Localization of abelian gauge fields on thick branes

In this work, we explore a mechanism for abelian gauge field localization on thick branes based on a five-dimensional Stueckelberg-like action. A normalizable zero mode is found through the identification of a suitable coupling function between the brane and the gauge field. The same mechanism is studied for the localization of the abelian Kalb--Ramond field.


Introduction
In the brane-world scenario, our universe is depicted as a four-dimensional sub-manifold (3-brane) embedded in a higher-dimensional space-time. Within this framework, gravity is able to propagate in all dimensions, but matter fields are restricted to live on the 3-brane. Among the most attractive schemes constructed under this hypothesis stand the proposals of Randall and Sundrum [1,2] (resp. RS1, RS2), which involve only one extra-dimension and a non-trivial warp factor due to the underlying anti-de Sitter (AdS) geometry.
Branes in RS models and its generalizations are very idealized, they are introduced as infinitely thin (singular) hyper-surfaces. Besides, such thin branes are static, with no dynamical mechanism responsible for their formation. In order to avoid the use of singular branes, thick branes modeled as domain wall configurations can be implemented in extra-dimensional theories (for a detailed review on thick brane solutions see [3]). The key feature of this approach is that thick branes are dynamically generated by one or several background scalar fields coupled with gravity.
In the pursuit of a more realistic picture of the brane-world, it is also important to provide a natural localization mechanism of bulk fields on domain walls. As shown in [4,5], the graviton can be successfully localized on a thick brane built with a single background scalar in an asymptotic 5D AdS space-time. Massless scalar fields can also be localized in this minimal setup, but unfortunately vector gauge fields cannot. This phenomenon can be observed clearly in the singular limit [6]. A significant amount of work has been devoted to the problem of vector gauge field localization in the context of 5D AdS space-time, in both singular and thick domain wall branes. The models available in literature show a wide variety of ideas, which include: mass terms for the vector boson [7][8][9], coupling between the dilaton and the kinetic term of gauge fields [10,11], insertion of kinetic terms induced by localized fermions [12] and a smearing out dielectric function inspired in the Friedberg-Lee model for hadrons [13].
The inherent difficulty of trapping gauge fields in a domain wall can be also observed in the anti-symmetric Kalb-Ramond (KR) tensor field. The corresponding zero-mode of the KR field in 5D is known to be non-localizable on the simplest thick brane, generated by a single real scalar field [14]. Analogously to gauge vector localization, the problem concerning the localization of the K-R field has been considered by several authors [13][14][15][16][17][18].
In this work, a new mechanism for gauge field localization on thick branes is explored. The model can be seen as a domain wall regularized version of the mechanisms presented in [7,8], within a smooth RS2 scenario. The basic setup of the model is a thick brane embedded in an asymptotically 5D AdS bulk space-time, described by a single real background scalar field. In this framework, localization is achieved by the introduction of Stueckelberg-compensating fields in the 5D action of a gauge field, where the quadratic coefficient of the gauge fields is modeled as a Yukawa-like brane-gauge coupling. It is important to notice that the use of Stueckelberg-compensating fields restricts the application of this mechanism to abelian gauge fields.
The structure of the paper is the following: In section 2 a brief review of the brane-world generated by a single real scalar is presented. Then, in section 3 it is shown that, starting from the Stueckelberg-compensated action of a vector gauge field, a normalizable zero-mode can be localized on the brane whenever a suitable brane-gauge coupling function is fixed. A similar procedure to localize the zero-mode of the KR field is presented in section 4. Finally, the conclusions of the work are presented in section 5.

