<math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi mathvariant="normal">cb</mi></mrow></msub><mo stretchy="false">|</mo></math> from the semileptonic decay <math altimg="si2.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo stretchy="false">→</mo><mi>D</mi><mi>ℓ</mi><msub><mrow><mover accent="true"><mrow><mi>ν</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mi>ℓ</mi></mrow></msub></math> and the properties of the D -meson distribution amplitude

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 <math altimg="si3.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo stretchy="false">→</mo><mi>D</mi></math> TFF <math altimg="si4.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>f</mi></mrow><mrow><mo>+</mo></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy="false">)</mo></math> 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 ( <math altimg="si5.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>ϕ</mi></mrow><mrow><mn>3</mn><mi>D</mi></mrow></msub></math> ), whose longitudinal behavior is dominantly determined by a parameter B . Moreover, we find its second Gegenbauer moment <math altimg="si6.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>D</mi></mrow></msubsup><mo>∼</mo><mi>B</mi></math> . 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 <math altimg="si3.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo stretchy="false">→</mo><mi>D</mi></math> 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 <math altimg="si4.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>f</mi></mrow><mrow><mo>+</mo></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy="false">)</mo></math> , we study the CKM matrix element <math altimg="si7.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi mathvariant="normal">cb</mi></mrow></msub><mo stretchy="false">|</mo></math> together with its uncertainties by adopting two types of processes, i.e. the <math altimg="si8.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>B</mi></mrow><mrow><mn>0</mn></mrow></msup><mo stretchy="false">/</mo><msup><mrow><mover accent="true"><mrow><mi>B</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mn>0</mn></mrow></msup></math> -type and the <math altimg="si9.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>B</mi></mrow><mrow><mo>±</mo></mrow></msup></math> -type. It is noted that a smaller <math altimg="si10.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>≾</mo><mn>0.20</mn></math> shows a better agreement with the experimental value on <math altimg="si7.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi mathvariant="normal">cb</mi></mrow></msub><mo stretchy="false">|</mo></math> . For example, for the case of <math altimg="si11.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mn>0.00</mn></math> , we obtain <math altimg="si12.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>c</mi><mi>b</mi></mrow></msub><mo stretchy="false">|</mo><mo stretchy="false">(</mo><msup><mrow><mi>B</mi></mrow><mrow><mn>0</mn></mrow></msup><mo stretchy="false">/</mo><msup><mrow><mover accent="true"><mrow><mi>B</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mn>0</mn></mrow></msup><mo>−</mo><mrow><mi mathvariant="normal">type</mi></mrow><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msubsup><mrow><msubsup><mrow><mn>41.28</mn></mrow><mrow><mo>−</mo><mn>4.82</mn></mrow><mrow><mo>+</mo><mn>5.68</mn></mrow></msubsup></mrow><mrow><mo>−</mo><mn>1.16</mn></mrow><mrow><mo>+</mo><mn>1.13</mn></mrow></msubsup><mo stretchy="false">)</mo><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup></math> and <math altimg="si13.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>c</mi><mi>b</mi></mrow></msub><mo stretchy="false">|</mo><mo stretchy="false">(</mo><msup><mrow><mi>B</mi></mrow><mrow><mo>±</mo></mrow></msup><mo>−</mo><mrow><mi mathvariant="normal">type</mi></mrow><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msubsup><mrow><mn>40.44</mn></mrow><mrow><mo>−</mo><mn>4.72</mn></mrow><mrow><mo>+</mo><mn>5.56</mn></mrow></msubsup><msubsup><mrow><mtext> </mtext></mrow><mrow><mo>−</mo><mn>1.00</mn></mrow><mrow><mo>+</mo><mn>0.98</mn></mrow></msubsup><mo stretchy="false">)</mo><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup></math> , whose first (second) uncertainty comes from the squared average of the mentioned theoretical (experimental) uncertainties