Kalb【-逻*辑*与-】ndash;Ramond field localization on a thick de Sitter brane
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Chen Yang1,2, Zi-Qi Chen1,2, Li Zhao,1,2,31Institute of Theoretical Physics, Lanzhou University, 222 South Tianshui Road, Lanzhou 730000, China 2Research Center of Gravitation, Lanzhou University, 222 South Tianshui Road, Lanzhou 730000, China
First author contact:3Author to whom any correspondence should be addressed. Received:2020-02-26Revised:2020-03-20Accepted:2020-03-23Online:2020-06-18
Abstract In this paper, we explore the localization condition of Kalb–Ramond (KR) tensorial gauge field on a thick de Sitter (dS) brane. Following the localization mechanism in the work by Chumbes et al (2012 Phys. Rev. D 85 085003), we analyze the localization of KR tensorial gauge field on a non-flat three-brane. We propose three kinds of coupling methods and two of them support the localization of zero mode. In addition, there exist resonant Kaluza–Klein modes on the thick dS brane. The effects of three parameters on the localization and the resonant mode for the KR field are also discussed. Keywords：Kalb–Ramond field;localization;braneworld
PDF (709KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Chen Yang, Zi-Qi Chen, Li Zhao. Kalb–Ramond field localization on a thick de Sitter brane. Communications in Theoretical Physics, 2020, 72(7): 075801- doi:10.1088/1572-9494/ab8a1b
1. Introduction
In the past two decades, the braneworld theory has attracted much attention since it gives a possible resolution to the long-existing gauge hierarchy problem [1–6] and cosmology constant problem [7–10]. The theory has proposed a new idea that we live on a membrane embedded into a higher-dimensional bulk [1, 2, 11]. In the original Randall–Sundrum (RS) (respectively, RS1, RS2) models [2, 11], the bulk geometry is allowed to be curved, and the branes are endowed with a tension. However, the brane is singular as the bulk curvature scalar is divergent at its location. In the most fundamental theory, a brane should have a minimal length scale and inner structure. A smooth generalization of the RS2 model has been proposed where five-dimensional (5D) gravity is coupled to scalar fields. For realizing the thick branes, the scalar fields are introduced to generate the topological defects (such as kinks and domain walls), which are essential for the appearance of thick brane [12, 13] (for reviews see [14, 15]).
In the braneworld scenario, the issue of localization of various fields is an important subject, as it can guide us to know which kind of brane structure is more acceptable. In general, the massless scalar field and graviton are localized in the RS thin brane and its generalizations [16–19]. The fermion field can be localized by introducing the scalar-fermion coupling [20, 21] or a derivative geometrical coupling [22]. Whereas for a free massless vector field, the localization is more difficult. It can be localized on some non-RS2 brane models [23, 24], and on the thick brane if one adds a dilaton scalar field [25], a 5D Stueckelberg-like action [26], a topology term to the 5D vector action [27]. More recently, the authors of [28] have proposed a new kind of non-minimal coupling between the vector field and the gravity in the 5D vector field action.
Motivated by the previous researches on the localization of vector field, our main subject is to answer the following question: how to localize the higher-spin gauge field on a thick brane? The simplest case of higher rank gauge fields to be studied is a rank-2 antisymmetric field. A usual two-form field is Kalb–Ramond (KR) field, which emerges as a mode of massless excitations of closed strings [29], and describes the torsion of a Riemannian–Cartan manifold as well [30]. For the free KR field, it is failed to localize it in a thick brane with the standard Lagrangian. In order to localize the KR field, the mechanisms for localization are mainly three-fold: a general Kehagias–Tamvakis mechanism with a coupling between the gauge field and an additional dilaton field [25, 31–33]; a Chumbes–Hoff–Hott mechanism to localize gauge field on thick brane directly coupled to the background scalar field [13]; the approach for gauge field localization on thick branes based on a 5D Stueckelberg-like action [26].
To the best of our knowledge, the localization of the KR field has been investigated based on a flat three-brane [32, 34, 35]. In this paper, following the work of [13], we introduce the non-minimal coupling between the 5D KR gauge field and the background scalar field on a non-flat three-brane. In detail, the brane possesses de Sitter (dS) symmetry instead of the Poincaré one. For a positive scale factor parameter β, the dS brane has a expanding solution. Because the property of dS brane is different from that of Minkowski brane, the localization of a 5D KR field on dS brane is an interesting topic.
