A few years ago, Campos investigated the critical phenomena of thick branes in warped spacetimes [Phys. Rev. Lett. {\bf 88} (2002) 141602]. Inspired by his work, we consider a toy model of thick branes generated by a real scalar field with the potential $V(ϕ)=aϕ^2-bϕ^4+cϕ^6$, and investigate the variation of the mass parameter $a$ on the branes as well as the localization and resonances of fermions. An interesting result is found: there is a critical value for the mass parameter $a$, and when the critical value of $a$ is reached the solution of the background scalar field is not unique and has the shape of a double kink. This happens in both cases with and without gravity. It is also shown that the numbers of the bound Kaluza-Klein modes of fermions on the gravity-free brane and the resonant states of fermions on the brane with gravity increase with the value of $a$.
We investigate the problem of localization of gauge fields within the 5D standing wave braneworld model and show that in the case of increasing warp factor there exist normalizable vector field zero modes on the brane.
Three types of thick branes, i.e., Poincaré, de Sitter and Anti-de Sitter brane are considered. They are realized as the non-singular solutions of the Einstein equations with the non-trivial dilatons and the potentials. The scalar perturbations of these systems are also investigated. We find that the effective potentials of the master equations of the scalar perturbations are positive definite and consequently these systems are stable under the small perturbations.
We introduce new 6D standing wave braneworld model generated by gravity coupled to a phantom-like scalar field and investigate the problem of pure gravitational localization of matter fields. We show that in the case of increasing warp factor spin 0, 1, 1/2 and 2 fields are localized on the brane.
We consider a 5-dimensional brane world model with a single brane which is distinct from the well known Randall-Sundrum model. We discuss the similarities and differences between our brane model and the Randall-Sundrum brane model. In particular we focus on the localization of 5D fields with different spins -- spin 0, spin 1/2, spin 1 -- to the brane, and a self-consistent mass generation mechanism. We find that the brane model studied here has different (and in some cases superior) localization properties for fields/particles with different spins to the brane, as compared to the original 5-di…
This work deals with thick braneworld models, in an environment where the Ricci scalar is changed to accommodate the addition of two extra terms, one depending on the Ricci scalar itself, and the other, which takes into account the trace of the energy-momentum tensor of the scalar field that sources the braneworld scenario. We suppose that the scalar field engenders standard kinematics, and we show explicitly that the gravity sector of this new braneworld scenario is linearly stable. We illustrate the general results investigating two distinct models, focusing on how the brane profile is chang…
Generalized Friedmann equations governing the cosmological evolution inside a thick brane embedded in a five-dimensional Anti-de Sitter spacetime are derived. These equations are written in terms of four-dimensional effective brane quantities obtained by integrating, along the fifth dimension, over the brane thickness. In the case of a Randall-Sundrum type cosmology, different limits of these effective quantities are considered yielding cosmological equations which interpolate between the thin brane limit (governed by unconventional brane cosmology), and the opposite limit of an ``infinite'' b…
In this work, we present new soliton solutions for thick branes in $4+1$ dimensions. In particular, we consider brane models based on the sine-Gordon ($SG$), $φ^{4}$ and $φ^{6}$ scalar fields, which have broken $Z_{2}$ symmetry in some cases, and are responsible for supporting and stabilizing the thick branes. The origin of the symmetry breaking in these models resides in the fact that the modified scalar field potential may have non-degenerate vacuua. These vacuua determine the cosmological constant on both sides of the brane. We also study the geodesic equations along the fifth dimension, in…
In this paper we investigate the localization of fermions on asymmetric thick branes generated by two scalars $ϕ$ and $χ$. In order to trap fermions on the asymmetric branes with kink-like warp factors, the couplings with the background scalars $η\barΨF(χ,ϕ)Ψ$ are introduced, where $F(χ,ϕ)$ is a function of $ϕ$ and $χ$. We find that the coupling $η\barΨχϕΨ$ do not support the localization of 4-dimensional fermions on the branes. While, for the case $η\barΨχΨ+η'\barΨϕΨ$, which is the kink-fermion coupling corresponding to one-scalar-generated brane scenarios, the zero mode of left-handed fermio…
