Localization of Spinor Fields in Higher-Dimensional Braneworlds

If the existence of extra dimensions is assumed, how can a concise, self-consistent, and unified theory consistent with the current theoretical framework be constructed?

The idea of extra dimensions originated from attempts to unify different forces in nature. For example, in the Kaluza-Klein (KK) theory Kaluza:1921tu ; Klein:1926tv , the $4$-dimensional gravity and electromagnetism are described by a $5$-dimensional metric. This led to the idea of explaining fundamental interactions in terms of spacetime geometry. However, when the extra dimension is compactified into a loop, the reduction of the $5$-dimensional Fermion field does not result in a chiral theory in $4$ dimensions, which conflicts with the experimental observation that only left-handed neutrinos exist. Therefore, it is necessary to modify the KK theory in order to produce chiral fermions. The most common solution is to use orbifold compactification Cheng:2010pt . For models with infinitely large extra dimensions or thick branes, an appropriate localization mechanism should also be used to obtain a $4$-dimensional chiral theory.

In theories of extra dimensions, it is possible that extra dimensions are non-compact and infinitely large, e.g., there is an infinitely large extra dimension in the 5-dimensional Randall-Sundrum (RS) braneworld model Randall:1999vf . However, in this case, to ensure consistency with 4-dimensional observations, a localization mechanism must exist to confine matter fields within a specific spatial range, namely the brane in which we live. For such a model to be realistic, this mechanism could take forms such as a domain wall Rubakov:1983bb ; Akama:1982jy .

It is known that it is not possible to simultaneously localize free scalar and vector fields in a $5$-dimensional RS-like braneworld model. The most natural approach is the minimal coupling between matter field and gravity by introducing additional compact dimension to allow for the localization of the $U(1)$ gauge field Freitas:2018iil ; Wan:2020smy . This prompts us to consider $6$ or higher-dimensional spacetime. However, the localization of fermions in braneworld models with spacetime dimension greater than $5$ remains an area requiring systematic investigation.

In Ref. Budinich:2001nh , Budinich studied the relationship between high-dimensional pure spinor and $4$-dimensional spinor using Clifford algebra in flat spacetime. This work provides some insight into how fermions can be described in higher-dimensional spacetime. However, the spacetime with a braneworld is not flat. This means that, in addition to studying the algebraic structure of spinors, we also need to consider the generalized Dirac equation in higher-dimensional curved spacetime in order to construct a consistent theory of fermion dynamics.

In order to obtain $4$-dimensional free massless fermions, the KK zero mode must be localized on the brane. Reference Randjbar-Daemi:2000lem proved that a $5$-dimensional massless Dirac fermion is generally non-normalizable with the use of minimal coupling of gravity and gauge fields under quite general assumptions about the geometry and topology of the internal manifold. The localization mechanism has been widely studied in the context of $5$-dimensional braneworld models Ringeval:2001cq ; Melfo:2006hh ; Flachi:2009uq ; Chumbes:2010xg ; Castillo-Felisola:2012fpz ; Liu:2013kxz ; Guo:2014nja ; Li:2017dkw ; MoazzenSorkhi:2018lvb .

To ensure the localization of the spinor zero mode, in a $5$-dimensional braneworld model, the introduction of the Yukawa coupling allows for the localization of the fermion field on the brane. Matter, gauge, and Higgs fields are allowed to propagate in the bulk, and the Standard Model fields and their interactions can be reproduced by the corresponding zero KK modes Dubovsky:2000am ; Smolyakov:2015zsa . By including a Yukawa-type coupling to a scalar field of a domain-wall type, we can ensure the chirality as well as the localization of the fermions. However, these mechanisms do not work well in higher-dimensional braneworld models. In higher dimensions, if one carefully considers each component of a higher-dimensional spinor field, the localization of fermions presents a general difficulty. The left and right chiral massless KK modes are decoupled, but the massive KK modes are not. The coupling between the higher-dimensional left and right chiralities cannot be avoided as long as a mass term or a Yukawa coupling term is present; the interaction between $4$-dimensional fermions results from this coupling. This means that a mass term for higher-dimensional fermions prevents one from obtaining a $4$-dimensional free fermion. However, we hope that a reasonable effective theory should include a description of the free field, that is, a description of the propagation process of free particles.

The aim of this paper is to provide general discussions applicable to higher-dimensional spacetime, rather than constructing a specific model. We hope to construct a localization mechanism that meets the following physical requirements:

• •

  (a) The high-dimensional spinor field and its interactions satisfy the Lorentz invariance.

• •

  (b) Localization of the spinor zero mode, resulting in a low-dimensional effective theory.

• •

  (c) In the effective theory, a description of an effective low-dimensional free field is required.

We will study whether fermions can be naturally localized and obtain $4$-dimensional free massless fermions by introducing a coupling term into the higher-dimensional fundamental theory and decouple the left and right chiralities of higher-dimensional spinors. We require that this decoupling should not break the $SO(n,1)$ symmetry. We hope to obtain a $4$-dimensional chiral theory through a reasonable localization mechanism. Therefore, we take the background of different topologies in the $6$-dimensional braneworld as an example to study the distinction in fermion chirality.

This paper is organized as follows. In section 2, we introduce a general localization mechanism, study the kinetic equations of high-dimensional spinors, and present the results. In section 3, we take $6$-dimensional spacetime as an example. Under some specific coupling forms, we study the localization conditions of the fermion zero modes. Differences in the chirality of fermions under different spacetime topological backgrounds are discussed. The last section 4 is devoted to conclusions and outlook. In the appendix, we will provide the mathematical foundations of Clifford algebra and high-dimensional spinors that are needed for this paper, for reference.

