Two-Loop Four-Gluon Amplitudes from Numerical Unitarity

Abreu, S.; Cordero, F. Febres; Ita, H.; Jaquier, M.; Page, B.; Zeng, M.

doi:10.1103/physrevlett.119.142001

Title: Two-Loop Four-Gluon Amplitudes from Numerical Unitarity

Abstract

We present the first numerical computation of two-loop amplitudes based on the unitarity method. As a proof of principle, we compute the four-gluon process in the leading-color approximation. We discuss the new method, analyze its numerical properties, and apply it to reconstruct the analytic form of the amplitudes. The numerical method is universal, and can be automated to provide multiscale two-loop computations for phenomenologically relevant signatures at hadron colliders.

Authors:

Abreu, S. ^[1]; Cordero, F. Febres ^[1]; Ita, H. ^[1]; Jaquier, M. ^[1]; Page, B. ^[1]; Zeng, M. ^[2]

Univ. of Freiburg (Germany). Inst. of Physics
Univ. of California, Los Angeles, CA (United States). Mani L. Bhaumik Inst. for Theoretical Physics. Dept. of Physics and Astronomy

Publication Date:: Thu Oct 05 00:00:00 EDT 2017

Research Org.:: Univ. of California, Los Angeles, CA (United States)

Sponsoring Org.:: USDOE Office of Science (SC), High Energy Physics (HEP)

OSTI Identifier:: 1541230

Alternate Identifier(s):: OSTI ID: 1398151

Grant/Contract Number:: SC0009937

Resource Type:: Accepted Manuscript

Journal Name:: Physical Review Letters

Additional Journal Information:: Journal Volume: 119; Journal Issue: 14; Journal ID: ISSN 0031-9007

Publisher:: American Physical Society (APS)

Country of Publication:: United States

Language:: English

Subject:: 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS

Citation Formats


                    Abreu, S., Cordero, F. Febres, Ita, H., Jaquier, M., Page, B., and Zeng, M. Two-Loop Four-Gluon Amplitudes from Numerical Unitarity.  United States: N. p., 2017. 
Web.  doi:10.1103/physrevlett.119.142001.

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                    Abreu, S., Cordero, F. Febres, Ita, H., Jaquier, M., Page, B., & Zeng, M. Two-Loop Four-Gluon Amplitudes from Numerical Unitarity.  United States.  https://doi.org/10.1103/physrevlett.119.142001

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                    Abreu, S., Cordero, F. Febres, Ita, H., Jaquier, M., Page, B., and Zeng, M. Thu .  
"Two-Loop Four-Gluon Amplitudes from Numerical Unitarity".  United States.  https://doi.org/10.1103/physrevlett.119.142001.  https://www.osti.gov/servlets/purl/1541230.

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@article{osti_1541230,

  title        = {Two-Loop Four-Gluon Amplitudes from Numerical Unitarity},

  author       = {Abreu, S. and Cordero, F. Febres and Ita, H. and Jaquier, M. and Page, B. and Zeng, M.},

  abstractNote = {We present the first numerical computation of two-loop amplitudes based on the unitarity method. As a proof of principle, we compute the four-gluon process in the leading-color approximation. We discuss the new method, analyze its numerical properties, and apply it to reconstruct the analytic form of the amplitudes. The numerical method is universal, and can be automated to provide multiscale two-loop computations for phenomenologically relevant signatures at hadron colliders.},

  doi          = {10.1103/physrevlett.119.142001},

  journal      = {Physical Review Letters},

  number       = 14,

  volume       = 119,

  place        = {United States},

  year         = {Thu Oct 05 00:00:00 EDT 2017},

  month        = {Thu Oct 05 00:00:00 EDT 2017}

}

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Splitting amplitudes are universal functions governing the collinear behavior of scattering amplitudes for massless particles. We compute the two-loop g {yields} gg splitting amplitudes in QCD, N = 1, and N = 4 super-Yang-Mills theories, which describe the limits of two-loop n-point amplitudes where two gluon momenta become parallel. They also represent an ingredient in a direct x-space computation of DGLAP evolution kernels at next-to-next-to-leading order. To obtain the splitting amplitudes, we use the unitarity sewing method. In contrast to the usual light-cone gauge treatment, our calculation does not rely on the principal-value or Mandelstam-Leibbrandt prescriptions, even though the loopmore »« less
https://doi.org/10.2172/826914

