Mass hierarchy and mass gap on thick branes with Poincaré symmetry
Nandinii Barbosa–Cendejas
Instituto de Física, Universidad de Guanajuato, Loma del Bosque 103, Frac. Lomas del Campestre, C.P. 37150 León, Guanajuato, México.
E-mail: Also at Alfredo Herrera–Aguilar, Ulises Nucamendi
Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo.
Edificio C–3, Ciudad Universitaria, C.P. 58040, Morelia, Michoacán, México.
, Israel Quiros
Depto. de Física, Universidad Central de las Villas, CP 54830, Santa Clara, Cuba.
February 20, 2001May 1, 2001November 27, 2001
We consider a scalar thick brane configuration arising in a 5D theory of gravity coupled to a self–interacting scalar field in a regular Riemannian manifold. We take as a starting point a known classical solution of the corresponding field equations and elaborate on the physics of the transverse traceless modes of linear fluctuations of the classical background, which obey a Schrödinger-like equation. We further consider a special case in which this equation can be solved analytically for any massive mode with m2≥0, allowing us to study in closed form the massive spectrum of Kaluza–Klein (KK) excitations. We obtain two bound states – the massless 4D graviton free of tachyonic instabilities and a massive KK excitation – as well as a continuous spectrum of massive KK modes. Between the massless and the excited modes there is a mass gap defined by the inverse of the brane thickness, getting rid of the potentially dangerous multiplicity of arbitrarily light KK modes. It is shown that due to this lucky circumstance, the solution of the mass hierarchy problem is much simpler and transparent than in the (thin) Randall–Sundrum (RS) two–brane configuration. Corrections to Newton's law in the thin brane limit are finally quoted.
Mass hierarchy, mass gap, thick branes, corrections to Newton's law, analytical approach
Even if large extra dimensions – in opposition to the KK picture where they are compact and small – have a spatial nature, they are quite similar to time in the sense that a 3–brane has only one point available along them. When considering time evolution of an object, it has no access to past or future, it can exist just at one instant along the arrow of time.
The idea of living in an Universe with extra dimensions is quite old and has evolved in several directions, depending on the problem under consideration: unification of gravity and electromagnetism [1, 2] , the cosmological constant problem [3] , dark matter [4] , the mass hierarchy problem [5, 6] , localization of gravity on thin [7] – [9] , and thick branes [10] – [14] , etc. Moreover, localization of 4D gravity [15] – [18] as well as matter fields [19] on scalar thick brane configurations was presented in the framework of Weyl geometries.
Within this context, an alternative mechanism to compactification and a solution to the mass hierarchy problem were proposed in [6] – [8] using a 5D world described by gravity, a cosmological constant and a pair of thin branes. However, this setup leads to a singular 5D manifold of orbifold type. Quite soon there have appeared several models that attempted to solve this crucial problem from the gravitational point of view by smoothing out the brane configurations in several ways [10] – [14] , [20] .
In this paper we continue previous work done on localization of 4D gravity on thick branes with Poincaré symmetry and we show that the mass hierarchy problem can be solved in a much simpler and more transparent way than in the Randall–Sundrum (combined with Lykken–Randall) thin brane approach [7, 8] . The novel feature is an analog quantum–mechanical potential of linearized (around a 4D Poincaré–invariant background) 5D gravity, that asymptotes to a positive constant value and leads to a mass gap in the KK-spectrum of the gravitational fluctuations. Besides, the present thick brane approach represents a better alternative to solve the mass hierarchy than the RS two–brane model [7] , since the TeV brane can have positive tension, contrary to RS.