To achieve a localization mechanism, one might consider the coupling between the spinor and the background dynamic field. Such a coupling can greatly facilitate localization, particularly in highly symmetric spacetimes, where the interaction term ensures the material field distribution is consistent with the symmetry of the brane.

To ensure the invariance of the Lagrangian under a Lorentz transformation, a spinor and its adjoint should be combined to form the Lorentz scalar $\bar{\Psi}\Psi$ or the Lorentz vector $\bar{\Psi}\Gamma^{A}\Psi$, as well as other combinations. First, it is natural to consider the following action for the spinor in a $(2n+2)$-dimensional spacetime with a Yukawa-like coupling Fu:2011pu ; Castillo-Felisola:2012fpz ; Barbosa-Cendejas:2015qaa ; Guo:2014nja ; Flachi:2009uq ; Salvio:2007qx ; Chumbes:2010xg ; Slatyer:2006un ; Kodama:2008xm ; Ringeval:2001cq ; Melfo:2006hh . We introduce

$\displaystyle{S_{1}}=\int{d}^{2n+2}{x}~{}\sqrt{-{g}}\left[\bar{\Psi}\Gamma^{M}{D}_{{M}}\Psi+\eta F(\phi,R,R^{\mu\nu}R_{\mu\nu},\cdots)\bar{\Psi}\Psi\right],$

(1)

where $F(\phi,R,R^{\mu\nu}R_{\mu\nu},\cdots)$ represents a scalar function dependent on the scalar field $\phi$, the curvature scalar $R$, and other geometric scalars.

The introduction of the mass term or Yukawa coupling couples the left- and right-handed chiralities of the fermion. The left- and right-handed chiralities of the fermion transform as independent entities under the Lorentz transformation (155). This prohibits decoupling the left- and right-handed chiralities through coordinate selection. In order to obtain a $4$-dimensional effective free field, we aim to find the $4$ components that have independent equations of motion from other components of the high-dimensional spinor. The Yukawa term appears to introduce a contradiction between high-dimensional Lorentz symmetry and $4$-dimensional free field theory.

Given a coupling between left and right chiral spinors in a $(2n+2)$-dimensional fundamental theory, a $2n$-dimensional free field cannot be derived through action reduction from either the left or right chiral spinors. One potential solution involves inserting an odd number of Gamma matrices between $\bar{\Psi}$ and $\Psi$, as exemplified by

$\displaystyle\bar{\Psi}\Gamma_{A_{1}}\Gamma_{A_{2}}\cdots\Gamma_{A_{2k+1}}\Psi,$

(2)

where the indices are fixed as $A_{1},A_{2},\cdots,A_{2k+1}$, such as $\bar{\Psi}\Gamma_{2n}\Psi$ or $\bar{\Psi}\Gamma_{0}\Gamma_{1}\Gamma_{2}\Psi$. The coupling forms (2) do not possess Lorentz invariance. A spinor and its adjoint must combine to form a Lorentz scalar. Next, we introduce the form of the interaction

$$\bar{\Psi}\Gamma^{M}\Gamma^{N}\Gamma^{P}\cdots T_{MNP\cdots}\Psi$$

to resolve this contradiction.

A possible solution involves imposing the type of coupling $\bar{\Psi}\Gamma^{M}\xi_{M}\Psi$, where $\xi_{M}$ is a vector. We can also extend the coupling to an odd-order tensor field, for example, $\bar{\Psi}\Gamma^{M}\Gamma^{N}\Gamma^{P}T_{MNP}\Psi$. Based on the preceding analysis, we focus on coupling with a vector field $\xi_{M}$ as opposed to a scalar function $F$. The action takes the form

$\displaystyle{S_{2}}=\int{d}^{2n+2}{x}~{}\sqrt{-{g}}\left[\bar{\Psi}\Gamma^{M}{D}_{{M}}\Psi\right.+\left.\varepsilon\bar{\Psi}\Gamma^{M}\xi_{M}\Psi\right].$

(3)

The Dirac equation is given by

$\displaystyle\left[\partial_{M}+\Omega_{M}+\varepsilon\xi_{M}\right]\Gamma^{M}\Psi\left(x^{N}\right)=0.$

(4)

Consider the ’Weyl’ representation of the Gamma matrices and note that $\Gamma_{A}\Gamma_{B}$ are block diagonal matrices. We have

$\displaystyle\Omega_{M}=\frac{1}{4}\Omega_{M}^{AB}\Gamma_{A}\Gamma_{B}=\left(\begin{array}[]{cc}\omega_{1M}&0\\ 0&\omega_{2M}\end{array}\right),$

(7)

where the spin connection $\Omega_{M}^{AB}$ is defined in Eq. (152). From Eq. (46), we find that the left and right chirality fermions satisfy independent equations of motion with the coupling term $\bar{\Psi}\Gamma^{M}\xi_{M}\Psi$.

