  Inspired by the recently observed tensions between the experimental data and
the theoretical predictions, based on QCD factorisation, for several
colour-allowed non-leptonic $B$-meson decays, we study the potential size of
new physics (NP) effects in the decay channels $b \to c \bar{u} d(s)$. Starting
from the most general effective Hamiltonian describing the $b \to c \bar{u}
d(s)$ transitions, we compute NP contributions to the theoretical predictions
of $B$-meson lifetime and of $B$-mixing observables. The well-known lifetime
ratio $\tau (B^+)/\tau (B_d)$ and the experimental bound on the semi-leptonic
CP asymmetry $a_{sl}^d$, provide strong, complementary constraints on some of
the NP Wilson coefficients.
