*p*-Forms non-minimally coupled to gravity in Randall–Sundrum scenarios

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## Abstract

In this paper we study the coupling of *p*-form fields with geometrical tensor fields, namely Ricci, Einstein, Horndeski and Riemann in Randall–Sundrum scenarios with co-dimension one. We consider delta-like and branes generated by a kink and a domain wall. We begin by a detailed study of the Kalb–Ramond (KR) field. The analysis of KR field is very rich since it is a tensorial object and more complex non-minimal couplings are possible. The generalization to *p*-forms can provide more information about the properties and structures that can possibly be universal in the geometrical localization mechanism. The zero mode is treated separately and conditions for localization of zero modes of *p*-forms are found for all the cases above and with this we arrive at the above conclusion about vector fields. Another property that can be tested is the absence of resonances found in the case of vector fields. For this we analyze the possible unstable massive modes for all the above cases via transmission coefficient. Our conclusion is that we have more probability to observe massive unstable modes in the Ricci and Riemann coupling.

## 1 Introduction

Since Kaluza and Klein introduced extra dimensions in high energy physics to unify electromagnetism and gravitation [1, 2, 3], it has been the subject of many developments [4]. In order to recover the four dimensional physics it was imposed that the extra dimension should be compact. However, at the end of the last century, Randall and Sundrum (RS) proposed an alternative to compactification using the concept of brane-words [5]. In this scenario the extra dimension is not compact and gravity is trapped to the four dimensional membrane by the introduction of a non-factorisable metric. Since the extra dimension is not compact all the matter fields, and not only gravity, must be trapped on the brane to provide a realistic model. However unlike gravity and the scalar field, the vector field is not trapped on the brane, what becomes a drawback to the RS model. To circumvent this problem some authors introduces a dilaton coupling [6], while others proposes that a strongly coupled gauge theory in five dimensions can generate a massless photon in the brane [7]. Most of these models introduces other fields or nonlinearities to the gauge field [8]. Some years ago Ghoroku et al. proposed a mechanism that does not includes new degrees of freedom and trap the gauge field to the membrane. This is based on the addition of two mass terms, one in the bulk and another on the brane [9]. Despite working, the mechanism has the undesirable feature of possessing two free parameters, from which one is left after imposing the boundary conditions. Beyond this, the mechanism is not covariant since in principle it introduces a four dimensional mass term which does not come from the five dimensional bulk.

An important point about the presence of more extra dimensions is that it provides the existence of many antisymmetric tensor fields. In five dimensions for example we have the two, three, four and five forms. From the physical viewpoint, they are of great interest because they may have the status of fields describing particles other than the usual ones. As an example we can cite the spacetime torsion [10] and the axion field [11, 12] that have separated descriptions by the two-form. Besides this, String Theory shows the naturalness of higher rank tensor fields in its spectrum [13, 14]. Other applications of these kind of fields have been made showing its relation with the AdS/CFT conjecture [15]. In the RS scenario much has been considered on these tensors. Localization of the zero mode of *p*-forms in delta like branes was first studied in Ref. [16] where it was claimed that, in *D* spacetime dimensions, only forms with \(p<(D-3)/2\) have a zero mode localized. However, it is well known that in the absence of a topological obstruction, the field strength of a *p*-form is Hodge dual to the \((D-p-2)\)-form [17]. Using this property is was shown that in fact only for the 0-form and its dual, the \((D-2)\)-form, the fields are localized [18, 19]. Recently the authors of Refs. [20, 21] showed that this is also related to the gauge fixing of the form fields. This make the problem of localization worse, since the vector field is not localized for any spacetime dimension. Beyond the zero mode, massive modes are important to be considered. Despite the fact that they are not localized, unstable massive modes can be found over the brane by using, for example, the transfer matrix method [22, 23, 24, 25]. Resonances of form fields has been found to exist for thick and thin branes [26, 27, 28, 29, 30, 31, 32, 33, 34]. Recently the Ghoroku mechanism was used to trapp the zero mode of *q*-form field [35]. The point is that the introduction of the mass terms break the Hodge duality and the argument of Refs. [18, 19, 20, 21] is not valid anymore. However this solution keeps the above cited undesired features of the Ghoroku mechanism.

