Leptonic and semileptonic decays of $B$ mesons

Semileptonic decays are ideally suited to study the weak interaction as well as strong interaction effects in $B$-meson decays. In the last decade, precision studies of semileptonic $B$ decays have been made possible by the large samples of $B$ mesons collected at the $B$ factories KEKB in Japan and PEP-II in the USA. Measurements of the charged-current semileptonic transitions $b\to q\ell \nu$ ($q=u$, $c$) allow for a determination of the magnitude of the Cabibbo-Kobayashi-Maskawa matrix elements $V_{cb}$ and $V_{ub}$ and the masses of the $b$ and $c$ quarks, which are fundamental parameters of the standard model of particle physics. The values of $|V_{cb}|$ and $|V_{ub}|$ are determined from measurements of inclusive $B$ decays in combination with calculations of partial decay rates or from exclusive decays combined with theoretical predictions of hadronic form factors. Purely leptonic $B$ decays $B\to \ell \nu$ ($\ell =e$, $\mu$, $\tau$) also provide access to $|V_{ub}|$. They are theoretically simpler, but the available signal samples are still small. Decays involving a $\tau$ lepton, $B\to \tau \nu$ and $B\to D^{(*)}\tau \nu$, are sensitive to new physics, in particular, to charged Higgs bosons in models with an extended Higgs sector, and provide a window to the physics of the third generation. In this article, the measurements and theoretical descriptions of charged-current leptonic and semileptonic $B$ decays and the status of $|V_{cb}|$ and $|V_{ub}|$ determinations are reviewed. An overview of the theoretical approaches and the experimental techniques used in the study of these decays is also provided.

Figure 1

(a) A leptonic $B$ decay ($B\to \ell \nu$), and (b) a semileptonic $B$ decay ($B\to X\ell \nu$).

Figure 2

Allowed kinematic region in the $q^2\text{−}{E}_{\ell }$ plane for $B\to D^{*}\ell \overline{\nu }$ decays. From [156].

Figure 3

Lepton momentum distributions for semileptonic $B$ decays: (a) $B\to X_c\ell \nu$ and (b) $B\to X_u\ell \nu$. From [39].

Figure 4

$B$-meson production in $e^{+}{e}^{-}$ collisions at the $\mathrm{\Upsilon }(4S)$ resonance.

Figure 5

Side views of (a) the Belle and (b) BABAR detectors. The acronyms used for the subdetector components of Belle are SVD = silicon vertex detector, CDC = central drift chamber, PID = particle identification system, TOF = time-of-flight counter, CsI = CsI crystal calorimeter, $\mathrm{KLM}=K_L^0/\mathrm{m}\text{uon system}$, and EFC = extreme forward calorimeter. From [4] and [33].

Figure 6

Event display of a $B^{+}\to {\rho }^0{\mu }^{+}\nu$ (${\rho }^0\to {\pi }^{+}{\pi }^{-}$) decay in the BABAR detector. The neutrino is not detected and can be only indirectly reconstructed; its flight direction is indicated by the dashed arrow. The second $B$ meson in the event decayed hadronically.

Figure 7

Distribution of $|\cos \mathrm{\Delta }{\theta }_T|=|{\widehat{T}}_{\mathrm{cand}}-{\widehat{T}}_{\mathrm{ROE}}|$ for $B\overline{B}$ events (open histogram) and continuum events (shaded histogram).

Figure 8

Observed distributions of (a) $m_{\mathrm{bc}}$ ($=m_{\mathrm{ES}}$) and (b) $\mathrm{\Delta }E$ from an analysis of hadronic $B$ decays ([46]). The data are compared with the result of a fit of two functions representing correctly reconstructed $B$ candidates (dashed lines) and the total continuum and combinatorial $B\overline{B}$ background (dash-dotted lines). The sum of the signal and background functions is shown as solid lines.

