kaluza-klein structure associated with a fat brane

PHYSICAL REVIEW D 69, 064003 ~2004!

Kaluza-Klein structure associated with a fat brane

Pham Quang Hung* and Ngoc-Khanh Tran†

Department of Physics, University of Virginia, 382 McCormick Road, Charlottesville, Virginia 22904-4714, USA~Received 30 September 2003; published 4 March 2004!

It is known that the imposition of orbifold boundary conditions on a background scalar field can give rise toa nontrivial vacuum expectation value along extra dimensions, which in turn generates fat branes and associ-ated unconventional Kaluza-Klein~KK ! towers of fermions. We study the structure of these KK towers in thelimit of one large extra dimension and show that normalizable~bound! states of massless and massive fermionscan exist at both orbifold fixed points. A closer look, however, indicates that orbifold boundary conditions actto suppress at least half of the bound KK modes, while periodic boundary conditions tend to drive high-lyingmodes to a conventional structure. By investigating the scattering of fermions on branes, we analyticallycompute the masses and wave functions of KK spectra in the presence of these boundary conditions up to theone-loop level. The implication of KK-number nonconservation couplings for the Coulomb potential is alsoexamined.

DOI: 10.1103/PhysRevD.69.064003 PACS number~s!: 04.50.1h, 11.25.Mj

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I. INTRODUCTION

Conventionally, Kaluza-Klein~KK ! towers of modes arisefrom the compactification of extra dimensions, i.e., the iposition of periodic boundary conditions on those finite exdimensions, to render effectively the low-energy 4D physfrom a bigger spacetime with finite spatial extra dimensio~see, e.g.,@1#!. Given that the standard model~SM! is thestandard low-energy effective theory of particle interactioone requires that any higher-dimensional scenario reducethe SM at the electroweak scale. From the KK perspectthis requires the lowest modes in KK towers of fermionsbe chiral because these are the only ones we observe aenergy. To this end, the simple periodic compactificatneeds to be replaced by an orbifold compactification andthe case of one extra dimension, the common choice isS1/Z2compactification.

A novel mechanism has been proposed to localize sdard model chiral fermions differently along extra dimesions making use of the Yukawa interaction of these fiewith a background scalar field@2# ~see also@3#!. This ap-proach is phenomenologically very attractive, because itlows for easy control of the 4D effective couplings by reglating the overlap of particle wave functions in the exdimensions. At the core of this mechanism, the necesbackground scalar field of a nontrivial vacuum expectatvalue ~VEV! profile along the extra dimension has been eplicitly realized through the imposition of various orbifolboundary conditions@4# ~see also@5#!. Because of the naturof this mechanism, one sees that the KK modes obtainedintimately related to the structure of this scalar VEV in ttransverse direction. At the same time, as in theories wfinite extra dimensions, the usual periodic boundary contions ~i.e., periodic compactification! certainly drive KKtowers back to the conventional structure. It is then of intest to analyze in detail the interplay between these two

*Electronic address: [email protected]†Electronic address: [email protected]

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dencies revealed in the KK structure of related fields.For this purpose, we find it particularly useful to treat t

dependence of fields on the extra dimensions as wave ftions subject to the potential along the extra dimension asciated with the background scalar kink solution~or ‘‘bulkpotential’’ in short!, whose centers are located at fixed poinof the orbifold. In what follows, these defects will be referreto as fat branes. In this view, we find that in the limit oflarge extra dimension, bound~i.e., normalizable! KK statesof massive fermions can actually exist at both fixed poinbecause the bulk potential possesses local minima~or ‘‘po-tential wells’’! at those points~see Fig. 2 below!. This differsfrom the conclusion of@4#, where in the same limit of a largeextra dimension, normalizable KK states are found onlyone of the fixed points. This approach further allows usmake a direct connection between the boundary conditiimposed on the wave functions and the transmission amtude in the related scattering process of particles onbranes.

Our presentation is structured as follows. Section IIviews the fermion localization mechanism@4,2# in a modelwith one infinite extra dimension, where the extra dimesional wave function of the fields and the corresponding Kmass can be exactly determined from the bulk potenviewpoint. In this limit, the periodic boundary condition~PBCs! do not exist, so very distinct KK towers, partialldiscretized, originating from orbifold boundary condition~OBCs! can readily be seen. Section III generalizes the cculation to the case of a large, but finite, extra dimensiHere the competition between the two types of boundconditions becomes apparent. We show that, in this limitOBCs essentially require wave functions along the extramension of any single KK mode to have the same parityall fixed points, whereas bulk potentials tend to make thdifferent at these points~see Fig. 2!. As a result, this paritymismatch strongly affects KK light modes, while for thheavier KK modes, the same boundary conditions canequivalently translated into the requirement that thesefermions cross the branes without reflection. Hence, incases, OBCs generally suppress many bound and low-l

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P. Q. HUNG AND N.-K. TRAN PHYSICAL REVIEW D69, 064003 ~2004!

KK modes. Section IV will highlight the fact that symmetrbreaking effects in the transverse dimension are actuallycoded in the effective couplings, which in turn generate nuniversal and KK-number nonconservating interaction veces of fermion–gauge boson and fermion-scalar. SectIV A and IV B examine the implication of these effects fothe Coulomb potential and the one-loop correction to themasses, respectively, from a 4D effective viewpoint. Indifferent but related problem and in order to clarify thphysical meaning of the KK mass equation obtained in ofold compactification, Sec. V studies the scattering of fmion on branes. It is found that low-lying KK modes of thfermion indeed survive only when resonant transmissthrough the branes occurs, which in turn is achieved onlysome very particular values of the brane’s parameters. Stion VI summarizes our main results and offers some olook. The Appendix classifies properties of the general sotions of a hypergeometric-related differential equatiencountered throughout the paper.

II. FERMION LOCALIZATION IN EXTRA DIMENSIONS

Consider the case of a massless fermion and a real sfield in 411 dimensions. We usey to denote the fifth dimen-sion’s coordinate,yP@0,L#. In this section we will eventu-ally let L→`. The Lagrangians

Lc5c~x,y!@ igm]m2g5]y2 f f~x,y!#c~x,y!, ~1!

Lf51

2]mf~x,y!]mf~x,y!2

1

2]yf~x,y!]yf~x,y!

2l

4@f2~x,y!2v2#2, ~2!

L5Lc1Lf ~3!

are invariant underZ2 symmetry

f~x,y!→F~x,y![2f~x,L2y!, ~4!

c~x,y!→C~x,y![g5c~x,L2y!, ~5!

which in turn allows the imposition of the following orbifoldand periodic boundary conditions:

f~x,2y!5F~x,L2y!5f~x,2L2y!, ~6!

c~x,2y!5C~x,L2y!5c~x,2L2y!. ~7!

