Local Integrand Representations of All Two-Loop Amplitudes in Planar SYM

Abstract

We use generalized unitarity at the integrand-level to directly construct local, manifestly dual-conformally invariant formulae for all two-loop scattering amplitudes in planar, maximally supersymmetric Yang-Mills theory (SYM). This representation separates contributions into manifestly finite and divergent terms---in a way that makes manifest the exponentiation of infrared divergences at the integrand-level. These results perfectly match the all-loop BCFW recursion relations, to which we provide a closed-form solution valid through two-loop-order. Finally, we describe and document a Mathematica package which implements these results, available as part of this work's source files on the arXiv.

Corrections made including an important erratum: merging finite integrands does not always result in finite integrals. Language modified to reflect this fact, but implications are left to future work. 58 pages, 85 figures, and a Mathematica package (with a demonstration notebook)

References in corpus (8)

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• The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM
• Yangian symmetry of scattering amplitudes in N=4 super Yang-Mills theory
• Bootstrapping an NMHV amplitude through three loops
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• Constructing d-log integrands and computing master integrals for three-loop four-particle scattering
• Amplitudes and Correlators to Ten Loops Using Simple, Graphical Bootstraps
• Evidence for a Nonplanar Amplituhedron
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• Prescriptive Unitarity
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• Landau Singularities from the Amplituhedron
• Landau Singularities and Symbology: One- and Two-loop MHV Amplitudes in SYM Theory
• Cutting Deep Into The Amplituhedron
• Local integrands for two-loop all-plus Yang-Mills amplitudes
• Stratifying On-Shell Cluster Varieties: the Geometry of Non-Planar On-Shell Diagrams
• Nonperturbative Negative Geometries: Amplitudes at Strong Coupling and the Amplituhedron
• All-Multiplicity Non-Planar MHV Amplitudes in sYM at Two Loops
• Prescriptive Unitarity for Non-Planar Six-Particle Amplitudes at Two Loops
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• Manifestly Dual-Conformal Loop Integration
• Two-loop $\mathcal{N}=2$ SQCD amplitudes with external matter from iterated cuts
• All-Mass $n$-gon Integrals in $n$ Dimensions
• Cluster Algebras and the Subalgebra Constructibility of the Seven-Particle Remainder Function
• UV cancelations in gravity loop integrands
• Building Bases of Loop Integrands
• Bootstrapping elliptic Feynman integrals using Schubert analysis
• What can we learn about QCD and collider physics from N=4 super Yang-Mills?
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• Prescriptive Unitarity with Elliptic Leading Singularities
• An Elliptic Yangian-Invariant, `Leading Singularity'
• Towards the Gravituhedron: New Expressions for NMHV Gravity Amplitudes
• Rooting Out Letters: Octagonal Symbol Alphabets and Algebraic Number Theory
• Multi-Particle Amplitudes from the Four-Point Correlator in Planar N=4 SYM
• Non-planar BCFW Grassmannian Geometries
• Conformally Regulated Direct Integration of the Two-Loop Heptagon Remainder
• A nice two-loop next-to-next-to-MHV amplitude in $\mathcal{N}=4$ super-Yang-Mills
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• On the finite basis topologies for multi-loop high-multiplicity Feynman integrals
• Illustrations of Integrand-Basis Building at Two Loops
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• Infrared and transcendental structure of two-loop supersymmetric QCD amplitudes
• Locally Finite Observables in sYM
• Positivity, Grassmannian Geometry and Simplex-like Structures of Scattering Amplitudes
• The spectrum of Feynman-integral geometries at two loops
• The Stratification of Rigidity