Integrated Correlators in N=4 Supersymmetric Yang-Mills Theory beyond Localization

We study integrated correlators of four superconformal primaries O_{p} with arbitrary charges p in N=4 super Yang-Mills theory. The ⟨O_{2}O_{2}O_{p}O_{p}⟩ integrated correlators can be computed by supersymmetric localization, whereas correlators with more general charges are currently not accessible from this method and, in general, contain complicated multiple zeta values. Nevertheless, we observe that, if one sums over the contributions from all different channels in a given correlator, then all the multiple zeta values (and products of ζ's) cancel, leaving only ζ(2ℓ+1) at ℓ loops. We then propose an exact expression of such integrated correlators in the planar limit, valid for arbitrary 't Hooft coupling. The expression matches with the known exact localization-based results for specific charges, as well as with all existing perturbative and strong-coupling results in the literature for more general charges. As an application, our result is used to determine certain seven-loop Feynman integral periods and fix previously unknown coefficients in the correlators at strong coupling.