Elliptic Feynman integrals and pure functions

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

doi:10.1007/jhep01(2019)023

Title: Elliptic Feynman integrals and pure functions

Abstract

We propose a variant of elliptic multiple polylogarithms that have at most logarithmic singularities in all variables and satisfy a differential equation without homogeneous term. We investigate several non-trivial elliptic two-loop Feynman integrals with up to three external legs and express them in terms of our functions. Here, we observe that in all cases they evaluate to pure combinations of elliptic multiple polylogarithms of uniform weight. This is the first time that a notion of uniform weight is observed in the context of Feynman integrals that evaluate to elliptic polylogarithms.

Authors:

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

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

Publication Date:: Thu Jan 03 00:00:00 EST 2019

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

Sponsoring Org.:: USDOE

OSTI Identifier:: 1493348

Report Number(s):: SLAC-PUB-17336
Journal ID: ISSN 1029-8479

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: 2019; Journal Issue: 1; 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 Feynman integrals and pure functions.  United States: N. p., 2019. 
Web.  doi:10.1007/jhep01(2019)023.

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                    Broedel, Johannes, Duhr, Claude, Dulat, Falko, Penante, Brenda, & Tancredi, Lorenzo. Elliptic Feynman integrals and pure functions.  United States.  https://doi.org/10.1007/jhep01(2019)023

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                    Broedel, Johannes, Duhr, Claude, Dulat, Falko, Penante, Brenda, and Tancredi, Lorenzo. Thu .  
"Elliptic Feynman integrals and pure functions".  United States.  https://doi.org/10.1007/jhep01(2019)023.  https://www.osti.gov/servlets/purl/1493348.

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@article{osti_1493348,

  title        = {Elliptic Feynman integrals and pure functions},

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

  abstractNote = {We propose a variant of elliptic multiple polylogarithms that have at most logarithmic singularities in all variables and satisfy a differential equation without homogeneous term. We investigate several non-trivial elliptic two-loop Feynman integrals with up to three external legs and express them in terms of our functions. Here, we observe that in all cases they evaluate to pure combinations of elliptic multiple polylogarithms of uniform weight. This is the first time that a notion of uniform weight is observed in the context of Feynman integrals that evaluate to elliptic polylogarithms.},

  doi          = {10.1007/jhep01(2019)023},

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

  number       = 1,

  volume       = 2019,

  place        = {United States},

  year         = {Thu Jan 03 00:00:00 EST 2019},

  month        = {Thu Jan 03 00:00:00 EST 2019}

}

Copy to clipboard

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Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes
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Analytic results for two-loop master integrals for Bhabha scattering I
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Henn, Johannes M.; Smirnov, Vladimir A.
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The elliptic dilogarithm for the sunset graph
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Henn, Johannes M.; Melnikov, Kirill; Smirnov, Vladimir A.
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Non-planar master integrals for the production of two off-shell vector bosons in collisions of massless partons
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Caola, Fabrizio; Henn, Johannes M.; Melnikov, Kirill
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The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms
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A Feynman integral via higher normal functions
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Three-loop master integrals for ladder-box diagrams with one massive leg
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Reducing differential equations for multiloop master integrals
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Notes on Motivic Periods
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Brown, Francis
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Bloch, Spencer; Kerr, Matt; Vanhove, Pierre
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Bonciani, Roberto; Di Vita, Stefano; Mastrolia, Pierpaolo
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Bootstrapping a Five-Loop Amplitude Using Steinmann Relations
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Caron-Huot, Simon; Dixon, Lance J.; McLeod, Andrew
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Henn, Johannes M.; Smirnov, Alexander V.; Smirnov, Vladimir A.
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Heptagons from the Steinmann Cluster Bootstrap
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Dixon, Lance J.; Drummond, James; Harrington, Thomas
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Feynman integrals and iterated integrals of modular forms
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Adams, Luise; Weinzierl, Stefan
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Two-Loop Master Integrals for the Planar QCD Massive Corrections to Di-photon and Di-jet Hadro-production
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Dixon, Lance J.
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Master integrals for the NNLO virtual corrections to $μe$ scattering in QED: the non-planar graphs
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Non-planar master integrals for the production of two off-shell vector bosons in collisions of massless partons
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Caola, Fabrizio; Henn, Johannes M.; Melnikov, Kirill
arXiv
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Three-loop corrections to the soft anomalous dimension in multi-leg scattering
text, January 2015

