Deviations from the 1/r 2 Newton law due to extra dimensions

Transcription

1 CERN-TH/ hep ph/ Deviatios fom the 1/ Newto law due to exta dimesios A. Kehagias ad K. Sfetsos Theoy Divisio, CERN CH-111 Geeva 3, Switzelad Abstact We systematically examie coectios to the gavitatioal ivese squae law, which ae due to compactified exta dimesios. We fid the iduced Yukawa-type potetials fo which we calculate the stegth α ad age. I geeal the age of the Yukawa coectio is give by the wavelegth of the lightest Kaluza Klei state ad its stegth, elative to the stadad gavitatioal potetial, by the coespodig degeeacy. I paticula, whe exta dimesios ae compactified o a -tous, we fid that the stegth of the potetial is α =, wheeas the compactificatio o a -sphee gives α = + 1. Fo Calabi Yau compactificatios the stegth ca be at most α = 0. CERN-TH/ May 1999

2 1 Itoductio ad discussio Recetly, deviatios of the ivese squae law fo gavity have eceived a lot of attetio [1] [5]. I geeal, these deviatios ae paametized by two paametes, α ad λ, coespodig to the stegth, with espect to the 1/ law, ad the age. Specifically, the fom of the potetial is (fo expeimetal aspects, see, fo istace, [5] ad efeeces theei) V () = G 4M( ) 1+αe /λ, (1) whee G 4 is the fou-dimesioal Newto costat ad M is the mass. This fom of the potetial is expected to be valid fo λ ad, i geeal, thee will be moe tems coectig the 1/ potetial, which ae evetheless subdomiat, as we will also see. Thee ae expeimetal bouds o the possible values of α ad λ, whichaeepeseted i a α λ diagam [1, 5], at the ed of the pape. The value of λ is esticted to be at most of ode 1 mm, leavig the possibility of ew foces i the submillimete egime [1, ]. O the othe had, as depicted i the figue, thee ae theoetical models that ca give diffeet values of the stegth α. Fo example, it was agued, i [1], that a Schek Schwaz supesymmety-beakig mechaism at 1 TeV gives ise to a scala adius modulus ad a potetial of the fom (1) with α 4/7, wheeas a mechaism ivolvig the dilato pedicts α 44 [4, 1]. I this lette we systematically examie coectios to the 1/ gavitatioal potetial due to exta dimesios. We coside Eistei gavity i + 4 dimesios, whee is the umbe of exta dimesios, ad we fid the Newtoia limit of the theoy. The we compactify the iteal dimesios i ode to obtai the fou-dimesioal effective gavitatioal potetial; to leadig ode, this is of the fom (1) with λ popotioal to the ivese mass of the lightest Kaluza Klei (KK) state ad α equal to its degeeacy. We explicitly deive this esult i sectio fo a geeal compactificatio maifold. I paticula, fo the case of a -dimesioal tous, we fid, sice the umbe of exta dimesios ca be =,3,...,7, that the stegth ca take the values α =4,6,...,14. 1 We also discuss the cases of sphee compactificatio, whee the stegth ca take the values α =3,4,...,8 ad Calabi Yau (CY) compactificatio, whee we ague that α 0. Note that thee exist othe possibilities such as tosio, massive gavitios, Bas Dicke scalas etc., which ae expected to poduce simila coectios to the Newto law ad it would be iteestig to study them. 1 Thee caot be oly oe exta dimesio ( = 1) because the deviatio fom Newtoia gavity would the have bee ove astoomical distaces []. The uppe value i the umbe of exta dimesios, = 7, coespods to a compactificatio fom the highest-dimesioal cosistet theoy, which is elevedimesioal supegavity. 1