We note that all fields are 2L periodic and the imposition othese boundary conditions actually transforms the extramension into an orbifoldS1/Z2 with two fixed points aty50 andy5L. As f(y) is antisymmetric at these points,lv2 is sufficiently large, the minimization of effective potential

V~f!51

2]yf~x,y!]yf~x,y!1

l

4@f2~x,y!2v2#2 ~8!

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gives rise to a kink solution of the VEV in the bulk~in thelimit L→`)

^f~x,y!&5h~y!5v tanhAlv2

2y. ~9!

By performing a chiral decomposition

c~x,y!5cR~x!jR~y!1cL~x!jL~y! ~10!

where the functions1 jR,L satisfy the OBCs~7!,

jR,L~2y!56jR,L~y!, jR,L~L2y!56jR,L~L1y!,~11!

we obtain the equations of motion for 4D chiral fields:

i ~gm]mcL!jL2cR~]y1 f h!jR50,

i ~gm]mcR!jR2cL~2]y1 f h!jL50. ~12!

The 4D Dirac mass can be explicitly recovered when we tjR,L→jmR,L with @4,2#

~]y1 f h!jmR5mjmL , ~2]y1 f h!jmL5mjmR. ~13!

As usual, whenm5” 0, these equations are coupled andcan combine them to make a second-order differential eqtion for each ofjR andjL . OncejR ~andm) is obtained, onecan put it back into Eq.~13! to solve forjL ~see the Appen-dix!. But here, for the purpose of visualizingjR ,jL as wavefunctions subject to different ‘‘bulk potentials’’VR

(1) ,VL(1) ,

we choose to work equally with both second-order equatias shown below

@2]y21VR

(1)~y!#jmR5m2jmR,

@2]y21VL

(1)~y!#jmL5m2jmL , ~14!

where we have defined by virtue of Eq.~9!

w[ f v, u[Alv2

2,

VR,L(1) ~y![7uw

1

cosh2uy1w2tanh2uy.

~15!

In this ‘‘Schrodinger-like equation,’’ evidentlyjmR,L and thesquared KK massm2 are, respectively, the eigenstates aeigenvalues subject to an analogue quantum-mechanproblem. The width of the underlying ‘‘potential wells’’ is;1/u;A1/lv2 and is the actual thickness of the domawall ~or fat brane! separating domains of differenasymptotic values of scalar VEV along the fifth dimensioFrom Fig. 1 we see that, ifu>w, thenVL

(1) becomes a ‘‘po-

1The decomposition~10! requires thatjR,L(y) be continuousfunctions because they represent the probability amplitude of fiing fermions in the extra dimension.

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KALUZA-KLEIN STRUCTURE ASSOCIATED WITH A . . . PHYSICAL REVIEW D69, 064003 ~2004!

tential hump’’ and it possesses no bound states; thus forsake of completeness, in the rest of this work we assumeu is essentially smaller thanw. Equations~14! are related tohypergeometric differential equation and can be exasolved@6#.

In the lower part of the spectrum (m2,w2), bound KKmasses are necessarily quantized, and their exact valuebe obtained as@see Eqs.~A1!, ~A10!, ~A25! of the Appendix#

mn252nRuw2nR

2u2 S 0<nR,w

u D , ~16!

mn252~nL11!uw2~nL11!2u2 S 0<nL,

w

u21D ,

~17!

where the ranges onnR,LPN come from the constraintsm2

,w2, mn>0. The corresponding eigenfunctions can be wten in terms of hypergeometric functions and are givenEqs.~A9!, ~A23!. The parity constraints~11! on these wavefunctions further requirenR andnL to be even and odd integers, respectively. The KK masses~16!, ~17! are already sur-prising as they indicate that the mass gap of the KK spectis finite even when the bulk volume is infinite (L→`). In-deed the KK spectrum is discretized for the first few lev(n,w/u) because the whole spectrum has been spontously distorted by nontrivial VEVs through Yukawa interation. In other words, the KK spectrum is now associated wthe internal structure of the fat brane, and not solely withcompactification.

In this bulk potential picture, for the zero modemn50,we havenR50 and no satisfying value ofnL , because~seeFig. 1! themn

250 level is even lower than the lower limit othe potentialVL

(1) , so only the right-handed zero modenormalizable and survives.2 This eventually can be identifiewith the standard model chiral fermions. Various interestphenomenological applications of fat brane models, suchthe fermion mass hierarchy,CP violation, baryogenesis, anproton decay suppression, etc., have been carried out inmerous works~see, e.g.,@2,5,7#!.

2By flipping the sign ofg5 in Eq. ~5!, one can invert the situationwhere now only the left-handed zero mode survives.

FIG. 1. PotentialsVR,L(1) experienced by fermions in the limitL

→`.

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The nonzero KK modes of massmn come in pairs of bothchiralities with the relation~16!, ~17!

nR5nL11. ~18!

Obviously, this assures the difference in parity ofjR,L asearlier required by the OBCs. In the case thatL is finite, therelation ~18! is so strict that it apparently acts to suppremany light KK modes as we will see in the next section.

We now proceed to the continuous spectrumm2>w2, k[A(m22w2)/u2. With the parity constraints~11! we canunambiguously construct the corresponding wave functi@see Eqs.~A19!, ~A32!#, whose asymptotic forms are

y→6`: jmR~y!→22 ikS e6 ikuy1D1

12D2e7 ikuyD ,

~19!

y→6`: jmL~y!→622 ikS e6 ikuy2D18

11D28e7 ikuyD ,

~20!

whereD1 ,D2 ,D18 ,D28 are given in Eqs.~A17!, ~A30!. Mani-festly, jmR and jmL are symmetric and antisymmetric functions undery→2y at y50. However, in Eqs.~19!, ~20! wealso see that matter waves propagate in both directions eas y→6`, signaling the existence of other fixed pointsinfinity, from which waves reflect. But in the limitL→`considered here, this reflection does not seem to be vreasonable. Either way, this is not more problematic, becathe model of the infinite extra dimension itself is not realisas soon as we introduce gauge fields that can propagathe bulk. For now we just note that the reflection from infiity is a consequence of the orbifold boundary condition~7!imposed on the fermion fields. The next sections will discuthe more realistic case of finiteL.

Similarly, one can analyze the KK expansion of bacground scalar fieldf(x,y) about its VEVh(y) @4#:

f~x,y!5h~y!1(n

fn~x! f n~y! ~21!

with f n(y) being antisymmetric at both fixed points. As bfore, the 4D Klein-Gordon mass of KK scalar modesfn isrecovered whenf n satisfies

@2]y21D2~y!# f n~y!5mn

2f n~y!, ~22!

where in the limitL→`

D2~y!5lv2~3 tanh2uy21!. ~23!