New soliton solutions for thick branes in 4 + 1 dimensions are considered in this article. In particular, brane models based on the sine-Gordon (SG), $φ^{4}$ and $φ^{6}$ scalar fields are investigated; in some cases $Z_{2}$ symmetry is broken. Besides, these soliton solutions are responsible for supporting and stabilizing the thick branes. In these models, the origin of the symmetry breaking resides in the fact that the modified scalar field potential may have non-degenerate vacuua and these non-degenerate vacuua determine the cosmological constant on both sides of the brane. At last, in order…
In this paper, we consider the localization of a five-dimensional gravitino field on $f(R)$-thick branes. We obtain the coupled chiral equations of the Kaluza-Klein (KK) modes of gravitinos with the gauge condition $Ψ_z=0$. The chiral equations of a gravitino's KK modes are found to be almost identical to those of the Dirac fermion. However, their chiralities are exactly opposite. The chiral KK modes of gravitinos could be localized in some types of $f(R)$-thick branes on introducing a coupling term. We investigate the localization of a gravitino on three types of $f(R)$-thick branes through a…
We investigate scalar and tensor fields in the brane model solution for the 5D space-time with standing gravitational waves in the bulk and show that even in the case of increasing warp factor there exist normalizable zero modes localized on the brane.
In this paper, we investigate the localization and the mass spectra of gravity and various bulk matter fields on a thick anti-de Sitter (AdS) brane, by presenting the mass-independent potentials of the Kaluza-Klein (KK) modes in the corresponding Schrödinger equations. For gravity, the potential of the KK modes tends to infinity at the boundaries of the extra dimension, which leads to an infinite number of the bound KK modes. Although the gravity zero mode cannot be localized on the AdS brane, the massive modes are trapped on the brane. The scalar perturbations of the thick AdS brane have been…
In this paper, we mainly investigate the problems of resonances of massive KK fermions on a single scalar constructed thick brane with a piecewise warp factor matching smoothly. The distance between two boundaries and the other parameters are determined by one free parameter through three junction conditions. For the generalized Yukawa coupling $η\barΨϕ^{k}Ψ$ with odd $k=1,3,5,...$, the mass eigenvalue $m$, width $Γ$, lifetime $τ$, and maximal probability $P_{max}$ of fermion resonances are obtained. Our numerical calculations show that the brane without internal structure also favors the appe…
A generalized version of the Randall-Sundrum model-2 with different cosmological constants on each side of a brane has been discussed. A possibility of replacing the singular brane by a configuration of a scalar field has been also considered, the Einstein equations for this setup were solved and stability of the solution discussed. It has been shown that under mild assumptions the relation between cosmological constants and the brane tension obtained in the brane limit does not depend on the particular choice of the regularizing profile of the scalar field.
In order to localize $U(1)$ gauge vector field on Randall-Sundrum-like braneworld model with infinite extra dimension, we propose a new kind of non-minimal coupling between the $U(1)$ gauge field and the gravity. We propose three kinds of coupling methods and they all support the localization of zero mode. In addition, one of them can support the localization of massive modes. Moreover, the massive tachyonic modes can be excluded. And our method can be used not only in the thin braneword models but also in the thick ones.
We study a codimension 2 braneworld in the Einstein Gauss-Bonnet gravity. We carefully examine the structure of possible singularities in the system which characterize the braneworld through matching conditions. Consequently, we find that the thickness of the brane can be incorporated as the distributional source, which we dub quasi-thickness. On the basis of our formalism, we analyze the linearized gravity and show the conventional Einstein gravity can be recovered on the brane. In the nonlinear regime, however, we find corrections due to the thickness and the bulk geometry. We also point out…
In this review, we give a brief introduction on the aspects of some extra dimension models and the five-dimensional thick brane models in extended theories of gravity. First, we briefly introduce the Kaluza-Klein theory, the domain wall model, the large extra dimension model, and the warped extra dimension models. Then some thick brane solutions in extended theories of gravity are reviewed. Finally, localization of bulk matter fields on thick branes is discussed.