Figure 9

The variable $\cos {\theta }_{BY}$ for $B^0\to D^{*-}{\ell }^{+}\nu$ decays: (a) reconstruction of the angle ${\theta }_{BY}$, and (b) distribution of $\cos {\theta }_{BY}$ for correctly reconstructed $B^0\to D^{*-}{\ell }^{+}\nu$ decays (topmost histogram) and various backgrounds (other histograms). From [121].

Figure 10

The selection and classification stages of the hadronic $B$-tagging algorithm in Belle. From [126].

Figure 11

Distributions from the untagged $B\to \mu \nu$ analysis by Belle: (a) lepton momentum in the $B$ rest frame $p_{\ell }^B$, (b) Fisher discriminant $F$ used for continuum suppression, and (c) $p_{\ell }^B$ after a cut on the Fisher discriminant ($F>0.3$). The arrows indicate the selection criteria applied on these variables. From [194].

Figure 12

Distribution of the lepton momentum in the $B$ rest frame $p_{\ell }^B$ for the hadronic-tag $B\to \mu \nu$ analysis by BABAR. The observed distribution (points) is compared with the fit of a signal (dotted line) and a background (solid line) function to data. The signal is shown with an arbitrary normalization. From [47].

Figure 13

Distributions of $E_{\text{extra}}^{\mathrm{ECL}}$ for the Belle $B\to \tau \nu$ analysis using (a) hadronic tags ([19]) and (b) semileptonic tags ([157]). The observed distributions (points) are compared with fits of simulated signal (topmost histogram) and background distributions (other histograms) to data.

Figure 14

Comparison of the $B\to \tau \nu$ results from Belle and BABAR. The inner error bars show the statistical uncertainty, and the outer ones the total experimental uncertainty. For averaged branching fractions, only the total uncertainty is shown.

Figure 15

Constraints on the parameters $m_{H^{\pm }}$ and $\tan \beta$ in the type-II 2HDM, derived from the $B\to \tau \nu$ branching fraction of the BABAR hadronic-tag analysis. The exclusion limits are shown for 90% (light-shaded) and 99% (dark-shaded) confidence level and two example $|V_{ub}|$ values: (a) $(3.23\pm 0.30)\times {10}^{-3}$ and (b) $(4.33\pm 0.28)\times {10}^{-3}$. From [163].

Figure 16

Inclusive $B\to X_c\ell \nu$ spectra measured by Belle ([195]; [204]) for (a) squared hadronic mass and (b) lepton momentum.

Figure 17

Results of the global fit to the various $E_{\ell }$ and $m_X$ moments (a)–(g) in $B\to X_c\ell \nu$ decays by [134]. The shaded bands correspond to the theoretical uncertainty of the HQE on the moments. The measured moments are shown as circles for BABAR, squares for Belle, and triangles for CDF, CLEO, and DELPHI. Measurements represented by filled symbols are included in the fit, and those with open symbols are not.

Figure 18

$\mathrm{\Delta }{\chi }^2=1$ contours for the $|V_{cb}|$, ${\mu }_{\pi }^2$, and $m_b$ results of the global $B\to X_c\ell \nu$ moment fit by [134]. The various ellipses correspond to different scenarios for the theoretical correlations. Shaded ellipses represent the fits without $m_c$ constraint and open ellipses those with $m_c$ constraint; see [134] for details.

Figure 19

Electron momentum distribution in the $\mathrm{\Upsilon }(4S)$ frame for the BABAR untagged $B\to X_{u}e\nu$ end point analysis. The distribution is shown for three stages of the analysis: (a) on-resonance data (open circles) and off-resonance data (filled circles). The line shows the parametrization of the continuum background fit to off-resonance data and to on-resonance data above 2.8 GeV, (b) continuum-subtracted on-resonance data (triangles) and $B\overline{B}$ background (histogram), and (c) on-resonance data after subtraction of all backgrounds (squares) compared with the simulated $B\to X_{u}e\nu$ signal (histogram). The hatched band illustrates the region that is combined into a single bin in the signal fit. From [39].