Almelid, Øyvind; Duhr, Claude; Gardi, Einan
arXiv
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An Elliptic Generalization of Multiple Polylogarithms
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Remiddi, Ettore; Tancredi, Lorenzo
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Works referencing / citing this record:

Elliptic polylogarithms and Feynman parameter integrals
journal, May 2019

Broedel, Johannes; Duhr, Claude; Dulat, Falko
Journal of High Energy Physics, Vol. 2019, Issue 5
DOI: 10.1007/jhep05(2019)120

Three-loop contributions to the ρ parameter and iterated integrals of modular forms
journal, February 2020

Abreu, Samuel; Becchetti, Matteo; Duhr, Claude
Journal of High Energy Physics, Vol. 2020, Issue 2
DOI: 10.1007/jhep02(2020)050

Cluster adjacency beyond MHV
journal, March 2019

Drummond, James; Foster, Jack; Gürdoğan, Ömer
Journal of High Energy Physics, Vol. 2019, Issue 3
DOI: 10.1007/jhep03(2019)086

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

Broedel, Johannes; Duhr, Claude; Dulat, Falko
Journal of High Energy Physics, Vol. 2019, Issue 9
DOI: 10.1007/jhep09(2019)112

Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers
conference, February 2020

Mandal, Manoj Kumar; Frellesvig, Hjalte Axel; Gasparotto, Federico
Proceedings of 14th International Symposium on Radiative Corrections — PoS(RADCOR2019)
DOI: 10.22323/1.375.0064

Cluster adjacency beyond MHV
text, January 2018

Drummond, James; Foster, Jack; Gurdogan, Omer
arXiv
DOI: 10.48550/arxiv.1810.08149

Elliptic polylogarithms and Feynman parameter integrals
text, January 2019

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

Monodromy relations from twisted homology
text, January 2019

Casali, Eduardo; Mizera, Sebastian; Tourkine, Piotr
arXiv
DOI: 10.48550/arxiv.1910.08514

Elliptic Double Box and Symbology Beyond Polylogarithms
journal, December 2021

Kristensson, Alexander; Wilhelm, Matthias; Zhang, Chi
Physical Review Letters, Vol. 127, Issue 25
DOI: 10.1103/physrevlett.127.251603

Modular Graph Forms and Scattering Amplitudes in String Theory
text, January 2020

Gerken, Jan E.
arXiv
DOI: 10.48550/arxiv.2011.08647

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 Feynman integrals and pure functions

Abstract

Citation Formats

Two-loop master integrals for the planar QCD massive corrections to di-photon and di-jet hadro-production journal, January 2018

Heptagons from the Steinmann cluster bootstrap journal, February 2017

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

Bootstrapping an NMHV amplitude through three loops journal, October 2014

Notes on motivic periods journal, January 2017

Master integrals for the NNLO virtual corrections to μe scattering in QED: the non-planar graphs text, January 2018

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

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

A direct evaluation of the periods of the Weierstrass zeta function journal, July 2013

Analytic results for two-loop master integrals for Bhabha scattering I journal, November 2013

The ρ parameter at three loops and elliptic integrals conference, October 2018

Reducing differential equations for multiloop master integrals journal, April 2015

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

Periods and Feynman integrals journal, April 2009

Two-loop planar master integrals for the production of off-shell vector bosons in hadron collisions journal, May 2014

The principle of maximal transcendentality and the four-loop collinear anomalous dimension journal, January 2018

