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Volume 183, number 1 PHYSICS LETTERS B 1 January 1987

DILATON TADPOLE FOR THE OPEN BOSONIC STRING "

Michael R. DOUGLAS and Benjamin GRINSTEIN 1

California Institute of Technology, Pasadena, CA 91125, USA

Received 15 September 1986

The amplitude for emission of a single dilaton for world-sheets with the topology of a disk is computed, using Polyakov's formulation of string theory. Combined with previous results this gives the complete amplitude to first non-trivial order in the string coupling constant. The total amplitude vanishes for gauge group SO(8192), while the vacuum energy is negative and independent of the gauge group.

One important unsolved issue in string theory is that of finiteness. It is currently believed that all divergences are infrared in nature, arising when massless particles go on-shell in tadpole contributions. In the open superstring these divergences can lead to anomalies [ 1,2]. In this letter we calculate the dilaton tadpole amplitude for the open bosonic string to lowest non-trivial order. Polyakov's path integral formulation [3-7] offers a convenient approach to string perturbation theory. Moreover it provides the means for calculating directly zero- and one-particle S-ma- trix elements. Using this method, a one-loop calculation of the vacuum energy in the closed bosonic string was recently carried out by Polchinski [8]. In the open stirng there are two topologies which contribute to the first order beyond tree level [2], the projective plane P2 and the disk D 2. The vacuum energy and one-dilaton ampli- tude were computed for world-sheets with the topology of the sphere and projective plane by Grinstein and Wise [9]. The ampfitudes for P2 are the same for the open string as for the unoriented closed string, so we need only calculate the amplitudes for D 2.

Our starting point is the gauge-fixed Polyakov path integral with a sum over all world-sheet topologies. The partition function is, to the order of interest,

Z ~ - ~ [det 'P?e] 1/2 = f [dx] exp [ - (S + Set)]. (1) S2,P2,D2 V(CKV)d

In eq. (1)P is a differential operator that maps vectors v d into second-rank tensors

+ ~a ~b -- gab gccl) Vc Od,

and det' denotes the determinant excluding the zero modes. The factor V(CKV) in the denominator ofeq. (1) is the volume of the group generated by the conformal killing vectors. ~ is the order of the group D of diffeomor- phism classes [8]. The elements of D represent the connected components of the group of conformal diffeomor- phi sms, including those which do not preserve orientation.

The action is

s= f d2}x/~gab +xu(})~xu(}) , (2a)

and the local counterterms are

Work supported in part by US Department of Energy under contract DE-AC 03-81-ER40050. a Tolman Fellow.

52 0370-2693/87/$ 03.50 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Volume 183, number 1 PHYSICS LETTERS B 1 January 1987

The renormalized value of X is the string coupling constant, which enters as X -x, where X is the Euler characteris- tic, so to first order only manifolds with X > 0 will contribute. The/a 2, p and p' counterterms are not conformal- ly invariant (ds is the length element along the boundary and k is its extrinstic curvature) and are chosen to cancel non-conformally invariant terms arising from the regularization of the quantum theory.

To d&me the general open string theory we must include Chan-Paton factors [10]. We define the sum over topologies for the open string as a sum over "topologies with indices". Each connected component of the boundary of a surface is given a group index, which ranges over the values 1 toN. In the sum over topologies we include each surface with specified indices once. This procedure reproduces the Chan-Paton rules for the gauge group SO(N) :as defined in the operator formalism and in string field theory. Boundaries are considered distinguishable, even for a surface with automorphisms which permute the boundaries. This gives the right answer because in the case of a ~urface with symmetry, we have already included the proper symmetry factor in order (D).

The coefficients which weigh the contributions to the partition function from different world-sheet topologies are uniquely determined by requiring a unitary S-matrix. In eq. (1) we have assumed that these coefficients are unity for each topology. One can argue heuristically as follows [11]. Any dependence of these coefficients on the world-sheet topology not already contained in (2) would correspond t0non-local terms in the action. Such a modi- fication would spoil unitarity in the two-dimensional field theory on the world-sheet.

To compute the scattering amplitudes for on.shell strings, we insert vertex operators on the world-sheet. These are to be regarded not as fundamental elements of the theory but as convenient shortcuts to doing the path integral with specified boundary conditions on incoming and outgoing strings. The coupling constants for the vertex operators are determined uniquely in terms of X by requiring factorization of tree amplitudes [12,13].