Figure 20

Measured distributions of (a) $m_X$, (b) $P_{+}$, (c) $q^2$ for $m_X<1.7\mathrm{GeV}$, and (d) $p_{\ell }^{*}$ for the BABAR hadronic-tag $B\to X_u\ell \nu$ analysis. Upper row: Comparison of the data with a fit of the simulated $B\to X_u\ell \nu$ signal (open histogram) and background (gray, shaded histogram) shapes to data. Lower row: Comparison of the background-subtracted data distributions with the fitted simulated signal distributions (histograms). From [161].

Figure 21

Projections of measured distributions in (a) $m_X$ and (b) $q^2$ for the Belle hadronic-tag $B\to X_u\ell \nu$ analysis ([205]). The observed distributions are compared with the result of a two-dimensional fit of the simulated $B\to X_u\ell \nu$ signal and background to data.

Figure 22

Distribution of $M_{\text{miss}}^2$ from the hadronic-tag $B\to D\ell \nu$ analysis by BABAR, for two $w$ intervals. From [55].

Figure 23

Dependence of the form factor on $w$ for the BABAR analysis of (a) $B\to D\ell \nu$ ([55]) and (b) $B\to D^{*}\ell \nu$ ([42]) decays. The distributions have been corrected for reconstruction efficiency. The results of a fit to the $w$ distributions are shown as lines. In (c), the differential width of $B\to D\ell \nu$ for the Belle hadronic-tag analysis ([137]) and the result of the combined fit of the BGL form factor parametrization to the experimental and lattice QCD (FNAL-MILC and HPQCD) data is shown.

Figure 24

The helicity angles ${\theta }_{\ell }$, ${\theta }_V$, and $\chi$ in the decay $B\to D^{*}\ell \nu$, $D^{*}\to D\pi$.

Figure 25

Distributions of the velocity transfer $w$, the cosine of the angles ${\theta }_{\ell }$ and ${\theta }_V$, and the magnitude of the angle $\chi$ for the Belle analysis of $B^0\to D^{*-}{\ell }^{+}\nu$ decays. From [121].

Figure 26

Excited charm-meson states with orbital angular momentum $L=0$ and $L=1$. The shaded boxes indicate the widths of the resonances. The lines show the possible decays involving the emission of one or two pions. The states collectively referred to as $D^{**}$ are $D_0^{*}(2400)$, $D_1^{*}(2430)$, $D_1(2420)$, and $D_2^{*}(2460)$. From [78].

Figure 27

Distributions of the mass difference $\mathrm{\Delta }m=m_{D^{(*)}\pi }-m_{D^{(*)}}$ for the BABAR analysis of $B\to D^{**}\ell \nu$ decays in four different final states: (a) $B^{-}\to D^{*+}{\pi }^{-}{\ell }^{-}\overline{\nu }$, (b) $B^{-}\to D^{+}{\pi }^{-}{\ell }^{-}\overline{\nu }$, (c) ${\overline{B}}^0\to D^{*0}{\pi }^{+}{\ell }^{-}\overline{\nu }$, and (d) ${\overline{B}}^0\to D^0{\pi }^{+}{\ell }^{-}\overline{\nu }$. From [43].

Figure 28

The form factor $f_{B\pi }^{+}$ as a function of $q^2$. Data points are from lattice QCD simulations; squares: HPQCD ([140]) and triangles: Fermilab and MILC ([56]). The value at $q^2=0$ is taken from a QCD sum rules calculation ([155]). The interpolation uses the BCL parametrization ([94]).

Figure 29

Observed distributions of $m_{\mathrm{bc}}$ and $\mathrm{\Delta }E$ for the untagged $B\to \pi \ell \nu$ analysis from BABAR ([160]) in two $q^2$ regions. The observed distributions (points) are compared with a fit of simulated signal (topmost histogram) and background (other histograms) to data.