The general solutions of Eq.~22! are given in the Appendix.

We find that, for the discrete spectrum,«[A(4u22m2)/u2

.0, there is only one bound eigenstate antisymmetric ay

50 ~A33! of quantum numbers~A34! n51, «51,

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f ~y!51

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corresponding to the KK massm5A3u.For the continuous spectrum, m2>4u2, k

[A(m224u2)/u25 i «, the wave functionf n(y) being anti-symmetric at y50 can be found@see Eq. ~A40!#. Itsasymptotic form is

y→6`:

f n~y!→622 i kS e6 i kuy2G~2 i k !G~12 i k !

G~2 i k22!G~2 i k13!e7 i kwyD

~25!

whereD is given in Eq.~A41!. This shows that, just as in thcase of the fermion described below Eq.~20!, the orbifoldboundary condition~6! gives rise to the nonphysical reflection from infinity of background scalar wave functions in thextra dimension. This point will be clarified below in a morealistic scenario where the other orbifold fixed point is takinto account by considering large, but not infiniteL.

III. FINITE LIMIT OF EXTRA DIMENSION

In the model of an infinite and flat extra dimension invetigated in the last section, the KK masses and wave functcan be solved exactly, but it may not be realistic as soonthe SM gauge bosons or graviton are introduced.3 In thissection we consider a more realistic situation whereL isfinite and much larger than the thickness of branesL@A2/lv251/u). In this limit, the VEV of the backgroundscalar field satisfying the OBCs~6! is given approximatelyby @4#

h~y!5v tanhSAlv2

2yD tanhSAlv2

2~L2y! D

1O~e2LAlv2!. ~26!

From this one obtains

@2]y21VR

(2)~y!#jmR5m2jmR,

@2]y21VL

(2)~y!#jmL5m2jmL , ~27!

with new composite potentials

VR,L(2) ~y!57uwS tanhu~L2y!

cosh2uy2

tanhuy

cosh2u~L2y!D

1w2tanh2uy tanh2u~L2y!. ~28!

3Theories with infinite but warped extra dimensions effectiveconsistent with 4D physics have been built~see@8#!.

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In Fig. 2 we sketch the potential~28! for the caseL510A2/lv2. It is clear that, in the limitL@1/u, each newpotential (VR

(2) or VL(2)) has the shape of a neatly separat

double-well potential, whose component wells are just offorms ~15!. More specifically, the newVR

(2) ~28! looks ex-actly like VR

(1) @of ~15!# at the fixed pointy50, and likeVL(1)

at y5L. The opposite holds forVL(2) . Consequently, the new

composite wave functions@solutions of~27!# can be obtainedby matching together the solutions of Eq.~14! found earlier.

For light ~bound! KK modes, m2,w2, «

[A(w22m2)/u2.0, the matching appears straightforwasince the wave functions given by Eqs.~A9!, ~A23! tend tozero at the matching regiony.L/2 @see Eqs.~A7!, ~A24!#.However, there exists a subtlety in the parity matching duethe shape interchange ofVR

(2) andVL(2) at the fixed points, as

we mentioned above. At the same squared mass levelm2,jmR behaves like a right-handed wave function aty50 and aleft-handed wave function aty5L, so in view of Eq.~11!, ifjmR is symmetric aty50, it must be antisymmetric aty5L, or vice versa. Meanwhile the OBCs~11! also requirejmR to be symmetric at all fixed points. So ifjmR were asolution satisfying these apparently contradictory conditioon parity and were nonzero in the vicinity ofy50, then itshould be zero aroundy5L, because a null function is thonly function that is both symmetric and antisymmetric wrespect to a given point. A similar parity mismatch also hofor left-handed states. As a result, we find that~Fig. 2! thetower of bound KK states with mode indicesn50,2, . . .will be localized aty50, while the tower withn51,3, . . .is localized aty5L with n being defined in Eq.~16!. Allthese modes are vectorlike~i.e., they possess both right- anleft-handed components!, except for the zero mode beinright-handed localized aty50.4 These towers do not essentially correlate, but in the limitL;1/u stronger tunnelingmay change the qualitative picture.

We can see this more clearly now by studying the uppart of the spectrum, wherem2>w2, k[A(m22w2)/u2

5 i«. Here the matching process is simple, because as nearlier, at the matching region in between the two wells,wave functions associated with each well already fully reatheir asymptotic forms. Using Eqs.~A19! and~A31! @or Eqs.~A20! and ~A32!#, the matching induces the following relation:

D1

12D25e2ikuL

12D28

D18~29!

4In @4#, after combining Eqs.~13! and ~26! to get the zero-modewave functionj0R5C@coshu(L2y)#fv/u;efv(L2y) close to the fixedpoint y5L, it is accordingly found that the zero and other massKK modes are non-normalizable aty5L. Although j0R increasesexponentially asy runs away fromy5L, we think that the matchingof this function with the zero-mode state~A9! bound to the fixedpoint y50, which decreases exponentially into the bulk, wounecessarily set the constantC to zero in order to preserve the nomalization of j0R in the limit of large L. As a result, the right-handed zero mode exists only aroundy50 while certain highernormalizable modes can exist around both fixed points as seethe bulk potential picture~Fig. 2!.

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whereD and D8 are given in Eqs.~A17!, ~A30!. Since allwave functions are 2L periodic@Eq. ~7!#, we can rewrite Eq.~29! as

e2ikuL51⇔ku5np

L, ~30!

D1D181D2251. ~31!

The first of these equations is the usual periodic boundconditions, while the second accounts for orbifold boundconditions and the effect of fat branes. Becausek, D8s are allfunctions ofm, Eqs.~30!, ~31! are actually equations detemining the KK masses.

Remarkably, the physical meaning of condition~31! canbe viewed as the requirement that the transmission ampliof fermions in the double-brane system needs to be exactA quantitative discussion will be given in Sec. V. Howevethe assertion’s outcome itself is readily plausible: oncetransmission amplitude is 1, the KK states propagate inbulk as if there were no potential at all, and subsequentlyperiodic boundary condition~30! drives them to the conventional structure. As a result, we expect high-lying modesthe KK spectrum to have a structure closer to that of periocompactification.

