Averages of $b$-hadron, $c$-hadron, and $\tau$-lepton properties as of 2021

This paper reports world averages of measurements of $b$-hadron, $c$-hadron, and $\tau$-lepton properties obtained by the Heavy Flavor Averaging Group using results available before April 2021. In rare cases, significant results obtained several months later are also used. For the averaging, common input parameters used in the various analyses are adjusted (rescaled) to common values, and known correlations are taken into account. The averages include branching fractions, lifetimes, neutral meson mixing parameters, $CP$ violation parameters, parameters of semileptonic decays, and Cabibbo-Kobayashi-Maskawa matrix elements.

Figure 1

Illustration of the possible dependence of a measured quantity $x$ on a nuisance parameter $y_i$. The plot compares the 68% confidence level contours of a hypothetical measurement’s unconstrained (large ellipse) and constrained (filled ellipse) likelihoods, using the Gaussian constraint on $y_i$ represented by the horizontal band. The solid error bars represent the statistical uncertainties $\sigma (x)$ and $\sigma (y_i)$ of the unconstrained likelihood. The dashed error bar shows the statistical uncertainty on $x$ from a constrained simultaneous fit to $x$ and $y_i$.

Figure 2

Illustration of the HFLAV combination procedure for correlated systematic uncertainties. Upper plots (a) and (b) show examples of two individual measurements to be combined. The large (filled) ellipses represent their unconstrained (constrained) iso-likelihood contours, while horizontal bands indicate the different assumptions about the value and uncertainty of $y_i$ used by each measurement. The error bars show the results of the method described in the text for obtaining $x$ by performing fits with $y_i$ fixed to different values. Lower plots (c) and (d) illustrate the adjustments to accommodate updated and consistent knowledge of $y_i$. Open circles mark the central values of the unadjusted fits to $x$ with $y$ fixed; these determine the dashed line used to obtain the adjusted values.

Figure 3

Illustration of the combination of two hypothetical measurements of $x$ using the method described in the text. The ellipses represent the unconstrained likelihoods of each measurement, and the horizontal band represents the latest knowledge about $y_i$ that is used to adjust the individual measurements. The filled small ellipse shows the result of the exact method using ${\mathcal{L}}_{\mathrm{comb}}$, and the hollow small ellipse and dot show the result of the approximate method using ${\chi }_{\mathrm{comb}}^2$.

Figure 4

The $B^0–{\overline{B}}^0$ oscillation frequency $\mathrm{\Delta }{m}_d$ as measured by the different experiments. The averages quoted for ALEPH, L3 and OPAL are taken from the original publications, while the ones for DELPHI, CDF, BABAR, Belle and LHCb are computed from the individual results listed in Table 15 without performing any adjustments. The time-integrated measurements of ${\chi }_d$ from the symmetric $B$ factory experiments ARGUS and CLEO are converted to a $\mathrm{\Delta }{m}_d$ value using $\tau (B^0)=1.519\pm 0.004\mathrm{ps}$. The two global averages are obtained after adjustments of all the individual $\mathrm{\Delta }{m}_d$ results of Table 15 (see text).

Figure 5

Contours of $\mathrm{\Delta }\ln L=0.5$ (39% CL for the enclosed 2D regions, 68% CL for the bands) shown in the $({\mathrm{\Gamma }}_s,\mathrm{\Delta }{\mathrm{\Gamma }}_s)$ plane on the left and in the $(1/{\mathrm{\Gamma }}_{s\mathrm{L}},1/{\mathrm{\Gamma }}_{s\mathrm{H}})$ plane on the right. The average of all the $B_s^0\to J/\psi \varphi$, $B_s^0\to J/\psi K^{+}{K}^{-}$ and $B_s^0\to \psi (2S)\varphi$ results is shown as the red contour, where ${\mathrm{\Gamma }}_s$ and $\mathrm{\Delta }{\mathrm{\Gamma }}_s$ are scaled by factors 2.56 and 1.72. The constraints given by the effective lifetime measurements of $B_s^0$ to flavor-specific, pure $CP$-odd and pure $CP$-even final states are shown as the blue, green and purple bands, respectively. The average taking all constraints into account is shown as the dark-filled contour. The light-gray bands are theory predictions. The horizontal band is $\mathrm{\Delta }{\mathrm{\Gamma }}_s=+0.091\pm 0.013{\mathrm{ps}}^{-1}$ [42, 45, 87, 88] that assumes no new physics in $B_s^0$ mixing. The vertical ${\mathrm{\Gamma }}_s$ band is calculated from Ref. [56] assuming the experimental world average for the $B^0$ lifetime, $1.519\pm 0.004\mathrm{ps}$.