First, we consider the general form of the diagonal metric to obtain mathematically general results. For such a metric, the line element $ds^{2}$ is expressed as:

$\displaystyle ds^{2}=-a_{0}^{2}(x^{N})dx_{0}^{2}+\sum_{k=1}^{2n+1}a_{k}^{2}(x^{N})dx_{k}^{2},$

(8)

where $a_{0}(x^{N})$ and $a_{k}(x^{N})$ are the warp factors for all spacetime coordinates. The spin connection $\Omega_{j}$ is given by

$\displaystyle\Omega_{j}=\frac{1}{2}\Gamma_{j}\sum_{i=0,i\neq j}^{2n+1}\frac{\Gamma^{i}\partial_{i}a_{j}(x^{N})}{a_{i}(x^{N})}.$

(9)

Furthermore, we have

$\displaystyle\Gamma^{M}\Omega_{M}=\frac{1}{2}\sum_{j=0}^{2n+1}\sum_{i=0,i\neq j}^{2n+1}\frac{\Gamma^{i}\partial_{i}a_{j}(x^{N})}{a_{i}(x^{N})a_{j}(x^{N})}.$

(10)

A vector field $G_{M}$ can be defined, its components $G_{i}$ are given by

$\displaystyle G_{i}=\frac{1}{2}\sum_{j=0,i\neq j}^{2n+1}\frac{\partial_{i}a_{j}(x^{N})}{a_{j}(x^{N})}.$

(11)

Although $G_{M}$ is not the spin connection and its components $G_{i}$ are functions depending on the choice of coordinates, $G_{M}$ serves as an alternative to the spin connection in the Dirac equation within the coordinate framework established by Eq. (8). Therefore, we have

$\displaystyle\Gamma^{M}\Omega_{M}=\Gamma^{A}E_{A}^{M}G_{M};$

(12)

subsequently, the Dirac equation takes the form

$$\left[\Gamma^{{A}}(E_{A}^{M}\partial_{M}+E_{A}^{M}G_{M})+\varepsilon\Gamma^{M}\xi_{M}\right]\Psi\left(\xi_{N}\right)=0.$$

(13)

As discussed in the previous section, $\bar{\Psi}\Gamma^{M}\xi_{M}\Psi$ does not couple the left and right chiralities. This decoupling enables the existence of two independent $4$-dimensional Dirac free fields in the $4$-dimensional effective theory. Utilizing the Weyl representation, we can decompose Eq. (13) into two independent components; these correspond to the left chiral spinor $\Psi_{L}$ and the right chiral spinor $\Psi_{R}$, as given by

	$\displaystyle\hat{D}^{(2\iota)}_{L}=P_{R}\Gamma^{{A}}E_{A}^{M}(\partial_{M}+\varepsilon\xi_{M}+G_{M}),$		(14)
	$\displaystyle\hat{D}^{(2\iota)}_{R}=P_{L}\Gamma^{{A}}E_{A}^{M}(\partial_{M}+\varepsilon\xi_{M}+G_{M}),$		(15)

where $\hat{D}_{L}$ and $\hat{D}_{R}$ are operators that act on $\Psi_{L}$ and $\Psi_{R}$, respectively, as indicated in Eq. (13). Employing these operators, we rewrite the equation as

	$\displaystyle\hat{D}^{(2\iota)}_{L}\Psi_{L}^{(2\iota)}=\hat{D}^{(2\iota)}_{L}\left(\begin{array}[]{l}\Psi_{1}^{(\iota)}\\ 0\end{array}\right)=0,$		(18)
	$\displaystyle\hat{D}^{(2\iota)}_{R}\Psi_{R}^{(2\iota)}=\hat{D}^{(2\iota)}_{R}\left(\begin{array}[]{l}0\\ \Psi_{2}^{(\iota)}\end{array}\right)=0.$		(21)

It is noteworthy that the matrix operators $P_{R}\Gamma^{{A}}$ and $P_{L}\Gamma^{{A}}$ are degenerate. As a result, the dimension can transition from $2\iota$ to $\iota$. We refer to the degenerate operators as $\hat{D}^{(\iota)}_{L}$ and $\hat{D}^{(\iota)}_{R}$, which operate solely on the two independent $2^{n}$-component spinors ${\Psi}^{(\iota)}_{1}$ and ${\Psi}^{(\iota)}_{2}$, respectively.

The action corresponding to this decomposition can be further split into its right and left chiral components as follows:

$\displaystyle{S}=S_{L}+S_{R},$

(22)

where $S_{L}$ and $S_{R}$ are defined by

$\displaystyle{S_{L}}=\int{d}^{2n}{x}\sqrt{-{g}}\left[\bar{\Psi}_{1}^{(\iota)}{\hat{D}}^{(\iota)}_{{L}}{\Psi}^{(\iota)}_{1}\right],$

(23)

and

$\displaystyle{S_{R}}=\int{d}^{2n}{x}\sqrt{-{g}}\left[\bar{\Psi}_{2}^{(\iota)}{\hat{D}}^{(\iota)}_{{R}}{\Psi}^{(\iota)}_{2}\right].$

(24)

We start by defining the operators

	$\displaystyle\tilde{D}^{(2\iota)}_{L}=P_{R}\Gamma^{{A}}E_{A}^{M}\partial_{M},$		(25)
	$\displaystyle\tilde{D}^{(2\iota)}_{R}=P_{L}\Gamma^{{A}}E_{A}^{M}\partial_{M}.$		(26)

For the diagonal metric, and further representing the frame using warp factors, we find

	$\displaystyle\tilde{D}^{(2\iota)}_{L}=P_{R}\Gamma^{{i}}\tilde{\partial}_{i},$		(27)
	$\displaystyle\tilde{D}^{(2\iota)}_{R}=P_{L}\Gamma^{{i}}\tilde{\partial}_{i},$		(28)

where $\tilde{\partial}_{i}\equiv a_{i}^{-1}\partial_{i}$ (no sum over $i$), and we define $\tilde{\xi}_{i}\equiv\varepsilon a_{i}^{-1}\xi_{i}$ (no sum over $i$) and $\tilde{G}_{i}={a_{i}}^{-1}{G}_{i}$. This transformation of the field allows us to describe the interaction of the field $\xi_{M}$ in curved spacetime in the same manner as in flat spacetime. It should be noted that these operators, $\tilde{D}^{(2\iota)}_{L}$ and $\tilde{D}^{(2\iota)}_{R}$, are applied only to what we will refer to as the "half-component" of the spinors. Thus, the non-zero elements of these operators can be denoted as $\tilde{D}^{(\iota)}_{L}$ and $\tilde{D}^{(\iota)}_{R}$.

