Advances in High Energy Physics

Volume 2018, Article ID 4596129, 7 pages

https://doi.org/10.1155/2018/4596129

## Lorentz Violation, Möller Scattering, and Finite Temperature

^{1}Instituto de Física, Universidade Federal de Mato Grosso, 78060-900, Cuiabá, MT, Brazil^{2}Department of Physics and Astronomy, University of Victoria, 3800 Finnerty Road, Victoria, BC, Canada^{3}Physics Department, Theoretical Physics Institute, University of Alberta, Edmonton, AB, Canada

Correspondence should be addressed to Alesandro F. Santos; rb.tmfu.acisif@arierrefordnasela

Received 24 January 2018; Revised 8 April 2018; Accepted 23 April 2018; Published 24 May 2018

Academic Editor: Edward Sarkisyan-Grinbaum

Copyright © 2018 Alesandro F. Santos and Faqir C. Khanna. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

Lorentz and CPT symmetries may be violated in new physics that emerges at very high energy scale, that is, at the Planck scale. The differential cross section of the Möller scattering due to Lorentz violation at finite temperature is calculated. Lorentz-violating effects emerge from an interaction vertex due to a CPT-odd nonminimal coupling in the covariant derivative. The finite temperature effects are determined using the Thermo Field Dynamics (TFD) formalism.

#### 1. Introduction

The Standard Model (SM) respects the Lorentz and CPT symmetries that are supported by many experiments. Although the SM has achieved a remarkable phenomenological success, there are still unresolved issues. Such problems emerge in a number of different scenarios, most of them related to new physics at the Planck scale ~. Small Lorentz and CPT violations emerge in theories that unify gravity with quantum mechanics such as string theory [1–3]. There are several other issues that lead to Lorentz violation such as loop quantum gravity [4, 5], geometrical effects such as noncommutativity [6, 7], torsion [8], and nonmetricity [9]. A theory that allows incorporation of all Lorentz-violating terms together with the SM and general relativity is the Standard Model Extension (SME) [10, 11]. The SME is an effective field theory that preserves the observer Lorentz symmetry, while the particle Lorentz symmetry is violated. This model is divided into two versions, a minimal extension which has operators with dimensions and a nonminimal version of the SME associated with operators of higher dimensions. Numerous possibilities have been investigated [12–14].

The SME framework is one way to investigate Lorentz and CPT symmetries violation. A different way is to modify the interaction vertex adding a new nonminimal coupling term to the covariant derivative. This new nonminimal coupling term may be CPT-odd or CPT-even. There are various applications for both terms [15–25]. In this paper, the CPT-odd term is chosen to calculate the Lorentz violation correction to the electron-electron scattering, known as Möller scattering. Corrections to the electron-electron scattering due to Lorentz violation have been studied [26]. In this work, the Z-boson exchange contribution to the amplitude is considered in addition to the electromagnetic interaction. In [26], the minimal version of the SME has been used. Here, the nonminimal CPT-odd term is used to calculate Lorentz violation corrections in the electron-electron scattering. In addition, the finite temperature effects are calculated. The Thermo Field Dynamics (TFD) formalism is used to introduce finite temperature.

TFD formalism is a thermal quantum field theory [27–32] where the statistical average of an arbitrary operator is interpreted as the expectation value in a thermal vacuum. The thermal vacuum, defined by , describes a system in thermal equilibrium, where , with being the temperature and the Boltzmann constant. This formalism depends on the doubling of the original Fock space, composed of the original and a fictitious space (tilde space), using Bogoliubov transformations. The original and tilde space are related by a mapping, tilde conjugation rules. The physical variables are described by nontilde operators. The Bogoliubov transformation is a rotation involving these two spaces. As a consequence, the propagator is written in two parts: and components.

This paper is organized as follows. In Section 2, a brief introduction to the TFD formalism is presented. In Section 3, the Lorentz-violating nonminimal coupling is considered. The transition amplitude at finite temperature is determined. The differential cross section for Möller scattering with Lorentz-violating parameter at finite temperature is calculated. In Section 4, some concluding remarks are presented.

#### 2. Thermo Field Dynamics (TFD)

Here, an introduction to TFD formalism is presented. It is a real-time formalism of quantum field theory at finite temperature where the thermal average of an observable is given by the vacuum expectation value in an extended Hilbert space. This is achieved by defining a thermal ground state . Then, the expectation value of an operator is given as . However, two main ingredients are required to achieve this: doubling of degrees of freedom in the Hilbert space and the Bogoliubov transformation. The doubling is defined by the tilde () conjugation rules, where the expanded space is , with being the standard Hilbert space and being the fictitious space. The mapping between the tilde and nontilde operators is defined by the following tilde (or dual) conjugation rules:with for bosons and for fermions. The Bogoliubov transformation is a rotation in the tilde and nontilde variables, thus introducing thermal quantities.

For bosons, Bogoliubov transformations arewhere and , with and being creation and annihilation operators, respectively. Algebraic rules for thermal operators areand other commutation relations are null.

For fermions, the Bogoliubov transformations arewhere and , with and being creation and annihilation operators, respectively. Here, algebraic rules areand other anticommutation relations are null.

As an important note, the propagator in TFD formalism is written in two parts: one describes the flat space-time contribution and the other displays the thermal effect.

#### 3. Lorentz-Violating Corrections to Möller Scattering at Finite Temperature

Our interest is in calculating the differential cross section for the process, , at finite temperature. This process is represented in Figure 1.