Antisymmetric Tensor Fields in Randall Sundrum Thick Branes
G. Alencar, R. R. Landim, M. O. Tahim, C. R. Muniz, R. N. Costa Filho
AAntisymmetric Tensor Fields in Randall Sundrum Thick Branes
G. Alencar a, ∗ , R. R. Landim b , M. O. Tahim a , C. R. Muniz c , R. N. Costa Filho b a Universidade Estadual do Cear´a, Faculdade de Educa¸c˜ao, Ciˆencias e Letras do Sert˜ao Central, Quixad´a,Cear´a, Brazil. b Departamento de F´ısica, Universidade Federal do Cear´a, Caixa Postal 6030, Campus do Pici, 60455-760, Fortaleza, Cear´a, Brazil. c Universidade Estadual do Cear´a, Faculdade de Educa¸c˜ao, Ciˆencias e Letras de Iguatu, Rua Deocleciano Lima Verde, s/n Iguatu,Cear´a,Brazil.
Abstract
In this article we study the issue of localization of the three-form field in a Randall-Sundrum-like scenario. We simulateour membrane by kinks embedded in D=5, describing the usual case (not deformed) and new models coming from aspecific deformation procedure. The gravitational background regarded includes the dilaton contribution. We show thatwe can only localize the zero-mode of this field for a specific range of the dilaton coupling, even in the deformed case.A study about resonances is presented. We use a numerical approach for calculations of the transmission coefficientsassociated to the quantum mechanical problem. This gives a clear description of the physics involved in the model. Wefind in this way that the appearance of resonances is strongly dependent on the coupling constant. We study the cases p = 1 , α = − .
75 and α = −
20. The first value of α give us one resonance peak for p = 1 and no resonancesfor p = 3 ,
5. The second value of α give us a very rich structure of resonances, with number deppending on the value of p . Keywords:
Randall-Sundrum, Resonances
PACS:
1. Introduction
From a mathematical viewpoint, anti-symmetrical ten-sors are natural objects living in differential manifolds.They play an important role in the construction of themanifold’s volume and therefore its orientation. The di-mension of the manifold defines the sort of possible anti-symmetrical tensors and this is used to define a specificspace only of forms [1]. Besides this, they are related tothe linking number of higher dimensional knots [2].From the physical viewpoint,they are of great interestbecause they may have the status of fields describing par-ticles other than the usual ones. As an example we cancite the space-time torsion [3] and the axion field [4, 5]that have separated descriptions by the two-form. Besidesthis, String Theory shows the naturalness of higher ranktensor fields in its spectrum [10, 11]. Other applicationsof these kind of fields have been made showing its relationwith the AdS/CFT conjecture [12].Topological defects soliton-like are studied with increas-ing interest in physics, not only in Condensed Matter, asin Particle Physics and Cosmology. In brane models, theyare used as mechanism of fields localization , avoiding theappearance of the troublesome infinities. Several kinds of ∗ Corresponding author.
