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Time-dependent $CP$ asymmetries in the decay rates of the singlyCabibbo-suppressed decays $D^0\rightarrow K^-K^+$ and $D^0\rightarrow\pi^-\pi^+$ are measured in $pp$ collision data corresponding to an integratedluminosity of 3.0 fb$^{-1}$ collected by the LHCb experiment. The $D^0$ mesonsare produced in semileptonic $b$-hadron decays, where the charge of theaccompanying muon is used to determine the initial state as $D^0$ or$\bar{D}^0$. The asymmetries in effective lifetimes between $D^0$ and$\bar{D}^0$ decays, which are sensitive to indirect $CP$ violation, aredetermined to be \begin{align*}  A_{\Gamma}(K^-K^+) = (-0.134 \pm 0.077 \; {}^{+0.026}_{-0.034})\% \,  A_{\Gamma}(\pi^-\pi^+) = (-0.092\pm 0.145 \; {}^{+0.025}_{-0.033})\% \,\end{align*} where the first uncertainties are statistical and the secondsystematic. This result is in agreement with previous measurements and with thehypothesis of no indirect $CP$ violation in $D^0$ decays.

Measurement of indirect $CP$ asymmetries in $D^0\rightarrow K^-K^+$ and $D^0\rightarrow \pi^-\pi^+$ decays using semileptonic $B$ decays