Higher Derivative Gravity, Causality and Positivity of Energy in a UV complete QFT

TL;DR

The paper investigates how causality constraints in higher-derivative gravity theories living in AdS bulk map to positivity of energy flux in the dual CFT. By analyzing black hole and shock-wave backgrounds, it connects bulk graviton and gauge-field propagation to the CFT data encoded in the central charges a and c, recovering known SUSY-based bounds on a/c and showing Gauss-Bonnet gravity uniquely aligns with these energy constraints at the level of three-point functions. The work also demonstrates that, in general higher-derivative theories, higher-point graviton interactions can spoil the simple bulk-to-boundary mapping, as illustrated by W^2 corrections. A field-theoretic argument is provided to justify energy positivity in any UV-complete QFT, reinforcing the robustness of these positivity constraints beyond holographic models. Overall, the paper clarifies the UV origin of energy positivity and delineates the limits of effective gravitational actions in reproducing CFT energy correlators.

Abstract

In this note we discuss the relation between the constraints imposed by causality in the bulk of $AdS$ and the condition of positivity of the energy measured in ideal calorimeters in a collider experiment in the dual CFT. We first extend the analysis in the literature and recover all bounds imposed by causality of the boundary theory in the bulk dynamics for all polarizations of the graviton and the gauge boson field. These results translate to specific bounds for the ratio of central charges $\frac{a}{c}$ in the dual CFT, already found by analyzing the energy one point function. Then, we generalize this discussion and we study shock wave backgrounds in which we make manifest the relation between causality in the bulk and the three point function in the dual field theory. We remark that particular care has to be given to the exponentiation procedure of the three point function when solving the classical equations of motion in the higher gravity theory, as it is not clear that every theory will present causality problems. Finally, we present a field theoretic argument explaining the positivity of energy condition in any UV complete QFT.

Figures (3)

• Figure 1: Plots of $c^2_2(r)$. (a) Plot for $\lambda=0.05<\frac{9}{100}$. (b) Plot for $\lambda=0.15>\frac{9}{100}$. Notice that $c^2_2>1$ for $r$ close enough to the boundary $r \rightarrow \infty$ when $\lambda>\frac{9}{100}$.
• Figure 2: Light cone structure of our boundary space time. The thick line represents the unchanged null geodesic away from the shock wave in transverse space $\{x,y\} \neq \{0,0\}$. The thin line is the displaced geodesic at the position of the shock wave $\{x,y\}=\{0,0\}$. Notice that if we have a $\delta$ function type shock wave this geodesic sits at $x^-=\infty$. If we find that $\Delta x^- < 0$ for the metric perturbation (dashed line), we will encounter a causality problem.
• Figure 3: Only the three point function contributes to the exponentiation given by the classical solution. Solid lines represent the perturbation while the dashed lines represents the shock wave.