Figure 6

Measurements of $\mathrm{\Delta }{m}_s$, together with their average.

Figure 7

Measurements of ${\mathcal{A}}_{\mathrm{SL}}^d$ and ${\mathcal{A}}_{\mathrm{SL}}^s$ listed in Table 20 ($B$-factory average as the gray band, D0 measurements as the green ellipses, LHCb measurements as the blue ellipse) together with their two-dimensional average (red hatched ellipse). The red point close to (0, 0) is the Standard Model prediction of Ref. [87] with error bars multiplied by 10. The prediction and the experimental world average deviate from each other by $0.5\sigma$.

Figure 8

68% CL regions shown in the $({\varphi }_s^{c\overline{c}s},\mathrm{\Delta }{\mathrm{\Gamma }}_s)$ plane on the top and in the $({\mathrm{\Gamma }}_s,\mathrm{\Delta }{\mathrm{\Gamma }}_s)$ plane on the bottom, for individual experiments and their combination. The left plots are obtained from all CDF [172], D0 [173], ATLAS [174, 175, 176], CMS [177, 178] and LHCb [93, 179, 180, 181, 182, 183, 227, 229] measurements of $B_s^0\to J/\psi \varphi$, $B_s^0\to J/\psi K^{+}{K}^{-}$, $B_s^0\to \psi (2S)\varphi$, $B_s^0\to J/\psi {\pi }^{+}{\pi }^{-}$ and $B_s^0\to D_s^{+}{D}_s^{-}$ decays, while the right plots are obtained from $B_s^0\to J/\psi \varphi$ measurements only [172, 173, 174, 175, 176, 177, 178, 179, 182, 183]. The expectation within the Standard Model neglecting penguin contributions [42, 45, 87, 88, 169] is shown as the white rectangle in the top plots. The ${\mathrm{\Gamma }}_s$ theory value in the bottom plots is calculated from Ref. [56] assuming the experimental world average for the $B^0$ lifetime, $1.519\pm 0.004\mathrm{ps}$.

Figure 9

The unitarity triangle.

Figure 10

Top: average of measurements of $S_{b\to c\overline{c}s}$, interpreted as $\sin (2\beta )$, with (bottom left) the same but excluding less precise measurements to allow inspection of the detail. Bottom right: more precise results and the world average for $C_{b\to c\overline{c}s}$.

Figure 11

Constraints on the $(\overline{\rho },\overline{\eta })$ plane, obtained from the average of $-\eta S_{b\to c\overline{c}s}$ and Eq. (147). Note that the solution with the smaller (larger) value of $\beta$ has $\cos (2\beta )>0$ ($<0$).

Figure 12

Averages of (left) $\sin (2\beta )\equiv \sin (2{\varphi }_1)$ and (right) $\cos (2\beta )\equiv \cos (2{\varphi }_1)$ from time-dependent analyses of $B^0\to J/\psi K^{*0}$ decays.

Figure 13

Averages of (left) $(J_c/J_0)$, (middle) $(2J_{s1}/J_0)\sin (2\beta )$ and (right) $(2J_{s2}/J_0)\cos (2\beta )$ from time-dependent analyses of $B^0\to D^{*+}{D}^{*-}{K}_S^0$ decays.

Figure 14

Averages of $\sin (2\beta )$ measured in color-suppressed $b\to c\overline{u}d$ transitions.

Figure 15

Averages of (left) $S_{b\to c\overline{c}d}$ and (right) $C_{b\to c\overline{c}d}$ for the mode $B^0\to J/\psi {\pi }^0$.

Figure 16

Averages of (left) $S_{b\to c\overline{c}d}$ and (right) $C_{b\to c\overline{c}d}$ for the mode $B^0\to D^{+}{D}^{-}$.

