Phenomenology, Astrophysics, and Cosmology of Theories with Large New Dimensions and Tev Scale Quantum Gravity

SLAC-PUB-7864 August 1998 SU-ITP-98/142 IC/98/44 hep-ph/9807344

Phenomenology, astrophysics, and cosmology of theories with large new dimensions and TeV scale quantum gravity

Nima Arkani-Hamed et al.

Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309 Work supported by Department of Energy contract DE–AC03–76SF00515. or 30 vi- IC/98/44 c .F y ts, for e el phe- ali C-PUB-7864 v v vit eV scale. 2 are safe SU-ITP-98/142 hep-ph/980734 4 SLA y tation of our times stronger n> vit 8 d, CA 94305, USA y problem not re- ts arise b oth from 10 h and Gia Dv wn to the T 2 new dimensions of b 6 ade all constrain eV scale, and more im- d, CA 94309, USA e also explore no 10 ydo e systematically study the ust b e pushed ab o n hnicolor. Instead, the prob- vit Constrain tum Gra Theories with ell as help stabilize the proton. eV. W ysics and Cosmology 1T k scale m e forces tum gra eccei-Quinn solution to the strong CP as Dimop oulos v s vide a natural dark matter candidate. rieste, 34100, Italy Abstract d University, Stanfor m eak scale axion in the bulk. Gauge elds ,Sa a , Astroph y the presence of vitons pro duced on our brane and captured on d University, Stanfor ertheless, the particular implemen y of this scenario. ICTP, T vitational e ects at the T yp e I string theory can ev c eV Scale Quan y bringing quan ed with a w ani-Hamed tly prop osed a solution to the hierarc eV suppressed couplings. tum gra artment, Stanfor y at sub-mm distances, as w C, Stanfor oid co oling SN1987A and distortions of the di use pho- w-energy sup ersymmetry or tec and T tal viabilit v t \fat" brane can pro 2, with string scale vit

hep-ph/9807344 12 Jul 1998 ulli ed b kground. Nev ork within t tly from the pro duction of massless higher dimensional gra SLA e recen n a Nima Ark W y = 2, the six dimensional Planc of Theories with Sub-Milli meter Dimensions eV to a Phenomenology an nomena resulting from thehigher existence dimensional of space. new The states P problem propagating is in reviv the in the bulk can mediate repulsiv Higher-dimensional gra a di eren than gra framew T ton bac This is accomplished b lem is n sub-millimeter size, withhigher the dimensional SM space. eldsexp lo erimen calised In on this a pap er 3-branestrong w quan in the p ortan tons with T due mainly to the infrared softness of higher dimensional gra n lying on lo Physics Dep b July 11, 1998

1 Intro duction

In a recent pap er [1], wehave prop osed a framework for solving the hierarchy

problem which do es not rely on sup ersymmetry or technicolor. Rather, the

problem is solved by removing its premise: the fundamental Planck scale,

where gravity b ecomes comparable in strength to the other interactions, is

taken to b e near the weak scale. The observed weakness of gravity at long

distances is due to the presence of n new spatial dimensions large compared

to the electroweak scale. This can b e inferred from the relation b etween the

Planck scales of the (4 + n) dimensional theory M and the long-distance

Pl(4+n)

4-dimensional theory M , which can simply b e determined by Gauss' law

Pl(4)

(see the next section for a more detailed explanation)

2 n n+2

(1) M r M

Pl(4) n

Pl(4+n)

where r is the size of the extra dimensions. Putting M 1TeV then

n Pl(4+n)

yields

30=n17

r 10 cm (2)

For n =1, r 10 cm, so this case is obviously excluded since it would

mo dify Newtonian gravitation at solar-system distances. Already for n =2,

however, r 1 mm, which is precisely the distance where our present ex-

p erimental measurement of gravitational strength forces stops. As n in-

creases, r approaches (TeV) distances, alb eit slowly: the case n = 6 gives

r (10MeV) . Clearly, while the gravitational force has not b een directly

measured b eneath a millimeter, the success of the SM up to 100 GeV

implies that the SM elds can not feel these extra large dimensions; that is,

they must b e stuckonawall, or \3-brane", in the higher dimensional space.

Summarizing, in our framework the universe is (4 + n)

dimensional with Planck scale near the weak scale, with n 2 new sub-

mm sized dimensions where gravity p erhaps other elds can freely propa-

gate, but where the SM particles are lo calised on a 3-brane in the higher-

dimensional space.

An imp ortant question is the mechanism by which the SM elds are

lo calised to the brane. In [1], we prop osed a eld-theoretic implementation of

our framework based on earlier ideas for lo calizing the requisite spin 0,1/2[2 ]

and 1 [3] particles. In [4] we showed that our framework can naturally b e

emb edded in typ e I string theory. This has the obvious advantage of b eing 1

formulated within a consistent theory of gravity, with the additional b ene t

that the lo calization of gauge theories on a 3-brane is automatic [5]. Further

interesting progress towards realistic string mo del-building was made in [6].

The most pressing issue, however, is to insure that this framework is not

exp erimentally excluded. This is a concern for two main reasons. First, quan-

tum gravity has b een broughtdown from 10 GeV to TeV. Second, the

structure of space-time has b een drastically mo di ed at sub-mm distances.

The main ob jective of this pap er is to examine the phenomenological, as-

trophysical and cosmological constraints on our framework. Subsequently,

we discuss a numb er of new phenomena which emerge in theories with large

extra dimensions.

The rest of the pap er is organized as follows. In section 2, we derive the

exact relationship b etween the Planck scales of the (4 + n) and 4-d theories

in three ways in order to gain some intuition for the physics of higher dimen-

sional theories. Of course, roughly sp eaking, if M 1TeV, we exp ect

Pl(4+n)

new physics resp onsible for making a sensible quantum theory of gravityat

the TeV scale. There is a practical di erence b etween the new physics o ccur-

ring at 1TeV versus 10 TeV, as far as accessibility to future colliders

is concerned. In section 3, we therefore give a more careful accountofthe

relationship b etween the scale of new physics and M in the particu-

Pl(4+n)

lar case where gravityisemb edded in typ e I string theory. As a set-up for

the discussion of phenomenological constraints,in section 4 we identify and

discuss the interactions of new light particles in the e ective theory b eneath

the TeV scale: higher dimensional graviton, and p ossibly Nambu-Goldstone

b osons of broken translation invariance. In section 5 we b egin the discussion

of phenomenological constraints in earnest, b eginning with lab oratory exp er-

iments. The most stringent b ounds are not due to strong gravitational e ects

at TeV energies, but rather due to the p ossibility of pro ducing massless

particles, the higher dimensional gravitons, whose couplings are only 1/TeV

suppressed. In section 5 we discuss p otential problems this can cause with

rare decays, and in sections 6 and 7 we consider astrophysical and cosmolog-

ical constraints. Remarkably, due primarily to the extreme infrared softness

of higher dimensional gravity,we nd that for n>2 all exp erimental limits

are comfortably satis ed. The case n = 2 is quite tightly constrained, with

alower b ound 30 TeV on the 6-d Planck scale. Nevertheless, precisely

for n = 2, this Planck mass can still b e consistent with string excitations

at the TeV scale, and therefore may still provide a natural solution to the 2

hierarchy problem. Not only are cosmological constraints satis ed, there are

new cosmological p ossibilities in our scenario. In particular, we discuss the

p ossibility that gravitons pro duced on our brane and captured on a di er-

ent, \fat" brane in the bulk, can form the dark matter of the Universe. The

following two sections illustrate further p ossibilities for new physics in this

framework. In section 9, we show that the Peccei-Quinn axion can solve the

Strong-CP problem and avoid the usual astrophysical b ounds if the axion

eld lives in the bulk. In section 10, we note that a gauge eld living in the

bulk can naturally have a miniscule gauge coupling 10 to wall states

and pick up a mass 1mm through sp ontaneous breaking on the wall. If

these gauge elds couple to B or B L, they can mediate repulsive forces

6 8

10 10 times stronger than gravity at sub-mm distances. This gives a

sp ectacular exp erimental signature may b e observed in the near future. Fi-

nally, in section 11, we turn to the imp ortant question of the determination

of the radii of the extra dimensions. While we do not o er any dynamical

prop osal, we parametrize the p otential for the radius mo dulus and consider

cosmological constraints coming from the requirement that the radius is not

signi cantly altered since b efore the era of Big-Bang Nucleosynthesis (BBN).

We draw our conclusions in section 12. App endix 1 discusses the somewhat

subtle issue of the Higgs phenomenon for sp ontaneously broken translational

invariance, and app endix 2 presentsatoy mo del illustrating some asp ects of

mo duli stabilization.

2 Relating Planck Scales

2.1 Gauss Law

Here we will derive the exact relationship b etween the Newton constants

G ;G of the full (4 + n) and compacti ed 4 dimensional theories,

N (4+n) N(4)

which are de ned by the force laws

m m

1 2

F (r ) = G

(4+n) N (4+n)

n+2

m m

1 2

F (r ) = G : (3)

(4) N (4)

We will carry out this simple exercise in three di erentways. The easiest

derivation is a trivial application of Gauss' Law. Let us compactify the n new 3

dimensions y by making the p erio dic identi cation y y + L. Supp ose

now that a p oint mass m is placed at the origin. One can repro duce this sit-

uation in the uncompacti ed theory by placing \mirror" masses p erio dically

in all the new dimensions. Of course for a test mass at distances r L from

m, the \mirror" masses make a negligible small contribution to the force and

wehave the (4 + n) dimensional force law. For r L, on the other hand, the

discrete distance b etween mirror masses can not b e discerned and they lo ok

like an in nite n spatial dimensional \line" with uniform mass density. The

problem is analogous to nding the gravitational eld of an in nite line of

mass with uniform mass/unit length, where cylindrical symmetry and Gauss'

law give the answer. Following exactly the same pro cedure, we consider a

\cylinder" C centered around the n dimensional line of mass, with side length

l and end caps b eing three dimensional sphere's of radius r .Wenow apply

the (4 + n) dimensional Gauss' law which reads

FdS = S G Mass in C (4)

(3+n) N (4+n)

surface C

D=2

where S =2 =(D=2) is the surface area of the unit sphere in D spatial

dimensions (recall that the usual Gauss lawhasa4 factor on the RHS). In

our case, the LHS is equal to F (r ) 4 l , while the total mass contained

n n 2

in C is m (l =L ). Equating the two sides, we nd the correct 1=r force

law and can identify

S G

(3+n) N (4+n)

G = (5)

N (4)

4 V

where V = L is the volume of compacti ed dimensions.

We can also derive this result directly by compactifying the Lagrangian

from (4 + n) to 4 dimensions, from whichwe can also motivate a de nition for

the \reduced" Planck scale in b oth theories. In the non-relativistic limit and

in (4 + n) dimensions, the action for the interaction of the (dimensionless)

gravitational p otential = g 1, with a mass density ,isgiven by

4+n n+2 2

+ + (6) I = d x M r

(4+n) (4+n)

(3+n)

(4+n)

where r is the D spatial dimensional Laplacian, and we de ne M as

(4+n)

the reduced Planck scale in (4 + n) dimensions. Note that if we wish to

1=2

M , and work with canonically normalized eld, we rewrite =

can

(4+n) 4

the Lagrangian b ecomes

1 1

4+n 2

I = d x r + + (7)

can can can

(4+n) (4+n)

(3+n)

n+2

2 ^

(4+n)

showing that the interaction of the canonically normalized eld are sup-

pressed by M .Uponintegrating out ,we generate the p otential

(4+n)

(3+n) (3+n) 2

dtd xd y (x)r (x y ) (y ): (8)

(4+n) (4+n)

(3+n)

n+2

(4+n)

Using

1 1

r (x y )= (9)

D 2

(D2)S jx y j

wehave for the force b etween two test masses

m m 1

1 2

(10) F (r )=

(4+n)

n+2 n+2

S M

(3+n)

(4+n)

from whichwe nd the relationship b etween the reduced Planck scale and

Newtons constant

N (4+n)

n+2

= M : (11)

(4+n)

(3+n)

We can compactify from (4 + n) to 4 dimensions by restricting all the elds

to b e constant in the extra dimensions; integrating over the n dimensions

then yields the 4 dimensional action

4 n+2 2

d x )r + + (12) I = (V M

3 4 n

(4+n)

and therefore the reduced Planck scales of the two theories are related ac-

cording to

2 2

M = M V ; (13)

(4) (4+n)

which using eqn.(11) repro duces the relation b etween the Newton constants

in eqn.(5). An interesting string theoretic application of this result was made

in [7], where it was used to low the string scale to the GUT scale, cho osing the

radius of 11'th dimension in M -theory to b e 10 cm. Attempts to reduce 5

the string scale much further were considered in [8], but their conclusions

were basically negative.

