Search for $\textit{CP}$ violation in the phase space of $D^{0} \rightarrow K_{S}^{0} K^{\pm} π^{\mp} $ decays with the energy test

A search for $\textit{CP}$ violation in $D^{0} \rightarrow K_{S}^{0} K^{+} \pi^{-} $ and $D^{0} \rightarrow K_{S}^{0} K^{-} \pi^{+} $ decays is reported. The search is performed using an unbinned model-independent method known as the energy test that probes local $\textit{CP}$ violation in the phase space of the decays. The data analysed correspond to an integrated luminosity of $5.4~$fb$^{-1}$ collected in proton-proton collisions by the LHCb experiment at a centre-of-mass energy of $\sqrt{s}=13$~TeV, amounting to approximately 950 thousand and 620 thousand signal candidates for the $D^{0} \rightarrow K_{S}^{0} K^{-} \pi^{+} $ and $D^{0} \rightarrow K_{S}^{0} K^{+} \pi^{-} $ modes, respectively. The method is validated using $D^{0} \rightarrow K^{-} \pi^{+} \pi^{-} \pi^{+} $ and $D^{0} \rightarrow K_{S}^{0} \pi^{+} \pi^{-} $ decays, where $\textit{CP}$-violating effects are expected to be negligible, and using background-enhanced regions of the signal decays. The results are consistent with $\textit{CP}$ symmetry in both the $D^{0} \rightarrow K_{S}^{0} K^{-} \pi^{+} $ and the $D^{0} \rightarrow K_{S}^{0} K^{+} \pi^{-} $ decays, with $p$-values for the hypothesis of no $\textit{CP}$ violation of 70% and 66%, respectively.