Behaviour of n-point scattering amplitudes at high energies

We study the on-shell scattering amplitudes in quantum gravity for high-energy collisions in the eikonal approximation. We first evaluate the $n$-loop 2-particle scattering amplitude in the high energy and low momentum transfer limit. We do so in a symmetrized manner by finding the contributions of each of the particle worldines to the scattering amplitude and gluing them together via the $n$ intermediate particle exchanges. In this limit on applying the eikonal approximation and summing over all $n$-loop Feynman diagrams we obtain a closed form for the 2 particle scattering amplitude. Finally, we extend this approach to obtain a generalized eikonal approximation for $N$-particle scattering at high energies and small momentum transfers. The generalised form of the scattering amplitude can then be used to evaluate the bound states of the system.