Let us now examine the compatibility of the two condtions ~30!, ~31!. For very heavy KK modes (k@w/u.1),D1 ,D18 and D2 ,D28 approach 1 and zero, respectively, athe condition~31! is automatically satisfied. Then Eq.~30!implies m25w21k2u25w21n2p2/L2'n2p2/L2, which isthe usual KK mass from periodic compactification. Fsmaller values ofk, it may be difficult to solve Eq.~31!analytically although a numerical approach is possible.Sec. V we will discuss the solution of this equation in tlimit of small k. Here we just mention that for ak that is notvery large the two conditions~30!, ~31! are not always com-patible. To illustrate this point, let us consider the speccase wheres[w/u is an integer. UsingG(z11)5zG(z) andEq. ~A18!, we findD25D2850 and

FIG. 2. PotentialsVR,L(2) and symbolic wave functionsjR,L of

fermion chiral components withL510/u. The solid curves representVR

(2) and its symmetric wave functionsjR , the dashed curvesrepresentVL

(2) and the antisymmetricjL .

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D15~ ik11!•••~ ik1s!

~ ik21!•••~ ik2s!, D185

~ ik11!•••~ ik1s21!

~ ik21!•••~ ik2s11!.

~32!

Equation ~31! becomes a polynomial equation of ord(k2)s21, so it has at most (s21) solutions ofk2 ~or squaredmassm2), which may or may not satisfy Eq.~30!. In anycase, the combination of Eqs.~30!, ~31! gives no more than(s21) different exact values ofm2 in the range of not-too-large k (k<s[w/u), while in the conventional compactification ~30! the number of KK states in the same range;wL/[email protected] Combined with the earlier consideration fobound KK states, we see clearly that the lower part ofKK spectrum is strongly distorted by spontaneous breakof the background scalar, to which the fermions are coupand also by orbifold boundary conditions, and the lattergenerally so strict that many of these states are effectivsuppressed. The higher levels are not essentially affectethe bulk potential~their transmission coefficient is close tunity!, and the usual periodic boundary conditions drive thmass structure to that of conventional compactification.

We now briefly investigate the KK scalar structure~21!,~22! in the limit L@1/u by a similar method. In this limit, thepotential~23! becomes@see Eq.~26!#

D2~y!5lv2@3 tanh2uy tanh2u~L2y!21#. ~33!

This potential is sketched in Fig. 3 forL510/u510A2/lv2. We see that the two component wells are idetical ~i.e., the potential isL periodic! and no parity mismatchoccurs, and we still have only one antisymmetric KK sta~24! with massm5A3u in the lower part of the spectrumFor the higher modesm2>4u2, the matching of the scalawave functionsf (y) @Eq. ~A40!# antisymmetric at all fixedpoints generates the following relations:

ku5np

L, ~34!

D25S ~11 i k !~21 i k !

~12 i k !~22 i k !D 2

51, ~35!

5However, we will see later that the case of the integralw/u is aspecial ‘‘resonant’’ case, where Eq.~31! is approximately satisfiedfor very small values ofk.

FIG. 3. PotentialsD2(y) experienced by scalar fields witL510/u.

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where k[A(m224u2)/u2. These equations have at motwo exact solutionsk250 and k252 ~the latter is a truesolution if only A2Lu/p is integer! corresponding tom2

54u2 and 6u2. However, for allk!1 or k@1, the condition~35! is approximately satisfied and in these ranges onlyperiodic boundary condition~34! effectively governs the KKmasses and gives them the conventional structure upconstant shift:

mn25u2~41 k2!5u2S 41

n2p2

u2L2 D , ~36!

wheren@uL/p or n!uL/p. As scalars are not chiral, thedo not suffer from parity mismatch and consequently thtransmission coefficient can reach unity not only for higlying KK levels, but also for levels immediately above thsurface of the potential well. This ‘‘resonant behavior’’ of thpotential~23! was discovered a long time ago~see@9#!. Wenote that, in contrast with the result obtained therein, herescalar fields are constrained to be antisymmetric by OBand then resonance occurs only in the two above rangemomentumk. This special structure of the KK scalar spetrum has important implications in practical calculations,we will see in the following fermion self-energy evaluatio

IV. FOUR-DIMENSIONAL EFFECTIVE COUPLINGSAND IMPLICATIONS

It is well known that in the universal extra dimensio~UED! scenario ~see, e.g.,@10#! where no localizationmechanism is invoked and all extra dimensions are acsible to all fields, momentum is conserved in both the lontudinal ~infinite! and transverse~finite! directions of spaceThis in turn implies the KK-number conservation of all vetex interactions, and there are no tree-level contributifrom KK excitation to standard model observables, whocontent fields are taken to be the zero modes in the KK topicture. The situation is quite different in the brane scenawhere Lorentz invariance is violated along transverse dimsions due to both the background kink and orbifold comptification. Consequently, in the reduced 4D picture, thereist KK-number nonconservation vertices characterizedeffective couplings. This is because the overlap integral oextra coordinates leading to 4D couplings actually measuthe effects of Lorentz invariance breaking on the wave futions of related fields along the extra dimension. These cplings may give rise to new interesting phenomenologsuch as tree-level flavor changing neutral currents, new ming of quark and lepton flavors, etc. The comparison of thnew contributions with experimental data will providbounds on various parameters of the model. In this sectjust for the purpose of illustration, we will present a tomodel which involves only one fermion flavor in orderstudy the new implications, if any, of the brane scenario

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the Coulomb interaction in a tree-level consideration andthe KK masses up to one-loop level.

A. Static Coulomb interaction

Charged fermions in QED interact with one anotherexchanging a photon. We now assume that the photonpropagate freely along the fifth dimension, otherwise a locization mechanism for the gauge boson in exact fashion sas we used for fermions encounters a serious complicarelated to the issue of universality of charge. TheS1/Z2OBCs turn the extra dimensional wave function of the phton KK modes into the form of cosine or sine functiondepending on their transformation property underZ2. TheZ2

symmetry of the termcgMAMc requires the photon’s firsfour componentsAm to be symmetric and the fifthA5 anti-symmetric aty50,L. This also eliminates the zero modeA5 as well as the contribution of its KK excitationsAn5 tothe zero-mode charged fermions interaction~Fig. 4!. Afterdimensional reduction, we obtain the following vertex copling of the fermion zero mode and photon KK modes:

E dyc~x,y!@2e5A” ~x,y!#c~x,y!

→2ec0~x!gmA0m~x!c0~x!

2 (n51

`

enc0~x!gmAnm~x!c0~x! ~37!

where theen’s are 4D effective couplings

en[e5

ALE

0

2L

dy@j0~y!#2cosS npy

L D ~38!

ande5e5 /A2L is the usual 4D charge of the fermion zemode, whose wave function isj0(y)5(coshuy)2w/u as givenin Eq. ~A9!. In the nonrelativistic limit, the potential betweetwo charged fermion zero modes can be found by workout the KK photon exchange process depicted in Fig. 4. Tresult is

FIG. 4. Effective fermion-photon vertex and tree-level fermiscattering diagram.

