Logarithmic forms and differential equations for Feynman integrals

Herrmann, Enrico; Parra-Martinez, Julio

doi:10.1007/jhep02(2020)099

Title: Logarithmic forms and differential equations for Feynman integrals

Abstract

We describe in detail how a d log representation of Feynman integrals leads to simple differential equations. We derive these differential equations directly in loop momentum or embedding space making use of a localization trick and generalized unitarity. For the examples we study, the alphabet of the differential equation is related to special points in kinematic space, described by certain cut equations which encode the geometry of the Feynman integral. At one loop, we reproduce the motivic formulae described by Goncharov [1] that reappeared in the context of Feynman integrals in [2–4]. The d log representation allows us to generalize the differential equations to higher loops and motivates the study of certain mixed-dimension integrals.

Authors:

^[1];

^[2]

SLAC National Accelerator Lab., Menlo Park, CA (United States)
Univ. of California, Los Angeles, CA (United States). Mani L. Bhaumik Inst. for Theoretical Physics

Publication Date:: Mon Feb 17 00:00:00 EST 2020

Research Org.:: SLAC National Accelerator Laboratory (SLAC), Menlo Park, CA (United States)

Sponsoring Org.:: USDOE

OSTI Identifier:: 1608837

Grant/Contract Number:: AC02-76SF00515; PHY-1607611

Resource Type:: Accepted Manuscript

Journal Name:: Journal of High Energy Physics (Online)

Additional Journal Information:: Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2020; Journal Issue: 2; Journal ID: ISSN 1029-8479

Publisher:: Springer Berlin

Country of Publication:: United States

Language:: English

Subject:: 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Differential and Algebraic Geometry; Scattering Amplitudes

Citation Formats


                    Herrmann, Enrico, and Parra-Martinez, Julio. Logarithmic forms and differential equations for Feynman integrals.  United States: N. p., 2020. 
Web.  doi:10.1007/jhep02(2020)099.

Copy to clipboard


                    Herrmann, Enrico, & Parra-Martinez, Julio. Logarithmic forms and differential equations for Feynman integrals.  United States.  https://doi.org/10.1007/jhep02(2020)099

Copy to clipboard


                    Herrmann, Enrico, and Parra-Martinez, Julio. Mon .  
"Logarithmic forms and differential equations for Feynman integrals".  United States.  https://doi.org/10.1007/jhep02(2020)099.  https://www.osti.gov/servlets/purl/1608837.

Copy to clipboard


                    
@article{osti_1608837,

  title        = {Logarithmic forms and differential equations for Feynman integrals},

  author       = {Herrmann, Enrico and Parra-Martinez, Julio},

  abstractNote = {We describe in detail how a d log representation of Feynman integrals leads to simple differential equations. We derive these differential equations directly in loop momentum or embedding space making use of a localization trick and generalized unitarity. For the examples we study, the alphabet of the differential equation is related to special points in kinematic space, described by certain cut equations which encode the geometry of the Feynman integral. At one loop, we reproduce the motivic formulae described by Goncharov [1] that reappeared in the context of Feynman integrals in [2–4]. The d log representation allows us to generalize the differential equations to higher loops and motivates the study of certain mixed-dimension integrals.},

  doi          = {10.1007/jhep02(2020)099},

  journal      = {Journal of High Energy Physics (Online)},

  number       = 2,

  volume       = 2020,

  place        = {United States},

  year         = {Mon Feb 17 00:00:00 EST 2020},

  month        = {Mon Feb 17 00:00:00 EST 2020}

}

Copy to clipboard

Journal Article:

Free Publicly Available Full Text

Accepted Manuscript (DOE)

Publisher's Version of Record

https://doi.org/10.1007/jhep02(2020)099

Other availability

Search WorldCat to find libraries that may hold this journal

Citation Metrics:

Cited by: 29 works

Citation information provided by
Web of Science

Save / Share:

Export Metadata

Save to My Library

Works referenced in this record:

Singularity Structure of Maximally Supersymmetric Scattering Amplitudes
journal, December 2014

Arkani-Hamed, Nima; Bourjaily, Jacob L.; Cachazo, Freddy
Physical Review Letters, Vol. 113, Issue 26
DOI: 10.1103/PhysRevLett.113.261603

