Hadronic $\mathit{D}$ and ${\mathit{D}}_{\mathit{s}}$ meson decays

A comprehensive review of hadronic decays of $D$ and $D_s$ mesons is provided. Current theoretical and experimental challenges and successes in understanding of hadronic transitions of those mesons are discussed. A brief overview of the theoretical and experimental tools is given before discussing the absolute branching fractions for $D$ and $D_s$ mesons. Cabibbo-suppressed and rare hadronic decays are discussed and compared with theory before discussing hadronic multibody decays.

Figure 1

The invariant mass distributions observed by the SLAC-LBL experiment for two and four hadrons in the final state. (a) ${\pi }^{+}{\pi }^{-}$ assigning pion mass to all tracks, (b) $K^{\mp }{\pi }^{\pm }$ assigning kaon and pion masses to all tracks, (c) $K^{+}{K}^{-}$ assigning kaon mass to all tracks, (d) ${\pi }^{+}{\pi }^{-}$ weighted by $\pi \pi$ time-of-flight probability, (e) $K^{\mp }{\pi }^{\pm }$ weighted by $K\pi$ time of flight, (f) $K^{+}{K}^{-}$ weighted by $KK$ time of flight, (g) ${\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$ weighted by $4\pi$ time-of-flight probability, (h) $K^{\pm }{\pi }^{\mp }{\pi }^{+}{\pi }^{-}$ weighted by $K3\pi$ time-of-flight probability, and (i) $K^{\pm }{K}^{\mp }{\pi }^{+}{\pi }^{-}$ weighted by $KK\pi \pi$ time-of-flight probability. From 161.

Figure 2

The measured cross sections for different $D\overline{D}$ final states. From 127.

Figure 3

The CLEO-c detector. The charged particle tracking system consists of an inner drift chamber near the interaction point and the main drift chamber for the momentum measurement. Radially outside the main drift chamber is the RICH detector for charged hadron identification followed by the CsI electromagnetic calorimeter. The instrumented flux return for muon detection is outside the superconducting solenoid coil.

Figure 4

Event display from CLEO-c showing a candidate $D_s^{*+}{D}_s^{-}$ event with $D_s^{*+}\to D_s^{+}\gamma$ and both $D_s^{\pm }$ candidates decaying to $K^{+}{K}^{-}{\pi }^{\pm }$. The charged particle trajectories, as determined from fits to the hits in the CLEO-c drift chambers, are shown as the curved lines.

Figure 5

The $M_{\mathrm{BC}}$ distribution. The dotted line shows the contribution from the ARGUS function that describes the combinatorial background. The solid and dashed lines show the contributions to the signal shape for two different detector resolution functions. The tail on the high side for the signal shape is due to initial-state radiation that lowers the energy of the produced $D$ mesons.

Figure 6

Histograms of and fits to $M_{\mathrm{miss}}^2$ distributions from $D^{+}\to K^{-}{\pi }^{+}{\pi }^{+}$ decays to determine the charged pion efficiency for $p_{{\pi }^{+}}>0.2\mathrm{GeV}$. (a), (c) Events in data; (b), (d) events in Monte Carlo simulation. (a), (b) From decays in which the pion was found, while (c), (d) are from decays in which the pion was not found. The solid curves are fits to the data or Monte Carlo sample; the dashed curves in (c) and (d) are background contributions. From 132.

Figure 7

Histograms of and fits to $M_{\mathrm{miss}}^2$ distributions to determine the $K_S^0$ efficiency. (a), (c) Events in data; (b), (d) events in Monte Carlo simulation. (a), (b) From decays in which the $K_S^0$ was found, while (c), (d) are from decays in which the $K_S^0$ was not found. The background peak and deficit are determined by searching for $K_S^0$ candidates in high and low sidebands of the $K_S^0$ mass. (a), (b) Dashed curves are the contributions from fake $K_S^0$ candidates. (c), (d)Dashed curve is the background, a linear function with a deficit due to events in which a fake $K_S^0$ candidate was found, and the solid curve is the total fit function including the signal peak. The area between the curves is proportional to the number of $K_S^0$ mesons not found. From 132.

Figure 8

The FOCUS (E831) spectrometer. Figure provided by the FOCUS Collaboration.

Figure 9

Basic topological amplitudes for $D$ meson decays. (a) Tree $T^{(\prime )}$; (b) color-suppressed $C^{(\prime )}$; (c) weak annihilation $A^{(\prime )}$;(d) weak exchange $E^{(\prime )}$; and (e) singlet weak exchange $S{E}^{\prime }$. CSC amplitudes are usually denoted by primes.

Figure 10

The $p_T^2$ distribution for data from the HRS Collaboration. (a) The fractional slow pion momentum in the range $0.03<x_F<0.06$; (b) the fractional momentum range $0.07<x_F<0.1$. In the low momentum range where we expect slow pions from $D^{*}$ decays a clear excess at low $p_t^2$ is seen. From 2.