Full Text Available
Analytic Form of Planar Two-Loop Five-Gluon Scattering Amplitudes in QCD

Journal Article Abreu, S. ; Dormans, J. ; Cordero, F. Febres ; ... - Physical Review Letters

We present the analytic form of the planar two-loop five-gluon scattering amplitudes in QCD for a complete set of independent helicity configurations of external gluons. These include the first analytic results for five-point two-loop amplitudes relevant for the computation of next-to-next-to-leading-order QCD corrections at hadron colliders. The results were obtained by reconstructing analytic expressions from numerical evaluations. The complexity of the computation is reduced by exploiting physical and analytical properties of the amplitudes, employing a minimal basis of so-called pentagon functions that have recently been classified.
https://doi.org/10.1103/physrevlett.122.082002

Full Text Available
Bootstrapping one-loop QCD amplitudes with general helicities

Journal Article Berger, Carola F ; Bern, Zvi ; Dixon, Lance J ; ... - Physical Review. D, Particles Fields

The recently developed on-shell bootstrap for computing one-loop amplitudes in nonsupersymmetric theories such as QCD combines the unitarity method with loop-level on-shell recursion. For generic helicity configurations, the recursion relations may involve undetermined contributions from nonstandard complex singularities or from large values of the shift parameter. Here we develop a strategy for sidestepping difficulties through the use of pairs of recursion relations. To illustrate the strategy, we present sets of recursion relations needed for obtaining n-gluon amplitudes in QCD. We give a recursive solution for the one-loop n-gluon QCD amplitudes with three or four color-adjacent gluons of negative helicity andmore »« less
https://doi.org/10.1103/PHYSREVD.74.036009
Caravel: A C++ framework for the computation of multi-loop amplitudes with numerical unitarity

Journal Article Abreu, S. ; Dormans, J. ; Cordero, F. Febres ; ... - Computer Physics Communications

We present the first public version of Caravel, a C++17 framework for the computation of multi-loop scattering amplitudes in quantum field theory, based on the numerical unitarity method. Caravel is composed of modules for the D-dimensional decomposition of integrands of scattering amplitudes into master and surface terms, the computation of tree-level amplitudes in floating point or finite-field arithmetic, the numerical computation of one- and two-loop amplitudes in QCD and Einstein gravity, and functional reconstruction tools. Here, we provide programs that showcase Caravel's main functionalities and allow to compute selected one- and two-loop amplitudes.
https://doi.org/10.1016/j.cpc.2021.108069

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Title: Two-Loop Four-Gluon Amplitudes from Numerical Unitarity

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Citation Formats

Analytical results for dimensionally regularized massless on-shell double boxes with arbitrary indices and numerators journal, January 2000

Two-loop integrand decomposition into master integrals and surface terms journal, December 2016

Explicit solutions of the 3-loop vacuum integral recurrence relations journal, September 1996

Integration-by-parts reductions from unitarity cuts and algebraic geometry journal, February 2016

Two-loop five-point all-plus helicity Yang-Mills amplitude journal, April 2016

LAPACK Users' Guide software, January 1999

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

The on-shell massless planar double box diagram with an irreducible numerator journal, October 2000

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

Integrand-level reduction of loop amplitudes by computational algebraic geometry methods journal, September 2012

Full one-loop amplitudes from tree amplitudes journal, April 2008

Reduze – Feynman integral reduction in C++ journal, July 2010

The two-loop scalar and tensor Pentabox graph with light-like legs journal, May 2000

The singular behaviour of QCD amplitudes at two-loop order journal, May 1998

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

Recursive calculations for processes with n gluons journal, September 1988

Regularization and renormalization of gauge fields journal, July 1972

Two-Loop Helicity Amplitudes for Gluon-Gluon Scattering in QCD and Supersymmetric Yang-Mills Theory journal, March 2002

