Semitauonic $b$-hadron decays: A lepton flavor universality laboratory

The study of lepton flavor universality violation (LFUV) in semitauonic $b$-hadron decays has become increasingly important in light of long-standing anomalies in their measured branching fractions, and the large datasets anticipated from the LHC experiments and Belle II. In this review, a comprehensive survey of the experimental environments and methodologies for semitauonic LFUV measurements at the $B$ factories and LHCb is undertaken, along with an overview of the theoretical foundations and predictions for a wide range of semileptonic decay observables. The future prospects of controlling systematic uncertainties down to the percent level, matching the precision of standard model (SM) predictions, are examined. Furthermore, new perspectives and caveats on combinations of the LFUV data are discussed and the world averages for the $\mathcal{R}(D^{(*)})$ ratios are revisited. Here it is demonstrated that different treatments for the correlations of uncertainties from $D^{**}$ excited states can vary the current $3\sigma$ tension with the SM within a $1\sigma$ range. Prior experimental overestimates of $D^{**}\tau \nu$ contributions may further exacerbate this. The precision of future measurements is also estimated; their power to exploit full differential information, and solutions to the inherent difficulties in self-consistent new physics interpretations of LFUV observables, are explored.

Figure 1

Left: definition of the ${\theta }_l$ and ${\varphi }_l$ helicity angles in the lepton pair rest frame. Center: definition of the ${\theta }_v$ and ${\varphi }_v$ helicity angles in the $D^{*}$ rest frame. Right: definition of the ${\theta }_h$ and ${\varphi }_h$ helicity angles in the $\tau$ rest frame for $B\to D^{(*)}(\tau \to h\nu )\overline{\nu }$ decay.

Figure 2

Left panel: side view of the Belle detector. See [20] for further details on the subdetectors and their acronyms. The BABAR detector has a similar configuration. Right panel: view perpendicular to the beam axis. The displayed event is reconstructed as a $\mathrm{\Upsilon }(4S)\to B^{+}{B}^{-}$ candidate, with $B^{-}\to D^0{\tau }^{-}{\overline{\nu }}_{\tau }$, $D^0\to K^{-}{\pi }^{+}$, and ${\tau }^{-}\to e^{-}{\nu }_{\tau }{\overline{\nu }}_e$, and the $B^{+}$ decaying to five charged particles (white solid lines) and two photons. The directions of undetected neutrinos are indicated as dashed lines. From [20], and [86].

Figure 3

Left panel: side view of the LHCb detector. See [43] and [3] for further details on the subdetectors and their acronyms. Right panel: side view of an event display for a $B^0\to D^{*+}{\tau }^{-}{\overline{\nu }}_{\tau }$ decay. The area around the interaction point is enlarged in the inset at the top. The trajectory of the $B^0$ meson is indicated with a thick dotted line, and the trajectories of the particles from the subsequent $D^{*+}\to D^0{\pi }^{+}$, $D^0\to K^{-}{\pi }^{+}$, and ${\tau }^{-}\to {\mu }^{-}{\overline{\nu }}_{\mu }{\nu }_{\tau }$ decays are illustrated with thick solid lines. Adapted from [3], and [86].

Figure 4

Examples of particle reconstruction performance for the BABAR detector; the performance for the Belle detector is similar. Left panel: electron reconstruction efficiency. Middle panel: muon reconstruction efficiency. Right panel: Cherenkov angle measurement for different particles species at BABAR’s detector of internally reflected Cherenkov light. Adapted from [104], and [41].

Figure 5

Examples of particle reconstruction performance for the LHCb detector. Left panel: electron reconstruction efficiency. Middle panel: muon reconstruction efficiency. Right panel: Cherenkov angle measurement for different particle species at LHCb’s Ring Imaging Cherenkov detector 1 (RICH1). Adapted from [3].

