Model-independent search for $CP$ violation in $D^{0} \to K^{-}K^{+}\pi^{-}\pi^{+}$ and $D^{0} \to \pi^{-}\pi^{+}\pi^{+}\pi^{-}$ decays

A search for $CP$ violation in the phase-space structures of $D^{0}$ and $\bar{D^{0}}$ decays to the final states $K^{-}K^{+}\pi^{-}\pi^{+}$ and $\pi^{-}\pi^{+}\pi^{+}\pi^{-}$ is presented. The search is carried out with a data set corresponding to an integrated luminosity of 1.0 fb$^{−1}$ collected in 2011 by the LHCb experiment in $pp$ collisions at a centre-of-mass energy of 7 TeV. For the $K^{-}K^{+}\pi^{-}\pi^{+}$ final state, the four-body phase space is divided into 32 bins, each bin with approximately 1800 decays. The $p$-value under the hypothesis of no $CP$ violation is 9.1 %, and in no bin is a $CP$ asymmetry greater than 6.5 % observed. The phase space of the $\pi^{-}\pi^{+}\pi^{+}\pi^{-}$ final state is partitioned into 128 bins, each bin with approximately 2500 decays. The p-value under the hypothesis of no $CP$ violation is 41 %, and in no bin is a $CP$ asymmetry greater than 5.5 % observed. All results are consistent with the hypothesis of no $CP$ violation at the current sensitivity.