Motivic cluster multiplication formulas in 2-Calabi-Yau categories

We introduce a notion of motivic cluster characters via virtual Poincar\'{e} polynomials, and prove a motivic version of multiplication formulas obtained by Chen-Xiao-Xu for weighted quantum cluster characters associated to a 2-Calabi-Yau triangulated category $\mathcal{C}$ with a cluster tilting object. Furthermore, a refined form of this formula is also given. When $\mathcal{C}$ is the cluster category of an acyclic quiver, our certain refined multiplication formula is a motivic version of the multiplication formula in [International Mathematics Research Notices, rnad172(2023)].