1. Introduction
The formulation of the standard model (SM) gives a field theory that describes the fundamental particles and unifies the interactions. It is a successful path of experimental verifications. However, besides SM does not take into account gravitation, there are problems in which need to be clarified, such as matter-antimatter imbalance [1]. A proposal to extend the SM appeared in the 1980s with Samuel and Kostelecký [2] in the context of string field theory. They had fixed which in the Lorentz symmetry is called as the spontaneously symmetry break (LSV) by a constant tensor field. This formulation is known as the standard model extension (SME) [3, 4]. In SME, there are various constant background tensors with different ranks that emerge from the condensation of primordial fields (there is a fixed background tensor for every sector) [3, 4]. Since then, it is possible to find several studies in the literature that involve LSV [5], as shown in [6-47]. Therefore, proposals have appeared with an explicit break of symmetry, but not necessarily a spontaneous symmetry breaking. Other examples of LSV are the noncommutative theory [48] and the doubly special relativity theory. In particular, in the doubly special relativity theory, a minimum length is introduced as a new fundamental constant in addition to the speed of light [49-55]. Another interesting proposal [56] explores SME by violating the Lorentz invariance, but it preserves the spacetime isotropy and homogeneity.
In this work, we investigate the effects of the violation of the Lorentz symmetry on a nonrelativistic neutral particle with a permanent magnetic dipole moment that interacts with a radial electric field. We consider a background of the Lorentz symmetry violation determined by a space-like vector field. We show that the effects of the violation of Lorentz symmetry can yield an effective scalar potential that plays the role of an attractive Coulomb-type potential. Then, we show that bound state solutions to the Schrödinger-Pauli equation can be achieved and the spectrum of energy is a function of the Aharonov-Casher geometric quantum phase [57]. Finally, we discuss the arising of persistent spin currents [58-60] in the system.
The structure of this paper is: in section 2 , we analyse the interaction of the permanent magnetic dipole moment of a nonrelativistic neutral particle with a radial electric field under the influence of a Lorentz symmetry violation background. Then, we show that the effects of the violation of the Lorentz symmetry yield an attractive Coulomb-type potential. Further, we analyse the Aharonov-Casher effect [57] in this system; in section 3 , we present our conclusions.
2. Aharonov-Casher system in a Coulomb-type potential
The quantum description of a neutral particle with a permanent magnetic dipole moment that interacts with external fields is made by introducing a nonminimal coupling iγμ∂μ → iγμ∂μ + $\tfrac{\mu }{2}$ Fμν(x)Σμν into the Dirac equation [57, 61]. Note that μ is the permanent magnetic dipole moment of the neutral particle, the tensor ${F}_{\mu \nu }\left(x\right)$ is the electromagnetic tensor, ${{\rm{\Sigma }}}^{{ab}}=\tfrac{{\rm{i}}}{2}\left[{\gamma }^{a},{\gamma }^{b}\right]$ and γa corresponds to the standard Dirac matrices [61]. On the other hand, it has been proposed in [62] that effects of violation of the Lorentz symmetry can be investigated by introducing a nonminimal coupling into the Dirac equation, where this coupling describes the presence of a fixed vector field that breaks the Lorentz symmetry. From this perspective, several proposals have been made in the literature with the focus on the violation of the Lorentz symmetry with fermions fields, such as, with fixed tensor fields [63-68] and fixed vector fields [42-45, 69-72].
