Local integrands for the five-point amplitude in planar N=4 SYM up to five loops

Abstract

Integrands for colour ordered scattering amplitudes in planar N=4 SYM are dual to those of correlation functions of the energy-momentum multiplet of the theory. The construction can relate amplitudes with different numbers of legs.By graph theory methods the integrand of the four-point function of energy-momentum multiplets has been constructed up to six loops in previous work. In this article we extend this analysis to seven loops and use it to construct the full integrand of the five-point amplitude up to five loops, and in the parity even sector to six loops.All results, both parity even and parity odd, are obtained in a concise local form in dual momentum space and can be displayed efficiently through graphs. We have verified agreement with other local formulae both in terms of supertwistors and scalar momentum integrals as well as BCJ forms where those exist in the literature, i.e. up to three loops.Finally we note that the four-point correlation function can be extracted directly from the four-point amplitude and so this uncovers a direct link from four-to five-point amplitudes.

Bootstrapping correlation functions in N = 4 $$ \mathcal{N}=4 $$ SYM

“…The duality (2.8) can be used to learn about amplitudes from the knowledge of correlation functions [1,2]. In particular, the predictions for the four-dimensional part of the amplitude integrands elaborated from (2.8) using available results for the correlation functions exactly agree with the results of the recursive all-loops procedure of [22].…”

“…, x n satisfying (2.15) and having only simple poles in the limit x 2 ij → 0. 2 As was demonstrated in [1,2,6], these properties alone fix the coefficient functions f (0) n (x) up to an overall normalization constant, e.g. for n = 5, 6 we have…”

“…In this way, the Born-level correlation functions define the all-loop integrands for G n . For example, in the special case of p = n − 4, the relation (2.7), combined with the uniqueness of the top nilpotent invariant N n,n−4 = 1, has been used in [1,2,6] to compute the four-point correlation function G 4 up to seven loops.…”

“…N = 4 superconformal symmetry fixes G 4 up to a single function of the conformal cross-ratios. At present, G 4 is known in planar N = 4 SYM theory at weak coupling up to seven loops in terms of scalar conformal integrals [1,2] whereas the integrated expressions have been worked out up to three loops [3][4][5][6][7]. At strong coupling, G 4 has been computed within the AdS/CFT correspondence in the supergravity approximation [8,9].…”

Evaluating four-loop conformal Feynman integrals by D-dimensional differential equations

“…On the field theory side, the integrand of the four-point function of stress-tensor multiplets has been elaborated in [8,[14][15][16] up to eight loops. It takes the form of a kinematic factor [17] times a sum of scalar conformal integrals in a propagator representation.…”