The plan of this paper is as follows. In section 2, we briefly review the action of the thick brane system and derive the equations of motion. Then, in section 3, a Schrödinger-like equation and the effective potential are obtained. Based on three kinds of coupling, the localization and resonance of KR tensor fields are discussed on this thick brane. Finally, our conclusion is given in section 4.
2. Brane setup and field equations
Here we consider a thick dS brane embedded into a 5D spacetime, and the thick dS brane is generated by one real scalar field coupled to gravity minimally. The action of our system is expressed as$ \begin{eqnarray}S=\int {{\rm{d}}}^{5}x\sqrt{-g}\left[\displaystyle \frac{1}{2{\kappa }_{5}^{2}}R-\displaystyle \frac{1}{2}{g}^{{MN}}{\partial }_{M}\phi {\partial }_{N}\phi -U(\phi )\right],\end{eqnarray}$where κ5 is the 5D coupling constant, and U(φ) is the scalar field potential. Here we set κ5=1 for simplicity. For a expanding and maximally symmetric dS brane, the 5D metric ansatz is assumed as$ \begin{eqnarray}{\rm{d}}{s}^{2}={{\rm{e}}}^{2A(z)}\left(-{\rm{d}}{t}^{2}+{{\rm{e}}}^{2\beta t}{\delta }_{{ij}}{\rm{d}}{x}^{i}{\rm{d}}{x}^{j}+{\rm{d}}{z}^{2}\right),\end{eqnarray}$where ${{\rm{e}}}^{2A(z)}$ is the warp factor and ${{\rm{e}}}^{2\beta t}$ is the scale factor of the brane. β is related to the positive four-dimensional cosmological constant Λ4 of dS brane with Λ4=3β2. For positive and vanishing scale factor parameter β, one will obtain the expanding and static solutions, respectively. The warp factor A(z) and the scalar field φ only depend on the extra coordinate z. With the metric ansatz(2), the variation of the action(1) with respect to the metric gMN and the scalar field φ yields the following field equations$ \begin{eqnarray}\phi {{\prime} }^{2}=3(A{{\prime} }^{2}-A^{\prime\prime} -{\beta }^{2}),\,\end{eqnarray}$$ \begin{eqnarray}\,U(\phi )=\displaystyle \frac{3}{2}{{\rm{e}}}^{-2A}(-3A{{\prime} }^{2}-A^{\prime\prime} +3{\beta }^{2}),\,\end{eqnarray}$$ \begin{eqnarray}\displaystyle \frac{{\rm{d}}U(\phi )}{{\rm{d}}\phi }={{\rm{e}}}^{-2A}(3A^{\prime} \phi ^{\prime} +\phi ^{\prime\prime} ),\,\end{eqnarray}$where the prime denotes the derivative with respect to z. A thick dS brane solution for this system was found in [36, 37]:$ \begin{eqnarray}U(\phi )=\displaystyle \frac{1+3\delta }{2\delta }\ 3{\beta }^{2}{\left(\cos \displaystyle \frac{\phi }{{\phi }_{0}}\right)}^{2(1-\delta )},\end{eqnarray}$$ \begin{eqnarray}{{\rm{e}}}^{2A}={\cosh }^{-2\delta }\left(\displaystyle \frac{\beta z}{\delta }\right),\end{eqnarray}$$ \begin{eqnarray}\phi ={\phi }_{0}\arctan \left(\sinh \displaystyle \frac{\beta z}{\delta }\right),\end{eqnarray}$where ${\phi }_{0}=\sqrt{3\delta (1-\delta )}$, 0<δ<1, β>0. The warp factor and the background scalar field are depicted in figure 1. It can be seen that the warp factor indicates a thick brane centered at the origin y=0, and the scalar field has a kink configuration with $\pm \tfrac{\pi }{2}{\phi }_{0}$ at $z\to \pm \infty $, corresponding to two consecutive minima of the potential U(φ). From the expressions of warp factor and scalar field, one can calculate the energy density ρ(z) along the extra dimension$ \begin{eqnarray}\rho (z)=\displaystyle \frac{3{\beta }^{2}(1+\delta )}{\delta }{\cosh }^{-2-2\delta }\left(\displaystyle \frac{\beta z}{\delta }\right),\end{eqnarray}$which is plotted in figure 2. This figure shows that the brane becomes thicker with the increase of the parameter δ. Thus δ parameterizes the thickness of the three-brane.
