Lepton Non-Universality in B Decays

• Lepton non-universality is defined by deviations in the R(D) and R(D*) ratios, where experimental values exceed SM predictions by 2.3σ and 2.7σ, indicating possible new physics.
• The analysis uses effective Hamiltonian frameworks and operator decompositions (e.g., C_{V1}, C_{S1}, C_T) to model potential contributions from new heavy particles.
• Advances from Belle II, LHCb upgrades, and lattice QCD improvements are pivotal for refining measurements and reducing uncertainties in B decay anomalies.

Lepton non-universality (LNU) in $B$ meson decays refers to observed deviations from the Standard Model (SM) principle that all charged leptons couple universally to the electroweak sector, with differences in decay rates arising solely from their masses and kinematic effects. Recent precision measurements in semileptonic $B$ decays, particularly those involving tau leptons, have produced anomalies—especially in the ratios $R(D)$ and $R(D^*)$—that challenge the SM’s assumption and have prompted extensive theoretical and experimental study of possible new physics in the flavor sector (Lüth, 2018).

1. Theoretical Basis and Standard Model Predictions

In the SM, all charged-lepton flavors couple identically to the $W$ boson. The only sources of non-universality are the lepton masses, which affect phase-space and helicity amplitudes in processes such as $\bar{B} \to D^{(*)} \ell^- \bar{\nu}_\ell$ for $\ell = e, \mu, \tau$. The SM thus predicts the lepton flavor ratios

$$R(D) = \frac{\mathcal{B}(\bar{B} \to D \tau^- \bar{\nu}_\tau)}{\mathcal{B}(\bar{B} \to D \ell^- \bar{\nu}_\ell)}\,, \quad R(D^*) = \frac{\mathcal{B}(\bar{B} \to D^* \tau^- \bar{\nu}_\tau)}{\mathcal{B}(\bar{B} \to D^* \ell^- \bar{\nu}_\ell)}\,,$$

are determined up to small uncertainties, with state-of-the-art predictions $R(D)_{\rm SM} = 0.299 \pm 0.003$ and $R(D^*)_{\rm SM} = 0.258 \pm 0.005$ (Lüth, 2018).

SM predictions for light-lepton universality are even tighter due to negligible $m_e, m_\mu$—for example, $R_K^{\rm SM} = 1.0003 \pm 0.0001$ in $B^+ \to K^+ \ell^+ \ell^-$ decays with $q^2$ in $[1, 6]\,\mathrm{GeV}^2$ (Lüth, 2018).

2. Experimental Status and Anomalies

Combined measurements by the BABAR, Belle, and LHCb collaborations yield

$$R(D)_{\rm exp} = 0.407 \pm 0.046,\quad R(D^*)_{\rm exp} = 0.306 \pm 0.015,$$

with $R(D_{\rm exp})$ and $R(D^*_{\rm exp})$ exceeding the SM by $2.3\,\sigma$ and $2.7\,\sigma$, respectively. The total significance of the combined deviation is $\sim4.0\,\sigma$ (Lüth, 2018). Key experimental inputs include different tagging and $\tau$ decay modes across the major flavor factories.

This LNU anomaly is not isolated: several ratios in rare $b \to s \ell^+ \ell^-$ decays, such as $R_K$ and $R_{K^*}$, have been measured as significantly below unity, challenging the SM's tightly constrained predictions (Lüth, 2018, 1804.02011).

3. Model-Independent Framework and Operator Analysis

To parameterize possible new physics underlying LNU, the low-energy effective Hamiltonian for $b \to c \tau \nu$ transitions is written as

$$\mathcal{H}_{\rm eff} = \frac{4 G_F V_{cb}}{\sqrt{2}} \Big[ (1+C_{V_1}) \mathcal{O}_{V_1} + C_{V_2} \mathcal{O}_{V_2} + C_{S_1} \mathcal{O}_{S_1} + C_{S_2} \mathcal{O}_{S_2} + C_T \mathcal{O}_T \Big] + \text{h.c.},$$

with operators

$$\mathcal{O}_{V_1} = (\bar{c} \gamma^\mu P_L b)(\bar{\tau} \gamma_\mu P_L \nu),\quad \mathcal{O}_{S_1} = (\bar{c} P_L b)(\bar{\tau} P_L \nu), \quad \mathcal{O}_T = (\bar{c} \sigma^{\mu\nu} P_L b)(\bar{\tau} \sigma_{\mu\nu} P_L \nu),$$

and analogous right-handed structures (Lüth, 2018). In the SM, only $C_{V_1} = 0$ (the unity in front), all other $C_i$ vanish.

