Unitarization of infinite-range forces: graviton-graviton scattering

Abstract

A method to unitarize the scattering amplitude produced by infinite-range forces is developed and applied to Born terms. In order to apply S-matrix techniques, based on unitarity and analyticity, we first derive an S-matrix free of infrared divergences. This is achieved by removing a divergent phase factor due to the interactions mediated by the massless particles in the crossed channels, a procedure that is related to previous formalisms to treat infrared divergences. We apply this method in detail by unitarizing the Born terms for graviton-graviton scattering in pure gravity and we find a scalar graviton-graviton resonance with vacuum quantum numbers (JPC = 0++) that we call the graviball. Remarkably, this resonance is located below the Planck mass but deep in the complex s-plane (with s the usual Mandelstam variable), so that its effects along the physical real s axis peak for values significantly lower than this scale. This implies that the corrections to the leading-order amplitude in the gravitational effective field theory are larger than expected from naive dimensional analysis for s around and above the peak position. We argue that the position and width of the graviball are reduced when including extra light fields in the theory. This could lead to phenomenological consequences in scenarios of quantum gravity with a large number of such fields or, in general, with a low-energy ultraviolet completion. We also apply this formalism to two non-relativistic potentials with exact known solutions for the scattering amplitudes: Coulomb scattering and an energy-dependent potential obtained from the Coulomb one with a zero at threshold. This latter case shares the same J = 0 partial-wave projected Born term as the graviton-graviton case, except for a global factor. We find that the relevant resonance structure of these examples is reproduced by our methods, which represents a strong indication of their robustness.

Unitarizing infinite-range forces: Graviton-graviton scattering, the graviball, and Coulomb scattering

“…Here, we follow and expand the ideas in the approach of Refs. [12][13][14] for unitarizing infinite-range forces. Compared to Refs.…”

“…Compared to Refs. [12,13], also dedicated to the study of graviton-graviton scattering, we go beyond the tree-level approximation and unitarize one-loop scattering amplitudes. Reference [14] considered also higher orders in Coulomb scattering, but was relegated to the non-relativistic case implying only Coulomb interactions (instantaneous interaction), while here the formalism is fully relativistic.…”

“…By studying the characteristic size of one-loop unitarity contributions and their variation under a change of scale, it was concluded in Ref. [12,13] that a typical scale for the low-energy graviton-graviton scale is Q 2 = πG −1 / ln a, instead of just G −1 . The parameter ln a emerges in the process of removing the infrared divergences.…”

“…An interesting argument was put forward in Ref. [12] to favor a value of ln a ≈ 1. This reference also emphasizes the stunning resemblance of the graviball pole with the lightest resonance in Quantum Chromodynamics, called the σ or f 0 (500) [24,25], for which its pole position |s σ |/(4πf π ) 2 ≃ 0.2 as well.…”

“…In our present study, the unitarization of the one-loop scattering amplitudes confirms the J = 0 graviball pole. Nonetheless, while keeping |s P | almost the same, now the real part of s P increases while its imaginary part decreases, giving rise, as a result, to a sharper resonance signal along the real s-axis as compared with the LO study [12,13]. The requirement of having a wellbehaved unitarized EFT for graviton-graviton scattering implies interesting constraints.…”