In order to solve the above issues, recently a new proposal called “geometrical localization mechanism” was born [36, 37]. Looking for a covariant version of the Ghoroku mechanism some of the present authors found that both mass terms can be obtained from a bulk action if the Ricci scalar is coupled to a quadratic mass term of the gauge field [36]. Beyond solving the covariantization problem, it also eliminated from the beginning one of the free parameters. The last one is fixed by the boundary conditions leaving no free parameters in the model. The mechanism also keep the advantage of do not adding any new degrees of freedom. Another good property is that it provides the trapping of the gauge fields for any smooth version of the RS scenario [36]. Soon latter many developments of the idea was put forward. The same non-minimal coupling with the Ricci scalar was proven to work for *q*-forms and ELKO spinors fields [38, 39, 40]. For non-abelian gauge fields it has been found that the non-minimal coupling with the field strength used in Refs. [41, 42] should also be introduced and the mechanism works [43]. A phenomenological prevision of the model has been found for branes with cosmological constant different of zero: a precise residual mass of the photon must exist and this is proposed to be a probe to extra dimensions [44]. Recently the same mechanism was shown to emerge from a conformal hidden symmetry of the Randal–Sundrum model [45, 46]. In this new scenario fermions are shown to be universally trapped to the membrane by adding a non-minimal coupling with torsion. A new phenomenological prevision was found: a minimum value for the torsion of the membrane [45]. Soon latter a smooth version of the model was constructed in [46]. Despite providing the solution to the localization problem, the mechanism raises some questions. When the non-minimal couplings to gauge fields was generalized to includes the Ricci and the Einstein [47], it has been found that the last one do not provides a localized solution. The coupling with metric tensor also do not provides a localized zero model and this suggests that tensors with null divergence do not provide a trapped gauge field. When massive modes are considered, a curious result is that for all smooth versions considered no resonances was found. This raises the question if this is an universal property of the mechanism [48].

In this paper we study the coupling of the Kalb–Ramond(KR) field with tensor fields. The analysis of this field is very rich since it is a tensorial object and more complex non-minimal couplings are possible. Beyond the above cited importance of the KR fields, this generalization can provide more information about the properties and structures that can possibly be universal in the geometrical localization mechanism. This paper is organized as follows. In the second section we make a briefly review the RS scenario in co-dimension one. In Sect. 3 we study the localization of the zero mode of KR field coupled to Ricci, Einstein, Horndeski and Riemann tensors. In the Sect. 4 we study the possible existence of unstable massives modes for KB field coupled to Ricci, Einstein, Horndeski and Riemann tensors in a RS, kink and domain wall scenarios. In the Sect. 5 we study the localization of the zero mode of the *p*-form field coupled to Ricci, Einstein, Horndeski and Riemann tensors. In the Sect. 6 we study the possible existence of unstable massive modes of the *p*-form field coupled to Ricci, Einstein, Horndeski and Riemann tensors in a RS, kink and domain wall scenarios. Finally, in the conclusions we discuss the results.

## 2 Co-dimension one Randall–Sundrum scenario

*D*-dimensional space-time and construct explicitly all the geometric tensors needed. The coordinates of the whole space-time are \(x^M\), \(M=0,1,2,\ldots D-1\) with \(x^{D-1}\equiv z\) the coordinate transverse to the brane and \(x^\mu , \mu =0,1,2,\ldots ,D-2\) is the usual Minkowski coordinates. The metric is \(ds^{2}=\text{ e }^{2A(z)}\eta _{MN}dx^{M}dx^{N}\) where \(\eta _{MN}=\text{ diag }(-++\cdots +)\) and the equations of motion are given by [5]

*V*is the brane tension. The conformal form of the metrics provides a simple way to obtain the needed geometrical quantities. Interestingly this will also provide a covariant description of the model. First we must remember that, under a conformal transformation \(\tilde{g}_{MN}=\text{ e }^{2\varphi }g_{MN}\), we have for the Christoffel symbols

## 3 The Kalb–Ramond zero mode case

In this section we will make a direct generalization of the geometric coupling, presented in Ref. [47], for the KR field. This is gonna be a prototype for the due generalization to the *q*-form field in the Sect. 5. Beyond this, due to its importance it is worthwhile to make a separate study. We will consider the coupling to tensors of order two and four.