Figure 1.
New window|Download| PPT slide Figure 1.Plots of ${{\rm{e}}}^{2A(z)}$ and φ(z), where ${{\rm{e}}}^{2A(z)}$ is the warp factor in equation (7) and φ(z) is the scalar field in equation (8). The parameters are set to δ=1/2, β=2.
Figure 2.
New window|Download| PPT slide Figure 2.Plots of the energy density ρ(z) in equation (9). The parameter is set to β=2.
3. Schrödinger-like equation and the effective potential
We consider the non-minimal coupling between the 5D KR field and the background scalar field φ with the action$ \begin{eqnarray}S=\int {{\rm{d}}}^{5}x\sqrt{-g}\left[f(\phi ){H}_{{MNL}}{H}^{{MNL}}\right],\end{eqnarray}$where ${H}_{{MNL}}={\partial }_{[M}{B}_{{NL}]}$ is the field strength, f(φ) is the coupling with the background scalar field. From the action and metric ansatz, the equation of motion for the KR field is expressed as$ \begin{eqnarray}{\partial }_{R}\left[\sqrt{-g}f(\phi ){H}^{{RPQ}}\right]=0.\end{eqnarray}$Considering the antisymmetry of the KR field strength, the equation of motion can be transformed into$ \begin{eqnarray}\begin{array}{rcl}f(\phi ){\partial }_{\mu }\left[\sqrt{-g}{H}^{\mu \nu \lambda }\right]+{\partial }_{5}\left[\sqrt{-g}f(\phi ){H}^{5\nu \lambda }\right] & = & 0,\\ f(\phi ){\partial }_{\mu }\left[\sqrt{-g}{H}^{\mu \nu 5}\right] & = & 0.\end{array}\end{eqnarray}$By choosing the gauge condition Bμ5=0, and making the Kaluza–Klein (KK) decomposition of the KR field as$ \begin{eqnarray}{B}^{\nu \lambda }=\sum _{n}{b}_{(n)}^{\nu \lambda }(x){\chi }_{(n)}(z){\rm{\Omega }}(z),\end{eqnarray}$the field strength becomes$ \begin{eqnarray}{H}_{\mu \nu \lambda }=\sum _{n}{{\rm{e}}}^{4A(z)}{h}_{\mu \nu \lambda (n)}(x){\chi }_{(n)}(z){\rm{\Omega }}(z),\end{eqnarray}$where ${h}_{\mu \nu \lambda (n)}(x)$ is the 4D part of KR field strength on the brane. Substituting equation (13) into equation (12), we get the following equations$ \begin{eqnarray}{\partial }_{\tau }\left(\sqrt{-\hat{g}}{h}_{(n)}^{\tau \mu \nu }\right)=\displaystyle \frac{1}{3}{m}_{n}^{2}{b}_{(n)}^{\mu \nu }\sqrt{-\hat{g}},\end{eqnarray}$$ \begin{eqnarray}\begin{array}{l}{\partial }_{5}\left({{\rm{e}}}^{-A(z)}f(\phi ){\partial }_{5}\left({{\rm{e}}}^{4A(z)}{\chi }_{(n)}(z\right){\rm{\Omega }}(z\right)\\ \quad =\,-{m}_{n}^{2}{{\rm{e}}}^{3A(z)}f(\phi ){\chi }_{(n)}(z){\rm{\Omega }}(z).\end{array}\end{eqnarray}$By eliminating the first derivative term of χ(n)(z), i.e.$ \begin{eqnarray}7A^{\prime} (z)f(\phi ){\rm{\Omega }}(z)+f^{\prime} (\phi ){\rm{\Omega }}(z)+2f(\phi ){\rm{\Omega }}^{\prime} (z)=0.\end{eqnarray}$Equation (16) can be transformed into a Schrödinger-like equation$ \begin{eqnarray}\left[-\displaystyle \frac{{{\rm{d}}}^{2}}{{\rm{d}}{z}^{2}}+{V}_{\mathrm{eff}}(z)\right]{\chi }_{(n)}(z)={m}_{n}^{2}{\chi }_{(n)}(z),\end{eqnarray}$where the effective potential Veff(z) of the Schrödinger-like equation is given by$ \begin{eqnarray}{V}_{\mathrm{eff}}(z)={\left(4A^{\prime} (z)+\frac{{\rm{\Omega }}^{\prime} (z)}{{\rm{\Omega }}(z)}\right)}^{2}-\left(4A^{\prime} (z)+\frac{{\rm{\Omega }}^{\prime} (z)}{{\rm{\Omega }}(z)}\right)^{\prime} .