Full colour for loop amplitudes in Yang-Mills theory journal, February 2017

Algorithm FIRE—Feynman Integral REduction journal, October 2008

Automated implementation of on-shell methods for one-loop amplitudes journal, August 2008

Erratum to “The two-loop scalar and tensor Pentabox graph with light-like legs” [Nucl. Phys. B 575 (2000) 416–436] journal, October 2000

HPL, a Mathematica implementation of the harmonic polylogarithms journal, February 2006

Scattering amplitudes over finite fields and multivariate functional reconstruction journal, December 2016

Automatic Integral Reduction for Higher Order Perturbative Calculations journal, July 2004

On the integrand-reduction method for two-loop scattering amplitudes journal, November 2011

Scattering amplitudes from multivariate polynomial division journal, November 2012

Subleading poles in the numerical unitarity method at two loops journal, May 2017

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

Two-Loop Six Gluon All-Plus Helicity Amplitude journal, August 2016

Application of the negative-dimension approach to massless scalar box integrals journal, January 2000

Reducing full one-loop amplitudes to scalar integrals at the integrand level journal, February 2007

Adaptive integrand decomposition in parallel and orthogonal space journal, August 2016

FIRE5: A C++ implementation of Feynman Integral REduction journal, April 2015

Singularities and Discontinuities of Feynman Amplitudes journal, September 1960

Generalized unitarity and one-loop amplitudes in super-Yang–Mills journal, October 2005

Local integrands for two-loop all-plus Yang-Mills amplitudes journal, August 2016

Towards a basis for planar two-loop integrals journal, February 2011

A numerical unitarity formalism for evaluating one-loop amplitudes journal, March 2008

A set of level 3 basic linear algebra subprograms journal, March 1990

Erratum: Analytic Form of the Two-Loop Planar Five-Gluon All-Plus-Helicity Amplitude in QCD [Phys. Rev. Lett. 116 , 062001 (2016)] journal, May 2016

Integration-by-parts reductions from unitarity cuts and algebraic geometry conference, October 2016

Integration-by-parts reductions from unitarity cuts and algebraic geometry text, January 2016

Analytic form of the two-loop planar five-gluon all-plus-helicity amplitude in QCD text, January 2016

Reduze – Feynman integral reduction in C++ text, January 2010

Full one-loop amplitudes from tree amplitudes text, January 2008

An Automated Implementation of On-Shell Methods for One-Loop Amplitudes text, January 2008

On the Integrand-Reduction Method for Two-Loop Scattering Amplitudes text, January 2011

Hepta-Cuts of Two-Loop Scattering Amplitudes text, January 2012

Integrand-Level Reduction of Loop Amplitudes by Computational Algebraic Geometry Methods text, January 2012

A Two-Loop Five-Gluon Helicity Amplitude in QCD text, January 2013

FIRE5: a C++ implementation of Feynman Integral REduction text, January 2014

A Complete Two-Loop, Five-Gluon Helicity Amplitude in Yang-Mills Theory text, January 2015

Two-loop Integrand Decomposition into Master Integrals and Surface Terms text, January 2015

Adaptive Integrand Decomposition in parallel and orthogonal space text, January 2016

Scattering amplitudes over finite fields and multivariate functional reconstruction text, January 2016

Full Colour for Loop Amplitudes in Yang-Mills Theory text, January 2016

The on-shell massless planar double box diagram with an irreducible numerator text, January 2000

Two-loop QCD corrections to gluon-gluon scattering text, January 2001

Two-Loop Helicity Amplitudes for Gluon-Gluon Scattering in QCD and Supersymmetric Yang-Mills Theory text, January 2002

Automatic Integral Reduction for Higher Order Perturbative Calculations text, January 2004

One-Loop n-Point Gauge Theory Amplitudes, Unitarity and Collinear Limits text, January 1994

