The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM

Abstract

We give an explicit recursive formula for the all L-loop integrand for scattering amplitudes in N=4 SYM in the planar limit, manifesting the full Yangian symmetry of the theory. This generalizes the BCFW recursion relation for tree amplitudes to all loop orders, and extends the Grassmannian duality for leading singularities to the full amplitude. It also provides a new physical picture for the meaning of loops, associated with canonical operations for removing particles in a Yangian-invariant way. Loop amplitudes arise from the "entangled" removal of pairs of particles, and are naturally presented as an integral over lines in momentum-twistor space. As expected from manifest Yangian-invariance, the integrand is given as a sum over non-local terms, rather than the familiar decomposition in terms of local scalar integrals with rational coefficients. Knowing the integrands explicitly, it is straightforward to express them in local forms if desired; this turns out to be done most naturally using a novel basis of chiral, tensor integrals written in momentum-twistor space, each of which has unit leading singularities. As simple illustrative examples, we present a number of new multi-loop results written in local form, including the 6- and 7-point 2-loop NMHV amplitudes. Very concise expressions are presented for all 2-loop MHV amplitudes, as well as the 5-point 3-loop MHV amplitude. The structure of the loop integrand strongly suggests that the integrals yielding the physical amplitudes are "simple", and determined by IR-anomalies. We briefly comment on extending these ideas to more general planar theories.

46 pages; v2: minor changes, references added

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• Stratifying On-Shell Cluster Varieties: the Geometry of Non-Planar On-Shell Diagrams
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• Symbology for elliptic multiple polylogarithms and the symbol prime
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• Three-loop octagons and n-gons in maximally supersymmetric Yang-Mills theory
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• Grassmannian Integral for General Gauge Invariant Off-shell Amplitudes in N=4 SYM
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• A Novel Algorithm for Nested Summation and Hypergeometric Expansions
• Illustrations of Integrand-Basis Building at Two Loops
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• All Two-Loop, Color-Dressed, Six-Point Amplitude Integrands in sYM
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• Differential equations for multi-loop integrals and two-dimensional kinematics
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• Yangian symmetry of light-like Wilson loops
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• Simplex-like Structures of Maximally Supersymmetric Scattering Amplitudes
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• Leading singularities in higher-derivative Yang-Mills theory and quadratic gravity
• The integrand form of infrared singularities of two-loop QCD scattering amplitudes
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