Figure 17

Averages of two $b\to c\overline{c}d$ dominated channels, for which correlated averages are performed, in the $S_{CP}$ vs. $C_{CP}$ plane. Contours at $S_{CP}^2+C_{CP}^2=1$ represent the physical boundary for the parameters. (Left) $B^0\to J/\psi {\pi }^0$ and (right) $B^0\to D^{+}{D}^{-}$.

Figure 18

Averages of (left) $S_{b\to c\overline{c}d}$ and (right) $C_{b\to c\overline{c}d}$ for the mode $B^0\to D^{*+}{D}^{*-}$.

Figure 19

Averages of (left) $-\eta S_{b\to c\overline{c}d}$ interpreted as $\sin (2{\beta }^{\mathrm{eff}})$ and (right) $C_{b\to c\overline{c}d}$. The $-\eta S_{b\to c\overline{c}d}$ figure compares the results to the world average for $-\eta S_{b\to c\overline{c}s}$ (see Sec. 6d1).

Figure 20

Compilation of constraints in the $-\eta S_{b\to c\overline{c}d}$, interpreted as $\sin (2{\beta }^{\mathrm{eff}})$, vs $C_{b\to c\overline{c}d}$ plane. The contours at $\sin (2{\beta }^{\mathrm{eff}}{)}^2+C_{b\to c\overline{c}d}^2=1$ represents the physical boundary for the parameters.

Figure 21

Top: averages of (left) $-\eta S_{b\to q\overline{q}s}$, interpreted as $\sin (2{\beta }^{\mathrm{eff}}$ and (right) $C_{b\to q\overline{q}s}$. The $-\eta S_{b\to q\overline{q}s}$ figure compares the results to the world average for $-\eta S_{b\to c\overline{c}s}$ (see Sec. 6d1). Bottom: same, but only averages for each mode are shown. More figures are available from the HFLAV web pages.

Figure 22

Averages of four $b\to q\overline{q}s$ dominated channels, for which correlated averages are performed, in the $S_{CP}$ $vs$. $C_{CP}$ plane, where $S_{CP}$ has been corrected by the $CP$ eigenvalue to give $\sin (2{\beta }^{\mathrm{eff}})$. Contours at $S_{CP}^2+C_{CP}^2=1$ represent the physical boundary for the parameters. Top left: $B^0\to \varphi K^0$, (top right) $B^0\to {\eta }^{\prime }{K}^0$, (bottom left) $B^0\to K_S^0{K}_S^0{K}_S^0$, (bottom right) $B^0\to {\pi }^0{K}_S^0$. More figures are available from the HFLAV web pages.

Figure 23

Compilation of constraints in the $-\eta S_{b\to q\overline{q}s}$, interpreted as $\sin (2{\beta }^{\mathrm{eff}})$, vs $C_{b\to q\overline{q}s}$ plane. The contours at $\sin (2{\beta }^{\mathrm{eff}}{)}^2+C_{b\to q\overline{q}s}^2=1$ represents the physical boundary for the parameters.

Figure 24

Averages of (left) ${\beta }^{\mathrm{eff}}\equiv {\varphi }_1^{\mathrm{eff}}$ and (right) $A_{CP}$ for the $B^0\to f_0{K}_S^0$ decay including measurements from Dalitz plot analyses of both $B^0\to K^{+}{K}^{-}{K}_S^0$ and $B^0\to {\pi }^{+}{\pi }^{-}{K}_S^0$.

Figure 25

Averages of (left) $S_{CP}$ and (right) $C_{CP}$ for the mode $B^0\to K_S^0{K}_S^0$.

Figure 26

Averages of (left) $S_{b\to s\gamma }$ and (right) $C_{b\to s\gamma }$. Recall that the data for $K^{*}\gamma$ is a subset of that for $K_S^0{\pi }^0\gamma$.

Figure 27

Averages of four $b\to s\gamma$ dominated channels in the $S_{CP}$ vs $C_{CP}$ plane. Contours at $S_{CP}^2+C_{CP}^2=1$ represent the physical boundary for the parameters. (Top left) $B^0\to K^{*}\gamma$, (top right) $B^0\to K_S^0{\pi }^0\gamma$ (including $K^{*}\gamma$), (bottom left) $B^0\to K_S^0\eta \gamma$, (bottom right) $B^0\to K_S^0{\rho }^0\gamma$.

Figure 28

Averages of (left) $S_{CP}$ and (right) $C_{CP}$ for the mode $B^0\to {\pi }^{+}{\pi }^{-}$.

