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We analyse the two purely isospin-violating decays $\Lambda_b\rightarrow\Sigma^0 \phi$ and $\Lambda_b\rightarrow \Sigma^0 J/\psi$, proceed merely viathe exchange topologies, in the framework of perturbative QCD approach.Assuming $\Sigma^0$ baryon belongs to the idealized isospin triplet with quarkcomponents of $usd$, the branching ratios of the two decay modes are predictedto be tiny, of the order $10^{-8}-10^{-9}$, leading to a difficulty inobserving them. We then extend our study to include the $\Sigma-\Lambda$mixing.It is found that the mixing has significant effect on the$\Lambda_b\rightarrow \Sigma$ decays, especially it can greatly increase therate of the $J/\psi$ process, by as much as two orders of magnitude, yield$10^{-7}$, which should be searchable in the future. We also estimate a set ofasymmetry observables with and without the mixing effect, which will be testedin coming experiments.

Estimates of the isospin-violating $\Lambda_b\rightarrow \Sigma^0 \phi, \Sigma^0 J/\psi$ decays and the $\Sigma-\Lambda$ mixing