Loop corrections for Kaluza-Klein AdS amplitudes

Recently we conjectured the four-point amplitude of graviton multiplets in ${\mathrm{A}\mathrm{d}\mathrm{S}}_5\times {\mathrm{S}}^5$ at one loop by exploiting the operator product expansion of $\mathcal{N}=4$ super Yang-Mills theory. Here we give the first extension of those results to include Kaluza-Klein modes, obtaining the amplitude for two graviton multiplets and two states of the first KK mode. Our method again relies on resolving the large N degeneracy among a family of long double-trace operators, for which we obtain explicit formulas for the leading anomalous dimensions. Having constructed the one-loop amplitude we are able to obtain a formula for the one-loop corrections to the anomalous dimensions of all twist five double-trace operators.