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

A search for time-dependent violation of the charge-parity ($C\!P$) symmetry in $D^0 \to K^+ K^-$ and $D^0 \to \pi^+ \pi^-$ decays is performed at the LHCb experiment using proton-proton collision data recorded from 2015 to 2018 at a centre-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 5.7 fb$^{-1}$. The $D^0$ meson is required to originate from a $D^*(2010)^+ \to D^0 \pi^+$ decay, such that its flavour at production is identified by the charge of the accompanying pion. The slope of the time-dependent asymmetry of the decay rates of $D^0$ and $\overline{D}^0$ mesons into the final states under consideration is measured to be \begin{align*} \Delta Y_{K^+ K^-} &= (-2.3 \pm 1.5 \pm 0.3) \times 10^{-4},\\ \Delta Y_{\pi^+ \pi^-} &= (-4.0 \pm 2.8 \pm 0.4)\times 10^{-4}, \end{align*} where the first uncertainties are statistical and the second are systematic. Neglecting possible final-state dependent contributions, these results are combined yielding \begin{equation*} \Delta Y = (-2.7 \pm 1.3 \pm 0.3 )\times 10^{-4}, \end{equation*} which is compatible with the $C\!P$-invariance hypothesis at the level of 2 standard deviations. This result improves the precision of the world average of $\Delta Y$ by a factor of two, and sets the strongest bound on the value of the phase $\phi^M_2$, which quantifies dispersive $C\!P$-violating contributions to $D^0$ mixing.