We consider, first, a thick brane classical solution, and make a suitable ansatz for its linear perturbations. Then, as customary, we recast the problem into the solution of a Schrödinger–like equation with an analog quantum–mechanical potential VQM defined by the curvature of the spacetime. Following [18] we consider a special case in which this potential constitutes a modified Pöschl--Teller one. We then show that 4D gravity can be recovered by solving this equation for the massless zero mode. A qualitative (physical) analysis of the behavior of the massive KK-modes is performed by studying the structure of the analog quantum mechanical potential. We point out that there are two bound states: the massless one corresponding to a localized 4D graviton free of tachyonic instabilities, and a massive excitation mode. These modes are separated by a mass gap that is completely defined by an integration constant inversely proportional to the thickness of the brane.111In [18] it was shown that this constant is indeed proportional to the square root of the self–interacting coupling constant of the scalar field. There is also a continuous spectrum of delocalized massive modes. We get more insight into the physics of these modes by analytically solving the corresponding Schrödinger-like equation. Explicit expressions for the massive bound state as well as for the continuous tower of massive KK-excitations are given. The latter are put in terms of associated Legendre functions of first and second kind, allowing us to analytically study the behaviour of massive modes, contrary to usual numerical approaches (see, e.g., [11] , [14] ).
By considering an scenario in which, in addition to the thick brane, a (in principle thin, positive tension) probe brane – where the Standard Model (SM) particles are trapped – is located at an appropriated distance away from the thick brane, we are able to solve the mass hierarchy problem by following the procedure of References [7, 8] . Due to the existence of the mass gap in the graviton KK–spectrum, the solution is much simpler and transparent than in thin RS brane configuration. Concluding remarks are finally given.
2 Poincaré–invariant Solution
Consider the 5–dimensional Riemannian action given by [10, 11] (we take M3∗=1/8 for the time being)
S5=∫d5x√|G|[14R5−12(∇ϕ)2−V(ϕ)], (1)
where ϕ is a bulk scalar field and V(ϕ) is a self–interacting potential for the scalar field. We shall study a solution which preserves 4–dimensional Poincaré invariance with the metric
ds25=e2A(y)ηnmdxndxm+dy2, (2)
where e2A(y) is the warp factor of the metric and depends just on the fifth dimension y, m,n=0,1,2,3. We do not go into details about obtaining solutions to this system (see [10, 11] for details), but just quote the following concrete solution
e2A={cos[a(y−y0)]}b,ϕ=√3b2ln{sec[a(y−y0)]+tan[a(y−y0)]}. (3)
In [17] it was also shown that if the range of the fifth dimension is [−π/2,π/2], such a solution describes a single thick brane located at y0, where a characterizes the width of the warp factor Δ∼1/a in the case when the constant b>0. Thus, we have considered just the positive definite part of the cos function in order to avoid sign changes in the signature of the metric. This solution corresponds to the following self–interacting potential
V=3a2b8{sec2[a(y−y0)]−2btan2[a(y−y0)]}, (4)
which, after inverting (3), can be explicitly written as a function of the scalar field:
V(ϕ)=3a2b24{1+1−2bbcosh2[2ϕ√3b]}. (5)
3 Fluctuations of the metric
Let us study the metric fluctuations hmn of (2) given by
ds25=e2A(y)[ηmn+hmn(x,y)]dxmdxn+dy2. (6)
In the general case one must also consider the fluctuations of the scalar field when treating fluctuations of the classical background metric since they are coupled, however, following [10] , we shall just study the transverse traceless modes of the background fluctuations hTmn since they decouple from the scalar perturbations.
We perform the coordinate transformation
dz=e−Ady, (7)
in order to get a conformally flat metric. Thus, the equation for the transverse traceless modes of the metric fluctuations hTmn reads [10, 11]
(∂2z+3A′∂z+□)hTmn=0, (8)
where A′=dA/dz. This equation supports a massless and normalizable 4D graviton given by hTmn=Cmneipx, where Cmn are constant parameters and p2=m2=0.