Figure 29

Averages of (left) $S_{CP}$ and (right) $C_{CP}$ for the mode $B^0\to {\rho }^{+}{\rho }^{-}$.

Figure 30

Averages of $b\to u\overline{u}d$ dominated channels, for which correlated averages are performed, in the $S_{CP}$ vs $C_{CP}$ plane. Contours at $S_{CP}^2+C_{CP}^2=1$ represent the physical boundary for the parameters. Left: $B^0\to {\pi }^{+}{\pi }^{-}$ and (right) $B^0\to {\rho }^{+}{\rho }^{-}$.

Figure 31

Averages of $CP$ violation parameters in $B^0\to a_1^{\pm }{\pi }^{\mp }$ in ${\mathcal{A}}_{a_1\pi }^{-+}$ vs ${\mathcal{A}}_{a_1\pi }^{+-}$ space.

Figure 32

Summary of the $U$ and $I$ parameters measured in the time-dependent $B^0\to {\pi }^{+}{\pi }^{-}{\pi }^0$ Dalitz plot analysis.

Figure 33

Averages of (left) $S_{b\to u\overline{u}d}$ and (right) $C_{b\to u\overline{u}d}$ for the mode $B^0\to {\rho }^0{\pi }^0$.

Figure 34

Averages of $b\to u\overline{u}d$ dominated channels, for the mode $B^0\to {\rho }^0{\pi }^0$ in the $S_{CP}$ vs $C_{CP}$ plane. The contour at $S_{CP}^2+C_{CP}^2=1$ represents the physical boundary for the parameters.

Figure 35

$CP$ violation in $B^0\to {\rho }^{\pm }{\pi }^{\mp }$ decays. Left: ${\mathcal{A}}_{CP}^{\rho \pi }$ vs $C_{\rho \pi }$ space, (right) ${\mathcal{A}}_{\rho \pi }^{-+}$ $vs$. ${\mathcal{A}}_{\rho \pi }^{+-}$ space.

Figure 36

World average of $\alpha \equiv {\varphi }_2$, in terms of 1−CL, split by decay mode.

Figure 37

Averages for $b\to c\overline{u}d/u\overline{c}d$ modes.

Figure 38

Averages of $A_{CP}$ and $R_{CP}$ from GLW analyses.

Figure 39

Averages of $R_{\mathrm{ADS}}$ and $A_{\mathrm{ADS}}$ for $B\to D^{(*)}{K}^{(*)}$ decays.

Figure 40

Averages of $R_{\mathrm{ADS}}$ and $A_{\mathrm{ADS}}$ for $B\to D^{(*)}\pi$ decays.

Figure 41

Contours in the $(x_{\pm },y_{\pm })$ from model-dependent analysis of $B^{+}\to D^{(*)}{K}^{(*)+}$, $D\to K_S^0{h}^{+}{h}^{-}$ ($h=\pi$, $K$). Left: $B^{+}\to D{K}^{+}$, (middle) $B^{+}\to D^{*}{K}^{+}$, (right) $B^{+}\to D{K}^{*+}$. Note that the uncertainties assigned to the averages given in these plots do not include model uncertainties.

Figure 42

Averages of $(x_{\pm },y_{\pm })$ from model-dependent analyses of $B^{+}\to D^{(*)}{K}^{(*)+}$ with $D\to K_S^0{h}^{+}{h}^{-}$ ($h=\pi$, $K$). Top left: $x_{+}$, (top right) $x_{-}$, (bottom left) $y_{+}$, (bottom right) $y_{-}$. The top plots include constraints on $x_{\pm }$ obtained from GLW analyses (see Sec. 6o1). Note that the uncertainties assigned to the averages given in these plots do not include model uncertainties.

Figure 43

Contours in the $(x_{\pm },y_{\pm })$ plane from model-independent analysis of $B^{+}\to D{K}^{+}$ with $D\to K_S^0{h}^{+}{h}^{-}$ ($h=\pi$, $K$).

Figure 44

World average of $\gamma \equiv {\varphi }_3$, in terms of 1−CL, split by decay mode.

Figure 45

World average of $\gamma \equiv {\varphi }_3$, in terms of 1−CL, split by analysis method.

Figure 46

World averages for the hadronic parameters $r_B$ in the different decay modes, in terms of 1−CL.

