Massless Positivity in Graviton Exchange

TL;DR

The paper extends positivity bounds for $2\to2$ scattering to include massless exchanges, notably gravitons, by developing a dispersion-relations framework that handles forward-limit divergences through Regge behavior and IR regularization. It shows that graviton exchange can cancel both the $t^{-1}$ pole and the $log(t)$ divergence, thanks to a universal sub-leading term, yielding well-defined bounds on the graviton-containing amplitudes. The authors derive explicit positivity functionals $\Sigma^{(j)}$ (and $\hat{\Sigma}^{(j)}$) that can be computed within EFTs and validate them on a free gravitating scalar (where bounds hold) and on scalar QED with a spectator (where nontrivial constraints imply new physics below the Planck scale for unitarity). These results provide a general framework to constrain gravity–matter EFTs and have potential implications for black hole physics and cosmology.

Abstract

We formulate Positivity Bounds for scattering amplitudes including exchange of massless particles. We generalize the standard construction through dispersion relations to include the presence of a branch cut along the real axis in the complex plane for the Maldestam variable $s$. In general, validity of these bounds require the cancellation of divergences in the forward limit of the amplitude, proportional to $t^{-1}$ and $\log(t)$. We show that this is possible in the case of gravitons if one assumes a Regge behavior of the amplitude at high energies below the Planck scale, as previously suggested in the literature, and that the concrete UV behaviour of the amplitude is uniquely determined by the structure of IR divergences. We thus extend previous results by including a sub-leading logarithmic term, which we show to be universal. The bounds that we present here have the potential of constraining very general models of modified gravity and EFTs of matter coupled to gravitation.

Figures (1)

• Figure 1: Integration contours in the complex plane for $s$. The zigzag line represents the branch cut. For points $s\pm i\epsilon$, the integration contour $\gamma_s$ in the corresponding half of the complex plane is shown in red. The equivalent contours used in \ref{['eq:newint']} are dotted in blue. The radius of the large semi-circumferences $\gamma_\infty^\pm$ is $|s|\rightarrow \infty.$