# Non-minimal couplings in Randall-Sundrum Scenarios

###### Abstract

In this paper we propose a new way of obtaining four dimensional gauge invariant gauge field from a bulk action. The results are valid for both Randall-Sundrum scenarios and are obtained without the introduction of other fields or new degrees of freedom. The model is based only in non-minimal couplings with the gravity field. We show that two non-minimal couplings are necessary, one with the field strength and the other with a mass term. Despite the loosing of five dimensional gauge invariance by the mass term a massless gauge field is obtained over the brane. To obtain this, we need of a fine tuning of the two parameters introduced through the couplings. The fine tuning is obtained by imposing the boundary conditions and to guarantee non-abelian gauge invariance in four dimensions. With this we are left with no free parameters and the model is completely determined. The model also provides analytical solutions to the linearized equations for the zero mode and for a general warp factor.

###### pacs:

64.60.ah, 64.60.al, 89.75.DaThe Randall Sundrum (RS) model appeared in the Physics of higher dimensions as an alternative to compactification that included the possibility of solving the Hierarchy problem Randall:1999ee ; Randall:1999vf . To solve the physical problem of dimensional reduction, the RS models should obtain fields with zero mode confined to the brane in order to recoverer the Physical models when the fields are properly integrated over the extra dimension. In the non-compact case, the gravity field localization is attained, however gauge fields as simple as minimally coupled to gravity are not localized and this was a problem to the theoryRandall:1999vf ; Bajc:1999mh . For the compact case the problem appears when we consider non-abelian gauge fields. After the Fourier decomposition of the zero mode and the normalization we have that , and the gauge invariance in four dimensions is lost Batell:2006dp . A number of extensions to RS were proposed in order to provide localized gauge fields for the non-compact case. A smooth warp factor was investigated?Batell:2006dp ; Dvali:1996xe ; Gremm:1999pj ; Kehagias:2000au ; Bazeia:2004dh ; Cvetic:2008gu ; Chumbes:2011zt ; German:2012rv , but also it did not yield localized gauge fields. Some models obtained the localization by the addition of new degrees of freedom such as a scalar or dilaton fields, but a more natural approach would be to obtain an extension of minimal couplings that would localize the gauge field without introducing other fields. A step in this direction was the introduction of a boundary interaction with the field strength Dvali:2000rx , but it is known that this produces only a quasi-localized zero mode.

Other idea is the breaking of the gauge invariance in five dimensions by the addition of a mass term. It was found that the only consistent way of getting a localized zero mode is again the introduction of a boundary term, this time with the mass term Ghoroku:2001zu . Soon it was shown that the boundary mass term was not enough to provide a consistent model in the case of Yang-Mills (YM) fields because non-abelian gauge invariance is lost in the membrane Batell:2006dp . The work Batell:2006dp shows that a combination of the above boundary terms are needed to solve the problem of YM localization. In the track of understanding the origin of the mass boundary term introduced in Ghoroku:2001zu , some of us identified that a non-minimal coupling of the Ricci scalar with the mass term could generate it Alencar:2014moa . More then this, it was discovered that this kind of coupling solves the localization of any form gauge field in co-dimension one brane worlds, valid for any warp factor Alencar:2014fga . However the mechanism does not work for Yang-Mills fields by the same reasons of Ref. Batell:2006dp . In fact, as shown by these authors, this is a problem even for the non-compact case since non-abelian gauge symmetry is lost over the brane if we depart from a massless gauge invariant action in the bulk. Other non-minimal couplings of gravity with the field strength seems to be needed also in this case. These kinds of couplings of gauge fields with gravity has been proposed and studied in 4D Horndeski:1976gi and recently it was shown that they generate the boundary term for the field strength in Ref. Germani:2011cv . However the model keeps the property of being only quasi-localized. Here we show that specific non-minimal couplings with gravity are enough to recover YM fields over the brane, moreover, the confined fields displays the expected properties, that is, are massless and therefore gauge invariant. Our approach also has the advantage of being valid for any extension of the RS model which recovers it asymptotically. This is obtained when we merge the coupling as proposed in Horndeski:1976gi with the non-minimal coupling proposed by us in Ref. Alencar:2014moa . Importantly, localization and gauge invariance depend on the tuning of two otherwise free parameters of the model.

The background metric of the RS model in its conformal form is given by , where . After considering a delta-like branes and an AdS vacuum a stable background solution to the Einstein equation is found, which is given by . The question of localizability of fields is resumed to find a square integrable solution() of a Schroedinger-like mass equation with potential which emerges from the process of dimensional reduction. At the end this provides a finite four dimensional action after the integration in the extra dimension is performed. For the non-abelian case the problem is reduced to finding a zero mode satisfying . The above warp factor generates effective potentials with Dirac delta functions Randall:1999ee ; Randall:1999vf . These singularities can be smoothed out through the introduction of kink-like membranes such as in Refs. Dvali:1996xe ; Gremm:1999pj ; Kehagias:2000au ; Bazeia:2004dh ; Cvetic:2008gu that recovers the RS case only asymptotically. Therefore the best approach is to consider a general warp factor and to look for a solution to the problem that does not depend on it. We call such solution warp independent for obvious reasons. For a minimally coupled gauge field the effective potential obtained is . Analyzing more carefully, for a thin brane, the solution has a convergent and a divergent component as . However this is not the only boundary condition to be imposed since depends on and we get a delta function in the potential. When we impose the proper boundary condition in the solution is completely fixed and we discover that it is not localized. The same conclusion is obtained for arbitrary and it is warp independent. The form of the above potential provides a general solution given by and this is not square integrable for asymptotic RS models. Also, this solution does not satisfy the condition needed for the compact case.

