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Lorentz Violation, Moller Scattering, and Finite Temperature.

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 ~[10.sup.19] GeV. 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 d [less than or equal to] 4 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 Moller 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 |0([beta])>, describes a system in thermal equilibrium, where [beta] = 1/[k.sub.B] T, with T being the temperature and [k.sub.B] 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: T = 0 and T [not equal to] 0 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 Moller 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 |0([beta])>. Then, the expectation value of an operator A is given as <A> = <0([beta])[absolute value of (A)]0([beta])>. However, two main ingredients are required to achieve this: (1) doubling of degrees of freedom in the Hilbert space and (2) the Bogoliubov transformation. The doubling is defined by the tilde (~) conjugation rules, where the expanded space is [S.sub.T] = S [cross product] [??], with S being the standard Hilbert space and [??] being the fictitious space. The mapping between the tilde [[??].sub.i] and nontilde [A.sub.i] operators is defined by the following tilde (or dual) conjugation rules:

[mathematical expression not reproducible], (1)

with [xi] = -1 for bosons and [xi] = +1 for fermions. The Bogoliubov transformation is a rotation in the tilde and nontilde variables, thus introducing thermal quantities.

For bosons, Bogoliubov transformations are

[mathematical expression not reproducible], (2)

where u ([beta]) = cosh [theta]([beta]) and v'([beta]) = sinh [theta]([beta]), with [a.sup.[dagger].sub.p] and [a.sub.p] being creation and annihilation operators, respectively. Algebraic rules for thermal operators are

[a (k, [beta]), [a.sup.[dagger]] (p, [beta])] = [[delta].sup.3] (k - p), [[??] (k, [beta]), [[??].sup.[dagger]] (p, [beta])] = [[delta].sup.3] (k - p), (3)

and other commutation relations are null.

For fermions, the Bogoliubov transformations are

[mathematical expression not reproducible], (4)

where u([beta]) = cos [theta]([beta]) and v([beta]) = sin [theta]([beta]), with [c.sup.[dagger].sub.p] and [c.sub.p] being creation and annihilation operators, respectively. Here, algebraic rules are

{c (k, [beta]), [c.sup.[dagger]] (p, [beta])} = [[delta].sup.3] (k - p), {[??] (k, [beta]), [[??].sup.[dagger]] (p, [beta])} = [[delta].sup.3] (k - p), (5)

and other anticommutation relations are null.

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

3. Lorentz-Violating Corrections to Moller Scattering at Finite Temperature

Our interest is in calculating the differential cross section for the process, [e.sup.-]([p.sub.1])[e.sup.-]([p.sub.2]) [right arrow] [e.sup.-]([p.sub.3])[e.sup.-]([p.sub.4]), at finite temperature. This process is represented in Figure 1.

The Lagrangian that describes the Moller scattering is given by

L = -[1/4] [F.sub.[mu][nu]][F.sup.[mu][nu]] + [bar.[psi]] (i[[gamma].sup.[mu]] [D.sub.[mu]] - m)[psi] - 1/2[xi] [([[partial derivative].sub.[mu]] [A.sup.[mu]]).sup.2]. (6)

To investigate the Lorentz violation in this scattering, an alternative procedure is used. It consists of modifying just the SME interaction part via a nonminimal coupling using the covariant derivative; that is,

[D.sub.[mu]] = [[partial derivative].sub.[mu]] + ie[A.sub.[mu]] + ig[b.sup.[nu]][[??].sub.[mu][nu]]], (7)

with e, g, and [b.sup.[mu]] being the electron charge, a coupling constant, and a constant four vector, respectively. [[??].sub.[mu][nu]] = (1/2)[[epsilon].sub.[mu][nu][alpha][rho]][F.sup.[alpha][rho]] is the dual electromagnetic tensor with [[epsilon].sub.0123] = 1 and [F.sup.[alpha][rho]] = [[partial derivative].sup.[alpha]] [A.sup.[rho]] - [[partial derivative].sup.[rho]] [A.sup.[alpha]]. The new interaction breaks the Lorentz and CPT symmetries. The interaction Lagrangian is given by

