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

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## Abstract

Amplitude models are applied to studies of resonance structure in $D^0\to K^0_S K^- \pi^+$ and $D^0\to K^0_S K^+ \pi^-$ decays using $pp$ collision data corresponding to an integrated luminosity of $3.0 \mathrm{fb}^{-1}$ collected by the LHCb experiment. Relative magnitude and phase information is determined, and coherence factors and related observables are computed for both the whole phase space and a restricted region of $100 \mathrm{MeV/}c^2$ around the $K^{*}(892)^{\pm}$ resonance. Two formulations for the $K\pi$ $S$-wave are used, both of which give a good description of the data. The ratio of branching fractions $\mathcal{B}(D^0\to K^0_S K^+ \pi^-)/\mathcal{B}(D^0\to K^0_S K^- \pi^+)$ is measured to be $0.655\pm0.004 (\textrm{stat})\pm0.006 (\textrm{syst})$ over the full phase space and $0.370\pm0.003 (\textrm{stat})\pm0.012 (\textrm{syst})$ in the restricted region. A search for $CP$ violation is performed using the amplitude models and no significant effect is found. Predictions from $SU(3)$ flavor symmetry for $K^{*}(892)K$ amplitudes of different charges are compared with the amplitude model results.

## Figures and captions

 SCS classes of diagrams contributing to the decays $D ^0 \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^\pm \pi ^\mp$ . The color-favored (tree) diagrams (a) contribute to the $K ^{*{\pm}}_{0,1,2} \rightarrow K ^0_{\rm\scriptscriptstyle S} \pi ^\pm$ and $( a ^{{}}_{0,2} ,\rho )^{\pm} \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^\pm$ channels, while the color-suppressed exchange diagrams (b) contribute to the $( a ^{{}}_{0,2} ,\rho )^{\pm} \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^\pm$ , $K ^{*{0}}_{0,1,2} \rightarrow K ^+ \pi ^-$ and $\overline{ K }{} ^{*{0}}_{0,1,2} \rightarrow K ^- \pi ^+$ channels. Second-order loop (penguin) diagrams (c) contribute to the $( a ^{{}}_{0,2} ,\rho )^{\pm} \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^\pm$ and $K ^{*{\pm}}_{0,1,2} \rightarrow K ^0_{\rm\scriptscriptstyle S} \pi ^\pm$ channels, and, finally, OZI-suppressed penguin annihilation diagrams (d) contribute to all decay channels. [Failure to get the plot] Mass (left) and $\Delta m$ (right) distributions for the $D ^0 \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^- \pi ^+$ (top) and $D ^0 \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^+ \pi ^-$ (bottom) samples with fit results superimposed. The long-dashed (blue) curve represents the \Dsttt signal, the dash-dotted (green) curve represents the contribution of real $D ^0$ mesons combined with incorrect \pislow and the dotted (red) curve represents the combined combinatorial and $D ^0 \rightarrow K ^0_{\rm\scriptscriptstyle S} \pi ^+ \pi ^- \pi ^0$ background contribution. The vertical solid lines show the signal region boundaries, and the vertical dotted lines show the sideband region boundaries. Fig_2.pdf [166 KiB] HiDef png [252 KiB] Thumbnail [160 KiB] *.C file tex code Dalitz plots of the $D ^0 \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^- \pi ^+$ (left) and $D ^0 \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^+ \pi ^-$ (right) candidates in the two-dimensional signal region. rd_tos[..].pdf [472 KiB] HiDef png [1 MiB] Thumbnail [772 KiB] *.C file rd_tos[..].pdf [527 KiB] HiDef png [1 MiB] Thumbnail [852 KiB] *.C file Efficiency function used in the isobar model fits, corresponding to the average efficiency over the full dataset. The coordinates \sKSpi and $m^{2}_{ K ^0_{\rm\scriptscriptstyle S} K }$ are used to highlight the approximate symmetry of the efficiency function. The $z$ units are arbitrary. efficiency.pdf [178 KiB] HiDef png [1 MiB] Thumbnail [339 KiB] *.C file Distributions of \sKpi (upper left), \sKSpi (upper right) and $m^{2}_{ K ^0_{\rm\scriptscriptstyle S} K }$ (lower left) in the $D ^0 \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^- \pi ^+$ mode with fit curves from the best \glass model. The solid (blue) curve shows the full PDF $P_{ K ^0_{\rm\scriptscriptstyle S} K ^- \pi ^+ }(\sKSpi,\sKpi)$, while the other curves show the components with the largest integrated fractions. [Failure to get the plot] [Failure to get the plot] [Failure to get the plot] [Failure to get the plot] Distributions of \sKpi (upper left), \sKSpi (upper