Localization of matter fields on a chameleon brane

The existence of hidden extra dimensions in our spacetime, particularly at ultraviolet scales, has been a topic of interest for a long time, starting from the proposal of the Kaluza-Klein (KK) theory in the 1920s. Based on this idea, the braneworld scenario was proposed, in which our universe could be a 3-brane embedded in a higher-dimensional space-time, called the bulk. This perspective offers a novel approach to addressing the gauge hierarchy problem and cosmological constant problem ArkaniHamed:1998rs ; ArkaniHamed:2000eg ; Randall:1999ee ; Randall:1999vf , which are two persistent challenges in particle physics and cosmology. The concept of a thick brane Gremm:2000dj ; Gass:1999gk ; Gremm:1999pj ; Afonso:2006gi extends the ideas of the Randall-Sundrum-2 (RS2) model Randall:1999vf , and the configuration of a domain wall Rubakov:1983bb ; Rubakov:1983bz serves as a topological defect in the bulk.

In the braneworld theory, the localization of matter fields is an important issue, as it not only allows the five-dimensional theory to reduce to a four-dimensional effective theory at low energy scales but also provides observable effects for probing the extra dimensions Bajc2000 ; Oda2000 ; Melfo:2006hh ; Liu:2009ve ; PhysRevD.79.125022 ; Liang2009 ; German:2013sk ; Vaquera-Araujo:2014tia ; Salvio:2007qx . In five-dimensional spacetime, free scalar fields can typically be localized on branes Bajc2000 ; Oda2000 , while fermion fields generally require Yukawa couplings with background scalar fields to achieve localization on branes Melfo:2006hh ; PhysRevD.78.065025 ; Liang2009 ; Alencar2014 ; Vaquera-Araujo:2014tia . However, it is generally challenging to localize vector field on branes in five dimensions. Some researchers have attempted to address this issue by assuming couplings between vector field and scalar field or spacetime geometry, or by considering models such as the Weyl geometry braneworld, de Sitter braneworld, and six-dimensional braneworld Chumbes:2011zt ; Alencar2014 ; Vaquera-Araujo:2014tia . See Refs. Dvali:2000rx ; Akhmedov:2001ny ; Guerrero:2009ac ; Delsate:2011aa ; Cruz:2012kd ; Dimopoulos:2000ej ; Belchior:2023gmr ; Zhao:2017epp for more works on the localization of vector field in braneworld models. Despite the extensive research conducted on this issue, there is still significance in providing a concise and natural new mechanism to effectively localize the vector field on the brane.

On the other hand, the chameleon gravity proposed by Khoury et al. in 2003 can address the dark energy problem PhysRevLett.93.171104 ; PhysRevD.69.044026 . This theory introduces a scalar field, known as the "chameleon" which exhibits mass variation depending on the surrounding matter density. The concept of chameleon gravity is closely related to string theory Brax2004 and scalar-tensor theory. The theory has been extensively tested through observations and experiments Burrage:2017qrf , showing promising prospects for resolving the Hubble constant problem PhysRevD.103.L121302 . In this theory, matter fields do not directly couple to the metric tensor $g_{\mu\nu}$ that describes spacetime curvature. Instead, they couple to a physical metric $\widetilde{g}_{\mu\nu}=b^{2}(\phi)g_{\mu\nu}$ resulting from a conformal transformation with the chameleon scalar field $\phi$. Thus, various matter fields are coupled to the chameleon scalar field $\phi$, providing a potential solution to the localization problem of vector field. Moreover, the coupling of scalar fields and fermion fields in matter to the chameleon scalar field $\phi$ results in novel characteristics within the braneworld model in chameleon gravity. Brane cosmology in chameleon gravity was investigated in Refs. PhysRevD.85.023526 ; Bisabr2017 , where thin brane models were considered. While thick brane model in chameleon gravity has not yet been explored, this study aims to address this gap. We will construct a thick chameleon brane and attempt to solve the localization problem of vector field in the chameleon thick braneworld. Furthermore, we will investigate the localization property of various matter fields in this model.

The layout of the paper is as follows: In Section 2, we construct a chameleon brane model within the background of a Sine-Gordon kink. In Section 3, we explore the localization and quasi-localization of three types of $q$-form fields. In Section 4, we examine the localization of a fermion field. Finally, we conclude with a discussion of our findings. Throughout this paper, $M,N,\cdots$ denote the indices of the five dimensional coordinates, and $\mu,\nu,\cdots$ denote the ones on the brane.

We start with the following five-dimensional action

which describes a canonical scalar field minimally coupled to gravity, and matter fields $\psi_{i}$ coupled to the Jordan frame metric $\tilde{g}_{MN}=b^{2}(\phi)g_{MN}$. It can be seen from the above action that the braneworld solution in this set up is the same with the standard case. However, there are some novel features regarding the localization of various matter fields are expected, since the matter fields couple to the Jordan frame metric $\tilde{g}_{MN}$ instead of the Einstein frame one $g_{MN}$.