### 3.1 Kalb–Ramond coupled with a rank two geometric tensor

*z*. First of all, the potential (21) can further be simplified if we note that \(H_{0}\) are combinations of \(A'^{2}\) and \(A''\)

*z*. Since all the smooth versions considered here recover RS for large

*z*, we can use this solution to test the localizability of the field. With these considerations we can use the Einstein equation (3), which will be valid for large

*z*, to obtain that

*z*

### 3.2 Kalb–Ramond coupled with a rank four geometric tensor

*p*-form fields. We do this in the next sections, but before we analyze the possible resonances for the cases considered here.

## 4 Kalb–Ramond massive modes

In this section we study the possible resonant modes with Kalb–Ramond field coupled to Ricci, Einstein, Horndeski and Riemann tensors, through the transmission coefficient. The resonant modes appears when the transmission coefficient *T* is equal to 1, i.e, \(Log(T)=0\). We analyze in three possible scenarios: Randall–Sundrum delta like brane, a brane generated by a domain wall and generated by a kink.

### 4.1 In Randall–Sundrum delta like scenario

*r*and

*t*are constants. The boundary conditions at \(z = 0\) imposes

### 4.2 In brane scenario generated by a domain-wall

*z*for \(n \in N^{*}\).

*n*.

The solution of massive modes of transversal of KB field can not be found analytically. To obtain information about this state we use the transfer matrix method to evaluate the transmission coefficient. The behavior of the transmission coefficient for Ricci, Einstein and Horndeski coupling is illustrated in Fig. 4a–c for some values of parameter *n*. As we can see, for Ricci tensor coupling, resonant peaks appears when we increase the values of the parameter *n* indicating the existence of unstable massive modes. The same occur for Einstein tensor coupling. In the Horndeski coupling we observe the absence of resonant peaks.

The behavior of the transmission coefficient for Riemann coupling is illustrated in Fig. 5a for some values of parameter *n* with \(\gamma = -2\) and in Fig. 5b for some values of coupling constant \(\gamma \) with \(n =1\). As we can see, when we increase the values of *n* and \(|\gamma |\) we observe the appearance of resonant peaks, indicating the existence of unstable massive modes.

*n*and the potential diverges at this same points (see Fig. 6a, b). These kind of divergence does not allow us to use the transfer matrix method to compute the transmission coefficient and to evaluate the existence of unstable massive modes.

### 4.3 In brane scenario generated by a kink

*y*relates with the conformal coordinate,

*z*, by

Like the previous case, the components \(H_0\) and \(H_1\) vanishes at regular points near to the origin and the potential diverges at this points. These kind of divergence does not allow us to use the transfer matrix method to compute the transmission coefficient of vector field and to evaluate the existence of unstable massive modes.

## 5 The *p*-form zero mode case

In this section we further develop the previous methods in order to generalize our results to the *p*-form field case in a \((D-1)\)-brane. We again must consider the coupling to tensors of order two and four.

### 5.1 The *p*-form coupled with a rank two geometric tensor

*p*-form field to rank two geometric tensors. The action is given by

*p*-form in

*D*-dimensions to a

*p*-form and a \((p-1)\)-form in \((D-1)\)-dimensions. For this we must expand Eq. (74) and use Eq. (8). We arrive at just two kinds of terms: one where none of the indexes are \(D-1\) and another where one of the indexes is \(D-1\)

*H*tensor will not be anti-symmetrized with any index of the field. In fact for this case the equation is given by

*p*-form does not has null divergence. In order to obtain a consistent zero mode over the brane we must decouple the longitudinal and transversal parts defined by

*p*-form field, the longitudinal part and the \((p-1)\)-form field. From Eq. (87) we see that the \((p-1)\)-form is coupled to the longitudinal part of the

