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J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, Elliptic polylogarithms and Feynman parameter integrals , JHEP 05
Original
1902
Earlier work this paper cites.
F. Brown, From the Deligne-Ihara conjecture to Multiple Modular Values , 1904.00178
Original
1904
Earlier work this paper cites.
C. Bogner, S. Müller-Stach and S. Weinzierl, The unequal mass sunrise integral expressed through iterated integrals on ℳ ¯ 1 , 3 \overline{\mathcal{M}}_{1,3} , 1907.01251
Original
1907
Earlier work this paper cites.
J. Broedel, C. Duhr, F. Dulat, R. Marzucca, B. Penante and L. Tancredi, An analytic solution for the equal-mass banana graph , JHEP 09
Original
1907
Earlier work this paper cites.
N. Nielsen, Der Eulersche Dilogarithmus und seine Verallgemeinerungen , Nova Acta Leopoldina (Halle) 90
1909
Earlier work this paper cites.
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 , 1911.06308
Original
1911
Earlier work this paper cites.
M. Caffo, H. Czyz, M. Gunia and E. Remiddi, BOKASUN: A Fast and precise numerical program to calculate the Master Integrals of the two-loop sunrise diagrams , Comput. Phys. Commun. 180
Original
1959
Earlier work this paper cites.
A. Sabry, Fourth order spectral functions for the electron propagator , Nucl. Phys. 33
1962
Earlier work this paper cites.
K. T. Chen, Iterated path integrals , Bull. Amer. Math. Soc. 83
1977
Earlier work this paper cites.
K. Chetyrkin and F. Tkachov, Integration by Parts: The Algorithm to Calculate beta Functions in 4 Loops , Nucl.Phys. B192
1981
Earlier work this paper cites.
F. V. Tkachov, A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions , Phys. Lett. B100
1981
Earlier work this paper cites.
D. J. Broadhurst, The Master Two Loop Diagram With Masses , Z. Phys. C47
1990
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.
O. V. Tarasov, Connection between Feynman integrals having different values of the space-time dimension , Phys. Rev. D54
1996
Earlier work this paper cites.
M. Caffo, H. Czyz, S. Laporta and E. Remiddi, The Master differential equations for the two loop sunrise selfmass amplitudes , Nuovo Cim. A111
1998
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.
T. Gehrmann and E. Remiddi, Numerical evaluation of harmonic polylogarithms , Comput. Phys. Commun. 141
2001
Earlier work this paper cites.
T. Gehrmann and E. Remiddi, Numerical evaluation of two-dimensional harmonic polylogarithms , Comput. Phys. Commun. 144
2002
Earlier work this paper cites.
Y. Yang, Transformation formulas for generlized dedekind eta functions , Bull. London Math. Soc. 36
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.

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2005
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Then

S. Borowka, G. Heinrich, S. P. Jones, M. Kerner, J. Schlenk and T. Zirke, SecDec-3.0: numerical evaluation of multi-scale integrals beyond one loop , Comput. Phys. Commun. 196
Original
2015
Later among the works it cites.
S. Bloch, M. Kerr and P. Vanhove, A Feynman integral via higher normal functions , Compos. Math. 151
Original
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Later among the works it cites.
H. Frellesvig, D. Tommasini and C. Wever, On the reduction of generalized polylogarithms to Li n \text{Li}_{n} and Li 2 , 2 \text{Li}_{2,2} and on the evaluation thereof , JHEP 03
Original
2016
Later among the works it cites.
E. Remiddi and L. Tancredi, Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral , Nucl. Phys. B907
Original
2016
Later among the works it cites.

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2019

Algorithms and tools for iterated Eisenstein integrals

Duhr, Claude, Tancredi, Lorenzo

Understand

We present algorithms to work with iterated Eisenstein integrals that have recently appeared in the computation of multi-loop Feynman integrals.

