Computing the R term at Two Super-string Loops

We use a previously derived integral representation for the four graviton amplitude at two loops in Super-string theory, whose leading term for vanishing momenta gives the two-loop contribution to the R term in the Effective Action. We find by an explicit computation that this contribution is zero, in agreement with a general argument implying the vanishing of the R term beyond one loop. As it is well known, Super-string perturbation theory at two loops implies computing a sigma-model functional integration on a genus two Riemann surface (meaning the functional integration over the Super-string fields Xμ(z) and ψμ(z), where z is a complex coordinate on the surface). We use the hyper-elliptic formalism, in which the genus two surface is represented as a two sheets covering of the complex plane, described by the equation: