# Fermion Scattering in a CPT-Even Lorentz Violation Quantum Electrodynamics

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Aspects of the Photonic SME Minimal Sector

#### Photon Propagator and a Lorentz Violation Gauge Choice

## 3. The Pair Annihilation ${\mathit{e}}^{+}\phantom{\rule{3.33333pt}{0ex}}+\phantom{\rule{3.33333pt}{0ex}}{\mathit{e}}^{-}\to {\mathit{\mu}}^{+}\phantom{\rule{3.33333pt}{0ex}}+\phantom{\rule{3.33333pt}{0ex}}{\mathit{\mu}}^{-}$

#### Sidereal Effects

## 4. The Scattering Process ${\mathit{e}}^{-}\phantom{\rule{3.33333pt}{0ex}}+\phantom{\rule{3.33333pt}{0ex}}{\mathit{\mu}}^{-}\to {\mathit{e}}^{-}\phantom{\rule{3.33333pt}{0ex}}+\phantom{\rule{3.33333pt}{0ex}}{\mathit{\mu}}^{+}$

#### Mott Scattering

**Pure parity-odd case:**when one considers only the parity-odd contribution, ${\kappa}_{tr}=0$ and ${\kappa}_{e-}=0$, the differential cross section becomes:

**Isotropic contribution:**The isotropic case is defined by ${\kappa}_{i}=0$ and ${\kappa}_{e-}=0$, for which the differential cross section is

**Anisotropic parity-even contribution:**now we present the differential cross section for the anisotropic parity-even case, with ${\kappa}_{i}=0$ and ${\kappa}_{tr}=0$. After integrating over $\varphi $, the differential cross section is given by