These algorithms allow one to analytically continue these integrals to all regions of the parameter space, and to obtain fast converging series representations in each region.
We illustrate our approach on the examples of hypergeometric functions that evaluate to iterated Eisenstein integrals as well as the well-known sunrise graph.

Built on

J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, Elliptic polylogarithms and Feynman parameter integrals , JHEP 05
Original
1902
Earlier work this paper cites.
F. Brown, From the Deligne-Ihara conjecture to Multiple Modular Values , 1904.00178
Original
1904
Earlier work this paper cites.
C. Bogner, S. Müller-Stach and S. Weinzierl, The unequal mass sunrise integral expressed through iterated integrals on ℳ ¯ 1 , 3 \overline{\mathcal{M}}_{1,3} , 1907.01251
Original
1907
Earlier work this paper cites.
J. Broedel, C. Duhr, F. Dulat, R. Marzucca, B. Penante and L. Tancredi, An analytic solution for the equal-mass banana graph , JHEP 09
Original
1907
Earlier work this paper cites.
N. Nielsen, Der Eulersche Dilogarithmus und seine Verallgemeinerungen , Nova Acta Leopoldina (Halle) 90
1909
Earlier work this paper cites.
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 , 1911.06308
Original
1911
Earlier work this paper cites.
M. Caffo, H. Czyz, M. Gunia and E. Remiddi, BOKASUN: A Fast and precise numerical program to calculate the Master Integrals of the two-loop sunrise diagrams , Comput. Phys. Commun. 180
Original
1959
Earlier work this paper cites.
A. Sabry, Fourth order spectral functions for the electron propagator , Nucl. Phys. 33
1962
Earlier work this paper cites.
K. T. Chen, Iterated path integrals , Bull. Amer. Math. Soc. 83
1977
Earlier work this paper cites.
K. Chetyrkin and F. Tkachov, Integration by Parts: The Algorithm to Calculate beta Functions in 4 Loops , Nucl.Phys. B192
1981
Earlier work this paper cites.
F. V. Tkachov, A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions , Phys. Lett. B100
1981
Earlier work this paper cites.
D. J. Broadhurst, The Master Two Loop Diagram With Masses , Z. Phys. C47
1990
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.
O. V. Tarasov, Connection between Feynman integrals having different values of the space-time dimension , Phys. Rev. D54
1996
Earlier work this paper cites.
M. Caffo, H. Czyz, S. Laporta and E. Remiddi, The Master differential equations for the two loop sunrise selfmass amplitudes , Nuovo Cim. A111
1998
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.
T. Gehrmann and E. Remiddi, Numerical evaluation of harmonic polylogarithms , Comput. Phys. Commun. 141
2001
Earlier work this paper cites.
T. Gehrmann and E. Remiddi, Numerical evaluation of two-dimensional harmonic polylogarithms , Comput. Phys. Commun. 144
2002
Earlier work this paper cites.
Y. Yang, Transformation formulas for generlized dedekind eta functions , Bull. London Math. Soc. 36
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.

Similar

A. Goncharov, Galois symmetries of fundamental groupoids and noncommutative geometry , Duke Math.J. 128
2005
Cited alongside, same era.
B. A. Kniehl, A. V. Kotikov, A. Onishchenko and O. Veretin, Two-loop sunset diagrams with three massive lines , Nucl. Phys. B738
2006
Cited alongside, same era.
Y. I. Manin, Iterated integrals of modular forms and noncommutative modular symbols , in Algebraic geometry and number theory , vol. 253 of Progr. Math. , (Boston), pp. 565–597, Birkhäuser Boston, 2006, math/0502576
2006
Cited alongside, same era.
R. S. Maier, On Rationally Parametrized Modular Equations , 2006, math/0611041
2006
Cited alongside, same era.
D. Zagier, Elliptic Modular Forms and Their Applications , in The 1-2-3 of Modular Forms , Springer, 2008
2008
Cited alongside, same era.