An Automated Implementation of On-Shell Methods for One-Loop Amplitudes

Abstract

We present the first results from BlackHat, an automated C++ program for calculating one-loop amplitudes. The program implements the unitarity method and on-shell recursion to construct amplitudes. As input to the calculation, it uses compact analytic formulae for tree amplitudes for four-dimensional helicity states. The program performs all related computations numerically. We make use of recently developed on-shell methods for evaluating coefficients of loop integrals, introducing a discrete Fourier projection as a means of improving efficiency and numerical stability. We illustrate the numerical stability of our approach by computing and analyzing six-, seven- and eight-gluon amplitudes in QCD and comparing against previously-obtained analytic results.

38 pages, 10 figures, RevTex

References in corpus (67)

• Direct Proof Of Tree-Level Recursion Relation In Yang-Mills Theory
• New Recursion Relations for Tree Amplitudes of Gluons
• Generalized Unitarity and One-Loop Amplitudes in N=4 Super-Yang-Mills
• Multi-Parton Amplitudes in Gauge Theories
• Reducing full one-loop amplitudes to scalar integrals at the integrand level
• Electroweak gauge boson production at hadron colliders through O(alpha_s^2)
• Reduction schemes for one-loop tensor integrals
• CutTools: a program implementing the OPP reduction method to compute one-loop amplitudes
• Electroweak corrections to charged-current e+e- --> 4 fermion processes - technical details and further results
• Scalar one-loop integrals for QCD
• Direct extraction of one-loop integral coefficients
• Full one-loop amplitudes from tree amplitudes
• On-Shell Methods in Perturbative QCD
• Fully differential Higgs boson production and the di-photon signal through next-to-next-to-leading order
• A Numerical Unitarity Formalism for Evaluating One-Loop Amplitudes
• On the Rational Terms of the one-loop amplitudes
• D-dimensional unitarity cut method
• Automating dipole subtraction for QCD NLO calculations
• Higgs boson production at hadron colliders: differential cross sections through next-to-next-to-leading order
• Two-Loop g -> gg Splitting Amplitudes in QCD
• Recursion Relations for Gauge Theory Amplitudes with Massive Particles
• On-Shell Recurrence Relations for One-Loop QCD Amplitudes
• An algebraic/numerical formalism for one-loop multi-leg amplitudes
• One-Loop Amplitudes Of Gluons In SQCD
• The W boson production cross section at the LHC through O(alpha_s^2)
• Bootstrapping Multi-Parton Loop Amplitudes in QCD
• The Cut-Constructible Part of QCD Amplitudes
• All Non-Maximally-Helicity-Violating One-Loop Seven-Gluon Amplitudes in N=4 Super-Yang-Mills Theory
• Bootstrapping One-Loop QCD Amplitudes with General Helicities
• Unitarity cuts and reduction to master integrals in d dimensions for one-loop amplitudes
• Loop Amplitudes in Pure Yang-Mills from Generalised Unitarity
• Numerical integration of one-loop Feynman diagrams for N-photon amplitudes
• Complete electroweak O(alpha) corrections to charged-current e+e- --> 4fermion processes
• The Last of the Finite Loop Amplitudes in QCD
• Computing One-Loop Amplitudes From The Holomorphic Anomaly Of Unitarity Cuts
• Recursive numerical calculus of one-loop tensor integrals
• QCD corrections to tri-boson production
• Recursive Calculation of One-Loop QCD Integral Coefficients
• On Triple-Cut of Scattering Amplitudes
• One-Loop Gluon Scattering Amplitudes in Theories with $N<4$ Supersymmetries
• Unitarity Cuts with Massive Propagators and Algebraic Expressions for Coefficients
• N=1 Supersymmetric One-loop Amplitudes and the Holomorphic Anomaly of Unitarity Cuts
• Algebraic evaluation of rational polynomials in one-loop amplitudes
• Optimizing the Reduction of One-Loop Amplitudes
• One-loop phi-MHV amplitudes using the unitarity bootstrap
• Integral Coefficients for One-Loop Amplitudes
• All One-loop Maximally Helicity Violating Gluonic Amplitudes in QCD
• Semi-Numerical Evaluation of One-Loop Corrections
• Numerical evaluation of loop integrals
• Recursive Construction of Higgs-Plus-Multiparton Loop Amplitudes: The Last of the Phi-nite Loop Amplitudes
• On-shell recursion relations for all Born QCD amplitudes
• Virtual QCD corrections to Higgs boson plus four parton processes
• The one-loop amplitude for six-gluon scattering
• All-Multiplicity One-Loop Corrections to MHV Amplitudes in QCD
• The rational parts of one-loop QCD amplitudes I: The general formalism
• One-loop MHV Rules and Pure Yang-Mills
• Amplitudes in Pure Yang-Mills and MHV Diagrams
• Analytic Structure of Three-Mass Triangle Coefficients
• Supersymmetric Ward Identities and NMHV Amplitudes involving Gluinos
• QCD radiation effects on the H->WW->l nu l nu signal at the LHC
• Scattering amplitudes with massive fermions using BCFW recursion
• The rational parts of one-loop QCD amplitudes III: The six-gluon case
• Numerical Evaluation of One-Loop Diagrams Near Exceptional Momentum Configurations
• Combining Monte Carlo generators with next-to-next-to-leading order calculations: event reweighting for Higgs boson production at the LHC
• How accurately can we measure the W cross section?
• Evaluation of the Theoretical Uncertainties in the Z to ll Cross Sections at the LHC
• Massive Quark-Gluon Scattering Amplitudes at Tree Level

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