The metric ansatz of a flat brane embedded in a five-dimensional anti-de Sitter (AdS) spacetime is

where $a(z)=e^{A(z)}$ is the warped factor. Varying the action with respect to the Einstein frame metric $g_{MN}$ and the scalar field $\phi$, and let $S_{m}=0$, we obtain the following equation of motion,

Assuming the sine-Gordon potential $V(\phi)=\frac{3}{4}k^{2}\left[5\cos\left(\frac{2\phi}{\sqrt{3}}\right)-3\right]$, the brane solution is solved as Koley:2004at

where the parameter $k$ is associated with the inverse of the five-dimensional AdS radius. Since the brane world solution is the same with the standard case, the KK gravitons generate a correction, which is proportional to ${1}/{kr^{3}}$, to the Newtonian potential Csaki2000 ; Bazeia2009 ; Liu:2012rc ; Yang:2022fed . Conversely, a recent experiment has demonstrated that the length scale deviating from the gravitational inverse square law is at most $48\mu\text{m}$ PhysRevLett.124.051301 . Thus the parameter $k$ can be roughly estimated to be $k\gtrsim 10^{-3}\text{eV}$.

In the thick brane models, the extra dimension is non-compact, and the matter fields are localized on the brane by the warped geometry, which is described by the Jordan frame metric

where we have defined an effective warped factor $\tilde{a}(z)=\text{e}^{\tilde{A}(z)}=\text{e}^{B(\phi)+A(z)}$ and $b(\phi)=\text{e}^{B(\phi)}$ for convenience, and all the functions deduced from $\tilde{g}_{MN}$ are with tilde in the following.

In this section, we consider the localization of $q$-form field $X_{M_{1}M_{2}\ldots M_{q}}$ on the brane, which represents a scalar field for $q=0$, a vector field for $q=1$, and Kalb-Ramond (KR) field for $q=2$. We will first investigate a generic $q$-form field, and then discuss each of the scalar, vector and Kalb-Ramond fields. The action of a massless $q$-form field in the bulk is given by

where $\tilde{g}$ is the determinant of the Jordan frame metric $\tilde{g}_{MN}$, the indices are also raised and lowered by $\tilde{g}_{MN}$, and the strength of the $q$-form field is defined as $Y_{M_{1}M_{2}\ldots M_{q+1}}=\partial_{[M_{1}}X_{M_{2}M_{3}\ldots M_{q+1}]}$. By varying action (9) with respect to $X_{M_{1}M_{2}\ldots M_{q}}$, the equations of motion of the $q$-form field can be obtained,

In this section, we investigate the localization of a spin-$\frac{1}{2}$ fermion field on the brane. The Dirac structure of the fermion fields in the bulk is described by $\tilde{\Gamma}^{M}=\tilde{e}^{M}_{\bar{M}}\tilde{\Gamma}^{\bar{M}}$ with $\{\tilde{\Gamma}^{M},\tilde{\Gamma}^{N}\}=2\tilde{g}^{MN}$. Here $\bar{M},\bar{N},\cdots$ denote the local Lorentz indices and $\tilde{\Gamma}^{M}$ are the gamma matrices corresponding to the Jordan frame metric $\tilde{g}^{MN}$. In this set up, it is easy to see that all the equations are the same with the standard case in general relativity except that all the quantities deduced from $\tilde{g}^{MN}$ should be marked by tilde. With the metric (8), we can obtain $\tilde{\Gamma}^{M}=\text{e}^{-B}\Gamma^{M}=(\text{e}^{-\tilde{A}}\gamma^{\mu},\text{e}^{-\tilde{A}}\gamma^{5})$. We consider the following action of a massless spin-$\frac{1}{2}$ fermion field coupled to the background chameleon scalar field in a Yukawa-type interaction,

where $f(\phi)$ is a function of the chameleon scalar field, and the generic covariant derivative $\tilde{D}_{M}=\partial_{M}+\tilde{\omega}_{M}$ is defined with the spin connection $\tilde{\omega}_{M}=\frac{1}{4}\tilde{\omega}_{M}^{\bar{M}\bar{N}}\tilde{\Gamma}_{\bar{M}}\Gamma_{\bar{N}}$, with

Obviously, the fermion field couples to the scalar field not only through the Yukawa interaction, but also through the determinant $\sqrt{-\tilde{g}}$, the Gamma matrix $\tilde{\Gamma}$ and the generic covariant derivative $\tilde{D}_{M}$, which are all related to the Jordan frame metric $\tilde{g}_{MN}$. With the metric (8), we obtain the non-vanishing components of the spin connection $\omega_{M}$,

The variation of (41) with respect to $\bar{\Psi}$ yields the five-dimensional Dirac equation,

Substituting Eqs. (43) into the above equation, one can obtain the explicit form of the Dirac equation in the metric (8)