Figure 11

The $p_t^2$ distribution for data from the ALEPH experiment. (a) The transverse slow pion momentum squared in six equal momentum bins from 1.0 to 4.0 GeV. The 1.0 to 1.5 GeV momentum bin is the uppermost and the 3.5 to 4.0 GeV bin is the lowest. The slow pion from $D^{*+}\to D^0{\pi }^{+}$ is clearly visible in the lower momentum range. (b) The transverse momentum distributions from different sources of $D^{*}$ mesons. From 79.

Figure 12

The $|\cos \theta |$ distribution for data from the ARGUS experiment. (a) The distribution for the slow pion momentum in the range 0.2 to 0.3 GeV, and (b) for the range 0.4 to 0.5 GeV. From 31.

Figure 13

The distribution of the missing mass squared $M_{\nu }^2$ for (a) right-sign events and (b) wrong-sign events. The wrong-sign events show that the simulation of the background shape is good. From 68.

Figure 14

The $\Delta M$ distribution for the reconstructed $D^0\to K^{-}{\pi }^{+}$ candidates in events with a ${\overline{B}}^0\to D^{*+}{\ell }^{-}\overline{\nu }$ tag. From 68.

Figure 15

Distributions of measured $M_{\mathrm{BC}}(D)$ and $M_{\mathrm{BC}}(\overline{D})$ values for single tag $D^0$ and $D^{+}$ candidates with $D$ and $\overline{D}$ candidates combined in each plot. The points are data and the curves are fits to the data. In each plot, the dashed curve shows the background contributions and the solid curve shows the sum of the background and signal function. The number of events is shown on a square-root scale. From 132.

Figure 16

Scatter plot of $M_{\mathrm{BC}}(\overline{D})$ vs $M_{\mathrm{BC}}(D)$ for $D^0{\overline{D}}^0$ double tag candidates. Signal candidates are concentrated at $M_{\mathrm{BC}}(\overline{D})=M_{\mathrm{BC}}(D)=M_D$. The signal shape and different background contributions are discussed in the text. From 132.

Figure 17

Projections of the sum of the double tag candidate mass on the $M_{\mathrm{BC}}(D)$ axis for (a) the nine $D^0{\overline{D}}^0$ double tag modes and (b) the 36 $D^{+}{D}^{-}$ double tag modes. The points show the data and the curves show the projection of the fit results. The dashed lines show the background contributions and the solid lines show the signal shape plus the background. From 132.

Figure 18

Missing mass squared distribution for all six tag modes for $D^{+}\to X{\pi }^{+}$. Events with extra tracks or ${\pi }^0$ candidates have been removed. From 169.

Figure 19

The recoil mass against a $D$ or $D^{*}$ in $B$ decays. Top left: $B^{-}\to D^0{D}_X$; top right: ${\overline{B}}^0\to D^{+}{D}_X$; lower left: $B^{-}\to D^{*0}{D}_X$; and lower right: ${\overline{B}}^0\to D^{*+}{D}_X$. From 63.

Figure 20

The recoil mass against a $D_s$ or $D_s^{*}$. Top left: ${\overline{B}}^0\to D_S^{-}{D}_X$; top right: $B^{-}\to D_S^{-}{D}_X$; lower left: ${\overline{B}}^0\to D_s^{*-}{D}_X$; and lower right: $B^{-}\to D_s^{*-}{D}_X$. From 63.

Figure 21

Single tag yields for $D_s$ modes used in the CLEO-c analysis. Charge conjugate modes are combined. From 34.

Figure 22

Double tag yields for $D_s$ modes used in the CLEO-c analysis. The signal region is indicated by the rectangle in the center and the two sideband regions are the diagonally offset rectangles. There are 1089 double tag candidates in the signal region and 339 candidates in the background region. With the signal-to-background region size of $1:3$ this gives a signal-to-background ratio close to $9:1$. From 34.

Figure 23

The $K^{-}{K}^{+}$ invariant mass near the $\varphi$ resonance in $D_s^{+}\to K^{-}{K}^{+}{\pi }^{+}$ events from the CLEO-c double tag analysis. The single tag fit procedure used in the CLEO-c analysis is applied to extract the yield in each $M(K^{-}{K}^{+})$ bin; hence backgrounds are subtracted and the yields shown are for the $D_s^{+}\to K^{-}{K}^{+}{\pi }^{+}$ signal. The $\varphi$ resonance is clear above an additional broad component. Indicated in the plot are the different mass windows considered by CLEO-c for their partial branching fractions. From 34.

Figure 24

Yields for (a) $D^{+}\to \eta {\pi }^{+}$, (b) $D^{+}\to {\eta }^{\prime }{\pi }^{+}$, (c) $D^0\to \eta {\pi }^0$, (d) $D^0\to {\eta }^{\prime }{\pi }^0$, (e) $D^0\to \eta \eta$, and (f) $D^0\to \eta {\eta }^{\prime }$. From 47.

Figure 25

Yields for (a) $D^0\to \eta {\pi }^{+}{\pi }^{-}$, (b) $D^{+}\to \eta {\pi }^{+}{\pi }^0$, (c) $D^0\to {\eta }^{\prime }{\pi }^{+}{\pi }^{-}$, and (d) $D^{+}\to {\eta }^{\prime }{\pi }^{+}{\pi }^0$. From 47.

