New Higher-Derivative $R^4$ Theorems

The non-minimal pure spinor formalism for the superstring is used to prove two new multiloop theorems which are related to recent higher-derivative $R^4$ conjectures of Green, Russo and Vanhove. The first theorem states that when $0<n<12$, $\partial^n R^4$ terms in the Type II effective action do not receive perturbative contributions above $n/2$ loops. The second theorem states that when $n\leq 8$, perturbative contributions to $\partial^n R^4$ terms in the IIA and IIB effective actions coincide.