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

Henn, Johannes; Herrmann, Enrico; Parra-Martinez, Julio

doi:10.1007/jhep10(2018)059

Title: Bootstrapping two-loop Feynman integrals for planar $$ \mathcal{N}=4 $$ sYM

Abstract

We derive analytic results for the symbol of certain two-loop Feynman integrals relevant for seven- and eight-point two-loop scattering amplitudes in planar $$ \mathcal{N}=4 $$ super-Yang-Mills theory. We use a bootstrap inspired strategy, combined with a set of second-order partial differential equations that provide powerful constraints on the symbol ansatz. When the complete symbol alphabet is not available, we adopt a hybrid approach. Instead of the full function, we bootstrap a certain discontinuity for which the alphabet is known. Then we write a one-fold dispersion integral to recover the complete result. At six and seven points, we find that the individual Feynman integrals live in the same space of functions as the amplitude, which is described by the 9- and 42-letter cluster alphabets respectively. Starting at eight points however, the symbol alphabet of the MHV amplitude is insufficient for individual integrals. In particular, some of the integrals require algebraic letters involving four-mass box square-root singularities. We point out that these algebraic letters are relevant at the amplitude level directly starting with N2MHV amplitudes even at one loop.

Authors:

Henn, Johannes ^[1];

^[2];

^[3]

Johannes Gutenberg University, Mainz (Germany); Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, Munchen (Germany)
SLAC National Accelerator Lab., Menlo Park, CA (United States); Stanford Univ., CA (United States)
Univ. of California, Los Angeles, CA (United States)

Publication Date:: Tue Oct 09 00:00:00 EDT 2018

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

Sponsoring Org.:: USDOE

OSTI Identifier:: 1490625

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: 10; Journal ID: ISSN 1029-8479

Publisher:: Springer Berlin

Country of Publication:: United States

Language:: English

Subject:: 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 1/N Expansion; Scattering Amplitudes; Supersymmetric Gauge Theory

Citation Formats


                    Henn, Johannes, Herrmann, Enrico, and Parra-Martinez, Julio. Bootstrapping two-loop Feynman integrals for planar $ \mathcal{N}=4 $ sYM.  United States: N. p., 2018. 
Web.  doi:10.1007/jhep10(2018)059.

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                    Henn, Johannes, Herrmann, Enrico, & Parra-Martinez, Julio. Bootstrapping two-loop Feynman integrals for planar $ \mathcal{N}=4 $ sYM.  United States.  https://doi.org/10.1007/jhep10(2018)059

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                    Henn, Johannes, Herrmann, Enrico, and Parra-Martinez, Julio. Tue .  
"Bootstrapping two-loop Feynman integrals for planar $ \mathcal{N}=4 $ sYM".  United States.  https://doi.org/10.1007/jhep10(2018)059.  https://www.osti.gov/servlets/purl/1490625.

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@article{osti_1490625,

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

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

  abstractNote = {We derive analytic results for the symbol of certain two-loop Feynman integrals relevant for seven- and eight-point two-loop scattering amplitudes in planar $ \mathcal{N}=4 $ super-Yang-Mills theory. We use a bootstrap inspired strategy, combined with a set of second-order partial differential equations that provide powerful constraints on the symbol ansatz. When the complete symbol alphabet is not available, we adopt a hybrid approach. Instead of the full function, we bootstrap a certain discontinuity for which the alphabet is known. Then we write a one-fold dispersion integral to recover the complete result. At six and seven points, we find that the individual Feynman integrals live in the same space of functions as the amplitude, which is described by the 9- and 42-letter cluster alphabets respectively. Starting at eight points however, the symbol alphabet of the MHV amplitude is insufficient for individual integrals. In particular, some of the integrals require algebraic letters involving four-mass box square-root singularities. We point out that these algebraic letters are relevant at the amplitude level directly starting with N2MHV amplitudes even at one loop.},

  doi          = {10.1007/jhep10(2018)059},

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

  number       = 10,

  volume       = 2018,

  place        = {United States},

  year         = {Tue Oct 09 00:00:00 EDT 2018},

  month        = {Tue Oct 09 00:00:00 EDT 2018}

}

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Table 1: Number of unconstraint coefficients in the heptagon Steinmann ansatz after applying various constraints for both two-loop heptagon integrals.

