Search for $CP$ violation in $D^{0} \to \pi^{-} \pi^{+} \pi^{+} \pi^{-}$ decays

We perform the first model-independent search for $CP$ violating variations in the phase space distribution of a four-body decay. We study the singly Cabibbo suppressed decay $D^{0} \to \pi^{-} \pi^{+} \pi^{+} \pi^{-}$, using approximately 180k signal events in LHCb's 2011 dataset of 1.0 fb^{-1}. No evidence for $CP$ violation is observed.