I will discuss the sharp rate of convergence of the value functions of N-particle stochastic control problems to their mean field limit, for merely Lipschitz mean field costs. For d ≥ 2, the optimal rate is that of the empirical measures of i.i.d. samples in the 1-Wasserstein distance, as conjectured by Daudin, Delarue, and Jackson, and it persists with additive common noise. In dimension one, this benchmark can surprisingly be beaten: cooperating particles outperform independent samples, and the optimal exponent is 4/7, strictly between the accuracy of independent sampling (N^{-1/2}) and that of quantization (N^{-1}). The proofs rely on a new control-theoretic technique of recoupled shadow flows and, in dimension one, on a Gibbs law implementing the cooperation and a Schrödinger ground-state estimate.