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Dalitz plot analysis of the $D^+ \rightarrow K^- K^+ K^+$ decay with the isobar model

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Abstract

This note presents a study of the $K^-K^+$ S-wave amplitude in doubly Cabibbo-suppressed $ D ^+ \!\rightarrow K ^- K ^+ K ^+ $ decays performed using $2 \,fb^{-1} $ of data collected by the LHCb detector in $pp$ collisions at 8 TeV centre-of-mass energy. The Dalitz plot is studied under the assumption of the isobar model for resonance scattering. Models with combinations of resonant states are tested. Fits of comparable quality are obtained for different $K^-K^+$ S-wave parameterizations. The results obtained indicate that a variation of the S-wave phase at both ends of $K^-K^+$ spectrum is needed to describe the data. Further studies beyond the na\"\i ve isobar model are foreseen to understand the $K^-K^+$ S-wave.

Figures and captions

Annihilation diagram for $ D ^+ \!\rightarrow K ^- K ^+ K ^+ $ via $\phi$, $a_0$ and $f_0$ resonances (left); tree diagram for $D^+\rightarrow f_0/a_0 K^+$ (right).

diagra[..].jpg [42 KiB]
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diagram_DKKK.jpg
diagra[..].jpg [39 KiB]
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diagram_Df0K.jpg

The $K^-K^+K^+$ mass spectrum of selected candidates (data points), with the fit result overlaid (solid blue line). The green and yellow dashed lines indicate the two Gaussian functions representing the signal. The red line represents the background.

fit_mass.pdf [62 KiB]
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fit_mass.pdf

The Dalitz plot of the final $ D ^+ \!\rightarrow K ^- K ^+ K ^+ $ sample, in linear (left) and logarithm scale (right).

Dalitz[..].pdf [36 KiB]
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DalitzSignalAfterCuts.pdf
Dalitz[..].pdf [39 KiB]
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DalitzSignalAfterCuts_log.pdf

Distribution of the final efficiency for the $ D ^+ \!\rightarrow K ^- K ^+ K ^+ $ decay in arbitraty units, including geometrical acceptance, reconstruction and selection, trigger and PID.

effTot[..].pdf [279 KiB]
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effTotal_10bin_spline.pdf

The Dalitz plot of the background model.

Dalitz[..].pdf [258 KiB]
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DalitzBackground.pdf

Projections of the data onto $s_{\rm high}$ (left) and $s_{\rm low}$ axes (right). The fit result (blue line) for Model 1 is superimposed. The contribution from the different amplitudes are also shown.

Model1[..].pdf [165 KiB]
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Model1_result.pdf

Projection of the data onto $s_{23}$ axis, with the fit result for Model 1 superimposed (left). Distribution of the $\chi^2$ over the folded Dalitz plot (right). A minus sign is assigned to the $\chi^2$ when there is a deficit of data compared to model.

Model1[..].pdf [165 KiB]
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Model1_result.pdf

Projections of the data onto $s_{\rm high}$ (left) and $s_{\rm low}$ axes (right). The fit result (blue line) for Model 2 is superimposed.

Model2[..].pdf [166 KiB]
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Model2_result.pdf

Projection of the data onto $s_{23}$ axis, with the fit result for Model 2 superimposed (left). Distribution of the signed $\chi^2$ over the Dalitz plot (right).

Model2[..].pdf [166 KiB]
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Model2_result.pdf

Projections of the data onto $s_{\rm high}$ (left) and $s_{\rm low}$ axes (right). The fit result (blue line) for Model 3 is superimposed.

Model3[..].pdf [166 KiB]
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Model3_result.pdf

Projection of the data onto $s_{23}$ axis (left), with the fit result for Model 3 superimposed. Distribution of the signed $\chi^2$ over the Dalitz plot (right).

Model3[..].pdf [166 KiB]
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Model3_result.pdf

The magnitude and phase variation of the S-wave amplitude for the three models.

swave_[..].pdf [23 KiB]
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swave_model123.pdf

Moment $t^0_0$ of angular distribution for data (blue dots) and ToyMC of Model2 (bars). From 0.9--1.05 $ {\mathrm{\,GeV\!/}c^2}$ , on left panel, and from 1.05 $ {\mathrm{\,GeV\!/}c^2}$ up to 1.9 $ {\mathrm{\,GeV\!/}c^2}$ on right panel.

Model2[..].pdf [74 KiB]
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Moment $t^0_1$ of angular distribution for data (blue dots) and ToyMC of Model2 (bars). From 0.9--1.05 $ {\mathrm{\,GeV\!/}c^2}$ , on left panel, and from 1.05 $ {\mathrm{\,GeV\!/}c^2}$ up to 1.9 $ {\mathrm{\,GeV\!/}c^2}$ on right panel.

Model2[..].pdf [74 KiB]
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Model2_left_comp.pdf
Model2[..].pdf [72 KiB]
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Moment $t^0_2$ of angular distribution for data (blue dots) and ToyMC of Model2 (bars). From 0.9--1.05 $ {\mathrm{\,GeV\!/}c^2}$ , on left panel, and from 1.05 $ {\mathrm{\,GeV\!/}c^2}$ up to 1.9 $ {\mathrm{\,GeV\!/}c^2}$ on right panel.

Model2[..].pdf [74 KiB]
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Model2_left_comp.pdf
Model2[..].pdf [72 KiB]
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Moment $t^0_3$ of angular distribution for data (blue dots) and ToyMC of Model2 (bars). From 0.9--1.05 $ {\mathrm{\,GeV\!/}c^2}$ , on left panel, and from 1.05 $ {\mathrm{\,GeV\!/}c^2}$ up to 1.9 $ {\mathrm{\,GeV\!/}c^2}$ on right panel.

