Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series

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

doi:10.1007/jhep08(2018)014

Title: Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series

Abstract

We present a generalization of the symbol calculus from ordinary multiple polylogarithms to their elliptic counterparts. Here, our formalism is based on a special case of a coaction on large classes of periods that is applied in particular to elliptic polylogarithms and iterated integrals of modular forms. We illustrate how to use our formalism to derive relations among elliptic polylogarithms, in complete analogy with the non-elliptic case. We then analyze the symbol alphabet of elliptic polylogarithms evaluated at rational points, and we observe that it is given by Eisenstein series for a certain congruence subgroup. We apply our formalism to hypergeometric functions that can be expressed in terms of elliptic polylogarithms and show that they can equally be written in terms of iterated integrals of Eisenstein series. Finally, we present the symbol of the equal-mass sunrise integral in two space-time dimensions. The symbol alphabet involves Eisenstein series of level six and weight three, and we can easily integrate the symbol in terms of iterated integrals of Eisenstein series.

Authors:

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

^[4]; Tancredi, Lorenzo ^[4]

Humboldt Univ. of Berlin (Germany). Inst. fur Mathematik und Institut fur Physik
European Organization for Nuclear Research (CERN), Geneva (Switzerland). Theoretical Physics Dept.; Univ. Catholique de Louvain, Louvain-La-Neuve (Belgium). Center for Cosmology, Particle Physics and Phenomenology (CP3)
SLAC National Accelerator Lab., Menlo Park, CA (United States)
European Organization for Nuclear Research (CERN), Geneva (Switzerland). Theoretical Physics Dept.

Publication Date:: Tue Aug 07 00:00:00 EDT 2018

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

Sponsoring Org.:: USDOE; European Research Council (ERC)

OSTI Identifier:: 1468725

Report Number(s):: SLAC-PUB-17240
Journal ID: ISSN 1029-8479; PII: 8736

Grant/Contract Number:: AC02-76SF00515; 637019

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: 8; Journal ID: ISSN 1029-8479

Publisher:: Springer Berlin

Country of Publication:: United States

Language:: English

Subject:: 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; NLO Computations; QCD Phenomenology

Citation Formats


                    Broedel, Johannes, Duhr, Claude, Dulat, Falko, Penante, Brenda, and Tancredi, Lorenzo. Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series.  United States: N. p., 2018. 
Web.  doi:10.1007/jhep08(2018)014.

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                    Broedel, Johannes, Duhr, Claude, Dulat, Falko, Penante, Brenda, & Tancredi, Lorenzo. Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series.  United States.  https://doi.org/10.1007/jhep08(2018)014

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                    Broedel, Johannes, Duhr, Claude, Dulat, Falko, Penante, Brenda, and Tancredi, Lorenzo. Tue .  
"Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series".  United States.  https://doi.org/10.1007/jhep08(2018)014.  https://www.osti.gov/servlets/purl/1468725.

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@article{osti_1468725,

  title        = {Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series},

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

  abstractNote = {We present a generalization of the symbol calculus from ordinary multiple polylogarithms to their elliptic counterparts. Here, our formalism is based on a special case of a coaction on large classes of periods that is applied in particular to elliptic polylogarithms and iterated integrals of modular forms. We illustrate how to use our formalism to derive relations among elliptic polylogarithms, in complete analogy with the non-elliptic case. We then analyze the symbol alphabet of elliptic polylogarithms evaluated at rational points, and we observe that it is given by Eisenstein series for a certain congruence subgroup. We apply our formalism to hypergeometric functions that can be expressed in terms of elliptic polylogarithms and show that they can equally be written in terms of iterated integrals of Eisenstein series. Finally, we present the symbol of the equal-mass sunrise integral in two space-time dimensions. The symbol alphabet involves Eisenstein series of level six and weight three, and we can easily integrate the symbol in terms of iterated integrals of Eisenstein series.},

  doi          = {10.1007/jhep08(2018)014},

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

  number       = 8,

  volume       = 2018,

  place        = {United States},

  year         = {Tue Aug 07 00:00:00 EDT 2018},

  month        = {Tue Aug 07 00:00:00 EDT 2018}

}

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journal, October 2018

Adams, Luise; Chaubey, Ekta; Weinzierl, Stefan
Physical Review Letters, Vol. 121, Issue 14
DOI: 10.1103/physrevlett.121.142001

