# Dalitz Plot analysis of the $D^+ \to K^- K^+ K^+$ decay

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## Abstract

The resonant structure of the doubly Cabibbo-suppressed decay $D^+ \to K^-K^+K^+$ is studied for the first time. The measurement is based on a sample of pp-collision data, collected at a centre-of-mass energy of 8 TeV with the LHCb detector and corresponding to an integrated luminosity of 2 fb$^-1$. The amplitude analysis of this decay is performed with the isobar model and a phenomenological model based on an effective chiral Lagrangian. In both models the S-wave component in the $K^-K^+$ system is dominant, with a small contribution of the $\phi(1020)$ meson and a negligible contribution from tensor resonances. The $K^-K^+$ scattering amplitudes for the considered combinations of spin (0,1) and isospin (0,1) of the two-body system are obtained from the Dalitz plot fit with the phenomenological decay amplitude.

## Figures and captions

 Invariant-mass spectrum of the $K^-K^+K^+$ candidates with the fit result overlaid (solid blue line). The orange and green dashed lines indicate the two Gaussian functions representing the signal and the red dashed line is the background. Fig1.pdf [35 KiB] HiDef png [208 KiB] Thumbnail [157 KiB] *.C file (left) Dalitz plot of the selected sample, including background. (right) Dalitz plot projections for candidates from regions I (blue) and II (red), above and below $s_{K^-K^+} = 1.5$ $\mathrm{ Ge V^2}$ . The interference between the S- and P-wave amplitudes cause the asymmetry in the number of candidates in the two regions, as well as the shift in the peak position. Both figures include all candidates in the selected mass range. Fig2a.pdf [20 KiB] HiDef png [234 KiB] Thumbnail [227 KiB] *.C file Fig2b.pdf [14 KiB] HiDef png [163 KiB] Thumbnail [152 KiB] *.C file Total efficiency, normalised to unity, for the $D ^+ \rightarrow K ^- K ^+ K ^+$ signal over the Dalitz plot, including the geometrical acceptance and the reconstruction, selection, PID and trigger efficiencies. Fig3.pdf [178 KiB] HiDef png [1 MiB] Thumbnail [541 KiB] *.C file Projection onto $s_{K^+K^-}$ of $K^-K^+K^+$ candidates with invariant mass in the range 1820--1830 MeV. Fig4.pdf [25 KiB] HiDef png [223 KiB] Thumbnail [173 KiB] *.C file High-resolution histogram representing the background model used in the Dalitz plot fits. Fig5.pdf [184 KiB] HiDef png [977 KiB] Thumbnail [466 KiB] *.C file Projections of the Dalitz plot onto (top left) $s_{K^+K^-}$, (top right) $s_{K^+K^+}$, (bottom left) $s_{K^+K^-}^{\rm high}$ and (bottom right) $s_{K^+K^-}^{\rm low}$ axes, with the fit result with model A overlaid (red histogram). The histogram in magenta represents the contribution from the background, whereas the dashed green line is the phase-space distribution weighted by the efficiency. Fig6a.pdf [22 KiB] HiDef png [214 KiB] Thumbnail [158 KiB] *.C file Fig6b.pdf [22 KiB] HiDef png [269 KiB] Thumbnail [195 KiB] *.C file Fig6c.pdf [22 KiB] HiDef png [259 KiB] Thumbnail [189 KiB] *.C file Fig6d.pdf [21 KiB] HiDef png [202 KiB] Thumbnail [159 KiB] *.C file (left) Normalised residuals $\Delta_i$ across the Dalitz plot, from the result of isobar fit. (right) Distribution of the normalised residuals with the fit result overlaid. The distribution is fitted with a Gaussian function and the fit result is consistent with the standard normal distribution. Fig7a.pdf [32 KiB] HiDef png [295 KiB] Thumbnail [281 KiB] *.C file Fig7b.pdf [15 KiB] HiDef png [166 KiB] Thumbnail [141 KiB] *.C file (left) Magnitude and (right) phase of the total S-wave from the result of the Dalitz plot fit with the isobar model. The black line corresponds to model A and the green band represents the statistical and systematic uncertainties added in quadrature. For comparison, the results of models B and C are shown as the blue solid and dashed thick red lines. Uncertainties on the S-wave magnitude and phase for models B and C are similar to those from model A and are not shown. Fig8a.pdf [24 KiB] HiDef png [535 KiB] Thumbnail [225 KiB] *.C file Fig8b.pdf [23 KiB] HiDef png [421 KiB] Thumbnail [193 KiB] *.C file Diagrams representing the two quark-level topologies for the $D ^+ \rightarrow K ^- K ^+ K ^+$ decay. In the Triple-M [3], diagram ($a$) is assumed to be the dominant mechanism of the decay, whereas diagram ($b$) is suppressed since the production of a $K^+K^-$ pair from a $d\bar d$ pair requires rescattering. Fig9.png [157 KiB] HiDef png [40 KiB] Thumbnail [14 KiB] *.C file Diagrams contributing to the amplitude $\mathcal{T}$ for the decay $D^+ \rightarrow K^- K^+ K^+$: (a) the final state kaons are produced directly from the weak vertex; (b) a bare resonance is produced directly from the weak vertex; (c) particles produced at the weak vertex undergo final state interactions; (d) final state interactions endow finite widths to the resonances. The full circle represents the unitary $ab\rightarrow K^+K^-$ scattering amplitude with angular momentum $J$ and isospin $I$, and $ab=K\overline{K},\ \pi\pi,\ \eta\pi$ and $\eta\eta$. Fig10.png [182 KiB] HiDef png [99 KiB] Thumbnail [50 KiB] *.C file Projections of the Dalitz plot onto (top left) $s_{K^+K^-}$, (top right) $s_{K^+K^+}$, (bottom left) $s_{K^+K^-}^{\rm high}$ and (bottom right) $s_{K^+K^-}^{\rm low}$ axes, with the fit result with the Triple-M amplitude superimposed, whereas the dashed green line is the phase space distribution weighted by the efficiency. The magenta histogram represents the contribution from the background. Fig11a.pdf [22 KiB] HiDef png [214 KiB] Thumbnail [158 KiB] *.C file Fig11b.pdf [22 KiB] HiDef png [269 KiB] Thumbnail [195 KiB] *.C file Fig11c.pdf [22 KiB] HiDef png [259 KiB] Thumbnail [189 KiB] *.C file Fig11d.pdf [21 KiB] HiDef png [202 KiB] Thumbnail [159 KiB] *.C file (left) Two-dimensional distribution of the normalised residuals for the Triple-M fit. (right) Distribution of normalised residuals of each bin. Fig12a.pdf [32 KiB] HiDef png [294 KiB] Thumbnail [280 KiB] *.C file Fig12b.pdf [15 KiB] HiDef png [165 KiB] Thumbnail [140 KiB] *.C file Phase of the $J = 0$ component of the decay amplitude (blue) $T^S=T^{00}+T^{01}+T_{\rm NR}^S$, compared to the phases of the scattering amplitudes, (red) $A_{K^+K^-}^{ 00 }$ and (magenta) $A_{K^+K^-}^{01}$ as a function of the $K^+K^-$ invariant mass. Fig13.pdf [83 KiB] HiDef png [220 KiB] Thumbnail [168 KiB] *.C file (top) Phase-shifts $\delta^{0I}_{K^+K^-}$ and (bottom) inelasticities $\eta^{0I}$ as a function of the $K^+K^-$ invariant mass, for both isospin states. Fig14a.pdf [21 KiB] HiDef png [133 KiB] Thumbnail [111 KiB] *.C file Fig14b.pdf [21 KiB] HiDef png [132 KiB] Thumbnail [120 KiB] *.C file Fig14c.pdf [20 KiB] HiDef png [124 KiB] Thumbnail [110 KiB] *.C file Fig14d.pdf [20 KiB] HiDef png [117 KiB] Thumbnail [102 KiB] *.C file Animated gif made out of all figures. PAPER-2018-039.gif Thumbnail