Figure 30

Distributions of $M_{\text{miss}}^2$ for the hadronic-tag analysis of exclusive charmless semileptonic $B$ decays from Belle ([200]): (a) $B^0\to {\pi }^{-}{\ell }^{+}\nu$, (b) $B^{+}\to {\pi }^0{\ell }^{+}\nu$ with ${\pi }^0\to \gamma \gamma$, (c) $B^0\to {\rho }^{-}{\ell }^{+}\nu$ with ${\rho }^{-}\to {\pi }^{-}{\pi }^0$, (d) $B^{+}\to {\rho }^0{\ell }^{+}\nu$ with ${\rho }^0\to {\pi }^{-}{\pi }^{+}$, (e) $B^{+}\to \omega {\ell }^{+}\nu$ with $\omega \to {\pi }^{-}{\pi }^{+}{\pi }^0$, and (f) $B^{+}\to \omega {\ell }^{+}\nu$ with $\omega \to {\pi }^0\gamma$. The data (points) are compared with a fit of the simulated signal (topmost histogram) and background (other histograms) distributions to data.

Figure 31

Fit of the BCL parametrization to the $B\to \pi \ell \nu q^2$ spectrum: (a) Combined BCL fit to data and FNAL-MILC and LCSR predictions. The shaded band shows the fit uncertainty. The theoretical predictions are scaled according to the $|V_{ub}|$ value derived from the fit. (b) Shape comparison of the BCL fit with various form factor predictions, shown as solid lines in their regions of validity ($q^2<12{\mathrm{GeV}}^2$ for LCSR, $q^2>16{\mathrm{GeV}}^2$ for lattice QCD, all $q^2$ for ISGW2), where they have been normalized to the BCL fit. Extrapolations to the full $q^2$ range are shown as dashed lines and have large uncertainties (the corresponding uncertainty bands are not shown for a better visibility).

Figure 32

A $B\to D^{(*)}\tau \nu$ decay. In models with an extended Higgs sector, the exchange of a charged Higgs boson instead of a $W$ boson is also possible.

Figure 33

Fit projections for the $D^{(*)}\ell$ samples in the $B\to D^{(*)}\tau \nu$ analysis of (a) BABAR ([164]) and (b) Belle ([147]). For the BABAR analysis, the distributions of the two fit variables $M_{\text{miss}}^2$ and $p_{\ell }^{*}$ after the fit are shown. For the Belle analysis, the $M_{\text{miss}}^2$ distribution is shown for $M_{\text{miss}}^2>0.85{\mathrm{GeV}}^2$.

Figure 34

History of (a) $R(D)$ and (b) $R(D^{*})$ measurements. The gray bands indicate the SM predictions.

Figure 35

Expected shapes of the (a) $M_{\text{miss}}^2$ and (b) $p_{\ell }^{*}$ distributions in $B\to D\tau \nu$ decays for four values of $\tan \beta /m_{H^{\pm }}$. (c) The $R(D)$ and $R(D^{*})$ results for the BABAR measurement reweighted to different $\tan \beta /m_{H^{\pm }}$ values (light-shaded, blue band) is compared with the type-II 2HDM prediction (dark-shaded, red band). The bands indicate the $1\sigma$ uncertainty. From [164].

Figure 36

(a) The selection strategy and (b) the lepton momentum spectrum in the Belle analysis of $B_s\to X^{-}{\ell }^{+}\nu$ decays. In (a), the dashed lines indicate $B_s\overline{B_s}$ oscillations. The spectrum in (b) is obtained from $K^{+}{K}^{-}{\pi }^{+}$ mass fits in bins of lepton momentum. The signal is the topmost, white histogram. From [188].

Figure 37

Comparison of the determinations of (a) $|V_{cb}|$ and (b) $|V_{ub}|$ based on exclusive and inclusive decays. Because of the marginal compatibility of the exclusive and inclusive results, the uncertainty on the average has been scaled by a factor of $\sqrt{{\chi }^2/\mathrm{ndf}}=2.8$ for $|V_{cb}|$ and 3.8 for $|V_{ub}|$. The error bars on the average correspond to the unscaled uncertainties, while the bands represent the scaled uncertainties.