\end{eqnarray}$In fact, the Schrödinger-like equation (18) can also be written as$ \begin{eqnarray}{ \mathcal H }{\chi }_{(n)}(z)={m}_{n}^{2}{\chi }_{(n)}(z),\end{eqnarray}$where$ \begin{eqnarray}\begin{array}{rcl}{ \mathcal H } & = & {Q}^{+}Q=\left(-{\partial }_{z}+4A^{\prime} (z)+\displaystyle \frac{{\rm{\Omega }}^{\prime} (z)}{{\rm{\Omega }}(z)}\right)\\ & & \times \left({\partial }_{z}+4A^{\prime} (z)+\displaystyle \frac{{\rm{\Omega }}^{\prime} (z)}{{\rm{\Omega }}(z)}\right),\end{array}\end{eqnarray}$with $Q={\partial }_{z}+4A^{\prime} (z)+\tfrac{{\rm{\Omega }}^{\prime} (z)}{{\rm{\Omega }}(z)}$. As the operator ${ \mathcal H }$ is positive definite, there are no tachyonic modes with negative mn2. Considering the KK decomposition of the KR field (13), we derive the 5D action as $ \begin{eqnarray*}\begin{array}{rcl}{S}_{\mathrm{KR}} & = & \displaystyle \int {{\rm{d}}}^{4}x\sqrt{-\hat{g}}\left[\Space{0ex}{3.0ex}{0ex}f(\phi ){h}_{\mu \nu \lambda (l)}{h}_{(m)}^{\mu \nu \lambda }\right.\\ & & \left.+\displaystyle \frac{1}{3}{m}_{l}^{2}\sqrt{-\hat{g}}{b}_{\mu \lambda (l)}(x){b}_{(m)}^{\mu \lambda }\right]\\ & & \times \displaystyle \int {{\rm{e}}}^{7A(z)}f(\phi ){\chi }_{(l)}(z){\chi }_{(m)}(z){{\rm{\Omega }}}^{2}(z){\rm{d}}z.\end{array}\end{eqnarray*}$ By imposing the orthonormality condition between different massive modes$ \begin{eqnarray}\int {{\rm{e}}}^{7A(z)}f(\phi ){\chi }_{(l)}(z){\chi }_{(m)}(z){{\rm{\Omega }}}^{2}(z){\rm{d}}z={\delta }_{{lm}},\end{eqnarray}$the 5D action can be reduced to the 4D effective one $ \begin{eqnarray*}\begin{array}{l}{S}_{\mathrm{KR}}\propto {S}_{\mathrm{eff}}\\ =\,\displaystyle \int {{\rm{d}}}^{4}x\sqrt{-\hat{g}}\left[f(\phi ){h}_{\mu \nu \lambda (m)}{h}_{(m)}^{\mu \nu \lambda }+\displaystyle \frac{1}{3}{m}_{m}^{2}\sqrt{-\hat{g}}{b}_{\mu \lambda (m)}(x){b}_{(m)}^{\mu \lambda }\right].\end{array}\end{eqnarray*}$
3.1. Case I: $f(\phi )={\phi }^{k}$
Firstly, we consider the simple case of $f(\phi )={\phi }^{k}$, where φ is the background scalar field and k describes the power of the coupling function. From equation (17), the function Ω(z) for this case reads$ \begin{eqnarray}\begin{array}{rcl}{\rm{\Omega }}(z) & = & {{\rm{e}}}^{-\tfrac{7}{2}A(z)}f{\left(\phi \right)}^{-\tfrac{1}{2}}\\ & = & {\cosh }^{\tfrac{7\delta }{2}}\left(\displaystyle \frac{\beta z}{\delta }\right){\left[\sqrt{3(1-\delta )\delta }{\tan }^{-1}\left(\sinh \left(\displaystyle \frac{\beta z}{\delta }\right)\right)\right]}^{-\tfrac{k}{2}}.