Investigating the ratio of CKM matrix elements $|V_{ub}|/|V_{cb}|$ from semileptonic decay $B_s^0\to K^-μ^+ν_μ$ and kaon twist-2 distribution amplitude
Abstract: In this paper, we calculate the ratio of Cabibbo-Kobayashi-Maskawa matrix elements, *
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In the paper, we investigate the moments $$\langle \xi _{2;a_1}^{\Vert ;n}\rangle $$ ⟨ ξ 2 ; a 1 ‖ ; n ⟩ of the axial-vector $$a_1(1260)$$ a 1 ( 1260 ) -meson distribution amplitude by using the QCD sum rules approach under the background field theory. By considering the vacuum condensates up to dimension-six and the perturbative part up to next-to-leading order QCD corrections, its first five moments at an initial scale $$\mu _0=1~{\mathrm{GeV}}$$ μ 0 = 1 GeV are $$\langle \xi _{2;a_1}^{\Vert ;2}\rangle |_{\mu _0} = 0.223 \pm 0.029$$ ⟨ ξ 2 ; a 1 ‖ ; 2 ⟩ | μ 0 = 0.223 ± 0.029 , $$\langle \xi _{2;a_1}^{\Vert ;4}\rangle |_{\mu _0} = 0.098 \pm 0.008$$ ⟨ ξ 2 ; a 1 ‖ ; 4 ⟩ | μ 0 = 0.098 ± 0.008 , $$\langle \xi _{2;a_1}^{\Vert ;6}\rangle |_{\mu _0} = 0.056 \pm 0.006$$ ⟨ ξ 2 ; a 1 ‖ ; 6 ⟩ | μ 0 = 0.056 ± 0.006 , $$\langle \xi _{2;a_1}^{\Vert ;8}\rangle |_{\mu _0} = 0.039 \pm 0.004$$ ⟨ ξ 2 ; a 1 ‖ ; 8 ⟩ | μ 0 = 0.039 ± 0.004 and $$\langle \xi _{2;a_1}^{\Vert ;10}\rangle |_{\mu _0} = 0.028 \pm 0.003$$ ⟨ ξ 2 ; a 1 ‖ ; 10 ⟩ | μ 0 = 0.028 ± 0.003 , respectively. We then construct a light-cone harmonic oscillator model for $$a_1(1260)$$ a 1 ( 1260 ) -meson longitudinal twist-2 distribution amplitude $$\phi _{2;a_1}^{\Vert }(x,\mu )$$ ϕ 2 ; a 1 ‖ ( x , μ ) , whose model parameters are fitted by using the least squares method. As an application of $$\phi _{2;a_1}^{\Vert }(x,\mu )$$ ϕ 2 ; a 1 ‖ ( x , μ ) , we calculate the transition form factors (TFFs) of $$D\rightarrow a_1(1260)$$ D → a 1 ( 1260 ) in large and intermediate momentum transfers by using the QCD light-cone sum rules approach. At the largest recoil point ($$q^2=0$$ q 2 = 0 ), we obtain $$ A(0) = 0.130_{ - 0.013}^{ + 0.015}$$ A ( 0 ) = 0 . 130 - 0.013 + 0.015 , $$V_1(0) = 1.898_{-0.121}^{+0.128}$$ V 1 ( 0 ) = 1 . 898 - 0.121 + 0.128 , $$V_2(0) = 0.228_{-0.021}^{ + 0.020}$$ V 2 ( 0 ) = 0 . 228 - 0.021 + 0.020 , and $$V_0(0) = 0.217_{ - 0.025}^{ + 0.023}$$ V 0 ( 0 ) = 0 . 217 - 0.025 + 0.023 . By applying the extrapolated TFFs to the semi-leptonic decay $$D^{0(+)} \rightarrow a_1^{-(0)}(1260)\ell ^+\nu _\ell $$ D 0 ( + ) → a 1 - ( 0 ) ( 1260 ) ℓ + ν ℓ , we obtain $${\mathcal {B}}(D^0\rightarrow a_1^-(1260) e^+\nu _e) = (5.261_{-0.639}^{+0.745}) \times 10^{-5}$$ B ( D 0 → a 1 - ( 1260 ) e + ν e ) = ( 5 . 261 - 0.639 + 0.745 ) × 10 - 5 , $${\mathcal {B}}(D^+\rightarrow a_1^0(1260) e^+\nu _e) = (6.673_{-0.811}^{+0.947}) \times 10^{-5}$$ B ( D + → a 1 0 ( 1260 ) e + ν e ) = ( 6 . 673 - 0.811 + 0.947 ) × 10 - 5 , $${\mathcal {B}}(D^0\rightarrow a_1^-(1260) \mu ^+ \nu _\mu )=(4.732_{-0.590}^{+0.685}) \times 10^{-5}$$ B ( D 0 → a 1 - ( 1260 ) μ + ν μ ) = ( 4 . 732 - 0.590 + 0.685 ) × 10 - 5 , $${\mathcal {B}}(D^+ \rightarrow a_1^0(1260) \mu ^+ \nu _\mu )=(6.002_{-0.748}^{+0.796}) \times 10^{-5}$$ B ( D + → a 1 0 ( 1260 ) μ + ν μ ) = ( 6 . 002 - 0.748 + 0.796 ) × 10 - 5 .
