Infrared features of gravitational scattering and radiation in the eikonal approach

Following a semiclassical eikonal approach—justified at transplanckian energies order by order in the deflection angle ${\mathrm{\Theta }}_s\sim \frac{4G\sqrt{s}}{b}\equiv \frac{2R}{b}$—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 $\frac{\mathrm{d}{E}^{\mathrm{GW}}}{\mathrm{d}\omega }$ 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 nonleading 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.

Figure 1

The scattering amplitude of two transplanckian particles (solid lines) in the eikonal approximation. Dashed lines represent (reggeized) graviton exchanges. The fast particles propagate on shell throughout the whole eikonal chain. The angles ${\mathbf{\Theta }}_j\simeq \sum _{i=1}^{j-1}{\mathit{\theta }}_i$ denote the direction of particle 1 with respect to the $z$-axis along the scattering process.

Figure 2

Center-of-mass view of the collision at impact parameter $\mathit{b}$ of particles 1 and 2 with associated emission of a graviton $q$. The polar angles ${\mathrm{\Theta }}_s$ and $\theta$ are related to the 2D vectors ${\mathbf{\Theta }}_s$ and $\mathit{\theta }$ as described in Eq. (2.1) and footnote 6.

Figure 3

Single-exchange emission diagram in $\mathit{Q}$-space with deflection angles (a), and its transverse-space counterpart with final-state variables $\mathit{b}$, $\mathit{x}$ and the shifted impact parameter $\mathit{b}-\frac{\omega }{E}\mathit{x}$ (b).

Figure 4

The H diagram providing the first subleading correction to the eikonal phase.

Figure 5

Picture of the polar and azimuthal angles in the transverse plane. ${\mathbf{\Theta }}_s$ and $\mathit{\theta }$ correspond respectively to the projections of the unit-vectors ${{\widehat{p}}_1}^{\prime }$ and $\widehat{q}$ on the $⟨x,y⟩$-plane of Fig. 2. In this configuration, all azimuthal angles ${\varphi }_j$ and $\psi$ are positive.

Figure 6

(a) The (reduced) graviton frequency spectrum against $\omega R$ for three values of ${\mathrm{\Theta }}_s$. Dots represent the full spectrum, while the solid lines represent the values obtained by using the analytic approximation (6.2)+(6.5) of the amplitude. The orange-dashed lines represent the leading approximation. (b) The (reduced) graviton frequency spectrum vs $\omega b$ and with ZFL subtracted out, for two values of ${\mathrm{\Theta }}_s$. The meaning of dots and lines is as in (a).

Figure 7

Behavior of the spectrum in the soft limit $\omega b\to 0$ for ${\mathrm{\Theta }}_s=0.01$. The full spectrum (empty and full dots) is compared with the one obtained from the analytic approximation (6.2)+(6.5) (solid green) and with the leading approximation (6.2) (dashed orange). The violet dotted line represents the function obtained by fitting the ten leftmost data of the full spectrum.