families as neighbors in extra dimension

2 March 2000

Ž .Physics Letters B 475 2000 295–302

Families as neighbors in extra dimension

G. Dvali a, M. Shifman b

a Department of Physics, New York UniÕersity, New York, NY 10003, USAb Theoretical Physics Institute, UniÕersity of Minnesota, Minneapolis, MN 55455, USA

Received 10 January 2000; accepted 17 January 2000Editor: M. Cvetic

Abstract

We propose a new mechanism for explanation of the fermion hierarchy without introducing any family symmetries.Instead, we postulate that different generations live on different branes embedded in a relatively large extra dimension,where gauge fields can propagate. The electroweak symmetry is broken on a separate brane, which is a source ofexponentially decaying Higgs profile in the bulk. The resulting fermion masses and mixings are determined by anexponentially suppressed overlap of the fermion and Higgs wave functions and are automatically hierarchical even if allcopies are identical and there is no hierarchy of distances. In this framework the well known pattern of the ‘‘nearestneighbor mixing’’ is predicted due to the fact that the families are literally neighbors in the extra space. This picture mayalso provide a new way of a hierarchically weak supersymmetry breaking, provided that the combination of three familybranes is a non-BPS configuration, although each of them, individually taken, is. This results in exponentially weaksupersymmetry breaking. We also discuss the issue of embedding identical branes in the compact spaces and localization ofthe fermionic zero modes. q 2000 Elsevier Science B.V. All rights reserved.

1. Introduction

Large extra dimensions may help in understand-ing the hierarchy between the Planck and weak

w xscales 1 . In the present paper we will concentrateon the hierarchy of the fermion masses which is

Ž .another mystery in the standard model SM . OneŽ .possible approach relies on spontaneously broken

flavor symmetries. But this does not really answerthe question, rather brings it at a different level.Instead of explaining the hierarchy of the Yukawacouplings, now one has to explain the hierarchy ofthe breaking scales.

In the present paper we will adopt a differentattitude. We assume that three SM families are iden-tical, the difference in their masses is simply becausethey happen to live in different places in the extra

space. More precisely, we assume that the originalhigher dimensional theory admits, as its solution, abrane with localized fermions with quantum numbersof one SM generation. Multiple brane states will thengenerate n identical copies of fermions, n genera-tions of the standard model. Due to obvious reasonswe will take ns3 in our discussion. It is clear that ifthe quarks and leptons are to come from differentbranes, then the gauge fields must freely propagatein the interbrane space. The transverse volume cov-ered by gauge fields may be a world-volume of a‘‘fatter’’ brane, or simply a compactified dimension.In any case there is an upper bound on a linear scale

Ž .associated with this volume, L;1r 1 TeV .The crucial question in this picture is where does

the electroweak symmetry breaking happens? WeŽ .postulate that the vacuum expectation value VEV

0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 00 00083-6

( )G. DÕali, M. ShifmanrPhysics Letters B 475 2000 295–302296

of the Higgs field is induced due to the presence of aseparate brane. The latter acts as a source for theHiggs VEV in the perpendicular direction, so that theHiggs VEV decays exponentially away from thesource,

H;eyr r r0 , 1Ž .where r is the distance from the source brane.

Thus, there is a nonzero Higgs profile in the bulk,and this will generate masses of the SM fermionslocalized on other branes. In this way the mass of theSM fermions will be determined by the overlap of its

Ž .wave function squared with the Higgs profile. Thiscan be of order one for the nearest brane, but expo-nentially suppressed for more distant neighbors.

Note that there is no need to postulate a hierarchyof distances between immediate neighbors. In anycase, the hierarchy of the fermion masses is guaran-teed. Note also that there is an inevitable correlationbetween the masses and mixings. The mixing be-tween the fermions is suppressed by the overlap of

Ž .two distinct wave functions with the Higgs profile.As a result the nearest neighbors will mix strongerthan the next-to-nearest, and so on. This pattern iswell-known experimentally.