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

Hidding, Martijn; Moriello, Francesco
ETH Zurich
DOI: 10.3929/ethz-b-000322319

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

Adams, Luise; Chaubey, Ekta; Weinzierl, Stefan
arXiv
DOI: 10.48550/arxiv.1804.11144

Rationalizing Loop Integration
text, January 2018

Bourjaily, Jacob L.; McLeod, Andrew J.; von Hippel, Matt
arXiv
DOI: 10.48550/arxiv.1805.10281

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

Adams, Luise; Chaubey, Ekta; Weinzierl, Stefan
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DOI: 10.48550/arxiv.1806.04981

From modular forms to differential equations for Feynman integrals
preprint, January 2018

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arXiv
DOI: 10.48550/arxiv.1807.00842

Cluster Algebras and the Subalgebra Constructibility of the Seven-Particle Remainder Function
text, January 2018

Golden, John; McLeod, Andrew J.
arXiv
DOI: 10.48550/arxiv.1810.12181

Elliptic polylogarithms and Feynman parameter integrals
text, January 2019

Broedel, Johannes; Duhr, Claude; Dulat, Falko
arXiv
DOI: 10.48550/arxiv.1902.09971

PolyLogTools - Polylogs for the masses
text, January 2019

Duhr, Claude; Dulat, Falko
arXiv
DOI: 10.48550/arxiv.1904.07279

Generalised power series expansions for the elliptic planar families of Higgs + jet production at two loops
journal, January 2020

Moriello, F.
Journal of High Energy Physics, Vol. 2020, Issue 1
DOI: 10.1007/jhep01(2020)150

Elliptic Double Box and Symbology Beyond Polylogarithms
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Kristensson, Alexander; Wilhelm, Matthias; Zhang, Chi
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DOI: 10.1103/physrevlett.127.251603

A double integral of dlog forms which is not polylogarithmic
conference, February 2022

Duhr, Claude; Brown, Francis
Proceedings of MathemAmplitudes 2019: Intersection Theory & Feynman Integrals — PoS(MA2019)
DOI: 10.22323/1.383.0005

The complete set of two-loop master integrals for Higgs + jet production in QCD
text, January 2020

Hjalte, Frellesvig,; Martijn, Hidding,; L., Maestri,
ETH Zurich
DOI: 10.3929/ethz-b-000425377

Tropical Monte Carlo quadrature for Feynman integrals
journal, January 2023

Borinsky, Michael
Annales de l’Institut Henri Poincaré D
DOI: 10.4171/aihpd/158

Recent Developments in Gluon Fusion Higgs Calculations
preprint, January 2018

Neumann, Tobias
arXiv
DOI: 10.48550/arxiv.1810.01800

Evaluating two-loop non-planar master integrals for Higgs + jet production with full heavy-quark mass dependence
text, January 2019

Bonciani, R.; Del Duca, V.; Frellesvig, H.
arXiv
DOI: 10.48550/arxiv.1907.13156

Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space
text, January 2019

Bourjaily, Jacob L.; McLeod, Andrew J.; Vergu, Cristian
arXiv
DOI: 10.48550/arxiv.1910.01534

Rationalizability of square roots
preprint, January 2020

Besier, Marco; Festi, Dino
arXiv
DOI: 10.48550/arxiv.2006.07121

Analytic representation of all planar two-loop five-point Master Integrals with one off-shell leg
text, January 2020

Canko, Dhimiter D.; Papadopoulos, Costas G.; Syrrakos, Nikolaos
arXiv
DOI: 10.48550/arxiv.2009.13917

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 symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series