3 Gavitatioal potetial ad exta dimesios I this sectio we coside the coectios to the gavitatioal potetial due to exta dimesios. We will fist calculate the potetial fo the case of compactificatio o a -dimesioal tous ad the o othe spaces, icludig o the -dimesioal sphee ad CY maifolds..1 Tooidal compactificatio We assume that the space-time is ( + 4)-dimesioal, whee the exta dimesios x i, i =1,,...,, ae compactified o cicles, each with adius R i. The Newtoia limit of a(+ 4)-dimesioal black hole will give the gavitatioal potetial of a massive object. Sice, to ou kowledge, highe-dimesioal black-hole solutios with some dimesios compactified ae ot kow, we will examie the Newtoia limit of highe-dimesioal gavity ad we will impose compactificatio o the solutio. Pesumably, the esult ca also be obtaied i the Newtoia limit of a, yet ukow, highe-dimesioal black hole. The gavitatioal potetial of a massive object with mass M at a distace = ( + x 1 + x +...+x )1/, whee = x + y + z is the thee-dimesioal adial distace, satisfies the ( + 3)-dimesioal Laplace equatio, ad it is give by V +4 = G +4 M ( m Z + ) (+1)/. () i=1 (x i πr i m i ) Hee, G +4 is the Newto costat i + 4 dimesios ad m =(m 1,m,...,m )is a vecto i a -dimesioal lattice. This potetial satisfies the appopiate bouday coditios, amely, it vaishes at spatial ifiity ad it is peiodic i the exta dimesios sice it is ivaiat ude the shifts x i x i +πr i. Fo vey lage R i s oly the tem with m = 0 suvives i the sum ad we ecove the familia Newto law i +4 dimesios: V +4 G +4M. (3) +1 O the othe had, if the R i s ae small, we may appoximate the sum by a itegal as V +4 G +4M Σ d 1 x ( + x ) = Ω G +4 M 1 (+1)/ Σ, (4) whee the volume of the -dimesioal tous Σ sphee Ω ae give by ad that of the -dimesioal uit Σ =(π) i=1 R i, Ω = +1 π Γ ( +1 ). (5) A peiodic-black-hole solutio i fou space-time dimesios has bee costucted i [6].

4 By compaig (4) with the potetial i fou space-time dimesios V 4 fou-dimesioal Newto costat is idetified as = G 4M,the G 4 = Ω G +4. (6) Σ This elatio, ad the obseved value of the fou-dimesioal Plack scale G 1/ cm, leads to a uificatio of the Plack-scale i +4 space-time dimesios (with ) with the electoweak iteactios scale 1 TeV (o 10 0 m), povided that the typical compactificatio adius of the cicles is R 1 mm o smalle [], thus ealizig pevious poposals fo lage iteal dimesios [7]. I tu, that suggests a ovel esolutio of the hieachy poblem [, 8]. I ode to discuss deviatios fom Newtoia gavity, we must compute the fist coectios to (4). This is doe by Poisso esumig () ad we obtai 3 V +4 = G +4M d i m x e x Σ ( m Z + ) +1 i=1 (x i πr i m i ) = Ω G +4 M Σ e i m x ρ 1 dρ ( + ρ ) (+1)/ m Z dx cos( m ρx)(1 x ) 3 = Ω / πγ ( ) 1 G +4 M i m x e dρ ρ/ J 1 ( m ρ), (7) Σ m / 1 m Z 0 ( + ρ )(+1)/ ( whee J 1 is the Bessel fuctio of ode 1ad m= m 1 R 1,..., m R ). Note that ( m 1 1/ m = + m R m 1 R R ae the masses of the KK-modes. Afte pefomig the ) last itegal i (7) we fid V +4 = G 4M e m e i m x, (8) m Z whee G 4 is defied i (6). Next we omit the iteal space depedece sice all poit paticles i the fou-dimesioal space-time ca be take to have x = 0. Hece, the foudimesioal gavitatioal potetial, i the pesece of exta dimesios compactified o a -dimesioal tous, is give by 4 V 4 = G 4M e m. (9) It is clea fom the above expessio that the Newto 1/ potetial esults fom the tem i the sum with m = 0. The fist coectio to it comes fom the lightest KK states. Thus, we fid that the gavitatioal potetial is appoximately of the type (1), amely V 4 G 4M( 1+0 e ) /R 0, (10) 3 The two itegals below ae computed usig the fomulae 8.411(8) ad 6.565(3) of [9]. 4 This esult has bee also obtaied i [10] ad by E. Floatos ad G. Leotais (to appea). We thak R. Rattazzi fo bigig [10] ito ou attetio. 3 m Z