Other input assumptions are more or less standardŽ .for the approach with the large compact extra

dimensions. Yet it is worth discussing them in brief.We will consider one extra dimension, so that the

space has the topology of M =S. The size of the4

extra dimension L is assumed to be much larger thanMy1 and the brane width d . Gravity is weak at thesePl

distances, and plays essentially no role providedŽ .there are other passive extra dimensions in which

our construction is embedded. 1 A microscopicPlanckean theory descends to distances L in theform of some field theory and the given geometry ofspace-time. This field theory is responsible for thebuild-up of the branes, with the zero modes as

Ž .discussed above. The field s that ‘‘build’’ the wallsare distinct from the matter and Higgs fields, theyhave to be introduced for the wall-building purpose.

1 The graviphoton is eliminated by the overall zero modeassociated with the breaking of the translational invariance in thefifth direction.

Presumably, the characteristic size of our extradimension, should be somewhat smaller than inverseTeV, due to reasons associated with the flavor viola-tion. In the theories with a low fundamental scaleŽ .M , there is a potential danger of higher-dimen-Pf

sional operators that lead to a flavor violation in thelow-energy processes through the higher dimensionaloperators suppressed by powers of M . These canPf

be, in principle, controlled by gauging non-Abelianw xflavor symmetries in the bulk 2 . Flavor-violating

exchange by the bulk flavor gauge fields or by thew xscalar flavons can be adequately suppressed 2 . In

our framework, however, there can be an additionalsource of flavor violation due to the exchange of the

w x 2ordinary gauge fields 3 . This exchange is sup-pressed by the size of extra dimension versus thelocalization width of the fermions, and may requirethe size of extra dimension to be below

Ž .1r 1000 TeV . In our discussion, we will assumethis bound to be satisfied, and keep the size as a freeparameter. Of course, making L small implies in-creasing the cutoff of the theory, and at some point itwill reintroduce the hierarchy problem. Therefore,

Žthe issue of the low-energy supersymmetry broken.at the scale <M may become important in ourPf

framework.In this respect it is interesting that localization of

families on the identical branes may automaticallylead to a novel mechanism of the exponentially weaksupersymmetry breaking, provided that each individ-ual family brane is a BPS state, whereas their combi-

Žnation is not. Supersymmetry breaking on non-BPSw x .branes was suggested in 5 . In Section 6 we will

formulate a general sufficient condition for such abreaking.

Before proceeding to a more detailed considera-tion let us summarize crucial differences of ourscenario from the existing alternative high-dimen-sional mechanisms of the fermion mass generation.

w xIn 6 the hierarchy of the fermion masses wasgenerated by invoking global flavor symmetries bro-ken on a set of distant branes. Each of the distant

2 Ž .This is in contrast to other unbroken global symmetries ofŽ .the standard model such as the baryon number , which, in

principle, can be protected by separating quarks from leptons inw xthe extra space 4 .

( )G. DÕali, M. ShifmanrPhysics Letters B 475 2000 295–302 297

branes was responsible for the breaking of a particu-lar subgroup of the full flavor group. This breaking

Ž .then was communicated ‘‘shined’’ to the standardmodel fermions via a set of bulk messenger fields, insome representation of the flavor group. Althoughthe SM fields were localized on the same brane, theresulting pattern of flavor symmetry breaking canstill be hierarchical if the branes responsible fordifferent breakings are located at different distances.

Note that in this picture it is essential to have aflavor symmetry, as well as a set of the flavor-break-ing branes with a variety of sets of the VEV’s, plus asector of the bulk messenger fields charged under theflavor group. In our scenario there is no need topostulate any flavor symmetry at all. No messengerfields are needed. The only Higgs field that acquiresan expectation value is the standard model elec-troweak Higgs. It is impossible to avoid the hierar-chical pattern of fermion masses, except for theunlikely case when all the three branes are stabilizedright on top of each other. As will be discussedbelow, such stabilization is very difficult to achievein practice, unless an unnatural distinction among thefamily branes is introduced.

2. Fermion hierarchies from extra dimensions

In this section we will discuss some model-inde-pendent features of the fermion masses in our frame-work and show why a hierarchical pattern is in-evitable. Although this will not be crucial for ourpurposes, for simplicity and economy, we assumethat all the standard model fermions are generatedfrom a single progenitor family in the original five-dimensional theory. The corresponding Yukawa cou-plings in the five-dimensional action are

5Ss d xg Hff , 2Ž .H f c

where H is the five-dimensional Higgs field and fand f are the five-dimensional fermions which givec

Ž . Ž .rise to the four-dimensional chiral SU 2 mU 1 -doublet and singlet fermions, respectively.