Abstract

Citation Formats

Heptagons from the Steinmann cluster bootstrap journal, February 2017

Higgs boson gluon–fusion production at threshold in N 3 LO QCD journal, October 2014

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

Bootstrapping an NMHV amplitude through three loops journal, October 2014

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

Analytic result for the two-loop six-point NMHV amplitude in $ \mathcal{N} = {4} $ super Yang-Mills theory journal, January 2012

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

Bootstrapping a Five-Loop Amplitude Using Steinmann Relations journal, December 2016

Multiloop Integrals in Dimensional Regularization Made Simple journal, June 2013

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

The ε -form of the differential equations for Feynman integrals in the elliptic case journal, June 2018

Galois symmetries of fundamental groupoids and noncommutative geometry journal, June 2005

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

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

An elliptic generalization of multiple polylogarithms journal, December 2017

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

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

The four-loop six-gluon NMHV ratio function journal, January 2016

Bootstrapping the three-loop hexagon journal, November 2011

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

Relations between elliptic multiple zeta values and a special derivation algebra journal, March 2016

Iterated elliptic and hypergeometric integrals for Feynman diagrams journal, June 2018

Soft triple-real radiation for Higgs production at N3LO journal, July 2013

The elliptic dilogarithm for the sunset graph journal, March 2015

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

The master two-loop diagram with masses journal, March 1990

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

Feynman integrals and iterated integrals of modular forms journal, January 2018

Harmonic Polylogarithms journal, February 2000

Twisted elliptic multiple zeta values and non-planar one-loop open-string amplitudes journal, June 2018

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

Multiple zeta values and periods of moduli spaces $\overline{\mathfrak{M}}_{0,n}$ journal, January 2009

Multiple polylogarithms, cyclotomy and modular complexes journal, January 1998

Mathematical Aspects of Scattering Amplitudes conference, November 2015

Elliptic polylogarithms and basic hypergeometric functions journal, February 2017

Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case journal, December 2017

Elliptic polylogarithms and iterated integrals on elliptic curves. II. An application to the sunrise integral journal, June 2018

Higgs boson gluon-fusion production beyond threshold in N3LO QCD journal, March 2015

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

Scalar one-loop integrals for QCD journal, February 2008

Soft expansion of double-real-virtual corrections to Higgs production at N3LO journal, August 2015

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

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

Real-virtual contributions to the inclusive Higgs cross-section at N3LO journal, December 2013

From polygons and symbols to polylogarithmic functions journal, October 2012

Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism journal, May 2018

The two-loop soft current in dimensional regularization journal, December 2013

Higgs Boson Gluon-Fusion Production in QCD at Three Loops journal, May 2015

Two-loop integrals for CP-even heavy quarkonium production and decays: elliptic sectors journal, April 2018

Elliptic associators journal, September 2013

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

The four-loop remainder function and multi-Regge behavior at NNLLA in planar $ \mathcal{N} $ = 4 super-Yang-Mills theory journal, June 2014

Local integrals for planar scattering amplitudes journal, June 2012

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

Heptagons from the Steinmann Cluster Bootstrap text, January 2017

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

Soft triple-real radiation for Higgs production at N3LO text, January 2013

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

Soft expansion of double-real-virtual corrections to Higgs production at N3LO text, January 2015

Higgs boson gluon-fusion production beyond threshold in N3LO QCD text, January 2015

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

Scalar one-loop integrals for QCD text, January 2007

Elliptic associators preprint, January 2010

Classical Polylogarithms for Amplitudes and Wilson Loops text, January 2010

Bootstrapping the three-loop hexagon text, January 2011

From polygons and symbols to polylogarithmic functions text, January 2011

Analytic result for the two-loop six-point NMHV amplitude in N=4 super Yang-Mills theory text, January 2011

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

Uniqueness of two-loop master contours text, January 2012

Heptagons from the Steinmann cluster bootstrap
journal, February 2017

Higgs boson gluon–fusion production at threshold in N 3 LO QCD
journal, October 2014