Email address: [email protected] (G. Alencar ) these defects in brane scenarios are considered in the liter-ature [6, 7, 8]. In these papers, the authors consider braneworld models where the brane is supported by a soliton so-lution to the baby Skyrme model or by topological defectsavailable in some models. As an example, in a recent pa-per, a model is considered for coupling fermions to braneand/or antibrane modeled by a kink antikink system [9].The subject of this article is to study aspects of thethree-form field in a scenario of extra dimensions. Specif-ically we will study the issue of localization using solitonskink-like and the possibility of resonant KK modes in aRandall-Sundrum-like model (a model with thick branes)[13]. The core idea of extra dimensional models is toconsider the four-dimensional universe as a hyper-surfaceembedded in a multidimensional manifold. The appealof such models is the determination of scenarios wheremembranes have the best chances to mimic the standardmodel’s characteristics. In particular, the standard modelpresents interesting topics to study such as the hierarchyproblem, and the cosmological constant problem that canbe treated by the above-mentioned scenarios. For example,the Randall-Sundrum model [14, 15] provides a possiblesolution to the hierarchy problem and show how gravity istrapped to a membrane.In mathematical terms, the presence of one more extradimension ( D = 5) provides the existence of many an-tisymmetric fields, namely the two, three, four and fiveforms. However, the only relevant ones for our brane are Preprint submitted to Physics Letters B October 25, 2018 a r X i v : . [ h e p - t h ] A ug he two and the three form. This is due to the fact thatwhen the number of dimensions increase, also increasesthe number of gauge freedom. This can be used to cancelthe dynamics of the field in the visible brane. The massspectrum of the two and three form have been studied, forexample, in Refs. [3] and [16]. Posteriorly, the couplingbetween the two and three forms with the dilaton was stud-ied, in different contexts, in [17, 18, 19]. The study of thiskind of coupling, inspired in string theory, is importantin order to produce a process that, in principle, could beseen in LHC. This is a Drell-Yang process in which a pairquark-antiquark can give rise to a three(two)-form field,mediated by a dilaton. In the present case, the coupling isneeded for a different reason. As we will see in this work,the three form field without a dilaton do not has its mass-less mode localized. When we introduce a dilaton, thismode can be localized and therefore this is a more inter-esting case to be considered. The localization of fields ina framework that consider the brane as a kink has beenstudied for example in [20, 21, 22, 23].This work is organized as follows. The second section isdevoted to the study of the membrane as a kink. A solu-tion of Einstein equation is found for two scalar fields, oneof them describing the kink solution and the other repre-senting the dilaton. In the third section we analyze thelocalization of the three form field with and without thedilaton, and observe that the localization is only possiblein the framework with a dilaton coupling under a specificcondition. In the next section we study the gravitationalbackground when a deformation procedure is carried outin our original framework. We show that the masslessmode of the three form is also localized only if we regardthe dilaton coupling with the same condition cited above.In the fifth section we make some considerations aboutthe massive modes and the possibility of resonances. Fi-nally, in the last section, we discuss the conclusions andperspectives.
2. The Kink as a membrane
