Searching for $T$, $\mathrm{CP}$, $\mathrm{CPT}$, and $\Delta S=\Delta Q$ rule violations in the neutral $K$ meson system: A guide

It is known that the existing experimental limit for $\mathrm{CPT}$ violation is rather poor; the $\mathrm{CPT}$-violating amplitude can be as large as 10% of the $\mathrm{CP}$-violating one. The $\Delta S=\Delta Q$ rule is tested only up to 2%. In this paper, we discuss the possibility of measuring $T$, $\mathrm{CP}$, $\mathrm{CPT}$, and $\Delta S=\Delta Q$ rule violations in the neutral $K$ meson system. It is shown that not all the symmetry-breaking parameters can be determined at a symmetric $\phi$ factory. Experiments with $K^0$ and ${\overline{K}}^0$ beams are needed to test all the symmetries.