skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

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:
 [1];  [2];  [3]; ORCiD logo [4];  [4]
  1. Humboldt Univ. of Berlin (Germany). Inst. fur Mathematik und Institut fur Physik
  2. 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)
  3. SLAC National Accelerator Lab., Menlo Park, CA (United States)
  4. European Organization for Nuclear Research (CERN), Geneva (Switzerland). Theoretical Physics Dept.
Publication Date:
Research Org.:
SLAC National Accelerator Lab., 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.
Broedel, Johannes, Duhr, Claude, Dulat, Falko, Penante, Brenda, & Tancredi, Lorenzo. Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series. United States. doi:10.1007/jhep08(2018)014.
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. doi:10.1007/jhep08(2018)014. https://www.osti.gov/servlets/purl/1468725.
@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 = {2018},
month = {8}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Citation Metrics:
Cited by: 11 works
Citation information provided by
Web of Science

Save / Share:

Works referenced in this record:

Heptagons from the Steinmann cluster bootstrap
journal, February 2017

  • Dixon, Lance J.; Drummond, James; Harrington, Thomas
  • Journal of High Energy Physics, Vol. 2017, Issue 2
  • DOI: 10.1007/JHEP02(2017)137

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

  • Broedel, Johannes; Mafra, Carlos R.; Matthes, Nils
  • Journal of High Energy Physics, Vol. 2015, Issue 7
  • DOI: 10.1007/JHEP07(2015)112

Bootstrapping an NMHV amplitude through three loops
journal, October 2014

  • Dixon, Lance J.; von Hippel, Matt
  • Journal of High Energy Physics, Vol. 2014, Issue 10
  • DOI: 10.1007/JHEP10(2014)065

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

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

  • Dixon, Lance J.; Drummond, James M.; Henn, Johannes M.
  • Journal of High Energy Physics, Vol. 2012, Issue 1
  • DOI: 10.1007/JHEP01(2012)024

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


NNLO QCD corrections to γ + η c (η b ) exclusive production in electron-positron collision
journal, January 2018

  • Chen, Long-Bin; Liang, Yi; Qiao, Cong-Feng
  • Journal of High Energy Physics, Vol. 2018, Issue 1
  • DOI: 10.1007/JHEP01(2018)091

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


Modular forms in quantum field theory
journal, January 2013


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

  • Adams, Luise; Bogner, Christian; Schweitzer, Armin
  • Journal of Mathematical Physics, Vol. 57, Issue 12
  • DOI: 10.1063/1.4969060

A symbol of uniqueness: the cluster bootstrap for the 3-loop MHV heptagon
journal, March 2015

  • Drummond, J. M.; Papathanasiou, G.; Spradlin, M.
  • Journal of High Energy Physics, Vol. 2015, Issue 3
  • DOI: 10.1007/JHEP03(2015)072

An elliptic generalization of multiple polylogarithms
journal, December 2017


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

Iterated path integrals
journal, January 1977


Analytical and numerical methods for massive two-loop self-energy diagrams
journal, January 1995


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

  • Dixon, Lance J.; von Hippel, Matt; McLeod, Andrew J.
  • Journal of High Energy Physics, Vol. 2016, Issue 1
  • DOI: 10.1007/JHEP01(2016)053

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

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

  • Broedel, Johannes; Matthes, Nils; Schlotterer, Oliver
  • Journal of Physics A: Mathematical and Theoretical, Vol. 49, Issue 15
  • DOI: 10.1088/1751-8113/49/15/155203

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

  • Ablinger, J.; Blümlein, J.; De Freitas, A.
  • Journal of Mathematical Physics, Vol. 59, Issue 6
  • DOI: 10.1063/1.4986417

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

  • Anastasiou, Charalampos; Duhr, Claude; Dulat, Falko
  • Journal of High Energy Physics, Vol. 2013, Issue 7
  • DOI: 10.1007/JHEP07(2013)003

The elliptic dilogarithm for the sunset graph
journal, March 2015


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


A non-planar two-loop three-point function beyond multiple polylogarithms
journal, June 2017

  • von Manteuffel, Andreas; Tancredi, Lorenzo
  • Journal of High Energy Physics, Vol. 2017, Issue 6
  • DOI: 10.1007/JHEP06(2017)127

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

  • Broadhurst, D. J.
  • Zeitschrift f�r Physik C Particles and Fields, Vol. 47, Issue 1
  • DOI: 10.1007/BF01551921

