Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism

Broedel, Johannes; Duhr, Claude; Dulat, Falko; Tancredi, Lorenzo

doi:10.1007/jhep05(2018)093

Title: Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism

Abstract

We introduce a class of iterated integrals, defined through a set of linearly independent integration kernels on elliptic curves. As a direct generalization of multiple polylogarithms, we construct our set of integration kernels ensuring that they have at most simple poles, implying that the iterated integrals have at most logarithmic singularities. We study the properties of our iterated integrals and their relationship to the multiple elliptic polylogarithms from the mathematics literature. On the one hand, we find that our iterated integrals span essentially the same space of functions as the multiple elliptic polylogarithms. On the other, our formulation allows for a more direct use to solve a large variety of problems in high-energy physics. As a result, we demonstrate the use of our functions in the evaluation of the Laurent expansion of some hypergeometric functions for values of the indices close to half integers.

Authors:

Broedel, Johannes ^[1]; Duhr, Claude ^[2]; Dulat, Falko ^[3]; Tancredi, Lorenzo ^[4]

Humboldt-Univ. zu Berlin, Berlin (Germany)
European Organization for Nuclear Research (CERN), Geneva (Switzerland); Univ. Catholique de Louvain, Louvain-La-Neuve (Belgium)
Stanford Univ., Stanford, CA (United States)
European Organization for Nuclear Research (CERN), Geneva (Switzerland)

Publication Date:: Tue May 15 00:00:00 EDT 2018

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

Sponsoring Org.:: USDOE

OSTI Identifier:: 1458523

Grant/Contract Number:: AC02-76SF00515

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: 2018; Journal Issue: 5; Journal ID: ISSN 1029-8479

Publisher:: Springer Berlin

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; NLO Computations; QCD Phenomenology

Citation Formats


                    Broedel, Johannes, Duhr, Claude, Dulat, Falko, and Tancredi, Lorenzo. Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism.  United States: N. p., 2018. 
Web.  doi:10.1007/jhep05(2018)093.

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                    Broedel, Johannes, Duhr, Claude, Dulat, Falko, & Tancredi, Lorenzo. Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism.  United States.  https://doi.org/10.1007/jhep05(2018)093

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                    Broedel, Johannes, Duhr, Claude, Dulat, Falko, and Tancredi, Lorenzo. Tue .  
"Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism".  United States.  https://doi.org/10.1007/jhep05(2018)093.  https://www.osti.gov/servlets/purl/1458523.

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@article{osti_1458523,

  title        = {Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism},

  author       = {Broedel, Johannes and Duhr, Claude and Dulat, Falko and Tancredi, Lorenzo},

  abstractNote = {We introduce a class of iterated integrals, defined through a set of linearly independent integration kernels on elliptic curves. As a direct generalization of multiple polylogarithms, we construct our set of integration kernels ensuring that they have at most simple poles, implying that the iterated integrals have at most logarithmic singularities. We study the properties of our iterated integrals and their relationship to the multiple elliptic polylogarithms from the mathematics literature. On the one hand, we find that our iterated integrals span essentially the same space of functions as the multiple elliptic polylogarithms. On the other, our formulation allows for a more direct use to solve a large variety of problems in high-energy physics. As a result, we demonstrate the use of our functions in the evaluation of the Laurent expansion of some hypergeometric functions for values of the indices close to half integers.},

  doi          = {10.1007/jhep05(2018)093},

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

  number       = 5,

  volume       = 2018,

  place        = {United States},

  year         = {Tue May 15 00:00:00 EDT 2018},

  month        = {Tue May 15 00:00:00 EDT 2018}

}

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Frellesvig, Hjalte; Vergu, Cristian; Volk, Matthias
arXiv
DOI: 10.48550/arxiv.2102.02769

The Path forward to N$^3$LO
preprint, January 2022

Caola, Fabrizio; Chen, Wen; Duhr, Claude
arXiv
DOI: 10.48550/arxiv.2203.06730

A GPU compatible quasi-Monte Carlo integrator interfaced to pySecDec
text, January 2018

