The Casimir energy of a stiff p-brane

Volume 247, number 1

PHYSICS LETTERS B

6 September 1990

The Casimir energy of a stiffp-brane Sergei D. Odintsov Department of Physics and Mathematics, PedagogicalInstitute, SU-634 041 Tomsk, USSR and Department of Physics, Lancaster University, Bailrigg LA 1 4 YB, UK Received 13 March 1990; revised manuscript received 8 June 1990

The one-loop Casimir energy for a stiff p-brane is calculated and compared with that of the standard bosonic p-brane. We observe that the effective potential for a toroidal still 2-brane has maxima rather than minima, while the effective potential of a standard toroidal 2-brane or pure stiff 2-brane has neither maxima nor minima.

A generalization o f the pure area N a m b u - G o t o action for a string to an action containing a stiffness t e r m has been suggested [ 1 ], both with a view to applications in q u a n t u m c h r o m o d y n a m i c s a n d to investigations o f the fluctuations o f physical surfaces. It is also possible to construct a bosonic m e m b r a n e ( o r p - b r a n e ) with rigidity, giving a stiff m e m b r a n e or stiffp-brane. ( N o t e that a review o f usual m e m b r a n e properties is given, for example, in ref. [ 2 ]. ) Some o f the classical properties o f a s t i f f p - b r a n e were investigated in ref. [ 3 ]. In the present letter we want to discuss some o f the q u a n t u m properties o f a stiffp-brane, in particular calculating its C a s i m i r energy. The action we consider is -2P - 7 [ A ( g ) X i] z ,

( 1)

where i = 1..... d, the metric is the i n d u c e d one gap = 0~ X i 0pX i, ot = 0 ..... p a n d the scalar laplacian is given by

A ( g ) =g-'/20~,g'/Zg~'P Op.

(2)

The constant k is the string tension and we recover the usual bosonic p-brane action when the stiffness coupling 1/p2 is zero. It can be shown that there are d - p - 1 physical degrees o f freedom. We shall consider the following classical solution o f the field equations (which is the same as for the bosonic p-brane [4,5] ): X ° l : ~o,

Xc~:0,

Xcdl- ' = ~1 . . . . .

Xcd-p:~p,

(3)

where

X~=(X

l ..... X d - P - ~ ) ,

(~1 ..... ~ p ) e R = [ O , a , ] × . . . × [ O , a v ] .

(4)

We now i m p o s e the b a c k g r o u n d gauge X

0 ~.-.~X c0l ,

xd-I

d--I

-- Xcl

,

...,

yd--p__yd--p ix

- - 1~. cl

(5)

which does not give rise to any F P ghosts [ 4 ]. We shall consider both fixed-end b o u n d a r y conditions (following refs. [4,5 ] ) X ± (0, ~,, ..., ~ p ) = X l (T, ~, .... , ~ p ) = 0 ,

(6)

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Volume247,number1

PHYSICSLETTERSB

6 September1990

and X ± (~0, O, ~2 ..... ~p) . . . . .

X ± (~0, ~1 , ---, ~p--l, O)

= X ± (~o, al, ~2, ..., ~p) . . . . . X ± (~o, ~1, ..-, ~p--l,

ap) = 0 ,

(7)

and those appropriate for a toroidal p-brane, where eq. (7) is replaced by x I (~o, o, ~2, ..., G ) = x i

(~o, a , , ~2, ..., G ) ,

X ± (~o, ~ 1 , 0 ..... ~p) r e X ± (~o, ¢1, a2, ..., ~ p ) , X J- ( ~ o , ~ 1 , . . . , ~ p - 1 , O ) = X

J- ( ~ o , ~ 1 , . . . , ~ p - l ,

(8)

ap) .

