Challenges of D=6N=(1,1) SYM theory

Maximally supersymmetric Yang–Mills theories have several remarkable properties, among which are the cancellation of UV divergences, factorization of higher loop corrections and possible integrability. Much attention has been attracted to the N=4D=4 SYM theory. The N=(1,1)D=6 SYM theory possesses similar properties but is nonrenormalizable and serves as a toy model for supergravity. We consider the on-shell four point scattering amplitude and analyze its perturbative expansion within the spin-helicity and superspace formalism. The integrands of the resulting diagrams coincide with those of the N=4D=4 SYM and obey the dual conformal invariance. Contrary to 4 dimensions, no IR divergences on mass shell appear. We calculate analytically the leading logarithmic asymptotics in all loops. Their summation leads to a Regge trajectory which is calculated exactly. The leading powers of s are calculated up to six loops. Their summation is performed numerically and leads to a smooth function of s. The leading UV divergences are calculated up to 5 loops. The result suggests the geometrical progression which ends up in a finite expression. This leads us to a radical point of view on nonrenormalizable theories.