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 superstring theories is vanishing point-wise in (even) moduli space at two loops. 
Zhu-Jun Zheng, Jun-Bao Wu, Chuan-Jie Zhu