Imprecated, yet Impeccable: On the Theoretical Evaluation of Gamma(B -> X_c \ell ν)

TL;DR

This work develops a robust heavy-quark expansion framework for inclusive semileptonic B decays, linking the width $\\Gamma_{\rm sl}$ to $|V_{cb}|$ and a controlled set of heavy-quark parameters with a Wilsonian cutoff near $1$ GeV. It provides master OPE formulae up to $\\mathcal{O}(1/m_b^3)$, including perturbative corrections captured by $A^{\rm pert}$ and nonperturbative inputs such as $\\mu_\\pi^2$, $\\mu_G^2$, $\\rho_D^3$, and $\\rho_{LS}^3$, plus potential four-quark charm contributions $H_c$. The authors show that the perturbative series converges well when using short-distance mass schemes, quantify uncertainties from perturbative, duality, and charm-related effects, and argue that the overall theoretical error on $|V_{cb}|$ is around $\\pm 2\%$. They also outline how to determine the HQ parameters from independent observables (moment analyses) and then combine them with the width to achieve a precise extraction of $|V_{cb}|$, with implications for $|V_{ub}|$ as well. Overall, the work presents an open, checkable framework for high-precision flavor physics in the B system, suitable for near-term progress with experimental input on lepton and hadronic moments.

Abstract

We present a detailed evaluation of the total semileptonic B meson width in terms of |V_{cb}| and heavy quark parameters (quark masses and the expectation values of local heavy quark operators). Special attention is given to perturbative corrections which can precisely be calculated in a scheme with a hard Wilsonian cutoff at a scale around 1GeV appropriate for the OPE, and to the potential impact of higher-order power corrections. We point out that the latter require control over possible contributions from four-quark operators containing charm quark fields. Analytical expressions are given which allow evaluating the width with various choices of parameters; ready-to-use expressions showing the dependence on the heavy quark parameters are presented as well. We illustrate these results by commenting on how these parameters can be extracted and what accuracy is likely to be achievable in the near future.

Figures (3)

• Figure 1: Example of the interference contribution to the decay width suppressed by a single power of $1/m_c$. Shown gluon is hard. All cuts should be included to maintain the correct scaling.
• Figure 2: Nonperturbative charm-field effects in the OPE for the inclusive decays. With charm fields being low-momentum, the kinematics is close to two-body for leptons, with $\,\sqrt{q^2}\!\simeq\!m_b\!-\!m_c$ .
• Figure 3: Gluon momentum scale distribution in $\Gamma_{\rm sl}(b\!\to\!c)$. Solid, dashed and dot-dashed lines correspond to $\mu\!=\!1\,\hbox{GeV}$, $\,1.5\,\hbox{GeV}\,$ and $\,2\,\hbox{GeV}$, respectively; lighter short-dashed line illustrates the case of $\mu\!=\!0$ (pole masses). The area under each curve gives the first-order perturbative coefficient.