Thick brane generated by a single scalar
Consider a model of 5D gravity coupled to a single scalar field φ: with M * being the fundamental Planck scale in five dimensions and V (φ) the scalar potential.
Here we have rescaled the scalar field in units of M * , such that φ is dimensionless and V (φ) has mass dimension 2. We are interested in background solutions where the metric displays four-dimensional Poincaré symmetry Here x M = (x µ , y), where x µ (µ = 0, . . . , 3) stand for the usual coordinates of four-dimensional Minkowski space-time with η µν = diag(−1, +1, +1, +1), and y denotes an infinitely extended extra dimension. Assuming that the scalar field φ depends exclusively on y, the equations of motion of Eq. (2.1) become where the prime denotes derivative with respect to y.
With the aid of the super-potential method [4], a solution for Eqs. (2.3,2.4,2.5) can be found through the first-order differential equations if the scalar potential can be written in terms of some super-potential function W (φ) as An example of such super-potential is given by the sinusoidal function where b > 0. It determines the following background solution for the scalar field that yields a domain wall with positive tension and the smooth warp factor Here y 0 denotes the center of the domain wall and A 0 the value of the warp factor at this point. Without loss of generality, we can set y 0 = 0, A 0 = 0. The background described above is asymptotically Anti-de Sitter A(|y| → ∞) ∼ −k|y| with AdS scale k = b|a|. Therefore, it constitutes a regularized version of the RS2 model. It is often convenient to work in a conformally flat frame, defined by the transformation dy = e A dz, where the metric takes the form It is important to notice that this transformation cannot be written in closed analytical form for arbitrary values of the parameters associated with brane thickness (b) and curvature (k) for the backgrounds of the form of Eqs. (2.9,2.11). However, particular cases are tractable such as b = 1, which yields In this work we do not adopt a particular form of the background, but only assume the existence of a domain-wall solution generated by some super-potential W (φ) with asymptotic AdS behavior A(|y| → ∞) ∼ −k|y| and well defined transformation dy = e A dz. We refer to Eq. (2.13) for a concrete realization of this setup.

Abelian gauge vector field localization
Our starting point is the Stueckelberg-like five-dimensional U (1) gauge field action Where A M is the 5D gauge vector field and B is a dynamical scalar field (see [19] for a review of the Stueckelberg field). This action is gauge invariant under the simultaneous transformation The coupling of the gauge field and the brane is described by a Yukawa-like interaction G(φ). The aim of this section is to find out whether the model defined by S A can lead to the localization of a gauge field zero-mode on thick branes, through the adoption of a particular functional form for G(φ). From the 4D low energy perspective, the model defined by Eq. (3.1) can be understood in terms of massless and massive sectors. The localization mechanism is expected to give rise to a massless sector, from the zero mode of S A , and massive sectors coming from the continuum of non-zero modes 1

Equations of Motion and Gauge fixing
Varying the action S A , we obtain the following 5D equations of motion for A M and B: where the equation of motion for B (cfr. Eq. (3.4)) is consistent with the Noether's identity obtained by taking the divergence of Eq. (3.3). For simplicity, in the present work we assume that the back reaction of G(φ) into the geometry is negligible, and thus, the minimum energy solutions for φ(y) and A(y) are determined by Eq. (2.6).
The first important task in our analysis is to fix the gauge. Instead of imposing an explicit gauge fixing condition, we perform an analogous analysis to the one done in [7,8], where the 5D field A M is parametrized as with A µ and ϕ as the transverse (∂ µ A µ = 0) and longitudinal components of A µ , respectively. The behavior of these components under the gauge transformation Eq.
This suggests that it is potentially useful to redefine the scalar degrees of freedom as such that, under Eq. (3.6), the new fields λ and ρ remain invariant: Therefore, the parameterization defined by Eqs. (3.5,3.7) is roughly equivalent to the gauge fixing condition ∂ µ A µ = 0, but incorporates the advantage of working directly with gauge invariant fields. Let us now write the 5D equations of motion in terms of the components A µ , λ and ρ. In first place, taking N = ν in Eq. (3.3), we obtain The left hand side of this equation is purely transverse, while its right hand side is purely longitudinal. Thus, each side must vanish independently for a nontrivial solution. Taking this fact into account, in this parameterization, Eqs. (3.3,3.4) become equivalent to the following system of equations: Note also that Eq. (3.11) is already satisfied in a weaker form by a combination of (3.12) and (3.13) Further insight can be gained writing S A in gauge invariant components. Substituting Eqs. (3.5,3.7) into Eq. (3.1), it can be shown that the transverse vector A µ decouples from the scalar fields (up to vanishing surface terms) and the action can be written as

Brane coupling and gauge field localization
Once the gauge has been fixed, we can study the role played by the brane-gauge coupling G in the gauge vector field localization. Decomposing the gauge field as Eq. (3.10) reduces to with a µ n (x) = m 2 n a µ n (x). In order to proceed further, it is now necessary to specify the functional form of the brane-gauge coupling G(φ). At this point we do not attempt to study the most general form of G(φ), but instead we explore if there is a particular choice for this function with physical significance that ensures the existence of a normalizable zero-energy ground state for the gauge field. As a first approach, we adopt the following set of requirements for the construction of an Ansatz: (a) G(φ) is an even function of y, (b) it has mass dimension 2, (c) it is determined by the super-potential W (φ) and its derivatives.
From Eq. (2.7), it is clear that the potential V (φ) -which can be written as a combination of W (φ) 2 and [∂ φ W (φ)] 2 -does indeed satisfy these three requirements. Thus, taking V (φ) as a guideline, we propose the following Ansatz for the brane-gauge field coupling: where c 1 , c 2 are arbitrary real constants. Using Eqs. (2.5,2.6), this functional can be written as where the dot denotes derivative with respect to z. This equation can be cast into the form of a typical quantum mechanical problem through the substitution where the auxiliary wave function ψ n satisfies the Schrödinger equation with QM potential As a last step in the determination of G(φ), in order to guarantee an effective fourdimensional theory with a normalizable zero-energy ground state, we require the Schrödinger equation (3.24) to be rewritten as which is of the form of a supersymmetric quantum mechanics problem, with for a positive real parameter ξ. This restriction imposes the following tuning among the constants c 1 , c 2 and ξ: c 1 = ξ, c 2 = ξ 2 + 2ξ, (3.28) such that the functional G[φ(y)] and the QM Potential U (z) become For a background like that of Eq. (2.13), the masses of the modes are distributed in a continuous spectrum, as U ξ (z) → 0 when |z| → ∞. The hermiticity and positive definiteness of Q † ξ Q ξ in Eq. (3.26) ensure that no normalizable negative energy modes are allowed. On the other hand, the zero-energy wavefunction annihilated by Q ξ is clearly normalizable ψ 0 (z) = k 0 e (ξ+ 1 2 )A(z) , (3.31) and the corresponding the zero-mode profile α 0 (y) turns out to be α 0 (y) = k 0 e ξA(y) , (3.32) with k 0 as a normalization constant.
however, a straightforward calculation shows that the integral over y vanishes Thus, despite being normalizable, the scalar zero-mode fails to be localized in the brane. It turns out to be a null state. For the n = 0 scalar modes, Eq. (3.14) imply that ρ n can be eliminated in favor of λ n as Inserting this relation into Eq. (3.12) and using Eqs. (3.33), the equation for the mode profiles becomes Summarizing, the model defined by Eqs. (3.1,3.29) contains a normalizable zero-energy ground state described by a massless 1-form in 4D and a continuous tower of massive 1-forms and massive real scalars. There is one additional massless scalar with normalizable profile in the spectrum (instead of the two scalars anticipated by the naive counting of d.o.f.), but this scalar field is hidden from the 4D perspective by dynamical effects induced through the coupling G ξ (φ).

Localization of the Kalb-Ramond field
The Kalb-Ramond (K-R) Field can also be localized on the brane using an analogous procedure to the one presented in the previous section. We start with the five-dimensional Stueckelberg-like formulation of the K-R action with the K-R field strength defined as Here C M plays the role of a Stueckelberg compensator and the function F(φ) models the coupling between the K-R field and the domain wall. Again, the back reaction of F(φ) to the geometry is neglected in the following analysis.

