CP violation in the hyperon decays $\Sigma\to N\pi$

The study of $CP$ violation in hyperon transitions has a long history. In the early 2000s the HyperCP experiment made a major effort to seek $CP$-odd signals in the decay sequence $\Xi^-\to\Lambda \pi^-$ and $\Lambda\to p\pi^-$, which motivated more searches. Most recently the BESIII and LHCb Collaborations have acquired or improved the upper bounds on $CP$ violation in a variety of hyperon nonleptonic processes, including $\Sigma^+\to n\pi^+$ and $\Sigma^+\to p\pi^0$. These measurements have not reached the standard-model level yet, but have stimulated a renewed interest in $CP$-violating new physics in strange-quark decay beyond what is constrained by the parameters $\varepsilon$ and $\varepsilon^\prime$ from the kaon sector. In this paper, after updating the standard-model expectations for $CP$-odd observables in the modes $\Sigma^\pm\to N\pi$, we revisit new-physics scenarios that could enhance the corresponding quantities in $\Lambda\to N\pi$ and $\Xi\to\Lambda\pi$ and apply them to the $\Sigma^\pm$ modes. We find that the $CP$ asymmetries in the latter can be significantly increased over the standard-model expectations, at levels which may be tested in the ongoing BESIII experiment and in future endeavors such as PANDA and the Super Tau Charm Facility.