Now, the fact that the SM fermions are localizedzero modes on the branes, means that the five-di-

mensional fermionic fields allow for the expansionof the form

fs V yyy f x q . . . , 3Ž . Ž . Ž .Ý i i i m

i

Ž .where y is the fifth coordinate and V yyy arei i

localized functions at y , with an exponentially de-iŽ < <caying profile. It is assumed that y yy <L. Ati j

< < Ž .y y y ; Lr2 the exponential decay regimei.changes, see below.

The functions f are zero modes of the four-di-i

mensional Dirac operator. In this way the expansionŽ .3 describes the zero-mode fermions of three stan-dard model generations localized at the hyperspacesysy in the bulk.i

However, because of the finite distance betweenthe branes these states are not completely orthogonal– there is a nonzero overlap between the wavefunctions. This amounts to a nonzero but smallmixing between the fermions. What is interesting,there is a nontrivial correlation between the fermionmixing and their masses. This is clear from the waythey are generated. The source of the masses is theexpectation value of the Higgs field, which we as-sume is induced on a separated ‘‘source’’ brane,located at some point ys0. The Higgs VEV ismaximal at the brane and decays exponentially in thebulk. Outside the Higgs VEV-generating brane, i.e.

< <at y )d

LLr2H y sÕ 2e cosh a yy , 4Ž . Ž . Ž .ž /2

where a is the mass scale defining the inverse brane‘‘thickness’’ and Õ is the Higgs VEV on the brane.Assuming that all three family branes sit in the

Ž .domain of the exponential decay of H y , thefermion masses will be determined by the overlap ofthe fermion wave functions with the exponentialHiggs tail in the extra dimension, and, thus, by thedistance from the source brane. Consider a situationwhen the family branes are located on the same side

Žfrom the source brane i.e. the Higgs VEV-gener-.ating brane . Then, even if they are placed in equal

intervals, the hierarchical pattern of masses and mix-ings is guaranteed.

For illustrative purposes we can approximatefermionic wave-functions by exponential profiles

< <V sexp y yyy b , 5Ž .Ž .i i


where as in the case of the Higgs field, b is a massscale that sets the ‘‘thickness’’ of the fermionic

Ž .profile s , and we assume that all fermionic branesŽare identical we neglect a small distortion of the

.wave functions because of the non-zero overlap .Then, the fermion masses are given by the followingoverlap integrals:

a< < < < < <m s dy Õexp yb y q yyy q yyy .Hi j i j½ 5ž /b

6Ž .

Note that, equivalently, we could have obtainedthe same result by going to the effective low-energy

Ž .picture. The existence of the Higgs profile 4 meansthat there is a four-dimensional Higgs state localizedon the brane. This mode corresponds to vibrations ofthe condensate and, therefore, is localized on thesource brane. Thus, integrating out the extra dimen-sion we will be left with a SM-like pattern, with thehierarchically suppressed Yukawa interactions.

3. Higgs profiles

In this and the next sections we will considersome technical details such as the generation of theHiggs condensate on a brane as well as the embed-ding of the multiple fermionic branes in the compactdimension. The generation of the Higgs condensateson the brane is a frequent phenomenon, wheneverthere is a nontrivial coupling of the bulk scalars withthe brane. For instance, we will consider a simple

w xexample of the domain wall studied in 7 . In thisexample there is a domain wall created by a realscalar field x , and another field H charged underthe gauge group G. In our case this will be assumed

Ž . Ž .to be electroweak SU 2 mU 1 . It is essential thatthe Lagrangian contains interaction among these

Ž .fields. In the simplest form the scalar sector of thefive-dimensional action can be taken as

22 25 2 2< < < <Ss d x D H q E x y a x ymŽ .H žm m

2 42 2 < < < <q bx ym H q H . 7Ž .Ž . /For bm2 )m2 the system has two ground states,

Ž .with xs"m and Hs0 for both . Due to simpleŽ .topological arguments there is a wall brane interpo-

lating between the two. For Hs0 the wall profilehas a simple form

'xsmtanh ym 4a . 8Ž .Ž .Since x goes through zero in the middle of the wall,H can become unstable and condense on the brane.This can be simply seen by examining small pertur-bations HsHe i v t in the wall background. The lin-earized Schrodinger equation takes the form¨

2 2 2 2 2'E Hy bm tanh ym 4a ym Hsv H , 9Ž .Ž .Ž .y

Žwhich clearly has an unstable eigenmode with imag-.inary v for some range of parameters.

