The ultracold state-to-state chemistry for three-body recombination (TBR) in realistic systems has recently been experimentally investigated with full quantum state resolution. However, many detected phenomena remain challenging to be explored and explained from the theoretical viewpoints because this generally requires computational powers beyond state of the art. Here, the product-state distributions after TBR of He-alkaline-earth-metal systems, i.e., after the processes of He + He + X → HeX + He with X being Be, Mg, Ca, Sr, or Ba, in the zero-collision-energy limit are theoretically studied. Two propensity rules for the distribution of the products found in current experiments have been checked, and the mechanism underlying these product-state distributions is explored. Particularly, two main intriguing transition pathways are identified, which may be responsible for the nonlinear distribution of the products vs their respective rotational quantum number. In addition, the TBR rates of these systems are also influenced by details of the interaction potential and relevant nonadiabatic couplings.
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