Abstract
The overline{Q} equations, rooted in the dual superconformal anomalies, are a powerful tool for computing amplitudes in planar mathcal{N} = 4 supersymmetric Yang-Mills theory. By using the overline{Q} equations, we compute the symbol of the first MHV amplitude with algebraic letters — the three-loop 8-point amplitude (or the octagon remainder function) — in this theory. The symbol alphabet for this amplitude consists of 204 independent rational letters and shares the same 18 algebraic letters with the two-loop 8-point NMHV amplitude.
Highlights
Considering scattering amplitudes with more than 7 external particles: firstly, transcendental functions beyond multiple polylogarithms occur in the scattering amplitudes, such as elliptic polylogarithms [24, 25] and beyond [26], secondly, even for MHV and NMHV cases, we lose control of symbol alphabets from cluster algebras since i) the algebraic letters, which are not rational in the Plücker coordinates anymore, start to appear in two-loop NMHV amplitudes [27], and ii) the corresponding cluster algebras Gr(4, n) with n > 7 are all of infinite type [28]
With the two-loop 9-point NMHV amplitude as the input, we computed the symbol of the three-loop octagon remainder function R8(3,0), which is the first threeloop amplitudes containing algebraic letters, by using the Qequations
The alphabet of the symbol S(R8(3,0)) consists of 18 algebraic letters and 204 rational ones, where 24 of the rational letters are new compared with the symbol alphabet of the two-loop 8-point NMHV amplitude
Summary
In N = 4 sYM theory, instead of the usual scattering of particles, we are more interested in the superamplitudes A(Φ1, · · · , Φn) of on-shell superfields. For the planar limit of the theory, the scattering amplitudes have dual superconformal symmetries in addition to usual superconformal symmetries. In terms of super momentum twistors, we further define the basic SL(4)-invariant ijkl := abcdZiaZjbZkcZld (or the Plücker coordinates of Gr(4, n)), and the basic R invariant [40, 46]. The infrared divergences and the dual conformal anomalies of planar N = 4 sYM amplitudes are captured by the BDS ansatz ABnDS [47], and BDS-subtracted amplitudes Rn,k = An,k/ABnDS are finite and dual conformally invariant (DCI). As we will see below, this amplitude can be computed from one-fold integrals of R9(2,1) where Yn, are R invariants (2.3)
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