The Subleading Eikonal in Supergravity Theories

TL;DR

The paper investigates the subleading eikonal in type II supergravity for massless states scattering off a D$p$-brane stack at large impact parameter, showing that elastic processes predominantly govern both leading and subleading eikonals, while inelastic channels contribute via a prefactor to the exponentiated elastic piece. It develops two complementary computational frameworks: (i) an on-shell bulk-vertex approach using only three- and four-point amplitudes to derive leading and subleading eikonals, and (ii) a one-point-amplitude method that extracts the classical backreaction and matches the geodesic deflection in the D$p$-brane background; both approaches agree. The analysis covers elastic dilaton scattering and inelastic dilaton-to-RR transitions, and extends to both Einstein gravity and gravity with additional fields (dilaton and RR), demonstrating that in many setups the subleading eikonal can be fully determined by elastic data, with inelastic effects encapsulated in a prefactor or canceling after appropriate subtraction. The work bridges quantum amplitude techniques with classical gravitational observables (deflection angle, backreaction) and provides a practical algorithm for extracting classical contributions from high-energy scattering, with potential implications for gravitational-wave effective actions and string-theoretic generalizations.

Abstract

In this paper we study the subleading contributions to eikonal scattering in (super)gravity theories with particular emphasis on the role of both elastic and inelastic scattering processes. For concreteness we focus on the scattering of various massless particles off a stack of D$p$-branes in type II supergravity in the limit of large impact parameter $b$. We analyse the relevant field theory Feynman diagrams which naturally give rise to both elastic and inelastic processes. We show that in the case analysed the leading and subleading eikonal only depend on elastic processes, while inelastic processes are captured by a pre-factor multiplying the exponentiated leading and subleading eikonal phase. In addition to the traditional Feynman diagram computations mentioned above, we also present a novel method for computing the amplitudes contributing to the leading and subleading eikonal phases, which, in the large $b$ limit, only involves knowledge of the onshell three and four-point vertices. The two methods are shown to give the same results. Furthermore we derive these results in yet another way, by computing various one-point amplitudes which allow us to extract the classical solution of the gravitational back reaction of the target D$p$-branes. Finally we show how our expressions for the leading and subleading eikonal agree with the calculation of the metric and corresponding deflection angle for massless states moving along geodesics in the relevant curved geometry.

Figures (5)

• Figure 1: A schematic diagram showing our procedure for calculating effective one-loop amplitudes. The circular blob represents the four-point effective vertex and the two oval blobs represent the D-branes. The four-point vertex is sewed with the D-branes by using the appropriate propagator and boundary coupling.
• Figure 2: The various topologies of diagrams that contribute to $\mathcal{A}^{\rm ddRR}$. In \ref{['fig:2a']} we have the t- and u-channels, in \ref{['fig:2b']} we have the s-channel diagram and finally in \ref{['fig:2c']} we have the contact diagram. The solid lines represent dilatons, wavy lines represent gravitons and the dashed lines represent RR fields.
• Figure 3: The various topologies of diagrams that contribute to $\mathcal{A}^{\rm ddgg}$. In \ref{['fig:3a']} we have the t- and u-channels, in \ref{['fig:3b']} we have the s-channel diagram and finally in \ref{['fig:3c']} we have the contact diagrams. The solid lines represent dilatons and the wavy lines represent gravitons.
• Figure 4: A schematic diagram showing our procedure for calculating the effective one-loop amplitude for dilaton to RR inelastic scattering. The circular blob represents the four-point effective vertex $\mathcal{A}_{bulk}^{\rm dRgR}$ and the two oval blobs represent the D-branes. As before the solid lines correspond to dilatons, the wavy lines correspond to gravitons and the dashed lines correspond to RR fields.
• Figure 5: The various contributions to the one-point function at subleading order used to construct the classical solution. Figure \ref{['fig:5a']} is the contribution with the 3-graviton vertex and figures \ref{['fig:5b']} and \ref{['fig:5c']} are the contributions with RR fields and dilaton sources respectively. As before the solid lines correspond to dilatons, the wavy lines correspond to gravitons and the dashed lines correspond to RR fields.