Abstract
A measurement is presented of decay-time-dependent $CP$ violation in the
decays $B^0\rightarrow J/\psi K^0_\mathrm{S}$ and $B^0\rightarrow\psi(2S)
K^0_\mathrm{S}$, where the $J/\psi$ is reconstructed from two electrons and the
$\psi(2S)$ from two muons. The analysis uses a sample of $pp$ collision data
recorded with the LHCb experiment at centre-of-mass energies of 7 and 8 TeV,
corresponding to an integrated luminosity of 3 $\mathrm{fb}^{-1}$. The
$CP$-violation observables are measured to be \begin{equation*} \begin{aligned}
C\left(B^0\rightarrow J/\psi
K^0_\mathrm{S}\right)&=\phantom{+}\,0.12\,\pm\,0.07\pm\,0.02\,,
S\left(B^0\rightarrow J/\psi
K^0_\mathrm{S}\right)&=\phantom{+}\,0.83\,\pm\,0.08\pm\,0.01\,,
C\left(B^0\rightarrow\psi(2S)
K^0_\mathrm{S}\right)&=-\,0.05\,\pm\,0.10\pm\,0.01\,,
S\left(B^0\rightarrow\psi(2S)
K^0_\mathrm{S}\right)&=\phantom{+}\,0.84\,\pm\,0.10\pm\,0.01\,, \end{aligned}
\end{equation*} where $C$ describes $CP$ violation in the direct decay, and $S$
describes $CP$ violation in the interference between the amplitudes for the
direct decay and for the decay after $B^0$-$\overline{B}^0$ oscillation. The
first uncertainties are statistical and the second are systematic. The two sets
of results are compatible with the previous LHCb measurement using
$B^0\rightarrow J/\psi K^0_\mathrm{S}$ decays, where the $J/\psi$ meson was
reconstructed from two muons. The averages of all three sets of LHCb results
are \begin{equation*} \begin{aligned}
C\left(B^0\rightarrow[c\bar{c}]K^0_\mathrm{S}\right)&=-0.017 \pm 0.029\,,
S\left(B^0\rightarrow[c\bar{c}]K^0_\mathrm{S}\right)&=\phantom{+}0.760 \pm
0.034\,, \end{aligned} \end{equation*} under the assumption that higher-order
contributions to the decay amplitudes are negligible. The uncertainties include
statistical and systematic contributions.