Equations of Motion and Gauge fixing
The action S KR in Eq. (4.1) is gauge invariant under the transformation and its 5D equations of motion are Paralleling the analysis of the previous section, we parametrize the 5D field B M N as where B µν are the transverse components of B µν (∂ µ B µν = 0, H µνρ = ∂ µ B νρ +∂ ν B ρµ +∂ ρ B µν ), while ϕ µ stand for the corresponding vector components (∂ µ ϕ νρ + ∂ ν ϕ ρµ + ∂ ρ ϕ µν = 0, with ϕ µν = ∂ µ ϕ ν − ∂ ν ϕ µ ). Their behavior under the gauge transformation Eq. (4.3) is given by Again, upon integration by parts, the transverse field B µν decouples from the vector fields and the action S KR can be written as together with the field redefinitions where the new fields λ µ and ρ µ are now invariant under Eqs. (4.3,4.7): In terms of the gauge invariant fields B µν , λ µ , and ρ µ , Eqs. (4.4,4.5) read (4.14) Notice here that Eqs. (4.13,4.14) follow from after isolating its transverse and vector parts, and that Eq. (4.14) is again satisfied in a weaker form by a combination of Eqs. (4.15,4.16):

Brane coupling
Decomposing the antisymmetric field B µν as Eq. Taking an analogous course of action as that for the gauge vector case, we propose as an Ansatz for F(φ) the following uni-parametric family of functions that admit a normalizable zero-energy ground state: Now the zero-energy wavefunction annihilated by Q κ is and the corresponding the zero-mode profile η 0 (y) becomes which is normalizable for κ > 0.

Vector Sector
The fate of the vector sector can be studied decomposing the vector fields as λ µ = n λ µ n (x)u n (y), ρ µ = n ρ µ n (x)v n (y), (4.28) and defining the modes as From Eq. (4.15), the vector zero-modes satisfy but again this this mode fails to be localized in the brane, as its effective action vanishes upon integration over the extra dimension: On the other hand, from Eq. (4.18) the following relation holds for the n = 0 modes: Plugging this relation into Eq. (4.15) we obtain the definitive equation for the mode profiles As a summary of the results obtained in this section, we can state that the model defined by Eqs. (4.1,4.21) contains a normalizable zero-energy ground state described by a massless 2-form in 4D and a continuous tower of massive 2-forms and 1-forms. The spectrum also includes a massless 1-form, which is dynamically removed from the 4D effective low energy action.
Before closing this section, let us point a possible natural connection between the functions G ξ and F κ . If we require the brane-gauge coupling to be universal, imposing Thus, in this special case, the localization of abelian gauge fields is driven by the very same function that determines the background geometry.

Conclusions
We have proposed a new mechanism for abelian gauge field localization on thick branes. The key feature of the model is the presence of Stueckelberg-compensating fields, which allow for the introduction of Yukawa-like interactions in a gauge-invariant (and Einstein-covariant) way.
In the vector case, the interaction between the brane and the gauge field is modeled by a function G(φ) that depends on the classical background responsible for the brane formation. Identifying the brane with a domain-wall solution generated by a single real scalar φ through some super-potential W (φ), with asymptotic AdS behavior in 5D, we have shown that there is a whole family of functions G ξ (φ) -constructed from W (φ) and its derivatives-which guarantee the existence of a normalizable zero-energy vector ground state in the theory. We have also studied the fate of the scalar sector of the model, concluding that despite of being normalizable, the scalar zero-mode is not trapped by the brane, but instead it is dynamically removed from the effective 4D low-energy theory by action of the brane-gauge coupling G ξ (φ).
The same localization mechanism can be straightforwardly applied to the abelian Kalb-Ramond field. In this case, we have also found a one-parameter family of brane-gauge coupling functions F κ compatible with the presence of a normalizable zero-mode.
Finally, we have shown that if the brane-gauge coupling is universal, then it must be proportional to the scalar potential V (φ), the same function that triggers the brane formation.