Extraction of $|V_{ub}|$ with reduced dependence on shape functions

Using BABAR measurements of the inclusive electron spectrum in $B\to X_{u}e\nu$ decays and the inclusive photon spectrum in $B\to X_s\gamma$ decays, we extract the magnitude of the Cabibbo-Kobayashi-Maskawa matrix element $V_{ub}$. The extraction is based on theoretical calculations designed to reduce the theoretical uncertainties by exploiting the assumption that the leading shape functions are the same for all $b\to q$ transitions ($q$ is a light quark). The results agree well with the previous analysis, have indeed smaller theoretical errors, but are presently limited by the knowledge of the photon spectrum and the experimental errors on the lepton spectrum.

Figure 1

The differential branching fraction for $B\to X_{u}e\nu$ decays [9] as a function of the charged lepton momentum, measured in the ${\rm Y}(4S)$ rest frame. The error bars indicate the statistical errors; the histogram represents the theoretical prediction [11].

Figure 2

The differential branching fraction of inclusive $B\to X_s\gamma$ decays as a function of the photon energy in the $B$ rest frame. The error bars indicate the statistical errors. The data are compared to the theoretical prediction [11] using shape function parameters $m_b=4.67\mathrm{GeV}$, ${\mu }_{\pi }^2=0.16{\mathrm{GeV}}^2$ that have been obtained from the fit to $E_{\gamma }$ spectrum [10].

Figure 3

The weight functions for Method I (LLR) for two values of $E_0=2.0\mathrm{GeV}$ and $E_0=2.3\mathrm{GeV}$, corresponding to different values of $\rho =0.9983$ (solid lines), $\rho =0.995$ (dashed lines), and nonresummed weight function (dotted lines).

Figure 4

The weight function (in units of ${\mathrm{ps}}^{-1}$) for Method II (LNP), for $E_0=2.0\mathrm{GeV}$ (upper curve) and $E_0=2.3\mathrm{GeV}$ (lower curve).

Figure 5

Comparison of $|V_{ub}|$ values extracted from the two different calculations as a function of the lepton energy cutoff, $E_0$. The errors bars represent the experimental and the total error.