It is shown that negative tension branes in higher dimensions may lead to an effective lower dimensional theory where the gauge-invariant vector fields associated with the fluctuations of the metric are always massless and localized on the brane. Explicit five-dimensional examples of this phenomenon are provided. Furthermore, it is shown that higher dimensional gauge fields can also be localized on these configurations with the zero mode separated from the massive tower by a gap.
In this paper, thick branes generated by mimetic scalar field are investigated. Three typical thick brane models are constructed and the linear tensor and scalar perturbations are analyzed. These branes have different inner structures, some of which are absent in general relativity. For each brane model, the solution is stable under both tensor and scalar perturbations. The tensor zero modes are localized on the branes, while the scalar perturbations do not propagate and they are not localized on the brane. As the branes split into multi sub-branes for specific parameters, the potentials of th…
It is shown that two gravitating scalar fields may form a thick brane in 5D spacetime. The necessary condition for the existence of such a regular solution is that the scalar fields potential must have local and global minima.
We consider Standard Model fields living inside a thick four dimensional flat brane embedded in a (possibly warped) five dimensional space-time and estimate the electromagnetic corrections to the anomalous magnetic moment (g-2) of the electron and muon by including virtual massive fermion and gauge boson states. Constraints on the mass of the ``excited'' states (or thickness of the brane-world) are obtained.
We study localization properties of fundamental fields which are coupled to one another through the gauge mechanism both in the original Randall-Sundrum (RS) and in the modified Randall-Sundrum (MRS) braneworld models: scalar-vector, vector-vector, and spinor-vector configuration systems. For this purpose we derive conditions of localization, namely the finiteness of integrals over the extra coordinate in the action of the system considered. We also derive field equations for each of the systems and then obtain their solutions corresponding to the extra dimension by a separation of variable me…
Thick branes obtained from a bulk action containing Gauss-Bonnet self-interactions are analyzed in light of the localization properties of the various modes of the geometry. The entangled system describing the localization of the tensor, vector and scalar fluctuation is decoupled in terms of variables invariant for infinitesimal coordinate transformations. The dynamics of the various zero modes is discussed and solved in general terms. Provided the four-dimensional Planck mass is finite and provided the geometry is everywhere regular, it is shown that the vector and scalar zero modes are not l…
I investigate models with scalar fields in 5 dimensions that exhibit thick-brane configurations with a non-trivial metric. I show that an appropriate coupling to the scalar curvature allows for periodic configurations, which, however, are unstable under small harmonic perturbations. A model for stabilizing these configurations is proposed and discussed.
In this paper, we investigate the localization of a five-dimensional vector field on a pure geometrical thick brane. In previous work, it was shown that a free massless vector field cannot be localized on such thick brane. Hence we introduce the interaction between the vector field and the background scalar field. Two types of couplings are constructed as examples. We get a typical volcano potential for the first type of coupling and a finite square-well-like potential for the second one. Both of the two types of couplings ensure that the vector zero mode can be localized on the pure geometric…
In this paper, we study the structure and gravitational resonances of thick branes generated by a mimetic scalar field in $f(R)$ gravity. We obtain several typical thick brane solutions for $f(R)=R+αR^2$. To study their stability, we analyze the tensor perturbation of the metric. It is shown that any thick brane model with $df/dR>0$ is stable and the graviton zero mode can be localized on the brane for each solution, which indicates that the four-dimensional Newtonian gravity can be restored. The effect of the parameter $α$ on the gravitational resonances is studied. As a brane splits into mul…
This work deals with the presence of thick branes in a model with two source scalar fields that interact with one another in a very specific way. The model is new, capable of generating kinklike configurations that engender important modifications in the energy density of the system, inducing the presence of symmetric and asymmetric structures inside the brane.
