  We give a comprehensive account of the flavour physics of
Beyond-Standard-Model (BSM) effects in $b \to c \bar{c} s$ transitions,
considering the full set of 20 four-quark operators. We discuss the
leading-order structure of their RG mixing with each other as well as the
QCD-penguin, dipole, and FCNC semileptonic operators they necessarily mix with,
providing compact expressions. We also provide the first complete results for
BSM effects in the lifetime observables $\Delta \Gamma_s$ and
$\tau(B_s)/\tau(B_d)$, as well as for the semileptonic CP-asymmetry $a_{sl}^s$.
From a global analysis, we obtain stringent constraints on 16 of the 20 BSM
operators, including the 10 operators $Q^{c\prime }_{1 \dots 10}$ involving a
right-handed strange quark. Focussing on CP-conserving new physics, the
constraints correspond to NP scales of order 10 TeV in most cases, always
dominated by exclusive and/or radiative $B$-decays via RGE mixing. For the
remaining four operators, including the two Standard-Model (SM) ones, larger
effects are experimentally allowed, as previously noted in arXiv:1701.09183. We
extend that paper's scope to the CP-violating case, paying attention to the
impact on the decay rate and time-dependent CP-violation in $B_d \to J/\psi
K_S$. We show that quantifiable constraints arise for new physics in either of
the two SM operators, with the uncertain non-perturbative matrix element of the
colour-suppressed operator determined from the data. For new physics in the SM
suppressed coefficient $C^c_1$ we find two perfectly viable, narrow bands of
complex Wilson coefficients. Somewhat curiously, one of them contains a region
where the fitted matrix element for the colour-suppressed operator is in
agreement with naive factorization, contrarily to a widely held belief that
large non-factorizable contributions to $B_d \to J/\psi K_S$ are implied by
experimental data.