Finally,we can understand this result purely from the 4-dimensional p oint

of view as arising from the sum over the Kaluza-Klein excitations of the

graviton. From the 4-d p oint of view, a (4 + n) dimensional graviton with

momentum (q ; ;q ) in the extra n dimensions lo oks like a massive particle

1 n

of mass jq j. Since the momenta in the extra dimensions are quantized in units

of 2=L, this corresp onds to an in nite tower of KK excitations for eachof

the n dimensions, with mass splittings 2=L. While each of these KK mo des

is very weakly coupled ( 1=M ), their large multiplicity can give a large

(4)

enhancementtoany e ect they mediate. In our case, the p otential b etween

two test masses not only has the 1=r contribution from the usual massless

graviton, but also has Yukawa p otentials mediated by all the massive mo des

as well:

(2 jk j=L)r

V (r ) e

: (14) = G

N (4)

m m r

1 2

(k ;;k )

Obviously, for r L, only the ordinary massless graviton contributes and

wehave the usual p otential. For r L,however, roughly (L=r ) KK mo des

make unsuppressed contributions, and so the p otential grows more rapidly

n n+1

as L =r . More exactly, for r L,

V (r ) L

n n juj

) d ue ! G r (

N (4)

m m 2R

1 2

G V S (n)

N (4) n n

= : (15)

n+1 n

r (2 )

This yields the same relationship b etween G and G found in eqn.(5)

N (4) N (4+n)

up on using the Legendre duplication formula

n n 1

( (n): (16) )( + )=

2 2 2 2

We will encounter this phenomenon rep eatedly in this pap er: the interaction

of the higher dimensional gravitons can b e understo o d in twoways. Directly

from the (4+n) dimensional p oint of view, the graviton couplings are sup-

pressed 1= M , which can b e understo o d from the 4 dimensional p ointof

4+n

view as arising from a sum over a large multiplicity of KK excitations each 6

of which has couplings suppressed by1=M . Note that for higher n, the

(4)

couplings of (4 + n) dimensional gravitons are suppressed by more p owers of

the (4 + n) dimensional Planck scale M , and so their interactions b ecome

(4+n)

increasingly soft in the infrared (the ip-side of the worse UV problems!). As

already mentioned, and as will b e seen in detail in many examples, this IR

softness is crucial to the survival of the theory when the fundamental Planck

scale is taken to b e near the TeV scale.

We make one last commenton2 factors. If we express V = L =

n 1

(2r ) , the rst KK excitation has a mass r , and r (not L ) more cor-

n n n

rectly describ es the physical size of the extra dimensions. For instance, the

p otential b etween two test masses a distance L apart is only mo di ed at

2 2

O (e ) 10 , whereas the change is O (1) for a distance r apart. In terms

of r , the relation eqn.(13) b ecomes

(n+2)

n n+2 2 n n+2

: (17) (2 ) M M = M r ;M

(4) n

(4+n) (4+n) (4+n)

We will see b elow that the exp erimental b ounds most directly constrain

M , and that it is M which is required to b e close to the weak scale

(4+n) (4+n)

for solving the hierarchy problem. Putting in the numb ers, we nd for r

1+2=n

1TeV

31=n16

r =210 mm (18)

(4+n)

For n = 2 and M =1 TeV, r 1 mm, which is precisely the distance at

(6)

which gravity is currently measured directly.For n =6, r (10 Mev) ,

. and for very large n, r approaches M

(4+n)

3 Relating the Planck scale to the String Scale

In this subsection we wish to b e more precise ab out the various scales in our

problem. Namely,we wish to quantify what exactly what we mean by

\gravity gets strong at the weak scale". Of course, we are really inter-

ested in relating the scale m at which the new physics resp onsible for

grav

making a sensible quantum theory of gravity app ears, to parameters of the

low-energy theory such as e.g. G or M . There is a practical reason

N (4+n) (4+n)

for nding determining this relationship. Both theoretically and exp erimen-

tally, m 1TeV is most desirable, on the other hand, the most stringent

grav

exp erimental constraints we will discuss directly constrain the interactions

of the (4 + n) dimensional gravitons and hence put a b ound on M .Itis

(4+n)

therefore imp ortant to determine how this b ound translates into a constraint

on m .

grav

Without a sp eci c theory in mind, it is dicult to relate m to M ,

grav

(4+n)

other than the exp ectation that they are \closeby". To b e more concrete,

we supp ose that the theory ab ove m is a string theory, sp eci cally the

grav

realization of our scenario within typ e I string theory outlined in [4], which

we brie y review here. The low-energy action of typ e-I string theory in 10-

dimensions reads

8 6

m 1 m

10 s 2 s

S = d x R + F + : (19)

7 2 7

(2 ) 4 (2 )

where e is the string coupling, and m is the string scale, whichwe can

identify with m . Compactifying to 4 dimensions on a manifold of volume

grav

2 2

V ,we can identify the resulting co ecients of R and (1=4)F with M and

(4)

1=g , from whichwe can nd

(2 )

M =

(4)

V m g

2 6

g V m

4 s

: (20) =

(2 )

Putting m 1TeV and g 1 xes a very small value for and a com-

s 4

pacti cation volume much smal ler than the string scale. A more appropriate

description is obtained by T dualising, where we compactify on a manifold

0 0

of volume V with a new string coupling given by

(2 )

V = ;

V m

2 6

g (2 )

4 0

= : (21) =

m V 2

In this T dual description, the KK excitations of the op en strings in the

typ e-I picture b ecome winding mo des of typ e-I' op en strings stucktoa D3

brane, while only the closed string (gravitational) sector propagates in the 8

bulk. Thus our scenario for solving the hierarchy problem can naturally b e

emb edded in this picture. The 4-dimensional Planck scale

2 V

2 8

M = (22) m

4 s

g (2 )

can then b e much larger than the string scale if V is much bigger than m .

To make contact with our framework, we assume that of the six compact

dimensions, (6 n)have a size L =(2r ) with the \physical" size

(6n) (6n)

r m , while the remaining n dimensions of size L =2r are

(6n) n n

(6n)

6 n

the \large" ones we previously discussed. Then V =(2 ) = r r , and

(6n)

combining eqns(22,17) we obtain

n+2

(4+n)

n+2

(r m ) = : (23)

(6n)

m g

It is clear that for the higher values of n, the p ossible enhancements of

M =m from the rst two factors are negligible and we should exp ect

(4+n)

M m 1TeV. For the case n =2,however, the rst factor can

(4+n) s

range from 2 to 3 dep ending on which of the SM gauge couplings are chosen

to represent g , and we can cho ose r to b e somewhat larger than the string

4 4

scale p erhaps as lowas (300GeV) . These factors can b e enough to push

M to somewhat higher values 10 TeV while keeping m 1TeV. As

(4+n)

will see later, the strongest constraints o ccur for the lowest values of n and

in some cases will indeed push M ab ove 10 TeV. It is reassuring to

(4+n)

know that even in this case, new string physics may b e seen at 1TeV.

4 Couplings of Bulk Gravitons and Nambu-

Goldstones of Broken Space-Time Symme-

tries.

In this section, we wish to describ e the light degrees of freedom which exist

in the e ective theory b eneath the scale of quantum gravity m and the

grav

tension f of the wall. In our scenario it is most natural to assume f m .

grav

This sort of e ective theory is interesting b ecause some states (such as the

SM elds) liveona wall in the extra dimensions, while other elds (such 9

as the gravitons) can freely propagate in the higher dimensional space. Of

course, the presence of the wall breaks translational invariance in the extra n

dimensions. Part of our discussion dep ends on whether this is a sp ontaneous

or explicit breaking of the (4 + n)dPoincare symmetry. Let the p osition

of a p oint x on the wall, in the higher dimensions a =4; 3+n, b e given

by y (x). In the case where the breaking is sp ontaneous, wall con gurations

a a

which di er from each other by a uniform translation y (x) ! y (x)+c are

degenerate in energy. The y (x) are then dynamical elds, Nambu-Goldstone

b osons of sp ontaneously broken translation invariance. The elds in the

e ective theory consist of the y (x), together with the SM elds on the wall

and gravity in the full higher dimensional bulk. The interactions of this

e ective theory are constrained by the requirement that the full (4 + n) d

Poincare invariance b e realized non-linearly on these elds. Avery nice

analysis of the structure of this e ective theory together with the leading

terms in its energy expansion has recently b een given by Sundrum [11]. We

will not rep eat this analysis here, as many of the details are unimp ortant for

phenomenological constraints we consider. We will instead study the form of

the least suppressed interactions to the y and the bulk gravitons.

Before turning to this, we raise a puzzling question not addressed in [11 ].

Since gravity can b e thought of as gauging translation invariance, and since

translation invariance is sp ontaneously broken, why are the y (x) not \eaten"

by the corresp onding \gauge eld" g , whichwould b ecome massive? We

analyze this question in app endix 1. The conclusion is that the y (x) are in-

deed eaten and the corresp onding 4-d \gauge" eld gets a mass f =M

(4)

(1 mm) for f 1TeV. Notice that, if M is held xed and r !1,

(4+n)

this mass go es to zero since M !1, and so that the analysis of [11], which

(4)

was implicitly done in this limit, is una ected. Furthermore,this mass is so

small that almost pro cesses we consider will involve energies 1(mm) ,

and so by the equivalence theorem, it is much more convenient to think in

terms of the original picture of massless gravitons and Nambu-Goldstone

elds. Nevertheless, as we will see later, a (mm) mass is generically b e

generated for any \bulk" gauge eld when the gauge symmetry is broken on

the wall, and can lead to very interesting exp erimental consequences.

Wenow turn to the leading couplings, rst to goldstone elds ignoring

gravity, then to gravity ignoring the goldstones. To b egin with, note that the

y have mass dimension 1, and are therefore written in terms of the canon-

a a a 2

ically normalized goldstone elds as y = =f . This is in analogy to the 10

usual case of Goldstone b oson of internal symmetries, where the analogue of

a a

y is an angle of the group transformation, related to the physical pion

a a

elds as = =f . This immediately means that the interactions with the

y , for f 1TeV, are always weaker than neutrino interactions which are

suppressed only by 1=m . In fact, it is easy to see that for interactions

with scalars or vectors or a single Well fermion, the leading op erators must

0 4

involve two y s and are therefore even more suppressed 1=f . This follows

from a completely straightforward op erator analysis, but can also b e simply

understo o d as follows. The uctuations in the wall given by @ y (x) induce

a non-trivial metric on the wall, inherited from the metric of the bulk space.

Ignoring gravity, the bulk metric is at and the induced metric on the wall

a a

g = + @ y @ y (24)

a a

which is symmetric under y !y , so the interactions of y which result

from non-trivial g involve pairs of y s.Following [11 ], an op erator involving a

single y interacting with vector-likeWell fermions ( ; ) can also b e written

O = c @ @ y (25)

Of course, since this op erator violates chirality,we exp ect that the co ecient

c is suppressed by m =m , up to a mo del-dep endent co ecient.

grav

Next consider the coupling of the SM elds to gravitation, but without

exciting the y . If the bulk metric is G where M; N =0; ;3+n, the

induced metric on the wall is trivially

g (x)= G (x; y =0): (26)

The bulk gravitons are the p erturbations of G ab out ,

MN MN

G = + (27)

MN MN

n+2

(4+n)

and the linear interactions with SM wall elds are given by

H (x; y =0)

d xT (28)

n+2

(4+n)

where T is the 4-d energy momentum tensor for the SM elds. Two things

are immediately obvious from this coupling. First, there is no coupling to 11

the H ;H gravitons. This is intuitively clear: without changing the shap e

a ab

of the wall (i.e. exciting the y ), the wall elds make zero contribution to

a ab 0

T ;T and the the couplings to the corresp onding H svanish. Second, the

interaction clearly violates translation invariance in the extra dimensions, and

therefore the extra dimensional momenta p need not b e conserved however

in the interactions b etween the wall and bulk states, while energy is still

conserved b ecause time translational invariance still holds. More intuitively

we can think of the wall as b eing in nitely heavy, so that it can recoil to

absorb extra-dimensional momentum without absorbing energy. This can

also b e seen explicitly by expanding H (x; y =0)into Kaluza-Klein mo des

a n

H (x; y =0)= H (29)

which shows that the wall T couples to all KK mo des with equal strength

1=M . Of course, there are many other couplings involving combinations

(4)

of the y and gravitons, but they are all suppressed by further p owers of

1=M and/or 1=f .