Figure 26

Graphical representation of Eq. (92) used to determine $S{E}^{\prime }$ from the amplitude analysis for $D\to {\eta }^{(\prime )}\eta ({\pi }^0)$. Circles represent absolute values of the decay amplitudes. The intersection points provide two possible solutions for $S{E}^{\prime }$ (see text). From 47.

Figure 27

The two quark diagrams that contribute to the decay $D^0\to K_S^0{K}_S^0$. Since $V_{cd}=-V_{us}$ the two amplitudes represented by these diagrams largely cancel. In the limit in which the $d$ and $s$ quark masses were the same the cancellation would have been exact.

Figure 28

The missing mass distribution for all $D_s^{+}\to p^{+}\overline{n}$ candidates. CLEO-c saw 13 signal candidates. From 53.

Figure 29

The left plot shows the $D^0\to K^{-}{\pi }^{+}{\pi }^0$ reconstructed mass distribution. The right plot shows the $M^2({\pi }^{+}{\pi }^0)$ vs $M^2(K^{-}{\pi }^{+})$ Dalitz plot for the 7070 $D^0\to K^{-}{\pi }^{+}{\pi }^0$ candidates. From 176.

Figure 30

The $D^0\to K^{-}{\pi }^{+}{\pi }^0$ Dalitz fit. From 176.

Figure 31

BABAR $D^0\to K_S^0{\pi }^{+}{\pi }^{-}$ Dalitz-plot analysis. From 67.

Figure 32

Belle $D^0\to K_S^0{\pi }^{+}{\pi }^{-}$ Dalitz-plot analysis. (a) $m^2(K_S^0{\pi }^{+})$, (b) $m^2(K_S^0{\pi }^{-})$, and (c) $m^2({\pi }^{-}{\pi }^{+})$ distributions, and (d) Dalitz-plot distribution. The points with error bars show the data and the smooth curve is the result of the fit. From 210.

Figure 33

BABAR Dalitz plot of $s_{+}=m_{{\pi }^{+}{\pi }^0}^2$ vs $s_{-}=m_{{\pi }^{-}{\pi }^0}^2$ for the $D^0\to {\pi }^{+}{\pi }^{-}{\pi }^0$ decay. From 66.

Figure 34

BABAR $D^0\to K^{+}{K}^{-}{\pi }^0$ Dalitz-plot analysis. From 64.

Figure 35

Dalitz-plot distribution from the BABAR analysis of $D^0\to K^{+}{K}^{-}{K}_S^0$. The invariant mass squares plotted are $m_0^2=m^2(K^{+}{K}^{-})$ and $m_{+}^2=m^2(K^{+}{K}_S^0)$ for a $D^0$ decay and the charge conjugate particles are used for the ${\overline{D}}^0$ decay. From 67.

Figure 36

CLEO $D^0\to K_S^0\eta {\pi }^0$ Dalitz-plot analysis. From 214.

Figure 37

Projections of the Dalitz-plot fit in the CLEO-c Dalitz-plot analysis of $D^{+}\to K^{-}{\pi }^{+}{\pi }^{+}$ in (a) for $m^2(K\pi )$ (two entries per candidate), and (b) for $m^2({\pi }^{+}{\pi }^{+})$. The data are shown as points with error bars. The insets on top show the residuals between the data and as points with error bars. The small contributions in the fit from the $K^{*}(1680)$ and $K_2^{*}(1430)$ resonances are also shown in the insets enhanced by a factor of 10. From 104.

Figure 38

CLEO-c $D^{+}\to {\pi }^{+}{\pi }^{-}{\pi }^{+}$ Dalitz-plot analysis. From 102.

Figure 39

CLEO-c $D^{+}\to K^{+}{K}^{-}{\pi }^{+}$ Dalitz-plot analysis. (a) The Dalitz-plot distribution. (b)–(d) The projections of the fit (solid) lines and the data (points). The dashed lines show the background contribution. From 216.

Figure 40

The $K^{+}{K}^{-}{\pi }^{+}$ invariant mass for the signal candidates in the CLEO-c Dalitz-plot analysis of $D_s^{+}\to K^{+}{K}^{-}{\pi }^{+}$. From 199.

Figure 41

The Dalitz plot for $D_s^{+}\to K^{+}{K}^{-}{\pi }^{+}$ candidates in the CLEO-c analysis of $D_s^{+}\to K^{+}{K}^{-}{\pi }^{+}$. From 199.

Figure 42

The CLEO-c Dalitz-plot fit for $D_s^{+}\to K^{+}{K}^{-}{\pi }^{+}$ candidates. From 199.

Figure 43

The ${\pi }^{+}{\pi }^{-}{\pi }^{+}$ invariant mass for the signal candidates in the BABAR Dalitz-plot analysis of $D_s^{+}\to {\pi }^{+}{\pi }^{-}{\pi }^{+}$. From 69.

Figure 44

The Dalitz plot for signal candidates in the BABAR Dalitz-plot analysis of $D_s^{+}\to {\pi }^{+}{\pi }^{-}{\pi }^{+}$. From 69.