黑洞对中微子和光子的透镜效应：辐射透镜方程和后牛顿贡献
Neutrino and Photon Lensing by Black Holes: Radiative Lens Equations and Post-Newtonian Contributions
摘要：
我们将以前的光子透镜现象在外部引力背景下的分析推广到无质量中微子的情况，并提出了一种方法，将辐射效应纳入中微子和光子的经典透镜方程。该研究是针对Schwarzschild度规进行的，该度规由点状源产生，并在一阶牛顿势中展开。我们使用半经典的方法，微扰修正中微子散射，在一个循环中的标准模型进行评估，比较与爱因斯坦公式的偏转使用碰撞参量制定。为此，我们使用的引力子/费米子/费米子顶点在以前的研究中提出的重整化表达式。我们显示的经典和半经典的配方之间的协议，为$b_h\sim 20$的顺序的中微子的碰撞参量$b_h$的值，测量的史瓦西半径的单位。分析，然后扩展与包括后牛顿修正的外部引力场，表明这种扩展发现应用的情况下，散射的中微子/光子关闭原始黑洞。的能量依赖性的偏转，产生的量子校正，然后结合标准制定的经典透镜方程。我们通过详细的数值研究来说明我们的方法，同时使用薄透镜和非线性Virbhadra-Ellis透镜作为参考。
英文摘要：
We extend a previous phenomenological analysis of photon lensing in an external gravitational background to the case of a massless neutrino, and propose a method to incorporate radiative effects in the classical lens equations of neutrinos and photons. The study is performed for a Schwarzschild metric, generated by a point-like source, and expanded in the Newtonian potential at first order. We use a semiclassical approach, where the perturbative corrections to neutrino scattering, evaluated at one-loop in the Standard Model, are compared with the Einstein formula for the deflection using an impact parameter formulation. For this purpose, we use the renormalized expression of the graviton/fermion/fermion vertex presented in previous studies. We show the agreement between the classical and the semiclassical formulations, for values of the impact parameter $b_h$ of the neutrinos of the order of $b_h\sim 20$, measured in units of the Schwarzschild radius. The analysis is then extended with the inclusion of the post Newtonian corrections in the external gravity field, showing that this extension finds application in the case of the scattering of a neutrino/photon off a primordial black hole. The energy dependence of the deflection, generated by the quantum corrections, is then combined with the standard formulation of the classical lens equations. We illustrate our approach by detailed numerical studies, using as a reference both the thin lens and the nonlinear Virbhadra-Ellis lens.
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