Thick brane solitons breaking Z 2 symmetry
aa r X i v : . [ g r- q c ] S e p October 9, 2018 9:20 WSPC Proceedings - 9.75in x 6.5in MG14˙AT2˙Peyravi2 page 1 Thick brane solitons breaking Z symmetry Marzieh Peyravi , Nematollah Riazi and Francisco S. N. Lobo Department of Physics, School of Sciences, Ferdowsi University of Mashhad, Mashhad91775-1436, IranE-mail: [email protected] Physics Department, Shahid Beheshti University, Evin, Tehran 19839, IranE-mail: n [email protected] Instituto de Astrof´ısica e Ciˆencias do Espa¸co, Faculdade de Ciˆencias da Universidade deLisboa, Edif´ıcio C8, Campo Grande, P-1749-016 Lisbon, PortugalE-mail: [email protected]
New soliton solutions for thick branes in 4 + 1 dimensions are considered in this article.In particular, brane models based on the sine-Gordon (SG), ϕ and ϕ scalar fields areinvestigated; in some cases Z symmetry is broken. Besides, these soliton solutions areresponsible for supporting and stabilizing the thick branes. In these models, the originof the symmetry breaking resides in the fact that the modified scalar field potential mayhave non-degenerate vacuua and these non-degenerate vacuua determine the cosmologicalconstant on both sides of the brane. At last, in order to explore the particle motion in theneighborhood of the brane, the geodesic equations along the fifth dimension are studied. Keywords : Domail wall, Brane, Soliton
1. Introduction
Since there is no known fundamental principle requiring spacetime to be (3 +1) − dimensional , our observable universe might be a (3 + 1) − dimensional branein a higher dimensional bulk . Most models suggest that there are one or moreflat 3-branes embedded discontinuously in the ambient geometry . Furthermore,almost all of the extra-dimensional models require the existence of scalar fields, forinstance, to generate a domain-wall which localizes matter fields . Scalar fieldsserve to stabilize the size of the compact extra dimensions , and can support themodification of the Randall-Sundrum warped-space to a smoothed-out version ,or to cut off the extra dimension at a singularity . In addition to this, the non-linearity in the scalar field and, in particular, the existence of disconnected vacuuain the self-interaction of the scalar field lead to the appearance of a stable localizedsolution, which is a good motivation for building thick brane models . It is fre-quently invoked to replace an infinitely thin brane with a thick one, by supposinga scalar field with soliton behavior. Based on this point of view, in this paper weinvestigate diverse models, namely, the sine-Gordon (SG), ϕ and ϕ brane modelswhich have broken Z symmetry in some cases. Note that the Z symmetry may berestored by a proper choice of model parameters. The origin of symmetry breakingin our models reside in the fact that the modified scalar field potential may havenon-degenerate vacuua . The cosmological constant on both sides of the brane aredetermine by these vacuua. ctober 9, 2018 9:20 WSPC Proceedings - 9.75in x 6.5in MG14˙AT2˙Peyravi2 page 2 We consider a thick brane, embedded in a five-dimensional (5D) bulk spacetime,modelled by the following action: S = Z d x p | g | (cid:20) R − ∂ α ϕ∂ α ϕ − V ( ϕ ) (cid:21) . (1)The simplest line element of the brane, embedded in the five-dimensional bulkspace-time can be written as : ds = g AB dx A dx B = dw + e A ( dx + dy + dz − dt ) , (2)where A is the warp factor which depends only on the 5D coordinate w .The 5D gravitational and scalar field equations take respectively the followingforms 3 A ′′ + 6 A ′ = − κ e − A T = − κ (cid:20) ϕ ′ + V ( ϕ ) (cid:21) , (3)6 A ′ = κ T = κ (cid:20) ϕ ′ − V ( ϕ ) (cid:21) , (4) ϕ ′′ + 4 A ′ ϕ ′ = dV ( ϕ ) dϕ , (5)where the prime denotes