The effective potential can now be calculated from V=-lim-ln

if

T~oo r

~X ±exp(-S)

(9)

and if we limit ourselves to a one-loop calculation we need only expand the action in eq. ( 1 ) to quadratic order in the fields,

S(2)=

fdp+,4(k_k

A X ± + - -±A1X 2 p

~X

AX ± ) ,

(10)

where A=0~0~. The one-loop approximation to the bosonic m e m b r a n e (i.e. without stiffness) was considered in ref. [4] and to the bosonic p-brane in ref. [5]. We can integrate out X ± and use the boundary conditions to evaluate the resulting t r l n A terms (see, for example ref. [ 5 ] ) giving, for the fixed-end stiffp-brane (boundary conditions eqs. (6), (7) ) Vfixedend=kA.t.

d-p-1

[

2

where A = a i ×... ×

nl,...,np=I

ap, and

Vtoroidal= kd + d - p - ~ l

2

{ g2n12

~

R2n2x~1/2

\ a~-l +'"+ a2 I

+

~

{ g2n12 1 k ~-p

7~2n2 -I-kp2)1/21 ap

"Jf-'""}- ~

nl,...,np=

(11) '

for the toroidal stiffp-brane (boundary conditions eqs. (6), ( 8 ) )

~

nl,...,np= --oo

(492n2 - 2n p2 ,"1/2 ~4~r ~x~-12 +'"+ ~ - p 2 ] q-

~

nh...,np=

4:R'an2 +kp2~'/2]. k ~ 1 2 q-..-q- ~ (41r2n 2

~

ap

,I

J

(12) If k = 0 then only the rigidity term is present and if 1/p2 = 0 we recover the usual bosonic p-brane as considered in ref. [5]. Let us consider the simple case where al . . . . . ap = R. When 1/p 2 = 0 we find the effective potential for a usual bosonic p-brane (see ref. [ 5 ] )

Vv' T =kRP+ zc(d-p2R 1 ) f ~ , v ( 1 ,

..., 1 ) ,

(13)

where the subscripts refer to fixed-end or toroidal boundary conditions and f ~ ( 1 ..... 1 ) =

[n2+...+n2] '/2,

~

fer(1 ..... 1 ) = 2

n± ,...,rip = 1

~

[nZ+...+n2] '/2

(14)

n l ,...rip= --oo

are calculated in ref. [ 6 ] (for p ~<8 ) with the help of ~-regularization [ 7 ]. The corresponding values are f~=0.026, fer=-0.46, 22

0.052, -0.53,

0.048, -0.59,

0.036, -0.65,

p = 2 , 3, 4, 5, p=2,3,4,5.

(15)

Volume 247, number 1

PHYSICS LETTERS B

6 September 1990

Thus, for a usual toroidal p-brane ( 1/p2 = 0) f e r < 0 and the p-brane tends to collapse. For a fixed-end usual p-brane (with 1/p 2= 0) there is an equilibrium between the attractive classical force and the repulsive Casimir force at a radius [ 5 ] Ro=[(d-p-1)nf~(1

.... ,1)(2kp)-~] l/p+l.

(16)

The same analysis (as well as the calculation of the sums in ( 11 ), (12) ) is more complicated for the stiffpbrane. Let us now consider a pure stiffp-brane (without N a m b u - G o t o term). Then

Vv,w-

n(d-p-1) R f~,T(1,...,1)

(17)

(the classical force is absent). The potential VF > 0 and Vx < 0. The potential has neither m a x i m a nor minima. Let us consider a stiff p-brane with N a m b u - G o t o term (k ¢ 0). For simplicity we limit ourselves to even p only. The regularized sums can be defined by using analytic continuation [ 8 ]:

(M2 + n~ +... + n2p)-'/2 nl,...,np= --oo

K(p_,)/z(2z~M[n 2 +...+ -

r(r/2)

F

+2

.......=-oo

nZl 1/2)] j'

(18)

where K(p_r)/2 is a modified Bessel function, F i s the Euler g a m m a function and M2=kp2R2/4zc 2. We thus find (for even p)

VT=kRP + n ( d - p - l ) f % ( 1 , . . . , 1 ) 2R

[ (~f__~)

(dIp-1)Mp+I

+

RF(-½)

zc'p+2)/2 F

~'

+2,,,...,no=_oo

g(p+l)/2(27cM[I'12[-"'31-n2]l/2)

~--M--(~[~(Tgi)/2

1

.

