PHYSICAL REVIEW D VOLUME 35, NUMBER 2 15 JANUARY 1987

Vacuum energy and dilaton tadpole for the unoriented closed bosonic string

Benjamin Grinstein and Mark B. Wise California Institute of Technology, Pasadena, California 91125

(Received 25 August 1986)

In oriented-closed-bosonic-string theory a dilaton tadpole (and vacuum energy) first develops at the one-loop level from performing the path integration over manifolds with the topology of the torus. We show that in the case of the unoriented closed bosonic string the leading contribution to the dilaton tadpole arises at the tree level from the path integral over manifolds with the topology of the projective plane. We explicitly compute the vacuum energy and the dilaton tadpole using Polyakov's formulation of string theory.

~ o l ~ a k o v ' s ' path-integral formulation for string theory provides a convenient method for computing string-theory S-matrix elements. Recently considerable progress has been made in understanding the appropriate measure for the integration over metric^.^-' polchinski6 has used the path-integral formulation to compute the leading contri- bution to the vacuum energy for the closed oriented bo- sonic string. It arises from performing the functional in- tegration over manifolds with the topology of the torus. ohm' has also computed the one-loop vacuum energy using the operator formalism. In this paper we note that in the case of closed-unoriented-bosonic-string theory the leading contribution to the vacuum energy occurs at the tree level from performing the functional integration over manifolds with the topology of the projective plane (i.e., the sphere with opposite points identified). The vacuum energy and the resulting dilaton tadpole are explicitly computed.

The path integral for closed-unoriented-bosonic-string theory8 contains a sum over both orientable and unorient- able two-manifolds. For our calculation only the contri- butions from manifolds with the topology of S2 and P 2 need to be considered. Then the partition function is

and the local counterterms are

The constant p2 is chosen so that conformal invariance is preserved. The second term in (3b) gives rise in Z to a factor of for S2 and a factor of h-' for P2 since

is the Euler number of the manifold. Connected parts of S-matrix elements are computed by

inserting the appropriate vertex operators into the path in- tegral. Conditions that the vertex operators must satisfy in the oriented-string case have been given by ~ e i n b e r ~ . ~ To these we must add the condition that the vertex opera- tors be definable on P2. This condition forbids the vertex operator for the antisymmetric tensor field, for example.'' Therefore in the unoriented bosonic string the states with mass squares less than or equal to zero are the tachyon,

( d e t ' ~ ~ P ) ' / ~ dilaton, and graviton. Z? 2 S [ d x ] e x p [ ( s + s c ~ ) ] ( I ) The leading contribution to tachyon scattering ampli- s 2 , p 2 ZvcKv tudes comes from the contribution of S2 to Z . To set our

conventions we briefly review the calculation of the In Eq. (1) P i s a differential operator that maps vectors vd sphere's contribution to tachyon scattering amplitudes. into second-rank tensors, The vertex operator for the tachyon is

and det' denotes the determinant excluding the zero ~ ~ ( ~ ) = ~ ~ J d ~ f g i e ' ~ ' * . modes. The factor V,,, in the denominator of Eq. (1) is the of the group generated the Kil- The contribution of S2 to the n-point tachyon scattering ling vectors, 2 is the order of the group6 5 of diffeomor- amplitude is given by phism classes. The elements of 5 represent the connected components of the group of conformal diffeomorphisms, including those which do not preserve orientation. For An(p l , . . . ,pn ) =

( d e t ' ~ 'P)' I2

the sphere 2 = 2 while for the projective plane a= 1. The v c ~ v action is

656 BENJAMIN GRINSTEIN AND MARK B. WISE

We perform the functional integration over x p by expand- ing in normalized eigenfunctions of the Laplacian on the unit sphere:

using

There is arbitrariness in the choice of measure that can- cels out in physical quantities.3.6'" Since the action is quadratic in the variable x p the functional integral over [dx] can be evaluated yielding

where

18b) In Eq. (8b) ( x p ( C i )x,(C, ) is the propagator excluding the zero modes

(9) It is convenient at this point to map the sphere onto the

complex plane by stereographic projection onto a plane cutting through the sphere's equator. The northern hemi- sphere is mapped onto the region / z > 1 and the south- ern hemisphere is mapped onto the region z 1 < 1. In complex coordinates the propagator is

