JHEP06(1999)014 June 11, 1999 Accepted: 10 TeV on the fundamental − April 20, 1999, Received: Superstrings and Heterotic Strings, Superstring Vacua. Recently, a number of authors have challenged the conventional assump- [email protected] [email protected] [email protected] HYPER VERSION Physics Department, University ofSeattle, Washington, WA 98195-1550 E-mail: Santa Cruz Institute forSanta Particle Cruz Physics, CA 95064 E-mail: Physics Department, Rutgers University, Piscataway, NJ 08855-0849 E-mail: energy scale. We notenatural that framework large for quintessence, dimensionscosmology. and with make bulk some supersymmetry other might tentative be remarks a about Keywords: Abstract: tion that the string scale,It Planck mass, has and been unification scale suggested arewe that roughly explore the comparable. constraints string on scalehave these could a scenarios. be fundamental as scale WeMeV. low argue of We as that at show a the that least TeV. a most 10gravitational In radial experiments plausible TeV dilaton this and cases arises mass note, five naturally inrequire dimensions in the huge of these range values inverse of of scenarios. size proposedsion flux 10 Most experiments millimeter and put scale other may a not scenarios conservative be lower bound realizable of in 6 M Theory. Existing preci- Ann E. Nelson Michael Dine Tom Banks Constraints on theories withdimensions large extra JHEP06(1999)014 .Itisthus branes . With the recent discovery d V 1 Ref. [2] pointed out a difficulty with this sort of picture. In a theory where at 2.1 Non-supersymmetric2.2 bulk Supersymmetric stabilization and quintessence 11 6 that all string theories areassumptions different have been limits called of into acoupled question. larger Type Indeed, theory, called I they M do theory.unification not Theory, these even suggests Witten hold argued that in weakly thatcoupled the heterotic the proper string, phenomenon in description which offication the of coupling scale, fundamental nature with Planck constant scale the might was eleventhit be of dimension has order about as the been 40 uni- times realized a larger thatsional strongly [1]. in submanifolds M More of Theory, generally, the nonabelian internalquite gauge space, natural fields which arise to have on been havescale, lower named an dimen- which would internal then manifold giveaffecting somewhat a the large larger gauge effective than four couplings. dimensional the Planck fundamental mass, without least five dimensions areradii. large, If it the is underlyingat theory difficult least is to that supersymmetric, understand of the a theradii. bulk five supersymmetry, stabilization This dimensional which means theory, of is that requires the the that radii the must potential be stabilized vanish at for values large of order the fundamental 1. Introduction It has been traditional,and in the string string theory, scale tofrom are the assume both belief that large, that the comparableerotic sensible compactification to string string. scale the phenomenology Planck could In scale.scales only is emerge weakly This inevitable, from stemmed coupled since the the heteroticproportional observed het- four string to dimensional gauge theory, the couplings the volume are inversely coincidence of of the compact these space, 3. Flavor, CP and precision4. electroweak constraints Cosmology5. Conclusions 16 22 24 Contents 1. Introduction2. Fixing the moduli 1 5 JHEP06(1999)014 , p 25 / M over w This suggests . M ) P φ/M ( v 4 GUT M 100 may be troubling, the only enormous − 2 The high fundamental scale also makes it easier to 2 100 was described in [3] and [2]. Note also, that in Witten’s − is the fundamental scale and that the inflaton is a bulk modulus in a brane 1 GUT M More recently, a number of authors have considered the possibility that the com- By contrast, the radii, in units of the fundamental Planck scale or string tension, While explaining numbers of order 10 As it is conventionallyWe understood. note See also however [4, that 5]. the simplest way to understand the magnitude of density fluctuations as 1 2 -theory type picture is valid. A possible explanation why numbers normally of and the smallness ofexplained the through exponentially cosmological small constant.problem low energy is The field a first theory feature problem effects,field of is and theory. any the description traditionally latter of the world in terms of low energy effective suppress violations of baryon number,with lepton standard number and model flavor, superpartnerssymmetries though in to around a ensure the theory that TeVthe these scale, requisite approximate accuracy. one conservation also laws needs are preserved discrete with pactification (energy) scale isbeingaslowasaTeV[6,7,8,9,10].Theyargue,first,thatthismightbedesirable. far lower, with theScalar fundamental masses scale would of then string naturallytional theory be fine of tuning order problem 1forbidden associated TeV by or with discrete so, the symmetries, eliminating Higgs much thecase, as boson. conven- in though Proton the more more decay conventional couldwere intricate supersymmetric be suggested symmetries in may [11]). be required Flavor (bulk problems gauge can symmetries be suppressed if one supposes that scale. This is similarIn to the the case argument of that the theinstead gauge gauge of coupling, coupling the one, must problem be and istheory. of to why For order explain a the one. why case weak the ofwhy coupling coupling the is there expansion 1 scales should are of hold ratiosM a in supersymmetric of the theory, scales one loworder of must energy one understand order might be 40 10 rather than 1, and why a large radius scenario with SUSY broken only on branes. measured by COBE, invokes an inflationary potential of the form are at most numbers ofin order generalizations a of few it (perhaps withthe 70 more basic in couplings dimensions the are a Witten numbers bit proposal,the of bigger and order GUT than smaller one. scale the The GUTneutrino allows coincidence scale), masses of us and the as to radial probes size understandscale of with arises the the from evidence fundamental scale the forno of product coupling fundamental the of significance. unification theory, a while and large the Planck number of numbers of order one, and has picture there is only oneas large dimension, seven. whereas in This generalthe would there explanation further might of be reduce the as the discrepancy many size between the of unification the and internal Planck manifold scales. necessary to hierarchies in these pictures which require explanation are the ratio of again that JHEP06(1999)014 3 3 . When SUSY is (approximately) preserved in the bulk, the mecha- 5 The problem of fixingcording the to moduli: the manner This inSUSY problem which consistent SUSY takes with observation. is on broken. It differenttal can TeV There forms scale) be are on ac- broken the two (at brane waysbulk. where or to the near break Or standard the it model fundamen- lives, can butcase, be unbroken there in broken the is everywhere an at essentiallyto the unique fundamental balance way scale. the to In stabilize energyas the the of proposed second radial a by dilaton, large Arkani-Hamed, whichearlier Dimopoulos flux is suggestions and against by March-Russell the Sundrum [15], [16]. bulkin following (Other cosmological [17].) discussions constant, We of show stabilization thatthan appear in 10 this case, the requisite fluxes are huge, always larger nisms of stabilization are moreis intricate. extremely In small, many and cases it thethere is radial is dilaton likely a mass to scenario leadof with to order observational 5 10 problems. relatively TeV, large However, where dimensions the and flux a needed fundamental to scale stabilize the radius is of order one. The possibility that we might observe violations of Newton’s law of gravity in In this note, we will address this question from several points of view. We should Our discussion will focus on three issues: We must also invoke a plausible weak coupling factor and a scale of SUSY breaking of order 1 • 3 there are sufficiently elaborateare non-abelian posed flavor by symmetries. thethors production show More of that serious the provided issues then light the production Kaluza-Klein compactification of (KK) radius these states.ments is particles set is But smaller limits not these than in a the au- alimits TeV serious [12]. millimeter, range problem [13], In [12]. while particular, astrophysics in Acceleratormight in the be experi- some scenario observable cases with in two sets sensitive largedimensional stronger gravitational dimensions, experiments, Planck where the scale KK bounds are states on pushed the up higher to 30 TeV. sensitive tabletop experiments, or probe extraexciting. dimensions The in fact accelerators is that extremely suchBut a it possibility is is natural not to a ask: priori ridiculous how is plausible very isstress interesting. it from that the the start extra thatand dimensions we large will be compact not so dimensions be large? which able are the to fundamental impossible. demonstrate scale that is WeSome a within will of low the our concentrate string reach remarks on of and scale of scenarios techniques planned may accelerator the in be experiments. wide applicable todimensions spectrum the [14] analysis of but of we choices other willof parts which not laboratory discuss have these constraints been morescenarios. is general explored scenarios. definitely for Our not the discussion relevant sizes to of most extra of these more general TeV on the brane. JHEP06(1999)014 4 violation. Without supposing additional structure or very weak CP couplings, one obtainstroubling a from limit the of pointthe at of Higgsmass-squared least view is 10 of 100 scalar TeV.This to masses. criticism These 1000 is One certainly times scales needsmetric not smaller are models. to decisive, than already explain and its why it expected also value. applies to many supersym- In this case the radialCavendish dilaton experiment. is within range ofWe emphasize proposed however improvements of that the insolved most in of the these low models, energyconserved the effective flux hierarchy theory on problem by a is introducingallows compact a such manifold. large large There integer fluxes,flat valued is and compact no there dimensions guarantee are which that support no MFinally, we such known Theory note large supersymmetric that values if vacua ofwe we with the might do flux. be not led stabilize to the anpossibility radial interesting is dilaton model in potentially of the exciting quintessence SUSY [18,natural for case, 19, understanding several 20, reasons. 