The resonant structure of the doubly Cabibbosuppressed decay $D^+ \to K^K^+K^+$ is studied for the first time. The measurement is based on a sample of ppcollision data, collected at a centreofmass 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 Swave 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 twobody system are obtained from the Dalitz plot fit with the phenomenological decay amplitude.
Invariantmass 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. 
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(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 Pwave 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. 
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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. 
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Projection onto $s_{K^+K^}$ of $K^K^+K^+$ candidates with invariant mass in the range 18201830 MeV. 
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Highresolution histogram representing the background model used in the Dalitz plot fits. 
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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 phasespace distribution weighted by the efficiency. 
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(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. 
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(left) Magnitude and (right) phase of the total Swave 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 Swave magnitude and phase for models B and C are similar to those from model A and are not shown. 
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Diagrams representing the two quarklevel topologies for the $ D ^+ \rightarrow K ^ K ^+ K ^+ $ decay. In the TripleM [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. 
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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 $\eta\eta$. 
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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 TripleM 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. 
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(left) Twodimensional distribution of the normalised residuals for the TripleM fit. (right) Distribution of normalised residuals of each bin. 
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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. 
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(top) Phaseshifts $\delta^{0I}_{K^+K^}$ and (bottom) inelasticities $\eta^{0I}$ as a function of the $K^+K^$ invariant mass, for both isospin states. 
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Animated gif made out of all figures. 
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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. 
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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. 
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Results of the $D^+\rightarrow K^K^+K^+$ Dalitz plot fit with the TripleM amplitude. 
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Relative fractions (%) of the various components of the TripleM amplitude. The uncertainties correspond to the combined statistical and systematic uncertainties. 
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Systematic uncertainties (%) on the results of the isobar model fit. For comparison, the statistical uncertainties are listed in the last column. 
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Systematic uncertainties (%) on the results of the TripleM fit. For comparison, the statistcal uncertainties are listed in the last column. 
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BlattWeisskopf form factors for angular momentum $L=0,1,2$ with two distinct formulations. 
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Created on 20 February 2021.