Symbols of One-Loop Integrals From Mixed Tate Motives
text, January 2011

Spradlin, Marcus; Volovich, Anastasia
arXiv
DOI: 10.48550/arxiv.1105.2024

Integration by parts: The algorithm to calculate β-functions in 4 loops
journal, November 1981

Chetyrkin, K. G.; Tkachov, F. V.
Nuclear Physics B, Vol. 192, Issue 1
DOI: 10.1016/0550-3213(81)90199-1

Feynman integrals via hyperlogarithms
conference, August 2014

Panzer, Erik
Proceedings of Loops and Legs in Quantum Field Theory — PoS(LL2014)
DOI: 10.22323/1.211.0049

One-loop amplitudes for e+e− to four partons
journal, March 1998

Bern, Zvi; Dixon, Lance; Kosower, David A.
Nuclear Physics B, Vol. 513, Issue 1-2
DOI: 10.1016/S0550-3213(97)00703-7

Hexagon functions and the three-loop remainder function
journal, December 2013

Dixon, Lance J.; Drummond, James M.; von Hippel, Matt
Journal of High Energy Physics, Vol. 2013, Issue 12
DOI: 10.1007/JHEP12(2013)049

Gluing Ladder Feynman Diagrams into Fishnets
journal, August 2017

Basso, Benjamin; Dixon, Lance J.
Physical Review Letters, Vol. 119, Issue 7
DOI: 10.1103/PhysRevLett.119.071601

Multiloop Integrals in Dimensional Regularization Made Simple
journal, June 2013

Henn, Johannes M.
Physical Review Letters, Vol. 110, Issue 25
DOI: 10.1103/PhysRevLett.110.251601

The Amplituhedron
journal, October 2014

Arkani-Hamed, Nima; Trnka, Jaroslav
Journal of High Energy Physics, Vol. 2014, Issue 10
DOI: 10.1007/JHEP10(2014)030

Feynman integrals and intersection theory
journal, February 2019

Mastrolia, Pierpaolo; Mizera, Sebastian
Journal of High Energy Physics, Vol. 2019, Issue 2
DOI: 10.1007/JHEP02(2019)139

The double pentaladder integral to all orders
journal, July 2018

Caron-Huot, Simon; Dixon, Lance J.; von Hippel, Matt
Journal of High Energy Physics, Vol. 2018, Issue 7
DOI: 10.1007/JHEP07(2018)170

Singular terms of helicity amplitudes at one loop in QCD and the soft limit of the cross sections of multi-parton processes
journal, June 1994

Kunszt, Zoltan; Signer, Adrian; Trócsányi, Zoltán
Nuclear Physics B, Vol. 420, Issue 3
DOI: 10.1016/0550-3213(94)90077-9

Grassmannian Geometry of Scattering Amplitudes
book, May 2016

Arkani-Hamed, Nima; Bourjaily, Jacob; Cachazo, Freddy
DOI: 10.1017/CBO9781316091548

Wave Equations in Conformal Space
journal, April 1936

Dirac, P. A. M.
The Annals of Mathematics, Vol. 37, Issue 2
DOI: 10.2307/1968455

A geometrical angle on Feynman integrals
journal, September 1998

Davydychev, A. I.; Delbourgo, R.
Journal of Mathematical Physics, Vol. 39, Issue 9
DOI: 10.1063/1.532513

Vector Space of Feynman Integrals and Multivariate Intersection Numbers
journal, November 2019

Frellesvig, Hjalte; Gasparotto, Federico; Mandal, Manoj K.
Physical Review Letters, Vol. 123, Issue 20
DOI: 10.1103/PhysRevLett.123.201602

Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes
journal, August 2012

Duhr, Claude
Journal of High Energy Physics, Vol. 2012, Issue 8
DOI: 10.1007/JHEP08(2012)043

Evidence for a nonplanar amplituhedron
journal, June 2016

Bern, Zvi; Herrmann, Enrico; Litsey, Sean
Journal of High Energy Physics, Vol. 2016, Issue 6
DOI: 10.1007/JHEP06(2016)098

Decomposition of Feynman integrals on the maximal cut by intersection numbers
journal, May 2019