Thus, the SM symmetry is spontaneously brokenon the brane but is restored in the bulk in the infinitevolume limit. For the finite extra dimension, thegauge fields will get nonzero masses by interactingwith the brane condensate, and the gauge symmetrywill be spontaneously broken in the effective low-en-ergy theory.

Finally, let us remark on a technical point regard-Žing the embedding of a brane or several branes, as

.opposed to the antibrane in the compact extra di-mension. This issue was discussed in great detail inw x8 . The wall one may have on the cylinder is slightlydifferent from the standard kink in the non-compactspace, which interpolates between distinct vacua ofthe theory. To have walls on the cylinder one mustassume that the fields of which the walls are built aredefined on manifolds with noncontractible cycles.For instance assume that x is an angular variableŽ .an ‘‘axionic’’ type field , defined modulo 2p . Theappropriate interaction potential is

4 2 < < 2 < < 4Vs2 M cos xq bsin xym H q H . 10Ž .Ž .The resulting brane is a sine-Gordon soliton

xs4tany1exp ym , 11Ž . Ž .and can be embedded in the compact space. Since x

changes by 2p through the soliton, sin x becomeszero and destabilizes H, much in the same way asfor the kink.

w xMoreover, as it was shown in 8 , the isolatedwall on the cylinder may be BPS saturated, i.e.leading to supersymmetric low-energy theory of thezero modes. But we have several branes: three‘‘fermion’’ and one Higgs VEV-generating. It isnatural to assume that, being considered in isolation,


each wall is BPS saturated. Taken all together theyneed not necessarily be BPS. Hence, one can get asupersymmetry breaking exponentially small in the

< <parameter y yy rd where d is the wall ‘‘thick-i j

ness.’’ In this way, one gets an exponential suppres-sion of the SUSY breaking scale without any inputhierarchy.

To reiterate, had we just one generation, super-symmetry will be unbroken. It is the intergenera-

Ž .tional interbrane interference that makes the wallconfiguration non-BPS, and breaks SUSY.

This effect is independent of the other argumentspresented above regarding the fermion mass hierar-chy. The original masslessness of the fermions is notdue to SUSY, but, rather, due to the topological

Žproperty of the brane or due to a mechanism to be.discussed in Section 4 . Note that the matter fields

need not be chiral at distances <L, when ourintermediate field theory flows to a fundamental one.The chirality of the trapped zero modes occurs as aresult of the winding of the solution under considera-tion in the extra dimension. In this picture it isnatural that the matter fermions are lighter than thesfermions.

4. Multiple branes in compact spaces

The fermionic fields must be localized on a num-ber of identical stable branes, admitting the fermioniczero modes. Stability of such branes, in general, isdue to some charge Q. This may be either a topolog-

Ž .ical charge e.g. in the case of the kink or soliton orŽa charge with respect to some higher forms e.g. as

.in the case of D branes . The corresponding fluxthen guarantees the stability of the brane. The sameflux conservation then often forbids the embeddingof the branes in the compact space, since the flux canend nowhere. One is then forced to introduce an-tibranes on which the flux lines may end, to balancethe total charge. In our scenario we would like toavoid antibranes.

One possible way out then is to consider topologi-cal charges compatible with the compact boundary

Ž w x.conditions, as above see 8 . For instance, we canput arbitrary number of solitonic branes of the formŽ .11 in the compact space.

Here we will consider an alternative way fordealing with this issue in the cases when the fluxconservation is incompatible with periodicity of thespace. Let such charge be Q. Then, instead of intro-ducing an antibrane with the charge yQ, we canassume that the charge in question is Higgsed. Thenthe flux lines will be absorbed by the ‘‘medium,’’much in the same way as the conductor absorbs theelectric flux.

