Swampland Conditions for Higher Derivative Couplings from CFT

TL;DR

The paper establishes precise swampland-type constraints on higher-derivative scalar interactions in AdS via the dual CFT bootstrap. By analyzing Lorentzian Regge four-point correlators in the CFT and leveraging analyticity, unitarity, and crossing symmetry, it derives necessary conditions on EFT couplings of the form $\phi^2\Box^k\phi^2$, notably positivity, monotonicity, and log-convexity for even $k\ge 2$ in the absence of gravity. Inclusion of dynamical gravity relaxes some bounds, allowing a negative $\lambda_2$ bounded by gravity-dependent terms proportional to $G_N$, while leaving higher-even-$k$ bounds intact; the framework naturally extends to multi-field EFTs and points toward constraints on AdS graviton scattering. These AdS/CFT-derived bounds align with flat-space dispersive bounds in appropriate limits and illuminate how CFT bootstrap data encodes swampland restrictions on low-energy EFTs. The results have potential implications for constraining EFT landscapes in holographic and gravity-coupled settings.

Abstract

There are effective field theories that cannot be embedded in any UV complete theory. We consider scalar effective field theories, with and without dynamical gravity, in $D$-dimensional anti-de Sitter (AdS) spacetime with large radius and derive precise bounds (analytically) on the coupling constants of higher derivative interactions $φ^2\Box^kφ^2$ by only requiring that the dual CFT obeys the standard conformal bootstrap axioms. In particular, we show that all such coupling constants, for even $k\ge 2$, must satisfy positivity, monotonicity, and log-convexity conditions in the absence of dynamical gravity. Inclusion of gravity only affects constraints involving the $φ^2\Box^2φ^2$ interaction which now can have a negative coupling constant. Our CFT setup is a Lorentzian four-point correlator in the Regge limit. We also utilize this setup to derive constraints on effective field theories of multiple scalars. We argue that similar analysis should impose nontrivial constraints on the graviton four-point scattering amplitude in AdS.

Figures (4)

• Figure 1: The coupling constants $\lambda_k$ for even $k\ge 2$, without dynamical gravity, must obey the conditions (\ref{['intro:condition1']})-(\ref{['intro:condition3']}). There is always some choice of the scale $M$ (and $\mu$) for which the coupling constants have this generic structure. When gravity is included, only bounds on $\lambda_2$ become weaker.
• Figure 2: Lorentzian four-point correlator (\ref{['corr']}) where all operators are restricted to a $2$d subspace.
• Figure 3: Analytic structure of the correlator (\ref{['corr']}) – branch cuts appear only when two operators become null separated. The Regge limit is obtained from the Euclidean correlator by analytically continuing $\rho$ along the path shown.
• Figure 4: The tree-level Witten diagrams that are relevant in the Regge limit for $G_N=0$. Of course, the exchange diagram should be summed over all channels.