In order to recast equation (8) into a Schrödinger's equation form, we adopt the ansatz
hTmn=eipxe−3A/2Ψmn(z)
and get the equation
[∂2z−VQM(z)+m2]Ψ=0, (9)
where we have dropped the subscripts in Ψ, m is the mass of the KK excitation modes, and the analog quantum mechanical potential, which is completely defined by the curvature of the manifold, reads
VQM(z)=32∂2zA+94(∂zA)2. (10)
In the particular case when b=2 we can invert the coordinate transformation (7) and explicitly express the variable y in terms of z yielding the equality:
cos[a(y−y0)]=sech(az),
practically decompactifying the fifth dimension and sending the singularities that were present at −π/2a and π/2a to spatial infinity.
In terms of z, the function A(z) adopts the following form
A(z)=lnsech(az),
while the analog quantum mechanical potential becomes [18]
VQM(z)=3a24{3−5sech2(az)}. (11)
Within the framework of the Quantum Mechanics, the spectrum of eigenvalues m2 parameterizes the spectrum of graviton masses that a 4–dimensional observer standing at z=0 sees. First of all we note that by looking at the asymptotical behavior of the effective quantum mechanical potential, it approaches the positive value VQM(∞)=9a2/4, a fact that guaranties the existence of a mass gap between the zero mode bound state and the first KK massive mode [12] . This situation is quite generic when considering de Sitter 3–branes (see [21] ). In fact, the Schrödinger equation can be solved analytically for massless and massive modes m2≥0, yielding two bound states [18] : a normalizable eigenfunction in the form
Ψ0=C0 sech3/2(az) (12)
and an excited massive mode given by the expression
Ψ1=C1 sinh(az)sech3/2(az), (13)
where C0 and C1 are normalization constants. The function (12) is the lowest energy eigenfunction of the Schrödinger equation (9) since it has no zeros, and can be interpreted as a massless 4D graviton with no tachyonic instabilities from modes with m2<0, whereas the function (13) represents a normalizable massive graviton with mass m=√2a localized on the brane as well. Thus, there exists a mass gap between these two bound states, a fact that eliminates the dangerous presence of arbitrarily light KK excitations, sending them to energies corresponding to the scale of a (see below).
There is also a continuous spectrum of massive modes with m≥3a/2. Surprisingly, in this special case, the Schrödinger equation can be integrated as well for these massive KK fluctuations and we get their explicit expression in terms of the associated Legendre functions of first and second kind of degree 3/2 and imaginary order μ=iρ [18] :
Ψ=k1 Piρ3/2[tanh(az)]+k2 Qiρ3/2[tanh(az)], (14)
where k1 and k2 are integration constants and ρ reads
ρ=√(m/a)2−9/4.
This fact allows us to analytically study the behaviour of the massive modes of the spectrum of KK excitations, a scarce phenomenon when considering smooth brane configurations with Poincaré symmetry since usually the coordinate transformation (7) cannot be explicitly inverted and then, the Schrödinger equation cannot be integrated for m2≠0 in the language of the coordinate z and one is forced to make use of numerical analysis (see [10, 11] , and [14] , for instance).
When the mass reaches the critical value 2m=3a, the order vanishes μ=0 and the associated Legendre functions simply become Legendre functions of first and second kind, respectively. Legendre functions of first kind are finite and normalizable at spatial infinity |z|⟶∞, hence we can set k2=0 when considering normalizability of this massive mode of KK fluctuations, getting the following expression for them
Ψ=k1 P3/2[tanh(az)]. (15)
Finally, in the case when 2m>3a we have a continuous spectrum of eigenfunctions that describe plane waves as they approach spatial infinity [18] :
These are precisely the massive modes that give rise to small corrections to the Newton law in 4D flat spacetime [12] .
Thus, the analytical study of the linear metric perturbations reflects the fact that there exists a zero mode separated from a massive excitation and a continuous spectrum of massive KK modes by a mass gap determined by the value of the constant m=√2a, a fact that, indeed, guarantees the normalizability of the 4D graviton zero mode.
