A puzzle in $$\varvec{\bar{B}_{(s)}^0 \rightarrow D_{(s)}^{(*)+} \lbrace \pi ^-, K^-\rbrace }$$ B ¯ ( s ) 0 → D ( s ) ( ∗ ) + { π - , K - } decays and extraction of the $$\varvec{f_s/f_d}$$ f s / f d fragmentation fraction

We provide updated predictions for the hadronic decays $$\bar{B}_s^0\rightarrow D_s^{(*)+} \pi ^-$$ and $$\bar{B}^0\rightarrow D^{(*)+} K^-$$ . They are based on $${\mathcal {O}}(\alpha _s^2)$$ results for the QCD factorization amplitudes at leading power and on recent results for the $$\bar{B}_{(s)} \rightarrow D_{(s)}^{(*)}$$ form factors up to order $$\mathcal{O}(\Lambda _\mathrm{QCD}^2/m_c^2)$$ in the heavy-quark expansion. We give quantitative estimates of the matrix elements entering the hadronic decay amplitudes at order $$\mathcal{O}(\Lambda _\mathrm{QCD}/m_b)$$ for the first time. Our results are very precise, and uncover a substantial discrepancy between the theory predictions and the experimental measurements. We explore two possibilities for this discrepancy: non-factorizable contributions larger than predicted by the QCD factorization power counting, and contributions beyond the Standard Model. We determine the $$f_s/f_d$$ fragmentation fraction for the CDF, D0 and LHCb experiments for both scenarios.