Abstract
We present in detail two resummation methods emerging from the application of the Simplified Differential Equations approach to a canonical basis of master integrals. The first one is a method which allows for an easy determination of the boundary conditions, since it finds relations between the boundaries of the basis elements and the second one indicates how using the x → 1 limit to the solutions of a canonical basis, one can obtain the solutions to a canonical basis for the same problem with one mass less. Both methods utilise the residue matrices for the letters {0, 1} of the canonical differential equation. As proof of concept, we apply these methods to a canonical basis for the three-loop ladder-box with one external mass off-shell, obtaining subsequently a canonical basis for the massless three-loop ladder-box as well as its solution.
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].
A.V. Kotikov, Differential equations method: The Calculation of vertex type Feynman diagrams, Phys. Lett. B 259 (1991) 314 [INSPIRE].
A.V. Kotikov, Differential equation method: The Calculation of N point Feynman diagrams, Phys. Lett. B 267 (1991) 123 [Erratum ibid. 295 (1992) 409] [INSPIRE].
T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [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].
J. Klappert, F. Lange, P. Maierhöfer and J. Usovitsch, Integral Reduction with Kira 2.0 and Finite Field Methods, arXiv:2008.06494 [INSPIRE].
A.V. Smirnov and F.S. Chuharev, FIRE6: Feynman Integral REduction with Modular Arithmetic, Comput. Phys. Commun. 247 (2020) 106877 [arXiv:1901.07808] [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].
S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].
K.J. Larsen and Y. Zhang, Integration-by-parts reductions from unitarity cuts and algebraic geometry, Phys. Rev. D 93 (2016) 041701 [arXiv:1511.01071] [INSPIRE].
D. Bendle et al., Integration-by-parts reductions of Feynman integrals using Singular and GPI-Space, JHEP 02 (2020) 079 [arXiv:1908.04301] [INSPIRE].
D.A. Kosower, Direct Solution of Integration-by-Parts Systems, Phys. Rev. D 98 (2018) 025008 [arXiv:1804.00131] [INSPIRE].
P. Mastrolia and S. Mizera, Feynman Integrals and Intersection Theory, JHEP 02 (2019) 139 [arXiv:1810.03818] [INSPIRE].
H. Frellesvig et al., Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers, JHEP 05 (2019) 153 [arXiv:1901.11510] [INSPIRE].
H. Frellesvig, F. Gasparotto, M.K. Mandal, P. Mastrolia, L. Mattiazzi and S. Mizera, Vector Space of Feynman Integrals and Multivariate Intersection Numbers, Phys. Rev. Lett. 123 (2019) 201602 [arXiv:1907.02000] [INSPIRE].
H. Frellesvig et al., Decomposition of Feynman Integrals by Multivariate Intersection Numbers, arXiv:2008.04823 [INSPIRE].
J. Klappert and F. Lange, Reconstructing rational functions with FireFly, Comput. Phys. Commun. 247 (2020) 106951 [arXiv:1904.00009] [INSPIRE].
J. Klappert, S.Y. Klein and F. Lange, Interpolation of Dense and Sparse Rational Functions and other Improvements in FireFly, arXiv:2004.01463 [INSPIRE].
J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].
J.M. Henn, Lectures on differential equations for Feynman integrals, J. Phys. A 48 (2015) 153001 [arXiv:1412.2296] [INSPIRE].
J.M. Henn, K. Melnikov and V.A. Smirnov, Two-loop planar master integrals for the production of off-shell vector bosons in hadron collisions, JHEP 05 (2014) 090 [arXiv:1402.7078] [INSPIRE].
R.N. Lee, Reducing differential equations for multiloop master integrals, JHEP 04 (2015) 108 [arXiv:1411.0911] [INSPIRE].
M. Prausa, epsilon: A tool to find a canonical basis of master integrals, Comput. Phys. Commun. 219 (2017) 361 [arXiv:1701.00725] [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].
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].
S. Abreu, B. Page and M. Zeng, Differential equations from unitarity cuts: nonplanar hexa-box integrals, JHEP 01 (2019) 006 [arXiv:1807.11522] [INSPIRE].
D. Chicherin, T. Gehrmann, J.M. Henn, P. Wasser, Y. Zhang and S. Zoia, All Master Integrals for Three-Jet Production at Next-to-Next-to-Leading Order, Phys. Rev. Lett. 123 (2019) 041603 [arXiv:1812.11160] [INSPIRE].
C. Dlapa, J. Henn and K. Yan, Deriving canonical differential equations for Feynman integrals from a single uniform weight integral, JHEP 05 (2020) 025 [arXiv:2002.02340] [INSPIRE].
J. Henn, B. Mistlberger, V.A. Smirnov and P. Wasser, Constructing d-log integrands and computing master integrals for three-loop four-particle scattering, JHEP 04 (2020) 167 [arXiv:2002.09492] [INSPIRE].
S. Abreu, H. Ita, F. Moriello, B. Page, W. Tschernow and M. Zeng, Two-Loop Integrals for Planar Five-Point One-Mass Processes, JHEP 11 (2020) 117 [arXiv:2005.04195] [INSPIRE].
C.G. Papadopoulos, Simplified differential equations approach for Master Integrals, JHEP 07 (2014) 088 [arXiv:1401.6057] [INSPIRE].
C.G. Papadopoulos, D. Tommasini and C. Wever, Two-loop Master Integrals with the Simplified Differential Equations approach, JHEP 01 (2015) 072 [arXiv:1409.6114] [INSPIRE].