4 Mass hierarchy problem
In order to be able to aim at the mass hierarchy problem within the present thick brane scenario, one has to add a (in principle thin, positive tension) probe brane some distance away from the location of the thick (Planck) brane222Since there is no precise meaning to the term "thick brane location", one could think of it as the location z=0 (y=y0) where the graviton wave–function is peaked. – where four–dimensional gravity is bound – in the z–direction. It is supposed, besides, that the SM particles are trapped in the (thin, positive tension) probe brane. In order to get the four-dimensional effective Planck scale, we replace the Minkowski metric by a four-dimensional metric ¯¯¯gmn in equation (6), leading to an effective four-dimensional action after integrating over z (y) in (1). As customary we look at the curvature term from which one can derive the scale of the gravitational interactions
Seff⊃2M3∗∫d4x∫∞−∞dz√|¯¯¯g|e3A¯¯¯¯R, (17)
where we performed the coordinate transformation dy=eAdz, and ¯¯¯¯R is the four–dimensional Ricci scalar. The effective Plack mass can be straightforwardly calculated:
M2pl=2M3∗∫∞−∞dze3A=πaM3∗, (18)
so that, if we take M∗∼a∼Mpl, then the zero mode Ψ0 is coupled correctly to generate four–dimensional (Newtonian) gravity.
Suppose the TeV probe brane were located at some definite position z0 in the extra–space, then the SM particles see the metric:
gSMab=e2A(z0)¯¯¯gab=sech2(az0)¯¯¯gab. (19)
Hence, following the approach of Reference [7] , the physical mass scales are set by a symmetry–breaking scale
m=sech(az0)m0. (20)
To produce TeV physical mass scales from fundamental Planck mass parameters, we need
sech(az0)≈TeV/Mpl,
i.e., az0≈ln2+16ln10≈38. Recalling that the thickness of the Planck (thick) brane is ∼1/a≈M−1pl, then the probe (TeV) brane has to be placed 40 times the thickness of the Planck brane away from the origin z=0, where the graviton wave-function is peaked. This means that even experiments probing energies far bigger than TeV scale would not be able to resolve the above mentioned separation.
There is another aspect of the problem that needs to be discussed. That standard 4D (Newtonian) gravity is reproduced on the thick (Planck) brane, is clear from the fact that there is a mass gap of energy m=√2a∼Mpl between the stable (ground state) graviton Ψ0 and the normalizable massive graviton Ψ1, that is also bound to the Planck brane. The continuous modes have masses bigger than Ψ1 and, since these are delocalized, the corresponding amplitudes are suppressed at the origin with respect to the amplitudes Ψ0 and Ψ1. However, since the probe brane where the SM particles live is located away from the origin, one could think that the massive continuous modes could play an important role in modifying gravitational interactions at the TeV brane, as long as the ratio of the corresponding amplitudes to Ψ0 (and Ψ1) grows as one recedes from z=0 in the extra-space (an effect related with the localization of Ψ0 and Ψ1 on the thick brane and with the delocalization of the KK–continuum). This is true (even if we need only a separation z0≈40/Mpl from the Planck brane to generate the correct hierarchy), but recall that the masses of the corresponding continuous modes are of the order of the Planck mass and bigger. Consequently, a 4D observer placed at the TeV brane would not be able to detect those modes. As the probe brane position recedes from the origin the energies accessible to probe brane observers decreases, as well as the possibility to detect massive modes of the KK–continuum. The consequence is that, as long as correct Newtonian gravity is achieved at the Planck brane, gravity on a probe brane located at any position in the extra–space will be Newtonian as well, confirming the result obtained in [8] .
5 KK corrections to Newton's law
Let us consider the thin brane limit a→∞, where we can locate two test bodies at the center of the brane in the transverse direction and compute the corrections to Newton's law in 4D flat spacetime coming from the fifth dimension that are generated by the massive gravitons (see [18] for details) and can be expressed as follows [7, 12] :
U(r)∼M1M2r(G4+M−3∗e−m1r|Ψ1(z0)|2+M−3∗∫∞m0dme−mr|Ψμ(m)(z0)|2)
=M1M2r(G4+ΔG4), (21)
where the thin brane is located at z=z0 and represents the physical universe, G4 is the four–dimensional gravitational coupling, Ψ1 represents the wave function of the excited state, and Ψμ(m) denotes the continuous eigenfunctions.
