Weak Decay Process of $B \to \rho \ell \bar\nu_\ell$: A Varying External Field Approach in QCD Sum Rules

Abstract

A varying external field approach in QCD sum rules is formulated in a systematic way to treat the weak decay form factors and their $q^2$ dependence in the process of $B \to \rho \ell \bar\nu_\ell$. From the form factor sum rules, we can also obtain the mass sum rules for B and $\rho$ mesons, which can help us determine the reliable Borel windows in studying the relevant form factor sum rules. In this way, we thus demonstrate that some QCD sum rule calculations in the literature are less reliable. We also include induced condensate contributions, which have been ignored, into the relevant sum rules. We obtain the ratios $\Gamma(\bar B^0\to\rho^+ e^- \bar\nu_e)$/ $\Gamma(\bar B^0\to\pi^+ e^- \bar\nu_e)$$\approx$ 0.94 and $\Gamma(\bar B^0\to\rho^+ \tau^- \bar\nu_\tau)$/ $\Gamma(\bar B^0\to\pi^+ \tau^- \bar\nu_\tau)$ $\approx$ 1.15. We apply this approach to re-examine the case of the $D$ meson decay.