Borowka, S.; Heinrich, G.; Jahn, S.
University of Zurich
DOI: 10.5167/uzh-161945

Massive kite diagrams with elliptics
text, January 2020

Bezuglov, M. A.; Onishchenko, A. I.; Veretin, O. L.
Universität Regensburg
DOI: 10.5283/epub.45398

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Title: Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism

Abstract

Citation Formats

The Massless Higher-Loop Two-Point Function journal, February 2009

The elliptic dilogarithm for the sunset graph journal, March 2015

Two-loop sunset diagrams with three massive lines journal, March 2006

Master integrals with 2 and 3 massive propagators for the 2-loop electroweak form factor—planar case journal, October 2004

Elliptic multiple zeta values and one-loop superstring amplitudes journal, July 2015

Iterated binomial sums and their associated iterated integrals journal, November 2014

Fourth order spectral functions for the electron propagator journal, May 1962

Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph journal, August 2017

Local mirror symmetry and the sunset Feynman integral journal, January 2017

The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms journal, October 2014

Elliptic Double-Box Integrals: Massless Scattering Amplitudes beyond Polylogarithms journal, March 2018

On the algebraic structure of iterated integrals of quasimodular forms journal, January 2017

Harmonic Polylogarithms journal, February 2000

Numerical evaluation of multiple polylogarithms journal, May 2005

Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral journal, June 2016

The iterated structure of the all-order result for the two-loop sunrise integral journal, March 2016

Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals journal, March 2015

Elliptic polylogarithms and basic hypergeometric functions journal, February 2017

On the generalized harmonic polylogarithms of one complex variable journal, June 2011

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

Analytic treatment of the two loop equal mass sunrise graph journal, January 2005

Simple one-dimensional integral representations for two-loop self-energies: the master diagram journal, July 1995

Harmonic sums and polylogarithms generated by cyclotomic polynomials journal, October 2011

The kite integral to all orders in terms of elliptic polylogarithms journal, December 2016

An elliptic generalization of multiple polylogarithms journal, December 2017

Classical Polylogarithms for Amplitudes and Wilson Loops journal, October 2010

Geometry of Configurations, Polylogarithms, and Motivic Cohomology journal, September 1995

Uniqueness of two-loop master contours journal, October 2012

Schouten identities for Feynman graph amplitudes; The Master Integrals for the two-loop massive sunrise graph journal, March 2014

A Feynman integral via higher normal functions journal, August 2015

The two-loop sunrise graph with arbitrary masses journal, May 2013

The two loop crossed ladder vertex diagram with two massive exchanges journal, January 2008

The two-loop sunrise integral around four space-time dimensions and generalisations of the Clausen and Glaisher functions towards the elliptic case journal, July 2015

Two-loop planar master integrals for Higgs → 3 partons with full heavy-quark mass dependence text, January 2016

Schouten identities for Feynman graph amplitudes; The Master Integrals for the two-loop massive sunrise graph text, January 2014

The two loop crossed ladder vertex diagram with two massive exchanges text, January 2008

Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral text, January 2016

Maximal cuts and differential equations for Feynman integrals : An application to the three-loop massive banana graph text, January 2017

A non-planar two-loop three-point function beyond multiple polylogarithms text, January 2017

Elliptic multiple zeta values and one-loop superstring amplitudes text, January 2015

Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals dataset, January 2019

Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes text, January 2012

A non-planar two-loop three-point function beyond multiple polylogarithms text, January 2017

Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals dataset, January 2019

Classical Polylogarithms for Amplitudes and Wilson Loops text, January 2010

Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes text, January 2012

Uniqueness of two-loop master contours text, January 2012

The elliptic dilogarithm for the sunset graph text, January 2013

The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms text, January 2014

A Feynman integral via higher normal functions text, January 2014

Elliptic multiple zeta values and one-loop superstring amplitudes text, January 2014

The two-loop sunrise integral around four space-time dimensions and generalisations of the Clausen and Glaisher functions towards the elliptic case preprint, January 2015

The iterated structure of the all-order result for the two-loop sunrise integral text, January 2015

