[Paper Review] Standard Model predictions for Lepton Flavour Universality ratios of inclusive semileptonic $B$ decays

Muslem Rahimi, K. Vos|arXiv (Cornell University)|Jul 7, 2022

Particle physics theoretical and experimental studies31 references26 citations

TL;DR

This paper presents updated Standard Model predictions for lepton flavour universality (LFU) ratios in inclusive semileptonic B decays, including full mass effects for the tau lepton and higher-order power corrections up to $1/m_b^3$. The authors correct a previous calculation by including missing terms in the $\rho^3_D$ coefficient, yielding refined predictions for $R_{\tau/\mu}$, $R_{\tau/e}$, and the inclusive $B \to X_c\tau\bar{\nu}_\tau$ branching ratio, which show slight tension with an unpublished Belle measurement and highlight the need for future experimental verification to probe potential new physics in the tau sector.

ABSTRACT

We present Standard Model predictions for lepton flavour universality ratios of inclusive $B o X_{(c)} \ell \bar u_\ell$. For the $\ell=\mu,e$, these ratios are very close to unity as expected. For the $ au$ mode, we update the SM prediction for the branching ratio including power-corrections in the heavy-quark expansion up to $1/m_b^3$. These inclusive ratios serve as an important cross-check of the exclusive $B o D^{(*)}\ell\bar u_\ell$ modes, in which tensions exists between the predictions and measurements in those modes.

Motivation & Objective

• To provide precise Standard Model predictions for lepton flavour universality (LFU) ratios in inclusive semileptonic B decays, particularly for the tau mode where experimental data is limited.
• To correct a previous calculation in [10] by including missing terms in the $\rho^3_D$ coefficient of the heavy quark expansion (HQE), improving theoretical accuracy.
• To update the SM prediction for the inclusive $B \to X_c\tau\bar{\nu}_\tau$ branching ratio with higher-order HQE parameters up to $1/m_b^3$, including power corrections.
• To provide a critical cross-check for the observed tensions in exclusive $B \to D^{(*)}\ell\bar{\nu}$ decays, especially in the $\tau$-mode, by offering a theoretically robust inclusive benchmark.
• To support future searches for new physics in the tau lepton sector by supplying precise, up-to-date SM predictions for upcoming experimental measurements.

Proposed method

• The study employs the heavy quark expansion (HQE) with an operator product expansion (OPE) to compute the triple-differential decay rate in lepton energy, neutrino energy, and dilepton invariant mass.
• Massive lepton effects are fully included in the calculation, particularly for the $B \to X_c\tau\bar{\nu}_\tau$ mode, where phase space boundaries and contributions from structure functions $W_4$ and $W_5$ are explicitly accounted for.
• The inclusive decay rate is expanded in inverse powers of $m_b$, including perturbative corrections ($\alpha_s/\pi$), and non-perturbative matrix elements such as $(\mu^2_\pi)_\perp$, $(\mu^2_G)_\perp$, $(\rho^3_D)_\perp$, and $(\rho^3_{LS})_\perp$ up to $1/m_b^3$.
• The authors derive and explicitly compute all coefficient functions in the HQE expansion, correcting discrepancies with [10]—especially in the $\rho^3_D$ term—through careful analytic integration involving delta functions and phase space constraints.
• Numerical results are obtained using updated input parameters from [1], including $m_b^{\text{kin}}$, $m_c$, and HQE parameters in the kinetic scheme at $\mu = 1$ GeV, with uncertainties combined in quadrature.
• The predictions are cross-checked via multiple methods: direct OPE calculation, multiplication with measured light-lepton branching ratios, and comparison with exclusive mode sums, ensuring consistency across approaches.

Experimental results

Research questions

• RQ1 What is the corrected Standard Model prediction for the $B \to X_c\tau\bar{\nu}_\tau$ branching ratio when higher-order power corrections up to $1/m_b^3$ are included, particularly correcting for missing terms in the $\rho^3_D$ coefficient?
• RQ2 How do the lepton flavour universality ratios $R_{\tau/\mu}(X_c)$, $R_{\tau/e}(X_c)$, and $R_{\mu/e}(X_c)$ compare to unity in the SM, and what is the impact of massive lepton effects and $1/m_b^3$ corrections?
• RQ3 Is there a discrepancy between the updated SM prediction for $B \to X_c\tau\bar{\nu}_\tau$ and the unpublished Belle measurement of $R_{\tau/(e,\mu)}(X)$, and what does this imply for new physics?
• RQ4 How do the direct OPE-based prediction and the prediction derived from multiplying the light-lepton branching ratio with the LFU ratio compare, and which is more robust?
• RQ5 Does the sum of exclusive $B \to D^{(*)} \ell\bar{\nu}$ and $B \to D^{**} \ell\bar{\nu}$ branching fractions saturate the fully inclusive $B \to X_c\tau\bar{\nu}_\tau$ rate predicted here?

Key findings

• The corrected SM prediction for the inclusive $B \to X_c\tau\bar{\nu}_\tau$ branching ratio is $2.34 \pm 0.13\%$, with the dominant uncertainty arising from $V_{cb}$, $m_b$, and $m_c$, and a significant contribution from the $\rho^3_D$ parameter.
• The lepton flavour universality ratios are predicted as $R_{\mu/e}(X_c) = 99.445 \pm 0.006 \times 10^{-2}$, $R_{\tau/\mu}(X_c) = 21.965 \pm 0.420 \times 10^{-2}$, and $R_{\tau/e}(X_c) = 21.843 \pm 0.419 \times 10^{-2}$, with $\rho^3_D$ contributing significantly despite being a $1/m_b^3$ term.
• The direct OPE calculation yields $B(B \to X_c\tau\bar{\nu}_\tau) = (2.34 \pm 0.13)\%$, which is in excellent agreement with the value obtained by multiplying the measured light-lepton branching ratio with the LFU ratio: $2.30 \pm 0.05\%$.
• The prediction from the Belle collaboration’s unpublished measurement, when combined with the measured $B(B \to X_c\ell\bar{\nu}_\ell)$ rate, yields $B(B \to X_c\tau\bar{\nu}_\tau) = (3.12 \pm 0.23)\%$, which shows a slight tension with the SM prediction of $2.34 \pm 0.13\%$, suggesting potential discrepancies in the data or analysis.
• The sum of exclusive $B \to D^{(*)} \ell\bar{\nu}$ and $B \to D^{**} \ell\bar{\nu}$ branching fractions for the $\tau$ mode is $2.14 \pm 0.06\%$, which is lower than the fully inclusive prediction, indicating that the exclusive modes do not fully account for the inclusive rate and highlighting the importance of inclusive theory as a cross-check.
• The authors confirm that their $\rho^3_D$ coefficient correction resolves a discrepancy with the earlier calculation in [10], and the two now agree after re-evaluation, validating the current results.