KKdecomposition_kalbramond

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spin-2, spin-1/2 and spin-3/2 zero mode elds localization on the brane on aglobal string-like defect [7, 13]. The KK massive modes for spin-0 and spin-1elds have been studied on a local string-like defect [12].

We aims to analyze the mechanisms of eld localization for the Kalb-Ramondtensor eld in a string-like defect, AdS 6 space. This will be a natural extensionof the works done in AdS 5 space.

2 String-like Defect: a briey review

The rst works on brane world in six dimensions [1], [17], [19] followed theusual Kaluza-Klein compactication . The emergence of the so called ADDmodel [2, 3, 4] and Randal-Sundrum model [14, 15] give us other classes of alternatives to one.

The work of Randal-Sundrum was extended to more than ve dimension assoon as it appears in literature. Particularly, in six dimension, we can citesome authors that contributed in a space of less than a year from the originalRandal-Sundrum work appearance [5, 8, 7, 13]. We now summarizes the string-like solution of the Einsteins equation.

We begin by considering the general metric ansatz in D-dimensional space-time

ds2 = gMN dxM dxN

= g dxdx + gabdxadxb

= eA(r )g dxdx + dr 2 + eB (r )d2n1 (1)

Where M,N,... denote D-dimensional space-time indices, ,,..., p-dimensionalbrane ones (we assume p 4), and a,b,... denote n-extra spatial dimensionsones.

The action we assume in this work is given by

S =1

22D dD

xg(R 2) + dD

xgLm (2)where D is the D-dimensional gravitational constant, is the bulk cosmologicalconstant and Lm is some matter eld Lagrangian.

The Einsteins equations are obtained by variation of the action (2) withrespect to the D-dimensional metric tensor gMN

RMN 12

gMN R = gMN + 2D T MN (3)

We adopt, for the energy-momentum tensor, T MN , the following ansatz,T =

t0(r ), T

rr = t r (r ), T

22 = T

33 = ... = T

nn = t(r ) (4)

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where t i(i = o,r, ) are function only of r , the radial coordinate. With thisansatz to the energy-momentum tensor we keep spherical symmetry.

The next step is to use the ansatzs (1) and (4) to rewrite the Einsteinsequations (3). A straightforward calculation give us the following results

eA(r )R p(n 1)2 A (r )B (r ) p( p1)4 (A (r ))2+

(n 1)(n 2)

4(B (r ))2 + ( n 1)(n 2)e

B (r )2 + 2

2D t r = 0 (5)

eA(r )R + ( n 2)B (r ) p(n 2)

2A (r )B (r )

p( p + 1)

4(A (r ))2

(n 1)(n 2)4

(B (r ))2

+( n 2)(n 3)eB (r ) + pA (r ) 2 + 22D t = 0 (6) p 2

peA(r )R + ( p 1) A (r )

n 12

A (r )B (r )

p( p1)

4(A (r ))2 +

+( n 1) B (r ) n(B (r ))2

4+ ( n 2)e

B (r )

2 + 22D t0 = 0 (7)Another equation is provided by the energy-momentum tensor conservation law

t r =p2

A (r )( t r (r ) t0(r )) +n 1

2B (r )( t r (r ) t(r )) (8)

where prime denotes differentiation with respect to r . The scalar curvature Rassociated with the brane metric g and the brane cosmological constant pare related by

R 12g = pg (9)It is possible to derive the functions A(r ) and B(r ) from scalar elds, see

[10], for example, but here we assuming the ansatz A(r ) = c.r , where c is aconstant, for the warp factor. Now restricting ourselves to six dimensions andsetting n = 2 the equations (5), (6), (7) and (8) assume the respective simplerforms

ecr R p2

cB (r ) p( p 1)

4(c)2 + 2 + 2

2D t r = 0 (10)

ecr R p( p + 1)

4(c)2 2 + 2

2D t = 0 (11)

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p 2 p

ecr R p 1

2cB (r )

p( p1)4

(c)2

+ B (r ) (B (r ))2

2 2 + 22D t0 = 0 (12)

t r = p2c(t r (r ) t0(r )) + 12B (r )( t r (r ) t(r )) (13)From this equation we can get the general solution for the metric:

ds2 = ecr g dxdx + dr 2 + eB (r )d2 (14)

whereB(r ) = cr +

4 pc

2D dr (t r t) (15)c

2

=1

p( p + 1) (8 + 82D ) (16)

andt = ecr + (17)

where and are constants and has to satisfy the inequality 8+82D > 0.This is a general result. It is possible to derive two special cases from thegeneral solution above. One of them, the global string-like defect, occur whenthe spontaneous symmetry breakdown , t r = t is present

ds2 = ecr g dxdx + dr 2 + R20ec1 r d2 (18)where R20 is a constant. In this case we have

c1 = c 8

pc2D t (19)

c2 =1

p( p + 1)(8 + 8

2D ) > 0 (20)

In the case t = 0 we a found a more simplied solution

ds2 = ecr g dxdx + dr 2 + R20ecr d2 (21)

withc2 = 8

p( p + 1)(22)

This solution, that can be found setting t i(r ) = 0, was rst found by [7, 8].Note that the general solution (5) and the special cases (9) and (12) representa 4-brane embedded in a six dimensional space-time. The on-brane dimension

, 0 2 is compact and assumed to be sufficiently small to realize the3-brane world. The other extra dimension r extend to the innite, 0 r .