Frellesvig, Hjalte; Gasparotto, Federico; Laporta, Stefano
Journal of High Energy Physics, Vol. 2019, Issue 5
DOI: 10.1007/JHEP05(2019)153

Magic identities for conformal four-point integrals
journal, January 2007

Drummond, James M.; Henn, Johannes; Smirnov, Vladimir A.
Journal of High Energy Physics, Vol. 2007, Issue 01
DOI: 10.1088/1126-6708/2007/01/064

Cuts from residues: the one-loop case
journal, June 2017

Abreu, Samuel; Britto, Ruth; Duhr, Claude
Journal of High Energy Physics, Vol. 2017, Issue 6
DOI: 10.1007/JHEP06(2017)114

Bootstrapping two-loop Feynman integrals for planar N = 4 $$ \mathcal{N}=4 $$ sYM
journal, October 2018

Henn, Johannes; Herrmann, Enrico; Parra-Martinez, Julio
Journal of High Energy Physics, Vol. 2018, Issue 10
DOI: 10.1007/JHEP10(2018)059

An approach toward the numerical evaluation of multi-loop Feynman diagrams
journal, December 2001

Passarino, Giampiero
Nuclear Physics B, Vol. 619, Issue 1-3
DOI: 10.1016/S0550-3213(01)00528-4

Scattering Amplitudes from Intersection Theory
journal, April 2018

Mizera, Sebastian
Physical Review Letters, Vol. 120, Issue 14
DOI: 10.1103/PhysRevLett.120.141602

Bootstrapping the three-loop hexagon
journal, November 2011

Dixon, Lance J.; Drummond, James M.; Henn, Johannes M.
Journal of High Energy Physics, Vol. 2011, Issue 11
DOI: 10.1007/JHEP11(2011)023

Generalized unitarity and one-loop amplitudes in super-Yang–Mills
journal, October 2005

Britto, Ruth; Cachazo, Freddy; Feng, Bo
Nuclear Physics B, Vol. 725, Issue 1-2
DOI: 10.1016/j.nuclphysb.2005.07.014

Eliminating spurious poles from gauge-theoretic amplitudes
journal, May 2013

Hodges, Andrew
Journal of High Energy Physics, Vol. 2013, Issue 5
DOI: 10.1007/JHEP05(2013)135

One-loop corrections for e+e− annihilation into μ+μ− in the Weinberg model
journal, November 1979

Passarino, G.; Veltman, M.
Nuclear Physics B, Vol. 160, Issue 1
DOI: 10.1016/0550-3213(79)90234-7

Iterative structure of finite loop integrals
journal, June 2014

Caron-Huot, Simon; Henn, Johannes M.
Journal of High Energy Physics, Vol. 2014, Issue 6
DOI: 10.1007/JHEP06(2014)114

Volumes of hyperbolic manifolds and mixed Tate motives
preprint, January 1996

Goncharov, Alexander
arXiv
DOI: 10.48550/arxiv.alg-geom/9601021

Fusing gauge theory tree amplitudes into loop amplitudes
journal, February 1995

Bern, Zvi; Dixon, Lance; Dunbar, David C.
Nuclear Physics B, Vol. 435, Issue 1-2
DOI: 10.1016/0550-3213(94)00488-Z

The massless hexagon integral in <mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:mi>D</mml:mi><mml:mo>=</mml:mo><mml:mn>6</mml:mn></mml:math> dimensions
journal, September 2011

Del Duca, Vittorio; Duhr, Claude; Smirnov, Vladimir A.
Physics Letters B, Vol. 703, Issue 3
DOI: 10.1016/j.physletb.2011.07.079

Two-loop amplitudes of gluons and octa-cuts in Script N = 4 super Yang-Mills
journal, November 2005

Buchbinder, Evgeny I.; Cachazo, Freddy
Journal of High Energy Physics, Vol. 2005, Issue 11
DOI: 10.1088/1126-6708/2005/11/036

The one-loop six-dimensional hexagon integral and its relation to MHV amplitudes in $ \mathcal{N} = 4 $ SYM
journal, June 2011

Dixon, Lance J.; Drummond, James M.; Henn, Johannes M.
Journal of High Energy Physics, Vol. 2011, Issue 6
DOI: 10.1007/JHEP06(2011)100