21, of 22] the First, This of large it supersymmetry value might breaking of provide and the a as of radius. in the more In vacuum addition, conventional energybulk pictures. the are moduli scale not fields As so invariation we closely of a will tied the explain, brane fine world quintessencevariation of structure does based Newton’s constant. on not constant havedrastically require Unfortunately, during a that post bounds problem BBN the on cosmological with radial historynation the (so dilaton the for that time did the time a large not cosmological size expla- change this of the time). extra dimensions In mustpossesses addition, involve cosmology Brans-Dicke the before couplings radial dilaton ofastronomy. is order extremely one, light, which and will generically causeProblems problems of for flavor:higher Here dimension we flavor-conserving includeorder operators. not 6 TeV just These on flavor, suggesthypothesize the but scale lower mechanisms of also limits for the the of fermiontextures, theory. and mass effects As these generation of explained mechanisms whichprocesses. can in suppress lead ref. many [11, to There dangerous 23], suitable are flavor-changing nomenological. one two The can difficulties, flavor proposals however,be of one [11, broken theoretical 23] on require andmodel “other, that one flavor faraway” fields phe- symmetries branes live.try from one the But, expects one asnot on that we likely which there will to the is be explain,but standard a widely it given potential separated. is the betweensense. still the lack This Second, one branes, flaw, of these more as and supersymme- mechanismsness we coincidence they do of will not required are see, account for by is the themselves hardly for whole fatal, the picture small- to make • JHEP06(1999)014 1TeV ∼ M ,with 4 M is the number of large compact n ,where 5 n − ) RM Cosmology. Here our statementsthe will be cosmology quite of tentative. suchzles. We high know Most little dimension about of theories.its these way are into Still, the connected there correcttionary with are vacuum. era the several Essentially [24, question all puz- 27, ofhas of 28, how them not 26]. lead the been Inflationary back theorysimply to cosmology sufficiently finds an exhibit in explored infla- the a to formidable braneabout problems warrant world attempts to scenario definite be to conclusions. solved solvemillimeter and them. extra make We dimensions a In will were fewthe particular, compatible remarks authors in with of Big arguing [12] Bangabout that assume 1 Nucleosynthesis, scenarios MeV, initial while with conditions bulkWe modes in suggest of which much that the lower it energy brane may areenergy in is be which their excited very ground leave difficult to state. thethough to system find Dvali in plausible and this conditions Tyeon at rather [29] the higher bizarre have brane state. proposed dominatestemperature an the In they explicit energy particular, obtain density model is al- atthis in too some model. which high, point a and in We field thepoint time, suspect bulk out the that that will reheat models become this withplings overexcited is only in between two a large brane quite dimensions and havegive general much bulk, rise stronger feature. cou- since to In homogeneous fieldsmodels addition excitations with which a we on do KK the threshold notbe somewhat brane viable. fall above an off MeV in are much the more bulk. likely to Our conclusion is that Our treatment of the non-supersymmetric case differs from that of [15] in the • way in which weconstant. treat The curvature latter and is the theenergy cosmological scale constant cancellation in and of the is effective the the theorya parameter below effective bulk actually the term, cosmological KK measured a by boundary observation.that term and It the a is bulk radiative the cosmological correction sum term.constant. constant of was In For tuned [15] large it to radius was cancelterm this assumed the implies is boundary that extremely cosmological the small, dimensionless of coefficient order of ( the bulk 2. Fixing the moduli It is natural towhich SUSY divide is this broken. discussionbrane Note at that into a supersymmetry two scale is comparable parts,bring almost to according us certainly a TeV. broken to into on A conflict theof the much with lower manner preserving scale direct in SUSY for searches. SUSY incharacter breaking in the This would this bulk. however case. stillthe We The leaves bulk begin us stabilization as with the well the problem as case option has on where a the SUSY brane. rather is completely different broken in dimensions. The boundary cosmological constant is of order JHEP06(1999)014 .By n ,thatpa- R in the SUSY R 4 . The Compton wavelength of the radial 6 /R we can use the techniques of low energy effective field R smaller than its naive expectation, in any number of dimen- 2 − ) RM We argue that the only plausible mechanism for stabilizing a large radius is to Our treatment of the supersymmetric case is quite different from that of [15]. In We believe that a more plausible mechanism for fine tuning the effective cosmo- This conclusion follows from equation (2.26) of [15] but was not emphasized in that paper. 4 dilaton is of orderfollows a from mm. the factpotential independent that is the of set dilaton the by is number the a of boundary bulk cosmological spatial modulus term. dimensions. and the This overall scale of its in the most ambitiousterms models. in the In effective this actionequations case because in it the the is curvatures presence obtained of consistentto the by to solving bulk the neglect cosmological the cosmological the constant Einstein are term curvature and smaller the itself. than KK or modes equal The have masses resulting of bulk order 1 geometry is approximately flat case. We find anraising acceptable the value fundamental for scale, thebrane, and we dilaton invoking find mass a SUSY only scenariowithin broken for with at reach rather only the of large modest TeV gravitational valuesinverse scale of size experiments. on the is fluxes, the of This and order model a 10 radial has MeV. dilaton five2.1 dimensions, Non-supersymmetric whose bulk We will concentrate onrameterizes a the particular overall modulus, scale calledof of the very the radial large internal dilaton, radial geometry. dilaton In order to study the regime balance a large flux againstsuggested in the [15]. bulk In cosmological this constant, casethat and the the internal curvature manifold only terms, will case as not bescale. with Ricci Cavendish flat. This signals Thus is scenario we find thatnot requires with seem a two to huge dimensions be flux, ofscale millimeter a which for likely although the vacuum technically bulkis state natural, vacuum much for energy, does larger and M than thus theory. that for found We the in also flux [15] find which inparticular, that stabilizes all it the dimensions the is above radius, natural two. notcosmological correct constant. that The SUSY potential breaking always in falls the to zero brane at induces infinite a small bulk sions. Of course,reader since may we feel are that talkingarguments our about against choice a is this single purely position, finecurvature a terms tuning but matter in in will of place. either nonetheless taste. case, present We the do our not results have with strong the logical constant is obtainedto by have allowing the the order curvaturefine of terms tune the in magnitude cosmological the suggested constantthe effective by by leading action canceling dimensional curvature the analysis. term. bulka cosmological The Then factor term coefficient one against of of can ( the bulk term now only needs to be JHEP06(1999)014 a R a can R ,with , can occur -cycle of the +4 R p n is incorrect in this and gives a four aM is a dimensionless /R 1 e = − M +4 n ∼ .Here is large enough. The next p 2 R n − n ) is the number of large compact which is of order one and may . Higher orders in the curvature c n 2 − MR n ( 4 R . However, we know that effective field 4 M +2 as a small parameter, consistency requires that 2 n M ,where , be small. It is a simple exercise to show that if b expansion of the effective potential for b Q n 7 2 M R R − would lead to particles of mass 1 e ). This question leads to precisely the constraint found +4 2 field strength tensor, threading a n R , and comes from terms quadratic in curvature. ∼ 4 p − b /R 2 − n If all dimensionless coefficients are of order one, and there is a stable minimum Higher order terms in the expansion of the effective potential for large The appearance of a bulk cosmological constant raises the specter of inflation in the extra dimen- o − Λ R √ is a numerical constant, and the boundary cosmological constant is a term ( d ∼ The careful reader may wonder why we have not considered a cancellation of the We can obtain a similar cancellation with a negative integrated Ricci curvature With the conventional sign for the Einstein action, a manifold with positive in- A conservative, or at least conventional, way to deal with this problem is to This viewpoint places a strong upper bound on the bulk cosmological constant. n 4+ boundary Here, Λ these terms are muchsuch smaller a than double indicated fine by tuning dimensional unacceptable and analysis. will We notcosmological consider consider constant which such involves scenarios a further. bulksome term term of lower natural order orderterm of in magnitude, in and a moment). Such a cancellation is possible, but the resulting mass for the Λ is very large ( and a negativecurvature cosmological and constant. the bulk cosmological Notesign,and term if however contribute that to the if potential both with the the same integrated constant requires us to takewill into