An algebraic/numerical formalism for one-loop multi-leg amplitudes

Abstract

We present a formalism for the calculation of multi-particle one-loop amplitudes, valid for an arbitrary number N of external legs, and for massive as well as massless particles. A new method for the tensor reduction is suggested which naturally isolates infrared divergences by construction. We prove that for N>4, higher dimensional integrals can be avoided. We derive many useful relations which allow for algebraic simplifications of one-loop amplitudes. We introduce a form factor representation of tensor integrals which contains no inverse Gram determinants by choosing a convenient set of basis integrals. For the evaluation of these basis integrals we propose two methods: An evaluation based on the analytical representation, which is fast and accurate away from exceptional kinematical configurations, and a robust numerical one, based on multi-dimensional contour deformation. The formalism can be implemented straightforwardly into a computer program to calculate next-to-leading order corrections to multi-particle processes in a largely automated way.

71 pages, 7 figures, formulas for rank 6 pentagons added in Appendix C

References in corpus (53)

• Matching NLO QCD computations and parton shower simulations
• Generalized Unitarity and One-Loop Amplitudes in N=4 Super-Yang-Mills
• Matching NLO QCD and parton showers in heavy flavour production
• Next-to-leading order calculation of three-jet observables in hadron-hadron collision
• Electroweak corrections to charged-current e+e- --> 4 fermion processes - technical details and further results
• NLO QCD corrections to t anti-t H production in hadron collisions
• Next-to-leading order corrections to W+2 jet and Z+2 jet production at hadron colliders
• Reduction of one-loop tensor 5-point integrals
• Three-jet cross sections in hadron-hadron collisions at next-to-leading order
• One-Loop Gauge Theory Amplitudes in N=4 Super Yang-Mills from MHV Vertices
• Recursion Relations for Gauge Theory Amplitudes with Massive Particles
• Numerical evaluation of multi-loop integrals by sector decomposition
• On-Shell Recurrence Relations for One-Loop QCD Amplitudes
• Next-to-Leading Order Results for t-tbar-h Production at the Tevatron
• One-Loop Amplitudes Of Gluons In SQCD
• Gluon-fusion contributions to Higgs plus two-jet production
• Twistor Space Structure Of One-Loop Amplitudes In Gauge Theory
• All Non-Maximally-Helicity-Violating One-Loop Seven-Gluon Amplitudes in N=4 Super-Yang-Mills Theory
• General subtraction method for numerical calculation of one-loop QCD matrix elements
• Separation of soft and collinear singularities from one-loop N-point integrals
• Complete electroweak O(alpha) corrections to charged-current e+e- --> 4fermion processes
• Computing One-Loop Amplitudes From The Holomorphic Anomaly Of Unitarity Cuts
• A Twistor Approach to One-Loop Amplitudes in N=1 Supersymmetric Yang-Mills Theory
• Recursive numerical calculus of one-loop tensor integrals
• All Next-to-Maximally-Helicity-Violating One-Loop Gluon Amplitudes in N=4 Super-Yang-Mills Theory
• Matching Parton Showers to NLO Computation
• Reduction method for dimensionally regulated one-loop N-point Feynman integrals
• All-Purpose Numerical Evaluation of One-Loop Multi-Leg Feynman Diagrams
• An Approach Toward the Numerical Evaluation of Multi-Loop Feynman Diagrams
• Loops in Twistor Space
• Gluon-induced WW background to Higgs boson searches at the LHC
• A Calculational Formalism for One-Loop Integrals
• Multi-jet Production in Hadron Collisions
• Automated computation of one-loop integrals in massless theories
• A numerical evaluation of the scalar hexagon integral in the physical region
• Electroweak radiative corrections to e+ e- -> nu nubar H
• Full one-loop electroweak radiative corrections to single Higgs production in e+ e-
• Full order_alpha electroweak and order alpha_s corrections to e^+e^- --> t t H
• Algebraic-Numerical Evaluation of Feynman Diagrams: Two-Loop Self-Energies
• Full order alpha electroweak corrections to double Higgs-strahlung at the linear collider
• Next-to-leading order numerical calculations in Coulomb gauge
• Electroweak radiative corrections to e+ e- -> t anti-t H
• Electroweak radiative corrections to single Higgs-boson production in e+e- annihilation
• Coplanarity In Twistor Space Of N=4 Next-To-MHV One-Loop Amplitude Coefficients
• Radiative corrections to Higgs-boson production in association with top-quark pairs at e+ e- colliders
• Numerical Evaluation of One-Loop Diagrams Near Exceptional Momentum Configurations
• Calculation of 1-loop Hexagon Amplitudes in the Yukawa Model
• A compact representation of the 2 photon 3 gluon amplitude
• $\mathcal{O}({\alpha }_s)$ QCD and $\mathcal{O}({\alpha }_{ew})$ electroweak corrections to $t\bar{t}{h}^0$ production in $\gamma \gamma$ collision
• Multi-leg calculations with the GRACE/1-LOOP system. --Toward Radiative Corrections to e+e- -> mu nu-bar u d-bar --
• Dimensionally regularized one-loop tensor-integrals with massless internal particles
• Progress in calculating hexagon amplitudes at one-loop
• Five-parton amplitudes with two-quark and two-photon at Next-to-Leading Order