\end{array}\end{eqnarray}$According to equation (19), the corresponding effective potential is expressed as$ \begin{eqnarray}\begin{array}{rcl}{V}_{\mathrm{eff}}(z) & = & \displaystyle \frac{{\beta }^{2}{{\rm{sech}} }^{2}\left(\tfrac{\beta z}{\delta }\right)}{8{\delta }^{2}{\tan }^{-1}{\left(\sinh \left(\tfrac{\beta z}{\delta }\right)\right)}^{2}}\left[2{k}^{2}\right.\\ & & +4(\delta -1)k\sinh \left(\displaystyle \frac{\beta z}{\delta }\right){\tan }^{-1}\left(\sinh \left(\displaystyle \frac{\beta z}{\delta }\right)\right)-4k\end{array}\end{eqnarray}$$ \begin{eqnarray}\begin{array}{rcl} & & -{\delta }^{2}{\tan }^{-1}{\left(\sinh \left(\displaystyle \frac{\beta z}{\delta }\right)\right)}^{2}+{\delta }^{2}\cosh \left(\displaystyle \frac{2\beta z}{\delta }\right){\tan }^{-1}\\ & & \left.\times {\left(\sinh \left(\displaystyle \frac{\beta z}{\delta }\right)\right)}^{2}+4\delta {\tan }^{-1}{\left(\sinh \left(\displaystyle \frac{\beta z}{\delta }\right)\right)}^{2}\right].\end{array}\end{eqnarray}$The values of Veff(z) at z=0 and $z\to \pm \infty $ read$ \begin{eqnarray}\begin{array}{rcl}{V}_{\mathrm{eff}}(0) & = & \left\{\begin{array}{l}+\infty ,\,0\lt k\lt 2\\ -\infty ,\,k\gt 2\\ \end{array}\right.\\ {V}_{\mathrm{eff}}(\pm \infty ) & = & 0.\end{array}\end{eqnarray}$We can find the effective potential tends to be zero when $z\to \pm \infty $. In the case of 0<k<2, the values of ${V}_{\mathrm{eff}}(z)\to +\infty $ when $z\to 0$, and the potential has a potential barrier. Otherwise, ${V}_{\mathrm{eff}}(z)\to -\infty $ when $z\to 0$, and the potential has a potential well. From the Schrödinger-like equation (20), the solutions of the KR field zero modes are given by$ \begin{eqnarray}{\chi }_{(0)}(z)\propto {{\rm{e}}}^{-4A(z)}{\rm{\Omega }}{\left(z\right)}^{-1}.\end{eqnarray}$In order to localize the zero modes on the brane, they should satisfy the normalization condition$ \begin{eqnarray}{\int }_{-\infty }^{+\infty }{{\rm{e}}}^{7A(z)}f(\phi ){\chi }_{(0)}(z){\chi }_{(0)}(z){{\rm{\Omega }}}^{2}(z){\rm{d}}z\lt \infty ,\end{eqnarray}$which can be converted into$ \begin{eqnarray}{\int }_{-\infty }^{+\infty }{{\rm{e}}}^{-8A(z)}{{\rm{\Omega }}}^{-2}(z){\rm{d}}z\lt \infty .\end{eqnarray}$The integrand function is expressed as$ \begin{eqnarray}\begin{array}{l}{{\rm{e}}}^{-8A(z)}{{\rm{\Omega }}}^{-2}(z)\\ =\,{{\rm{sech}} }^{-\delta }\left(\displaystyle \frac{\beta z}{\delta }\right){\left[\sqrt{-3\delta (\delta -1)}{\tan }^{-1}\left(\sinh \left(\displaystyle \frac{\beta z}{\delta }\right)\right)\right]}^{k}.\end{array}\end{eqnarray}$When $z\to \infty $, ${\rm{sech}} {\left(\tfrac{\beta z}{\delta }\right)}^{-\delta }\to 0$ and ${\tan }^{-k}\left(\sinh \left(\tfrac{\beta z}{\delta }\right)\right)\to {\left(\tfrac{\pi }{2}\right)}^{k}$ with δ<0. The integral in equation (29) is finite when δ<0, however, the background solution requires 0<δ<1. Thus, the value of δ does not satisfy the normalization condition and this coupling fails to localize the KR field on the thick dS brane.
