Measurements of $\sin 2\,\beta$ using charmonium and open charm decays at LHCb

The $C\!P$ violation observables $S_f$ and $C_f$ in the decays of $B^0$ and $\kern 0.18em\overline{\kern -0.18em B}{}^0$ mesons to the $J\!/\!\psi K^0_S$ final state and to the $D^+ D^-$ final state are measured with a data sample corresponding to an integrated luminosity of $3 fb^{-1}$ collected with the LHCb experiment in proton-proton collisions at centre-of-mass energies of $7$ and $8 ~TeV$. The analysis of the decay-time evolution of $41560 ~B^0\!\rightarrow J\!/\!\psi K^0_S$ decays yields \begin{align*} S_{J\!/\!\psi K^0_S} &= \phantom{-}0.731 \, \pm 0.035 \, \text{(stat)} \pm 0.020 \, \text{(syst)}\,, \\ C_{J\!/\!\psi K^0_S} &= - 0.038 \, \pm 0.032 \, \text{(stat)} \pm 0.005 \, \text{(syst)}\,, \end{align*} which is consistent with the current world averages and with the Standard Model expectations. In a flavour-tagged, decay-time-dependent analysis of $1410 ~B^0\!\rightarrow D^+ D^-$ decays the following results are determined: \begin{align*} S_{D^+ D^-} &= -0.54 \, ^{+0.17}_{-0.16} \, \text{(stat)} \pm 0.05 \, \text{(syst)}\,, \\ C_{D^+ D^-} &= \phantom{-}0.26 \, ^{+0.18}_{-0.17} \, \text{(stat)} \pm 0.02 \, \text{(syst)}\,. \end{align*} With these results high-order Standard Model corrections, which could lead to differences between the absolute values of the obtained observables of the two decay modes, are constrained to be small.