Localization of $q-$form fields on $AdS_{p+1}$ branes

In this paper, we investigate localization of a free massless $q-$form bulk field on thin and thick $AdS_{p+1}$ branes with codimension one. It is found that the zero mode of the $q-$form field with $q>(p+2)/2$ can be localized on the thin negative tension brane, which is different from the flat brane case given in [JHEP 10 (2012) 060]. For the thick $AdS_{p+1}$ branes, the $q-$form field with $q>(p+2)/2$ also has a localized zero mode under some conditions. Furthermore, we find that there are massive bound KK modes of the $q-$form field, which are localized on this type $p-$branes.

In this paper, we will consider an interesting brane-world model, the AdS p+1 branes with codimension one, which have p spacial dimensions and an effective non-zero cosmological constant. These branes could have some realistic applications if the branes have 3 infinite large dimensions (which are those we can feel) and p − 3 finite small enough dimensions with topology S 1 × S 1 × ... × S 1 = T p−3 (thus can not be seen. According to string/M theory, there are 6 or 7 extra dimensions). It is known that the AdS space has many unusual properties [29], and plays a center role in the AdS /CFT correspondence [30,31].
In the brane-world theory, investigating the KK modes of various fields is an important and interesting work [32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49]. Because the information of extra dimensions is encoded in these KK modes. And the most important thing is that the massless modes (zero modes) are in fact the fields which have existed on the brane. Thus, the zero modes of ordinary fields (gravity, scalar, vector, and fermion et al.) localized on the brane can rebuild the effective gravity and standard model in our world [35].
It is known that, in the string/M theory, the q−form fields are related to closed strings, which play an important role in the D− or M−brane [50,51]. As the brane-world theory is motivated by the string/M theory, we naturally ask what about the q−form fields on the AdS p+1 branes. On the other hand, although the q−form fields with q > 1 are dual to the scalar field in four-dimensional spacetime, they denote new types of particles in higher-dimensional spacetime. Thus, we investigate in this paper the KK modes of the q−form fields on the AdS p+1 branes.
In fact, the q−form fields have been investigated in some literatures. For example, in Refs. [52][53][54][55][56], the KK modes of the 2−form and 3−form fields were discussed in the RS brane model, and it was found that the zero modes of these fields cannot be localized on the brane, unless they are coupled with a dilaton field. In Ref. [57], some authors of the current work considered any form (q−form) field on the flat p−branes with p the spatial dimensions of the brane, and found that only the lower form fields (with q < (p − 1)/2 or q < p/2) can be localized on the flat p−branes.
In this work we will find that the higher form fields (with q > (p + 2)/2) can be localized on the thin negative tension AdS p+1 brane and the thick AdS p+1 brane, which is an interesting result. For example, on the thin AdS 4 brane with negative tension and the thick one, there is a 3−form field that can be localized. This may be useful for some unknown problems such as the cosmological constant problem or dark energy problem [58]. In fact, in Ref. [7], the authors had considered a 3-form field with a nonstandard action to solve the cosmological constant problem in the five-dimensional RS model. This paper is organized as follows. We first simply review the localization mechanism for the KK modes of the q−form fields in Sec. 2. Then we try to find the solutions of AdS p+1 branes in Sec. 3, and investigate the localization of the KK modes on these branes. Finally we give a brief conclusion.