3-6

hsasdan

hn-hoorKK

ner-

rn

one

lev-ure

on-on,one

ive

nd

the

c

a-

c-heul

gy


V~r !5e2

r1 (

n51

` en2

re2npr /L

5e2

r1

2e2

r E0

2L

dy@j0~y!#2E0

2L

dy8@j0~y8!#2

3 (n51

`

cosS npy

L D cosS npy8

L De2npr /L ~39!

wherer denotes the spatial separation in the 4D picture. Tfirst term of Eq.~39! represents the contribution of the masless photon, while the remaining terms come from its msive KK modes. By transforming the sum over the moindex n into a sum over elements of a geometric series,approximatingj0(y) with a Gaussian function~A11!, weobtain the following expression for the potentialV(r ):

V~r !5e2

r1

2e2

r F2111

12e2pr /L

23p2

4wuL2

e2pr /L~11e2pr /L!

~12e2pr /L!31OS 1

w2u2L4D S L

r D 5G .

~40!

If r @L ~and r .1/MZ so that the contribution from theZboson can be neglected!, the potential~40! takes the form

V~r !.e2

r1S 12

3p2

4wuL2D 2e2

re2pr /L1•••, ~41!

where we have kept only a few terms of higher orders. Tfirst term in Eq.~41! is the usual Coulomb interaction potetial arising from the exchange of the zero mode of the pton, whereas the second term has a Yukawa potential fbecause it is mediated by an infinite tower of massivemodes. This correction is considerable only whenr ap-proaches the size of the extra dimension, and the braexplicit contribution to this correction is at most a few pecent in the limitL@u, so it is unlikely that we could findsome phenomenological bounds on the model’s parameteuand w by just looking at the static Coulomb interactiopotential.6

B. One-loop correction to KK mass

It is known that in theories with just one extra dimensithe sum over an infinite tower of KK modes for tree-lev

6For r<L,1/MZ , one needs to take into account the full eletroweak contribution. But if we neglected the contribution from tZ boson and its KK modes as an illustrative computation, we woobtain to leading orders ~40! V(r ).2e2L/pr 21pe2/6L23e2L/pwur41•••. It is interesting that, forr<L, the model-independent leading order of the potential is;1/r 2, which agreeswith the classical result obtained by the Gaussian theorem@1# in apurely geometric approach.

06400

e--

ed

e

-m

’s

s

l

diagrams converges as we have seen above. At the loopels this is no longer true and some renormalization procedis needed. To see specifically how the KK-number noncservating couplings contribute in an actual loop computatilet us evaluate, as an example, the fermion self-energy atloop and its corresponding mass shift in the 4D effectpicture.

Decomposingc(x,y) and f(x,y) as in Eqs.~10!, ~21!and performing the integration over they coordinate wetransform the LagrangianLc(1) into its 4D version:

E Lcdy5(n

cn~x!~]”2mn!cn~x!2 (n, r ,s

f cn~x!f r~x!

3~gRLnrsPL1gLR

nrsPR!cs~x!,

where themn’s are the tree-level masses of KK fermions athe g’s are the effective 4D couplings~Fig. 5!:

gRLnrs5E jnR~y! f r~y!jsLdy, gLR

nrs5E jnL~y! f r~y!jsRdy.

~42!

We note that ingRLnrs the OBCs requiren ands to be even and

odd integers, respectively. IngLRnrs , however, these parity

constraints are reversed. The couplings are related by

equationgRLnrs5gLR

srn . The modified propagator of a fermioniKK mode n is

1

p”2mn2Sn~p” !

where ~using Feynman’s parametrization and Wick’s rottion!

2 iSn~p” !5(r ,s

f 2EL d4k

~2p!4

1

k22mr2 ~gRL

nrsPL

1gLRnrsPR!

1

~p”2k” !2ms

~gLRnrsPL1gRL

nrsPR!

→ i f 2

16p2 (r ,s

S 1

2@~gRL

nrs!2PR1~gLRnrs!2PL#p”

1gRLnrsgLR

nrsmsD lnS L2

ms2D . ~43!

d

FIG. 5. Effective fermion-scalar vertex and fermion self-enerdiagram.

3-7


FIG. 6. CouplinggRLnrs @Eq. ~42!# as a function ofn,s with u51, w/u510.3, uL540. The background scalar fieldf r is in the

normalizable~bound! KK mode ~a!, or r 50 mode~b!, or r 55 mode~c!. The position of the peaks clearly indicatesn5s11 enhancement.We note also that the last two graphs are very similar, which justifies our approximation leading to Eq.~44!.

y

fi

tref-

sr-u

e

no

re

r-

.bs

go-K

a-we

rre-

i-

og a

and, asec-mear

L is a cutoff scale above which the physics is governed bmore fundamental theory, and the sum over KK modesaccordingly limited by the relationsmr

2 ,ms2<L2.

If L2>w2, heavy KK modes (mr2>u2,ms

2>w2) will con-tribute to the sum~43! and drive its value to that of the UEDscenario. In this work we assumeL2<w2 to investigate thecontribution, if any, solely from the distinctive lower part othe KK spectrum of the brane picture. This assumptionself-consistent because by tuningu andw we can pushL toa sufficiently high scale, and the nonrenormalizable conbution is expected to be cutoff by quantum gravitationalfects @10#. With this assumption, from the orbifolding constraints ~34! and ~35!, it follows that only the lower KKscalar modes (r !uL) appear in the sum~43! as their massesare below the cutoff. The number of these relevant modeNs;O(1) in the limit L@1/u considered here. This obsevation allows a rough evaluation of the nonuniversal co

plings gRLnrs .

Specific solutions ofjnR,L @Eqs.~A9!, ~A23!# and f r @Eqs.~A33!, ~A40!# suggest that for KK modes relevant to thself-energy diagram (n,s<w/u; r !uL), the characteristicwidth of the normalized wave function of the KK fermiojnR,L(y) in the extra dimension is much smaller than thatf r(y) ~of the scalar field! asw is sufficiently larger thanu.7

For a rough estimation we neglect the variation off r(y) overthe extent ofjnR,L(y); then for each relevant moder the

couplingsgRLnrs are most enhanced for a few modess closest

to n ~i.e., when n.s11), because this is whenjnR re-semblesjsL most closely~i.e., 2*0

LjnRjsLdy;1). In this ap-proximation, the fermionic KK zero-mode mass shouldceive a rather small one-loop correction, because it hascorresponding ‘‘closest-neighbor’’ modejsL with s521,

i.e., gRL0r 215gLR

21r050. Our estimation is verified by a pa

ticular numerical evaluation of the couplinggRLnrs as a func-

tion of the mode indicesn,s. The result is presented in Fig6, where one can see clearly the effect of closest-neighenhancement on the values of the couplings. The value

7This assertion can be less rigorous for KK modess→w/u in theupper limit of the sum, but contributions from these modesactually suppressed by the factor ln(L/ms).