right) and $m^{2}_{ K ^0_{\rm\scriptscriptstyle S} K }$ (lower left) in the $D ^0 \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^- \pi ^+$ mode with fit curves from the best \lassc model. The solid (blue) curve shows the full PDF $P_{ K ^0_{\rm\scriptscriptstyle S} K ^- \pi ^+ }(\sKSpi,\sKpi)$, while the other curves show the components with the largest integrated fractions. [Failure to get the plot] [Failure to get the plot] [Failure to get the plot] [Failure to get the plot] Distributions of \sKpi (upper left), \sKSpi (upper right) and $m^{2}_{ K ^0_{\rm\scriptscriptstyle S} K }$ (lower left) in the $D ^0 \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^+ \pi ^-$ mode with fit curves from the best \glass model. The solid (blue) curve shows the full PDF $P_{ K ^0_{\rm\scriptscriptstyle S} K ^+ \pi ^- }(\sKSpi,\sKpi)$, while the other curves show the components with the largest integrated fractions. [Failure to get the plot] [Failure to get the plot] [Failure to get the plot] [Failure to get the plot] Distributions of \sKpi (upper left), \sKSpi (upper right) and $m^{2}_{ K ^0_{\rm\scriptscriptstyle S} K }$ (lower left) in the $D ^0 \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^+ \pi ^-$ mode with fit curves from the best \lassc model. The solid (blue) curve shows the full PDF $P_{ K ^0_{\rm\scriptscriptstyle S} K ^+ \pi ^- }(\sKSpi,\sKpi)$, while the other curves show the components with the largest integrated fractions. [Failure to get the plot] [Failure to get the plot] [Failure to get the plot] [Failure to get the plot] Decay rate and phase variation across the Dalitz plot. The top row shows $|\mcM_{ K ^0_{\rm\scriptscriptstyle S} K ^\pm \pi ^\mp }(\sKSpi, \sKpi)|^2$ in the best \glass isobar models, the center row shows the phase behavior of the same models and the bottom row shows the same function subtracted from the phase behavior in the best \lassc isobar models. The left column shows the $D ^0 \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^- \pi ^+$ mode with $D ^0 \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^+ \pi ^-$ on the right. The small inhomogeneities that are visible in the bottom row relate to the \glass and \lassc models preferring slightly different values of the $K ^{*}_{{}}({892})^{{\pm}}$ mass and width. [Failure to get the plot] [Failure to get the plot] [Failure to get the plot] [Failure to get the plot] [Failure to get the plot] [Failure to get the plot] Comparison of the phase behavior of the various \kpswave parameterizations used. The solid (red) curve shows the \lassc parameterization, while the dashed (blue) and dash-dotted (green) curves show, respectively, the \glass functional form fitted to the charged and neutral $S$-wave channels. The final two curves show the \glass forms fitted to the charged \kpswave in $D ^0 \rightarrow K ^0_{\rm\scriptscriptstyle S} \pi ^+ \pi ^-$ decays in Ref. \cite{Aubert:2008bd} (triangular markers, purple) and Ref. \cite{delAmoSanchez:2010xz} (dotted curve, black). The latter of these was used in the analysis of $D ^0 \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^\pm \pi ^\mp$ decays by the CLEO collaboration \cite{Insler:2012pm}. lass_g[..].pdf [168 KiB] HiDef png [282 KiB] Thumbnail [228 KiB] *.C file Smooth functions, $c_{ K ^0_{\rm\scriptscriptstyle S} K ^\pm \pi ^\mp }( m^{2}_{ K ^0_{\rm\scriptscriptstyle S} K } , \sKSpi)$, used to describe the combinatorial background component in the $D ^0 \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^- \pi ^+$ (left) and $D ^0 \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^+ \pi ^-$ (right) amplitude model fits. [Failure to get the plot] [Failure to get the plot] Two-dimensional quality-of-fit distributions illustrating the dynamic binning scheme used to evaluate $\chi^2$ . The variable shown is $\frac{d_i - p_i}{\sqrt{p_i}}$ where $d_i$ and $p_i$ are the number of events and the fitted value, respectively, in bin $i$. The $D ^0 \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^- \pi ^+$ ( $D ^0 \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^+ \pi ^-$ ) mode is shown in the left (right) column, and the \glass (\lassc) isobar models are shown in the top (bottom) row. [Failure to get the plot] [Failure to get the plot] [Failure to get the plot] [Failure to get the plot] Animated gif made out of all figures. PAPER-2015-026.gif Thumbnail