q−form fields and their KK modes
In Ref. [57], some authors of the current paper proposed a localization mechanism for a free massless q−form field, which can be applied to any RS-like braneworld model. Here we briefly review the localization mechanism.
The action for a free massless q−form field X M 1 M 2 ···M q , which is completely antisymmetric, in a curved D−dimensional spacetime is where The q−form field may have some localized KK modes on lower dimensional branes (such as the p−brane . This can be found by the KK decomposition and KK reduction. There exists a gauge freedom δX for the q−form field. So we can choose a simple gauge X µ 1 ···µ q−1 w = 0. Decomposing the remained components of the q−form field as whereX (n) µ 1 µ 2 ···µ q (x µ ) is only relative to the brane coordinates and U (n) (w) is the function of the extra dimension, and substituting the decomposition into the equation of motion for the q−form field, one will find that the KK mode U (n) (w) satisfies the following Schrödinger-like equation: where m n is the mass of the KK mode, and the effective potential V q (w) takes the form For the above Schrödinger-like equation (3), we need an orthonormality condition to seek the exact solution for U (n) and m n . As our aim is to find the KK modes which can be localized on the brane, the orthonormality condition can be obtained by the KK reduction for the action (1): If U (m) (w) satisfy the following orthonormality condition: we will get the effective action for a series of q−form fields on the brane: This effective action is just for the KK modes localized on the brane, and the localization condition is the orthonormality condition (6). Thus with the Schrödinger-like equation (3) and the orthonormality condition (6), we can investigate the KK modes of a q−form field on branes. There are usually two kinds of KK modes localized on the brane, i.e., the massless (zero mode) and the massive bound KK modes. The zero mode is actually irrelative to the extra dimension (from , and so it can be regarded as the matter that has existed on the brane. While the massive bound modes are relative to the extra dimension, and the mass spectrum carries the information of the extra dimension. Thus, we should find the mass spectrum of the KK modes. This requires to discuss the effective potential V q (4), which is decided by the geometry of the background spacetime. In this paper, we will consider AdS p+1 branes and investigate the KK modes of q−form fields on them.
We will first find solutions of the thin and thick AdS p+1 branes with codimension one. For the thin brane, we can get analytic solutions. For the thick ones, the solutions are depended on the special background scalar potential V(φ). In this paper, we will consider the simple but typical φ 4 potential, and use the numerical method to obtain the solutions with the initial and boundary conditions. However, we will not give the particular results in this paper, because the initial and boundary conditions are enough for us to analyze the behavior of the effective potentials for the KK modes.

AdS p+1 branes
We assume that the line element for the D = p + 2 dimensional spacetime is where r denotes the physical extra dimension perpendicular to the p−brane, andĝ µν is the induced metric on the brane. In this work, we will focus on the AdS p+1 branes, for which the metricĝ µν is given bŷ Here, H is a parameter associated with the effective cosmology constant Λ p on the p−brane by Λ p = −p H 2 [59]. Then the nonvanishing components of the Einstein tensor are turned out to be In this paper, the prime denotes the derivative with respect to the physical extra dimension coordinate r.
Then with the matter field we can get the Einstein equations. In the following, we will consider the thin and thick branes, in which the "matters" are made up by the brane tension σ (the energy density per unit area) and a scalar field φ, respectively.

