Graviton scattering and a sum rule for the c anomaly in 4D CFT

4D CFTs have a scale anomaly characterized by the coefficient c, which appears as the coefficient of logarithmic terms in momentum space correlation functions of the energy-momentum tensor. By studying the CFT contribution to 4-point graviton scattering amplitudes in Minkowski space we derive a sum rule for c in terms of TTO OPE coefficients. The sum rule can be thought of as a version of the optical theorem, and its validity depends on the existence of the massless and forward limits of the < TTTT > correlation functions that contribute. The finiteness of these limits is checked explicitly for free scalar, fermion, and vector CFTs. The sum rule gives c as a sum of positive terms, and therefore implies a lower bound on c given any lower bound on TTO OPE coefficients. We compute the coefficients to the sum rule for arbitrary operators of spin 0 and 2, including the energy-momentum tensor.