We will use the conventions of ref. [9]. As shown there, for a tachyon vertex

VT@) = f V xptip'x( )],

the three- and four-point tachyon scattering amplitudes are

An(P1 ..... Pn) = (27026 ~ (171 + "" +Pn)an, (3a)

a3(Pl ,P,2 ,P3) = X-2 @Te2)3 Qs, (3b)

a4(Pl,P2,P3,P4 ) = 7rX_2(KT e2~4 ,q i"(3 -- (1/8r 0 (t + s))P(t/8rr -- 1)I'(s/87r" 1) ' ~esp((1/S~r) (s + t) 2 23r(2 - t/8~r)r(2 - slS~r) ' (3c)

where s = (Pl +P2) 2 and t = (Pl +P3) 2,

Qs = 212 (3/16zr) 3 [det'P?P] 1/2 [det'(_V2)]-13

is a normalization factor including the volume of conformal Killing vectors, zero mode contributions, and a factor of 1/2 for d', and e is a short-distance cutoff. Qs depends on our conventions for the functional measures, but physical quantities only depend on the combination Xp = XQ~ 1/2 . The normalization of/~T iS determined by com- paring (3b) with the tachyon pole in (3@

~T = (87r2) 112 XQs 112 e -2 .

The dilaton vertex operator is

( ~__~_.24 R x /~ exp(ip .x ) , (4) xu ~b xv 16rr 7re-2 ]

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Volume 183, number 1 PHYSICS LETTERS B 1 January 1987

where [14]

e~V (p) = (1/X/r~ (~v _ p~V _ pV~U ) ,

and p /) = 1,p2 = if2 = 0. The counterterms in V D are needed to preserve conformal invariance in scattering am- plitudes involving the dilaton. A calculation similar to the tachyon case gives

/~D = 8~'2 ~Q~l /2

To define thepath integral on the disk, we must specify the boundary conditions. The physical condition that momentum does not flow out of the boundary leads to Neumann boundary conditions on the coordinates x~,

na l)aX~t [ aM = O,

where n a is the normal vector to the boundary. In terms of the tangent vector to the boundary t a,

n a=eabgbct c .

Note that n a depends on the world-sheet metric. Unless we choose our boundary conditions on the metric varia- tions correctly, the boundary conditions will couple x~ and gbc. The correct choice is

8gab [aMnat b = O .

This insures that metric variations do not change the definition of the normal vector n a . This condition can also be expressed as a boundary condition on v a, the vector fields which generate infinitesimal diffeomorphisms. Since ~gab = V (ab ), we have

nat b V(aOb) = O.

To completely define the operator P~P we need a boundary condition for each component of v a. The second boundary condition comes from the requirement that the infinitesimal diffeomorphisms preserve the boundary

naoa = O .

These two boundary conditions ensure that the operator P~P is self-adjoint and positive [5 ]. If we work in the critical dimension, the conformal factor of the metric drops out, and we do not need a boundary condition for it [151.

The calculation of scattering amplitudes for the disk now proceeds in parallel to that for the projective plane. In fact the two calculations are so similar that we will be able to produce a single explicit expression for the ampli- tude which contains both possibilities simultaneously. We represent the disk as the Riemarm sphere with points identified if they are images under reflection in a plane passing through the equator. Thus the equator corresponds to the boundary of the disk. In complex coordinates this map is z ~ 1/~. We can represent the projective plane in a similar way by identifying z with - l [ z . In both cases we use the metric induced by stereographic projection from the sphere in euclidean three-space,

ds 2 = 4dzdz/(1 + Iz 12) 2

We will use the symbol D 2 to refer to the disk, P2 to the projective plane, and M tO refer to either. The parameter k will be +1 for D 2 and -1 for P2, so our identification is now z with k/'~.

We first discuss the group of globally defined conformal transformations. For P2 this is SO(3) [9]. For the D 2 case this is well known to be SL(2;R). This is a non-compact group, and has infinite volume. We therefore have the result that the zero-point amplitude, or vacuum energy, due to the disk is zero. Therefore, the entire contribu- tion to the vacuum energy in order X is from the P2 topology, and we f'md that the vacuum energy is non-zero and independent of the choice of gauge group.

We now consider the one-point dilaton amplitude. The relevant group here is the group of globally defined conformal transformations which preserve the location of a single point, the location of the vertex operator. If we

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PHYSICS LETTERS B 1 January1987 Volume 183, number 1

place the vertex at the center of the disk, we see that the remaining transformations are simple rotations, a group with finite volume. We thus reach the