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Colladay, D.; Kostelecký, V.A. Lorentz-violating extension of the standard model. Phys. Rev. D
**1998**, 58, 116002. [Google Scholar] [CrossRef] - Kosteleckỳ, V.A.; Russell, N. Data tables for Lorentz and C P T violation. Rev. Mod. Phys.
**2011**, 83, 11. [Google Scholar] [CrossRef] - Carroll, S.M.; Field, G.B.; Jackiw, R. Limits on a Lorentz- and parity-violating modification of electrodynamics. Phys. Rev. D
**1990**, 41, 1231–1240. [Google Scholar] [CrossRef] - Kostelecký, V.A.; Mewes, M. Cosmological Constraints on Lorentz Violation in Electrodynamics. Phys. Rev. Lett.
**2001**, 87, 251304. [Google Scholar] [CrossRef] [PubMed] - Kostelecký, V.A.; Mewes, M. Signals for Lorentz violation in electrodynamics. Phys. Rev. D
**2002**, 66, 056005. [Google Scholar] [CrossRef] - Kostelecký, V.A.; Mewes, M. Sensitive Polarimetric Search for Relativity Violations in Gamma-Ray Bursts. Phys. Rev. Lett.
**2006**, 97, 140401. [Google Scholar] [CrossRef] [PubMed] - Kostelecký, V.A.; Mewes, M. Electrodynamics with Lorentz-violating operators of arbitrary dimension. Phys. Rev. D
**2009**, 80, 015020. [Google Scholar] [CrossRef] - Escobar, C.A.; Garcia, M.A. Full C P T-even photon sector of the standard model extension at finite temperature. Phys. Rev. D
**2015**, 92, 025034. [Google Scholar] [CrossRef] - Martín-Ruiz, A.; Escobar, C. Casimir effect between ponderable media as modeled by the standard model extension. Phys. Rev. D
**2016**, 94, 076010. [Google Scholar] [CrossRef] [Green Version] - Andrianov, A.A.; Soldati, R. Lorentz symmetry breaking in Abelian vector-field models with Wess-Zumino interaction. Phys. Rev. D
**1995**, 51, 5961–5964. [Google Scholar] [CrossRef] [Green Version] - Andrianov, A.A.; Soldati, R. Patterns of Lorentz symmetry breaking in QED by CPT-odd interaction. Phys. Lett. B
**1998**, 435, 449–452. [Google Scholar] [CrossRef] [Green Version] - Andrianov, A.A.; Soldati, R.; Sorbo, L. Dynamical Lorentz symmetry breaking from a (3+1)-dimensional axion-Wess-Zumino model. Phys. Rev. D
**1998**, 59, 025002. [Google Scholar] [CrossRef] - Alfaro, J.; Andrianov, A.; Cambiaso, M.; Giacconi, P.; Soldati, R. Bare and induced lorentz and cpt invariance violations in qed. Int. J. Mod. Phys. A
**2010**, 25, 3271–3306. [Google Scholar] [CrossRef] - Zhukovsky, V.C.; Lobanov, A.E.; Murchikova, E.M. Radiative effects in the standard model extension. Phys. Rev. D
**2006**, 73, 065016. [Google Scholar] [CrossRef] - Chen, W.F.; Kunstatter, G. Constraint from the Lamb shift and anomalous magnetic moment on radiatively induced Lorentz and CPT violation in quantum electrodynamics. Phys. Rev. D
**2000**, 62, 105029. [Google Scholar] [CrossRef] - Adam, C.; Klinkhamer, F. Causality and CPT violation from an Abelian Chern–Simons-like term. Nucl. Phys. B
**2001**, 607, 247–267. [Google Scholar] [CrossRef] [Green Version] - Adam, C.; Klinkhamer, F. Photon decay in a CPT-violating extension of quantum electrodynamics. Nucl. Phys. B
**2003**, 657, 214–228. [Google Scholar] [CrossRef] [Green Version] - Baêta Scarpelli, A.P.; Belich, H.; Boldo, J.L.; Helayël-Neto, J.A. Aspects of causality and unitarity and comments on vortexlike configurations in an Abelian model with a Lorentz-breaking term. Phys. Rev. D
**2003**, 67, 085021. [Google Scholar] [CrossRef] - Casana, R.; Ferreira, M.M.; Gomes, A.R.; dos Santos, F.E.P. Feynman propagator for the nonbirefringent CPT-even electrodynamics of the standard model extension. Phys. Rev. D
**2010**, 82, 125006. [Google Scholar] [CrossRef] - Colladay, D.; McDonald, P.; Potting, R. Gupta-Bleuler photon quantization in the standard model extension. Phys. Rev. D
**2014**, 89, 085014. [Google Scholar] [CrossRef] - Casana, R.; Ferreira, M., Jr.; dos Santos, F. Gupta-Bleuler quantization of the anisotropic parity-even and C P T-even electrodynamics of a standard model extension. Phys. Rev. D
**2014**, 90, 105025. [Google Scholar] [CrossRef] - Casana, R.; Ferreira, M., Jr.; dos Santos, F. Gupta-Bleuler’s quantization of a parity-odd C P T-even electrodynamics of the standard model extension. Phys. Rev. D
**2016**, 94, 125011. [Google Scholar] [CrossRef] - Belich, H.; Costa-Soares, T.; Ferreira, M.M., Jr.; Helayël-Neto, J. Non-minimal coupling to a Lorentz-violating background and topological implications. Eur. Phys. J. C
**2005**, 41, 421–426. [Google Scholar] [CrossRef] - Belich, H.; Colatto, L.; Costa-Soares, T.; Helayël-Neto, J.; Orlando, M. Magnetic moment generation from non-minimal couplings in a scenario with Lorentz-symmetry violation. Eur. Phys. J. C
**2009**, 62, 425–432. [Google Scholar] [CrossRef] [Green Version] - Bakke, K.; Belich, H. Abelian geometric phase for a Dirac neutral particle in a Lorentz. symmetry violation environment. J. Phys. G
**2012**, 39, 085001. [Google Scholar] [CrossRef] - Bakke, K.; Belich, H.; Silva, E. Relativistic Landau-Aharonov-Casher quantization based on the Lorentz symmetry violation background. J. Math. Phys.
**2011**, 52, 063505. [Google Scholar] [CrossRef] [Green Version] - Bakke, K.; Silva, E.; Belich, H. He–McKellar–Wilkens effect and scalar Aharonov–Bohm effect for a neutral particle based on the Lorentz symmetry violation. J. Phys. G
**2012**, 39, 055004. [Google Scholar] [CrossRef] [Green Version] - Bakke, K.; Belich, H.; Silva, E. Relativistic Anandan quantum phase in the Lorentz violation background. Ann. Phys. (Berl.)