Then we employ the chiral decomposition

where $\psi_{Ln}(x)=-\gamma^{5}\psi_{Ln}(x)$ and $\psi_{Rn}(x)=\gamma^{5}\psi_{Rn}(x)$ are left-handed and right-handed components of four-dimensional Dirac fields respectively. By substituting this decomposition into the five-dimensional Dirac equation (44), the four-dimensional Dirac parts $\psi_{Ln}(x)$ and $\psi_{Rn}(x)$ satisfy the four-dimensional Dirac equations

and the five-dimensional Dirac parts $\alpha_{L}(z)$ and $\alpha_{R}(z)$ satisfy the coupled equations

After reassembling the two coupled equations, one ultimately achieves the Schrödinger-like equations

with the effective potentials given by

Further, to obtain the effective action of a massless and a series of massive four-dimensional fermions on the brane, the following orthonormal conditions have to be imposed,

The chiral zero modes can be solved by applying the factorizing method, given by

with $N_{L0,R0}$ the normalization constants. Since we need at least the massless fermion to be localized on the brane, the normalization condition for the zero modes must be satisfied, i.e.,

where the plus or minus sign is corresponding to the zero mode of left or right-handed four-dimensional fermion. Obviously, only one of the zero modes can be localized on the brane depending on the value of coupling constant $\eta$.

It is well known that the standard model is a chiral theory, where the left and right-handed fermions transform differently under the electroweak gauge group. To generate the chiral fermions on the brane, one can include a left-handed $SU(2)_{L}$ doublet $\Psi_{L}$ and a right-handed singlet $\Psi_{R}$ in the five-dimensional bulk, then the corresponding chiral zero modes are picked up by properly setting the coupling constants $\eta_{L/R}$ PhysRevD.62.084025 . Here, we take the localization of left-handed four-dimensional fermion as an example, i.e., $\int^{+\infty}_{-\infty}\text{e}^{-2\eta_{L}\int\text{e}^{A}b(\phi)f(\phi)dz}dz<\infty$. Note that the brane is embedded in a five-dimentional asymptotic AdS spacetime, we have $\text{e}^{A(\pm\infty)}=\pm\frac{1}{kz}$, and the kink configuration of the scalar field yields $\phi(\pm\infty)=\pm\frac{\sqrt{3}\pi}{2}\equiv\pm v_{0}$. Therefore,

Then the normalization condition (56) is reduced to that: $f(\phi)$ must be an odd function of $\phi$, $b(v_{0})f(v_{0})$ is a non-vanishing constant, and $\eta_{L}>\frac{k}{b(v_{0})f(v_{0})}$. As two examples, we can assume that $f(\phi)=\frac{\phi}{\cos^{p}(\frac{\phi}{\sqrt{3}})}$ or $f(\phi)=\frac{\sin(\phi)}{\cos^{p}(\frac{\phi}{\sqrt{3}})}$, then the localization of left-handed zero mode is realized for $\eta_{L}>\frac{k}{v_{0}}$ or $\eta_{L}>\frac{k}{\sin v_{0}}$, respectively.

It is noted that, although the fermion field couples to the chameleon scalar field $\phi$ in several places in the action (41), the Yukawa interaction is still essential to localizing the fermion zero mode.

In this work, we investigated the localization of $q$-form fields and fermion field on the thick brane construct by a chameleon scalar field. We had demonstrated that the localization problem of the vector field can be solved by choosing an appropriate conformal factor $b(\phi)$ in this model. The conditions for localization of various matter fields were obtained. For the $q$-form fields, the zero modes of the scalar and vector fields can be localized, in the condition that $b[\phi(\infty)]=z^{r}$ with $r<0$, while the zero mode of KR field can not. Furthermore, the quasi-localization of $q$-form fields was also considered. It was found that the relative probability, mass, and life-time of the resonances increase with the parameter $p$. The parity of first resonances of scalar and vector fields is odd, and the one of the KR field is even. In order to be consistent with both the experimental constraints of the photon mass and the deviation from gravitational inverse-square law, the parameters were constrained as $0<p\lesssim 7$ and $k\gtrsim 10^{-3}\text{eV}$.

For the fermion field, it couples to the chameleon scalar field not only through the Yukawa interaction, but we found that the term of Yukawa interaction is still necessary in order to localize the fermion zero modes. The condition for localizing the left-handed fermion zero mode was found to be, $f(\phi)$ an odd function of $\phi$, $b(v_{0})f(v_{0})$ a non-vanishing constant, and $\eta>\frac{k}{b(v_{0})f(v_{0})}$.

Y. Zhong acknowledges the support of the Natural Science Foundation of Hunan Province, China (Grant No. 2022JJ40033), the Fundamental Research Funds for the Central Universities (Grants No. 531118010195) and the National Natural Science Foundation of China (No. 12275076). He also thanks the generous hospitality offered by the Center of Theoretical Physics at Lanzhou University where part of this work was completed. K. Yang acknowledges the support of the National Natural Science Foundation of China under Grant No. 12005174.