# Probing Lorentz violation effects via a laser beam interacting with a high-energy charged lepton beam

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## Abstract

In this work, the conversion of linear polarization of a laser beam to circular one through its forward scattering by a TeV order charged lepton beam in the presence of Lorentz violation correction is explored. We calculate the ratio of circular polarization to linear one (Faraday conversion phase \(\Delta \phi _{\mathrm{FC}}\)) of the laser beam interacting with either electron or the muon beam in the framework of the quantum Boltzmann equation. Regarding the experimentally available sensitivity to the Faraday conversion \(\Delta \phi _{\mathrm{FC}}\simeq 10^{-3}-10^{-2}\), we show that the scattering of a linearly polarized laser beam with energy \(k_0\sim 0.1\) eV and an electron/muon beam with flux \({\bar{\epsilon }}_{e,\mu }\sim 10^{10}/10^{12}\) TeV cm\(^{-2}\) s\(^{-1}\) places an upper bound on the combination of lepton sector Lorentz violation coefficients \(c_{\mu \nu }\) components \((c_{TT}+1.4~c_{(TZ)}+0.25(c_{XX}+c_{YY}+2~c_{ZZ})\). The obtained bound on the combination for the electron beam is at the \(4.35\times 10^{-15}\) level and for the muon beam at the \(3.9\times 10^{-13}\) level. It should be mentioned that the laser and charged lepton beams considered here to reach the experimentally measurable \(\Delta \phi _{\mathrm{FC}}\) are currently available or will be accessible in the near future. This study provides a valuable supplementary to other theoretical and experimental frameworks for measuring and constraining Lorentz violation coefficients.

## 1 Introduction

Usually, radiation can be both linearly and circularly polarized. It is well known that when an initially unpolarized photon scatters off a free electron through Compton scattering, it results in linear polarization but not circular polarization of the scattered radiation. However, it is shown that Compton scattering in the presence of external background fields, similar to strong magnetic fields [1, 2, 3, 4] or theoretical (non-trivial) backgrounds such as non-commutativity in space-time [2, 5, 6] and Lorentz symmetry violation [2], can produce circular polarization. Moreover, nonlinear effects such as the nonlinear Euler–Heisenberg effect can cause converting of photons from linear polarization to circular polarization [3, 7, 8]. In this paper, we consider Compton scattering through the collision of laser photons and high-energy charged lepton beams in the presence of Lorentz violation (LV) effects to study the generation of circular polarization in an Earth-based laboratory. Lorentz symmetry is a fundamental symmetry of the standard model in flat space-time and quantum field theory. However, it can be violated by an underlying theory at the Planck scale [9, 10]. There are many theories in which the Lorentz symmetry is violated spontaneously, such as string theory [11, 12, 13], quantum gravity [14, 15, 16, 17] and non-commutative space-time [18, 19, 20]. Meanwhile, it is also possible to study LV in a general model-independent way in the context of effective field theory known as the Standard Model extension (SME). In the SME Lagrangian, the observer Lorentz symmetry (i.e. change of coordinate) is obeyed, while the particle Lorentz symmetry (i.e. boosts on particles and not on background fields) is violated [21, 22]. The SME contains all feasible Lorentz breaking operators created by known fields of the standard model of dimension three or more. These operators are contracted with coefficients representing backgrounds and preferred directions in space-time [21, 23, 24] and can describe a small Lorentz symmetry violation at available energies. Generally, the number of coefficients is infinite. By the way, it is possible to choose a minimal subset of the SME with finite coefficients. The minimal SME contains renormalizable operators which are invariant under the gauge group of the standard model, \(SU(3)\times SU(2)\times U(1)\). In recent years, new studies have provided new types of constraints on the LV parameters [25, 26, 27, 28, 29]. Among them astrophysical [30, 31, 32] and Earth [33, 34, 35] systems have shown stronger bounds on the LV parameters [36]. Observation of circularly polarized photons in lepton and photon scattering can be a proof of LV and might result in new physics beyond the standard model. In contrast, constraints on circular polarization might improve the available bounds on the parameters of the SME.

The paper is organized as follows: in Sect. 2 we briefly introduce the Stokes parameter formalism and the generalized Boltzmann equation. In Sect. 3 we study the effect of LV on the collision of the relativistic lepton beam (electron/muon) and the laser. In Sect. 4 we give the value of the FC phase of the laser beam through this interaction. Finally, we discuss the results in the last section.

## 2 Stokes parameters and quantum Boltzmann equation

*I*,

*Q*,

*U*and

*V*.

*I*denotes the intensity of the laser beam,

*V*shows the difference between left- and right-circular polarization

*Q*and

*U*indicate the linear polarization.

*Q*is defined by the intensity difference between the polarized components of the electromagnetic wave in the direction of the

*x*and

*y*axes.

