Abstract
We study several multiscale one-loop five-point families of Feynman integrals. More specifically, we employ the Simplified Differential Equations approach to obtain results in terms of Goncharov polylogarithms of up to transcendental weight four for families with two and three massive external legs and massless propagators, as well as with one massive internal line and up to two massive external legs. This is the first time this computational approach is applied to cases involving internal masses.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. Heinrich, Collider Physics at the Precision Frontier, Phys. Rept. 922 (2021) 1 [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 2019), Les Houches France (2019) [arXiv:2003.01700] [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.M. Henn, Lectures on differential equations for Feynman integrals, J. Phys. A 48 (2015) 153001 [arXiv:1412.2296] [INSPIRE].
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].
J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].
A.V. Kotikov, Differential equations and Feynman integrals, in Antidifferentiation and the Calculation of Feynman Amplitudes, Zeuthen Germany (2021) [arXiv:2102.07424] [INSPIRE].
A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998) 497 [arXiv:1105.2076] [INSPIRE].
C. Duhr, H. Gangl and J.R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP 10 (2012) 075 [arXiv:1110.0458] [INSPIRE].
C. Duhr, Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes, JHEP 08 (2012) 043 [arXiv:1203.0454] [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. (2015), pg. 419 [arXiv:1411.7538] [INSPIRE].
E. Remiddi and L. Tancredi, An Elliptic Generalization of Multiple Polylogarithms, Nucl. Phys. B 925 (2017) 212 [arXiv:1709.03622] [INSPIRE].
J. Broedel, C. Duhr, F. Dulat and L. Tancredi, Elliptic polylogarithms and iterated integrals on elliptic curves. Part I. General formalism, JHEP 05 (2018) 093 [arXiv:1712.07089] [INSPIRE].
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].
J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series, JHEP 08 (2018) 014 [arXiv:1803.10256] [INSPIRE].
J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, Elliptic Feynman integrals and pure functions, JHEP 01 (2019) 023 [arXiv:1809.10698] [INSPIRE].
J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, Elliptic polylogarithms and Feynman parameter integrals, JHEP 05 (2019) 120 [arXiv:1902.09971] [INSPIRE].
C. Duhr and L. Tancredi, Algorithms and tools for iterated Eisenstein integrals, JHEP 02 (2020) 105 [arXiv:1912.00077] [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].
T. Gehrmann, J.M. Henn and N.A. Lo Presti, Analytic form of the two-loop planar five-gluon all-plus-helicity amplitude in QCD, Phys. Rev. Lett. 116 (2016) 062001 [Erratum ibid. 116 (2016) 189903] [arXiv:1511.05409] [INSPIRE].
D. Chicherin, J. Henn and V. Mitev, Bootstrapping pentagon functions, JHEP 05 (2018) 164 [arXiv:1712.09610] [INSPIRE].
T. Gehrmann, J.M. Henn and N.A. Lo Presti, Pentagon functions for massless planar scattering amplitudes, JHEP 10 (2018) 103 [arXiv:1807.09812] [INSPIRE].
D. Chicherin, T. Gehrmann, J.M. Henn, N.A. Lo Presti, V. Mitev and P. Wasser, Analytic result for the nonplanar hexa-box integrals, JHEP 03 (2019) 042 [arXiv:1809.06240] [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].
D. Chicherin and V. Sotnikov, Pentagon Functions for Scattering of Five Massless Particles, JHEP 12 (2020) 167 [arXiv:2009.07803] [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].
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].
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].
M. Hidding, DiffExp, a Mathematica package for computing Feynman integrals in terms of one-dimensional series expansions, Comput. Phys. Commun. 269 (2021) 108125 [arXiv:2006.05510] [INSPIRE].
C.G. Papadopoulos, Simplified differential equations approach for Master Integrals, JHEP 07 (2014) 088 [arXiv:1401.6057] [INSPIRE].
