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We show that for a class of two-loop diagrams, the on-shell part of the integration-by-parts (IBP) relations correspond to exact meromorphic one-forms on algebraic curves.
Integration by Parts: The Algorithm to Calculate beta Functions in 4 Loops
K.G. Chetyrkin and F.V. Tkachov · 1981
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LARGE LOOP INTEGRALS
W. L. van Neerven and J. A. M. Vermaseren · 1984
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Riemann Surfaces: With 27 Figures
H.M. Farkas and I. Kra · 1992
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Two loop two point functions with masses: Asymptotic expansions and Taylor series, in any dimension
David J. Broadhurst, J. Fleischer, and O. V. Tarasov · 1993
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One loop n point gauge theory amplitudes, unitarity and collinear limits
Zvi Bern, Lance J. Dixon, David C. Dunbar, and David A. Kosower · 1994
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Closed expressions for specific massive multiloop selfenergy integrals
Frits A. Berends, M. Buza, M. Bohm, and R. Scharf · 1994
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Calculation of two loop selfenergies in the electroweak standard model
S. Bauberger, M. Bohm, G. Weiglein, Frits A. Berends, and M. Buza · 1994
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Fusing gauge theory tree amplitudes into loop amplitudes
Zvi Bern, Lance J. Dixon, David C. Dunbar, and David A. Kosower · 1995
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Algebraic Curves and Riemann Surfaces
R. Miranda · 1995
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Analytical and numerical methods for massive two loop selfenergy diagrams
S. Bauberger, Frits A. Berends, M. Bohm, and M. Buza · 1995
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Simple one-dimensional integral representations for two loop selfenergies: The Master diagram
S. Bauberger and M. Bohm · 1995
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The Master differential equations for the two loop sunrise selfmass amplitudes
Michele Caffo, H. Czyz, S. Laporta, and E. Remiddi · 1998
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A New technique for computing the spectral density of sunset type diagrams: Integral transformation in configuration space
S. Groote, J. G. Korner, and A. A. Pivovarov · 1998
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On the evaluation of sunset - type Feynman diagrams
S. Groote, J. G. Korner, and A. A. Pivovarov · 1999
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High precision calculation of multiloop Feynman integrals by difference equations
S. Laporta · 2000
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Numerical evaluation of the general massive 2 loop sunrise selfmass master integrals from differential equations
Michele Caffo, H. Czyz, and E. Remiddi · 2002
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Automatic integral reduction for higher order perturbative calculations
Charalampos Anastasiou and Achilleas Lazopoulos · 2004
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Holomorphic anomaly of unitarity cuts and one-loop gauge theory amplitudes
Freddy Cachazo · 2004
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Generalized unitarity and one-loop amplitudes in N=4 super-Yang-Mills
Ruth Britto, Freddy Cachazo, and Bo Feng · 2005
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New recursion relations for tree amplitudes of gluons
Ruth Britto, Freddy Cachazo, and Bo Feng · 2005
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Computing one-loop amplitudes from the holomorphic anomaly of unitarity cuts
Ruth Britto, Freddy Cachazo, and Bo Feng · 2005
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Direct proof of tree-level recursion relation in Yang-Mills theory
Ruth Britto, Freddy Cachazo, Bo Feng, and Edward Witten · 2005
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One-loop amplitudes of gluons in SQCD
Ruth Britto, Evgeny Buchbinder, Freddy Cachazo, and Bo Feng · 2005
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Analytic treatment of the two loop equal mass sunrise graph
S. Laporta and E. Remiddi · 2005
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An Algorithm to construct Grobner bases for solving integration by parts relations
A.V. Smirnov · 2006
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Precise numerical evaluation of the two loop sunrise graph master integrals in the equal mass case
S. Pozzorini and E. Remiddi · 2006
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Two-loop sunset diagrams with three massive lines
B. A. Kniehl, A. V. Kotikov, A. Onishchenko, and O. Veretin · 2006
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D-dimensional unitarity cut method
Charalampos Anastasiou, Ruth Britto, Bo Feng, Zoltan Kunszt, and Pierpaolo Mastrolia · 2007
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Unitarity cuts and Reduction to master integrals in d dimensions for one-loop amplitudes
Charalampos Anastasiou, Ruth Britto, Bo Feng, Zoltan Kunszt, and Pierpaolo Mastrolia · 2007
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Reducing full one-loop amplitudes to scalar integrals at the integrand level
Giovanni Ossola, Costas G. Papadopoulos, and Roberto Pittau · 2007
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On the evaluation of a certain class of Feynman diagrams in x-space: Sunrise-type topologies at any loop order
FIRE4, LiteRed and accompanying tools to solve integration by parts relations
A.V. Smirnov and V.A. Smirnov · 2013
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Critical points and number of master integrals
Roman N. Lee and Andrei A. Pomeransky · 2013
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Two-Loop Maximal Unitarity with External Masses
Henrik Johansson, David A. Kosower, and Kasper J. Larsen · 2013
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Global Residues and Two-Loop Hepta-Cuts
Mads Sogaard · 2013
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Maximal Unitarity for the Four-Mass Double Box
Henrik Johansson, David A. Kosower, and Kasper J. Larsen · 2013
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Multivariate Residues and Maximal Unitarity
Mads Søgaard and Yang Zhang · 2013
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S. Groote, J. G. Korner, and A. A. Pivovarov · 2007
Cited alongside, same era.