06400

ais

s

i--

is

-

f

-no

orof

these couplings indeed decrease very quickly off the dianal n5s11. Putting all observations together, for the Knonzero mode of the fermion we have

2 iSn~p” !5C i

16p2

f 2

2LlnS L2

mn2D S 1

2p”1mnD , ~44!

whereC;O(1) accounts for the number of relevant KK sclar modes in the sum and some crude estimations thathave made. This one-loop divergence requires the cosponding counterterms

L→L1dL5(n

~Zccni ]”cn2Zmmncncn!

with renormalization scaling factors

Zc512C

64p2

f 2

2LlnS L2

mn2D ,

Zm511C

32p2

f 2

2LlnS L2

mn2D .

After transforming to the canonical basisc→AZcc we fi-nally obtain a one-loop correction to the mass of the fermonic KK nonzero mode~in the leading order!:

dmn5mnS Zc

Zm21D.mnS 3C

64p2

f 2

2Lln

L2

mn2D , ~45!

where themn’s are given by Eq.~16!. Interestingly, the sameresult is obtained in@11#, where the one-loop correction tthe KK mass in the UED scenario was computed usinfundamental 5D approach~see @12#!. This would indicatethat, at the tree level, KK mass structures in the fat braneUED scenarios are very different at least in the lower partwe saw in previous sections; however, their radiative corrtions may scale somewhat similarly. We now turn to soresonant effects on the KK spectrum of fermions.

e

3-8

ats

erup

ee

d

sucheo

iote

ahe

aroa

on

-

pli-ell-al

the

ualoreis-here-ndi-

in

al-er-rel-is,oplaythe

n

ding


V. RESONANCES AND KALUZA-KLEIN SPECTRUMOF FERMIONS

The considerations of Sec. III led to the conclusion thorbifold boundary conditions, which are crucial ingredienin achieving single-chirality standard model fermions as zmodes of fields in a higher-dimensional theory, act to spress many bound (m2,w2) fermionic KK levels. Each ofthese levels’ wave functions has a definite parity with respto a fixed point, and if this parity is not identical to the onimposed by OBCs, the corresponding level is suppresseparity mismatch. Higher levels (m2>w2) are twofold degen-erate and can be arranged to have the desired parities~see theAppendix!, so the impact of OBCs on these levels is notobvious. In this section we discuss in more detail the strture of this upper part of the KK spectrum and clarify tclose connection between periodic boundary conditions,bifold boundary conditions, and the complex transmissamplitude of free fermions through the double-brane sysmentioned in Sec. III.

We first consider the related problem of one noncompdimension where a fermion, being free at infinity, approaca single double-well potentialVR

(2) @Eq. ~28!# ~Fig. 2! withwave vectorku[Am22w2.0. In this problem, the wavefunction of the particle is not constrained by any boundconditions since the dimension is noncompact. The mgeneral particle wave function and its asymptotic formsreferred to the fixed point aty50 are, respectively@see Eqs.~A12!, ~A13!#,

jmR~y!5aR1~y!1bR2~y!, ~46!

y→`: jmR~y!→aeikuy1be2 ikuy, ~47!

y→2`: jmR~y!→~aC11bC2!e2 ikuy

1~aD11bD2!eikuy. ~48!

As referred to the fixed point aty5L, similarly we have

jmR8 ~y!5a8L1~y8!1b8L2~y8!, ~49!

y8→`: jmR8 ~y!→a8eikuy81b8e2 ikuy8, ~50!

y8→2`: jmR8 ~y!→~a8C181b8C28!e2 ikuy8

1~a8D181b8D28!eikuy8, ~51!

wherey85y2L anda,b,a8,b8 are constant coefficients.For the wave traveling from left to right and scattering

the double brane, we letb850 and the matching of Eqs.~47!and ~51! gives a/b5(D18/C18)exp(22ikuL) and a8/b51/C18 . Next, using the relations~A18! C152D2 , C2

5D1* , andD2852D25D2* , we obtain the transmission amplitude

tR5a8

aD11bD25

1

D1D18exp~22ikuL!1D22

. ~52!

06400

t

o-

ct

by

o-

r-nm

cts

ysts

Now we see clearly that conditions~30!, ~31! imply thephysical requirement that the complex transmission amtude of a fermion propagating through a single double-wpotentialVR

(2) is precisely 1. We will come back to this observation below. A similar consideration gives an identictransmission amplitude for a wave function subject toother type of single double-well potentialVL

(2) @Eq. ~28!#:tL5tR . From here on we drop all indicesR,L as well as thefactor exp(22ikuL) in the expression oft by virtue of Eq.~30!.

For k@w/u.1, we saw earlier thatmn2.n2p2/L2, i.e.,

the higher KK structure is always dominated by the usperiodic compactification. Now we can see this result mphysically: the high-lying KK modes have a large transmsion amplitudet→1 because they are not sensitive to tunderlying potential and can cross it without significantflection, and on these modes, the periodic boundary cotions ~30! are the more influential ones. However, as seenthe previous section, the couplingsf and l in Eq. ~3! havenegative dimensions of mass, so the theory is not renormizable. It may effectively describe physics only under a ctain mass scale, and our calculation may no longer beevant for heavy KK modes above that scale. Apart from ththe smaller range of values ofk deserves special interest alsbecause it is where the fat brane structure is expected toa dominant role. To see this specifically, we now examinemass quantization equationt51 @Eq. ~31!# for low-lying fer-mionic KK states,k!1,w/u. Using the product expansio@13#

G~z1!G~z2!

G~z11z3!G~z22z3!5 )

q50

` S 11z3

z11qD S 12z3

z21qDwe can expandD1 as follows:

D15k2 iw/u

k )q51

` S 12w/u

q2 ik D S 11w/u

q2 ik D5uD1uexpH 2 i

p

21 i Fw

u Gp1 iku

w

1 ik (q51

` S 2

q2

1

q2w/u2

1

q1w/uD1O~k3!J5uD1uexpH 2 i

p

21 i Fw

u Gp1 iku

w

1 ikF2g1CS w

u D1CS 2w

u D G1O~k3!Jwhere @w/u# is the maximal integer not larger thanw/u,C(z)[(d/dz)ln G(z) is the PolyGamma function, ang'0.577 is the Euler-Mascheroni constant. After expandD18 in a similar way and using the recursion formulaC(s11)5C(s)11/s we obtain

3-9

r

esot

olva

l-se

o-

d

te

theofryatec-ry

is-e isom-rs ofeak-ex-

thisbleti-iontheon-ton of

tobi-ve

thehe.

ryf-s-the5-

lu--e,


D1D185uD1D18uexpF ikPS w

u D1O~k3!G5uD1

2uexpF ikPS w

u D1O~k3!G ~53!

with

PS w

u D[2F2g1CS w

u D1CS 2w

u D G . ~54!

