  We study applications of spectral positivity and the averaged null energy
condition (ANEC) to renormalization group (RG) flows in two-dimensional quantum
field theory. We find a succinct new proof of the Zamolodchikov $c$-theorem,
and derive further independent constraints along the flow. In particular, we
identify a natural $C$-function that is a completely monotonic function of
scale, meaning its derivatives satisfy the alternating inequalities
$(-1)^nC^{(n)}(\mu^2) \geq 0$. The completely monotonic $C$-function is
identical to the Zamolodchikov $C$-function at the endpoints, but differs along
the RG flow. In addition, we apply Lorentzian techniques that we developed
recently to study anomalies and RG flows in four dimensions, and show that the
Zamolodchikov $c$-theorem can be restated as a Lorentzian sum rule relating the
change in the central charge to the average null energy. This establishes that
the ANEC implies the $c$-theorem in two dimensions, and provides a second,
simpler example of the Lorentzian sum rule.