(4+n)

We should also mention that if translation invariance is explicitly broken

in the extra dimensions, as in the case where the wall is \stuck" to a p oint

in the higher dimensions, the mo des y corresp onding to the uctuations of

the wall b ecome massive and are irrelevanttolow energy physics.

5 Lab b ounds

5.1 Macroscopic gravity

Given that the gravitational interaction is unchanged over distances bigger

than the size of the extra dimensions, and that gravity is only signi cant

on much larger scales, the change in gravity at distances smaller than 1

mm is harmless. One maywonder ab out systems where gravity is known to

b e imp ortant, but where the typical inter-particle separation is smaller than

1 mm, e.g. in the sun. It is clear, however, that all e ects due to the new

gravity b eneath 1mmmust b e suppressed bypowers in the ratio of the

size of the new dimensions over the typical size R of the gravitating b o dy.

The reason is that, if we divide the b o dy into 1 mm balls, these balls have

normal gravitational interactions. Since imp ortant gravitational e ects are 12

bulk e ects, the only error incurred in splitting the b o dy into mm sized

balls can at most b e p ower suppressed in (1mm=R). For instance, let us

compute for the gravitational self energy p er unit mass of a ball of radius R

and density :

Z Z

r R

E G G

grav N (4) N (4+n)

3 3

+ d r (30) d r

3 (n+1)

R r r

r 0

where the rst integral uses the (4 + n) dimensional gravitational p otential

and the second is the usual piece. Now, the usual piece is dominated by

.However, for n =2, large distances and gives a contribution G R

N (4+n)

the new contribution is log divergent and is cuto o at short distances by

the typical inter-particle separation r and at long distances by R, and for

min

n>2, the new contribution is dominated by short distances and is cuto

by r . The fractional change in the gravitational energy due to the new

min

interaction is then

E r

grav N (4+n)

: (31)

n2 n2

2 2

G r R r R

grav

N (4)

min min

(n+2) 1

Note that for G (TeV) and for r larger than (TeV) ,

min

N (4+n)

this contribution is largest for n = 2, and

E 1mm

grav

: (32)

E R

grav

which is completely irrelevant for the sun. The smallest ob jects for which the

gravitational self-energy plays any role is the neutron star which has R 10

km, giving an unobservably small fractional change 10 in gravitational

energy.

5.2 Mesoscopic gravity

While the normal Newtonian gravitation is una ected on distances larger

than r , the gravitational attraction b etween two ob jects grows much more

n+2

quickly 1=r at distances smaller than r . This is of course a re ection of

the fact that, in this scenario, gravity \catches up" with the other interactions

3 19

at 10 GeV rather than at 10 GeV. The ip side of this is that, even

though gravityismuch stronger than b efore, it is still muchweaker than the 13

other forces at distances appreciably larger than the weak scale. Consider

for instance the ratio of the new gravitational force to the electromagnetic

force b etween a proton and an electron a distance r apart

G m m

F 10 cm

e p

grav N (4+n)

10 : (33)

F r r

The smallest value of r where electromagnetic e ects are dominant are atomic

sizes r 10 cm, and even then for the worst case n = 2, the ab ove ratio

is unobservably small 10 . Of course on larger distances the electro-

magnetic interactions are screened due to average charge neutrality, while

gravity is not. Even here, however, the residual electromagnetic forces still

dominate over the new gravity. As an example consider the Van der Waals

(VdW) force b etween twohydrogen atoms, in their ground state, a distance

r r apart from each other. This arises due to the dip ole-dip ole in-

b ohr

teraction p otential, i.e. the energy of the dip ole-moment of atom 1 in the

electric eld set up by the dip ole moment of atom 2:

d d

1 2

V : (34)

int:

The rst order energy shift due to this interaction vanishes in the ground state

since the ground state exp ectation value of each dip ole momentvanishes by

rotational invariance. The second order p erturbation then gives the usual

VdW 1=r p otential,

2 2

d d 1

10n 20n

V (r )

0 2 6

2E E E 16 r

0 n

n;n

r 1

bohr

: (35)

r r

The ratio of this VdW force to the ordinary gravitational attraction b etween

the hydrogen atoms is

F 1mm

VdW

; (36)

F r

ord. grav.

and we see that while electrostatic e ects are irrelevant for distances larger

than 1 mm, the VdW force dominates over ordinary gravity at sub-mm 14

distances. This is in fact the central obstacle to the sub-mm measurements of

gravitational strength forces. Even in our scenario with much stronger grav-

ity, VdW forces dominate down to atomic scales, (where the electromagnetic

e ects are no longer even shielded). For the case of n = 2 new dimensions,

the new dimensions op en up near the mm scale, and the gravitational force

only increases as 1=r at smaller distances, which is still overwhelmedby

VdW. Already for n = 3, the new dimensions op en at 10 cm and VdW

dominates still further.

2 4

Of course in the case n =2,we exp ect a switch from 1=r to 1=r gravity

roughly b eneath r 1 mm. There are no direct measurements of gravity

at sub-mm distances. The b est current b ound on sub-mm 1=r p otentials

actually comes from exp eriments measuring the Casimir forces at 5 microns

[9]. Parametrizing the force b etween two ob jects comp osed of N ;N nucleons

1 2

separated by a distance r as

15 2

(10 m)

V (r )=CN N (37)

1 2

the b est current b ound is C 7 10 [9]. If we assume that the only

gravitational strength forces b eneath r is the 6d Newtonian p otential, this

corresp onds to

15 1 2

G (m 10 m )

1GeV

N (6)

C = =

3 50M

(6)

! M 4:5TeV: (38)

(6)

If we take this indirect b ound seriously, then from eqn.(18), r shrinks to

30 microns, whichishowever still well within the reach of the planned

exp eriments directly measuring gravity at sub-mm distances. There maybe

contributions to the long-range force b eneath r beyond those from the KK

excitations of the ordinary graviton, whichmay comp ensate the gravitational

force and the and the force at the 5 micron distances prob ed in the Casimir

exp eriments may not b e as strong wehave considered, with corresp ondingly

weaker b ounds. If the 1=r force is canceled at short distances, a sub-leading

3 2 3

1=r force may remain. In this case, the transition from 1=r ! 1=r could

b e observed for r as large as :5mm. It is interesting that this p otential

could also b e interpreted as Newtonian gravity in 5 space-time dimensions,

with a new dimension op ening up at the millimeter scale! 15

5.3 \Comp ositeness" b ounds

We next discuss lab oratory b ounds. Since wehave quantum gravityatthe

TeV scale, in theory ab ovea TeV will generate higher dimension op erators in-

volving SM elds, suppressed bypowers of TeV. Of course, op erators such

as these which lead to proton decay or large avor-violations in the Kaon

system must somehow b e adequately suppressed as wehave discussed in

previous pap ers[1 , 4 ]. However, the ma jority of higher dimension op erators

suppressed by TeV are safe. Their e ects can show up either in mo di-

fying SM cross-sections (and are therefore constrained by \comp ositeness"

searches), or they can give corrections to precisely measured observables such

as the electron/muon (g 2) factors or the S-parameter. Since we do not

know the exact theory ab oveaTeV, the co ecients of these higher dimen-

sion op erators are unknown, but we will estimate their order of magnitude

e ects to show that they do not provide signi cant constraints on the frame-

work. We discuss \comp ositeness" constraints rst. The strongest b ounds

on 4-fermion op erators of the form

O = ( ) (39)

4fermi

are from LEP searches in the lepton sector, which require at most 3:5

TeV. If the this op erator is generated with co ecient1=m , it is safe for

grav

m 1TeV.

grav

While most of these op erators have unknown co ecients, some have con-

tributions from physics b eneath the scale m which are in principle cal-

grav

culable. For instance, the tree-level exchange of the (4 + n) dimensional

gravitons can give rise to lo cal 4fermion op erators[4 ]. We can understand

this from the 4-dimensional viewp oint as follows. If the typical external en-

ergy for the fermions is E , then the exchange of a KK excitation of the

1 1

graviton lab eled by momenta (k ; ;k )r with mass jk jr E generates

1 n

n n

a lo cal 4-fermion op erator. Summing over the KK mo des yields an op erator

of the form

E 1

O = C ( ) (40)

1 2

M jkr j

(4)

jk jr E

where C is an O (1) co ecient to b e determined by an exact computation.

For n = 2, the sum over KK mo des is log-divergent in the UV, while for n>2 16

it is p ower divergent. Of course, this sum must b e cuto for the KK mo des

heavier than m , where new physics sets in. For n = 2, the logarithm is

grav

not large enough to signi cantly enhance the op erator; however, for n>2,

2 4

the p ower divergence changes the 1=M suppression to an E =m e ect:

(4)

grav

n+2

E m

grav

O = C ( ) : (41)

M m

(4+n)

grav

Of course the precise b ound on m dep ends on the relationship b etween

grav

m and M .Ifwe take the string scenario and identify m with m ,

grav grav s

(4+n)

then this relationship is given in eqn.(23). Even in the worst case where the

the \small" radii are not larger than the string scale (r m = 1, the b ound

6n s

on m coming from equating the co ecient of the four-fermi op erator with

2 2

2 = yields

1=4

m > E: (42)

Since the strongest b ounds on come from LEP where the energy is at most

100 GeV, we are safe for all n as long as m TeV.

5.4 Cosmic rays

While colliders have not yet attained the energies required to prob e new

strong quantum gravitational e ects at the TeV scale, one can wonder whether

very high energy cosmic rays place any sort of b ounds on our scenario. In-

deed, there are very high energy cosmic rays (nucleons) of energies up to

20 8

10 eV = 10 TeV, eight orders of magnitude more energetic than the

fundamental Planck scale. Furthermore, when these nucleons impinge on

a stationary nucleon, the center of mass energy can b e as high as 1000

TeV. This raises two questions. First, is there anything wrong with having

a particle with energy so much larger than the fundamental Planck scale?

And second, do interactions with such high energies prob e p ost-Planckian

physics? The answer to b oth questions is no, and we address them in turn.

It is obvious that there is nothing wrong with having a particle of arbi-

trarily high energy, since energy is not Lorentz invariant. The question is

however, whether a particle can b e accelerated from rest to a Post-Planckian

energy. There is certainly no problem with accelerating a particle to p ost-TeV 17

energies, as long as the acceleration is suciently small (but over large enough

distances) so that energy loss to ordinary radiation is negligible. Note that

relevant acceleration scales will b e so much smaller than the weak scale that

the couplings to ordinary radiation vastly dominate the coupling to higher

dimension gravitons, so that as long as ordinary radiation is negligible, the

gravitational radiation energy loss is even smaller. It is interesting to note

that, in the context of normal gravitational theory, there have b een sp ecu-

lations that it may b e imp ossible to accelerate a particle to p ost-Planckian

energies; at least many acceleration mechanisms fail for a variety of reasons

[10]. As a typical example, supp ose that the acceleration is provided bya

constant electric eld E acting over a region of size R. In order to accelerate

acharge e to energy E ,wemust have eE R E. On the other hand, there is

2 3

an energy V E R stored inside the region, whichwould give a black hole

2 2

of event horizon size R V=M l =(E=M ) R.For EM , the hori-

P Pl Pl

hor

zon size is much larger than R and the system would collapse into a black

hole. These sorts of arguments have led to sp eculations that p erhaps for

reasons related to fundamental short-distance physics, p ost-Planckian ener-

gies are inaccessible. Our example suggests otherwise: while may b e dicult

to accelerate to energies b eyond the e ective four dimensional Planck scale,

energies b eyond the fundamental short-distance Planck scale can easily b e

attained.