## Tables and captions

 Signal yields and estimated background rates in the two-dimensional signal region. The larger mistag rate in the $D ^0 \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^+ \pi ^-$ mode is due to the different branching fractions for the two modes. Only statistical uncertainties are quoted. Table_1.pdf [52 KiB] HiDef png [47 KiB] Thumbnail [23 KiB] tex code Blatt-Weisskopf centrifugal barrier penetration factors, $B_J(q, q_0, d)$ \cite{BlattWeisskopf}. Table_2.pdf [47 KiB] HiDef png [121 KiB] Thumbnail [43 KiB] tex code Angular distribution factors, $\Omega_J(p_{ D ^0 } + p_{ C }, p_{ B } - p_{ A })$. These are expressed in terms of the tensors $T^{\mu\nu} = -g^{\mu\nu} + \frac{p^\mu_{ A B }p^\nu_{ A B }}{\mRsq}$ and $T^{\mu\nu\alpha\beta} = \frac{1}{2}(T^{\mu\alpha}T^{\nu\beta} + T^{\mu\beta}T^{\nu\alpha}) - \frac{1}{3}T^{\mu\nu}T^{\alpha\beta}$. Table_3.pdf [49 KiB] HiDef png [45 KiB] Thumbnail [21 KiB] tex code Particle ordering conventions used in this analysis. Table_4.pdf [48 KiB] HiDef png [42 KiB] Thumbnail [21 KiB] tex code Modulus and phase of the relative amplitudes between resonances that appear in both the $D ^0 \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^- \pi ^+$ and $D ^0 \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^+ \pi ^-$ modes. Relative phases are calculated using the value of $\delta_{ K ^0_{\rm\scriptscriptstyle S} K \pi }$ measured in $\psi (3770)$ decays \cite{Insler:2012pm}, and the uncertainty on this value is included in the statistical uncertainty. The first uncertainties are statistical and the second systematic. Table_5.pdf [77 KiB] HiDef png [226 KiB] Thumbnail [119 KiB] tex code Values of $\chi^2/\mathrm{bin}$ indicating the fit quality obtained using both \kpswave parameterizations in the two decay modes. The binning scheme for the $D ^0 \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^- \pi ^+$ ( $D ^0 \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^+ \pi ^-$ ) mode contains 2191 (2573) bins. Table_6.pdf [49 KiB] HiDef png [53 KiB] Thumbnail [22 KiB] tex code Coherence factor observables to which the isobar models are sensitive. The third column summarizes the CLEO results measured in quantum-correlated decays \cite{Insler:2012pm}, where the uncertainty on $\delta_{ K ^0_{\rm\scriptscriptstyle S} K \pi } - \delta_{ K ^* K }$ is calculated assuming maximal correlation between $\delta_{ K ^0_{\rm\scriptscriptstyle S} K \pi }$ and $\delta_{ K ^* K }$. Table_7.pdf [58 KiB] HiDef png [42 KiB] Thumbnail [22 KiB] tex code $\mathrm{SU}(3)$ flavor symmetry predictions \cite{Bhattacharya:2012fz} and results. The uncertainties on phase difference predictions are calculated from the quoted magnitude and phase uncertainties. Note that some theoretical predictions depend on the $\eta$ -- $\eta ^{\prime}$ mixing angle $\theta_{\eta -\eta ^{\prime} }$ and are quoted for two different values. The bottom entry in the table relies on the CLEO measurement \cite{Insler:2012pm} of the coherence factor phase $\delta_{ K ^0_{\rm\scriptscriptstyle S} K \pi }$, and the uncertainty on this phase is included in the statistical uncertainty, while the other entries are calculated directly from the isobar models and relative branching ratio. Where two uncertainties are quoted the first is statistical and the second systematic. Table_8.pdf [68 KiB] HiDef png [103 KiB] Thumbnail [50 KiB] tex code Nominal values for isobar model parameters that are fixed in the model fits, or used in constraint terms. These values are taken from Refs. \cite{PDG2014,Abele:1998qd,Bargiotti:2003ev,Dunwoodie} as described in Sect. ???. Table_9.pdf [71 KiB] HiDef png [371 KiB] Thumbnail [177 KiB] tex code Matrices $\mathbf{U}$ relating the fit coordinates $\mathbf{b'}$ to the \lassc form factor coordinates $\mathbf{b}=\mathbf{Ub'}$ defined in Sect. ???. Table_10.pdf [62 KiB] HiDef png [51 KiB] Thumbnail [25 KiB] tex code Additional fit parameters for \glass models. This table does not include parameters that are fixed to their nominal values. The first uncertainties are statistical and the second systematic. Table_11.pdf [63 KiB] HiDef png [210 KiB] Thumbnail [108 KiB] tex code Additional fit parameters for \lassc models. This table does not include parameters that are fixed to their nominal values. The first uncertainties are statistical and the second systematic. Table_12.pdf [62 KiB] HiDef png [193 KiB] Thumbnail [99 KiB] tex code Change in \ntll value when removing a $\rho$ resonance from one of the models. Table_13.pdf [52 KiB] HiDef png [77 KiB] Thumbnail [37 KiB] tex code Listing of abbreviations required to typeset the systematic uncertainty tables. Table_14.pdf [75 KiB] HiDef png [298 KiB] Thumbnail [138 KiB] tex code Lookup table filenames. Table_15.pdf [59 KiB] HiDef png [42 KiB] Thumbnail [21 KiB] tex code