account involve many further terms fine inbut tuning the will effective of not potential. the change This the coefficient order of of the magnitude bulk estimate cosmological of its constant size. Naive dimensional analysis estimates it to be of order fully understand this scenariocan (in be particular, we satisfied do inand not the will understand not presence how explore of Gauss’ itin a law here. a vast SUSY phenomenologically number on the of viablewould brane branes definitely theory, must be so with be an a the broken extra near same small fine the charge) boundary TeV tuning. scale, cosmologicaltegrated constant Ricci curvature, willbulk cosmological make constant a is negativecosmological positive, contribution constant we to between can these the obtainmensionless two a energy. coefficient terms. cancellation of of If the In the bulk the order cosmological effective for constant must this be to very occur, small if the di- is of this order incrystallization” most mechanism of for what stabilization follows.idea in The is which authors to it of have might a [15] be large have proposed much number larger. of a branes, “brane The each with tension of order assume that the effectiveon four a dimensional tiny cosmological nonzero constantin vanishes value effective (or consistent field takes with theories, observation) andof by pursue the some the physical mechanism problems consequences unknown at of hand. this constraint on theThe rest effective four dimensional cosmological constant is JHEP06(1999)014 . p 5 (2.4) (2.3) (2.2) is small. 2 e =4andthe , p # . Remembering and seven large n 2 ) − 6 2 0 p 2 Q ) 2 − MR e , wrapped around some MR with fluxes of order 10 Q ( 10 TeV 0 + R ∼ 2 , − ) n a ) M a. − b ( MR 2 ( b − )=2 p a 2 − ) 0 9 n − ) b 1 in order to satisfy these equations. Since is always greater than ( MR 2)( MR 2 ( . Note further that eleven dimensional SUGRA p> 2 − 4 =( Q − 2 p 2 ) − 0 Q e (2 2 0and − we find that even for e in seven large dimensions. We can do somewhat better in MR ( a 13 /n 2 " ) 4 b>a> M ) may be written as : /M 0 P )= M ( are positive constants of order one. MR R ( b ∼ V 0 and a MR The equations for a minimum with zero cosmological constant (up to corrections One might have imagined that, having made the two largest terms in the effec- Thus, some other term must come into the solution of the variational equations, We choose this scale both because it is of order the phenomenological bounds we derive in the 6 Thus we must have flux would be of order 10 is an integer, this means that and where sub-leading in dimensions,the flux is of order 10 In this case one might hope to do better if the dimensionless coupling that string theoretic limits in sixhave large bulk dimensions. two Heterotic form and field Type strengths I and and IIA yield strings suitable In Type IIA theory the two form is a Ramond-Ramond field and we do not expect does not have a two form field strength. Thus, in M Theory, we have radial dilaton and KK modesfor is large very extra large dimensions and disappears most in of the this phenomenology scenario. expected tive action of the same orderjust of from magnitude, these that two we terms.variational could obtain equations This a is are suitable not the minimum thestant euclidean case. and Einstein that It equations is the with easilynegative action a verified number (which cosmological that is of con- the the order relevant effective energy the cosmological in cosmological constant the constant. is non-compact incompatiblethese dimensions) Thus, two with is the terms. solving a cancellation the of equations the with just which means that this termthe must basis have of a naive coefficientto much dimensional obtain larger analysis. than such is a Therep expected term. is cycle on a One of unique, can the technically have internal a natural manifold. large way The p corresponding form effective flux, potential is next section, and becausedimensions and it flux was of claimedon order in the one assumption [15] when of that the a one fundamental very could scale small bulk is obtain cosmological 10 a constant. TeV. minimum Their with claims 6 were based large JHEP06(1999)014 ) 0 MR ( ln .Toobtain cycle of the 2 p − . 0 and the flux is very large, /R p 7. Thus, one cannot probe the extra dimensions until of order a TeV, the radial dilaton Compton wavelength is ∼ M ) 0 MR ( , and for ln 4 q might lead to fine tuning problems for the potentials of moduli other M p ∼ is the only one where the analysis we have made above really works. Other n As we have emphasized, for Actually, one should be somewhat careful in the two dimensional case. The A word should be said about the other moduli of the internal space, which Let us also discuss briefly the scenario of [15] in which the dimensionless coef- = values even this smallnotion from of string Kahler dynamics stabilization [3].of , the We one also dimension, has note the to that radial invoke in dilaton the these mass scenarios, is ill of independent understood order 1 one gets upthe to fundamental these scale higher whichdiscouraging energy we for the will scales. prospects derive ofon in testing Combined gravity the the at two with next dimensional millimeter the scenario section, scales. in this lower experiments result bounds may on will be be important.constant In implies their a presence, muchhave larger the just bulk cancellation discussed. energy of density the In than effective these in cosmological curved the scenarios almost we flat find case a we radial dilaton mass of the within reach of proposed improvements ofof the large Cavendish experiment, compact for dimensions. anyaffect number The the cosmological masses constant of is the so KK small modes. that it doesbrane tension not (boundary cosmological constant) is enhanced by a factor of because long range fieldseffective do cosmological not constant, fall in offconsequence, turn at the enhances infinity. masses the This, bulk offactor via cosmological both of the term. radial fine dilaton As tuning and a of the KK modes are enhanced by a such a factor, butfor magnetic in fluxes the at heterotic/Typevery weak I coupling. small we theories Note are however we introducinga that would modulus another if get and we stabilization it make an problem. this isWe enhancement The difficult believe parameter string to that coupling understand it is how is it unreasonable is to stabilized expect at a very small value. we have been presuming fixed. In general, if than the radial dilaton.a We further will indication continue of to thetheory ignore delicate with this balancing large problem, act dimensions. but which we must consider be it performed toficient find in a the bulkorder cosmological as term is the taken boundarypotential is so cosmological small constant. that the In term this is case, of the the overall same scale of the this assumption is probablyoverwhelmedbyahugefluxwhichreallywantstoblowuponlyone untenable. The potentialmanifold. for On the a other torus oneof moduli can cycles will (e.g. “solve put ”this be two problem formclear by flux that putting on this both flux is the on possible 12 ap on and complete a 34 set general cycles of curved manifold. a fourvalues Thus, torus). it It may of be is not that the case JHEP06(1999)014 .Thus 0 /R 2) by including n> is to have a large integer flux, as M 0 . The hierarchy problem is to a large 5 R 11 . The overall coefficient of this term might be either 4) − n ( ) MR ( 4 M much smaller than the constant which determines the four dimensional apriori It is also natural to impose a Ricci flatness condition on the bulk geometry, since The leading terms in the effective action that cannot be set to zero by making In the low energy effective theory, large fluxes appear to be technically natural cosmological constant. this is the simplestwe way might to preserve imagine SUSY asupercharges in IIB except the on string low a or energy threebranewhere F theory. (which various is gauge theory To point-like fields be compactification in live. specific, thebe which The compact broken preserves remaining dynamically dimensions) four eight by supercharges gauge dynamics. onin the the The brane only context might unnatural of thingbreaking about a scale this very idea to low be fundamentalbreaking close dynamics scale to by is low the the energyusing fundamental fact effective this scale, field that scenario theory so we merely may the want to be treatment the fix suspect. of SUSY ideas, Since the we we will SUSY technically are not natural explore discrete this choices, issue. are now come of from order curvature squared terms. They the estimates of productionnot be cross substantially sections changedscenarios. by and our discovery reevaluation One criteria of concludes cited theonly stability then in way of to that [13] large stabilize will effective dimension the field system theory at analysis large shows that the extent solved by hand. because they are quantizedinvestigate and the conserved. possibility At thatflux a there more by microscopic are popping level quantumthis branes one processes scenario should out becomes which of a canexternal the question screen dimensions of vacuum. exist. the whether vacuum WeFluxes For states know have M of with a tendency no Theorists, large to SUSY fluxconsiderations the appear of vacuum and anomaly in plausibility states flat cancellation. Chern with of Simons However, thisnon-supersymmetric the like present property. vacuum terms subsection states and is and devoted to to our be understanding bounded of by those2.2 is practically Supersymmetric nil. stabilization and quintessence If we assume that thethe system becomes story supersymmetric is in quite theMany large different. of radius limit, the In then termsnow this in case the the potential bulk which cosmological determine constant the vanishes. value of the radius, are positive or negative, but isa naturally technically of natural order stabilization one. at large As radius before, is to the introduce only a way term to with achieve large flux same order as the KK mass, which is of order its naive flat space value 1 originally proposed by [15]. Our analysis differs from theirs (for curvature terms for the internal manifold,a which consequence, dominate even the a effective large potential.scale number As still of large require dimensions, integer with a fluxes 10 of TeV fundamental order 10 JHEP06(1999)014 . . 