To illustrate the reduction from $6$ to $4$ dimensions, we examine the following action terms. The action $S_{L}$ is given by

	$\displaystyle S_{L}$	$\displaystyle=\int{d}^{6}{x}\sqrt{-{g}}\left[\bar{\Psi}_{1}^{(\iota)}\tilde{D}^{(\iota)}_{{L}}{\Psi}^{(\iota)}_{1}+\bar{\Psi}_{1}^{(\iota)}(\tilde{\xi}_{a}+\tilde{G}_{a})\gamma^{a}\Psi_{1}^{(\iota)}\right.$		(29)
		$\displaystyle\left.+\bar{\Psi}_{1}^{(\iota)}(\tilde{\xi}_{5}+\tilde{G}_{5})\gamma^{5}\Psi_{1}^{(\iota)}+i\bar{\Psi}_{1}^{(\iota)}(\tilde{\xi}_{6}+\tilde{G}_{6})\Psi_{1}^{(\iota)}\right],$

while the action $S_{R}$ is

	$\displaystyle S_{R}$	$\displaystyle=\int{d}^{6}{x}\sqrt{-{g}}\left[\bar{\Psi}_{2}^{(\iota)}\tilde{D}^{(\iota)}_{{R}}{\Psi}^{(\iota)}_{2}+\bar{\Psi}_{2}^{(\iota)}(\tilde{\xi}_{a}+\tilde{G}_{a})\gamma^{a}\Psi_{2}^{(\iota)}\right.$		(30)
		$\displaystyle\left.+\bar{\Psi}_{2}^{(\iota)}(\tilde{\xi}_{5}+\tilde{G}_{5})\gamma^{5}\Psi_{2}^{(\iota)}-i\bar{\Psi}_{2}^{(\iota)}(\tilde{\xi}_{6}+\tilde{G}_{6})\Psi_{2}^{(\iota)}\right],$

where $\gamma^{a}$ are generators representing the subalgebra of the Clifford algebra. This action is partitioned into $S_{L}$ and $S_{R}$, which correspond to ${\Psi}^{(\iota)}_{1}$ and ${\Psi}^{(\iota)}_{2}$, respectively.

Though the actions $S_{L}$ and $S_{R}$ appear as $4$-dimensional actions, it should be noted that ${\Psi}_{1}$ and ${\Psi}_{2}$ remain as spinor fields in the bulk. For a complete reduction to $4$-dimensional effective actions, the KK decomposition is also required.

The diagonal metric Eq. (8) is a relatively common situation. However, in the context of the brane-world scenario, the space of extra dimensions and the four-dimensional space can be decomposed independently. This decomposition facilitates the formulation of a 4-dimensional effective theory. Notably, the four-dimensional space is conformally flat. By assuming a flat brane, our focus is directed towards the localization of the matter field on the brane, avoiding the complexities of the brane’s internal structure. In the context of braneworld models within high-dimensional spacetimes, the additional dimensions are typically characterized by a pronounced symmetry. For scenarios involving two extra dimensions, this manifests as a specific orientation in the additional dimensional space, denoted as $x^{5}$. Considering the metric:

$$ds^{2}=a_{4}(x^{5})\eta_{\mu\nu}dx^{\mu}dx^{\nu}+a_{5}(x^{5})dx^{5}dx^{5}+a_{6}(x^{5})dx^{6}dx^{6},$$

(31)

The integral of the Lagrangian can be decomposed into $4$ dimensional and extra dimensional

$$\begin{split}S_{L,R}=&\int{d^{4}}x\,\,\bar{\psi}_{1,2}^{(\iota)}(x^{a})\gamma^{a}\partial_{a}\psi_{1,2}^{(\iota)}(x^{a})\\ &+\tilde{\alpha}_{5}\frac{\partial_{5}\phi_{1,2}}{\phi_{1,2}}\int{d^{4}}x\,\,\bar{\psi}_{1,2}^{(\iota)}(x^{a})\gamma^{5}\psi_{1,2}^{(\iota)}(x^{a})\\ &\pm\tilde{\alpha}_{6}\frac{\partial_{6}\phi_{1,2}}{\phi_{1,2}}\int{d^{4}}x\,\,\bar{\psi}_{1,2}^{(\iota)}(x^{a})i\,\,\psi_{1,2}^{(\iota)}(x^{a})\\ &+\sum_{a=0}^{3}{\tilde{\alpha}_{4}\left[\varepsilon\xi_{4}+{G}_{4}\right]\int{d^{4}}x\,\,i\,\,\bar{\psi}_{1,2}^{(\iota)}(x^{a})\gamma^{a}\psi_{1,2}^{(\iota)}(x^{a})}\\ &+\tilde{\alpha}_{5}\left[\varepsilon\xi_{5}+{G}_{5}\right]\,\,\int{d^{4}}x\,\,i\,\,\bar{\psi}_{1,2}^{(\iota)}(x^{a})\gamma^{5}\psi_{1,2}^{(\iota)}(x^{a})\\ &\pm\tilde{\alpha}_{6}\left[\varepsilon\xi_{6}+{G}_{6}\right]\int{d^{4}}x\,\,i\,\,\bar{\psi}_{1,2}^{(\iota)}(x^{a})i\,\,\psi_{1,2}^{(\iota)}(x^{a}),\end{split}$$