*U*quantifies the discrepancy between \(45^{\circ }\) and \( 135^{\circ }\) counted from the positive

*x*axis, to the reference plane [37]. The linear polarization can also be shown by vector \(P\equiv \sqrt{Q^2+U^2}\) [38]. The Stokes parameters can be specified by a superposition of two opposite, right- and left-hand circular polarization contributions, \((\hat{\mathbf{R}})\) and \((\hat{\mathbf{L}})\):

*Q*and

*U*as follows:

## 3 The generation of circular polarization in the SME

*c*coefficients. In fact, the anti-symmetric part at leading order is equivalent to the redefinition in the representation of the Dirac matrices [40]. Therefore the physical quantities are independent of the anti-symmetric part of the c coefficient at leading order. Then the time evolution of the Stokes parameters given in Eq. (5) is obtained as follows: [2]:

*q*and

*k*are the momenta of leptons and photons, respectively. In particular, available bounds on coefficients of the SME are given in the Standard Sun-centered non-rotating inertial reference frame. However, the experimental set up is usually managed based on the Earth. Therefore, for parameterizing our result, it is necessary to introduce Cartesian coordinates on the Earth frame and suitable basis of vectors for a non-rotating frame. Let us define a coordinate system in three-dimensional space. \(({{\hat{X}}},{{\hat{Y}}},{{\hat{Z}}})\) shows the non-rotating basis in which \({{\hat{Z}}}\) points along the Earth’s axis on the north direction. Then the non-relativistic transformation to a lab basis \(({{\hat{x}}},{{\hat{y}}},{{\hat{z}}})\) at any time

*t*is given by [41]

## 4 Set up of laser and charged beams collision

*z*-axis and the 4-vector momentum of charged lepton

*q*has been assumed to be in the \({\hat{z}}\)-direction.

Experimental bounds on components of c

\(c_{\mu \nu } \) | Experimental bounds | System |
---|---|---|

\(c_{TT}\) | \(~~~~~~~~~2\times 10^{-15}\) | Collider physics [42] |

\(c_{YY}\) | \(~~~~~~~~~3\times 10^{-15}\) | Astrophysics [40] |

\(c_{ZZ}\) | \(~~~~~~~~~5\times 10^{-15}\) | Astrophysics [40] |

\(c_{(XY)}\) | \(~~~~~~~~~3\times 10^{-15}\) | Astrophysics [40] |

\(c_{(YZ)}\) | \(~~~~~~~~~1.8\times 10^{-15}\) | Astrophysics [40] |

\(c_{(XZ)}\) | \(~~~~~~~~~3\times 10^{-15}\) | Astrophysics [40] |

\(c_{(TX)}\) | \(~~~~~~-30\times 10^{-14}\) | Collider physics [42] |

\(c_{(TY)}\) | \(~~~~~~-80\times 10^{-15}\) | Collider physics [42] |

\(c_{(TZ)}\) | \(~~~~~~-11\times 10^{-13}\) | Collider physics [42] |

The basic picture of our experiment is fairly simple. We measure the generation of circular polarization for the laser beam via its forward scattering with the charged lepton beam in the presence of LV effects. Equation (22) shows that in order to generate a wide range of circularly polarized photons, the intensity of the lepton and laser beam should be large enough. However, in high-intensity laser, the effects of the magnetic part of the Lorentz force on the charged lepton become important. In this case, the magnetic field as a trivial background will produce circular polarization [1, 2]. Moreover, at strong electromagnetic fields, the nonlinear Compton scattering effect become significant [48]. In addition to the collision of the laser beam with an electron high-energy beam, photons would possibly obtain high energy by back-scattering off the high-energy electron beam, i.e. we have inverse Compton scattering. Therefore, pair production is possible as a result of the Breit–Wheeler reaction. However, the mentioned effects using a low-intensity laser beam are negligible and can safely be ignored. Another point is that low-intensity lasers are available sources of monochromatic radiation. They are lightweight and low-cost devices.

While electromagnetic fields and the coherent time duration of the laser pulse increase, the number of laser photons colliding with an electron beam will increase as well. Moreover, forward scattering of linearly polarized photons on polarized lepton such as the electron can cause the circular polarization of photons [49]. To reduce unwanted backgrounds effects as explained above, which are particularly important at high frequencies and intensities, we assume the lepton beam is nearly unpolarized and the laser beam has a low intensity and energy. As shown in Eq. (14), in addition to FC, FR also results by forward scattering in the presence of LV effects.