3) /n /R − 4) p (2.5) ( − ) p 2(2 at which 10 TeV. ) /M P 4. If it is, R 2. We note ∼ independent /M M ( P M R independent, term 3 (there are also p> ∼ M n< ( R 6 − ∼ p> which are now cubic p when 2 7 ) 4) 10 − p M eV and the force coming independent and are not 0 (2 ) expansion, the couplings is the value of . The radial dilaton mass 8 ) R − R 0 15 ( M M 0 R 0 in the effective potential term ∼ R R expansion is the ( c 2 ( / . Q ) 2 ∼ e /n 2 RM (where . Ifweallowthistermtobeofits of order 10 TeV this gives a Compton (4 =3,andis10 ) Q R  2 n n M e − P (4 M 12 ) M  expansion. This means that M = 4 and equals 10 0 , and a large flux term with M 2 R p =3and − =3,thisisoforder10 MR ∼ ) p . Even for 2 n term coming from the various quadratic curvature / m 3 MR ) 4) P − n ( ) no longer give rise to KK states with masses of order 1 M/M ( MR R M 4, the quadratic curvature terms in the effective action give rise should be of order ( 4) = 4 the terms quadratic in curvature are n> − n n ≥ ( ) 3 the formula is RM ( For 6 When The details of the stabilization depend crucially on whether The mass of the radial dilaton is much smaller in these supersymmetric scenarios. n< This is because it can be viewed as the fine tuning of a much larger, but 4 7 boundary cosmological term.cM Thus, the coefficient because the large termthermore, in the next the largest action term gives in a the large inverse mass to all of the modes. Fur- natural order ofDimensions magnitude, of then size the whole extra dimension picture is modified. then the fine tuningparameter of in the the higher 4d dimensional Lagrangian effectiveStabilization to is cosmological take decoupled on constant a from does particularly the smallit, problem not value. of require the the any cosmological sign constant. of To achieve the ( wavelength of order a kilometer.tional Thus, experiments, this since scenario there ismuch ruled is smaller out no than by way that existing to of gravita- tune gravity. the radial dilatonto couplings terms to in be the potential which grow with invariants must be negative. Furthermore, comes out of order The smallest value is obtained for The smallest flux is obtained for curvature terms (scaling like ( Even for a scale of 10 TeV and from exchange of thisthat particle SUSY should will have imply been that to seen leading in order existing in the experiments. 1 Note of this particle are universal. useful in the stabilization program.stant The again fine has tuning no of effect the observed on cosmological the con- relevant terms in the potential, which is sub-leading inthat the in large the presenceRicci of tensor flux, term the scales manifoldchange just is the like no argument the longer substantially. flux Ricci term flat, which but produced the it. additional Thus it does not terms with covariant derivatives ofbic curvature which ones). have the The same conditions scaling for as stabilization the give cu- in the potential. For JHEP06(1999)014 . 7 × / 2 1) ) − for all and it .Ifwe p/n 4 /M 7) (2 P / ) M (6 M 4. The only 4 ) SUSY ( P /M = 5 this is of ≥ M P ∼ p n 7 M / ( 6) M/M ( − ∼ with p M p (2 2 ) Q − 7 =3and ) /M p = 4. This also gives the low- P p RM M ( ( 4 . In order to find a stable mini- 2 .For ∼ − M 1) ) 2 for all scenarios in this group. Thus, Q − . In order to cancel the cosmological Q M P is replaced by 0 R p/n or R (2 = 6), to the action. Stabilization can be /M Q ) 10. Taking into account the weak coupling 6) 2 − n 13 . Note also that there is no plausible weak can be obtained by adding a large flux term p 4 ∼ M /M (2 R ) P Q =5. 3if M M n ( 0 R p> ∼ ( 2 2( / ∼ ) n 2 =3and is 10 TeV, while SUSY is broken on the brane at around − Q p p> p (2 the magnitude of the effective potential is of order ) M n M with 0 ) R p ( = 5 as above, we get 2 − , because the scale of the potential will be lowered to n ∼ n 2 ( ) ) Q 1 TeV. Then the equation for = 7 both the quadratic and cubic curvature terms in the effective action 10 TeV, this is larger than 10 /M expected from eleven dimensional SUGRA is , the smallest value we have yet seen, when the fundamental scale is of order RM n ∼ 3 p ( ∼ 4 =3and is of order 1 TeV we have a Compton wavelength in the range of proposed SUSY p M M We can make things even better by invoking SUSY on the brane. Suppose We have seen that there are a variety of radial stabilization schemes whose plau- For For this range of 2 M M SUSY ( Q , so the radial dilaton mass is of order est value of flux, namely the fundamental scale improvements of short distanceattractive tests scenario with of gravity. In particular, this is true for the order 10 cannot be seen in Cavendish experiments. sibility depends on whether the underlying dynamics has stable vacua with large flux, × take constant with only one finecient tuned of parameter, the we quadratic must term take the to dimensionless be coeffi- of order ( factor mentioned above, this couldthe be radial an dilaton integer mass flux in ofAs this order far case one. as would Note we be furtherin can in that the it, see, range which this of still is Cavendish gives the experiments. exciting model near with term the phenomenology. give fewest terms large in dimensionless the numbers potential which grow with 10 TeV. Indeed, if thea coupling standard of model the coupling,likely three the to form amelioration make gauge of field thethan the strength necessary the flux is integer other bound as of examples weak mentioned orderstrength as we above 100, in have is which found. weakly is coupled Ifnatural. perhaps IIB the more three string One palatable form theory, wouldregime, is then have however. a a to Neveu-Schwarz large explain field weak the coupling stabilization factor of is the string dilaton in this For value of coupling enhancement in thission case, parameter. since 11D The SUGRA radial has dilaton no mass dimensionless comes expan- out of order mum, we must add a large flux term of the form n the potential is minimized)observed in cosmological order constant. to obtainsame This the is fine right of tuning order coursetechnically which of a natural magnitude sets minimum fine for at the tuning, the large cosmological but constant, since we it is should the discount it. A , M if achieved if JHEP06(1999)014 . /yr. 11 − 10 .Thereis 1TeV,and itself is time M ∼ R . This translates M ˙ term comes from R/R < 4 /yr 11 /R − , simply as a result of the 10 R < N cannot be so very different than /G N has changed by at most a factor of ˙ R then the system is driven to infinity. G as a consequence of the large age of , which implies that R R R RM which must be canceled by fine tuning. as possible. The 1 . In the presence of such a term, even the 4 1 2 14 − − M ) R RM is smaller than the conventional contributions to R . Thus, in the context of a theory with cm 3 − to infinity during the expansion of the three dimensional part 10 only the scenario with two large dimensions is viable. However, would be a form of quintessence [18]. Such a scenario would be a R ∼ 8 M R 1 2 − P n/ dominating the universe at previous eras. On the other hand it means ) M R M 30 0 10 R ( > ∼ The bound that the current Casimir energy density not exceed the critical density To determine whether the bound on the radial potential was satisfied in the early We will assume that SUSY is broken on our brane, at the scale In order to achieve this goal, it is obviously best to have a leading term in the 0 For an alternative model of quintessence in large radii models see ref. [22]. 8 R P directly into a bound on the time variation of order one. On thedensity one in hand, this implies that there is no problem with the energy that our hope to explain the large value of Thus, during the time since nucleosynthesis the universe, cannot work. The initial value of its value today. This is certainly disappointing. A cosmological explanation for the M the Casimir energy, andvalue its of coefficient this is coefficientfields expected depends to to on the the be SUSY nature ofmust breaking of demand order system that the one. the on coupling coefficient the of The be wall. the precise positive. massless For bulk theis quintessence scenario, we Thus, we might hope to explain a large current value for and a single example whereradial we dilaton can with get a fluxes relatively ofof low order bulk fundamental SUSY, one. scale is and A simply athe much that light potential more the is exciting radius positive possibility is and not in vanishes stabilized the at at case large all. Ifslow the migration leading of term in two dimensional case wouldexplore seem the to other be features of ruled the out. radial quintessence Let scenario. ushistory of agree the to universe, we ignore invokeof another this bound, Newton’s and the constant. constraint This on the is time usually variation stated as We must also require that the additional, time dependent (because then a cosmological constant of order it is also hardcurvature to terms, which see scales how like to ( avoid a term in the potential coming from quadratic of the universe. realization of Dirac’s oldis idea old. that Newton’s Although constant weproblems is think with small it. that because this the is universe an attractive possibility, we will find several dependent), potential energy of the energy density ofbe the universe a throughout finite most fraction of of closure cosmic density. history. Today itpotential can which is as high a power of JHEP06(1999)014 in (2.7) (2.6) RM ,from /yr 11 ,thisisless − N 10 G ˙ R/R < coming from the effect of 3 =2andthateveninthat , 2 − n and from the Oklo natural nuclear 14 . α ) q 10 − ] q − ] RM ([ RM RM > [ 15 O ˙ R R + 0 ∼ 1 α ˙ α α = 1 α would have to come from an inflationary era preceding the time 9 RM . Their coupling comes only through radiative corrections in which KK R field is really compatible with the bounds. That is, is the radial dilaton an field on astronomy. In this respect it behaves like a classic Brans-Dicke field. Another appealing feature of this picture is that the scale of the potential is