The four-loop six-gluon NMHV ratio function

Dixon, Lance J.; von Hippel, Matt; McLeod, Andrew J.

doi:10.1007/JHEP01(2016)053

Title: The four-loop six-gluon NMHV ratio function

Abstract

We use the hexagon function bootstrap to compute the ratio function which characterizes the next-to-maximally-helicity-violating (NMHV) six-point amplitude in planar N=4 super-Yang-Mills theory at four loops. A powerful constraint comes from dual superconformal invariance, in the form of a Q¯ differential equation, which heavily constrains the first derivatives of the transcendental functions entering the ratio function. At four loops, it leaves only a 34-parameter space of functions. Constraints from the collinear limits, and from the multi-Regge limit at the leading-logarithmic (LL) and next-to-leading-logarithmic (NLL) order, suffice to fix these parameters and obtain a unique result. We test the result against multi-Regge predictions at NNLL and N³LL, and against predictions from the operator product expansion involving one and two flux-tube excitations; all cross-checks are satisfied. We study the analytical and numerical behavior of the parity-even and parity-odd parts on various lines and surfaces traversing the three-dimensional space of cross ratios. As part of this program, we characterize all irreducible hexagon functions through weight eight in terms of their coproduct. As a result, we also provide representations of the ratio function in particular kinematic regions in terms of multiple polylogarithms.

Authors:

Dixon, Lance J. ^[1]; von Hippel, Matt ^[2]; McLeod, Andrew J. ^[3]

SLAC National Accelerator Lab., Menlo Park, CA (United States); California Inst. of Technology (CalTech), Pasadena, CA (United States)
Perimeter Institute for Theoretical Physics, Waterloo, ON (Canada)
SLAC National Accelerator Lab., Menlo Park, CA (United States)

Publication Date:: Mon Jan 11 00:00:00 EST 2016

Research Org.:: SLAC National Accelerator Laboratory (SLAC), Menlo Park, CA (United States); California Institute of Technology (CalTech), Pasadena, CA (United States)

Sponsoring Org.:: USDOE Office of Science (SC), High Energy Physics (HEP); National Science Foundation (NSF); Betty Moore Foundation

OSTI Identifier:: 1239131

Alternate Identifier(s):: OSTI ID: 1598894

Grant/Contract Number:: AC03-76SF00515; SC0011632; 1066293; 776

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: 2016; 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; scattering amplitudes; supersymmetric gauge theory; 1/N expansion

Citation Formats


                    Dixon, Lance J., von Hippel, Matt, and McLeod, Andrew J. The four-loop six-gluon NMHV ratio function.  United States: N. p., 2016. 
Web.  doi:10.1007/JHEP01(2016)053.

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                    Dixon, Lance J., von Hippel, Matt, & McLeod, Andrew J. The four-loop six-gluon NMHV ratio function.  United States.  https://doi.org/10.1007/JHEP01(2016)053

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                    Dixon, Lance J., von Hippel, Matt, and McLeod, Andrew J. Mon .  
"The four-loop six-gluon NMHV ratio function".  United States.  https://doi.org/10.1007/JHEP01(2016)053.  https://www.osti.gov/servlets/purl/1239131.

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@article{osti_1239131,

  title        = {The four-loop six-gluon NMHV ratio function},

  author       = {Dixon, Lance J. and von Hippel, Matt and McLeod, Andrew J.},

  abstractNote = {We use the hexagon function bootstrap to compute the ratio function which characterizes the next-to-maximally-helicity-violating (NMHV) six-point amplitude in planar N=4 super-Yang-Mills theory at four loops. A powerful constraint comes from dual superconformal invariance, in the form of a Q¯ differential equation, which heavily constrains the first derivatives of the transcendental functions entering the ratio function. At four loops, it leaves only a 34-parameter space of functions. Constraints from the collinear limits, and from the multi-Regge limit at the leading-logarithmic (LL) and next-to-leading-logarithmic (NLL) order, suffice to fix these parameters and obtain a unique result. We test the result against multi-Regge predictions at NNLL and N3LL, and against predictions from the operator product expansion involving one and two flux-tube excitations; all cross-checks are satisfied. We study the analytical and numerical behavior of the parity-even and parity-odd parts on various lines and surfaces traversing the three-dimensional space of cross ratios. As part of this program, we characterize all irreducible hexagon functions through weight eight in terms of their coproduct. As a result, we also provide representations of the ratio function in particular kinematic regions in terms of multiple polylogarithms.},

  doi          = {10.1007/JHEP01(2016)053},

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

  number       = 1,

  volume       = 2016,

  place        = {United States},

  year         = {Mon Jan 11 00:00:00 EST 2016},

  month        = {Mon Jan 11 00:00:00 EST 2016}

}

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Title: The four-loop six-gluon NMHV ratio function

Abstract

Citation Formats

The Multi-Regge limit of NMHV amplitudes in N=4 SYM theory journal, January 2013

Polylogarithmes multiples uniformes en une variable journal, April 2004

Space-time S-matrix and flux-tube S-matrix III. The two-particle contributions journal, August 2014