Algorithm FIRE – Feynman Integral REduction
A.V. Smirnov · 2008
Cited alongside, same era.
Group structure of the integration-by-part identities and its application to the reduction of multiloop integrals
R.N. Lee · 2008
Cited alongside, same era.
CutTools: A Program implementing the OPP reduction method to compute one-loop amplitudes
Giovanni Ossola, Costas G. Papadopoulos, and Roberto Pittau · 2008
Cited alongside, same era.
Elliptic integral evaluations of Bessel moments
David H. Bailey, Jonathan M. Borwein, David Broadhurst, and M. L. Glasser · 2008
Cited alongside, same era.
BOKASUN: A Fast and precise numerical program to calculate the Master Integrals of the two-loop sunrise diagrams
Michele Caffo, Henryk Czyz, Michal Gunia, and Ettore Remiddi · 2009
Cited alongside, same era.
Towards all-order Laurent expansion of generalized hypergeometric functions around rational values of parameters
Mikhail Yu. Kalmykov and Bernd A. Kniehl · 2009
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The classification of two-loop integrand basis in pure four-dimension
Bo Feng and Rijun Huang · 2013
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Integrand-Reduction for Two-Loop Scattering Amplitudes through Multivariate Polynomial Division
Pierpaolo Mastrolia, Edoardo Mirabella, Giovanni Ossola, and Tiziano Peraro · 2013
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Multiloop Integrand Reduction for Dimensionally Regulated Amplitudes
Pierpaolo Mastrolia, Edoardo Mirabella, Giovanni Ossola, and Tiziano Peraro · 2013
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A Two-Loop Five-Gluon Helicity Amplitude in QCD
Simon Badger, Hjalte Frellesvig, and Yang Zhang · 2013
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On Genera of Curves from High-loop Generalized Unitarity Cuts
Rijun Huang and Yang Zhang · 2013
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The two-loop sunrise graph with arbitrary masses
Luise Adams, Christian Bogner, and Stefan Weinzierl · 2013
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The elliptic dilogarithm for the sunset graph
Spencer Bloch and Pierre Vanhove · 2013
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Integration-by-parts identities from the viewpoint of differential geometry
Yang Zhang · 2014
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Unitarity Cuts of Integrals with Doubled Propagators
Mads Sogaard and Yang Zhang · 2014
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Massive Nonplanar Two-Loop Maximal Unitarity
Mads Sogaard and Yang Zhang · 2014
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Schouten identities for Feynman graph amplitudes; The Master Integrals for the two-loop massive sunrise graph
Ettore Remiddi and Lorenzo Tancredi · 2014
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The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms
Luise Adams, Christian Bogner, and Stefan Weinzierl · 2014
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A novel approach to integration by parts reduction
Andreas von Manteuffel and Robert M. Schabinger · 2015
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Elliptic Functions and Maximal Unitarity
Mads Søgaard and Yang Zhang · 2015
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Cross-Order Integral Relations from Maximal Cuts
Henrik Johansson, David A. Kosower, Kasper J. Larsen, and Mads Søgaard · 2015
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Global Structure of Curves from Generalized Unitarity Cut of Three-loop Diagrams
Jonathan D. Hauenstein, Rijun Huang, Dhagash Mehta, and Yang Zhang · 2015
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The two-loop sunrise integral around four space-time dimensions and generalisations of the Clausen and Glaisher functions towards the elliptic case
Luise Adams, Christian Bogner, and Stefan Weinzierl · 2015
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Two-loop Maximal Unitarity in Dimensional Regularization
Larsen Kasper and Yang Zhang · 2015
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