Differential equations method. New technique for massive Feynman diagram calculation
journal, January 1991

Kotikov, A. V.
Physics Letters B, Vol. 254, Issue 1-2
DOI: 10.1016/0370-2693(91)90413-K

Logarithmic singularities and maximally supersymmetric amplitudes
journal, June 2015

Bern, Zvi; Herrmann, Enrico; Litsey, Sean
Journal of High Energy Physics, Vol. 2015, Issue 6
DOI: 10.1007/JHEP06(2015)202

New differential equations for on-shell loop integrals
journal, April 2011

Drummond, James M.; Henn, Johannes M.; Trnka, Jaroslav
Journal of High Energy Physics, Vol. 2011, Issue 4
DOI: 10.1007/JHEP04(2011)083

A theorem on analytical calculability of 4-loop renormalization group functions
journal, March 1981

Tkachov, F. V.
Physics Letters B, Vol. 100, Issue 1
DOI: 10.1016/0370-2693(81)90288-4

Dual-conformal regularization of infrared loop divergences and the chiral box expansion
journal, January 2015

Bourjaily, Jacob; Caron-Huot, Simon; Trnka, Jaroslav
Journal of High Energy Physics, Vol. 2015, Issue 1
DOI: 10.1007/JHEP01(2015)001

Analytic structure of Schläfli function
journal, November 1977

Aomoto, Kazuhiko
Nagoya Mathematical Journal, Vol. 68
DOI: 10.1017/S0027763000017839

Dimensionally regulated one-loop integrals
journal, March 1993

Bern, Zvi; Dixon, Lance; Kosower, David A.
Physics Letters B, Vol. 302, Issue 2-3
DOI: 10.1016/0370-2693(93)90400-C

Exponential suppression with four legs and an infinity of loops
journal, August 2010

Broadhurst, David J.; Davydychev, Andrei I.
Nuclear Physics B - Proceedings Supplements, Vol. 205-206
DOI: 10.1016/j.nuclphysbps.2010.09.014

Star integrals, convolutions and simplices
journal, May 2013

Nandan, Dhritiman; Paulos, Miguel F.; Spradlin, Marcus
Journal of High Energy Physics, Vol. 2013, Issue 5
DOI: 10.1007/JHEP05(2013)105

Hyperlogarithmic expansion and the volume of a hyperbolic simplex
journal, January 1992

Aomoto, K.
Banach Center Publications, Vol. 27, Issue 1
DOI: 10.4064/-27-1-9-21

Classical Polylogarithms for Amplitudes and Wilson Loops
journal, October 2010

Goncharov, A. B.; Spradlin, M.; Vergu, C.
Physical Review Letters, Vol. 105, Issue 15
DOI: 10.1103/PhysRevLett.105.151605

Differential equations for two-loop four-point functions
journal, July 2000

Gehrmann, T.; Remiddi, E.
Nuclear Physics B, Vol. 580, Issue 1-2
DOI: 10.1016/S0550-3213(00)00223-6

From polygons and symbols to polylogarithmic functions
journal, October 2012

Duhr, Claude; Gangl, Herbert; Rhodes, John R.
Journal of High Energy Physics, Vol. 2012, Issue 10
DOI: 10.1007/JHEP10(2012)075

Exact results for three- and four-point ladder diagrams with an arbitrary number of rungs
journal, May 1993

Ussyukina, N. I.; Davydychev, A. I.
Physics Letters B, Vol. 305, Issue 1-2
DOI: 10.1016/0370-2693(93)91118-7

Wave Equations in Conformal Space
journal, April 1936

Dirac, P. A. M.
The Annals of Mathematics, Vol. 37, Issue 2
DOI: 10.2307/1968455

Dimensionally-regulated pentagon integrals
journal, January 1994

Bern, Zvi; Dixon, Lance; Kosower, David A.
Nuclear Physics B, Vol. 412, Issue 3
DOI: 10.1016/0550-3213(94)90398-0

Algebraic Structure of Cut Feynman Integrals and the Diagrammatic Coaction
journal, July 2017

Abreu, Samuel; Britto, Ruth; Duhr, Claude
Physical Review Letters, Vol. 119, Issue 5
DOI: 10.1103/PhysRevLett.119.051601