Three-Loop Corrections to the Soft Anomalous Dimension in Multileg Scattering journal, October 2016

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

Analytic results for planar three-loop integrals for massive form factors text, January 2016

A Feynman integral via higher normal functions journal, August 2015

Two-loop master integrals for $ q\overline{q}\to VV $ : the planar topologies journal, August 2013

Master integrals for the NNLO virtual corrections to μe scattering in QED: the non-planar graphs journal, September 2018

A K3 in $\phi^{4}$ journal, July 2012

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

Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series journal, August 2018

Two-loop master integrals for $ q\overline{q}\to VV $ : the planar topologies text, January 2013

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

Bootstrapping the three-loop hexagon journal, November 2011

Master integrals for the NNLO virtual corrections to μe scattering in QED: the planar graphs journal, November 2017

The elliptic dilogarithm for the sunset graph journal, March 2015

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

The last of the simple remainders journal, August 2014

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

Pentagon functions for massless planar scattering amplitudes journal, October 2018

On genera of curves from high-loop generalized unitarity cuts journal, April 2013

CHAPLIN—Complex Harmonic Polylogarithms in Fortran journal, October 2014

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

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

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

Harmonic Polylogarithms journal, February 2000

The three-loop cusp anomalous dimension in QCD and its supersymmetric extensions journal, January 2016

Multiple polylogarithms, cyclotomy and modular complexes journal, January 1998

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

Mathematical Aspects of Scattering Amplitudes conference, November 2015

Three-loop master integrals for ladder-box diagrams with one massive leg journal, September 2014

Two-loop master integrals for the mixed EW-QCD virtual corrections to Drell-Yan scattering journal, September 2016

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

A planar four-loop form factor and cusp anomalous dimension in QCD journal, May 2016

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

Feynman integrals and iterated integrals on moduli spaces of curves of genus zero journal, January 2015

Uniqueness of two-loop master contours journal, October 2012

A planar four-loop form factor and cusp anomalous dimension in QCD text, January 2016

Heptagons from the Steinmann Cluster Bootstrap text, January 2017

Transforming differential equations of multi-loop Feynman integrals into canonical form journal, April 2017

Fuchsia : A tool for reducing differential equations for Feynman master integrals to epsilon form journal, October 2017

Single-valued harmonic polylogarithms and the multi-Regge limit journal, October 2012

Two-loop master integrals for the leading QCD corrections to the Higgs coupling to a W pair and to the triple gauge couplings ZW W and γ∗ W W journal, April 2017

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

Fuchsia : A tool for reducing differential equations for Feynman master integrals to epsilon form text, January 2017

Analytic results for planar three-loop integrals for massive form factors journal, December 2016

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

Dressing and wrapping journal, October 2007

Transcendentality and crossing journal, January 2007

Local integrals for planar scattering amplitudes journal, June 2012

Numerical evaluation of multiple polylogarithms journal, May 2005

Classical Polylogarithms for Amplitudes and Wilson Loops journal, October 2010

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

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

Analytic and algorithmic aspects of generalized harmonic sums and polylogarithms journal, August 2013

Lectures on differential equations for Feynman integrals journal, March 2015

On the reduction of generalized polylogarithms to Li n and Li2,2 and on the evaluation thereof journal, March 2016

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

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

Two-loop master integrals for the planar QCD massive corrections to di-photon and di-jet hadro-production
journal, January 2018

Heptagons from the Steinmann cluster bootstrap
journal, February 2017

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

Bootstrapping an NMHV amplitude through three loops
journal, October 2014

Notes on motivic periods
journal, January 2017

Master integrals for the NNLO virtual corrections to μe scattering in QED: the non-planar graphs
text, January 2018

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

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

A direct evaluation of the periods of the Weierstrass zeta function
journal, July 2013

Analytic results for two-loop master integrals for Bhabha scattering I
journal, November 2013

The ρ parameter at three loops and elliptic integrals
conference, October 2018

Reducing differential equations for multiloop master integrals
journal, April 2015

