The $a_1$ factorisation coefficient in ${\overline{B}^0} \to D^{(*)+}M^{-}$ and ${\overline{B}^0} \to D^{(*)+}D_s^{(*)-}$ decays: measurements versus theory
Using recent measurements of exclusive B-meson decays we extract the $a_1$ factorization parameter in ${\overline{B}^{0}_{d}} \to D^{(*)^+} K^-/\pi^-$ decay channels. { The values obtained in the four channels are very similar, in agreement with the theoretical expectations obtained in the $m_Q \to \infty$ limit, but the measured values differ definitely from the expected central values. } Such differences have already been observed. Using recent data, we improve the accuracy by {a factor close to two} in this comparison. We study possible interpretations for such a difference and conclude that the claimed accuracy of corrections due to soft gluon contributions and finite mass effects has to be revisited before arguing for possible "New-Physics" effects. We observe that the corrections to the $m_Q \to \infty$ limit are larger in the spectator topology than effects from exchange amplitude contributions. We discuss also the way expected ratios of $a_1$ values, involving hypotheses from theory, are used to obtain the fraction of $B^0_s$ meson production in jets at LHC and conclude that the uncertainties attached to this approach are not well established. Finally, the $a_1$ values extracted from $\overline{B}^{0}_{d} \to D^{(*)+} D_s^-$ decays are found to be similar, within uncertainties, to the ones obtained with $K^-$ emission, once expected penguin contributions are corrected. Using the same approach, we note that $a_1$ values measured in channels with a $D_s^{*-}$ emission are about two standard deviations lower than $a_1(D^{(*)+}K^-$).