*p*-form field. As in the case of the KR field, we should expect that we have to uncouple the effective massive equations for the gauge fields \(X_{T}^{\mu _{1}\mu _{2}\ldots \mu _{p}}\) and \(X^{\mu _{2}\ldots \mu _{p}}\) since both satisfy the null divergence condition in \((D-1)\) dimensions. Lets walk along and prove this now. First of all note that using \(\partial _{\mu _2}X^{\mu _{2}\ldots \mu _{p}}=0\) we can show that

*p*-form we must impose the separation of variables in the form \( X_{T}^{\mu _{1}\ldots \mu _{p}}(z,x) = \tilde{X}_{T}^{\mu _{1}\ldots \mu _{p}}(x)\text{ e }^{-\alpha _{p}A/2}\psi (z)\) to obtain

### 5.2 The *p*-form coupled with a rank four geometric tensor

*p*-form field to rank four geometric tensors. Since we have already constructed and defined all the relevant quantities before this will be very direct. The action is given by

*p*-form and \((p-1)\)-forms in \((D-1)\)-dimensions

*p*-form fields. As we can see Eqs. (103), (104) and (105) are identical to Eqs. (79), (80) and (81). Therefore this can be seem as a master equation that governs the

*p*-form fields non-minimally coupled to gravity in RS scenarios. By separating the variables we also arrive at the following master equations that drives the spectrum of the reduced

*p*-form and \((p-1)\)-form fields.

*p*-form we find that the field is localized if \(\lambda _0\ne 0\) and

## 6 The *p*-form massive modes

*p*-form field is coupled to the Riemann tensor has the form

In the following sections we study the possible existence of unstable massive modes for the Ricci, Einstein, Horndeski and Riemann tensors couplings in Randall–Sundrum delta like, domain wall and kink scenarios.

### 6.1 In Randall–Sundrum delta like scenario

*p*-form field, the potential of Schrödinger equation, Eq. (116), is given by

*p*-form field, the potential of Schrödinger equation, Eq. (117), is given by

### 6.2 In brane scenario generated by a domain-wall

Now we analyze all tensor coupling in a brane scenario generated a domain-wall. As in the Kalb–Ramond case, the warp factor used will be given by Eq. (69). As we can see in Fig. 12a, no resonant peaks appear in Einstein and Horndeski coupling. However we have a resonant peak \(m_T^2=10.5\) for the Ricci tensor coupling. For the Riemann coupling, in Fig. 12b, we observe three resonant peaks.

### 6.3 In brane scenario generated by a kink

## 7 Conclusion

*p*-form field in co-dimension one brane scenarios non-minimally coupled to gravity. First we consider the Kalb–Ramond field coupled to rank two and four geometric tensors. We show that the reduced fields can be decoupled in a similar way as in the vector field case. We analyze the localization of zero mode of the transversal part of KR field for a generic geometric tensor and found the conditions to localize it. For the vector component of the KR field, the study of localization is more complicated, due to the potential of the Schrödinger equation and the analise could be done only asymptotically. We find that for both, the reduce KR and vector component, the fields are localized for the Ricci, Einstein and Horndeski tensors but not for the Riemann tensor. We find that the value of the coupling constant is the same for both and therefore consistent. To analyze the localizability for general

*p*, we use the values of \(\beta _0/\lambda _0\) and substitute in Eqs. (112) and (114) to obtain Table 1. From this we can see that for some values of the parameters both components of the reduced

*p*-form can be localized. For example, for the Ricci tensor and for \(D=5\) we have that the reduced

*p*-form is localized for \(p>-1\) and therefore for any value of

*p*. For the \((p-1)\)-form in the same case the condition is given by \(p<11/2\), what means all the cases, since in five dimensions the bigger value of

