$B$ to $\pi$ Form Factors in collinear factorization approach

The form factors for semi-leptonic B decays, $\bar{B}\to \pi l{\bar{\nu }}_l$, are calculated under collinear factorization approach. The end-point divergences are regularized by a $\xi$-regularization, where $\xi$ means the collinear fraction of the spectator anti-quark of the $\bar{B}$ meson. The form factors are calculated up-to $O({\alpha }_s\mathrm{/}{m}_B)$. The complete $O(1\mathrm{/}{m}_B)$ contributions from the $\pi$ meson are calculated explicitly by a collinear expansion method. A well-defined power expansion scheme is built such that the $O(1\mathrm{/}{m}_B)$ contributions are about 30% of the leading order contributions. A small value $F_{+}(0)=0.164$ is found. This confirms the SCET result $F_{+}(0)=0.17$ from $B\to \pi \pi$ decays. The form factors are calculated for {\normalsize $0\le q^2\le 16{\text{GeV}}^2$}, where $q^2$ is the invariant mass of the lepton pair in $\bar{B}\to \pi l{\bar{\nu }}_l$. An extrapolation of the form factors to $q^2>16{\text{GeV}}^2$ is made to obtain {\normalsize $|V_{ub}|^{-2}\int_{0}^{26.42 $\text{GeV}$^{2}}dq^{2}(d\Gamma/dq^{2}){$\text{th}$}=(5.71\pm 0.91)ps^{-1}$}. We determine $|V{ub}|=(3.95\pm 0.13_{$\text{exp}$}\pm 0.32_{$\text{th}$})\times 10^{-3}$fromtheworldaveragedbranchingratio$Br(\bar{B}\to\pi l\bar{\nu}_{l})=(1.36\pm 0.09)\times 10^{-4}$.