Let us consider the issue of the stability of such asystem. As a prototype toy model consider a branewhich is a source of a massless scalar field f. Thecorresponding coupling of f to the world-volume ofthe brane is

Q dx ndx ndx ndx e f . 12Ž .H a b g d a bgd

In this way the brane ‘‘shines’’ a massless scalarfield. If there are no lighter states in the theorycharged under Q, the brane will be stable due to theflux conservation. The massless field f satisfies theclassical equation with the delta-function source in

Ž .the fifth coordinate y transverse to the brane

E 2fsQd y . 13Ž . Ž .y

The solution of the above equation is evident,

< <fsQ y . 14Ž .This solution shows that it is impossible to put suchfield on the cylinder.

Imagine now that we give a small mass m to thescalar field. This will guarantee that the flux isscreened at large distances y4my1 as

< <f;Qexp ym y . 15Ž .Ž .The exponential solution above is valid for the non-

Žcompact fifth dimension. If it is compactified a.circle , the solution with the appropriate boundary

conditions rather takes the form

Lf;Qcosh m yy 0FyFL . 16Ž .ž /2

The existence of such a brane is compatible withcompactification.

Let us consider the issue of the fermionic zeromodes on such branes. Consider the following cou-pling of a bulk fermion to f

g E f cg c . 17Ž . Ž .A A


Ž .This creates a u y function type mass term, whichchanges the sign across the brane

gQu y cg c . 18Ž . Ž .5

Due to the index theorem there is a localized zero< <mode on the brane, which at y <Lr2 takes the

form

< <cs f x exp yg y Q . 19Ž . Ž .Ž .m

This zero mode will persist for the nonzero mass ofthe f as well.

Note that in the above discussion we could haveused antisymmetric 4-form field A instead ofa . . . b

Žthe scalar f. Such fields are present in many braneŽ w x.constructions e.g. see 9 . To the best of our knowl-

edge, however, this is the first attempt of using them.for localizing the chiral fermionic zero modes . The

whole discussion would go through, except that thefermion couplings now would be modified as fol-lows. The coupling to the brane is

Q dx ndx ndx ndx A , 20Ž .H a b g d a bgd

and the fermions now couple to the 5-form fieldstrength

F sE A , 21Ž .abgdv w a bgdv x

Ž .which changes the sign across the brane F;u y .The fermions coupled to F

abgdvF e cc 22Ž .abgdv

will develop a zero mode on the brane.

5. Prototype model

Now we are in a position to write down a simpleprototype five-dimensional Lagrangian giving rise tothe desired structure. The important interaction termsare

< 3 < 2 4 2 < < 2Ls LyX q2 M cos xq bsin xym HŽ .4 2 ) 2< <q H qX r ffqX r f f qg Hff 23Ž .f f c c f cc

plus the standard kinetic and gauge terms. Here X isa gauge-singlet scalar that breaks Z symmetry and3

produces three walls on a compactified dimension. X

changes the phase by 2pr3 through each of thewalls and creates a single zero mode from eachspecies of fermions f , f . These fermions have quan-c

tum numbers of one SM generation. Moreover, fŽ . Ž .stands for SU 2 mU 1 -doublet, while f stands forc

Ž . Ž .SU 2 m U 1 -singlet states, respectively. TheYukawa couplings with H and X are assumed to

Ž . Ž .have the SU 2 mU 1 doublet and singlet struc-tures, respectively. In conventional notations the zeromodes coming from f are the left-handed quark andlepton doublets Q, L and the ones coming from fc

are the left-handed antisinglets u ,d ,e . The pre-c c c

cise form of the interaction terms between X and x

is not very important, provided that the x-wall getsstabilized on top of one of the X-walls. This can be

w xachieved, for instance, by adding 10

X < < 2b sin x X 24Ž . Ž .

with a positive bX. The above Lagrangian reproducesall the desired features discussed above. It has threeidentical branes compatible with periodic boundarycondition, with one SM generation localized perbrane. Plus a separate brane that breaks the elec-troweak symmetry.

