Theory of charmless inclusive $B$ decays and the extraction of $V_{ub}$

We present “state-of-the-art” theoretical expressions for the triple differential $\overline{B}\to X_u{l}^{-}\overline{\nu }$ decay rate and for the $\overline{B}\to X_s\gamma$ photon spectrum, which incorporate all known contributions and smoothly interpolate between the “shape-function region” of large hadronic energy and small invariant mass, and the “OPE region” in which all hadronic kinematical variables scale with $M_B$. The differential rates are given in a form which has no explicit reference to the mass of the $b$ quark, avoiding the associated uncertainties. Dependence on $m_b$ enters indirectly through the properties of the leading shape function, which can be determined by fitting the $\overline{B}\to X_s\gamma$ photon spectrum. This eliminates the dominant theoretical uncertainties from predictions for $\overline{B}\to X_u{l}^{-}\overline{\nu }$ decay distributions, allowing for a precise determination of $|V_{ub}|$. In the shape-function region, short-distance and long-distance contributions are factorized at next-to-leading order in renormalization-group improved perturbation theory. Higher-order power corrections include effects from subleading shape functions where they are known. When integrated over sufficiently large portions in phase space, our results reduce to standard OPE expressions up to yet unknown $O({\alpha }_s^2)$ terms. Predictions are presented for partial $\overline{B}\to X_u{l}^{-}\overline{\nu }$ decay rates with various experimental cuts. An elaborate error analysis is performed that contains all significant theoretical uncertainties, including weak annihilation effects. We suggest that the latter can be eliminated by imposing a cut on high leptonic invariant mass.

Figure 1

The hadronic phase space in $P_{+}$ and $P_{-}$. The light gray region contains background from $\overline{B}\to X_c{l}^{-}\overline{\nu }$ decays, while the dark gray region is only populated by $\overline{B}\to X_u{l}^{-}\overline{\nu }$ events. The line separating the two regions is the contour where $M_X^2=P_{+}{P}_{-}=M_D^2$. Each point represents a $\overline{B}\to X_u{l}^{-}\overline{\nu }$ event in a Monte-Carlo simulation using the results of this paper. While the shape-function region of large $P_{-}$ and small $P_{+}$ is highly populated, there is not a single event with $P_{+}$ larger than 3 GeV out of the 1300 events generated.

Figure 2

Left: Different functional forms for the leading shape function. We show $F^{(\exp )}(\widehat{\omega },\Lambda ,2)$ (solid), $F^{(\mathrm{gauss})}(\widehat{\omega },\Lambda ,2)$ (dotted), and $F^{(\mathrm{hyp})}(\widehat{\omega },\Lambda ,2)$ (dash-dotted) as functions of the ratio $\widehat{\omega }/\Lambda$. Right: The same functions with the parameters $\Lambda$ and $b$ tuned such that $m_b({\mu }_{*},{\mu }_{*})=4.61\mathrm{GeV}$ and ${\mu }_{\pi }^2({\mu }_{*},{\mu }_{*})=0.2{\mathrm{GeV}}^2$. See text for explanation.

Figure 3

Nine models for the subleading shape function $\widehat{u}(\widehat{\omega })$ obtained by adding or subtracting one of the four functions $h_n(\widehat{\omega })$ to the default model in (51), shown as the thick central line. See text for explanation.

Figure 4

Left: Theoretical prediction for the double differential decay rate. The light area represents a large decay rate. Black regions denote areas where the decay rate is close to zero. The dotted line is given by $P_{+}{P}_{-}=M_D^2$, which means that charm background is located in the upper wedge. See text for further explanation. Right: The $P_{+}$ spectrum extended to large values of $P_{+}$. The thin solid line denotes the leading-power prediction, the dashed line depicts first-order power corrections, the dash-dotted line shows second-order power corrections, and the thick solid line is their sum.