$V_{cb}$ from the semileptonic decay $B\to D\ell \bar{\hat{I}}{½}_{\ell }$ and the properties of the $D$ meson distribution amplitude

Abstract

The improved QCD light-cone sum rule (LCSR) provides an effective way to deal with the heavy-to-light transition form factors (TFFs). Firstly, we adopt the improved LCSR approach to deal with the $B\to D$ TFF $f^{+}(q^2)$ up to twist-4 accuracy. Due to the elimination of the most uncertain twist-3 contribution and the large suppression of the twist-4 contribution, the obtained LCSR shall provide us a good platform for testing the $D$-meson leading-twist DA. For the purpose, we suggest a new model for the $D$-meson leading-twist DA ($\ddot{I}{•}_{3D}$), whose longitudinal behavior is dominantly determined by a parameter $B$. Moreover, we find its second Gegenbauer moment $a_2^D\sim B$. Varying $B$ within certain region, one can conveniently mimic the $D$-meson DA behavior suggested in the literature. Inversely, by comparing the estimations with the experimental data on the $D$-meson involved processes, one can get a possible range for the parameter $B$ and a determined behavior for the $D$-meson DA. Secondly, we discuss the $B\to D$ TFF at the maximum recoil region and present a detailed comparison of it with the pQCD estimation and the experimental measurements. Thirdly, by applying the LCSR on $f^{+}(q^2)$, we study the CKM matrix element $\Vcb$ together with its uncertainties by adopting two types of processes, i.e. the $B^0/{\bar{B}}^0$-type and the $B^{\pm }$-type. It is noted that a smaller $B\precsim 0.20$ shows a better agreement with the experimental value on $\Vcb$. For example, for the case of $B=0.00$, we obtain $|V_{cb}|(B^0/{\bar{B}}^0-\mathrm{type})=(41.28{}_{-4.82}^{+5.68}{}_{-1.16}^{+1.13})\times 10^{-3}$ and $|V_{cb}|(B^{\pm }-\mathrm{type})=(40.44{}_{-4.72}^{+5.56}{}_{-1.00}^{+0.98})\times 10^{-3}$, whose first (second) uncertainty comes from the squared average of the mentioned theoretical (experimental) uncertainties.

13 pages, 10 figures. Reference updated and discussion improved. To be published in Nucl.Phys.B