We investigate the localization problem of matter fields within the 5D standing wave braneworld. In this model the brane emits anisotropic waves into the bulk with different amplitudes along different spatial dimensions. We show that in the case of increasing warp factor there exist the pure gravitational localization of all kinds of quantum and classical particles on the brane. For classical particles the anisotropy of the background metric is hidden, brane fields exhibit standard Lorentz symmetry in spite of anisotropic nature of the primordial 5D metric.
Universe is considered as a brane in infinite (2+4)-space.It is shown that zero modes of all kinds of matter fields and 4-gravity are localized on the brane by increasing transversal gravitational potential.
Thickness corrections to static, axisymmetric Dirac-Nambu-Goto branes embedded into spherically symmetric black hole spacetimes with arbitrary number of dimensions are studied. First, by applying a perturbative approximation, it is found that the thick solutions deviate significantly in their analytic properties from the thin ones near the axis of the system, and perturbative approaches around the thin configurations can not provide regular thick solutions above a certain dimension. For the general case, a non-perturbative, numerical approach is applied and regular solutions are obtained for a…
We discuss the backreaction of a massless, minimally coupled, quantized scalar field on a thick, two-dimensional de Sitter (dS) brane as an extension of our previous work. We show that a finite brane thickness naturally regularizes the backreaction on the brane. The quantum backreaction exhibits a quadratic divergence in the thin wall limit. We also give a theoretical bound on the brane thickness, in terms of the brane self-consistency of the quantum corrected Einstein equation, namely the requirement that the size of the backreaction should be smaller than that of the background stress-energy…
Hints on the possible localization of fermions on double thick D-branes (Domain Walls) are found by analyzing the moduli space of parameters. Deeper analysis toward this direction might help to select phenomenologically plausible models. A new kind of condition for fermion localization is proposed. This might be useful in multi-brane-world scenarios, which are important when symmetry breaking is considered in the AdS/CFT formalism, as well as in curved brane-worlds.
We consider a thick brane model supported by two interacting scalar fields in 7D and 8D general relativity. Using the special type of a potential energy, we obtain numerically the regular asymptotically flat vacuum solutions. A possibility of obtaining the similar solutions for an arbitrary number of the extra spatial dimensions is estimated.
We study the linearization of a class of thick K-branes, namely, four-dimensional domain walls generated by a scalar field with particular nonstandard kinetic terms. The master equations for linear perturbations are derived from the point of view of both dynamical equations and quadratic action. The spectra of the canonical normal modes are studied using supersymmetric quantum mechanics. Our results indicate that the scalar perturbation is nonlocalizable in general. Conditions for stable $K$-branes are also found.
In this letter, we first investigate the gravity localization and mass spectrum of gravity KK modes on de Sitter and Anti-de Sitter thick branes. Then, the effective Newtonian gravitational potentials for these bent branes are discussed by the two typical examples. The corrections of the Newtonian potential turns out to be $ΔU(r)\sim 1/r^{2}$ at small $r$ for both cases. These corrections are very different from that of the Randall-Sundrum brane model $ΔU(r)\sim 1/r^{3}$.
We consider a spherical thick 3-brane immersed in a five-dimensional bulk spacetime. We demonstrate how the thick brane equation of motion expanded in powers of the thickness of the brane can be obtained from the expected junction conditions on the boundaries of thick brane with the two embedding spacetimes. It is shown that the finite thickness leads to a faster collapse of the spherical shell.
We study bulk fields in various multi-brane models with localized gravity. In the case of spin 0, 1/2 and 3/2 fields, the non-trivial background geometry itself can induce (multi-) localization of the zero modes along the extra dimension. The addition of appropriate mass terms controls the strength or/and the position of this localization and can induce (multi-) localization of spin 1 fields as well. We show that multi-brane models may also give rise to anomalously light KK modes which are also localized. The possibility of multi-localization in the context of supersymmetric brane world models…
In this letter we introduce a novel way to spontaneously localize particles (including gauge bosons) and gravitons kinetic terms on a four-dimensional brane via a gravitational mechanism. The model underlying this localization mechanism can be considered as a high-energy resolution of the so-called braneworlds scenario. In particular, we show how to construct a braneworld with induced gravity without pathologies. Finally, we argue that the brane is only stable if its own mass does not exceed a critical value related to the particles-gravity couplings.