Model2[..].pdf [74 KiB]
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Model2_left_comp.pdf
Model2[..].pdf [72 KiB]
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Moment $t^0_4$ of angular distribution for data (blue dots) and ToyMC of Model2 (bars). From 0.9--1.05 $ {\mathrm{\,GeV\!/}c^2}$ , on left panel, and from 1.05 $ {\mathrm{\,GeV\!/}c^2}$ up to 1.9 $ {\mathrm{\,GeV\!/}c^2}$ on right panel.

Model2[..].pdf [74 KiB]
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Model2_left_comp.pdf
Model2[..].pdf [72 KiB]
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Moment $t^0_0$ of angular distribution for data (blue dots) and ToyMC of Model3 (bars). From 0.9--1.05 $ {\mathrm{\,GeV\!/}c^2}$ , on left panel, and from 1.05 $ {\mathrm{\,GeV\!/}c^2}$ up to 1.9 $ {\mathrm{\,GeV\!/}c^2}$ on right panel.

Model3[..].pdf [73 KiB]
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Model3_left_comp.pdf
Model3[..].pdf [72 KiB]
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Model3_right_comp.pdf

Moment $t^0_1$ of angular distribution for data (blue dots) and ToyMC of Model3 (bars). From 0.9--1.05 $ {\mathrm{\,GeV\!/}c^2}$ , on left panel, and from 1.05 $ {\mathrm{\,GeV\!/}c^2}$ up to 1.9 $ {\mathrm{\,GeV\!/}c^2}$ on right panel.

Model3[..].pdf [73 KiB]
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Model3_left_comp.pdf
Model3[..].pdf [72 KiB]
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Model3_right_comp.pdf

Moment $t^0_2$ of angular distribution for data (blue dots) and ToyMC of Model3 (bars). From 0.9--1.05 $ {\mathrm{\,GeV\!/}c^2}$ , on left panel, and from 1.05 $ {\mathrm{\,GeV\!/}c^2}$ up to 1.9 $ {\mathrm{\,GeV\!/}c^2}$ on right panel.

Model3[..].pdf [73 KiB]
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*.C file
Model3_left_comp.pdf
Model3[..].pdf [72 KiB]
HiDef png [155 KiB]
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Model3_right_comp.pdf

Moment $t^0_3$ of angular distribution for data (blue dots) and ToyMC of Model3 (bars). From 0.9--1.05 $ {\mathrm{\,GeV\!/}c^2}$ , on left panel, and from 1.05 $ {\mathrm{\,GeV\!/}c^2}$ up to 1.9 $ {\mathrm{\,GeV\!/}c^2}$ on right panel.

Model3[..].pdf [73 KiB]
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Thumbnail [53 KiB]
*.C file
Model3_left_comp.pdf
Model3[..].pdf [72 KiB]
HiDef png [155 KiB]
Thumbnail [38 KiB] *.C file
Model3_right_comp.pdf

Moment $t^0_4$ of angular distribution for data (blue dots) and ToyMC of Model3 (bars). From 0.9--1.05 $ {\mathrm{\,GeV\!/}c^2}$ , on left panel, and from 1.05 $ {\mathrm{\,GeV\!/}c^2}$ up to 1.9 $ {\mathrm{\,GeV\!/}c^2}$ on right panel.

Model3[..].pdf [73 KiB]
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Model3_left_comp.pdf
Model3[..].pdf [72 KiB]
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Model3_right_comp.pdf

Background subtracted and efficiency corrected $ D ^+ \!\rightarrow K ^- K ^+ K ^+ $ Dalitz plot. The left (right) plot is in linear (log) scale.

signal[..].pdf [585 KiB]
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signaldalitz.pdf

Two 3D views of the background subtracted and efficiency corrected $ D ^+ \!\rightarrow K ^- K ^+ K ^+ $ Dalitz plot.

signal[..].pdf [585 KiB]
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signaldalitz.pdf

Animated gif made out of all figures.

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Tables and captions

Blatt-Weisskopf barrier factors for angular momentum $L=0,1,2$, used for the form factors $F_D^L$ and $F_R^L$.

Table_1.pdf [53 KiB]
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Table_1.pdf

Summary of all isobar models considered in this analysis.

Table_2.pdf [36 KiB]
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Table_2.pdf

Values of the parameters describing the lineshape of the resonances, from PDG [11].

Table_3.pdf [38 KiB]
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Table_3.pdf

Dalitz plot fit results for Models 1, 2 and 3.

Table_4.pdf [59 KiB]
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Table_4.pdf

Direct and interference fit fractions (%) for Models 1, 2 and 3.

Table_5.pdf [29 KiB]
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Table_5.pdf

Systematic uncertainties on the mass and width of $f_0(X)$ in Model 2. The total systematic uncertainty is calculated from the quadrature of all sources. The statistical uncertainty is also shown for comparison.

Table_6.pdf [40 KiB]
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Table_6.pdf

Systematic uncertainties on magnitudes, phases and fractions of Model 2, from the various sources described in the text. The total systematic uncertainty is calculated adding all components in quadrature. The statistical uncertainty is also shown for comparison.

Table_7.pdf [47 KiB]
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Table_7.pdf

Systematic uncertainties on magnitudes, phases and fractions for Model 3 from the various sources described in the text. The total systematic uncertainty is calculated adding the various components in quadrature. The statistical uncertainty is also shown for comparison.

Table_8.pdf [48 KiB]
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Table_8.pdf

Dalitz plot fit results for Models 2 and 3. For each entry, the first uncertainty is due to statistics and the second due to systematics.

Table_9.pdf [56 KiB]
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Table_9.pdf

Created on 17 December 2018.Citation count from INSPIRE on 17 December 2018.