We start our analysis by studying the space-time back-ground. We go right to the point and give an explicit ex-ample through the localization of a zero mode of the three-form mater gauge field in a four-dimensional thick mem-brane embedded in five dimensions (a Randall-Sundrumlike scenario). It is well known that vector gauge fieldsin these kind of scenarios are not localizable: in four di-mensions the gauge vector field theory is conformal and allinformation coming from warp factors drops out necessar-ily rendering a non-normalizable four dimensional effectiveaction. However, in the work of Kehagias and Tamvakis[13], it is shown that the coupling between the dilaton andthe vector gauge field produces localization of the later.In analogy with the work of Kehagias and Tamvakis weintroduce here the coupling between the dilaton and thethree-form field. As we comment at the end of the nextsection, this coupling is also necessary to localize the three form field. Before analyzing this coupling, it is necessaryto obtain a solution of the equations of motion for thegravitational field in the background of the dilaton andthe membrane. For such, we introduce the following ac-tion [13]: S = (cid:90) d x √− G [2 M R −
12 ( ∂φ ) −
12 ( ∂π ) − V ( φ, π )] . (1)Note again that we are working with a model containingtwo real scalar fields. The field φ plays the role of to gener-ate the membrane of the model while the field π representsthe dilaton. The potential function now depends on bothscalar fields. It is assumed the following ansatz for thespace-time metric: ds = e A ( y ) η µν dx µ dx ν + e B ( y ) dy . (2)The equations of motion are given by12 ( φ (cid:48) ) + 12 ( π (cid:48) ) − e B ( y ) V ( φ, π ) = 24 M ( A (cid:48) ) , (3)12 ( φ (cid:48) ) + 12 ( π (cid:48) ) + e B ( y ) V ( φ, π ) = − M A (cid:48)(cid:48) − M ( A (cid:48) ) + 12 M A (cid:48) B (cid:48) , (4) φ (cid:48)(cid:48) + (4 A (cid:48) − B (cid:48) ) φ (cid:48) = ∂ φ V, (5)and π (cid:48)(cid:48) + (4 A (cid:48) − B (cid:48) ) π (cid:48) = ∂ π V. (6)In order to solve that system, we use the so-called super-potential function W ( φ ), defined by φ (cid:48) = ∂W∂φ , followingthe approach of Kehagias and Tamvakis [13]. The partic-ular solution regarded follows from choosing the potential V ( φ, π ) and super-potential W ( φ ) as V = exp ( π √ M ) {
12 ( ∂W∂φ ) − M W ( φ ) } , (7)and W ( φ ) = vaφ (1 − φ v ) . (8)In this way it is easy to obtain differential equations offirst order whose solutions are solutions of the equationsof motion above, namely π = −√ M A, (9) B = A − π √ M , (10)and A (cid:48) = − W M . (11)The solutions for these set of equations are the following: φ ( y ) = v tanh( ay ) , (12) A ( y ) = − v M (cid:0) ay ) + tanh ( ay ) (cid:1) (13)2nd π ( y ) = v √ M (cid:0) ay ) + tanh ( ay ) (cid:1) . (14)Following the argumentation in Ref. [13], it is possible tosee that, by the linearization of the geometry described inthis section, this model supports a massless zero mode ofthe gravitational field localized on the membrane, even inthe dilaton background.Now the dilaton contribution makes the space-time sin-gular. However this kind of singularity is very common inD-brane solutions in string theory (the dilaton solution isdivergent). The Ricci scalar for this new geometry is nowgiven by R = − (cid:0) A (cid:48)(cid:48) + 18 A (cid:48) (cid:1) e π √ M (15)where the dilaton has an important contribution. Whatis interesting here is that this singularity disappears if welift the metric solution to D = 6. In this case the dilatonrepresents the radius of the sixth dimension [13].
3. Antisymmetric Tensor Fields and the DilatonCoupling
In this section we study antisymmetric tensor fields inthe gravitational background with the dilaton field. Wemust look for localization of these fields in this framework.As the number of antisymmetric tensor fields increase withdimensions, these fields should be considered. In fact thesefields have been taken in account in the literature. Thefact is that, when the number of dimension increases, thenumber of gauge freedom also increases, and this can beused to cancel the degrees of freedom in the visible brane.Therefore the only antisymmetric tensors relevant to thevisible brane are that of rank two and three[16]. We mustfocus here in the second case. We must specifically studythe string inspired coupling of the three form field to thedilaton. As commented before this will give rise to the lo-calization of the field. As commented in the introduction,this kind of coupling can give an interaction that in prin-ciple could be seen at LHC. Despite this, we focus here onthe localization properties of this framework.We have for the action S = (cid:90) d x √− G [ − e − λπ Y MNLP Y MNLP ] (16)and Y MNLP = ∂ [ M X NLP ] is the field strength for thethree form X . We can use gauge freedom to fix X µνy = ∂ µ X µνα = 0 and we are left with the following terms Y MNAB = ∂ [ µ X ναβ ] (17) Y MNAB = ∂ [ y X ναβ ] . (18)Using the above facts we obtain for the action S X = (cid:90) d x (cid:90) dy {− e − λπ e − A + B Y αµλγ Y αµνλ +4 e − A − B ˙ X αµλ ˙ X αµλ ] } , (19)with the equations of motion ∂ α Y αµλγ + e λπ +4 A − B ∂ y [ e − λπ − A − B ˙ X αµλ ] = 0 . (20)We can now separate the variables with the followingansatz X µνα ( x α , y ) = B µνα ( x α ) U ( y ) = B µνα (0) e ip α x α U ( y ) , (21)where p = − m . We now write Y αµλγ = ˜ Y αµλγ U , where˜ Y stands for the four dimensional field strength and weget for the EM U (cid:48)(cid:48) ( y ) − ( λπ (cid:48) + 2 A (cid:48) + B (cid:48) ) U (cid:48) ( y ) = − m e B − A ) U ( y ) . (22)A solution is found in the case m = 0, where we findthat U = cte solves the above equation. In this case, theeffective action can be found easily to give us S X = (cid:90) dye − λπ − A + B U (cid:90) d x [ − Y αµλγ ˜ Y αµνλ ] . (23)Using now our solution for A , B and π we have that,when λ > / √ M , the integration in the extra dimen-sion is finite. Therefore the massless mode of the threeform field is possibly localized in a framework with a dila-ton coupling.It is worthwhile to analyze the case without the dilatoncoupling. In this case the metric is given by ds = e A ( y ) η µν dx µ dx ν + dy . (24)The gravitational solution has been found in [13] andthe warp factor is the same as in the case with dilatoncoupling. Therefore following the same steps as before, wecan find the equation of motion U (cid:48)(cid:48) ( y ) − AU (cid:48) ( y ) = − m e − A U ( y ) . (25)Again a direct solution is found in the case m = 0, and U = cte solves the above equation. In this particular case,the effective action can be found easily to give us S X = (cid:90) dye − A U (cid:90) d x [ − Y αµλγ ˜ Y αµνλ ] . (26)Using now our solution for A , we have that the integra-tion in the extra dimension is not finite. Therefore we seethat the dilaton coupling is really essential to the localiza-tion of the zero mode.3 . The Dilatonic Deformed Brane In this section we analyze a special class of solutionsby defining a deformation of the λφ potential [24]. It ispossible to solve the equations of motion by the super-potential method. This formalism was initially introducedin studies about non-super-symmetric domain walls in var-ious dimensions by [25, 26]. The first step in our analysisis to find the Einstein’s equations for the coupled system ofthe scalar-dilaton-gravity system that composes the back-ground space-time. We repeat here the same action al-ready discussed in the sections above now with the super-potential W p ( φ ) = 2 p p − φ p − p − p p + 1 φ p +1 p , (27)where p is an odd integer.By solving ∂W p ∂φ = φ (cid:48) we arrive at φ p ( y ) = tanh p ( yp )and we see that for p = 1 we get the usual kink solution.For p = 3 , , ... we can construct the so called two-kinksolutions, describing internal structures inside the mem-brane [24]. The parameter p introduced in the procedurecontrols characteristics such as thickness and matter en-ergy density of the membrane [27]. We must note that theequations (9), (10) and (11) are left unchanged. From thesuper-potential cited we get the first order equations π p = −√ M A p , (28) B p = A p − π √ M , (29)and A (cid:48) p = − W M . (30)From the last equation we can find A p ( y ) [27], A p ( y ) = − v M p p + 1 tanh p (cid:18) yp (cid:19) − v M (cid:18) p p − − p p + 1 (cid:19) (31) × (cid:26) ln (cid:20) cosh (cid:18) yp (cid:19)(cid:21) − p − (cid:88) n =1 n tanh n (cid:18) yp (cid:19)(cid:27) and all the steps for obtaining the equations of motionand the effective action are basically identical. For theequation of motion we therefore obtain after a separationof variables U (cid:48)(cid:48) ( y ) − (cid:0) λπ p (cid:48) + 2 A p (cid:48) + B p (cid:48) (cid:1) U (cid:48) ( y ) = − m e B p − A p ) U ( y ) , (32)and we can see that, again, we have a trivial solution forthe case m = 0. The effective action in the present case iseasily found to be S X = (cid:90) dye − λπ − A p + B p U (cid:90) d x [ − Y αµλγ ˜ Y αµνλ ] . (33) Using now our solution for A p , B p and π p we arrive inthe same situation as before and have that, when λ > / √ M and for finite p , the integration in the extradimension is finite. Therefore the massless mode of thethree form field is possibly localized in a framework witha dilatonic deformed brane.