We study the brane world scenario of a single brane (or a single stack of branes) with codimension higher than one. When the extra dimensions are not small, localization of gravity around the brane is needed in order to reproduce the observable four-dimensional gravity. We focus on the case of noncompact extra dimensions, where the possibility of localized gravity becomes non-trivial. We show that in large class of gravity models, localization of massless gravity is not possible for codimension-2 branes with at least one noncompact extra dimension. With additional mild assumptions on field bac…
We study brane worlds in an anisotropic higher-dimensional spacetime within the context of $f(R)$ gravity. Firstly, we demonstrate that this spacetime with a concrete metric ansatz is stable against linear tensor perturbations under certain conditions. Moreover, the Kaluza-Klein modes of the graviton are analyzed. Secondly, we investigate thick brane solutions in six dimensions and their properties. We further exhibit two sets of solutions for thick branes. At last, the effective potential of the Kaluza-Klein modes of the graviton is discussed for the two solved $f(R)$ models in higher dimensi…
In this letter we consider a specific model of braneworld with nonstandard dynamics diffused in the literature, specifically we focus our attention on the matter energy density, the energy of system, the Ricci scalar and the thin brane limit. As the model is classically stable and capable of localize gravity, as a natural extension we address the issue of fermion localization of fermions on a thick brane constructed out from one scalar field with nonstandard kinetic terms coupled with gravity. The contribution of the nonstandard kinetic terms in the problem of fermion localization is analyzed.…
We study two exactly solvable five-dimensional thick brane world models in pure metric $f(R)$ gravity. Working in the Einstein frame, we show that these solutions are stable against small linear perturbations, including the tensor, vector, and scalar modes. For both models, the corresponding gravitational zero mode is localized on the brane, which leads to the four-dimensional Newton's law; while the massive modes are nonlocalized and only contribute a small correction to the Newton's law at a large distance.
We investigate quantum fluctuations on a de Sitter (dS) brane, which has its own thickness, in order to examine whether or not the finite thickness of the brane can act as a natural cut-off for the Kaluza-Klein (KK) spectrum. We calculate the amplitude of the KK modes and the bound state by using the zeta function method after a dimensional reduction.We show that the KK amplitude is finite for a given brane thickness and in the thin wall limit the standard surface divergent behavior is recovered. The strength of the divergence in the thin wall limit depends on the number of dimensions, e.g., l…
We present thick de Sitter brane solutions which are supported by two interacting {\it phantom} scalar fields in five-, six- and seven-dimensional spacetime. It is shown that for all cases regular solutions with anti-de Sitter asymptotic (5D problem) and a flat asymptotic far from the brane (6D and 7D cases) exist. We also discuss the stability of our solutions.
Discrete symmetries are studied in warped space-times with one extra dimension. In particular, we analyze the compatibility of five- and four-dimensional charge conjugation, parity, time reversal and the orbifold symmetry Z_2 with localization of fermions on the four-dimensional brane-world and Lorentz invariance. We then show that, when a suitable topological scalar field (the ``kink'') is included, fermion localization is a consequence of (five-dimensional) CPT invariance.
In this paper we apply the five-dimensional $f(T)$ gravity with $f(T)=T+k T^n$ to brane scenario to explore the solutions under a given warp factor, and we find that the analytic domain wall solution will be a double-kink solution when the geometric effect of spacetime torsion is strongly enhanced. We also investigate the localization of fermion fields on the split branes corresponding to the double-kink solution.
