# Localization of five-dimensional Elko spinors with non-minimal coupling on thick branes

## Abstract

It has been found that the zero mode of a five-dimensional Elko spinor could be localized on branes by introducing a Yukawa-type coupling between the Elko spinor and the background scalar field or the Ricci scalar. However, the Yukawa-type coupling is not appropriate for all brane models. In this paper, we explore other localization mechanism for the Elko spinor by introducing the non-minimal coupling \(f(\phi )\mathfrak {L}_{Elko}\) between the five-dimensional Elko spinor and the background scalar field. We give the general expressions of the Elko zero mode and the function \(f(\phi )\). Through two thick brane models and three concrete examples, we show that the Elko zero mode can be localized on the branes by this new mechanism. This provides us more possibilities of localizing the Elko zero mode.

## 1 Introduction

Brane-world models have attracted more and more interests since they were brought up [1, 2, 3]. They not only give a novel viewpoint of our world, but also open a new way to interpret many outstanding issues that the Standard Model (SM) can not interpret sufficiently, such as the hierarchy problem [1, 2, 4, 5, 6, 7], cosmological constant problem [8, 9, 10, 11, 12, 13, 14, 15], the nature of dark matter and dark energy [16, 17, 18, 19] and so on. Most of the early brane world models like the famous Randall-Sandrum (RS) branes were thin ones [1, 2, 3] neglecting the thicknesses of branes. Subsequently, some more realistic branes with thickness (i.e., thick brane models) were proposed, whose energy densities have a distribution along the extra dimension. Usually, a thick brane can be generated dynamically by bulk matter fields [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], or by pure gravity [39, 40, 41]. More details about various kinds of thick branes could be found in Refs. [28, 42].

In brane world scenarios, one of the most important and interesting questions is to investigate the localization mechanism of various matter fields in a higher-dimensional spacetime on a 3+1 brane. The investigation of the Kaluza–Klein (KK) modes of various bulk matter fields can give us the way to probe the extra dimensions by considering their coupling with particles on the brane and the corrections to Newton’s law and Coulomb’s law [3, 43, 44, 45, 46]. In general, the zero modes of all five-dimensional matter fields, which correspond to the four-dimensional massless particles on a brane, must be localized on the brane to make sure the four-dimensional SM can be rebuilt on the brane at least at low energy. There were many works that studied the localization of various matter fields on branes [46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66].

The localization of fields beyond SM have also been investigated. Elko spinor field, which was proposed by Ahluwalia and Grumiller [67, 68] in 2005, is a new quantum field with spin-1/2 beyond the four-dimensional SM and is the eigenspinor of the charge conjugation operator. However, Elko spinor field has some different properties compared with Dirac spinor: (1) it satisfies the Klein–Gordon (KG) equation rather than the Dirac equation, (2) for the five-dimensional Elko spinor, its mass dimension is one, instead of 3/2 that is suit for Dirac spinor. Hence the interactions between Elko spinor and gauge fields in SM will be suppressed strongly (at least one order of Plank scale), which means that Elko spinor only interacts with itself, graviton and Higgs fields [67, 68, 69, 70, 71, 72, 73, 74, 75]. Elko spinor has been considered as a first-principle candidate of dark matter [67, 68, 69, 70, 71] and has also been investigated extensively in cosmology [76, 77, 78, 79, 80, 81, 82, 83, 84, 85] and mathematical physics [86, 87, 88, 89, 90, 91, 92].

The localization of Elko spinor in five-dimensional brane world models has been investigated in Refs. [93, 94, 95]. It was found that the zero mode of Elko spinor can be localized on branes thought a Yukawa-type coupling with the background scalar field [93, 95], or the Ricci scalar [95]. However, the Yukawa-type coupling is only appropriate for some kinds of thick branes, and the coupling constant must be taken as some particular expression determined by the parameters in the brane models. This motivates us to consider the other kinds of couplings to release the limits of coupling constants and brane models. In Ref. [42], a general coupling between the five-dimensional Dirac fermion and the background scalar fields is introduced, which includes the non-minimal coupling between the Dirac fermion and the scalar fields. Inspired by this, in this paper, we will introduce this new type of coupling, i.e., the non-minimal coupling \(f(\phi )\mathfrak {L}_{Elko}\) between the five-dimensional Elko spinor and the background scalar field, where \(f(\phi )\) is a function of the backgound scalar field and \(\mathfrak {L}_{Elko}\) is the Lagrangian of the five-dimensional Elko spinor. We will study the localization of the Elko spinor with this new coupling on the Minkowski thick branes which have been considered in Yukawa-type coupling case [93] and compare the characteristics between the Yukawa-type coupling and this new coupling. We will show that localized Elko zero mode on the branes can be realized for different forms of \(f(\phi )\), which means that this new coupling can provide us more possibilities of localizing the Elko zero mode. By the way, the localization of the Elko spinor in a six-dimensional string-like model was considered in Ref. [96], recently.