Now putting this back into the expression~52! of the trans-mission amplitude, we find

t51

D1D181D22

51

uD1u2exp@ ikP~w/u!1O~k3!#2uD2u2

~55!

whereuD1u2,uD2u2 are calculated in the Appendix:

uD1u25sinh2pk1sin2pw/u

sinh2pk, uD2u25

sin2pw/u

sinh2pk.

~56!

First, if w/u has values such thatP(w/u) in Eq. ~54! or(sin2pw/u) in Eq. ~56! vanishes, one can see thatt511O(k), i.e., the condition~31! is approximately satisfied folow-lying KK modes k!1. Using the condition~30!, thismeansn!uL, and since we are considering the limituL@1, this approximation holds for a certain number of modThis again can be seen as a resonance behavior of the ptials ~28! such that for certain values ofw/u, even low-lyingparticles can essentially go through it, and the condition~30!then determines their massesmn

2'w21n2p2/L2. The reso-nance values ofw/u are positive integers@for sin(pw/u)50] and others that are solutions of the equationP(w/u)50. From Fig. 7 we see that there is exactly one such stion between any two successive integers. For all otherues ofw/u, we find thatt5O(k) and the equationt51 doesnot have solutions fork!1. In other words, branes are amost ‘‘opaque’’ for low-lying modes and consequently thefermion modes are absent in the KK spectrum.

FIG. 7. Sketch of function P(w/u)[2@2g1C(w/u)1C(2w/u)#.

06400

.en-

u-l-

Overall, except for some particular values of the bulk ptential parameterw/u, we see that the lower part of the KKspectrum of fermions is strongly distorted~i.e., suppressed!by orbifold compactification and the VEV of the backgrounscalar field.

VI. CONCLUSION

In this work we attempt to investigate the effect of finisize of the extra dimension~and of the brane! on the KKmode structure of fermions and scalar fields by invokinganalogous bulk potential. The orbifold compactificationthe extra dimension involves different types of boundaconditions: the usual periodic boundary conditions dominthe high-lying KK modes and give them the familiar struture of conventional compactification, the orbifold boundaconditions ~along with nontrivial fat branes! dominate thebound and low-lying KK modes and give them a more dtinctive structure. The observation being emphasized herthat these conditions on wave functions are not always cpatible, depending on the specific values of the parametethe model through some resonance effects. Roughly sping, the limit separating the two very different parts of thspectrum is of the order of the potential barrier’s height eperienced by particles along the extra dimension, andcould serve as the cutoff scale of the nonrenormalizahigher-dimensional theory. The effects of orbifold compacfication and symmetry breaking in the transverse dimensare now embedded in the 4D effective couplings, andcorresponding vertex interactions do not necessarily cserve the KK number. This allows KK modes to contributetree- and higher-level processes. For an extra dimensioarbitrary size, certain numerical techniques are requiredsolve for the background scalar VEV subject to given orfold boundary conditions. The matching of component wafunctions is not simple, but we believe that the use ofbulk potential would remain the right approach to find tKK masses and their state functions in this general case

ACKNOWLEDGMENTS

P.Q.H. would like to thank Gino Isidori and the TheoGroup at LNF~Frascati! for hospitality during the course othis work. N.-K.T. would like to thank Dr. A. Soddu, Professor V. Celli, and Professor P. Arnold for many helpful dicussions and advice. This work is supported in part byU.S. Department of Energy under Grant No. DE-A5089ER40518.

APPENDIX

In this appendix, we solve and classify the different sotions of the differential Eqs.~14!, which have been accordingly employed with different physical constraints in thmain text. First, let us introduce some shorthand notation

3-10

on

v-

ate

de-


w[ f v, u[Alv2

2, z[tanhAlv2

2y5tanhuy,

«[Aw22m2

u2. ~A1!

Dividing both sides of the first equation in~14! by u2(12z2), we obtain

d

dzS ~12z2!djmR

dz D1Fw

u S w

u11D2«2

1

12z2GjmR50.

~A2!

Again, using some new notation,z1[ 12 (12z) and jmR(z)

[(12z2)«/2p(z) we transform Eq.~A2! into the standardhypergeometric differential equation

z1~12z1!d2p

dz12

1~«11!~122z1!dp

dz1

2S «2w

u D S «1w

u11D p50. ~A3!

Equation~A3! has two linearly independent solutions@13#

r 1~z1!5FS «2w

u,«1

w

u11,«11;z1D

r 2~z1!5z12«FS 2

w

u,w

u11,2«11;z1D ,

whereF is the hypergeometric function

F~a,b,c;z!511ab

c

z

1!1

a~a11!b~b11!

c~c11!

z2

2!1•••.

~A4!

Two corresponding solutions of Eq.~A2! are

R1~y!5~12z2!«/2r 1~z!

5S 1

coshuyD«

FS «2w

u,«1

w

u11,«11;

1

11e2uyD ,

~A5!

R2~y!5~12z2!«/2r 1~z!

5~2euy!«FS 2w

u,w

u11,2«11;

1

11e2uyD .

~A6!

First, consider the discrete spectrumm2,w2; «.0. Theasymptotic forms of Eqs.~A5!, ~A6! are

06400

y→`: R1~y!→2«e2«uy,

y→2`: R1~y!→2«e«uyFS «2w

u,«1

w

u11,«11;1D ,

~A7!

y→`: R2~y!→2«e«uy,

y→2`: R2~y!→2«e«uyFS 2w

u,w

u11,2«11;1D .

~A8!

WhenL→` as considered in Sec. II, we see from Eq.~A8!thatR2 blows up at infinity and cannot be a physical solutifor bound states, and

jmR~y!5R1~y!

5S 1

coshuyD«

FS «2w

u,«1

w

u11,«11;

1

11e2uyD~A9!

is the sought physical solution ofjmR with the followingcondition@otherwise,F in Eq. ~A7! and thenjmR in Eq. ~A9!blows up asy→2`]:

«2w

u52nR ~nRPN!. ~A10!

From this follows the mass quantization of KK discrete leels @Eq. ~16!#. As far as the zero mode (m50) is concerned,in some computations it is more convenient to approximj0R with a normalized Gaussian function

j0R~y!5S wu

p D 1/4

e2wuy2/2. ~A11!