Local integrals for planar scattering amplitudes
journal, June 2012

Arkani-Hamed, N.; Bourjaily, J.; Cachazo, F.
Journal of High Energy Physics, Vol. 2012, Issue 6
DOI: 10.1007/JHEP06(2012)125

Jumpstarting the all-loop S-matrix of planar $ \mathcal{N} = {4} $ super Yang-Mills
journal, July 2012

Caron-Huot, S.; He, Song
Journal of High Energy Physics, Vol. 2012, Issue 7
DOI: 10.1007/JHEP07(2012)174

Differential equations for Feynman graph amplitudes
journal, December 1997

Remiddi, E.
Il Nuovo Cimento A, Vol. 110, Issue 12
DOI: 10.1007/BF03185566

Similar Records in DOE PAGES and OSTI.GOV collections:

Lectures on differential equations for Feynman integrals

Journal Article Henn, Johannes M. - Journal of Physics. A, Mathematical and Theoretical

Over the last year significant progress was made in the understanding of the computation of Feynman integrals using differential equations (DE). These lectures give a review of these developments, while not assuming any prior knowledge of the subject. After an introduction to DE for Feynman integrals, we point out how they can be simplified using algorithms available in the mathematical literature. We discuss how this is related to a recent conjecture for a canonical form of the equations. We also discuss a complementary approach that is based on properties of the space–time loop integrands, and explain how the ideas ofmore »« less
Cited by 182
https://doi.org/10.1088/1751-8113/48/15/153001
Bootstrapping two-loop Feynman integrals for planar $$ \mathcal{N}=4 $$ sYM

Journal Article Henn, Johannes ; Herrmann, Enrico ; Parra-Martinez, Julio - Journal of High Energy Physics (Online)

We derive analytic results for the symbol of certain two-loop Feynman integrals relevant for seven- and eight-point two-loop scattering amplitudes in planarmore »« less
Cited by 25
https://doi.org/10.1007/jhep10(2018)059

Full Text Available
Loop-by-loop differential equations for dual (elliptic) Feynman integrals

Journal Article Giroux, Mathieu ; Pokraka, Andrzej - Journal of High Energy Physics (Online)

We present a loop-by-loop method for computing the differential equations of Feynman integrals using the recently developed dual form formalism. We give explicit prescriptions for the loop-by-loop fibration of multi-loop dual forms. Then, we test our formalism on a simple, but non-trivial, example: the two-loop three-mass elliptic sunrise family of integrals. We obtain an ε-form differential equation within the correct function space in a sequence of relatively simple algebraic steps. In particular, none of these steps relies on the analysis of q-series. Then, we discuss interesting properties satisfied by our dual basis as well as its simple relation to themore »« less
https://doi.org/10.1007/jhep03(2023)155

Full Text Available
Differential equations from unitarity cuts: nonplanar hexa-box integrals

Journal Article Abreu, Samuel ; Page, Ben ; Zeng, Mao - Journal of High Energy Physics (Online)

We compute ϵ-factorized differential equations for all dimensionally-regularized integrals of the nonplanar hexa-box topology, which contribute for instance to 2-loop 5-point QCD amplitudes. A full set of pure integrals is presented. For 5-point planar topologies, Gram determinants which vanish in 4 dimensions are used to build compact expressions for pure integrals. Using unitarity cuts and computational algebraic geometry, we obtain a compact IBP system which can be solved in 8 hours on a single CPU core, overcoming a major bottleneck for deriving the differential equations. Alternatively, assuming prior knowledge of the alphabet of the nonplanar hexa-box, we reconstruct analytic differentialmore »« less
Cited by 31
https://doi.org/10.1007/jhep01(2019)006

Full Text Available
Differential equations on unitarity cut surfaces

Journal Article Zeng, Mao - Journal of High Energy Physics (Online)

We reformulate differential equations (DEs) for Feynman integrals to avoid doubled propagators in intermediate steps. External momentum derivatives are dressed with loop momentum derivatives to form tangent vectors to unitarity cut surfaces, in a way inspired by unitarity-compatible IBP reduction. For the one-loop box, our method directly produces the final DEs without any integration-by-parts reduction. We further illustrate the method by deriving maximal-cut level differential equations for two-loop nonplanar five-point integrals, whose exact expressions are yet unknown. Finally, we speed up the computation using finite field techniques and rational function reconstruction.
Cited by 21
https://doi.org/10.1007/JHEP06(2017)121