Local mirror symmetry and the sunset Feynman integral preprint, January 2016

Elliptic Polylogarithms and Basic Hypergeometric Functions text, January 2016

On the algebraic structure of iterated integrals of quasimodular forms text, January 2017

An Elliptic Generalization of Multiple Polylogarithms text, January 2017

Numerical evaluation of multiple polylogarithms text, January 2004

Two-loop sunset diagrams with three massive lines text, January 2005

Harmonic Polylogarithms text, January 1999

Relations between elliptic multiple zeta values and a special derivation algebra text, January 2015

Numerical evaluation of multiple polylogarithms text, January 2004

Cluster algebras and the subalgebra constructibility of the seven-particle remainder function journal, January 2019

Heterotic-string amplitudes at one loop: modular graph forms and relations to open strings journal, January 2019

All orders structure and efficient computation of linearly reducible elliptic Feynman integrals journal, January 2019

Feynman integrals and intersection theory journal, February 2019

Evaluating ‘elliptic’ master integrals at special kinematic values: using differential equations and their solutions via expansions near singular points journal, July 2018

Systematic approximation of multi-scale Feynman integrals journal, August 2018

An analytic solution for the equal-mass banana graph journal, September 2019

Analytic results for the planar double box integral relevant to top-pair production with a closed top loop journal, October 2018

Planar Double Box Integral for Top Pair Production with a Closed Top Loop to all orders in the Dimensional Regularization Parameter journal, October 2018

The cosmic Galois group and extended Steinmann relations for planar N $$ \mathcal{N} $$ = 4 SYM amplitudes text, January 2019

All orders structure and efficient computation of linearly reducible elliptic Feynman integrals text, January 2019

Evaluating ‘elliptic’ master integrals at special kinematic values: using differential equations and their solutions via expansions near singular points text, January 2018

Systematic approximation of multi-scale Feynman integrals text, January 2018

The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N} = 4$ SYM Amplitudes text, January 2019

The double pentaladder integral to all ordersC text, January 2018

The Massless Higher-Loop Two-Point Function
journal, February 2009

The elliptic dilogarithm for the sunset graph
journal, March 2015

Two-loop sunset diagrams with three massive lines
journal, March 2006

Master integrals with 2 and 3 massive propagators for the 2-loop electroweak form factor—planar case
journal, October 2004

Elliptic multiple zeta values and one-loop superstring amplitudes
journal, July 2015

Iterated binomial sums and their associated iterated integrals
journal, November 2014

Fourth order spectral functions for the electron propagator
journal, May 1962

Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph
journal, August 2017

Local mirror symmetry and the sunset Feynman integral
journal, January 2017

The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms
journal, October 2014

Elliptic Double-Box Integrals: Massless Scattering Amplitudes beyond Polylogarithms
journal, March 2018

On the algebraic structure of iterated integrals of quasimodular forms
journal, January 2017

Harmonic Polylogarithms
journal, February 2000

Numerical evaluation of multiple polylogarithms
journal, May 2005

Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral
journal, June 2016

The iterated structure of the all-order result for the two-loop sunrise integral
journal, March 2016

Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals
journal, March 2015

Elliptic polylogarithms and basic hypergeometric functions
journal, February 2017

On the generalized harmonic polylogarithms of one complex variable
journal, June 2011

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

Analytic treatment of the two loop equal mass sunrise graph
journal, January 2005

Simple one-dimensional integral representations for two-loop self-energies: the master diagram
journal, July 1995

Harmonic sums and polylogarithms generated by cyclotomic polynomials
journal, October 2011

The kite integral to all orders in terms of elliptic polylogarithms
journal, December 2016

An elliptic generalization of multiple polylogarithms
journal, December 2017

Classical Polylogarithms for Amplitudes and Wilson Loops
journal, October 2010

Geometry of Configurations, Polylogarithms, and Motivic Cohomology
journal, September 1995

Uniqueness of two-loop master contours
journal, October 2012

Schouten identities for Feynman graph amplitudes; The Master Integrals for the two-loop massive sunrise graph
journal, March 2014

