1.Introduction

The five-dimensional braneworld scenarios require standard model fields to be confined to the four-dimensional sector of the theory. In particular, when the electromagnetic field localization is considered, a non-normalizable gauge field is found ¹. This issue has been addressed in several opportunities, and some proposals have emerged in order to solve the problem. For instance, in ², massive bulk vector fields are coupled to the Randall-Sundrum (RS) brane ³ through a quadratic interaction term, in such a way that the photon is the bounded state of a vector fields spectrum. On the other hand, in ⁴, where the brane is generated by a domain wall solution to the Einstein scalar field system, the coupling between the bulk gauge fields and a dilaton is required to find a normalizable gauge boson on the thick brane. In both mechanisms, the localized state generates the standard electromagnetic interaction. Outside of this framework, other proposals can be found in Refs. ^5-10.

In this paper, we consider the vector field localization on self-gravitating domain walls via the coupling of the bulk vectors with the scalar field of the wall. We propose an interaction term defined by the scalar potential of the wall. As a result, the generalization of the Ghoroku-Nakamura mechanism ² to thick walls is obtained. We also show that the four-dimensional degrees of freedom of the bulk vectors are determined by a supersymmetric quantum mechanics problem where the ground state yields the standard electrostatic potential on the wall.

We apply the mechanism on the RS scenario and regularized versions of it, and the so-called singular domain walls ¹¹, where the scalar field interpolates between the lower values of the scalar potential. Whereas in the first case, standard electromagnetism on the wall is obtained, in the second one, the electrostatic interaction is the five-dimensional way.

Finally, we discuss coordinate covariance and gauge the symmetry of the model.

2.Vector field coupled to the wall

Consider the five-dimensional coupled Einstein-scalar field system (Latin and Greek indices correspond to five and four-dimensions, respectively)

We are interested in domain wall geometries, i.e., smooth scenarios where the scalar field ϕ interpolates between the minima of the self-interaction potential V(ϕ). Besides consider the coupling of the bulk vectors with the scalar field of the wall, namely

where Q is the five-dimensional coupling constant between two charged particles on the wall.

Before moving forward, it is necessary to point out two aspects regarding (2): i) we will assume that the vector fields do not modify the gravitation of the scenario and ii) the term V(ϕ)AaAa will not be justified. We want to show that it leads to standard electromagnetism in the four-dimensional sector of the wall.

In conformal coordinates,

from (1), we have

and

and from (2), we get

and

We want to study the electromagnetic interaction in the four-dimensional sector of the five-dimensional spacetime. To do this we will calculate the electrostatic potential, which can be determined from the following generating functional

with Gab in the bulk propagator. In the momentum space, (8) can be written as

where, tilde indicates four-dimensional Fourier transform. So, if we assume that the electrostatic sources are at z=z'=0, namely J̃a(p⃗,z)=δ(z)δμaj̃μ(p⃗), we find that the potential is given by

where Gμν are the four-dimensional components of propagator Gab.

By choosing

from (6) and (7), in the momentum space, we find a coupled system of equations for the components of G̃ab given by

and

where p¯α≡ηαβpβ. Now, in (12, 13), the transversal and longitudinal decomposition of the four-dimensional components of the propagator is considered, i.e.

where G₁ and G₂ satisfy

and

On the other hand, for b = z, we find

Due to that pαj̃α=0, only the traversal sector of (14) contributes with the electrostatic potential (10). Therefore, we will focus on finding a solution to (15).

Let us consider a continuous set of states ψm(z) satisfying the Schrödinger like the equation

and the following orthonormality and closure relationships

Notice that, for V(ϕ) given by (5), the eigenvalues equation (20, 21) can be written as a supersymmetric quantum mechanics problem,

and hence, in the eigenfunctions spectrum, there are no states with negative mass ^12,13. On the other hand, the orthonormality condition (22) is divergent for all m=m', which is a technical problem that can be solved by introducing two regularity branes at ±zr ¹⁴, in such a way that in limit zr→∞ the initial scenario is recovered. As a consequence of the regulatory branes, the basis is discretized, and the orthonormality relationships satisfied in the sense of Kronecker

Now, by expanding G₁ in the discrete basis

we find that (26) satisfies automatically the Eq. (15) and that the electrostatic potential, in the coordinates space, between two charged particles q1 and q2, i.e., jiμ(x⃗)=qiδ(x⃗-xi⃗)δ0μ, is determined by

where r=x2⃗-x1⃗. Notice that, in the presence of regulatory branes, the system resembles a well of the infinite potential of width 2zr, so it is estimated that m is quantized in units of 1/zr. Thus, for zr→∞, the potential (27) can be written as

where κ is a proportionality constant defined by boundary conditions at ±zr.

For a domain wall spacetime, we expect a zero-mode localized around z = 0, ψ0∼exp(5a/2) in correspondence with standard electromagnetism, and a tower of unbounded KK modes generating corrections to the Coulomb law.

Next, we will determine the electrostatic potential ([VR]) on three scenarios: RS brane, regular domain wall, and singular domain wall.

3.Vector fields on RS brane world

The RS scenario ³ in the conformal coordinates ([3) is determined by

and

with τ=6α the brane’s tension and Λ=6α2 the bulk cosmological constant.

In this scenario, the localization mechanism defined by action (2) takes the form

which is similar to the proposed in Ref. ²: a five-dimensional Proca theory with generic massive terms in five and four dimensions. However, in (31), the bulk mass of the vector fields and the four-dimensional coupling parameter are determined by the cosmological constant and the brane’s tension, respectively.

To evaluate the electrostatic potential (28), from (21) and (29) we obtain

such that the eigenstates of the problem (20, 32) are given by a normalizable zero mode

and a discrete spectrum of unbounded eigenstates

with J and Y, the Bessel functions of the first and second kind and A, B, and N_m the integration and normalization constants.