(32)

where, we define

$$\tilde{\alpha}_{4,5,6}=\sqrt{-g}\left(\phi_{1}^{*}\phi_{1}\right)a_{4,5,6}^{-1}.$$

(33)

Under the premise of metric (31), we have

	$\displaystyle G_{4}$	$\displaystyle\equiv G_{0}=G_{1}=G_{2}=G_{3}=0$		(34)
	$\displaystyle G_{5}$	$\displaystyle=\frac{1}{2}\sum_{\begin{subarray}{c}j=1\\ j\neq 5\end{subarray}}^{2n}{\partial_{5}\ln a_{j}},$		(35)
	$\displaystyle G_{6}$	$\displaystyle=0.$		(36)

Corresponding to the kinetic energy term in $4$-dimensional effective theory, the localization condition is

$$\int{dx^{5}}dx^{6}\tilde{\alpha}_{4}=1.$$

(37)

In the context of high-dimensional spacetime, we aim to derive $4$-dimensional free fields while preserving the underlying symmetries. A common approach is to couple bulk fermion and vector fields in the form $\bar{\Psi}\Gamma^{M}\xi_{M}\Psi$. However, brane-world models often utilize a scalar field to characterize the dynamical background. One can construct a vector field from the derivative of such a scalar field, allowing us to use the covariant derivative of the scalar field or a scalar function based on geometric background quantities as the vector field coupling to the fermion. This setup provides a mechanism for localizing matter fields through derivative coupling, a topic that has been explored in literature Liu:2013kxz ; Li:2017dkw . It is worth noting that the couplings used in these localization mechanisms can be considered as special cases of the more general coupling scheme $\bar{\Psi}\Gamma^{M}\Gamma^{N}\Gamma^{P}\cdots T_{MNP\cdots}\Psi$.

In the proposed model, the action takes the form

$$S_{3}=\int{d}^{6}x\sqrt{-g}\left[\bar{\Psi}\Gamma^{M}{D}_{M}\Psi+\varepsilon\bar{\Psi}\Gamma^{M}\partial_{M}F(\phi,R,R^{\mu\nu}R_{\mu\nu},\ldots)\Psi\right],$$

(38)

where the vector field $\xi_{M}(x^{5})$ is expressed as the derivative of a scalar function $F(x^{5})$ . This scalar function $F$ is a function of the scalar field $\phi$, the Ricci scalar $R$, and other curvature invariants like $R^{\mu\nu}R_{\mu\nu}$, among other possible terms ²²2 The $\xi^{M}$ in equation is actually a vector function constructed from the background dynamical fields and geometric scalars, rather than a vector constructed from the spinor field. It does not act as an independent dynamical field or degree of freedom. This vector field appears as a spacetime background field, and its dynamical terms are unrelated to the spinor field under consideration, and are only related to the dynamical description of the spacetime background. When considering the localization of fermions, we did not take into account the back-reaction of fermions on the spacetime background. As a specific example, the dynamics of the spacetime background are described by (81) and (80).. Under this assumption, we have $\xi_{4}=0$ and the action can be written as

	$\displaystyle S_{L,R}$	$\displaystyle=$	$\displaystyle\int{dx^{5}}dx^{6}\tilde{\alpha}_{4}\,\,\int{d^{4}}x\,\,\bar{\psi}_{1,2}^{(\iota)}(x^{a})\gamma^{a}\partial_{a}\psi_{1,2}^{(\iota)}(x^{a})$		(39)
			$\displaystyle+\int{dx^{5}}dx^{6}\tilde{\alpha}_{5}\,\,\left(\frac{\partial_{5}\phi_{1}}{\phi_{1}}\,+\varepsilon\partial_{5}F+{G}_{5}\right)\int{d^{4}}x\,\bar{\psi}_{1,2}^{(\iota)}(x^{a})\gamma^{5}\psi_{1,2}^{(\iota)}(x^{a})$
			$\displaystyle{\mp\int{dx^{5}}dx^{6}\tilde{\alpha}_{6}\frac{\partial_{6}\varphi_{1}}{\varphi_{1}}\int{d^{4}}x\,\bar{\psi}_{1,2}^{(\iota)}(x^{a})\psi_{1,2}^{(\iota)}(x^{a})}.$

Note that the left- and right-handed parts of the high-dimensional spinor corresponds to independent Dirac spinors in the reduced effective action. The action not only includes zero-mass Dirac fermions but also a series of massive ones. The introduction of higher-dimensional spacetime provides us with an explanation for the generation mechanism of particle masses, which is distinct from the Higgs mechanism. This stands in contrast to the decomposition of a Dirac spinor into Weyl spinors, where the Weyl spinor is massless. In the effective four-dimensional theory, there are multiple massive particles, which are four-dimensional Dirac spinors, not Weyl spinors, even though we employ chiral representations in the study of higher-dimensional spinors.

