  We study three-point correlation functions of local operators in planar
$\mathcal{N}=4$ SYM at weak coupling using integrability. We consider
correlation functions involving two scalar BPS operators and an operator with
spin, in the so called SL(2) sector. At tree level we derive the corresponding
structure constant for any such operator. We also conjecture its one loop
correction. To check our proposals we analyze the conformal partial wave
decomposition of known four-point correlation functions of BPS operators. In
perturbation theory, we extract from this decomposition sums of structure
constants involving all primaries of a given spin and twist. On the other hand,
in our integrable setup these sum rules are computed by summing over all
solutions to the Bethe equations. A perfect match is found between the two
approaches.
