Localization and mass spectra of various matter fields on Weyl thin brane
It has been shown that the thin brane model in a five-dimensional Weyl gravity can deal with the wrong-signed Friedmann-like equation in the Randall–Sundrum-1 (RS1) model. In the Weyl brane model, there are also two branes with opposite brane tensions, but the four-dimensional graviton (the gravity zero mode) is localized near the negative tension brane, while our four-dimensional universe is localized on the positive tension brane. In this paper, we consider the mass spectra of various bulk matter fields (i.e., scalar, vector, and fermion fields) on the Weyl brane. It is shown that the zero modes of those matter fields can be localized on the positive tension brane under some conditions. The mass spectra of the bulk matter fields are equidistant for the higher excited states, and relatively sparse for the lower excited states. The size of the extra dimension determines the gap of the mass spectra. We also consider the correction to the Newtonian potential in this model and it is proportional to .
Over the last decades, there was a great interest to construct a unified theory in higher-dimensional spacetime. The higher-dimensional theory dates back to the work of Kaluza and Klein KK26 in the 1920s. They tried to unify electromagnetism with gravity by assuming that the component of the five-dimensional metric represents the electromagnetic potential KK26 . Subsequently, the brane world theories appear as an alternative theory to solve the gauge hierarchy and cosmological constant problems (see e.g., Refs. Rubakov ; RSmodel1 ; Randall2 ; Arkani-Hamed1998 ) with a fundamental idea that the visible universe is localized on a 3-brane embedded in a higher-dimensional bulk.
Most research as regards about brane worlds is based on a Riemann geometry Bajc2000 ; Fuchune ; Hatanaka1999 ; Das2008 ; Volkas2007 ; RandjbarPLB2000 ; Guo1103 ; Liu2009uca ; Shiromizu2000 . The non-Riemannian models can be classified within the language of metric-affine gravity Liu0708 ; Puetzfeld2005 ; Hehl1995 ; Hammond2002 . Generally, there are two types of generalized Riemann geometries, Riemann–Cartan geometry and Weyl geometry Puetzfeld2005 . Cartan introduced the torsion to represent the antisymmetric piece of the connection, which may be related to spin. When the connection includes a geometric scalar , there will be a nonmetricity part called , which is related to the scalar . In Riemann–Cartan geometry, the torsion , and the nonmetricity part . When , the Riemann–Cartan geometry can degenerate into Riemann geometry. The Weyl geometry is an affine manifold described by a pair (, ), where is the metric and is the “gauge” vector as the gradient of a scalar function . In the Weyl geometry, and . When , the Weyl geometry can also degenerate into Riemann geometry. As a generalization of Riemann, Weyl geometry has been received much attention in the studies in field theory and cosmology Barbosa2006 ; ChengPRL1998 ; Hochberg1991 ; MoonJCAP2010 . Moreover, in Refs. Barbosa2008 ; LiuJHEP2008 ; Correa1607 ; Arias0202 ; Cendejas0603 ; Liu0708 ; Liu0911 , the authors have studied the Weyl brane world scenario and obtained various solutions of Minkowski thick branes.
In the brane world scenario with all matter fields (scalar, vector, and fermion fields) propagating in the bulk, the zero modes of these bulk fields, i.e., the four-dimensional matter fields in the Standard Model, should be localized on the brane complying with the present observations. So, localization of various bulk
matter fields on a brane is an important and interesting issue.
References Bajc2000 ; Fuchune ; Furlong1407 showed that the zero mode of a free massless scalar field can be localized on the RS brane or its generalized branes.
In order to ensure that the zero mode of vector field can be localized on the brane, the case of generalized RS branes with more extra dimensions was considered Oda2000 ; Sakamura1607 , a dynamical mass term was included in gauge field localization zhao1406 ; alencer1409 , and in Refs. Araujo1406 ; Costa1304 ; Aguilar1401 ; Costa1501 ; sui1703 ; Chumbes1108 ; Cruz1211 ; Zhao1402 ; Alencar1005 ; Alencar1008 the authors introduced the interaction with the background scalar field.
Localization of the fermion zero mode on a brane is especially important. It has been demonstrated that localization mechanisms, e.g. Yukawa coupling Liu0708 ; Volkas2007 ; RandjbarPLB2000 ; Guo1103 ; Fuentevilla1412 ; Cendejas1503 ; Rocha1606 ; Brito1609 ; KoleyCQG2005 ; D. Bazeia0809 ; MelfoPRD2006 ; Almeida0901 ; Almeida2009 ; Davies0705 ; LBCastro1 ; LBCastro2 ,
derivative scalar Cfermion coupling
Liu1312 , and derivative geometrical–fermion coupling
Liu1701 , can ensure that the fermion zero mode can be localized on the brane.
