From Six to Four and More: Massless and Massive Maximal Super Yang-Mills Amplitudes in 6d and 4d and their Hidden Symmetries

Abstract: A self-consistent exposition of the theory of tree-level superamplitudes of the 4d N = 4 and 6d N = (1, 1) maximally supersymmetric Yang-Mills theories is pro-vided. In 4d we work in non-chiral superspace and construct the superconformal and dual superconformal symmetry generators of the N = 4 SYM theory using the non-chiral BCFW recursion to prove the latter. In 6d we provide a complete derivation of the stan-dard and hidden symmetries of the tree-level superamplitudes of N = (1, 1) SYM theory, again using the BCFW recursion to prove the dual conformal symmetry. Furthermore, we demonstrate that compact analytical formulae for tree-superamplitudes in N = (1, 1) SYM can be obtained from a numerical implementation of the supersymmetric BCFW recursion relation. We derive compact manifestly dual conformal representations of the five- and six-point superamplitudes as well as arbitrary multiplicity formulae valid for cer-tain classes of superamplitudes related to ultra-helicity-violating massive amplitudes in 4d. We study massive tree superamplitudes on the Coulomb branch of the N = 4 SYM theory from dimensional reduction of the massless superamplitudes of the six-dimensional N = (1, 1) SYM theory. We exploit this correspondence to construct the super-Poincaré and enhanced dual conformal symmetries of massive tree superamplitudes in N = 4 SYM theory which are shown to close into a finite dimensional algebra of Yangian type. Finally, we address the fascinating possibility of uplifting massless 4d superamplitudes to 6d mass-less superamplitudes proposed by Huang. We confirm the uplift for multiplicities up to eight but show that finding the uplift is highly non-trivial and in fact not of a practical use for multiplicities larger than five