N. Syrrakos, Pentagon integrals to arbitrary order in the dimensional regulator, JHEP 06 (2021) 037 [arXiv:2012.10635] [INSPIRE].
S. Badger, H.B. Hartanto and S. Zoia, Two-Loop QCD Corrections to W\( b\overline{b} \) Production at Hadron Colliders, Phys. Rev. Lett. 127 (2021) 012001 [arXiv:2102.02516] [INSPIRE].
C.G. Papadopoulos and C. Wever, Internal Reduction method for computing Feynman Integrals, JHEP 02 (2020) 112 [arXiv:1910.06275] [INSPIRE].
D.D. Canko and N. Syrrakos, Resummation methods for Master Integrals, JHEP 02 (2021) 080 [arXiv:2010.06947] [INSPIRE].
A. Georgoudis, K.J. Larsen and Y. Zhang, Azurite: An algebraic geometry based package for finding bases of loop integrals, Comput. Phys. Commun. 221 (2017) 203 [arXiv:1612.04252] [INSPIRE].
J. Klappert, F. Lange, P. Maierhöfer and J. Usovitsch, Integral reduction with Kira 2.0 and finite field methods, Comput. Phys. Commun. 266 (2021) 108024 [arXiv:2008.06494] [INSPIRE].
M. Heller, A. von Manteuffel and R.M. Schabinger, Multiple polylogarithms with algebraic arguments and the two-loop EW-QCD Drell-Yan master integrals, Phys. Rev. D 102 (2020) 016025 [arXiv:1907.00491] [INSPIRE].
M. Besier, D. Van Straten and S. Weinzierl, Rationalizing roots: an algorithmic approach, Commun. Num. Theor. Phys. 13 (2019) 253 [arXiv:1809.10983] [INSPIRE].
M. Besier, P. Wasser and S. Weinzierl, RationalizeRoots: Software Package for the Rationalization of Square Roots, Comput. Phys. Commun. 253 (2020) 107197 [arXiv:1910.13251] [INSPIRE].
M. Bonetti, E. Panzer, V.A. Smirnov and L. Tancredi, Two-loop mixed QCD-EW corrections to gg → H g, JHEP 11 (2020) 045 [arXiv:2007.09813] [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].
A.V. Smirnov, FIESTA4: Optimized Feynman integral calculations with GPU support, Comput. Phys. Commun. 204 (2016) 189 [arXiv:1511.03614] [INSPIRE].
T. Huber and D. Maître, HypExp 2, Expanding Hypergeometric Functions about Half-Integer Parameters, Comput. Phys. Commun. 178 (2008) 755 [arXiv:0708.2443] [INSPIRE].
C. Duhr and F. Dulat, PolyLogTools — polylogs for the masses, JHEP 08 (2019) 135 [arXiv:1904.07279] [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].
L. Naterop, A. Signer and Y. Ulrich, handyG — Rapid numerical evaluation of generalised polylogarithms in Fortran, Comput. Phys. Commun. 253 (2020) 107165 [arXiv:1909.01656] [INSPIRE].
J. Vollinga and S. Weinzierl, Numerical evaluation of multiple polylogarithms, Comput. Phys. Commun. 167 (2005) 177 [hep-ph/0410259] [INSPIRE].
F. Brown and C. Duhr, A double integral of dlog forms which is not polylogarithmic, arXiv:2006.09413 [INSPIRE].
S. Abreu, H. Ita, B. Page and W. Tschernow, Two-Loop Hexa-Box Integrals for Non-Planar Five-Point One-Mass Processes, arXiv:2107.14180 [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: 2107.02106
Supplementary Information
ESM 1
(TGZ 217 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
Syrrakos, N. One-loop Feynman integrals for 2 → 3 scattering involving many scales including internal masses. J. High Energ. Phys. 2021, 41 (2021). https://doi.org/10.1007/JHEP10(2021)041
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2021)041