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3 Localization of the Kalb-Ramond eld

In this section we will study the mechanism of eld localization for the Kalb-Ramond eld in the background geometry (18). We will analyze both the zeromode and massive modes cases.

3.1 The zero-mode case

In this subsection we will show that, for the Kalb-Ramond eld, the zero modeis localized on a string-like defect since the constants c and t respect somespecic inequality conditions.

From the action of the Kalb-Ramond tensor eld

S m = 112

dD xgg

MQ gNR gLS H MNL H QRS , (23)

we derive the equation of motion

Q[gH MNL gMQ gNR gLS ] = 0 (24)

A straightforward calculation give us the following form for the equation of motion

P 1(r ) H + P p/ 2+2 (r )Q1/ 2(r ) r [P p/ 22(r )Q1/ 2(r )H r ]+ Q1(r ) H = 0

(25)

where we redene g = , eA(r ) = P (r ) and R20eB (r ) = Q(r ). Although weknow that A(r ) = cr and B (r ) = c1r , we keep the forms A(r ) and B(r ) becausethis is more general.

Let us assume the gauge condition B r = B = 0 and decomposition

B (xM ) = b (x) m(r )eil (26)

B r (xM ) = br (x) m(r )eil (27)

By choosing h

= m20b

we get the following equation for the radial variable

2r m(r ) + ( p2 2)

P (r )P (r )

+Q (r )2Q(r )

r m(r ) +1

P (r )m20

1Q(r )

l2 m(r ) = 0

(28)This equation has the zero mode ( m0 = 0) and s-wave ( l = 0) and br =constant 1 solution. If we substitute the constant solution in the action (23),and remember the forms of A(r ) and B (r ), the r integral reads

I 0 drP p/ 2

3Q

1/ 2

dre [( p/ 2

3)c+1 / 2c1 ]r

(29)1 Note that b r need not to be constant. This is a special case to get localization similar to the ones in scalar and vector elds

[12]

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We need I 0 to be nite. It is possible to rewrite this condition as an inequality,for c > 0,

12D

< t < ( p5)62D (30)and

t > ( p5)62D (31)for c > 0.

3.2 The massive modes case

Now we are going to analyze the massive modes for the Kalb-Ramond eld.We will study only the local defect (21) leaving the case of global defect (18)to another work.

We begin by considering equation (28) and performing the change of variablesz = 2 /cM nP 1/ 2 and = P ( p3)/ 4h. This give us the following Bessel equationof order ( p3)/ 2

d2hdz2

+1z

dhdz

+ 1 1z2

p32

2

h = 0 (32)

The solution to this equation is directly, so the radial function is given by

(z) = P ( p3)/ 4 1J

( p3)/ 2(z) +

2Y

( p3)/ 2(z) (33)

where 1 and 2 are constants.We now impose the boundary conditions

(0) = () = 0 . (34)For z >> 2 1/ 4 the asymptotically form of the Bessel function is (citararfken)

J (x)

2

xcos x

4(2 + 1) (35)

By other side, the KK masses can be derived from the formula [12]

J ( p5)/ 2(z(r )) = 0 (36)

where r is the infrared cutoff which will be extended to innity in the end of calculations. Finally using (35) and (36) we get the approximate mass formula

M n = c2

n +p4

32

ecr/ 2 (37)

From (37) we see that in the limit r , the KK masses of the Kalb-Ramond eld depends on l2/R 2o, so the only massless mode is the s-wave l = 0,consequently the other are massive.

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4 Discussions

In this work we study the Kalb-Ramond tensor eld in a string like model.Beginning by the action for the KR eld we derived the equation of motionand study the possibility to to have zero mode localization on the brane and

additionally we calculate the KK mass spectra.Unlike the case studied in ve dimensions where the Kalb-Ramond is not

localized only by gravitational interactions [18], in the case in study we havezero mode localization. However, while for the scalar and gauge elds thelocalization, in the local defect, occur for c > 0, in the KR eld case it isnecessary to have c < 0 to obtain zero mode localization.

The mass spectra for the KR eld is similar to the one encountered forthe scalar and gauge elds [12] differing from them only by 1 and 1/ 2,respectively, in the quantity between parenthesis, in equation (37).Our next next step is to look for the extension of this work, particularly, thecalculations of the mass spectra, to the global defect case. It is important tosay that in this context there is no similar work, so it is interesting to performthis work, not only to the Kalb-Ramond eld but for the other SM elds.

5 Acknowledgments

We wish to thank Departamento de Fsica of the UFC, LASSCO, CAPES andSEDUC-CE.

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