Dispersive contributions toK0−K¯0mixing andCPviolation

We consider the role of dispersive contributions in the analysis of the $K_L-K_S$ system. These long-distance pieces can be present in both the real and imaginary parts of the mixing element $M_{12}$. We provide an estimate of one such effect in Im $M_{12}$, the ${\eta }^{\prime }$ pole, which turns out to be potentially significant. These terms can substantially modify phenomenological studies, as we demonstrate for several cases. Improved experimental bounds on $\frac{{\epsilon }^{\prime }}{\epsilon }$ can provide a strong limit on dispersive effects in Im $M_{12}$, and may in the near future remove the uncertainties caused by these contributions.