Built on

N. Nielsen, Der Eulersche Dilogarithmus und seine Verallgemeinerungen , Nova Acta Leopoldina (Halle) 90
1909
Earlier work this paper cites.
A. Sabry, Fourth order spectral functions for the electron propagator , Nucl. Phys. 33
1962
Earlier work this paper cites.
H. Cheng and T. T. Wu, Expanding protons: scattering at high-energies . The MIT Press, 1987
1987
Earlier work this paper cites.
D. J. Broadhurst, The Master Two Loop Diagram With Masses , Z. Phys. C47
1990
Earlier work this paper cites.
A. V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation , Phys. Lett. B254
1991
Earlier work this paper cites.
Z. Bern, L. J. Dixon and D. A. Kosower, Dimensionally regulated pentagon integrals , Nucl.Phys. B412
1994
Earlier work this paper cites.
S. Bauberger, F. A. Berends, M. Bohm and M. Buza, Analytical and numerical methods for massive two loop selfenergy diagrams , Nucl. Phys. B434
1995
Earlier work this paper cites.
S. Bauberger and M. Bohm, Simple one-dimensional integral representations for two loop selfenergies: The Master diagram , Nucl. Phys. B445
1995
Earlier work this paper cites.
E. Remiddi, Differential equations for Feynman graph amplitudes , Nuovo Cim. A110
1997
Earlier work this paper cites.
T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions , Nucl.Phys. B580
2000
Earlier work this paper cites.
E. Remiddi and J. A. M. Vermaseren, Harmonic polylogarithms , Int. J. Mod. Phys. A15
2000
Earlier work this paper cites.
T. Gehrmann and E. Remiddi, Two loop master integrals for γ ∗ → 3 \gamma^{*}\to 3 jets: The Planar topologies , Nucl.Phys. B601
2001
Earlier work this paper cites.
U. Aglietti and R. Bonciani, Master integrals with 2 and 3 massive propagators for the 2 loop electroweak form-factor - planar case , Nucl. Phys. B698
2004
Earlier work this paper cites.
J. Vollinga and S. Weinzierl, Numerical evaluation of multiple polylogarithms , Comput. Phys. Commun. 167
2005
Earlier work this paper cites.
S. Laporta and E. Remiddi, Analytic treatment of the two loop equal mass sunrise graph , Nucl. Phys. B704
2005
Earlier work this paper cites.
B. A. Kniehl, A. V. Kotikov, A. Onishchenko and O. Veretin, Two-loop sunset diagrams with three massive lines , Nucl. Phys. B738
2006
Earlier work this paper cites.
U. Aglietti, R. Bonciani, L. Grassi and E. Remiddi, The Two loop crossed ladder vertex diagram with two massive exchanges , Nucl. Phys. B789
Original
2008
Earlier work this paper cites.
F. Brown, The Massless higher-loop two-point function , Commun. Math. Phys. 287
Original
2009
Earlier work this paper cites.
M. Czakon and A. Mitov, Inclusive Heavy Flavor Hadroproduction in NLO QCD: The Exact Analytic Result , Nucl. Phys. B824
Original
2010
Earlier work this paper cites.
L. J. Dixon, J. M. Drummond and J. M. Henn, Bootstrapping the three-loop hexagon , JHEP 1111
Original
2011
Earlier work this paper cites.
C. Duhr, H. Gangl and J. R. Rhodes, From polygons and symbols to polylogarithmic functions , JHEP 1210
Original
2012
Earlier work this paper cites.
C. Duhr, Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes , JHEP 08
Original
2012
Earlier work this paper cites.
L. J. Dixon, J. M. Drummond and J. M. Henn, Analytic result for the two-loop six-point NMHV amplitude in N=4 super Yang-Mills theory , JHEP 1201
Original
2012
Earlier work this paper cites.
S. Caron-Huot and K. J. Larsen, Uniqueness of two-loop master contours , JHEP 10
Original
2012
Earlier work this paper cites.

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2019

Elliptic polylogarithms and Feynman parameter integrals

Broedel, Johannes, Duhr, Claude, Dulat, Falko et al.

Understand

In this paper we study the calculation of multiloop Feynman integrals that cannot be expressed in terms of multiple polylogarithms.

