A functional approach to the next-to-eikonal approximation of high energy gravitational scattering

The Fradkin–Schwinger functional methods to represent a Green function in an external gravitational field are used study the eikonal and next-to-eikonal limit, including nonlinear interactions, of scattering amplitudes ultra-relativistic scalar particle on static super-massive target nearly forward limit. approach confirms exponentiation leading which also applies first non-leading power energy light particle, moreover includes interaction at impact parameter much larger than Schwarzschild radius associated with center mass