A Feynman integral via higher normal functions
journal, August 2015

The two-loop sunrise graph with arbitrary masses
journal, May 2013

The two loop crossed ladder vertex diagram with two massive exchanges
journal, January 2008

The two-loop sunrise integral around four space-time dimensions and generalisations of the Clausen and Glaisher functions towards the elliptic case
journal, July 2015

Two-loop planar master integrals for Higgs → 3 partons with full heavy-quark mass dependence
text, January 2016

Schouten identities for Feynman graph amplitudes; The Master Integrals for the two-loop massive sunrise graph
text, January 2014

The two loop crossed ladder vertex diagram with two massive exchanges
text, January 2008

Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral
text, January 2016

Maximal cuts and differential equations for Feynman integrals : An application to the three-loop massive banana graph
text, January 2017

A non-planar two-loop three-point function beyond multiple polylogarithms
text, January 2017

Elliptic multiple zeta values and one-loop superstring amplitudes
text, January 2015

Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals
dataset, January 2019

Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes
text, January 2012

A non-planar two-loop three-point function beyond multiple polylogarithms
text, January 2017

Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals
dataset, January 2019

Classical Polylogarithms for Amplitudes and Wilson Loops
text, January 2010

Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes
text, January 2012

Uniqueness of two-loop master contours
text, January 2012

The elliptic dilogarithm for the sunset graph
text, January 2013

The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms
text, January 2014

A Feynman integral via higher normal functions
text, January 2014

Elliptic multiple zeta values and one-loop superstring amplitudes
text, January 2014

The two-loop sunrise integral around four space-time dimensions and generalisations of the Clausen and Glaisher functions towards the elliptic case
preprint, January 2015

The iterated structure of the all-order result for the two-loop sunrise integral
text, January 2015

Local mirror symmetry and the sunset Feynman integral
preprint, January 2016

Elliptic Polylogarithms and Basic Hypergeometric Functions
text, January 2016

On the algebraic structure of iterated integrals of quasimodular forms
text, January 2017

An Elliptic Generalization of Multiple Polylogarithms
text, January 2017

Numerical evaluation of multiple polylogarithms
text, January 2004

Two-loop sunset diagrams with three massive lines
text, January 2005

Harmonic Polylogarithms
text, January 1999

Relations between elliptic multiple zeta values and a special derivation algebra
text, January 2015

Numerical evaluation of multiple polylogarithms
text, January 2004

Cluster algebras and the subalgebra constructibility of the seven-particle remainder function
journal, January 2019

Heterotic-string amplitudes at one loop: modular graph forms and relations to open strings
journal, January 2019

All orders structure and efficient computation of linearly reducible elliptic Feynman integrals
journal, January 2019

Feynman integrals and intersection theory
journal, February 2019

Evaluating ‘elliptic’ master integrals at special kinematic values: using differential equations and their solutions via expansions near singular points
journal, July 2018

Systematic approximation of multi-scale Feynman integrals
journal, August 2018

An analytic solution for the equal-mass banana graph
journal, September 2019

Analytic results for the planar double box integral relevant to top-pair production with a closed top loop
journal, October 2018

Planar Double Box Integral for Top Pair Production with a Closed Top Loop to all orders in the Dimensional Regularization Parameter
journal, October 2018

The cosmic Galois group and extended Steinmann relations for planar N $$ \mathcal{N} $$ = 4 SYM amplitudes
text, January 2019

All orders structure and efficient computation of linearly reducible elliptic Feynman integrals
text, January 2019

Evaluating ‘elliptic’ master integrals at special kinematic values: using differential equations and their solutions via expansions near singular points
text, January 2018

Systematic approximation of multi-scale Feynman integrals
text, January 2018

The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N} = 4$ SYM Amplitudes
text, January 2019

The double pentaladder integral to all ordersC
text, January 2018

Systematic approximation of multi-scale Feynman integrals
text, January 2018

The planar double box integral for top pair production with a closed top loop to all orders in the dimensional regularisation parameter
text, January 2018

Rationalizing Loop Integration
text, January 2018