## Supplementary Material [file]

 Supplementary material full pdf supple[..].pdf [508 KiB] This .zip archive contains supplementary material for the publication LHCb-PAPER-2015-026. The files contained are: supplementary.pdf : An overview of the extra figures *.pdf, *.png, *.eps : The figures in various formats Fig1-S.pdf [200 KiB] HiDef png [541 KiB] Thumbnail [354 KiB] *C file Fig10-S.pdf [201 KiB] HiDef png [522 KiB] Thumbnail [353 KiB] *C file Fig11-S.pdf [199 KiB] HiDef png [482 KiB] Thumbnail [313 KiB] *C file Fig12-S.pdf [201 KiB] HiDef png [477 KiB] Thumbnail [318 KiB] *C file Fig2-S.pdf [198 KiB] HiDef png [471 KiB] Thumbnail [305 KiB] *C file Fig3-S.pdf [204 KiB] HiDef png [555 KiB] Thumbnail [370 KiB] *C file Fig4-S.pdf [200 KiB] HiDef png [530 KiB] Thumbnail [350 KiB] *C file Fig5-S.pdf [197 KiB] HiDef png [448 KiB] Thumbnail [290 KiB] *C file Fig6-S.pdf [202 KiB] HiDef png [524 KiB] Thumbnail [349 KiB] *C file Fig7-S.pdf [203 KiB] HiDef png [563 KiB] Thumbnail [379 KiB] *C file Fig8-S.pdf [200 KiB] HiDef png [506 KiB] Thumbnail [330 KiB] *C file Fig9-S.pdf [203 KiB] HiDef png [524 KiB] Thumbnail [350 KiB] *C file

Created on 19 October 2019.