In this limit, since the Ψ1(z0) wave function is odd, it does not contribute to the corrections, moreover, the integral of (21) is dominated by the small - ρ region and it can be well–approximated by expanding the prefactor of the exponential at ρ=0. In this way one obtains the following expression for the corrections ΔG4 to 4D Newton's law
ΔG4∼ M−3∗1∣∣Γ(−14)Γ(74)∣∣21re−32ar(1+O(1ar)). (22)
It is worth noticing that away from the thin brane limit, the correction will involve the massive bound state and the form of Newton's law will depend on the precise location of the two test bodies along the extra dimension. Corrections to Newton's law were obtained quite recently for brane models with a non–minimally coupled bulk scalar field in [22] .
6 Concluding Remarks
We have considered a particular scalar smooth brane generalization of the RS model in which 4D gravity is localized at a certain value of the fifth dimension. These field configurations do not restrict the 5–dimensional spacetime to be an orbifold geometry by smoothing out the singularities, and avoid the introduction of thin branes in the action by hand.
When studying linear metric perturbations, we obtain an analytic expression for the lowest energy eigenfunction which represents a single bound state that can be interpreted as a stable 4D graviton free of tachyonic modes with m2<0. There is also an excited KK massive mode separated from the massless graviton state by a mass gap determined by the scale of the constant a, inversely proportional to the thickness of the brane. The appearance of the mass gap eliminates the dangerous low–energy KK graviton excitations from the theory under consideration (in the considered special case). The gap is a consequence of the fact that the quantum analog potential of the Schrödinger-like equation, obtained after linearizing 5D gravity, asymptotes to a positive constant, unlike most part of related models, where the potential asymptotes to zero (see, for instance, [10] – [14] and [20] ).
We get as well a continuous spectrum of massive KK modes that starts at m=3a/2 and turn into continuum plane waves as they approach spatial infinity. The massive modes of KK excitations are explicitly given in terms of associated Legendre functions of first and second kind, allowing for an analytical computations of corrections to Newton's law along the lines of [12] and [18] , in contraposition to implemented numerical approaches [11, 14] .
One of the main achievements of the present paper is that, following the procedure of [7, 8] , a resolution of the mass hierarchy is given by placing a positive–tension probe brane – where the SM particles are trapped – some distance away from the thick (Planck) brane in the extra–space. As already discussed, there is a mass gap of order of the Planck mass scale, that provides a clue to solving the mass hierarchy while recovering Newtonian gravity on the TeV brane in a clear and simple way.
7 Acknowledgements
Two of the authors (AHA and IQ) are really grateful to C. Germani, R. Maartens, and S.L. Parameswaran for fruitful and illuminating discussions while this investigation was carried out. This research was supported by grants CIC–UMSNH–4.8 Y 4.16, CONACYT 42949–F and 60060–J, COECYT, Instituto Avanzado de Cosmología (IAC), and the MES of Cuba. NBC acknowledges a PhD grant from CONACYT.
[1] G. Nordstrom, On the possibility of unifying the electromagnetic and the gravitational fields, Physik Z. 15 (1914) 504; On a theory of electricity and gravitation, ArXiv: physics/0702222.
[2] T. Kaluza, On the Problem of Unity in Physics, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math.Phys.) 1921 (966) ; O. Klein, Quantum Theory and Five–Dimensional Theory of Relativity, Z. Physik 37 (1926) 895; M. Gogberashvili, Four dimensionality in noncompact Kaluza-Klein model, Mod. Phys. Lett. A (14) 19992025.