5 whee 1/R 0 is the lightest KK mass ad 0 is its degeeacy ad 0 is the umbe of equal adii, i.e. R 1 =... = R 0 R 0. Thus we see that the stegth equals the degeeacy of the lightest KK state ad the age is its wavelegth.. Compactificatio o othe maifolds Let us coside a space-time of the fom Mikowski M,wheeM is a -dimesioal compact maifold; let {Ψ m } be a set of fuctios i M obeyig the othogoality coditio Ψ (x)ψ m(x) =δ,m, (11) M as well as the completeess elatio Ψ m (x)ψ m (x )=δ () (x,x ). (1) m The fuctios {Ψ } ae eigefuctios of the -dimesioal Laplace opeato with eigevalues µ m Ψ m = µ mψ m. (13) I the Newtoia limit, the gavitatioal potetial V +4 satisfies the Poisso equatio i + 3 spatial dimesios: +3V +4 =(+1)Ω + G +4 Mδ (+3) (x), (14) whichissolvedbyv +4 = G +4M.Focompact dimesios, we may expad V i tems of the complete basis of eigefuctios of the Laplace opeato o M, {Ψ m } as V +4 = m Φ m ()Ψ m (x). (15) The the Φ m s obey with solutio so that (8) chages to 3Φ m µ mφ m =(+1)Ω + Ψ m(0)g +4 Mδ (3) (x), (16) Φ m () = Ω G +4 MΨ m(0) V +4 = Ω G +4 M 1 e µm, (17) Ψ m (0)Ψ m(x) e µm. (18) m As explaied befoe we may omit the iteal space depedece ad set x = 0 i the above fomula. The we may futhe simplify it by ealizig that the sum ove m is ove all possible allowed ieducible epesetatios of the symmety goup of the compact maifold M ad the, fo each such epesetatio, ove all epesetatives. Howeve, the eigevalue of the Laplace opeato µ m depeds oly o the epesetatio ad ot 4

6 o the paticula epesetative that was used to compute it i (13). The, fom (18) we obtai the fou-dimesioal gavitatioal potetial i the pesece of exta dimesios compactified o a geeal maifold M,as V 4 = G 4M d mi e µmi, (19) m i whee we sum up ove all possible ieducible epesetatios m i,add mi deotes the coespodig degeeacy. The fou-dimesioal Newto costat G 4 is defied as i (6), whee Σ is the volume of the compact maifold M. I passig fom (18) to (19) we have also used the goup theoetical esult that the sum of Ψ mi ove all epesetatives of a give ieducible epesetatio equals d mi /Σ. Usig this geeal fomula we see that, to leadig ode fo lage, the gavitatioal potetial is of the fom (1), with age ivesely popotioal to the mass of the lightest KK state ad stegth equal to its degeeacy. The geeal esult (19) educes to (8) fo the case of compactificatio o a -tous. I that case, the symmety goup is abelia ad d mi =1...1 Compactificatio o sphees Let us illustate these by fist cosideig the -dimesioal sphee of adius R as ou compactificatio maifold. A geeal KK state has mass ad degeeacy give by [11] m(m + 1) µ m =, m =0,1,..., R d m = (m + 1)(m + )! ( 1)!m!. (0) The (19) takes the fom V 4 = G 4M d m e µm, (1) m=0 whee, the Newto costat is G 4 = G +4 R. () This potetial is appoximately, fo lage, V 4 G 4M ( ) 1+(+1)e /R. (3) Note that the age of the iduced Yukawa potetial is give by the mass of the lightest KK state, wheeas its stegth is its degeeacy, which is + 1, amely the dimesio of the vecto epesetatio of SO( + 1). It is istuctive to compae the stegths of the Yukawa-type coectio fo compactificatios o the -sphee ad o the -dimesioal tous. Fo the -sphee, the stegth of the Yukawa-type coectio is α = 3,4,...,8, wheeas fo the -tous α = 4,...,14. Hece, the topology of the compactificatio maifold of the exta dimesios seems to be had to detect expeimetally, sice the stegths ae compaable. 5

