Search for time-dependent $CP$ violation in $D^0 \to K^+ K^-$ and $D^0 \to \pi^+ \pi^-$ decays

A search for charge-parity ($CP$) violation in the time-dependent decay rates of $D^0 \to K^+ K^-$ and $D^0 \to \pi^+ \pi^-$ decays is reported. The data correspond to an integrated luminosity of $1.9\,{\rm fb}^{-1}$ collected by the LHCb detector in 2015--2016 in $pp$ collisions at a centre-of-mass energy of $13\,{\rm TeV}$. The flavour of the $D^0$ meson is inferred from the charge of the pion in $D^{\ast}(2010)^+\to D^0 \pi^+$ decays. The asymmetries of the time-dependent decay rates of $D^0$ and $\overline{D}^0$ decays, sensitive to $CP$ violation in the mixing and in the interference between mixing and decay, are measured to be $A_\Gamma(K^+ K^-) = (1.3 \pm 3.5 \pm 0.7)\times 10^{-4}$ and $A_\Gamma(\pi^+ \pi^-) = (11.3 \pm 6.9 \pm 0.8)\times 10^{-4}$, where the first uncertainties are statistical and the second are systematic.