Fusing Gauge Theory Tree Amplitudes Into Loop Amplitudes text, January 1994

The Singular Behaviour of QCD Amplitudes at Two-loop Order text, January 1998

The two-loop scalar and tensor pentabox graph with light-like legs text, January 1999

Automatic Integral Reduction for Higher Order Perturbative Calculations journal, July 2004

Towards a Basis for Planar Two-Loop Integrals text, January 2010

Scattering amplitudes over finite fields and multivariate functional reconstruction text, January 2016

Full Colour for Loop Amplitudes in Yang-Mills Theory text, January 2016

Subleading Poles in the Numerical Unitarity Method at Two Loops text, January 2017

A minimal approach to the scattering of physical massless bosons journal, May 2018

Extracting analytical one-loop amplitudes from numerical evaluations journal, July 2019

PolyLogTools — polylogs for the masses journal, August 2019

Planar two-loop five-parton amplitudes from numerical unitarity journal, November 2018

Four-particle scattering amplitudes in QCD at NNLO to higher orders in the dimensional regulator journal, December 2019

Determining arbitrary Feynman integrals by vacuum integrals journal, April 2019

Two-loop five-point massless QCD amplitudes within the integration-by-parts approach journal, April 2019

Analytical results for dimensionally regularized massless on-shell double boxes with arbitrary indices and numerators
journal, January 2000

Two-loop integrand decomposition into master integrals and surface terms
journal, December 2016

Explicit solutions of the 3-loop vacuum integral recurrence relations
journal, September 1996

Integration-by-parts reductions from unitarity cuts and algebraic geometry
journal, February 2016

Two-loop five-point all-plus helicity Yang-Mills amplitude
journal, April 2016

LAPACK Users' Guide
software, January 1999

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

The on-shell massless planar double box diagram with an irreducible numerator
journal, October 2000

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

Integrand-level reduction of loop amplitudes by computational algebraic geometry methods
journal, September 2012

Full one-loop amplitudes from tree amplitudes
journal, April 2008

Reduze – Feynman integral reduction in C++
journal, July 2010

The two-loop scalar and tensor Pentabox graph with light-like legs
journal, May 2000

The singular behaviour of QCD amplitudes at two-loop order
journal, May 1998

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

Recursive calculations for processes with n gluons
journal, September 1988

Regularization and renormalization of gauge fields
journal, July 1972

Two-Loop Helicity Amplitudes for Gluon-Gluon Scattering in QCD and Supersymmetric Yang-Mills Theory
journal, March 2002

Full colour for loop amplitudes in Yang-Mills theory
journal, February 2017

Algorithm FIRE—Feynman Integral REduction
journal, October 2008

Automated implementation of on-shell methods for one-loop amplitudes
journal, August 2008

Erratum to “The two-loop scalar and tensor Pentabox graph with light-like legs” [Nucl. Phys. B 575 (2000) 416–436]
journal, October 2000

HPL, a Mathematica implementation of the harmonic polylogarithms
journal, February 2006

Scattering amplitudes over finite fields and multivariate functional reconstruction
journal, December 2016

Automatic Integral Reduction for Higher Order Perturbative Calculations
journal, July 2004

On the integrand-reduction method for two-loop scattering amplitudes
journal, November 2011

Scattering amplitudes from multivariate polynomial division
journal, November 2012

Subleading poles in the numerical unitarity method at two loops
journal, May 2017

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

Two-Loop Six Gluon All-Plus Helicity Amplitude
journal, August 2016

Application of the negative-dimension approach to massless scalar box integrals
journal, January 2000

Reducing full one-loop amplitudes to scalar integrals at the integrand level
journal, February 2007

Adaptive integrand decomposition in parallel and orthogonal space
journal, August 2016

FIRE5: A C++ implementation of Feynman Integral REduction
journal, April 2015

Singularities and Discontinuities of Feynman Amplitudes
journal, September 1960

Generalized unitarity and one-loop amplitudes in super-Yang–Mills
journal, October 2005

Local integrands for two-loop all-plus Yang-Mills amplitudes
journal, August 2016