7 .. Compactificatio o Calabi Yau maifolds Theoies with N = 1 supesymmety i fou-dimesios ae obtaied by CY compactificatios i stig theoy. The CY maifolds ae Ricci-flat Kähle maifolds with o cotiuous isometies ad the explicit metic fo them is ot kow. Hece, it is ot possible to eve attempt solvig the eigevalue equatio (13). Howeve, we may compute the degeeacy of the eigestates usig well kow goup theoetical esults. Typically, fo CY maifolds with a (discete) global symmety goup thee exists a symmety facto cotaiig poducts of the pemutatio goup S ad of the cyclic goup Z( l Z... Z ). 5 The ieducible epesetatios of S ae labelled my a set of o-egative iteges {m i }, subject to the costait [1] m 1 m... m 0, m 1 +m +...+m =. (4) The dimesioality of a ieducible epesetatio is give by d m =! h 1!h!...h! (h i h j ), h i m i + i. (5) i<j The lowest dimesioal massless state coespods to the solutio of (4) with m 1 = ad m = m 3 =...= m = 0. It is a siglet ude both S ad Z l cosistet with the fact that the Hodge umbe h 0,0 = 1 fo all CY spaces. The fist massive state is i the lowest o-tivial epesetatio of S, which coespods to m 1 = m =... = m =1, ad it is oe-dimesioal as ca be see fom (5). This state is degeeate i Z l so that its degeeacy is at least l. I the cases of CY maifolds with o symmeties at all, the lowest boud fo the degeeacy of the fist massive state is of couce (we bea i mid accidetal degeeacies) d lowe = O(1). (6) Fo a maximally symmetic isotopic quitic, the symmety goup is isomophic to the semi-diect poduct of S 5 ad Z 4 5. Usig this model, we obtai the uppe boud fo the degeeacy of the fist massive state as d uppe =0. (7) Hece, the stegth of the Yukawa-type coectio to the ivese squae law associated with the CY compactificatio ca be at most α = 0 which is a bit lage, but evetheless compaable, to the values α =1adα= 7 fo the tous T 6 ad sphee S 6, espectively. Simila coectios to the Newto law ae expected to aise fom othe souces such as tosio, massive gavitios, Bas Dicke scalas etc., which howeve have ot discussed hee. We have collected ou esults i fig A way to costuct maifolds with SU(3) holoomy is to stat with the N-dimesioal complex pojective space (CP N ) ad place eough costaits that educe its complex dimesios to thee. Fo example, fo CP 4 we put 4 i=1 z5 i =0,foCP 3 CP 3 we put 3 i=1 z3 i =0, 3 i=1 w3 i =0ad 3 i=1 z iw i =0. 6

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9 10 15 Λ [TeV] s α 10 5 Dilato 10 0 Exta dimesios (α>0) λ [ Figue 1: Plot of the α λ diagam whee the expeimetal bouds ae idicated by solid lies accodig to [1, 5]. Ou pedictios fo the stegth i the case of exta dimesios age betwee α =1,...,9,...,196,...,400 as idicated by the two ( ) lies. We also idicated the value α 000, which coespods to the dilato cotibutio. m] 8