[3] V.A. Rubakov and M.E. Shaposhnikov, Extra space–time dimensions: towards a solution of the cosmological constant problem, Phys. Lett. B 125 (1983) 139; M. Visser, An exotic class of Kaluza–Klein models, Phys. Lett. B 159 (1985) 22; E.J. Squires, Dimensional reduction caused by a cosmological constant, Phys. Lett. B 167 (1986) 286; M. Gogberashvili, Localizing gravity in extra dimensions, Eur. Phys. J. C 47 (2006) 857.
[4] J.E. Lidsey, T. Matos and L.A. Urena-Lopez, The inflaton field as selfinteracting dark matter in the brane world scenario, Phys. Rev. D 66 (2002) 023514; J.A.R. Cembranos, A. Dobado and A.L. Maroto, Brane world dark matter, Phys. Rev. Lett. 90 (2003) 241301; X. Chenand S.-H.Henry Tye, Heating in brane inflation and hidden dark matter, J. Cosmology Astroparticle Phys. 0606 (2006) 011; C.G. Boehmer and T. Harko, Galactic dark matter as a bulk effect on the brane, Class. and Quant. Grav. 24 (2007) 3191.
[5] N. Arkani–Hamed, S. Dimopoulos and G. Dvali, The hierarchy problem and new dimensions at a millimeter, Phys. Lett. B 429 (1998) 263; I. Antoniadis, N. Arkani–Hamed, S. Dimopoulos and G. Dvali, New dimensions at a millimeter to a Fermi and superstrings at a TeV, Phys. Lett. B 436 (1998) 257.
[6] M. Gogberashvili, Hierarchy problem in the shell universe model, Int. J. Mod. Phys. D 11 (2002) 1635.
[7] L. Randall and R. Sundrum, An alternative to compactification, Phys. Rev. Lett. 83 (1999) 4690; A large mass hierarchy from a small extra dimension, Phys. Rev. Lett. 83 (1999) 3370.
[8] J. Lykken and L. Randall, The shape of gravity, J. High Energy Phys. 0006 (2000) 014.
[9] J. Garriga and T. Tanaka, Gravity in the brane world, Phys. Rev. Lett. 84 2778 (2000); A. Kehagias and K. Tamvakis, Localized gravitons, gauge bosons and chiral fermions in smooth spaces generated by a bounce, Phys. Lett. B504, 38 (2001); R. Maartens, Brane world gravity, Living Rev. Rel. 7, 07 (2004); L.A. Gergely and R. Maartens, Asymmetric brane-worlds with induced gravity, Phys. Rev. D 71 (2005) 024032.
[10] O. De Wolfe, D.Z. Freedman, S.S. Gubser and A. Karch, Modeling the fifth dimension with scalars and gravity, Phys. Rev. D 62 (2000) 046008.
[11] M. Gremm, Four–dimensional gravity on a thick domain wall, Phys. Lett. B 478 (2000) 434; Thick domain walls and singular spaces, Phys. Rev. D 62 (2000) 044017.
[12] C. Csaki, J. Erlich, T. Hollowood and Y. Shirman, Universal Aspects of gravity localized on thick branes, Nucl. Phys. B 581 (2000) 309.
[13] R. Guerrero, A. Melfo and N. Pantoja, Selfgravitating domain walls and the thin wall limit, Phys. Rev. D 65 (2002) 125010; A. Melfo, N. Pantoja and A. Skirzewski, Thick domain wall space–time with and without reflection symmetry, Phys. Rev. D 67 (2003) 105003; D. Bazeia, F.A. Brito and A.R. Gomes, Locally localized gravity and geometric transitions, J. High Energy Phys. 0411 (2004) 070; O. Castillo–Felisola, A. Melfo, N. Pantoja and A. Ramirez, Localizing gravity on exotic thick three–branes, Phys. Rev. D 70 (2004) 104029; N. Barbosa–Cendejas and A. Herrera–Aguilar, Localizing gravity on thick branes: A Solution for massive KK modes of the Schroedinger equation, J. Phys. Conf. Ser. 68 (2007) 012021.
[14] D. Bazeia, A.R. Gomes and L. Losano, Gravity localization on thick branes: a numerical approach, arXiv:0708.3530 [hep-th].
