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 R4 term in type II superstring theories is vanishing
point-wise in (even) moduli space at two loops.Comment: v1, LaTex file, 33 pages; v2, 34 pages, add references and minor
correction