Next we turn to the second issue: do cosmic ray collisions with center of

mass energies far ab ove the TeV signi cantly prob e the physics at distances

smaller than (TeV) ? The answer to this is obviously no; the huge frac-

tion of the cross-section for nucleon-nucleon scattering is di ractive, arising

from the nite size of the nucleon, giving a typical cross section 30 mil-

libarn. The p oint is of course that it is not enough for the c.o.m. energy

to b e large, after all two particles traveling in opp osite direction with large

energies but in nitely far apart havehuge c.o.m. energy but do not interact!

In order to prob e short distance physics at distances r , it is necessary to

have a momentum transfer r ; but the vast ma jorityofnucleon-nucleon

interactions only involve GeV momentum transfers. In fact, cosmic rays

lose energy in the atmosphere not through di ractive QCD scattering but

by creating electromagnetic showers, where the e ective momentum transfer

per interaction is still smaller. 18

5.5 Precision observables

Corrections to electron and muon (g 2) are exp ected to b e naturally small

foravery general and well-known reason. The higher dimension op erators

which can contribute to e.g. the electron (g 2) are of the form

e F e + higher dimensional op erators : (43) L

g 2

grav

Since the lowest dimension op erator violates electron chirality,we parametrize

c = d m =m , and since the QED contribution to (g 2) generates the

5 5 e grav

same op erator with co ecient =(m ), the fractional change in (g 2) is of

order

(g 2) m

d (44)

g 2 m

grav

whicheven for d 1 and m 1TeV is 10 , smaller than the

5 grav

exp erimental uncertainty 10 . The contribution to the muon (g 2) is

similarly safe. Of course, there are contributions to d which can b e computed

in the low energy theory involving lo ops of the light(4+n) dimensional

graviton, in which case d is further suppressed by a lo op factor, and the

fractional change in (g 2) is corresp ondingly smaller. Furthermore, since

all other op erators have higher dimension, they will at most make comparable

contribution to (g 2). Note that the chirality suppression of the dimension

5 op erator was crucial: a c 1 is grossly excluded. The correct estimate

given ab ove indicates why the anomalous magnetic moment measurements,

in spite of their high precision, do not signi cantly constrain new weak scale

physics.

Similar arguments apply to the corrections to precision electroweak ob-

servables. Consider the graviton lo op correction to the S parameter. Again

from the 4d viewp oint, we are summing over the contributions of the towers

of KK gravitons. We consider contributions from mo des heavier and lighter

than m resp ectively. Recall that each KK mo de has 1=M suppressed cou-

Z (4)

plings. For the mo des lighter than m , each contributes (m =M ) to S.

Z Z

(4)

We therefore estimate

2 n

S (m =M ) (m r )

(4)

KK Z

n+2

(m =M ) (45)

Z (4+n) 19

3 4

which is a tiny 10 10 contribution even for the worst case n =2,

M =1TeV. For the contribution from a KK mo de heavier than m , S also

(4) Z

vanishes in the limit m !1, so the contribution to S from eachmodeis

2 4 2

m ). Therefore, the contribution to S from these states is m =(M

KK Z

(4)

S = : (46)

m >m

KK Z

1 2

M (jk jr )

(4)

>m jk jr

n Z

This is precisely the same sum as was encountered in the comp ositeness

section, and it is p ower divergent in the UV for n 3. Cutting the p ower

divergence o at m ,we nd

grav

S ( ) (47)

m >m

KK Z

grav

whicheven for n =6 is 10 for m 1TeV.

grav

5.6 Rare decays to higher dimension gravitons

A far more imp ortant set of constraints follow from the fact that the (4 + n)

dimensional graviton is a massless particle with couplings to SM elds sup-

pressed bypowers of 1=TeV. In this resp ect, it is similar to other light

particles like axions or familons. These are known to b e in disastrous con-

ict with exp eriment for decay constants in the TeV region, for familons

b ecause they give rise to large rates for rare avor-changing pro cesses, for

axions b ecause they can takeawaytoomuch energy from stellar ob jects

through their copious pro duction. Wemust check that the analogous pro-

cesses do not rule out a (4 + n) dimensional graviton with 1=TeV-suppressed

couplings. Another way of stating the problem is as follows. As wehave

remarked several times, from the 4-dimensional p oint of view, the graviton

sp ectrum consists of the ordinary massless graviton, together with its tower

of KK excitations spaced by r . While the coupling of each of these KK

mo des is suppressed by1=M , there is an enormous number (Er ) of

(4)

them available with mass lower than energy E , and there combined e ects

are much stronger than suppressions of 1=M l . This large multiplicity

factor is resp onsible for converting 1=M e ects to stronger =M e ects,

(4) (4+n)

2 n+2

as wehave already seen explicitly in the conversion b etween 1=r to 1=r

Newtonian force law. However, as wehave mentioned, the infrared softness 20

of higher dimension gravity will allow this scenario to survive. We b egin with

b ounds from rare decays of SM particles involving the emission of gravitons

into the extra dimensions, b eginning with the decay K ! + graviton (the

analogous familon pro cess K ! + familon puts the strongest b ound on

familon decay constants 10 GeV). Recall that even though the emission

of a single graviton into the extra dimensions violates conservation of extra-

dimensional momentum,itisnevertheless allowed, since the presence of the

wall on which SM elds is lo calised breaks translational invariance in the

extra dimensions. However, since time translational invariance is still go o d,

energy must still b e conserved. Notice also that this pro cess will pro ceed

through e.g. the spin-0 comp onent of the massive KK excitations of the

graviton in order to conserve angular momentum. A tree-level diagram for

the pro cess can b e obtained by attaching a (4 + n) dimensional graviton to

any of the legs of the Fermi interactionsd ud . Again, on dimensional grounds,

the decay width for the decayinto any single KK mo de is at most

2 5

1 m m

K K

(48)

4 2

16 M M

(4)

where the rst factor has b een isolated as roughly the total KK decay width.

However, there is a large multiplicity factor from the number of KK mo des

with mass m which are energetically allowed, (m r ) . The total

K K n

width to gravitons is then

! !

n+2

1 m m

(49)

K ! +graviton

16 M M

4+n

yielding a branching ratio

n+2

B (K ! + graviton) (m =M ) (50)

(4+n)

Even in the most dangerous case n =2;M 1TeV , this branching ratio

(6)

is 10 and is safely smaller than the b ound, although a more careful

calculation is required for this case. As we will see in the next sections,

10 TeV for n =2,in astrophysics and cosmology seem to require M

(6)

which case the branching ratio in Kaon decaygoesdown another four orders

of magnitude to 10 . Note that the scaling for the branching ratio could

have also b een derived directly from the (4 + n) dimensional p oint of view. 21

As wehave remarked earlier, the couplings of the graviton are dimensionless

when expressed in terms of the (4 + n) dimensional metric G , which can

b e expanded ab out at space-time as

G = + (51)

MN MN

(n+2)=2

(4+n)

where h is the canonically normalized eld (of mass dimension 1 + n=2)

(n+2)=2

in (4 + n) dimensions. Therefore, there is a factor of 1=M in the

(4+n)

(n+2)

amplitude and 1=M in the rate. Inserting factors of the only other scale,

(4+n)

m to make a dimensionless branching ratio, we arrive at the same estimate

for B (K ! + graviton). We see explicitly that it is the infrared softness

of the interactions of the higher-dimensional theory which is resp onsible for

insuring safety, although this was certainly not guaranteed for relatively low

Analogous branching fractions for avor-conserving and violating decays

for B quarks are also safe, with branching ratios 10 for the worst case

n =2;M =1TeV, and further down to 10 for the M 10TeV

(6) (6)

favored by astrophysics and cosmology. The largest branching fractions are

for the heaviest particles, the most interesting b eing for Z decays. The decay

Z ! f f + graviton can o ccur at tree-level, with a branching fraction

n+2

B (Z ! f f + graviton ) ( ) (52)

4+n

which can b e as large as 10 , still not excluded by Z -p ole data. Other

decays like Z ! + graviton are only generated at lo op level, with unob-

servably small branching ratios.

6 Astrophysics

Wenow turn to astrophysical constraints on our scenario, analogous to

b ounds on the interaction of other light particles such as axions. In our

case, the worry is that, since the gravitons are quite strongly ( 1=)TeV

coupled, they are pro duced copiously and escap e into the extra dimensions,

carrying away energy.Having escap ed, the gravitons havea very small prob-

ability to return and impact with the wall elds: this is intuitively obvious 22

since the wall only o ccupies a tiny region of the extra dimensions. We can

also understand this from the p oint of view of pro ducing graviton KK exci-

tations. As usual, even though eachKKmodeis1=M coupled, signi cant

(4)

energy can b e dump ed into the KK gravitons b ecause of their large multi-

plicity.However, each single KK mo de, once pro duced, has only its 1=M

(4)

coupled interactions with wall elds. In the next section we will quantify

this corresp ondence, nding that the higher dimensional gravitons havea

mean-free time for interaction with wall exceeding the age of the universe

for all graviton energies relevant here. the upshot is that the gravitons carry

away energy without returning energy, thereby mo difying stellar dynamics

in an unacceptable way. For the axion, the strongest such b ounds come

from SN1987A, which constrain the axion decay constant f 10 GeV.

This naively sp ells do om for our 1= TeV coupled gravitons. However, since

the gravitons propagate in extra dimensions and haveinteractions that are

softer in the infrared, our scenario survives the astrophysical constraints.

We will do a more detailed analysis b elow; however, in order to get an

idea of what is going on we estabilsh a rough dictionary b etween rates for

axion and graviton emission. Since any axion vertex is suppressed by1=f ,

any rate for axion emission is prop ortional to

Rate of axion pro d. / (53)

Now consider graviton pro duction. The rst p oint is that if the temp erature

T of the star is much smaller than r , none of the KK excitations of the

graviton can b e pro duced and the only energy loss is the miniscule one to

the ordinary graviton. If T r , on the other hand, a very large number

(Tr ) of KK mo des can b e pro duced. Since each of these mo des has

couplings suppressed by1=M , the rate for graviton pro duction go es like

(4)

1 T

Rate of graviton pro d. / (Tr ) (54)

n+2

(4)

(4+n)

Note that this is exactly analogous to what happ ened with e.g. K ! +

graviton, and that this dep endence could have b een inferred directly from

the (4 + n)-dimensional viewp oint just as in eqn.(51). We can now establish

the rough dictionary b etween f and M :

(4+n)

T 1

! : (55)

n+2

(4+n) 23

This dictionary contains the essence of what will b e found by more detailed

analysis b elow. The strongest b ounds come from the hottest systems (where

the b ounds on f are also the strongest). However, even for the SN where

the average kinetic energy corresp onds T 30 MeV, f 10 GeV requires

that for n = 2 that M 10 TeV, whereas already for n 3, M can

(6) (4+n)

be 1TeV. Recall also that an M 10 TeV can b e consistent with new

(6)

physics (for instance string excitations) at the 1TeV scale and is therefore

not unnatural as far as the gauge hierarchy is concerned. The constraints

from other systems such as the Sun (where T KeV or Red giants (where

T 100 KeV) are weaker and are satis ed for M 1TeV.