As said before, a model was constructed which introduces a boundary term through an interaction with a delta function Dvali:2000rx , that is, besides the standard gauge action a contribution given by was introduced, where are four dimensional indices and is a mass parameter. Although the origin of this term is not explained, the authors argue that it is needed to guarantee that there is no current in the extra dimension. Then a quasi-localized zero mode is obtained. Some time latter the possible gravitational origin of such an interaction was discovered to come from a non-minimal coupling with the field strength given by Germani:2011cv . This coupling has been proposed early as an extension of the four dimension gauge field which preserve current conservation and recovers Maxwell equations in the flat limit Horndeski:1976gi . This is obtained if for any of the indices and the tensor posses the same symmetries of the curvature tensor. A satisfying this is given by

(1) |

By computing explicitly the above coupling for the thin brane case Germani has obtained the correct boundary term Germani:2011cv . Despite this clear advance the model does not provide a fully localized solution.

In the direction of five dimension breaking of gauge invariance a topological mass term was added through a three form field with a localized zero mode Oda:2001ux . However this generates a massive gauge field in four dimension, beyond introducing a form field as a new degree of freedom. To solve this problem Ghoroku et al proposed to introduce a mass term directly in the action Ghoroku:2001zu . Although this term generates a localizable solution as the boundary condition at fixes . This is solved in a similar way as in Ref.Dvali:2000rx by the introduction of a boundary mass term such that the boundary condition is satisfied with . Therefore the model is consistent if we use a interaction term given by where and should be chosen such that the boundary condition are satisfied and to obtain a localized solution. A range in the parameters is obtained which satisfy the boundary conditions leaving a free parameter to be determined. However it was shown in Batell:2006dp that this is not enough when we consider YM fields. The reason is that now we have quartic terms in the action and even if the zero mode is integrable we generally have causing the lost of gauge invariance after the integration is performed. However Batell et al showed that this can be solved by the use of both boundary terms of Refs. Dvali:2000rx ; Ghoroku:2001zu and the final action which was found to be

(2) | |||||

where is the five dimensional gauge coupling and . Since the model posses three parameters: and to guarantee the localization and a third parameter is introduced to guarantee the gauge invariance in four dimensions. However, we show here that this action must be corrected to preserve the expected symmetries of the system.

Searching for the origin of the above action, some of us discovered in a series of previous papers that a non-minimal coupling with the Ricci scalar given by consistently provides a solution to the localization of gauge field in co-dimension one brane worldsAlencar:2014moa ; Alencar:2014fga ; Jardim:2014vba . This term modifies the potential that now is given by where is the order of the form considered. The zero mode for this potential has solution which is square integrable for any warp factor that asymptotically recovers RS, being therefore a warp independent solution. Another advantage is that the exigence of general covariance determines one of the parameters leaving only which is fixed by boundary conditions, and at the end we get a model with no free parameters. However the problem for YM fields is not solved and other non-minimal coupling seems to be needed.

With the above ingredients we propose a model defined by the following action

(3) | |||||

where and are parameters that will be fixed by the boundary condition and the demand of gauge invariance in . From now on we will not write explicitly the group indices. The linearized equations of motion are

(4) |

and from the above we get the five dimensional divergenceless condition , or in components

(5) |

where we have defined . By computing explicitly we obtain the equations of motion in components. For we get

(6) |

and for

(7) |

where and . Therefore our system is defined by the three equations above, which include a vector and a scalar field in four dimensions. From Eq.(5) we see that our four dimensional vector field does not satisfy the four dimensional divergenceless condition. The strategy, as explained before in Ref. Alencar:2014moa is to split the field in longitudinal and transverse parts 5) as an equation relating the longitudinal part of and the scalar field. The main question posed at this point is if we can decouple the transverse component in order to obtain a well defined YM field in four dimensions. For the case and for , with only the coupling to the mass term, we have shown that this is the case. Now we have the additional complication of the non-minimal coupling with the field strength, however we will see that it is also valid for the present case. For this end first we define , from where we get the identity . Beyond this we also have and Eq. (6) becomes and to interpret(

(8) |

Now by using Eq. (7), from the definition of and our identity for we can show that

where we have used the fact from the tracelessness of . Therefore we successfully get our decoupled equation for the transverse YM field given by

Now performing the standard separations of variables , with and by considering the zero mode we get the Schroedinger equation with

Now the resolution of the problem is translated in finding a square integrable solution to the above potential

(9) |

such that

(10) |

leaving us with a consistent four dimensional theory.