not There probably is a fine tuning problem of order 10 To summarize: models with SUSY in the bulk can have similar stabilization It is worth pointing out that, in contrast to many models of quintessence, the However, it is clear that in the best of all possible scenarios, we can construct None of these estimates touches on the question of whether the dynamics of R R We note a recent paper, [30], in which an attempt is made to explain the large value of 9 reactor [31]. necessarily connected to the scalemore of conventional pictures supersymmetry [32], breaking. whereconstrained. This these relations is are in tightly contrast (and to unacceptably) Since we have already established that excitations are exchanged between thefine lines structure in constant has a the vacuum polarization form diagram. The the bounds on the cosmological constant and the time variation of Thus than the strongest bound on the time variation of of nucleosynthesis. large value of case the large currentsometime value before of the theuniverse era radius cosmology of and has inflation nucleosynthesis. to can be give It us put a remains in naturalmechanisms to as explanation to be an of non this initial seen number. supersymmetricin condition whether models. the early SUSY Often case,with the and five radial the large dilaton models dimensions is andor are small a very ruled parameters 10 light out. and TeV has fundamental There a scale, radial is which dilaton however requires which a might no SUSY be large found model in sub-millimeter the terms of a thermal effective potential in a pre BBN era. current model is notstructure seriously constant. constrained by In the thisdirectly cosmological to model variation gauge of fields the live fine on the brane and do not couple acceptable dynamical model ofany quintessence. other We cosmological will question not in attempt the to current answer paper. thisa or viable model of radial dilaton quintessence only for the JHEP06(1999)014 GeV, proton stability 15 = 2, but the explanation of n 16 via the cosmological time variation of this parameter runs R , but close when compared to the scale of compactification. The hierar- M as quintessence might be constructed in the case Of course, there are many other sorts of flavor violation which must be sup- In the rest of this section, however, we will adopt the viewpoint of refs. [11, 23], R must be protected bystandard symmetries. model such However as in themetries many MSSM, in it models is order of already to physics necessarylepton suppress to beyond violating dangerous impose the operators. renormalizable additional sym- More andviolating elaborate dimension operators symmetries 5 up can baryon to suppress and terms baryon-number of very high dimension. pressed, and refs.interesting [11, proposal. 23] They suggested discussedlarge that some flavor the symmetry, underlying and, aspects theory in might addition, ofthe possess several scale this additional some branes, problem, far fromchy and “ours” of on quark made and an lepton massesbranes. then arises Explaining because this of hierarchy ait will hierarchical certainly raise separation provides of many a the way ofas of the parameterizing for issues the the discussed breaking of above, bulkat chiral but extreme moduli symmetries. values we Just is have problematic.approximate discussed supersymmetry We in have above, these already pictures. fixing arguedis In these that no weakly there potential “separation” coupled string cannot between moduli theory, branes be there mass at any grows the with classical level. thea separation However, there cancellation of are between the states bosonic branes. whose perturbatively, and In if fermionic supersymmetric there modes, theories, is andfor there enough perturbatively non-supersymmetric is (and supersymmetry) theories, non- generically therealready there is is at no already the potential. a onethis force loop force between However, corresponds branes level. to thethe In exchange branes. of weakly massless Again, coupled particles then, string (gravitons, onethe etc.) expects theory, fundamental stabilization between at scale, only if large for at brane distances, separations all. of order and suppose that the solution of the flavor problems lies in a large separation of at A traditional argument for aflavor large changing fundamental scale processes. has been Thetion. the most Unless absence dramatic the of of certain fundamental scale these is is higher baryon than number around conserva- 10 3. Flavor, CP and precision electroweak constraints into the observational bounds onof the variation of Newton’s constant. A viable model gravitational experiments. Thewhich SUSY the case radial also dilatonCompared allows is to us not most to stabilized quintessence construct models,time and variation this a might of scenario model act the has as cosmological in lessing constant. a of the However, form a large the problem of appealing value with quintessence. of idea the of explain- its current value would have to come from some very early inflationary era. JHEP06(1999)014 (3.1) (3.2) (3.4) 6TeV. M/g > operators, which, can be found by considering the ) (3.3) H . i µ , j f flavor conserving q j µ D M/g ` µ † 003 places a constraint γ µ i γ ¯ j γ f H ¯ q j ) i ¯ ` )( ` i µν 5 ` 17 H Z µ γ µ ν µ can be found by considering other precisely γ i δρ/ρ < . γ D ¯ D ` i † ( ¯ ` 2 2 2 2 H 2 2 g ( M g M/g g M 2 M 2 g M , which can be affected by operators such as Z (1) (one might argue that they should be larger). With this assump- parameter. Requiring ρ O 3TeV[34]. Direct searches for 4-fermi couplings at LEPII. Atomic parity violation. Limits from precision measurements ofdimension properties 6 of operators. the weak gauge bosons on Other dimension 6 operators such as A comparable or slightly stronger limit on In analyzing the effectiveness of this scheme in suppressing flavor changing pro- • • • M/g > of effects of operators such as on atomic parity violation [35]. can affect the due to recent progresscome in from precision processes electroweak such physics, as can be quite severe. These For instance, data from95% the C.L. recently limit completed on high 4-lepton couplings luminosity such LEPII as run place a tion, one should first examine constraints from least some of the branes.be We will of see that order this 6 still TeV,assumptions, requires and that a the suppression higher fundamental of scale scale, CP or violating both. effects requires eithercesses, additional we will imposestring a theory strict that operators notionlevel. permitted of In by naturalness. symmetries addition, in are Inthe accord generated particular, couplings with already one should the at expects arguments be tree be of weak. in ref. present As at [33], a one result, doesn’t operators expect allowed that by symmetries should Similarly strong constraints on measured electroweak observables such astotal the hadronic total width width, of total the leptonic width, and JHEP06(1999)014 3) , M/g (3.5) is constrained 3), respectively, are heavier than , M 1 χ , transforming as (3 ` χ vev). For example, one can 1) and (3 χ quark, another, farther away, , ,and 3 b d , symmetry. The first assumption , 5 i f U(3) µ of about 2 TeV. γ × i ¯ u f ) H M/g 18 µ U(3) , which is affected by operators such as b ¯ D × † q + H ( . transforming as (3 b 2 L 2 3 g q M u,d µ → χ γ L 3 Z ¯ q ). There are some bulk fields which transform under these L 1 3 − q . are in the representations necessary to give quark and lepton µ M . In this picture, some masses are smaller than others because γ H χ e expectation values on our brane are exponentially suppressed by L µ 3 q χ D mass and the distance to the source of . The simplest way to suppress flavor changing neutral currents is † mixing, which, after the effects of CKM missing are considered, is U(3) χ χ d H ¯ L B × 3 ` q µ quark, another for the electron mass, etc. The smallness of mixing angles → γ is any fermion. . The question is by how much. To analyze the amount of suppression of s L d 5 3 f ¯ q The partial width B affected by ¯ In addition, any possible flavor symmetry involving the top quark and left handed These processes give a constraint on We turn now to flavor violating processes in the first two generations. Clearly, In such a picture, flavor violation is clearly suppressed by the discrete subgroup • • under the discrete subgroups of U(3) others (since the masses, i.e. there are various is that a large discreteunderlying subgroup theory. of (These this chiral symmetry symmetriesNambu-Goldstone is must bosons be in on discrete, fact the in a branes.symmetries.) order good Also, to symmetry This in avoid of symmetry light M the is Theoryour we assumed own do to (in not be units expect broken of global on branes which are far from bottom quark must be maximally broken, and one can also find constraints on where from processes such as reasonably low dimension operators, we assume the discrete flavor symmetry is large under U(3) without some assumption of underlying flavorto symmetries, be the scale farwhich larger provide than maximal a suppressionignoring few of Yukawa unwanted TeV. couplings, processes. there The is In authors a the of global standard [11] U(3) model, made a set of assumptions and the product of some of the branes are farther away than others, or some of the of U(3) could arise from the particular alignmentfor of the example symmetry (given breaking the on assumption different walls, of large discrete groups). for the to assume that the symmetries and which communicate thisgenerically breaking by to our wall. We will denote these imagine a nearby wall responsible for the mass of the JHEP06(1999)014 , u .It (3.8) (3.9) (3.7) (3.6) χ .The M mixing is ¯ K and by short 2 s K replaced by . n d d /m 2 K nor the expectation symmetry. First, it m d 5 χ = 1 operators also provide s to the right handed down-type . is presumably of order d R , λ µ χ s . c L L q R ˜ ¯ of order 1 TeV provides suppression G d † d λ R d ν s b χχ χ L L L ¯ G d ¯ ¯ q q M/g will have the same order of magnitude as ν L R µ ! 19 = 2 processes are reasonably safe, provided q d a 2 † d cd s = 2 operator of order G u,d,` 2 V χ s 2 s v χχ χ L abc ¯ q L m f ¯ q 2 2 2 2