Cited by in corpus (136)

• Collier: a fortran-based Complex One-Loop LIbrary in Extended Regularizations
• Reduction schemes for one-loop tensor integrals
• CutTools: a program implementing the OPP reduction method to compute one-loop amplitudes
• General-purpose event generators for LHC physics
• An Automated Implementation of On-Shell Methods for One-Loop Amplitudes
• OneLOop: for the evaluation of one-loop scalar functions
• Quest for precision in hadronic cross sections at low energy: Monte Carlo tools vs. experimental data
• Hard Interactions of Quarks and Gluons: a Primer for LHC Physics
• Direct extraction of one-loop integral coefficients
• GoSam-2.0: a tool for automated one-loop calculations within the Standard Model and beyond
• On-Shell Methods in Perturbative QCD
• On the Rational Terms of the one-loop amplitudes
• Scalar one-loop 4-point integrals
• Next-to-Leading Order QCD Predictions for W+3-Jet Distributions at Hadron Colliders
• SecDec-3.0: numerical evaluation of multi-scale integrals beyond one loop
• Towards a Basis for Planar Two-Loop Integrals
• NLO QCD corrections to pp -> t anti-t b anti-b + X at the LHC
• Full top quark mass dependence in Higgs boson pair production at NLO
• Two-loop Integrand Decomposition into Master Integrals and Surface Terms
• Electroweak Radiative Corrections for Collider Physics
• Sector Decomposition
• NLO QCD corrections to top anti-top bottom anti-bottom production at the LHC: 2. full hadronic results
• Recursive generation of one-loop amplitudes in the Standard Model
• Golem95: a numerical program to calculate one-loop tensor integrals with up to six external legs
• Next-to-leading order QCD corrections to t-tbar-Z production at the LHC
• Polarizing the Dipoles
• NLO QCD corrections to off-shell top-antitop production with leptonic decays at hadron colliders
• The Cut-Constructible Part of QCD Amplitudes
• Automation of the Dipole Subtraction Method in MadGraph/MadEvent
• Bootstrapping One-Loop QCD Amplitudes with General Helicities
• Gluon-induced W-boson pair production at the LHC
• SecDec: A general program for sector decomposition
• FIESTA 3: cluster-parallelizable multiloop numerical calculations in physical regions
• Feynman Rules for the Rational Part of the QCD 1-loop amplitudes
• Direct Extraction Of One Loop Rational Terms
• Collider Physics at the Precision Frontier
• NLO QCD corrections to tri-boson production
• Numerical integration of one-loop Feynman diagrams for N-photon amplitudes
• Numerical evaluation of multi-loop integrals for arbitrary kinematics with SecDec 2.0
• Hadronic top-quark pair production in association with a hard jet at next-to-leading order QCD: Phenomenological studies for the Tevatron and the LHC
• On the Integrand-Reduction Method for Two-Loop Scattering Amplitudes
• NLO QCD corrections to top anti-top bottom anti-bottom production at the LHC: 1. quark-antiquark annihilation
• Next-to-leading order QCD predictions for W+W+jj production at the LHC
• QCD corrections to tri-boson production
• S@M, a Mathematica Implementation of the Spinor-Helicity Formalism
• Golem95C: A library for one-loop integrals with complex masses
• Multi-Higgs boson production in the Standard Model and beyond
• Next-to-leading order QCD corrections to pp --> b b_bar b b_bar + X at the LHC: the quark induced case
• Algebraic evaluation of rational polynomials in one-loop amplitudes
• Numerical Evaluation of Six-Photon Amplitudes
• Tensorial Reconstruction at the Integrand Level
• NLO QCD corrections to ZZ+jet production at hadron colliders
• Evaluating multi-loop Feynman diagrams with infrared and threshold singularities numerically
• Numerical NLO QCD calculations
• Optimizing the Reduction of One-Loop Amplitudes
• spinney: A Form Library for Helicity Spinors
• Event Generators for High-Energy Physics Experiments
• Loop induced interference effects in Higgs Boson plus two jet production at the LHC
• All One-loop Maximally Helicity Violating Gluonic Amplitudes in QCD
• Semi-Numerical Evaluation of One-Loop Corrections
• Antenna subtraction for the production of heavy particles at hadron colliders
• NLO QCD corrections for W^+W^- pair production in association with two jets at hadron colliders
• Standard Model radiative corrections in the pion form factor measurements do not explain the $a_{\mu }$ anomaly
• Numerical evaluation of loop integrals
• Recursive Construction of Higgs-Plus-Multiparton Loop Amplitudes: The Last of the Phi-nite Loop Amplitudes
• Complete Nagy-Soper subtraction for next-to-leading order calculations in QCD
• Multi-gluon one-loop amplitudes using tensor integrals
• NLO QCD corrections to pp/ppbar --> WW+jet+X including leptonic W-boson decays