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Works referencing / citing this record:

Prescriptive Unitarity for Non-Planar Six-Particle Amplitudes at Two Loops
text, January 2019

Bourjaily, Jacob L.; Herrmann, Enrico; Langer, Cameron
arXiv
DOI: 10.48550/arxiv.1909.09131

Figures / Tables found in this record:

Table 1 (p. 13)

Table 2 (p. 14)

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Heptagons from the Steinmann cluster bootstrap

Journal Article Dixon, Lance J. ; Drummond, James ; Harrington, Thomas ; ... - Journal of High Energy Physics (Online)

We reformulate the heptagon cluster bootstrap to take advantage of the Steinmann relations, which require certain double discontinuities of any amplitude to vanish. These constraints vastly reduce the number of functions needed to bootstrap seven-point amplitudes in planarmore »« less
Cited by 102
https://doi.org/10.1007/JHEP02(2017)137

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A symbol of uniqueness: the cluster bootstrap for the 3-loop MHV heptagon

Journal Article Drummond, J. M. ; Papathanasiou, G. ; Spradlin, M. - Journal of High Energy Physics (Online)

Seven-particle scattering amplitudes in planar super-Yang-Mills theory are believed to belong to a special class of generalised polylogarithm functions called heptagon functions. These are functions with physical branch cuts whose symbols may be written in terms of the 42 cluster A-coordinates on Gr(4, 7). Motivated by the success of the hexagon bootstrap programme for constructing six-particle amplitudes we initiate the systematic study of the symbols of heptagon functions. We find that there is exactly one such symbol of weight six which satisfies the MHV last-entry condition and is finite in the 7 ll 6 collinear limit. This unique symbol ismore »« less
Cited by 106
https://doi.org/10.1007/JHEP03(2015)072

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Bootstrapping a stress-tensor form factor through eight loops

Journal Article Dixon, Lance J. ; Gürdoğan, Ömer ; McLeod, Andrew J. ; ... - Journal of High Energy Physics (Online)

We bootstrap the three-point form factor of the chiral stress-tensor multiplet in planarmore »« less
https://doi.org/10.1007/jhep07(2022)153

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Title: Bootstrapping two-loop Feynman integrals for planar $$ \mathcal{N}=4 $$ sYM

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Feynman Diagrams and Differential Equations journal, September 2007

Heptagons from the Steinmann cluster bootstrap journal, February 2017

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

Boundaries of amplituhedra and NMHV symbol alphabets at two loops journal, April 2018

A cluster bootstrap for two-loop MHV amplitudes journal, February 2015

On analytic properties of vertex parts in quantum field theory journal, October 1959

Bootstrapping an NMHV amplitude through three loops journal, October 2014

N=4 supersymmetric Yang–Mills scattering amplitudes at high energies: the Regge cut contribution journal, December 2009

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

Four-Gluon Scattering at Three Loops, Infrared Structure, and the Regge Limit journal, October 2016

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

Rationalizing loop integration journal, August 2018

Multiloop Integrals in Dimensional Regularization Made Simple journal, June 2013

Landau singularities from the amplituhedron journal, June 2017

The Amplituhedron journal, October 2014

The double pentaladder integral to all orders journal, July 2018

A periodic table of effective field theories journal, February 2017

Evidence for a nonplanar amplituhedron journal, June 2016

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

Gluon scattering amplitudes at strong coupling journal, June 2007

Into the amplituhedron journal, December 2014

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

On-shell structures of MHV amplitudes beyond the planar limit journal, June 2015

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

On hyperlogarithms and Feynman integrals with divergences and many scales journal, March 2014

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

Landau singularities and symbology: one- and two-loop MHV amplitudes in SYM theory journal, March 2016

Iterative structure of finite loop integrals journal, June 2014

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

Dual superconformal symmetry of scattering amplitudes in super-Yang–Mills theory journal, March 2010

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

All-helicity symbol alphabets from unwound amplituhedra journal, May 2018

Superconformal symmetry and two-loop amplitudes in planar N = 4 super Yang-Mills journal, December 2011

From multiple unitarity cuts to the coproduct of Feynman integrals journal, October 2014

Simple recursion relations for general field theories journal, June 2015

On-Shell Recursion Relations for Effective Field Theories journal, January 2016

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

Logarithmic singularities and maximally supersymmetric amplitudes journal, June 2015

Bootstrapping pentagon functions journal, May 2018

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

Motivic amplitudes and cluster coordinates journal, January 2014

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

Cluster Adjacency Properties of Scattering Amplitudes in N = 4 Supersymmetric Yang-Mills Theory journal, April 2018

Optical theorems and steinmann relations journal, April 1975

One-loop n-point gauge theory amplitudes, unitarity and collinear limits journal, August 1994

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

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

Scattering Amplitudes from Superconformal Ward Identities journal, July 2018

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

Automatized analytic continuation of Mellin–Barnes integrals journal, October 2006

Harmonic Sums, Mellin Transforms and Integrals journal, May 1999

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

Classical Polylogarithms for Amplitudes and Wilson Loops journal, October 2010

On the resolution of singularities of multiple Mellin–Barnes integrals journal, May 2009

From polygons and symbols to polylogarithmic functions journal, October 2012

Gravity on-shell diagrams journal, November 2016

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

Grassmannian Geometry of Scattering Amplitudes book, May 2016

Mathematical Aspects of Scattering Amplitudes conference, November 2015

Feynman integrals via hyperlogarithms conference, August 2014

Cluster ensembles, quantization and the dilogarithm journal, January 2009

On Analytic Properties of Vertex Parts in Quantum Field Theory book, January 1965

