Two-body non-leptonic heavy-to-heavy decays at NNLO in QCD factorization

We evaluate in the framework of QCD factorization the two-loop vertex corrections to the decays $\bar{B}_{(s)}\to D_{(s)}^{(\ast)+} \, L^-$ and $\Lambda_b \to \Lambda_c^+ \, L^-$, where $L$ is a light meson from the set $\{\pi,\rho,K^{(\ast)},a_1\}$. These decays are paradigms of the QCD factorization approach since only the colour-allowed tree amplitude contributes at leading power. Hence they are sensitive to the size of power corrections once their leading-power perturbative expansion is under control. Here we compute the two-loop ${\cal O}(\alpha_s^2)$ correction to the leading-power hard scattering kernels, and give the results for the convoluted kernels almost completely analytically. Our newly computed contribution amounts to a positive shift of the magnitude of the tree amplitude by $\sim 2$\%. We then perform an extensive phenomenological analysis to NNLO in QCD factorization, using the most recent values for non-perturbative input parameters. Given the fact that the NNLO perturbative correction and updated values for form factors increase the theory prediction for branching ratios, while experimental central values have at the same time decreased, we reanalyze the role and potential size of power corrections by means of appropriately chosen ratios of decay channels.