The 5D Standing Wave Braneworld With Real Scalar Field

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1 The 5D Stnding Wve Brneworld With Rel Sclr Field rxiv:0.9v [hep-th] 0 Dec 0 Merb Gogbershvili Andronikshvili Institute of Physics, 6 Tmrshvili Street, Tbilisi 077, Georgi nd Jvkhishvili Stte University, Chvchvdze Avenue, Tbilisi 08, Georgi E-mil: gogber@gmil.com Pvle Midodshvili Ili Stte University, /5 Cholokshvili Avenue, Tbilisi 06, Georgi E-mil: pmidodshvili@yhoo.com December, 0 Abstrct We introduce the new 5D brneworld with the rel sclr field in the bulk. The model represents the brne which bounds collective oscilltions of grvittionl nd sclr field stnding wves. These wves re out of phse, i.e. the energy of oscilltions psses bck nd forth between the sclr nd grvittionl wves. When the mplitude of the stnding wves is smll the brne width nd the size of the horizon in extr spce re of sme order of mgnitude, nd mtter fields re loclized in extr dimension due to the presence of the horizon. When oscilltions re lrge trpping of mtter fields on the brne is provided minly by the pressure of bulk wves. It is shown tht in this cse the mss of the lightest KK mode is determined by the smller energy scle corresponding to the horizon size, i.e. these modes cn be creted in ccelertors t reltively low energies, which gives chnce to check the present model. PACS numbers: h,.5.-w,.7.+d Introduction Brneworld models involving lrge extr dimensions [ 6] hve been very useful in ddressing severl open questions in high energy physics (e.g. the hierrchy problem nd the nture of

2 flvor. Most of the brneworlds re relized s time-independent field configurtions. However, mostly within the frmework of cosmologicl studies, there were proposed models with timedependent metrics nd mtter fields, s well s brnes with tensions vrying in time[7 0]. One of the 5D brneworld models with non-sttionry metric coefficients ws proposed by one of us in [ ] (for the generliztion to 6D cse see [5, 6]. In this scenrio the brneworld ws generted by stnding grvittionl wves coupled to phntom-like bulk sclr field, rpid oscilltions of these wves provide universl grvittionl trpping of zero modes of ll kinds of mtter fields on the brne [7 ]. In this pper we introduce the new non-sttionry 5D brneworld generted by stnding wves of the grvittionl nd rel sclr fields, insted of the phntom-like sclr fields of [ ]. The model lso hs new fetures: it does not require bulk cosmologicl constnt nd the oscilltion frequency of the stnding wves cn be smll. The Model We consider 5D spce-time without bulk cosmologicl constnt contining brne nd nonself intercting rel sclr field coupled to grvity: S = d 5 x ( M g R+ gmn M ϕ N ϕ+l B. ( Here L B is the brne Lgrngin nd M is the 5D fundmentl scle, which reltes to the 5D Newton constnt, G = /(8πM. Cpitl Ltin indexes numerte the coordintes of 5D spce-time with the signture (+,,,,, nd we use the units where c = =. Vrition of the ction ( with respect to g AB leds to the 5D Einstein equtions: R AB g ABR = M (σ AB +T AB. ( Here the source term is the combintion of the energy-momentum tensors of the brne, σ AB, nd of the bulk sclr field, T AB = A ϕ B ϕ g AB C ϕ C ϕ. ( Using ( the Einstein equtions ( cn be rewritten in the form: R AB = ( σ M AB g ABσ + A ϕ B ϕ, ( where σ = g AB σ AB. To solve the equtions ( we tke the metric nstz: ds = e S ( r / ( dt dr ( r /( e u dx +e u dy +e u dz, (5 where is positive constnt nd S = S( r, u = u(t, r re some functions.

3 The metric (5 is some combintion of the 5D generliztions of the known domin wll solution [ ] (when S = u = 0 nd of the colliding plne wve solutions [5 7] (when S = = 0. Similr nstz in the D cse ws considered by one of us in [8]. To find solution to the system of Einstein nd bckground rel sclr field equtions let us ssume tht the 5D sclr field, ϕ ϕ(t, r, (6 depends only on time, t, nd on bsolute vlue of the extr coordinte, r. Then its eqution of motion: g M ( gg MN N ϕ = 0, (7 where g = ( r / e S, (8 is the determinnt of the bckground metric (5, reduces to [ ϕ + δ(r r ] ϕ ϕ = 0, (9 where overdots nd primes denote derivtives with respect to t nd r, respectively. For the nstz (5 the Einstein equtions ( split into the system of the equtions for the metric functions: u + S ( r S = M ϕ, u u = ϕ, M ϕ ( r (u ü u = 0, (0 u S ( r S =, M ϕ nd for the brne tensions: ( M δ(r +S = σ tt e S ( r /σ, ( M δ(r u e S+u = σ xx + ( r / e u σ, ( M δ(r u e S+u = σ yy + ( r / e u σ, ( ( M δ(r +u e S u = σ zz + ( r / e u σ, ( M δ(r S = σ rr + e S ( r /σ.