derivative with respect to w .In order to obtain the first-order equation, one can introduce the auxiliary func-tion W in such a way that: A ′ = − W ( ϕ ) ,ϕ ′ = 12 ∂W ( ϕ ) ∂ϕ , (6)while V ( ϕ ) achieves the special form : V ( ϕ ) = 18 (cid:18) ∂W ( ϕ ) ∂ϕ (cid:19) − W ( ϕ ) . (7)Moreover, it may be instructive to calculate the geodesic equation along the fifthdimension in a thick brane, in order to investigate the particle motion near thebrane . To this end, we start with the geodesic equation: d x dτ + Γ AB dx A dτ dx B dτ = 0 ⇒ ddτ (cid:0) − e A ˙ t (cid:1) = 0 ,d x dτ + Γ AB dx A dτ dx B dτ = 0 ⇒ ¨ w + A ′ e A ˙ t = 0 , (8)which leads to ¨ w + c f ( w ) = 0 , (9)where c is a constant of integration and the function f ( w ) is defined as f ( w ) = A ′ ( w ) e − A ( w ) . (10) ctober 9, 2018 9:20 WSPC Proceedings - 9.75in x 6.5in MG14˙AT2˙Peyravi2 page 3
2. Thick brane models
First we consider the Sine-Gordon (SG) system, where the self-interaction potentialof this model reads: ˜ V ( ϕ ) = ab [1 − cos ( bϕ )] , (11)where a and b are free parameters of the model. The SG system has the followingexact static kink solution : ϕ ( w ) = 4 b arctan (cid:16) e √ abw (cid:17) , (12)which is plotted in Fig.1(a), for various values of parameters a and b , which corre-spond to branes with different thickness. When considered as the brane potential,however, this potential should be modified to become consistent with the Einsteinequations. The corresponding potential for this model is given by: V ( ϕ ) = 2 ab sin (cid:18) bϕ (cid:19) − a b (cid:20) (cid:18) bϕ (cid:19)(cid:21) , (13)which is depicted in Fig.2(a). Notice that this potential has two series of non-degenerate vacuua, as in the DSG (double sine-Gordon) system potential . How-ever, in the limit of b ≫ a these vacuua tend to the same value (become degenerate),such as the potentials used in .In this case the energy density is localized at the brane and the thickness of thelatter is △ = √ ab . It can be seen that there is no singularity in the Ricci scalarand/or Kretschmann scalar. Moreover, in the limits of w → ±∞ , the Ricci scalarand all the components of the Einstein in the right and left sides of the brane aredifferent (Fig.3(a)). However, in the limit of w →
0, the Einstein tensor is givenby G µν =
83 (16 − b ) ab δ µν (for µ = ν = 0 , , , G ij ∝ Λ δ ij .Therefore, we have a broken Z -symmetry in the bulk, as the two sides of thebrane differ completely. On the right and in the limit of w → + ∞ , the Einsteintensor and consequently the cosmological constant of the bulk vanish, so the bulk isasymptotically Minkowski. However, on the other side of the brane, these quantitiesare nonzero and equal to the constant value 512 a/ (3 b ), and as a result the bulkwould be de Sitter. Besides, by checking the geodesic equation of a test particlemoving only in the direction of the extra dimension, the confining effect of the scalarfield is proved.The second brane model is the ϕ -based model, for which we have :˜ V ( ϕ ) = β α (cid:0) ϕ − α (cid:1) , (14)where α and β are constants. The kink solution reads ϕ ( w ) = α tanh( βw ) , (15) ctober 9, 2018 9:20 WSPC Proceedings - 9.75in x 6.5in MG14˙AT2˙Peyravi2 page 4 −4 −2 0 2 40123456 w ϕ −4 −2 0 2 4−1−0.500.51 w ϕ −4 −2 0 2 400.20.40.60.81 w ϕ β =1 β =5 β =10 β =1 β =5 β =10a=1a=5a=10 (c)(b)(a) Fig. 1. Soliton solutions as a function of the fifth dimension for the following models: (a) SG for b = 1, (b) φ for α = 1 and (c) φ for α = 1 systems. Dashed, dotted-dashed, and continuouscurves correspond to solitons with decreasing brane thickness. In the limit of an infinite a/β parameter, the soliton approaches the step function. which is depicted in Fig.1(b). The potential is obtained as V ( ϕ ) = 12 α β (cid:18) − ϕ α (cid:19) − ϕ α β (cid:18) − ϕ α (cid:19) , (16)which is plotted in Fig.2(b) and the brane thickness becomes △ = β − . As forthe previous SG model, we determine the limits w → ±∞ for all the componentsof the Einstein tensor components, which are the same at different sides and givenby α β (Fig.3(b)) and in the limit of w → µ = ν = 0 , , ,