In the coupling form $\bar{\Psi}\Gamma^{M}\xi_{M}\Psi$, both the left- and right-handed components of the higher-dimensional fermionic field satisfy independent equations of motion. Therefore, the covariant derivative can be defined with respect to either the left- or right-handed part of the higher-dimensional spinor. Furthermore, the equations of motion can be separated into $4$-dimensional and extra-dimensional parts. We can then separate the variables as follows:

	$\displaystyle\hat{D}^{(2\iota)}_{L}$	$\displaystyle=\hat{D}^{(2\iota)}_{Lbrane}+\hat{D}^{(2\iota)}_{Lextra},$		(40)
	$\displaystyle\hat{D}^{(2\iota)}_{R}$	$\displaystyle=\hat{D}^{(2\iota)}_{Rbrane}+\hat{D}^{(2\iota)}_{Rextra},$		(41)

where

	$\displaystyle\hat{D}^{(2\iota)}_{Lbrane}$	$\displaystyle=P_{L}\Gamma^{{A}}E_{A}^{M}(\partial_{M}+\varepsilon\xi_{M}+G_{M}),~{}M=0,1,2,3,$		(42)
	$\displaystyle\hat{D}^{(2\iota)}_{Lextra}$	$\displaystyle=P_{L}\Gamma^{{A}}E_{A}^{M}(\partial_{M}+\varepsilon\xi_{M}+G_{M}),~{}M=5,6,$		(43)
	$\displaystyle\hat{D}^{(2\iota)}_{Rbrane}$	$\displaystyle=P_{R}\Gamma^{{A}}E_{A}^{M}(\partial_{M}+\varepsilon\xi_{M}+G_{M}),~{}M=0,1,2,3,$		(44)
	$\displaystyle\hat{D}^{(2\iota)}_{Rextra}$	$\displaystyle=P_{R}\Gamma^{{A}}E_{A}^{M}(\partial_{M}+\varepsilon\xi_{M}+G_{M}),~{}M=5,6.$		(45)

In light of the $4$-dimensional effective action (39), $\hat{D}_{\text{extra}}$ can be considered as a ’mass’ operator. $\psi_{1}$ and $\psi_{2}$ serve to represent the $4$-dimensional part of higher-dimensional left- and right-hand particles, respectively. These correspond to a set of $4$-dimensional massless and massive fermions in the effective theory.

The $6$-dimensional Dirac equation can be expressed as

$\displaystyle\left[a_{4}^{-1}\Gamma^{\mu}\partial_{\mu}+a_{5}^{-1}\Gamma^{5}\left(\partial_{5}+G_{5}+\varepsilon\partial_{5}F\right)+a_{6}^{-1}\Gamma^{6}\partial_{6}\right]\Psi\left(x^{N}\right)=0,$

(46)

where $G_{5},G_{6},$ and $F$ are functions of $x^{5}$ and $x^{6}$. Through the decomposition of (154) under the Weyl representation, the $6$-dimensional Dirac equation can be split into two components:

$$\begin{array}[]{l}\left[a_{4}^{-1}\gamma^{\mu}\partial_{\mu}+a_{5}^{-1}\gamma^{5}\left(\partial_{5}+G_{5}+\varepsilon\partial_{5}F\right)+a_{6}^{-1}i\partial_{6}\right]\Psi_{1}^{(4)}=0,\\ \left[a_{4}^{-1}\gamma^{\mu}\partial_{\mu}+a_{5}^{-1}\gamma^{5}\left(\partial_{5}+G_{5}+\varepsilon\partial_{5}F\right)-a_{6}^{-1}i\partial_{6}\right]\Psi_{2}^{(4)}=0.\end{array}$$

(47)

At this point, the $8$-component spinor decomposes into two independent $4$-component spinors. Each $4$-component spinor further decomposes into two $2$-component spinors by the following decomposition:

	$\displaystyle\Psi_{1}^{(4)}=\left(\begin{array}[]{l}\Psi_{11}^{(2)}\\ \Psi_{12}^{(2)}\end{array}\right)$	$\displaystyle=\sum_{m_{1}}\left(\begin{array}[]{l}\psi_{11{m_{1}}}^{(2)}(x)\phi_{11{m_{1}}}^{(2)}e^{il\Theta}\\ \psi_{12{m_{1}}}^{(2)}(x)\phi_{12{m_{1}}}^{(2)}e^{il\Theta}\end{array}\right),$		(52)
	$\displaystyle\Psi_{2}^{(4)}=\left(\begin{array}[]{l}\Psi_{21}^{(2)}\\ \Psi_{22}^{(2)}\end{array}\right)$	$\displaystyle=\sum_{m_{2}}\left(\begin{array}[]{l}\psi_{21{m_{2}}}^{(2)}(x)\phi_{21{m_{2}}}^{(2)}e^{il\Theta}\\ \psi_{22{m_{2}}}^{(2)}(x)\phi_{22{m_{2}}}^{(2)}e^{il\Theta}\end{array}\right).$		(57)

As analyzed in the previous section, the reduction of the action of the Dirac spinor field in higher-dimensional spacetime yields two independent $4$-dimensional Dirac spinors. If we take the $4$-dimensional Weyl representation, then we have

