An ansatz for a non-perturbative four-graviton amplitude in type IIB superstring theory

A natural SL(2,Z)-invariant generalization of the Veneziano amplitude in type IIB superstring theory is investigated. It includes certain perturbative and non-perturbative (D-instanton) contributions, and it reduces to the correct expressions in different limits. The singularities are poles in the s-t-u channels, corresponding to the exchange of particles with a mass spectrum coinciding with that of (p, q) string states. We describe the general structure of the associated perturbative corrections to the effective action. July 1997 ∗ e-mail address: [email protected] A number of interesting aspects of M-theory have recently been clarified [1,2]. Its fundamental degrees of freedom are not yet completely understood, and an adequate formulation is still elusive. Such degrees of freedom could be revealed if an exact scattering amplitude is determined. In conventional quantum theories, scattering amplitudes exhibit, through their analytic structure, the physical states of the theory (appearing as poles in the different channels), and the residue of the poles gives important information about the couplings. In M-theory, the simplest non-trivial process that one can investigate is the four-graviton amplitude. Some progress in this direction was made in refs. [3-6]. In ref. [3], an exact function of the coupling multiplying the R4 term in type II superstring theories was proposed. Such function, which was further motivated in refs. [4] and [5], represents the leading term in a low-energy expansion of the four-graviton scattering amplitude. In ref. [5] it was shown that this function can also be obtained from the one-loop four graviton amplitude in eleven dimensional supergravity with space-time topology R × T . The one-loop graviton amplitude including all powers of external momenta was then calculated in [6]. In this paper we shall concentrate on ten-dimensional type IIB superstring theory. A generalization of the present result to type IIB on R9×S1 would allow, by string dualities, the determination of the corresponding four-graviton amplitude in M-theory. Here the basic idea is to exploit the SL(2,Z) symmetry of type IIB superstring theory, which requires that the effective action must be invariant under SL(2,Z) transformations to all orders in the α expansion. In the Einstein frame, a term of given order in derivatives involving the metric must be multiplied by a modular function of the coupling, as happens in theR4 example investigated in [3,4]. Since there is a one-to-one correspondence between certain ( SL(2,Z) invariant) terms in the effective action (see eq. (20) ) and the terms of the momentum expansion of the four-graviton amplitude, the same modular functions appear in the four-graviton amplitude, which must therefore be invariant under SL(2,Z) transformations. This symmetry property constrains its allowed structure. It will be seen that there is a simple SL(2,Z)-invariant scattering amplitude that in both weak coupling (gB ≪ 1) and low-energy (α′(momentum) ≪ 1) regimes reproduces the known expressions. Its analytic structure is consistent with the expected particle spectrum. Our starting point is the tree-level scattering amplitude: A4 = κ KA4(s, t, u) , (1) A4(s, t, u) = 1 s̄t̄ū Γ(1− s̄)Γ(1− t̄)Γ(1− ū) Γ(1 + s̄)Γ(1 + t̄)Γ(1 + ū) , (2) s̄ = 14α s , t̄ = 14α t , ū = 1 4α u , s̄+ t̄+ ū = 0 , 1 where K is the usual kinematic factor [7], depending on the momenta and polarization of the external states (K ∼ (momentum)). We will find convenient to work with the logarithm of the amplitude, which has a simple expansion in powers of s̄, t̄, ū, i.e. lnA4(s, t, u) = − ln s̄t̄ū+ 2 ∞