3.2. Case II: $f(\phi )={\cos }^{k}\phi $
Next we consider $f(\phi )={\cos }^{k}\phi $, for this coupling the effective potential is given by$ \begin{eqnarray}\begin{array}{l}{V}_{\mathrm{eff}}(z)\\ =\,\displaystyle \frac{{\beta }^{2}(\delta -k){{\rm{sech}} }^{2}\left(\tfrac{\beta z}{\delta }\right)\left(-\delta +(\delta -k)\cosh \left(\tfrac{2\beta z}{\delta }\right)+k+4\right)}{8{\delta }^{2}},\end{array}\end{eqnarray}$which has the asymptotic behavior of a finite square-well like potential: ${V}_{\mathrm{eff}}(z\to \pm \infty )\to \mathrm{constant}$ and ${V}_{\mathrm{eff}}(0)=\tfrac{{\beta }^{2}(\delta -k)}{2{\delta }^{2}}$. Since Veff(0)<0 with δ<k and the value of ${V}_{\mathrm{eff}}(z\to 0)$ is a positive constant at infinity (this kind of potential is also called Pöschl–Teller (PT)-like), the effective potential provides a mass gap to separate the zero mode from the KK modes. Figure 3 gives a detailed description of the effect of the parameters β, δ and k on the effective potentials. It can be seen that, with the increase of β and k, or the decrease of δ, the effective potential becomes deeper and thinner.
Figure 3.
New window|Download| PPT slide Figure 3.Plots of the effective potentials Veff(z) for different values of β, δ and k.
The expression of the zero mode χ(0)(z) in this case reads$ \begin{eqnarray}{\chi }_{(0)}(z)\propto {\cosh }^{\tfrac{\delta -k}{2}}\left(\displaystyle \frac{\beta z}{\delta }\right),\end{eqnarray}$and the shapes of the effective potential Veff and the zero mode χ0(z) are shown in figure 4. Similar to the case I, the normalization condition is$ \begin{eqnarray}\int {\chi }_{(0)}^{2}(z){\rm{d}}z\lt \infty .\end{eqnarray}$When $z\to \infty $, ${\cosh }^{\delta -k}\left(\tfrac{\beta z}{\delta }\right)\to 0$ with the condition $\delta -k\lt 0$. We have known the background solution requires 0<δ<1. Therefore, in order to localize the KR field on the brane, we require the parameter k satisfies the condition $k\gt \delta \in (0,1)$.
Figure 4.
New window|Download| PPT slide Figure 4.Plots of the effective potentials Veff(z) (the blue line) and the zero mode χ(0)(z) (the dashed red line) for coupling II. The parameters are set to β=0.2, δ=0.2, k=3.
Next, by introducing the parameters $h=\beta /\delta ,\alpha \,=k-\beta $, the Schrödinger-like equation in equation (18) is derived as:$ \begin{eqnarray}\left[-{\partial }_{z}^{2}-\left(\displaystyle \frac{\alpha }{2}+\displaystyle \frac{{\alpha }^{2}}{4}\right){h}^{2}{{\rm{sech}} }^{2}({hz})\right]{\chi }_{(n)}(z)={E}_{n}{\chi }_{(n)}(z),\end{eqnarray}$where the energy is ${E}_{n}={m}_{n}^{2}-{\alpha }^{2}{h}^{2}/4$. If we further perform the following rescaling of the fifth coordinate v=hz, and suppose $\tau =\alpha /2+{\alpha }^{2}/4$, equation (34) can be written in the canonical form:$ \begin{eqnarray}\left[-{\partial }_{v}^{2}-{\tau }^{2}{{\rm{sech}} }^{2}(v)\right]{\chi }_{(n)}(z)={E}_{n}{\chi }_{(n)}(z),\end{eqnarray}$where the energy now reads ${E}_{n}={m}_{n}^{2}/{h}^{2}-{\alpha }^{2}/4$. If the Schrödinger equation has a modified PT potential, it has a exactly energy spectrum of bound states with $\tau =n(n+1)$, where n is the number of bound states. In this case the energy spectrum of bound states reads$ \begin{eqnarray}{E}_{n}=-{h}^{2}{\left(\displaystyle \frac{\alpha }{2}-n\right)}^{2},\end{eqnarray}$or, in terms of the squared mass:$ \begin{eqnarray}{m}_{n}^{2}=n(\alpha -n){h}^{2}.