UV graviton scattering and positivity bounds from IR dispersion relations

TL;DR

This paper investigates how IR singularities in graviton scattering amplitudes affect positivity bounds, revealing a non-trivial connection between UV and IR behaviors and the importance of arc integrals in gravitational contexts.

Contribution

It introduces an integral transform approach to relate UV and IR behaviors of graviton scattering amplitudes and highlights the significance of arc integrals in gravitational positivity bounds.

Findings

IR singularities obstruct standard positivity bounds in gravity.

Arc integrals can contribute non-trivially in gravitational dispersion relations.

Positivity bounds require assumptions about UV completion in gravitational theories.

Abstract

Scattering amplitudes mediated by graviton exchange display IR singularities in the forward limit. This obstructs standard application of positivity bounds based on twice subtracted dispersion relations. Such divergences can be cancelled only if the UV limit of the scattering amplitude behaves in a specific way, which implies a very non-trivial connection between the UV and IR behaviors of the amplitude. We show that this relation can be expressed in terms of an integral transform, obtaining analytic results when $t \log{s}\rightarrow 0$. Carefully applying this limit to dispersion relations, we find that infinite arc integrals, which are usually taken to vanish, can give a non-trivial contribution in the presence of gravity, unlike in the case of finite negative $t$. This implies that gravitational positivity bounds cannot be trusted unless the size of this contribution is estimated in some way, which implies assumptions on the UV completion of gravitational interactions. We discuss the relevance of these findings in the particular case of QED coupled to gravity.