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.