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

中文翻译：

---

高能引力散射的近似 eikonal 近似的函数方法

用 Fradkin-Schwinger 泛函方法来表示外部引力场中的格林函数，用于研究超相对论标量粒子在静态超大质量标量目标处于近前极限。泛函方法证实了领先 eikonal 的指数化，这也适用于轻粒子能量中的第一个非领先幂，此外还包括在撞击参数处的相互作用，该相互作用远大于与质心能量中心相关的史瓦西半径。超相对论极限。