Figure 47

Contributions to the combination from different input measurements, shown in the plane of the relevant $r_B$ (left) or ${\delta }_B$ (right) parameter $vs$. $\gamma \equiv {\varphi }_3$. From top to bottom: $B^{+}\to D{K}^{+}$, $B^{+}\to D^{*}{K}^{+}$, $B^{+}\to D{K}^{*+}$ and $B^0\to D{K}^{*0}$. Contours show the two-dimensional 68% and 95% CL regions.

Figure 48

Summary of the constraints on the angles of the unitarity triangle.

Figure 49

Branching fractions of exclusive semileptonic $B$ decays: (a) ${\overline{B}}^0\to D^{*+}{\ell }^{-}{\overline{\nu }}_{\ell }$ (Table 69) and (b) $B^{-}\to D^{*0}{\ell }^{-}{\overline{\nu }}_{\ell }$ (Table 70).

Figure 50

Illustration of (a) the average and (b) the dependence of ${\eta }_{\mathrm{EW}}\mathcal{F}(1)|V_{cb}|$ on ${\rho }^2$. The error ellipses correspond to $\mathrm{\Delta }{\chi }^2=1$ ($\mathrm{C}\mathrm{L}=39\%$). Figure (c) is a zoomed in view of the Belle and BABAR measurements.

Figure 51

Branching fractions of exclusive semileptonic $B$ decays: (a) ${\overline{B}}^0\to D^{+}{\ell }^{-}{\overline{\nu }}_{\ell }$ (Table 72) and (b) $B^{-}\to D^0{\ell }^{-}{\overline{\nu }}_{\ell }$ (Table 73).

Figure 52

Illustration of (a) the average and (b) dependence of ${\eta }_{\mathrm{EW}}\mathcal{G}(w)|V_{cb}|$ on ${\rho }^2$. The error ellipses correspond to $\mathrm{\Delta }{\chi }^2=1$ ($\mathrm{CL}=39\%$). Figure (c) is a zoomed in view of the Belle and BABAR measurements.

Figure 53

Average branching fraction of exclusive semileptonic $B$ decays (a) ${\overline{B}}^0\to D^0{\pi }^{+}{\ell }^{-}{\overline{\nu }}_{\ell }$, (b) ${\overline{B}}^0\to D^{*0}{\pi }^{+}{\ell }^{-}{\overline{\nu }}_{\ell }$, (c) $B^{-}\to D^{+}{\pi }^{-}{\ell }^{-}{\overline{\nu }}_{\ell }$, and (d) $B^{-}\to D^{*+}{\pi }^{-}{\ell }^{-}{\overline{\nu }}_{\ell }$. The corresponding individual results are also shown.

Figure 54

Rescaled individual measurements and their averages for (a) $\mathcal{B}(B^{-}\to D_1^0{\ell }^{-}{\overline{\nu }}_{\ell })\times \mathcal{B}(D_1^0\to D^{*+}{\pi }^{-})$ and (b) $\mathcal{B}(B^{-}\to D_2^0{\ell }^{-}{\overline{\nu }}_{\ell })\times \mathcal{B}(D_2^0\to D^{*+}{\pi }^{-})$.

Figure 55

Rescaled individual measurements and their averages for (a) $\mathcal{B}(B^{-}\to D_1^{\prime 0}{\ell }^{-}{\overline{\nu }}_{\ell })\times \mathcal{B}(D_1^{\prime 0}\to D^{*+}{\pi }^{-})$ and (b) $\mathcal{B}(B^{-}\to D_0^{*0}{\ell }^{-}{\overline{\nu }}_{\ell })\times \mathcal{B}(D_0^{*0}\to D^{+}{\pi }^{-})$.

Figure 56

Fit to the inclusive partial semileptonic branching fractions and to the lepton energy moments in the kinetic mass scheme. In all plots, the gray band is the theory prediction with total theory error. BABAR data are shown by circles, Belle by squares and other experiments (DELPHI, CDF, CLEO) by triangles. Filled symbols mean that the point was used in the fit. Open symbols are measurements that were not used in the fit.

Figure 57

Same as Fig. 56 for the fit to the hadronic mass moments in the kinetic mass scheme.

Figure 58

The $B\to \pi \ell \nu q^2$ spectrum measurements and the average spectrum obtained from the likelihood combination (shown in black).

