Studies of the resonance structure in $D^{0} \rightarrow K^\mp \pi ^\pm \pi ^\pm \pi ^\mp $ decays

Amplitude models are constructed to describe the resonance structure of ${D^{0}\rightarrow K^{-}\pi ^{+}\pi ^{+}\pi ^{-}}$ and ${D^{0} \rightarrow K^{+}\pi ^{-}\pi ^{-}\pi ^{+}}$ decays using pp collision data collected at centre-of-mass energies of 7 and 8 TeV with the LHCb experiment, corresponding to an integrated luminosity of 3.0 $fb^{-1}$ . The largest contributions to both decay amplitudes are found to come from axial resonances, with decay modes $D^{0} \rightarrow a_1(1260)^{+} K^{-}$ and $D^{0} \rightarrow K_1(1270/1400)^{+} \pi ^{-}$ being prominent in ${D^{0}\rightarrow K^{-}\pi ^{+}\pi ^{+}\pi ^{-}}$ and $D^{0}\rightarrow K^{+}\pi ^{-}\pi ^{-}\pi ^{+}$ , respectively. Precise measurements of the lineshape parameters and couplings of the $a_1(1260)^{+}$ , $K_1(1270)^{-}$ and $K(1460)^{-}$ resonances are made, and a quasi model-independent study of the $K(1460)^{-}$ resonance is performed. The coherence factor of the decays is calculated from the amplitude models to be $R_{K3\pi } = 0.459\pm 0.010\,(\mathrm {stat}) \pm 0.012\,(\mathrm {syst}) \pm 0.020\,(\mathrm {model})$ , which is consistent with direct measurements. These models will be useful in future measurements of the unitary-triangle angle $\gamma $ and studies of charm mixing and $C\!P$ violation.