Full Text Available

Similar Records

Title: Logarithmic forms and differential equations for Feynman integrals

Abstract

Citation Formats

Singularity Structure of Maximally Supersymmetric Scattering Amplitudes journal, December 2014

Symbols of One-Loop Integrals From Mixed Tate Motives text, January 2011

Integration by parts: The algorithm to calculate β-functions in 4 loops journal, November 1981

Feynman integrals via hyperlogarithms conference, August 2014

One-loop amplitudes for e+e− to four partons journal, March 1998

Hexagon functions and the three-loop remainder function journal, December 2013

Gluing Ladder Feynman Diagrams into Fishnets journal, August 2017

Multiloop Integrals in Dimensional Regularization Made Simple journal, June 2013

The Amplituhedron journal, October 2014

Feynman integrals and intersection theory journal, February 2019

The double pentaladder integral to all orders journal, July 2018

Singular terms of helicity amplitudes at one loop in QCD and the soft limit of the cross sections of multi-parton processes journal, June 1994

Grassmannian Geometry of Scattering Amplitudes book, May 2016

Wave Equations in Conformal Space journal, April 1936

A geometrical angle on Feynman integrals journal, September 1998

Vector Space of Feynman Integrals and Multivariate Intersection Numbers journal, November 2019

Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes journal, August 2012

Evidence for a nonplanar amplituhedron journal, June 2016

Decomposition of Feynman integrals on the maximal cut by intersection numbers journal, May 2019

Magic identities for conformal four-point integrals journal, January 2007

Cuts from residues: the one-loop case journal, June 2017

Bootstrapping two-loop Feynman integrals for planar N = 4 $$ \mathcal{N}=4 $$ sYM journal, October 2018

An approach toward the numerical evaluation of multi-loop Feynman diagrams journal, December 2001

Scattering Amplitudes from Intersection Theory journal, April 2018

Bootstrapping the three-loop hexagon journal, November 2011

Generalized unitarity and one-loop amplitudes in super-Yang–Mills journal, October 2005

Eliminating spurious poles from gauge-theoretic amplitudes journal, May 2013

One-loop corrections for e+e− annihilation into μ+μ− in the Weinberg model journal, November 1979

Iterative structure of finite loop integrals journal, June 2014

Volumes of hyperbolic manifolds and mixed Tate motives preprint, January 1996

Fusing gauge theory tree amplitudes into loop amplitudes journal, February 1995

Two-loop amplitudes of gluons and octa-cuts in Script N = 4 super Yang-Mills journal, November 2005

The one-loop six-dimensional hexagon integral and its relation to MHV amplitudes in $ \mathcal{N} = 4 $ SYM journal, June 2011

Differential equations method. New technique for massive Feynman diagram calculation journal, January 1991

Logarithmic singularities and maximally supersymmetric amplitudes journal, June 2015

New differential equations for on-shell loop integrals journal, April 2011

A theorem on analytical calculability of 4-loop renormalization group functions journal, March 1981

Dual-conformal regularization of infrared loop divergences and the chiral box expansion journal, January 2015

Analytic structure of Schläfli function journal, November 1977

Dimensionally regulated one-loop integrals journal, March 1993

Exponential suppression with four legs and an infinity of loops journal, August 2010

Star integrals, convolutions and simplices journal, May 2013

Hyperlogarithmic expansion and the volume of a hyperbolic simplex journal, January 1992

Classical Polylogarithms for Amplitudes and Wilson Loops journal, October 2010

Differential equations for two-loop four-point functions journal, July 2000

From polygons and symbols to polylogarithmic functions journal, October 2012

Exact results for three- and four-point ladder diagrams with an arbitrary number of rungs journal, May 1993

Wave Equations in Conformal Space journal, April 1936

Dimensionally-regulated pentagon integrals journal, January 1994

Algebraic Structure of Cut Feynman Integrals and the Diagrammatic Coaction journal, July 2017

Local integrals for planar scattering amplitudes journal, June 2012

Jumpstarting the all-loop S-matrix of planar $ \mathcal{N} = {4} $ super Yang-Mills journal, July 2012

