2026/07/20 by Alex Prygarin, Alexander Prygarin, Claudelle Capasia Madjuogang Sandeu
#hep-th #hep-ph
The eigenvalue of the next-to-next-to-leading order BFKL kernel of planar N=4 super Yang-Mills is presented in closed form at odd conformal spin n, as a finite combination of nested harmonic sums, rational functions of the single variable z=(|n|-1)/2 + i nu, and transcendental constants. Each odd spin n>=3 is extracted exactly, in rational arithmetic, from the Caron-Huot-Herranen three-loop integrand; a shipped command regenerates the atom table for any such spin, with n=1 a supplied boundary block. The coefficients are rational multiples of 1, pi2 and zeta3; their dependence on the spin and on the position along the ladder of integer-shifted arguments is given, on the computed range in a fixed elementary basis, for forty of the forty-seven coefficient slots by harmonic sums of the two ladder distances (all-spin form conjectural), that split being a property of the basis, and for one of the remaining families by a finite-data holonomic recurrence, with an all-n rule for the rest still open. Beyond the explicit harmonic sums the coefficient- weighted ladder part collapses onto depth-one digamma and trigamma functions at integer and half-integer shifts. That a single resummed master reproduces the Caron-Huot-Herranen integrand at every spin, a conjecture in the companion, is verified by exact finite-order checks through n=17 and spin by spin at 9,11,13,15,21,31. At nu=0 the intercepts reproduce the independent Quantum Spectral Curve values at the evaluated odd spins through n=91. At each fixed odd spin the eigenvalue is strictly product-free, the additive case of the Kotikov-Lipatov hermitian-separable form; the next-to-leading anomalous term carrying 1+(-1)n is absent at odd n. No n-uniform form is claimed. Derivations (eleven of sixteen closures proven, five verified numerically) and complete expressions are given in a companion paper.