Averaged Null Energy and the Renormalization Group

Understand

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.