[L.sub.I] = -e[bar.[psi]][[gamma].sup.[mu]][psi][A.sub.[mu]] - g[b.sup.[nu]][bar.[psi]][[gamma].sup.[mu]][psi][[partial derivative].sup.[alpha]] [A.sup.[rho]] [[epsilon].sub.[mu][nu][alpha][rho]]. (8)

The second term represents the new interaction produced by the nonminimal coupling that leads to Lorentz violation, while the first term is the usual QED interaction. These vertices are represented as

* [right arrow] [V.sup.[mu]] = -ie[[gamma].sup.[mu]] x [right arrow] g[V.sub.[rho]] = - g[b.sup.[nu]][[gamma].sup.[mu]][q.sup.[alpha]] [[epsilon].sub.[mu][nu][alpha][rho]], (9)

where [q.sup.[alpha]] is the momentum operator. The diagrams in Figure 1 represent the usual QED when the vertex [V.sup.[mu]] is used and the Lorentz violation corrections are represented by g[V.sub.[rho]] vertex. The total contribution is the sum of all diagrams.

In order to calculate the differential cross section for Moller scattering, the transition amplitude,

M([beta]) = <f, [beta] [absolute value of ([[??].sup.(2)])] i, [beta]>, (10)

is determined. Here, [[??].sup.(2)] is the second-order term of the [??]-matrix that is defined as

[mathematical expression not reproducible], (11)

where T is the time ordering operator and [mathematical expression not reproducible] describes the interaction. Here, [L.sub.I] (x) and [[??].sub.I](x) are interaction Lagrangian in usual and tilde Hilbert space, respectively. The thermal states are

[mathematical expression not reproducible], (12)

with [mathematical expression not reproducible] and [mathematical expression not reproducible] being the creation and annihilation operators, respectively. Then, the transition amplitude becomes

[mathematical expression not reproducible], (13)

where

[mathematical expression not reproducible] (14)

An important note is that the [M.sub.3]([beta]) contribution will be ignored since it is proportional to the second order of Lorentz-violating parameter. Let us consider the lowest order in this parameter. There are similar equations for the transition amplitude that include tilde operators. The fermion field is written as

[psi] (x) = [integral] dp [N.sub.p] [[c.sub.p] u (p) [e.sup.-ipx] + [d.sup.[dagger].sub.p] v(p) [e.sup.ipx]], (15)

where [c.sub.p] and [d.sub.p] are annihilation operators for electrons and positrons, respectively, [N.sub.p] is the normalization constant, and u(p) and v(p) are Dirac spinors. There are two Feynman diagrams that describe the Moller scattering, the t-channel and the u-channel, as represented in Figure 1. For simplicity, consider

[M.sub.1] ([beta]) = [M.sup.t.sub.1]([beta]) + [M.sup.u.sub.1]([beta]), (16)

where [M.sup.1.sub.1]([beta]) and [M.sup.u.sub.1]([beta]) are contributions due to the t-channel and the u-channel diagrams, respectively. Then, using (15) and Bogoliubov transformations, we get

[mathematical expression not reproducible]. (17)

The photon propagator at finite temperature is defined as

<0 ([beta])| T[A.sub.[mu]] (x)[A.sub.[nu]] (y)| 0 ([beta])> = i[integral] [[d.sup.4]/[(2[pi]).sup.4]] [e.sup.-iq(x-y)] [[DELTA].sup.[mu][nu]](q, [beta]), (18)

with [[DELTA].sub.[mu][nu](q, [beta]) [equivalent to] [[DELTA].sup.(0).sub.[mu][nu]](q) + [[DELTA].sup.([beta]).sub.[mu][nu]](q), where [[DELTA].sup.(0).sub.[mu][nu]](q) and [[DELTA].sup.([beta]).sub.[mu][nu]](q) are zero and finite temperature components, respectively. Explicitly, these are

[mathematical expression not reproducible]. (19)