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

Bootstrapping an NMHV amplitude through three loops
journal, October 2014

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

Analytic result for the two-loop six-point NMHV amplitude in $ \mathcal{N} = {4} $ super Yang-Mills theory
journal, January 2012

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

Bootstrapping a Five-Loop Amplitude Using Steinmann Relations
journal, December 2016

Multiloop Integrals in Dimensional Regularization Made Simple
journal, June 2013

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

The ε -form of the differential equations for Feynman integrals in the elliptic case
journal, June 2018

Galois symmetries of fundamental groupoids and noncommutative geometry
journal, June 2005

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

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

An elliptic generalization of multiple polylogarithms
journal, December 2017

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

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

The four-loop six-gluon NMHV ratio function
journal, January 2016

Bootstrapping the three-loop hexagon
journal, November 2011

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

Relations between elliptic multiple zeta values and a special derivation algebra
journal, March 2016

Iterated elliptic and hypergeometric integrals for Feynman diagrams
journal, June 2018

Soft triple-real radiation for Higgs production at N3LO
journal, July 2013

The elliptic dilogarithm for the sunset graph
journal, March 2015

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

The master two-loop diagram with masses
journal, March 1990

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

Feynman integrals and iterated integrals of modular forms
journal, January 2018

Harmonic Polylogarithms
journal, February 2000

Twisted elliptic multiple zeta values and non-planar one-loop open-string amplitudes
journal, June 2018

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

Multiple zeta values and periods of moduli spaces $\overline{\mathfrak{M}}_{0,n}$
journal, January 2009

Multiple polylogarithms, cyclotomy and modular complexes
journal, January 1998

Mathematical Aspects of Scattering Amplitudes
conference, November 2015

Elliptic polylogarithms and basic hypergeometric functions
journal, February 2017

Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case
journal, December 2017

Elliptic polylogarithms and iterated integrals on elliptic curves. II. An application to the sunrise integral
journal, June 2018

Higgs boson gluon-fusion production beyond threshold in N3LO QCD
journal, March 2015

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

Scalar one-loop integrals for QCD
journal, February 2008

Soft expansion of double-real-virtual corrections to Higgs production at N3LO
journal, August 2015

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

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

Real-virtual contributions to the inclusive Higgs cross-section at N3LO
journal, December 2013

From polygons and symbols to polylogarithmic functions
journal, October 2012

Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism
journal, May 2018

The two-loop soft current in dimensional regularization
journal, December 2013

Higgs Boson Gluon-Fusion Production in QCD at Three Loops
journal, May 2015

Two-loop integrals for CP-even heavy quarkonium production and decays: elliptic sectors
journal, April 2018

Elliptic associators
journal, September 2013

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

The four-loop remainder function and multi-Regge behavior at NNLLA in planar $ \mathcal{N} $ = 4 super-Yang-Mills theory
journal, June 2014

Local integrals for planar scattering amplitudes
journal, June 2012

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

Heptagons from the Steinmann Cluster Bootstrap
text, January 2017

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

Soft triple-real radiation for Higgs production at N3LO
text, January 2013

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

Soft expansion of double-real-virtual corrections to Higgs production at N3LO
text, January 2015

Higgs boson gluon-fusion production beyond threshold in N3LO QCD
text, January 2015

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

Scalar one-loop integrals for QCD
text, January 2007

Elliptic associators
preprint, January 2010

Classical Polylogarithms for Amplitudes and Wilson Loops
text, January 2010

Bootstrapping the three-loop hexagon
text, January 2011

From polygons and symbols to polylogarithmic functions
text, January 2011

Analytic result for the two-loop six-point NMHV amplitude in N=4 super Yang-Mills theory
text, January 2011

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

Uniqueness of two-loop master contours
text, January 2012

Multiloop integrals in dimensional regularization made simple
text, January 2013

The elliptic dilogarithm for the sunset graph
text, January 2013

Real-Virtual contributions to the inclusive Higgs cross-section at N3LO
text, January 2013