All elastic amplitudes in the (black hole) eikonal phase

Abstract

In this article we calculate the eikonal scattering amplitude for an arbitrary number of in- and out-particles, using covariant quantization in a spherical harmonics basis on the Schwarzschild background. We extend prior results to resummation over all partial waves, restoring contributions from transverse separation and correctly taking into account the particle masses in the pole structure. We consider leading order interactions mediated by scalar-scalar-graviton vertices and scalar electrodynamics. We perform our calculations in the black hole eikonal phase. The $2\to 2$ eikonal amplitude is measured by the transverse Green's function $(-Δ_Ω+a)G(Ω,Ω')=δ^{(2)}(Ω-Ω')$. As a consistency check, we use our formalism in flat space to find an exact match with the known flat space eikonal $2\to 2$ amplitude in literature. We then extend the eikonal amplitude to arbitrarily many particles for the first time in both flat space and on the black hole background. We show that the black hole amplitude matches the black hole S-matrix as derived by 't Hooft. We conclude that this amplitude provides the most general elastic contribution one can achieve in the eikonal phase.

Figures (23)

• Figure 1: The Penrose diagram for the maximally extended Schwarzschild black hole. The four different regions are labelled in the convention of 't Hooft. The direction of the coordinates $x,y$ and the definition of the horizons are shown, as well as the conventional notation for null infinity. The momentum direction is orthogonal to the coordinate direction because of the off-diagonal metric.
• Figure 2: An illustration of the different harmonic modes that split off from $A_{\mu}$, denoted by black dots. In general the equal parity modes have interactions indicated by the solid line. In the first gauge choice $A_+$ is removed explicitly (replaced by an empty set), leaving only the two decoupled modes. In the last gauge choice all 3 modes remain, however the gauge choice breaks the coupling in the even sector.
• Figure 3: An illustration of the different harmonic modes (denoted by black dots) and gauges for gravity. Here $\tilde{\mathbf{H}}_{ab}$ is the traceless version of $\mathbf{H}_{ab}$. In both the even and odd sector there are many couplings between the modes. In the Regge-Wheeler (RW) gauge three modes are explicitly set to zero (replaced by empty sets in the image), also removing many couplings. In the eikonal gauge only two modes are removed, however the even scalar modes are combined into one, and the coupling between $\tilde{H}_{ab}$ and $H$ is broken, effectively reducing in less couplings than the RW gauge.
• Figure 4: The necessary propagators in the black hole eikonal phase. For all different spin fields only a single relevant mode survives.
• Figure 5: The interaction vertices for the different fields. The gauge field only interacts with the complex scalar, whereas the graviton interacts with both in an identical fashion. One could symmetrize the graviton vertices over the indices, however the fact that the graviton propagator is symmetric automatically takes care of this.