In the RS1 model RSmodel1 , the authors assumed that there are two thin branes located at the boundaries of the extra dimension. One of them is the negative tension brane (called Tev brane, where our universe is resided), and the other is the positive tension brane (called the Planck brane, where the spin-2 gravitons are localized). The warping of the extra dimension in RS1 model can solve the famous gauge hierarchy problem, but the fundamental assumption that the our universe is localized on the negative tension brane can lead to a wrong-signed Friedmann-like equation Csaki99 ; Cline99 ; Shiromizu99 . The authors of Ref. Yang:2011pd presented a generalized RS1 model in the Weyl integrable geometry, which can give a correct Friedmann-like equation since our universe is assumed to be located on the positive tension brane. It was shown that the brane system is stable under the tensor perturbation and the gravity zero mode is localized near the negative tension brane, whereas the matter fields should be confined on the positive tension brane, since our universe is located on this brane.
In this paper, we will do further research on the localization of various bulk matter fields in this Weyl brane model and check whether the zero modes of these fields can be confined on the positive tension brane. We will consider a dilaton coupling between the bulk matter fields and the background scalar field. It will be found that the coupling is necessary for most of the bulk fields. For the fermion case, we consider the usual Yukawa coupling to obtain analytical wave functions of the fermion KK modes. This paper is organized as follows. In Sect. II, we give a brief review of the Weyl brane model proposed in Ref. Yang:2011pd . In Sect. III, we investigate localization and mass spectra of the scalar, vector, and fermion fields on the Weyl brane. In Sect. IV, gravitational fluctuations and correction to the Newtonian potential are discussed. Finally, we give a brief conclusion and summary in Sect. V.
Ii Review of Weyl brane model
The brane model we consider here is based on the five-dimensional Weyl geometric action where the gravity is non-minimally coupled with a background scalar field . The action is given by Yang:2011pd
Here is the five-dimensional Weyl Ricci scalar, is the fundamental scale of gravity, the Latin letters , and is a gradient of the Weyl scalar . In what follows it is convenient to denote the physical four coordinates by , and the extra dimension by .
In this frame the Weyl Ricci tensors are constructed from in a standard way,
where the Weyl affine connection is expressed as
with representing the Riemannian Christoffel symbol. Now with some algebra, the Weyl Ricci tensor can be separated into a Riemannian Ricci tensor part plus some other terms with respect to the Weyl scalar function, i.e.,
where the hatted magnitudes and operators are defined by the Christoffel symbol. Then from the action (1), the equations of motion are
Further, we can write the field equation (5) in the form of the Einstein equation by moving all Weyl scalar terms to the right-hand side to compose an effective energy-momentum tensor:
The authors in Refs. Barbosa2006 ; Barbosa2005 ; Barbosa2008 ; Vladimir0904 ; Vladimir0912 ; D. Bazeia1502 ; Marzieh1504 studied thick brane solutions, by considering the metric that satisfies four-dimensional Poincar invariance. The localization and mass spectrum problems of bulk matter fields on the Weyl Minkowski thick branes were discussed in Refs. Barbosa2005 ; Barbosa2008 ; LiuJHEP2008 ; Correa1607 ; Arias0202 ; Cendejas0603 ; Liu0708 ; Liu0911 . We are interested in the Weyl Minkowski thin branes with space , for which the metric with four-dimensional Lorentz invariance in a five-dimensional Weyl spacetime is given by
where is the conformal coordinate of the extra dimension with and it relates to the physical coordinate by a coordinate transformation . One assumes that the background scalar field only depends on the extra dimension.
According to the metric (8), the field equations in the bulk can be expressed in terms of the warp factor , and the scalar field ,as in the following set of equations:
where the prime denotes the derivative with respect to . The above field equations admit the following brane solution Yang:2011pd :
where , the parameter and .
The five-dimensional effective energy-momentum tensor is given as usual by
with the effective matter action. In this case the effective energy density is given by the null-null component of the energy-momentum tensor,
This expression directly shows that there are two delta functions appearing at each boundary, so they stand for two thin branes, namely a positive tension brane at the origin and a negative one at the boundary . These tensions compensate for the effects produced by the bulk component and hence ensure the existence of four-dimensional flat branes. So the brane configuration of this case is similar to that of the RS1 model. However, localization of massless graviton in these two brane scenarios will be quite different, and in Ref. Yang:2011pd one can suppose that the standard model fields are confined on the positive tension brane at , so this is crucial to overcome the severe cosmological problem of the RS1 model.