Thin AdS p+1 branes
We first consider the thin brane case, for which the action is where Λ is a negative cosmology constant and g (b) µν is the induced metric on the brane. From the action (11) we obtain the Einstein equations: i.e., Considering the expressions of the Einstein tensor for the AdS p+1 brane, the above Einstein equations are rewritten as: where b = −Λ/(p + 1). From the above equations, we can obtain the solution of the AdS p+1 brane, which will be shown below.
Our aim is to investigate the KK modes of the q−form field; to this end we need to discuss the effective potential V q for the KK modes. With the coordinate transformation dw = e −A dr, the potential V q in Eq. (4) can be expressed as the function of r: It is clear that with the Einstein equations (14), the behavior of the effective potential V q (w(r)) at r = 0 and r → r bou is where r bou is the boundary of the extra dimension. If V q (r → 0) < 0 and V q (r → r bou ) > 0, there will exist a zero KK mode U 0 ∝ e (p−2q)A/2 . We then can check whether it can be localized on the brane by the orthonormality condition (6): In order to check the above condition, we will look for the solution of the warp factor by solving the Einstein equations. And from (17), we see that for the AdS p+1 branes with negative tension σ, V q (r → 0) < 0 can be satisfied for q > p/2, which means that there may exist the q−form fields with q > p/2 on the branes, and this is different with that on the flat p−branes [57]. While for branes with positive tension, the q−form fields on the branes are the ones with q < p/2, which is similar to that on the flat p−branes. So in the following we only focus on the thin AdS p+1 with σ < 0, and discuss the existence and localization conditions for the zero mode of the q−form fields in detail.
For the AdS p+1 brane with negative tension p−brane, the solution is Here c = arccosh(b/H), and the brane cosmology constant Λ AdS p+1 brane is given by where the bulk cosmology constant should satisfy the following condition: From the warp factor, it can be seen that the physical extra dimension for AdS p+1 is infinite, i.e., r bou → ∞, and the behavior of the warp factors at infinity is A(r → ∞) → e b r . According to Eqs. (17) and (18), the values of the effective potential at r → 0 and at infinity are obtained: which determines whether a localized zero mode exists. If there is a zero mode, we can check whether it can be localized on the brane: Then it is clear that only for the above integral is finite, and the zero mode can be localized on the brane. With the localization condition (27), we seek for the existence condition for the zero mode, i.e., V q (r → 0) < 0 and V q (r → ∞) > 0, which requires that Remember that only the q−form fields with q < p + 1 play a role in the p−brane. Thus finally the conditions for the zero mode to be existence and localized are: Thus for the thin AdS p+1 brane with negative tension, there exists a localized zero mode with the condition (29). For example, on the thin AdS 4 brane, a 3−form field can be localized. This result is different with that in the flat thin p−brane [57].

Thick AdS p+1 branes
In this subsection, we consider the thick AdS p+1 branes generated by a scalar field with scalar potential, which are more realistic. The action of such a system is where φ = φ(w) is the scalar field, and V(φ) is a potential which generates the brane. We can obtain the Einstein equations using the metric ansatz (8): which read as With the expressions of the Einstein tensor (10), we get The field equation for the scalar field is In order to find the solutions of these equations (34)-(36), we use the numerical method. Because the three equations are not independent, we suppose the form of the background scalar potential is where v 0 , g 1 , and g 2 are constants. As the equations of motion are second order, we need four initial or boundary conditions. We choose the conditions as Here the r bou is the boundary of the extra dimension, which may be infinite or finite. What should be noted from Eq. (34) is that with the initial conditions A(0) = 0 and A ′ (0) = 0, there is a constraint between the parameters v 0 , H, and the spacial dimension p: where the constant v 0 can be negative or positive for AdS p+1 brane. Also with the initial and boundary conditions, we can analyze the behavior of the effective potential for the KK modes of the q−form field, V q (w(r)), at r = 0 and at r → r bou from the equations of motion of the background spacetime (34)- (36). As the warp factor is A(0) = 0, A ′ (0) = 0, and A ′′ (0) = (−p H 2 − 2v 0 )/p at r = 0, the value of the effective potential at r = 0 is found to be with ξ = − p H 2 − 2v 0 p . At r → r bou , as φ ′ (r → r bou ) = 0, it is easy to get where k = − 2V(φ(r → r bou )) p(p + 1) .
From (40), it can be seen that for ξ > 0 and q > p/2, V q (r → 0) < 0, which suggests that there may be higher form fields with q > p/2 on the branes, and this is new result different with that on the thick flat p−branes [57]. With the conditions (39), we find that for the AdS p+1 brane, the value of ξ can be positive with v 0 < 0.
For the AdS p+1 case, as r → r bou Eq. (34) can be reduced to A ′2 (φ(r → r bou )) = −e −2A(r bou ) H 2 + k, from which it is seen that k must be positive. The behavior of the warp factor at r bou is We see that the physical extra dimension is infinite. For a zero mode of the q−form filed, we can check whether the localization condition (19) is satisfied. As the integral is it will be finite for q > p − 1 2 .