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A measurement of charm mixing and $CP$-violating parameters is reported,using $\bar{B} \rightarrow D^0 ( \rightarrow K_S^0 \pi^+ \pi^-) \mu^-\bar{\nu}_\mu X$ decays reconstructed in proton-proton collisions collected bythe LHCb experiment during the years 2016 to 2018, corresponding to anintegrated luminosity of 5.4 fb$^{-1}$. The measured mixing and $CP$-violatingparameters are $$x_{\rm CP} = [ \,\,\,\,\,4.29 \pm 1.48 \mbox{(stat)} \pm 0.26\mbox{(syst)} ] \times 10^{-3}\,,$$ $$y_{\rm CP} = [ \,\,12.61 \pm 3.12\mbox{(stat)} \pm 0.83 \mbox{(syst)} ] \times 10^{-3} \,,$$ $$\Delta x \,\,= [-0.77 \pm 0.93 \mbox{(stat)} \pm 0.28 \mbox{(syst)} ] \times 10^{-3} \,,$$$$\Delta y \, = [ \,\,\,\,\,3.01 \pm 1.92 \mbox{(stat)} \pm 0.26 \mbox{(syst)}] \times 10^{-3} \,.$$ The results are complementary to and consistent withprevious measurements. A combination with the recent LHCb analysis of $D^{*+}\rightarrow D^0 ( \rightarrow K_S^0 \pi^+ \pi^-) \pi^+ $ decays is reported.

Model-independent measurement of charm mixing parameters in $\bar{B} \rightarrow D^0 ( \rightarrow K_S^0 \pi^+ \pi^-) \mu^- \bar{\nu}_\mu X$ decays