merit we used is defined as: F OM = S √$_{S}$ + N , (1) where S is the number of signal events and N is the number of cross-feed background events. For the τ − → K $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ signal ( S ), the cross-feed background component ( N ) includes contributions from τ − → π $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ and τ − → K $^{−}$K $^{+}$π $^{−}$ν$_{τ}$ decays. The result of FOM optimization for the τ − → K $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ decay is shown in Fig. 4, where one can see that FOM is indeed maximal for the particle identification criteria used in this analysis. The same FOM analysis was performed for the τ − → π $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ and τ − → K $^{−}$K $^{+}$π $^{−}$ν$_{τ}$ decay modes and resulted in similar particle identification criteria. For electron identification, the likelihood variable is calculated based on the track extrapolation to the ECL, the ratio of the energy deposited in the ECL to the momentum measured in the CDC, the measured dE/dx in the CDC, the shower shape in the ECL, and light yield in the ACC [16]. The extrapolation of tracks from the CDC and SVD to the KLM is used to construct the likeli- hood variable for muon identification [17]. The efficiencies and systematic uncertainties for lepton identification are evaluated by using a γγ → e $^{+}$e $^{−}$/µ $^{+}$µ − control sample. FIG. 4: Figure-of-merit as a function of the likelihood ratio L ( K ) / ( L ( K )+ L ( π )) requirement, used to separate the kaon and pion hypotheses. The uncertainties are evaluated from the comparison of the efficiencies and fake rates of kaon identification with those of the D ∗ + → D $^{0}$( → K $^{−}$π $^{+}$) π + con- trol sample. To reduce the feed-down background and to estimate the remaining backgrounds that 9 )) L(K)/(L(K)+L( 0.7 0.75 0.8 0.85 0.9 0.95 1 Figure of Merit 37 38 39 40 41 42 43 44 45