**2011**, 523, 910–918. [Google Scholar] [CrossRef] - Anacleto, M.A. Lorentz violation correction to the Aharonov-Bohm scattering. Phys. Rev. D
**2015**, 92, 085035. [Google Scholar] [CrossRef] - Bakke, K.; Belich, H. Relativistic Landau–He–McKellar–Wilkens quantization and relativistic bound states solutions for a Coulomb-like potential induced by the Lorentz symmetry breaking effects. Ann. Phys.
**2013**, 333, 272–281. [Google Scholar] [CrossRef] [Green Version] - Ribeiro, L.; Furtado, C.; Passos, E. An analogy of the quantum hall conductivity in a Lorentz-symmetry violation setup. J. Phys. G
**2012**, 39, 105004. [Google Scholar] [CrossRef] - Charneski, B.; Gomes, M.; Maluf, R.V.; da Silva, A.J. Lorentz violation bounds on Bhabha scattering. Phys. Rev. D
**2012**, 86, 045003. [Google Scholar] [CrossRef] - Gazzola, G.; Fargnoli, H.; Scarpelli, A.B.; Sampaio, M.; Nemes, M.C. QED with minimal and nonminimal couplings: On the quantum generation of Lorentz-violating terms in the pure photon sector. J. Phys. G
**2012**, 39, 035002. [Google Scholar] [CrossRef] - Scarpelli, A.B. QED with chiral nonminimal coupling: aspects of the Lorentz-violating quantum corrections. J. Phys. G
**2012**, 39, 125001. [Google Scholar] [CrossRef] - Brito, L.; Fargnoli, H.; Scarpelli, A.B. Aspects of quantum corrections in a Lorentz-violating extension of the Abelian Higgs model. Phys. Rev. D
**2013**, 87, 125023. [Google Scholar] [CrossRef] - de Brito, G.; Junior, J.G.; Kroff, D.; Malta, P.; Marques, C. Lorentz violation in simple QED processes. Phys. Rev. D
**2016**, 94, 056005. [Google Scholar] [CrossRef] [Green Version] - Santos, A.F.; Khanna, F.C. Lorentz Violation, Möller Scattering, and Finite Temperature. Adv. High Energy Phys.
**2018**, 2018, 7. [Google Scholar] [CrossRef] - Casana, R.; Ferreira, M.M.; Passos, E.; dos Santos, F.E.P.; Silva, E.O. New CPT-even and Lorentz-violating nonminimal coupling in the Dirac equation. Phys. Rev. D
**2013**, 87, 047701. [Google Scholar] [CrossRef] - Araujo, J.B.; Casana, R.; Ferreira, M.M. General CPT-even dimension-five nonminimal couplings between fermions and photons yielding EDM and MDM. Phys. Lett. B
**2016**, 760, 302–308. [Google Scholar] [CrossRef] - Ding, Y.; Kosteleckỳ, V.A. Lorentz-violating spinor electrodynamics and Penning traps. Phys. Rev. D
**2016**, 94, 056008. [Google Scholar] [CrossRef] [Green Version] - Mewes, M. Optical-cavity tests of higher-order Lorentz violation. Phys. Rev. D
**2012**, 85, 116012. [Google Scholar] [CrossRef] - Schreck, M. Quantum field theoretic properties of Lorentz-violating operators of nonrenormalizable dimension in the photon sector. Phys. Rev. D
**2014**, 89, 105019. [Google Scholar] [CrossRef] - Kosteleckỳ, V.A.; Mewes, M. Fermions with Lorentz-violating operators of arbitrary dimension. Phys. Rev. D
**2013**, 88, 096006. [Google Scholar] [CrossRef] - Altschul, B. Vacuum Čerenkov Radiation in Lorentz-Violating Theories Without C P T Violation. Phys. Rev. Lett.
**2007**, 98, 041603. [Google Scholar] [CrossRef] [PubMed] - Cambiaso, M.; Lehnert, R.; Potting, R. Massive photons and Lorentz violation. Phys. Rev. D
**2012**, 85, 085023. [Google Scholar] [CrossRef] - Cambiaso, M.; Lehnert, R.; Potting, R. Asymptotic states and renormalization in Lorentz-violating quantum field theory. Phys. Rev. D
**2014**, 90, 065003. [Google Scholar] [CrossRef] - Bailey, Q.G.; Kosteleckỳ, V.A. Lorentz-violating electrostatics and magnetostatics. Phys. Rev. D
**2004**, 70, 076006. [Google Scholar] [CrossRef] - Colladay, D.; Kosteleckỳ, V.A. Cross sections and Lorentz violation. Phys. Lett. B
**2001**, 511, 209–217. [Google Scholar] [CrossRef] - Kosteleckỳ, V.A.; Lane, C.D. Constraints on Lorentz violation from clock-comparison experiments. Phys. Rev. D
**1999**, 60, 116010. [Google Scholar] [CrossRef] - Bluhm, R.; Kosteleckỳ, V.A.; Lane, C.D.; Russell, N. Clock-comparison tests of Lorentz and CPT symmetry in space. Phys. Rev. Lett.
**2002**, 88, 090801. [Google Scholar] [CrossRef] [PubMed] - Bluhm, R.; Kosteleckỳ, V.A.; Lane, C.D.; Russell, N. Probing Lorentz and CPT violation with space-based experiments. Phys. Rev. D
**2003**, 68, 125008. [Google Scholar] [CrossRef] - Derrick, M.; Fernandez, E.; Fries, R.; Hyman, L.; Kooijman, P.; Loos, J.S.; Musgrave, B.; Price, L.E.; Schlereth, J.; Sugano, K.; et al. New results on the reaction e
^{+}+ e^{−}→ μ^{+}+ μ^{−}, at $\sqrt{s}$ = 29 GeV. Phys. Rev. D**1985**, 31, 2352. [Google Scholar] [CrossRef] - Bender, D.; Derrick, M.; Fernandez, E.; Gieraltowski, G.; Hyman, L.; Jaeger, K.; Klem, R.; Kooijman, P.; Kooijman, S.; Loos, J.; et al. Tests of QED at 29 GeV center-of-mass energy. Phys. Rev. D
**1984**, 30, 515. [Google Scholar] [CrossRef] - Levi, M.; Blocker, C.; Strait, J.; Abrams, G.; Amidei, D.; Baden, A.; Boyarski, A.; Breidenbach, M.; Burchat, P.; Burke, D.; et al. Weak Neutral Currents in e+ e- Collisions at s = 29 GeV. Phys. Rev. Lett.
**1983**, 51, 1941. [Google Scholar] [CrossRef] - Casana, R.; Ferreira, M., Jr.; Maluf, R.; dos Santos, F. Effects of a C P T-even and Lorentz-violating nonminimal coupling on electron-positron scattering. Phys. Rev. D
**2012**, 86, 125033. [Google Scholar] [CrossRef]

