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 known expressions contain nested harmonic sums only at the two conjugate points z=(|n|-1)/2+i nu and zbar. At three loops the sums are evaluated on a ladder of integer-shifted arguments from the reflected point -zbar up to z, with one point past it, the coefficient at a rung fixed by its two distances to the ends. Seven families are the exception, their coefficients written through ladder sums whose summand shifts with the summation index. For one of the seven the coefficient lies outside the fixed-coefficient algebra of nested harmonic sums of the two distances, an arithmetic obstruction in the denominators. That exclusion reaches the eigenvalue coefficient only through an agreement of rule and coefficient verified and not proved. Rules uniform in the conformal spin produce all forty-seven families at every odd n>=3; no per-spin coefficient is tabulated in the closed form, 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 together carrying transcendental weight five. Along nu=0 the intercepts match the Quantum Spectral Curve values at each computed odd spin through n=91, the extent of those values. The comparison tests the integrand and the reduction together at one point of each spin, 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.