Renormalization and Factorization Scale-Invariant Predictions for the $e^+e^-\to J/ψ+η_c$ process at $B$ factories

We perform a NNLO perturbative QCD analysis for exclusive double charmonium production $e^+e^-\to J/ψ+η_c$ within the NRQCD factorization formalism using the Principle of Maximum Conformality (PMC). It is well known that this charmonium process suffers poor pQCD convergence, as well as large renormalization scale uncertainty. The PMC is used to remove conventional renormalization scale uncertainty. Based on two fundamental physical processes of charmonium, $J/ψ\rightarrow e^+e^-$ and $η_c\rightarrowγγ$, and using the NNLO perturbative results together with experimental measurements, we extract the factorization-scale-dependent long-distance matrix elements for $J/ψ$ and $η_c$. Combining these two matrix elements with the factorization-scale-dependent short-distance coefficient, we obtain predictions that are independent of the factorization scale for the $e^+e^-\to J/ψ+η_c$ process. Our predictions agree with the measurement from the BaBar collaborations within errors. Remarkably, we provide a new way to eliminate conventional renormalization and factorization scale ambiguities for the $e^+e^-\to J/ψ+η_c$ process. We further demonstrate that the slow convergence of the pQCD series is an intrinsic property of this double-charmonium process.