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

Amplitude models are constructed to describe the resonance structure of $D^0\to K^{-}{\pi }^{+}{\pi }^{+}{\pi }^{-}$ and ${D^{0} \to K^{+}\pi^{-}\pi^{-}\pi^{+}}$decaysusing$pp$ collision data collected at centre-of-mass energies of 7 and 8 TeV with the LHCb experiment, corresponding to an integrated luminosity of $3.0{\mathrm{f}\mathrm{b}}^{-1}$. The largest contributions to both decay amplitudes are found to come from axial resonances, with decay modes $D^0\to a_1(1260{)}^{+}{K}^{-}$ and $D^{0} \to K_1(1270/1400)^{+} \pi^{-}$beingprominentin${D^{0}\to K^{-}\pi^{+}\pi^{+}\pi^{-}}$and$D^{0}\to K^{+}\pi^{-}\pi^{-}\pi^{+}$, 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{s}\mathrm{t}\mathrm{a}\mathrm{t}$) \pm 0.012,($\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}$) \pm 0.020,($\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}$)$, 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 $C\text{\hspace{0.17em}⁣}P$ violation.