4 The solution to (9 nd (0 is ( ω u(t, r = Asin(ωtJ 0 ω r, M ( ω ϕ(t, r = Acos(ωtJ 0 ω r, ( S( r = ω ( r [ ( ω ( ω A J 0 ω r +J ω r ( ω ( ω ] ω( r J 0 ω r J ω r B, where A nd B re some dimensionless constnts, J 0 nd J re Bessel functions of the first kind nd the integrtion constnt ω corresponds to the frequency of stnding wves. Using ( from the equtions ( one cn esily find the brne energy-momentum tensor: where the brne tensions re: σ B A = M δ(rdig [ τ t t,τx x,τy y,τz z,0], ( τ t t =, ( ω τx x = τy y = +B +Aωsin(ωtJ τz z = ( ω +B Aωsin(ωtJ., ( To interpret the solution ( s describing the brne t r = 0, which bounds the sclr nd grvittionl bulk stnding wves, one needs to ssume tht the oscilltory metric functions nd 5D sclr field vnish t the origin, i.e. u r=0 = 0, ϕ r=0 = 0, S r=0 = 0. (5 These conditions cn be chieved ssuming the reltion between ω nd : ω = Z(J 0 n, (6 where Z (J 0 n is the n-th zero of the function J 0, nd simultneously fixing the constnt B: B = ω A ( ω J. (7 In fct, the conditions (5 led to the quntiztion, (6, of the rtio of the stnding wve frequency, ω, to the curvture scle,. From ( one cn lso see tht the metric function, u(t, r, nd the sclr field, ϕ(t, r, hving similr dependence on sptil coordintes, re oscillting π/ out of phse in time, i.e. the energy of the oscilltions is pssing bck nd forth between the grvittionl nd sclr field stnding wves bounded by the brne. There re three free prmeters in our model: the constnt A (defining the mplitude of oscilltions, the curvture scle (giving the size of extr spce for observers on the brne nd the rtio Z (J 0 n = ω/ (the n-th zero of Bessel function J 0. Below we consider two limiting cses corresponding to the smll, A, nd the lrge, A, mplitudes of bulk stnding wves.

5 The Smll Extr Spce In the first limiting cse the mplitude, A, nd consequently the energy of the oscilltions is smll. Then, neglecting the terms contining the constnts A nd B A in the brne tensions (, one gets the brne energy-momentum tensor, σa B M δ(rdig [,,, ],0, (8 which obeys the eqution of stte E = P, with E nd P being the brne energy density nd pressure, respectively. Moreover, from ( it is cler tht in this cse the functions u, S nd ϕ does not ply significnt role nd from the very beginning one cn ssume: S( r = u(t, r = ϕ(t, r = 0, (9 nd consider the metric nstz without oscilltory metric functions: ds = ( r / ( dt dr ( r /( dx +dy +dz. (0 This metric is 5D generliztions of the D domin wll solution of [ ]. Due to the presence of the bsolute vlue of the extr coordinte, r, the Ricci tensor t r = 0 hs δ- function-like singulrity which corresponds to the brne tension, see (. The solution (0 hs lso new fetures, since, in contrst to D domin wlls [ ], the prmeter hs the opposite sign. Due to this fct the metric (0 hs the horizon t r = / in the bulk. At these points R tt nd R rr components of the Ricci tensor get infinite vlues, while ll grvittionl invrints, e.g. Ricci sclr, [ ( R = e S ( r / u u ] ( +S B + 8 δ(r, ( re finite there. It resembles the sitution with the Schwrzschild Blck Hole, however, in contrst the determinnt of our metric (0 becomes zero t r = /. As the result nothing cn cross the horizon of (0, nd mtter fields re confined inside of the -brne of the width / in the extr spce. To provide experimentlly cceptble locliztion of mtter fields the ctul size of the extr dimension must be sufficiently smll, /M H, where M H denotes the Higgs scle. So in the limiting cse A the curvture scle must be lrge M H M, where M is the 5D fundmentl scle. As regrds the brne, its width, in fct, is defined by the horizon size. The Lrge Extr Spce In the second lrge mplitude limiting cse it s obvious tht A B. ( 5

