Electroweak corrections to photon scattering, polarization and lensing in a gravitational background and the near horizon limit

Abstract

We investigate the semiclassical approach to the lensing of photons in a spherically symmetric gravitational background, starting from Born level and include in our analysis the radiative corrections obtained from the electroweak theory for the graviton/photon/photon vertex. In this approach, the cross section is related to the angular variation of the impact parameter (b), which is then solved for b as a function of the angle of deflection, and measured in horizon units (b h ≡ b/(2GM )). Exact numerical solutions for the angular deflection are presented. The numerical analysis shows that perturbation theory in a weak background agrees with the classical Einstein formula for the deflection already at distances of the order of 20 horizon units (∼ 20 b h ) and it is optimal in the description both of very strong and weak lensings. We show that the electroweak corrections to the cross section are sizeable, becoming very significant for high energy gamma rays. Our analysis covers in energy most of the photon spectrum, from the cosmic microwave background up to very high energy gamma rays, and scatterings with any value of the photon impact parameter. We also study the helicity-flip photon amplitude, which is of O(α 2 ) in the weak coupling α, and its massless fermion limit, which involves the exchange of a conformal anomaly pole. The corresponding cross section is proportional to the Born level result and brings to a simple renormalization of Einsten's formula.

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

“…These are obtained for impact parameters b h of the order of 2 × 10 4 . Even larger deflections, of 1 to 2 degrees or a significant fraction of them, are generated in scatterings which proceed closer to the event horizon [5]. In fact, as we are going to show, for closer encounters, with the beam located between 20 and 100 b h , such angular deflections are around 10 −2 radians in size, as predicted by classical GR.…”

“…In such extreme cases the beam circulates around the source one or more times before escaping to infinity, generating a set of relativistic images [6]. This is also the region where the simple Newtonian approach, discussed in [5], fails to reproduce the classical GR prediction, as expected.…”

“…This is also the radial distance from the black hole center at which the scattering angle diverges. A simple expansion of the Einstein formula for the deflection shows that this singularity is logarithmic [5]. In such extreme cases the beam circulates around the source one or more times before escaping to infinity, generating a set of relativistic images [6].…”

“…The classical deflections of photons, as pointed out in the past and in a recent work [5], can be compared at classical and quantum levels by equating the classical gravitational cross section, written in terms of the impact parameter of the incoming photon beam, to the perturbative cross section. The latter is expanded in ordinary perturbation theory with the inclusion of the corresponding radiative corrections.…”

“…In the fermion case (f), the corresponding vertex is the T f f , with f denoting a neutrino. The comparison between the classical and the semiclassical formula for the deflection derived by this method can then be performed at numerical level, as shown in [5] for the photons. The energy dependence of the bending angle, for a given impact parameter of the photon beam, though small, is found to become more pronounced at higher energies, due to the logarithmic growth of the electroweak corrections with the energy.…”