Prescriptions for Off-Shell Bosonic String Amplitudes 1

We give, in the framework of the bosonic string theory, simple prescriptions for computing, at tree and one-loop levels, off-shell string amplitudes for open and closed string massless states. In particular we obtain a tree amplitude for three open strings that in the field theory limit coincides with the three-gluon vertex in the usual covariant gauge and two-string one-loop amplitudes satisfying the property of transversality. Talk delivered by one of the authors (R. M.) at the Mid-Term TMR meeting, Corfù (Greece), 20-26 September 1998. Work partially supported by the European Commission TMR programme ERBFMRXCT96-0045 (Nordita-Copenhagen and Università di Napoli). e-mail: Name.Surname@na.infn. it e-mail: marotta@nbivms.nbi. dk Perturbative calculations in string theories involve scattering amplitudes with onshell physical external states. The underlying conformal (and superconformal) invariance imposes that vertex operators have to be primary fields and this condition constrains the external states to be on-shell. However, on-shell conditions, even if very important for the internal consistency of the theory, sometimes turn out to be a serious limitation in practical calculations. One of these limitations appears in the field theory limit or zero slope limit (α ′ → 0) of string amplitudes. It is well-known that in the zero-slope limit the various string models have to reproduce perturbative aspects of ordinary gauge field theories of the fundamental interactions, including gravity. In particular, closed strings can be used to get information on perturbative quantum gravity and its divergences [1], while open strings, with a SU(N) gauge symmetry embodied by Chan-Paton factors, give informations on non-abelian gauge theories. Indeed, instead of computing field theory amplitudes by conventional methods, which are known to be algebraically very complex expecially at high loop order and in quantum gravity, one can compute the corresponding string amplitudes, that have more compact expressions and are much fewer, and then evaluate their low-energy limit [2]÷[6]. In field theory, however, one is also interested in calculating off-shell, gauge dependent quantities such as anomalous dimensions or general Green functions. It is then clear that a reliable prescription for off-shell string amplitudes has to be formulated. Furthermore, once an off-shell prescription has been defined, one can also use it to compute processes involving interactions among D-branes [7, 8, 9]. Therefore, there are many reasons of interest in studying off-shell string amplitudes. Off-shell extensions have been studied in great deal until now [10]÷ [16] and it can be shown that while on-shell string amplitudes are independent from the choice of local coordinate systems defined around the punctures [17, 18], their off-shell extension depends on how that choice is performed. In the following, in the framework of the operatorial formalism, we will give a general prescription for computing off-shell string amplitudes at 1 tree and one-loop levels. We will examine the general M-string amplitude for M massless states and we will propose a simple choice of local coordinates at the punctures on the Riemann surface, such that none of the on-shell conditions on the external states has to be kept. The tree three-gluon amplitude obtained with this procedure, reproduces, in the low energy limit, the usual three-gluon vertex written in the usual covariant gauge while the one-loop string amplitudes obtained in the case M = 2 turn out to satisfy the transversality condition. Finally, we would like also to stress that in the following it is not necessary to use a consistent model of string expecially if the final aim is the exploration of the field theory limit. In fact, as shown in Ref. [19], in order to extract field theory results from string theories one can use the bosonic model and disregard by hand tachyons in calculations. Keeping this in mind we will consider the simplest, but rich enough for further extensions, the bosonic string model and, by using the operatorial formalism, we will obtain off-shell bosonic string amplitudes. The starting point of the operatorial formalism we use, both in closed and open string, is the M-string g-loop Vertex [17] VM ;g, which can be considered as a generating functional for scattering amplitudes among arbitrary string states, at all order of the perturbative expansion. In fact, by saturating the operator VM ;g on M external states |α1 >, · · · , |α2 >, one obtains the corresponding amplitude: A(α1, · · ·αM) = VM ;g|α1 >, · · · , |α2 > . (1) Let us restrict to the case of external massless states of the bosonic closed string. Such states depend both on the polarization tensor ǫ, that we decompose as ǫμν = ξμ ⊗ ξ̄ν , and on the momentum p: |ǫ; p >= N0ǫμνα −1ᾱ −1|p > . (2) where N0 = κ/π, with κ being the gravitational constant in d dimensions [20]. The state defined in (2), according to the symmetry properties of the polarization 2 tensor ǫμν , describes the antisymmetric tensor or a combination of gravitons and dilatons. We first consider the one-loop case. When the operator V1,M is saturated on M massless states (2), it gives the following amplitude [20]: AM ;1 = 1 (2πα′)d/2 ∫