Precision determination of |V_{ub}| from inclusive decays

TL;DR

This work tackles the challenge of precisely determining |V_{ub}| from inclusive $B$ decays by introducing a dual-cut strategy on the dilepton invariant mass $q^2$ and the hadronic invariant mass $m_X$. By combining cuts, the authors interpolate between pure $q^2$ and pure $m_X$ selections, reducing both the ${ m O}(rac{\, m \Lambda_{QCD}^3}{m_b^3})$ and light-cone distribution-function uncertainties, while increasing the signal fraction. They derive the partially integrated rate $G(q_{ m cut}^2,m_{ m cut})$ within the OPE framework, quantify perturbative, mass, and ${ m O}(rac{\, m \Lambda_{QCD}^3}{m_b^3})$ uncertainties (including weak annihilation), and show that a theoretical precision of 5–10% on $|V_{ub}|$ is attainable with realistic cuts. The approach yields sensitivity to up to ~45% of $B\to X_u\ell\nu$ decays—roughly double the yield of a pure $q^2$ cut and allows cuts well away from the $B\to X_c\ell\nu$ threshold, enhancing experimental feasibility and robustness against modeling assumptions.

Abstract

We propose determining |V_{ub}| from inclusive semileptonic B decay using combined cuts on the leptonic and hadronic invariant masses to eliminate the b->c background. Compared to a pure dilepton invariant mass cut, the uncertainty from unknown order (Λ_{QCD}/m_b)^3 terms in the OPE is significantly reduced and the fraction of b->u events is roughly doubled. Compared to a pure hadronic invariant mass cut, the uncertainty from the unknown light-cone distribution function of the b quark is significantly reduced. We find that |V_{ub}| can be determined with theoretical uncertainty at the 5-10% level.

Figures (7)

• Figure 1: The Dalitz plot for $b\to u$ semileptonic decay, indicating the regions corresponding to $b\to c$ decay (shaded), the lepton invariant mass cut $q^2 > (m_B-m_D)^2$ (vertically striped), and the hadron invariant mass cut $m_X< m_D$ (horizontally striped).
• Figure 2: The thin dashed lines show the location of the perturbative singularity of ${\rm d}\Gamma_c(m_{\rm cut})/{\rm d}q^2$, given by Eq. (\ref{['qsqblowup']}), for $m_b =4.6,\ 4.7$ and $4.8\,{\rm GeV}$. The thick dashed lines correspond to $m_{\rm cut}=1.5,\ 1.7$ and $1.86\,{\rm GeV}$. The intersection of the thick and thin dashed lines give qualitatively, for a given value of $m_{\rm cut}$, the value of $q_{\rm cut}^2$ below which the effects of the distribution function become large.
• Figure 3: (a) The $O(\epsilon)$ and $O(\epsilon^2_{\rm BLM})$ contributions to $G(q_{\rm cut}^2,m_{\rm cut})$ (normalized to the tree level result) for hadronic invariant mass cut $m_{\rm cut}=1.86\,{\rm GeV}$ (solid lines), $1.7\,{\rm GeV}$ (short dashed lines) and $1.5\,{\rm GeV}$ (long dashed lines). (b) Scale variation of the perturbative corrections: The difference between the perturbative corrections to $G(q_{\rm cut}^2,m_{\rm cut})$, normalized to the tree level result, for $\mu=4.7\,{\rm GeV}$ and $\mu=1.6\,{\rm GeV}$.
• Figure 4: The fractional effect of a $\pm 80\,{\rm MeV}$ and $\pm 30\,{\rm MeV}$ uncertainty in $m_b^{1S}$ on $G(q_{\rm cut}^2,m_{\rm cut})$ for $m_{\rm cut}=1.86\,{\rm GeV}$ (solid line), $1.7\,{\rm GeV}$ (short dashed line) and $1.5\,{\rm GeV}$ (long dashed line).
• Figure 5: Estimate of the uncertainties due to dimension-six terms in the OPE as a function of $q_{\rm cut}^2$ from weak annihilation (WA) (solid line) and other operators (dashed line).