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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Corianò, L. Delle Rose, M.M. Maglio and M. Serino, Electroweak Corrections to Photon Scattering, Polarization and Lensing in a Gravitational Background and the Near Horizon Limit, JHEP 01 (2015) 091 [arXiv:1411.2804] [INSPIRE].
V. Bozza, S. Capozziello, G. Iovane and G. Scarpetta, Strong field limit of black hole gravitational lensing, Gen. Rel. Grav. 33 (2001) 1535 [gr-qc/0102068] [INSPIRE].
R. Delbourgo and P. Phocas-Cosmetatos, Radiative corrections to the photon graviton vertex, Phys. Lett. B 41 (1972) 533 [INSPIRE].
F.A. Berends and R. Gastmans, On the High-Energy Behavior in Quantum Gravity, Nucl. Phys. B 88 (1975) 99 [INSPIRE].
O. Mena, I. Mocioiu and C. Quigg, Gravitational Lensing of Supernova Neutrinos, Astropart. Phys. 28 (2007) 348 [astro-ph/0610918] [INSPIRE].
A. Gould, Femtolensing of Gamma-Ray Bursters, Astrophys. J. 386 (1992) L5.
K.S. Virbhadra and G.F.R. Ellis, Gravitational lensing by naked singularities, Phys. Rev. D 65 (2002) 103004 [INSPIRE].
C. Corianò, L. Delle Rose, E. Gabrielli and L. Trentadue, One loop Standard Model corrections to flavor diagonal fermion-graviton vertices, Phys. Rev. D 87 (2013) 054020 [arXiv:1212.5029] [INSPIRE].
C. Corianò, L. Delle Rose, E. Gabrielli and L. Trentadue, Fermion Scattering in a Gravitational Background: Electroweak Corrections and Flavour Transitions, JHEP 03 (2014) 136 [arXiv:1312.7657] [INSPIRE].
G. Degrassi, E. Gabrielli and L. Trentadue, Flavor Changing Fermion-Graviton Vertices, Phys. Rev. D 79 (2009) 053004 [arXiv:0812.3262] [INSPIRE].
C.G. Callan Jr., S.R. Coleman and R. Jackiw, A New improved energy-momentum tensor, Annals Phys. 59 (1970) 42 [INSPIRE].
C. Corianò, L. Delle Rose and M. Serino, Gravity and the Neutral Currents: Effective Interactions from the Trace Anomaly, Phys. Rev. D 83 (2011) 125028 [arXiv:1102.4558] [INSPIRE].
D.Z. Freedman, I.J. Muzinich and E.J. Weinberg, On the Energy-Momentum Tensor in Gauge Field Theories, Annals Phys. 87 (1974) 95 [INSPIRE].
A. Accioly and R. Paszko, Quantum mechanics versus equivalence principle, Phys. Rev. D 78 (2008) 064002 [INSPIRE].
A. Accioly and R. Paszko, Photon mass and gravitational deflection, Phys. Rev. D 69 (2004) 107501 [INSPIRE].
A. Accioly and H. Blas, Gravitational rainbow, Phys. Rev. D 64 (2001) 067701 [gr-qc/0107003] [INSPIRE].
S. Chang, C. Corianò, R.D. Field and L.E. Gordon, Spin dependent Drell-Yan in QCD to O(α 2 S ). 1. The nonsinglet sector, Nucl. Phys. B 512 (1998) 393 [hep-ph/9705249] [INSPIRE].
K. Greisen, End to the cosmic ray spectrum?, Phys. Rev. Lett. 16 (1966) 748 [INSPIRE].
G.T. Zatsepin and V.A. Kuzmin, Upper limit of the spectrum of cosmic rays, JETP Lett. 4 (1966) 78 [INSPIRE].
F.A. Berends and R. Gastmans, Quantum Electrodynamical Corrections to Graviton-Matter Vertices, Annals Phys. 98 (1976) 225 [INSPIRE].
P. Amore and S. Arceo Diaz, Analytical formulas for gravitational lensing, Phys. Rev. D 73 (2006) 083004 [gr-qc/0602106] [INSPIRE].
C.R. Keeton and A.O. Petters, Formalism for testing theories of gravity using lensing by compact objects. I. Static, spherically symmetric case, Phys. Rev. D 72 (2005) 104006 [gr-qc/0511019] [INSPIRE].
S. Mao, Introduction to Gravitational Microlensing, arXiv:0811.0441 [INSPIRE].
K.S. Virbhadra and G.F.R. Ellis, Schwarzschild black hole lensing, Phys. Rev. D 62 (2000) 084003 [astro-ph/9904193] [INSPIRE].
V. Bozza, A Comparison of approximate gravitational lens equations and a proposal for an improved new one, Phys. Rev. D 78 (2008) 103005 [arXiv:0807.3872] [INSPIRE].
E.F. Eiroa, G.E. Romero and D.F. Torres, Reissner-Nordstrom black hole lensing, Phys. Rev. D 66 (2002) 024010 [gr-qc/0203049] [INSPIRE].
B.J. Carr and S.W. Hawking, Black holes in the early Universe, Mon. Not. Roy. Astron. Soc. 168 (1974) 399 [INSPIRE].
B.J. Carr, Primordial Black Holes and Quantum Effects, arXiv:1402.1437
M.Yu. Khlopov, Primordial Black Holes, Res. Astron. Astrophys. 10 (2010) 495 [arXiv:0801.0116] [INSPIRE].
S. Clesse and J. García-Bellido, Massive Primordial Black Holes from Hybrid Inflation as Dark Matter and the seeds of Galaxies, Phys. Rev. D 92 (2015) 023524 [arXiv:1501.07565] [INSPIRE].
A. Barnacka, J.F. Glicenstein and R. Moderski, New constraints on primordial black holes abundance from femtolensing of gamma-ray bursts, Phys. Rev. D 86 (2012) 043001 [arXiv:1204.2056] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1504.01322
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Corianò, C., Costantini, A., Dell’Atti, M. et al. Neutrino and photon lensing by black holes: radiative lens equations and post-Newtonian contributions. J. High Energ. Phys. 2015, 160 (2015). https://doi.org/10.1007/JHEP07(2015)160
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2015)160