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

Periods and Feynman integrals
journal, April 2009

Two-loop planar master integrals for the production of off-shell vector bosons in hadron collisions
journal, May 2014

The principle of maximal transcendentality and the four-loop collinear anomalous dimension
journal, January 2018

Three-Loop Corrections to the Soft Anomalous Dimension in Multileg Scattering
journal, October 2016

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

Analytic results for planar three-loop integrals for massive form factors
text, January 2016

A Feynman integral via higher normal functions
journal, August 2015

Two-loop master integrals for $ q\overline{q}\to VV $ : the planar topologies
journal, August 2013

Master integrals for the NNLO virtual corrections to μe scattering in QED: the non-planar graphs
journal, September 2018

A K3 in $\phi^{4}$
journal, July 2012

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

Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series
journal, August 2018

Two-loop master integrals for $ q\overline{q}\to VV $ : the planar topologies
text, January 2013

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

Bootstrapping the three-loop hexagon
journal, November 2011

Master integrals for the NNLO virtual corrections to μe scattering in QED: the planar graphs
journal, November 2017

The elliptic dilogarithm for the sunset graph
journal, March 2015

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

The last of the simple remainders
journal, August 2014

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

Pentagon functions for massless planar scattering amplitudes
journal, October 2018

On genera of curves from high-loop generalized unitarity cuts
journal, April 2013

CHAPLIN—Complex Harmonic Polylogarithms in Fortran
journal, October 2014

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

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

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

Harmonic Polylogarithms
journal, February 2000

The three-loop cusp anomalous dimension in QCD and its supersymmetric extensions
journal, January 2016

Multiple polylogarithms, cyclotomy and modular complexes
journal, January 1998

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

Mathematical Aspects of Scattering Amplitudes
conference, November 2015

Three-loop master integrals for ladder-box diagrams with one massive leg
journal, September 2014

Two-loop master integrals for the mixed EW-QCD virtual corrections to Drell-Yan scattering
journal, September 2016

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

A planar four-loop form factor and cusp anomalous dimension in QCD
journal, May 2016

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

Feynman integrals and iterated integrals on moduli spaces of curves of genus zero
journal, January 2015

Uniqueness of two-loop master contours
journal, October 2012

A planar four-loop form factor and cusp anomalous dimension in QCD
text, January 2016

Heptagons from the Steinmann Cluster Bootstrap
text, January 2017

Transforming differential equations of multi-loop Feynman integrals into canonical form
journal, April 2017

Fuchsia : A tool for reducing differential equations for Feynman master integrals to epsilon form
journal, October 2017

Single-valued harmonic polylogarithms and the multi-Regge limit
journal, October 2012

Two-loop master integrals for the leading QCD corrections to the Higgs coupling to a W pair and to the triple gauge couplings ZW W and γ∗ W W
journal, April 2017

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

Fuchsia : A tool for reducing differential equations for Feynman master integrals to epsilon form
text, January 2017

Analytic results for planar three-loop integrals for massive form factors
journal, December 2016

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

Dressing and wrapping
journal, October 2007

Transcendentality and crossing
journal, January 2007

Local integrals for planar scattering amplitudes
journal, June 2012

Numerical evaluation of multiple polylogarithms
journal, May 2005

Classical Polylogarithms for Amplitudes and Wilson Loops
journal, October 2010

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

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

Analytic and algorithmic aspects of generalized harmonic sums and polylogarithms
journal, August 2013

Lectures on differential equations for Feynman integrals
journal, March 2015

On the reduction of generalized polylogarithms to Li n and Li2,2 and on the evaluation thereof
journal, March 2016

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

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

Three Loop Cusp Anomalous Dimension in QCD
journal, February 2015

Next-to-leading order QCD corrections to the decay width H → Zγ
journal, August 2015

Analytic Form of the Two-Loop Planar Five-Gluon All-Plus-Helicity Amplitude in QCD
journal, February 2016