Updated analysis of $a_1$ and $a_2$ in hadronic two-body decays of B mesons

Using the recent experimental data of $\overrightarrow{B}{D}^{\left(*\right)}(\pi ,\rho ),$ $\overrightarrow{B}{D}^{\left(*\right)}{D}_s^{\left(*\right)},$ $\overrightarrow{B}J/\psi K^{\left(*\right)}$ and various model calculations on form factors, we reanalyze the effective coefficients $a_1$ and $a_2$ and their ratio. QCD and electroweak penguin corrections to $a_1$ from $\overrightarrow{B}{D}^{\left(*\right)}{D}_s^{\left(*\right)}$ and $a_2$ from $\overrightarrow{B}J/\psi K^{\left(*\right)}$ are estimated. In addition to the model-dependent determination, the effective coefficient $a_1$ is also extracted in a model-independent way as the decay modes $\overrightarrow{B}{D}^{\left(*\right)}h$ are related by factorization to the measured semileptonic distribution of $\overrightarrow{B}{D}^{\left(*\right)}l\overline{\nu }$ at $q^2{=m}_h^2$. Moreover, this enables us to extract model-independent heavy-to-heavy form factors, for example, $F_0^{\mathrm{BD}}{(m}_{\pi }^2)=0.66\mathrm{}\pm 0.06\pm 0.05$ and $A_0^{{\mathrm{BD}}^{*}}{(m}_{\pi }^2)=0.56\mathrm{}\pm 0.03\pm 0.04$. The determination of the magnitude of $a_2$ from $\overrightarrow{B}J/\psi K^{\left(*\right)}$ depends on the form factors $F_1^{\mathrm{BK}},$ $A_{1,2}^{{\mathrm{BK}}^{*}}$ and $V^{{\mathrm{BK}}^{*}}$ at $q^2{=m}_{J/\psi }^2$. By requiring that $a_2$ be process insensitive (i.e., the value of $a_2$ extracted from $J/\psi K$ and $J/\psi K^{*}$ states should be similar), as implied by the factorization hypothesis, we find that $\overrightarrow{B}{K}^{\left(*\right)}$ form factors are severely constrained; they respect the relation $F_1^{\mathrm{BK}}{(m}_{J/\psi }^2\left)\approx {1.9A}_1^{{\mathrm{BK}}^{*}}{(m}_{J/\psi }^2\right)$. Form factors $A_2^{{\mathrm{BK}}^{*}}$ and $V^{{\mathrm{BK}}^{*}}$ at $q^2{=m}_{J/\psi }^2$ inferred from the measurements of the longitudinal polarization fraction and the P-wave component in $\overrightarrow{B}J/\psi K^{*}$ are obtained. A stringent upper limit on $a_2$ is derived from the current bound on $B^0\to D^0{\pi }^0$ and it is sensitive to final-state interactions.