Relativistic corrections to $\psi (nS)\to \rho \pi$ exclusive decays and their role in the understanding of the $\rho \pi$-puzzle

We study relativistic corrections to exclusive $S$-wave charmonium decays into $\rho \pi$ and $\gamma \pi$ final states. The contribution of relative order $v^2$ and the set of associated higher order corrections are calculated using NRQCD and collinear factorisation framework. Numerical estimates show that the dominant effect is provided by the corrections of relative order $v^2$. The numerical values of these contributions are of the same order as the leading-order ones. These results suggest a scenario where the sum of relativistic and radiative QCD corrections could explain the $\rho \pi$-puzzle.