In this work, our focus is on the effects of the violation of the Lorentz symmetry on a nonrelativistic neutral particle with a permanent magnetic dipole moment that interacts with an electric field. Our interest is in the Lorentz symmetry breaking through a fixed vector field background. Therefore, the background of the Lorentz symmetry violation is introduced into the Dirac equation through a nonminimal coupling ${\rm{i}}{\gamma }^{\mu }{\partial }_{\mu }\to {\rm{i}}{\gamma }^{\mu }{\partial }_{\mu }-g\,{b}^{\mu }\,{F}_{\mu \nu }\left(x\right){\gamma }^{\nu }$ proposed in [70-72], where g is a constant and ${b}^{\mu }$ is a fixed four-vector that plays the role of the vector field that breaks the Lorentz symmetry. From now on, we work with the units ℏ = 1and c = 1. We also work with the line element of the Minkowski spacetime written in cylindrical coordinates:
$\begin{eqnarray}{{\rm{d}}{s}}^{2}=-{{\rm{d}}{t}}^{2}+{{\rm{d}}{r}}^{2}+{r}^{2}\,{\rm{d}}{\varphi }^{2}+{{\rm{d}}{z}}^{2}.\end{eqnarray}$
Observe that the line element (1 ) is written in curvilinear coordinates, therefore, it is worth pointing out that the Dirac equation must be written in terms of the covariant derivative as follows [73]:1 ), the scale factors are h0 = 1, h1 = 1, h2 = r and h3 = 1 [74]. Hence, the Dirac equation becomes:3 ) corresponds to the nonminimal coupling that describes the background of the violation of the Lorentz symmetry proposed in [70-72].
$\begin{eqnarray}{\rm{i}}{\gamma }^{\mu }\,{{\rm{\nabla }}}_{\mu }={\rm{i}}\,{\gamma }^{\mu }\,{D}_{\mu }+\displaystyle \frac{{\rm{i}}}{2}\,\displaystyle \sum _{k=1}^{3}\,{\gamma }^{k}\left[{D}_{k}\,\mathrm{ln}\left(\displaystyle \frac{{h}_{1}\,{h}_{2}\,{h}_{3}}{{h}_{k}}\right)\right].\end{eqnarray}$
Then, we have that ${D}_{\mu }=\tfrac{1}{{h}_{\mu }}\,{\partial }_{\mu }$ is the derivative of the corresponding coordinate system. The parameter hμ corresponds to the scale factors of this coordinate system. With respect to the line element ( $\begin{eqnarray}m\,{\rm{\Psi }}={\rm{i}}{\gamma }^{\mu }\,{{\rm{\nabla }}}_{\mu }{\rm{\Psi }}+\displaystyle \frac{\mu }{2}\,{F}_{\mu \nu }\left(x\right){{\rm{\Sigma }}}^{\mu \nu }\,{\rm{\Psi }}-g\,{b}^{\mu }\,{F}_{\mu \nu }\left(x\right){\gamma }^{\nu }\,{\rm{\Psi }},\end{eqnarray}$
where the last term of equation (We shall consider the components of the electromagnetic tensor to be given by ${F}_{0i}={E}_{i}$ and ${F}_{{ij}}=-{\epsilon }_{{ijk}}\,{B}^{k}$, where the vectors $\vec{E}$ and $\vec{B}$ are the electric and magnetic fields, respectively. From now on, let us consider a scenario of the Lorentz symmetry violation to be defined by an electric field and a space-like vector background as follows:4 ), the Dirac equation (3 ) becomes
$\begin{eqnarray}\vec{E}=\displaystyle \frac{\lambda }{r}\,\hat{r};\,\,\,\vec{b}=b\,\hat{r}.\end{eqnarray}$
This radial electric field is produced by linear distribution of electric charges along the z-axis, where λ is a constant associated with this distribution of electric charges. Moreover, b is a constant. In this scenario defined by a radial electric field and the space-like fixed vector as given in equation ( $\begin{eqnarray}\begin{array}{rcl}{\rm{i}}\displaystyle \frac{\partial {\rm{\Psi }}}{\partial t} & = & m\hat{\beta }\,{\rm{\Psi }}-{\rm{i}}\,{\hat{\alpha }}^{1}\left(\displaystyle \frac{\partial }{\partial r}+\displaystyle \frac{1}{2r}\right){\rm{\Psi }}-{\rm{i}}\,\displaystyle \frac{{\hat{\alpha }}^{2}}{r}\displaystyle \frac{\partial {\rm{\Psi }}}{\partial \varphi }\\ & & -{\rm{i}}\,{\hat{\alpha }}^{3}\,\displaystyle \frac{\partial {\rm{\Psi }}}{\partial z}-{\rm{i}}\hat{\beta }\,\vec{\alpha }\cdot \vec{E}\,{\rm{\Psi }}-g\,\vec{b}\cdot \vec{E}\,{\rm{\Psi }}.\end{array}\end{eqnarray}$
Our interest is on the effects of the background of the Lorentz symmetry violation on the nonrelativistic neutral particle in the presence of the radial electric field (4 ). The nonrelativistic limit of the Dirac equation (5 ) can be obtained, for instance, by writing the solution to the Dirac equation in the form [61]:5 ) is given by8 ) stems from the curvilinear coordinates system, i.e. it stems from the covariant derivative written in equation (2 ) for cylindrical coordinates.