Moreover, we determine the FC phase for different components of \(c_{\mu \nu }\) with the intention of exploring the effect of each component of \(c_{\mu \nu }\) coefficients on FC phase. To do that we assume that only one specific component of \(c_{\mu \nu }\) contributes in the FC phase. Therefore, the FC phase \(\Delta \phi |_e^{\text {LV}}\) for different components of \(c_{\mu \nu }\) in forward scattering of the linearly polarized laser beam with the energy \( k_0\sim 0.1\) eV and electron beam as a function of scattering angle \(\theta \) for four scenarios of assumptions on time free parameter: (time: 1, 6, 12, 18 o’clock) are plotted in Fig. 2. Besides, Fig. 3 represents the FC phase \(\Delta \phi |_\mu ^{\text {LV}}\) versus \(\theta \) with the same assumptions for the laser beam interacting with the muon beam. For simplicity we set \(\varphi =\chi =\pi /4\) and polarization of incoming laser beam as \(U_0 = Q_0=I_0/2\) in Figs. 2 and 3. \(I_0\) is the intensity of the laser beam.

## 5 Discussion and conclusion

We finally review how the laser beam interacting with the charged lepton beams can provide a new situation to constrain \(c_{\mu \nu }\) coefficients. The \(d_{\mu \nu }\) coefficient, as noted previously, has no contribution for the generation of circular polarization via forward scattering of laser and charged lepton beams.

As our results depend on the direction of the beams, the time of performing the experiment and location of the lab, there might be an optimal beam direction, performing time and position for the lab to observe the \(c_{\mu \nu }\) effects. As an example, the FC phases are obtained for four scenarios with assumptions on time with \(\phi =\chi =\pi /4\). The energy of the linearly polarized laser beam is set 0.1 eV and the energy of the charged lepton beam is assumed \(q_0=1\) TeV, the number densities of the electron and the muon beam are \(n_e=10^{10}\,\hbox {cm}^{-3}\) and \(n_\mu =10^{12}\,\hbox {cm}^{-3}\), respectively. Note these suggested charged beams are experimentally proposed [44, 45, 46]. The results are illustrated in Figs. 2 and 3 for four different times during the day–night. Plots show that all FC phases \(\Delta \phi |_{e,\mu }^{\text {LV}}\) get their maximum values at specific scattering angles for different components of \(c_{\mu \nu }\). This could be one of the most important characters of our results to distinguish the contribution of the LV effect from other sources of circular polarization.

Based on the current constraint on the \(c_{\mu \nu }\) components [36] and the available sensitivity level to detect circular polarization or FC phase [50, 51, 53], the estimated total FC phase is in the range of current experimental precision. It should be emphasized that we do not need to use very high-intensity laser beams which would help us to avoid other background effects.

Location dependence of the independent components of \(c_{\mu \nu }\)

Parameters | Location dependence |
---|---|

\(c_{00}\) | \(c_{TT}\) |

\(c_{22}\) | \(\frac{c_{XX}+c_{YY}}{2}\) |

\(c_{33}\) | \(\frac{1}{2} \left( (c_{XX}+c_{YY}) \sin ^2(\chi )+2 c_{ZZ} \cos ^2(\chi )\right) \) |

\(c_{(12)}\) | 0 |

\(c_{(13)}\) | \(\sin \chi \cos \chi (c_{XX}+c_{YY}-2 c_{ZZ})\) |

\(c_{(23)}\) | 0 |

\(c_{(01)}\) | \( - c_{(TZ)}\sin \chi \) |

\(c_{(02)}\) | 0 |

\(c_{(03)}\) | \( c_{(TZ)} \cos \chi \) |

We estimate the bounds on a combination of \(c_{\mu \nu }\) components contributing to the FC phase based on the available accuracy on the FC phase \(\Delta \phi \sim 10^{-3}{-}10^{-2}\) rad [52]. Assuming the same typical linearly polarized laser beam interacting with the electron beam as mentioned above and \(\phi =\chi =\pi /4\), we find a bound of \(4.35\times 10^{-15}\) on \([c_{TT}+1.4~c_{(TZ)}+0.25(c_{XX}+c_{YY}+2~c_{ZZ})]\). For the muon beam we obtain a looser bound of \(3.9\times 10^{-13}\) on \([c_{TT}+1.4~c_{(TZ)}+0.25(c_{XX}+c_{YY}+2~c_{ZZ})]\). While the LV components in Eq. (18) depend on both location of laboratory and time of the experiment, it would be possible to find various constraints on a combination of the \(c_{\mu \nu }\) components at different locations and times. Moreover, by improving the sensitivity of the experiment to FC in the future, the obtained bounds will be improved.

We should also mention that any backgrounds can be a possible source to generate circular polarization. For example, in addition to the LV correction to Compton scattering, interaction of photon with a LV background can also create circular polarization which is linearly proportional to the LV coefficient for the photon sector of \(k_F\) [2] and \(k_{AF}\) [50]. This effect does not modify the Compton scattering but the dynamics of the laser beam can be influenced [2]. Briefly we ensure that our study provides a valuable supplement to other theoretical and experimental frameworks for improving the available constraints on the LV coefficients.

## Notes

### Acknowledgements

We are thankful to I. Motie for collaboration in the initial stages of this work. S. Tizchang is grateful to S. Aghababaei for fruitful discussions on LV, and to M.Aslaninejad for helpful discussion on charged lepton beams.

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