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

We compute explicitly the four-particle amplitude in superstring theories by using the hyperelliptic language and the newly obtained chiral measure of DʹHoker 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.