Notes on the scattering amplitude — Wilson loop duality journal, July 2011

Conformal properties of four-gluon planar amplitudes and Wilson loops journal, May 2008

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

On planar gluon amplitudes/Wilson loops duality journal, May 2008

The Amplituhedron journal, October 2014

MHV amplitudes in super-Yang–Mills and Wilson Loops journal, May 2008

When does the gluon reggeize? journal, May 2015

Adjoint BFKL at finite coupling: a short-cut from the collinear limit journal, January 2015

Review of AdS/CFT Integrability: An Overview journal, October 2011

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

Fermionic pentagons and NMHV hexagon journal, May 2015

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

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

Gluon scattering amplitudes at strong coupling journal, June 2007

Non-MHV Tree Amplitudes in Gauge Theory journal, July 2004

Into the amplituhedron journal, December 2014

Bootstrapping six-gluon scattering in planar N=4 super-Yang-Mills theory conference, August 2014

An analytic result for the two-loop hexagon Wilson loop in $ \mathcal{N} = 4 $ SYM journal, March 2010

Factorization in one-loop gauge theory journal, August 1995

Generalized unitarity for super-amplitudes journal, April 2013

Bootstrapping the three-loop hexagon journal, November 2011

Nonsinglet pentagons and NMHV amplitudes journal, July 2015

Light-cone superspace and the ultraviolet finiteness of the N=4 model journal, March 1983

Spacetime and Flux Tube S -Matrices at Finite Coupling for N = 4 Supersymmetric Yang-Mills Theory journal, August 2013

What is the simplest quantum field theory? journal, September 2010

Eliminating spurious poles from gauge-theoretic amplitudes journal, May 2013

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

The complete planar S-matrix of $ \mathcal{N} = 4 $ SYM as a Wilson loop in twistor space journal, December 2010

Space-time S-matrix and flux tube S-matrix II. Extracting and matching data journal, January 2014

Generating tree amplitudes in 𝒩 = 4 SYM and 𝒩 = 8 SG journal, September 2008

From d logs to dilogs; the super Yang-Mills MHV amplitude revisited journal, January 2014

Dual superconformal invariance, momentum twistors and Grassmannians journal, November 2009

Quantum mechanics of null polygonal Wilson loops journal, May 2014

The two-loop hexagon Wilson loop in $ \mathcal{N} = 4 $ SYM journal, May 2010

Supersymmetric Yang-Mills theories journal, March 1977

Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes journal, February 2010

Harmonic Polylogarithms journal, February 2000

Numerical evaluation of multiple polylogarithms journal, May 2005

Motivic amplitudes and cluster coordinates journal, January 2014

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

Miraculous ultraviolet cancellations in supersymmetry made manifest journal, April 1984

Scattering amplitudes, Wilson loops and the string/gauge theory correspondence journal, November 2008

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

The all-loop integrand for scattering amplitudes in planar $ \mathcal{N} = 4 $ SYM journal, January 2011

Two-loop master integrals for jets: the planar topologies journal, May 2001

The soft-collinear bootstrap: $ \mathcal{N} = {4} $ Yang-Mills amplitudes at six- and seven-loops journal, March 2012

The ultra-violet finiteness of the N=4 Yang-Mills theory journal, April 1983

Exciting the GKP string at any coupling journal, April 2012

Space-time S-matrix and flux-tube S-matrix IV. Gluons and fusion journal, September 2014

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

A current algebra for some gauge theory amplitudes journal, November 1988

Classical Polylogarithms for Amplitudes and Wilson Loops journal, October 2010

Pulling the straps of polygons journal, December 2011

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

From polygons and symbols to polylogarithmic functions journal, October 2012

Hexagon Wilson loop = six-gluon MHV amplitude journal, July 2009

Introduction to the GiNaC Framework for Symbolic Computation within the C++ Programming Language journal, January 2002

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

Jumpstarting the all-loop S-matrix of planar $ \mathcal{N} = {4} $ super Yang-Mills journal, July 2012

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

Motivic amplitudes and cluster coordinates text, January 2014

Single-valued harmonic polylogarithms and the multi-Regge limit text, January 2012

Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes text, January 2007

Hexagon Wilson loop = six-gluon MHV amplitude text, January 2008

Generating Tree Amplitudes in N=4 SYM and N=8 SG text, January 2008

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

What is the Simplest Quantum Field Theory? text, January 2008

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

An Operator Product Expansion for Polygonal null Wilson Loops text, January 2010

Classical Polylogarithms for Amplitudes and Wilson Loops text, January 2010

Exciting the GKP string at any coupling text, January 2010

The Multi-Regge limit of NMHV amplitudes in N=4 SYM theory
journal, January 2013

Polylogarithmes multiples uniformes en une variable
journal, April 2004

Space-time S-matrix and flux-tube S-matrix III. The two-particle contributions
journal, August 2014

