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A<sc>bstract</sc> 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 Zamolodchikovc-theorem, and derive further independent constraints along the flow. In particular, we identify a naturalC-function that is a completely monotonic function of scale, meaning its derivatives satisfy the alternating inequalities (–1)ⁿC⁽ⁿ⁾(μ²) ≥ 0. The completely monotonicC-function is identical to the ZamolodchikovC-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 Zamolodchikovc-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 thec-theorem in two dimensions, and provides a second, simpler example of the Lorentzian sum rule. 

A<sc>bstract</sc> We establish a connection between the averaged null energy condition (ANEC) and the monotonicity of the renormalization group, by studying the light-ray operator ∫duT_uuin 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 thea-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 thea-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. 

A bstract We study effective field theories (EFTs) enjoying (maximal) biform symmetries. These are defined by the presence of a conserved (electric) current that has the symmetries of a Young tableau with two columns of equal length. When these theories also have a topological (magnetic) biform current, its conservation law is anomalous. We go on to show that this mixed anomaly uniquely fixes the two-point function between the electric and magnetic currents. We then perform a Källén-Lehmann spectral decomposition of the current-current correlator, proving that there is a massless mode in the spectrum, whose masslessness is protected by the anomaly. Furthermore, the anomaly gives rise to a universal form of the EFT whose most relevant term — which resembles the linear Einstein action — dominates the infrared physics. As applications of this general formalism, we study the theories of a Galileon superfluid and linearized gravity. Thus, one can view the masslessness of the graviton as being protected by the anomalous biform symmetries. The associated EFT provides an organizing principle for gravity at low energies in terms of physical symmetries, and allows interactions consistent with linearized diffeomorphism invariance. These theories are not ultraviolet-complete — the relevant symmetries can be viewed as emergent — nor do they include the nonlinearities necessary to make them fully diffeomorphism invariant, so there is no contradiction with the expectation that quantum gravity cannot have any global symmetries.