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

  • Bloch, Spencer; Kerr, Matt; Vanhove, Pierre
  • Advances in Theoretical and Mathematical Physics, Vol. 21, Issue 6
  • DOI: 10.4310/ATMP.2017.v21.n6.a1

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

  • Adams, Luise; Bogner, Christian; Weinzierl, Stefan
  • Journal of Mathematical Physics, Vol. 55, Issue 10
  • DOI: 10.1063/1.4896563

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


Harmonic Polylogarithms
journal, February 2000

  • Remiddi, E.; Vermaseren, J. A. M.
  • International Journal of Modern Physics A, Vol. 15, Issue 05
  • DOI: 10.1142/S0217751X00000367

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

  • Broedel, Johannes; Matthes, Nils; Richter, Gregor
  • Journal of Physics A: Mathematical and Theoretical, Vol. 51, Issue 28
  • DOI: 10.1088/1751-8121/aac601

Numerical evaluation of multiple polylogarithms
journal, May 2005


The two-loop helicity amplitudes for q q ¯ ′ → V 1 V 2 → 4 $$ q\overline{q}^{\prime}\to {V}_1{V}_2\to 4 $$ leptons
journal, September 2015

  • Gehrmann, Thomas; von Manteuffel, Andreas; Tancredi, Lorenzo
  • Journal of High Energy Physics, Vol. 2015, Issue 9
  • DOI: 10.1007/JHEP09(2015)128

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

  • Adams, Luise; Bogner, Christian; Weinzierl, Stefan
  • Journal of Mathematical Physics, Vol. 57, Issue 3
  • DOI: 10.1063/1.4944722

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

  • Brown, Francis C. S.
  • Annales scientifiques de l'École normale supérieure, Vol. 42, Issue 3
  • DOI: 10.24033/asens.2099

Multiple polylogarithms, cyclotomy and modular complexes
journal, January 1998


Mathematical Aspects of Scattering Amplitudes
conference, November 2015

  • Duhr, Claude
  • 2014 Theoretical Advanced Study Institute in Elementary Particle Physics, Journeys Through the Precision Frontier: Amplitudes for Colliders
  • DOI: 10.1142/9789814678766_0010

Elliptic polylogarithms and basic hypergeometric functions
journal, February 2017


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

  • Abreu, Samuel; Britto, Ruth; Duhr, Claude
  • Journal of High Energy Physics, Vol. 2017, Issue 12
  • DOI: 10.1007/JHEP12(2017)090

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

  • Anastasiou, Charalampos; Duhr, Claude; Dulat, Falko
  • Journal of High Energy Physics, Vol. 2015, Issue 3
  • DOI: 10.1007/JHEP03(2015)091

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

  • Anastasiou, Charalampos; Duhr, Claude; Dulat, Falko
  • Journal of High Energy Physics, Vol. 2015, Issue 8
  • DOI: 10.1007/JHEP08(2015)051

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

  • Caron-Huot, Simon; Larsen, Kasper J.
  • Journal of High Energy Physics, Vol. 2012, Issue 10
  • DOI: 10.1007/JHEP10(2012)026

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

  • Adams, Luise; Bogner, Christian; Weinzierl, Stefan
  • Journal of Mathematical Physics, Vol. 54, Issue 5
  • DOI: 10.1063/1.4804996

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

  • Anastasiou, Charalampos; Duhr, Claude; Dulat, Falko
  • Journal of High Energy Physics, Vol. 2013, Issue 12
  • DOI: 10.1007/JHEP12(2013)088

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

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

  • Broedel, Johannes; Duhr, Claude; Dulat, Falko
  • Journal of High Energy Physics, Vol. 2018, Issue 5
  • DOI: 10.1007/JHEP05(2018)093

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

  • Chen, Long-Bin; Jiang, Jun; Qiao, Cong-Feng
  • Journal of High Energy Physics, Vol. 2018, Issue 4
  • DOI: 10.1007/JHEP04(2018)080

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

  • Adams, Luise; Bogner, Christian; Weinzierl, Stefan
  • Journal of Mathematical Physics, Vol. 56, Issue 7
  • DOI: 10.1063/1.4926985

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

  • Dixon, Lance J.; Drummond, James M.; Duhr, Claude
  • Journal of High Energy Physics, Vol. 2014, Issue 6
  • DOI: 10.1007/JHEP06(2014)116

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

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