We show in detail how certain types of two- and three-point functions at two loops, which appear in the calculation of higher order corrections in QED, QCD and in the electroweak theory (EW), can naturally be expressed in terms of a recently introduced elliptic generalisation of multiple polylogarithms by direct integration over their Feynman parameter representation.
Moreover, we show that in all examples that we considered a basis of pure Feynman integrals can be found.

Built on

N. Nielsen, Der Eulersche Dilogarithmus und seine Verallgemeinerungen , Nova Acta Leopoldina (Halle) 90
1909
Earlier work this paper cites.
A. Sabry, Fourth order spectral functions for the electron propagator , Nucl. Phys. 33
1962
Earlier work this paper cites.
H. Cheng and T. T. Wu, Expanding protons: scattering at high-energies . The MIT Press, 1987
1987
Earlier work this paper cites.
D. J. Broadhurst, The Master Two Loop Diagram With Masses , Z. Phys. C47
1990
Earlier work this paper cites.
A. V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation , Phys. Lett. B254
1991
Earlier work this paper cites.
Z. Bern, L. J. Dixon and D. A. Kosower, Dimensionally regulated pentagon integrals , Nucl.Phys. B412
1994
Earlier work this paper cites.
S. Bauberger, F. A. Berends, M. Bohm and M. Buza, Analytical and numerical methods for massive two loop selfenergy diagrams , Nucl. Phys. B434
1995
Earlier work this paper cites.
S. Bauberger and M. Bohm, Simple one-dimensional integral representations for two loop selfenergies: The Master diagram , Nucl. Phys. B445
1995
Earlier work this paper cites.
E. Remiddi, Differential equations for Feynman graph amplitudes , Nuovo Cim. A110
1997
Earlier work this paper cites.
T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions , Nucl.Phys. B580
2000
Earlier work this paper cites.
E. Remiddi and J. A. M. Vermaseren, Harmonic polylogarithms , Int. J. Mod. Phys. A15
2000
Earlier work this paper cites.
T. Gehrmann and E. Remiddi, Two loop master integrals for γ ∗ → 3 \gamma^{*}\to 3 jets: The Planar topologies , Nucl.Phys. B601
2001
Earlier work this paper cites.
U. Aglietti and R. Bonciani, Master integrals with 2 and 3 massive propagators for the 2 loop electroweak form-factor - planar case , Nucl. Phys. B698
2004
Earlier work this paper cites.
J. Vollinga and S. Weinzierl, Numerical evaluation of multiple polylogarithms , Comput. Phys. Commun. 167
2005
Earlier work this paper cites.
S. Laporta and E. Remiddi, Analytic treatment of the two loop equal mass sunrise graph , Nucl. Phys. B704
2005
Earlier work this paper cites.
B. A. Kniehl, A. V. Kotikov, A. Onishchenko and O. Veretin, Two-loop sunset diagrams with three massive lines , Nucl. Phys. B738
2006
Earlier work this paper cites.
U. Aglietti, R. Bonciani, L. Grassi and E. Remiddi, The Two loop crossed ladder vertex diagram with two massive exchanges , Nucl. Phys. B789
Original
2008
Earlier work this paper cites.
F. Brown, The Massless higher-loop two-point function , Commun. Math. Phys. 287
Original
2009
Earlier work this paper cites.
M. Czakon and A. Mitov, Inclusive Heavy Flavor Hadroproduction in NLO QCD: The Exact Analytic Result , Nucl. Phys. B824
Original
2010
Earlier work this paper cites.
L. J. Dixon, J. M. Drummond and J. M. Henn, Bootstrapping the three-loop hexagon , JHEP 1111
Original
2011
Earlier work this paper cites.
C. Duhr, H. Gangl and J. R. Rhodes, From polygons and symbols to polylogarithmic functions , JHEP 1210
Original
2012
Earlier work this paper cites.
C. Duhr, Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes , JHEP 08
Original
2012
Earlier work this paper cites.
L. J. Dixon, J. M. Drummond and J. M. Henn, Analytic result for the two-loop six-point NMHV amplitude in N=4 super Yang-Mills theory , JHEP 1201
Original
2012
Earlier work this paper cites.
S. Caron-Huot and K. J. Larsen, Uniqueness of two-loop master contours , JHEP 10
Original
2012
Earlier work this paper cites.