C.G. Papadopoulos, D. Tommasini and C. Wever, The Pentabox Master Integrals with the Simplified Differential Equations approach, JHEP 04 (2016) 078 [arXiv:1511.09404] [INSPIRE].
A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998) 497 [arXiv:1105.2076] [INSPIRE].
S. Di Vita, P. Mastrolia, U. Schubert and V. Yundin, Three-loop master integrals for ladder-box diagrams with one massive leg, JHEP 09 (2014) 148 [arXiv:1408.3107] [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].
D. Binosi, J. Collins, C. Kaufhold and L. Theussl, JaxoDraw: A Graphical user interface for drawing Feynman diagrams. Version 2.0 release notes, Comput. Phys. Commun. 180 (2009) 1709 [arXiv:0811.4113] [INSPIRE].
C. Duhr and F. Dulat, PolyLogTools — polylogs for the masses, JHEP 08 (2019) 135 [arXiv:1904.07279] [INSPIRE].
E. Panzer, Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals, Comput. Phys. Commun. 188 (2015) 148 [arXiv:1403.3385] [INSPIRE].
M. Beneke and V.A. Smirnov, Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys. B 522 (1998) 321 [hep-ph/9711391] [INSPIRE].
V.A. Smirnov, Problems of the strategy of regions, Phys. Lett. B 465 (1999) 226 [hep-ph/9907471] [INSPIRE].
A. Pak and A. Smirnov, Geometric approach to asymptotic expansion of Feynman integrals, Eur. Phys. J. C 71 (2011) 1626 [arXiv:1011.4863] [INSPIRE].
B. Jantzen, Foundation and generalization of the expansion by regions, JHEP 12 (2011) 076 [arXiv:1111.2589] [INSPIRE].
B. Jantzen, A.V. Smirnov and V.A. Smirnov, Expansion by regions: revealing potential and Glauber regions automatically, Eur. Phys. J. C 72 (2012) 2139 [arXiv:1206.0546] [INSPIRE].
T.Y. Semenova, A.V. Smirnov and V.A. Smirnov, On the status of expansion by regions, Eur. Phys. J. C 79 (2019) 136 [arXiv:1809.04325] [INSPIRE].
A.V. Smirnov, FIESTA4: Optimized Feynman integral calculations with GPU support, Comput. Phys. Commun. 204 (2016) 189 [arXiv:1511.03614] [INSPIRE].
S. Borowka et al., pySecDec: a toolbox for the numerical evaluation of multi-scale integrals, Comput. Phys. Commun. 222 (2018) 313 [arXiv:1703.09692] [INSPIRE].
N. Syrrakos, Ph.D. Thesis, in preparation.
D.D. Canko, C.G. Papadopoulos and N. Syrrakos, Analytic representation of all planar two-loop five-point Master Integrals with one off-shell leg, JHEP 01 (2021) 199 [arXiv:2009.13917] [INSPIRE].
C.G. Papadopoulos and C. Wever, Internal Reduction method for computing Feynman Integrals, JHEP 02 (2020) 112 [arXiv:1910.06275] [INSPIRE].
C. Duhr, Mathematical aspects of scattering amplitudes, in Theoretical Advanced Study Institute in Elementary Particle Physics: Journeys Through the Precision Frontier: Amplitudes for Colliders, Boulder U.S.A. (2014), pg. 419 [arXiv:1411.7538] [INSPIRE].
F. Moriello, Generalised power series expansions for the elliptic planar families of Higgs + jet production at two loops, JHEP 01 (2020) 150 [arXiv:1907.13234] [INSPIRE].
R. Bonciani et al., Evaluating a family of two-loop non-planar master integrals for Higgs + jet production with full heavy-quark mass dependence, JHEP 01 (2020) 132 [arXiv:1907.13156] [INSPIRE].
H. Frellesvig, M. Hidding, L. Maestri, F. Moriello and G. Salvatori, The complete set of two-loop master integrals for Higgs + jet production in QCD, JHEP 06 (2020) 093 [arXiv:1911.06308] [INSPIRE].
M. Hidding, DiffExp, a Mathematica package for computing Feynman integrals in terms of one-dimensional series expansions, arXiv:2006.05510 [INSPIRE].
C. Anastasiou and G. Sterman, Removing infrared divergences from two-loop integrals, JHEP 07 (2019) 056 [arXiv:1812.03753] [INSPIRE].
C. Anastasiou, R. Haindl, G. Sterman, Z. Yang and M. Zeng, Locally finite two-loop amplitudes for off-shell multi-photon production in electron-positron annihilation, arXiv:2008.12293 [INSPIRE].
G. Heinrich, Collider Physics at the Precision Frontier, arXiv:2009.00516 [INSPIRE].
S. Amoroso et al., Les Houches 2019: Physics at TeV Colliders: Standard Model Working Group Report, in 11th Les Houches Workshop on Physics at TeV Colliders: PhysTeV Les Houches, Les Houches France (2019) [arXiv:2003.01700] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2010.06947
Supplementary Information
ESM 1
(ZIP 367 kb)
Rights and permissions
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.
About this article
Cite this article
Canko, D.D., Syrrakos, N. Resummation methods for Master Integrals. J. High Energ. Phys. 2021, 80 (2021). https://doi.org/10.1007/JHEP02(2021)080
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2021)080