Search for $\mathit{\text{CP}}$ violation in the phase space of $D^0\to K_S^0{K}^{\pm }{\pi }^{\mp }$ decays with the energy test

A search for $\mathit{\text{CP}}$ violation in $D^0\to K_S^0{K}^{+}{\pi }^{-}$ and $D^0\to K_S^0{K}^{-}{\pi }^{+}$ decays is reported. The search is performed using an unbinned model-independent method known as the energy test that probes local $\mathit{\text{CP}}$ 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 $D^0\to K_S^0{K}^{-}{\pi }^{+}$ and $D^0\to K_S^0{K}^{+}{\pi }^{-}$ modes, respectively. The method is validated using $D^0\to K^{-}{\pi }^{+}{\pi }^{-}{\pi }^{+}$ and $D^0\to K_S^0{\pi }^{+}{\pi }^{-}$ decays, where $\mathit{\text{CP}}$-violating effects are expected to be negligible, and using background-enhanced regions of the signal decays. The results are consistent with $\mathit{\text{CP}}$ symmetry in both the $D^0\to K_S^0{K}^{-}{\pi }^{+}$ and the $D^0\to K_S^0{K}^{+}{\pi }^{-}$ decays, with $p$-values for the hypothesis of no $\mathit{\text{CP}}$ violation of 70% and 66%, respectively.