(4+n)

Wenowmove to a somewhat more detailed analysis. This is necessary

b ecause there are some qualitative di erences b etween the axion and gravi-

ton couplings; for instance, the axion coupling to photons is suppressed not

only by1=f but also by an \anomaly factor" =4 , while there is no cor-

resp onding anomaly price for gravitons. Furthermore, there are some e ects

that can not b e determined from dimensional analysis alone, for instance, in

some systems, most of the gravitational radiation comes from non-relativistic

particles, and the energy emission rate dep ends on the small ratio = v=c

in a way that can not b e xed by dimensional analysis. It is easy to deduce

the dep endence on from the couplings to the physical gravitons. A non-

relativistic particle of mass m,moving with some velo city has an energy

momentum tensor T = m(dx =d )(dx =d ) which in the non-relativistic

limit 1 b ecomes

p p

i j

T = m; T = p ;T = : (56)

00 0i i ij

Therefore, the coupling to the physical graviton p olarizations, which come

ij 2

from the transverse,traceless comp onents h , has a factor p =m T in

the amplitude. Therefore, there is no dep endence on from the gravita-

tional vertex. The situation is di erent for couplings to photons; there the

fundamental coupling is eA (dx =d ), and in the non-relativistic limit the

coupling to the physical photons (the transverse part of A ) is suppressed by

. Of course, for relativistic particles 1 and dimensional analysis is all

that is needed to estimate the relevant cross sections.

Since we are concerned with the energy lost to gravitons escaping into the

extra dimensions, it is convenient and standard [14] to de ne the quantities

_ which are the rate at which energy is lost to gravitons via the

a+b!c+grav 24

pro cess a + b ! c + g r av iton, p er unit time p er unit mass of the stellar

ob ject. In terms of the cross-section the numb er densities n

a+b!c+grav a;b

for a,b and the mass density ,_is given by

hn n v E i

a b a+b!c+grav rel grav

_ = (57)

a+b!c+grav:

where the brackets indicate thermal averaging. We can estimate the cross-

sections for all graviton pro duction pro cesses as follows. From the graviton

n+2

vertex alone, we get the usual T =M dep endence which already has the

(4+n)

correct dimensions for a cross-section. The dep endence on dimensionless

gauge couplings etc. are trivially obtained, while the appropriate factors of

for non-relativistic particles are dealt with as in the previous paragraph.

Finally,we insert an overall factor 1=16 to approximately account for

the phase-space. The relevant pro cesses and estimated cross-sections are

shown b elow

Gravi-Compton scattering: + e ! e + grav

2 2

v e (58)

n+2

(4+n)

Gravi-brehmstrahlung: Electron- Z nucleus scattering radiating a gravi-

ton (e + Z ! e + Z + grav)

2 2

v Z e (59)

(4+n)

Graviton pro duction in photon fusion: + ! grav

(60) v

n+2

(4+n)

Gravi-Primako pro cess: + EM eld of nucleus Z ! grav

v Z (61)

n+2

(4+n)

Nucleon-Nucleon Brehmstrahlung: N + N ! N + N + grav (relevant for

the SN1987A where the temp erature is comparable to m and so the strong

interaction b etween N's is unsuppressed)

n+2

v (30 millibarn) ( ) (62)

(4+n) 25

Armed with these cross-sections, we can pro ceed to discuss the energy-loss

problems in the Sun, Red Giants and SN1987A.

6.1 Sun

The temp erature of the sun is 1 KeV, and the relevant particles in equilib-

rium are electron, protons and photons. The numb er densities n = n and

e p

n are roughly comparable, n ( Kev) . The electrons and protons

e;p;

are non-relativistic. The observed rate at which the sun releases energy p er

unit mass p er unit time is

1 1 45

_ 1erg g s 10 TeV: (63)

nor mal

Wemust therefore demand that the rate of energy loss to gravitons is less

than this normal rate. We will consider the pro cesses in turn. Begin with

the Gravi-Compton scattering. Using n = = n = =1=m and n T ,

e p p

sun

we nd

n+5

sun

_ 4 (64)

n+2

m m M

p e

(4+n)

and therefore

166n

n+2

M 10 GeV: (65)

(4+n)

Even the worst case n = 2 only requires M 10 GeV. Gravi-brehmstrahlung

(4+n)

is not relevant since there are no high-Z nuclei present in the sun. Photon

pair fusion into graviton is more imp ortant than the the analogous pro cess

+ ! axion, which is highly suppressed by the \anomaly price" =4 .For

the case of graviton, the rate is given by

n+7

sun

_ : (66)

n+2

(4+n)

This places a lower b ound on M ,

(4+n)

186n

n+2

M 10 GeV : (67)

(4+n)

For n = 2, this is a stronger b ound M 30 GeV, but certainly no problem.

(6)

The Gravi-Primako (with photons scattering o the electric eld of the

protons) is sub-dominant to the last b ound b ecause, while protons and pho-

tons have roughly equal numb er density, the electric eld surrounding a pro-

ton is prop ortional to the electric charge e=4 and so the Gravi-Primako

rate is suppressed relative to the photon-photon fusion by rate by .

Finally,nucleon-nucleon brehmstrahlung is is irrelevant b ecause at these

temp eratures, the collisions of nucleons can not prob e the strong interaction

core.

It is clear that the situation with the sun is so safe b ecause it's temp era-

ture is so low. Because electrons, protons and photons o ccur in equal abun-

dance, but the cross-sections involving photons and electrons are suppressed

by and e ects, the dominant pro cess is the photon-photon fusion, which

yields even for the worst case n =2,M 30 GeV. For red giants, the tem-

(6)

p erature is somewhat larger, T 10 KeV, and the constraints are somewhat

di erent, but the temp erature is still so low that certainly M 1TeV is

(4+n)

safe for all n. Clearly the strongest b ounds will come from SN1987A where

the temp erature is signi cantly higher 30 MeV. We turn there now.

6.2 SN1987A

During the collapse of the iron core of SN1987A, ab out 10 ergs of gravi-

tational binding energy was released in a few seconds; the resulting neutron

star had a core temp erature 30 MeV. Wemust ensure that the graviton

53 1

luminosity do es not exceed the lib erated 10 erg s :

53 1 16 2

L =_M 10 er g s (10 GeV) (68)

grav SN

There are two dominant pro cesses here: nucleon-nucleon brehmstrahlung

(which is the dominant pro cess for axions), together with the Gravi-Primako

pro cess (which is again sub-dominant in the axion case b ecause the \anomaly

factor" =4 ). The graviton luminosity from the nucleon-nucleon brehm-

strahlung is roughly

n+2

n T

L M 30millibarn : (69)

grav SN

4+n)

57 3 3 3 4

For M 1:6M 10 GeV, n 10 GeV and 10 GeV ,we

SN sun N

nd the following b ound on M

(4+n)

154:5n

n+2

TeV: (70) M 10

(4+n) 27

For n = 2, this is quite a strong b ound, requiring M 30 TeV. We next

(6)

estimate the graviton luminosity from the Gravi-Primako pro cess. Using

n = 1=m and Z 50, wehave

Fe Fe

n+4

L 10 GeVZ (71)

grav

n+2

(4+n)

which requires

124:5n

n+2

TeV: (72) M 10

(4+n)

This is a somewhat weaker b ound than for nucleon-nucleon brehmstrahlung.

The basic reason is that while again in the SN, nucleon and photon abun-

dances are comparable (actually nucleons are somewhat more abundant),

the nucleon-nucleon brehmstrahlung cross-section is enhanced by strong-

interaction e ects.

In summary,wehave found as exp ected that the strongest astrophysi-

cal b ounds come from the hottest system, SN1987A. The b ounds for n =2

were quite strong, requiring M 30 TeV. This illustrates that the phe-

(6)

nomenological viability of our scenario is not an immediate consequence of

lo calizing the SM particles on a wall. Nevertheless, for n>2, the infrared

softness of higher dimensional gravitywas enough to evade the constraints

for M 1TeV. Even for n =2, M 30 TeV is consistent with a

(4+n) (6)

string scale few TeV, and therefore this case is still viable for solving the

hierarchy problem and accessible to b eing tested at the LHC.

7 Cosmology

It is clear that in our scenario, early universe cosmology is drastically di er-

ent than the current picture. Since the fundamental short distance scale is

1TeV, the highest temp erature at whichwe can conceivably think ab out

a reasonable space-time where the universe is b orn is m TeV rather

grav

than M 10 GeV. Even b eneath these temp eratures, however, the dy-

(4)

namics of the extra dimensions is critical to the b ehavior of the universe on

the wall. In the absence of any concrete mechanism for stabilizing the radius

of the extra dimensions, we can not track the history of the universe start-

ing from TeV temp eratures. Of course, nothing is known directly ab out the

universe at TeV temp eratures. The only asp ect of the early universe which 28

we know ab out with some certainty is the era of Big-Bang Nucleosynthesis

(BBN) which b egins at temp eratures 1 MeV. The successful predictions

of the light element abundances from BBN implies that the expansion rate

of the universe during BBN can not b e mo di ed by more than 10%. Since

the size of the extra dimensions determines G and hence the expansion

N (4)

rate of the 4 d universe on the wall, we know that whatever the mechanism

for stabilizing the extra-dimensional radii, they must have settled to their

current size b efore the onset of BBN. Note that the radii must b e xed with

size mm, whichismuch smaller than the Hubble size 10 cm at BBN.

Therefore, the expansion of the 4-d universe can b e describ ed by the usual

4-d Rob ertson-Walker metric. This is analogous to the analysis of macro-

scopic gravity in section 5.1, where wesaw that even when inter-particle

separations are smaller than r , the large-distance gravitational energetics

are una ected. Furthermore, the extra-dimensions must b e relatively empty

of energy-density, since this would also contribute to the expansion rate of

the 4-d universe.

This leads us to parametrize our ignorance ab out the physics determining

the radius as follows. Extrap olating back in time from BBN, we assume that

the universe is \normal" from BBN up to some maximum temp erature T

for the wall states. By \normal" we mean that the extra dimensions are

essentially frozen and empty of energy density. One p ossible way this initial

condition can come ab out is if T is the re-heating temp erature after a p erio d

of in ation on the wall. The in aton is a eld lo calised on the wall and

its decays re-heat predominantly wall-states while not pro ducing signi cant

numb ers of gravitons.

We will test the consistency and cosmological viability of such a starting

p oint. The main reason this will b e non-trivial is due again to the presence of

light mo des other than SM particles- namely the extra-dimensional gravitons

and, for the case where the wall is free to move, the goldstones describing

the p osition of the wall.

It is easy to see that the goldstones are not esp ecially problematic: they

havea very small mass 10 eV , and since they are their own antipar-

ticles, they would countasn=2 extra neutrinos during Nucleosynthesis if

they have thermal abundance. For n = 2, this is marginally consistent with

BBN, whereas for n>2wehave to insure that they are not thermal during

BBN. This puts some upp er b ound on the \normalcy" temp erature T .If

c 2

the (mo del-dep endent) coupling @ @ g=f is resp onsible for thermal-

ization, the goldstone drops out of equilibrium when

4=3

1=3 2=3

M (T )

max

2 2=3

10 GeV (73) ! (T )T

where (T ) is the largest Yukawa coupling of a SM particles thermal

max

1 GeV. If instead the at temp erature T .This roughly translates to T

mo del-indep endent couplings suppressed by1=f are keeping equilibrium,

decoupling happ ens when

8=7

T 10GeV : (74)

1=7

This is a weak b ound for obvious reasons: the goldstones are essentially

massless, with smaller interaction cross-sections than neutrinos, and so it

is guaranteed that they decouple b efore BBN, where neutrinos decouple.

Furthermore, since they are so light, these goldstones can not over-close the

universe.

Gravitons provide further cosmological challenges.

Expansion dominated co oling

The energy density of the radiation on the wall co ols in twoways. The

rst is the normal co oling due to the expansion of the universe:

d T

j 3H 3 (75)

expansion

dt M

The second is co oling by \evap oration" into the extra dimensions, by pro-

ducing gravitons which escap e into the bulk. Notice again that this sort of

co oling do es not o ccur if the SM elds couple to some generic 1= TeV coupled

but 4-dimensional particle X, since the rates for the forward and backward

reactions would pro ceed at the same rate and X would thermalize. The rate

n+2

for graviton pro duction is prop ortional to the usual factor 1=M , and the

(4+n)

rate for evap orative co oling can b e determined by dimensional analysis to b e

n+7

T d

j (76)

evap.

n+2

(4+n) 30

The expansion rate of the universe can only b e normal if the rate for normal

expansion by co oling is greater than the that from evap oration. This put an

upp er b ound on the temp erature T at where the universe can b e thoughtof

as normal:

0 1

1=(n+1)

n+2 (n+2)=(n+1)

(6n9)

(4+n) (4+n)

< (n+1)

@ A

T 10 MeV (77)

M l 1TeV

10 MeV for M 1TeV. However, For the worst case n = 2, this is T

Pl(4+n)

the astrophysical constraints prefer M 10 TeV, in which case T moves

(4+n)

< <

up to 100 MeV, while for n =6,T 10 GeV. Of course,as n !1;T !