Differential equations for Feynman graph amplitudes journal, December 1997

Singularity Structure of Maximally Supersymmetric Scattering Amplitudes
journal, December 2014

Symbols of One-Loop Integrals From Mixed Tate Motives
text, January 2011

Integration by parts: The algorithm to calculate β-functions in 4 loops
journal, November 1981

Feynman integrals via hyperlogarithms
conference, August 2014

One-loop amplitudes for e+e− to four partons
journal, March 1998

Hexagon functions and the three-loop remainder function
journal, December 2013

Gluing Ladder Feynman Diagrams into Fishnets
journal, August 2017

Multiloop Integrals in Dimensional Regularization Made Simple
journal, June 2013

The Amplituhedron
journal, October 2014

Feynman integrals and intersection theory
journal, February 2019

The double pentaladder integral to all orders
journal, July 2018

Singular terms of helicity amplitudes at one loop in QCD and the soft limit of the cross sections of multi-parton processes
journal, June 1994

Grassmannian Geometry of Scattering Amplitudes
book, May 2016

Wave Equations in Conformal Space
journal, April 1936

A geometrical angle on Feynman integrals
journal, September 1998

Vector Space of Feynman Integrals and Multivariate Intersection Numbers
journal, November 2019

Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes
journal, August 2012

Evidence for a nonplanar amplituhedron
journal, June 2016

Decomposition of Feynman integrals on the maximal cut by intersection numbers
journal, May 2019

Magic identities for conformal four-point integrals
journal, January 2007

Cuts from residues: the one-loop case
journal, June 2017

Bootstrapping two-loop Feynman integrals for planar N = 4 $$ \mathcal{N}=4 $$ sYM
journal, October 2018

An approach toward the numerical evaluation of multi-loop Feynman diagrams
journal, December 2001

Scattering Amplitudes from Intersection Theory
journal, April 2018

Bootstrapping the three-loop hexagon
journal, November 2011

Generalized unitarity and one-loop amplitudes in super-Yang–Mills
journal, October 2005

Eliminating spurious poles from gauge-theoretic amplitudes
journal, May 2013

One-loop corrections for e+e− annihilation into μ+μ− in the Weinberg model
journal, November 1979

Iterative structure of finite loop integrals
journal, June 2014

Volumes of hyperbolic manifolds and mixed Tate motives
preprint, January 1996

Fusing gauge theory tree amplitudes into loop amplitudes
journal, February 1995

Two-loop amplitudes of gluons and octa-cuts in Script N = 4 super Yang-Mills
journal, November 2005

The one-loop six-dimensional hexagon integral and its relation to MHV amplitudes in $ \mathcal{N} = 4 $ SYM
journal, June 2011

Differential equations method. New technique for massive Feynman diagram calculation
journal, January 1991

Logarithmic singularities and maximally supersymmetric amplitudes
journal, June 2015

New differential equations for on-shell loop integrals
journal, April 2011

A theorem on analytical calculability of 4-loop renormalization group functions
journal, March 1981

Dual-conformal regularization of infrared loop divergences and the chiral box expansion
journal, January 2015

Analytic structure of Schläfli function
journal, November 1977

Dimensionally regulated one-loop integrals
journal, March 1993

Exponential suppression with four legs and an infinity of loops
journal, August 2010

Star integrals, convolutions and simplices
journal, May 2013

Hyperlogarithmic expansion and the volume of a hyperbolic simplex
journal, January 1992

Classical Polylogarithms for Amplitudes and Wilson Loops
journal, October 2010

Differential equations for two-loop four-point functions
journal, July 2000

From polygons and symbols to polylogarithmic functions
journal, October 2012

Exact results for three- and four-point ladder diagrams with an arbitrary number of rungs
journal, May 1993

Wave Equations in Conformal Space
journal, April 1936

Dimensionally-regulated pentagon integrals
journal, January 1994

Algebraic Structure of Cut Feynman Integrals and the Diagrammatic Coaction
journal, July 2017

Local integrals for planar scattering amplitudes
journal, June 2012

Jumpstarting the all-loop S-matrix of planar $ \mathcal{N} = {4} $ super Yang-Mills
journal, July 2012

Differential equations for Feynman graph amplitudes
journal, December 1997