	$\displaystyle\gamma^{\mu}\partial_{\mu}\psi_{L}^{(4)}$	$\displaystyle=m\psi_{R}^{(4)},$	$\displaystyle\gamma^{\mu}\partial_{\mu}\psi_{R}^{(4)}$	$\displaystyle=m\psi_{L}^{(4)},$
	$\displaystyle\gamma^{5}\psi_{L}^{(4)}$	$\displaystyle=\psi_{L}^{(4)},$	$\displaystyle\gamma^{5}\psi_{R}^{(4)}$	$\displaystyle=-\psi_{R}^{(4)},$

and

	$\displaystyle a_{4}^{-1}m_{1}\phi_{12}+a_{5}^{-1}\left[\partial_{5}+H_{5}\right]\phi_{11}+ia_{6}^{-1}\partial_{6}\phi_{11}$	$\displaystyle=0,$		(58a)
	$\displaystyle a_{4}^{-1}m_{1}\phi_{11}-a_{5}^{-1}\left[\partial_{5}+H_{5}\right]\phi_{12}+ia_{6}^{-1}\partial_{6}\phi_{12}$	$\displaystyle=0,$		(58b)
	$\displaystyle a_{4}^{-1}m_{2}\phi_{22}+a_{5}^{-1}\left[\partial_{5}+H_{5}\right]\phi_{21}-ia_{6}^{-1}\partial_{6}\phi_{21}$	$\displaystyle=0,$		(58c)
	$\displaystyle a_{4}^{-1}m_{2}\phi_{21}-a_{5}^{-1}\left[\partial_{5}+H_{5}\right]\phi_{22}-ia_{6}^{-1}\partial_{6}\phi_{22}$	$\displaystyle=0.$		(58d)

where we have defined

$\displaystyle H_{5}(x^{5},x^{6})$

$\displaystyle=G_{5}(x^{5},x^{6})+\varepsilon\partial_{5}F(x^{5},x^{6}).$

(59)

We can always do a coordinate transformation

$\displaystyle a_{4}^{-1}\tilde{\partial}_{5}=a_{5}^{-1}\partial_{5}.$

(60)

which gives $a_{5}=a_{4}$. Then, the equation can be written as

	$\displaystyle m_{1}\phi_{12}+\left[\partial_{5}+H_{5}\right]\phi_{11}+ia_{4}a_{6}^{-1}\partial_{6}\phi_{11}$	$\displaystyle=0,$		(61a)
	$\displaystyle m_{1}\phi_{11}-\left[\partial_{5}+H_{5}\right]\phi_{12}+ia_{4}a_{6}^{-1}\partial_{6}\phi_{12}$	$\displaystyle=0,$		(61b)
	$\displaystyle m_{2}\phi_{22}+\left[\partial_{5}+H_{5}\right]\phi_{21}-ia_{4}a_{6}^{-1}\partial_{6}\phi_{21}$	$\displaystyle=0,$		(61c)
	$\displaystyle m_{2}\phi_{21}-\left[\partial_{5}+H_{5}\right]\phi_{22}-ia_{4}a_{6}^{-1}\partial_{6}\phi_{22}$	$\displaystyle=0.$		(61d)

This includes the case where the bulk spacetime is conformally flat. The effect of $H_{5}$ can be equivalently described by a field transformation. The field is transformed by the conformal factor as follows:

$\displaystyle\phi_{ij}=\Upsilon(x^{5})\tilde{\phi}_{ij},$

(62)

where the conformal transformation factor of the field is given by

$\displaystyle\Upsilon(x^{5})=\exp\left(-\int H_{5}(x^{5})dx^{5}\right)=\exp\left(-\int G_{5}(x^{5})dx^{5}\right)\exp\left(-\varepsilon F(x^{5})\right).$

(63)

With

$\displaystyle\exp\left(-\int G_{5}(x^{5})dx^{5}\right)=\frac{1}{\sqrt{a_{1}a_{2}a_{3}a_{4}a_{6}}},\quad a_{5}=a_{4}.$

(64)

Then, $\tilde{\alpha}_{4}$ is reduced to

$\displaystyle\tilde{\alpha}_{4}(x^{5},x^{6})=\left(e^{-\varepsilon F(x^{5})}\tilde{\phi}\right)^{*}\left(e^{-\varepsilon F(x^{5})}\tilde{\phi}\right).$

(65)

The localization condition requires convergence of the integral of $\tilde{\alpha}_{4}$. When $F$ is a real function, the localization condition can be written as

$\displaystyle\int\exp\left(-2\varepsilon F(x^{5})\right)\tilde{\phi}\tilde{\phi}^{*}dx^{5}dx^{6}.$

(66)

Following the field transformation described above, the integrated function in the localization condition can be decomposed into two parts. Here, the field $\tilde{\phi}$ satisfies the Schrödinger-like equation, and the effective potential is determined by the warp factor, representing the contribution of the minimal coupling between fermions and gravity. Furthermore, $e^{-2\varepsilon F(x^{5})}$ depends only on the coupling term $F$ rather than the warp (conformal) factor.

The minimal coupling to gravity can be analyzed independently. Thus, this allows us to focus on the geometry of spacetime. For example, one can study the difference in localization behavior under different topologies of spacetime. In contrast, the contributions of other interactions can be analyzed independently. Consequently, the localization condition can work under relatively relaxed conditions, and a universal localization mechanism can be constructed.