Extensions to the effective Lagrangian allow new heavy particles to modify $C_{V_1}$, $C_{S_1}$, $C_{T}$, etc., generally resulting in enhanced $R(D^{(*)})$ relative to the SM.

4. Phenomenological Implications and Constraints on New Physics

A variety of mediators have been proposed to explain semitauonic anomalies:

• Charged Higgs ($H^-$) in Two-Higgs-Doublet Models (2HDM): Type II 2HDM yielding significant scalar contributions is disfavored by joint $R(D^{(*)})$ fits and direct LHC searches (which give $m_{H^\pm} \gtrsim 800$ GeV for $\tan\beta = \mathcal{O}(1)$). Less constrained Type III 2HDM with flavor-off-diagonal couplings can accommodate data for some parameter ranges (Lüth, 2018).
• Leptoquarks (LQ): Scalar $S_1$ or vector $U_1$ leptoquark models allow tree-level $b \to c \tau \nu$ couplings fit to excesses in $R(D^{(*)})$, provided the LQ mass is in the TeV range and couplings to first and second generations are suppressed. Direct LHC constraints are evaded in this setup (Lüth, 2018).

An effective Hamiltonian parameterization relates constraints on LNU to the Wilson coefficients of these operators, which can be extracted from joint fits to all $b \to c \tau \nu$ and $b \to s \ell \ell$ observables. For instance, the Type II 2HDM is excluded, while more general models remain viable (Lüth, 2018).

5. Experimental and Theoretical Future Prospects

Experimental Outlook

• Belle II aims to collect $\sim50\,\mathrm{ab}^{-1}$ of $e^+ e^-$ data, a factor of 40 larger than Belle, enabling precision measurements of $R(D)$, $R(D^*)$, $R(D^{**})$, the $q^2$ spectra, $\tau$ polarization, and angular observables.
• LHCb Upgrades (Run 3 and Run 4) target a $B$-meson yield increased by factors of $3$–$5$, permitting improved measurements of $R(D^*)$ in new $\tau$ modes, first $R(D)$ results at hadron colliders, and complementary channels ($B_s\to D_s\tau\nu$, $\Lambda_b\to\Lambda_c\tau\nu$, $B_c\to J/\psi\,\tau\nu$, $B^-\to\tau^-\bar\nu$) (Lüth, 2018).

Theoretical Developments

• Lattice QCD—offers prospects for more precise form-factor calculations for $B\to D^{(*)}$ at nonzero recoil, which are critical to reducing hadronic uncertainties.
• Inclusive Theory—improvements in modeling charm resonances $D^{**}$ and higher-order QCD corrections will aid the interpretation of backgrounds.
• Global Fits—will enable integrated effective Hamiltonian analyses that use all available $b \to c \tau \nu$ and $b \to s \ell \ell$ datasets, including angular and polarization observables and exploration of correlations with other FCNC anomalies (e.g., $R_K$, $R_{K^*}$) (Lüth, 2018).

6. Summary and Outlook

The world averages

$$R(D)_{\rm exp} = 0.407 \pm 0.046, \quad R(D^*)_{\rm exp} = 0.306 \pm 0.015$$

compared to

$$R(D)_{\rm SM} = 0.299 \pm 0.003, \quad R(D^*)_{\rm SM} = 0.258 \pm 0.005$$

establish a joint $4\,\sigma$ deviation in semitauonic $B$ decays, independently established by BABAR, Belle, and LHCb (Lüth, 2018). Given the double-ratio structure of $R(D^{(*)})$, experimental and lattice-theory uncertainties are sufficiently under control that, if confirmed with the higher statistics of Belle II and LHCb upgrades and augmented by differential and polarization measurements, these results would constitute compelling evidence for physics beyond the SM, with major implications for the electroweak and flavor structure of new physics.