The six-point remainder function to all loop orders in the multi-Regge limit

We present an all-orders formula for the six-point amplitude of planar maximally supersymmetric $ \mathcal{N}=4 $ Yang-Mills theory in the leading-logarithmic approximation of multi-Regge kinematics. In the MHV helicity configuration, our results agree with an integral formula of Lipatov and Prygarin through at least 14 loops. A differential equation linking the MHV and NMHV helicity configurations has a natural action in the space of functions relevant to this problem — the single-valued harmonic polylogarithms introduced by Brown. These functions depend on a single complex variable and its conjugate, w and w ^* , which are quadratically related to the original kinematic variables. We investigate the all-orders formula in the near-collinear limit, which is approached as |w| → 0. Up to power-suppressed terms, the resulting expansion may be organized by powers of log |w|. The leading term of this expansion agrees with the all-orders double-leading-logarithmic approximation of Bartels, Lipatov, and Prygarin. The explicit form for the sub-leading powers of log |w| is given in terms of modified Bessel functions.