Figure 59

Fit of the BCL parametrization to the averaged $q^2$ spectrum from BABAR and Belle and the LQCD and LCSR calculations. The error bands represent the $1\sigma$ (dark green) and $2\sigma$ (light green) uncertainties of the fitted spectrum.

Figure 60

(a) Summary of exclusive determinations of $\mathcal{B}(B^{+}\to {\rho }^0{\ell }^{+}\nu )$ and their average. Measurements of $B^0\to {\rho }^{-}{\ell }^{+}\nu$ branching fractions have been scaled by $0.5{\tau }_{B^{+}}/{\tau }_{B^0}$ in accordance with isospin symmetry. (b) Summary of exclusive determinations of $B^{+}\to \omega {\ell }^{+}\nu$ and their average.

Figure 61

The averaged $q^2$ spectrum of the measurements listed in the text for the $\rho$ (left) and $\omega$ (right) final state on top of the latest Belle and BABAR measurements. The isospin transformation is applied to the $B^0\to {\rho }^{-}{\ell }^{+}\nu$ measurements. In the right figure we also show the model (green band) which was used to split the bins in the averaging procedure.

Figure 62

(a) Summary of exclusive determinations of $\mathcal{B}(B^{+}\to \eta {\ell }^{+}\nu )$ and their average. (b) Summary of exclusive determinations of $\mathcal{B}(B^{+}\to {\eta }^{\prime }{\ell }^{+}\nu )$ and their average.

Figure 63

Measurements of $|V_{ub}|$ from inclusive semileptonic decays and their average based on the BLNP (a) and DGE (b) prescription. The labels indicate the variables and selections used to define the signal regions in the different analyses.

Figure 64

Measurements of $|V_{ub}|$ from inclusive semileptonic decays and their average based on the GGOU (a) and ADFR (b) prescription. The labels indicate the variables and selections used to define the signal regions in the different analyses.

Figure 65

Measurements of $|V_{ub}|$ from inclusive semileptonic decays and their average in the BLL prescription.

Figure 66

Combined average on $|V_{ub}|$ and $|V_{cb}|$ including the LHCb measurement of $|V_{ub}|/|V_{cb}|$, the exclusive $|V_{ub}|$ measurement from $B\to \pi \ell \nu$, and the $|V_{cb}|$ average from $B\to D\ell \nu$, $B\to D^{*}\ell \nu$ and $B_s\to D_s^{(*)}\mu \nu$ measurements. The dashed ellipse corresponds to a $1\sigma$ two-dimensional contour (68% of CL). The point with the error bars corresponds to the inclusive $|V_{cb}|$ from the kinetic scheme (Sec. 7b2), and the inclusive $|V_{ub}|$ from GGOU calculation (Sec. 7d3).

Figure 67

Measurements of $\mathcal{R}(D)$ and $\mathcal{R}(D^{*})$ listed in Table 97 and their two-dimensional average. Contours correspond to $\mathrm{\Delta }{\chi }^2=1$, i.e., 68% CL for the bands and 39% CL for the ellipses. The black and blue points with error bars, are two recent SM prediction for $\mathcal{R}(D^{*})$ and $\mathcal{R}(D)$. The SM predictions reported are based on results from Refs. [610, 613, 615]. More information is given in the text. An average of these predictions and the experimental average deviate from each other by about $3.3\sigma$. The dashed ellipse correspond to a $3\sigma$ contour (99.73% CL).

Figure 68

(a) Measurements of $\mathcal{R}(D)$ and (b) $\mathcal{R}(D^{*})$. The green bands are the averages obtained from the combined fit. The red bands are the averages of the theoretical predictions obtained as explained in the text.

Figure 69

A selection of high-precision charmless mesonic $B$ meson branching fraction measurements.

Figure 70

Branching fractions of charmless baryonic $B^{+}$ and $B^0$ decays into nonstrange baryons.

Figure 71

Branching fractions of charmless baryonic $B^{+}$ and $B^0$ decays into strange baryons.

Figure 72

Branching fractions of charmless ${\mathrm{\Lambda }}_b^0$ decays.

Figure 73

Branching fractions of charmless leptonic $B_s^0$ decays.

Figure 74

Branching fractions of charmless nonleptonic $B_s^0$ decays.