$\begin{eqnarray}{\rm{\Psi }}={{\rm{e}}}^{-{\rm{i}}{m}{t}}\left(\begin{array}{c}\psi \\ \chi \end{array}\right),\end{eqnarray}$
where ψ and χ are 2-spinors, with ψ as being the ‘large' component and χ as being the ‘small' component. After some calculations, then, the nonrelativistic limit of the Dirac equation ( $\begin{eqnarray}{\rm{i}}\displaystyle \frac{\partial \psi }{\partial t}=\displaystyle \frac{{\hat{\pi }}^{2}}{2m}\,\psi -\displaystyle \frac{{\mu }^{2}{E}^{2}}{2m}\,\psi +\displaystyle \frac{\mu }{2m}\left(\vec{{\rm{\nabla }}}\cdot \vec{E}\right)\psi -g\,\vec{b}\cdot \vec{E}\,\psi ,\end{eqnarray}$
where σi are the Pauli matrices that satisfy the relation $\left({\sigma }^{i}\,{\sigma }^{j}+{\sigma }^{j}\,{\sigma }^{i}\right)=2\,{\delta }^{{ij}}$. Besides, the operator $\hat{\pi }$ is defined as $\begin{eqnarray}{\hat{\pi }}_{k}=-{\rm{i}}\,{D}_{k}-\displaystyle \frac{1}{2r}\,{\sigma }^{3}\,{\delta }_{\varphi k}+\mu {\left(\vec{\sigma }\times \vec{E}\right)}_{k},\end{eqnarray}$
where the second term of the right-hand side of equation (Observe that the third term of the right-hand side of equation (8 ) plays the role of an effective vector potential ${\vec{A}}_{\mathrm{AC}}=\left(\vec{\sigma }\times \vec{E}\right)$. It was introduced by Aharonov and Casher [57], where they showed that the wave function of a neutral particle with a permanent magnetic dipole moment can acquire a geometric quantum phase due to the interaction of the magnetic dipole moment with the radial electric field (4 ). The appearance of this phase shift in the wave function of the neutral particle has became known as the Aharonov-Casher effect [57]. Hence, the Aharonov-Casher geometric quantum phase is given by8 ), then, we have that the third term of the right-hand side plays the role of an effective vector potential, where we can write it as4 ) into the last term of equation (7 ), we have the effective scalar potential:11 ) plays the role of an attractive Coulomb-type potential when λ > 0. Hence, we shall deal with the attractive Coulomb-type potential from now on. Note that it stems from the Lorentz symmetry violation background. It is worth mentioning that the space-like vector $\vec{b}$ given in equation (4 ) can be considered as a special case of the tensor ${a}^{(5)\mu \alpha \beta }$ in the systematic classification of operator $-\tfrac{1}{2}{a}^{(5)\mu \alpha \beta }{F}_{\alpha \beta }\bar{\psi }{\gamma }_{\mu }\psi $ [75, 76], where they are related by ${a}^{(5)\mu \alpha \beta }=g\left({\eta }^{\mu \beta }{b}^{\alpha }-{\eta }^{\mu \alpha }{b}^{\beta }\right)$.