6 Now, ssuming tht curvture scle is reltively smll, the width of the brne, locted t the origin of the lrge (of the size / but finite extr spce, is determined by the metric function S( r in (5. Indeed, for smll ω the time-dependent terms in the brne tensions ( re negligible, σ B A M δ(rdig[,b,b,b,0], ( nd the brne width is of the order of /(B. So trpping of mtter fields is cused by the pressure of the bulk oscilltions nd not by the existence of the horizon in the extr spce. As n illustrtive exmple of this trpping mechnism let us consider locliztion of rel mssless sclr field, Φ(x A, on the brne in the bckground metric (5. From the 5D ction, S Φ = dx 5 gg MN M Φ N Φ, ( we obtin the Klein-Gordon eqution for Φ(x A : Using (8 the eqution (5 cn be written s: [ t e S ( ] e u ( r / x +e u y +eu z Φ = g M ( gg MN N Φ = 0. (5 ( r r[( r r Φ]. (6 In ddition, it s esy to find tht the ssumption ( leds to the following reltion between the metric functions: u(t, r S( r. (7 Then in the sclr field eqution (6 we cn drop the function u(t,r in the exponents nd rewrite it s: [ t e S ( ] ( r / x + y + z Φ = We look for the solution in the form: ( r r[( r r Φ]. (8 Φ(t,x,y,z,r = φ(x µ ρ(r, (9 where Greek indexes numerte D coordintes, nd the D fctor of the sclr field wvefunction φ(x µ obeys the eqution: η νβ ν β φ(x µ = m φ(x µ. (0 In wht follows we ssume φ(x µ = e i(et pxx pyy pzz. ( Then extr dimension fctor ρ( r of the sclr field wvefunction obeys the eqution: ρ sgn(r [ r ρ E e S ]ρ = m e S ( r / ( r /ρ, ( 6

7 with the boundry conditions: ρ r =0 = 0, ρ r / = 0. ( For the sclr field zero mode wvefunction, ρ 0 (r, with the dispersion reltion: the eqution ( reduces to ρ 0 sgn ( r ρ 0 E E = p x +p y +p z, ( [ ] e S ρ ( r / 0 = 0. (5 To show tht the eqution (5 hs the solution loclized on the brne we investigte the eqution in two limiting regions: close to the brne ( r 0 nd close to the horizon in the extr spce ( r /. In the first limiting region, r 0, [ ] u r 0 = sin(ωt B r +O( r, S r 0 = B r +O ( r, (6 ( g r 0 = B + r +O ( r, the eqution (5 hs the following symptotic form: ρ 0 sgn(rρ 0 +BE r ρ 0 = 0. (7 This eqution hs the generl solution: ( ( ] ρ 0 (r = e [C r / BE Ai B E r BE +C Bi B E r, (8 where C nd C re integrtion constnts, nd Ai nd Bi re Airy functions. To fulfill the conditions ( the constnts C nd C must obey the reltion: ( ( BE Ai Ai B E B E C = ( ( C, (9 BE Bi Bi B E B E where Ai nd Bi denote the first derivtives of Airy functions. Then t the origin of the extr spce, i.e. on the brne, the function ρ 0 ( r will hve the following series expnsion: ρ 0 (r r 0 = C ( 6 BE r +O ( r, (0 7

8 where we introduced the constnt: ( Ai BE C = C B E Bi ( BE Bi ( B E Ai ( Bi B E ( Bi B E ( B E. ( B E In the second limiting region, r /, u r / = sin(ωt [A Aω +O ( ] ( r ( r, S r / = B + Aω ( r +O ( ( r, ( g r / = e B ( r / +O ( ( r 7/, the eqution (5 hs the following symptotic form: where The generl solution to this eqution is: ρ 0 sgn(r c r ρ 0 ( r ρ / 0 = 0, ( c = E e B. ( ( ( c c ρ 0 (r = C I 0 r +C K 0 r, (5 where C nd C re some integrtion constnts nd I 0 nd K 0 re zero-order modified Bessel functions of the first nd second kind, respectively. To fulfill the conditions ( we must choose C = 0 (the function K 0 is unbounded function t r /. Then for the series expnsion of ρ 0 (r t r / we get: [ ] ρ 0 (r r / = C + 9c +O ( ( r /. (6 ( r / According to (7 the ction ( of the 5D sclr filed zero mode reduces to: S 0 = dx dr [ Q (r t φ Q (r ( x φ + y φ + z φ Q (rφ ], (7 where integrtion over r is performed on the finite integrtion region [ /,+/], nd Q i functions re defined s: Q (r = ( r ρ 0, Q (r = ( r / e S ρ 0, (8 Q (r = ( r ρ 0. 8

