The next-to-next-to-leading order BFKL eigenvalue at odd conformal spin in planar N=4 super Yang-Mills: a reconstructed closed form, coefficient structure, and arithmetic

Abstract

We give the colour-singlet eigenvalue of the three-loop BFKL kernel of planar N=4 super Yang-Mills in closed form at odd conformal spin n. It splits into a holomorphic block and its conjugate, chi_2(n,nu)=(1/4)[F_n(z)+F_n(zbar)] with z=(|n|-1)/2+i nu, and the block is a finite combination of nested harmonic sums of one variable, rational functions and transcendental constants, of uniform transcendental weight five. At the two lower orders the sums are evaluated at the two conjugate points only. At three loops they are evaluated at every point of the integer ladder running from the reflected endpoint -zbar up to z, and one step beyond it, and for all but seven of the slots the coefficient at each rung is fixed by the two distances to the ends. Summing over the one-variable functions before summing along the ladder replaces the rules of those slots by one function of one variable under each ladder weight. The remaining seven are carried by a sum along the ladder whose summand moves with the summation index. All forty-seven coefficient slots follow from rules uniform in the conformal spin at every odd n>=3, with no per-spin coefficient tabulated in the definition; 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. Along nu=0 the intercepts agree with the Quantum Spectral Curve values at every computed odd spin through n=91. That comparison tests the three-loop integrand and the reduction performed here together, at one point of each spin, and for most of those spins it is the first comparison of its kind.