## Tables and captions

 Results from the $D ^+ \rightarrow K ^- K ^+ K ^+$ Dalitz plot fit with the isobar models A, B and C. Magnitudes, $|c_k|$ , phases, $\arg(c_k)$ (in degrees), and fit fractions (in %) are given with statistical uncertainties only. Table_1.pdf [59 KiB] HiDef png [121 KiB] Thumbnail [42 KiB] tex code Fit results with model A, given in terms of the magnitudes $|c_k|$, phases, $\arg(c_k)$ (in degrees), and fit fractions (in %). For each measurement, the first uncertainty is statistical, the second systematic and the third is a systematic uncertainty due to model. Table_2.pdf [56 KiB] HiDef png [43 KiB] Thumbnail [21 KiB] tex code Results of the $D^+\rightarrow K^-K^+K^+$ Dalitz plot fit with the Triple-M amplitude. Table_3.pdf [84 KiB] HiDef png [158 KiB] Thumbnail [73 KiB] tex code Relative fractions (%) of the various components of the Triple-M amplitude. The uncertainties correspond to the combined statistical and systematic uncertainties. Table_4.pdf [51 KiB] HiDef png [17 KiB] Thumbnail [9 KiB] tex code Systematic uncertainties (%) on the results of the isobar model fit. For comparison, the statistical uncertainties are listed in the last column. Table_5.pdf [69 KiB] HiDef png [58 KiB] Thumbnail [29 KiB] tex code Systematic uncertainties (%) on the results of the Triple-M fit. For comparison, the statistcal uncertainties are listed in the last column. Table_6.pdf [63 KiB] HiDef png [81 KiB] Thumbnail [37 KiB] tex code Blatt--Weisskopf form factors for angular momentum $L=0,1,2$ with two distinct formulations. Table_7.pdf [69 KiB] HiDef png [63 KiB] Thumbnail [28 KiB] tex code

Created on 20 February 2021.