Iii Localization of various matter fields
In this section, we investigate the localization and mass spectra of various bulk matter fields, including the spin-0 scalar, spin-1 vector and spin-1/2 fermion fields on the Weyl brane. A natural mechanism for localizing spin-0 scalar and spin-1 vector fields on the brane involves the couplings with the background scalar field. For the case of spin-1/2 fermions, we will investigate the usual Yukawa coupling .
iii.1 Spin-0 scalar field
Firstly, we explore the localization condition and the mass spectrum for a massless scalar field on the Weyl brane by considering two types of coupling with the background scalar fields, dilaton coupling and Higgs potential coupling Liu2009uca .
iii.1.1 The dilaton coupling
Let us consider a massless scalar field coupled with the dilaton field via the following five-dimensional action:
where is a dimensionless coupling constant. For convenience, we rewrite the warp factor so that the metric (8) takes the form
Making the KK decomposition
with the quantity , and demanding that satisfies the four-dimensional massive Klein–Gordon equation , we obtain the equation of the extra-dimensional part of the scalar KK mode, which can be converted into the following Schrödinger-like equation:
where is the mass of the scalar KK mode and the effective potential reads
It is clear that the effective potential is only dependent on the warp factor. In fact, the Schrödinger-like equation (17) can be written as , where the Hamiltonian operator is given by with . Since the operator is positive definite, there are no KK modes with negative .
we get the effective low-energy theory in a four-dimensional familiar form
where we have used Eq. (17).
Due to the explicit forms of and , the expression of the effective potential corresponding to Eq. (18) is given by
where . The value of at is
In order to localize the scalar zero mode on the positive tension brane, the effective potential should be negative at , which results in the following condition:
By setting in Eq. (17), we easily get the normalized zero mode of the scalar field for :
which is indeed localized on the positive tension brane under the condition (23). When the extra dimension is infinite, the normalization condition means that the localization condition for the scalar zero mode is much stronger: . For , the normalized scalar zero mode reads as
and it is also localized on the positive tension brane. But if the extra dimension is infinite, the scalar zero mode is not normalized and hence we cannot get a localized scalar zero mode. When there is no coupling between the scalar and dilaton fields, i.e., the parameter , one can verify that the scalar zero mode is also localized on the positive tension brane when is finite.
In order to investigate the mass spectrum of the scalar KK modes, we redefine the following dimensionless parameters:
with which the Schrdinger-like equation (17) is rewritten as
where the effective potential reads as
The general solution of Eq. (27) is given in terms of the combination of Bessel functions
where is the normalization coefficient, and are the -dependent parameters, and are the Bessel functions of the first and second kinds.
Then we impose the two kinds of boundary condition (the Neumann boundary condition and the Dirichlet boundary condition) to calculate the mass spectra of scalar KK modes, respectively. For the the Neumann boundary condition , we get the mass spectrum which is determined by following condition:
where and are defined as:
And, for the Dirichlet boundary condition at the boundaries and , we get the mass spectrum of the scalar KK modes which satisfies the following relation:
where and are given by
From Eqs. (30) and (32), we see that the Neumann and Dirichlet boundary conditions give the similar expressions which determine the mass spectrum, whereas the orders of the Bessel functions are different as shown in (30) and (31). By tuning the parameters and , the same orders of Bessel functions can be obtained, and thus the expression of the mass spectrum derived by the Neumann boundary condition is equivalent to that derived by the Dirichlet boundary condition. Therefore, we only consider the Neumann boundary condition for later discussion.
According to Eq. (30), we plot the relation between and in Figs. 1, 2, 3, where the zero points represent the mass of the KK modes. Figure 1 shows that the effect of the parameter on the mass of the first massive scalar KK mode. It can be seen that the mass increases with . On the other hand, there are some singular (vertical) lines in these figures, which come from the zero points of in (31). Since the gaps between two arbitrary adjacent singular lines are almost the same for a set of parameters, we can call the average gap the period . For example, the periods for the three cases shown in Fig. 1 are 3.13, 3.22 and 3.42, respectively. The period for a special mass spectrum increases with . Figure 2 states the effect of the size of the extra dimension, , on the gap of the massive KK modes. It shows that the number of excited states in a single period increases with , and the gap of massive KK modes decreases with . An interesting phenomenon will appear if we set (and and at the same time). The massive KK modes will not appear in every period anymore for this case; however, they will appear after severe periods, as shown in Fig. 3. The mass spectra of the scalar KK modes are shown in Fig. 4 for different parameters. The mass gap between the zero mode and the first massive KK mode increases with the dilaton coupling constant and decreases with the size of the extra dimension . On the other hand, the mass spectrum is almost equidistant for the higher excited states, and relatively sparse for the lower excited states.
iii.1.2 The Higgs potential coupling
Next, we consider another coupling, the Higgs potential type coupling between a real scalar and the background scalar field . The five-dimensional action is assumed as Liu2009uca ,
where the Higgs potential is given by
Then, making the KK decomposition
and demanding that satisfies the four-dimensional massive Klein–Gordon equation , we obtain the Schrdinger-like equation of the extra–dimensional part
Here is the mass of the -th KK mode, and the effective potential is given by
The fundamental five-dimensional action (34) can be reduced to the effective four-dimensional action of the massive scalars, when satisfy the following orthonormality conditions:
The explicit expression of the effective potential for the brane solution (10) is
At , the value of is
Considering that , and , we know that the effective potential is negative at . Therefore, the scalar zero mode can be localized on the positive tension brane.