(19)

Let us represent the sums on the right-hand side of eq. ( 18 ) in the form where the n~ change from 1 to ~ . Then one can obtain the expression for VF. For example, i f p = 2 then

(M2+n2+n2)l/2=4 nl ,n2 = --oo

~

(M2+n2)l/2+4 ~ (M2+n21+r122)'/2+M.

n= 1

(20)

hi,n2= 1

Making use of ( 18 ), (20) in ( 11 ) we get

Vv=kR2+ r e ( d - 3) f2v(1, 2R

1) "'"

+ -( d -- 3 ) r c [ 87rM 3 ( F ( _ 3 ) + 2 8R LF(-1)

~'

K3/2 (47~M[ n2 +___/'/22]_1/2)'~

.... 2=-oo (2zrM[n~+n~]'/2)3/2,] - 2 M - 4

n=l

]

~ (n2+4M2)l/2 "

(21)

The last sum in (21 ) can be easily calculated by the ff-regularization method [ 7 ]. In the same way as above one can obtain VF for other p. Let us analyse V-r(R) and present the corresponding graphs for p = 2. (We do not consider such an analysis for Vv. ) The result of this analysis is given schematically in fig. 1. The first graph (I) corresponds to the usual 23

Volume 247, number 1

PHYSICS LETTERS B

6 September 1990

V-

]qT a

>lq

Fig. 1.

bosonic m e m b r a n e ( 1 / p 2 = 0). The second graph (II) corresponds to the pure stiff m e m b r a n e ( k = 0). The graph IIIa corresponds to the stiff m e m b r a n e ( k s 0, p - 2 ~ 0 ) if k~/2p 3( d - 3 ) < 12zc and IIIb corresponds to the stiff p-brane ( k s 0, p - 2 ~ 0) if k ~/2p 3( d - 3 ) > 12re. We see that the graph for a stiff m e m b r a n e can contain a global m a x i m u m (if k~/2p 3 ( d _ 3 ) > 12re) rather than a global m i n i m u m . (There is a local m i n i m u m only for the IIIa graph. ) It follows from the above analysis that the v a c u u m is unstable (but does not contain tachyons). Consequently the classical solution (6), (8) considered here is not a ground state for the usual bosonic or stiff membrane. In conclusion we note that it would be interesting to construct supersymmetric stiff p-branes a n d consider their classical a n d q u a n t u m properties: consistency (cf. ref. [ 9 ] ) and the calculation of the static potential. I would like to thank Professor R.W. Tucker and Dr. D. Wiltshire for discussions and Dr. D. Johnston for reading the manuscript. I am grateful to the referee of this paper for very useful remarks.

References [ 1] A.M. Polyakov,Nucl. Phys. B 268 (1986) 406. [2] E. Bergshoeff, E. Sezginand P.K. Townsend, Ann. Phys. (NY) 185 (1988) 330; M.J. Duff, T. Inami, C.N. Pope, E. Sezginand K.S. Stelle, Nucl. Phys. B 297 (1988) 515; M.J. Duff, Class. Quantum Grav. 6 (1989) 1577. [3] D.H. Hartley, M. Onder and R.W. Tucker, Class. Quantum Grav. 6 (1989) 30. [4 ] E.G. Floratos, Phys. Lett. B 220 ( 1989) 61 ; E.G. Floratos and G.K. Leontaris, Phys. Lett. B 223 (1989) 137. [ 5 ] S.D. Odintsov and D.L. Wiltshire, The static potential for bosonic p-branes, Newcastle preprint NCL-TP-90/2. [6 ] J.S. Dowker, Zeta-functions on spheres and in cubes (Manchester, 1983), unpublished. [ 7 ] S.W. Hawking,Commun. Math. Phys. 55 (1977 ) 133; J.S. Dowker and R. Critchley, Phys. Rev. D 13 (1976) 3224. [8] D. Birmingham and S. Sen, Ann. Phys. 161 (NY) ( 1985 ) 121. [9] A. Achucarro,J.M. Evans, P.K. Townsend and D.L. Wiltshire, Phys. Len. B 198 (1987) 441.

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