5. The Massive Modes and Resonances
Now we must analyze the possibility of localization forthe massive modes in both, deformed and usual cases. Asthe usual case can be obtained just by putting p = 1, wemust study only the more general deformed case. Thebest way to analyze it is to transform the equation (32) ina Schr¨odinger type. It is easy to see that an equation ofthe form (cid:18) d dy − P (cid:48) ( y ) ddy (cid:19) U ( y ) = − m Q ( y ) U ( y ) , (34)can be transformed in a Schr¨odinger type (cid:18) − d dz + V ( z ) (cid:19) ¯ U ( z ) = m ¯ U ( z ) , (35)through the transformations dzdy = f ( y ) , U ( y ) = Ω( y ) ¯ U ( z ) , (36)with f ( y ) = (cid:112) Q ( y ) , Ω( y ) = exp( P ( y ) / Q ( y ) − / , (37)and V ( z ) = ( P (cid:48) ( y )Ω (cid:48) ( y ) − Ω (cid:48)(cid:48) ( y )) / Ω f (38)where the prime is a derivative with respect to y . Nowusing the equation (32) we obtain V ( z ) = e A p / (cid:18) ( α −
964 ) A (cid:48) p ( y ) − ( α A (cid:48)(cid:48) p ( y ) (cid:19) ,f ( y ) = e − Ap , Ω( y ) = e ( α + ) A p , (39)where α = 9 / − λ √ M , and we take y of function of z through z ( y ) = (cid:82) y f ( η ) dη . Therefore, the eingenvaluesgive us the masses for which the localization is possible. Itis important to note that, after the above transformations,the condition for localization becomes (cid:90) dz ¯ U = f inite (40)and this is exactly the square integrable condition of thewave function in quantum mechanics. This enforce theargument that we have a schroedinger-like equation.Despite its importance, the analytic solution for theabove equation has not been found. The strategy there-fore is to plot numerically the potential and analyze itsshape. The appearance of resonances depends strongly on4 (cid:45) z (cid:45) (cid:45) (cid:45) V (cid:72) z (cid:76) (cid:45) (cid:45) z (cid:45) V (cid:72) z (cid:76) Figure 1: Potential of the Schroedinger like equation for p = 1 (lined)scaled by 1/10, p = 3 (dashed), p = 5 (dotted) for α = − .
75 (top)and α = −
20 (bottom). the shape of the potential and therefore of the couplingconstant. This plot is given by fig. 1 for p = 1 , ,
5. Forthis we choose v M = 1 and v = a = 1.Note that the shape of the potential changes signifi-cantly and this will give us a very different resonance struc-ture. Another thing we can observe is that the potentialbecomes null for z = ∞ .Therefore we do not have a dis-crete spectrum for m >
0. Beyond this, it is a known factfrom quantum mechanics that any solution for positive m must posses a oscillatory contribution, and therefore thewave function is not normalizable. The conclusion is thatthe only localized mode is the massless one. This meansthat this is the unique mode living inside the membrane.Another important possibility is the appearance of res-onances. As we have a volcano potential, we can ask forthe possibility that the massive modes, besides living inthe extra dimension, has a peak of probability to be foundat the location of the membrane. This analysis has beendone extensively in the literature [23, 28]. If we compareour Schroedinger equation with that found by these au-thors, we see that they are basically the same, so thatwe can extract the same information about resonances ofthe model. The authors found that, after normalize thewave function in a truncated region, there is a resonancevery close to m = 0, and this indicates that as lighter isthe mass, bigger is the probability of interacting with themembrane. Them we must corroborate with the analysisand the same happens in our model. From our viewpoint the wave function has an oscilla-tory part and can not be normalized. To scape of this,the authors define a relative probability, but for us it isnot clear how it solves the problem. Another very impor-tant point is the truncation of the integration region. Thisis equivalent to have a potential that do not falls to zeroat infinity, therefore it is natural to have a peak of reso-nance very close to m = 0. This could leads to the wronginterpretation that very light modes can interact with themembrane. From our analysis, it is not true that as lighteris the massive modes, bigger is the probability of findingthem inside the membrane. In fact, this is corroboratedonly when p = 1 and for the value of the coupling constantused by the authors in [28]. In this case we find, as can beseen in fig. 2, a resonance for the specific value m = 0 .
