Heavy-to-light form factors in the final hadron large energy limit of QCD

We argue that the large energy effective theory (LEET), originally proposed by Dugan and Grinstein, is applicable to exclusive semileptonic, radiative, and rare heavy-to-light transitions in the region where the energy release E is large compared to the strong interaction scale and to the mass of the final hadron, i.e., for $q^2$ not close to the zero-recoil point. We derive the effective Lagrangian from the QCD one, and show that in the limit of heavy mass M for the initial hadron and large energy E for the final one, the heavy and light quark fields behave as two-component spinors. Neglecting QCD short-distance corrections, this implies that there are only three form factors describing all the pseudoscalar to pseudoscalar or vector weak current matrix elements. We argue that the dependence of these form factors with respect to M and E should be factorizable, the M dependence $\left(\sqrt{M}\right)$ being derived from the usual heavy quark expansion while the E dependence is controlled by the behavior of the light-cone distribution amplitude near the end point $u\sim 1.$ The usual expectation of the $\sim \left(1\mathrm{}-u\right)$ behavior leads to a ${1/E}^2$ scaling law, that is a dipole form in $q^2.$ We also show explicitly that in the appropriate limit the light-cone sum rule method satisfies our general relations as well as the scaling laws in M and E of the form factors, and obtain very compact and simple expressions for the latter. Finally we note that this formalism gives theoretical support to the quark model-inspired methods existing in the literature.