2 2 g M of 900 GeV. Similar operators, with g g 2 M 2 M g M M/g mixing. However a ¯ D D fields are just those necessary to generate quark and lepton masses. = 2 operators arise from terms such as can be diagonalized simultaneously with the expectation value of s 2 χ and its derivatives. Thus the operator (3.7) need not be diagonal in the to either the flavon field or to derivatives of the field with respect to ) d χ d χ χ refers generically to the left handed quarks, or small CKM angles, and are less dangerous. ∆ q c Other ∆ Consideration of CP violating operators provides, potentially, more stringent is very big, CP violating operators such as m Dangerous dimension six operators can arise from terms such as with various contractions ofof the indices. But these are suppressed by four powers down quark mass basis since in general neither derivatives of quark masses also actually receiveinvolving contributions from an infinite number of operators will make huge contributions to the dipole moment of the neutron The matrix element of the operator (3.7) is enhanced by a factor distance QCD renormalization group effects. However the real part of can contribute to consistent with current limits. those of the correspondingquark Yukawa matrices. mass basis, With one these could assumptions, find in a the ∆ down Here quarks, value of ( weaker limits. constraints. To explain the smallnessof of the CP bulk, violation, CP violatedM must spontaneously be on a a good distant symmetry brane. Otherwise, unless the scale should be noted that CP conserving ∆ the lightest enough so that we may proceed as if we have the full U(3) coordinates transverse to thesuppression of brane. such derivatives, There since is the no mass of reason to expect any additional adequately suppressed for is however true that inentries models of such the as matrices those indicated suggested by by refs. [11, 23], the various JHEP06(1999)014 n d (3.11) (3.10) unless n d field is rather χ from the strange n d , which is minimized d χ will be complex in the quark H R i d d χ µν H † u σ R d (and its derivatives) are considered. χ d u χ † u µν χ u is anywhere near the electroweak scale. σ h χ d χ u χ ) which diagonalize 20 which still could arise from the strong CP M χ L ¯ R q L n ¯ q d µν µν 10 TeV. The latter may give a too large eF is not light, its derivatives are real and diagonal in eF 2 2 d g ! M χ )andU( 2 2 L g M M/g >

. The contribution from the down quark is not dangerous, so that contributions of its derivatives are suppressed. It is n d M unless . Recall that we are assuming that the bulk physics preserves a d K χ  may refer to a derivative of itself, not necessarily real in the same More restricted assumptions about CP violation can ameliorate this subgroup of the flavor symmetries. Thus the bulk physics provides a 1 TeV in order to suppress the contribution to χ 10 . ¯ θ 40 TeV (assuming the contribution of the down quark dipole moment to discrete M/g > Note that suppression of the contribution of the operator (3.10) still does not Assuming the CKM mechanism of CP violation, it is necessary that CP be This strong CP problem might still be solved by an invisible axion in the bulk, see ref. [12], or 10 M/g > mass basis and so operators such as However it will still be expected that, e.g. for discrete values. Itsince is thus to plausible do that so thesebasis, matrices might in are cost which not too the spatiallywere varying, much down good potential masses symmetries energy. were until real There the and would effects diagonal, therefore of and be CP a and strangeness the same basis as parameter quark electric dipole moment. completely explain the small size of and even if the strangesional quark analysis matrix and elements indicatedhave are by order 1 several as experiments given [37], by then naive dimen- one would have to is given by naivebe dimensional substantially analysis suppressed [36]). if the The scale contributions of the latter must One way to suppress the contribution of the operator (3.10) is if the light compared with give a contribution to problem as well asthere is the no constraints CP previously violation mentioned. either in One the alternative bulk is or on that the branes which serve as sources large potential for the matrices U( a massless up quark. violated on some ofphase the is branes not responsible possiblegeneration for masses. if quark If CP masses. CP isquark is only violated masses, An generically then violated order on operators on one the such the the CKM as branes branes (3.7) which and responsible provide also the for such the as first (where again basis as the quarkgives too masses) large may an have complex coefficients. The former potentially also quite possible that even if JHEP06(1999)014 is 12 (3.13) (3.14) (3.12) M/g unless e 3 → µ or µ = 2 CP violating operator 3 s . Lepton flavor violation from H. ν , field transforming as a 27 under → R χ L q d τ ` χ µ µν . γ σ L L d ¯ ` d χ µ L † ` u γ µ L χ fields which combine into a complex flavor γ 21 ¯ s u † L χ ν χ d q χ µ χ ν γ L χ L ¯ q L ¯ s ¯ ` µν 2 2 g eF expectation values are very small and not dangerous. M mixing. 2 2 and dangerous CP violating operators such as (3.9) could ¯ χ B g M ¯ θ B mixing. For instance there could be another distant wall, on ¯ K This could have the advantage of solving the strong CP problem. − . The expectation value of a heavy 11 K u,d χ . Then the quark mass matrix is nearly real and the phase in the CKM matrix u,d Lepton number and lepton flavor violation also must be highly suppressed. It χ For another discussion of neutrino masses and large extra dimensions see ref. [40]. Unlike in the standard model case [38], current data still allows a real CKM matrix if there are 12 11 singlet have enough suppressiona of few the years product such of aprovide their direct solution expectation will evidence values. be for a definitely Within nonzero tested CKM by phase. the Bis factories, not which reasonable might simplysince to there assume that is the goodOne individual might evidence lepton assume for flavors that areref. violation small conserved [11, of Dirac 39]. lepton neutrino Neutrinothe flavor masses masses left in arise then handed neutrino from leptons, imply oscillations. which large the we violation mechanism parameterize of of by the U(3) symmetries of be sufficiently small provided that any very large. However byor choosing a different different properties mechanism for forthat the neutrino lepton right masses flavor handed violating one neutrinos clearly has the option of assuming higher dimension operators such as The strong CP parameter could lead to visible nonstandard decays such as Quark electric dipole momentsoperators would such as now arise only from additionally suppressed Because there are many possiblelepton options flavor for violating the constraints neutrino further masses, in we this do paper. not consider for which CP and, say, thebut flavor symmetry the acting other onsource flavor the for left symmetries handed are quarks conserved. is broken, Thus this wall cannot serve as a However, one then mustviolation hypothesize in more complicated mechanisms to provide CP the SU(3) of the left handed quarks could provide a ∆ is very small. nonstandard contributions to JHEP06(1999)014 (3.15) from 13 .Theremight M H. R e µν σ ` χ 22 L ¯ ` µν eF 2 . If this small term arises by accident, a fine tuning 2 1TeV. 2 g M , above which thermal production on the branes will n T M/g > –(800 GeV) 2 We find this condition quite puzzling. For example, if the inflaton is a bulk Even so, cosmology is likely to pose serious problems for theories with such light First, as noted in [12], it is necessary that the bulk moduli are in their ground However even with suppression of lepton flavor violation, a constraint comes from We have seen that consideration of the effects of flavor conserving higher dimen- A similar bound may also be placed by considering the explicit contributions from KK 13 over-produce the bulk modes.dimensions it This is barely temperature“inflation” above is only nucleosynthesis reheat quite temperatures. to low. temperaturestemperature So very is it In slightly orders above is the of an necessary magnitude case MeV. that below In of the all two cases, fundamental the modulus, scale. it will contribute to the bulk cosmological constant and inevitably displace This is too large unless is required which is greaterfor than some at mysterious reason least the abe natural part size some in of approximation 100. the in Higgs Of whichAlso, mass course, the is there it Higgs not is is might possible light, beerators. and that receives very its However small mass avoiding couplings radiatively. additional substantial nontrivial that fine constraints enter on tuning in the of underlying the the theory. higher weak dimension scale op- clearly places moduli. Lacking a detailedbrief cosmological remarks model, in we will this content section. ourselves with a few states to a high“normalcy degree temperature”, of accuracy at early times. Indeed, these authors define a 4. Cosmology The cosmology of theoriesrich. with several At large very dimensions earlytheir could times, present potentially the values. be gravitational Thequite quite and different conventional gauge form, horizon and couplings and might couldassumed. be flatness amenable be problems to far quite might from different take solutions, a than usually modes [41]. a possible contribution to the anomalous magnetic moment of the muon, sion operators suggest thatmany the models fundamental scale of shouldflavor CP be symmetry violation at is the least required scale 6are TeV, somewhat which must and troubling must be in from be atHiggs the particle, very least perspective which judiciously 10 requires of some broken. TeV.of understanding mechanism order Also the to (200 These suppress a lightness GeV) its scales of peculiar mass the squared term to JHEP06(1999)014 . or M 1 − eV, and 4) or (3 − 23 = 2, there are potentially efficient production n = 2 there is enormous energy stored in the gravitational n . This would be the case in the brane separation transition described 0 R . Smaller scales seem to require fine tuning. M We are not