  We establish a connection between the averaged null energy condition (ANEC)
and the monotonicity of the renormalization group, by studying the light-ray
operator $\int du T_{uu}$ in quantum field theories that flow between two
conformal fixed points. In four dimensions, we derive an exact sum rule
relating this operator to the Euler coefficient in the trace anomaly, and show
that the ANEC implies the a-theorem. The argument is based on matching
anomalies in the stress tensor 3-point function, and relies on special
properties of contact terms involving light-ray operators. We also illustrate
the sum rule for the example of a free massive scalar field. Averaged null
energy appears in a variety of other applications to quantum field theory,
including causality constraints, Lorentzian inversion, and quantum information.
The quantum information perspective provides a new derivation of the
$a$-theorem from the monotonicity of relative entropy. The equation relating
our sum rule to the dilaton scattering amplitude in the forward limit suggests
an inversion formula for non-conformal theories.
