Measurement of the branching fraction and search for $CP$ violation in $D^0\rightarrow K^0_S\,K^0_S\,\pi^+\pi^-$ decays at Belle

We measure the branching fraction for the Singly Cabibbo-suppressed decay $D^0\rightarrow K^0_S\,K^0_S\,\pi^+\pi^-$, and we search for $CP$ violation via a measurement of the $CP$ asymmetry $A^{}_{CP}$ and also the $T$-odd triple-product asymmetry $a_{CP}^{T}$. The later two measurements are complementary. We use 922~fb$^{-1}$ of data recorded by the Belle experiment, which ran at the KEKB asymmetric-energy $e^+ e^-$ collider. The branching fraction is measured relative to the Cabibbo-favored normalization channel $D^0\rightarrow K^0_S\,\pi^+\pi^-$. Singly Cabibbo-suppressed charm decays are expected to have especially good sensitivity to new physics effects.