  We study some analytic properties of the BFKL ladder at next-to-leading
logarithmic accuracy (NLLA). We use a procedure by Chirilli and Kovchegov to
construct the NLO eigenfunctions, and we show that the BFKL ladder can be
evaluated order by order in the coupling in terms of certain generalised
single-valued multiple polylogarithms recently introduced by Schnetz. We
develop techniques to evaluate the BFKL ladder at any loop order, and we
present explicit results up to five loops. Using the freedom in defining the
matter content of the NLO BFKL eigenvalue, we obtain conditions for the BFKL
ladder in momentum space at NLLA to have maximal transcendental weight. We
observe that, unlike in moment space, the result in momentum space in N = 4 SYM
is not identical to the maximal weight part of QCD, and moreover that there is
no gauge theory with this property. We classify the theories for which the BFKL
ladder at NLLA has maximal weight in terms of their field content, and we find
that these theories are highly constrained: there are precisely four classes of
theories with this property involving only fundamental and adjoint matter, all
of which have a vanishing one-loop beta function and a matter content that fits
into supersymmetric multiplets. Our findings indicate that theories which have
maximal weight are highly constrained and point to the possibility that there
is a connection between maximal transcendental weight and superconformal
symmetry.