$\begin{eqnarray}{\phi }_{\mathrm{AC}}=\mu \oint \left(\vec{\sigma }\times \vec{E}\right)\cdot {\rm{d}}\vec{r}=\pm 2\pi \mu \lambda ,\end{eqnarray}$
where the ± signs correspond to the projections of the magnetic dipole moment on the z-axis. Returning to equation ( $\begin{eqnarray}\mu \,{\vec{A}}_{\mathrm{AC}}=\mu \left(\vec{\sigma }\times \vec{E}\right)=\pm \displaystyle \frac{{\phi }_{\mathrm{AC}}}{2\pi \,r}\,\hat{\varphi },\end{eqnarray}$
with $\hat{\varphi }$ as being a unit vector in the azimuthal direction. Furthermore, by substituting the fields ( $\begin{eqnarray}{V}_{\mathrm{eff}}=-\displaystyle \frac{g\,b\,\lambda }{r}.\end{eqnarray}$
Observe that the effective scalar potential (Henceforth, let us consider the neutral particle to be placed into the region where $r\ne 0$ and assume that the wave function of the neutral particle is well-behaved at the origin, then, the term $\vec{{\rm{\nabla }}}\cdot \vec{E}$ is null. By assuming that the magnetic dipole moment of the neutral particle is aligned in the z-direction, therefore, the Schrödinger-Pauli equation (7 ) becomes
$\begin{eqnarray}\begin{array}{rcl}{\rm{i}}\displaystyle \frac{\partial \psi }{\partial t} & = & -\displaystyle \frac{1}{2m}\left[\displaystyle \frac{{\partial }^{2}\psi }{\partial {r}^{2}}+\displaystyle \frac{1}{r}\displaystyle \frac{\partial \psi }{\partial r}+\displaystyle \frac{1}{{r}^{2}}\displaystyle \frac{{\partial }^{2}\psi }{\partial {\varphi }^{2}}+\displaystyle \frac{{\partial }^{2}\psi }{\partial {z}^{2}}\right]\\ & & +\displaystyle \frac{{\rm{i}}}{2m}\displaystyle \frac{{\sigma }^{3}}{{r}^{2}}\displaystyle \frac{\partial \psi }{\partial \varphi }-\displaystyle \frac{1}{2{{mr}}^{2}}\displaystyle \frac{{\phi }_{\mathrm{AC}}}{2\pi }\psi \\ & & -\displaystyle \frac{{\rm{i}}}{m}\,\displaystyle \frac{{\phi }_{\mathrm{AC}}}{2\pi }\displaystyle \frac{{\sigma }^{3}}{{r}^{2}}\,\displaystyle \frac{\partial \psi }{\partial \varphi }+\displaystyle \frac{1}{2{{mr}}^{2}}{\left(\displaystyle \frac{{\phi }_{\mathrm{AC}}}{2\pi }\right)}^{2}\,\psi \\ & & +\displaystyle \frac{1}{8{{mr}}^{2}}\,\psi -\displaystyle \frac{g\,b\,\lambda }{r}\,\psi .\end{array}\end{eqnarray}$
In search of the bound state solutions to the Schrödinger-Pauli equation (12 ), we should observe that ${\sigma }^{3}\psi =\pm \psi =s\psi $ (s = ±1), which means that ψ is an eigenfunction of σ3. Besides, the operators ${\hat{J}}_{z}=-{\rm{i}}{\partial }_{\varphi }$ [73], ${\hat{p}}_{z}=-{\rm{i}}{\partial }_{z}$ and the Hamiltonian operator $\hat{H}$ (given in the right-hand side of equation (12 )) satisfy the relations:12 ) in terms of the eigenfunctions of ${\hat{J}}_{z}=-{\rm{i}}{\partial }_{\varphi }$ and ${\hat{p}}_{z}=-{\rm{i}}{\partial }_{\varphi }$ as follows:14 ) into the Schrödinger-Pauli equation (12 ), and thus, we obtain the radial equation:
$\begin{eqnarray}\left[\hat{H},{\hat{J}}_{z}\right]=0;\,\left[\hat{H},{\hat{p}}_{z}\right]=0.\end{eqnarray}$
Therefore, we can write the solution to the Schrödinger-Pauli equation ( $\begin{eqnarray}\psi \left(t,r,\varphi ,z\right)={{\rm{e}}}^{-{\rm{i}}{ \mathcal E }t}\,{{\rm{e}}}^{{\rm{i}}\left(l+\tfrac{1}{2}\right)\varphi }\,{{\rm{e}}}^{{{ip}}_{z}z}\,G\left(r\right),\end{eqnarray}$