In this paper, we investigate thick branes with a nonminimally coupled background scalar field, whose solution is a single-kink or a double-kink. The effects of the nonminimal coupling constant $ξ$ on the structure of the thick branes and the localization of gravity, fermions, scalars and vectors are discussed. It is shown that each brane will split into two sub-branes as increasing the nonminimal coupling constant $ξ$. By investigating the tensor perturbation equations of gravity and the general covariant Dirac equation of fermions, we find that both the gravity zero mode and left-chiral ferm…
We investigate the problem of pure gravitational localization of matter fields within the 5D standing wave braneworld generated by gravity coupled to a phantom-like scalar field. We show that in the case of increasing warp factor there exist normalizable zero modes of spin-0, -1/2, -1, and -2 fields on the brane.
In this work we study braneworlds generated by several scalar fields. The investigation describes the necessary formalism to examine models and evaluate the conditions for the stability of the gravitational sector under linear perturbations. In particular, we develop a mechanism that help us investigate distinct situations controlled by two and by three fields, focusing on how the fields can be used to modify the internal structure of the brane.
We examined 5d thick brane worlds constructed by a real scalar field. The solutions are obtained in terms of a simple form of smooth warp factor. For the case of dS brane, we found a disease of the solutions. For example, it is impossible to construct thick-brane worlds with our simple smoothing. This result is independent of supersymmetry or other symmetries.
We present the generalized Friedmann equations describing the cosmological evolution of a finite thick brane immeresed in a five-dimensional Schwarzschild Anti-de Sitter spacetime. A linear term in the density in addition to a quatratic one arises in the Friedmann equation, leading to the standard cosmological evolution at late times without introducing an ad hoc tension term for the brane. The effective four-dimensional cosmological constant is then uplifted similar to the KKLT effect and vanishes for a brane thickness equal to the AdS curvature size, up to the third order of the thickness. T…
It is known that by introducing the Yukawa coupling between the fermion and the background scalar field, a bulk spin-half fermion can be localized on general Randall-Sundrum braneworlds generated by a kinklike background scalar. However, this localization mechanism does not work anymore for Randall-Sundrum braneworlds generated by a scalar whose configuration is an even function of the extra dimension. In this paper, we present a new localization mechanism for spin-half fermions for such a class of braneworld models, in which extra dimension has the topology $S^1/Z_2$. By two examples, it is s…
We explore thick branes in $f(R,ϕ,X)$ gravity. We obtain the linear tensor perturbation equation of $f(R,ϕ,X)$ branes and show that the branes are stable against the tensor perturbations under the condition of $\frac{\partial f(R,ϕ,X)}{\partial R}>0$. In order to obtain thick brane solutions of the fourth-order field equations in this theory, we employ the reconstruction technique. We get exact solutions of the specific $f(R,ϕ,X)$ thick brane generated by a non-canonical scalar field. It is shown that the zero mode of the graviton for the thick brane is localized under certain conditions. This…
This work deals with new results on the Kalb-Ramond (KR) field localization in braneworld models. We consider a five-dimensional warped spacetime with an embedded 4D thick brane which is generated by two real scalar fields coupled with gravity (the so called Bloch brane). We find a KR field zero mode localized with the inclusion of the dilaton coupling. Analyzing the massive spectrum, we detected a series of resonant modes that arise from the solutions of the Schrödinger-like equation for KR field. The effects of the brane thickness and of the dilaton coupling over the resonance structures are…
In this work we investigate the issue of fermion localization and resonances in $(4,1)$-deformed branes constructed with one scalar field coupled with gravity. Such models provide us branes with internal structures that turns the gravitational interaction more effective for fermions aside the brane, increasing their lifetime. The coupling between the scalar field and spinors is a necessary condition for fermions to be localized on such branes. After performing a chiral decomposition of the five-dimensional spinor we found resonances with both chiralities. The correspondence between the spectra…
In this work we present a simple thick braneworld model that is generated by an intriguing interplay between a 5D cosmological constant with a de Sitter metric induced in the 3-brane without the inclusion of scalar fields. We show that 4D gravity is localized on this brane, provide analytic expressions for the massive Kaluza-Klein (KK) fluctuation modes and also show that the spectrum of metric excitations displays a mass gap. We finally present the corrections to Newton's law due to these massive modes. This model has no naked singularities along the fifth dimension despite the existence of a…