For convenience, we redefine some new dimensionless parameters:
and the Schrdinger-like equation (37) is changed as
where the expression of the dimensionless effective potential is
At the boundary , the effective potential (44) has a positive function (just as an infinite high barrier), and it will yield infinite discrete bound KK modes. Due to the non-analytical solution of the Schrdinger-like equation (43), we can only solve numerically Eq. (43) by fixing the parameter and choosing the proper value , to ensure the zero mode can be localized on the brane.
The wave functions of the lower bound KK modes are plotted in Fig. 5 and the corresponding mass spectrum is listed as follows:
where the parameters are set to , and . From the numerical solution above, we know that the zero mode can be trapped on the brane by fine tuning the parameters. We can see that the mass spectrum would get denser for the higher excited states while they get sparser for the lower excited states, and the gap of the higher excited states is almost the same for a set of parameters, on account of the dimensionless effective potential at the boundary having two positive functions, and the effective potential just like an infinite high barrier. Therefor, the solutions at the high excited states should be trigonometric functions with the same mass gap. By increasing the value of the parameter , the bound state solution with will appear. Therefore, in order to obtain nonnegative eigenvalues , the mass parameter will have an upper limit when the other parameters are fixed. For , as we have chosen here, the upper limit of is .
iii.2 Spin-1 vector field
Secondly, we turn to the spin- vector field. A typical five-dimensional action of a vector field coupled with a dilaton field is given as
where is the field strength as usual and is a dimensionless coupling constant. Considering the explicit form of the metric (14), the equations of motion read as
In order to be consistent with the gauge invariant equation , we use the gauge freedom to choose . Under the KK decomposition of the vector field
where , and the orthonormality conditions
the action (46) can be reduced to an effective one including the four-dimensional massless vector field (the zero mode) and a set of massive vector fields (the massive KK modes):
where is the field strength of the four-dimensional vector field. In addition, the extra–dimensional part satisfies the following Schrödinger-like equation:
where is the mass of the vector KK mode and the effective potential is
The explicit expression of the effective potential reads
where b=. The value of the effective potential at is
For localizing the vector zero mode on the positive tension brane, the effective potential should be negative at , which is equivalent to the following condition:
By setting in Eq. (51), we can get the normalized zero mode of the vector field for :
where . It is localized on the positive tension brane when the extra dimension is finite under the condition (55). It is easy to see that the vector zero mode can also be localized on the brane when if the extra dimension is finite. If the parameter , the normalized vector zero mode is
The vector zero mode (57) can be localized on the positive tension brane only for the case of a finite extra dimension. However, we note here that if the extra dimension is infinite, the zero mode can also be localized on this brane for .
By defining the dimensionless potential
Eq. (51) can be rewritten in a dimensionless form,
where the dimensionless parameters and are defined as in Eq. (26). For simplicity, we only require that the KK modes satisfy the Neumann boundary condition at the boundaries and . Using the boundary condition at , we get the general solution of Eq. (51),
where is the normalization coefficient and
With another boundary condition at , we obtain the spectrum of the vector KK modes determined by the following condition:
We plot the relation between and In Figs. 6 and 7 for large and small , respectively. Figure 6(a) and Fig. 6(b) imply that the mass of the first excited state increases with the coupling constant of dialton, . And Fig. 6(b) and Fig. 6(c) show that in a single cycle the number for the excited states increases with the size of the extra dimension, . In Fig. 7, we adjust the size of the extra dimension and and let at the same time; the result shows that the excited states do not exist in each cycle and they only emerge after multiple periods. In addition, Fig. 8 shows that the spectrum interval approaches a constant for the higher excited states, while for the lower excited states it is relatively sparse.
iii.3 Spin-1/2 fermion field
Furthermore, we investigate the localization and mass spectrum of spin-1/2 fermions on the Weyl brane. For studying the localization of fermions on thick branes, Refs. Liu0708 ; Volkas2007 ; Fuentevilla1412 ; Cendejas1503 ; Rocha1606 ; Brito1609 ; KoleyCQG2005 ; D. Bazeia0809 ; MelfoPRD2006 ; Almeida0901 ; Almeida2009 ; Davies0705 ; LBCastro1 ; LBCastro2 have introduced the Yukawa coupling. Here, we consider the following five-dimensional Dirac action for a fermion coupled with the background scalar field :
where is the spin connection defined as
with given by