In agreement with ¹⁵, each eigenvalue of (20) is associated with a pair of eigenfunctions, ψmc and ψmd; which, for the particular case (32), where VQM(z)=VQM(-z), corresponds to odd and even eigenstates of the problem (20, 32). Hence, on the brane, they satisfy

and

while, on the regular branes

Therefore, the constants of integration set for ψmc and ψmd can be determined by (36, 37) and (25) in the first case; and by (38, 39) and (25) in the second one. On the other hand, the boundary condition (40) induces the discretization of the mass, i.e., m∼nπ/zr with n∈Z+.

Thus, for the integration constants, we have

and

Finally, on the brane and for m≪α, we find

such that the electrostatic interaction

is in correspondence with the standard four-dimensional potential for r≫α-1.

4.Regular Domain Walls

Let a(z)be the metric factor of a domain wall corresponding to a regularized version of the RS brane ^16-21. Following , it is possible to estimate at least the order of corrections as follows. Asymptotically, (where the effects of wall thickness are negligible) the metric factor of a regularized wall resembles the RS solution: a(z)∼-ln(1+α|z|). Hence, for z≫α-1 the quantum mechanics potential (21)

In Ref. ²², it is shown that for a VQM with asymptotic behavior similar to (46), the density of the state on the wall is determined by ψm(0)∼(m/α)(5/2)-1 in such a way that, the corrections to the electrostatic potential (28) go as r-5 and they are negligible compared to the Coulomb term from a critical radius determined by the parameters of the regular scenario.

5.Singular Domain Walls

The domain walls are understood as solutions to the coupled Einstein-scalar field system where ϕ interpolates between the minimums of the scalar potential. However, there is another family of walls where V(ϕ) does not have minimums, but ϕ interpolates among the lower values of the scalar potential. These scenarios are called singular domain walls, and like standard walls, they can locate gravity ¹¹.

Next, let us explore the four-dimensional effective behavior of the vector field on the singular scenario reported in wherein in conformal coordinate (3), the warp factor, the scalar field, and potential are given by

and

Notice that the scalar field interpolates between ±∞, and the scalar potential has a maximum in ϕ=0. On the other hand, the scalar curvature for this geometry is determined by

which diverges for z→±∞. Thus, the solution represents a wall embedded in a five-dimensional spacetime that interpolates between two subspace with naked singularities in the horizon.

To calculate the electrostatic potential (28), the density of states associated to (20) with

is required. In this case, we can see that the zero mode is separated from continuous modes by a mass gap defined by m0¯.

By considering the change of variable ξ=αz, the equation (20, 51) takes the form of a Schrödinger equation with a potential of Pöschl-Teller ,

where

with bounded states determined by

and

such that

Regarding the free states, under the change of variable

the differential equation (52) takes the form of a Legendre equation

whose solutions are the associated Legendre functions of the first and second kind, of 5/2 grade and order μ=±-2E, given by

The functions P5/2μ and Q5/2μ are orthogonal for |1-u|<2 and |u|>1, respectively; in particular, the change (57) satisfies |1-u|<2. Hence

where Γ-=Γ(1-μ) and Γ+=Γ(1+μ).

The solution is doubly degenerate and satisfies the following boundary conditions

Therefore, the density of the state in z = 0 is determined by

Now, the integral in the electrostatic potential saturates for m≫m0¯, and in this case, the states ([Modomfull]) reduces to

By substituting the modes (54), (55) y (65) in (28), we obtain

where the dominant term for r≪α-1 is the order of r^-2 such that the electrostatic interaction on the four-dimensional sector of the scenario is defined for a five-dimensional potential.

6.Discussion

In this paper, the mechanics of Ghoroku-Nakamura ² for confining vector fields on the RS brane was extended to self-gravitating domain walls in a five-dimensional bulk.

This was achieved by considering the coupling of the vector boson with the scalar field of the wall. We found that the four-dimensional degrees of freedom of the vector field can be expanded in terms of a basis of eigenstates of the Schrödinger equation, free of tachyonic states. In the electrostatic potential, the ground state defines the Coulomb interaction on the wall, while the massive state density generates corrections to the potential.

When mechanism (2) is applied on the RS brane, the bulk cosmological constant plays the role of mass for the vector field while the brane tension defines the parameter coupling with the wall. The corrections generated by the massive states in the electrostatic potential (28) are exponentially suppressed, and the standard electromagnetism can be recovered on the brane from a critical radius similar to the gravitational case.

In the case of thick brane, the mechanism was applied on regularized versions of the RS scenario and singular walls. In the first case, in agreement with the Csaki theorem ²², we find that deviations to the Coulomb law are the order of r^-5. In the second case, the deviations are not negligible, and the electrostatic interaction goes to r^-2 on the four-dimensional sector of the wall.

Notice that theory (2) is covariant but not gauge-invariant due to the presence of the quadratic term in the action. This can also be seen in the four-dimensional effective theory (see Appendix),

with

where we have considered Ab→e-a/2Ab and

such that φn=(-∂z+5/2 ∂za)ψn/mn is the supersymmetric partner of ψn for mn≠0 ¹². In (67), the first two terms correspond to the Maxwell action for aμ and the Stueckelberg action for the massive vector aμn. The absence of gauge symmetry is due to the final two terms; the second to last is a term of interaction between aμp and a5n via the dynamics of the scalar modes ∂μa5n, and the last one corresponds to massive terms for (a5n)2 when n = p and to interaction terms between a5n and a5p when n≠p.

In a forthcoming paper, we look forward to reporting a gauge-invariant generalization of (2) and justify the term of interaction between vector fields.

This work was supported by IDI-ESPOCH under the project FCPI167.