Single-valued harmonic polylogarithms and the multi-Regge limit - Dixon, Lance J.; Duhr, Claude; Pennington, Jeffrey (The journal of high energy physics, 2012)
A bstract We argue that the natural functions for describing the multi-Regge limit of six-gluon scattering in planar super Yang-Mills theory are the single-valued harmonic polylogarithmic functions introduced by Brown. These functions depend on a single complex variable and its conjugate, ( w , w ∗ ). Using these functions, and formulas due to Fadin, Lipatov and Prygarin, we determine the six-gluon MHV remainder function in the leading-logarithmic approximation (LLA) in this limit through ten loops, and the next-to-LLA (NLLA) terms through nine loops. In separate work, we have determined the symbol of the four-loop remainder function for general kinematics, up to 113 constants. Taking its multi-Regge limit and matching to our four-loop LLA and NLLA results, we fix all but one of the constants that survive in this limit. The multi-Regge limit factorizes in the variables ( ν , n ) which are related to ( w , w ∗ ) by a Fourier-Mellin transform. We can transform the single-valued harmonic polylogarithms to functions of (ν, n ) that incorporate harmonic sums, systematically through transcendental weight six. Combining this information with the four-loop results, we determine the eigenvalues of the BFKL kernel in the adjoint representation to NNLLA accuracy, and the MHV product of impact factors to N 3 LLA accuracy, up to constants representing beyond-the-symbol terms and the one symbol-level constant. Remarkably, only derivatives of the polygamma function enter these results. Finally, the LLA approximation to the six-gluon NMHV amplitude is evaluated through ten loops.