\begin{document}$ \tilde\Lambda $\end{document}CDM), in which the vacuum energy interacts with matter and radiation, and test this model using the current cosmological observations. Using the CMB+BAO+SN (CBS) dataset to constrain the model, we find that \begin{document}$ H_0 $\end{document} and \begin{document}$ S_8 $\end{document} tensions are relieved to \begin{document}$ 2.87\sigma $\end{document} and \begin{document}$ 2.77\sigma $\end{document}, respectively. However, in this case, the \begin{document}$ \tilde\Lambda $\end{document}CDM model is not favored by the data, compared with ΛCDM. We find that when the \begin{document}$ H_0 $\end{document} and \begin{document}$ S_8 $\end{document} data are added to the data combination, the situation is significantly improved. In the CBS+\begin{document}$ H_0 $\end{document} case, the model relieves the \begin{document}$ H_0 $\end{document} tension to \begin{document}$ 0.47\sigma $\end{document}, and the model is favored over ΛCDM. In the CBS+\begin{document}$ H_0 $\end{document}+\begin{document}$ S_8 $\end{document} case, we obtain a synthetically best situation, in which the \begin{document}$ H_0 $\end{document} and \begin{document}$ S_8 $\end{document} tensions are relieved to \begin{document}$ 0.72\sigma $\end{document} and \begin{document}$ 2.11\sigma $\end{document}, respectively. In this case, the model is most favored by the data. Therefore, this cosmological model can greatly relieve the \begin{document}$ H_0 $\end{document} tension and simultaneously effectively alleviate the \begin{document}$ S_8 $\end{document} tension."> Dark energy and matter interacting scenario to relieve <i>H</i><sub>0</sub> and <i>S</i><sub>8</sub> tensions -
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