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 $O(\alpha_{ s}/m_{ B})$. The complete $O(1/m_{ B})$ contributions from the $\pi$ meson are calculated explicitly by a collinear expansion method. A well-defined power expansion scheme is built such that the $O(1/m_{ B})$ 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 $0\leq q^{ 2}\leq16 \text{ GeV}^{ 2}$}, 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 $|V_{ ub}|=(3.95\pm 0.13_{ \text{ exp}}\pm 0.32_{ \text{ th}})\times 10^{ -3}$ from the world averaged branching ratio $Br(\bar{ B}\to\pi l\bar{ \nu}_{ l})=(1.36\pm 0.09)\times 10^{ -4}$.
Categories: hep-ph