09. When we consider α = −
20, as shown in fig. 3 we finda very rich resonance structure and that heavy modes canalso resonate. We must point here that the Schroedingereq. used by them is very similar to ours but have a dif-ferent multiplicative factor in the potential because theystudy the two form. In principle this could change theresults, therefore in a separate paper we analyze carefullythat case [29].In other to analyze resonances, we must compute trans-mission coefficients( T ), which gives a clearer and cleanerphysical interpretation about what happens to a free wavethat interact with the membrane. The idea of the existenceof a resonant mode is that for a given mass the potentialbarrier is transparent to the particle, i. e., the transmissioncoefficient has a peak at this mass value. That means theamplitude of the wave-function has a maximum value at z = 0 and the probability to find this KK mode inside themembrane is higher. In order to obtain these results nu-merically, we have developed a program to compute trans-mission coefficients for the given potential profile. A moreextensive and detailed analysis of resonances with trans-mission coefficients will be given in a separate paper by theauthors [29]. In fig. 2 we give the plots of T for α = − . , and considering the case p = 1 , , α only for p = 1. In fig. 3 we show the plot for α = −
20 andwe can see how this alter the existence of resonances. Forthis values we have an interesting structure of resonances,witch is similar to that found in quantum mechanical prob-lems. As we stress in our separate paper, the existence ofresonances depends strongly of the shape of the potential.In our case, therefore, the resonance is driven by the choiceof the parameter α .
6. Conclusions and Perspectives
In this paper we have studied the issue of localizationfor a three form field in a Randall-Sundrum-like model.We approach this in a smooth space scenario, where theextra dimension is described by a kink. We show that, forthe usual and the deformed cases, we have the localizationof the zero mode only for a specific range of the coupling5 .2 0.4 0.6 0.8 m (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:72) T (cid:76) m (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:72) T (cid:76) m (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:72) T (cid:76) Figure 2: Logarithm of the transmission coefficient for p = 1 (top), p = 3 (middle), p = 5 (bottom) for α = − .
75 . constant given by λ > / √ M . We study the possibil-ity of resonances in the model. This is done throughoutthe numerical computations of the transmission coefficientand we find that the appearance of peaks depends stronglyon the value of the dilaton coupling constant. For highvalues of this constant we find an interesting structure ofresonances, very similar to what is common in quantummechanical problems. For the value α = −
20 and p = 2,for example, we found eight peaks of resonances. Thereforea bigger value of the coupling constant favor resonances ofmassive modes. This should seems natural, as the dilatonis responsible for localizing the zero mode and a biggervalue of the constant would augment the possibility forexistence of resonances. It should be noted that from ouranalysis heavy modes can also resonate.We thank A. A. Moreira for providing a computer for
50 100 150 200 m (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:72) T (cid:76) m (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:72) T (cid:76) m (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:72) T (cid:76) Figure 3: Logarithm of the transmission coefficient for p = 1 (top), p = 3 (middle), p = 5 (bottom) for α = −
20 . numerical calculations. The authors would like to ac-knowledge the financial support provided by Funda¸c˜aoCearense de Apoio ao Desenvolvimento Cient´ıfico e Tec-nol´ogico (FUNCAP) and the Conselho Nacional de Desen-volvimento Cient´ıfico e Tecnol´ogico (CNPq).This paper is dedicated to the memory of my wife IsabelMara (R. R. Landim)
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