Neutrino and photon lensing by black holes: radiative lens equations and post-Newtonian contributions

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 ∼ 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 Virbhadra-Ellis lens.

Gravitational Lensing Time Delays with Massive Photons

“…Massive neutral particles which play a role in astrophysics include neutrinos (Escribano et al 2001;Corianò et al 2015). In the standard model of particle physics, photons are massless.…”

Spin effects in neutrino gravitational scattering

“…We also mention, that we cannot integrate Eqs. (16) and (17) to the turn point r m and then automatically reconstruct ψ`8, as we made in Sec. (2) to find α`8.…”

“…Thus, to obtain ψ`8, one should integrate Eqs. (16) and (17), first, in the interval`8 ą r ą r m and, then, for r m ă r ă`8, with the solutions being stitched at r m . This fact significantly reduces the accuracy of the numerical simulation compared to Sec.…”

“…In this section, we present the numerical solutions of Eqs. (16) and (17) for the neutrino scattering off SMBH surrounded by an accretion disk. We discuss different orientations of neutrino trajectories with respect to the disk plane.…”

“…The neutrino gravitational scattering was studied recently [16], mainly in connection with the determination of the BH shadow produced by these particles [17]. In our work, we shall focus on the analysis of spin oscillations in the neutrino gravitational scattering, which effectively reduce the flux of neutrinos measured in a detector.…”