Measurements of Branching Fractions of $τ$ Lepton Decays with one or more $K^{0}_{S}$
We report measurements of branching fractions of $τ$ lepton decays to final states with a $K^{0}_{S}$ meson using a 669 fb$^{-1}$ data sample accumulated with the Belle detector at the KEKB asymmetric-energy $e^{+}e^{-}$ collider. The inclusive branching fraction is measured to be $\mathcal{B}(τ^{-} \to K^{0}_{S}\ X^{-} ν_τ)=(9.15 \pm 0.01 \pm 0.15) \times 10^{-3}$, where $X^{-}$ can be anything; the exclusive branching fractions are $\mathcal{B}(τ^{-} \to π^{-} K^{0}_{S} ν_τ) = (4.16 \pm 0.01 \pm 0.08) \times 10^{-3}$, $\mathcal{B}(τ^{-} \to K^{-} K^{0}_{S} ν_τ) = (7.40 \pm 0.07 \pm 0.27) \times 10^{-4}$, $\mathcal{B}(τ^{-} \to π^{-} K^{0}_{S} π^{0} ν_τ) = (1.93 \pm 0.02 \pm 0.07) \times 10^{-3}$, $\mathcal{B}(τ^{-} \to K^{-} K^{0}_{S} π^{0} ν_τ) = (7.48 \pm 0.10 \pm 0.37)\times 10^{-4}$, $\mathcal{B}(τ^{-} \to π^{-} K^{0}_{S} K^{0}_{S} ν_τ) = (2.33 \pm 0.03 \pm 0.09) \times 10^{-4}$, $\mathcal{B}(τ^{-} \to π^{-} K^{0}_{S} K^{0}_{S} π^{0} ν_τ) = (2.00 \pm 0.22 \pm 0.20) \times 10^{-5}$, where the first uncertainty is statistical and the second is systematic. For each mode, the accuracy is improved over that of pre-$B$-factory measurements by a factor ranging from five to ten. In $τ^{-} \to π^{-} K^0_S K^0_S π^{0} ν_τ$ decays, clear signals for the intermediate states $τ^{-} \to π^- f_{1}(1285)ν_τ$ and $τ^{-} \to K^{*-}K^{0}_{S} π^{0} ν_τ$ are observed.