M . It is reassuring that in all cases, T 1 MeV, so that BBN will not

(4+n)

b e signi cantly p erturb ed.

We can understand this constraint in another way. The rate of pro duction

of (4 + n) dimensional gravitons pro duced p er relativistic sp ecies (\photons")

on the wall, is given by

n+3

d n T

grav

(78) = hn v i

!grav

n+2

dt n

(4+n)

so that the total numb er density of gravitons pro duced during a Hubble time

starting at temp erature T is

n+1

n T M

grav Pl

(79)

n+2

(4+n)

The \co oling" b ound wehave given corresp onds to requiring n << n .

grav

BBN constraints

Wemust ensure that the pro duced gravitons do not signi cantly a ect

the expansion rate of the universe during BBN. The energy density in gravi-

3 4

tons red-shifts awayasR rather than R . This is b ecause, from the 4

dimensional p oint of view, the gravitons pro duced at temp erature T are mas-

sive KK mo des with mass T . Alternately, from the (4 + n) dimensional

p oint of view, while the graviton is massless, the extra radii are frozen and

not expanding, so the comp onent of the graviton momentum in the extra

dimensions is not red-shifting. The ratio of the energy density in gravitons

versus photons by the time of BBN is then

n+1

T T M

grav: Pl

j (80)

n+2

BBN

1MeV

(4+n) 31

Therefore, to insure normal expansion rate during BBN, the b ound on T is

slightly stronger

6n9

(4+n)

n+2

T 10 : (81)

1TeV

Over-closure by gravitons

The constraints wehave discussed ab ovewould equally well apply to the

pro duction of purely purely 4-d particles with 1=TeV suppressed couplings

of the appropriate p ower. The pro duction of- gravitons is, however, qualita-

tively di erent since they escap e into the bulk, with a very low probability

of returning to interact with the SM elds on the wall. Consider the width

for a graviton propagating with energy E in the bulk, to decayinto two

photons on the wall. This interaction can only take place if the graviton is

within its Compton wavelength E from the wall. The probability that

this is the case in extra dimensions of volume r is

P (Er ) (82)

grav. near wall

On the other hand, when it is near close to the wall, it decays into photons

(n+2)=2

with a coupling suppressed by M , and therefore the width is

(4+n)

n+3

(83)

near wall

n+2

M )

(4+n)

The total width is

3 3

E E

=P (84)

n+2

grav. near wall near wall

r M

(4)

(4+n)

This simple result could have also b een understo o d directly from the KK

p oint of view: the coupling of any KK mo de is suppressed by1=M ), so

the width for any individual KK mo de to go into SM elds is suppressed

by1=M and the ab ove width follows from dimensional analysis. Of course

(4)

signi cant amounts of energy can b e lost to these KK mo des, despite their

weak coupling, for the usual reason of their enormous multiplicity. Among

other things, eqn.(84) implies that the gravitons can b e very long-lived, since

they can not decay in the empty bulk . This is b ecause, as long as the

momenta in the extra dimensions is conserved, the graviton (which is massless

from the (4+n) dimensional p oint of view) can not decayinto two other 32

massless particles. Of course, interaction with the wall breaks translational

invariance and allows momentum non-conservation in the extra dimensions,

but this requires that the decay take place on the wall. The lifetimeofa

graviton of energy E is then

100MeV

(4)

(E ) 10 yr : (85)

E E

The gravitons pro duced at temp eratures b eneath 100 MeV have life-

times of at least the present age of the universe. The ratio n =n whichwas

grav

constrained to b e 1 in the ab ove analysis must b e in fact much smaller in

order for the gravitons not to over-close the universe. As wehave mentioned,

most of the gravitons are \massive" with mass T from the 4-d p oint, they

dramatically over-close the universe if their abundance is comparable to the

photon abundance at early times.

The energy density stored in the gravitons pro duced at temp erature T

n+5

T M

T n (86)

grav grav

n+2

(4+n)

3 3

which then red-shifts mostly as R . The ratio =T is invariant. The

grav

3 9

critical density of the universe to day corresp onds to to ( =T ) 3 10

cr it

GeV. For the gravitons not to over-close the universe, we therefore require

for critical density at the present age of the universe. We therefore require

n+2

T M

3 10 GeV =T

grav

n+2

(4+n)

6n15

(4+n)

n+2

! T 10 (87) MeV

TeV

This is a serious constraint. For n =2,wehave to push M to the

(4+n)

astrophysically preferred 10 TeV, to even get T 1 MeV, although of

course in this case a much more careful analysis has to b e done. For n =6,

we need T 300 MeV.

Late decays to photons

Finally,we discuss the b ounds coming from the late decay of gravitons

into photons whichwould showuptoday as distortions of the di use photon

sp ectrum. For T 100 MeV, the graviton lifetime is longer than the age of

3 3

the universe by (100 MeV=T ) , but a fraction (T =100 MeV) of them

have already have already decayed, pro ducing photons of energy T . The

ux of these photons (i.e. the numb er passing through a given solid angle

d p er unit time) is then roughly

dF (T ) T

1 3

n H ( ) : (88)

0grav

d 100 MeV

This is to b e compared with the observational b ound on the di use back-

ground radiation at photon energy E , which can b e t approximately by

dF (E ) 1MeV

2 1 1

cm sr s (89)

d E

Using the previously derived expressions for the present n , this gives us

grav

a b ound on T ,

n+2

n+5

6n15

(4+n)

n+5

T 10 (90) MeV

TeV

Again, for n =2,even pushing M to 10 TeV pushes T up to only

(4+n)

< <

1 MeV. On the other hand, for n = 6 and M 1TeV, T 100 MeV

is safe.

Notice that the b ound from photons always demands a T whichislower

than that which critically closes the universe. Therefore, in this minimal sce-

nario, the KK gravitons can not account for the dark matter of the universe.

Of course this is not a problem, the dark matter can b e accounted for by

other states in the theory. Given the inevitability of graviton pro duction,

however, graviton dark matter would certainly b e attractive. There is a way

out of the b ound from decay to photons which can make this p ossible.

Fat-branes in the bulk

The problem arose b ecause we assumed that, once the graviton is emitted

into the extra dimensions, it must eventually return to our 4-d wall in order to

decay. Supp ose however that there was another brane in the bulk, of p erhaps

a di erent dimensionality. Since gravity couples to everything, it could in

particular couple to the matter on this new wall and lower the branching ratio

for decaying on our wall. In fact, if the new wall has more than three spatial

dimensions, the branching ratio to decayinto photons on our wall would b e

drastically reduced. This can b e seen in a number of ways. Supp ose that the

new wall has (3+p) spatial dimensions with p n. Note that since the extra

dimensions are compacti ed, the extra p spatial dimensions are not in nite 34

but have size r .We will call this new wall a fat-branes. Now, a graviton

propagating in the bulk with energy E r cannot resolve the di erence

p 1

between this new wall and stacks of (Er ) normal 3 d walls spaced E

apart. But then, the branching ratio for the graviton to decay on our wall

is reduced greatly by(Er ) . The width for gravitons to decay on the new

wall is

3 p+3

T T

(91) (Tr )

p+2

Pl(4+p)

where wehave used the relationship b etween Planck scales of di erent di-

mensionalities in the nal expression. This also gives another interpretation

of the result. From the viewp oint of a graviton of energy E r , the

fat-brane mayaswell b e in nite in all 3 + p dimensions. Therefore, just as

the width to decay on our wall is small b ecause the interaction of any single

graviton KK mo de is suppressed by1=M , so the width to decay on the

(4)

(p+2)=2

other wall is suppressed by M . The branching ratio is then bigger

(4+p)

b ecause the higher dimensional Planck scale relevant to the (3 + p)-brane is

smaller. The lifetime for the graviton to decay on the fat-brane can easily

be much smaller than the age of the universe. What is the fate of gravitons

which decay on the fat-brane?

Dark matter on the fat-brane

In order to understand the evolution of the universe after the decayof

gravitons on the fat-brane, it is imp ortant to understand the cosmology of

the fat-brane itself. There are two imp ortant p oints. First, just as for our

3-brane, at distances larger than r gravity on the fat-brane is normal and

four-dimensional. This is b ecause on scales larger than r , the \thickness" of

the fat-brane can not b e resolved. Second, the energy densities on all branes

contribute to the 4-d expansion rate of b oth our brane and the fat-brane.

Therefore, there is e ectively a single energy density and a common 4-d

expansion rate for the two branes. Consequently, the way that the expansion

rate is a ected after the gravitons are captured on the fat-brane dep end on

the nature of the decay pro ducts there. If they are non-relativistic, their

energy density red-shifts away like R and they may provide a dark matter

candidate. Notice that this dark matter may actually \shine" on its own

brane; it is only dark to us. This allows any mass range for the dark matter

candidates, since they can never into ordinary photons. 35

8 TeV Axion in the bulk and the strong CP

problem

As wehave remarked, the main reason our scenario remains phenomenolog-

ically viable is that the couplings to states that can propagate in the bulk

are suppressed. This observation can also b e used to revive the TeV axion

as a solution to the strong CP problem, if the axion is taken to b e a bulk

eld. Without sp ecifying the origin of the axion, the relevant terms in the

low-energy e ective theory are

Z Z

a(x; x =0)

4+n 2 4

L d x(@a) + d x F F: (92)

ef f

(n+2)=2

where a =4; ;3+n runs over the extra dimensions. Just as always,

QCD will generate a p otential for a(x; x = 0). In order to minimize energy,

a(x; x = 0) will prefer to sit at the minimum of this p otential, solving the

strong CP problem on the wall. Furthermore, in order to minimize kinetic

energy, a will take on this vev uniformly everywhere in the bulk. From the 4

dimensional p oint of view, we can expand a into KK excitations. After going

to canonical normalization, each of these has 1=M suppressed couplings

(4)

to F F for f M TeV. The p otential that is generated by QCD is

(4+n)

then minimized with the zero mo de acquiring the appropriate vev and all the

massive mo des having zero vev.

An explicit eld theoretic mo del pro ducing such an axion eld can b e

c c

easily constructed. Let u ;d and Q b e the weak doublet and singlet quark

elds resp ectively and H b e a electroweak Higgs doublet. In our theory

these states are the four-dimensional mo des on the 3-brane. Let b e a bulk

complex scalar eld whose spatially constantvev will break PQ symmetry.

h iM (93)

(4+n)

The 4 + n dimensional axion eld is de ned as

a (x )

+ KK- mo des (94) a = h iar g =

where wehave expanded into KK mo des. As already mentioned, the zero

mo de a is a genuine four-dimensional axion eld, with the 1=r insuring

4 36

that its 4-d kinetic term is canonically normalized. The coupling of with

matter on the 3-brane can b e written as

n+4 a c c

d x (x ) (HQu + H Qd ) (95)

(4+n)

It is straightforward to see that an e ective coupling of the genuine axion to

F F is

a a

4 4

~ ~

F F (96) F F

n=2

h ir

and thus from the p oint of view of the four-dimensional theory it is e ectively

a Planck-scale axion. While the bulk axion eld a has only 1/TeV suppressed

couplings, it is safe from all astrophysical constraints wehave considered for

the same reason gravitons are safe. Of course, 4-d axions with such high decay

constants ordinarily su er from the usual cosmological mo duli problem [15 ];

wehave nothing to add to the early cosmology which needs to drive the axion

to the origin. However,as long as the axion is at its origin at temp erature

T , it will not b e signi cantly excited during the subsequentevolution of the

universe, again for the same reason gravitons were not signi cantly excited.

