From Trees to Loops and Back

Abstract

We argue that generic one-loop scattering amplitudes in supersymmetric Yang-Mills theories can be computed equivalently with MHV diagrams or with Feynman diagrams. We first present a general proof of the covariance of one-loop non-MHV amplitudes obtained from MHV diagrams. This proof relies only on the local character in Minkowski space of MHV vertices and on an application of the Feynman Tree Theorem. We then show that the discontinuities of one-loop scattering amplitudes computed with MHV diagrams are precisely the same as those computed with standard methods. Furthermore, we analyse collinear limits and soft limits of generic non-MHV amplitudes in supersymmetric Yang-Mills theories with one-loop MHV diagrams. In particular, we find a simple explicit derivation of the universal one-loop splitting functions in supersymmetric Yang-Mills theories to all orders in the dimensional regularisation parameter, which is in complete agreement with known results. Finally, we present concrete and illustrative applications of Feynman's Tree Theorem to one-loop MHV diagrams as well as to one-loop Feynman diagrams.

52 pages, 17 figures. Some typos in Appendix A corrected

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• The Complete Planar S-matrix of N=4 SYM as a Wilson Loop in Twistor Space
• The Cut-Constructible Part of QCD Amplitudes
• Analytic two-loop form factors in N=4 SYM
• Form Factors in N=4 Super Yang-Mills and Periodic Wilson Loops
• One-loop Gravitational Bremsstrahlung and Waveforms from a Heavy-Mass Effective Field Theory
• Harmony of Super Form Factors
• On Triple-Cut of Scattering Amplitudes
• Loop amplitudes in gauge theories: modern analytic approaches
• Gravity, Twistors and the MHV Formalism
• Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case
• Loops and trees
• MHV Diagrams in Momentum Twistor Space
• Integration Rules for Loop Scattering Equations
• Amplitudes in the beta-deformed Conformal Yang-Mills
• Two loop correction to interference in $gg\to ZZ$
• Iterated amplitudes in the high-energy limit
• An Introduction to On-shell Recursion Relations
• Four-point Amplitudes in N=8 Supergravity and Wilson Loops
• The rational parts of one-loop QCD amplitudes I: The general formalism
• One-loop MHV Rules and Pure Yang-Mills
• Structure of the MHV-rules Lagrangian
• Amplitudes in Pure Yang-Mills and MHV Diagrams
• Scattering amplitudes with massive fermions using BCFW recursion
• Tree-Level Formalism
• The rational parts of one-loop QCD amplitudes III: The six-gluon case
• One-Loop Gluonic Amplitudes from Single Unitarity Cuts
• Superspace formulations of the (super)twistor string
• One-loop monodromy relations on single cuts
• A quantization of twistor Yang-Mills theory through the background field method
• S-Matrix Equivalence Theorem Evasion and Dimensional Regularisation with the Canonical MHV Lagrangian
• MHV Lagrangian for N=4 Super Yang-Mills
• Propagators, BCFW Recursion and New Scattering Equations at One Loop
• Comments on gluon 6-point scattering amplitudes in N=4 SYM at strong coupling
• Numerical integration of loop integrals through local cancellation of threshold singularities
• Form factor recursion relations at loop level
• On the Factorisation of the Connected Prescription for Yang-Mills Amplitudes
• The rational parts of one-loop QCD amplitudes II: The five-gluon case
• Null Wilson loops with a self-crossing and the Wilson loop/amplitude conjecture
• Multi-soft theorems in Gauge Theory from MHV Diagrams
• One-loop N=8 Supergravity Amplitudes from MHV Diagrams
• Integrability and MHV diagrams in N=4 supersymmetric Yang-Mills theory
• One-Loop Soft Theorems via Dual Superconformal Symmetry
• A scattering amplitudes approach to hard thermal loops
• A Note on Dual MHV Diagrams in N=4 SYM
• A Surprise in the Amplitude/Wilson Loop Duality
• Application of the Feynman-tree theorem together with BCFW recursion relations
• Twistor and Polytope Interpretations for Subleading Color One-Loop Amplitudes
• Perturbative Gravity and Twistor Space
• Generating MHV super-vertices in light-cone gauge
• S-matrix equivalence restored
• Holonomies of gauge fields in twistor space 4: functional MHV rules and one-loop amplitudes
• Application of BCFW-recursion relations and the Feynman-tree theorem to the four gluon amplitude with all plus helicities
• Recursive structures in string field theory
• Forward scattering amplitudes in the imaginary time formalism
• Feynman diagrams in terms of on-shell propagators