Observation of the suppressed ADS modes $B^\pm \to [\pi^\pm K^\mp \pi^+\pi^-]_D K^\pm$ and $B^\pm \to [\pi^\pm K^\mp \pi^+\pi^-]_D \pi^\pm$

An analysis of $B^{\pm}\to DK^{\pm}$ and $B^{\pm}\to D\pi^{\pm}$ decays is presented where the $D$ meson is reconstructed in the four-body final state $K^{\pm}\pi^{\mp} \pi^+ \pi^-$. Using LHCb data corresponding to an integrated luminosity of $1.0{\rm \,fb}^{-1}$, first observations are made of the suppressed ADS modes $B^{\pm}\to [\pi^{\pm} K^{\mp}\pi^+\pi^-]_D K^{\pm}$ and $B^{\pm}\to [\pi^{\pm} K^{\mp} \pi^+\pi^- ]_D\pi^{\pm}$ with a significance of $5.1\sigma$ and greater than $10\sigma$, respectively. Measurements of $CP$ asymmetries and $CP$-conserving ratios of partial widths from this family of decays are also performed. The magnitude of the ratio between the suppressed and favoured $B^{\pm}\to DK^{\pm}$ amplitudes is determined to be $r^K_B = 0.097 \pm{0.011}$.