• Targeting Multi-Loop Integrals with Neural Networks
• Automated Squark and Gluino Production to Next-to-Leading Order
• Mixed QCD-electroweak contributions to Higgs-plus-dijet production at the LHC
• Virtual QCD corrections to Higgs boson plus four parton processes
• Towards pp -> VVjj at NLO QCD: Bosonic contributions to triple vector boson production plus jet
• Six-Photon Amplitudes
• A complete algebraic reduction of one-loop tensor Feynman integrals
• The rational parts of one-loop QCD amplitudes I: The general formalism
• Simplicity in the Structure of QED and Gravity Amplitudes
• Tools for NLO automation: extension of the golem95C integral library
• Efficient Numerical Evaluation of Feynman Integral
• WZ production in association with two jets at NLO in QCD
• Analytic Structure of Three-Mass Triangle Coefficients
• $ZH$ production in gluon fusion: two-loop amplitudes with full top quark mass dependence
• A complete reduction of one-loop tensor 5- and 6-point integrals
• NLO QCD corrections to graviton production at hadron colliders
• Susy-QCD corrections to neutralino pair production in association with a jet
• Virtual Color-Kinematics Duality: 6-pt 1-Loop MHV Amplitudes
• Automated computation of spin- and colour-correlated Born matrix elements
• Next-to-leading order QCD corrections to $W^{+}{W}^{+}$ and $W^{-}{W}^{-}$ production in association with two jets
• Six-Photon Amplitudes in Scalar QED
• Numerical multi-loop integrals and applications
• Complete QED NLO contributions to the reaction $e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-}\gamma$ and their implementation in the event generator PHOKHARA
• Numerical integration of loop integrals through local cancellation of threshold singularities
• String-inspired representations of photon/gluon amplitudes
• One-Loop Calculations with BlackHat
• Automated third generation squark production to next-to-leading order
• Next-to-leading order QCD corrections to ZZ production in association with two jets
• Next-to-leading order QCD corrections to $W\gamma$ production in association with two jets
• One-loop hexagon integral to higher orders in the dimensional regulator
• On the tensor reduction of one-loop pentagons and hexagons
• Precise predictions for LHC using a GOLEM
• A recursive reduction of tensor Feynman integrals
• Dissecting polytopes: Landau singularities and asymptotic expansions in $2\to 2$ scattering
• Random polarisations of the dipoles
• Automated Evaluation of One-Loop Six-Point Processes for the LHC
• Recent Progress in the Golem Project
• A Fully Numerical Approach to One-Loop Amplitudes
• Gluon Fusion Contribution to $VHj$ Production at Hadron Colliders
• A solution for tensor reduction of one-loop N-point functions with N >= 6
• Towards LHC phenomenology at the loop level: A new method for one-loop amplitudes
• On-Shell Unitarity Bootstrap for QCD Amplitudes
• Direct numerical evaluation of multi-loop integrals without contour deformation
• ZZ production at high transverse momenta beyond NLO QCD
• Flow-oriented perturbation theory
• SecDec: A tool for numerical multi-loop calculations
• One-loop tensor Feynman integral reduction with signed minors
• Electroweak corrections to $gg\to H^{-}t\bar{b}$ at the LHC
• New developments in PJFry
• A novel approach to the computation of one-loop three- and four-point functions. I -- The real mass case
• On the finite basis topologies for multi-loop high-multiplicity Feynman integrals
• Stable One-Dimensional Integral Representations of One-Loop N-Point Functions in the General Massive Case: I - Three Point Functions
• Three photon production to NLO+PS accuracy at the LHC
• Automated calculations for multi-leg processes
• NLO QCD corrections to the production of a weak boson pair associated by a hard jet
• Supersymmetric QCD corrections to $e^{+}{e}^{-}\to t\bar{b}{H}^{-}$ and the Bernstein-Tkachov method of loop integration
• On Higgs boson plus gluon amplitudes at one loop
• A novel approach to the computation of one-loop three- and four-point functions. III - The infrared divergent case
• QCD at hadron colliders
• An Automated Approach for $q\bar{q}\to b\bar{b}b\bar{b}$ at Next-to-Leading Order QCD
• Symmetric reduction of high-multiplicity one-loop integrals and maximal cuts
• New one-loop techniques and first applications to LHC phenomenology
• Numerical evaluation of multi-loop integrals
• Gluon Fusion Processes at One-loop within the Standard Model and Beyond
• Accurate backgrounds to Higgs production at the LHC
• Recurrence Relations and Dispersive Techniques for Precision Multi-Loop Calculations
• Applications of higher order QCD
• Two-loop scalar functions with $I$ internal lines, $I\le 5$