Figure 75

Branching fractions of $B^{+}$ and $B^0$ decays of the type $b\to s{\ell }^{+}{\ell }^{-}$.

Figure 76

Branching fractions of $B^{+}$ and $B^0$ decays of the type $b\to u{\ell }^{+}{\ell }^{-}$, purely leptonic and leptonic radiative.

Figure 77

Compilation of $R_K^{(*)}$ ratios in the low dilepton invariant-mass region. These are ratios between branching fractions of $B$-meson decays to $K^{(*)}{\mu }^{+}{\mu }^{-}$ and $K^{(*)}{e}^{+}{e}^{-}$, which provide information on lepton universality.

Figure 78

Limits on branching fractions of lepton-flavor-violating $B^{+}$ and $B^0$ decays.

Figure 79

Limits on branching fractions of lepton-number-violating $B^{+}$ and $B^0$ decays.

Figure 80

Branching fractions of charmless $B$ decays with neutrinos.

Figure 81

A selection among the most precise direct $CP$ asymmetries ($A_{CP}$) measured in charmless $B^{+}$ and $B^0$ decay modes.

Figure 82

Longitudinal polarization fraction in charmless $B$ decays.

Figure 83

Longitudinal polarization fraction in charmless $B_s^0$ decays.

Figure 84

World average value of $R_M=(x^2+y^2)/2$ as calculated from $D^0\to K^{+}{\ell }^{-}\overline{\nu }$ measurements [1202, 1203, 1204, 1205]. The confidence level from the fit is 0.97.

Figure 85

World average value of $y_{CP}$ as calculated from $D^0\to K^{+}{K}^{-},{\pi }^{+}{\pi }^{-},K^{+}{K}^{-}{K}_S^0$, and $K_S^0\omega$ measurements [1222, 1223, 1224, 1225, 1226, 1227, 1228, 1229, 1230]. The confidence level from the fit is 0.21.

Figure 86

World average value of $A_{\mathrm{\Gamma }}$ calculated from $D^0\to K^{+}{K}^{-},{\pi }^{+}{\pi }^{-}$ measurements [1226, 1228, 1231, 1232]. The confidence level from the fit is 0.73.

Figure 87

Two-dimensional contours for theoretical parameters $(x_{12},y_{12})$ (top left), $(x_{12},{\varphi }_{12})$ (top right), and $(y_{12},{\varphi }_{12})$ (bottom), under the assumption of no direct $CPV$ in DCS decays (Fit 3).

Figure 88

Two-dimensional contours for parameters $(x,y)$ (upper) and $(|q/p|-1,\varphi )$ (lower), allowing for $CPV$ (fit 4).

Figure 89

The function $\mathrm{\Delta }{\chi }^2={\chi }^2-{\chi }_{\min }^2$ for fitted parameters $x,y,\delta ,{\delta }_{K\pi \pi },|q/p|$, and $\varphi$. The points where $\mathrm{\Delta }{\chi }^2=3.84$ (denoted by dashed horizontal lines) determine 95% C.L. intervals.

Figure 90

Plot of all data and the fit result. Individual measurements are plotted as bands showing their $\pm 1\sigma$ range. The no-$CPV$ point (0,0) is shown as a filled circle, and the best fit value is indicated by a cross showing the one-dimensional uncertainties. Two-dimensional 68% C.L., 99.7% C.L., and 99.99997% C.L. regions are plotted as ellipses.

Figure 91

Comparison of the results of $f_{+}^K(0)|V_{cs}|$ measured by the Belle [1335], BABAR [1336], CLEO-c [1337], and BESIII [1269, 1334, 1339, 1340] experiments.

Figure 92

Decay angles ${\theta }_V$, ${\theta }_{\ell }$ and $\chi$. Note that the angle $\chi$ between the decay planes is defined in the $D$-meson reference frame, whereas the angles ${\theta }_V$ and ${\theta }_{\ell }$ are defined in the $V$ meson and $W$ reference frames, respectively.

Figure 93

WA values for $f_D|V_{cd}|$ (left) and $f_{D_s}|V_{cs}|$ (right). For each point, the first error is statistical and the second error is systematic. BESIII(a) represents results based on $0.48{\mathrm{fb}}^{-1}$ of data recorded at $\sqrt{s}=4.009\mathrm{GeV}$ [1412], and BESIII(b) represents results based on $6.32{\mathrm{fb}}^{-1}$ of data recorded at $\sqrt{s}=4.178–4.226\mathrm{GeV}$ [1405, 1406, 1407].