Figure 6

Resolution on the $q^2$ reconstruction in simulated $B\to D^{*}\tau \nu$ decays for the different methods for estimating the $p_{B_{\mathrm{sig}}}$ momentum. The $\tau$ vertex and rest-frame approximation (RFA) methods used at LHCb are described in Secs. 3c2 and 3c3, respectively. Each value in parentheses corresponds to the rms of the distribution. The various curves were extracted from [140] and [5], [10].

Figure 7

Schematic illustration of the fei algorithm. From [135].

Figure 8

Reconstructed topologies for the $B\to D^{*}\tau \nu$ decays in the hadronic-$\tau$ ([10]) (top) and muonic-$\tau$ ([5]) (bottom) measurements of $\mathcal{R}(D^{*+})$ at LHCb. The filled circles correspond to the reconstructed vertices, and solid lines to reconstructed particles. “PV” refers to the primary vertex, $\mathrm{\Delta }z$ indicates the distance in the $z$ direction between the ${\overline{B}}^0$ (or $D^{*+}$) and ${\tau }^{-}$ vertices, and $\alpha$ refers to the angle between the beam axis and the momentum of the ${\overline{B}}^0$ meson.

Figure 9

Projections of the signal fits for the BABAR ([139]) and Belle ([122]) measurements of $\mathcal{R}(D^{(*)})$ with hadronic tagging. (a),(b) Full $m_{\mathrm{miss}}^2$ projections of the BABAR fit showing the normalization components for the $D\ell$ and $D^{*}\ell$ samples (a combination of $D^{(*)0}\ell$ and $D^{(*)+}\ell$). (c),(d) $m_{\mathrm{miss}}^2$ projections of the BABAR fit focusing on the signal contributions at high $m_{\mathrm{miss}}^2$. (e)–(h) Full projections of the fit to the neural network output $o_{\mathrm{NB}}^{\prime }$ by Belle in the region $m_{\mathrm{miss}}^2>0.85{\mathrm{GeV}}^2$ for the four $D^{(*)}\ell$ samples.

Figure 10

Checks on the kinematic distributions for events in the signal enhanced high-$m_{\mathrm{miss}}^2$ region [$m_{\mathrm{miss}}^2>1.5{\mathrm{GeV}}^2$ for (a)–(d) and $m_{\mathrm{miss}}^2>2{\mathrm{GeV}}^2$ for (e)–(h)]. The solid histograms correspond to the simulation scaled to the fit results. Adapted from [140], and [122].

Figure 11

Efficiency-corrected $q^2$ distributions for $B\to D\tau \nu$ (top panels) and $B\to D^{*}\tau \nu$ events (bottom panels) with (a),(b) $m_{\mathrm{miss}}^2>0.85{\mathrm{GeV}}^2$ and (c)–(f) $m_{\mathrm{miss}}^2>1.5{\mathrm{GeV}}^2$. The shaded distributions correspond to the SM expectations in (a)–(d) and a type-II 2HDM with $\tan \beta /m_{H^{\pm }}=0.45{\mathrm{GeV}}^{-1}$, the value that reproduces that of $\mathcal{R}(D)$ measured by BABAR, in (e) and (f). The ${\chi }^2$ values are calculated based on the statistical uncertainties only. Adapted from [140], and [122].

Figure 12

Signal fit for the Belle measurement of $B\to \pi \tau \nu$ decays. The $E_{\mathrm{ECL}}$ distributions for the three reconstructed $\tau$ decay modes are shown: (left panel) $\tau \to e\nu \overline{\nu }$, (middle panel) $\tau \to \pi \nu$, and (right panel) $\tau \to \rho \nu$. Adapted from [112].

Figure 13

Projection of the signal fit for the Belle measurement of $\mathcal{R}(D^{(*)})$ using semileptonic tagging. The four panels correspond to the four reconstruction categories: (top left panel) $D^0\ell$, (top right panel) $D^{*0}\ell$, (bottom left panel) $D^{+}\ell$, (bottom right panel) $D^{*+}\ell$. The signal-enriched regions, which were obtained using a cut on a multivariate classifier, are shown in the insets. The uncertainties are only statistical. Adapted from [82].