Taking u([beta]) = cos [theta]([beta]) and v([beta]) = sin [theta]([beta]), we get [([u.sup.2] - [v.sup.2]).sup.2] = [tanh.sup.2] ([beta][absolute value of ([q.sub.0])]/2), where [q.sub.0] = [omega], and using the definition of the four-dimensional delta function,

[mathematical expression not reproducible], (20)

the transition amplitude after carrying out the q integral becomes

[mathematical expression not reproducible], (21)

where [N.sub.p] = 2 has been used. The propagator is

[[DELTA]'.sub.[mu][nu]] (q) [equivalent to] [DELTA]'(q) [[eta].sub.[mu][nu]] (22)

with

[mathematical expression not reproducible]. (23)

The remaining delta function, which has the overall four-momentum conservation, is taken out. In addition, the center of mass is considered so that

[mathematical expression not reproducible], (24)

where [mathematical expression not reproducible], and s = [(2E).sup.2] = [E.sup.2.sub.CM]; we get [absolute value of ([([p.sub.1] - [p.sub.2]).sub.0])] = [absolute value of ([([p.sub.3] - [p.sub.2]).sub.0])] = [E.sub.CM]. Then,

[mathematical expression not reproducible]. (25)

In a similar way, the linear term in the Lorentz-violating parameter becomes

[mathematical expression not reproducible]. (26)

The differential cross section is given by

[(d[sigma]/d[OMEGA]).sub.[beta]] = 1/64[[pi].sup.2][E.sup.2.sub.CM] x [1/4] [summation over (spins)] [[absolute value of (M([beta]))].sup.2]. (27)

Now, an average over the spin of the incoming particles and sum over the spin of the outgoing particles are taken. Then, the square transition amplitude is determined:

[[absolute value of (M([beta]))].sup.2] = [[absolute value of ([M.sub.1] ([beta]) + [M.sub.2] ([beta]))].sup.2]. (28)

In addition, the relation

[bar.u] ([p.sub.2]) [[gamma].sub.[alpha]] u ([p.sub.1]) [bar.u] ([p.sub.1]) [[gamma].sup.[alpha]] u ([p.sub.2]) = tr [[[gamma].sub.[alpha]] u ([p.sub.1]) [bar.u] ([p.sub.1]) [[gamma].sup.[alpha]] u ([p.sub.2]) [bar.u] ([p.sub.2])] (29)

and the completeness relation

[summation]u ([p.sub.i]) [bar.u] ([p.sub.i]) = [[??].sub.i] + m (30)

with i = 1, 2, 3, 4 are used. Now, the electron mass is ignored since the momenta are large; that is, the ultra relativistic limit ([p.sup.2] [much greater than] [m.sup.2]) is used.

The differential cross section (see (27)) at finite temperature for a time-like four-vector [b.sup.[nu]] = (b, 0) becomes

[mathematical expression not reproducible], (31)

where

[mathematical expression not reproducible] (32)

[C.sub.2] ([beta]) [equivalent to] 16[E.sup.4] {[[DELTA].sup.2] ([p.sub.1] - [p.sub.2]) (cos 2[theta] + 3) + [4[[DELTA].sup.2] ([p.sub.3] - [p.sub.2]) [cos.sup.4][theta]]/2} (33)

with

[mathematical expression not reproducible]. (34)

This result shows that corrections for Moller scattering due to Lorentz violation at finite temperature are altered. Even if the Lorentz symmetry is conserved, there are corrections due to temperature for the usual result for electron-electron scattering.

An important note is that, in the Born approximation, the interaction potential may be calculated using the Fourier transform of the scattering amplitude; that is,

V(r) = 1/[(2[pi]).sup.3] [integral] M([beta]) [e.sup.ipxx] [d.sup.3]p, (35)

with M([beta]) being the total scattering amplitude given in (13). In the nonrelativistic limit, some studies have been carried out about electron-electron interaction potential [33-35]. Our result (see (31)) has been obtained in the ultrarelativistic limit, i.e., [p.sup.2] [much greater than] [m.sup.2]. However using the nonrelativistic limit (i.e., [p.sup.2] [much less than] [m.sup.2]), in (27), a different cross section at finite temperature and with Lorentz-violating term is obtained. This result may be compared with the nonrelativistic and Lorentz invariant differential cross section at zero temperature that is given as

(d[sigma]/d[OMEGA]) = [e.sup.4]/4[E.sup.2] [(1 - [cos.sup.2][theta]) + 1/[1 + [cos.sup.2][theta]] - 1/[(1 - cos [theta]).sup.2]], (36)

where E is the total kinetic energy. In our case, the potential is altered directly by Lorentz violation and temperature effects.

