of HPL to complex arguments, Comput. Phys. Commun. 183 (2012) 846 [hep-ph/0703052] [INSPIRE].
T. Hahn, Generating Feynman diagrams and amplitudes with FeynArts 3, Comput. Phys. Commun. 140 (2001) 418 [hep-ph/0012260] [INSPIRE].
E. Hairer, Solving ordinary differential equations on manifolds, Lecture notes, University of Geneva (2011).
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1401.2979
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Argeri, M., Di Vita, S., Mastrolia, P. et al. Magnus and Dyson series for Master Integrals. J. High Energ. Phys. 2014, 82 (2014). https://doi.org/10.1007/JHEP03(2014)082
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2014)082