where $l=0,\pm 1,\pm 2,\ldots $ and pz is a constant. Let us consider pz = 0 hereafter. In this way, we simplify our discussion by dealing with the system in the plane z = 0. Next, let us substitute equation ( $\begin{eqnarray}\left[\displaystyle \frac{{{\rm{d}}}^{2}}{{{\rm{d}}{r}}^{2}}+\displaystyle \frac{1}{r}\displaystyle \frac{{\rm{d}}}{{\rm{d}}{r}}-\displaystyle \frac{{\nu }^{2}}{{r}^{2}}+\displaystyle \frac{2{mg}\,b\,\lambda }{r}+2m{ \mathcal E }\right]G=0,\end{eqnarray}$
where we have defined the parameter ν as $\begin{eqnarray}\nu =l+\displaystyle \frac{1}{2}\left(1-s\right)+s\displaystyle \frac{{\phi }_{\mathrm{AC}}}{2\pi }.\end{eqnarray}$
The bound state solutions associated with the effective attractive potential (11 ) can be achieved by considering ${ \mathcal E }=-\left|{ \mathcal E }\right|$. Then, let us define the parameter $\zeta =\sqrt{-2m{ \mathcal E }}$, and thus, we can write equation (15 ) in the form:17 ) becomes
$\begin{eqnarray}\left[\displaystyle \frac{{{\rm{d}}}^{2}}{{{\rm{d}}{r}}^{2}}+\displaystyle \frac{1}{r}\displaystyle \frac{{\rm{d}}}{{\rm{d}}{r}}-\displaystyle \frac{{\nu }^{2}}{{r}^{2}}+\displaystyle \frac{2{mg}\,b\,\lambda }{r}-{\zeta }^{2}\right]G=0.\end{eqnarray}$
We can go further by defining the dimensionless parameter: x = 2ζ r; and thus, equation ( $\begin{eqnarray}\left[\displaystyle \frac{{{\rm{d}}}^{2}}{{{\rm{d}}{x}}^{2}}+\displaystyle \frac{1}{x}\displaystyle \frac{{\rm{d}}}{{\rm{d}}{x}}-\displaystyle \frac{{\nu }^{2}}{{x}^{2}}+\displaystyle \frac{{mg}\,b\,\lambda }{\zeta \,x}-\displaystyle \frac{1}{4}\right]G=0.\end{eqnarray}$
By imposing that $G\left(x\right)\to 0$ when $x\to \infty $ and $x\to 0$. Thereby, the function $G\left(x\right)$ can be given in the form:
$\begin{eqnarray}G\left(x\right)={x}^{\left|\nu \right|}\,{{\rm{e}}}^{-x/2}{}_{1}{F}_{1}\left(-\displaystyle \frac{{mg}\,b\,\lambda }{\zeta }+\left|\nu \right|+\displaystyle \frac{1}{2},2\left|\nu \right|+1;x\right).\end{eqnarray}$
The function ${}_{1}{F}_{1}\left(-\tfrac{{mg}\,b\,\lambda }{\zeta }+\left|\nu \right|+\tfrac{1}{2},2\left|\nu \right|+1;x\right)$ is the confluent hypergeometric function [74, 77]. The asymptotic behaviour of a confluent hypergeometric function for large values of its argument is given by [77] $\begin{eqnarray}{}_{1}{F}_{1}\left(a,b;x\right)\approx \displaystyle \frac{{\rm{\Gamma }}\left(b\right)}{{\rm{\Gamma }}\left(a\right)}\,{{\rm{e}}}^{x}\,{x}^{a-b}\left[1+{ \mathcal O }\left({\left|x\right|}^{-1}\right)\right].\end{eqnarray}$
Hence, it diverges when $x\to \infty $. With the purpose of having $G\left(x\right)\to 0$ when $x\to \infty $, thus, we must impose the condition: $a=-n$ (n = 0, 1, 2, 3, …). With $a=-\tfrac{{mg}\,b\,\lambda }{\zeta }\,+\tfrac{\left|\nu \right|}{2}+\tfrac{1}{2}$, the condition $a=-n$ yields the energy levels: $\begin{eqnarray}{{ \mathcal E }}_{nl,s}=-\displaystyle \frac{m{\left(gb\lambda \right)}^{2}}{2{\left[n+\left|l+\tfrac{1}{2}\left(1-s\right)+s\tfrac{{\phi }_{\mathrm{AC}}}{2\pi }\right|+\tfrac{1}{2}\right]}^{2}}.\end{eqnarray}$
