An amplitude analysis of ${B^0\to (\pi^+\pi^) (K^+\pi^)}$ decays is performed in the twobody invariant mass regions ${300 < m(\pi^+\pi^)<1100}$ MeV/$c^2$, accounting for the $\rho^0$, $\omega$, $f_0(500)$, $f_0(980)$ and $f_0(1370)$ resonances, and ${750 < m(K^+\pi^)<1200}$ MeV/$c^2$, which is dominated by the $K^{*}(892)^0$ meson. The analysis uses 3 fb$^{1}$ of protonproton collision data collected by the LHCb experiment at centreofmass energies of 7 and 8 TeV. The $CP$ averages and asymmetries are measured for the magnitudes and phase differences of the contributing amplitudes. The $CP$averaged longitudinal polarisation fractions of the vectorvector modes are found to be $\tilde{f}^0_{\rho K^{*}} = 0.164 \pm 0.015 \pm 0.022$ and $\tilde{f}^0_{\omega K^{*}} = 0.68 \pm 0.17 \pm 0.16$, and their $CP$ asymmetries, $\mathcal{A}^0_{\rho K^{*}} = 0.62 \pm 0.09 \pm 0.09$ and $\mathcal{A}^0_{\omega K^{*}} = 0.13 \pm 0.27 \pm 0.13$, where the first uncertainty is statistical and the second systematic.
Leading Feynman diagrams in the $ B ^0 \rightarrow \rho ^0 K ^{*0} $ decay, from left to right: doubly Cabibbosuppressed tree, gluonicpenguin and electroweakpenguin diagrams. 
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Fit to the invariantmass distribution of selected (left) $ B ^0$ and (right) $\overline{ B }{} {}^0$ candidates after the subtraction of $ B ^0_ s \rightarrow K ^{*0} \overline{ K }{} {}^{*0} $ background decays. The four trigger and datataking year categories are aggregated in the figures. The contributions due to the $ B ^0 \rightarrow (\pi ^+ \pi ^ )( K ^+ \pi ^ )$ signal, $\overline{ B }{} {}^0_ s \rightarrow (\pi ^+ \pi ^ )( K ^+ \pi ^ )$ background and combinatorial background are represented by the solid green, red and grey lines, respectively. Data are shown using black dots and the overall fit is represented by the solid blue line. 
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Definition of the helicity angles in the $ B ^0 \rightarrow \rho ^0 K ^{*0} $ decay. 
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Projections of the amplitude fit to the (left) $ B ^0$ and (right) $\overline{ B }{} {}^0$ data samples. The four trigger and datataking year categories are aggregated in the figures. Data are shown by black points with uncertainties and the overall fit is represented by the solid blue line. The contributions of the partial waves sharing the same angular dependence are shown as ($VV$) solid green, ($VS$) dashdotted violet, ($SV$) dashed dark magenta and ($SS$) dotted orange lines. Direct $ C P$ violating effects are most visible in the projections of the $VV$ component over $\cos\theta_{K\pi}$ and $\cos\theta_{\pi\pi}$ and in the different oscillation frequency in $\phi$. 
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Legend for the plots. The partial waves sharing the same angular dependence are represented as ($VV$) solid green, ($VS$) dashdotted violet, ($SV$) dashed dark magenta and ($SS$) dotted orange lines. The overall fit is shown by a solid blue line. 
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Animated gif made out of all figures. 
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Yields obtained in the extended simultaneous fourbody invariant mass fit to the four categories and for the two final states. The quoted uncertainties are statistical only. 
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Contributions to the total amplitude and their angular and mass dependencies. 
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Numerical fit results for the $ C P$ averages and asymmetries in the (top) modulus and (bottom) phase differences of all the contributing amplitudes and among the $VV$ polarisation fractions. For the numbers in the table, the first and second uncertainties correspond to the statistical and total systematic, respectively. The total systematic uncertainty is obtained from the sum in quadrature of the individual sources detailed in Sect. ???, accounting for 100% correlation of the common systematic uncertainties for $ B ^0$ and $\overline{ B }{} {}^0$ . 
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Comparison of theoretical predictions for the $ B ^0 \rightarrow \rho (770)^0 K ^* (892)^0$ mode with the results obtained from this analysis. It should be noted that the theoretical predictions involving the $ C P$ averaged value of $\delta^\perp_{\rho K ^* }$ have been shifted by $\pi$ on account of the different phase conventions used in the theoretical and experimental works. 
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Central values of the masspropagator parameters and their uncertainties, used to estimate the corresponding systematic uncertainties. The values of the parameters used to describe the $f_0(500)$ and $f_0(1370)$ resonances were taken from Ref. \cite{Aaij:2014emv} and the rest, from Ref. \cite{PDG2018}. 
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Created on 15 June 2019.Citation count from INSPIRE on 15 June 2019.