6. Hierarchical SUSY breaking from multiplebranes

In this section we will argue that the presence ofthe identical ‘‘family branes’’ may result to an expo-nentially weak supersymmetry breaking, even thougheach individual brane, in isolation, may be super-

Ž .symmetry preserving i.e. BPS saturated . This is thecase if the multiple-brane state is a non-BPS config-uration. The idea that the observable SUSY breakingmay be due to the fact that we live on a non-BPS

w xbrane, was first put forward in 5 . Here we showthat in the present circumstances this breaking maybe exponentially weak.

Before formulating a very general sufficient con-dition for such weak supersymmetry breaking, let usillustrate the main point in a toy example. We arelooking for a model that gives rise to a BPS brane,

Ž .but in which the states with two or more suchbranes are are non-BPS. The simplest model of this


type is the one with the spontaneously broken RŽ .symmetry the symmetry is Z . In four-dimensionsN

w xthe superpotential can be chosen as 11

cX Nq1

WsL Xy , 25Ž .Nq1

where X is a chiral superfield.This theory has stable domain wall solutions

across which the phase of X changes by 2prN.Clearly, having N such domain walls is compatiblewith periodicity of the transverse coordinate. Be-cause the superpotential changes through the wall,

w xthis system admits a nontrivial central extension 5 ;it can be shown that the elementary wall is BPSsaturated. For N™`, the corresponding solutions

w xcan be found explicitly 11 . However, the N-wallstate on the cylinder is not BPS saturated, generallyspeaking. 3 Thus, the N-wall state, breaks all super-symmetries.

Let us assume that the transverse coordinate isŽ .compactified on a circle of radius R sLr2p . The

equilibrium state in such a case corresponds to Nbranes around the circle at equal distances betweenthe neighbors. What is the strength of the resultingsupersymmetry breaking? It is exponentially sup-pressed by the inter-brane distance

;eyR m , 26Ž .

where m is a mass of the X quanta, the scale thatŽsets the width of the brane we ignore factors of

order N, which may be important, however, in the.large N case . This weakness is not difficult to

understand. The wall is a field configuration, thatapproaches the vacuum state exponentially rapidlyin the transverse coordinate. Thus, unless there aremassless fields that can ‘‘carry away’’ the messageabout its presence, all the influence of the brane isexponentially suppressed at large distances. So is theresulting supersymmetry breaking.

3 Note, however, that the central charge can still be defined, itw xdoes not vanish for N-wall states 12 . Due to this reason, in

Žparticular, the junction of N domain walls in the noncompacti-. w xfied space can be BPS saturated 13,12 .

This gives us a very general sufficient conditionfor exponentially suppressed supersymmetry break-ing:

.1 the presence of the BPS brane, whose stabilityis not due to massless fields in the theory;

.2 the N-brane states should not be BPS saturated.Note that it is very important that the stability of

the brane is due to the topological charge, which isnot a source of any massless bulk field. If thisstability were due to some other charge coupled to

Žsome massless bulk field e.g. as in the D brane.case , the corresponding field would serve as a mes-

senger between the branes, and the resulting SUSYbreaking would be power suppressed.

This is the crucial difference which differentiatesour mechanism from the conventional schemes, inwhich the SUSY breaking gets transmitted between

Žthe branes by some bulk messenger interactions e.g.w x.see 15,14 .

Once again, it is important to understand thatmassless fields in question are those coupled to thestabilizing charge, and not other massless fields. Forinstance, in any realistic theory, there is at least onemassless field, the graviton, coupled to the energy-momentum tensor of the brane. This, however, willnot serve as a messenger for the SUSY breaking,since individually all branes are SUSY preserving,and only their exponentially suppressed interactionbrakes supersymmetry.

Acknowledgements

We would like to thank Savas Dimopoulos, Gre-gory Gabadadze, Alex Pomarol and Massimo Porratifor useful discussions. A part of this work was doneat ITP, Santa Barbara, where we were participants ofthe program ‘‘Supersymmetric Gauge Dynamics andString Theory.’’ We are grateful to the ITP staff forhospitality. This work was supported in part by DOEunder Grant No. DE-FG02-94ER40823, by NationalScience Foundation under Grant No. PHY94-07194,and by David and Lucile Packard Foundation Fel-lowship for Science and Engineering.

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