[15] O. Arias, R. Cardenas and Israel Quiros, Thick brane worlds arising from pure geometry, Nucl. Phys. B 643 (2002) 187.
[16] N. Barbosa–Cendejas and A. Herrera–Aguilar, 4D gravity localized in non Z2–symmetric thick branes, J. High Energy Phys. 0510 (2005) 101.
[17] N. Barbosa–Cendejas and A. Herrera–Aguilar, Localization of 4D gravity on pure geometrical thick branes, Phys. Rev. D 73 (2006) 084022.
[18] N. Barbosa–Cendejas, A. Herrera–Aguilar, M.A. Reyes and C. Schubert, Mass gap for gravity lozalized on Weyl thick branes, arXiv:0709.3552 [hep-th].
[19] Y.–X. Liu, X.–H. Zhang, L.–D. Zhang, and Y.–S. Duan, Localization of Matters on Pure Geometrical Thick Branes, arXiv:0708.0065 [hep-th]; X.–H. Zhang, Y.–X. Liu and Y.–S. Duan, Localization of fermionic fields on braneworlds with bulk tachyon matter, arXiv:0709.1888 [hep-th] .
[20] D. Bazeia, F.A. Brito and J.R. Nascimento, Supergravity brane worlds and tachyon potentials, Phys. Rev. D 68 (2003) 085007; R. Koley and S. Kar, Scalar kinks and fermion localisation in warped spacetimes, Class. and Quant. Grav. 22 (2005) 753; D. Bazeia and L. Losano, Deformed defects with applications to braneworlds, Phys. Rev. D 73 (2006) 025016; S. Ghassemi, S. Khakshournia and R. Mansouri, Generalized Friedmann equations for a finite thick brane, J. High Energy Phys. 0608 (2006) 019; Thick planar domain wall: Its thin wall limit and dynamics, Int. J. Mod. Phys. D 16 (2007) 629; C. Bogdanos, A. Dimitriadis and K. Tamvakis, Brane models with a Ricci–coupled scalar field, Phys. Rev. D 74 (2006) 045003; V. Dzhunushaliev, Thick brane solution in the presence of two interacting scalar fields, gr–qc/0603020; K. Farakos and P. Pasipoularides, Brane world scenario in the presence of a non–minimally coupled bulk scalar field, J. Phys. Conf. Ser. 68 (2007) 012041; C. Bogdanos and K. Tamvakis, Brane Cosmological Evolution With Bulk Matter, Phys. Lett. B 646 (2007) 39; C. Bogdanos, Exact Solutions in 5–D Brane Models With Scalar Fields, J. Phys. Conf. Ser. 68 (2007) 012045; E. Abdalla, B. Cuadros–Melgar, A.B. Pavan, and C. Molina, J. Phys. Conf. Ser. 68 (2007) 012043; A.A. Saharian, A.L. Mkhitaryan, Wightman function and vacuum densities for a Z(2)–symmetric thick brane in AdS spacetime, J. High Energy Phys. 0708 (2007) 063; V. Dzhunushaliev, V. Folomeev, K. Myrzakulov and R. Myrzakulov, Thick brane in 7D and 8D spacetimes, arXiv:0705.4014 [gr-qc].
[21] M.K. Parikh and S.N. Solodukhin, De Sitter brane gravity: From close–up to panorama, Phys. Lett. B 503 (2001) 384; A. Wang, Thick de Sitter 3 branes, dynamic black holes and localization of gravity, Phys. Rev. D 66 (2002) 024024; M. Minamitsuji, W. Naylor and M. Sasaki, Quantum fluctuations on a thick de Sitter brane, Nucl. Phys. B 737 (2006) 121.
[22] K. Farakos, G. Koutsoumbas and P. Pasipoularides, Graviton localization and Newton's law for brane models with a non-minimally coupled bulk scalar field, Phys. Rev. D 76 (2007) 064025.