Reanalysis of the decay ${\mathit{K}}_{\mathit{L}}$→${\mathrm{\pi }}^0$${\mathit{e}}^{+}$${\mathit{e}}^{\mathrm{-}}$

The decay ${\mathit{K}}_{\mathit{L}}$→${\mathrm{\pi }}^0$ ${\mathit{e}}^{+}$ ${\mathit{e}}^{\mathrm{-}}$ is searched for as a signal of direct ΔS=1 CP violation. We provide a thorough updating of the analysis of the three components of the decay: (1) direct CP violation, (2) CP violation through the mass matrix, and (3) CP-conserving (two photon) contributions. First the chiral calculation of the ${\mathit{K}}_{\mathit{S}}$→${\mathrm{\pi }}^0$ ${\mathit{e}}^{+}$ ${\mathit{e}}^{\mathrm{-}}$ rate, due to Ecker, Pich and, de Rafael, is updated to include recent results on the nonleptonic amplitude. Then we systematically explore the uncertainties in this method. These appear to be so large that they will obscure direct CP violation unless it is possible to measure the ${\mathit{K}}_{\mathit{S}}$→${\mathrm{\pi }}^0$ ${\mathit{e}}^{+}$ ${\mathit{e}}^{\mathrm{-}}$ rate. The CP-conserving amplitude remains somewhat uncertain, but present indications are such that there may be a sizable CP-violating asymmetry in the ${\mathit{e}}^{+}$,${\mathit{e}}^{\mathrm{-}}$ energies from the interference of the CP-conserving and CP-violating amplitudes and this may potentially be useful in determining whether direct CP violation is present.