\end{eqnarray}$By solving equation (34), we get the general solution of massive KK modes as$ \begin{eqnarray}{\chi }_{(n)}={C}_{1}{P}_{\tfrac{\alpha }{2}}^{u}(\tanh ({hz}))+{C}_{2}{Q}_{\tfrac{\alpha }{2}}^{u}(\tanh ({hz})),\end{eqnarray}$where C1 and C2 are arbitrary constants and $u=\sqrt{\tfrac{{a}^{2}}{4}-\tfrac{{m}_{n}^{2}}{{h}^{2}}};$${P}_{\tfrac{\alpha }{2}}^{u}$ and ${Q}_{\tfrac{\alpha }{2}}^{u}$ are associated Legendre functions of first and second kind, respectively. Changing α and h, we can get a series of KK mode massive spectrum. Here the mass spectra mn2 of the KK mode are listed as:$ \begin{eqnarray}{m}_{n}^{2}=(0,3),\,\mathrm{for}\,\alpha =4,h=1;\end{eqnarray}$$ \begin{eqnarray}{m}_{n}^{2}=(0,5,8),\,\mathrm{for}\,\alpha =6,h=1;\end{eqnarray}$$ \begin{eqnarray}{m}_{n}^{2}=(0,7,12,15),\,\mathrm{for}\,\alpha =8,h=1;\end{eqnarray}$$ \begin{eqnarray}{m}_{n}^{2}=(0,28,48,60),\,\mathrm{for}\,\alpha =8,h=2;\end{eqnarray}$$ \begin{eqnarray}{m}_{n}^{2}=(0,63,108,135),\,\mathrm{for}\,\alpha =8,h=3.\end{eqnarray}$As an example, the shapes of the effective potential and the mass spectra for α=8, h=2 are shown figure 5, which reveal that the massive KK modes asymptotically turn into continuous plane waves when m2>60.
Figure 5.
New window|Download| PPT slide Figure 5.Plots of the effective potential of scalar KK modes and the mass spectra (dashed red lines) for KR field with the parameter α=8, h=2.
The wave functions of four bound KK modes are shown in figure 6 for the parameters α=8 and h=2.
Figure 6.
New window|Download| PPT slide Figure 6.Plots of the bound KK modes for KR field. The parameters are set to α=8 and h=2.
The effective potentials for the former two couplings do not produce the resonant structure of KR field. Thus an appropriate form of f(φ) is needed to generate the resonant mode of KR field. In order to obtain the resonant state we choose a relatively complicated coupling f(φ),$ \begin{eqnarray}f(\phi )={\left({\sinh }^{2}\left(\displaystyle \frac{\beta z}{\delta }\right)+1\right)}^{-\tfrac{\delta }{2}}/{\left({\sinh }^{-1}{\left(\sinh \left(\displaystyle \frac{\beta z}{\delta }\right)\right)}^{2}+\displaystyle \frac{1}{{\delta }^{2}}\right)}^{-k},\end{eqnarray}$which will bring us some different results. From equation (17), the function Ω(z) for this case reads$ \begin{eqnarray}{\rm{\Omega }}(z)={{\rm{e}}}^{-\tfrac{7A(z)}{2}}{{\rm{sech}} }^{-\tfrac{k}{2}}\left(\displaystyle \frac{{az}}{b}\right),\end{eqnarray}$and the effective potential of the Schrödinger-like equation has the form:$ \begin{eqnarray}\begin{array}{rcl}{V}_{\mathrm{eff}}(z) & = & {\left(4A^{\prime} (z)+\displaystyle \frac{{\rm{\Omega }}^{\prime} (z)}{{\rm{\Omega }}(z)}\right)}^{2}-\left(4A^{\prime} (z)+\displaystyle \frac{{\rm{\Omega }}^{\prime} (z)}{{\rm{\Omega }}(z)}\right)^{\prime} \\ & = & \displaystyle \frac{{\beta }^{2}k\left({\beta }^{2}(k-1){z}^{2}+1\right)}{{\left({\beta }^{2}{z}^{2}+1\right)}^{2}}.\end{array}\end{eqnarray}$It should be noted that the effective potential behaves as a potential well with negative k, and is independent of the brane thickness parameter δ. The shape of effective potential is plotted in figure 7. As shown, with the increase of β or the decrease of k, the potential well becomes thinner and deeper. The potential is a modified volcano-type potential since it has the following asymptotic behaviors: it tends to zero at $z\to \infty $, and at z=0, the potential reaches its minimum ${\beta }^{2}k$ when k<0. For this type of volcano-like potential, there is no mass gap to separate zero mode from the KK massive modes, but there may exist some resonant modes of KR field. For the zero mode, the normalization condition is$ \begin{eqnarray}{\int }_{-\infty }^{+\infty }{\left(1+{z}^{2}{\beta }^{2}\right)}^{k}{\rm{d}}z\lt \infty ,\end{eqnarray}$which is finite for $k\lt -\tfrac{1}{2}$. So the zero mode can be localized on the brane under the condition $k\lt -\tfrac{1}{2}$.
