Studies of the resonance structure in

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