Towards a basis for planar two-loop integrals
journal, February 2011

A numerical unitarity formalism for evaluating one-loop amplitudes
journal, March 2008

A set of level 3 basic linear algebra subprograms
journal, March 1990

Erratum: Analytic Form of the Two-Loop Planar Five-Gluon All-Plus-Helicity Amplitude in QCD [Phys. Rev. Lett. 116 , 062001 (2016)]
journal, May 2016

Integration-by-parts reductions from unitarity cuts and algebraic geometry
conference, October 2016

Integration-by-parts reductions from unitarity cuts and algebraic geometry
text, January 2016

Analytic form of the two-loop planar five-gluon all-plus-helicity amplitude in QCD
text, January 2016

Reduze – Feynman integral reduction in C++
text, January 2010

Full one-loop amplitudes from tree amplitudes
text, January 2008

An Automated Implementation of On-Shell Methods for One-Loop Amplitudes
text, January 2008

On the Integrand-Reduction Method for Two-Loop Scattering Amplitudes
text, January 2011

Hepta-Cuts of Two-Loop Scattering Amplitudes
text, January 2012

Integrand-Level Reduction of Loop Amplitudes by Computational Algebraic Geometry Methods
text, January 2012

A Two-Loop Five-Gluon Helicity Amplitude in QCD
text, January 2013

FIRE5: a C++ implementation of Feynman Integral REduction
text, January 2014

A Complete Two-Loop, Five-Gluon Helicity Amplitude in Yang-Mills Theory
text, January 2015

Two-loop Integrand Decomposition into Master Integrals and Surface Terms
text, January 2015

Adaptive Integrand Decomposition in parallel and orthogonal space
text, January 2016

Scattering amplitudes over finite fields and multivariate functional reconstruction
text, January 2016

Full Colour for Loop Amplitudes in Yang-Mills Theory
text, January 2016

The on-shell massless planar double box diagram with an irreducible numerator
text, January 2000

Two-loop QCD corrections to gluon-gluon scattering
text, January 2001

Two-Loop Helicity Amplitudes for Gluon-Gluon Scattering in QCD and Supersymmetric Yang-Mills Theory
text, January 2002

Automatic Integral Reduction for Higher Order Perturbative Calculations
text, January 2004

One-Loop n-Point Gauge Theory Amplitudes, Unitarity and Collinear Limits
text, January 1994

Fusing Gauge Theory Tree Amplitudes Into Loop Amplitudes
text, January 1994

The Singular Behaviour of QCD Amplitudes at Two-loop Order
text, January 1998

The two-loop scalar and tensor pentabox graph with light-like legs
text, January 1999

Automatic Integral Reduction for Higher Order Perturbative Calculations
journal, July 2004

Towards a Basis for Planar Two-Loop Integrals
text, January 2010

Scattering amplitudes over finite fields and multivariate functional reconstruction
text, January 2016

Full Colour for Loop Amplitudes in Yang-Mills Theory
text, January 2016

Subleading Poles in the Numerical Unitarity Method at Two Loops
text, January 2017

A minimal approach to the scattering of physical massless bosons
journal, May 2018

Extracting analytical one-loop amplitudes from numerical evaluations
journal, July 2019

PolyLogTools — polylogs for the masses
journal, August 2019

Planar two-loop five-parton amplitudes from numerical unitarity
journal, November 2018

Four-particle scattering amplitudes in QCD at NNLO to higher orders in the dimensional regulator
journal, December 2019

Determining arbitrary Feynman integrals by vacuum integrals
journal, April 2019

Two-loop five-point massless QCD amplitudes within the integration-by-parts approach
journal, April 2019

Analytic Two-Loop Higgs Amplitudes in Effective Field Theory and the Maximal Transcendentality Principle
journal, September 2018

Analytic Form of Planar Two-Loop Five-Gluon Scattering Amplitudes in QCD
journal, February 2019

A numerical evaluation of planar two-loop helicity amplitudes for a W-boson plus four partons
text, January 2019