A Dalitz plot analysis of $B^0 \to \eta_c(1S) K^+\pi^$ decays is performed using data samples of $pp$ collisions collected with the LHCb detector at centreofmass energies of $\sqrt{s}=7, 8$ and $13$ TeV, corresponding to a total integrated luminosity of $4.7 \text{fb}^{1}$. A satisfactory description of the data is obtained when including a contribution representing an exotic $\eta_c(1S) \pi^$ resonant state. The significance of this exotic resonance is more than three standard deviations, while its mass and width are $4096 \pm 20 ^{+18}_{22}$ MeV and $152 \pm 58 ^{+60}_{35}$ MeV, respectively. The spinparity assignments $J^P=0^+$ and $J^{P}=1^$ are both consistent with the data. In addition, the first measurement of the $B^0 \to \eta_c(1S) K^+\pi^$ branching fraction is performed and gives $\displaystyle \mathcal{B}(B^0 \to \eta_c(1S) K^+\pi^) = (5.73 \pm 0.24 \pm 0.13 \pm 0.66) \times 10^{4}$, where the first uncertainty is statistical, the second systematic, and the third is due to limited knowledge of external branching fractions.
Feynman diagrams for \subref{B2etacKstar} $B^0 \rightarrow \eta_cK^{*0}$ and \subref{B2ZK} $B^0 \rightarrow Z_c^K^+$ decay sequences. 
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Distribution of the $ p $ $\overline p $ $ K ^+$ $\pi ^$ invariant mass. The solid blue curve is the projection of the total fit result. The components are shown in the legend. 
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Distribution of the $ p $ $\overline p $ invariant mass in (left) linear and (right) logarithmic verticalaxis scale for weighted $ B ^0 \rightarrow p \overline p K ^+ \pi ^ $ candidates obtained by using the sPlot technique. The solid blue curve is the projection of the total fit result. The full azure, tightcrosshatched red and dashedblack line areas show the $\eta _ c $ , $ { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu}$ and NR $ p $ $\overline p $ contributions, respectively. 
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Results of the 2D mass fit to the joint [$ m( p \overline p K ^+ \pi ^ )$, $ m( p \overline p )$] distribution for the (a) Run 1 $ m( p \overline p K ^+ \pi ^ )$ projection, (b) Run 1 $ m( p \overline p )$ projection, (c) Run 2 $ m( p \overline p K ^+ \pi ^ )$ projection, and (d) Run 2 $ m( p \overline p )$ projection. The legend is shown in the top left plot. 
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SDP distributions used in the DP fit to the Run 2 subsample for (a) combinatorial background and (b) NR $ B ^0 \rightarrow p \overline p K ^+ \pi ^ $ background. 
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Backgroundsubtracted (top) DP and (bottom) SDP distributions corresponding to the total data sample used in the analysis. The structure corresponding to the $K^*(892)^0$ resonance is evident. The veto of $ B ^0 \rightarrow \eta _ c K ^+ \pi ^ $ decays in the $\overline{ D }{} {}^0$ region is visible in the DP. 
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$ B ^0 \rightarrow \eta _ c K ^+ \pi ^ $ signal efficiency across the SDP for the (a) Run 1 and (b) Run 2 samples. 
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Projections of the data and amplitude fit using the baseline model onto (a) $ m( K ^+ \pi ^ )$, (c) $ m(\eta _ c \pi ^ )$ and (e) $ m(\eta _ c K ^+ )$, with the same projections shown in (b), (d) and (f) with a logarithmic verticalaxis scale. The veto of $ B ^0 \rightarrow p \overline p \overline{ D }{} {}^0 $ decays is visible in plot (b). The $ K ^+ \pi ^ $ Swave component comprises the LASS and $K^*_0(1950)^0$ meson contributions. The components are described in the legend at the bottom. 
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Projections of the data and amplitude fit using the nominal model onto (a) $ m( K ^+ \pi ^ )$, (c) $ m(\eta _ c \pi ^ )$ and (e) $ m(\eta _ c K ^+ )$, with the same projections shown in (b), (d) and (f) with a logarithmic verticalaxis scale. The veto of $ B ^0 \rightarrow p \overline p \overline{ D }{} {}^0 $ decays is visible in plot (b). The $ K ^+ \pi ^ $ Swave component comprises the LASS and $K^*_0(1950)^0$ meson contributions. The components are described in the legend at the bottom. 
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Comparison of the first four $ K ^+ \pi ^ $ Legendre moments determined from backgroundsubtracted data (black points) and from the results of the amplitude fit using the baseline model (red triangles) and nominal model (blue triangles) as a function of $ m( K ^+ \pi ^ )$. 
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Comparison of the first four $\eta _ c \pi ^ $ Legendre moments determined from backgroundsubtracted data (black points) and from the results of the amplitude fit using the baseline model (red triangles) and nominal model (blue triangles) as a function of $ m(\eta _ c \pi ^ )$. 
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Comparison of the first four $\eta _ c K ^+ $ Legendre moments determined from backgroundsubtracted data (black points) and from the results of the amplitude fit using the baseline model (red triangles) and nominal model (blue triangles) as a function of $ m(\eta _ c K ^+ )$. 
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2D pull distribution for to the baseline model. 
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2D pull distribution for to the nominal model. 
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Animated gif made out of all figures. 
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Relative systematic uncertainties on the ratio $R$ of Eq. \eqref{ratio}. The total systematic uncertainty is obtained from the quadratic sum of the individual sources. 
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Yields of the components in the 2D mass fit to the joint [$ m( p \overline p K ^+ \pi ^ )$, $ m( p \overline p )$] distribution for the Run 1 and 2 subsamples. 
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Resonances included in the baseline model, where parameters and uncertainties are taken from Ref. \cite{PDG2016}. The LASS lineshape also parametrise the $ K ^+ \pi ^ $ Swave in $ B ^0 \rightarrow \eta _ c K ^+ \pi ^ $ NR decays. 
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Complex coefficients and fit fractions determined from the DP fit using the nominal model. Uncertainties are statistical only. 
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Significance of the $Z_c(4100)^$ contribution for the systematic effects producing the largest variations in the parameters of the $Z_c(4100)^$ candidate. The values obtained in the nominal amplitude fit are shown in the first row. 
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Rejection level of the $J^P=0^+$ hypothesis with respect to the $J^P=1^$ hypothesis for the systematic variations producing the largest variations in the parameters of the $Z_c(4100)^$ candidate. The values obtained in the nominal amplitude fit are shown in the first row. 
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Fit fractions and their uncertainties. The quoted uncertainties are statistical and systematic, respectively. 
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Branching fraction results. The four quoted uncertainties are statistical, $ B ^0 \rightarrow \eta _ c K ^+ \pi ^ $ branching fraction systematic (not including the contribution from the uncertainty associated to the efficiency ratio, to avoid double counting the systematic uncertainty associated to the evaluation of the efficiencies), fit fraction systematic and external branching fractions uncertainties, respectively. 
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Symmetric matrix of the fit fractions (%) from the amplitude fit using the nominal model. The quoted uncertainties are statistical and systematic, respectively. The diagonal elements correspond to the values reported in Table ???. 
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Created on 19 February 2021.