Abstract
We initiate an exploration of on-shell functions in N = 4 SYM beyond the planar limit by providing compact, combinatorial expressions for all leading singularities of MHV amplitudes and showing that they can always be expressed as a positive sum of dierently ordered Parke-Taylor tree amplitudes. This is understood in terms of an ex- tended notion of positivity in G(2;n), the Grassmannian of 2-planes in n dimensions: a single on-shell diagram can be associated with many dierent \positive regions, of which the familiar G+(2;n) associated with planar diagrams is just one example. The decom- position into Parke-Taylor factors is simply a \triangulation of these extended positive regions. The U(1) decoupling and Kleiss-Kuijf (KK) relations satised by the Parke-Taylor amplitudes also follow naturally from this geometric picture. These results suggest that non-planar MHV amplitudes inN = 4 SYM at all loop orders can be expressed as a sum of polylogarithms weighted by color factors and (unordered) Parke-Taylor amplitudes.
Highlights
The main obstacle to finding an on-shell representation of N = 4 SYM beyond the planar limit is a simple, almost kinematical one: that the notion of “the” integrand for scattering amplitudes becomes ambiguous beyond the planar limit
If all non-planar loop amplitude integrands in N = 4 SYM are logarithmic as conjectured in ref. [9], this suggests that all MHV loop amplitudes of N = 4 SYM can be expressed in terms of polylogarithms with all coefficients being color factors times planar tree amplitudes involving differently ordered sets of external legs
Any MHV (k = 2) on-shell diagram corresponding to an ordinary function of the external data with kinematical support will involve precisely (n 2) black vertices, each of which is attached to exactly three external legs via white vertices
Summary
Leading singularities correspond to on-shell diagrams obtained by taking successive residues of loop amplitude integrands — putting some number of internal particles on-shell. 2.2 Classification of reduced, MHV on-shell functions and diagrams We begin by showing that any generically non-vanishing, reduced MHV on-shell diagram is naturally labeled by a list of (n 2) triples of external legs — corresponding to the labels of the (precisely) three external legs attached (via white vertices) to each of the (n 2) black vertices in the diagram This labeling is illustrated in the following example involving. Any MHV (k = 2) on-shell diagram corresponding to an ordinary function of the external data (nδ = 0) with kinematical support will involve precisely (n 2) black vertices, each of which is attached to exactly three external legs via white vertices. While we have chosen a particular ordering for each triple in the examples above, these choices should be viewed as completely arbitrary
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