*p*is five. However beyond this there is a second consistence condition. The coupling constants (113) and (115) must be the same. By imposing that \(\gamma _p=\gamma _{(p-1)}\) we find that \(p=(D-1)/2\). Therefore the condition for having both components localized is universal and independent of the kind of coupling used and just depends on the dimension of spacetime. In five dimensions for example the Kalb–Ramond field generates a Kalb–Ramond plus a vector field trapped over the membrane for any kind of coupling. Another important result of the Table 1 is that, as said in the introduction, we can test some previous hypothesis of previous works. In Ref. [47] the coupling of the vector field with the Ricci and the Einstein tensor has been studied. It has been found that for the second case it is not possible to localize the field. The hypothesis was that tensors with zero divergence do not provides a localized field. However from the above table we can see that the Einstein tensor can trap any

*p*-form in any dimension with only one exception: the case \(p=1\). Therefore somehow the hypothesis is right but is only valid for the vector field. The Ricci tensor also can trap any

*p*-form in any

*D*with one exception: the gauge field in \(D=2\). The Riemann tensor can not localize any field. The Horndeski tensor can trap any field since this coupling is possible for \(p>1\) and the localization condition is given by \(p>3/2\).

The localizability condition for the *p*-forms fields

Tensor | \(\beta _0/\lambda _0\) | \(p>\) | \(p<\) |
---|---|---|---|

Einstein | \((D-3)/2\) | 1 / 2 | \((2D-3)/2\) |

Horndeski | \((D-4)/2\) | 1 | \((D-2)\) |

Ricci | \((D-2)\) | \(-(D-2)/2\) | \((3D-4)/2\) |

Riemann | – | – | – |

The plus signal means appearance of resonant peaks, the minus absence for *p*-forms

Tensor | Delta like | Domain wall | Kink |
---|---|---|---|

Einstein | − | − | − |

Horndeski | − | − | − |

Ricci | − | + | + |

Riemann | + | + | + |

Now we will consider massive modes. This is done for all geometric tensors for many kinds of smooth branes. In the case of massives modes, we used the transmission coefficient to observe possible unstable massive modes. The emergence of resonant peaks was observed when we increased the coefficients of \(A''(z)\) and \(A'(z)\). From Fig. 2a, b we observed the absence of resonance for KR field in RS delta like branes for all tensor coupling. The same occur for *p*-forms field in \(D=10\), for Einstein, Horndeski and Ricci coupling as we can see from Fig. 11a. In delta like brane, we observe the appearance of resonance only in Riemann coupling in \(D=10\) (see Fig. 11b). For domain wall branes, for KR field, we concluded from Figs. 4a–c, 5a, b, the resonance appear only for Ricci and Riemann coupling with increasing value of parameter *n*. The same occur for *p*-forms field in \(D=10\). The conclusion is the same for kink branes. This can be explaining as follows. As we can see from Eq. (116), when we increase \(\sigma _p\), \(A'(z)\) predominates over \(A''(z)\). The same occur for \(\gamma _p<0\) in Eq. (117). The behavior of *U*(*z*) looks like a double barrier, for domain wall and kink like branes (see the Figs. 3, 7). When we increase the values of the parameters, the width of the barrier increase and we have more probability to see resonant peaks. For the domain wall brane, for large *n*, \(A'(z)\sim |z|^{-2}\) for \(|z|>1\) and \(A'(z) \sim 0\) for \(|z|<1\). That is, for large *n* and higger dimension, we have a Schrödinger potential like a double delta barrier with two deltas located at \(z=\pm 1\). As pointed in the section 6, for the Einstein and Horndeski coupling, the Schrödinger potential does not depend on dimension of space. Consequently, the resonant peaks will appear for greater values of the form *p* in Einstein and Horndeski coupling. Since in the Ricci and Riemann tensor coupling, the potential depends on the dimension of space-time, it was observed, that these cases are more sensitive to the presence of unstable massive modes as showed in Table 2.

## Notes

### Acknowledgements

The authors would like to thanks Alexandra Elbakyan and sci-hub, for removing all barriers in the way of science. We acknowledge the financial support provided by Fundação Cearense de Apoio ao Desenvolvimento Científico e Tecnológico (FUNCAP) through PRONEM PNE-0112-00085.01.00/16 and the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) [305961/2015-2].

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