Here c2 is an invariant short-distance cutoff and we have written the metric as g,=e*. Dependence on the choice of conformal gauge cancels if the tachyons are on the mass shell:

p i2=8v, i = l , . . . , n

Then we find that

The integration over three of the complex z's is equivalent to an integration over the invariant measure for the group generated by the conformal killing vectors. This is the group SL(2,C) of transformations

The integrand in Eq. (1 1) is a scalar density with respect to these transformations. For infinitesimal transforma- tions

Eq. (12) becomes

The infinitesimal diffeomorphisms generated by the con- formal Killing vectors u p take the form SF= ~ , a , u ~ The volume of the group generated by the conformal Kil- ling vectors VCKV is defined to be the integral over the in- variant group volume normalized so that when the up are

I

chosen orthonormal it reduces to 11, d a , near the identi- ty. Using the metric

I u j 2 = J d 2 < ~ F g a b ~ 0 ~ b 11 5 )

we find that the six properly normalized conformal Kil- ling vectors, using the notation

are

It follows that the invariant group measure near the iden- tity is

3 5 - VACUUM ENERGY AND DILATON TADPOLE FOR T H E . . . 657

Trading the integrations over z , , z2, and z3 for an in- (17) tegration over dg (which cancels the factor VCKV in the

denominator) we find that

where

and the values of z , , z2, and z 3 are arbitrary. In particular the three- and four-point tachyon scattering amplitudes are

Q , ( P I , P ~ , P , ) = A - 2 ( ~ ~ ~ 2 ) 3 ~ ~ , (20a)

where s = ( p l +p2 ) 2 and t = ( p +p3 12. The four-point amplitude has a tachyon pole

which implies by factorization that

A = 1 a3 1 ~ ~ ~ / ~ ( 8 n - ~ ) - ~ ' ~ . The dilaton vertex operator is

and p .p= 1, p 2 = p 2 = 0 . The counterterms in VD are needed to preserve conformal invariance in scattering am- plitudes involving the dilaton. The normalization of K D is determined by computing the dilaton-tachyon-tachyon scattering amplitude and comparing it with the dilaton pole in Eq. (20b). We find that

To compute the contribution of the manifold P2 to scattering amplitudes it is necessary to expand xp in nor- malized eigenfunctions of the Laplacian on P2, the unit sphere with antipodal points identified. Since x p is a sca- lar field, it must take the same value at antipodal points, and for P2 we expand

1 xp=fl2 2 cLYlrn(g)

1 even m = - 1

and use the measure in E q (7) for the functional integral over xp. The factor of d 2 in Eq. (26) occurs because the Yl, ( 6) have normalization one-half on P2 .

A vector field u a on S2 is defined on P2 if each of its integral curves is mapped into itself by the identification of antipodal points. Suppose in some coordinate system points p on the unit sphere have coordinates and their antipodal points - p have coordinates ga(&). Then the condition for a vector field u a ( & j to be well defined on P2 is

where the matrix D,b(c) is given by

For the complex coordinates z the map g takes z into - l f i a n d so

In these coordinates a vector field is defined on P2 if

The group generated by the conformal Killing vectors is not the same on P 2 as on S2 . Only those transforma- tions in (12) which satisfy ( - 1 f i ' ) = ( - 1 D)' are defined on P2. This restricts the transformations (12) to those that satisfy

Hence on P 2 the group generated by the conformal Kil- ling vectors is SO(3) (Ref. 13). This corresponds to the group of rotations on the sphere which clearly leave an- tipodal points identified. Writing A =( 1 - 1 B / 2)1/2eia the invariant group measure is proportional to d 2 ~ d a .

65 8 BENJAMIN GRINSTEIN AND MARK B. W I S E - 3 5

Near the identity the transformations (30) are

6 z = ~ ~ ( l + z ~ ) + i ~ ~ ( l - z ~ ) + 2 i a z , (31)

where B = B + iB2. The normalized conformal Killing vectors are

and so the group volume is

The finite volume for SO(3) gives rise to a qualitative difference in the contribution to amplitudes from S 2 and P2. The manifold P, will give a nonzero contribution to the vacuum energy (the zero-point amplitude) and to the dilaton one-point function.