sure that these problems are insurmountable, and existing models of Another issue that will have to be resolved in these theories is an analog of The potential difficulty posed by this last constraint is illustrated by the other- Finally, particularly for the case field surrounding the brane,this spread energy over cannot a be millimeter. dissipatedmodes. If adiabatically; the it will transition be is principally too radiated rapid, in bulk inflation also have their difficulties.be Still, skeptical absent of a the concreteearly possibility universe. model, of one It arranging is is such interesting entitled that a to models delicate with sequence dimensions of of order events a in (10 MeV) the smaller avoid most of the difficultiesmodes with are Big Bang unexcited Nucleosynthesis, atwe because consider nucleosynthesis the this KK temperatures. the most This plausible is realization of another the reasonthe large why extra cosmological dimension moduli scenario. problem.dilaton will We be have a argued field with quite gravitational generally couplings, that mass the of order radial 10 above. Recall that for the radial dilaton farin from the its bulk, true it minimum.So is On hard one the to must other understand have hand,Furthermore, how inflation without the in in inflation bulk order the system tonucleosynthesis, got bulk account into the with for its a bulk ground the reheatin inflation state. excitation temperature [12], (or of below leave some the over anentirely other standard some MeV. into mechanism) model standard excited must, model during field degreesscenarios as on with of discussed only the freedom. two large brane This extraon which dimensions. seems dumps particularly the In this difficult its brane case, in homogeneous energy haveassuming excitations almost logarithmically that all growing of couplingsthe these to brane other must bulk criteria be modes have lower been [25]. than the met, normalcy Finally, the temperaturewise reheat of attractive temperature model [12]. on of inflationappears proposed in in [29] [26]). (another modelscenario These for there authors brane are inflation make natural thethe candidate very separation inflatons. interesting of point Theseexchanges, the that are which branes. in the fall the fields Thesebranes brane rapidly which are fields to describe nearby, have they zero are potentials, when expected arising the to from branes have potentials massless are with separated. curvature of order When the Thus if the ground stateseparated has at some branes early close times,difficulties together, with and the such yet system a they picture.have can start sufficiently out First, inflate. large well as fluctuations. pointed However, outorder Second, there by the the are natural authors, at reheating it is temperature least difficult is two to of mechanisms for the bulkfor modes example , that which thecompared have to universe not undergoes been a carefully phase studied. transition Suppose on a time scale small JHEP06(1999)014 .Itisnot 5 10 TeV. We believe − Our analysis assumed that 14 . Thus, a mechanism for setting this 4 24 These problems appear formidable to us, but the cosmology of brane worlds has Probably the most severe problems with models of large dimensions were asso- For a slightly different and more explicit flavor proposal see ref. [42]. 14 field at its minimummatter before dominated or universe at duringdimensions nucleosynthesis inflation there energies. must will In be be scenarios similar found problems with in with two order the large to KKmany modes. avoid potential a sources ofand surprise. a rich The spectrum interplay of of energy inflation scales on seems and quite off complicated. the brane clear whether M Theory allowstimes such large a fluxes compact in manifold. spacesare which When are five the Minkowski large bulk space dimensions, theoryleast obeys and 10 one SUSY, TeV we is (which saw willing anyway one thatiments), to must if while push do there keeping the to the satisfy fundamental boundaryby constraints scale cosmological from invoking up constant precision SUSY scale to exper- fixed on at atvalues the 1 of brane), TeV the (e.g. then flux.bilization one mechanisms The can suggest combination stabilize that of theof this experimental radius large kind constraints with dimension of and moderate model scenarios. plausiblelarge is values sta- There the of are most flux many and likely other have realization radial SUSY dilaton scenarios masses which which require contradict experiment. The possibility that thetroweak scale, fundamental while scales the ofIn scales nature this of note are compactification we comparablenomenological are have to considerations, argued large, it the that is is while elec- highly extremelywork this constrained, exciting. of possibility particularly within is M the not Theory. frame- from ruled precision out We electroweak by argued measurements any thatpush as phe- the purely well lowest as phenomenological allowed value flavor constraints, of violating the coming rare fundamental scale processes, up to 6 5. Conclusions a potential energy density of order a (TeV) models incorporating these ideas willare eventually appear. probably If stronger. not, thewith We flavor respect constraints have to made CP similar, violation. maximally mitigating, assumptions ciated with the stabilizationnecessary that of there the be radius. large, quantized In fluxes the of non-supersymmetric magnitude at case, least it 10 is that we have been veryassumptions conservative in proposed deriving by these bounds, otheralthough and authors used the with ameliorating authors a of largelarge [11] degree discrete propose of nonabelian faith. a symmetries,with resolution broken all In on the of particular, distant required the properties branes, flavor have no problem been explicit constructed. employing models JHEP06(1999)014 or so. And bounds on 4 − This scalar must couple strongly to 15 25 for which the bulk KK spectrum is below 1 MeV, n The inflationary cosmology of brane worlds is only beginning to be explored [29] There are in fact a large number of interesting questions about large dimension An interesting possibility, which we had hoped would provide more motivation T.B. thanks Nima Arkani-Hamed for explaining this scenario to us. 15 the standard modelother in hand, order its toorder reheat dump to most temperature avoid of excitation muststandard of its be model. the energy low bulk on (not Furthermore,squeezed through the into baryogenesis more its this brane. and rather well than abbreviated understood structurebulk a cosmic On is formation couplings history. few particularly the to The must hard GeV) absence the allhomogeneous to of in excitations understand be coupling on in to the models the branethe with give bulk. two rise large to dimensions, logarithmically where growing effects in and it remainsconstructed. to Here be we seen noteliorated (once whether in again) models only scenarios that which with many a meet of large these all problems number the are of challenges ame- dimensions can with be inverse size of order the time variation oflarge Newton’s value constant of imply that thethe one radius long cannot period as explain of a thesuggestive, time consequence and current since of also BBN. the Still, the highmodels given evolution degree that [20], of the of it the is fine rest probably tuning ofwould universe worth have the required during to exploring numerology by understand this is existing brane intriguing so quintessence of world idea models such further. during scenarios To inflationary is do eras. only so, just The we analysis beginning. scenarios that can onlyparticular, be for understood those in valuesthe the of initial context conditions of one inflationaryBBN must cosmology. are assume In quite in bizarre. orderthe bulk to The is make brane in the its ismechanism ground model excited state. which consistent to put This with a could both onlysome temperature bulk be above and stretched accounted 1 for boundary scalar by MeV into some field while their inflationary ground on states the apart brane. from for the idea ofthe quintessence large field. dimensions, isplausible: In that the assuming the case a radialis of non-zero dilaton naturally two Casimir large might within potential, play dimensions, aconstant. the the the However, few one role scales mass orders must fine at of of ofquadratic tune least in the magnitude the coefficient are curvatures of radial of (or a thehave find dilaton term the a present in Casimir energy model the value dominate effective with the of action constant potential. vanishing between the bulk The boundary cancellation curvature) Hubble and of in bulk theof order effects cosmological is to the still mysterious, radius butproblems dependent at with least terms this the idea. value isof The of order Brans-Dicke coupling the one, of whereas right the observation radial order restricts dilaton of its is value naturally magnitude.There to are 10 two JHEP06(1999)014 ]. B D50 ]; (1998) 257 (1990) 377; (1999) 180 ]. (1999) 046004 hep-th/9603133 B 246 B 436 New dimensions at B 548 D59 hep-ph/9803315 hep-th/9803132 Phys. Lett. (1996) 3693 [ (1983) 136; , The hierarchy problem and new di- ]. D54 B 125 (1998) 263 [ (1999) 4971 [ Effective field theory, black holes, and the 82 26 B 429 Brane world, Nucl. Phys. ]. hep-th/9605136 ]. Dynamical compactification as a mechanism of spontaneous Coping with strongly coupled string theory, Phys. Rev. Couplings and scales in strongly coupled heterotic string theory, (1996) 173 [ ]. ]. ]. hep-th/9602070 A possible new dimension at a few TeV hep-th/9406132 B 479 Strong coupling expansion of Calabi-Yau compactification, Nucl. Phys. M Theory as a holographic field theory, Phys. Rev. Weak scale superstrings, Phys. Rev. (1996) 135 [ hep-th/9712130 hep-ph/9804398 hep-th/9809147 Nucl. Phys. 471 cosmological constant, Phys. Rev. Lett. (1994) 7454 [ [ J. Lykken, mensions at a millimeter, Phys. Lett. I. Antoniadis, N.a Arkani-Hamed, millimeter S. to Dimopoulos[ a and Fermi G. and Dvali, superstrings at a TeV, Phys. Lett. [ Do we live insideG. a Dvali and domain M. wall?, Shifman, Phys. Lett. [2] T. Banks and M. Dine, [1] E. Witten, [3] T. Banks and M. Dine, [4] A. Cohen, D.B. Kaplan and A.E. Nelson, [5] P. Horava, [6] I. Antoniadis, [7] N. Arkani-Hamed, S. Dimopoulos and G. Dvali, [8] Z. Kakushadze and S.