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

Abstract

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