According to Eq. ( 1 ) the contribution o f P2 to the vacu- u m energy density is

3 / 2

vac

Since the determinants and h are positive the vacuum en- ergy is negative. Using Eqs. (19) and (22) the dependence

on A. can be replaced by dependence on the physical three tachyon scattering amplitude a,. This gives

( d e t ' ~ ' ~ ) ~ ~

6 vac =-L[:] l a3 I [ ( d e t ' ~ ~ ~ ) ~ ~ ] ~ / ~

where the subscripts have been placed on the determinants to signify what manifold the operator is defined on.

T o compute the ratio o f determinants o f P I P we note

It follows that on S 2 the eigenvectors o f P'P are

with eigenvalues 2( 1 - 1)(1+2). In Eq. (37) cab is the an- tisymmetric symbol. For P2 only vector fields which satisfy u e ( p ) = - u '( -p and ) = u4( -p ) are adrnissi- ble. Hence for odd I only

is allowed and for even 1 only

is allowed. On P2 every eigenvalue o f P I P that occurs on S 2 is present but its multiplicity (21 + 1) is only half what it is on S 2 . Therefore

( d e t ' ~ ' ~ ) ~ , = [ ( d e t ' ~ ~ P ) ~ , ] ~ / ~ . (38)

Next we consider the ratio

It is straightforward to relateI4 the logarithm o f r to the Riemann 5 function S R ( s ) and its first derivative evaluated at s=o:

Thus r = b'7T/2 and our expression for the vacuum energy density simplifies to

1 / 2

- 2 x 4 x 6 X X X . . . 1 ~ 3 X 5 X 7 X . . .

[detl( - ~ ~ ) ] p ~ - - r =

{[det1( - v ~ ) ] ~ , ) ~ / ~

T o compute the dilaton tadpole the propagator ( ~ ~ ( 6 ~ ) x v ( c j ) is needed on P2. It satisfies

n [ 1 ( 1 + 1 ) l 2 ' + ] 1 even

[ I ( , + 1 ) I 2 ' + ' I odd

The factor o f - 1 / 2 ~ occurs because the zero-mode contribution is omitted. In complex coordinates

3 5 - VACUUM ENERGY AND DILATON TADPOLE FOR T H E . . .

Performing the functional integral with a single insertion of the dilaton vertex operator gives

Using Eqs. (251, (38), and (40) this becomes 3 84 I /2 yeff= IF] J d 2 6 x d z ) ~ ( ~ ) . (47) 1 /2

(46) There is a sign ambiguity fo r the coefficient which can be absorbed into the sign of the dilaton field.

Therefore, the effective action for the dilaton field D con- We thank S. Das, J. Polchinski, S . Wadia, and our advi- tains a termI5 sor R . R o h m for useful discussions.

'A. M. Polyakov, Phys. Lett. 103B, 207 (1981). 2D. Friedan, in Recent Advances in Field Theory and Statistical

Mechanics, 1982 Les Houches Lectures, edited by J. Zuber and R. Stora (North-Holland, Amsterdam, 1984).

30. Alvarez, Nucl. Phys. B216, 125 (1983). 4G. Moore and P. Nelson, Nucl. Phys. B266, 58 (1986). 5E. d'Hoker and D. Phong, Nucl. Phys. B269, 205 (1986). 6J. Polchinski, Commun. Math. Phys. 104, 37 (1986). 'R. Rohm, Nucl. Phys. B237, 553 (1984). 8C. Burgess and T. Morris, Princeton report, 1986 (unpublish-

ed). 9S. Weinberg, Phys. Lett. 156B, 309 (1985). I0For the coordinates xp to be defined on P, they must satisfy

xp(z)=xp( - 1E) . Then

"W. Weisberger, Report No. ITP-SB-86-22, 1986 (unpublished). I2J. Scherk and J. Schwarz, Nucl. Phys. B81, 118 (1974). I3This conclusion disagrees with Ref. 8. I4E. Whittaker and G. Watson, A Course on Modern Analysis

(Cambridge University Press, London, 1965). ISThe dilaton kinetic term is assumed to be normalized in the

conventional fashion.