Sharp Boundaries for the Swampland

TL;DR

This work develops a dispersive-S-matrix bootstrap for gravity to bound higher-derivative EFT couplings. By localizing dispersion relations at small impact parameter, it avoids the graviton pole and reveals correct EFT scaling with the UV cutoff $M$, enabling precise two- and multi-parameter bounds on coefficients like $g_2,g_3,g_4,\\ldots$ in various dimensions. The authors implement a numerical linear-programming approach using impact-parameter–localized functionals, obtaining consistent bounds in non-gravitational and gravitational theories, and providing explicit results for maximal supergravity that confirm positivity and suppression as gravity decouples. They compare to string-theory UV completions, discuss $D=4$ infrared subtleties, and outline extensions to AdS and loop corrections, highlighting a pathway to sharp swampland constraints in flat and curved spacetimes.

Abstract

We reconsider the problem of bounding higher derivative couplings in consistent weakly coupled gravitational theories, starting from general assumptions about analyticity and Regge growth of the S-matrix. Higher derivative couplings are expected to be of order one in the units of the UV cutoff. Our approach justifies this expectation and allows to prove precise bounds on the order one coefficients. Our main tool are dispersive sum rules for the S-matrix. We overcome the difficulties presented by the graviton pole by measuring couplings at small impact parameter, rather than in the forward limit. We illustrate the method in theories containing a massless scalar coupled to gravity, and in theories with maximal supersymmetry.

Figures (10)

• Figure 1: A schematic representation of the low-energy amplitude \ref{['Mlow param']}. The external scalar particles interact through graviton exchange, scalar exchange and a series of higher-derivative contact interactions.
• Figure 2: Contour deformation leading to the dispersive sum rule \ref{['eq:DispersiveSumRule1']}. We start from an integral over a large circle \ref{['starting']}, which vanishes due to the spin-2 boundedness assumption \ref{['eq:ReggeBound']}. After the contour deformation, we end up with an IR contribution, represented by the small circle on the right, and a UV contribution, represented by an integral over the cuts starting at $s'=M^2,-M^2-u$. In the presence of EFT loops, the IR contribution would also include cuts stretching between $s'=-M^2-u$ and $s'=M^2$, but these are subleading under our assumptions. Since the theory is assumed weakly coupled at scale $M$, the heavy cut is well approximated by a discrete set of poles for $s'$ not much larger than $M^2$. However, this will play no role in our reasoning.
• Figure 3: Allowed region for $g_3$ and $g_4$ in a non-gravitational theory in $D=6$ dimensions, with heavy mass scale $M$. We show results using two different methods: the blue region uses derivatives around the forward limit as in Caron-Huot:2020cmc (with a 33-dimensional space of functionals), while the yellow region uses small impact parameter wavepackets (built from the 17-dimensional space of functionals listed in Table \ref{['tab:parameters']}, together with $\mathcal{C}_{4,u=0}$). The two regions are essentially identical and appear overlapping in the plot. We give more details on our numerical computations in Appendix \ref{['app:flatspacenumerics']}.
• Figure 4: Allowed regions for $g_2$ and $g_3$ in a theory of a scalar coupled to gravity in flat space in dimensions $D=5,\dots,12$, with heavy mass scale $M$. For each curve, the region to the right is allowed and the region to the left is disallowed. Each bound was computed using a 17-dimensional space of functionals, listed in Table \ref{['tab:parameters']}. We give more details on the numerical computation in Appendix \ref{['app:flatspacenumerics']}. The inequalities plotted here are listed in Table \ref{['eq:inequalitiesforfigure1']}.
• Figure 5: The impact parameter wavefunction $\widehat{f}(b)$ defined by (\ref{['eq:impactparametertransform']}) for the extremal functional that minimizes $g_2$ in $D=6$. As discussed in the text, it is localized near $b\sim 1/M$. The wavefunction is normalized by $\widehat{f}(0)=1/2$ so that the contribution of $g_2$ in (\ref{['eq:fulllinearprogram']}) is precisely $g_2$. The fact that it has zero slope at $b=0$ guarantees that the contribution of $g_3$ vanishes.