  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 $TT\mathcal{O}$ 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 $\langle TTTT \rangle$
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 $TT\mathcal{O}$ OPE coefficients. We compute the
coefficients to the sum rule for arbitrary operators of spin 0 and 2, including
the energy-momentum tensor.
