Averaged null energy and the renormalization group

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)(μ2) ≥ 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.

Null energy constraints on two-dimensional RG flows

Hartman,

Mathys

2024

Light-ray sum rules and the c-anomaly

Hartman,

Mathys

2024

In a four-dimensional quantum field theory that flows between two fixed points under the renormalization group, the change in the conformal anomaly ∆a has been related to the average null energy. We extend this result to derive a sum rule for the other anomaly coefficient, ∆c, in terms of the stress tensor three-point function. While the sum rule for ∆a is an expectation value of the averaged null energy operator, and therefore positive, the result for ∆c involves the off-diagonal matrix elements, so it does not have a fixed sign.

Explicit entropic proofs of irreversibility theorems for holographic RG flows

Deddo,

Liu,

Pando Zayas

et al. 2024

We revisit the existence of monotonic quantities along renormalization group flows using only the Null Energy Condition and the Ryu-Takayanagi formula for the entanglement entropy of field theories with anti-de Sitter gravity duals. In particular, we consider flows within the same dimension and holographically reprove the c-, F -, and a-theorems in dimensions two, three, and four. We focus on the family of maximally spherical entangling surfaces, define a quasi-constant of motion corresponding to the breaking of conformal invariance, and use a properly defined distance between minimal surfaces to construct a holographic c-function that is monotonic along the flow. We then apply our method to the case of flows across dimensions: there, we reprove the monotonicity of flows from AdSD+1 to AdS3 and prove the novel case of flows from AdS5 to AdS4.