Observe that n = 0, 1, 2, 3, …is the radial modes quantum number.Hence, the energy levels (21 ) are achieved when Aharonov-Casher system [57] is subject to the Coulomb-type potential induced by Lorentz symmetry breaking effects. We can see that these energy levels depend on the Aharonov-Casher geometric quantum phase φAC. This dependence of the geometric quantum phase yields a quantum effect analogous to the Aharonov-Casher effect for bound states [78, 79], because there are no classical forces that act on the neutral particle. In this sense, it can be considered to be a geometrical effect due to the local nature of the Aharonov-Casher effect [57]. Moreover, with ${\phi }_{0}=\pm 2\pi $, by taking ${\phi }_{\mathrm{AC}}\to {\phi }_{\mathrm{AC}}+{\phi }_{0}$, we have21 ) have a periodicity ${\phi }_{0}=\pm 2\pi $ [78].
$\begin{eqnarray}{{ \mathcal E }}_{nl,s}\left({\phi }_{\mathrm{AC}}+{\phi }_{0}\right)={{ \mathcal E }}_{nl+1,s}\left({\phi }_{\mathrm{AC}}\right).\end{eqnarray}$
This means that the energy levels (Another quantum effect related to the dependence of the energy levels on the Aharonov-Casher geometric quantum phase φAC is the arising of the persistent spin currents in the system. By following [58-60], the persistent spin currents associated with confinement of the neutral particle to the Coulomb-type potential induced by Lorentz symmetry breaking effects are given by
$\begin{eqnarray}{ \mathcal I }=-\displaystyle \sum _{n,l}\displaystyle \frac{\partial {{ \mathcal E }}_{n,l,s}}{\partial {\phi }_{\mathrm{AC}}}=-\displaystyle \frac{s}{2\pi }\displaystyle \sum _{n\,l}\displaystyle \frac{\nu }{\left|\nu \right|}\times \displaystyle \frac{m{\left(gb\lambda \right)}^{2}}{{\left[n+\left|\nu \right|+\tfrac{1}{2}\right]}^{3}}.\end{eqnarray}$
Finally, we can observe that the persistent spin currents (23 ) are a periodic function of the Aharonov-Casher geometric quantum phase φAC.
3. Conclusions
We have analysed the interaction of the permanent magnetic dipole moment of a neutral particle with a radial electric field in a background of the Lorentz symmetry violation. We have seen that the interaction of the radial electric field with the permanent magnetic dipole moment gives rise to the Aharonov-Casher geometric quantum phase [57]. In addition, we have shown that the effects of the violation of Lorentz symmetry determined by a space-like vector field and the radial electric field yield an attractive Coulomb-type potential.
Hence, when the neutral particle is confined to the attractive Coulomb-type potential, we have shown that bound state solutions to the Schrödinger-Pauli equation can be achieved. In addition, the spectrum of energy depends on the Aharonov-Casher geometric quantum phase φAC. Due to this dependence on φAC, we have an analogue of the Aharonov-Casher effect for bound states [78, 79], since there are no classical forces that act on the neutral particle. Finally, we have discussed the arising of persistent spin currents in the system, which is a quantum effect that also stems from the dependence of the energy levels on the Aharonov-Casher geometric quantum phase φAC.