Abstract
Following a semi-classical eikonal approach --- justified at transplanckian energies order by order in the deflection angle $\Theta_s\sim\frac{4G\sqrt{s}}{b} \equiv \frac{2 R}{b}$ --- we investigate the infrared features of gravitational scattering and radiation in four space-time dimensions, and we illustrate the factorization and cancellation of the infinite Coulomb phase for scattering and the eikonal resummation for radiation. As a consequence, both the eikonal phase $2\delta(E,b)$ and the gravitational-wave (GW) spectrum $\frac{\mathrm{d}E^{GW}}{\mathrm{d}\omega}$ are free from infrared problems in a frequency region extending from zero to (and possibly beyond) $\omega =1/R$. The infrared-singular behavior of $4$-D gravity leaves a memory in the deep infrared region ($\omega R \ll \omega b < 1$) of the spectrum. At $\mathcal{O}(\omega b)$ we confirm the presence of logarithmic enhancements of the form already pointed out by Sen and collaborators on the basis of non leading corrections to soft-graviton theorems. These, however, do not contribute to the unpolarized and/or azimuthally-averaged flux. At $\mathcal{O}(\omega^2 b^2)$ we find instead a positive logarithmically-enhanced correction to the total flux implying an unexpected maximum of its spectrum at $\omega b \sim 0.5$. At higher orders we find subleading enhanced contributions as well, which can be resummed, and have the interpretation of a finite rescattering Coulomb phase of emitted gravitons.
Highlights
The recent discovery of gravitational waves (GW) in black hole and neutron-star mergers [1,2] has revived interest in gravitational phenomena at the level of elementary-particle processes
V we present the basic starting point for our study of soft gravitational bremsstrahlung in the form of an infrared-finite unitary S-matrix which agrees, in the appropriate limit, with the classical calculation obtained earlier by completely different techniques
We show how this regime connects smoothly with a logarithmically decreasing one in the region 1=b < ω < 1=R leading to a peak in the flux around ωb ∼ 0.5
Summary
The recent discovery of gravitational waves (GW) in black hole and neutron-star mergers [1,2] has revived interest in gravitational phenomena at the level of elementary-particle processes. Since such a coupling basically occurs as an overall factor in the effective action (in ħ units) this suggests the validity of a semiclassical approximation This eikonal approach to high-energy gravitational scattering was developed further by Amati, Ciafaloni and Veneziano (ACV) [11,14,15,16] in a series of papers by deriving, in particular, higher order corrections to the eikonal function. The possible emergence of large soft logarithms (in D 1⁄4 4) has been recently emphasized in [26,27] as subleading contributions to soft theorems and a possible source of memory effects This approach, unlike the eikonal one, is not limited to high energy or to small deflection angles, but only covers a tiny region of frequencies VII we discuss our results and point to possible directions for future research
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