Figure 7.
New window|Download| PPT slide Figure 7.Plots of the effective potential for KR field with different parameters k and β.
To find the resonant states of KR field, we use the numerical method given in [38–40] where a function as the relative probability was proposed:$ \begin{eqnarray}P=\displaystyle \frac{{\int }_{-{z}_{c}}^{{z}_{c}}| {\chi }_{n}(z){| }^{2}{\rm{d}}z}{{\int }_{-{z}_{\max }}^{{z}_{\max }}| {\chi }_{n}(z){| }^{2}{\rm{d}}z},\end{eqnarray}$where 2zc is about the width of the thick brane and zmax is set to ${z}_{\max }=10{z}_{c}$. For instance, we set the parameters as β=0.2, k=−21 and find some numerical results. When the wave functions are either even-parity or odd-parity, the shapes of the relative probability are plotted respectively in figure 8. Corresponding to even-parity or odd-parity, there exist two or one resonant modes, and the masses of resonances are marked as m2=2.908, 4.524 or 3.839. The corresponding wave functions of resonances for even-parity or odd-parity KR field are plotted in figure 9. If we choose another two sets of parameters as β=0.32, k=−21, and β=0.2, k=−15, the shapes of the relative probability for even-parity state are plotted in figure 10. Comparing figure 8 with figure 10, we find the mass of the first resonant mode increases with β and $| k| $. In more detail, we list the numerical results of the mass spectra for the first even-parity resonance in table 1.
Figure 8.
New window|Download| PPT slide Figure 8.Plots of the relative probability for even-parity or odd-parity KR field with the parameters k=−21 and β=0.2.
Figure 9.
New window|Download| PPT slide Figure 9.Plots of the wave function of even-parity or odd-parity resonances with k=−21 and β=0.2.
Figure 10.
New window|Download| PPT slide Figure 10.Plots of the relative probability of resonances for even-parity KR field with two sets of parameters k and β.
Table 1. Table 1.The numerical results for the mass of first even-parity resonance with different β and k.
In this paper, in order to localize the KR field on a non-flat dS brane, we propose three kinds of coupling function, only two of them support the localization of zero mode, and the third one supports the resonant structure of massive modes. There are three parameters β, δ and k in this model, which parameterize respectively the expanding of the dS brane, the thickness of the brane, and the power of the coupling function. We find that the two parameters δ and k control the localization of zero modes, namely,the first coupling does not guarantee the localization of KR tensorial zero mode because δ breaks the localization condition δ<0, for the second case, the effective potential is PT-like potential, and the zero mode of KR field can be localized on the brane under the condition k>δ, for the third case, the zero mode of KR field can be localized on this thick brane under the condition k<−1/2. Moreover, another two parameters β and k determine the resonant state of KR tensor field. When k<0, the effective potential is a volcano-like one which indicates the possibility of appearance of resonant modes, and the volcano-like effective potential gets thinner and deeper when the parameters β and $| k| $ become larger. The numerical analysis of resonant state shows the mass of the even-parity resonant mode also increases with β and $| k| $ as shown in table 1.
Acknowledgments
We sincerely thank Professor Yu-Xiao Liu and Drs Tao-tao Sui, Qin-Tan and Zheng-quan Cui for useful discussions. This work was supported by the National Natural Science Foundation of China (Grant No. 11 705070) and the Fundamental Research Fund for Physics of Lanzhou University (No. Lzujbky-2019-ct06).
,1,2,31Institute of Theoretical Physics, 
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