9 Gauge Fields in the Bulk

Foravariety of reasons, it seems unlikely that SU (3) SU (2) U (1) is the

only gauge group under which the SM elds are charged. Normally, the non-

observation of additional gauge particles is attributed to a very high scale

of symmetry breaking TeV and comparably high masses for the gauge

b osons. The impact of these heavy gauge b osons on low energy physics is

then very limited. By contrast, in this section we will see that the situa-

tion can change dramatically in theories with large extra dimensions. This

can happ en if if the new gauge b osons can freely propagate in the bulk ,

while matter charged under the gauge group, including scalars whichmay

sp ontaneously break the symmetry, live on a 3-brane. The following features

emerge: indep endent of the numb er of extra dimensions, the gauge eld can

mediate a repulsive force more than a million times stronger than gravityat

distances smaller than a millimeter. This raises the exciting p ossibility that

The gravi-photons are mo del-indep endent examples of this sort. 37

these forces will b e discovered in the measurements of sub-mm gravitational

strength forces [16].

Consider for simplicitya U(1) gauge eld propagating in the bulk. The

free action is

4+n 2

L = d x F ;F = @ A @ A (97)

MN M N N M

4+n

where we take the scale of the dimensionful (4+n) dimensional gauge coupling

to b e the ultraviolet cuto M :

(4+N )

2 n

(98) g M

(4+n)

This gauge eld interacts with matter elds living on a 3-brane via the in-

duced covariant derivative on the brane

D =(@ +iq A (x; x = 0)): (99)

Expanding A in KK mo des, only the zero mo de A (x) transforms under

i (x)

wall eld gauge transformations ! e ; the rest of the KK mo des are

1 1

massive starting at r . At distances much larger than r , only the zero

n n

mo de is relevant, and the action b ecomes

4 0 0 2 0

S = d x (@ A @ A ) + L (; D ) (100)

matter

where the e ective 4-dimensional gauge coupling is

(4+n)

g : (101)

n 2

r M M

(4+n) (4)

The rst interesting p oint is that this is a miniscule gauge coupling g 10

for M 1TeV, indep endent of the numb er of extra dimensions n.

(4+n)

Supp ose this gauge eld couples to protons or neutrons. The ratio of the

repulsive force mediated by this gauge eld to the gravitational attraction is

g g

gauge

10 (102)

2 16

F G m 10

grav N

Clearly, the corresp onding gauge b oson can not remain massless. If the gauge

symmetry is broken by the vev of a eld on the wall, the gauge b oson will 38

get a very small mass, whichishowever exactly in the interesting range

exp erimentally:

! !

1TeV 10

1 1

m =(g q h i) 1mm (103)

g q h i

Of course there are a numb er of undetermined parameters so a hard pre-

diction is dicult Nevertheless, it is reasonable that g 10 isalower

6 8

b ound, and so we can exp ect repulsive forces b etween say10 10 times

gravitational strength at sub-mm distances. For all n>2, the mass of the

KK excitations of the gauge eld are to o large to give a signal at the distances

prob ed by the next generation sub-mm force exp eriments. The case n =2

has still richer p ossibilities since the KK excitations will have comparable

masses to the lowest mo de, and may contribute signi cantly to the measured

long-range force.

The most interesting p ossibility is to relate this new gauge eld with the

global Baryon (B ) or Lepton (L)numb er symmetries of the standard mo del.

The gauging of the anomaly-free B L symmetry has a de nite exp erimental

signal: since atoms are neutral, the B L charge of an atom is its neutron

numb er. Thus, the hydrogen atom will not feel this force, while it will b e

isotop e dep endent for other materials. Gauging other combinations of B

and L, e.g. either B or L separately,isvery interesting as well. Let us

consider the case of gauging baryon numb er. Of course wehavetoworry

ab out canceling anomalies; the most straightforward way out is to add chiral

fermions canceling the anomaly which b ecome massive when SU (2) U (1)

L Y

is broken. For instance, we can add three extra generations with opp osite

baryon numb ers (ignoring the obvious problem with the S -parameter). The

interest in this exercise is that it may provide a mechanism for suppressing

proton decay. Although baryon number must b e broken, dangerous proton

decay op erators may b e tremendously suppressed if the higgs that breaks B

lives on a di erent brane.

10 Cosmological Stability of Large Radii

Wehave said nothing ab out what xes the radii of the extra dimensions

at their large values, this is an outstanding problem. The largeness of the 39

extra b egs another, more dramatic question: why is our 4-dimensional uni-

verse so much larger still? It is not considered a failing of the SM that it

o ers no explanation of why the universe is so much larger than the Planck

scale. Indeed, this is equivalent to the cosmological constant problem. If the

density of the universe at the Planck time was O (1) in Planck units, there

would b e no other time scale than the Planck time and the universe would

not growtobe10 years old. This was only p ossible b ecause the energy

density is so miniscule compared to M , which is precisely the cosmological

constant problem. It may b e that once the cosmological constant problem

is understo o d, whatever makes the enormity of our 4-d universe natural can

also explain the (much milder) largeness of the extra n dimensions.

In this context wewould like to make the side remark that the usual

cosmological constant problem is in some sense less severe in our frame-

work. Supp ose for instance that there is a string theory with string tension

m TeV, but where the SUSY is primordially broken only on our wall at the

scale m .Aswe argued in [4], the SUSY breaking mass splittings induced

for bulk mo des is then highly suppressed (1 mm) at most. However,

there is nothing that can b e done ab out the (TeV) vacuum energy on the

wall, and wehave to imagine canceling it by ne-tuning it away against a

bare cosmological constant

Z Z

p p

4+n 4 n

d x g ! d x g (r ): (104)

0(4+n) 0(4+n)

1=(4+n)

Since the radii r are large compared to (TeV) , the mass scale

0(4+n)

do es not have to b e as large as the TeV scale to cancel the cosmological

constant. Note that since the SUSY splittings in the bulk are so small, there

4+n

is no worry of an (TeV) cosmological constant b eing generated.

We do not, however, have to hide b ehind our ignorance ab out the cosmo-

logical constant problem. It may b e that the radii are large for more mundane

reasons: for some reason, some of the radius mo duli have a p otential energy

with a minimum at very large values of rm .Even without knowing any-

grav

thing ab out the origin on such a p otential, we can place phenomenological

constraints on V (r )by requiring that the eld was not signi cantly p erturb ed

from its minimum byinteracting with the hot universe from the time of BBN

to the present. Since the mo dulus is a bulk eld, V (r ) should b e a bulk en-

ergy density, and U (r )=V(r)r should have a minimum at large r . Supp ose

that at temp erature T , the mo dulus was already stabilised at its minimum

r .How signi cantly is it p erturb ed as the wall elds dump energy into the

extra dimensions? We estimate this by rst computing the total amount

of energy dump ed into the extra dimensions. Any bulk eld must haveat

(n+2)=2

least a 1=M suppression for its coupling, and so the maximum rate for

(4+n)

dumping energy into the extra dimensions, p er unit time is

n+7

E V (105)

w all 3

n+2

(4+n)

where V is the three-volume of the region of the wall losing the energy.At

worst, this energy gets entirely transfered to changing the p otential of the

radius mo dulus,

_ _ _

E = E = U (r )V (106)

w all r ad: 3

and so the change in U (r )over a Hubble time is

n+5

T M

U (r) (107)

n+2

(4+n)

2 00

Translating this change as U (r) (r) U (r )=2, we obtain a b ound on

00 1

U (r ) from the requirement that r=r 10 :

n+5

T M

00 2

U (r )r (108)

n+4

Any theory where this inequality can not b e satis ed for T 1 MeV is ruled

out by cosmology during and after BBN.

Of course we do not have a theory predicting a U (r ) which naturally

generates a large radius. Nevertheless, we can sp eculate on what sort of

U (r ) can pro duce minima at large value r . In analogy with dimensional

transmutation, a large hierarchy can b e generated if logr is determined to b e

,say,O(10). We can in any case parametrize U (r ) so that

U (r )=g(r;m )f (log (rm )) (109)

grav grav

where f (x) is a dimensionless function, and g (r;m ) has dimensions mass .

grav

One natural assumption on the form of g is that the fully decompacti ed

theory rm !1should b e a minimum of the p otential; certainly in

grav

string theory, there is a vacuum as the string coupling go es to zero with all 41

ten dimensions large. In that case, it must b e that g (r;m ) ! 0 at least

grav

as fast as a p ower lawas r!1.We will also consider the case where g (r )

is essentially at in analogy with the \geometric hierarchy" p otential. The

question is now whether the theory can develop a minimum, not for in nitely

large radius but for nite but large values of rm . Since we are interested

grav

in the limit of a large rm anyway,we can approximate g (r ) at large r

grav

with its leading p ower law b ehavior

grav

g (r;m ) ! c : (110)

grav

Requiring U (r ) to b e stationary then gives

grav

(f af )= 0 (111)

a+1

so there is a minimum at a value x =log(r m ) where

grav

f (x )

= a: (112)

f (x )

Note that the condition for the existence of a lo cal minimum at large r is

completely determined by f . It is certainly not implausible that the there

are dimensionless ratios of O (10) in f , leading to a value of x also of O (10),

leading to a very large (but not in nite) radius.

Consider rst the \geometric hierarchy" scenario where a =0. In this

case,

4 00

m f (x )

grav

U (r )= ; (113)

and the b ound from eqn.(108) translates to

n+6

n+5

1 3n

(4+n)

n+5 n+5

GeV f (114) T 10

TeV

We do not exp ect f (x ) to b e larger than O (10), and even if it is larger,

it is raised to small fractional p ower. For n = 2 and M 10 TeV, this

(6)

requires T 100 GeV, certainly the weakest of all cosmological b ounds we

have considered. For all n>2, the b ound is easily met with T 10 GeV

and M 1TeV.

(4+n) 42

The cases of intermediate \hardness", 4 >a > 0 are also less constraining

00 2

than the other cosmological b ounds for obvious reasons: U (r )r is enhanced

by a p ositivepower of m , so it cost signi cant energy to excite uctuations

grav

in the radius.

Finally, consider the \soft" case of n = 4. Here, the scale of the p otential

is determined by its very size, making it \soft" for large radii. Here,

f (x ) 16f (x )

00 2

U (r )r = (115)

and T is b ounded as

6+n+8=n

n+5

3n120=n

) (

n+5

GeV T 10 (116)

TeV

Clearly for n = 2, and even for M 10 TeV, T 100 eV,and so the radius

(6)

could no have settled by the time of BBN, where T 1 MeV.The case n =3

is marginal T 1 MeV for M 1TeV, but is ne already for M 10

(7) 7

TeV. For n 4, however, even this \soft" scenario can b e accomadated with

T 10 MeV and M 1TeV.

(4+n)

Other examples of situations where a large vev for a eld can b e generated

while the excitations ab out the minimum have a mass uncorrelated with the

vev can b e constructed. In app endix 2, we present a sup ersymmetrictoy

example of this typ e.

11 The large n limit

Finally,we wish to commentonaninteresting limit of our framework, where

the numb er of new dimensions b ecomes very large. This case may b e ex-

cluded by theoretical prejudices ab out string theory b eing the true theory

of gravity, which seems to limit n 6 or 7, but we will ignore this prej-

udice here. The n !1limit is interesting for many reasons. The main

p oint is that in this limit, the size of the new dimensions do es not haveto

be much larger than M , solving the remaining \hierarchy" problem in

(4+n)

our framework. For instance, for n = 100 new dimensions, the correct M

(4)

can b e repro duced with M 2TeV and extra dimensional radii (1

Pl(4+n)

TeV ) in size. Since the extra dimensions are now in the TeV range, no 43

sp ecial mechanism is required to con ne SM elds to a wall in the extra

dimensions. Furthermore, all the KK excitations of the graviton are at 1

TeV and are therefore irrelevanttolow-energy physics. All the low-energy

lab and astrophysical constraints involving emission of gravitons into the ex-

tra dimensions are gone. Indeed, the theory b eneath a TeV is literally the

SM, and all that is required is that dangerous higher-dimension op erators

suppressed bya TeV are forbidden, a feature required even for small n. The

cosmology of this framework is also completely normal at temp eratures 1

TeV, with no worries ab out losing energy by emitting gravitons into the extra

dimensions. Apart from the usual strong-gravitational signals at colliders, in

this limit the KK excitations of SM elds for each of the n new dimensions

may also b e observed.