9 It is esy to see tht for the extr dimension fctor of the sclr field zero mode, ρ 0 (r, hving symptotes (0 nd (6, the integrl over extr coordinte r in (7 is finite. This mens tht the sclr field zero mode wvefunction is loclized on the brne. Also note tht on the brne, r = 0, due to the boundry conditions (, the Lgrngin in the ction (7 gets the stndrd D form for the mssless sclr field. To estimte the msses of KK excittions on the brne we consider the sclr prticles with zero -momentum, p = 0. Then the eqution ( for the mssive modes reduces to: ρ sgn(r r ρ +m ρ = 0. (9 The exct generl solution to this eqution is: ρ(r = D J 0 ( m ( r +D Y 0 ( m ( r, (50 where D nd D re some constnts. Imposing the boundry conditions ( we get D = 0, (5 ( m J = 0, (5 from which we get the KK mss spectrum of sclr field on the brne: m n = Z (J n, (5 where Z (J n is the n-th zero of the Bessel function J. So the mss gp between the zero nd the first mssive modes will be: m = m = Z (J.8. (5 In the limiting cse of this section the curvture scle, which determines the size of extr dimension, is smller thn the scle ssocited with the width of the brne, B, where B. So KK modes, hving the msses.8, cn be creted in ccelertors t reltively low energies, wht gives chnce to check this model. 5 Conclusion In this pper we hve introduced the new non-sttionry 5D brneworld model with the rel sclr field in the bulk. The model represents single brne which bounds collective bulk oscilltions of grvittionl nd sclr field stnding wves. These wves re out of phse, i.e. the energy of oscilltions psses bck nd forth between the sclr nd grvittionl wves. The metric of the model hs the horizon in the extr spce, nd consequently, the extr spce is finite for the observer on the brne. We hve investigted limiting cses of lrge nd smll extr spce, depending on the two dimensionl prmeters of the model - the curvture scle nd the energy scle ssocited with the brne. 9

10 In the limiting cse of smll extr spce, when the mplitude of the stnding wves is smll, these two prmeters re of sme order of mgnitude, nd mtter fields re loclized on the brne due to the presence of the metric horizon. In the cse of lrge oscilltions the distnce to the horizon is reltively lrge s compred with the brne width, nd trpping of mtter fields on the brne is cused minly by the pressure of bulk oscilltions. The mss of the lightest KK mode in this cse is determined by the smller energy scle ssocited with the horizon size, nd therefore such prticles cn be creted in future ccelertors t reltively low energies. Acknowledgments MG ws prtilly supported by the grnt of Shot Rustveli Ntionl Science Foundtion #DI/8/6 00/. The reserch of PM ws supported by Ili Stte University. References [] N. Arkni-Hmed, S. Dimopoulos nd G. Dvli, Phys. Lett. B 9 (998 6, rxiv: hep-ph/9805. [] I. Antonidis, N. Arkni-Hmed, S. Dimopoulos nd G. Dvli, Phys. Lett. B 6 (998 57, rxiv: hep-ph/ [] M. Gogbershvili, Int. J. Mod. Phys. D (00 65, rxiv: hep-ph/9896. [] M. Gogbershvili, Mod. Phys. Lett. A (999 05, rxiv: hep-ph/9908. [5] L. Rndll nd R. Sundrum, Phys. Rev. Lett. 8 (999 70, rxiv: hep-ph/9905. [6] L. Rndll nd R. Sundrum, Phys. Rev. Lett. 8 ( , rxiv: hep-th/ [7] M. Gutperle nd A. Strominger, JHEP 00 (00 08, rxiv: hep-th/000. [8] M. Kruczenski, R.C. Myers nd A.W. Peet, JHEP 005 (00 09, rxiv: hepth/00. [9] V.D. Ivshchuk nd D. Singleton, JHEP 00 (00 06, rxiv: hep-th/007. [0] C.P. Burgess, F. Quevedo, R. Rbdn, G. Tsinto nd I. Zvl, JCAP 00 (00 008, rxiv: hep-th/00. [] M. Gogbershvili nd D. Singleton, Mod. Phys. Lett. A 5 (00, rxiv: [hep-th]. [] M. Gogbershvili, A. Herrer-Aguilr nd D. Mlgón-Morejón, Clss. Qunt. Grv. 9 ( , rxiv: 0.5 [hep-th]. 0

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