Figure 14

Projections of the control sample fits in the highest $q^2$ bin for the LHCb measurement of $\mathcal{R}(D^{*+})$ involving muonic $\tau$ decays. (a),(b) $D^{*+}{\mu }^{-}{\pi }^{-}$ sample enriched in $B\to D^{**}l\nu$ decays and (c),(d) a $D^{*+}{\mu }^{-}{K}^{\pm }$ sample enriched in $\overline{B}\to D^{*+}{H}_{c}X$ decays. Adapted from [5].

Figure 15

Projections of the signal fit for the LHCb measurement of $\mathcal{R}(D^{*+})$ involving muonic $\tau$ decays ([5]). Left panel: full $q^2$ projection. Middle panel: $m_{\mathrm{miss}}^2$ projection in the highest $q^2$ bin. Right panel: $E_{\ell }^{*}$ projection in the highest $q^2$ bin.

Figure 16

Distribution of the distance between the ${\overline{B}}^0$ vertex and the $\tau$ vertex along the beam direction (Fig. 8, top panel) divided by its uncertainty in simulated events for the LHCb measurement of $\mathcal{R}(D^{*+})$ involving $\tau \to {\pi }^{-}{\pi }^{+}{\pi }^{-}\nu$ decays ([10]). The vertical line shows the $4\sigma$ requirement used in the analysis to separate signal events in red (dark gray) from the prompt background component in medium gray.

Figure 17

Control sample fits for the LHCb measurement of $\mathcal{R}(D^{*+})$ involving $\tau \to {\pi }^{-}{\pi }^{+}{\pi }^{-}\nu$ decays ([10]) employed to evaluate the composition of the various double-charm background contributions. (a)–(d) Low-BDT sample. (e),(f) $B\to D^{*+}{D}_s^{-}(\to {\pi }^{-}{\pi }^{+}{\pi }^{-})X$ sample.

Figure 18

Projections of the signal fit for the LHCb measurement of $\mathcal{R}(D^{*+})$ involving $\tau \to {\pi }^{-}{\pi }^{+}{\pi }^{-}\nu$ decays ([10]). The four rows correspond to the four BDT bins for increasing values of the BDT response.

Figure 19

Projections of the signal fit for the LHCb muonic measurement of $\mathcal{R}(J/\psi )$ ([9]). Left panel: full $m_{\mathrm{miss}}^2$ projection. Middle panel: $m_{\mathrm{miss}}^2$ projection in the highest $q^2$ and lowest $E_{\ell }^{*}$ bins. Right panel: decay time projection in the highest $q^2$ and lowest $E_{\ell }^{*}$ bins.

Figure 20

Signal fit for the measurement of the $\tau$ polarization fraction $P_{\tau }(D^{*})$ by Belle ([119]). The fits to the neutral and charged $B$ candidates as well as the $\tau \to \pi \nu$ and $\tau \to \rho \nu$ decay modes and the two $\cos {\theta }_h$ bins are combined.

Figure 21

Values of $\mathcal{R}(D^{*})$ and $P_{\tau }(D^{*})$ (white star) and the $1\sigma$, $2\sigma$, and $3\sigma$ contours as measured by Belle ([119]). The SM expectations ([167]; [38]) are shown as a white triangle. The gray band shows the then-world-average measurement of $\mathcal{R}(D^{*})$.

Figure 22

Signal fit to the lowest $\cos {\theta }_v$ bin, $[-1,-0.67]$, in the ${\overline{D}}^0\to K^{+}{\pi }^{-}{\pi }^0$ channel for the measurement of the longitudinal $D^{*}$ polarization fraction by Belle ([23]). The red (gray dash-dotted) curve corresponds to the signal contribution, and the blue (dark gray dashed) and green (light gray dotted) curves display the nonresonant and resonant background contributions, respectively.

