Gravity-Matter Sum Rules in models with a single extra-dimension

We prove a set of sum rules needed for KK-graviton pair production from matter in orbifolded extra-dimensional models. The sum rules can be found in full generality by considering the properties of solutions to the Sturm-Liouville problem, which describes the wave functions and the masses of the KK-gravitons in four dimensions. They ensure cancellations in the amplitudes of the processes mentioned above which considerably reduce their growth with $s$ in the high-energy limit. This protects extra-dimensional theories from the low-scale unitarity problems that plague other theories with massive spin-2 particles. We argue that such relations are valid for a broader category of models thus generalizing our previous results that were limited to the large $\mu$ limit of the Randall-Sundrum model.