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We study rare three-body decays of the Omega baryon using SU(3) chiralperturbation theory, the successful effective field theory of quantumchromodynamics at low energies. At leading order, we calculate the branchingfractions of the decay $\Omega^- \to \Xi \pi \pi$ for all possible combinationsof pions. For one channel we find an order-of-magnitude discrepancy betweentheory and experiment. This tension is known to exist in the non-relativisticlimit, and we confirm that it remains in the relativistic calculation. Fairlyindependent of the values of the low-energy constants we establish lower limitsfor the branching fractions of these three-body Omega decays, which reaffirmthe gap between theory and experiment. We point out that this discrepancy isclosely tied to the $\Delta I =1/2$ selection rule. In turn, this means thatthe three-body decays constitute an interesting tool to scrutinize theselection rule. Using next-to-leading order calculations we also providepredictions for the decay $\Omega^- \to \Xi^0 \mu^- \bar{\nu}_\mu$. We showthat fully-differential distributions will provide access to low-energyconstants needed in the axial-vector transitions from a decuplet to octetbaryon. Since data for all of these rare three-body Omega decays are scarce(fully differential data are nonexistent), we recommend that they be remeasuredat running and upcoming experiments, such as BESIII, LHCb, Belle-II, and PANDA.

Estimates for rare three-body decays of the Omega baryon using chiral symmetry and the $\Delta I = 1/2$ rule