-H.H. Tye, [9] Some precursors of these ideas can be found in V.A. Rubakov and M.E. Shaposhnikov, References Acknowledgments We thank Sean Carroll,This S. work Dimopoulos of and M.D. Nimawork was Arkanani-Hamed of T.B. for supported was discussions. in supported in partDE-FG02-96ER40559. part by by The the the work Department of U.S. of Energyof A.N. Department under Energy was grant of under number supported grant Energy. no. in DE-FG03-96ER40956. The and part A.N. CERN by would theory the like to groups Department thank for the hospitality. UCSC 10 MeV. The KKthe modes cosmological are point now ofsions above view nucleosynthesis as and temperatures. well, a models 10large Thus, with dimension from TeV a scenarios. fundamental large These scale numbergravitational models of and seem could large accelerator have like exciting dimen- experiments the phenomenology and both we most in believe plausible they deserve realization further of study. JHEP06(1999)014 ]; B ]; ]. ,ZZ D58 − ]; (1999) Domain W 82 + ]; Getting to ]; W → ]; − hep-ph/9810535 e hep-ph/9811323 + hep-th/9612128 e ]; hep-ph/9811501 ]; hep-th/9611213 ]; ]. ]; hep-ph/9812486 (1999) 257 [ ]; (1997) 64 [ Black holes and submillimeter di- . hep-ph/9811291 Phenomenology, astrophysics and (1999) 105002 [ (1999) 343 [ ]. B 447 Collider signatures of new large space B 396 hep-ph/9811337 (1996) 127 [ Large extra dimensions and deep inelas- (1999) 115 [ D59 Aspect of type I string phenomenology, Strings at the intermediate scale, or is the Quantum gravity and extra dimensions at hep-ph/9901209 B 450 ahler potential and extra space-time dimen- (1999) 3 [ hep-th/9808138 hep-th/9901103 hep-ph/9811350 27 hep-ph/9902263 B 504 On Kaluza-Klein states from large extra dimen- ]; Out of this world supersymmetry breaking, B 455 New origin for approximate symmetries from dis- hep-ph/9811353 B 544 hep-ph/9807344 (1999) 2236 [ Phys. Rev. ]; ]; Limits on low scale gravity from , (1998) 96 [ 82 . Effects of SM Kaluza-Klein excitations on electroweak Some remarks on theories with large compact dimensions The Standard model from extra dimensions, Phys. Lett. (1999) 60 [ (1999) 115010 [ The effective K¨ Extra space-time dimensions and unification, Phys. Lett. (1999) 025005 [ (1999) 105006 [ TeV scale superstring and extra dimensions, Phys. Rev. ]; B 441 D59 B 456 hep-ph/9806263 D60 D59 (1999) 086004 [ ; ; hep-ph/9803466 hep-th/9805157 Indirect collider signals for extra dimensions, Phys. Rev. Lett. More and more indirect signals for extra dimensions at more and more hep-ph/9902467 D59 (1998) 255, [ , Phys. Lett. hep-ph/9811356 (1998) 55 [ γγ and supersymmetry breaking, Nucl. Phys. Fermi scale dual to the Planck scale?, Phys. Lett. hep-th/9810155 C.P. Burgess, L.E. Ibanez and F. Quevedo, M. Masip andobservables, A. Pomarol, P.C. Argyres, S. Dimopoulosmensions, and J. Phys. March-Russell, Lett. (1998) 106007 [ das and T. Gherghetta, walls in strongly coupled theories, Phys. Lett. 436 G. Shiu and S.H. Tye, hep-ph/9812397 A. Pomarol and M.B Quiros, 438 T.E. Clark and S.T. Love, L.E. Ibanez, C. Munoz and S. Rigolin, sions, Phys. Rev. tant breaking in extra dimensions, cosmology of theories withPhys. submillimeter Rev. dimensions and TeV scale quantum gravity, high energy colliders, Nucl. Phys. T. Han, J.D. Lykkensions, and Phys. R. Rev. Zhang, J.L. Hewett, 4765 [ E.A. Mirabelli, M. Perlstein and M.E. Peskin, dimensions, Phys. Rev. Lett. P. Mathews, S. Raychaudhuri andtic K. Srihar, scattering at HERA, Phys. Lett. the top with extra dimensions, Phys. Lett. S. Nussinov and R.E. Shrock, and TeV scale quantum gravity T.G. Rizzo, colliders, Phys. Rev. K. Agashe and N.G. Deshpande, [14] L. Randall and R. Sundrum, [10] Among the papers exploring the possibility of large dimensions are K.R. Dienes, E. Du- [11] N. Arkani-Hamed, S. Dimopoulos, [12] N. Arkani-Hamed, S. Dimopoulos and G. Dvali, [13] G.F. Giudice, R. Rattazzi and J.D. Wells, JHEP06(1999)014 B . D56 ]. (1999) B 450 ]. (1999) 42 ]; (1998) 1582 D59 . 80 . Phys. Rev. B 548 , hep-ph/9809582 (1992) 1226; astro-ph/9505060 hep-ph/9812486 ]; hep-ph/9901443 Stabilization of submillimeter D46 hep-th/9809124 (1989) 25; hep-ph/9811375 Cosmological imprint of an en- ]. Phys. Rev. Lett. Cosmology with ultra-light pseudo- ]; 19 , (1999) 5 [ Large extra dimensions and deep in- (1999) 115 [ (1995) 2077 [ ]. 75 Cosmology with a time variable cosmologi- Dynamical lambda models of structure for- Cosmological structure formation from soft B 550 B 455 Quintessence and the rest of the world, Phys. astro-ph/9608122 28 Late time cosmological phase transitions. 1. Par- hep-ph/9811378 (1988) L17; hep-ph/9810280 ]. ]; Cosmological consequences of a rolling homogeneous Branes and the gauge hierarchy, Phys. Lett. Low scale closed strings and their duals, Nucl. Phys. (1988) 3406; D-branes and cosmology, Nucl. Phys. Baryogenesis in models with a low quantum gravity scale, 325 CDM models with a smooth component Flavor violation in theories with TeV scale quantum grav- ]. astro-ph/9806099 Anisotropic type I string compactification, winding modes Phys. Rev. Lett. (1997) 1851 [ ]. Quintessential difficulties, , D37 (1999) 24 [ ]; D55 Comments Nucl. Part. Phys. (1999) 02500, [ , ]. B 450 hep-th/9902055 Astrophys. J. Compactification for a three-brane universe, Phys. Rev. (1998) 3067 [ , astro-ph/9701138 Phys. Rev. Phenomenology of low quantum gravity scale models D60 hep-th/9812093 , 81 Phys. Rev. hep-ph/9807348 , (1999) 41 [ astro-ph/9708069 hep-th/9811089 I. Antoniadis and B. Pioline, 550 dimensions, the new guise of the hierarchy problem, 085010 [ and large extra dimensions, Nucl. Phys. P. Mathews, S. Raychaudhurielastic and scattering K. at Sridhar, HERA, Phys. Lett. scalar field K. Coble, S. Dodelsonmation and J.A. Frieman, cal ‘constant’ M.S. Turner and M. White, C.T. Hill, D.N. Schramm,topological defects and J.N. Fry, [ (1997) 4439 [ R.R. Caldwell,ergy R. component Dave with and general equation P.J. of Steinhardt, state J. Frieman, C. Hill, and R. Watkins, ticle physics models and cosmic evolution Phys. Rev. J. Frieman, C.Nambu-Goldstone Hill, bosons A. Stebbins and I. Waga, Rev. Lett. ity, Phys. Lett. concerning quintessence in large radius scenarios. We thank them for discussions. Phys. Rev. M. Maggiore and A. Riotto, [ (1999) 83 [ [15] N. Arkani-Hamed, S. Dimopoulos and J. March-Russell, [16] R. Sundrum, [17] A. Donini and S. Rigolin, [18] B. Ratra and P.J.E. Peebles, [19] For a recent review, see S. M. Carroll, [20] C. Kolda and D.H.Lyth, [21] J. Frieman, J. Lykken, M. Pietroni and[22] A. K. Benakli Riotto have made[23] similar Z. observations Berezhiani and G. Dvali, [24] K. Benakli and S. Davidson, [25] I. Antoniadis and C. Bachas, JHEP06(1999)014 , , , Nucl. D59 (1985) B 448 B 168 , B 275 (1996) 37 ;A.Lukas (1999) 72 (1999) 387 B 162 Phys. Lett. B 480 Phys. Lett. . B 450 , , B 543 (1987) 163; 17 hep-th/9902071 The Oklo bound on the time , Report to LEP experiments Cosmological phase transitions Nucl. Phys. hep-th/9806022 , ] for a recent review. , (1988) 527; Chiral symmetry and the spin of the proton 29 Boundary inflation, L3 report to LEPC. B 310 The quark content of the proton Brane inflation, Phys. Lett. Strange matrix elements in the proton from neural ]. hep-ph/9705435 (1997) 1759; ]. Chiral quarks and the nonrelativistic quark model The pattern of chiral symmetry breaking and the strange (1988) 357. Flavor conserving CP violation in invisible axion models , Comments Nucl. Part. Phys. Inflation and large internal dimensions, Phys. Rev. 275 Microscopic knowledge from macroscopic physics in string Is the superstring weakly coupled?, Phys. Lett. Nucl. Phys. B 301 (1986) 241. , (1997) 277 [ (1988) 309; Science Atomic parity violation in cesium and implications for new physics ]. ]. ]. hep-th/9811141 Strong evidence for a new strange matrix element (1984) 189; B 276 hep-ph/9810320 Inflation with TeV scale gravity needs supersymmetry, Phys. Lett. B 409 B 206 B 234 hep-ph/9606486 hep-ph/9809406 hep-ph/9812483 Cosmological solutions of Horava-Witten theory (1999) 191 [ (1999) 101303 [ [ theory, Nucl. Phys. variation of the fine structure constant revisited, Nucl. Phys. 299. [ [ and radius stabilization in higher dimensions Phys. Lett. see A. Deandrea, committee, Nov 19, 1998. Phys. H. Georgi and L. Randall, Nucl. Phys. (1986) 105; R.L. Jaffe and C.L.quark Korpa, content of the proton D.B. Kaplan andcurrent A. experiments Manohar, S.J. Brodsky, J. Ellis and M. Karliner, Phys. Lett. D.B. Kaplan, (1992) 137. [26] A. Lukas, B. Ovrut and D. Waldram, [27] D.H. Lyth, [28] N. Kaloper and A. Linde, [29] G. Dvali and S.-H. H. Tye, [32] M. Dine and N. Seiberg, [33] M. Dine and N. Seiberg, [34] R. Clare, for the L3 collaboration [31] For a recent discussion see T. Damour and F. Dyson, [30] K.R. Dienes, E. Dudas, T. Ghergetta, A. Riotto, [35] C.S. Wood et al., [36] A. Manohar and H. Georgi, [37] J.F. Donoghue and C. R. Nappi. JHEP06(1999)014 . . (1999) 549 Light neutrinos hep-ph/9811428 , B 551 hep-ph/9901418 ]; ; , Presented at SUSY 98 Conference, Flavor ’conservation’ and hierarchy ]; A real CKM matrix?, hep-ph/9810333 30 . hep-ph/9811448 Phenomenological issues in TeV scale gravity with light hep-th/9812163 (1999) 113011 [ Indirect measurement of the vertex and angles of the unitarity tri- D59 hep-ph/9901299 ]. Extra dimensions and the muon anomalous magnetic moment, TeV scale supersymmetric standard model and brane world, Nucl. ; . (1999) 3 [ Experimental constraints on the Cabibbo-Kobayashi-Maskawa matrix, B 552 Neutrino masses from large extra dimensions hep-th/9902080 angle, Phys. Rev. hep-ph/9808411 P. Checchia, E. Piotto and F. Simonetto, Oxford, England, 11-17 Jul 1998, 8014, Phys. neutrino masses, hep-ph/9902310 without heavy mass scales: a higher-dimensional seesaw mechanism in TeV[ scale supersymmetric standard model, Nucl. Phys. K.R. Dienes, E. Dudas, and T. Gherghetta, report CERN- TH-98-370, [38] S. Mele, [39] N. Arkani-Hamed, S. Dimopoulos, G. Dvali and J. March-Russell report SLAC-PUB- [42] Z. Kakushadze, [40] A.E. Faraggi and M. Pospelov, [41] M.L. Graesser,