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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Cited by in corpus (16)

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• Notes on the scattering amplitude / Wilson loop duality
• A Proof of the Explicit Minimal-basis Expansion of Tree Amplitudes in Gauge Field Theory
• New differential equations for on-shell loop integrals
• Recursion Relations for AdS/CFT Correlators
• Dual Conformal Properties of Six-Dimensional Maximal Super Yang-Mills Amplitudes
• Harmonic R-matrices for Scattering Amplitudes and Spectral Regularization
• Simple loop integrals and amplitudes in N=4 SYM
• Recursion Relations for Tree-level Amplitudes in the SU(N) Non-linear Sigma Model
• Bonus Yangian Symmetry for the Planar S-Matrix of N=4 Super Yang-Mills
• Pomerons and BCFW recursion relations for strings on D-branes
• On tree amplitudes with gluons coupled to gravitons
• Some analytic results for two-loop scattering amplitudes
• Manifest SO(N) invariance and S-matrices of three-dimensional N=2,4,8 SYM
• Heat kernel expansion and induced action for the matrix model Dirac operator
• On Closed Twistor String Theory