The organization of the paper is listed as follows. In Sect. 2, we give a briefly review of Elko spinor field. In Sect. 2, we consider the localization of a five-dimensional massless Elko spinor with a non-minimal coupling to the background scalar field and give the equation of the Elko zero mode. Then in Sect. 3, we discuss the localization of the Elko zero mode on various thick branes. Finally, the conclusion is given in Sect. 4.

## 2 Review of Elko spinor field

*C*is defined as

*K*is the complex conjugation operator, and \(\Theta \) is the spin one half Wigner time reversal operator which satisfies \(\Theta (\mathbf {\sigma }/2)\Theta ^{-1}=-(\mathbf {\sigma }/2)^{*}\). Elko spinors are eigenspinors of the charge conjugation operator: \(C\lambda (k^\mu )=\pm \lambda (k^\mu )\) with \(k^\mu \) a polarization vectorx. The plus sign generates the self-conjugate spinors which are denoted by \(\varsigma (k^\mu )\) and the minus sign generates the anti-self-conjugate spinors which are denoted by \(\tau (k^\mu )\). Both of the two kinds of spinors have two possible helicities \(\chi _{\pm }(k^\mu )\), which can be expressed as

*E*, \(p_x\), \(p_y\), \(p_z\)). Here the \(\Gamma \) is given by [71]

## 3 Localization of five-dimensional Elko spinor with non-minimal coupling

*R*has been considered in Refs. [93, 95]. However, the Yukawa-type coupling is not appropriate for all brane models and the expression of the coupling constant will be determined exactly by the parameters in the models. Therefore, we will focus on the non-minimal coupling between a five-dimensional Elko spinor and the background scalar field in this section. The five-dimensional line-element is assumed as

*B*(

*z*) is arbitrary, which means that we can choice an appropriate

*B*(

*z*) to eliminate the term \(A'^2\alpha _0\). To this end, we introduce two new functions

*C*(

*z*) and

*D*(

*z*) that satisfy the following equations

*C*(

*z*) and

*B*(

*z*)

*C*(

*z*) is required to be an odd one. We introduce

*C*(

*z*) can be expressed by the function

*K*(

*z*) according to Eq. (41): \(C(z)=\frac{\text {e}^{\int ^z_1 K(\bar{z})d\bar{z}}}{\text {constant}-\int _1^z\text {e}^{\int ^{\hat{z}}_1 K(\bar{z})d\bar{z}}d\hat{z}}\). Thus Eqs. (42) and (43) give the general expressions of the zero mode \(\alpha _0\) and the function \(f(\phi )\). For a given

*K*(

*z*) the zero mode \(\alpha _0\) is obtained by integrating the Eq. (42). Then, the scalar field function \(f(\phi (z))\) is determined by integrating the Eq. (43). We will find that the role of

*K*(

*z*) is similar to the auxiliary superpotential \(W(\phi )\), which is introduced in order to solve the Einstein equations in thick brane models. As the superpotential \(W(\phi )\) does in thick brane models, different

*K*(

*z*) could lead to different solutions. To illustrate this, next we will review two kinds of thick brane solutions and obtain the zero mode by considering different forms of

*K*(

*z*), for example.

## 4 Localization of the Elko zero mode with non-minimal coupling on thick branes

*K*(

*z*) is similar to the superpotential \(W(\phi )\). By choosing a simple superpotential [28]

*b*and

*c*are parameters related to the brane thickness. The potential approaches the negative values \(V(\pm \infty )=-6c^2\). Thus the bulk is asymptotically AdS. For simplicity, we define the parameters \(\frac{1}{3b^2}=\bar{b}\), \(cb^2=a\). Then, the brane solution becomes

*R*approaches a negative value when \(y\rightarrow \infty \). It means that the bulk is asymptotically AdS.

*a*is an arbitrary constant parameter and

*b*a positive real constant. For the first brane model it means \(\bar{b}=b\) according to Eq. (52) and for the second one it means \(\gamma =b\) according to Eq. (55). It is easy to check that it is the solution to the Einstein Eqs. (46) and (54) by bringing the solution back to Eqs. (46) and (54). With the coordinate transformation Eq. (13), we can get the conformal coordinate

*F*is the hypergeometric function

*z*. As what we have discussed in the previous section, the role of

*K*(

*z*) is similar to the superpotential \(W(\phi )\) in thick brane models. Thus for different forms of

*K*(

*z*), there exist different zero mode solutions and configurations of \(f(\phi )\). For example, next we will focus on three kinds of

*K*(

*z*) and investigate the localization of the zero mode Eq. (42).