**Figure 4.**The differential cross section in arbitrary units. The red line corresponds to the usual case, while green and blue lines represent small Lorentz violation scenarios with ${k}_{Z}\left|\mathbf{v}\right|cos\beta sin\chi $ assuming positive and negative values. The angles $\beta $ and $\chi $ are fixed, while $\theta $ goes from $\pi /2$ to $\pi $.

**Figure 5.**The differential cross section in arbitrary units. The red line corresponds to the usual case while green and blue lines represent small Lorentz violation situations with ${\left({\kappa}_{e-}\right)}_{ZZ}\left(1-3{cos}^{2}\beta {sin}^{2}\chi \right)$ assuming positive and negative values. The angles $\beta $ and $\chi $ are fixed, while $\theta $ goes from $\pi /2$ to $\pi $.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Santos, F.E.P.d.; Ferreira, M.M.
Fermion Scattering in a CPT-Even Lorentz Violation Quantum Electrodynamics. *Symmetry* **2018**, *10*, 302.
https://doi.org/10.3390/sym10080302

**AMA Style**

Santos FEPd, Ferreira MM.
Fermion Scattering in a CPT-Even Lorentz Violation Quantum Electrodynamics. *Symmetry*. 2018; 10(8):302.
https://doi.org/10.3390/sym10080302

**Chicago/Turabian Style**

Santos, Frederico E. P. dos, and Manoel M. Ferreira.
2018. "Fermion Scattering in a CPT-Even Lorentz Violation Quantum Electrodynamics" *Symmetry* 10, no. 8: 302.
https://doi.org/10.3390/sym10080302