Firstly, we can see that the only effect of the non-minimal coupling with the field strength is the changing of the measure in the above integrals. However since for large we have that and are constant, the solution without the mass coupling Alencar:2014moa does not give a localized solution. This is consistent with the previous results of Refs. Dvali:2000rx ; Germani:2011cv . However, as stressed before what will guarantee the full localized solution is the non-minimal coupling with the mass term. Since the equation of motion is not affected by the coupling with the field strength, the solution is obtained by fixing with solution as found in Ref. Alencar:2014moa , where is a normalization constant. This therefore can be plugged directly in Eq. (9). With this and after some manipulations we find a normalization constant given by

which is convergent for any warp factor recovering RS for large . Therefore our solution is localized and warp independent. The next problem to be solved is about the non-abelian gauge invariance in four dimensions. Since we yet have the parameter and Eq. (10) is a scalar equation this is in principle possible. In fact when we plug our solution in Eq. (10), after some manipulations we get

(11) |

The above integral is also convergent. The numerator is trivially convergent since for large it converges faster than (9). The denominator is weighted by which is a constant for large and therefore it is convergent for any warp factor. Therefore the problem is completely solved and warp independent. The above results are also valid for the RS1 model. This is due to the fact that our analytical solution is integrable for any range of the integration in the extra dimension sector. The case of RS1 is particularly interesting. In Ref. Batell:2005wa the model of Ref. Ghoroku:2001zu , with boundary and mass term, has been used to analyze phenomenological consequences of localizing gauge fields. Since with this model the relation between boundary and mass term is not fixed, the authors can choose to localize the zero mode of the gauge field in either brane. In our model the solution to the zero mode is completely fixed to and therefore we can only have gauge fields localized in the UV brane. The holographic interpretation of such result has also been clarified in Ref Batell:2005wa . The authors showed that when the mode is localized on the UV brane, the photon eingenstate in the dual theory is primarily composed of the source field.

As concluding remarks we should point out that our model solves the long standing problem of consistently obtaining non-abelian YM fields in RS scenarios from a bulk action. The strong points are that it solves the problem for arbitrary warp factors beyond does not adding any other fields or degrees of freedom. Although non-minimal couplings with gravity has not been much exploited in RS models the resolution of this problem poses this kind of couplings in a central position. Many questions arise such as the origin of this coupling or if there are other couplings which can solve the problem. If this is true we should also answer what kind of restriction must be imposed to get a desirable theory in four dimensions. In fact in separate works we have analyzed resonances of the modelJardim:2015vua . The results point that it does not possess resonances of massive modes, what seems at least curious. Soon later we generalized the case of gauge field to include interactions with other geometrical objectsAlencar:2015oka . The result is that there are some kinds of tensors which do not provide a localized zero mode. In these cases we also do not found resonances for the cases considered. From another viewpoint, the fact that our five dimensional action is not gauge invariant points to the possibility of considering other interactions which break this symmetry. Another question is if these kind of couplings work for other fields. In this direction some of us has shown that the non-minimal coupling with the mass term can be used to localize ELKO spinors and form fields in a warp independent way Alencar:2014moa ; Alencar:2014fga ; Jardim:2014cya ; Jardim:2014xla . Despite the fact that the problem seems to be punctual, the way used to solve it points to many interesting directions. It is clear for us that a wide range of possibilities has been opened. It is also important to understand the reason why these kind of couplings generate solutions which solve the problems independently of a specific form for the warp factor. In particular, spin 1/2 fields are specifically interesting, but this seems to be an yet more difficult problem since basically all the models and extensions of RS give only one chirality of the field localized on the brane. However, since our model provides a localized gauge field we must also have a localized current, what seems to imply that the fermion field is also localized. However this is very speculative, since in the present model we have not specified if the SM fields are localized near the IR, or if SM fields are allowed to propagate in the bulk (specially fermions). Possibly localized gauge field might be problematic in this context due to FCNC constraints. We should also point that the presence of two fine tuned parameters maybe a shortcoming that can be solved if more symmetries are added to the model, such as supersymmetryGherghetta:2000qt . Finally, in another direction, the coupling to the Ricci scalar with gauge fields will change the phenomenology of the RS model. The present model changes the coupling between the radion graviscalar and the gauge fields and this can be phenomenologically relevant at the LHC and also to cosmology of RS modelsChacko:2013dra ; Csaki:1999mp . All the above points are under consideration by the present authors.

## Acknowledgment

Geova Alencar would like to thanks Moreira, A. A. for fruitful discussions. We also thank the referee for very fruitful criticism of the manuscript. We acknowledge the financial support provided by Fundação Cearense de Apoio ao Desenvolvimento Científico e Tecnológico (FUNCAP), the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and FUNCAP/CNPq/PRONEX.

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