Massless positivity in graviton exchange

Mario Herrero-Valea,Raquel Santos-Garcia,Anna Tokareva

doi:10.1103/physrevd.104.085022

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.

Highlights

Positivity bounds [1,2,3,4,5] have become standard tools in assessing the validity of low-energy effective field theories (EFT)
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. Validity of these bounds requires the cancellation of divergences in the forward limit of the amplitude, proportional to t−1 and logðtÞ. 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 behavior of the amplitude is uniquely determined by the structure of IR divergences
The bounds (16) and (19) are completely general and valid in the case of massless particles in the spectrum of the theory, they are meaningless in the presence of divergences in the forward limit, as is the case when gravitons are exchanged in the t channel

Summary

INTRODUCTION

Positivity bounds [1,2,3,4,5] have become standard tools in assessing the validity of low-energy effective field theories (EFT). The scattering amplitude contains pathologies that impede one from taking the forward limit—a pole t−1 and a logarithmic divergence logðtÞ, due to exchange and production of massless particles. This is relevant in the presence of gravity, since gravitons couple to all forms of matter. It was suggested that forward divergences in graviton exchange can be cancelled by assuming a Regge form for the high-energy limit of the scattering amplitude [27], which is expected to hold from string theory [28,29]. We discuss the robustness of our result by showing agreement with previous works in the literature, formally deriving the bounds recently proposed by [25,26]

DISPERSION RELATIONS

REGULARITY IN THE FORWARD LIMIT

GRAVITATING SCALAR FIELD

SCALAR QED

CONCLUSIONS