Time dependent $C\!P$ asymmetries in $B^0 \rightarrow D_{C\!P} \pi^+ \pi^-$ decays at LHCb

This thesis principally presents a time-dependent Dalitz plot analysis of $B^0\rightarrow D_{C\!P}\pi^+\pi^-$ decays, completed using the data collected by the LHCb experiment between 2011 and 2018. Here, $D_{C\!P}$ refers to a neutral $D$-meson reconstructed from a $C\!P$ eigenstate: here, $K^+K^-$ or $\pi^+\pi^-$. The physics parameters of interest are $\cos(2\beta)$ and $\sin(2\beta)$, the central values of which remain blinded, as of the publication of this thesis, with statistical uncertainties given by: \begin{align*} \sigma_{\rm stat}[\sin(2\beta)] &= 0.065\,, \\ \sigma_{\rm stat}[\cos(2\beta)] &= 0.087\,. \end{align*} The systematic uncertainties are yet to be fully evaluated. If, as expected, the analysis is found to be statistically dominated, this novel analysis method will yield the highest precision measurement of $\sin(2\beta)$ from any analysis of a $b\rightarrow c \bar{u} d$ transition, as well as the highest precision measurement of $\cos(2\beta)$ ever made.