Neutrino and Photon Lensing by Black Holes: Radiative Lens Equations and Post-Newtonian Contributions

TL;DR

This paper develops a unified framework combining classical and quantum approaches to analyze how neutrinos and photons are deflected by black holes, including radiative and post-Newtonian effects, with applications to primordial black hole lensing.

Contribution

It introduces a method to incorporate quantum radiative corrections and post-Newtonian effects into classical gravitational lens equations for neutrinos and photons.

Findings

Classical and semiclassical deflections agree for impact parameters around 20 Schwarzschild radii.

Quantum corrections introduce energy dependence in the deflection angles.

The approach applies to scattering off primordial black holes, enhancing lensing models.

Abstract

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.