12 Discussion and Outlo ok

Over the last twentyyears, the hierarchy problem has b een one of the central

motivations for constructing extensions of the SM, with either new strong dy-

namics or sup ersymmetry stabilizing the weak scale. By contrast, in [1] we

prop osed that the problem simply do es not exist if the fundamental short-

distance cuto of the theory, where gravity b ecomes comparable in strength

to the gauge interactions, is near the weak scale. This led immediately to the

requirement of new sub-mm dimensions and SM elds lo calised on a brane

in the higher-dimensional space. Unlike the other solutions to the hierarchy

problem, our scenario do es not require any sp ecial dynamics to stabilised the

weak scale. On the other hand, it leads to one of the most exciting p ossi-

bilities for new accessible physics, since in this scenario the structure of the

quantum gravity can b e exp erimentally prob ed in the near future. Given the

amount of new physics broughtdown to p erhaps dangerously accessible ener-

gies, it is crucial to check that this framework is not already exp erimentally

excluded.

In this pap er, wehave systematically studied exp erimental constraints

on our framework from phenomenology, astrophysics and cosmology. Be-

cause of the p ower law decoupling of higher-dimension op erators, there are

no signi cant b ounds from \comp ositeness" or precision observables, which

in any case do not tightly constrain generic new weak scale physics. Rather,

the most dangerous pro cesses involve the pro duction of unavoidable new 44

massless particles in our framework{the higher dimensional gravitons{whose

couplings are only suppressed by1=TeV. Analogous light 4-dimensional parti-

cles with 1=TeV suppressed couplings, such as axions or familons, are grossly

excluded. Nevertheless, we nd that for all n>2, the extreme infrared soft-

ness of higher dimension gravity allows the (4 + n) dimensional Planck scale

M to b e as lowas 1TeV. The exp erimental limits are not trivially

(4+n)

satis ed, however, and for n = 2, energy loss from SN1987A and distortions

of the di use photon background by the late decay of cosmologically pro-

duced gravitons force the 6 dimensional Planck scale to ab ove 30 TeV.

For precisely n =2,however, there can b e an O (10) hierarchybetween the

6-dimensional Planck scale and the true cuto of the low-energy theory,as

was discussed in section 3 for the particular implementation of our scenario

within typ e I string theory. A natural solution to the hierarchy problem to-

gether with new physics at the accessible energies can still b e accomadated

even for n =2.

Of course it is p ossible that wehaveoverlo oked some imp ortant e ects

which exclude our framework. Nevertheless, these theories haveevaded the

quite strong exp erimental limits wehave considered in a quite general way.

The strongest b ounds were evaded since higher-dimensional theories are soft

in the infrared. Alternately,as n grows, r decreases and the number of

available KK excitations b eneath a given energy E also decreases. In fact,

as n !1,r ! TeV, all KK excitations are at a TeV, and e ectivelow

energy theory is simply the SM with no additional light states. This shows

that this sort of new physics can not b e excluded simply b ecause it is exotic,

the theory b ecomes safest in the limit of in nitely extra dimensions.

On the theoretical front, p erhaps the most imp ortant issues to address

are the mechanism for generating large radii, and early universe cosmology.

Other issues include proton stability, SUSY breaking in the string theory

context, and gauge coupling uni cation. The latter is the one piece of in-

direct evidence that suggests the existence of a fundamental energy scale

far ab ove the weak scale [17]. It has b een p ointed out that the existence

of intermediate scale dimensions larger than the weak scale, into which the

SM elds can propagate, can sp eed up gauge coupling uni cation due to the

power law running of gauge couplings in higher dimensional theories [18]. In

[4], the prop osal of [1] was combined with the mechanism of [18 ] in a string

context, therebyachieving b oth gravity and gauge uni cation near the TeV

scale. A similar prop osal was later made in [19 ]. Alternately, [6] suggested 45

that di erent gauge couplings may arise from di erent branes, leading to a

p ossibly di erent picture for gauge coupling uni cation.

Exp erimentally, this framework can b e tested at the LHC, and on a

shorter scale, can b e prob ed in exp eriments measuring sub-mm gravitational

strength forces. If the scale of quantum gravity is close to a TeV as motivated

by the hierarchy problem, at least twotyp es of signatures will b e seen at the

LHC [1, 4 ]. The rst involve the unsuppressed emission of gravitons into the

higher dimensional bulk, leading to missing energy signatures. The second

involve the pro duction of new states of the quantum gravitational theory,

such as Regge-recurrences for every SM particle in the string implementa-

tion of our framework. Clearly the detailed characteristics of these signatures

must b e studied in greater detail.

There is also the exciting p ossibility that the up coming sub-mm measure-

ments of gravity will uncover asp ects of our scenario. There are at least three

typ es of e ects that may b e observed.

2 4

Transition from 1=r ! 1=r Newtonian gravity for n = 2 extra dimen-

sions. In view of our astrophysical and cosmological considerations, which

push the 6-dimensional Planck mass to 30 TeV, the observation of this

transition will b e esp ecially challenging.

On the other hand, particles with sub-mm Compton wavelengths can

naturally arise in our scenario, for instance due to the breaking of SUSY on

our 3-brane [4]. These will mediate gravitational strength Yukawa forces.

A new p ossibility p ointed out in this pap er is that gauge elds living in

the bulk and coupling to a linear combination of Baryon and Lepton number

6 8

can mediate repulsive forces which are 10 10 times gravity at sub-mm

distances.

Acknowledgments: Wewould like to thank I. Antoniadis, M.Dine,

L. Dixon, G. Farrar, L.J. Hall, Z. Kakushadze, T. Moroi, H. Murayama, A.

Pomarol, M. Peskin, M. Porratti, L. Randall, R. Rattazzi, M. Shifman, E.

Silverstein, L. Susskind, M. Suzuki, S.Thomas and R. Wagoner and for useful

discussions and comments. NAH and SD would like to thank the ICTP high

energy group, and GD would like to thank the ITP at Stanford, for their

hospitality during various phases of this pro ject. NAH is supp orted by the

Department of Energy under contract DE-AC03-76SF0051 5. SD is supp orted

by NSF grant PHY-9219345-004 . 46

App endix 1

In this app endix we consider the Higgs e ect for the breaking of \gauged"

translation invariance, that is sp ontaneous translation invariance breaking

in the presence of gravity. In the usual KK picture (some of the) g ( =

0; ;3, a =4; ;3+n comp onents of the higher-dimensional metric are

viewed as massless gauge elds in 4 dimensions, with the gaugess ymmetry

b eing translations in the extra n dimensions. The result we nd is simple

and easy to as the exact analogue of the result we found for the small mass

of bulk gauge elds when the gauge symmetry is broken on the wall. The

a a

zero mo des of the g eat the y goldstone b osons to get a mass

m (117)

(4)

where f is the wall tension. Intuitively, the large radius means that the

zero mo de of the \gauge eld" g has a very small \gauge coupling"

1=M . The \vev" which breaks translation invariance is nothing but the

(4)

lo calised energy density of the wall f , and the ab ove formula follows. For

completeness, however, we will consider this e ect in somewhat more detail.

We will refer to the `wall' as to a lo calized, stable con guration indep endent

of the co ordinate (x ), that minimizes the action. One can imagine the wall

as some sort of top ological defect in higher dimensions.

First turn o gravity and let (x ) b e the vev of the real scalar eld

forming the wall. Consider the action S for the eld con guration (x +

a a

y (x)). Translation invariance in x demands that

a a 4+n a a a

S [(x + y (x))] = d xf (x )@ y @ y + (118)

where no linear term is present since S is stationary at , and f (x ) is some

function lo calised around the p osition of the wall x =0.At distances large

a 4 a

compared to the \thickness" of the wall, we can approximate f (x )=f (x )

where f has units of mass, and

a a 4+n a 4 a a

S [(x + y (x))] ! d x (x )f @ y @ y : (119)

As exp ected, the y are massless dynamical degrees of freedom living on

the wall, the Nambu-Goldstone b osons of sp ontaneously broken translation 47

a a

invariance. Global translations in x are realized non-linearly on the y via

a a 4

y (x) ! y (x)+ c. The quantity f can b e interpreted as the tension of the

wall.

Now turn on gravity, sp eci cally the g \gauge" elds which gauge lo cal

translations in x :

a a a a

y (x) ! y (x)+c(x);g ! g + @ c: (120)

As usual, we can go to a unitary gauge where y (x) are everywhere set to

zero. In this gauge, the g obtain a p osition dep endent mass term

4+n a 4 a 2

L = d x (x )f (g ) : (121)

mass

That the mass term should b e p osition dep endentisintuitively obvious. Far

from the wall, no lo cal observer knows that translation invariance has b een

sp ontaneously broken; the graviton masses should therefore vanish away from

the wall.

a a

Let us expand g in canonically normalized KK mo des h , recalling that

each individual KK mo de will come suppressed by1=M . The KK mo des

(4)

have already have a mass (n=r ) , and the p osition dep endent mass term

from symmetry breaking b ecomes

4 a 2

L = d x ( h ) : (122)

break 2

(4)

As long as f =M is smaller than 1=r , the masses of the heavy KK excita-

(4)

tions are not signi cantly p erturb ed by the breaking term. The zero mo de

do es not haveany mass in the absence of symmetry breaking, however, so it

gets a mass

m = : (123)

(4)

Note that, for f TeV, this mass is (mm) , and at least for n>2 the

assumption than the mass is much smaller than r is justi ed. For n =2,

the rst few KK mo des can not b e completely decoupled, and some linear

combination of them eat the y .Wehowever still exp ect the lightest graviton

mo de to have mass (mm) in this case as well. 48

App endix 2

As discussed in the text, a vague worry ab out having very large dimensions

comes from the impression that the p otentials resp onsible fro stabilizing the

radius mo dulus will b e very \soft", and therefore the mo dulus will b e very

light, p ossibly giving cosmological problems. In this example we presentan

explicit counter-example to this intuition, alb eit in a toy mo del. We will

write down a theory where (a) eld S is a at direction to all orders in

p erturbation theory, b) a p otential for S is generated by non-p erturbative

e ects leading to distinct minima very far separated from each other, while

uncorrelated with the sizes of hS i. The mo del is sup ersymmetric and the

these features will b e generated without any ne-tuning.

Consider rst, an SU (N ) QCD with N avors Q; Q and a singlet eld S ,

coupled with a tree-level sup erp otential

W = S T r (QQ) (124)

tr ee

This mo del has b een discussed many times and has found a variety of appli-

cations. At the classical level and to all orders in p erturbation theory, S 6=0

and Q; Q = 0 is a at direction. For S , Q; Q can b e integrated out, and

gaugino condensation in the low energy theory gives

W (S )= S (125)

ef f

This is of course the only sup erp otential consistent with all the symmetries,

and gives rise (at lowest order) to an exactly at p otential V (S )=j j .

Of course, the p otential is mo di ed by corrections to the Kahler p otential of

S , and most generally

2 2

j j

V (S )= (126)

Z(S)

where Z is the wavefunction renormalisation of S .For S , the p otential

remains approximately at since the corrections to Z (S ) are p erturbative,

however for S , this description breaks down. We are guaranteed, how-

ever, that there is a supersymmetric minimum at S = 0. The exact sup erp o-

tential including the quantum mo di ed constraint (see [20] for a review) for

this case is

): (127) W = S T r (M )+X(detM B B 49

This sup erp otential admits sup ersymmetric vacuum with hS i = 0, while the

curvature of the e ective p otential for S is .

We can nd variations on this mo del with copies of the gauge group

and matter to pro duce multiple minima for S . Consider e.g. the group

0 0

SU (N ) SU (N ) with resp ectively N; N avors, and still a single singlet S ,

and consider the tree sup erp otential

0 0

W = S T r (QQ)+(S m)Tr(Q Q ) (128)

where m is some arbitrary dimensionful scale. It is easy to see that, for

S m, the p otential lo oks like what we discussed previously, with a SUSY

minimum around S = 0 with curvature , while there is also a SUSY

minimum at S = m with curvatures , with a at p otential separating the

minima. In this example, the (classically at) eld S can obtain an arbitrary

vev, completely uncorrelated with the curvatures of the p otential around the

minima.

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