### 4.1 \(K(z)=k\frac{C'}{C}\)

### 4.2 \(K(z)=k z\)

*C*(

*z*) reads

*z*. So we have \(\alpha _0(z)\varpropto {\exp }\left[ -\frac{k}{4}z^{2}\right] \) as \(|z| \rightarrow \infty \). The zero mode Eq. (69) is plotted in Fig. 3 with \(a=k=1\).

*K*(

*z*). At the same time, it is worth pointing out that the parameter

*k*is related with the coupling constant, and its range is larger than that in the previous case in this paper. For this case, the zero mode can be localized on branes for any positive

*k*.

### 4.3 \(K(z)=k\tanh (kz)\)

*C*(

*z*) can be solved as

*k*. The function \(f(\phi )\) for the two brane models are

## 5 Conclusion and discussion

In this paper, we reconsidered the localization of a five-dimensional Elko spinor on thick branes. In our previous work [93], the Yukawa-type coupling between the Elko spinor and the background scalar field has been introduced to realize the localization of the Elko zero mode. It was shown that the localized Elko zero mode can just be obtained for some kinds of thick branes, only if the coupling constant is taken as some particular expression determined by the parameters in the brane models. Here, we introduced new localization mechanism, i.e., the non-minimal coupling \(f(\phi )\mathfrak {L}_{Elko}\) between the five-dimensional Elko spinor and the background scalar field. The Schrödinger like equation for the Elko zero mode was derived. The result shows that the effective potential depends on the form of \(f(\phi )\) and the warp factor \(\text {e}^{2A}\). For convenience, we introduced a series of new functions, especially the function *K*(*z*), to obtain the Elko zero mode. We gave the general expressions of the zero mode \(\alpha _0\) and the function \(f(\phi )\) by using *K*(*z*). It was found that the role of *K*(*z*) is similar to the superpotential \(W(\phi )\) in thick brane models. The zero mode \(\alpha _0\) and the scalar field function \(f(\phi (z))\) can be obtained by a given *K*(*z*). Thus an appropriate form of *K*(*z*) is needed to confine the Elko zero mode on branes and different forms of *K*(*z*) will lead to different zero mode solutions and configurations of \(f(\phi )\).

Next we reviewed two kinds of Minkowski thick brane models. One is for a standard canonical scalar field and the other is for a non-minimally coupled scalar field. In Ref. [93] we had investigated the localization of the Elko spinor with Yukawa-type coupling on these thick branes. For these branes, the warp factor has the same form while the solutions of the background scalar field are different. In order to illustrate the effect of *K*(*z*) on the zero mode \(\alpha _0\) and the function \(f(\phi )\), we took three different forms of *K*(*z*) as examples. Firstly, we focused on the case of \(K(z)=k\frac{C'}{C}\) with \(k\ge 0\). Here the parameter *k* is related with the coupling constant. It was found that the Elko zero mode can be localized on the branes with the limit of \(0<k<1\). It is very interesting that a split at the point \(z=0\) divides the Elko zero mode into two halves, which comes from the absolute value in the form of the zero mode as well as the function \(f(\phi )\). The forms of \(f(\phi )\) are very complex and different for the two different thick brane models. It is clear that there will exist different non-minimal couplings for different brane models, even though the zero modes have the same form.

Secondly, we took \(K(z)=kz\) with \(k>0\). In this case, the zero mode is so complex that it is hard to be written as an explicit function. Fortunately, the complex part of the zero mode just acts an unimportant role and the zero mode has the same behaviors as \({\exp }\left( -\frac{k}{4}z^{2}\right) \) at the boundaries of the extra dimension. Thus the zero mode can also be localized in this case. The absolute value in the zero mode and the function \(f(\phi )\) disappears and the range of the parameter *k* is larger than that in the first case. The result in this case shows that different choices of *K*(*z*) will lead to different configurations of solutions.

Finally, we adopted \(K(z)=k\tanh (kz)\) with \(k>0\). In this case, the zero mode also has a complex form and can not be written as an elementary function. As the second case, the complex part does not affect the asymptotic behavior of the Elko zero mode and the zero mode has the same property of \(\text {sech}^{\frac{1}{2}}(kz)\), i.e., \(\alpha _0(z) \propto \text {sech}^{\frac{1}{2}}(kz)\) at the boundaries of the extra dimension. Such zero mode can be localized on both thick branes with any positive *k*. The forms of the function \(f(\phi )\) are still very complex and they are inequable for different kinds of thick branes.

In this paper, we considered two kinds of Minkowski thick branes and obtained three different expressions of bounded Elko zero modes. The result shows that the non-minimal coupling can provide us more possibilities of localizing the Elko zero mode. In addition, it may give us new way to solve the localization problem of Elko massive KK modes, which will be investigated in our further work.

## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11522541, 11705106, and 11305095), and the Fundamental Research Funds for the Central Universities (Grant No. lzujbky-2016-k04).

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