Figure 94

Comparison of magnitudes of CKM matrix elements $|V_{cd}|$ (left) and $|V_{cs}|$ (right), as determined from leptonic and semileptonic $D_{(s)}$ decays. Also listed are results from neutrino scattering, from $W$ decays, and from a global fit of the CKM matrix assuming unitarity [242].

Figure 95

The $K\pi$ invariant mass distribution for $D^0\to K^{-}{\pi }^{+}(n\gamma )$ decays. The three curves correspond to three different configurations of photos for modeling FSR: version 2.02 without interference (blue/gray), version 2.02 with interference (red dashed) and version 2.15 with interference (black). The true invariant mass has been smeared with a typical experimental resolution of $10\mathrm{MeV}/c^2$. Inset: The corresponding spectrum of total energy radiated per event. The arrow indicates the $\sum E_{\gamma }$ value that begins to shift kinematic quantities outside of the range typically accepted in a measurement.

Figure 96

FOCUS data (dots), original fits (blue) and toy MC parametrization (red) for $D^0\to K^{-}{\pi }^{+}$ (left), $D^0\to {\pi }^{+}{\pi }^{-}$ (center), and $D^0\to {\pi }^{+}{\pi }^{-}$ (right).

Figure 97

Comparison of measurements of $\mathcal{B}(D^0\to K^{-}{\pi }^{+})$ (blue) with the average branching fraction obtained here (red, and yellow band). For these measurements only, the partial ${\chi }^2$ is 4.9 in the final fit.

Figure 98

The $\mathcal{B}(D^0\to K^{+}{K}^{-})$ (left) and $\mathcal{B}(D^0\to {\pi }^{+}{\pi }^{-})$ (right) values obtained either from absolute measurements or by scaling the measured branching ratios with the $\mathcal{B}(D^0\to K^{-}{\pi }^{+})$ branching fraction average obtained here. For the measurements (blue points), the error bars correspond to the statistical, systematic and either the $K\pi$ normalization uncertainties or, in case of an absolute measurement, the FSR modeling uncertainty. The average obtained here (red point, yellow band) lists the statistical, systematics excluding FSR, and the FSR systematic. For the measurements related to $\mathcal{B}(D^0\to K^{+}{K}^{-})$ [$\mathcal{B}(D^0\to {\pi }^{+}{\pi }^{-})$] only, the partial ${\chi }^2$ is 15.7 [6.0] in the final fit.

Figure 99

(a) Average masses for excited $D_s$ mesons; (b) average masses for excited $D$ mesons; (c) average widths for excited $D_s$ mesons; (d) average widths for excited $D$ mesons. The vertical shaded regions distinguish between different spin-parity states.

Figure 100

Level diagram for multiplets and transitions for excited charm baryons.

Figure 101

Upper limits at 90% C.L. for $D^0$ decays. The top plot shows flavor-changing neutral current and radiative decays, and the bottom plot shows lepton-flavor-changing (LF), lepton-number-changing (L), and both baryon- and lepton-number-changing (BL) decays.

Figure 102

Upper limits at 90% C.L. for $D^{+}$ (top) and $D_s^{+}$ (bottom) decays. Each plot shows flavor-changing neutral current and rare decays, lepton-flavor-changing decays (LF), and lepton-number-changing (L) decays.

Figure 103

Upper limits at 90% C.L. for ${\mathrm{\Lambda }}_c^{+}$ decays. Shown are flavor-changing neutral current decays, lepton-flavor-changing (LF) decays, and lepton-number-changing (L) decays.

Figure 104

$|V_{us}|$ determinations. In the CKM-unitarity evaluation, $|V_{ud}|$ is taken from a 2020 experimental update [1658]. The value ${|V_{us}|}_{K\ell 3}$ from $K^{-}\to {\pi }^0{\mu }^{-}{\overline{\nu }}_{\mu }$ decays is taken from a 2021 update [1665]. The value ${|V_{us}|}_{K\ell 2}$ from $K^{-}\to {\mu }^{-}{\overline{\nu }}_{\mu }$ decays is taken from Ref. [9].