3) takes the form − α β δ µν . Therefore, in this model thecosmological constant on the brane Λ would be − α β . For this model a testparticle oscillate around the brane by Ω = q c αβ . −2 −1 0 1 2−0.100.10.2 ϕ V ( ϕ ) −2 −1 0 1 2−1135 ϕ V ( ϕ ) −1 0 1−0.10.20.50.8 ϕ V ( ϕ ) (c)(b)(a) Fig. 2. The plots depict the modified soliton potential as a function of the fifth dimension for (a)SG with a = b = 10, (b) φ with α = β = 1 and (c) φ with α = β = 1 systems. The potential ofthe SG and φ systems have non-degenerate vacuua. In contrast, the φ potential has degeneratevacuua and this leads to stable, topological solitonic brane. ctober 9, 2018 9:20 WSPC Proceedings - 9.75in x 6.5in MG14˙AT2˙Peyravi2 page 5 The last model is based on the ϕ soliton. For this model, we have the followingpotential: ˜ V ( ϕ ) = β α ϕ (cid:0) ϕ − α (cid:1) , (17)where α and β are constant(as in the ϕ model) and as a result, the kink solutionis given by : φ ( w ) = α p e ( −√ αβw ) . (18)The kink solution is depicted in Fig.1(c). Thus, for the ϕ system the self-interactionpotential takes the form (Fig.2(c)): V ( ϕ ) = 14 β ϕ (cid:0) α − ϕ (cid:1) α − β ϕ (cid:0) α − ϕ (cid:1) α , (19)The thickness of this brane is given by △ = ( √ αβ ) − . Moreover, for this modeland in the limit of w −→ ±∞ the mixed Einstein tensor components reduce to α β and zero respectively see Fig.3(c). and in the limit of w →
0, the Einsteintensor is G µν = α β (cid:0) α − (cid:1) δ µν . −10 −5 0 5 10−202060100140180 w G −10 −5 0 5 10−2−1.5−1−0.500.511.5 w G −10 −5 0 5 10−0.15−0.1−0.0500.050.1 w G (a) (b) (c) Fig. 3. The plots depict the Einstein tensor component G for the (a) SG with a = b = 1, (b) φ with α = β = 1 and (c) φ with α = β = 1 systems. Note that this quantity approaches differentconstant values for the SG and ϕ systems, while the ϕ system is Z -symmetric.
3. Conclusion
In this work, we obtained exact thick brane models inspired by well-known nonlinearsystems, namely, the sine-Gordon ( SG ), ϕ and ϕ models. The confining effect ofthe scalar field in all these three models were confirmed by examining the geodesicequation for a test particle moving normal to the brane. In particular, it turns outthat the modified potential for the SG system resembles that of the double sine-Gordon ( DSG ) system, while those of ϕ and ϕ became ϕ and ϕ , respectively. ctober 9, 2018 9:20 WSPC Proceedings - 9.75in x 6.5in MG14˙AT2˙Peyravi2 page 6 In the case of the SG model, the resulting brane does not have Z symmetry,in general, where the center of the brane may be displaced from w = 0 and thepotential will not be an odd function of w in general. However, by a suitable choiceof the model parameters it is possible to make the vacuua of the effective potentialdegenerate, in which case the Z symmetry is restored. In the case of the φ model,however, we could not restore this symmetry via re-parametrization.
4. Acknowledgments
M.P. acknowledges the support of Ferdowsi University of Mashhad via the pro-posal No. 32361. N.R. acknowledges the support of Shahid Beheshti UniversityResearch Council. F.S.N.L. acknowledges financial support of the Funda¸c˜ao para aCiˆencia e Tecnologia through an Investigador FCT Research contract, with referenceIF/00859/2012, funded by FCT/MCTES (Portugal), and the grant EXPL/FIS-AST/1608/2013.
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