Zero-slope limit of the compactified closed bosonic string.

In the framework of the compactified closed bosonic string theory with the extra spatial coordinates being circular with radius R, we perform both the zero-slope limit and the R → 0 limit of the tree scattering amplitude of four massless scalar particles. We explicitly show that this double limit leads to amplitudes involving scalars which interact through the exchange of a scalar, spin 1 and spin 2 particle. In particular, this latter case reproduces the same result obtained in linearized quantum gravity. ∗This work is carried out in the framework of the E.C. Research Programme “Gauge Theories, Applied Supersymmetry and Quantum Gravity”, under contract SCI-CT92-0789. 1. String theories have to reproduce, at the “low-energy” limit in which the slope α′ goes to zero, ordinary field theories formulated in a number of spacetime dimensions, which coincides with the dimensionality D of the space-time in which the string is embedded (D = 26 for the bosonic string). In order to reduce this dimension to D = 4 a compactification scheme must be adopted. Our work just goes in the direction to show explicitly how, with a suitable compactification procedure, string amplitudes lead to the ones of four-dimensional field theories in the above mentioned limit. We will consider here the so-called “toroidal compactification” [1] [2] [3] of the bosonic closed string, in which twenty-two spatial coordinates are compactified into circles with radius R and the compactified space becomes a lattice which is required to be Lorentzian, self-dual and even in order to have consistency with the properties of the bosonic string theory. Constructing such a lattice yields the introduction of Lie algebra lattices where the massless states which arise from the toroidal compactification lie on the root lattice and belong to the adjoint representation of the gauge group relative to the algebra. After having endowed the theory with the above compactification procedure, the limit α′ → 0 makes all the massive modes uncouple giving rise to a description in terms of only massless states. In particular we consider tree scattering amplitudes of massless scalar particles in the compactified bosonic closed string theory. We perform the double limit α′ → 0 and R → 0, keeping the ratio a = R/ √ α′ fixed; in so doing, we obtain amplitudes corresponding to the exchange of a scalar, spin 1 and spin 2 particle. In particular this latter amplitude is coincident with the one obtained in the framework of linearized quantum gravity. An analogous computation was performed in the context of the generalized dual Virasoro model [4]. 2. The toroidal compactification consists in associating the d internal extracoordinates of the string to d circles having radius Ri with i = 1, . . . , d; this can be done by identifying the points of the internal space as follows: X ≡ X + 2πL where I = 1, . . . , d and L = √