In the paper, we investigate the moments $$\langle \xi _{2;a_1}^{\Vert ;n}\rangle $$ ⟨ ξ 2 ; a 1 ‖ ; n ⟩ of the axial-vector $$a_1(1260)$$ a 1 ( 1260 ) -meson distribution amplitude by using the QCD sum rules approach under the background field theory. By considering the vacuum condensates up to dimension-six and the perturbative part up to next-to-leading order QCD corrections, its first five moments at an initial scale $$\mu _0=1~{\mathrm{GeV}}$$ μ 0 = 1 GeV are $$\langle \xi _{2;a_1}^{\Vert ;2}\rangle |_{\mu _0} = 0.223 \pm 0.029$$ ⟨ ξ 2 ; a 1 ‖ ; 2 ⟩ | μ 0 = 0.223 ± 0.029 , $$\langle \xi _{2;a_1}^{\Vert ;4}\rangle |_{\mu _0} = 0.098 \pm 0.008$$ ⟨ ξ 2 ; a 1 ‖ ; 4 ⟩ | μ 0 = 0.098 ± 0.008 , $$\langle \xi _{2;a_1}^{\Vert ;6}\rangle |_{\mu _0} = 0.056 \pm 0.006$$ ⟨ ξ 2 ; a 1 ‖ ; 6 ⟩ | μ 0 = 0.056 ± 0.006 , $$\langle \xi _{2;a_1}^{\Vert ;8}\rangle |_{\mu _0} = 0.039 \pm 0.004$$ ⟨ ξ 2 ; a 1 ‖ ; 8 ⟩ | μ 0 = 0.039 ± 0.004 and $$\langle \xi _{2;a_1}^{\Vert ;10}\rangle |_{\mu _0} = 0.028 \pm 0.003$$ ⟨ ξ 2 ; a 1 ‖ ; 10 ⟩ | μ 0 = 0.028 ± 0.003 , respectively. We then construct a light-cone harmonic oscillator model for $$a_1(1260)$$ a 1 ( 1260 ) -meson longitudinal twist-2 distribution amplitude $$\phi _{2;a_1}^{\Vert }(x,\mu )$$ ϕ 2 ; a 1 ‖ ( x , μ ) , whose model parameters are fitted by using the least squares method. As an application of $$\phi _{2;a_1}^{\Vert }(x,\mu )$$ ϕ 2 ; a 1 ‖ ( x , μ ) , we calculate the transition form factors (TFFs) of $$D\rightarrow a_1(1260)$$ D → a 1 ( 1260 ) in large and intermediate momentum transfers by using the QCD light-cone sum rules approach. At the largest recoil point ($$q^2=0$$ q 2 = 0 ), we obtain $$ A(0) = 0.130_{ - 0.013}^{ + 0.015}$$ A ( 0 ) = 0 . 130 - 0.013 + 0.015 , $$V_1(0) = 1.898_{-0.121}^{+0.128}$$ V 1 ( 0 ) = 1 . 898 - 0.121 + 0.128 , $$V_2(0) = 0.228_{-0.021}^{ + 0.020}$$ V 2 ( 0 ) = 0 . 228 - 0.021 + 0.020 , and $$V_0(0) = 0.217_{ - 0.025}^{ + 0.023}$$ V 0 ( 0 ) = 0 . 217 - 0.025 + 0.023 . By applying the extrapolated TFFs to the semi-leptonic decay $$D^{0(+)} \rightarrow a_1^{-(0)}(1260)\ell ^+\nu _\ell $$ D 0 ( + ) → a 1 - ( 0 ) ( 1260 ) ℓ + ν ℓ , we obtain $${\mathcal {B}}(D^0\rightarrow a_1^-(1260) e^+\nu _e) = (5.261_{-0.639}^{+0.745}) \times 10^{-5}$$ B ( D 0 → a 1 - ( 1260 ) e + ν e ) = ( 5 . 261 - 0.639 + 0.745 ) × 10 - 5 , $${\mathcal {B}}(D^+\rightarrow a_1^0(1260) e^+\nu _e) = (6.673_{-0.811}^{+0.947}) \times 10^{-5}$$ B ( D + → a 1 0 ( 1260 ) e + ν e ) = ( 6 . 673 - 0.811 + 0.947 ) × 10 - 5 , $${\mathcal {B}}(D^0\rightarrow a_1^-(1260) \mu ^+ \nu _\mu )=(4.732_{-0.590}^{+0.685}) \times 10^{-5}$$ B ( D 0 → a 1 - ( 1260 ) μ + ν μ ) = ( 4 . 732 - 0.590 + 0.685 ) × 10 - 5 , $${\mathcal {B}}(D^+ \rightarrow a_1^0(1260) \mu ^+ \nu _\mu )=(6.002_{-0.748}^{+0.796}) \times 10^{-5}$$ B ( D + → a 1 0 ( 1260 ) μ + ν μ ) = ( 6 . 002 - 0.748 + 0.796 ) × 10 - 5 .
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