Another outcome of conformally transforming the field is that it restores the equations of motion to their flat spacetime form. With the relationship

$\displaystyle{\partial}_{5}{\phi}_{ij}=\Upsilon{\partial}_{5}\tilde{\phi}_{ij}-{H}\tilde{\phi}_{ij},$

(67)

this allows Eq. (61) to be further simplified to

	$\displaystyle m_{1}\tilde{\phi}_{12}$	$\displaystyle=-\partial_{5}\tilde{\phi}_{11}-ib\partial_{6}\tilde{\phi}_{11},$		(68a)
	$\displaystyle m_{1}\tilde{\phi}_{11}$	$\displaystyle=+\partial_{5}\tilde{\phi}_{12}-ib\partial_{6}\tilde{\phi}_{12},$		(68b)
	$\displaystyle m_{2}\tilde{\phi}_{22}$	$\displaystyle=-\partial_{5}\tilde{\phi}_{21}+ib\partial_{6}\tilde{\phi}_{21},$		(68c)
	$\displaystyle m_{2}\tilde{\phi}_{21}$	$\displaystyle=+\partial_{5}\tilde{\phi}_{22}+ib\partial_{6}\tilde{\phi}_{22},$		(68d)

with the expression for $b\left(x^{5}\right)$ given by

$\displaystyle b\left(x^{5}\right)=a_{4}a_{6}^{-1}.$

(69)

It is worth noting that the warp factor $a_{4}a_{6}^{-1}$ appears in pairs. This indicates that the bulk’s overall conformal transformation does not affect the form of the equations. It is also important to note that just like the asymptotically AdS bulk and the flat bulk, the localization of fermions cannot be achieved through minimal coupling to gravity. Eq. (68) can subsequently be transformed into a set of independent second-order differential equations

	$\displaystyle{m_{1}}^{2}\tilde{\phi}_{11}$	$\displaystyle=-\partial_{5}\partial_{5}\tilde{\phi}_{11}-b^{2}\partial_{6}\partial_{6}\tilde{\phi}_{11}-i\partial_{5}b\partial_{6}\tilde{\phi}_{11},$		(70a)
	$\displaystyle{m_{1}}^{2}\tilde{\phi}_{12}$	$\displaystyle=-\partial_{5}\partial_{5}\tilde{\phi}_{12}-b^{2}\partial_{6}\partial_{6}\tilde{\phi}_{12}+i\partial_{5}b\partial_{6}\tilde{\phi}_{12},$		(70b)
	$\displaystyle{m_{2}}^{2}\tilde{\phi}_{21}$	$\displaystyle=-\partial_{5}\partial_{5}\tilde{\phi}_{21}-b^{2}\partial_{6}\partial_{6}\tilde{\phi}_{21}+i\partial_{5}b\partial_{6}\tilde{\phi}_{21}.$		(70c)
	$\displaystyle{m_{2}}^{2}\tilde{\phi}_{22}$	$\displaystyle=-\partial_{5}\partial_{5}\tilde{\phi}_{22}-b^{2}\partial_{6}\partial_{6}\tilde{\phi}_{22}-i\partial_{5}b\partial_{6}\tilde{\phi}_{22}.$		(70d)

In polar coordinates $x^{6}=\theta$, the field function $\tilde{\phi}$ must satisfy a periodic condition. The field function can be decomposed into the following variables

$\displaystyle\tilde{\phi}=u_{r}(r)u_{\theta}(\theta),$

(71)

with $u_{\theta}(\theta)=e^{il_{6}\theta}$. Upon separation of variables, the equations of motion take the form

	$\displaystyle{m_{1}}^{2}u_{11r}$	$\displaystyle=-\partial_{5}\partial_{5}u_{11r}+V_{11}u_{11r},$		(72a)
	$\displaystyle{m_{1}}^{2}u_{12r}$	$\displaystyle=-\partial_{5}\partial_{5}u_{12r}+V_{12}u_{12r},$		(72b)
	$\displaystyle{m_{2}}^{2}u_{21r}$	$\displaystyle=-\partial_{5}\partial_{5}u_{21r}+V_{21}u_{21r},$		(72c)
	$\displaystyle{m_{2}}^{2}u_{22r}$	$\displaystyle=-\partial_{5}\partial_{5}u_{22r}+V_{22}u_{22r},$		(72d)

with the corresponding potentials given by

	$\displaystyle V_{11}(r)=V_{22}(r)={l_{6}}^{2}b^{2}(r)+l_{6}\partial_{r}b(r),$		(73a)
	$\displaystyle V_{12}(r)=V_{21}(r)={l_{6}}^{2}b^{2}(r)-l_{6}\partial_{r}b(r).$		(73b)

If spacetime is conformally flat, we can find a coordinate such that $a_{4}=a_{6}$, thereby implying that $b=1$. ³³3 In the next section, we will see that for two conformally flat topologies in six dimensions, we may not always make the choice of the coordinate systems such that $a_{4}=a_{6}$. Therefore, we cannot assert that the spinor field propagates freely along the extra dimensions. The shape of the effective potential will depend on the choice of coordinates, but the judgment of whether the spinor field is localized is independent of the coordinate choice. In more specific cases, we need to further clarify the topology of spacetime. In this case, the equations become simplified, and the four two-component fermions satisfy

$\displaystyle m^{2}\tilde{\phi}=-\partial_{5}\partial_{5}\tilde{\phi}-\partial_{6}\partial_{6}\tilde{\phi}.$

(74)

The above equation shares a similar form with the Klein-Gordon equation in flat spacetime, so we have

$\displaystyle\tilde{\phi}_{1}=e^{il_{5}x^{5}}e^{il_{6}x^{6}},$

(75)

where $l_{5}$ and $l_{6}$ represent the quantized momenta associated with the fermion’s movement in the extra dimensions. This yields the energy-momentum relation

$\displaystyle m^{2}=l_{5}^{2}+l_{6}^{2}.$

(76)

If $l_{5}$ and $l_{6}$ are real, then the localization condition depends only on the integral of the background scalar field:

$\displaystyle\int e^{-2\varepsilon F(x^{5})}dx^{5}.$

(77)