Figure 23

Measured $\cos {\theta }_v$ distribution in $B^0\to D^{*-}{\tau }^{+}{\nu }_{\tau }$ decays for the determination of the longitudinal $D^{*}$ polarization fraction by Belle. The red (dark gray) solid curve shows the best fit of the longitudinal polarization fraction and the yellow (light gray) band corresponds to the SM expectation ([121]). Adapted from [23].

Figure 24

Measurements of $\mathcal{R}(D^{(*)})$, $\mathcal{R}(D^{(*)}{)}_e$, and $\mathcal{R}(D^{(*)}{)}_{\mu }$ from BABAR ([104]) and Belle ([83]).

Figure 25

Measurements of $\mathcal{R}(D^{(*)})$ as a function of paper submission time. Green (light gray) refers to BABAR, dark blue (dark gray) refers to Belle, light blue (medium gray) refers to LHCb, and violet refers to the SM predictions. Circular markers refer to hadronic tagging, triangles refer to semileptonic tagging, diamonds refer to inclusive tagging, and squares refer to untagged measurements. Filled markers refer to measurements using muonic decays of the $\tau$ lepton, while hollow markers refer to hadronic decays. Some of the earlier results measured $\mathcal{B}(B\to D^{(*)}\tau \nu )$ instead of $\mathcal{R}(D^{(*)})$. In those cases, the values for $\mathcal{R}(D^{(*)})$ were obtained by normalizing the $\tau$ branching fraction with the latest world averages for $\mathcal{B}(B\to D^{(*)}\ell \nu )$ ([174]).

Figure 26

Left panel: $\mathcal{R}(D^{(*)})$ world averages with different assumptions for the unknown correlation ${\rho }_{D^{**}}$. The average with ${\rho }_{D^{**}}=0$ (light blue or light gray dotted curves) is based on assumptions similar to those made by [38] and shows a compatibility with the SM expectation of 3.2 standard deviations while taking into account the small uncertainties of the theoretical predictions; the scenarios ${\rho }_{D^{**}}=1$ (red or medium gray dashed curves) and ${\rho }_{D^{**}}=-1$ (orange or light gray dash-dotted curves) agree with the SM expectation within 2.9 and 3.7 standard deviations, respectively. In our quoted average we profile the unknown correlation and obtain ${\widehat{\rho }}_{D^{**}}=-0.88$ (heather gray solid curves) with a compatibility with the SM of 3.6 standard deviations. Right panel: our world average of $\mathcal{R}(D)$ and $\mathcal{R}(D^{*})$ (black solid curves) compared to the various measurements of $\mathcal{R}(D^{(*)})$. The unknown correlation ${\rho }_{D^{**}}$ is treated as a free but constrained parameter of the average (see the main text for more details).

Figure 27

Saturation of the inclusive SM prediction (red or medium gray band) for $\mathcal{B}(B\to X_c\tau \nu )$ by the sum of the measured exclusive branching fractions that are implied by the $\mathcal{R}(D)$ and $\mathcal{R}(D^{(*)})$ world averages (blue or dark gray square). By comparison, the SM prediction for the sum of the $B\to D^{(*,**)}\tau \nu$ exclusive branching fractions (blue or dark gray band), is compatible with, and does not saturate, the inclusive prediction. Also shown are (i) the measured inclusive branching fraction measurements for $b\to X\tau \nu$ from LEP ([174]) (open square), which is normalized against the total number of tagged $b\overline{b}$ events. Assuming that the hadronization effects cancel, this can be interpreted as $\mathcal{B}(B\to X\tau \nu )$, and (ii) the unpublished inclusive measurement taken by [116] using Belle data (red or medium gray filled square), which shows a large excess.