Notes on the scattering amplitude — Wilson loop duality
journal, July 2011

Conformal properties of four-gluon planar amplitudes and Wilson loops
journal, May 2008

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

On planar gluon amplitudes/Wilson loops duality
journal, May 2008

The Amplituhedron
journal, October 2014

MHV amplitudes in super-Yang–Mills and Wilson Loops
journal, May 2008

When does the gluon reggeize?
journal, May 2015

Adjoint BFKL at finite coupling: a short-cut from the collinear limit
journal, January 2015

Review of AdS/CFT Integrability: An Overview
journal, October 2011

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

Fermionic pentagons and NMHV hexagon
journal, May 2015

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

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

Gluon scattering amplitudes at strong coupling
journal, June 2007

Non-MHV Tree Amplitudes in Gauge Theory
journal, July 2004

Into the amplituhedron
journal, December 2014

Bootstrapping six-gluon scattering in planar N=4 super-Yang-Mills theory
conference, August 2014

An analytic result for the two-loop hexagon Wilson loop in $ \mathcal{N} = 4 $ SYM
journal, March 2010

Factorization in one-loop gauge theory
journal, August 1995

Generalized unitarity for super-amplitudes
journal, April 2013

Bootstrapping the three-loop hexagon
journal, November 2011

Nonsinglet pentagons and NMHV amplitudes
journal, July 2015

Light-cone superspace and the ultraviolet finiteness of the N=4 model
journal, March 1983

Spacetime and Flux Tube $S$ -Matrices at Finite Coupling for $N = 4$ Supersymmetric Yang-Mills Theory
journal, August 2013

What is the simplest quantum field theory?
journal, September 2010

Eliminating spurious poles from gauge-theoretic amplitudes
journal, May 2013

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

The complete planar S-matrix of $ \mathcal{N} = 4 $ SYM as a Wilson loop in twistor space
journal, December 2010

Space-time S-matrix and flux tube S-matrix II. Extracting and matching data
journal, January 2014

Generating tree amplitudes in 𝒩 = 4 SYM and 𝒩 = 8 SG
journal, September 2008

From d logs to dilogs; the super Yang-Mills MHV amplitude revisited
journal, January 2014

Dual superconformal invariance, momentum twistors and Grassmannians
journal, November 2009

Quantum mechanics of null polygonal Wilson loops
journal, May 2014

The two-loop hexagon Wilson loop in $ \mathcal{N} = 4 $ SYM
journal, May 2010

Supersymmetric Yang-Mills theories
journal, March 1977

Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes
journal, February 2010

Harmonic Polylogarithms
journal, February 2000

Numerical evaluation of multiple polylogarithms
journal, May 2005

Motivic amplitudes and cluster coordinates
journal, January 2014

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

Miraculous ultraviolet cancellations in supersymmetry made manifest
journal, April 1984

Scattering amplitudes, Wilson loops and the string/gauge theory correspondence
journal, November 2008

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

The all-loop integrand for scattering amplitudes in planar $ \mathcal{N} = 4 $ SYM
journal, January 2011

Two-loop master integrals for jets: the planar topologies
journal, May 2001

The soft-collinear bootstrap: $ \mathcal{N} = {4} $ Yang-Mills amplitudes at six- and seven-loops
journal, March 2012

The ultra-violet finiteness of the N=4 Yang-Mills theory
journal, April 1983

Exciting the GKP string at any coupling
journal, April 2012

Space-time S-matrix and flux-tube S-matrix IV. Gluons and fusion
journal, September 2014

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

A current algebra for some gauge theory amplitudes
journal, November 1988

Classical Polylogarithms for Amplitudes and Wilson Loops
journal, October 2010

Pulling the straps of polygons
journal, December 2011

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

From polygons and symbols to polylogarithmic functions
journal, October 2012

Hexagon Wilson loop = six-gluon MHV amplitude
journal, July 2009

Introduction to the GiNaC Framework for Symbolic Computation within the C++ Programming Language
journal, January 2002

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

Jumpstarting the all-loop S-matrix of planar $ \mathcal{N} = {4} $ super Yang-Mills
journal, July 2012

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

Motivic amplitudes and cluster coordinates
text, January 2014

Single-valued harmonic polylogarithms and the multi-Regge limit
text, January 2012

Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes
text, January 2007

Hexagon Wilson loop = six-gluon MHV amplitude
text, January 2008

Generating Tree Amplitudes in N=4 SYM and N=8 SG
text, January 2008

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

What is the Simplest Quantum Field Theory?
text, January 2008

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

An Operator Product Expansion for Polygonal null Wilson Loops
text, January 2010

Classical Polylogarithms for Amplitudes and Wilson Loops
text, January 2010

Exciting the GKP string at any coupling
text, January 2010

Review of AdS/CFT Integrability: An Overview
text, January 2010

Pulling the straps of polygons
text, January 2011

Bootstrapping the three-loop hexagon
text, January 2011