The all-loop integrand for scattering amplitudes in planar $ \mathcal{N} = 4 $ SYM
By N. Arkani-Hamed, J. Bourjaily, F. Cachazo, S. Caron-Huot, J. Trnka. 2010
313 citations recorded by the source corpus.
We give an explicit recursive formula for the all ℓ-loop integrand for scattering amplitudes in $ \mathcal{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.
Subfields: Nuclear and High Energy Physics
View Paper: https://www.semanticscholar.org/paper/40803a75e0362d9a9c1ed4e7f6adfc6c87e2f206
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In Nuclear and High Energy Physics
Citations in the corpus
- Hexagon functions and the three-loop remainder function (2013, unknown): “where Ω(1) is a finite, four-dimensional one-loop hexagon integral [74, 75], Ω(u, v, w) = lnu ln v + Li2(1− u) + Li2(1− v) + Li2(1− w)− 2ζ2 .”
- Lectures on differential equations for Feynman integrals (2014, background): “We already mentioned that leading singularities were also used to construct loop integrands having desirable physical properties, such as good behavior in the infrared [12–14].”
- Motivic amplitudes and cluster coordinates (2013, background): “This Grassmannian, describing the external kinematic data of an amplitude, may look unrelated to those which star in [27–32] and involve also ‘internal’ data related to loop integration variables.”