In addition, the result obtained here may be compared with the nonrelativistic result; see (36). Let us consider the following comparisons: (i) b = 0 and T [not equal to] 0--in this case, there are corrections due to temperature effects that become relevant in the high temperature limit since the estimates defined in (34) become large. Then, the dependent temperature part becomes dominant. This result leads to a new motivation: what are the modifications due to Lorentz violation at high temperature for electron-electron scattering? Then, these estimates will give us a reasonable idea of the role of SME parameters at finite temperatures. (ii) b [not equal to] 0 and T = 0--when the temperature is small, the differential cross section ismodified only by Lorentz-violating parameter. Thus, while small, Lorentz-violating terms do not contradict any experimental measurements of the Moller scattering at zero temperature. However, there is a motivation to perform such study at finite temperature. Although Moller scattering at high energy has been investigated in experiments, these are still at zero temperature. There is certainly no investigation at extremely high energy with nonzero temperature that may indicate any role for Lorentz violation.

4. Conclusion

Lorentz and CPT symmetries are foundations of SM and general relativity (GR). However, violations of these symmetries could emerge in new physics beyond the SM, at the Planck scale. The SME is an effective field theory that contains SM, GR, and all Lorentz- and CPT-violating operators. In parallel to SME exists an alternative procedure that consists in modifying the interaction part using a nonminimal coupling via covariant derivative. In this paper, the nonminimal coupling term, CPT-odd term, is used to calculate the Lorentz-violating corrections to Moller scattering at finite temperature. Finite temperature effects are introduced using the TFD formalism. Our results show that Lorentz-violating operators at finite temperature contribute to the differential cross section of the electron-electron scattering. This is important since there is no investigation at extremely high energy with nonzero temperature for this scattering. Here, a theoretical study is developed such that relevant effects may arise for processes at very high energies. These results may modify the results that are anticipated for astrophysical processes. In addition, it is shown that the differential scattering cross section for Moller scattering depends on temperature. The cross section also changes with Lorentz violation. Then, this result may be useful to understand the role of Lorentz violation term depending on temperature. The interior of stars is a region where there is a variation of the temperature; then stars may be used to test this result. At present, this result is of theoretical interest and does not provide a direct way to measure an upper limit on the magnitude of the Lorentz-violating nonminimal coupling. Constraints on Lorentz-violating parameter can be obtained if measurements at finite temperature are realized in the future. Although corrections due to squared Lorentz-violating parameters avoid the reach to stronger upper bounds, there are ways to constrain Lorentz violation. The dependence of the cross section on the second-order Lorentz-violating parameters has been obtained also in other scatterings, such as Bhabha scattering and pair annihilation in the presence of a CPT-odd nonminimal coupling [18, 19] and electron-positron scattering with nonminimal CPT even coupling term [20]. Good upper bounds for Lorentz violating parameters have been obtained using experimental data from [36].

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

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work by Alesandro F. Santos is supported by CNPq Project 308611/2017-9.

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Alesandro F. Santos (iD) (1) and Faqir C. Khanna (2, 3)

(1) Instituto de Fisica, Universidade Federal de Mato Grosso, 78060-900, Cuiaba, 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; alesandroferreira@fisica.ufmt.br

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

Academic Editor: Edward Sarkisyan-Grinbaum

Caption: Figure 1: Moller scattering.
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Title Annotation:Research Article
Author:Santos, Alesandro F.; Khanna, Faqir C.
Publication:Advances in High Energy Physics
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