Heptagons from the Steinmann Cluster Bootstrap text, January 2017

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

Motivic amplitudes and cluster coordinates text, January 2014

The double pentaladder integral to all ordersC text, January 2018

Feynman Diagrams and Differential Equations text, January 2007

N=4 supersymmetric Yang Mills scattering amplitudes at high energies: the Regge cut contribution text, January 2008

On the Resolution of Singularities of Multiple Mellin-Barnes Integrals text, January 2009

Eliminating spurious poles from gauge-theoretic amplitudes preprint, January 2009

Classical Polylogarithms for Amplitudes and Wilson Loops text, January 2010

Superconformal symmetry and two-loop amplitudes in planar N=4 super Yang-Mills text, January 2011

Feynman Diagrams and Differential Equations
journal, September 2007

Heptagons from the Steinmann cluster bootstrap
journal, February 2017

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

Boundaries of amplituhedra and NMHV symbol alphabets at two loops
journal, April 2018

A cluster bootstrap for two-loop MHV amplitudes
journal, February 2015

On analytic properties of vertex parts in quantum field theory
journal, October 1959

Bootstrapping an NMHV amplitude through three loops
journal, October 2014

N=4 supersymmetric Yang–Mills scattering amplitudes at high energies: the Regge cut contribution
journal, December 2009

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

Four-Gluon Scattering at Three Loops, Infrared Structure, and the Regge Limit
journal, October 2016

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

Rationalizing loop integration
journal, August 2018

Multiloop Integrals in Dimensional Regularization Made Simple
journal, June 2013

Landau singularities from the amplituhedron
journal, June 2017

The Amplituhedron
journal, October 2014

The double pentaladder integral to all orders
journal, July 2018

A periodic table of effective field theories
journal, February 2017

Evidence for a nonplanar amplituhedron
journal, June 2016

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

Gluon scattering amplitudes at strong coupling
journal, June 2007

Into the amplituhedron
journal, December 2014

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

On-shell structures of MHV amplitudes beyond the planar limit
journal, June 2015

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

On hyperlogarithms and Feynman integrals with divergences and many scales
journal, March 2014

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

Landau singularities and symbology: one- and two-loop MHV amplitudes in SYM theory
journal, March 2016

Iterative structure of finite loop integrals
journal, June 2014

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

Dual superconformal symmetry of scattering amplitudes in super-Yang–Mills theory
journal, March 2010

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

All-helicity symbol alphabets from unwound amplituhedra
journal, May 2018

Superconformal symmetry and two-loop amplitudes in planar N = 4 super Yang-Mills
journal, December 2011

From multiple unitarity cuts to the coproduct of Feynman integrals
journal, October 2014

Simple recursion relations for general field theories
journal, June 2015

On-Shell Recursion Relations for Effective Field Theories
journal, January 2016

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

Logarithmic singularities and maximally supersymmetric amplitudes
journal, June 2015

Bootstrapping pentagon functions
journal, May 2018

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

Motivic amplitudes and cluster coordinates
journal, January 2014

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

Cluster Adjacency Properties of Scattering Amplitudes in $N = 4$ Supersymmetric Yang-Mills Theory
journal, April 2018

Optical theorems and steinmann relations
journal, April 1975

One-loop n-point gauge theory amplitudes, unitarity and collinear limits
journal, August 1994

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

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

Scattering Amplitudes from Superconformal Ward Identities
journal, July 2018

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

Automatized analytic continuation of Mellin–Barnes integrals
journal, October 2006

Harmonic Sums, Mellin Transforms and Integrals
journal, May 1999

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

Classical Polylogarithms for Amplitudes and Wilson Loops
journal, October 2010

On the resolution of singularities of multiple Mellin–Barnes integrals
journal, May 2009

From polygons and symbols to polylogarithmic functions
journal, October 2012

Gravity on-shell diagrams
journal, November 2016

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

Grassmannian Geometry of Scattering Amplitudes
book, May 2016

Mathematical Aspects of Scattering Amplitudes
conference, November 2015

Feynman integrals via hyperlogarithms
conference, August 2014

Cluster ensembles, quantization and the dilogarithm
journal, January 2009

On Analytic Properties of Vertex Parts in Quantum Field Theory
book, January 1965

Heptagons from the Steinmann Cluster Bootstrap
text, January 2017

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

Motivic amplitudes and cluster coordinates
text, January 2014

The double pentaladder integral to all ordersC
text, January 2018

Feynman Diagrams and Differential Equations
text, January 2007

N=4 supersymmetric Yang Mills scattering amplitudes at high energies: the Regge cut contribution
text, January 2008

On the Resolution of Singularities of Multiple Mellin-Barnes Integrals
text, January 2009

Eliminating spurious poles from gauge-theoretic amplitudes
preprint, January 2009

Classical Polylogarithms for Amplitudes and Wilson Loops
text, January 2010

Superconformal symmetry and two-loop amplitudes in planar N=4 super Yang-Mills
text, January 2011

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

Multiloop integrals in dimensional regularization made simple
text, January 2013