Two-loop superstrings in hyperelliptic language III: the four-particle amplitude

We compute explicitly the four-particle amplitude in superstring theories by using the hyperelliptic language and the newly obtained chiral measure of D'Hoker and Phong. Although the algebra of the intermediate steps is a little bit involved, we obtain a quite simple expression for the four-particle amplitude. As expected, the integrand is independent of all the insertion points. As an application of the obtained result, we show that the perturbative correction to the R-4 term in type II supersoing theories is vanishing point-wise in (even) moduli space at two loops. (C) 2003 Elsevier B.V. All rights reserved.