Sampling the Graviton Pole and Deprojecting the Swampland

TL;DR

This paper introduces a primal bootstrap method for effective field theories with a graviton pole, enabling stronger bounds and new insights into the spectra and scale of EFT couplings.

Contribution

It develops a novel sampling-based bootstrap framework that reproduces known bounds and derives new non-projective bounds on EFT couplings in various dimensions.

Findings

Reproduces known projective bounds in D≥6

Yields stronger bounds in D=5

Finds that EFT cutoff is limited to near the Planck scale in D=5

Abstract

We develop a primal bootstrap framework for effective field theories in the presence of a graviton pole, based on finite-resolution sampling rather than smearing, while also allowing direct control over the number of subtractions. We show that this approach reproduces the known projective bounds obtained from smearing in $D{\ge}6$, while yielding slightly stronger bounds in $D{=}5$. This method also makes it straightforward to impose linearized unitarity directly and provides an access to the extremal spectra. Applying the method to crossing-symmetric dispersion relations, we derive new non-projective bounds that fix the overall scale of the EFT couplings. In $D{=}5$, for example, we find that $\frac{M}{M_{\rm P}}{\lesssim}7.8$, showing that the EFT cutoff cannot be taken parametrically larger than the Planck scale. At the extremal values of the couplings, the spectra exhibit a surprising structure: for projective bounds, they exhibit peaks around quadratic Regge-like trajectories, while for the non-projective bounds they form sharp quadratic bands. In the latter case, the leading coefficients further display an inverse-quadratic dependence on the band number.