The next-to-next-to-leading order BFKL eigenvalue at odd conformal spin in planar N=4 super Yang-Mills

Abstract

We give the next-to-next-to-leading order color-singlet BFKL eigenvalue of planar N=4 super Yang-Mills at odd conformal spin n in closed form. At the two lower orders the nested harmonic sums appear only at the two conjugate points z=(|n|-1)/2+i nu and zbar. At three loops they are evaluated on a ladder of integer-shifted arguments from the reflected point -zbar up to z, with one further point past it, the coefficient at a rung fixed by its two distances to the ladder ends. Seven families are the exception: their coefficients need a ladder sum whose summand shifts with the summation index. Rules uniform in the conformal spin produce all forty-seven families at every odd n>=3; no per-spin coefficient is tabulated in the definition, and the n=1 boundary block is supplied separately. Those rules are a reconstruction from the computed spins, exact at every one of them, verified over the range n<=99 and at the holdout spin n=101, and not proved at arbitrary odd n. Each atom-table coefficient is a rational combination of 1, pi^2 and zeta_3, coefficient and atom sharing the transcendental weight five. Along nu=0 the intercepts match the Quantum Spectral Curve values at each computed odd spin up to n=91, which tests the integrand and the reduction together at one point of each spin, and for most of them for the first time. Away from that line the closed form gives the collinear behavior of the block uniformly in the spin, which the intercepts alone do not fix.

Comments: 17 pages, 1 table. Derivations and validation record: arXiv:2607.17613. Supplementary Material: a module carrying every coefficient rule the printed equations use, a program that checks them against the extracted atom tables in exact rational arithmetic, an evaluator, the exact atom tables for odd n <= 33, the rows of Table 1, and the Quantum Spectral Curve intercepts through n=91