Perturbative QCD Analysis of B to pi and B to rho Transitions

We calculate the form factors of $B\to \pi$ and $B\to \rho$ heavy to light transition matrix elements by using the factorization formalism of perturbative QCD. We obtain them at $q^2=0$ and show their dependences on the parameter $ϵ$ of the B meson distribution amplitude. We also obtain the form factors as functions of $q^2$ in the region $0\le q^2\le M_B^2\mathrm{/}2$. The relations among the form factors are found in the limit of $m_{\pi }\mathrm{/}{M}_B=0$, $m_{\rho }\mathrm{/}{M}_B=0$, $M_b\mathrm{/}{M}_B=1$ and $(1-x)<<1$.