Figure 28

Top row: typical variation of experimental acceptances for the 2HDM, the leptoquark models $R_2$ and $S_1$, and a pure tensor current, normalized with respect to the SM acceptance ${ϵ}_{\mathrm{SM}}$, for $B\to D\tau \nu$ (thin blue lines) and $B\to (D^{*}\to D\pi )\tau \nu$ (thick red lines), with $\tau \to e\nu \nu$. The dotted, solid, and dashed lines show the resulting acceptances for the $q^2$ resolutions (see text) of 0.8, 1.2, and $1.6{\mathrm{GeV}}^2$, respectively. Bottom row: variation in $\mathcal{R}(D^{(*)})/{\mathcal{R}(D^{(*)})}_{\mathrm{SM}}$ for the same models.

Figure 29

Top row: color map of the percent variation per bin in the reconstructed $m_{\mathrm{miss}}^2$ normalized distribution for $\overline{B}\to D\tau \nu$ comparing the SM to a range of couplings for the 2HDM, the leptoquark models $R_2$ and $S_1$, and a pure tensor current. The variations for $B\to D^{*}\tau \nu$ are similar but somewhat smaller, ranging up to the 1% to 2% level. Bottom row: examples of normalized $m_{\mathrm{miss}}^2$ distributions for the SM (solid lines) vs NP (dashed lines) for $\overline{B}\to D\tau \nu$ (thin blue lines) and $\overline{B}\to D^{*}\tau \nu$ (thick red lines). The chosen NP coupling for each model is shown as a dashed line in the corresponding top row panel.

Figure 30

Projections for the expected precision on the measurement of selected $\mathcal{R}(H_c)$ ratios at LHCb as a function of the year in which the corresponding data sample becomes available. The order of the curves in the legend corresponds to the order of the curves on the plot for the year 2026. Left panel: pessimistic scenario for an irreducible systematic uncertainty of 3% on $\mathcal{R}(D^{(*)})$ and 5% on the other ratios. Right panel: optimistic scenario for an irreducible systematic uncertainty of 0.5% on $\mathcal{R}(D^{(*)})$ and 2% on the other ratios. These extrapolations are based on the current muonic-$\tau$ measurements of $\mathcal{R}(D^{(*)})$ and $\mathcal{R}(J/\psi )$, as well as the forthcoming hadronic-$\tau$ measurement of $\mathcal{R}(D_1^0)$ for the $\mathcal{R}(D^{**})$ curve. The symbol $\mathcal{R}(D_s^{[*]})$ refers to the sum of the $D_s$ and $D_s^{*}$ yields, as described in the text. The $\mathcal{R}({\mathrm{\Lambda }}_c^{*})$ entry in the legend refers to $\mathcal{R}\mathbf{(}{\mathrm{\Lambda }}_c^{*}(2625)\mathbf{)}$. The shaded regions correspond to the long shutdowns during which there is no data taking at the LHC and have been updated including the latest estimates ([53]).

Figure 31

Projections for the expected precision on the measurements of $\mathcal{R}(D^{(*)})$, $\mathcal{R}(X)$, and $\mathcal{R}(\pi )$ at Belle II as a function of the year in which the corresponding data sample will become available. The order of the curves in the legend corresponds to the order of the curves on the plot for the year 2022. The “$\mathcal{R}(D^{(*)})$ (SL FEI, lep $\tau$)” curve sits under the “$\mathcal{R}(X)$ (had FEI, lep $\tau$)” curve because their projected uncertainties are extremely similar. An irreducible systematic uncertainty of (left panel) 3% for the pessimistic scenario and (right panel) 0.5% for the optimistic one is assumed. The optimistic scenario also assumes a 50% increase in the reconstruction efficiency of the exclusive tagging algorithms. The shaded regions indicate years in which significant downtime is expected due to upgrades of the detector and/or the accelerator.

Figure 32

The 68%, 95%, and 99% C.L. allowed regions for the $R_2$ simplified model coupling $c_{SL}=8c_T$ fitting to an Asimov dataset with $c_{SL}=8c_T=0.25(1+i)$. The best fit recovered points are shown as gray dots.