The fact that here onlyR1 is a physical solution is readilyunderstood, as all bound states of a 1D potential are nongenerate. Further, we see from Eq.~A9! that jmR is an evenor odd function, respectively, whennR is an even or oddinteger. For the continuous spectrum,m2>w2, k

[A(m22w2)/u25 i«, Eqs.~A5!, ~A6! read

R1~y!5S 1

coshuyD2 ik

FS 2 ik2w

u,

2 ik1w

u11,2 ik11;

1

11e2uyD , ~A12!

R2~y!5~2euy!2 ikFS 2w

u,w

u11,ik11;

1

11e2uyD ,

~A13!

whose asymptotic forms are

3-11

alon

-

iain

s.dd

on

ndpec-


y→`: R1~y!→22 ikeikuy,

y→2`: R1~y!→22 ik~C1e2 ikuy1D1eikuy!,~A14!

y→`: R2~y!→22 ike2 ikuy,

y→2`: R2~y!→22 ik~C2e2 ikuy1D2eikuy!,~A15!

where for the limity→2`, we have used a fundamentanalytic continuation formula of the hypergeometric functi@13#

F~a,b,c;z!5G~c!G~c2a2b!

G~c2a!G~c2b!F~a,b,a1b112c;12z!

1~12z!c2a2bG~c!G~a1b2c!

G~a!G~b!

3F~c2a,c2b,c112a2b;12z! ~A16!

and

C15G~ ik !G~12 ik !

G~w/u11!G~2w/u!,

D15G~2 ik !G~12 ik !

G~2 ik2w/u!G~2 ik1w/u11!,

C25G~ ik !G~11 ik !

G~ ik2w/u!G~ ik1w/u11!,

D25G~2 ik !G~11 ik !

G~w/u11!G~2w/u!. ~A17!

Using the Gamma function relations@14# G(z11)5zG(z), G(z)G(12z)5p/sin(pz), we obtain some relations useful for our calculation:

C15i sinpw/u

sinhpk⇒uC1u25

sin2pw/u

sinh2pk

C152D2 , C25D1* ⇒uC1u25uD2u2,

uC2u25uD1u2,

uD1u22uD2u25uD1u21D2251⇒uD1u2

5sinh2pk1sin2pw/u

sinh2pk. ~A18!

We note especially that whenw/u is integer D250 anduD1u51.

In contrast with bound states, neither of~A12!, ~A13! hasdefinite parity. However, continuous levels of 1D potentare always two-fold degenerate, so we can linearly comb

06400

le

R1 , R2 to produce a function of the desired parity. From Eq~A14!, ~A15! we can respectively build up the even and oright-handed wave functions:

Reven~y!5R1~y!1D1

12D2R2~y!, ~A19!

Rodd~y!5R1~y!2D1

11D2R2~y!. ~A20!

A similar consideration applies to the left-handed fermiwave functions, in place of Eqs.~A5!, ~A6! we have

L1~y!5S 1

coshuyD«

FS «2w

u11,«1

w

u,«11;

1

11e2uyD ,

~A21!

L2~y!5~2euy!«FS 2w

u11,

w

u,2«11;

1

11e2uyD .

~A22!

The key observation here, when comparing Eqs.~A5!, ~A6!with Eqs. ~A21!, ~A22!, is that left-handed solutionsL1 ,L2are effectively the same asR1 ,R2 after changingw/u tow/u21. Then the corresponding physical solutions amass quantization equation for a left-handed discrete strum are@see Eqs.~A9!, ~A7!, ~A10!#

jmL~y!5L1~y!5S 1

coshuyD«

3FS «2w

u11,«1

w

u,«11;

1

11e2uyD~A23!

y→`: L1~y!→2«e2«uy, ~A24!

«2w

u1152nL~nLPN!. ~A25!

In the continuous spectrum, instead of Eqs.~A12!–~A15!, wehave

L1~y!5S 1

coshuyD2 ik

FS 2 ik2w

u11,

2 ik1w

u,2 ik11;

1

11e2uyD , ~A26!

L2~y!5~2euy!2 ikFS 2w

u11,

w

u,ik11;

1

11e2uyD ,

~A27!

3-12


y→`: L1~y!→22 ikeikuy,

y→2`: L1~y!→22 ik~C18e2 ikuy1D18e

ikuy!,

~A28!

d

e

ar

at

li,

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y→`: L2~y!→22 ike2 ikuy,

y→2`: L2~y!→22 ik~C28e2 ikuy1D28e

ikuy!,

~A29!where

C185G~ ik !G~12 ik !

G~w/u!G~2w/u11!, D185

G~2 ik !G~12 ik !

G~2 ik2w/u11!G~2 ik1w/u!,

C285G~ ik !G~11 ik !

G~ ik2w/u11!G~ ik1w/u!, D285

G~2 ik !G~11 ik !

G~w/u!G~2w/u11!. ~A30!

By making the changew/u→w/u21 in Eq. ~A18! we canfind similar properties ofC18 ,C28 ,D18 ,D28 . In particular, wehaveD2852D2 , uD18u5uD1u. The even and odd left-handewave functions now are

Leven~y!5L1~y!1D18

12D28L2~y!, ~A31!

Lodd~y!5L1~y!2D18

11D28L2~y!. ~A32!

Finally, for a scalar field, Eq.~22! can be solved by the sammethod. For a discrete spectrum,m2,4u2, «

[A(4u22m2)/u2.0, we obtain, respectively, the scalwave function and mass quantization equation

f n~y!5S 1

coshuyD«

FS «22,«13,«11;1

11e2uyD ,

~A33!

«2252n~ nPN!. ~A34!

Since«.0, there are only two discrete levelsn50,1 and thecorresponding states are symmetric and antisymmetricy50. In the continuous spectrum, m2>4u2, k

[A(m224u2)/u25 i «, in place of Eqs. ~A12!–~A15!,~A19!, ~A20!, we have

S1~y!5S 1

coshuyD2 i k

FS 2 i k22,

2 i k13,2 i k11;1

11e2uyD , ~A35!

S2~y!5~2euy!2 i kFS 22,3,i k11;1

11e2uyD~A36!

y→`: S1~y!→22 i keikuy,

y→2`: S1~y!→22 i k~Deikuy!, ~A37!

y→`: S2~y!→22 i ke2 i kuy,

y→2`: S2~y!→22 i kS 1

De2 ikuyD , ~A38!

Seven~y!5S1~y!1DS2~y!, ~A39!

Sodd~y!5S1~y!2DS2~y!, ~A40!

where

D5G~2 i k !G~12 i k !

G~2 i k22!G~2 i k13!5

~11 i k !~21 i k !

~12 i k !~22 i k !.

~A41!

ys.

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kaluza-klein structure associated with a fat brane

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